Fluid Flow in Oil Reservoirs

23 Nov 2004 1:23 AR AR235-FL37-09.tex AR235-FL37-09.sgm LaTeX2e(2002/01/18) P1: IBD10.1146/annurev.fluid.37.061903.175748

Annu. Rev. Fluid Mech. 2005. 37:211–38doi: 10.1146/annurev.fluid.37.061903.175748

Copyright c© 2005 by Annual Reviews. All rights reserved

MODELING FLUID FLOW IN OIL RESERVOIRS

Margot G. Gerritsen1 and Louis J. Durlofsky1,21Department of Petroleum Engineering, Stanford University, Stanford,California; email: [email protected] Energy Technology Company, San Ramon, California

Key Words reservoir simulation, heterogeneity, multiscale methods, upscaling,compositional modeling

■ Abstract Efficiently and accurately solving the equations governing fluid flowin oil reservoirs is very challenging because of the complex geological environmentand the intricate properties of crude oil and gas at high pressure. We present thesechallenges and review successful and promising solution approaches. We discuss indetail the modeling of fluid flow in reservoirs with strongly varying rock properties.This requires subgrid-scale models that accurately represent the flow physics due tofine-scale fluctuations. A second focus is on the complex multiphase, multicomponentsystems that describe miscible gas injection processes for enhanced oil recovery andCO2 sequestration.

1. INTRODUCTION

The field of fluid flow simulation in petroleum reservoirs has seen major advancesin the last few decades. These advances partly reflect the increasing computerpower that has facilitated the application of more sophisticated and detailed mod-els; but mostly the advances are the result of intense research in reservoir fluidflow modeling driven by the petroleum industry. The decline of the easiest-to-produce petroleum accumulations motivates the industry to optimize productionfrom existing reservoirs using enhanced oil recovery (EOR) processes, and tomove to high-risk developments of reservoirs in more challenging physical envi-ronments. In EOR processes fluids that may not naturally occur in the reservoirare injected. The injected fluids alter the flow properties of the reservoir and pro-mote oil production. Examples of high-risk developments are offshore productionin deep water and oil production in the Arctic. To maximize the production andreduce the risks of failure of these projects, the oil industry increasingly relies oncomputational performance prediction and optimization tools, of which reservoirfluid flow simulation is an integral part.

Efficiently and accurately solving the reservoir fluid flow equations is very chal-lenging because of the complex geological environment and the intricate proper-ties of crude oil and gas at high pressure. The aim of this review is to provide

0066-4189/05/0115-0211$14.00 211

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23 Nov 2004 1:23 AR AR235-FL37-09.tex AR235-FL37-09.sgm LaTeX2e(2002/01/18) P1: IBD

212 GERRITSEN � DURLOFSKY

an understanding of the physics of reservoir fluid flows, present the challengesin simulating them, and review successful and promising solution approaches.Because of the breadth of the field, and the limited space in this review, we nec-essarily limit detailed discussion to two main topics. The first topic is the efficientmodeling of flow in reservoirs with strongly varying heterogeneity, which requiressubgrid-scale models that accurately represent the flow physics due to fine-scalefluctuations. This topic presents interesting technical challenges and has a signifi-cant impact on reservoir performance prediction. Second, we discuss the complexand strongly nonlinear multiphase, multicomponent systems that describe produc-tion of light oil reservoirs and miscible gas injection processes for EOR and CO2

sequestration. We emphasize the challenges in modeling transport of componentsin miscible gas injection processes that have great economic and environmentalrelevance. We note that with these two choices, we touch on key challenges in thetwo main parts of a reservoir simulation: flow and transport.

1.1. Fluid Flow in Oil Reservoirs

Typically, a petroleum reservoir is a body of deep underground sedimentary rockthat contains a mixture of fluids in the interstitial spaces (about 10–100 µm across)between grains (see Figure 1a). The volume fraction of void space, referred toas porosity, generally ranges from 0.1–0.3. Generally, three fluid phases exist;we designate them aqueous (the phase containing predominantly water), liquid(the phase containing liquid hydrocarbons, which we also refer to as oil), andthe vapor or gas phase, which contains gaseous hydrocarbons. Each phase mayconsist of many chemical components. For example, oil is a very complex mixtureof hundreds of hydrocarbons with different chemical properties. Due to differencesin density, the phases are frequently found segregated in the reservoir with highergas saturations in the upper reservoir rock, followed by oil, with water primarilypresent in the lower layers. At the pore scale, surface tension controls the fluidconfiguration. Reservoir pressure and temperature are generally very high (of theorder of thousands of psia and hundreds of degrees centigrade, respectively).

Both microscopic and macroscopic effects control the movement of fluids inthe reservoir. The microscopic effects include viscosity and interfacial tension(IFT). For light crudes, the viscosity is comparable to or less than that of water,whereas heavy oils may have viscosities comparable to that of tar. Interfacialtension between reservoir fluids may cause fluids with lower saturation to becomedisconnected in the pores and unable to establish continuous flow paths. Also,the saturation of the immobile fluid in the crooks and crevices of the rock can besubstantial. The presence of one fluid can inhibit the flow of another because ofthe resistance to change in interfacial geometry.

Macroscopically, fluid flow is controlled by reservoir heterogeneity and mo-bility differences between the fluids. Reservoir rocks are usually formed fromnonuniform deposits that vary in time and space (Figure 1b). Erosion, fault-ing, and fracturing further complicate the heterogeneity. Consequently, the rock

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FLUID FLOW IN OIL RESERVOIRS 213

Figure 1 (a) Outcrop showing rock heterogeneity with a superimposed typical gridcell used in reservoir simulations. (b) Slice of berea sandstone reservoir rock. Actualdimensions of section shown 400 × 280µ2.

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permeability may vary by orders of magnitude [typically of the order of a milli-darcy to a darcy with 1 darcy (d) = 10−12 m2 = the permeability that gives a flowof 1 cm s−1 to a fluid with a viscosity of 1 centipoise under a pressure gradient of1 atm cm−1]. Mobility differences may severely impact the efficiency of gas orwater injection. The more mobile displacing fluid seeks pathways formed by con-nected high-permeability zones, faults, and fractures. Because these pathways mayextend over large distances, large areas of oil may be bypassed. In gas injectionprocesses gravity segregation can also greatly affect the global sweep.

Viscosities, IFT, and mobility differences vary throughout the reservoir anddepend on phase saturations, phase interactions at the prevailing pressure andtemperature, and molecular composition of the phases. Chemical components maytransfer between contacting phases, altering the fluid properties of both.

Interactions between the fluids and the reservoir rock may also impact perfor-mance. For example, pressure reduction during depletion can cause (local) rockcompaction that affects porosity as well as permeability.

Thermal effects are generally very small due to the large heat capacity of therock. However, in thermal recovery methods, such as steam flooding and in situcombustion (described below), temperature effects must be included. In situ com-bustion is further complicated by the presence of fast chemical reactions.

