Per Arne Slotte, and Carl Fredrik Berg Department of Geoscience and Petroleum NTNU
Lecture notes in well-testing
Copyright notes: These present lecture notes are © Per Arne Slotte 2017–2018 The text is distributed under the “Creative Commons Attribution-ShareAlike 4.0 International” (CC BY-SA 4.0) https://creativecommons.org/licen commons.org/licenses/by-sa/4.0 ses/by-sa/4.0 licence: https://creative The above mentioned licence does not apply to any of the figures or illustrations, as the authors do not have copyright to some of these. Including such material is allowed under the rules that apply to educational material at NTNU, but creates limitations on the possible distribution of this book in electronic or paper form.
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Foreword In June June 2016 2016 I was was aske asked d to lect lectur ure e the the well well test testin ing g part part of cours ourse e TPG4115 Reservoir Reservoir Property Property Determination Determination by Core Analysis and Well Testing at NTNU. My knowledge of well testing at that point was very limited, as I have had no formal training in reservoir technology, and have mainly been working with reservoir modell modelling ing and data data integr integrati ation. on. Thus, the the well well known known princi principle ple of learning by teaching did did apply apply,, and these these notes does, to a certain degree, sum up what I have learnt during the fall-semester 2016. In the previous year (2015) students had used the books “T.A. Jelmert: Introductory well testing ” (bookboon boon 2013) 2013) and and “J.W “J.W. Lee: Lee: Well Testing Testing ” (Socie (Society ty of Petro Petroleu leum m Engine Engineers ers 1981) 1981) as their their main main study study mater material ial.. The first of these does not cover all of the material students are expected to learn in the course, and additional ditional material material is thus needed. needed. The second second covers covers most of the needed subjects, subjects, but it is old and the perspective is to a certain degree outdated. The book has the additional drawback that it uses oilfield units and merged numeric conversion factors factors throughout, which in my view only creates confusion. You can always derive the right numeric factors from an equation given consistent units. While still mentioning the books of T.A. Jelmert and J.W. Lee as possible material for the students, I decided instead to use the book “Well Well test design and analysis analysis” (PennWell Books 2011) by George Stewart as main reference material. This book cover everything the students are expected to learn, and has a more modern perspective. However, However, it covers a lot more than needed and it is totally out of the question that students should be required to buy this 1484 page monster. However, students have access to the book in electronic form through the NTNU library, so it was possible to run the course this way. The present notes cover the material that was taught in lectures and homework assignments in the fallsemest semester er 2016, 2016, and very very little little else. else. This This is approx approxima imatel tely y equiva equivalen lentt to what what has has been been taught taught in the previo previous us years (2014 and 2015). There is much to be said about a textbook with such a limited scope, but I have decided that it in this case will be useful for the students to have some extended material (in addition to lecture slides) that actually follow the progression of the lectures. Skipping back and forth in a large book, where in many cases only minor parts of chapters are directly relevant for the course, is not optimal. I do, on the other hand, have no actual practical experience in well-testing, neither in operations nor in research, so writing a full-fledged textbook that could accompany the course is out of the question. Well testing theory is quite heavy on mathematics compared to what petroleum students tend to be exposed to in other courses. The notation that are used in these notes may also be unfamiliar to some, as derivations and equations for the most part is presented using coordinate free mathematical objects and operators. A chapter with mathematical mathematical notes have been included (page 139 (page 139), ), and the reader is referred there whenever in doubt about the meaning of an equation. The present lecture notes are written in the hope that they will be useful. Trondheim August 3, 2017 Per Arne Slotte
Contents
Introduction
7
Basic theory
13
Drawdown test
23
Buildup test
39
Finite reservoir
49
Reservoir Re servoir boundaries Horizontal wells
63 77
Fractured wells
85
Naturally fractured reservoirs Gas reservoirs
91
105
Multiphase flow
113
Numerical methods
121
The role of well testing in reservoir characterization Mathematical notes Relevant literature Nomenclature References
139 143
145 149
135
Introduction The The well well test testin ing g that that is the the subj subjec ectt matt matter er of the the curr curren entt lect lectur ures es are are a number number of method methodss where wherein in rates rates and pressu pressures res are manip manipula ulated ted and measured in one or more wells in order to obtain information about the sub surface reservoir . Thus it is, in spite of the name, not the well or well well produc productio tion n that that is tested tested,, but but the reserv reservor or.. Note, Note, howe howeve ver, r, that that the term well test is also used in production technology for tests that actually test the well and the well production, but these tests are not of interest here. Well testing is also known as pressure transient tests , which arguably gives a better description of the test. Well testing is important in many disciplines in addition to petroleum engineering. engineering. Examples Examples are groundwa groundwater ter hydrology, hydrology, geology geology,, waste disposal, and pollution control. The theory and methods are in principle the same in all diciplines, although nomenclature may vary somewhat. In this course we will concentrate concentrate on petroleum engineering applications. The purpose of reservoir characterisation in general is to provide data for describing and modelling the reservoir in order to estimate reserves, forecast future performance, and optimize production. The testing of wells is especially important in exploration when reservoir data is scarce. scarce. The data from well test contibute contibute to reserve estimaestimation and are used to determine if reservoirs and reservoir zones are ecomomic. Well testing is also used in reservoir monitoring, by pro viding average and local reservoir pressure. These pressure data are import important ant input input to produc productio tion n optim optimiza izatio tion, n, but but also also contr contribu ibute te indiindirectly rectly to the reserv reservoir oir chara characte cteriz rizati ation on as input input to model model condi conditio tionin ning g (history matching). In production productionengine engineering ering welltestin well testing g also contribu contribute te by providing providing data on the state of the near-wel near-welll reservoir volume. volume. These These data are used to answer questions about near-well formation formation damage, and the need for and the effect of well stimulation treatments. The basic concept of well testing is described in Fig. 1: A signal signal is sent into the reservoir from the well by changing well production rate rate or pressu pressure, re, and the respon response se (press (pressure ure/ra /rate te chang change) e) is measur measured ed at the well. The analysis of the response is used to estimate reservoir proper propertie ties. s. Since Since the respon response se is the result result of a distur disturba banc nce e that that trave travell away from the well, the early responses are determined by the property erty in the the near near well well regi region on,, whil while e late laterr resp respon onse sess dete detect ct more more dist distan antt reserv reservoir oir featu features res.. The The respon response se may may also also be record recorded ed in anothe anotherr well well in order to investigate reservoir communication, this type of test is
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called an interference test, Typical information derived from well tests include permeability, distan distance ce to bounda boundarie riess and faults faults,, size size and shape shape of sand sand bodies bodies,, near near wellbore damage or stimulation (Skin), and length of induced fractures. tures. An example example of a well test interpretatio interpretation n procedure procedure (Horner analysis, see page 40 page 40)) used to estimate the permeability of the formation is shown in Fig. 2 Fig. 2..
Figure 1: The well test concept: A signal is sent into the reservoir, and a response is recorded. Early response give information about the formation, later responses detect reservoir boundaries, and responses in observation wells depend on reservoir communication.
Figure 2: Horner analysis, an example of well test interpretation: The bottom hole pressure record from build-up is plotted and used for estimating permeability and skin.
introduction
9
Measuring pressure and rates For well testing it is the pressure and the production rate (equivalently, injection rate) that are the most important measured quantities. The pressure pressure measured measured at the bottom of the well is referred to as the the bottom-hole bottom-hole pressure pressure (BHP). (BHP). This This is the prefe preferre rred d pressu pressure re meameasurement, as it is closest to the formation. Using the wellhead pressure involves back-calculating the BHP based on a well flow model, and this typically introduces too much uncertainty to be reliable for well test analysis. In modern wells, the pressure sensor is typically connected to the surface surface by cable. cable. This enables enables continuou continuouss ( every second) surface read readou outt of the the BH BHP P. When When seve severa rall rese reserv rvoi oirr zone zoness are are prod produc uced ed at dif different bottom hole pressures, it is common to have a pressure transducer in each zone. It is also common with pressure sensors both inside the tubing and in the annulus. The pressure sensor is normally placed at the top of the perforated zone, thus a hydrostatic correction is required to obtain the reservoir pressure pressure at different different heights in the reservoir reservoir (e.g. the depth at the middle middle of the perforati perforation). on). Transien Transientt pressure pressure tests utilize utilize the relative change in pressure, thus a constant correction factor will not influence influence the well test analysis analysis.. Howeve However, r, if the fluids between the pressure sensor and the height of interest is changing, this will influence the hydrostatic correction. As an example, consider a gas-liquid interface slowly moving up the tubing. This could severely influence the pressure readings, and thereby the well test analysis. Earlier Earlier pressure pressure gauges gauges included included mechanic mechanical al gauges gauges (based (based on chanchanges ges in stra strain in of a meta metall due due to pres pressu sure re chan change ges, s, e.g. e.g. a Bour Bourdo don n gaug gauge) e) and strain gauges (based on the change in resistance of an conductor due to pressure pressure changes). changes). While While mechanical mechanical gauges gauges might be preferred for extreme well conditions, present-day pressure gauges are mostly quartz gauges. A quartz gauge employs that the resonant frequency quency of a quartz crystal crystal changes with pressure. pressure. As the resonant frequency is sensitive to temperature in addition to pressure, it is common common to run two quartz quartz pressure gauges gauges in parallel. parallel. Only one of them is exposed to the surrounding pressure, while both are exposed to the temper temperatu ature. re. The temper temperatu ature re is measur measured ed by the gauge gauge solely solely exposed to temperature, and this temperature is used to correct the pressure reading from the gauge exposed to pressure. The flow rate is controlled either at surface or down-hole. For surface control it is important to distinguish the flow rates observed at the surface from the flow rates experienced by the reservoir downhole. Typical Typical flow gauges are turbines, turbines, Coriolis Coriolis meters meters and multiphase flow meters applying gamma ray attenuation at different energy levels. levels. In general, rate measurement measurementss has much lower lower quality than pressure measurements.
∼
Figure 3: The cable connections from two down-hole gauges coming out of the Christmas tree.
Figure 4: A schematic of the placement of down hole pressure gauge and flow gauge.
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Types of tests The main classes of well tests are drawdown test, buildup test, and interference test. In addition we have injection tests and falloff tests, which are the equivalents of drawdown and buildup tests for injectors. The Dill Stem Test (DST), is a special drawdown test that is often performed in exploration wells and newly drilled wells. In a drawdown test, a static, stable and shut-in well is opened to flow. For traditional analysis, the flow rate should be constant. Typical objectives for a drawdown test are to obtain an average permeabili ability ty ( ) of the the drai draina nage ge area area,, to esti estima mate te the the skin skin ( ), to obta obtain in pore pore volume of the reservoir reser voir,, and to detect reservoir heterogeneity. During a buildup test, a well which is already flowing (ideally constant rate) is shut in, and the downhole pressure is measured as the pressure pressure builds up. The objectives objectives includes includes obtaining obtaining average average permeability and skin , as with the drawdown drawdown test. In addition, addition, the buildup test is conducted to obtain initial reservoir pressure during the transient state ( ), and to obtain the average reservoir pressure ( ) over the drainage area during pseudo-steady state. In an injection test, a static, stable and shut-in well is opened to (water-)injection. Thus, an injection test is conceptually similar to a drawdown drawdown test, test, except except flow is into the well rather rather than out of it. In most cases the objectives of the injection test is the same as those of a production test (e.g. , ), but the test can also be used to map the injected water. For a pressure pressure falloff test, a well already already injection injection (ideally (ideally at at a constant rate) is shut in, and the pressure drop during the falloff period is measured as the pressure declines. Thus, the pressure falloff test is similar similar to the pressure pressure buildup buildup test. A pressure pressure falloff test is usually proceeded by an injectivity test of a long duration.
Drawdown test
Buildup test
Injection test
Falloff test
Obtained properties and time Due to the transient nature of a pressure front moving through the reserv reservoir oir,, the differ different ent classe classess of obtai obtained ned reserv reservoir oir proper propertie tiess are intrinsically linked to the time after the change in well rates. The different ferent reserv reservoir oir proper propertie tiess are organi organized zed acco accordi rding ng to time time in Tabl Table e 1.
Early time
Middle time
Late time
Near wellbore
Reservo rvoir
Reservo rvoir boundaries
Skin
Permeability
Reservoir volume
Wellbore storage
Heterogeneity
Faults
Fra Fractur tures
Dual poros orosit ity y
(sea sealing ling/n /non on-s -sea eali ling ng))
Dual Dual permea permeabi bilit lity y
Bounda Boundary ry pressu pressure re
Table 1: Time of measurement versus type of measurements
introduction
11
Other well tests In an interference test, one well is produced (rate change) and pressure (response) (response) is observed observed in a different different well. The main objective objective of an interference test is to investigate reservoir communication and continuity, continuity, including communication communication over faults and barriers. The specialised drill stem test (DST) is commonly used to test a newly drilled drilled well. well. The well is opened to flow by a valve valve at the base of the test tool, and reservoir fluid flows up the drill string. Analysis of the DST requires special techniques, since the flow rate is not constant as the fluid rises in the drill string. This is a severe example of a wellbore storage effect (see page 35 page 35). ).
Interference test
Drill stem test
Homogeneity and scales All reserv reservoir oir model modellin ling g assum assumes es that that the sub surfac surface e can can be descri described bed in terms of model elements.1 Each element have homogenous (constant) or slowly varying properties, a characteristic size and shape, and a corresponding length scale, or representative elementary volume (REV), over whitch property variations are averaged. Well Well testing provides data for determining the properties of these model elements. The relation between reservoir characterisatin and geological reservoir modelling is discussed in the corresponding chapter on pages 135 pages 135– –139 139.. In general each measurement type probes a certain reservoir volume, and is associat associated ed with a correspon corresponding ding measurem measurement ent scale. scale. Some measurement types and their corresponding scales (depth of investigation gation)) are shown shown in Tabl Table e 2. The invest investiga igatio tion n scales scales may may or may may not
Measuremen Measurementt type type
Approxima Approximate te length length scale scale (m)
Core
0.1
Well log
0.5
DST/RFT
1–10
Well test
0.1–500
Production data
100–1000
correspond to geological modelling scales (REV). In contrast to most other methods, a well test does not probe properties at a fixed length scale: scale: The early early time time part part of the test test probe probe small small scale scale proper propertie tiess near near the well, and later later part of the test probe larger larger scale scale properties properties further away from the well. Note also that, apart from the production production data itself, well testing is the only mesurement type that supply data on the scales that are directly relevant for reservoir simulation.
1
Mark Bentley Philip Ringrose. Reservoir Model Design. A Practitioner’s Guide. Springer, 2015. Model elements are three-dimensional three-dimensional rock bodies which are petrophysically and/or geometrically distinct from each other in the specific context of the reservoir fluid system.
Table 2: Some measurement types and their corresponding measurement length scales
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Inverse problem It should be noted that well test interpretation in isolation is actually an ill-posed inverse inverse problem problem . Well testing testing tries to describe describe an unknown system by matching parameters in a model to measurement. The model parameters hopefully correspond to a reality in the subsurface, but the solution is generally non-unique both in terms of modeland model and modelpara model parameters meters.. Severe Severe simplific simplification ations/ass s/assump umptions tionsare are often made to obtain a unique solution, and a requirement for any useful well test analysis is that it is consistent with realistic geological concepts for the reservoir at hand.
An inverse problem starts with the results and then calculates the causes. A well-posed problem has an unique solution that changes continuously with the initial conditions.
Basic theory In this chapter we will derive the linear hydraulic diffusivity equation, which is the fundamental fundamental equation in well test analysis. analysis. There are a number number of impor importan tantt approx approxima imatio tions ns and assum assumpti ptions ons involv involved ed when deriving this equation. These assumptions includes that flow is isothermal and single phase, permeability is isotropic and independent of pressure, the fluid viscosity and compressibility is pressure indepe independe ndent, nt, the fluid fluid compr compress essibi ibilit lity y is low, low, andthat the well well is comcompleted across the full formation thickness.
The diffusivity equation The starting point for deriving the diffusivity equation is is the continuity equation for single phase flow which is an expression of conservation of mass in a volume element:
= ()) . −∇⋅ −∇ ⋅ ) = ( ) = − ∇ ∇ ⋅ (∇) ∇) = ()) . ∇∇ ⋅ ∇ + ∇ = + . ”mass in”
−
”mass out” ”change in mass”
(1)
Here is the fluid density, is the porosity, and is the volumetric fluid flux2 . The volumetri volumetricc flux is related related to the gradient gradient in pore pressure via Darcys law: (2)
Inserting Eq. (2 (2) into Eq. (1 (1), and assuming constant permeability, , and pressure independent viscosity, , we get (3)
We may may expa expand nd the the deri deriva vati tive vess of the the prod produc uctt on both both side sidess of Eq. Eq. (3), which gives:
(4)
We will first investigate the left hand side of Eq. 4 4:: Compressibility is a measure of the relative volume change as a response to a pressure change:
= −1 ,
(5)
Conservation of mass in a volume element For a general introduction to the mathematical notation, see the mathematical notes at page 139 page 139.. The nabla nabla symbol represents the del operator
∇ ∇
. The Gauss theorem,
which is used in the derivation of the continuity equation (1 (1) is discussed on page 140 page 140.. 2
Volumetric flux has the dimension of a velocity, and is often called Darcy velocity. velocity. Note that the Darcy velocity is always lower than the (interstitial) velocity of the fluid flowing through the pores.
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where is the volume, and the liquid compressibility may be used to conv convert ert deriva derivativ tives es of densit density y into into deriv derivati ative vess of pressu pressure. re. For a liquid liquid the density is is the fraction of mass divided by the volume of the liquid . Rewriting, we also have . We can then derive the liquid compressibility compressibility as: as:
= / = / = −1 = − / = − −1 = 1 . ∇ ⋅ ∇ = ∇ ⋅ ∇ = |∇| ∝ , ∇ . + = 1 + 1 . = + = 1 , = + . + = . ∇ = . = [] = [] = = , [][][] ( ⋅ )()
Liquid compressibility:
(6)
.
Thus, we see that the first term on the left hand side of Eq. (4 ( 4) is (7)
and since this term is proportional to the compressibility it may be ignored in the low compressibility limit. The left hand side of Eq. ( 4) is then simply
Low compressibility limit:
→
(8)
The The righ rightt hand hand side side of Eq. Eq. (4) can can be expr expres esse sed d in term termss of the the time time derivative of pressure by applying the chain rule,
The formation compressibility, compressibility,
(9)
, is defin defined ed3 as
Formation compressibility: 3
(10)
(11)
and
is called total compressibility, so Eq. 9 Eq. 9 is is simply
It is left to the reader to show, assuming that the sand grains themselves are incompressible, that the formation compressibility as defined here is also equal to the compressibility of a porous rock sample with total volume subject to an increased pore pressure: .
Total Total compressibility:
(12)
Equating left (Eq. 8 (Eq. 8)) and right (Eq. 12 (Eq. 12)) hand side and dividing by we get
(13)
This is the (hydraulic) diffusivity equation which is the fundamental equation in well testing, and the quantity
(14)
Hydraulic diffusivity
is called the (hydraulic) diffusivity. The unit for hydraulic diffusivity is
(15)
where we use that porosity is dimensionless. Using the notation for the diffusivity, we get the following simplified diffusivity equation:
The (hydraulic) diffusivity equation
basic theory
∇ = .
15
(16)
The diffusivity determines how fast pressure signals move through the reservoir, but it is important to note that the signal moves in a diffus diffusion ion proce process, ss, where where the actua actuall speed speed decrea decreases ses as it spread spreads. s. This This is very very differ different ent from from a seismi seismicc pressu pressure re wave wave which which move move at const constant ant velocity. velocity. By investiga investigating ting the different different elements elements of the diffusivit diffusivity y inEq.(14 inEq.(14), ), we observe that a pressure disturbance moves moves faster in a high permeable reservoir than in a low permeable reservoir. On the other hand, increased porosity, viscosity or compressibility reduces the speed of the pressure signal.
Diffusivity equation in oil reservoirs The diffusivity equation (Eq. 13 (Eq. 13)) was derived under the following assumptions: • Isotherm Isothermal al flow • A single single fluid fluid phase phase • Constant isotropic permeability • Fluid viscosity viscosity independent of pressure • Compressibility Compressibility independent independent of pressure • Low fluid compressibility compressibility As deriv derived, ed, the equati equation on is thus thus only only valid valid for for reserv reservoir oirss that that conta contain in a single single low-com low-compress pressible ible fluid phase, phase, that is for water water reservoirs. reservoirs. HowHowever, the validity of the equation may be extended to oil reservoirs at irreducibl irreducible e water water saturatio saturation, n, . The The irredu irreducib cible le water water does does not flow flow, but it influences the total compressibility, so the diffusivity equation is valid for oil reservoirs at provided the following:
= + +
• Permeability, Permeability, , is replaced by oil permeability 4 , • Viscosity Viscosity,, , is the oil viscosity,
= ()
.
• Total Total compressibility compressibility is defined as: as:
In this case the equation takes the form
∇ =
(17)
Gas reserv reservoir oirs, s, which which may may conta contain in a highly highly comp compres ressib sible le fluid, fluid, and reservoirs with several flowing fluids, can not be analyzed based on the simple hydraulic diffusivity equation. Well test analysis in these reserv reservoir oirss will will be discus discussed sed in separa separate te chap chapter terss (see (see page page 105 and 113 113))
4
The oil permeability at irreducible water saturation is typically very close to the absolute permeability, , and can thus be interchanged.
≃
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Figure 5: Vertical fully penetrating well in a reservoir of constant thickness
Vertical Vertical fully penetrating well – Radial flow We will will cons consid ider er a vert vertic ical al full fully y pene penetr trat atin ing g well well in a rese reserv rvoi oirr of cononstant thickness, as illustrated in Fig. 5 Fig. 5.. In this case it is natural to use cylinder coordinates. The general form of the Laplace operator on the pressure field in cylinder coordinates is
∇ = 1 + 1 + , = 0 = 0 1 = . − ) ( ) = () , = 2ℎ 2ℎ = (2ℎ) = () , ℎ () = . = + = + = + ≃ .
∇
(18)
where is the radius, is the angle, and is the height. height. For a fully penetrating well in an isotropic medium flow is independent of angle and height so that we have and . The diffusiv diffusivity ity equation (16 (16)) is then
(19)
Alternatively, it is illustrative to derive the radial diffusivity equation (19 (19)) starting from the radial form of the continuity equation (1 ( 1):
where we obtain
(20)
is the cylinder area. Employing Darcy’s law, Eq. (2 ( 2),
(21) (22)
where we have used that , , and are constants. From Eq. (12 (12)) we have
(23)
For the left hand side we obtain
(24) (25)
The radial diffusivity equation. That is the diffusivity equation for a vertical fully penetrating well in cylinder coordinates
basic theory
Here we used Eq. (6 (6) for the third equality, and applied the low compressibility limit (small liquid compressibility ) for the last similarity. Combining these results, we get the equation for radial flow:
1 = .
(26)
Flow regimes At different times in a well test the solution of the diffusivity will be in one of three possible stages of flow regimes: Initially we have a general unsteady state, or transient, flow situation where pressure change differently with time dependent on position: (27)
(,) = (,, )
Analysis of the transient flow period is the main concern of well testing. Very late late in the test test the reserv reservoir oir reach reaches es a semi semi steady steady state state where where the the pres pressu sure re profi profile le is cons consta tant nt and and pres pressu sure re chan change gess at the the same same rate rate 5 everywhere :
(,) =
(28)
Analysis of the time to reach semi steady state and the rate of pressure sure chang change e in this this period period give give inform informati ation on about about the reserv reservoir oir shape, shape, area, and volume. In the case that the reservoir pressure is supported by a strong aquifer, or by pressure maintenance operations (water or gas injection), pseudo steady state is replaced by true steady state flow:
(, ) = 0 .
(29)
Steady state solution We will will first first invest investiga igate te the steadystate steadystate soluti solution on of radia radiall flow flow (Eq. (Eq. 19 19). ). Except for early times in a well test, this solution describes the pressure sure profilearoun profilearound d a verti vertica call fully fully penetr penetrati ating ng well. well. Steadystatemeans Steadystatemeans that that the the righ rightt hand hand side side of Eq. Eq. (19 19)) is zero zero,, so the the stea steady dy stat state e pres pressu sure re profile can be found by solving
= 0 . = 1 , − = .
(30)
We may integrate Eq. (30 ( 30)) and get
(31)
where is an integration constant. Furthermore, using integration by substitut substitution ion to integrate integrate both sides from the well radius radius to , we get
where
ln
is the well pressure.
(32)
Transient flow
Semi steady state flow 5
In some contexts the term pseudo steady state is used for semi steady state flow.
Steady state flow
17
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Darcys law tells us that the volumetric flux is proportional to the pressure gradient:
= − ∇ . = 2ℎ ,
(33)
The volumetric fluid flux, , is defined as volumetric rate per area, so given the total down hole (reservoir) well production rate 6 , , we have
ℎ
(34)
where is the the perf perfor orat atio ion n heig height ht (whi (which ch in our our case case equa equals ls the the heig height ht of the reserv reservoir oir). ). The integr integrati ation on const constant ant is determ determine ined d by insert inserting ing the flux from Eq. (34 (34)) and the pressure derivative, , from Eq. (31 (31)) into Darcys law (33 (33): ):
∇ =
= 2ℎ , = + 2ℎ .
(35)
which when inserted into Eq. (32 32)) gives the general steady state solution for radial flow: ln
6
In well testing literature it is most common to derive equations in terms of surface rates. In the present text all equations are derived using down hole (reservoir) rates. The apparent only difference between the two approaches is that here is replaced by , where is the formation volume factor and is the surface well production rate, in those other texts. However, representing the relation between a reservoir rate and a surface rate, which involves for instance equipment such as test separators, in general by a simple constant factor underplays the non-triviality of the conversion.
The integration constant is determined by production rate via Darcys law
General steady state solution for radial flow
(36)
We will now discuss some of the characteristics of the solution (Eq. 36 (Eq. 36). ). The pressure profile has the form of a logarithmic singularity at at the well location, and the pressure changes increasingly fast close to the well and very slowly at larger distances. This means that most of the pressure drop from the reservoir into the well is located in the near well region, and any changes in the permeability in this region will have a significant influence on well productivity. In general, the difference in pressure (pressure drop) from to is given by
()−() = 2ℎ , ( ) − = 2ℎ . ln
and we see that the pressure profile, characteristic pressure scale :
(37)
is proportional to a Characteristic pressure scale
(38)
Note that is proportional to the rate and inversely proportional to permeability.
Skin Theformatio Theformation n volum volume e close close to thewellbore thewellbore typica typicallyhas llyhas altere altered d propproperties compared to the surrounding reservoir. Of highest importance for well productivity is an altered permeability, and the effect of this alteration on productivity is called skin. Skin is typically caused by formation damage as a result of drilling and production, production, but can also be the result of reduced mobility mobility due multipha multiphase se flow. flow. Intenti Intentional onal improve improved d permeabi permeability lity due to well treattreatments and hydraulic fracturing also contribute to skin, but the positive results of these treatments result in a negative skin in contrast
Skin is typically caused by formation damage.
basic theory
19
Figure 6: Pressure profile around a well with skin. Actual pressure profile in red, ideal profile without skin in blue.
to the normal normal positive positive skin due to formation formation damage. damage. Flow restricrestrictions in the wellbore itself including scale buildup, wax, and asphaltene deposits are not referred to as formation damage, but are quite often often includ included ed as part part of the skin. skin. Exampl Examples es of factor factorss that that contr contribu ibute te to skin are given below: • Formation Formation damage damage due due to drilling drilling mud filtrate invasion – Solids plugging from mud Clay-particle swelling or dispersion – Clay-particle – Emulsion blockage
• Formation damage due due to production – Fines migration – Deposition of paraffins or asphaltenes asphaltenes – Deposition of scale scale minerals
• Reduced Reduced mobility due to phase phase behavior behavior – Condensate banking (Production (Production below dewpoint) – Free gas (Production below bubble point)
As shown in Fig. 6 Fig. 6,, the effect of the skin is an additional pressure drop compared compared to a well without without skin. The effect effect can be described described quantitatively quantitatively by the dimensionless skin factor,
= 2ℎ = , = + + . >5
(39)
and the steady state solution (ref. Eq. (36 (36)) )) with skin is ln ln
Skin factor, .
Note:
(40)
< −3.5
Typically skin factors are considered bad and factors are viewed as excellent. excellent. The effect of skin can alternatively be described in terms of the equivalent (or effective) wellbore radius, . The conc concept ept is illusillustrated in Fig. 7 Fig. 7:: The additional additional pressure pressure drop due to skin can be ex-
20
l e c t u r e no no t e s i n we we l l - t e s t i ng ng
Figure 7: The concept of effective wellbore radius: The red curve shows the pressure profile with a positive skin (formation damage), corresponding to a reduced effective wellbore radius. The green curve shows the pressure profile with negative skin, corresponding to an increased effective radius. The radius is scaled by the well radius .
pressed in terms of the effective wellbore radius by noting that the ideal pressure profile is
() = + + , () = . = = . ln
ideal
(41)
and that by definition we have
ideal
(42)
Theextrapressure Theextrapressure drop drop dueto skin skin is then then found found bycombini bycombining ng Eqs. Eqs. (41 41)) and (42 (42): ): ln
Effective wellbore radius,
(43)
.
By compa comparin ring g Eqs. Eqs. (39 39)and( )and(43 43)) we see that that the equiva equivalen lentt wellb wellbore ore radius can be expressed in terms of the skin factor:
(44)
Productivity Productivity index The productivity index, PI, is a measure of well productivity, and is conceptually conceptually defined through 7
= ⋅ ( − ) , PI
(45)
where is the surface production rate, and is the reservoir pressure. Since the the pressure is not constant throughout the reservoir, equation (45 (45)) does not uniquely define as it depends on how and where the reservoir pressure is defined. For steady state, and semi steady state flow, the productivity index will be constant if the reser voir pressure is identified as the average reservoir pressure, so a natural choice is
= = 1 (,,,), ) ,
7
(46)
where is the reservoir volume We define define the equivalent equivalent radius, , as the radius radius where where the reserv reservoir oir
( )
The pressure difference is the driving force, and is called drawdown. Note that the definition of the productivity index vary in the literature, and some text alternatively define PI based on the pressure at the outer boundary or the reservoir.
Equivalent radius
basic theory
() = = 2ℎ + − .
21
pressure equals the average reservoir pressure, . Inserting Inserting this into Eq. (40 (40), ), and solving for the production rate gives
(47)
ln
Thus, production is proportional to the difference between reservoir pressure and well pressure, and introducing the formation volume factor (48) we have PI
= / = 2ℎ 1 + .
(49)
ln
As would be expected, we see that long wells, with a large radius and small skin, in a high permeability formation, have high productivity. For transient non-steady state flow, flow, the equivalent radius increase with time, corresponding to a decline in productivity, productivity, while for steady and and semi semi stea steady dy stat state e flow flow , and conse conseque quentl ntly y , is const constant ant.. To To get a measure for how depend on reservoir size, we will investigate the semi steady state solution for a finite circular reservoir with an outer radius . This will also allow us to quantify the relative importance of skin on productivity. The average reservoir pressure is given by the integral
2 1 = 2()) = + + , = + − 21 + , = − 21 , = ≈ 0.6 . = 2ℎ 1− + , || ≪ − 21 . = 1 = . − 2 ≈ 8 , >5 ln
(50)
that is
ln
where we have ignored the small terms proportional to
(51) that
originate from the lower lower integration limit. Comparing (51 (51)) and (40 (40)) we see that ln ln (52) or
(53)
We can insert (52 (52)) into (49 (49)) to get PI
(54)
ln
and and we see see that that we may may igno ignore re the the effe effect ct of skin skin on prod produc ucti tivi vity ty when when ln
As an example; if we have
500 m and and
(55)
0 1 m, we we get
ln
which can also be compared compared to the criterion for bad skin ( tioned on page 19 page 19..
) men-
Get average pressure by integrating the pressure profile.
Drawdown test In a drawdown test a non-producing well is opened for production at an ideally ideally constant constant rate, and the flowing bottom hole pressure pressure is recorded. recorded. An ideal drawdown drawdown rate schedule, schedule, with correspon corresponding ding pressure response, is shown in Fig. 8 Fig. 8.. In this this chap chapter ter,, a specia speciall case case of thesolution thesolution to the diffus diffusivi ivity ty equaequation will be derived. This solution, known as the infinitely acting line source fundamental solution, is directly relevant for analyzing a drawdown down test, test, and the inspec inspectio tion n of this this soluti solution on gives gives import important ant insigh insightt into into how how pressu pressure re signal signalss move move in the reserv reservoir oir.. Furth Further er,, we will will show show how a drawdown test can be used to derive reservoir and well properties such as permeability and skin factor.
