..

.

A. S. Odeh, SPE-AIME,Mobil Research& DevelopmentCorp.

IntroductionReservoir simulation is based on well known reservoirengineering equations and techniques the sameequations and techniques the reservoir engineer hasbeen using for years.

In generai, sirmukition .e.w.. .-.-f m-c tO Lie representationof some process by either a theoretical or a physicalmodel. Here, we limit ourselves to the simulation ofpetroleum reservoirs. Our concern is the developmentand use of models that describe the reservoir perform-ance under various operating conditions.

Reservoir simulation itself is not really new. Engi-neers have long used mathematical models in per-forming reservoir engineering calculations. Before thedevelopment of modem digital computers, however,the models were relatively simple. For example, whencalculating the oil in place volumetrically, the engi-neer simulated the reservoir by a simple model inwhich average values for the porosity, saturation, andthickness were used.

Although simulation in the petroleum industry isnot new, the new aspects are that more detailed reser-voir features, and thus more accurate simulations,have become practical because of the capability af-forded by the computers now available. The more de-tailed description, however, requires complex mathe-matical expressions that are difficult to understand,and this difiicuhy has caused some engineers to shyaway from using simulators, and others to misusethem.

We in the petroleum industry are in the reservoirsimulation revolution. As time goes on, simulators willbe used more and more, so a basic understanding of

. ..

The engineer! espe-reservolr modelmg is esseiitia,.cially, must become competent in setting up simula-tion problems, in deciding on appropriate input data,and in evaluating the results.

. 7-Z3t-

Basic AnalysisIf a reservoir is fairly homogeneous, average valuesof the reservoir properties, such as porosity, are ade-quate to describe it. The average pressure, time, andproduction behavior of such a reservoir under a solu-

. .

tion gas drive, for ~Acwy.-, -----+ nre MNTI-MUy calculatedby the familiar methods of Tamer, Muskat, or Tracy.All of these methods use the material balance equa-tion normally referred to as the MBE. A simple ex-pression for the oil MBE is the following

(cumulative net withdrawal in STB) = (originaloil in place in STB) (oil remaining in placein STB)

The cumulative net withdrawal is the difference be-tween the oil that leaves the reservoir and the oil thatenters it. In this basic analysis, there is no oil enteringthe reservoir since the boundaries are considered im-permeable to flow. Thus, the MBE reduces to itssimplest form. Such a reservoir model is called thetank model (Fig. 1). It is zero dimensional becauserock, fluid properties, and pressure values do not varyfrom point to point. Instead, they are calculated asaverage values for the whole reservoir. This tankmodel is the basic building block of reservoir simu-lators.

Now let us consider a reservoir represented by asandbar. Let the two halves of the sandbar vary inlithology. The sandbar as a whole cannot be repre-sented by average properties, but each half can. Thus,the sandbar consists of two tank units, or cells, as theyare normally called. The MBE describes the fluidbehavior in each ceU as in the previous tank model.

.,M. . . . ~] te~ of the MBE isHowever, the net wl,,,d,~.v-. .+...-more complicated because there can be migration offluid from one cell to another, depending on the aver-

1383

age pressure values of the two cells. This fluid transferbetween the two cells is calculated by Darcys law.The MBE, together with Darcys law, describes thebehavior of each cell. This model is not a zero-dimen-sional reservoir simulator since reservoir parametersmay vary between the two cells. Instead it is a one-dimensional model, because it consists of more thanone cell in one direction and of only one cell in theother two directions (Fig. 2).

This analysis can be extended to reservoirs whereproperties as well as pressure values vary in twodimensions, and to others where the variation occursin three dimensions. The simulators representing thesereservoirs are called, respectively, two-dimensionaland three-dimensional simulators, as illustrated inFigs. 3 and 4. Thus, a two-dimensional reservoir simu-lator consists of more than one cell in two dimensionsand of one cell in the thhxi dimension. And a three-dimensional simulator consists of more than one cellin all of the three dimensions.

Regardless of the number of dimensions used, theMBE is the basic equation describing the fluid be-havior within a cell; and Darcys law describes theinteraction between the cells. In one-, two-, and three-dimensional models each cell, except the boundarycell, interacts respectively with 2, 4, and 6 cells. Sincea simulator can consist of hundreds of cells, keepingaccount of the MBE for each cell is a formidablebookkeeping operation ideally suited to digital com-~utation. But we emphasize once again that the prin-ciples and equations used in reservoir simulation arenot new. They only appear so because of the complex-ity of the bookkeeping.

