on the numerical simulation of waterflooding of ... · on the numerical simulation of waterflooding...

On the Numerical Simulation of

Waterflooding of Heterogeneous

Petroleum Reservoirs

Jim Douglas, Jr.∗ Frederico Furtado† Felipe Pereira‡

Abstract. We present a new, naturally parallelizable, accurate numerical method forthe solution of transport-dominated diffusion processes in heterogeneous porous media.For the discretization in time of one of the governing partial differential equations, weintroduce a new characteristics-based procedure which is mass conservative, the modifiedmethod of characteristics with adjusted advection (MMOCAA). Hybridized mixed finiteelements are used for the spatial discretization of the equations and a new strip-baseddomain decomposition procedure is applied towards the solution of the resulting algebraicproblems.

We consider as a model problem the two-phase immiscible displacement in petroleumreservoirs. A very detailed description of the numerical method is presented. Follow-ing that, numerical experiments are presented illustrating the important features of thenew method and comparing computed results with ones derived from previous, relatedtechniques.

1 Introduction

In this paper we discuss a new, naturally parallelizable, accurate, Eulerian-Lagrangian numerical pro-cedure for solving the transport-dominated diffusion process generated by two-phase, incompressible,immiscible displacement in heterogeneous porous media. The technique has applications in manyother areas; in particular, miscible displacement in porous media, miscible-immiscible displacement,compressible flows of these types in porous media, and several types of simulations of semiconductordevices can be treated with methods closely related to those given in this paper. Also, because of itsaccuracy and efficiency, this procedure promises to be very appropriate for ensemble studies of fluidflows in heterogeneous porous media that arise from stochastic geology [19]. This numerical procedurecombines:

• The modified method of characteristics with adjusted advection (MMOCAA), a new, compu-tationally effective, mass conserving time-discretization procedure which inherits the competi-tiveness of the usual MMOC (see [26, 42, 28, 23]);

• Hybridized mixed finite elements for accurate velocity field computation (see [16, 17, 28]);

• A new domain decomposition iteration based on parallel iterative procedures introduced by[14, 15, 38] and the authors and collaborators [20, 21, 25] for solving the algebraic problemsarising at each time step of a simulation.

The resulting numerical scheme has several distinctive properties:

∗Department of Mathematics, Purdue University, West Lafayette, IN 47907-1395, U.S.A.†Department of Mathematics, University of Campinas, 13081-970 Campinas, SP, Brazil‡Laboratorio Nacional de Computacao Cientıfica/CNPq, Rua Lauro Muller, 455, 22290 Rio de Janeiro,

RJ, Brazil

On the Numerical Simulation of Waterflooding of Heterogeneous PetroleumReservoirs

• A two-stage operator-splitting allows the use of time steps for the global pressure calculationto be longer than the steps used for the diffusive part of the saturation calculation, which inturn can be longer than those used in the transport part of the saturation calculation. In theexperiments reported, the diffusive time step was taken equal to the pressure step.

• In contrast to the method in [19], the new method conserves mass (to within the toleranceimposed on the iterative solution of the diffusive part of the saturation solution). The alterationin the procedure is associated solely with the transport part of the saturation calculation; thediffusive part does conserve mass in both the original method [19] and the method of this paper.

• The scheme is naturally parallelizable in distributed memory machines, because the iterativeprocedure requires exchange only of information pertaining to elements adjacent to interfacesbetween subdomains.

• The procedure has very low storage requirements, since the large linear systems of equationswhich appear in traditional numerical techniques for this problem do not have to be constructedor solved. The fluxes and Lagrange multipliers normally arising from hybridized mixed finite ele-ment methods for both the pressure and the saturation need not be stored in the implementationdiscussed below; however, they can be evaluated when needed from the stored information.

• Upwinding techniques do not have to be used in order to stabilize the numerical solution. Thisreduces the effect of numerical diffusion on the computed solution, including grid orientationeffects and excessive smoothing of sharp fronts.

• The implementation of the code is quite simple, because the domain decomposition into stripsof elements of the finite element method localizes the computations. Other choices for clumpingelements also lead to simple codes.

Although designed for distributed memory machines, the above properties make the new methodcompetitive even when used in serial machines.

Domain decomposition techniques for mixed methods distinct from the one used here can be foundin [13, 30, 29, 34, 35]. Other Eulerian-Lagrangian approaches leading to conservative schemes differentfrom ours can be found in [2, 4, 8, 9, 37]. A brief history of forerunners of the technique given in [23]and here can be found in [23].

The model problem we have taken to test our numerical method corresponds physically to thewaterflooding of a petroleum reservoir (two-phase immiscible flow) [11, 40, 22]. We consider thesystem of partial differential equations governing the flow written in terms of the global pressure[1, 11], which leads to a system consisting of an elliptic-parabolic coupled pair of equations. Since ourmain concern is to introduce and justify the new numerical method, we will limit ourselves to relativelysimple cases. Specifically, we shall treat waterflooding of horizontal, two-dimensional reservoirs. Themethod is essentially unaltered in three space, and the inclusion of gravitational terms is not difficult.We expect to repeat and extend the stochastic geology results of [19] using the new method elsewherein the immediate future.