1.2. Oil Production

There are several mechanisms for primary production in which the natural drivemechanisms present in the reservoir are exploited. In dissolved gas drives, oil isdisplaced toward production wells by gas that evolves from the oil as productionlowers the reservoir pressure. Gas cap drives exist if the reservoir initially containsfree gas overlying the oil. Expansion of this gas cap pushes the oil toward thewells. In natural water drives, it is expanding brine that underlies the reservoir thatdisplaces the oil. It is very common to combine primary production with waterflooding, also referred to as secondary production, where water (possibly producedalong with the oil) is injected back into the reservoir. Water flooding is relativelyinexpensive and helps maintain reservoir pressure, but capillary forces betweenthe water and oil phases generally act to retain oil in the rock, causing a substantialfraction of the oil to remain undisplaced.

Despite the very high reservoir pressures and temperatures, the percentage ofoil originally in place that is recovered by primary and secondary production isusually only 20%–40%. In the United States alone this means 300 billion barrelsof the estimated 400 billion barrels of original oil in place will not be recoveredusing these production techniques. A substantial fraction of the residual oil may berecovered using EOR processes. In thermal recovery, heating reduces the viscosityof the oil. Steam flooding has had long-standing commercial success, but can not beused effectively in deep reservoirs. In situ combustion, in which heat is generatedby burning part of the oil in the reservoir, is an economically attractive alternativein recovering heavy oil from shallow reservoirs and lighter oil from deep reservoirs

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(Gerritsen et al. 2004, Sarathi 1999). It is also an ideal process for producing oilfrom thin formations, being most effective in 10–50-ft thick sandbodies. An addedadvantage is that elevated temperatures ahead of the combustion zone cause theoil to crack into lighter components, increasing its value. It is now well establishedthat injecting gases such as CO2, methane, enriched hydrocarbon gases, or nitrogeninto an oil reservoir can lead to efficient local displacements and small residual oilsaturations if the displacement pressure is sufficiently high such that miscibilitydevelops and IFT is (near) zero. CO2 injection has proven to be one of the mostefficient EOR methods since it was first tried in Texas in 1972.

EOR techniques have been hampered by their relatively high cost and, in somecases, by the unpredictability of their effectiveness. Today, EOR produces less than15% of petroleum in the United States, with gas injection accounting for about 40%of this EOR production. Worldwide, only 1% (or approximately 1 million barrelsof oil per day) of world production comes from gas injection projects. But thepotential for EOR techniques, in particular CO2 injection and in situ combustionprojects, is high. For example, in the United States alone there are roughly 125billion barrels of heavy oil (corresponding to about 18 years of U.S. petroleumconsumption) that could be targeted by in situ combustion. CO2 injection canpotentially be applied to a large number of reservoirs. Also, CO2 injection is apotential means to sequester carbon dioxide from power plants and other energyfacilities.

1.3. Challenges in Reservoir Fluid Flow Simulation

Clearly, accurate and efficient simulation of reservoir fluid flows presents manychallenges. Feasible grid-block sizes are generally large compared to at least someof the length scales associated with the geological characterization or fluid flowprocesses, and multiscale approaches are necessary (see section 3). Geologicalmodels may include O(107–108) cells for a typical reservoir, whereas practical in-dustrial models typically contain O(105–106) grid blocks (depending on the typeof model), with the model size often determined such that the simulation can berun in a reasonable time frame (i.e., overnight) on the available hardware. Also,rock properties are uncertain. Geostatisticians create a statistical ensemble of pos-sible permeability distributions that reflect measurements from a range of scalesand resolutions (such as seismic studies, drill core samples, production data, andmodels of the geological structure). Monte-Carlo or stochastic approaches arenecessary to assess the effects of this uncertainty on performance, dramaticallyincreasing the computational costs of a reservoir simulation study. In composi-tional simulation, component transfer between phases introduces very strong non-linearities in the equations describing transport and increases the computationalrequirements significantly, as discussed in section 4. But, despite the modeling andcomputational challenges and inherent uncertainties, reservoir fluid flow simula-tion has been widely adopted by the petroleum industry as a reservoir managementtool.

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2. MODELING RESERVOIR FLUID FLOW

Crude oil generally contains some amount of dissolved gas and invariably occursin conjunction with water. In many cases it is acceptable to assume that the oil andgas compositions are fixed, and that the solubility of the gas in the oil depends onpressure only. Then it is possible to consider a single oil “pseudo-component” anda single gas “pseudo-component.” If oil and gas equilibrium compositions varystrongly as a function of space and time, a compositional formulation is neededthat includes a larger number of (pseudo-) components and appropriate equationsof state. We first consider the general case of multiple components and then discusssome simplified representations.

We label the n p phases present in the reservoir with the subscript j , and mayrefer to specific phases using the subscripts a, l, or v for aqueous, liquid, andvapor. The local volume fraction of phase j is referred to as the saturation Sj .Components are designated by the subscript i . A key quantity is the mass fractionof component i in phase j , which we designate yi, j .

2.1. Compositional and Black-Oil Models

Fully compositional models must be used when the fluid flow depends stronglyon component transfer between phases. In fact, many EOR processes, includingmiscible gas injection, are specifically designed to take advantage of the phase be-havior of multicomponent fluid systems. Compositional modeling is also requiredin modeling depletion and/or cycling of retrograde reservoirs and reservoirs withhighly volatile oils. In these cases, the phase compositions are away from thecritical point, which simplifies the behavior of the fluid system.

If we consider transport to occur by both convection and dispersion, the massbalance for component i in an isothermal system gives (Acs et al. 1985)

∂

∂t

∑

j

φρ j yi, j S j + ∇·∑

j

(ρ j yi, j u j + ρ j S j Di, j∇ yi, j ) + mc = 0, i = 1, . . . , nc,

(1)

where φ is porosity (volume fraction of void space); ρ j is the phase density; u j isthe phase Darcy velocity, defined as the superficial velocity (volumetric flow ratedivided by area) of phase j ; Di, j is the dispersivity of component i in phase j ;mc is the mass source term (where the tilde indicates per unit volume); and nc isthe number of components. The equations in Equation 1 are also referred to as thetransport equations. The Darcy velocity is related to the phase pressure gradientvia the semiempirical representation

u j = −kr j

µ jk∇ p j , j = 1, . . . , n p. (2)

In Equation 2, kr j is the relative permeability to phase j , µ j is the viscosity ofphase j , p j is the pressure of phase j , and n p is the number of phases. The

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absolute permeability tensor k appears in Equation 2 describing flow, and not inthe transport equations. The phase pressures can differ as a result of capillarypressure effects. Equation 2 can be generalized to include the effects of gravitythrough the introduction of the phase potential � j in place of p j .

For single-phase flow, kr j = 1 and Equation 2 defaults to the single-phaseDarcy’s law. For multiphase flows, kr j < 1, due to the interference between phasesand the resulting reduction in total flow for a given ∇p. Relative permeability isgenerally a strong function of one or more of the phase saturations but does notdepend directly on fluid flow properties (Lake 1989). However, when miscibilitydevelops, relative permeabilities vary with IFT. Extending the single-phase Darcy’slaw to multiphase flow using relative permeability functions is generally acceptedby the reservoir simulation community, although, as pointed out in a previousAnnual Reviews article (Adler et al. 1988), it is an incomplete model that is validonly at small capillary numbers. Alternative models have been proposed in recentyears (Gray & Hassanizadeh 1998), but these have yet to be explored for reservoirsimulation applications.

