Comparison of Solutions to aThree-Dimensional Black-OilReservoir Simulation ProblemAziz S. Odeh, SPE, Mobil Resemch and Development Corp.

SummaryA comparison of solutions to a three-dimensionalblack-oil reservoir simulation problem is presented.The test of the problem and a brief description of thesevensimulators used in the study are given.

IntroductionSeven companies participated in a reservoirsimulation project to compare the results obtained bydifferent black-oil simulators. The companies werechosen to give a good cross section of the solutionmethods used in the industry. The participants wereAmoco Production Co., Computer Modelling Groupof Calgary (CMG), Exxon Production Research Co.,Intercomp Resource Development and EngineeringInc., Mobil Research and Development Corp., ShellDevelopment Co., and Scientific Software Corp.(SSC). The paper presents the text of the problem, acomparison of results in graphical form, and a briefdescription of each model. The descriptions weresupplied by the participants.

A variety of computers was used. Amoco usedIBM 3033, IBM 370/168, and Amdahl V/6. CMGused Honeywell 6000 DPS, and Exxon used Amclahl470/V5 and IBM 370/168. Intercomp used Cray-1and Harris/7. Mobil and SSC used CDC Cyber 175,and Shell used Univac 1110/2C Level 36. Thenumber of time steps and the central processor timesvaried considerably. Those interested in the actualvalues should contact the individual companies.

Except for Shell, all the participants used single-point upstream mobility weighting. Shell used twopoints upstream. Constraints and data are given inthe text.

Statement of the ProblemAreal and cross-section views of the reservoir aregiven in Figs. 1 and 2. The grid system is given in Fig.

0149.2136/81 /0001 .9722S00.25Copyright 1981 Society of Petroleum Engineers of AlME

JANUARY 1981

1. Stratification and reservoir properties are given inFig. 2. The reservoir is initiall~ undersaturated. A gasinjection well is located at Grid Point (1, 1), and aproducing well is located at Grid Point (10, 10).Pertinent data and constraints are given in Table 1.PVT properties and relative permeabilities are givenin Tables 2 and 3. The participants were asked tomake the runs and report the results described below.

Runs To Be Madecase 1Let the bubble-point (saturation) pressure be con-stant with a value equal to the original value.

Case 2Let the saturation pressure vary with gassatiation —i.e., this is a variable saturation-pressurecase. The PVT lines at pressures above the calculatedsaturation pressures are parallel to the original line.

Results To Be ReportedThe followingresults are to be reported.

1.Plots Ota. Oil rate vs. time.b. GOR vs. time.

2.Report annually and at abandonment:

3

a. ‘The pressures of the cell where the injectorand producer are located. *b. Gas saturation at Grid Points (1, 1, 1), (1, 1,2), (1, 1, 3), (lo, 1, 1), (lo, 1, 2), (lo, 1, 3), (lo,10, 1), (10, 10,2), and (10, 10,3).Report at the end of 8 years:a. Tables of gas saturation.b. Tables of cell pressures.*c. Tables of saturation pressures for the variablesaturations-pressure case.*

“Report of all pressures referred to a depth of 8,325 ft or at the center of theraspectlva blocks. If both are available, report both.

13

x—QA5I

1WELL

tooo FT.

Fig, 1 - Reservoir and grid system.

QAS INJECTION OIL PRODUCTION100 MM SCFID WELL

k + H, FT. KX Ky (LI~K) SW GO8326

I .3 20 600 600 .12 .88 (I 833650

2 .3 30 60 50 .12 .aa #~S360

28

3 .3 60 200 200 .12 .88

1: J=123450 78

FT.FT.LAYER

LAYER FT.

LAYER FT.

FT.

Fig. 2 - Diagonal cross section.

