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DEVELOPMENT OF FULLY IMPLICIT OIL-WATER MODEL RESERVOIR SIMULATOR

Final Project Submitted as one of the requirements TM 6011 Advanced Reservoir Simulation

By Billal Maydika Aslam (NIM 22216008)

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EXECUTIVE SUMMARY DEVELOPMENT OF FULLY IMPLICIT OIL-WATER MODEL RESERVOIR SIMULATOR

Reservoir Simulator has became a robust tool to predict hydrocarbo

reservoir performance. Its widespread use has been attributed to th

advance in computing technology and advances in numeric

techniques which is used to solve partial differential equation for flow

in porous media. The formulation of reservoir simulator s imulator is mainly base

on combination of continuity equation, fluid flow equation and equatio

of state. These equations are then formulated using finite differenc

method with unique boundary condition based on operation scenari and reservoir geology.

A reservoir simulator has been developed to account for two phase (oi

water) flow in 2-D cartesian grid using MATLAB ® programmin

language. The effect of capillary pressure and gravity is neglected i

formulation. Fully implicit method is used to formulate the multiphas

flow equation. The simulator is capable to handle multiple wells cas

Master your semester with Scribd  Read Free For 30this Days Signand/or up to vote on title as producer or injector with rate constraint bottomhole pressur  & The New York Times  Useful  Not useful Cancel anytime. constraint. Peaceman Well Model is used to treat well effect.

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TABLE OF CONTENT EXECUTIVE SUMMARY ........................................... ................................................................. ........................................... ..................... i TABLE OF CONTENT ............................................ .................................................................. ............................................ ........................ .. ii LIST OF FIGURES ........................................ .............................................................. ............................................ .................................. ............ iv LIST OF TABLES ............................................. ................................................................... ............................................ ............................... ......... v CHAPTER I.

INTRODUCTION .......................................... ................................................................. ............................... ........ 6

I.1

Background .......................................... ................................................................ ............................................ ............................... ......... 6

I.2

Problem Statement ............................................................ .................................................................................. ........................ .. 7

I.3

Purpose of This Study ............................................ .................................................................. ................................... ............. 7

I.4

Importance of This Study .......................................... ................................................................. ............................... ........ 7

CHAPTER II.

THEORETICAL BACKGROUND ........................................... ........................................... 8

II.1

Gridding ........................................... ................................................................. ............................................ ................................... ............. 8

II.2

Formulation of Governing Equation ............................................ ....................................................... ........... 10

II.3

Well Representation ........................................ .............................................................. ........................................ .................. 12

II.4

Solution Method ......................................... ............................................................... ............................................ ...................... 14

II.5

Rock and Fluid Model ........................................... ................................................................. ................................. ........... 16 16

II.5.1

Relative Permeability ............................................. .................................................................... ......................... .. 16

II.5.2

Formation Compressibility ................................................... .............................................................. ........... 17

Master your semester with Scribd  Read Free Foron 30this Days Sign up to vote title II.5.3 Formation Volume Factor ......................................... ............................................................... ...................... 17  & The New York Times  Useful  Not useful Cancel anytime.

Fluid Viscosity ..................................... ........................................................... ............................................ ...................... 18 Special offer for students:II.5.4 Only $4.99/month.

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IV.1.2

Multiple Well Producer ............................................ ................................................................... ....................... 3

IV.1.3

Multiple Well Producer and Injector............................................... Injector............................................... 3

CHAPTER V.

CONCLUSION......................................... ............................................................... ............................... .........

V.1

. Conclusion ................................ ...................................................... ............................................ ...................................... ................

V.2

. Recommendation & Future Works........................................... ....................................................... ............ 3

REFERENCES..................................................... ........................................................................... ............................................. ........................... .... APPENDIX ........................................... ................................................................. ............................................ .......................................... ....................



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LIST OF FIGURES Figure II.1 Regular Cartesian Grid – Block Centered ............................................. 8 Figure II.2 Five Point Formulation Transmissibility Calculation ........................... 9 Figure II.3 Parallel and Diagonal Flow Path in Five Point Formulation................. 9 Figure II.4 Control Volume with Evaluated Cell in Middle.................................. 10 Figure II.5 Pressure Distribution in Well Block (Carlson,M.R.) .......................... 12 Figure II.6 Newton Iteration Method .................................................................... 15 Figure II.7 Quasi Newton Iteration Method .......................................................... 16 Figure II.8 Oil-Water Relative Permeability Data................................................. 16 Figure II.9 Typical Oil FVF Data (CMG Manual) ................................................ 18 Figure II.10 Typical Oil Viscosity Data (CMG Manual) ...................................... 18 Figure III.1 Main Workflow in Developed Reservoir Simulator (Trastour,S.)..... 20 Figure III.2 Detailed Workflow in Developed Reservoir Simulator (Trastour,S.) 22 Figure III.3 Linear Solver Selection Algorithm in function mldivide (MATLAB® R2016b documentation) ........................................................................................ 23 You're Reading a Preview Figure IV.1 Calculated Oil Viscosity fortrial. Developed Simulator input Unlock fullfrom accessCMG with a free data ........................................................................................................................ 25 Download With Free Trial Figure IV.2 Calculated Oil Formation Volume Factor from CMG for Developed Simulator input data .............................................................................................. 26 Figure IV.3 Calculated Grid Pressure (psi) from CMG after 180 days ................. 27 Figure IV.4 Calculated Grid Pressure (psi) from developed simulator after 180 days ............................................................................................................................... 27

Master your semester with Scribd  Figure IV.5 Difference in Calculated Grid Pressure Read (psi) between Developed Sim Free Foron 30 Days Sign up to vote this title  & The New York Times Useful  Not useful  to CMG .................................................................................................................. 27 Special offer for students: Only $4.99/month.

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Figure IV.6 Calculated Grid Water Saturation from CMG after 180 days ........... 28

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Figure IV.11 Comparison of Well Rates Performance ......................................... 3

Figure IV.12 Comparison of Well Pressure Performance .................................... 3

Figure IV.13 Grid Pressure and Saturation at day 60 (Well 1 & 2 Producer) ...... 3

Figure IV.14 Well-1 Performance (BHP constraint) ............................................ 3

Figure IV.15 Grid Pressure and Saturation at day 60 ........................................... 3

Figure IV.16 Well-1 Performance (BHP constrain) ............................................. 3

LIST OF TABLES

Table IV.1 Case Definition used for Validation ................................................... 2

Table IV.2 Numerical Setting Used in Simulation ............................................... 2

Table IV.3 In Place Calculation Comparison ....................................................... 2

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CHAPTER I. INTRODUCTION

I.1 Background

Reservoir Simulator has became a robust tool to predict hydrocarbon reservo

performance. Its widespread use has been attributed to the advance in computin

technology and advances in numerical techniques which is used to solve parti differential equation for flow in porous media.

The formulation of reservoir simulator is mainly based on combination o

continuity equation, fluid flow equation and equation of state. These equations ar

then formulated using finite difference method – in order to be solved numericall

with unique boundary condition based on operation scenario and reservoir geolog

This approach has enabled reservoir simulator to analyze complex process whic otherwise cannot be solved analytically.

Reading a Previewsimulator is limited to simula However, it is importantYou're to note that reservoir Unlock access with a free trial.this imply that simulator is process that is embedded on its full formulation. Hence

suitable based on assumption that is used. Moreover, due to its numerical natur Download With Free Trial reservoir simulator cannot produce an exact solution but rather an approximatio

which bound to introduce errors. Some of these err ors could give misleading resu

on the forecast, thus it is necessary to consider these errors and validate the mod performance based on analytical model/historical performance.

