Petroleum Engineering Handbook

Petroleum Reservoir Simulation A Basic A p p r o a c h Jamal H Abou-Kassem Professor of Petroleum Engineering United Arab Emirates University AI-Ain, The United Arab Emirates S M Farouq Ali Petroleum Engineering Consultant PERL Canada Ltd Edmonton, Alberta, Canada M Rafiq Islam Professor and Killam Chair in Oil and Gas Dalhousie University Halifax, Nova Scotia, Canada @ Petroleum Reservoir Simulation: A Basic Approach Copyright © 2006 by Gulf Publishing Company, Houston, Texas All rights reserved No part of this publication may be reproduced or transmitted in any form without the prior written permission of the publisher HOUSTON, TX: Gulf Publishing Company Greenway Plaza, Suite 1020 Houston, T X 77046 AUSTIN, TX: 427 Sterzing St., Suite 104 Austin, T X 78704 1098765432 I Library of CongressCataloging-in-PublicationData Petroleum Reservoir Simulation: A Basic Approach/ Jamal H Abou-Kassem [et al.] p cm Includes bibliographical references and index ISBN 0-9765113-6-3 (alk paper) I Petroleum Simulation methods, manuals, etc Petroleum Mathematical models, manuals, etc Hydrocarbon reservoirs Simulation methods, manuals, etc Hydrocarbon reservoirs Mathematical models, manuals, etc Petroleum engineering Mathematics, manuals, etc I Abou-Kassem~ 3amal H (lama1 Hussein) TN870.57.R47 2006 553.2'8015118 dc22 2005O29674 Printed in the United States of America Printed on acid-free paper, oo Text design and composition by TIPS Technical Publishing, Inc We dedicate this book to our parents Preface The "Information Age" promises infinite transparency, unlimited productivity, and true access to knowledge Knowledge, quite distinct and apart from "know-how," requires a process of thinking, or imagination the attribute that sets human beings apart Imagination is necessary for anyone wishing to make decisions based on science Imagination always begins with visualization actually, another term for simulation Under normal conditions, we simulate a situation prior to making any decision, i.e., we abstract absence and start to fill in the gaps Reservoir simulation is no exception The two most important points that must not be overlooked in simulation are science and multiplicity of solutions Science is the essence of knowledge, and acceptance of the multiplicity of solutions is the essence of science Science, not restricted by the notion of a single solution to every problem, must follow imagination Multiplicity of solutions has been promoted as an expression of uncertainty This leads not to science or to new authentic knowledge, but rather to creating numerous models that generate "unique" solutions that fit a predetermined agenda of the decision-makers This book reestablishes the essential features of simulation and applies them to reservoir engineering problems This approach, which reconnects with the o l d - - o r in other words, time-tested concept of knowledge, is refreshing and novel in the Information Age The petroleum industry is known as the biggest user of computer models Even though space research and weather prediction models are robust and are often tagged as "the mother of all simulation," the fact that a space probe device or a weather balloon can be launched while a vehicle capable of moving around in a petroleum reservoir cannot-makes modeling more vital for tackling problems in the petroleum reservoir than in any other discipline Indeed, from the advent of computer technology, the petroleum industry pioneered the use of computer simulations in virtually all aspects of decision-making This revolutionary approach required significant investment in long-term research and advancement of science That time, when the petroleum industry was the energy provider of the world, was synonymous with its reputation as the most aggressive investor in engineering and science More recently, however, as the petroleum industry transited into its "middle age" in a business sense, the industry could not keep up its reputation as the biggest sponsor of engineering and long-term research A recent survey by the United States Department of Energy showed that none of the top ten breakthrough petroleum technologies in the last decade could be attributed to operating companies If this trend continues, major breakthroughs in the petroleum industry over the next two decades are expected to be in the areas of information technology and materials science When it comes to reservoir simulators, this latest trend in the petroleum industry has produced an excessive emphasis on the tangible aspects of modeling, namely, the number of blocks used in a simulator, graphics, computer speed, etc For instance, the number of blocks used in a reservoir model has gone from thousands to millions in just a few years Other examples can xi xii Preface be cited, including graphics in which flow visualization has leapt from 2D to 3D to 4D, and computer processing speeds that make it practically possible to simulate reservoir activities in real time While these developments outwardly appear very impressive, the lack of science and, in essence, true engineering render the computer revolution irrelevant and quite possibly dangerous In the last decade, most investments have been made in software dedicated to visualization and computer graphics with little being invested in physics or mathematics Engineers today have little appreciation of what physics and mathematics provide for the very framework of all the fascinating graphics that are generated by commercial reservoir simulators As companies struggle to deal with scandals triggered by Enron's collapse, few have paid attention to the lack of any discussion in engineering education regarding what could be characterized as scientific fundamentals Because of this lack, little has been done to promote innovation in reservoir simulation, particularly in the areas of physics and mathematics, the central topical content of reservoir engineering This book provides a means of understanding the underlying principles of petroleum reservoir simulation The focus is on basic principles because understanding these principles is a prerequisite to developing more accurate advanced models Once the fundamentals