Petroleum Engineering Handbook

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 engi- n

P e t r o l e u m

R e s e r v o i r S i m u l a t i o n

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 N o part of this publication may be reproduced or transmitted in any form without the prior written permission of the publisher

H O U S T O N , TX:

Gulf Publishing Company

2 Greenway Plaza, Suite 1020

Library of Congress Cataloging-in-Publication Data

Petroleum Reservoir Simulation: A Basic Approach/ Jamal H Abou-Kassem [et

a l ]

p cm

Includes bibliographical references and index

ISBN 0-9765113-6-3 (alk paper)

I Petroleum Simulation methods, manuals, etc 2 Petroleum Mathematical models, manuals, etc 3 Hydrocarbon reservoirs Simulation methods, manuals, etc 4 Hydrocarbon reservoirs Mathematical models, manuals, etc 5 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 Imagina- tion 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 multi- plicity of solutions Science is the essence of knowledge, and acceptance of the multi- plicity 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 p r o m o t e d as an expression of uncertainty This leads not to science or to new authentic knowledge, but rather to creating numerous models that generate "unique" solu- tions 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 c a n n o t - - 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 i n v e s t m e n t in l o n g - t e r m 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 engi- neering 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 big- gest 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 technolo- gies 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 reser- voir 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 res- ervoir 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 gener- ated by commercial reservoir simulators As companies struggle to deal with scandals trig- gered by E n r o n ' 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 reser- voir engineering

This book provides a means of understanding the underlying principles of petroleum res- ervoir 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 treat- ment 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 con- fusing In keeping up with the same rigor of treatment, this book presents the discretiza- tion 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 reser- voir simulation For instance, we discuss the effect of boundary conditions on material- balance-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 sci- ence 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

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 engi- neering 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 mathemat- ical 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 inter- prets boundary conditions as fictitious wells that transfer fluids across reservoir bound- aries 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 9 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-cylin- drical 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 7 (Secs 2.7, 4.5, 6.2.2)

Chapter 1 provides an overview of reservoir simulation and the relationship between the mathematical approach presented in simulation books and the engineering approach pre- sented in this book In Chapter 2, we present the derivation of single-phase, multidimen- sional 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 block- centered 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 6 deals with wells completed in both single and multiple layers and presents fluid flow rate equations for different well operating conditions Chapter 7 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 9 presents the basic direct and iterative solution methods of linear algebraic equations used in reservoir simula- tion Chapter 10 is entirely devoted to multiphase flow in petroleum reservoirs and its sim- ulation The book concludes with Appendix A, which presents a user's manual for a single- phase simulator The CD that accompanies the book includes a single-phase simulator

XV

IAI = square coefficient matrix

b L, = reservoir east boundary

b L = reservoir lower boundary

b N = reservoir north boundary

b s = reservoir south boundary

b U = reservoir upper boundary

b w = reservoir west boundary

b = coefficient o f unknown x,_,~,, r ,

defined b y Eq 9.46a

B = parameter, defined b y Eq 9.29 in

Bei = formation v o l u m e factor of phase p

c = fluid compressibility, psi -1 l k P a - l l

c¢ = porosity compressibility, psi -1 [kPa -1]

c a = rate o f fractional viscosity change with pressure change, psi -1 [kPa -1]

C = parameter, defined b y Eq 9.30 in Tang' s algorithm

CMe = cumulative material balance check, dimensionless

xvii

xviii Nomenclature

Cop = coefficient o f pressure change over

time step in expansion of oil

accumulation term, STB/D-psi

[std m3/(d.kPa)]

Cow = coefficient o f water saturation

change over time step in expansion

o f oil accumulation term, STB/D

[std m3/d]

C,~p = coefficient o f pressure change over

time step in expansion o f water

accumulation term, B/D-psi

[ std m3/(d.kPa) ]

Cww = coefficient o f water saturation

change over time step in expansion

of water accumulation term, B/D

between two successive iterations

d = RHS o f equation for gridblock n,

f , l p,,,~ = nonlinearity, defined b y Eq 8.17

F(t) = argument o f an integral at time t

F~ = ratio o f w e l l b l o c k i area to theoretical area from which well withdraws its fluid (in Chapter 6), fraction

F m = argument o f an integral evaluated

at time t m F(t m) = argument o f an integral evaluated at time t"

F ° = argument o f an integral evaluated

at time t"

F(t") = argument o f an integral evaluated at time t"

F n+l = argument o f an integral evaluated

at time t n+l F(t n+l) = argument of an integral evaluated at time t "+~

F n+1/2 = argument o f an integral evaluated at time t "~ln

F(t n+l/z) = argument o f an integral evaluated at time t "+1/2

g = gravitational acceleration, ft/sec 2 [m]s 2]

gi = element i o f a temporary vector (~) generated in T h o m a s ' algorithm

Gy i = interblock geometric factor between b l o c k i and b l o c k i -T- 1

Nomenclature xix

along the x direction, defined by

Eq 8.4

G = interblock geometric factor

between blocks 1 and 2 along the

x direction

Gy2 6 = interblock geometric factor

between blocks 2 and 6 along the

y direction

G~,,,2.j k = interblock geometric factor

between block (i,j,k) and

block (i-y-l,j,k) along the

r direction in radial-cylindrical

coordinates, defined in Table 4-2,

4-3, 5-2, and 5 - 3

Gx i+ll2,j,k = interblock geometric factor

between block (i,j,k) and

block (i-y-l,j,k) along the

x direction in rectangular

coordinates, defined in Table 4-1

and 5-1

Gy~j = interblock geometric factor

between block (i,j,k) and

block (i,j-y-l,k) along the

y direction in rectangular

coordinates, defined in Table 4-1

and 5-1

G = interblock geometric factor

between block (i,j,k) and

block (i,j, kT-1) along the

z direction in rectangular

coordinates, defined in Table 4-1

and 5-1

G = interblock geometric factor

between block (i,j,k) and

block (i,j, kT-1) along the

h = thickness, ft [m]