Figure 8: The ideal drawdown schedule.
Transient solution for drawdown test In this section we will derive a solution to the transient behavior of a drawdown drawdown test in an infinite infinite reservoir. reservoir. Since Since this solution solution is not influenced by any boundaries it is called an infinite acting solution. In the derivation we will treat the well as a line source, i.e. we will ignore ignore any effects due to the finite well radius. radius. The infinite infinite acting line-s line-sour ource ce soluti solution on is the most most fundam fundament ental al soluti solution on in well well testin testing. g. and the analysis of the spatial and temporal behavior of this solution gives valuable valuable insights. The line source solution is accurate accurate for distances far away from the well at all times, but is not accurate for the well pressure at short times. At the end of this subsection we will compare well pressures from the approximate solution to the exact finite wellbore solution to establish its range of validity. Thediffusiv Thediffusivityequat ityequationin ionin cylin cylinder der coord coordina inates tes was was given given byEq. (19 19). ). Usin Using g the the prod produc uctt rule rule for for diff differ eren enti tiat atio ion, n, we can can writ write e out out Eq. Eq. (19 19)) as
1 = 1 + = 1 1 + − 1 = 0 .
,
Note:
(56)
and rearranging the terms, this gives the following equation:
(57)
We are searching the transient solution for a constant rate drawdown test in an infinite infinite reservoir. reservoir. This solution solution will also be valid for the early times of a test in a finite reservoir, i.e. for times shorter than the time needed for the pressure signal to travel to the outer (nearest) est) bounda boundary ry.. We will will assum assume e a line line sourc source, e, which which means means that that we will will
(hydraulic diffusivity).
24
l e c t u r e no no t e s i n we we l l - t e s t i ng ng
=0
take the inner constant flow boundary condition at and not at . This This is equi equiva vale lent nt to the the corr correc ectt boun bounda dary ry cond condit itio ion n as long long as we consider large distances. Since a semi steady state pressure profile is established in the near well area after a relatively short time, the line source solution will also be valid for times larger than a characteristic time, and the skin effect will amount to an additional constant pressure drop, , except for short times. In order to derive the solution we will assume that it has a special form form and then then proce proceed ed to show show that that this this form form satisfi satisfies es the differ different entia iall equation equation and the boundary condition conditions. s. The assumptio assumption n is that the solution is on the form
(,) = 4 = () . / = /(4) = 4 = 42 = 2 = 4 = 24 = 2 , = 4 = −4 = − 2 1 = 1 = = + = 2 + 2 . −1 = −1 = = 4 4 + 4 + 4 = 0 , ((1 + ) + = 0 . = /(4) = 0= 0 (, 0)0) = (∞,) = , =∞
(58)
As the hydraulic diffusivity has unit m2 s, s, we see that is a dimensionless sionless variable. variable. We will now perform perform a change change of variable variable from and to to in the diffusivity equation (eq. 57 in 57). ). Observing Observing that
(59)
we get the following equalities:
(60)
Collecting these terms, the diffusivity equation (eq. 57 (eq. 57)) has the form
(61)
which can be simplified to
(62)
Since Eq. (62 (62)) is an equation of only, the assumed form (Eq. 58 (Eq. 58)) is a possible solution. Given this form of , a boundary condition at (line source) corresponds to an inner boundary condition at . We also have a specified initial pressure in the whole reservoir and a constant pressure at infinity, infinity, (63) where the constant pressure at infinity corresponds to an outer boundary condition at .
The source at the well is approximated approximated by a line source.
drawdown test
25
It can be shown, by back substitution, that the solution to Eq. ( 62 62)) has the form (64)
( ) ) = + ( ) ) , () = − , ∞ () = () =
where
is a function that satisfy the condition is
(65)
and and are constants to be determined by the boundary conditions. The function E 1
(66)
is an exponential integral, which is tabulated and available as a special function in math packages. 8 Since
∞ = ∞ −
,
(67)
we can apply the first fundamental theorem of calculus to obtain Eq. (65 (65). ). The exponential integral function E1 is plotted in Fig. 9. The plot indicates that E1 , which we will use to determine the constant : Since E1 , the outer boundary condition implies implies (68)
( ) ∞ = 0 ( ) ( ∞ ) = ∞ = 0 ( ) (∞) = = , () = + ()) = ( + ()) = = − . =0 2ℎ = = = = − 2 = − = 0 = −4ℎ .
( )
8
The exponential integral E 1 is available as expint in Matlab, scipy.special.exp1(x) in Python, and Ei(1,x) in Maple. Note that traditionally the function
∞ Ei
is used in well testing literature. Using this function we have Ei
which gives gives a r ather awkward notation. The two functions give identical results for positive real values of , so tabulated values for Ei found in the literature may be used for E1 .
so . The inner boundary condition is used to determine the constant . We first note that
(69)
A constant total rate at then translates to a condition on the pressure gradient via Darcys law:
Figure 9: The exponential integral
(70)
In the last equality we invoke the first equality in Eq. (59 ( 59). ). The inner boundary condition is at , so Eq. (70 (70)) gives
(71)
Substituting the integration constants from Eqs. (68 (68)) and (71 (71)) into Eq. (64 (64)) gives the final expression for the infinite acting line source solution for a constant rate drawdown test:
Infinite acting line source solution.
26
l e c t u r e no no t e s i n we we l l - t e s t i ng ng
= − 4ℎ 4 . = − 21 41 . E1
Using the notation for hydraulic diffusivity and teristic pressure, this equation simplifies to:
(72)
for the charac-
E1
(73)
After a small example, we will investigate some of the properties of this solution.
Example We will employ Eq. (72 ( 72)) using the reservoir data in the margin, Table 3 ble 3.. In Fig. Fig. 10 10 we we have plotted Eq. (72 (72)) for afte afterr 1 to 4 days. days. As we see from the left figure, the first days witness a large change in the pressure profile, while the subsequent days yields subsequently smalle smallerr change changess from from the previo previous us day. day. We also also observ observe e that that the prespressure profile at a radius of 1000 m begins to significantly deviate deviate from the initia initiall pressu pressure re after after approx approxima imatel tely y 4 days. days. This This indica indicates tes that that the infinite acting solution will be invalid at this distance after 4 days. In the the righ rightt figur figure e we have have plot plotte ted d the the same same grap graphs hs in a loga logari rith thmi micc plot. This plot shows clearly that the change in the pressure profile diminish with time. It also indicates that the pressure profile is close to linear linear when when plotte plotted d on a logari logarithm thmic ic scale. scale. This This will will be invest investiga igated ted further in the next section.
SPE Metric
SI
0 1 mD 10 m
9 869 869 10 m
100 100 m3 d
1 16 16
/ ×
× × / × ××
0 12 m
0 12 12 m
1 cP
1
0.2 1
10
200 bar
4
bar
1
14
10
10
3
m3 s
10
3
Pa s
1
10
9
Pa
2
107 Pa
0.2
Table 3: Basic data for example.
Figure 10: Vertical fully penetrating well in a reservoir of constant thickness
Logarithmic approximation
)) ( ) 0 () ≈ 1 − , = 0.5772… 5772… () (1/1/) − 14 < < 0.01 ,
The exponential integral E1 has a logarithmic singularity at , and for small it can be approximated by a logarithm: E1
ln
= (74)
where is the Euler(–Mascheroni) constant. A comparison between E1 and ln is shown in Fig. 11 Fig. 11.. In this this plot, plot, the functi functions ons are simila similarr for for value valuess smalle smallerr than than . This indicates that the logarithmic approximation is fair for
10 = 0.01
(75)
Figure 11: A plot comparing the exponential integral E 1 in blue and ln in green.
//
m2
1
drawdown test
(,) = − 2 4 − . ()−() = .
thus the approximation is fair for small radii and late times . Employing the approximation given by Eq. 74 Eq. 74 we we obtain ln
(76)
27
The exponantial integral is replaced by the logarithmic approximation for small (large or or small )
By subtracting the pressure given by Eq. (76 (76)) at two different distances, and we see that ln
(77)
independent independent of time. This is identical identical to the expressi expression on for steady state state (Eq. (Eq. 37 37), ), whic which h mean meanss that that a loga logari rith thmi micc semi semi stea steady dy stat state e pres pres-sure profile has developed around the well. The exponential integral is exponentially small for large values of , and we have the approximation:
() ≈ . () / 10 () ≃ 0 = E1
(78)
Fig. 12 Fig. 12 is is a comparison between E 1 and and , and this plot shows that the functions are similar for values larger than . As E1 for large values of , the pressure is undisturbed ( ) at large distances or short times.
Figure 12: A plot comparing the exponential integral E 1 in blue and in green.
/
Figure 13: Pressure profile at a fixed time (blue line). A logarithmic semi steady state pressure profile has de veloped around the well, and pressure beyond a certain distance is essentially unchanged. (In this figure is given in units of , and is given in units of
)
The pressure profile at a fixed time is illustrated in Fig. 13 13.. For small small distan distance cess the pressu pressure re profile profile is match matched ed by the logari logarithm thmic ic approxim proximate ated d given given by Eq. 76 76.. For For larg large e dist distan ance cess the the pres pressu sure re is esse essenntially unchanged and still at initial pressure. The time–spa time–space ce depend dependenc ency y of the pressu pressure re is given given by E 1 , shown in Fig. 14 Fig. 14.. We see that the pressure is essentially unchanged (E1 0 01) 01) wherev wherever er , which which corre correspo sponds nds to a distan distance ce :
( ) < .
>3 = √12 √12 . = = 3 ,
The speed at which the pressure front is moving,
( ) >
(79)
, is given by Figure 14: The exponential integral
(80)
28
l e c t u r e no no t e s i n we we l l - t e s t i ng ng
The speed of the pressure front is diverging at early times, and is diminishing as the front moves away from the well. As mentioned earlier the line source solution is not valid for early times close to the wellbore, so the divergence at is unproblematic. unproblematic. In addition to the location of the pressure front (Eq. 79 79), ), we are interested in the radius within which the pressure is significantly in fluenced by by the test. This radius is called radius of investigation, inv inv , and is somewhat arbitrarily defined as the radius where :
→0
=1
= √ 44 . = 4 .
inv inv
(81)
Eq. 81 Eq. 81 can be inverted in order to determine the time needed to in vestigate reservoir features at a given distance from the well: inv
inv inv
(82)
Well Well pressure The primary measurement in a well test is the bottom hole well pressure. The line source solution is not valid at the well radius for short times, but for late times we can get the well pressure by inserting the well radius in Eq. (76 76). ). Additional Additionally ly,, the effect effect of skin must be accounted for. for. Since a steady state pressure profile develops in the near well area, the skin effect amounts to an additional constant pressure drop except for very short times. From the steady state solution (Eq. 39 (Eq. 39)) we have (83)
which gives
= , () = − 2 4 − + 2 . ln
(84)
Dimensionless analysis Dimensionless variables simplify our equations by embodying well and reservoir parameters that are assumed constant, e.g. well radius, permeability, viscosity, porosity, compressibility, perforation height etc. Using dimensionless variables then yields solutions that are independ dependent ent of such such well well and and reserv reservoir oir param paramete eters, rs, which which enable enabless comcomparisons of pressure behavior from different wells. Additionally, Additionally, dimensio mensionle nless ss varia variable bless yields yields soluti solution on that that are indepe independe ndent nt of the unit unit system. We have already seen that the infinitely acting solution is characterized by the dimensionless group
= 4 .
(85)
The diffusivity equation itself can also be expressed on dimensionless less form form using using dimens dimension ionles lesss variab variables les for space space,, time, time, and and pressu pressure. re.
Radius of investigation
drawdown test
Dimensionless variables are variables measured in terms of a characcharacteristic scale for that variable. In the well testing context the characteristic length is the well radius with corresponding dimensionless radius (86)
= .
= =//
We introduce a similar dimensionless time variable is a characteris characteristic tic time. Using a general direction direction have the following equality’s: equality’s:
= = 1 . = = 1 ∇ = , 1 ∇ = 1 ,
29
In well testing, the well radius is a characteristic length.
, where , we
(87)
If we use the dimensionless space variables into the diffusivity equation (Eq. 16 (Eq. 16), ),
we obtain
∇
where (e.g.
(88)
(89)
use derivation with respect to the dimensionless variable ). From Eq. (89 (89)) we see that the diffusivity equation can be
written on a simple form, the dimensionless diffusivity equation:
∇ = , = . = 2ℎ , = − = ( − ) 2ℎ .
(90)
if the characteristic time is defined as
/
(91)
The characteristic time is defined such that the dimensionless diffusivity equation has a simple form.
From Eq. (15 (15)) we have that the unit for the hydraulic diffusivity is m2 s, s, thus we observer that is actually dimensionless. We see from from the general general steady steady state solution solution for for radial radial flow (Eq. 36 (Eq. 36),that ),that the characteristic scale for pressure is
and we may also use use the initial reservoir pressure, pressure. The dimensionless dimensionless pressure pressure is then
(92) , as a datum
(93)
Note that the derivativ derivative e of the dimensionl dimensionless ess pressure and the true pressure have opposite sign. Summ Summin ing g up Eqs. Eqs. (86 86), ), (91 91),and( ),and(93 93), ), we have have the follo followin wing g dimendimensionless variables:
= = = −
.
Dimensionless pressure is measured relative to a datum pressure .
(94)
These are the dimensionless variables most commonly used in the well testing context.
30
l e c t u r e no no t e s i n we we l l - t e s t i ng ng
The general steady state solution for radial flow (Eq. 36 (Eq. 36), ),
= + , − = − − . = + () . ln
(95)
can be rearranged as:
ln
(96)
Substi Substitut tuting ing with with dimens dimension ionles lesss variab variables les,, we then then obtai obtain n the dimendimensionless form of the general steady state solution for radial flow: ln
(97)
To transl translate ate equati equations ons with with partia partiall deriva derivativ tives es into into the dimens dimension ion-less variables given above, we need to replace the partial derivatives with their dimensionless versions:
= 1 = = − . ∇ = ∇ ⋅ ∇ = ∇ ⋅(−)∇ = − . ∇ = . = = 1 (−)1 = − . 1 = .
(98)
From the diffusivity equation using dimensionless dimensionless radius and time Eq. (90 (90), ), we can change of variable to dimensionless pressure:
(99)
By dividing out the (scalar) characteristic pressure on both sides, we obtain the dimensionless diffusivity equation as
(100)
Substituting Eq. (99 (99)) into the diffusivity equation in cylindrical coordinates we obtain
(101)
By rearranging this equation, we obtain the dimensionless diffusivity equation with cylindrical coordinates:
(102)
Similarly, the infinite acting line source solution for the pressure in the reservoir (Eq. 72 (Eq. 72)) is:
( , ) = 21 44 , E1
(103)
drawdown test
31
and the logarithmic approximation for well pressure in the infinitely acting drawdown test (Eq. 84 (Eq. 84)) is
( ) = 21 ( (4 4) − + 2) ln ln
.
(104)
Note Note that that dimens dimension ionles lesss varia variable bless are not unique unique,, and the ones ones defined in Eq. (94 (94)) are not the only set used even in the context of well testing. When setting up the characteristic dimensionless equations for for dete detect ctin ing g the the dist distan ancce to a seal sealin ing g faul faultt it is for for inst instan ancce more more nat natural ural to use use the the dist distan ance ce to the the faul faultt as a char charac acte teri rist stic ic leng length th than than the the well radius.
Validity Validity of equations for well pressure The logarithmic expression for the well pressure given by Eq. 104 Eq. 104 is is only valid for times longer than some characteristic time, dependent on the wellbore radius, . The error has two sources: first, the logarithm is an approximation to the exponential integral, E 1 , and second, the constant rate boundary condition is incorrectly placed at instead of at the sand face, . We will show below below that that the requirement that the effect of these two approximations shall be neglig negligibl ible e essent essential ially ly lead lead to the same same criter criterion ion for for select selecting ing the charcharacteristic time. The exac exactt infinit infinite e actin acting g soluti solution on (with (with the const constant ant rate rate bound boundary ary 9 condition at ) for the pressure in the well is
=0
(⋅ )
=
= ∞ ( ) = 2 1(−() (+− ())) , (⋅) (⋅) exp
J
Y
(105)
where and and are are Bessel functions10 . In Fig. Fig. 15 15 we we have plotted the infinite acting solution for well pressure given by equation Eq. (103 (103)) (red line), together with the expressions sions logari logarithm thmic ic appro approxim ximati ation on in Eq. (104 104)) (blu (blue e line line), ), and and the the exac exactt solution given by Eq. (105 (105). ).
9
A. F. Van Everdingen and W. Hurst. “The Application of the Laplace Transformation to Flow Problems in Reser voirs”. In: Petroleum Transactions, AIME (Dec. (Dec. 1949), pp. 305–324. 10
The integral in Eq. 105 Eq. 105 is is well behaved and can be evaluated using a standard integration routine in most math packages. It should be noted, however, however, that a the more general expression for that appear in A. F. Van Everdingen and W. Hurst is not that well behaved.
>
Figure 15: Comparing the exact solution for well pressure (blue line) with the line source solution (red line) and the logarithmic approximation (green line). and are dimensionless pressure and time as defined in Eq. 94 Eq. 94..
32
l e c t u r e no no t e s i n we we l l - t e s t i ng ng
The infinite acting solution for well pressure given by equation Eq. (103 (103)) is accurate within 1% for
which corresponds to
> 100100 , > ⋅ . 100
(106)
(107)
This is reflected in the plot in Fig. 15 15.. This validity range is in good agreement with the validity range obtained for the logarithmic approximation, given by Eq. (75 ( 75), ), which for gives
= > ⋅ 4 , > 25 . 100
which corresponds to
(108)
(109)
We observe from the plots in Fig. 15 that the validity range for the logari logarithm thmic ic appro approxim ximati ation on is in fair fair agreem agreement ent with with thevalidityrange thevalidityrange of the infinite acting solution. Note that the early time behavior is also influenced by skin and wellbore-effects, wellbore-effects, and it will be shown (see page 37 page 37)) that the so called wellbore storage effect in most cases last for times longer that the time defined by Eq. (107 ( 107). ). Having determined the lower time limit for the validity of Eq. 84 84,, we now turn to the upper time for its validity. The well pressure behaves as if the reservoir were infinite as long as the pressure signal is not yet reflected back from the outer (nearest) boundary. Let the distance to the outer boundary be , then the reflected pressure change must travel to influence influence pressures pressures near the wellbore. wellbore. The time– space dependency of the pressure is given by E 1 , and the infinite acting solution is valid to within 1% as long as E 1 0 01, 01, that is for or
>4
2
< 4
((()(2) < .
(110)
Eq. 104 Eq. 104 is is valid (eqs. 107 (eqs. 107 and and 110 110)) for dimensionless times
where
100 < < 4 ,
(111)
is the dimensionless distance to the outer boundary.
Permeability and skin In this this sectio section n we will will descri describe be how how the reserv reservoir oir permea permeabil bility ity and the skin factor of the well can be found by analyzing the well test data. The The pres pressu sure re resp respon onse se in an idea ideall draw drawdo down wn test test was was deri derive ved d in the the previous section (Eq. 84 (Eq. 84); );
() = − 2 4 − + 2 . ln
(112)
Validity Validity of infinite acting solution
33
drawdown test
= /(2ℎ) 4 () = −4ℎ ()++ − 4ℎ − + 2 = () + , = −/(4ℎ) ()
Rearranging this equation equation and writing writing out the the characteristic characteristic pressure , we get ln
ln
(113)
ln
where and are constants independent of time. We see from Eq. (113 (113)) that if we plot the well pressure as a function of ln (semilog (semilog plot), we should see a straight line with slope . Figure 16: Semilog plot of a drawdown test
The early part of the data will deviate from the straight line due to wellbore and near wellbore effects, and the late time part of the data will deviate due to the effect of boundaries or large scale heterogene geneit itie ies, s, but but perm permea eabi bili lity ty and and skin skin may may be esti estima mate ted d usin using g Eq. Eq. (112 112)) based on the part of the data that fall on a straight line. Permeability may be estimated based on the slope, of the line, . As illustrated in fig. 16 fig. 16::
= −4ℎ 1 .
(114)
Once the permeability is known, the skin can be estimated from the fitted straight line intercept or value on the straight line at any other point in time :
= 21 () − − 4 + . ln
(115)
Example We will go through an example to illustrate the procedure. Basic data for the well test are given in Table 4 Table 4,, and the recorded pressure data are given in Table 5 Table 5.. A semilog plot of the data are shown in Fig. 17 Fig. 17.. The time between 0 1 and 10 h can be fitted to a straight line. line. Based Based on the slope, , of this line, the permeability can be estimated using Eq. (114 ( 114). ). We have
.
4ℎ = . × / 9 23 23
10
9
m2 Pa
(116)
SPE Metric
SI
10 m
10 m
/ ×
100 100 m3
d
× / × ×
1 16 16
0 12 m
0 12 12 m
1 cP
1
0.2 1
10
4
bar
1
10
3
m3 s
10
3
Pa s
10
9
Pa
0.2 1
Table 4: Basic data for the example well test.
1
34
l e c t u r e no no t e s i n we we l l - t e s t i ng ng
time time (h) (h)
pres pressu sure re (bar (bar))
time time (s) (s)
pres pressu sure re (Pa) (Pa)
ln(t ln(tim ime) e)
0
100.0
0.0
10000000
-∞
0.002
98.6
7.2
9860000
1.974081026
0.01
93.4
36
9340000
3.5835189385
0.02
89.15
72
8915000
4.276666119
0.05
87.3
180
8730000
5.1929568509
0.1
86.6
360
8660000
5.8861040315
0.2
86
720
8600000
6.579251212
0.3
85.6
1080
8560000
6.9847163201
0.4
85.3
1440
8530000
7.2723983926
0.5
85.1
1800
8510000
7.4955419439
1
84.5
3600
8450000
8.1886891244
1.5
84.1
5400
8410000
8.5941542326
2
83.9
7200
8390000
8.881836305
3
83.5
10800
8350000
9.2873014131
4
83.2
14400
8320000
9.5749834856
5
83
18000
8300000
9.7981270369
6
82.8
21600
8280000
9.9804485937
7
82.7
25200
8270000
10.1345992735
8
82.6
28800
8260000
10.2681306661
9
82.5
32400
8250000
10.3859137018
10
82.4
36000
8240000
10.4912742174
11
82.3
39600
8230000
10.5865843972
12
82.2
43200
8220000
10.6735957742
13
82.1
46800
8210000
10.7536384819
14
82.1
50400
8210000
10.8277464541
15
82
54000
8200000
10.8967393255
16
81.9
57600
8190000
10.9612778467
17
81.9
61200
8190000
11.0219024685
18
81.8
64800
8180000
11.0790608823
19
81.8
68400
8180000
11.1331281036
20
81.7
72000
8170000
11.184421398
21
81.7
75600
8170000
11.2332115622
22
81.6
79200
8160000
11.2797315778
23
81.6
82800
8160000
11.3241833404
24
81.6
86400
8160000
11.3667429548
36
81
129600
8100000
11.7722080629
48
80
172800
8000000
12.0598901354
60
78.4
216000
7840000
12.2830336867
72
77.2
259200
7720000
12.4653552435
84
75.9
302400
7590000
12.6195059233
96
74.6
345600
7460000
12.7530373159
Table 5: Example pressure data for estimating permeability and skin
drawdown test
Figure 17: Plot of example data
Ideally we should estimate the slope by fitting all points in the linear interv interval al to a strai straight ght line line (linea (linearr regres regressio sion), n), but but we may may also also estima estimate te the slope by using the endpoints of interval:
) = () = ()−( = − . × (−100). × =. × = .. ×× = . × = = = × = 4 ⋅ ( ) = . , ( − () = × − . × = . × . = 21 . . ×× − . + . = . ln
ln
106 ln
8 66 66
8 24 24
104 Pa
9 12 12
106
Pa
(117)
1
1
Inserting Eqs. (116 (116)) and (117 (117)) into Eq. (114 (114)) gives the permeability estimate: 9 23 23 10 9 2 m 1 01 01 10 13 m2 9 12 12 104 (118) 101mD The skin can be estimated based on the initial pressure at 100bar 100 105 Pa and any pressure on the straight line. We will use the pressure at 1 h: ln
3600s 3600s
13 1
(119)
105 Pa
(120)
and the pressure difference is 100
105 Pa
84 5
105 Pa
15 5
We get a skin factor estimate by inserting Eqs. (119 ( 119)) and (120 (120)) into Eq. (115 (115): ): 15 5 9 12 12
105 104
13 1
0 5772 5772
2 2
(121)
Wellbore storage effect Due Due to the the pres presen ence ce of the the well wellbo bore re,, the the idea ideall cons consta tant nt flow flow boun bounda dary ry conditions are never obtained. The effect is especially important for the early time part of surface controlled well tests, and it is crucial
35
36
l e c t u r e no no t e s i n we we l l - t e s t i ng ng
to distinguish the effects of wellbore storage from the interpretable reservoir response. Actual down hole boundary conditions are illustrated in Fig. 18 Fig. 18.. When a well is opened there is a lag time before the flow through to sand sand fac face reac reach h a const onstan antt rate rate,, and and when when a well ell is shut shut in, in, the the sand sand face flow rate does not go to zero instantaneously. In the following a simple model for the wellbore storage effect will be presented. presented. The model is based on the assumption assumption of a wellbore wellbore filled with a liquid with constant compressibility, . . The initial production at constant surface rate is due to expansion of liquid in the wellbore. The situation is illustrated in Fig. 19 Fig. 19..
Figure 18: Non-ideal flow schedule due to wellbore storage
()
Figure 19: The liquid filled wellbore and the corresponding model for the wellbore storage effect
If is the production rate from the reservoir, and is the con is trolled production rate at the surface, we may express the mass balance in the wellbore volume,
“Mass in” - “Mass out” = “Change in mass” as
(()− ) = () = , =
,
/
Note: volume factor
(122)
where the total wellbore volume is assumed constant, and is the liquid compressibility as defined in Eq. (6 ( 6).
=
The equation governing the well pressure is
()− = , = . () () ≈ 0 = −
Wellbore storage constant.
(123)
Here is defined as the wellbore storage constant. The reservoir production rate serve as a boundary condition serve for the diffusivity equation that that governs the pressure in the reservoir. We have have a set of coupled equations for reservoir and well. However, However, for small most most of the production is due to the expansion of the fluid in the wellbore, and we have :
is the formation
(124)
Initially, the well pressure varies linearly with time due to wellbore storage.
drawdown test
37
This early time wellbore storage equation can be utilized to estimate the wellbore storage constant coefficient from a time versus pressure plot. When the liquid liquid compress compressibil ibility ity is is known, known, this this wouldthen wouldthen give give an estim estimate ate of the wellb wellbore ore volum volume e . A large large misma mismatch tch betwe between en the tubing tubing and casing casing volum volume e below below the test test valv valve e and the estima estimated ted wellwellbore volume from the early time wellbore storage equation could be an indica indicatio tion n of e.g. e.g. fractu fractures res around around the well, well, as such such fractu fractures res would would effectively contribute contribute to the wellbore volume in our equations. We may introduce the dimensionless time and well pressure variables (Eq. 94 (Eq. 94))
= = ( − ) 2ℎ , = 1 ,
to express Eq. (124 (124)) as
(125)
where is a dimensionless dimensionless wellbore wellbore storage storage constant constant.. We see by inserting the dimensionless variables into Eq. (124 ( 124)) that the dimensionless storage constant is
= 21 1ℎ . = 2 ℎ
(126)
From Eq. (126 (126)) we observe that the is proportional to the ratio of total wellbore volume to the wellbore volume in the completed interval. The dimensionless wellbore storage constant is therefore always larger than one, and can be very large in long and deep wells. Value aluess clos close e to one one can can only only be obta obtain ined ed in a test test cont contro roll lled ed by down down-hole rates and build up tests controlled by downhole valves. Inspection of the solution to the fully coupled wellbore–reservoir system have shown 11 that the wellbore storage effect must be taken into account for times 60 (127)
≫ 1
<
.
Dimensionless wellbore storage constant.
Since typically , wellbore storage tend to obscure the region where we need to take into account account the difference between a finite wellbore solution and the line-source solution (Eq. 107 (Eq. 107). ). The model for wellbore storage that has been presented in this chapter is very simplified, and the validity of the model in real world situa situatio tions ns may be questi questione oned. d. The main main compl complica icatio tion n may may be that that oil wells tend to produce two phase at surface, and that prior to a buildup the wellbore usually contains a two-phase mixture which segregate after shutdown. In a drawdown test the situation is reversed, and in an appraisal well test drawdown there is a period of rising liquid level until fluid reaches the wellhead.
Downhole control valves minimize wellbore storage. 11
R. Agarwal R. G. Al-Hussainy and H. J Ramey. “An Investigation of Wellbore Storage and Skin in Unsteady Liquid Flow - I. Analytical Treatment”. Treatment”. In: SPE Journal 10.3 (Sept. 1970), pp. 279–290.
Figure 20: Segregating two-phase mixture in well-bore
Buildup test In a buil buildu dup p test test,, a well well that that idea ideall lly y have have been been prod produc ucin ing g at a cons consta tant nt rate is shut, and the bottom hole pressure pressure is recorded. recorded. Well tests in exploration wells and new wells are often performed as a drawdown– buildup sequence as shown in Fig. 21 21.. In additi addition on to provid providing ing data data for for reserv reservoir oir charac character teriza izatio tion, n, such such as permeability and skin, buildup tests can also provide data for reser voir monitoring, in particular reservoir pressure data. Planned buildup tests have a cost in terms of lost production, and this will limit limit the amount amount of availabl available e late time shut in data. HowHowever, production wells normally do not produce at 100% efficiency, and in wells with permanent downhole pressure gauges the periods of unplanned shut-in may provide valuable well test data at no extra cost. It is easier to get high quality data from buildup than from drawdown since maintaining zero rate is trivial compared to maintaining a fixed rate, and the effect of wellbore storage is minimized by using a dowhhole valve. Ideally rates should be stable before shut-in, and the determination of the effective production time, which is used in the analysis, can be challenging.
Superposition principle In this section we will present the superposition principle. Let a partial differential operator, such that
ℒ = 0 , = = ℒ ℒ (ℒ+()) == ℒℒ + ,ℒ
ℒ
be
(128)
is a partial differential equation for functions . Further, let
ℒ
(129)
be a operator giving the boundary conditions. In our case, would be the diffusivity equation in the region given by the reservoir, while would be an operator that restricts to the boundary of the reser voir (e.g. the boundary given by ), and would be the function that is required required to equal equal on the boundary boundary of the reservoir reservoir (e.g. a constant rate in the well). We say that an operator is linear if if
(130)
Figure 21: Ideal drawdown–buildup sequence
40
l e c t u r e no no t e s i n we we l l - t e s t i ng ng
ℒ
where is a scala scalar. r. The linear linearity ity of an operat operator or makes makes it impossib impossible le to do operations such as taking the square of the function , but it allow allowss for for taking taking the second second deriva derivativ tive. e. Thus, Thus, the diffus diffusivi ivity ty equati equation on (Eq. 13 (Eq. 13)) is a linear partial differential equation. The superpositio superposition n principle principle states states that that for for all linear linear system systems, s, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. We observe that solutions to linear differential equations with linear boundary conditions conditions obey obey superposit superposition: ion: Given Given two functions functions and and that that are solutions to with boundary conditions and (131)
(,) (,) ℒ = 0 = (,) = (,) , ℒ = + ℒ = 0 = (,)+(,) .
where both to
and
is linear linear operators. operators. Then
Superposition can be used to add solutions to linear differential equations with linear boundary conditions.
is a solu soluti tion on
(132)
with boundary condition
(133)
In this chapter we will use the superposition principle in time to obtain solutions to buildup in terms of the fundamental solution to drawdown derived in the previous chapter (Eq. 72 (Eq. 72). ). In later chapters we will show how the superposition principle can be used in space to obtain solutions that take into account the effect of reservoir boundaries (See page 63 page 63). ). The pressu pressure re soluti solution on for for a drawd drawdow own-b n-buil uildup dup sequen sequence ce in a well well is the the same same as the the sum sum of the the solu soluti tion onss for for two two well wellss in the the same same loca locati tion on that that is star starte ted d at diff differ eren entt time times. s. Note Note that that due due to the the oute outerr boun bounda dary ry condition, , it is the the pres pressu sure re that that has this this proper property ty.. An ideal buildup test has the rate history (boundary condition) shown in Fig. 22 Fig. 22.. By superposition, this can be replaced by two wells in the same location that is started at different times with opposite rates as shown in Fig. 23 Fig. 23..
