Heriot-Watt University - Reservoir Simulation

Introduction and Case Studies

1

CONTENTS 1

WHAT IS A SIMULATION MODEL? 1.1 A Simple Example of a Simulation Model 1.2 A Note on Units

2

WHAT IS A RESERVOIR SIMULATION MODEL? 2.1 The Task of Reservoir Simulation 2.2 What Are We Trying To Do and How Complex Must Our Model Be?

3

FIELD APPLICATIONS OF RESERVOIR SIMULATION 3.1 Reservoir Simulation at Appraisal and in Mature Fields 3.2 Introduction to the Field Cases 3.3 Case 1: The West Seminole Field Simulation Study (SPE10022, 1982) 3.4 Ten Years Later - 1992 3.5 Case 2: The Anguille Marine Simulation Study (SPE25006, 1992) 3.6 Case 3: Ubit Field Rejuvenation (SPE49165,1998) 3.7 Discussion of Changes in Reservoir Simulation; 1970s - 2000 3.8 The Treatment of Uncertainty in Reservoir Simulation

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STUDY EXAMPLE OF A RESERVOIR SIMULATION

5

TYPES OF RESERVOIR SIMULATION MODEL 5.1 The Black Oil Model 5.2 More Complex Reservoir Simulation Models 5.3 Comparison of Field Experience with Various Simulation Models

6

SOME FURTHER READING ON RESERVOIR SIMULATION

APPENDIX A - References APPENDIX B - Some Overview Articles on Reservoir Simulation 1. Reservoir Simulation: is it worth the effort? SPE Review, London Section monthly panel discussion November 1990. 2. The Future of Reservoir Simulation - C. Galas, J. Canadian Petroleum Technology, 36, January 1997. 3. What you should know about evaluating simulation results - M. Carlson; J. Canadian Petroleum Technology, Part I - pp. 21-25, 36, No. 5, May 1997; Part II - pp. 52-57, 36, No. 7, August 1997.

LEARNING OBJECTIVES: Having worked through this chapter the student should:

2

•

Be able to describe what is meant by a simulation model, saying what analytical models and numerical models are.

•

Be familiar with what specifically a reservoir simulation model is.

•

Be able to describe the simplifications and issues that arise in going from the description of a real reservoir to a reservoir simulation model.

•

Be able to describe why and in what circumstances simple or complex reservoir models are required to model reservoir processes.

•

Be able to list what input data is required and where this may be found.

•

Be able to describe several examples of typical outputs of reservoir simulations and say how these are of use in reservoir development.

•

Know the meaning of all the highlighted terms - or terms referred to in the Glossary - in Chapter 1 e.g. history matching, black oil model, transmissibility, pseudo relative permeability etc.

•

Be able to describe and discuss the main changes in reservoir simulation over the last 40 years from the 60's to the present - and say why these have occurred.

•

Know in detail and be able to compare the differences between what reservoir simulations can do at the appraisal and in the mature stages of reservoir development.

•

Have an elementary knowledge of how uncertainty is handled in reservoir simulation.

•

Know all the types of reservoir simulation models and what type of problem or reservoir process each is used to model.

•

Know or be able to work out the equations for the mass of a phase or component in a grid block for a black oil or compositional model.


1

CONTENTS 1

WHAT IS A SIMULATION MODEL? 1.1 A Simple Example of a Simulation Model 1.2 A Note on Units

2

WHAT IS A RESERVOIR SIMULATION MODEL? 2.1 The Task of Reservoir Simulation 2.2 What Are We Trying To Do and How Complex Must Our Model Be?

3

FIELD APPLICATIONS OF RESERVOIR SIMULATION 3.1 Reservoir Simulation at Appraisal and in Mature Fields 3.2 Introduction to the Field Cases 3.3 Case 1: The West Seminole Field Simulation Study (SPE10022, 1982) 3.4 Ten Years Later - 1992 3.5 Case 2: The Anguille Marine Simulation Study (SPE25006, 1992) 3.6 Case 3: Ubit Field Rejuvenation (SPE49165,1998) 3.7 Discussion of Changes in Reservoir Simulation; 1970s - 2000 3.8 The Treatment of Uncertainty in Reservoir Simulation

4

STUDY EXAMPLE OF A RESERVOIR SIMULATION

5

TYPES OF RESERVOIR SIMULATION MODEL 5.1 The Black Oil Model 5.2 More Complex Reservoir Simulation Models 5.3 Comparison of Field Experience with Various Simulation Models

6

SOME FURTHER READING ON RESERVOIR SIMULATION

APPENDIX A - References APPENDIX B - Some Overview Articles on Reservoir Simulation 1. Reservoir Simulation: is it worth the effort? SPE Review, London Section monthly panel discussion November 1990. 2. The Future of Reservoir Simulation - C. Galas, J. Canadian Petroleum Technology, 36, January 1997. 3. What you should know about evaluating simulation results - M. Carlson; J. Canadian Petroleum Technology, Part I - pp. 21-25, 36, No. 5, May 1997; Part II - pp. 52-57, 36, No. 7, August 1997.

LEARNING OBJECTIVES: Having worked through this chapter the student should:

2

•

Be able to describe what is meant by a simulation model, saying what analytical models and numerical models are.

•

Be familiar with what specifically a reservoir simulation model is.

•

Be able to describe the simplifications and issues that arise in going from the description of a real reservoir to a reservoir simulation model.

•

Be able to describe why and in what circumstances simple or complex reservoir models are required to model reservoir processes.

•

Be able to list what input data is required and where this may be found.

•

Be able to describe several examples of typical outputs of reservoir simulations and say how these are of use in reservoir development.

•

Know the meaning of all the highlighted terms - or terms referred to in the Glossary - in Chapter 1 e.g. history matching, black oil model, transmissibility, pseudo relative permeability etc.

•

Be able to describe and discuss the main changes in reservoir simulation over the last 40 years from the 60's to the present - and say why these have occurred.

•

Know in detail and be able to compare the differences between what reservoir simulations can do at the appraisal and in the mature stages of reservoir development.

•

Have an elementary knowledge of how uncertainty is handled in reservoir simulation.

•

Know all the types of reservoir simulation models and what type of problem or reservoir process each is used to model.

•

Know or be able to work out the equations for the mass of a phase or component in a grid block for a black oil or compositional model.


1

BRIEF DESCRIPTION OF CHAPTER 1 A brief overview of Reservoir Simulation is first presented. This module then develops this introduction by going straight into three field examples of applied simulation studies. This is done since this course has some reservoir engineering pre-requisites which will have made the student aware of many of the issues in reservoir development. In these literature examples, we introduce many of the basic concepts that are developed in detail throughout the course e.g. gridding of the reservoir, data requirements for simulation, well controls, typical outputs from reservoir simulation (cumulative oil, watercuts etc.), history matching and forward prediction etc. After briefly discussing the issue of uncertainty in reservoir management, some calculated examples are given. Finally, the various types of reservoir simulation model which are available for calculating different types of reservoir development process are presented (black oil model, compositional model, etc.). PowerPoint demonstrations illustrate some of the features of reservoir simulation using a dataset which the student can then run on the web (with modification if required) and plot various quantities e.g. cumulative oil, watercuts etc. This module also contains a Glossary which the student can use for quick reference throughout the course.

1 WHAT IS A SIMULATION MODEL? 1.1 A Simple Example of a Simulation Model A simulation model is one which shows the main features of a real system, or resembles it in its behaviour, but is simple enough to make calculations on. These calculations may be analytical or numerical . By analytical we mean that the equations that represent the model can be solved using mathematical techniques such as those used to solve algebraic or differential equations. An analytic solution would normally be written in terms of “well know” equations or functions (x2, sin x, ex etc). For example, suppose we wanted to describe the growth of a colony of bacteria and we denoted the number of bacteria as N. Now if our growth model says that the rate of increase of N with time (that is, dN/dt) is directly proportional to N itself, then:

 dN  = α.N  dt 

(1)

where α is a constant. We now want to solve this model by answering the question: what is N as a function of time, t, which we denote by N(t), if we start with a bacterial colony of size No. It is easy to show that, N(t) is given by:

Institute of Petroleum Engineering, Heriot-Watt University

3

N( t ) = N o .e α . t

(2)

which is the well-known law of exponential growth. We can quickly check that this analytical solution to our model (equation 1), is at least consistent by setting t = 0 and noting that N = No, as required. Thus, equation 1 is our first example of a simulation model which describes the process - bacterial growth in this case - and equation 2 is its analytical solution. But looking further into this model, it seems to predict that as t gets bigger, then the number N - the number of bacteria in the colony - gets hugely bigger and, indeed, as t →∞, the number N also →∞. Is this realistic ? Do colonies of bacteria get infinite in size ? Clearly, our model is not an exact replica of a real bacterial colony since, as they grow in size, they start to use up all the food and die off. This means that our model may need further terms to describe the observed behaviour of a real bacterial colony. However, if we are just interested in the early time growth of a small colony, our model may be adequate for our purpose; that is, it may be fit-for-purpose. The real issue here is a balance between the simplicity of our model and the use we want to make of it. This is an important lesson for what is to come in this course and throughout your activities trying to model real petroleum reservoirs. In contrast to the above simple model for the growth of a bacterial colony, some models are much more difficult to solve. In some cases, we may be able to write down the equations for our model, but it may be impossible to solve these analytically due to the complexity of the equations. Instead, it may be possible to approximate these complicated equations by an equivalent numerical model. This model would commonly involve carrying out a very large number of (locally quite simple) numerical calculations. The task of carrying out large numbers of very repetitive calculations is ideally suited to the capabilities of a digital computer which can do this very quickly. As an example of a numerical model, we will return to the simple model for colony growth in equation (1). Now, we have already shown that we have a perfectly simple analytical solution for this model (equation 2). However, we are going to “forget” this for a moment and try to solve equation 1 using a numerical method. To do this we break the time, t, into discrete timesteps which we denote by Δt. So, if we have the number of bacteria in the colony at t = 0, i.e. No, then we want to calculate the number at time Δt later, then we use the new value and try to find the number at time Δt later and so on. In order to do this systematically, we need an algorithm (a mathematical name for a recipe) which is easy to develop once we have defined the following notation: Notation:

the value of N at the current time step n is denoted as Nn the value of N at the next time step, n+1 is denoted as Nn+1

Clearly, it is the Nn+1 that we are trying to find. Going back to the main equation that defines this model (equation 1), we approximate this as follows:

N n +1 − N n ≈ α .N n ∆t

(3)

where we use the symbol, "≈", to indicate that equation 3 is really an approximation, or that it is only exactly true as Δt → 0. Equation 3 is now our (approximate) numerical

4


1

model which can be rearranged as follows to find Nn+1 (which is the “unknown” that we are after):

N n +1 = (1 + α.∆t ).N n

(4)

where we have gone to the exact equality symbol, “=”, in equation 4 since, we are accepting the fact that the model is not exact but we are using it anyway. This is our numerical algorithm (or recipe) that is now very amenable to solution using a simple calculator. More formally, the algorithm for the model would be carried out as shown in Figure 1. Set,

t=0

Choos e the time step size, ∆t Specify the initial no. of bacteria at t = 0 i.e. No and set the first value (n=0) of N n to N o No = No

Print n, t and N (N n)

Set

N n+1 = (1 + α.∆t). N n

Set

No

Figure 1 Example of an algorithm to solve the simple numerical “simulation” model in the text

N n = N n+1 n = n+1 t = t + ∆t

Time to stop ? e.g. is t > tmax or n > nmax

Yes End

The above example, although very simple, explains quite well several aspects of what a simulation model is. This model is simple enough to be solved analytically. However, it can also be formulated as an approximate numerical model which is organised into a numerical algorithm (or recipe) which can be followed repetitively. A simple calculator is sufficient to solve this model but, in more complex systems, a digital computer would generally be used.


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1.2 A Note on Units

Throughout this course we will use Field Units and/or SI Units, as appropriate. Although the industry recommendation is to convert to SI Units, this makes discussion of the field examples and cases too unnatural. EXERCISE 1. Return to the simple model described by equation 1. Take as input data, that we start off with 25 bacteria in the colony. Take the value α = 1.74 and take time steps Δt = 0.05 in the numerical model. (i) Using the scale on the graph below, plot the analytical solution for the number of bacteria N(t) as a function of time between t = 0 and t = 2 (in arbitrary time units). (ii) Plot as points on this same plot, the numerical solution at times t = 0, 0.5, 1.0, 1.5 and 2.0. What do you notice about these ? (iii)Using a spreadsheet, repeat the numerical calculation with a Δt = 0.001 and plot the same 5 points as before. What do you notice about these?

1000

N(t) 500

0

(i) (ii)

6

1

Time

2


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2 WHAT IS A RESERVOIR SIMULATION MODEL? In the previous section, we introduced the idea of a simulation model applied to the growth of a bacterial colony. Now let us consider what we want to model - or simulate - when we come to developing petroleum reservoirs. Clearly, petroleum reservoirs are much more complex than our simple example since they involve many variables (e.g. pressures, oil saturations, flows etc.) that are distributed through space and that vary with time. In 1953, Uren defined a petroleum reservoir as follows: “ ... a body of porous and permeable rock containing oil and gas through which fluids may move toward recovery openings under the pressure existing or that may be applied. All communicating pore space within the productive formation is properly a part of the rock, which may include several or many individual rock strata and may encompass bodies of impermeable and barren shale. The lateral expanse of such a reservoir is contingent only upon the continuity of pore space and the ability of the fluids to move through the rock pores under the pressures available.” L.C. Uren, Petroleum Production Engineering, Oil Field Exploitation, 3rd edn., McGraw-Hill Book Company Inc., New York, 1953. This fine example of old fashioned prose is not so easy on the modern ear but does in fact “say it all”. And, whatever it says, then it is precisely what the modern simulation engineer must model!

2.1 The Task of Reservoir Simulation Let us consider the possible magnitude of the task before us when we want to model (or simulate) the performance of a real petroleum reservoir. Figure 2 shows a schematic of reservoir depositional system for the mid-Jurassic Linnhe and Beryl formations in the UK sector of the North Sea. Some actual reservoir cores from the Beryl formation are shown in Figure 3. It is evident from the cores that real reservoirs are very heterogenous. The air permeabilities (kair) range from 1mD to almost 3000 mD and it is evident that the permeability varies quite considerably over quite short distances. It is common for reservoirs to be heterogeneous from the smallest scale to the largest as is evident in these figures. These permeability heterogeneities will certainly affect both pressures and fluid flow in the system. By contrast, a reservoir simulation model which might be used to simulate waterflooding in a layered system of this type is shown schematically in Figure 4. This model is clearly hugely simplified compared with a real system. Although the task of reservoir simulation may appear from this example to be huge, it is still one that reservoir engineers can - and indeed must - tackle. Below, we start by listing in general terms the activities involved in setting up a reservoir model. One way of approaching this is to break the process down into three parts which will all have to appear somewhere in our model: (i) Choice and Controls: Firstly, there are the things that we have some control over. For example: Institute of Petroleum Engineering, Heriot-Watt University

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• • • •

Where the injectors and producer wells are located The capability that we have in the well (completions & downhole equipment) How much water or gas injection we inject and at what rate How fast we produce the wells (drawdown)

We note that certain quantities such as injection and production rates are subject to physical constraints imposed on us by the reservoir itself. (ii) Reservoir Givens: Secondly, there are the givens such as the (usually very uncertain) geology that is down there in the reservoir. There may or may not be an active aquifer which is contributing to the reservoir drive mechanism. We can do things to know more about the reservoir/aquifer system by carrying out seismic surveys, drilling appraisal wells and then running wireline logs, gathering and performing measurements on core, performing and analysing pressure buildup or drawdown tests, etc. (iii) Reservoir Performance Results: Thirdly, there is the observation of the results i.e the reservoir performance. This includes well production rates of oil, water and gas, the field average pressure, the individual well pressures and well productivities etc.

SSW

Fluvial mud/sand supply FC

OM/CS

CRS

E

TC SM

SM TC

ay

eB

rin

a stu

L

L

OM/CS L

v

Flu

lain

dp

oo

Fl ial/

TC TF

TF

FTD

BM

TS

SM TF

TC

SS

SS

r

rrie

Ba

SM

o Sh

a ref

ce

SS

TCI ETD

Fluvial/Floodplain Facies Asociation FC: Fluvial channel sandstones CRS: Crevasse channel/splay sandstones OM/L: Overbank/lake mudstone CS: Coal swamp/marsh mudstone and coal Estuarine Bay-Fill Facies Association TC: Tidal channel sandstones TF: Lower intertidal flat sandstones TS: Tidal shoal sandstone SM: Salt marsh/upper intertidal flat mudstones BM: Brackish bay mudstones FTD: Flood tidal delta Tidal Inlet-Barrier Shoreline Facies Association TCI: Tidal inlet/ebb channel sandstones SS: Barrier shoreline sandstone ETD: Ebb tidal delta

8

.15

12

km

Block diagram illustrates the gradual infilling of the Beryl Embayment by fluvial/floodplain (Linnhe l), estuarine-bay fill (Linnhe ll) and tidal inlet-barrier shoreline facies sequences (Beryl Formation). Coal Fluvial/crevasse channel-fills Tidal channel-fills Tidal inlet-fills Shoal/bars Flood-oriented currents Ebb-oriented currents Longshore currents

Figure 2 Conceptual depositional model for the Linnhe and Beryl formations from the middle Jurassic period (UK sector of the North Sea). (G. Robertson in Cores from the Northwest European Hydrocarbon Provence, edited by C D Oakman, J H Martin and P W M Corbett, Geological Society, London. 1997).


Slab 1 Top 15855 ft

Slab 2 Top 15852 ft

Slab 3 Top 14591 ft

Slab 4 Top 14361 ft

1

Slab 5 Top 14358 ft

Medium-grained Carbonate cemented sandstone (φ =14%, ka = 2mD) - some thin clay and carbonate rich lamination

1m

Figure 3 Cores from the midJurassic Beryl formation from UK sector of the North Sea. φ is porosity and ka is the air permeability. (G. Robertson in Cores from the Northwest European Hydrocarbon Provence, edited by C D Oakman, J H Martin and P W M Corbett, Geological Society, London. 1997).

Medium-grained ripple-laminated and bioturbated carbonate cemented sandstone (φ =10%, ka = 1mD)

Pyritic mudstone (pm)→ fine-grained bioturbated sandstone (φ =16%, ka = 29mD)

Medium to coarse-grained cross-stratified sandstone (φ =21%, ka =1440mD) - in fining-up units

15858 ft Base

15855 ft Base

14594 ft Base

14364 ft Base

14361 ft Base

Coarse-grained carbonaceous sandstone (φ =20%, ka =2940mD) - in cross-stratified, fining-up units

Producer

Water Injector

Figure 4 A schematic diagram of a waterflood simulation in a 3D layered model with an 8x8x5 grid. The information which is input for a single grid block is shown. Contrast this simple model with the detail in a geological model (Figure 2) and in the actual cores themselves (Figure 3).

∆z

∆x Inp φ, ut: cr kx, ock, n S ky, k et t og w, P i krw(z, ros c (S S s w), w) kr w(S w ),

∆y

Approximate Size of Core vs. Grid Size


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2.2 What Are We Trying To Do and How Complex Must Our Model Be? Therefore, at its most complex, our task will be to incorporate all of the above features (i) - (iii) in a complete model of the reservoir performance. But we should now stop at this point and ask ourselves why we are doing the particular study of a given reservoir? In other words, the level of modelling that we will carry out is directly related to the issue or question that we are trying to address. Some engineers prefer to put this as follows: •

What decision am I trying to make?

•

What is the minimum level of modelling - or which tool can I use - that allows me to adequately make that decision?

This matter is put well by Keith Coats - one of the pioneers of numerical reservoir simulation - who said: “The tools of reservoir simulation range from the intuition and judgement of the engineer to complex mathematical models requiring use of digital computers. The question is not whether to simulate but rather which tool or method to use.” (Coats, 1969). Therefore, we may choose a very simple model of the reservoir or one that is quite complex depending on the question we are asking or the decision which we have to make. Without giving technical details of what we mean by simple and complex, in this context, we illustrate the general idea in Figure 5 which shows three models of the same reservoir. The first (Figure 5a), shows the reservoir as a tank model where we are just concerned with the gross fluid flows into and out of the system. In Chapter 2, we will identify models such as those in Figure 5a as essentially material balance models and will be discussed in much more detail later. The particular advantage of material balance models is that they are very simple. They can address questions relating to average field pressure for given quantities of oil/water/gas production and water influx from given initial quantities and initial pressure (within certain assumptions). However, because the material balance model is essentially a tank model, it cannot address questions about why the pressures in two sectors of the reservoir are different (since a single average pressure in the system is a core assumption). The sector model in Figure 5b is somewhat more complex in that it recognises different regions of the reservoir. This model could address the question of different regional pressures. However, even this model may be inadequate if the question is quite detailed such as: in my mature field with a number of active injector/ producer wells where should I locate an infill well and should it be vertical, slanted or horizontal ? For such complicated questions, the model in Figure 5c would be more appropriate since it is more detailed and it contains more spatial information. This schematic sequence of models illustrates that there is no one right model for a reservoir. The simplicity/complexity of the model should relate to the simplicity/ complexity of the question. But there is another important factor: data. It is clear that to build models of the types shown in Figure 5, we require increasing amounts of data as we go from Figure 5a→5b→5c. It is also evident that we should think carefully before building a very detailed model of the type shown in Figure 5c, if we have almost no data. There are some circumstances where we might build quite 10


1

a complicated model with little data to test out hypotheses but we will not elaborate on this issue at this point. The simplicity/complexity of the model should relate to the simplicity/complexity of the question, and be consistent with the amount of reliable data which we have. (a) "Tank" Model of the Reservoir

Average Pressure Average Saturations

Wells Offtake

= =

P So , Sw and Sg

Aquifer

(b) Simple Sector Model Producer - West Flank

Producer - East Flank Oil Leg

Aquifer

(c) Fine Grid Simulation Model of a Waterflood

Figure 5 Schematic illustrations of reservoir models of increasing complexity. Each of these may be suitable for certain types of calculation (see text).

Injector

Producer

200ft

2000ft

We are now aware that various levels of reservoir model may be used and that the reservoir engineer must choose the appropriate one for the task at hand. We will assume at this point that building a numerical reservoir simulation model is the correct approach for what we are trying to achieve. If this is so, we now address the issue: What do we model in reservoir simulation and why do we model it ? There are, as we have said, a range of questions which we might answer, only some of which require a full numerical simulation model to be constructed. Let us now say what a numerical reservoir simulation model is and what sorts of things it can (and cannot) do. Definition: A numerical reservoir simulation model is a grid block model of a petroleum reservoir where each of the blocks represents a local part of the


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reservoir. Within a grid block the properties are uniform (porosity, permeability, relative permeability etc.) although they may change with time as the reservoir process progresses. Blocks are generally connected to neighbouring blocks are fluid may flow in a block-to-block manner. The model incorporates data on the reservoir fluids (PVT) and the reservoir description (porosities , permeabilities etc.) and their distribution in space. Sub-models within the simulator represent and model the injection/producer wells. An example of numerical reservoir simulation gridded model is shown in Figure 6, where some of the features in the above definition are evident. We now list what needs to be done in principle to run the model and then the things which a simulator calculate, if it has the “correct” data. To run a reservoir simulation model, you must: (a) Gather and input the fluid and rock (reservoir description) data as outlined above; (b) Choose certain numerical features of the grid (number of grid blocks, time step sizes etc); (c) Set up the correct field well controls (injection rates, bottom hole pressure constraints etc.); it is these which drive the model; (d) Choose which output (from a vast range of possibilities) you would like to have printed to file which you can then plot later or - in some cases - while the simulation is still running. The output can include the following (non-exhaustive) list of quantities: • • • • • • •

The average field pressure as a function of time The total field cumulative oil, water and gas production profiles with time The total field daily (weekly, monthly, annual) production rates of each phase: oil, water and gas The individual well pressures (bottom hole or, through lift curves, wellhead) over time The individual well cumulative and daily flowrates of oil, water and gas with time Either full field or individual well watercuts, GORs, O/W ratios with time The spatial distribution of oil, water and gas saturations throughout the reservoir as functions of time i.e. So(x,y,z;t), Sw(x,y,z;t) and Sg(x,y,z;t)

Some of the above quantities are shown in simulator output in Figure 7. This field example is for a Middle East carbonate reservoir where the structural map is shown in Figure 7(d). Figure 7(a) shows the field and simulation results for total oil and water cumulative production over 35 years of field life. Figure 7(b) shows the actual and modelled average field pressure. The type of results shown in Figures 7(a) and 7(b) are very common but the modelling of the RFT (Repeat Formation Tester) pressure shown in Figure 7(c) is less common. The RFT tool measures the local pressure at a given vertical depth and, in this case, it can be seen that the reservoir comprises of three zones each of ~ 100 ft thick and each is at a different pressure. This indicates that 12


1

pressure barriers exist (i.e. flow is restricted between these layers). This is correctly modelled in the simulation. This is an interesting and useful example of how reservoir simulation is used in practice.

Figure 6 An example of a 3D numerical reservoir simulation model. The distorted 3D grid covers the crestal reservoir and a large part of the aquifer which is shown dipping down towards the reader. Oil is shown in red and water is blue and a vertical projection of a cross-section at the crest of the reservoir is shown on the x/z and y/z planes on the sides of the perspective box. Two injectors can be seen in the aquifer as well as a crestal horizontal well. Two faults can be seen at the front of the reservoir before the structure dips down into the aquifer. The model contains 25,743 grid blocks.

Note that a vast quantity of output can be output and plotted up and the post-processing facilities in a reservoir simulator suite of software are very important. There is no point is doing a massively complex calculation on a large reservoir system with millions of grid blocks if the output is so huge and complex that it overwhelms the reservoir engineerʼs ability to analyse and make sense of the output. In recent years, data visualisation techniques have played on important role in analysing the results from large reservoir simulations.

600 500

Observed Oil

400

Modelled Oil

300 200

Observed Water Modelled Water

100 0

0

5

10

15

20

25

30

35

Year of Production (a) Full field history match of cumulative oil and water production 3500 Institute of Petroleum Engineering, Heriot-Watt University ure (psia)

Figure 7 (a) to (d) Example of some typical reservoir simulator output. From SPE36540, “Reservoir Modelling and Simulation of a Middle Eastern Carbonate Reservoir”, M.J. Sibley, J.V. Bent and D.W. Davis (Texaco), 1996.

Cumulative Production (MMB)

700

3000

13

Cumulative Pro

Modelled Oil 300 200

Observed Water Modelled Water

100 0

0

5

10

15

20

25

30

35

Year of Production

Average Pressure (psia)

(a) Full field history match of cumulative oil and water production 3500 3000 2500

Observed Data Modelled Data

2000 1500

0

5

10

15

20

25

30

35

Year of Production

Figure 7b

(b) Full field history match of volume weighted pressure

Datum

Observed Modelled Depth (ft.)

-100

-200

-300 1000

1500

1000

2500

3000

(c) Match of RFT pressure data by reservoir simulation model at Year 30

Figure 7c

• A Lower Cretaceous

Carbonate Reservoir in the Arabian Peninsula

C

• Most wells drilled in 1955-1962

C

• > 600 MMBO produced by early 1980s

C

• -this study 1992 C

C

Drilled New Location Injector Location

1 Mile

(d) Field structural map with 50' contour interval

14

C

Convert to Injector

Figure 7d


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How some of this output might be used is illustrated schematically in Figure 8. This is an imaginary case where the reservoir study is to consider the best of four options in Field A: Option 1 - to continue as present with the waterflood; Option 2 - upgrade peripheral injection wells; Option 3 - upgrade injectors and drill six new injectors; Option 4 - drill four new infill wells. Clearly, it is much cheaper to model these four cases than to actually do one of them. The important quantities are the oil recovery profiles for each case compared with the scenario where we simple proceed with the current reservoir development strategy (Option 1). Of course, we do not know whether the forward predictions which we are taking as what would happen anyway, are actually correct. Likewise, we may be unsure of how accurate our forward predictions are for each of the various scenarios. In fact, an important aspect of reservoir simulation is to assess each of the various uncertainties which are associated with our model. This would ideally lead to range of profiles for any forward modeling but we will deal with this in detail later. We discuss the handling of uncertainties in rather more detail in Section 3.8. of this Chapter. In the schematic case shown in Figures 8(a) - 8(g) we note that: (i) The areal plan of the reservoir is given showing injector and producer well location in Figure 8(a); (ii) The corresponding stratification/lithology of the field is shown along the well A-B-C-D transect in Figure 8(b); (iii) Figures 8(c) and 8(d) show the areal grid and the vertical grid, respectively; (iv) The forward predictions of cumulative oil for the various options are shown in Figure 8(f). Note that Option 3 produces most oil (but it involves drilling six additional injection wells); (v) The economic evolution of each option using the predicted oil recovery profiles in Figure 8(f) is shown in Figure 8(g) (where NPV = Net Present Value; IRR = Interval Rate of Return: these are economic measures explained in the economics module of the Heriot-Watt distance learning course). Note that option 4 emerges in the most economic case although it produces rather less oil than option 3. Injector Producer A C

Figure 8 Schematic example of how reservoir simulation might be used in a study of four field development options (see text).

B

D

(a) Field A areal plan showing injector and producer well locations; lithology is given from wells A, B, C and D


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Sand 1

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Sand 2 Sand 3 Sand 4

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Figure 8 (b)

(b) Schematic vertical cross-section showing the lithology across the field through 4 wells A, B, C and D A

A B

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Figure 8 (c)

(c) Reservoir simulation (areal) grid showing current well locations.

A A B

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NZ = 8 NZ = 8

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(d) Reservoir simulation vertical cross-sectional grid showing current well locations.

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Figure 8 (d)


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The grid has 8 blocks in the z- direction representing the thickness of the formation as shown below; NZ = 8. Note that the vertical grid is not uniform. Periferal Injectors A

Periferal Injectors C Periferal Injectors

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(e) Option 1- continue as at present; Option 2 - upgrade peripheral injection wells; Option 3- upgrade injectors + add 6 new injectors; Option 4 - drill four new infill wells. Option 4 Infill Wells Option 3

Infill Wells

Option 4 Option 3 Option 4

Cumulative Oil Oil Cumulative

Cumulative Oil

Figure 8 (e)

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Option 3 Continue as at present (do nothing) Option 1 Option 2

Time Continue as at present (do nothing) Option 1 Option 2 Continue as at present (do nothing) Option 1

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Figure 8 (f)

Simulated oil recovery results for various4options 3 1

NPV NPV or IRR or IRR

NPV or IRR

(f)

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3 3

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Figure 8 (g) Institute of Petroleum Engineering, Heriot-Watt University

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(g) Economic evaluation of options - NPV or IRR Now consider what we are actually trying to do in a typical full field reservoir simulation study. There is a short answer to this is often said in one form or another: it is that the central objective of reservoir simulation is to produce future predictions (the output quantities listed above) that will allow us to optimise reservoir performance. At the grander scale, what is meant by “optimise reservoir performance” is to develop the reservoir in the manner that brings the maximum economic benefit to the company. Reservoir simulation may be used in many smaller ways to decide on various technical matters although even these - for example the issue illustrated in Figure 8 - are usually reduced to economic calculations and decisions in the final analysis as indicated in Figure 8(g).

3 FIELD APPLICATION OF RESERVOIR SIMULATION 3.1 Reservoir Simulation at Appraisal and in Mature Fields

Up to this point, we have considered what a numerical reservoir simulation model is and we have touched on some of the sorts of things that can be calculated. Rather than continue with a discussion of the various technical aspects of reservoir simulation one by one, we will proceed to three field applications of reservoir simulation. These studies will raise virtually all of the technical terms and concepts and many of the issues that will be studied in more detail later in this course. The important terms and concepts will be italicised and will appear in the Glossary at the front of this chapter. Reservoir simulation may be applied either at the appraisal stage of a field development or at any stage in the early, middle or late field lifetime. There are clearly differences in what we might want to get out of a study carried out at the appraisal stage of a reservoir and a study carried out on a mature field. Appraisal stage: at this stage, reservoir simulation will be a tool that can be used to design the overall field development plan in terms of the following issues: •

The nature of the reservoir recovery plan e.g. natural depletion, waterflooding, gas injection etc.

•

The nature of the facility required to develop the field e.g. a platform, a subsea development tied back to an existing platform or a Floating Production System (for an offshore fileld).

•

The nature and capacities of plant sub-facilities such as compressors for injection, oil/water/gas separation capability.

•

The number, locations and types of well (vertical, slanted or horizontal) to be drilled in the field.

•

The sequencing of the well drilling program and the topside facilites.

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It is during the initial appraisal stage that many of the biggest - i.e. most expensive - investment decisions are made e.g. the type of platform and facilities etc. Therefore, it is the most helpful time to have accurate forward predictions of the reservoir performance. But, it is at this time when we have the least amount of data and, of course, very little or no field performance history (there may be some extended production well tests). Therefore, it seem that reservoir simulation has a built-in weakness in its usefulness; just when it can be at its most useful during appraisal is precisely when it has the least data to work on and hence it will usually make the poorest forward predictions. So, does reservoir simulation let us down just when we need it most? Perhaps. However, even during appraisal, reservoir simulation can take us forward with the best current view of the reservoir that we have at that time, although this view may be highly uncertain. As we have already noted, if major features of the reservoir model (e.g. the stock tank oil initially in place, STOIIP) are uncertain, then the forward predictions may be very inaccurate. In such cases, we may still be able to build a range of possible reservoir models, or reservoir scenarios, that incorporate the major uncertainties in terms of reservoir size (STOIIP), main fault blocks, strength of aquifer, reservoir connectivity, etc. By running forward predictions on this range of cases, we can generate a spread of predicted future field performance cases as shown schematically in Figure 9. How to estimate which of these predictions is the most likely and what the magnitude of the “true” uncertainties are is very difficult and will be discussed later in the course.

Cumulative Oil Recovery (STB)

"Optimistic" Case

Figure 9 Spread of future predicted field performances from a range of scenarios of the reservoir at appraisal.

2005

"Pessimistic" Case Most Probable Case

2010

Time (Year)

2015

For example, scenarios for various cases may involve: •

Different assumptions about the original oil in place (STOIIP; Stock Tank Oil Originally In Place).

•

Different values of the reservoir parameters such as permeability, porosity, net-to-gross ratio, the effect of an aquifer, etc..

•

Major changes in the structural geology or sedimentology of the reservoir e.g. sealing vs. “leaky” faults in the system, the presence/absence of major fluvial channels, the distribution of shales in the reservoir etc..


19

Mature field development: we define this stage of field development for our purposes as when the field is in “mid-life”; i.e. it has been in production for some time (2 - 20+ years) but there is still a reasonably long lifespan ahead for the field, say 3 - 10+years. At this stage, reservoir simulation is a tool for reservoir management which allows the reservoir engineer to plan and evaluate future development options for the reservoir. This is a process that can be done on a continually updated basis. The main difference between this stage and appraisal is that the engineer now has some field production history, such as pressures, cumulative oil, watercuts and GORs (both field-wide and for individual wells), in addition to having some idea of which wells are in communication and possibly some production logs. The initial reservoir simulation model for the field has probably been found to be “wrong”, in that it fails in some aspects of its predictions of reservoir performance e.g. it failed to predict water breakthough in our waterflood (usually, although not always, injected water arrives at oil producers before it is expected). By the way, if the original model does turn out to be wrong, this does not invalidate doing reservoir simulation in the first place. (Why do you think this is so?) At this development stage, typical reservoir simulation activities are as follows: •

Carrying out a history match of the (now available) field production history in order to obtain a better tuned reservoir model to use for future field performance prediction

•

Using the history match to re-visit the field development strategy in terms of changing the development plan e.g. infill drilling, adding extra injection water capability, changing to gas injection or some other IOR scheme etc.

•

Deciding between smaller project options such as drilling an attic horizontal well vs. working over 2 or 3 existing vertical/slanted wells

•

It may be necessary to review the equity stake of various partner companies in the field after some period of production although this typically involves a complete review of the engineering, geological and petrophysical data prior to a new simulation study

•

The reservoir recovery mechanisms can be reviewed using a carefully history matched simulation model e.g. if we find that, to match the history, we must reduce the vertical flows (by lowering the vertical transmissibility), we may wish to determine the importance of gravity in the reservoir recovery mechanism. (Coats (1972) refers to this as the “educational value of simulation models” and it is a part of good reservoir management that the engineer has a good grasp of the important reservoir physics of their asset.)

There are many reported studies in the SPE literature where the simulation model is re-built in early-/mid-life of the reservoir and different future development options are assessed (e.g. see SPE10022 attached to this chapter). Late field development: we define this stage of field development as the closing few years of field production before abandonment. A question arises here as to whether the field is of sufficient economic importance to merit a simulation study at this stage. 20


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A company may make the call that it is simply not worth studying any further since the payback would be too low. However there are two reasons why we may want to launch a simulation study late in a fieldʼs lifetime. Firstly, we may think that, although it is in far decline, we can develop a new development strategy that will give the field “a new lease of life” and keep it going economically for a few more years. For example, we may apply a novel cheap drilling technology, or a program of successful well stimulation (to remove a production impairment such as mineral scale) or we may wish to try an economic Improved Oil Recovery (IOR) technique. Secondly, the cost of field abandonment may be so high - e.g. we may have to remove an offshore structure - that almost anything we do to extend field life and avoid this expense will be “economic”. This may justify a late life simulation study. However, there are no general rules here since it depends on the local technical and economic factors which course of action a company will follow. In some countries there may be legislation (or regulations) that require that an oil company produces reservoir simulation calcualtions as part of their ongoing reservoir management.

3.2 Introduction to the Field Cases

Three field cases are now presented. We reproduce the full SPE papers describing each of these reported cases. In the text of each of these papers there are margin numbers which refer to the Study Notes following the paper. We use these to explain the concepts of reservoir simulation as they arise naturally in the description of a field application. In fact, you may very well understand many of the term immediately from the context of their description in the SPE paper. The three field examples are as follows: Case 1: “The Role of Numerical Simulation in Reservoir Management of a West Texas Carbonate Reservoir”, SPE10022, presented at the International Petroleum Exhibition and Technical Symposium of the SPE, Beijing, China, 18 - 26 March 1982, by K J Harpole and C L Hearn. Case 2: “Anguille Marine, a Deepsea-Fan Reservoir Offshore Gabon: From Geology Toward History Matching Through Stochastic Modelling”, SPE25006, presented at the SPE European Petroleum Conference (Europec92), Cannes, France, 16-18 November 1992, by C.S. Giudicelli, G.J. Massonat and F.G. Alabert (Elf Aquitaine) Case 3: “The Ubit Field Rejuvenation: A Case History of Reservoir Management of a Giant Oilfield Offshore Nigeria”, SPE49165, presented at the SPE Annual Technical Conference and Exhibition, New Orleans, LA, 27-30 September 1998, by C.A. Clayton et al (Mobil and Department of Petroleum Resources, Nigeria) These cases were chosen for the following main reasons: •

They are all good technical studies that illustrate “typical” uses of reservoir simulation as a tool in reservoir management (we have deliberately taken all cases at the middle and the mature stages of field development since much more data is available at that time);

•

They introduce virtually all of the main ideas and concepts of reservoir simulation in the context of a worked field application. As these concepts


21

and specialised terms arise, they are explained briefly in the study notes although more detailed discussion will appear later in the course. Compact definitions of the various terms are given in the Glossary at the front of this module; •

They are all well-written and use little or no mathematics;

•

By choosing an example from the early 1980s, the early/mid 1990s and the late 1990s, we can illustrate some of the advances in applied reservoir simulation that have taken place over that period (this is due to the availability of greater computer processing power and also the adoption of new ideas in areas such as geostatistics and reservoir description).

How you should read the next part of the module is as follows: •

Read right through the SPE paper and just pay particular attention when there is a Study Note number in the margin;

•

Go back through the paper but stop at each of the Study Notes and read through the actual point being made in that note.

As noted above, all the main concepts that are introduced can also be found in the Glossary which should be used for quick reference throughout the course or until you are quite familiar with the various terms and concepts in reservoir simulation. See SPE 10022 paper in Appendix

3.3 Case 1: The West Seminole Field Simulation Study (SPE10022, 1982)

Case 1: “The Role of Numerical Simulation in Reservoir Management of a West Texas Carbonate Reservoir”, SPE10022, presented at the International Petroleum Exhibition and Technical Symposium of the SPE, Beijing, China, 18 - 26 March 1982. by K J Harpole and C L Hearn. Summary: This paper presents a study from the early 1980s where a range of reappraisal strategies for a mature carbonate field are being evaluated using reservoir simulation. For example, possible development strategies include the blowdown of the gas cap or infill drilling. They explicitly state in their opening remarks that their central objective is to “optimise reservoir performance” by choosing a future development strategy from a range of defined options. The structure of the study is very typical of the work flow of a field simulation study, viz Introduction; Reservoir Description; Simulation Model; History Matching; Future Performance; Conclusions and recommendations. Although this paper is almost 20 years old, it introduces the reader in a very clear way to virtually all the concepts of conventional reservoir simulation. Location maps and general reservoir structure shown in Figures 1 and 2 of SPE 10022.

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Study Notes Case 1:

1. States explicitly that the objective of the study is to “optimise reservoir performance” as discussed in the introductory part of this module. 2. Raises the issue of an accurate reservoir description being required and this is emphasised throughout this paper. 3. An interesting point is raised comparing the carbonate reservoir of this study broadly to sandstone reservoirs. It notes that the post-depositional diagenetic effects are of major importance in the West Seminole field in that they affect the reservoir continuity and quality i.e. the local porosity and permeability. In contrast, it is noted that sandstone reservoir are mainly controlled by their depositional environment and tend to show less diagenetic overprint. However, a point to note is that the broad outline and work flow of a numerical reservoir simulation study are quite similar for both carbonate and sandstone reservoirs. 4. Carbonate diagenetic processes include dolomitisation (dolomite = CaMg(CO3)2), recrystallisation, cementations and leaching. This geochemical information is not directly used in the simulation model but it is important since it leads to identification of reservoir layer to layer flow barriers (see below). 5. Strategy: Previous gas re-injection into the cap + peripheral water injection => not very successful. They want to implement a 40 acre, 5-spot water flood; see Fig. 3. A “5-spot” is a particular example of a “pattern flood” which is appropriate mainly for onshore reservoirs where many wells can be drilled with relatively close spacing (see Waterflood Patterns in the Glossary). 6a. They raise the issue of vertical communication between the oil and gas zones. This is an excellent example of an uncertain reservoir feature that can be modelling using a range of scenarios from free flow between layers to zero interlayer flow + all cases in between. Therefore, we can run simulations of all these cases and see which one agrees best with the field observations (which is what they do, in fact). 6b. The vertical communication - or lack of it - will affect flow between the oil and gas zones which may lead to loss of oil to the gas cap; see Figure 4. 7. States the structure of the simulation study work flow: Accurate reservoir description - Develop the simulation model (perform the history match - see below - use model for future predictions - evaluate alternative operating plans). A history match is when we adjust the parameters in the simulation model to make the simulated production history agree with the actual field performance (expanded on below). 8a. A lengthy geological description of the reservoir is given where the depositional environment is described - reference is made to extensive core data (~7500 ft. of core). 8b. The impact of the geology/diagenesis in the simulation model is discussed here. There is evidence of field wide barriers due to cementation with anhydrite which may reduce vertical flows. This is important since it gives a sound geological interpretation to the existence of the vertical flow barriers. Therefore, if we need to include this to Institute of Petroleum Engineering, Heriot-Watt University

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match the field performance, we have some justification or explanation for it rather than it simply being a “fiddle factor” in the model. 9. Figure 5 shows the 6 major reservoir layers where the interfaces between the layers are low φ, low k anhydrite cement zones. Again, these may be explained from the depositional environment and the subsequent diagenetic history of the reservoir. 10. 7500 ft. of whole core analysis for the W. Seminole field was available which was digitised for computer analysis (not common at that time, late 1970s). This is very valuable information and is often not available. 11. Permeability distributions in the reservoir are shown in Fig. 6 and these data are vital for reservoir simulation. Dake (1994; p.19) comments on this type of data: “What matters in viewing core data is the all-important permeability distribution across the producing formations; it is this, more than anything else, that dictates the efficiency of the displacement process.” 12. They note that no consistent k/φ correlation is found in this system (which is quite common in carbonates). Often some approximate k/φ correlation can be found for sandstones (e.g. see k/φ Correlations in the Glossary). 13. The W. Seminole field does “exhibit a distinctly layered structure” and the corresponding permeability stratification in the model is shown in Fig. 7. 14a. Pressure transient work - again gives important ancillary information on the reservoir. The objectives of this work were to determine whether there was (i) directional permeability effects, directional fracturing or channelling; (ii) the degree of stratification in the reservoir; (iii) evaluation of the pay continuity between the injectors and producers 14b. No evidence of “channelling or obvious fracture flow system” 14c. Distinct evidence was seen for: (a) the presence of a layered system; (b) restricted communication between layers (ΔP ≈ 200 - 250 psi between layers). This is vital information since it gives an immediate clue that there is probably not completely free flow between layers i.e. there are barriers to flow as suspected from the geology. 14d. Finally on this issue, there is pressure evidence of “arithmetically averaged permeabilities”. This is again typical of layered systems. 15. Native state core tests are referred to from which they obtained steady-state relative permeabilities. These could be very valuable results but no details given here. NB it appears that only one native state core relative permeability was actually measured. This is probably too little data but reflects the reality in many practical reservoir studies that often the engineer does not have important information; however, we just have to “get on with it”. 16. In this study the reservoir simulator which they used was a commercial Black Oil Model (3D, 3 phase - oil/water/gas). Modelling carried out on the main dome portion of the reservoir. This is done quite often in order to simplify the model and to focus 24


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on the region of the field of interest (and importance in terms of oil production). A no flow boundary is assumed in the model on the saddle with the east dome (justified by different pressure history). Again, this is supported by field evidence but it may also be a simplifying judgement to avoid unnecessary complication in the model. 17a. The grid structure used in the simulations is shown in Fig 8. The particular grid that is chosen is very important in reservoir simulation. An areal grid of 288 blocks ( 16 x 18 blocks) - about 10 acre each is taken along with six layers in the vertical direction; i.e. a total of 1728 blocks. This would be a very small model by todayʼs standards and could easily be run on a PC - this was not the case in late 1970s. 17b. They refer to changing the transmissibilities between grid blocks in order to reduce flows. (See Glossary for exact definition of transmissibility.) 18. The following three concepts are closely related (see Pseudo-isation and Upscaling in the Glossary): 18a. Grid size sensitivity: Refers to the introduction of errors due to the coarsness of the grid known as numerical dispersion. 18b. The very important concept of pseudo--relative permeability is introduced here (Kyte and Berry, 1975). “Pseudos” are introduced in order to control numerical dispersion and account for layering. In essence, the use of pseudos can be seen as a fix up for using a coarse grid structure. 18c. Corresponding coarse and fine grid reservoir models are shown in Fig. 9. They note that the fine grid model uses rock relative permeabilities while the coarse grid model uses pseudo relative permeabilities. 19. History Matching: The basic idea of history matching is that the model input is adjusted to match the field pressures and production history. This procedure is intended as being a way of systematically adjusting the model to agree with field observations. Hopefully we can change the “correct” variables in the model to get a match e.g. we may examine the sensitivity to changes in vertical flow barriers in order to find which level of vertical flow agrees best with the field (indeed, this is done in this study). See History Matching in the Glossary. 20a. “Early” mechanism identified as solution gas drive and assistance from expansion. Some initial discussion of field experience and numerical simulation conclusions is presented and developed in these points. 20b. They note some problems with data from early field life. (i) Complicated by free gas production; (ii) channelling due to poor well completions; (ii) no accurate records on gas production for the first 6 years. 20c. The actual field history match indicates that approx. 8 - 10 BCF of gas must have been produced over this early period in order to match the field pressures. This is a use of a material balance approach in order to find the actual early STOIIP (STOIIP = Stock Tank Oil Initially In Place).


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21a. They present a description of some adjustments to the history match - but overall it is very good (which they attribute to extensive core data). 21b. Some highlighting of problems with earlier water injection . 21c. The actual history match of reservoir pressure and production is shown in Fig. 10. This is a good history match but think of which field observable - gas production, water production or average field pressure - is the easiest/most difficult to match? 22. A good description of their study of the sensitivity to vertical communication is given at this point. This is examined by adjusting the vertical transmissibilities. They look at the following cases: (i) no barriers; (ii) moderate barrier; (iii) strong barriers and (iv) no-flow barriers. Most of the sensitivties are for the moderate and strong barrier cases. 23a. Results showed that => strong barrier case is best but some problem high GOR wells are encountered randomly spaced through the field. They diagnosed and simulated this as “behind the pipe” gas flow in these wells to explain the anomalies in the field observations. This is quite a common explanation that appears in many places. 23b. Layer differential pressures up to 200 - 250 psi can only be reproduced for the strong barrier case. In simulation terms, this is probably the strongest evidence that this is the best case match. 24. The strong barrier case was chosen as the base case and this was used for the predictive runs. The base case predictions refer to the cases which essentially continue the current operations and these are shown in Fig. 11. 25. The strategies looked at for the future sensitivities are listed as follows: (i) change rate of water injection; (ii) management of gas cap voidage i.e. increase of gas and blowdown at different times; (iii) infill drilling. 26a. Outlines the problems/issues for various strategies as follows: (i) shows vertical communication is very importance - it has a major impact on predicted reservoir performance; (ii) shows that can avoid high future ΔP between gas cap and oil zone by high water injection or early blowdown; (iii) shows better development strategy is to keep low ΔP e.g. increase gas injection or infill drill. Finally, shows infill drilling is the most attractive option and the forward prediction for this case is shown in Figure 12. 26b. Table showing some alternatives in text. 27a. A brief summary of the best future development option is given which is: (i) infill drilling as the best option; (ii) water injection increased concurrently with the drilling program to maintain voidage replacement (but prevent the over-injection of water). 27b. For completeness, it is explained why other plans are not as attractive; i.e. blowdown of gas cap before peak in waterflood production rate would significantly reduce oil recovery. 26


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28. A reasonably good initial forward prediction from 1978 - 1981 is shown in Figs. 13 and 14. 29. Conclusions are given which, in summary, are as follows: 1.

Detailed reservoir description essential for numerical modelling.

2.

Carbonate - both primary and post-depositional diagenetic factors are important.

3.

Waterflood in W. Seminole very sensitive to vertical permeability.

4.

Vertical permeability is broadly characterised using 3D numerical simulation.

5.

Understanding of reservoir response (mechanism) essential to good management.

6.

Management of W. Seminole field best if minimum _P between oil zone and gas cap (lower losses of oil --> gas cap) by: (i) infill drilling; (ii) controlling water injection rates to maintain voidage replacement - donʼt over-inject; (iii) careful management of voidage replacement into gas cap.

Important terms and concepts introduced in SPE10022: Specific to Reservoir Simulation: history match, permeability distribution, black oil model, grid structure, transmissibility, numerical dispersion, pseudo--relative permeabilites. General terms: 5-spot water flood, permeability distribution, k/φ correlation, steadystate relative permeability, rock relative permeabilities, solution gas drive, material balance, infill drilling, voidage replacement.

3.4 Ten Years Later - 1992

An interesting snapshot of where reservoir simulation technology had reached 10 years after the West Seminole study can be seen in the following papers: From the proceedings of the SPE 67th Annual Technical Conference, Washington, DC, 4-7 October 1992: SPE24890: “From Stochastic Geological Description to Production Forecasting in Heterogeneous Layered Systems”, K. Hove, G. Olsen, S. Nilsson, M. Tonnesen and A. Hatloy (Norsk Hydro and Geomatic) Summary: This paper describes the transfer of data from a detailed gridded stochastic geological model to a more coarsely gridded reservoir simulation model. It is essentially a field application of a methodology described in a previous paper from the same company (Damsleth et al, 1992; Damsleth, E., Tjolsen, C.B., Omre, H. and Haldonsen, H.H., “A Two Stage Stochastic Model Applied to a North Sea Reservoir”, J. Pet. Tech., pp. 402-408, April 1992). The two step procedure involves a first step of constructing the geological architecture of the reservoir followed by a Institute of Petroleum Engineering, Heriot-Watt University

27

second stage where the petrophysical values are assigned to each building block in the geological model. The consequences of making various assumptions in the gridding are evaluated for water, gas and WAG (water-alternating-gas) injection. They note that is it very important to represent the main geological features in the gridded model. It was also noted that, when a regular coarse grid was used, the contrast in properties of this heterogeneous reservoir were “smoothed out” by the averaging process and in most cases led to a more optimistic predicted production performance. That is, the more stochastic models led to a reduction in predicted recovery compared with conventional coarse gridded models. In the proceedings of the SPE European Petroleum Conference, Cannes, France, 16-18 November 1992. A session at this conference produced the following selection of reservoir simulation papers: SPE25008: “Reservoir Management of the Oseberg Field After Four Years”, S. Fantoft (Norsk Hydro) Summary: The Oseberg Field (500x106 Sm3 oil; 60x106 Sm3 gas) comprises of seven partly communicating reservoirs. Both water and gas are being injected and modelled in this study and results indicate over 60% recovery in the main reservoir units. The modelling results indicate that the plateau production will be extended by the use of horizontal wells. The objective of the simulation study was exactly this - i.e. to maximise the plateau and improve ultimate oil recovery. This is a very competent simulation study and - although details are not given - it is stated that the geological model is updated annually based on information from new wells. It establishes several aspects of the reservoir mechanics and makes a number of recommendations for operating practice in the future. In other respects, this is quite a “conventional” study. SPE25057: “The Construction and Validation of a Numerical Model of a Reservoir Consisting of Meandering Channels”, W. van Vark, A.H.M. Paardekam, J.F. Brint J.B. van Lieshout and P.M. George (Shell) Summary: This study focuses on a reservoir which has low sandbody connectivity and which is interpreted as a meandering channel fluvial system. Two years of depletion data is available and one of the aims of the study was to evaluate the possibility of performing a waterflood in this field. They identified a problem in that the sandbody connectivity was lower than might be expected from the sedimentology alone and it was conjectured that this might be due to minor faulting with throws of a few meters. This study again emphasises the importance of the reservoir geology and tries to relate the performance back to this. The geological model is also an early practical example of using a “voxel” representation of the system - approx. 128,000 voxels were used in the model. They noted that the original (sedimentological) models gave over optimistic connectivity. An acceptable match to observed field pressures by including some level of smaller scale faulting. SPE25059: “Development Planning in a Complex Reservoir: Magnus Field UKCS Lower Kimmeridge Clay Formation (LKCF)”, A.J. Leonard, A.E. Duncan, D.A. Johnson and R.B. Murray (BP Exploration Operating Co.) Summary: This simulation study was carried out on the geologically complex, low net to gross LKCF (rather than on higher net to gross Magnus sands studied 28


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previously). The objective was to formulate a development plan for the LKCF which would accelerate production from these sands. Stochastic modelling techniques were integrated into more conventional “deterministic” models and various options were screened for inherent uncertainty and risks. The study concluded that a phased water injection scheme was the best way forward with the phasing being used to manage and offset the considerable geological risks. Ranges of expected recovery were generated and an incremental recovery of 60 MMstb was predicted increasing the total reserve of the LKCF by a factor of x2.4. This study also demonstrated the importance of inter-disciplinary team work to overcome the previously inhibiting high risks involved. The proceedings of Europec92 also included the following paper: SPE25006: “Anguille Marine, a Deepsea-Fan Reservoir Offshore Gabon: From geology Toward History Matching Through Stochastic Modelling”, C.S. Giudicelli, G.J. Massonat and F.G. Alabert (Elf Aquitaine) This paper is such a good example of contemporary studies at that time, that this is chosen as our Case 2 example and is presented in some detail in the next section.

3.5 Case 2: The Anguille Marine Study (SPE25006,1992)

Case 2: “ Anguille Marine, a Deepsea-Fan Reservoir Offshore Gabon: From geology Toward History Matching Through Stochastic Modelling”, SPE25006, presented at the SPE European Petroleum Conference (Europec92), Cannes, France, 16-18 November 1992, by C.S. Giudicelli, G.J. Massonat and F.G. Alabert (Elf Aquitaine) See SPE 25006 paper in Appendix Summary: The Anguille Marine Field in Gabon has 25 years of production history. The waterflood performance indicated severe sedimentary heterogeneity as the field is known to have been deposited in a deep water fan sedimentary environment. This paper is one of the first to refer to the multi-scale nature of the heterogeneity (5 scales were studied) and to refer this back to the sequence stratigraphy of the depositional environment. The sequence stratigraphic approach allowed the field to be divided into the main types of turbiditic geometries (channels, lobes, slumps, laminated facies). Fine scale models (> 2 million grid blocks) were generated using geostatistical techniques and several issues were raised concerning both the geological model and the upscaling process itself. This is a very good example of an early integrated geology(sedimentology)/engineering study in reservoir simulation. The multi-scale nature of the heterogeneity is well related back to the geology.

Study Notes Case 2:

1. Depositional environment: the Anguille Marine field is a deep sea fan environment (i.e a turbidite) with a low sand/shale ratio. This geological description opens the discussion (unusual for previous simulation studies) and the geology features heavily in the flow properties and hence in the geological and reservoir models of this field. 2. Sequence stratigraphy: A more modern feature of reservoir simulation is that the five identified scales of heterogeneity are recognised and some attempt is made to Institute of Petroleum Engineering, Heriot-Watt University

29

incorporate them into the 3D simulation model. These scales are also firmly linked to the geology (sedimentology) through the principles of sequence stratigraphy. 3. Geostatistics: Reference is made to how the geological features constrain the fine scale 3D models (of > 2 million blocks - which was large for the time) using geostatistical techniques. By the early 90s, the use of geostatistical methods was becoming more widespread and how it has been applied in this case is covered better by Refs. 1 - 5 in this paper. Location and structure maps of Anguille Marine are given in Figures 1 - 4. 4. Brief field facts: Discovery 1962; primary depletion commenced in 1966 but reservoir pressure fell rapidly over the next 2 - 3 years and GOR increased; waterflooding from 1971 restored pressure support but channelling led to early water breakthrough; infill drilling not very successful due to lack of current understanding of complex reservoir geology; new approach in 1990 focused more strongly on the reservoir geology of this heterogeneous low sand/shale ratio system recognising the characteristic geometries of a tubditic fan - lobes channels, levees, slumps, laminated facies etc. 5. The approach: It is important in all reservoir simulation studies to have a clear logic to how we approach the simulation of a large complex reservoir system. Here they describe their general methodology although details are in Refs. 1 - 4 at the end of the paper. Basically they: describe and model upper reservoir/ extend to the whole reservoir/ try to translate the geological model to a practical simulation model. On the latter issue they describe the use of “partial models” where just a smaller sector of the reservoir is studied but lessons are taken back into the full model. 6. Reservoir description: Section 2 of the paper gives a sedimentological description of the reservoir as a “slope-apron fan” of complex lithology (depositional model Figure 3) in which 14 (simplified) facies were retained; criteria of composite log recognition of various facies shown in Figure 5. Some contradictory water breakthrough observations were noted. Table 1 gives sedimentary body dimensions (lengths and widths) for channels, lobes. levees/crevasse-splay, slumps, channels (Upper Anguille); Table 2 gives mean petrophysical characteristics. A very important final result for reservoir simulation is the identification of five scales of heterogeneity - Figures 6 and 7; this makes the geological analysis and information numerically useable. 7. Sedimentary history: In earlier reservoir simulation studies, and indeed up to the present time, it is rare to see sedimentary history discussed in terms of a sequence stratigraphic analysis (even mentioning the pioneering work on sea level changes of P. R. Vail et al, “Seismic stratigraphy and global changes of sea level”, in Seismic Stratigraphy, Applications to Hydrocarbon Exploration, AAPG Memoir 26, pp. 49-212, 1977). Chronostratigraphic correlations refer to the “timelines” of simultaneous deposition. This analysis underpins much of the reservoir description but we will not elaborate on it here. 8. Geostatistical modelling: Mainly discussed in Refs. 1 and 2 of this paper. Firstly, focus on geostatistical modelling of the 3D distributions of the major flow units (channels and lobes) and barriers (laminated facies or slumps) for the entire reservoir. This is done as a “conditional” simulation where the distribution is constrained 30


1

(or conditioned) to the observed facies and reservoir quality observed at the wells. Secondly, the smaller scale heterogeneities are “unconditionally” simulated (synthetically) to yield average properties within the major flow units (see Ref. 2 in paper). The geostatistical simulation method used was “indicator simulation” (Refs. 1 and 8) which require the average frequency and variogram information. 9. Reservoir zonation: Six unit vertical reservoir zonation shown in Figure 10. Simulations of lateral continuity within each of the units (five - not Middle Anguille, Figure 10) performed independently since they correspond to separate sedimentary phases. For horizontal zonation, Figure11 shows lateral zonation on LA2 and UA2 units showing directional trends and thus variograms with spatially variable anisotropy direction used in final model. Figures 12 - 15 show resulting correlation structures of the various units. Ends up with >2 million grid blocks in the full field 3D model. 10. Flow simulations: Discusses details of upscaling from fine grid stochastic model (>2 million blocks) to coarse grid simulation model (11,000 grid blocks). 11 vertical layers are retained to represent the reservoir layering with more blocks being used in the best reservoir units. Upscaling of absolute permeability at some “aggregation rate” (e.g. 4x4) is applied leading to areal block sizes of 200m x 200m - see Figure14. Relative permeabilitiees were upscaled “on a typical block configuration” (details in Refs. 2 and 4). Additionally: Three major zero-transmissibility faults included in model; some WOC variation across field; depth varying bubble points assigned; 25 years of injection/production for history matching. 11. Simulation results: Initial pressure depletion results shown in Figure 16 - where 14 out of 17 wells show satisfactory pressure behaviour. Pressure behaviour and water breakthrough are poorly predicted during injection stage - Figure 17; water saturations around injectors shown in Figure18 - upscaling has “washed out” the finer scale strong anisotropy. 12. Model changes: Table 8 lists a number of sometimes quite radical changes to the model in order to achieve a better fit to observed field performance - Figure 15 shows differences in upscaled permeability maps. Continuing problems with injection predictions => - is geological model correct? - what is the real effect of upscaling? 13. Partial models: “Thin” model - Figure 19 shows the “thin” partial field model to verify reservoir geology; well AGM18 good water breakthrough match (Figure 20) - early breakthrough for well AGM29 (Figure 21). When thin model upscaled as in full field model (abs. k upscale + rel perm as before) - results in Figures 21 and 22 - breakthrough delayed in both wells but shape of BSW is satisfactory. Conclusions: Thin model partly validates geological model; Some problems with upscaling not supressing breakthrough, making reservoir too connected and eliminating strong anisotropy. Test model - (50 x 20 x 56) model extracted from full field model. Figure 23 shows that an optimum upscaling aggregation rate (2 x 2 x 7) is found - they warn caution on this point. We note that if very reliable and general upscaling techniques were available, then this should be eliminated (more work has been done on this issue since 1992 - much of it at Heriot-Watt!).


31

14. Conclusions: • Sedimentology controls heterogeneity analysis when very wide variation in sandbody geometries is found (as in this case) •

Link understanding of reservoir history to sequene stratigraphy

•

Litho-interpretation of seismic canʼt give paleo-direction when there is techtonic activity during sedimentation

•

Multi-scale heterogeneity analysis essential to quantify sub-grid petrophysical properties

•

Geostatistical indicator simulation is a good tool for modelling this multi- scale heterogeneity - trends can also be included

•

Stochastic model for Anguille Marine constrained by geology gives hopeful first results

•

If aggregation rate in upscaling is optimised, history matching is possible with the use of strictly controlled geological parameters

3.6 Case 3: Ubit Field Rejuvenation (SPE49165,1998)

Case 2: “ The Ubit Field Rejuvenation: A Case History of Reservoir Management of a Giant Oilfield Offshore Nigeria”, SPE49165, presented at the SPE Annual Technical Conference and Exhibition, New Orleans, LA, 27-30 September 1998, by C.A. Clayton et al (Mobil and Department of Petroleum Resources, Nigeria) See SPE 49165 paper in Appendix Summary: This is another good example of where integrated reservoir management has greatly contributed to the success of a field redevelopment plan. In particular, a clearer understanding of the reservoir structural geology has been central to this process. The reinterpretation of the structural geology of the field (the fault blocks, compartments and slump blocks) was achieved using seismic data in a range of complementary ways. The Ubit reservoir is a prograding shallow marine system which has been tectonically disturbed. The downslope movements of the youngest sand sequences resulted in large scale slumping and block sliding although reservoir quality in these sediments is good to excellent. Important facts on the Ubit reservoir and this study are: STOIIP = 2.1 billion bbl oil; 37ºAPI black oil, Bo = 1.38, GOR 612 scf/stb, μo = 0.64 cp and μg = 0.16 cp; production from a relatively thin oil column (160 ft.) and a fairly thick gas cap (50 - 550 ft.). Previous average production = 30 MBD; after implementation of study recommendations (many horizontal wells etc.), expected production ≈ 140 MBD. The notes on this SPE paper will not be very extensive and only a few of the main novel points will be discussed below.

Study Notes Case 3:

1. New data and techniques: The study is a very good example of the close integration of (especially) 3D seismic data used in several ways, computer mapping 32


1

and reconstruction of the slump blocks, advanced reservoir simulation procedures, visualisation etc. 2. Recommendations: These have been quite clearly established and stated. The study shows that the key strategies are • • • • • •

Implement horizontal well drilling (approx. 57 wells). Full field simulation defining drilling placement and timing. Balancing a non-uniform gas cap. Maintaining a stable gas cap (gravity stable displacement) and pressure. Establish field plateau rate. Minimising free gas production.

3. Uncertainties: there was an initially erroneous view of certain aspects of the reservoir geology and the key uncertainties at the start of the study were • • • •

Geological complexities in reservoir architecture, particularly structural deformation. Sandbody geometries. Petrophysical rock and fluid properties. Distribution of flow units.

4. Structural reinterpretation: Figures 2a and 2b show both the original and current interpretations of the structure. The original “rubble beds” are reinterpreted as being techtonically disturbed downslope movements of the youngest sand sequences resulting in large scale slumping and block sliding. The older interpretation saw these facies as being essentially “chaotic” whereas they are now thought to be more ordered and predictable. 3D seismic data is of central importance in the definition of the structural geometries where the “bedded” and “disturbed” strata are shown on a seismic section in Figure 4. Several seismic techniques were applied including attribute analysis, rock physics and amplitude analysis, seismic facies analysis of time slices and conventional reflector mapping. The resulting 70 internal slump and fault blocks are shown in Figure 7. 5. Petrophysics-based facies: Seven flow controlling depositional facies were identified as shown in Figure 8 with rock properties related to grain size (lithology, typical log response, net/gross, k vs. φ, Pc and kro-krw). Depositional facies types present are - marine turbidites and debris flow sands; lower delta plain tidal channels and lagoonal sands; shallow marine upper shoreface and lower shoreface sands and shelf shales (best are turbidites, shoreface and channel sands - comprise 80% pore volume in oil column). 6. Layering and Reservoir Simulation Model: Vertical layering is shown schematically in Figure 9 and the areal grid is given in Figure 12, with a set of rock property maps for a single simulation layer given in Figure10. Grid is Nx x Ny x Nz = 93 x 40 x 18 (67,000 blocks) with most oil leg cells being Δz = 10ft. to resolve the gravity stable gas front. Rock property slices were loaded into the 3D modelling software to “connect up” the stratigraphic layers (using new but unclear developments by authors) as shown in Figure 11.


33

7. Relative permeability: An interesting point is made concerning the oil relative permeability (kro) close to its end point - see Figure 14. Although kro is low in this region (kro ~10-6 - 10-7), it significantly affects the tail of the reservoir oil production profile - see Figure 15. The adjusted curve in Figure 14 is used to get more realistic longer time recoveries (Figure 15). This is important because gravity stable gas cap expansion and downward displacement is the principal oil recovery mechanism - see schematic view in Figure 9. 8. History match and forward prediction: Average field pressure and GOR are history matched - see Figures 15 and 16. Matching field pressure is not so difficult since Ubit shows good pressure communication (high k - lack of sealing faults). Some discussion of field management is presented. Figure 22 shows the initial part of the improved productivity (up to 140 MBD from 37 horizontal wells) and future predictions. See Recommendations (point 2 ) above.

3.7 Discussion of Changes in Reservoir Simulation; 1970s - 2000

From the above field examples (Cases 1 -3), there is clearly a progression in the engineering approach, the degree of reservoir description and the computational capabilities as we go from reservoir simulation in the late 1970s to the present time. The main changes are as follows: Computer power: There has been a vast increase in computer processing power over this period because of : (a) CPU: The growth of powerful CPU (central procesing units - i.e. chips) especially as implemented in Unix machines (workstations) and RISC technology and more recently by the development of modern PCs. The corresponding cost of computing has fallen dramatically. A graph of processing power (Mflops/s) vs. time and a corresponding graph of maximum practical model size vs. time is shown in Figure 10:

1000000

Gridblocks

Mflops/s

1000

100000

100 10 1

1000

0.1 1970

10000

1975

1980

1985

1990

1995

Year (a) State of art CPU performance

2000

100 1960

1970

1980

Year

1990

2000

(b) Maximum practical simulation model size

(b) Parallel Processing: Part of the increase in computing power referred to above is the growth of parallel processing in reservoir simulation. The central idea here is to distribute the simulation calculation around a number of processors ( or “nodes”) which perform different parts of the computational problem simultaneously. A bank of such processors is shown in Figure 11 (from the 34

Figure 10 (a) CPU performance (Mflops/s) vs. time and (b) maximum practical model size vs. time; Mflop/s = mega-flops per second = million floating point operations per second; from J.W. Watts, “Reservoir Simulation: Past, Present and Future”, SPE Reservoir Simulation Symposium, Dallas, TX, 5-7 June 1997.


1

work of Dogru, SPE57907, 2000). The general impact of parallel simulation is shown according to Dogru (2000) in Figure 12. If the problem gets linearly faster with the number of parallel processors, then it is said to be “scalable” and the closeness to an ideal line is a measure of how well the process “parallelises” (reaches the ideal scaling line); an example is shown in Figure 13. Finally, the type of fine scale calculation that can now be performed using megacell simulation is shown in Figure 14 where it is shown that there is a lengthscale of remaining oil that is missed in the coarser (but still quite fine) simulation. A table of what types of calculation can be performed and some timings for these is also included (although these numbers will probably be out of date very quickly!). For further details, see Megacell Reservoir Simulation - A.H. Dogru - SPE Distinguished Author Series, SPE57907, 2000 and the references therein.

Figure 11 A cluster of parallel processors; from “Megacell Reservoir Simulation” A.H. Dogru - SPE Distinguished Author Series, SPE57907, 2000.

1.2 1

Parallel Simulators

0.8

Figure 12 The impact of parallel reservoir simulation; from “Megacell Reservoir Simulation” - A.H. Dogru - SPE Distinguished Author Series, SPE57907, 2000.

0.6 0.6 Conventional Simulators

0.2 0

1988

1990

1992

1994

1996


1997

1998

35

0.6 Conventional Simulators

0.2 0

1988

1990

1992

1994

1996

1997

1998

Speedup

16

12

8

4

0

0

4

12

8

Figure 13 A cluster of parallel processors; from “Megacell Reservoir Simulation” A.H. Dogru - SPE

16

Number of Nodes Cluster

SP2

Ideal

Distinguished Author Series, SPE57907, 2000.

Fig. 7—Comparison of conventional simulation (40,500 cells, 5 la yers) and megacell simulation 2.45 ( million cells,67 layers).

Reservoir

Model Size (Millions of Gridblocks)

History Length (Years)

CPU Hours On 64 Nodes On 128 Nodes

Carbonate

1.2

27

3

1.7

Sandstone

1.3

49

4.5

2.5

Carbonate (With gas cap)

3.9

10

-

2.0

Carbonate

2.5

2.5

-

4.0

(c) Visulisation: Huge improvements in visualisation capabilities have taken which allow us to evaluate vast quantities of numerical data in a more convenient manner. This, in turn allows us to apply our intuitive engineering judgement to reservoir development problems. In addition, the visual representation of output allows a more fruitful communication to take place between geologists and engineers; (d) Integrated software: The availability of more integrated software suites that handle all the initial data from seismic processing, to petrophysical analysis and data generation, geocellular modelling and upscaling through to the actual simulators themselves. 36

Figure 14 The type of reservoir simulation that becomes more possible with parallel processing. Comparison with fine grid and megacell simulation which identifies the scale of remaining oil in a reservoir displacement process; from “Megacell Reservoir Simulation” A.H. Dogru - SPE Distinguished Author Series, SPE57907, 2000.


1

(e) Linked to this increased power, is the ability to handle huge geocellular models and somewhat smaller but still very large reservoir simulation models. Geostatistics: there have been significant advances in the application of geostatistical techniques in reservoir modelling. Such approaches were quite well known in the mining, mineral processing and prospecting industries but only in the last 10 to 15 years have they been specifically adapted for application in petroleum reservoir modelling. Introductory texts are now available such as: •

“An Introduction to Applied Geostatistics” by E.H. Isaaks and R.M. Srivastava, Oxford University Press, 1989

•

“Introduction to Geostatistics: Applications in Hydrogeology” by P.K. Kitanidis, Cambridge University Press, 1997

Both pixed-based point geostatistical techniques and object based modelling have been developed and applied in various reservoirs. Upscaling: There have been a number of advances in approaches to upscaling (or pseudo-isation) from fine geocellular model → the reservoir simulation model. This still an area of active research and the debate is still in progress on the question: •

Will increasing computing power remove the need for upscaling?

The basic idea of upscaling has been introduced in the SPE examples. Upscaling is dealt with in much more detail in Chapter 7. Organisational changes in the oil industry: A number of major organisational changes have occurred in the oil industry since the 1970s which have affected the practice of reservoir simulation. The main ones are as follows: (a) Many companies have taken a more integrated geophysics/geology/engineering view of reservoir development and many studies have made a central virtue of this by organising reservoir studies within more multi-disciplinary asset teams (e.g. SPE25006 clearly shows a strong integration of geology and engineering); (b) There have been significant organisational changes in the structure of the industry given the sucessive rounds of “downsizing” and “outsourcing” that have occurred. For example, see the short article by Galas, “The future of Reservoir Simulation”, JCPT, p.23, Vol. 36 (1), January 1997, which is reproduced in Appendix B. This article has an interesting slant from the point of view of the smaller consultant and it makes a number of interesting observations; (c) There have been a number of major mergers and take-overs recently which have formed some very large companies e.g. BP (BP - Amoco - ARCO), ExxonMobil, Total-Fina-Elf. Likewise, a number of very “low cost” operations have grown up which may specialise in the successful (i.e. profitable) exploitation of mature assets e.g. Talisman, Kerr-McGee etc. How these changes will affect the future of reservoir simulation remains to be seen.


37

(d) Up until the late 1970s, almost every major and medium sized oil company had a “research centre” where programmes of applied R&D were carried out by oil company personnel. The view was essentially that this in-house technology development would give the company a competitive and commercial edge in reservoir exploration and development. Most companies have greatly reduced the amount of in-house “R” that takes place and have focused much more heavily on shorter term asset related “D”. Many companies do support research in universities and other independent outside organisations - they also ally themselves with service companies in order to have their R&D needs met in certain areas. Again, the situation is in flux and the longer term effects of this change is yet to be seen. (e) More specific to reservoir simulation is the fact that, in the 1970s, most companies would have had their own numerical reservoir simulator which was built (programmed up) and maintained in-house. To this day, a few companies still do. However, most oil companies use specialised software service companies to supply their reservoir simulation (and visualisation, gridding etc.) software. Again, the relative merits and demerits of this will emerge in the coming years. Detailed technical advances: In addition to the changes discussed above, many advances have been made over the past 50 years on how we perform the simulations i.e. on the formulation and numerical methods etc. Our practical capabilities have also expanded greatly as discussed above. Table 1 presents a list of capabilities and major technical advances in reservoir simulation over the last 50 years; this table was adapted from two tables in J.W. Watts “Reservoir Simulation: Past, Present and Future”, SPE38441, SPE Reservoir Simulation Symposium, Dallas, TX, 5-7 June 1997. This article is well worth reading. Most of the technical details in the advances listed in Table 1 are beyond the scope of this course and the introductory student does not need to have any in-depth knowledge on these.

38


Decade

Capabilities

References

1950’s

Two dimensions Two incompressible phases Simple Geometry

Aronofsky and Jenkins radial gas model Alternating-direction implicit (ADI) procedure

Aronofsky, and Jenkins (1954) Peaceman and Rachford (1955)

1960’s

Three dimensions Three phases Black-oil fluid model Multiple wells Realistic geometry Well coning

IMPES computational method Upstream weighting Understanding of numerical dispersion Strongly-implicit procedure (SIP) Implicit computational method Additive correction to line-successive overrelaxation

Sheldon et al (1960) Stone and Garder (1961) Lantz (1971) Stone (1968)

Compositional Miscible Chemical Thermal

Stone relative permeability models Vertical equilibrium concept Todd-Longstaff miscible displacement computation Two-point upstream weighting D4 direct solution method Total velocity sequential implicit method Pseudofunctions Variable bubblepoint black-oil treatment Conjugate gradients and ORTHOMIN

Stone (1970, 1973) Coats et al (1971) Todd and Longstaff (1972)

1970’s

Table 1 Capabilities and major technical advances in reservoir simulation over the last 50 years (adapted from two tables in J.W. Watts “Reservoir Simulation: Past, Present and Future”, SPE38441, SPE Reservoir Simulation Symposium, Dallas, TX, 5-7 June 1997; (all references are given in Appendix A)

Technical Advances

1

Iterative methods based on approximate factorizations Peaceman well correction Nine-point method for grid orientation effect

MacDonald and Coats (1970) Watts (1971)

Todd et al (1972) Price and Coats (1974) Spillette et al (1973) Kyte and Berry (1975) Thomas et al (1976) Meijerink and Van der Vorst (1977) Vinsome (1976) Peaceman (1978) Yanosik and McCracken (1979)

1980’s

Complex well management Fractured reservoirs Special gridding at faults Graphical user interfaces

Code vectorization Nested factorization Volume balance formulation Young-Stephenson formulation Adaptive implicit method Constrained residuals Local grid refinement Cornerpoint geometry Geostatistics Domain decomposition

Appleyard and Cheshire (1983) Acs et al (1985); Watts (1986) Young and Stephenson (1983) Thomas and Thurnau (1983) Wallis et al (1985) Ponting (1992)

1990’s

Improved ease of use Geologic models and upscaling Local grid refinement Complex geometry Integration with non-reservoir computations

Code parallelization Upscaling Voronoi grid

Heinemann et al (1991) Palagi and Aziz (1994)

3.8 The Treatment of Uncertainty in Reservoir Simulation

Figure 15 A single forward prediction of the oil recovery production profile for a given reservoir.

Cumulative Oil Recovery

It is well recognised in modern reservoir development that when we calculate the future oil recovery profile of a reservoir, it is not “accurate”. Suppose a particular reservoir simulation model for Field X - a new development which is currently under appraisal - gives the forecast in Figure 15.

2000

2005

2010 Time (Year)


39

Clearly, we cannot trust this single curve since there is a considerable amount of uncertainty associated with it for various easily appreciated reasons. The main contributors to this uncertainty are to do with lack of knowledge about the input data although the modelling process itself is not error free. A list of possible sources of error is as follows: •

Lack of knowledge or wide inaccuracies in the size of the reservoir; its areal extent, thickness and net-to-gross ratios

•

Lack of knowledge about the reservoir architecture i.e. its geological structure in terms of sandbodies, shales, faults, etc.

•

Uncertainties in the actual numerical values of the porosities (φ) and permeabilities (k) in the inter-well regions (which make up the vast majority of the reservoir volume)

•

Inaccuracy in the fluid properties such as viscosity of the oil (μo), formation volume factors (Bo, Bw, Bg), phase behaviour etc., or doubts about the representativity of these properties

•

Lack of data - or very uncertain data- on the multiphase fluid/rock properties, particularly relative permeability and capillary pressure, and on knowledge as to how these curves vary from rock type within the reservoir volume away from the wells

•

Because the representational reservoir simulations model may be poor, e.g. the numerical errors due to the coarse grid block model may significantly affect the answer in either an optimistic or pessimistic manner.

The above list of uncertainties for a given reservoir, especially at the appraisal stage, is really quite realistic and is by no means complete. As we have noted elsewhere, it is at the appraisal stage when, although the future reservoir performance is at its most uncertain, we must make the biggest decisions about the development and hence speed most of our investment money. At a first glance, the task of doing something useful with reservoir simulation may seem quite hopeless in the face of such a long list of uncertainties. No matter how bleak things look, the only two options are to give up or do something, and reservoir engineers never give up! We must produce an answer - even if it is an educated guess (or even just a guess) - and some estimate of the sort of error sound that we might expect. Before considering what we can do in practice, let us first consider what the answer might look like for the case above in Figure 15. Figure 16 gives some idea of what is required:

40


1

Figure 16 Outcome of reservoir simulation calculations showing a range of recoveries for various reservoir development scenarios.

Cumulative Oil Recovery

Most probable case

Range of cases with a 50% probability

Range of cases with a 90% probability

2000

2005

2010 Time (Year)

The results in Figure 16 can be understood qualitatively without worrying about how we actually obtain them right now. Our single curve in Figure 15 may becomes a “most probable” (or “base case”) future oil recovery forecast. The closer set of outer curves is the range of future outcomes that can be expected with a 50% probability. That is, there is a probability, p = 0.5, that the true curve lives within this envelope of curves shown in Figure 16. Such results allow economic forecasts to be made with the appropriate weights being given to the likelihood of that particular outcome. A company can then estimate its risk when it is considering various field development options. In fact, here we will just discuss doing some simple sensitivities to various factors in the simulation model. We can think of a given calculation as a scenario. Therefore, we can set up various scenarios based on our beliefs about the various input values in our model and we simply compare the recovery curves for each of the cases. For example, suppose we have a layered reservoir as shown in Figure 17 which we think has a field-wide high permeability streak set in background of 100 mD rock. INJECTOR

Figure 17 This shows a layered reservoir where we have some uncertainties in the various parameters such as the permeability (khi ), the thickness (ΔZhi ) and the porosity (φhi ) in the high permeability layer.

PRODUCER 1000ft 1000ft

klow = 100mD

100ft φ = 0.18 High Permeability Streak, khigh, φhi

∆Zhi

The reservoir is being developed by a five-spot waterflood as shown in Figure 17 However, we are uncertain as to the actual thickness (ΔZhi), the permeability (khi) and the porosity (φhi) of the high permeability streak. From various sources, we derive mean, high and low estimates of each of these quantities as follows:


41

Permeability, khi Thickness, ∆Zhi Porosity, φhi

Low Value

Mean Value

High Value

400 mD

800 mD

1600 mD

20 ft 0.18

30 ft 0.22

40 ft 0.26

Even with just the three uncertainties in this single model, we can see that there are 3x3x3 = 27 possible scenarios or combinations of input data for which we could run a reservoir simulation model. Alternatively, we could conclude that some input combinations are unlikely (e.g. lower permeability with higher value of porosity) and we could reduce the number. We could simply keep the mean value of two of the factors while varying only the third factor, leading to 7 scenarios to simulate. Taking this view, we can take some measure of the oil recovery e.g. cumulative oil produced (predicted) at year 2010. The notional results could be entered in Table 2 Changed Input Value

Oil Initially in

Cumulative

% Change in

Change in Recovery

Place (OIIP) (res. bbl)

Recovery at Year 2010 (stb)

Input Value Relative to Base Case

from Base Case

Base Case khi = 400 mD khi = 1600 mD

Table 2 Results of sensitivity simulations described in the text.

∆Zhi = 20 ft Zhi = 40 ft φhi = 0.18 φhi = 0.25

Note: The OIIP will vary somewhat from case to case since the thickness of the high permeability layer and its porosity both change. In Table 2, we have noted the % change in the varied parameters relative to its base case value. Not that different physical quantities such as k and φ, vary by different percentages for realistic min./max. values. A useful way to plot the variation in recovery is against this % change in input value since all three factors can be represented on the same scale in a so-called “spider diagram”. Such a plot is shown in Figure 18.

Porosity

Permeability

X X

Layer Thickness

% Change in Parameter

Change in Recovery From Base Case (STB)

42

Figure 18 “Spider diagram” showing the sensitivity of the cumulative oil to various uncertainties in the reservoir model parameters (khi; ΔZhi; φhi) in Figure 17.


1

This type of spider plot is very useful since it displays the effect of the different uncertainties on the outcome. It clearly highlights which is the most important input quantity (of those considered) and has the most impact on the result. Thus, if we were going to spend time and effort on reducing the uncertainty in our predictions, then this tells us which quantity to focus on first. Indeed it ranks the effects of the various uncertainties. There are more sophisticated ways to deal with uncertainty in reservoir performance but these are beyond the scope of the current course. The basic ideas presented above give you enough to go on with in this course.

4 STUDY EXAMPLE OF A RESERVOIR SIMULATION The examples presented in the above SPE papers should give you a good idea of what reservoir simulation is all about. By this point you should also be familiar with many of the basic concepts that are involved in reservoir simulation from the study notes and the Glossary. However, the reservoir simulation of say a waterflood or a gas displacement is a dynamic process. That is, as time progresses, the water or gas front should be moving through the reservoir interacting with the underlying geological structure in quite a complicated manner. The pressure distribution through the system should also be evolving with time. For example, the water may possibly be advancing preferentially through a high permeability channel between an injector and producer pair. The sequence of the saturation fronts in the series of “snapshots in time” shown in Figure 19 give some idea of the progression of the flood with time. However, this can better be illustrated by looking at an animated sequence of saturation distributions as the flood front moves through the reservoir. An example of such a sequence is shown in file Res_Sim_D1.ppt This can be run on your PC from the CD supplied with this course and double clicking on the PowerPoint presentation.


43

Figure 19 The sequence of saturation distributions as the flood front moves through the reservoir. From Res_ Sim_D1.ppt Down arrow injector, up arrow producer.

44


1

5 TYPES OF RESERVOIR SIMULATION MODEL Until now, we have confined our discussion to relative simple reservoir recovery processes such as natural depletion (blowdown) and waterflooding. However, there are many more complex reservoir recovery processes which can also be carried out. Dry gas (methane, CH4) injection, for example, would generally result in the flow of three phases (gas, oil and water) in the reservoir which is more complicated than two phase flow. Another process is where we alternately inject water and gas in repeating sequence - this is water-alternating-gas or WAG flooding. If the injected gas was carbon dioxide (CO2), then quite complex phase behaviour may occur and this requires some particular steps to be taken in order to model this. More exotic Improved Oil Recovery (IOR) processes can also be carried out where we inject chemicals (polymers, surfactants, alkali or foams) into the reservoir to recover oil that is left behind by primary and secondary oil recovery processes.

5.1 The Black Oil Model

Different types of simulator are available to model these different types of reservoir recovery process. Throughout the chapters of this course we will focus on the simplest of these (which is quite complex enough!) known as the "Black Oil Model". However, for completeness, we will also list the others and present a table comparing experience of these various models. The Black Oil Model: This model was used in the three SPE field case studies above and is the most commonly used formulation of the reservoir simulation equations which is used for single, two and three phase reservoir processes. It treats the three phases - oil, gas and water - as if they were mass components where only the gas is allowed to dissolve in the oil and water. This gas solubility is described in oil and water by the gas solubility factors (or solution gas-oil ratios), Rso and Rsw, respectively; typical field units of Rso and Rsw are SCF/STB. These quantities are pressure dependent and this is incorporated into the black oil model. A simple schematic of a grid block in a black oil simulator is presented in Figure 20 showing the amounts of mass of oil, water and gas present. Note that, because the gas is present in the oil and water there are extra terms in the expression for the mass of gas. These mathematical expressions for the mass of the various phases are important when we come to deriving the flow equations (Chapter 5). Reservoir processes that can be modelled using the black oil model include: •

Recovery by fluid expansion - solution gas drive (primary depletion).

•

Waterflooding including viscous, capillary and gravity forces (secondary recovery).

•

Immiscible gas injection.

•

Some three phase recovery processes such as immiscible water-alternatinggas (WAG).

•

Capillary imbibition processes.


45

Vp.ρosc Mass oil = Bo

So;

Mass water =

Mass gas =

Vp.ρwsc Bw

Sw

Vp.ρgsc (Sg + So.Rso + Sw.Rsw) Bg

Rock Gas, Sg Oil + Gas, So Water + Gas, Sw

free gas gas in oil gas in water

5.2 More Complex Reservoir Simulation Models

The Compositional Model: A compositional reservoir simulation model is required when significant inter-phase mass transfer effects occur in the fluid displacement process. It can be considered as a generalisation of the black oil model. This model usually defines three phases (again gas, oil and water) but the actual compositions of the oil and gas phases are explicitly acknowledged due to their more complicated PVT behaviour. That is, the separate components (C1, C2, C3, etc.) in the oil and gas phases are explicitly tracked as is indicated in Figure 21 (which should be compared with Figure 20). The mass conservation is applied to each component rather than just to “oil”, “gas” and “water” as in the black oil model. For example, in a nearcritical fluid where small changes in say pressure can result in large compositional changes of the “oil” and “gas” phases which, in turn, strongly affects their physical properties (viscosity, density, interfacial tensions etc.). Examples of reservoir processes that can be modelled using a compositional model include: •

Gas injection with oil mobilisation by first contact or developed (multi- contact) miscibility (e.g. in CO2 flooding).

•

The modelling of gas injection into near critical reservoirs.

•

Gas recycling processes in condensate reservoirs.

46

Figure 20 Schematic of a grid block in a black oil simulator showing the amounts of mass of oil, water and gas present. Note that, because the gas is present in the oil and water there are extra terms for the mass of gas; pore volume = Vp = block vol. x φ; ρosc, ρwsc. and ρgsc are densities at standard conditions (60°F and 14.7 psi); Bo, Bw and Bg are the formation volume factors; Rso and Rsw are the gas solubilities (or solution gas/oil ratios).


Component concentrations in each phase:

ROCK

Figure 21 The view of phases and components taken in compositional simulation. Cij - is the mass concentration of component i in phase j (j = gas, oil or water) - dimensions of mass/ unit volume of phase; pore volume = Vp = block vol. x φ

Phase Labels: j = 1 = Gas j = 2 = Oil j = 3 = Water

1

GAS Sg OIL + GAS So WATER + GAS Sw

Gas:

C11, C21, C31....

Oil:

C12, C22, C32....

Water: C13, C23, C33....

3

Mass of component in block = Vp. Σ Sj.Cij j=1

The Chemical Flood Model: This model has been developed primarily to model polymer and surfactant (or combined) displacement processes. Polymer flooding can be considered mainly as extended waterflooding with some additional effects in the aqueous phase which must be modelling e.g. polymer component transport, the viscosification of the aqueous phase, polymer adsorption, permeability reduction etc. Surfactant, flooding however, involves strong phase behaviour effects where third phases may appear which contain oil/water/surfactant emulsions. Specialised phase packages have been developed to model such processes. For economic reasons, activity on field polymer flooding has continued at a fairly low level world wide and surfactant flooding has virtually ceased in recent years. However, if economic factors were favourable (a very high oil price), then interest in these processes may revive. Extended chemical flood models are also used to model foam flooding. Examples of reservoir processes that can be modelled using a chemical flood model include: •

Polymer flooding which can be thought of as an “enhanced waterflood” to improve the mobility ratio and hence improve the microscopic sweep efficiency and also to reduce streaking in highly heterogeneous layered systems;

•

Polymer/surfactant flooding where the main purpose of the surfactant is to lower interfacial tension (IFT) between the oil and water phases and hence to “release” or “mobilise” trapped residual oil; the polymer is for mobility control behind the surfactant slug;

•

Low-tension polymer flooding (LTPF) where a more viscous polymer containing injected solution also contains some surfactant to reduce IFT; the combined effect of the lower IFT and viscous drive fluid improves the sweep and also helps to mobilise some of the residual oil;

•

Alkali flooding where a solution of sodium hydroxide is injected into the formation. The sodium hydroxide may react with certain conponents in the oil to produce natural "soaps" which lower IFT and which may help to mobilise some of the residual oil;


47

•

Foam flooding where a surfactant is added during gas injection to form a foam which has a high effective viscosity (lower mobility) in the formation than the gas alone which may then displace oil more efficiently.

Another near-wellbore process that can be modelled using such simulators in water shut-off using either polymer-crosslinked gels or so-called “relative permeabilty modifiers”. Thermal Models: In all thermal models heat is added to the reservoir either by injecting steam or by actually combusting the oil (by air injection, for example). The purpose of this is generally to reduce the viscosity of a heavy oil which may have μo of order 100s or 1000s of cP. The heat may be supplied to the reservoir by injected steam produced using a steam generator on the surface or downhole. Alternatively, an actual combustion process may be initiated in the reservoir - in-situ combustion - where part of the oil is burned to produce heat and combustion gases that help to drive the (unburned) oil from the system. Examples of reservoir processes that can be modelled using thermal models include: •

Steam “soaks” where steam in injected into the formation, the well is shut in for a time to allow heat dissipation into the oil and then the well is back produced to obtain the mobilised oil (because of lower viscosity). This is known as a “Huff nʼ Puff” process.

•

Steam “drive” where the steam is injected continuously into the formation from an injector to the producer. Again, the objective is to lower oil viscosity by the penetration of the heat front deep into the reservoir.

•

In situ combustion where - as noted above - an actual combustion process is initiated in the reservoir by injecting oxygen or air. Part of the oil is burned (oxidised) to produce heat and combustion gases that help to drive the (unburned) oil from the system. This is not a common improved oil recovery method but a number of field cases showing at least technical success have been reported in the SPE literature.

The above more complex reservoir simulation models are really based on the fluid flow process. However, there are also other types of simulator that are more closely defined by their treatment of the rock structure or the rock response. These include: Dual-Porosity Models of Fractured Systems: These models have been designed explicitly to simulate multiphase flow in fractured systems where the oil mainly flows in fractures but is stored mainly in the rock matrix. Such models attempt to model the fracture flows (and sometimes the matrix flows) and the exchange of fluids between the fractures and the rock matrix. They have been applied to model recovery processes in massively fractured carbonate reservoir such as those found in many parts of the Middle East and elsewhere in the world. There is quite considerable field experience of modelling such systems in certain companies but there are also doubts over the validity of such models to model flow in fractured systems. 48


1

Coupled Hydraulic, Thermal Fracturing and Fluid Flow Models: These simulators are still essentially at the research stage although there have been published examples of specific field applications. The main function of these is to model the mechanical stresses and resulting deformations and the effects of these on fluid flow. This is beyond the scope of this course although, in the future, these will be important in many systems.

5.3 Comparison of Field Experience with Various Simulation Models

We now consider what field experience exists in the oil industry with the various models from the black oil model through to more complex fracture models and in situ combustion models etc. The vast majority of simulation studies which are carried out involve the black oil model. However, there are pockets of expertise with the various other types of simulation model, depending on the asset base of the particular oil company or regional expertise within regional consultancy groups. For example, there is (or until recently, was) a concentration of expertise in both California and parts of Canada on steam flooding since this process is applied in these regions to recover heavy oil; in the Middle East (and within the companies that operate there) there is great competence in the dual-porosity simulation of fractured carbonate reservoirs.

Table 3 This is an adapted version of a table in Chapter 11 of Mattax and Dalton (1990). This gives some idea of the problems and issues encountered in applying advanced simulation models relative to applying a black oil simulator. The view about the difficulties and computer time consuming these are is somewhat subjective.

Degree of Difficulty

Relative Computing Costs

Processes Modelled

Black Oil Model

• Primary depletion • Waterflooding • Immiscible gas injection • Imbibition

Routine

Cheap = 1

• Huge • But there are still challenges with upscaling of large models • >90% of cases

Any of the books on reservoir simulation listed in Section 7 (Chapter 1)

Compositional Model

• Gas injection • Gas recycling • CO2 injection • WAG

Difficult Specialisd

Expensive (x3 - x20)

• Moderate • High in certain companies

Coats, (1980a), Acs et al (1985), Nolen (1973), Watts (1986), Young and Stephenson (1983).

Compositional Model- Near Crit.

• Gas injection near crit. • Condensate development • MWAG

Difficult

Very expensive (x5 - x30)

• Low to moderate

Chemical Model - Polymer

• Polymer flooding • Near-well water shut-off

Not too difficult

Moderate (x2 - x5)

• Moderate to large

Bondor et al (1972), Vela et al (1976), Sorbie (1991)

Chemical Model - Surfactant

• Micellar flooding • Low tension polymer flooding

Difficult Specialisd


• Low • Mainly “research type” pilot floods

Todd and Chase (1979), Todd et al (1978), Van Quy and Labrid (1983); Pope and Nelson (1978)

Thermal Model - Steam

• Steam soak (Huff n’ Puff) • Steam flooding

Not too difficult


• Moderate • High in limited geographical areas

Coats (1978), Prats (1982), Mathews (1983)

Thermal Model In Situ Combustion

• In situ combustion processes

Very difficult Very specialised


• Very low

Crookston et al (1979), Youngren (1980), Coats (1980b)


Amount of Industrial Experience

Example References2

Simulator Type

as above

49

6 SOME FURTHER READING ON RESERVOIR SIMULATION A full alphabetic list of References which are cited in the course is presented in Appendix A. Here, we briefly review some good texts which cover Reservoir Simulation from various viewpoints. The authors have learned something from each of these and we would recommend anyone who wishes to specialise in Reservoir Simulation to consult these. Archer, J S and Wall, C: Petroleum Engineering: Principles and Practice, Graham and Trotman Inc., London, 1986. This book is not a specialised reservoir simulation text. However, it offers a good overview of petroleum engineering and it contexts reservoir simulation very well within the overall picture of reservoir development. This book is also one of the earliest proponents of the importance of integrating the reservoir geology within the simulation model. Aziz, K. and Settari, A.: Petroleum Reservoir Simulation, Elsevier Applied Science Publishers, Amsterdam, 1979. This is a classic text on the discretisation and numerical solution of the reservoir simulation flow equations. It is quite mathematical with a focus on the actual difference equations that arise from the flow equations and how to solve these. Crichlow, H B: Modern Reservoir Engineering: A Simulation Approach, PrenticeHall Inc., Englewood Cliffs, NJ, 1977. This book gives a fairly good introduction to reservoir simulation from the viewpoint of it being a central part of current reservoir engineering. Dake, L P: The Practice of Reservoir Engineering, Developments in Petroleum Science 36, Elsevier, 1994. Again, this book is not about reservoir simulation but it makes a number of interesting and controversial observations on reservoir simulation (not all of which the authors agree with!). An interesting lengthy quote from this book on the relationship between material balance and reservoir simulation is reproduced in Chapter 2. Fanchi, J R: Principles of Applied Reservoir Simulation, Gulf Publishing Co., Houston, TX, 1997. This recent book provides a good elementary text on reservoir simulation. It is a based around the BOAST4D black oil simulation model which is supplied on disk and can be run on your PC. The software makes this a very attractive way to familiarise yourself with reservoir simulation if you donʼt have ready access to a simulator. Mattax, C C and Dalton, R L: Reservoir Simulation, SPE Monograph, Vol. 13, 1990. This is an excellent SPE monograph which covers virtually every aspect of traditional reservoir simulation. It is has been put together by a team of Exxon reservoir engineers between them have vast experience of all areas of reservoir simulation. Peaceman, D W: Fundamentals of Numerical Reservoir Simulation, Developments in Petroleum Science No. 6, Elsevier, 1977. 50


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This book presents an excellent treatment of the mathematical and numerical aspects of reservoir simulation. It discusses the discretisation of the flow equations and the subsequent numerical methods of solution in great detail. SPE Reprint No. 11, Numerical Simulation I (1973) and SPE Reprint No. 20, Numerical Simulation II (19**). These two collections present some of the classic SPE papers on reservoir simulation. All aspects of reservoir simulation are covered including numerical methods, solution of linear equations, the modelling of wells and field applications. Most of this material is too advanced or detailed for a newcomer to this field but the volumes contain excellent reference material. They are also relatively cheap! Thomas, G W: Principles of Hydrocarbon Reservoir Simulation, IHRDC, Boston, 1982. This short volume is written - according to Thomas - from a developers viewpoint; i.e. someone who is involved with writing and supplying the simulators themselves. The treatment is quite mathematical with quite a lot of coverage of numerical methods. The treatment of some areas is rather brief; for example, there are only 7 pages on wells.

APPENDIX A:

REFERENCES

NOTE: SPEJ = Society of Petroleum Engineers Journal - there was an early version of this and it stopped for a while. Currently, there are SPE Journals in various subjects but reservoir simulation R&D appears in SPE (Reservoir Engineering and Evaluation). Acs, G., Doleschall, S. and Farkas, E., “General Purpose Compositional Model”, SPEJ, pp. 543 - 553, August 1985. Allen, M.B., Behie, G.A. and Trangenstein, J.A.: Multiphase Flow in Porous Media: Mechanics, Mathematics and Numerics, Lecture Notes in Engineering No. 34, Springer-Verlag, 1988. Amyx, J W, Bass, D M and Whiting, R L: Petroleum Reservoir Engineering, McGrawHill, 1960. Appleyard, J.R. and Cheshire, I.M.: “Nested Factorization,” paper SPE 12264 presented at the Seventh SPE Symposium on Reservoir Simulation, San Francisco, CA, November 16-18, 1983. Archer, J S and Wall, C: Petroleum Engineering: Principles and Practice, Graham and Trotman Inc., London, 1986. Aronofsky, J.S. and Jenkins, R.: “A Simplified Analysis of Unsteady Radial Gas Flow,” Trans., AIME 201 (1954) 149-154 Aziz, K. and Settari, A.: Petroleum Reservoir Simulation, Elsevier Applied Science Publishers, Amsterdam, 1979.


