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

Introduction Reservoir simulation is based on well known reservoir Basic Analysis engineering equations and techniques — the same If a reservoir is fairly homogeneous, average values equations and techniques the reservoir engineer has of the reservoir properties, such as porosity, are ade- been using for years. quate to describe it. The average pressure, time, and .-f m-c tO Lie representation production behavior of such a reservoir under a solu- In generai, sirmukition .e.w.. .- . . ---+ nre MNTI-MUy calculated of some process by either a theoretical or a physical tion gas drive, for ~Acwy.-, -- model. Here, we limit ourselves to the simulation of by the familiar methods’ of Tamer, Muskat, or Tracy. petroleum reservoirs. Our concern is the development All of these methods use the material balance equa- and use of models that describe the reservoir perform- tion normally referred to as the MBE. A simple ex- ance under various operating conditions. pression for the oil MBE is the following Reservoir simulation itself is not really new. Engi- (cumulative net withdrawal in STB) = (original neers have long used mathematical models in per- oil in place in STB) — (oil remaining in place forming reservoir engineering calculations. Before the in STB) development of modem digital computers, however, the models were relatively simple. For example, when The cumulative net withdrawal is the difference be- calculating the oil in place volumetrically, the engi- tween the oil that leaves the reservoir and the oil that neer simulated the reservoir by a simple model in enters it. In this basic analysis, there is no oil entering which average values for the porosity, saturation, and the reservoir since the boundaries are considered im- thickness were used. permeable to flow. Thus, the MBE reduces to its Although simulation in the petroleum industry is simplest form. Such a reservoir model is called the not new, the new aspects are that more detailed reser- tank model (Fig. 1). It is zero dimensional because voir features, and thus more accurate simulations, rock, fluid properties, and pressure values do not vary have become practical because of the capability af- from point to point. Instead, they are calculated as forded by the computers now available. The more de- average values for the whole reservoir. This tank tailed description, however, requires complex mathe- model is the basic building block of reservoir simu- matical expressions that are difficult to understand, lators. and this difiicuhy has caused some engineers to shy Now let us consider a reservoir represented by a away from using simulators, and others to misuse sandbar. Let the two halves of the sandbar vary in them. lithology. The sandbar as a whole cannot be repre- We in the petroleum industry are in the reservoir sented by average properties, but each half can. Thus, simulation revolution. As time goes on, simulators will the sandbar consists of two tank units, or cells, as they be used more and more, so a basic understanding of are normally called. The MBE describes the fluid . .. reservolr modelmg is esseiitia,. behavior in each ceU as in the previous tank model. “ ‘ The engineer! espe- .,M. .. . ~] te~ of the MBE is cially, must become competent in setting up simula- However, the net wl,,,d,~.v-. .+...- tion problems, in deciding on appropriate input data, more complicated because there can be migration of and in evaluating the results. fluid from one cell to another, depending on the aver- . 7-Z3t- 1383 age pressure values of the two cells. This fluid transfer several two-dimensional geometries, the most popu- between the two cells is calculated by Darcy’s law. lar of which is the horizontal (x-y) geometry; but the The MBE, together with Darcy’s law, describes the vertical (x-z) and the radial (r-z) geometries are also behavior of each cell. This model is not a zero-dimen- used quite often. sional reservoir simulator since reservoir parameters Simulators can be classified also according to the may vary between the two cells. Instead it is a one- type of reservoir or process they are intended to simu- dimensional model, because it consists of more than late. There are, for example, gas, black oil, gas con- one cell in one direction and of only one cell in the densate, end miscible displacement reservoir simulat- other two directions (Fig. 2). ors. Moreover, there are one-, two- and three-phase This analysis can be extended to reservoirs where reservoir models. Furthermore, any of these simula- properties as well as pressure values vary in two tors may or may not account for gravitational or dimensions, and to others where the variation occurs capillary forces. It is not enough to choose the proper in three dimensions. The simulators representing these simulator with respect to dirnensionality; the simu- reservoirs are called, respectively, two-dimensional lator must represent the type of hydrocarbon and the and three-dimensional simulators, as illustrated in fluid phases present. Figs. 3 and 4. Thus, a two-dimensional reservoir simulator consists of more than one cell in two dimensions Simulation Steps Downloaded from http://onepetro.org/JPT/article-pdf/21/11/1383/2223263/spe-2790-pa.pdf by guest on 23 September 2021 and of one cell in the thhxi dimension. And a three- Preparation of Data dimensional simulator consists of more than one cell After the type of model to use in a study has been in all of the three dimensions. selected, the next step is to divide the reservoir into a Regardless of the number of dimensions used, the number of cells, as illustrated in Figs. 2 through 4. MBE is the basic equation describing the fluid be- This is accomplished by laying out a grid system for havior within a cell; and Darcy’s law describes the the reservoir. In a two-dimensional study, the grid is interaction between the cells. In one-, two-, and three- established by drawing lines on a map of the reser- dimensional models each cell, except the boundary voir. All grid lines must extend across the reservoir. cell, interacts respectively with 2, 4, and 6 cells. Since Each cell is identified by its x, y, z coordinates. Then a simulator can consist of hundreds of cells, keeping the flow conditions around the perimeter of the res- account of the MBE for each cell is a formidable ervoir are established. Normally the reservoir bound- bookkeeping operation ideally suited to digital com- ary is considered sealed, but influx or efflux at an ~utation. But we emphasize once again that the prin- assigned pressure or rate may also be specified. m. . ciples and equations used in reservoir simulation are 1rie next s’fip is to ass:gn the. .. -following- for each not new. They only appear so because of the complex- celk rock properties, geometry, initial fluid distribu- ity of the bookkeeping. tion, and fluid properties. The rock properties consist of specific permeability, porosity, relative permeabili- Types of Reservoir Skmihiiim Cj ~d ~~m~e&~.e$~Apillary pressure. The Cell geome- There are several types of reservoir simulators. Choice try includes the depth, thickness and locations of of the proper simulator to represent a particular res- wells. Usually the wells are assumed to be located at ervoir requires an understanding of the reservoir and the centers of ‘&e cells in which *Aeyfal!a The initial a careful examination of the data available. A model fluid distribution consists of the oil, water and gas that fits Reservoir A may not be appropriate for Res- saturations at the beginning of simulation. Also, the ervoir B, in spite of apparent similarities between average pressure of the cell at that time is assigned or Reservoirs A and B. A reservoir model is useful only calculated from known data. Fluid properties are when it fits the field case. specitied by the usual PVT data. In addition, for each One basis for classif@g models, as discussed ear- well it is necessary to provide a production schedule lier, is the number of dimensions. The two-dimen- and a productivity index or a skin value (i.e., damage sional model is the most commordy used. There are or ,improvement).

