Download - An ELLAM Approximation for Highly Compressible Multicomponent Flows in Porous Media

Computational Geosciences 6: 227–251, 2002. 2002 Kluwer Academic Publishers. Printed in the Netherlands.

An ELLAM approximation for highly compressiblemulticomponent flows in porous media

H. Wang a, D. Liang b, R.E. Ewing a,c, S.L. Lyons d and G. Qin d

a Department of Mathematics, University of South Carolina, Columbia, SC 29208, USAb Institute for Scientific Computation, Texas A&M University, College Station, TX 77843-3404, USA

c School of Mathematics and System Sciences, Shandong University, Jinan, Shandong, 250100, China;currently visiting the Department of Mathematics, University of South Carolina,

Columbia, SC 29208, USAd ExxonMobil Upstream Research Company, Houston, TX 77252-2189, USA

Received 6 March 2001; accepted 11 December 2001

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 Eulerian–Lagrangianlocalized 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: advection–diffusion equations, characteristic methods, compressible flows, Euler-ian–Lagrangian methods, multicomponent flow in porous media, Peng–Robinson equation ofstate, point sources and sinks

1. Introduction

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

228 H. Wang et al. / An ELLAM approximation for multicomponent flows

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.

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.

Due to the hyperbolic nature of advective transport, characteristic methods havebeen investigated extensively for the numerical solution of advection–diffusion 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]

H. Wang et al. / An ELLAM approximation for multicomponent flows 229

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.

The ELLAM concept was first presented by Celia et al. in solving one-dimensional(constant-coefficient) advection–diffusion PDEs [7,26]. The ELLAM methodology pro-vides a general characteristic solution procedure for advection–diffusion 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 advection–diffusion 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.

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.

2. A mathematical model

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 � ⊂ R

d .

2.1. Mass balance of individual constituents and the fluid mixture

Let ρ be the mass density of the fluid mixture and φ be the porosity of the porousmedium. For α = 1, 2, . . . , N , let cα(x, t) and uF

α(x, t) = (uFα,1(x, t), . . . , u

Fα,d(x, t))

be the concentration (mass fraction) and the velocity of component α, respectively. Let


Vα = Vα(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 α over thematerial volume Vα is given by [1]

d

dt

∫Vα(t)

φρcα dx =∫Vα(t)

ρ∗c∗αQ dx,

where Q is the volumetric flow rate per unit volume injected into (or produced from)the material volume Vα(t). At sources, ρ∗ and c∗

α are the prescribed mass density of thefluid mixture and concentration of the component α. At sinks, ρ∗ = ρ and c∗

α = cα .Applying the Reynolds transport theorem to the previous equation leads to the

following equation∫Vα(t)

Dα

Dt(φρcα)+ (∇ · uF

α

)φρcα dx =

∫Vα(t)

ρ∗c∗αQ dx, (2.1)

where

Dαf

Dt= ∂f

∂t+ uF

α · ∇fis the material derivative of f .

Equation (2.1) can be written as the following local mass balance law since (2.1)holds for arbitrary material volume Vα(t)

∂

∂t(φρcα)+ ∇ · (uF

αφρcα) = ρ∗c∗

αQ. (2.2)

Using the relations for the concentrations and the interstitial velocity uF, we obtain

N∑α=1

cα = 1,N∑α=1

c∗α = 1, uF =

N∑α=1

cαuFα. (2.3)

Summing (2.2) for α = 1, 2, . . . , N , we obtain a mass balance equation for the fluidmixture

∂

∂t(φρ)+ ∇ · u = ρ∗Q, (2.4)

with u = φρuF 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

∂

∂t(φρcα)+ ∇ · (ucα)+ ∇ · jα = ρ∗c∗

αQ. (2.5)

jα = ρ(uFα − uF)φcα is the diffusive flux and will be discussed in the next subsection.


2.2. Empirical correlations of porous medium properties

In analytical models of fluid flow in porous media, Darcy’s law [1,16]

u = −ρ

µK(∇p − ρg∇d), (2.6)

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 µ 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 the

permeability tensor of the medium that quantifies the ability of the porous medium totransmit a fluid.

The diffusive flux jα can be expressed as jα = −D∇cα with the hydrodynamicdispersion tensor D being defined by [1,2,16]

D = dmρφI + dt|u|I + dl − dt

|u|

u1u1 . . . u1ud...

...

udu1 . . . udud

, (2.7)

with dm being the molecular diffusion coefficient, I being the d × d identity tensor, anddt and dl being the transverse and longitudinal dispersivities, respectively.

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]

dφ

dp= Cφ(x)φ, φ

∣∣p=pr = φr(x),

where Cφ(x) is the compressibility of the medium and φr(x) is the reference porosityof the medium at the reference pressure pr . Solving this problem leads to the followingformulation

φ(x, p) = φr(x) exp(Cφ(x)(p − pr)

). (2.8)

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


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.

2.3. Equation of state

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

ρ = p

ZRT. (2.9)

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.Extensive studies have been carried out on how to evaluate the compressibility

factor Z. We use the Peng–Robinson equation [21]

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].

After obtaining the compressibility factor Z and the density ρ, we compute theviscosity µ = µ(c1, c2, . . . , cN , p) of the fluid mixture by the Lohrenz–Bray–Clarkcorrelation [19].

2.4. A system of PDEs modeling multicomponent compressible flow

We differentiate (2.8) and (2.9) with respect to p and cα to obtain the followingrelations

∂ρ

∂p= ρ

(1

p− 1

Z

∂Z

∂p

),

∂ρ

∂cα= − ρ

Z

∂Z

∂cα,

∂φ

∂p= φCφ.

Furthermore, the first equation in (2.3) shows that there are only N − 1 independentconcentrations cα and that

∂cN

∂t= −

N−1∑α=1

∂cα

∂t.