The design of three-phase relative permeability models is an area of activeresearch. Blunt (2000) presents a good overview of the most commonly usedempirical models for three-phase relative permeability. Most simulators still useempirical correlations for three-phase relative permeability based on two-phaserelative permeability data. Besides the IFT effect, three-phase relative permeabil-ities may also depend strongly on hysteresis between imbibition and drainage(Lenhard & Parker 1987). A unified theory for handling the effects of IFT andhysteresis on three-phase relative permeabilities does not currently exist.

Phase viscosities are generally given by empirical correlations that considerthe viscosity a function of the mole fractions and molar density for the phase.The polynomial Lohrenz-Bray-Clark correlation (Lohrenz et al. 1964) is popularbecause of its simplicity and ease of implementation. It was developed primarilyfor low-density gases, but alternative (and generally more complex) models areavailable (Aziz & Settari 1979).

If we consider a system of nh hydrocarbon components plus a water component(total number of components nc = nh +1) existing in n p phases (typically n p = 3),there are a total of n p(nc + 2) equations and unknowns. The unknowns in eachgrid block include a total of ncn p values for yi, j and n p values each of Sj andp j . The governing equations include nc conservation equations of the form ofEquation 1, n p − 1 capillary pressure relationships (which relate the pressure inone phase to that in a reference phase), n p phase constraints (

∑i yi, j = 1 for

each phase j), one saturation constraint (∑

j S j = 1), and nc(n p − 1) equilibriumrelationships describing how each component partitions between phases. Of then p(nc + 2) unknowns, only nc are primary unknowns. The remaining unknownscan be computed algebraically once the primary unknowns are determined. Thereare many possible sets of primary variables that can be used for compositionalproblems. Cao (2002) reviews several of these formulations.

To calculate the mole fractions of the components in each phase, we assume thatchemical equilibrium is achieved instantaneously. An adequate thermodynamic

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description requires a cubic equation of state (EOS) such as the Soave-Redlich-Kwong equation (Soave 1972) or the Peng-Robinson equation (Peng & Robinson1976). Typically, solving these EOSs comprises the bulk of the computational workin a compositional simulation. Extensive research in the last decade resulted in amuch-increased efficiency of the algorithms. For a thorough discussion of thermo-dynamic models and their computational complexity, see Michelsen & Mollerup(2004).

When the oil is not overly volatile, the phase behavior representation can besimplified and the full compositional model is not required. The system can thenbe represented in terms of two hydrocarbon pseudo-components, designated oiland gas. These are defined as the phases at stock tank (standard) conditions. Thisformulation is called a black-oil model. In its most general form this model caninclude the effects of mass transfer of all components between phases (e.g., oilcan vaporize into the vapor phase), although in many cases the only interactionconsidered is gas dissolved in the oil phase. Refer to Aziz & Settari (1979) for fulldetails of this formulation.

2.2. Oil-Water Model

The compositional and black-oil models described above can be further simplifiedfor two-phase immiscible displacements (no mass transfer between phases). Theresulting formulation is useful both for describing some real cases (“dead-oil” sys-tems) and for developing and studying numerical solution procedures. Additionalsimplification arises if we neglect the effects of fluid and rock compressibility(generally acceptable for liquid systems) as well as capillary pressure. The latterassumption is often appropriate in models describing displacements at the lengthscales associated with reservoir simulation grid blocks.

By summing the conservation equations for oil and water, we form the pressureequation as

∇ · (kλt (Sw)∇ p) = qt , (3)

where we have introduced the total mobility λt , defined via

λt (Sw) = krw(Sw)

µw

+ kro(Sw)

µo. (4)

In Equation 3, q j = m j/ρ j and the total volumetric source term is qt = qw +qo. The pressure equation as written is elliptic in the absence of compressibility.Because the total mobility depends on saturation, the pressure field changes as thedisplacement evolves. The total velocity ut is defined as the sum of the water andoil Darcy velocities (ut = uw + uo) and can be computed from the pressure viaut = −λt k∇ p.

The water Darcy velocity can be expressed as uw = f (Sw)ut , where f (Sw)is the Buckley-Leverett fractional flow function given by f (Sw) = λw/λt , withλw = krw/µw. This yields the water conservation equation

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φ∂Sw

∂t+ ∇ · [ut f (Sw)] = −qw, (5)

also called the water saturation equation or the water transport equation. It de-scribes the movement of saturation fronts in the reservoir and is purely hyperbolicas written. Including capillary pressure effects renders this equation parabolic.Representing more general compositional or black-oil systems in this way is alsopossible, as we discuss below.

2.3. Numerical Formulations

Practical models contain of the order of 106 (black-oil model) or 105 (composi-tional simulation) grid blocks, with the model size often determined such that thesimulation can be run in a reasonable time frame (i.e., overnight). In composi-tional simulation, the system of appropriate primary variables can be solved usingvarying degrees of implicitness. We now briefly discuss various numerical formu-lations (our intent is not to provide a detailed review of numerical procedures forthe general system). The most stable approach is a fully implicit solution techniquebut this leads to large matrices (containing nc primary unknowns per grid block),which are often ill-conditioned. Consequently, fully implicit methods are limitedforemost by the number of components. To reduce the level of implicitness, andthus solve smaller linear systems, the system can be reformulated into a pressureequation (which can be viewed as an overall volume balance) plus a sequenceof nc − 1 component conservation equations (Coats 2000, Trangenstein & Bell1989, Wong et al. 1990). The conservation equations have a strongly hyperboliccharacter, and are therefore amenable to explicit solution. The pressure equationis always solved implicitly. The overall technique is called an IMPEC procedure(implicit in pressure, explicit in composition). The IMPEC methods are limitedby, often severe, stability restrictions on the time step size, but the solutions donot suffer less smoothing than fully implicit methods, which can strongly affectperformance prediction of compositional problems (see section 4).

Intermediate degrees of implicitness are also possible and often provide en-hanced efficiency. One such technique is an implicit pressure and saturation(IMPSAT) procedure, in which pressure and n p − 1 saturations (but not com-positions) are determined implicitly (Cao & Aziz 2002). Adaptive implicit (AIM)approaches enable varying degrees of implicitness, with some grid blocks treatedmore implicitly than others (Thomas & Thurnau 1983). For example, blocks nearfronts might be treated fully implicitly (or via IMPSAT), whereas the bulk of thedomain is treated using IMPEC.

Computational efficiency can also be improved through parallelization, or re-ducing the number of (pseudo-) components in the simulation. However, the lattermay lead to inaccurate representation of the physics. Recently, efforts were madeto include adaptive mesh refinement (AMR) in compositional simulation (e.g.,Sammon 2003). AMR focuses computational effort in regions near displacementfronts to accurately capture the local displacement efficiency.

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Although there has been much progress in recent years, conventional simulatorsbased on finite difference or finite element discretization of the three-dimensionalgoverning system of equations may not provide solutions for realistic composi-tional problems in reasonable time frames, even after code parallelization andadding AMR capability. We discuss recently developed solution methods for com-positional simulation of gas injection processes based on Euler-Lagrangian-typeapproaches in section 4.