h i = thickness of wellblock i , ft [m]

h I = thickness of wellblock l, ft [m] IMB = incremental material balance check, dimensionless

k = horizontal permeability, md [~tm 2]

k., = horizontal permeability of wellblock i , md [Ixm 2]

k r = permeability along the r direction in radial flow, md [~m 2]

kr~ = relative permeability to gas-phase, dimensionless

kro = relative permeability to oil-phase, dimensionless

k = relative permeability to oil-phase

at irreducible water saturation, dimensionless

krog = relative permeability to oil-phase

in gas/oil/irreducible-water system, dimensionless

k o w = relative permeability to oil-phase

in oil/water system, dimensionless krp = relative permeability to phase p, dimensionless

k~p = relative permeability phase p between point i and point i-T- 1 along the x axis, dimensionless

kx Ix,~1,2 = permeability between point i

and point i -T 1 along the x axis,

log e = natural logarithm

L = reservoir length along the x axis, ft

[m]

[L] = lower triangular matrix

L = reservoir length along the x axis, ft

m i = mass of component c entering

reservoir from other parts of

reservoir, lbm [kg]

m~ ]x,_l,~ = mass of component c entering

block i across block boundary

xi_ m , l b m [kg]

m~o [~, = mass of component c leaving

block i across block boundary

xi+l/2 , lbm [kg]

mcs, = mass of component c entering (or leaving) block i through a well, lbm [kg]

m n ,, = mass of component c per unit volume o f block i at time t n, lbm/ft 3 [kg/m 3]

m "+1 = mass of component c per unit c v i

volume of block i at time t n+l, lbm/ft 3 [kg/m 3]

rhcx = x-component of mass flux of component c, lbm/D-ft 2 [kg/(d.m2)]

mig~ = mass of free-gas-component per unit volume of reservoir rock, lbm/ft 3 [kg/m 3]

rhig~ = x-component o f mass flux o f free- gas-component, lbm/D-ft 2

[kg/(d.m2)]

m i = mass of fluid entering reservoir from other parts of reservoir, lbm [kg]

mi Ix = mass of fluid entering control volume boundary at x, lbm [kg]

milr = mass of fluid entering control w)lume boundary at r, lbm [kg]

mi [x, = mass of fluid entering block i across block boundary xi_l/2 , l b m [kg]

mi 10 = mass o f fluid entering control volume boundary at O, lbm [kg]

m o = mass of fluid leaving reservoir to other parts of reservoir, lbm [kg]

mo[r+Ar = mass of fluid leaving control volume boundary at r + A r , lbm [kg]

N o m e n c l a t u r e xxi

mov = mass of oil-component per unit

volume of reservoir rock, lbm/ft 3

lkg/m 3]

rhox = x-component of mass flux of oil-

component, lbm/D-ft 2 lkg/(d.m2)]

mo ]x+~ = mass of fluid leaving control

volume boundary at x + A x , Ibm

[kg]

mo [x~ = mass of fluid leaving block i

across block boundary xi,1/2 , Ibm

Ikg]

mo [0+A0 = mass of fluid leaving control

w)lume boundary at 0 + A 0 , Ibm

[kg]

m = mass of fluid entering (or leaving)

reservoir through a well, Ibm [kg]

m = mass of fluid entering (or leaving)

block i through a well, lbm [kg]

m = mass of fluid per unit volume of

m = mass of water-component per unit

volume of reservoir rock, lbm/ft 3

[kg/m 3 ]

rhwx = x-component of mass flux of water-component, lbm/D-fl 2 [kg/(d.m2)]

rh x = x-component of mass flux, lbm/D-

ft 2 [kg/(d.m2)]

rhx [x = x-component of mass flux across control volume boundary at x, lbm/D-ft 2 [kg/(d.m2)l

r h [x+~ = x-component of mass flux across control volume boundary at

x + A x , lbm/D-ft 2 [kg/(d.m 2) l

r h [x~ = x-component of mass flux across block boundary xi+l/2, lbm/D-ft 2 [kg/(d.m2)]

M = gas molecular weight, lbm/lb mole [kg/kmole]

Mp, = mobility of phase p in wellblock

i , defined in Table 10-4

n = coefficient of unknown x,+,, x , defined by Eq 9.46e

n r = number of reservoir gridblocks (or gridpoints) along the r direction

N = number of blocks in reservoir

P = pressure, psia [kPa]

p° = reference pressure, psia [kPa]

xxii Nomenclature

= average value pressure, defined by

Eq 8.21, psia [kPa]

Pb = oil bubble-point pressure, psia

[kPa]

Pi = pressure of gridblock (gridpoint) or

wellblock i , psia [kPa]

p~' = pressure of gridblock (gridpoint)

i at time t m, psia lkPal

m

Pi~l = pressure o f gridblock (gridpoint)

i -Y- 1 at time t m, psia [kPa]

( i,j,k ) at time t m, psia [kPa]

p~ i~l,j.k = pressure of gridblock (gridpoint)

( iT 1,j,k ) at time t m, psia [kPa]

r n

Pgj~l,k = pressure of gridblock (gridpoint)

( i, j T- l, k ) at time t m, psia [kPa]

p~ i,j,k~l = pressure of gridblock (gridpoint)

( i,j,k -Y- 1 ) at time t m, psia [kPa]

pn = pressure of gridblock (gridpoint) i

at time t n, psia [kPa]

p n 4 1 = pressure of gridblock (gridpoint)

i at time t n+l, psia [kPa]

( e ~ l )

p y l = pressure of gridblock (gridpoint)

i at time level t n+l and iteration

v + 1, psia [kPa]

{ ~ - t )