This paper is organized as follows. In §2, a brief description of the model problem consideredhere is given. The spatial discretization (with continuous time) of the governing system of partialdifferential equations in terms of hybridized mixed methods along with a description of the domaindecomposition procedure we use is the content of §3. Next, in §4, we specialize the discussion to thelowest index Raviart-Thomas space over rectangles. The time discretizations of the pressure and sat-uration equations are described in §5. The new method for discretizing the time, the MMOCAA, isintroduced and applied to the saturation equation. The elimination of Lagrange multipliers resultingfrom the use of hybridized mixed finite elements is presented in §6. In §7 a detailed description ofour domain decomposition iterations designed for solving the algebraic problems posed by the sat-uration and pressure equations is given, followed by a summary of the algorithm employed in ourtime-dependent computations in §8. Numerical simulations are analyzed in §10, where a systematic

2


comparison between the usual MMOC and the new MMOCAA is performed. The experimental cal-culations described in §10 work are restricted to one- and two-dimensional porous media; however, thenumerical method described in this work applies to general three-dimensional media. The algorithmis extended in §9 to include the effects of gravity; the experimental calculations including gravity willappear in a paper [24] on fractured reservoirs, where the gravitational effects are critical.

2 The Two-Phase Flow System

The standard Muskat equations [11, 40] describing two-phase incompressible, immiscible displacementin a gravity-free environment are the two conservation equations

φ∂so∂t−∇ · (kkro(sω)µ−1

o ∇po) = qo, (1)

φ∂sω∂t−∇ · (kkrω(sω)µ−1

ω ∇pω) = qω, (2)

and the equation of statepc(sω) = po − pω.

Here, the subscripts o and w refer to oil and water, respectively, and si denotes the saturation, pi thepressure, the function kri(si) the relative permeability, µi the viscosity, and qi the external volumetricflow rate, each with respect to the ith phase. The functions φ(x) and k(x) are the porosity andabsolute permeability, respectively, of the medium and pc is the capillary pressure, assumed, as usual,a function of the saturation sw only. The equations are defined on a bounded domain Ω in R2 for thespatial variables and a finite time interval. The medium is assumed to be saturated:

so + sω = 1;

thus, we can set s = sω = 1− so. Let

λ =kroµo

+krωµω

, λi =kriµiλ

, i = o, w,

represent a total mobility of the two-phase fluid and the relative mobilities of the two phases, respec-tively. Introduce a global pressure [1, 10, 11] by setting

p =1

2

po + pω +

∫ pc

0

[λo(p

−1c (ζ))− λω(p−1

c (ζ))

]dζ

,

and the total external volumetric flow rate

q = qo + qω.

Then, a bit of algebraic manipulation with equations (1) and (2) leads to the (global) pressure equation

∇ · u = q, where u = −k(x)λ(s)∇p, (3)

and the saturation equation (assuming, when q > 0, that only water is injected and, when q < 0, thatthe water production is proportional to its local mobility)

φ∂s

∂t+ λ′ωu · ∇s+∇ · [k(λλωλop

′c)(s)∇s] = qω − λq =

(1− λω)q, q > 0,0, q < 0,

(4)

For simplicity, we suppose the periphery of the reservoir to be impermeable, so that the normalcomponents of both water and oil fluxes across the boundary vanish. This is equivalent to requiringthat

u · n |∂Ω= kλλωλop′c∇s · n |∂Ω= 0, (5)

3


where n is a unit vector normal to the reservoir boundary. Compatibility to the incompressibility ofthe fluids requires that ∫

Ω

q dx = 0. (6)

Finally, it is necessary to specify the initial saturation:

s(x, 0) = s0(x). (7)

The time discretization for the saturation equation to be discussed below is based on time steppingalong the characteristics related to the transport in the equation. Let us consider the differentialproblem first and rewrite the saturation equation in a more convenient form (see also [42]). If

b(s, u) = λ′ω(s)u, d(x, s) = k(x)(λλωλop′c)(s), (8)

and we generalize the right-hand side of the saturation equation slightly, (4) is equivalent to

φ∂s

∂t+ b(s, u) · ∇s+∇ · (d(x, s)∇s) = g(s). (9)

For any reasonable water injection rate and capillary dissipation strength, the flow is essentially alongthe characteristic curves of the transport operator φ∂/∂t + b(s, u) · ∇, so that it is appropriate tointroduce differentiation in this characteristic direction. Let

ψ(x, s, u) =√φ(x)2 + |b(s, u)|2,

ψ∂/∂τ = φ∂/∂t+ b(s, u) · ∇,

and note that the direction τ depends on x as well as on the saturation and fluid velocity, which varyin space and time. It follows easily that the saturation equation can be written in the form