Types of Reservoir SkmihiiimThere are several types of reservoir simulators. Choiceof the proper simulator to represent a particular res-ervoir requires an understanding of the reservoir anda careful examination of the data available. A modelthat fits Reservoir A may not be appropriate for Res-ervoir B, in spite of apparent similarities betweenReservoirs A and B. A reservoir model is useful onlywhen it fits the field case.

One basis for classif@g models, as discussed ear-lier, is the number of dimensions. The two-dimen-sional model is the most commordy used. There are

/

/

several two-dimensional geometries, the most popu-lar of which is the horizontal (x-y) geometry; but thevertical (x-z) and the radial (r-z) geometries are alsoused quite often.

Simulators can be classified also according to thetype of reservoir or process they are intended to simu-late. There are, for example, gas, black oil, gas con-densate, end miscible displacement reservoir simulat-ors. Moreover, there are one-, two- and three-phasereservoir models. Furthermore, any of these simula-tors may or may not account for gravitational orcapillary forces. It is not enough to choose the propersimulator with respect to dirnensionality; the simu-lator must represent the type of hydrocarbon and thefluid phases present.

Simulation StepsPreparation of DataAfter the type of model to use in a study has beenselected, the next step is to divide the reservoir into anumber of cells, as illustrated in Figs. 2 through 4.This is accomplished by laying out a grid system forthe reservoir. In a two-dimensional study, the grid isestablished by drawing lines on a map of the reser-voir. All grid lines must extend across the reservoir.Each cell is identified by its x, y, z coordinates. Thenthe flow conditions around the perimeter of the res-ervoir are established. Normally the reservoir bound-ary is considered sealed, but influx or efflux at anassigned pressure or rate may also be specified.

m. . the following for each1rie next sfip is to ass:gn . .. - -celk rock properties, geometry, initial fluid distribu-tion, and fluid properties. The rock properties consistof specific permeability, porosity, relative permeabili-Cj ~d ~~m~e&~.e$~Apillary pressure. The Cell geome-try includes the depth, thickness and locations ofwells. Usually the wells are assumed to be located atthe centers of &e cells in which *Aeyfal!a The initialfluid distribution consists of the oil, water and gassaturations at the beginning of simulation. Also, theaverage pressure of the cell at that time is assigned orcalculated from known data. Fluid properties arespecitied by the usual PVT data. In addition, for eachwell it is necessary to provide a production scheduleand a productivity index or a skin value (i.e., damageor ,improvement).

Flow

Flow

5TY?l--Fig. 24ne-dimensional simulator.

Fig. lTsmk model. Fig. 3-Two-dimensional simulator.

JOURNAL OF PETROLEUM TECHNOLOGY

The engineer should scrutinize carefully these basicdata for consistency and accuracy. For example, ifpressure buildup data are available on a well, thepermeability-thickness product of the cell where thewell is located and the flow rate assigned to the well,should be compatible with the buildup data. The timespent in examining the basic data is well spent, for itcan lead to fewer simulation runs. Moreover one mustalways remember that theanswer is only as good asthe input data.

History Matchfng and Performance PredktionThe rn-fi ~urpose of reservoir simulation is to prediCtthe rate of hydrocarbon recovery for dtierent meth-

.*.-.- CI,.J A+.ods of field operation. d adeq-ua LG ~eiu -a~ ex!st~

reasonably accurate performance predictions can bemade. If data are incomplete or suspect, simulatorsmay be used only to compare semi-quantitatively theresults of dtierent ways of operating the reservoir. Ineither case, the accuracy of the simulator can be imp-roved by history matching.

The first step in a history match is to calculate res-ervoir performance using the best data available. Theresults are compared with the field recorded historiesof the wells. If the agreement is not satisfactory, suchdata as permeability, relative permeabiUty, and po-ro@ are va~~ed from one computer run to anotheruntil a match is achieved. The s&ndator is then usedto predict performance for alternative plans of oper-ating the reservoir.

The behavior of the reservoir is influenced by manyfactors permeability, porosity, thickness, satura-tion distributions, relative permeability, etc. thatare never known precisely aU over the reservoir. Whatthe engineer arrives at is only a combination of thesevariables, which results in a match. Tis colmbiiationis not unique, so it may not represent precisely thecondition of the reservoir. When the simulator, aftera match, is used to predict, it is not certain that thephysical picture of the reservoir described in the simu-lator will give predictions sufficiently close to the ac-tual reservoir performance. In generaI, the longer thematched history period, the more reliable the pre-dicted performance wiU be. It behooves the engineerto monitor periodically the predicted vs the actualperformance and to update his physioal picture of

Flow

Fig. 4-Three-dimensional simulator.

the reservoir.

