Recent Developments in Reservoir SimulationRichard E. Ewing�AbstractThree basic problem areas have dominated much of the recent research in reservoir simulation. First,one must obtain an e�ective model to describe the complex uid/ uid and uid/rock interactions thatcontrol recovery processes. This includes the problem of obtaining accurate reservoir descriptions atvarious length scales and including the e�ects of this heterogeneity in the reservoir simulators. Next,one must develop accurate discretization techniques that retain the important physical properties of thecontinuous models. Finally, one should develop e�cient numerical solution algorithms that utilize thepotential of the emerging computing architectures. We will discuss recent advances in each of these threeareas.Simulators are severely hampered by the lack of knowledge of reservoir properties, heterogeneities,and relevant length scales and important mechanisms like di�usion, dispersion, and viscous instabilities.Recent developments have been made in homogenization, scaled averaging and the use of the simulatoras an experimental tool to develop methods to model the interrelations between localized and larger scalemedia e�ects.Discretization techniques must be developed for both the pressure and transport equations. Mixed�nite element and �nite volume methods can be used to yield accurate mass-conservative approximationsto the pressure and Darcy velocity of the uid. Eulerian-Lagrangian techniques have been developedthat conserve mass but also take advantage of the computed ow of the uids to accurately model theimportant transport phenomena and simultaneously symmetrize the model equations. A posteriori errorestimators can motivate adaptive local grid re�nement in space and time. Then multigrid or multileveliterative techniques can be used to e�ciently solve the discrete systems.Major potential advances in computing lie in the emerging parallel and vector/parallel computerarchitectures. Techniques like domain decomposition that naturally split up the problem into pieces tobe addressed separately on distinct processors, also allow modularized local grid re�nement and can playa signi�cant role in developing e�cient simulation codes.1 IntroductionThe objective of reservoir simulation is to understand su�ciently well the complex chemical, physical, and uid ow processes occurring in a petroleum reservoir so as to be able to optimize the recovery of hydrocarbon.To do this, one must be able to predict the performance of the reservoir under various exploitation schemes.In order to predict reservoir performance, a series of models of reservoir processes is constructed which yieldinformation about the complex phenomena accompanying di�erent recovery methods.There are four major stages to the modeling process for reservoir simulation. First, a physical modelof the ow processes is developed incorporating as much geology and physics as is deemed necessary todescribe the essential phenomena. Second, a mathematical formulation of the physical model is obtained,usually involving coupled systems of nonlinear partial di�erential equations. Third, once the properties ofthe mathematical model, such as existence, uniqueness, and regularity of the solution, are su�ciently wellunderstood and the properties seem compatible with the physical model, a discretized numerical model ofthe mathematical equations is produced. A numerical model is determined that has the required propertiesof accuracy and stability and which produces solutions representing the basic physical features as well aspossible without introducing spurious phenomena associated with the speci�c numerical scheme. Finally, acomputer program capable of e�ciently performing the necessary computations for the numerical model issought. The total modeling process encompasses aspects of each of these four intermediate steps.�Institute for Scienti�c Computation, Texas A&M University, College Station, TX 77843{3404.1

Finally, the modeling process is not complete with one pass through these four steps. Once a computerprogram has been developed which gives concrete quantitative results for the total model, this output shouldbe compared with measured observations of the physical process. If the results do not compare extremelywell, one should iterate back through the complete modeling process, changing the various intermediatemodels in ways to obtain better correlation between the physical measurements and the computationalresults. Usually, many iterations of this modeling loop are necessary to obtain reasonable models for thehighly complex physical phenomena describing petroleum production processes.The trends in reservoir modeling are dominated by three broad topics: 1) obtaining better reservoirdescriptions and incorporating these descriptions in reservoir simulators, 2) understanding increased com-plications due to modeling of complex enhanced oil recovery processes, developing accurate discretizationschemes to address these complexities, and 3) development of algorithms which can exploit the potentialparallelism of the parallel/vector architecture computers. We will brie y discuss major trends in each ofthese areas.2 Reservoir Characterization and DescriptionSimulation of complex uid ow processes in heterogeneous porous media is an important aspect of petroleumreservoir engineering and the modeling and control of contaminant transport through porous media. We�rst address techniques for treating the di�culties caused in simulation by heterogeneities in the reservoir,di�usion and dispersion, viscous �ngering, and other highly-localized ow properties such as ow aroundwells. The common di�culty in modeling these various ow regimes is the inability to treat highly varyinglength scales in large-scale reservoir simulation.The processes of both single- and multiphase ow involve convection, or physical transport, of the uidsthrough a heterogeneous porous medium. The equations used to simulate this ow at a macroscopic level arevariations of Darcy's law. Darcy's law has been derived [68, 69] via a volume averaging of the Navier-Stokesequations, which govern ow through the porous medium at a microscopic or pore-volume level. The lengthscale for Navier-Stokes ow (10�4{10�3 meters) is very di�erent from the scale required by normal reservoirsimulation (102{103 meters). Reservoirs themselves have scales of heterogeneity ranging from pore-level to�eld scale. In the standard averaging process for Darcy's law, many important physical phenomena whichmay eventually govern the macroscopic ow are lost. The further averaging of reservoir and uid propertiesnecessary to use grid blocks of the size of 10{102 meters in �eld-scale simulators further complicates themodeling process. We discuss certain techniques that are beginning to address these scaling problems.Many of the enhanced recovery processes are characterized by the chemical and physical interaction ofthe uids. Therefore, di�usion and dispersion are often critical to the ow processes and must be understoodand modeled. Molecular di�usion is typically quite small. However, dispersion, or the mechanical mixingcaused by velocity variations and ow through heterogeneous rock, can be extremely important and shouldbe incorporated in some way in our models. We must also develop accurate discretization schemes that canbe applied e�ectively to problems with large transport and small dispersion.Since the mixing and velocity variations are in uenced at all relevant length scales by the heterogeneousproperties of the reservoir, much work must be done in volume averaging of terms like porosity and perme-ability. Renormalization group techniques [53] have been e�ective. Statistical methods have shown promisein this area [3, 45]. Statistical techniques are currently being considered to obtain e�ective permeabilitytensors for large-scale models of ow through highly fractured media. However, if the fractures or �eld-scaleheterogeneities are su�ciently large and can be identi�ed, they should be incorporated in the model viaspecial gridding and high permeability variations. If the fractures are su�ciently large, perhaps Darcy's lawwill not be a good model for ow through these fractures. The adaptive local grid-re�nement techniquespresented in this paper can be very valuable in these applications.The e�ects of dispersion in various ow processes have been discussed extensively in the literature [55,60, 70, 71, 74]. Russell and Wheeler [65] and Young [78] have given excellent surveys of the in uence ofdispersion and attempts to incorporate it in present reservoir simulators. Various terms which a�ect thelength of the dispersive mixing zone include viscosity and velocity variations and reservoir heterogeneity.Much work is needed to quantify these e�ects and to obtain useful e�ective dispersion coe�cients for �eld-scale simulators. The dispersion tensor has strong velocity dependence [24, 65]. The longitudinal dispersion2