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

Flow

Fig. l—Tsmk model. Fig. 3-Two-dimensional simulator. JOURNAL OF PETROLEUM TECHNOLOGY The engineer should scrutinize carefully these basic the reservoir. data for consistency and accuracy. For example, if Mathematical Considerations pressure buildup data are available on a well, the permeability-thickness product of the cell where the Derivation of Equations well is located and the flow rate assigned to the well, For the engineer to adequately understand reservoir should be compatible with the buildup data. The time simulation, he should be acquainted with the equa- spent in examining the basic data is well spent, for it tions used. These are basicaUy material balances about can lead to fewer simulation runs. Moreover one must cells for each phase, and Darcy’s law, which describes always remember that theanswer is only as good as the interactions between cells. For illustration, we the input data. derive here the fundamental equations for a black oil system. and exp!ain thei: physical significance. History Matchfng and Performance Predktion For the sake of simphclty, consider a cdl in a one- The rn-fi ~urpose of reservoir simulation is to prediCt dimensional reservoir simulator, as shown in Fig. 5. the rate of hydrocarbon recovery for dtierent meth- Tine sane mia!jjsis is applicable to a cell in two- and .* .-.- CI,.J A+. ods of field operation. d adeq-ua LG ~eiu -a~ ex!st~ th.me-dimensional models. (An expression for the oil reasonably accurate performance predictions can be material balance of the cell was given eariier.) made. If data are incomplete or suspect, simulators (Oil volume entering the ceU during a time incre- Downloaded from http://onepetro.org/JPT/article-pdf/21/11/1383/2223263/spe-2790-pa.pdf by guest on 23 September 2021 may be used only to compare semi-quantitatively the ment At, in STB) minus (oil volume leaving the ceU results of dtierent ways of operating the reservoir. In during the same time increment, in STB) equals (the either case, the accuracy of the simulator can be imp- change in oil volume in the ceU, in STB). roved by history matching. Volume of oil entering the cell during At, in STB, The first step in a history match is to calculate res- equals QinAt. ervoir performance using the best data available. The Volume of oil leaving the cell during At, in STB, results are compared with the field recorded histories equals (At + dA~. of the wells. If the agreement is not satisfactory, such Change in volume of oil in the cell during At, in data as permeability, relative permeabiUty, and po- STB, equals ro@ are va~~ed from one computer run to another until a match is achieved. The s&ndator is then used to predict performance for alternative plans of operating the reservoir. where Qi. is the average flOWmte of ofl into the cell The behavior of the reservoir is influenced by many during At in STB/unit time, Q..t is the avenge flow factors — permeability, porosity, thickness, satura- rate of oil leaving the cell to its neighbors during At tion distributions, relative permeability, etc. — that in STB/unit time, and QOis the oil production rate are never known precisely aU over the reservoir. What from the cell, if it contains a well, in STB/unit time; the engineer arrives at is only a combination of these AxAyh@. represents the volume of oil in the cell at variables, which results in a match. ‘T’is colmbiiation l?. is not unique, so it may not represent precisely the any time, n+ 1 refers to the end of the time step, and condition of the reservoir. When the simulator, after n to the beginning. a match, is used to predict, it is not certain that the Substitution in the oil MBE, after dividing through physical picture of the reservoir described in the simu- by At, gives lator will give predictions sufficiently close to the actual reservoir performance. In generaI, the longer the Qin – Qcmt – go= matched history period, the more reliable the pre- %w’)w%)rnl dicted performance wiU be. It behooves the engineer .,...... (1) to monitor periodically the predicted vs the actual performance and to update his physioal picture of However, by Drircy’s&.w, assuming the flaw to be