51

Bondor, P.L., Hirasaki, G.J and Tham, M.J., “Mathematical Simulation of Polymer Flooding in Complex Reservoirs”, SPEJ, pp. 369-382, October 1972. Clayton, C.A., et al, “The Ubit Field Rejuvenation: A Case History of Reservoir Management of a Giant Oilfield Offshore Nigeria”, SPE49165, presented at the SPE Annual Technical Conference and Exhibition, New Orleans, LA, 27-30 September 1998. Coats, K.H., .......... 1969 - tools of res sim Coats, K.H., “A Highly Implicit Steamflood Model”, SPEJ, pp. 369-383, October 1978. Coats, K.H., “An Equation of State Compositional Model”, SPEJ, pp. 363-376, October 1980a; Trans. AIME, 269. Coats, K.H., “ In-Situ Combustion Model”, SPEJ, pp. 533-554, December 1980b; Trans. AIME 269. Coats, K.H., Dempsey, J.R., and Henderson, J.H.: “The Use of Vertical Equilibrium in Two-Dimensional Simulation of Three-Dimensional Reservoir Performance,” Soc. Pet. Eng. J. 11 (March 1971) 63-71; Trans., AIME 251 Craft, B C, Hawkins, M F and Terry, R E: Applied Petroleum Reservoir Engineering, Prentice Hall, NJ, 1991. Craig, F F: The Reservoir Engineering Aspects of Waterflooding, SPE monograph, Dallas, TX, 1979. Crichlow, H B: Modern Reservoir Engineering: A Simulation Approach, PrenticeHall Inc., Englewood Cliffs, NJ, 1977. Crookston, R.B., Culham, W.E. and Chen, W.H., “A Numerical Simulation Model for Thermal Recovery Processes”, SPEJ, pp. 35-57, February 1979; Trans. AIME 267. Dake, L P: The Fundamentals of Reservoir Engineering, Developments in Petroleum Science 8, Elsevier, 1978. Dake, L P: The Practice of Reservoir Engineering, Developments in Petroleum Science 36, Elsevier, 1994. Fanchi, J R: Principles of Applied Reservoir Simulation, Gulf Publishing Co., Houston, TX, 1997. Fantoft, S., “Reservoir Management of the Oseberg Field After Four Years”, SPE25008, proceedings of the SPE European Petroleum Conference, Cannes, France, 16-18 November 1992. Giudicelli, C.S., Massonat, G.J. and Alabert, F.G., “Anguille Marine, a DeepseFan Reservoir Offshore Gabon: From Geology Toward History Matching Through 52


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Stochastic Modelling”, SPE25006, proceedings of the SPE European Petroleum Conference, Cannes, France, 16-18 November 1992. Harpole, K.J. and Hearn, C.L., “The Role of Numerical Simulation in Reservoir Management of a West Texas Carbonate Reservoir”, SPE10022, presented at the International Petroleum Exhibition and Technical Symposium of the SPE, Beijing, China, 18 - 26 March 1982. Heinemann, Z.E., Brand, C.W., Munka, M., and Chen, Y.M.: “Modeling Reservoir Geometry with Irregular Grids,” SPERE 6 (1991) 225-232. Hove, K., Olsen, G., Nilsson, S., Tonnesen, M. and Hatloy, A., “From Stochastic Geological Description to Production Forecasting in Heterogeneous Layered Systems”, SPE24890, the proceedings of the SPE 67th Annual Technical Conference, Washington, DC, 4-7 October 1992. Katz, D.L., “Methods of Estimating Oil and Gas Reserves”, Trans. AIME, Vol. 118, p.18, 1936 (classic early ref. on Material Balance) Kyte, J.R. and Berry, D.W.: “New Pseudo Functions to Control Numerical Dispersion,” Soc .Pet. Eng. J. 15 (August 1975) 269-276. Lantz, R.B.: “Quantitative Evaluation of Numerical Diffusion (Truncation Error),” Soc .Pet. Eng. J. 11 (September 1971) 315-320; Trans., AIME 251. Leonard, A.J., Duncan, A.E., Johnson, D.A. and Murray, R.B., SPE25059: “Development Planning in a Complex Reservoir: Magnus Field UKCS Lower Kimmeridge Clay Formation (LKCF)”, SPE25059, proceedings of the SPE European Petroleum Conference, Cannes, France, 16-18 November 1992. Mathews, C.W., “Steamflooding”, J. Pet. Tech., pp. 465-471, March 1983; Trans. AIME 275. MacDonald, R.C. and Coats, K.H.: “Methods for Numerical Simulation of Water and Gas Coning,” Soc. Pet. Eng. J. 10 (December 1970) 425-436; Trans., AIME 249. Mattax, C C and Dalton, R L: Reservoir Simulation, SPE Monograph, Vol. 13, 1990. Meijerink, J.A. and Van der Vorst, H.A.: “An Iterative Solution Method for Linear Systems of Which the Coefficient Matrix is a Symmetric M-Matrix,” Mathematics of Computation 31 (January 1977) 148. Nolen, J.S., “Numerical Simulation of Compositional Phenomena in Petroleum Reservoirs”, SPE4274, proceedings of the SPE Symposium on Numerical Simulation of Reservoir Performance, Houston, TX, 11-12 January 1973. Palagi, C.L. and Aziz, K.: “Use of Voronoi Grid in Reservoir Simulation,” SPE Advanced Technology Series 2 (April 1994) 69-77. Peaceman, D.W.: “Interpretation of Well-Block Pressures in Numerical Reservoir Institute of Petroleum Engineering, Heriot-Watt University

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Simulation,” Soc. Pet. Eng. J. 18 (June 1978) 183-194; Trans., AIME 253. Peaceman, D.W. and Rachford, H.H.: “The Numerical Solution of Parabolic and Elliptic Differential Equations,” Soc Ind. Appl. Math. J. 3 (1955) 28-41 Peaceman, D W: Fundamentals of Numerical Reservoir Simulation, Developments in Petroleum Science No. 6, Elsevier, 1977. Ponting, D.K.: “Corner point geometry in reservoir simulation,” in The Mathematics of Oil Recovery - Edited proceedings of an IMA/SPE Conference, Robinson College, Cambridge, July 1989; Edited by P.R. King, Clarendon Press, Oxford, 1992. Pope, G.A. and Nelson, R.C., “A Chemical Flooding Compositional Simulator”, SPEF, pp.339-354, October 1978. Prats, M., “Thermal Recovery” SPE Monograph Series No. 7, SPE Richardson, TX, 1982. Price, H.S. and Coats, K.H.: “Direct Methods in Reservoir Simulation,” Soc. Pet. Eng. J. 14 (June 1974) 295-308; Trans., AIME 257 Robertson, G., in Cores from the Northwest European Hydrocarbon Provence, C D Oakman, J H Martin and P W M Corbett (eds.), Geological Society, London. 1997. Schilthuis, R.J., “Active Oil and Reservoir Energy”, Trans. AIME, Vol. 118, p.3, 1936; (original ref. on Material Balance) Sheldon, J.W., Harris, C.D., and Bavly, D.: “A Method for Generalized Reservoir Behavior Simulation on Digital Computers,” SPE 1521-G presented at the 35th Annual SPE Fall Meeting, Denver, Colorado, October 1960. Sibley, M.J., Bent, J.V. and Davis, D.W., “Reservoir Modelling and Simulation of a Middle Eastern Carbonate Reservoir”, SPE36540, proceedings of the SPE 71st Annual Conference and Exhibition, Denver, CO, 6-9 October 1996. Sorbie, K.S., “Polymer Improved Oil Recovery”, Blakie and SOns & CRC Press, 1991. SPE Reprint No. 11, Numerical Simulation I (1973) and SPE Reprint No. 20, Numerical Simulation II (19**). Spillette, A.G., Hillestad, J.H., and Stone, H.L.: “A High-Stability Sequential Solution Approach to Reservoir Simulation,” SPE 4542 presented at the 48th Annual Fall Meeting of the Society of Petroleum Engineers of AIME, Les Vegas, Nevada, September 30-October 3, 1973. Stone, H.L.: “Iterative Solution of Implicit Approximations of Multidimensional Partial Differential Equations,” SIAM J. Numer.Anal. 5 (September 1968) 530-558 Stone, H.L.: “Probability Model for Estimating Three-Phase Relative Permeability,” J. Pet. Tech. 24 (February 1970) 214-218; Trans., AIME 249. 54


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Stone, H.L.: “Estimation of Three-Phase Relative Permeability and Residual Oil Data,” J. Can. Pet. Tech. 12 (October-December 1973) 53-61. Stone, H.L. and Garder, Jr., A.O.: “Analysis of Gas-Cap or Dissolved-Gas Drive Reservoirs,” Soc .Pet. Eng. J. 1 (June 1961) 92-104; Trans., AIME 222. Thomas G.W. and Thurnau, D.H.: “Reservoir Simulation Using an Adaptive Implicit Method,” Soc. Pet. Eng.. J. 23 (October 1983) 759-768. Thomas L.K., Lumpkin, W.B., and Reheis, G.M.: “Reservoir Simulation of Variable Bubble-point Problems,” Soc. Pet. Eng. J. 16 (February 1976) 10-16; Trans., AIME 261. Todd, M.R. and Longstaff, W.J.: “The Development, Testing, and Application of a Numerical Simulator for Predicting Miscible Flood Performance, “ J. Pet. Tech. 24 (July 1972) 874-882; Trans., AIME 253. Todd, M.R., OʼDell, P.M., and Hirasaki, G.J.: “Methods for Increased Accuracy in Numerical Reservoir Simulators,” Soc. Pet. Eng. J. 12 (December 1972) 515-530. Thomas, G W: Principles of Hydrocarbon Reservoir Simulation, IHRDC, Boston, 1982. Todd, M.R. and Chase, C.A., “A Numerical Simulator for Predicting Chemical Flood Performance”, SPE7689, proceedings of the SPE Symposium on Reservoir Simulation, Denver, CO, 1-2 February 1979. Todd, M.R. et al , “Numerical Simulation of Competing Chemical Flood Designs”, SPE7077, proceedings of the SPE Symposium on Improved Methods for Oil Recovery, Tulsa, OK, 16-19 April 1978. Uren, L.C., Petroleum Production Engineering, Oil Field Exploitation, 3rd edn., McGraw-Hill Book Company Inc., New York, 1953. Van Quy, N. and Labrid, J., “A Numerical Study of Chemical Flooding - Comparison with Experiments”, SPEJ, pp.461-474, June 1983; Trans. AIME 275. van Vark, W., Paardekam, A.H.M., Brint, J.F., van Lieshout, J.B. and George, P.M., “The Construction and Validation of a Numerical Model of a Reservoir Consisting of Meandering Channels”, SPE25057, proceedings of the SPE European Petroleum Conference, Cannes, France, 16-18 November 1992. Vela, S., Peaceman, D.W. and Sandvik, E.I., “Evaluation of Polymer Flooding in a Layered Reservoir with Crossflow, Retention and Degradation”, SPEJ, pp. 82-96, April 1976. Vinsome, P.K.W.: “Orthomin, an Iterative Method for Solving Sparse Banded Sets of Simultaneous Linear Equations,” paper SPE 5729 presented at the Fourth SPE Symposium on Reservoir Simulation, Los Angeles, February 19-20, 1976. Institute of Petroleum Engineering, Heriot-Watt University

55

Wallis, J.R., Kendall, R.P., and Little, T.E.: “Constrained Residual Acceleration of Conjugate Residual Methods,” SPE 13536 presented at the Eighth SPE Reservoir Simulation Symposium, Dallas, Texas, February 10-13, 1985. Watts, J.W.: “An Iterative Matrix Solution Method Suitable for Anisotropic Problems,” Soc Pet. Eng .J. 11 (March 1971) 47-51; Trans., AIME 251. Watts, J.W., “A Compositional Formulation of the Pressure and Saturation Equations”, SPE (Reservoir Engineering), pp. 243 - 252, March 1986. Watts, J.W., “Reservoir Simulation: Past, Present and Future”, SPE Reservoir Simulation Symposium, Dallas, TX, 5-7 June 1997. Yanosik, J.L. and McCracken, T.A.: “A Nine-Point Finite Difference Reservoir Simulator for Realistic Prediction of Unfavorable Mobility Ratio Displacements,” Soc. Pet. Eng. J. 19 (August 1979) 253-262; Trans., AIME 267. Young, L.C. and Stephenson, R.E., “A Generalised Compositional Approach for Reservoir Simulation”, SPEJ, pp. 727-742, October 1983; Trans. AIME 275. Youngren, G.K., “Development and Application of an In-Situ Combustion Reservoir Simulator”, SPEJ, pp. 39-51, February 1980; Trans. AIME 269.

APPENDIX B - Some Overview Articles on Reservoir Simulation 1. Reservoir Simulation: is it worth the effort? SPE Review, London Section monthly panel discussion November 1990. This one pager summarises a panel discussion that was held in London in 1990. Given the brevity of the article, it is packed with some genuine wisdom - and some things to disagree with - from a really excellent group of front line “users” of the technology. Briggs comes closest to capturing the principal authorsʼ particular prejudices! 2. The Future of Reservoir Simulation - C. Galas, J. Canadian Petroleum Technology, 36, January 1997. This short viewpoint from a Canadian independent consultant is interesting since it contexts reservoir simulation in the current “outsourced” and “downsized” oil industry. He notes that virtually everyone can have PC based powerful simulation technology on their desk tops. However, he concludes that the overall demand for simulation will rise and that, for the sake of efficiency, this will be performed by specialists. At the same time, he promotes a teamwork environment for the simulation engineer where he or she will be involved in the preceding reservoir characterisation process and the subsequent decision making process. This is a well argued position but not all of his conclusion would be generally accepted. 3. What you should know about evaluating simulation results - M. Carlson; J. Canadian Petroleum Technology, Part I - pp. 21-25, 36, No. 5, May 1997; Part II - pp. 52-57, 36, No. 7, August 1997. 56


1

This very interesting pair of articles gives a very good broad brush commentary on a range of technical issues in reservoir simulation e.g. gridding, handling wells, pseudo-relative permeability, error analysis and “consistency checking”. The views are clearly those of someone who has been deeply involved in applied reservoir simulation. They are well presented and quite individual although again there are issues that would provoke disagreement. Read this and decide for yourself what you accept and what you donʼt.


57

2

Basic Concepts in Reservoir Engineering

1.

INTRODUCTION

2.

MATERIAL BALANCE 2.1 Introduction to Material Balance (MB) 2.2 Derivation of Simplified Material Balance Equations 2.3 Conditions for the Validity of Material Balance

3.

SINGLE PHASE DARCY LAW ∇ 3.1 The Basic Darcy Experiment 3.2 Mathematical Note: on the Operators “gradient” ∇ and “divergence” ∇ 3.3 Darcy’s Law in 3D - Using Vector and Tensor Notation 3.4 Simple Darcy Law with Gravity ∇ 3.5 The Radial Darcy Law

.

.

4.

TWO-PHASE FLOW 4.1 The Two-Phase Darcy Law

5.

CLOSING REMARKS

6.

SOME FURTHER READING ON RESERVOIR ENGINEERING

LEARNING OBJECTIVES: Having worked through this chapter the student should: • be familiar with the meaning and use of all the usual terms which appear in reservoir engineering such as, Sw, So, Bo, Bw, Bg, Rso, Rsw, cw, co, cf, kro, krw, Pc etc. • be able to explain the differences between material balance and reservoir simulation. • be aware of and be able to describe where it is more appropriate to use material balance and where it is more appropriate to use reservoir simulation. • be able to use a simple given material balance equation for an undersaturated oil reservoir (with no influx or production of water) in order to find the STOOIP. • know the conditions under which the material balance equations are valid. • be able to write down the single and two-phase Darcy Law in one dimension (1D) and be able to explain all the terms which occur (no units conversion factors need to be remembered). • be aware of the gradient (∇) and divergence (∇.) operators as they apply to the generalised (2D and 3D) Darcy Law (but these should not be committed to memory). • know that pressure is a scalar and that the pressure distribution, P(x, y, z) is a scalar field; but that ∇P is a vector. • know that permeability is really a tensor quantity with some notion of what this means physically (more in Chapter 7). • be able to write out the 2D and 3D Darcy Law with permeability as a full tensor and know how this gives the more familiar Darcy Law in x, y and z directions when the tensor is diagonal (but where we may have kx ≠ ky ≠ kz). • be able to write down and explain the radial Darcy Law and know that the pressure profile near the well, ΔP(r), varies logarithmically.

2

2


REVIEW OF BASIC CONCEPTS IN RESERVOIR ENGINEERING Brief Description of Chapter 2

This module reviews some basic concepts of reservoir engineering that must be familiar to the simulation engineer and which s/he should have covered already. We start with Material Balance and the definition of the quantities which are necessary to carry out such calculations: φ, co , cf , Bo , Swi etc. This is illustrated by a simple calculator exercise which is to be carried out by the student. The same exercise is then repeated on the reservoir simulator. Alternative approaches to material balance are discussed briefly. The respective roles of Material Balance and Reservoir Simulation are compared. The unit then goes on to consider basic reservoir engineering associated with fluid flow: the single phase Darcy law (k), tensor permeabilities, k , two phase Darcy Law - relative permeabilitites (kro , krw) and capillary pressures (Pc). Note that many of the terms and concepts reviewed in this section are summarised in the Glossary at the front of this chapter.

1. INTRODUCTION

It is likely that you will have completed the introductory Reservoir Engineering part of this Course. You should therefore be fairly familiar with the concepts reviewed in this section. The purpose of doing any review of basic reservoir engineering is as follows: (i) Between them, the review in this section and the Glossary make this course more self-contained, with all the main concepts we need close at hand; (ii) This allows us to emphasise the complementary nature of “conventional” reservoir engineering and reservoir simulation; (iii) We would like to review some of the flow concepts (Darcy’s law etc.), in a manner of particular use for the derivation of the flow equations later in this course (in Chapter 5). An example of point (ii) above concerns the complementary nature of Material Balance (MB) and numerical reservoir simulation. At times, these have been presented as almost opposing approaches to reservoir engineering. Nothing could be further from the truth and this will be discussed in detail below. Indeed, a MB calculation will be done by the student and the same calculation will be performed using the reservoir simulator. In addition to an introductory review of simple material balance calculations, we will also go over some of the basic concepts of flow through porous media. These flow concepts will be of direct use in deriving the reservoir simulation flow equations in Chapter 5. Again, most of the concepts are summarised in the Glossary. Exercises are provided at the end of this module which the student must carry out.


3

The following concepts are defined in the Glossary and should be familiar to you: viscosity (μo, μw, μg), density (ρo, ρw, ρg), phase saturations (So, Sw and Sg), initial or connate water saturation (Swi or Swc), residual oil saturation (Sor). In addition, you should also be familiar with the basic reservoir engineering quantities in Table 1 below: Symbol

Name

Field Units Meaning / Formulae

Bo, Bw, Bg

Formation volume factors (FVF) for oil, water and gas

bbl/STB or RB/STB

Bo = Vol. oil + dissolved gas in reservoir Vol. oil at STC STC = Stock Tank Conditions (60°F; 14.7 psi). Likewise for water (usually const.) and gas; Pb = bubble point pressure below.

FVF

Bg

Bo Bw Pb

P Rso, Rsw

Gas solubility factors or solution gas oil ratios

SCF/STB

Rso =

Vol. dissolved gas in reservoir Vol. gas at STC

Rso Rso P co, cw, cg

Isothermal fluid compressibilities of oil water and gas

Pb

-1

psi

ck = 1 ρk

∂ρk ∂P

=-

1 Vk

∂Vk ∂P

ρk and Vk - density and volume of phase k; k = o, w, g

2. MATERIAL BALANCE 2.1 Introduction to Material Balance (MB)

The concept of Material Balance (MB) has a central position in the early history of reservoir engineering. MB equations were originally derived by Schilthuis in 1936. There are several excellent accounts of the MB equations and their application to different reservoir situations in various textbooks (Amyx, Bass and Whiting, 1960; Craft, Hawkins and Terry, 1991; Dake, 1978, 1994). For this reason, and because this subject is covered in detail in the Reservoir Engineering course in this series, we only present a very simple case of the material balance equation in a saturated reservoir case. The full MB equation is presented in the Glossary for completeness. Our objectives in this context are as follows: •

4

To introduce the central idea of MB and apply it to a simple case which we will then set up as an exercise for simulation;

Table 1: Basic reservoir engineering quantities to revise

2


• To demonstrate the complementary nature of MB and reservoir simulation calculations. Material balance has been used in the industry for the following main purposes: 1. Determining the initial hydrocarbon in place (e.g. STOIIP) by analysing mean reservoir pressure vs. production data; 2. Calculating water influx i.e. the degree to which a natural aquifer is supporting the production (and hence slowing down the pressure decline); 3. Predicting mean reservoir pressure in the future, if a good match of the early pressure decline is achieved and the correct reservoir recovery mechanism has been identified. Thus, MB is principally a tool which, if it can be applied successfully, defines the input for a reservoir simulation model (from 1 and 2 above). Subsequently, the mean field pressure decline as calculated in 3 above can be compared with the predictions of the numerical reservoir simulation model. Before deriving the restricted example of the MB equations, we quote the introduction of Dake’s (1994) chapter on material balance.

Material Balance Applied to Oilfields

(from Chapter 3; L. P. Dake, The Practice of Reservoir Engineering, Developments in Petroleum Science 36, Elsevier, 1994.) Dake says: It seems no longer fashionable to apply the concept of material balance to oilfields, the belief being that it has now been superseded by the application of the more modern technique of numerical reservoir simulation modelling. Acceptance of this idea has been a tragedy and has robbed engineers of their most powerful tool for investigating reservoirs and understanding their performance rather than imposing their wills upon them, as is often the case when applying numerical simulation directly in history matching. As demonstrated in this chapter, by defining an average pressure decline trend for a reservoir, which is always possible, irrespective of any lack of pressure equilibrium, then material balance can be applied using simply the production and pressure histories together with the fluid PVT properties. No geometrical considerations (geological models) are involved, hence the material balance can be used to calculate the hydrocarbons in place and define the drive mechanisms. In this respect, it is the safest technique in the business since it is the minimum assumption route through reservoir engineering. Conversely, the mere act of construction of a simulation model, using the geological maps and petrophysically determined formation properties implies that the STOIIP is “known”. Therefore, history matching by simulation can hardly be regarded as an investigative technique but one that merely reflects the input assumptions of the engineer performing the study.


5

There should be no competition between material balance and simulation, instead they must be supportive of one another: the former defining the system which is then used as input to the model. Material balance is excellent at history matching production performance but has considerable disadvantages when it comes to prediction, which is the domain of numerical simulation modelling. Because engineers have drifted away from oilfield material balance in recent years, the unfamiliarity breeds a lack of confidence in its meaningfulness and, indeed, how to use it properly. To counter this, the chapter provides a comprehensive description of various methods of application of the technique and included six fully worked exercises illustrating the history matching of oilfields. It is perhaps worth commenting that in none of these fields had the operators attempted to apply material balance, which denied them vital information concerning the basic understanding of the physics of reservoir performance. Notes on Dake’s comments 1. The authors of this Reservoir Simulation course would very much like to echo Dake’s sentiments. Performing large scale reservoir simulation studies does not replace doing good conventional reservoir engineering analysis - especially MB calculations. MB should always be carried out since, if you have enough data to build a reservoir simulation model, you certainly have enough to perform a MB calculation. 2. Note Dake’s comments on the complementary nature of MB in defining the input for reservoir simulation, as we discussed above. 3. Take careful note of Dake’s comment on where a reservoir simulation model is used for history matching. The very act of setting up the model means that you actually input the STOIIP, whereas, this should be one of the history matching parameters. The reservoir engineer can get around this to some extent by building a number of alternative models of the reservoir and this is sometimes, but not frequently, done.

2.2 Derivation of Simplified Material Balance Equations

Material balance (MB) is simply a volume balance on the changes that occur in the reservoir. The volume of the original reservoir is assumed to be fixed. If this is so, then the algebraic sum of all the volume changes in the reservoir of oil, free gas, water and rock, must be zero. Physically, if oil is produced, then the remaining oil, the other fluids and the rock must expand to fill the void space left by the produced oil. As a consequence, the reservoir pressure will drop although this can be balances if there is a water influx into the reservoir. The reservoir is assumed to be a “tank” - as shown in Figure 5 Chapter 1. The pressure is taken to be constant throughout this tank model and in all phases. Clearly, the system response depends on the compressibilities of the various fluids (co, cw and cg) and on the reservoir rock formation (crock). If there is a gas cap or production goes below the bubble point (Pb), then the highly compressible gas dominates the system response. Typical ranges of fluid and rock compressibilities are given in Table 2:

6

2


Table 2: Typical rock and fluid compressibilities (from Craft, Hawkins and Terry, 1991)

-6

Fluid or formation

Compressibility (10

Formation rock, crock Water, cw Undersaturated oil, co Gas at 1000psi, cg Gas at 5000psi. cg

3 - 10 2-4 5 - 100 900 - 1300 50 - 200

-1

psi )

The simple example which we will take in order to demonstrate the main idea of material balance is shown in Figure 1 where the system is simply an undersaturated oil, with possible water influx. Initial conditions pressure = po

Figure 1. Simplified system for material balance (MB) in a system with an undersaturated oil above the bubble point and possible water influx.

After production (Np) pressure = p

Oil

Oil

N

(N - Np)Bo

NBoi = Vf.(1-Swi)

NBoi = Vf.(1-Swi)

Water, Swi

Water, Swi

W = Vf.Swi

W + We - Wp

Water influx

Oil, Np

Water, Wp

Water influx We

Definitions: N = initial reservoir volume (STB) Boi = initial oil formation volume factor (bbl/STB or RB/STB) Np = cumulative produced oil at time t, pressure p (STB) Bo = oil formation volume factor at current t and p (bbl/STB) W = initial reservoir water (bbl) Wp = cumulative produced water (STB) Bw = water formation volume factor (bbl/STB) We = water influx into reservoir (bbl) cw = water isothermal compressibility (psi-1) Δ P = change in reservoir pressure, p - po Vf = initial void space (bbl); Vf = N.Boi/(1- Swi); W = Vf.Swi Swi = initial water saturation (of whole system) cf

= void space isothermal compressibility (psi-1); c f =

1  ∂Vf    Vf  ∂p 

(NB: (i) bbl = reservoir barrels, sometimes denoted RB; and (ii) in the figures above, the oil and water are effectively assumed to be uniformly distributed throughout the system)


7

Definitions of the various quantities we need for our simplified MB equation for the depletion of an undersaturated oil reservoir above the bubble point (Pb) are given in Figure 1. (NB a more extensive list of quantities required for a full material balance equation in any type of oil or gas reservoir is given in the Glossary for completeness). In going from initial reservoir conditions shown in Figure 1 at pressure, po, to pressure, p, volume changes in the oil, water and void space (rock) occur, ΔVo, ΔVw, ΔVvoid (ΔVvoid = - ΔVrock). The pressure drop is denoted, Δ P = p - po. The volume balance simply says that:

∆Vo + ∆Vw + ∆Vrock = ∆Vo + ∆Vw − ∆Vvoid = 0

(1)

Each of these volume changes can be calculated quite __ straightforwardly as W (W W B + W + . . p ∆ V = ∆ W c follows: w p w e w Oil volume = WΔVBo ∆Vchange, w

p

w

__

- We - W.c w .∆ p

Initial oil volume in reservoir

= __

- (Vt,f pressure - Vf .c f .p∆ p ) = Vtf time Oil volume

= (N - Np). Bo

__

= V .c .∆ p

f volume, f Change in oil

N.Boi

(bbl = RB) (bbl)

= N.Boi - (N - Np). Bo (bbl) (1)

ΔVo

__

∆Vrockchange, ∆Vvoid = - Vf .c f .∆ p = - ΔV Water volume w Initial reservoir water volume

= W

Cumulative water production at time = Wp Reservoir volume of cumulative water production at time = Wp.Bw Volume of water influx into reservoir

= We

Water volume change due to compressibility = W.cw. Δ P Change in water volume,

ΔVw

(bbl) (STB) (bbl) (bbl) (bbl)

= W - (W - Wp Bw + We + W.cw. Δ P ) (bbl)

ΔVw

= Wp Bw - We - W.cw. Δ P

(2)

= Vf

(bbl)

Change in the void space volume, ΔVvoid Initial void space volume

8

2


Change in void space volume, ΔVvoid

= Vf - (Vf - Vf.cf. Δ P ) = Vf.cf. Δ P

Change in rock volume, ΔVrock = - ΔVvoid = - Vf.cf. Δ P

(3)

Now adding the volume changes as follows: __

__

∆Vo + ∆Vw + ∆Vrock = N . Boi + ( N − N p ). Bo + Wp . Bw − We − W .cw .∆ P − Vf .c f .∆ P = 0 (4)

 Swi .cw + cf 

__

__

__

+VN p . Bo=−4N Wand P =V 0 = N.B /(1-S ), we Rearranging V .S  .∆and o∆ p .(B .eB+oiW+noting ∆VNo. B+oi∆−VNw . B+equation Nw −−Nthat N. Bpoi).W Bo 1+=−W rock Swipf . Bwiw − We −f W .cw .∆oiP − Vwif .c f .∆ P = 0 obtain:

 Swi .cw + c f  __  S .c + cf  __ − N−. BW  .∆ P = 0 wi w p . Bo. B oi  + W . B − NN..BBoioi −− NN. B . Bo o++NN  .∆ P = 0 p o e 1 −pS w  N . Boi  wi  1 − Swi  (5)  Swi .cw + crock  __ __ __ N . Boi − N . Bo + N p . Bo + N . Boi   S .c + .c∆ P = 0__ wi wbalance fN ). . ( ∆ + ∆ + ∆ = + − + . − − . . ∆ − . . ∆ P V V V N B N B W B W W c P V c 1 − S   Equation 5 is the (simplified) material expression for the undersaturated wi o w oi p o p w e w rock f f .∆ P = 0 N.B − N.B + N .B − N.B oi

o

p

o

oi





Swi above its bubble point). system given in Figure 1 (as longas it1 − remains  S .c + c



__

w rock  Swiexample, .cw + cf let __ To Nillustrate in an us assume .∆ Peven = 0 simpler . Boi − N . Bthe N p . Bof N . Boi  wibalance o + use o +material N . B − N . B + N . B − W + W . B N − . B − S 1    .∆ P =the0 MB oi water oinfluxp (W o =0) e wi pc w  oi  = 0). Therefore, __ that there is no or production (W S c . + wi w rock e p 1 − Swi  + N . Boito: N . Boi −simplifies N . Bo + Neven   .∆ P = 0 p . Bo further equation  1− S 

wi

 S .c + c  __ N . Boi − N . Bo + N p . Bo − N . Boi  wi w __f  .∆ P = 0 S .c + c − S .∆ P=0 __ N . Boi − N . Bo + N p . Bo + N . Boi  wi w S 1rock  wi (6) wi .cw+ crock 1 − S  N B N B N B N B . − . + . + . . ∆ P = 0 wi   oi o p o oi Note that we can divide through equation (the 6Sby.cN +  __reservoir oil volume, c initial N .obtain: Boi − N . Bo + N p . Bo + N . Boi  wi1 −wSwi rock  .∆ P = 0 bbl = RB) to  S .c1 −+ Scwirock   __ N . Boi − N . Bo + N p . Bo + N . Boi  wi w  .∆ P = 0  1 − Swi  __ Np  Swi .cw + c f  Boi − Bo + N . Bo − Boi   S .c + c .∆ P__= 0 (7) Boi − Bo +N p . Bo − Boi  1wi− Swwi f  .∆ P = 0 N  1 − Swi .c  + crock  __ N . Boi − N . Bo +to:N p . Bo + N . Boi  wi w  .∆ P = 0 which rearranges easily  1 −__Swi  Boi Boi  Swi .cw + c f   Np  +   N = 1 − B   Swi .cw + cf .∆ P__= 0  N p = 1 −Bo oi +BB 1 − Swi  .∆ P = 0 oi o     (8)  N Bo Bo  1 − Swi  where the __quantity (Np/N) is the Recovery Factor (RF) as a fraction of the STOIIP. ∆ p__= 0 It is seen from equation 8 that, at t = 0, Bo = Boi and and therefore (Np/N)= 0, as ∆p=0 expected. Note also in equation 8 that ∆P is negative in depletion ( ∆P = p-po, where po. is the k  ∂for P depletion). Q higherkinitial ∆P pressure

u=  =− . = − .   AQ  µk L∆P µk ∂∂xP  u= =− . = − .   A µ L µ  ∂x 


9

It is convenient to rearrange equation 8 above as follows:

N p  Boi Boi  Swi cw + c f   − 1 − =   ∆P  N  Bo Bo  1 − Swi 

(9)

We then identify 1-(Np/N) as the fraction of the initial oil still in place. We can then plot this quantity vs. - ∆P shown in Figure 2 (we take - ∆P since it plots along the positive axis, since ∆P is negative). "almost" straight line for w/o systems

1 1-

Np N

Figure 2 Plot of remaining oil,

0

Np   1 −  vs. − ∆P  N

-∆P

As noted in Figure 2, this decline plot is not necessarily a straight line but for oil water systems, it is very close in practice. Figure 2 suggests a way of applying a simple material balance equation to the case of an undersaturated oil above the bubble point (with no water influx or production). This is a pure depletion problem driven by the oil (mainly), water and formation compressibilities. Suppose we know the pressure behaviour of B0 (i.e. B0(P)) as shown in Figure 3. 1.4

Bo(P) = m.P + c

Oil FVF Bo 1.3

4000

P (psi)

5500

Figure 3 B0 as a function of pressure for a black oil.

If we draw the reservoir pressure down by an amount ∆P (known or measured) and we know that to do this we had to produce a volume Np (STB) of oil. This point of depletion is shown in Figure 4. 1 1-

X

Np N Y 0

10

-∆P

Figure 4 Reservoir depletion on a plot following equation 9.

2


We know Y ( it is ∆P ), we can calculate X (the RHS of equation 9). X is equal to 1(Np/N) and we know Np (the amount of oil we had to produce to get drawdown ∆P ). Hence, we can find N the initial oil in place. An exercise to do this is given below.

2.3 Conditions for the Validity of Material Balance

The basic premise for the material balance assumptions to be correct is that the reservoir be “tank like” i.e. the whole system is at the same pressure and, as the pressure falls, then the system equilibrates immediately. For this to be correct, the pressure communication through the system must at least be very fast in practice (rather than instantaneous which is strictly impossible). For a pressure disturbance to travel very quickly through a system, we know that the permeability should be very high and the fluid compressibility should be low (pressure changes a re communicated instantaneously through and incompressible fluid). Indeed, we will show later (Chapter 5) that pressure equilibrates faster - or “diffuses” through the system faster - for larger values of the “hydraulic diffusivity”, which is given by k/(φµc) (Dake, 1994, p.78). Dake (1994, p.78), also points out two “necessary” conditions to apply material balance in practice as follows: (i) We must have adequate data collection (production/pressures/PVT); and (ii) we must have the ability to define an average pressure decline trend i.e. the more “tank like”, the better and this is equivalent to having a large k/(φμc) as discussed above.

EXERCISE 1. Material Balance problem for an undersaturated reservoir using equation 8 above. This describes a case where production is by oil, water and formation expansion above the bubble point (Pb) with no water influx or production. Exercise: Suppose you have a tank - like reservoir with the fluid properties given below (and in Figure 4). Plot a figure of

Np   1 −  vs. - ∆P over the first 250 psi of depletion  N

of this reservoir. Suppose you find that after 200 psi of depletion, you have produced 320 MSTB of oil. What was the original oil in place in this reservoir? Input data: The initial water saturation, Swi = 0.1. The rock and water compressibilities are, as follows: cf = 5 x 10-6 psi-1;

cw = 4 x 10-6 psi-1.

The initial reservoir pressure is 5500 psi at which Boi = 1.3 and the bubble point is at Pb = 4000 psi where Bo = 1.4. That is, the oil swells as the pressure drops as shown in Figure 4.


11

3. SINGLE PHASE DARCY LAW We review the single phase Darcy Law in this section in order to put our own particular “slant” or viewpoint to the student. This will prove to be very useful when we derive the flow equations of reservoir simulation in Chapter 5. We also wish to extend the idea of permeability (k) somewhat further than is covered in basic reservoir engineering ∇ we wish to introduce the idea of permeability as ∇ texts. In particular, a tensor property, denoted by k . Some useful mathematical concepts will also be introduced in this section associated with vector calculus; in particular, the idea of gradient ∇ and divergence ∇ • will be discussed in the context of the generalised formulation of the single phase Darcy law. Note for reference,  __ many of the terms Swi .cthat w + crock N . Boiare− also N . Bsummarised N .the Boi Glossary.  .∆ P = 0 o + N p . Bo + in discussed here  1− S 

.

.

wi

3.1 The Basic Darcy Experiment Darcy in 1856 conducted through __ packs of sands which he took S .ctests N a series of flow w + cf  . ∆ 0 ground water supply at Boi  wiof an Boi − experimental Bo + p . Bo −models as approximate aquifer for=the  P 1 − Sexperiment  wi Dijon. A schematic ofNthe essential Darcy is shown in Figure 5 where

we imagine a single phase fluid (e.g. water) being pumped through a homogeneous sand pack or rock core. (Darcy used a gravitational head of water as his driving force __ c f  normally B B  S .cw +would p whereas, inNmodern = 1 −coreoilaboratories, + oi wi we .∆ P = 0 use a pump.)

   N

Bo

 Bo  1 − Swi

 

The Darcy law given in Figure 5, is in its “experimental” form where a conversion factor, β, is indicated that allows us to work in various units as may be convenient __ to the problem ∆ p =at0hand. In differential form, a more useful way to express the Darcy Law and introducing the Darcy velocity, u, is as follows:

k ∆P k  ∂P  Q u=  =− . = − .   A µ L µ  ∂x 

(9)

where the minus sign in equation 9 indicates that the direction of fluid flow is down the pressure gradient from high pressure to low pressure i.e. in the opposite direction to the positive pressure gradient. ∆P

Q

Q L Q = β.

k.A ∆P . µ L

Definitions: Symbol Dimensions

Meaning

Consistent Units c.g.s lab.

field

SI field

Q

L3/T

Volumetric flow rate

cm3/s

cm3/s

bbl/day

m3/day

L

L

Length of system

cm

cm

ft.

m

L2

Cross - sectional cm2

cm2

ft.2

m2

12 A

2

∆P

Q

Q Basic Concepts in Reservoir Engineering L Q = β.

k.A ∆P . µ L

Definitions: Symbol Dimensions

Consistent Units c.g.s lab.

field

SI field

Q

L3/T

Volumetric flow rate

cm3/s

cm3/s

bbl/day

m3/day

L

L

Length of system

cm

cm

ft.

m

A

L2

Cross - sectional cm2 area

cm2

ft.2

m2

Viscosity

cP

cP

cP

Pa.s

µ

Figure 5. The single phase Darcy Law

Meaning

∆P

M.L.T.2 (Force/Area)

Pressure drop

atm

dyne/cm2

psi

Pa

k

L2

Permeability#

darcy

darcy

mD

mD

β

dimensionless Conversion factor

1.00

9.869x10

-6

-3

1.127x10

8.527 x10-3

# permeability - dimensions L2; e.g. units m2, Darcies (D), milliDarcies (mD); 1 Darcy = 9.869 x 10-9 cm2 = 0.98696 x 10-12 m2 ≈ 1 µm2.

Note on Units Conversion for Darcy’s Law: the various units that are commonly used for Darcy’s Law are listed in Figure 2 above. Sometimes, the conversion between various systems of units causes confusion for some students. Here, we briefly explain how to do this using in the previous figure; that is, we go from c.g.s. k.A the ∆examples P Q -=gram β. - second) .  units where β = 1, indeed, the Darcy was defined such (centimetre µ  L that β = 1. Starting k.Afrom  ∆Pthe  Darcy Law in c.g.s. units:

Q = β.

. µ  L

k (Darcy) . A (cm 2 )  ∆P (atm )  .  µ (cp) 2  L ( cm )  k (Darcy) . A (cm )  ∆P (atm )  . Q (cm 3 / s) = 1.00   L (cm )  µ (cp) Suppose we now wish to convert to field units as follows: k (Darcy) . A (ft 2 )  ∆P ( psi)  Q ( bbl / day) = ?? .  µ (cp) 2  L ( ft.)  k (Darcy) . A (ft )  ∆P ( psi)  Q ( bbl / day) = ?? .   L (ft.)  µ (cp) k ∆P ( mD) . A (ft.2 ).30.482  ( psi)   bbl 1.58999x10 5   1000 for these new units? Essentially, .7we How do we Q find the . correct conversion . 14  = factor 4 k ∆ P 2 2  8starting .64 x10 from cp 30 L (βft=.).(1.  day ( mD)µ. (A ()ft. ).we .48 that 30know psi.48 )  5  the c.g.s. expression convert it unit by unit where  bbl 1.58999 x10    1000 14 . 7  Q to know . a few conversion . (exact), We do need  = factors as follows: 1 ft. = 30.48 cm 4  8.64 day 1 bbl µ 3(cp ) 30.48  x10 14.7 psi = 1 atm., = 5.615 ft3= 5.615 x 30.483 cm = 1.58999 x 105 cm day  L3,(1ft.).   = 24 x 3600 s = 8.64 x 104 s. Thus, we now 4 convert 2 everything in the field units to2 ∆      .64which bbl x10 . 30.48 c.g.s. units for 8cp. are the same):  k ( Darcy) . A ( ft. ) . P( p Qas follows   = (except  L (ft µ (cp)  day   1000 . 1.58999x410 5. 14.27 . 30.48   bbl    k (Darcy) . A (ft.2 )  ∆P( p 8.64 x10 . 30.48 Q .  =   L (ft µ (cp)  day   1000 . 1.58999x10 5. 14.7 . 30.48  2  bblEngineering,  k (Darcy) . A (ft. )  ∆P( psi)  Institute of Petroleum Heriot-Watt 13 -3 University Q .   = 1.126722x10 µ (cp)  day  2  L ( ft.)  ∆P psi   Q (cm 3 / s) = 1.00

k.Ak.A∆P ∆P Q= Qβ=. 3βµ. . L.   k (Darcy) . A (cm 2 )  ∆P (atm )  µ  L

Q (cm / s) = 1.00

µ (cp)

.   L (cm ) 

k (Darcy) . A (cm 2 )  ∆2P (atm )  . )  ∆P (atm )  k (Darcy) . A 2(cm µ (cp ) )  Q (cm / s) = 1.00 ( psi  k (Darcy ) ). A (ft )  L∆P(.cm

Q (cm 3 / s3) = 1.00

µ (cp) µ (cp)

Q ( bbl / day) = ??

.  L (cm  )   L (ft.) 

k (Darcy) . A (ft 2 )  ∆P ( psi)  2 k (Darcy) . A .(ft L ) (ft∆ P ( psi)  Q ( bbl / day) = ?? µ (cp)k . .)  2 

Q ( bbl / day) = ??

∆P  (Lft.(ft)..)30.482  µ (cp)( mD) . A ( psi)   bbl 1.58999x10    1000 14 . 7 Q . . = k 4  µ ((ft cp.2)).30.482  ∆PL (psi ft.).30  day 8.64 x10   .48  ( ) . mD A ( ) 5  bbl 1.58999x10    1000 k 142.7 ∆P 2 Q . . .48  =5 4 ( ) . ( . ). mD A ft 30 psi ( )   64 x10 x10 µ (cp) 30.48  day    bbl 8.1.58999   L (ft.). Q . = 1000 . 14.7   4  Thus, collecting thex10 numerical day 8.64 (cpobtain: ) L (ft.).30.48   factors togetherµwe  bbl    k (Darcy) . A (ft.2 )  ∆P( psi)  8.64 x10 4. 30.482 Q .   =  5  L (ft.)  µ (cp)2  day   1000 . 1.58999 4 x10 . 14 2 .7 . 30.48  5

 bbl    k (Darcy) . A (ft. )  ∆P( psi)  8.64 x10 . 30.48 Q .   =  5    L (2ft.)  day 1000 . 1.58999 x 10 14 7 30 48 . . . . µ (cp)   4 2

 bbl    k (Darcy) . A (ft. )  ∆P( psi)  8.64 x10 . 30.48 2 Q . ∆P( psi)     =  5  bbl  ( ) . ( . ) k Darcy A ft -3     ( .) L ft 1000 . 1.58999 x10 . 14.7 . 30.48 µ cp ( )  Q   day = 1.126722 x 10 .    which simplifies to 2   day µ cp ( ) ( .) L ft  bbl   -3 k ( Darcy ) . A ( ft. )  ∆P( psi )  Q  = 1.126722x10  day 

µ (cp)

.   L2 (ft.) 

 bbl  -3 k ( Darcy ) . A ( ft. )  ∆P( psi )  Q .   = 1.126722x10  L (ft.)  µ (cp)  day 

.

and hence β = 1.127 x 10-3 for these units (as given in Figure 5).∇

.

∇

3.2 Mathematical Note: on the Operators “gradient” ∇ and “divergence” ∇ •

Before generalising the Darcy Law to 3D, we first make a short mathematical digression to introduce the concepts of gradient and divergence operators. These will be used to write the generalised flow equation of single and two phase flow in Chapter 5. Gradient (or grad) is a vector operation as follows:

∂∂ ∂∂ ∂∂ ∇ ∇ = = ∂∂x ii + + ∂∂y jj + + ∂∂z kk ∇ = ∂x i + ∂y j + ∂z k ∂x ∂y ∂z where i, ij, and k are the unit vectors which point in the x, y and z directions, i , jj and and kk respectively. The operation can be carried out on a scalar field such as i , j and gradient k pressure, P, as follows:

∂∂P P i + ∂∂P P j + ∂∂P P P ∇ = P i + j + ∂∂Pz kk ∇ = ∂ P ∂ P ∇ ∂ x ∂ y ∇P = ∂x i + ∂y j + ∂z k ∂x ∂y ∂z

.

.

∇

where ∇ P is sometimes written as grad P. The quantity ∇ P is actually a vector of ∇ ∇P P the pressure ∇Pgradients in the three directions, x, y and z as follows:

 ∂∂P P i   ∂∂P x i   ∂∂xx i  P j   ∂∂P ∇ ∇P P = =  ∂∂P j ∇P =  ∂yy j   ∂y  P k  ∂∂P  ∂∂∂Pzz kk  ∂z  14

 ∂∂P   ∂P   ∂∂P P   P  ii,,  ∂P  jj,, and and   kk

.

∇

 ∂P  i  ∂x    Basic ∂P Concepts  ∇P = j  ∂y     ∂P k  ∂z 

2

in Reservoir Engineering

This is shown schematically in Figure 6 where the three components of the vector

∂P ∇ P, i.e.   i,  ∂x 

 ∂P   ∂P    j, and   k , and are shown by the dashed lines. ∂z  ∂y 

Figure 3: The definition of grad P or

∆

z

P

Unit vectors k

i

j y

.

∇

Figure 6 The definition of grad P or ∇P

x

Divergence (or div) is the dot product of the gradient operator and acts on a vector to produce a scalar. The operator is denoted as follows:

. . . . . . . . . .. . . .. .. . . .. . .

. . . . . . . . .. . . . . . .. . . . .. .. .. . .

. . .

 ∂ ∂ ∂  ∇ = i j k ∂∂y ∂∂z   ∂∂x ∇ = i j k ∂y ∂z   ∂x For example, taking velocity vector, u, gives the  ∂ the divergence ∂ ∂ of the Darcy   u i x ∇ = i j k following:   ∂  z ∂∂ i   ∂ j  ∂∇ u ∂=∂x ∂ i ∂∂y ∂∇j ∂= k k   uu yx ij   j ∇ = i k ∂z  ∂∂zx   ∂y ∂y ∂∂x ∂z ∂∂y  ∂x   ∂ ∇ u = i j k   u zy kj   ∂y ∂z   u i   ∂x x  ∂ ∂ ∂   u z k  ux i  ∇ u = i j  u x i k ∂ u y j  ∂  ∂     u i ∂∂y∇  u ∂= z   i x   ∂∂x  ∂ j∂u k ∂uu y j   ∇ = u i j k u j     ∂ ∂ ∂ ∂u  y the  u k   y  ∂equation where we∇can x ∂z ∂by y multiplying u =  ∂the k x uu zxy ij  = i i +  out j j +first z k k  y i RHS∂zofj the  ∂xexpand   above    matrix” (1x3) matrix by the second which ∂∂x (3x1) ∂∂y matrix zkobtain   a “1x1  u∂∂zto ∂∂uyuyisz kascalar    ∂∂ux ∂∂uzz ∇ u = i j k   u zy kj  =  x i i + jj + k k as follows: ∂y ∂z   u i   ∂x ∂y ∂z  ∂x  x ∂uuy i   ∂  ∂ ∂   u z k  ∂u x ∂ u ∇i i =u j =j = k ik = 1 j  u x i k ∂ u y j =∂  i i∂ +  xj j +  ∂uz k k  ∂∂y∇  u ∂= z   ∂u   ∂x∂u y k ∂yu∂ujz =  ∂zx i i +  ∂∂x  ∂  ∇ u = i j k  u y j  ∂=xui z kxi ∂iy+j j j +  y k k ∂x ∂ z   i i∂x= j j =∂ky k = 1 ∂z    ∂z   ∂y   ∂x  u z k u z k  ∂u y  ∂u x ∂u z  ∇i i =u j =j = k k =+1 + ∂ux ∂uy ∂uz   ∂  ∂ ∂ y where i i += j j z= k k = 1 , to obtain: uk =relationships, i i =we j j∇use = kthe = 1 x + ∂ x ∂ y ∂z   ∂u y ∂u z  ∇P ∇ ∇P∂u x ∇ u =  + + ∂u y ∂u z  ∂z ∂u x  ∂u x ∇P∂∂uxy ∂∂uy  ∇ u =∇P ∇ + + ∇z  u =  ∂x + ∂y + ∂z    ∂y ∂z   ∂x   ∂P   i ∇P ∇ ∇P   ∂x   Institute of Petroleum Engineering, Heriot-Watt University   ∂ 15 P  ∇P ∇ ∇P    i  ∇P ∇ ∇P 2 2  ∂  ∂ P ∂ ∂ ∂    ∂P x    ∂ P 

.

. .

.

.

.

. .

.. . . . . . . . . . . . . . . . . . . . .. .

. .

.

.

. . . .

. u =  ∂∂x i.

∇

.  u j  =  ∂∂ux i.i +

.

∂ j ∂y

. . .

∂u y

x

y

∂y

i i = j j = k k = 1  u z k

. . .

i i = j j = k k =1

∇

∇

x

∂ k ∂z

.. .

. u =  ∂∂ux

+

x

∂u y ∂u z∂ + ∂y∇ = ∂ z∂x i

.

∂u y  ∂u ∂u z  u =  x + + ∂y ∂z  ∇  ∂x

∇

.

jj +

.

 ∂u z k k ∂z 

.

∂  k ∂z 

∂ j ∂y

.

.

∇

.

.

∇ ∇P

∇P

 uquantity,  Likewise, we can take the divergence of the grad P vector, ∇ P, to obtain the xi   ∇P ∇ ∇P ∂ ∂ ∂ ∇ • ∇ P, (sometimes denoted div grad ∇ P), = follows: u as i j k   uy j 

.

∇

.

.

. .

. . . .

.

. . .

.

. . .. .

.

. .

.

.

2

2

2

2

2

. .

.

.

i i = j j = k k = 1 , we obtain the familiar

. ∇P =∇. ∂∂x∇PP += ∂∂y∂P P + + ∂∂z∇P∂.P=u∇+=P ∂∂uP  += ∂∇u P +   ∂x   ∂y    ∇. ∂x   ∂z    ∂y   2

.

.

. . .

.

=j1jrelationships, i i = j using j =i ki k=the Again = k k =1 expression: ∇

.

 ∂z    ∂x  ∂P  ∂y  i  u z k   ∂x     ∂P   i     ∂x    ∂    ∂P    ∂ 2 P   ∂  ∂2 P   ∂2 P  ∂ ∇ ∂ ∇P =∂ i ∂   j∂P   k ∂2 P   j  ∂=2 P 2  i ∂i2 uP+ i  2  j j +  2  k k  xk k ∂y  j ∂z=    ∂iyi +  2  ∂ j xj + ∇P =  i  ∂z∂u  j∂x  ∂ky   ∂u z  ∂ ∂y  ∂ ∂z 2     ∂u x ∂y ∂z    ∂y    ∂x 2   ∂x y ∇  u =  i j k u j = + + i i j j k k  y     ∂ P  ∂y   ∂P    ∂x  ∂z   ∂y ∂z    ∂x    k    k u k     z   ∂z   ∂z  

2

2

2

2

2

2

y 2

x

2

∂u z  ∂z 

.

where, in summary, ∇ 2 is the Laplacian ∇P operator: ∇ ∇P

∇2 = ∇

. ∇ = div . grad = ∂∂x

2

+

∂2 ∂2 + ∂y 2 ∂y 2

  ∂P   i   ∂x   The final issue we wish to discuss in this mathematical note is the rule for  taking  now    ∂P of  ∂ 2 P   ∂2 P   ∂2 P  ∂ omit∂the explicit ∂ inclusion the dot product of a tensor and a vector. N.B. We k xx k xy k xz ∇ ∇ P = i j k j = + + i i j j    2 k k        the unit vectors,  ∂z  ∂y ∂z    ∂y    ∂x 2   ∂x  ∂y 2   i, j and k in the  following developments.   k =  k yx k yy k yz    ∂P    k in  ∂ 2 and Tensor ∂2 ∂2 3.3 Darcy’s 3D k- zzUsing Vector Notation 2 Law k    k zx zy 2 2 2 ∇ =have ∇ a∇tensor = div k grad = ∂ 23D+ can + ∂2 a 3∂zx 3 2be represented ∂ Suppose we which ∂in by 2 ∇ = ∇ ∇ = div grad = x 2 + ∂y 2 + ∂y 2 matrix as follows: ∂x ∂y ∂y ∇P k ∇P i i = j j = k k =1  k xx k xy k xz   k k k xz  k =  k xx k yz  k xy yx yy   ∂P   2 k =  k yx k yy k yz   ∂2 P   ∂ 2 P  ∇ 2    ∂ P   k k k   ∇ ∇ P = + + ∂ x zx zy zz  2  = ∇ P  2  2     ∂x   ∂z    ∂y  k xx k zzk xy k xz   ∇  k zx kzy      ∂P   k product k = k k ∇Pwish    of Suppose we ∇Pnow k ∇P  toyxtake yya dot yz  ∂y  this tensor, k , with the vector ∇ P;   that is k •∇∇ product of a tensor and a vector is a vector and the operation ∇Pdot P P. kThe  k zx k zy k zz  ∂P  is carried out like a matrix multiplicationas follows:      ∂Pz      ∂∂P x     k xx k xy k xz   ∂x    k k xz   ∂P     ∂P  k  ∂P   ∂P    k yz   ∂∂P k xy k ∇P =  k xx   k xx   + k xy   + k xz     yx yy P   ∂z     ∂y  k ∇P =  k yx k yy k yz   ∂∂yx     ∂x  k k k ∂ y   zx zy zz  kk xx kk xy kk xz   ∂P      zx zy zz      ∂P   ∂P    ∂∂P  ∂P     P k ∇P =  k yx k yy k yz   ∂z   = k yx   + k yy   + k yz      ∂y     ∂x   ∂z     ∂y  k      ∂ z      zx k zy k zz  ∂P           k zx  ∂P  + k zy  ∂P  + k zz  ∂P  ∂P  ∂P    ∂Pz    ∂P  16 k xx  ∂∂xP  + k xy ∂∂yP + k xz  ∂∂Pz       ∂∂P x   k xx  ∂x  + k xy  ∂y  + k xz  ∂z            2

.

..

.

.

. . . .. .

.

.

.

.

. .

. . . .

.

.

.

.

.

k∇∇PP k ∇P

∇P

2

Basic Concepts Engineering   ∂P   in P   ∂Reservoir

  ∂x     ∂x    k xx kxyk k xz k  k    xx  xy ∂P  xz    ∂P    k k yx k k ∇Pk = ∇      P = yyk yx yz kyy ∂yk yz k     ∂y    zx kzyk k zz k ∂Pk   zx zy zz           ∂P     ∂z       which multiplies out as follows:   ∂z  

.

k

.

.

  ∂P   k  ∂P  + k  ∂P  + k  ∂P     xx xy  xz   ∂x   ∂P ∂x  k  ∂P∂y  + k ∂∂Pz   + k  ∂P          xx xy xz   k xx k xy k xz      ∂x   ∂z   ∂y      ∂P   ∂x ∂P      k yx    + k yy  ∂P  + k yz  ∂P    ∇P =  k yx kyyk xxk yz k xy k xz = ∂ y    ∂P ∂x    ∂P∂y  ∂∂Pz     ∂P  k zx k ∇ P =kzyk yxk zz kyy k yz    = k yx   + k yy   + k yz     ∂P   ∂y    ∂x   ∂z   ∂y   k   k  k zx  ∂P   + k zy  ∂P  + k zz  ∂P   k   zy ∂z zz   zx  ∂z   ∂y  ∂P  ∂x    ∂P    ∂P   ∂P 

.

    ∂z  

k zx     ∂x 

+ k zy    ∂y 

giving the final result:  ∂P   ∂P     ∂P  k xx   + k xy   + k xz      ∂z     ∂y    ∂x      ∂ P ∂ P      ∂P      ∂Pk xx   ∂P+ k xy   ∂P + k xz       ∂z    k ∇P = k yx   + ∂kxyy  + k∂yzy      ∂x   ∂z     ∂y        ∂P ∂P   ∂P      ∂P  ∂P  ∂P   + k k ∇ P = k + k k zx   yx+ ∂kxzy  yy+ k∂zzy   yz  ∂z    ∂x ∂z   ∂y    

+ k zz    ∂z 

.

.

  ∂P  k zx     ∂x 

 ∂P  + k zy    ∂y 

 ∂P   + k zz     ∂z   

Using the above concepts from vector calculus (div. and grad), we can extend the Darcy Law (in the absence of gravity) to 3D as follows by introducing the tensor permeability, k :

u= -

.

1 k µ u

  ∂P   ∂P    ∂∂PP     ∂P   k xz    + k xx   ∂P+ k xy   +k xx     ∂x    ∂∂xz      ∂x∂x    ∂y    k xx k xy k xz       ∂P   k∂xx 1 P   k xy1 k xz∂P   ∂P   ∇P = 1-  k yx k yy k yz1   = - k yx  ∂P+ k yy  1 + k yz ∂P    = - µkk ∇kP = k- k∂yxy   k yyµ k yz∂x   = -∂y  k yx   ∂z    +   ∂x   µ  zx zy zzµ ∂P    ∂P∂y    ∂Pµ   ∂P    k zx   k zy kkzz + k zy   + k zz    zx      ∂z      ∂z   ∂ x ∂ y   ∂ P       ∂ P 

.

    ∂z  

which we maywrite  ∂Pas: 

 ∂P   ∂P    k xx   + k xy   + k xz      ux     ∂z    ∂ x ∂ y         ∂P∂ P   ∂P  ∂P    1   ∂P  k xx   + k+yz  k xy   k yy u =  u y  = - kuyx   +  ∂z  ∂y  µ  x  ∂x      ∂y ∂ x         ∂P         ∂ P ∂ P    uz  k zy   ∂P + k zz   ∂P  k zx   1+     x u =  u y ∂= k ∂y  + ∂kzyy    yx  

    1   ∂P u  u x = - k xx   z+ µ

 ∂x 

µ   ∂x   ∂y    ∂P   ∂P  ∂P∂P   k xy  k+zx k xz    + k zy     ∂ z  ∂y   ∂y    ∂x  

 ∂P  1   ∂P   ∂P   u y = - k yx   + k yy   + k yz     ∂z ∂P  µ   ∂x  1   ∂∂yP  u =- k + k 

 xx  Heriot-Watt  + x xy  Institute of Petroleum Engineering, University 1

 ∂P 

µ

∂x

 ∂P 

 ∂y 

 ∂P  

k zx     ∂x 

 ∂P k yy   ∂y

 ∂P  + k zy    ∂y 

 ∂P    + k xz      ∂z      ∂P    + k yz     ∂z      ∂P   + k zz     ∂z   

 ∂P   k xz     ∂z  

 ∂P k xy   ∂y

17

 ∂z 

  ∂x    ∂P   ∂P   ∂P    k + k + k         xx xy xz   ux   ∂x   ∂z     ∂y      ∂P  ∂P   ∂P     k + k + k         xx xy xz  u x   ∂Pz   x  y   ∂P 1   ∂P u =  u y  = - k yx   + k yy   + k yz      ∂z    µ   ∂x   ∂y    1 ∂P  ∂P   ∂P   u =  u y  = - k yx   + k yy   + k yz       ∂zP   µ   ∂Px   ∂∂Py   uz  k + k + k       zx    zy  zz      ∂ x ∂ y ∂ z         ∂P   ∂velocity P  ∂P   and we can identify components of the as follows:  u z  the three k zx   + k zy   + k zz     ∂z    ∂y    ∂x    ∂P  1   ∂P   ∂P   u x = - k xx   + k xy   + k xz     ∂z   µ   ∂x   ∂y  1 ∂P  ∂P   ∂P   u x = - k xx   + k xy   + k xz     ∂z   µ   ∂x   ∂y   ∂P  1   ∂P   ∂P   u y = - k yx   + k yy   + k yz     ∂z   µ   ∂x   ∂y  1 ∂P  ∂P   ∂P   u y = - k yx   + k yy   + k yz     ∂z   µ   ∂x   ∂y   ∂P  1   ∂P   ∂P   u z = - k zx   + k zy   + k zz     ∂z   µ   ∂x   ∂y  1 ∂P  ∂P   ∂P   u z = - k zx   + k zy   + k zz     ∂z   µ   ∂x   ∂y  If the permeability tensor is diagonal i.e. the cross-terms are zero as follows:  k xx 0 0    k = 0k k0yy 00  xx 0  k = 0 k0yy k0zz  0 0 k zz   then the various components of the Darcy law revert to their normal form and :

1  ∂P  k xx    ∂x  µ 1  ∂P  u x = - 1 k xx  ∂P  1 u = -∂µP k xx  ∂x  u x = - kxxx  µ   x  µ u y = -∂1x k yy  ∂P  µ  ∂y   ∂P  1 u y = - 1 k yy  ∂P  1 u = -∂µP k yy  ∂y  u y = - kyyy  µ 1   ∂∂Py  µ u z = -∂y k zz    ∂z  µ 1  ∂P  u z = - 1 k zz  ∂P  1 u = -∂µ P k zz  ∂z with Gravity 3.4 - kzzz Darcy u z =Simple  ∂zP µ1  Law ∂z  µ u x = -∂of z gravity gρDarcy k xx  the-1D  Law becomes: In the presence  ∂x µ ∂x  1 ∂z   ∂P u x = - 1 k xx  ∂P - gρ ∂z  1 u = -∂µP k  ∂∂xz - gρ ∂x  u x = - kx∂xxz µ -xxgρ∂x  ∂x  µ   ∂=x cos θ ∂x   ∂x  ∂zcase  of a simple inclines system at a slope of θ, as shown in Figure 7, where, inthe  ∂∂xz  = cos θ θ  ∂z    , =ascos  ∂xθ 1 shown   = cos ∂Pin the figure above  and:  ∂x  u x = - k xx  - g.ρ. cosθ  ∂x  µ 1  ∂P  u x = - 1 k xx  ∂P - g.ρ. cosθ  1 u = -∂µP k  ∂x - g.ρ. cosθ u x = - kxxx 2 πµkhr-xxgdP .ρ∂.xcosθ    µ Q = ∂x µ  dr  2 πkhr  dP  18 Q = 2 πkhr  dP  Q = dPµ  dr  2 πkhr ux = -

 ∂y 

 ∂z   

2


Note that: ∂z ∂x

= cos θ

x

Figure 7 Radial form of the singlephase Darcy Law

θ

3.5 The Radial Darcy Law

In the above discussion, in both 1D and 3D we considered the Darcy Law in normal Cartesian coordinates (x, y and z). In Chapter 6, we will explain how wells are treated in reservoir simulation. Because a radial (r/z) geometry is appropriate for the near-well region, it is useful to consider the Darcy Law in radial coordinates. In 1D, this simply involves the radial coordinate, r. In fact, the radial form of the Darcy law can be derived from the linear form as shown in Figure 8.

1  ∂P  k xx    ∂∂P  µ 1  xQ u x = - k xx    ∂x  µ ux = -

 ∂P  1 k yy   µ y   ∂∂P 1 = u k   yy h y µ  ∂y 

dr

uy = -

Figure 8 Single phase Darcy Law in an inclines system - effect of gravity

Area, A = 2π.r.h

r

Radial Darcy Law is:

1  ∂P  u z = - k zz   2πkhr dP k.A dP Q= =  ∂∂Pz  µ µ dr µ dr 1   u z = - k zz    ∂z  µ 1Q ∂z flow rate of fluid into well Notation:  ∂=Pvolumetric - gρ  u x = - k xx  r radial distance  ∂∂=P µ ∂∂xz  from well 1  x - gρ of formation u x = - h k xx  = height  µ ∂x  pressure drop from r→ (r + dr) i.e. over dr dP  ∂=xincremental = area of surface at r = 2π.r.h  ∂z  A   =μ cos θ = fluid viscosity  ∂∂xz    k   = cos θ = formation permeability  ∂x  r = wellbore radius w

1dr  ∂=Pincremental radius  u x = - k xx  - g.ρ. cosθ   µ x 1radial form ∂∂P  Starting from the of the Darcy u x = - k xx  - g.ρ. cosθ Law, as follows:   µ ∂x 2 πkhr  dP  Q= dr  2 πµkhr  dP Q= µ  dr  we can rearrangeµthis Q todrobtain:

  2µπQ kh  dr r  dP = 2 πkh  r  dP =


19

Taking rw as the wellbore radius and r some appropriate radial distance, we can easily integrate the above equation to obtain the radial pressure profile in a radial system as follows: r

r

µQ dr = µQ ∫r  dr  ∫r dP   r  r∫w ∫rw dP = 22ππkh kh rw  r  w r

which gives:

µQ  r  ∆P( r ) = µQ ln r  ∆P( r ) = 2 πkh ln rw  2 πkh  rw 

∂Pw ∂z  k.k rw the where weuhave denoted - gpressure ρw ∂z  drop (or increase for a producer) from rw to  ∂Pradial w = - k.k  ∂xw the µ rw unlike x  Law, the pressure profile is logarithmic r as, ΔP(r). u wNote = - that, - linear gρw ∂Darcy µ means ∂x that pressure ∂x  drops are much higher closer to the well. in the radial case. This This is exactly what we expect physically since the area is decreasing with r as we k.k ro Q∂Piso the same;∂ztherefore,  approach uthe=well the pressure drop, dP, over a given - k.kand  ∂P - gρo ∂z  o ro o   dr is higher. This is shown schematically for an injector and a producer in Figure 9. u o = - µ  ∂x - gρo ∂x    ∂x developed ∂x here will be used later in Chapter 4 on well The formulae and µthe ideas modelling in reservoir simulation and we will not discuss this further here.  ∂Pw   ∂P   ∂P  and  ∂Po   ∂xw  and  ∂xQo   ∂x   ∂x Injector

Q Producer

Pwf

Pc (Sw ) = Po − Pw Pc (Sw ) =∆P(r) Po =−PP wfw- P(r)

∆P(r)

∆P(r) ∆P(r) = P(r) - Pwf

Pc (Sw ) = Pnon − wett . − Pwett . Pc (Sw ) = Pnon − wett . − Pwett . k rww = k k rwr k w = k k rw 4. TWO-PHASE k o = k kFLOW ro k o = k k ro 4.1 The Two-Phase Darcy Law

Pwf rw

r

Darcy’s Law was originally applied to single phase flow only. However, in reservoir k w and k o engineering, it hask been convenient to extend it to describe the flows of multiple k w and phases such as oil, owater and gas. To do this, the Darcy Law has been modified empirically to include a term - the relative permeability - which is intended to describe the impairment of the flow of one phase due to the presence of another. A schematic representation of a steady-state two phase Darcy type (relative permeability) experiment is shown in Figure 10, where all of the quantities are defined. Examples of the relative permeability curves which can be measured in this way are also shown schematically in Figure 10 and actual experimental examples are given for rock curves of different wettability states in the Glossary.

20

Figure 9 Pressure profiles, ΔP(r), in radial single-phase flow; Pwf is the well flowing pressure (at rw)

2


At steady - state flow conditions, the oil and water flow rates in and out, Qo and Qw, are the same:

∆Po

∆Pw

Qw Qo

Qw Qo

L

The two - phase Darcy Law is as follows:

Qw =

Schematic of relative permeabilities, krw and kro

k.krw.A ∆Pw . L µw

1

Figure 10 The two-phase Darcy Law and relative permeability

kro

Rel. Perm.

k.kro.A ∆Po Qo = . L µo

0

krw

0

Sw

1

Where: Qw and Qo A L µw and µo k ΔPw and ΔPo

= volumetric flow rates of water and oil; = cross-sectional area; = system length; = water and oil viscosities; = absolute permeabilities; = the pressure drops across the water and oil phases at steady-state flow conditions r r µthe Q water drand   r r krw and kdP = oil relative permeabilities = µQ  dr  ro

∫

2 πkh r∫w  r 

rw

NB the Units for the two-phase Darcy Law 5.

µQ

∫ dP = 2πkh ∫  r 

arerwexactly

rw the same as those in Figure

 r

∆P( r ) =form of ln µQ  including r   phase Darcy ∆Law The differential gravity ln  P( r )in= 1D, again 2 πkh the rtwo w which is taken to act in the z-direction, is as follows: 2 πkh  rw  uw = -

∂z  k.k rw  ∂Pw - gρw   µ  ∂x ∂x 

uw = -

∂z  k.k rw  ∂Pw - gρw   µ  ∂x ∂x 

uo = -

∂z  k.k ro  ∂Po - gρo    µ ∂x ∂x 

uo = -

∂z  k.k ro  ∂Po - gρo   µ  ∂x ∂x 

where we note ∂P that the flow ∂P of the two phases (water and oil, in this case) depends

 w     and  o  ∂P ∂P on the pressure  ∂x gradient  ∂inx that  phase; i.e. on  w  and  o  .  ∂x   ∂x  Pc (Sw ) = Po − Pw

Pc (Sw ) = Po − Pw


P (S

)=P

− P

21





= µQ∫  dr  ∫ dP k ∫∂P ∫ dP =2kπ.kh r  

rw

khrwrw wr - gρw ∂z  urw w = - 2 πrw µ  ∂x ∂x 

µQ  r r  ∆P∆(Pr()r=) k=.kµQln   ∂z  ln ro ∂P u o = - 2 π2kh  orwr-w gρo  π kh µ  ∂x ∂x 

uw = -

∂z  k.k rw  ∂Pw - gρw   µ  ∂x ∂x 

uo = -

∂z  k.k ro  ∂Po - gρo   µ  ∂x ∂x 

∂z∂z k.k  ∂PwP u wu ∂=P=-w - k.rwk rw ∂∂P w- gρw  o g ρ P∂and w wx  x ∂  ∂SPw (S  = 1 - ∂SPo), are generally  µand  The phase pressures, P , at a given ∂x  saturation,  w  oand  w ∂x  µ o ∂∂xx w  ∂x  pressure,  ∂x as  follows: not equal. However, they are related through the capillary ∂z∂z k.kk.rok  ∂P∂oP   u oP co(S u= =- w-)µ= Proo ∂−x Po w- -gρgoρ∂o x   µ  ∂x ∂x 

Pc (Sw ) = Po − Pw

More strictly, the capillary pressure is the difference between the non-wetting P S = P ∂P− wett . − Pwett .  ∂Pc∂w(Pw)andnon phase pressure the pressure; Pc (Sw ) = Pnon − wett . − Pwett .. We Po   ∂owetting-phase  of the w and and ∂x pressure  can think as a constraint on the phase pressures. That is, if  ∂x  capillary   ∂x  ∂x we know kthe capillary pressure function - from experiment , say - then, if we have w = k k rw k rw pressure curves k w =of kcapillary Po at a given saturation, we can calculate Pw. Examples are alsoPshown Glossary. (SwS) =in=Pthe cP oP − −PwP c

( w)

o

k o = k k ro

w

Note that, as in the single-phase Darcy Law, we maykgeneralise the two-phase Darcy o = k k ro expressions 3D. Defining the combination of absolute permeability in its full = − PcP(Swto P P ) S = non P − wett . − wett P.

( )

− wett . wett . c kw, with non tensor form, k w and k o the phase relative permeabilities gives:

k wk = =k kk rwk w rw k ok = =k kk rok o

k w and k o

ro

k o are the effective phase permeability tensors of water and oil, where k wk and w and k o respectively. Using this notation, the Darcy velocity vectors for the water and oil, uw and uo, may be written in 3D as follows: uw = −

1 k w .(∇Pw − ρw g∇z) µw

uo = −

1 k o .(∇Po − ρo g∇z) µo

This form of these equations is particularly useful in deriving the two-phase flow equations in their most general form (this will done in Chapter 5).

5. CLOSING REMARKS The purpose of Chapter 2 is to review some key concepts in reservoir engineering which impact directly on the subject matter of reservoir simulation. The topics reviewed specifically involved: - Material balance and its particular relationship with reservoir simulation; - The single-phase Darcy law and its extension using vector calculus terminology to a 3D version of the Darcy Law including tensor permeabilities;

22

2


- The two-phase Darcy Law and the related concepts that arise in two-phase flow e.g. relative permeabilities (kro and krw), phase pressures (Po and Pw), capillary pressure (Pc(Sw) = Po - Pw), etc. Ideas and concepts developed here will be used in other parts of this course.

6. SOME FURTHER READING ON RESERVOIR ENGINEERING A full alphabetic list of References which are cited in the course is presented in Appendix A. Many excellent texts have appeared over the years covering the basics of Reservoir Engineering. Some of these are listed below, although this list is far from comprehensive. Amyx, J W, Bass, D M and Whiting, R L: Petroleum Reservoir Engineering, McGrawHill, 1960. This is still an excellent petroleum engineering text although the coverage in some areas a little old fashioned. It has a very good chapter on material balance. Archer, J S and Wall, C: Petroleum Engineering: Principles and Practice, Graham and Trotman Inc., London, 1986. This book offers a good overview of petroleum engineering and covers many of the basics of reservoir engineering. This book is also one of the earliest proponents of the importance of integrating the geology within the reservoir model. Craft, B C, Hawkins, M F and Terry, R E: Applied Petroleum Reservoir Engineering, Prentice Hall, NJ, 1991. The original text by Craft and Hawkins was already an early classic. This was revised and updated by Terry and reissued in 1991. This has very good clear coverage of material balance and its application in various reservoir systems. Craig, F F: The Reservoir Engineering Aspects of Waterflooding, SPE monograph, Dallas, TX, 1979. This text is confined to the underlying principles and reservoir engineering applications of waterflooding. It is an excellent monograph on the subject and an essential reference text for the reservoir engineer who is interested in the traditional analytical methods for assessing waterflooding. Dake, L P: The Fundamentals of Reservoir Engineering, Developments in Petroleum Science 8, Elsevier, 1978. This has become a modern classic on the basics of reservoir engineering. It is very widely referenced and draws on Dake’s vast experience of teaching reservoir engineering basics. It has particularly good coverage of material balance and Buckley-Leverett theory. Dake, L P: The Practice of Reservoir Engineering, Developments in Petroleum Science 36, Elsevier, 1994. This book is a modern plea for the continued application traditional reservoir engineering principles and techniques in performance analysis and prediction. It gives central place to the interpretation of well testing, the application of material balance and the use of Buckley Leverett theory. It has many examples from the hundreds of reservoirs that Dake himself worked on. This book also makes a number of interesting and controversial observations on reservoir simulation (not all of which the authors agree with!). Institute of Petroleum Engineering, Heriot-Watt University

23

Solution To Exercises EXERCISE 1: Material Balance problem for an undersaturated reservoir using equation 8 above. This describes a case where production is by oil, water and formation expansion above the bubble point (Pb) with no water influx or production. Exercise: For the input data below, do the following: (i) Plot the function (1 - N/Np) as calculated by equation 8 vs. -ΔP.

 

As a reminder equation 8 is 1 −

Np  B B  Swi + c f   = 1 − oi + oi   ∆P N Bo Bo  1 − Swi 

This is shown below (1-Np/N) vs. -DP 0.999 0.997

(1-Np/N)

0.995 0.993

Series 1

0.991 0.989 0.987 0.985 0

50

100

150

200

250

300

-∆p (psi)

(ii) Note from the graph (or from your numerical calculation when plotting the graph) that, at - ΔP = 200 psi, then (1 - Np/N) = 0.991. However, we know by field observation that this 200 psi drawdown was caused by the production of 320 MSTB. That is, we know that Np = 320 MSTB. Hence, (1 - 320/N) = 0.991 => N = 35555.5 MSTB ≈ 35.6 MMSTB Answer: the STOOIP = 35.6 MMSTB. Input data: The initial water saturation, Swi = 0.1. The rock and water compressibilities are, as follows: crock = 5 x 10-6 psi-1;

cw = 4 x 10-6 psi-1.

The initial reservoir pressure is 5500 psi at which Boi = 1.3 and the bubble point is at Pb = 4000 where Bo = 1.4. That is, the oil swells as the pressure drops as shown below:

24

2


1.4

Bo(P) = m.P + c

Oil FVF Bo 1.3

4000

P (psi)


5500

25

26

3

Reservoir Simulation Model Set-Up

CONTENTS 1.

INTRODUCTION TO CHAPTER 3

2. SETTING UP A RESERVOIR SIMULATION MODEL 2.1. Defining The Objectives Of A Simulation Study

3. DATA INPUT AND OUTPUT 4.

EXAMPLE INPUT DATA FILE 4.1. Reservoir System to be Modelled 4.2. ECLIPSE Syntax 4.3. Model Dimensions 4.4. Grid and Rock Properties 4.5. Fluid Properties 4.6. Initial Conditions 4.7. Output Requirements 4.8. Production Schedule

5.

RUNNING ECLIPSE AND FILE NAME CONVENTIONS 5.1. Running ECLIPSE on a PC 5.2. File Name Conventions

6.

CLOSING REMARKS

LEARNING OBJECTIVES Having worked through this chapter and the associated tutorials the student should be able to: Simulation Input • Identify what questions the simulation is expected to address. • Identify what data is required as input to perform the desired calculations. • Format data correctly, taking account of keyword syntax and required units. Simulation Output • Select required output of calculations. • Quality check output data to check for errors in input. • Identify purpose of each output file and use post-processors to analyse data. Analysis of Results • Identify impact of reservoir engineering principles in calculation performed. • Identify numerical effects and impact of grid block size and orientation on results. • Perform simple upscaling calculation to address numerical diffusion.

2

3


BRIEF DESCRIPTION OF CHAPTER 3 In this module, a step-by-step approach is given to setting up a 3D reservoir simulation model. This is done by working through an actual case which is complex enough to demonstrate most of the basic ideas, can demonstrate various sensitivities and can also show the effects of well controls. The simulator used in this course is ECLIPSE which is a software product of Schlumberger GeoQuest. However, the general approach and methodology for setting up a field simulation calculation is very similar for other commercial simulators. This example will be used to illustrate the power of reservoir simulation in understanding reservoir recovery mechanisms.

1. INTRODUCTION TO CHAPTER 3 In this section of the course, we set up a practical 3D, two phase (oil/water) reservoir simulation model using the ECLIPSE reservoir simulator. This is proprietary software of Schlumberger GeoQuest. The central objective of this exercise is to get you actually applying reservoir simulation to a realistic (but quite simple) case. However, there are also some tasks in the study itself - you can think of these as the “objectives” if this were a real field case. One of the tasks in the exercise is as follows: in your calculation, you should observe initial short-term rise in BHP (bottom hole pressure) in the injection well and drop in BHP in the production well. You are asked to explain these trends. This is good example of where you may have run a calculation without necessarily thinking about what was going to happen to the BHP (or it could be watercut at the producer, or field average pressure etc.). However, when you study the simulator output - usually as graphs and figures - you would notice the BHP trends. This would catch the attention of a good engineer who would not be happy just to note it and move on. She or he would immediately stop and think and ask a few questions, “What’s going on here?”, “Is this something physical that I should expect or is there something wrong with the calculation?”. The engineer would stop and work it out from their basic reservoir engineering knowledge ... just as you are going to! The engineer would conclude that although I possibly didn’t expect it - this behaviour is perfectly understandable and predictable. From the above discussion, you can see that it is not just the mechanics of running a numerical simulator and getting the results out that we want you to achieve in this course. We want you to be able to formulate the right questions for a given reservoir application, carry out the appropriate simulations and then interpret the results correctly. The mechanics of running a simulation - if this was all you did - is really a technician’s job, the important job of correctly formulating the simulation problem, understanding the results and predicting reservoir performance is an engineer’s job and this course is intended for the latter (or for the former strongly intent on becoming the latter in the future!).


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2. SETTING UP A RESERVOIR SIMULATION MODEL This section will not be very long since there are many excellent examples of reservoir simulation studies elsewhere in this course e.g. Cases 1 - 3 in Chaper 1, the SPE field cases. Some generalities on how to set up a reservoir simulation study were also discussed in Chaper 1. Here, we will lay out more formally, the general procedues broadly following the workflow of a typical simulation study.

2.1 Defining The Objectives Of A Simulation Study

Defining the objectives is a vitally important stage of any field simulation study. The general spirit which is suggested for approaching this (in Chaper 1) is to correctly formulate the question you are trying to ask in order to make a particular decision. For example, the decision may be: “do I need to infill drill in this field in order to significantly (i.e. economically) improve reservoir performance?”. This is like the schematic example in Chaper 1, Figure 8 where the question was not “will I get more oil by ...”, since you could get more oil but at too great a cost - the decision must be economically based. Having said this, some reservoir decisions are made that may not in themselves be economic; however, they may be strategic or may lead to some knowledge or experience which will be economic in the future. The important matter in that you know what sort of decision you are trying to make.

3. DATA INPUT AND OUTPUT When run, every reservoir simulator will require input data that defines the system to be modelled, and should generate output data that represents the results of the calculations which have been performed. Although different reservoir simulation codes have different formats for entering data, they all have some basic components in common. Most will read data from an input file. The input file must therefore be set up before starting the simulation run. The data file may be set up by manual editing (if it is in ASCII format), or by using a Graphical User Interface (GUI). Whichever method is used, most data files will be divided into certain key sections that define: • Model dimensions • Grid and rock properties • Fluid properties • Initial conditions • Output requirements • Production schedule Additional optional sections may allow for manipulation of an imported grid structure and for subdivision of the grid into regions. We will find that there are a huge number of possible refinements in all of the above general sections representing special models for particular applications but we will focus on the simpler common features of most black oil simulations. Individual parts of the input data may be set up by other programs that may be supplied by the same supplier as the simulation code, or by other companies. These are referred to as pre-processors: They are used to perform calculations that set up the model in

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advance of the actual reservoir fluid flow calculation. Typically, a pre-processor is used to set up the grid and to define the rock properties (permeability, porosity, etc.) of each cell. Setting up a model with more than a few hundred cells would be very laborious if performed by hand. These gridding packages are usually designed to generate output that conforms to the input format of several of the more widely used simulators. Thus, the reservoir simulation input data file need only refer to these grid files by means of a simple “include” statement, and no further manipulation of the grid is required by the flow simulator. Pre-processors may be used to: • Define grid and rock properties • Define fluid properties • Convert the results of special core analysis data to a form that can be used in the simulation • Upscale rock data so that it is appropriate for the size of grid cells being used • Define vertical flow performance tables • Set up the production schedule Indeed, any software that is used for setting up a part or the whole of an input data file is termed a pre-processor. The output of the reservoir flow calculations usually comes in two forms, which in both cases results in the creation of files that can be stored and read at a later date. The first category of output data is typically referred to as “summary” data and the second type of output consists of grid data. The two types are as follows: (a) Summary data: this consists of calculated parameters such as oil, water and gas production rates, well bottom hole or tubing head pressures, etc. These data may be plotted as line charts, usually as a function of time, either by using specialised post-processing software, or by using standard graphing software such as Microsoft Excel or Lotus 1,2,3. (b) Grid data: in this type of data, values such as pressure or saturation to be plotted for each cell at a given time step. These files are typically in binary format, which means that they may only be read by appropriate post-processing software. The reason that ASCII format is not generally used is one of disk space usage. For example, a 100,000 cell model, with output data for 20 time steps, would generate 2 million values of pressure (usually to eight significant figures) during the course of the simulation, and similarly for every other property such as phase saturations, etc. Two major, and usually understated, elements of good reservoir simulation practice are thus: • Keeping a record of what each calculation represents (by choosing sensible file names and inserting comments) • Minimising disk usage (by outputting only data that is actually required). Current software developments are addressing automatic report generation and minimising the time taken to obtain a good history match by automatically varying specified parameters. Institute of Petroleum Engineering, Heriot-Watt University

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4. EXAMPLE INPUT DATA FILE 4.1 Reservoir System to be Modelled

In the first tutorial exercise, Tutorial 1A, the reservoir system to be modelled consists of a five-spot pattern of four production wells surrounding each injection well. However, the symmetry of the system allows us to model a single injection production pair, which will be located at opposite corners of a grid, as shown in Figure 1. The system is initially at connate water saturation (Swc), and a waterflood calculation is to be performed to evaluate oil production and water breakthrough time. The reservoir will be maintained above the bubble point pressure (Pb) at all times, and thus there is no need to perform calculations for a free gas phase (in the reservoir). Reservoir and fluid properties, such as layer permeabilities, porosity, oil and water PVT and relative permeability data, are provided, as are the initial reservoir conditions and production schedule (proposed injection and production rates). The input data file, whether generated using a text editor or by GUI, consists of various sections that incorporate all of these components just described. Here we will go through the input file, TUT1A.DATA, used for this calculation. TUT1A.DATA will also be used as a base case for other tutorial sessions associated with this course. The data format is that required by the Schlumberger GeoQuest Reservoir Technologies model, ECLIPSE 100, but other than syntactical differences, the style of data entry is similar for most other simulators. While the data file may be set up using a GUI, it is useful in the first instance to set up a simple model using a text editor, thus ensuring by the end of the exercise that every line of data is familiar and relatively well understood. This file can then be used as a starting point for other models, which may be set up by modifying the appropriate parts of this data file.

Production well Injection well Two well quarter five-spot grid

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Figure 1. A five spot pattern consists of alternating rows of production and injection wells. The symmetry of the system means that the flow between any two wells can be modelled by placing the wells at opposite corners of a Cartesian grid, and is referred to as a quarter fivespot calculation.

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4.2 ECLIPSE Syntax

Each item of data, such as porosities or relative permeabilities, are identified by use of set keywords. The individual sections are also designated by keywords. The syntax, format and function of each keyword may be found in the online manuals. These also give examples of how the keywords may be used. There are certain rules governing data entry for any simulator, and effort must be made at the outset to get these right; otherwise setting up the input data file can be very frustrating and as time consuming as performing the calculations themselves. ECLIPSE uses free format. This means that, with a few exceptions, as many or as few spaces, tabs and new lines may be used as desired. However, arranging the file appropriately, such as by lining data up in columns, etc., can improve readability, reducing unnecessary typographical mistakes, and saving time in the long run. The following additional rules should be noted • Each section starts with a keyword • There must be no other characters (or spaces) on the same line as a keyword (i.e. each keyword must start in column 1, and be immediately followed by a new line keystroke) • All data associated with a keyword must appear on the subsequent lines • Data entry is terminated by a forward slash symbol (/) • Lines beginning with two dashes (--) are ignored, and treated as comment lines • Blank lines are ignored To illustrate the use of keywords, data and comments, the following style conventions, illustrated in Figure 2, will be used here. Figure 2. Example of conventions used to identify components of input data file. It should be noted that the actual input file should be in ASCII text format only (as produced by Notepad, WordPad or other basic text editor), and should not contain italic, bold or coloured letters.

FEATURE

EXAMPLE FROM DATA FILE

Comments NX

KEYWORDS Data (followed by /)

Number of cells NY NZ

DIMENS 5

5

3/

4.3 Model Dimensions

The first step in setting up a model is to define: • Title of run • Type of geometry to be used (Cartesian or radial, though Cartesian is often the default) • Number of cells in each direction (x, y, z, or r, θ, z) • Phases to be modelled (oil, water, gas, vapourised oil in the gas, dissolved gas in the oil) Institute of Petroleum Engineering, Heriot-Watt University

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• • •

Units to be used (field, metric or lab) Number of wells Start date for simulation (usually corresponding to the date of first oil production)

The model set up in Tutorial 1A consists of a Cartesian grid of 5 x 5 x 3 cells, each cell having dimensions of 500 ft x 500 ft x 50 ft, as shown in Figure 3. 500

1

500

2

500

3

500

4

500

5

1 2 3

50 50 50 1

500

2

500

3

500

4

500 500

5

Cartesian is the default geometry used by ECLIPSE, so it does need to be specified explicitly. A two-phase oil and water calculation is to be performed in this model, with the injection and production wells to be located at opposite corners, and completed in all three layers. ECLIPSE allows wells to be grouped together so that the cumulative production or injection rates may be specified or calculated. Here, we will assume that the injection and production wells are in two separate groups. Field units are to be used throughout the input data file. (Note that once a choice of units has been made, it must be used consistently for all data entry. This precludes, for example, using feet (field units) for depths and bars (metric units) for pressures in the same run.) The title for this calculation will be “3D 2-Phase”, and it is assumed that first oil was on 1st January 2001. Generated output should be written to a single unified output file. (The ECLIPSE default is to create a separate output file for every time step, which has the advantage that not all the data output data is lost if one file is in some way corrupted, but this may result in an unmanageable number of files being generated.) The above information is all that is required for defining the dimensioning data that goes in the first section of an ECLIPSE data file, referred to as the RUNSPEC section. The form in which this data should be entered is shown in Figure 4. The following keywords are used: RUNSPEC DIMENS OIL WATER FIELD WELLDIMS 8

Section header Number of cells in X, Y and Z directions Calculate oil flows Calculate water flows Use field units throughout (i.e. feet, psi, lb, bbl, etc.) Number of wells, connections per well, groups, wells per group

Figure 3. Cartesian grid of 5 x 5 x 3 cells used to represent reservoir system to be modelled in Tutorial 1A.

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UNIFOUT START

Unified output file Start date of simulation (1st day of production)

Every time an unfamiliar keyword is encountered, it is well worth looking it up in the online manual, and this is probably as good a point to start as any! Particular attention should be paid to units. For example, when using field units gas rates are entered in MSCF/day. Entering a value in SCF/day would be allowed by the simulator, but would lead to completely wrong results. TUT1A. DATA Base case for tutorials

RUNSPEC TITLE 3D 2-Phase Number of cells NX NY NZ DIMENS 5

5

3/

Phases OIL WATER

Units FIELD Maximum well / connection / group values #wells #cons/w #grps #wells/grp WELLDIMS 2

3

2

1/

Unified output files UNIFOUT

Figure 4. RUNSPEC Section of input data file.

Simulation start date START 1 JAN 2001 /


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4.4 Grid and Rock Properties

Having identified the number of grid cells in the X, Y and Z directions required to model the reservoir or part of the reservoir being studied, the following grid properties must be defined: • Dimensions of each cell • Depth of each cell (or at least the top layer) • Cell permeabilities in each direction (x, y, z, or r, θ, z) • Cell porosities If a Cartesian grid is being used, as here, then the size of each cell may be specified by providing data on the length, width and height of each cell. The current grid has 75 cells (5 x 5 x 3), and thus 75 values must be specified for each property. Each cell in the model is to be 500 ft long, by 500 ft wide, by 50 ft thick. There are three layers, the top layer being at a depth of 8,000 ft. All three layers are assumed to be continuous in the vertical direction, so there is no need to specify the depths of the second and third layers - the simulator can calculate these implicitly from the depth of the top layer and the thickness of the top and middle layers. The formation has a uniform porosity of 0.25, and the layer permeabilities in each direction are given below.

Layer 1 2 3

Permeability (mD) Horizontal Vertical X direction Y direction Z direction 200 150 20 1000 800 100 200 150 20

This represents the minimum information that is required for defining the grid and rock properties for the second section of an ECLIPSE data file, referred to as the GRID Section. It is useful to output a file that allows these values to be viewed graphically by one of the post-processors. This enables a quick visual check that the grid data has been entered correctly. The following keywords are used: GRID DX DY DZ TOPS PERMX PERMY PERMZ PORO INIT

Section header Size of cells in the X direction Size of cells in the Y direction Size of cells in the Z direction Depth of cells Cell permeabilities in the X direction Cell permeabilities in the Y direction Cell permeabilities in the Z direction Cell porosities Output grid values to .INIT file

ECLIPSE normally assumes that grid values, such as DY, DZ, PERMX, PORO, etc., are being entered for the whole grid. If values are only being entered for a subsection of the grid, then the BOX and ENDBOX keywords may be used to identify this subsection (an example is given later). If no BOX is defined, or after an ENDBOX 10

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keyword, ECLIPSE assumes that cell values are being defined for the entire grid. The values of each grid property, such as cell length, DX, are read in a certain order. If the co-ordinates of each cell are specified by indices (i, j, k), where i is in the X direction, j is in the Y direction, and k is in the Z direction, then the values are read in with i varying fastest, and k slowest. The first value that is read in is for cell (1, 1, 1), and the last one is for cell (NX, NY, NZ), where NX, NY and NZ are the number of cells in the X, Y and Z directions respectively. Thus, in this (NX=5, NY=5, NZ=3) model, the values of DX (and every other grid value such as DY, DZ, PERMX, PORO, etc.) will be read in in the order shown in Figure 5.

Figure 5. Order in which cell property values are read in by ECLIPSE, starting at (1, 1, 1), and finishing at (5, 5, 3), with the i index varying the fastest, and the k index the slowest.

No 1 2 3 4 5 6 7 8 . . 24 25 26 27 . . 75

i 1 2 3 4 5 1 2 3 . . 4 5 1 2 . . 5

j 1 1 1 1 1 2 2 2 . . 5 5 1 1 . . 5

500

k 1 1 1 1 1 1 1 1 . . 1 1 2 2 . . 3

1

500

2

500

3

500

4

500

5

1 2 3

50 50 50 1

500

2

500

3

500

4

500

Y

5

500

X Z

The length (DX) of each of the 75 cells in Tutorial 1A is the same: 500 ft. Thus the data may be entered as: DX 500 500 500 500 500

500 500 500 500 500

500 500 500 500 500

500 500 500 500 500

500 500 500 500 500

500 500 500 500 500

500 500 500 500 500

500 500 500 500 /

500 500 500 500

500 500 500 500

500 500 500 500

500 500 500 500

500 500 500 500

500 500 500 500

500 500 500 500

500 500 500 500

500 500 500 500

Most simulators will allow the definition of multiple cells, each with the same size, to be lumped together. In ECLIPSE this is done by prefixing the value (cell size) by the number of cells to be assigned that value, and separating these two numbers by a “*”. Thus, since all 75 cells in the model have a length of 500 ft, this may be entered as: DX 75*500 / Note that the multiplier comes first, then the “*” operator, then the value. There should be no spaces on either side of the “*”. Institute of Petroleum Engineering, Heriot-Watt University

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This convention may be used for all other grid parameters. If cell depths are only to be defined for the top layer of cells using the TOPS keyword, then a box must be used to identify this top layer as the only section of the grid for which depths are being defined. The box that encompasses the top layer is defined as from 1 to 5 in the X direction, 1 to 5 in the Y direction, but only 1 in the Z direction. Instead of 75 cells for the whole model, there are only 25 cells in this section of the model, and thus only 25 values of TOPS need be defined: BOX

1

5

1

5

1

1 /

TOPS 25*8000 / ENDBOX The GRID Section of the Tutorial 1A input data file should be as shown in Figure 6.

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GRID Size of each cell in X, Y and Z directions DX 75*500 /

DY 75*500 /

DZ 75*50 TVDSS of top layer only X1 X2 Y1

Y2

Z1

Z2

5

1

1/

BOX 1

5

1

TOPS 25*8000 /

ENDBOX Permeability in X, Y and Z directions for each cell PERMX 25*200 25*1000 25*200 / PERMY 25*150 25*800 25*150 / PERMZ 25*20 25*100 25*20 / Porosity of each cell PORO 75*0.2 / Figure 6. GRID Section of input data file.

Output file with geometry and rock properties (.INIT) INIT


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4.5 Fluid Properties

Having defined the grid and rock properties such as permeability and porosity, the following Pressure/Volume/Temperature (PVT), viscosity, relative permeability and capillary pressure data must be defined: • Densities of oil, water and gas at surface conditions • Formation factor and viscosity of oil vs. pressure • Pressure, formation factor, compressibility and viscosity of water • Rock compressibility • Water and oil relative permeabilities, and oil-water capillary pressure vs. water saturation The densities of the three phases ρoil, ρwater and ρgas are given below. (Note that all three densities must be supplied, even though free gas is not modelled in the system.) Oil (lb/ft3) 49

Water (lb/ft3) 63

Gas (lb/ft3) 0.01

The oil formation volume factor (Bo) and viscosity (μo) is provided as a function of pressure (P). Pressure (psia) 300 800 6000

Oil FVF (rb/stb) 1.25 1.20 1.15

Oil Viscosity (cP) 1.0 1.1 2.0

At a pressure of 4,500 psia, the water formation volume factor (Bw) is 1.02 rb/stb, the compressibility (cw) is 3 x 10-6 PSI-1 and the viscosity (μw) is 0.8 cP. Water compressibility does not change with pressure within the pressure ranges encountered in the reservoir, and thus viscosibility (∂μw/∂P) is 0. The rock compressibility at a pressure of 4,500 psia is 4 x 10-6 PSI-1. Water and oil relative permeability data and capillary pressures are given as functions of water saturation below. Sw

krwater

kroil

0.25 0.50 0.70 0.80

0.00 0.20 0.40 0.55

0.90 0.30 0.10 0.00

capillary pres. (psi) 4.0 0.8 0.2 0.1

This data should be inserted in the third section of the ECLIPSE data file, the PROPS section. The form in which this data should be entered is shown in Figure 7. The following keywords are used: PROPS DENSITY PVDO PVTW ROCK 14

Section header Surface density of oil, water and gas phases PVT data for dead oil relating FVF and viscosity to pressure PVT data for water relating FVF, compressibility and viscosity to pressure Compressibility of the rock

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SWOF

Table relating oil and water relative permeabilities and oil-water capillary pressure to water saturations

PROPS Densities in lb/ft3 Oil Water

Gas

DENSITIES 49

63

0.01 /

PVT data for dead oil P Bo

Vis

300 800 6000

1.0 1.1 2.0 /

PVDO 1.25 1.20 1.15

PVT data for water P Bw Cw

Vis

Viscosibility

4500

0.8

0.0 /

PVTW 1.02

3e-06

Rock compressibility P Cr ROCK 4500

4e-06 /

Water and oil rel perms and capillary pressure Sw Krw Kro Pc SWOF

Figure 7. PROPS Section of input data file.

0.25 0.5 0.7 0.8

0.0 0.2 0.4 0.55

0.9 0.3 0.1 0.0

4.0 0.8 0.2 0.1 /


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4.6 Initial Conditions

Once the rock and fluid properties have all been defined, the initial pressure and saturation conditions in the reservoir must be specified. This may be done in one of three ways: 1) Enumeration 2) Equilibration 3) Restart from a previous run 1) Enumeration. In this method of initialising the model, the pressure, oil and water saturations in each cell at time = 0 are set in much the same way as permeabilities and porosities are set. This method is the most complicated and least commonly used. A failure to correctly account for densities when setting the pressures in cells at different depths will result in a system that is not initially in equilibrium. 2) Equilibration. This is the simplest and most commonly used method for initialising a model. A pressure at a reference depth is defined in the input data, and the model then calculates the pressures at all other depths using the previously entered density data to account for hydrostatic head. The depths of the water-oil and gas-oil contacts are also specified if they are within the model, and the initial saturations can then be set depending on position relative to the contacts. (In a water-oil system, above the oil-water contact the system is at connate water saturation, below the contact Sw = 1.) 3) Restart from a previous run. If a model has already been run, then one of the output time steps can be used to provide the starting fluid pressures and saturations for a subsequent calculation. This option will typically be used where a model has been history matched against field data to the current point in time, and various future development scenarios are to be compared. A restart run will use the last time step of the history-matched model as the starting point for a predictive calculation, which may then be used to assess future performance. Time is saved by not repeating the entire calculation. In this example the model is being set up to predict field performance from first oil, and thus there is no previous run to use as a starting point. The equilibration model is to be used, with an initial pressure of 4,500 psia at 8,000 ft. The model should initially be at connate water saturation throughout. To achieve this, the water-oil contact should be set at 8,200 ft, 50 ft below the bottom of the model. The water saturation in each cell will be set to the first value in the relative permeability (SWOF) table, which is 0.25. (If any cells were located below the water-oil contact, they would be set to the last value in the relative permeability table, which would thus have to include relative permeability and capillary pressure values for Sw = 1.) An output file containing initial cell pressures and saturations for display should be requested so that a visual check can be made that the correct initial values of these properties have been calculated. This initialisation data should be inserted in the fourth section of the ECLIPSE data file, the SOLUTION section. The form in which this data should be entered is shown in Figure 8. The following keywords are used: SOLUTION Section header 16

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EQUIL RPTRST

Equilibration data (pressure at datum depth and contact depths) Request output of cell pressures and saturations at t = 0

SOLUTION Initial equilibration conditions Datum Pi@datum

WOC

Pc@WOC

8200

0/

EQUIL 8075

4500

Output to restart file for t=0 (.UNRST) Restart file Graphics for init cond only Figure 8. SOLUTION Section of input data file.