is often an order of magnitude larger than the transverse dispersion. This variation enhances unstable owregimes induced by viscosity di�erences and reservoir heterogeneity. Initial work on correlation of dispersioncoe�cients presented with statistical simulations in [40] will be summarized below.In both miscible and immiscible displacements, the process of pushing a heavy, viscous oil through aheterogeneous porous medium with a lighter, less viscous uid can be a very unstable one. If the ow rateis su�ciently high, the interface between the resident petroleum and the invading uid becomes unstableand tends to form long �ngers which grow in length toward the production wells, bypassing much of thehydrocarbon. Once a path consisting of the injected uid has extended from an injection well to a productionwell, that production well will henceforth produce primarily the injected uid, which ows more easily due toits lower viscosity and higher mobility. The production of petroleum from that well is then greatly reduced.This phenomenon, termed viscous �ngering, is well known [5, 11, 52, 61, 66, 67, 73, 76, 77] and is a seriousproblem in hydrocarbon recovery. This problem and di�erent techniques for understanding and modeling itwere surveyed in [28].The equations used to describe uid ow in porous media are obtained by applying various averagingtechniques to the Navier-Stokes equations which describe the macroscopic ow properties. The averaging isdone over tubes of varying geometry and tortuosity. Clearly, the Navier-Stokes equations do not describethe ow on a macroscopic level. Analysis must be applied to the Navier-Stokes equations to understand theconditions under which small ow perturbations caused by tortuous ow in the porous media will becomeunstable and will grow into large �ngers which a�ect the ow on a macroscopic level. The understandingof this instability is crucial, both in attempts to stabilize the ow via polymers, etc., and to model the�ngering phenomenon when it cannot be controlled. There are three distinct problems associated with theviscous �ngering phenomenon. Knowledge of the conditions causing the onset of the instability is one majorgoal. Then the understanding of the nonlinear e�ects which cause certain �ngers to grow preferentially andcoalesce to form larger �ngers is essential. The rate of this �nger growth must also be determined. Once thegrowth of the �ngers is understood, from a microscopic to a macroscopic level, statistical methods shouldbe used to incorporate the large-scale e�ects of �ngering into the mathematical models used to simulate the�eld scale process without trying to treat the individual �ngers.An important in uence upon the �nger growth is the velocity dependence of the dispersion tensor. Once a�nger is initiated, ow is preferentially directed along the �nger axis. Then, since the longitudinal dispersion(in the direction of ow) is considerably larger than the transverse dispersion, the dispersive e�ects tendto increase �nger growth. Therefore, a complete understanding of the viscous �ngering phenomena is notpossible without a better model for the dispersion tensor and the relative length scales of the mixing zoneand the �nger size.When viscous �ngering occurs, the �nger wavelengths are generally much smaller than the typical compu-tational cells which are used in a simulator, and therefore cannot be represented on a standard numerical grid.In order to correctly model the physics of the �ngering phenomenon, one would have to add a prohibitivenumber of grid points around the moving front. If the �ngering e�ect is not modeled, an overly optimisticvalue of uid recovery is generally predicted. Realizing the inability to realistically model the physics of�ngering with reasonable grids, a lumped-parameter approach, similar in principle to the averaging of theNavier-Stokes ow equations which was previously discussed, has been tried. The original attempt alongthese lines was due to Koval [54], utilizing a global Buckley-Leverett approach. This approach was extendedby Todd and Longsta� [72] via introduction of a mixing parameter !.Other attempts to incorporate the e�ects of �ngering in a simulator without trying to model the physicson its smaller scale are discussed by Gardner and Ypma [44] and by Fishlock and Rodwell [43]. See [28] fora survey of some of this work together with a method for incorporating velocity variation in the modeling ofviscous �ngering. None of these attempts is completely satisfactory and the author feels this is an importantarea for future research. Attempts to use statistics of the ow to empirically determine the longitudinal andtransverse dispersion coe�cients in the dispersion tensor to incorporate �ngering e�ects were presented in[40].In [40], Ewing et al. indicated the ability of coarse-grid dispersion models to match results of both labo-ratory experiments and �ne-grid simulations on highly heterogeneous meshes. The use of a dispersion tensoravoided the optimistic recovery predictions often attributed to standard convection-di�usion models. Het-erogeneity, by itself, was shown to be less of an in uence on recovery than viscous �ngering from adversemobility ratios. The mixing parameter approach [72] predicted too early breakthrough and optimistic ulti-3

mate recoveries compared to experiments and the dispersion models. More research is needed in this areaand is in progress.Studies are currently underway to extend the global e�ective dispersion concept to multiphase owin an analogous manner. Volume averaging [75] or stochastic approaches have been used e�ectively todevelop governing equations for immiscible two phase ow. The dispersion tensor plays a key role in thesedevelopments. Langlo and Espedal [56] have shown that small scale spatial variations in multiphase ow canbe modeled by a saturation dependent dispersion term.Although dispersion by itself is important for modeling �ne-scale properties, e�ects of heterogeneitieswith higher correlation length require �rst-order e�ective transport parameters; Fayers and Hewett [42] havea nice survey of these concepts. Also Glimm, Lindquist, and Sharp [46, 47] have argued that the dispersionis not fully Fickian but can be described by macrodispersion concepts. Fractals [51] and multi-fractals [46]are being used to model the e�ects of the heterogeneities.Given the importance of dispersion and viscous �ngering in the modeling of most enhanced recoveryprocesses, research must progress in several directions. First, the averaging processes used to change lengthscales must be improved, perhaps via more statistical techniques or renormalization, to obtain better e�ectivereservoir coe�cients for the macroscopic models presented in the next section. Simultaneously, the e�ectivelength scales of dispersion and its e�ect upon viscous �ngering must be better understood. Next, bettermacroscopic techniques for including the e�ects (perhaps statistical) of viscous �ngering and dispersion mustbe developed. Finally, since the uid interactions are so critical to enhanced recovery procedures, we cannota�ord to continue to use coarse grids and standard upstream-weighting techniques around the uid interfacesin our simulators, because these methods introduce an arti�cial di�usion [24], which dominates the physicaldi�usion/dispersion and smears fronts over coarse grid blocks. This involves higher resolution of the uidinterfaces which, one hopes, can be achieved through adaptive, local grid-re�nement techniques presented inthis paper.3 Model Equations for Porous Media FlowAlthough the techniques that we discuss will apply equally well to the recovery of hydrocarbon and thetransport of contaminants through the saturated or unsaturated soil zones, we will describe the multiphase ow processes in the terminology of miscible and immiscible displacement of resident uids by injected uids.The miscible displacement of one incompressible uid by another, completely miscible with the �rst, in ahorizontal porous reservoir � R2 over a time period J = [T0;T1], is given in Aziz and Settari [1] and Ewing[24] by �r ��k�rp� � r � u = q; x 2 ; t 2 J; (3.1)�@c@t �r � (Drc� uc) = q~c; x 2 ; t 2 J; (3.2)where p and u are the pressure and Darcy velocity of the uid mixture, � and k are the porosity and thepermeability of the medium, � is the concentration-dependent viscosity of the mixture, c is the concentrationof the invading uid, q is the external rate of ow, and ~c is the inlet or outlet concentration. The form ofthe di�usion/dispersion tensor D is given byD = �(x) fdmI + juj [dlE(u) + dt(I �E(u))]g (3.3)where Eij(u) = uiujjuj2 ; (3.4)dm is the molecular di�usion coe�cient, and dl and dt are the longitudinal and transverse dispersion coef-�cients, respectively. In general, dl � 10dt, but this may vary greatly with di�erent soils, fractured media,etc. Also, the viscosity � in Equation (3.1) is assumed to be determined by some mixing rule, such as�(c) = �o h1 + c�(�o�s)1=4 � 1�i�4 ; (3.5)4

where �o is the viscosity of the resident uid and �s is the viscosity of the invading uid. In addition toEquations (3.1) and (3.2), initial and no- ow boundary conditions are speci�ed. The ow at injection andproduction wells is modeled in Equations (3.1) and (3.2) via point sources and sinks.The equations describing two phase, immiscible, incompressible displacement in a horizontal porousmedium are given by �@Sw@t �r ��k krw�w rpw� = qw; x 2 ; t 2 J; (3.6)�@So@t �r � �k kro�o rpo� = qo; x 2 ; t 2 J; (3.7)where the subscripts w and o refer to water and oil, respectively, Si is the saturation, pi is the pressure, kriis the relative permeability, �i is the viscosity, and qi is the external ow rate, each with respect to the ithphase.The saturation constraint is given by Sw + So = 1: (3.8)From Equation (3.8), we see that one of the saturations can be eliminated. Let S = Sw = 1 � So. Thepressure between the two phases is described by the capillary pressurepc(S) = po � pw: (3.9)Although formally, the equations presented in (3.1) and (3.2) seem quite di�erent from those in (3.6) and(3.7), the latter system may be rearranged in a form which very closely resembles the former system. Inorder to use the same basic simulator in our sample computations to treat both miscible and immiscibledisplacement, we brie y discuss a miscible/immiscible ow analogy.Let �(S) = kro�o + krw�w (3.10)denote a total mobility of the two-phase uid, and de�ne relative mobilities by��i(S) = kri(S)�i�(S) ; i = w; 0: (3.11)We will try to rearrange the immiscible system using the pressure variable given byp = po + pw2 + 12 Z pc0 ���o(�)� ��w(�)� d�: (3.12)Note that ��w + ��o = 1.Adding Equations (3.6) and (3.7) and performing some simple calculations, we obtain:�r � (k�(S)rp) = qw + q0 = qt; (3.13)vt = �k�(S)rp: (3.14)Taking the di�erence of Equations (3.6) and (3.7), the following equation is obtained:�@S@t +r � �k�(S)��o dpcdS rS�+r � ���wvt� = qw: (3.15)For the sake of completeness, the three-phase, black-oil, ow equations that are used in our computationalexamples will be listed below. These are simple examples of multiphase ow with the additional complicationof some mass transfer between phases and gravity e�ects. The equations for conservation of the oil and waterphases are given by (i = o; w) 5