ψ∂s

∂τ+∇ · (d(x, s)∇s) = g(s). (10)

Our space and time discretization of equations (3) and (10) rests on mixed formulations of bothequations. Introduce flux vectors in both equations to obtain the equations

∇ · u = q, u = −k(x)λ(s)∇p, (11)

ψ∂s

∂τ+∇ · v = g(s), v = d(x, s)∇s. (12)

The development of our numerical method will start from equations (11) and (12). We will applya two-stage operator splitting technique involving possibly three distinct time steps. First, we willsplit the pressure calculation from the saturation calculation, as has been done for more than a decade[18]. In addition, we will split the transport and the diffusive parts of the saturation equation. Indoing so, we will be led to time steps satisfying

∆tp ≥ ∆tsd ≥ ∆tst,

where the subscripts p, sd and st refer to pressure, diffusion and transport, respectively. Clearly,practicality requires integral ratios between the time steps. The originalMMOC procedure [26, 42, 27]implicitly splits the transport and the diffusion, though with the same time step; we must explicitlyrecognize that splitting in formulating the MMOCAA procedure. Experiments, as we will reportbelow, have shown an advantage to taking ∆tsd > ∆tst.

4


3 Continuous-Time Mixed Finite Element Approximations

Parallelization of the solution of the global problems (11) and (12) is achieved through a spatialdecomposition, which we now describe, first when only spatial discretization is considered.

Let Ω ⊂ R2 be a bounded domain with a Lipschitz boundary ∂Ω. Let Ωj , j = 1, . . . ,M be apartition of Ω:

Ω = ∪Mj=1Ωj ; Ωj ∩ Ωk = ∅, j 6= k.

Assume that ∂Ωj , j = 1, . . . ,M , is also Lipschitz and that Ωj is star-shaped. In practice, with theexception of perhaps a few Ωj ’s along ∂Ω, each Ωj would be convex with a piecewise-smooth boundary.Let

Γ = ∂Ω, Γj = Γ ∩ ∂Ωj , Γjk = Γkj = ∂Ωj ∩ ∂Ωk.

Let us consider decomposing (11) and (12) over the partition Ωj. In addition to requiring thatuj , pj (where pj = p|Ωj , etc.) be a solution of (11) and sj , vj a solution of (12) for x ∈ Ωj ,j = 1, . . . ,M , it is necessary to impose the consistency conditions

pj = pk, sj = sk, x ∈ Γjk , (13)

anduj · nj + uk · nk = 0, vj · nj + vk · nk = 0, x ∈ Γjk, (14)

where nj is the unit outer normal to Ωj .Let Vj = H( div ,Ωj) and Wj = L2(Ωj) for j = 1, . . . ,M . The weak formulations of (11) and

(12) corresponding to the domain decomposition above is given by seeking pj , uj ∈ Wj ×Vj andsj , vj ∈Wj ×Vj , j = 1, . . . ,M , such that (after an integration by parts step),

(∇ · u, p)Ωj = (q, p)Ωj ,

(k−1λ−1u, u)Ωj − (p,∇ · u)Ωj +∑

k 6=j〈p, u · nj〉Γjk = 0,

and

(ψ∂s/∂τ, s)Ωj + (∇ · v, s)Ωj = (g, s)Ωj ,

(d−1v, v)Ωj + (s,∇ · v)Ωj −∑

k 6=j〈s, v · nj〉Γjk = 0,

for all u, v ∈ Vj and p, s ∈Wj .Initially, we shall treat the case in which Ωj is a partition of Ω into individual elements (simplices,

rectangles, prisms), though an inspection of the procedure would indicate that larger subdomains arepermissible. Let Vh ×Wh be a mixed finite element space over Ωj; any of the usual choices is

acceptable [5, 6, 7, 12, 39, 41, 43]. Each of these spaces is defined through local spaces Vhj ×Wh

j =

Vh(Ωj)×Wh(Ωj) and the setting of

Vh = v ∈ H( div ,Ω) : v|Ωj ∈ Vhj and Wh = w : w|Ωj ∈Wh

j .

In each space Wh in the various families of mixed elements referenced above, the functions w ∈Wh are allowed to be discontinuous across each Γjk. As a consequence, attempting to impose theconsistency conditions (13) would force continuity of p and s; i.e., (13) would not be satisfied unlessthe approximate solutions ph and sh to the discrete analogues of (11) and (12) are constant, a totallyuninteresting case. So, let us introduce Lagrange multipliers [3, 31] `αjk (α = g for the global pressuresystem (11) and α = s for the saturation system (12)) on the edges Γjk. Assume that, when

qhj = qh|Ωj , qh ∈ Vh, its normal component qhj · nj on Γjk is a polynomial of some fixed degree

σ = degree(qhj · nj), where for simplicity we shall assume σ to be independent of Γjk. Set

Λh = ` : `|Γjk ∈ Pσ(Γjk) = Λjk , Γjk 6= ∅;