RPTRST BASIC=2

NORST=1 /

4.7 Output Requirements

Clearly there is no point in performing a reservoir simulation if no results are output. The parameters that should be calculated are specified in the SUMMARY section by the use of appropriate keywords, but for this section only the keywords are not found in the main section of the manual, but in the Summary Section Overview. Most of the summary keywords consist of four letters that follow a basic convention. 1st letter: F - field R - region W - well C - connection B - block 2nd letter: O - oil (stb in FIELD units) W - water (stb in FIELD units) G - gas (Mscf in FIELD units) L - liquid (oil + water) (stb in FIELD units) V - reservoir volume flows (rb in FIELD units) T - tracer concentration S - salt concentration C - polymer concentration N - solvent concentration


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3rd letter: P - production I - injection 4th letter: R - rate T - total Thus, use of the keyword FOPR requests that the Field Oil Production Rate be output, and WWIT represents Well Water Injection Total, etc. Keywords beginning with an F refer to the values calculated for the field as a whole, and require no further identification. However, keywords beginning with another letter must specify which region, well, connection or block they refer to. Thus, for example, a keyword such as FOPR requires no accompanying data, but WWIT must be followed by a list of well names, terminated with a /. If no well names are supplied, and the keyword is followed only by a /, the value is calculated for all wells in the model. An example would be FOPR WWIT Inj / WBHP / Here, the following will be calculated: • Oil production rate for the entire field • Cumulative water injection for well “Inj” • Well bottomhole pressure for all wells in the model In Tutorial 1A the following parameters should be calculated and output: • Field average pressure • Bottomhole pressure of all wells • Field oil production rate • Field water production rate • Field oil production total • Field water production total • Water cut in well PROD • CPU usage In addition, the output Run Summary file (.RSM) should be defined such that it can easily be read into MS Excel. The form in which this data should be entered is shown in Figure 9. following keywords are used: SUMMARY Section header FPR Field average pressure 18

The

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WBHP FOPR FWPR FOPT FWPT WWCT CPU EXCEL

Well Bottomhole pressure Field Oil Production Rate Field Water Production Rate Field Oil Production Total Field Water Production Total Well Water Cut CPU usage Create summary output as Excel readable Run Summary file

SUMMARY Field average pressure FPR Bottomhole pressure of all wells WBHP / Field oil production rate FOPR Field water production rate FWPR Field oil production total FOPT Field water production total FWPT Water cut in PROD WWCT PROD / CPU usage TCPU Figure 9. SUMMARY Section of input data file.

Create Excel readable run summary file (.RSM) EXCEL

4.8 Production Schedule

Having defined the initial conditions (t = 0) in the SOLUTION Section, the final part of the input data file defines the well controls and time steps (t > 0) in the SCHEDULE Section. The main functions that are performed here are: • Specify grid data to be output for display or restart purposes • Define well names, locations and types • Specify completion intervals for each well • Specify injection and production controls for each well for each given period


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(time step) Basic pressure and saturation data should be generated at every time step to enable a 3-D display of the model to be viewed at each time step. A production well, labeled PROD and belonging to group G1, is to be drilled in cell position (1,1), and a water injection well, INJ belonging to group G2, is to be drilled in cell (5,5). Both wells will have 8 inch diameters, and should be completed in all three layers. They both have pressure gauges at their top perforation (8,000 ft). Both will be open from the start of the simulation, enabling production of 10,000 stb/day of liquid (oil + water) from the system, and pressure support provided by injection of 11,000 stb/day of water. The simulation should run for 2,000 days, outputting data every 200 days. A number of keywords in the SCHEDULE section will be able to read in data that the user may wish to default or not supply all. This can be done by using the “*” character, with the number of values to be defaulted or ignored on the left, and the space to the right left blank. Thus “1* “ ignores one value, “2* “ ignores the next two values, etc. For example, in the COMPDAT keyword, we may wish to specify values for items 1 to 6, and item 9 (which is the wellbore diameter), but items 7 and 8 should remain unspecified. This may be achieved as follows: Completion interval Well Location name I J

COMPDAT Item number 1 2 PROD 1 /

3 1

Interval K1 K2

4 1

5 3

Status O or S

6 OPEN

Well ID

7 8 2*

9 0.6667 /

The keywords to be used are: SCHEDULE Section header RPTRST Request output of cell pressures and saturations at all time steps (t > 0) WELSPECS Define location of wellhead and pressure gauge COMPDAT Define completion intervals and wellbore diameter WCONPROD Production control WCONINJ Injection control TSTEP Time step sizes (for output of calculated data) END End of input data file These keywords should appear as the last section of the data file as shown in Figure 10. Although not the case in this simple example, this section will typically be the longest, containing flow rates for each well on a monthly basis for the history of the field. It should be noted that the time steps input here refer to time intervals at which 20

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data are output. The simulator will try to use these time step sizes as numerical time step also, but if the calculations do not converge, it will automatically cut the numerical time step sizes. SCHEDULE Output to Restart file for t > 0 (.UNRST) Restart file Graphics only every step RPTRST BASIC=2

NORST=1 /

Location of wellhead and pressure gauge Well Well Location BHP name group I J datum WELSPECS PROD INJ /

G1 G2

1 5

Completion interval Well Location name I J COMPDAT PROD INJ /

1 5

1 5

Production control Status Control Well name mode WCONPROD PROD OPEN /

Figure 10. SCHEDULE Section of input data file.

WATER

8000 8000

OIL / WATER /

Interval K1 K2

Status 0 or S

1 1

OPEN OPEN

3 3

Oil rate

LRAT

Injection control Fluid Status Well Name TYPE WCONINJ INJ /

1 5

Pref. phase

Wat rate

Gas rate

3*

Control mode

OPEN RATE

Well ID 2* 2*

Liq. rate 10000

Surf rate 11000

Resv rate

0.6667 / 0.6667 /

Resv rate 1*

BHP lim 2000 /

Voidage frac flag

BHP lim

3*

20000 /

Number and size (days) of timesteps TSTEP 10*200 / END


21

5. RUNNING ECLIPSE AND FILE NAME CONVENTIONS 5.1 Running ECLIPSE on a PC

Once the input data file has been edited, it should be saved with the file extension “.DATA”. For Tutorial 1A choose a file name such as “TUT1A.DATA”. Care should be taken that the text editor does not append the suffix “.txt” onto the file name, as this will render the file unreadable to ECLIPSE. This can be avoided by using Menu->File->Save As and selecting “All Files” instead of “Text Documents” as the “Save as type”. Having saved the input data file, the GeoQuest Launcher may be used to run ECLIPSE, and the user will be prompted to locate the input file. The simulation will then start, and will run by reading every keyword in the order in which they appear in the input file.

5.2 File Name Conventions

If any of the keywords or data are incorrectly entered, the run will stop without performing the required flow calculations. If the simulator is satisfied that all data has been entered correctly, then it will perform the requested flow calculations, and various output files will be generated during the run, as follow: TUT1A.PRT The .PRT file is an ASCII file that is generated for every successful and unsuccessful run. It contains a list of the keywords, and will indicate if any keywords have been incorrectly entered. If the simulation fails, this file should be checked for the cause of the failure. A search for an ERROR in this file will usually reveal which keyword was the culprit. If the run was successful, this file will contain summary data such as field average pressure and water cut for each time step. TUT1A.GRID The .GRID file is a binary file that contains the geometry of the model, and is used by post processors for displaying the grid outline. TUT1A.INIT The .INIT file is a binary file that contains initial grid property data such as permeabilities and porosities. These may be displayed using a post-processor to check that the data have been entered correctly, and to display a map of field permeabilities, etc. TUT1A.UNRST The .UNRST file is a unified binary file that contains pressure and saturation data for each time step. These may be displayed using a post-processor, or may be used as the starting point for an ECLIPSE restart run. TUT1A.RSM The .RSM file is an ASCII file that can be read into MS Excel to display summary data in line chart format. This file is only created once the run has completed. During the run the summary data is stored in file TUT1A.USMRY, which is a binary file readable only by the GeoQuest post-processors. The GeoQuest post-processors are Graf and FloViz. During this course FloViz is used for 3D displays of the model, showing, for example, progression of the water flood 22

3


by displaying saturations varying with time (Figure 11). Excel is used to display line charts such as water cut vs. time, etc. Both grid and line charts may be displayed in Graf, which is a powerful though more complicated post-processor than FloViz. The functionality of Graf is being replaced by ECLIPSE Office, which is a GUI that may be used for setting up data files and viewing results, and may optionally be used for subsequent tutorial sessions. However, students are encouraged to use a basic text editor for pre-processing, and Excel and FloViz for post-processing for Tutorial 1A, since this will give a better understanding of the calculations being performed.

Figure 11 FloViz visualisation of water saturation for four time steps, showing progression of water flood in Tutorial 1A. The injection well is on the left and the production well on the right of the 5 x 5 x 3 model.

6. CLOSING REMARKS In this section of the course, we have presented the working details of how to set up a a practical reservoir simulation model. We have used the Schlumberger GeoQuest ECLIPSE software for the specific case presented here. However, the general procedures are very similar for most other commercially available simulators. The various input data that are required should be quite familiar to you from the discussion in the introductory chapter of the course (Chapter 1). However, how these are systematically organised as input for the simulator should now be clear. The vast possibilities for simulation output have also been discussed in this section and you should know be aware of how to choose this output, organise it is files and then visualise it later. The issue of visualisation was also discussed previously but its value should be better appreciated by the student.


23

ECLIPSE TUTORIAL 1 (A 3D 2-Phase Reservoir Simulation Problem) A. Prepare an input data file for simulating the performance of a two-phase (water/oil) three dimensional reservoir of size 2500’ x 2500’ x 150’, dividing it into three layers of equal thickness. The number of cells in the x and y directions are 5 and 5 respectively. Other relevant data are given below, using field units throughout: Depth of reservoir top: Initial pressure at 8075’: Porosity:

8000 ft 4500 psia 0.20

Permeability in x direction:

200 mD for 1st and 3rd layers and 1000 mD for 2nd layer. 150 mD for 1st and 3rd layers and 800 mD for 2nd layer. 20 mD for 1st and 3rd layers and 100 mD for 2nd layer.

Permeability in y direction: Permeability in z direction:

500

1

500

2

500

3

500

4

500

5

1 2 3

50 50 50 1

500

2

500

3

500

Figure 1 Schematic of model.

4

500 500

5

Water and Oil Relative Permeability and Capillary Pressure Functions.

24

Water Saturation

krw

kro

0.25* 0.5 0.7 0.8 1.0

0.0 0.2 0.4 0.55 1.00

0.9 0.3 0.1 0.0 0.0

Pcow (psi) 4.0 0.8 0.2 0.1 0.0

* Initial saturation throughout.

3


Water PVT Data at Reservoir Pressure and Temperature. Pressure (psia)

Bw (rb/stb)

cw (psi-1)

μw (cp)

Viscosibility (psi-1)

4500

1.02

3.0E-06

0.8

0.0

Oil PVT Data, Bubble Point Pressure (Pb) = 300 psia. Pressure (psia)

Bo (rb/stb)

Viscosity (cp)

300 800 6000

1.25 1.20 1.15

1.0 1.1 2.0

Rock compressibility at 4500 psia: Oil density at surface conditions: Water density at surface conditions: Gas density at surface conditions:

4E-06 psi-1 49 lbs/cf 63 lbs/cf 0.01 lbs/cf

The oil-water contact is below the reservoir (8,200 ft), with zero capillary pressure at the contact. Drill a producer PROD, belonging to group G1, in Block No. (1, 1) and an injector INJ, belonging to group G2, in Block No. (5, 5). The inside diameter of the wells is 8”. Perforate both the producer and the injector in all three layers. Produce at the gross rate of 10,000 stb liquid/day and inject 11,000 stb water/day. The producer has a minimum bottom hole pressure limit of 2,000 psia, while the bottom hole pressure in the injector cannot exceed 20,000 psia. Start the simulation on 1st January 2000, and use 10 time steps of 200 days each. Ask the program to output the following data: •

Initial permeability, porosity and depth data (keyword INIT in GRID section)

•

Initial grid block pressures and water saturations into a RESTART file (keyword RPTRST in SOLUTION section)

•

Field Average Pressure Bottom Hole Pressure for both wells Field Oil Production Rate Field Water Production Rate Total Field Oil Production Total Field Water Production Well Water Cut for PROD CPU usage

(FPR) (WBHP) (FOPR) (FWPR) (FOPT) (FWPT) (WWCT) (TCPU)

to a separate Excel readable file (using keyword EXCEL) in the SUMMARY section. Institute of Petroleum Engineering, Heriot-Watt University

25

•

Grid block pressures and water saturations into RESTART files at each report step of the simulation (keyword RPTRST in SCHEDULE section)

Procedure: 1

Edit file TUT1A.DATA in folder \eclipse\tut1 by dragging it onto the Notepad icon, fill in the necessary data, and save the file.

2

Activate the ECLIPSE Launcher from the Desktop or the Start menu.

3

Run ECLIPSE and use the TUT1A dataset.

4

When the simulation has finished, use Excel to open the output file TUT1A.RSM, which will be in the \eclipse\tut1 folder. You will need to view. “Files of type: All files (*.*)” and import the data as “Fixed width” columns.

5

Plot the BHP of both wells (WBHP) vs. time and the field average pressure (FPR) vs. time on Figure 1.

6

Plot the water cut (WWCT) of the well PROD and the field oil production rate (FOPR) vs. time on Figure 2.

7

Plot on Figure 3 the BHP values for the first 10 days in the range 3,500 psia to 5,500 psia.

Explain the initial short-term rise in BHP in the injection well and drop in BHP in the production well. Account for the subsequent trends of these two pressures and of the field average pressure, relating these to the reservoir production and injection rates, water cut and the PVT data of the reservoir fluids. B. Make a copy of the file TUT1A.DATA called TUT1B.DATA in the same folder tut1. By modifying the keyword TSTEP change the time steps to the following: 15*200 Modify the WCONINJ keyword to operate the injection well at a constant flowing bottom hole pressure (BHP) of 5000 psia, instead of injecting at a constant 11,000 stb water/day (RATE). Add field volume production rate (FVPR) to the items already listed in the SUMMARY section. Run Eclipse using the TUT1B.DATA file, and then plot the two following pictures in Excel: 26

3


Figure 4: Figure 5:

Both well bottom hole pressures and field average pressure vs. time, showing pressures in the range 3,700 psia to 5,100 psia. Field water cut and field volume production rate vs. time.

Account for the differences between the pressure profiles in this problem and Tutorial 1A. To assist with the interpretation, calculate total mobility as a function of water saturation for 4 or 5 saturation points, using:

MTOT (Sw ) =

K ro (Sw ) Krw (Sw ) + µo µw

and show how this would change the differential pressure across the reservoir as the water saturation throughout the reservoir increases. From Figure 5, explain the impact of the WWCT profile (fraction) on the FVPR (rb/day). C. Copy file TUT1B.DATA to TUT1C.DATA in the same folder. This time, instead of injecting at a constant flowing bottom hole pressure of 5000 psi, let the simulator calculate the injection rate such that the reservoir voidage created by oil and water production is replaced by injected water. To do this, modify the control mode for the injection well (keyword WCONINJ) from BHP to reservoir rate (RESV), and use the voidage replacement flag (FVDG) in item 8. Set the upper limit on the bottom hole pressure for the injection well to 20,000 psia again. Note the definitions given in the manual for item 8 of the WCONINJ keyword. Based on the definition for voidage replacement, reservoir volume injection rate = item 6 + (item 7 * field voidage rate) Therefore, to inject the same volume of liquid as has been produced, set item 6 to 0, and item 7 to 1. Run Eclipse using the TUT1C.DATA file, and then run Floviz, to display the grid cell oil saturations (these displays need NOT be printed). Discuss the profile of the saturation front in each layer, and explain how it is affected by gravity and the distribution of flow speeds between the wells.


27

TUT1A.DATA RUNSPEC TITLE NDIVIX

NDIVIY

NDIVIZ

NCWMAX

NGMAXZ

Y1

Z1

DIMENS OIL WATER FIELD NWMAXZ WELLDIMS START GRID DX DZ PORO X1 BOX TOPS ?DBOX PERMX PERMY PERMZ INIT

28

X2

Y2

Z2

NWGMAX

3


PROPS OIL

WAT GAS

DENSITY P

Bo

Vis

P

Bw

Cw

Vis

P

Cr

Sw

Krw

Kro

Pc

PVDO Viscosibility

PVTW

ROCK

SWOF SOLUTION DATUM

Pi@DATUM

WOC

Pc@WOC

GOC

Pc@GOC

EQUIL Block Block Create initial P Sw restart file RPTSOL SUMMARY

RPTSMRY


29

SCHEDULE Block P

Block Sw

Create restart file at each time step

WELL GROUP

LOCATION I J

RPTSCHED WELL NAME

BHP DATUM

PREF. PHASE

WELSPECS WELL NAME

LOCATION I J

INTERVAL K1 K2

STATUS O or S

WELL ID

COMPDAT WELL STATUS CONTROL OIL NAME MODE RATE

WAT RATE

GAS RATE

LIQ RESV RATE RATE

BHP LIMIT

WCONPROD WELL FLUID STATUS NAME TYPE WCONINJ DAYS TSTEP END

30

CONTROL MODE

SURF RATE

RESV RATE

VOIDAGE BHP FRAC FLAG LIMIT

Gridding And Well Modelling

CONTENTS 1. INTRODUCTION 2. GRIDDING IN RESERVOIR SIMULATION 2.1. Introduction 2.2. Accuracy of Simulations and Numerical Dispersion 2.3. Grid Orientation Effects 2.4 Local Grid Refinement (LGR) λ TGrids or λ (So ) i +1/ 2 (So ) iand 2.5 Distorted −1 / 2 Cornero Point Geometry 2.6 Issues in Choosing a Reservoir Simulation  Grid     λ λ ∆ P  2.7 Streamline Simulationp  p 

(

)

(

)

 Pi − Pi −1  Q p = − kA.  . = − kA.  .   Bp   ∆x   Bp    ∆x i −1 + ∆x i     2

3. THE CALCULATION OF BLOCK TO BLOCK FLOWS IN RESERVOIR SIMULATORS 3.1. Introduction to Averaging of Block to kA Block Flows 3.2. Averaging of the Two-Phase Mobility Term, λ p 4. WELLS IN RESERVOIR SIMULATION 4.1. Basic Idea Bp of a Well Model 4.2. Well Models for Single and Two-Phase Flow 4.3 Well Modelling in a Multi-Layer System ∆x 4.4. Modelling Horizontal Wells 4.5 Hierarchies of Wells and Well Controls

=

∆x i −1 + ∆x i 2

5. CLOSING REMARKS - GRIDDING AND WELL MODELLING

k rp µp

4

LEARNING OBJECTIVES: Having worked through this chapter the student should: • understand and be able to describe the basic idea of gridding and of spatial and temporal discretisation. • be aware of all the main types of grid in 1D, 2D and 3D used in reservoir simulation and be able to describe examples of where it is most appropriate to use the different grid types. • be able to give a short description with simple diagrams of the phenomena of numerical dispersion and grid orientation and to explain how these numerical problems can be overcome. • be familiar with more sophisticated issues in gridding such as the use of local grid refinement (LGR), distorted, PEBI and corner point grids • given a specific task for reservoir simulation, the student should be able to select the most appropriate grid dimension (1D, 2D, 3D) and geometry/structure (Cartesian, r/z, corner point etc.). • be able to discuss the issues of grid fineness/coarseness (i.e. how many grid blocks do we need to use) in terms of some examples of what can happen if an inappropriate number of grid blocks are used in a reservoir simulation calculation. • be able to describe the basic ideas behind streamline simulation and to compare it with conventional reservoir simulation in terms of its advantages and disadvantages. • be familiar with the different types of average used for single phase kA , two phase relative permeabilities (krp) and for μp and Bp (p = o, w, g phase) when calculating the block to block flows (Qp) in a reservoir simulator. • be able to describe the physical justification for using the upstream value of two phase relative permeabilities when calculating the block to block flows (Qp) in a reservoir simulator. • understand the origin of all the pressure drops that are experienced by the reservoir fluids from deep in the reservoir, through to the wellbore and then to the surface facilities and beyond • know what a well model is and what productivity index (PI) is, including knowing the radial Darcy Law and how this gives a mathematical expression for PI for single phase flow (know the expression from memory). • be able to describe the main issues in relating the pressure in the reservoir, Pe, at some drainage radius, re, to an average grid block pressure and how this leads to the Peaceman formula (Δr = 0.2 Δx) which is then used to calculate PI. • be able to extend PI to the concept of multi-phase flow to calculate PIw and PIo. 2


4

• describe a well model for a multi layer system where there is two phase flow into the wellbore. • understand and be able to describe the various types of well constraint that can be applied e.g. injection volume constrained wells, well flowing pressure constraints and voidage replacement constraints.

GRIDDING AND WELL MODELLING IN RESERVOIR SIMULATION This chapter deals with the related issues of grid selection and well modelling in reservoir simulation. Gridding: Examples of various grid geometries are presented including 1D linear, 2D areal, 2D cross-sectional , 2D radial (r/z) and 3D Cartesian cases. Selection of grid geometry, how fine a grid to take and potential problems are discussed. Non-mathematical introductions to the concepts of numerical dispersion and grid orientation are given along with some examples of the consequences of these effects. More sophisticated gridding such as PEBI grids, distorted grids and local grid refinement (LGR) are illustrated with examples. A brief discussion of streamline simulation is also presented. Treatment of the block to block flows in reservoir simulation is presented and it shown how the various inter-block quantities such as the permeability and relative permeabilities are averaged. Well Modelling: All interactions between the surface facilities and the reservoir takes place through the injector and producer wells. It is therefore very important to model wells accurately in the reservoir simulation model. The central issue with a “well model” in a simulator is that it must represent a near singular line source within (usually) a very large grid block. The basic ideas of well modelling are explained and how simple well controls are applied is introduced. Modelling of horizontal wells and the control of well hierarchies are also briefly discussed.

2

GRIDDING IN RESERVOIR SIMULATION

2.1 Introduction

By this point in the course, you will be familiar with the idea of gridding since it has been discussed in Chapter 1 both in general terms and with reference to the SPE examples (Cases 1 - 3; Section 1.3). You will also have seen how to set up a 3D grid in Chapter 3. Basically, the gridding process is simply one of chopping the reservoir into a (large) number of smaller spatial blocks which then comprise the units on which the numerical block to block flow calculations are performed. More formally, this process of dividing up the reservoir into such blocks is known as spatial discretisation. Recall that we also divide up time into discrete steps (denoted Δt) and this related process is known as temporal discretisation. The numerical details of how the discretisation process is carried out using finite difference approximations of the governing flow equations is presented in Chapter 6. However, we would point out that the grid used for a given application is a user choice - it is certainly not a Institute of Petroleum Engineering, Heriot-Watt University

3

reservoir “given” - although, as we will see, there are some practical guidelines that help us to make sensible choices in grid definition. In a Cartesian grid, which to date has been the mot common type of grid used, we denote the block size as Δx, Δy and Δz and these may or may not be equal. This grid can also be in one dimension (1D), two dimensions (2D) or three dimensions (3D). Some typical examples of 1D, 2D and 3D Cartesian grids are shown in Figure 1. This is the most straightforward type of grid to set up and typical application of such grids are as follows: - 1D linear grids may be used to simulate 1D Buckley-Leverett type water displacement calculations (x-direction) or for single column vertical displacements (z-direction) such as gravity stable gas displacement of oil (Figures 1(a) and 1(b)); - 2D Cartesian grids: 2D cross-sectional (x/z) grids may be used to; (a) study vertical sweep efficiency in a heterogeneous layered system; (b) calculate water/oil displacements in a geostatistically generated cross-section; (c) generate pseudo-relative permeabilites (can be used to collapse a 3D calculation down to a 2D system); (d) to study the mechanism of a gas displacement process - e.g. to determine the importance of gravity etc. (Figure 1(e)); 2D areal (x/y) grids may be used to; (a) calculate areal sweep efficiencies in a waterflood or a gas flood; (b) to examine the stability of a near-miscible gas injection within a heterogeneous reservoir layer; (c) examine the benefits of infill drilling in an areal pattern flood etc. (Figures 1(c) and 1(d)); - 3D (x/y/z) Cartesian grids are used to model a very wide range of field wide reservoir production processes and would often be the default type of calculation for a typical full field simulation of waterflooding, gas flooding, etc. (Figure 1(f)). Cartesian grids are clearly quite versatile but they are not appropriate for all flow geometries that can occur in a reservoir. For example, close to the wellbore (of a vertical well), the flows are more radial in their geometry. For such systems, an r/z - geometry may be more appropriate as shown in Figure 2. An r/z grid is frequently used when modelling coning either of water or gas into a producer. The pressure gradients near the well are very steep and, indeed, we know from the discussion in Chapter 2, section 3.5 that the pressure varies as ln(r/rw), where rw is the well radius. For this reason, in coning studies, a logarithmically spaced grid is often used for the grid block size, Δr; a logarithmic spacing divides up the grid such that (ri/ ri-1) is constant where Δri = (ri- ri-1) as shown in Figure 2. For example, if we take rw = 0.5 ft and the first grid block size is, Δr1 = 1ft, then this sets (ri/ ri-1) = 3 since r0 = rw = 0.5ft and r1 = 1.5ft.; hence r2 = 3x1.5ft. = 4.5ft. and Δr2 = 3ft., r3 = 13.5ft and Δr3 = 9ft., and so on.

4


(a) 1D horizontal grid

4

∆x

(c) 2D Areal grid - top view showing injectors ( ) and producers ( )

(b) 1D vertical grid 1D vertical displacement e.g. 1D vertical gravity drainage calculations ∆z

W2

(d) 2D areal grid

W3

W1 y x

∆z

∆x ∆y

Perspective view of a 2D areal (x/y) reservoir simulation grid: W = well

Just 1 x z - block in 2D areal grid

(e) 2D (x/z) cross-sectional model showing a waterflood

∆z

∆x

5


Water Injector

Producer

∆z

∆x

(f) a 3D Cartesian grid with variable vertical grid (Δz varies from layer to layer)

Water Injector

Producer

Figure 1 Examples of 1D, 2D and 3D Cartesian grids

∆y ∆z ∆x

Q

top view

∆r

∆ri rw

z h r

Notation:

r ∆r z ∆zi h rw

∆zi

ri ri-1

= radial distance from well = radial grid size (can vary ∆r1, ∆r2, ...) = vertical coordinate = vertical grid size (can vary ∆z1, ∆z2, ...) = height of formation = wellbore radius

2.2 Accuracy of Simulations and Numerical Dispersion

The issue of numerical dispersion was touched upon briefly in Chapter 1 in connection with Case 1 (SPE10022). Here we expand on the concept in a non-mathematical manner. Numerical dispersion is essentially an error due to the fact that we use a grid block approximation for solving the flow equations. A more mathematical description of numerical dispersion is presented in Chapter 6, Section 8 but here we focus on explaining physically how it arises and what its consequences are. We also

6

Figure 2 r/z grid geometry - more

appropriate for modelling flows in the near well region even in a heterogeneous layered system as shown.


4

discuss how to reduce this source of error in our simulations or to make it a minor effect. The balance, as we will see, is between accuracy (usually by taking more grid blocks) and computational cost. Ideally, we would like to capture all the main reservoir processes (e.g. frontal displacement, crossflow, gravity segregation etc.) and accurately forecast recovery to some acceptable percentage error, for the minimum number of grid blocks. A simple schematic of the way that the numerical dispersion error arises is shown step by step in Figure 3. This figure illustrates a simple linear waterflood and we imagine that each of the sub-figures shown from (a) to (e) represents a time step, Δt, of the water injection process. Block i = 1 contains the injector well which is injecting water at a constant volumetric rate of Qw, and block i = 5 contains the producer. The system is initially at water saturation, Swc, and this water is immobile i.e. the relative permeability of water is zero, krw(Swc) = 0. Each block has constant pore volume, Vp = Δx.A.φ where φ is the porosity. In Figure 3(a), we see that after time, t = Δt, some quantity of fluid has been injected into block i = 1; the volume of water injected is Qw. Δt and this would cause a water saturation change in grid block i = 1, ΔSw1 = (Qw. Δt)/Vp i.e. the new water saturation in this block is now, Swc+ (Qw. Δt)/Vp. Over the first time step, no fluid flowed from block to block since the relative permeability of all blocks was zero (krw(Swc) = 0). However, the relative permeability in block i = 1 is now krw(Sw1) > 0. The second time period of water injection is shown in Figure 3(b). Another increment of water, Qw. Δt, is injected into block i = 1 causing a further increase in water saturation Sw1. However, over the second time period, krw(Sw1) > 0 and therefore water can flow from block i = 1 to i = 2, increasing the water saturation such that Sw2 > Swc making the relative permeability in this block, krw(Sw2) > 0. In the third time step, shown in Figure 3 (c), the same sequence occurs except that fluid can now from block 1 → 2 and also 2 → 3, where for the same reasons as explained, krw(Sw3) > 0. In the fourth and fifth time steps (Figures 3(d) and 3(e)), flow can now go from block 3 → 4 and from 4 → 5 where, since krw(Sw5) > 0, then it can be produced from this block, although the relative permeability in block 5 will be very small. Hence, after only five time steps to time , t = 5Δt, the water has reached the producer in block 5 from whence it can be produced (although not a very high rate because the relative permeability is very small) and this is an unsatisfactory situation. If we had taken 10 grid blocks, then clearly a similar argument would apply and water would be produced after 10 time steps - with an even lower relative permeability in block i = 10 - and this is more satisfactory. Indeed, this underlies why we take more grid blocks. This simple illustration explains in a quite physical way the basic idea of numerical dispersion.


7

1.0

Relative Permeability to Water

krw Swc = 20%

0

0

20

Water Injection

Qw

Sor = 30%

40 60 Sw %

80

100 Oil Production

Mixing due to numerical dispersion & relative permeability effects

Time Step (a)

∆x

∆Sw1

i=1

i=2

i=3

Sw = Swc i=4

1 t = ∆t

i=5

(b)

2 t = 2.∆t

(c)

3 t = 3.∆t

(d)

4 t = 4.∆t

(e)

5 t = 5.∆t

The frontal spreading effect of numerical dispersion can be seen when we try to simulate the actual saturation profile, Sw(x,t), in a 1D waterflood. Under certain conditions, this may have an analytical solution, e.g. the well known Buckley-Leverett solution described in Chapter 2, Section 4.2. This often has a shock front solution for the advancing water saturation profile, Sw(x,t) as shown in Figure 4. Clearly, this sharp front may be “lost” in a grid calculation since, at time t, the front will have a definite position, x(t). However, in a grid system with block size Δx, any saturation front can only be located within Δx as shown in Figure 4. If we take more grid blocks (Δx decreases), then we will locate the front more accurately. Indeed, taking more and more blocks we will gradually get closer to the analytical (correct) solution. Hence, one method of reducing numerical dispersion is to increase the number of grid blocks. An alternative is to use a numerical method which has inherently less dispersion in it but we will not pursue this here. Another approach is to use pseudo functions to control numerical dispersion - as we briefly introduced in Chapter 1 - and this is discussed further below.

8

Figure 3 Effect of grid on water breakthrough time - numerical dispersion.


4

1.0 Numerical Grid block size = ∆x Water saturation = Sw(x,t1)

time = t1

Front moving in this direction with velocity Vw Analytical solution

Sw

Figure 4 The frontal spreading of a Buckley-Leverett shock front when calculated using a 1D grid block model

xf(t1)

0

∆x

x

2.3 Grid Orientation Effects

Figure 5 Flow between an injector (I) and 2 producers (P1 and P2) where the injectorproducer separations are identical but flow is either oriented with the grid or diagonally across it illustrating the grid orientation effect.

Another numerical problem arising in 2D and 3D grids is the grid orientation effect. This is illustrated in Figure 5 where the distance between wells I - P1 and I - P2 are the same. However I - P1 are joined by a row of cells oriented to the flow as shown. The flow between I - P2 is rather more tortuous as also shown in Figure 5. The Grid Orientation effect arises when we have fluid flow both oriented with the principal grid direction and diagonally across this grid. Numerical results are different for each of the fluid “paths” through the grid structure. This problem arises mainly due to the use of 5-point difference schemes (in 2D) in the Spatial Discretisation. It may be alleviated by using more sophisticated numerical schemes such as 9-point schemes (in 2D).

P1 I

I = Injector P = Producer

P2

Flow arrows show the fluid paths in oriented grid and diagonal flow leading to grid orientation errors

The effects on the breakthrough time and in recoveries of these two flow orientations are shown in Figure 6. The I-P1 orientation tends to lead to somewhat earlier breakthrough and a less optimistic recovery that the I-P2 orientation. The reasons for this are intuitively fairly obvious.


9

diagonal flow result "true" recovery

Oil recovery

Time

grid aligned result

Figure 6 Oil recoveries for “true”, aligned and diagonal flows in 2D grid

The grid orientation error can be emphasised for certain types of displacement. For example, in gas injection, gas viscosity is much less than the oil viscosity (μg << μo) leading to viscous fingering instability. This is exaggerated when flow is along the grid as in the I-P1 orientations. Again, as for numerical dispersion, some grid refinement can help to reduce the grid orientation effect. Alternative numerical schemes can also be devised to reduce this source of error. In particular, in a 2D grid the flows are usually worked out using the neighbouring grid blocks shown in Figure 7. The 5-point numerical scheme uses the neighbours shown in Figure 7(a). If information from the blocks in Figure 7(b) are used, the 9-point scheme that emerges helps greatly to reduce the grid orientation error. We will not go into further technical details here. (a)

(b)

Figure 7 5 - point and 9 - point schemes for discretising the grid - the latter helps to reduce grid orientation effects

2.4 Local Grid Refinement (LGR)

In a reservoir, the changes in pressure, saturations and flows tend to be quite different in different parts of the system. For example, close to a well which is changing production rate every day or so, there will be large pressure and saturation changes. On the contrary, on a flank of the field which is connected to, but is remote from, the active wells, the pressure may be quite slowly changing and the saturations may hardly be changing at all. To represent regions with rapidly changing waterfronts will require a finer grid than will be required for relatively stagnant regions of the system. Thus, a single uniform grid with fixed Δx, Δy and Δz will often not be suitable to represent all regions of an active reservoir. Instead, the application of some local grid refinement (LGR) may be much more appropriate. LGR options are supported by most major simulation models and the simplest version is shown in Figure 8 (indeed, this simpler version is sometimes not referred to as LGR). “True” LGR is shown in Figure 9 where the refined grid is clearly seen.

10


4

Figure 8 A simple version of local grid refinement where the grid is finer in the (central) area of interest in the reservoir

Figure 9 “True” local grid refinement (LGR) where the refined grid is embedded in the coarser grid.

In addition to conventional LGR (Figure 9), we can also define Hybrid Grid LGR as shown in Figure 10. Hybrid Grids are mixed geometry combinations of grids which are used to improve the modelling of flows in different regions. The most common use of hybrid grids are Cartesian/Radial combinations where the radial grid is used near a well. Hybrid Grid LGR can be used in a similar way to other LGR scheme. Schematic of Local Grid Refinement (LGR)

injector producer

Figure 10 A simple example of LGR and Hybrid Grid structure

Coarse grid in aquifer

Hybrid Grid

2.5 Distorted Grids and Corner Point Geometry

In recent years, many studies have used grids that have tried to distort either to reservoir geometry or to the particular flow field and an examples of a distorted grid is shown in Figure 11.


11

Figure 11 An example of a distorted grid

An interesting class of non-Cartesian grids are PEBI grids; PEBI = Perpendicular Bisector. In the PEBI grid the chosen grid points are blocked off into volumes using a geometrical construction which is not shown here. PEBI grids have been developed very extensively by Aziz and co-workers at Stanford University and by Heinemann in Austria (Heinemann, et al 1991; Palagi and Aziz, 1994). An example of a study using PEBI grids is shown in Figure 12 where the particularly flexible form of this grid is used to model faults in this particular case.

PEBI grids can be orientated to follow major reservoir faults This example from:

R.E Phelps, T. Pham and A.M. Shahri, “Rigorous Inclusion of Faults and Fractures in 3D Simulation”, SPE59417, 2000 SPE Asia Pacific Conference, Yokohama, Japan, 25-26 April 2000

Figure 12 An example of a study using a PEBI grid

Another way of building distorted grids where the individual blocks retain some broad relationship with an underlying Cartesian form is using corner point geometry (Ponting, 1992). This is shown in Figure 13. This scheme is implemented in the reservoir simulator Eclipse (GeoQuest, Schlumberger) where it has been applied quite widely. In corner point geometry it appears rather tedious to build up a grid by specifying all 8 corners of every block (although some are shared with neighbours). However, if this approach is used, the engineer would virtually always have access to grid building software although building complex grids can still be time consuming. The engineer may be reluctant to use corner point geometry if there is a high likelihood 12


4

that the reservoir model will change radically. In future, this may be overcome by auto-generating the corner point mesh directly from the geo-model (although some upscaling may also be necessary in this process). At present, corner point geometry is probably more common in fairly fixed base case reservoir models that the engineer has fairly high confidence in. The model shown in Figure 14 is constructed using corner point geometry since it has a major fault in it between the aquifer and the main reservoir. Corner Point Geometry

Coordinates of vertices ( ) specified. Block centres ( )

Figure 13 Corner point geometry

Highly distorted grid blocks

Block <-> Block Transmissibility

Figure 14 Complex reservoir model constructed using corner point geometry

2.6 Issues in Choosing a Reservoir Simulation Grid

The main issues in choosing a grid for a given reservoir simulation calculation are as follows: (i) Grid Dimension: Refers to whether we should use a 1D, 2D or 3D grid structure;


13

(ii) Grid Geometry/Structure: The next issue is whether we should use a simple Cartesian grid (x, y, z) or some other grid structure such as r/z. This choice also includes where local grid refinement, a distorted grid or corner point geometry is appropriate; (iii) Grid Fineness/Coarseness: How many grid blocks do we need to use? This asks whether a few hundred or thousand is adequate or whether we need 10s or 100s of thousands for an adequate simulation calculation. We remind the student of the advice in Chapter 1. This was to carry out the reservoir simulation calculation keeping firmly in mind the question which had to be answered or the decision which had to be taken. Therefore, the issues of grid dimension, type and fineness are directly related to the appropriate question/decision. However, as we will see, there are several technical considerations that can guide us in these choices. Essentially, all 3 choices (grid dimension, type and fineness) depend strongly on the problem we are trying to solve. Consider the issues of grid dimension and type together. A 2D x/z cross-sectional model (with dip if necessary) may be used to study the effects of vertical heterogeneity - layering for example - on the sweep efficiency or water breakthrough time. For a near-well coning study, an r/z grid is usually more appropriate since it more closely resembles the geometry of the near well radial flow. 2D x/z grids are also used to generate pseudo relative permeabilities for possible use in 2D areal models. For full field simulations, 3D grids are generally used which in most models are still probably Cartesian with varying grid spacing in all three dimensions. In recent years, other types of grid such as distorted or corner point grids are being applied - especially if a geocellular model has been generated as part of the reservoir description process. Such guides are also applied in some studies to model major faults in reservoirs. Flow through major faults can lead to communication between non neighbour blocks and this can be modelled in some simulators by defining nonneighbour grid block connections as shown schematically in Figure 15.

Fault L1 L2

L2

L3

L1 L4

L3 L4

The issue of grid fineness/coarseness, or how many grid blocks to use in a given simulation, can sometimes be quite subtle as we will show below. However, in many practical calculations, some “reasonable” and practical number of grid block is chosen by the engineer. Then, this can be checked by refining the grid and seeing if the answers are close enough to the coarser calculation. If, as we carry out this grid

14

Figure 15 Grid system at a fault which may have non-neighbour connections


4

refinement, the answer - e.g. recovery profiles and water etc. - no longer changes, the calculation is said to be “converged” and is probably quite reliable. By “reliable”, we mean that the grid errors are probably a small contribution of to the overall uncertainties of the whole calculation. If the grid is far from being converged, then comparisons between different sensitivity calculations may be masked by numerical errors. These various points are illustrated by the two examples below. The two examples used to show the importance of the number of grid blocks are: (i) Example 1: the effect of vertical grid fineness (i.e. NZ blocks) on a miscible water-alternating-gas (MWAG) process. (ii) Example 2: resolving the vertical equilibrium (VE) limit of a gas displacement calculation. Example 1: Figures 16(a) and 16(b) shows the recovery results (at a given time or pore volume throughput) for both a waterflood and an MWAG flood in the same system each as a function of 1/NZ. The difference between the two calculations is the incremental oil recovered by the MWAG process. The economics of performing MWAG depends on how large this difference is. The purpose of plotting this vs. (1/NZ) is that we can extrapolate this to zero i.e., effectively to NZ → ∞. Taking the results at NZ = 2 (1/NZ = 0.5) shows an incremental recovery of (72% - 36%) = 36% of STOIIP which is a huge increase and would make such a project very attractive. However, as we refine the vertical grid, the waterflood recovery increases while the MWAG recovery decreases, i.e. the calculations move closer together and the incremental oil is greatly reduced. Indeed, as we extrapolate to (1/NZ) = 0, we see that the incremental oil is only (47.5% - 47%) = 0.5% which is well within the error band of the calculation. So, rather than having a very attractive project, we appear to have a completely marginal or non-existent improved oil recovery scheme. Certainly, performing just one coarse grid calculation and taking the results at face value would be very misleading in this case. Example 2: The vertical equilibrium (VE) condition in a gas flood is where the gas if fully segregated by gravity from the oil. This limit has a simple analytical form (not discussed here) which can be written down without doing a grid block calculation. However, we can test the numerical simulation by seeing how many blocks (NZ again) we need to correctly reproduce the VE limit. The answer is rather surprising as shown by the results in Figure 17. These results show that 200 layers are needed to fully resolve the gas “tongue” at the top of the reservoir. Clearly, if we just guessed that 5 vertical blocks would be enough and did not check, then our calculation would be significantly in error. The two examples above illustrate how the number of blocks chosen for a simulation can strongly affect the results. It shows the need to check that a calculation has converged or that changing the number of grid blocks does not significantly change the answers.


15

Figure 16: (a) Extrapolation of Predicted Waterflood Recovery Efficiency for 2D Stratified Model C Sand Base Case 100

Process ==> Waterflooding

90

Vertical Grid Refinement NX

Recovery Efficiency, Recovery %ooIPEfficiency, %ooIP

80 70 100

Process ==> Waterflooding

60 90

Vertical Grid Refinement As vertical gridNX is refined 1 --> 0 NZ

50 80 40 70 30 60 Extrapolated RE = 27.6%

20 50

Water

1 2 3

Oil NZ

1 2 3

Gas

Oil

Homogeneous Model, kv/kh = 0.1 Stratified Model, 1.2 HCPVI, 0.02 NZkv/kh = Gas Homogeneous 1 Model, kv/kh = 0.01

As vertical grid is refined

NZ

--> 0

10 40

2D cross-sectional model

0 30 0.00 0.05

0.10

0.15

0.20 0.25 0.30 1

Homogeneous = 0.1 0.55 0.35 0.40 Model, 0.45kv/kh 0.50 Stratified Model, 1.2 HCPVI, kv/kh = 0.02

0.60

/nz grid blocksHomogeneous Model, kv/kh = 0.01

Extrapolated RE = 27.6%

20

Recovery Efficiency, Recovery %ooIPEfficiency, %ooIP

10 Vertical Grid Refinement 100 NX 0 (b) Extrapolation of Predicted MWAG Recovery Efficiency for 2D Stratified 1 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 Water 90 Case 2 1 C Sand Base /nz grid blocks 3 Oil 80 70 100 60 90 50 80 40 70 30 60 20 50

NZ Vertical Grid Refinement As vertical grid isNX refined 1 --> 0 NZ 1 2 3

As vertical is refined Extrapolated REgrid = 35.3%

1 --> NZ

Gas Water Oil

Homogeneous Model, kv/kh = 0.1 NZ Gas Model, side solver Variable Width Homogeneous 0Stratified Model, 25% slug, 1.2 HCPVI, kv/kh = 0.02 Homogeneous Model, kv/kh = 0.01

Process ==> Wiscible Water - Alternating - Gas (MWAG)

10 40 Extrapolated RE = 35.3%

0 30 0.00 0.05 20

0.10

0.15

0.20

1

Homogeneous Model, kv/kh = 0.1 Variable Width Homogeneous Model, side solver Stratified Model, 25% slug, 1.2 HCPVI, kv/kh = 0.02 kv/kh 0.45 = 0.01 0.50 0.55 0.60 0.25Homogeneous 0.30 0.35Model, 0.40

grid blocks Process ==> Wiscible/nzWater - Alternating - Gas (MWAG)

10 0

0.00 0.05

16

Figure 16 The effect of vertical grid refinement on recovery in (a) a waterflood and (b) a MWAG displacement in a

Water

0.10

0.15

0.20 0.25 0.30 1

0.35

/nz grid blocks

0.40

0.45 0.50 0.55 0.60

Model

Figure 17 Resolving the gas “tongue” in the Vertical Equilibrium (VE) limit in a gas - oil displacement by increasing the number of vertical grid blocks (from Darman et al, 1999)

Recovery factor


0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 x

0

Gravity Dominated

x

x

x

x

x

x x

x x x

x

0.2

0.4

Gas

x x x x x x x x x x x

Oil

VE Limit

x

x

x x

4

fine grid : 5 layers fine grid : 25 layers fine grid : 50 layers fine grid : 200 layers coarse grid

PVI

0.6

0.8

1

2.7 Streamline Simulation

The problem of numerical dispersion was discussed above. One approach to have a more accurate transport calculation is to use streamlines. We will describe this qualitatively in a non-mathematical manner with reference to Figure 18 from the work of Gautier et al (1999). The basic procedure in streamline simulation for a given permeability field (Figure 18(a)) is to calculate the pressure distribution by solving a conventional pressure equation (see Chapter 5 and 6). From this the iso-potentials (pressure contours) can be calculated as shown in Figure 18(b); the gradient of the pressures locally perpendicular to the iso-potentials are the streamlines as shown in Figure 18(c). These streamlines are essentially the “paths” of the injected fluid from the injectors (sources) to the producers (sinks). Since the velocity along these paths is known (from Darcy’s Law using the calculated ∇P), we can work out how far the saturation front moves along the streamline, Δl = v.Δt, where v is the (local) velocity at that point on the streamline. Since v is known quite accurately, the advance of the front along the streamline can be calculated accurately without the problem of block to block numerical dispersion. After we propagate the front along the streamlines, the saturations will change over the reservoir domain. These saturation changes are then projected back onto the Cartesian grid as shown in Figure 18(d), hence changing fluid mobilities. These updated mobilities can be used to recalculate the pressures which, in turn, can be used to update the streamline pattern. This process can be continued throughout the calculation. However, the calculation of the pressure equation is what takes most computational time in most reservoir simulation.


17

Outline of Streamline Method Prod a) Permeability map with an injector and a producer wells

Inj

b) Solve the pressure field c)

Compute the velocity field and trace streamlines

(b)

(a)

d) Move saturation along streamlines and compute the values of the saturation on the grid.

(d)

(c)

From Gautier et al (1999)

Figure 18 Schematic of streamline simulation from the work of Gautier et al (1999)

2D Areal Displacements with Streamlines I1 • Pair Injector / Producer

P1

• IW flows faster in a direct line between the wells and slower in the corners • Arrival of different streamlines at producer at the same time.

Injected water concentration

An example of a streamline calculation in a five-spot pattern is shown in Figure 19. In streamline simulation, it may be possible to recalculate the pressures after many transport (saturation update) time steps. This relies on the assumption that the streamlines do not vary too rapidly as the flood progresses. This is a good assumption for many applications. Clearly, if the wells change very significantly, or we switch off some wells and add new ones, it will almost certainly be necessary to recalculate the pattern of streamlines in the reservoir domain. Streamline simulation has gained some popularity in recent years since 3D streamline codes have been developed e.g. by (Blunt and coworkers at Imperial College in London) and are available commercially. Streamline simulation is fastest compared with conventional simulation for viscous dominated flow where the assumption of slowly changing streamlines is probably best. For flow where gravity effects are very prominent, there tends to be “side flow” between streamlines and hence it is necessary to recalculate the pressure field quite often. This slows the streamline simulation down quite significantly in many cases although it can sometimes remain competitive with conventional simulation. At the present time, streamline simulation has a place in our simulation “toolbox” but it 18

Figure 19 An example of a streamline calculation in a five-spot pattern


4

is not suitable for all types of reservoir calculation. It is strictly for incompressible flow and compressibility effects can cause some errors. Also, streamline codes tend not to be as developed in terms of “bells and whistles” as conventional simulators e.g. complex well models, “difficult” PVT oil behaviour etc.

3

THE CALCULATION OF BLOCK TO BLOCK FLOWS IN RESERVOIR SIMULATORS

3.1 Introduction to Averaging of Block to Block Flows

In reservoir simulation, block to block flow terms arise between blocks which we often denoted by mobility terms such as (λT(S0))i-1/2 or (λ0(S0))i+1/2 etc, where the (i+1/2) and (i-1/2) denote the block boundary as the location where that term is evaluated. However, we do not specify properties directly on the boundaries, instead we define them within the grid blocks themselves e.g. the saturations (Sw and So), the permeability, porosity, relative permeability etc. A typical block-to-block flow is shown in Figure 20 where the appropriate terms in Darcy’s Law are also shown:

(

)

(

)

Applying λ Darcy’s law to the inter block flow shown in Figure 20 and using the T (So ) i −1 / 2 or λ o (So ) i +1 / 2 notation in that figure, we obtain:

(λ (S )) T

o

or (λ o (So ))λi +1(/ S ( T 2 o ))i −1/ 2 λ  ( T (So ))i −1/ 2 or (λ o (So )

i −1 / 2

  λ p   ∆P   λ p   Pi − Pi −1  Q p = − kA.  . (λ =(S− kA )) . 2 por .(λo∆(Sxoi −)1)i++1/∆2 x i    Bp   ∆x  T o i −1/ B      . λ p . ∆P = −kA. λp   Q p2 = − kA λ  B  (1) ∆PB. x . Q p p= .− kA Q pp = −∆kA  p      Bp  ∆x  where thekA overbar denotes an average of that  λ pquantity.   ∆P  The issue  λ phere   is: Pwhich  i − Pi −1 Q kA = − . . kA . . = − average should we take? The twopmain specific are:   B questions    p   ∆x   Bp    ∆x i −1 + ∆x i   kA   kA 2 What isλ the correct average for the permeability-area product, ? λ S ( ) ( ) T o i −1 / 2 or ( λ o (So ))kA p i +1 / 2 What is the correct average for kAthe phase mobility, λ p ?

λp  Further issues involve (i) whether we should average the terms k and   λ p separate λ ∆ P  Pi − P rp Q p = − kA. Bp  .  = − kA. p  . λ μp - within the phase mobility, p ; and (ii) which average use for  Bp B  Bp  we∆xshould  p  ∆x i −1 + Bp from ∆x formation volume factor does not usually vary although the very rapidly block  2 Bp

λp

to block.

∆x Bp ∆x ∆x i −1 + ∆x i ∆x 3.2 Averaging of the k-A Product, kA = We first examine 2the kA averaging by considering what ∆ this be for singlex i −should 1 + ∆x i ∆x + ∆x ∆ x = and the related phase flow. Consider the volumetric flow, Q, of a single phase x i i −1 ∆x i −pressure 1 + ∆= 2 = drops as shown 2 λp k rp in Figure 21: 2 ∆x i −1 + ∆x i µp k rp = k rp 2 Bp k rp µp µp µp k rp ∆x µp Institute of Petroleum Engineering, Heriot-Watt University

=

∆x i −1 + ∆x i 2

19

(λ (S )) T

(λ (S )) Permeability = ki-1

Qp

kA

Area, A i-1

∆x

Notation: Qp

i −1 / 2

or (λ o (So ))i +1/ 2

or (λ o (So ))i +1/ 2

   λ p   ∆P   λ p   Pi − Pi −1  Q p = − kA.  . = − kA.  .   Bp   ∆x    Bp    ∆ x i −1 + ∆x i   ∆x i   λ p   Pi − Pi −1    2 Permeability =λkpi   ∆P  Q p = − kA.  . = − kA.  .   Bp   ∆x   Bp    ∆x i −1 + ∆x i     2 kA T

∆x i-1

o

o

i −1 / 2

Area, A i

λp

Figure 20 Block to block flow in a simulator

λp Bp

= volumetric flow of phase p (p = o, w, g)

Bp

∆x

Δxi-1, Δxi

= sizes of the (i-1) and i blocks (may not be equal)

∆x

= distance between grid block centres =

Ai-1, Ai

= areas of the (i-1) and i blocks ∆x (may + ∆xnot be equal)

∆x

=

i −1

∆x i −1 + ∆x i 2

i

k rp 2 = permeabilites of the (i-1) and i blocks (usually not equal) µp k = mobility of phase p = rp µp

ki-1, ki λp krp

= relative permeability of phase p

ΔP

= pressure drop between grid block centres

μp, Bp

= viscosity and formation volume factors of phase p

Ai

Ai-1

Pi-1

Q

Pi

ki-1

ki

∆xi-1

∆xi

Pi-1

Pressure P

Pf = face pressure between blocks

Pf Pi

x

We now consider flows from Block (i-1) to the interface of the two blocks, where we denote the interface pressure as Pf (see Figure 21).

20

Figure 21 Single-phase flow between blocks to determine the correct kA average to use in flow simulation.


k .A P −P ∆x i −1 . − k A P P Q = − i −1 i −1µ. µf −1x i −1 Q=− . ∆x i −1i∆ µ ∆x i −1 2 µ 2 k .A 1 Pf −2Pi −1 2 from which difference (Pf - Pi-1): Q =we−cani −1findi −the . pressure ∆x i −1 µ k .A P − P 1 ∆x  1  Q = − ( Pi −1 − Pi −1 .) f= 2∆−ix−Q ∆.x i −1i −11 1  ∆µ− x. ∆ i −.µ 1 iQ −1xµ   k.A )i −1  ((PPf −−(PPPif−1−f)) µP==i −1i)−−−1 =Q i −1  2 2 1 ) i −1  Qµ.2 2  ( k.A )(k(.A f i −1 2  ( k.A )ii −−11  ∆x 1 (Pf − Pi −1 ) = − Qµ. i −1  2 Pf  ( k.A )i −1  k.Ai .A iP P−i − k P i i i =− i− Pi −.Pf. ∆x fi −1  1  In block (i): QkkQ =i ..A A . P − P . x∆i x i  = − − P P i (Q )  Q f= − i −1µ i=. µi−∆µxQi µf ∆ ∆x i 22  ( k.A )i −1  µ 2 k .A i Pi −2 Pf Q =we−cani find . the2 pressure difference (P - P ): from which ∆x i i f µ k i .A i Pi −2 Pf   Q = − ( P − P. ) = ∆−xQµ∆.x∆i1xi 1 1  xµi−. ∆Qxµi . 1   ((PPi −−(PPPfi))−µi ==Pf )−−f =∆Q k.)A)i  i  2 2 (k(.A  Q µ . i i f 2 22  (( kk..A A ))i  ∆x i  1 i  (Pi − Pf ) = − Qµ.   2 Q  (µk .A ∆)xi∆x i −1 ∆x∆x i  µ Q  ∆x−to∆eliminate Adding equations( P3i and 5 above  i gives that: ∆xP+ term x .1 i −1the µ=.µ (PPPii −−−(PPPPifi−)1−)=−P==i −P1−i)−−−1Q)=Q µ−. i ∆2.xii −−11 ( k+.A )∆+x ii f ( k.A)  Q k.)A+)i −k1i .−A 1 ( k.A )i  i ( i i −1 ) 2 .2 (2k .(Ak .)(A 2  ( k.A )ii −−11 i (( k.A ))ii  ∆x Qµ ∆x (Pi − Pi −1 ) = − . i −1 + i  2 ( k.A ) ( k.A)  Qµ  ∆x i−1  ∆ x i   i −1 (Pi − Pi1Q−1 )= =−1 1−. 2 . 2 2  +   . iP − P .  22∆xi (−1k.A )i−.1∆ (P.PAi(i−1)−)i Pi −1i)−1 ) Darcy Law: xi i(−k.single-phase Q = −Q1=to.−the ∆x i −1 ∆form ∆ofx.((iP which rearranges the + Q = − µ . µ∆µxfollowing − P x  +x i ( k.Ai ) Pi −1 ) .+)A )∆ k+ µ  ∆x ii(−−11k(.A i − 1 i ( k.A )i i  1  ( k.A )i −1 2 i(−k1 .A ) Q = − . ( k.A )i −1 ( k.A )ii  .( Pi − Pi −1 ) µ  ∆x i −1 + ∆x i   1  2 Q = − . ( k.A )i −1 ( k.A )i  .( Pi − Pi −1 ) µ  ∆x i −1 + ∆x i   ( k.A )i −1 ( k.A )i  1 i −1i −1Pf − f Pi −1i −1 k i −1i.−A In block (i-1): QkQ − 1 .A i −1 Pf −. P.i −1 =i −=−

4

(2)

(3) (4)

(5)

(6)

(7)

But, taking kA as the correct average, then, by definition, the single-phase Darcy Law is of the form:

kA ( P − Pi −1 ) Q=− . i x µ ∆( xPi −−1 +Pi∆ kA −1 )i Q=− . i µ ∆xi −12+ ∆xi 2

(8)

Comparing identical terms in equations 7 and 8 above gives that:

kA ∆xi −kA 1 + ∆xi ∆xi −12+ ∆xi 2

    2 =   ∆xi −1 2+ ∆xi   =  xi)−i1−1 (k∆. A xi)i   (k∆. A +    ( k . A) i − 1 ( k . A) i 

which easily rearranges to:

   ∆xi −1 + ∆xi  kA =  ∆∆ xix−1i −1 + Heriot-Watt ∆x  Institute of Petroleum Engineering, + ∆xi i  University kA =  xi)−i1−1 (k∆. A xi)i   (k∆. A +  

(9)

21

kA ( Pi − Pi −1 ) . µ ∆xi −1 + ∆xi 2 (λ T (So ))i −1/ 2 or (λ o (So ))i +1/ 2     ∆x i−1 + ∆xi   kA =  ∆x  ∆x     i −1 i +    kA  2  λ p   ∆P   λ p   Pi − Pi −1  (k.A=) (k.A)  ∆xpi −=1 i−kA.∆ xBi . ∆x  = − kA. B  .  ∆(10) ∆xi −1 +∆xi i−1 Q  p   x i −1 + ∆x i   +  p   2  ( k . A) i − 1 ( k . A) i    2

Q=−

Note: This is an important result and in particular, we observe the following:

(i) The appropriate  average kA is not the arithmetic average, it is the harmonic average weighted  by∆the x grid + ∆xblocksizes. i −1 i kA =  ∆xi −1 ∆xi   average+gives  more weighting to the lower permeability λ much (ii) This harmonic k. A)i −1 (kp. A)i  (  value. If the grid sizes are equal, this reduced to the exact harmonic average, k , H

of the permeabilities as follows:

(λ (S ))

Bp 1 1 1 1 =  +  2  ki −1 ki  kH ∆x

T

o

i −1 / 2

or (λ o (So ))i +1/ 2

(11)





3.9∆P  λ   Pi − Pi −1  This can be seen for the following example: k1 = 200 mD, k2 = 2 mD ⇒ kHλ = krp Q p = − kA. p  .  = − kA. p  .  mD. We λwould expect the flows∆xto be+much by the  ∆x  ∆xλi T (more p =  Blower  Bp    ∆x i −1 + ∆x i   p So ) i −strongly or λaffected i −1 o ( So ) i + 1 / 2 µ = 1 2 / permeability since, p if one of the permeabilities were zero, then the flow would be   2 2 zero, no matter how large the other permeability was.

(

µ

+µ

B

)

(

)

+B



 Pi − Pi −1    Bp    ∆x i −1 + ∆x i     2

i −1 i kBrpp =overi −1into ithe averaging of kA for multi-phase andcarry (iii) Theµabove arguments p = 2 2 Q = − kA. λ p  . ∆P  = − kA. λ p  . flow. µp p    

 Bp   ∆x  λp

EXERCISE 1. 1. For the two grid blocks below, calculate

kA (in mD.ft.2)

Bp 15 ft.

20 ft.

λp

(λ (S )) T

o

i −1 / 2

or (λ o (So ))i +1/ 2

∆x λ S ( T ( o ))i −1/ 2 or (λ o (So ))i +1/ 2

      λ λ ∆P  Pi − Pi −1  15 ft. k2 25 ft. Qi −p1 + =∆ −xkA . p  .  = − kA. p  . x ∆ B k1  i p   or (λ o (So ))i +=1/ 2 (λ T (So ))i −1/ 100  Bp  ∆x  Bp    ∆x i −1 + ∆x i  2 ft. 2  λ p   ∆P   λ p     Pi − P2i −1   120 ft. Q p = − kA.  . = − kA.  .  ∆x Bp   ∆x     Bp    ∆x i −1 + ∆x i    k rp   2 kA  λ pthe  answer  λ pcalculated   Pi − Pi −1  (i) For k1 = 200 mD and k1 = 185 mD. Compare ∆P  with µ Q p = − kA.  . = − kA p . as the arithmetic average.   . ∆x + ∆ x   Bp    ∆x i −1 + ∆x i   = i −B1 p  ∆i x   2 answer with kA 2 (ii) For k1 = 200 mD and k1 = 5 mD. Compare the λ pcalculated as the arithmetic average.

2. If k1 ≈ k2, and A1 ≈ A2, show that average.

k rp kA is approximately equal to the λ pBarithmetic p µp λp

Bp∆x

Bp

∆x ∆x

22

+ ∆x

P−P (λ T (So ))i −1/ 2  kAkA . ( i  i −1 ) 2 Q=− =  ∆∆xxi ∆xi  ∆xi −1 +µ∆xi∆xi −1 + i −1 λ S or ( ) + ( )  T o 2 k. A i −1/ 2 k. A(λ o(So ))i +1/ 2 Q = − kA. λ p  . ∆P  = − kA. 2  ( )i − 1 ( )i  p  B   ∆x    p  Gridding And Well Modelling λ Qp =     − kA. B     λ p   Pi − Pi −1   kA 2 λ p   ∆ P  = +p ∆=xi− kA.  .  = − kA.  .  ∆xi −1  B∆xi ∆x  ∆xi −=1 + ∆xi∆xi −1Q kA kA  Bp    ∆x i −1 + ∆x i    ∆x + p   ∆ x i −1  2  +(k. A)ii −1 (k. A)i  2  ( k . A) i − 1 ( k . A) i  kA 3.3 Averaging of the Two-Phase Mobility Term, λ p To recap, the single-phase kA term is taken as a weighted harmonic average and this   1 1 1 1  averages to take for the two-phase flow term, λ p . leaves us with the issue of what =   ∆xi −1++ ∆xi  kA = In fact, it is from the viscosity p 2 ∆  relative permeabilityBterm ki ∆xi the kHconvenient  kxii −−to11 separate λ+p as follows:  separately  and consider these  ( k . A) i − 1 ( k . A) i  Bp ∆ x k λ p = rp Bp 1 µ p1  1 1 (12) ∆x =  +  x x ∆ ∆ + i i −1 2  ki −1 ki  kH = 2 variable than We do this since the relative ∆x permeability of phase p is far more + + µ µ B B i i i i − 1 − 1 ∆x i −1 + ∆x the phaseµviscosity. If fact, that andwithout Bp = further discussion, we will simply note = p = 2 2 are very accurately calculated as the 2 the viscosity and volume factors krp formation k rp = x x ∆ ∆ + arithmeticλaverages, since they usually do p i −1 i not vary very much from block to block = µp µp i.e. the averages between grid blocks 2 (i-1) and i are: k rp µp µ + µi k Bi −1 + Bi rp B = µ p = i −1 and p 2 2 (13) µp

4

Therefore the situation is summarised in Figure 22:

Qp

Figure 22 Which average should be taken for the relative permeability of phase p in the averaging of block to block flows?

Qp = - kA.

Harmonic average

krp µp.Bp

.

∆P

Which average??

∆x

Arithmetic averages

We will determine which relative permeability average to take by considering the physical situation of two-phase oil/water flow from block i → (i+1) as shown in Figure 23. Consider the situation in Figure 23 where: Block i is at Sw = (1-Sor) i.e. only water can flow; krw > 0 and kro = 0; Block (i+1) is at Sw = Swc i.e. only oil can flow; krw = 0 and kro > 0.


23

Sor

1 - Sor Sw

krw > 0

krw = 0

kro = 0

kro > 0

Swc Block i

Block i + 1

Physically, it is clear in Figure 23 that water can flow from i → (i+1) but oil cannot. Therefore, for flow in this direction, the average water relative permeability must be non-zero but the average oil relative permeability must be zero. Let us consider the different averages that are possible in turn: Harmonic Average: First consider if the harmonic average can be used since this was appropriate for the single-phase permeability averaging. Harmonic average of water relative permeabilities = Harm. Av. {krw i >0; krw i+1 = 0} =0 Likewise, for oil, Harm. Av. {kro i = 0; kro i+1 > 0} = 0 Thus, the harmanic average gives zero flow for both water (incorrect) and oil (correct). Therefore, it cannot be the harmonic average which is correct for the relative permeability. Arithmetic Average: Now consider if the arithmetic average can be used since this is a natural thing to try and it is certainly the simplest. Arithmetic average of water relative permeabilities

(k

> 0) + ( k

rw i rw i +1 = ( krw i > 0) + ( krw i +1 = 0) > 0 2

= 0)

> 0

2

So, the arithmetic average could be physically correct for the water phase since it gives the average krw > 0 and thus allows water to flow.

krw > 0

But, for oil the oil phase, we find that : Arith. Av. {kro i = 0; kro i+1 > 0} > 0 and this is physically incorrect, since oil cannot flow from i → (i+1). Therefore, it cannot be the arithmetic average which is correct for the relative permeability. What average does this leave? We simply state the answer and then give some physical justification. Upstream Value: In fact, it turns out that the physically correct value of the relative permeability is simply the upstream value. The upstream value refers to the block from 24

Figure 23 Flow of water from block i → (i+1) and the corresponding relative permeability values for water and oil.


4

which the flow is coming i.e. in the flow from left to right in Figure 23. This would be block i. This can be seen to be consistent with the physical situation since: Flowing from i → (i+1), the “average” water relative permeabilities = Upstream {krw i} > 0, as required. Likewise, flowing from i → (i+1), the “average” oil relative permeabilities = Upstream {kro i } = 0, as required. Now reversing the flows from (i+1) → i, we find that only oil should flow and this is again consistent since: Flowing from (i+1) → i, the “average” water relative permeabilities = Upstream {krw i+1} = 0, as required since water cannot flow in this direction. Likewise, flowing from (i+1) → i, the “average” oil relative permeabilities = Upstream {kro i+1} > 0, as required since only oil can flow in this direction. The situation is summarised in Figure 24.

Qp

Figure 24 The correct inter-block averages for all terms in the two-phase block-toblock flows in a reservoir simulator.

Qp = - kA.

krp µp.Bp

Harmonic average

4

.

∆P ∆x

Upstream value of krp

Arithmetic averages

WELLS IN RESERVOIR SIMULATION

4.1 Basic Idea of a Well Model

The only way fluids can be produced from or injected into a reservoir is through the wells and we must therefore include them in our reservoir simulation model. As you may know, the area of Well Technology is vast and in addition to the long wellbore between the reservoir and the surface, there are many other technical features of wells that can have a major impact on the flows into and out of the reservoir. For example, there will be safety valves at the surface and many different types of completion in the well construction itself. Here, we will simplify things as much as possible in order to extract the central functions of the well that we will have to model in the simulator. A schematic of the total well is shown in Figure 25 where the details of the near well formation are shown inset. The near wellbore flows are thought to be radial in an ideal vertical well and this will have some relevance in modelling the near-well pressure behaviour, as discussed in Chapter 2 and elaborated upon below. In addition to these near-well pressure drops, there are several other identifiable


25

pressure drops between the fluids in the reservoir and the surface oil storage facilities and we may have to model at least some of these. Indeed, it is this topside pressure behaviour that “links” or “couples” the surface with the pressure and flows that we are trying to model in the reservoir using reservoir simulation. The main decision is to determine how much of the formation to surface well assembly we will actually have to model. The main pressure drops are shown in Figure 26 (based on Figure 25 of whole well + ΔPs) and are associated with: (i) Formation → wellbore flow, ΔPf→w: where fluids flow from a “drainage radius”, re, at pressure, Pe, to the wellbore. Figure 26 shows the near-well pressure profile, in the near-sandface region with bottom hole flowing well pressure (BHFP), Pwf. Thus the formation → wellbore pressure drop, ΔPf→w , is:

(

∆Pf → w = Pe − Pwf

)

(14)

(ii) The pressure drop, ΔPwell, that may occur along the completed region of the = Pthe ∆Pr → s of the well (or the “toe” of a horizontal well) to the atm + wellborePwf from bottom wellbore just at the top of the completed interval. In very long wells, this pressure drop along the wellbore due to friction may be quite significant although there will Pe −bePwfignored; o = PI be casesQwhere it. can

(

Well head Q

o max .

)

Surface facilities (separator etc...)

= PI .Pe

To storage / export

∆P(r ) =

r Qµ ln  2π (k.h)  rw 

Well tubulars

(

∆P(re ) = Pe − Pwf

)

r  Qµ = ln e  2π (k.h)  rw 

2π (k.h) . Pe − Pwf  re  µ. ln  Reservoir showing  rw 

(

Q=

)

two geological layers

PI =

26

2π (k.h) r  µ.ln e   rw 

Well

Well completed in reservoir

Fluid flow from reservoir layers to wellbore

Figure 25 Schematic of the fluid flows into a well in a grid block model of the reservoir through to their storage or export from the field.


4

∆Pwh -> export

Surface facilities (separator etc...)

Well head

To storage / export

Well tubulars

Near wellbore formation to wellbore ∆Pf -> w

Well

Pe

∆Pr -> s pressure drop from reservoir top to surface

Figure 26 Schematic of the fluid flows from the well through to storage or export showing the associated pressure drops that occur in the system.

∆Pf -> w Pwf rw

r

re

Fluid flow from reservoir layers to wellbore

Reservoir showing two geological layers

Well completed in reservoir ∆Pwell pressure drop along the well within reservoir section

(iii) Reservoir → surface pressure drop, ΔPr→s : the pressure drop from the well at the top of the completed formation just above the reservoir to the wellhead which is at pressure, Pwh . This ΔP is quite significant and, locally in any sector of the well, there will be a local pressure drop vs. flow rate/fluid composition relationship. This may be calculated from models (often correlations) of multi-phase flow in pipes. As the fluids move up the wellbore, the pressure drops in oil/water production and free gas may also appear; thus, we can have three phase flow in the well tubular to the surface and we may have to incorporate this flow rate/pressure drop behaviour in our modelling. (iv) There will frequently be further pressure drops as the fluids flow from the wellhead through the surface facilities such as the separators, various chokes, etc. We will not consider this in detail here although it can be an important consideration in some field cases e.g. if the well is feeding into a network gathering system which other wells are also feeding into. This could be a complex surface gathering system network or a multiple-well manifold of a subsea production system. In this section, we will mainly focus on the formation to wellbore pressure drops. Thus, our main task is to either calculate or set the well flowing pressure (Pwf) although we will return briefly to the issue of calculating the pressure drops between the reservoir and the surface in the discussion below. To set the scene in modelling wells in a simulator, we will first consider a very simple model well producing only oil into the wellbore. How do we decide what Institute of Petroleum Engineering, Heriot-Watt University

27

the volumetric flow rate, Qo, of this well is? Indeed, do we decide or is it set for us by the reservoir and well properties? We will start with the simplest case where we basically “take what we can get” by drawing the wellhead pressure, Pwh, down as low as possible. Suppose that we simple open it up such that the oil pressure drops essentially to atmospheric. There is then the additional reservoir to surface pressure drop, ΔPr→s, to consider. Thus, the well flowing pressure, Pwf, would be given by:

(

∆Pf → w = Pe − Pwf

)

Pwf = Patm + ∆Pr → s

(15)

So, what happens ? Clearly, if this value of Pwf > Pe (the reservoir pressure), then no − Pwf oil can beQproduced. if Pwf < Pe, then some oil will flow into the well and o = PI . Pe However, we can now calculate how much. As we will see, this will depend on the physical properties of the system such as the permeability of the rock, the viscosity of the oil, .P−ethePwell etc. However, in our simple conceptual well, we will o max . == PI the preciseQ P Pof ∆geometry e f →w wf take all of these quantities as “givens” for the moment. Suppose the well does flow at a volumetric flow rate, Qo, for reservoir pressure, Pe, and well flowing pressure, r Pwf . We can index, PI, of the well as follows: =r )P=atmdefine +Q∆µPar →productivity PwfP(then sln ∆

(

)

(

)

  2π (k.h)  rw 

(16) ( ) r  Qµ ∆P(r ) = ( P − P ) = ln  where possible units of PI could2π be( kbbl/day/psi, .h)  r  for example. The above equation

Qo = PI . Pe − Pwf e

e

e

wf

w

= PImuch .Pe oil is produced per psi of drawdown. This simple equation Qo max .how basically states takes us back to our original question on “what/who decides on Qo?”. In our simple case, the ∆ answer .he )− clear; Pf → w2π=is(know P Pwf i.e. some things are “givens” - e.g. PI and Pe in a = .P e- −  Prwfsome  Q µ virgin oil Q producing system and we can set within limits - e.g. Pwf by setting ∆P(r ) =  re  ln  µ. ln2π (kHowever, the wellhead pressure. you may be able to set a flowrate, Qo, by installing .h)  rw  r   w a downhole ESP = Patm(an +∆ Pr →-s electrical submersible pump). In that case, you would Pwfpump set Qo and then calculate Pwf where we are still considering the reservoir pressure (Pe) as a given. But clearly we cannot any arbitrarily large value since the Qµ set Qo rto e 2 π . k h ( ) ∆ = − = ln P r P P ( ) lowest possible value vacuum),which  would then set a maximum value PIo = ePI Q − PPwfwf = 0 (a . Peeof 2π (k.h)  rw   re  wf of Qo given by:

(

( )

((

))

)

µ.ln   rw  (17) Qo max .2=π (PI k.h.P)e Q= . Pe − Pwf  re  set either a pressure or a flowrate but (a) not both and (b) So, in summary, µ. we ln can  in either case, within limits.  rQ r µ w ∆P(r ) = ln  2π (k.h)  rw 

(

)

But, can’t we affect the well PI or the reservoir pressure, Pe? We can actually affect π (“stimulating” k.h) the PI of a well 2by it possibly by locally hydraulically fracturing the PI = well or by acidising it rtoincrease the effective permeability of the near well region. e  re  Qµ .lnPincrease In addition, more commonly Pwf (or ∆Pwe = − = ln (re µ)can   maintain) the reservoir pressure e rw  2π (k.h) to some  rw  extent by injecting a fluid - usually (which relates to the “reservoir energy”) water or gas in another injector well. However, the basic well controls are either setting pressure or flow rate and this must be kept in mind when we model wells in 2π (k.hWe ) elaborate on these ideas in the following section where we reservoir Q simulation. = . Pe − Pwf introduce the central  ridea  of a well model. e

(

µ. ln   rw 

28

PI =

2π (k.h)

)

(

)

( ) = (P − P )


∆Pf → w = Pe − Pwf ∆Pf → w

e

4

wf

Pwf = Patm + ∆Pr → s = Patm +for ∆PSingle PwfModels r→s 4.2 Well and Two-Phase Flow

( (

) )

We now consider how a well model can be developed, firstly in our simple conceptual Qo = PI . Pe − Pwf reservoir producing only oil. Figure 27 shows the local pressure profiles in a simple Qo = well PI . Psystem homogeneous e − Pwf in single phase flow (see Chapter 2, section 3.5). The pressure profile close to the wellbore, assuming radial flow, was derived in Chapter Qo max . = PI .Pe 2 and is given by (section 3.5):

Qo max . = PI .Pe

r Qµ ln  2πQ(kµ.h)  rrw  ∆ = P r ( ) ∆Pf → w = 2πPe(k−.hP)wfln r  ∆P(r ) =

(18)

( ) Taking the∆P pressure P radius = (at − P )r as being the reservoir pressure, P , then gives: r  Qµ ∆ = − = ln P r P P ( ) ( )   P = P + ∆P 2πQ(kµ.h)  r  ∆P(r ) = ( P − P ) = ln  P = P + ∆P 2π (k.h)  r  (19) Q = PI 2π.((kP.h−) P ) Q= .( P − P )   r Q = PI P − . P k h 2 π . ( ) Q ( (a) (b) Q = µ. ln  .( P) − P ) Grid block r     Well at BHFP P w

f →w

wf wf

Figure 27 Schematic showing (a) the near well pressure profiles that occur in a radial system and (b) corresponding quantities in a Cartesian grid block

e

P

e e

atm atm

e

wf

e

wf r→s

e

wf r→s

o

e

wf

o

ee

wf

Qo maxµ. .=lnPI .wePe    r.wP = PI Q o max .

e

e

e

w e w

e

wf

e

wf

pressure

Pwf

e

2π (kQ .hµ) r PIP(=r )∆P(r) = P(r) - ln Pwf ∆ = h 2π2(π k.r(hek).h)  r  rw PI = µ.lnQµ  rw  ln  Pwf ∆P(r ) =  µ.ln2π (ek.h)  rw  ∆x rw r  rw  re r  Qµ ∆P(re ) = Pe − Pwf = ln e  2πQ(kµ.h)  rrwe  − = ln  P(re ) =be P P which can∆easily arranged to obtain: e wf 2π (k.h)  rw 

( (

) )

P

Pwf

∆y (assume ∆x = ∆y)

2π (k.h) . Pe − Pwf k.rhe ) 2 π ( Q = µ. ln  . Pe − Pwf (20)  rrw  µ. ln e   rw  and hence from equation 16 above, we can identify the productivity index 2π (k.h) (PI) of the well as: PI = 2π (k.rhe ) PI = µ.ln   rrw  µ.ln e  (21)  rw  Q=

( (

) )

This now demonstrates exactly how the quantities k, h, μ, rw and re affect the well productivity. All of these factors behave as we might expect them to physically e.g. as k↑, PI↑; as μ ↑, PI↓ etc. Now consider how this relates to the pressures in the simulation block shown in Figures 27 and 28. In a grid block, the pressure is thought of as being constant throughout the block although we know that it should be varying continuously across the block; we will refer to this as the average block pressure, P . The size of the grid block in Institute of Petroleum Engineering, Heriot-Watt University

29

our example is (Δx, Δy) and, for simplicity, we will assume that, Δx = Δy. Looking at the expression for PI in equation 21 and the quantities we have in the grid block, it is easy to make direct relations for some of them - obviously k, μ and h and also possibly rw and Pwf , although these latter two do not seem to appear in the grid model. The drainage radius, re, and the reservoir pressure, Pe, which appear in the radial model do not appear in the grid model - instead, the block size (Δx, Δy) and average block pressure ( P ) appear. This immediately suggests the following 2 questions: 1. what is the relation between re , and the block size (Δx, Δy)? 2. what is the relation between Pe and the average grid block pressure, P ? Well at BHFP = Pwf

Grid block pressure P

re

Pwf

re

=>

P

e = 

P

h ∆y

∆x

Pe

How do we choose re such that

P

Pe =P ??

Pwf

Relation re <-> ∆x, ∆y ??

rw

r

re

(assume ∆x = ∆y)

re rw

∆y

re is where P(re) =P and re = re (∆x, ∆y) but what is the formula ?

Figure 29 The relation between re and the block dimensions, Δx and Δy.

∆x

The issue is defined quite clearly in Figure 28. From this figure, we would like to choose re such that Pe coincides with the average grid block pressure, P . The latter quantity ( P ) is calculated in the simulation itself. In fact, we need to know how to calculate re from the quantities Δx and Δy as indicated in Figure 29 where we show the re as function of Δx and Δy, i.e. re(Δx, Δy). If we know the formula for re(Δx, Δy), then we can calculate the PI (equation 21) and use this inPthe simulation to couple the quantities Qo (oil flow rate) and P (average grid block pressure); i.e.

Q o = PI.( P − Pwf )

30

 r  2π( k.h ) ( P( r ) − Pwf ) ln  = . µ Q  rw 

Figure 28 Schematic indicating how the near well pressures relate to the corresponding quantities in a Cartesian grid block

(22)


4

Here, we can set either Qo or Pwf and then calculate the other one from P (and the known PIP). This was achieved in a very simple but ingenious way in a classic paper by Donald Peaceman (1978), another pioneer of numerical reservoir simulation. Peaceman did this by carrying out a 2D numerical solution of the pressure equation on a Cartesian . Pgrid − Pfor Q o =(x, PIy) wf a quarter five-spot configuration as shown in Figure 30. But, from equation 19, we know that:

(

)

P

 r  2 π( k.h ) ( P( r ) − Pwf ) ln  = . rw PI Q  .( P − µPwf ) Q o =

(23)

Hence, if we plot the pressure at grid blocks away from the well block vs. the well re ≈r 0.2 ∆x ( r ) −asPshown  on a logarithmic block spacing in Figure 31, then we can extrapolate 2 π( k.h ) Pscale wf ln = .   back to find the equivalent radius where = P in terms of the well block dimension rw  µ Q P e (Δx). It turns out that the2 simple2 relation is (for Δx = Δy):

(

)

P re ≈ 0.14 ∆x + ∆y

re ≈ 0.2 ∆x Q oo = PI.o(.P( P−−PPwfwf) );

Q w = PI w .( P − Pwf )

(24)

Therefore, we have a very simple way of calculating the PI or “well connection 2 2 r x 0 14 ≈ ∆ + ∆ . y e factor” as it is sometimes called of a well in a simulation grid block.

 r 2 πkh 2 π.k( kro.(hS)o )( P( r ) − Pwf )

ln PIPeaceman o =  = formula. applies to a well in a radial environment (the five-spot The simple rw PI  . P −µPre  ; QQ = PI . P − P Q o = w to radial w wf a common 2D Cartesian grid) configuration is asµoclose aswfwe using  can get o .ln  r   and for Δx = Δy. In fact,wsome modification to the simple formula is required for wells in “corner” or set close to a boundary and these are shown in Figure re ≈ 0.locations 2 ∆x 2π khbut .k rothe So well ( ) is isolated (radial flow), then: 32. Also,PI if Δx ≠ Δy, =

(

)

(

)

o 2πkh.k (S ) PI w = µ .ln rwre  w o re ≈ 0.14 ∆x2rwr+e∆y 2 µ w .ln    rw 

Figure 30 The 2D areal grid used to compare pressures with the expected radial profiles (from Peaceman, 1978)

(25)

Q o = PI2π .kh (P.−k P(S); ) Q w = PI w .(P − Pwf ) PI w = o 11 rwwf w 10  re  µ w .ln 9   w ) 2 πkh.8k ro (rS o 7 PI o = 6 r  µ o .ln5 e  4 rw 

PI w =

3 2 2πkh.10k rw Sw -1  r 

( )

µ w .ln-1e 0 1  rw 

2 3 4 5 6 7 8 9 10 11


31

0.6 • On Edge: i = 0 or j = 0 • On Diagonal: i = j •i≠j≠0

0.5

pij - po qµ / kh

0.4

(2.2) (2.1) (2.0)

0.3

(1.1) (1.0)

0.2 0.1

r Ao / ∆x From areal average pressure

ro / ∆x

0 0.1

0.2

0.4

0.6

1

2

3

4

5 6

Figure 31 Log plot of the pressures from the 2D areal grid used to find re (from Peaceman, 1978)

Factor in terms of (re / ∆x) (re / ∆x) = 0.2 (Peaceman's equation)

(re / ∆x) = 0.196 implies no flow boundary

(re / ∆x) = 0.433

(re / ∆x) = 0.193

(re / ∆x) = 0.72

For anisotropic permeabilities kx ≠ ky

32

re = 0.28

ky kx

1/2

2

.∆x +

kx ky

1/2

2

.∆y

ky 1/4 k 1/4 + x kx ky

1/2

Figure 32 Well factors for wells in various positions relative to the boundaries; after Kuniansky and Hillstad (1980)


4

We may find that, in a given simulation of a field case, that we input all the known or estimated data but the well in our simulation does not perform like the real case. It may produce more (higher PI) or it may produce less (lower PI) than expected. The former case may be due to the well being stimulated and the second case may be due to well damage. Within limits, we may adjust the calculated well PI in the simulation model in order to reproduce the observed field behaviour. However, we should think carefully before making such changes since the simulated well productivity may be wrong because some (or several) other aspects of the reservoir simulator input data are wrong. It is now relatively straightforward to extend our discussion on PI and simple well models in a homogeneous single layer system to the flow of two phases - say, oil and water - as shown in Figure 33. Since two phases are being produced, then the saturations of both oil and water (So and Sw) must be at values where their relative permeabilities are > 0 (i.e. = > So > Sor ; Sw > Swc). Well radius = rw

P

Qw

Qo

P P Q o = Saturation PI.( P − Pwf )

Figure 33 Near well two-phase flows in a Cartesian grid block.

w .and Q o = SPI P −SoPwf ) ( Q o = PI.( P − Pwf h )  r  2 π( k.h ) ( P( r ) − Pwf ) ln  = .  rw  2 π(µk.h ) (∆xP( r )Q− Pwf ) ln r  = 2 π( k.h ) . ( P( r ) − Pwf ) µ . Q ln rw  = µ Q  rw  (assume ∆x = ∆y) re ≈ 0.2 ∆x

∆y

r ≈ 0.2 ∆x

e We now apply were developed above for the single phase case. re ≈ the 0.2 same ∆x 2ideas as 2 From the rradial two phase Darcy Law, the volumetric production rates of oil and e ≈ 0.14 ∆x + ∆y water are given by: 2 2

re ≈ 0.14 ∆x 2 + ∆y 2 re ≈ 0.14 ∆x + ∆y Q o = PI o .( P − Pwf ); Q w = PI w .( P − Pwf )

(26)

Q o = PI o .( P − Pwf ); Q w = PI w .( P − Pwf ) − Pthe PI oPI .(wPare Q w water = PI wproductivity .( P − Pwf ) indices, respectively, and where theQPIo 0=and wf );oil and 2 πkh .k ro (So ) PI o = 2 πkh .kror(eSo ) PI o = 2µ o .ln πkh .kror(So )  rw  PI o = µ o .ln re  µ o .ln rwe  (27) r  2πkh.k rww(Sw ) and PI w = 2π .krwr(eSw ) .ln µkh PI w = 2π w kh.krwr(Sw )  rw  PI w = µ w .ln re  µ w .ln rwe   rw  (28) Institute of Petroleum Engineering, Heriot-Watt University

33

where, again, re is calculated using Peaceman’s formula. In the above equation, we have not incorporated the separate phase pressure, Po and Pw, in the well block but these may be used in a given calculation.

4.3 Well Modelling in a Multi-Layer System

The most common case which is modelled is where we have multi-phase (e.g. two phase oil/water) flow in a layered system where the layers are of different permeability. This situation is shown for a simple four layer system in Figure 34. Clearly, there are additional issues in this system since all four layers may be producing both oil and water and the proportions of each phase may be changing as the saturations (and hence relative permeabilities) change. In addition, there is also a gravitational potential in each layer which we may have to take into account. Using the notation in Figure 34, we note that the oil flows in layer k (k = 1, 2, 3, 4, in this example) are:

Q ok = PI ok ( Pk − Pwfk ) =

2 π( khk ro (So ))k

(Pk − Pwfk )  rek  µ o .ln  2 π( khk )) ro (rS w o k (29) Q ok = PI ok ( Pk − Pwfk ) = (Pk − Pwfk )  rek  µ o .ln  and a similar expression applies for water. 3 3  rw The total oil and water flows in the well are: Q T = ∑ (Q ok + Q wk ) = ∑ ( PI ok + PI wk )( Pk − Pwfk ) k =1 3

k =1 3

Q T = ∑ (Q ok + Q wk ) = ∑ ( PI ok + PI wk )( Pk − Pwfk ) 1 k =1 Q Tk =k(=Q ok + Q wk ) = ( PI ok + PI wk )( Pk − Pwf )

( ( ) )

)

(30)

where we have taken the mean block pressure and also the well flowing pressure in Q Tk = (Q ok + Q wk ) = ( PI ok + PI wk ) Pk − Pwf layer k asQbeing the same for both oil water phases. Again, in the layered case, Pwf πand i ,khk j, k ro (So ) o i , j, k = PI o i , j, k . P i , j, k − 2 k we can specify the flowing bottom hole and then we can Q ok = the PI oktotal Pk flow − Pwfkor = Pk − pressure Pwfk   r calculate the other one using the above (In fact, we can also specify ek µ .lnrelations.  find the flow of the other and the P ior Q oeither PIQ , j, k − Pwf i ,oj, k  o i , j,-k .oil i , j, k =B the flow of phase water and then r   o o w Q wflowing + Q w For example, suppose inj = bottom hole we specify the total flow, QT. Bpressure). w We need to decide how this total flow is made up - i.e. what are the separate Qo and Qw 3 Bo Q o (QT = Qo +QQ ) and how this is3allocated from each of the layers in the system. = +QQflow w w inj w = = + − Pwfkwell flowing pressure, Q Q PI wkis) P ( ) (PIthat T ok the assumption wk ok +there BQ For simplicity, weBmake a ksingle w o k =1 o k =1 Q Q > + Pwf (i.e. Pwf1w=injPwf2 =BPwf3 = Pwf4w). Hence, for each layer, k :

(

( (

∑

)

)

(

(

∑

)

)

w

B Q Q wTkinj=>(Q oko +oQ+wkQ) w= ( PI ok + PI wk )( Pk − Pwf ) Bw

(31)

Everything is known in the above equation which allows us to determine the allocation Pwfcan Q o from PI o i , layer , j, k −we j, k . P i and i , j, k calculate the corresponding bottom hole i , j, k = each of all fluids flowing pressure, Pwf.

(

)

Bo Q o we could specify the well flowing pressure, P , and then In a very Q similar= manner, + Qw wf w inj Bw flows, Qok and Qwk etc. in each layer. calculate the individual

34

Q w inj >

Bo Q o + Qw Bw


Well completion

(a)

4

Layer 1 2 3 4

(b)

QT = Qo + Qw Qo1 Qw1 k1, h1, P1, Sw1 Qo2 k ,h ,P ,S Qw2 2 2 2 w2

h

Figure 34 Well modelling of twophase flow in a multilayered system

Qo3 Qw3 k3, h3, P3, Sw3

Q ok = PI ok ( Pk − Pwfk ) =

2 π( khk ro (So ))k

r  µ o .ln ek  4.4 Modelling Horizontal Wells rw 

(P

k

Qo4 Qw4 k4, h4, P4, Sw4

− Pwfk )

Figure 35 shows the trajectory of a “horizontal” well in a reservoir simulation model. This is not well represented by the purely radial r/z model grid discussed 3 3 in Section 2.1 above in the context of a vertical well. Hence, it is less likely that Q T = (Q ok + Q wk ) = ( PI ok + PI wk ) Pk − Pwfk the well connection factors calculated as shown in previous sections will apply for k =1 k =1 a horizontal well. This is broadly true although the basic principle is very similar. That is, each well sector intersects a grid block (i,j,k) even although the well may be Q going through this block in say the x(i) direction and the flows between the well Tk = (Q ok + Q wk ) = ( PI ok + PI wk ) Pk − Pwf sector and the grid are given by an expression of the form:

∑

(

∑

(

(

Q o i , j, k = PI o i , j, k . P i , j, k − Pwf i , j, k

)

)

)

(32)

where the actual of the productivity index expression, PIoi,j,k , may be rather more Bo Qform o Q w inj = + Qwell complex since (a) the may intersect the block in a more complex way and (b) w B w the aspect ratio of the block is rather different when a horizontal well intersects it in that the x-direction well is very close to the z-boundaries since Δz is often smaller than Δx or Δy. B Q

Q w inj >

o

Bw

o

+ Qw


35

Figure 35 Cartesian grid cut from a 3D reservoir model showing two horizontal wells going through the system; two vertical wells also shown.

4.5 Hierarchies of Wells and Well Controls

Simple well control can be understood in terms of the well models discussed above. For a single well, we can essentially set the well flowing pressure and then calculate the flows or vice versa but not both and with certain constraints (see above). Alternatively, we may set the wellhead pressure and then calculate the Pwf from the - calculated or input - well formation to surface pressure drops etc. We now consider controls on pairs, then groups and then clusters of groups of wells in a field - indeed, we can even couple together the wells from several reservoirs and set more global controls and this will be described briefly. For a simple injector/producer well pair, for example in a 2D x/z cross-section, we can apply a range of well controls. Suppose this is a simple waterflood in an oil/water system. One of the most common controls is to fix the water injection rate at the injector but with (upper) limits on the well flowing pressure. The corresponding producer is then controlled by setting the bottom hole flowing pressure and then allowing the calculation of the oil and water phase flows (Qo and Qw). The volumetric production will be approximately balanced with the total production volume being of the order of the injected water volume - but not quite the same. Do you know why this is? Clearly the formation volume factors (Bo and Bw) will affect the exact production volumes; when we are injecting water and producing 100% oil, the reservoir volume of injected water per day is Qw.Bw and this will displace (virtually) the same reservoir volume of oil. The volume of oil produced per day is Qo stb which is actually Qo.Bo reservoir bbls, equating these reservoir volumes shows that if we inject water at a rate of Qw (stb/day), we produce oil at a rate of Qw.Bw/Bo stb/day. Since Bo > Bw, then the volumetric production rate of oil (in stb/day) is lower than the injected water injection rate (in stb/day). This must be taken into account in considering well control by voidage replacement as discussed below. If we wish to set injected Qw to precisely voidage replace whatever is produced, then we can do so to a good approximation by noting that if the production rate of oil

36

ok

ok

(

k

wfk

)

r  µ o .ln ek   rw 

(

k

wfk

)

Gridding And Well Modelling 3

3

Q T = ∑ (Q ok + Q wk ) = ∑ ( PI ok + PI wk )( Pk − Pwfk ) k =1

k =1

4

Q Tk = (Q ok + Q wk ) = ( PI ok + PI wk )( Pk − Pwf )

(

)

and water in our simulation is currently, Qo and Qw. What volume of injected water 2 π khk ro (So ) k must Q we inject to exactly replace the reservoir volume these two phases? This is Pk − Pof ok = PI ok Pk − Pwfk = wfk   r now quiteQstraightforward since and, from the above discussion, it is evident that the = . P − P PI , , i j k ek o i , j, k wf i , j, k o i , j, k µ o .ln Q  (in stb/day) is given by: quantity of water that must be injected,  rw inj

(

)

(

)

(

)

w

B Q Q w3inj = o o + Q w 3 B Q T = ∑ (Q ok + wQ wk ) = ∑ ( PI ok + PI wk )( Pk − Pwfk ) k =1

(33)

k =1

Hence, we would gradually adjust the volume of water injected in the simulation Bo Q o Q w inj + Q wjust produced (at the last time step say) to the above model based on>what we have Bwwk ) = replace. Q Tkin=order Q Pk − Pwf the most common option would (Q okto+voidage (PI ok + PI quantity Atwk the) producer, be to constrain by bottom hole flowing pressure as described above.

(

(

)

)

P i ,less Pwf i , j, k to constrain all wells by volumetric injection/ PI o i , j, k .but Note Q that j, k −common o i , it j, kis=possible production rate. We can see why if we consider an incompressible fluid where it is clearly impossible for the injection and production rates to be different since the B go Q to + ∞ or - ∞, depending on whether we over- or under-injected, pressure would Q w inj = o o + Q w respectively. Although it is possible to specify different volumetric flow rates at Bw injector and producer for a compressible fluid, this can only be done within very tight limits and the pressures tend to go to unrealistic limits e.g. if we over-inject Bo Q o

+ Q w ), the pressure tends to rise to unphysically high levels (i.e. Q w inj > B w well above fracture pressure of the reservoir rock. 4.5 CLOSING REMARKS - GRIDDING AND WELL MODELLING In this section, the student has been presented with a largely non-mathematical description of gridding and well modelling in reservoir simulation. A more mathematical treatment of these issues will be given as we develop the flow equations and consider their numerical solution in Chapter 5 and 6, respectively.


37

5

The Flow Equations

CONTENTS 1

INTRODUCTION

2

THE SINGLE PHASE PRESSURE EQUATION 2.1 The Physics of Single Phase Compressible Systems 2.2. The Single Phase Pressure Equation 2.3 The Simplified Compressible Pressure Equation 2.4 Extension of the Single Phase Pressure Equation to 2D and 3D 2.5 Mathematical Shorthand for the 3D SinglePhase Pressure Equation

3

THE TWO-PHASE FLOW EQUATIONS 3.1 Review of Two-Phase Flow Concepts 3.2 Derivation of the Two-Phase Conservation Equations 3.3 The Two-Phase Pressure Equation 3.4 Schematic Strategy for Solving the TwoPhase Pressure and Saturation Equation 3.5 The Simplified Pressure and Saturation Equations

4

CLOSING REMARKS

LEARNING OBJECTIVES: Note that in this chapter, there are a number of lengthy derivations of the flow equations. The student should work through all of these in detail and understand each step in the process. In the learning objectives below, we clarify what parts of this the student should know and be able to reproduce and/or apply in an exam question and what parts need only be appreciated or understood at a general level.

Having worked through this chapter the student should: • know that the central principles in deriving the flow equations in porous media are (a) to apply material balance to the flow and accumulation terms; and (b) to then apply Darcy’s Law. • know how to apply material balance to a control volume with flow and accumulation terms, recognising the difference between mass flow rates and volumetric flow rates. • understand be able to describe the basic physics of single phase compressible flow through porous media. • be able to derive the (pressure) equation for single phase compressible flow. • know physically, in the context of the single phase compressible flow equation, what a non-linear partial differential equation (PDE) is and why it is difficult to solve. • be able to work through all of the simplifying assumptions on the single phase compressible flow equation to arrive at the much simpler pressure equation (linear PDE) for slightly compressible flow involving the hydraulic diffusivity, Dh =k/(μφcf) • understand the extension of the single phase pressure equation to 2D. • appreciate (but not memorise) the use of the gradient (∇) and divergence (∇.) operators from vector calculus in writing the multi-phase flow equations. • be able to apply conservation + Darcy’s law in the two phase case to arrive at the two phase flow equations for compressible fluids and rock. • follow the argument on how the full two phase pressure equation is derived. • be able to identify the relation of each of the terms in the two phase pressure equation to viscous, gravity and capillary forces. • be able to reproduce (in words and diagrams) the outline solution scheme for the two phase pressure equations for both the full equations and for the simplified pressure and saturation equations.

2

5

The Flow Equations

BRIEF DESCRIPTION OF CHAPTER 5 The central activity of reservoir simulation is the numerical solution of the multiphase flow equations in real reservoir systems. This chapter introduces the underlying equations of flow through porous media which are solved in a reservoir simulation code. We start with the single phase flow equation for a compressible fluid and go on to develop the two-phase flow equations. The same principle is used in the derivation of these equations viz. the application of mass (material) balance between flow and accumulation and then the application of Darcy’s Law. We show that the flow equations are non-linear partial differential equations (PDEs) which can usually not be solved analytically. However, some simplified cases of the equations are considered in order to establish some important physical insights into these equations.

1. INTRODUCTION In the course so far, we have used many of the concepts of day to day numerical simulation as it is applied in reservoir engineering. However, we have not yet studied the mathematical equations which underpin the whole subject of reservoir simulation. This section will introduce you to these equations in a step-by step manner. You should be able to follow all the details of the derivations of these equations since only elementary undergraduate mathematics is involved. The general approach to the flow equations is basically the same for both single and two-phase flow. We first derive a mass balance between the flows and the accumulation (local mass build up or decline) in a local control volume. A control volume can be thought of as a typical isolated grid block in the system, as will be evident below. We note that the mass balance equation that we will set up must be correct. That is, it is simply a mathematical way of expressing something that is a fact. Mass balance simply states that over a given time period (say, Δt), then the sum of all the mass that flows into (+) the system and out of it (-) is the change of mass in that block. Once we have set up the mass balance between the flows and the accumulations, we then need some expressions to describe these mass flows. We do not usually think of fluid flow laws as being mass flow laws. Within a porous medium, the principal flow law is Darcy’s law (either for one or two phases) which is a volumetric flow law. That is, in Darcy’s law, the volumetric flow rate, usually denoted Q, is proportional to the pressure gradient; Q ~ -(dP/dx) for a single one dimensional system oriented along the x- direction. (Note the minus sign here since fluids flow down the pressure gradient, from high P to low P). So, to derive the flow equations we simply use mass balance + Darcy’s law. This is shown schematically for single phase fluid flow in Figure 1, where the operator, Δt, implies the difference in the quality (pφΔxΔyΔz) between times t and t + Δt (of course, only the p and φ quantities can change).


