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AL"M&RHOUN; MUHAMMAD ALI OPTIMAL NUMERICAL PROCEDURE TO 80LVE two- d im e n s io n a l t h r e e - phase PETROLEUM RESERVOIR SIMULATOR, THE UNIVERSITY OF OKLAHOMA# PH.D.# 1976

UniversiN

/vOoOTlms

IntematiCXVll

300n zeebroad, ann arbor, m i 48106

©

1978 MUHAMMAD A L I

A

L

L

R

I

G

H

T

S

AL-M AR HO UN

R

E

S

E

R

V

E

D

THE UNIVERSITY OF OKLAHOMA. GRADUATE COLLEGE

OPTIMAL NUMERICAL PROCEDURE TO SOLVE TWO-DIMENSIONAL THREE-PHASE PETROLEUM RESERVOIR SIMULATOR

A DISSERTATION SUBMITTED TO THE GRADUATE FACULTY in p a rtia l f u lf illm e n t o f the requirements fo r the degree o f DOCTOR OF PHILOSOPHY

By MUHAMMAD ALI AL-MARHOUN Norman, Oklahoma 1978

OPTIMAL NUMERICAL PROCEDURE TO SOLVE TWO-DIMENSIONAL THREE-PHASE PETROLEUM RESERVOIR SIMULATOR A DISSERTATION APPROVED FOR THE DEPARTMENT OF PETROLEUM ENGINEERING

By

/

ACKNOWLEDGEMENTS The author wishes to express his appreciation to the Chairman o f his Graduate Committee,

Dr. H.B. Crichlow, D ire cto r o f the department o f

Petroleum and Geological Engineering, fo r his assistance in connection w ith th is work. Appreciation is also extended to Dr. D.E. Menzie, Dr. A.W. McCray o f Petroleum Engineering Department, Dr. J.A. Payne o f Computing Sciences Department, and Dr. A.A. Aly o f In d u s tria l Engineering Department fo r th e ir re v is io n o f th is d is s e rta tio n and fo r serving as members o f the a uthor's Graduate Committee. P a rtic u la r thanks are due to the U n ive rsity of Petroleum and Minerals o f Dhahran, Saudi Arabia, fo r th e ir fin a n c ia l support. The author also wishes to make a special recognition o f his parents f o r th e ir constant support and in s p ira tio n .

ni

TABLE OF CONTENTS Page ACKNOWLEDGEMENTS ................................................................................... i i i LIST OF TABLES..........................................................................................vi LIST OF ILLUSTRATIONS............................................................................ v f i Chapter I. II. III.

INTRODUCTION ..........................................................................

RESERVOIR MODEL...........................................................................9 FINITE DIFFERENCE 3.1 3.2 3.3 3.4

IV.

Im p lic it Solution o f Pressure ............................... 18 E x p lic it Solution o f S a tu ratio ns..............................25 D is c re tiz a tio n o f Flow C o e ffic ie n t (hX) . . . . 27 Boundary Conditions .....................................................32

Successive Overrelaxation ...................................... 34 A lte rn a tin g D irection I m p lic it ................................. 38 Strongly Im p lic it ...................................................... 47

DIRECT METHODS.........................................................................62 5.1 5.2 5.3 5.4

VI.

SYSTEM .............................................. 18

ITERATIVE METHODS.................................................................. .3 3 4.1 4.2 4.3

V.

1

LU F a c to riz a tio n ............................................................ 62 Ordering Schemes............................................................ 66 Generate-And-Solve Algorithms ............................... 77 Band M atrix T echnique.................................................77

A NEW APPROACH TOTHE APPLICATION OFDIRECT METHODS IN PETROLEUM RESERVOIRSIMULATION........................ 80 6.1 6.2 6.3

Restricted A lte rn a tin g Diagonal Ordering. . . . 80 LU Factorization Applied to RAD..............................84 Implementation o f RAD Scheme in to the Simulation Package......................................................... 92

IV

Chapter V II.

Page

SIMULATION PACKAGE ................................................................ 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9

V III.

