# An ELLAM Approximation for Highly Compressible Multicomponent Flows in Porous Media

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<ul><li><p>Computational Geosciences 6: 227251, 2002. 2002 Kluwer Academic Publishers. Printed in the Netherlands.</p><p>An ELLAM approximation for highly compressiblemulticomponent flows in porous media</p><p>H. Wang a, D. Liang b, R.E. Ewing a,c, S.L. Lyons d and G. Qin da Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA</p><p>b Institute for Scientific Computation, Texas A&M University, College Station, TX 77843-3404, USAc School of Mathematics and System Sciences, Shandong University, Jinan, Shandong, 250100, China;</p><p>currently visiting the Department of Mathematics, University of South Carolina,Columbia, SC 29208, USA</p><p>d ExxonMobil Upstream Research Company, Houston, TX 77252-2189, USA</p><p>Received 6 March 2001; accepted 11 December 2001</p><p>We develop an ELLAM-MFEM approximation to the strongly coupled systems of time-dependent nonlinear partial differential equations (PDEs) and constraining equations, whichdescribe fully miscible, highly compressible, multicomponent flows through heterogeneousand compressible porous media with singular sources and sinks. An EulerianLagrangianlocalized adjoint method (ELLAM) is presented to solve the transport equations for concen-trations. A mixed finite element method (MFEM) is used to solve the pressure PDE for thepressure and Darcy velocity simultaneously, which generates accurate fluid velocities andminimizes the numerical difficulties occurring in standard methods caused by differentiationof the pressure and then multiplication by rough coefficients. The ELLAM-MFEM solutiontechnique symmetrizes and stabilizes the governing transport PDEs and greatly reduces non-physical oscillation and/or excessive numerical dispersion present in many large-scale sim-ulators. Computational experiments show that the ELLAM-MFEM solution technique cangenerate stable and physically reasonable numerical simulations even if coarse spatial gridsand very large time steps are used.Keywords: advectiondiffusion equations, characteristic methods, compressible flows, Euler-ianLagrangian methods, multicomponent flow in porous media, PengRobinson equation ofstate, point sources and sinks</p><p>1. Introduction</p><p>The basic goal of subsurface fluid flow modeling is to simulate complex flowprocesses occurring in porous media sufficiently well to optimize the recovery of hy-drocarbon in the petroleum industry or to accurately predict and thoroughly remediatecontamination in groundwater transport processes among other applications. In orderto fulfill these objectives, one must build mathematical models to describe the essen-tial phenomena and the fundamental laws, and design numerical methods to discretize</p></li><li><p>228 H. Wang et al. / An ELLAM approximation for multicomponent flows</p><p>these models and to represent the basic features without introducing serious nonphysicalphenomena. However, the mathematical models are typically strongly coupled systemsof time-dependent, nonlinear PDEs and constraining equations. Additional difficultiesinclude (1) the advection dominance of and the moving steep fronts present in the solu-tions to these PDEs; (2) strong effect of sources and sinks; (3) the compressibility of thefluid mixture; (4) the heterogeneity and the compressibility of the porous media; (5) theinteraction between the fluid mixture and the porous media; (6) the anisotropic disper-sion in tensor form; and (7) the enormous size of field-scale application and the requiredlong time period of prediction.</p><p>Extensive research has been conducted in the effort to generate accurate, robust,stable, and physically reasonable numerical simulations to these systems with reasonablecomputational effort [12,16,29]. For example, Douglas et al. presented and analyzed asequential solution procedure for the miscible displacement of one incompressible fluidby another in a porous medium [9], in which an MFEM procedure [5,24] was used toapproximate the pressure, and the Darcy velocity and a Galerkin finite element methodwas used to approximate the concentration. However, standard finite difference andfinite element methods (FDMs, FEMs) tend to generate numerical solutions with severenonphysical oscillations. In industrial applications, upstream weighting techniques arecommonly used to stabilize the numerical approximations in large-scale simulators. Butthey yield excessive numerical diffusion and grid orientation effects [12,14,29]. Twogeneral classes of improved methods can be identified from the literature: the Eulerianmethods that use the standard temporal discretization and the characteristic methods thatcarry out the temporal discretization by a characteristic tracking. Eulerian methods tendto require small time steps (and often fine spatial grids), due to the stability requirementfor explicit methods or accuracy constraint for implicit methods [14,34]. Furthermore,the explicit treatment of the diffusion term in high-resolution methods could potentiallyintroduce a more restrictive stability constraint of |D|t/h2 = O(1), when diffusionvaries significantly.