are understood, further development of more useful simulators is only a matter of time The book takes a truly engineering approach and elucidates the principles behind formulating the governing equations In contrast to cookbook-type recipes of step-by-step procedures for manipulating a black box, this approach is full of insights To paraphrase the caveat about computing proposed by R W Hamming, the inventor of the Hamming Code: the purpose of simulation must be insight, not just numbers The conventional approach is more focused on packaging than on insight, making the simulation process more opaque than transparent The formulation of governing equations is followed by elaborate treatment of boundary conditions This is one aspect that is usually left to the engineers to "figure out" by themselves, unfortunately creating an expanding niche for the select few who own existing commercial simulators As anyone who has ever engaged in developing a reservoir simulator well knows, this process of figuring out by oneself is utterly confusing In keeping up with the same rigor of treatment, this book presents the discretization scheme for both block-centered and point-distributed grids The difference between a well and a boundary condition is elucidated In the same breadth, we present an elaborate treatment of radial grid for single-well simulation This particular application has become very important due to the increased usage of reservoir simulators to analyze well test results and the use of well pseudo-functions This aspect is extremely important for any reservoir engineering study The book continues to give insight into other areas of reservoir simulation For instance, we discuss the effect of boundary conditions on materialbalance-check equations and other topics with unparalleled lucidity Even though the book is written principally for reservoir simulation developers, it takes an engineering approach that has not been taken before Topics are discussed in terms of science and mathematics, rather than with graphical representation in the backdrop This makes the book suitable and in fact essential for every engineer and scientist engaged in modeling and simulation Even those engineers and scientists who wish to limit their Preface xiii activities to field applications will benefit greatly from this book, which is bound to prepare them better for the Information Age Acknowledgments We, the authors, are grateful to many colleagues, friends, and students who have contributed to this book We thank Dr M S Osman, of the Kuwait Oil Company, for his contributions, fruitful discussions, and critiques at the various stages of writing this book, during his tenure at the UAE University We are indebted to Dr T Ertekin, of The Pennsylvania State University, and Dr R Almehaideb, of UAEU, for their reviews and comments on chapters of the book We thank the many students at UAEU who took the undergraduate reservoir simulation course during writing the book We thank Mr Othman N Matahen, of the UAEU Center of Teaching and Learning Technology, for his most skillful computer drafting of all figures in this book We are most deeply thankful to all our "teachers," from whom we have learned all that we know, and to members of our families, for their encouragement, their support, and most importantly their patience and tolerance during the writing of this book J H Abou-Kassem S M Farouq Ali M R Islam Introduction In this book, the basics of reservoir simulation are presented through the modeling of single-phase fluid flow and multi-phase flow in petroleum reservoirs using the engineering approach This text is written for senior-level B.S students and first-year M.S students studying petroleum engineering and aims to restore engineering and physics sense to the subject In this way, it challenges the misleading impact of excess mathematical glitter that has dominated reservoir simulation books in the past The engineering approach employed in this book uses mathematics extensively but injects engineering meaning to differential equations and to boundary conditions used in reservoir simulation It does not need to deal with differential equations as a means for modeling, and it interprets boundary conditions as fictitious wells that transfer fluids across reservoir boundaries The contents of the book can be taught in two consecutive courses The first, an undergraduate senior-level course, includes the use of a block-centered grid in rectangular coordinates in single-phase flow simulation This material is mainly included in Chapters 2, 3, 4, 6, 7, and The second, a graduate-level course, deals with a block-centered grid in radial-cylindrical coordinates, a point-distributed grid in both rectangular and radial-cylindrical coordinates, and the simulation of multiphase flow in petroleum reservoirs This material is covered in Chapters 5, 8, and 10 in addition to specific sections in Chapters 2, 4, 6, and (Secs 2.7, 4.5, 6.2.2) Chapter provides an overview of reservoir simulation and the relationship between the mathematical approach presented in simulation books and the engineering approach presented in this book In Chapter 2, we present the derivation of single-phase, multidimensional flow equations in rectangular and radial-cylindrical coordinate systems In Chapter 3, we introduce the Control Volume Finite Difference (CVFD) terminology as a means to writing the flow equations in multidimensions in compact form Then we write the general flow equation that incorporates both (real) wells and boundary conditions, using the blockcentered grid (in Chapter 4) and the point-distributed grid (in Chapter 5), and present the corresponding treatments of boundary conditions as fictitious wells and the exploitation of symmetry in practical reservoir simulation Chapter deals with wells completed in both single and multiple layers and presents fluid flow rate equations for different well operating conditions Chapter presents the explicit, implicit, and Crank-Nicolson formulations of single-phase, slightly compressible, and compressible flow equations and introduces the incremental and cumulative material balance equations as internal checks to monitor the accuracy