This work attempts to give a brief introduction on the important theoretical aspec Master your semester with Scribd  Read Free Foron 30this Days Sign up to vote title  in developing reservoir simulation, present a practical computer program f & The New York Times  Useful  Not useful Cancel anytime.

simulation under some assumptions, and finally validate the develope Special offer for students: reservoir Only $4.99/month.

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I.2 Problem Statement 1. How to create a practical reservoir simulator program under some realistic

assumptions, which are : a) Insignificant capillary pressure and gravity effect , b) Only two phase flow of oil and water exist c) 2-D direction of flow

I.3 Purpose of This Study

Based on problem statement, the purpose of this study are: 1. To understand the important aspects on the mechanics of reservoir simulation 2. To understand how to design and create computer program on the defined

reservoir simulator 3. To understand the performance of developed program by comparison with

standard commercial simulator

You're Reading a Preview I.4 Importance of This Study Unlock full access with a free trial.

To provide a practical reservoir simulator program that can be used to predict Download With Free Trial reservoir performance under predefined assumptions.



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CHAPTER II. THEORETICAL BACKGROUND

II.1 Gridding

In general, the partial differential equation that describe fluid flow cannot be solv

analytically but it can be solved numerically by replacing differential equat ion wi

difference equation. Thus, the aim of gridding in reservoir simulation is to turn th

geological model of the field into a discrete system on which the fluid flow equatio can be solved. Related

issues

concerning

gridding

in

reservoir

simulation

has

bee

comprehensively discussed by Azis, et.al. (SPE 25233), the most importan consideration for gridding are accuracy with which the geological description

matched and how the grid represent the flow phenomena that is intended to b studied.

You're Reading a Preview One of the simplest and most used gridding scheme is Cartesian Grid in which th Unlock with a freeas trial. grid geometry is defined in x,y full andaccess z direction seen in f igure II.1




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Figure II.2 Five Point Formulation Transmissibility Calculation The consequence of using five point formulation is that no diagonal flow is allowed. Instead the flow path of diagonal flow will follow a zigzag pattern. Figure II.3 illustrate the flow path for parallel and diagonal flow in five point formulati on. You're Reading a Preview Unlock full access with a free trial.




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Figure II.3 Parallel and Diagonal Flow Path in Five Point Formulation

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Formulation of Governing Equation

This section will discuss the derivation of the equation through gridbloc

(discretization of partial differential equation) using discrete approach. W begin with a a statement of the continuity, or material balance, equation:

     =    …1

We can consider the gridblock as control volume with in and out flow

direction based on grid connection. For 2-D problems, it can be illustrate as fig. II.4.

Δx You're Reading a Preview Δy Unlock full access with a free trial.


Figure II.4 Control Volume with Evaluated Cell in Middle The pore volume of each gridblock is

 =∆.∆.ℎ.…2

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    =                ,…5 where u is flow velocity and A is flow area. Based on continuity equation (1), equating (4) and (5)

+     1       = ∆    −   …6 flow rate of each direction can be described as (taking flow from E or right as example)

 = ..∆…7 expand


…(8)

Download Here we define Transmissiblity as With Free Trial

…(9) and Mobility as

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Hence for 2-D problem, the material balance equation in final form is

…(12)

Note that in this formulation, the equations assume that capillary pressu

and gravity effect can be ignored and no solution gas present. The

assumptions are justified when the reservoir is in undersaturated conditio

(P>Pb), Transition zone is not present or the initial condition is not i

equilibrium, and also the reservoir layer is thin and has insignificant d (horizontal layer). The same approach using similar equation (1) to (12) also used for water phase II.3

Well Representation

You're Reading a Preview The ultimate goal of reservoir simulation study is to forecast well flow rat

Unlock full access with a free trial. and/or flowing bottomhole pressure accurately and to esti mate pressure an

saturation distributions. Well treatment Download With Free Trialin reservoir simulator prese difficulties that require special consideration. In general, these difficulti can be divided into :

1) The pressure at wellblock is not the same as the wellbore pressure sin

the wellblock sizes are significantly larger than wellbore radius. A

Master your semester within fig Scribd illustrated II.5 below & The New York Times Special offer for students: Only $4.99/month.


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2) Allocating phase production rate in multiphase flow when single phase or total fluid rate is used as constraint In 1983, Peaceman found that the pressure calculated for a well block is the same as the flowing pressure at an equivalent radius, r o. Peaceman defined r o as “the radius at which the steady-state flowing pressure for the actual well is equal to the numerically calculated pressure for the well block ”, or put in mathematical expression p (r )  p0 

q w   r   ln  2 kh  r o  …(13)

Knowing that wellblock pressure is steady state flowing pressure at r=Δx (for regular grid), we can get r o analytically as

 =0.208∆…14

You're Reading a Preview Introducing Well Index as Unlock full access with a free trial.

2  k h Download With Free Trial r ln( o )  S r w …(15) WI 

where S is Van Everdingen & Hurst Skin Factor to account for formation damage or stimulation. Substitute equation (15) to (13) and rearrange, we get

Master your semester with Scribd P p(r ) & The New York Times wf




q




  WI


Useful

…(16)

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Solution Method

From the nature of material balance equation, it can be inferred that th pressure and fluid saturation value in each grid block can only satisfy t

material balance if we get the same result for both left hand side (net flo

rate in) and right hand side (accumulation rate). Hence to get the corre

value of pressure and fluid distribution in each time step, the materi balance can be rearrange to become a root finding problem.

For each phase, material balance equation for whole reservoir model can b written as Ncells

 e 1

 N  V S  ( n1)  V S  n   1  p p n 1 n 1     …(19)    T ie M ie  P ie   t  B   B  e    i 1 e 

Here we only consider oil and awater phase for simplicity. Note th You're Reading Preview superscript n represent current time step properties and n+1 represent ne Unlock full access with a free trial.

time step. Here we use fully-implicit formulation (also known Download With Free Trial simultaneous solution method) where both pressure and saturation a

simultaneously solved for the next timestep. It has been found that full implicit method is unconditionally stable technique.

Rearrange equation (19) for oil and water phase and including well term we get the residual functions :

Master your semester with Scribd  & The New York Times r    T M  Special offer for students: Only $4.99/month. n 1 o

Ncells e 1

N

 i 1

i e

n 1

n 1

o i e

o i e



Read Free Foron 30this Days Sign up to vote title ( n 1) n  S o    V puseful S o    V pUseful 1  Not

 

 Cancel  anytime.    qosc     t  Bo  e  Bo  e   

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Our goals is now to minimize the residual functions by ‘guessing’ correct value for pressure and saturation in each grid. Initial properties is used for initial guess. Iterative method can be used to refine our ‘guess’ until some acceptable tolerance (defined as Maximum Material Balance Error from whole gridblock). In order to solve the equations iteratively, we should linearize the problem by computing the Jacobian matrix, J of those equations.

  =  [ 

  …22  ]

The problem is then become a linear problem, with

 ∆+ = − …23 The superscript of k indicates k th inner iteration. Δm is defined as You're Reading a Preview

∆ = …24

Unlock full access with a free trial.