</p><p>Due to the hyperbolic nature of advective transport, characteristic methods havebeen investigated extensively for the numerical solution of advectiondiffusion PDEs[3,11,22,31]. Because the solutions are much smoother along the characteristics thanthey are in the time direction, characteristic methods allow large time steps to be usedin numerical simulations while still maintaining their stability and accuracy. However,characteristic methods usually require extra implementational effort. Traditional parti-cle methods advance the grids forward following the characteristics. They greatly re-duce temporal errors, but often distort the evolving grids severely and complicate thesolution procedure considerably. Backtracking characteristic methods, such as the mod-ified method of characteristics (MMOC) [11], follow the flow direction by tracking thecharacteristics backward from a fixed grid at the current time step and avoid the griddistortion problems present in forward tracking methods. These methods symmetrizeand stabilize the governing transport PDEs and generate accurate numerical solutionseven if large time steps are used. Ewing et al. replaced the Galerkin FEM scheme in [9]by an MMOC scheme [11] and developed an MMOC-MFEM solution procedure [13]</p></li><li><p>H. Wang et al. / An ELLAM approximation for multicomponent flows 229</p><p>for the miscible displacement of one incompressible fluid by another. The application ofthe MMOC allows large time steps to be used in solving the transport PDE without lossof accuracy and greatly reduces the excessive numerical dispersion and grid orientationeffects [12,29]. However, the MMOC scheme fails to conserve mass and has difficultyin treating boundary conditions, which seriously restricted its application.</p><p>The ELLAM concept was first presented by Celia et al. in solving one-dimensional(constant-coefficient) advectiondiffusion PDEs [7,26]. The ELLAM methodology pro-vides a general characteristic solution procedure for advectiondiffusion PDEs and aconsistent framework for treating general boundary conditions and maintaining massconservation. Thus, it overcomes the two principal shortcomings of previous character-istic methods while maintaining their numerical advantages. Computational experimentsshow that the ELLAM schemes are very competitive with many widely used and wellregarded methods in the context of linear advectiondiffusion transport PDEs [33,34]and of coupled systems [4,15] (at least in this case the computational grids are rectan-gular; since this is the case, the computational results of ELLAM schemes have beenreported). Optimal-order error estimates were also proved for ELLAM schemes [32,35].It was shown that ELLAM schemes conserve mass even if the characteristics and allthe integrals in the schemes are computed approximately [28]. Finite-volume ELLAMschemes were also developed [6,15], which were shown to conserve mass locally. Bin-ning and Celia [4] found that both ELLAM and finite-volume ELLAM schemes arerobust and computationally efficient, with the standard ELLAM scheme being preferredto the finite-volume ELLAM scheme.</p><p>So far, the ELLAM schemes have been applied to the incompressible (or slightlycompressible) flow of a single component, in which the pressure-density relation is rela-tively simple. In this paper we develop an ELLAM-MFEM solution technique for fullymiscible, highly compressible, multicomponent fluid flow processes through heteroge-neous and compressible porous media with singular sources and sinks. The rest of thispaper is organized as follows. In section 2, we present a mathematical model. In sec-tion 3, we develop an ELLAM-MFEM solution technique. In section 4, we performnumerical experiments to demonstrate the strength of the ELLAM-MFEM solution pro-cedure. In section 5, we summarize the results and draw conclusions.</p><p>2. A mathematical model</p><p>In this section, we present a mathematical model for describing fully miscible,highly compressible, multicomponent isothermal fluid flow processes through a com-pressible porous medium reservoir Rd .2.1. Mass balance of individual constituents and the fluid mixture</p><p>Let be the mass density of the fluid mixture and be the porosity of the porousmedium. For = 1, 2, . . . , N , let c(x, t) and uF(x, t) = (uF,1(x, t), . . . , uF,d(x, t))be the concentration (mass fraction) and the velocity of component , respectively. Let</p></li><li><p>230 H. Wang et al. / An ELLAM approximation for multicomponent flows</p><p>V = V(t) be a representative material volume, which is defined by a particularset of material points (rather than by any explicit spatial coordinates) and which couldmove as time t evolves. Then the global mass balance law for the species over thematerial volume V is given by [1]</p><p>ddt</p><p>V(t)</p><p>c dx =V(t)</p><p>cQ dx,</p><p>where Q is the volumetric flow rate per unit volume injected into (or produced from)the material volume V(t). At sources, and c are the prescribed mass density of thefluid mixture and concentration of the component . At sinks, = and c = c .</p><p>Applying the Reynolds transport theorem to the previous equation leads to thefollowing equation</p><p>V(t)</p><p>DDt</p><p>(c)+( uF)c dx =</p><p>V(t)</p><p>cQ dx, (2.1)</p><p>whereDfDt</p><p>= ft</p><p>+ uF fis the material derivative of f .