of generated solutions In Chapter 8, we introduce the space and time treatments of nonlinear terms encountered in single-phase flow problems Chapter presents the basic direct and iterative solution methods of linear algebraic equations used in reservoir simulation Chapter 10 is entirely devoted to multiphase flow in petroleum reservoirs and its simulation The book concludes with Appendix A, which presents a user's manual for a singlephase simulator The CD that accompanies the book includes a single-phase simulator XV xvi Introduction written in F O R T R A N 95, a compiled version, a users' manual, and data and output files for four solved problems The single-phase simulator provides users with intermediate results as well as a solution to single-phase flow problems so that a user's solution can be checked and errors are identified and corrected Educators m a y use the simulator to make up new problems and obtain their solutions Nomenclature a n = coefficient of unknown Xn,n.,,, , B i = fluid formation volume factor in defined by Eq 9.46f A block i , RB/STB [m3/std m 3] = parameter, defined by Eq 9.28 in Bo Tang' s algorithm RB/STB [m3/std m 3] IAI = square coefficient matrix Bob = cross-sectional area normal to x- Ax [m3/std m 3] Axlx = cross-sectional area normal to x- Bei = formation volume factor of phase p direction at x, ft [me1 in block i = cross-sectional area normal to Bw x-direction at x + A x , ft [m 2] Ax Ix,_,,2 = = water formation volume factor, RB/B [m3/std m 3] cross-sectional area normal to B ° = fluid formation volume factor at x-direction at block boundary reference pressure p° and x~+l/2 , ft [m 2] b = oil formation volume factor at bubble-point pressure, RB/STB direction, ft [m 2] Axlx+x = oil formation volume factor, reservoir temperature, RB/STB [m3/std m 3] = reservoir boundary b L, = reservoir east boundary c = fluid compressibility, psi -1 l k P a - l l b L = reservoir lower boundary c e = coefficient of unknown of block i bN = reservoir north boundary bs = reservoir south boundary in Thomas' algorithm c n = coefficient of unknown x , , defined by Eq 9.46g b U = reservoir upper boundary bw b = reservoir west boundary cu = coefficient of unknown x N in Thomas' or T a n g ' s algorithm = coefficient of unknown x,_,~,,r , defined by Eq 9.46a co = oil-phase compressibility, psi -1 [kPa -1 ] B = parameter, defined by Eq 9.29 in c¢ = porosity compressibility, psi-1 [kPa-1] Tang' s algorithm c a = rate of fractional viscosity change B = fluid formation volume factor, RB/STB [m3/std m ] with pressure change, psi -1 [kPa -1] = average fluid formation volume factor C in wellbore, RB/STB [m3/std m 3] = parameter, defined by Eq 9.30 in Tang' s algorithm Bg = gas formation volume factor, CMe RB/scf [m3/std m 3] = cumulative material balance check, dimensionless xvii Nomenclature xviii Cop = coefficient of pressure change over F~ = ratio of wellblock i area to time step in expansion of oil theoretical area from which well accumulation term, STB/D-psi withdraws its fluid (in Chapter 6), [std m3/(d.kPa)] fraction Cow = coefficient of water saturation F m = argument of an integral evaluated at time t m change over time step in expansion of oil accumulation term, STB/D F(t m) = argument of an integral [std m3/d] evaluated at time t" C,~p = coefficient of pressure change over F ° = argument of an integral evaluated time step in expansion of water accumulation term, B/D-psi at time t" F(t") = argument of an integral [std m3/(d.kPa) ] evaluated at time t" Cww = coefficient of water saturation F n+l = argument of an integral evaluated at time t n+l change over time step in expansion of water accumulation term, B/D F(t n+l) = argument of an integral [std m3/d] evaluated at time t "+~ ~¢ = vector of known values D F n+1/2 = argument o f an integral = parameter, defined by Eq 9.31 in Tang' s algorithm evaluated at time t "~ln F(t n+l/z) = argument o f an integral di = known RHS of equation for evaluated at time t "+1/2 block i in Thomas' algorithm g between two successive iterations d gi = element i of a temporary vector = RHS of equation for gridblock n, (~) generated in Thomas' defined by Eq 9.46h algorithm e~ = coefficient of unknown of G block i + in Thomas' algorithm e,, = coefficient o f unknown Xn+1 = G wellblock i , defined by Eq 6.32, Tang' s algorithm fp = the pressure dependent term in transmissibility f ,p,,,~ l = nonlinearity, defined by Eq 8.17 F(t) = argument of an integral at time t = well geometric factor, RB-cp/D-psi Gw, = well geometric factor for coefficient of unknown x~ in f ( ) = function of = geometric factor [m3.mPa.s/(d.kPa)] , defined by Eq 9.46d eu = gravitational acceleration, ft/sec [m]s 2] dm~ = m a x i m u m absolute difference RB-cp/D-psi [m3.mPa.s/(d.kPa)] G* = well geometric factor of the theoretical well for wellblock i , RB-cp/D-psi [m3.mPa.s/(d.kPa)] Gyi = interblock geometric factor between block i and block i -T-1 426 Appendix A User's Manual for Single-Phase Simulator DATA 06A Reservoir Description and Initial Pressure Distribution to DATA 21A Lower and upper limits in the xdirection of a parallelepiped region or the rdirection of a I1, 12 reservoir region in single-well simulation Lower and upper limits in the ydirection of a parallelepiped region; for single-well simulaJ1,J2 tion, set J1 = J2 = Lower and upper limits in the zdirection of a parallelepiped region or a reservoir region in K1, K2 single-well simulation IACTIVE DX DY DZ KX KY KZ DEPTH Block indicator for active and inactive blocks = O, inactive gridblock or gridpoint = -1, inactive gridblock or gridpoint to identify constant pressure block = 1, active gridblock or gridpoint Block size in the xdirection for block-centered grid (or gridpoint spacing in the xdirection for point-distributed grid), ft [m] Block size in the ydirection for block-centered grid (or gridpoint spacing in the ydirection for point-distributed grid), ft [m] Block size in the zdirection for block-centered grid (or gridpoint spacing in the zdirection for point-distributed grid), ft [m] Block permeability in the xor r direction, md [pm 2] Block permeability in the y direction if NY > O, md [pm 2] Block permeability in the z direction, md [pm 2] Elevation of top of gridblock for block-centered grid (or elevation of gridpoint for pointdistributed grid) below selected datum, ft [m] PHI P Block porosity, fraction Block pressure, psia [kPa] RATIO Property modifier, dimensionless = 0.0, property is not modified > 0.0, property is increased by that ratio < 0.0, property is decreased by that ratio NOTE A number of gridblocks (or gridpoints) that are part of the reservoir are deactivated on purpose in order to simulate a specified gridblock (or gridpoint) pressure DX, DY, and DZ are supplied for all gridblocks (or gridpoints) whether active or inactive Ratio is the desired fractional change of a property value entered by the user or internally calculated by the simulator Modifiers can be applied to the block porosity, block elevation, block bulk volume, and transmissibilities in the x, y, and z directions For a point-distributed grid, define the gridpoint spacing in a given direction (DX in i direction, DY in jdirection, DZ in kdirection) by setting the upper limit of a parallelepiped region in that direction only equal to the coordinate of the upper limit gridpoint in the same direction minus one A.3 Description of Variables Used in Preparing a Data File 427 DATA22B Rock Data and Fluid Density CPHI Porosity compressibility, psi-1 [kPa-1] PREF RHOSC Reference pressure at which porosities are reported, psia [kPa] Fluid density at reference pressure and reservoir temperature, Ibm/ft [kg/m 3] D~A~B Type of Fluid in the Reservoir Type of fluid indicator LCOMP = 1, incompressible fluid = 2, slightly compressible fluid = 3, compressible fluid (natural gas) IQUAD Interpolation within gas property table = 1, linear interpolation = 2, quadratic interpolation DATA 24B Fluid Properties for LCOMP = (Incompressible Fluid) FVF Formation volume factor at reservoir temperature, RB/STB MU Fluid viscosity, cp [mPa.s] DATA 24B Fluid Properties for LCOMP = (Slightly Compressible Fluid) FVFO Formation volume factor at reference pressure and reservoir temperature, RB/STB MUO CO CMU PREF MBCONST Fluid viscosity at reference pressure and reservoir temperature, cp [mPa.s] Fluid compressibility, psi-1 [kPa-1] Rate of relative change of viscosity with respect to pressure, psi-1 [kPa-1] Reference pressure at which FVFOand MUO are reported, psia [kPa] Handling of liquid FVF and liquid viscosity in transmissibility terms = 1, constant values independent of pressure = 2, values that depend on pressure DATA 24C PRES FVF MU NOTE Fluid Properties for LCOMP = (Natural Gas) Pressure, psia [kPa] Gas formation volume factor, RB/scf [m3/std m 3] Gas viscosity, cp [mPa.s] Gas FVF and viscosity are supplied in a table form The pressure is entered in increasing order using equal intervals DATA 25A Boundary Conditions I1, 12 Lower and upper limits in the xdirection of a parallelepiped region or the rdirection of a reservoir region in single-well simulation J1,J2 Lower and upper limits in the ydirection of a parallelepiped region; for single-well simulation, set J1 = J2 = 428 Appendix A User's Manual for Single-Phase Simulator K1, K2 Lower and upper limits in the zdirection of a parallelepiped region or a reservoir region in single-well simulation IFACE Block boundary subject to boundary condition = 1, block boundary in the negative direction of zaxis = 2, block boundary in the negative direction of yaxis = 3, block boundary in the negative direction of xaxis or rdirection = 5, block boundary in the positive direction of xaxis or r direction = 6, block boundary in the positive direction of yaxis ITYPBC = 7, block boundary in the positive direction of zaxis Type of boundary condition = 1, specified pressure gradient at reservoir boundary, psi/ft [kPa/m] = 2, specified flow rate across reservoir boundary, STB/D or scf/D [std m3/d] = 3, no-flow boundary = 4, specified pressure at reservoir boundary, psia [kPa] = 5, specified pressure of the block on the other side of reservoir boundary, psia [kPa] SPVALUE Specified value of boundary condition ZELBC Elevation of center of boundary surface for block-centered grid (or elevation of boundary node for point-distributed grid) below selected datum, ft [m] RATIO Property modifier for area open to flow or geometric factor between reservoir boundary and boundary gridblock (or gridpoint), dimensionless = 0.0, property is not modified > 0.0, property is increased by that ratio < 0.0, property is decreased by that ratio NOTE All reservoir boundaries are assigned a no-flow boundary condition as a default Therefore, there is no need to specify no-flow boundaries For ITYPBC = 5, ZELBC is the elevation of the point (node) that represents the block whose pressure is specified DATA 25A has no option identifier at the beginning of the parameter sequence line (line 2) For single-well simulation using point-distributed grid, a specified FBHP must be simulated as a specified pressure boundary condition DATA26D Well Recursive Data NOW Number of wells that will change operational conditions SIMNEW DELT PWFMIN = O, no change in well operations > O, number of wells that change operational conditions Time specification signaling user's new request, D [d]; well data entered here will be active starting from previous time specification until this time specification and beyond Time step to be used, D [d] Minimum BHP allowed for production well, psia [kPa] A.3 Description of Variables Used in Preparing a Data File 429 PWFMAX Maximum BHP allowed for injection well, psia [kPa] IDW Well identification number; each well must have a unique IDW =1,2,3,4 IW, JW, KW L j, k location of wellblock IWOPC Well operating condition IWOPC for production well = -1, specified pressure gradient at well, psi/ft [kPa/m] = -2, specified production rate, STB/D or scf/D [std m3/d] = -3, shut-in well -4, specified bottom-hole pressure, psia [kPa] IWOPC for injection well = 1, specified pressure gradient at well, psi/ft [kPa/m] = 2, specified injection rate, STB/D or scf/D [std m3/d] = = 3, shut-in well = 4, specified bottom-hole pressure, psia [kPa] GWl Wellblock/geometric factor, RB-cp/D-psi [m3.mPa.s/(d.kPa)] SPVALUE Specified value of the operating condition RADW NOTE Well radius, ft [m] The NOW