Mathematical ConsiderationsDerivation of EquationsFor the engineer to adequately understand reservoirsimulation, he should be acquainted with the equa-tions used. These are basicaUy material balances aboutcells for each phase, and Darcys law, which describesthe interactions between cells. For illustration, wederive here the fundamental equations for a black oilsystem. and exp!ain thei: physical significance.

For the sake of simphclty, consider a cdl in a one-dimensional reservoir simulator, as shown in Fig. 5.Tine sane mia!jjsis is applicable to a cell in two- andth.me-dimensional models. (An expression for the oilmaterial balance of the cell was given eariier.)

(Oil volume entering the ceU during a time incre-ment At, in STB) minus (oil volume leaving the ceUduring the same time increment, in STB) equals (thechange in oil volume in the ceU, in STB).

Volume of oil entering the cell during At, in STB,equals QinAt.

Volume of oil leaving the cell during At, in STB,equals (At + dA~.

Change in volume of oil in the cell during At, inSTB, equals

where Qi. is the average flOWmte of ofl into the cellduring At in STB/unit time, Q..t is the avenge flowrate of oil leaving the cell to its neighbors during Atin STB/unit time, and QOis the oil production ratefrom the cell, if it contains a well, in STB/unit time;AxAyh@.

l?. represents the volume of oil in the cell at

any time, n+ 1 refers to the end of the time step, andn to the beginning.

Substitution in the oil MBE, after dividing throughby At, gives

Qin Qcmt go= %w)w%)rnl.,. . . . . . . . . . (1)

However, by Drircys&.w, assuming the flaw to be

Qout

Qin

Fig. 5-Cell in a one-dimensional simulator.

1385.

from left to right as shown in Fig. 5,

and

where Ayh is the cross-sectional area of the cell, Axis the length of the cell, 00 is the flow potential in theoil phase, i refers to the cell of interest, 11 refers tothe left-hand neighbor, and i+ 1 refers to the right-hand neighbor. The flow potential @oequals pressureplus capillary pressure plus gravitational potential,and its Id$eat the (n+ 1)-time level is explained later.Substituting Eq. 2 in Eq. 1 and dividing through byAyAx gives

[(1 hko *+Oi-1 )*F hko .Ax p.. B. Ax /.LOBa

Eq. 3 is rearranged to give

(@y@% )1 qoAx AxAyS*[(*)+l-(*)] , ~ (4)

.-

where A = * , and the subscripts i+% and i 1%~B

indicate that the quantity is evaluated as an averagefor the (i+ 1, i) and (i, i 1) cells, respectively. DiEer-ent investigators use different averaging techniques.The upstream value for A is the most commonly used.

Eq. 4 is the oil mass balance equation in one di-mension, in difference form, which is used in the simu-lation calculations. In two and three dimensions, y-and z-&ection terms identical with the x-directionterm are added.

Eq. 4 may be written in differential form as

()a+. ()qo _ a +JW& Ao~ . .AxAy 2t B. (4a)and in vector notation as

These three forms of the MBE are used inter-changeably in the literature. Because of its compact-ness, Eq. 4b is the most commonly used.

In deriving Eq. 4 we used the value of % at the(n+ 1)-time level, i.e., at the end of the time step.This diflerencing technique is called the implicit2 orbackward difference method and is the most com-monly used. The Crank-Nicholson methodz uses an

average value for @oi.e., at the (n+ %2 )-time levelwhile the forward difference method uses @. at thebeginning of the time step i.e., at the n-time level.The implicit method is the most stable of the three.The time at which hkJBop~ is evaluated was left un-specified. Most authors use the n-time level, but someuse the (n+ 1)-time level. This will be discussed later.

Similar derivations can be made for the water andgas. The water and gas equations in vector notationform are, respectively,

and

v (Afv@g) + v O&AJv%)

+ V l (R,,oA,oV@t.) *Y

where in Eq. 6 the gas dissolved in the oil and wateris accounted for.

Eqs. 4b, 5, and 6 are the MB eqiaaticms for three-phase immiscible flow in a black oil system, and werederived by Muskat.3 Written in difference form, theseare virtually the only equations used in the most com-mon type of simulation, that of a black oil reservoir.

Method of SolutionEq. 4 and its comparable forms for the water and gasgive the relationships, for each cell, among pressure;and oil, water, and gas saturations; and time. If thereare m cells, then we have m equations for each phase,giving a tdai of 3m equations. The solution of theseequations is the major chore of reservoir simulation.