TABLE 1- DATA AND CONSTRAINTS

Initial reservoir pressure, psia at 8,400 ftGas injection rate, MMscf/DMaximum oil production rate, STWDMinimum oil rate, STB/DMinimum flowing bottomhole pressure, PSIMaximum saturation change during time stepRock compressibility, I/psiPorosity value of 0.3 was measured at a base pressure of 14.7 PSIWellbore radius, ftSkinCapillary pressureFleservoir temperature, &FGas specific gravityRuns are terminated either at the end of 10 years or when

GOR%20,000 scf/STB or when the oil production ratesl ,000STB/Q whichever occurs first terminates the run.

4,800100

20,0001,0001,0000.05

3X1 O-6

0.2500

2000.792

TABLE 2- PVT PROPERTIES

Saturated Oil PVT Functions Saturated Water PVT FunctionsReservoirPressure FVF Viscosltv Densitv Gas/Water Rat io

ReSeNOlrPressure IWF viscosity Density Solution GOR

(psla) (RBISTB) (Cp) (Ibm/cu ft) (scflstb)

14.7 1.0620 1.0400 46.244- 1.0(psia) (RBlbbl) (Cp) - (ibm/cu it) (scflbbl)

14.7 1.0410264.7 1.0403514.7 1.0395

1014.7 1.03802014.7 1.03502514.7 1.03353014.7 1.03204014.7 1.02905014.7 1.02589014.7 1.0130

0.3100 62.2380.3100 62.2830.3100 62.3280.3100 62.4180.3100 62.5990.3100 62.6900.3100 62.7810.3100 62.9640.3100 63.1600.3100 63.959

0.00.00.00.00.00.00.0 “0.00.00.0

264.7514.7

1014.72014.72514.73014.74014.75014.79014.7

1.15001.20701.29501.43501.50001.56501.69501.82702.3570

0,97500.91000.83000.69500.64100.59400.51000.44900.2030

4il.54442.28741.00438.99538.30437.78137.04636.42434.482

90:5180.0371.0636.0775.0930.0

1270.01618.02984.0

Undersaturated Oil PVT FunctionsGas PVT FunctionsRf3St3NOlr

Pressure(psia)4014.79014.7

FVF(RBlbbl)

0.1666660.0120930.0062740.0031970.0016140.0012940.0010800,0008110.0006490.000386

FVF Viscosity Density(RBISTB) (Cp) (ibm/cu ft)

1.6950 0.5100 37.0461,5790 0.7400 39.768

Ff(3St3NOlrPressure viscosity

(Cp)

0.0080000,0096000.0112000.0140000.0189000.0208000.0228000.0268000.0309000.047000

Density(ibm/cu ft)

Pseudo Gas PotentialM(P)

(psia) (psia2/cp)

14.7264.7514.7

1014.72014.72514.73014.74014.75014.79014.7

0.06470.89161.71853.37276.88068,33269.9837

13.295216.613927.9483

0.0.777916 E+ 070.267580 E+ 080.875262 E+ 080.270709 E+ 090.386910 E+ 090.518118 E+090.803963 E+ 090.112256 E+ 100.251845 E+ 10

Undersaturated Water PVTFunctions

ReservoirPressure

(psla)

4014.79014.7

14

FVF Viscosity Density(RBlbbl) (Cp) (ibm/cu ft)

