H. K. van Poolen

H. K. van Poolen

Marathon Oil Co.

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Paper presented at the Numerical Simulation Symposium, Dallas, Texas, April 1968.

Paper Number:
SPE-2021-MS

Published:
April 22 1968

SOCIETY OF PETROLEUM ENGINEERS OF AIME

6200 North Central Expressway

PAPER

NUMBER

SP E 2021

Dallas, Texas

75206

THIS IS

A PREPRINT --- SUBJECT TO CORRECTION

Solution of the

Immiscible Fluid Flow

Simulation Equations

By

E. A. Breitenbach, Junior Member AIME, D. H. Thurnau and H. K. van Poolen, Members AIME,

Marathon Oil Co., Littleton, Coio.

Â©

Copyright 1968

American Institute of Mining, Metallurgical and Petroleum Engineers, Inc.

This paper was prepared for the Symposium on Numerical Simulation of Reservoir Performance,

to be held in Dallas, Texas, April 22-23, 1968. Permission to copy is restricted to an abstract

of not more than 300 words. Illustrations may not be copied. The abstract should contain conâ€¢

spicuous acknowledgment of where and by whom the paper is presented. Publication elsewhere after

publication in the JOURNAL OF PETROLEUM TECHNOLOGY or the SOCIETY OF PETROLEUM ENGINEERS JOURNAL

is usually granted upon .request to the Editor of the appropriate journal provided agreement to

give proper credit is made.

Discussion of this paper is invited. Three copies of any discussion should be sent to the

Society of Petroleum Engineers office. Such discussion may be presented at the above meeting and,

with the paper, may be considered for publication in one of the two SPE magazines.

ABSTRACT

breakdown requires definition of the numerical

solution to be used. This paper describes the

breakdown and solution processes most often

used in the MUFFS program. Sufficient detail

is given so that computer programming can be

done.

This paper presents the methods used to

solve the finite difference equations which were

developed in a companion paper (1).

Various possible methods of solution are

discussed. Experience has narrowed the number

of suitable numerical methods that are practiâ€¢

cal to three: Gauss elimination, successive

overrelaxation, and the iterative alternating

direction implicit process.

Contrary to popular opinion, economic simâ€¢

ulation has been found to require the developâ€¢

ment of several solution methods, rather than

relying on a single one. This requires that

the computer subprogram for generating coeffiâ€¢

cients (A's and O's) be written as

a distinct,

The final sections of the paper are devoted separate entity to supply the coefficients in

presentation of computational techniques

to

a

Equation (1). Furthermore, it is necessary to

be able to obtain these coefficients automatiâ€¢

cally in column-by-column, row-by-row, or pointâ€¢

by-point form, in any order required by a numerâ€¢

ical solution. Columns, rows, and points refer

to the columns, rows, and points of the finite

difference grid. A program that can generate

which are vital to actual use of each of the

above-mentioned methods.

FllUTE DIFFERENCE EQUATIONS, THE MATRIX, AND

DEFINITIONS

coefficients in several forms is

a simple but

The final finite difference equation for

pressure developed in Reference (1) is:

important concept, for it allows the easy inserâ€¢

tion and modification of experimental methods.

The computing inefficiencies that may be incurred

+

+

tl_{-p}to

within

a general coefficient generator are small

O

+ A9 =

y

y

in comparison to the computing time saved by

using the fastest of several solution techniques.

â€¢ . â€¢(1)

Final Finite Difference Forms

All the terms are defined in the paper. Here,

however, we have dropped the subscript denoting

the pressure, p, as an oil pressure. Further

Equation (1), when confined to

space dimension, becomes:

a single

References at end of paper.