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 cα of the fluid mixture as follows

Sp∂p

∂t+ ∇ · u = ρ∗Q+

N−1∑α=1

(Scα − ScN )∂cα

∂t,

u = −ρ

µK(∇p − ρg∇d),

∂(φρcα)

∂t+ ∇ · (

ucα − D(u, p)∇cα) = ρ∗c∗

αQ,

x ∈ �, t ∈ (0, tf ], α = 1, 2, . . . , N − 1,

(2.10)

with Sp and Scα being defined by

Sp = φρ

[Cφ +

(1

p− 1

Z

∂Z

∂p

)], Scα = φρ

1

Z

∂Z

∂cα.

Different boundary conditions could be imposed to the system (2.10) dependingon the application. For instance, in petroleum reservoir simulation, the boundary of thereservoir is often modeled to be impermeable and leads to noflow boundary conditions

u · ν = 0, (x, t) ∈ ∂�× [0, tf ],(D∇cα) · ν = 0, (x, t) ∈ ∂�× [0, tf ], α = 1, 2, . . . , N − 1.

(2.11)

Finally, initial conditions are provided for the pressure p and the concentrations cα

p(x, 0)= p0(x), x ∈ �,cα(x, 0)= c0,α(x), x ∈ �, α = 1, 2, . . . , N − 1.

(2.12)

In summary, the mathematical model for describing fully miscible, highly com-pressible, multicomponent isothermal fluid flow processes through compressible porousmedia is a strongly coupled system of time-dependent nonlinear advection–diffusiontransport PDEs (2.10), which is closed by the constraining equations (2.8) and (2.9) thatprovide the constitutive relationship between the pressure p, density ρ, temperature T ,and composition cα . The diffusion–dispersion term indicated by the size of the coeffi-cients dm, dl, and dt is usually very small. The effective Peclet number of these problemsis large, and traveling interfaces between the injected and resident fluids diffuse slowlyas they move through the reservoir.

3. An improved numerical simulation technique

In this section we develop an ELLAM-MFEM simulation technique for system(2.8)–(2.10). The ELLAM and MFEM schemes can be derived and implemented formultidimensional partial differential equations over a general domain. In this paper,we restrict our development for two-dimensional rectangular domain. This is for thesimplicity of the development of the ELLAM-MFEM simulator.


3.1. An ELLAM reference equation for advection–diffusion transport PDEs

We derive an ELLAM reference equation for the advection–diffusion transportPDEs in (2.10) for cα under the assumption that the pressure p and the Darcy’s ve-locity u in these equations are known. Let I and J be the numbers of grid blocksin the x and y directions, and K be the number of time steps on the interval [0, tf ].We define a temporal partition on [0, tf ] and a spatial partition on � = [ax, bx] ×[ay, by] by

ax = x0 < x1 < · · · < xi < · · · < xI−1 < xI = bx,

ay = y0 < y1 < · · · < yj < · · · < yJ−1 < yJ = by, (3.1)

0 = t0 < t1 < · · · < tk < · · · < tK−1 < tK = tf ,

with �xi = xi − xi−1 for i = 1, . . . , I , �yj = yj − yj−1 for j = 1, . . . , J , and�tk = tk − tk−1 for k = 1, . . . , K.

We define the concentration trial and test function spaces to be the space of con-tinuous and piecewise bilinear polynomials on � with respect to the spatial partitionin (3.1)

S(�) = M01 [ax, bx ] ⊗M0

1 [ay, by ],with

Mγ

β [ax, bx] = {v(x) ∈ Cγ [ax, bx] | v(x) ∈ Pβ[xi−1, xi], i = 1, 2, . . . , I

},

Mγ

β [ay, by] = {v(x) ∈ Cγ [ay, by] | v(x) ∈ Pβ[yj−1, yj ], j = 1, 2, . . . , J

},

(3.2)

where C0[a, b] and C−1[a, b] are the spaces of continuous and piecewise continuousfunctions, respectively. Pβ is the space of univariate polynomials of degree not exceed-ing β.

The ELLAM formulation uses a time-marching algorithm, so we only need to fo-cus on the current time interval [tk−1, tk] defined by (3.1). Multiplying the transportPDE in (2.10) by space–time test functions z(x, t) that are continuous and piecewisesmooth, vanish outside the space–time strip � × (tk−1, tk], and are discontinuous intime at time tk−1, we obtain a space–time weak formulation for the transport PDE forα = 1, 2, . . . , N − 1

∫�

φ(·, p(·, tk))(ρcαz)(·, tk) dx +

∫ tk

tk−1

∫�

(∇z · D(u, p)∇cα)(·, ·) dy dθ

−∫ tk

tk−1

∫�

cα

[ρφ

∂z

∂θ+ u · ∇z

](·, ·) dy dθ

=∫�

φ(·, p(·, tk−1)

)(ρcα)(·, tk−1)z

(·, t+k−1

)dx +

∫ tk

tk−1

∫�

(ρ∗c∗αQz)(·, ·) dy dθ. (3.3)


Here z(x, t+k−1) = limt→tk−1, t>tk−1 z(x, t) takes into account that z(x, t) is discontinuousin time at time tk−1. Note that the noflow boundary conditions (2.11) are used to removeall boundary terms from (3.3).

In the ELLAM framework [7], an appropriate operator splitting of the adjoint equa-tion of the transport PDE in (2.10) concludes that the test functions z(y, θ) should bechosen to satisfy the hyperbolic part of the adjoint equation of the governing transportPDE

φρ∂z

∂θ+ u · ∇z = 0, y ∈ �, θ ∈ [tk−1, tk].