r � [�i (rpi � �igrz)] + qi = @ (�BiSi)@t (3.16)in the paper by Ewing, Boyett, Babu, and Heinemann [27]. The gas equation isr � [�g (rpg � �ggrz)] +r � [Rs�o (rpo � �ogrz)] + qq +Rsqo = @ (�BqSq)@t + @(Rs�BoSo)@t : (3.17)The saturations are summed to one and the pressures are related to the capillary pressures as in Equation(3.9). These capillary pressure relationships are strongly nonlinear functions of the phase saturations. Darcy'slaw is given by vi = ��ir�i � �k kriBi�ir�i; (3.18)with i = o; ; w; ; g, where k is the permeability tensor, vi is the ith phase velocity, and the ow potentialgradient is de�ned, for i = o; ; w; ; g; as r�i = rpi � �igrz: (3.19)Again, the relative permeabilities kri, which describe the competition of the uid phases for ow, are nonlinearfunctions of the phase saturations. Also, due to the homogeneities in �eld problems, a need for more accurateapproximation of the phase velocities has been identi�ed. In Section 4, we discuss mixed �nite elementmethods that accurately approximate the phase velocities directly.The ow rate across any boundary surface A may be expressed in terms of �i asQi ����A = Z ZA k kriBi�i (r�i � n) dA (3.20)for i = o; ; w; ; g: Ewing and Lazarov [34] obtained accurate �nite di�erence formulae by conserving massacross grid interfaces through matching Q from Equation (3.20) on both sides of the face.The equations used to describe uid ow in porous media are obtained by applying various averagingtechniques to the Navier-Stokes equations which describe the macroscopic ow properties. The averaging isdone over tubes of varying geometry and tortuosity. Clearly, the Navier-Stokes equations do not describethe ow on a macroscopic level. Analysis must be applied to the Navier-Stokes equations to understand theconditions under which small ow perturbations caused by tortuous ow in the porous media will becomeunstable and will grow into large �ngers which a�ect the ow on a macroscopic level. The understanding ofthis instability is crucial, both in attempts to stabilize the ow via polymers, etc., and to model the �ngeringphenomenon when it cannot be controlled. Once the growth of the �ngers is understood, from a microscopicto a macroscopic level, statistical methods should be used to incorporate the large-scale e�ects of �ngeringinto the mathematical models used to simulate the �eld-scale process without trying to treat the individual�ngers. Use of supercomputers is essential in testing the e�ectiveness of the nonlinear scale-up of complexlocal phenomena of this type.The equations presented above describe multiphase and multicomponent ow in porous media. Theycan be used to simulate various production strategies in an attempt to understand and quite possiblyoptimize hydrocarbon recovery. However, in order to use these equations e�ectively, parameters that describethe rock and uid properties for the particular reservoir application must be input into the model. Therelative permeabilities, which are nonlinear functions of water saturation, can be estimated via laboratoryexperiments using reservoir cores and resident uids. Similarly, uid viscosities are relatively easy to obtain.However, the permeability tensor k, the porosity �, the capillary pressure curve pc(S), and the di�usion anddispersion coe�cients are e�ective values that must be obtained from local properties via scaling techniques.In addition, the inaccessibility of the reservoir to measurement of even the local properties increases thedi�culties. See Ewing [24], Ewing and George [29], Lin and Ewing [57], and the references contained thereinfor a survey of parameter estimation and history-matching techniques which have been applied.6

Even if complete information is known about the reservoir properties in a highly heterogeneous medium,the problem of how to represent this medium on coarse-grid blocks of di�erent length scales still remains. Thepower of supercomputers must be brought to bear for simulation studies using homogenization and statisticalaveraging to represent �ne-scale phenomena on coarser grids. The estimated viscosity, permeability, ordispersion coe�cients must be modi�ed to incorporate the important e�ects of viscous �ngering. Describingviscous �ngering on a grid size that resolves the �ngering is impossible, even on the largest supercomputers,but since its presence can dominate ow, its e�ects must be included in large-scale simulators on coarsegrids. Ewing, Russell, and Young [40] are developing e�ective equations and e�ective parameters to modelthis important e�ect.4 Mixed Methods for Accurate Velocity ApproximationsThere are two major sources of error in the methods currently being utilized for �nite di�erence discretizationof Equations (3.13){(3.17). The �rst occurs in the approximation of the uid pressure and velocity. Thesecond comes from the techniques for up stream weighting to stabilize Equation (3.15). In this section,we describe mixed �nite element methods for the accurate approximation of the total velocity vt. Somealternate upstream-weighting techniques developed from a �nite element context were presented in Ewinget al. [32].Since the transport term and the di�usion/dispersion term in Equation (3.15) are governed by the uidvelocity, accurate simulation requires an accurate approximation of the velocity vt. Because the lithology inthe reservoir can change abruptly, causing rapid changes in the ow capacities of the rock, the coe�cient kin Equations (3.13) and (3.15) can be discontinuous. In this case, in order for the ow to remain relativelysmooth, the pressure changes extremely rapidly. Thus, standard procedures of solving Equation (3.13) as anelliptic partial di�erential equation for pressure, di�erentiating or di�erencing the result to approximate thepressure gradient, and then multiplying by the discontinuous k� can produce very poor approximations tothe velocity vt. In this section, a mixed �nite element method for approximating vt and p simultaneously,via a coupled system of �rst-order partial di�erential equations, will be discussed. Similar techniques can beused to approximate the phase velocities from Equation (3.20) directly. This formulation allows the removalof singular terms as in Ewing et al. [33] and accurately treats the problem of rapidly changing ow propertiesin the reservoir.The coupled system of �rst-order equations used to de�ne our methods arises from Darcy's Law andconservation of mass. The system for incompressible uids is given byu = �k�rp; x 2 ; (4.1)r � u = q; x 2 ; (4.2)subject to the boundary condition u � n = 0; x 2 @: (4.3)Clearly, Equations (4.1){(4.3) will determine p only to within an additive constant. Thus, a normalizingconstraint such as R p(x)dx = 0 or p(xs) = 0, for some xs 2 , is required in the computation to preventa singular system. We next de�ne certain function spaces and notation. Let W = L2() be the set of allfunctions on whose squares are �nitely integrable. LetH(div; ) be the set of vector functions v 2 �L2()�2such that r � v 2 L2() and let V = H(div; ) \ fv � n = 0 on @g: (4.4)Let (v; w) = Z vwdx; hv; wi = Z@ wvds; and kvk2 = (v; v)7

be the standard L2 inner products and norm on and @. We obtain the weak solution form of Equations(4.1){(4.3) by dividing each side of Equation (4.1) by k=�, multiplying by a test function v 2 V , andintegrating the result to obtain ��ku;v� = (p;rv); v 2 V: (4.5)The right-hand side of Equation (4.5) was obtained by further integration by parts and use of Equation (4.4).Next, multiplying Equation (4.2) by w 2 W and integrating the result, we complete our weak formulation,obtaining (r � u; w) = (q; w); w 2W: (4.6)For a sequence of mesh parameters h > 0, we choose �nite-dimensional subspaces Vh and Wh with Vh � Vand Wh �W and seek a solution pair (Uh;Ph) 2 Vh �Wh satisfying��kUh;vh�� (Ph; div vh) = 0; vh 2 Vh; (4.7)(divUh; wh) = (q; wh); wh 2 Wh: (4.8)One could then complete the description of our mixed �nite element methods with a discussion of particularchoices of Vh and Wh. Examples of these spaces are presented by Ewing and Wheeler [41].For problems with smooth coe�cients and smooth forcing functions, Douglas et al. [16, 17] use standardapproximation theory to show that, by using higher-order basis functions, correspondingly higher-orderconvergence rates can be obtained. For the uid ow in porous media applications, the source and sinkterms q are not smoothly distributed, but are sums of Dirac delta functions. As shown by Ewing andWheeler [41], the resulting smoothness of vt is reduced; vt is not contained in the space L2 and thus usingthe mixed methods, the velocity approximations would not converge at the wells. This result was obtainedtheoretically by Ewing and Wheeler [41] and Douglas et al. [16, 17] and computationally by Ewing et al.[33]. By removing the leading term of each of the singularities (the logarithm terms), the remaining partsof the velocities are now in H2�� for any � > 0. Thus, the approximations to these parts will now convergeat the wells since we have regained su�cient regularity for convergence.We use mixed method techniques for accurately approximating the total velocity vt. The transportEquation (3.15) requires a phase velocity in regimes where two-phase ow occurs. Ewing and Heinemann[30, 31] discussed �nite element weightings of the pressure obtained from standard �nite di�erence codeswhich resulted in signi�cantly better phase velocities than via standard upstream-weighting methods. These�nite element weightings are similar to the mixed method ideas presented above, but can be implementedeven more easily in existing reservoir simulation codes.Clearly, the accuracy of the uid velocities that govern the advection of the uids is a limiting factor inlarge-scale simulations. Ewing, Lazarov, and Wang [37] have shown that along certain lines, the uid veloci-ties are considerably more accurate than in general, and have quanti�ed this superconvergence phenomenon.Since these loci of higher accuracy correspond to Gauss points which are used for quadrature points in thesimulation codes, we can take advantage of this greater accuracy without any extra computation from postprocessing. This observation allows considerably greater accuracy on coarse grids.Since many ow phenomena are highly localized, the coarse grids dictated by the storage requirements ofthe supercomputer are often not capable of resolving the physics of ow, and local grid re�nement techniques,which are discussed in detail in Section 6, must be utilized. Special methods are required to maintain thee�ciency of the code while incorporating the local re�nement. Ewing, Lazarov, Russell, and Vassilevski[35] have analyzed the accuracy and e�ciency of the use of local grid re�nement in the context of mixedmethods. Some superconvergence is retained (theoretically and computationally), again along predictablelines associated with Gauss points.5 Operator-Splitting TechniquesIn �nite di�erence simulators, the convection is stabilized via upstream-weighting techniques. In a �niteelement setting, we use a possible combination of a modi�ed method of characteristics and Petrov-Galerkintechniques to treat the transport separately in an operator-splitting mode.8