6p~ *~ = change in pressure of gridblock

(gridpoint) i over an iteration at

time level n + 1 and iteration

P.1 = pressure of gridblock (gridpoint)

i + 1 at time t n, psia [kPa]

p l = pressure of gridblock (gridpoint) i + 1

i + 1 at time t n+l, psia lkPa] p.+l i-T-1 ~-~ pressure of gridblock (gridpoint) i-T-1 at time t n+l, psia [kPa] Pij,k = pressure of gridblock (gridpoint)

or wellblock ( i,j,k ), psia [kPa] p~ = pressure of neighboring gridblock (gridpoint) l, psia [kPa]

p = pressure of gridblock (gridpoint) or wellblock n, psia lkPa]

pO = initial pressure of gridblock (gridpoint) n, psia [kPal p" = pressure of gridblock (gridpoint) or wellblock n at time level n, psia [kPal

pn+l = pressure of gridblock (gridpoint)

i at time level t n+l and iteration

v , psia [kPal p~+l = pressure of gridblock (gridpoint)

or wellblock n at time level n + 1 , psia [kPa]

p ~ = pressure of gridblock (gridpoint) n

at old iteration v , psia [kPa] p~V+l~ = pressure of gridblock (gridpoint)

n at new iteration v + 1, psia [kPa]

Pp, = pressure of phase p in gridblock (gridpoint) i , psia [kPa]

Pp~;, = pressure of phase p in gridblock (gridpoint) i -Y- 1, psia [kPa]

Po = oil pressure, psia [kPa]

Pr# = pressure at reference datum, psia [kPa]

Psc = standard pressure, psia [kPa]

Pw = water-phase pressure, psia [kPa] Pws = flowing well bottom-hole pressure, psia lkPa]

Nomenclature xxiii

Pw~., = estimated flowing well bottom-

hole pressure at reference depth,

psia [kPa]

Pw~ = flowing well bottom-hole pressure

opposite wellblock i , psia [kPa]

P.~.: = flowing well bottom-hole

pressure at reference depth, psia

[kPa]

Pw:,~ = specified flowing well bottom-

hole pressure at reference depth,

qc.,, = mass rate of component c entering

block i through a well, lbm/D

[kg/d]

q:g = production rate of free-gas-

component at reservoir conditions,

RB/D [std m3/d]

q~,~m = mass production rate of free-gas-

component, lbm/D [kg/d]

q:~c = production rate of free-gas-

component at standard conditions,

qom = mass production rate of oil-

component, lbm/D [kg/d]

qosc = production rate of oil-phase at

standard conditions, STB/D [std m3/d]

qsc = well production rate at standard conditions, STB/D or scf/D [std m3/d]

qsc, = production rate at standard conditions from wellblock i , STB/D or scf/D [std m3/d]

qm = production rate at standard sc i conditions from wellblock i at time t " , STB/D or scf/D [std m3/d]

q" scn = production rate at standard conditions from wellblock n at time t m , STB/D or scf/D lstd m3/d]

qm SCi,i,k = production rate at standard conditions from wellblock (i, j , k )

at time t ' , STB/D or scf/D [std m3/d]

qn+l sci = production rate at standard conditions from wellblock i at time level n + 1 , STB/D or scf/D [ std m3/d]

(v)

qn+l s,., - - production rate at standard conditions from wellblock i at time :÷1 and iteration v , STB/D

or scf/D [std m3/d]

qm SCl,(i,j,k) = volumetric rate of fluid at standard conditions crossing

= volumetric rate of fluid at

standard conditions crossing

reservoir boundary l to block n,

STB/D or scf/D lstd m3/dl

= volumetric rate of fluid at

standard conditions crossing

reservoir boundary I to block n at

time t ~ , STB/D or scf/D

[std m3/d]

= production rate at standard

conditions from wellblock n,

STB/D or scf/D [std m3/dl

= interblock volumetric flow rate

at standard conditions between

gridblock (gridpoint) i and

gridblock (gridpoint) i -Y- 1, STB/D

or scf/D [std m3/d]

= volumetric flow rate at standard

conditions across reservoir

boundary to boundary gridblock

bB, STB/D or scf/D [std m3/d]

= volumetric flow rate at standard

conditions across reservoir

boundary to boundary gridpoint

bP, STB/D or scf/D [std m3/d]

= volumetric flow rate at standard

conditions across reservoir west

boundary to boundary gridblock

(gridpoint) 1, STB/D or scf/D

[std m3/d]

= volumetric flow rate at standard

conditions across reservoir east

boundary to boundary gridblock

(gridpoint) n x , STB/D or scf/D [std m3/d]

gas-component, lbm/D [kg/d]

qsesc = specified well rate at standard conditions, STB/D or scf/D [std m3/dl

q,,m = mass production rate o f water- component, lbm/D [kg/d]

qwsc = production rate of water-phase at standard conditions, B/D [std m3/dl

qx = volumetric rate at reservoir conditions along the x axis, RB/D [m3/d]

r = distance in the r direction in the radial-cylindrical coordinate system, ft lm]

r e = external radius in D a r c y ' s Law for radial flow, ft [m]

req ~-~ equivalent wellblock radius, ft [m]

which the theoretical well for block n withdraws its fluid, ft [m] r~l = r-direction coordinate of point i-T-I, ft [m]

calculations, defined by Eqs 4.82a and 4.83a (or Eqs 5.75a and 5.76a), ft [ml

r~;~/2 = radii squared for bulk volume calculations, defined by Eqs 4.84a and 4.85a (or Eqs 5.77a and 5.78a), ft 2 [m 2]

r,, = residual for block n, defined by

Eq 9.61

r = well radius, ft [m]

s = skin factor, dimensionless

S = fluid saturation, fraction

Sg = gas-phase saturation, fraction

Siw = irreducible water saturation,

fraction

s = coefficient of unknown x_.~ ,

defined by Eq 9.46b

S O = oil-phase saturation, fraction

S w = water-phase saturation, fraction

t = time, day

T = reservoir temperature, °R [K]