Thus, the test functions z(y, θ) should be constant along the characteristics y =r(θ; x, tk) defined by the initial-value problem of the ordinary differential equation

drdθ

= u(r, θ)ρφ(r, p(r, θ))

, r(θ; x, t)∣∣θ=t = x, θ ∈ [tk−1, tk]. (3.4)

Because of (2.11), for any (y, θ) ∈ � × [tk−1, tk], there exists an x ∈ � suchthat y = r(θ; x, tk). We use the Euler method to evaluate the source/sink term on theright-hand side of (3.3) to obtain∫ tk

tk−1

∫�

(ρ∗c∗αQz)(·, ·) dy dθ =

∫�

∫ tk

tk−1

(ρ∗c∗αQ)

(r(·; x, tk), ·

)z(x, tk)

∣∣∣∣∂r(·; x, tk)∂x

∣∣∣∣ dθ dx

=�tk

∫�

ρ∗c∗α(·, tk)Q(·, tk)z(·, tk)dx + EQ(ρ

∗c∗α, z),

where |∂r(·; x, tk)/∂x| = 1 + O(tk − θ) is the Jacobian of the transformation from xto r(θ; x, tn), and EQ(ρ∗c∗

α, z) is the truncation error due to the application of the Eulerquadrature. Similarly, we evaluate the diffusion–dispersion term and have∫ tk

tk−1

∫�

(∇z · D(u, p)∇cα)(·, ·) dy dθ = �tk

∫�

(∇z · D(u, p)∇cα)(·, tk) dx +ED(cα, z),

where ED(cα, z) is the truncation error term.Substituting these two equations in (3.3), we obtain a reference ELLAM weak

formulation for the transport equation in (2.10) for α = 1, 2, . . . , N − 1 as follows∫�

φ(·, p(·, tk))(ρcαz)(·, tk) dx +�tk

∫�

(∇z · D(u, p)∇cα)(·, tk) dx

=∫�

φ(·, p(·, tk−1)

)(ρcα)(·, tk−1)z

(·, t+k−1

)dx

+�tk∫�

(ρ∗c∗αQz)(·, tk) dx + E(ρ, cα, z), (3.5)

with E(ρ, cα, z) = ∫ tktk−1

∫�cα[ρφ(∂z/∂θ)+ u · ∇z] dy dθ − ED(cα, z)+ EQ(ρ

∗c∗α, z).


3.2. An MFEM scheme for the pressure equation

One important issue in the numerical simulation of system (2.10) is how the Darcyvelocity u is calculated, which governs the advection and diffusion–dispersion termsin the transport PDEs in (2.10). The porous media often change abruptly with sharpchanges in lithology, which are accompanied by large changes in the pressure gradi-ent ∇p. Nevertheless, the Darcy velocity u is usually a smooth quantity. The stan-dard FDMs or FEMs solve the pressure equation in (2.10) for p, which is differentiatednumerically and then multiplied by a possibly rough coefficient K/µ to determinethe Darcy velocity via equation (2.6). Because p is usually not smooth due to therough coefficients in the pressure PDE, the approximate Darcy velocity u obtained inFDMs and FEMs is rough and often inaccurate, which in turn reduces the accuracyof the approximation to the transport PDEs in (2.10). MFEM schemes approximateboth p and u in the pressure PDE simultaneously, and so yield accurate Darcyvelocity u and conserve mass [9,12,29]. We introduce the Sobolev spaces

H(div;�)= {v(x) ∈ (

L2(�))2 | ∇ · v ∈ L2(�)

},

H0(div;�)= {v(x) ∈ H(div;�) | v(x) · ν(x) = 0, x ∈ ∂�}

,

L2(0, tf ;X)= {w(x, t) | w(·, t) : (0, tf ) �−→ X,

∥∥w(·, t)∥∥X

∈ L2(0, tf )},

with X being a Sobolev space defined on �.Then, the pressure equation in (2.10) can be rewritten as a saddle-point mixed weak

formulation as follows [5]: Find a pair (u, p) ∈ L2(0, tf ;H0(div;�)×L2(�)) such that

∫�

µ

ρK−1u · v dx −

∫�

p∇ · v dx =∫�

ρg∇d · v dx,

∫�

w∇ · u dx +∫�

Sp∂p

∂tw dx =

∫�

ρ∗Qw dx +∫�

N−1∑α=1

(Scα − ScN )∂cα

∂tw dx,

∀(v, w) ∈ H0(div;�)× L2(�), t ∈ (0, tf ].

We introduce the lowest-order Raviart–Thomas space on �

Su(�)= (M0

1 [ax, bx ] ⊗M−10 [ay, by]

) × (M−1

0 [ax, bx] ⊗M01 [ay, by]

),

Su0 (�)=

{v(x) ∈ Sp(�) | v(x) · ν(x) = 0, x ∈ ∂�}

,

Sp(�)=M−10 [ax, bx] ⊗M−1

0 [ay, by],

with the space Mγ

β [a, b] being defined in (3.2). Then we formulate an MFEM schemefor the parabolic pressure equation (2.10) as follows: For k = 1, 2, . . . , K, find


a pair (uh(x, tk), ph(x, tk)) ∈ Su0 (�)× Sp(�), such that

∫�

µ(c, p̄h)(·, tk)ρ(c, p̄h)(·, tk)K−1(·)uh(·, tk) · vh(·) dx −

∫�

ph(·, tk)∇ · vh(·) dx

=∫�

ρ(c, p̄h)(·, tk)g∇d(·) · vh(·) dx,

�tk

∫�

wh(·)∇ · uh(·, tk) dx +∫�

(Sp(c, p̄h)ph

)(·, tk)wh(·) dx

=∫�

Sp(c, p̄h)(·, tk)ph(·, tk−1)wh(·) dx +�tk

∫�

(ρ(c, p̄h)Q

)(·, tk)wh(·) dx

+∫�

N−1∑α=1

(Scα (c, p̄h)− ScN (c, p̄h)

)(·, tk)

(cα(·, tk)− cα(·, tk−1)

)wh(·) dx,

∀(vh,wh) ∈ Su0 (�)× Sp(�).

(3.6)

Here c(x, tk) = (c1(x, tk), . . . , cN−1(x, tk)) is assumed to be known. ph(x, 0) ∈ Sp(�)

is an approximation to p0(x) and p̄h(x, tk) is a projected approximation of ph(x, tk),which are to be described in section 3.3.