Download WithtoFree Trialequation (23) is Newton Some of available iterative method solve

Iteration and Quasi Newton Iteration. The latter method only calculate the jacobian matrix in the first iteration and use the same jacobian matrix for the subsequent iteration, while in Newton Iteration the Jacobian Matrix is calculated in each iteration. These methods are illustrated in Fig II.6 and Fig

II.7 Master your semester with Scribd & The New York Times r (x) Special offer for students: Only $4.99/month.


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r (x)

 r  x

k



0

r (k=0)

 r  x

r (k=1) Xn+1

Xk=1

k



Xk=0

0

X

Figure II.7 Quasi Newton Iteration Method

The Jacobian Matrix element is evaluated by perturbing the pressure an

saturation value independently in turns for each evaluated grid and the

connected grid. The perturbed value is usually the predicted chang

II.5

according to timestep size You're Reading a Preview Rock and Fluid Model Unlock full access with a free trial. II.5.1 Relative Permeability Download With Free Trial

A straight line relative permeability was used on the model. The relativ

permeability data was input as table. The graph of relative permeability da is given on Fig II.8. 1

Master your semester 0.9 with Scribd 0.8 y t i & The New York Times l i 0.7 b a Special offer for students: Only $4.99/month. e 0.6 m r


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The developed simulator uses linear interpolation method to get rel-perm value at various saturation. However the base simulator (CMG) can only uses quadratic interpolation method. By including closely spaced data points, the different in interpolation method is insignificant. CMG default rel-perm input was based on end-point data with following equations k rw  0

if S w  S wc nw

 S w  S wc   k rw  k rwro  otherwise 1 S S   wc orw   k row  0 if (1  S w  S orw )  0 k row

 1  S w  S orw    k rocw  1 S S   wc orw  

now

otherwise …(25)

where n is relperm exponent. II.5.2 Formation Compressibility

Formation compressibility value is useda Preview to update porosity change due to You're Reading pressure change in reservoir (i.e. compaction from depletion). Both CMG Unlock full access with a free trial. and developed simulator use following equation Download With Free Trial

∅ = ∅ ×(1−)…26

where P is pore pressure (psi), Cpor rock compressibility (1/psi) at a reference pressure, Prpor (psi). The input porosity data at initial condition is measured on the reference pressure.

Master your semester with Scribd II.5.3 Formation Volume Factor & The New York Times For oil phase, the formation volume Special offer for students: Only $4.99/month.

undersaturated condition is




factor


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(dimensionless, rb/stb) in

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Figure II.9 Typical Oil FVF Data (CMG Manual) For water phase, similar equation is used

 

Bw P





Bwi 1





C w P



P rw

 …(28)

where Cw is water compressibilit y (1/psi) and Prw is reference pressure at which formation volume factor Bwi is calculated. II.5.4 Fluid Viscosity

For oil phase, the oil viscosity (cp) in undersaturated condition is governe by following equation

     P   C  P  P  …(29)

 o P

o

B

 o

B

You're Reading a Preview

where

 o curve above P c o  slopeUnlock of the full access with a free trial. B

(cp/psi) typical oil viscosity

curve is given on fig II.10 Download With Free Trial

Figure II.10 Typical Oil Viscosity Data (CMG Manual)

Master your semester with Scribd Whereas for water phase & The New York Times Special offer for students: Only $4.99/month.

   

 w P

wi


  C  w


Useful

 P  P  rw

…(30)

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Errors in Reservoir Simulation

In any type of computer simulation work, it is important to determi ne the accuracy of the solution. There are seven type of errors that can affect the solution from reservoir simulation 1. Round off error, occur when using single precision accuracy when double

precision is required or by mixing single and double precision variables 2. Material Balance Error (MBE) is a measure of consistency, not accuracy. Its

primary cause is from non-linear error. MBE is an indication of how well the equation J.m=-r is solved. One method to solve MBE's is to use a Newton or outer iteration loop that would eventually converge on zero MBE. 3. Non-Linear Error occurs when using a linear approximation (chord slope) to

find a value at the n+ 1 time level of a non-linear function such as formation volume factors. 4. Instability Error is caused by explicit saturation dependent variables (kr and Pc) You're Reading a Preview in IMPES. A solution for instability is to take smaller timesteps or go to a fully

implicit model.


5. Truncation Error is caused by truncating the Taylor series. A solution to Download With Free Trial truncation error is to vary timestep size (Δt) and gridblock size (Δx) by trial and

error until the solution converges. 6. Numerical Dispersion is caused by saturation discontinuity within a cell. The

solutions for numerical dispersion are: (1) smaller Δx, (2) modify kr (3) select proper Δt and ( 4) use pseudo kr. Master yourcalculations, semester with Scribd  Free Foron 30this Days Sign up to vote Grid orientation istitle generally 7. Grid orientation can change the final answer. Read  & The New York Times  Useful  Not useful important in calculating saturation distributions in a waterflood. Typically, a Special offer for students: Only $4.99/month.

Cancel anytime.

diagonal grid system will result in better recoveries (more optimistic).

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CHAPTER III.

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RESERVOIR SIMULATOR PROGRAM STRUCTURE

III.1

Program Flowchart

The reservoir simulator program is mainly divided into three major tasks : 1) Read input data and initialize rock and fluid properties in res ervoir grids.

2) Preprocessing input to complete the flow equation (check connections, calcula transmissibility and mobility) 3) Solving non-linear problems in each timestep until the time of end run In brief, these three main tasks can be illustrated in Fig III.1





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The developed simulator uses Max Material Balance Error convergence criteria of 0.0001 (equivalent to MATBALTOL in CMG) for ti mestep passing. Quasi-Newton iterative method is used along with MATLAB® built-in function mldivide (\) to solve linear problem (eq(23)). The number of Quasi-Newton iteration is set to be limited to 10 iteration (predefined in initialization, ITERMAX). If the solver failed to converge within the limit, the timestep is repeated with halved timestep size. If the failure persist, the timestep will be continued to be halved until it exceed the minimum time step size limit (dtmin = 0.1) and the run will be terminated with message of timestep limit reached. However if the t imestep converged, the solution will be checked against the secondary well constraint if it exist (i.e.minimum bottom hole pressure). If the secondary well constraint is violated, the well constraint is switched to secondary constrain (calculation in residual function is changed according to constraint) and the timestep is repeated. At the end of all converged time step which did not violate well constraints, the well rates, well bhp, grid pressures and saturations are saved. These detail ed process in time step solving You're Reading a Preview is illustrated in Fig III.2. Unlock full access with a free trial.

Algorithm that is used behind the linear solver mldivide is actually composed of Download With Free Trial many linear solver that is selected based on suitability with the matrix involved. In this case, from linear equation s ystem A.x = B, A is jacobian matrix which is sparse and B is residual matrix which is full. The algorithm for mldivide is shown in Fig III.3. Actually there are many linear solvers that are embedded in MATLAB® and can also be used such as gmres, bicgstab, pcgstab, etc. Some of these solvers are

Master your with Scribd for the matrix  robustsemester and already include preconditioner to bevote easier to title be solved. Read Free For 30this Days Sign up to on  & The New Yorkmldivide Timesis the most common function  Useful  Not usefulto solve However, used in MATLAB® Special offer for students: Only $4.99/month.

system of linear equation.

Cancel anytime.

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CHAPTER IV.