At = time step, day

t ~ = time at which the argument F of

integral is evaluated in Eq 2.30,

day

At, = mth time step, day

t ~ = old time level, day

A t = old time step, day

At.+ 1 = current (or new) time step, day

TY b,b = transmissibility between reservoir

boundary and boundary gridblock

at time t ~

m

boundary and boundary gridpoint

at time t m

T b m ,bP* = transmissibility between

reservoir boundary and gridpoint

immediately inside reservoir

boundary at time

x direction, scf/D-psi [ std m3/(d.kPa)]

gridblocks (gridpoints) l and

T," = transmissibility between l,n gridblocks (gridpoints) 1 and n at time t m

Tox = oil-phase transmissibility along the

x direction, STB/D-psi [std m3/(d.kPa)]

~1,2,:,~ = transmissibility between point

the r direction, STB/D-psi or scf/D-psi [std m3/(d.kPa) ]

Lm

~,2~,, = transmissibility between point

( i , j , k ) and point ( i - T - l , j , k ) along the r direction at time t m , STB/D- psi or scf/D-psi [std m3/(d.kPa)]

[std m3/(d.kPa) l

Tx n'-I i~l/2 ~ - transmissibility between point i and point i -Y- 1 along the x axis at time ~+1, STB/D-psi or scf/D-psi

xxvi Nomenclature

T x,:m,~.j., = transmissibility between point

( i , j , k ) and point (i-y- 1,j,k) along

the x axis, STB/D-psi or scf/D-psi

[std m3/(d.kPa)]

,,~,~.j.~ = transmissibility between point

( i , j , k ) and point (i-y- 1,j,k) along

the x axis at time t r" , STB/D-psi or

scf/D-psi [std m3/(d.kPa)]

Ty,.j = transmissibility between point

( i , j , k ) and point (i,j-T-l,k) along

the y axis, STB/D-psi or scf/D-psi

[std m3/(d.kPa)]

r ; ,.j+~,2., = transmissibility between point

( i , j , k ) and point (i,j-y-l,k) along

t h e y axis at time t m , STB/D-psi or

scf/D-psi [std m3/(d.kPa)]

T j = transmissibility between point

( i , j , k ) and point (i,j,k-T-1) along

the z axis, STB/D-psi or scf/D-psi

[std m3/(d.kPa)]

T " z,.j.~,,2 = transmissibility between point

( i , j , k ) and point (i,j,k-y-1) along

the z axis at time t m , STB/D-psi or

scf/D-psi [std m3/(d.kPa)]

T% = transmissibility between point

( i , j , k ) and point (i,j-y-l,k) along

the 0 direction, STB/D-psi or

scf/D-psi [std m3/(d.kPa)]

To"j = transmissibility between point

( i , j , k ) and point (i,j-y-l,k) along

the 0 direction at time t m ,

Uo~ = x-component of volumetric

velocity o f oil-phase at reservoir conditions, RB/D-ft 2 [m3/(d m2)]

Up x x 1,2 = x-component of volumetric

t

velocity of phase p at reservoir conditions between point i and point i-T- 1, RB/D-ft 2 [m3/(d m2)]

Uwx = x-component of volumetric

velocity of water-phase at reservoir conditions, RB/D-ft 2 [m3/(d m2)]

u x = x-component of volumetric velocity

at reservoir conditions, RB/D-ft 2 [m3/(d m2)]

V b = bulk volume, ft 3 [m 3]

Vb, = bulk volume of block i , ft 3 [m 3] Vb,.j.~ = bulk volume of block (i, j , k ) , ft 3

[m 3 ]

Vb = bulk volume of block n, ft 3 [m 3]

Wci x~ 1/2

entering block i across block boundary xi_l/2 , lbm/D [kg/d] weilx,.~,z = mass rate of component c

leaving block i across block

w~ x = x-component o f mass rate of component c, lbm/D [kg/d]

wi = coefficient of unknown of block i - 1 in T h o m a s ' algorithm

defined by Eq 9.46c

entering (or leaving) block i

across block boundary xi~1/2 ,

lbm/D [kg/d]

x = distance in the x direction in the

Cartesian coordinate system, ft [m]

Ax = size o f block or control v o l u m e

along the x axis, ft [m]

= vector o f unknowns (in Chapter 9)

x i = x direction coordinate o f point i , ft

6x_ = distance between gridblock

(gridpoint) i and block boundary

in the direction o f decreasing

i along the x axis, ft [ml

6x~ = distance between gridblock

(gridpoint) i and b l o c k boundary

in the direction o f increasing

i along the x axis, ft [m]

x~l = x direction coordinate o f point

i -T- 1, ft lm]

xi~ 1 = unknown for b l o c k i -Y- 1 in

T h o m a s ' algorithm (in Chapter 9) Axirl = size o f b l o c k i-T- 1 along the

x = unknown for b l o c k n (in Chapter 9)

x (v) n = unknown for b l o c k n at old iteration v (in Chapter 9)

x (v+l~ = unknown for block n at new

n

iteration v + 1 (in Chapter 9)

x x = x direction coordinate o f gridblock (gridpoint) n x, ft [m]

Y = distance in the y direction in the Cartesian coordinate system, ft [m]

Ay = size o f b l o c k or control v o l u m e along the y axis, ft [m]

Ayj = size o f b l o c k j along the y axis, ft Ira]

z = gas compressibility factor, dimensionless

z = distance in the z direction in the Cartesian coordinate system, ft [m]

Az = size of block or control v o l u m e along the z axis, ft [m]

Az~ = size o f b l o c k k along the z axis, ft Ira]

Azi,j,~ = size o f b l o c k ( i , j , k ) along the

~Pxb = pressure gradient in the

x direction evaluated at reservoir

boundary, psi/ft [kPa/m]

x direction evaluated at block

boundary x~1/2 , psi/ft [kPa/m]

r direction evaluated at well radius,

"73~x b = elevation gradient in the

x direction evaluated at reservoir

boundary, dimensionless

numerical value is given in

Table 2-1

alg = logarithmic spacing constant, defined by Eq 4.86 (or Eq 5.79), dimensionless

tic = transmissibility conversion factor whose numerical value is given in Table 2-1

fli = element i of a temporary vector (2) generated in Tang's algorithm (in Chapter 9)