(∞,) =
−
Figure 22: Buildup test rate history with production time and buildup time
Horner analysis
= 0 − = − = −4ℎ 4( + )) − + 2 − = 4ℎ 4 − + 2 ,
We will now derive the well pressure for a well test where both the production time and buildup time is short enough to employ the transient infinitely acting solution. Due to superposition, the expressi pression on for for well well pressu pressure re during during build buildup up will will conta contain in two two terms: terms: One for for a well well with with cons consta tant nt rate rate starti starting ng at time time , and and one one for a well ell with constant rate starting at time . Apart from the period domina dominated ted by wellb wellbore ore storag storage e and skin skin effect effectss just just after after start start of shutshutin, we can use the logarithmic approximation approximation (Eq. 84 (Eq. 84)) for both terms:
ln
ln ln
(134)
Figure 23: Equivalent two-well rate history for build up using superposition. One well (in blue) has rate and the other well (in red) has the opposite rate . The sum of the rates is the same as the buildup test rate in Fig. 22
buildup test
− ℒ = ℒ = 0 − ℒ (∞, ) = − = (− )+(4( − +)) 4 = −4ℎ − = −2 + . = − 2 + + = + , = − / 2 (() + )// ( + )//
41
where the well with pressure has rate and the well with pressure has rate . Note that when we superposition two solutions to the partial differential operator , we also add the boundary conditions . Since for any scalar when is the operator for the diffusivity equation, we can use to ensure that the outer boundary condition stays zero, thus for for all solutions. Applying the superposition principle, where we note that any effect of skin is equal in both terms and cancel out, we obtain:
ln
ln
(135)
ln
This can be rearranged as
Characteristic pressure
ln
(136)
ln
where and are const constant antss indepe independe ndent nt of time. time. Note that that equals the constant constant in Eq. (113 ( 113)) for the drawdown test. However, However, while the pressure was a function of ln in in the drawdown test, we observe that pressure is a function of ln in in this buildup test. The corresponding plot of versus is ln is is called the Horner plot (see Fig. 24 Fig. 24). ). Figure 24: An example of a Horner plot, where pressure is plotted as a function of ln . Permeability and skin is estimated from the straight line fit. Note that increasing time goes from right to left on the x-axis.
//
(( ++)//)/ → ∞ ( + )/ → 1 = = − 2 = −4ℎ ,
The time is sometimes referred to as Horner time. Ob is serve that decreases decreases when increases. increases. Further, infinite time corresponds to . Similar to the drawdown test, the permeability can be estimated based on the derivative of well pressure as a function of ln . The slope, , of the straight line on the Horner plot (Eq. 135 (Eq. 135)) is
ln
which gives the permeability
(137) Permeability
42
l e c t u r e no no t e s i n we we l l - t e s t i ng ng
= −4ℎ 1 . < ( + )/ > ( + )/ ) −− () = ( − = () − () . = −4ℎ 1 = 4ℎ 1− ⎡ ⎢ = 4ℎ ⎢⎣ () − ()⎤⎥⎥⎦ . ≈ . = . ℎ (() − () . ( + ) = − 2 + , () = − 2 4 − + 2 , ( + )−) − () = 2 4 + − + 2 . 1 = ( + )−) − () − 21 4 + + 2 = −21 ( + )−) − () − 21 4 + + 2 . = 1 ∗
(138)
The slope should be determined by fitting a straight line to the pressure points in the linear region. If we, for simplicity, use two points on the straight line corresponding to and , where . Then , so
ln
ln
(139)
ln
We then have
ln
(140)
Plots are often log based, and it is natural to select one decade apart. Since
ln
and
0 183 183 we then have
0 183 183
(141)
The Horner Horner analysis analysis also provide provide an estimate for the skin. The pressure after shut-in (Eq. 135 (Eq. 135), ), ln
(142)
is independent of skin, while the pressure just before shut-in, ln
(143)
is skin dependent. By subtracting Eqs. (142 ( 142)) and (143 (143)) we get ln
(144)
Thus Thus,, when when the the perm permea eabi bili lity ty and and pres pressu sure re at star startt of shut shut in is know known, n, the skin factor may me estimated based on the pressure at a specific time after after shut in. ln
(145)
ln
Typically the straight line extrapolated pressure at infinite time), , (see Fig. 24 Fig. 24)) is used:
(i.e.
buildup test
= −21 ∗ − () − 21 4 + 2 . ln
Skin
(146)
In the derivations above we have assumed a constant production rate in the whole production period. This may not always be possible to obtain, especially for unplanned unplanned shut down periods. In these cases we may use the effective production time (also known as equivalent constant rate drawdown time), defined as
∫ = (()) .
(147)
The basis for the approximation is not rigorous, but it is adequate if the most recent flow rate is reasonably stable and maintained long enough. The The Horn Horner er anal analys ysis is is base based d on the the infin infinit ite e acti acting ng solu soluti tion on,, so it is only valid if the production and drawdown period is short (Eq. 110 110): ):
+ < 4 = 4 ,
max max
(148)
where is the outer radius of the reservoir (distance to closest barrier). Note, Note, howev however, er, that the Horner analysis analysis can be approximately extended so that it may be used for analyzing transient buildups outside of this strict area of validity (see page 48 48). ). When goes goes to infinity, we have
→ ∞ ⇒ + → 1 ⇒ + → 0 . ∗ ∗ ≫ + ≃ = + ′+ ≃ + = − () + , ln
(149)
We thus see from Eq. (135 (135)) that the extrapolated pressure at infinite time should equal equal wheneve wheneverr the Horner Horner approxima approximation tion is strictly strictly valid. Any deviation from this is an indication of depletion, and we will see later (page 56 (page 56)) that plays a role when estimating reservoir pressure in the generalized Horner analysis. When we we have . Using this approximation for long production times , then Eq. (136 (136)) simplifies to ln
ln
(150)
ln
thus thus a plot plot of pressu pressure re versu versuss would would yield a straight line from which one could estimate permeability and skin. However, a large might violate the validity range of the infinite acting solution. In the next sectio section n will will will will consi consider der a more more genera generall treatm treatment ent of long long produc productio tion n periods.
Example In this example section we will employ Horner analysis on a buildup test. We assume that a vertical discovery well is produced at a reser voir rate of 100 0 m3 d for a period of 12 hours prior to a closure for
. /
Effective production time
43
44
l e c t u r e no no t e s i n we we l l - t e s t i ng ng
an initial initial pressure pressure buildup buildup survey survey.. The productio production n data and estimated estimated reservoir and fluid properties are summarized in Table 6 Table 6.. We assume that the well is completed across the entire formation. The plot of the pressure development is shown in Fig. 25 25.. We can now estimate the permeability and skin based on Eq. ( 138 138)) and Eq. (146 (146), ), respectively. We start with the permeability.
SPE Metric
SI
10 m
10 m
/ ×
100 100 m3
d
0 12 m 0 6 cP
10
4
bar
3
10
0 12 12 m 6 10
0.2
26
× / × ×
1 16 16
4
Pa s
0.2 1
26
10
9
Table 6: Basic data for example.
Figure 25: The pressure data from the buildup test in a Horner plot.
For estimating the permeability we need the slope of the straight part part of the pres pressu sure re curv curve e in the the Horn Horner er plot plot.. We obse observ rve e that that the curve curve is appro approxim ximate ately ly strai straight ght betwee between n and , 7 which corresponds to pressures of approximately approximately 2 798 798 10 Pa and 7 2 793 793 10 Pa, respectively. respectively. We can then calculate the slope of the straight part as
10 10 . × . × ≃ . × (10)−− . 10× = − . × . = −4ℎ . / ⋅ . × = . × / ⋅ 4 ⋅ . ⋅ . × = . × () = 1 ∗ () ∗≃=. . × × ∗ ( + )/ = 1 = −21 1∗ − () − 21 4 + 2 = 2 ⋅ . × 4(⋅ .. ×× − ⋅ . ⋅× . × ) / −0.5∗ . × ⋅ . ⋅ . × / ⋅ ( . ) + 2 = 0.46 2 793 793
107 Pa ln
2 798 798 ln
107 Pa
2 17 17
104 Pa
(151)
Employing Eq. (138 (138)) we can then estimate the permeability as
100 0 m3 d 6 0 10 4 Pa s 104 s d 10 0 m 2 17 17 104 Pa
8 64 64
2 55 55
This corresponds to approximately 255 mD. For Eq. (146 (146)) we need the pressure at shut in,
10
13 m2
(152)
, and the pres-
sure sure at (i.e. (i.e. infinit infinite e time), time), , (see (see Fig. Fig. 24 24). ). By extrapol extrapolatin ating g the curve in Fig. 25 Fig. 25 to to high values, we estimate the pressure at shut in to be 2 766 766 107 Pa. The The pressure pressure at is estimated as 2 804 804 107 Pa. This yields the following estimate for the skin from Eq. (146 (146): ): ln
2 24 24
ln
104
Pa
60
2 804 804
107 Pa
2 766 766
107 Pa
2 47 47 10 13 m2 12 h 3 6 103 s h 10 4 Pa s 0 2 2 6 10 9 1 Pa Pa 0 07 07 m
Thus the well appears to have little skin.
(153)
m3 s
Pa
1
buildup test
45
Miller–Dyes–Hutchinson (MDH) analysis The Horner analysis assumes infinite acting flow at shut-in, that is a short production period. The Miller–Dyes–Hutchinson (MDH) anal ysis is based on the opposite assumption: That the stable production period before shut-in is very long, so that pressure changes which originates from production can be ignored compared to the pressure changes due to shut-in. Rates for a general buildup test are presented in Fig. 26 26.. Based on the superposition principle, this rate history may be replaced by two wells with the rate history shown in Fig. 27 Fig. 27 The The well pressure before shut in ( ) is
≤ ()− = −(4ℎ) ( [ () + 2]2] , ( ) − ()− = 4ℎ 4 − + 2 . > () = − 4ℎ ( ( + ) − 4 + .
Figure 26: Rate history for a general buildup test
(154)
where is is some function. function. If we assume infinite infinite acting transient transient solution for the second well producing at a constant rate , the well pressure is given by ln ln
(155)
Using Using the superp superposi ositio tion n princi principle ple,, then then the well well pressu pressure re after after shut shut in ( ) is ln
Figure 27: Equivalent rate history for two wells in the same location
(156)
Note that, just as in the Horner case, the well pressure after shutin is independent independent of skin. Subtract Subtracting ing Eqs. (154 (154)) and (156 (156), ), and assuming that pressure changes which originates from production can be ignored compared to the pressure changes due to shut in (that is , thus ), ), we get
( ( + ) ≈ () ( + ) ≃ (4) = () ()−() = 4ℎ − + 2 . () ln
We see that the pressure is linear in ln
(157)
, as illustrated in Fig. 28 Fig. 28..
Figure 28: MDH analysis. The pressure is linear in ln .
The permeability can be estimated based on the slope, straight line
= 4ℎ 1 .
, of a fitted (158)
A plot of pressure versus the logarithm of time (semilog plot), such as in Fig. 28 Fig. 28,, is commonly denoted a MDH plot.
46
l e c t u r e no no t e s i n we we l l - t e s t i ng ng
= 4ℎ () − () . = 12 (() − () − 21 4 − .
We may alternatively alternatively select data at two times
and
on the line
Permeability
Skin
ln
(159)
By solving for in Eq. (157 (157), ), we see that the skin factor can be estimated based on the well pressure at a specific time after shut-in:
ln
(160)
Slider analysis and desuperposition As shown in Fig. 27 Fig. 27,, a buildup test is analyzed in terms of superposition tion of the the cont contri ribu buti tion on from from two two well wells: s: One One well well has has the the real real prod produc uc-tion history up to shut-in, and a constant rate at later times, and another other well well has a const constant ant negati negative ve rate rate after after shutshut-in. in. In MDH analys analysis is we are assuming that the contribution from the first well is zero, so the shut-in period that can be analyzed analyzed using MDH is limited. Any rate changes just prior to shut-in, which will contribute to the transient behav behavior ior are also also ignore ignored. d. In this this sectio section n we will will first first discus discusss how how the investigation period can be extended by taking into account the projected jected (semi (semi steady steady state) state) contr contribu ibutio tion n from from the first first well well (Slide (Sliderr analanal ysis), and secondly possible strategies for taking rate changes into accoun countt will will be disc discus usse sed. d. In both both case cases, s, the the effe effect ctss from from the the first first well well is incorp incorpora orated ted into into an effect effectiv ive e pressu pressure, re, and the analys analysis is is perfo performe rmed d in terms of . This approach is called desuperposition. In the next chapter we will see that pressure falls linearly in semi steady steady state state (Eq. (Eq. 184 184), ), and in Slider Slider analys analysis is the pressu pressure re differ differenc ence e in Eq.(157 Eq.(157)) is replac replaced ed by thelinearlyprojec thelinearlyprojected ted pressu pressure re develo developme pment nt as shown shown in Fig. Fig. 29 29.. If we expan expand d the pressu pressure re contr contribu ibutio tion n from from the first first
Figure 29: Slider analysis
( + ) ()−()−∗ = 4ℎ 4 − + 2 ,
well ( we get
) to first order in
, and subtract Eqs. (154 (154)) and (156 (156), ),
ln
(161)
buildup test
∗ ∗ = − || . () = ()−()−∗ ,
where the pressure derivative
at time is
(162)
So by defining the effective pressure difference, ex
Effective pressure difference by desuperposition.
(163)
we may extend the applicability applicability of MDH type analysis. This effective pressure can also be used in the diagnostic plots that will be introduced on page 54 page 54,, and used in several context thereafter. Note that permanent dowhhole gauges are needed for obtaining the pressure derivative. Alternatively the derivative can be provided by reserv reservoir oir simula simulatio tion, n, howe howeve verr extre extreme me cauti caution on should should be observ observed ed if doin doing g so sinc since e it is very very easy easy to end end up with with self self confi confirm rmin ing g assu assump mp-tions (circle (circle arguments arguments). ). The projected projected pressure pressure will also compensa compensate te for the effect of aquifer, communication with neighboring compartments, and injection/production in neighboring wells. Desuperposition based on the overall pressure derivative, , assume sumess that that the the well well has has a stab stable le rate rate befo before re shut shut-i -in. n. This This may may be diffi diffi-cult to achieve, in particular for producing wells experiencing an unplanned shut-in, and in this case the transients resulting from rate variations just prior to shut-in, , may be calcul calculate ated d numer numer-sim ically, ically, or based on analytical models. The pressure difference used in the analysis is then
∗
() ∗ () = ()−()+ (()−)−∗ , ex
where
47
sim
(164)
is the overa overall ll pressu pressure re trend trend measur measured ed by downho downhole le gauges gauges..
Similarity between drawdown and buildup responses In this this sect sectio ion n we will will disc discus usss in what what sens sense e the the pres pressu sure re resp respon onse sess in buildu buildup p are mirror mirror images images of the corre correspo spondi nding ng drawdo drawdown wn respon responses ses,, and how this enables the use of drawdown analysis tools, such as the semilog plot, for buildup analysis. For short short produc productio tion n and buildu buildup p (infini (infinitel tely y actin acting) g) the buildu buildup p response is (Eq. 144 (Eq. 144))
()−() = 4ℎ 4 + − + 2 , ()−(0) = − 4ℎ 4 − + 2 . = + . ln
(165)
and the corresponding drawdown test response is (Eq. 112 (Eq. 112)) ln ln
(166)
Thus, buil buildu dup p is the the mirro mirrorr image image of drawd drawdow own n when expressed in terms of an effective time (Agarwal time)
(167)
Numerical desuperposition
48
l e c t u r e no no t e s i n we we l l - t e s t i ng ng
For a long production period and short buildup the buildup response is (Eq. 157 (Eq. 157))
()−() = 4ℎ 4 − + 2 . ln ln
(168)
And the corresponding short time drawdown test response is again given by Eq. (166 (166). ). Thus, Thus, that in this case buildup is the mirror image of drawdown in terms of real time, but also with respect to Agarwal time, since
= + = 1 +1 ≈ , = + .
(169)
in this case. We may conclude that, at least approximately, the radial transi transient ent build buildup up is the mirror mirror image image of drawdo drawdown, wn, Irresp Irrespect ectiv ive e of the length of the production time, , provided that drawdown time is is replaced by the Agarwal time.
(170)
Based on this we also see that approximately we may employ an extended Horner analysis to long production periods where the infinite acting solution is not applicable. The Agarwal time transformation includes wellbore storage and skin effects, but non radial flow data (as in wells with hydraulic fractures) will not be transformed accurately. Similarly, late time behavior for drawdown and buildup are not similar under the transformation. As will be show in the next two chapters, the late time behavior of a well well test test is used used to infe inferr info inform rmat atio ion n rela relate ted d to rese reserv rvoi oirr boun bounda dari ries es and heterogeneities. The theory and procedures will be discussed in terms terms of drawd drawdow own. n. Since Since the Agarw Agarwal al time time transf transform ormati ation on is not applicable, some sort of desuperposition is needed if buildup data is to be used in this context. context. Examples Examples of desuperpositi desuperposition on is Eqs. (163 163)) and (164 (164). ). Correct Correct desuperposi desuperposition tion includes includes all effects, so that the late time behavior for drawdown and buildup are similar.
When analyzed in terms of AgarAgar wal time, the radial infinite acting transient buildup is equivalent to drawdown.
Horner analysis may be extended to long production periods.
Finite reservoir Real reservoirs are not infinite, and in this chapter we will discuss how the finite size of the reservoir influence a well test and which properties may be inferred. We will also discuss how well testing may be used in reservoir monitoring by giving information related to the pressure in the reservoir at the time of the test.
Semi steady state Late Late in a draw drawdo down wn test test,, the the flow flow reac reache hess semi semi stea steady dy stat state e in a clos closed ed reservoir. At semi steady state the pressure pressure profile is constant over time, while the average reservoir pressure falls due to production. The situation is illustrated in Fig. 30 Fig. 30 Figure 30: Semi steady state pressure profile
.
In the following we will discuss semi steady state flow in a cylindrical reservoir. First, the steady state profile will be derived, and by adding adding the condi conditio tion n that that the avera average ge pressu pressure re falls falls linear linearly ly with with time time we will derive the equation for well pressure as a function of time. The The crit criter erio ion n for for semi semi stea steady dy stat state e is that that pres pressu sure re fall fallss with with a cononstant rate in the whole drainage area. This gives the following equation for the pressure profile, (see (see Eqs. 19 Eqs. 19 and and 28 28): ):
( ) 1 = 1 = . ℎ2 = , = 2ℎ = ,
(171)
The inner boundary condition on Eq. (171 ( 171)) is given by Darcys law,
that is
(172)
(173)
50
l e c t u r e no no t e s i n we we l l - t e s t i ng ng
while the outer boundary condition is no flow, flow, which using Darcys law transform to zero pressure gradient:
= 0 ⇒ = 0 , = 2 + . = − = 2 − . = 2 = 22 − = − − ≈ − . ≫ − ≃ = − − . − (,)−)−() = − 2 . ≪
(174)
where is the outer radius of the reservoir. Integrating Eq. (171 (171)) with respect to gives
(175)
We may apply the outer boundary condition (Eq. 174 (Eq. 174)) to to eliminate the integration constant
from Eq. (175 (175)) (
) which gives
(176)
The constant is found by applying the inner boundary condition (Eq. 173 (Eq. 173)) to Eq. (176 (176): ):
(177)
In the last similarity we us that Eq. (178 (178)) into Eq. (176 (176)) gives
, thus
(178)
. Inserting
We may now integrate both sides of Eq. (179 (179)) from pressure profile:
(179)
to to get the
ln
Pressure profile
(180)
As long as the last term in Eq. (180 ( 180)) can be ignored, so the pressure profile near the well has the familiar general steady state logarithmic form (Eq. 36 (Eq. 36). ). In a well test we are primarily interested in the well pressure, and remains an arbitrary integration constant in Eq. (180 (180). ). We may may however integrate the pressure profile over the entire reservoir in order to get an expression for the average reservoir pressure, and by equating this pressure with the average pressure found by applying material balance, an expression for well pressure can be found. The average reservoir pressure is given by the integral
1 = 2( 2() . () = ()+ − 43 ,
(181)
Substitut Substituting ing the reservoir reservoir pressure pressure profile profile (Eq.180 (Eq. 180)intoEq.( )intoEq.(181 181), ), and and integrating gives the average pressure ln
Get average pressure by integrating the pressure profile.
(182)
For the integration yielding Eq. (182 182)) we use that
ln
ln
finite reservoir
≫ () = − , () = − − − 43 .
where again the condition has been used to eliminate small terms. Since the compressibility compressibility is constant, we may also calculate calculate the average reservoir pressure from material balance:
(183)
where is bulk volume, and is total compressibility. compressibility. The average pressure in the two expressions (Eqs. 182 (Eqs. 182 and 183 183)) should be the same, so we may eliminate it and solve for the semi steady state well pressure: ln
(186)
Our Eq. (184 (184)) is valid for a very special (cylindrical) reservoir never seen in real life, and we have ignored the effect of skin. We will now derive a generalization of Eq. (184 (184)) valid for reservoirs of constant thickness of any shape. For a cylindrical reservoir the spatial correction depend on reservoir size through the outer radius . It is is not not obvious what the corresponding characteristic length should be in a reserv reservoir oir of genera generall shape shape so we will will use the reserv reservoir oir area, area, , instea instead: d:
(187)
It is most convenient to work with dimensionless variables:
(188)
The semi steady state solution for a cylindrical reservoir on dimensionless form is then ln
Reservoir limit test
(185)
so, if total compressibility is known, the reservoir pore volume is
= = . = 1 = = − 1 = 1 ℎ = 1 ℎ = = 2 + 21 − 23 .
Semi steady state well pressure
(184)
The well well pressu pressure re (Eq. (Eq. 184 184)) has two contr contrib ibuti utions ons:: The avera average ge reser reser- voir pressure, which decrease linearly with time, and a geometric correctio rection, n, which which depend depend logari logarithm thmica ically lly on reserv reservoir oir size. size. The geomet geomet-ric correction is determined by the pressure profile and will vary with reserv reservoir oir geomet geometry ry and well well place placemen ment. t. The The well well pressu pressure re in Eq. (184 184)) is valid for a cylindrical reservoir with a well at the center. A reservoir limit test is a test that is run for a time long enough for the test to investigate the whole reservoir (see Eqs. 82 Eqs. 82 and and 204 204). ). The purpose of the test is to get information on reservoir size and shape. Reservoir pore volume can be estimated from the derivative of pressu pressure re at semi semi steady steady state. state. In semi steady steady state (Eq. 184 184)) the derivative of well pressure is
= − , = − .
Get average pressure from material balance.
(189)
Reservoir pore volume
51
52
l e c t u r e no no t e s i n we we l l - t e s t i ng ng
Since Since the near near well well pressu pressure re profile profile has the famili familiar ar time time indepe independe ndent nt logarithmic form, the effect of skin can be represented by the dimensionless skin factor. We may may also incorporate the effect of reservoir shape and relative well placement into a dimensionless shape factor, , with a corresponding contribution to the pressure12 . Introduc Introduc-ing skin and shape factor, the general semi steady state solution is written
= 2 + 21 4 − + 2
,
ln
12
The shape factor Dietz shape factor.
is called the
General semi steady state well pressure
(190)
and we see that the shape factor is defined such that the spatial pressure correction is similar to the temporal infinitely acting solution (Eq. 104 (Eq. 104))
= 21 ( (4 4) − + 2) ln ln
.
(191)
By comparing Eqs. (189 ( 189)) and (190 (190), ), we see that the shape factor for a cylindrical reservoir with a central well is
= 4 (3/2−/2 − ) = . … . < . … . exp
31 6
(192)
For any other geometry we have
31 6
(193)
Example In this example we will estimate the reservoir volume by employing Eq. (186 (186)) on a drawdown test. We assume that a vertical well is produced at a reservoir rate of 100 0 m3 d for a prolonged prolonged period. period. The relev relevant ant produc productio tion n data data and estima estimated ted reserv reservoir oir and fluid fluid proper propertie tiess are summarized in Table 7 Table 7.. The plot of the pressure development is shown in Fig. 31 Fig. 31.. We can then estimate the reservoir volume based on Eq. (186 ( 186). ). We then need to estimate the slope of the linear part of the plot.
. /
SPE Metric
/ ×
SI
100m3 d 10 m 0.2
26
10
4
× / × 3
1 16 16 10 10 m
bar
1
0.2
26
10
9
Pa
Table 7: Basic data for example.
Figure 31: The pressure data from the drawdown test in a plot with Cartesian scales.
×. × × × . × . × × − . × = = × − × = − .
We observ observe e that that the curve curve is approx approxima imatel tely y straig straight ht betwe between en 7 7 7 the the time timess 1 10 s a n d 6 10 s. At 1 10 s the the pres pressu sure re is 7 approximately approximately 2 01 01 10 Pa, while while the pressu pressure re is approx approxima imatel tely y 1 65 65 107 Pa at time 6 107 s. This gives a slope of 1 65 65
107 Pa 6 107 s
2 01 01 107 Pa 1 107 s
0 072 072 Pas
1
.
(194)
m3 s
1
finite reservoir
53
Then, employing Eq. (186 (186), ), we get estimate the reservoir pore volume as
= − = − . × . × ⋅− . / = . × . = . =×ℎ /0.2 = . × = ℎ = . × ≃ . 26
1 16 16 10 10 9 Pa 1
3 m3
s 0 072Pas 072Pas
106 m3
6 2
1
(195)
With the given porosity, this yields a reservoir volume of 6 2 106 m3 3 1 107 m3 . For a cylindr cylindrica icall reservoir reservoir,, were were , this corresponds to an outer radius of 31
107 m3 10 m
1000 1000 m
(196)
Exact solution for cylindrical reservoir There exist an exact expression for the well pressure in a cylindrical reservoir. 13 That is, a solution to the diffusivity equation with constant rate boundary conditions on inner radius
|| = , | | = 0 . () = / ≫ = / = − 22 + () − 43 + ( , ) , ∞ ( , ) = 2 (( ( )−)() , ( )() − ()( ) = 0 .
13
Everdingen and Hurst, “The Hurst, “The Application of the Laplace Transformation Transformation to Flow Problems in Reservoirs”. Reservoirs”.
(197)
and no flow on outer radius
(198)
The late-time behavior of this solution may be analyzed to get the semi steady state well pressure (Eq. 184 (Eq. 184): ): Expressed in dimensionless variables ( and ) the exact solution for the well pressure when when is ln
Exact expression for the well pressure in a cylindrical reservoir
(199)
where
and
(200)
are roots of the equation
(201)
Inspecting the exact solution (Eq. 199 (Eq. 199), ), we see that it is equal to the semi stable solution (Eq. 184 (Eq. 184)) plus a time dependent correction, . The correction is a sum over terms (modes) that fall off exponentially tially with time, so the late time behavio behaviorr is described described by the semi semi stable solution as derived derived earlier. The longest living mode has time dependency exp , where is the smallest smallest root. In Fig. 32 Fig. 32 are are plots of the function
(− ) () = ( )()−()( ) ≈ 4 .
(Eq. 201 201)) for for two two diff differ eren entt valu values es of est root is
(202)
, and and we we see see tha thatt the the smal smalll
(203)
The correction to semi steady state fall off exponentially with time.
54
l e c t u r e no no t e s i n we we l l - t e s t i ng ng
=
=
10
≈
100 000 000
Figure 32: The function
two different values of the smallest root is cases.
for . we see that in both
The time constant for the longest living mode is thus
= 16 .
(204)
Note the similarity with “time of investigation” and “radius of investigation” (Eq. 82 (Eq. 82). ). Figure 33 shows shows the spatia spatiall shape shape of the three three longes longestt living living modes modes of the correction correction to the semi steady state state profile. profile. We see that the sum in Eq. (200 (200)) is similar to a Fourier expansion, and that the “high frequency” modes die off first.
Figure 33: Spatial shape of longest living modes of correction to SSS solution,
Log–log derivative diagnostic plot
′
In well test analysis, the log–log plot of the logarithmic derivative, , versus time, , is extensivel extensively y used for diagnostic diagnostic purposes. purposes. This log–log derivative diagnostic plot is used in general to find flow regimes, and it will be discussed and used extensively in the following chapters. The plot has
on the –axis, and
(′) = () ()
ln
ln
ln
(205)
ln
on the –axis. An example diagnostic plot is shown in Fig. 34 34.. Since details and variability are hidden, log–log plots can be deceptive, and it is often said that any data can be fitted to anything on a log–log plot. However, However, as will be demonstrated below, this removal of detail can also be set to good use. From Eq. (113 (113), ), the infinitely acting transient has the form
= () + , ln
(206)
so the logari logarithm thmic ic deriva derivativ tive e is a const constant ant.. Thus, Thus, the time time interv interval al for for infinitely acting flow can be identified as a period with zero slope or a plateau value. Such a plateau is indicated in Fig. 34 34.. In semi steady state, the well pressure is linear in time (Eq. 184 184), ),
= + ,
(207)
Figure 34: Log–log derivative diagnostic plot, showing infinite acting and semi steady state flow regimes.
finite reservoir
(′) = () = = = ( ) + ( ) , ( ) (′) ()
so the logarithmic derivative is ln
ln ln
ln
ln
ln
ln
55
(208)
ln
(209)
and the plot (ln vs. vs. ln ) has unit slope. Thus, the the time inter val for semi steady state flow can be can be identified as a period with unit slope. Such a unit slope period is indicated in Fig. 34 34.. In general, if a flow regime is characterized by by an exponent, ,
we have
= + , (′) = ()+ () .
ln
=
ln
ln
(210)
(211)
Thus, Thus, the time time interva intervall for for a flow flow regim regime e with with chara characte cteris ristic tic expon exponent ent can be identified as a period on the log–log diagnostic plot with slope . If we ignore the early time effects of wellbore storage and skin, the reservoir limit test has two flow regimes: infinitely acting radial flow and semi steady state flow. flow. The first will be characterized by by a time interval with zero slope, and the second will be characterized by an interval with unit slope. This is illustrated in Fig. 34 Fig. 34.. As has been discussed in the previous chapters, permeability and skin skin are are esti estima mate ted d base based d on fitti fitting ng a stra straig ight ht line line to the the infin infinit itel ely y act acting flow period on a a semilog plot ( vs. ln ). ). Reservoir pore volume and shape factor is found from the semi steady state flow period using a Cartesian plot ( vs. ). ). This This is illustrated in Fig. 36 Fig. 36.. If the straight line fit is
()
Figure 35: When analyzing a reservoir limit test a semilog plot ( vs. ln ) is used to find permeability and skin
Figure 36: A cartesian plot ( vs. ) is used to find reservoir volume and shape factor
= − , = − . = ℎ ,
then the reservoir volume is estimated from the slope
(212) :
Estimating reservoir volume
(213)
In a reservoir with constant thickness, the reservoir area is
(214)
56
l e c t u r e no no t e s i n we we l l - t e s t i ng ng
= 0 4 − ( ) = − + 2 − 2 .
so, if permeability and skin is known, the Dietz shape factor may be found by extrapolating the straight line (Eq. 212 (Eq. 212)) to and solving Eq. (190 (190)) for : ln
ln
(215)
Reservoir monitoring with buildup tests In this this sect sectio ion n we will will show show that that the the buil buildu dup p test test can can be used used for for monmonitoring reservoir pressure by employing an extended version of the Horner Horner analys analysis. is. In the strict strict Horner Horner analys analysis is (page (page 40 40), ), vali valid d for for shor shortt production times, , the extrapolated pressure at , , corresponds responds to the initial initial pressure. pressure. Due to the short production production time, the initia initiall pressu pressure re is approx approxima imatel tely y equal equal the avera average ge reserv reservoir oir prespressure. For longer longer productio production n times, times, is no long longer er equa equall to the the aver averag age e reserv reservoir oir pressu pressure, re, but the strai straight ght line line extra extrapol polati ation on can can still still be used used for estimating this pressure. There are two methods for estimating the average reservoir pressure for buildup tests: In the Ramey–Cobb Ramey–Cobb method, which is applicable for long production times (semi steady state), a time is found where the pressure on the straight line correspond to the average pressure, and in the Matthews-Brons-Hazebroek method, which is universally applicable, a correction to is found. Both methods require knowledge of the reservoir shape. We have seen (page 47 (page 47)) that irrespective of the length of the production time, , the radial transient buildup is approximately the mirror image of drawdown provided provided that drawdown time is is replaced by the effective (Agarwal) time
= 1 ∗
∗
∗
= + = . = ( + ) / → ∞ → 1 () → 0 () − ( + ) = 21 (4 ) − ()− )− + 2 .