1.0290 0.3100 62.9641.0130 0.3100 63.959

.IOURNAL OF PETROLEUM TECHNOLOGY

‘! 6

14

4

2

0

— MOBIL

— SHELL

— AMOCO

— INTERCOMP

— EXXON

k -“-”- SSc

EXXON AGREES WITH MOBIL BETWEEN 3-7 YEARS

EXXON AGREES WITH INTERCOMP BETWEEN 8-10 YEARS

AMOCO AGREES WITH SHELL AFTER 7YEARS

CMG AGREES WITH MoBILTHRolJGH 7 YEARSANDWITH iNTERCOMP BETWEEN 8-10 YEARS

~~~234567 89 10 TIME,YEARS

Fig. 3 - Case 1- oil rate vs. time.

ResultsA comparison of the results is given in Figs. 3through 18.No comparison of saturation pressures isgiven because the values reported by the sevencompanies werewithin 20 psi of each other.

Description of the ShuulatorsAmoco’s ModelThe IMPES method was used, with semi-implicitadjustments in well rates. This method proved quitesatisfactory; additional computations for implicithandling of interlock flow werenot needed.

Maximum time-step size can vary with time and isinput. The model determines internal time-step sizesto satisfy both the current maximum At and themaximum saturation change for any grid block (5070PV). A sequence of runs using maximum At of 0.25,0.5, 1, and 2 months yielded virtually identicalresults, confirming the applicability of the IMPESmethod. The final results are for a maximum At of1.0month.

For each internal time step the computationsequencewas as follows.

1. Wellrates.2. Coefficients including terms for semi-implicit

production rates.3. Iterative computation of grid-block pressure

JANUARY 1981

TABLE 3- RELATIVE PERMEABILITY DATA*

Oil-Gas

k3- ‘[9 —o 1.0’00.001 ;:;0.02 0.0 :::970.05 0.005 0.9800.12 0.025 0.7000.2 0.075 0.3500.25 0.125 0.200

0.190 0.090:: 0.410 0.0210.45 0.60 0.010

0.72 0.001l:: 0.87 0.00010.7 0.94 0.0000.65 0.98 0,0001.0 1.0 0.000

“This Is a two.phaae, gas/oil problem. Set the relative permeability to waterequal to zero for all values of water saturations.

1

5500

4500

— MOBIL

— SHELL

— AMOCO

—INTERCOMP

— EXXON

-“-”- Ssc

EXXON AGREES WITH MOBIL UP TO 3 YEARS

SSC AGREES WITH MOBIL BETWEEN 2-7 YEARS

AMOCO AGREES WITH INTERCOMP AFTER 6 YEARS

CMG AGREES WITH MOBIL THROUGH 7 YEARSAND WITH INTERCOMP BETWEEN 7-10 YEARS

t t i

3500

2500

2345676 9 10 TIME, YEARS

Fig, 4 - Case 1- GOR vs. time.

* INTERCOMP.

— MOBIL — INTERCOMP

— SHELL — EXXON

— AMOCO -o-” Ssc

AMOCO AGREES WITH SHELL UP TO 3 YEARS

EXXON AGREES WITH AMOCO BETWEEN 6-10 YEARS

/ \~ SSC AGREES WITH AMOCO BETWEEN 3-5 YEARS

MOBIL, lNTERCOMP~

CMGAGREES WiTH MOBIL BETWEEN I-4, AND$-10 YEARS,AND WITH AMOCO BETWEEN 5-8 YEARS

I t I , ,1 , 1 4

1 2 34 7 8 9 10TIME,5YEARS6

Fig. 5 - Case 1- pressure vs. time for producing well Cell 10,10,3.

JOURNAL OF PETROLEUM TECHNOLOGY

,.

changes using slicesuccessiveoverrelaxation.4. Noniterative computation of grid-block sat-

uration changes and of semi-implicit adjustments inproduction rates.

5. If any saturation change exceeds the maximum,reduce At and go to Step 2.

6. Noniterative computation of grid-block bubble-point pressure changes (for variable saturation-pressure case only).

Production rates for each step were the sum ofrates at the start of the step plus semi-implicit ad-justments as saturations changed in the well block.h important exception is that the oil rate was heldconstant if, at the start of the step, the well hadexcess computed productivity (i.e., if the computedbottomhole pressure exceeded the minimum value of1,000psi).