2" = fluid gravity, psi/ft [kPa/m]

~i = element i of a temporary vector (7) generated in Tang' s algorithm (in Chapter 9)

~ = gravity conversion factor whose numerical value is given in Table 2-1

Vg = gravity of gas-phase at reservoir conditions, psi/ft [kPa/ml

~m/2 = fluid gravity between point i and point i -y- 1 along the x axis, psi/ft [kPa/m]

m

V~l/2,j,k = fluid gravity between point (i, j, k) and neighboring point (i-T- 1, j , k ) along the x axis at time

t m , psi/ft [kPa/m]

r n

V~,j+I/2,~ = fluid gravity between point

(i, j g 1, k) along the y axis at time

~/~'ti j k~ = fluid gravity between point

time t " , psi/ft [kPa/m]

Nomenclature xxix

m

y~,,, = fluid gravity between point n and

neighboring point 1 at time t m ,

psi/ft [kPa/m]

Yz,(i,j,k~ = fluid gravity between point

(i,j,k) and neighboring point I,

psi/ft [kPa/m]

y~, = fluid gravity between point n and

neighboring point l, psi/ft [kPa/ml

Yo = gravity of oil-phase at reservoir

conditions, psi/ft [kPa/m]

Yp = gravity of phase p between point

i and point i -y- 1 along the x axis,

psi/ft lkPa/m]

Yp,, = gravity of phase p between point l

and point n, psi/ft [kPa/m]

y., = gravity of water-phase at reservoir

conditions, psi/ft [kPa/m]

7wb = average fluid gravity in wellbore,

psi/ft [kPa/m]

e = convergence tolerance

t/i,,j = set of phases in determining

mobility of injected fluid = { o, w,g }

rlprd = set of phases in determining

mobility of produced fluids,

defined in Table 10-4

0 = angle in the Odirection, rad

A0j = size of block (i,j,k) along the

0 direction, rad

and point (i,j-y-l,k) along the

O" iT-1

lff~n t i i-T1

~°

kt~

= porosity at reference pressure p°, fraction

= potential, psia [kPa]

= potential of gas-phase, psia [kPa]

= potential of gridblock (gridpoint)

i , psia [kPa]

= potential of gridblock (gridpoint)

i at time t m, psia [kPa]

= potential of gridblock (gridpoint)

i at time t n, psia [kPa]

= potential of gridblock (gridpoint)

i at time t n+l, psia [kPa]

= potential of gridblock (gridpoint) i-T- 1, psia [kPa]

= potential of gridblock (gridpoint)

i T- 1 at time t m, psia [kPal

= potential of gridblock (gridpoint)

i -Y- 1 at time t n, psia lkPa]

= potential of gridblock (gridpoint)

i -Y- 1 at time t ~+ 1, psia [kPa]

= potential of gridblock (gridpoint)

(i,j,k) at time t m, psia [kPa]

= potential of gridblock (gridpoint)

1 at time t m, psia [kPa]

= potential of oil-phase, psia lkPa]

= potential of phase p in gridblock (gridpoint) i , psia [kPa]

= potential at reference depth, psia [kPa]

= potential of water-phase, psia [kPa]

= fluid viscosity, cp [mPa.s]

= viscosity of fluid in gridblock (gridpoint) i , cp [mPa.sl

= fluid viscosity at reference pressure

p ° , cp [mPa.s]

= gas-phase viscosity, cp lmPa.s]

xxx Nomenclature

/~P x,:,,2 = viscosity of phase p between

point i and point i T 1 along the

x axis, cp [mPa.s]

/'to = oil-phase viscosity, cp [mPa.s]

/~ob = oil-phase viscosity at bubble-point

pressure, cp [mPa.s]

/~w = water-phase viscosity, cp [mPa.s]

/~1 = fluid viscosity between point i

and point i -T- 1 "along the x axis, cp

[mPa.sl

~0 = a set containing gridblock (or

gridpoint) numbers

~0 b = the set of gridblocks (or gridpoints)

sharing the same reservoir

boundary b

~,:,~ = the set of existing gridblocks (or

gridpoints) that are neighbors to

gridblock (gridpoint) (i, j , k )

~p, = the set of existing gridblocks (or

gridpoints) that are neighbors to

gridblock (gridpoint) n

~Pro = the set of existing gridblocks (or

gridpoints) that are neighbors to

gridblock (gridpoint) n along the

r direction

~P~o = the set of existing gridblocks (or

gridpoints) that are neighbors to

gridblock (gridpoint) n along the

x axis

gridpoints) that are neighbors to

gridblock (gridpoint) n along the

y axis

~P:,, = the set of existing gridblocks (or

gridpoints) that are neighbors to

gridblock (gridpoint) n along the

z axis

~P0,, = the set of existing gridblocks (or gridpoints) that are neighbors to gridblock (gridpoint) n along the 0direction

~Pw = the set that contains all wellblocks

pg = gas-phase density at reservoir conditions, lbm/ft 3 [kg/m 3]

Pgsc = gas-phase density at standard conditions, lbm/ft 3 [kg/m 3]

Po = oil-phase density at reservoir conditions, lbm/ft 3 [kg/m 3]

Pos, = oil-phase density at standard conditions, lbm/ft 3 [kg/m 3] Psc = fluid density at standard conditions, lbm/ft 3 [kg/m 3]

Pw = water-phase density at reservoir conditions, lbm/ft 3 [kg/m 3] Pw.~c = water-phase density at standard conditions, lbm/ft 3 lkg/m 3]

Pwb = average fluid density in wellbore, lbm/ft 3 [kg/m 31

= summation over all members of set ~p

= summation over all members of

lE~)i,j,k

set ~Pi,j,k

= summation over all members of set ~0

g r i d p o i n t , or p o i n t a l o n g the x or

r direction

i T - l / 2 = b e t w e e n i and i-T-1 i,i -y- 1 / 2 = b e t w e e n b l o c k (or point) i and b l o c k b o u n d a r y i-T- 1 / 2 a l o n g the x d i r e c t i o n