(216)
Here
is the Horner Horner time. time. Reme Remembe mberr that that infinit infinite e time time yields , thus ln . Using dimensionless variables, the buildup response (Eq. 144 144)) is then ln ln
ln
(217)
This implies that the Horner analysis can be extended to long productio duction n period periodss to get estim estimate atess for for permea permeabil bility ity (from (from the slope) slope) and skin skin as show shown n in Fig. Fig. 37 37.. Permeability Permeability is estimated from the slope, :
= 4ℎ 1 . =1 = 21 ∗ −() − (4 ) + .
∗
(218)
Extrapolation of the straight line to defines the pressure . And, with known permeability, skin can be found using pressure difference ference at any point on the straight straight line. If we use the extrapol extrapolated ated pressure the expression is ln
(219)
Estimating the shape factor
finite reservoir
57
Figure 37: Extending the Horner anal ysis. Note that the x-axis is reversed in this plot, this is often the case for Horner plots as it makes time run from left to right.
() =
We will now derive the Ramey–Cobb method for estimating average reservoir reservoir pressure: pressure: First, First, remember remember from Eq. (183 (183)) that . Then the dimensionless dimensionless average average reservoir reservoir pressure is
− /() () = −() = 2ℎ = 2 = 2 () = 2 +121 44 − + 2 = () + 2 − + 2 . ( ) = ( ) > > = ( + ) − 21 = =
(220)
If flow is in semi-steady-state at end of the production period (see Eq. 190 Eq. 190)) we have ln ln
ln ln
(221)
(222)
Due Due to shut shut in afte afterr , we hav have for , and and can can thus thus write just for all . If we insert Eq. (222 (222)) into the expression for the linear region of the Horner plot (Eq. 217 (Eq. 217)) and solve for we get ln
(223)
We see that by selecting a time such that
(224)
the logari logarithm thmic ic term term vanish vanish and the corre correspo spondi nding ng pressu pressure re is the avaverage reservoir pressure. The Ramey–Cobb Ramey–Cobb method is illustrated in Fig. 38 Fig. 38,, and can be summarized as follows: First fit a straight line to the infinitely acting radial transient part of the Horner plot
= ∗ − ( ) . ∗ = . (∗) = ∗ − (∗) = . ln
Then, Then, using using the Dietz Dietz shape shape factor factor that that corre correspo spond nd to the actua actuall reser reser- voir shape, determine
(225)
The corresponding pressure on the straight line is the average reser voir pressure: ln (226)
58
l e c t u r e no no t e s i n we we l l - t e s t i ng ng
Figure 38: Ramey–Cobb method: A time is selected on the straight line such that the corresponding pressure is .
e r u s s e r p
Note that in order to apply this method, permeability, and reservoir area and shape, must be known from a separate analysis. The Ramey–Cobb method is only applicable if the reservoir has reached semi steady state before shut in. We see from Eq. (226 ( 226)) that the Ramey–Cobb Ramey–Cobb method alternatively may be expressed as a correction to , and in the Matthews–Brons–Hazebroek (MBH) method, an expression for this correction is sought that is valid also for production times shorter than the time needed to reach steady state. A general explicit expression for the correction does not exist, but if the reservoir shape and area is known, the correction can be found by solving drawdown–buildup in dimensionless form for a given geometry ometry.. The corre correcti ction on has has been been tabula tabulated ted for for a number number of simple simple geometries, and may be found by solving the flow numerically for more complex geometries. Figure. 39 Figure. 39 shows shows four examples. The correction
∗
Figure 39: Matthews-Brons-Hazebroek Matthews-Brons-Hazebroek correction. (© SPE, 1977, after Earlougher, R. Advances in Well Test Test Analysis)
=
is a functi function on of dimens dimension ionles lesss time time and area area throug through h the time time varia variable ble .
Since Since both both the Ramey Ramey–C –Cobband obband the MBHmethod MBHmethod can can be expre expresse ssed d as a corre correcti ction on to the extra extrapol polate ated d pressu pressure, re, there there should should be a conne connecction between the two. We see from Fig. 39 Fig. 39 that that the MBH correction is line linear ar in ln for for large large produc productio tion n times. times. Large Large times times corre correspo spond nd to
( )
finite reservoir
semi steady state, which is where the Ramey–Cobb method is applicable, and we see from Eqs. (225 ( 225)) and (226 (226)) that this linear behavior is consistent when the MBH correction for large is
= 211 1 = 2 ( ) + 2
MBH correction for large tion of
ln ln
1/ 2
is a func-
(227)
ln
We see from Eq. (227 (227)) that all MBH correction curves have the same slop slope e ( ), and and that that the the offs offset et is half half the the loga logari rith thm m of the the Diet Dietzz shap shape e 14 factor, . The Horner analysis is based on a constant production rate over a certain time, and all results are expressed in terms of this production time . For cases where where the production production has been varying, we have seen earlier that we may use an effective production time (Eq. 147 147)) inst instea ead, d, as long long as ther there e is a stab stable le peri period od just just befo before re shut shut in. in. Anot Anothe herr complication is that for long production times we will in Eqs. ( 219 219)) and (227 (227)) need to subtract two large numbers; this will introduce an increasingly large numeric error for increasing . Luckily Luckily,, it can be show shown n that that for for time timess larg larger er than than the the time time need needed ed to reac reach h semi semi stea steady dy state, SSS SSS , the results are independent of the choice of as long as This is due due to the the fact fact that that depen depends ds linear linearly ly on the choic choice e SSS SSS . This of when . We should should therefore therefore preferab preferably ly use use SSS SSS whenever the production time is longer than the time to reach semi steady state.
≥ ≪
59
∗
=
14
In the paper where the Dietz shape factor were first introduced, it was actually defined from eq. 227 eq. 227..
For long production times: use SSS SSS .
Drainage areas In our analysis we have assumed that there is only one well in the reservoir, and the question of whether any of this is applicable to situations with more than one well obviously arises. The answer to this lies partly in the concept of drainage areas: A well test in a well in a multi well reservoir can often be analyzed in terms of a single well in a reservoir shaped like the wells drainage area. In general each wells drainage area is defined in terms of streamlines. lines. The streamlines streamlines are defined by the fluid flow vectors, vectors, and are coupl coupled ed to the pressu pressure re gradie gradients nts via Darcy Darcyss law. law. The stream streamlin lines es end in wells, and space is divided into regions where flow is towards different wells. These regions are called drainage areas, and in general the size size and shape shape of these these areas areas change change over over time. time. Each Each produc producer er will will define its own drainage area, while injectors may divide its flow between several producers, and a producers drainage area may contain (parts of) several injectors. In a reservoir with only producers producing at constant rates, a semi semi steady steady state state deve develop lopss where where draina drainage ge areas areas are const constant ant in time, time, and the pore volume of each area is proportional to the production rate. rate. Each Each draina drainage ge area area has has the same same semi semi steady steady state state pressu pressure re proprofile, and average pressure, as a corresponding single well reservoir with no-flow boundaries. In this case a buildup test can be analyzed analyzed
Figure 40: Example of streamline pattern that partition the reservoir into drainage areas. There is one drainage area for each producer. producer.
60
l e c t u r e no no t e s i n we we l l - t e s t i ng ng
using the Horner plot. The calculatio calculation n of average average pressure pressure should be based on the shape and area of the drainage area at shut in, and the resulting pressure estimate will be an average of pressure in the drainage area. Note that the drainage area concept has limited applicability, and that that in gene genera rall the the anal analys ysis is of a buil build d up well well test test in a mult multii well well rese reserr voir is best performed in terms of some sort of desuperposition (see page 46 page 46). ).
Reservoir pressure vs. local pressure The MBH and Ramey–Cobb methods provide estimates for average pressu pressure re in a reserv reservoir oir region region or draina drainage ge area. area. In terms terms of condi conditio tionn15 ing reservoir models to monitoring data , this may actually not be what is most useful. The two methods also require knowledge of the reservoir shape, which sometimes is among the unknown reservoir proper propertie tiess the monito monitorin ring g data data is suppos supposed ed to contr contrib ibute ute in determ determinining. ing. For the condi conditio tionin ning g of reserv reservoir oir model modelss the local local pressu pressure re in the near well region may be more relevant. The corresponding measurement is the well pressure at a fixed time after shut-in This measurement ment is often often referr referred ed to as “one “one hour hour shutshut-in in pressu pressure re””, but but the actua actuall relevant time does depend on permeability (radius of investigation). For the conditioning of reservoir models the local pressure near the well is most relevant. In a reservoir simulation model pressure is defined defined in grid-b grid-bloc locks, ks, and flowing flowing pressu pressure re near near a well well is repres represent ented ed by the pressure pressure in the block where the well is perforated. perforated. This gridblock pressure is equal to the pressure at Peaceman equivalent wellblock radius16 0 2 (228)
≈ . ,
where is the linear block size. The concept of an equivalent radius is illustrated in fig. 41 fig. 41:: The flow into a grid block with a well is deter-
15
Conditioning of reservoir models to time dependent data, such as measured pressure, is often referred to as “history matching”. matching”.
16
D. W. Peaceman. “Interpretation of Well-Block Pressures in Numerical Reservoir Simulation”. In: Society of Petroleum Engineers Journal 18.03 (June 1978), pp. 183–194.
Figure 41: The Peaceman equivalent well-block radius
mined by the pressure difference between an upstream block and the grid-bloc grid-block k pressure. pressure. This is a linear linear pressure pressure profile, while the real profile towards the well is a logarithmic singularity. As a result, well pressu pressure re is lower lower than than wellwell-gri grid-b d-bloc lock k pressu pressure, re, and the distan distance ce from from the well at which reservoir pressure is equal to grid-block pressure is the equivalent well-block radius.
finite reservoir
≈ . (, ) = ()+ + , + () = ()+ 2 4 −− + 2 . = 4 ≈ . .
61
What is the time it takes for well pressure to reach flowing pressure at Peacem Peaceman an equivalent equivalent well-bloc well-block k radius radius 0 2 ? The flowin flowing g pressure at is given by the steady state profile (Eq. 40 40)) ln
and the shut-in well pressure at time Eq. 157 Eq. 157))
(229)
is (MDH approximation,
ln ln
(230)
By combining Eq. (229 (229)) and (230 (230), ), and solving for we we see that the time after shut in at which the well block pressure es equal to flowing grid block pressure is 0 018 018
(231)
This is the time when shut in pressure should be measured so that it can can be compa compared red with with simula simulated ted grid grid block block pressu pressure re for condi conditio tionin ning. g.
Time at which shut-in pressure is equal to flowing grid-block pressure.
Reservoir boundaries In this chapter we will discuss how well testing can detect reservoir boundaries and heterogeneities. As shown in Fig. 42 42,, pressure signals are reflected at reservoir boundaries, and these reflections can, when measured at the well, be used to infer quantities such as distance to barriers and reservoir shape. Examples of situations where well test data can offer valuable information formation are shown shown in Fig. 43 Fig. 43.. The well test test data can be analyz analyzed ed quantitatively to get estimates for the distance to reservoir boundaries, the distance to and the strength of an aquifer, the distance to a gas cap, the distance to an advancing water front, and the shape and size of sand bodies. bodies. The data can also give give qualitative qualitative informati information on
Figure 42: Pressure signal reflected at reservoir boundary
Figure 43: Examples of boundary like reservoir features that may be probed in a well test: A fault (top), a water injection front (bottom left), and shape of sand bodies (bottom right)
relating to the nature of the reservoir and the flow in the near well region. For the latter purpose, the derivative diagnostic plot plot briefly introduced on page 54 page 54 is is particularly useful, and several examples of its use will be shown.
Well close to a linear boundary The The pres pressu sure re resp respon onse se in a well well clos close e to line linear ar boun bounda dary ry can can be foun found d usin using g supe superp rpos osit itio ion n in spac space. e. The The meth method od is call called ed the the meth method od of imimages, since the boundary condition at a barrier or reservoir boundary may be satisfied by placing imaginary image wells outside the reser voir area as shown in Fig. 44 Fig. 44.. The sum of the unconstrained pressure
64
l e c t u r e no no t e s i n we we l l - t e s t i ng ng
Figure 44: Method of images: Imaginary image wells are placed outside the reservoir in order to satisfy the boundary conditions.
field from the real well and the image wells should satisfy the boundary condition, and as shown in Fig. 45 45;; the no-flow boundary condition is obtained by placing an image well with the same rate, while a Figure 45: Method of images for no– flow and constant–pressure boundary condition. For the red and yellow pressure curves, their difference to the initial pressure are given by the sum of the difference between the dashed curves and the initial pressure.
const constant ant pressu pressure re bounda boundary ry condi conditio tion n is obtai obtained ned by placin placing g an image image well with opposite rate. The pressure response in a well close to a single linear boundary, such as a sealing fault, can be derived by the method of images: The bounda boundary ry condi conditio tion n is satisfi satisfied ed by placin placing g an image image well well with with the same same rate on the other side of the boundary as shown in Fig. 46 Fig. 46.. The well pressure in a constant rate drawdown test is the sum of two infinitely acting terms, one from the real well, and another from the image well. We will ignore the short time effects of wellbore storage age and and skin skin,, so we can can use use the the loga logari rith thmi micc appr approx oxim imat atio ion n for for the the real real well, as given by Eq. (84 84): ):
− () = 2 4 − + 2 . ln
(232)
The distance to the image well is large compared to the well radius, so this contribution is given by the exponential integral at all times, as in Eq. (73 (73): ):
1 1 − (,,) = 2 4 . = 2 − (2,2, ) = 21 1 . () = − 2 4 + 2 − + . E1
For radius
(233)
(234)
, this gives:
E1
Employing the superposition principle in space to add the pressure equations Eqs. (232 (232)) and (234 (234), ), we obtain the well pressure as: ln
E1
(235)
Note that the skin simply adds an extra contribution to the well pressure, so there is no skin contribution contribution from the image well. well. We may
Figure 46: A well close to a linear boundary, modelled using an image well on the other side of the boundary
r e s e r v o i r b o u n da da r i e s
65
write Eq. 235 Eq. 235 in in dimensionless form:
( , ) = 21 (4 4) − + 2 + . → 0 / → ∞ / → 0 ( , ) = 21 ( (4 4) − + 2) , ln
E1
(236)
When , then , so E1 . This reflects reflects that at early times the reflected signal has not reached the well, thus the exponential integral contribution is zero: ln ln
(237)
which will give a slope of half when plotting and on a semi-log plot. At late times, when the reflection has passed the well, the exponential integral may be approximated by a logarithm:
( , ) = 21 (( (4 4) − + 2) + − = 2 − + . ln ln
ln
ln
(238)
We observe that this gives a slope of half when plotting and on a semi-log plot. The fault will therefore manifest itself as a doubling of slope on a semilog plot, going from a slope of half at early times to a slope slope of 1 at later later times times.. Note Note that that if the fault fault is is very very close close to boundary, the line with the first slope will be hidden by wellbore and and near near well wellbo bore re effe effect ctss on one one side side and and the the tran transi siti tion on to the the seco second nd slope on the other, so that we will only see a single straight line. This may the be wrongly be interpreted as a reduced permeability. As illust illustrat rated ed in Fig. Fig. 47 47,, the the dist distan ancce to the the faul faultt can can be found ound from from the the time time of cros crossi sing ng of the the two two fitte fitted d stra straig ight ht line liness (Eqs (Eqs.. 237 and 238 238): ):
12 (4 4) − 21 = 2 − . = √ , = . ln
ln
(239)
Solving Eq. (239 (239)) for gives
or
0 749 749
A fault manifests itself as a doubling of slope on a semilog plot.
Figure 47: The distance to the fault can be found from the time of crossing of the two fitted straight lines
(240)
The The pres pressu sure re resp respon onse se in a well well clos close e to a corn corner er can can also also be solv solved ed in terms of image wells, as shown in Fig. 48 Fig. 48.. In this case three image wells are needed to satisfy the no-flow boundary conditions along the two two reserv reservoir oir walls walls,, and the expre expressi ssion on for for well well pressu pressure re has four four concontributions:
( , , ) = 21 ( (4 4) − + 2) + + + + ln ln
E1
E1
E1
.
(241) At early times, the contribution from the image wells are again zero, and the well pressure is given by Eq. (237 (237), ), while at late times
Figure 48: A well in a corner, and the corresponding image wells
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l e c t u r e no no t e s i n we we l l - t e s t i ng ng
the exponential integrals may be approximated by logarithms:
2 − 2 + ( , , ) = 2 ( ) + + ln
ln
.
(242) Thus, the straight angled corner will manifest itself as a four times increase of the slope on a semilog plot 17 . Simi Simila larr to the the situ situat atio ion n with with a sing single le faul fault, t, if the the well well is clos close e to the the corner we may only see a single straight line, and this can wrongly be interpreted interpreted as a reduced reduced permeability permeability.. On the other hand, if the distance distance to the two two reservoir reservoir walls walls are well well separated separated , we we will will see three straight lines: The first representing representing the infinitely infinitely acting acting radial flow, a second with double slope representing the semi radial flow due to the reflection from the nearest wall, and the third with four times the slope.
17
In general, if the corner has angle the slope will increase by a factor .
≫
Flow regimes – Diagnostic plots Until now we have seen two fundamental flow types or flow regimes: radial flow, which is characterized by a linear slope on a semilog plot and a constant value on the log–log derivative plot, and depletion, which is characterized by a unit slope on the log–log derivative plot. The unit slope is seen early in the test when all production is from the wellbore (wellbore storage effect, see Eq. 124 Eq. 124), ), and late for semi steady state flow (Eq. 190 (Eq. 190). ). In this this sect sectio ion n wewilldiscus wewilldiscusss vari variou ouss flow flow regi regime mess that that can can be seen seen in a well well test, test, and their their signat signature ures. s. A const constantrate antrate well well test test probes probes the volume around the well at an increasing distance from the well, consistent with the concept of a radius of investigation, and the derivative of the pressure response reflects the properties at the pressure front. The test probes the volume in the transition zone between the essentially semi steady state region closer to the well and the undisturbed outer regions, as illustrated in Fig. 13 Fig. 13 on page 27 page 27.. Thus, the early part of the test reflects properties and front movement close to the well, well, while while the later later parts parts reflect reflect front front propag propagati ation on and proper propertie tiess at increasingly larger distances. As illust illustrat rated ed in Fig. Fig. 49 49,, the the time time peri period odss of a well well test test are are typi typica call lly y Figure 49: Flow periods on a log–log plot
r e s e r v o i r b o u n da da r i e s
characterized as: “early time”, time”, dominated by wellbore effects, skin, and near well heterogeneities, “middle time”, where the flow has not yet seen the whole reservoir, and “late time”, where flow is boundary dominated or in semi steady state. Flow regimes are identified using the log–log derivative diagnostics plot introduced on page 54 page 54.. This This plot plot has has
(′) = () = − () (′) = ()+ () . = ln
ln
67
Identifying flow regimes
ln
on the –axis, and ln on on the –axis. If the pressure response have the form then we have ln
ln
ln
Thus, Thus, the time time interva intervall for for a flow flow regim regime e with with chara characte cteris ristic tic expon exponent ent can be identified as a period on the log–log diagnostic plot with slope . We will see below that the characteristic exponent depend on the dimension in which the pressure front is propagating.
Spherical flow When the pressure front propagates in three dimensions we have spherical flow. Two examples of a situation where we may see a flow period with a spherical flow regime is shown in Fig. 50 Fig. 50:: If the well Figure 50: Examples of spherical flow due to an incompletely perforated perforated reservoir zone
is only partially perforated in a reservoir zone, there will be a period where the pressure front propagates in a spherical or hemispherical pattern until it reaches the top and bottom of the zone. For spherical flow we have
which gives ln
() ∝ √ 1 = , () = −12 () + , −1/2 ln
ln
thus the slope on the derivative plot will be
.
(243)
(244)
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Radial flow We have seen earlier that for radial flow, where the pressure front moves in a 2-dimensional radial pattern, the pressure has a characteristic logarithmic time behavior:
() = + () , () = . ln
(245)
which gives the derivative
ln
(246)
The plateau height corresponds to the constant slope on a semi-log plot, and depends on permeability and geometry. Several examples of radial flow are shown in Fig. 51 51.. Figure 51: Examples of radial flow
We have seen the signature of a well test with a single radial flow period period seve several ral times times alread already y, and Fig. Fig. 52 illustrat illustrates es how this signature signature is affected by permeability and skin.
Figure 52: How permeability and skin effects the radial flow diagnostic plot.
Linear flow When When the pressu pressure re front front propa propagat gates es in an essent essential ially ly one dimens dimension ional al pattern we have linear flow. flow. Examples of situations with linear flow periods are flow in narrow, channel shaped, reservoirs (see page 70 70), ), and flow in hydraulically induced fractures (see page 85 85). ). Channe Channell flow occur at middle to late times, while early time linear flow is indicative of fractures. Linear flow has the following characteristic (see page 89 89): ):
() ∝ √ = ,
(247)
r e s e r v o i r b o u n da da r i e s
which give the derivative ln
() = 21 () + , ln
ln
(248)
thus thus this this flow flow regi regime me ischara ischaract cter eriz ized ed bya slop slope e of a half half onthe deri deriva va-tive plot. Figure 53 Figure 53 illustrates illustrates how channel width and fracture length influence the corresponding corresponding derivative plots. This will be discussed
in more detail later.
Figure 53: Derivative plots for two examples of linear flow.
Summary of flow regimes Each flow regime is identified by a time period with a characteristic slope on the log–log derivativ derivative e diagnostic diagnostic plot. Between Between each period there will be a transition period, typically covering more than a decade. Two subsequent flow flow regimes may be of different or similar type. The sequence sequence of flow periods is called called a fingerprint fingerprint,, and galleries of these are called fingerprint libraries. An example of part of such a library is shown in Fig. 54 54..
There is a close relation between the dimension in which the pressure front spreads and the slope in the log–log derivative diagnostic
Figure 54: Examples of fingerprints from a fingerprint library.
69
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l e c t u r e no no t e s i n we we l l - t e s t i ng ng
1−/2 1 − /2 −0 1
plot over the corresponding time period: If the front spreads in dimensions, then the slope will be . This relationship is summarized in Table 8 Table 8.. Flow Flow regi regime me
dime dimens nsio iona nali lity ty
Spherical
3
Radial
2
Bilinear
1½
Linear
1
Depletion
Table 8: The connection between flow dimensionality and slope in the log–log derivative diagnostic plot.
slop slope e
Finite-conductive Finite-conductive fractures
0
Channel sands and narrow fault blocks In this section we will investigate systems with middle-time linear flow, that is systems with channelized flow. Examples of such reser voirs are systems of tilted fault blocks that form narrow rectangular compartments as illustrated in Fig. 55 Fig. 55.. Another example are systems Figure 55: A system of tilted fault blocks forming narrow rectangular compartments.
of stacked channels forming channel belts in a low-permeable background. These channels form narrow compartments of sand as illustrated in Fig. 56 Fig. 56.. Figure 56: Stacked channel sands in a low-permeable background. Narrow compartments of sand are formed at low sand/background ratios.
A system system with with two two parall parallel el bounda boundarie riess canbe analy analyzed zed using using super super-positi position. on. As shown shown in Fig. Fig. 57 an infinit infinite e number number of images images are needed needed in this case. The first flow regime, valid for times where the radius of invest investiga igatio tion n is shorte shorterr than than the distan distance ce to the neares nearestt wall, wall, is radial radial flow: ln ln (249)
= 21 [ (4 ) − + 2]
.
Figure 57: A well between to parallel boundaries and its images
r e s e r v o i r b o u n da da r i e s
71
At late times, when the distance to the pressure front is much larger than the channel width, we have linear flow
= / ≫
= 2 √ + +
,
(250)
where is the dimensionless channel width, and is pseudoskin, which is due to limited entry as illustrated in Fig. 58 Fig. 58.. 18 19 As long as , the pseudoskin is ,
= 2 , ln
sin
(251)
18
where is the the dist distan ance ce from from the the well well to the the near neares estt boun bounda dary ry.. We will will later see that a similar expression for pseudoskin (Eq. 282 (Eq. 282)) is important for the understanding of the productivity of horizontal wells. As shown in Fig. 59 Fig. 59,, the derivative plot for a channel system will have a plateau, characteristic of radial flow, at early times, and a line with slope for later later times. The radial radial flow period period is used to get estima estimates tes for permea permeabil bility ity and skin, skin, and the chann channel el width width and and pseupseudoskin can be found based on a fitted straight line on pressure versus plot. We see from Eq. (250 plot. ( 250)) that the slope, , of the straight line is
1/2
√
M. S. Hantush and C. E. Jacob. “Non-steady green’s functions for an infinite strip of leaky aquifer”. In: Eos, Transactions American Geophysical Union 36.1 (1955), pp. 101–112. 19 Madhi S Hantush. “Hydraulics of wells”. In: Advances in hydroscience 1 (1964), pp. 281–432.
= ℎ ℎ , = 1 ℎ , = ( − ∗)2ℎ − − , =0
which gives
Figure 58: Linear flow into a well in a channel. The convergence of streamlines towards the well result in an increased pressure drop compared to fully linear flow.
(252)
(253)
and that the pseudoskin is
∗
(254)
where is the extrapolated straight line pressure at . Actual Actual pseudoskin depends on reservoir heterogeneity, and well test based estimates (Eq. 254 (Eq. 254)) is preferred to estimates based on the theoretical formula (Eq. 251 (Eq. 251). ). Linear flow
plot
Slope Slope m
For cases cases where where the channe channels ls are narrow narrow,, the radial radial flow flow period period will will be suppressed, and it will be impossible to to use it to get good estimates for permeability and skin. If we have other data that can give
/
Figure 59: Plots for a channel system: The time period with slope on the derivative plot corresponds to linear flow, and the channel width is estimated based on the slope of the straight line on the plot.
√
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estimates for the channel width, such as seismic data for fault blocks and and outc outcro rop p data data for for chan channe nell belt belts, s, we can can in thes these e case casess use use Eq. Eq. (252 252)) to get a permeability estimate:
= ℎ .
(255)
Leaking boundaries In this this sect sectio ion n we will will inve invest stig igat ate e the the effe effect ct of boun bounda dari ries es that that are are not not completely sealing. sealing. Two examples examples will be discussed in more detail: a partly sealing fault with good reservoir sand on both sides, and a boundary between good sand and a low permeable background. Fig. 60 Fig. 60 illustrates illustrates a typical situation where a fault runs through a rese reserv rvoi oirr. We may may have have good good rese reserv rvoi oirr sand sand on both both side sidess of the the faul fault, t, and at the fault location there is a zone with reduced permeability. Both the faulting itself, and subsequent diagenetic processes contribute to the permeability reduction. The permeability in the fault zone is typically not zero, so communication is possible across the fault. In general the communication across the fault will be between different reservoir zones, with different permeability and thickness. A general model that could be used for investigating the effect of a non sealing fault on a well test is shown in Fig. 61 61..
Figure 60: Communication across a fault
Figure 61: Simplified model for a non-sealing fault
We will will deri derive ve the the well well test test resp respon onse se for for a mode modell that that has has been been simsimplified even further: First, the fault zone is modelled as as a plane with characteristic property . This is the standard representation
=
of faults in reservoir reservoir simulation. simulation. Second, Second, we will assume that the reservoir on both sides have the same properties and height (essentially the same zone). Last, we will model the flow in two dimensions only, that is we will ignore the effect of the flow convergence convergence towards the limited height contact zone. Fig. 62 shows shows an aerial aerial view view of the model. model. An activ active e well well is place placed d at a distance from the fault, and we will also investigate the response in an observation well on the far side of the fault. In this analysis we will use as characteristic length for dimensionless parameters:
= = =
,
(256)
The fault zone is modelled as a plane.
Figure 62: Areal view of a non-sealing fault with an observation well on the far side
r e s e r v o i r b o u n da da r i e s
= ℎℎ .
73
and the fault is characterized by the dimensionless quantity :
(257)
If we igno ignore re the the shor shortt time time effe effect ctss due due to the the finit finite e well wellbo bore re radi radius us,, the the dime dimens nsio ionl nles esss pres pressu sure re in the the faul faultt bloc block k with with a well well prod produc ucin ing g at at a constant rate is described by the following diffusivity equation:
+ + 2( 2 ( + 1)(()) = , = −1 = 0 + = . = = 0 = 0 (0, )) = (0,) , (0,)) = ((0,)−(0, )) . = −1 1= 0 1 1 1 ( , )) = 2 44 , ) , + 2 −( √ ( , )) = ᵆ 2√ 2√ + 1√ √ . 1/2
(258)
where is the Dirac Dirac delta delta functi function on20 which represent a point sink (the well at and ). The diffusivity equation that describe the pressure on the other side of the fault is
∫
20
The delta function is zero every where, but
(259)
The initial condition, and the boundary condition at infinity, is that pressu pressure re is initia initiall pressu pressure: re: . In addi additi tion on we have have two two cononditions at the fault that couple the two equations: Flow is continuous across the fault ( )
(260)
and the flow through the fault is proportional to the pressure difference across the fault
(261)
The equation system (258 (258)–( )–(261 261)) may be solved using a Laplace transform in time and a Fourier transform in .21 The well pressure ( and ) is E1
E1
(262)
where
erfc
(263)
We see that the well test response is the sealing fault response (Eq. 235 (Eq. 235)) minus a correction that is proportional to the transmissibility of the fault ( ). The corresponding derivative diagnostic diagnostic plot is shown in Fig. 63 Fig. 63.. For long enough enough times, times, the plot falls back back to a plateau value of , so we have full-circle radial radial flow. The effect of the fault is seen in an intermediate period, and the fault transmissibility can be estimated based on type-curve matching (see page 128 page 128). ). The fault properties can also be inferred from the response in an observation observation well well on the far side of the fault. fault. If the observation observation well is directly opposed to the active well as shown in Fig. 62 62,, the observation-well pressure (pressure at and ) is
= = 0 ( , )) = (+, ) = = +
(264)
21
L.M. Yaxley. “Effect of a Partially Communicating fault on Transient Pressure Behavior”. In: SPE Formation Evaluation (Dec. 1987), pp. 590–598.
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Figure 63: Log derivative diagnostic plot for a well near a non-sealing fault
(, ))
where
is the correction function defined in Eq. (263 is (263). ). The cor-
Figure 64: Log derivative diagnostic plot for the observation well on the far side of a non-sealing fault
responding derivative diagnostic plot is shown in Fig. 64 Fig. 64.. The time time it takes for the pressure signal to reach the observation well depend on the fault transmissibility, and again the fault transmissibility can be estimated based on type-curve matching. For long enough times, the plot has a plateau value of , corresponding to full-circle radial flow. The next example of a leaking boundary is a system where the well is placed placed in a sand body body of limited extent. extent. If the well is close to the edge edge of the sand-b sand-body ody,, and theboundary theboundary is linear linear,, this this syste system m is called called linear-composite. The system is illustrated in Fig. 65 Fig. 65.. The equations that govern the pressure response of this system is very similar to the equations for a leaky fault (Eqs. 258 258– –260 260), ), and and they they can can be solve solved d with with the same same method methods. s. The fingerp fingerprin rintt diagno diagnosti sticc plot plot corresponding to the linear composite are shown in Fig. 66 Fig. 66.. The characteristics of the plot is an initial plateau, corresponding to radial radial flow flow, follo followe wed d by a transi transitio tion n to a second second higher higher platea plateau. u. The first plateau plateau reflects reflects the permeabil permeability ity in the sand body where where the well is located, while the second reflects the average permeability in the
1/2
Figure 65: A well close to the edge of a sand-body
Figure 66: Fingerprint of a linear– composite system
r e s e r v o i r b o u n da da r i e s
sand and the low permeability background:
/ ⪅ 0.01
= 4ℎ = 4 ℎ
(265)
If the permeability of the background is much lower than the sand ( ), then we will see a doubling just as for a completely closed barrier.
