--J cPT- oj - D7

., ;

" \-

1:'-.
,
Numerical Methods for Three-Dimensional
,..i
j
, Reservoir Models
,
j

~""-
.j By K. AZIZ" [: ;'

:?J , (18th Annual Technical Meeting, The Pet"a/en'" Society of elM, Banff. Alta" May, 1967)
J,\

Downloaded from http://onepetro.org/JCPT/article-pdf/7/01/41/2166452/petsoc-68-01-07.pdf/1 by guest on 01 August 2024

,"
'~'"·1
"
'I ABSTRAC:r SURROUNDING
".1
.J This paper presents techniques for the mathematical
J!l-0deling of ph:rsical systems which are described by
S ft~i
-,- -

lInear and, non-linear partial differential equations. The .

;1 most popular numerical methods for the solution of non-
',-

."" linear and even some linear problems are based on finite-
,.j
:~;
difference approximations for the derivatil·es. This paper
presents finite-difference methods which are most suitable
DOMAIN
R
w.hen ch~nge~ are taki~g place in all three space coor-
dmate directions and In the time direction. The main
, concern here is '\-'ith physical systems - such as petroleum
"I" res~rvoirs - whic!I are often described by a set of para-
", bohc and/or elliptIc partial differential equations.
~1
BOUNDARY r
INTRODUCTION Figun~ l.-Hypothetical Physical System_

IN THE LAST FIVE YEARS OR SO) a great deal has hap- Note: (1) Laws of conservation of mass, momentum and
energy must be satisfied in R.
pened on the technological front. The engineer has be- (2) l~t,eraction with surroundIng, S. must be spe,
come more and more sophisticated in the use of mathe- cifled by a set of boundary conditions on r
matical tools, and the computers have become faster
and bigger to satisfy engineers' evel'-increasing de-
mands. As a result of these significant changes, the all three space dimensions with or without transient
partnership between engineer and computer is now effects wiII be referred to in this paper as three-di=-
perhaps stronger than the traditional partnership of mensional models. Often changes in one or two of the
engineer and slide rule. Engineers are more than mere space dimensions may be neglec.ted and we are left
,users of this new technology. Their contributions in with' a simplified one- or two-dimensional problem.
the areas of computer hard\'mre, software and numer- For example, changes in the z-direction are often
ical techniques are monumental. It is interesting to ne.glected to give a two-dimensional model of the
note that some of the greatest numerical analysts of reservoir_ Even simpler models are employed in the
our age received their first degree in engineering. analysis of build-up analysis or back pressure tests
These people were motivated by the need for the solu- where only changes in one direction (r-direc.tion in a
c~..lindrical coordinate system) are considered.
tion to difficult practical problems and not just by
mathematical beauty. Their efforts have resulted in
giant, put in some cases crude, steps_ The mathemati- BASIC STEPS FOR MODELING
~'
cians have made contributions of their own and in Regardless of the problem being stUdied, certain
addition, in many cases have tied up the loose ;nds steps are alwa}'s necessary in constructing mathemat- "
, left b}r the engineer regarding mathematical rigour. ical models of physical systems. Let us consider a
This is not intended to underrate the work of mathe- hypothetical system, as shown in Figure 1. The steps
maticians, but to put engineering contributions in to be followed in modeling are: '
their rightful place. , '
The computers of this generation are now at a stage (1) Gather as much information about the system
where engineers may no longer have to assume that as is practically possible and economically justi-
their physical system is one- or two-dimensional when fied.
in fact this assumption is not justified. All petroleum (2) On the basis of (1). make assumptions which are
reservoirs are three-dimensional in space, and phe- justified on physical grounds. For example, one "
"

~omena of interest are always changing with time, so

can assume that the system can be represented by
In fact there are four dimensions to be considered - either a one- or a two-dimensional model at this
x, YJ z and t*. The models which consider changes in stag-e.