3

Mass in block (i,j) = ρ.φ.∆x.∆y.∆z

i,j+l

Change in mass over ∆t

 4  ∆m =  ∑ (ρ.Q) m .∆t m =1 

Qj+l/2 Qi-l/2

i,j

where m counts over the 4 neighbours. And Darcy's law is:

Qi+l/2

i-1,j

i+1,j Qj-l/2 i,j-l

Qi −1/ 2

 kA ( Pi , j − Pi −1, j ) in x direction = −   .  µ  i−1/ 2 ∆x

Therefore:

 4  ∆t (ρφ∆x∆y∆z) =  ∑ (ρ.Q)m .∆t m=1  where we then substitute Darcy's law

In Section 2, we start by deriving the single-phase flow equation for a compressible system. This is essentially a pressure equation since this is the only quantity we need to find. The pressure distribution in space is the main unknown in the system and we need to find this as a function of time as the system evolves; pressure is denoted, P(x,y,z;t), for a three-dimensional (3D) system. If we have a way of finding P(x,y,z;t), we can then find the flow rates from the gradient of this quantity i.e. we use Darcy’s law. This sort of “completes the circle” since we use Darcy’s law to derive the pressure equation and, once we solve for the pressure, we find the flows from Darcy’s law etc. Although we can obtain the single-phase pressure equation for a compressible fluid/ rock system in 1D, it turns out to be a non-linear partial differential equation (PDE). There is no known analytical solution to this equation for the general case. That is, even the apparently simple single phase pressure equation cannot be solved using “well known” mathematical functions. You may remember that we discussed the idea of an analytical model in the opening section of this course (Chapter 1 Section 1). If we are faced with this difficult non-linear PDE for the pressure there are two things we might do to solve it as follows: (i) we could possibly use a numerical method to solve it; or (ii) we may be able to simplify the equation by making various assumptions that then allow us to solve the equation analytically. In fact, we will take the latter course of action at this point. (We return to the topic of numerical solution of the flow equation in Chapter 6). Section 2 describes how we simplify the full equation for a compressible fluid in order to obtain an equation that we can solve. Indeed, the simplified equation that arises is none other than the main equation used in single phase well testing. In Section 3, we go on to derive the equations of two-phase flow in their complete form i.e. for a compressible fluid/rock system. Clearly, if the single-phase pressure equation is not analytically soluble in general, then the two-phase equivalent will certainly not be soluble since it is even more complicated (again, except for some special simple cases). In the two-phase equations, we also have a pressure equation but, in addition, there is also a saturation equation. Other features are also different from single phase flow, 4

Figure 1 The basic principles of Mass Balance + Darcy’s Law as the basis for all equations for flow through porous media; the Qs are volumetric fluid flows.

5

The Flow Equations

such as the capillary pressure, Pc (Sw), leading to two different phase pressures, Po and Pw (where Pc (Sw) = Po-Pw). The two-phase Darcy law also introduces the concept of relative permeability into the flow equations. These concepts have been fully reviewed in Chapter 2 and they are also explained in the Glossary.

2. THE SINGLE PHASE PRESSURE EQUATION 2.1 The Physics of Single Phase Compressible Systems

Before deriving the single phase pressure equation, consider first the “physics” of what is happening in a compressible single-phase system. Figure 2 shows a long thin “reservoir” containing compressible fluid and rock, which we can consider as being essentially one-dimensional (1D). The fluid and the rock have compressibilities, cf and cr (where cf >> cr; see Glossary). Imagine the “reservoir” is horizontal and has two wells in it at either end as shown in Figure 2a. We take the location of Well 1 as being at x = 0, and Well 2 is located at x = L. If the wells are both shut-in and the reservoir is left to achieve steady-state, then clearly the pressure in the system will reach a constant value, Po (ignoring the gravitational potential since the reservoir is also “thin” in the z-direction). The bottom hole pressure also equalises in both wells and will also be Po. This is shown as the dashed line (at t = 0) in Figure 2b where the “pressure profile” through the reservoir, P(x)=Po, is also shown. Injector Well

(a)

Producer Well Long thin (1D) reservoir

Figure 2 (a) Schematic of a long thin “reservoir” containing compressible fluid and rock, which we can consider as being essentially one-dimensional (1D); (b) pressure profile along the system at various times from t = 0, t1 , t2 etc. - this shows the pressure disturbance travelling (“diffusing”) along the system.

t=t1

(b) Pin

t=t2

t=t3

P Po

Pout = Po 0

x

L

t=0

Now consider what happens to the pressure profile if we cause the pressure to rise suddenly at Well 1, say by injecting fluid (identical to that already in the reservoir). Is this pressure disturbance immediately “felt” at Well 2 at x = L? The answer is of course “no” since the pressure wave will take some time to transmit from Well 1


5

to Well 2. At a short time after this disturbance, at Well 1 say at t = t1, the pressure profile, P(x,t1), is shown schematically in Figure 2b. Likewise, at a later time, t2, it is shown as, P(x,t2). But what happens at Well 2? This depends on what we do at Well 2: Suppose, we leave it shut-in, what will eventually happen? If we were injecting a volumetric flow rate, q1(t), into Well 1 such that it maintained a constant pressure, P1(x = 0,t); i.e. we would be varying (reducing) q1 with time, the outcome is quite predictable. We would keep pumping in at (decreasing) rate q1(t) until we had “blown-up” the pressure in the reservoir – like pumping up a car tyre – to pressure P1. Hence, the q1 value would reduce to q1 = 0 and the reservoir pressure – and the pressure in both wells – would settle to P(x,t) = P1. On the other hand, suppose we simply pumped into Well 1 at constant rate q1(now fixed) and we held the bottom hole pressure of Well 2 to be constant at P(x =L) = Po (the original reservoir pressure). What would happen for this case? With a little thought, you can probably visualise that the sequence of events would be as follows: • Firstly, at early times up to t2 and a little later, nothing would happen at Well 2. It would still have pressure Po and would not be flowing; • The pressure “wave” would propagate or “diffuse” through the reservoir as shown in Figure 2b; the “speed” of this wave is governed by the “diffusivity” which is quantified by k/(cfφμ) as we will show later in this chapter. Note that the larger the fluid compressibility (cf) or porosity (φ), the “slower” the wave propagates. • At a certain time, denoted, t3, the pressure disturbance just reaches the produced Well 2 as shown in Figure 2b; • At t = t3, Well 2 must start to flow in order to maintain the pressure at P = Po (this works like a back-pressure regulator); • As time proceeds (t → ∞), the pressure wave starts to be felt all across the reservoir such that the flowrate at Well 1 is (still fixed) at q1 and the pressure settles to a constant value P1; • Likewise, as t → ∞, Well 2 (still at fixed pressure Po) will go to steady-state flow exactly at flowrate, q1; • At t → ∞, the pressure profile P(x,t → ∞) will tend to a straight line as shown by the inclined dashed line in Figure 2b. This “thought experiment” is extremely useful to appreciate the physics of flow in a compressible system. It also introduces the ideas of boundary conditions. That is, not only do we need a way of modelling the development of pressure in our reservoir (i.e. finding an equation for P(x,t)), we also need to appreciate what boundary conditions to apply in a given situation. We have been quite explicit here in identifying the boundary conditions with different well constraints. For example, for the main case considered above, Well 1 is a rate-constrained injector and Well 2 is a pressure–constrained producer. 6

5

The Flow Equations

EXERCISE 1. Can you explain the sequence of events that would occur in the simple 1D system of Figure 2 if both wells were pressure constrained to Pin (Well 1) and Po (Well 2) with Pin > Po?

2.2 The Single Phase Pressure Equation

We now return to the task of deriving the single-phase flow equations for a compressible system. Throughout this derivation we refer to the control volume shown as block i in Figure 3 where certain terminology is also explained. In Figure 3, we have divided up the x-axis into increments of (constant) size, Δx, and constant cross-sectional area, A. Flow is considered to be in the positive x-direction (i increasing). The fluid has density, ρ, which may depend on pressure i.e. ρ(P). The porosity is denoted as φ and it too may depend on pressure, φ(P); in block i, the porosity is, φi. The volumetric flows across the boundaries of block i are given by: qi-1/2 and qi+1/2 as shown in Figure 3; the dimensions of these quantities are volume/time and typical units might be STB/day, m3/sec etc. Let us now apply the mass conservation conditions as were applied in the introductory section of this chapter. Mass conservation states that: The mass accumulation (increase or decrease) in block i over a time step, ∆t

=

The mass that flows IN over time ∆t

-

The mass that flows OUT over time ∆t

(1)

We can easily write mathematical expressions for each of the terms in equation 1 as follows:  Volumetric rate flowing IN to block i =  of in flow to    block i over time ∆t   The mass of fluid

 density of  x   x ∆t = (q.ρ)i- 1 2 .∆t (2a)  the fluid 

Llikewise: The mass of fluid flowing OUT of block i over time Δt

= (qρ)i+1/2.Δt


(2b)

7

Boundary i-1/2 Boundary i+1/2

i-1

Blo

i

ck

q

i-1 /

2

i+1

q

i+1 /2

∆x

x ∆x ∆x Notation: 1

The boundaries of block i are denoted (i-1/2) between blocks (i-1)and i (i+1/2) between blocks i and i+1

2

q(i-1/2) denotes the volumetric flow rates across the (i-1/2) boundary q(i+1/2) denotes the volumetric flow rates across the (i+1/2) boundary 3 Note – units bbl/day, m /s etc…

3

The porosity of block i = φi, the permeability is ki etc..

Figure 3 Control volume grid block for application of material balance for the single phase flow equation; notation is given below.

Therefore, the change in mass of fluid in block i over time Δt is given by: Change in mass over Δt in block i =

(qρ)i - 12 .∆t - (qρ)i + 12 .∆t

[(qρ) = − [(qρ) =

] ]∆t

i − 12

− (qρ)i+ 1 .∆t

i + 12

− (qρ)i - 1

2

2

(3)

Thus, equation 3 expresses the change in mass due to flow that occurs in block i using quantities – volumetric flow rates and densities – defined on the boundaries. Note that we have changed the signs consistently in the final step of equation 3 for convenience below. Which quantity do you think we mean when we refer to ρi-1/2 : the density “on the boundary”, i-1/2? This seems a bit strange. However, you can think of ρi-1/2 being some sort of “average” density between ρi-1 and ρi, the densities in blocks (i-1) and i, respectively. In fact, we have discussed this matter of grid block to grid block average in Chapter 4. We now turn to the alternative way of expressing the mass of fluid in a grid block i at the different times, t and t+Δt. This is shown in Figure 4 where we denote the mass of fluid in grid block i at times t and (t + Δt) by (ρφΔxA)t and (ρφΔxA)t+Δt , respectively. Therefore:

8

5

The Flow Equations

The change of mass of fluid in block i over the time from t to t+Δt

=

The mass at - The mass (t + Δt) at t [(ρφ)t+Δt - (ρφ)t].ΔxA

=

(4)

where we have used the fact that Δx and A are constants in equation 4.

Ro ck Po Sp re ac e ∆x

Area = A

The mass of fluid in block i

Figure 4 Expressions for the mass of single-phase fluid in a grid block

=

(Pore volume of block i) x density

=

(Volume of block x porosity) x density

=

(∆x.A.φ.)ρ

=

ρφ.∆x.A

We now apply the material balance condition (equation 1) by equating the expressions in equations 3 and 4 as follows:

[(ρφ )

t + ∆t

]

[

]

− ( ρφ )t .∆xA = − (qρ )i+ 1 − (qρ )i- 1 .∆t 2

2

(5)

Now divide through equation 5 by the (constant) Δx.AΔt to obtain:

[(ρφ )

t + ∆t

- ( ρφ )t

∆t

]

 qρ   qρ −      A i+ 1 2  A  i − 1 2   =− ∆x

(6)

where we have taken the (constant) area, A, to the inner parenthesis on the RHS of equation 6. As it stands, equation 6 is exactly true since it is simply a statement of mass balance. There are no assumptions in this equation. Indeed, we can simplify equation 6 a little more by noting that q/A is the Darcy velocity, u. This becomes:

[(ρφ)

t + ∆t

- (ρφ)t

∆t

] = − [(u.ρ)

i+ 1 2

- ( u.ρ)i- 1

∆x


2

]

(7)

9

which is still an exact statement of fact. We can take the limits of the difference equation 7 at Δt → 0 and Δx → 0 to obtain the equivalent differential equation. Clearly:

Lim. ∆t → 0

[(ρφ )

t + ∆t

- ( ρφ )t

∆t

]

=

∂( ρφ ) ∂t

(8)

and

[

]

(u.ρ )i+ 12 - (u.ρ )i − 12 ∂( u.ρ ) Lim. − = − ∂x ∆x → 0 ∆x

(9)

and therefore:

∂(ρφ) ∂( u.ρ) = − ∂t ∂x

(10)

Equation 10 is the differential form of the conservation equation and again, it is “exact” in the sense discussed above. Before going on to use Darcy’s law (for u), look at the structure of equation 10; in particular look at the symmetry between ρ and φ. These two quantities appear in an identical manner in the LHS of this equation. Hence, if the mass in a grid block stayed the same with time, i.e. ∂(ρφ)/∂t = 0, then this could be because the fluid density went down (the fluid expands) and the rock porosity went up (the rock contracts or compresses). We now assume Darcy’s law for u, that is:

u=-

k  ∂P  .  µ  ∂x 

(11)

(as given in the Glossary etc.). We may substitute this form of the Darcy law directly into equation 10 (taking care with the signs) to obtain:

∂(ρφ) ∂  ρk  ∂P   =   ∂t ∂x  µ  ∂x  

(12)

This equation is now inexact, in that its validity depends on whether Darcy’s law is or is not a good assumption. However, it was necessary to use a flow law such as Darcy’s law since pressure (i.e. P(x,t)) does not explicitly appear in equation 10. Thus, Darcy’s law is our “link” between fluid velocities and pressure gradients. Equation 12 still does not have pressure (P(x,t)) explicitly shown on the LHS; it simply has a term ∂(φρ)/∂t. However, we know that φ and ρ depend locally only on pressure i.e. they can be written ρ(P) and φ(P). We therefore manipulate the LHS of equation 12 as follows using the chain rule of differentiation:

∂(ρφ) ∂(ρφ)  ∂P  = .  ∂t ∂P  ∂t  10

(13)

5

The Flow Equations

Thus, substituting equation 13 back into equation 12 we obtain a “true” single phase pressure equation as follows:

 ∂( ρφ )   ∂P  ∂  ρk  ∂P   .    .  =  ∂P   ∂t  ∂x  µ  ∂x  

(14)

where we can consider the term ∂(φρ)/∂P as a generalised fluid and rock compressibility, term, C(P). Equation 14 would then become:

 ∂P  ∂  ρk  ∂P   .  C( P ).  =  ∂t  ∂x  µ  ∂x  

(15)

Some points should be noted about equation 15, as follows: (i) It is a non-linear partial differential equation (PDE). By “non-linear” we mean that the coefficients in the equation – the “input”, if you like- depend on the quantity we are trying to find, the unknown pressure, P(x,t). In other words, quantities such as C(P), ρ(P), μ(P) etc. “depend on the answer”. We will not go into mathematical detail but such problems are notoriously difficult to solve analytically for general cases. (ii) Following on from point (i) above, such equations can only usually be solved by one or two approaches: • By solving them numerically where we can handle the non-linearities using certain types of iterative methods (see Chapter 6). • We may simplify the equation to the extent that it becomes soluble. Clearly, by its nature, a numerical solution will be approximate although it is an approximation to the full equation (equation 15, in this case). With a simplified equation, we may have an exact solution, but the simplifications may have “thrown away” some of the important physics. In fact, in the following section, we will take the second approach. (iii) Going back to equation 10, we could include gravity effects by taking the Darcy equation with gravity as follows:

u=−

∂z  k  ∂P    − ρg  µ  ∂x  ∂x 

(16)

(instead of equation 11). If the 1D system is at constant incline, z(x), then (∂z/∂x) = cos θ where θ is the angle of incline, therefore, we obtain equation 17 below:

u=−

k  ∂P     − ρg cos θ  µ  ∂x  


(17)

11

This could simply lead to the generalised single phase equation as follows:

 ∂( ρφ )   ∂ρ  ∂  ρk  ∂ρ     − ρg cos θ    .  =  ∂x  µ  ∂x   ∂P  ∂t 

(18)

but this is fairly straightforward.

2.3 The Simplified Compressible Pressure Equation

In this section, we will take equation 15 as our starting point and introduce some simplifying assumptions. Our hope is to get to a simpler equation which will have analytical solutions for given boundary conditions. Indeed, for this to be the case, we would normally aim to derive a simplified linear PDE. In fact, to make clear what is happening to the LHS of the equation we start from equation 14, which is repeated below:

 ∂( ρφ )   ∂P  ∂  ρk  ∂P   .    .  =  ∂P   ∂t  ∂x  µ  ∂x  

(14)

We first list the assumptions which we will make in the above equation and then we will go on to examining their consequences. Simplifying Assumptions 1 2 is 3

Viscosity, μ, is constant (with x and P); Permeability and porosity, k and φ, are constant, (with x and P), i.e. the system homogeneous and the rock is incompressible; That pressure gradients, (∂P/∂x), are “small” such that: 2

 ∂P    ≈0  ∂x 

4 The fluid has a constant compressibility, cf, i.e. cf =

1  ∂ρ    ≈ cons tan t . ρ  ∂P 

Assumption 1 is probably quite reasonable since viscosity does not vary greatly for most oils (or water) over small pressure ranges. Assumption 2 is quite drastic since it says that permeability (k) is constant through the reservoir i.e. that the system is homogeneous in k (and φ). For a real system, this is indeed a very simplifying assumption. However, we will indicate below how we can turn this assumption round, in a sense. The second part of Assumption 2 is that the rock is incompressible and this is quite reasonable (usually, cf >> crock). Assumption 3 is rather odd: it is clearly designed to get rid of “difficult” terms with terms like (∂P/∂x)2 in them. Assumption 4 – fluid compressibility, cf, does not vary with pressure, is again quite reasonable for most (non-critical) reservoir oils. Applying Assumptions 1 and 2 results in the following simplification of equation 14:

12

5

The Flow Equations

 ∂ρ   ∂P  k ∂   ∂P   φ   .  = ρ   ∂P   ∂t  µ ∂x   ∂x  

(19)

which rearranges quite simply to:

 µφ   ∂ρ  . ∂P  = ∂ ρ. ∂P         k   ∂P   ∂t  ∂x   ∂x  

(20)

Now expand the RHS of equation 20 as follows using the product rule:

 ∂2 P  ∂   ∂P    ∂ρ   ∂P  ρ = . + ρ       2  ∂x  ∂x   ∂x    ∂x   ∂x 

(21)

and use the fact that ρ is a function of pressure only, i.e. ρ(P), to further expand the (∂ρ/∂P) in equation 21 as follows:

 ∂2 P  ∂   ∂P    ∂ρ   ∂P   ∂P  ρ  =   .  .  + ρ 2   ∂x  ∂x   ∂x    ∂P   ∂x   ∂x 

(22)

Equation 22 can be then simplified as follows: 2

 ∂2 P  ∂   ∂P    ∂ρ   ∂P  ρ = . + ρ        2  ∂x  ∂x   ∂x    ∂P   ∂x 

(23)

Now use Assumption 3 to eliminate the first term on the RHS of equation 23 above 2

 ∂P   ≈ 0 ) to obtain:  ∂x 

(i.e. 

2  µφ   ∂ρ   ∂P  = ρ ∂ P      2  k   ∂P   ∂t   ∂x 

(24)

Note that we can divide both sides of 24 by ρ and use the fact that the fluid has constant compressibility, cf (assumption 4), to obtain: 2  µφc f   ∂P  =  ∂ P     2  k   ∂t   ∂x 

(25)

where the coefficient (μφcf/k) is a constant. This is more commonly written with the constant on the RHS as follows: 2  ∂P   k   ∂ P  .   =  ∂t   (µφc f )   ∂x 2 

(26)

where the constant in this form is normally referred to as the hydraulic diffusivity, Dh=k/(μφcf).


13

Equation 26 is now a linear PDE and has the form of a linear diffusion equation as follows:

 ∂2 P   ∂P  = D    h  ∂t   ∂x 2 

(27)

Note the following about the above equations (26 or 27): (i) It is the simplified (slightly) compressible 1D flow equation in a linear porous media i.e. in Cartesian form. (ii) Analytical solutions are available for a range of boundary conditions (see Crank, The Mathematics of Diffusion, 2nd edition, Oxford Clarendon Press, 1975). (iii) In its radial form (i.e. in radial coordinates, r, rather than x) equation 27 becomes the following:

 ∂P  D h ∂  ∂P   = r   ∂t  r ∂r  ∂r 

(28)

which is a well known equation of well testing (Stanislav and Kabir, 1990). Likewise, this equation has a number of well known analytical solutions for various boundary conditions. (iv) The reason these equations have many ready-made analytical solutions available is because these diffusion equations are well-known and are identical in form to the equations of heat conduction which have been studied for many years (Carslaw and Jaeger, 1959).

2.4 Extension of the Single Phase Pressure Equation to 2D and 3D

For the 2D case, the control volume is now grid block (i,j), as shown in Figure 5. The flows are shown at the boundaries as before and are labelled as (i- 1/2) for flow from (i-1) → i in the x-direction, (j + 1/2) for the flow from block (i,j) → (i,j + 1) etc. as listed on Figure 5. The flow areas in the x- and y-directions are given by, Ax = (Δy.h) and Ay = (Δx.h), respectively, as shown in Figure 5. In addition, we show a source/sink term due to a (single phase) well. This well again injects or produces exactly the same fluid as is in the reservoir already. The well flow rate into block (i,j) is described by q˜ ij which is the volumetric flow rate per unit volume of block (i,j); possible units for q˜ ij are m3/ s or bbl/day etc. For this definition, the volume rate of flow into (sign +) or out of (sign -) the well in block (i,j) is as follows:  density of injected or Mass rate of flow into or out of block (i, j) = q˜ ij ×   × volume block (i, j)  produced fluid, ρij  = q˜ ij × ρij ∆x∆y.h

(29)

14

5

The Flow Equations

NOTATION i, j control volume i - 1/2 = (i-1) → i i + 1/2 = i→ (i+1) j - 1/2 = (j-1) → j j + 1/2 = j→(j+1)

Well Mij

y (j)

x (i)

i-1

i,j+

,j i,j

i,j-1

1

i+1

Thickness = h

,j

Note for the control volume, block (i,j), the flow areas in the x and y directions, Ax and Ay are given by:

h

Figure 5 The 2D x/y Grid Showing the Control Volume, Block (i,j)

Ay a,

x.h =∆

∆x

Are

a, A

∆y

x=

Are

∆y .h

We now apply the material balance equation as follows: Change in mass in block (i,j) over time Δt due to flows across boundaries and the well

[

] [

]

= (qρ )i − 1 − (qρ )i + 1 .∆t + (qρ ) j − 1 − (qρ ) j + 1 ∆t + q ij ρij ∆x∆y.h.∆t 2

2

(the x - flows)

2

2

(the y - flows)

(well source / sink term) (30)

The accumulation term is the same as previously (Figure 4) as follows: Change in mass in block (i,j) between time t and t+Δt

[

]

= (φρ ∆x∆y.h )t + ∆t −(φρ ∆x∆yh )t = (φρ )t + ∆t − (φρ )t ∆x∆yh (31) We then equate expressions in equations 30 and 31 to obtain:

[(φρ )

t + ∆t

[

]

− (φρ )t ∆x∆y.h

] [

]

= − (qρ )i + 1 − (qρ )i − 1 .∆t − (qρ ) j + 1 − (qρ ) j − 1 .∆t + q˜ ij ρij ∆x∆y.h∆t 2

2

2

2

(32) Institute of Petroleum Engineering, Heriot-Watt University

15

where we note that the signs have been adjusted slightly on the RHS. Dividing through by Δx.Δy.h.Δt and cancelling gives the following:

[(φρ)

t + ∆t

− (φρ)t

∆t

] = − [(qρ)

− (qρ)i − 1

i + 12

∆x.A x

2

] − [(qρ)

j+ 12

− (qρ) j − 1

∆y.A y

2

] + (q˜ ρ)

ij

(33) where we have used the fact that ΔxΔyh = ΔxAx and ΔxΔyh = Δy.Ay. The flow areas, Ax and Ay, may then be divided into the q’s to give Darcy velocity terms (e.g.

 qρ  ) as follows:   Ax  i+ 1

(uρ)i + 12 = 

[(φρ)

] = − [(u.ρ) 2

t + ∆t

− (φρ)t

∆t

− ( u.ρ)i − 1

i + 12

∆x

2

] − [(u.ρ)

j+ 12

− ( u.ρ) j − 1 ∆y

2

]

+ (q˜ ρ)ij (34)

Again, taking limits as Δt, Δx, Δy and → 0, gives:

( )

∂ u yρ (q˜ ρ) ∂( u xρ) ∂(φρ) − + =− ∂t ∂x ∂y

(35)

This is the 2D mass conservation equation is still exact in that no approximations have yet been made. Clearly, we can easily generalise this to 3D by simply adding a z-flow term (for vertical flows, uz) to give:

( )

∂ u yρ ∂(φρ) =− ∂t ∂x

−

(

)

∂ u y .ρ ∂y

−

∂( u z .ρ) + (q˜ ρ) ∂z

(36)

As before, the pressure P(x,y;t) (in 2D) or P(x,y,z;t) (in 3D) does not yet appear. We must manipulate the LHS of equation 35 (or 36) to yield, exactly as before:

∂(φρ) ∂(φρ)  ∂P  = .  ∂t ∂P  ∂t 

(37)

And on the RHS of equation 35 (or 36) we must use Darcy’s law. Darcy’s law in each of the 3 directions, x, y, and z, is given as follows, where we choose z as the vertical direction (of gravity):

ux = −

16

∂z  k x   ∂P     − ρg  µ  ∂x ∂x 

(38a)

5

The Flow Equations

uy = −

k y   ∂P  ∂z    − ρg   µ   ∂y  ∂y 

(38b)

uz = −

k z   ∂P      − ρg  µ  ∂z

(38c)

where we note that the permeability may be anisotropic i.e. kx ≠ ky ≠ kz. Using the above expressions (equations 38a-c) for the Darcy velocity along with equation 37 for the LHS in the 3D equations 36 gives:

∂(φρ)  ∂P  ∂  ρk x   ∂P   ∂z    ∂  ρk y   ∂P   ∂z    .  = − ρg      − ρg    +        ∂x    ∂P  ∂t  ∂x  µ   ∂x  ∂y  µ   ∂y   ∂y    +

 ∂  ρk z   ∂P     − ρg  + (q˜ ρ)  ∂z  µ   ∂y    (39)

Equation 39 is the 3D generalisation of equation 14 including gravity and also a well (source/sink) term. Again, this is a 3D non-linear partial differential equation (PDE) which cannot generally be solved analytically. Note also that the equation is rather lengthy and it would be useful if there was a more shorthand way of writing this equation. This is dealt with in the following section.

2.5 Mathematical Shorthand for the 3D Single-Phase Pressure Equation

In order to write equation 39 in a more compact form we must use the notation of vector calculus. Before doing this, we review the following mathematical concepts: Mathematical Concepts Review: Review the meaning of: (i)

The divergence operator, ∇., on a vector V:

∂V ∂V ∂V ∇.V = x + y + x ∂V∂x ∂V∂y ∂V∂z ∇.V = x + y + x ∂x ∂y  V ∂z x

 V  where vector V = V  x y     where vector V = V y z  V    V  z

(ii) The gradient operator, ∇, on a scalar, such as pressure, P

 ∂P   ∂P   ∂P  ∇P =   i +   j +   k  ∂ P∂x   ∂ P∂y   ∂ P∂z  ∇P =   i +   j +   k  ∂x   ∂y   ∂z 


17

where i, j and k are the unit vectors in the x-, y- and z-directions. (iii) see also operations on a tensor (Chapter 2, Section 3.2). Using this notation, equation 35 (the exact conservation equation) becomes:

∂(φρ) = −∇.(ρu) + (q˜ ρ) ∂t

(40)

or expanding the LHS of equation 40 as before:

 ∂(φρ)   ∂P      = −∇.(ρu) + (q˜ ρ)  ∂P   ∂t 

(41)

After the Darcy law has been used – as in equation 38 – the vector calculus version of this equation becomes:

 ∂(φρ)   ∂P  ρ    .  = ∇. k.(∇P − ρg∇z) + (q˜ ρ)  ∂P  ∂t µ 

(42)

We will not expand on equation 42 here but it can be seen that it does give a compact way of writing the 3D pressure equation. This approach can also be used to write the 3D multi-phase flow equations, as we will develop below. Exercise 2 The following equation is the full 2D equation for a compressible fluid (in the absence of gravity and with no well terms):

∂(φρ)  ∂P  ∂  ρk x  ∂P   ∂  ρk y  ∂P   .  +  =    ∂P  ∂t  ∂x  µ  ∂x   ∂y  µ  ∂y   Simplify this equation as far as possible by making all the Assumptions (1) - (4) in Section 2.3 but keep kx ≠ ky (although both are constant).

3. THE TWO-PHASE FLOW EQUATIONS 3.1 Review of Two-Phase Flow Concepts

We now consider the equations which govern the flow of two phases through a porous medium e.g. oil-water, gas-oil, air-water. In certain respects, the approach is very similar to that used for single-phase flow in Section 2 above. We apply the mass conservation equation to each of the phases separately. However, there is a little more “two-phase physics” that we must be aware of before proceeding. For this reason, we first review some of the key concepts from two-phase flow (see Chapter 2 or the Glossary).

18

5

The Flow Equations

Key Concepts: Ensure you are familiar with the following main ideas on two-phase flow (where we assume oil/water in the examples below): • Phase saturations, So and Sw; where So + Sw = 1 • Formation volume factors, Bo and Bw (units RB/STB) where, for example, ρo = ρosc/Bo and ρosc is the oil density at standars conditions (likewise for ρw). • The two-phase Darcy Law (with gravity) and relative permeabilities, krw(Sw) and kro(So)

uo = -

k k ro µo

 ∂Po   ∂z    ∂x  − ρo g ∂x    

uw = -

k k rw µw

 ∂Pw   ∂z    ∂x  − ρw g ∂x    

• Phase pressures, Po and Pw, and the concept of capillary pressure, Pc (Sw) = Po–Pw, as a constraint of the phase pressure difference at various saturations, Sw. Exercise 3 Calculate the mass of oil and water phases for the control volume (grid block) shown using the usual notation.

∆z

Ans: Mass oil = Volume of oil x density of oil = (∆x∆y∆z φ So) x ρo

Ro c

k Oil ,S o Wa ter ,S

∆x

w

Mass oil = ∆x∆y∆z ( φSoρo) = ∆x∆y∆zρosc (φ So/Bo)

∆y

Likewise: Mass water = ∆x∆y∆z (φSwρw) = ∆x∆y∆zρwsc (φ Sw/Bw)

3.2 Derivation of the Two-Phase Conservation Equations

As for the single phase case, we will use the material balance in a control volume as shown in Figures 6 and 7. Definition: A very useful quantity to define is the oil flux, Jo, and the water flux, Jw. The oil (water) flux is the mass rate of flow of oil (water) per unit cross sectional area. Dimensions are M.L-2.T-1 and possible units would be kg.m-2.s-1 or lbs.ft-2.day-1. The oil flux, Jo (and Jw) can easily be related to other more familiar quantities as follows: consider the flow area across the block area shown below.


19

Oil Flux:

a,

Are

A

ρ .q

z y.∆ =∆

Jo= oA o =

il ic Oe r t e at lum w R o V Flo qo

∆z

∆y

ρosc.uo Bo Figure 6 Definition of oil (water) flux, Jo .

∆x

The volumetric flow rate of oil, qo, can be used to obtain an expression for the flux, Jo. Clearly: ∆y ∆z

i-1 Are =A a =∆ z. ∆

i i+1

Oil

y

Wa te

∆x

Bo

x

S

r

o

S

w

un (i-1 dary /2)

Oil

∆x

Wa te

∆x r Bo un (i+ dary 1/2 )

Flu id (φ)

Pe r Po mea ros bili ity ty = =φ k

Mass flow rate of oil  volumetric flow rate  density of  =  x  = Q o .ρo (43) Across area, A  of oil across A   oil, ρo  u ρ  qρ ∴ Mass flow rate of oil per unit area =  o o  = u oρo =  o osc  (44)  A   Bo  But, by definition the above quantity is the flux, Jo (Figure 6). (Where we note that qo/A = uo, the Darcy velocity of oil). Thus, the flux expressions are:

J o = ( u o ρo )

20

;

J w = ( u w ρw )

(45)

Figure 7 Control Volume (block i) for Two-Phase Flow

5

The Flow Equations

However, we wish to work with reference to standard conditions (sc ⇒ standard conditions; 60°F and 14.7 psia; ~15.6°C and 1 bar). Therefore, introducing the densities of oil and water at standard conditions, ρosc and ρwsc, we obtain (by definition above):

u ρ  J o =  o osc   Bo 

(46)

u ρ  J w =  w wsc   Bw 

Now use the flux expression to perform the mass balance on control block i in Figure 7. The flow terms are as follows: Mass of oil flowing into block i over time step Δt

Mass of oil flowing out of block i over time step ∆t  Flux of   =   .Area.∆t  = ( J o .A.∆t )i- 1 2  oil   i- 1 2

(47)

Mass of oil flowing out of block i over time step Dt = (Jo.A.Δt)i-1/2

[

(48)

]

Change in mass of oil in block over ∆t due to flow = − ( J o )i+ 1 - ( J o )i- 1 A.∆t (49) 2

2

(where we note in Figure 7 that A = ΔyΔz) The accumulation - i.e. change in static mass of oil in block i - is calculated. The expression for this is given in the Exercise above: Change of mass in block over Δt (accumulation) = Mass in block at t+Δt – Mass in block at t

[

]

= ( ∆x∆y∆zφSo ρo )t + ∆t − ( ∆x∆y∆zφSo ρo )t = (φSo ρo )t + ∆t − (φSo ρo )t ∆x∆y∆z (50) Equate the expressions in equations 49 and 50 to obtain

[(φS ρ )

o o t + ∆t

[

]

]

− (φSoρo )t ∆x∆y∆z = - ( J o )i + 1 − ( J o )i- 1 A.∆t (51) 2

2

Divide through the above equation by ΔxΔyΔzΔt to obtain:

[(φS ρ )

o o t + ∆t

− (φSoρo )t

∆t

] = - [(J )

o i+1 2

− ( J o )i- 1

∆x


2

]

(52)

21

Taking limits as Δt, Δx → 0, we obtain the differential form of the mass conservation equation:

∂(φSo ρo )  ∂J  = − o  ∂x  ∂t Using the fact that ρo =

(53)

ρosc in the LHS of the above equation gives: Bo

∂  φSo ρosc   ∂J o     = − ∂t  Bo  ∂x 

(54)

 u oρosc   gives:  Bo 

Substituting for J o = 

∂  φSoρosc  ∂  u oρosc   = −   ∂t  Bo  ∂x  Bo 

(55)

However, ρosc is a constant reference density (at standard conditions) which cancels to yield:

∂  φSo  ∂  uo   = −   ∂t  Bo  ∂x  Bo 

(56)

Likewise,

∂  φSw  ∂  uw   = −   ∂t  Bw  ∂x  Bw 

(57)

The above equations are the (exact) differential forms of the oil and water mass conservation equations. They involve no assumptions; they simply arise as a result of the definition of the various terms. However, the equations are of little practical use in this form since they do not mention pressure, and it is this that we measure most directly in a reservoir – not local (spatially distributed) velocities, uo and uw. Clearly, it is now time to introduce Darcy’s Law. The expressions for the two phase Darcy Law are reviewed at the start of this subsection. Substituting for uo and uw in equations 56 and 57 above gives:

∂  φSo  ∂  k k ro  = ∂t  Bo  ∂x  µ o Bo

∂z    ∂Po   ∂x  − ρo g ∂x    

∂  φSw  ∂  k k rw  = ∂t  Bw  ∂x  µ w Bw

22

∂z    ∂Pw  − ρ g   w  ∂x  ∂x   

(58)

(59)

5

The Flow Equations

These equations now express mass conservation of oil and water, on the assumption that the 2-phase Darcy Law applies. These are our “working equations” for the two phase flow (in their 1D form). At a first glance, it appears that equations 58 and 59 present us with two equations in four unknowns. So, Sw, Po and Pw. But, as we have noted above in the review of key concepts, there are two constraints on these quantities; viz So + Sw = 1 and Pc(Sw) = Po – Pw. Thus, only two of these quantities e.g. Po, Sw or So, Pw etc. are truly independent. It is now clear why we described the Pe as a constraint equation earlier in this chapter. The above equations still require some manipulation since we would like to have a pressure equation which was free of terms of the type (∂So/∂t) and (∂Sw/∂t). This is derived in the next section.

3.3 The Two-Phase Pressure Equation

In order to eliminate the saturation derivatives with respect to time in equations 58 and 59 above, we proceed as follows. Firstly, we decide to retain the oil pressure, Po, (P = Po) as our reference pressure (this is arbitrary). Next we expand the LHS of the oil equation (equation 58) as follows:

∂  φSo  ∂  1  φ  ∂So  So  ∂φ   +     = φSo   + ∂t  Bo  ∂t  Bo  Bo  ∂t  Bo  ∂t  = φSo =

∂  1   ∂Po  φ  ∂So  So  ∂φ   ∂Po  . . + +    .  . ∂Po  Bo   ∂t  Bo  ∂t  Bo  ∂Po   ∂t 

(60)

φ  ∂So   ∂  1  So  ∂φ   ∂Po    + φSo   +    Bo  ∂t   ∂Po  Bo  Bo  ∂Po   ∂t 

where the underlined term is the one we wish to eliminate. Likewise, for the LHS of the water equation (equation 59):

∂  φSw  φ  ∂Sw   ∂  1  Sw  ∂φ   ∂Po   + φSw  =  +   ∂t  Bw  Bw  ∂t   ∂Po  Bw  Bw  ∂Po   ∂t

(61)

Note that the underlined terms in equations 60 and 61 are to be eliminated and the following terms are essentially of the form = (compressibility terms) x (∂P/∂t). To eliminate the time derivative terms of the saturations above, then note that

 ∂Sw   ∂So    +  = 0  ∂t   ∂t  by definition (since Sw + So = 1∴

(62)

∂Sw ∂So + = 0) ∂t ∂t


23

Therefore, multiply equation 60 (oil) by Bo/φ on both sides and multiply 61 (water) by Bw/φ on both sides to obtain:

and

Bo ∂  φSo  ∂So  ∂  1  So ∂φ  ∂Po  + BoSo .    =  + φ ∂t  Bo  ∂t  ∂Po  Bo  φ ∂Po  ∂t 

(63)

Bw ∂  φSw  ∂Sw  ∂  1  Sw ∂φ  ∂Po  + BwSw .    =  + φ ∂t  Bw  ∂t  ∂Po  Bw  φ ∂Po  ∂t 

(64)

Now ADDING the above equations 63 and 64, we see that the ∂S/∂t terms vanish (equation 62) as follows: Bo ∂  φSo  Bw ∂  φSw   ∂  1 ∂  1  So  ∂φ  Sw  ∂φ   ∂Po  +   = BoSo .   + BwSw .  +  +   φ ∂t  Bo  φ ∂t  Bw   ∂Po  Bo  ∂Po  Bw  φ  ∂Po  φ  ∂Po   ∂t

 ∂  1 ∂  1  1  ∂φ   ∂Po = BoSo .   + BwSw .  +   ∂Po  Bo  ∂Po  Bw  φ  ∂Po   ∂t 

(65)

where we have used So + Sw = 1 on the RHS in the last simplification in equation 65. The expression above (equation 65) is now equal to the RHS of equation 58 multiplied by Bo/φ + RHS of equation 61 multiplied by Bw/φ. First, we simplify a little by denoting all of the compressibility terms above by, α(So,Po) - that is:

 ∂  1 ∂  1  1 ∂φ  α(So ;Po ) = BoSo .    + BwSw .  + P B P ∂ ∂    Bw  φ ∂Po  o o o 

(66)

Therefore:

 ∂P  B ∂  k k ro  ∂Po   ∂z   B ∂  k k rw   ∂Pw   ∂z    α(So ;Po ) o  = o   − ρo g   + w   − ρ w g        ∂t  φ ∂x  µ o Bo  ∂x   ∂x   φ ∂x  µ w Bw   ∂x   ∂x   

(67) Now use capillary pressure, Pc(Sw)=Po-Pw, to eliminate the water pressure as follows:

∂Pc (Sw )  ∂Po   ∂Pw   ∂P   ∂P ∂P  =  −  i.e.  w  =  o − c   ∂x   ∂x   ∂x   ∂x ∂x  ∂x Substituting this expression from (∂Pw/∂x) into equation 67, we obtain

24

(68)

5

The Flow Equations

 ∂P  B ∂  k k ro  ∂Po   ∂z   α(So ;Po ) o  = o   − ρo g     ∂t  φ ∂x  µ o Bo  ∂x   ∂x   +

Bw ∂  k k rw   ∂Po   ∂Pc   ∂z     −  − ρ w g        ∂x    φ ∂x  µ w Bw  ∂x   ∂x 

(69)

This is the 1D pressure equation for a two-phase compressible (fluids and rock) system. Each of the terms is physically interpretable and we expand this out to see each of the contributions more clearly.

∂  k k  ∂P   ∂  k k rw  ∂Po    ∂P  φα(So ;Po ) o  = Bo .  ro  o   + Bw .     ∂t  ∂x  µ o Bo  ∂x   ∂x  µ w Bw  ∂x   OIL FLOW − Bo .

WATER FLOW

∂  k k roρo g  ∂z   ∂  k k ρ g  ∂z      − Bw .  rw w     ∂x  µ o Bo  ∂x   ∂x  µ w Bw  ∂x   GRAVITY TERMS

− Bw .

∂  k k rw  ∂Pc     ∂x  µ w Bw  ∂x  

CAPILLARY PRESSURE TERM

(70)

Notes on equation 70 (i)

The two-phase pressure equation for the fully compressible system is clearly very complex. Again, it is a non-linear partial differential equation (PDE) which cannot be solved analytically for the general case;

(ii) Despite its complexity, all of the terms in equation 70 have a clear physical interpretation in terms of viscous, gravity and capillary forces; (iii) Following from (i) above, our two alternatives are either to use numerical methods to solve equation 70 or to simplify it greatly such that an analytical solution may be possible. We will consider the simplified pressure equation in the next section; (iv) Recall that in two-phase flow, the dependent variables (the unknown we want to find) were chosen to be: Po(x,t) and So(x,t) Therefore, once we solve the pressure equation 70, we must then calculate the saturation. In fact, ideally we would like to solve the pressure equation at the same time as solving for the saturation. Where is our saturation equation in any case? In fact, we have already met this – as equation 59 above.


25

3.4 Schematic Strategy for Solving the Two-Phase Pressure and Saturation Equations In fact, we can write the two equations for pressure and saturation in the following schematic way:

 ∂P 

∂  k k  ∂P  

ro PRESSURE φα(So ;Po )  o   + gravity terms + cap. terms  = Bo .    ∂ t ∂ x µ B  o o  ∂x   etc. (70)

∂  φSo  ∂  k k ro  = ∂t  Bo  ∂x  µ o Bo

SATURATION

∂z    ∂Po  − ρ g   o  ∂x  ∂x   

(58)

Clearly, these two equations are coupled. That is, the coefficients in the pressure equation e.g. ko, etc. depend on So which is the unknown we are trying to find. Likewise, in the saturation equation, flow terms appear with (∂Po/∂x) in them – we are also trying to find Po. Hence, if we solve these equations “one at a time”, we face the problem that there are unknowns which we don’t yet know. We will go into more detail in Chapter 6 where we discuss the numerical solution of the flow equations. However, we will just think in general terms about how we may go about solving the pair of pressure/saturation equations above. Think of the unknowns in our grid block i, Poi and Soi. We imagine the time levels n meaning time = t or now where we know Poi and Soi. We denote this as follows: n Time level n (time = t) (KNOWN) Poi n Time level n (time = t) (KNOWN) Poi n

Soi n Soi We are trying to find Poi and Soi at the next time, t = t +Δt or n + 1 denoted n+1

Time level n+1 (time = t+∆t) (UNKNOWN) Time level n+1 (time = t+∆t) (UNKNOWN)

Poi n+1 Poi n+1

Soi n+1 Soi The problem is shown schematically if Figure 8 below as follows: GRID BLOCK i At time = t (time step n)

Time step, ∆t

Rock n

S oi n

Rock n

P oi n

(S wi ) (Pwi ) 26

GRID BLOCK i At time = t+∆t (time step n+1)

Solve equations 70 and 59 to find ⇒

n+1

S oi

n+1

n+1

P oi

n+1

(S wi ) (Pwi )

Figure 8 Schematic showing update of the pressure and saturation in a grid block (i) over a time step, Δt.

5

The Flow Equations

Now the pressure equation has lots of terms that depend on So, so strictly we need to n +1 know Soi to solve the pressure. BUT suppose we try the following strategy: (a) Use Snoi i.e. the known saturation value to calculate all the coefficients in the pressure equation (equation 70). (b) Solve the pressure equation to get a first estimate of Poin +1 (c) Now solve the saturation equation (equation 58) using the latest Poin +1 for n +1 the pressure dependent flow terms, to get ⇒ Soi (d) We now have Poin +1 and Soin+1 as we required – but we may be able to do a bit better than this by going back to step a and using our (from this step) to get a better solution to the pressure equation, and hence an “improved” Poin +1 (e) Clearly, we could iterate through steps a to d until our process “converges” i.e. the newly calculated Soin+1 and Poin +1 don’t change any more (or change only by a tiny amount). We would then accept these as our “accurate” new time level values and then go on to the next time level. The description above outlines – in words – an algorithm or numerical strategy for solving the pressure and saturation equations that arise in two-phase flow. We used no mathematics since it is just the basic idea that we want to get across just at this point.

3.5 The Simplified Two-Phase Pressure and Saturation Equations

In the case of single phase flow, we found it quite useful to simplify the (analytically insoluble) compressible flow equation. This allowed us to see its structure quite clearly and it turned out that the simplified single phase equation was a diffusion equation. This diffusive process underlay how pressure “waves” spread across an oil reservoir. For the two-phase flow case, we will make some slightly different assumptions when we simplify the equations as follows. Assumptions: 1 The viscosity of the oil and water are constant μo and μw (with x and P); 2 Both the rock and the fluids are incompressible (φ constant; Bo = Bw = 1); 3 We neglect both capillary pressure (Pc = 0; Po = Pw = P) and gravity (g = 0). It is easiest to see the consequences of the above assumptions by going back to the conservation equations 58 and 59. In the absence of gravity and capillarity (Assumption 3), these become as follows:

∂  φSo  ∂  k k ro  ∂P      = ∂t  Bo  ∂x  µ o Bo  ∂x  

(71)

∂  φSw  ∂  k k rw  ∂P      = ∂t  Bw  ∂x  µ w Bw  ∂x  

(72)


27

Now using Assumptions 1 and 2, we obtain:

 ∂S  ∂  k k ro  ∂P   φ o  =    ∂t  ∂x  µ o  ∂x  

(73)

 ∂S  ∂  k k rw  ∂P   φ w  =    ∂t  ∂x  µ w  ∂x  

(74)

Clearly, it is now very simple to eliminate the (∂S/∂t) terms since we simply need to add the above two equations to obtain the simplified pressure equation:

∂  k k ro  ∂P   ∂  k k rw  ∂P     +   =0 ∂x  µ o  ∂x   ∂x  µ w  ∂x  

(75)

which simplifies to:

∂   ∂P   λ o + λ w )   = 0 (   ∂x   ∂x 

(76)

where λo and λw are the phase mobilities given by:

λo =

k k ro k k rw and λ w = µo µw

(77)

These phase mobilities are strong functions of the saturation So (So = 1 – Sw) through the relative permeabilities i.e. λo(So); λw(So). Indeed, we can define the total mobility, λT(S0), as:

λ T ( So ) = λ w + λ o

(78)

thus simplifying the pressure equation to:

∂  ∂P  λ T ( So )  = 0  ∂x  ∂x 

(79)

The simplified saturation equation is simply equation 73 above which, in the above notation, becomes:

φ

∂So ∂  ∂P  = λ o ( So )   ∂t ∂x  ∂x 

(80)

In summary, the simplified (incompressibility no Pc, no gravity) two-phase pressure and saturation equations are as follows:

∂  ∂P  λ S =0 ( ) T o ∂x  ∂x  28

(81)

5

The Flow Equations

∂P   ∂S  ∂  φ o  = λ S ( ) o o  ∂t  ∂x  ∂x 

(82)

It is probably clearer for these equations how we would apply our schematic strategy (Section 3.4) to these two equations. We illustrate this in the flowchart in Figure 9. n

n

Time level n ⇒ KNOW Soi P i Calculate mobilities at these n n n λT( So )=λo(So )+λw(So )

n+1

n+1

Set the latest P i S oi to "current" values and ITERATE through calculation again

Solve PRESSURE equation with  ∂P   ∂  "current"  λ T (Sno )   = 0 ∂x   ∂x   ⇒obtain Pin +1

Keep them and set to "current" values. Take next time step

n

Use Pin +1 and λo(So ) To solve SATURATION equation  ∂P  n+1  ∂  ∂S φ o =  λ o (Sno )   ∂t ∂x   ∂x   ⇒obtain Soin +1

Figure 9 Schematic Strategy for the Iterative Solution of the (simplified) Pressure and Saturation Equations for Two-Phase Flow

NO

n+1

n+1

Are these P i ; S oi satisfactory? (i.e. converged)

YES

4. CLOSING REMARKS The purpose of this module is to familiarise the student with the fundamental flow equations of single- and two-phase flow. We have shown that these can be derived in a unified way by applying: MATERIAL BALANCE + DARCY’S LAW ⇒ FLOW EQUATIONS In the two-phase case, the mass conservation equation was applied to each of the two phases – oil and water, in the example here. There is also a little more “two-phase physics” to be dealt with due to capillary pressure and relative permeability. For both single- and two-phase flow, the resulting full equations for the compressible system were non-linear PDE’s (indeed, 2 non-linear, completed PDE’s for two-phase flow). These could not be solved analytically but, in simplified form, they illustrate some interesting features of the processes. The numerical solution of these equations is discussed in some detail in Chapter 6. Institute of Petroleum Engineering, Heriot-Watt University

29

Solution to Exercise 2:

∂(φρ )  ∂ρ  = φ   ∂P  ∂(φρ )  ∂∂ρP Expand the LHS (φ constant) ⇒ = φ   ∂(φρ )  ∂ρ   ∂P  ∂P ⇒ ∂(φρ ) = φ ∂ρ  P ) = φ  ∂∂ρP(φρ ) ⇒ ∂∂(∂φρ  ρ  ) k x ∂∂ρ   ∂P   k y ∂   ∂P   (∂φρ  ∂∂Pρassumptions)  P ) = φφ⇒ φ  ∂(φρ ⇒ = φ  ρ   + The RHS⇒ ⇒becomes  ∂P∂P =⇒ ∂∂PP =(using ρ      ∂P  ∂P ∂ P k x ∂   ∂µP ∂x∂P ky∂x∂   ∂µP ∂y   ∂y   ρ   k ∂   ∂P   ⇒k ∂  ρ∂P    + ⇒ k xx ∂  ρ ∂P   + k yy µ∂ ∂xρ ∂P∂x  µ ∂y   ∂y   The x- and form just need to expand up one  ∂so we ∂∂x terms xkare ykand ⇒ yρ  ∂∂∂PP +∂ kkµy ∂∂∂y∂Pin  ∂∂∂PP kkµxRHS   identical ρ  ∂    y ∂  k ∂ ∂ P P   ∂ρ      x   y   ⇒ ρ + ρ kP∂y2P∂  ρ ∂∂P  ∂2 P  ∂ ∂ P   x x ∂x will ∂bex very µ ∂ y  ρ of these (the other similar).   ∂∂+yyρρ   ρ ⇒ µ +     ∂+x µ ρ⇒  = + = ρ ρ        µµ ∂∂xx  ρ⇒ ∂∂xxµ ∂ y  x   ∂∂yµ∂Px∂2y∂ρ  ∂∂xy  ∂∂2xP  ∂x 2    ∂µ ∂y∂∂xµP  ∂∂xx∂yµ∂∂∂2xyP     ∂22 P ∂x ρ∂P∂x  ∂=ρρ  ∂x 2 ∂22+P∂x   ∂x  = ρ  ∂x 2  ∂  ∂P  ∂ ρ ∂P = ρ ∂ P  +  ∂P   ∂ρ  = ρ ∂ 2 P2   2  ∂2 P  ∂∂x ρ ∂∂Px  = ρ∂ ∂2x2P22∂P+  ∂∂Px∂2∂P ∂x = ρ ∂∂∂2x2PPP2∂ρ ∂PP  ∂∂∂xP∂∂∂∂Pρ ∂ρ  ∂ P ρ  ∂2 P  ∂P   ∂ρ      =    ∂∂x ρρ ∂∂xP  == ρρ ∂∂ρxP2  ++ ∂ x x   2ρ + = ρ  = + ρ    = ρ = = ρ ρ ρ           ρ = 2 2  2 ∂∂xx  ∂∂xx  ∂x ∂∂xx 2∂x  ∂∂x∂xx∂x∂∂∂xxx  ∂2x∂∂∂xxx2 ∂2x   ∂2 x  ∂x∂x  ∂x    ∂  ∂P  y  ∂∂yP  ∂y  Likewise 2 ρ ρ =  2   ∂ P  ∂  ∂P   ∂y  ∂  ρ ∂P  = ρ ∂ 22 P2 ∂y  ∂y  2 ∂∂y ρ ∂∂Py  = ρ∂ ∂ y2P2∂P  PP ∂ρ  ∂∂2 PP k y ∂2 P k x ∂2 P ∂∂y ρρ ∂∂yP  == ρρ ∂∂ρyP2  =∂ρρ∂ ∂∴ ρ = φ = ρ + ρ  ∂∂yy  ∂∂yy  ∂y ∂∂yy 2∂y  ∂y  ∂y∂2y  ∂y 2 2    ∂ρ   ∂P ∂P k x∂t∂ P µk y∂x∂22 P µ ∂y 2   2 ρ +ρ 2 k∴ φ∂2 P   k ∂= P  ∂ρ   ∂P  µ ∂y 2 ∴ φ ∂ρ  ∂P  = ρ k xx ∂22 P∂2 P+ρ k∂tyy ∂ 22 P2 µ ∂x 2 P ∂∂Pt  = ρ µky ∂∂y2P ∴ φ  ∂∂ρρ ρρkµ k y∂∂k22PxPk y  k y1 ∂ 2∂Pρ  2P ∂ρkkµµxx∂∂∂∂xx2PPP2+ kµ ρ P . k ∂ ∂ P  ∂P  ∂∂P     yx∂   ∴ φ = + ρ   t y 2φ 2 = + ρρk =x   +=ρρ   ∂where ∴ φ ∂P  ∂t∴φ= ρ µ ∂∴ + ρ   = cf  ∂P   ∂t   ∂Pµµ.∂∂xxt2  ∂Pρµ µkµ ∂∂yytx2k2 x k y  µµ 1∂∂xy222∂ρ  µρ ∂y∂2P  µ. 1  ∂ρk =  2  ρ  ∂P  = c f  k x kρy k where where k = k k   µ.  k x k y  1  ∂ρ  = c f ∂Pk = c f 1 k∂ρ ρµk. where k µ = 1.ρ . kxx22kyy  µρ 1  ∂∂kρ ∂Pxk that  ∂2∂Pρ kto ∂2 P  y==ckfccand y k x 1 k = y obtain: = Pρx µφ ρµk. where f  note multiply both sides by where k = = c  where k = where = c   = +   f          ρ  ∂P2  f ρ  ∂2P    22 f  ρρkk ρ k 22 µφ ρ ∂P y ∂x∂∂PP  k  ∂y 2  k x ∂∂t 2P  k kρ  2 ρckf   ∂P  k= 2 +  µφc f   ∂P  = k x ∂2 Pk + k∂y t∂2 P k  ∂x 2  k  ∂y 2    k  ∂ P  c ∂P k  ∂ P   µφ k f  ∂t  µφkckxx  ∂∂2x2PP22  + kkkyy  ∂∂2y22P22  k  ∂2 2 P  k 2  µφ µφkccff  ∂∂∂PPt  = c x y y  ∂ P == kkxf ∂∂x 2µφ ++ k=∂PP k  + k y /∂zP   k  ∂t  kk ∂x∂t2 = kkf   ∂∂yyx222 x=+ xky  ∂y2 2+  2  k  ∂t  k k∂x= k k+kk ∂/yt z  k ∂x  k  ∂y  x y k = kx + ky / z k general form of the simplified We haveksimply = k x introduced + k y / z k = (kx+ky)/2 to retain the Dh = == kkx ++ The kky k/hydraulic zz k + kdiffusivity = / z pressure kkequation. in this case is therefore given by: / k = k + k / z x y x y y ( µφc f ) kx Dh = k ( µφcf ) Dh = k D h = ( µφkkc f ) D c f ) D h = k D = k xk/ k Dhh == (( µφ µφ ( µφk xcf/)kh ( µφcf ) ( µφccff )) The coefficients k x / k on the RHS of the simplified pressure equation here are kx/ k and ky / k k /reflect k ky/ k which the anisotropy of the problem. Obviously, if kx = ky then these kkxx // kk k / k k /kkx / k terms would x be unity. x y ky / k ky / k kky // kk k /k k /k

( (( (

y

30

) )) ) (

y

⇒

(

)

) (

y

(

)

)

5

The Flow Equations


31

Numerical Methods in Reservoir Simulation

6

CONTENTS 1.

INTRODUCTION

2.

REVIEW OF FINITE DIFFERENCES

3.

APPLICATION OF FINITE DIFFERENCES TO PARTIAL DIFFERENTIAL EQUATIONS (PDEs) 3.1. Explicit Finite Difference Approximation of the Linear Pressure Equation 3.2. Implicit Finite Difference Approximation of the Linear Pressure Equation 3.3. Implicit Finite Difference Approximation of the 2D Pressure Equation 3.3.1 Discretisation of the 2D Pressure Equation 3.3.2 Numbering Schemes in Solving the 2D Pressure Equation 3.4. Implicit Finite Difference Approximation of Non-linear Pressure Equations

4.

APPLICATION OF FINITE DIFFERENCES TO TWO-PHASE FLOW 4.1 Discretisation of the Two-Phase Pressure and Saturation Equations 4.2 IMPES Strategy for Solving the Two-Phase Pressure and Saturation Equations

5.

THE NUMERICAL SOLUTION OF LINEAR EQUATIONS 5.1. Introduction to Linear Equations 5.2. General Methods for Solving Linear Equations 5.3. Direct Methods for Solving Linear Equations 5.4. Iterative Methods for Solving Linear Equations 5.5. A Comparison of Iterative and Direct Methods for Solving Linear Equations

6.

DIRECT SOLUTION OF THE NON-LINEAR EQUATIONS OF MULTI-PHASE FLOW 6.1. Introduction to Sets of Non-linear Equations 6.2. Newton’s Method for Solving Sets of Nonlinear Equations 6.3. Newton’s Method Applied to the Non-linear Equations of Two-Phase Flow

7.

NUMERICAL DISPERSION - A MATHEMATICAL APPROACH 7.1. Introduction to the Problem 7.2. Mathematical Derivation of Numerical Dispersion

8.

CLOSING REMARKS

APPENDIX A: Some Useful Matrix Theorems.

LEARNING OBJECTIVES: Having worked through this chapter the student should be able to: • write down from memory simple finite difference expressions for derivatives, (∂P/∂x), (∂P/∂t) and (∂2P/∂x2) explaining your spatial (space) and temporal (time) n n +1 n +1 n +1 notation ( Pi , Pi , Pi +1 , Pi −1 etc. ); for (∂P/∂x), the student should know the meaning of the forward difference, the backward difference and the central difference and the order of the error associated with each, O(Δx) or O(Δx2). • apply finite difference approximations to a simple partial differential equation (PDE) such as the diffusion equation and explain what is meant by an explicit and an implicit numerical scheme. • write a simple spreadsheet to solve the explicit numerical scheme for a simple PDE for given boundary and initial conditions and be able to describe the effect of time step size, Δt. • show how the implicit finite difference scheme applied to a simple linear PDE leads to a set of linear equations which are tridiagonal in 1D and pentadiagonal in 2D. • derive the structure of the pentadiagonal A-matrix in 2D for a given numbering scheme going from (i, j) notation to m-notation where m is an ordered numbering scheme e.g. for the natural numbering scheme, m = (j - 1).NX + i • describe a solution strategy for the non-linear single phase 2D pressure equation where the fluid and rock compressibility (and density, ρ, and viscosity, μ) are functions of the dependent variable, pressure, P(x,y,t). • write down the discretised form of both the pressure and saturation equation for two-phase flow given the governing equations (in simplified form in 1D), and be able to explain why these lead to sets of non-linear algebraic equations. • outline with an explanation and a simple flow chart the main idea behind the IMPES solution strategy for the discretised two-phase flow equations. • write down the expanded expressions for a set of linear equations which, in compact form are written A.x = b, where the matrix A is an nxn matrix of (known) coefficients (aij; i = rows, j = columns), b is a vector of n (known) values and x is the vector of n unknowns which we are solving for. • explain clearly the main differences between a direct and an iterative solution method for the set of linear equations, A.x = b. • write down the algorithm for a very simple iterative scheme for solving A.x = b, and be able to describe the significance of the initial guess, x(0) , what is meant by iteration (and iteration counter, ν), the idea of convergence of x(ν) as ν → ∞; and be able to comment on the number of iterations required for convergence, Niter.

2


6

• explain how to apply (without derivation) the Newton-Raphson method for solving a single non-linear algebraic equation, f ( x ) = 0 ; Newton-Raphson scheme => x (ν +1) = x (ν ) −

f ( x (ν ) ) . f ' ( x (ν ) )

• extend the application of the Newton-Raphson to sets of non-linear algebraic equations such as those arising from the discretisation of the two-phase pressure and

S  P

saturation equations; F( X ) = 0, where X =   and S and P are the (unknown) vectors of saturation and pressure; (given the Newton-Raphson expression,

[

]

X (ν +1) = X (ν ) − J ( X (ν ) ) .F( X (ν ) )). −1

• understand, but not be able to reproduce the detailed derivation of, the more mathematical explanation of numerical dispersion.

NUMERICAL METHODS IN RESERVOIR SIMULATION As noted in Chapter 5, the multi-phase flow equations for real systems are so complex that it is not remotely possible to solve them analytically. In practice these equations can only be solved numerically. The most commonly applied numerical methods are based on finite difference approximations of the flow equations and this approach will be followed in this chapter. After a brief review of finite differences, we go on to apply them to very simple systems such as for the simplified 1D pressure equation (derived in Chapter 5). This equation does not need to be solved numerically but it demonstrates how explicit and implicit finite difference solution can be developed. We then show how sets of linear equations arise in solving implicit equations and we consider solution of the linear equations in an elementary manner. We then consider how the 2D pressure equation is solved. In the numerical solution of the multi-phase flow equations, we need to solve for pressure and flow. An outline of how this is done is presented but we do not go into great detail.

1.

INTRODUCTION

At the very start of this course, we considered a very simple “simulation model” for a growing colony of bacteria. The number of bacteria, N, grew with a rate proportional to N itself i.e. (dN/dt) = α.N, where α is a constant. We saw that this simple equation α.t had a well-known analytical solution, N( t ) = N o .e , where No is the number of bacteria at time, t =0. This exponential growth law provides the solution to our model. However, we also introduced the idea of a numerical model where, even although we could solve the problem analytically, we formulated it this way, in any case. The numerical version of the model came up with the algorithm (or recipe) N n +1 = (1 + α.∆t ).N n . In the exercise in Chapter 1, you should have compared the results of the analytical and numerical models and found out that, they get closer as Institute of Petroleum Engineering, Heriot-Watt University

3

we take successively smaller time steps, Δt. In this case, we say that the numerical model converges to the analytical model. In fact, in many areas of science and engineering, we often apply a numerical technique to a problem we can already solve analytically i.e. where we “know the answer”. Why would we do this? The answer is that we might be testing the numerical method to see how closely it gives the right answer. More commonly, we might test several - maybe 3 or 4 - numerical techniques to determine which one works best. The phrase “works best” in the context of a numerical method usually means gives the most accurate numerical agreement with the analytical answer for the least amount of computational work. Note the importance of this balance between accuracy and work for a numerical method. There may be no point in having a numerical method that is “twice” as accurate (in some sense) for ten times the amount of computational work. In Chapter 5, we already met the equations for single-phase and two-phase flow of compressible fluids through porous media. These turned out to be non-linear partial differential equations (PDEs). Recall that a non-linear PDE is one where certain coefficients in the equation depend on the answer we are trying to find e.g. Sw(x,t), P(x,y,z,t) etc. For example, for single phase compressible flow, the equation for pressure, P(x,t), in 1D is given by:

 ∂P  ∂  k.ρ  ∂P   c( P)  =    ∂t  ∂x  µ  ∂x  

(1)

In this equation, both the generalised compressibility term, c(P) (of both the rock and the fluid), and the density, ρ(P), terms depend on the unknown pressure, P, which we are trying to find. As noted previously, such non-linear PDEs are very difficult to solve analytically and we must usually resort to numerical methods. The main topic of this module is on how we solve the reservoir flow equations numerically. This process involves the following steps: (i) Firstly, we must take the PDE describing the flow process and “chop it up” into grid blocks in space. This is known as spatial discretisation and, in this course, we will exclusively use finite difference methods for this purpose. (ii) When we apply finite differences, we usually end up with sets of non-linear algebraic equations which are still quite difficult to solve. In some cases, we do solve these non-linear equations. However, we usually linearise these equations in order to obtain a set of linear equations. (iii) We then solve the resulting sets of linear equations. Many numerical options are available for solving sets of linear equations and these will be discussed below. This is often done iteratively by repeatedly solving them until the numerical solution converges. This module will deal successively with each of the parts of the numerical solution process, (i) - (iii) above.

4


6

2. REVIEW OF FINITE DIFFERENCES Definition:

A finite difference scheme is simply a way of approximating

2 derivatives of a function (e.g.  dP  ,  ∂P  ,  ∂ P  etc.) 2

 dx   ∂t   ∂x 

numerically from either point or block values of the function. The main concept of finite difference approximation is best illustrated by the following simple example where we refer to Figure 1. Study the Notation in this figure, since it is the basis of that used throughout this chapter. The main task of the finite difference approach is to represent the derivatives of the function, P(x), in an approximate manner 2  dP  ,  ∂P  ,  ∂ P  etc.    2 i.e.  dx   ∂t   ∂x 

First consider how we might approximate (dP/dx) at xi using the quantities in Figure 1. In fact, it is easily seen that there are three ways we may do this as follows: Approximation 1 -Forward Differences (fd): we may take the slope between Pi and Pi+1 as being approximately  dP  at xi to obtain:

 dx  i

 dP  = Pi +1 − Pi  dx  i fd ∆x

(2)

Approximation 2 -Backward Differences (bd): we may take the slope between Pi-1

dP  at x to obtain: and Pi as being approximately  i  dP  = Pi − Pi −1  dx  i bd ∆x

 dx  i

(3)

Approximation 3 -Central Differences (cd): thirdly, we could take the average of the forward difference (fd) and backward difference (bd) approximations to give us

 dP  at x . This is known as central differences (cd) and is given by: i  dx  i

 dP  = 1  dP  +  dP   = 1  Pi +1 − Pi + Pi − Pi −1   dx  i cd 2  dx  i fd  dx  i bd  2  ∆x ∆x  P −P  dP  =  i +1 i −1   dx  i cd  2.∆x  (4)


5

Pi+1 P(x) Pi Pi-1 ∆x = constant

∆x xi-1

x

∆x xi

xi-1

Notation: ∆x = the x- grid spacing or distance between grid points (i - 1), i, (i + 1) = the x - label (subscript) for the grid point or block numbers Pi-1, Pi, Pi+1 = the corresponding function values at grid point i-1, i, i+1 etc.

Each of the above approximations to

 dP  is shown graphically in Figure 2.  dx  i

You may wonder why we bother with three different numerical approximations to

 dP   dx  i and ask: which is “best”? There is not an unqualified answer to this question

just yet, but let us take a simple numerical example where we know the right answer and examine each of the forward, backward and central difference approximations. The values given in Figure 3 will illustrate how the methods perform. Firstly, simply calculate the values given by each methods for the data in Figure 3:

 dP  ≈  3.0042 − 2.7183  = 2.859 [err ≈ +0.14]   dx  i fd  0.1  dP  ≈  2.7183 − 2.4596  = 2.587 [err ≈ −0.13]   dx  i bd  0.1  dP  ≈  3.0042 − 2.7183  = 2.723 [err ≈ −0.005]   dx  i cd  2 x 0.1

(5)

The quantity in square brackets after each of the finite difference approximations above is the error i.e. the difference between that method and the right answer which is 2.7183. As expected from the figure for this case, the forward difference answer is a little too high (by ≈ +0.14) and the backward difference answer is a little too low (by ≈ -0.13). The central difference approximation is rather better than either of the previous ones. In fact, we note that the fd and bd methods give an error of order Δx, the grid spacing (where, as engineers, we are saying 0.14 ≈ 0.1 !). The cd approximation, on the other hand, has an error of order Δx2 (where again we are 6

Figure 1 Notation for the application of finite difference methods for approximating derivatives.


6

saying 0.005 (0.1x0.1 = 0.01). More formally, we say that the error in the fd and bd approximations are "of order Δx" which we denote, O(Δx), and the cd approximation has an error "of order Δx2" which we denote, O(Δx2).

 ∂2 P   2 Now consider the finite difference approximation of the second derivative,  ∂x   ∂P    Going back to Figure 1, the definition of second derivative at  ∂x  xi is the rate of

change of slope (dP/dx) at xi. Therefore, we can evaluate this derivative between xi-1 and xi (i.e. the bd approximation) and then do the same between xi and xi+1 (i.e. the fd approximation) and simply take the rate of change of these two quantities with respect to x, as follows:

 ∂ P  2 ≈  ∂x  2

 dP  −  dP   dx  i fd  dx  i bd ∆x  Pi +1 − Pi  −  Pi − Pi −1   ∆x  i fd  ∆x  i bd

≈  ∂2 P   2 ≈  ∂x 

∆x

( Pi +1 + Pi −1 − 2 Pi ) ∆x 2

(6)

 ∂2 P   2 Calculating the numerical value of  ∂x  for the example in Figure 3 (where again  ∂2 P   2  = 2.7183 since the example is the exponential function), gives:  ∂x   ∂ 2 P  3.0042 + 2.4596 − 2 x 2.7183 = 2.7200 [err ≈ 0.0017]  2 ≈  ∂x  0.12 (7) Thus, we can see that the error in this case (err ≈ 0.0017 ≈ 0.12) is actually O(Δx2). In the introductory section of Chapter 1, we already applied the idea of finite differences (although we didn’t call it that at the time) to the simple ordinary differential equation

dN  (ODE);  = α .N .  dt 

Here, N (number of bacteria in the colony) as a function of time, N(t), is the unknown we want to find. Denoting Nn and Nn+1 as the size of the colony at times t (labelled n) and t+Δt (labelled n+1), we applied finite differences to obtain:


7

n +1 n  dN  ≈ N − N ≈ α . N n  dt  ∆t

N n +1 ≈ (1 + α .∆t ) N n

(8)

This gave us our very simple algorithm to explicitly calculate Nn+1. We then took this as the current value of N and applied the algorithm repeatedly. dP dx i cd

=

Pi+1 - Pi-1 2.∆x

Pi+1 P(x)

dP dx i fd

Pi

=

Pi+1 - Pi ∆x

Pi-1

dP ∆x xi-1

x

= dx i bd

∆x

xi

xi+1

dP = 2.7183 dx x=1.0

d2P dx2

Pi - Pi-1

∆x = constant

∆x

Figure 2 Graphical illustration of the finite difference derivatives calculated by backward differences (bd), forward differences (cd) and central differences.

= 2.7183 x=1.0

Pi+1

3.0042

P(x) Pi

2.7183

Pi-1

2.4596

0.1 0.9

8

x

0.1

1.0

1.1

Figure 3 A numerical example where the function values and derivative values are known (P(x) = ex)


6

EXERCISE 1.

Apply finite differences to the solution of the equation:

 dy  = 2. y 2 + 4  dt  where, at t = 0, y(t = 0) = 1. Take time steps of Δt = 0.001 (arbitrary time units) and step the solution forward to t = 0.25. Use the notation yn+1 for the (unknown) y at n+1 time level and yn for the (known) y value at the current, n, time level. Plot the numerically calculated y as a function of t between t = 0 and t = 0.25 and plot it against the analytical value (do the integral to find this). Answer: is given below where the working is shown in spreadsheet CHAP6Ex1.xls. This gives the finite difference formula, a spreadsheet implementing it and the analytical solution for comparison.

3. APPLICATION OF FINITE DIFFERENCES TO PARTIAL DIFFERENTIAL EQUATIONS (PDEs) 3.1 Explicit Finite Difference Approximation of the Linear Pressure Equation

We have seen in Chapter 5 that the flow equations are actually partial differential equations (PDEs) since the unknowns, P(x,t) and Sw(x,t) say, depend on both space and time. As an example of a linear PDE, we will take the simplified pressure equation (equation 26; Chapter 5) as follows:

k  ∂2 P   ∂P   =  ∂t  cφµ  ∂x 2 

(9)

k where the constant cφµ is the hydraulic diffusivity, which we denoted previously

by Dh. As we noted in Chapter 5, this PDE is linear which has known analytical solutions for various boundary conditions. However, we will again neglect these and apply numerical methods as an example of how to use finite differences to solve PDEs numerically. To make things even simpler, we will take Dh = 1, giving the equation: 2  ∂P   ∂ P    = 2  ∂t   ∂x 

(10)

This is the pressure equation for a 1D system where 0 ≤ x ≤ L, where L is the length of the system. We can visualise this physically - much as we did in Chapter 5, Section 2.1 - using Figure 4. After the system is held constant at P = Po, the inlet pressure is raised (at x = 0) instantly to P = Pin while the outlet pressure is held at Pout = Po. Institute of Petroleum Engineering, Heriot-Watt University

9

Pin t = t1

P

t = t2 Pout = Po

Po t=0 0

x

L

These pressures, Pin and Pout, represent the constant pressure boundary conditions (Mathematically these are sometimes called Dirichlet boundary conditions). In other words, these are set, as if by experiment, and the system between 0 < x < L must “sort itself out” or "respond" by simply obeying equation 10 above. We now approach this problem using finite differences as follows: • discretise the x-direction by dividing it into a numerical grid of size Δx; • choose a time step, Δt; • use the following notation

Pin Pinn +1 P Piin n +1 P Pin +1 i

→ time level n = 0, 1, 2 ... → x-grid block label, i = 1, 2, 3 ... NX (at x = L) current (known) P at time level new (unknown) P at time level

• fix the boundary conditions which, in this case, are as follows (see Figure 4): P1 = P in and PNX = Pout = Po which are fixed for all t. • apply finite differences to equation 10 using the above notation to obtain:

P n +1 − Pi n  ∂P    ≈ i  ∂t  i ∆t

(11)

and

 ∂2 P  Pi ??+1 + Pi ??−1 − 2 Pi ??  2 ≈  ∂x  i ∆x 2

(12)

However, an issue arises in equation 12 above as shown by the question marks on the spatial derivative time levels. It is simply: which time level should we take for the spatial derivative terms in equation 12? This is important and we will return to this matter soon. However, for the moment, let us take these spatial derivatives at time level n (the “known” level) since this will turn out to be the simplest thing we can do. Thus, we obtain:

10

Figure 4 Physical picture of pressure propagation in a 1D (compressible) system described by.  ∂P   ∂ 2 P 

  = 2  ∂t   ∂x 


 ∂2 P  Pi n+1 + Pi n−1 − 2 Pi n  2 ≈  ∂x  i ∆x 2

6

(13)

Equating the numerical finite difference approximations of each of the above derivatives as required by the original PDE, equation 10 (i.e. equating the expressions in equations 11 and 13) gives:

Pi n +1 − Pi n Pi n+1 + Pi n−1 − 2 Pi n ≈ ∆t ∆x

(14)

which easily rearranges to obtain an explicit expression for, Pi n +1, the only unknown in the above equation:

Pi n +1 = Pi n +

∆t ( Pi n+1 + Pi n−1 − 2 Pi n ) ∆x 2

(15)

In words, we can interpret this above algorithm as saying: New (n+1 level) value of Pi = Old (n level) value of Pi + a “correction term” Equation 15 gives the algorithm for propagating the solution of the PDE forward in time from the given set of initial conditions. We now consider how to set the initial conditions. The initial conditions are the values of all the P0i (i = 1, 2, 3 ... , NX) at t = 0. From Figure 4, these are clearly:

P10 P10 P1 = 1 for all time (boundary condition) Pi 00 Pi 0 = 0 at time t = 0 for 2 < i < NX-1 Pi 0 = Po for all time (boundary condition) PNX 0 PNX 0 PNX Let us now apply the above algorithm to the solution of equation 10. For this problem, suppose we take the following data: 0 ≤ x ≤ 1.0 Δx = 0.1 => implies 11 grid points, P1 , P2 , P3, .... , P11 (NX = 11). Δt = 0.001 n +1

The problem is then to calculate the solution, P(x,t), - that is Pi for all i (at all grid points) and all future times up to some final time (possibly when the equation comes to a steady-state as will happen in this example). This can be done by filling in the “solution chart” Table 1 - see Exercise 2 below.


11

Grid Blocks Time x= t

n

0

0

0.1 100

i= P0 i → (IC) P100 i

0.2 200

0.0 0.1 (BC↓) 1 2

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

3

4

5

6

7

8

9

10

1.0 (BC↓) 11

1

0

0

0

0

0

0

0

0

0

0

1

0

P200 i

1

0

0.3 300

P300 i

1

0

0.4 400

P400 i

1

0

0.5 500

P500 i

1

0

0.6 600

P600

1

0

0.7 700

P700 i

1

0

0.8 800 P800 i

1

0

0.9 900

P900 i

1

0

1 1000

P1000 i

1

0

Note:

BC = Boundary Conditions - these points are fixed; IC = Initial Conditions, i.e. values of P(x, t = 0) for all i at t= 0

i

EXERCISE 2.

Fill in the above table using the algorithm (where (Δt/Δx2)=0.1):

Pi n +1 = Pi n +

∆t P n + Pi n−1 − 2 Pi n ) 2 ( i +1 ∆x

Hint: make up a spread sheet as above and set the first unknown block (shown grey shaded in table above) with the above formula. Copy this and paste it into all of the cells in the entire unknown area (surrounded by bold border above). Answer: If you get stuck, look at spreadsheet CHAP6Ex2.xls on the disk.

12

Table 1. Solution chart for the solution of the simple pressure equation


6

The assumption we made in equation 12 was that the spatial derivative was taken at the n (known) time level. This allowed us to develop an explicit formula for Pi n +1 (for all i). This method is therefore known as an explicit finite difference method. We can learn about some interesting and useful features of the explicit method which is encapsulated in equation 15 by simply experimenting with the spreadsheet CHAP6Ex2.xls. Three features of this method can be illustrated by “numerical experiment” as follows: (i)

the effect of time step size, Δt;

(ii) the effect of refining the spatial grid size, Δx (or number of grid cells, NX); (iii) the effect of running the calculation to steady-state as t → ∞ (in practice, until the numerically computed solution stops changing). It is best if you do these yourself by modifying the spreadsheet (CHAP6Ex2.xls).

EXERCISE 3.

Experiment with the spreadsheet in CHAP6Ex2.xls to examine the effects of the three quantities above - Δt, Δx (or NX) and the solution as t → ∞.

HINTS for Exercise 6.3: • Sensitivity to Δt: When you make Δt too big, the predicted numerical solution of the PDE goes badly wrong - indeed negative pressures can occur which is physically impossible. In fact, the solution has become unstable for the larger time steps. This means that this explicit numerical method does have some time step limitations which we must be careful of. • Sensitivity to Δx: the effects of grid refinement are that, as the grid blocks get smaller (i.e. Δx → 0 or NX → ∞). The answer should get more accurate although, to make this happen, you need to reduce the time step as well. • Behaviour as t → ∞ : Finally, considering the long-time behavior of the solution of the PDE, you should find that P(x, t → ∞) tends to a straight line. Some other intermediate pressure profiles can also be plotted using CHAP6Ex2.xls. In fact, a little bit of analysis shows us that this is quite expected. As t → ∞, then if steady-state is reached, then:

 ∂2 P   ∂P    = 0, which implies =>  2  = 0  ∂t   ∂x  But the “curve” with a zero second derivative (i.e. a first derivative which is constant) is a straight line. Therefore, this result is physically reasonable and our numerical model appears to be behaving correctly. Institute of Petroleum Engineering, Heriot-Watt University

13

Another way to represent this explicit finite difference solution to the PDE is shown in Figure 5, where we indicate the time levels for the solution of the equations and we also show the dependency of the unknown pressure ( Pi n +1). n

P i-1

Time

n

∆x

Pi

n

∆x

P i+1 Time level n

i-1

i

i+1

n+1

Pi

n+1

P i+1

∆t Time level n + 1

P i-1

n+1

3.2 Implicit Finite Difference Approximation of the Linear Pressure Equation

We now return to the original finite difference equation 12, where we had to

 ∂2 P   2 make a choice of time level for the spatial derivative,  ∂x  . We now examine the consequences of taking this derivative at the (n+1) time level - this is the “unknown” time level. The finite difference equation for this case is as follows: The time derivative is the same, i.e.

P n +1 − Pi n  ∂P    ≈ i  ∂t  i ∆t

(16)

but the spatial derivative now becomes:

 ∂2 P  Pi n+1+1 + Pi n−1+1 − 2 Pi n +1  2 ≈  ∂x  i ∆x 2

(17)

As before, we now equate the numerical finite difference approximations of each of the above derivatives (as required by the original PDE, equation 10) to obtain:

Pi n +1 − Pi n Pi n+1+1 + Pi n−1+1 − 2 Pi n +1 ≈ ∆t ∆x

(18)

n +1 This equation shouldPibe compared with equation 14. In the previous case, this could n easily be rearranged into Pi +1 an explicit equation for Pi n +1 (equation 15). However, equation 18 above cannot be rearranged to give a simple expression for the pressure n +1 n +1 Pi n+1+1 there now appear to be three unknowns at at the new time step P (n+1), .and Indeed, i −1 , Pi 1 n +1 and Pi n−1+1 , Ptoi n +be time level (n+1), viz. Pi n−1+1 , Pi n +1 and Pi n+1+1 . This appears a bitPiof +1 a paradox: how do we find three unknowns from a single equation (equation 18 above)? The answer is not really too difficult: Basically, we have an equation - like equation 18 - at every grid point. We will show below that this leads to a set of linear equations where we have exactly the same number of unknowns as we have linear equations.

14

Figure 5. Schematic of the explicit finite difference algorithm for solving the simple pressure equation (a PDE).


6

Therefore, if these equations are all linearly independent, then we can solve them numerically. This is a bit more trouble than our earlier simple explicit finite difference method. Because, we do not get an explicit expression for our unknowns - instead we get an implicit set of equations - this method is known as an implicit finite difference method. To see where these linear equations come from, rearrange equation 18 above to obtain:

  ∆x 2  n ∆x 2  n +1 n +1 n + 1 Pi n−1+1 −  2 + − P P + =  i  P i +1   ∆t  i ∆t Pi

(19)

where all the unknowns ( Pi n−1+1 , Pi n +1 and Pi n+1+1 ) are on the LHS of the equation and the term on the RHS is “known”, since it is at the old time level, n. We can write this equation as follows:

ai −1 Pi n−1+1 + ai Pi n +1 + ai +1 Pi n+1+1 = bi

(20)

  ∆x 2  n ∆x 2  and where ai −1 = 1; ai = − 2 + ; 1 and − a b = =  i +1  P i   ∆t  i ∆t 

are all constants. The ai do not change throughout the calculation but the quantity bi is updated at each time step as the newly calculated Pn+1 is set to the Pn for the next time step. We can see how this works for the 5 grid block system in Figure 6 below: 1

2

n

P1

3 n

Time level n

P2

4 n

P3

5

n

P4

n

P5

∆x ∆t

Time level n=1

Figure 6. Simple example of a 5 grid block system showing how the implicit finite difference scheme is applied

P n+1 1

P n+1 2

P n+1 3

Boundary condition (fixed)

P n+1 4

P n+1 5 Boundary condition (fixed)

At each grid point, then (a) we know the value from the boundary condition (i = 1 and i = 5), (b) it uses a boundary condition (i=2 and i=4) or (c) it is specified completely by equation 20 (only i = 3, in this case - but it would be most points for a large number of grid points). Consider each grid point in turn as follows: i = 1 a boundary; therefore P1 is fixed, say as P1


15

P1

+ a2 P2n +1

+

P3n +1 =

=>

a2 P2n +1

+

P3n +1 =

i=2

i=3

i=4

P2n +1

+ a3 P3n +1

P3n +1

+ a4 P4n +1

=>

P3n +1 +

+

 ∆x 2  n − P  ∆t  2  ∆x 2  n −  P − P1  ∆t  2

b2 =

P4n +1 =

+

P5 =

a4 P4n +1

=

b3 =

 ∆x 2  n − P  ∆t  3

 ∆x 2  n − P  ∆t  4  ∆x 2  n −  P − P5  ∆t  4

b4 =

iP=5 5 a boundary; therefore P5 is fixed, say as P5 Therefore there are only three unknowns in the above set of linear equations 1 n +1 ( P2n +as ,P and P4n +1 ) follows: ( P2n +1 , P3n +1 and P4n +1 ) which can be summarised 3

i = 2:

a2 P2n +1

n +1 2

P

i = 3:

+

P3n +1

n +1 3 3

+ aP

n +1 3

P

i = 4:

=

+

n +1 4

=

P

n +1 4 4

+ aP

=

 ∆x 2  n − P −  ∆t  2

P1

 ∆x 2  n − P  ∆t  3  ∆x 2  n − P −  ∆t  4

P5

(21) Note that the set of linear equations above can be represented as a simple matrix equation as follows:

 a2  1  0

1 a3 1

0  1 a4 

 P2n +1  n +1  P3  n +1  P4

    

=

  ∆x 2  n  P2 − −  ∆t    ∆x 2  n −  P3   ∆ t   2 − ∆x  P n −   ∆t  4

 P1       P5   (22)

16


6

The structure of this matrix equation clearer when there are more equations P1 and Pis 12 involved. For example, it is quite easy to show that, if we take 12 grid points instead of the 5 above, we obtain 10 equations (using the two fixed boundary conditions, P1 and P12 ) for the quantities, P2n +1 , P3n +1 , P4n +1 ....P11n +1, of the form: n +1 2a2

P 1  0  0 0  0 0  0  0 0 

n +1 13

, P

n +1 4 0

,0P

n +1 110

....P

0

0

0

0

a3

1

0

0

0

0

0

0

1

a4

0

0

0

0

0

0

0

1

a5

1

0

0

0

0

0

0

1

a6

1

0

0

0

0

0

0

1

a7

1

0

0

0

0

0

0

1

a8

1

0

0

0

0

0

0

1

a9

1

0

0

0

0

0

0

1

a10

0

0

0

0

0

0

0

1

0  0   0   0  0   0  0   0   1  a11 

 P2n +1  n +1  P3  n +1  P4  P n +1  5  P6n +1  n +1  P7  n +1  P8  P n +1  9  P10n +1  n +1  P11

                

 − ( ∆x 2   − ( ∆x 2   − ( ∆x 2   − ( ∆x 2   − ( ∆x 2 =   − ( ∆x 2   − ( ∆x 2   − ( ∆x 2   − ( ∆x 2   − ( ∆x 2 

∆t ) P2n − P1    ∆t ) P3n   ∆t ) P4n  n  ∆t ) P5   ∆t ) P6n   ∆t ) P7n  n  ∆t ) P8   ∆t ) P9n   ∆t ) P10n  ∆t ) P11n − P12 

(23) And 20 grid points in 1D would lead to the following set of 18 equations:

a2 1  0  0 0  0 0  0  0 0  0 0  0  0 0  0 0  0

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  a3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0   1 a4 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0   0 1 a5 1 0 0 0 0 0 0 0 0 0 0 0 0 0  0 0 1 a6 1 0 0 0 0 0 0 0 0 0 0 0 0   0 0 0 1 a7 1 0 0 0 0 0 0 0 0 0 0 0  0 0 0 0 1 a8 1 0 0 0 0 0 0 0 0 0 0   0 0 0 0 0 1 a9 1 0 0 0 0 0 0 0 0 0   0 0 0 0 0 0 1 a10 1 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 1 a11 1 0 0 0 0 0 0 0   0 0 0 0 0 0 0 0 1 a12 1 0 0 0 0 0 0  0 0 0 0 0 0 0 0 0 1 a13 1 0 0 0 0 0   0 0 0 0 0 0 0 0 0 0 1 a14 1 0 0 0 0   0 0 0 0 0 0 0 0 0 0 0 1 a15 1 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 1 a16 1 0 0   0 0 0 0 0 0 0 0 0 0 0 0 0 1 a17 1 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 a18 1   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 a19 

 P2n +1  n +1  P3  n +1  P4  P n +1  5  P6n +1  n +1  P7  n +1  P8  P n +1  9  P10n +1  n +1  P11  n +1  P12  P n +1  13  P n +1  14  P15n +1  n +1  P16  n +1  P17  P n +1  18  P19n +1

                              

 − ( ∆x 2   − ( ∆x 2   − ( ∆x 2   − ( ∆x 2   − ( ∆x 2   − ( ∆x 2   − ( ∆x 2   − ( ∆x 2   − ( ∆x 2 =  −( ∆x 2   − ( ∆x 2   − ( ∆x 2   − ( ∆x 2   − ( ∆x 2   − ( ∆x 2   − ( ∆x 2   − ( ∆x 2   − ∆x 2  (

∆t ) P2n − P1    ∆t ) P3n  n  ∆t ) P4   ∆t ) P5n  n  ∆t ) P6   ∆t ) P7n   ∆t ) P8n   ∆t ) P9n   ∆t ) P10n   ∆t ) P11n   ∆t ) P12n   ∆t ) P13n   ∆t ) P14n   ∆t ) P15n   ∆t ) P16n   ∆t ) P17n   ∆t ) P18n  n ∆t ) P19 − P20 

(24) Institute of Petroleum Engineering, Heriot-Watt University

17

For this simple 1D PDE, it is clear that the matrix arising from our implicit finite difference method has the following properties: (i) It is tridiagonal - that is, it has a maximum of three non-zero elements in any row and these are symmetric around the central diagonal; (ii) It is very sparse - that is, most of the elements are zero. In an MxM matrix, there are only 3M non-zero terms but M2 actual elements. If M = 100, then the matrix is only (300/1002)x100% = 3% filled with non-zero terms. As it happens, a very simple computational procedure, called the Thomas algorithm, can be used to solve tridiagonal systems very quickly. The FORTRAN code for this is shown for interest in Figure 7. Note that it is very compact and quite simple in structure. You are not expected to know this or to necessarily understand how this algorithm works.

18


6

THE THOMAS ALGORITHM

Figure 7. The Thomas algorithm for the solution of tridiagonal matrix systems.

subroutine thomas c Thomas algorithm for tridiagonal systems + (N ! matrix size + ,a ! diagonal + ,b ! super-diagonal + ,c ! sub-diagonal + ,s ! rhs + ,x ! solution + ) c---------------------------------------c This program accompanies the book: c C. Pozrikidis; Numerical Computation in Science and Engineering c Oxford University Press, 1998 c-----------------------------------------------c Coefficient matrix: c | a1 b1 0 0 ... 0 0 0 | c | c2 a2 b2 0 ... 0 0 0 | c | 0 c3 a3 b3 ... 0 0 0 | c | .............................. | c | 0 0 0 0 ... cn-1 an-1 bn-1 | c | 0 0 0 0 ... 0 cn an | c-----------------------------------------Implicit Double Precision (a-h,o-z) Dimension a(200),b(200),c(200),s(200),x(200) Dimension d(200),y(200) Parameter (tol=0.000000001) this is a measure of how close to coverage we are c prepare Na = N-1 c reduction to upper bidiagonal d(1) = b(1)/a(1) y(1) = s(1)/a(1) DO i=1,Na i1 = i+1 Den = a(i1)-c(i1)*d(i) d(i1) = b(i1)/Den y(i1) =(s(i1)-c(i1)*y(i))/Den End DO c Back substitution x(N) = y(N) DO i=Na,1,-1 x(i)= y(i)-d(i)*x(i+1) End DO c Verification and alarm Res = s(1)-a(1)*x(1)-b(1)*x(2) If(abs(Res).gt.tol) write (6,*) “ thomas: alarm” DO i=2,Na Res = s(i)-c(i)*x(i-1)-a(i)*x(i)-b(i)*x(i+1) If(abs(Res).gt.tol) write (6,*) “ thomas: alarm” End DO Res = s(N)-c(N)*x(N-1)-a(N)*x(N) If(abs(Res).gt.tol) write (6,*) “ thomas: alarm” c Done 100 Format (1x,f15.10) Return End Institute of Petroleum Engineering, Heriot-Watt University

19

Finally, we note that the implicit finite difference method can be viewed as shown in Figure 8 below. The choice of the (n+1) time level for the spatial discretisation leads to a set of linear equations which are somewhat more difficult to solve than the simple algebraic equation that arises in the explicit method (equation 15). However, there are exactly the same number of linear equations as there are unknowns and so it is possible to solve these. n

P i-1

Time

n

n

∆x

Pi

∆x

P i+1 Time level n

i-1

i

i+1

n+1 P i-1

n+1 Pi

n+1 P i+1

∆t Time level n + 1

Figure 8. Schematic of the implicit finite difference algorithm for solving the simple pressure PDE

3.3 Implicit Finite Difference Approximation of the 2D Pressure Equation

Before we go on to discuss how to solve the sets of linear equations that arise in the finite difference approximation of PDEs arising in reservoir simulation, we will first consider the discretisation of the 2D single-phase pressure equation. This raises some additional important issues which occur when we try to solve more complicated systems, as follows: (i)

the complication of the more “connected” grid block system in a 2D domain;

(ii) the possibility of heterogeneity in the permeability field which leads us to the matter of how to take average properties in grid-to-grid flows between blocks of different permeability (dealt with in Chapter 4); (iii) the issue of non-linearity for a compressible system e.g. c(P) is clearly a function of P(x,t), which is the “unknown”.

3.3.1 Discretisation of the 2D Pressure Equation

We take as the basic pressure equation for single phase slightly compressible flow, the following (see the solution for exercise 2 at the end of Chapter 5):

cµφ  ∂P  ∂  ˜  ∂P   ∂  ˜  ∂P   kx   +  = k y    k  ∂t  ∂x   ∂x   ∂y   ∂y   k˜ x and k˜ y  cφµ   k 

(25)

 cφµ  k˜ x and k˜ y   (a k simplified form of equation 39, Chapter 5) where the term k is a constant which we will denote as β below, k is an average permeability of the entire reservoir and k˜ x and k˜ y are the local permeabilities normalised by k ; i.e. k non-linearities for k˜ x = k x / k and k˜ y = ky / k. Note that we have avoided the the moment (the quantities c(P), μ and ρ usually depend on pressure). However, the

k

20

k˜ x = k x / k and k˜ y = ky / k.

k˜ x = k x / k and k˜ y = ky / k.


6

system may be anisotropic (kx ≠ ky) and heterogeneous (the permeability may vary from grid block to grid block). Equation 25 above can be discretised in a similar way to that applied to the 1D linear pressure PDE discussed above. However, we will need to be quite clear about our notation in 2D and, for this purpose, we refer to Figure 9. This shows the discretisation grid for the above PDE - note that this is essentially the opposite of what we did when we derived the equation in the first place! In Chapter 5, we used a control volume (or grid block) to express the mass conservation and then inserted Darcy’s law for the block to block flows; we then took limits as Δx, Δy and Δt → 0. Here, we are starting with the PDE and going back to the local conservation of flows and introducing finite size Δx and Δy.

i, j + 1

(j + 1/2) ∆y

i - 1, j

i, j

i + 1, j

(j - 1/2)

Figure 9. Discretisation and notation for the 2D pressure equation.

i, j - 1

y (j)

∆x

(i x (i)

1/ ) 2

(i + 1/2)

Discretising the following equation using the above notation

 ∂P  ∂   ∂P   ∂   ∂P   β   = k˜x    + k˜y     ∂t  ∂x   ∂x   ∂y   ∂y   we obtain:  ˜  ∂P     ∂P   −  k˜ x     k x  ∂x   P −P    i +1/ 2   ∂x   i −1/ 2 β +  ≈   ∆t ∆x n +1

n

(26)  ˜  ∂P     ∂P   −  k˜ y    k y      ∂y   j +1/ 2   ∂y   j −1/ 2 ∆y

(27) where the (i ± 1/2) and (j ± 1/2) subscripts refer to quantities at the boundaries as shown in Figure 9. We can now expand these boundary flows as follows:


21

 P n +1 − Pi ,nj   ˜ β  i, j  =  kx ∆t   

( )

 Pi n+1+,1j − Pi ,nj+1    ˜    −  kx i +1 / 2  ∆x 2   

( )

 +  k˜ y 

( )

 Pi ,nj+1 − Pi n−1+,1j     i −1 / 2  ∆x 2  

 Pi ,nj++11 − Pi ,nj+1    ˜    −  ky j +1 / 2  ∆y 2   

( )

 Pi ,nj+1 − Pi ,nj+−11     j −1 / 2  ∆y 2  

(28)

(˜ )

(˜ )

(˜ )

(˜ )

where the quantities k x i +1/ 2 , k x i −1/ 2 , k y j +1/ 2 and k y j −1/ 2 are some type of average permeabilities between the two neighbouring grid blocks - as discussed in Chapter 4. Note also we have chosen the spatial discretisation terms at the new (n+1) time level making the this an implicit finite difference scheme. We can rearrange equation 28 above by taking all the unknown terms (at n+1) to the LHS and the known terms (at time level n) to the RHS. This gives the following:

( )

( )

( )

( )

( )

( )

( )

( )

 k˜   k˜ x   k˜ x  k˜ y k˜ y k˜ x x β n +1 j −1 / 2 j +1 / 2 n +1 n +1 i −1 / 2 i +1 / 2 i −1 / 2 i +1 / 2   Pi , j    − − P − P + + + + i −1, j i +1, j 2 ∆t  ∆x 2 ∆y 2 ∆y 2  ∆x 2   ∆x 2   ∆x        k˜ x   k˜ y  β n j −1 / 2 j +1 / 2 n +1    Pi ,nj++11 = − P − Pi , j i , j −1 2 2 ∆t  ∆y   ∆y     

(29) Since the coefficients in equation 29 above are constants, then this defines a set of linear equations similar to those found in 1D. However, here we have up to five non-zero terms per grid block to deal with rather than the three we found for the 1D system. The matrix which arises in this 2D case is known as a pentadiagonal matrix. This set of linear equations can be written as follows:

ai −1, j .Pi n−1+,1j + ai +1, j .Pi n+1+,1j + ai , j .Pi ,nj+1 + ai , j −1 .Pi ,nj+−11 + ai , j +1 .Pi ,nj++11 = bi , j (30) where the constant coefficients, ai −1, j , ai +1, j , ai , j , ai , j −1 and ai , j +1 , are given by the coefficients in equation 29; bi, j is also a (know) constant.

3.3.2 Numbering Schemes in Solving the 2D Pressure Equation

It is quite convenient to label the pressures as Pi, j when we are working out the discretisation of the equations, but this is not helpful when we are arranging the linear equations. Here, it is useful first to consider the numbering scheme for the 2D system that allows us to dispense with the (i,j) subscripting in equations 29 or 30 above. The structure of the A - matrix in equation 30 can be made clearer by working out a specific 2D example as shown in Figure 10.

22


j=5 j=4 j=3 j=2 j=1

Figure 10. Numbering system for 2D grid conversion from (i,j) → m counter.

m = 17 m = 13 m=9 m=5 m=1 i=1

m = 18 m = 14 m = 10 m=6 m=2 i=2

m = 19 m = 15 m = 11 m=7 m=3 i=3

6

m = 20 m = 16 m = 12 m=8 m=4 i=4

Notation: NX = maximum number of grid blocks in x - direction, i = NX; NY = maximum number of grid blocks in y - direction, j = NY m = grid block number in the natural ordering scheme shown m = (j - 1).NX + i e.g. for i = 3, j = 4 and NX = 4, m = (4 - 1).4 + 3 = 15 (as above)

In the m-notation shown in Figure 10 (m = (j-1)NX +i), the reordered equations 30 become the following:

a˜ m−1 .Pmn−+11 + a˜ m+1 .Pmn++11 + a˜ m .Pmn +1 + a˜ m− NX .Pmn−+1NX + a˜ m+ NX .Pmn++1NX = bm (31) where the a˜ m are the reordered coefficients where the subscript is calculated from the m-formula in Figure 10. For example:

ai , j −1 → a˜ ( j −1−1) NX +i =( j −1) NX +i − NX → a˜ m− NX

(32)

ai , j +1 → a˜ ( j +1−1) NX +i = ( j −1) NX +i + NX → a˜ m+ NX

(33)

Note that, when we apply the above equation numbering scheme to the example in Figure 10 (NX = 4, NY = 5 and therefore, 1 ≤ m ≤ 20), certain “neighbours” are “missing” since a block is at the boundary (or in a corner where two neighbours are missing). For example, for block (i = 3; j = 3), that is block m = 9, the “j-1 block” is not there. Therefore, the coefficient Am-1 =0 in this case. This is best seen by writing out the structure of the 20 x 20 A-matrix by referring to Figure 10; the A-matrix structure is shown in Figure 11. Note that the A-matrix structure in Figure 11 is sparse and has a maximum of five non-zero coefficients in a given row - it is a pentadiagonal matrix. All implicit methods for discretising the pressure equation lead to sets of linear equations. These have the general matrix form:

A.x = b

(34)

where A is a matrix like the examples shown above, x is the column vector of unknowns (like the pressures) and b is a column vector of the RHSs. This is just like equation 31 but it is in shorthand form. We will discuss methods for solving these equations later in this chapter. For the meantime, we will just assume that it can be solved. We next consider when the PDEs describing a phenomenon are non-linear PDEs.