75
Horizontal wells
Today most new production wells are horizontal or highly deviated. Compared to a vertical well, a horizontal well has an increased reser voir contact and an increased drainage area. As long as vertical communication is sufficient, a horizontal well will typically also have increase creased d produc productiv tivity ity.. In reserv reservoir oirss with with an aquif aquifer er,, gas cap or both, both, or with water or gas injection, horizontal wells can be judiciously placed to avoi avoid d gas gas and and wate waterr coni coning ng and and earl early y brea break k thro throug ugh h of wate waterr or gas, gas, which gives higher oil rates and increased reservoir sweep. The object objectiv ives es for for testin testing g of horizo horizonta ntall wells wells are basic basicall ally y the same same as for horizonta horizontall wells: wells: Determine Determine permeabil permeability ity (horizont (horizontal al and vertivertical), skin (formation damage) and pseudoskin (completion effectiveness), and detecting reservoir and sand body boundaries.
Figure 67: Horizontal vs. vertical well
Flow regimes The main ideal flow regimes in a horizontal well are illustrated in Fig. 68 Fig. 68.. The flow regimes are vertical radial flow, where the top and botto bottom m of the reserv reservoir oir is not yet yet seen, seen, interm intermedi ediate ate linear linear flow flow, which which is simi simila larr to flow flow in a chan channe nel, l, and, and, when when the the leng length th of the the well ell is negnegligibl ligible e compa compared red to the radius radius of invest investiga igatio tion, n, horiz horizont ontal al radial radial flow flow. In addi additi tion on ther there e will will be an init initia iall peri period od domi domina nate ted d by well wellbo bore re stor stor-age and skin, and a late time boundary dominated flow period. The ideal log–log diagnostic plot showing these flow regimes are shown in Fig. 69 Fig. 69.. It should be noted that it in practice can be difficult to identify all flow regimes.
Vertical radial flow.
Intermediate linear flow
Horizontal radial flow. Figure 68: Ideal flow regimes in a horizontal well.
Figure 69: Ideal log–log diagnostic and flow regimes for a horizontal well
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Anisotropy We have previously assumed that permeability is isotropic and can be treated as a scalar. However, the permeability is usually different for for verti vertical cal and horiz horizont ontal al flow flow, so for the descri descripti ption on of flow flow around around a horizo horizonta ntall well, well, where where we have have flow flow in both both the horiz horizont ontal al andvertica andverticall direction, this assumption is normally not valid. Permeability is actually a tensor 22 , K , and Darcys law should be expressed as
= −1 ⋅ ∇ , = −1 ,
K
22
Permeability is a linear operator that operates on a gradient to produce a vector. Such operators are called (2,0) tensors, or more simply tensors of order 2. A tensor of order 2 can be represented by a matrix in a given coordinate system, and the operation as a matrix–vector matrix–vector multiplication.
(266)
or, on component form
=
(267)
where K . Since Since permeability permeability is a tensor, tensor, flow is in general general not parallel parallel to the pressure pressure gradient. gradient. Howeve However, r, the permeabil permeability ity is associated with certain certain principal directions in space. If the pressure gradient is along any of these principal directions the flow will be in that direction, and the matrix that represent the permeability tensor is diagonal in a coordinate system that follows the principal directions. There are three orthogonal principal directions, and the directions tions are determ determine ined d by the geolog geology y. In genera generall both both the permea permeabil bility ity along along the princi principal pal direct direction ionss and the direct direction ionss themse themselv lves es are space space dependent, however, the main anisotropy is usually vertical vs. horizontal, in which case the principal directions are
• Orthogonal to bedding (“vertical” • Parallel to bedding (“horizontal”
or
or
Principal directions
)
and
)
Vertica erticall permea permeabi bilit lity y can can be seve several ral orders orders of magnit magnitude ude smalle smallerr than than horizo horizonta ntall permea permeabil bility ity.. The ratio ratio is calle called d KVKH-r KVKH-rati atio o and is an important property for correct reservoir modelling. The main source of anisotropy is small scale heterogeneities. geneities. Since Since the nature nature of deposited deposited material material is not constant constant over time, there will always, in a sandstone reservoir, be a layered or semi-layered structure. structure. The spatial frequency of these structures structures vary, and can be as high as on the mm scale in the case of tidally influenced enced deposi deposits ts as illust illustrat rated ed in Fig. Fig. 70 70.. On larger larger scales scales (dm–m (dm–m), ), both both depositional and diagenetic processes produce horizontal sheets of shales and cemented barriers that restrict vertical flow as illustrated in Fig. 71 Fig. 71.. In contrast, in fractured reservoirs (see page 91 page 91)) the main fracture direction is often vertical so that the permeability is largest in the vertical direction.
<
Quantitative analysis In this section we will discuss how the vertical–radial and linear flow regimes may be analyzed. analyzed.
KVKH-ratio
Figure 70: Small scale heterogeneities
Figure 71: Barriers and baffles restricting vertical flow
horizontal wells
Vertical–radial Vertical–radial flow The vertical radial flow regime (Fig. 72 72)) can be analyzed using a straight line fit on a semilog plot. That is, permeability and skin can be found in the same way as for a vertical vertical well. well. But since since the permeability is anisotropic it is unclear which permeability is actually estimated by this procedure, and what the measured skin factor represents. The flow around a vertical well is described by the anisotropic diffusivity equation
+ =
,
Figure 72: Vertical radial flow
(268)
where the well runs along the x-direction. Due to the permeability anisotropy, the pressure front travels faster in the horizontal direction than in the vertical, and the lines of equal pressure are ellipses. We can transf transform orm the anisot anisotrop ropic ic equati equation on (Eq. (Eq. 268 268)) into into an equi equivvalent isotropic equation by scaling lengths with , so that . Then we have
′ √ / ′ = √ ′/ = ′ ′ = ′ ′ . ′ ′ + ′ = ′′ , ′ ′ = ′ ′ ′ ′ = ′ ′ = ′ . ′ = 1 ⟹ ′ = , ′ ′ + ′ = . ′ = 4 1 ,
(269)
Applying the above transformation to Eq. (268 (268), ), then the resulting equation is
(270)
where we have replaced , and with , and to indicate that these parameters may not be invariant under the coordinate transform. form. Since Since the parameters parameters are volumetric volumetric they will be invariant invariant if the coordinate transform preserve volume, i.e. .We have
(271)
Thus, under the condition
(272)
the parameters and will will be invariant under the coordinate transform. Flow governed by the anisotropic diffusivity equation can then be described by an equivalent isotropic equation:
(273)
This implies that the permeability estimate found by fitting to a straight line,
(274)
79
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where is the well length, corresponds to the equivalent isotropic permeability
′ = .
(275)
In order to understand how the estimated skin is to be interpreted we will have a closer look at how the shape of the bore-hole and its surroundin surroundings gs are transform transformed. ed. The anisotrop anisotropy y is characte characterized rized by the KVKH–ratio, :
permeability
.
= . ′ = , ′ = √ √ ′ = 1 ′ . ′ < < 1 < >
(276)
In the equivalent isotropic system lengths in the scaled with
direction are
thus
An anisotropic reservoir behaves like an isotropic reservoir with effective
(277)
(278)
Typically, , so that . This yields and , thus due to the scaling of lengths (Eq. 278 278), ), the effective thickness of the reservoir is increased. As a result, the duration of the vertical radial flow period is increased in an anisotropic reservoir. In spite of this, the flow regime may often not be observed with realistic well paths. Vertical positioning of horizontal wells can be poor even when geo-steering tools are used, and wells are also intentionally drilled through layering to create improved improved reservoir contact. Examples of Figure 73: Typical well paths of a real horizontal well
well paths are shown in Fig. 73 Fig. 73,, and it is unlikely that vertical radial flow is reached in these wells. We see from Eq. (278 (278)) that the circular wellbore is transformed to an ellipse as shown in Fig. 74 Fig. 74,, and the damaged volume around the wellbore is also transformed in a similar manner. The permeability of the damaged zone does typically not have the same anisotropy as the reservoir, so we will have an anisotropic permeability in the skin region region of the equivale equivalent nt model. model. Any near wellbore wellbore heterogeneit heterogeneities ies will also be transformed, and will influence pressure profile differently than the bulk – model. As a consequence consequence,, the measured measured skin skin has no direct direct physic physical al interp interpret retati ation on in terms terms of permea permeabil bility ity and thic thickn knes esss of a dama damage ged d zone zone.. Predi Predict ctin ing g the the skin skin base based d on a mode modell of the damaged zone is non-trivial. non-trivial. The skin is therefore estimated estimated by the differ differenc ence e betwe between en thepressure thepressure in a isotro isotropicmodel picmodel with with effect effectiv ive e
Figure 74: The circular wellbore is transformed to an ellipse.
horizontal wells
permeability and a circular wellbore with radius , and the actual extra extra pressu pressure re drop drop due to a damage damaged d zone zone in an anisot anisotrop ropic ic reserv reservoir oir.. The coordinate transform is volume preserving, so it does not influence the wellbore storage effect.
Intermediate–linear flow As shown in Fig. 75 Fig. 75,, the intermediate flow situation is analogous to linear flow in a channel (see page 71 page 71), ), and the well pressure is given by
= ℎ √ + 2√ ( + ( , , ℎ, ℎ , ) . 2 √ = = √ √ = 1 . ℎ √ ∗ = 0 2√ ∗ − . = 2√ = / ℎ/ /ℎ ℎ √ ( ) , (,ℎ,) = 1 + /ℎ
(279)
Here is the length of the well. The first term is the solution to the 1-D 1-D diff diffus usiv ivit ity y equa equati tion on with with a poin pointt sour source ce,, and and sinc since e the the flow flow is hor horizontal it only depend on the horizontal permeability . The second term is the pressure drop due to skin, which depend on the effective isotropic permeability . The third third term is prespressure drop due to convergence of flow into the well in two dimensions (pseudoski (pseudoskin). n). Here is the height from the formation bottom, as illustrated in Fig. 75 Fig. 75.. The linear flow regime can be analyzed to get estimates for horizontal permeability, permeability, , and pseudoskin, . We see from from Eq. Eq. (279 (279)) that if we plot pressure as a function of and and fit the linear flow period riod to a straig straight ht line. line. Then Then the horiz horizont ontal al permea permeabil bility ity is found found from from the slope, , as
(280)
If the effective permeability, , and skin is known, the pseudoskin can be found using the extrapolated pressure at
(281)
The ideal pseudoskin in a homogeneous system depend on the three ratios , , and as ln
sin
(282)
where is the relative position (depth) of the well in the zone. In theory, one can use a measured pseudoskin to evaluate the well positioning. In practice this may be impossible due to near wellbore heterogeneities or variation in the relative vertical position along wellbore
Near well heterogeneities All well testing is sensitive to near well heterogeneities, and a horizontal well is even more sensitive to near well heterogeneities than a
Figure 75: Intermediate linear flow. The additional pressure drop due to flow convergence towards the well is called pseudoskin.
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vertical well. An illustration of a horizontal well in a heterogeneous environment is shown in Fig. 76 Fig. 76.. The heterogeneities will influence Figure 76: A horizontal well in a heterogeneous environment
skin and pseudoskin, so the measured skin and pseudoskin will contain information relating to the heterogeneity pattern. On the other hand, the presence of these heterogeneities will influence the pressure response in a way that may mask the vertical radial flow regime. The ideal idealize ized d analys analysis is assum assume e that that the inflow inflow is const constant ant along along the well. This may not hold due to a heterogeneous formation, uneven formation damage, or well friction, as illustrated in Fig. 77 Fig. 77..
Figure 77: Uneven inflow along the well
Well tip effects If the well does not penetrate the whole reservoir there will be a heel and toe effect that may mask the vertical radial flow or linear flow. The drainage pattern around the central part of the well will be radial, while the drainage at the tip will have a (hemi)-spherical shape. This is illustrated in Fig. 78 Fig. 78.. To get a rough estimate for when the spherical flow pattern start influ influen enci cing ng the the pres pressu sure re resp respon onse se in the the well well we may may assu assume me that that the the contribution from each flow regime (radial vs. spherical) is proportional tional to the area of the correspon corresponding ding pressure pressure front. The area of the spherical front is
≈ 4 ∝ , = √ 44 ≈ 2 ∝ √ . spherical
inv
(283)
where inv is the radius of investigation (Eq. 81 (Eq. 81). ). The area of inv the radial front is radial
inv inv
(284)
The spherical contribution is negligible for short times, but due to the different time dependency of the two terms, spherical flow will ultimately be dominant. The tip effect is negligible when spherical radial :
≪ 21 , inv inv
≪
(285)
Figure 78: Effect of finite well length
horizontal wells
which gives
≪ 16 = 16′ ′ ,
(286)
where Eq. (286 (286)) is expressed in terms of lengths in an isotropic reser voir. We will need to take into account account anisotropy. anisotropy. By repeating the procedure outlined on page 79 page 79 (scaling (scaling lengths with ), but now in three dimensions, we see that we need to make the following substitutions23 :
√ ′/ ′ → , ′ → ℎ′ → ℎ , ℎ ≪ 16 = 16 . and
(287)
where is the reserv reservoir oir thickn thickness ess.. We may may concl conclude ude that that the tip effect effect can be neglected as long as
(288)
The time where the tip effect can be ignored depend only on the horizontal permeability, and Eq. (288 ( 288)) can also be obtained by just considering Eq. (285 (285)) as an inequality in one dimension along the wellbore. In short horizontal wells the tip effect may come into play before the pressure pressure front reaches reaches the top and bottom bottom of the reservoir reservoir.. In these cases we will not see the intermediate linear flow regime. If the well is in the middle of the reservoir, the time to reach top and bottom is determined by
= 21ℎ′ ,
inv inv
(289)
where lengths are measured in the equivalent isotropic reservoir. Using the definition of radius of investigation, solving for time, and making the substitutions (Eq. 287 (Eq. 287)) we get
= 16′ ℎ′ = 16 ℎ = 16 ℎ
,
(290)
where is the the time time to reac reach h the the top top of the the rese reserv rvoi oirr. This This time time depe depend nd only on the vertical permeability, permeability, and Eq. (290 ( 290)) can also be derived by just considering Eq. (289 (289)) as an equality in the vertical direction. If we say that Eq. (288 ( 288)) is equivalent to
< 0.01 16 , < . 16 16 ℎ < . 16 ℎ <
(291)
then by comparing Eq. (291 ( 291)) and Eq. (290 (290)) we see that intermediate linear flow can be observed when 0 01 01 0 01 01
100
83
(292)
23
The reader might find it confusing that the effective permeability is different in two and three dimensions (Eqs. 275 (Eqs. 275 and and 287 287). ). Note that the equation for the permeability estimate (Eq. 274 (Eq. 274)) contain the well length, , which has to be scaled for three dimensional flow. When taking this into account, the permeability estimated in the well test is just as in the two dimensional case. Note also that the relative scaling of horizontal and vertical lengths is the same.
√
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This yields the following criterion for observing intermediate linear flow:
>
10
ℎ .
(293)
Criterion for observing intermediate linear flow
Fractured wells Hydraulic fracturing is a popular and effective stimulation method. A fracture is defined as a single crack initiated from the wellbore by hydraulic fracturing, that is by injecting a fluid (typically water with additive additives) s) at high pressure. pressure. The fracture fracture is kept open by injecting injecting a proppant (sand or similar particulate material) with the fracturing fluid. A fractured well has an increased increased productivity since since the fracture provides an increased surface for the reservoir fluid to enter the wellbore. The orientation of the fracture plane is determined by the minimum stress direction, and since the maximum stress is always the overb overburd urden en in deep deep reserv reservoir oirs, s, fractu fractures res are verti vertica call in deep deep reser reser- voirs. Note that hydraulically hydraulically induced fractures are different from naturally occurring fractures, which will be discussed in a separate chapter (page 91 91). ). Natural Natural fractures fractures are often often called “fissures “fissures”” to avoid confusion. In addition to the usual reservoir characterization goals, well tests are performed in order to investigate the efficiency of hydraulic fracturing turing jobs, jobs, and to monito monitorr any possib possible le degrad degradati ation on of fractu fracture re propproperties due to production. production. In order to reach these goals, goals, a separate separate well test should preferentially be performed prior to fracturing of the reservoir to determine permeability and skin.
Laplace transform in well testing In this and in subsequent chapters we will apply the Laplace transform. The Laplace transform is useful for solving differential differential equations, as it can transform partial differential equations into ordinary differential equations we are able to solve. The transform was in fact develo developed ped to solve solve the diffus diffusion ion equati equation, on, i.e. i.e. an equati equation on on the form form of Eq. (14 (14). ). Joseph Fourier had introduced a method to solve the diffusion equation using using the Fourier transform. Laplace recognized recognized that the Fourier transform was only applicable for a limited space, and introduced the Laplace transform to find solutions in indefinite space. The Laplace transform is an integral transform, and it is usually applied as a transform in time:
ℒ ∶ → ̃ (̃ () = (∞ ) .
(294)
The partial differential operator we used in the superposition principle (see page 39 page 39)) was linear. linear. Also the Laplace Laplace transform transform is a linear linear
Figure 79: The concept of hydraulic fracturing
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operator:
ℒ (ℒ +()) == ℒℒ +=ℒ̃ =. ̃+̃
(295)
∞ ∞ 1 ℒ(1) = ∞ = − ∞= 1 ∞ ℒ() == = − − −1 = 1 ℒ(1) = 1 , ∞ ℒ = ∞ = ̃ ℒ = = (̃ (,,)−) − (,,0) , ∫ (, ) = ∫ (,) ∇ = , ∇̃ = ̃ − .
Calculating the integral, we observe that the Laplace transform of unity and of are are
(296)
where we are applying integration by parts when solving the transform in time . We observe that that time and and the variable has an in verse correspondence, correspondence, hence late times corresponds corresponds to small . This correspondence will be utilized in the next chapter. The transformation of the partial derivatives are
(297)
where we use the basic rule for for the first first equality, equality, and integration by parts for the second equality. Using Using the above above rules, rules, we see that that the diffus diffusivi ivity ty equati equation on (Eq. (Eq. 16 16), ),
(298)
is transformed into the form
(299)
Vertical Vertical fractures As discussed in the introduction, fractures in deep reservoirs are typically in the vertical direction. In this chapter we will therefore conside siderr the the anal analys ysis is of a well well test test in a vert vertic ical al well well with with a vert vertic ical al frac fractu ture re that covers the whole reservoir thickness. A simpli simplified fied model model for for this this situat situation ion is illust illustrat rated ed in Fig. Fig. 80 80,, and and the the fracture is characterized by fracture half length, , fracture width, , and fracture permeability, permeability, .
fractured wells
87
Figure 80: Characteristic parameters for a vertically fractured well
Flow periods and flow regimes The possible flow regimes in a vertically fractured well are shown in Fig. 81 Fig. 81:: Initially Initially there is flow only in the fractures fractures (a), this is called called Figure 81: Possible flow regimes in a vertically fractured well
fracture linear flow. This regime is, however, never observed in practice, and the effect of the fracture is instead seen as an increased volumefor wellb wellbore ore storag storage. e. In the second second regime regime,, (b), (b), the pressu pressure re front front extends both linearly along the fracture and linearly into the reser voir close to the fracture. This is called bilinear flow. This regime is only observed when the fracture has a low conductivity. In the third regime, (c), the pressure front moves linearly out from the fracture into the reservoir. reservoir. This is formation formation linear flow. When the distance distance from the well to the pressure front is large compared to the fracture length, we are in the fourth regime, (d), where the flow is infiniteacting radial flow, called formation radial flow.
Fracture linear flow
Bilinear flow
Formation linear flow
Formation radial flow
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Formation radial flow Theradial Theradial flow flow canbe analy analyzedis zedis using using thestandardmetho thestandardmethods ds for for nonnonfractu fractured red wells wells.. In parti particul cular ar this this means means that that permea permeabil bility ity and skin skin is found found by fitting to a straight straight line on semi-log plot. plot. As illustrated illustrated in Fig. 82 Fig. 82,, there will be a reduced convergence of flow lines into a fractured tured well well comp compare ared d to a non-fr non-frac actur tured ed well. well. A succe successf ssfull ully y fractu fractured red well will thus have a negative skin, reflecting the desired improved productivity.
Formation linear flow In this section we will derive the equation for linear (1-dimensional) flow, and show how the analysis of the formation linear flow period can provide an estimate for how far the fracture has propagated into the reservoir, that is the fracture half-length. As long as the propagating pressure front is close enough to the fracture such that we can ignore tip effects, the problem is one dimensional. The diffusivity equation is then
where
= 1 , = − = . (,0), 0) = 0 (∞, ) = 0 . || = −4ℎ () ,
and
Figure 82: Due to the reduced flow convergence, convergence, the effect of the fracture will be a negative skin or increased defective wellbore radius.
(300)
(301)
Pressure is at initial pressure at infinity, giving the boundary conditions and (302) Darcy’s law gives the boundary condition for the pressure gradient at the fracture: (303) where we have used that half of the flow rate goes into each flow direction (see Fig. 80 Fig. 80). ). Equations (300 (300– –303 303)) can be solved by Laplace transform in time: Similar to the transform into Eq. (299 (299), ), the partial differential equation (Eq. 300 (Eq. 300)) is then transformed into an ordinary differential equation in x:
̃ = ̃ . ( ̃ ,, ) = + − . = 0
(304)
where we have applied the initial condition (Eq. 302 (Eq. 302). ). The gene general ral solution to Eq. (304 (304)) is exp
exp
(305)
Due to the bounda boundary ry condi conditio tion n at infinit infinity y (Eq. (Eq. 302 302)) we hav have , and and B is determ determine ined d by applyi applying ng the bounda boundary ry condi conditio tion n on the deriva derivativ tive e (Eq. 303 (Eq. 303): ):
̃ || = ∞ −4ℎ () = − 4ℎ (̃ () .
(306)
Solving by Laplace transform.
89
fractured wells
() = = 0 (̃ () = ⋅ ⋅̃ 1 = , 1/ − = ̃ || = −4ℎ = 4√ ℎ ( ̃ ,, ) = 4√ ℎ − . √ (,, ) = 4ℎ 2 − √ 2√ . =0 √ ℎ√ √ . () = − 2ℎ√
Assuming a constant well rate
starting at
then gives
where we use that the Laplace transform of unity is Thus
(307)
fromEq. fromEq. (296 296). ).
(308)
This give the following solution in Laplace space: exp
(309)
Applying Applying the inverse inverse transform transform24 onEq.(309 onEq.(309)) give givess the the pres pressu sure re as a function of time and distance from the fracture: erfc
24
The inverse transform can be found in good tables of Laplace transforms, or by using a computer algebra program such as Maple or Mathematica.
(310)
For well testing purposes we are only interested in the well pressure, and inserting into Eq. (310 (310)) gives
(311)
We see from Eq. (311 (311)) that the well pressure response in the formation linear flow regime is proportional to the square root of time. The flow regime is identified on the log-derivative diagnostic plot as a peri period od with with slop slope e , and and on a plot plot of pres pressu sure re as a func functi tion on of the the period with formation linear flow will show a straight line with slope :
√ √
√ ℎ√ = 2ℎ . = 2ℎ√
(312)
If permeability is known, for instance from analysis of the formation radial flow period or from a test performed prior to fracturing, the slope can be used to estimate the fracture half length:
2ℎ
2 . ℎ = 2
Note that the property actually measured is the fracture area, .
(313)
√
=
Bilinear flow If the fracture has a finite conductivity, an early flow period with bilinear linear flow flow, as illust illustrat rated ed in Fig. Fig. 83 83,, may be observ observed. ed. This This flow flow period period can be analyzed to obtain an estimate for fracture conductivity.
Fracture half-length is estimated based on the slope of a straight line on a vs. plot.
Figure 83: Bilinear flow
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The early time behavior with a finite permeability fracture can be shown to be25 (314)
= − √ √ , = ℎ √ √ .
where 26
(315)
25
H. Cinco-Ley and Samaniego-V. Samaniego-V. F. “Transient Pressure Analysis for Fractured Wells”. In: Journal of Petroleum Technology (Sept. 1981), pp. 1749– 1766. 26 is the gamma function.
⋅
We see see fromEq. fromEq. (314 314)) that that the the pres pressu sure re resp respon onse se for for bili biline near ar flow flow is proportional to the 4th root of time. Thus, the flow regime is identified on the log-de log-deriv rivati ative ve diagno diagnosti sticc plot plot as a period period with with slope slope , and
√ √ 1 . = ℎ √ ≈ √ ℎ .
on a plot of pressure as a function of , the period with formation linear flow will show a straight line with slope (see Fig. 84 Fig. 84). ). If permea permeabil bilityis ityis known, known, the slope slope can can be used used to estima estimate te the fracfracture conductivity conductivity :
0 152 152
√ √
Figure 84: Special plot used to estimate fracture conductivity
(316)
Naturally fractured reservoirs Naturally fractured reservoirs constitute constitute a huge portion of petroleum reservoirs throughout the world, especially in Middle East. A natural fracture are created when stresses exceed the rupture strength of the rock, rock, and the fractu fracturin ring g proce process ss is more more preva prevalen lentt in brittl brittle e rocks rocks such such as limestone, as opposed to sandstone. The presence of fractures is crucial for the productivity of low permeable meable rocks, rocks, as the highlycondu highlyconducti ctive ve fractu fractures res increa increase se the effect effectiv ive e permeability of the formation. A naturally fractured formation is generally represented by a tight matrix rock broken up by highly permeable fractures (see Fig. 85 85). ). Figure 85: Representations of a dual porosity reservoir
The fractures tend to form a continuous fracture network throughout throughout the formation, formation, and they represent the dominant dominant flow paths. If only the fractures are connected, and there is virtually no long distance transp transport ort in the matrix matrix system system,, the forma formatio tion n may be denote denoted d as dual porosity . If there is some flow through the matrix system, the formation is denoted as dual porosity – dual permeability . Note that macroscopic behavior of these systems in many cases are single porosity, where the presence of fractures only effect effective permeability and porosity. In these notes we will only discuss dual porosity systems.
Dual porosity model In a dual porosity system, the macroscopic fluid flow is only in the fractures fractures.. A characteri characteristic stic feature feature of dual porosity porosity systems systems is that the effective permeability of the formation, as found by well testing and production, is much larger that the permeability measured on core material:
≫ = . formation
This is illustrated in Fig. 86 Fig. 86..
matrix
core
(317)
Figure 86: A cored well in a fractured reservoir. The permeability of the core plugs represent the matrix and is not representative for formation permeability which describes flow in the fractures.
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In many cases the number of fractures penetrated by a well is limited, and extra fractures must be created in the near well volume by hydraulic fracturing fracturing to sufficient productivity. The wells produce from the fracture system only, but the bulk of the oil is stored in the matrix: (318) matrix matrix fracture fracture
≫ .
As a result result of produc productio tion, n, a pressu pressure re diffe differen rence ce betwe between en matri matrix x and fracture develops, and the matrix will supply fluid to the fractures. This oil is subsequently produced through the fracture system as illustrated in Fig. 87 Fig. 87 In well testing, dual porosity systems are analyzed in terms of the dual dual porosi porosity ty model model comm commonl only y used used in reserv reservoir oir simula simulatio tion. n. The dual dual porosity model is a macroscopic model where the representative elementary volume (REV) contains many matrix blocks. The fractures are thus not explicitly modelled, in stead each point in space (REV) is modelled as having two sets of dynamic variables, one set for matrix, and one set for fractures:
• Two pressures, matrix pressure
, and fracture pressure
• Two Two set of satura saturatio tions, ns, matrix matrix satura saturatio tions ns ( fracture saturations ( , , and ).
,
, and and
.
), and and
• The oil and gas in matrix matrix and fracture can can also have have different fluid composition (in the BO-model: and ). In terms of static variables the dual porosity model is described using one permeability (the bulk permeability of the fracture system ), two porosities (the bulk fracture porosity , and the bulk matrix porosity ), and a matri matrix x–fract –fracture ure coupl coupling ing term, term, , which which descri describe be the the abil abilit ity y of the the matr matrix ix to supp supply ly fluid fluid to the the frac fractu ture res. s. Not Note e that that the the permeability is not the permeability in the fractures, it is the effectiv fective e permea permeabil bility ity of the forma formatio tion, n, and that that the porosi porositie tiess are bulk bulk porosities, i.e.
= =
Pore volume in fractures Total volume Pore volume in matrix Total volume
.
(319)
How the dual porosity parameters are actually specified in various reservoir simulation software does however differ between simulators. The dual porosity model is often visualized in terms of the Warren and Root “sugar cube” model, where the matrix is a set of equal rectangular cuboids as illustrated in Fig. 85 Fig. 85.. The validity of the dual porosity model is however not limited to this picture.
The diffusivity equation for dual porosity In this section a diffusivity equation will be developed for the dual porosity model, and the main dimensionless parameters that characterize the solutions to this equation will be identified.
Figure 87: Flow in a dual porosity system.
naturally fractured reservoirs
93
The equation for the flow in the fracture system is similar to the ordinary single porosity hydraulic diffusivity equation (Eq. 13 13), ), but the flow from the matrix into the fractures must be accounted for:
∇ + = .
(320)
Here is the volume of liquid flowing from matrix to fracture per time time and and bulk bulk volu volume me,, and and Eq. Eq. (320 320)) expre expresse ssess the mass mass balan balance ce in the fracture system. There is no bulk flow in the matrix system, so there the mass balance is simply expressed as
= − . = ( − ) ,
(321)
The two equations (Eq. 320 (Eq. 320 and 321 321)) are coupled through an equation for matrix–fra matrix–fracture cture flow. flow. In the standard standard dual porosity model model this flow is proportional to the pressure difference:
Flow from matrix to fracture is proportional to the pressure difference.
(322)
where is the matrix–fra matrix–fracture cture coupling. coupling. If the permeabi permeability lity in the matrix blocks is homogeneous, and the matrix blocks are of similar shape and size, it follows from Darcys law that is proportional to the matrix permeability. Eq. 322 Eq. 322 is is therefore usually written as
= ( − ) , = = + . = ′ ′ ℎ = = ( + ) . = 2 ℎ ( − ) = = +
(323)
where
Shape factor
(324)
is called the shape factor, and
(325)
where is the the volu volume me of matr matrix ix,, is the fractu fracture re volum volume, e, and and is the permeability permeability of the matrix matrix rock. The matrix–fracture matrix–fracture coupling, , is dimensionless, while the shape factor, has dimension of inverse area area.. We will will see see late laterr (on (on page page 101 101,, see see Eq. Eq. 362 362)) that that the the shap shape e fact factor or can be interpreted as , where is a pure geometric factor representing the shape of the matrix blocks, while is the typical matrix block size. For further analysis we will express Eqs. (320 ( 320– –322 322)) on dimensionless form using the following variables and parameters:
(326)
Now Now Eq. 322 can can be subs substi titu tute ted d in Eq. Eq. 321 321,, and and then then we have have two two coucoupled equations: One for the flow in the fracture system
∇ + ( − ) = ,
(327)
Dimensionless variables for the dual porosity model
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and one for the mass balance in the matrix.
(1−) = −( − ) .
(328)
Note that these equations holds for pressures both on normal and dimensionless form. Compared to a single porosity system, the dual porosity system is characte characterized rized by two additiona additionall dimensionl dimensionless ess parameter parameters. s. These These parameters are the storativity ratio
= + , = = .
Storativity ratio
(329)
and the inter-porosity flow parameter
Inter-porosity Inter-porosity flow parameter
(330)
The storativity ratio express how much of the total compressibility can be attributed to the fractures, and in a fractured reservoir is small: . In other systems with dual porosity like properties, such such as certa certain in high high contr contrast ast layere layered d forma formatio tions, ns, can can be larger larger.. The inter-porosity flow parameter express the strength of the fracture– matrix coupling, that is the ability of the matrix to supply fluid to the fracture system. Typical values for is in the range to .
< 10
10 109
Solution in the Laplace domain We will will analy analyze ze theequation theequation syste system m Eq.(327 Eq.( 327)and( )and(328 328)intermsofthe )intermsofthe Laplace transform (see page 85 page 85). ). The Laplace Laplace transformed equation equation for flow in the fracture system (Eq. 327 (Eq. 327)) is
∇̃ + (̃ − ̃) = ̃ , (1 − )̃ = −(̃ − ̃ ) .
(331)
and the equation for mass balance in the matrix (Eq. 328 (Eq. 328)) is
(332)
To simplify simplify notation, notation, we always always assume assume dimension dimensionless less numbers numbers in the Laplace domain. Using Eq. 332 Eq. 332,, the matrix pressure can be expressed in terms of the fracture pressure as
̃ = (1 − ) + ̃ , ∇ − ((11−−))+++ ̃ = 0 .