* A cartesian coordinate system is employed in this dis- ;~ Associate Professor) Dept. Of Che1nical Engineer-
cussion. ing, The UniveTsity of CalgaTJI, Calgary, Alta.
Technology, Jonuary-March, 1968, Montreal 41
(3) "\Vrite the equations that describe the laws of The number of equations to be solved will be equal
consenration of momentum, energy and mass for to the number of independent variables. Usually, in
the system. Also, determine the phenomenological the solution of a transient problem, the s:~rstem of
relations (for example) shear stress - shear rate equations will include parabolic, elliptic and algebraic
relations) and necessarJr equations of state for equations. In certain problems) hyperbolic equations
the fluids in the system. Make the equations di- may also be encountered. Some convenient notation
mensionless and reduce the number of parameters will be introduced before presenting typical partial
to a minimum_ differential equations for systems involving more than
(4) Describe the interaction of the system with its one dependent variable. Let Lm be an operator defined
surroundings for all times, specif.y the initial con- by
ditions and make these relations dimensionless.
(5\ Make the assumptions that must be made in
order to get a solution for the problem. The ne-
cessanr assumptions will depend upon (a) the
competence of the person solving the problem and ••• (2)
(b) the accuracJ' desired.
(6") Solve the problem bJ· the "best"-suited method. Then equation (1) may be written as
All of the steps given above are important and L (Ul =
;;~
t - ~• • (3)
often difficult_ No attempt will be made here to an-

Downloaded from http://onepetro.org/JCPT/article-pdf/7/01/41/2166452/petsoc-68-01-07.pdf/1 by guest on 01 August 2024

swer all these points. Indeed, it will require a great in this new notation. The superscript ,~ has been omit-
deal of work and large number of pages to do this. ted in the case of a single equation, If aU/at = 0 in
Often the problems are such that the desired accuracy eqllati(J1~ (1), the problem becomes elliptic and may
can only be obtained if t.he movement of fluid in all be expressed as
three space directions is c·onsidered in step (6). Meth-
ods that can be used in this step are the topic of this L CU) '" ;:, • •• (4 I
paper,
A typical set of 1'1 differential equations for a com-
plex flow .system may be written as
TYPICAL EQUATIONS OF RESERVOIR
i'vIECHANICS
L!"I eu"l J u~·') ,
For convenience, typical equations of reservoir me-
chanics are considered in two parts. Simple flo,..' sys- rn = I, =, ,.. , s 5 : I.' · ,. 15i!)

tem where on!].. a single dependent variable is in- M

volved are considered first- L~ (if) '" ;P I >" ',',~, T, Ul, U~, U ~,

(a) Simple FloIV p '" 5 +- !, s .;. 2, . . , :·1. • • _ (50 )

The most general form of the equation used in iso- The first s equations (:ja) are parabolic and the re-
thermal, single~component, single-phase studies in maining M-s equations (.5b) are elliptic. The equations
porous media is are coupled if any of the coefficients

b
~
;j
-
~,
(I.,
..
~u
-) - to

in the mth equation contains an~r dependent variables

• . . (II other than UOI. Solution of a coupled system of non-
linear equations introduces additional complications.
where U maJ' represent pressure. The coefficients k",
k~, k•. bl<, b), b., a~, a." and a~ are either constants (in-
(c) BoundCLTY and Initial Conditions
cluding zero) or known functions of x, J', z, t and U. Interaction of the system with the surroundings is
The function ~ includes illl remaining terms (usually described bJr a set of boundarJr c.onditions. Let l' be
source terms in single-phase flow), and it may depend the bounding' surface of the domain of interest and
upon the independent variables x, J', z and t and the let R be the region inside this domain. For each equa-
dependent variable U in a known but arbitrarily com- tion, a condition of the type
plicated manner, The equation is non-linear if, and
only if, the coefficients are a function of U and/or E If'-I. If'''"!.
is a non-linear function of U. Analytical solutions of
even the lineal' problem described by l:qilation (1) rn = I, 2, ... I' • . (6)

are complicated, except for some highly simplified on r must be specified. The function of fm may also
cases. The analytical solution of the corresponding invoh'e any of the derivatives of U I. U\ ... V'R-I, ...
non-linear problem is impossible b~r kno\\'n mathemat- U"'- " ... UOI or it could be a constant. Here ii is n
ical techniques. direction normal tu the surface. In addition, for the
parabolic equation we must also spedfy, in R values
I b) Comple:t: Plow of all independent variables at some specified time,
t = O.
The system is described by a system of partial dif-
ferential equations if the problem involves
(i) non-isothermal flow, ... (.')