Input S e c tio n ........................... 94 One-Time C a lcu la tio n Section................................... 99 Time Control S e ctio n ...................................................... 101 C o e ffic ie n t M a trix C alculation Section................... 103 Pressure S o lu tio n Section ....................................... 104 Saturation C a lcu la tio n Section...................................104 Mass Balance S e ctio n ...................................................... 104 Output S e ctio n .................................................................. 106 A uxiluary Section ...................................................... 106

COMPARATIVE EVALUATION ........................................................ 8.1 8.2 8.3 8.4 8.5

IX.

94

110

D if f ic u lt ie s o f Ite r a tiv e Methods ....................... 110 Storage Requirements...................................................... I l l CPU Time Com parison...................................................... 112 Mass Balance E r r o r s ...................................................... 112 Average Reservoir Pressure.......................................... 125

CONCLUSIONS....................................................................................128

NOMENCLATURE ..........................................................................................

130

REFERENCES.................................................................................................. 136 APPENDICES A.

Units and ConversionFactors ......................................

140

B.

Flow Chart Convention....................................................... 141

C.

Constant Values Used in This S tu d y .................... 142

D.

Geometry and Properties o f the Reservoir Model

E.

Computer L is tin g o f the Numerical Procedures

F.

A Sampleo f Computer Simulation Output .................

.

144

. . 158 176

LIST OF TABLES Page

Table 5.1

Work and Storage Requirements fo r ordering schemes......................74

8.1

Computer Storage Requirem ents........................................................... I l l

8.2

CPU Time fo r Homogeneous C ase........................................................... 113

8.3

CPU Time fo r Heterogeneous Case ...............................................

115

8.4

Total Mass

Balance R elative Error

fo r Homogeneous Case. .

117

8.5

Total Mass

BalanceR elative Error

fo r Heterogeneous Case.

118

8.6

O il Mass Balance R elative Error fo r Homogeneous Case. .

.

119

8.7

Oil Mass Balance R elative Error fo r Heterogeneous Case.

.

120

8.8

Water Mass

BalanceR ela tive Error fo r Homogeneous Case. .

121

8.9

Water Mass

BalanceR e lative Error

122

8.10

Gas Mass Balance R elative Error fo r Homogeneous Case. .

.

123

8.11

Gas Mass Balance R elative Error fo r Heterogeneous Case.

.

124

8.12

Average Reservoir Pressure fo r Homogeneous Case ................

126

8.13

Average Reservoir Pressure fo r Heterogeneous Case . . . .

127

VI

fo r Heterogeneous Case.

LIST OF ILLUSTRATIONS Figure

Page

1.1

Stone's Comparative Study. . .....................................................

4

1.2

Breitenbach^et a l . 's Comparative S tu d y .......................................5

1.3

Watts' Comparative Study .............................................................

1.4

Welnsteln^et a l . 's Comparative S tu d y .......................................... 7

3.1

D iscretized Reservoir System ...................................................... 19

3.2

Cell System D e f in it io n .................................................................... 20

3.3

P ro file o f Two Adjacent C e lls ........................................................ 28

3.4

Flow Chart fo r the C alculation of In te rb lo ck R elative

6

Perm eability ...................................................................................... 31 4.1

PSOR Flow Chart....................................................................................35

4.2

LSOR Flow Chart....................................................................................37

4.3

Typical P lot o f Relaxation Parameter a t Fixed Tolerance. . 39

4.4

Flow Chart fo r Close-Band Thomas Algorithm ........................... 42

4.5

Flow Chart fo r Wide-Band Thomas Algorithm ................................. 45

4.6

ADI Flow C h a r t................................................................................... 46

4.7

Cell Arrangement................................................................................48

4.8

M atrix

4.9

L and U Matrices fo r Odd-Numbered I t e r a t io n s ..........................52

4.10

B M atrix fo r Odd-Numbered Ite ra tio n s ....................................... 53

4.11

L and U Matrices fo r Even-Numbered Ite ra tio n s ..........................56

4.12

B M atrix fo r Even-Numbered Ite ra tio n s ......................................... 57

4.13

SIP Flow C h a rt.................................................................................... 61

In SIP....................................................................................51

v11

Figure

Page

5.1

J ï P= d System o f Equations.................................................................. 63

5.2

LU Fa ctorization o f % .......................................................................... 65

5.3

Standard Ordering........................................................................................68

5.4

Diagonal Ordering. .