(333)
and this can be substituted in Eq. 331 Eq. 331 to to get the equation in Laplace space for the fracture pressure:
(334)
Since the wells produce from the fracture system, and there is no macroscopic matrix flow, this is the equation that is relevant for well testing.
Equations in the Laplace domain are on dimensionless form.
naturally fractured reservoirs
(, ,, )) = ((11−−))+++ , ∇ − ̃ = 0 . (,,) + 1 − ()) ̃ = 0 . + − ̃ = 0 ,
95
We can define the following function
(335)
then Eq. (334 (334)) is simplified to the following form:
(336)
The limiting behavior of the function will play an important will role in the well test interpretation. For a well test in a fully penetrating vertical well we are interested in the radial version of Eq. (336 (336): ):
(337)
This can be rearranged to the following equation:
(338)
= √() ̃(,) = (() + (() , (⋅) (⋅) (⋅) (()⋅) ( ) ( ∞ , ) = 0 =0 =1 ̃ || = −1 . () = − () , ̃ = (() = − (() (() ̃ || = − (() (() = − 1 = √(1) √() , ̃(() = √()√(√()) .
where . Equatio Equation n (338 338)) is the modifie modified d Bessel Bessel’s ’s equati equation on27 with constant equal 0. The general solution to Eq. (338 338), ), or equivalently to Eq. (337 (337), ), is I
K
27
For the modified Bessel’s equation and its solutions, see https://en.wikipedia.org/wiki/ Bessel_function.
(339)
where and are constants determined by the boundary conditions, and I and and K are modified Bessel functions of first and second are kind. A plot of the Bessel functions I and K is shown in Fig. 88 Fig. 88.. From this plot we observe that the function I grows and K decreases with increasing . Then the boundary condition implies that . Darcys law determines the boundary condition at the well ( , remember that we are on dimensionless form in the Laplace domain) which in the Laplace domain is (see Eq. 307 Eq. 307))
(340)
By combining Eq. (340 (340)) and (339 (339), ), and using K
K
(341)
we can solve for :
K
K
K
K
(342)
which gives the following general solution for the pressure: K
K K
(343)
Figure 88: The modified Bessel functions I and K .
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l e c t u r e no no t e s i n we we l l - t e s t i ng ng
=1
As we have have atthe well, ell, Eq. Eq. (343 343)) reduc reduce e to the follo followi wing ng soluti solution on for the well pressure:
̃() = √()√(√() ) . K
K
(344)
To get an expression for the well test response, we need to get the inverse transform of Eq. 344 Eq. 344.. An explicit expression for the inverse, and and thusfor thusfor the the the the pres pressu sure re asa func functi tion on of time time,, has has not not been been foun found. d. Howe Howeve ver, r, the early early time time and late late time time behavi behavior or can be analy analyze zed d based based on the large and small limits of . The middle middle time behavio behaviorr can in a similar way be approximately analyzed under some extra assumptions on and . The inverse transform can also be evaluated numerically. We will in the following analyze the early and late time behavior and investigate how the well test may be analyzed. From Eq. (296 (296)) we have that the Laplace transform of time is , thus thus early early times times (small (small ) corre correspo spond nd to large large . Furt Furthe herr, we see see from from Eq. (335 (335)) that for large we we have
()
1/
(, ,, )) ≈ . →0 (, ,, )) ≈ . = (∇ − )̃ = 0 ∇ = . = (∇ − )̃ = 0 ∇ = . = ( + ) + = = =
Similarly, for late times (
(345)
(346)
The early time and late time behavior of the well test can be analyzed based on the large and small limits of (Eq. 335 (Eq. 335). ).
) we have
For , we see that Eq. (336 ( 336)) is , which corresponds to the following inverse of the the Laplace transform:
(347)
Similarly, for , we see that Eq. (336 (336)) is , which corresponds to the following inverse of the the Laplace transform:
(348)
Thus, at early and late time we are back to the dimensionless diffusivity sivity equati equation, on, repres represent enting ing a infinit infinite e actin acting g system system.. These These early early and late time systems are similar up to a scaling of dimensionless time by the constant . We see from the definition of (Eq. 329 (Eq. 329)) and the dimensionless time (Eq. 326 (Eq. 326)) that
(349)
thus a division of dimensionless time with switches between a system system with with the total total compr compress essib ibili ility ty of fractu fractures res and matri matrix x to a syssystem with only fracture compress compressibil ibility ity.. Also from the definition definition of 28 the Laplace Laplace transform transform , multiply multiplying ing with with correspon corresponds ds to dividing dividing time with . Accordingly, Accordingly, represents the infinitely acting sys represents tem with total compressibility, and represents the infinitely acting system with fracture compressibility. compressibility.
̃ ∫ ∫∞
28
Laplace transform:
naturally fractured reservoirs
= 0 (, 0,0,) = . =
97
=0
If we have have no matr matrix ix–f –fra ract ctur ure e flow flow ( ) the the syst system em will will be sing single le porosity with only the fracture system active, and by setting in Eq. (335 (335)) we get (350) This gives the same pressure equation as for the early time behavior, thus the early time behavior ( ) is single porosity infinitely acting with fracture compressibility compressibility.. Similarly, Similarly, the late time behavior ( ) is single porosity infinitely acting with total compressibility. We see from the dimensionless expression for the infinitely acting single porosity well test (Eq. 104 (Eq. 104), ), and Eq. (347 (347)) that the early time dimensionless well pressure is
=
= 21 + + 2 , = (4) − ≈ . = 21 ( ( ) + + 2) . (1 − ) ≫ ⟹ ≪ ( 1 − ) , (1 − ) ≪ ⟹ ≫ 1 − .
where ln pressure is
early
ln
Both early and late time in an infinitely acting well test in a dual porosity system show single porosity behavior.
(351)
0 8091. 8091. Similarly, from Eq. (348 ( 348)) the late time
late
ln ln
(352)
The time when cross-over from early to late time behavior occurs is related to . By inspecting Eq. (335 (335)) we see that we have early time behavior when
(353)
and late time behavior when
(354)
Ideally dual porosity behavior will manifest itself as two parallel straight lines in a semilog plot, as illustrated in Figs. 89 Figs. 89 and and 90 90,, and a dip on the log–log derivative diagnostic plot. Combining Eq. (351 (351)) and (352 (352), ), we see that the separation of the two lines in dimensionless time, that is at fixed pressure early for dimensionless times , gives for late
()
() =
= ,
( ) − ( ) = − 12
(355)
Figure 89: Dimensionless semilog plot of dual porosity with varying from in red to in blue.
as can be seen in Fig. 89 89.. Likewise, the separation separation in dimensionless pressure at a fixed time gives early
late
() . 1− ≈ 1 < ln
(356)
In a fractured reservoir, the time where the late time behavior starts (Eq. 354 (Eq. 354)) is (almost) independent of since . A small , i.e. poor matrix fracture coupling, corresponds to late start of late time behavior, and a large corresponds to early start. Note also that , sotha sothat ln , whic which h mean meanss that that a smal smalll corre correspo sponds nds to a large large separation of the two lines. As mentioned earlier, no explicit expression for the well pressure as a function of time exist. Finding the inverse of the Laplace transform for arbitrary times require numerical numerical inversion. The inversion of Laplace transforms is in general a hard problem, and there exist
1
() < 0
Figure 90: Dimensionless semilog plot of dual porosity with varying from in green to in blue.
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l e c t u r e no no t e s i n we we l l - t e s t i ng ng
no general algorithm for the inversion. However, since we here have an expli explicit cit expre expressi ssion on for the Lapla Laplace ce transf transform ormed ed pressu pressure, re, and since since the pressure has a nice monotonic behavior, a number of possible algorithms gorithms do exist. In cases where we only know the Laplace Laplace transformed pressure for a number of discrete values of , the selection of algorithms is much more restricted, but an algorithm that is suitable for the kind of functions that occur in well testing is the Stehfest algorithm. gorithm. This algorithm algorithm will be discussed discussed in more detail later later (see page 127 page 127). ).
Flow periods In this section we will give a qualitative presentation of the flow regimes and corresponding flow periods that were identified in the preceding section. The The first first flow flow peri period od is radi radial al frac fractu ture re flow flow, that that is radi radial al flow flow in the the fracture fracture system. system. The pressure pressure front spreads spreads radially from the well, and the matrix–fracture flow is negligible since the necessary pressure difference difference has not yet been developed developed.. Since Since diffusivity diffusivity of the radial flow in this period is determined by the small fracture storativety, , the pressure pressure front is spreading spreading relatively relatively fast. Radial Radial fracture flow is illustrated in Fig. 91 Fig. 91.. The late flow period is radial bulk flow, where fractures and matrix are in quasiquasi-equ equili ilibri brium um and the pressu pressure re front front spread spreadss with with the same same speed speed in both both fractu fractures res and matri matrix. x. Since Since diffus diffusivi ivity ty of the radial radial flow flow in this period is determined by the larger total storativety, , the pressure front is spreading relatively slower than in the first period29 . Radial bulk flow is is illustrated in Fig. 92 92.. The intermediate flow period is a transition period with matrix– fracture fracture equilibrati equilibration. on. The spread of the pressure front slows down in the the tran transi siti tion on peri period od,, and and the the matr matrix ix–f –fra ract ctur ure e flow flow is in a tran transi sien entt state as illustrated in Fig. 93 Fig. 93.. The transitio transition n will show up as a pronounced dip on the log-derivative diagnostic plot, and in the period after the dip minimum, the slope on the diagnostic plot will be close to , similar to constant drainage of a finite volume (see Fig. 94 94))30 .
Figure 91: Radial fracture flow
+
1
Figure 92: Radial bulk flow 29
The absolute speed of the pressure
front is proportional (eq. 80 (eq. 80), ), so it is always slowing with time. The relative speed discussed here refers to the proportionality constant.
Analyzing a drawdown test Both early and late time behavior of a dual porosity system is similar to a single porosity system, which means that in these two time periods the pressure will be a straight line on a semilog plot. We will now show show how how thes these e two two stra straig ight ht line liness and and thei theirr sepa separa rati tion on can can be used used to get estimates for formation permeability, , matrix–fracture coupling, , and storativity ratio, . In practice the first straight line is often often hidden hidden by wellb wellbore ore storag storage e and near near near near well well effect effects. s. Howeve However, r, a dual porosity system will also show a dip on the log–log derivative diagnostic plot, and a fit to log–log derivative data, a variant of type curve matching, can also give estimates for and .
Figure 93: Transition period with matrix–fracture matrix–fracture equilibration 30
It should be noted that the nature of the transition period is dependent on how the matrix–fracture matrix–fracture coupling is modelled. In alternative “transient models” the dip in log-derivative will not be as pronounced (see George Stewart:“Well test design and analysis ” pages 582-586 for a discussion of this).
naturally fractured reservoirs
99
We will first discuss the analysis based on semilog plot where, ideally ally, dual dual porosi porosity ty behav behavior ior will will manif manifest est itself itself as two two parall parallel el straig straight ht lines as shown in Fig. 94 Fig. 94.. Figure 94: Semilog plot with dual porosity behavior
= . = . ⋅ (+ ) = . ⋅ . = 4(+) .
The second straight line is used in the usual manner to get estimates for bulk permeability, permeability, , and skin, . The length of the radial fracture flow period depends on (see Fig. 90 Fig. 90), ), and the first straight line line is ofte often n only only seen seen when when is very very smal small. l. In the the case casess it is appa appare rent nt,, the semilog plot can be used to find and (see Fig. 94 Fig. 94). ). As shown in Fig. 94 Fig. 94 two dimensionless times and , can be found based on the pressure pressure in the middle of the transition transition.. The storativity storativity ratio ratio is estimated based on the separation of the two lines (Eq. 355 355 and and 356 356): ):
(357)
and an estimate for is found based on either of the two times: 31 0 561 561
0 561 561
(358)
The numerical factor (0 561) 561) is based on an analysis of the intermediate period and on numerical inversion of Eq. (344 (344). ). One should not expe expect ct Eq. Eq. (358 358)) to give give very very accur accurate ate estima estimates tes for for . A thir third d time time is defined as the start of the late time period based on visual inspection of the log–derivative plot. The time can also used for estimating :
(359)
The numerical factor (4) is again based on numerical inversion.
Derivative analysis Early time dual-porosity behavior is usually suppressed by wellbore storage and near wellbore effects. However, However, the log–derivative log–derivative will have a dip with a minimum which depends on . For large times (and
31
Alain C. Gringarten. “Interpretation of Tests in Fissured and Multilayered Reservoirs With Double-Porosity Behavior: Theory and Practice”. In: Journal of Petroleum Technology (Apr. (Apr. 1984), pp. 549–564.
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l e c t u r e no no t e s i n w e ll ll - t e s t i ng ng
small ) the dimensionless pressure log–derivative is a function of . Based on these observations a number of type curves can be created by numerically inverting Eq. (344 (344)) for various combinations of these parameters. Examples of such type curves are shown in Fig. 95 Fig. 95.. Figure 95: Example of a type curve for log–log derivative diagnostic plot with dual porosity behavior.
With known permeability, permeability, and can be found by fitting the end of transition period to a type-curve by adjusting the two parameters as shown in Fig. 96 Fig. 96.. This is an an example example of a general general method much much used
Figure 96: Finding and by fitting to a log–log derivative type curve: Early time data that are influenced by wellbore storage and skin are removed. Data points are moved horizontally horizontally by adjusting , and is found based on the minimum.
in well test analysis: fitting to type curves. Several other examples of type curves for the analysis of dual porosity systems are given in the review article by Alain C. Gringarten. 32 In a dual porosity simulator, the parameters that characterize the system are , , , and , and based on the estimates for and we have
= = 1 − , and
(360)
Fitting data to a type curve is a much used method in well test analysis.
32
Gringarten, “Interpretation Gringarten, “Interpretation of Tests in Fissured and Multilayered Reservoirs With Double-Porosity Behavior: Theory and Practice”. Practice”.
naturally fractured reservoirs
101
where is based based on core core plug plug measur measureme ements nts.. It It should should howe howeve verr be noted noted that that the waythese param paramete eters rs are actua actually lly specifi specified ed in diffe differen rentt simulators will vary. vary. In the “industry standard” simulator simulator ECLIPSE the matrix–fracture coupling is for instance specified by giving the matrix permeability and the shape factor separately (Eq. 323 323). ). Physically the matrix–fracture flow depend on matrix block permeability, , matrix block geometry, and the characteristic matrix block size, . Given an estimate for and some prior knowledge of matrix block geometry, it should therefore be possible to say something about matrix block size. If all matrix blocks have the same size and shape, we could express Eq. (323 ( 323)) as
ℎ = ( − ) = ℎ′ (ℎ−) . ′ ℎ = ℎ ′ , ℎ = √ ′ . ′
(361)
The first term is proportional to matrix-block area per volume, and is the product of a pure geometric factor and , and the last term is the pressure gradient, which is the driving force for flow. flow. Eq. 361 Eq. 361 gives
(362)
′
Slabs or strata
so the characteristic matrix block size is
(363)
Exampl Examples es of the geomet geometric rical al factor factor for for variou variouss matri matrix x block block geomegeometries are shown in Fig. 97 Fig. 97.. Based on Eq. (363 (363)) it is in principle possible to estimate matrix block size from if we know something about block shape. Since actual tual block block sizes sizes vary vary throug throughou houtt the forma formatio tion, n, this this estima estimate te is in any case only an indication of a typical size, and due to diagenesis, fracture ture walls walls are additi additiona onally lly often often cove covered red with with calci calcite te cemen cement. t. In these these cases, the measured from well-testing is insufficient for estimating block size.
Skin in fractured reservoirs
′ Sticks
′ Cubes
Since it is assumed that a representative elementary volume contain many many fractu fractures res,, the dual dual porosi porosity ty model model is a macro macrosc scopi opicc model model which which is valid only on length scales much larger than the matrix block size. This This means means that that the model model is not not valid valid for near near wellb wellbore ore flow flow. This This fact fact contribute to the suppression of early time dual–porosity behavior in real well tests, and is also important for the interpretation of the physical significance of a measured skin in the well. If the well intersects the fracture system, pressure drops will be significantly less than predicted by the dual porosity model. The reason for this is that the flow follows the fractures so that the near well flow pattern will not have the logarithmic convergence close to the well that is responsible for much of the pressure drop in an equivalent single porosity model (see Fig. 98 Fig. 98). ). Based Based on flow in a network, network,
′
Figure 97: Geometrical factor for for various matrix block geometries
Figure 98: A well located in a fracture system.
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l e c t u r e no no t e s i n w e ll ll - t e s t i ng ng
wells with fracture system contact should have a negative skin. The negative skin is larger for large matrix block sizes, and for a square network, as the one shown in Fig. 98 Fig. 98,, an approximate formula is
≈ 2 − (ℎ ) . ln
ℎ > 5
(364)
This approximation is only valid for matrix block sizes . Wells that do not intersect fractures have larger pressure drops than predicted, and a corresponding large positive skin. The missing fractu fracture– re–we well ll conta contact ct introd introduc uce e an area area around around the well well with with reduc reduced ed permeability (matrix permeability ) compared to the effective permeability of the formation (bulk fracture permeability ), and if we use the formula for a circular damage area we get the following approximate expression for skin
≈ − 1 (2ℎ ) . ln
(365)
It should be noted that wells in fractured reservoirs are usually acidized or hydraulically fractured to obtain good communication with the fracture system. A successful well stimulation will result in a large negative skin.
Validity Validity of model The dual porosity model is a macroscopic theory based on the representativ resentative e elementary elementary volume volume (REV) concept. concept. The short and early middle time behavior of a well test in a fractured reservoir will however often correspond to length scales than are smaller than the REV. This This impl implie iess that that any any anal analys ysis is of thes these e time time peri period odss base based d on the the dual dual porosity model is approximate at best. The relevance of type curves used for well storage and skin correction can for instance be question tioned ed,, as it is not not expe expect cted ed that that the the dual dual poro porosi sity ty mode modell can can desc descri ribe be the corresponding time periods well. Fracture systems are often localized. Two examples are shown shown in Figure 99: Examples of localized fracture systems. Fractures near a fault zone, and located in a high stress zone at the top of an anticline.
Fig. 99 99;; fractures that are clustered in a fracture zone near a large fault, and fractures that are located in regions of high stress such as the top of an anticline. Such systems can not be analyzed in terms of a simple simple dual dual porosi porosity ty model model with with homoge homogeneo neous us proper propertie ties. s. A well well in a frac fractu ture red d zone zone on the the top top of an anti anticl clin ine e shou should ld for for inst instan ance ce be anaanalyzed lyzed in terms terms of a chana chanaliz lized ed syste system m (see (see page page 70 70), ), and and in this this cont contex extt the well test can be used to estimate the extent of the fractured zone.
naturally fractured reservoirs
Some fracture systems are highly irregular in the sense that fracture ture and matri matrix x block block sizes sizes have have a large large varia variatio tion. n. Typic Typicall ally y the varia varia-tion tion can can span span seve several ral orders orders of magnit magnitude ude and the system system approa approach ches es fractal behavior 33 . To capture the behavior behavior of such systems, systems, large fractu fractures res must must be expli explicit citly ly modell modelled, ed, while while the smalle smallerr fractu fractures res may may be included as dual porosity. The dual porosity model has a single variable set representing the inner state of the matrix blocks. This is a good approximation if the pressu pressure re profile profile in the matri matrix x blocks blocks have have reache reached d a semi semi steady steady state, state, and due to this the model is often called the pseudo-semi-steadystate (PSSS) inter-porosity flow model. In reality the real profile will have have a tran transi sien entt beha behavi vior or as show shown n in Fig. Fig. 100 100.. Multi Multi porosi porosity ty models models is an atte attemp mptt to incl includ ude e thes these e tran transi sien ents ts by desc descri ribi bing ng the the inne innerr stat state e of the matrix with more variables, each describing layers within matrix trix blocks blocks.. In partic particula ular, r, the model model often often known known as the “trans “transien ientt interpor terporosi osity ty model” model” is a conti continuo nuous us varia variant nt of a multi multi-po -poros rosity ity model. model. In real systems, matrix blocks have a varying shape and size, while multi multi porosity porosity models assume a given given shape shape and size for all blocks. blocks. Thus, Thus, the actua actuall transi transient ent behav behavior ior maynot be well well descri described bed by these these models. Also due to diagenesis, the fracture walls are often covered with calcite cement, and the matrix–fracture matrix–fracture pressure drop will be localize localized d over over these barriers. barriers. In these cases, a semi-stea semi-steady-s dy-state tate is developed early. The dual–porosity and dual–porosity–dual–permeability model may also be applicable in layered systems. systems. Layer communication communication is governed by diagenetic or depositional barriers of unknown extent and strength. Dual porosity layer communication communication ( ) can be obtained from well test data.
33
Fractals are systems with no characteristic length scale.
Multi porosity models.
Figure 100: Pressure transients in a matrix block.
Final comments Dual porosity effects are elusive and may not be readily accessible by well testing for all systems where they are important in production. A number of heterogeneous systems show well test responses that can be mis-interprete mis-interpreted d as dual porosity porosity effects. The occurrenc occurrence e of natural fractures must be established from the inspection of whole core or well logs before applying the dual porosity model for well test interpretation. Note, however, however, that high-contrast high-contrast heterogeneous or layered systems may also show dual-porosity dual-permeability behavior and the use of these models need not be limited to naturally fractured reservoirs. Fractured reservoirs with large large matrix–fracture matrix–fracture coupling are are effectively tively single single porosity porosity,, but the effectiv effective e permeabil permeability ity can only be measured by well testing.
103
Figure 101: Layered system
Gas reservoirs The objectives of well testing in gas wells are the same as in oil and water wells. The well test gives estimates for formation formation permeability, permeability, and probe reservoir structure such as the location of faults. The test can also contribute to reservoir and well monitoring by giving reser voir pressure and skin. Due to high flow rates, the skin in gas wells is however often rate dependent, and multi rate testing is needed for skin estimation. The analysis methods used in oil-well testing can however not be directly applied to the testing of gas wells. The reason for this is that the fluid properties have a stronger pressure dependency; compressibility and viscosity can not be treated as a constants, and constant surface surface volume volume rates correspond correspondss to variable down-hole down-hole rates. We will in this chapter show that some smart tricks, in particular the concept cept know known n a s pseu pseudo do-p -pre ress ssur ure, e, can can be empl employ oyed ed so that that the the fami famili liar ar analysis methods can, at least approximately, still be applied. Before Before any major approxima approximations tions have been made, made, the general general equation for the pressure is (see page 13 page 13))
∇ ⋅ (()) ∇∇ = ()() .
(366)
Since the parameters, , , and depend on pressure, Eq. (366 (366)) is non-linear (see page 39 39 for for the definition of a linear partial differential ential equation). equation). All of the methods methods developed developed over over the preceding preceding chapte chapters rs are deriv derived ed under under the condi conditio tion n that that the gove governi rning ng equati equation on (Eq. 13 (Eq. 13)) is linear. Thus, in order to take advantage of these methods, Eq. (366 (366)) must be approximately approximately linearized by a change of variables or . The new variable replacing pressure is called pseudo-pressure, and and the the use use of this this vari variab able le inst instea ead d of pres pressu sure re in the the anal analys ysis is is stan stan-dard procedure in gas well testing. The introduction of an additional variable change for time, that is pseudo-time, may reduce non linearities further in some cases, and can be introduced in cases when the total pressure change in the test is large.
Pseudo pressure In order to take advantage of the methods developed for the testing of oil wells in gas well testing, Eq. (366 ( 366)) must be linearized. We will show that the equation can be approximately approximately linearized by a variable change change as long as the total total pressure change change is not too large. Thus, Thus,
In gas reservoirs, the diffusivity equation is non-linear.
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if the total pressure change in the test period is small, well test in gas reservoirs can be analyzed using the same methods and equations tions as foroil reserv reservoir oirss provid provided ed that that pressu pressure re is repla replace ced d by pseudo pseudo-pressure in the analysis. Pseudo-pressure is defined as
′ ( () = (′)) ′ , . ∇ = ()() .
Pseudo-pressure
(367)
where is some reference pressure, and and are viscosity and density density at initial initial reservoir reservoir pressure. pressure. The differentia differentiall equation for pseudo pressure is
(368)
Note Note that that the choic choice e for for refere referenc nce e pressu pressure re is arbi arbitr trar ary y, as Eq. Eq. (367 367)) only contains derivatives of the pseudo-pressure. Applying the fundamental theorem of calculus, we obtain the following:
∇( ) = ∇ = ( (()))∇ () = = () .
(369)
Using the equality’s above, we can show that Eq. ( 368 368)) is equivalent with Eq. (366 (366). ). The formulation using pseudo pressure (Eq. 368 (Eq. 368)) does not introd introduc uce e any extra extra appro approxim ximati ations ons to the full full nonlin nonlinear ear formu formulalation tion (Eq. (Eq. 366 366), ), and and it is stil stilll nonl nonlin inea earr as the the righ rightt hand hand side side is pres pressu sure re dependent. Pseudo pressure, , has dimension pressure and is a monotonous function of pressure. Note also that is a pure fluid property, by that we mean that it is a characteristic of a given reservoir fluid. The same pseudo-pressure function, , applies to all tests in wells in a given reservoir. To obtain the pseudo pressure a good model for the density and viscosity as a function of pressure is needed. This model should be based based on on some some key key measur measureme ements nts perfor performed med on the reserv reservoir oir fluid, fluid, including compositional compositional analysis. These measurements can be used to create a tuned equation of state and viscosity correlation. In many many textbo textbooks oks the pseudo pseudo pressu pressure re is expre expresse ssed d using using the comcompressibility factor, factor, :
()
() ′ () = (′)(′) ′ . () ∝ //() ( ) ′ () = 22 (′)(′) ′ .
The pseudo-pressure function, a pure fluid property. property.
(370)
Since , this definition is identical to the definition in Eq. (367 (367). ). Note, however, that it is also common in many text to use the so called un-normalized pseudo pressure, , which is defined as
Figure 102: Pseudo pressure is a monotonous function of pressure
(371)
Un-normalized pseudo pressure
, is
gas reservoirs
()
107
does does not have the dimension of pressure, and numerical values are very different from the corresponding true pressures. In all other aspects, and and are are interchangeable.
() ()
Pseudo pressure in low pressure reservoirs (
analysis)
If thereservoir thereservoir pressu pressure re is low low it can can be modell modelled ed as an ideal ideal gas. gas. Thus, Thus, the density is determined by the ideal gas law,
(, )) = , () = 12 ( − ) . ∇ = () .
(372)
and the viscosity is independent of pressure. The pseudo pressure is then (373) Inserting Eq. (373 (373)) into Eq. (368 (368)) then yields
(374)
Well Well test analysis in the low pressure regime thus amounts to replacing pressure with the pressure squared, and is therefore called analysis. We can further simplify Eq. (374 ( 374)) by assuming that the total compressibility for a gas can be approximated by the liquid compressibility, , and that the liquid compressibility for an ideal gas can be approximated as :
≃ ≃ 1/ ∇ = 2 . ° <
(375)
At reservoir temperatures in the range 50–150 C, C, the approximations above are accurate for pressures 140 bar. bar.
Analyzing tests using pseudo pressure
If the total pressure change in the test period is small the product may approximately be treated as a constant during the test. The differential equation for pseudo pressure is then our well known linear diffusivity equation.
where the diffusivity, diffusivity,
∇ = ,
=
(376)
, is a const constant ant evalu evaluate ated d at the pressu pressure re
at the start of the test. The notion notion of a const constant ant rate rate bounda boundary ry condi conditio tion n is differ different ent when when pseudo pressure replace pressure in the analysis. analysis. We will will show that well tests should be analyzed in terms of (constant) mass rates when pseudo pressure is used, in contrast to reservoir volume rates when pressure is used. Note that a constant mass rate, , corresponds to a constant surface volume rate, , as , while a constant reservoir rate corresponds to a variable surface rate as .
∇
Diffusivity is a constant evaluated at conditions at the start of the test.
= =
The variable volume rate which should be used for determining the inner boundary condition for via Darcys law 34 is
34
The boundary conditions are discussed on page 18 page 18 and and 25 25
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l e c t u r e no no t e s i n w e ll ll - t e s t i ng ng
() = () = ( ) . ∇ = ∇ , = ∇ = 2ℎ , = () () = = = .
(377)
By rearranging the first equation in Eq. (369 ( 369)) we see that
(378)
and Eq. (377 (377)) and (378 (378)) may be combined to give the boundary condition at the sand face ( ). The result is
where
(379)
(380)
We see from Eq. (367 (367), ), (376 ( 376), ), and (379 (379)) that a well test in a gas reser voir can be analyzed using the same methods as for oil wells by replacing pressure with pseudo pressure, and using the values at initial reservoir pressure for viscosity and gas reservoir volume factor. The diffus diffusivi ivity ty is, on the other other hand, hand, evalu evaluate ated d at the pressu pressure re at the start start of the test, which may be different from the initial pressure.
Oil well test are analyzed using reser voir volume rates. Gas well test are analyzed using mass rates.
Two-rate drawdown–buildup sequences
′
Flow rates in gas wells are often so large that non-Darcy (turbulent) pressure drops must be accounted for. The measured skin, , is then
′ = + ,
(381)
where is the normal skin factor, while is a rate-dependent skin term term with with a coeffi coefficie cient nt for the rate rate . In orde orderr to dete determ rmin ine e the the two two skin components, a multi-rate test is needed. An example of a multirate test is two drawdown–buildup sequences with different rates as illustrated in Fig. 103 Fig. 103.. If the first buildup is long enough for the presFigure 103: Two rate test with independent drawdown–buildup sequences.
> ,
sure to recover to the initial value
, that is
(382)
we have two build-ups that can be analyzed analyzed independently using the Horner Horner plot plot (page (page 40 40). ). The Horner Horner analys analysis is give give two two indepe independe ndent nt perpermeability estimates and two skin estimates
′ = + ′ = + . and
(383)
gas reservoirs
109
Given these two skin estimates the two components of the skin are
′ ′ − = ′ − −′ = − .
(384)
Step rate test In gas wells the mobility ratio of gas and drilling-mud is highly unfa vorable. As a result, gas have a tendency to finger through the mud filtrat filtrate e (see (see Fig. Fig. 104 104), ), and and well well clea clean n up by prod produc ucti tion on may may take take a long long time. The resulting time dependent skin may invalidate well test results, in particular skin estimates. A multi-rate test with more than two rates provides data redundancy which can be used for detecting insufficient clean up and improve data quality. A step rate test is a test where the rate is increased in steps, as shown in Fig. 105 105,, and at least three steps are needed in order to get the necessary necessary redundancy redundancy in the skin estimates. estimates. Below Below we will derive a method for analyzing the four flow periods in a three-rate step rate test. The method involve involvess plots that are similar similar to Horner Horner plots plots (see (see page page 40 40), ), and the deriva derivatio tion n is based based on superp superposi ositio tion. n. The Horner plot is actually a special case of these variable rate plots, and the method can easily be generalized to any number of rate steps. e r u s s e r p l l e W
e t a R
()−() )−() = + + 2′ , = 4ℎ , = 4 − , ℎ= ()−() = − − + + 2′ . ln
(385)
where
(386)
(387)
and is an arbitrary time unit conventionally set to 1 3600 . Note that, as discussed on page 107 page 107,, all rates are surface rates, and the product in Eq. (387 (387)) is evaluated at the start of the test, while the product product inEq. inEq. (386 386)) is evalu evaluate ated d at initia initiall reserv reservoir oir pressu pressure. re. The contribution for the other pseudo-wells are ln
Variable Variable rate plots are similar to Horner plots.
Figure 105: Step rate test
The expected infinite acting pressure for each of the four test periods is found by superposition, that is the variable rates in the well is replaced replaced by superposi superposition tion of four four constant constant-rate -rate pseudo-we pseudo-wells lls starting starting at different times. The first pseudo-well has a contribution (Eq. 84 84): ):
ln
Figure 104: Fingering of gas during well clean up.