23

m 1 2 x x 1 x x 2 x 3 4 5 x x 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

3 x x x

4

5 x

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20

x x

x x

x x

x

x

x x

x x

x x x x

x x x

x

x x x x

x x

x x x

x

x x

x x x

x

x x

x

x x x

x x

x x x

x

x x

x x x

x

x x

x

x x x

x x

x x x

x

x x x

x x

m - ordering scheme used in figure 10 - shown here for reference. j=5 j=4 j=3 j=2 j=1

m = 17 m = 13 m=9 m=5 m=1 i=1

m = 18 m = 14 m = 10 m=6 m=2 i=2

m = 19 m = 15 m = 11 m=7 m=3 i=3

Figure 11. Structure of the A-matrix for the m-ordering scheme in the table shown below for reference.

m = 20 m = 16 m = 12 m=8 m=4 i=4

3.4 Implicit Finite Difference Approximation of Non-linear Pressure Equations

We have noted previously that using numerical methods are usually the only way which we can solve non-linear PDEs of the type that arise in reservoir simulation. We will use the single-phase 2D equation for the flow of compressible fluids (and rocks) as the example for discussing the numerical solution of non-linear PDEs. Equation 35 below was derived in Chapter 5 and has the form:

 ∂P  ∂  kx ρ  ∂P   ∂  ky ρ  ∂P   c( P)  =   +  ∂t  ∂x  µ  ∂x   ∂y  µ  ∂y  

(35)

where the non-linearities arise in this equation due to the dependence of the generalised compressibility, c(P), the density, ρ(P) and the viscosity, μ(P), on pressure, P(x,t). It is the pressure that is the main unknown in this equation and, hence, if these quantities depend on it, then they introduce difficulties into the equations. In fact, we will see shortly that the equations that arise are not linear equations - they are non-linear algebraic equations.

24


6

Proceeding as before, we can discretise the above equation as follows:  ky ρ  ∂P    k ρ  ∂P    kx ρ  ∂P    kx ρ  ∂P   −  y   −              P n +1 − P n   µ  ∂x   i +1/ 2  µ  ∂x   i −1/ 2  µ  ∂y   j +1/ 2  µ  ∂y   j −1/ 2 c( P) + =   ∆t ∆x ∆y

(36) where we have not yet decided on the time level of the non-linearities (i.e. the time level of the pressure, Pn or Pn+1, at which we evaluate c, ρ and μ). The most difficult case would be if these were set at the “unknown” time level Pn+1 and this is what we will do as follows:

 k ρ  n +1  ∂P   −  x     n +1 n  µ   ∂x      − P P  i +1 / 2 c( P n +1 ) =   ∆t ∆x

 k ρ  n +1  ∂P    x      µ   ∂x   i −1/ 2

 ky ρ  n +1  ∂P   −      µ   ∂y   j +1/ 2 + ∆y

 ky ρ  n +1  ∂P        µ   ∂y   j −1/ 2 (37)

 ∂P   ∂P   and   terms gives:  ∂x   ∂y 

Now expanding up the 

 k ρ  n +1  Pi n+1+,1j − Pi ,nj+1    k ρ  n +1  Pi ,nj+1 − Pi n−1+,1j   x  x     −  µ    n +1 n µ ∆ x ∆x         − P P    i +1 / 2 i −1 / 2  i, j i, j n +1    c( P ) =  ∆t ∆x    ky ρ  n +1  Pi ,nj++11 − Pi ,nj+1    ky ρ  n +1  Pi ,n1+1 − Pi ,nj+−11        −  µ    ∆y ∆y       µ  j +1/ 2  j −1 / 2  + ∆y

(38)

which can be simplified to the following: n +1

c( P

n +1

n +1

 P n +1 − Pi ,nj   kx ρ   kρ  Pi n+1+,1j − Pi ,nj+1 −  x 2  Pi ,nj+1 − Pi n−1+,1j ) i , j =  2 ∆t  µ.∆x  i −1/ 2    µ.∆x  i +1/ 2

(

n +1

)

 kρ  + y 2 Pi ,nj++11 − Pi ,nj+1  µ.∆y  j +1/ 2

(

)

(

)

n +1

 Pi ,nj+1 − Pi ,nj+−11   kρ  −  y 2  ∆y  µ.∆y  j −1/ 2  

(39)


25

This can be treated as before to gather together similar unknown P-terms as follows: n +1

n +1

 kρ   kρ  −  x 2 .Pi n−1+,1j −  y 2  .Pi ,nj+−11  µ.∆x  i −1/ 2  µ.∆y  j −1/ 2 n +1  c( P n +1 )   k ρ  n +1  k ρ  n +1  ky ρ  n +1  n +1  ky ρ  x x + + +  + + .Pi , j      2 2 2 2  ∆t   µ.∆x  i +1/ 2  µ.∆x  i −1/ 2  µ.∆y  j +1/ 2  µ.∆y  j −1/ 2 

n +1

n +1

 c( P n +1 ).Pi ,nj   kρ   kρ  .Pi ,nj++11 =  − x 2 .Pi n+1+,1j −  y 2   ∆t  µ.∆y  j +1/ 2  µ.∆x  i +1/ 2  

(40) As before, we can write this in a compact form as follows:

α in−+11, j .Pi n−1+,1j + α in++11, j .Pi n+1+,1j + α in, +j 1 .Pi,nj+1 + α in, +j −11 .Pi,nj+−11 + α in, +j +11 .Pi,nj++11 = βin, +j 1 (41) where the elements of theA-matrix (now denoted α in−+11, j , α in, +j −11 , α in, +j 1 , α in++11, j and α in, +j +11 ) are not constants since they depend on the (unknown) value of Pn+1. How do we go about solving the non-linear set of algebraic equations in 39 or 40 above? It turns out that we have two choices in tackling this more difficult problem as follows: (i) We can actually use a numerical equation solver which is specifically designed to solve more difficult non-linear problems. An example of this type of approach is in using the Newton-Raphson method. We will return to this method later (once we have seen how to solve linear equations). (i) We can choose to apply a more pragmatic algorithms such as the following: (a) Although our α-terms in equation 41 are strictly at time level (n+1), simply take them at time level n, as a first guess. So we approximate equation 41 by the following first guess:

α in−1, j .Pi n−1+,1j + α in+1, j .Pi n+1+,1j + α in, j .Pi,nj+1 + α in, j −1 .Pi,nj+−11 + α in, j +1 .Pi,nj++11 = βin, j n n n α in−1, j , α in+1, j , α in, j , α i(42) , j −1 , α i , j +1 and β i , j

Pi ,nj+1

n +1

Pi , at j the where the quantities α in−1, j , α in+1, j , α in, j , α in, j −1 , α in, j +1 and βin, j are evaluated n (known) time level Pi ,nj+1 α ν n - i.e. at values of pressure = Pi, j . i, j

n +1

αν

Pi , 42 j (b) Solve the now linear above to obtain a first estimate - or ia, j first Pi ,nj equations

(

)

ν

iteration - of the Pi ,nj+1at each grid point. We will use the following notation ⇒ ν

(

α iν, j

where ν is the iteration counter and α i , j is the value of the a-coefficient at the Pi ,nj+1 value after ν iterations; that is:

[(

α iν, j = α i , j Pi,nj+1

26

)] ν

(P )

n +1 ν i, j

(43)

)

ν

(P )

n +1 ν i, j


6

Pi ,nj+1 ν (c) Use the latest iterated values of the α i , j to solve the (now linear) equations to

go from

(P )

n +1 ν i, j

(

to Pi ,nj+1

)

ν +1

.

(

)

n +1 ν

(d) Keep iterating the above scheme untilPit i , j converges; i.e. the difference between the sum of two successive iterated values of the pressure (Err.) is sufficiently small (< Tol, which is an acceptably small value)

Err =

∑ (P )

n +1 ν +1 i, j

(

− Pi ,nj+1

all i , j

)

ν

(44)

Stop if Err < Tol. Otherwise continue through steps above. The algorithm outlined in (ii) above for solving the non-linear pressure equation is represented in Figure 12. Set the iteration counter ν = 0

Calculate the α iν−1, j , α iν+1, j , α iν, j , α iν, j −1 , α iν, j +1 and β iν, j at

( )

n values of pressure = Pi , j

[(

α iν, j = α i , j Pin, j+1

ν = ν +1

)] ν

ν

where ν = current iteration number.

Solve the set of linear equations: α iν−1, j .Pin−+11, j + α iν+1, j .Pin++11, j + α iν, j .Pin, j+1 + α iν, j −1 .Pin, j+−11 + α iν, j +1 .Pin, j++11 = β iν, j

(

to obtain Pin, j+1 = Pin, j+1

)

ν +1

Calculate Err =

∑ (P )

n +1 ν +1 i, j

all i , j

No - continue iterations

Figure 12. Algorithm for the numerical solution of the non-linear 2D pressure equations:

(

− Pi ,nj+1

)

ν

Err < Tol ?

Yes (the method has converged)

Stop


27

4 APPLICATION OF FINITE DIFFERENCES TO TWO-PHASE FLOW 4.1 Discretisation of the Two-Phase Pressure and Saturation Equations

We now consider how finite difference methods are applied to the two-phase flow equations. We have seen how these equations were derived in Chapter 5. Recall that, in two-phase flow, we have two coupled equations to solve - a pressure equation and a saturation equation. For example, we may solve for the oil pressure, Po(x,t) and the oil saturation, So (x,t); we would then find the water saturation and water pressure by using the constraint equations, So+Sw = 1 and the capillary pressure relation, Pc(Sw) = Po - Pw, respectively. From Chapter 5, we use the highly simplified form of the 1D pressure and saturation equations, where we choose to solve for ⇒ P(x,t) and So (x,t) as shown in Chapter 5, equations 81 and 82. Note that we have taken zero capillary pressure (therefore P = Po = Pw) and zero gravity which gives: PRESSURE EQUATION

SATURATION EQUATION

∂   ∂P   λ T ( So )   = 0   ∂x   ∂x   ∂S  ∂   ∂P   φ  o  = λ o ( So )    ∂t  ∂x   ∂x  

(45)

(46)

(Equations 45 and 46 are the same as equations 81 and 82 from Chapter 5, respectively). λ (S ) T

o

The quantity λ T ( So ) is the total mobility and is the sum of the oil and water mobilities;

λ T ( So ) = λ o ( So ) + λ w ( So ) . The above two equations are clearly coupled together since

the oil saturation appears in the non-linear coefficient of the pressure equation. Likewise, = λthe λ T ( So ) in (So ) + λterm (Soon) the RHS of the saturation equation. the pressure appears o flow w We can now apply finite differences to each of these equations in the same way as discussed above to obtain, for the pressure equation (see notation in Figure 8):

   ∂P    ∂P   − λ T ( So )   λ T ( So ) ∂x    ∂x   i −1/ 2   i +1 / 2  =0 ∆x

(47)

which can be expanded further to give:

   Pi n+1+1 − Pi n +1    Pi n +1 − Pi n−1+1   λ S − λ S ( ) ( )      T o    i +1/ 2  T o    i −1 / 2 ∆x ∆x  =0 ∆x (48)

28


(λ (S )) T

(λ (S )) T

o

o

(

)

n +1 and λSTo(So )

i +1 / 2

i +1 / 2

(

)

and λ T (So )

i −1 / 2

i −1 / 2

(

)

(

6 )

In equation 48 above, we must now specify the time non-linear λ Tlevel λ T (Somobility (So )of the and )

(

( ( ) )

)

and λ T (Sλon +T1()So )

terms, Son +1 λ T (So )

i +1 / 2

i −1 / 2

; we must choose whether we take these terms i +1 / 2 i +1 / 2i −1 / 2 at time n or at time level (n+1)? Again, we go straight to the most difficult case by n +1 defining these terms at the later time level i.e. at So . These terms are denoted as,

(λ (S S )) and (λ (S )) , in equation 48 above to obtain:  (λ ( S ))   (λ ((λS (S)) ))  λ S ) (   ( ) (λ (S )) ∆x ( P − P ) −  ∆x ( P − P ) = 0     (λ (S )) (49) (λ (S )) T

n +1 n +1 o o

T

n +1 n +1 o T i +1 / 2 o n +1 T i +21 / 2 o i −1 / 2

T

T

i +1 / 2

n +1 o

n +1 o

i −1 / 2

n +1 i +1

i

n +1 nT+1 o i +1 / 2 o n +1 i −1 / 2 i 2

T

n +1

T

n +1 o

n +1 i −1

i −1 / 2

i −1 / 2

The above equation can now be arranged into the usual order as follows:

(λ (S )) T

n +1 o

∆x n +1 i −1

P

i −1 / 2

2

, Pi

n +1

n +1 i −1

P

(

)

 λ ( Son +1 ) T −  ∆x 2 

i −1 / 2

(λ (S )) + T

n +1 o

∆x

i +1 / 2

2

(

)

 λ ( S n +1 )  P n +1 + T o  i ∆x 2 

i +1 / 2

Pi n+1+1 = 0

(50)

n +1 i +1

and P

This is a non-linear set of algebraic equations for the unknown pressures, Pi n−1+1 , Pi n +1 and Pi n+1+1 , but the coefficients depend on the - also unknown - saturations, Son +1. Note that equation 50 represents one of a set of equations since there is one at each grid point (and at the ends of the 1D system the values may be set by the boundary conditions). Son +1 We now consider the discretisation of the saturation equation 46. Finite differences may be applied to this equation as follows:

   ∂P    ∂P   − λ o ( So )   λ o ( So ) ∂x    ∂x   i −1/ 2 S −S    i +1 / 2  φ =  ∆t  ∆x n +1 oi

n oi

(51)

Again, we can expand the derivative terms at the (i+1/2) and (i-1/2) boundaries (Figure 9) and we can take the mobility terms at the (n+1) time level to obtain:

(

)

n +1   Soin+1 − Soin   λ o ( So ) φ =  ∆t   ∆x 2 

i +1 / 2

(P

n +1 i +1

− Pi

n +1

(

)

  λ ( Son +1 ) ) −  o ∆x 2  

i −1 / 2

(P i

n +1

n +1 i −1

−P

 ) 

(52) The above non-linear algebraic set of equations can be written in various ways. Two particularly useful ways to write the above equations are as follows:


29

Form A: n +1 oi

S

(

)

n +1  ∆t  λ o ( So ) =S +  φ  ∆x 2  n oi

i +1 / 2

(P

n +1 i +1

− Pi

n +1

(

)

  λ ( Son +1 ) ) −  o ∆x 2  

i −1 / 2

(P i

n +1

−P

S

 )  

(53)

or Form B: n +1 oi

n +1 i −1

(

)

n +1  ∆t  λ o ( So ) −S −  φ  ∆x 2   n oi

i +1 / 2

(P

n +1 i +1

− Pi

n +1

(

)

  λ ( Son +1 ) ) −  o ∆x 2  

i −1 / 2

 n +1 n +1   − P P (i i −1 )  = 0   (54)

The reason for writing the above two forms of the saturation equation is that each is useful, depending on how we intend to approach the solution of the coupled pressure (equation 45) and saturation (equation 46) equations. As with the case of the solution of the non-linear single phase pressure equation for a compressible system, we have two strategies which we can use to solve the twophase equations above, as follows: (i) We can actually use a numerical equation solver which is specifically designed to solve more difficult non-linear problems; e.g. the Newton-Raphson method. We will return to this method later (once we have seen how to solve linear equations). (i) We can again choose to apply a more pragmatic algorithm and this is the subject of section 4.2 below.

4.2 IMPES Strategy for Solving the Two-Phase Pressure and Saturation Equations

The form of the discretised pressure and saturation equations which we will take are as follows (equations 50 and 53): Pressure:

(λ (S )) T

n +1 o

∆x

i −1 / 2

2

n +1 i −1

P

(

)

 λ ( Son +1 ) T −  ∆x 2 

i −1 / 2

(λ (S )) + T

n +1 o

∆x

i +1 / 2

2

(

)

 λ ( S n +1 )  P n +1 + T o  i ∆x 2 

i +1 / 2

Pi n+1+1 = 0

(50) Saturation: Soin+1 + Soin −

30

(

)

n +1  ∆t  λ o ( So )  φ  ∆x 2 

i +1 / 2

(

)

  λ ( Son +1 ) n +1 n +1  − P P ( i +1 i ) −  o ∆x 2  

i −1 / 2

 n +1 n +1   − P P (i i −1 )    (53)


6

Pressure equation 50 would be a set of linear equations if the coefficients were known at the current time step (n), rather than being specified at the (n+1) - i.e. the unknown - time level. However, we could solve equation 50 above as if it were Pi n−1+1 , Pi n +1 and Pi n+1+1 n +1 1 n +nthe n +(see a linear system of equations by time-lagging the coefficients Pi n−1+1as , Pbefore and P +i1 1+1 and Pi n+1+1 P i i −1 , iP algorithm in Figure 13). This would give us a “first guess” (or first iteration) to find Pi n−n1++11, Pi nn++11and Pi n+n1++11 the unknowns i.e. the quantities Pin−+11 , Pi and Pi +1 . The same problem exists P n +1 forPthe equation 53, if wei had the first guess at the Pi n +1 (by thePprocedure n +1saturation n +1 n +1 i , P and P i −1 i i +1 +1 just mentioned), then we could use latest values of pressure and still time lag Pi nthese n +1 the coefficients (the oil mobility Sterms) Pni +1 and use the saturation equation 53 as if it +1 o n an +1 explicit expression. This would give us an updated Sonvalue were of Son +1 which Pi could then be used back in the pressure Sonn++11 equation and the whole process could be ν ν So n +1 νpreviously. iterated until convergence, as discussed A flow chart of this procedure Pi −1 , Pi n +1 and Pi n+1+1 .n +1 ν n +1n +ν1 ν n +1n +ν1 ν n + 1 hasSalready been outlined in Chapter 5 (Figure 9) but is elaborated Figure and ,here . P Pi Pin , Pi Pi +1 and i −1 i −1 o 13. Note that by taking time-laggednvalues ofn +the +1 ν 1 ν saturations, n +1 νthe pressure equation Pi −n1 +1 ν, Pi n +1 ν and Pi +n1 +1 ν. is linearlised and can then be solved for the pressure for that iteration, ν Pin−+implicitly 11 ν , Pin +1 ν and Pin++11 ν .

( ) ( ) ( () ) ( ) ( ) (( )) (( )) (( )) (( )) ( P ) , ( P ) and ( P )P obtained ⇒ ( P ) , ( P ) and ( P ) . The saturations can then (be ) , ( P( Pexplicitly ) )and, ( P( P ) )and using the latest pressures (i.e. the ( P ) , ( P ) and ( P ) ) and the most recent ( P ) , ( P ) and ( P ) iteration of the saturation (i.e. the ( S ) , ( S ) etc.). Therefore, this approach is , ( PIMPES ) , (S(S Explicit ) )etc. , ( S ) etc. ( P as) the ) and ( P ) which stands for IMplicit(Sin Pressure, known approximation, S )it,converges in Saturation. This can be iterated(until although there are limitations in ((SS )) etc. , etc. S ( ) the size of time step, Δt, which can be taken. If Δt is too large, the IMPES method may (Sbecome ) , (Sunstable ) etc.and give unphysical results like the example in Exercise 3 n +1 ν i −1

i

n +1 ν i −1

i

n +1 ν i

above.

n +1 ν

n +1 ν

n +1 ν i +1

n +1 ν i +1

n +1 ν i +1

n +1 ν i −1

i −1

i

i +1

n +1 ν i −n1 +1 ν ni −+1 ν i

n +1 ν i n +1 ν in +1 ν i +1

n +1 ν i n +1 ν i

n +1 ν i +n1+1 ν i +1

n +1 ν i +n1 +1 ν i +1 n +1 ν i


i

ν ν n +1n + 1 i −1

ν n +1n +ν1 i i +1

n +1n +ν1 ν i +1i

n +1 ν i +1

31

Set the iteration counter to zero, ν = 0 Take current values of Son and P n as the ν level iteration values: ⇒ Soν and P ν

PRESSURE EQUN. Set the total mobilities ⇒ λ T (Soν )i +1/ 2 and λ T (Soν )i −1/ 2 obtain the linearised pressure equation:

(λ (S ))

(

)

(

)

(

)

 λ (Soν )  λ T (Soν ) λ T (Soν ) T i −1 / 2 i +1 / 2  n +1 i +1 / 2 Pin−+11 −  + Pi + Pin++11 = 0 2 2   ∆x ∆x ∆x ∆x 2   Solve this linear implicit pressure equation for pressure at iteration, ν, to obtain: T

ν o

i −1 / 2

2

⇒ ( Pin−+11 ) , ( Pin +1 ) and ( Pin++11 ) ν

ν

ν

SATURATION EQUN. Now update the saturation equation as if it were explicit by taking ν ν (i) the oil mobility terms at the latest iteration, ν ⇒ λ 0 (So )i λ 0 (So )i +1 , etc. ν ν ν (ii) the latest values of the pressures just calculated implicity ⇒ ( Pin−+11 ) , ( Pin +1 ) and ( Pin++11 ) Update saturation explicity as follows: Soin+1 = Soin +

(

)

ν  ν ν ∆t  λ o (So ) i +1/ 2 (Pin++11 ) − (Pin +1 )  φ  ∆x 2 

(

to obtain latest iteration of saturations

)

(

)

  λ (Soν ) ν ν i −1 / 2 − o (Pin +1 ) − (Pin−+11 )   ∆x 2  

(

)

    

⇒ (Sin +1 ) , (Sin++11 ) etc. ν

ν

ν = ν +1

Calculate the error, Err., by comparing the change in pressure or saturations (or both as here) at the current (ν) and last (ν-1) iterations: NX

Err. = ∑ ( Pin +1 ) − ( Pin +1 ) i =1

ν

ν −1

NX

+ ∑ (Sin +1 ) − (Sin +1 ) ν

ν −1

i =1

Note that the Err. term above would have some scaling factors weighing the pressure and saturation terms because of the units of pressure.

No - continue iteration

32

Is Err. < Tol ?

Yes

Stop

Figure 13. IMPES Algorithm for the numerical solution of the two-phase pressure and saturation equations (IMPES ⇒ IMplicit in Pressure, Explicit in Saturation)


5

6

THE NUMERICAL SOLUTION OF LINEAR EQUATIONS

5.1 Introduction to Linear Equations

In all implicit finite difference methods, we end up with a set of linear equations to solve. In fact, we have shown above that the equations involved - e.g. the discretised pressure equation - may be a set of non-linear algebraic equations. However, these can usually be linearised (say, by time lagging the coefficients as discussed above) in order to obtain a set of linear equations. The solution to the original non-linear equations may then be found by repeating or iterating through the cycle of solving the linear equations. In the following section, we will address the issue of solving the non-linear equations which arise in the discretised two-phase flow equations directly. A generalised set of linear equations may be written in full as follows:

a11 . x1 + a12 . x2 + a13 . x3 + a14 . x4 + a15 . x5 ... + a1n . xn = b1 a21 . x1 + a22 . x2 + a23 . x3 + a24 . x4 + a25 . x5 ... + a2 n . xn = b2 a31 . x1 + a32 . x2 + a33 . x3 + a34 . x4 + a35 . x5 ... + a3n . xn = b3 ..... .....

... ...

..... ... an1 . x1 + an 2 . x2 + an 3 . x3 + an 4 . x4 + an 5 . x5 ... + ann . xn = bn

(55)

where the aij and bi are known constants and the xi (i = 1, 2, ... , n) are the unknown quantities which we are trying to find. The xi in our pressure equation, for example, would be the unknown pressures at the next time step. The set of linear equations can be written in matrix form as follows:

a11   a21   a31  ..... .....  .....  an1

a12

a13

a14

a22

a23

a24

a32

a33

a34

an 2

an 3

an 4

a15 ... a1n   x1  b1          a25 ... a2 n   x2  b2          a35 ... a3n   x3  = b3      ...  .  .  ...  .  .      ...  .  .   an 5 ... ann   xn  bn 


(56)

33

We can write the matrix A and the column vectors x and b as follows:

A

a11   a21   = a31  ..... .....  .....  an1

a12

a13

a14

a22

a23

a24

a32

a33

a34

an 2

an 3

an 4

a15 ... a1n    a25 ... a2 n    a35 ... a3n  ;  ...  ...   ...   an 5 ... ann 

 x1       x2      x =  x3 ; .  .    .  x   n

and

b1      b2      b = b3    .  .    .  b   n (57)

and the set of linear equations can be written in very compact form as follows:

A.x = b

(58)

where A in an n x n square matrix and x and b are n x 1 column vectors. A very simple example of a set of linear equations is given below:

4. x1 + 2. x2 + 1. x3 = 13 1. x1 + 3. x2 + 3. x3 = 14 2. x1 + 1. x2 + 2. x3 = 11

(59)

How do we solve these? In the above case, you can do a simple trial and error solution to find that x1 = 2, x2 = 1 and x3 = 3, although this would be virtually impossible if there were hundreds of equations. Remember, there is one equation for every grid block in a linearised pressure equation in reservoir simulation (although there are lots of zeros in the A matrix). This implies that we need a clear numerical algorithm that a computer can work through and solve the linear equations. The overview of approaches to solving these equations is discussed in the next section.

5.2 General Methods for Solving Linear Equations

Firstly, let us note that there is a vast literature on solving sets of linear equations and many books on the underlying theory and on the numerical techniques have appeared. Indeed, petroleum reservoir simulation has led to the development of some of these numerical techniques. We will not cover much of this huge subject but we will cover enough that the student can appreciate - rather than understand in detail - how the linear equation are solved in a simulator.

34


6

Basically, there are two main approaches for solving sets of linear equations involving direct methods and iterative methods as follows: (i) Direct Methods: this group of methods involves following a specific algorithm and taking a fixed number of steps to get to the answer (i.e. the numerical values of x1, x2 .. xn). Usually, this involves a set of forward elimination steps in order to get the equations into a particularly suitable form for solution (see below), followed by a back substitution set of steps which give us the answer. An example of such a direct method is Gaussian Elimination. (ii) Iterative Methods: in this type of method, we usually start off with a first guess (or estimate) to the solution vector, say x(o) . Where we take this as iteration zero, v = 0. We then have a procedure - an algorithm - for successively improving this guess by iteration to obtain, x(1) → x(2) → x(3) .... → x(ν) etc. If it is successful, then this iterative method should converge to the correct x as ν increases. The solution should get closer and closer to the “correct” answer but, in many cases, we cannot say exactly how quickly it will get there. Therefore, iterative methods do not have a fixed number of steps in them as do direct methods. Examples of iterative methods are the Jacobi iteration, the LSOR (Line Successive Over Relaxation) method, etc. The following two sections will discuss each of these approaches for solving sets of linear equations in turn.

5.3 Direct Methods for Solving Linear Equations

In fact, we will not present the details of any direct method for solving linear equations. Instead, we will discuss a general outline of how these methods work. Starting with the basic linear equation in matrix form:

A.x = b

(60)

where A in an n x n square matrix and x and b are n x 1 column vectors. We can write the n x (n+1) augmented matrix of A and b coefficients as follows:

a11   a21   a31  ..... .....  .....  an1

a12

a13

a14

a22

a23

a24

a32

a33

a34

an 2

an 3

an 4

a15 ... a1n b1    a25 ... a2 n b2    a35 ... a3n b3   ... ... ... ...  ... ...  an 5 ... ann bn 

(61)

Remember that any mathematical operation we perform on one of the linear equations (e.g. multiplying through by x2) must be performed on both sides of the equation i.e. Institute of Petroleum Engineering, Heriot-Watt University

35

on the aij and the bi. Therefore, suppose we can transform the above augmented matrix into the following form (we do not say how this is done, just that it can be done):

c11   0   0  ..... .....  ..... 0 

c12

c13

c14

c22

c23

c24

0

c33

c34

0

0

0

c15 ... c1n e1    c25 ... c2 n e2    c35 ... c3n e3   ... ... ... ...  ... ... 0 cnn en  ...

(62)

The C-matrix above is in upper triangular form i.e. only the diagonals and above are non-zero (cij ≠ 0, if j ≥ i) and all elements below the diagonal are zero (cij = 0, if i > j), as shown above. Now consider why this particular form is of interest in solving the original equations. The reason is that, if we have formed the augmented matrix correctly (i.e. doing the same operations to the A and b coefficients), then the following matrix equation is equivalent to the original matrix equation: i.e.

A.x = b

<=>

C.x = e

c12

c13

c14

c22

c23

c24

0

c33

c34

0

0

(63)

Therefore,

c11   0   0  ..... .....  ..... 0 

0

c15 ... c1n   x1  e1          c25 ... c2 n   x2  e2          c35 ... c3n   x3  = e3      ..  .  .  ..  .  .      ..  .  .  0 cnn   xn  en  ...

(64)

This matrix equation is very easy to solve, as can be seen by writing it out in full as follows:

36


6

c11 . x1 + c12 . x2 + c13 . x3 + c14 . x4 + c15 . x5 ... + c1n . xn = e1 c22 . x2 + c23 . x3 + c24 . x4 + c25 . x5 ... + c2 n . xn = e2 c33 . x3 + c34 . x4 + c35 . x5 ... + c3n . xn = e3 c44 . x4 + c45 . x5 ... + c4 n . xn = e4 ...... ......

... ...

cn −1, n −1 . xn −1 + cn −1, n . xn = en −1 cnn . xn = en (65) We can easily solve this equation by back-substitution, starting from the equation n which only has one term to find xn, this xn is then used in the (n-1) equation (which has 2 terms) to find xn-1 etc. as follows:

cnn . xn = en

cn −1, n −1 . xn −1 + cn −1, n . xn = en −1

=>

=>

xn = en / cnn now use xn in ..  e − c .x  xn −1 =  n −1 n −1, n n  use xn −1 , xn in .. cn −1, n −1  

cn − 2, n − 2 . xn − 2 + cn − 2, n −1 . xn −1 + cn − 2, n . xn = en − 2

e - cn − 2, n −1 . xn −1 − cn − 2, n . xn  => xn − 2 =  n − 2  etc. cn − 2, n − 2   (66) Hence, by working back through these equations in upper triangular form, we can calculate xn, xn-1, , xn-2.... back to x1.


37

EXERCISE 4.

Solve triangular matrix: + 1example = upper 4 x -the1following x − 2 x simple x + 1of x an 5 1

2

3

4 x1 - 1x2 − 2 x 2 − 2 x3 + 2 x2 − 3 x3 +

4

2 x3 1x4 2 x3 4 x4 3 x3 2 x4

5

+ 1 x 4 + 1x 5 = 5 + 2 x5 = 12 + 1x4 + 2 x5 = 12 + 1x5 = 30 + 4 x4 + 1x5 = 30 − 1x 5 = 3 2 x 4 − 1x 5 = 3 3 x5 = 15 3 x5 = 15

Answer Answer:

Answer x1 = ..........; x2 = ..........; x3 = ..........; x4 = ..........; x5 = .......... x1 = ..........; x2 = ..........; x3 = ..........; x4 = ..........; x5 = ..........

5.4 Iterative Methods for Solving Linear Equations

As noted above, the idea in an iterative method for solving a set of linear equations is to make a first guess at the solution and then to refine it in a stepwise manner using a suitable algorithm. This procedure should gradually converge to the correct answer. We illustrate the idea of such a method using a very simple iterative scheme. Note that this simple point iterative scheme will work for some of the examples we try here but it is not one that is recommended for use in reservoir simulation. However, it adequately illustrates the main ideas which you need to know for the purposes of this part of the course. Our simple scheme starts with the longhand version of a set of linear equations and, for our purposes, we will just take a set of four linear equations as follows:

a11 . x1 + a12 . x2 + a13 . x3 + a14 . x4 = b1 a21 . x1 + a22 . x2 + a23 . x3 + a24 . x4 = b2 a31 . x1 + a32 . x2 + a33 . x3 + a34 . x4 = b3 a41 . x1 + a42 . x2 + a43 . x3 + a44 . x4 = b4 We can rearrange the above set of equations as follows:

38

(67)


x1 =

1 b1 − ( a12 x2 + a13 x3 + a14 x4 ) a11

x2 =

1 b2 − ( a21 x1 + a23 x3 + a24 x4 ) a22

x3 =

1 b3 − ( a31 x1 + a32 x2 + a34 x4 ) a33

x4 =

1 b4 − ( a41 x1 + a42 x2 + a43 x3 ) a44

[

6

]

[

]

[

]

[

]

(68)

The resulting equations are precisely equivalent to the original set although, if anything, they look a bit more complicated. Why would we deliberately complicate the situation? In fact, the above reorganised equations forms the basis for an iterative scheme, which we can use to solve the equations numerically. Firstly, we need to introduce some notation as follows: Notation: (ν ) xi(v)

xi

-

the solution for xi at iteration ν; where ν is the iteration counter.

x ( 0 ) , x (ν )

-

the first guess and νth iteration of the solution vector, x.

Err.

-

an estimate of the error in the iteration scheme from the ν to the (ν+1) iteration

Tol

-

some “small” quantity which determines the acceptable error in the iteration scheme; if Err. < Tol, then the scheme has converged.

Using the above notation in equations 68 above, gives the following simple iteration scheme:


39

[

]

[

]

[

]

[

]

x1(ν +1) =

1 b1 − ( a12 . x2(ν ) + a13 . x3(ν ) + a14 . x4(ν ) ) a11

x2(ν +1) =

1 b2 − ( a21 . x2(ν ) + a23 . x3(ν ) + a24 . x4(ν ) ) a22

x3(ν +1) =

1 b3 − ( a31 . x1(ν ) + a32 . x2(ν ) + a34 . x4(ν ) ) a33

x4(ν +1) =

1 b3 − ( a41 . x1(ν ) + a42 . x2(ν ) + a43 . x3(ν ) ) a44

(69)

This iteration scheme may now be applied as follows: (i)

Make an initial guess at the solution, iteration ν = 0: x ( 0 ) = x1( 0 ) , x2( 0 ) , x3( 0 ) , x4( 0 )

(ii)

Update the solution to the next iteration ν+1 using equations 69 4

Err. = ∑ xik +1 − x ki

(iii) Estimate an Error term (Err.) by comparing the latest with the previous i =1 iteration of the unknowns, e.g.: 4

Err. = ∑ xiν +1 − xiν i =1

(iv) Is Err. < Tol. If yes - the scheme has converged; if no - go back to step (ii) and continue the iterations. This scheme is now illustrated with a practical example. Example: Solve the following set of equations using the iterative scheme above.

3.1x1 0.2 x1

- 0.32 x2 + 0.5 x3 + 2.1x2

= 13.92

+ 0.33 x3 + 0.21x4 = 5.63

0.23 x1 - 0.32 x2 + 4.0 x3 0.42 x1 + 0.22 x2 +

40

+ 0.1x4

0. 5 x 3

+ 0.3 x4 + 5.2 x4

= 15.19 = 16.76


6

(0) (0) (0) (0) Take the first guess: x1 = 2, x2 = 2, x3 = 2, x4 = 2

Hint: reorganise the above equations as follows:

(

x1(ν +1) = (1 / 3.1) * 13.92 - (- 0.32 x2(ν ) + 0.5 x3(ν )

(

(0.20 x

(

(0.23x

(

( 0.42 x

x2(ν +1) = (1 / 2.1) * 5.63 x3(ν +1) = (1 / 4.0) * 15.19 x4(ν +1) = (1 / 5.2) * 16.76 -

( ν + 1) 1

)

+ 0.1x4(ν ) )

+ 0.33 x3(ν ) + 0.21x4(ν )

(ν +1) 1

(ν +1) 1

-

0.32 x2(ν +1)

))

+ 0.30 x4(ν )

+ 0.22 x2(ν +1) +

0.5 x3(ν +1)

))

))

EXERCISE 5.

Fill in the table below for 10 iterations using a calculator or a spreadsheet: POINT ITERATIVE SOLUTION OF LINEAR EQUATIONS Iteration counter ν First guess = 0 1 2 3 4 5 6 7 8 9 10

x at iter. ν x1 x2 2.0 2.0

x3 2.0

x4 2.0

Iteration counter x at iter. ν ν x1 x2 x3 x4 In some First cases, it may guess = 0be possible 2.0 to develop 2.0 improved 2.0 iteration 2.0 schemes by using the latest information 1that is available. For example, in the above iteration scheme, 4.309677 1.97619 3.6925 2.784615 (ν+1) 2 4.008926 1.411795 3.498943 2.436331 we could use the very latest value of x1 when we are calculating x2(ν+1) since we 3 3.993119 1.505683 3.497206 already have this quantity. Likewise, we can use both x1(ν+1) 2.503112 and x2(ν+1) when we 4 4.000937 1.500783 3.500617 2.500584 (ν+1) calculate x3 etc as shown below: 5 3.999963 1.499755 3.499965 2.499832 6 3.999986 1.500026 3.499995 2.500017 7 4.000003 1.5 3.500002 2.500001 8 4 1.499999 3.5 2.5 Converged ⇒ 9 4 1.5 3.5 2.5 10 4 1.5 3.5 2.5 Note - converges fully at k = 9 iterations.


41

(

x1(ν +1) = (1 / 3.1) * 13.92 - (- 0.32 x2(ν ) + 0.5 x3(ν )

(

(0.20 x

(

(0.23x

(

( 0.42 x

x2(ν +1) = (1 / 2.1) * 5.63 x3(ν +1) = (1 / 4.0) * 15.19 x4(ν +1) = (1 / 5.2) * 16.76 -

( ν + 1) 1

(ν +1) 1

(ν +1) 1

)

+ 0.1x4(ν ) )

+ 0.33 x3(ν ) + 0.21x4(ν )

-

0.32 x2(ν +1)

))

+ 0.30 x4(ν )

+ 0.22 x2(ν +1) +

0.5 x3(ν +1)

))

))

x1(ν +1) , x2(ν +1) , x3(ν +1) => underlined terms, imply they are at the latest time

available).

When this is done, we find the following results: IMPROVED POINT ITERATIVE SOLUTION OF LINEAR EQUATIONS (Using latest information) - see spreadsheet CHAP6Ex5.xls - Sheet 2 Iteration counter ν First guess = 0 1 2 3 4 5 Converged ⇒ 6 7 8 9 10

x at iter. ν x1 2 4.309677 4.021247 3.999376 3.999973 3.999998 4 4 4 4 4

x2 2 1.756221 1.495632 1.500005 1.499974 1.5 1.5 1.5 1.5 1.5 1.5

x3 2 3.540191 3.501408 3.500161 3.499997 3.5 3.5 3.5 3.5 3.5 3.5

x4 2 2.460283 2.498333 2.500035 2.500004 2.5 2.5 2.5 2.5 2.5 2.5

Note - converges fully at ν = 6 iterations.

Clearly, comparing this table with the previous one using the simple point iterative method, we see that this method does indeed converge quicker. Although this is just one example, it is generally true that using the latest information in an iterative scheme improves convergence. Notes on iterative schemes: there are several point to note about iterative schemes, which we have not really demonstrated or explained here. However, you can confirm some of these points by using the supplied spreadsheets. These points are: (i) Iterative solution schemes for linear equations are often relatively simply to apply - and to program on a computer (usually in FORTRAN). If you study the supplied spreadsheets (CHAP6Ex5.xls), you can see that they are quite simple in structure for this set of equations. 42


6

(ii) The convergence rate of an iterative scheme may depend on how good the initial guess (x(0)) is. If you want to demonstrate this for yourself, run the spreadsheet (CHAP6Ex5.xls) with a more remote initial guess. Here is the same example as that presented above with a completely absurd initial guess: POINT ITERATIVE SOLUTION OF LINEAR EQUATIONS - Bad First Guess (you can use CHAP6Ex5.xls to confirm these results) Iteration counter ν First guess = 0 1 2 3 4 5 6 7 8 9 10 11 12 Converged ⇒ 13 14 15

x at iter. ν x1 900 -1017.45 63.79526 55.59796 1.890636 4.326953 4.090824 3.987986 4.001064 4.000117 3.999963 4.000004 4 4 4 4

x2 -2000 -2454.69 406.2002 -20.1756 -2.67691 2.668945 1.389409 1.49622 1.502872 1.499593 1.500013 1.500006 1.499999 1.5 1.5 1.5

x3 456 -1932.95 -131.922 4.491703 -0.53117 3.538073 3.519613 3.491895 3.500729 3.500025 3.499982 3.500003 3.5 3.5 3.5 3.5

x4 23000 -28.7 375.1144 -6.43019 0.84584 3.234699 2.420476 2.495457 2.50191 2.499722 2.500005 2.500004 2.499999 2.5 2.5 2.5

(iii) We cannot tell in advance how many iterations may be required in order to converge a given iterative scheme. Clearly, from the results above, a good initial guess helps. In reservoir simulation, when we solve the pressure equation, a good enough guess of the new pressure is the values of the old pressures at the last time step. If nothing radical has changed in the reservoir, then this may be fine. However, if new wells have started up in the model or existing wells have changed rate very significantly, then the “pressure at last time step” guess may not be very good. However, with a very robust numerical method, convergence should still be achieved. (iv) It helps in an iterative method to use the latest information available. This was demonstrated in the iterative schemes in spreadsheet CHAP6Ex5.xls.

5.5 A Comparison of Iterative and Direct Methods for Solving Linear Equations

In this Chapter, we have described two general methods - direct and iterative - for solving the linear equations which arise when we discretise the flow equations of reservoir simulation. We have not indicated which of these methods is usually “best” for reservoir simulation problems. This is because, it depends on the problem. The final numerical problem which is solved in a reservoir simulator could be quite small and simple, or it could be very large and have some intrinsically difficult numerical problems within it.


43

In any numerical computational method, such as those for solving a set of linear equations, we need to have some way of calculating the amount of “work” involved. To some extent this depends on the computer architecture since new generation parallel processing machines are becoming available as discussed in Chapter 1. However, we will take quite a simple view based on a conventional serial processing machine and will define computational work as simply the number of multiplications and divisions (addition and subtraction is cheap!). In this way, we can define the amount of work required for any given direct and iterative scheme for solving linear equations. We present results without any proof for two such schemes called the BAND (a direct method) and LSOR (Line Successive Over-Relaxation; an iterative method) schemes. For a 2D problem with NX and NY grid blocks in the x and y directions, respectively (where we choose NY < NX) BAND (direct),

Work, WB

≈ NX.NY3

(70)

LSOR (iterative),

Work, WLSOR

≈ NX.NY. Niter

(71)

where Niter is the number of iterations until convergence is reached. It is already quite clear that the amount of work for the direct method - which only has to be done once - is larger than that for a single iteration of the iterative method. However, we do not know in advance the number of iterations, Niter. We illustrate some typical results for these two methods with a simple exercise below.

EXERCISE 6.

Which is best method (i.e. that requiring the lowest amount of computational work), BAND or LSOR, for the following problems? (i) NX = 5,

NY = 3,

(ii) NX = 20, NY = 5,

Niter = 50 (a small problem) Niter = 50

(iii) NX = 100, NY = 20, Niter = 70 (iv) NX = 400, NY = 100, Niter = 150

44

NX

NY

Niter

5 20 100 400

3 5 20 100

50 50 70 150

BAND WB

LSOR, WLSOR

Comment


6

We can summarise our comparison between direct and iterative methods for solving the sets of linear equations that arise in reservoir simulation as follows: (i) A direct method for solving a set of linear equations has an algorithm that involves a fixed number of steps for a given size of problem. Given that the equations are properly behaved (i.e. the problem has a stable solution), the direct method is guaranteed to get to the solution in this fixed number of steps. No first guess is required for a direct method. (ii) An iterative method on the other hand, starts from a first guess at the solution (x(o)) and then applied a (usually simpler) algorithm to get better and better approximations to the true solution of the linear equations. If successful, the method will converge in a certain number of iterations, Niter, which we hope will be as small as possible. However, we cannot usually tell what this number will be in advance. Also, in some cases the iterative method may not converge for certain types of “difficult” problem. We may need to have good first guess to make our iterative method fast. Also, it often helps to use the latest computed information that is available (see example above). (iii) Usually the amount of “work” required for a direct method is smaller for smaller problems but iterative methods usually win out for larger problems. For an iterative method, the amount of work per iteration is usually relatively small but the number of iterations (Niter) required to reach convergence may be large and is usually unknown in advance.


45

EXERCISE 7.

The linear equations which arise in reservoir simulation may be solved by a direct solution method or an iterative solution method. Fill in the table below:

Direct solution method

Iterative solution method

1.

1.

2.

2.

1.

1.

2.

2.

Give a very brief description of each method

Main advantages

Main disadvantages

46


6

6. DIRECT SOLUTION OF THE NON-LINEAR EQUATIONS OF MULTI-PHASE FLOW 6.1 Introduction to Sets of Non-linear Equations

We noted in Section 4 above that the equations which we obtain when we discretise the two-phase flow equations are actually non-linear in nature. However, the strategy we discussed above (the IMPES method) involved tackling the problem almost as if it were a linear set of equations since we time-lagged the coefficients to reduce the problem to a linear set of equations. Then we repeated this process - we applied repeated iterations - until it (hopefully) converged. Therefore, we took a non-linear problem and solved it as if it were a series of linear problems. In this section, we will introduce the general idea of solving sets of non-linear equations numerically and indicate how this can be applied to the two-phase flow equations. We will do this in a simplified manner that shows the basic principles without going into too much detail. Firstly, compare the difference between solving the following two sets of two equations, a set of 2 linear equations:

2 x1 + x2 = 10 3 x1 − x2 = 5

(72)

and a set of 2 non-linear equations:

x1

2 x2 .e − x1 = 2

+

x2 .( x1 )2 − x1 .( x2 ) = −5 2

(73)

It is immediately obvious that the second set of non-linear equations is more difficult. The first set of linear equations can be rearranged easily to show that x1 = 3 and x2 = 4. However, it is not as straightforward to do the same thing for the non-linear set of equations. As a first attempt to solve these, we might try to develop an iterative scheme by rearranging the equations as follows (ν )

x1(ν +1) = 2 − 2 x2(ν ) .e − x1 x

(ν +1) 2

=

x2(ν ) .( x1(ν ) )2 +5 x1(ν )

(74)

where, as before, ν denotes the iteration counter. The actual solution in this case is, x1 =2.51068, x2 = 3.144089. Applying the above iterative scheme with a first guess, x1(0) = x2(0) = 1.0, gives the results shown in Table 2 below (where we have removed some of the intermediate iterations). Table 2: Non-linear scheme applied to solution of equations 73 using the point iterative scheme of equation 74 (spreadsheet, CHAP6Ex5.xls - Sheet 3)


47

Iteration counter ν First guess = 0 1 2 3 4 .... 9 10 .... 25 26

x at iter. ν x1 1 2.735759 2.317673 2.575338 2.454885 .... 2.513337 2.508958 .... 2.510682 2.510681

x2 1 2.44949 2.920421 2.987627 3.104133 .... 3.141814 3.144173 .... 3.144089 3.144089

You can check the results in Table 2 or experiment with other first guesses using Sheet 3 of spreadsheet CHAP6Ex5.xls. We note that the convergence rate in Table 2 is not very fast but, in this case, it does reach a solution. In general, it is usually very difficult to establish for certain whether a given scheme will converge for nonlinear equations although there is a large body of mathematics associated with the solution of such systems (which is beyond the scope of this course).

6.2 Newton’s Method for Solving Sets of Non-linear Equations

We take a step back to the solution of a single non-linear equation such as:

x 2 .e − x = 0.30

(75)

which has the solution x = 0.829069 (you can verify this by calculating x 2 .e − x and making sure it is 0.30 - to an accuracy of ~ 4.6x10-8). This equation can be represented as:

f ( x ) = 0 where f ( x ) = x 2 .e − x − 0.30

(76)

and the solution we require is the value of x for which the function f(x) is zero. Expand f(x) as a Taylor series as follows:

f ( x ) ≈ f ( xo ) + δx. f ' ( x0 ) +

δx 2 ' ' . f ( x0 ) + ... 2

(77)

Thinking in terms of an iteration scheme (x(ν) → x(ν+1)) as a basis for calculating better and better guesses of the solution of f(x) = 0, we can rewrite the Taylor series above as:

f ( x ) ≈ f ( x (ν ) ) + ( x (ν +1) − x (ν ) ). f ' ( x (ν ) )

(78)

where we have neglected all the second order and higher terms. Since we require the solution for f(x) = 0, then we obtain:

f ( x (ν ) ) + ( x (ν +1) − x (ν ) ). f ' ( x (ν ) ) = 0

48

(79)


6

which can be rearranged to the following algorithm to estimate our updated guess, x(ν+1) as follows:

x (ν +1) = x (ν ) −

f ( x (ν ) ) f ' ( x (ν ) )

(80)

This equation is the basis of the Newton-Raphson algorithm for obtaining better and better estimates of the solution of the equation, f(x) = 0. Note that we need both a first guess, x(0), and also an expression for the derivative f ' (x(ν)) at iteration ν. In the simple example in equation 76, we can obtain the derivative analytically as follows:

df ( x ) = f ' ( x ) = 2 x.e − x − x 2 .e − x dx

(81)

Therefore the Newton-Raphson algorithm for solving equation 75 above is as follows:

x

( ν + 1)

=x

(ν )

 ( x (ν ) )2 .e − x (ν ) − 0.30   -  (ν ) − x ( ν ) (ν ) 2  2 x .e − ( x (ν ) ) .e − x 

(82)

The point iterative method of the previous section and the Newton-Raphson method of equation 82 have both been applied to the solution of equation 75 (see Sheet 4 of spreadsheet CHAP6Ex5.xls for details). The results are shown in Table 3 for a first guess of x(0) = 1. Table 3: Comparison of the point iterative and Newton-Raphson methods for solving non-linear equation 75; (see spreadsheet CHAP6Ex5.xls - Sheet 4 for details) Iteration Number

Value of x at iteration ν

ν

Point Iteration method x(ν)

NewtonRaphson method x(ν)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 0.903042 0.860307 0.84212 0.834497 0.831322 0.830003 0.829456 0.82923 0.829136 0.829097 0.82908 0.829074 0.829071 0.82907 0.829069 0.829069

1 0.815485 0.825272 0.82806 0.828805 0.829 0.829051 0.829064 0.829068 0.829069 0.829069 0.829069 0.829069 0.829069 0.829069 0.829069 0.829069


49

The correct solution to equation 75 is x = 0.829069. The results in Table 3 show that the Newton-Raphson method converges to this solution (to an accuracy better than 10-6) in 9 iterations, whereas it takes 15 iteration for the point iterative method to converge at the same level accuracy. This performance can vary quite a bit from problem to problem but the Newton-Raphson method is generally better if the derivative term, f '(x(ν)), is not too close to zero. When this derivative gets too small, it can be seen that the second term in equation 80 would start to get very large or “blow up”, as it is sometimes described. We will not go into details about the convergence properties of the Newton-Raphson but it is meant to be “quadratic” meaning that the error should decrease quite rapidly. We now go on to see how the Newton-Raphson method can be applied to sets of non-linear equations. Returning to the set of two non-linear equations in the previous section, we note that another way of writing this set of equations in a general way is as follows:

F1 ( x1 , x2 ) = 0 F2 ( x1 , x2 ) = 0

(83)

where, in the case of equation 73 above, these functions would be given by:

F1 ( x1 , x2 ) = x1

+

2 x2 .e − x1 − 2

F2 ( x1 , x2 ) = x2 .( x1 )2 − x1 .( x2 ) + 5 2

(84)

The problem is then to find the values of x1 and x2 that make F1 = F2 = 0. In this section, we present Newton’s method for the solution of sets of non-linear equations. Basically, we will simply state the Newton-Raphson algorithm without proof (although it will resemble the simple form of the Newton-Raphson method above) for sets of non-linear equations. We will then illustrate what this means for the example of the set of two non-linear equations above (equation 73). Definitions: As before, (ν ) andx (νx+(ν1)+1=) the solution vectors at iteration levels ν and (ν+1) x (νx) and

New terms are: (ν )

(ν x) ) ) (ν +1) F(F xx((ν ) and x

N

(ν )

νx) x (F ( x (ν ) )

= the number of non-linear equations (and hence unknowns, x1, x2, .... xN) = the vector of function values, F1, F2 .. FN at x(ν)

the"true" "true"solution solutionofofthe thesetsetofofnon non - linearequations equations - linear F(Fx((ν)x) =) =0 0forforthe x 50

F( x ) = 0 for the "true" solution of the set of non - linear equations


That is:

 F1 ( x1ν , x2ν , x3ν ,....., x νN )     F2 ( x1ν , x2ν , x3ν ,....., x νN )     F3 ( x1ν , x2ν , x3ν ,....., x νN )  (ν ) F( x ) =   .....     .....    F ( x ν , x ν , x ν ,....., x ν ) N   N 1 2 3

J ( x (ν ) )

6

(85)

= the NxN Jacobian matrix defined as follows:

 ∂F1 ( x (ν ) ) ∂F1 ( x (ν ) ) ∂F1 ( x (ν ) ) ∂F1 ( x (ν ) )  ......   ∂x2 ∂x3 ∂x N  ∂x1       ∂F2 ( x (ν ) ) ∂F2 ( x (ν ) ) ∂F2 ( x (ν ) ) ∂F2 ( x (ν ) )  ......   ∂x2 ∂x3 ∂x N  ∂x1     J ( x (ν ) ) =   ∂F3 ( x (ν ) ) ∂F3 ( x (ν ) ) ∂F3 ( x (ν ) ) ∂F3 ( x (ν ) )  ......   ∂ x ∂ x ∂ x ∂x N 1 2 3     .....   .....    (ν ) (ν ) (ν ) (ν )   ∂FN ( x ) ∂FN ( x ) ∂FN ( x ) ...... ∂FN ( x )   ∂x1 ∂x2 ∂x3 ∂x N  (86)

[ J ( x )] (ν )

−1

= the inverse of the Jacobian matrix. (Recall that the inverse of a matrix A is denoted by A-1 and by definition, A-1 .A = I, where I is the identity matrix - all diagonals 1 and all other elements zero - see below). The above matrix is also an NxN matrix with the property:

[ J ( x )] .[ J ( x )] = I, the identity matrix [ J ( x )] .[ J ( x )] = I, the identity matrix (ν )

(ν )

−1

−1

(ν )

(ν )

1 0 0 0 0 10 0 1 0 0 0 0 0 0 0 1 0 0 0  0 0 1 0 0 00 0 0 1 0 0 1 0 0 −1 That is → J ( x (ν ) ) . J ( x (ν ) ) = I = 0 0 0 1 0 −1 0 0 0 0 1 (ν ) (ν ) J(x ) . J(x ) = I =   0 ....... 0 0 0 1  ....... .......  ....... 0 0 0 0 0  0 0 0 0 0

[

[

] [ ] [

]

]


...... 0  ...... ...... 0 0   ...... ...... 0 0  ...... ...... 0 0  (87)  ...... ...... 0 0  ...... 0      ...... 1   ...... 1 

51

Using all of the above definitions, the Newton-Raphson algorithm for a set of nonlinear equations, F( x ) = 0, is given by the following expression:

[

]

x (ν+1) = x (ν) − J ( x (ν) ) .F( x (ν) ) −1

(88)

We first demonstrate how this is applied to a simple example (equations 73) before going on to show how it is applied to the more complicated non-linear sets of equations which arise in the fully implicit discretisation of the reservoir simulation equations. Example: Returning to the simple example, where N = 2 (rearranged equations 73):

F1 ( x1 , x2 ) = x1

+

2 x2 .e − x1 − 2

F2 ( x1 , x2 ) = x2 .( x1 )2 − x1 .( x2 ) + 5 2

(89)

In the notation developed above, these equations become:

 x1 + 2 x2 .e − x1 − 2   F1 ( x1 , x2 )  F( x ) =    =  2  x2 .( x1 )2 − x1 .( x2 ) + 5  F2 ( x1 , x2 )

(90)

We note that the 2x2 Jacobian matrix is given by:

 ∂F1 ( x (ν ) )   ∂x1 (ν ) J(x ) =    ∂F2 ( x (ν ) )   ∂x1

∂F1 ( x (ν ) )   ∂x2    ∂F2 ( x (ν ) )   ∂x2 

(91)

where each of the elements can be evaluated analytically to obtain:

(

)

 1 − 2 x (ν ) .e − x1(ν ) 2  J ( x (ν ) ) =    2 x (ν ) . x (ν ) − x (ν ) 2 (2 )  2 1

(

(2.e ) − x1( ν )

) (( x

) − 2x

(ν ) 2 1

(ν ) 1

. x2(ν )

)

     

Hence, the Newton-Raphson method for this simple system becomes:

52

(92)


(

x ( ν + 1) = x ( ν )

)

 1 − 2 x (ν ) .e − x1(ν ) 2     2 x (ν ) . x (ν ) − x (ν ) 2 (2 )  2 1

−

(

(2.e ) − x1( ν )

) (( x

) − 2x

(ν ) 2 1

(ν ) 1

. x2(ν )

)

     

−1

6

 x (ν ) + 2 x (ν ) .e − x1(ν ) − 2  2  1      2  x2(ν ) .( x1(ν ) )2 − x1(ν ) .( x2(ν ) ) + 5

(93)

[ ( )] (ν )

−1

which we could solve if we knew how to invert the Jacobian matrix to find J x This is a detail which we don’t need to know for this course but it can be done. Indeed, for a 2x2 matrix - such as in the above case - it is quite easy to work out the inverse. The inverse of a simple 2 x 2 matrix A given by: A =  a b

   c d

is well known to be A-1 =

1  d − b   ( ad − bc)  − c a 

This can easily be proven by multiplying out there matrices and showing that A-1 .A = I. The factor (ad-bc) is known as the determinant of the matrix and it must be non-zero for A-1 to exist. Rather than using equation above to write out the analytical form of the J ( x (ν ) ) matrix we first simply numerically evaluate the matrix J at a −1 given iteration and apply equation above no get J . This is done in the spreadsheet Chap6Exxx.xls where results are shown.

[

]

[ J ( x )] (k )

−1

= ***

[]

(94)

which allows us to apply the Newton-Raphson method directly to our simple example. For details see Sheet 5 spreadsheet ***.xls. The results are shown in Table 6.xx. (NOTES & SPEADSHEET STILL UNDER CONSTRUCTION)


53

6.3 Newton’s Method Applied to the Non-linear Equations of TwoPhase Flow

We now return to the discretised equations of two phase flow which we noted at the time, formed a set of non-linear equations. The simplified form of these equations is as follows (see Section 4.1): Pressure (equation 50):

(λ (S )) T

n +1 o

∆x

i −1 / 2

2

n +1 i −1

P

(

)

 λ (Son +1 ) T −  ∆x 2 

i −1 / 2

(λ (S )) + n +1 o

T

∆x

i +1 / 2

2

(

)

 λ (S n + 1 )  P n +1 + T o  i ∆x 2 

i +1 / 2

Pin++11 = 0

Saturation (Form B - equation 54): Soin+1 − Soin −

(

)

n +1  ∆t  λ o ( So )  φ  ∆x 2  

(

)

  λ ( Son +1 ) n +1 n +1  − P P ( i +1 i ) −  o ∆x 2  

i +1 / 2

i −1 / 2

Given that the unknowns that we are trying to find are Pi then we can write the above equations in general form as:

 n +1 n +1   − P P (i i −1 )  = 0  

n +1

, Sin +1 i = 1,2, NX. ,

FP, i ( S n +1 , P n +1 ) = 0

FS , i ( S n +1 , P n +1 ) = 0

(95)

FP, i ( S n +1 , P n +1 ) andn +F1S , i (nS+n1+1 , P n +1 ) n +1 n +1 where the two non-linear equations, FP, i ( S , P ) and FS , i ( S , P ) , arise from the pressure (P) and saturation (S) equations, respectively, as given above. The vectors of unknowns, S n +1 and P n +1, nare +1 given nby +1 :

S

and P

n +1 1

S n +1

54

S   n +1  S2   n +1  S3  ...    ...   S n +1  =  i −1  Sin +1   n +1  Si +1    ...  ...    ...   S n +1   NX 

and

P n +1

 P1n +1   n +1   P2   n +1   P3  ...    ...   P n +1  =  i −1   Pi n +1   n +1   Pi +1    ...  ...    ...   P n +1   NX 

(96)


6

FP, i ( S n +1 , P n +1 ) and FS , i ( S n +1 , P n +1 )

(

n +1

n +1

)

(

n +1

n +1

)

and FS , i S , P However, the given FP, i S , P at grid block i only depend on the quantities Sin−+11 , Sin +1 , Sin++11 and Pi n−1+1 , Pi n +1 , Pi n+1+1 and, rather than on all of the other saturations and pressures in the system, since these are the nearest neighbours +1 n +1 n +1 Sin−+11equations , Sin +1 , Sinabove. , Pi n+1+1 coupled together in the +1 and Pi −1 , Pi We may also write one total unknows vector Xn+1 using a combination of the saturation and pressure vectors as follows:

X n +1

 S1 n +1   n +1   P1   S n +1   2   P2 n +1      =  S n +1   i   Pi n +1       S n +1   NX   n +1   PNX 

and the equation to be solved is then F(Xn+1)=0 The saturations and pressures are coupled together to their nearest neighbours through the discretisation equations (50 and 54 above). The general form of the solution using the Newton iteration is then to take a starting guess (iteration, ν=0) Xn+1(0) and then apply the formulation as above to obtain:

[

X n +1 (ν +1) = X n +1(ν ) + J ( X )(ν )

]

−1

. F ( X (ν ) )

It is rather more complex to constuct the Jacobian matrix J ( X ) of derivatives but −1 it can be done. We also need to invert this matrice to obtain J in order to apply the above algorithm. However, in practice, there are various methods that try to −1 construct the J matrix more directly, often in an approximate manner. Note that the Jacobian matrix is very sparse since there are just nearest neighbour interaction (as was the case for the matrices associated with single phase flow discussed earlier in this Chapter). A consequence of this is that the inverse matrix is also quite sparse and there are many numerical techniques available to solve such problems.

[]

[]

In this course, we will not give any more detail on the solution of the non-linear equations which arise in reservoir simulation.


55

7 NUMERICAL DISPERSION - A MATHEMATICAL APPROACH 7.1 Introduction to the Problem

In Chapter 4, we discussed the physical idea of numerical dispersion, which is also sometimes referred to as numerical diffusion. Numerical dispersion is essentially the artificial spreading or “diffusion” of a front - e.g. a waterfront in a water/oil displacement - due to the coarse grid used in the simulation. It is a numerical effect and it can be reduced by taking a finer grid. We note that in real reservoirs there are some real dispersive physical mechanisms which result in the spreading of fronts. These arise due to the effects of capillary pressure and also from the interaction of the fluid flow with small scale (cm - m) permeability heterogeneity of the reservoir rock. Indeed, there are also some quite complex interactions between the capillary forces and the small scale heterogeneity that also lead to types of physical spreading in the reservoir. In an ideal calculation, we would take a sufficiently fine grid that the level of physical (i.e. real) spreading or diffusion was correctly represented by our grid; i.e. the numerical diffusion would be less that the physical diffusion. This is almost never possible in a field scale simulation, although it can be achieved in modelling laboratory scale experiments on flows through small rock samples or bead packs. For such a fine scale simulation, the level of numerical diffusion can realistically be made much less than that from physical sources. In this section, we return to the issue of numerical dispersion - a term which we will now use interchangeably with “numerical diffusion”. Indeed, “diffusion” is often referred to as quite a specific physical effect which is described by certain well-known equations which are, not surprisingly, known as diffusion equations. For example, the simplified pressure equation derived in Chapter 5, equation 27, is an example of a diffusion equation. This equation has the form:

 ∂2 P   ∂P    = Dh  2   ∂t   ∂x 

(97)

where Dh is the hydraulic diffusivity (Dh = k/( cf .φ.μ)) and this is a standard form of the classical diffusion equation. We now consider how the effects of a grid can lead to “diffusion-like” terms when we try to solve the flow equations numerically.

 ∂2 P 

In the above equations, it is specifically the  2  term that is the “dispersive” or  ∂x  “diffusive” part.

7.2 Mathematical Derivation of Numerical Dispersion

In order to show mathematically how a “diffusive” term arises when we solve certain transport equations numerically, we use a simple form of the two-phase saturation equation. In 1D, the transport equation for a simple waterfront (with Pc = 0 and zero gravity) is the well-known fractional flow equation (see Chapter 2) as follows:

 ∂S   ∂f  φ  w  = − v w   ∂t   ∂x 

(98)

where Sw is the water saturation and fw ( Sw ) is the fractional flow of water

56

 qw  fw ( Sw ) =    qw + qo 


6

fw ( Sw ) 

qw   ) which is a function only of water saturation, Sw. We can  qw + qo 

( fw ( Sw ) = 

differentiate fw ( Sw ) by the chain rule to obtain:

fw ( Sw )

 ∂f   ∂S   ∂S   ∂f  φ w  = − v w  = − v w   w   ∂ft S =  ∂xqw  ∂Sw   ∂x    w( w)  qw + qo 

(99)

 ∂f  − v w  where the termfw ( Sw)∂Sw  is a non-linear term giving the water velocity, v w (Sw ) , which is also a function of water saturation: that is:

 ∂f  v w (S w ) = − v  w   ∂Sw 

fw (Sw )

(100) To simplify the problem even further for our purposes here, we take a straight line

 ∂fw    fractional flow, fw (Sw ) , which means that  ∂Sw  is a constant and, hence, so is the water velocity, Vw. Hence, the simple 1D transport equation for a convected waterfront is:

 ∂fw  ∂ S  w  ∂S   ∂Sw    = −wv w    ∂t   ∂x 

(101)

where, as noted above, vw is now constant. This equation describes the physical situation illustrated schematically in Figure 14. t1

Governing equation ∂Sw ∂Sw = - Vw ∂t ∂x

t2

1

Figure 14 The advance of a sharp saturation front governed by the transport equation with a constant velocity vw.

Sw Water 0

Vw x1

Oil x2

x

Vw = constant Vw =

x2 - x1 t2 - t1

Starting from the transport equation 101 above, we can easily apply finite differences using our familiar notation (Chapter 5) to obtain: n  Sn +1 − Swi   Swi − Swi −1  φ wi  = − vw   + Error terms     ∆t ∆x

(102)

where we have used the backward difference (sometimes referred to as the upstream

 ∂Sw    difference) to discretise the spatial term,  ∂x  . Note that we have not yet specified the time level of the spatial terms. If we take these at the n time level (known), then it


57

would be an explicit scheme and if we take them at the n+1 time level (unknown), it would be an implicit scheme. It does not matter for our current purposes here, since we are principally interested in the “Error terms” which are indicated in equation 102. To determine what these terms look like, we need to go back to the original finite differences based on Taylor expansion of the underlying function, Sw(x,t), in this case. We can expand Sw (x,t) either in space (x) or in time (t) as follows: SPACE

2 2  ∂S  δx  ∂ Sw  Sw (x, t ) = Sw (x 0 ) + δx. w  +   + higher order terms  ∂x  2  ∂x 2  (t fixed)

 ∂S 

(103)

δt 2  ∂ 2S 

w w + TIME Sw ( x, t ) = Sw ( t 0 ) + δt.  2  + higher order terms  ∂t    2 ∂ t (x fixed) (104)

We can rearrange each of the above equations 103 and 104 to obtain the finite

 ∂S   ∂S    and  

difference approximations for the  ∂x   ∂t  terms with the leading error terms as follows (now ignoring the higher order terms):

 S(x, t ) − S(x 0 )  δx  ∂ 2S  ∂S SPACE   ≈   −  2  ∂x 

TIME



δx



2  ∂x 

2  ∂S   S(x, t ) − S( t 0 )  δt  ∂ S    ≈  − 2  ∂t 2   ∂t   δt 

(105)

(106)

To return to our usual notation, we make the identities:

S( x, t ) ⇔ Sin +1 S( x 0 ) ⇔ Si −1 S( t 0 )

⇔ Sin

δx

⇔ ∆x

δt

⇔ ∆t

(107)

Equation 105 now becomes: 2  ∂S   Si − Si −1  ∆x  ∂ S  −   ≈    ∂x   ∆x  2  ∂x 2 

(108)

and equation 106 becomes: n +1 n 2  ∂S   Si − Si  ∆t  ∂ S  −   ≈  ∂t   ∆t  2  ∂t 2 

58

(109)


6

We now substitute the above finite difference approximations into the governing equation 102 with their leading error terms as follows:

 Sn +1 − Sin  ∆t  ∂ 2S   Si − Si −1  v w .∆x  ∂ 2S  φ i − v ≈ −    +   w  ∆t  2  ∂t 2   ∆x  2  ∂x 2  (110) Collecting the error terms together on the RHS gives:

 Sn +1 − Sin   Si − Si −1  v w .∆x  ∂ 2S  ∆t  ∂ 2S  φ i v ≈ −  +  +   w  ∆t   ∆x  2  ∂x 2  2  ∂t 2  Error term

(111)

The error term in the finite difference scheme is now clear and is shown in equation 111. However, it still needs some simplifying since it is a strange mixed term with

 ∂ 2S   ∂ 2S   ∂ 2S   2  and  2  both  ∂x   ∂t  terms in it. We now want to eliminate the  ∂t 2  term and this is done by using the original governing equation.

 ∂ 2S   ∂Sw   ∂S    = − vw  w  2  , first note from the original equation 101 that   ∂x  ∂t   ∂t   ∂Sw   with respect to t as follows: and we can then differentiate   ∂t  To obtain 

∂  ∂Sw   ∂ 2Sw  ∂  ∂S    =  2  = − vw  w  ∂t  ∂t   ∂t  ∂t  ∂x 

(112)

Thus, we can rearrange the RHS of equation 112 as follows:

− vw

∂  ∂Sw  ∂  ∂S    = − vw  w  ∂t  ∂x  ∂x  ∂t 

(113)

 ∂S 

We now return again to the governing equation (equation 101) and use it for  w   ∂t  in equation 113 to obtain:

− vw

2 ∂  ∂Sw  ∂  ∂Sw  2  ∂ Sw  = − − = v v v    w   w w  ∂x 2  ∂x  ∂t  ∂x  ∂x 

(114)

and hence: 2  ∂ 2 Sw  2  ∂ Sw  = v  2   w  ∂t   ∂x 2 

(115)

We now substitute this expression into the error term in equation 111 above to obtain the following: Institute of Petroleum Engineering, Heriot-Watt University

59

v w .∆x  ∂ 2S  ∆t  ∂ 2S  v w .∆x  ∂ 2S  v 2w ∆t  ∂ 2S   +  =  +   2  ∂x 2  2  ∂t 2  2  ∂x 2  2  ∂x 2   v .∆x v 2w ∆t   ∂ 2S  = w +    2 2   ∂x 2 

(116)

The finite difference equation with its error term in equation 111 now becomes:

 Sin +1 − Sin   Si − Si −1   v w .∆x v 2w ∆t   ∂ 2S  v + ≈ −    +   w  ∆t   ∆x   2 2   ∂x 2 

(117)

In this equation, we now see that the form of the error term is exactly like a “diffusive” term i.e. it multiplies a (∂2Sw/∂x2) term. Hence we identify the level of numerical dispersion or diffusion, Dnum, arising from our simple finite difference scheme as:

 v .∆x v 2w ∆t   ∆x v w ∆t  D num =  w + +  = v w .   2  2 2  2   ∆x

If D num = v w . + For this case:  2

D num v w ∆t << ∆x≈

 ∆x v w ∆t  D num = v w . +  2  (118)  2

v w ∆t   , we can take such a small time step that v w ∆t << ∆x 2 

v w .∆x 2

(119)

We can now solve two problems as follows (i) Firstly, we may apply an explicit finite difference method to obtain an accurate solution of the following convection-dispersion equation.

 ∂ 2 Sw   ∂Sw   ∂Sw  = − v + D .     2  w  ∂t   ∂x   ∂x 

(120)

That is, where the numerical dispersion is much less than the physical dispersion due to the explicit D term. If this solution is converged (i.e. does not change on refining Δx and Δt) then we can plot this at a suitable time, t1, as shown in Figure 15.

Governing equation:

1

Sw = - Vw t

t = t1

Sw

2

Vw = water velocity

0

60

Sw +D x

x

Sw x2

Figure 15. Dispersed frontal displacement with a known and converged level of dispersion.


6

n +1 n n n  Swi   Swi − Swi − Swi −1  (ii) Now take only the explicit finite difference φapproximation tov wthe  =−  convection      ∆t ∆x equation only, that is: n n n  Sn +1 − Swi   Swi − Swi  v .∆t  n −1  n +1 n n φ wi  = − vw   which gives ⇒ Swi = Swi −  w  (Swi − Swi −1 )     ∆t ∆x  ∆x.φ 

Solve this with a relatively coarser grid, Δx, but with a fine time step thus predicting x n  vwv.∆ n +1 n n w .∆t  = − Swi − Swi S S  Dwinum to be level of dispersion to deliberately match that in wi  −this 1) 2x.φ.( Choose ∆ part (i) above i.e. choose Δx such that Dnum = D. Plot out a saturation profile like that in Figure 15 at the same time t1 and compare these.

8

CLOSING REMARKS

In this Chapter, we have introduced the student to finite difference approximations of the partial differential equations (PDEs) that describe both single- and two-phase flow through porous media. These discretised equations have led to systems of either linear or non-linear equations which are then solved numerically. Methods for solving these equations have been discussed in principle and some idea has been given of how these are applied in practical reservoir simulation. At the end of the unit, some discussion was presented on grid-to-grid flows and how these inter-gridblock properties are averaged. A more mathematical of numerical dispersion was also given.


61

APPENDIX A: Some useful Matrix Theorems This lists (without proof) some useful theorems from matrix algebra which underpin much of the practical application work which we have described in this section.

A.x = b

The central problem which we are interested in is:

A.x = b A.x = b where A is an NxN square matrix 1. If A.x = b actually has a solution, then A is said to be non-singular or invertible Aand the following five conditions apply and are equivalent (i.e. if one of them hold, they all hold):

I A (i) A −1 exists where A −1 . A = I ; I is the identity matrix which has 1s on the diagonal and zeros elsewhere. I is like the number “1” in that I.A = A.I = A . (ii) det( A ) ≠ 0 ; the determinant of the matrix is non-zero. This is a number found A a=specific A.I = Away. by “multiplying out” the matrixI.in (iii) There is no non-zero A.x = 0vector x such that, A.x = 0 . In other words, if A.x = 0 , then the vector x must be zero - have 0s for all its elements.

A.x = 0

(iv) The rows of A are linearly independent.AAnd ..

A

(v) the columns of A are linearly independent. Where (iv) and (v) say that we cannot get one Aof .x the = 0rows or columns by making some combination of the existing ones. 2. A matrix A which is square and symmetric T ( A = A T , whereAA . x =is0the transpose of A; i.e. a ij = a ji ) is said to be positive T definite if: x .A.x > 0 3. If the matrixAA=isGpositive definite then it can be decomposed in exactly one .G T way into a product: A = G.G T such that matrix G is lower triangular and has positive entries on the main diagonal. G decomposition. This is known as Cholesky

62


6

SOLUTIONS TO EXERCISES

EXERCISE 1. Apply finite differences to the solution of the equation:

 dy  = 2. y 2 + 4  dt  where, at t = 0, y(t = 0) = 1. Take time steps of Δt = 0.001 (arbitrary time units) and step the solution forward to t = 0.25. Use the notation yn+1 for the (unknown) y at n+1 time level and yn for the (known) y value at the current, n, time level. Plot the numerically calculated y as a function of t between t = 0 and t = 0.25 and plot it against the analytical value (do the integral to find this). Answer: is given below where the working is shown in spreadsheet CHAP6Ex1.xls. This gives the finite difference formula, a spreadsheet implementing it and the analytical solution for comparison. SOLUTION 1. Discretise the equation using the suggested notation as follows:

 y n +1 − y n  n 2   = 2.( y ) + 4  n +1∆t n  y −y  n 2   = 2.( y ) + 4  ∆t  which is rearranged to the explicit formula: y n +1 = y n + ∆t (2.( y n )2 + 4)

 ny+n1+1 − yn n  nn 22 y = y + ∆=t (22.(.(yy )) ++ 44)  n +du ∆t   y n +1 − y n  n 2  y 1 − y n = 1 tan −n1 2 u    = 2.( y ) + 4 = 2.( y ) +4 ∫ u 2 + α 2  α   ∆ t  1 easily α ∆t up quite which can be set on a spreadsheet. du −1n 2u  n +1 n = 2.( y ) + 4) ∫y 2 += αy 2 + ∆αt(tan α might need the standard form of the following To find the uanalytical answer, you 1n +1 dy n n1 2 y n++1C= y n + ∆t (2.( y n )2 + 4) −1  y  integral: y =2 y + ∆t2(2=.( y ) +tan 4 dt t = = )  2 ∫ 2 ∫ ydu+ ( 2 1) 2 2 1 dy= tan −11 u  −1  y  ∫2 u∫2y+2 α+ 2( 2α)2 = 2 2α tan   2  = ∫ dt = t + Cdu 1 u du 1 u  −1  = tan −1   2 2 ∫ = tan α α ∫ u=2 +2α 2 α u +α α α Using this1standard form gives: α = 2 2 dy 2 = 1 tan −1  y  = dt = t + C  2 ∫ 2 1∫ y + (−1 2 )y  2 2 1 dy 1  y  1 tandy 1+C −1  y  t = = tan −1 = 2 2 ∫ = tan dt t + C = =   22 ∫ 2y 2 + ( 2 )22  2 2 ∫ ( ) y + 2 2 2 2 2   2 1 −1  y  α2 =2 tan 2  2  = t +C where C isy(the constant of integration and we identify α = 2 above. Therefore, α t=) = 2 2 tan 2 2 (t + C )

[ ]  y2 2 (t + C ) y(1t ) =tan2 tan [ = t +C ]  2 2 2

this becomes:

−1

1  y−1  1  C 1= tan −1tan = t+=C0.2176049 2  2 2Heriot-Watt  Institute of Petroleum University 2 22 1Engineering, −1  1  C = tan = 0 . 2176049 y(t ) = 2 tan 2 2 (t + C ) 2 2  2

[

]

1 2 2

tan −1

 y  = t +C  2 63

∫ u + α = α tan  α  du 1  u ∫ u + α = α tan  α  du 1  u 1  ∫1 u + αdy= α tan α tan = ( 2) 21 ∫ y +dy 2 12 = tan ( 2) 21 ∫ y +dy 2 12 α2 ∫= y 2+ ( 2 ) = 2 2 tan 2

2

2

2

2

2

−1 −1

2

2

2

2

2

2

  −1   −1   −1

y  = ∫ dt = t + C y2  = dt = t + C  ∫ 2 y  = dt = t + C 2 ∫

α= 2 α 1= tan 2 −1  y  = t +C  y2  2 12   tan −1 = t +C   2 2 12  −1  yto: which easily rearranges = t +C y2(t )2=tan2 tan 2  2  2 (t + C ) y(t ) = 2 tan 2 2 (t + C )

[ ] [ ] y(t=) = 1 2 tan tan[2 12 (t += C0).2176049 ] C  conditions since y(t = 0) = 1; it is easy to see that: We now find C from the initial −1

 tan −1  2 12  C= tan −1  2 2  C=

2 12

12   = 0.2176049 2 1  = 0.2176049 2 

which is then used in the analytic solution for y(t) above. This is implemented in the spreadsheet CHAP6Ex1.xls. Note that at t > 0.3, both the analytical and numerical solutions go a bit strange (very large and they start to disagree) since the tan function has a singularity y(t) → ∞. EXERCISE 2. Fill in the above table using the algorithm:

Pi n +1 = Pi n +

∆t ( Pi n+1 + Pi n−1 − 2 Pi n ) ∆x 2

Hint: make up a spread sheet as above and set the first unknown block (shown grey shaded in table above) with the above formula. Copy this and paste it into all of the cells in the entire unknown area (surrounded by red border above). SOLUTION 2. If you get stuck, look at spreadsheet CHAP6Ex2.xls on the disk. EXERCISE 3. Experiment with the spreadsheet in CHAP6Ex2.xls to examine the effects of the three quantities above - Δt, Δx (or NX) and the solution as t → ∞.

64


6

EXERCISE 4. Solve the following simple example of an upper triangular matrix:

4 x1 - 14xx21 −- 21xx32 −+ 21xx43 2 x − 22xx2 −+ 21xx3 2

3

3 x3

4

+ + ++

11xx4 =+ 5 1x5 = 5 5 12xx4 = +122 x5 = 12 5

+ 34xx34 ++ 41xx54 =+301x5 = 30 2 x − 21xx4 =− 3 1x5 = 3 4

5

3 x5 = 153 x5 = 15

SOLUTION 4.

Answer Answer x1 = ..........; x2 = ..........; x3 = ..........; x4 = ..........; x5 = .......... x1 = ..........; x2 = ..........; x3 = ..........; x4 = ..........; x5 = ..........

EXERCISE 5. fill in the table below for 10 iterations using a calculator or a spreadsheet: POINT ITERATIVE SOLUTION OF LINEAR EQUATIONS Iteration counter ν First guess = 0 1 2 3 4 5 6 7 8 9 10

x at iter. ν x1 x2 2.0 2.0

x3 2.0

x4 2.0

SOLUTION 5.

Iteration counter x at iter. ν ν x1 x2 x3 x4 Answer First - Exercise CHAP6Ex5.xls 1 guess =5: 0see spreadsheet 2.0 2.0 2.0 - Sheet2.0 1 4.309677 1.97619 3.6925 2.784615 2 4.008926 1.411795EQUATIONS 3.498943 2.436331 POINT ITERATIVE SOLUTION OF LINEAR 3 3.993119 1.505683 3.497206 2.503112 4 4.000937 1.500783 3.500617 2.500584 5 3.999963 1.499755 3.499965 2.499832 6 3.999986 1.500026 3.499995 2.500017 7 4.000003 1.5 3.500002 2.500001 8 4 1.499999 3.5 2.5 Converged ⇒ 9 4 1.5 3.5 2.5 10 4 1.5 3.5 2.5 Note - converges fully at k = 9 iterations.


65

1 2 3 4 5 6 7 8 9 10

Iteration counter ν First guess = 0 1 2 3 4 5 6 7 8 Converged ⇒ 9 10

x at iter. ν x1 2.0 4.309677 4.008926 3.993119 4.000937 3.999963 3.999986 4.000003 4 4 4

x2 2.0 1.97619 1.411795 1.505683 1.500783 1.499755 1.500026 1.5 1.499999 1.5 1.5

x3 2.0 3.6925 3.498943 3.497206 3.500617 3.499965 3.499995 3.500002 3.5 3.5 3.5

x4 2.0 2.784615 2.436331 2.503112 2.500584 2.499832 2.500017 2.500001 2.5 2.5 2.5

Note - converges fully at k = 9 iterations.

EXERCISE 6. Which is best method (i.e. that requiring the lowest amount of computational work), BAND or LSOR, for the following problems? (i) NX = 5,

NY = 3,

Niter = 50 (a small problem)

(ii) NX = 20, NY = 5,

Niter = 50

(iii) NX = 100, NY = 20, Niter = 70 (iv) NX = 400, NY = 100, Niter = 150 NX

NY

Niter

5 20 100 400

3 5 20 100

50 50 70 150

BAND WB

LSOR, WLSOR

Comment

SOLUTION 6.

66

NX

NY

Niter

5

3

50

BAND WB 135

20

5

50

2500

100

20

70

8x105

400

100

150

4x108

LSOR, WLSOR 750

Comment

For a very small problem like this, a direct method is usually better, although both would take very little time even on a PC. 5000 Again a direct method is somewhat better for a small problem. 1.4x105 As the problem grows in size, iterative methods start to overtake direct method. 6x106 For very large problems, iterative methods virtually always win over direct methods by more than an order of magnitude.


6

EXERCISE 7. The linear equations which arise in reservoir simulation may be solved by a direct solution method or an iterative solution method. Fill in the table below:

Direct solution method

Iterative solution method

1.

1.

2.

2.

1.

1.

2.

2.

Give a very brief description of each method

Main advantages

Main disadvantages


67

68



6

69

7

Permeability Upscaling

CONTENTS 1

SINGLE-PHASE FLOW 1.1 INTRODUCTION 1.2 Upscaling Porosity and Water Saturation 1.3 Averaging Permeability 1.3.1 Flow Parallel to Uniform Layers 1.3.2 Flow Across Uniform Layers 1.3.3 Flow through Correlated Random Fields 1.3.4 Additional Averaging Methods 1.3.5 Summary of Permeability Averaging 1.4 Numerical Methods 1.4.1 Recap on Flow Simulation 1.4.2 Boundary Conditions 1.5 Upscaling Errors 1.5.1 Correlated Random Fields 1.5.2 Evaluating the Accuracy of Upscaling 1.5.3 Upscaling of a Sand/Shale Model 1.6 Summary of Single-Phase Upscaling

2

TWO-PHASE FLOW 2.1 Introduction 2.2 Applying Single-Phase Upscaling to a Two-Phase Problem 2.3 Improving Single-Phase Upscaling 2.3.1 Non-Uniform Upscaling 2.3.2 Well Drive Upscaling 2.4 Introduction to Two-Phase Upscaling 2.5 Steady-State Methods 2.5.1 Capillary-Equilibrium 2.6 Dynamic Methods 2.6.1 Introduction 2.6.2 The Kyte and Berry Method 2.6.3 Discussion on Numerical Dispersion 2.6.4 Disadvantages of the Kyte and Berry Method 2.6.5 Alternative Methods 2.6.6 Example of the PVW Method 2.7 Summary of Two-Phase Flow

3

ADDITIONAL TOPICS 3.1 Upscaling at Wells 3.2 Permeability Tensors 3.2.1 Flow Through Tilted Layers 3.2.2 Simulation with Full Permeability Tensors 3.3 Small-Scale Heterogeneity 3.3.1 The Geopseudo Method 3.3.2 Capillary-Dominated Flow

3.3.3 Geopseudo Example 3.3.4 When to use the Geopseudo Method 3.4 Uncertainty and Upscaling 3.5 Upscaling Summary 4

REFERENCES

Learning Objectives

After reading through this Chapter, the Student should be able to do the following: •

Appreciate why upscaling is necessary.

•

Know how to calculate effective permeability in simple models by averaging.

•

Understand how to perform numerical upscaling of single-phase flow.

•

Be aware of the effects of heterogeneity on two-phase flow.

•

Realise the limitations of applying single-phase upscaling to a two-phase problem.

•

Know how to carry out steady-state, capillary-equilibrium upscaling for twophase flow.

•

Become familiar with two-phase dynamic upscaling (the Kyte and Berry Method), and understand the advantages and disadvantages of applying dynamic upscaling.

•

Understand how to upscale around a well.

•

Appreciate that permeability is a full tensor property.

•

Know how to upscale from the core-scale to the scale of a geological model, taking account of fine-scale structure and capillary effects.

2

7


1 SINGLE-PHASE FLOW 1.1 Introduction

Reservoir modelling often involves generating multi-million cell models, which are too large for carrying out flow simulations using conventional techniques. The number of cells must therefore be reduced by “upscaling” (Figure 1). Some quantities, such as porosity and water saturation, are easy to upscale, because they may be averaged arithmetically. However, other quantities – notably permeability – are much more difficult to upscale. Full-field model

Geological model

Figure 1 Upscaling Example

We usually refer to the upscaled permeability as the effective permeability. The effective permeability is defined as the permeability of a single homogeneous cell which gives rise to the same flow as the fine-scale distribution when the same pressure gradient is applied. We assume that Darcy’s Law holds at the coarse scale:

Q=−

Ak eff ∆P µ ∆x

(1)

where Q = total flow, A = area, keff = effective permeability, μ = viscosity, and ΔP/Δx is the pressure gradient. True effective permeability is an intrinsic property of the model and ought to be independent of the applied boundary conditions (Section 1.4). However, in practice, the effective permeability often does depend on the boundary conditions, and on the method used for calculation. Upscaling must always be carried out with care in order to obtain “sensible” results. In Figure 1, the geological model on the left is a fine-scale model with 20 million cells, and the coarse-scale model on the right consists of about 300,000 cells. Each of the coarse-scale cells contains an effective permeability. An example of fine-scale and coarse-scale grids is shown in the 2D model in Figure 2. An effective permeability is calculated for each coarse-scale cell, either by averaging the fine-grid values, or by performing a numerical simulation.


3

fine grid

coarse grid one coarse cell

Figure 2 The upscaling procedure

In this section of the upscaling course, we assume that there is only one phase present – water or oil, and that we have steady-state linear flow. We show how simple averaging may sometimes be used to estimate upscaled parameters, and then move on to methods which involve numerical simulation. This is followed by a set of examples which demonstrate how errors may arise, and how to avoid them. 1.2 Upscaling Porosity and Water Saturation

We start by averaging porosity and water saturation, using a simple model (Figure 3). (Note that the water saturation is not required for a single-phase problem. However, we include it here because it is simple to upscale.) There are 10 grid blocks of size 1 m3, 4 of which have a porosity of 0.15, and 6 of which have a porosity of 0.20. φ = 0.15

φ = 0.20

Sw = 0.50

Sw = 0.40

Figure 3 Example for averaging porosity and water saturation

The average porosity, φ , is given by:

φ=

total pore volume total volume

(2)

Therefore, in this case:

φ=

4 × 0.15 + 6 × 0.20 = 0.18. 10

When averaging the water saturation, we need to take the porosity into account. In the previous example, suppose the water saturation was 0.5 in the blocks with porosity of 0.15, and 0.4 in the blocks with porosity 0.2, then the average water saturation is:

Sw =

total amount of water total pore volume

Here, the average water saturation is:

4

(3)

7


4 × 0.15 × 0.5 + 6 × 0.20 × 0.4 4 × 0.15 + 6 × 0.20 0.3 + 0.48 0.78 = = 0.433. = 1.8 1.8

Sw =

1.3 Averaging Permeability

In some simple models, such as parallel layers or a random distribution, the effective permeability may be calculated by averaging. 1.3.1 Flow Parallel to Uniform Layers

P1

P2

Qi

ki, ti

Figure 4 Along-layer flow

∆x Consider a set of (infinite) parallel layers of thickness, ti and permeability ki, where i = 1, 2, .. n (the number of layers). The effective permeability of these layers is given by the arithmetic average, ka. n

k eff = k a =

∑t k i =1 n

i

∑t i =1

i

i

(4)

(Equation (4) may be proved by applying a fixed pressure gradient along the layers.) Example 1 x Figure 5 A simple, two-layer model

z

1

t1 = 3 mm, k1 = 10 mD

2

t2 = 5 m mm, k2 = 100 mD

Suppose we have two layers as shown in Figure 5. The effective permeability for flow in the x-direction is given by Equation (4), and is:

ka =

3 × 10 + 5 × 100 530 = = 66.25 m mD 3+5 8


5

1.3.2 Flow Across Uniform Layers

Q

ki, ti

∆Pi

Figure 6 Across-layer flow

∆x For flow perpendicular to the layers, the effective permeability is given by the harmonic average, kh: n

k eff = k h =

∑t i =1 n

i

ti ∑ i =1 k i

. (5)

(Equation (5) may be proved by assuming a constant flow rate through each layer.) Example 2 Equation (5) may be used to calculate the effective permeability for flow across the two layers in the model shown in Figure 5, i.e. flow in the z-direction.

kh =

8 3+5 = = 22.86 mD 3 10 + 5 100 0.35

From Examples 1 and 2, we see that the permeability is different in different directions. In reservoirs with approximately horizontal layers, the arithmetic average may be used for calculating the effective permeability in the horizontal direction, and the harmonic average may be used for calculating the effective permeability in the vertical direction. 1.3.3 Flow through Correlated Random Fields

Figure 7 shows an example of a correlated random permeability distribution. Correlated random fields are described in Section 1.5.1. Basically, “correlated” means that areas of high or low permeability tend to be clustered, so that the spatial distribution is smoother than a totally random one. The “correlation length” is approximately the size of patches of high or low permeability. The longer the correlation length, the longer will be the range of the semi-variogram for the permeability distribution. Assuming that we are averaging over many correlation lengths, permeability should be isotropic (same in the x-, y- and z-directions). The effective permeability for a random permeability distribution is proportional to the geometric average, which is given by:

6

7


 n  ln( k i ) ∑  k g = exp i =1  n    

(6)

where i = 1, 2, .. n is the number of cells in the distribution.

Correlation Length

Figure 7 A correlated, random permeability distribution (white = high permeability, dark = low permeability)

The results given below have been derived theoretically for log-normal distributions, with a standard deviation of σY, where Y = ln(k). The results depend on the number of dimensions:

k eff = k g (1 − σ 2Y 2)

in 1D

k eff = k g

in 2 D

k eff = k g (1 + σ 2Y 6)

in 3D

}

(7)

These formulae are approximate, and assume σY is small (< 0.5). (You are not required to know the proof.) The 1D result is an approximation of the harmonic average. Note that the results do not depend on the correlation length of the field, provided it is much smaller than the system size. Also note that ka > kg > kh, and the effective permeability always lies between the two extremes: ka and kh. Example 3

Figure 8 A random arrangement of the permeabilties in the simple example

75 cells of 10 mD 125 cells of 100mD


7

Suppose that the permeability values in the simple example are jumbled up, so that there are 75 small cells of 10 mD, and 125 small cells of 100 mD. See Figure 8. The effective permeability of this model is:

k eff = k g = 10 = 10

 75 × log(10 ) +125 × log(100 )    200

 75 + 250   200 

= 101.625 = 42.17 mD. 1.3.4 Additional Averaging Methods

Since averaging is very quick (compared with numerical simulation), many engineers use this technique in more complex models. Sometimes engineers increase the accuracy by using power averaging. The power average is defined as: 1/ α

 n α  ∑ ki  k p =  i =1   n   

, (8)

where α is the power. The value of the power depends on the type of model, and must be calibrated against numerical simulation (Section 1.4). Also, sometimes, engineers use a combination of the arithmetic and harmonic averages, e.g. they take the arithmetic average of the permeabilities in each column and then calculate the harmonic average of the columns. 1.3.5 Summary of Permeability Averaging

To summarise, there are two types of simple model in which we can calculate the effective permeability by averaging: • Parallel layers • Correlated random fields Since averaging is very quick, it is frequently used as an approximation for the effective permeability in more complex models. 1.4 Numerical Methods

In general, the permeability distribution will not be simple enough for us to be able to calculate the effective permeability analytically (i.e. by averaging), and we will have to perform a numerical simulation. We can use a finite difference method to calculate the pressures. 1.4.1 Recap on Flow Simulation

The continuity equation tells us that there is no net accumulation or loss of fluid within a grid block:

8

7


q xin + q zin = q xout + q zout .

(9)

(We are assuming incompressible rock and fluids, here.) qzin

x

i,j-1

z

qxin

i-1,j

qxout

Figure 9 Recap on numerical flow simulation

i,j

i+1,j

i,j+1 qzout

Darcy’s law is used to express the flows in terms of the pressures and permeabilities. For example, if the grid blocks in Figure 9 are of length Δx and height Δz (and unit width in the y-direction), then:

q xin = −

(

k x, i −1/ 2, j ∆z Pi , j − Pi −1, j µ

∆x

) (10)

where kx,i-1/2,j is the harmonic average of the permeabilities in the x-direction in blocks (i-1,j) and (i,j). (You now should know now why the harmonic average is used here.) The other flows are calculated in a similar manner. It is useful to use the transmissibilities, Tx = kxΔz/Δx and Tz = kzΔx/Δz. (Assume the width, Δy = 1.) We can therefore derive the pressure equation:

(T

x , i −1 / 2 , j

)

+ Tx, i +1/ 2, j + Tz, i , j −1/ 2 + Tz, i , j +1/ 2 Pi , j

− Tx, i −1/ 2, j Pi −1, j − Tx, i +1/ 2, j Pi +1, j − Tz, i , j −1/ 2 Pi , j −1 − Tz, i , j +1/ 2 Pi , j +1 =0

(11)

An equation is set up for each Pij, i = 1, 2, .. nx and j = 1, 2, .. nz. The transmissibilities are known, and using the appropriate boundary conditions, we can solve this set of linear equations to obtain the pressure in each grid block. The effective permeability is than calculated from the total flow and the total pressure drop, as described below. Note that the boundary conditions are applied to each coarse grid cell in turn, and they may not be a good approximation to the pressures which would arise in a finegrid simluation. This leads to errors in the results. Upscaling errors are discussed in Section 1.5. 1.4.2 Boundary Conditions

Boundary conditions are required to specify what happens at the edges of the model.


9

a) No-Flow Boundaries no flow through the sides

P1

P2 Figure 10 Fixed pressure, or no-flow, boundary conditions

no flow through the sides

The pressure is fixed on two sides of the model, and no flow is allowed through the others sides of the model. This type of boundary condition is suitable for models where there is little cross-flow: for example, models with approximately horizontal layers, or a random distribution. These are the most commonly applied boundary conditions. Figure 11 illustrates how an effective permeability may be calculated in the x-direction. Pressure= P1 on left face

Pressure= P2 on right face

y x

Area, A

Flow Rate, Q

z

Figure 11 The calculation of effective permeability using no-flow boundary conditions

L

1. Solve the steady-state equation to give the pressures, Pij, in each grid block. 2. Calculate the inter-block flows in the x-direction using Darcy’s Law. (See Equation 10.) 3. Calculate the total flow, Q, by adding the individual flows between two y-z planes. (Any two planes will do, because the total flow is constant.) 4. Calculate the effective permeability for flow in the x-direction, using the equation:

Q=

k eff , x A( P1 − P 2) µL

(12)

Repeat the calculation for flow in the y- and z-directions, to obtain keff,y and keff,z. (b) Periodic Boundary Conditions Periodic boundary conditions are useful for calculating the effective permeabilities in models where there are infinitely repeated geological structures in each direction. (See Section 3.2.) The use of periodic boundary conditions ensures that we also

10

7


have an infinitely repeated pattern of flows and pressure gradients. In the example shown in Figure 12, there is a net pressure gradient in the x-direction. The blocks are numbered i = 1, 2, ..nx in the x-direction, and j = 1, 2, .. nz in the z-direction. x

z

P(i,0) = P(i,nz)

Figure 12 Periodic boundary conditions

P(nx+1,j) = P(1,j)-∆P

P(0,j) = P(nx,j)+∆P

P(i,nz+1) = P(i,1)

One advantage of using periodic boundary conditions, is that fluid can flow through the sides of the model. This method can be used to calculate a full tensor effective permeability (Section 3.2). (c) Linear Pressure Boundary Conditions In linear pressure boundary conditions (Figure 13), the pressure is fixed at each end, as in the fixed-pressure boundary conditions. Then, the pressure at the edges of the model is interpolated linearly from one side to the other. Like the periodic boundary conditions, the linear pressure boundary conditions allow flow through the edges. P1

P2

P1

Figure 13 Linear pressure boundary conditions

P2

P1

P2

(d) Flow Jacket, or Skin To reduce the effect of boundary conditions when calculating the effective permeability, some engineers perform the simulations on a larger grid than necessary. The extra grid blocks round the edges are referred to as a “jacket” or “skin”. See Figure 14.


11

boundary conditions applied to outer edges of model

keff calculated for this block

Figure 14 Example of a flow jacket round a model. In this case the jacket is 4 cells thick

1.5 Upscaling Errors

In the process of upscaling, information about the fine-scale structure is lost, and upscaling usually gives rise to errors. However, in some cases, the errors will be larger than in others. In this section, we examine a series of models which show examples of where upscaling is successful and where it is not. 1.5.1 Correlated Random Fields

We introduce the concept of a correlated random field. Although the permeability distribution in real rocks may not follow this type of model, it is a useful way to parameterise heterogeneity. We assume that the probability density function (pdf) of the model is normal or log normal, as shown in Figure 15. In an isotropic model (i.e. same in all directions), the field is then characterised by three parameters: the mean, μ, the standard deviation, σ, and correlation length, λ. The standard deviation determines the width of the pdf (i.e. the permeability contrast), and the correlation length determines approximately the distance over which the permeability values are similar. Figure 16 shows examples of the 4 models with varying σ and λ. 0.06

a)

0.06 0.05 Frequency

Frequency

0.05 0.04 0.03 0.02

0.03 0.02 0.01

0.00 40

0.00 0.5

60

80 100 120 140 Perm eabil ity (m D)

160

c)

0.05 Frequency

0.04

0.01

0.06 0.04 0.03 0.02 0.01 0.00

12

b)

0

500 1000 1500 Per m eab ility (m D)

2000

1.0

1.5 2.0 2.5 3.0 log (p erm eabil ity)

3.5

Figure 15 Normal and log-normal permeability distributions. a) Normal distribution with mean = 100 mD, and standard deviation = 20 mD. b) Log-normal distribution with mean = 2.0, and standard deviation = 0.5. c) Log-normal distribution as above, but with permeability plotted on the x-axis, rather than log(permeability)

7


Figure 16 Models with different standard deviations and correlations lengths

a) small σ, small λ

b) large σ, small λ

a) small σ, large λ

b) large σ, large λ

Permeability (mD) 0

50

100

150

200

1.5.2 Evaluating the Accuracy of Upscaling

One way to evaluate the accuracy of upscaling is to compare upscaling in two stages,

k 2eff , with upscaling in a single stage, k1eff , as shown in Figure 17. If upscaling is 2 1 accurate, then k eff = k eff . This is the case for upscaling along and across parallel layers in the idealised models of Sections 1.3.1 and 1.3.2. fine-scale model

k1eff

single cell

k2eff

Figure 17 Comparison of one-stage and two-stage upscaling

The accuracy of scale-up is affected by the correlation length and standard deviation of the distribution, and we use the method shown in Figures 10 and 11 to demonstrate this effect. Instead of generating many fine-scale models with different correlation lengths, we create 1 fine-scale model, but upscale by different factors – so that the


13

coarse block size varies relative to the correlation length. Figure 18a shows a finescale model with a correlated random permeability distribution. The model has 400 x 400 grid cells, each of size 1 m3. The permeability distribution is ln-normal, with a mean of 4.6 (corresponding to 100 mD), and a correlation length of 10 m. Three different versions of the model were created with different standard deviations: 0.5, 0.75 and 1.0. The following scale-up factors were tested:

4 × 4, 5 × 5, 8 × 8, 10 × 10, 16 × 16, 20 × 20, 40 × 40, 80 × 80 In terms of the correlation length, this gives coarse-scale cells of size: 0.4, 0.5, 0.8, 1.0, 1.6, 2.0, 4.0, 8.0. Figure 18b and c show examples of coarse-scale models with scale-up factors of 5 and 50. In each case the ratio of two-stage upscaling to single-stage upscaling was calculated. The results are plotted in Figure 19. The results are least accurate when the scale-up factor is 10 – 50, i.e. when the coarse block size is 1 – 5 times the correlation length. Also, the error increases with the standard deviation of the model, as one might expect. a)

b)

c)

Permeability (mD) 0.1

1

10

100

1000

10000

Figure 18 a) Fine-scale model with 400 x 400 cells; b) coarsescale model with 80 x 80 cells; c) coarse-scale model with 8 x 8 cells

keff2/keff1

1.03

1.02

sigma = 1.00 sigma = 0.75 sigma = 0.50

1.01 1. 01

1.00 0

20

40

60

80

Scale-up Factor

The conclusions from these examples are: • •

14

Upscaling will be least accurate when the coarse cell size is comparable to, or slightly larger than the correlation length. Upscaling errors increase as the standard deviation of the model increases.