In miscible or multicomponent ow models, the convective, hyperbolic part is a linear function of thevelocity. An operator-splitting technique has been developed to solve the purely hyperbolic part by timestepping along the associated characteristics [18, 38, 39, 62]. We �rst obtain the non-divergence form of (3.2)by using the product rule for di�erentiation on the r � uc term and applying (3.1) to obtain�@c@t + u � rc�r �Drc = q(~c� c): (5.1)Next, the �rst and second terms in Equation (5.1) are combined to form a directional derivative along whatwould be the characteristics for the equation if the tensor D were zero. The resulting equation isr � (Drc) + q(~c� c) = �@c@t + u � rc � � @c@� : (5.2)The system obtained by modifying Equations (3.1){(3.2) in this way is solved sequentially. An approximationfor u is �rst obtained at time level t = tn from a solution of Equation (3.1) with the uid viscosity � evaluatedvia some mixing rule at time level tn�1. Equation (3.1) can be solved as an elliptic equation for the pressure,p, or via a mixed �nite element method for a more accurate uid velocity. Let Cn(x) and Un(x) denote theapproximations of c(x; t) and u(x; t), respectively, at time level t = tn. The directional derivative is thendiscretized along the \characteristic" mentioned above as� @c@� (x; tn) � �Cn(c)� Cn�1(�xn�1)�t ; (5.3)where �xn�1 is de�ned for an x as �xn�1 = x� Un(x)�t� (5.4)This technique is a discretization back along the \characteristic" generated by the �rst order derivatives fromEquation (5.2). Although the advection-dominance in the original Equation (5.2) makes it non-self-adjoint,the form with directional derivatives is self-adjoint and discretization techniques for self-adjoint equationscan be utilized. This modi�ed method of characteristics can be combined with either �nite di�erence or�nite element spatial discretizations.In immiscible or multiphase ow, the convective part is nonlinear. A similar operator-splitting techniqueto solve this equation needs reduced time steps because the pure hyperbolic part may develop shocks.Recently, an operator-splitting technique has been developed for immiscible ows [12, 13, 14, 19, 20] whichretains the long time steps in the characteristic solution without introducing serious discretization errors.The operator splitting gives the following set of equations:�@ �S@t + ddS fm( �S) � r �S � � dd� �S = 0; (5.5)�@S@� +r � (bm(S)S)� �r � (D(S)rS) = q(x; t); (5.6)tm � t � tm+1, together with proper initial and boundary conditions. As noted earlier, the saturationS is coupled to the pressure/velocity equations, which will be solved by mixed �nite element methods[2, 16, 17, 30, 33, 38].The splitting of the fractional ow function into two parts: fm(S) +b(S)S, is constructed [20] such thatfm(S) is linear in the shock region, 0 � S � S1 < 1, and b(S) � 0 for S1 � S � 1. Further, Equation (5.5)produces the same unique physical solution as@S@t +r � (fm(S) + b(S)S) = 0 (5.7)with an entropy condition imposed. This means that, for a fully developed shock, the characteristic solutionof Equation (5.5) always will produce a unique solution and, as in the miscible case, we may use long timesteps �t without loss of accuracy. 9

The solution of Equation (5.6) via variational methods leads to the following Petrov-Galerkin equations:B(Smh ; �i) � (Sm+1h ; �i)���t� b(x; tm)Sm+1h ;r�i�+���t� D(x; tm)rSm+1h ;r�i�= (gmh (x; tm); �i); i = 1; 2; � � � ; N; Smh 2Mh; �i 2 Nh (5.8)where Mh and Nh are the trial and test spaces spanned by f�ig and f�ig, i = 1; 2; � � � ; N , respectively.B(�; �) given by Equation (5.8) is an unsymmetrical bilinear form with spatially-dependent coe�cients.In order to obtain Equation (5.8), we have used the characteristic solution from Equation (5.5) to ap-proximate (@=@�)S and the nonlinear coe�cients in Equation (5.6). The nonsymmetry in the bilinear formB(�; �) is caused by the nonlinearity of the convective part of the equation, represented by the term b(S)S.This term balances the di�usion forces in the shock region after a traveling front has been established.We want to use numerical techniques which work well for the symmetric, coercive, bilinear forms to solveEquation (5.8). We consider a procedure, developed by Barrett and Morton [4], which symmetrizes thebilinear form B(�; �) by de�ning a new set of test functions as follows:B(Sm; �i) = �akl @@xk Sm; @@xl �i� � B�(Sm; �i); 0 < akl < K: (5.9)The test functions �i de�ned by Equation (5.9) have nonlocal support and would thus cause seriouscomputational di�culties for large-scale problems. However, a localization procedure was developed byDemkowitz and Oden [15] which allows e�cient computational procedures. The procedure has been modi�ed[12, 13, 14] and will be presented below.Let �i denote the usual linear basis functions in one dimension:�i(x) = 8>>><>>>:0; x0 � x � xi�1(x� xi)=(xi � xi�1); xi�1 � x � xi(xi+1 � x)=(xi+1 � xi); xi � x � xi+10; xi+1 � x � x1; (5.10)and let �i be the second order polynomial given by�i(x) = ((x� xi�1)(x � xi)=(xi � xi�1)2; xi�1 � x � xi�(x� xi)(x� xi+1)=(xi+1 � xi)2; xi � x � xi+1:Dahle [12] demonstrated that a suitable choice of test functions associated with the trial space spanned by(5.10) was given by �i =8>>><>>>:0; x0 � x � xi�1�i + ci�1�i; xi�1 � x � xi�i + ci�i; xi � x � xi+10; xi+1 < x; (5.11)where ci = 3� 2�i � coth��i2 �� ; (5.12)and �i is a local mesh P�eclet number de�ned over element i:�i = hbi�Di :Di and bi denote averages over element i of the di�usion coe�cient and the transport coe�cient respectively.10

If we choose bilinear elements spanned by the trial functions �ij(x) = �i(x)�j(y), the obvious extensionof these test functions to two spatial dimensions is given by�ij(x) = ��i(x) + cI1�i(x)� � ��j(y) + cI2�j(y)� ; (5.13)where cIk; k = 1; 2, is de�ned by (5.12) with respect to the local mesh P�eclet number determined from thecomponents of b = [b1; b2] and D = �D11 00 D22�such that �Ik = hbIi�DIkk ; k = 1; 2; (5.14)and (�)I again denotes some sort of average over element I .We observe that these test functions are severely skewed in the shock region. Away from the shock,b(x) = 0, and the trial and test functions coincide. We emphasize that the test function (5.13) is constructedto stabilize the solution around sharp shocks, where the solution is mainly determined by B(�; �). Away fromthe shock region, where the asymmetric transport term is zero, it may be necessary to construct optimal testfunctions with respect to the complete symmetric bilinear form B�(�; �), given by equation (5.9). Demkowitzand Oden [15] have constructed optimal localized test functions for such problems. Although we will notpursue the problem further here, we note that their test functions may be convenient to use away from theshock region.Since the bilinear form B(�; �) is coercive, we obtain optimal approximation properties in the normsde�ned by the form. For computational reasons, it may be better to use an approximate form of the optimaltest function �i. An estimate for the error introduced by an approximate symmetrization of B(�; �) is givenby Barrett and Morton [4].It seems natural to relate the size of the coarse domains to the solution of the pressure-velocity equationas in [19], since the velocity varies slowly and de�nes a natural long space scale compared to the variation ofthe saturation S at a front. A local error estimate which determines if a coarse-grid block must be re�ned,is given in reference [19]. Normally, local re�nement must be performed if a uid interface is located withinthe coarse-grid block in order to resolve the solution there. A slightly di�erent strategy is to make the regionof local re�nement big enough so that we can use the same re�nements for several of the large time stepsallowed by the method. The local grid-re�nement strategy combined with the operator splitting is de�ned inthe literature [12, 13, 14, 20, 21, 22]. The solution at each of the coarse-grid vertices and the local re�nementcalculation may be sent to separate processors to achieve a high level of parallelism in the solution process.The di�cult problem with these techniques is the communication of the solution between the �ne andcoarse grids. The domain decomposition technique described in [56] gives accurate and e�cient treatmentof the communication problem.Unfortunately, the modi�ed method of characteristics techniques described above generally do not con-serve mass. Also, the proper method for treating boundary conditions in a conservative and accurate mannerusing these techniques is not obvious. Recently, M.A. Celia, T.F. Russell, I. Herrera, and the author havedevised Eulerian-Lagrangian localized adjoint methods (ELLAM) [9, 50], a set of schemes that are de�nedexpressly for conservation of mass properties.The ELLAM formulation was motivated by localized adjoint methods [8, 48], which are one form of theoptimal test function methods discussed above [4, 14, 15, 20]. We brie y describe these methods. LetLu = f; x 2 or (x; t) 2 ; (5.15)denote a partial di�erential equation in space or space-time. Integrating against a test function �, we obtainthe weak form Z Lu�d! = Z f�d!: (5.16)11