A basic issue in any simulator is the numerical representation of the flow terms inthe governing equations. The block-to-block flow in simulators is determined by thetransmissibility, which relates interblock flow to the pressure difference betweenneighboring blocks. Standard finite difference (or finite volume) techniques arebased on diagonal-tensor permeabilities and orthogonal coordinate systems. Underthese assumptions, the calculation of interblock transmissibilities, which accountfor varying permeability, is straightforward. For example, transmissibility in thex-direction between two blocks i and i + 1 is given as (Tx )i+1/2 = 2(kx )i+1/2�yi �zi

�xi+1+�xi,

where �xi , �yi , and �zi represent the grid-block dimensions and (kx )i+1/2 is theweighted harmonic average of (kx )i and (kx )i+1. Extensions to handle more generalsystems, such as those with discontinuous full-tensor permeability fields, werepresented, within a finite volume context, by Aavatsmark et al. (1996), Edwards& Rogers (1998) and Lee et al. (2002).

Numerous finite element procedures have also been developed for solving thereservoir flow equations. Mixed finite element methods are well suited for reservoirsimulation because they provide local conservation over the primary grid. Many ofthe original applications of these and other finite element methods are described inthe extensive review by Russell & Wheeler (1983). Some recent work has focusedon the use of multiblock models (e.g., Wheeler et al. 2002) and multiscale methods(discussed below in section 3.2).

Alternate procedures, based on the solution of stochastic equations (in whichporosity and permeability are treated as random variables), are widely studiedwithin the context of groundwater hydrology and are also under study for reser-voir simulation (Jarman & Russell 2003, Zhang et al. 2000). These approaches are,however, not yet sufficiently developed for application in practical reservoir simu-lation studies. Outstanding issues include difficulties encountered when applyingthese approaches to general permeability fields (e.g., fields characterized by multi-point geostatistics) and the inherent complexity associated with the incorporationof additional physics into the stochastic formulation.

3. COARSE-SCALE MODELS OF HETEROGENEOUSRESERVOIRS

The equations presented above are strictly valid when fine-scale effects are fullyresolved. However, we are interested in solutions over grid blocks that are largecompared to at least some of the length scales associated with the geologicalcharacterization (as is evident from the fine-scale detail illustrated in Figure 1b).

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The coarse-scale modeling procedures considered here fall within two broad (butrelated) classes: upscaling techniques and multiscale methods. We view upscalingtechniques as methods in which coarse-scale equations of a prescribed analyticalform are solved. This form may differ from the underlying fine-scale equations. Itis either derived via homogenization or volume-averaging procedures, developedbased on physical arguments, or, as is often the case in practice, simply assumedto be the same (or nearly so) as the fine-scale equations. The coefficients in theseupscaled equations appear explicitly and are typically computed in a preprocessingstep from appropriate solutions over regions of the fine-scale model. By contrast,in multiscale methods, the coarse-scale equations are not expressed analytically,but rather are formed and solved numerically. Fine-scale information may be car-ried throughout the simulation and used at various stages, and different grids aregenerally used for flow and transport computations.

3.1. Computation of Upscaled Parameters

Upscaling within the context of reservoir simulation has been the subject of inten-sive research. Several reviews of numerical upscaling procedures have appearedin recent years (Christie 1996, Farmer 2002, Renard & de Marsily 1997, Wen& Gomez-Hernandez 1996). These reviews describe a wide range of proceduresand should be consulted for an exhaustive coverage of recent work. Many relatedissues are also discussed in the hydrology and applied physics literatures.

Upscaling methods can be classified in terms of the types of parameters that areupscaled (single- or two-phase flow parameters) and in terms of the way in whichthe upscaled parameters are computed (e.g., using local or global calculations). It isoften possible to develop coarse models of reasonable accuracy for multiphase flowwith only the single-phase flow parameters (e.g., absolute permeability) upscaled.In other cases, two-phase flow parameters [e.g., kr j and pc (capillary pressure)]are also effectivized. In any upscaling method, the two main issues that must beaddressed are the analytical form of the coarse-scale equation and the way in whichthe parameters in the coarse-scale model are computed.

3.1.1. COARSE-SCALE MODEL FOR SINGLE-PHASE FLOW Here we consider steady,single-phase incompressible flow with no source terms. The system is character-ized via an idealized two-scale model for permeability k(x, y) that varies on twodistinct scales: the slow scale x and the fast scale y. Homogenization procedures,applied within the context of reservoir simulation by, for example, Bourgeat (1984)allow this fully resolved k(x, y) to be replaced by an effective permeability tensork∗(x) in certain cases. This means that effects on the y-scale need not be resolvedin solving the global pressure equation, which leads to significant computationalsavings.

The upscaling of k(x, y) to k∗(x) is mathematically valid only when the regionover which k∗ is computed is large relative to the fast (y) scale of variation (e.g.,the correlation length of the heterogeneity field). In addition, special treatmentis required near boundaries or sources. In cases where these restrictions are not

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satisfied, upscaled permeability can still be computed, but it is more properlyreferred to as the equivalent grid-block permeability tensor rather than an effectivepermeability [see Durlofsky (1991) for more discussion of this point].

In purely local upscaling methods, the problem solved to determine k∗(x) in-volves only the fine cells that reside in the coarse block of interest. In general, a localproblem must be solved for each coarse block in the domain. Common choices forlocal boundary conditions are fixed pressure—no flow conditions or periodicity.The coarse-scale permeability k∗ can then be computed through 〈u〉 = −k∗〈∇ p〉,with the averaged velocity 〈u〉 and averaged pressure gradient 〈∇ p〉. The averagescan be computed as either outlet averages or volume averages. If outlet aver-ages are used, the cross terms of k∗ are undetermined if fixed pressure—no flowconditions are applied. Using volume averages enables the determination of theoff-diagonal terms of k∗ for any boundary condition, although the symmetry of k∗

is not generally guaranteed. Symmetry can be approximately recovered by usinga least square procedure (Wen et al. 2003a). Note that k∗ may also be computedbased on averages of other quantities (e.g., Zijl & Trykozko 2001).

Periodic boundary conditions have some desirable features in that the result-ing k∗ is guaranteed to be symmetric and positive definite (Boe 1994, Durlofsky1991, Pickup et al. 1994). In addition, the computed k∗ is the same with eitheroutlet or volume averaging. Any local method can alter large-scale permeabilityconnectivity, which can sometimes lead to error (as is illustrated below).

Upscaled transmissibility (T ∗) can be computed directly during the upscalingstep (as opposed to being determined from the k∗ in a subsequent calculation).In this case, the purely local problem may be shifted such that it is centered onthe target interface rather than the target block. The upscaled transmissibility canthen be calculated as the ratio of the total flux through the interface to the imposedpressure drop.