5.5

M atrix X Corresponding

to Standard Ordering.............................. 69

5.6

M atrix X Corresponding

to Diagonal Ordering.............................. 70

5.7

A lte rn a tin g Point Ordering ................................................................

5.8

A lte rn a tin g Diagonal Ordering................................................................ 71

5.9

M atrix X Corresponding

5.10 M atrix X Corresponding

........................................

68

71

to A lte rn a tin g P oint Ordering . . . .

72

to A lte rn a tin g Diagonal Ordering. . .

73

5.11 A lte rn a tin g Line Ordering........................................................................ 75 5.12 M atrix X Corresponding to A lte rn a tin g Line Ordering......................76 6.1

RAD Ordering Along the Shortest Diagonal ...................................

6.2

RAD Ordering Along the Longest Diagonal........................................... 81

81

6.3

M a tr ix X Corresponding

to RAD Along the Shortest Diagonal. .

82

6.4

M atrix X Corresponding

to RAD Along the Longest Diagonal . .

83

6.5

General Form o f M atrix X o f RAD............................................................. 86

6.6

The Composite M atrix C} o f RAD...............................................................87

6.7

The Lower Quarter o f M atrix C| o f Semi Bandwidth m........................89

6.8

RAD Ordering Enclosing Standard Ordering ...................................

93

7.1

Simulator Flow C h a r t ................................................................................ 95

7.2

The Type o f Well I d e n t if ie r .....................................................................98

7 .3

Simulation Numbering Scheme................................................................... 100

8.1

CPU Time vs. Real Time (Homogeneous Case-345 C e lls )...................114

8.2

CPU Time vs. Real Time (Heterogeneous Case-345 C e lls ). . . . 116

V lll

Figure

Page

E .l

D iscretized Reservoir System.............................................................. 148

E.2

Reservoir Thickness ............................................................................ 149

E.3

D ig itiz e d Reservoir Thickness ........................................................ 150

E.4

Top o f the Reservoir Depth.................................................................. 151

E.5

D ig itiz e d Middle o f the Reservoir Depth .................................... 152

E.6

D ig itiz e d Reservoir I n i t i a l Pressure............................................... 153

E.7

Reservoir Heterogeneous P e rm e a b ility............................................... 154

E.8

D ig itiz e d Reservoir Heterogeneous P e rm e a b ility ........................... 155

E.9

C onfiguration o f W e lls ..........................................................................156

E.IQ Production - In je c tio n FlowR a te s .....................................................157

IX

CHAPTER I INTRODUCTION Mathematical models fo r petroleum re servoirs incorporate the re s e rv o ir physical p rop e rtie s and the in te ra c tio n o f natural and a r t i f i c i a l forces to sim ulate re s e rv o ir behavior.

Therefore,

mathematical sim u la tio n helps in understanding re s e rv o ir behavior. Such inform ation leads to the most economically de sirable form o f e x p lo ita tio n . Reservoir sim ulators are used to design the most economical secondary recovery program and to la y out the complete re s e rv o ir management from discovery to d e pletio n. Natural re se rvo irs con sist o f one, two or three phases ( o i l , water, and gas) w ith various geometries. sim ulators were developed.

Therefore, various re s e rv o ir

They s ta r t w ith one-dimensional one-phase

simple sim ulators and end up w ith complex three-dimensional th ree phase ones.

Although three-dimensional sim ulators are now a v a ila b le ,

the most popular sim u la tor is the two-dimensional one because a w e llorganized two-dimensional sim ulator can be used to approximate thre edimensional one a t a lower co st. The e ffic ie n c y o f a sim ulator as a working and economical tool hinges upon the a b i l i t y o f it s algorithm s to solve the pressure equations e ffic ie n tly .G

Most o f the computer time is spent in performing pressure 1

s o lu tio n s .

For th is reason, the best sim ulator is the one which

requires the minimum work during the pressure so lu tio n s. The pressure equations can be solved e ith e r by ite r a tiv e or d ire c t methods. Ite r a tiv e methods are by fa r the most common methods o f solving the pressure equations used in sim u la tio n .