If we choose test functions � to satisfy the formal adjoint equation L�� = 0 and � = 0 on the boundary,except at @ certain nodes or edges denoted by `i, then integration by parts (the divergence theorem inhigher dimensions) yields Xi Z`i uL��d! = Z f�d!: (5.17)Various di�erent test functions can be used to focus upon di�erent types of information. Herrera has builtan extensive theory around this concept; see [48] for references. The theory is quite general and can dealwith situations where distributions do not apply, such as when both u and � are discontinuous.As in the work of Demkowitz and Oden [15], we want to localize these test functions to maintain sparsematrices. Certain choices of space-time test functions which are useful for linear equations of the form (3.2)have been described in [15, 63]. For examples of nonlinear applications of the form (3.15), see [26, 49].We next extend the ELLAM techniques to the nonlinear multiphase ow equations. We consider thedivergence form of the multiphase ow equation given by (3.13) with � assumed constant in time and vt � v :LS � �@S@t +r � (�wv)�r �DrS = qw; x 2 ; t 2 J; (5.18)(�wv �DrS) � � = h; x 2 @; t 2 J; (5.19)where � is the outward unit normal to the boundary @. Let � = � J denote the space-time domain.Then we obtain a weak formulation of (5.18) by integrating against a test function w = w(x; t). This yieldsa weak form, R�(LS)w dxdt = R� qw dxdt. We obtain the speci�c equationZ Zj�(Sw)tdtdx+ ZJ Zr � (�wv �DrS)wdtdx + Z�DrS � rwdxdt� Z�(�Swt + �wv � rw)dxdt = Z� qww dxdt: (5.20)Then, as in [64], we begin to study the time dependence of the potentially useful test functions by looking ata semidiscrete scheme on the time interval Jn+1 = [tn; tn+1] or over the space time regionPn+1 = �Jn+1.By applying the divergence theorem to (5.20), we obtainZ �S(x; tn+1)w(x; tn+1)dx+ Z�n+1 DrS � rw dxdt+ ZJn+1 Z@(�wv �DrS) � �wd�dt � Z�n+1(�Swt + �wv � rw)dxdt= Z �S(x; tn)w(x; tn)dx + Z�n+1 qwwdxdt: (5.21)In order to consider the ELLAM formulation from [9] directly, we should look for solutions of the adjointto treat the term of the form Z�n+1 SL�wdxdt = 0: (5.22)Since L is not a linear operator, we must perform some linearizations before we apply the analogue of (5.22)to treat the fourth term in (5.21).Motivated by [20], we de�ne �f(S)S �8><>:d�wds (S1)S; 0 � S � S1;(1� r)(1� S1)S + c; S1 � S � 1; (5.23)where S1 is the top saturation of an established front,r = d�wds (S1)S1; (5.24)12

and c = 1� (1� S1)(1� r) : (5.25)This is the piecewise linearization of �w using the top saturation of the established front and its value �w(S1).Then, we de�ne b(s) by the di�erence of �w and �fS. Thus,�w = �f(S)S + b(S)S: (5.26)For 0 � S � S1; b(S)S is an antidi�usive term causing the fronts to tend to sharpen. For S1 � S �1; b(S)S is a di�usive term. Using these de�nitions, the fourth term in (5.21) can be written asZ�n+1 S ��wt + � �f(S) + b(S)v � rw� dxdt = Z�n+1 S ��wt + �fv � rw� dxdt + Z�n+1 Sbv � rw dxdt:(5.27)We cannot, in general, determine a test function w that satis�es �wt+ �fv �rw = 0, even locally within eachsmall space-time element. However, we will make a choice of test functions that will make this term small.Analysis of the size of this term will be presented elsewhere.By choosing a test function w(x; t) that is constant in time along the characteristics that de�ne themoving Lagrangian frame of reference, we can make the �rst term in (5.27) small. If the test function were astandard chapeau basis function in the x-direction like that pictured in Figure 1, it would also make secondterm in (5.21) small. This would be an e�ective test function if the second term on the right side of (5.27)were zero or were small. However, in many multiphase ow problems, the b(S)v term is not small and theuse of characteristics has not symmetrized the form which is analagous to the form in Equation (5.9). Asabove, the use of an upwinded form of the test function for constant x will e�ciently treat the b term from(5.27) together with the D term from (5.21).We thus arrive at a choice of w(x; t) that is of the form of the test functions �ij described in Equation(5.13) for constant x and which are also constant along the characteristics determined by the directionalderivative along � with �f de�ned in (5.23). Using these test functions, our approximation scheme can bede�ned in the interior of the region on prisms as in [64]. For example, let (xi; yi) be a grid point in twodimensions; let ij = [xi�1; xi+1]� [yi�1; yi+1]be the large rectangle surrounding (xi; yi). De�ne wi;j(x; tn+1) as the tensor product of upstream-weightedtest functions �i(x) � �j(y) where �i(x) and �i(y) are given by (5.11. Let �n+1ij be the prism obtained bytracing ij backward in time in the Lagrangian coordinates from tn+1 (head of the characteristic) to tn (footof the characteristic). The basis test function wij(x; tn+1) will be one at (xi; yi; tn+1), zero on the lateralboundary of �n+1ij , �i(x) � �j(y) on ij at tn+1, and constant along backtracked characteristics.Therefore, if �n+1ij does not meet the boundary of the total space-time computational region, the approx-imation procedure can be de�ned in �n+1ij as follows:Zij�S(x; tn+1)w(x; tn+1)dx + Z�n+1ij (DrS � b( ~Sn+1)vS) � rw(x; tn+1)dxdt= Z�ij �S(x; tn)w(x; tn)dx+ Z�n+1ij qww dxdt; (5.28)where �ij is the region at tn spanned by the feet of the characteristics whose heads are in ij and ~Sn+1is some approximation of Sn+1 obtained by extrapolating values from earlier time levels. As discussed in[63, 64], if the time integrals along the characteristics are approximated by a one-point backward Euler ruleat tn+1, with �t = tn+1 � tn, we obtain the MMOC formulation presented in Equation (5.5).Di�erences between ELLAM and MMOC for linear partial di�erential operators have been discussedin [63, 64]. These comparisons also apply in the nonlinear problems considered here. Reference [64] also13

ttn+1tn xi�3 xi�2 xi�1 xi xi+1 x

i1 i2xil(t) xic(t) xir(t)x�i�1 x�i x�i+1 -6

� � � � �� � � � �ttn+1

tn xi�3 xi�2 xi�1 xi xi+1 xx�i�1 x�i x�i+1Wn+1i (x; t)�����HHHHH

�����HHHHH-

6 (a)

(b)

Figure 1:

14

contains excellent discussions of the errors involved in numerical integration along the characteristics viavarious tracking algorithms when the coe�cients are spatially dependent, and for the terms arising whenthe adjoint equation is not completely satis�ed.Since one motivation for considering ELLAM instead of MMOC techniques was to obtain more accuratetreatment of the boundary conditions, we next extend our previous treatment of these terms for constantcoe�cients to the nonlinear case.As was discussed earlier, the number of test functions intersecting the computational boundaries dependupon the Courant number; for this exposition, we assume it is between zero and one. At both in ow andout ow boundaries, this means that the third term on the left-hand side of (5.21) is non zero. At in owboundaries, part of the integration at the base given by the �rst term on the right-hand side of (5.28) is goneand has been replaced by the in ow boundary term from (5.21). For out ow conditions, this replacementoccurs with the �rst term on the left-hand side of (5.28), which involves unknowns. Thus, we will be requiredto obtain values for these degrees of freedom along the out ow boundary if mass is to be conserved. Theanalogue of (5.28) at an in ow boundary isZij�S(x; tn+1)w(x; tn+1)dx + Z�n+1ij (DrS � b( ~Sn+1vS) � rw(x; tn+1)dxdt+ ZJn+1 Z@+(�wv �DrS) � �w dsdt= Z1�ij �S(x; tn)w(x; tn)dx+ Z�n+1ij qww dxdt; (5.29)where @+ is the in ow boundary, and 1�ij corresponds to the analogue of the interval from x0 to x�2 inFigure 2. This is the intersection of the prism for the test function tracked back along the characteristicswith ij at time level t = tn. Again, an analogue for linear coe�cients and a one-point temporal integrationrule is presented in [64] along with speci�cs for various forms of the boundary conditions at both in ow andout ow boundaries.� �� �ttn+1t�1tn x0 x�2 x1 x2 x