CMG’SBlack-Oil ModelCMG’S black-oil simulator models three-phasewater/oil/gas systems or two-phase water/oilsystems. The model includes the effects of gravityand capillary pressure. It can be run in the one-, two-or three-dimensional mode. Variable grid spacingcan be used. The nonlinear equations are solved byNewtonian iteration with the derivatives of theJacobian matrix evaluated numerically. The modelcontains several possible options for the weighting ofnobilities. These include single-point upstream, two-

point upstream, and centralized upstreamweighings. The time discretization is by backwarddifferences with a modified Crank-Nicholsonmethod included as an option. The well modelpermits the placing of wells at various positions in agrid block. Multiblock completion wells are includedand are modeled in a manner which does not increasethe matrix bandwidth. Finally, an efficient solutionroutine is included in the model. This routineprovides Gaussian elimination with block D4 or-dering, a bandwidth-reducing option, and twodifferent iterative solutions methods: AB andCOMBINATIVE.1

The model is fully implicit in its basic formulation.It becomes highly implicit, not fully implicit, whenthe options for two-point upstream or centralizedupstream weighings are used or when multiblockcompletion wellsare modeled.

Disappearance of the gas phase is not handled bythe conventional variable substitution technique butby a novel pseudo solution-gas formulation.2 Thepseudo solution-gas formulation allows both variablebubble-point problems and fixed bubble-pointproblems to be handled in a simplemanner.

For this problem the simulator was run in three-phase, three-dimensional mode. The basic fullyimplicit formulation was used. The timediscretization was backward differences. The matrixproblem was solved by the AB iterative routine. 1

— MOBIL

— SHELL, AMOCO

— lNTERC(WIP

“-”-” SSC

“’ INTERCOMP, SHELL, AMOCO

EXXON AGREES WITH SHELL, AMOCO THROUGH 6 YEARS,AND WITH INTERCOMP BETWEEN 7-10 ?’EARS

CMG AGREES WITH MOBIL THROUGH 3 YEARS, ANDWITH INTERCOMP BETWEEN 4-10 YEARS

SSC AGREES WITH SHELL BETWEEN 4-9 YEARS

o 1 2 34 6 7 8 9 10TIME, !EARS

Fig. $ - Case 1- gas saturation vs. time for producing well Cell 10,10,3.

JANUARY 1981 17

7500T

6500

— MOBIL

— SHELL

— INTERCOMP

-—- EXXON

*“o”*oDw~=CMG

AMOCO AGREES WITH SHELL UPT06 YEARS,AND WITH EXXON AFTER 6 YEARS

SSC AGREES WITH SHELL BETWEEN 1-2YEARS, AND4-5 YEARS,WITH MOBIL BETWEEN 3-4YEARS, ANDWITH EXXON BETWEEN 6-10 YEARS—

MOBIL-IN’TERCOMP

4500” -

MOBIL-INTERCOMP

39004 * , I t *012345 6709

i10 TIME, YEARS

Fig. 7 -Casel- pressurevs,timeforinjection wellCelll, l,l,

4500

3000

18

---- -1

)----

—

.—------ .

---- 1-----

—.-....—.,-----

-—.— . ..-

.- .-*-

— MOBIL .—-.—-..4-s-. -1

— SHELL I 1. .. .. ....1

— Nhloco

— INTERCOMP %‘-— EXXON I I“-”-” Ssc I

Q----..-CMG AGREES WITH AMOCO

I=M

,1,1 ‘ 2,2 ‘ 3,3 ‘ 4,4 ‘ 5,5 ‘ 6,6 ‘ T,7 ‘ 8,8 ‘ qg ‘ 10,10’

GRID POINT LOCATIONFig. 8 - Case 1- pressure vs. grid-point location, time= 8 years, top layer.

JOURNAL OF PETROLEUM TECHNOLOGY

SyM

v..—.- ...- 1 A, S,E=M— MOBIL

— 1=s — SHELL““”””t””’:J&ti — AMOCO

sSC, E= A1

— INTERCOMP‘=— EXXON

m“’””””‘“”””- 1=s“-”-” SSC-=~””=~m”=CMG