This is because ite r a tiv e

methods are easy to program and re q u ire less storage and less computation compared to e x is tin g d ir e c t method a lg orithm s.

Hence during the early

years o f work in th is d is c ip lin e , the d ire c t s o lu tio n techniques were almost completely abandoned as a s o lu tio n process.

However, in

recent years w ith the development o f sparse m atrix techniques, the d ire c t so lu tio n process has become less c o s tly , w hile producing as good a re s u lt.

For th is reason, d ir e c t s o lu tio n techniques are once

again being considered in re s e rv o ir sim ulation areas. The e x is tin g numerical methods ( d ir e c t or ite r a tiv e ) to solve the mathematical models o f petroleum reservoirs are time consuming. Those methods consume d iffe r e n t computer time when applied on reservoirs w ith d iffe re n t c h a ra c te ris tic s . The computer time is expensive, th e re fo re , a tool to choose the fa s te s t, most e f f ic ie n t methods fo r a given re s e rv o ir is needed. A comparative study helps to show the features o f each method.

Such

inform ation leads to the choice o f the most economically s u ita b le method. The comparative studies done in lite r a tu r e are lim ite d to two or three methods and most o f these comparative evaluations are done on a system o f simulaneous equations not as p a rt of a complete sim ulation. Also most o f these comparative studies used idealized model o f fixe d rates and square re s e rv o ir w ith pre-determined flow c o e ffic ie n ts .

Stone^^ has presented a comparison among the fo llo w in g numerical procedures:

a lte rn a tin g d ire c tio n Im p lic it (AD I), stro n g ly Im p lic it

(S IP ), p o in t successive ove rre laxa tion (PSOR), and p o in t Jacobi. study was in terms o f computational work and re s id u a l.

His

He used an

Idealized square re s e rv o ir w ith fix e d flow rates and a combination o f homogeneous regions.

His re s u lts are presented In Figure 1.1.

Stone

concluded th a t both ADI and SIP possess convergence rates a great deal fa s te r than those o f the other two methods. Is s lig h t ly fa s te r than SIP.

For homogeneous case, ADI

However, SIP Is s ig n ific a n tly fa s te r

than ADI In case o f heterogeneous re se rvo irs.

Also he concluded th a t

PSOR Is more rapid than p o in t Jacobi method. Breltenbach, et a l.^ gave a very b r ie f comparison between Gauss e lim in a tio n , PSOR, and ADI.

For his model, he showed th a t PSOR and

ADI are more e f f ic ie n t than Gauss e lim in a tio n fo r more than 9x26 c e ll system.

His re s u lts are shown In Figure 1.2.

W a t t s h a s presented a comparison among ADI, SIP, lin e successive o ve rre la xa tio n (LSOR), and corrected LSOR In terms o f accuracy. re s u lts are shown In Figure 1.3.

He concluded th a t fo r two-dimensional

s tro n g ly a n iso tro p ic problems, the corrected other a v a ila b le techniques.

His

LSOR Is fa s te r than

However, In homogeneous Is o tro p ic square

problem, the corrected LSOR Is slower than ADI, SIP, and LSOR. W einstein, e t a l.^ ° have compared SIP w ith ADI fo r a two-dimensional two-phase model.

Their re s u lts are shown in Figure 1.4.

They found th a t

the computational work r a t io , which Is the r a tio o f ADI computer time to SIP computer tim e, g e nerally Increases w ith Increasing tra n s m ls s lb llIty (Pjj^ax^*

T^^Gy also found th a t ADI f a lls to converge fo r higher

tra n s m ls s lb llIty .

rPOINT-JACOQI

^/O V E R R E LA X A TIO N

g 10

ADI

STRONGLY IMPLICIT

FI XED R A T E j KY = 0. 01 31 X

31 G R I D

20 .30 40 50 COMPUTATIOMAL W ORK ( O R N O . O F I T E R A T I O N S FOR S T R ON GL Y I MP L I C I T M E T H O D )

POINT-JACCDI

Q IC

OVCRRTLAXATION

10“= ,- S T R O N G L Y I MP L I C I T

ADI

10-

i

20

30

40

SO

COMPUTAIlOrJAL W ORK ( OP N O . OF I T T R A T I O N S F O R S T R O N G L Y I MP L I C I T M E T H O D )