11 12-

6� �� �ttn+1t�1tn x0 x�2x1 x2 x-

6 (a) (b)

Figure 2:In summary, we have presented the extensions of the ELLAM ideas discussed in [9, 26, 50, 63, 64] tothe nonlinear equations needed to model multiphase ow. The one-point integration rules in time make thisan extension of the MMOC ideas with more accurate treatment of the boundary conditions. More accuratetemporal integration rules involving more complex approximation procedures are under development.6 Adaptive Grid-Re�nement TechniquesMany time-dependent uid ow problems involve both large-scale processes and highly localized phenomenathat are often critical to the overall chemical and physical behaviors of the ows. For large-scale applications,it is frequently impossible to use a uniform grid which is su�ciently �ne to resolve the local phenomena15

without yielding numbers of unknowns that will overburden even the largest of today's supercomputers.Since these local processes are often dynamic, e�cient numerical simulation requires the ability to performdynamic self-adaptive local grid re�nement. The need for adaptive techniques has provided the impetus forthe development of local grid-re�nement software tools, some of which are used in day-to-day applicationsfor small- to mid-size problems. Software and engineering tools capable of dynamic local grid re�nementneed to be developed for large-scale, uid ow applications. The adaptive grid-re�nement algorithms mustalso be closely matched with the architecture features of the new advanced computers to take advantage ofpossible vector and parallel capabilities.The local patch re�nement techniques [6, 36, 58, 59] have proven to be very e�ective for obtaining localresolution around �xed singular points such as wells in a reservoir. We will discuss the patch approximationtechnique in the context of local re�nement around a point like a well. The major input and output froma reservoir in various production procedures is through wells. Hence, it is important to obtain an accurateapproximation to the ow nearby.We will consider a simple example problem to illustrate our local re�nement techniques. We want toapproximate the pressure p of the uid described by either Equation (3.1) or Equation (3.13).First, we consider the matrix Ac, generated by a �nite element or �nite di�erence approximation of theEquations (3.1) or (3.13) using a coarse quasi-uniform mesh. Let the solution P of the original coarse gridproblem be decomposed in the form P = (P1; Pb; P2)T , where P1; P2; and Pb are the parts of the coarse-gridsolution in two separate domains 1;2 and the intersection of the boundary of 1 and 2, respectively.The corresponding decomposition of the matrix Ac can be described inAc0@P1PbP21A = 0@Ac11 Ac1b 0Acb1 Acbb Acb20 Ac2b Ac221A0@P1PbP21A : (6.1)We assume that a code exists or can be easily written to solve (6.1) for a quasi-uniform grid which canbe highly vectorized to take advantage of the banded structure of the matrix �Ac which is equivalent to Acexcept utilizing a standard lexicographical ordering of the unknowns.Next, assume that due to some identi�ed localized process, grid re�nement is desired in 2. Let Pr be thenew approximation on the re�ned grid in 2 and Arr be the local matrix on 2. Let Abr and Arb be the newconnection matrices between the interface between 1 and 2 and the re�ned grid on 2. Then, in order tomaintain the sparsity of the composite grid matrix and a simple data structure obtained by concatonatingPr to P , we can write the composite matrix problem in the form~A ~P = 0BB@Ac11 Ac1b 0 0Acb1 Acbb 0 Abr0 0 I 00 Arb 0 Arr1CCA0BB@P1PbP2Pr1CCA = 0BB@f1f20f31CCA : (6.2)We note that the I on the diagonal of (6.2) and the zeroes in the corresponding row, column, and right-handside enforce the removal of P2 from the system without destroying the relationship of�Ac11 Ac1bAcb1 Acbb� (6.3)to Ac and hence �Ac.Bramble et al. [6] de�ned a preconditioned conjugate gradient iterative procedure for the e�cient solutionof the composite problem (6.2). The preconditioner involves two local solutions on the re�ned grid in 2 andone on the original coarse grid and yields of symmetric preconditioner. Below, we will describe a simpleriterative process which utilizes similar inversions on the re�ned grid 2 and the coarse grid .Given a previous iterate for Pb, denoted Pnb , we solve the local problem on 2 with Dirichlet conditionson the interface between 1 and 2 (given by ArbPb):Pnr = A�1rr (f3 �ArbPnb ): (6.4)This problem can be solved exactly or approximately by some iterative technique. This step could beconsidered as the �rst part of a block Gauss-Seidel iterative procedure for the solution of (6.1). The next16

step would be to use the approximation for Pnr and then invert the block (6.3) to obtain an approximationfor Pn1 and Pnb . Since this block involves a complex region and may not be well-conditioned, we use analternate solution method which involves a preconditioner, denoted by B, for the composite matrix ~A.Using B, we de�ne, for each iterate n and an iteration parameter � ,~Pn+1 = ~Pn + �B�1( ~f � ~A ~Pn): (6.5)Let Q be the residual vector given by~f � ~A ~Pn = 0BB@ f1 �Ac11Pn1 �Ac1bPnbf2 �Acb1Pn1 �AcbbPnb �AbrPnr0f3 �ArbPnb �ArrPnr 1CCA � 0BB@Qn1Qn20Qn41CCA : (6.6)Next, we solve the original coarse grid problemAc0@Wn+11Wn+1bWn+12 1A = 0@ Qn1Qn2 �AbrA�1rr Qn40 1A (6.7)(or its rearranged equivalent problem using �Ac to take advantage of banding of �Ac) for Wn+11 and Wn+1b .We have simply inverted (6.3) in an e�cient and vectorizable manner. Then, using Wn+1b , we complete theblock Gauss-Seidel analogy on (6.2) and obtain Wn+1r by solvingArrWn+1r = Q4 �ArbWn+1b : (6.8)Finally, from (6.5), we set ~Pn+1 = 0BB@Pn1Pnb0Pnr 1CCA+ � 0BB@Wn1Wnb0Wnr 1CCA :Since this algorithm only requires two separate solutions of mixed problems on the subregions (eachsubregion problem possibly being solved via a di�erent parallel processor) and one solution on the original,uniform coarse grid, it is relatively easy to perform. Similarly, no complex data structure is required, and thealgorithm can be implemented in existing large-scale codes without severely disrupting the solution process.Promising numerical results for the algorithm have appeared [6, 36]. These results have also been extendedto more general reservoir simulation problems in a paper by Ewing, Boyett, Babu, and Heinemann [27].As stated, the algorithm in its most general form involves two separate solutions on the subregions ateach step. This iterative procedure is uniformly well-conditioned for �nite element procedures such as thoseused in this paper or point-centered �nite di�erence methods, but not for cell-centered �nite di�erences[36]. For discretizations arising from cell-centered �nite di�erence methods, a scaling of the iteration viathe parameter � in (6.5) may be necessary. The use of the algorithm as a preconditioner [6] for anotheriterative procedure such as conjugate gradient also involves two distinct solutions on the subregions at eachstep. This comes from the desire to have a symmetric preconditioner of the form B(�; �) which is importantfor conjugate gradient methods. As has been pointed out in [58], the FAC algorithm [59] involves only onesubregion solution per iteration. See Mandel and McCormick [58] for a comparison of FAC, the symmetricBEPS [6] preconditioner and the algorithm presented here and their theories.By considering the domain decomposition techniques presented by Bramble, Pasciak, and Schatz [7] thatled to this algorithm, we can see that if the subregion problems ((6.4) and its sequels with updated guessesfor Pnb ) are solved exactly, then Qn4 in (6.6) and (6.7) is identically zero and the iterative method presentedhere requires only one subregion solution per iteration (from (6.8)). Preliminary computations indicatethat if the subregion problem is solved iteratively with its own preconditioner, the full algorithm with twosubregion solves will converge faster than the version with one subregion solve for some problems. Iterativesolution of the unre�ned region causes no di�culty with either version of the algorithm. This is an important17