(ii) multi-component mixtures with non-uniform

concentration, Note that f'" and g'" are gi\"en functions.
(iii) multi-phase flow, or For a single equation. condition (6) reduces to (i)
(iv) any combination of the above three. the specification of the dependent variables, (ii) the

42 The Jgurnal of Canadian Petroleum

 --'----'~-- _.
, .'

specification of the normal derivative of the depend-

ent variable, or (iii) a' combination of (i) and (in.
Clearly, the boundary con-ditions and the partial dif-
ferential equations must form a boundar}r value prob-
lem that p'J"OPBTly describes the s,ystem being investi-
gated.
x-axis
NUMERICAL METHODS o i=o i=1 i=2
There are rnan~r different finite-difference methods
by which the solution of equations (3) (4) or (5)
;, may be attempted, Any of the methods used for the
,I,
,,, linear diffusion equation or the linear Poisson's equa-
, tion may be extended for use in the solution of the /--+-"7"'"---".1 k=2
1 problem on hand. Because of the complexity of the
.'
problem on hand, most of the methods will be imprac-
ticaL A second-order method is presented which is es-
., (
pecially suitable for the solution of systems of equa-
tions of the form bei ng considered here. It '''auld be

Downloaded from http://onepetro.org/JCPT/article-pdf/7/01/41/2166452/petsoc-68-01-07.pdf/1 by guest on 01 August 2024

necessar}r to introduce some nomenclature before go-
ing into the details of the method. As usual, the do-
main of interest is divided into a grid network. This
is shown in Figure 2 for the simple case of a cube_ FigU1'C 2.-GJ·id Netwo-rk for a eu.b,e.
The exact solution to the boundary value problem
Ie,n) and (i+l,i,Ie,n). The same points are utilized
(5) is the vector
when the second derivative a2 u/ox 2 is approximated
"'u' by a central difference approximation. Similar argu-
ments hold for the derivatives in the y and z direc-
U =

[ .

ul.\
tions_ This suggests that the alternating direction
methods developed for the linear diffusion equation
could be extended for use with the parabolic problems
of this investigation. Similarly, all of the iterative
methods used for the Laplace equation or Poisson
The value of anyone component of this vector at a equation could be extended for the solution of the more
grid point is given b)r c9mplicated elliptic equations being considered here.
This is precisely the method that has had the greatest
t '" 71 tl 1
Lf'l(x.v,~,t), x'" i tl x amount of success in several problems investigated
V ~ j tl Y by the author (of. Aziz and Hellums, 1967).
z " k tl z

The solution of the finite difference problem is in- PARABOLIC EQUATIONS

dicated by u""',j,I.:,n (the effect of round-off error is The method to be proposed is a modification of the
negligible in stable methods and will not be considered alternating direction implicit (ADI) method of Brian
here). The following definitions will be useful: (1961) and Douglas (1962). Consider a finite differ-
T ot ence approximation for equation (3). The solution is ': j

il 6x obtained in three steps: ." i.

tl)( ui ui;'1 - ui _ 1 (1) The problem is solved implicitly in the x-direction

2h and an intermediate solution, u~:-, is obtained:
k {u·l-u.>-k (u,:-U~.il'
'<i-l:\: 1-- :. xi~!~ - ~
"-------
... _ (B<I)

nx o., •
x 0,
,
(2) The problem is solved implicitly in the y-direc-
Note that in the above definitions, indices i,1.: and n tion and an intermediate solution, u**, is ob-
have been sllpressed from each term, i_e., instead of tained: <,

Ul+l, J, k} n simply UIH is used.

Ih With, the above definitions, it is 'easy to show that

Ui~,?,k,I'I+1 -
, ui .:i,k,1'I ;. F;:::,.f,k.n+!r •• (!3b) ,-, -
-" ,
I
1 2. 2- (3) The problem is solved implicitly in the z-direction
=~ ~\ [U":,J.k.n'H + UiJ .j • k ,1!] -I- 0 (;/ + T ). and the final solution, u, is obtained:
Similar equations may be obtained for other space -de-
rivatives. The time derivative is approximated by :lfn (u·_- +u._ l+~o (U\,I. +u __ )
x 'I..J. k ,11+1 ':.,J,k.J1 Y 1-,.1 •.21:.1'1+1 :.,,1,k,"

" ui,.i,k,tl+1 ui.d...~ ~

,- 0 ('[2)
;. ~ °z {Ui,j,k,tl+J ;. Ui,j,k,n) = ui ..i.l::. n + 1 - ui ,,?,k,11 1
T

}\.. point 'Worth noting here is that the Q" operator (Be)
involves the use of only three points, (i-l,i 1 k,u)J (i,j,