Q out Flow

Qin

Fig. 4-Three-dimensional simulator. Fig. 5-Cell in a one-dimensional simulator.

. 1385 from left to right as shown in Fig. 5, average value for @o—i.e., at the (n+ %2 )-time level— while the forward difference method uses @. at the beginning of the time step — i.e., at the n-time level. The implicit method is the most stable of the three. and The time at which hkJBop~ is evaluated was left un- specified. Most authors use the n-time level, but some use the (n+ 1)-time level. This will be discussed later. Similar derivations can be made for the water and gas. The water and gas equations in vector notation where Ayh is the cross-sectional area of the cell, Ax form are, respectively, is the length of the cell, 00 is the flow potential in the oil phase, i refers to the cell of interest, 1—1 refers to the left-hand neighbor, and i+ 1 refers to the right- hand neighbor. The flow potential @oequals pressure and plus capillary pressure plus gravitational potential, and its Id$eat the (n+ 1)-time level is explained later. ‘ (Afv@g) + “ O&AJv%) Substituting Eq. 2 in Eq. 1 and dividing through by v v Downloaded from http://onepetro.org/JPT/article-pdf/21/11/1383/2223263/spe-2790-pa.pdf by guest on 23 September 2021 AyAx gives 1 hko *“+’ + V ● (R,,oA,oV@t.) – *Y —— Oi-1 –*F .— hko Ax p.. B. /.LOBa [( Ax )

where in Eq. 6 the gas dissolved in the oil and water is accounted for. Eqs. 4b, 5, and 6 are the MB eqiaaticms for three- phase immiscible flow in a black oil system, and were Eq. 3 is rearranged to give derived by Muskat.3 Written in difference form, these are virtually the only equations used in the most common type of simulation, that of a black oil reservoir. Method of Solution @y—@% ——qo Ax )1 AxAy Eq. 4 and its comparable forms for the water and gas ( give the relationships, for each cell, among pressure; S*[(*)”+l-(*)”] , ~ (4) and oil, water, and gas saturations; and time. If there .- are m cells, then we have m equations for each phase, giving a tdai of 3m equations. The solution of these where A = * , and the subscripts i+% and i —1% equations is the major chore of reservoir simulation. ~B Two methods of solution are generally used; these indicate that the quantity is evaluated as an average are the implicit-implicit” 3 and the implicit-explicit.” for the (i+ 1, i) and (i, i – 1) cells, respectively. DiEer- They are similar in one respect. Given a value for the ent investigators use different averaging techniques. saturations and pressure at each cell at the beginning The upstream value for Ais the most commonly used. of a time step, new saturations and pressure values Eq. 4 is the oil mass balance equation in one di- are found at the end of the time step. These values in mension, in difference form, which is used in the simu- turn represent the starting point for the next time step. lation calculations. In two and three dimensions, y- This stepwise process is continued until the desired and z-&ection terms identical with the x-direction amount of time elapses. term are added. The implicit-implicit”’ method solves Eqs. 4 and Eq. 4 may be written in differential form as the difference forms of Eqs. 5 and 6 directly. The solution usually involves an iterative procedure. Cap- a+. qo _ a +JW’ & Ao~ –—–— — . . (4a) illary pressure occasionally tames instability prob- () AxAy 2t()B. lems.* The implicit-implicit method overcomes the and in vector notation as problem by expressing saturation as a function of capillary pressure. To start the calculations, values of saturations are assumed, and the pressures in the oil, .~,ater, ~d gas phasesarecalculated.These calculated These three forms of the MBE are used inter- pressures result in new capillary pressures, which are changeably in the literature. Because of its compact- used to calculate saturations. These are compared ness, Eq. 4b is the most commonly used. with the assumed values, and if necessary the calcula- In deriving Eq. 4 we used the value of % at the tions are repeated. (n+ 1)-time level, i.e., at the end of the time step. Using the fact that the oil, water, and gas satura- This diflerencing technique is called the implicit2 or tions add up to one, we can manipulate the three MB backward difference method and is the most com- “For definition, refer to the section on Computetionel Consid- monly used. The Crank-Nicholson methodz uses an eration.