Figure

1 .1 -S to n e ’ s Comparative Study

Method Grid

Go u ss

SOR

ADTPIT

*** 3X5

2

5

3 x 5 x 2

5

12

9 X 10

8

11

-

18

25

19

21

40

21

21

22

13

X

11

9 X 26 9

*

26

X

5 **

**

-

-

26

X

23

94

59

62

34

X

36

302

122

111

20

X

24

564

110

X

2

-

*■ Vertical cross section ** Line SOR *** Time in second

Figure

1.2 - Breltenbach, e t a l . ' s Comparative Study

C o m o u la tio n jI Work per

C cm piilational Work per Ite ra tio n

Ite ra tio n

I SOR

I SOR

10

ISORC S IP ADI

ISORC AOI

2

10?

3

101

n n

a 10

s

II}

10

6 0

100

10

a

s

6 0

C o r.p u ta litn a l W on tor num ber of iterations

«

60

C om putational Work

lor IS O S l

tor num ber

— C O N V K R r.r .N C E CO M PA R ISO N ’ S. H O M O G E ­ N E O U S S Q U A R E 3 1 x 3 1 G R ID , t / t y .

Figure

?0

B

c l iln ra lio n s

1er ISORl

— C O N V E R G E N C E C O M P A R IS O N S . H E T E R O G E ­ N E O U S S Q U A R E 31 X 31 G R ID . 100 ty .

1 . 3 - Wat t s ' Comparative Study

Number P o ro m e te r P r o b le m D is s o lv e d G D rive C a s e 1*

Ite ra tio n P rocedure

''m o . qs

D i s sol v e d G as D riv e

G o S' 0 i 1

W o le r« O il R a d io l C o n in g

. of N um ber of

A ve ro g o Ite ra tio n s /

I t e r o t io n » /

T o to l T im e

W or k

St o p »

ite r o tio n s

T i m e Step

B lo c k )

(m in u te .)

R o tio

3

189

23.6

2.41

3.83

8

247

30.9

2.26

4.69

8

193

24.2

2.40

3.89

M ax SIP \

V M in A D I P /

SI P

0.9 9 9 9 4 2 ,

A D IP

0.5 6 9 X

SI P

0.9 9 9 9 9 4 2 ,

A D IP

0.231

SI P

0 .9 9 9 9 9 8

A D IP

0.2 0 3 X

SI P

0 .9995

9 "

A D IP

0.001

9**

I

66 lO "', 0 1

1 .2 2

660

C o s e 11*

In c o m p re s s ib le

/

I t e r of io n T im e ( m ill i l e c /

X

oo

lo " *, 0 25

77

3. 1

1.97

0.9 9 3

25

389

1 5 .6

1. 4 8

3.75

36

9 .6

1 .9 0

0.65

3,600

7.05 X

1 0 '^

3 .7 8

10^

* A l l K^, o f C o s o 11 a r e 10 t i m e s t h o s e o f C o s e 1» * * S i m u l o t i o a f r o m 3 0 . 4 t o 3 65 d a y s , ( T h e 0 —3 0 . 4

d o y t im e stop w o s r o t in c lu d e d ,

s in c e b o t h SIP end A O IP f o i l e d to c o n v e rg e .)

Figure 1.4-Weinstein,et al.'s Comparative Study

Price and Coats^^ have presented a comparison among fo ur d iffe r e n t ordering schemes o u tlin e d in chapter V.

A d ir e c t so lu tio n method is

used w ith these ordering'schemes to solve id ea lized re s e rv o ir problems. The Price and Coats study produces impressive re s u lts in terms o f computational work re duction.

However, these re s u lts in computational

work reduction are lim ite d to the computer time involved in e lim in a tio n p o rtio n o f non-zero elements o f the equations system and do not include the a d d itio n a l work required to reorder the c e ll system from standard ordering to the appropriate ordering. From the afore-mentioned survey, a development o f a b e tte r d ire c t method approach is needed. using a r e a lis t ic petroleum address these two areas.

A more comprehensive comparative evaluation re s e rv o ir is also needed.