A=MS,Af M

1I 3 E=lt

T -----CMG = A , + ..............

E=M

E=M

1 v Y L

40411:22:33:44: ~5~ 66:77:88:9,9s 9 Y 9 # ‘ 10,10’

GRID P&NT L06ATION ‘Fig. 9 - Case 1- gas saturation vs. grid-point location, time= 8 years, top layer..

I,Ez A

TG=M -. .

CMG = M— MOBIL

SSC = A — SHELL

&1

.........*.E = CMG

, 81,1 ‘ 2,2 ‘ 3,3

— AMOCO— INTERCOMP— EXXON“-’-” SSC● “””=s”=””-CMG

GAS SAT. <.005 AT:

1) 4,4- SHELL, EXXON, AMOCO

2) 5,5 -ALL PARTICIPANTS

3) 6,6- ALL PARTICIPANTS EXCEPT MOBIL,

4) 7,7-SHELL, AMOCOCMG

5) 8,8- SHELL

CMG : MI

GRID POINT LOCATION

1

Ssc,CMG = I

10,10 ‘

Fig. 10- Case 1- gas saturation vs. grid.point location, time= 8 years, middle layer.

JANUARY 1981 19

Exxon’s General Purpose SimulatorExxon’s general purpose reservoir simulator (GP-SIM) uses a sequential implicit solution procedure.3The first step in this approach is the solution of a setof pressure equations, This set consists of a singleequation for each grid block, and solving it yields acomplete new pressure distribution at the end of atime step.

This pressure distribution then is used to calculatethe sum of the velocities of all phases at eachbrxm!ary between grid blocks, and these totalvclncities we used in a set of saturation equations. Ifeither capillary pressure or relative permeability isbeing treated semi-implicitly, this set consists of twocoupled equations per grid block and is solvedsimultaneously to yield saturation distributions at thenew time, Otherwise, the equations are uncoupledand can be solved point by point explicitly, in thenormal IMPES fashion.

Several options are available for solving thematrices involved. In the problem discussed here, apreconditioned conjugate gradient method4 was usedto solve for pressures, and strongly implicitprocedure (SIP) was used to solve for saturations.(The full saturation solution was needed becausenobilities were treated semi-implicitly.)

As is common in modern reservoir simulators,GPSIM can account for reservoir heterogeneity, rockcompressibility, and solution of gas in both oil and

water. Less common features it can model are vapo-rization of oil into the gas phase and hysteresis in thecapillary pressure and relative permeability data.

GPSIM has only minor restrictions on the numberof grid blocks that it can use; large problems can berun using only relatively modest amounts of centralmemory. This desirable feature is accomplished byusing disks to store data temporarily by planes forthree-dimensional problems or by rows for two-dimensional ones. If the central memory madeavailable is sufficiently large, the programautomatically will eliminate the temporary datastorage, keeping all data within core.

Intercomp’s Black-OilSimulatorlntercomp’s BETA 11black:oil model is designed tosimulate numerically two- or three-phase com-pressible flow in heterogeneous hydrocarbonreservoirs. Gas is assumed to be soluble in oil but notin water; neither oil nor water can exist in any phaseother than its own. Solutions are obtained in one,two, or three spatial dimensions using either rec-tangular or cylindrical coordinates. Alternate solu-tion procedures provide for efficient modeling of allclasses of black-oil reservoir problems, ranging fromindividual well behavior (coning simulations or welltest analysis) to large, multireservoir fields. Tocomplement the three-phase simulation capabilities,BETA 11 contains two distinct segments of code