Two methods of solution are generally used; theseare the implicit-implicit 3 and the implicit-explicit.They are similar in one respect. Given a value for thesaturations and pressure at each cell at the beginningof a time step, new saturations and pressure valuesare found at the end of the time step. These values inturn represent the starting point for the next time step.This stepwise process is continued until the desiredamount of time elapses.

The implicit-implicit method solves Eqs. 4 andthe difference forms of Eqs. 5 and 6 directly. Thesolution usually involves an iterative procedure. Cap-illary pressure occasionally tames instability prob-lems.* The implicit-implicit method overcomes theproblem by expressing saturation as a function of cap-illary pressure. To start the calculations, values ofsaturations are assumed, and the pressures in the oil,.~,ater, ~d gas phasesarecalculated.These calculatedpressures result in new capillary pressures, which areused to calculate saturations. These are comparedwith the assumed values, and if necessary the calcula-tions are repeated.

Using the fact that the oil, water, and gas satura-tions add up to one, we can manipulate the three MB

For definition, refer to the section on Computetionel Consid-eration.

JOURNAL OF PETROLEUM TECHNOLOGY

..

equations in such a way as to result in a pressureequation. The pressure equation in symbolic formand vector notation is

v k,up Zpq=c-z (7)

.&,KeKe).~ is the total effective mobility of the threephases, q is the total production, and c is the totaleffective compressibili~. (Capillary and gravity forceshave been neglected.) In the implicit-explicit method,at any given time, the pressure equation @q. 7) issolved first, giving the pressure distribution at eachcell. Then the saturations are determined from thesolution of the three MB equations.

To illustrate the method of solving Eq. 7 we writeit in difference form in one dimension. We also assumethat k = c = 1, and that q* = O. The implicit dif-ference formulation is

where i, i 1, and i+ 1 refer to the cell of interest andits two neighbors, and n refers to the time level. Eq.8 gives P ~, the pressure tO be dete~ed> as a finc-tion of two unknowns PY1 and P U. TM We cannotsolve for p ~1 with this equation alone. For this reasonwe call this an implicit equation in pressure. However,similar equations can be written for all cells, resultingin m equations with m unknowns.

Several methods have been devised to solve the mpressure equations. The simplest are tbe relaxationtechniques. The pressures at i 1 and i+ 1 are as-sumed, and the pressure p~l is calculated. This tnal-and-error process is repeated at each point in turnuntil a sufficiently accurate solution to the m equa-tions is found. Given this pressure solution, we thensolve explicitly for the saturations, using the threeMB equations.

The coeflkients ~ and CTcontain effeCtiVeperme-~~i~te$, @-e~i~e~3 and fo~ation volume factors, SOthey are functions of saturation and pressure. Untilnow we have ignored this fact. However, if we wantto account for i.hs ~eYwL,uw..- --dI=nYy which is sometimesnecessary due to instability, then an iterative methodis used. The method is summarized by the following ~~steps, which are symbolically correct. In actuality thecalculations are more involved. 7

1. Begin with known pressure and saturation dis-tribution at time n and, using the pressure equation,and the values for ATand CTcalculated from the sat-uration distribution and the pressure values at timen, solve for pressures at time n+ 1.

2. Solve for the saturation diSifibiitiOtl at time n + 1using the three MB equations.

3. Using the new saturations and the pressures cal-culated in Step 1, recalculate & and cT vaiues.

4. Repeat Steps 1 through 3 until the convergencecriterion is achieved. In repeating Step 1, the valuesOf& and c Of Step 3 are used.

5. Proceed with the next time step.Methods that cycle between pressure and satura-

tion equations-are called fully implicit or iterative,whereas those that do not, and in which essentially

only Steps 1 and 2 and then 5 are executed, are calledmixed. 7 Mixed methods are extensively used becausethey require less computer time.

One criterion for dete rmining the compatibility ofthe pressure and saturation values is the material bal-ance error. One form of the material balance is thesummation of the stock tank oil at the beginnimg andat tiie end of the time step. The ditlerence betweenthe values should be equal to the totai production dtii-ing the time step. The incremental error is calculatedby the following equation.

incremental MBE error =

where V is the volume of the cell and the summationis taken over the m ceils.

Some authors use cumulative MBE error, which isgiven by the following equation.

cumulative MBE error =

initial oil in place-! [v@ln:_l.

cumulative total productionA low value for MBE error is a necessary but not

a sufficient criterion for a correct solution. In essence,low error indicates that the total oil in the reservoir attime n+ 1 is correct, but it does not guarantee thatthe oil is distributed properly.

Computational ConsiderationsComputing TimeFor a given computer, the time required for a par-ticular reservoir simulation depends primarily upon(1) the number of cells, and (2) the number of timesteps.