3.3. A fully discretized ELLAM-MFEM approximation to system (2.8)–(2.10)

We begin by defining the following extrapolation operators

E1f (x, tk)=f (x, tk−1), k = 1,(

1 + �tk

�tk−1

)f (x, tk−1)− �tk

�tk−1f (x, tk−2), k = 2, 3, . . . , K,

E2c(x, tk)= c(x∗, tk−1), k = 1, 2, . . . , K,

(3.7)

where x∗ = r(tk−1; x, tk). We also define a weighted iteration operator Fω

Fωf(m)(x, tk) =

{f (0)(x, tk), m = 1,

(1 − ω)f (m−2)(x, tk)+ ωf (m−1)(x, tk), m � 2,

where 0 < ω < 2 is a weighting parameter.We now define an ELLAM-MFEM solution technique for system (2.8)–(2.10) as

follows:


A. Initialization: k = 0.

A1: Compute (uh(x, 0), ph(x, 0)) ∈ Su0 (�)×Sp(�) from a stationary analogue of (3.6)

∫�

µ(c0, p0)(·)ρ(c0, p0)(·)K−1(·)(uh(·, 0)− u0(·)

) · vh(·) dx

−∫�

(ph(·, 0)− p0(·)

)∇ · vh(·) dx = 0,∫�

wh(·)∇ · (uh(·, 0)− u0(·))

dx = 0,∀(vh,wh) ∈ Su

0 (�)× Sp(�),

(3.8)

with c0(x) = (c0,1(x), . . . , c0,N−1(x)), and

u0(x) = −ρ(c0(x), p0(x))µ(c0(x), p0(x))

K(x)(∇p0(x)− ρ

(c0(x), p0(x)

)g∇d(x)).

In the MFEM scheme (3.8), the pressure ph(x, 0) can only be determined up to anarbitrary constant. To uniquely determine the pressure ph(x, 0) while maintaining theconservation of mass, we impose the following condition

∫�

ph(·, 0) dx =∫�

p0(·) dx.

A2: For α = 1, 2, . . . , N − 1, define cαh(x, 0) ∈ S(�) to be the L2-projection of c0α(x)∫�

cαh(·, 0)zh(·) dx =∫�

c0α(·)zh(·) dx, ∀zh ∈ S(�).

B. Time Stepping Procedure:for k = 1, 2, . . . , K do

B1. Projection step:

A. Find the solution pair (u(0)h (x, tk), p(0)h (x, tk)) ∈ Su

0 (�)× Sp(�) such that

∫�

µ(Ech, Eph)(·, tk)ρ(Ech, Eph)(·, tk)K−1(·)u(0)h (·, tk) · vh(·) dx −

∫�

p(0)h (·, tk)∇ · vh(·) dx

=∫�

ρ(Ech, Eph)(·, tk)g∇d(·) · vh(·) dx,

�tk

∫�

wh(·)∇ · u(0)h (·, tk) dx +∫�

(Sp(Ech, Eph)p

(0)h

)(·, tk)wh(·) dx

= �tk

∫�

(ρ(Ech, Eph)Q

)(·, tk)wh(·) dx (3.9)


+∫�

Sp(Ech, Eph)(·, tk)ph(·, tk−1)wh(·) dx

+∫�

N−1∑α=1

(Scα(Ech, Eph)− ScN (Ech, Eph)

)(·, tk)

× (cαh(·, tk−1)− cαh(·, tk−2)

)wh(·) dx,

∀(vh,wh) ∈ Su0 (�)× Sp(�),

where cαh(·, t−1) = cαh(·, t0); and Ech = E2ch and Ep = E1p in (3.7).

B. Find the solution c(0)αh (x, tk) ∈ S(�) such that for α = 1, 2, . . . , N − 1

∫�

φ(·, p(0)h (·, tk))(ρ(

p(0)h , E2ch

)c(0)αhzh

)(·, tk) dx

+�tk∫�

(ρ(p(0)h , E2ch

)∇zh · D(u(0)h , p

(0)h

)∇c(0)αh)(·, tk) dx

=∫�

φ(·, ph(·, tk−1)

)(ρ(ph, ch)cαh

)(·, tk−1)zh

(·, t+k−1

)dx

+�tk∫�

(ρ∗(p(0)h , E2ch

)c(0),∗αh Qzh

)(·, tk) dx, ∀zh(·, tk) ∈ S(�).

(3.10)

Here c(0),∗αh (x, tk) = c∗α(x, tk) and ρ∗(x, tk) are specified at sources or c(0),∗αh (x, tk)

= c(0)αh (x, tk) and ρ∗(p(0)h , E2ch) = ρ(p

(0)h , E2ch) at sinks. ρ is computed with the

equation of state (2.9). We compute x̃(0) = r(tk; x, tk−1) by tracking the characteristicdefined by the discrete analogue of (3.4)

drdθ

= u(0)h (r, tk)

ρφ(r, p(0)h (r, tk)), r(θ; x, tk−1)

∣∣θ=tk−1

= x, θ ∈ [tk−1, tk], (3.11)

forward from time tk−1 to tk and then evaluate zh(x, t+k−1) = zh(x̃(0), tk).

B2. Iteration step:

if ERROR > TOLERANCE then m = m+ 1

A. Find the solution pair (u(m)h (x, tk), p(m)h (x, tk)) ∈ Su

0 (�)× Sp(�) such that

∫�

µ(Fωc(m)h , Fωp(m)h )(·, tk)

ρ(Fωc(m)h , Fωp(m)h )(·, tk)

K−1(·)u(m)h (·, tk) · vh(·) dx −∫�

p(m)h (·, tk)∇ · vh(·) dx

=∫�

ρ(Fωc(m)h , Fωp

(m)h

)(·, tk)g∇d(·) · vh(·) dx,


�tk

∫�

wh(·)∇ · u(m)h (·, tk) dx +∫�

(Sp

(Fωc(m)h , Fωp

(m)h

)p(m)h

)(·, tk)wh(·) dx

= �tk

∫�

(ρ(Fωc(m)h , Fωp

(m)h

)Q

)(·, tk)wh(·) dx (3.12)

+∫�

Sp(Fωc(m)h , Fωp

(m)h

)(·, tk)ph(·, tk−1)wh(·) dx

+∫�

N−1∑α=1

(Scα

(c(m−1)h , p

(m−1)h

) − ScN(c(m−1)h , p

(m−1)h

))(·, tk)

× (c(m−1)αh (·, tk)− cαh(·, tk−1)

)wh(·) dx, ∀(vh,wh) ∈ Su

0 (�)× Sp(�),

with c(m)h (x, tk) = (c(m)1,h (x, tk), c

(m)2,h (x, tk), . . . , c

(m)N−1(x, tk)).