This study w ill

CHAPTER I I RESERVOIR MODEL The in tro d u ctio n o f Darcy's law in to the two dimensional c o n tin u ity equation fo r each o f the three immiscible f lu id phases in petroleum reservoirs leads to the fo llo w in g system o f p a rtia l d iffe r e n tia l equations :

n 3$^ . 30^ S3T (hPo^o 37"^ ^ 37 (hOo^o

Cl Pn* ^ 7 t (^^o^o)

fo r the o il phase,

h (hPw^w - 37) + &

(^Vw -3y) +

^ I t (OPwfw)

(2.2)1

fo r the water phase, and

a

3$.

3$n

3$w

3 7(h P g ^ g"sf '^ s o ^ P o ^ o17^^ s w ^ P w ^ w 17 ^ a

30

30

+ 3 ÿ (hPg^g 1 y

+ '^so^^o^o 1 y

" ^

f It

30 '^sw^^w^w 1 ÿ ^

^ Rso^o^o +

'2.3)

10

fo r the gas phase, where

X, = k ^ "j

■J =

0,

g, w

(2.4)

These equations have been derived making the fo llo w in g assumptions: 1) Darcy's law applies 2) Composition o f each f lu id phase Is constant The generalized multiphase flow equation fo r the unsteady-state flow o f o i l , gas, and water In a porous medium Is developed by combining the three single-phase equations In to one basic equation.

In order to

do t h is , we need to express the fo llo w in g a u x llu ry equations: The p o te n tial terms are * 0

"

'’o

+ P o 9 0

( 2

- 5

)

♦ g

'

P g

+ P

g 9 0

( 2

- 6

)

*w “ P«

(2.7)

the c a p illa ry pressure terms are Pew - Po - P«

'2 .8 )

Peg = Pg - Pg

3y '

,

3y

"

* W

ï ; (*Po

■

+4^0

(

%

<

'

* Pq^o ^ - 3 Î >

&

•

^

w 3x'

' ^

(^Pw %

- &T ChX , 3x ' w

3x

'

^

S: >

The water equation (2.2) is divided by

& 3x '

-

«

and expanded by d iffe r e n tia tio n :

(hX„ w

3x

3x

'

^

The gas equation (2.3) is divided by p^ and expanded by d iffe r e n tia tio n :

.

&

(h x ,

,

12

^ &

x ( ^

pg

'

^

V

w

■" PgSg

^ i

'

3»g— ^

^ ax'

( 5 ) P 5 . +(hpQ^o)y ( Ü 2. /

.(

- jf )

="g + C’ V o ’ x ( ®*of

3X '

Pg ‘ s y '

34g

- ly )

— Og

(

5«so + (h P „x „), , 3*wf

34,- 7 ^

ay'

!V

'

®''swj

3x'

34*

4 (h p „» „)y( !îw f ! V . j ,2 1 4 , 34, ' 3y' 3 4 * '( 2 - ''"

where

^ ° ■"fe ^ ^ ^

+ 1; :

h C'Po^o “ si*

I t (PPo^o’ -

(h I t ('*’Pw\> ■

h (hPw^w “ 3T> ■ ly

• & C’Po^o 1 7 ) '

C’Pw^w

(^-'®>

Incorporating equations (2.1) and (2 .2 ) in to (2 .1 5 ), we obtain

s

=

- (

P g P g *

-

R

s

o

^

P

q

*

-

V

V

w

*

'

/

( P g A )

( 2

- ' 6

)

Combining equations (2 .1 0 ), (2 .1 1 ), (2 .1 2 ), (2.15) and (2 .1 6 ), we obtain

a 3Pn 37 (h ^t - 3 7 )

2 3y

9P^ (h^t

~hÿ'^ "

3p. ^1 - a t + Bg +

(2.17)

13

Equation (2.17) has been derived using u n ifie d SI u n its , but i f p ra c tic a l u n its (Appendix A) are used, equation (2.17) becomes

^

(h ^t " w )

â ÿ (h ^t

=

3Pn 11.57407408

+ 11.57407408 Bg + Bg + 0.001

+ Eg

(2.18)

where the to ta l m o b ility term is

•'t = -'o

"

^

(2-19)

the PVT term is

Bj =

(S„C^ + S„C„ + SgCg + C ,)

( 2 .2 0 )

I r