Figure 19 2 1 The ratio of k eff k eff for different scale-up factors

7


1.5.3 Upscaling of a Sand/Shale Model

Upscaling errors are largest in models where there are high permeability contrasts. Unfortunately, high permeability contrasts frequently occur in reservoir rocks. For example, we often require to model the following: • Low permeability shales in a high permeability sandstone • Low permeability faults in a high permeability sandstone • High permeability channels in a low net/gross reservoir • High permeability fractures in a low permeability reservoir All these cases are difficult to model. As an example, we consider a sand/shale model, where the shale has zero permeability. Figure 20 shows the fine-scale model, which has to be upscaled to 3 coarse blocks, as shown. Since there is a shale lying across each coarse block, each coarse block will have zero permeability in the z-direction (vertical). However, fluid can flow through the model vertically, as shown. This error arises because the coarse block size is similar to the characteristic length of the shales. Upscaling would be more accurate, if the coarse block size was much larger, or much smaller than the shales. Alternatively, using a “skin” or “flow jacket” will increase the accuracy of upscaling (Section 1.4.2, Figure 14).

Figure 20 Sand/shale model 1.6 Summary of Single-Phase Upscaling

The main points to remember from this section are: • • • •

• • • •

Fine-scale geological models usually require upscaling for full-field simulation. Upscaled permeability is generally referred to as effective permeability. Some quantities, such are porosity and water saturation are easy to upscale, because they may be averaged arithmetically. In some simple models, permeability may also be upscaled using averaging, as follows: − the arithmetic average for along-layer flow; − the harmonic average for across-layer flow; − the geometric average for a random distribution. In more complex models, the effective permeability is calculated using a numerical simulation. Different boundary conditions may be used when calculating the effective permeability numerically: constant pressure (or no-flow), periodic and linear. Upscaling is least accurate when the coarse cell size is comparable to, or slightly larger that the correlation of the permeability distribution. Upscaling errors increase as the standard deviation of the model increases.


15

2 TWO-PHASE FLOW 2.1 Introduction

Often we need to simulate two-phase systems, e.g. a water flood or a gas flood of an oil reservoir, or an oil reservoir with a gas cap or an aquifer. The aim of upscaling in this case is to calculate a coarse-scale model which can reproduce the flow rates of the different fluids. The coarse model should also provide a good approximation to the saturation distribution in the reservoir with time. The paths which the injected fluid takes through the reservoir depends on the forces present: • Viscous – due to injection of a fluid • Capillary • Gravity Therefore, the balance of forces should be taken into account during upscaling. Before learning how to upscale two-phase flow, we show the effects which geological heterogeneity may have on hydrocarbon recovery. Consider the following simple model (Figure 21), with alternating horizontal layers of 100 mD and 10 mD (referred to as facies 1 and facies 2). We assume that the model is filled with oil and connate water initially, and simulate a water flood, by injecting at uniform rate at the left side, and producing from the right side (at constant bottomhole pressure). The density of the two fluids is the same for this example, so that there are no gravity effects. Figure 22 shows the relative permeabilities and capillary pressures, and Table 1 lists the properties of the 3 cases simulated with this model. 100 mD 10 mD

12

0.9

krw 1 kro 1 krw 2 kro 2

8 6

Pc 1 Pc 2

0.8 0.7 0.6

Rel Perm

Cap Pressure

10

0.5 0.4 0.3

4

0.2

2

0.1 0

0 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.2

0.3

Case 1 2 3

Porosity facies 1 facies 2 0.2 0.2 0.2 0.05 0.2 0.1

0.4

0.5

0.6

Water Saturation

Water Saturation

16

Figure 21 Simple layered model for demonstrating viscous and capillary effects

Rel Perm/Pc Curve No. facies 1 facies 2 1 1 1 1 1 2

0.7

0.8

Figure 22 The relative permeability and capillary pressure curves

Flow Regime viscous viscous visc+capillary

Table 1 Properties of the first set of examples

7


Figure 23 shows the distribution of oil saturation after the injection of 0.2 PV, for Cases 1 and 2. Both cases use only a single relative permeability curve and the flow regime is viscous-dominated. Water flows faster along the high permeability layers, as one would expect. However, notice that this effect is reduced when the porosity of facies 2 is reduced.

Figure 23 The oil saturation for Cases 1 and 2, after the injection of 0.2 PV

Oil Saturation

0.3

0.4

0.5

0.6

0.7

In Case 3, both relative permeability and capillary pressure tables are used. These curves are typical of a water-wet rock: the capillary pressure is much higher in the low permeability facies, and the connate water saturation is higher. In this case, water is imbibed along the low permeability layers, and also there is cross-flow from the high permeability layers to the low permeability ones (Figure 24). Due to the effects of capillary pressure, the front is nearly level in the two facies.

Figure 24 The oil saturation for Case 3, after the injection of 0.2 PV

Oil Saturation

0.3

0.4

0.5

0.6

0.7

Figure 25 shows the recovery as a function of pore-volumes injected, and demonstrates that models may have the same effective absolute permeability, but different recoveries.


17

0.5 0.4 0.3

Case 1 Case 2 Case 3

0.2 0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

Pore Volumes Injected

Figure 25 Cumulative recovery and watercut for Cases 1 - 3

In the next example, graded models are used, as shown in Figure 26. There are two versions: Case 4 with permeability increasing upwards (referred to as coarsening-up) and Case 5 with permeability decreasing upwards (referred to as fining-up). The permeabilities range from 200 mD to 1000 mD, the porosity was kept constant at 0.2, and the first relative permeability curve was used. In this model, however, the densities of the fluids were different: the density of water was set to 1000 kg/m3 and that of oil was set to 200 kg/m3. a) Coarsening-up

b) Fining-up

1000 mD

200 mD

200 mD

Figure 26 The graded layer models for Cases 4 and 5

1000 mD

Again a waterflood was performed, and the results are shown in Figure 27. Since water is more dense than oil, water has a tendency to slump down. In Case 4 (coarsening-up), this tendency is reduced by the fact that the viscous forces tend to move the fluid faster in the upper layers. However, in Case 5 (fining-up), the slumping effect is reinforced by the viscous force moving fluid faster along the lower layers. This means that the breakthrough time is earlier in Case 5 than in Case 4, as shown in Figure 27. The effective absolute permeability in these two models is the same, but the two-phase flow effects are different, due to the effect of gravity. (If the density of water equalled the density of oil, the recovery and watercut curves would be identical.) a) Coarsening-up

a) Fining-up

Oil Saturation

0.3

18

0.4

0.5

0.6

0.7

Figure 27 The oil saturation after the injection of 0.2 PV in the graded layer models

7

Figure 28 Cumulative recovery and watercut for the graded layer models

0.5

1.0

0.4

0.8

Water Cut

Fractional Recovery


0.3 0.2 0.1 0.0 0.0

0.6

Case 4 Case 5

0.4 0.2

0.1

0.2

0.3

0.4

0.5

0.0

0.6

0.0

0.1


0.2

0.3

0.4

0.5

0.6


In summary, in a viscous-dominated flood, permeability heterogeneity disperses the flood front (Cases 1 and 2), so that breakthrough occurs earlier, and the water cut curve rises less steeply. However, the effect of heterogeneity also depends on the balance of fluid forces. In a water-wet system, capillary pressure can help the front to advance more evenly along the layers (Case 3). (More information on capillary effects is given in Section 3.3.) Gravity effects may increase or reduce the viscous effects, depending on permeabilities in the model (Cases 4 and 5). 2.2 Applying Single-Phase Upscaling to a Two-Phase Problem

Most engineers only perform single-phase upscaling although, as shown above, heterogeneities give rise to a variety of effects in two-phase flow. The reason for this is that two-phase upscaling is time-consuming and the results are not always robust (i.e. they may contain large errors). We deal with two-phase upscaling in Sections 2.4 – 2.6. Figure 29 (left side) shows a very heterogeneous 2D model, with correlated random permeabilities. The permeability distribution was ln-normal (i.e. natural logs), with a standard deviation of 2.0 The model is assumed to be in the horizontal plain. The details of the model are given in Table 2. The model was upscaled using the pressure solution method with no-flow boundaries. Three different scale-up factors were used, and the coarse-scale models 1 and 3 are also shown in Figure 29 (middle and right side).

Figure 29 Fine- and coarsescale models used for demonstrating the effects of applying single-phase upscaling to a two-phase problem

Table 2

log10(k)

-2

Model Fine Coarse 1 Coarse 2 Coarse 3

-1

0

Number of Cells 105 x 105 21 x 21 15 x 15 7x7

1

2

3

4

5

Cell dimension (m) 5 25 35 75


6

Scale-up Factor 5x5 7x7 15 x 15

19

A waterflood with a quarter 5-spot well pattern was performed (i.e. 2 wells in diagonally opposite corners). The same relative permeability curve was used for the whole model (Figure 30), and the capillary pressure was set to zero. The flood was therefore viscous-dominated. The viscosity of water was 0.3 and that of oil was 3.0. The resulting recovery curves are shown in Figure 31. As one might expect, the error increases with the scale-up factor.

Relative Permeability

1 0.8 0.6

krw kro

0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

Water Saturation

Figure 30 The relative permeability curve used for the random model. (Capillary pressure was set to zero)

0.6

Fractional Recovery

0.5 0.4

fine ups 5x5 ups 7x7 ups 15x15

0.3 0.2 0.1

Figure 31 Comparison of recovery for different scale-up factors

0 0

0.2

0.4

0.6

0.8

1


The model was modified by reducing the standard deviation to 0.2 (lowering the permeability contrast), and the simulations were repeated with the low heterogeneity model. The recovery is shown in Figure 32 for the scale-up factor of 15 x 15. As expected, the errors are smaller for the low heterogeneity model.

Fractional Recovery

0.6 0.5 0.4

lo het fine lo het coarse hi het fine hi het coarse

0.3 0.2 0.1 0

0

0.2

0.4

0.6

0.8

1


In addition, the errors caused by using only single-phase upscaling, are larger when the coarse block size similar to the correlation length. Also, the upscaling error tends 20

Figure 32 Comparison of recovery for models with different levels of heterogeneity

7


to be larger in unstable floods (injected fluid is of lower viscosity than the in situ fluid) than in stable floods. In summary, single-phase upscaling may be adequate for upscaling two-phase systems, provided that: • • • • • •

The scale-up factor is small The permeability contrasts are small The correlation length is very large, or very small compared to the coarse cell size The flood is stable (favourable mobility ratio) The flood is not capillary-dominated (See Section 3.3) The flood is not gravity-dominated

2.3 Improving Single-Phase Upscaling

There are two approaches which may make single-phase more accurate when applying it to two-phase problems. The first is to use non-uniform upscaling, and the second is to perform a global single-phase simulation (i.e. over the whole fine-scale model) using the correct boundary conditions, including wells. We refer to this second method as Well Drive Upscaling (WDU). 2.3.1 Non-Uniform Upscaling

Consider a model with horizontal layers, as shown in Figure 33. There is a high permeability streak running across the model. The model details are given in Table 3. In both coarse-scale models there are 3 coarse cells in the vertical direction. In model Coarse 1, the cells are each 5 m thick. However, in model Coarse 2, the thicknesses are: 7 m, 1 m, 7 m, so that the high permeability streak is still resolved. b) C Coa oars oa rse rs e1

c) Coar c) Coar oarse se 2

Figure 33 Model with a high permeability streak

Table 3

Model Fine Coarse 1 Coarse 2

No. of cells 100 x 15 20 x 3 20 x 3

Cell size (m) 1x1 5x5 variable

Figure 34 shows the recovery for these models. It can be seen that the model with uniform coarse cells (Coarse 1) gives very inaccurate results, and the model which maintains the high permeability streak (Coarse 2) is much more accurate.


21

Fractional Recovery

0.25 0.20 0.15

fine coarse 1 coarse 2

0.10 0.05 0.00

0.0

0.1

0.2

0.3

Figure 34 Recovery and watercut for fine- and coarse-scale models

0.4


In a practical upscaling application, much attention is paid to upgridding the model – i.e. deciding how to amalgamate the layers, so that upscaling is as accurate as possible. The coarse grid may also be non-uniform in the x- and y-directions. Several methods for performing non-uniform upscaling have been developed. For example, Durlofsky et al. (1996, 1997) first carry out a single-phase simulation. Then they use the inter-block flows to determine the coarse block boundaries. Smaller coarse blocks are assigned to regions where there are high flow rates. 2.3.2 Well Drive Upscaling

When upscaling using numerical simulation, boundary conditions are applied to each coarse-scale cell in turn (Figure 2.) These are called local boundary conditions. However, the boundary conditions described in Section 1.4.2 may be quite different from the pressures which actually occur in a fine-scale simulation, leading to inaccuracies in the upscaled model. To overcome this problem, we may use global boundary conditions. Figure 35 demonstrates the difference between local and global boundary conditions. Boundary conditions applied to coarse cell

P1

Injector

P2

Producer

Local

Global

In the Well-Drive Upscaling method (WDU), a single-phase simulation is performed on the whole fine grid (Zhang et al, 2005). (It is feasible to perform a single pressure solve on grids with several million grid cells.) Then the effective transmissibility between coarse-scale cells is calculated, rather than the effective permeability. The upscaled transmissibility is give by:

22

Figure 35 Local and global boundary conditions

7


T=

∑q

PΙ − PΙΙ

(13)

Where q denotes the fine-scale flows (Figure 36) and PI and PII are the (pore-volume weighted) average pressures in coarse cells I and II.

Figure 36 Upscaling transmissibility

Ι

ΙΙ

Upscaling is also performed at the wells, using the method described in Section 3.1. This method produces very accurate single-phase upscaling, which leads to an increase in the accuracy of two-phase flow at the coarse scale. Many tests on upscaling methods have been carried out using the model generated for the 10th SPE comparative solution project, which was on upscaling (Christie and Blunt, 2001). This model is referred to as the SPE 10 model (Figure 37a). We use layer 59 (Figure 37b) as an example of well drive upscaling, because this layer is particularly heterogeneous. There is an injection well at the centre and production wells in each corner. a)

Figure 37 The SPE 10 model, and layer 59

b)

P1

P2

P4

P3 log10(k)

-3

-2

-1

0

1

2

3

4

5

Layer 59 was upscaled using the WDU method and also the conventional method with local boundary conditions. Figure 38 shows the oil saturation distribution. It can clearly be seen that the WDU method reproduces the results of fine-scale model much better than the conventional approach. This is because appropriate boundary conditions have been applied to the model.


23

fine

local

WDU

Figure 38 Comparison of oil saturation distribution in fine- and coarse-scale simulations of Layer 59 of the SPE 10 model

Soil

0.20

0.35

0.50

0.65

0.80

2.4 Introduction to Two-Phase Upscaling

So far, when performing upscaling, we have assumed that there is only one phase present, and that the flow is in a steady state. We only need to upscale the absolute permeability. However, when there are two phases flowing, such as water displacing oil, the system is not, in general, in a steady state. We need to simulate finescale floods in order to upscale relative permeability and capillary pressure. This is referred to as dynamic upscaling, and the upscaled relative permeabilities are known as pseudo relative permeabilities, or pseudos. Pseudos can be calculated to take account of physical dispersion, and also to compensate for numerical dispersion (Section 2.6.3). When upscaling, we should use the phase permeabilities:

k f = k abs k rf

(14)

Where “f” stands for fluid – oil, gas or water. Generally, we assume that both the absolute and the relative permeabilities are homogeneous and isotropic at the smallest scale ( k x = k z ). As we upscale, the absolute and relative permeabilities may become anisotropic ( k rx ≠ k rz ). To obtain effective (or pseudo) relative permeabilities, the absolute permeability must be scaled-up separately. Then the pseudo relative permeability is calculated as follows:

k rf , x = k f , x k abs, x

(15)

Similar equations are used for flow in the y- and z- directions. 2.5 Steady-State Methods

If fluids are in a steady state, the saturation does not change with time and the fractional flow (flow of water/total flow) is constant. Although floods are dynamic processes, sometimes a flood may approach a steady state. For example, over small scales (20 cm, or less), oil and water may come into capillary equilibrium.

24

7


In a steady-state upscaling method, we assume that within a short interval of time the zone of interest is in a steady-state, but we allow the fluid saturation to change gradually, so that a full range of saturation is obtained. At steady-state, the water saturation does not change with time, i.e. ∂Sw/∂t = 0, so the continuity equation becomes:

∇ ⋅ u f = 0,

(16)

where u is Darcy velocity, and f is fluid. From Darcy’s law:

∇ ⋅ ( k f ⋅ ∇Pf ) = 0.

(17)

There are several steady-state methods, depending on the balance of forces: • Capillary equilibrium, • Vertical equilibrium (gravity-dominated flood) • Viscous-dominated steady-state We concentrate here on the capillary equilibrium method. The advantage of steady-state methods is that they turn two-phase upscaling into a series of single-phase upscaling calculations. This means that steady-state methods are feasible for models with large numbers of grid cells. (See, for example, Pickup and Stephen, 2000; and Pickup et al, 2000.) 2.5.1 Capillary-Equilibrium

Assume that the injection rate is very low, gravity forces are negligible, and that the fluids have come into capillary equilibrium with a coarse-scale cell. This means that the saturation distribution is determined by the capillary pressure curves. The method is as follows: 1. Choose a Pc level. 2. Determine the water saturations, and then the relative permeabilities. 3. Calculate the pore volume-weighted average water saturation. 4. Calculate the phase permeabilities: ko = kabskro, kw = kabskrw. 5. Calculate the effective water phase permeability, kw 6. Calculate the effective oil phase permeability, ko 7. Calculate the relative permeabilities, krw = kw/kabs, etc. 8. Repeat the process with another value of Pc. Steps 5 and 6 may be carried out analytically or numerically, depending on the distribution. Institute of Petroleum Engineering, Heriot-Watt University

25

Example 4 Consider a model with two layers of equal thickness, as shown in Figure 39. The absolute permeabilities are 100 mD and 20 mD. Assume that the porosity in each layer is equal to 0.2. The relative permeability and Pc curves for each layer are shown in Figure 40.

kabs (mD) 100

Figure 39 Model with horizontal layers

20

0 8 6 4

0.8

lo

Rel Perm

Cap Pressure

12

hi

2 0 0.2

0.3

0.4

0.5

0.6

Water Saturation

0.7

0.8

0.6

hi

lo

0.4

hi

0.2 0

0.2

0.3

0.4

0.5

0.6

Water Saturation

lo 0.7

0.8

Figure 40 Relative permeability and capillary pressure curves

Using the arithmetic and harmonic averages (Section 1.3), the effective permeability is:

k x = 60.00

k z = 33.33

Suppose we choose a capillary pressure of Pc = 0.45. In the high perm layer: Sw = 0.34, krw = 0.0013, kw = 0.13, kro = 0.5, ko = 50. In the low perm layer: Sw = 0.44, krw = 0.0016, kw = 0.032,

kro = 0.48, ko = 9.6.

Figure 41 shows the phase permeabilities.

kw (mD)

ko (mD)

0.13

50.0

0.032

9.6

Since the layers are of equal width, the average saturation is Sw = 0.39. The effective phase permeabilities are then calculated using the arithmetic and harmonic averages. Then the relative permeabilities are calculated using Equation 15.

26

Figure 41 Phase permeabilities

7


k wx = 0.081

k rwx = 0.081 / 60.00 = 0.00135

k wz = 0.051

k rwz = 0.051 / 33.33 = 0.00153

k ox = 29.8

k rox = 29.8 / 60.00 = 0.50

k oz = 16.1

k roz = 16.1 / 33.33 = 0.48

Note that the kv/kh ratio ( = k z k x ) is different for oil and water:

k w, z k w, x = 0.63

k o, z k o, x = 0.54

Effective relative permeability curves may be derived by repeating this calculation for a range of capillary pressure values (Figure 42). The capillary-equilibrium method is useful as a quick method for upscaling small-scale models (Section 3.3). However, it is only valid in cases where the flow rate is very low. 0.9 0.8

Rel Perm

0.7

krox

0.6 0.5 0.4

kroz

0.3

krwz

0.2

Figure 42 Effective relative permeability curves

0.1 0 0.2

0.3

0.4

0.5

0.6

Water Saturation

krwx 0.7

0.8

2.6 DYNAMIC METHODS 2.6.1 Introduction

For dynamic (or non steady-state methods), we need to perform a two-phase flow simulation on a fine grid. There are basically two types of dynamic method: a) Weighted Pressure Methods As in single-phase numerical upscaling, a common approach in two-phase upscaling is to sum the flow, average the pressure gradient and use Darcyʼs Law to obtain the pseudo phase permeability. However the pressure may be averaged in different ways. Here, we shall concentrate on the Kyte and Berry (1975) method. b) Total Mobility Methods In total mobility methods, we avoid averaging the pressure, and scale-up the total mobility. Then the average fractional flow is used to calculate the pseudo relative permeabilities. The total mobility is:

λt = λo + λw =

k ro k rw + . µo µw


(18) 27

The fractional flow is the flow of water divided by the total flow:

fw =

qw q = w qo + qw qt

(19)

Again there are a number of variations of this method, the most commonly used being that of Stone (1991). 2.6.2 The Kyte and Berry Method

A simple version of the Kyte and Berry (1975) method is presented here, using the grid shown in Figure 43. i=1

2

3

4

5

6

7

8

9 10

j=1 2 3 4 5

∆z ∆x

Pseudo calculated for this coarse block

DZ ∆Z

DX ∆X

Figure 43 Model used for describing the Kyte and Berry Method. The thickness of the model is Δy Δ

The diagram shows two coarse grid blocks, each of which is made up of 5 x 5 fine blocks. The equations below show how to calculate the pseudo relative permeabilities and capillary pressure for the left coarse block. The first step is to perform a fine-scale, two-phase simulation (e.g. in ECLIPSE), saving the pressures and inter-block flows at specified intervals of time (in the re-start files). The method proceeds as follows: 1. Calculate the effective absolute permeability in the area shown in Figure 44, i.e. half way between the two coarse blocks. i=3

i=7

j=1

j=5

Kyte and Berry approximate the effective permeability using the arithmetic average in each column, and then taking the harmonic average of the columns. The area between the two coarse blocks is used, for reasons explained below.

28

Figure 44 The area used for calculating the effective absolute permeability

7


5

ki =

∑ ∆z k j =1

j

ij

∆Z

(20)

where Δzzj and ΔZ are the thicknesses of the fine and coarse blocks, respectively. (In this case, all the blocks are of equal size.)

kI =

∆X

7

∑ ∆x i=3

i

ki

(21)

where Δxi and ΔX are the lengths of the fine and coarse blocks, and k I is the required effective absolute permeability. The pseudos are then calculated, at certain times during the simulation. (These are the times at which the restart files are written in the Eclipse simulation.) 2. Calculate the average water saturation: 5

Sw =

5

∑∑S j =1 i =1 5 5

φ ∆x i ∆z j

w , ij ij

∑ ∑ φ ∆x ∆z j =1 i =1

ij

i

j

(22)

where φij is the porosity. 3. Calculate the total flow of oil and water out of the left coarse block (Figure 45). 5

q f = ∑ q f 5, j , j =1

(23)

where qf5,j is the flow of fluid “f” from fine block number (5,j). i=5 j=1

Figure 45 Calculation of the total flow

j=5

4. Calculate the average phase pressures in the central column of each coarse block. In this example, we use the fine blocks in columns 3 and 8, the shaded areas in Figure 46.


29

i=3

i=7

j=1

j=5

I

Figure 46 The cells used for averaging the phase pressures

II

In the Kyte and Berry method, the pressures are weighted by the phase permeabilities times the height of the cells (which in this case are all the same size). This is so that more weight is given to regions where there is greater flow. However, there is no scientific justification for using this weighting. In the first coarse block (numbered, I), the average pressure is: 5

P fI =

∑k j =1

3j

(

k rf 3 ∆z 3 j Pf 3 j − gρf (D3 j − D) 5

∑k j =1

3j

)

k rf 3 ∆z 3 j

(24)

where D3j is the depth of cell (3,j) and D is the average depth of coarse cell I. The term gρf(D3j - D ) is to normalise the pressure to the grid block centre. The average pressure for coarse block II is calculated in the same manner, but using column 8 instead of column 3. The pressure difference is then calculated as:

∆P f = P fI − P fII .

(25)

5. The pseudo rel perms are then calculated using Darcy’s law. Firstly, calculate the pseudo potential difference. (Potential is defined as Φ = P-ρgz, so that the flow rate is proportional to ∇Φ .)

∆Φ f = ∆P f − gρf ∆ ∆D,

(26)

where ΔD is the depth difference between the two coarse grid centres. Then:

k rf =

−µ f q f ∆X ∆Zk I ∆Φ I

(27)

6. Calculate the pseudo capillary pressure using:

P c = P oI − P wI

(28)

The Eclipse PSEUDO package can be used for calculating Kyte and Berry pseudos.

30

7


2.6.3 Discussion on Numerical Dispersion

One advantage of pseudo-isation methods, such as that of Kyte and Berry is that they can take account of numerical dispersion. When a simulation is carried out using a larger grid, the front between the oil and water becomes more spread out. However, the Kyte and Berry method counteracts this effect by calculating the flows on the down-stream side of the coarse block, instead of the middle. This is illustrated by a simple example. Figure 47 shows an example of input relative permeability curves (“rock” curves). 0.9 0.8 0.7

Perm Rel

0.6 0.5 0.4 0.3 0.2 0.1 0

Figure 47 Example of “rock” curves

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Water Saturation

If the water saturation is Sw = 0.5, the rock curves show that there is a small amount

of oil and water flowing. However, when the average saturation, Sw , is 0.5 in the coarse block, the distribution could be as shown in Figure 48.

oil Figure 48 Example of the water saturation in a coarse block

water

coarse block

Since the water has reached only half way across the coarse block, there should be no water flowing out of the right side. The Kyte and Berry method calculates the pseudo relative permeabilities using the flow on the downstream side of the coarse block, to prevent water breaking through too soon. The pseudo water relative permeability curve is moved to the right, relative to the rock curves, as shown in Figure 49.


31

1

pseudos - solid lines ave. rock curves - dashed lines


0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.2

0.4

0.6

0.8

Water Saturation

2.6.4 Disadvantages of the Kyte and Berry Method

There are certain problems with the Kyte and Berry method. • • •

• •

Negative rel perms are produced, if ∆Φ f has the same sign as q f . Infinite rel perms occur if ∆Φ f is zero. The method of averaging the pressures, using relative permeability as a weighting function, may cause errors when the fluids are separated due to gravity. For floods which are gravity-dominated, the TW method works better (Section 2.6.5). Non-zero pseudo capillary pressure may be produced, even if there is no capillary pressure in the fine-scale simulation. This is because a different weighting is used for calculating the average pressure in each phase. The capillary pressure may be different in different directions, because only the central column is used for averaging the pressures.

Because of the first two disadvantages, i.e. negative, or infinite rel perms, pseudos obtained from packages like the PSEUDO must be vetted before using at the coarse scale. Often “odd” values of relative permeability are set to zero. Good reviews of various methods for calculating pseudos are presented in Barker and Dupouy (1999) and Barker and Thibeau (1997). Note that dynamic upscaling methods, such as that of Kyte and Berry are difficult to apply in practice. Ideally, a fine-scale two-phase flow simulation is required for each coarse-scale cell (plus a “flow jacket”), and this is time consuming. Also, it is difficult to determine the correct boundary conditions to use, so the results may not be accurate. If a pseudo is calculated for each coarse cell, in each direction, there may be 10,000s of pseudos in the coarse-scale model. The number of pseudos must be reduced, by grouping similar pseudos together.

32

Figure 49 Example of pseudo relative permeability curves

7


Pseudo relative permeability curves depend on a number of factors, including: (a) The balance of forces The shape and end points of a pseudo depend on the ratio of viscous/capillary and viscous/gravity forces. These ratios may be different in different parts of the reservoir. (b) The well locations The wells determine the flow rate and direction. If a new well is drilled, the pseudos ought to be re-calculated. Because of these problems, two-phase upscaling is rarely used for upscaling from a geological model to a full-field simulation. 2.6.5 Alternative Methods

There are a number of similar methods to the Kyte and Berry Method. (a) The Pore-Volume Weighted Method The problems of non-zero capillary pressure and directional capillary pressure, mentioned in Section 2.6.4, may be overcome by using a pore volume weighted average of the pressures over the entire coarse block. ECLIPSE uses this method for calculating the average capillary pressure in the Kyte and Berry method. Also, pore volume weighting may be used for averaging the pressures when calculating the pseudo relative permeabilities. In this case, the method is called the Pore Volume Weighted Method. It is available in the Eclipse PSEUDO package. (b) The TW Method This method was developed by Nasir Darman at Heriot-Watt University (Darman et al, 1999). It is similar to the Kyte and Berry method, except transmissibility weighting is used when calculating the average pressure. The method works better than the Kyte and Berry method in cases where gravity effects are significant (e.g. a gas flood). Both these methods share the same problems discussed in Section 2.6.4, namely, they are difficult to apply in practice. 2.6.6 Example of the PVW Method

In two-phase dynamic upscaling methods, pseudo relative permeabilities are calculated, so that (hopefully) the results of a coarse-scale simulation provide a good approximation to the fine-scale results. Layer 59 of the SPE 10 upscaling study (Figure 37b) is used here as an example. A global simulation was performed on the fine grid, i.e. the whole of the fine grid was included in the fine-scale flow simulation. (Note that this would not be done, in practice, because there is no point in upscaling, if you can simulation the whole fine grid.) The fine-scale model had 60 x 220 cells and the coarse-scale model had 10 x 22, which corresponded to a scale-up factor of 6 x 10. Pseudo relative permeabilities were calculated for each coarse cell, in each direction. Figure 50 shows the oil saturation for the fine-scale simulation, a coarse-scale simulation using the WDU method (Section 2.3.2), and a coarse-scale simulation using pseudos


33

from the PVW method. Both the WDU and the PWV methods give reasonable oil saturation distributions. The oil recovery rate, for well P4, for these models is shown in Figure 51, along with the oil rate for a coarse-scale simulation using single-phase upscaling with local boundary conditions (Sections 2.2. and 2.3). fine

WDU

PVW

Figure 50 The oil saturation distribution for the finescale model of layer 59 and the coarse-scale models obtained using the WDU and the PVW methods

Soil

0.20

0.35

0.50

0.65

0.80

Well Rate (m3/day)

60 50 40

Fine Local WDU PVW

30 20 10 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8


It can be seen in Figure 51 that both the WDU and the PVW methods agree well with the fine-scale simulation. However, the results from the single-phase upscaling with local boundary conditions are poor. This example shows that two-phase upscaling is not necessarily always more accurate than single-phase upscaling. 2.7 Summary of Two-Phase Flow

The main points on two-phase flow are: • • • •

34

Two-phase upscaling is time consuming and not always robust, so is rarely used by engineers. Usually, only single-phase upscaling is performed. But, heterogeneity interacts with two-phase flow, and tends to produce dispersion of the flood front, which is not taken into account using single-phase upscaling. So, single-phase upscaling may give rise to errors, especially when there is a large scale-up factor, and the reservoir model is very heterogeneous (large standard deviation).

Figure 51 The oil recovery rate for well P4 for the fine-scale model and coarse-scale models obtained using the WDU, PVW and local upscaling methods

7


Errors in single-phase upscaling may be reduced by using non-uniform upscaling, or well-drive upscaling. Ideally, two-phase upscaling should be performed to take account of two-phase flow. Steady-state upscaling is relatively quick to apply, and is feasible for large models. However, it is only valid in limited cases, e.g. when the fluids are approximately in capillary equilibrium. Dynamic methods are potentially more accurate. The Kyte and Berry (1975) Method was described as an example. Dynamic methods can compensate for the effects of numerical dispersion. Dynamic methods are difficult to apply in practice.

• • • • • • •

3 ADDITIONAL TOPICS This course has, so far, focussed mainly on common methods for upscaling a geological model for full-field simulation. Most of the single-phase upscaling methods presented may be found in geological packages, such as IRAP/RMS and Petrel. (The WDU and TW methods which were developed at Heriot-Watt are not available in commercial packages.) However, there are a number of other important issues which should be taken into account when upscaling. In this section, we cover these issues in a variety of additional topics: • Upscaling as Wells • Permeability Tensors • The Geopseudo Method • Uncertainty and Upscaling 3.1 Upscaling at Wells

In the single-phase upscaling methods described in Chapter 1, we assumed that the flow was linear. This means that the upscaling methods were not appropriate for regions containing wells, where there is radial flow. We start with a brief overview of simulation in blocks containing a well. A grid block in the simulator is much larger than the diameter of a well, and the pressure calculated for a block containing a well is different from the actual bottom hole pressure. These are related by:

q=

Iw (Pw − Pb ) µ

(29)

where Pw is the well-bore pressure and Pb is the pressure of the block. Iw is the well index, given by:

Iw =

2πk∆z ln( ro rw )


(30)

35

where rw is the well-bore radius and ro is the equivalent radius, given by Peacemanʼs equation (Peaceman, 1978; Peaceman, 1983). Iw is also referred to as the well connection factor, or the connection transmissibility factor. Durlofsky et al. (2000) put forward a method for upscaling in the near-well region. Others have put forward similar methods. The method is only approximate, but improves the accuracy of coarse-scale simulations. The first step is to calculate effective single-phase permeabilities, using one of the conventional methods (e.g. periodic boundary condition applied to each block in turn). Then, a fine-scale singlephase simulation of the well block and surrounding blocks is carried out (Figure 52). From the results, the total flows out of the coarse-scale well block, and the average pressures in the coarse blocks are calculated. These are used to calculate upscaled transmissibilities between the coarse-scale well block and the surrounding blocks, and a coarse-scale well index. q

T4

T3

T1

T2 Figure 52 Near-well upscaling (after Durlofsky et al., 2000)

This method improves the accuracy of upscaling at well, and it is also incorporated into the well drive upscaling method (WDU), described in Section 2.3.2. 3.2 Permeability Tensors

Suppose that we have layers which are tilted at an angle to the horizontal, as in Figure 53. net flow in z-dir

x net flow in x-dir

z

∆P

A pressure gradient has been applied in the x-direction. This will obviously give rise to a flow in the x-direction. The fluid takes a path through the medium, so that it expends a minimum amount of energy. There will be a component of flow up the high permeability, and only a small amount of flow across the low permeable layer, as shown. This gives rise to a net flow in the z-direction, or cross-flow. Here, the term cross-flow is used to describe flow perpendicular to the applied pressure gradient. When calculating the effective permeability of this model, we need to take this cross-flow into account. This may be done using a tensor effective permeability, k , where: 36

Figure 53 Cross-flow due to tilted layers. Light-coloured layers represent high permeability and darkcoloured layers represent low permeability

7


 k xx  k =  k yx  k zx 

k xz   k yz  k zz 

k xy k yy k zy

(31)

The first index applies to the flow direction, and the second to the direction of the pressure gradient. For example kxy is the flow in the x-direction caused by a pressure gradient in the y-direction. The terms kxx, kyy, kzz are known as the diagonal terms. These are the terms which are usually considered – the horizontal and vertical permeabilities, kh and kv, respectively. The other terms, which describe the crossflow, are the off-diagonal terms. With tensor permeabilities, Darcyʼs Law becomes:

u=−

k ⋅ ∇P, µ

(32)

where u is the Darcy velocity (vector) and P is pressure (scalar).

1 ∂P ∂P ∂P  u x = −  k xx + k xy + k xz  µ ∂x ∂y ∂z  1 ∂P ∂P ∂P  u y = −  k yx + k yy + k yz  µ ∂x ∂y ∂z  1 ∂P ∂P ∂P  u z = −  k zx + k zy + k zz  µ ∂x ∂y ∂z 

(33)

In Sections 1.3.1 and 1.3.2, we studied flow along and across horizontal layers. The model in Figure 5 is repeated here (Figure 54), showing the arithmetic average for the effective permeability for along-layer flow and the harmonic average for across layer flow. x

Figure 54 The simple two-layer model

z

finefi ne-sca nescale sca le

coar co arse ar se--sca se scale le

t1 = 3 mm, k1 = 10 m mD D t2 = 5 mm mm,, k 2 = 100 mD mD

66.2 66 .25 .2 5 mD 22..86 mD 22 mD

In this model, there is no cross-flow, so we may write the effective permeability in tensor form with zero off-diagonal terms, as follows:

0  66.25 k= 222.886  0 or in 3D:


37

0 0  66.25  k= 0 66.25 0     0 0 22.86 3.2.1 Flow Through Tilted Layers θ

x z

z'

Figure 55 Layers tilted at an angle of θ to the horizontal

x'

This model is essentially the same as the one in Figures 4 and 6, although the layers have are repeated, and they have been tilted. In the frame of reference defined by the x ′ and z′ axes, the effective permeability may be calculated using the arithmetic and harmonic averages as before. However, in the x-z co-ordinate system, the effective permeability should be represented by a full tensor. The terms of the tensor may be calculated from the arithmetic and harmonic averages, as follows:

 k a cos2 θ + k h sin 2 θ ( k a − k h ) sin θ cos θ  k= 2 2   ( k a − k h ) sin θ cos θ k a sin θ + k h cos θ

(34)

This formula is obtained by rotating the co-ordinate axes through an angle θ. (You are not required to know the proof.) This example is in 2D, so only the kxx, kxz, kzx and kzz are shown. Further rotations may be carried out around the x ′ or z ′ axes to obtain a full 3D tensor. Note that: • The tensor is symmetric (kxz = kzx). • Depending on the sign of θ, the off-diagonal terms may be positive or negative. Example 5 Suppose the example in Figure 54 is rotated by 30º (Figure 56), and calculate the effective permeability tensor.

38

7


'z x

Figure 56 Layered model tilted by 30º

z

cos230 = 0.75,

30o

sin230 = 0.25,

sin30.cos30 = 0.433.

From before, ka = 66.25 mD, and kh = 22.86 mD.

k xx = 66.25 × 0.75 + 22.86 × 0.25 = 55.40 mD, k xz = (66.25 − 22.86) × 0.433 = 18.79 mD, k zz = 66.25 × 0.25 + 22.86 × 0.75 = 33.71 mD. 55.40 18.79 k= . 18.79 33.71 Full tensor permeabilities may also be calculated from numerical simulations. It is useful to use periodic boundaries, as described in Section 1.4.2. When a pressure gradient is applied in the x-direction, there will be flow in the x-direction, and also flow in the z-direction due to internal heterogeneity. These flows can be used to calculate the kxx and kzx tensor terms. Then a pressure gradient is applied in the zdirection to obtain the kzz and kxz terms. (In 3D, a pressure gradient should also be applied in the y-direction.) 3.2.2 Simulation with Full Permeability Tensors

Having calculated full effective permeability tensors, we need special software to handle them at the larger scale. Conventional finite difference simulators use a 5-point scheme in 2D and a 7-point scheme in 3D, and only take diagonal tensors – e.g. when running ECLIPSE, you usually specify PERMX, PERMY and PERMZ. Simulation with full tensors is more complicated and more time-consuming, but some packages allow the user to input full tensors. In Eclipse, there is a full tensor option which allows you to specify terms such as PERMXY. In 2D, a 9-point scheme is required to take account of cross-flow. This means that there are 9 terms in each of the pressure equations, as illustrated in Equation (35).

a1Pi , j − a 2 Pi −1, j − a 3 Pi +1, j − a 4 Pi , j −1 − a 5 Pi , j +1 −a 6 Pi −1, j −1 − a 7 Pi −1, j +1 − a 8 Pi +1, j −1 − a 9 Pi +1, j +1 = 0.

(35)

The coefficients, ai, in Equation (35) depend on the transmissibilities between the blocks. There are several different methods of discretisation which give slightly difInstitute of Petroleum Engineering, Heriot-Watt University

39

ferent results. To extend this to 3D, we need either a 19-point scheme or a 27-point scheme. See Figure 57. (The 19-point scheme leaves out the 8 corners of the cube.) Obviously, it takes longer to solve equations with a larger number of terms. a)

b)

x i-1,j-1

i,j-1

i+1,j-1

i-1,j

i,j

i+1,j

i-1,j+1

i,j+1

i+1,j+1

z

Figure 57 a) 9-point scheme for 2D. b) 27-point scheme for 3D

Often the off-diagonal elements of the permeability tensor (kxy, etc) are negligible, so the limitations of using a 5-point (2D) or a 7-point (3D) scheme are not serious. In layered systems, the size of the off-diagonal term may be gauged from Equation (34) in Section 3.2.1:

k xz = ( k a − k h ) sin θ cos θ.

(36)

This is a maximum for θ = 45º, and increases as (ka – kh) increases. Therefore, full permeability tensors become more important as the angle of the lamination or bedding increases, and as the permeability contrast increases. 3.3 Small-Scale Heterogeneity

Most reservoirs are modelled using, what is commonly termed a “fine-scale geological model”. This is a stochastic model with grid cells of size approximately 50 m in the horizontal directions, and about 0.5 m in the vertical. There are typically about 107 such cells in a full field model. These cells must be reduced in number to about 104 for full-field simulation. However, each of the grid cells in the geological model is likely to be heterogeneous, containing, for example, sedimentary structures. Petrophysical data (permeabilities, relative permeabilities, and capillary pressures) are acquired from core plugs, which are only a few cm long. When small-scale structure is present, petrophysical data should be upscaled before being applied to the grid blocks of the geological model. Figure 58 shows the ranges of scales of sedimentary structures, along with the scales of measurements and typical sizes of models.

Vertical thickness (m)

100

Log

1

Flow model Geological model

Core

0.1

0.01

Parasequences

Seismic data

10 0

Beds Probe Laminae

0.001 0.001

0.01

0.1

1.0

1

10

Horizontal length (m)

40

100

1000

10000

Figure 58 Length scales

7


For convenience, we consider upscaling as two separate stages (Figure 59). Stage 1 is upscaling from the smallest scale at which we may treat the rock as a porous medium (rather than a network of pores), up to the scale of the stochastic geological model, i.e. from the mm – cm scale to the m – Dm scale. Stage 2 is upscaling from the stochastic geological model to the full-field simulation model, which has already been described. Core plug

Sedimentary Structure

Stage 1 Upscaling Geological Model ~ 107 blocks

Figure 59 Two separate stages of upscaling. (Geological model taken from “Tenth SPE Comparative Solution Project: A Comparison of Techniques”, by Christie and Blunt, 2001.)

Stage 2 Upscaling

Simulation Model ~ 104 blocks

3.3.1 The Geopseudo Method

Upscaling from the core-scale to the scale of the geological model (Stage 1 in Figure 59) is frequently ignored by engineers, who apply core plug permeabilities and “rock” relative permeability curves directly to the geological model. However, work carried out at Heriot-Watt University has demonstrated that small-scale structures, such as sedimentary lamination may have a significant effect on oil recovery (Corbett et al., 1992; Ringrose et al., 1993; Huang et al., 1995). For example, in a waterflood of a water-wet rock, water is imbibed into the low permeability laminae, and oil may become trapped in the high permeability laminae. The Geopseudo Method is an approach, where upscaling is carried out in stages, using geologically significant length-scales (Figure 60). Models of typical sedimentary structures are created and permeability values are assigned to the laminae (from probe permeameter measurements, or by analysing core plug data). Relative permeabilities and capillary pressure curves are also assigned to each lamina-type (by history matching SCAL experiments on core plugs). Flow simulations are carried out to calculate the effective single-phase permeability and the two-phase pseudo parameters. Additional stages of modelling and upscaling may be required – e.g. upscaling from beds to bed-sets. In the finest-scale model, the grid cells may be a mm cube, or less. If we upscale to blocks of 50 m x 50 m x 0.5 m, we are upscaling by a factor of at least 5 x 104 in the horizontal directions and 500 in the vertical.


41

Low Perm

High Perm Individual Rel. Perm Curves

Effective Perm

Pseudo Rel. Perm Curves

Figure 60 Illustration of the Geopseudo Method

3.3.2 Capillary-Dominated Flow

At small scales the flow is often capillary-dominated. Figure 24 showed a moderate capillary effect: the imbibition of water along the low permeability layer made the flood front approximately level in the two layers (instead being ahead in the high permeability layer, in the case of a viscous-dominated flood). In that model (Figure 21), the layers were 1 m thick, and the grid cells were 10 cm square. If the size of the model is reduced by a factor 100, so that the layers are 1 cm thick, and represent sedimentary laminae, the effects of capillary pressure are much stronger, as shown in Figure 61. In this case, strong capillary imbibition draws water into the low permeability layers (black) so that the front advances faster in this layer. Notice that, in this figure, there is little lateral variation in the shading, showing that the water saturation is almost constant in each layer. This is because the front has been spread out by the effects of capillary pressure, and the model is almost in capillary equilibrium.

Oil Saturation 0.3

0.4

0.5

0.6

0.7

When the flow is across the layers, as in Figure 62, the effects of capillary pressure are even more striking. This figure shows the same small-scale model of sedimentary lamination. When the injection rate is low (average frontal advance rate of 0.3 m/day), the flood is capillary-dominated (Figure 62a), water (black) has been imbibed into the low permeability layers leaving oil trapped in the high permeability laminae (grey). As the injection rate is increased, the oil has more viscous force and can overcome the capillary forces leading to less trapping of oil (Figures 61b). In the case of Figure 62c, the flood is viscous dominated and all the movable oil has been displaced by water.

42

Figure 61 Example of capillarydominated flood in a layered model

7


b)

a)

Figure 62 Examples of acrosslayer flow. a) capillary dominated, b) intermediate, c) viscous-dominated

c)

Oil Saturation 0.3

0.4

0.5

0.6

0.7

The examples shown in Figures 61 and 62 demonstrate the significance of capillary effects at the small-scale. When upscaling from the lamina-scale, these effects should not be ignored, and two-phase upscaling should be performed. 3.3.3 Geopseudo Example

All the upscaling methods described in the previous sections may be used in the Geopseudo approach, depending on the type of heterogeneities and the fluids flowing – averaging, single-phase numerical methods, two-phase steady-state methods, or two-phase dynamic methods. Since a flood is often capillary-dominated, as shown above, steady-state upscaling using the capillary equilibrium method is often appropriate. Two-phase dynamic upscaling may also be used, and we show an example of the Kyte and Berry method below. Figure 63 shows a model of sedimentary ripples. Kyte and Berry pseudos were calculated for the model using four different flow rates. There is a factor of 10 between each flow rate, with rate 1 being the fastest. Figure 64 shows the resulting pseudos (from Pickup and Stephen, 2000). Figure 63 A model of ripples (based on the Ardross Outcrop, near St. Monance in Fife, Scotland)

10 mD

3 cm, 54 cells

200 mD

1 cm, 18 cells

0.9

0.9 0.7 0.6



Figure 64 Pseudo relative permeabilities for different flow rates, for oil (left) and water (right)

0.8

rate 1 rate 2 rate 3 rate 4

0.8

0.5 0.4 0.3 0.2 0.1 0.0 0.2

0.3

0.4

0.5

0.6

0.7

0.8

rate 1 rate 2 rate 3 rate 4

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.2

Water Saturation


0.3

0.4

0.5

0.6

0.7

0.8

Water Saturation

43

Note the following: 1. At high rates, the pseudos are shifted to the right. This is to compensate for numerical dispersion. 2. At very low flow rates (rate 4), the flood is capillary-dominated, and the oil is trapped. The pseudo oil relative permeability goes to zero around Sw = 0.46. 3.3.4 When to use the Geopseudo Method

Geopseudo upscaling may be time-consuming, and there is no point in upscaling from the smallest scales, unless cores are available for the field. Cores must be studied to identify the sedimentary structures present, and probe permeability measurements should be taken to populate the small-scale models. Additionally, reliable SCAL data is also required. Ringrose et al. (1999) give a list of guidelines for when Geopseudo upscaling may be necessary: 1) Are immiscible fluids flowing? 2) Are significant small-scale heterogeneities present? Specifically: • Is the permeability contrast greater than 5:1? • Is the layer thickness less than 20 cm? • Is the mean permeability less than 500 mD? 3) What is the large-scale structure of the reservoir? In many cases, large-scale connectivity may be the dominant issue, in which case, small-scale structure may have to be ignored. The Weber and van Geuns (1990) classification may be used to describe the large-scale structures: • Layer cake reservoirs – small-scale structure will usually have primary importance. • Jigsaw puzzle reservoirs – small-scale structure may be important. • Labyrinth reservoirs – small-scale structure will usually be of secondary importance. 3.4 Uncertainty and Upscaling

During the 1990s, reservoir modelling developed (along with computing power) so that geologists could create models containing millions of grid cells. Such models are often time-consuming to generate, and only a few are created for each reservoir. These detailed models are too large for full-field flow simulation, and must be upscaled to reduce the number of cells to, about 104 or 105. Research into upscaling has focussed on trying to develop methods to accurately upscale these types of models. However, it is now recognised that there are many uncertainties in the reservoir modelling, and instead of concentrating on a few detailed models, geologists and engineers are starting to generate thousands of models in order to characterise the effects of uncertainty. These models must be coarse so that the simulations can run very quickly. These changes mean that, in future, people are less likely to follow the “traditional” upscaling approach. However, if the effects of fine-scale structure are ignored, this

44

7


will lead to errors in the predicted recovery. It is therefore very important to understand the effects of possible sub-grid heterogeneity on absolute and relative permeability, and to include these effects, when necessary. This is an area of active research at Heriot-Watt University. 3.5 Upscaling Summary

Several reviews have been published on upscaling. These give an overview of some of the methods described in this chapter: e.g. Christie (1996), Renard and Marsily (1997) and Christie (2001). Here is a summary of the main points: •

The effective permeability of simple permeability models (layered or random) may be calculated using averaging.

•

In general, effective permeability should be calculated using a numerical simulation, along with suitable boundary conditions.

•

Permeability upscaling is often inaccurate, particularly when the coarse cell size is comparable, or slightly larger than the correlation length of the permeability distribution, and when the standard deviation is large.

•

Usually only single-phase upscaling is used in two-phase systems. However, this can give rise to errors, especially when the scale-up factor is large and when the standard deviation of the permeability distribution is large.

•

Single-phase upscaling for two-phase systems may be made more accurate by using a non-uniform coarse grid, or by using the Well Drive Upscaling method, which increases the accuracy of single-phase upscaling by using the “correct” boundary conditions.

•

The capillary equilibrium method is useful, particularly for small-scale models. It is feasible, even for models with a relatively large number of grid cells.

•

Two-phase dynamic upscaling methods should be able to reproduce two-phase flow on a coarse scale. The Kyte and Berry (1975) method is an example of a pressure averaging approach.

•

In general, two-phase upscaling is difficult to apply. It is more time consuming than single-phase upscaling, and the results are not robust (negative or infinite values may be obtained).

•

The regions around wells should be treated as a special case, because the flow is radial. The well index and the transmissibilities around the well block should eb upscaled.

•

Permeability is actually a tensor quantity (4 terms in 2D, 9 terms in 3D). Full tensors may be used to take account of cross-flow within a grid cell. However, in general, only the diagonal terms are used (kxx, kyy and kzz, often referred to as kx, ky and kz).


45

•

It is important to take account of small-scale (mm – m) heterogeneity in some reservoirs. This may be done using the Geopseudo Method, in which models of sedimentary structures are generated and upscaled.

•

Capillary effects are often significant at small-scales, and it is important to take these into account using two-phase upscaling (steady-state or dynamic).

46

7


4 REFERENCES Barker, J. W. and Thibeau, S., 1997. “A Critical Review of the Use of Pseudo Relative Permeabilities for Upscaling”, SPE Reservoir Engineering, May, 1997, 138-143. Barker, J. W. and Dupouy, P., 1999. “An Analysis of Dynamic Pseudo-Relative Permeability Methods for Oil-Water Flows”, Petroleum Geoscience, 5 (4), 385 - 394. Christie, M. A., 1996. “Upscaling for Reservoir Simulation”, J. Pet. Tech., November 1996, 48, 1004-1008. Christie, M. A., 2001. “Flow in Porous Media – Scale Up of Multiphase Flow”, Current Opinion in Colloid and Interface Science”, 6, 23 – 241. Christie, M. A. and Blunt, M. J., 2001. “Tenth SPE Comparative Solution Project: A Comparison of Upscaling Techniques”, presented at the SPE Reservoir Simulation Symposium, Houston Texas, 11 – 14 February, 2001. Corbett, P. W. M., Ringrose, P. S., Jensen, J. L. and Sorbie, K. S., 1992. “Laminated Clastic Reservoirs - The Interplay of Capillary Pressure and Sedimentary Architecture”, SPE 24699, presented at the 67th Annual Technical Conference of the SPE, Washington, DC, 4 - 7 October, 1992. Darman, N. H., 2000. “Upscaling of Two-Phase Flow in Oil-Gas Systems”, Ph.D. Thesis, Heriot-Watt University. Darman, N. H., Pickup, G. E. and Sorbie, K. S., 2002. “A Comparison of Two-Phase Dynamic Upscaling Methods Based on Fluid Potentials”, Computational Geosciences, 6, 5 – 27. Durlofsky, L. J., Behrens, R. A., Jones, R. C. and Bernath, A., 1996. “Scale Up of Heterogeneous Three Dimensional Reservoir Descriptions”, SPEJ, 1, 313-326. Durlofsky, L. J., Jones, R. C. and Milliken, W. J., 1997. “A Nonuniform Coarsening Approach for the Scale Up of Displacement Processes in Heterogeneous Porous Media”, Advances in Water Resources, 20, 335 – 347. Durlofsky, L. J., Milliken, W. J. and Bernath, A., 2000. “Scaleup in the Near-Well Region”, SPEJ, 5 (1), 110 – 117. Huang, Y., Ringrose, P. R. and Sorbie, K. S., 1995. “Capillary Trapping Mechanisms in Water-Wet Laminated Rocks”, SPE RE, 10 (4), 287 – 292. Kyte, J. R. and Berry, D. W., 1975. “New Pseudo Functions to Control Numerical Dispersion”, SPEJ, August 1975, 269-276. Peaceman, D. W., 1978. “Interpretation of Well-Block Pressures in Numerical Reservoir Simulation”, SPEJ, June 1978, 183-194.


47

Peaceman, D. W., 1983. “Interpretation of Well-Block Pressures in Numerical Reservoir Simulation with Nonsquare Grid Blocks and Anisotropic Permeability”, SPEJ, June 1983, 531-543. Pickup, G. E. and Stephen, K. D., 2000. “An Assessment of Steady-State Scale-Up for Small-Scale Geological Models”, Petroleum Geoscience, 6 (3), 203 – 210. Pickup, G. E., Ringrose, P. S. and Sharif, A., 2000.”Steady-State Upscaling: From Lamina-Scale to Full-Field Model”, SPEJ, 5 (2), 208 – 217. Renard, P. and de Marsily, G., “Calculating Equivalent Permeability: A Review”, Advances in Water Resources, 20 (5/6), 253 – 278. Ringrose, P. S., Sorbie, K. S., Corbett, P. W. M. and Jensen, J. L., 1993. “Immiscible Flow Behaviour in Laminated and Cross-bedded Sandstones”, J. Petroleum Science and Engineering, 9(2), 103-124. Ringrose, P. S., Pickup, G. E., Jensen, J. L. and Forrester, M. M., 1999. “The Ardross Reservoir Gridblock Analog: Sedimentology, Statistical Representivity, and Flow Upscaling”, in Reservoir Characterization – Recent Advances, eds R. Schatzinger and J. Jordan, AAPG Memoir 71, p 256 – 276. Stone, H. L. 1991. “Rigorous Black Oil Pseudo Functions”, SPE 21207, presented at the 11th SPE Symposium on Reservoir Simulation, Anaheim, CA, February, 1720, 1991. Weber, K. J. and van Geuns, L. C., 1990. “Framework for Constructing Clastic Reservoir Simulation models”. JPT, October 1990, p 1248 – 1297. Zhang, P., Pickup, G. E. and Christie, M. A., 2005. “A New Upscaling Approach for Highly Heterogeneous Reservoirs”, presented at the SPE Reservoir Simulation Symposium , February 2005.

48

Petrophysical Input

1 INTRODUCTION

7 CONCLUDING REMARKS

2 MODELLING SINGLE-PHASE FLOW AT THE PORE-SCALE - A BRIEF OVERVIEW 2.1 Deviations from Darcy’s Law 2.2 Empirical Models 2.3 Probabilistic Models 2.4 Capillary Bundle Models 2.5 First Principles Derivation of CarmenKozeny Model 2.6 Network Modelling Techniques

8 APPENDIX A: and Concepts

3 MODELLING MULTIPHASE FLOW AT THE PORE-SCALE 3.1 Capillary Pressure — What Does it Mean and When is it Important? 3.2 Steady-and Unsteady-State Flow 3.3 Drainage at the Pore-Scale 3.4 Imbibition at the Pore-Scale 3.5 The Pore Doublet Model 3.6 Introduction to Percolation Theory 3.7 Network Modelling of Multiphase Flow 4 EXPERIMENTAL DETERMINATION OF PETROPHYSICAL DATA 4.1 Laboratory Measurement of Capillary Pressure 4.2 Laboratory Measurement of Relative Permeability 5 EMPIRICAL AND THEORETICAL APPROACHES TO GENERATING PETROPHYSICAL PROPERTIES FOR RESERVOIR SIMULATION 5.1 Methods for Generating Capillary Pressure Curves and Pore Size Distributions 5.2 Methods for Generating Relative Permeabilities 5.3 Hysteresis Phenomena 6 WETTABILITY - CONCEPTS AND APPLICATIONS 6.1 Introductory Concepts 6.2 Wettability Measurement and Classification 6.3 The Impact of Wettability on Petrophysical Properties 6.4 Network Modelling of Wettability Effects

8

Some Useful Definitions

9 APPENDIX B: Unsteady-State Relative Permeability Calculations 10APPENDIX C: Details of the Heriot-Watt MixWet Simulator

LEARNING OBJECTIVES Having worked through this chapter the students should be able to... Modelling Single Phase Flow • Appreciate the different types of model used to predict single phase permeability • Use a Carman-Kozeny equation to calculate absolute permeability given values for the remaining variables Modelling Multiphase Flow • Explain the meaning of capillary pressure and use Laplace’s equation to relate capillary pressure to pore entry radius, contact angle and interfacial tension • Identify the difference between steady- and unsteady-state flow • Describe the pore-scale physics characterising drainage processes in porous media • Describe the pore-scale physics characterising imbibition processes in porous media • Describe a network model and explain how it can be used to investigate multiphase flow in porous media Experimental Determination of Petrophysical Data • Identify several different methods for measuring capillary pressure and relative permeability Generating Petrophysical Properties for Reservoir Simulation • Calculate oil-water capillary pressure curves from mercury injection data • Derive a pore size distribution from mercury injection data • Generate several capillary pressure curves from a single curve using a LeverettJ function • Determine the Brooks and Corey _-parameter from capillary pressure data and use this to predict relative permeabilities given the relevant equations • Identify three causes of capillary pressure hysteresis Wettability - Concepts and Applications • Explain how wettability variations affect waterflooding at the pore-scale • Identify two wettability measures used routinely in industry • List “Craig’s Rules of Thumb” in the context of water-wet and oil-wet relative permeability curves

2

Petrophysical Input

8

1 INTRODUCTION Outline of the purpose of this chapter: • To inform the student of the types of petrophysical data that are used in reservoir simulation (k, φ, kro, krw, Pc). k and φ are generally measured or determined by correlation or model, so, the central focus here will be on multi-phase properties (kro, krw, Pc); • To briefly review relative permeability and capillary pressure are measured experimentally. • To explain the underlying pore-scale physics of two-phase flow and show how this behaviour leads to the results we see at the macroscopic scale. • To review empirical and theoretical models used to generate relative permeabilities and their application in Reservoir Simulation; • To review wettability measures (such as USBM and Amott tests) and to explain how wettability modifies relative permeability and capillary pressure. The notion of a "porous medium" immediately conjures up an intuitive picture: put in its crudest terms, a porous medium may be thought of as a solid with holes in it. Unfortunately, such a superficial definition is of little use when trying to describe such materials objectively, and a more precise formulation must be attempted. A cylindrical pipe, for example, would not generally be considered a porous medium, nor would a solid containing isolated holes. There is a tacit understanding that "real" porous media should be capable of sustaining fluid transport, implying a certain degree of interconnectedness within the underlying pore structure. In short, a truly porous material should have a specific permeability associated with it. There are countless examples of porous materials in everyday life, each with its own particular pore structure and transport potential. These range from leather, wood, paper and textiles, to bricks, concrete and sand; even animal tissue and bones contain intricate pore networks. The need to understand such a vast array of permeable materials has consequently fostered a great deal of scientific interest from many diverse fields: soil mechanics, groundwater hydrology, industrial filtration, and petroleum engineering, to name but a few. Although the theme of fluid flow through porous media is a common feature of all of the disciplines listed above, each has its own technical terminology associated with the subject. For example, "dewetting", "desaturation" and "drainage" are all terms synonymous with the displacement of a wetting phase from the interstices of a porous material by a nonwetting phase. Throughout this section, however, the terminology used will be that generally encountered in the petroleum industry. A variety of fundamental concepts relating to both pore structure and solid/fluid interactions can be found in Appendix A.


3

2 MODELLING SINGLE-PHASE FLOW AT THE PORE-SCALE - A BRIEF OVERVIEW Examination of any photomicrograph immediately demonstrates why the modelling of fluid flow through a porous medium is such a formidable task: the underlying pore structure is extremely complex, with tortuous channels embedded throughout the solid matrix. Nevertheless, over the years there have been many attempts to encapsulate this structure into a simple, idealised analogue. Such models can generally be divided into four broad categories; (i) those which attempt to reduce the porous medium to a single representative conduit, (ii) probabilistic models, (iii) empirical correlations, and (iv) network models, where the medium is approximated by a lattice of connected conduits with distributed radii. A brief discussion of such analogues will be presented below; more detailed descriptions can be found in the monographs of Scheidegger (1963) and Dullien (1979). 2.1 Deviations from Darcy’s Law

Although Darcy’s Law has been validated countless times experimentally, we should nevertheless be aware of some possible difficulties. For example, liquid permeabilities can be greatly affected by clay distribution, brine composition, and brine pH. This is evident in Table 1, which shows variations in absolute permeability with increased brine salinity — in some cases, decreased salinity leads to a decreased permeability estimate, whilst other samples exhibit the reverse behaviour. There is also experimental evidence for so-called lubrication effects, where oil permeability measured at Swi actually exceeds kabs (this is rather surprising when one considers the fact that isolated water islands within a sample should actually form effective baffles to flow).

4

Field

Zone

Ka

K1000

K500

K300

K200

K100

KW

S S S S S

34 34 34 34 34

4080 24800 40100 39700 12000

1445 11800 23000 20400 5450

1380 10600 18600 17600 4550

1290 10000 15300 17300 4600

1190 9000 13800 17100 4510

885 7400 8200 14300 3280

17.2 147 270 1680 167

S S S S S

34 34 34 34 34

4850 22800 34800 27000 12500

1910 13600 23600 21000 4750

1430 6150 7800 15400 2800

925 4010 5460 13100 1680

736 3490 5220 12900 973

326 1970 3860 10900 157

5.0 19.5 9.9 1030 2.4

S S S S T

34 34 34 34 36

13600 7640 111000 6500 2630

5160 1788 4250 2380 2180

4640 1840 2520 2080 2140

4200 2010 1500 1585 2080

4150 2540 866 1230 2150

2790 2020 180 794 2010

197 119 6.2 4.1 1960

T T T T T

36 36 36 36 36

3340 2640 3360 4020 3090

2820 2040 2500 3180 2080

2730 1920 2400 2900 1900

2700 1860 2340 2860 1750

2690 1860 2340 2820 1630

2490 1860 2280 2650 1490

2460 1550 2060 2460 1040

Table 1 Effect of water salinity on permeability of natural cores (grains per gallon of chloride ion as shown). Ka means permeability to air; K500 means permeability to 500 grains per gal chloride solution; Kw means permeability to fresh water

Petrophysical Input

8

There are two other well-known limitations of Darcy’s Law: (i) At high injection rates, inertial effects can become important and Darcy’s Law should be replaced with the Forcheimer Equation:

dp = aµq + bρq 2 dx where η is the fluid viscosity and the second term on the right-hand side corresponds to inertial effects. (ii) Low pressure gas measurements must be corrected by using the Klinkenberg Equation:

 α k app = k 1 +  p  where kapp is the apparent (measured) permeability, k the actual permeability, p the gas pressure, and α a parameter that depends upon the properties of the gas being used. The correction is needed because the mean free path of a gas molecule at low pressure is of the order of a pore radius and the continuum concept begins to break down. 2.2 Empirical Models

There have been numerous attempts to derive correlations between permeability and other sample properties, such as porosity, capillary pressure, grain size distribution, and electrical properties. The porosity/permeability relationship has perhaps been the most widely studied (see, for example, Mavis and Wilsey, 1936; Büche, 1937; Rose, 1945; Habesch, 1990), but correlations vary so much that no universally accepted formula can be adopted successfully. Consequently, most empirical correlations contain "geometrical factors" which serve to fit experimental measurements without any real consideration of the underlying pore structure. 2.3 Probabilistic Models

As their name suggests, probabilistic models involve the use of some kind of probability law. One of the most popular of these is the "cut-and-random-rejoin" model of Childs and Collis-George (1950), which has been further extended by Marshall (1958) and Millington and Quirk (1961). The underlying theory involves the sectioning of the porous sample into two parts perpendicular to the direction of flow. These are then joined together again in a random fashion, and statistical analysis is used to approximate the permeability of the subsequent hybrid. 2.4 Capillary Bundle Models

Capillary bundle models characterise the porous medium using systems of capillary tubes with well-defined properties (Figure 1). Each different model contains tubes with different characteristics: they may be uniform and identical, for example, or uniform but with distributed radii, or periodically constricted and identical, etc.


5

a

b

k=φ

D2 32

c

Dmax

D2 =

∫s

2

f(s)ds

Dmin 2

k=φ

D

96

<> D

2

k=

φ

[∫ f(D) dD / D ]

96

∫ f(D) dD / D6

+ σ 2D

2

2

If all tubes are identical (diameter=D) and lie parallel to the flow direction, then a combination of Poiseuille's law and Darcy's law gives:

k=φ

D2 32

(1)

as a permeability predictor, where φ is the porosity (see Appendix B for mathematical details). An interesting aspect of this simple result is that the quantity (k/φ)1/2 can be thought of as a sort of average pore diameter. 2.5 First Principles Derivation of Carmen-Kozeny Model

This popular approach, based upon hydraulic radius theory, relies upon two main assumptions: (i) that a porous medium can be adequately characterised by a single tortuous channel having a characteristic radius, usually called the hydraulic radius, and (ii) that the effect of the interconnected pore structure can be contained within an empirical constant known as the tortuosity factor. The mathematical details of this approach are given in Appendix B. Carmen-Kozeny Model - One of the more notable correlations based upon conduit flow is that developed by Carmen and Kozeny (Carman, 1937, 1938, 1956; Kozeny, 1927). The basic premise of this modelling approach is that particle transit times in the actual porous medium and the equivalent tortuous rough conduit must be the same. After some analysis (see Appendix B), we arrive at the relationship:

k=

φ3 2 T 2 (1 − φ)2 Ss2

(2)

See Appendix A for definitions The permeability can be written in terms of an average grain diameter (Dp) by noting that, for spherical particles, Ss=6/Dp. Hence,

6

Petrophysical Input

k=

φ3D 2p 72 T 2 (1 − φ)2

8

(3)

Although a whole family of similar models exist, they differ only in the method of calculating an hydraulic radius RH and shape factors. 2.6 Network Modelling Techniques

Models that account for the interconnected nature of porous media constitute a group of analogues which can truly be referred to as network models. Here, the medium is modelled using a system of interconnected capillary elements, which generally configure to some known lattice topology. A variety of lattices are shown in Figure 2. Although these network structures are somewhat idealised, the capillary radii are assigned randomly from a realistic pore size distribution in an effort to partially reconstruct the actual porous medium under investigation.

Figure 2

Hexagonal

Square

Kagome

Triagonal

Cubic

Crossed square

A fully interconnected network was first used by Fatt (1956) for primary drainage studies. Although the two-dimensional lattice used was extremely small (200-400 tubes), the novelty of the approach encouraged great interest in the subject, and improvements on Fatt's primitive model were soon forthcoming (Rose ,1957; Dodd & Kiel ,1959). However, simulations using small lattices could never hope to capture the full behaviour of microscopic flow processes. With the advent of high speed computers, however, much larger 3D systems can now be constructed, with the result that microscopic flow behaviour can be more accurately simulated. Figure 3 shows a capillary dominated drainage process using an 80 x 80 square lattice, the details of which will be presented later: the resulting picture is somewhat more reassuring. The fractal capillary fingering is in excellent agreement with experimental observation (see Lenormand et al, 1988).


7

Figure 3

Only recently has it been computationally feasible to carry out studies using large three-dimensional networks (Figure 4).

Figure 4

The Basic Model Many network models attempt to distinguish between "pores" and "throats", by building networks consisting of hollow spheres connected by thin capillary tubes. In such models, all of the liquid volume is assumed to be contained in the spherical pores with pressure differences being maintained by the throats (Lenormand, 1986). The approach taken here is somewhat more straightforward. The porous medium is initially modelled using a three-dimensional cubic network of what will be referred to as pore elements; unlike many previous studies, no distinction is made here between pores and throats. The model consists of a three dimensional cubic network of 8

Petrophysical Input

8

capillary elements. This simple lattice has dimensions Nx x Ny x Nz where Nx, Ny, Nz are the number of nodes in the x, y and z directions respectively. Periodic boundary conditions are assumed in the y and z directions in order to simulate larger systems and eliminate surface effects. The pressure gradient is taken to be in the x direction. Now, for a single element of radius r and length L, the flow Q is given by Poiseuille’s law:

Q=

π r 4 ∆P 8µ L

(4)

where µ is the viscosity and ∆P the pressure difference acting across the capillary. At each node (i, j, k), the sum of the flows Qi must add up to zero (conservation of mass), and so: 6

∑Q

i

=0

i =1

Consideration of the whole network leads to a set of Nx.Ny.Nz linear pressure equations, the solutions to which can then used to calculate the elemental flows. Summing the outlet flows yields the total network flow, which can then be substituted into Darcy's equation to give a value for the total network permeability. If the system contains more than one fluid, then the process is carried out for each fluid in turn; the resulting phase conductivities now being referred to as effective permeabilities. The modelling of multiphase flow is discussed more fully in later sections. 3 MODELLING MULTIPHASE FLOW AT THE PORE-SCALE 3.1 Capillary Pressure - What Does it Mean and When is it Important?

For many, the term "capillary pressure" is a rather difficult concept to grasp, especially in the context of a flowing hydrocarbon reservoir. For example, we could be shown the schematic in Figure 5(a) and wonder why no oil flow occurs even though Poil>Pwater. Athough capillary pressure may seem problematic, we shall soon see that it is actually a very straightforward measure to interpret.


9

(a)

Poil>Pwater - why no flow?

Oil

Water

Projected area = πr2 (b)

(c)

P0 Pi

(Pi - P0)πR2 = 2πRΣ

r => (Pi - P0) =

2Σ R

Figure 5

Let us first consider a rubber balloon that has been inflated to a certain pressure (Pi) and then tied (we will assume that this "experiment" is taking place in an atmosphere at pressure Po, usually atmospheric pressure). If a force balance is considered for one half of the balloon (Figure 5(c)), then we can show that, at equilibrium, the elastic force acting around the circular perimeter of the balloon must counterbalance the difference in pressure projected onto the shaded cross section. This leads to a relationship between the pressure difference across the balloon surface and the radius of the balloon:

( Pi − Po ) =

2Σ R

(5)

where Σ is the elastic tension characterising the balloon wall (dimensions of Force/ Length). Here is an example where a pressure difference exists between two regions of fluid but no flow occurs — the elastic membrane of the balloon counteracts this. Now, instead of thinking of a rubber balloon (where Σ is actually a function of R itself), we can carry out a similar analysis for a gas bubble at pressure Pgas floating in oil at pressure Poil. We can now write immediately:

( Pgas − Poil ) =

2σ go R

(6)

where σgo represents the interfacial tension between gas and oil. This relationship tells us that the pressure difference across a small spherical bubble is larger than that across a large spherical bubble. This type of analysis can be applied to the more general case of an interface characterised by two different principal radii of curvature (eg a sausage-shaped balloon Figure B2). The details are slightly more involved (see Dullien (1979), but the final result is simply: 10

Petrophysical Input

( Pgas − Poil ) = σ go (

1 1 + ) R1 R 2

8

(7)

where R1 and R2 are the principal radii of curvature characterising the interface. Return now to the idealized (and some what unrealistic) situation where we have two fluids at equilibrium in a capillary tube separated by a curved interface (this curved interface appears at the microscopic scale when one of the fluids preferentially wets a solid surface, Figure 6). The pressure difference across the fluid-fluid interface is known as the capillary pressure. Note the difference between tube radius (rA) and the interface radius (RA) when the contact angle is non-zero. RA

rA

θ

Figure 6

A little bit of trigonometry can be used to derive an equation for two fluids at equilibrium in a circular cylinder, known as the Young-Laplace equation. This relates the pressure difference across a curved interface (i.e. capillary pressure Pc) in terms of the associated contact angle, interfacial tension and pore (tube) radius:

Pc = ( Pgas − Poil ) = σ go cos θ(

2σ go cos θ 1 1 + )= R R R

(8)

Although we have used gas displacing oil in our example, results for any nonwetting fluid displacing a wetting fluid can be inferred immediately. Another consequence of the equation is that larger pressure differences are needed for a nonwetting phase to displace a wetting phase from smaller tubes ((Pgas-Poil)∝1/R). We can therefore go on to examine the very simple drainage process shown in Figure 7, where oil (dark) displaces water (light) from a set of parallel tubes (R1>R2>R3) as oil pressure is gradually increased. Once the oil pressure exceeds the water pressure by an amount 2σowcosθ/R1, then the largest tube fills and the system settles down to a new equilibrium configuration. Subsequent displacements occur at (PoilPwat)>2σowcosθ/R2, and (Poil-Pwat)>2σowcosθ/R3. The corresponding plot of water saturation vs capillary pressure is shown in Figure 8. This could be thought of as a very basic capillary pressure curve and we will return to this concept later.


11

P1

P2

P3

P4

Figure 7

Pc 2σ cos θ R3 2σ cos θ R2 2σ cos θ R1 Sw

Simulators use capillary pressure curves to relate oil and water pressures at a given water saturation (see earlier chapters regarding this). Generally, capillary pressure is mostly important at the small scale (~cm), although it should also be included in reservoir-scale models involving transition zones. 3.2 Steady-and Unsteady-State Flow

The aim of this section is to broaden the understanding of two-phase flow in porous media; more specifically, to the simultaneous flow of oil and water through reservoir rock. To this end, concepts from percolation theory will be introduced towards the end of this section, where the physical porous medium will be approximated using an interconnected capillary network and each element may contain a different fluid: either water or oil. Before dealing directly with percolation issues, however, it will be beneficial to first discuss some of the more general aspects of two-phase flow, such as phase distributions and relative permeabilities. When dealing with any process involving multiphase flow, it is important to distinguish between steady-state and unsteady-state regimes (Figure 9). With regard to flow in porous media, the former is characterised by phase saturations that are invariant with time; that is, the volume flux of each fluid entering the system is the same as that leaving. Experimentally, this would be achieved by fixing the inlet flows of oil and water at a certain ratio and leaving the system to equilibrate (such that the individual fluid fluxes exiting the sample are the same as those entering). There are consequently no pore-scale displacements of any kind once steady-state has been reached - each 12

Figure 8

Petrophysical Input

8

phase tends to flow within its own tortuous network of pores. Once measurements have been completed at this ratio of fluxes, a new ratio could be set and the process repeated. Core-Scale

Steady-state

Unsteady-state Pore-Scale

Figure 9

Steady-state

Unsteady-state

In contrast to this, unsteady-state flow is accompanied by almost continuous saturation changes, implying continuous displacement of one fluid by another. This would occur if one fluid was injected into a sample containing a second "resident" phase. Note that the two flow regimes are seldom independent, however, as steady-state conditions are only achieved once some degree of transient unsteady-state displacement has taken place to redistribute the phases. Although the "channel flow concept" of steady-state flow outlined above appears to be generally valid (Craig, 1971), there are certain conditions under which this assumption may be questioned. If there is no strong wetting preference for the matrix, for example, or if the interfacial tension between the two fluids is very small, then a slug-like flow regime may develop. Alternatively, if the flow channels are rough and have an irregular cross-section (which is almost always the case in natural rock material), then the wetting phase will tend to line the channel walls and the nonwetting phase will usually reside in the centre of pores. Such phenomena can play a vital role in determining phase distributions during a variety of displacements, and their effects cannot always be overlooked. We shall return to these Issues later. 3.3 Drainage at the Pore-Scale

In order to better understand the following discussion on two phase displacements, we begin with the idea of a pore size distribution, PSD — a schematic representation of the range and frequency of pore radii characterising a given sample (Cf probability distributions from statistics). A schematic PSD and 3D network are shown in Figure 10, where the network of "pores" are simply capillary tubes of different radius (but equal length in this case). The radius r of each pore is the capillary entry radius (i.e. the radius characterising the pressure difference required for a nonwetting fluid to invade a tube containing a wetting fluid, see Section 3.1). We will now discuss drainage capillary pressure and relative permeability in two different arrays of capillary tubes. First, we will return to a capillary bundle model and then we will go on to examine drainage in an interconnected network.


13

(a)

(b) PSD(r)

r

Single Pore

Figure 10

Drainage in a Capillary Bundle Suppose we now start with a water wet (θ = 0) porous medium - and return to the case of an ideal parallel bundle of tubes - filled with 100% water (Sw = 1) and then consider the physics of oil (the non-wetting fluid) displacing this water. For simplicity, we will take a fully connected capillary bundle of tubes with a uniform PSD and a minimum radius, Rmin, and a maximum pore size, Rmax (Figure 11). Oil cannot spontaneously invade water-wet pores and requires an increase in pressure for a displacement to occur (see earlier discussion).

1 Rmax - Rmin

PSD

f(r)

Rmin

radius, r ->

Rmax Figure 11

The steps in a Primary Drainage process - and the corresponding drainage capillary pressures - would be as follows and these are illustrated schematically in Figures 12(c). From the pore occupancies, we calculate the water saturation Sw by summing the volume of the water-filled pores, divided by the volume of all pores. Similarly, we may calculate the relative permeabilities (described in a later section). Step 1: To enter the water filled porous network the oil pressure must be such that Po1 - Pw > Pc,1 = 2σ/Rmax - this is the minimum entry pressure before the oil can displace the water from the largest pore where r = Rmax. Figure 12(a). Because Pc,1 is the smallest of all entry pressures, oil enters the biggest pore first.

14

Petrophysical Input

8

Step 2: The oil pressure increases such that, Pc1 < Pc2 = 2σ/r2 where r2 < Rmax. At this higher capillary pressure, the oil displaces water from all the pores that have Rmax > r > r2. This leads to a finite oil saturation in the (fully accessible) capillary bundle. Figure 12(b). Step 3: The oil pressure increases again such that, Pc1 < Pc2 < Pc3 = 2σ/r3 where r3 < r2 < Rmax. At this even higher capillary pressure, the oil displaces more water from all the pores that have Rmax > r > r3. This leads to a increased oil saturation in the (fully accessible) capillary bundle. Figure 12(c). Step 4 (not shown): In principle, for a fully accessible system, we can increase the capillary pressure to Pcmax = 2σ/ Rmin and this would displace all the water with 100% oil (So = 1). For this Primary Drainage process, the corresponding drainage relative permeabilities (rel perms) are shown in Figures 13(a) – 13(c). Step 1: At this point, there is only water flow (Sw = 1) and therefore krw = 1 and kro = 0. Figure 13(a). Step 2: Now, the water saturation is Sw = Sw2 and krw falls quite rapidly since the water is now flowing in the smaller pores. Correspondingly, kro rises more rapidly since the oil is flowing in the larger pores. Figure 13(b). Step 3: The water saturation is now Sw = Sw3 and krw is relatively low since the water is now flowing in the smallest pores. Again, kro rises very rapidly since the oil is flowing in the larger pores as shown in Figure 13(c). In fact, there are analytical expressions for the relative permeabilities of a capillary bundle model and these will be introduced later in the course. Also, for a capillary bundle, the sum of the relative permeabilities turns out to be unity, but this is a result specific to simple fully accessible models and is not of great importance for real porous media.


15

(a) Step 1: primary drainage, Pc1 = 2σ/Rmax

PSD f(r)

Pc

water

oil

Pc1 = 2σ Rmax Rmin

Rmax

radius, r ->

Sw -->

(b) Step 2: primary drainage, Pc1 < Pc2 = 2σ/r2 (r2 < Rmax)

PSD f(r)

Pc

water

oil r2

Pc2 = 2σ

r2 Rmin

radius, r ->

Sw -->

Rmax

(c) Step 3: primary drainage, Pc1 < Pc2 < Pc3 = 2σ/r3 (r3 < r2 < Rmax)

PSD f(r)

Pc

water

oil

Pc3 = 2σ

r3

r3

Rmin

radius, r ->

Rmax

Sw -->

Figure 12

16

Petrophysical Input

8

(a) Step 1: corresponding drainage relative permeabilities

PSD f(r)

krw

kr

water

kro Rmin

Rmax

radius, r ->

Sw -->

(b) Step 2: drainage relative permeabilities at Sw = Sw2

krw

PSD f(r)

water

kr

oil r2

kro Rmin

radius, r ->

Sw -->

Rmax

Sw2

(c) Step 3: drainage relative permeabilities at Sw = Sw3

PSD f(r)

kro

kr

water

oil krw

r3

Sw --> Rmin

radius, r ->

Sw3

Rmax

Figure 13

Drainage in connected networks Drainage physics: We now illustrate what happens when drainage occurs in a connected network of pores such as that shown in Figure 10(a). Without loss of generality we can again assume a uniform pore size distribution. The Young-Laplace equation still applies and the steps shown in Figures 12 and 13 still broadly occur but there are some important differences from the (fully connected) capillary bundle model as follows. Although a pore could be occupied by the oil (non-wetting phase) at a given capillary pressure, Pc1 say where Pc1 =2σ/r 1, there are two reasons why the oil may be prevented from invading this “target” pore:


17

(i) The particular pore may be inaccessible to the invading oil, i.e. the invading oil phase may not be able to "see" the pore of radius r1, (this would be the case if the invading oil cluster had not yet reached the target pore). This is the issue of accessibility and a water filled pore may not be occupied by oil unless the Pc is above the entry pressure and the oil can access the pore in question; (ii) The water residing in the target pore could be trapped. Water can be trapped in a pore when two conditions are satisfied; (a) when. there is no chain of water-filled pores from the target pore to the outlet of the network (the target pore is then hydraulically disconnected) and (b) when there are no wetting films coating the pore surfaces that could allow water to "leak away" from the pore. These films fill the "corners" of pores as shown for the example of a simple triangular pore in Figure 14. water oil

The drainage process in an interconnected network as described above is illustrated in Figure 15 and it is known as invasion percolation (with or without trapping). In this figure, the oil (yellow) is displacing the water (blue) from the left. As noted above, this is governed by the Young-Laplace equation, but for the oil to displace the water from a given pore, this pore must also be accessible. There are probably pores in the large areas of trapped water which are large enough to be occupied by oil but they are inaccessible. A sequence of invasion percolation (drainage) calculations in a 2-D network showing the fluid distributions are shown in Figure16 for co-ordination number, z = 2.667 and in Figure17 for z = 4 (here, red is oil, blue is water and spaces denote missing pores i.e. lower co-ordination number). The characteristics of the overall displacement pattern in these two cases are quite similar but in the lower z case, the initial percolating capillary finger of non-wetting phase is somewhat more "spindly" or "ramified" because of lower accessibility. Just at breakthrough, a spanning cluster of non-wetting phase is formed, i.e. a continuous cluster that goes from the inlet to the outlet. This spanning cluster has a flowing "backbone" but it also has some dead end branches. After, breakthrough, the main cluster of non-wetting phase continues to grow and water is displaced. In the primary drainage simulations shown, water is first displaced directly - the water is displaced by oil and escapes through a direct route of bulk-filled pores to the outlet of the network. Later in the flood, at higher oil saturations, the oil "surrounds" some water filled pores, such that the water appears trapped (see for example Figures 16 (d) and 17(d)). However, it is evident from the later figures (Figures 16 (e) and 17(e)) that some of this hydraulically disconnected water has escaped. This water has escaped through the water films in the corners of the oil-filled pores (e.g. see the triangular pore in figure above).

18

Figure 14 Invasion percolation where oil (yellow) displaces the water

Petrophysical Input

8

Figure 15 Invasion percolation where oil (yellow) displaces the water

Figure 16 Oil/water drainage flood at various stages in a 2-D (20 x 20) network for a water wet system with coordination number, z = 2.667. This is essentially invasion percolation with periodic boundary conditions (i.e. if the invading oil leaves the top of the network, it reenters at the bottom). Observe that water films in oil-filled pores are not visualised

(a) drainage; z = 2.667, So =0.1;

(b) drainage; z = 2.667, So =0.3;

(c) drainage; z = 2.667, So =0.5;

(d) drainage; z = 2.667, So =0.7;

(e) drainage; z = 2.667, So =0.9.


19

(a) drainage; z = 4, So =0.1;

(b) drainage; z = 4, So =0.3;

(c) drainage; z = 4, So =0.5;

(d) drainage; z = 4, So =0.7;

(e) drainage; z = 4, So =0.9.

Accessibility and the accessibility function A(r): The meaning of accessibility and the definition of the accessibility function, A(r), are shown in Figure 18, together with the pore filling sequence for a connected network. Compare this with the drainage process in a fully accessible (capillary bundle) model.

20

Figure 17 Oil/water drainage flood at various stages in a 2-D (20 x 20) network for a water wet system with coordination number, z = 4. This is essentially invasion percolation

Petrophysical Input

(a)

p(r)

Accessibility A(r)=A2/(A1+A2)

Pores filled with Hg

M e r c u r y

(b)

3

10 5

12 2

8

13 6

4

8

7 11

A i r

M e r c u r y

9

1

3

10 5

12 2

8

13 6

4

7 11

A i r

9

1

A1 (c) A2

Figure 18 The idea of accessibility and the accessibility function for mercury (black) invasion into air

Rmin

r

(d) Shielded

Rmax

M e r c u r y

3

10 5

6

4

12 2

8

13

Shielded

7 11

1

9

A i r

M e r c u r y

3

10 5

8

13 6

4

12 2

7 11

1

A i r

9

Shielded

The accessibility function is defined as A(r) = A2/(A1+A2) and we can explain this concept physically as follows..Consider a mercury (Hg)→ air invasion percolation process (Figure 18). For the mercury to displace the air from a given pore, this pore must be accessible. In the sequence of Hg-pore filling in (a) to (d), at certain stages (e.g. (c)) some pores are “shielded”, i.e. they are large enough to be invaded but can not (yet) be seen by the invading mercury. How accessibility is related to the accessibility function for a drainage displacement is shown in Figure 19(a) – 19(c). where the “low occupancy” value of the accessibility is seen to be 0 and the high occupancy is 1 i.e. the phase is at a sufficiently high saturation that it can effectively “see” all pores that can be entered at a given (high) capillary pressure. The underlying theory of the drainage process in an interconnected network is a topic known as Percolation Theory (See Stauffer and Aharony, Introduction to Percolation Theory, 2nd Edition, Taylor and Francis, 1992) and we will give a fuller exposition of this topic later in this chapter.


21

(a) Accessibility in a connected network - above percolation radius, rp rp

f(r)

Accessibility function

◆

PSD

1

water

A(r)

A(r) = 0 for Rmax > r > rp

Rmin

radius, r ->

◆

0

Rmax

rp

r -->

Rmax

(b) Accessibility in a connected network - just below percolation radius, r < rp r < rp

◆

PSD f(r)

water

Not occupied = A1

Accessibility function 1 A(r)

A(r) > 0 for r < rp

◆ Oil occupied = A2 Rmin

radius, r ->

Rmax

0

rp

r -->

Rmax

(c) Accessibility in a connected network - well below percolation radius, r << rp Accessibility function

r << rp

f(r)

◆

◆

PSD

water

1

oil

Oil occupied = A2

A(r)

A(r) = 1 for r << rp

(A1 = 0) => A(r) = 1

Rmin

a)

b)

22

radius, r ->

Rmax

0

r -->

rp

Rmax

Figure 19 The meaning of accessibility and the accessibility function, A(r), in a primary drainage process

Figure 20 Comparison of a drainage flood (oil displacing water) in (a) a 2-D water wet glass micromodel where oil is the light fluid and the etched “geometrical” pore pattern can be seen; and (b) the corresponding 2-D theoretical network model calculation of the same flood where oil is in lighter blue and the red pores are water filled (McDougall and Sorbie, Heriot-Watt U., unpublished)

Petrophysical Input

8

A comparison of an experimental drainage flood (oil displacing water) in a glass micromodel and the corresponding network model calculation is shown in Figure 20. Figure 20(a) shows the 2D water wet glass micromodel where oil is the light fluid and the etched “geometrical” pore pattern can be seen. The corresponding 2D theoretical network model calculation of the same flood is shown in Figure 20(b) where oil is in red and the lighter blue pores are water filled. The agreement between these two figures is sufficiently good to be confident that we have captured the main pore scale physics of the drainage process in our network model. 3.4 Imbibition at the Pore-Scale

We now consider waterflooding of the same strongly water-wet 2D network models used in the primary drainage processes described above (i.e. cosθ = 1 in all pores). A process where the wetting phase increases such as water injection in this system is known as imbibition. The pore scale physics of imbibition is not simply the reverse of the drainage process although there are some features that are common to each. We noted that in drainage (oil → water; or o→w), the oil displaced the water by a pistonlike displacement mechanism governed by the Young-Laplace equation. At the pore level, imbibition - or water displacing oil (w→o) in this case - can actually take place by two distinct mechanisms viz. by piston-like displacement and by snap-off. Piston-like displacement of oil by water is the reverse of drainage except that it occurs when one of the phase pressures change such that Po - Pw < Pc = 2σ/r. The second mechanism, snap-off, is associated with the flow of wetting phase (water) through films, which swell around the oil in a pore to form a “collar” which eventually - at an appropriate capillary pressure - causes the oil to snap off thus occupying the space with water. This snap-off process is shown schematically in Figure 21 for a single pore and in Figure 22 for a 2D interconnected network. A given waterflood will generally consist of a mixture of the two displacement mechanisms outlined above. Swelling of wetting phase to form "collar"

Non-wetting fluid

Σ

Figure 21

Wetting fluid


23

(a)

W a t e r

(b)

3

10 5

12 2

8

13 6

4

7

O i l

11

W a t e r

9

1

3

10 5

12 2

8

13 6

4

7 11

9

1

(c)

O i l

(d)

Trapped

W a t e r

3

10 5

8

13 6

4

12 2

7 11

1

9

O i l

W a t e r

3

10 5

8

13 6

4

12 2

7

O i l

11

1

9

How the snap-off process relates to oil recovery is shown schematically in Figure 23, which shows a “ganglion” of oil being snapped-off by water in a water-wet rock. The oil could only escape through the connected cluster of oil filled pores (since there are no oil films in a water -wet porous medium). We also note that the isolated blob of oil left behind in this process in Figure 23 is “residual oil” since it is “trapped” and cannot now move (unless viscous rather than capillary forces are invoked). Without proof, we note here that the capillary pressure for snap-off is lower than that for piston-like displacement. Indeed, in a strongly water-wet circular capillary (cosθ =1), the snap-off capillary pressure is approximately, Pc = σ/r i.e. half the value for piston like displacement. Hence, if a capillary is occupied by oil and the oil pressure is lowered, the capillary entry pressure for piston-like displacement will be reached first and - if water is freely available, in adjacent pores for example - then piston-like displacement will occur first. On the other hand, if the capillary entry pressure drops below the piston-like entry pressure but no water front is available, then it will not fill with water. However, if the oil phase pressure, Po, drops sufficiently (or Pw increases sufficiently) that the snap-off capillary pressure is reached and there are water films to carry water to that pore, then snap-off will occur. Hence, imbibition is a more complex process than drainage and the balance between piston-like and snap-off events that occurs depends on a range of factors such as the range of pore sizes, poregeometry (aspect ratios), the connectivity of the network (z) and the presence/absence of wetting films (wettability). Clearly, if we have a wide range of pore sizes, then as we drop the oil phase pressure, Po, the pore that fills next with water is that for which Po - Pw < Pc; i.e. where the Pc refers to either Pc piston-like in an accessible pore or snap-off in a smaller but isolated pore which can be supplied wetting phase through films.

24

Figure 22 Schematic of the snap-off mechanism in imbibition. A 3-D view of a pore with wetting films in the corners is given in Figure 14

Petrophysical Input

Trapped Oil

8

this oil filament is unstable and "snaps"

continous oil Flow oil escapes through continous oil phase

Sand Grains

Oil Trapping by Filament Snap Off

Trapped Oil

"snap-off"

continous oil Flow oil escapes through continous oil phase

Figure 23 Schematic of snap-off of an oil ganglion within a porous medium

Sand Grains

Oil Trapping by Filament Snap Off

Such a model of imbibition has been implemented in the 2D network of water wet pores discussed above where we take: => Pc for piston-like displacement = 2σ/r => Pc for snap-off events = σ/r The phase distribution patterns for imbibition in this network is shown for z = 2.667 in Figures 24(a) - (d) and for z = 4 in Figures 25(a) - (d). Observe that in this case no direction of flow can be observed, as pores fill simultaneously throughout the network. In the two cases above, we can determine that the percentages of snap-off and piston like events in each flood are: ♦ z = 2.667 (Figure 24) Piston-like = 79 % ♦ z = 4 (Figure 25) Piston-like = 92.5%

Snap-off = 21 %

Snap-off = 7.5%


25

These results are as we expect since, if we considered a fully accessible capillary bundle model (as discussed earlier in this section), then only piston like displacements would occur (for the reasons discussed above). For such a case, the imbibition capillary pressure for a parallel bundle of tubes would be identical to the drainage curve i.e. no hysteresis should be observed. For z = 4, the system is moderately accessible and hence there is only a small fraction of snap-off events (7.5%). For z = 2.667, we find that the fraction of snap-off events grows to 21%. The corollary of this is that the more snap-off events we observe in the imbibition process, the greater should be the hysteresis between the primary drainage and the imbibition Pc curves i.e. the lower the imbibition Pc curve should drop below the primary drainage curve. This is precisely what is seen in the simulations (Figure 26). Notice that there are more sources for hysteresis than just snap-off — the lower coordination number also leads to more residual oil trapping and so magnifies the hysteresis effect. Obviously, we can also derive the relative permeabilities for the drainage and imbibition processes in the connected networks (see later), which show hysteresis effects similar to those for the Pc curves of Figure 26.

(a) imbibition z = 2.667, Sw = 0.05;

(c) imbibition z = 2.667, Sw = 0.25;

26

(b) imbibition, z = 2.667, Sw =0.15

(d) imbibition, z = 2.667, Sw =0.281

Figure 24 Oil/water imbuition flood at various stages in a 2-D (20 x 20) network for a water wet system with coordination number, z = 2.667

Petrophysical Input

Figure 25 Oil/water imbuition flood at various stages in a 2-D (20 x 20) network for a water wet system with coordination number, z = 4

(a) imbibition z = 4, Sw = 0.05;

(b) imbibition, z = 4, Sw =0.15

(c) imbibition z = 4, Sw = 0.25;

(d) imbibition, z = 4, Sw =0.366

10000

10000 Pc(Pa)

15000

Pc(Pa)

15000

3 2

0000

8

drainage 0000

1 imbibition 0

0 0

0.25

0.5 Sw

0.75

1

0

0.25

0.5 Sw

0.75

1

Figure 26

In reality, of course, pore geometries in rock samples are far more complicated than those characterising capillary networks and imbibition is somewhat more complex (drainage is still relatively straightforward). Micromodel studies by Lenormand and co-workers (Lenormand and Zarcone, 1984) has uncovered other possible mechanisms, as illustrated in Figure 27. Their different mechanisms have been termed I1, I2, I3, etc, where the integer corresponds to the number of pores surrounding a junction that are filled with nonwetting fluid. We can determine the stability of each meniscus in Figure 27 as the capillary pressure is lowered — remember, that the imbibition process is characterized by increasing meniscus radii as the flood proceeds. Hence, as long as the next stage of the waterflood results in an increase in meniscus radius, the displacement remains stable. So, in Figure 27(a) (an I1 mechanism), the displacement Institute of Petroleum Engineering, Heriot-Watt University

27

begins steadily as the capillary pressure is decreased, with the meniscus moving gradually from position 1 to position 2 to position 3. However, the next step involves the meniscus leaving three—“grain” corners and the radius of curvature”decreases”— this is unstable and the water immediately flows to position 4 and continues to displace oil from the corresponding pore. Similar reasoning holds for the case shown in— Figure 27(b) (an I2 mechanism) — here, however, the meniscus remains stable up to position 4 and only becomes unstable after grain edge A has been reached and the meniscus separates. The relative importance of each mechanism during an imbibition displacement once again depends upon pore geometry, pore size distribution, flowrate and supply. x 1

p0

nw

x'

4

p0 3

2

1

01

w

02

1

(a)

x' = √2x

x 1 2 5

3 A

1 (c)

R

4

5 (b)

(d)

Figure 27

3.5 The Pore Doublet Model

The simplest connected pore system is known as the pore doublet (see Appendix B, Figure B3). Here, the displacing fluid is introduced to the inlet of the doublet at a characteristic flowrate q. When the displacing fluid is nonwetting, the widest branch of the doublet fills first as expected (lowest capillary entry pressure) and wetting phase is trapped in the thin branch. The situation is far more interesting however when we consider imbibition in the doublet. A little analysis shows that the frontal velocity in each branch depends upon the ratio of branch radii (β), the supply rate (q), and a dimensionless number known as the capillary number (Nvc=µLq/πR13σcosθ), which is a ratio of viscous to capillary forces. At low flow rates (small q and small Nvc), the velocity ratio v2/v1 shows that the thinnest tube fills first and the meniscus in the wider tube actually retracts. At high rates (large Nvc), we see that v2/v1=R22/R12 and the wider tube fills first. Fuller treatments of the problem can be found in Moore and Slobod (1956), Chatzis and Dullien (1983), Laidlaw and Wardlaw (1983), and Sorbie, Wu and McDougall (1995). 28

Petrophysical Input

8

3.6 Introduction to Percolation Theory

In the previous two sections, the process of fluid flow in a porous medium has been considered from the point of view of the fluid. However, it is also possible to consider the process as being determined by the geometry of the porous medium itself. One approach, which has since become known as percolation theory, was first used by Broadbent and Hammersley (1957) to investigate the flow of gas through carbon granules (for the design of gas masks to be used in coal mines). As well as describing the flow of fluids and gases through porous media, their theory has subsequently been used to describe many other diverse processes; such as, the electrical properties of amorphous semiconductors, the behaviour of crystalline semiconductors containing impurities, and magnetic phase transitions. Phenomena that are best described by percolation theory are critical phenomena: characterised by a critical point at which some property of the system changes abruptly. In the present context of fluid flow, percolation theory emphasises the topological aspects of problems, dealing with the connectivity of a very large number of elemental pores and describing the size and behaviour of connected phase clusters in a welldefined manner. The primary focus of this section is to discuss how ideas from percolation theory can be applied more specifically to flow in a porous medium. As a simple, instructive analogue, consider first a two-dimensional square lattice of capillary elements as shown in Figure 28(a). The most pertinent concepts from percolation theory will now be discussed using this geometry. Consider the critical behaviour of the network. Assume that, initially, all of the tubes are blocked and that they are then opened at random. For any given geometry there is a unique fraction of tubes that must be open before flow across the network can commence; this critical fraction is called the percolation threshold (Pth) and for a simple square lattice has the value Pc=0.5 exactly (Figure 28(b) shows the distribution of closed and open tubes and Figure28(c) shows the flowing, spanning cluster of open tubes). One of the most incredible aspects of this result is that it is independent of the radius distribution; it only depends upon the topological structure of the network (actually, the co-ordination number (z) and the Euclidean dimension (d)). In fact, the percolation threshold and system topology are linked by the equation:

z.Pc =

d (d − 1)

(9)

(see Stauffer and Aharony, 1992). Table B1 in Appendix B shows percolation thresholds for a variety of two- and three-dimensional geometries.


29

(a)

(b)

(c)

Figure 28

Now, if instead of random opening, the pores are opened systematically beginning with those of largest radius (top-down filling), it is clear that flow will commence once a cluster of large open pores spans the system. The radius at which this occurs is known as the percolation radius, Rp, and is defined implicitly by the equation:

Pth = ∫R Pmax f ( r )dr R

(10)

where f(r) is the normalised tube radius distribution function and Pc the percolation threshold. However, the simulation of low-rate drainage processes is carried out using a top-down invasion percolation model with hydraulic trapping of the wetting phase. In this case, the injected nonwetting phase first fills the largest pores connected to the inlet face of the network, and then proceeds along progressively narrower pathways, sequentially occupying smaller and smaller pores. Although this process appears to be very different from the pure top-down pore filling, the resulting flowing clusters are, in fact, identical. Hence, the invasion percolation spanning cluster also appears at R=Rp. How does this apply to flow in porous media? Well, many imbibition and drainage processes exhibit critical behaviour: porous media contain clusters of oil-filled, water-filled and gas-filled pores and we would only see flow of a particular phase when the corresponding phase clusters span the system (Figure 29).