B. Find the solution c(m)αh (x, tk) ∈ S(�) such that for α = 1, 2, . . . , N − 1∫�

φ(·, p(m)h (·, tk)

)(ρ(p(m)h , Fωc(m−1)

h

)c(m)αh zh

)(·, tk) dx

+�tk

∫�

(ρ(p(m)h , Fωc(m−1)

h

)∇zh · D(u(m)h , p

(m)h

)∇c(m)αh

)(·, tk) dx

=∫�

φ(·, ph(·, tk−1)

)(ρ(ph, ch)cαh

)(·, tk−1)zh(·, t+k−1) dx

+�tk

∫�

(ρ∗(p(m)h , Fωc(m−1)

h

)c(m),∗αh Qzh

)(·, tk) dx, ∀zh(·, tk) ∈ S(�).

(3.13)

Here c(m),∗αh (x, tk) = c∗α(x, tk) and ρ∗ are specified at sources or c(m),∗αh (x, tk) =

c(m)αh (x, tk) and ρ∗(p(m)h , Fωc(m−1)

h ) = ρ(p(m)h , Fωc(m−1)

h ) at sinks. ρ is computed withthe equation of state (2.9).We compute x̃(m) = r(tk; x, tk−1) by tracking the characteristics defined by

drdθ

= u(m)h (r, tk)

ρφ(r, p(m)h (r, tk)), r(θ; x, tk−1)

∣∣θ=tk−1

= x, θ ∈ [tk−1, tk] (3.14)

to compute x̃(m) = r(tk; x, tk−1) and then evaluate zh(x, t+k−1) = zh(x̃(m), tk).

else Define

ph(x, tk)= p(m)h (x, tk), uh(x, tk) = u(m)h (x, tk),

cαh(x, tk)= c(m)αh (x, tk), α = 1, 2, . . . , N − 1.

(3.15)

endifend do

Remark 3.1. Through a characteristic tracking, the ELLAM schemes (3.10) and (3.13)symmetrize and stabilize the governing transport PDEs in (2.10). Previous numericalexperiments show that the ELLAM schemes are very competitive with widely used


and well regarded methods in the context of linear advection–diffusion transport PDEsand generate accurate and stable numerical solutions even if very large time steps andcoarse spatial grids are used [33,34]. Optimal-order error estimates were also derived forELLAM schemes [32,35]. Note that the transport PDEs in (2.10) are linear in their pri-mary unknowns cα , which explains why the ELLAM schemes have generated accurateand efficient solutions in the context of subsurface contaminant transport of incompress-ible or slightly compressible fluid flow processes where the relationship between thedensity and pressure is relatively simple [4,15]. In this section we develop an ELLAM-MFEM solution technique for system (2.8)–(2.10), which bears stronger nonlinearitiesand couplings due to the higher compressibilities and heterogeneities.

Remark 3.2. All except the first terms on the right-hand side of equations (3.10) and(3.13) are standard integrals in FEMs. The evaluation of the first terms on the right-handside of (3.10) and (3.13) is very challenging, because cαh(x, tk−1) and zh(x, t

+k−1) are

defined at different time steps and each cell [xi−1, xi] × [yj−1, yj ] is deformed as thegeometry is backtracked from time step tk to time step tk−1. An inappropriate treatmentof these terms could cause serious problems [20]. In a backtracked characteristic method[11], these terms are rewritten as integrals at time tk in which the test function zh(x, tk)is standard but the trial function cαh(x∗, tk−1) are evaluated by a backtracking of char-acteristics. This could potentially lose mass unless this term is evaluated exactly [10],which is very difficult to guarantee numerically.

The forward tracking algorithm proposed in [28] would enforce an integrationquadrature at time step tk−1 with respect to the spatial grid (3.1) on which cαh(x, tk−1) isdefined. For the test function z(x, t+k−1), discrete quadrature points chosen on the fixedgrid at time step tk−1 will be be tracked forward to time step tk, where evaluation ofzh(x, t) is straightforward [33]. The forward tracking algorithm avoids the complicationof distorted grids in conventional particle methods or the complication of geometry inbacktracking methods.

It was proved in [28] that with the forward tracking algorithm an ELLAM schemeconserves mass globally when it is used to solve a linear advection–diffusion PDEs inmultiple space dimensions, even if the characteristics are tracked approximately [28].A locally mass-conservative finite volume ELLAM (FVELLAM) scheme was presentedin [6,15] for linear advection–diffusion PDEs. Binning and Celia [4] carried out care-ful study and comparisons on both the (finite element based) ELLAM scheme and the(finite volume based) FVELLAM schemes. Both schemes were found to be robust andcomputationally efficient, and the standard ELLAM was found to be preferred to theFVELLAM. Hence, we use the standard (finite element version of the) ELLAM schemein this paper, although we could also use the FVELLAM scheme.

Remark 3.3. For a general velocity field u and porosity φ(x, p), the initial-value prob-lems (3.11) and (3.14) cannot be solved analytically and so numerical means have tobe used. Because uh(x, tn) ∈ Su

0 (�) in (3.11) and (3.14) is piecewise smooth, solving(3.11) or (3.14) by numerical quadratures, which has been successful in the context of


linear transport PDEs with a smooth velocity field [33,34], could introduce large numer-ical errors. Note that φ is constant on each cell in applications and that uh in (3.11) and(3.14) is a Raviart–Thomas solution. We can analytically solve (3.11) and (3.14) to trackthe characteristics on a cell-by-cell basis [15,18,23,30]. This way, we greatly improvethe accuracy of the characteristic tracking and minimize the effect of the source andsink singularities. Furthermore, because uh satisfies the noflow boundary condition (thefirst equation in (2.11)) that is treated as an essential boundary condition in the MFEMschemes (3.9) and (3.12), the resulting approximate characteristics never run out of thephysical domain �, avoiding potential numerical difficulties. This is another advantageof the ELLAM-MFEM solution technique when applied to the system. Finally, the verystrong effect of the singular injection and production wells requires a very careful treat-ment of the source and sink terms. We treat the injection and production wells as sourcesand sinks defined on the entire cells as in [27,36] and enforce mass conservation nearthe wells.