consideration for the reservoir simulation applications when iterative solution of the unre�ned problem isessential due to their size, since direct solution of the re�ned region problems is usually not possible.Adaptive grid-re�nement techniques utilizing patch techniques have been presented in several surveys(e.g., see [25]). The patch techniques have been incorporated e�ciently in existing multiphase industrialreservoir simulation codes. Results for the SPE Comparison projects number 1 and 2 were presented in [27].The local re�nement was both e�cient and e�ective since excellent results were obtained without destroyingthe e�ciency of the original codes.References[1] K. Aziz and A. Settari, Petroleum Reservoir Simulation, Applied Science, New York, 1979.[2] M.B. Allen, R.E. Ewing, and P. Lu, Well conditioned iterative schemes for mixed �nite element modelsof porous-media ows, SIAM Journal of Scienti�c and Statistical Computing 13(3) (1992), 794{814.[3] A.A. Baker, L.W. Gelhar, A.L. Gutjahr, and J.R. Macmillan, Stochastic analysis of spatial variability insubsurface ows, I. Comparison of one- and three-dimensional ows, Water Resour. Res. 14(2) (1978),263{271.[4] J.W. Barrett and K.W. Morton, Approximate symmetrization and Petrov-Galerkin methods fordi�ussion-convection problems, Comp. Meth. in Appl. Mech. and Eng. 45 (1984), 97{122.[5] A.L. Benham and R.W. Olson, A model study of viscous �ngering, Soc. Pet. Eng. J. June (1963),138{144.[6] J.H. Bramble, R.E. Ewing, J.E. Pasciak, and A.H. Schatz, A preconditioning technique for the e�-cient solutino of problems iwth local grid re�nement, Computer Methods in Applied Mechanics andEngineering 67 (1988), 149{159.[7] J.H. Bramble, J.E. Pasciak, and A.H. Schatz, An iterative method for elliptic problems on regionspartitioned into substractures, Math. Comp. 46 (1986), 361{370.[8] M.A. Celia, I. Herrera, E. Bouloutas, and J.S. Kindred, A new numerical approach for the advective-di�usive transport equation, Numerical Methods for PDE's 5 (1989), 203{226.[9] M.A. Celia, T.F. Russell, I. Herrera, and R.E. Ewing, An Eulerian-Lagrangian localized adjoint methodfor the advection-di�usion equation, Advances in Water Resources 13(4) (1990), 187{206.[10] C. Chardair-Riviere, G. Chavent, and J. Ja�r�e, Multiscale representation for simultaneous estimation ofrelative permeabilities and capillary pressure, SPE 20501, Proceedings 65th Annual Technical ConferenceSPE, New Orleans, LA, September 23{26, 1990.[11] R.L. Chuoke, P. Van Meurs, and C. Van Der Poel, The instability of slow, immiscible, viscous, liquid-liquid displacements in permeable media, Trans. AIME 216 (1959), 188{194.[12] H.K. Dahle, Adaptive characteristic operator splitting techniques for convection-dominated di�usionproblems in one and two space dimensions, IMA Volumes in Mathematics and Its Applications II,Springer Verlag, 1988, 77{88.[13] H.K. Dahle, M.S. Espedal, and R.E. Ewing, Characteristic Petrov-Galerkin subdomain methods forconvection di�usion problems, IMA Volume 11, Numerical Simulation in Oil Recovery (M.F. Wheeler,ed.), Springer-Verlag, Berlin, 1988, 77{88.[14] H.K. Dahle, M.S. Espedal, R.E. Ewing, and O. S�vareid, Characteristic adaptive sub-domain methodsfor reservoir ow problems, Numerical Methods for Partial Di�erential Equations 6 (1990), 279{309.(charact1) 18

[15] L. Demkowitz and J.T. Oden, An adaptive characteristic Petrov-Galerkin �nite element method forconvection-dominated linear and nonlinear parabolic problems in two space variables, Comp. Meth. inAppl. Mech. and Eng. 55 (1986), 63{87.[16] J. Douglas, Jr., R.E. Ewing, and M.F. Wheeler, The approximation of the pressure by a mixed methodin the simulation of miscible displacement, R.A.I.R.O. Analyse Numerique 17 (1983), 17{33.[17] J. Douglas, Jr., R.E. Ewing, and M.F. Wheeler, A time-discretization procedure for a mixed �niteelement approximation of miscible displacement in porous media, R.A.I.R.O. Anal. Numer. 17 (1983),249{265.[18] J. Douglas, Jr. and T.F. Russell, Numerical methods for convection dominated di�usion problems basedon combining the modi�ed method of characteristics with �nite element or �nite di�erence procedures,SIAM J. Numer. Anal. 19 (1982), 871{885.[19] M.S. Espedal and R.E. Ewing, Characteristic Petrov-Galerkin subdomain methods for two-phase im-miscible ow, Comp. Meth. Appl. Mech. and Eng. 64 (1987), 113{135.[20] M.S. Espedal, R.E. Ewing, and T.F. Russell, Mixed methods, operator splitting, and local re�nementtechniques for simulation on irregular girds, Proceedings 2nd European Conference on the Mathematicsof Oil Recovery (D. Guerillot and O. Guillon, eds.), Editors Technip, Paris, 1990, 237{245.[21] M.S. Espedal, R.E. Ewing, T.F. Russell, and O. S�vareid, Reservoir simulation using mixed methods, amodi�ed method of characteristics, and local grid re�nement, Proceedings of Joint IMA/SPE EuropeanConf. on the Mathematics of Oil Recovery, Cambridge University, July 25{27, 1989.[22] M.S. Espedal, R. Hansen, P. Langlo, O. S�vareid, and R.E. Ewing, Heterogeneous porous media anddomain decomposition methods, Proceedings 2nd European Conference on the Mathematics of Oil Re-covery (D. Guerillot and O. Guillon, eds.), Editors Technip, Paris, 1990, 157{163.[23] M.S. Espedal, P. Langlo, O. S�vareid, E. Geslifosa, and R. Hansen, Heterogeneous reservoir models,local re�nement, and e�ective parameters, SPE 21231, 11th SPS Symposium, Anaheim, California,1991.[24] R.E. Ewing, Problems arising in the modeling of processes for hydrocarbon recovery, in The Mathematicsof Reservoir Simulation (R.E. Ewing, ed.), Frontiers in Applied Mathematics 1, SIAM, Philadelphia,1983, 3{34.[25] R.E. Ewing, E�cient adaptive procedures for uid ow applications, Comp. Meth. Appl. Mech. Eng.55 (1986), 89{103.[26] R.E. Ewing, Operator splitting and Eulerian-Lagrangian localized adjoint methods for multiphase ow,The Mathematics of Finite Elements and Applications, MAFELAP 199 (J. Whiteman, ed.), AcademicPress, Inc., San Diego, CA, 1991, 215{232.[27] R.E. Ewing, B.A. Boyett, D.K. Babu, and R.F. Heinemann, E�cient use of locally re�ned grids formultiphase reservoir simulation, SPE 18413, Proceedings of Tenth Society of Petroleum Engineers Sym-posium on Reservoir Simulation, Houston, Texas, February 6{8, 1989, 55{70.[28] R.E. Ewing and J.H. George, Viscous �ngering in hydrocarbon recovery processes, Mathematical Meth-ods in Energy Research (K.I. Gross, ed.), SIAM, Philadelphia, 1984, 194{213.[29] R.E. Ewing and J.H. George, Identi�cation and control of distributed parameters in porous media ow,Distributed Parameter Systems (F. Kappel, K. Kunisch, W. Schappacher, eds.), Lecture Notes in Controland Information Sciences 75, Springer-Verlag, Berlin, 1985, 145{161.[30] R.E. Ewing and R.F. Heinemann, Mixed �nite element approximation of phase velocities in compo-sitional reservoir simulation, Computer Methods in Applied Mechanics and Engineering (R.E. Ewing,ed.), 47 (1984), 161{176. 19