(388)
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l e c t u r e no no t e s i n w e ll ll - t e s t i ng ng
We will get the best permeability estimate from the final buildup period. By adding all the contributions to the well pressure (Eq. 385 (Eq. 385 and 388 and 388)) we get (389)
() − ()) = () , () = − + −− + −− . () =(∞ )4ℎ=0 1 =. 0 ∗
where
ln
ln
ln
(390)
WeseefromEq.(389 WeseefromEq.(389), ), that that if weplot weplot the the pseu pseudo do pres pressu sure re as a func functi tion on of the log-time like variable , we get a straight line with slope m, as shown in Fig. 106 Fig. 106.. Note the similarity with the Horner plot. The permeability is estimated from the slope (Eq. 386 (Eq. 386): ):
(391)
We see see from from Eq. Eq. (390 390)) that that , so correspon corresponds ds to infinite infinite time. Thus, Thus, just as for the Horner plot, plot, the average average reservoir reservoir pressure can be found based on the extrapolated pressure, , using the MBH-corr MBH-correcti ection on as explaine explained d on page58 page 58.. For a gas gas reserv reservoir oir,, reserv reservoir oir volume may also be estimated estimated based on the corresponding depletion, and known total production, using mass balance. By adding all the active contributions to the well pressure we get the following expression for the nth drawdown period
with
((()−() )−()) = (() + + 2) , () = − () = − + − − . () = − + − + ′ = +((()−() ) −()) () ′ = + ′ 4 = 2 − + .
Figure 106: Variable rate buildup plot is used for permeability estimate. The time axis is a transformed time given by Eq. (390 (390). ).
(392)
ln
ln ln
ln
ln
(393)
ln
Each of the three drawdown periods will give estimates for total effective skin . We see from from Eq. (392 (392)) that the scaled pseudo-pressures, , shou should ld be stra straig ight ht line liness with with the the same slope, slope, , when when plot plotte ted d as a func functi tion on of . The slop slope e is know known n from analyzing the buildup, thus each line should be fitted with one param paramete eterr (the (the interc intercept ept)) while while forci forcing ng the slope. slope. All period periodss are typtypically plotted on the same plot, as shown in Fig. 107 Fig. 107,, and if the actual slope on early periods is not it indicates insufficient clean-up, or other problems. The effective skins, , are estimated from the intercepts, (Eq. 392 (Eq. 392 and and 387 387): ): ln
Figure 107: Step rate semilog plot for estimating rate dependent skin. All drawdown periods are fitted to straight lines with a common slope.
(394)
The The effe effect ctiv ive e skin skin is a line linear ar func functi tion on of rate rate (Eq. (Eq. 381 381)), soif we plot lot the the effective skin as a function of rate as in Fig. 108 108,, and fit to a straight straight line, the intercept is and the slope is . Again, if the first period(s) are not on a line it is an indication of insufficient clean-up, or other problems.
Figure 108: Skin vs. rate plot for estimating the components of rate dependent skin
gas reservoirs
111
Pseudo time
In the analys analysis is with with pseudo pseudo pressu pressure re above above (see (see 107 107), ), we assum assumed ed that that the the pres pressu sure re chan change ge duri during ng the the test test peri period od was was so smal smalll that that could be treated as a constant. If the maximum pressure change in the test period is to large, then the product cannot be treated as a constant. stant. In partic particula ular, r, this this could could be a proble problem m for exten extended ded tests tests in tight tight reservoirs. In the case when cannot be treated as a constant, the differential equation for pseudo pressure can approximately be expressed as
∇ = , = (′)1(′) ′ .
when time is replaced by pseudo time,
(395)
, which is defined as
(396)
The linear equation for pseudo pressure using pseudo time (Eq. 395 (Eq. 395)) is simply derived by inserting Eq. (396 (396)) into Eq. (368 (368). ). The equation is at best only approximately correct since is a function of pressure history, and thus different at different points in space, while the spatial derivative on the left hand side in Eq. (395 ( 395)) is evaluated at constant time. Fortunately, numerical experiments show that the approximation may be accurate for well testing purposes 35 . The The pseu pseudo do time time is a mono monoto tono nous us func functi tion on of time time,, but but it is a func func-tion of pressure history, history, and thus not a pure fluid property property.. Unlike Unlike pseudo pressure, the pseudo time function is different for each well test, and since is space dependent, the question of which pressure history should be used arise. In addition, the formulation formulation does not honor material balance. balance. A full numerical simulation simulation should al ways be performed in order to quality check results obtained using the pseudo time formulation, and matching the measurements with simulation may be more appropriate in cases with significant pressure changes. Estimates based on pseudo time will in any case serve as good starting values for the final matching process.
()
The pseudo time function, function of pressure history.
35
, is a
It is claimed on page 862 in “ Well test design and analysis ” that the pseudo time transformation is exact. This is unfortunate, as it is not correct.
Multiphase flow In reservoirs with pressure support from an aquifer or gas cap, and when well testing is used for reservoir monitoring of water or gas flooded reservoirs, the understanding of multiphase flow effects will be important important for the interpretat interpretation ion of well tests. In this chapter chapter we will, however, however, not investigate these situations. Here we will discuss the testing of wells where a two-phase region develops in the near well region. This situation will usually occur occur in gas condensate reser voirs and in oil reservoirs at or near the bubble point. We will also not investigate well testing in situations were, during the test or as a result of prior production, the reservoir is depleted such that a twophase situation develops in the whole reservoir. The phase diagram of the reservoir fluid in a gas condensate reser voir is shown in Fig. 109 Fig. 109.. A two phase region will develop close to the Figure 109: Phase diagram of gascondensate reservoir fluid. At initial pressure the fluid is in a supercritical single phase. The two phase region is entered and liquid drops out when the pressure is reduced to the dew point pressure.
well when well pressure falls below the dew point pressure. Since flow depend depend on relati relative ve permea permeabil bility ity,, the two two phase phase region region has has reduc reduced ed total mobility, which introduce an extra contribution to skin. However, However, the two phase region is also highly dynamic; it grows in size, saturations tions are changi changing, ng, and and the comp composi ositio tion n of gas and liqu liquid id will will chang change e while producing. Relative Relative permeability in the two phase region will change due to the dynamic saturation, and in a condensate reservoir relative permeability may also depend on composition which is also changing. The pressure and saturation profile around a well in a gas condensate reservoir is illustrated in Fig. 110 Fig. 110.. The The phas phase e diag diagra ram m of the the rese reserv rvoi oirr fluid fluid in an oil oil rese reserv rvoi oirr is show shown n inFig. 111 111.. A two two phas phase e regi region on will will deve develo lop p clos close e to the the well well when when well well
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Figure 110: Pressure and saturation profile around a well below dew point pressure.
Figure 111: Phase diagram of oil reservoir fluid. At initial pressure the fluid is one phase liquid. The two phase region is entered and gas appears when the pressure it reduced to the bubble point pressure.
pressure falls below the bubble point pressure. The two phase region is also in this case dynamic, and additionally mobile gas outside the immediate near well well region may segregate. The composition of the the produced fluids (surface gas oil ratio, and of the reservoir liquid) will consequently not be constant during the test. Due to the complications mentioned above, results based on well test analysis should always be checked with a “full-physics” simulation tion (see (see page page 121 121). ). These These simula simulatio tions ns must must allow allow for for possib possible le gravit gravity y segregation for oil and compositional changes for condensate.
Radial composite Build-up tests and multi rate tests can often be analyzed in terms of the simple radial composite composite model. As shown in Fig. 112 Fig. 112,, the radial comp compos osit ite e mode modell has has an inne innerr regi region on near near the the well well with with alte altere red d prop prop-erties, representing the two-phase region, and an outer region with the original single-phase properties. The fundamental response of a radial composite is shown figure 113 113.. At early early times times the pressu pressure re front front will will trave travell throug through h the inner inner region, and on the pressure vs. ln plot we will see a straight line with a slope that reflects the inner region properties. At late times the front travels in the outer region, and we will see a straight line
()
multiphase flow
115
Figure 112: Radial composite model
Figure 113: The fundamental response of a radial composite model with mobility ratio .
>
with a reduced slope reflecting the properties of the outer region. The corre correspo spondi nding ng log-lo log-log g deriva derivativ tive e plot plot for for the fundam fundament ental al respon response se is shown in Fig. 114 Fig. 114.. 10
2
10
1
10
0
Figure 114: Log-log derivative plot of the fundamental response of a radial composite model.
-1
10 -2 10
10
-1
10
0
10
1
10
2
The The init initia iall draw drawdo down wn peri period od in any any draw drawdo down wn test test has has a high highly ly dydynamic namic inner zone and does not behave behave as a radial radial composite. composite. Due to the changing composition, it is also difficult to maintain a constant rate. rate. On the other hand, we will see below that that buildup and well designed step-rate tests may see radial composite at the pressure front. Other tests may also show behavior that are reminiscent of radial composite, but the model can not be used for quantitative analysis.
Drawdown does not behave as a radial composite.
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Pressure profiles during a build up test are illustrated in Fig. 115 Fig. 115.. Figure 115: Pressure profiles during buildup. Note that the pressure front travels through the two-phase region as long as well pressure is below bubble-point pressure, and through the single phase region at later times.
We will assume that the production period, , is of sufficient length so that a near steady state profile has developed in the near well region, and that the changes in the pressure profile during the test is dominated by the shut-in (that is by the negative rate well in the superposition picture). Since the pressure front travels travels through the inner two-phase region at early times, the early-time step response reflects the properties of the two-phase region. Similarly, Similarly, since the pressure front travels through the outer single-phase region at later times, the late-time response reflects the single phase properties. The fact that the two phase-region disappears when well pressure rise above bubble point pressure will represent a reduced skin at this point, but this is irrelevant since the rate is zero. Anal Analys ysis is of the the earl early y part part of the the buil buildd-up up test test will will giv give esti estima mate tess for for the effective mobility in the two phase region, the time of the transition tion can can be used used to esti estima mate te the the size size of the the twotwo-ph phas ase e regi region on,, and and the the late late part part yield yield estima estimates tes for for forma formatio tion n permea permeabi bilit lity y and effect effectiv ive e skin. skin. The reduced mobility in the near well two-phase region contribute to an increased skin, so the measured effective skin has three contributions: The altered permeability in the near wellbore region (normal skin), reduced mobility in the near well two-phase region, and, in gas conde condensa nsate te reserv reservoir oirs, s, an extra extra pressu pressure re drop drop due to high-r high-rate ate turbuturbulent flow close to the well. The last contribut contribution ion is rate dependent, and the second is dynamic, so any detailed analysis of skin require a multi rate test. An example of a step rate test is shown in Fig. 116 116,, and this test has been discussed for gas reservoirs on page 109 page 109.. We will analy analyze ze the response to a rate change, and again assume that the previous production period is of sufficient length so that a steady state pressure profile profile has been develope developed d in the near well region. The develdevelopment of the pressure profile is illustrated in Fig. 117 Fig. 117.. Initially the front travels in the two-phase region, and the response will reflect the properties of this region, and the late-time response reflects single phase with effective skin. Note that, since the rates in this case is
Effective skin include reduced mobility in the near well two-phase region.
e t a R
Figure 116: Example of a step rate schedule.
multiphase flow
117
Figure 117: Pressure profiles during step rate test. Initially the pressure front travels through the two-phase region. The front travel through the single phase region at later times. Relative change in two-phase zone is small during the test provided that the initial production time is sufficiently long.
not zero, the change in two-phase region properties and size due to the pressu pressure re change changess repres represent ent a non-c non-cons onstan tantt effect effectiv ive e skin skin that that will will influen influence ce the pressu pressure re respon response. se. The respon response se is thus thus not compl complete etely ly that that of a radia radiall compo composit site. e. Howe Howeve ver, r, the dynam dynamic ic skin skin effect effect will will be of the order
, and the rate schedule may be designed to minimize
its influence.
Well tests in water injectors Water ater inje inject ctor orss may may be test tested ed in orde orderr to inve invest stig igat ate e how how the the inje inject cted ed water displace oil. The composite radial model can in many cases be used in order to understand and analyze these tests, as can be seen from from the mobili mobility ty distri distribu butio tion n shown shown in Fig. Fig. 118 118.. Ideal Ideally ly the near near well well region region will will compr comprise ise three three region regionss with with diffe differen rentt total total mobili mobility: ty: In the outer unflooded region the mobility will be that of oil at initial water saturation, and since the flooding is of Buckley-Leverett type36 there will be a sharp front with a step change in water saturation. Behind this front we have a two-phase two-phase region with reduced mobility. mobility. Since typically the injected water is much colder than the reservoir the volume close to the well will be cooled, and as cold fluids have higher viscosity than hot fluids the total mobility is reduced in the near well region. Typically the viscosity is increased by a factor for both oil and water water at NorthNorth-sea sea condi conditio tions. ns. In ideal ideal one-di one-dimen mensio sional nal or radial radial displa displace cemen mentt the temper temperatu ature re distri distribu butio tion n will will be a sharp sharp front front which which typically travels with a speed of the saturation front. The temperature is constant (at injection temperature) behind the front. Due to heat flow to the rock above and below the reservoir zone the real temperature front will however typically be zone with gradual temperature change. We have seen that the region around a water injector can be described in terms of three zones with different mobility, and ideally the mobility in each zone is approximately constant, so that a well test may be analyzed in terms of a radial composite model. The sat-
≈ 1/3
≈4
Figure 118: Sketch of the ideal relative mobility in the region around a water injector.
36
For a discussion of Buckley-Leverett Buckley-Leverett displacement, see for instance L.P. Dake. Fundamentals of Reservoir Engineering . Developments in Petroleum Science 8. Elsevier, 1978, page 356– 362, where the theory is developed for 1-dimensional linear flow. The radial displacement near a water injector is described by the same equations where the distance is replaced by .
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uration behind the saturation front is however not exactly constant, even even for for ideal ideal Bucley Bucley-Le -Leve veret rettt displa displace cemen mentt (see (see Fig. Fig. 118 118), ), and and in real real cases there will be some dispersion in the front location due to reser voir heterogeneities. Since the temperature front is also typically a zone with gradual temperature change, the radial composite analysis should not be expected to give very accurate results, but it can serve as a good starting point for more detailed analysis.
Pseudo pressure and pseudo time Well tests in gas reservoirs can in many cases be analyzed using the methods developed for oil reservoirs provided the analysis is performed in terms of pseudo-pressure (see page 105 105). ). We will will now now show show that that the the conc concep eptt of pseu pseudo do-p -pre ress ssur ure, e, in the the form form of the the so call called ed steady-state pseudo pressure, is approximately valid for both single and two-phase flow. flow. Before Before any major approxima approximations tions have been made, made, the general general equation for the pressure in the single phase case is Eq. (366 ( 366), ), which, in the two-phase situation, may be generalized to
where
∇ = ()() , ∇ ⋅ + ∇ = 1 + 1 . = +
(397)
(398)
Eq. (397 (397)) is just an expression for the conservation of total mass. If we assum assume e that that the total composi compositio tion n is const constant ant,, we may may uniquely define the following pseudo pressure
′ , () = +
(399)
since density, saturation, and viscosity in that case is a function of pressure only. The relative permeabilities are also functions of pressure through the pressure dependent saturations. In practice we will use an equation of state to define the pressure dependency of saturations and densities, and the pressure dependency of viscosity is determined by a tuned viscosity correlation. The diffusivity equation for the pseudo pressure defined in Eq. 399 Eq. 399 is where
∇ = ()() , = +
(400)
(401)
is the effective viscosity. This can be verified by substituting the definition of the pseudo pressure (Eq. 399 (Eq. 399)) into Eq. (400 (400). ). The concept of pseudo pressure in a two-phase system assumes constant constant compositi composition on in both time and space. space. This implies implies a conconstan stantt surf surfac ace e GOR, GOR, and and a stea steady dy stat state e profi profile le wher where e the the satu satura rati tion on is a
Pseudo pressure is defined provided that total composition is constant.
multiphase flow
functi function on of distanc distance e from from the well well while while the total total comp composi ositio tion n is conconstant. Both of these conditions are at best only approximately valid. The right hand side of Eq. (400 (400)) is also, just as in the single-phase gas case, a function of pressure, and we need to assume that the product changes little during the test so that it may be treated as a changes constant. Pseudo Pseudo pressure, pressure, effectiv effective e viscosity viscosity,, and total total compress compressibil ibility ity should be evaluated using the composition of the production stream at the start of the analyzed period. As a consequence, in contrast to the single phase case, the steady state pseudo pressure is process dependent. Note also that depend on relative permeability, permeability, which, in particular for gas condensate, is difficult to measure with high accuracy. Relative permeability may also depend on rate, which translate into a variation with distance from the well. All these concerns calls calls for cauti caution on when when apply applying ing pseudo pseudo pressu pressure. re. Howe Howeve ver, r, it has has been been show shown n that that the the anal analys ysis is of twotwo-ph phas ase e buil buildu dup p test testss can can ofte often n be made made to suffic sufficien ientt accu accurac racy y using using pseudo pseudo-pr -press essure ure and replac replacing ing 37 with a constant. If the the maxi maximu mum m pres pressu sure re chan change ge in the the test test peri period od is not not smal smalll then then the product cannot be treated as a constant. In these cases we may also introduce pseudo time (see page 111 page 111). ). The two two phase phase pseudo-time is defined as
119
()()
()
()() = (′)1(′) ′ .
()()
(402)
Like in gas well testing, pseudo time is an uncontrolled approximation. For well tests producing below bubble- or dew-point pressure, results sults based based on pseudo pseudo pressu pressure re and pseudo pseudo time, time, or any other other approx approx-imate analytical theory, should always be quality checked using full physics numerical simulations.
Two phase pseudo pressure is process dependent.
37
J.R. Jones and R. Raghavan. “Interpretation of Flowing Well Response in Gas-Condensate Wells”. Wells”. In: SPE Formation Evaluation (Sept. 1988), pp. 578–594.
Numerical methods Analysis based on analytical solutions of simple models should al ways be the first pass in well test interpretation. However, However, analytical models have limited applicability in real world situations with complex well paths, non trivial geology, multiple interacting wells, and multi-phase flow. In this chapter we will discuss how a well test can be simula simulated ted,, and the way way such such numeri numerical cal simula simulatio tions ns can be utiliz utilized ed in the analysis in order to obtain estimates for reservoir parameters. In genera generall numeri numerica call simula simulatio tions ns are necess necessary ary in a number number of concontexts: • In the context of unknown unknown and complex geometry, geometry, direct simulasimulations is used to test “what if” scenarios. • Given Given the scenario, scenario, that is a geologica geologicall concept concept that can be described by a set of numerical numerical parameters, simulation simulation is combined with automatic or manual parameter estimation. • Simulatio Simulation n on models with increased increased complex complexity ity and additiona additionall physic physicss is used used to valida validate te result resultss from from param paramete eterr estima estimatio tion n based based on simplified analytic analytic or numerical numerical models, or the model paramparameters from these simple models are used as starting points for parameter estimation using the more complex simulation model. Model validation is particularly important for multiphase flow.
Scenario testing.
Parameter estimation.
Model validation.
• The analysis of well well tests on wells with complex complex paths.
Complex well paths.
• Situation Situationss involvin involving g several several wells, and in particula particularr interfere interference nce testing, can usually only be be adequately analyzed analyzed using numerical simulations.
Interference testing
• Fina Finall lly y, it shou should ld be ment mentio ione ned d that that simu simula lati tion onss are are used used to obta obtain in type curves for a complex geological geological concepts. concepts. Traditional typecurve curve match matching ing can then then be perfo performe rmed d using using these these type type curve curves, s, see page 100 page 100 for for an example.
Well tests simulators Commercial software packages for well test interpretation typically include a well test simulator and a module for building suitable simulation ulation grids. The complex complexity ity of the models that can be built, and the ease of use may vary, but the first choice will be to use the pro vided simulator as it is well integrated with the other visualization
Type curves.
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and interpretation tools. Conventional reservoir simulators can also be used for simulating well test. These simulators are the most complete with regards to reservoir flow-physics and phase behavior, but are not tuned to solving well-test problems, and the creation of suitable grids, and providing correct boundary conditions can be challenging. lenging. On the other other end of the scale scale we have have production production technology oriented “multi-physics” simulators, which can also be used to simulate simulate well tests. These simulators simulators are coupled coupled well–near well–near-wel -welllreservoir simulators that give an accurate description of flow in the completions and wellbore, including details in the inflow to the well, but they typically have limited capabilities regarding the representation of reservoir geology, and additionally they tend to be very computationally inefficient. Based on which differential equations that are solved, simulators may be divided into four classes: Simulators that solve the linear diffusivity fusivity equation, equation, thosethat those that solve solve the non-linear non-lineardiffu diffusivi sivity ty equation, equation, black-oil simulators, and compositional simulators. A given simulator program may include more than one of these in the same package. The simulators simulators may also be categorized categorized accordin according g to the way the equati equations ons are discre discretiz tized, ed, either either by finite finite elemen elements ts or by contr controlol volume finite difference methods. The linear diffusivity equation,
()
∇ ⋅ ( ⋅ ∇)∇) − = 0 ,
Linear diffusivity equation.
(403)
is the equation that all the analytical analysis methods are based on, and, with being the pseudo pressure, it is universally applicable being for single phase well tests as long as the total total pressure pressure change during the the test is not too too large. large. In eq. 403 403,, is the space dependent permeability tensor, and the purpose of the well test is, in a general sense, to characterize , i.e. the reservoir. reservoir. Numerica Numericall solution of eq. 403 eq. 403 is is needed whenever the reservoir has a complex shape or permeability distribution, or when the well path or completion pattern is non-trivial. Finite element discretisation is more suited than finite difference for complex geometries, as it is easier to create grids that follow general complex shapes, so simulators that solve the linear (and also non-linear) diffusivity equation are are often finite element based. A simple example of a finite element grid is shown in fig. 119 fig. 119.. In a finite element discretisation the solution is approximately expressed as a sum of basis functions, ,
Finite elements vs. finite difference.
(,) = ()() .
(404)
Each basis function is localized around a node in the grid as shown in fig. 120 fig. 120,, and different finite element methods differ in terms of the shape shape of the select selected ed basis basis functi functions ons.. If we insert insert eq. 404 into into eq. 403 403,, we get
− 1 ∇ ⋅ ( ⋅ ∇) = 0 . ∗
We now introduce a second set of basis functions,
(405)
. Note that the
In finite element methods, the solution is expressed in terms of local basis functions, often called trial-functions.
numerical methods
123
Figure 119: An example of a finiteelement grid.
Figure 120: Example of basis functions (elements) for finite element discretisation. Note that the simple pyramid shaped elements shown here may be to simple for accurate simulation, and higher order elements are used in actual simulators.
the two sets are usually identical, but they need not be and sometimes lower order functions are used for this second set than for the first set. If we multiply multiply eq. 405 eq. 405 with with each function in the second set and integr integrate ate over over all all space space,, we get the follow following ing system system of equati equations ons expressed on matrix form
where
and
⋅ ⋅ − 1 ∗ ⋅ = 0 , = ∗ , ∗ = ∗ ∇ ⋅ ( ⋅ ∇∗) .
(406)
(407)
(408)
The above expression for the matrix (eq. 408 (eq. 408), ), contains derivatives of the permeability, and second derivatives of the basis functions. tions. Howeve However, r, since the reservoir reservoir is modelled modelled in terms of regions with different properties, and since the derivative of the basis functions are usually not continuous (see for instance fig. 120 fig. 120), ), the integrand grand will will conta contain in -funct -function ion type type terms. terms. We We would would prefe preferr an expre expresssion that contain first derivatives only, and we will also see that as an additional bonus we will get an expression where flow boundary conditions can be implemented in a very natural manner.
Projections onto a second set of local basis functions, the test functions, transform the partial differential equation into a system of ordinary differential equations.
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We can apply the divergence theorem to transform a volume integral to a surface integral over the outer boundaries
∇ ⋅ (∗ ⋅ ⋅ ∇) = ∗ ⋅ ∇ ⋅ . ∇⋅(∗ ⋅ ⋅ ∇) = ∇∗ ⋅⋅∇ +∗ ∇⋅ ( ⋅ ∇) . ∗ ∇ ⋅( ⋅ ∇) = − ∇∗ ⋅⋅∇ + ⋅ ∗ ⋅∇∇⋅ , . ⋅ ⋅ + 1 ⋅ = , = 1 ∗ 1 ⋅ ∇ ⋅ = 1 ∗ ⋅ , = −∇ − ∇∗ ⋅ ⋅ ∇ , {()} {∗()} ()
(409)
We may also apply the chain rule for derivation:
(410)
Combining eqs. 409 eqs. 409 and and 410 410 gives gives
(411)
and substituting (411 (411)) for (408 (408)) in eq. 406 eq. 406 gives gives
(412)
where
(413)
and
(414)
We have seen that by introducing two sets of localized basis functions, and , the partial differential equation (eq. 403 (eq. 403)) is transformed into an system of ordinary differential equations for the coefficients (eq. 412 412). ). The The righ rightt hand hand side side of this this syst system em of equa equa-tions tions are determ determine ined d by the fluxacross fluxacross the bounda boundarie ries. s. Flux Flux bound boundary ary conditions, such as a given well rate, are thus straight forward to implement. Other boundary conditions, such as constant pressure, can of cour course se also also be impl implem emen ente ted, d, but but we will will not not go into into any any deta detail il here here.. The system of differential equations (eq. 412 412)) is solved by timestepping, that is the coefficient vector at time is is calculated lated by solvin solving g a systemof systemof linear linear equati equations ons where where thecoefficie thecoefficients nts are calcu calcula lated ted based based on thevector thevector at . An implic implicit it time time steppi stepping ng scheme scheme has to be applied, and we will use the Crank–Nicholson scheme as an example. In that case the time-discretized equation 412 equation 412 is is
= +
⋅ ⋅ ()−() + 1 21 ⋅ (()+() = () , + + 2 () = − − 2 () + . ∗
(415)
which correspond to the following set of linear equations
(416)
The matrix elements and are non-zero only if the functions and overlap, so since the basis functions are local, these matrices are sparse, and the number of non-zero elements is proportionalto tionalto the number number of grid grid nodes. nodes. The matri matrice cess are also also indepe independe ndent nt
The pressure change during a time step is calculated by solving a sparse linear system of equations.
numerical methods
of time time,, impl implyi ying ng that that they they only only need need to be eval evalua uate ted d onc once for for all all time time steps. The finite element method is extremely flexible in terms of grids, Note Note also also that that the reserv reservoir oir proper propertie tiess ( ) is not not expr expres esse sed d in term termss of the basis functions, so they can have an independent grid representation, adding an additional level of flexibility. If the maximum pressure change in the test period is not small then the product cannot cannot be treated as a constant. constant. This is especially acute for extended tests in gas reservoirs. Introducing pseudo time can provide some insight, but analysis that employ full numerical cal simula simulatio tion n should should be consi consider dered ed as manda mandatory(See tory(See page page 111 111). ). It It is clearly necessary to perform the simulations using pseudo pressure. The diffusivity equation,
125
Flexible gridding to complex geometry and well paths is the major strength of the finite element discretisation.
∇ ⋅ ( ⋅ ∇)∇) − ()() = 0 , ()()
Non-linear diffusivity equation
(417)
is now non-linear non-linear.. However However,, if if the -produ -product ct does does not chang change e much over short time periods, which seems to be a good approximation tion in many many cases cases since since the pseudo pseudo-pr -press essure ure analys analysis is works works for for short short tests, tests, it can can be treate treated d expli explicit citly ly in time time steppi stepping. ng. Note Note that that the prespressure should be solved using some level of implicit time stepping, this is uncoupled to the treatment of the product.38 If we in addition assume that the product varies little over each overlap area it can be treated as a constant in the integrations and eq. 416 416 becomes becomes
+ + 2 ′ () = − − 2 ′ ()+ )+ . 1 ) . ′ () = = (,)(, () ()
38
This is reminiscent of much used IMPES (Implicit pressure Explicit saturation) type formulations is reservoir simulation.
(418)
where
and
(419)
We see that a finite element simulation is just marginally more computationally demanding in the case of a pressure dependent product compared to the linear case, provided that pseudo-pressure is used and we have an efficient way of calculating and and . In situat situation ionss with with dynam dynamic ic two-p two-pha hase se flow flow, such as produc productio tion n below the bubble- or dew-point pressure in oil or condensate reser voirs, simulations based on the diffusivity equation are not sufficient. Formulations the take changes in saturation or composition into account, implemented in “full physics” physics” simulators, are needed. These simulators are either black-oil, which is sufficient for oil reservoirs, or compositional, which are usually needed in condensate reservoirs. Full physics simulators invariably employ finite difference control volume discretisation, which is the class of discretisation used by all major commercial commercial reservoir simulators. In this formulation space space is divided into grid blocks (control volumes), volumes), and a mass balance balance equation is written for each each of these. Each grid grid block is assumed assumed to be in thermodynamic equilibrium so that the properties are spatially constant, and the saturation and composition of each phase is a function of total composition and pressure through an equation of state.
Full physics simulators
Finite difference control volume discretisation
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Reservoir properties are also discretized using using the same grid, with a single value of permeability and porosity for each block. The flux of each each phas phase e betw betwee een n grid grid bloc blocks ks are are norm normal ally ly dete determ rmin ined ed by the the pres pres-sure sure differ differenc ence e betwe between en the blocks blocks and by the satura saturatio tion n (relat (relativ ive e perpermeabil meability ity)) in the upstre upstream am block. block. This This is called called the two-p two-poin ointt flux flux approximation with upstream weighting. weighting. Multi-point flux approximaapproximations, tions, where where the flux betwe between en two two grid grid blocks blocks depend dependss on the pressu pressure re in additional neighboring blocks, are sometimes also implemented. These multi-point flux approximations reduce the grid-orientation effect which can severely damage the accuracy of multi-phase simulations, but are nevertheless in little use and not all simulators implement them. The finite difference discretisation is much more restricted with respect respect to gridding gridding than the finite finite element element methods. In particular particular,, grid blocks need to be orthogonal, that is that grid block faces must be orthogonal to the line connecting grid centers. This is illustrated in fig. fig. 121 121.. Often Often simple simple shoe-b shoe-box ox shaped shaped grid-b grid-bloc locks ks are used, used, but in general general these orthogonal orthogonal grids are called called PEBI-grids PEBI-grids or Voronoi-gri oronoi-grids. ds. In spite spite of the limita limitatio tions ns on griddi gridding ng impose imposed d by the finite finite differ differenc ence e discretisation, tools delivered with specialist well-test simulators are in many cases able to create good orthogonal grids as illustrated in fig. 122 fig. 122.. Some vendors supply a unified simulation framework where
The grid consist of grid blocks, and all reservoir properties, both static and dynamic live on the same grid.
Figure 121: Example illustrating orthogonality of grid blocks.
Figure 122: Example of a finite difference PEBI-grid with refinements around wells.
finite difference discretisation is used throughout, even for problems where the linear diffusivity equation is applicable.
Laplace finite element, and the Stehfest algorithm As an alternative to time stepping, the linear diffusivity equation can be solved in Laplace space:
∇ ⋅ ⋅ ∇(˜(,) − (˜(,) = 0 . + 1 ⋅ () = () .
(420)
If we discre discretiz tize e byfinite elemen elements ts in space space,, we get the Laplac Laplace e space space analogue of eq. 412 eq. 412::
(421)
numerical methods
˜ ( ( , )
This This spar sparse se line linear ar syst system em must must be solv solved ed to get get the the solu soluti tion on for for a give given n . If we were to numeric numericall ally y invert invert by numerically evaluating an integral for each needed , then this method would have been forbiddingly numerically numerically demanding. Luckily Laplace Laplace transforms of monotonously decaying functions (especially functions with exponential nential decay) decay) can be accurat accurately ely inverted inverted for a given given based based on transtransformed function values at a small number of . by the Stehfest algorithm. rithm. In well well testing testing the fundam fundamental ental soluti solutions ons have have the the desired desired properties, and the Stehfest algorithm have extensively been applied to well test problems were the Laplace transform is known on closed form, but no explicit inverse transform has been found. The Laplace finite element in conjunction with the Stehfest algorithm can thus be an attractive option at least for obtaining fundamental solutions in complex geometries and geologies. More complex well test response can then be found by superposition of fundamental solutions. A number of algorithms for numerical inversion of the Laplace transform do exist, but no single algorithm is suitable for all problems. lems. In the Lapla Laplace ce finite finite elemen elementt method method,, the calcu calculat lation ion of require quire solvin solving g a large large system system of linear linear equati equations ons,, imply implying ing that that any in version method that require a large number of values values to be evaluated ated will will be highly highly ineffic inefficien ient. t. Values alues for for is also also only only avail availab able le for for real , Numeri Numerica call invers inversion ion of the Lapla Laplace ce transf transformis ormis a hard hard proble problem, m, and it is especially difficult if we are seeking algorithms where only a small number of values for real suffice suffice to calculate a reasonable in verse for a given . A family of algorithm algorithmss for this type of problems problems 39 are based on the so called Gaver functionals, and among these it is the Stehfest algorithm40 that is commonly used in the well-testing context context.. We will not even attempt attempt to derive the Stehfest Stehfest algorithm algorithm here here,, but but the the resu result lt is that that the the solu soluti tion on at a give given n is is foun found d as a line linear ar combination of the Laplace solution at a series of specific :
( ) ()
ln
The inverse Laplace transform is calculated using the Stehfest algorithm.