3.1.2. NONLOCAL CALCULATION OF COARSE-SCALE SINGLE-PHASE FLOW PARAMETERS

The purely local upscaling approach discussed above can be improved on byusing nonlocal information when calculating k∗ or T ∗. Extended local proce-dures introduce the effects of neighboring regions in the upscaling computations.Boundary conditions are imposed on the edges of the extended region (target blockplus neighboring cells), although averaged quantities and the resulting upscaledparameters are usually computed only over the target coarse block or interface.Extended local upscaling procedures have been applied by numerous researchers(e.g., Gomez-Hernandez & Journel 1994, Holden & Lia, 1992, Wen et al. 2003a,Wu et al. 2002). Methods of this type have the advantage of reducing the ef-fect of the assumed boundary conditions on the upscaled quantities and provideimproved coarse-scale descriptions over purely local methods in many cases. How-ever, for some types of geological models, such as channelized systems with highlydiscontinuous permeability descriptions, these methods still may not provideresults of great accuracy. In such cases it may be necessary to introduce globalflow information into the upscaling procedure.

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There are various global single-phase parameter upscaling methods (Holden& Nielsen 2000, Moulton et al. 1998, Pickup et al. 1994, White & Horne 1987).These methods have the apparent limitation of requiring a global fine-scale solu-tion for the particular flow scenario. However, this may not pose a major problemin cases for which the ultimate goal is a two-phase or multiphase flow simula-tion because the global computation we are considering here is a single-phasesteady state solution. A limitation of global procedures is that the coarse-scaleparameters found are not always positive. This problem can be circumvented byapplying optimization procedures with the coarse-scale parameters constrained tobe positive (Holden & Nielsen 2000). An approach that avoids some of the dif-ficulties associated with global methods, while still effectively using global flowinformation, is the recently developed local-global method of Chen et al. (2003).This method uses global coarse-scale pressure information to provide boundaryconditions for the extended local calculation of k∗ or T ∗. The method is iterateduntil self-consistency between the upscaled properties and the global flow field isachieved, which typically requires only a few iterations.

In reservoir simulation, global flow is almost always driven by wells ratherthan fixed-pressure or fixed-rate boundary conditions. Therefore, it is importantthat near-well effects be captured in coarse-scale models. Near-well upscaling iscomplicated by the fact that the pressure gradient, which is assumed constant inlocal upscaling procedures, is not constant in the well region. Several techniqueshave been presented to address near-well upscaling (Ding 1995, Mascarenhas &Durlofsky 2000, Muggeridge et al. 2002) and they are effective in practice. Thelocal-global procedure described above has also been extended to capture theeffects of wells on the coarse scale.

An example of the performance of local-global upscaling on a complex per-meability field is shown in Figure 2. The permeability field (Figure 2a) derivesfrom a channelized geological model (see Christie & Blunt 2001 for details).This is a particularly difficult example because the complex connectivity in thehigh permeability channels is easily lost with simpler upscaling techniques. Thefine-scale model in this case is 220 × 60 grid blocks and the upscaled model is22 × 6. We specify fixed pressures over portions of the left and right edges of themodel (this boundary specification is representative of partially penetrating wells)and compute the total flow rate Q through the system. The initial estimate for Q,computed on a coarse model generated using an extended local transmissibilityupscaling method, is in error by 58%. After two iterations of the local-global pro-cedure, this error is reduced to 2.3%. This illustrates the upscaling errors that canresult from the application of local procedures in highly heterogeneous systemsand the improvement that can be obtained by introducing global coupling into theupscaling calculation.

Using flow-based and other gridding procedures can significantly improve thequality of the coarse model. A number of procedures based on single-phase flowcalculations (e.g., Portella & Hewett 2000, Wen et al. 2003b and references therein)enhance solution accuracy. Alternate procedures based on the solution of elliptic

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Figure 2 (a) Channelized permeability field and inflow regions (red indicates high perme-ability, blue indicates low permeability). (b) Total flow rate through the system for coarsemodels generated using extended local (iteration 0) and local-global upscaling procedures.The fine-scale result is matched closely after only 1 or 2 iterations.

grid generation equations have not been extensively investigated within the contextof reservoir simulation, although such techniques may be useful.

3.1.3. COARSE-SCALE MODELS FOR TWO-PHASE FLOW A key requirement for accu-rate coarse-scale models of two-phase or multiphase systems is that they be builton top of accurate single-phase upscaling procedures. This is because absolutepermeability (or transmissibility) appears in two-phase flow problems in muchthe same way as in single-phase systems. The simplest two-phase parameter up-scaling entails the calculation of so-called pseudo-relative permeability functions.Procedures based on pseudo-relative permeability involve the use of coarse-scaleequations of the same form as the fine-scale equations. The upscaled relativepermeability and capillary pressure functions, designated k∗

r j (Sc) and p∗c (Sc), are

generally functions of Sc (saturation on the coarse scale) only. The k∗r j may vary

in each coarse grid block and may also be direction dependent. Because relativepermeability (and pseudo-relative permeability) functions are typically stored astables of data, the additional information that must be carried when using pseudo-functions can be substantial.

A number of different local and nonlocal procedures for computing upscaledtwo-phase flow parameters have been developed. Barker & Dupouy (1999) andDarman et al. (2002) presented descriptions and comparisons of various proce-dures. The upscaling of two-phase parameters may be especially important inhomogenizing small-scale effects (e.g., cm scale) up to geocellular-scale descrip-tions (e.g., 10 m). At very small scales capillary forces are important, and thevariation of capillary pressure with rock type (facies) can have a dominant effecton the upscaled quantities. Numerous investigators have addressed this issue; see,e.g., Pickup & Stephen (2000) and Virnovsky et al. (2004).

Complications arise in two-phase parameter upscaling because the k∗r j (Sc) are

often sensitive to the local boundary conditions used in the upscaling computation.

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This lack of robustness has been addressed by using extended local and globalmethods as well as “effective flux” boundary conditions, which attempt to ap-proximate the local velocity field using analytical approximations (Wallstromet al. 2002). Portella & Hewett (2000) and Aarnes & Espedal (2002) applied globalsingle-phase flow solutions to determine appropriate boundary conditions for cal-culating upscaled relative permeabilities. These and other methods have been suc-cessful in some cases, but the potentially strong boundary condition dependence ofupscaled relative permeabilities represents an important practical limitation. Thelack of robustness may also be due in part to the limited functionality of the coarse-scale equation (i.e., the form of the coarse model is incomplete). Recently, someinvestigators attempted to address this issue. Within stochastic frameworks, Langlo& Espedal (1994) and Lenormand & Fenwick (1998) introduced models that con-tain dispersive terms in addition to a modified convective flux function f ∗(Sc).Efendiev & Durlofsky (2003) recently extended these ideas to coarse-scale deter-ministic models. The dispersive terms represent small-scale subgrid effects and ap-pear even in systems with no diffusion on the fine scale. This model was more robustthan standard procedures for the coarse grid modeling of transport, in some cases.