line can be repeated for different times, but each subsequent line must have a time specification larger than the previous time specification 2.The NOW line can be used to specify new values for DELT, PWFMIN, or PWFMAX at desired times during simulation 3.The specified value of PWFMIN and PWFMAX must be within the range of the pressure specified in the PVT table For realistic simulation of slightly compressible and compressible fluids, these two parameters need to be specified However, setting PWFMIN < -10 and PWFMAX > 106 deactivates the function of these two parameters 4.This data group terminates with a line o zero entries Each IWD line enters specifications for r~ne well This line must be repeated NOW times if NOW > O Both IWOPC and the specified rate are positive for injection and both are negative for production For single-well simulation using point-distributed grid, a specified FBHP must be simulated as a specified pressure boundary condition RADW is specified here to handle options IWOPC = or -1 430 A.4 AppendixA User'sManual for Single-Phase Simulator Instructionsto Run Simulator The user of the simulator is provided with a copy of a reference data file (e.g., REFDATA.TXT) similar to the one presented in Section A.6 The user first copies this file into a personal data file (e.g., MY-DATA.TXT) and then follows the instructions in Section A.2 and observes the variable definitions given in Section A.3 to modify the personal data file such that it describes the constructed model of the reservoir under study The simulator can be run by clicking on the compiled version (SinglePhaseSim.exe) that is provided on the accompanying CD The computer responds with the following statement requesting file names (with file type) of one input and four output files: ENTERNAMESOF INPUTAND OUTPUTFILES 'DATA.TXT' '0UT1.LIS' '0UT2.LIS' '0UI-3.LIS' '0UT4.LIS' The user responds using the names of five files, each enclosed within single quotes separated by a blank space or a comma as follows and then hits the "Return" key 'MY-DATA.TXT','MY-0UT1.LIS','MY-0UT2.LIS','MY-OUT3ilS','MY-0UT4.LIS' The computer program continues execution until completion Each of the four output files contains specific information 'MY-OUT1.LIS' contains debugging information of the input data file if requested and a summary of the input data, block pressure, production and injection data including rates and cumulative rates of fluid across reservoir boundaries, and material balance checks for all time steps ' M Y OUT2.LIS' reports intermediate results, equations for all blocks, and details specific to the linear equation solver every iteration in every time step 'MY-OUT3.LIS' contains concise reporting in tabular form of block pressures at various times 'MY-OUT4.LIS' contains concise reporting in tabular form of reservoir performance as well as individual well performances The accompanying CD contains four output files generated by running the computer program for each of the four data files ex7-1.txt, ex7-7.txt, ex7-12.txt, and ex55.txt available on the CD A.5 Limitations Imposed on the Compiled Version The compiled version of SinglePhaseSim is provided here for demonstration and student training purposes The critical variables were therefore restricted to the dimensions given below: Number of gridblocks (or gridpoints) in x or r direction < 20 Number of gridblocks (or gridpoints) in y direction < 20 Number of gridblocks (or gridpoints) in z direction < 10 Number of entries in PVT table < 30 Number of wells = well/block Unrestricted number of times wells change operational conditions Maximum number of time steps = 1000 (precautionary_measure) A.6 Example of a Prepared Data File A.6 Example of a Prepared Data File The following data file was prepared as a benchmark test problem '*DATA 01E* T i t l e o f Simulation Run' 'TITLE' ].H, Abou-Kassem, Input data f i l e f o r Example 7,1 in Chap 7.Chap 7-1 '*DATA 02B* Simulation Time Data' 'IPRDAT TMTOTAL TMSTOP' 36O 10 '*DATA 03B* U n i t s ' 'MUNITS' '*DATA 04B* Control Integers f o r P r i n t i n g Desired Output' 'BORD MLR BASIC OBC EOS PITER ITRSOL' 1 1 1 '*DATA 05B* Reservoir D i s c r i t i s a t i o n and Method o f Solving Equations' 'IGRDSYS NX NY NZ RW RE NONLNRLEQSM TOLERSP DXTOLSP' 1 0.25 526.604 0.0 0.0 '*DATA O6A* RESERVOIR REGION WITH ACTIVE OR INACTIVE BLOCK IACTIVE' 1~'I1 12 J1 J2 K1 K2 IACTIVE' 14 11 11 O0 O0 O0 '*DATA 07A* RESERVOIR REGION HAVING BLOCK SIZE DX IN THE X-DIRECTION' I , ' 1 12 ]1 J2 K1 K2 DX (FT)' 1 1 300 O0 O0 O0 0.0 ' * D ~ A OBA* RESERVOIR REGION H~ING BLOCK SIZE DY 1N THE Y-DIRECTION' 0,'11 12 J1 32 K1 K2 DY (FT)' 350 350 350 350 '*DATA 09A* RESERVOIR REGION HAVING BLOCK SIZE DZ IN THE Z-DIRECTION' 0,'Ii 12 ]1 ]2 K1 K2 DZ (FT)' 4*40 '*DATA IOA* RESERVOIR REGION H~ING PERMEABILITY KX IN THE X-DIRECTION' , ' 1 12 31 32 K1 K2 KX (MD)' 1 i 270 O0 O0 O0 0.0 '*DATA 11A* RESERVOIR REGION H~ING PERMEABILITY KY IN THE Y-DIRECTION' I , ' I I 12 J l ]2 K1 K2 KY (MD)' 14 11 11 O0 O0 O0 0.0 '*DATA 12A* RESERVOIR REGION HAVING PERMEABILITY KZ IN THE Z-DIRECTION' , ' 1 12 ]1 ]2 K1 K2 KZ (MD)' 14 11 11 O0 O0 O0 0.0 '*DATA 13A* RESERVOIR REGION HAVING ELEVATION Z' I , ' 1 12 ] I 32 KI K2 DEPTH (FT)' 14 11 11 0.0 O0 O0 O0 0.0 '*DATA 14A* RESER~IR REGION HAVING POROSITY PHI' 1,' 11 12 31 ]2 K1 K2 PH1 (FRACTION)' 1 1 0.27 O0 O0 O0 0.0 431 Appendix A 432 '*DATA 15A* , ' I 12 0 '*DATA 16A* i~'11 I2 0 '*DATA 17A* , ' I 12 0 '*DATA 18A* , ' I 12 o '*DATA 19A* I~'Ii 12 o o User's Manual for Single-Phase Simulator RESERVOIR REGION HAVING INITIAL PRESSURE P' ]1 ]2 K1 K2 P (PSIA)' I 1 o 0 0 0.0 RESERVOIR REGION WITH BLOCK POROSITY MODIFICATION RATIO' ]I ]2 K1 K2 RATIO' 0 0 0.0 RESERVOIR REGION WITH BLOCK ELEVATION MODIFICATION RATIO' 31 ]2 K1 K2 RATIO' 1 1 0.0 0 0 0o0 RESERVOIR REGION WITH BLOCK VOLUME MODIFICATION RATIO' 31 ]2 K1 K2 RATIO' 1 1 0.0 0 0 0.0 RESERVOIR REGION WITH X-TRANSMISSIBILITY MODIFICATION RATIO' ]i ]2 K1 K2 RATIO' 1 1 0.0 o o o o O.O '*DATA 20A* RESERVOIR REGION WITH Y-TRANSMISSIBILITY MODIFICATION RATIO' I,'Ii 12 31 32 K1 K2 RATIO' 1 1 0.0 0 0 0 0.0 '*DATA 21A* RESERVOIR REGION WITH Z-TRANSMISSIBILITY MODIFICATION RATIO' , ' I 12 31 32 K1 K2 RATIO' 1 1 0.0 0 0 0 0.0 '*DATA 