(i, j , k ) = i n d e x for g r i d b l o c k , gridpoint,

or p o i n t in x-y-z (or r-O-z) s p a c e

j -Y- 1 / 2 = b e t w e e n j and j -T- 1

j , j -Y- 1 / 2 = b e t w e e n b l o c k (or point) j and b l o c k b o u n d a r y j-T- l / 2 a l o n g the y d i r e c t i o n

k = i n d e x for g r i d b l o c k , gridpoint, or

p o i n t a l o n g the z d i r e c t i o n

z d i r e c t i o n

0 = 0 d i r e c t i o n 0j~l/2 = b e t w e e n j and j-T- 1 a l o n g the

(v) = o l d iteration v (v + 1) = current iteration v + 1

* = i n t e r m e d i a t e v a l u e b e f b r e S O R

a c c e l e r a t i o n

References

compositional, three-phase steam model Ph.D diss., University of Calgary, Alberta

thermal recovery J Pet Sci Eng 15: 281-90

simulation Soc Pet Eng J 25 (4): 573-79

inactive blocks in reservoir simulation J Can Pet Tech 31 (2): 25-31

dimensional modeling of one-eighth of confined five- and nine-spot Patterns J

Pet Sci Eng 5: 137-49

of reservoir simulation equations SPE # 17072, Paper presented at the 1987 SPE Eastern Regional Meeting, Pittsburgh, Pennsylvania, 21-23 October

engineering approach versus the mathematical approach in developing reservoir simulators Under review

interblock geometric factors and bulk volumes in single-well simulation," J Pet

Sci Tech In press

to the treatment of constant pressure boundary condition in block-centered grid in reservoir simulation Under review

multipurpose simulator J Pet Sci Eng 16: pp 221-35

presented at the 1983 SPE Reservoir Simulation Symposium, San Francisco, 15-18 November

12 Aziz, K 1993 "Reservoir simulation grids: opportunities and problems," SPE #

25233, Paper presented at the1993 SPE Symposium on Reservoir Simulation, New Orleans, LA, 28 February-3 March

434 References

thermal reservoir behavior Paper presented at Canada-Venezuela Oil Sands Symposium, 27 May 27-4 June, Edmonton, Alberta

Pet Tech 25 (1): 51-56

simulation SPE Textbook Series, vol 7 Richardson, Tex.: SPE

course notes, Petroleum Engineering, Mineral Engineering Department, University

of Alberta

McGraw-Hill

Trans AIME 237: 997-1000

multidimensional polymer injection simulator Ph.D diss., The Pennsylvania State University

Extension of Modified Brinkman Formulation for Fluid Flow through Porous

(IMECE), Orlando, Florida, USA, November 5-10

Solution to the Modified Brinkman Model (MBM) for a 2-Dimensional, 1-Phase

Engineering (CSCE), 33rdAnnual Conference, Toronto, ON, Canada, June 2-4

435

semi-implicit reservoir simulation techniques Trans AIME 253: 253-66

hydrocarbon reservoirs SIAM Rev 24 (3): 263

simulation with nonsquare gridblocks and anisotropic permeability SPEJ 23 (3):

531-34

and elliptic equations J SIAM 3 (1): 28-41

solution of partial differential equations for multiphase flow in porous media Int J

Multiphase Flow 1: 8 1 7 4 4

capillary forces SPEJ 9 (3): 255-69

solution approach to reservoir simulation SPE # 4542, Paper presented at the 48th Annual Fall Meeting, Las Vegas, Nevada, 30 Sept.-3 Oct

Zeitschrifi fur angewandte Mathematik und Mechanik 8 (49): 508

implicit method SPEJ 23 (5): 759-68

sets of simultaneous linear equations SPE # 5729, Paper presented at the 1976 SPE Symposium on Numerical Simulation of Reservoir Performance, Los Angeles, 19-

20 February

techniques in reservoir simulation SPE # 4544, Paper presented at the 48th Annual Fall Meeting, Las Vegas 30 Sept-3 Oct

1.2 Milestones for the Engineering Approach 2

1.3 Importance of the Engineering and Mathematical

2.2 Properties of Single-Phase Fluid 7

2.3 Properties of Porous Media 8

2.4 Reservoir Discretization 8

2.5 Basic Engineering Concepts 9

2.6 Multidimensional Flow in Cartesian Coordinates 11

2.7 Multidimensional Flow in Radial-Cylindrical Coordinates

3.2 Flow Equations Using CVFD Terminology 43

3.3 Flow Equations in Radial-Cylindrical Coordinates

Using CVFD Terminology 54

vii

viii Contents

3.4 Flow Equations Using CVFD Terminology

in any Block Ordering Scheme 57

4.3 Flow Equation for Boundary Gridblocks 66

4.4 Treatment of Boundary Conditions 73

5.3 Flow Equation for Boundary Gridpoints 126

5.4 Treatment of Boundary Conditions 135

Practical Considerations Dealing with

Modeling Well Operating Conditions 203

Summary 204

Exercises 204

Contents ix

10.2 Reservoir Engineering Concepts in Multiphase Flow

10.3 Multiphase Flow Models 375

10.4 Solution of Multiphase Flow Equations 393

10.5 Material Balance Checks 414

10.7 Summary 416

10.8 Exercises 416

365

A User's Manual for Single-Phase Simulator 421

A.2 Data File Preparation 421

A.3 Description of Variables Used in Preparing a Data File 423 A.4 Instructions to Run Simulator 430

A.5 Limitations Imposed on the Compiled Version 430

A.6 Example of a Prepared Data File 431

References 433

Author Index 437

Subject Index 439

art of combining physics, mathematics, reservoir engineering, and computer programming

to develop a tool for predicting hydrocarbon reservoir performance under various oper- ating strategies Figure 1-1 depicts the major steps involved in the development of a reser- voir simulator: formulation, discretization, well representation, linearization, solution, and

inherent to the simulator, states these assumptions in precise mathematical terms, and applies them to a control volume in the reservoir The result of this step is a set of coupled, nonlinear partial differential equations (PDEs) that describes fluid flow through porous media