30

Petrophysical Input

8 Isolated clusters

8

3 Isolated clusters + 1 spanning cluster

Figure 29

Moreover, we see no flow below certain critical saturations (Sor, Swi) and these critical saturations will largely be determined by the connectedness of the porous medium under investigation (see equation 10 above, where Pth is a function of z). 3.7 A Brief Introduction to Network Modelling of Multiphase Flow

The last section described the concept of percolation theory. The aim of this section is to broaden the understanding of two-phase flow in porous media; more specifically, to the simultaneous flow of oil and water through reservoir rock. To this end, concepts from percolation theory are utilised more fully. The physical porous medium is once again simulated using a capillary network, but now each element may contain a different fluid: either water or oil, according to the pore-scale physics of drainage and imbibition discussed earlier. In the parlance of percolation theory, this type of analogue is commonly known as a bond percolation model. The porous medium is initially modelled using a three-dimensional cubic network of what will be referred to as pore elements; unlike many previous studies, no distinction is made here between pores and throats. We consider it debatable as to what constitutes a pore and a throat in a real porous medium, and propose a more abstract approach based upon effective flow cylinders (the 3R’s approach; McDougall, 1994). In this formulation, which closely follows the analysis of Heibaet al (1982), the“radius” derived from the pore-size distribution is the radius governing the capillary entry pressure of the pore element and is related to the capillary pressure by:

p c ( r )α

1 r

(11)

On the basis of geometrical analysis, Heiba et al postulated an approach for the estimation of volume and conductance of each pore element in the network. They reported that the volume, v(r), and conductance, g(r), of each pore element can be related to the radius governing the pore capillary entry pressure by the following relationships.

v(r) α r ν

0 ≤ ν ≤ 3


(12)

31

g(r) α r λ

1 ≤ λ ≤ 4

(13)

Different combinations of these exponents correspond to different “facies”; e.g.ν=3 andλ=1 would be most appropriate for unconsolidated media, whilst ν=1 and λ=4 would apply primarily to consolidated samples. These ideas are central to the 3R’s approach. One can further dispense with pore/throat arguments by incorporating effective contact angles into the model formulation (see Dixit et al, 1997). Different contact angle ranges can be used to mimic the effects of different pore/throat aspect ratios and the competition between snap-off and pistonlike displacement during imbibition. In addition, the rich variety of hysteresis phenomena observed during multiphase flow in consolidated and unconsolidated porous media can be reproduced and interpreted. When more than one fluid is flowing through a network, phase occupancy during a given process (e.g. primary drainage, secondary imbibition etc.) may be characterised by a set of rules (based upon the physics discussed earlier) which, when combined with topological considerations (accessibility), give realistic saturation distributions throughout a displacement. This approach is particularly well-suited to the modelling of capillary-dominated flow. However, since any increase or decrease in the saturation of a particular phase depends upon the spatial distribution of that phase, the computational effort saved in dispensing with unsteady state calculations is more than offset by the implementation of a clustering algorithm (after Hoshen and Kopelman, 1976). This algorithm is essentially a “book keeping” exercise which locates and labels the phase clusters which are distributed throughout the network. These clusters are continually changing their structure during a displacement, and so the efficacy of the simulation as a whole is intrinsically linked to the efficacy of the clustering algorithm itself. A great deal of time and effort has been invested in achieving the optimum performance of this element of the network simulator. The precise computational details are dealt with more fully in McDougall (1994). Although we have rules that tell us how a given displacement proceeds as a function of capillary pressure, we still need a method of calculating the flow of each phase at any given stage. In fact, this turns out to be fairly straightforward: we can set a certain capillary pressure in the model, determine the phase occupancies, effectively “freeze” each phase, and use the numerical approach from Section 2.6 to calculate the fractional flows . This facilitates the calculation of effective permeabilities for the two intertwined networks (one for each phase) over a range of saturation values. Inherent in this is the assumption that the flow is stationary, thus steady-state relative permeabilities are considered here; for work on dynamic relative permeabilities see Blunt and King, 1990. Measured relative permeabilities depend upon the saturation histories, saturation of the fluids, pore space morphology, wetting characteristics of the fluids, the ratio of the fluid viscosities and the capillary number. Many of these factors are already included in the model and will be described later in Section 5.2 when we discuss the use of network models in calculating capillary pressure curves and relative permeabilities. To summarise, network modelling is a powerful tool for increasing our understanding of multiphase flow in porous media ——the key element of such an approach lies in the nterconnected nature of the underlying model. It is particularly well-suited to 32

Petrophysical Input

8

qualitative sensitivity studies of petrophysical parameters (examining the effects of connectivity, pore size distribution, dead-end pore space, co-operative filling events, &c. upon Pc and Krel), although more quantitative studies can also be considered.

f(R)

Imbibition 0.02 0.015 0.01

Pores filled with wetting phase

0.005

Pores filled with nonwetting phase

20

40

50

R 80

100

80

100

f(R)

Drainage 0.02 0.015 z y

x

0.01

Pores filled with wetting phase

0.005 Nonwetting phase

R 20

40

50

Figure 30

4 EXPERIMENTAL DETERMINATION OF PETROPHYSICAL DATA 4.1 Laboratory Measurement of Capillary Pressure

There are a number of ways in which capillary pressure can be measured experimentally and a few will be briefly discussed here — almost all rely upon highly specialised equipment of varying degrees of sophistication. Mercury Injection — Perhaps the most straightforward approach to capillary pressure measurement utilises a mercury porosimeter (although manual pumps are also available, Figure 31(a)(iii). Typically, a 1” diameter cylindrical plug of the porous sample is inserted into a glass penetrometer (essentially a close-fitting glass container), which is then put into the machine, evacuated and contacted with mercury at zero pressure. A table of pressure values ranging from 0-60,000psia is then input into the porosimeter and the pressure of the mercury in contact with the porous plug is raised sequentially in a number of discrete steps. The volume of mercury penetrating the sample is measured at each pressure step (once a given equilibration interval has elapsed) and the resulting mercury-vacuum capillary pressure curve is output to a file. Mercury extrusion curves can be similarly measured, although the sub-atmospheric part of the curve is difficult to obtain without experimental artefact. Note, that the mercury-vacuum curve cannot be used directly in a reservoir simulator — it must be re-scaled appropriately (the procedure for doing this is described in section 5.1). Porous Diaphragm — The porous diaphragm method is shown in Figure 31(a)(ii). A water-filled porous plug is placed in an oil-filled (or gas-filled) container, with its lower face in hydraulic contact with a water-wet glass frit (a glass disc containing Institute of Petroleum Engineering, Heriot-Watt University

33

micron-scale holes). The pressure in the surrounding nonwetting phase (oil or gas) is then increased incrementally, and the displaced water leaves the container via the frit. Saturations are measured at each pressure (once the system has equilibrated) and the oil-water or gas-water Pc curve can subsequently be determined. Note that the topology of the invading nonwetting clusters could be different from those obtained during mercury intrusion, as nonwetting fluid is unable to enter the lower face of the sample. Note also that a residual wetting phase saturation is also measured using this method, whereas mercury injection ends with the sample completely filled with mercury (there is no wetting phase in this case, as the sample is initially evacuated). Unfortunately, the porous diaphragm method is rather slow (and therefore expensive), as equilibration times for an oil-water system are generally high. This problem can be ameliorated somewhat by using water/air systems and rescaling the resulting data. One final point of note relates to the operating range of the experimental equipment. The maximum possible driving pressure is determined by the displacement pressure of the glass frit (diaphragm) upon which the sample is placed. Once the displacement pressure of the diaphragm has been exceeded, the wetting phase is no longer able to leave the sample and the experiment ends (Figure 31(b)). Centrifuge Method — The centrifuge method is quick but relies on highly specialist equipment and analytical treatment of the data. The sample is put into a centrifuge surrounded by a displacing phase (Figure 31(a)(ii)). The centrifuge is spun at different speeds and the volume of displaced fluid measured on each occasion. Whilst an average capillary pressure can be inferred corresponding to each rotational speed, the local capillary pressure and saturation values vary with distance from the centre of rotation. Hence, some analysis is required to back-out an appropriate average Pc curve. Dynamic Method — This method is used in conjunction with steady-state relative permeability measurements (see below). Here, semi-permeable membranes can be used to measure wetting-phase and non-wetting phase pressures independently at a number of different fixed fractional flows. Hence, a direct measure of Pc can be achieved over a range of saturation values.

34

Petrophysical Input

(i)

I

(ii)

Nitrogen Pressure

8

Saran Tube Crude Oil Neoprene Stopper

H Scale of Squared Paper

P Nickel-Plated Spring Core Kleenex Paper

Seal of Red Oil

M N

D

L

F

Ultra-Fine Fritted Glass Disk

G

E

J

K

Brine

C

Porous Diaphragm

Centrifuge 0-200 psi Pressure Guage 0-2,000 psi Pressure Guage

(iii)

Regulating Valve Lucite Window

To Atmosphere

Cylinder

U-Tube Manometer Lucite Window

Figure 31a

Mercury Pump


35

Pc Pc diaphragm

Pc core

100% 0

Operating range

σW

Figure 31b

4.2 Laboratory Measurement of Relative Permeability

Relative permeabilities can be measured using either steady-state or unsteady-state methods: unsteady-state measurements can usually be completed in a much shorter time, and have consequently become the oil industry standard. The question as to whether such rapid measurements are representative of conditions in the reservoir, however, is a matter of some debate. Steady-State Methods Steady-state methods are characterised by simultaneous injection of the two phases at a fixed ratio and known flowrates. Steady-state conditions are assumed to have been reached once the inlet and outlet fluxes of each phase have equilibrated and/or a constant pressure drop is seen to exist across the sample. This may take many hours or even days, depending upon the type of material under investigation and the measurement technique being used. Once equilibration has been reached, Darcy’s law can be used for each phase in turn, resulting in a pair of relative permeability values valid at that particular saturation. The fluid flux ratio is then changed (whilst keeping the total flowrate constant), yielding a second set of data once a new steady-state has been achieved. Similar measurements for a number of different flux ratios can then be used to give a set of relative permeability curves which span the entire saturation range. Although the time involved in extracting such data is clearly an important concern, steady-state measurements performed at low rates should be considered most indicative of reservoir behaviour. Typical laboratory relative permeability curves are shown in Figures32 and 33. Some of the more popular experimental techniques will now be discussed.

36

Petrophysical Input

8

100 kg ,

k0 , Penn State

kg ,

k0 , single-core dynamic

★ kg , ★ k0 , dispersed feed

Relative permeability, %

80

★

kg ,

k0 , Hafford technique

kg , kg ,

k0 , gas-drive technique k0 , Hassler method

60 ★

40

★

20

0

★ ★

Core No. 0-2-A Berea outcrop K = 120 md L = 2.30 cm

0

20

★ ★

★ ★ ★ ★★ 80 100

40 60 Oil saturation, %

Figure 32

100 kg ,

k0 , Penn State

kg ,

k0 , single-core dynamic

★ kg , ★ k0 , dispersed feed

Relative permeability, %

80

kg ,

k0 , Hafford technique

kg , kg ,

k0 , gas-drive technique k0 , Hassler method

★ ★

60 ★ ★ ★

40

Core No. 0-2 Berea outcrop K = 118 md L = 7.23 cm

20

0

★★ ★ ★

0

20

★

40 60 Oil saturation, %

★

80

★ ★

100

Figure 33


37

The Penn-State Method — Originally designed by Morse et al (1947), this technique has recently become one of the most popular. A typical apparatus is shown in Figure 85a and essentially consists of three similar core plugs mounted in a core holder with a pair of pressure tappings. Only the central plug is considered for measurement purposes; one of the two outer units acting as a mixer at the inlet, whilst the other serves to alleviate capillary end effects at the outlet. The three-plug assembly also facilitates dismantling of the test section for saturation measurements to be carried out. The method can be used for both liquid-liquid and liquid-gas measurements and has been used over a wide range of wettability conditions. The Hassler Method — The laboratory apparatus used for this type of steady-state test is shown in Figure 35d. Semi-permeable membranes are positioned at both the inlet and outlet ends of the core sample, which serve to separate the test fluids outwith the sample whilst permitting two-phase flow to take place within it. The importance of this is that allows the two pressure drops to be regulated independently, enabling equilibration of the capillary pressure at both ends of the core. This procedure is designed to provide a uniform saturation distribution throughout the system, and thus eliminate any capillary end effects. The Dispersed-Feed Method — This technique was devised by Richardson et al (1952) in an attempt to introduce the fluids to the test sample in a more appropriate manner. It utilises an upstream dispersing medium in order to spread the wetting phase uniformly across the face of the test sample before entering it. The nonwetting phase is injected into radial grooves that are machined into the downstream end of the dispersing section, at its boundary with the core plug. A similar concept lies behind the Hafford apparatus, which injects nonwetting fluid directly into the sample, whilst the wetting phase must first pass through a central semi-permeable membrane. Again, the idea is to obtain more realistic injection behaviour. Richardson et al (1952) have compared a variety of steady-state methods, and conclude that all four of the techniques outlined above should give very similar drainage relative permeability-saturation curves (Figures 32 and 33). Unsteady-State Methods Although unsteady-state relative permeability measurements can be made much more rapidly than those requiring steady-state equilibration, analysis of the resulting data is more difficult and open to a certain degree of ambiguity. In unsteady-state tests, one phase is displaced directly from the core sample by the injection of another, at a rate high enough such that capillary pressure remains negligible. An analysis of the resulting production data, based upon the Buckley-Leverett frontal advance theory (Buckley and Leverett, 1942), then permits the determination of a relative permeability ratio (krw/kro). The relevant theory was given by Welge (1952), who demonstrated that:

fo =

38

1 µ k 1 + o rw µ w k ro

(14)

Petrophysical Input

8

where fo is the fraction of oil in the outlet stream, and the mi are viscosities. This was later extended by Johnson et al (1959), who developed a technique for calculating the individual relative permeabilities from the permeability ratio. Several other methods also exist (Saraf and McCaffery, 1982; Jones and Roszelle, 1978; inter alia). A sample calculation is shown in Figure 34 —note that we only get relative permeability after breakthrough of the displacing phase. 1.0 krw kro

0.8

kri

0.6 0.4 0.2 0.0 0.0

Figure 34

0.2

0.4

0.6

0.8

1.0

Sw

In the past, much work has been carried out to compare these unsteady-state results with their steady-state equivalents. Unfortunately, although the outcomes appear to be broadly in agreement, there have been a number of exceptions (e.g. Schneider and Owens, 1970; Owens et al, 1965; Loomis and Crowell, 1962; Archer and Wong, 1971). The limitations of the unsteady-state approach for determining water-oil relative permeabilities have been discussed more fully by Craig (1971): the main concerns relate to the high pressure differentials involved (in excess of 50psi), and the fact that large viscosity ratios are often used in an attempt to extend the range of twophase flow. The general applicability of the resulting data must consequently be called into question. Centrifuge Methods Centrifuge techniques can provide useful results extremely rapidly, with the added bonus that they are thought to be free of the viscous fingering problems that accompany many other unsteady-state methods. They are, however, subject to problems associated with capillary end effect and cannot be used to quantify the relative permeability of the displacing phase. Such methods involve monitoring the production from samples that are initially saturated with one or two phases, with analytical techniques being used to back-out relative permeability values (see Van Spronsen, 1982, for a fuller account).


39

Gas

(a)

Thermometer Packing nut

Copper orifice plate

Electrodes

Inlet

(b) Gas pressure guage

Porous end plate

End section

Test section

Mixing section

Oil pressure

Differential pressure taps Bronze screen

Oil pressure pad

Highly permeable disk

Outlet

Oil

Gas meter Intlet

Penn-State

Hafford Oil burette

(c)

Gas meter

Gas-pressure guage

(d)

Gas

vacuum

Lucite

Core material

Core Lucite-mounted core

Dispersing section

Oil

Oil burette

Dispersing section face

Dispersed Feed

Flowmeter

Hassler

The Effects of Flowrate, Viscosity Ratio, and Interfacial Tension. At sufficiently low flowrates, microscopic flow behaviour should always be capillarydominated if the system has a strong wetting preference. At such rates, relative permeabilities in the medium saturation range should consequently be independent of the viscosity ratio, and this conclusion appears to have been borne out by many subsequent investigations (Leverett et al, 1939, 1941; Wyckoff and Botset, 1936; Saraf and Fatt, 1967; Levine, 1954). Near the endpoints, however, and in cases where the wetting phase is flowing only through thick films, there sometimes exists a strong hydraulic coupling between the two phases. As a result, the nonwetting phase may experience hydraulic slip and any analysis using Darcy’s Law becomes invalid. If Darcy’s law is applied regardless, however, nonwetting phase relative permeabilities greater than unity can often be the result: the experimental results of Odeh (1959) clearly demonstrate this effect (Figure 36a). If high rates are used in relative permeability measurements, the subsequent flow behaviour will no longer remain capillary-dominated, and viscous forces will tend to take over. Moreover, if the 40

Figure 35

Petrophysical Input

8

associated viscosity ratio is high, then the less viscous invading fluid will begin to finger through the sample and a condition of uniform saturation will be impossible to achieve. With this in mind, it is clear that such considerations should not be overlooked when attempting to interpret a wide variety of unsteady-state coreflood data. The effect of interfacial tension upon relative permeabilities can also be significant. In cases where the experimental flowrate is high and the interfacial tension is small, the capillary forces become less significant and slugs of both fluids may begin to flow through the same network of pores. In fact, as the interfacial tension approaches zero, the relative permeability curves actually become straight diagonal lines: i.e. the total effective mobility of the system remains constant over the entire saturation range. Experimental results showing this effect are reproduced in Figure 36b. (a)

(b)

240 1.0

???? m=74.5

200

???? m=82.2

.75

???? m=42.0

????

???? m=5.7

100

.5

????

???? m=5.2

.25

???? m=0.9

????

???? m=0.6

0

0

50

100

0 0

.25

Sw

Figure 36

???? .5 Sq

.75

1.0

QUESTIONS WE SHOULD NOW BE ABLE TO ANSWER • • • • • • •

What is capillary pressure? At what scale is it most important? What things affect meniscus curvature? How do drainage and imbibition processes differ at the pore scale? Why can the roughness of pore walls be important? Is flowrate important in laboratory tests and, if so, why? What are the advantages of network models over other pore-scale models?

Now, how do we use the knowledge we’ve obtained so far ?


41

5 EMPIRICAL AND THEORETICAL APPROACHES TO GENERATING PETROPHYSICAL PROPERTIES FOR RESERVOIR SIMULATION 5.1 Methods for Generating Capillary Pressure Curves and Pore Size Distributions

Background In this section, we will describe a number of approaches to generate capillary pressure curves for use in reservoir simulation. We often have to undertake a simulation study without possessing all of the data we would ideally require — perhaps we have only one experimental capillary pressure curve at our disposal and perhaps it comes from a part of the reservoir not directly related to our current study. We need some way of inferring reasonable data from those available to us at the time and, although we may have to make some gross assumptions, we should be able to invoke our knowledge of flow at the pore-scale to help us. Let us begin by reminding ourselves of the drainage case (drainage curves are frequently used in simulators — often erroneously). Remember that, in drainage, we are increasing the pressure difference between the nonwetting and wetting phases. As Pc increases, the radius of interface curvature decreases and, using a capillary tube analogue, the Young-Laplace equation tells us that the nonwetting phase begins to invade the porous medium when Pc>2σcosθ/Rmax — i.e. at the so-called displacement pressure of the sample. As Pc continues to increase, the nonwetting phase invades progressively smaller pores. Hence, the nonwetting phase saturation gradually increases with Pc. The resulting plot of Pc vs Sw is the drainage capillary pressure curve (Figure 37).

Pc

Water wet sand

Pc

D.P. Displacement pressure

%100 0

Funicular Pendular

Insular Funicular

0 σ0 100% σw

Figure 37

42

Petrophysical Input

8

We can now use this discussion to infer the type of capillary pressure curve that may result from different samples. For example, how would the pore size distribution affect the curve? We already know that the displacement pressure is affected by Rmax — the largest pore in the sample — but it should also be clear that the range of pore sizes in a sample can greatly affect the shape of the corresponding Pc curve (Figure 88). Flat plateaux indicate samples that are fairly homogeneous at the pore scale, whilst steeper curves indicate a large variance in the distribution. Moreover, fine textured rocks with small cemented grains can be expected to exhibit higher capillary pressures at a given saturation that coarse-textured media — also higher displacement pressures. So, from our basic understanding of capillary pressure at the microscopic scale, we have been able to infer a great deal about how we would expect capillary pressure curves to vary at the continuum (macroscopic) scale.

Pc

Water wet sand

Irreducible water saturation

3 2 1

D.P. All same radii

0 100%

σw σ0

100% 0

Figure 38

Rescaling Mercury Injection Data One of the most common ways to derive oil-water or gas-oil capillary pressure curves is to rescale mercury injection data (which is routinely measured and relatively cheap to obtain). Consider mercury injection into a porous medium containing a large exterior pore of radius Rextmax. The capillary pressure required for this pore to be invaded by mercury is given by the Young-Laplace equation:

(Pce )mercury / air =

2(σ cos θ)mercury / air max R ext


(15)

43

If we now consider the same medium, filled with water, undergoing oil injection, then the requisite oil-water capillary pressure is now:

(Pce )oil / water =

2(σ cos θ)oil / water max R ext

(16)

Now, eliminating Rextmax from equations 15 and 16we arrive at the scaling relationship:

(σ cos θ)

(Pce )oil / water = (Pce )mercury / air (σ cos θ) oil / water

(17)

mercury / air

This clearly holds for any Pc-value, and so a complete oil-water capillary pressure curve can be obtained from the mercury injection data by applying equation 17at each saturation. Experience shows that the ratio of interfacial tensions and cosines on the right hand side of 17(often difficult to measure accurately) should have a value of approximately 6. However, different ratios have sometimes been needed to reconcile experimental mercury-air and oil-water data from different rock-types (Figure 39 — the ratio is referred to as—“Factor” in these plots). Such differences may be due to interactions between contact angle and pore geometry.

290

Mercury injection

40

232

30

174

20

Porosity - 23.0% Permeability - 3.36 md Factor - 5.8

10 0

116

Limestone core

0

20

40

100

80

60

58

60

80

0 100

40

20

0

225.0 Restored state

25

187.5

Mercury injection

20

150.0 Sandstone core Porosity - 28.1% Permeability - 1.43 darcys Factor - 7.5

15

112.5

10

75.0

5

37.5

0

0

20

40

100

80

60

H20

60

80

0 100

40

20

0

Mercury capiliary pressure, psi

Restored state

50

30 Water/nitrogen capiliary pressure, psi

348

Mercury capiliary pressure, psi

Water/nitrogen capiliary pressure, psi

60

H20 Hg

Liquid saturation, %

Hg Liquid saturation, %

Figure 39

44

Petrophysical Input

8

Derivation of Pore Size Distributions In addition to providing raw data for the derivation of capillary pressure curves, mercury injection data can also be used to derive so-called pore-size distributions (PSDs). The theory for deriving PSDs from intrusion data comes from a paper by Ritter and Drake (1945), which makes a number of simplifying assumptions: (i) the pores are circular cylinders in shape, and (ii) all pores of a given radius are accessible to the mercury (i.e. no accessibility issues). By using the Young-Laplace equation to convert Pc to pore radius, the PSD (D(r)) can be derived from the capillary pressure data via the relationship:

D( r ) =

Pc  dSmercury    r  dPc 

(18)

In fact, some algebra shows that this can be rewritten as:

dS D( r ) =  air   dr 

(19)

The procedure can be tested using the data shown in Table 2 and you are encouraged to follow the example. Unfortunately, the distributions that are produced by this method (and therefore by commercial porosimeters) are not pore size distributions but pore volume distributions. The spike (very typical) is caused by accessibility effects prevalent during drainage processes and the volume associated with some large “shielded” pores can be incorrectly assigned to small pores (see Figure 40). To get a better idea of the true PSD (that is the frequency distribution of pore radii), the Ritter and Drake distribution D(r) should be divided by r2 for each r-value — this generally shows that there are far more smaller pores than D(r)would suggest. That said, the volume-weighted distributions provided by commercial porosimeters are still useful lithological fingerprints.

Table 2

(Pc) mercury/air 0 5 6 8 9 11 13 16 18 25 50

sair 0 0.97 0.92 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.15

1. Plot the mercury capilliary pressure curve 2. Derive the oil-water curve (σcosθ=40mN/m for oil-water and 375mN/m for mercury-air) 3. Use Y-L equation to derive Sw vs r plot (this is the cumulative intrusion curve) 4. Plot PSD (D(r)) using Ritter and Drake formula (Hint: use Y-L equation to get a formula for dSw/dr)


45

De

De

D

(a)

Figure 40

(b)

The Leverett J-Function The foregoing discussion has shown that Pc curves are clearly affected by the sample pore-size distribution, interfacial tension, contact angle, and pore structure. Leverett (1941) developed a dimensionless group — called a J-function—— to account for these effects. The J-function has become an important tool for capillary pressure interpolation and extrapolation and increases the utility of a single capillary pressure curve. It allows us to adapt a single data set to other areas of a reservoir where data may be unavailable. We saw in section 3.4 (Capillary Bundle Models) that a “representative pore-radius” can be inferred from sample permeability and porosity via the relation:

˜ ~ R

k φ

(20)

Hence, a dimensionless group can be formed by defining: 1

Pc  k  2 J=   σ cos θ  φ 

(21)

This can be applied at a number of different saturation values — at each saturation, Pc(Sw) is determined and (21) used to find J(Sw). Often, the contact angle is difficult to ascertain with any accuracy, and the cosθ term is often set to unity. Having found J(Sw), capillary pressure curves corresponding to a range of different permeability and porosity values (and different fluid combinations via the σcosθ term) can be determined by simply inverting the J-function, viz:

Pc (Sw ) =

46

J(Sw ).σ cos θ  k    φ

1 2

(22)

Petrophysical Input

8

Example J-functions are shown in Figure41 together with a derived set of Pc-curves. The procedure can also be used to examine the relationship between permeability (and/or porosity and/or fluid combinations) and water saturation at different capillary pressure cut-off values ——in general, at a given Pc, we find that lower k => higher Sw (Figure 42).

4

1.5

1.0

Alundum (consolidated) Leverett (unconsolidated)

2

1

3

k φ

0.371 0.419

√ (Sw) =

Pc σ

Hawkins

1.1

0.347 0.151 0.18 0.315 0.116 0.114

()

2

1.2

Capilliary pressure function, √ (Sw) =

Kinsella

0.9

0

0.5

Leverett

20

40 60 Liquid saturation, % (d)

80

100

0

20

40 60 Liquid saturation, % (c)

80

100

2

1

() k φ

30

40 50 60 70 Water saturation, Sw

80

90

100

√ (Sw) =

20

Pc σ cos θ

Kinsella Shale

10

10 md

50 md

100 md

200 md

500 md

90

160

72

140

63

120

54

100

45

80

36

60

27

2

1

0

81

4

2

k φ

()

1

3

40

18

3

20

9

0

0

0

10

20

30

40

50

60

70

80

90

0 100

Water saturation, %

Figure 41


Pc σ cos θ

Oil water capilliary pressure, psi (reservoir conditions)

0

3

Katie

900 md

180 Height above zero capilliary pressure, ft

27

5

100

Rangely

0.3

After Amyx Bass and Whiting 1960

9

80

4

Leduc Morano

200

12

40 60 Liquid saturation, % (b)

0.447

0.4

30

15

20

Theoretical limiting value for regular packed spheres

0.6

0 0

18

0

Alundum

0.1

21

1

0.7

0.2

24

2

0.8

√ (Sw) =

k φ

()

1

1.3

Lim Sw-1√(Sw)

Formation Woodbine Weber Moreno Viking Deese Devonian

Pc σ cos θ

Reservoir Hawkins Rangely El Robie Kinsella Katie Leduc

1.4

2

1

0

47

log k log permeability, md

Capillary pressure = 5 psi

Brine saturation S

20 25 30 Porosity φ

Transition Zones We conclude this section on capillary pressure curves with a brief discussion of transition zones. Transition zones are usually described using the water-wet vertical bead pack idealization shown in Figure 43. A water-saturated bead pack is allowed to drain under gravity until capillary equilibrium is reached. At the top of the column, the water is in the form of discrete rings connected via thin wetting films over grains — called the pendular regime. Moving down the column, the rings become larger and ultimately coalesce — the funicular regime. Near the bottom of the pack, water is continuous.

48

Figure 42

Petrophysical Input

8

σ σ θ

R θ

r r

(a)

R θ

r

r

y

d

y

(i)

(ii)

b

b δ

δ θ

θ

c

θ

c

a

a

(i)

(ii)

(b) b b δ δ θ c

Figure 43

a

c

(iii)

a

(iv)

Whilst this idealization gives some limited insight into transition zone behaviour, a more enlightening picture emerges if we use our knowledge of drainage capillary pressure at the pore scale — this can give us a mental picture of fluid distributions as we move through a transition zone. Referring to Figure 44a, we see that the hydrostatic oil and water gradients meet at the OWC (Po=Pw). Above this contact, Po>Pw and capillary pressure increases as we go further up the reservoir. Figure 44b shows how the phases would be distributed in the pore network, with oil present in smaller and smaller pores as we move up through the transition zone.


49

Exploration Well

Water

Oil

Gas GOC 5200' Test Results at 5250 ft po = 2402 psia dpo = 0.35 psi/ft dD

Oil OWC 5500' Water Pressure (psia) 2250 5000

2375 2265

Cappilary Pressure Increasing

2500

2369

Pcgo

Depth (feet)

GOC: po = pg = 2385 5250

Pcow 5500

OWC: po = pw = 2490

(a)

Figure 44

(b)

Capillary pressure data can also be used to infer fluid saturations as a function of depth within a transition zone. Assuming that oil and water pressures remain continuous throughout the entire height of the zone (doubtful under some circumstances), we can write equations for each hydraulic gradient (z measures height above OWC and increases vertically upwards):

dPw = − gρ w dz

(23)

dPo = − gρ o dz

(24)

Subtracting (23) from (24) leads to:

dPo dPw dPc − = = g(ρw − ρo ) dz dz dz which can be integrated to give:

50

(25)

Petrophysical Input

Pc = g(ρw − ρo )z

8

(27)

Hence, any capillary pressure curve can be re-plotted to give saturation with depth information. An example is shown in Figure 45.

830 840 Minimum of 22% connate water 850 860 870 880 Data derrived from capillary pressure Data obtained from seismic logs

Depth, M below sea level

890 900 910 920 930 940 950 960 970 980

Approximate gas-oil contact 990 1000 1010 1020

Figure 45

0

10

20

30

40

50

60

70

80

90

100

Water saturation scale %

5.2 Methods for Generating Relative Permeabilities

Multiphase flow experiments are costly, difficult and time-consuming to carry out and it is often infeasible to perform a large number of laboratory sensitivity studies. It would therefore be highly desirable if a cheap, predictive model for relative permeability could be developed that only required cheap, easily-accessible data as inputs. In fact, a number of models are already currently available (although some still lack a degree of refinement) and we will discuss three such approaches below. Purcell Model The Purcell model seeks to utilise cheap capillary pressure data in order to predict more expensive relative permeabilities. The model revisits the capillary bundle model and initially uses a little algebra to develop a predictor for absolute permeability. Institute of Petroleum Engineering, Heriot-Watt University

51

The development is shown in Figure B4 in Appendix B. Extending the approach to deal with relative permeabilities is relatively straightforward — we simply need to replace the summation in (equation B6 with integrals over water saturation. The wetting phase relative permeability integral runs from 0 to Swt (the wetting phase saturation of interest), whilst the nonwetting integral runs from Swt to 1. The equations to be applied at successive values of water saturation are given in Figure B5. A graphical “recipe” for relative permeability prediction via capillary pressure data is essentially the following: (i) Obtain capillary pressure data Pc(Sw) (ii) Plot the curve 1/Pc2 vs Sw (see Figure 46) (iii)Find the area under the entire curve (=k) (iv) For any chosen value of water saturation (Swt, say), calculate the area under the 1/Pc2 curve from 0 to Swt. This gives kwt (v) For the same value of water saturation, calculate the area under the 1/Pc2 curve from Swt to 1. This gives knwt

14

0.56

12

0.48

Capillary pressure, Pc, atm

10

8

0.40

Pc

0.32

1 (Pc)2

6

0.24

4

0.16

2

0.08

0 100

1/(capillary pressure)2, 1/Pc2, atm-2

(vi) Use the values obtained from (iii) – (v) to determine relative permeabilities

0 80 60 40 20 Percent of total pore space occupied by mercury

0

Figure 46

There is one clear drawback using this approach, however: the relative permeabilities add up to unity over the full saturation range and are therefore completely symmetrical. This limitation led Burdine to add the following improvement.

52

Petrophysical Input

8

Burdine Model The Burdine model simply takes the Purcell model and re-scales the endpoints through the prefactors:

λ w (Sw ) =

Sw − Swi 1 − Swi − Sor

λ nw (Sw ) =

1 − Sw − Sor 1 − Swi − Sor

(27)

The curves are therefore defined over the wetting-phase saturation range (1-Sor) to (1Swi). The full equations are given in Appendix B (Figure B6). Although a number of gross assumptions have been made along the way, the Burdine method offers a cheap alternative to laboratory measurement of relative permeabilities. It is easily coded into a spreadsheet and is often carried out by practicing reservoir engineers — experience appears to show that the wetting phase prediction is often fairly good, whilst the nonwetting curve is less well reproduced. Brooks and Corey Model Although the models described above yield relative permeability curves that are qualitatively reasonable, reproduction of experimental data is only achieved via the introduction of some form of empiricism. Many studies have subsequently relied entirely upon empirical curve fitting techniques. One of the most popular empirical correlations is that due to Corey (1954), who proposed the following: 4 k rw ∝ Seff

(28)

k rnw ∝ (1 − Seff )2 (1 − S2eff )

(29)

However, it was soon discovered (not surprisingly) that the exponents in Corey’s original equations would have to be varied in order to fit different materials. They were consequently generalised by Brooks and Corey (1964) to:

k rw = S(eff2 + 3λ ) λ

(30)

k rnw = (1 − Seff )2 (1 − Seff )( 2 + λ ) λ

(31)

Seff = ( Pcb Pc )

(32)

λ

( Pc ≥ Pcb )

with Pcb representing the breakthrough capillary pressure and λ the “pore size distribution index”. Both of these parameters have to be determined experimentally: Seff vs Pc data is plotted on a log-log scale and a straight line fitted, the slope gives λ and the intercept with Seff=1 is assumed to give Pcb. Network Models It is quite apparent from this discussion that no satisfactory predictive model currently exists which can adequately account for the complex jumble of channels that pervade a porous medium. One of the most promising developments of recent years, however, has been the possibility of applying interconnected network models to the study of Institute of Petroleum Engineering, Heriot-Watt University

53

microscopic flow. A detailed discussion now follows to determine the extent to which the capillary network model is an analogue for two-phase relative permeability experiments. McDougall and Sorbie have developed an approach for “predicting” relative permeability based on “anchoring” to capillary pressure data. They have recently developed a PC-based software package known as MixWet, which allows a wide range of multiphase simulations to be undertaken at the pore-scale under a variety of different wettability conditions — the interface is shown in Figure 47. Certain pore scale parameters required as input to MixWet are estimated using the inverse mercury capillary pressure curve - which we call the R-plot since 1/Pc ~ R, where R is the pore radius. The full methodology is explained by McDougall et al (SCA, Edinburgh, 2001).

Figure 47

The model calculates the total flow through two intertwined networks (one for each phase) over a range of saturation values. Inherent in this is the assumption that the flow is stationary, thus steady-state relative permeabilities are considered here. In all that follows, the wetting phase is assumed to be water and the non-wetting phase oil. Each pore in the network is assigned a capillary entry radius from a distribution. At each stage of the displacement, the total flow is calculated separately for each of the 54

Petrophysical Input

8

intertwined networks. The results are then converted into normalised permeabilities as functions of saturation. Additional details can be found in the earlier discussion presented in section 2.6. Primary Displacements — Consider first the case of a strongly water-wet rock which is initially 100% saturated with oil. Several possible displacement mechanisms have been identified in two dimensions (Lenormand and Zarcone, 1984), one of which is known as “snap-off”. When brought into contact with the water phase, a strongly water-wet rock will spontaneously imbibe the wetting fluid via film flow along irregularities on the pore surfaces. In effect, the water slowly wets all internal grain surfaces and is essentially present everywhere in the matrix. As this imbibition process continues, the thin films begin to swell. Eventually, the thinnest pores will become completely filled with water and the original oil will be displaced if an escape route exists (see earlier discussion in section 3.4). This continues with the gradual filling of progressively wider pores until no further displacement is possible since the oil phase becomes disconnected. The snap-off mechanism is thought to be the most prevalent imbibition mechanism governing capillary-dominated waterfloods of consolidated media (low aspect ratio pores). More pistonlike displacements may occur in media containing high aspect ratio pores (e.g. unconsolidated beadpacks) but such systems are not considered here (the competition between snap-off and pistonlike displacements is discussed more fully in Dixit et al, 1997). A typical set of imbibition relative permeability curves obtained from 3D network modelling are shown in Figure 48 and are seen to correlate well with experimental observations 1.0

0.8

Kri

0.6

0.4

0.2

Figure 48 Imbibition relative permeability curves from pore-scale simulation

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Sw

The simulation of low-rate drainage processes is carried out using an invasion percolation model with hydraulic trapping of the wetting phase. In this case, the injected non-wetting phase first fills the largest pores connected to the inlet face of the network, and then proceeds along progressively narrower pathways, occupying successively smaller pores (see section3.3). The drainage displacement is terminated at a pre-determined limiting capillary pressure — if an unrealistically high capillary pressure were applied to the network, no irreducible water saturation would remain at the end of the flood. This drainage model, first proposed by Chandler et al (1982) and Wilkinson and Willemson (1983), is basically the displacement mechanism governing mercury porosimetry experiments. A set of simulated drainage relative Institute of Petroleum Engineering, Heriot-Watt University

55

permeability curves are shown in Figure 49 and again correlate well with experimental observation. 1.0

0.8

Kri

0.6

0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Sw

Secondary Displacement Processes ——Secondary displacement processes are displacements carried out on a porous medium which is already partially saturated with the displacing phase. For example, if a primary drainage experiment is followed by a waterflood, the process is called secondary imbibition. This cycle exhibits an hysteresis effect between the primary drainage and secondary imbibition non-wetting relative permeability curves, but very little variation is evident in the wetting phase curves and the hysteresis effect is usually considered to be negligible. Various theories relating to pore size distribution and matrix cementation have been put forward in an attempt to explain this phenomenon, but network modelling of the various secondary processes provides important clues as to the real causes of hysteresis in porous media. Both secondary drainage and secondary imbibition have been studied, but for brevity only the primary drainage-secondary imbibition scenario will be discussed here. The simulation is shown in Figure 50. The similarity to experimental curves from consolidated media is striking and the hysteresis effect is clearly duplicated in the case of the non-wetting phase. The wetting phase curve shows little sign of deviation which is also in general agreement with experimental findings. In order to explain this phenomenon, a step by step analysis of the distribution of invaded pores must be considered. It is found that the hysteresis effect is due to the different physics governing imbibition and drainage processes; in particular, the associated issue of accessibility during drainage (McDougall and Sorbie, 1992). Different hysteresis patterns are exhibited by unconsolidated media and possible causes of this have been discussed by Jerauld and Salter (1990). More recently, the full range of experimentallyobserved hysteresis phenomena has been modelled and interpreted by Dixit et al (1997). Both studies show that pore aspect ratio and the competition between snapoff and pistonlike displacement affect the hysteresis trend, although different methodologies were used to capture the effects.

56

Figure 49 Drainage relative permeability curves from pore-scale simulation

Petrophysical Input

8

1.0 krwint Primary Drainage

0.8

Secondary Imbibition krwsec

Kri

0.6

0.4

0.2

Figure 50 Non-wetting phase hysteresis after a primary drainage -> secondary imbibition cycle

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Sw

Comparison with Experiment — Although we have presented a number of idealised relative permeability simulations, the acid test of such a model is to compare relative permeability prediction with experiment. This is an on-going exercise but some preliminary examples are shown in Figure 51 together with the matching inverse capillary pressure data (R-Plots)..


57

Sample F with best-fit 7

0.9

6

0.8 0.7

Kro sim

0.6

Krg sim

0.5

Kro expt

0.4

Krg expt

5 R, microns

kr

Comparing Sample F with z3.3nu0.9n0lmb4 1

4 3 2

0.3

1

0.2 0

0.1

0

0.2

0.4

0.1 0

0.2

0.4

0.6

0.8

Experimental

Sg compensated

Comparing Sample G with z3.3nu1.2n0lmb3

1

Analytic

40

0.9

35

0.8 0.7

Kro sim

0.6

Krg sim

0.5

Kro expt

0.4

Krg expt

30 R, microns

kr

0.8

Sample F with best-fit

1

25 20 25 10

0.3 0.2

5

0.1

0 0

0.1 0

0.2

0.4

0.6

0.2

0.8

Comparing Sample H with z4nu0.6n0lmb3 90 80

0.8 0.7

Kro sim

0.6

Krg sim

0.5

Kro expt

0.4

Krg expt

0.3 0.2 0.1

R, microns

0.9

0.6

1

Analytic

70 60 50 40 30 20 10 0 0

0.1 0.4

0.8

Sample H with best-fit 100

0.2

0.6

Experimental

1

0

0.4 SHg

Sg compensated

kr

0.6 SHg

0.8

0.2

0.4

0.6

0.8

1

SHg Experimental

Analytic

Sg compensated

5.3 Hysteresis Phenomena

We have already seen that both capillary pressure and relative permeability curves exhibit hysteresis — that is, they depend upon saturation history (Figure 52 and 53). There are a number of possible causes for hysteresis but the three main effects are the following: (i) Contact angle hysteresis — advancing and receding contact angles differ (reflected in Pc through the Young-Laplace equation)

58

Figure 51 Gas-oil relative permeabilities (prediction vs experimental) and inverse capillary pressure data for reservoir samples

Petrophysical Input

8

(ii) Pore structure hysteresis — sloping pore walls mean that pores fill and empty at different capillary pressures (iii)Topological hysteresis — imbibition and drainage processes are different topologically (film-flow versus fingered invasion) There is often such a difference between imbibition and drainage curves (both capillary pressure and relative permeability) that we must make sure that we are using the correct set of curves in our reservoir simulation studies (for instance, we should not be using drainage curves as simulation input when modelling a waterflood in a water-wet reservoir). Built-in numerical models are available in Eclipse and other commercial reservoir simulators to account for flow reversals at intermediate saturations.

Pc

a b

d

c Sw

Pc

Swi

1

1.0

1.0 Sor

kr

kro Drainage

Secondary drainage Intermediate path

Imbibition

0.0 0.0

0.0 0.0 Swi

Sw

Sw

1.0

1.0

Figure 52


59

48 Water Wet

Capillary Pressure - Cm of Hg

40 Venango Core VL-2 k = 28.2 md 32

24

16 2

1

8

0

0

20

40

60

80

100

Water Saturation - percent

6

Figure 53

WETTABILITY — CONCEPTS AND APPLICATIONS

6.1 Introductory Concepts

What is Wettability? The term wettability refers to the wetting preference of a solid substrate in the presence of different fluid combinations (liquids and/or gases). We have already seen in Section 3,1 and Appendix A6 that the wetting preference of a solid can be characterized by a contact angle, as shown in Figure 54. The wetting phase is defined as the fluid that contacts the solid surface at an angle less that 90o. θ = 158º θ = 35º θ = 30º

θ = 30º

Water

Water Silica Surface

Isooctane

Isooctane + 5.7% Isoquinoline

Isoquinoline θ = 54º

θ = 30º

θ = 106º

θ = 48º Water Calcite Surface

60

Naphthenic Acid

Water

Figure 54

Petrophysical Input

8

This figure clearly demonstrates an important issue, namely, that the wetting preference of a rock depends not only upon the fluids involved but also upon the mineralogy of the rock surface. For instance, we see that, in the presence of isoquinoline, water is nonwetting on a silica substrate but wetting on a calcite substrate. In cases when a solid has no wetting preference, (i.e. when the angle separating the two fluid interfaces is close to 90o) the system is said to be of “neutral” wettability. The importance of rock wettability cannot be over-emphasised, as it affects almost all types of core analysis: capillary pressure, relative permeability, waterflood behaviour, and electrical properties (see Anderson, 1987, for an excellent series of review articles). The simple reason for this is that wettability affects the location, flow, and distribution of fluids in a porous medium — hence, most measured petrophysical properties must be affected. Moreover, we shall see later how some core handling procedures can drastically alter the wettability state of a core: this would invariably lead to the measurement of SCAL data inappropriate to the reservoir under investigation. In most of what has been described in this chapter, the assumption has been made that the system under consideration was water-wet. Indeed, historically, all reservoirs were believed to be strongly water-wet and almost all clean sedimentary rocks are in a water-wet condition. An additional argument for the validity of the water-wet assumption was the following: the majority of reservoirs were deposited in an aqueous environment, with oil only migrating at a later time. The rock surfaces were consequently in constant contact with water and no wettability alterations were possible as connate water would prevent oil contacting the rock surfaces. However, Nutting (1934) realised that some producing reservoirs were, in fact, oil-wet (the rock surface was preferentially wetted by oil in the presence of water) and it is now generally accepted that water-wet reservoirs are the exception rather than the rule (Table 3 and 4).

Table 3

Table 4

Water-wet Intermediate wet Oil-wet Strongly oil-wet

Water-wet Intermediate wet Oil-wet Total

Contact Angle (degrees) 0 to 75 75 to 105 105 to 180

Contact Angle (degrees) 0 to 80 80 to 100 100 to 160 160 to 180

Percent of Reservoirs 8 12 65 15

Silicate Reservoirs 13 2 15 30

Carbonate Reservoirs 2 1 22 25

Total Reservoirs 15 3 37 55

We now know that wettability is a rather complicated issue (still actively being researched) and that there are a number of different factors affecting reservoir wettability, including:


61

(i)

Surface-active compounds in the crude oil. These are generally believed to be polar compounds (polar head and hydrocarbon tail), mostly prevalent in the heavier crude fractions — resins and asphaltenes — that form an organic film or adsorb onto pore walls

(ii)

Brine chemistry

(iii) Brine salinity (iv) Brine pH (v)

The presence of multivalent metal cations (Ca2+, Mg2+, Cu2+, Ni2+, Fe3+)

(vi) Pressure and temperature (vii) Mineralogy (including clays)

In short, surface chemistry determines the wettability state of a reservoir (or, more correctly, any given region of a reservoir; as wettability can be expected to vary spatially — and possibly temporally——throughout a reservoir). Having discovered that most reservoirs are not water-wet, we now have to modify everything that we have learned so far — pore-scale physics, capillary pressure models, relative permeability models, and network models. However, this is not as difficult as it may first appear; we already understand drainage and imbibition processes at the pore scale, we know how to define capillary pressure, and we appreciate the dynamics underlying relative permeability measurements. The following discussion should help you apply your previous water-wet knowledge to systems that are not water-wet. Pore-Scale Effects The effect of wettability at the pore-scale is shown in Figure 55. If a rock is water-wet, we have already seen that there is a tendency for water to reside in the tighter pores and to form a film over the grain surfaces. Oil (the nonwetting phase) resides in the larger pores. In this case, the term “imbibition””— a process whereby a wetting phase displaces a nonwetting phase — would refer to the displacement of oil by water. The term “drainage” would apply to oil displacing water . In an oil-wet system, however, the situation is reversed— oil now forms a thin film over the grain surfaces and water fills the larger pores. Consequently, in an oil-wet medium,“imbibition” refers to the displacement of water by oil, whilst “drainage” refers to the displacement of oil by water. These differences clearly have major implications for waterflooding reservoirs (if a reservoir is oil-wet, a waterflood is a drainage displacement, oil is flushed from small pores, and oil production via film-flow also becomes important).

62

Petrophysical Input

Oil

(a) Water Water

Oil

(b) Water Figure 55

Water

Oil

Oil

Water

Water

Oil

Rock Grains

Oil

Oil

Water

Water

Oil

8

Rock Grains

The foregoing discussion has clarified some of the pore-scale physics affecting waterwet and oil-wet media. However, given the complex nature of wettability alterations in reality, a more realistic representation of wettability at the pore scale may be that shown in Figure 56. Here, a combination of water-wet and oil-wet pathways exists that allows film-flow access to the displacing phase and film-flow escape for the displaced phase. Hence, a waterflood would now consist of a combination of imbibition and drainage events: water initially imbibing along water-wet pathways (displacing oil from small pores) and then requiring an overpressure to drain oil from the larger oilwet pores. It is clear that the underlying mineralogy and surface chemistry of the system would determine the connectivity and topology of the “wettability pathways” but how can we determine these pathways”— in short, how can we quantify the wettability of a porous medium?


63

Water-Wet

Oil-Wet

6.2 Wettability Classification and Measurement

When we come to define the type of wettability associated with a particular porous sample, we come up against a minefield of inconsistent terminology. It will be beneficial, therefore, to give some definitions that will appear periodically throughout this section. Wettability can be classed as being uniform or non-uniform as follows: Uniform — the wettability of the entire porespace is the same (100% water-wet, 100% oil-wet, or 100% “intermediate-wet”) and the contact angle is essentially the same in every pore; Non-Uniform — this is more characteristic of hydrocarbon reservoirs. The porespace exhibits “heterogeneous” wettability, with variations in wetting from pore to pore (and possibly within a pore) — say 70% water-wet pores and 30% oil-wet pores. We can introduce 2 subdivisions (Figure 57): Mixed-wet — a certain fraction of the largest pores are oil-wet (there are valid depositional arguments for how this may come about). Fractionally-wet — no size preference for oil-wetness (there are valid mineralogical arguments for this). Given our knowledge of pore-scale displacements, it is clear that each type of wettability distribution will yield different capillary pressures and relative permeabilities (and recoveries), and we should be aware of this when we come to interpret SCAL and wettability test data.

64

Figure 56

Petrophysical Input

8

F(R)

Mixed-Wet 0.02

0.015

0.01

α Water-Wet

0.005

Oil-Wet R 20

40

60

80

100

80

100

F(R)

Fractionally-Wet 0.02

0.015

Water-Wet

α

0.01

0.005

Oil-Wet R 20

40

60

Figure 57

Wettability Measurement Core wettability can be determined in a number of ways, although three main quantitative methods are used most frequently: (i) Contact angle measurement — this measures the wettability of a mineral surface and is mainly used by specialist researchers (ii) Amott method — this measurement gives valuable information regarding the extent and connectivity of wettability pathways of a core (iii)U S Bureau of Mines (USBM) method — yields an average wettability measurement, is less informative, but much faster (and therefore cheaper) than the Amott method Table 5 gives some expected values for water-wet, neutral-wet, and oil-wet media from each test (we will explain each of these shortly).


65

Contact angle Minimum Maximum USBM wettability index Amott wettability index Displacement-by-water ratio Displacement-by-oil ratio Amott-Harvey wettability index

Water-Wet

Neutrally Wet

Oil-Wet

0º 60 to 75º W near 1

60 to 75º 105 to 120º W near 0

105 to 120º 180º W near -1

Positive Zero 0.3 ≤ / ≤ 1.0

Zero Zero -0.3 < / < 0.3

Zero Positive -1.0 ≤ / ≤ -0.3

Table 5

Other qualitative methods exist, such as rate-of-imbibition tests, NMR methods and dye adsorption methods, but these are not routinely undertaken. Let us now examine the three main methodologies in more detail. Contact angle method This is shown in Figure 58 and is mainly applicable to pure fluids and isolated mineral surfaces. A sessile drop (single surface) or modified sessile drop (two surfaces) can be used. A drop of fluid (oil, say) is generally placed between the mineral surfaces in a bath of a second fluid (water). The surfaces are then moved in opposite directions parallel to one another and advancing and receding contact angles can be determined. These angles usually differ from one another (hysteresis effect) and this is one component of the hysteresis observed in capillary pressure and relative permeability curves. Notice also that the early-time behaviour of the measurement is often misleading — you have to wait for the system to equilibrate. Whilst this technique is often used in wettability research laboratories, it is not carried out routinely in the industry.

Crystal

Water

Oil

Water Advancing Contact Angle

Oil

Crystal

180 Curve "E" Kareem Contact Angle (degrees)

150 Curve "D" San Andres 120

90 Curve "B" Deosol

60

Curve "C" Tertiary Kenai

30

Curve "A" Pure Grade C10 0

0

200

400

600

800

1000

1200

Age of the oil-mineral interface (hours)

66

1400

1600

1800

Figure 58

Petrophysical Input

8

Amott Test The Amott test is slow but informative. It combines both imbibition and forced displacement and is usually carried out via the following “recipe”: (1) Centrifuge core down to Sor (2) Immerse in oil and measure volume of oil imbibed after 20 hours (Voi) (3) Centrifuge the core down to Swi and measure the total oil volume (Vot) (4) Immerse in water and measure volume of water imbibed after 20 hours (Vwi) (5) Centrifuge the core down to Sor and measure the total water volume (Vwt) (6) Calculate the displacement-by-oil ratio (Io) and the displacement-by-water ratio (Iw), i.e.

Io =

Voi Vot

Iw =

Vwi Vwt

If Iw ->1 and Io->0, the core is water-wet If Io ->1 and Iw->0, the core is oil-wet If Iw=0 and Io=0, the core is said to be neutrally-wet (neither phase has imbibed) If Iw>0 and Io>0, the core is said to be heterogeneously-wet (mixed- or fractionally-wet) Often, the indices are combined to give a single index; the Amott-Harvey Index =IwIo, yielding a number between -1 (oil-wet) and 1 (water-wet). If possible, however, it is better to have access to both Io and Iw, as these individual indices tell far more than the single Amott-Harvey index. Unfortunately, there are two main drawbacks to the Amott test: firstly, it takes a long time for fluids to equilibrate, making the test timeconsuming and costly; and secondly, the ratios used in the calculations are very sensitive to errors in saturation measurement. A far quicker, but less informative, measure of wettability is given by the USBM test. USBM Test The USBM test uses thermodynamic arguments to ascertain the approximate work done on the system durig centrifuge displacements. This essentially boils down to calculating areas under Pc curves (= work done during a displacement) as follows (Figure 59).


67

*

Capillary Pressure, PSI

10

Water Wet Log A1/A2 = 0.79

II A1 0 I A2

a -10 0

Average Water Saturation, Percent

*

Capillary Pressure, PSI

10

100

Oil Wet Log A1/A2 = -0.51

II A1 0 A2 I b

-10 0

Average Water Saturation, Percent

100

(1) Core driven down to Swi by centrifuge (2) Core is centrifuged under brine at incremental speeds until Pc=-10psi (NB. water pressure greater than oil pressure, so Pc=Po-Pw is negative). Measure expelled oil volume (3) Core is centrifuged under oil at incremental speeds until Pc=+10psi. Measure expelled water volume (4) Plot the two capillary pressure curves (5) Calculate the areas under the oil-drive (A1) and water-drive(A2) curves (6) USBM Index (W) = log(A1/A2)

68

Figure 59

Petrophysical Input

8

Although W theoratically goes from -infinity to +infinity, -10 indicates some degree of water-wetness, W<0 some degree of oil-wetness). Note, that other methods exist that combine elements of the Amott and USBM tests (see the literature for details, an example is shown in Figure 60). 6.3 The Impact of Wettability on Petrophysical Properties

Core Handling The previous discussion has highlighted the importance of wettability at the porescale and its implications for oil recovery (as demonstrated by the Amott and USBM tests). We should therefore endeavour to preserve the natural state of a core as much as possible if we wish to derive SCAL data that has relevance to the reservoir under consideration. There are a number of issues of which we should be aware: (i) wettability alterations can occur during drilling due to complex mud chemistry, (ii) pressure and temperature change as a core is brought to surface, (iii) asphaltenes may subsequently precipitate and light ends may be lost. Pressure coring can help alleviate some of these problems but this is not always available. Uniform wetting-contact angle (Morrow and McCafferty, 1976)

180º

133º oil wet

62º intermediate

0º water wet

Amott-IFP Index (Cuiec, 1991)

-1

-0.3

-0.1

+0.1

+0.3

+1

slightly neutral slightly oil wet water wet

Figure 60

oil wet

intermediate

water wet

Three types of core are generally used in core analysis: Native-state core — ideal if available as it minimises losses, wrap cores in polyethylene film and foil, then seal in liquid paraffin Cleaned core — should only be used if reservoir is strongly water-wet (rare). Cleaning procedure should depend upon the crude oil/brine/rock system under investigation Restored-state core — Only course of action if wettability has been altered. Clean the core, flow reservoir fluids in correct order, age the core (1000 hours/40 days) and hope that reservoir conditions have been re-established. Often, of course, this is not the case and recoveries and relative permeabilities are badly affected (Figure 61).


69

Contaminated

Swi k0 452, mO 27.6%

φ ka 25.8% 780, mO

Cleaned

254, mO 31.8%

24.8% 338, mO

Restored-state

202, mO 29.2%

24.8% 338, mO

1 60

NATIVE CORE CRUDE OIL CLEANED CORE CRUDE OIL CLEANED CORE REFINED OIL

.8


Oil Recovery, Percent PV

50

40

30

20

.4

.2

10

0 0.01

.6

0.1

1.0

Water Injected, Pore Volumes

10

100

0

0

.2

.4 .6 .8 Water Saturation, PV

1

The following discussion, regarding the sensitivity of a number of petrophysical properties to wettability, will highlight the importance of proper core handling and cleaning procedures still further. The Effect of Wettability Upon Electrical Properties We begin our discussion of petrophysical parameters with a brief look at Archie’s equation (Figure 62). This is routinely used by petrophysicists to determine water saturation in a formation from wireline log data and the exponent (n) has been determined experimentally from plugs and has a value of about 2 for water-wet/ cleaned cores. Under non-water-wet conditions, however, the fluids are distributed differently in the pore-space (fewer water films are also present) and we should not be surprised to learn that Archie’s equation breaks down (although it is often used regardless). Consequently, if n=2 is used regardless of the wettability condition pertaining in the reservoir, then saturation predictions will be wrong (see Pirson and Fraser, 1960, for an example of how expensive this assumption has been in the past). In order to demonstrate how inappropriate it would be to assume n=2 for non-waterwet material, we could take reservoir core and actually measure saturations directly. When taken together with direct resistivity measurements, we could then use Archie’s equation to infer what the appropriate exponent should be over a range of saturation values. The table in Figure 62 shows just such a case: we see that the exponent is not even constant, implying the breakdown of the assumptions underpinning the model itself - so use the Archie equation with care

70

Figure 61

Petrophysical Input

8

Archie Saturation Exponents as a function of Saturation for a Conducting Nonwetting Phase Air/NaCi Solution Brine Saturation n (% PV) 66.2 1.97 65.1 1.98 63.2 1.92 59.3 2.01 51.4 1.93 43.6 1.99 39.5 2.11 33.9 4.06 30.1 7.50 28.4 8.90

_ Sw n

Rt

=

Ro

≡

Oil/NaCi Solution Brine Saturation n (% PV) 2.35 64.1 63.1 2.31 60.2 2.46 55.3 2.37 50.7 2.51 44.2 2.46 40.5 2.61 36.8 2.81 34.3 4.00 33.9 7.15 31.0 9

IR

where:

Figure 62

Sw

=

brine saturation in the porous medium

Rt

=

Ro

=

resistivity of the porous medium at saturation S w, and resistivity of the 100% brine-saturated formation

The Effect of Wettability Upon Capillary Pressure We have already seen an oil/water interface will become curved in order to balance the pressure jump across it opposing interfacial tension forces. The pressure jump at which this balance is attained is given by Laplace’s equation:

 1 1 pc = po − pw = σ  +   r1 r2 

(33)

where s is the interfacial tension between the two fluids, and r1 and r2 are the two principal radii of curvature (see section 3.1). The pressure difference Pc is the capillary pressure and, in oil-water systems, is conventionally taken to be the pressure in the oil phase minus the pressure in the water phase. For drainage of a water-wet circular capillary (radius R) and zero contact angle, this relationship becomes:

pc = po − pw =

2σ R

(34)

Notice, however, that for water to invade an oil-wet capillary, the capillary pressure must become negative (i.e. pw must become greater than po) The combination of the words “negative” and “pressure” may at first seem confusing, but the term is merely an artefact of conventional terminology. The concept of negative capillary pressure is central to displacements in heterogeneously-wet media.


71

When discussing displacements in porous media of heterogeneous wettability, the terms imbibition and drainage become somewhat confused. For example, waterfloods in heterogeneously-wet media may be a combination of conventional imbibition and drainage processes. When the pore network contains a mixture of water-wet and oilwet pores, the waterflood initially proceeds as an imbibition, with water spontaneously imbibing along water-wet pathways displacing oil from the smallest pores. Eventually, however, the water has to be forced into oil-wet pores and the displacement becomes one of drainage.

Primary Drainage

Swi (Initial Water)

Pc(+ve)

32

A CURVE 1.DRAINAGE 2.SPONTANEOUS IMBIBITION 3.FORCED IMBIBTION

24

16 Age + Change Wettability

1 8

0

2

100

Sw%

0

B

Primary Drainage

Swi (Initial Water) Pc(+ve)

Forced Oil Drive (Secondary Drainage)

BEREA CORE k = 184.3 md

-8

Water Imbibition (Spontaneous)

Pc(-ve) Forced Water Drive

0

Oil Imbibition (Spontaneous)

3

POINT A. IRREDUCIBLE WETTING SATURATION B. ZERO - CAPILLARY-PRESSURE NONWETTING SATURATION C. IRREDUCIBLE NONWETTING SATURATION

-16

C -24 0

20

60

40

100

WATER SATURATION, PERCENT P.V.

100

Sw%

48

-48

WATER WET

OIL WET -40

CAPILLARY PRESSURE, CM HG

40

CAPILLARY PRESSURE - Cm of Hg

80

VENANGO CORE VL - 2 k = 28.2 md

32

24

16 2

1

-32

-24

-16 1 -8

8

2 0

0

20 40 80 60 WATER SATURATION - PERCENT

16

24

12

16

8

4

20 40 60 80 WATER SATURATION, PERCENT

100

Drainage capillary pressure characteristics (after Ref.30)

CAPILLARY PRESSURE - Cm of Hg

32

0

60

40

100

80

Oil-water capillary pressure characterisitics. Ten-sleep sandstone. oil-wet rock (after Ref.29). Curve 1 - drainage. Curve 2 - Imbition.

20

0

20

OIL SATURATION, PERCENT

Capillary pressure characteristics, strongly water-wet rock. Curve 1 - Drainage. Curve 2 - Imbition.

CAPILLARY PRESSURE, CM.HG

0 0

100

INTERMEDIATE WET

1 8

2

0

BEREA CORE 2-MO16-1 k - 184.3 md

-8

3

-16

-24

0

20 40 50 60 WATER SATURATION - PERCENT

100

Oil-water capillary pressure characteristics, intermediate wettability. Curve 1 - drainage. Curve 2 - spontaneous imbition. Curve 3 - forced imbition.

72

Figure 63 Experimental capillary pressures in cores for various processes ( drainage, imbibition) and wettability conditions ( water-wet, oil-wet, intermediate-wet )

Petrophysical Input

8

So, in general, a negative leg in the capillary pressure curve indicates some fraction of oil-wet porespace. The concept is shown both schematically and for experimental data in Figure 63. The Effect of Wettability Upon Relative Permeability As wettability controls the distribution of phases within the porespace, it is hardly surprising that wettability has a majorimpact upon relative permeability curves and subsequent reservoir performance. We can use our pore-scale modeling knowledge to help explain the differences seen between the slopes of water-wet and oil- wet curves (Figure 64). In the water-wet case, water imbibes via the smallest pores in the system, leaving oil resident in large, fast-flowing pores. Consequently, we would expect the oil relative permeability curve to decrease slowly with increasing water saturation and the water relative permeability endpoint to remain low — this is exactly what is observed. Conversely, waterflooding an oil-wet medium should lead to a rapid decrease in oil relative permeability, together with a high water endpoint—— once again, this is what is observed.

RELATIVE PERMEABILITY, PERCENT

100

OIL WET WATER WET

80

CRAIG'S RULES OF THUMBS FOR CETERMINING WETTABILITY

OIL

Water-Wet

Oil-Wet

Interstitial water saturation

Usually greater than 20 to 25% PV.

Generally less than 15% PV. Frequently less than 100%.

Saturation at which oil and water relative permeabilities are equal.

Greater than 50% water saturation.

Less than 50% water saturation.

Relative permeability to water at the maximum water saturation (i.e., floodout): based on the effective oil permeability at reservoir interstitial water saturation.

Generally less than 30%

Greater than 50% and approaching 100%.

WATER

60 OIL 40

WATER 20

0 0

20

40

60

80

100

WATER SATURATION, PERCENT P.V.

Relative permeability, % of air permeability

100

100 Water-wet reservoir

Oil-wet reservoir

Oil Oil 60

60

20 S wi 0 0

Water 40 80 Water saturation %

In water-wet system: S w mostly > 20% At point A: kro = k rw ;Sw > 50% k rw at S or / k ro at W wi < 30%

A

20

A

S wi 100

0 0

ter

Wa

40 80 Water saturation %

100

In oil-wet system: S w < 15% At point A: kro = k rw ;Sw < 50% k rw at S or / k ro at S wi> 50%

Influence of wettability on relative permeability; after Fertl, OGJ, 22 May 1978

Figure 64


73

In fact, the key features of such curves were presented by Craig (1971) who indicated the differences between the two in the form of several rules of thumb (see Table in Figure 64). Many subsequent experimental studies have agreed with these ideas (Donaldson and Thomas, 1971; Shankar and Dullien, 1981; inter alia) (Figure 65 ). However, as always, there are a few exceptions that prove the rule(s) — pore geometry and connectivity can change things quite a lot. Nevertheless, the trends are often useful [Influence of wettability on relative permeability; after Fertl, OGJ, 22 May 1978]

10

10 2

.9 3 4 5

RELATIVE PERMEABILITY TO OIL

.8

1.0

108

UP TO 49 o

o

USBN WETTABILITY

1

0

0.649

2

0.02

0.176

3

.2

-0.222

4

2.0

-1.250

5

10.0

-1.333

0.5

7

.6

6

.5

5 5

4

4

.3

o

131

8

.7

.4

DISPLACED PHASES

9

3

3

RELATIVE PERMEABILITY, FRACTION %

CORE PERCENT NO. SILANE

RELATIVE PERMEABILITY TO WATER

1

138 o AND GREATER

DISPLACING PHASES 0.10

0.05

2

.2

2 DISPLACING PHASE

1

.1

1 Nitrogen

5

0 10

0.01

20

30

40

50

60

70

0 80 0

74

q up to 49 o 108 o

Nitrogen

Water

Dioclyl Ether

Nitrogen

131o

Nitrogen

138 and greater

Heptone, Dodecone

WATER SATURATION, PERCENT

DISPLACED PHASE Heptone, Dodecone Dioclyl Ether

0.2 0.4 0.6 0.8 DISPLACED PHASE SATURATION, FRACTION P.V.

1.0

Figure 65

Petrophysical Input

8

OIL RECOVERY, PERCENT OIL-IN PLACE

80 CONTACT ANGLE 0o WATER-WET 47 o 90 oo 138 180 o OIL-WET 0 WOR= 25

70

60

50

40

30

20

10

0

0

0.2

0.4

0.6

0.8

10

OIL SATURATION. % P.V.

WATER INJECTED, PORE VOLUMES 100 70 50 30 20 10 6 3

1

2

5

10 20 50 100 200 500 1000 2000 5000 PORE VOLUMES OF FLOOD WATER

Figure 66

The Effect of Wettability Upon Waterflood Performance Clearly, relative permeability affects waterflooding performance through the fractional flow equation (as does the viscosity ratio) but what type of wettability distribution leads to the optimum recovery? In the relatively sparse literature on non-uniform systems there is much disagreement regarding this question (Figures 66 and 67).


75

OIL RECOVERY AT 24 P.V. THROUGHPUT, % ORIGINAL

70

60

50

OHIO SANDSTONE

70

60

50 WATER-WET

OIL-WET

1.0

0.5

0.0

DISPLACEMENT BY WATER RATIO

1.0

0.5 DISPLACEMENT BY OIL RATIO

RESIDUAL WATER SATURATION, PERCENT P.V.

AMOTT WETTABILITY INDEX

40

20

0

60

80

100

WEIGHT - PERCENT OIL - WET SAND

Figure 67

1.0 krw 1/0 100% WW 75% WW 50% WW 25% WW 0% WW krw1 kro2/25 krw1/25 krw2 kro2

0.8

Kri

0.6 0.4 0.2 0.0 0.0

76

0.2

0.4

0.6 SW

0.8

1.0

Figure 68 Relative permeability curves from mixed-wet systems: a is the oil-wet pore fraction

Petrophysical Input

8

Donaldson et al (1969) and Emery et al (1970) performed waterflood experiments using core plugs which had been aged in crude for varying periods of time. Both studies showed that the more water-wet the rock, the more efficient the displacement. Conversely, Kennedy et al (1955), Amott (1959) and Salathiel (1973) have all shown that the most efficient recovery takes place at close to neutral conditions. The precise wettability details are unclear from study to study however and so some sort of modelling approach is appropriate to understand the issue more fully. In the next section, we therefore go on to examine how network modelling techniques may be applied in this context. 6.4 Network Modelling of Wettability Effects

Introduction The wettability characteristics of a porous medium play a major role in a diverse range of measurements including: capillary pressure data, relative permeability curves, electrical conductivity, waterflood recovery efficiency and residual oil saturation. This section describes the development and implementation of a pore-scale simulator capable of modelling multiphase flow in porous media of nonuniform wettability. This has been achieved by explicitly incorporating pore wettability effects into the steady-state models described earlier. Results are presented which show how α (the fraction of pores which are assigned oilwet characteristics) affects resulting relative permeability curves. These have been used to calculate waterflood displacement efficiencies for a range of wettability conditions, and recovery is shown to be maximum at close to neutral conditions. Moreover, simulated capillary pressure data have demonstrated that standard wettability tests (such as Amott-Harvey and free imbibition) may give spurious results when the sample is fractionally-wet in nature (McDougall and Sorbie, 1993a). Here, attention is restricted to mixed-wet systems. The term “mixed” wettability was first introduced by Salathiel (1973) to describe systems where the oil-wet pores correspond to the largest in the sample, the small pores remaining water-wet. Such situations may arise when oil migrates to water-wet reservoirs and preferentially fills the larger interstices. The wettability characteristics of these pores may then be altered by the adsorption of polar compounds and/or the deposition of organic matter from the original crude, thereby rendering them oil-wet. Fractional wettability, however, is generally related to the rock matrix itself and is due to the differences in surface chemistry of the constituent minerals. Because of these variations, crude oil components may adsorb onto some pore walls whilst ignoring others. This, in effect, means that fractionally-wet rock contains oil-wet pores of all sizes. Attention here, however, will be restricted to mixed-wet systems. Waterflood Simulation Details One of the advantages of using microscopic network simulators is that physical properties can easily be ascribed to each pore individually. Here, the wettability of each pore is controlled so that some are preferentially wetted by water and others by oil; the fraction of pores wetted by oil is denoted by α. The wetting phase contact angle is taken to be zero in both oil-wet and water-wet pores, i.e. pore walls are very strongly wetted by the corresponding wetting phase. In all that follows, the term “cluster” refers to any group of connected pores containing the same phase, whilst a “spanning Institute of Petroleum Engineering, Heriot-Watt University

77

cluster” is a cluster which spans the network — connecting the inlet face of the network to the outlet face. Each simulation begins with the network 100% saturated with oil, and water is then introduced at the inlet face. Each waterflood consists of the following two stages: (1) Water is first allowed to spontaneously imbibe via film flow but only along continuous water-wet pathways which have access to the inlet. Accessible water-wet pores are then assumed to become filled via a “snap-off” mechanism, whereby the smallest pores are filled first followed by the next smallest and so on. The defending oil phase can escape from a pore by draining along a pathway of oil-filled pores which connect it to the outlet. (2) Once spontaneous imbibition has ceased, the invading water is over-pressured (i.e. a negative capillary pressure is applied) and now acts as a nonwetting fluid. The displacement is modelled using an invasion percolation process, and the water next fills the largest oil-wet pores connected to either the inlet face of the network or the invading water cluster. If, at any time during the forced imbibition, water-wet pores are contacted by the invading cluster, then they are filled spontaneously if the defending oil can escape. Throughout forced imbibition, oil may escape in two different ways: either (a) by draining along a pathway of oil-filled pores which connect it to the outlet, or (b) by draining via film flow along a pathway of oil-wet pores to the outlet. Clustering algorithms (following Hoshen and Kopelman, 1976) have been developed which permit the labelling of both oil clusters and water clusters as well as clusters of oil-wet and water-wet pores. Note that the fluid clusters are a dynamic phenomenon, whilst the “wettability clusters” remain static during a given process. The relative permeability curves from mixed-wet networks, computed for a variety of α values, are shown in Figure 68. It is apparent that the oil curve loses curvature and the water curve gains curvature as the oil-wet pore fraction increases. Furthermore, the crossover point does not steadily move towards lower water saturations (as is often supposed). For a between 0 and 0.5, it actually shifts to higher saturations; only when α> 0.5 does it begin to moves back towards lower values. The precise structure of relative permeability curves plays a vital role in determining reservoir performance and efficiency. The results described above show that experiments performed on unrepresentative core samples may yield inaccurate curves and subsequently lead to incorrect field predictions. The precise effect of reservoir wettability on waterflood performance is now examined in more detail. Modelling Waterflood Performance The relative permeability curves described above can now be used in the conventional fractional flow equations enabling the construction of a family of fractional flow curves. Buckley-Leverett analyses can then be carried out to uncover how the microscopic displacement efficiency is expected to be influenced by the wettability of the system. The results from the pore-scale simulations are shown in Figure 69 and 78

Petrophysical Input

8

support the conclusion that optimum recovery occurs at some intermediate wettability state. Indeed, comparisons with a laboratory study on wettability effects (Figure 70) are extremely encouraging: the simple rule-based simulator implemented here reproduces the experimental observations very satisfactorily.

Recovery Efficiency

0.8 20PV

0.7 3PV

0.6 BT

0.5 -20

Figure 69 Recovery efficiency vs % water-wet pores using a mixed-wet simulator

0

20

40

60

80

100

% Water-Wet Pores

Recovery Efficiency

0.8

Figure 70 Experimental observations from Jadhunandan and Morrow (1991)

0.7 0.6 0.5

20PV

3PV

0.4 0.3 -1.0

BT

-0.6

-0.2

0.2

0.6

1.0

Amott-Harvey Wettability Index

Current Research Recent development work in the Institute of Petroleum Engineering at Heriot-Watt University has focussed upon on a new PC-based Visual C++ mixed-wet simulator. The advantages of the Visual C++ approach are twofold: (i) PC-based material is far more accessible to the general user than raw Unix-based research code, and (ii) C++ facilitates the creation of dynamic arrays that can be created “on the heap” and deleted at the end of function calls — this utilises memory far more efficiently, leading to the possibility of producing far larger networks than before. The latest version of the mixed-wet simulator (MixWet) is now fully-functional (Appendix C) and a large number of new features are available. The full cycle of primary drainage — aging — water imbibition — water drainage — oil imbibition — oil drainage are all included. Aging after primary drainage is controlled via the Institute of Petroleum Engineering, Heriot-Watt University

79

“Wettability Parameters” group, which can be used to vary the percentage and size distribution of oil wet pores. Additional checkboxes are available to turn film flow off and on, calculate relative permeabilities, change boundary conditions from unidirectional flooding to intrusion from all sides, and calculate NMR T2 signals automatically. Moreover, the random number seed can be set explicitly by the user in order to examine different realisations of statistically similar networks. A step-by-step description of the new interface is given in Appendix C. 7 CONCLUDING REMARKS We conclude with some final thoughts as to the importance of understanding multiphase flow at the microscopic scale: •

The continuum approach fails to explain a great many observations

•

A lot of the confusion surrounding petrophysics and petrophysical simulator input can be cleared up by considering the associated small-scale physics

•

Remember — all displacements ultimately occur pore-by-pore

•

So, by understanding the controlling physics at the pore-scale, we can look at ways of improving recovery in the future (IFT reduction, depressurisation, gravity drainage, something more novel?)

•

The underlying physics can be complicated and a number of controlling phenomena are intrinsically coupled (wettability, pore structure, capillarity, etc)

•

BUT, if we don’t attempt to understand petrophysics at a fundamental level, then we run the risk of ever increasing uncertainty in our reservoir predictions

80

Petrophysical Input

8

REFERENCES Baker, L. E., 1988, “Three-Phase Relative Permeability Correlations”, SPE17369, Proceedings of the Sixth SPE/DOE Symposium on Enhanced Oil Recovery, Tulsa, OK, April 1988. Bartell, F.E. and Osterhof, H.J., 1927,”“Determination of the Wettability of a Solid by a Liquid”, Ind. Eng. Chem., 19 (11), 1277-1280. Blunt, M.J., 1999,”“An Empirical Model for Three-phase Relative Permeability”, SPE56474, Proceedings of the SPE Annual Technical Conference and Exhibition, Houston, TX, October 1999. Cuiec, L., 1991, “Evaluation of Reservoir Wettability and Its Effects on Oil Recovery”, in Interfacial Phenomena in Oil Recovery, N.R. Morrow, ed., Marcel Dekker, Inc., New York City, 319. Dixit, A.B., McDougall, S.R., Sorbie, K.S. and Buckley, J.S., 1999, “Pore-Scale Modelling of Wettability Effects and their Influence on Oil Recovery,“SPE Res. Eval. and Eng., 2(1), 1-12. Hui, M.-H. and Blunt, M.J., 2000, “Pore-Scale Modeling of Three-Phase Flow and the Effects of Wettability”, SPE59309, Proceedings of the SPE/DOE Improved Oil Recovery Symposium, Tulsa, OK, April 2000. Johnson, R.E. and Dettre, R.H., 1993, “Wetting of Low Energy Surfaces”, in”Wettability, Surfactant Science Series, Volume 49, J. C. Berg (editor), Marcel Dekker, New |York. Kalaydjian, F. J.-M., 1992, “Performance and Analysis of Three-Phase Capillary Pressure Curves for Drainage and Imbibition in Porous Media”, SPE24878, Proceedings of the 67th Annual Technical Conference and Exhibition of the SPE, Washington, DC, October 1992. Landau, L.D. and Lifschitz, E.M., 1958, Statistical Physics, Pergamon, London, 471473. Li, K. and Firoozabadi, A., 2000, “Experimental Study of Wettability Alteration to Preferentially Gas-Wetting in Porous Media and Its Effects”, SPE Res. Eval. and Eng., 3 (2), 139-149. Morrow, N.R., 1990, “Wettability and Its Effects on Oil Recovery”, J. Pet. Tech., 42, 1476-1484. Morrow, N.R. and McCaffery, F., 1978, “Displacement Studies in Uniformly Wetted Porous Media ”, in”Wetting, Spreading and Adhesion, G.F. Padday (ed.), Academic Press, New York City, 289-319.


81

flren, P.E and Pinczewski, W.V., 1995, “Fluid Distributions and Pore-Scale Displacement Mechanisms in Drainage Dominated Three-Phase Flow”, Transport in Porous Media, 20, 105-133. Rowlinson, J.S. and Widom, B., 1989, Molecular Theory of Capillarity, Clarendon Press, Oxford. Stone, H.L., 1970, “Probability Model for Estimating Three-Phase Relative Permeability”, J. Pet. Tech., 20, 214-218. Stone, H.L., 1973, “Estimation of Three-Phase Relative Permeability and Residual Data”, J. Can. Pet. Tech., 12,53-61. van Dijke, M.I.J., Sorbie K.S. and McDougall, S.R., 2000, “A Process-Based Approach for Capillary Pressure and Relative Permeability Relationships in MixedWet and Fractionally-Wet Systems”, SPE59310, Proceedings of the SPE/DOE Improved Oil Recovery Symposium, Tulsa, OK, April 2000 van Dijke, M.I.J. and Sorbie K.S., 2000, “A Probabilistic Model for Three-Phase Relative Permeabilities in Simple Pore Systems of Heterogeneous Wettability”, Proceedings of ECMOR 7, Baveno, Italy, September 2000. van Dijke, M.I.J., Sorbie K.S. and McDougall, S.R., 2001a, “Saturation-Dependencies of Three-Phase Relative Permeabilities in Mixed-Wet and Fractionally-Wet Systems”, Adv. Water Resour., 24, 365-384. Dixit, A. B., McDougall, S. R., Sorbie, K.S. and Buckley, J.S.: “Pore-Scale Modeling of Wettability Effects and their Influence on Oil Recovery”, SPE Reservoir Eval. and Eng., 2 (1), pp. 25-36, February 1999. Fenwick, D.H. and Blunt, M.J.: 1998, “Three-Dimensional Modeling of Three Phase Imbibition and Drainage, Advances in Water Resources, 21, 121-143. Hui, M.-H. and Blunt, M.J.: 2000, “Pore-Scale Modeling of Three-Phase Flow and the Effects of Wettability”, SPE59309, Proceedings of the SPE/DOE Improved Oil Recovery Symposium, Tulsa, OK, April 2000 Lerdahl, T.R., FLren, P.E. and Bakke, S.: “A Predictive Network Model for ThreePhase Flow in Porous Media”, SPE59311, Proceedings of the SPE/DOE Conference on Improved Oil Recovery, Tulsa, OK, April 2000. Mani, V. and Mohanty, K.K.: 1997, “Effect of Spreading Coefficient on Three-Phase Flow in Porous Media”, J. Colloid and Interface Science, 187, 45. Mani, V. and Mohanty, K.K.: 1998, “Pore-Level Network Modeling of Three-Phase Capillary Pressure and Relative Permeability Curves”, SPE Journal, 3, 238-248. Moulu, J.-C., Vizika, O., Egermann, P. and Kalaydjian, F.: 1999 “A New Three-Phase Relative Permeability Model for Various Wettability Conditions”, SPE56477,

82

Petrophysical Input

8

Proceedings of the SPE Annual Technical Conference and Exhibition, Houston, TX, October 1999. McDougall, S.R., Dixit, A.B. and Sorbie, K.S.: 1996,”“The Use of Capillarity Surfaces to Predict Phase Distributions in Mixed-Wet Porous Media”, Proceedings of ECMOR V conference, Loeben, Austria, September 1996. Flren, P.E. and Pinczewski, W.V.: 1995, “Fluid Distributions and Pore-Scale Displacement Mechanisms in Drainage Dominated Three-Phase Flow”, Transport in Porous Media, 20, 105-133. Pereira, G.G., Pinczewski, W.V., Chan, D.Y.C., Paterson, L. and FLren, P.E.: 1996, “Pore-Scale Network Model for Drainage-Dominated Three-Phase Flow in Porous Media”, Transport in Porous Media, 24, 167-201. Pereira, G.G.: 2000, “Numerical Pore-Scale Modelling of Three-Phase Fluid Flow: Comparsion between Simulation and Experiment”, Phys. Rev. E., 59, 4229-4242. WAG Report 6: “Water Alternating Gas (WAG) Injection Studies Progress Report No. 6, Heriot-Watt University, Edinburgh, December 2000. WAGrep5:““Water Alternating Gas (WAG) Injection Studies Progress Report No. 5, Heriot-Watt University, Edinburgh, June 2000. WAGrep6:““Water Alternating Gas (WAG) Injection Studies Progress Report No. 6, Heriot-Watt University, Edinburgh, December 2000. Sohrabi, M., Henderson, G.D., Tehrani, D.H. and Danesh, A.: “Visualisation of Oil Recovery by Water Alternating Gas (WAG) Injection using High Pressure Micromodels - Water-Wet System”, SPE63000, Proceedings of the 2000 SPE Annual Technical Conference, Dallas TX, October 2000. From AWR paper Aleman, M.A. and Slattery, J.C.: 1988, “Estimation of three-phase relative permeabilities”, Transport in Porous Media, 3, 111-131. Aziz, K. and Settari, T.: 1979, Petroleum Reservoir Simulation, Applied Science Publishers, London. Baker, L. E.: “Three-Phase Relative Permeability Correlations”, SPE17369, Proceedings of the 1988 Sixth SPE/DOE Symposium on Enhanced Oil Recovery, Tulsa, OK, April 1988. Baker, L. E.: “Three-Phase Relative Permeability of Water-Wet, Intermediate-Wet and Oil-Wet Sandstone,” Proceedings of the 7th European IOR -Symposium, Moscow, October 1993. Bear, J.: 1972, Dynamics of fluids in porous media, Elsevier, New York, 1972.


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Bradford, S.A., Abriola, L.M. and Leij, F.J.: 1997 “Wettability Effects on Two- and Three-Fluid Relative Permeabilities”, J. Contaminant Hydrology, 28, 171-191. Burdine, N.T.: 1953 “Relative Permeability Calculations from Pore-Size Distribution Data”, Trans AIME, 198, 71-77. Blunt, M.J.: “An Empirical Model for Three-phase Relative Permeability”, SPE56474, Proceedings of the SPE Annual Technical Conference and Exhibition, Houston, TX, October 1999. Corey, A.T., Rathjens, C.H., Henderson, J.H. and Wyllie, M.R.J.: 1956 ”Three-Phase Relative Permeability”, J. Pet. Tech., 8, 3-5; Trans. AIME, 207, 349-351. Cuiec, L.: 1991, “Evaluation of Reservoir Wettability and Its Effects on Oil Recovery”, in Interfacial Phenomena in Oil Recovery, N.R. Morrow, ed., Marcel Dekker, Inc., New York City, 319. Delshad, M. and Pope, G.A.: 1989, “Comparison of Three-Phase Oil Relative Permeability Models”, Transport in Porous Media, 4, 59-83. DiCarlo, D.A., Sahni, A., and Blunt M.J.: “Effect of Wettability on Three-Phase Relative Permeability”, SPE40567, Proceedings of the SPE Annual Technical Conference and Exhibition, New Orleans, September 1998. Dixit, A.B., Buckley, J.S., McDougall, S.R. and Sorbie, K.S.: 2000,”“Empirical Measures of Wettability in Porous Media and the Relationship Between Them”, Transport in Porous Media, in press. Fayers, F.J. and Mathews, J.D.: 1984, ”Evaluation of Normalised Stone’s Methods for Estimating Three-Phase Relative Permeabilities”, SPE Journal, 20, 224-232. Fayers, F.J.: “Extension of Stone’s Method 1 and Conditions for Real Characteristics in Three-Phase Flow”, SPE Reservoir Engineering, 4, 437-445, November 1989. Heiba, A.A., Davis, H.T and Scriven, L.E.: “Effect of Wettability on Two-Phase Relative Permeabilities and Capillary Pressures”, SPE12172, Proceedings of the SPE Annual Technical Conference and Exhibition, San Francisco, October 1983. Heiba, A.A., Davis, H.T and Scriven, L.E.:”“Statistical Network Theory of ThreePhase Relative Permeabilities”, SPE12690, Proceedings of the 4th DOE/SPE Symposium on Enhanced Oil Recovery, Tulsa, OK, April 1984. Hustad, O.S. and Hansen, A.-G.: 1996 “A Consistent Formulation for Three-Phase Relative Permeabilities and Phase Pressures Based on Three Sets of Two-Phase Data”, in RUTH: A Norwegian Research Program on Improved Oil Recovery Program Summary, S.M. Skjaeveland, A. Skauge and L. Hinderacker (Eds.), Norwegian Petroleum Directorate, Stavanger. Jerauld, G.R. and Rathmell, J.J: ”Wettability and Relative Permeability of Prudhoe

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Petrophysical Input

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Bay: A Case Study in Mixed-Wet Reservoirs”, SPE Reservoir Engineering, 12, 58, February 1997. Jerauld, G.R.: “General Three-Phase Relative Permeability Model for Prudhoe Bay“, SPE Reservoir Engineering, 12, 255-263, November 1997. Kalaydjian, F. J.-M.: “Performance and Analysis of Three-Phase Capillary Pressure Curves for Drainage and Imbibition in Porous Media”, SPE24878, Proceedings of the 67th Annual Technical Conference and Exhibition of the SPE, Washington, DC, October 1992. Kalaydjian, F. J.-M., Moulu, J.-C., Vizika, O., and Munkerud, P.K.: “Three-phase Flow in Water-Wet Porous Media: Determination of Gas/Oil Relative Permeabilities Under Various Spreading Conditions”, SPE26671, Proceedings of the 68th Annual Technical Conference and Exhibition of the SPE, Houston, TX, October 1993. Killough, J.E.:”“Reservoir Simulation with History-Dependent Saturation Function”, SPE Journal, February 1976; Trans. AIME, 261, 37-48. Land, C.S.: 1968, “Calculation of Imbibition Relative Permeability in Two- and Three-Phase Flow”, SPE Journal, June 1968; Trans. AIME, 243, 149-156. Larsen, J.A. and Skauge, A.: “Methodology for Numerical Simulation with CycleDependent Relative Permeabilities”, SPE Journal, 3, 163-173, June 1998. Leverett, M.S. and Lewis, W.B.: 1941, ”Steady Flow of Gas-Oil-Water Mixtures Through Unconsolidated Sands”, Trans. AIME, 142, 107. Mani, V. and Mohanty, K.K.: 1997, “Effect of Spreading Coefficient on Three-Phase Flow in Porous Media”, J. Colloid and Interface Science, 187, 45. Morrow, N.R.: 1990, “Wettability and Its Effects on Oil Recovery”, J. Pet. Tech., 42, 1476-1484. Moulu, J.-C., Vizika, O., Kalaydjian, F. and Duquerroix, J.-P.: “A New Model for Three-Phase Relative Permeabilities Based on a Fractal Representation of the Porous Medium”, SPE38891, , Proceedings of the SPE Annual Technical Conference and Exhibition, San Antonio, TX, October 1997. Moulu, J.-C., Vizika, O., Egermann, P. and Kalaydjian, F.: “A New Three-Phase Relative Permeability Model for Various Wettability Conditions”, SPE56477, Proceeding of the SPE Annual Technical Conference and Exhibition, Houston, TX, October 1999. Naar, J. and Wygal, R.J.: 1961, “Three-Phase Imbibition Relative Permeability”, SPE Journal, 1, 254-258, 1961; Trans. AIME, 222, 254-258. Oak, M.J.: “Three-Phase Relative Permeability of Intermediate-Wet Berea Sandstone”, SPE22599, Proceedings of the SPE Annual Technical Meeting and Exhibition, Dallas, TX, October 1991. Institute of Petroleum Engineering, Heriot-Watt University

85

Flren, P.E. and Pinczewski, W.V.: 1995, “Fluid Distributions and Pore-Scale Displacement Mechanisms in Drainage Dominated Three-Phase Flow”, Transport in Porous Media, 20, 105-133. Parker, J.C., Lenhard, R.J. and Kuppusamy, T.: 1987, “A Parametric Model for Constitutive Properties Governing Multiphase Flow in Porous Media”, Water Resources Research, 23, 618-624. Soll, W.E. and Celia, M.A.: 1993 “A modified percolation approach to simulating three-fluid capillary pressure-saturation relationships”, Advances in Water Resources, 16, 107-126. Stone, H.L.: 1970, “Probability Model for Estimating Three-Phase Relative Permeability”, J. Pet. Tech., 20, 214-218. Stone, H.L.: 1973, “Estimation of Three-Phase Relative Permeability and Residual Data”, J. Can. Pet. Tech., 12, 53-61. Temeng, K.O.: “Three-Phase Relative Permeability Model for Arbitrary Wettability Systems”, Proceedings of the 6th European IOR-Symposium, Stavanger, May 1991. Van Dijke, M.I.J., McDougall, S.R. and Sorbie K.S.: 2000a, “Three-Phase Capillary Pressure and Relative Permeability Relationships in Mixed-wet Systems”. Transport in Porous Media, in press. Van Dijke, M.I.J., Sorbie K.S. and McDougall, S.R.: 2000b, “A Process-Based Approach for Three-Phase Capillary Pressure and Relative Permeability Relationships in Mixed-Wet Systems”, SPE59310, Proceedings of the SPE/DOE Symposium on Improved Oil Recovery, Tulsa, OK, April 2000. Vizika, O. and Lombard, J.-M.: 1996, “Wettability and Spreading: Two Key Parameters in Oil Recovery with Three-Phase Gravity Drainage”, SPE Reservoir Engineering, 11, 54-60. Zhou, D. and Blunt, M.: 1997, “Effect of spreading coefficient on the distribution of light non-aqueous phase liquid in the subsurface”, J. Contam. Hydrol. 25,1-1

86

Petrophysical Input

8

8 APPENDIX A: SOME USEFUL DEFINITIONS AND CONCEPTS A1Sphere Packings Sphere packings are very useful for qualitative understanding of pore shape and can also give some insight into porosity measurements. Examples are shown in Figure A.1 — the cubic packing gives the largest porosity (47.6%) whilst the rhombohedral packing gives the smallest (26%). Let us analyse a simple packing in more detail.

Case 1

Figure A1

Case 4

Case 2

Case 5

Case 3

Case 6

A2Specific Surface An important measure characterising the grain surface of a porous medium is known as the specific surface. This plays an important role in the adsorption capacity of a sample and affects a number of petrophysical measures, including electrical resistivity, initial water saturation, and absolute permeability. There are two definitions of specific surface; (i) The surface area of the pores per unit volume of solids (Ss), and; (ii) The surface area of the pores per unit bulk volume (Sv). Mathematical relationships for (i) and (ii) can be derived by noting that the surface area of a sphere is 4πR2. For this idealized system, we find that Ss=3/R and Sv=π/2R — hence, the specific surface of a porous medium is inversely proportional to the (mean) grain size. A3The Pore Size Distribution Photomicrographs show that pore structure is extremely complex and we may ask ourselves whether there is any point in trying to impose conformity by attempting analysis using idealised pore geometries (eg cylinders, etc.). Although it is possible to partially reconstruct real pore geometries using microtomographical techniques, these methods are extremely computer-intensive and we really have no option but to use idealized geometries if we wish to model petrophysical parameters in a costeffective way.


87

In fact, mercury porosimetry analysis still relies upon such models to define so-called pore-size distributions — probability distribution functions defined for the radius range (Rmin, Rmax) that act as fingerprints for different rock samples. However, we need to be careful about what we mean by “pore-size”. Mercury porosimeters assume pores to be circular cylinders and experimental pore size distribution functions are derived under these assumptions (in fact, these distributions are actually volume-weighted throat-size distributions — see notes).— A4 The Coordination Number One measure of the interconnectedness of pore structure is the so-called co-ordination number (z): it is defined as the average number of branches meeting at a point (Figure A2). The co-ordination number plays an important role in determining such things as breakthrough and residual saturations during multiphase displacements, and is probably one of the most important parameters governing flow processes at the porescale. Co-ordination numbers can vary greatly from system to system: from z=6 for a simple cubic sphere packing to z ≈ 2.8 for Berea sandstone (Doyen, 1988; Dullien, 1979). This wide variation should clearly be taken into account when attempting to interpret flow behaviour.

x z y

Z=6

Z=4

Z=2

A5 Surface and Interfacial Tension It is well-documented that the molecules of a liquid are closely bound together by forces of molecular attraction, which serve to keep it as one cohesive assemblage of particles. Although these forces of cohesion act to cancel one another in the interior of the liquid, the situation is somewhat different at the surface: at an air/liquid interface the cohesive forces of the underlying liquid far exceed those of the competing air molecules, resulting in a net inward pull. The system then behaves as if the liquid and air were separated by a uniformly stretched membrane, characterised by a surface tension (σ). If, instead of a liquid/air system, a liquid/liquid system is considered, the tensile force is referred to as interfacial tension and is one of the most important parameters governing multiphase flow in porous media. A6 Wettability, Contact Angle and Spreading Phenomena If a solid surface is contacted by a pair of fluids, one of them will tend to have a greater affinity for that surface than the other. This phase is identified as the wetting phase, whilst the other is known as the nonwetting phase. The wetting preference of a flat solid surface can be quantified by inspecting the contact angle and the associated force balance (Figure A3). This is due to the fact that the magnitude of the contact angle at equilibrium is intrinsically linked to the free surface energies of the system via Young’s equation: 88

Figure A2

Petrophysical Input

8

σ S1 − σ S2 = σ12 cos θ where σS1 represents the solid/fluid 1 surface free energy, σS2 the solid/fluid 2 surface free energy, and σ12 the fluid-fluid interfacial tension. The value of the contact angle may lie anywhere from 0o to 180o, and is strongly dependent upon the fluid pair and surface material involved (Figure A4). If θ=90o, then σs1=σs2 and neither fluid is wetting; the system is then described as neutral. σ12 σs1 Figure A3

σs1 - σ s2= σ12 cos θ

σs2

θ

θ = 158º θ = 35º θ = 30º

θ = 30º

Water

Water Silica Surface

Isooctane

Isooctane + 5.7% Isoquinoline

Isoquinoline θ = 54º

θ = 30º

Naphthenic Acid θ = 106º

θ = 48º Water

Water

Calcite Surface

Figure A4

A7Spreading Phenomena Consideration of the trigonometric term in Young’s equation shows that mechanical equilibrium between two fluids and a solid surface is only possible if:

σ S1 − σ S2 < σ12 i.e. cosθ must always be <1. If this condition is violated, however, the system is unstable and the wetting phase will spontaneously spread on the solid. Although quantification of this “spreadability” in fluid/solid systems is not possible at present (there is no current technique available for measuring either σS1 or σS2), this is not a problem if the solid is replaced by a third fluid or a gas.


89

The spreadability of an oil can be quantified with recourse to a spreading co-efficient, defined by:

So = σ wa − (σ ow + σ oa ) and so initial spreading occurs if this coefficient is either zero or positive. It is evident from the above discussion that the spreading characteristics of oil/water/gas systems must have serious implications for a wide range of hydrocarbon recovery processes. The effect of the spreading coefficient upon three-phase displacements is dealt with more fully in the notes.

A8Capillarity Consider the situation where a liquid is in contact with a glass capillary tube. If the adhesive forces of the liquid to the glass are greater than the cohesive forces in the liquid, then the interface will curve upwards towards the tube, forming a meniscus which intersects the tube wall at an angle θ (Figure A5). The fact that the meniscus is curved, means that there is now a non-zero vertical component of surface tension, which acts to pull liquid up the capillary. This continues, until the vertical component of surface tension is exactly balanced by the weight of fluid below, i.e. when:

πR 2 hρg = 2 πRσ cos θ Total Upward Force = 2πRσWA cosθ

σWA θ

2R

where h is the height of the liquid column, R the capillary radius, and r the density of the fluid. Although the application of this simplistic example to flow in porous media may not be immediately obvious, it should serve to demonstrate how surface and interfacial tension forces can play a crucial role in determining fluid distributions at the pore scale. The full implications of capillary phenomena are apparent in the notes.