22B* Rock and Fluid Density' ' CPHI PREF RHOSfi' 0.0 14.7 50.0 '*DATA 23B* Type of Fluid in the Reservoir' ' LCOMP IOUAD' 1 '*DATA 24B* FOR LCOMP= AND OR *DATA 24C* FOR LCOMP= ENTER FLUID PROP' 'LCOMP=I:FVF,MUjLCOMP=2:FVFO~MUO,CO~CMU~PREF~MBCONST;LCOMP=3:PRES,FVF,MU TABLE' 1.0 0.5 '*DATA 25A* Boundary Conditions' RATIO' '11 12 ]1 ]2 K1 K2 1FACE ITYPEBC SPVALUE Z E L B C 4OOO 2O 0.0 1 1 1 3 20 0,0 4 1 1 O, 0,0 0.0 0 0 0 '*DATA 26D* Well Recursive Data' ' NOW SIMNEW DELT PWFMIN PWFMAX' ' IDW IW ]W KW IWOPC GWI SPVALUE RADW' 10.0 -10000000.0 100000000,0 10.0 1 -2 11,0845 -600 0.25 0,0 0,0 0.0 0,0 A u t h o r Index Abou-Kassem, J.H., 2, 4-5, 12, 43, 79, 87, 88, 91-92, 111,153-154, 169, 183, 194, 198, 204, 236-337,360-361,365, 395,400, 414, 421,422, 433-434 Appleyard, J.R., 4, 361,433 Aziz, K., 2, 3, 4, 18, 43, 91,118, 153, 154, 183, 198, 327, 337, 339, 350, 360, 365, 433,435 Babu, D.K., 186, 433 Bear, J., 3,433 Behie, A., 4, 361,434 Belhaj, H., 17, 434 Bentsen, R.G., 209, 434 Berry, D.W., 289, 407, 435 Biazar, J., 17, 434 Breitenbach, E.A., 4, 434 Cheshire, I.M., 4, 361,433 Coats, K.H., 4, 12, 237, 266, 289, 299, 323,372, 383,407, 420, 434-435 Dranchuk, P.M., 209, 434 Eakin, B.E., 209, 434 Ertekin, T., 4-5, 12, 43, 87, 88, 91-92, 111,153-154, 169, 183, 186, 194, 236-337, 360-361,365, 395,400, 414, 421,433-434 Farouq Ali, S.M., 2-4, 43, 90, 91,433-434 George, W.D., 237, 288, 372, 420, 434 Gonzalez, M.H., 209, 434 Gupta, A.D., 4, 434 Gustavson, S.G., 12, 435 Hillestad, J.G., 4, 435 Hoffman, J.D., 238, 434 Islam, M.R., 2, 17, 79, 209, 433-434 Keast, P., 238,434 King, G.R., 4-5, 43, 87, 92, 153, 169, 183, 194, 237, 337, 361,365,395,414, 421, 433-434 Lee, A.L., 209, 434 Leverett, M.C., 437, 434 Lewis, W.B., 437,434 Lutchmansingh, P.M., 43, 111,169, 433-434 Ma, F., 17,434 MacDonald, R.C., 289, 407, 434 Marcum, B.E., 237, 288, 372, 420, 434 McDonald, A.E., 12, 434 Mitchell, A.R., 238,434 Mustafiz, S., 17,434 Nolen, J.S., 289, 407, 435 Odeh, A.S., 1, 2, 186, 433,435 Osman, M.E., 2, 79, 421,433 Peaceman, D.W., 185, 199, 358, 435 Pedrosa, Jr., O.A., 153, 154, 435 Price, H.S., 12, 435 Rachford, Jr., H.H., 358,435 Ramesh, A.B., 4, 299, 323,434 Roberts, S.J., 12, 435 Saad, N., 4, 435 Settari, A., 2, 3, 4, 18, 43, 91,118, 153, 154, 183, 198, 327, 337, 339, 350, 360, 365,433, 435 Sheffield, M., 4, 435 Spillette, A.G., 4, 435 Stone, H.L., 4, 370-373,435 Tang, I.C., 325,329-334, 361,425,435 Thomas, G.W., 4, 435 Thurnau, D.H., 4, 434, 435 Trimble, R.H., 12, 434 van Poollen, H.K., 4, 434 Vinsome, P.K.W., 4, 361,434, 435 Winestock, A.G., 4, 299, 323, 434 Woo, P.T., 12, 435 Zaid, A.M., 421,433 437 Subject Index radial grid and, 92 rectangular grid and, 75 reservoir discretization and, 63 well geometric factor and, 184, 186 block identification, 11 block ordering, 12 engineering notation, 11 block successive over-relaxation (BSOR), 356-358 boundary conditions, 2, 3, 6, 63, 66, 68, 123, 126, 128, 182, 207, 209, 211, 365, 428 linearization of, 293,407 material balance check and, 212, 240-241, 267-268, 414 multiphase flow and, 393,403-405 no-flow, 77-78, 139, 405 single-phase flow and, 75-79, 136-142 single-well simulation and, 75, 136, 183 specified flow rate, 76-77, 138,404 specified boundary gridblock pressure, 79 specified boundary gridpoint pressure, 141-142 specified boundary pressure, 78-79, 139-141,405 specified pressure gradient, 75-76, 136-138, 403-404 symmetry and, 111,116, 169, 170 A accumulation terms, 2, 3, 18, 19, 22, 66, 127, 167, 207, 210, 241,365, 393, 416 block pressure and, 235,264-265,284, 289, 294-295 coefficient of, 294-295 expansion of, 299-300, 393-396 linearization and, 293 solution method and, 408-410 alternating-direction-implicit procedure (ADIP), 358-360 anisotropic permeability, 8, 23, 36, 153, 185, 186, 199 geometric factor and, 87, 90, 153 wellblock geometric factor, 185-186 aquifer, 74, 76, 78, 136, 138, 403-405 areal discretization See reservoir discretization areal grids radial-cylindrical, 28-29 rectangular See Cartesian grid B backward-central-difference, 3, 235,265 See also implicit formulation backward-difference, 5,235 black oil model, 3, 376, 414, 415-416, 422 boundary conditions and, 403-405 capillary pressures and, 373-374 flow equations and, 390-391 fluid properties and, 366-367 formulation of, 383 initial conditions and, 414 material balance check and, 414 relative permeability and, 370-371 solution and, 414-415 block-centered grid, 63-65, 90-91, 116, 123, 128, 212, 421,424, 426, 428 geometric factors and, 36, 88, 285-286, 288, 290 C capillary pressure, 365, 373-374, 383-384, 404, 415,416 1MPES method and, 410 411 SS method and, 411 Cartesian coordinates, 7, 11, 28, 36, 54-55, 63, 67, 123 boundary conditions and, 75, 136 CVFD and, 43 flow equation and, 15 transmissibility and, 78, 153 439 440 Subject Index Cartesian grid block-centered, 63-64 point-distributed, 123-125 central-difference, 5, 235 compressible fluid, 208-209, 276, 338, 425 accumulation term and, 264-265 flow equation and, 266-267 fluid properties and, 208, 287 formulation and, 265-267 material balance check and, 267-268 nonlinearity and, 287-288 pressure solution and, 267 conservation of mass See mass conservation constitutive equation See Darcy's Law control volume, 2-5 control volume finite difference (CVFD) method, 43, 127, 207, 209, 334, 365, 382 block ordering and, 49, 57 engineering notation and, 44 flow equation and, 44, 50-51, 54-57 natural ordering and, 49-50 Crank-Nicolson formulation, 1, 20, 36, 207, 236 material balance check and, 242 multiphase flow equations and, 382 single-phase flow equations and, 237-238, 266-267 cylindrical coordinates, 28-29 D Darcy's Law, 3, 7, 17, 36, 91,380 multiphase flow and, 370, 374-376 radial flow and, 154, 184 single-phase flow and, 10 density, 7, 366 dimensionality, 43, 58, 68-69, 129, 181 direct solution methods, 325 1D rectangular flow problem and See Thomas' algorithm 1D tangential flow problem and See Tang' s algorithm 2D and 3D flow problem and See sparse matrices Gaussian elimination and, 336-337 discretization, 1-5, 7, 8-9, 28, 63-65, 98, 116, 123-125, 157, 376, 424 grid systems and See reservoir discretization E elementary volume See control volume engineering approach, 2-5, 