The PDEs derived during the formulation step, if solved analytically, would give reservoir pressure, fluid saturations, and well flow rates as continuous functions of space and time Because of the highly nonlinear nature of the PDEs, however, analytical techniques cannot

be used and solutions must be obtained with numerical methods In contrast to analytical solutions, numerical solutions give the values of pressure and fluid saturations only at dis-

PDEs into algebraic equations Several numerical methods can be used to discretize the PDEs; however, the most common approach in the oil industry today is the finite difference method The most commonly used finite difference approach essentially builds on Taylor series expansion and neglects terms that are considered to be small when small difference in space parameters is considered This expanded form is a set of algebraic equations Finite element method, on the other hand uses various functions to express variables in the gov- erning equation These functions lead to the development of an error function that is minimized in order to generate solutions to the governing equation To carry out discretiza- tion, a PDE is written for a given point in space at a given time level The choice of time level (old time level, current time level, or intermediate time level) leads to the explicit, implicit, or Crank-Nicolson formulation method The discretization process results in a system of nonlinear algebraic equations These equations generally cannot be solved with linear equation solvers, and the lincarization of such equations becomes a necessary step

2 Chapter 1 Introduction

C ) C ) C ) ( C ° 3

& WELL RATES

C- 3

Figure 1-1 Major steps used to develop reservoir simulators

Redrawn from Odeh (1982),

nonlinear terms (transmissibilities, production and injection, and coefficients of unknowns

in the accumulation terms) in both space and time Linearization results in a set of linear algebraic equations Any one of several linear equation solvers can then be used to obtain the solution, which comprises pressure and fluid saturation distributions in the reservoir and

"after which the simulator can be used for practical field applications The validation step is necessary to make sure that no errors were introduced in the various steps of development or

in computer programming This validation is distinct from the concept of conducting experi- ments in support of a mathematical model Validation of a reservoir simulator merely involves testing the numerical code

There are three methods available for the discretization of any PDE: the Taylor series method, the integral method, and the variational method (Aziz and Settari, 1979) The first two methods result in the finite-difference method, whereas the third results in the varia- tional method The "mathematical approach" refers to the methods that obtain the non- linear algebraic equations through deriving and discretizing the PDEs Developers of simulators relied heavily on the mathematics in the mathematical approach to obtain the nonlinear algebraic equations or the finite-difference equations However, Abou-Kassem, Farouq Ali, Islam, and Osman (2006) presented a new approach that derives the finite-dif- ference equations without going through the rigor of PDEs and discretization This approach also uses fictitious wells to represent boundary conditions This new tactic is termed the "engineering approach" because it is closer to the engineer's thinking and to the physical meaning of the terms in the flow equations The engineering approach is simple and yet general and rigorous, and both the e n g i n e e r i n g and m a t h e m a t i c a l approaches treat b o u n d a r y conditions with the same accuracy if the m a t h e m a t i c a l approach uses second-order approximations In addition, the engineering approach results

in the same finite-difference equations for any hydrocarbon recovery process Because the engineering approach is independent of the mathematical approach, it reconfirms the use

of central differencing in space discretization and highlights the assumptions involved in choosing a time level in the mathematical approach

1.2 Milestones for the Engineering Approach

The foundations for the engineering approach have been overlooked all these years Tradi- tionally, reservoir simulators were developed by first using a control volume (or elemen- tary volume), such as that shown in Figure 1-2 for 1D flow or in Figure 1-3 for 3D flow, that was visualized by mathematicians to develop fluid flow equations Note that point x in 1D and point (x, y, z) in 3D fall on the edge of control volumes The resulting flow equa-

1.2 Milestones for the Engineering Approach 3

,I Ax I,

Figure 1-2 Control volume used by mathematicians for 1D flow

(x+Ax,y,z+Az

zj T

x

(x+Ax,y+Ay, z)

Figure 1-3 Control volume used by mathematicians for 3D flow

Redrawn from Bear (1988)

tions are in the form of PDEs Once the PDEs are derived, early pioneers of simulation looked to mathematicians to provide solution methods These methods started with the description of the reservoir as a collection of gridblocks, represented by points that fall within them (or gridpoints representing blocks that surround them), followed by the replacement of the PDEs and boundary conditions by algebraic equations, and finally the solution of the resulting algebraic equations Developers of simulators were all the time occupied by finding the solution and, perhaps, forgot that they were solving an engi- neering problem The engineering approach can be realized should one try to relate the terms in the discretized flow equations for any block to the block itself and to all its neigh- boring blocks A close inspection of the flow terms in a discretized flow equation of a given fluid (oil, water, or gas) in a black-oil model for a given block reveals that these terms are nothing but Darcy' s Law describing volumetric flow rates of the fluid at stan- dard conditions between the block and its neighboring blocks The accumulation term is the change in the volume at standard conditions of the fluid contained in the block itself at two different times

More than 30 years ago, Farouq Ali was the first to observe that flow terms in the dis- cretized form of governing equations are nothing but Darcy's Law describing volumetric flow rate between any two neighboring blocks Making use of this observation coupled with an assumption related to the time level at which flow terms are evaluated, he devel- oped the forward-central-difference equation and the backward-central-difference equation

4 Chapter I Introduction

without going through the rigor of the mathematical approach in teaching reservoir simula- tion to undergraduate students (Farouq Ali, 1986) Ertekin, Abou-Kassem, and King (2001) were the first to use a control volume represented by a point at its center in the mathemat- ical approach as shown in Figure 1-4 for 1D flow and Figure 1-5 for 3D flow This control volume is closer to an engineer's thinking of representing blocks in reservoirs The obser- vation by Farouq Ali in the early seventies and the introduction of the new control volume

by Ertekin et al have been the two milestones that contributed significantly to the recent development of the engineering approach