4. Numerical experiments

In this section, we carry out numerical simulations to the system (2.8)–(2.12) byusing the ELLAM-MFEM solution technique developed in this paper. The numericalexamples include test cases with anisotropic dispersion in tensor form; highly compress-ible light or heavy components in the fluid flow through compressible, substructuredporous media with discontinuous permeabilities and porosities; and point sources andsinks.

The numerical experiments simulate multicomponent flows within a horizontalreservoir of one unit thickness over a period of ten years (3600 days) for one-quarterof a regular five-spot pattern with injection and production wells at the corners. The spa-tial domain is� = (0,1000)×(0,1000) ft2, and the time period [0, tf ] = [0, 3600] days.The injection well is located at the upper-right corner of the domain with a volumetricinjection rate of Q = 30 ft2/day. The production well is located at the lower-left cornerwith a production rate of Q = −30 ft2/day. Other common data used in the numericalexperiments in this section is given in table 1 below. We understand that a simulationon a nonuniform partition with finer cells around wells could generate more accurate

Table 1Common data used in the numerical simulations.

Compressibility of the medium cφ = 0.000001Reference pressure pr = 1 atm. = 14.696 psiaInitial pressure p0(x, y) = 3000 psiaSpatial grid size �x = �y = 50 ftTime step size �t = 360 days = 1 year


solutions, and our simulator has this capability. We also use a very large time step ofone year. Note that longitudinal and transverse Peclet numbers are 2000 and 20,000,respectively.

4.1. Example 1: Transport of light components in homogeneous media

We perform numerical experiments to simulate the transport of three light hydro-carbon components (methane, ethane, and propane) in a homogeneous porous medium.The critical data used in the Peng–Robinson equation for methane, ethane, and propaneare given in table 2 with other additional data presented in table 3.

The surface and contour plots for the concentrations of methane and propane att = 3, 5, 7, and 10 years are presented in figures 1(a) and (b). The surface and contourplots for ethane are very similar to those for propane and are omitted. It is observedthat the solution at t = 3 years is a family of concentric circles as expected, since thepermeability is homogeneous and the mobility of the methane, ethane, and propane issimilar. In addition, the large differences in longitudinal versus transverse dispersionlevels force the fluid flow to move much faster along the diagonal (flow) direction fromthe injection well to the production well as time evolves. These are reflected in the plotsin figures 1(a) and (b) for t = 5, 7, and 10 years. Furthermore, these plots have shown avery satisfactory recovery of the resident hydrocarbon compared to the results presentedin [12,13], because the mobility of the methane, ethane, and propane is fairly close toeach other. This numerically demonstrates the importance of reducing the mobility ratiosin petroleum reservoir recovery.

Table 2Critical data for methane, ethane, and propane.

CH4 (methane) C2H6 (ethane) C3H8 (propane)

Molecular weight MW 16.043 30.070 44.094Critical temperature Tc (K) 190.4 305.4 369.8Critical pressure Pc (psia) 667.2 708.3 615.8Critical volume Vc (ft3/lb.mol) 1.589 2.376 3.252Critical Z-factor Zc 0.288 0.285 0.281Acentric factor ωc 0.011 0.099 0.153

Table 3Additional data used in example 1.

Reference porosity φr = 0.1Permeability tensor k11 = k22 = 60 md and k12 = k21 = 0 mdMolecular diffusion coefficient Dm = φrdm = 0.0 ft2/dayMechanical dispersion coefficients Dl = φrdl = 0.5 ft, Dt = φrdt = 0.05 ftConcentrations at the injection well c∗methane = 0.6, c∗ethane = 0.25, c∗propane = 0.15Initial concentrations cmethane(x, 0) = 0.1, cethane(x, 0) = 0.5, cpropane(x, 0) = 0.4


Figure 1(a). The concentrations of methane at t = 3, 5, 7 and 10 years in example 1.


Figure 1(b). The concentrations of propane at t = 3, 5, 7 and 10 years in example 1.


Table 4Critical data for dodecane, n-tetradecane, and n-heptadecane.

C12H26 (dodecane) C14H30 (n-tetradecane) C17H36 (n-hetradecane)

Molecular weight MW 170.340 198.394 240.475Critical temperature Tc (K) 658.2 693.0 733.0Critical pressure Pc (psia) 264.5 208.7 188.1Critical volume Vc (ft3/lb.mol) 11.42 13.30 16.02Critical Z-factor Zc 0.24 0.23 0.22Acentric factor ωc 0.575 0.581 0.77

Table 5Additional data used in example 2.

Common data in �

Concentrations at the injection well c∗dodecane = 0.6, c∗n-tetradecane = 0.25, c∗

n-heptadecane = 0.15

Initial concentrations cdodecane(x, 0) = 0.1, cn-tetradecane(x, 0) = 0.5,cn-heptadecane(x, 0) = 0.4

Data in subdomain �(1)

The subdomain �(1) = (150, 600) × (150, 600) ft2

The permeability tensor k11 = k22 = 20 md, k12 = k21 = 0 mdThe reference porosity φr = 0.09The molecular diffusion coefficient Dm = φrdm = 0.0 ft2/dayThe mechanical dispersion coefficients Dl = φrdl = 0.54 ft, Dt = φrdt = 0.054 ft

Data in subdomain �(2)

The subdomain �(2) = �−�(1)

Permeability tensor k11 = k22 = 60 md, k12 = k21 = 0 mdReference porosity φr = 0.1Molecular diffusion coefficient Dm = φrdm = 0.0 ft2/dayMechanical dispersion coefficients Dl = φrdl = 0.6 ft, Dt = φrdt = 0.06 ft

4.2. Example 2: Transport of heavy components in substructured media

We perform numerical experiments to simulate the transport of three heavy hy-drocarbon components (dodecane, n-tetradecane, and n-heptadecane) in a substruc-tured medium. The critical data used in the Peng–Robinson equation for dodecane,n-tetradecane, and n-heptadecane is given in table 4, while additional data is given intable 5.