90

Figure A5

Petrophysical Input

8

9 APPENDIX B: MATHEMATICAL BACKGROUND AND DERIVATIONS B1Capillary Bundle Permeability This can be derived as follows: Consider the system shown in Figure 1a, which consists of n capillary tubes per unit cross-sectional area. The length of the system is taken to be L. The flow through a single cylindrical capillary of radius R is given by Poiseuille’s law:

q=

πR 4 ∆P 8µL

where µ is the fluid viscosity and ∆P the pressure drop across the tubes. Hence, the total flow (Q) through the porous medium is:

Q = nq =

nπR 4 ∆P 8µL

Now, the porosity (φ) of the medium is nπR2L/(AL)= nπR2 (as cross-sectional area A=1). Hence,

Q=

φR 2 ∆P 8µL

Setting this equal to Q given by Darcy’s Law finally leads to:

R2 D2 k=φ =φ 8 32 An interesting aspect of this simple result is that the quantity (k/φ)1/2 can be thought of as a sort of average pore diameter. B2 Carman-Kozeny Equation The basic premise of this modelling approach is that particle transit times in the actual porous medium and the equivalent tortuous rough conduit must be the same. Particle velocity in a rough conduit (vt) is given by the equation:

vt =

R 2 ∆P 2µL t

where Lt is the conduit length and particle velocity through the porous medium (vr) is given by:


91

vr =

k∆P φµL t

For travel times to be the same in both systems, we require Lt/vt=Lr/vr, and this leads to the relationship:

k=φ

R 2H 2T 2

where T is called the tortuosity of the sample (Lt/Lr). We now need to determine a hydraulic radius. The theory utilises established hydraulic practice, in that the “equivalent” conduit is assumed to have a radius, RH which takes the form:

RH =

Volume of pores φ = Surface area of pores (1 − φ)Ss

and hence we can write:

k=

φ3 2 T 2 (1 − φ)2 Ss2

The permeability can be written In terms of an average grain diameter (Dp) by noting that, for spherical particles, Ss=6/Dp. Hence, Lr

T=

Lt Lr

Lt

vt =

R 2 ∆P 2µL t

ur =

ur = vr φ

vr =

k∆P µLr k∆P µL rφ

For travel times to be equal

L t Lr = vt vr

92

Rφ 2T 2 2

=>

k=

What do we take for R H ? Volume of pores φ RH = = Surface area of pores (1 - φ)Ss Ss= specific surface / solid volume =>

k=

φ3

2(1- φ)2 T 2 Ss2

Figure B1 Carman-Kozeny details

Petrophysical Input

8

D B

P

ρ

ρ

δl

C

A σδl

R2 φ

Figure B2 Generalised Derivation of Capillary Pressure Across a Curved Interface

R1

N

a

b

L

L

r2

q2

r2

qw r1

q1

Figure B3 Pore Doublet Details

r1

Currently accepted values of the percolation thresholds of some two-dimensional networks Network

Z

pcb

Bc = Zpcb

pcs

Honeycomb Square Kagomé Triangular

3 4 4 6

1-2 sin(π/18) ~ 0.6527* 1/2* 0.522 2 sin(π/18) ~ 0.3473*

1.96 2 2.088 2.84

0.6962 0.5927 0.652 1/2*

*Exact result

Currently accepted values of the percolation thresholds of some three-dimensional networks

Table B1 Percolation Thresholds For a Variety of Network Geometries

Network

Z

pcb

Bc = Zpcb

pcs

Diamond Simple cubic BCC FCC

4 6 8 12

0.3886 0.2488 0.1795 0.198

1.55 1.49 1.44 1.43

0.4299 0.3116 0.2464 0.119


93

Figure B4 Purcell Method for Absolute Permeability Prediction

S = S wt

dS/ (Pc ) 2 kwt ∫ s=0 = krwt = S =1 k dS/ (P ) 2

∫

s=0

c

S =1

dS/ (Pc ) 2 ∫ knwt s = s wt = krnwt = S =1 k dS/ (P ) 2

∫

s=0

94

c

Figure B5 Purcell Extension to Relative Permeability Prediction

Petrophysical Input

S = Sw

dS 2 ∫ 2 S = 0 Pc k rw (Sw ) = ( λ w (Sw )) S =1 dS ∫S= 0 Pc2

8

S =1

dS 2 2 S = Sw Pc k rnw (Sw ) = ( λ nw (Sw )) S =1 dS ∫S= 0 Pc2

∫

where the prefactorsare defined by: Figure B6 Burdine Extension to Purcell Model of Relative Permeabilities

λ w ( Sw ) =

Sw − Swi 1 − Swi − Sor

λ nw (Sw ) =


1 − Sw − Sor 1 − Swi − Sor

95

10 APPENDIX C: DETAILS OF THE HERIOT-WATT MIXWET SIMULATOR Description of Interface Controls (1)-(3)

The number of nodes (junctions) in the X- Y- and Z-direction

(4)

Coordination Number (Z): the average number of pore elements meeting at a node

(5)

Distortion Factor used to distort the network, leading to a distribution of pore lengths (under development). 0.0 gives a regular cubic (3D) or square (2D) network. 0.5 is the maximum allowed value

(6)

Fraction of oil-wet pores: this refers to the fraction of the total number of pores in the network that become oil-wet after primary drainage

(6a)

Theta button —used to read in water-wet and oil-wet contact angle ranges

(7)-(8)

The mixed-wet and fractionally-wet radio buttons are mutually exclusive. Mixed-wet assigns oil-wet characteristics to a fraction of the largest pores containing oil after primary drainage. Fractionally-wet assigns oil-wet characteristics to a size-independent fraction of the pores containing oil after primary drainage

(9)

Primary drainage (PD) checkbox: oil displaces water from a 100% waterwet network. This can also be used to examine other generic 2-phase incompressible drainage displacements (e.g. gas-oil drainage). Successively higher (positive) capillary pressures are applied to the system and this drives the displacement

(10)

Water imbibition (WI) checkbox: water imbibes along water-wet pathways in the system and snaps-off in the smallest oil-filled pores. Aging will already have taken place before this part of the cycle. The displacement is controlled by reducing the pressure in the oil phase (i.e. successively lower (positive) capillary pressures are applied to the system

(11)

Water drainage (WD) checkbox: successively higher (negative) capillary pressures are applied to the system and water is forced into successively narrower oil-wet pores

(12)

Oil imbibition (OI) checkbox: oil imbibes along oil-wet pathways, snappingoff the smallest water-filled pores. This is driven by a gradual reduction in water pressure and this drives the oil imbibition

(13)

Oil drive (OD) checkbox: oil pressure is increased once again and oil is forced into water-wet pores

(14)

Random number seed — changing this value produces a new network realisation with the same average properties as others that use the same

96

Petrophysical Input

8

parameter set. In order to get statistically meaningful results, several runs should be performed using different random number seeds and the results averaged (15)

PSD refers to the pore-size distribution exponent (n), where f(r)~rn. n=0 refers to a Uniform distribution, n=3 gives a Cubic distribution, etc. For a Log-Uniform distribution, this parameter should be set to n=–0.999 (not n=–1, as this leads to a singularity). N=10 gives a truncated normal distribution

(16)

Volume exponent (n) — the volume of a pore element is taken to be V(r)~r n. n =2 gives cylinders

(17)

Conductivity exponent (λ) — the conductance of a pore element is taken to be G(r)~r l . l =4 gives cylinders

(18)

Rmin is the minimum capillary entry radius of the sample

(19)

Rmax is the maximum capillary entry radius of the sample

(20)

Graphics Options: these radio buttons allow the user to view different aspects of the simulation

(21)

Calculate Kri — 2-phase relative permeabilities are calculated when this checkbox is checked. An SOR algorithm is used to solve the pressure field and elemental flows can be subsequently calculated

(22)

Mercury? — this checkbox is used to allow invasion of mercury from all sides of the network instead of unidirectional flooding

(23)

Water Films? — if this is checked, then water will leave the system through thin films. This, in effect, leads to no trapping of the water phase during oil invasion in a 100% water-wet network. This option is checked when simulating mercury injection, as the intrusion in this case is essentially mercury-vacuum

(24)

Oil Films? — if this is checked, then oil will leave the system through thin films. This, in effect, leads to no trapping of the oil phase during water imbibition in a 100% oil-wet network

(25)

“NMR Calculations” GroupBox — partial T2 signals are calculated if this is checked. This information is then used to calculate the Accessibility Function (A(R)). Values for “Rho” and”“Mag. Density” should be left unchanged at present as the NMR section is still an area of active research

(26)

RUN — this button is pressed to set the simulation running


97

(27)

Exit—— used to leave the package. At present, an error box appears upon termination: this should simply be cancelled and does not affect performance

The main output file contains information regarding nonwetting phase saturation (Snw) (oil or gas), wetting phase relative permeability (krw) (water or oil), nonwetting phase relative permeability (krnw) (oil or gas), current capillary entry radius (R), number fraction of pores filled with nonwetting phase (p). Ultimately, it should be possible to simply import an experimental capillary pressure data set, interpret this automatically and “instantiate” a network that would serve as a pre-anchored numerical representation of an experimental sample. This could then be used for a wide variety of sensitivity studies.

1 2 3 4 5 6 6a

7 8 20 9 10 11 12 13 14

98

26

15 16

27

17 25 18

21

23

19

22

24

Figure 78 Annotated interface (see text for details of functions)

Glossary of Terms

CONTENTS 1

GLOSSARY OF COMMON TERMS AND CONCEPTS IN RESERVOIR SIMULATION AND FLOW THROUGH POROUS MEDIA 1.1 Some General Definitions 1.2 Reservoir Fluid Properties 1.3 Single Phase Rock Properties 1.4 Multi-Phase Rock/Fluid Properties 1.5 Wettability and Fluid Displacement Processes 1.6 Oil Recovery Methods, Waterflood Patterns and Sweep Efficiency 1.7 Terms Used in Numerical Reservoir Simulation 1.8 Numerical Solution of the Flow Equations in Reservoir Simulation 1.9 Pseudo-Isation and Upscaling 1.10 Numerical Simulation of Flow in Fractured Systems 1.11 Miscellaneous - Vertical Equilibrium, Miscible Displacement and Dispersion

2

Glossary of Terms

1 GLOSSARY OF COMMON TERMS AND CONCEPTS IN RESERVOIR SIMULATION AND FLOW THROUGH POROUS MEDIA This glossary is intended for use by the reader as a quick reference to terms used commonly in reservoir engineering in general and in reservoir simulation in particular. The student is not expected to work through this from begining to end in a systematic manner. However, the students should make sure that he or she is quite familiar with all the technical terms that appear in the main text of this unit. It is hoped that this is of particular use for distance learning students who may have studied the reservoir engineering distance leasrning unit some time ago but hopefully it will also be of use to our residential students.

1.1 Some General Definitions

Oilfield Units volumes in oilfield units are barrels (bbl or B); 1 bbl = 5.615 ft3 or 0.159 m3. A stock tank barrel (STB) is the same volume defined at some surface standard conditions (in the stock tank) which are usually 60oF and 14.7 psi. A reservoir barrel (RB) is the same volume defined at reservoir conditions which can range from ~ 90oF and 1500 psi for shallow reservoirs to > 350oF and 15,000 psi for very deep (high temperature - high pressure, HTHP) reservoirs. Note that when 1RB of oil is produced it gives a volume generally less than 1B at the surface since it loses its gas. (See formation volume factor.) Oil Types: Dry gas; Wet gas; Gas Condensate; Volatile oil; “Black” oil; Heavy (viscous) oil; Tar - see Tables 1 and 2 below.

t o in eP

2500

C1

l bb 80% Bu

De w

Po i

B2 D

2000 40% 20% 1500

u Liq

i

m olu dV

e

10% 5%

A2

A1

0%

B3

of Pr oducti on

B1

1000

Figure 1 Pressure Temperature Phase Diagram of a Reservoir Fluid

Critical Point

Single Phase Gas Reservoirs A

Path

C

B

nt

Reservoir Pressure, PSIA

3000

Tc

= 127 ºF F

3500

Dew Point or Retrograde Gas-Condensate Reservoirs

Path of Reservoir Fluid Flui

Bubble Point or Dissolved Gas Reservoirs

Cricondentherm = 250 ºF

4000

500 0

50

100

150

200

250

300

350

Reservoir Temperature, T ºF


3

Reservoir Fluid

Surface Appearance

GOR Range

API Gravity

Typical Composition, Mole %

C1

C2

C3

C4

C5

C6+

Dry gas

Colourless gas

Almost no liquids

-

100

Wet gas

Colourless gas -

>100 Mscf/bbl some clear or straw-coloured liquid

60o -70o

96

2.7

0.3

0.5

0.1

0.4

Condensate

Colourless gas significant amounts of light coloured liquid

3-100 Mscf/bbl (900-18000 m3/m3)

50o-70o

87

4.4

2.3

1.7

0.8

3.8

"3000 scf/bbl (500 m3/m3)

40o-50o

64

7.5

4.7

4.1

3.0

16.7

“Volatile” or Brown liquid high shrinkage various yellow, red, oil or green hues

“Black” or low shrinkage oil

Dark brown to black viscous liquid

100- 2500 scf/bbl (20 - 450 m3/m3)

30o-40o

49

2.8

1.9

1.6

1.2

43.5

Heavy oil

Black viscous liquid

Almost no gas in solution

10o-25o

20

3.0

2.0

2.0

12.0

71

Tar

Black substance

No gas viscosity > 10,000cp

< 10

-

-

-

-

-

90+

o

Component

Black Oil

Volatile Oil

Gas-Condensate

Dry Gas

Gas

C1 C2 C3 C4 C5 C6 C7+

48.83 2.75 1.93 1.60 1.15 1.59 42.15

64.36 7.52 4.74 4.12 2.97 1.38 14.91

87.07 4.39 2.29 1.74 0.83 0.60 3.80

95.85 2.67 0.34 0.52 0.08 0.12 0.42

86.67 7.77 2.95 1.73 0.88 .... ....

100.00 225 625 34.3 Greenish Black

100.00 181.00 2000 50.1 Medium Orange

100.00 112 18,200 60.8 Light Straw

100.00 157 105,000 54.7 Water White

100.00 .... Inf. ....

Mol. Wt. C7+ GOR, SCF/bbl Tank gravity, 0 API Liquid color

1.2 Reservoir Fluid Properties

Phase: A chemically homogeneous region of fluid which is separated from another phase by an interface e.g. oleic (oil) phase, aqueous phase (mainly water), gas phase, solid phase (rock). There is no particular symbol but frequently subscripted o, w, g; phases are immiscible.

4

Table 1 Describing various oil types from dry gas to tar

Table 2 Mole Composition and Other Properties of Typical Single-Phase Reservoir Fluids

Glossary of Terms

Inter Facial Tension (IFT): The IFT between two phases is a measure of energy required to create a certain area of the interface. Indeed, the IFT is given in dimensions which are energy per unit area. The symbol for IFT is σ and units are dyne/cm in c.g.s. units and N/m (newtons per m) in S.I. units. For example, if both gas and oil are present in a reservoir then the gas/oil IFT may be in the range, σgo ~ 0.1-10 mN/m; likewise. The oil/water value may be in the range, σ0w ~ 15 - 40 mN/m. Note that numerically 1mN/m = 1dyne/cm. Component: A single chemical species that may be present in a phase; e.g. in the aqueous phase there are many components - water (H2O), sodium chloride (NaCl), dissolved oxygen (O2) etc.; in the oil phase there can be hundreds or even thousands of components - hydrocarbons based on C1, C2, C3, etc. Some of these oil components are shown in Table 2. Viscosity: The viscosity of a fluid is a measure of the (frictional) energy dissipated when it is in motion resisting an applied shearing force; dimensions [force/area.time] and units are Pa.s (SI) or poise (metric). The most common unit in oilfield applications is centiPoise (cP or cp). Typical example are:- water viscosity at standard conditions, μw ~ 1 cP; typical light North Sea oils have μo ~ 0.3 - 0.6 cP at reservoir conditions (T ~ 200oF ; P ~ 4000 - 6000 psi); at reservoir conditions, medium viscosity oils have μo ~ 1 - 6 cP; moderately viscous oils have μo ~ 6 - 50 cP; very viscous oils may have μo ~ 50 - 1000s cP and tars may have μo ~ up to 10000 cP. Formation Volume Factor: The factor describing the ratio of volume of a phase (e.g. oil, water) in the “formation” (i.e. reservoir at high temperature and pressure) to that at the surface; symbols Bw, Bo etc. For oil, a typical range for Bo is ~1.1 - 1.3 since, at reservoir conditions, it often contains large amounts of dissolved gas which is released at surface as the pressure drops and the oil shrinks; oilfield units [reservoir barrels/stock tank barrel (RB/STB)]. API Gravity (°API): Definition = Gas Solubility Factors (or Solution Gas/Oil Ratios): These factors describe the volume of gas (usually in standard cubic feet, SCF) per volume of oil (usually stock tank barrel, STB); symbol, Rso and Rsw; units SCF/STB. Compressibility: The compressibility (c) of a fluid (oil, gas, water) or rock formation can be defined in terms of the volume (V) change or density (ρ) change with pressure as follows:

c = −

1  ∂V  1  ∂ρ   =   V  ∂P  ρ  ∂P 

Note that this quantity is normally expressed in units of psi-1. Typical ranges of compressibilities are presented below (from Craft & Hawkins (Terry revision), 1991):


5

Compressibilities (units of 10-6 psi-1) Formation rock Water Undersaturated Oil Gas at 1000psi Gas at 5000psi

3 2 5 900 50

-

10 4 100 1300 200

Compressibilities are used in reservoir engineering for Material Balance Calculations. Material Balance Equations: Material Balance applied to a reservoir is simply a volumetric balance. It may be expressed as an equation which relates: • The quantities of oil, gas and water produced. • The reservoir (average) pressure. • The quantity of water influx (e.g. from the aquifer). • The initial oil and gas content of the reservoir. Essentially the material balance equations described how the energy of expansion and influx “drive” production in the reservoir. If there is a sufficiently low (or zero) fluid influx, the reservoir pressure will decline. One form of the Material Balance Equation is given below where each term on the left-hand side described a mechanism of fluid production (from Craft & Hawkins (Terry revision), 1991):

N .( Bt − Bti ) +

[

 c .S + c f N .m. Bti .( Bg − Bgi ) + (1 + m). N. N . Bti . w wi Bgi  1 − Swi

]

= N p . Bt + ( Rp − Rsoi ). Bg + Bw .Wp Where the terms have the following meaning: N = initial reservoir oil, STB; Np = cumulative produced oil, STB Boi = initial oil formation volume factor, bbl/STB Bo = oil formation volume factor, bbl/STB Bgi = initial gas formation volume factor, bbl/STB Bg = gas formation volume factor, bbl/STB Bw = water formation volume factor, bbl/STB Rsoi = initial solution gas-oil ratio, SCF/STB Rp = cumulative produced gas-oil ratio, SCF/STB Rso = solution gas-oil ratio, SCF/STB We = water influx into the reservoir, bbl Wp = cumulative produced water, bbl cw = water isothermal compressibility, psi-1 cf = formation isothermal compressibility, psi∆ p = change in average reservoir pressure, psi 6

 .∆pp + We 

Glossary of Terms

Swi = initial water saturation m = (Initial hydrocarbon vol. of gas cap)/(Initial hydrocarbon vol. of oil) In practice the material balance equation is often applied in the “linear form” of Havlena and Odeh (J. Pet. Tech., pp896-900, Aug. 1963; ibid, pp815-822, July 1964); see discussion in Craft & Hawkins (Terry revision, 1991). In the above formulation of the Material Balance Equation, the various terms have the following interpretation. Left-Hand Side of the Material Balance Equation • The following terms account for the expansion of any oil and/or gas zones that may be present in the reservoir:

N .(Bt − Bttii ) +

N.m.Bti .(Bg − Bgi ) Bgi

• The following term accounts for the change in void space volume which is the expansion of the formation and the connate water:

 c .S + c f (1 + m). ). N . Bti . w wi  1 − Swi

 .∆pp 

• The next term is the amount of water influx that has occurred into the reservoir: We Right-Hand Side of the Material Balance Equation •

The first term of the RHS represents the production of oil and gas:

[

= N p . Bt + ( Rp − Rsoi ). Bg •

]

The second term of the RHS represents the production of water: Bw.Wp

1.3 Single Phase Rock Properties

Porosity: the fraction of a rock that is pore space; common symbol, φ Porosity varies from φ ≈ 0.25 for a fairly permeable rock down to φ ≈ 0.1 for a very low permeability rock; there may be an approximate correlation between k and φ. Pores & pore throats: The tiny connected passages that exist in permeable rocks; typically of size 1μm to 200 μm; they are easily visible in s.e.m. (scanning electron microscopy). Pores may be lined by diagenetic minerals e.g. clays. The narrower constrictions between pore bodies are referred to as Pore Throats. See Figure 2: Institute of Petroleum Engineering, Heriot-Watt University

7

illite

illite quartz quartz

10mm ~1mm

Permeability: The fluid (or gas) conducting capacity of a rock is known as the permeability; symbol k ; units Darcy (D) or milliDarcy (mD); dimensions -> [L]2. Permeability is found experimentally using Darcy's Law (see below). Permeability can be anisotropic and show tensor properties (see below) - denoted k . Probably the most important quantity from the point of view of the reservoir engineer since its distribution dictates connectivity and fluid flow in a reservoir. Timmerman (p. 83, Vol. 1, Practical Reservoir Engineering, 1982) presents the rule: Classification poor to fair moderate good very good excellent

Permeability Range (mD) 1 - 15 15 - 50 50 - 250 250 - 1000 > 1000

k/φ Correlations: It has been found in many systems that there is a relationship between permeability, k, and porosity, φ. This is not always the case and much scatter can be seen in a k/φ crossplot. Broadly, higher permeability rocks have a higher porosity and some of the relationships reported in the literature are shown below. Some examples of k/φ correlations which have appeared in the literature are shown in Figure 3:

8

Figure 2

Glossary of Terms

Core Core Permeabilty Permeabilty (md) (md)

100.0 100.0 50.0 50.0

10.0 10.0 5.0 5.0

1.0 1.0 0.5 0.5

0.1 0.1 0.05 0.05

Figure 3 Permeability/Porosity Correlation for Cores from the Bradford Sandstone

0.01 0.01

6 6

8 8

10 10

12 14 16 12 14 16 Core Porosity (%) Core Porosity (%)

18 18

20 20

22 22

10,000 10,000

Core Core Permeabilty Permeabilty (md) (md)

1,000 1,000

100 100

10 10

Figure 4 Permeability/Porosity Correlation for Cores from the Brent Field

1 10 0

10 10

20 20 Core Porosity (%) Core Porosity (%)

30 30

Darcy’s Law: Originally a law for single phase flow that relates the total volumetric flow rate (Q) of a fluid through a porous medium to the pressure gradient (∂P/∂x) and the properties of the fluid (μ = viscosity) and the porous medium (k = permeability; A = cross-sectional area): Note that Darcy's law can be used to define permeability using the quantities defined in Figure 5 as follows:


9

 k.A   ∂P  Q = −    µ   ∂x  Note that the β in the equation in Figure 5 is a factor for units conversion (see Chapter 2). Darcy Velocity: This is the velocity, u, calculated as, u = Q/A; this may be expressed as,

u=

 k   ∂P  Q = −    A  µ   ∂x 

Pore Velocity: This is the fluid velocity, v, given by,

v=

Q u = A.φ φ

∆P Q

Q L

Q = β.

k.A  ∆P ∆  .  µ  L

Permeability Anisotropy: Since permeability can be directional, then it is possible for kx ≠ ky ≠ kz in a given system. This is often seen in practice when comparing the horizontal permeability, (kh), with the vertical permeability, (kv) - usually as the ratio, (kv/kh). It is often found (kv/kh) < 1, i.e. there is more resistance to vertical flow than horizontal flow. The value of (kv/kh) must be evaluated with respect to the scale (i.e. the size) of the sample or system. The value of this quantity will be different in a core plug or in a large simulator grid block in which the core plug was a small part. The origin of the anisotropy may be quite different in each case. At the small (core plug) scale, anisotropy may come from fabric anisotropy at the grain level or from lamination at the slightly larger scale (laminaset scale). At the larger scale (grid block), the anisotropy may arise from larger scale heterogeneity, even though locally the component rock facies are completely isotropic. This is illustrated in Figure 6.

10

Figure 5 Schematic of the Single Phase Darcy Law

Glossary of Terms

Core Plug

Small Scale

Fabric Anistropy

kh kv Hi k lamina

Rock Grains

Lo k lamina

Lamination

Grid Block Scale (100s m) kv

kh

Figure 6 Permeability anistropy at different scales

For Whole System

Heterogeneity Anistropy Low Perm Lenses or Shales (kv/kh) = 1 Low Perm Sand (kv/kh) = 1

1.4 Multi-Phase Rock/Fluid Properties

Saturation: The saturation of a phase (oil, water, gas) is the fraction of the pore space that it occupies (not of the total rock + pore space volume); symbols Sw, So and Sg ; saturation is a fraction, where Sw + So + Sg = 1. Multi-phase flow functions such as relative permeability and capillary pressure (see below) depend strongly on the local fluid saturations. Residual Saturation: The residual saturation of a phase is the amount of that phase (fraction pore space) that is trapped or is irreducible; e.g. after many pore volumes of water displace oil from a rock, we reach residual oil saturation, Sor; the corresponding connate (irreducible) water level is Swc (or Swi); the related trapped gas saturation is Srg; at the residual or trapped phase saturation the corresponding relative permeability (see below) of that phase is zero. Strictly, we should refer to the phases in terms of wetting and non-wetting phases - the residual phase of non-wetting phase is trapped in the pores by capillary forces. Typically, in a moderately water wet sandstone, Sor ~ 0.2 - 0.35. The amount of trapped or residual phase depends on the permeability and wettability of the rock and a large amount of industry data is available on this quantity: For example, the relationship for k vs. Swc (or Swi) is shown for a range of reservoir formations, Figure 7.


11

10,000 5,000

1

Air Permeability (mD)

1,000 3

500

2

12 13

100

8

10

50

7

11

10

9

11A

5 4

5

6

1.0 0

10

20

30

40

50

60

70

80

90 100

% Connate water 1 2 3 4 5 6 7 8

= = = = = = = =

Hawkins Magnolia Washington Elk Basin Tangely Creole Synthetic Alundum Lake St John

9

=

10 = 11 = 11A = 12 = 13 =

Louisiana Gulf Coast Miocene Age-Well A Ditto-Wells Band C North Belridge California North Belridge California Core Analysis Data Dominguez Second Zone Ohio Sandstone

Relative Permeability: A quantity (fraction) that describes the amount of impairment to flow of one phase on another. It is defined in the two phase Darcy law (see notes); depends on the Saturation of the phase; e.g. in two phase flow -> krw and kro depend on Sw (since Sw + So = 1). A schematic of the Two Phase Darcy Law showing the definition of Relative Permeability is presented in Figure 8. At steady-state flow conditions, the oil and water flow rates in and out, Qo and Qw, are the same:

12

Figure 7 Correlation between (air) permeability and the connate water (Swc) for a range of reservoir rocks

Glossary of Terms

At steady-state flow conditions, the oil and water flow rates in and out, Qo and Qw are the same ∆Po ∆Pw

Qw Qo

Qw Qo

L

The two-phase Darcy Law is as follows:

Qw =

k.k rw .A . ∆Pw  µw  L 

k.k ro .A . ∆Po  Qo = µo  L 

Figure 8 The two-phase Darcy Law and relative permeability

Scematic of relative permeability, krw and kro

1 kro

Rel. Perm. 0

krw

0

Sw

1

The two-phase Darcy Law is as follows: Where:

Qw and Qo = volumetric flow rates of water and oil Where: = cross-sectional area Qw and QoAL = Volumetric flow rates of water and oil; = system length A = µo Cross-sectional µw and = system lengtharea; k = absolute permeabilities L = System length; the pressure drops across the water and oil phases at o = μw and μo ∆Pw and = ∆PWater and oil viscosities; steady-state flow conditions k = kro Absolute permeabilities; krw and = the water and oil relative permeabilities ΔPw and ΔPo = The pressure drops across the water and oil phases at NB the Units for the two-phase Darcy are exactly the same as those in Figure 2 steady-state flowLaw conditions krw and kro = The water and oil relative permeabilities

Several further examples of relative permeabilities and capillary pressure are given later in this glossary. Note that the Units for the two-phase Darcy Law are given in Figure 2, Chapter 2. Capillary Pressure: The difference in pressure between two (immiscible) phases; defined as the non wetting phase pressure minus the wetting phase pressure; depends on the saturation - for two phases capillary pressure, Pc(Sw) = Po- Pw (for a water wet porous medium). The following figures show schematic figures for Capillary Pressure (Pc(Sw)) and Relative Permeability (krw(Sw) and kro(Sw)) for a water wet system:


13

Capillary Pressure Swc


Sor

Pc

Swc

Sor

krel kro krw

Sw

Sw

Mobility and Mobility Ratio: the mobility of a phase (e.g. λw or λo) is defined as the effective permeability of that phase (e.g. kw = k . krw ; ko = k . kro) divided by the viscosity of that phase;

 k.k rw   k.k ro  λw =  ; λo =    µw   µo 

Mobility ratio, M, is given by:

M=

λ o  k ro .µ w  =  λ w  k rw .µ o 

Fractional Flow: The Fractional Flow of a phase is the volumetric flow rate of the phase under a given pressure gradient, in the presence of another phase. The symbols for water and oil fractional flow are fw and fo and they depend on the phase saturation, Sw:

fw =

Qw Q ; fo = o ;;where Q T = Q o + Q w QT QT

The fractional flows play a central part in Buckley-Leverett (B-L) theory of linear displacement which starts from the conservation equation:

 ∂Sw   ∂f    = − w  ;  ∂t   ∂x 

 ∂So   ∂f    = − o   ∂t   ∂x 

Buckley-Leverett Theory: This mathematical theory of viscous dominated water → oil displacement is based on the fact that the velocity, vSw, of a fixed saturation value Sw is given by:

 ∂f  vSw = v. w   ∂Sw 

14

Figure 9 Schematics of capillary pressure and relative permeability for a water-wet system

Glossary of Terms

where v is the fluid velocity, v = Q/(Aφ)) and (dfw/dSw) is the slope of the fractional flow curve. The relationship between the fractional flow and Buckley Leverett theory is illustrated in Figure 10. Fractional Flow

Welge Tangent

Fractional Flow of Water,

"Buckley-Leverett" Saturation Profile

Sor

fw

Figure 10 Relationship between the fractional flow function and the Buckley-Leverett front height

Sw

Flood Front Height

Sor

Swf Swc

Length, (x/L) Swc

Swf

Water Saturation, Sw

1.5 Wettability and Fluid Displacement Processes

Wettability: This is a measure of the preference of the rock surface to be wetted by a particular phase - aqueous or oleic - or some mixed or intermediate combination. The Wettability of a porous medium determines the form of the relative permeabilities and capillary pressure curves; a very complex subject which is still the subject of much research. We refer to: Water wet, Oil wet and Intermediate wet systems in the following definitions. Water-Wet: Water-wet formation: Where water is the preferential wetting phase. Water occupies the smaller pores and forms a film over all of the rock surface - even in the pores containing oil. A Waterflood in such a system would be an imbibition process (see below). Water would spontaneously imbibe (see below) into a waterwet core containing mobile oil at Sor, hence displacing the oil. Oil-Wet: Oil-wet formation: Where oil is the preferential wetting phase. Oil occupies the smaller pores and forms a film over all of the rock surface - even in the pores containing water. A Waterflood in such a system would be a drainage process (see below). Oil would spontaneously imbibe into an oil-wet core containing mobile water at Swr, displacing the water. Intermediate-Wet: An Intermediate wet formation is where some degree of water wetness and oil wetness is shown by the same rock. Some different types of intermediately wet system have been identified known as Mixed wet and Fractionally wet. Both water and oil may spontaneously imbibe (see below) into such a system to some degree and indeed this forms the basis for certain Wettability Tests known as the Amott Test and the USBM Test (USBM => United States Bureau of Mines). Drainage: A Drainage displacement process is when the non-wetting phase is increasing. For example, in a water wet porous medium, drainage would be oil displacing water. The drainage and imbibition capillary pressure curves and relative permeabilities are different since these petrophysical functions depend on the saturation history. A simple schematic of a drainage process is shown in Figure 11. Institute of Petroleum Engineering, Heriot-Watt University

15

Qo Qo Oil Injection

Water Wet Core at 100% Water Oil Injection Water Wet Core Qo at 100% Water

Figure 11 Drainage

Imbibition: An Imbibition displacement process is when the wetting phase is increasing. For example, in a waterWater wet porous medium, drainage would be water Oil Injection Wet Core displacing oil as shown in Figure 12. at The drainage 100% Waterand imbibition capillary pressure curves and relative permeabilities are different since these petrophysical functions depend on the saturation Q history. w

Water Injection Q

Water Wet Core at sor Water Injection Water Wet Core at sor Qw w

Figure 12

Water Wet Core Spontaneous Imbibition: This process occurs at sorwhen a wetting phase invades a porous Water Injection

Imbibtion

medium in the absence of any external driving force. The wetting fluid is “sucked in” under the influence of the surface forces. For example, if we have a water wet core at irreducible water saturation, Swr, then water may spontaneously imbibe and displace the oil as shown in Figure 13. Oil

The observed behaviour in a system of Intermediate Wettability is shown in Figure 14 Oil where we seeatthat Core s both phases can spontaneously imbibe under certain conditions. wc

Core at swc Core at swc

Water

Water imbibes into core displacing Water imbibes oil-water wet or into core displacing intermediate wet oil-water wet or system intermediate wet Water imbibes into system core displacing oil-water wet or intermediate wet system

Oil

Water Water

Core at swc

Figure 13 Spontaneous Imbibition

Core at sor

Oil

Water

Primary and Secondary Recovery Processes: Primary and Secondary processes refer to the stage in the fluid displacement when one phase displaces another. For example, in a water wet porous medium (--> means displaces) :

16

Drainage and Imbibition Capillary Pressure

Drainage (d) and Imbibition (i) Relative Permeabilities Swc

Sor

Figure 14 Intermediate wettability. Both water and oil may spontaneously imbibe into the core displacing the other phase. Shows both water wet and oil wet character

Glossary of Terms

Primary drainage: is oil --> water from a core at 100% water saturation to Swr. Secondary imbibition: is water --> oil from a core at Swr and mobile oil to Sor. Examples: Figures 15 and 16 show schematics of typical Drainage and Imbibition capillary pressure (Pc) and relative permeability (krw and kro) curves for a water wet system. Primary Drainage (oil --> water from core at 100% water) and Secondary Imbibition (water --> oil from core at Swr) processes are illustrated: Drainage and Imbibition Capillary Pressure

Drainage (d) and Imbibition (i) Relative Permeabilities Swc

Sor

krel

Pc

d

Drainage kro

d

i

Figure 15 Drainage and imbibition capillary pressures

Imbibition

krw

i

Sw

Sw

Figure 16 Drainage and imbibition relative permeability char-

Relative Permeability, %

100 80 60

Drainage

40 Imbibition

20 0

0

20 40 60 80 100 Wetting Phase Saturation, %PV

acteristics

Examples: Further examples of experimental capillary pressures and relative permeabilities in cores are shown for various processes (Drainage and Imbibition) and wettability conditions (Water wet and Intermediate wet) in figure 17,18,19 and 20 on the following pages.


17

80 80 60 60

OilOil

Relative Relative Permeability, Permeability, Fraction Fraction

100 100

40 40 20 20 0 00 0

Water Water 20 40 60 80 20Water 40 60 %PV 80 Saturation, Water Saturation, %PV

100 100

Figure 17 Typical water-oil relative permeability characteristics, strongly water-wet rock

1.0 1.0

0.01 0.01 WaWtearter

Relative Relative Permeability, Permeability, Fraction Fraction

il il O O

0.1 0.1

0.001 0.001

0.0001 0.0001 0 0

20 40 60 80 20 Saturation, 40 60 %PV 80 Water Water Saturation, %PV

100 100

Figure 18 Typical water-oil relative permeability characteristics, strongly water-wet rock

80 60

r Wa te

40 20 0

18

Oil

Relative Permeability, Fraction

100

0

1.0

20 40 60 80 Water Saturation, %PV

O

100

Figure 19 Typical water-oil relative permeability characteristics, strongly oil-wet rock

il

Glossary of Terms

1.0

O

Wate r


il

Figure 20 Typical water-oil relative permeability characteristics, strongly oil-wet rock

0.1

0.01

0.001

0.0001

0


100

Examples: Experimental Capillary Pressures in cores for various processes (Drainage and Imbibition) and wettability conditions (Water wet, Oil Wet and Intermediate Wet). Figure 21.


19

Water Wet Venango core VL-2 k = 28.2 md


40 32 24 16 2

1

8 0 0

40

Capillary Pressure Characteristics, Strongly Water-Wet Rock. Curve 1 - Drainage Curve 2 - Imbibition

24 16

2

20

32

16

24

12 8 4 0

0

20 40 60 80 Water Saturation - %

100

Capillary Pressure Characteristics, (After Ref. 30)

1

8

20

40 60 80 100 Oil Saturation - %

Oil-Water Capillary Pressure Characteristics, Ten-Sleep Sandstone, Oil-Wet Rock (After Ref. 29). Curve 1 - Drainage Curve 2 - Imbibition



32

0 0

100


Oil Wet

48


48

Intermediate Wet

16 1 8

2

0 -8

3

-16 -24

0


100

Oil-Water Capillary Pressure Characteristics, Intermediate Wettability. Curve 1 - Drainage Curve 2 - Spontaneous Imbibition Curve 3 Forced Imbibition

20

Figure 21

Glossary of Terms

Examples: Relative Permeabilitites Examples of experimental relative permeabilities in cores for Water Wet and Oil Wet systems. Figure 22. 100 Relative Permeability, Fraction

40 20 0

Water

0


60

r

40 20 0

100

Typical Water-Oil Relative Permeability Characteristics, Strongly Water-Wet Rock

Oil

60

80

Wa te

80

Oil


100

0


100

Typical Water-Oil Relative Permeability Characteristics, Strongly Oil-Wet Rock

1.0

1.0

O il

0.001

0.0001

Figure 22

0.1

Wate r


0.01 Water


il O

0.1

0.01

0.001

0.0001 0


100

Typical Water-Oil Relative Permeability Characteristics, Strongly Water-Wet Rock


0


100

Typical Water-Oil Relative Permeability Characteristics, Strongly Oil-Wet Rock

21

EXAMPLES: Relative Permeabilitites

A simple table summarising the typical characteristics of water-wet and oil-wet relative permeabilities is given below. WATER WET

OIL WET

Swc

mostly > 20%

< 15% (Often < 10%)

Sw where krw = kro (Points A on Figure 23)

Sw > 50%

Sw < 50%

krw at Sro (Points B on Figure 23)

krw < 0.3

krw > 0.5 (approaching 1)

This is shown schematically in Figure 23: 100

100 Water-Wet Reservoir

80

Oil-Wet Reservoir

80

Relative Permeability, % of Air Permeability

Oil

Oil

60 40

40

20

20

0

Swi

0

20

B

60

r Wate

A

B

40 60 80 Water Saturation, %

100

0

A Swi 0

Wa

20

te

r

40 60 80 Water Saturation, %

In Water-Wet System Sw mostly > 20%

In Oil-Wet System Sw < 15%

At Point A: kro = krw ; Sw > 50% krw at Sor / kro at Swi < 30%

At Point A: kro = krw ; Sw < 50% krw at Sor / kro at Swi > 50%

100

1.6 Oil Recovery Methods, Waterflood Patterns and Sweep Efficiency Here we refer to the method used to develop the reservoir as follows:

• Primary Depletion - allowing the reservoir to produce under the original reservoir energy i.e. by natural expansion. If the pressure falls below the bubble point (Pb), then gas appears and the primary depletion process is known as solution gas drive: • Secondary Recovery - where reservoir pressure is supported by injection, usually of water in waterflooding but early gas injection may be considered also as secondary recovery In addition to supporting the pressure (maintaining reservoir energy), water or gas injection also displaces oil directly:

22

Figure 23 Influence of wettability on relative permeability (after Fertl, OGJ, 22 May 1978)

Glossary of Terms

• Tertiary Recovery or Enhanced Oil Recovery (EOR) or Improved Oil Recovery (IOR) - this refers to a range of methods which are designed to recover additional oil that would not be recovered by either Primary or Secondary Recovery. Such methods include: Thermal Methods Gas Injection Chemical Methods Microbial Methods

- steam, in-situ combustion,.. - N2, CO2, hydrocarbon gas injection (usually after a waterflood) - surfactant, polymer, alkali,.. - using bugs to recover oil!

Waterflood Pattern: On-land Waterflooding is often carried out with the producers and injectors in a particular pattern. This is known as pattern flooding and examples of such patterns are: Five Spot, Nine Spot, Line Drive etc. as shown schematically in Figure 24. Injection Well Production Well Pattern Boundary

Regular Four-Spot

Five-Spot

Figure 24 Examples of areal patterns of injectors and producers (pattern flooding)

Normal Nine-Spot

Skewed Four-Spot

Seven-Spot

Direct Line Drive

Inverted Nine-Spot

Inverted Seven-Spot

Staggered Line Drive

Areal Sweep Efficiency: The Areal Sweep Efficiency refers to the fraction of areal reservoir that is swept at a given pore volume throughput of displacing fluid as shown schematically in Figure 25. For example, the Areal Sweep Efficiency at Breakthrough for various processes (Waterflooding, Gas Displacement and Miscible flooding) is shown as a function of mobility ratio in Figure 26: Institute of Petroleum Engineering, Heriot-Watt University

23

High Areal Sweep

High Areal Sweep

Figure 25 Schematic of areal sweep efficiency

Poor Areal Sweep

Poor Areal Sweep

Breakthrough Areal Sweep Efficiency, %

100

Breakthrough Areal Sweep Efficiency, %

100 90

90 80 70

80 70

60 50 0.1

60

Water→Oil Gas→Oil Miscible

50 0.1 Water→Oil

1.0 Mobility Ratio

1.0 Mobility Ratio

Figure 26 Areal sweep efficiency at breakthrough in a five spot pattern for various displacement processes

10.0

10.0

Gas→Oil Vertical Sweep Effi ciency: The Vertical Sweep Efficiency refers to the fraction of vertical section (orMiscible cross-section) of reservoir that is swept at a given pore volume throughput of displacing fluid. This is function of the heterogeneity of the system (e.g. stratification), the fluid displacement process (e.g. waterflooding, gas injection) and the balance of forces (e.g. importance of gravity). It is shown schematically in Figure 27.

24

Glossary of Terms

Good Vertical Sweep

Figure 27 Schematic showing vertical sweep efficiency

Poor Vertical Sweep (By gravity over-ride or the presence of a high-k streak in this case)

1.7 Terms Used in Numerical Reservoir Simulation

Mass Conservation: This is a general principle which is used in many areas of computational fluid dynamics. It says that: (mass flow rate into a block) - (mass flow rate out) = (the rate of mass accumulation in that block) A reservoir simulation model (for 1, 2 or 3 phases) is basically: A mass conservation equation combined with Darcy’s law. Black Oil Model: Different types of formulation of the transport equations for multiphase/multicomponent flow are used to simulate the various recovery processes; by far the most common is the “Black Oil Model” which can simulate primary depletion and most secondary recovery processes. A black oil simulation model is one of the most common approaches to modelling immiscible two and three phase (o, w, g) flow processes in porous media; it treats the phases rather like components; it does not model full compositional effects; instead, it allows the gas to dissolve in the other two phases (described by Rso and Rsw); however, no “oil” is allowed to enter the gas phase. Grid Structure: This refers to the geometry of the grid being used in the numerical simulation of the system. This grid may be Cartesian, radial or distorted and may be 1D, 2D or 3D (see notes). Spatial Discretisation: This is the process of dividing the grid in space into divisions of Δx, Δy and Δz. In reservoir simulation, we always “chop up” the reservoir into blocks as shown in the gridded examples below and then we model the block → block flows.


25

Temporal Discretisation: This is the process of dividing up the time steps into divisions of Δt. 2D Areal Grid: This is a 2D grid structure as shown in Figure 28 which is imposed looking down onto the reservoir. For a Cartesian system, it would divide up the x and y directions in the reservoir into increments of Δx and Δy. y

W1

W2

W3

x

∆z

∆x ∆y Just 1 x z-block in 2D Areal Grid

Figure 28 Perspective view of a 2D areal (x/y) reservoir simulation grid: W = well

2D Cross-Sectional Model: This is a 2D grid structure which is imposed on a vertical slice down through the reservoir. For a Cartesian system, it would divide up the x and z directions in the reservoir into increments of Δx and Δz. Cross-sectional calculations are carried out to asses the effects of vertical stratification in the system and to generate pseudo-function for upscaling. (Figure 29).

Figure 29

3D Cartesian Grid The 3D Cartesian Grid is the most commonly used grid when constructing a relatively simple model of a reservoir or a setion of a reservoir. This is shown in Figure 30.

26

Glossary of Terms

Producer

∆y

Water Injector

Figure 30 A 3D Cartesian grid for reservoir simulation

∆x

∆z (Variable)

Transmissibility: The transmissibility between two grid blocks is a measure of how easily fluids flow between them. The mathematical expression for two phase flow between grid blocks i and (i+1) is (for water): (i+1/2) Boundary

Block i Sw i

Block i+1 Qw

krw i (kA)i

Figure 31

Sw i+1 krw i+1 (kA) i+1

 k  (P − Pi ) Q w = ( kA kA )i +1/ 2  rw  . i +1 ∆x  µ w Bw  i +1/ 2 where the inter-grid block quantities are averages at the interfaces (where i+1/2 denotes this block to block interface. The single phase Transmissibility,, Ti+1/2 , is given by:

Ti +1/ 2 =

( kA)i +1/ 2 ∆x

and the full Water Transmissibility,, Tw,i+1/2, between the two grid blocks is given by:

Tw, i +1/ 2 =

( kA)i +1/ 2  k rw  ∆x

 k  = Ti +1/ 2 . rw     µ w Bw  i +1/ 2  µ w Bw  i +1/ 2


27

Therefore, we may write:

Qw = Tw, i +1/ 2 ( Pi +1 − Pi ) The water transmissibility is clearly made up of two parts each of which is an average between the blocks. The single phase part is (k.A)av and the two phase part is [krw/(μw.Bw)]av (k.A)av - a Harmonic Average between blocks is taken for the single phase part of the transmissibility (see Chapter 4; Section 3.2). [krw/(μw.Bw)]av - this term is more complicated. For the average relative permeability term, [krw]av an Upstream Weighting is used; For [(μw.Bw)]av the Arithmetic Average between blocks is taken. (See Chapter 4; Section 3.3). Numerical Dispersion: The spreading of a flood front in a displacement process such as waterflooding, which is due to numerical effects, is known as Numerical Dispersion. It is due to both the spatial (Δx) and time (Δt) discretisation or truncation error that arises from the gridding. This spreading of flood fronts tends to lead to early breakthrough and other errors in recovery. How bad the error is depends on several factors including the actual fluid recovery process being simulated e.g. waterflooding, water-alternating-gas (WAG) etc. (See Chapter 4; Section 2.2). Grid Orientation: The Grid Orientation problem arises when we have fluid flow both oriented with the principal grid direction and diagonally across this grid as shown schematically in Figure 32. Numerical results are different for each of the fluid “paths” through the grid structure. This problem arises mainly due to the use of 5-point difference schemes (in 2D) in the Spatial Discretisation. It may be alleviated by using more sophisticated numerical schemes such as 9-point schemes (in 2D).

P

I

P

I = Injector P = Producer

Local Grid Refinement: Local Grid Refinement is when the simulation grid is made fine in a region of the reservoir where (LGR) quantities (such as pressure or saturations) are changing rapidly. The idea is to increase the accuracy of the simulation in the region where it matters, rather than everywhere in the reservoir. E.g. LGR ==> close to wells or in the flood pilot area but coarser grid in the aquifer. 28

Figure 32 Flow arrows show the fluid paths in oriented grid and diagonal flow leading to grid orientation errors

Glossary of Terms

Hybrid Grid LGR: Hybrid Grids are mixed geometry combinations of grids which are used to improve the modelling of flows in different regions. The most common use of hybrid grids are Cartesian/Radial combinations where the radial grid is used near a well. Hybrid Grid LGR can be used in a similar way to other LGR scheme. Examples: A simple example of LGR and Hybrid Grid structure is shown in figure 33. Injector

Coarse Grid in Aquifer

Producer

Figure 33 Schematic of local grid refinement (LGR)

Hybrid Grid

Distorted Grids: A Distorted Grid is a grid structure that is “bent” to more closely follow the flow lines or the system geometry in a particular case. Corner Point Geometry: In some simulators (e.g. Eclipse), the option exists to enter the geometry of the vertices of the grid blocks. This allows the user to define complex geometries which better match the system shape. This option is known as Corner Point Geometry and it requires that the block → block transmissibilities are modified accordingly. The idea of Corner Point Geometry is illustrated schematically in figure 34:


29

Corner Point Geometry Coordinates of Vertices ( ) Specified Block ( ) centres

Highly Distorted Grid Blocks

Block <-> Block Transmissibilty

Fault L1 L2 L3

L1 L2 L3 L4

L4

Distorted Grid

30

Figure 34 Grid structures for Faults and Distorted Grids

Glossary of Terms

1 Extented Refinement

2 Imbedded Refinement (Rectangular)

3 Variable Refinement (Radial)

4 Hybrid (Radial in Rectangular) Local Grid Refinement Around Vertical Wells

5 Hybrid Local Grid Refinement (Horizontal Wells)

Figure 35 Types of local Grids Institute of Petroleum Engineering, Heriot-Watt University

31

History Matching: History Matching in numerical simulation is the process of adjusting the simulator input in such a way as to achieve a better fit to the actual reservoir performance. Ideally, the changes in the simulation model should most closely reflect change in the knowledge of the field geology e.g. the permeability of a high perm streak, the presence of sealing faults etc. The observables which are commonly matched are the field and individual well cumulative productions, watercuts and pressures. Examples: Examples of history matching of pressure and water-oil-ratio (WOR) in two reservoirs are Figure 36. Note that in the WOR match the “first pass” was very inaccurate but that eventually a suitable match was found. A vitally important point is that a good history match must be obtained for the right reason. It may be possible to get a satisfactory match for the wrong reason i.e. by adjusting a variable that is not the primary cause of the mis-match (indeed, this is very often the case). However, such a model will eventually have very poor predictive properties.

(a)

1.0

3600 3400

0.8 Final Match Pressure (psi)

3000

Watercut

0.6

0.4

3

5 7 Time, Days x 10

2600 2400 2200 0

9

1

2

3

4

5

6 7 Year

8

9 10 11 12

(b)

Observed Data 1

2800

2000

First Trial

0.2

0.0 0

Calculated Field Data

3200

3600 11

3400


3200 Pressure (psi)

3000

Initial and Final Matches of WOR of a Well in a Highly Stratified Reservoir

2800 2600 2400 2200 2000 0

1

2

3

4

5

6 7 Year

8

9 10 11 12

(c) 3600 3400


3200 Pressure (psi)

3000 2800 2600 2400 2200 2000 0

1

2

3

4

5

6 7 Year

8

9 10 11 12

Final Pressure Matches of Typical Khursaniyah Field Wells: (a) Reservoir AB (b) Reservoir C (c) Reservoir D (MP)

1.8 Numerical Solution of the Flow Equations in Reservoir Simulation

Finite Differences: When the derivative in a differential equation is approximated as a difference equation as follows: 32

Figure 36 A field of a history match of watercut and well pressures; redraw from Mattax and Dalton (1990)

Glossary of Terms

 ∂S  (Si − Si −1 )   ≈  ∂x  i ∆x then this is referred to as a Finite Difference approximation. In this example, which is illustrated below, (∂S/∂x)i is the derivative of Saturation (S) with respect to x at grid point i ; Si and Si-1 are the discrete values of S at grid points i and i-1, respectively; Δx is the size of the spatial grid. Si-1

Slope = ∂s ∂x

i

S(x,t)

Si Si+1 ∆x

∆x

∆xx ....

i-1

i

i+1

x

Figure 37

Linear Equations: When finite difference methods are applied to the differential equations of reservoir simulation, a set of linear equations results. These have the form:

A. x = b where A is a matrix of coefficients, x is the vector of unknowns and b is the (known) “right hand side”. Expanded up, this set of linear equations has the form: a11 x1 a21 x1 a31 x1 a41 x1

+ + + +

a12 x2 a22 x2 a32 x2 a42 x2

+ + + +

a13 x3 .... a23 x3 .... a33 x3 .... a43 x3 ....

............ an1 x1 + an2 x2 + an3 x3 ....

+ a1n xn = + a2n xn = + a3n xn = + a4n xn =

b1 b2 b3 b4

+ ann xn = bn

Direct Solution Of Linear Equations: A Direct Solution method is when the linear equations are solved by an algorithm which has a fixed number of operations (given N, the number of linear equations [unknowns]). If the equations have a solution, then, in principle, a direct method will give the exact answer, x(true), to the machine accuracy. E.g. Gaussian Elimination Iterative Solution Of Linear Equations: An Iterative Solution method is when the linear equations are solved by an algorithm which has a variable number of operations. A first estimate of the solution vector x(0) is made and this is successively refined to converge to the true solution. In a convergent iterative method, then x(v) → x(true) as v → ∞. It is because of this iterative process that a variable number of steps may


33

be required depending on how accurate the answer must be. Normally, the iterative method would be carried out until: | x(v) - x(true)| < Tol. where Tol. is some small pre-specified tolerance. E.g. Line Successive Over-relaxation (LSOR) Grid Ordering Schemes: When the simulation grid blocks are ordered in various ways, the structure of the non-zeros in the sparse matrix, A, is different. Advantage can be taken of the precise structure when solving these equations. E.g. Schemes known as D2 and D4 ordering.

1.9 Pseudo-Isation and Upscaling

Upscaling: The process of reproducing the results of a calculation which is carried out on a “fine grid” on a coarser grid is known as Upscaling. The basic idea of upscaling is shown schematically in Figure 38. The input properties at the coarser scale must take into account the flow effects of the smaller scale structure. These coarser scale properties then become pseudo-properties.

34

Glossary of Terms

Fine Grid Layered Model High Sw

Low Sw Oil

Upscaling or Pseudo-Isation

Coarse Grid Layered Model

Fine Grid Coarse Grid

% OOIP

Oil Recovery

Figure 38 Basic idea of upscaling

Time

Pseudo-Property: This refers to the value of a property or function (e.g. permeability, relative permeability..) which is an average or effective value at a certain scale usually the grid block scale. For example, we might put the value kx = 150 mD in a simulator grid block which is 200 ft x 200 ft x 30 ft. Clearly, this incorporates a large amount of geological substructure and permeability may vary very significantly in different parts of this block. Pseudo-Relative Permeability: This is probably the most important pseudo-property that is used in reservoir simulation. It refers to the effective relative permeability in the simulation model at the grid block scale and is shown schematically in Figures 39 and 40. It must incorporate the effects of all the smaller scale geological heterogeneity, the balance of forces (viscous/capillary/gravity) and certain numerical effects (numerical dispersion). Methods for calculating the Pseudo-rel perm include: Jacks et al, Kyte and Berry, Stone etc. Newer methods are based on tensor pseudorelative permeabilities (Pickup and Sorbie, 1994). Institute of Petroleum Engineering, Heriot-Watt University

35

Geopseudos: When the fluid flow upscaling is performed in the correct context of the sedimentary structure up from the lamina, laminaset, bedform.. scales, then the approach is known as the Geopseudo Methodology. This has been developed in work at Heriot-Watt which has extended more conventional approaches by “putting in the geology”. Figure 40 shows a simple example of a pseudo relative permeability showing “holdup of fluid”.

Fine Grid Layered Model High Sw

"Rock" Relative Permeabilities

krel Low Sw

Sw

Oil

Upscaling or Pseudo-Isation

Pseudo-Relative Permeabilities

krel Sw Coarse Grid Layered Model

Fine Grid Coarse Grid

% OOIP

Oil Recovery

Time

36

Figure 39 Basic idea of upscaling or pseudo-isation

Glossary of Terms

Water Flow

Rel Perms. Sor

Sw Swc

Sw Rel Perms. ?

No Water Flow Figure 40 A simple example of a pseudo relative permeability

krw

kro

Sor Sw Swc

1.10 Numerical Simulation of Flow in Fractured Systems

Fractured System: In this context, we imply systems (such as in many carbonate reservoirs) where small scale conductive fractures occur but most of the oil is in the rock matrix. In certain non-porous fractured rock reservoirs (e.g. fractured volcanics), it is possible to have all the oil in the fractures but these are much less common. Typically, in porous fractured systems: fracture porosity, φf = 0.1 - 1% of bulk volume (i.e. as a fraction φf = 0.001 - 0.01). Features of Fracture Geometry: The main geometric features of fractures which are thought to affect fluid flow are the: fracture orientation, width, conductivity, connectivity and spacing (or fracture density). The “interface” between the fracture and the matrix will also play a very important role in multiphase flow and fluid displacement processes. Stylolites: Stylolites are frequently found in limestones. They are laterally extensive features formed by grain-to-grain sutured contact caused by the phenomenon of pressure solution. These features may significantly reduce vertical permeability thus causing systems containing them to have very low (kv/kh) ratios (at certain scales). Vugs: Vugs are dissolution “holes” in a carbonate rock caused by diagenetic reactions. Dual Porosity Models: These are the most widely used simulation models for modelling flow in fractured systems. They have separate conservation/flow equations for the matrix and the fractures and matrix → fracture flow is represented by Transfer Functions. They are most frequently used to model multiphase flow in fractured carbonates. Variants of this model allow for; (i) flow only in fractures and (ii) flow in both fractures and matrix. Discrete Models of Fracture Systems: A more recent approach to flow in fractured systems tries to represent the fractures explicitly as oriented planes with various shapes in 3D. Single phase (tracer) flow models of this type are used to model radioInstitute of Petroleum Engineering, Heriot-Watt University

37

nuclide transport in fractured media. However, multi-phase models of this type are not commercially available at present. Transfer Function: The function which describes the oil flow rate between the matrix and the fractures is known as the Transfer Function. Approximate analytical equations for this function have been suggested by Birk, Boxerman and Ahronovsky. Sudation: When oil is recovered from the matrix blocks in a fracture by a combination of gravity and capillary forces, the recovery mechanism is sometimes referred to as Sudation.

1.11 Miscellaneous-Vertical Equilibrium, Miscible Displacement and Dispersion

Vertical Equilibrium: The concept of Vertical Equilibrium (VE) is quite widely used in reservoir engineering. It takes several forms, two of which are listed below (and illustrated schematically in Figure 41: A: the pressure gradients in a particular direction (x, say), ∂P/∂x, are all equal locally in a long system. See Figure 41a. B: there is virtually instant crossflow vertically - nearly infinite - compared with the horizontal flow. See Figure 41b. The VE assumption is often made in order to simplify the mathematical analysis of certain fluid flow problems in reservoir engineering. Vertical Equilibrium (VE) is known to apply in “long thin” systems (where Δx >> Δz, in Figure 41b). More accurately, the VE limit is approached as the scaling group, RL → ∞ ; where:

∆x  k  R L =   . z   ∆z   k x 

1/ 2

and kz and kx are the vertical and horizontal permeabilities, respectively. In practice, if RL is > 10, then VE is a very good assumption.

38

Glossary of Terms

A: Equality of Layer Pressure Gradients (Where ∆x >> ∆z) Layer 1 Layer 2

∆z

Layer 3 ∆x

∂P ∂x

1

∂P ∂x

2

∂P ∂x

3

B: Instantaneous/Infinite Vertical Crossflow

∆z

Figure 41 Schematic views of vertical equilibrium

∆x

Miscible Displacement: Whereas oil and water are immiscible fluids (i.e. they do not mix and are separated by an interface), some fluids are fully Miscible (i.e. they mix freely in all proportions). When a gas (or other fluid) is injected into an oil reservoir and the fluids are miscible, this is referred to as a Miscible Displacement. When two fluids (e.g. gas and oil) are fully miscible (σgo = 0), the local pressure and the pressure gradients are the same (there is no capillary pressure since there is no interface). The mixing between the solvent and the oil can occur locally by Dispersion and by Fingering (see Viscous Fingering below). The displacement is described by a generalised Convection-Dispersion Equation where the mixing viscosity, μ(c) is a function of the concentration of the solvent, c (or oil). Often, the solvent viscosity is below that of the oil (i.e. μs < μo) which tend to cause an instability to develop in the displacement known as Viscous Fingering. The Miscible Flow Equations: These comprise of a Pressure Equation and a Transport Equation. The pressure equation is derived by inserting Darcy’s Law (with a viscosity dependent on solvent concentration) into the continuity equation. The transport equation is a generalised convection-dispersion (or convection-diffusion) equation. Continuity equation:

∇.u = 0 (assuming incompressible flow) Darcy Equation:

u=−

k .∇P ∇P µ( c)

where u is the Darcy velocity, c is the miscible solvent concentration and k is the Institute of Petroleum Engineering, Heriot-Watt University

39

permeability tensor. The pressure equation is then given by:

 k  ∇.u = −∇. .∇ ∇P P = q˜  µ (c )  or  k  ∇. .∇ ∇P P = − q˜  µ (c )  where q˜ represent any source/sink terms The transport equation is the generalised convection-dispersion (diffusion) equation:

 ∂c  ∇cc   = ∇.(D.∇c) − v.∇  ∂t  where D is the dispersion tensor and v is the pore velocity ( v =

u ). φ

Dispersion and Dispersivity: Hydrodynamic dispersion in a porous medium at the small (core) scale is a frontal spreading or mixing which is due to various flow paths which the fluid can flow along at the pore scale. This mixing is a “diffusive” process since the growth of the mixing zone, Lf , tends to grow in proportion to t . In a tracer core flood experiment, the Dispersion Coefficient, D, may be measured by fitting the effluent profile to an analytical solution of the Convection-Dispersion Equation (see figure 42). Units of D (cm2/s - at lab scale). Dispersive mixing behaviour can also be seen from the “mixing” effect of heterogeneities at larger scales in a porous medium. D, has been found to depend linearly on velocity through the relationship: D = α . v; where α is the Dispersivity and has dimensions of length. Dispersion is actually a tensor in rather the same way that permeability ( k ) in its general form is a tensor.

40

Glossary of Terms

Dispersive Mixing Zone C(x,t1)

1

Lf

Ce 1

Effluent Concentration Ce(1,t)

C 0 Figure 42 Schematic illustration of dispersion and the convection-dispersion equation for simple tracer flow

x

1

1 time (pv)

In situ concentration profile at time, t1; C(x,t1) 2

Described by Convection Dispersion Equation: ∂C = D ∂ C2 - v ∂C ∂t ∂x ∂x C = Dimensionless Concentration (C/Co) D = Dispersion Coefficient v = Pore Velocity (Q/Aφ)

Viscous Fingering: When a high mobility (lower viscosity) fluid displaces a lower mobility (higher viscosity) fluid, a type of instability may develop known as Viscous Fingering. For such a displacement, the Mobility Ratio (see above) is high and the process may be observed in either miscible or immiscible displacements, although it occurs more readily in miscible systems. Results from an experiment are shown in Figure 43 where the dark fluid is high viscosity and the light fluid is low viscosity. Clearly, viscous fingering leads to a poorer sweep efficiency in such floods.

Figure 43 Experimental demonstration of viscous fingering


41

PAPER NO. 28902B

HERIOT WATT UNIVERSITY DEPARTMENT OF PETROLEUM ENGINEERING Examination for the Degree of Meng in Petroleum Engineering Reservoir Simulation Monday 17th April 1995

09.30

- 12.30

( 70% of Total marks )

NOTES FOR CANDIDATES (1)

Answer ALL the questions and try to confine your answer to the space provided on the paper.

(2)

The amount of space and the relative mark for the question will give you some idea of the detail that is required in your answer.

(3)

If you need more space in order to answer a question then continue on the back of the same page indicating clearly (by PTO) that you have done so.

(4)

The total number of marks in this examination is 262; this will be rescaled to give an appropriate weighted percentage for the exam. The marks are relative and, together with the space available, should give an approximate guide to the level of detail required.

(5)

There is a compulsory 15 minute reading time on this paper during which you must not write anything.

(6)

You will be allocated 3 hours to complete this paper.

1

Q1.

List two uses of numerical reservoir simulation for each of the following stages of field development.

(i)

At the appraisal/early field development stage 1. (2) 2. (2)

(ii)

At a stage well beyond the maximum oil production in a large North Sea field: 1. (2) 2.

(2)

Q2.

At any stage in a reservoir development by waterflooding, the engineer may use material balance calculations and/or numerical reservoir simulation. Under which particular cir cumstances would you use each of these approaches? (i) Material Balance?

(3)

(ii) Reservoir simulation?

(3)

Q3. Given all the problems and inaccuracies, which are known to exist in the application of reservoir simulation, why do engineers still use it?

(2) 2

Q4 Water injector Water injector

Producer

High k

Continuous or Discontinuous Shales ???

Low k

OWC

2000 m

Vertical scale 100 m

Reservoir X is a light oil reservoir (35º API) being developed by waterflooding. The reservoir comprises a high average permeability massive sand overlying lower average permeability laminated sands. The thickness of each sand is approximately equal and there are shales at the interface of these two sands. However, the operator is uncertain if these shales are continuous or discontinuous. The following questions refer to Figure 1 above. (i)

What would the main differences be between the cases where the shales in the above reservoir were continuous and where they were discontinuous?

(4)

(ii)

Say briefly how you would go about investigating this using reservoir simulation.

(4) 3

(iii)

Suppose a reservoir engineer took 10 vertical grid blocks (NZ=10) in a simulation model of this system. Would you expect the local (kv.kh) values in each of the main reservoir sands to be similar or different? Explain your answer briefly.

Similar/different (1)

Explain

(4) (iv)

If this reservoir well had a gas cap, then gas coning might be a problem.

What is gas coning? Draw a rough sketch.

(4) How would you use reservoir simulation to investigate this problem?

(4) 4

(v)

In which two ways would the grid used to investigate gas coning be different from that which was used in the full field waterflooding simulation?

1. (2) 2. (2)

Q5.

Two of the main numerical problems/errors that arise in reservoir simulators are due to numerical dispersion and grid orientation.

Explain each of these terms briefly saying - what it means, its origin and how we might get round it. Draw a simple hand drawn sketch illustrating each. (i)

Numerical Dispersion: Sketch

(5) (ii)

Numerical Dispersion - Explanation?

(5) 5

(iii)

Grid Orientation - sketch

(4) (iv)

Grid orientation - Explanation?

(4) Q6.

A very simple single phase pressure equation is given by Eq. 1 below. 2 Ê ∂P ˆ Ê ∂ P ˆ Á ˜ =Á 2˜ Ë ∂t ¯ Ë ∂x ¯

(i)

6

Eq. 1

Write down how this equation is discretised in an explicit finite difference scheme - briefly explain your notation.

(4)

(ii)

If an implicit finite difference scheme was used to solve Eq. 1, then a set of linear equations would arise which could be solved using either a direct or an iterative linear equation solution technique. Briefly explain each of the bold terms above:

• Implicit finite difference scheme

(4) • Set of linear equations

(4) • Direct linear equation solution technique

(4) • Iterative linear equation solution technique

(4) 7

Q7.

Statement: “The Equations of Two Phase Flow can be derived easily simply by using Material Balance and Darcy’s Law”.

Explain this statement with reference to two phase flow - you do not need to actually derive the equations and, indeed, you may not use any equations other than Darcy’s law.

(8)

Q8.

(i)

Draw a schematic sketch of a single grid block of size Dx by Dy by Dz, showing the porosity f, the oil and water saturations So and Sw (only 2 phases present).

(2)

8

(ii)

(a)

Using the sketch in part (i) above, derive expressions for (a) the volume of oil in the grid block, Vo; (b) the mass of oil in the grid block, Mo, introducing the formation volume factor, Bo. Vo?

(3) (b)

Mo?

(3)

(iii)

Write an expression for the oil flux, Jo, saying briefly what it is, any units it might be expressed in and explaining any terms you introduce.

(6) 9

Q9.

(i)

Name three ways in which a Black Oil reservoir simulation differs from a Compositional simulation model.

1.

2.

3.

(6) (ii)

Draw a simple sketch of a single grid block showing what is meant by a component and a phase.

(4)

(iii)

Using the notation CIJ to denote the mass composition of component I per unit volume of phase J (dimensions of CIJ are mass/volume), derive an expression - based on the quantities labelled in your sketch in (ii) above - for the mass of component I in the grid block.

(6) 10

(iv)

Give one example of (a) where you would use a Black Oil model and (b) where you would use a Compositional simulation model.

(a)

Use Black Oil model?

(3) (b)

Use Compositional model?

(3)

Q10. Figure 2: The figure below shows the basic idea of “upscaling” or “Pseudo-isation”.

"Fine" Grid Cross-Sectional Model

OIL

"Rock" Propts.

"Coarse" Grid Upscaled or Pseudo-ised Model

"Pseudo-" Propts.

With reference to Figure 2 above, answer the following:

11

(i)

What is meant by “Upscaling” with reference to the modelling of say a waterflood.

(2) (ii)

What is the difference between “rock” relative permeabilities and pseudo-relative permeabilities?

(4) (iii)

In order to perform upscaling in reservoir simulation, we need both an Upscaling Methodology and Upscaling Mathematical Techniques. Explain very briefly the meaning of the bold terms.

• Methodology

(4) • Techniques?

(4)

12

Q11. (i)

Sketch (roughly) a semi variogram for each of the following permeability models:

(a)

a correlated random field with a range of 100m and a sill of 10,000 mD2; and

(b)

a laminated system where the laminae are of constant width of 1cm and where the high permeability = 2D and the low permeability = 1D. Label your sketches clearly.

(a)

(6) (b)

(6) (ii)

What can you deduce about the standard deviation of the correlated random field in (i)(a) above.

(3)

13

(iii)

Sketch the correlogram for case (i) above.

(5)

Q12. Figure 3 below shows the sketch of simple 3 layer model.

k = 0.5 D

2 cm

k = 2.0 D

5 cm

k = 1.0 D

1 cm

x

y

(i) Calculate the effective permeability of the above model in the x-direction; show your working.

(5)

14

(ii)

Calculate the effective permeability of the above model in the y-direction, show your working.

(6) (ii) Suppose we put a very find grid (say of size 0.1 cm x 0.1 cm) on the 3 layer model in Figure 3 above. If we jumbled up all the grid blocks randomly so that the new model had no discernable structure, would the new effective permeability be: greater than that in (i) and (ii)?; less than that in (i) and (ii)?; in between these values? Explain your answer.

(6)

15

Q13. In miscible flow in a random correlated field, explain how the mixing zone grows with time in each of the following cases (illustrate your answers with simple sketches): (i) dispersive flow

(4) (ii) fingering flow

(4) (iii) channelling flow

(4) (iv) On the same diagram below, sketch the expected type of fractional flow curve f(c) vs c) you would expect for each type of flow.

(6) 16

Q14. In the Kyte and Berry pseudo-isation (upscaling) method, describe briefly (using a diagram) how numerical dispersion is taken into account (no detailed mathematics is required).

(10) 17

Q15. (i)

List the main categories in the hierarchy of stratal sedimentary elements - give one short sentence explaining each of these.

(8) (iii)

Describe which of the above length scales of sedimentary heterogeneity are likely to have most significance for the following reservoir flow phenomena:

* Reservoir pay-zone connectivity:

(3) * Gravity slumping or water over-ride:

(3) * Vertical sweep efficiency:

(3) * Residual/Remaining oil saturation

(3)

18

Frequency

Q16. You have been given the following distribution of core-plug permeabilities in a particular reservoir unit which includes a higher permeability a fluvial channel sand overlying a lower permeability deltaic sand:

100 400 600 800 Permeability (md)

With reference to Figure 4 above: (a) explain the probable reason that the permeability distribution has the above form; (b) sketch the sort of permeability models (laminar, bed and formation scale) you might use for the flow simulation of this unit. (a)

(3) (b)

(continue on the back of this sheet if necessary)

(10) 19

Q17. (i) Draw a sketch of water displacement of oil across the laminae in a water-wet laminated system at (a) “low” flow rate (capillary dominated) and (b) “high” flow rate (viscous dominated); in this sketch show where the residual remaining oil is and give a sentence or two of explanation.

(a)

(8)

(b)

(8)

20

(ii)

Is the effective water permeability at residual (i) remaining oil saturation (across the laminae) higher in case (a) or (b) in part (i) above? Explain.

(6) (iii)

What are the implications of the results in (i) and (ii) above for upscaling in reservoir simulation?

(6)

21

yyyy ;;;; yyyy ;;;; yyyy ;;;; yyyy ;;;;

Model Solutions to Examination

O N t N O no TI o E: RA td M T bu ion A n N GIS SE: at io in ct RE UR E: se am : R s ex d hi CO AR TU e t l the ishe E A et N pl unti fin G m is l Co sea

Y

SI

.:

8 Pages

Date: Subject:

Reservoir Simulation INSTRUCTIONS TO CANDIDATES No.

Mk.

1. Complete the sections above but do not seal until the examination is finished. 2. Insert in box on right the numbers of the questions attempted. 3. Start each question on a new page. 4. Rough working should be confined to left hand pages. 5. This book must be handed in entire with the top corner sealed. 6. Additional books must bear the name of the candidate, be sealed and be affixed to the first book by means of a tag provided

PLEASE READ EXAMINATION REGULATIONS ON BACK COVER

1

Answer Notes #

2

=>

indicates one of several possible answers which are equally acceptable.

[…] =>

extra information good but not essential for full marks - may get bonus.


Q1

(i)

#

1. To perform broad scooping calculations which examine different development options e.g. waterflooding, gas flooding etc.

#

2. To extend initial material balance calculations by examining some other spatial factor such as well-placement or aquifer effect.

(ii) #

1. To assess additional field management options such as infill drilling, pressure blowdown etc.

#

2. To take the improved history match model which can be developed after same development twice and to use this to assess various IOR strategies e.g. gas injection, WAG or chemical flooding.

Q2 (i) #

Because of its inherent simplicity you would virtually always apply single material balance to assess your field performance - to see if DP decline tallies with estimated field size, sources of influx and production.

(ii) Would be used when a more complex development strategy requiring spatial information is essential e.g. well placement, assessment of shale effects, gravity segregation etc.

3

Q3 (#)

Because it is the only tool we have to tackle complex reservoir development/flow problems which extends material balance. Clearly, it is much better than simple material balance alone.

Q4 (i) The shale continuity strongly affects the hi/lo permeability layer vertical communication (both pressure and fluid flow). Thus, it will affect the effective kv/kh (lower or zero for continuous shales) and will strongly influence gravity slumping of water in a waterflood. In the situation above with high k on top, some vertical communication will help recovery.

(ii) Set up a simple 2D cross sectional model with , say, 50 blocks in the x direction and 10 vertical grid blocks - 5 in each layer. Run waterflood cases with and without shales - and some in-between cases with transmissibility modifiers set beween Tz = 0.0 Æ 0.01 Æ0.1 Æ0.5 Æ1.0. Compare water saturation fronts and recoveries as fraction of pv water throughput. Result will allow us to assess the effects of the shales in the waterflood.

(iii) Different

4


The high perm massive sand would have a small scale kv/kh ~1 which would result in a similar larger scale value. In the laminated sands, the “small” scale (say core plug scale) would have a low kv/kh of say 0.1 to 0.01 and this would result in a correspondingly lower kv/kh at the grid block scale.

(iv)

Well

Gas

Oil and Water

Gas

Oil and Water

= perforations Gas Coning It is the drawdown of the highly mobile (low mg) gas into the perforations. Pattern is shown here in figure. Causes high GOR production at a level well above the solution gas value.

5

Set up a near-well r/z geometry fine grid - possibly 50 layers and set reservoir near-well rock properties e.g. Layering, Tz modifiers, Rel. perms. etc.

Perform simulations to look at issues such as effect of rate, vertical communication, gas/oil/water Rel. perms. etc… Generally needs a fine Dr, Dz grid, often finer near the well where most rapid changes of Sg and pressure with time occur.

(v) 1. The geometry would be different: r/z for coning and cartesian or corner point for full field.

2. The fineness of the grid would be different. Very fine for nearwell; much coarser for full field.

#

6

(Dimensionality too 2D vs. 3D)


Q5 (i)

(ii) Numerical dispersion is the artificial spreading of saturation fronts due to the numerical grid block structure in the simulation. It arises because we take large grids to represent moving fronts. It can be improved by refining the grid (globally or locally) or by using improved numerical methods.

(iii) Wells same distance apart in Figs A and B L I

P

I

P

Fluid tends to flow along (parallel) to the grids

Fig A

L

Fig B I = injector ;

P = producer

7

(iv) The injected fluid tends to flow parallel with the grid from the injector (I) to the producer (P) - see previous page. This means that early breakthrough and poorer recoveries are seen in A then in B above. i.e. Fig B Actual Recovery Fig A %00IP Producer

Pv or Time

Q6 (i)

8


(ii) ∂2 P In this scheme the spatial term in Eq. 1 i.e. would be specified at ∂x 2 the new time level n+1

A set of linear equations is the following type of equation (e.g.3x3system)

a11 X1 + a12 X2 + a13 X3 = b1 a21 X1 + a22 X2 + a23 X3 = b2 a31 X1 + a32 X2 + a33 X3 = b3

where X1, X2, X3 are unknown - the a’s are a matrix of known coefficients and b’s are a known right-hand side.

A direct solution method (e.g. Gaussian Elimination Elimination) is an algorithm with a fixed number of steps which will solve these linear equations (under certain conditions). [Typically forward elimination is applied to get an upper triangular A* matrix and back substitution is then easily applied to get the X solution]

In contrast, an iterative technique starts with a first estimate of the unknown vector X (0) where the (o) denotes 0th iteration: This is then improved by some algorithm to a better and better solution of the original linear equations i.e. X(1) Æ X(2)… Æ X(V) until the method converges e.g.

9

/ X(V+1) - X(V)/ < small number TOL. [Methods such as the Jacobi, LSOR, etc. are examples of this]. Q7 In block (i, j), then material balance can be applied for each phase (e.g. oil and water) for 2-phase flow. mo mw

mo i, j

Mass Accumulation of

Amount that

=

-

Amount that

oil over time

flows in over

flows out over

Dt

Dt

Dt

But amount that flows in/out is given by the pressure differences between blocks i.e.

(

Ê A.k.kro ( So ) ˆ Qoil ( i -1, j ) Æ ( i , j ) Á Po - Poi -1 j ˜ mo Ë ¯ i - 1 ij 2

)

Thus the two phase Darcy Law supplies the relation for volumetric flow rate and pressure in the grid block. These volumetric flows can be converted to MASS flows (x by density) and then put into the material

10


balance equation to obtain Æ a conservation equation and in pressure equation for oil and water.

\

\ Material Balance + Darcy’s Law => 2-phase Flow Equation.

Q8 (i)

(ii)

(a)

Vo = Dx Dy Dz f So

(b)

11

(iii)

#

Q9

1. The black oil model essentially treats a phase (o,w,g) as the basic conserved unit or “pseudo component”

2. Compositional models are based more correctly on the conservation of components (CH4, C10, H2O etc.) - the black oil model simply treats gas dissolution in oil through Rso - gas solubility

3. The compositional models incorporate a full PVT description of the oil whereas the black oil model relies on the simple Rso type treatment.

12


(iii)

(iv) #

(a) Waterflood calculations in a low GOR - say 30∞ API - oil reservoir with pressure maintenance.

#

(b) CO2 injection in a - say 36∞ API - light oil system [Condensate system - gas recycling etc…]

Q10 (i) Upscaling in a waterflood essentially means getting the correct (effective) parameters (-e.g. rel. perm.) for the larger scale grid blocks which will reproduce a “correct” fine grid model.

(ii) “Rock” relative permeabilities are meant to be the intrinsic representative properties of a representative piece of reservoir rock at the “small” (i.e. core plug) scale.

Pseudo rel. perms. are effective properties at the “larger” (usually gridblock) scale which incorporate other effects and artefacts (e.g.

13

numerical dispersion, heterogeneity etc..) in addition to the intrinsic “rock” rel. perms.

(iii) Methodology This is a geologically consistent approach to the task of upscaling. i.e. data collection, sedimentological framework,…

[The function of the methodology is to get the geologically + fluid] mechanically “right” answer.

Techniques? These refer to the actual mathematical algorithm to go from a fine grid Æ coarse grid. E.g. Kyte and Berry, Stone’s method, two phase tensors etc…

[N.B. This just needs to reproduce the fine grid result - even if it is WRONG - at the coarse scale]

14


Q11

(i)

(ii) It takes a lag distance of about the “range” to see the field variability (standard dev. - i.e. ~ 100mD) of the field.

15

(iii)

1

lag

Q12 (i)

(ii) The effective permeability is clearly the harmonic (thickness weighted) average as follows:

16


(iii) The keff in the randomised model would be between the two answers in (i) and (ii) above (the answer in (ii) being the lower).

e.g. Strictly in a randomised distribution of permeability the average value tends to the geometric average (kg) in 2D kg - is less than the arithmetic (along layer) answer. kg - is greater than the harmonic (across layer) answer.

Q13 (i) Note - we take the same contour values (c= 0.1, 0.5, 0.8) in all sketches below.

17

(ii)

(iii)

(iv)

18


Q14 With no maths

Consider only ACTUAL flows of Qw and Qo across this interface interface only. Thus, if the fine grid water (say) flows have not reached the coarse grid interface

then Qw = 0. => set krw= 0

When there is oil or water flow e.g. water flow.

19

Q15 (i) Lamina Æ The simplest unit within which we can assume (almost) homogeneous k. (length mm Æ m) Lamina set Æ A collection of the above (cm Æ m) e.g. core.

hi k lo k

Bedform Æ How lamina sets are joined together geometrically to form 3D beds.

20


Eroded/? top sets

e.g.

Tabular cross-bedding

2m

~50cm

Bottom sets

or climbing ripples

Para-sequence/sequence-stacks of bedforms

(ii) •

Para sequence - sequence scale

•

At parasequence - sequence - also bed form influence

•

Para sequence - bed form

•

Lamina set - bed form

Q16 (a) There is a double peak - the bimodality probably arises from the lower perm plugs from deltaic sands, and the higher permeability plugs from the fluvial channel.

21

(b)

laminated sand pseudo

Tightly laminated deltaic sands A Crossbeddes fluvial channel -stacked crossbeds

B

2 Scale pseudo-isation - inclined cross bed pseudo A - bedform pseudo B

Q17 (i)

(a)

hi

lo

hi

lo

hi

lo

Slow Flow CAPILLARY DOMINATED

Water flow direction

High water Sw in LOW perms in a water-wet system

Sw

HIGH "remaining" oil in hi k Spontaneous water inhibition into the LOW k laminae occurs in Pc-dominated flow. This traps oil in the HIGH k laminae behind the front where it is well above "residual" but it can't move because the Rel. Perm. to oil in the low k water-filled laminae is so low.

22


(b)

hi

lo

hi

lo

hi

lo

VISCOUS DOMINATED WATERFRONT

"Fast" Flow of water

Water flow direction

High water Sw LOW perm in a water-cut system

Sw

Note at Viscous dominated conditions a water front goes through which reduces the oil in all layers to its local "residual" level. Recovery of oil is better in this case since it is not "stranded" by downstream capillary imbibition.

(ii) It is higher in case (a) for the reasons already explained. [I give a slightly over-detailed answer to part (a) and (b) above].

(iii) The central implications are twofold.

(a) The two phase pseudo relative perms. are highly anisotropic for such laminar systems. Along and across layer water displacement in laminar system gives widely different pseudos.

23

Along

Across

(b) The levels of “remaining” oil can be vastly different in laminar systems which, in simulation/upscaling, moves the pseudo rel. perm. end points. (see above).

24


25

Course:- 28117 Class:- 28912

HERIOT WATT UNIVERSITY DEPARTMENT OF PETROLEUM ENGINEERING Examination for the Degree of Meng in Petroleum Engineering Reservoir Simulation Tuesday 20th April 1999 09.30 - 13.30 ( 100% of Total marks )

NOTES FOR CANDIDATES 1.

This is a Closed Book Examination.

2.

15 minutes reading time is provided from 09.15 - 09.30.

3.

Examination Papers will be marked anonymously. See separate instructions for completion of Script Book front covers and attachment of loose pages. Do not write your name on any loose pages which are submitted as part of your answer.

4.

This Paper consists of 1 Section.

5.

Attempt all Questions.

6.

Marks for Questions and parts are indicated in brackets

7.

This Examination represents 70% of the Class assessment.

8

State clearly any assumptions used and intermediate calculations made in numerical questions. No marks can be given for an incorrect answer if the method of calculation is not presented. Be sure to allocate time appropriately.

1

2

Q.1: Consider the following statement which is made referring to a reservoir development plan for a field which has been in production for some time: “A reservoir engineer should always apply Material Balance calculations and should usually but not always - use Numerical Reservoir Simulation”. (i) Why should you always perform some sort of material Material Balance calculations ? .............................................................................................................................. .............................................................................................................................. .............................................................................................................................. ...........................................................................................................

(4)

(ii) What is the main disadvantage of using material balance calculations in reservoir development? .............................................................................................................................. ...........................................................................................................

(2)

(iii) In the above context, explain when you would use Reservoir Simulation and when you may not use it. Give an example of each case. When you would use Reservoir Simulation + Example: .............................................................................................................................. .............................................................................................................................. .............................................................................................................................. ...........................................................................................................

(4)

When you may not use Reservoir Simulation + Example: .............................................................................................................................. .............................................................................................................................. .............................................................................................................................. ...........................................................................................................

(4)

3

Q.2: Various types of 2D and 3D grids are used in reservoir simulation calculations. Describe what you think the best type of grid is for performing calculations on each of the reservoir processes described below and say why. Reservoir processes: (i) Modelling of likely gas and water coning and its effect on (vertical) well productivity in a light oil reservoir with a gas cap and an underlying aquifer; (ii) Simulating a large number of options in an injector/producer well pair in a gas injection scheme where the objective is to look at the effects of formation heterogeneity on gas - oil displacement and to develop some pseudo relative permeabilities to use in a full field model; (iii) Carrying out an appraisal of an entire flank of a complex faulted field which has several injector and producer wells. (i) Which grid?............................................................................................................................ .................................................................................................................................................... (4) Why?........................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... (4)

(ii) Which grid?............................................................................................................................ ..................................................................................................................................................... (4) Why?........................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... (4) (iii) Which grid?.......................................................................................................................... .................................................................................................................................................... (4) Why?.......................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... (4)

4

Q.3: Numerical dispersion and grid orientation are two of the main numerical problems that occur in reservoir simulation. Explain in your own words, with the help of a simple sketch, the meaning of each of these terms: (i) Numerical dispersion ? Sketch:

(5) Numerical dispersion ? Explanation: ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... (5) (ii) Grid orientation ? Sketch:

(5) Grid orientation ? Explanation: ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... (5) 5

Q4.

Figure 1 below shows the results of a series of 6 waterflooding and gas flooding calculations (labelled A - F) in a 2D vertical cross-sectional numerical simulation model. Results are plotted as “Oil Recovery at 1PV Injection” vs. 1/NZ , where NZ is the number of vertical grid blocks in the simulation. Assume (a) that the number of grid blocks in the x-direction (NX) is sufficiently large and is constant in all calculations; and (b) that the axes of the graph are “quantitative”. Figure 1

2D cross-section Key gas injection waterflood

No. vertical grid blocks = NZ

Oil recovery at 1PV injection

C

B

60% A 50%

D E

40% F 0.1

0.2

0.3

0.4

0.5

Inverse number of grid blocks (1/NZ) -->

Answer the following questions: (i) How many vertical grid blocks (NZ) were used in case F? .........

(3)

(ii) Do the simulated waterflood and gas flooding calculations become more “optimistic” or “pessimistic” as we take more vertical grid blocks in the calculation? ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... (4)

6

Q4. (continued) (iii) Extrapolate each of the calculations to NZ --> ∞ for both the waterflood and the gas flood on Figure 1. Estimate the % recovery for each and the incremental recovery of the gas flood compared with the waterflood. Comment on the implications of your result for carrying out a gas flooding project in this reservoir. Estimations........................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... (6) Comment: ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... (4)

————————————————— End of Q.4 ——————————————— Q5. on next page

7

Q. 5

Figure 2 below shows a “control volume” - block i - for single-phase compressible flow in 1D. The quantities qi+1/2 and qi-1/2 are the volumetric flow rates of the fluid at the boundaries of block i. All grid blocks are the same size in the x-direction (∆x) and the cross-sectional area is constant, A = ∆y.∆z (where ∆y and ∆z are the block sizes in the yand z-directions). The density and porosity are denoted by symbols - ρ and φ respectively. Figure 2

i-1 Area =A = ∆ y. ∆ z

i

i+1

q i-1/2

qi-1/2

∆x x With reference to the above figure, (i) Write a clear mathematical expression for the change in mass in block i over a time step ∆t due to flow: Change in mass due to flow over time step ∆t = ...................................................................................................................................................... ...................................................................................................................................................... (6)

(ii) Write a clear mathematical expression for the change in mass in block i from the beginning of the time step to the end (i.e. the accumulation): Difference in mass in block i over time step ∆t = ...................................................................................................................................................... ...................................................................................................................................................... (6)

(iii) Considering the above two expressions, what equation can you now write from material balance ? ...................................................................................................................................................... ...................................................................................................................................................... (6)

8

Q.5 (continued) (iv) Are there any assumptions in the equation you have just written in part (iii) above? Briefly explain. ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... (4) (v) Now use the single-phase Darcy Law in the equation you wrote in (iii) above and show how by taking Limits ∆x, ∆t --> 0 - you obtain the pressure equation for single-phase compressible flow: Show the steps you take.

(8) (vi) If you have written down the answer to part (v) above correctly, then you should have written down a non-linear partial differential equation (PDE). What does “non-linear” mean in this context and explain in physical terms what the main problem is with this sort of equation. Non-linear?................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... (4) The problem?............................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... (4) 9

Q.6: (i) From the four assumptions listed below, show clearly how Equation 1 arises from the non-linear equation you derived in Q.5 part (v) above. Indicate clearly where you use each of the assumptions in your derivation. 2  ∂P   ∂ P  = κ   2   ∂t   ∂x 

Equation 1

The term κ is a constant (Greek kappa - not permeability, k). Other quantities which may be used are Cf - the fluid compressibility, ρ - the density; k - the permeability; µ - the fluid viscosity; φ - the porosity. Assumptions:

1. The rock is incompressible. 2. Permeability (k) and viscosity (µ) are constant. 3. The fluid has a constant compressibility, C f =

1  ∂ρ   . ρ  ∂P 

 ∂P  4. Pressure gradients are small - hence   ≈ 0  ∂x  2

(i) Answer:

(12) (continue on the back of this page if necessary indicating that you have done so) 10

Q.6 (continued) (ii) From your answer to part (i) above, write down what constant κ is in terms of the other constants. κ = .............................................................................................................................................. (4) (iii) Using the notation in Figure 3 below, apply finite differences to Equation 1 above and define the discretised spatial derivative at the new time level, (n+1) (i.e. an implicit method). Show each step in your working and show clearly how this leads to a sparse set of linear equations. Figure 3

P in+1

n+1

∆x Time level

∆t

All ∆ x and ∆ t fixed.

P in

n i-1

i

i+1

Space --->

(12) (continue on the back of this page if necessary indicating that you have done so) 11

Q.6 (continued) (iv) We can write down the set of linear equations that arises from applying finite differences to the flow equations as follows: A.x = b

Equation 2

where A is a matrix, x is the vector of unknowns and vector b is known. Write out Equation 2 explicitly for three equations and rearrange these to show how a simple iterative scheme can be formulated. Say very briefly how this is solved. Give one advantage and one disadvantage of an iterative scheme.

(continue on the back of this page, if necessary, indicating you have done so) (10)

12

Advantage?......................................................................................

(2)

Disadvantage?.................................................................................

(2)

Q.7: The oil flux, Jo, into and out of a grid block is shown in Figure 4 below. Other quantities are denoted as follows: Oil saturation So; porosity, φ; Formation volume factor, Bo; Oil density at standard conditions, ρosc; Darcy velocity of oil, vo (similar quantities apply to the water phase). Figure 4

∆y

Oil Flux Jo

Jo ∆z

∆x x

x + ∆x

(i) Write an expression for the oil flux, Jo, giving any possible units.

(6) (ii) The concentration of oil, Co, is defined as the mass of oil per unit volume of reservoir. Write an expression for Co in terms of the quantities defined above.

(6) 13

Q.7 (continued) (iii) Prove that, in 1D two-phase flow, then for the oil phase: (∂Jo/∂x) = - (∂Co/∂t)

Equation 3

showing your working clearly.

(10) (continue on the back of this page, if necessary, indicating you have done so)

14

Q.7 (continued) (vi) State the two-phase Darcy’s law for oil using (∂Po/∂x) for the pressure gradient and kro for the relative permeability, and substitute this into Equation 3 above and derive the oil conservation equation. • Two-phase Darcy Law for the oil phase (in terms of Darcy velocity, vo)

(4)

• Oil conservation equation:

(8) ————————————————————End of Q.7 ——————————————

15

Q.8 In three-phase flow (oil, water and gas), we can define a gas flux, Jg, and a gas concentration, Cg, in exactly the same way as was defined for oil in Q. 7 above. (i) Explain physically why the gas flux and the gas concentration are more complicated than the corresponding quantities for the flow of oil or water. ......................................................................................................................................................... ......................................................................................................................................................... ...................................................................................................................................................... (4) (ii) Using Rso and Rº to denote the gas solubility factors, derive expressions for Jg and Cg showing your working.

(12) (continue on the back of this page, if necessary, indicating you have done so) 16

Q. 8 (continued) (iii) Use your expressions for Jg and Cg in the conservation Equation 3 (for gas) to write down the first step in obtaining the conservation equation for gas.

(6) ————————————————————End of Q.8 ——————————————— Q.9: Explain what history matching is in a reservoir simulation of a field saying briefly how it is done and what can go wrong. • History matching? How is it done? What can go wrong? ........................................................................................................................................................... ........................................................................................................................................................... ........................................................................................................................................................... ........................................................................................................................................................... ........................................................................................................................................................... ........................................................................................................................................................... ........................................................................................................................................................... ........................................................................................................................................................... ........................................................................................................................................................... ........................................................................................................................................................... ........................................................................................................................................................... ........................................................................................................................................................... ........................................................................................................................................................... ........................................................................................................................................................... ...........................................................................................................................................................(10) (continue on the back of this page, if necessary, indicating you have done so) 17

Q.10 (i) Write down the formulae for the arithmetic (ka), harmonic (kh) and geometric (kg) averages of a permeability field with permeabilities k1, k2, .... kM. The number of data points you have = M.

ka =

kh =

kg = (10) (ii) State how you would use these averages for calculating the effective permeabilities in the horizontal and vertical directions in the models in Figures 5(a) and 5(b) below. Figure 5 (a)

(b)

• For Figure 5(a): Horizontal keff & Vertical keff :

(8) 18

Q.10 (continued) • For Figure 5(b): Horizontal keff & Vertical keff :

(8) (iii) Calculate the effective permeability for flow across the laminae in Figure 6 below. Show your working.

20 mD, 1 cm

100 mD, 2 cm

10 mD, 1 cm

Figure 6

........................................................................................................................................................... ........................................................................................................................................................... ........................................................................................................................................................... ........................................................................................................................................................... ...........................................................................................................................................................(6)

————————————————————End of Q.10 ——————————————

19

Q.11 The diagram in Figure 7 below shows a grid consisting of 2 coarse grid blocks, with 7 x 3 fine blocks in each of these coarse blocks. Figure 7

i=1 2

3 4

5

6 7

8

9 10 11 12 13 14

j= 1 2 ∆z

3 ∆x

Suppose you are calculating the pseudo relative permeabilities for the left coarse block using the Kyte and Berry method. Assume that you have all the necessary information (saturations, pressures, flows etc.) from a fine-scale (3x14) simulation. (i) Show clearly on Figure 7 which part of the grid you would use for calculating the following quantities and give a brief sentence of explanation: • the average water saturation

(3)

........................................................................................................................................................... ...........................................................................................................................................................(4)

• the average pressure gradient

(3)

........................................................................................................................................................... ...........................................................................................................................................................(4)

• the total flows of oil and water

(3)

...........................................................................................................................................................

...................................................................................................................................................... (4) (ii) What is the formula for the average water saturation? Sw =

20

(5)

Q. 11 (continued)

(iii) What is the weighting for the average pressure? Give a brief sentence of explanation. (5)

..........................................................................................................................................................

........................................................................................................................................................ (3)

(iv) What is the formula for the total flow of water?

qw =

(5)

————————————————————End of Q.11 ——————————————

21

Q.12 (i) By means of a simple sketch, show how a cubic packing of spherical grains is arranged. Sketch:

(4) (ii) Use the sketch to help you calculate the specific surface of the sample per unit volume of solid, Ss (in m2/m3). Specific surface - working:

Specific surface per unit volume of solid, Ss = ..................... m2/m3 (6) (iii) If the grain radius is taken to be 10µm, determine the porosity (φ) of the sample

Porosity working:

Sample porosity (φ) = ......... 22

(6)

Q.12 (continued) (iv) If the grain radius was 100µm instead of 10µm, what would the porosity (φ) of the sample now be?

Porosity = .................................................................................................................................... (3)

(v) What do the results of parts (iii) and (iv) above suggest concerning the porosity of cubic sphere packs?

Ans.............................................................................................................................................

(2)

(vi) Write down the Carman-Kozeny equation in terms of the grain diameter (D), porosity (φ), and the specific surface per unit volume of solid, Ss Carman-Kozeny equation:

k=

(6) (vii) Taking the grain radius to be 10 µm and the tortuosity of the sample to be T=1, calculate an approximate permeability in Darcies for a cubic packing of spheres (NB Use the porosity found in Q.12 part (iii) in this calculation) . Working:

Permeability = ................. Darcies. (1 Darcy = 1x10-12 m2)

(6) 23

Q.13 (i) Describe the meaning of the term “contact angle” and draw a rough sketch to illustrate your answer

Contact angle?................................................................................................................................... ........................................................................................................................................................... ........................................................................................................................................................... (4) Sketch

(4) (ii) If an oil/water meniscus is at equilibrium in a cylindrical capillary tube, what is the equation that relates the capillary pressure, Pc, to the tube radius R and the contact angle θ?

Pc=

(5)

What is this reduced form of the equation called?

..................................................................................................................................................... (3)

24

Q. 13 (continued) (iii) Oil is introduced at the inlet face of the water-filled pore network shown in Figure 8 on the next page. The numbers on Figure 8 refer to pore radii (in microns, µm). The pores are taken to be capillary tubes and are water-wet, the contact angle is θ = 60o in every pore and the oil/water interfacial tension is 80 mN/m. The capillary pressure of the system is gradually increased and oil begins to invade the network. Shade in the pores that become oil-filled at each of the 4 capillary pressure values Pc1, Pc2, Pc3 and Pc4 in Figure 8 (NB 14.7 psi = 105 Newtons/m2).

Show any working out below:

25

Q. 13 (continued)

P

Figure 8: Shade in the pores that become oil-filled at each of the 4 capillary pressure values Pc1, , c 2 5 2 Pc3 and Pc4 below (NB 14.7 psi = 10 Newtons/m ).

Pc1= 0.6 psi

5

15 11

2 1

Oil

Water 10

20 3

3 12

4

12

8

Pc2= 1.2 psi

5

15 11

2 1

Oil

Water 10

20 3

3 12

4

12

8

Pc3= 3.0 psi

5

15 11

2 1

Oil

Water 10

20 3

3 12

4

12

8

Pc4= 10.0 psi

5

15 11

2 1

Oil

Water 10

20 3 12

3 12

4

8

(20) 26

Q.14 (i) Explain the differences between a drainage flood and an imbibition flood at the porescale, paying particular attention to the roles played by pore size, film-flow and accessibility to the inlet (sketch each displacement in the boxes provided below to illustrate your answer). ........................................................................................................................................................... ........................................................................................................................................................... ........................................................................................................................................................... ........................................................................................................................................................... ........................................................................................................................................................... ........................................................................................................................................................... ........................................................................................................................................................... ........................................................................................................................................................... ........................................................................................................................................................... ........................................................................................................................................................... ..............................................................................

(10)

(continue on the back of this page, if necessary, indicating you have done so)

Imbibition Sketch

(4) Drainage Sketch

(4) 27

Q. 14 (continued) (ii) The strongly water-wet network in Figure 9 below is initially completely filled with oil and the capillary pressure is so high that no water can currently imbibe. If the capillary pressure is slowly decreased, so that water can invade via film-flow and snap-off, how is the residual oil distributed at the end of imbibition (i.e. when Pc=0)? Shade in the residual oil using the network template provided in Figure 10. The numbers on the bonds again refer to the tube radii in microns (µm). Note this network is deliberately different from that in Figure 8. Also, oil cannot enter the water reservoir due to the presence of a water-wet membrane. Figure 9 Initial distribution of oil

5

15 11

2 1

Water

Oil 10

20 3

3 12

4

12

8

Figure 10

Answer: Template for final oil distribution (shade in appropriately)

5

15 11

2 1

Water

Oil 10

20 3 12

3 12

4

8

(8) ————————————————————End of Q.14 ——————————————

28

Q15. You have been supplied with the table of mercury injection data below.

Table 1 Mercury saturation 0

P c (air/mercury) (psia) 2

Oil saturation Pc(oil/water) (psia) 0

0.2 0.4 0.5

4 5 6

0.2 0.4 0.5

0.6 0.8 0.9

7 20 60

0.6 0.8 0.9

(i) Sketch the air/mercury capillary pressure curve using the axes provided

(6) (ii) Write down the equation used to re-scale mercury injection data to oil/water systems and use it to complete Table 1 (assume the following values for interfacial tensions and contact angles: σmercury/air = 360x10-3 N/m, σoil/water = 60x10-3 N/m, θmercury/air = θoil/water = 0o Equation:

(6) Now complete Table 1 at the top of this page

(8) 29

Q. 15 (continued) (iii) Plot the oil/water capillary pressure curve using the axes provided below Sketch.

(5)

(iv) Write down the equation that determines the Leverett J-function from capillary pressure datai.e. complete the following equation:

J(Sw) = Pc(Sw) x ..........................................

(Q. 15 is continued on the next page)

30

(5)

Q. 15 (continued) (v) Using the capillary pressure data shown below in Table 2, calculate an appropriate J-function and complete the table below : assume the values k = 100 mD, φ= 0.1, interfacial tension, σ =10 x10-3 N/m, and contact angle, θ = 0o. Choose any suitable units but label your sketch clearly. Sketch the J-function below.

Table 2 Complete the table Wa te r S a tu ra tio n P c (p s ia ) 1 0 0.8 2 0.6 4 0.5 6 0.4 10 0.2 20

J (S w)

(6) Write down the form of the J-function here:

J(Sw)= (4)

Sketch the J-function here:

(6) 31

Q.16 (i) Name the two most popular tests used to determine the wettability of a reservoir rock. The .................................................................. Test

(2)

The .................................................................. Test

(2)

(ii) Give the three “rules of thumb” that can often be used to distinguish between water-wet and oil-wet relative permeability curves. Rule 1 ................................................................................................................................................ .......................................................................................................................................................(4)

Rule 2 ................................................................................................................................................ .......................................................................................................................................................(4)

Rule 3 ................................................................................................................................................ .......................................................................................................................................................(4) (iii) Capillary pressure curves in Figure 11 below have been measured on two different core samples. The two plots are shown below. Use your knowledge of wettability variations at the pore scale to answer the following: Figure 11 Pc

Pc

Sw

(A)

Sw

(B)

(iv) Which sample is probably the more water-wet? Explain your choice. ........................................................................................................................................................... ........................................................................................................................................................... ........................................................................................................................................................... ........................................................................................................................................................... .......................................................................................................................................................(6) 32

Q.16 (continued) (v) Give a physical explanation for the negative leg shown in the capillary pressure curve in Figure 11(A). ........................................................................................................................................................... ........................................................................................................................................................... ........................................................................................................................................................... ........................................................................................................................................................... ........................................................................................................................................................... (6)

(End of examination paper)

Total number of marks possible = 462

33

Heriot-Watt University - Reservoir Simulation

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