Overlooking the engineering approach has kept reserw)ir simulation closely tied with PDEs From the mathematician's point of view, this is a blessing because researchers in reservoir simulation have devised advanced methods for solving highly nonlinear PDEs, and this enriched the literature in mathematics in this important area Contributions of res- ervoir simulation to solving PDEs include the following:

and Winestock 1977; Saad 1989; Gupta 1990)

(Breitenbach, Thurnau, and van Poollen 1969), SEQ (Spillette, Hillestad, and Stone, 1973; Coats 1978), Fully Implicit SS (Sheffield 1969), and Adaptive Implicit

(Thomas and Thurnau 1983) methods

equations, such as the Block Iterative (Behie and Vinsome 1982), Nested

Factorization (Appleyard and Cheshire 1983), and Orthomin (Vinsome 1976) methods

Figure 1-4 Control volume for 1D flow

The importance of the engineering approach lies in being close to the e n g i n e e r ' s mindset and in its capacity to derive the algebraic flow equations easily and without going through the rigor of PDEs and discretization In reality, the development of a res- ervoir simulator can do away with the mathematical approach because the objective of this approach is to obtain the algebraic flow equations for the process being simulated

In addition, the engineering approach reconfirms the use of central-difference approxi-

Figure 1-5 Control volume for 3D flow

mation of the second-order space derivative and provides interpretation of the approxi- mations involved in the forward-, backward-, and central-difference of the first-order time derivative that are used in the mathematical approach

The majority, if not all, of the available commercial reservoir simulators were developed without even looking at an analysis of truncation errors, consistency, convergence, or sta- bility The importance of the mathematical approach, however, lies within its capacity to provide analysis of such items Only in this case do the two approaches complement each other and both become equally important in reservoir simulation

The traditional steps involved in the development of a reservoir simulator include formu- lation, discretization, well representation, linearization, solution, and validation The mathematical approach involves formulation to obtain a differential equation, followed by reservoir discretization to describe the reservoir, and finally the discretization of the dif- ferential equation to obtain the flow equation in algebraic form In contrast, the engi- neering approach involves reservoir discretization to describe the reservoir, followed by formulation to obtain the flow equation in integral form, which, when approximated, pro- duces the flow equation in algebraic form The mathematical approach and engineering approach produce the same flow equation in algebraic form but use two unrelated routes The seeds for the engineering approach existed a long time ago but were overlooked by pioneers in reservoir simulation because modeling petroleum reservoirs has been consid- ered a mathematical problem rather than an engineering problem The engineering approach is both easy and robust [t does not involve differential equations, discretization

of differential equations, or discretization of boundary conditions

How does the engineering approach differ from the mathematical approach

in developing a reservoir simulator?

Name the major steps used in the development of a reservoir simulator using the engineering approach

Indicate the input and the expected output for each major step in Exercise 1-4 Draw a sketch, similar to Figure l - l , for the development of a reservoir simulator using the engineering approach

Using your own words, state the importance of the engineering approach in reservoir simulation

Fluid properties that are needed to model single-phase fluid flow include those that appear

in the flow equations, namely, density ( P ), formation volume factor (B), and viscosity (/~ ) Fluid density is needed for the estimation of fluid gravity ( Y ) using

where Yc = the gravity conversion factor and g = acceleration due to gravity In general, fluid properties are a function of pressure Mathematically, the pressure dependence of fluid properties is expressed as:

7

8 C h a p t e r 2 Single-Phase Fluid F l o w E q u a t i o n s in Multidimensional D o m a i n

The derivation of the general flow equation in this chapter does not require more than the general dependence of fluid properties on pressure as expressed by Eqs 2.2 through 2.4

In Chapter 7, the specific pressure dependence of fluid properties is required for the deri- vation of the flow equation for each type of fluid

2.3 Properties of Porous Media

Modeling single-phase fluid flow in reservoirs requires the knowledge of basic rock prop- erties such as porosity and permeability, or more precisely, effective porosity and absolute permeability Other rock properties include reservoir thickness and elevation below sea level Effective p o r o s i t y is the ratio of interconnected pore spaces to bulk volume of a rock sample Petroleum reservoirs usually have heterogeneous porosity distribution; i.e., porosity changes with location A reservoir is homogeneous if porosity is constant inde- pendent of location Porosity depends on reservoir pressure because of solid and pore compressibilities It increases as reservoir pressure (pressure of the fluid contained in the pores) increases and vice versa This relationship can be expressed as

where ~ ' = porosity at reference pressure ( p° ) and c¢ = porosity compressibility Per-

same fluid fills all the interconnected pores Permeability is a directional rock property If the reservoir coordinates coincide with the principal directions of permeability, then per- meability can be represented by k x, ky, and k z The reservoir is described as having iso- tropic permeability distribution if k = ky = k z ; otherwise, the reservoir is anisotropic if permeability shows directional bias Usually k x - - ky k H and k z = k v with k v < k H

because of depositional environments

2.4 Reservoir Discretization

gridpoints) whose properties, dimensions, boundaries, and locations in the reservoir are well defined Chapter 4 deals with reservoirs discretized using a block-centered grid and Chapter 5 discusses reservoirs discretized using a point-distributed grid Figure 2-1 shows reservoir discretization in the x direction as the focus is on block i

The figure shows how the blocks are related to each other block i and its neighboring blocks i - 1 and i + 1 - - b l o c k dimensions ( Ax i , ~ i - 1 , / ~ i ÷ 1 ), block boundaries ( xi_l/z , xi.,2 ), distances between the point that represents the block and block boundaries

The terminology presented in Figure 2-1 is applicable to both block-centered and point- distributed grid systems in 1D flow in the direction of the x axis Reservoir discretization in the y and z directions uses similar terminology In addition, each gridblock (or gridpoint) is assigned elevation and rock properties such as porosity and permeabilities in the x, y, and z directions The transfer of fluids from one block to the rest of reservoir takes place through