JOURNAL OF PETROLEUM TECHNOLOGY . .

equations in such a way as to result in a pressure only Steps 1 and 2 and then 5 are executed, are called equation. The pressure equation in symbolic form mixed. 7 Mixed methods are extensively used because and vector notation is they require less computer time. One criterion for dete rmining the compatibility of Zp (7) v “ k,up – the pressure and saturation values is the material bal- q’=c’-z ‘ “ “ “ “ ance error. One form of the material balance is the .&,KeKe).~ is the total effective mobility of the three summation of the stock tank oil at the beginnimg and

phases, q’ is the total production, and c’ is the total at tiie end of the time step. The ditlerence between effective compressibili~. (Capillary and gravity forces the values should be equal to the totai production dtii- have been neglected.) In the implicit-explicit’ method, ing the time step. The incremental error is calculated at any given time, the pressure equation @q. 7) is by the following equation. solved first, giving the pressure distribution at each cell. Then the saturations are determined from the incremental MBE error = solution of the three MB equations. To illustrate the method of solving Eq. 7 we write it in difference form in one dimension. We also assume that k’ = c’ = 1, and that q* = O. The implicit dif- Downloaded from http://onepetro.org/JPT/article-pdf/21/11/1383/2223263/spe-2790-pa.pdf by guest on 23 September 2021 ference formulation’ is where V is the volume of the cell and the summation is taken over the m ceils. Some authors use cumulative MBE error, which is given by the following equation. where i, i—1, and i+ 1 refer to the cell of interest and its two neighbors, and n refers to the time level. Eq. cumulative MBE error = 8 gives P ~“, the pressure tO be dete~ed> as a finc- tion of two unknowns Wecannot initial oil in place PY1 and P U. TM -! solve for p ~1 with this equation alone. For this reason cumulative total production[v’@ln:_l. we call this an implicit equation in pressure. However, similar equations can be written for all cells, resulting A low value for MBE error is a necessary but not in m equations with m unknowns. a sufficient criterion for a correct solution. In essence, Several methods have been devised to solve the m low error indicates that the total oil in the reservoir at pressure equations. The simplest are tbe relaxation time n+ 1 is correct, but it does not guarantee that techniques.’ The pressures at i– 1 and i+ 1 are as- the oil is distributed properly. sumed, and the pressure p~l is calculated. This tnal- Computational Considerations and-error process is repeated at each point in turn Computing Time until a sufficiently accurate solution to the m equations is found. Given this pressure solution, we then For a given computer, the time required for a par- solve explicitly for the saturations, using the three ticular reservoir simulation depends primarily upon MB equations. (1) the number of cells, and (2) the number of time The coeflkients ~’ and CTcontain effeCtiVeperme- steps. ~~i~te$, @-e~i~e~3 and fo~ation volume factors, SO The computing time required for a time step is they are functions of saturation and pressure. Until proportional to the number of mesh points. Doubling now we have ignored this fact. However, if we want the number of mesh points approximately cioutiles the to account for i.hs ~eYwL,uw..- computer time per time step. “ ‘ --”dI=n”Yy which is sometimes The ~ti,m&r Of time stepsrequired to simulate an necessary due to instability, then an iterative method is used. The method is summarized by the following ~~assigned number of years dep-nds on the allowed steps, which are symbolically correct. In actuality the length of the time step At. The maximum value At calculations are more involved. 7 may take is a function of the volume and shape of the 1. Begin with known pressure and saturation dis- cell. In a two-dimensional horizontal model, for ex- tribution at time n and, using the pressure equation, ample, the cell volume is AXAytimes thickness, and and the values for ATand CTcalculated from the sat- the shape is given by the ratio of Ax/Ay, where Ax uration distribution and the pressure values at time and Ay are the horizontal dimensions of the cell. The n, solve for pressures at time n+ 1. allowed time step decreases as AX and Ay decrease,