5


note that there are two copies of Pσ assigned to the set Γjk: Λjk and Λkj .The continuous-time hybridized mixed finite element method is given by (dropping the superscript

h and) seeking pj , uj , `gjk ∈Wj ×Vj ×Λjk and sj , vj , `sjk ∈Wj ×Vj ×Λjk, j = 1, . . . ,M, k =1, . . . ,M, such that

(∇ · u, p)Ωj = (q, p)Ωj , (15)

(k−1λ−1u, u)Ωj − (p,∇ · u)Ωj +∑

k 6=j〈`gjk , u · nj〉Γjk = 0, (16)

and

(ψ∂s/∂τ, s)Ωj + (∇ · v, s)Ωj = (g, s)Ωj , (17)

(d−1v, v)Ωj + (s,∇ · v)Ωj −∑

k 6=j〈`sjk , v · nj〉Γjk = 0, (18)

for all u, v ∈ Vj , p, s ∈Wj .In order later to define an iterative method for solving the above problems, it will be convenient

[14, 15] to replace (13) and (14) by equivalent Robin transmission boundary conditions. Consider theLagrange multipliers to be `αjk (α = g, s), as seen from Ωj , and `αkj , as seen from Ωk. Consistencyconditions for the global pressure system will be given by

−χgjkuj · nj + `gjk = χgjkuk · nk + `gkj , x ∈ Γjk ⊂ ∂Ωj , (19)

−χgjkuk · nk + `gkj = χgjkuj · nj + `gjk , x ∈ Γkj ⊂ ∂Ωk, (20)

and for the water saturation system by

−χsjkvj · nj + `sjk = χsjkvk · nk + `skj , x ∈ Γjk ⊂ ∂Ωj , (21)

−χsjkvk · nk + `skj = χsjkvj · nj + `sjk, x ∈ Γkj ⊂ ∂Ωk, (22)

where χαjk , α = g, s, is a positive function on Γjk .

4 Reduction to the Lowest Index Raviart-Thomas Space over Rectangles

Before discussing the time-discretization and the domain decomposition iterative procedures, we shallmake the choice of the mixed finite element space Wh

j ×Vhj to be the lowest index Raviart-Thomas

space [41] over rectangles. (We are using a singly-indexed notation for the rectangles in the partitionof Ω.) The natural degrees of freedom on the element Ωj , where the elements are squares of sidelength h, in the case of the pressure equation (15)-(16) are the (constant) value pj of the pressureover Ωj , which can be interpreted as the value at the center of the element, and the four (constant)values ujβ , β = L,R,B, T , of the outward normal component of the total flux across the edges of theelement. The Lagrange multipliers `gjk and `gkj on Γjk are also constants.

The same spaces will be employed for the saturation equation, so that the parameters associatedwith Ωj will be the constants sj and vjβ , β = L,R,B, T , and the parameters on Γjk will be theconstants `sjk and `skj .

We shall assume the porosity, the absolute permeability and the external flow rate q to be constanton each element and the capillary pressure and relative permeability functions to be independent ofposition. Thus, the coefficients in (15)-(18) are constant with respect to the space variable on eachΩj .

Suppress for the moment the subscript j. Then, (15) can be written as

uL + uR + uB + uT = qh.

In (16), consider the inner product with the test function being a basis function for vectors and applya trapezoidal rule to the (k−1λ−1u, u)Ωj -term, since it will allow us to write the resulting algebraic

6


problem in a particularly simple form. Consider u, for example, to be the basis function (1 − x, 0)(on the standard reference square). Then, with the mobility function λ being defined through theLagrange multiplier for the saturation on the interface,

uL −2

hkλL(p− `gL) = 0.

For convenience later, collect the five equations for the global pressure:

uL + uR + uB + uT = qh, (23)

uβ −2

hkλβ(p− `gβ) = 0, β = L,R,B, T. (24)

All variables which appear in the above system refer to a single element. In order to define an iterativescheme later, apply (19) to (24) to express all Lagrange multipliers in terms of Lagrange multipliers andfluxes of the adjacent elements. Introduce the superscript ˜ to denote variables of adjacent elements,and let β′ denote the edge of the adjacent element corresponding to the edge β of the element underconsideration. Then, the equations (24) take the form

(1 + χgβξgβ)uβ − ξgβp = −χgβξgβuβ′ − ξgβ ˜gβ′ , β = L,R,B, T, (25)

whereξgβ = 2kλβ/h. (26)

Similar considerations lead to the following equations for the saturation:

hψ∂s

∂τ+ vL + vR + vT + vB = g(s)h, (27)

(1− χsβξsβ)vβ + ξsβs = χsβξsβ vβ′ + ξsβ ˜sβ′ , β = L,R,B, T, (28)

whereξsβ = 2dβ/h. (29)

This completes the derivation of the continuous-time finite element equations; let us turn to theirdiscretization in time.