3.2. Multiscale Methods

Multiscale finite-element and finite-volume approaches make up an importantclass of techniques designed to account for fine-scale information in coarse-scalesolutions. Most methods discussed below provide a specialized treatment of theflow (pressure) equation only. In most cases the transport equations are solved onthe fine grid after reconstruction of the fine grid velocity field based on the solutionof the flow equation. Because the computational costs of the flow equation aregenerally (much) higher than those of the transport equations in noncompositionalmodels, owing to implicitness and ill-conditioning, fine-grid transport solves cangenerally be afforded in such cases. Thus, these methods require that both a coarseand fine grid be used, and that fine-scale data be retained during the computations.

Within the context of reservoir simulation, most of the multiscale methods todate have been based on finite element procedures. Hou & Wu (1997), in a key pa-per, presented a technique that entails the construction of multiscale basis functionsfrom local solutions of the pressure equation. They applied an “oversampling” pro-cedure in which the local basis functions are computed using a small amount offine-scale information from neighboring elements. This approach is somewhat akinto using border regions when calculating upscaled permeability. Efendiev (1999)coupled the multiscale finite element solution with a coarse-scale model for unitmobility ratio transport (i.e., transport of a passive scalar) to give a model that didnot require the global fine-scale solution of the saturation equation. The transportmodel was based on a moment expansion (Efendiev et al. 2000) and included adispersivity term that is driven by subgrid velocity fluctuations (determined fromthe multiscale pressure solution).

In subsequent work, Chen & Hou (2003) presented a mixed multiscale method.Unlike the method of Hou & Wu (1997), this method retains local mass

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conservation and is therefore better suited for transport calculations. Chen & Housimulated unit mobility ratio transport on the fine scale using the reconstructed ve-locity field as well as the coarse grid dispersivity model of Efendiev et al. (2000).Chen & Yue (2003) extended the multiscale method to account for well-drivenflow. Aarnes (2004) reported further developments involving a mixed multiscalemethod and well effects.

Arbogast (2002) and Arbogast & Bryant (2002) developed variational multi-scale finite element methods. Their approach differs from the methods discussedabove in that they represent the discrete solution as the sum of a coarse gridcomponent and a subgrid component. Different mixed finite element basis func-tions represent these two components. Arbogast & Bryant (2002) demonstratednumerical results of good accuracy for some highly heterogeneous permeabilitydescriptions. Another finite element technique with a multiscale component is themortar upscaling method of Peszynska et al. (2002). This approach was formu-lated within a multiblock context and can handle nonmatching subdomains as wellas different physical models in different domains. Jenny et al. (2003) introduceda multiscale finite volume procedure that represents a generalization of previousflux-continuous techniques (e.g., Lee et al. 2002) to account for subgrid effects.Flux-continuous finite volume techniques require local solutions, which enforcepressure and flux continuity between cells, to determine the finite volume sten-cil. Jenny et al. included fine-scale information in these local solutions to providecoarse-scale transmissibilities that incorporate the effects of the underlying fine-scale permeability.

Multiscale methods that incorporate velocity reconstruction and the subsequentsolution of the transport equation on the fine grid are in some ways analogous topreviously developed dual-grid procedures (e.g., Gautier et al. 1999, Guerillot &Verdiere 1995). These techniques entail an upscaling step, the coarse grid solutionof the pressure equation, velocity reconstruction, and the solution of the transportequation on the fine grid. In most dual-grid procedures, standard (local) upscalingtechniques are applied. Using a more sophisticated upscaling procedure leads tomore accurate reconstructed velocity fields and hence more accurate fine-scalesaturation solutions for highly heterogeneous systems (Chen et al. 2003).

In comparing the accuracy of upscaling and multiscale procedures, it is nec-essary to distinguish between the impact of the treatment of the flow (pressure)equation, whether multiscale or upscaled, and the effect of the scale on which thetransport equations are solved. In many cases it is the solution of the transportequations on the fine-scale rather than the multiscale treatment of the pressureequation that is largely responsible for the significant improvement in accuracyover procedures that use coarse grids for both flow and transport.

3.3. Key Challenges in Coarse-Scale Reservoir Modeling

Although there has been extensive progress in the efficient modeling of flowand transport in heterogeneous reservoirs, many outstanding issues remain.

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Specifically, robust and practical techniques for the coarse-scale representationof transport (i.e., the saturation equation) have yet to be presented. Much lesswork has been directed toward the upscaling of three-phase flow or compositionalproblems, although there is a clear need for such capabilities.

Applying upscaling and multiscale procedures in conjunction with complexunstructured grids, as might be required, for example, to resolve complex geologyand multilateral wells, is still in its early stages. Grids of this type are now used inmany reservoir simulation applications, so it is important that compatible upscalingtechniques be developed. Using dynamic coarse grids (either fully adaptive oroccasionally updated) in conjunction with upscaling procedures is another areathat might be usefully pursued.

It will also be of interest to effectively combine multiscale (reconstruction)techniques with fully coarse-scale models. Then upscaled representations couldbe applied in some portions of the domain, whereas a multiscale representationcould be used in key active regions. This sort of general framework was previouslysuggested (e.g., E et al. 2003), although the development of such methods withinthe context of reservoir simulation has not yet been attempted.

4. MODELING MISCIBLE GAS INJECTION PROCESSES

In this section we focus on the challenging physics and simulation of misciblegas injection processes. Because of the relatively high costs of such processes, asdiscussed in section 4.3, reliable performance prediction tools are of great interestto the petroleum industry.

4.1. Miscible Gas Injection

When gas injected at sufficiently high pressure contacts oil, components in theoil transfer into the gas (or vapor) phase, and components in the gas dissolve inthe oil to establish chemical equilibrium. The resulting liquid and vapor phasesmove through the porous medium at different flow velocities that depend on thelocal pressure gradient and the viscosity and saturation of each phase. The vaporphase encounters oil with a composition different from the equilibrium liquidcomposition created at first contact, and additional component transfers occur. Inthis way a series of composition changes may be induced, with the most volatilecomponents being displaced faster. In the transition zone the IFT and capillarypressure between the vapor and liquid phases are reduced, thus improving themobility of the oil. The residual oil saturation is reduced significantly as comparedto water flooding because the injected gas can vaporize and transport componentstrapped in the immobile oil that sits in the crooks and crevices. When the pressureis at or above the Minimal Miscibility Pressure (MMP), the two phases are fullymiscible. Then the IFT and capillary pressure are reduced to zero, and a very high,piston-like local displacement efficiency is achieved.

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We illustrate the behavior of miscible gas injection processes in one dimension.Because gas injection processes are generally very strongly advection dominated,we ignore physical diffusion and hydrodynamic dispersion. The conservation equa-tions (Equation 1) describing the transport of the components can be written as

∂Ci

∂τ+ ∂ (vD Fi )

∂ξ= 0, i = 1, . . . , nc. (6)

The total molar density of component i is given by Ci = ∑n p

j=1 yi, jρ j S j , withn p the number of phases, and the total molar flux is vD Fi = vD

∑n p

j=1 yi, jρ j f j .

Here ρ j , Sj , and f j refer to the molar density, saturation and fractional flow ofeach phase, respectively. The total dimensionless velocity is given by vD = v/vin j ,

where vin j denotes the constant injection velocity. The velocity is determined fromthe flow (pressure) equation. The dimensionless time and spatial variables are

τ = vin j t

φL, ξ = x

L,

with length of the medium L and distance from the inlet x . The injection takesplace at ξ = 0. The partitioning of the components over the phases is determinedby solving phase equilibrium equations in the multiphase regions.