()
( 2 ) () ≈ ()⋅ ˜ (2) , 22 − . ⌊/⌋ ! () = (−1) ln
127
39
P.P. Valkó and J. Abate. “Comparison of Sequence Accelrators Accelrators for the Gaver Method of Numerical Laplace Transform Inversion”. Inversion”. In: Computers and Mathematics with Applications 48 (2004), pp. 629–636. 40 The names Gaver-Stehfest Gaver-Stehfest algorithm and Salzer summation is also used.
(422)
where41
41
min
(423)
The series in eq. 422 eq. 422 converges to the exact solution with an increasing creasing number number of terms. The convergen convergence ce can, however, however, be very slow, slow, and additional additionally ly the coefficie coefficients nts have have altern alternati ating ng sign sign so that that extended precision arithmetic is necessary in the calculation if more than a few terms are needed. Using standard double precision arithmetic the maximum number of terms is 7. Extended precision arithmetic, which is implemented in software, is computationally much more demanding than double precision, which is implemented in the CPU hardware. However, However, it is easy to implement in the Stehfest algorithm itself, and for situations where the Laplace solution is explicitly known the inversion should always be performed using extended
⌊⌋
is the largest integer less than or equal to , and is the binomial coefficient.
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precis precision ion.. In Laplac Laplace e finite finite elemen elementt the Laplac Laplace e soluti solution on is only only avail avail-able at a precision given by the finite element code. This is typically at double double precis precision ion,, and implem implement enting ing the finite finite elemen elementt method method using extended precision is not feasible. Fortunately, experience show that the maximum of 7 terms available for double precision is usually sufficient at least for the monotonous fundamental solutions found in the well testing context. The main advantage of the Laplace finite element method is that the calculation cost is proportional to the number of time values we are actually actually interested interested in. When the diffusivity diffusivity equation equation is solved by traditional time stepping the calculation cost is proportional to the number of time steps. Since accuracy is determined by time step length, , efficient automatic time step control is an issue, but the numb number er of time time step stepss take taken n is typi typica call lly y much much larg larger er than than the the ones ones that that we actually need in order to perform the well test analysis. Also, in the initia initiall transi transient ent period period must must be very very smal small, l, and and if we are are main mainly ly intere intereste sted d in late late time time behav behavior ior we still still have have to time-s time-step tep the soluti solution on all the way from .
=0
Parameter estimation The The purp purpos ose e of the the well well test test,, and and the the well well test test anal analys ysis is is to char charac acte terrize the reserv reservoir oir.. In practi practice ce this this means means to find numeri numerica call param paramete eters, rs, such as permeability, permeability, distance to faults, and so on, related to conceptual models of the reservoir. Analysis based on analytical analytical solutions of simple models should always be the first pass, but more often than not, the reservoir itself and our conceptual models are too complex to be accurately represented by the models where we have analytical solutions. In these cases, simulations using a well test simulator will be an integral part of the well test analysis. The simulator can be utilized in two different approaches. The first approach is to run a number number of simula simulatio tions ns to genera generate te type type curve curvess for for a specifi specificc scena scenario rio.. The The seco second nd appr approa oach ch is to incl includ ude e simu simula lati tion on runs runs in the the inne innerr loop loop of an automatic parameter matching, matching, or optimization, procedure. The nume numeri ricc mode modell used used in the the well well test test anal analys ysis is must must corr corres espo pond nd to a gegeological concept, and this concept should have as few parameters as possible. The parameters should be transferable to parameters used, either directly or for conditioning, the full field geological model, but it is normally not recommended to use well test data directly in conditioning these models. Type curve matching is a proven method that historically has had a wide use in well test analysis, and the method is well supported in well test analysis software. Type curve matching is based on a model with a small number of parameters, and a set of curves, typically for pressure and pressure derivative, have been created for selected parameter rameter settings. settings. The curves curves are typically typically generated generated in terms of dimensionless time and pressure, and sometimes also in terms of other dimensionless groups, so that the actual measured time and pressure and must be scaled. scaled. This scaling scaling may include include some of the un-
Type curve matching is attractive for models with few parameters.
numerical methods
known known parameter parameters. s. The matching matching process process simply simply involves involves comparin comparing g the scaled measured data with the type curves, adjusting the scaling, and finding the curve curve that is closest closest to the measured. measured. An example example of type curve matching was briefly discussed on page 100 100,, and fig. 96 fig. 96.. Note Note that that,, just just as for for the the anal analyt ytic ic meth method ods, s, the the actu actual al resp respon onse se can can be divided into time periods and different type curves used for the anal ysis of each. The matching process can be manual, and purely visual, or a numerical matching measure can be defined and the matching process can be assisted by an optimization algorithm. Well testing software typically come with a large set of type curves preloaded, and matching to these are well integrated in the software. In addition, type curves can also be found in the literature, but in many cases no curves exist that correspond to the geological concept concept at hand. In these cases cases it can be an option to use a well test simulasimulator to generate generate type curve sets. This is particularl particularly y attractive attractive if the concept is expected to be valid for a number of well tests, so that the curves can be reused. If the situation is unique, then direct parameter estimation with the simulator simulator in the loop, as described described below, below, will probably be more effective. The principle behind automatic, or computer assisted, parameter matching matching with a well well test simulator simulator is simple: simple: We have have geologica geologicall model with a set of parameters, , and this model is implement implemented ed in a numerica numericall well test simulator simulator.. Run the simulator simulator with differdifferent parameter values until the simulated response matches the measured. While manual matching matching may be based on pure visual inspection, in order to involve any algorithms in the matching process the mismatch, that is the difference between the simulated and the measured, sured, must must be quanti quantified fied.. The measur measure e is calle called d a mismat mismatch ch functi function on or an objective function, and a typical form that is used is
129
{}
({}) = ( − ({}) ,
(424)
where is a measured quantity and is the corresponding simulated value. is the weight that is put on the -th measurement. When When the object objectiv ive e functi function on is defined defined,, the param paramete eterr estima estimatio tion n is a minimization process, that is a question of finding the parameter set that minimize the objective. Creating a good objective function, that is selecting which measured values to include and what weights to put on them, is highly non-trivial. It would seem that the definition is totally arbitrary and subjec subjectiv tive, e, but but if the object objectiv ive e functi function on is defined defined in a Baye Bayesia sian n frameframe work we have a language in which the question of what is a good objective function can be discussed. Bayes theorem 42 states
( { } |{ } ) ∝ ( { } ) ( { } |{ } ) , ({}) ∝ (−(−({}) , ({},{}) = ({})+({},{}) . posterior
prior
(425)
and if we define an objective function such that exp
(426)
Bayes theorem can written as posterior
prior
(427)
The mismatch function is a measure for the difference between the simulated and the measured response.
42
See any good book on statistics. Bayes theorem: The updated probability of a parameter set after we have made measurements is proportional to the product of the probability assigned to the set before the measurement and the probability that a measurement will give the measured values given the parameter set.
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We see that if we define the objective function by eq. 426 426,, minimizing the objective function corresponds the finding the most probable parameter set. Comparin Comparing g eqs. 424 and 427 427 we we can identify the mismatch function
({}, {}) = ( − ({}) , ({},{}) = ({})+ ( − ({}) .
(428)
which gives
posterior
prior
(429)
If we identify the best parameter set with the most probable posterior parameter set, we see from eq. 429 eq. 429 that that we need to minimize an objective function that contain two terms: a prior term and a mismatch match functi function. on. The The prior prior term term repres represent entss prior prior knowl knowledg edge e of possipossible parameter values, and is often set to zero reflecting that we make no prior assumptions. Note that if the number of parameters is large, the minimization problem without a prior term is ill posed, and the prior prior term term serves serves as a natura naturall regula regulariz rizati ation. on. The The misma mismatch tch term term reprepresents what in Bayesian statistics is called the measurement likelihood, that is the probability that a measurement will give the measured values given the parameter set. If we use a Gaussian error model for uncorrelated measurements, the likelihood is
− ( { } ) ( ( ({}|{}) ∝ − + , = 1+ . exp
model model
The objective function is the sum of a prior term and a mismatch function.
(430)
meas meas
where meas measureme ement nt error error and model the mode modell erro errorr. meas is the measur model is the From eq. 430 eq. 430,, we see that the weights in eq. 424 424 are are
model model
We seek the most probable posterior parameter set.
(431)
meas meas
By introducing the Bayesian framework, we have identified the need need for for a prio priorr term term in addi additi tion on to the the mism mismat atchterm chterm in the the obje object ctiv ive e function, and we have concluded that the weights in the mismatch function are given by the model and measurement errors. Determining which which measur measureme ements nts that that should should be includ included ed in the misma mismatch tch remain undetermined. One might think that using all actual actual pressure measurements would be the correct thing to do. However, However, the errors, in particular the model errors, are highly correlated in time, and we have assumed uncorrelated uncorrelated measurements. Introducing the error co variance matrix, , in the Gaussian error model:
({}, {}) ∝ − ( − ({}) () ( − ({}) , exp
(432) is sometimes proposed as a solution to this, but it is actually not a good idea. idea. There are two main main reasons reasons for this: first, first, it is very difdifficult to obtain good estimates for the correlations, and second, the inverse of the covariance matrix is highly sensitive to small errors in
The weights in the mismatch function are given by the model and measurement errors.
numerical methods
the covar covarian iance ce estima estimates tes.. The soluti solution on to the proble problem m of corre correlat lation ionss is to use as few measurements as possible. To obtain this, calculated quantities that are characteristic of the well test response, such as deriva derivativ tives, es, should should be used used in the misma mismatch tch functi function on in stead stead of many many direct pressure measurements.
131
The mismatch function should be defined using quantities that are characteristic of the well test response.
Interference testing An interf interfere erenc nce e test test involv involves es at least least two two wells wells.. The pressu pressure re respon response se from a rate change in one well is recorded in one or more non-producing wells. The main purpose of interference testing is to determine and quantify pressure communication. In some simple cases analytical models can be used for the analysis of these tests, and we have seen such an example in the section on non sealing faults (page 72 72), ), but numerical simulation is in general a necessary tool in the planning and analysis of interference tests. We will in this section give a brief presentation of two examples in order to illustrate this. Fig. 123 123 show showss two wells in a a fluvial fluvial reservoir reservoir.. Optimal Optimal reserreserFigure 123: Interference testing wells for estimating sand body communication in a fluvial reservoir
voir steering, and the need for and placement of future in-fill wells, depend on the permeability distribution which is determined by the densit density y of sand sand bodies bodies and the sand sand body body geomet geometrie ries. s. An An interf interfere erenc nce e test, using the two wells, would give valuable information relating to these parameters. Fig. 124 Fig. 124 shows shows two wells in different reservoir compartments. Due to the limited resolution of seismic, the location and extension of faults faults is uncertain. uncertain. When a fault with a small throw throw links up with a large fault, such as fault B with the thrust fault in fig. 124 124,, it is in particular often difficult to determine whether it extends all the way to the large faults. It is also unknown to what extent the faults are sealing. Well tests in each of the two wells might give information relating to some some of these these param paramete eters, rs, while while an interf interfere erenc nce e test test would would give give additional data. We will in the following discuss some simulated data that illustrates the possible use of an interference test in this situation. tion. Relat Relating ing to fault fault B, we are intere intereste sted d in discri discrimi minat nating ing three three difdifferent cases; A completely sealing fault that links up with the thrust fault, a sealing fault that does not link up, and a incompletely sealing fault. fault. In the first case, the observation observation well in isolated isolated from the producer, so this situation will be easy to detect by a single pressure
Figure 124: Interference testing wells for estimating fault patterns and fault communication
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Figure 125: Pressure distribution in two alternative reservoirs; a reservoir with a fault that does not extend to the bounding fault (left), and a reservoir where the same fault is not completely sealing(right).
measurement, while we in the two other cases will get a time dependent response response in both producer producer and observation observation well. well. The pressure pressure distribution after prolonged production is shown in fig. 125 fig. 125.. Figure 126: Log-derivative diagnostic plot for the drawdown test and different fault configurations.
As shown in fig. 126 fig. 126,, a well test might be able to differentiate between a completely sealing fault, and the two leaky-fault situations. A long long test test,, abou aboutt a mont month h in the the simu simula late ted d 300 300 mD rese reserv rvoi oirr, is howhowever needed, and the two other cases have an almost identical response. sponse. A log–log plot for the observation observation well response response is shown shown Figure 127: Log-log plot for the obser vation well response and different fault configurations.
in fig. 127 fig. 127.. We see that the late time response for the non-sealin non-sealingg-
numerical methods
fault and unlinked-fault case is almost identical. The pressure signal does on the other hand arrive earlier in the non-sealing case (a factor of two in arrival time), so that the observation well response can be used to discriminate the two scenarios. In the simulated case, a test running over just a few days is sufficient for this. The strength of the signal is proportional to the rate, and the simulation results can also be used as a guide to what is a sufficient test rate in order to receive a detect detectab able le early early respon response. se. The observ observati ation on well well respon response se is also also more more sensit sensitiv ive e to the fault fault seal seal param paramete eters rs such such as fault fault-z -zone one permea permeabil bility ity,, and will provide more accurate estimates.
133
The role of well testing in reservoir characterization The title of the course for which these lecture notes have been written contains the phrase: “Reservoir Property Determination by Well Testing”. Testing”. We have have seen that well testing is used in many additional context contexts, s, such as reservoir reservoir monitorin monitoring g and well productiv productivity ity monitor monitor-ing, but these additional contexts have been touched only briefly. So, what is the role of well testing in property determination, and which properties are measured for what purpose? In this chapter we will try to place well testing in the broader context of reservoir characterization and reservoir modelling. The simplistic perspective is that well testing gives us reservoir permeability, distance to faults, fault communication, and reservoir volume. A broader perspective is that well testing provides data for reservoir characterization. Reservoir characterization is a multidisciplina disciplinary ry task coordina coordinated ted by “geo-eng “geo-engineer ineers” s”,, that is people people with a broad knowledge of reservoir geology and reservoir engineering. The purpose of reservoir characterization is twofold: to provide qualitative data to improve the understanding of the reservoir, and quantitative tative data in order to model the reservoir reservoir.. Reservoi Reservoirr models play a major role in current development planning and reservoir management practices, Reservoir characterization has two levels; The conceptual model, which may include a number of alternative scenarios, and parameterparameterized models, or property models. Reservoir parameters parameters (properties) have no meaning outside an associated conceptual model. A conceptual model is a clearly defined concept of the subsurface in the sense that a geo-scientist could represent it as simple sketches. Scenarios are a set of reasonably plausible conceptual models.
Scenarios Scenarios are not incrementally different models based on changes in input data or parameter values, scenarios are a set of reasonably plausible conceptual conceptual models. Scenario uncertainty uncertainty is the main contributor to sub-surface uncertainty in early phase field development, and often remain highly important even after many years of production. In order to take into account the uncertainty in a reservoir response, such as the production in a given year, here represented by , we represent the response as a probability distribution . The
()
Scenarios are a set of reasonably plausible conceptual models
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probability can be expressed as
() = ()() ( )
()
(433)
where is the probability of a scenario member , and is the probability of a a certain response ( ) in scenario . Well testing can supply data that contribute to falsify a conceptual model, or change the probability associated with (plausibility of) each scenario member. In exploration, field development planning, and early phase production, when scenario uncertainty is the dominant uncertainty, uncertainty, it is important to obtain overview and understand the full set of possible geological concepts; concepts; the scenarios. Plan for well tests and well placements that can contribute to the reduction of scenario uncertainty. Below we will present two examples of scenario uncertainty. uncertainty. These examples, which are based on real field cases, are not worked out in much detail, and it is up to the reader to apply the knowledge of well testing obtained in previous chapters in order to evaluate how well a well test might contribute to reducing the uncertainty. The first example is a reservoir delineated by two large faults, as shown in figure 128 figure 128.. The structural interpretation is uncertain, and a
Well testing can reduce scenario uncertainty.
Figure 128: Alternative structural conceptual conceptual models based on poor seismic.
noisy noisy seismi seismicc may may be interp interpret reted ed based based on two two compe competin ting g conc concep epts: ts: A few intern internal al faults faults in the east– east–we west st direct direction ion,, or heavy heavy intern internal al fault fault-ing parallel to the eastern main faults. Optimal well placement in a water injection scheme will be very different for the two alternatives. The second example relates to two different concepts on how the reservoir reservoir material material has been deposited. deposited. The two concepts concepts are illusillustrated in fig. 129 fig. 129.. The first concept concept implies implies that the reservoir reservoir is deposited in two isolated zones, and that the upper zone pinches out to the west. west. In In the second second conc concept eptua uall model model,, the bounda boundary ry layer layer is heavheavily eroded so that the two zones are communicating, and the western part of the upper zone is deposited as a thin layer reaching west wards. Note that, as far as modelling for well test interpretation or reservoir simulation is concerned, the two concepts can be spanned out in a single model with continuous variables (extent of up-flank volume and sealing capacity of inter-zonal layer). They do, however, however,
the role of well testing in reservoir characterization
137
Figure 129: Alternative concepts concepts of reservoir pinch-out and vertical communication.
represent two different geological concepts, and the concept selection will also possibly influence other aspects of the modelling. This illustrates that the distinction between scenario uncertainty, uncertainty, and uncertainty described by parameter variation is sometimes not clear cut in practice.
Model elements A given conceptual model contains model elements. Model elements are rock bodies which are petrophysically and/or geometrically distinct in the context of reservoir fluid flow, flow, and are the building blocks of reservoir model, both conceptually and as realized in a computer. Model elements have properties, i.e. they can be described with a set of parameters. parameters. These parameters parameters are geometric geometric properties, properties, such as length, height, direction, and thickness, and volumetric properties (property fields), such as porosity, permeability, permeability, clay content, and facies fractions, The property fields of different model elements are associated with a homogeneity scale or representative elementary volume (REV). In general the model elements form a hierarchy, where large scale elements consist of elements at smaller scales (see fig. 130 130). ). The The property fields of large scale elements are effective properties which represent some form of averaging of the property fields of the constitutive elements. For permeability, permeability, which is a tensor field, this averaging is non-trivial. Well testing can provide data related to geometric properties of structural elements, but in terms of property fields a well test can only see permeability and porosity. porosity. Since, as described above, above, permeability is a scale dependent property associated with representative elementary volumes at certain homogeneity scales, the question of which which permeability the the well test is probing has to be answered. answered. In genera general, l, the early early time time data data see smallsmall-sca scale le proper propertie tiess (and (and short short time time scales scales), ), while while the late late time time data data see large large scale scale proper propertie tiess (and (and longer longer time scales). Additionally, in particular important for the short time data, there is always some along-wellbore along-wellbore “averaging” “averaging” involved. In any case, well testing is the only measurements that measure permeability at real reservoir conditions at scales directly relevant for reservoir simulation.
Figure 130: Example of geological features represented by model elements at different scales Large scale property fields, such as permeability, permeability, are effective properties
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Measurements The main main data data sourc sources es for for reserv reservoir oir chara characte cteriz rizati ation on are seismi seismic, c, outoutcrops and analogues, reservoir cores, well logs, well tests, and production data. All measurements, including well testing results, must be inte interp rpre rete ted d in the the cont contex extt of the the mode modell elem elemen ents ts of a rele releva vant nt cononceptual model, and a well test interpretation must be consistent with other other relevant data. data. Each measureme measurement nt is associat associated ed with a measurement scale, or probed volume, and thus measure properties of model elements on different levels of the model-element hierarchy. Note Note that, that, since since,, additi additiona onally lly,, the measur measureme ement nt scale scale assoc associat iated ed with with a given measurement often do not correspond to any of the homogeneity scales of the model elements we need to characterize, data consistency and consistent data integration is in general non trivial and involve both up- and down-scaling.
Final words Well ell test testin ing g does does not not live live in isol isolat atio ion, n, and and a well well test test must must be plan planne ned d and interpreted in the context of conceptual geological models and their model elements. A key to successful reservoir characterization is that everyone work with shared scenarios in a collective effort, and a planned well test should be performed in order to answer specific pre defined questions. questions. Remembe Rememberr also that scenario scenario uncertainty uncertainty is often the dominant uncertainty. We should always ask ourselves and our colleagues: Have we included all plausible conceptual models?
Data integration and data consistency involve up- and down-scaling.
Mathematical notes The notation that are used in these lecture notes may be unfamiliar to some. The coordinate free notation that is used is more common in physics texts than in engineering books. However, I am convinced that once the basics of the notation is mastered, equations are much easier to read and understand. Darcys law is for instance written as
= − 1 ⋅ ∇ 1 = −
in coordinate free notation, instead of
in the normal engineering type notation.
Scalars, vectors, and tensors The fundamental objects in any continium theory, including fluid flow in porous porous media, media, are scalar scalar-field -fields, s, vector vector-field -fields, s, and tensortensorfields. fields. This This text text folll folllow owss the comm common on lazy lazy tradit tradition ion of using using the terms terms scalar, vector, and tensor for these fields. All All read reader erss shou should ld be fami famili liar ar with with the the conc concep epts ts of scal scalar arss and and vec vectors, while tensors may be a less familiar object. In the context of the present text, only tensors of order 2 is encountered43 , and the most prominent is the permeability, . Permeabi Permeability lity is a linear operator operator that operates on a gradient (derivative of pressure) to produce a vector (volumetric flux). Such operators are called (2,0) tensors, or more simply tensors of order 2. The tensor concept can can be viewed as a generalization of a vector, and taking the so called tensor product of two vectors create a tensor of order 2 (see table 9 table 9). ). No special notation is used in order to distiguish between scalars, vecrtors, and tensors. However, However, lower case letters are typically used for scalars and vectors, and upper case letters for tensors. Scalars, vectors and tensors can be multipled, either using the dot product or the tensor product as shown in table 9 table 9..
Spatial derivatives and the gradient operator
∇
∇
The operator for spatial derivatives is the nabla, or gradient, operator, . In terms of notatio notation, n, behaves like a vector, but it must be
43
The term tensor is usually not used for scalars and vectors, but scalars can be viewed as tensors of order 0 and vectors as tensors of order 1.
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Coordina Coordinate te free notation notation
== ⋅ == ⋅⋅ == ⋅
∇
ᵆ
Table 9: Multiplication of tensors. is a scalar, and are vectors and , , and are tensors.
Index Index notation notation
==∑ == ∑∑ == ∑
remembered that operates on (takes the derivative of) the expression to the right. Some examples of expressions involving the nabla operator can be found in table 10 table 10.. The shorthand notation is used for the second derivative operator (the Laplace operator). The derivation derivationtakes takespresed presedence ence over over multiplic multiplicatio ation. n. Paranthesis Paranthesis are used to group.
∇⋅∇
Coordina Coordinate te free notation notation
∇∇ = ∇⋅∇ ∇⋅ ∇ ∇∇ = ∇ ==∇∇∇⋅ == ∇∇⋅ ⋅∇⋅∇ = ∇ ⋅ ( ⋅ ∇)∇) (, , )
∇
ᵆ∇
Table 10: Examples of expressions involving the operator. , , and are a scalars, and are vectors and is a tensor.
Index Index notation notation
== ∑ == = ∑ = ∑
= ∇⋅∇⋅ ∇= =+, +,
We see in particular that the divergence of a vector field is a scalar given by the dot product
∇
that that the gradi gradient ent of a scala scalarr field is a vect vector or field field and that the Laplace operator
,
,
sends a scalar to a scalar.
The Gauss theorem, and the continuity equation The Gauss divergence theorem:
∇⋅ ∇ ⋅ = ⋅ ⋅ .
Gauss divergence theorem.
(434)
is a vect vector or,, the the left left inte integr gral al is over over a volu volume me,, and and the the righ rightt inte integr gral al is over over the enclos enclosing ing surfac surface. e. Note Note that that is is an outwa outward rd pointi pointing ng vector normal to the surface element. The continuity equation for some entity involves three quantities: •
: Density, that is the amount per volume.
m a t h e m at at i c a l n o t e s
• •
141
: Flux, that is the amount that flows per area and time (a vector). : The amount that is created per volume and time.
= − ⋅
For any given volume we then have
Continuity equation on integral form.
(435)
This is the continuity equation on integral form. Numerical simulation often employ control volume discretisation, and eq. 435 eq. 435 is is then applied to the each control volume or grid-block. If we apply the Gauss theorem to the surface integral in eq. 435 eq. 435 we we get
= − ∇⋅ ∇⋅ , + ∇ ⋅ = ,
(436)
and if we let the volume be an infinitely small differential element, eq. 436 eq. 436 gives gives
(437)
which is the continuity equation on differential form.
Laplace transform
ℒ ∶ → ̃ (̃ () == (∞ ) . ℒ (ℒ +()) == ℒℒ +=ℒ̃ =. ̃+̃ ∞ 1 ∞ 1 ℒ(1) == ∞ = − ∞= ∞ ℒ() = = − − −1 = 1 ℒ(1) = 1 , ∞ ℒ = ∞ = ̃ ℒ = = (̃ (,,)−) − (,,0) , ∫ (, ) = ∫ (,)
The Laplace transform is an integral transform, and it is usually applied as a transform in time:
(438)
The Laplace transform is a linear operator:
(439)
Calculating the integral, we observe that the Laplace transform of unity and of are are
(440)
where we are applying integration by parts when solving the transform form in time time . We observ observe e that that time time and the variab variable le has has an invers inverse e correspondence, correspondence, hence late times corresponds corresponds to small . The transformation of the partial derivatives are
(441)
where we use the basic rule for for the first first equality and integration by parts for the second equality.
Continuity equation on differential form.
Relevant literature The follo followi wing ng books books are highly highly relev relevant ant for for the cours course e “Rese “Reservo rvoir ir Proper Property ty Determ Determina inatio tion n by Core Core Analys Analysis is and Well Testing”. Note that some of the web links are only accessible from within the NTNU network. • George George Stewa Stewart. rt. Well test design and analysis . PennWell Books, 2011.
url
: http://site.ebrary.com/
lib/ntnu/reader.action?docID=10607597
• T.A. Jelmert. Jelmert. Introductory Introductory well testing testing .
book bookbo boon on,, 2013. 2013.
url
http p : / / bookbo bookboon on . com / en / : htt
introductory-well-testing-ebook
• R.W. R.W. Zimmerman. Fluid Flow in Porous Media . The Imperial College lectures in petroleum engineering. World Scientific Publishing Company Pte Limited, 2018. i s b n : 9781786344991 9781786344991 • J.W J.W. Lee. Lee. Well Testing . Society of Petroleum Petroleum Engineers, 1981. 1981.
url
: http://site.ebrary.com/lib/
ntnu/reader.action?docID=10619584
Other material The following posters from Fekete Associates can serve as “cheat sheets”. • Fekete poster: poster: Well testing fundamentals fundamentals http://www.fekete.ca/SiteColle ete.ca/SiteCollectionDocuments/P ctionDocuments/Posters/Fekete_We osters/Fekete_WellTest_Fundament llTest_Fundamentals_ als_ (http://www.fek 5731_0614AA_LOW.png) • Fekete poster: poster: Well testing applications applications http://www.fekete.ca/SiteColle ete.ca/SiteCollectionDocuments/P ctionDocuments/Posters/Fekete_We osters/Fekete_WellTestApplicatio llTestApplications_5731_ ns_5731_ (http://www.fek 0614AA_LOW.png)
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Nomenclature
ℎ
shape factor (fracture systems)
ex
effective pressure difference (by desuperposition)
= / =
dimensionless radius,
dimensionless distance to the outer boundary (hydraulic) diffusivity, diffusivity,
=0.5772… 5772/( … )
Euler(–Mascheroni) Euler(–Mascheroni) constant, viscosity, viscosity, SI derived unit: Pa s
/
, SI derived unit: m2 s
kg m s
effective viscosity (multiphase flow) average reservoir pressure
porosity bulk fracture porosity bulk matrix porosity pseudo-pressure (gas or multiphase reservoirs) fluid density matrix–fracture matrix–fracture coupling volume of liquid flowing from from matrix to fracture per time and and bulk volume volume formation volume factor
/
compressibility, compressibility, SI derived unit: 1 Pa Pa
/
formation compressibility, compressibility, SI derived unit: 1 Pa Pa liquid compressibility compressibility wellbore storage constant Total compressibility compressibi lity total compressibility compressibility of fracture total compressibility compressibility of matrix (in a fracture system) perforation (reservoir) height permeability, permeability, SI derived unit: m2 fracture permeability
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permeability of matrix (in a fracture system) oil permeability at
bulk permeability of the fracture system relative oil permeability length of horizontal well
= / ( ) =
pressure, SI derived unit: Pa characteristic pressure, dimensionless pressure
kg m s2
fracture pressure matrix pressure well pressure (bottom-hole pressure) down hole (reservoir) well production rate volumetric fluid flux (Darcy velocity) surface well production rate
radius equivalent radius (
() =
)
outer reservoir radius
radius of pressure front wellbore radius radius of investigation
inv inv
equivalent (effective) wellbore radius skin factor, factor, dimensionless irreducible water saturation pseudo time (gas or multiphase reservoirs) dimensionless time effective time (Agarwal (Agarwal time) production time
volume volume of fracture liquid volume volume of matrix speed of pressure front total wellbore volume fracture width fracture half length
relevant literature
m
mass
Subscripts
dimensionless form
initial
irreducible
liquid oil
outer
relative
water
well
147
References This is a list of papers that are mentioned in the main text. Articl Articles es have have only only been been cited cited whene wheneve verr a method method or equati equation on has has been been presen presented ted withou withoutt proper proper deriva deriva-tion. The articles are cited so that interested readers can find a proper derivation. The selected articles are not necessarily the ones with the earliest derivation, as clarity in presentation has also been a criterion. —✻— Agarwal R. G. Al-Hussainy, Al-Hussainy, R. and H. J Ramey. “An “An Investigation of Wellbore Wellbore Storage and Skin in Unsteady Liquid Flow - I. Analytical Treatment”. Treatment”. In: SPE Journal 10.3 (Sept. 1970), pp. 279–290. Cinco-Le Cinco-Ley, y, H. and Samanieg Samaniegoo-V V. F. “Transien “Transientt Pressure Pressure Analysis Analysis for Fractured Fractured Wells”. ells”. In: Journal Journal of Petroleum Technology (Sept. (Sept. 1981), pp. 1749–1766. Everdingen, A. F. Van and W. Hurst. “The Application of the Laplace Transformation to Flow Problems in Reservoirs”. In: Petroleum Transactions, AIME (Dec. (Dec. 1949), pp. 305–324. Gringa Gringarte rten, n, Alain Alain C. “Inter “Interpre pretat tation ion of Tests ests in Fissur Fissured ed and Multi Multilay layere ered d Reserv Reservoir oirss With With Double Double-P -Poro orosit sity y Behavior: Theory and Practice”. Practice”. In: Journal of Petroleum Technology (Apr. (Apr. 1984), pp. 549–564. Hant Hantus ush, h, M. S. and and C. E. Jaco Jacob. b. “Non “Non-s -ste tead ady y gree green’ n’ss func functi tion onss for for an infin infinit ite e stri strip p of leak leaky y aqui aquife fer” r”.. In: In: Eos, Transactions American Geophysical Union 36.1 (1955), pp. 101–112. Hantush, Madhi S. “Hydraulics “Hydraulics of wells”. wells”. In: Advances in hydroscience 1 (1964), pp. 281–432. Jones, J.R. and R. Raghavan. “Interpretation of Flowing Well Response in Gas-Condensate Wells”. In: SPE Formation Evaluatio n (Sept. 1988), pp. 578–594. Peac Peacema eman, n, D. W. “Inter “Interpre pretat tation ion of Well-Bl ell-Block ock Pressu Pressures res in Nu Numer merica icall Reserv Reservoir oir Simula Simulatio tion” n”.. In: Society of Petroleum Engineers Journal 18.03 (June 1978), pp. 183–194. Valkó, P.P P.P.. and J. Abate. “Comparison “Comparison of Sequence Accelrators Accelrators for the Gaver Method of Numerical Laplace Transform Transform Inversion”. Inversion”. In: Computers and Mathematics with Applications 48 (2004), pp. 629–636. Yaxley, axley, L.M. “Effect “Effect of a Partially Partially Communi Communicati cating ng fault fault on Transien Transientt Pressure Pressure Behavior” Behavior”.. In: SPE Formation Evaluation (Dec. 1987), pp. 590–598.