5 Discretization in Time

As announced in the Introduction, we shall employ a somewhat complicated time-discretization pro-cedure based on operator splitting ideas. Assume that

∆tp = i1∆tsd, ∆tsd = i2∆tst,

where i1 and i2 are positive integers. Let tm = m∆tp and denote a function f evaluated at timetm by fm. Similarly, let tn = n∆tsd and tn,κ = tn + κ∆tst and let fn = f(tn) and fn,κ = f(tn,κ),respectively.

The global pressure will be approximated at times tm, m = 0, 1, . . .. The saturation will beapproximated at times tn, n = 1, 2, . . .; recall that it is specified at t = 0. However, there willbe values for the saturation computed at intermediate times tn,κ for tn < tn,κ ≤ tn+1 that take intoaccount the physical transport of the water but not the diffusive effects. The algorithm will be detailedin the following subsections.

7


5.1 The pressure equation

The initial pressure must be computed from the given initial saturation; the algorithm to be describedin §8 below will produce the approximate saturation sm at time tm before calling for computing pm.Thus, the equations to be solved for pm and um are

umL + umR + umB + umT = qmh, (30)

(1 + χgβξmgβ)umβ − ξmgβpm = −χgβξmgβ umβ′ − ξmgβ ˜m

gβ′ , β = L,R,B, T, (31)

where ξmgβ = 2kλ(`msβ)/h.

5.2 Transport

The saturation equation, written in the equivalent forms (4) and (10), seems to be in a classicalconvection-diffusion form, but this is not quite true, since the coefficient λ′ωu is not the volumetricflow rate of the water and the coefficient −d is not the diffusion coefficient for water. Each of thesecoefficients relates to both phases. The right-hand side of (4) also is influenced by both phases. Ourintention is to apply an operator-split version of the MMOC procedure, altered to take the remarksjust preceding into account and altered to conserve the mass (volume) of water in the medium. Letus begin by deriving a conservation identity associated with the transport part,

φ∂s

∂t+ λ′ωu · ∇s = ψ

∂s

∂τ,

of equations (4) and (10). Then, it follows from (3) and (5) that

∫

Ω

λ′ωu · ∇s dx =

∫

Ω

u · ∇λω(s) dx = −∫

Ω

λωq dx,

so that

∫ tn,κ

tn,κ−1

∫

Ω

(φ∂s

∂t+ λ′ωu · ∇s) dx = (32)

∫

Ω

φs(x, tn,κ) dx −∫

Ω

φs(x, tn,κ−1) dx−∫ tn,κ

tn,κ−1

∫

Ω

λωq dx dt.

In conformity with the notation above, we shall indicate the approximate saturation at time tn bysn and the approximate transport-stage saturation at time tn,κ = tn + κ∆tst, κ = 1, . . . , i2, by sn,κ.

Over the full diffusion time step ∆tsd, integration of the differential equation (4) for the saturationand taking into account the boundary conditions (5) and the relation (32) written over (tn, tn+1) leadsto the following equation for the conservation of mass:

∫

Ω

φsn+1 dx =

∫

Ω

φsn dx +

∫ tn+1

tn

∫

Ω

(q+ − λωq−) dx dt, (33)

where q+ = max(q, 0) and q− = max(−q, 0). Since we microstep over (tn, tn+1), the double integralin (33) will be interpreted as a discrete sum in the time direction. Moreover, in line with (32), weshall split the integrand in this term into the two parts

q+ − λωq− = λω(q+ − q−) + (1− λω)q+ = λωq + (1− λω)q+. (34)

Then, we shall ask that the net change in water volume over a transport-microstep (tn,κ−1, tn,κ) begiven by

∆tst

∫

Ω

λω(E(∆tst)sn,κ)q dx,

8


where E(∆t)f(t) = 2f(t − ∆t) − f(t − 2∆t); implicitly, we are assuming the function q(x) to beindependent of t over (tn, tn+1). The term (1 − λω)q+ in (34) will go to the diffusive stage of thecalculation of sn+1.

Before defining the transport stage of the saturation calculation, let us define another extrapolationoperator. Assume that tm ≤ t ≤ tm+1. Set

E1(m, t)f =

(1 +

t− tm∆tp

)fm − t− tm

∆tpfm−1, m ≥ 1;

if m = 0, set E1(0, t)f = f0.The transport stage of the MMOC procedure is given as follows. Let xj denote the center of the

element Ωj , and set

Uj,n,κ(x) = λ′ω (E(∆tst)sn,κ(x))E1(m, tn,κ)um(x)/φj .