Technology, January-March, 1968, Montreal 43

Equutions (Sb) and (8c) may be simplified by sub- From the above discussion, it is clear that u* un-
tracting (B",) from (Bb) and (Bb) from (Bc)~ This knowns are solved simultaneously for one x-grid line
yields at a time. Similarly, u'l ,."" unknowns al'e solved for one
y-grid line at a time and u unknowns are solved for
'~ny (uI:.1 ,1:,'I... 1 - ui.,.i,~,r:!) ui::i,k,n+l - Ui"j,k."d • (JtJ l) one z-gdd line at a time. Solution along anyone grid
, line involves the solution of n matrix equation with n
;Jnd tri-diagonal coefficient matrix (10), which is the type
of equation obtained in one-dimensional problems.
Hence, a three-dimensional problem has been reduced
EquLlti(lr~ (Sa), (8b l ) and (8c l ) may be arranged in to a series of one-dimensional problems. The addition-
al local discretization error introduced by this alter-
a more convenient form by putting the unknown quan-
tities on the left-hand side: nating direction procedure' is 0 (h~ -::l) for the lin- +
ear diffusion equation. It seems that the result should
hold for the non-linear problem on hand .
Cn, - .".,
, ui,.i,k,n+1 -- , • '" • 'n, + -'<--c
(!l'
). ui,;i,k,n The question of the stability of this procedure is
. 2\. ,j.:.:. •..,+ ~
(g,d
difficult to investigate analyticall,Y. The eqlIations (9)
involve coefficients ,...· hich are not known at the step
, n+ 1 until the solution at that step is obtained. Hence,
<0
,- , 01 ,"
;:,J ,k,l1+1
- n , ui ,J, 1;: ,1'1 - 2
U~ ,.1 ,k ,,,+I (9b) some iteration or "prediction correction" is always in-

Downloaded from http://onepetro.org/JCPT/article-pdf/7/01/41/2166452/petsoc-68-01-07.pdf/1 by guest on 01 August 2024

volved in the solution of such a non-linear system. For
-',-I
",
(Q - - , u~ ~.
" ~ ..i.k,,,+1 :, ,~~ ,k ,'1 t ,J,~ ,,,+1
( 9;:) this reason, no method can be uTIconditionall:y stable.
The step size to ensure stability must almost always
The unknowns at variolls grid points are ordered be determined by experimentation on the computer.
differently for each of the equations (9"'), (9b) and The author has found that the best approach is to
9c). For equation. (9a), the i-index changes most rap- e"aluate all coefficients at the old step in the first
idly to yield (J-11 X (K-l) matrix equations: iteration, then to use the previous iterate values for
the coefficients until the solution converges. Usually,
three or foul' iterations are required at each time
... .I-I ..lnj step.
f.-I. '01
ELLIPTIC EQUATIONS
~I""r" ,'r-I)',(I-II '''It'-I' H., IS ol lridic.'l:>r,;J1 n,~tJ'l~ ot I'lL' t,,-r",
.J,'.
There are two highly useful methods for the solu-
tion of elliptic problems of the form of equa.ti01l (4).
Approximating the derivatives by finite-difference
approximations, one can write
( 17)
, till

H ••
"" as an approximation for (~). An ADI scheme similar
to the one proposed for the parabolic equation mll~'
':1-.' be derived for the .solution of equation (12):

'-1-1

"'' ..,,', . .,
Ii, - " I ,
"i "i,~, "t I
y
" u.rJ,},v+1 y u .. ,.i.!r, "
" " 'OJ
co , -
"" I , -:.. "I,' _'+1
, , ".~,.1, ;', v
u
I,.i. ,
~.
. " " I

This method is similar to the one proposed by Douglas

(1962) fol' the Laplace equation. A major prohlem in
the use of this method is the selection of an optimum
sequence of iteration parameters, -'1'1. A crude method
The value of u'· at all grid points is obtained by solv- is to select the parameters for the Laplace equation
ing (J-1) X f,K-1) matrix equations with 1-1 un- (cf. Douglas, 1962; \Vachspress, 1966) and improve
knowns in each equation. these by experimentation on the computer. The equa-
tions are solved at each iteration level by the proce-
Similar procedure is followed fOl" the solution of dure outlined for eqnations (,?a). (9b) and (9c). the
(91J), where the unknowns ll"",ofi,i,r..,nT' are ordered bJ'
only difference being the substitution of TI'I for 2/T_
allowing the ,i-index to e-hange most rapidly. FinallY,
un1mowns in (l1-c) are ordered so that the k-index va- The ADI method i.5 recommended for the elliptic
ries most rapidly. The reason for the name "alternat- equation when the grid points being considered are
ing diredion" should now be cleaL The coefficient large. A very rough l'ule to follow is that this method
matrix in each case is of the form given in equation be used ,....henever the number of grid points along any
(10). It has been assumed here that the boundary of the grid lines is greater than 10.
conditions either yield the value of the unknO'''..n at For a coar1:ie grid network, a procedure which is
the boundar,}' or the value of this unknmvn in terms much simpler to program is recommended. This is the
of other unknowns ah'wdy in the particular scalar \ovell kno\,·m successive over-relaxation (SOR) in three-
equation being considered. dimensions. (Equation (12) in expanded form, may