-nz’lb-w...INTERCOMP :“CM

SHELL = MOBIL — MOBIL

— SHELL

— AMOCO

---- Ssc

INTERCOMP AGREES WITH MOBIL UP TO 7 YEARS,AND WITH AMOCO BETWEEN 7-10 YEARS

EXXON AND AMOCO AGREE FOR 10 YEARS

CM(3 AGREES WITH AMOCO BETWEEN 5-8 YEARS,AND WITH MOBIL BETWEEN 8-10 YEARS

, , 1 * I , , , I2345678 9 10 TIME, YEARS

Fig. 11- Case 2- oil rate vs. time,

JOURNAL OF PETROLEUM TECHNOLOGY

~ MOBIL

— SHELL

— hMOCO ●/*— EXXON

-“-” Ssc ●4

/ AMOCO AGREES WITH SHELL BETWEEN 8-10 YEARS

~ EXXON AGREES WITH SHELL UP TO 4 YEARS, AND! WITH MOBIL BETWEEN 5-8 YEARS

! INTERCOMP AGREES WITH MOBIL UP TO 7 YEARS,I AND WITH SHELL BETWEEN 8-10 YEARS~

CM(3 AGREES MOBIL THROUGH 9 YEARSAT 10 YEAR GOR= 2O,OOO

,, , d , * , , t

2345678 9 10 TIME, YEARSFig. 12- Case 2-GOR vs. time.

— MOBIL— SHELL— AMOCO-— EXXON-“-0 Ssc

•*.~~**~o==CMG

INTERCOMP AGREES WITH SHELL UP TO4 YEARS, AND WITH MOBIL AFTER THAT

CMCI AGREES WITH MOBIL THROUGH2 YEARS

* ,I

* ,, 4

1 2 3 4 5 6 7 8 9 10—TIME, YEARS

Fig. 13- Case 2-pressure vs. time for producing well Cell 10,10,3.

JANUARY 1981

which are designed specifically for optimai solutionof two-phase problems: one for water/oil (dissolved-gas content constant) problems and one forgas/water (no oil phase) problems. BET,.4II has alarge variety of user-oriented features mch as in-put/output options and wellcontrol options.

There is a one-equation implicit pressure for-mulation in which the equations are decoupled andsolved in this order: pressure, gas saturation, andwater saturation. For the one-equation formuiaticn,there are options to solve for both saturations ex-plicitly (IMPES) and either or both saturationsimplicitly (sequential). There is an alternative two-equation formulation in which the program solvesimplicitly for pressure and gas saturation and thensolves for water saturation. Similar to the previousformulation, water saturation can be treated eitherimplicitly or explicitly. In any of the formulations,muitiple outer iterations maybe taken to account forthe mmiinearity of the basic flow equations.However, if no vaporization or resolution of gas isoccurring, oniy one or two iterations are required toconvergethe noniinearities adequately.

The large systems of linear algebraic equationsmay be solved by a variety of methods, any one ofwhich may offer significant speed advantages on agiven problem: (1) direct solution by reduced band-width Gaussian elimination, (2) several forms of one-line, two-line, and planar successive overrelaxation(SOR), or (3) SIP.

Mobil’s Ail Purpose Reservoir Shnuiator (ALPURS)ALPURS is a three-dimensionai, three-phase,multiwell, black-oil reservoir simulator which uses astrongly coupled, fully implicit method to solvesimultaneously for ail unknowns.5 The noniinearintercell flow equations and weii-constraintequations are linearized and iterated to convergeusing Newton-Raphson iteration. Linear equationsare solved with block successive overrelaxation. Atypical block is an x-z, y-z, or x-y reservoir slice,which is solved by sparse elimination. The relaxationparameter is computed automatically using thepower method and Rayleigh quotients. ALPURSaccounts for reservoir heterogeneityy, rock com-pressibility, gravity, gas dissolved in both the oii andwater phases, constant or variable bubble-pointpressures, hysteresis in saturation-dependent data,tubing string pressure drop, and flash surfaceseparation. Modern concepts of well flow equationsare incorporated, including pseudo gas-potentiaifunction, skin factor to account for damage orimprovement, non-Darcy flow effect, and flowrestriction due to limited entry such as partialpenetration.