Then, follow the tangent to the characteristic curve related to the directional derivative ∂/∂τ through(xj , tn,κ) back to its location at time tn,κ−1 and denote this point by

xj,n,κ = xj − δj,n,κ, δj,n,κ = ∆tstUj,n,κ(xj). (35)

Then, evaluate s(xj,n,κ, tn,κ−1) as the bilinear interpolant of sn,κ−1 on the square through centers ofelements that contains xj,n,κ; if xj,n,κ is close enough to the boundary of Ω that it fails to lie in sucha square, extend sn,κ−1 by reflection outside of Ω and use the same definition. (See [27, 23].) Now,compare

Qn,κ = h2∑

j

φjs(xj,n,κ, tn,κ−1)

with the theoretical value

Qn,κ−1 = h2∑

j

φj sj,n,κ−1 + ∆tsth2∑

j

λω(E(∆tst)sj,n,κ)qj,n,κ. (36)

In general, Qn,κ 6= Qn,κ−1 and mass has not been conserved; in the waterflooding problem, the

usual case is that Qn,κ < Qn,κ−1. We wish to introduce an altered version of the MMOC thatwill preserve mass; it will be called the Modified Method of Characteristics with Adjusted Advection(MMOCAA). Let γ be a positive constant, and let

εj,n,κ = γhUj,n,κ(xj)/maxi|Ui,n,κ(xi)|,

x+j,n,κ = xj + εj,n,κ, x−j,n,κ = xj − εj,n,κ,δ+j,n,κ = ∆tstUj,n,κ(x+

j,n,κ), δ−j,n,κ = ∆tstUj,n,κ(x−j,n,κ),

x+j,n,κ = x+

j,n,κ − δ+j,n,κ, x−j,n,κ = x−j,n,κ − δ−j,n,κ.

Determine the various s-values by bilinear interpolation as above and the various u-values consistentlywith the definition of the vector variable in the Raviart-Thomas space.

Consider the case Qn,κ 6= Qn,κ−1. Let

s#(xj,n,κ, tn,κ−1) =

max(s(x+j,n,κ, tn,κ−1), s(x−j,n,κ, tn,κ−1)), Qn,κ ≤ Qn,κ−1,

min(s(x+j,n,κ, tn,κ−1), s(x−j,n,κ, tn,κ−1)), Qn,κ > Qn,κ−1.

Then, form

Q#

n,κ = h2∑

j

φjs#(xj,n,κ, tn,κ−1).

9


While it can happen that Q#

n,κ = Qn,κ for the first few microsteps at the beginning of a problem,

thereafter it can be assumed that Q#

n,κ 6= Qn,κ. If Q#

n,κ = Qn,κ, then we are forced to accepts(xj,n,κ, tn,κ−1) as sj,n,κ, along with possibly a small mass balance error that totals, at worst, O(∆tp)over all such steps. Otherwise, determine θn,κ such that

Qn,κ−1 = θn,κQn,κ + (1− θn,κ)Q#

n,κ; (37)

then, setsj,n,κ = θn,κs(xj,n,κ, tn,κ−1) + (1− θn,κ)s#(xj,n,κ, tn,κ−1), κ = 1, . . . , i2, (38)

and setsj,n+1 = sj,n,i2 .

5.3 Diffusion

The diffusive stage of the time step for the saturation is based on the approximation

(ψ∂s

∂τ

)

j,n+1

≈ φsj,n+1 − sj,n+1

∆tsd.

This is one proper generalization of the simple case in which ∆tst = ∆tsd.In the calculation of the quantity Qn,κ−1 in (36), some of the information that was needed to

evaluate the term related to λω(s)q should be saved, since it enters the diffusive stage equation. Weneed to have available

Sj,n+1 = ∆tsth2i2∑

κ=1

(1− λω(E(∆tst)sj,n,κ))q+j,n,κ.

Then, the diffusive-stage equation (in a form equivalent to that in (27)-(29)) is given on the j th-elementby

hφjsj,n+1 − sj,n+1

∆tsd+ vj,n+1,L + vj,n+1,R + vj,n+1,T + vj,n+1,B =

1

∆tsdhSj,n+1, (39)

(1− χsβξsβ)vj,n+1,β + ξsβsj,n+1 = χsβξsβ vβ′ + ξsβ ˜sβ′ , β = L,R,B, T, (40)

where, as above, ξsβ = 2dβ/h; the superscript ˜ and the subscript β ′ have the same meaning as in §4and the time index n+ 1 is omitted on ξsβ and on the right-hand side of (40).

5.4 Remarks

If, for κ = 1, . . . , i2, θn,κ can be found to satisfy (37), then mass is preserved over the time interval(tn, tn+1), and, consequently, over the entire time after it is possible to find θn,κ. Elsewhere, theauthors will show that, if θn,κ is bounded for all n, κ (except for the few initial steps where θn,κcannot be evaluated), the MMOCAA is stable and convergent when applied to a standard convection-diffusion equation, with an error that is O(h+ ∆t), where ∆t is the diffusive step.