44 The Journal of Canadian Pctroleum

 POOR IMAGE DUE TO ORIGINAL DOCUMENT QUALITY

._'- .L.-......J.-
- - - " - - ' , -,-~.. , .',

be written as

+ B••
"I.,J- , ,. . I k+ B7.,.1,i'.
k"'"I.,J-, . • k + Ii . . I k'
. . , u1"J,' 1,~.7+,
u •• , k
1, •.1+, TIME STEP COUNTER
MAXIMUM VALUE "'N
'" <!: • • ,
7.,.1,i'.
OUTER INTERATION
(III) COUNTER
wherE!

NUMBER OF EQUATION BEING

SOLVED MAXIMUM VALUE:M i .;
~~ '"
r:s"O"C"V:::E'-=F:::O:::R--' I.THIS STEP INVOLVES THE SOLUTION [-.-J.
OF m1h EQUATION BY THE
,,.
a i,d,h
. -- L
- (lIx)l b Ck + k I
um,q
I
'..'::.
xi,d,H.. xi-!:;,d,k Xit'-1.,.i,k APPROPRIATE METHOD.
i,l,k , " ~:.=
FOR ALL POINTS 2..VALUES OF DEPENDENT ,VARIABLES I: .
'Cb
(~2 k I+ '~b) la ) Y S
REQUIRED AT THIS STAGE BUT
NOT AVAILABLE AT q LEVEL OF
,! -.
: , Xi,.i,k xi:+'-i,.i,k -i,,i,k m<M?

Downloaded from http://onepetro.org/JCPT/article-pdf/7/01/41/2166452/petsoc-68-01-07.pdf/1 by guest on 01 August 2024

ITERATION ARE EVALUATED AT
I
Bi,d-I,k" (;[""yI2 (by k I
, NO THE LEVEL (q-IL
- 211ly) CONVERGENCE 3.ZERO LEVEL IS ASSUMED TO BE
i:,,i.k Yi,j-~,k NO , OF
PREVIOUS STEP.
OUTER ITERATION
5.. ,,-.!. 'Z b (k + k I
1..,a,k (lIy) Yi"i,k Yi,,i-!.:f,k Xi,j~,k

'--YL.!E""S,-o-( "< N ~

)"
'~.,.
.'
-
~: .
NO .,' ~

;\, ..
+
~.;!:
b (k k ) STOP
Zi,j,k Zi,j,k_lJ Zi,j,k+~
.' -.
Figm'e 8.-Logic Flow Cha1't.

Note that the above notation may be convenient in the The selection of iteration parameter w is relatively
solution of both equations (9) and (18). simple for a rectangular prism and a linear problem
Let (,) indicate the iteratio'n level ss in (13). Equa- (cf. Young, 1955):
tion (14) may then be solved by
." ,;:~.
w '" I + [ --'-I'
I +.rr+P"z • , _ (17)

"i,j, k ,'11+1 "---"-[-~

I'i,j,k . . ,.+a.,
t.,J,"-
.,.U.,
L- ,J,Y. '. ,
t.- ,,7,Y.:,\J~
'....here

, I
[(li"xl 1 eos

~
<l;I )2 cos (L I +
y y

••• (I S)
L", Ll • and L" are lengths in the x, Y, and z directions
respectively.
For 6x = 6Y 6Z and L" = Ly
1"

Note that for the Poisson equation p " cos ("hl

For an irregular domain, a first estimate of w may be

VLU :: t, with lIx" b.y " r.z = h obtained by using a rectangular prism which encloses
the volume being considered.

SYSTEMS OF EQUATIONS
=_3- Now that methods for a single parabolic and a
h'
single elliptic equation have been developed, methods
for systems of equations may be proposed. The order
in which the equations are solved will depend upon
~-,. '.
For this simple problem, equation (15) becomes the problem being considered_ One such example where
seven simultaneous partial differential equations are
solved is discussed by Aziz and Hellums (1967). Ideal-
l}~, equations should be ordered in increasing degree " ~'

of coupling with other equations, and the equation

\vith the weakest couplfng should be solved first. The
+ ui,,,i_I,k,v+1 + ui,d+l,k,u + Ui,,i,k-I,ui-/ + Ui,j.k~l.v
basic procedure is best illustrated by a logic flow
chart, as shown in Figu1"e 3_ Here, n.r equations are
, •• (16)
being solved and m = 1, 2, 3, . }1 has been selected !.