The surface and contour plots of dodecane and n-heptadecane at t = 3, 5, 7,and 10 years are presented in figures 2(a) and (b). The surface and contour plots forn-tetradecane are very similar to those for n-heptadecane and have been omitted. Be-cause of the presence of the low-permeability zone in the middle of the physical domain,the fluid flow moves more slowly in the diagonal direction. Consequently, the injected


Figure 2(a). The concentrations of dodecane at t = 3, 5, 7, and 10 years in example 2.


Figure 2(b). The concentrations of n-hetradecane at t = 3, 5, 7, and 10 years in example 2.


fluid sweeps over a wide area and so increases the efficiency of the petroleum recov-ery. An important technique in enhanced oil recovery is the use of polymers in floodingprocesses to alter the permeability of the reservoir porous medium to allow flow in cer-tain ways. Since the polymers are highly viscous, they can be used to selectively blockor reduce the permeabilities of certain pores or flow regions to direct the flow in a man-ner to optimize hydrocarbon recovery. The numerical results in this section could alsoserve as a demonstration for this technique.

5. Summary and conclusions

In this paper we develop an ELLAM-MFEM solution technique for the system ofPDEs and constraining equations that describe single-phase, highly compressible, multi-component fluid flow processes through heterogeneous and compressible porous mediawith singular sources and sinks. The ELLAM-MFEM solution technique symmetrizesand stabilizes the governing transport PDEs and greatly reduces nonphysical oscilla-tion and/or excessive numerical dispersion present in many large-scale simulators thatare widely used in industrial applications. Computational experiments illustrate that theELLAM-MFEM solution technique can treat highly compressible fluid flows throughcompressible media, discontinuous permeabilities and porosities, anisotropic dispersionin tensor form, heterogeneous media, and singular sources and sinks.

Previous numerical experiments showed that the ELLAM schemes are very com-petitive with widely used and well-regarded methods and generate very accurate nu-merical solutions in the context of linear advection–diffusion transport PDEs even ifvery coarse spatial grids and time steps are used [33,34]. Optimal-order error estimatesfor ELLAM schemes were also derived [32,35]. The ELLAM schemes were then devel-oped for the incompressible flow arising in groundwater contaminant transport (of singlecontaminant species) and were shown to outperform some widely used methods [4,15].In this paper we develop an ELLAM-MFEM solution technique for system (2.8)–(2.10),which has stronger couplings and nonlinearities. The numerical experiments show thatELLAM-MFEM solution techniques generate stable and physically reasonable resultsin the current context, even if coarse spatial grids and very large time steps, which arelarger than the time steps used in the MMOC-MFEM sequential solution procedure andone or two orders of magnitude larger than those used in many large-scale simulators,are used. In this manner, the ELLAM-MFEM solution technique has a greatly improvedcomputational efficiency over many other methods.

Acknowledgements

This research was supported in part by generous awards from Mobil Technol-ogy Company and ExxonMobil Upstream Research Company. In addition, the firsttwo authors would like to acknowledge the support from the South Carolina StateCommission of Higher Education: South Carolina Research Initiative Grant and the


NSF Grant DMS-0079549. The first author would also like to acknowledge the sup-port from the Institute for Scientific Computation of Texas A&M University during hisvisit. The third author would like to acknowledge the support from NSF Grants DMS-9626179, DMS-9706985, DMS-9707930, NCR9710337, DMS-9972147, INT-9901498;EPA Grant 825207; and Texas Higher Education Coordinating Board Advanced Re-search and Technology Program Grants 010366-168 and 010366-0336.

References

[1] M.B. Allen III, G.A. Behie and J.A. Trangenstein, Multiphase Flow in Porous Media, Lecture Notesin Engineering, Vol. 34 (Springer, Berlin, 1988).

[2] H. Aziz and A. Settari, Petroleum Reservoir Simulation (Applied Science Publishers, 1979).[3] J.P. Benque and J. Ronat, Quelques difficulties des modeles numeriques en hydraulique, in: Comput-

ing Methods in Applied Sciences and Engineering, eds. Glowinski and Lions (North-Holland, Ams-terdam, 1982) pp. 471–494.

[4] P.J. Binning and M.A. Celia, A three-dimensional forward particle tracking Eulerian–Lagrangian lo-calized adjoint method for solution of the contaminant transport equation, in: Computational Methodsin Water Resources, Vol. XIII, eds. L.R. Bentley, J.F. Sykes, C.A. Brebbia, W.G. Gray and G.F. Pinder(Balkema, Rotterdam, 2000) pp. 611–618.

[5] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Series in ComputationalMathematics, Vol. 15 (Springer, New York, 1991).

[6] M.A. Celia and L.A. Ferrand, A comparison of ELLAM formulations for simulation of reactivetransport in groundwater, in: Advances in Hydro-Science and Engineering, Vol. 1(B), ed. S.Y. Wang(Univ. of Mississippi Press, 1993) pp. 1829–1836.

[7] M.A. Celia, T.F. Russell, I. Herrera and R.E. Ewing, An Eulerian–Lagrangian localized adjointmethod for the advection–diffusion equation, Adv. in Water Resourc. 13 (1990) 187–206.

[8] H. Chen, R.E. Ewing, S.L. Lyons, G. Qin, T. Sun and D.P. Yale, Numerical solution of coupledgeomechanics and fluid flow in elastic porous media, in: Numerical Treatment of Multiphase Flowsin Porous Media, eds. Z. Chen, R.E. Ewing and Z. Shi, Lecture Notes in Physics, Vol. 552 (Springer,Berlin, 2000) pp. 80–92.