6 Elimination of the Lagrange Multipliers and Fluxes

6.1 The pressure equation

Consistency, as expressed by (13)-(14) or (19)-(22), requires that

uβ + uβ′ = 0, `gβ = ˜gβ′ , λβ = λβ′ . (41)

10


Thus, it follows from the two equations

uβ −2

hkλβ(p− `gβ) = 0, uβ′ −

2

hkβλβ′(pβ − ˜gβ′) = 0,

that

`gβ = ˜gβ′ =

kp+ kβ pβ

k + kβ. (42)

In the above, pβ and kβ indicate the values of p and k in the element across the β-interface, respectively.Then, if

keff,β = keff,β′ =2kkβ

k + kβ(43)

is the effective permeability between the elements,

uβ = keff,βλβ(p− pβ)/h, (44)

where λβ = λ(`sβ) if the relative permeability functions are independent of x. (Otherwise, k should bereplaced by kλ in (42) and (43) and the resulting effective (kλ)eff,β used in (44).) Thus, the pressureequation reduces to the cell-centered finite difference equation

∑

β

keff,βλβh

(p− pβ) = qh, (45)

which will be solved by a domain-decomposition iterative procedure.In the domain decomposition procedure, we apply the Robin transmission condition (19) on either

two or four of the interfaces of each element. Let us solve for uβ when this transmission condition is

used. First, substitute in (31) for umβ′ in terms of p and pβ using (44) and then for ˜mgβ′ using (42);in these substitutions, we will employ “old” values of p and pβ , denoted by p∗ and p∗β, respectively.Then, we solve for the new iterate uβ :

uβ =ξgβ

1 + χgβξgβ

(p− p∗ +

(kβ

k + kβ+χgβkeff,βλβ

h

)(p∗ − p∗β)

). (46)

If the domain decomposition is to be performed at the level of individual elements, then (46) shouldbe employed for all four uβ ’s and substituted into (30). Then, the following relation holds:

∑

β

ξgβ1 + χgβξgβ

(p− p∗ +

(kβ


h

)(p∗ − p∗β)

)= qh. (47)

A more efficient iteration can be based on decomposing the domain into strips. We shall describethe use of strips in the x-direction. Then, we use (44) for β = L and R and (46) for β = B and T ;this leads to a linear, tridiagonal system based on the equation

∑

β=L,R

keff,βλβh

(p− pβ) + (48)

∑

β=B,T

ξgβ1 + χgβξgβ

(p− p∗ +

(kβ


h

)(p∗ − p∗β)

)= qh

for each element along a constant y-position and interior to the domain. For an element next to theboundary, the boundary condition (5) calls for the replacement of uβ on any boundary face by zero;i.e., one or perhaps two of the terms in the sums will vanish. If the strip does not go from the leftboundary to the right (as might happen in a parallel implementation of the procedure), then theRobin condition (46) would replace the (44)-term at the left or right internal extremity of the strip.

11


6.2 The saturation equation

If the capillary pressure function pc(s) is independent of x, similar consistency considerations lead tothe relations

vβ + vβ′ = 0, `sβ = ˜sβ′ , (λλωλop

′c)β = (λλωλop

′c)∼β′ , (49)

vβ = − 2

hdβ(s− `sβ), vβ′ = − 2

hdβ′(sβ − ˜sβ′), (50)

`sβ = ˜sβ′ =

ks+ kβ sβ

k + kβ, deff,β = deff,β′ = keff,β(λλωλop

′c)(`sβ). (51)

It then follows thatvβ = deff,β(sβ − s)/h. (52)

Thus, the cell-centered equation for the saturation on the j th-element at time tn+1 corresponding to(45) is given by (with the time index n+ 1 on the capacity term being suppressed)

φsn+1 − sn+1

∆tsd+∑

β

deff,β(sβ − s)h2

=1

∆tsdh2Sn+1,

from (39) and (52). Equation (40) and the elimination of the saturation fluxes and the Lagrangemultipliers give

vβ = − ξsβ1− χsβξsβ

(s− s∗ +

(kβ

k + kβ− χsβdeff,β

h

)(s∗ − s∗β)

).

Since the fundamental solution of a parabolic equation decays rapidly away from its singularity, it willsuffice to consider just the element-by-element domain-decomposition iteration; thus, the iterationwill be based on

φsn+1 − sn+1

∆tsd− (53)

1

h

∑

β

ξsβ1− χsβξsβ

(s− s∗ +

(kβ

k + kβ− χsβdeff,β

h

)(s∗ − s∗β)

)=

1

∆tsdh2Sn+1.

6.3 Choices of χgβ and χsβ

The parameters χgβ and χsβ will lead to faster convergence of the iterations if they are properly chosen;in particular, it is advantageous to let them be defined on each interface in terms of local propertiesof the medium and saturation. Consider first the pressure iteration. Let cg be a (dimensionless)constant, and choose

χgβ =cgh

keff,βλβ. (54)

Then, it follows from the definition (26) of ξgβ and (54) that (46) takes the form

uβ =keff,βλβ

h

(p∗ − p∗ +

p− p∗cg + kβ/(k + kβ)

),

from which the equation (48) for the strip domain decomposition iteration for p reduces to

∑

β=L,R

keff,βλβh

(p− pβ) + (55)

∑

β=B,T

keff,βλβh

(p∗ − p∗β +

p− p∗cg + kβ/(k + kβ)

)= qh.