7, 36, 66, 127, 183, 237, 266, 283, 292, 425 boundary conditions and, 66, 127, 183 flow equations and, 15-20 formulation methods and See explicit formulation; implicit formulation; Crank-Nicolson formulation well rate linearization and, 283, 292-293 equation of state, 10 equations ADIP, 358-360 black-oil model, 390-391 block-centered cylindrical grid, 91-92, 98 block-centered rectangular grid, 63-64 boundary conditions, 75-79, 136-142, 403 405 BSOR, 356-358 capillary pressure, 373 compressible flow, 265-267 conservative expansion, 395-396 Crank-Nicolson formulation, 237-238, 266-267 Darcy's Law, 10, 374-375 explicit formulation, 236, 265-266 fluid gravity, 7, 207 fluid injection, 400403 fluid potential, 11,374 fluid production, 397-399 fluid viscosity, 208-209 formation volume factor, 208 gas/water flow model, 385-386 Ganssian elimination, 336-337 Gauss-Seidel iteration, 341-342 general flow, 67 IMPES method, 410411 implicit formulation, 237, 266 incompressible flow, 210 integral approximation, 18-19 geometric factors, 88, 91, 99, 153, 154, 157 IPR, 185-186 Jacobi iteration, 338-339 Subject Index linearized flow, 293-299 LSOR, 350-351 mass accumulation term, 31,377 material balance, 15, 376 material balance check, 211-212, 240-242, 267-268,414 multiblock wells, 192-195 multiphase flow, 382-383 oil/gas flow model, 387-388 oil/water flow model, 384 optimum over-relaxation parameter, 345-346 point-distributed cylindrical grid, 155, 156-157 point-distributed rectangular grid, 123-124 porosity, potential differences, 11,374 production rate linearization, 292-293 PSOR, 345-346 single-phase flow rate, 182-185 single-well simulation, 28-29, 35, 54-57, 98-99, 156-157, 183-185 slightly compressible flow, 235-238 SS method, 411-414 Stone' s three-phase model, 370-371 Tang's algorithm, 329-332 Thomas' algorithm, 325-328 transmissibility, 17, 23, 35, 87, 90, 152, 153, 383 transmissibility linearization, 289-291 viscosity, 208-209 wellblock geometric factor, 185-186, 197-200 well rate linearization, 292-293 explicit formulation, 236 multiphase flow equations and, 382 single-phase flow equations and, 236, 265-266 F fictitious we11,2,66-75,127-136,182-183, 211-212,239-240,283-288, 293-294, 297-299, 320, 365, 382,393,403,407,411,415, 417 See also boundary condifions 441 finite-difference equations, 2, 42, 79, 235 See also flow equation flow equation, 2-5, 7, 11, 15, 22-23, 30, 35-36 CVFD and, 43, 49, 54, 57 multiphase and, 381-383 flow geometry and, 28 flowing bottom-hole pressure (FBHP), 181, 183-185, 195, 212, 293, 296, 398, 399, 401-403, 428 multiblock wells and, 193-194, 203-204, 397 fluid injection/production multiblock wells, 192 single block wells, 181 fluid properties black oil description, 366-367 capillary pressure, 373-374 compressibility, 366 FVF, 366-367 multiphase flow, 375-376 potential, 374 relative permeability, 370-371 saturation, 365 single-phase, 7-8, 208-209 solution gas/oil ratio and, 367 three-phase relative permeability, 370-371 viscosity, 208-209, 366 fluid volumetric velocity, 10 multiphase flow and, 374-375 radial-cylindrical flow and, 33 single-phase flow and, 16, 33 formation volume factor (FVF), 36, 235, 241, 264, 285,427 multiphase flow and, 366-367 single-phase flow and, 7,208-209 slightly compressible flow and, 208 forward-central-difference, 3, 235, 265 See also explicit formulation forward-difference, 5,235 fully implicit method, 4, 289, 291-293, 297-300, 320, 365,408, 411 G g-band algorithm See Gaussian elimination gas, 376, 388 free, 376, 379, 382 FVF and, 208 442 Subject Index multiphase flow and, 376, 390 solution, 376, 379, 383 gas-cap, 403 gas/oil contact (GOC), 403, gas/oil ratio, 367 Gaussian elimination, 336-337 Gauss-Seidel iteration, 341-342 gravity, 7, 21, 51, 79, 141,207, 237, 266-267, 285 grid system, 8, 207, 424 block-centered, 63 boundary conditions and, 73-79, 135-142 radial coordinates and, 28, 90, 153 point-distributed, 123 properties of, 63-64, 90-91,124, 153-154 Cartesian coordinates and, 28, 123 transmissibility and, 87, 90, 152-153 H heterogeneous, 8, 110, 111,212, 423 homogeneous, 8, 17, 23, 35, 36, 111,242, 422 implicit-pressure-explicit-saturation (IMPES) method, 4, 365,408, 416, 410 implicit formulation, 36, 235, 237, 238, 240, 265, IMPES method and, 410 material balance check and, 240, 267,414 multiphase flow equations and, 382-383 single-phase flow equations and, 237, 266 SS method and, 411 transmissibility and, 291 incompressible fluid, 208 accumulation term and, 210 flow equation and, 210 fluid properties and, 208 material balance and, 211-212 pressure solution and, 211 inflow performance relationship (IPR), 185 initial conditions, 111,169, 240, 276 multiphase flow and, 414-415 single-phase flow and, 238, 267 injection, 1, 2, 36, 66, 75, 111,127, 136, 169, 204, 211,240, 365, 393 in multiblock wells, 400-403 in single block wells, 181-182 isotropic permeability, 8, 186, 212 iterative methods, 4, 325, 337-338 ADIP, 358-360 BSOR, 356-358 Jacobi, 338-339 Ganss-Seidel, 341-342 LSOR, 350 351 PSOR, 345-346 J K Jacobi's iterative method, 338-339 L law of mass conservation multiphase flow and, 376 single-phase flow and, 15 See also mass balance line successive over-relaxation (LSOR) method, 350 linear algebraic equation, 1-5, 36, 212, 283, 285-288, 421 linear algebraic equation solvers, 325-361 ADIP, 358 BSOR, 356 direct methods, 325 Ganss-Seidel iteration, 341 Gaussian elimination, 336 iterative methods, 337 Jacobi method, 338 LSOR, 350 PSOR method, 345 Tang' s algorithm and, 329 Thomas' algorithm and, 325 linear difference equation See linear algebraic equation linearization, 1-2, 87, 153, 266, 288-292 fictitious well rate terms and, 293 flow equations and, 293-300 flow terms and, 283 implicit formulation and, 297-300 in space, 290 in time, 266, 289, 291 of week nonlinearities, 283 of well rates, 292-293 transmissibility and, 289-291 Subject Index M mass accumulation term, 15, 377 mass balance, 376 See also material balance mass conservation, 9-10 mass flux, 15, 31,377 mass rate, 15, 31- 32, 376 material balance, 7, 15, 30, 36, 77, 138, 240, 242, 264, 289-300, 375-376, 393,411,424 mathematical approach, 2, 4-5, 79, 235-237, 266, 283, 292, 425 matrix, 327, 330, 337 