SSCModelSSC’Sblack-oil model employs an Adaptive ImplicitMethod (AIM). This technique, which recently wasdeveloped at SSC, seeks to achieve an optimum with

BETWEEN

I #1 I , , , ,1 2 34 6 7

18 9 10

TiME, tEAFiSFig. 14- Case 2-gas saturation vs. time for producing well Cell 10,10,3.

JOURNAL OF PETROLEUM TECHNOLOGY

— MOBiL

— SHELL, AMOCO

— iNTERCOMP

-=-” Ssc

IEXXON AGREES WiTH iNTERCOMP

3-10 YEARS

CMG AGREES WiTH INTERCOMP

———-— —..

7600

6500

4s00

w-Uamawaa

350a

3ooa

MOBIL-SHELL-EXXON\.

— MOBIL— SHELL_ AMOCO— INTERCOMP— EXXON-“-” Ssc9==o====MCMG

EXXON AGREES WITH AMOCOBETWEEN 6-10 YEARS

SSC AGREES WITH MOBILAFTER 6 YEARS

CMG AGREES WITH EXXONBETWEEN 6-10 YEARS

, I * t* 1 I

2 3 4 5 6 7 8 9 10Fig. 15-Case 2-pressure vs. tlmefor lnJectlon well Celll,l,l.

i

—.—.-. I 1

I 1

I

TIME, YEARS

— MOBIL

M-u-SHELL Y

_ AMOCO

—INTERCOMP1--z ---EXXON L-“-”” SSC

“==””==-== CMG HSSC AGREES WITH MOBIL AT 4,4 THROUGH 10,10

CMQ AGREES WITH EXXON AT 1,1 THROUGH 6,6

,1,1 ‘ 2,2 ‘ 3,3 ‘ 4,4 ‘ 5,5 ‘ 6,6 ‘ 7,7 ‘ 8,8

I9,9 ‘1 O,1O’

GRID POINT LOCATIONFig. 16- Case 2-pressure vs. gridpoint location, time =8 years, top layer.

JANUARY 1981 23

respect to stability, truncation errors, and computercosts. Typically, only a small fraction of the totalnumber of grid blocks during simulation experiencesufficiently large surges in pressure and/or saturationto justify implicit treatment, When it is needed,implicit treatment may not be required in all phasesor for long periods of time. Moreover, those cellsrequiring implicit treatment will change as thesimulation proceeds. Consequently, a model offeringa fixed degree of implicitness to all cells for afl timesteps is not always the most desirable. For example, afully implicit model, while ensuring stable answers,amounts to overkill in most of the cells most of thetime, while an IMPES model can cause underkill.With AIM there is no problem of over-or underkill.Various degrees of implicitness are invokedregionally or individually cell by cell- i.e. thesolution is advanced with adjacent cells havingdifferent degrees of implicitness. As the calculationsproceed, the degrees of implicitness locally anddynamically shift as needed-all automatically. Thewhole idea is to apply the right amount of im-plicitness where and when needed and for only aslong as needed.

The simulator also provides a wide variety of user-oriented features. For example, one can overrideAIM and operate in a fully implicit, partially im-plicit, or an IMPES mode. Variable bubble-pointproblems, such as that in Case 2, are handled by

55

50

45

4a

%

‘“””””””””* S= E

:M

~~s.O-

CMG k A

R

...*. ......----

variable substitution. The simulator is a generalpurpose package offering three-dimensional capabil-ity in both Cartesian and cylindrical coordinates.