IV.1

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RESERVOIR SIMULATOR VALIDATION

SIMULATOR VALIDATION & TESTING

IV.1.1 Oil Rate and Pressure Control Well (Comparison with CMG)

The developed simulator is compared with CMG-IMEX release 2015.10 using oil-water option. The case scenario that is used for comparison summarized in Table 4.1. Table IV.1 Case Definition used for Validation STATIC PROPERTIES

Value

Remark

PVT

porosity () Cpor (1/psi) Prpor (psi) permeability (md) pressure (psi) oil saturation, So () water saturation, Sw()

0.3 homogen 1E-05 constant 14.6596 constant 200 homogen You're Reading a Preview 2000 homogen Unlock with a free trial. 0.7full access homogen 0.3 homogen

Download With Free Trial GRID

Value

dx(ft) dy(ft) Nx Ny thickness (ft)

Remark

150 150 9 9 25

homogen homogen

0.2

homogen

Well Number

homogen

Master your semester with Scribd ROCKTYPE Value Remark & The New York Times Sorw 0.1 homogen Special offer for students: Only $4.99/month.

Swc

Pb [psi] Bo(Pb) Co [1/psi] µo(Pb) [cp] Cµo [cp/psi] Prw [psi] Bwi Cwi [1/psi] µwi [cp] Cµwi [cp/psi]

Value

500 1.12646 3.00E-05 0.58405 5.00E-03 14.696 1 0 1 0 Value

1 Type producer Perf UBA 5,5,1  Read Free Foron 30this Days Sign up to vote title rw (ft) 0.25  Not useful  Useful  skin 0 Cancel anytime. Qo (stb/d) 250

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Table IV.2 Numerical Setting Used in Simulation Numerical Setting dtmin

dt itermax matbaltol dP dSw

Value

Remark

0.25 1 10 0.0001 10 0.01

Min timestep size Timestep size Max Newton Iter. max residual error perturbed press perturbed Sw

The case is run for 180 days so that the well is experiencing both primary and secondary constraints. Both simulators were initialized without equilibration. To have an exactly same PVT and rockfluid model. Data input for developed simulator was taken from calculated FVF and viscosity from CMG, which are shown on fig IV.1 and fig IV.2


Oil Viscosity (cp)


12 10 ) p c ( y t i s o c s i V

8


y = 4.999989E-03x + 8.653630E-01 R² = 1.000000E+00

6 4 2

0 Master your semester with Scribd  Read Free For 30this Days Sign up to vote on title 2500 0 500 1000 1500 2000  Pressure (psi) & The New York Times  Useful  Not useful


Cancel anytime.

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Oil Formation Volume Factor 1.06 1.05

y = -3.203337E-05x + 1.067779E+00 R² = 1.000000E+00

1.04 o B

1.03 1.02 1.01 1 0

500

1000

1500

2000

25

Pressure (psi)

Figure IV.2 Calculated Oil Formation Volume Factor from CMG for Developed Simulator input data

Water viscosity and FVF hase been observed to be 1. Relative permeabili data calculated also has been found to be the s ame as input. You're Reading a Preview Unlock full access with a free trial. and developed simulator Fluid in place calculation from CMG

summarized in table IV.3., Developed simulator calculate 2% less in plac Download With Free Trial for oil and water compared to CMG Table IV.3 In Place Calculation Comparison CMG Total oil in place

STB

Total water in place STB Master your semester with Scribd Total gas in place SCF & The New York Times


SIM

error

1.73E+06 1.70E+06 -2% 7.4E+05 7.30E+05 -2% 5.46E+06 n/a n/a Read Free Foron 30this Days Sign up to vote title

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Comparison of grid pressure distribution between CMG and develope

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Figure IV.3 Calculated Grid Pressure (psi) from CMG after 180 days



Figure IV.4 Calculated Grid Pressure (psi) from developed simulator afte r 180 days



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Comparison of grid water saturation distribution between CMG an

developed simulator can be seen on Fig IV.6, and Fig IV.7 The differenc of forecasted pressure is shown on Fig IV.8

Figure IV.6 Calculated Grid Water Saturation from CMG after 180 days



Figure IV.7 Calculated Grid Water Saturation from developed simulator


after 180 days Read Free Foron 30this Days Sign up to vote title

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The water saturation distribution between developed simulator and CMG showed near perfect match. The difference may be attributed to round-off error since the error is very small and negligible To see the comparison in pressure and saturation change with respect to time. Corner grid result was used as comparison. Fig IV.9 and Fig IV.10 showed the comparison for changes of pressure and water saturation in corner grid between the two simulators 2500 CMG_Pcorner 2000

BMA_Pcorner

) i s p 1500 ( e r u s s 1000 e r P 500

You're Reading a Preview 0

full70 access a free 0 10 20 30 40Unlock 50 60 80 with 90 100 110trial. 120 130 140 150 160 170 180 190

time (day)


Figure IV.9 Comparison of Pressure Changes in Corner Grid w.r.t Time 0.302 0.3 0.298

0.296 Master your semester with Scribd 0.294 & The New York w Times S 0.292

Special offer for students: Only $4.99/month. 0.29

CMG_Sw_corner BMA_Sw_corner Read Free For 30this Days Sign up to vote on title

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Again the changes of pressure and water saturation with respect to time corner grid is a close match between the two simulators.

The next comparison is on the well performance. Fig IV.11 illustrate th

comparison between the well oil and water rates between CMG an developed simulator. 450 CMG_Qo

400

CMG_Qw

) 350 y a d / B300 T S ( e t 250 a R n o i t c 200 u d o r 150 P l l e W100

BMA_Qo BMA_Qw


50


0 0

With 20 Download 40 60

Free 80 Trial100

120

140

160

1

time (day)

Figure IV.11 Comparison of Well Rates Performance

Overall the comparison show a good match for both oil and water rate

Master your semester Scribd Howeverwith It is interesting to observe that the water rates shows a good  matc Read Free Foron 30this Days Sign up to vote title  in transient phase but become deviated  once pseudo-steady state flow beg & The New York Times Useful  Not useful Cancel anytime.

Special offer for students: Only $4.99/month. but the rates slowly converging again at late time. Since the primary we

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100

150

200

2000

2.500% CMG_wbp CMGcor_bhp

1800

BMA_wbp BMAcor_bhp

1600

2.000%

BHP error 1400

) 1200 i s p ( e r 1000 u s s e r P 800

1.500% ) % ( r o r r e P H 1.000% B

600 400

0.500%

200 0 0


50

100

0.000%

150

200

Unlock full access with a free trial. time (day)

Download With Free Trial Figure IV.12 Comparison of Well Pressure Performance

It can be seen that the well pressures also show a good match. It is also interesting to see that the difference between developed simulator and CMG on well pressure performance is showing similar pattern with the well rates performance. The same reason apply for the difference in well pressure Master your semester with Scribd  Read Free Foron 30this Days Sign up to vote title  performance. & The New York Times  Useful  Not useful Special offer for students: Only $4.99/month. IV.1.2 Multiple Well Producer

Cancel anytime.

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Figure IV.13 Grid Pressure and Saturation at day 60 (Well 1 & 2 Produce



Figure IV.14 Well-1 Performance (BHP constraint) Master your semester with Scribd  Read Free Foron 30this Days Sign up to vote title  It can be observed that due to production both wells pressures are & The New York Times Useful useful  in  Notthe Cancel anytime.