bitridiagonal, 412-413 sparse, 334, 336, 357 tridiagonal, 327, 336-337, 411-414 multiblock wells, 192, 204, 365, 398, 402 injection and, 400-403 production and, 397-399 vertical effects and, 192-193, 397-398 wellbock rate and, 193-195 multiphase flow, 7, 265, 289, 361,365-366, 370, 374, 376, 380, 383,394, 403,407,414, 416 basic models for, 384, 385,387, 390 boundary conditions and, 403-405 Crank-Nicolson formulation and, 382 explicit formulation and, 382 flow equations for, 381-383 fluid injection and, 400 403 fluid production and, 397-399 gas/water model, 385-386 implicit formulation, 382-383 initial conditions and, 414 linearization and, 407-408 mass balance and, 376 material balance check and, 414 oil/gas model, 387-388 oil/water model, 384 oil/water/gas model, 390-391 solution methods for, 408-414 N Newton's iteration, 297, 425 See also fully implicit method 443 no-flow boundaries, 73,77-78,111-116,136, 139,169-171,173,192,197, 199, 212,403,405,411-412, 428 O oil, 1, 3, 21,208, 365 FVF and, 10, 366 density and, 10, 366 viscosity and, 366, 367 oil/gas model 387-388 oil/water model, 384 IMPES method and, 410-411 SS method and, 410-414 oil/water/gas model, 390-391 optimum over-relaxation parameter, 345-346 P-Q partial differential equation (PDE), 1-4, 36, 235 discretization and, 1-2, 4-5 permeability, anisotropic, horizontal, 8, 185 isotropic, relative, 370 vertical, phase mobilities, 399 point-distributed grid, 123-124, 153-154 point Jacobi method, 338-339 point successive over-relaxation (PSOR) method, 345-346 pore volume, 98, 156, 365 porosity, heterogeneity and, incompressible porous media, 210 pressure boundary conditions and See specified pressure boundaries capillary, 373 coefficient of, 239, 283,285, 287, 293, 296, 425 compressibility and, 208, 209 explicit formulation and, 236 fluid properties and, 208 FVF and, 208 444 Subject Index pressure (continued) IMPES method and, 411 implicit formulation and, 237 injection and, 401 phase, 373 solution GOR and, 367 viscosity and, 366 pressure-specified wells, 183, 195 injection and, 403 production and, 399 pressure/volume/temperature (PVT), 365, 368, 423,430, 431 pressure gradient-specified wells, 182-183, 195 injection and, 402 production and, 399 production, 1, 2, 22, 36, 66, 75, 111,127, 136, 169, 203-204, 210-211, 240-241,283-284, 292-293, 327, 365, 397 in multiblock wells, 192-195, 397-399 in single block wells, 181-182 R rate-specified wells, 182, 193-195 injection and, 402 production and, 398-399 rectangular coordinates, 28, 36 grids and, 9, 29 reservoir discretization, 8, 63, 123, 376, 425 block-centered grid and, 63-64, 91-92, 98 irregular grid, 64, 125 point-distributed grid, 123-124, 154-155, 156-157 radial direction and, 98, 156-157 single-well simulation and, 28 terminology for, 8-9, 28-29, reservoir simulation, 1, 4-5, 18 algebraic equations, block ordering and, 12 Darcy's Law and, 10 engineering approach and, flow equations and, mathematical approach and, multiphase flow and, 365 single phase flow and, 7, 207 use of symmetry in, 110, 169 well representation and, 181 residual oil saturation, 371 rock properties, permeability, porosity, sandface pressure See FBHP saturation, 365 black oil description and, 365 gas/water model and, 386 IMPES method and, 411 multiphase flow and, 390 oil/gas model and, 387 oil/water model and, 384 sequential solution (SEQ) method, 4, 365 shut-in wells, 182, 193, 398, 402 simultaneous solution (SS) method, 4, 365, 408, 416 explicit linearization and, 411 fully implicit linearization and See fully implicit method single phase flow, 207 boundary conditions and, 73, 135 compressible fluid flow and, 264-268 explicit formulation and, 265 implicit formulation and, 266 incompressible fluid flow and, 210-212 initial conditions and, 238, 267 pressure and, 210, 238, 267 slightly compressible fluid flow and, 235-242 transmissibility and, 87, 90, 152-153 slightly compressible fluid, 208, 336 accumulation term and, 235 flow equation and, 235-238 fluid properties and, 208 source/sink terms See well representation sparse matrices, 334, 336, 357 specified pressure boundaries, 78-79, 116, 140-141,174, 428 specified pressure gradient boundaries, 73, 75-76, 116, 136-138, 174, 403-404, 428 specified rate boundaries, 76-77, 138,404 stability, 5, 266, 293 Crank-Nicolson formulation and, 238 explicit formulation and, 236 implicit formulation and, 237 Subject Index nonlinearity and, 291 of solution, 236 steady-state, 210, 267, 287 Stone' s three-phase model, 370-371 successive over-relaxation (SOR) methods, 345, 350, 356, 358 T Tang's algorithm, 329-332 temperature, 11, 21,208, 209, 235, 264, 287, 366-367 thickness, 8, 185,195 Thomas' algorithm, 327-328 time, 1-2, 4-5 transmissibility and, 291 transmissibility explicit formulation and, 236, 265-266 fully implicit method and, 291 IMPES method and, 410 implicit formulation and, 266, 237 incompressible flow and, 210 interblock, 87, 90, 152, 153 linearization of, 288-291 slightly compressible flow and, 285 U unsteady-state, 239, 267, 276, 287, 415 445 V vertical wells, 181 viscosity, 7, 208-209, 366 volumetric rate See Darcy's Law W-Z water, 376, 383, 384, 385, 390, 403,404 water/oil contact (WOC), 403 weighting, 17, 290, 381,408 average function value, 290 average component function value, 290 average pressure value, 290 single-point upstream, 290, 405,408,415 transmissibility, 291 two-point upstream, 380 well rate See injection; production wells multiblock, 192 single block, 181 well representation, 1, 181 horizontal effects, 185-186 multiblock model, 192 single block model, 181 vertical effects, 192 ... flow in petroleum reservoirs using the engineering approach This text is written for senior-level B.S students and first-year M.S students studying petroleum engineering and aims to restore engineering. .. in a petroleum reservoir cannot-makes modeling more vital for tackling problems in the petroleum reservoir than in any other discipline Indeed, from the advent of computer technology, the petroleum. .. the petroleum industry was the energy provider of the world, was synonymous with its reputation as the most aggressive investor in engineering and science More recently, however, as the petroleum