</p><p>Equation (2.1) can be written as the following local mass balance law since (2.1)holds for arbitrary material volume V(t)</p><p>t(c)+ </p><p>(uFc</p><p>) = cQ. (2.2)Using the relations for the concentrations and the interstitial velocity uF, we obtain</p><p>N=1</p><p>c = 1,N=1</p><p>c = 1, uF =N=1</p><p>cuF. (2.3)</p><p>Summing (2.2) for = 1, 2, . . . , N , we obtain a mass balance equation for the fluidmixture</p><p>t()+ u = Q, (2.4)</p><p>with u = uF being the (superficial) mass flow rate of the fluid mixture.We can also rewrite equation (2.2) in terms of the mass flow rate u as follows</p><p>t(c)+ (uc)+ j = cQ. (2.5)</p><p>j = (uF uF)c is the diffusive flux and will be discussed in the next subsection.</p></li><li><p>H. Wang et al. / An ELLAM approximation for multicomponent flows 231</p><p>2.2. Empirical correlations of porous medium properties</p><p>In analytical models of fluid flow in porous media, Darcys law [1,16]</p><p>u = </p><p>K(p gd), (2.6)</p><p>is probably the most widely used field equation for fluid velocities, which establishesthe basic relationship between the Darcy velocity u of the fluid mixture and the pressuregradient and can be derived as a special case from the general momentum balance law.Here is the viscosity of the fluid mixture, g is the magnitude of the acceleration dueto gravity, and d(x) measures depth below some datum level. K = (kij )di,j=1 is thepermeability tensor of the medium that quantifies the ability of the porous medium totransmit a fluid.</p><p>The diffusive flux j can be expressed as j = Dc with the hydrodynamicdispersion tensor D being defined by [1,2,16]</p><p>D = dmI + dt|u|I + dl dt|u|</p><p>u1u1 . . . u1ud...</p><p>...</p><p>udu1 . . . udud</p><p> , (2.7)</p><p>with dm being the molecular diffusion coefficient, I being the d d identity tensor, anddt and dl being the transverse and longitudinal dispersivities, respectively.</p><p>Due to the effect of large pressure changes involved in some fluid flow processes,the porous medium can be deformed. Conversely, the deformation of the porous mediumcauses pressure changes, which in turn affect the fluid flow processes. Thus, fluid flowand deformation of porous media are coupled processes. In this paper we adopt a fre-quently used empirical correlation to model the porosity, which addresses the volumetriceffect of the deformation of the porous medium caused by the pressure in the fluid flowprocess [2,16]</p><p>ddp</p><p>= C(x), p=pr = r(x),</p><p>where C(x) is the compressibility of the medium and r(x) is the reference porosityof the medium at the reference pressure pr . Solving this problem leads to the followingformulation</p><p>(x, p) = r(x) exp(C(x)(p pr)</p><p>). (2.8)</p><p>If the porous medium consists of stress-sensitive elasto-plastic material, a more accurate(and possibly more complicated) model should be introduced to address the interactionsbetween the fluid flow and the deformation of the porous medium. In this case, themethod developed in this paper still applies, but with stronger nonlinearities through</p></li><li><p>232 H. Wang et al. / An ELLAM approximation for multicomponent flows</p><p>the Sp(x, p) term defined in equation (2.10) and possibly an additional system of PDEsfor describing the deformation of the porous medium. We refer interested readers to[8,17] for more details.</p><p>2.3. Equation of state</p><p>In compressible fluid flow processes through porous media, the density, viscosity,and compressibility of the fluid mixture could change with the pressure and the composi-tion of the fluid. The following equation of state is often used to describe the relationshipbetween the density, the pressure, and the temperature of the fluid mixture</p><p> = pZRT</p><p>. (2.9)</p><p>Here R is the universal gas constant and T is the temperature. Z = Z(c1, c2, . . . ,cN , p, T ) is the compressibility factor of the fluid mixture.</p><p>Extensive studies have been carried out on how to evaluate the compressibilityfactor Z. We use the PengRobinson equation [21]</p><p>Z3 (1 B)Z2 + (A 3B2 2B)Z (AB B2 B3) = 0,which has been widely used in the analysis of fluids over large ranges of temperature,pressure, and compositions of fluid mixtures. In this equation, the A and B depend onpressure, temperature, and composition [25].</p><p>After obtaining the compressibility factor Z and the density , we compute theviscosity = (c1, c2, . . . , cN , p) of the fluid mixture by the LohrenzBrayClarkcorrelation [19].</p><p>2.4. A system of PDEs modeling multicomponent compressible flow</p><p>We differentiate (2.8) and (2.9) with respect to p and c to obtain the followingrelations</p><p>p= </p><p>(1p 1Z</p><p>Z</p><p>p</p><p>),</p><p>c= </p><p>Z</p><p>Z</p><p>c,</p><p>p= C.</p><p>Furthermore, the first equation in (2.3) shows that there are only N 1 independentconcentrations c and that</p><p>cN</p><p>t= </p><p>N1=1</p><p>c</p><p>t.</p></li><li><p>H. Wang et al. / An ELLAM approximation for multicomponent flows 233</p><p>We use these relations to rewrite (2.4)(2.6) as a system of PDEs for the pressure p, theDarcy velocity u, and the concentrations c of the fluid mixture as follows</p><p>S...</p></li></ul>