Shell Development ModelThe Shell reservoir simulation system operates anIMPES mode or an implicit mode. There are threepseudocomponents: water, stock-tank oil, andseparator gas. There are three phases: aqueous,hydrocarbon liquid, and hydrocarbon vapor. Theaqueous phase contains water. The hydrocarbonliquid and vapor phases can contain both oil and gas.A fourth component is also available for modelingpolymer or carbon dioxide. There are several indirectand direct solution methods as a user option. Ad-ditionally, two-point upstream weighting is used tocalculate phase nobilities.

References1. Behie, A. and Vinsome, P.K. W.: “Block Iterative Methods for

Fully Implicit Reservoir Simulation,” paper SPE 9303presented at the SPE 55th Annual Technical Conference andExhibition, Dallas, Sept. 21-24,1980,

2. Au, A. D. K., Behie, A,, Rubin, B., and V]nsome, P. K,W,:“Techniques for Fully Implicit Reservoir Simulation,” paperSPE 9302 presented at the SPE 55th Annual Technical Con-ference and Exhibition, Dallas, Sept. 21-24, 1980.

3. Spillette, A.G., Hillestad, J.G., and Stone, H. L.: “A HQh-Stability Sequential Solution Approach to ReservoirSimulation,”paperSPE 4542 presented at the SPE 48thAnnual Meeting, Las Vegas, Sept. 30-Ott. 3,1973.

— MOBIL

— SHELL

— AMOCO

— INTERCOMP

— EXXON

“-”-” SSC

● =o*-8nm9m*CMG

AiS

R-..- CMG = S

1I

SSC, CMG = A

CMG = A●............

1 * , I I1,1 ‘ 2,2 ‘ 3,3 4,4 5,5 6,6 7,7 8,8 9,9 10,10

I

GRID POINT LOCATIONFig. 17- Case 2-gas saturation vs. grid-point location, time =8 years, top layer.

JOURNAL OF PETROLEUM TECHNOLOGY24

I=LS=l

—=

,1,1 ‘ 2,2 * 3,3

— IVOBIL

— SHELL

— AMOCO

— INTERCOMP

— EXXON

-“-”-SSC

● 0=0===0==~CMG

GAS SAT. C .005” AT:

‘}) 4,4 AND 5,5-SHELL, AMOCO,Ssc, EXXON

2) 4,4-CMG

3) 6,6-SHELL, Ssc

S, 1,E, CM(3 = M

SSC, S = E I

E, CM(3 ~ M

%

-. .

A=l

r- Ss’ .CMG= ME=A L-

14,4 * !5,6 ‘ 6,6 ‘ 7,7 ‘ 8,8

GRID POINT LOCATIONFig. It) - Case 2- gas saturat ion vs. grid.point Iocatlon, time =8 years, middle layer.

4. Watts, J.W.: “A Conjugate Gradient-Truncated DirectMethod for the Iterative Solution of the Reservoir SimulationPressure Equation,” paper SPE 8252 presented at the SPE 54thAnnual Technical Conference and Exhibition, Las Vegas, Sept.23-26,1979.

5. Bansal, P, P., Harper, J. L., McDonald, A. E., Moreland, E. E,,Odeh, AS., and Trirnble, R. H.: “A Strongly coupled,FullyImplicit, Three-Dimensional, Three-Phase ReservoirSimulator,” paper SPE 8329 presented at the SPE 54th AnnualTechnieal Conference and Exhibition, Las Vegas, Sept. 23-26,1979,

Nomenclatureh =I =

=

L=~=kx =ky =

M(p) =

thicknessnumber of grid points in the ~ directionnumber of grid points in they directionpermeabilityrelative permeability togasreIativepermeability to oilpermeability in the x directionpermeability in they directionpseudo gas potential

Sg= g$s saturation= od saturation

:: = water saturationAt = time step@= porosity

S1Metric Conversion Factorsbbl x 1.589873 E-01 = m3Cp x 1.O* E-03 = Pans

tuft x 2.831 685 E-02 = m3‘F (eF-32)/l.8 = ‘cft X 3.048” E-01 = m

lbm x 4.535924 E-01 = kgpsi, psia x 6.894’757 E+ 00 = kPa

scf x 2.863 640 E– 02 = std m3“Conversionfaxrorisexact.

orlglrral rr?8rw8cript recleved In SOC18tyof Petroleum Engineers office Nov.18,1980. Paper (SPE 9723) acceptedtorpublication Nov. 26,1980.

25JANUARY 1981