[9] J. Douglas, Jr., R.E. Ewing and M.F. Wheeler, A time-discretization procedure for a mixed finite ele-ment approximation of miscible displacement in porous media, RAIRO Modél. Math. Anal. Numér.17 (1983) 249–265.

[10] J. Douglas, Jr., F. Furtado and F. Pereira, On the numerical simulation of waterflooding of heteroge-neous petroleum reservoirs, Comput. Geosci. 1 (1997) 155–190.

[11] J. Douglas, Jr. and T.F. Russell, Numerical methods for convection-dominated diffusion problemsbased on combining the method of characteristics with finite element or finite difference procedures,SIAM J. Numer. Anal. 19 (1982) 871–885.

[12] R.E. Ewing, ed., The Mathematics of Reservoir Simulation, Research Frontiers in Applied Mathemat-ics, Vol. 1 (SIAM, Philadelphia, PA, 1984).

[13] R.E. Ewing, T.F. Russell and M.F. Wheeler, Simulation of miscible displacement using mixed meth-ods and a modified method of characteristics, SPE 12241 (1983) 71–81.

[14] B.A. Finlayson, Numerical Methods for Problems with Moving Fronts (Ravenna Park, Seattle,1992).

[15] R.W. Healy and T.F. Russell, Solution of the advection–dispersion equation in two dimensions bya finite-volume Eulerian–Lagrangian localized adjoint method, Adv. in Water Resourc. 21 (1998)11–26.

[16] R. Helmig, Multiphase Flow and Transport Processes in the Subsurface (Springer, Berlin, 1997).


[17] R.W. Lewis and B.A. Schrefler, The Finite Element Methods in the Static and Dynamic Deformationand Consolidation of Porous Media (Wiley, New York, 1998).

[18] N. Lu, A semianalytical method of path line computation for transient finite-difference groundwaterflow models, Water Resour. Res. 30 (1994) 2449–2459.

[19] J. Lohrenz, B. Bray and C. Clark, Calculating viscosities of reservoir fluids from their composition,J. Petrol. Tech. 16 (1964) 1171–1176.

[20] K.W. Morton, A. Priestley and E. Süli, Stability of the Lagrangian–Galerkin method with nonexactintegration, RAIRO Modél. Math. Anal. Numer. 22 (1988) 123–151.

[21] D.Y. Peng and D.B. Robinson, A new two-constant equation of state, Indian Engrg. Chem. Fundam.15 (1976) 59–64.

[22] O. Pironneau, On the transport-diffusion algorithm and its application to the Navier–Stokes equations,Numer. Math. 38 (1982) 309–332.

[23] D.W. Pollock, Semianalytical computation of path lines for finite-difference models, Ground Water26 (1988) 743–750.

[24] P.A. Raviart and J.M. Thomas, A mixed finite element method for second order elliptic problems,in: Mathematical Aspects of Finite Element Methods, eds. Galligani and Magenes, Lecture Notes inMathematics, Vol. 606 (Springer, Berlin, 1977) pp. 292–315.

[25] R.C. Reid, J.M. Prausnitz and B.E. Poling, The Properties of Gases and Liquids, 4th ed. (McGraw-Hill, New York, 1987).

[26] T.F. Russell, Eulerian–Lagrangian localized adjoint methods for advection-dominated problems, in:Pitmann Research Notes in Mathematics Series, Vol. 228, eds. Griffiths and Watson (Longman Sci-entific & Technical, Harlow, UK, 1990) pp. 206–228.

[27] T.F. Russell and R.W. Healy, Analytical tracking along streamlines in temporally linear Ravirt–Thomas velocity fields, in: Computational Methods in Water Resources, Vol. XIII, eds. L.R. Bentley,J.F. Sykes, C.A. Brebbia, W.G. Gray and G.F. Pinder (Balkema, Rotterdam, 2000) pp. 631–638.

[28] T.F. Russell and R.V. Trujillo, Eulerian–Lagrangian localized adjoint methods with variable coef-ficients in multiple dimensions, in: Computational Methods in Surface Hydrology, eds. Gambolatiet al. (Springer, Berlin, 1990) pp. 357–363.

[29] T.F. Russell and M.F. Wheeler, Finite element and finite difference methods for continuous flows inporous media, in: The Mathematics of Reservoir Simulation, ed. R.E. Ewing (SIAM, Philadelphia,PA, 1984) pp. 35–106.

[30] A.L. Schafer-Perini and J.L. Wilson, Efficient and accurate front tracking for two-dimensionalgroundwater flow models, Water Resources Res. 27 (1991) 1471–1485.

[31] E. Varoglu and W.D.L. Finn, Finite elements incorporating characteristics for one-dimensionaldiffusion–convection equation, J. Comput. Phys. 34 (1980) 371–389.

[32] H. Wang, An optimal-order error estimate for an ELLAM scheme for two-dimensional linearadvection–diffusion equations, SIAM J. Numer. Anal. 37 (2000) 1338–1368.

[33] H. Wang, H.K. Dahle, R.E. Ewing, M.S. Espedal, R.C. Sharpley and S. Man, An ELLAM scheme foradvection–diffusion equations in two dimensions, SIAM J. Sci. Comput. 20 (1999) 2160–2194.

[34] H. Wang, R.E. Ewing, G. Qin, S.L. Lyons, M. Al-Lawatia and S. Man, A family of Eulerian–Lagrangian localized adjoint methods for multi-dimensional advection–reaction equations, J. Comput.Phys. 152 (1999) 120–163.

[35] H. Wang, R.E. Ewing and T.F. Russell, Eulerian–Lagrangian localized methods for convection-diffusion equations and their convergence analysis, IMA J. Numer. Anal. 15 (1995) 405–459.

[36] H. Wang, D. Liang, R.E. Ewing, S.L. Lyons and G. Gin, An approximation to miscible fluid flows inporous media with point sources and sinks by an Eulerian–Lagrangian localized adjoint method andmixed finite element methods, SIAM J. Sci. Comput. 22 (2000) 561–581.

Download - An ELLAM Approximation for Highly Compressible Multicomponent Flows in Porous Media

Top Related