2. Solve for the saturation diSifibiitiOtl at time n + 1 IAX and as 1’ incieases. For example, the ‘d- using the three MB equations. Fy – 3. Using the new saturations and the pressures calculated in Step 1, recalculate & and cT vaiues. Iowed At in a simulation study in which AX = Ay = 4. Repeat Steps 1 through 3 until the convergence 300 ft will be about four times the Al if AX= Ay = criterion is achieved. In repeating Step 1, the values 150 ft. Of& and c’ Of Step 3 are used. 5. Proceed with the next time step. Instability Methods that cycle between pressure and satura- Numerical techniques do not yield exact solutions. tion equations-are called fully implicit or iterative,’ There is an error associated with the answers. This whereas those that do not, and in which essentially error sometimes grows very rapidly, causing the solu-

1387 . .

tion to “blow up”; in other words, the solutions be- only the effect of the time-step size. This is done by come physicaiiy ‘-mrea!istic. The most common cause rerunning the simulation with reduced time steps and of this instability is excessively large changes in sat- compa~ng ~~e ~e~u!t~.The time stepisreduced until urations and pressures during the time step. Usually further reduction does not change the results signifi- this may be remedied by reducing the size of the cantly, thus indicating that the best solution has been time step. obtained for the chosen sizes of the cells. Nnmerfctd Dispersion Acknowledgment This is an inherent property of digital simulation. It I should liie to thank G. L. Smitt, J. W. Watts and is due to the representation of the reservoir by cells in J. E. Walraven for helpful comments, and Mobil Re- which properties are averaged. When a saturation search & Development Corp. for permission to pub- front enters the ceil, it is S~ir3Xl out over the cell to lish this paper. arrive at average satmdicm values. Numerical dispersion can be minimized by decreasing the dimensions References of the cells. However, this leads to increased com- 1.Craft, B. C. and Hawkins, M. F., Jr.: Applied Petroleum Reservoir Engineering, Prentice-HallInc.,EnglewoodCliffS, puter time. N. J. (1959).

2. Smith, G. D.: Numerical Solution of Partial Differential Downloaded from http://onepetro.org/JPT/article-pdf/21/11/1383/2223263/spe-2790-pa.pdf by guest on 23 September 2021 Validity of Solution Equations, Oxford U. Press, Inc., New York (1965). 3. Muskat, M.: Physical Principles of Oil Production, Mc- Once a simulation run has been made, the question Graw-Hill Book Co., Inc., New York (1949 ). ansex “How good is the solution?” Small MBE error 4. ~hss, Jim, Jr., Peacemen,D. W. and Rachford, H. H., indkates that the total fluid volumes are correct, but “ “A Method for Calculating Multi-Dimensional lm- ~ncible Displacement”, Trans., AIME ( 1959) 216, 297- does not guarantee that the fluid distribution is valid. If the resulting fluid distribution is questionable, a 5. ~oa~, K. H., Nielsen, R. L., Terhune, M. H. and Weber, systematic analysis is needed. Variables that influence “ “Simulation of Three-Dimensional, Two-Phase Fiow ;n Oil and Gas Reservoirs”, Sot. Pet. Eng. J. (Dec., the saturation distribution are the time step size At, 1967) 377-388. and the cell dimensions ~ and Ay. For a correct 6. Fagin, R. G. and Stewart, C. H., Jr.: “A New .Approach to the Two-Dimensional Multiphase Reservor Sumdator”, mathematical analysis, the sensitivity of the results to Sot. Pet. Eng. J. (June, 1966) 175-182. ti, Ay, and At should be examined. A change in Ax, 7. Blair, P. M. and Weinaug, C. F.: “Solution of. Two-Phase Ay values may require a major revision of the data, Flow Problems Using Implicit Difference Equations”, paper SPE 2185 presented at SPE 43rd Annual Fall Meeting, which is not practical. A common practice is to study Houston, Tex., Sept.29-Ott. 2, 1968. JPT

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