Special offer for students: Only $4.99/month. depleted especially in the corner wells and water saturation is increasing

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The case is also run until 60 days. Fig IV.15 shows the result for calculated grid pressure and water saturation at day 60, while Fig IV.16 shows the result for well performance

Figure IV.15 Grid Pressure and Saturation at day 60 (Well 1 Producer & Well 2 Injector) You're Reading a Preview Unlock full access with a free trial.


Master your semester with Scribd Read Free Foron 30this Days Sign up to vote title & The New York Times  Useful  Not useful Figure IV.16 Well-1 Performance (BHP constrain) Special offer for students: Only $4.99/month.

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It can be observed that the pressure is increasing near injection well and decreasin g

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CHAPTER V.

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CONCLUSION

V.1. Conclusion

1. A practical reservoir simulator that account only for oil-water pha

flow in 2-D cartesian grid under the assumptions of negligible capilla

pressure and gravity effect has been succesfully developed usin MATLAB® programming language

2. The developed reservoir has been validated with standard commerci

reservoir simulator (CMG) and shows similar performance und overall 1% difference. V.2. Recommendation & Future Works

1. Schedule features should be added to increase practicality in simulatin

field operating scenario You're Reading Preview 2. Continue the development to a3-D 3-phase flow Unlock full access withroutines a free trial. management 3. Add group/well

4. Include GUI and error catching to avoid mistake during data entr y an Download With Free Trial automation



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REFERENCES Calvin C. Mattax and Robert L. Dalton: Reservoir Simulation, SPE, Henry L. Doherty Series, 1990. Ertekin, T., Abou-Kassem, J., and King, G.R.: Basic Applied Reservoir Simulation, 2001 Mattax, C. C. and Dalton, R. L.: Reservoir Simulation, Monograph Series, SPE, Richardson, TX (1990) 13. Michael Carlson: Practical Reservoir Simulation, Penwell Corp., 2003. Syihab, Zuher, Advanced Reservoir Simulation Class Notes, ITB, 2017 Troster,S.Energy 223-Formulation of Oil Gas Reservoir Simulator., Stanford University.2016 Wattenberger & Lee., PETE 603 Reservoir Simulation Class Notes, Texas A&M University





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APPENDIX

[FINALPROJECTTM-6011]2D2PhaseBlackOilSimulator

Advisor : Dr. Zuher Syihab Code Written b y : Billal Aslam // Student ID : 2221600 Last Revision : 17 May 2017 %********************************************************************** % Purpose : Simulation of Oil-Water Flow in Porous Media areal 2D % Program Features : % No Pc or g effect % No flow boundary % uniform grid, homogeneous properties % multiple wells, injector and producer % auto time step %*********************************************************************** clc; clearvars;

CreateGrid


%gridb number

Unlock full access with a free trial. nx=9; ny=9; ngrid=nx*ny; Download With Free Trial nphase=2; %oil,water %gridb size, uniform gridb L=150*9; %ft, reservoir length dx=L/nx; dy=dx; thick=25; %ft %A=dx*thick; %sqft %gridb properties array input poro=zeros(ny,nx); Read Free Foron 30this Days Sign up to vote title perm=zeros(ny,nx);  Useful  Not useful for j=1:ny Cancel anytime. Special offer for students: Only for i=1:nx $4.99/month. poro(j,i)=0.3; %( )

Master your semester with Scribd & The New York Times

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'kx',[],'ky',[],... 'dx',[],'dy',[],... 'Vp',[],... 'thick',[],... 'pres',[],... 'So',[],'Sw',[],... 'propn',[],... 'resoi',[],'reswi',[],... 'resok',[],'reswk',[]);

NumericalSettings dt=1; %initial timestep size trun=60; %running time matbaltol=0.0001; itermax = 10; dp=10; %pressure change per timestep dsw=0.01; %saturation change per timestep dtmax=1; dtmin=0.25; warning=0; numflag=0; gmsetup.type='ilutp'; gmsetup.droptol=1e-3; gmsetup.udiag=1; You're Reading

a Preview


Specifygridbproperties


id=0; for j=1:ny for i=1:nx id=id+1; gridb(id).id=id; gridb(id).poro=poro(j,i); %r = -5 + (5+5)*rand(10,1) %In general, you can generate N random numbers in the interval (a,b) with the formula r = a + (b-a).*rand(N,1)  Read Free Foron 30this Days Sign up to vote title gridb(id).kx=200;  Not useful gridb(id).ky=gridb(id).kx;  Useful  Cancel anytime. gridb(id).dx=dx; Special offer for students: Only $4.99/month. gridb(id).dy=dy;


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if (i==1||i==nx)&&(j==1||j==ny) gridb(id).ncon=2; elseif ((i==1||i==nx)&&(j~=1&&j~=ny))||((i~=1||i~=nx)&&(j==1||j==ny)) gridb(id).ncon=3; else gridb(id).ncon=4; end end end

WELLMODELLING nwell=2; %set active wells well(nwell)=struct('blockid',[],'perf',[],'rw',[],'constrain',[],... 'type',[],'mode',[],'skin',[],'WI',[],'Qo',[],'Qw',[],'bhp',[]); well(1).perf.x=5; well(1).perf.y=5; well(1).constrain.Qo=-250; %bopd well(1).constrain.bhp=800; %psia well(1).mode=2; %1 for oil rate control well(1).rw=0.25; %ft well(1).type='producer'; well(2).perf.x=9; well(2).perf.y=9; well(2).constrain.Qw=250; You're %bopd Reading a Preview % well(2).constrain.bhp=800; %psia Unlock full access with a free trial. well(2).rw=0.25; well(2).type='injector'; well(2).mode=1; Download well(3).perf.x=5; well(3).perf.y=5; well(3).constrain.Qo=-100; %bopd well(3).constrain.bhp=500; %psia well(3).rw=0.5; well(3).mode=1;

With Free Trial

for k=1:nwell well(k).blockid=nx*(well(k).perf.y-1)+well(k).perf.x; Read Free Foron 30this Days Sign up to vote title well(k).skin=0; well(k).WI=WI(k,gridb,well);  Useful  Not useful Cancel anytime. end


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on(1); gridb(id).idcon(2)=id+nx; gridb(id).lcon(2)=dy; gridb(id).trans(2)=0.00633*dx*thick*0.5*(gridb(id).ky+gridb(id+nx).ky)/gridb(id).l con(2); elseif (i==1)&&(j==ny) %southwest corner gridb gridb(id).idcon(1)=id+1; gridb(id).lcon(1)=dx; gridb(id).trans(1)=0.00633*dy*thick*0.5*(gridb(id).kx+gridb(id+1).kx)/gridb(id).lc on(1); gridb(id).idcon(2)=id-nx; gridb(id).lcon(2)=dy; gridb(id).trans(2)=0.00633*dx*thick*0.5*(gridb(id).ky+gridb(idnx).ky)/gridb(id).lcon(2); elseif (i==1)&&(j~=1)&&(j~=ny) %grids near west boundary gridb(id).idcon(1)=id+1; gridb(id).lcon(1)=dx; gridb(id).trans(1)=0.00633*dy*thick*0.5*(gridb(id).kx+gridb(id+1).kx)/gridb(id).lc on(1); gridb(id).idcon(2)=id-nx; You're Reading a Preview gridb(id).lcon(2)=dy; gridb(id).trans(2)=0.00633*dx*thick*0.5*(gridb(id).ky+gridb(idUnlock full access with a free trial. nx).ky)/gridb(id).lcon(2);


gridb(id).idcon(3)=id+nx; gridb(id).lcon(3)=dy;

gridb(id).trans(3)=0.00633*dx*thick*0.5*(gridb(id).ky+gridb(id+nx).ky)/gridb(id).l con(3); elseif (i==nx)&&(j==1) %northeast corner gridb gridb(id).idcon(1)=id-1; gridb(id).lcon(1)=dx; gridb(id).trans(1)=0.00633*dy*thick*0.5*(gridb(id).kx+gridb(id 1).kx)/gridb(id).lcon(1); Read Free Foron 30this Days Sign up to vote title 