[31] R.E. Ewing and R.F. Heinemann, Incorporation of mixed �nite element methods in compositionalsimulation for reduction of numerical dispersion, SPE 12267, Proceedings of Seventh Society of PetroleumEngineers Symposium on Reservoir Simulation, San Francisco, November 15{18, 1983, 341{347.[32] R.E. Ewing, R.T. Heinemann, J.V. Koebbe, and U.S. Prasad, Velocity weighting techniques for uiddisplacement problems,Computer Methods in Applied Mechanics and Engineering 64 (1987), 137{151.[33] R.E. Ewing, J.V. Koebbe, R. Gonzalez, and M.F. Wheeler, Mixed �nite element methods for accurate uid velocities, Finite Elements in Fluids 4 (1985), Wiley, New York, 233{249.[34] R.E. Ewing and R.D. Lazarov, Adaptive local grid re�nement, SPE 17806, Proceedings of Society ofPetroleum Engineers Rocky Mountain Regional Meeting, Casper, Wyoming, May 11{13, 1988, 87{102.[35] R.E. Ewing, R.D. Lazarov, T.F. Russell, and P.S. Vassilevski, Local re�nement via domain decomposi-tion techniques for mixed �nite element methods with rectangular Ravirat-Thomas elements, DomainDecompositions for Partial Di�erential Equations (T. Chan, R. Glowinski, J. Periaux, and O. Widlund,eds.), SIAM, Philadelphia, 1990, 98{114.[36] R.E. Ewing, R.D. Lazarov, and P.S. Vassilevski, Local re�nement techniques for elliptic problems oncell-centered grids, II: Two-grid iterative methods, J. on Numer. Linear Algebra Appl., (to appear).[37] R.E. Ewing, R.D. Lazarov, and J. Wang, Superconvergence of the velocities along the Gaussian lines inthe mixed �nite element methods, SIAM Journal on Numerical Analysis 28(4) (1991), 1015{1029.[38] R.E. Ewing, T.F. Russell, and M.F. Wheeler, Simulation of miscible displacement using mixed methodsand a modi�ed method of characteristics, Proceedings Seventh SPE Symposium on Reservoir Simulation,SPE No. 12241, San Francisco, November 15{18, 1983, 71{82.[39] R.E. Ewing, T.F. Russell, and M.F. Wheeler, Convergence analysis of an approximation of miscible dis-placement in porous media by mixed �nite elements and a modi�ed method of characteristics, ComputerMeth. Appl. Mech. Eng. 47 (1984), 73{92.[40] R.E. Ewing, T.F. Russell, and L.C. Young, An anisotropic coarse-grid dispersion model of heterogeneityand viscous �ngering in �ve-spot miscible displacement that matches experiments and �ne-grid simu-lations, SPE 18441, Proceedings, 10th SPE Reservoir Simulation Symposium, Houston, Texas, 1989,447{466.[41] R.E. Ewing and M.F. Wheeler, Computational aspects of mixed �nite element methods, NumericalMethods for Scienti�c Computing (R.S. Stepleman, ed.), North-Holland, New York, 163{172.[42] F.J. Fayers and T.A. Hewett, A review of current trends in petroleum reservoir description and assessingthe impacts on oil recovery, Vol. 2, Mathematical Modeling in Water Resources, Computational Methodsin Water Resources IX (T.F. Russell, R.E. Ewing, C.A. Brebbia, W.G. Gray, and G.F. Pinder,eds.),Elsevier Applied Science, London, 1992, 3{34.[43] T.P. Fishlock and W.R. Rodwell, Improvements in the numerical simulation of carbon dioxide displace-ment, European Paris Conference, 1983.[44] J.W. Gardner and J.G.J. Ypma, Investigation of phase behavior-macroscopic bypassing interaction inCO2 ooding, SPE 10686.[45] L.W. Gelhar and C.L. Axness, Three-dimensional stochastic analysis of macro-dispersion in aquifers,Water Resour. Res. 19(1) (1983), 161{180.[46] J. Glimm and B. Lindquist, Scaling laws for macrodispersion, Vol. 2, Mathematical Modeling in WaterResources, Computational Methods in Water Resources IX (T.F. Russell, R.E. Ewing, C.A. Brebbia,W.G. Gray, and G.F. Pinder,eds.), Elsevier Applied Science, London, 1992, 35{50.[47] J. Glimm and D.H. Sharp, A random �eld model for anomalous di�usion in heterogeneous porous media,J. Statistical Physics 62 (1991), 415{424. 20

[48] I. Herrera, Uni�ed formulation of numerical methods I. Green's formula for operators in discontinuous�elds, Numerical Methods in PDE's 1 (1985), 25{44.[49] I. Herrera and R.E. Ewing, Localized adjoint methods: Application of multiphase ow problems, Pro-ceedings of the Fifth Wyoming Enhanced Oil Recovery Symposium, Casper, Wyoming, May 10{11, 1989,147{166.[50] I. Herrera, R.E. Ewing, M.A. Celia, and T.F. Russell, Eulerian-Lagrangian localized adjoint methods,SIAM J. Numer. Anal., (submitted).[51] T.A. Hewett and R.A. Behrens, Conditional simulation of reservoir heterogeneity with fractals, SPEFormation Evaluation, 1990, 217{225.[52] G.R. Jerauld, H.T. Davis, and L.E. Scriven, Frontal structure and stability in immiscible displacement,SPE/DOE Fourth Symposium on Enhanced Oil Recovery 2 (1984), 135{144.[53] P.R. King, The use of renormalisation in calculating e�ective permeability, Transport in Porous Media4 (1989), 37.[54] E.J. Koval, A method for predicting the performance of unstable miscible displacement in heterogeneousmedia, Soc. Pet. Eng. J. June (1963), 145{154.[55] L.W. Lake and G.J. Hirasaki, Taylor's dispersion in strati�ed porous media, Soc. Pet. Eng. J. 21 (1981),459{468.[56] O. Langlo and M. Espedal, Heterogeneous reservoir models, two-phase immiscible ow in 2-D, Vol. 2,Mathematical Modeling in Water Resources, Computational Methods in Water Resources IX (T.F. Rus-sell, R.E. Ewing, C.A. Brebbia, W.G. Gray, and G.F. Pinder,eds.), Elsevier Applied Science, London,1992, 71{80.[57] T. Lin and R.E. Ewing, Parameter estimation for distributed systems arising in in uid ow problemsvia time series methods, Proceedingts of Conference on \Inverse Problems" 77, Oberwolfach, WestGermany, Birkhauser, Berlin, 1986, 117{126.[58] J. Mandel and S. McCormick, Iterative solution of elliptic equations with re�nement: The two-levelcase, Domain Decomposition Methods (T.F. Chan, R. Glowinski, J. Periaux, and O. Widlunds, eds.),SIAM Publications, Philadelphia, PA, 81{92.[59] S. McCormick and J. Thomas, The fast adaptive composite grid methods for elliptic boundary valueproblems, Math. Comp. 46 (1986), 439{456.[60] T.K. Perkins, O.C. Johnston, and R.N. Ho�man, Mechanics of viscous �ngering in miscible systems,Soc. Pet. Eng. J. December (1965), 301{317.[61] E.J. Peters and D.L. Flock, The onset of instability during two-phase immiscible displacement in porousmedia, Soc. Pet. Eng. J. April (1981), 249{258.[62] T.F. Russell, The time-stepping along characteristics with incomplete iteration for Galerkin approxi-mation of miscible displacement in porous media, SIAM J. Numer. Anal. 22 (1985), 970{1013.[63] T.F. Russell, Eulerian-Lagrangian localized adjoint methods for advection-dominated problems, Pro-ceedings of 13th Biennial Conference on Numerical Analysis, Pitmann Publishing Company, Dundee,Scotland, June 27{30, 1989.[64] T.F. Russell and R.V. Trujillo, Eulerian-Lagrangian localized adjoint methods with variable coe�cientsin multiple divergences, Proceedings 7th International Conference on Computational Methods in WaterResources, Venice, Italy, (to appear).[65] T.F. Russell and M.F. Wheeler, Finite element and �nite di�erence methods for continuous ows inporous media, in The Mathematics of Reservoir Simulation (R.E. Ewing, ed.) Frontiers in AppliedMathematics, SIAM, Philadelphia, 1983. 21

[66] P.G. Sa�man, Fingering in porous media, Lecture Notes in Physics 154 (1982) (R. Burridge, ed.),208{215.[67] A.E. Scheidegger, Growth of instabilities on displacement fronts in porous media, Physics of Fluids 3(1960), 94.[68] J.C. Slattery, Single-phase ow through porous media, AIChE J. 15 (1969), 866{872.[69] J.C. Slattery, Two-phase ow through porous media, AIChE J. 16 (1970), 345{352.[70] G.I. Taylor, Dispersion of soluble matter in solvent owing slowly through a tube, Proc. Roy. Soc. A219(1953), 183{203.[71] G.I. Taylor, Conditions under which dispersion of a solute in a stream of solvent can be used to measuremolecular di�usion, Proc. Roy. Soc. A225 (1954), 473{477.[72] M.R. Todd and W.J. Longsta�, The development, testing and application of a numerical simulator forpredicting miscible ood performance, Jour. Pet. Tech. 253 (1972), 874{882.[73] P. Van Meurs and C. Van Der Poel, A theoretical description of water-drive processes involving viscous�ngering, Trans. AIME 213 (1958), 103{112.[74] J.E. Warren and F.F. Skiba, Macroscopic dispersion, Soc. Pet. Eng. J. 4 (1964), 215{230.[75] S. Whitaker, Flow in porous media II: The governing equations for immiscible two-phase ow, Transportin Porous Media 1 (1986), 102{125.[76] R.A. Wooding, The stability of an interface between miscible uids in a porous medium, Zeit. Fur Ang.Math. Und Physik 13 (1962), 255{265.[77] Y.C. Yortsos and A.B. Huang, Linear stability of immiscible displacement including continuously chang-ing mobility and capillary e�ects, SPE/DOE Fourth Symposium on Enhanced Oil Recovery 2 (1984),145{162.[78] L.C. Young, A study of spatial approximations for simulating uid dispaclements in petroleum reservoirs,Comp. Meth. in Appl. Mech. Eng. 47 (1984), 3{46.

22