The fractional flow functions depend on relative permeabilities and viscositiesof the phases. For two-phase flows, fl = 1 − fg, and the fractional flow of thegas is given by

f = krg(S)

krg(S) + µg

µlkrl(S)

,

where S is the gas saturation.We discussed the relative permeability functions in section 2.1. The commonly

used two-phase relative permeability functions retain the hyperbolic character ofthe two-phase system (Equation 6). However, most three-phase relative perme-ability models used today introduce regions with elliptic behavior when applied toone-dimensional immiscible three-phase flow (with capillary effects neglected).Juanes & Patzek (2004) give an overview of this interesting and controversialphenomenom.

In the last 10 years, much progress was made in formulating (semi-) analyticalsolutions for the hyperbolic system (Equation 6) with constant initial and boundaryconditions (the Riemann problem), computed by a characteristic decompositionof the system. Jessen et al. (2001) give solutions for two-phase, multicomponentsystems. Solutions for miscible three-phase flow are currently limited to threecomponents.

4.2. An Illustration: Two-Phase Three-Component System

For illustration purposes we show solutions for two-phase, three-component flowswith a constant partitioning of components between phases; that is, with constant

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equilibrium ratios Ki = yi,g

yi,l, which are also known as the K-values. For constant

K-values and under ideal mixing, vD is constant and equal to 1 (in the generalcase, vD varies in time and space due to volume change on mixing as a resultof component transfer between phases). We inject a mixture of C1 (methane)and C2 (ethane) into a mixture of C2 and C3 (propane). Figure 3a (left) showssolution profiles for C1 and C2. In Figure 3a (right) the composition path ofthe solution is plotted in a ternary phase diagram. The vertices of the trianglerepresent pure component mixtures and the opposite edges correspond to the binarymixtures lacking that component. Although phase space does not explicitly showthe time dependence of the solution, it lends insight into the behavior of the systemthat cannot be determined by examining profiles alone, and it offers an attractivevalidation of numerical methods as used in, for example, Mallison et al. (2003). Thesolutions generally consist of segments of shocks and rarefactions, whose numberincreases with nc. The eigenvectors of the system give candidate directions inphase space along which the composition path may vary, and the eigenvalues givethe propagation speeds. The correct composition path is determined by requiringthat the path satisfy certain physical rules, akin to the entropy condition for shocksin the Euler equations.

In Figure 3a (right) the dew and bubble point curves of the three-componentsystem are drawn as two key tie lines that extend through the initial composition(a) and the injection composition (f). Transitions to and from the two-phase regionoccur as shocks (ab and ef) along tie line extensions. In the two-phase region thesolution consists of short rarefactions along the key tie lines, a rarefaction connect-ing the key tie lines and a zone of constant state that appears as the point (d). Atthe transition point (c) the propagation speeds of both rarefactions are equal (equaleigenvalue point). Multicomponent systems are weakly hyperbolic because theeigenvectors associated with equal eigenvalue points are not independent, whichposes extra challenges to the numerics.

The number of key tie lines increases with nc. Each additional key tie lineintroduces an additional nontie line path, a zone of constant state, and possibly atie line rarefaction.

For a given gas/oil displacement, the lengths of the key tie lines dictate theefficiency of the displacement (in 1D). When all of the tie line lengths are long,two-phase flow dominates and the displacement is inefficient. If one of the key tielines is short, then the phases that form will be similar in composition and the overallefficiency of the displacement will increase. In the limit that a tie line has lengthzero, miscibility develops and an optimal, piston-like displacement is achieved.

4.3. Key Challenges in Simulating Gas Injection Processes

Although the local displacement efficiency can be very high in miscible gas in-jection processes, the injected gas may contact only a small portion of a reservoirbecause low-viscosity gas finds the high-permeability flow paths. In addition,gravity can cause low-density gas to override the oil in place. So, the global sweepefficiency of a gas flood may not be high. The process performance of gas injection

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Figure 3 Solution profiles (left) and phase diagrams (right) for a two-phase three-component illustrative example. (a, top) Exact solutions to the Riemann problem.(b, middle) Numerical solutions for a first-order upwind method (400 grid points).(c, bottom) Numerical solutions for the third-order ENO scheme (400 grid points).

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schemes depends on this balance between local displacement efficiency and globalsweep efficiency, and both need to be captured accurately by a performance pre-diction tool. This presents three substantial challenges for compositional reservoirsimulators.

First, because the global sweep depends foremost on the underlying heterogene-ity, high-resolution reservoir models are required. Solver efficiency is paramountbecause the computational costs per grid cell are high owing to the potential largenumber of unknowns and the high expense of thermodynamic equilibrium calcula-tions. Alternatively, subgrid models could be developed for compositional modelsas indicated in section 3.

Second, the component transfer between phases strongly affects the predictionof local displacement efficiency. Thus, the number and type of (pseudo-) com-ponents can strongly affect the accuracy of the simulation. Although 6 or morepseudo-components often suffice, 10 to 15 components are sometimes neededto match phase behavior. Physical models, such as viscosity and relative perme-abilities, can also affect predictions. This is especially true in three-phase flowswhere predicted recovery can range widely depending on the selection of relativepermeability models alone.

Third, the equation of state introduces a very strong nonlinear coupling betweenthe fluid flow equations, which renders the simulation sensitive to modeling anddiscretization errors. Figure 3b,c shows the solution profiles and the compositionpath of the example problem computed with a first-order upwind scheme (com-monly used in reservoir simulation) and a third-order essentially nonoscillatory(ENO) scheme, respectively (Mallison et al. 2003). The errors made by the diffu-sive first-order scheme in capturing the leading shock speed are mainly due to theerror in composition path, and originate at the equal eigenvalue point. The com-position of the gas phase at the leading shock is less rich in the second componentand hence displaces oil less efficiently. Oil predicted by the analytical solution tobe displaced is left behind and accumulates in the zone of constant state, as theobserved dip in the solution profiles in Figure 3b (left). The ENO scheme capturesthe phase behavior near the equal eigenvalue point more accurately, leading tomuch improved results. In this simple example the errors are not very large. Inrealistic cases, the nonlinearities in the system are much stronger and numericalerrors are greater (Mallison et al. 2004). Also, the number of grid points (400)used in these 1D computations are much larger than can be used in practice.

4.4. Novel Compositional Simulation Methodsfor Gas Injection

The combination of high grid resolution and accurate representation of phasebehavior required for reliable performance predictions leads to very high compu-tational expense. Parallelization and adaptive mesh refinement strategies improveefficiency but generally not to the required degree.

As an alternative to the more traditional Eulerian methods described in sec-tion 2.1, we mention two Euler-Lagrangian-type methods that show promise for

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gas injection processes: the Euler-Lagrange Localized Adjoint Method (ELLAM)and the compositional streamline method. For compositional problems, both ap-proaches are still in early stages of development.