Master your semester with Scribd & The New York Times gridb(id).idcon(2)=id+nx; Special offer for students: Only $4.99/month. gridb(id).lcon(2)=dy;

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nx).ky)/gridb(id).lcon(2); elseif (i==nx)&&(j~=1)&&(j~=ny) %grids near east boundary gridb(id).idcon(1)=id-1; gridb(id).lcon(1)=dx; gridb(id).trans(1)=0.00633*dy*thick*0.5*(gridb(id).kx+gridb(id1).kx)/gridb(id).lcon(1); gridb(id).idcon(2)=id-nx; gridb(id).lcon(2)=dy; gridb(id).trans(2)=0.00633*dx*thick*0.5*(gridb(id).ky+gridb(idnx).ky)/gridb(id).lcon(2); gridb(id).idcon(3)=id+nx; gridb(id).lcon(3)=dy;

gridb(id).trans(3)=0.00633*dx*thick*0.5*(gridb(id).ky+gridb(id+nx).ky)/gridb(id). con(3); elseif (j==1)&&(i>1)&&(i
gridb(id).trans(1)=0.00633*dy*thick*0.5*(gridb(id).kx+gridb(id+1).kx)/gridb(id). on(1); gridb(id).idcon(2)=id-1; gridb(id).lcon(2)=dx; You're Reading a Preview gridb(id).trans(2)=0.00633*dy*thick*0.5*(gridb(id).kx+gridb(id1).kx)/gridb(id).lcon(2);Unlock full access with a free trial. gridb(id).idcon(3)=id+nx; Download With gridb(id).lcon(3)=dy;

Free Trial

gridb(id).trans(3)=0.00633*dx*thick*0.5*(gridb(id).ky+gridb(id+nx).ky)/gridb(id). con(3); elseif (j==ny)&&(i>1)&&(i
gridb(id).trans(1)=0.00633*dy*thick*0.5*(gridb(id).kx+gridb(id+1).kx)/gridb(id).l Master your semester with Scribd  on(1); Read Free Foron 30this Days Sign up to vote title  & The New York Times Useful  Not useful  gridb(id).idcon(2)=id-1;

Special offer for students: Only $4.99/month. gridb(id).lcon(2)=dx;

Cancel anytime.

gridb(id).trans(2)=0.00633*dy*thick*0.5*(gridb(id).kx+gridb(id-

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on(1); gridb(id).idcon(2)=id-1; gridb(id).lcon(2)=dx; gridb(id).trans(2)=0.00633*dy*thick*0.5*(gridb(id).kx+gridb(id1).kx)/gridb(id).lcon(2); gridb(id).idcon(3)=id+nx; gridb(id).lcon(3)=dy; gridb(id).trans(3)=0.00633*dx*thick*0.5*(gridb(id).ky+gridb(id+nx).ky)/gridb(id).l con(3); gridb(id).idcon(4)=id-nx; gridb(id).lcon(4)=dy; gridb(id).trans(4)=0.00633*dx*thick*0.5*(gridb(id).ky+gridb(idnx).ky)/gridb(id).lcon(4); end end end

TimesteppingProcedure tstep=0; P_sol=zeros(ny,nx,trun/dt); %matYou're to store P result Reading a Preview Sw_sol=zeros(ny,nx,trun/dt); %mat to store Sw result full access with a free trial. So_sol=zeros(ny,nx,trun/dt); %matUnlock to store So result t_sol=zeros(trun/dt,1); t_sol(1,1)=dt; Download With Free Trial bhp_sol=zeros(trun/dt,nwell); Qo_sol=zeros(trun/dt,nwell); Qw_sol=zeros(trun/dt,nwell); tic %start timing simulation run OOIP=0; WIP=0; Np=zeros((trun/dt)+1,1); %fluid in place calculation for i=1:ngrid Read Free Foron 30this Days Sign up to vote title OOIP=OOIP+(1/5.615)*(gridb(i).Vp*gridb(i).So/Bon(i,gridb)); WIP=WIP+(1/5.615)*(gridb(i).Vp*gridb(i).Sw/Bwn(i,gridb));  Useful  Not useful Cancel anytime. end


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r=zeros(nphase*ngrid,1); for i=1:ngrid r(2*i-1,1)=gridb(i).resoi; r(2*i,1)=gridb(i).reswi; end if iter==1||iter==itermax %Solver Option Control J=calcjacobian(dt,dp,dsw,gridb,well,nphase,ngrid,nwell); iter=1; end

dm=J\-r;

if any(isnan(dm(:))) %convergence control numflag=1; propreset(ngrid,gridb); dt=dt/2; if dt

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% % % constrain']); % %

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elseif Pwf>well(i).constrain.bhp well(i).mode=1; disp(['well ' num2str(i) ' is changed to orat/lrat iter=0; continue end

end end if switchflag==1 mbe=matbaltol+1; continue end disp(['mat.bal.error = ' num2str(mbe)]);

end trun=trun-dt; disp(['runtimeleft = ' num2str(trun) ' days']); if warning==1 disp('Abnormal Termination, timestep size is too low'); break end You're Reading a Preview

% % % % % % % % % %

%TIME STEP CONTROL Unlock full access with a free trial. dtn=round(dt*matbaltol/mbe); if isnan(dtn) Download With Free Trial dt=dtmax; elseif dtndtmax&&dtmax
Master your semester with Scribd t_sol(tstep+1,1)=t_sol(tstep,1)+dt; & The New York Times %Write Grid Result Special offer for students: Only $4.99/month. id=0; for j=1:ny


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end %Write Well Result for i=1:nwell well(i).Qo=calcoratw(i,gridb,well)/5.615; well(i).Qw=calcwratw(i,gridb,well)/5.615; well(i).bhp=calcbhp(i,gridb,well); Np(tstep+1)=Np(tstep)+(1/5.615)*well(i).Qo*dt; Qo_sol(tstep,i)=well(i).Qo; Qw_sol(tstep,i)=well(i).Qw; bhp_sol(tstep,i)=round(well(i).bhp); end numflag=0; %display sim log disp('Pressure Profile = '); disp(P_sol(:,:,tstep)); disp('Sw Profile = '); disp(Sw_sol(:,:,tstep)); end t=toc; disp(['Elapsed Running time

= ' num2str(t) ' seconds']);

RemainingFluidVolumeCheck

You're Reading a Preview ROIP=0; RWIP=0; Unlock full access with a free trial. for i=1:ngrid ROIP=ROIP+(1/5.615)*(gridb(i).Vp*gridb(i).So/Bon(i,gridb)); RWIP=RWIP+(1/5.615)*(gridb(i).Vp*gridb(i).Sw/Bwn(i,gridb)); Download With Free Trial end

DisplayResultGrid imagesc(P_sol(:,:,tstep)); axis([1 nx 1 ny]); xticks(1:nx); yticks(1:ny); title('Grid Pressure'); x1=well(1).perf.x; Special offer for students: y1=well(1).perf.y; Only $4.99/month. z1=0;