Technology, January-March, 1968, Montreol 45

 POOR IMAGE DUE TO ORIGINAL DOCUMENT QUALITY

as the order in which the equations must be solved. NOMENCLATURE

Some of these equations are elliptic and some para-
bolic.
'rhe stability and convergence of such a scheme is
not al,',:ays assured. Sometimes, considerable experi-
mentation with various orderings of equations, time
step sizes and grid sizes are required before satis-
".;./ Id,li.'1' ·,,1 . - .. u I " '1. ' .",; , (10)'
factor}' answers can be obtained. Engineering intui- L,L" "e- •.>I,., ,.' r • \. '-'" I .. ,

tion, mathematical ability and 'a real feel for the ;<I'·r.·'o)" ';,10',
physical phenomena being investigated are important u·. ,," ;,,1, rr, Ii ,', ",'"

ingredients in a reliable solution. Often some conveni- d'._-,

,) ~;r~lh
, ,
,!"
"",",
,~I
.
" , , ' ; '..
I'· ,.• , , . I';,,,

ent transformation of variables reduces or eliminates

d-, , ' , , ' ", " I " oil". "11'-:'"
the degree of coupling between equations and thus
I;',;h'-lif'
makes the equations easier to solve. This was the case '" I, ' J ,,', ",

in the problem reported b~' Aziz and Hellums (1967)_ ., .

~,

;.,1,-,
S(',I,-
It,- , .. " , "
"
,I, ", • ,,,,,'"

CONCLUDING REMARKS 1,.,- .. ' Q' iJ - '; ,

"
l.--Solution:5 to three-dimensional problems in reser-

Downloaded from http://onepetro.org/JCPT/article-pdf/7/01/41/2166452/petsoc-68-01-07.pdf/1 by guest on 01 August 2024

yoir and fluid mechanks are lJossible with present- ,',• .1., .' ,I"" ", . '., ",,_
day computers.
2.--The amount of computer work required to do a , ~~"., ' ," ,,
three-dimensional problem as opposed to the cor- "h.,:;
responding t\"v'o-dimensional problem rna}' be 10 to
50 times l20 times for most problems) larger. ~,,:' 'I ,-
3.-The question of the stability and convergence of O. I . . • r, '"
i',
various schemes becomes more critical than it is
with two-dimensional models_ 0, I, • J , .. " ,I" "

4.-The competence of the engineer in both numerical

0, I, ,,', " "
analysis and in understanding the physical sys-
tem being investigated are extremely important. ", I, .' r"

Othen··..ise, the use of such elaborate procedures

,,,,ill not yield the desired result.
" "", , '.. h "'" ," ,[." "

~" ,
REFERENCES
(1) Aziz, K., and Hellums, J. D., "Numerical Solution of
the Three-Dimensional Equations of Motion for La-
minar Nautral ConYedion," Physics of FLUids, 10, Dr, Khalid Aziz: IS assistant professor
314, (1967). of chemical engineering at the Univers-
(2 ) Brian, P. L. T., HA Finite Difference Method of High ity of Calgary, He wos born in Pakistan
Order Accuracy for the Solution of Three-Dimensional and received his engineenng education
Transient Heat Conduction Problem," A.l,Ch,E..!.• 7, ot the University of Michigan, the Uni-
367, (1961). versity of Alberto ond Rice University.
(3 ) Douglas, J., Jr., "Alternating Direction Method for He has worked for the Korochi Gas Co,
Three SpaC'.e Variables I" Nll'nz. Ma.th .• .4, 41, (1962). in Pakiston in various ;;apacities. in-
(4 ) "\Vachspress, E. L., Ucra.tive Solllti.on of Elliptic cluding thot of chief engineer, and has
EquutioHS, Prentice Hall, N.J., (196G). taught at the University of Alberto for
(5 ) Young, D .• "Iterative Methods fOl" Solving' Partial three years.
Differential Equations of Elliptic T,,\'pe," T1·UHS.
A me?·. MaUL. Soc., 76, ~12, (1954).

46 The Journal of Canadian Petroleum