ELLAM was first developed by Celia et al. (1990) for solving constant-coeffi-cient advection-diffusion problems, and has subsequently been applied to variouslinear and nonlinear transport problems. It was specifically designed to conservemass while taking large time steps, and to allow systematic treatment of any typeof boundary condition. It also naturally accommodates spatial and temporal re-finements, which is attractive for problems with propagating fronts such as gasinjection processes, but efforts in this direction have so far been limited to onedimension. As Russell & Celia (2002) mention, there are a few obstacles to wideruse of ELLAM: the absence of a robust mechanism to control nonphysical oscil-lations and the need to improve efficiency of current implementations. Chen et al.(2000) and Qin et al. (2000) reported a first extension to compositional simulation.Applying ELLAM to the compositional problem requires linearization, which af-fects phase behavior. Finding a suitable linearization, especially in near-miscibledisplacements, is an outstanding research problem.

In heterogeneous reservoirs the time scale at which fluids flow along stream-lines is generally much shorter than the scale at which the streamline locationschange significantly. This observation motivates decoupling of the transport prob-lem into a sum of one-dimensional problems along streamlines that can be solvedindependently and very efficiently between pressure updates, as discussed in theoverview paper by King & Datta-Gupta (1998). Streamline methods have beenused successfully in predicting the global sweep of water floods in heterogeneousreservoirs, and have found applications as fast screening tools in optimization-and history-matching problems (see, e.g., Batycky et al. 1997). Cross-flow is ig-nored between pressure updates, but efficient corrections have been developedfor capillary and gravity effects (Bradvedt et al. 1996). Very recently, the stream-line methods were extended to compositional simulation for gas injection pro-cesses that are generally strongly dominated by heterogeneity (Jessen & Orr 2002,Mallison et al. 2003). The phase equilibrium equations are solved in points onthe streamlines. Because the streamline solves are independent and thus inher-ently parallel, the computational efficiency of compositional streamline simula-tion is significantly higher than that of Eulerian approaches. High-order meth-ods, such as the ENO scheme discussed in section 4.2, can be applied alongstreamlines, which, together with adaptive mesh refinement, further improve ef-ficiency. In contrast to the ELLAM methods, the full nonlinear formulation ofthe mass balance equations is used and phase behavior can be adequately cap-tured, but streamline methods are not mass conservative. So, in some sense, themethods are complementary. Intense research efforts are ongoing in both solutionapproaches.

Figure 4 shows a two-dimensional example of a near-miscible CO2 injec-tion into a heterogeneous reservoir simulated using compositional streamlinesimulation (Mallison 2004). In Figure 4a the two-dimensional permeability fieldis given, which was taken from layer 19 of the Tenth SPE Comparative Solution

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Figure 4 (a) Permeability field (in Darcy). The extent of the domain is 670 m (220grid cells) in the x-direction and 266 m (60 grid cells) in the y-direction. (b) Samplestreamline distribution during CO2 injection. (c) Mole fraction of CO2 after 1300 daysof injection.

Project (Christie & Blunt 2001). A sample streamline distribution duringsimulation is shown in Figure 4b, which illustrates that the highly mobile gasseeks high permeability flow paths. We injected CO2 along the left boundary x =0 at a total rate of 5m3 per day. Production takes places at the right boundary x =670 m, where the pressure was fixed at 235 atm. Zero flux boundary conditionswere applied at the other boundaries. The third-order ENO scheme was used tosolve the transport equations along the streamlines. The mole fraction of CO2 in

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the reservoir after 1300 days of injection (Figure 4c) shows high local displace-ment efficiency but a low global sweep as a result of the strongly heterogeneousmedium.

5. ADDITIONAL RESEARCH DIRECTIONS

There are many important practical aspects of reservoir simulation that are notdiscussed in this review. For example, using so-called “smart wells” (i.e., wellsequipped with downhole sensors and chokes) can substantially increase oil pro-duction while simultaneously decreasing the production of unwanted water orgas. Accurate representation of these wells in simulators poses a challenge, asdoes determination of optimum choke settings. The modeling of naturally frac-tured reservoirs is another active area of research. This problem is complicatedbecause of the wide range of length scales that appear in the problem and becauseof the need to accurately model transfer between the fractures and the matrix.In current practice, interacting continua (e.g., dual porosity) procedures are com-monly used, although discrete fracture modeling techniques are beginning to findsome application. Furthermore, the simulation of coupled multiphase flow andgeomechanics is important for predicting oil recovery as well as for determiningthe onset of well or facility failure. This problem is demanding because the flowand geomechanical effects, although tightly coupled, require different numericaltreatments.

We hope that this article has provided an understanding of the physics of fluidflow in oil reservoirs and an overview of challenges faced when modeling reservoirfluid flows. Finally, it is important to emphasize that research in reservoir simulationis largely driven by practical concerns. The future directions of the technology willtherefore be determined, at least in part, by advances in geological modeling andby developing new recovery processes and oil field technologies.

The Annual Review of Fluid Mechanics is online at http://fluid.annualreviews.org

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P1: JRX

November 24, 2004 12:5 Annual Reviews AR235-FM

Annual Review of Fluid MechanicsVolume 37, 2005

CONTENTS

ROBERT T. JONES, ONE OF A KIND, Walter G. Vincenti 1

GEORGE GABRIEL STOKES ON WATER WAVE THEORY,Alex D.D. Craik 23

MICROCIRCULATION AND HEMORHEOLOGY, Aleksander S. Popeland Paul C. Johnson 43

BLADEROW INTERACTIONS, TRANSITION, AND HIGH-LIFT AEROFOILSIN LOW-PRESSURE TURBINES, Howard P. Hodsonand Robert J. Howell 71

THE PHYSICS OF TROPICAL CYCLONE MOTION, Johnny C.L. Chan 99

FLUID MECHANICS AND RHEOLOGY OF DENSE SUSPENSIONS, JonathanJ. Stickel and Robert L. Powell 129

FEEDBACK CONTROL OF COMBUSTION OSCILLATIONS, Ann P. Dowlingand Aimee S. Morgans 151

DISSECTING INSECT FLIGHT, Z. Jane Wang 183

MODELING FLUID FLOW IN OIL RESERVOIRS, Margot G. Gerritsenand Louis J. Durlofsky 211

IMMERSED BOUNDARY METHODS, Rajat Mittaland Gianluca Iaccarino 239

STRATOSPHERIC DYNAMICS, Peter Haynes 263

THE DYNAMICAL SYSTEMS APPROACH TO LAGRANGIAN TRANSPORTIN OCEANIC FLOWS, Stephen Wiggins 295

TURBULENT MIXING, Paul E. Dimotakis 329

GLOBAL INSTABILITIES IN SPATIALLY DEVELOPING FLOWS:NON-NORMALITY AND NONLINEARITY, Jean-Marc Chomaz 357

GRAVITY-DRIVEN BUBBLY FLOWS, Robert F. Mudde 393

PRINCIPLES OF MICROFLUIDIC ACTUATION BY MODULATION OFSURFACE STRESSES, Anton A. Darhuber and Sandra M. Troian 425

MULTISCALE FLOW SIMULATIONS USING PARTICLES,Petros Koumoutsakos 457

vii

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