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imagesc(Sw_sol(:,:,tstep)); axis([1 nx 1 ny]); xticks(1:nx); yticks(1:ny); title('Grid Sw'); x1=well(1).perf.x; y1=well(1).perf.y; z1=0; txt1=' Well-1'; text(x1,y1,z1,txt1); x2=well(2).perf.x; y2=well(2).perf.y; z2=0; txt2=' Well-2'; text(x2,y2,z2,txt2); c = colorbar; c.Label.String = 'Water Saturation ()'; grid on

DisplayWellGraphs figure plot(t_sol(1:tstep),-Qo_sol(1:tstep)); hold on plot(t_sol(1:tstep),-Qw_sol(1:tstep)); % xticks(1:tstep); You're Reading a Preview % xlabel('timestep'); Unlock full access with a free trial. tend=round(t_sol(tstep,1)); xticks(1:10:tend); xlabel('time (days)'); Download ylabel('well production rate (stb/d)'); title('Well-1'); legend('oil rate', 'water rate'); hold off

With Free Trial

figure i=1:tstep; Pwblock=zeros(tstep,1); Pwblock(i,1)=P_sol(well(1).perf.x,well(1).perf.y,i); Read Free Foron 30this Days Sign up to vote title plot(t_sol(1:tstep),bhp_sol(1:tstep)); hold on plot(t_sol(1:tstep),Pwblock(1:tstep));  Useful  Not useful Cancel anytime. tend=round(t_sol(tstep,1)); Special offer for students: Only $4.99/month. xticks(1:10:tend);


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txt1=' Well-1'; text(x1,y1,z1,txt1) plot3(x1,y1,z1,'ko'); % % % % % %

x2=well(2).perf.x; y2=well(2).perf.y; z2=0; txt2=' Well-2'; text(x2,y2,z2,txt2) plot3(x2,y2,z2,'ko');

functioncalculateoilFVFnow function bo=Boi(id,grid) P_pvt1=[500,550,600,650,700,750,800,850,900,950,1000,1050,1100,1150,1200,... 1250,1300,1350,1400,1450,1500,1550,1600,1650,1700,1750,1800,1850,1900,... 1950,2000]; %psi Bo_pvt1=[1.05176215,1.050160465,1.04855878,1.046957095,1.04535541, ... 1.043753725,1.04215204,1.040550355,1.03894867,1.037346985,1.0357453,... 1.034143615,1.03254193,1.030940245,1.02933856,1.027736875,1.02613519,... 1.024533505,1.02293182,1.021330135,1.01972845,1.018126765,1.01652508,... 1.014923395,1.01332171,1.011720025,1.01011834,1.008516655,1.00691497,... 1.005313285,1.0037116]; %rb/stb switch grid(id).pvttype case 1 You're Reading a Preview bo=interp1(P_pvt1,Bo_pvt1,grid(id).pres); Unlock full access with a free trial. otherwise disp('assign PVT type') end end


functioncalculateoilFVF function bo=Bon(id,grid) P_pvt1=[500,550,600,650,700,750,800,850,900,950,1000,1050,1100,1150,1200,... 1250,1300,1350,1400,1450,1500,1550,1600,1650,1700,1750,1800,1850,1900,... 1950,2000];  Read Free Foron 30this Days Sign up to vote title Bo_pvt1=[1.05176215,1.050160465,1.04855878,1.046957095,1.04535541, ...  Not useful 1.043753725,1.04215204,1.040550355,1.03894867,1.037346985,1.0357453, ...  Useful  Cancel anytime. ... Special offer for students: Only 1.034143615,1.03254193,1.030940245,1.02933856,1.027736875,1.02613519, $4.99/month. 1.024533505,1.02293182,1.021330135,1.01972845,1.018126765,1.01652508,...


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functioncalculatewaterFVFnow function bw=Bwi(id,grid) P_pvt1=[500,550,600,650,700,750,800,850,900,950,1000,1050,1100,1150,1200,... 1250,1300,1350,1400,1450,1500,1550,1600,1650,1700,1750,1800,1850,1900, ... 1950,2000]; Bw_pvt1=[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]; switch grid(id).pvttype case 1 bw=interp1(P_pvt1,Bw_pvt1,grid(id).pres); otherwise disp('assign PVT type') end end

functioncalculatewaterFVFnext function bw=Bwn(id,grid) P_pvt1=[500,550,600,650,700,750,800,850,900,950,1000,1050,1100,1150,1200,... 1250,1300,1350,1400,1450,1500,1550,1600,1650,1700,1750,1800,1850,1900, ... 1950,2000]; Bw_pvt1=[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]; switch grid(id).pvttype case 1 You're Reading a Preview bw=interp1(P_pvt1,Bw_pvt1,grid(id).propn.pres); otherwise Unlock full access with a free trial. disp('assign PVT type') end Download With Free Trial end

functiontocalculatewellBHP %requires precalculated Qo (use calcoratw) function Pw=calcbhp(wid,gridb,well) Qo=well(wid).Qo; WI=well(wid).WI; idb=well(wid).blockid; Po=gridb(idb).pres; Special offer for students: Only $4.99/month. Mo=moboi(idb,gridb);



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WI=well(wid).WI; idb=well(wid).blockid; Po=gridb(idb).propn.pres; Mo=moboi(idb,gridb); Pw=Po+(Qo/(0.001127*WI*Mo)); end

subroutinetocalculatejacobianmatrix function Jmat=calcjacobian(dt,dp,dsw,grid,well,nphase,ngrid,nwell) Jmat=zeros(nphase*ngrid); %dp=10; dsw=0.01; for i=1:ngrid %perturbate pressure of itself grid(i).propn.pres=grid(i).propn.pres-dp; grid(i).resok=reso(dt,i,grid,nwell,well); grid(i).reswk=resw(dt,i,grid,nwell,well); droi_dpi=(grid(i).resoi-grid(i).resok)/dp; drwi_dpi=(grid(i).reswi-grid(i).reswk)/dp; Jmat(2*i-1,2*i-1)=droi_dpi; Jmat(2*i,2*i-1)=drwi_dpi;

You're Reading a Preview %perturbate Sw of itself, return the pressure Unlock full access with a free trial. grid(i).propn.pres=grid(i).propn.pres+dp; grid(i).propn.Sw=grid(i).propn.Sw+dsw; grid(i).propn.So=1-grid(i).propn.Sw; Download With Free Trial grid(i).resok=reso(dt,i,grid,nwell,well); grid(i).reswk=resw(dt,i,grid,nwell,well); droi_dswi=(grid(i).resok-grid(i).resoi)/dsw; drwi_dswi=(grid(i).reswk-grid(i).reswi)/dsw; Jmat(2*i-1,2*i)=droi_dswi; Jmat(2*i,2*i)=drwi_dswi; grid(i).propn.Sw=grid(i).propn.Sw-dsw; grid(i).propn.So=1-grid(i).propn.Sw;

Master your semester with Scribd end for i=1:ngrid & The New York Times Special offer for students: Only $4.99/month.