The computing time required for a time step isproportional to the number of mesh points. Doublingthe number of mesh points approximately cioutiles thecomputer time per time step.

The ~ti,m&r Of time stepsrequired to simulate anassigned number of years dep-nds on the allowedlength of the time step At. The maximum value Atmay take is a function of the volume and shape of thecell. In a two-dimensional horizontal model, for ex-ample, the cell volume is AXAytimes thickness, andthe shape is given by the ratio of Ax/Ay, where Axand Ay are the horizontal dimensions of the cell. Theallowed time step decreases as AX and Ay decrease,

IAXand as Fy 1 incieases. For example, the d-

Iowed At in a simulation study in which AX = Ay =300 ft will be about four times the Al if AX = Ay =150 ft.

InstabilityNumerical techniques do not yield exact solutions.There is an error associated with the answers. Thiserror sometimes grows very rapidly, causing the solu-

1387

..

tion to blow up; in other words, the solutions be-come ph ysicaiiy -mrea!istic. The most common causeof this instability is excessively large changes in sat-urations and pressures during the time step. Usuallythis may be remedied by reducing the size of thetime step.

Nnmerfctd DispersionThis is an inherent property of digital simulation. Itis due to the representation of the reservoir by cells inwhich properties are averaged. When a saturationfront enters the ceil, it is S~ir3Xl out over the cell toarrive at average satmdicm values. Numerical disper-sion can be minimized by decreasing the dimensionsof the cells. However, this leads to increased com-puter time.

Validity of SolutionOnce a simulation run has been made, the questionansex How good is the solution? Small MBE errorindkates that the total fluid volumes are correct, butdoes not guarantee that the fluid distribution is valid.If the resulting fluid distribution is questionable, asystematic analysis is needed. Variables that influencethe saturation distribution are the time step size At,and the cell dimensions ~ and Ay. For a correctmathematical analysis, the sensitivity of the results toti, Ay, and At should be examined. A change in Ax,Ay values may require a major revision of the data,which is not practical. A common practice is to study

Norriseal 3-way2-position

motor valves.

only the effect of the time-step size. This is done byrerunning the simulation with reduced time steps andcompa~ng ~~e ~e~u!t~.The time stepisreduced untilfurther reduction does not change the results signifi-cantly, thus indicating that the best solution has beenobtained for the chosen sizes of the cells.

AcknowledgmentI should liie to thank G. L. Smitt, J. W. Watts andJ. E. Walraven for helpful comments, and Mobil Re-search & Development Corp. for permission to pub-lish this paper.

References1.Craft, B. C. and Hawkins, M. F., Jr.: Applied Petroleum

Reservoir Engineering, Prentice-HallInc.,EnglewoodCliffS,N. J. (1959).

2. Smith, G. D.: Numerical Solution of Partial DifferentialEquations, Oxford U. Press, Inc., New York (1965).

3. Muskat, M.: Physical Principles of Oil Production, Mc-Graw-Hill Book Co., Inc., New York (1949 ).

4. ~hss, Jim, Jr., Peacemen, D. W. and Rachford, H. H., A Method for Calculating Multi-Dimensional lm-

~ncible Displacement, Trans., AIME ( 1959) 216, 297-5. ~oa~, K. H., Nielsen, R. L., Terhune, M. H. and Weber,

Simulation of Three-Dimensional, Two-PhaseFiow ;n Oil and Gas Reservoirs, Sot. Pet. Eng. J. (Dec.,1967) 377-388.

6. Fagin, R. G. and Stewart, C. H., Jr.: A New .Approachto the Two-Dimensional Multiphase Reservor Sumdator,Sot. Pet. Eng. J. (June, 1966) 175-182.

7. Blair, P. M. and Weinaug, C. F.: Solution of. Two-PhaseFlow Problems Using Implicit Difference Equations, paperSPE 2185 presented at SPE 43rd Annual Fall Meeting,Houston, Tex., Sept.29-Ott. 2, 1968. JPT

for controlled,high pressure automation!

Series 3023 withworking pressures up to 3600 psig;temperatures from -20 to 180 F.

aAvailable in 1, 2 and 3sizes, the NorriseaiSeries 3023 will he.ndlemetering, blending or divertingfor tough highdifferential end staticpressure applications. Cast

~-

steel body with hammer-

nut top closure .. . stainlesssteel plugs and seats ...selection of hardor soft cagegu ided trims . .. widerange of seals.

~ ,

Write for Bulletin 3-5 7-69. I

P. O. Box 9528, Houston, Taxss 77011 -(713) 928-2701

JOURNAL OF PETROLEUM TECHNOLOGY