for j=1:grid(i).ncon


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grid(idc).propn.pres=grid(idc).propn.pres+dp; grid(idc).propn.Sw=grid(idc).propn.Sw+dsw; grid(idc).propn.So=1-grid(idc).propn.Sw; grid(i).resok=reso(dt,i,grid,nwell,well); grid(i).reswk=resw(dt,i,grid,nwell,well); droi_dswc=(grid(i).resok-grid(i).resoi)/dsw; drwi_dswc=(grid(i).reswk-grid(i).reswi)/dsw; Jmat(2*i-1,2*idc)=droi_dswc; Jmat(2*i,2*idc)=drwi_dswc; grid(idc).propn.Sw=grid(idc).propn.Sw-dsw; grid(idc).propn.So=1-grid(idc).propn.Sw; end end end

functioncalculatewelloilrate function Qo=calcoratw(wid,gridb,well) idb=well(wid).blockid; if well(wid).type=='producer' if well(wid).mode==1 %oil rate constrain You're Reading a Preview Qo=5.615*well(wid).constrain.Qo; Unlock full access with a free trial. end if well(wid).mode==2 %bhp minimum constrain Jo=Joil(wid,well,gridb); Download With Free Trial Qo=-Jo*(gridb(idb).propn.pres-well(wid).constrain.bhp); end elseif well(wid).type=='injector' Qo=0; end end

Master your semester with Scribd functioncalculatewellwaterrate & The New York Times function Special offer for students: OnlyQw=calcwratw(wid,gridb,well) $4.99/month.


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end

subroutinecalculatefluidresidual function grid=calcresf(dt,grid,ngrid,well,nwell) for i=1:ngrid grid(i).resoi=reso(dt,i,grid,nwell,well); grid(i).reswi=resw(dt,i,grid,nwell,well); end end

functiontocalculateJoil function Jo=Joil(wid,well,gridb) idb=well(wid).blockid; Jo=0.00633*well(wid).WI*moboi(idb,gridb); end

functiontocalculatewellPIofwaterrate

You're Reading function Jw=Jwtr(wid,well,gridb)

a Preview

idb=well(wid).blockid; Unlock full access with a free trial. Jw=0.00633*well(wid).WI*mobwi(idb,gridb); end


functioncalculaterelpermoilnow

function kro=kroi(id,grid) %kro table for rock type 1 Sw_rpt1=[0.2,0.25,0.3,0.35,0.4,0.45,0.5,0.55,0.6,0.65,0.7,0.75,0.8,0.85,0.9]; Kro_rpt1=[1,0.92855,0.85712,0.78569,0.71426,0.64283,0.5714,0.49997,0.42854,0.3571  ,0.28568,0.21425,0.14282,0.07139,0]; Read Free Foron 30this Days Sign up to vote title  switch grid(id).rtype Useful  Not useful  case 1 Cancel anytime. Special offer for students: Only $4.99/month. kro=interp1(Sw_rpt1,Kro_rpt1,grid(id).Sw); otherwise


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,0.28568,0.21425,0.14282,0.07139,0]; switch grid(id).rtype case 1 kro=interp1(Sw_rpt1,Kro_rpt1,grid(id).propn.Sw); otherwise disp('assign grid rock type') end end

functioncalculaterelpermwaternext function krw=krwn(id,grid) %krw table for rock type 1 Sw_rpt1=[0.2,0.25,0.3,0.35,0.4,0.45,0.5,0.55,0.6,0.65,0.7,0.75,0.8,0.85,0.9]; Krw_rpt1=[0,0.071428571,0.142857143,0.214285714,0.285714286,0.357142857,0.42857142 9,0.5,0.571428571,0.642857143,0.714285714,0.785714286,0.857142857,0.928571429,1]; switch grid(id).rtype case 1 krw=interp1(Sw_rpt1,Krw_rpt1,grid(id).propn.Sw); otherwise disp('assign grid rock type') end end

You're Reading a Preview functioncalculaterelpermwaternow Unlock full access with a free trial. function krw=krwi(id,grid) %krw table for rock type 1 Download With Free Trial Sw_rpt1=[0.2,0.25,0.3,0.35,0.4,0.45,0.5,0.55,0.6,0.65,0.7,0.75,0.8,0.85,0.9]; Krw_rpt1=[0,0.071428571,0.142857143,0.214285714,0.285714286,0.357142857,0.42857142 9,0.5,0.571428571,0.642857143,0.714285714,0.785714286,0.857142857,0.928571429,1]; switch grid(id).rtype case 1 krw=interp1(Sw_rpt1,Krw_rpt1,grid(id).Sw); otherwise disp('assign grid rock type') end

Master your semester with Scribd & The New end York Times Special offer for students: Only $4.99/month.


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functiontocalc.watermobilityofevaluatedblock %use esp for wellblock function Mw=mobwi(id,gridb) Mw=krwn(id,gridb)/(Bwn(id,gridb)*viswn(id,gridb)); end %function to reset property when time step reset happens function propreset(ngrid,gridb) id=0; for id=1:ngrid gridb(id).propn.pres=gridb(id).pres; gridb(id).propn.Sw=gridb(id).Sw; gridb(id).propn.So=gridb(id).So; end end

functioncalculateoilresidual function ro=reso(dt,id,grid,nwell,well) fluxin=0; Cpor=1e-5; You're Reading a Preview Prpor=14.6923; Pen=grid(id).propn.pres; Unlock full access with a free trial. Pei=grid(id).pres; Vpi=grid(id).Vp*(1+Cpor*(Pei-Prpor)); %initial PV Download With Free Trial Vpn=grid(id).Vp*(1+Cpor*(Pen-Prpor)); for i=1:grid(id).ncon idc=grid(id).idcon(i); Pcon=grid(idc).propn.pres; grid(id).mobo(i)= moboil(id,idc,grid); fluxin=fluxin+(grid(id).trans(i)*grid(id).mobo(i)*(Pcon-Pen)); end accum = ((Vpn*grid(id).propn.So/Bon(id,grid))-(Vpi*grid(id).So/Boi(id,grid)))/d

Master your semester with Scribd for i=1:nwell if well(i).blockid==id & The New York Times Qo=calcoratw(i,grid,well); Special offer for students: Only ro=fluxin-accum+Qo; $4.99/month. break


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Mo=kro/(Bo*Uo); end

functioncalculateoilresidual function rw=resw(dt,id,grid,nwell,well) fluxin=0; Cpor=1e-5; Prpor=14.6923; Pen=grid(id).propn.pres; Pei=grid(id).pres; Vpi=grid(id).Vp*(1+Cpor*(Pei-Prpor)); %initial PV Vpn=grid(id).Vp*(1+Cpor*(Pen-Prpor)); for i=1:grid(id).ncon idc=grid(id).idcon(i); Pcon=grid(idc).propn.pres; grid(id).mobw(i)= mobwtr(id,idc,grid); fluxin=fluxin+(grid(id).trans(i)*grid(id).mobw(i)*(Pcon-Pen)); end accum = ((Vpn*grid(id).propn.Sw/Bwn(id,grid))-(Vpi*grid(id).Sw/Bwi(id,grid)))/dt; for i=1:nwell if id==well(i).blockid You're Reading a Preview Qw=calcwratw(i,grid,well); Unlock full access with a free trial. rw=fluxin-accum+Qw; break else Download With Free Trial rw=fluxin-accum; end end end function Mw=mobwtr(id,idc,grid) Bw=0.5*(Bwn(id,grid)+Bwn(idc,grid)); Uw=0.5*(viswn(id,grid)+viswn(idc,grid)); krw=0.5*(krwn(id,grid)+krwn(idc,grid));

Master your semester with Scribd & The New York Times Mw=krw/(Bw*Uw); Special offer for students: Only $4.99/month.


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