12


An analogous iterative relation can be written for the element-by-element domain procedure for p,but the strip procedure has proved to be more efficient computationally.

In the saturation iteration in the diffusive stage calculation, let us make the corresponding choicefor χsβ :

χsβ = − csh

deff,β,

where cs is again a (dimensionless) constant. Then,

vβ = −deff,βh

(s∗ − s∗β +

s− s∗cs + kβ/(k + kβ)

),

so that (53) reduces to

φsn+1 − sn+1

∆tsd−∑

β

deff,βh2

(s∗ − s∗β +

s− s∗cs + kβ/(k + kβ)

)=

1

∆tsdh2Sn+1. (56)

6.4 Remarks

In [21], convergence of domain decomposition iterations based on updating the Lagrange multipliers`gβ and `sβ in the mixed method formulations (30)-(31) and (39)-(40) for the global pressure p and thesaturation s, respectively, was proved when the parameters χgβ and χsβ were assumed to be constant;i.e., to be independent of the interfaces. However, if the definition of the term En is modified to be

En =∑

j

∑

k

| − χ1/2jk r

nj · nj + χ

−1/2jk µnjk |20,Γjk ,

the convergence proof (which is very similar to that given earlier by Despres [15, 14] for the Helmholtzproblem; see also [38]) in §4 of [21] goes through in the case of variable parameters.

The iterative procedures for pressure and saturation would reduce to point or line successiveoverrelaxation if the underlying difference equations relate to differential equations with constantcoefficients and the parameters χgβ and χsβ are taken to be constants. However, when the permeabilityis variable, permeability values both on the interior of elements and on their faces enter into thealgorithm, and the iterations are not equivalent to SOR.

7 Iterative Procedures

7.1 Pressure

The calculation of p0, the initial global pressure, can be made much more efficient by combining amultigrid iteration with the domain decomposition iteration to be defined below; see [23] for details.

We shall describe the iteration based on strip subdomains (“french fry” decomposition), with thestrips taken in the x-direction. If the strips end at the left and right boundaries of the domain, labelthe strips from the bottom to top alternately as “red” and “black”. If the strips do not so extend,label them as “red” and “black” in a (deformed) checkerboard manner.

The iteration for pm is as follows:

• Set pm,0 = E(∆tp)pm.

• For all red strips, compute pm,α+1 by solving the sets of tridiagonal equations generated by (48)or, better, (55); p∗ refers to pm,α on all subdomains.

• For all black strips, compute pm,α+1; now, p∗ means pm,α for p-values on black elements andpm,α+1 on red elements.

• Check for convergence.

13


7.2 Saturation

The iteration for the saturation at times tn, n > 0, is based on an element-by-element domaindecomposition and the equation (53). Here, use an ordinary checkerboard ordering to separate thesubdomains into “red” and “black” subsets. Then, (with n replacing n+ 1) iterate as follows:

• Set s0n = sn.

• For all red elements, compute sα+1n using (53) or, again better, (56); s∗ means sαn on all elements.

• For all black elements, compute sα+1n ; now, s∗ means sαn on black elements and sα+1

n on redelements.

• Check for convergence.

8 The Algorithm

A precise statement of the overall algorithm is as follows:

• Given the initial saturation s0 = s0 and the function q at t = 0, compute the initial globalpressure p0. Then, evaluate u0 by (44).

• For m ≥ 0 and n ≥ 0, compute sn,κ, κ = 1, . . . , i2.

• If κ = i2, set sn+1 = sn,i2 . Then, compute sn+1.

• If n+ 1 = 0 (mod i1), compute pm, where tm = tn+1.

• If tm < T , where T is the final time of the desired simulation, return to the second item; if not,stop.

9 Inclusion of Gravity

The inclusion of gravity introduces several complications into the differential system for the water-flooding problem, and they lead to differences in the numerical treatment, though the philosophy willremain the same. The differential equations (see [36]) will be given below. First,

−∇ ·(k(x)λ(s)[∇p − (λωρω + λoρo)g∇z]

)= q.

The total volumetric flow rate is now given by

u = −k(x)λ(s)[∇p− (λωρω + λoρo)g∇z].

Then, the saturation equation can be written as

φ∂s

∂t+ [λ′ωu+ k(λλωλo)

′(ρω − ρo)g∇z] · ∇s+∇ · (kλλωλo∇pc)+λλωλo(ρω − ρo)g∇k · ∇z = (1− λω)q+. (57)

We again assume the boundary of the reservoir to be impermeable and require that the normalcomponents of both the water flux and the oil flux across the boundary to vanish; thus, we ask that

u · n |∂Ω= 0,

and(∇pc + (ρω − ρo)g∇z) · n |∂Ω= 0. (58)

14


Compatibility to the incompressibility of the fluids still requires the relation (6) to hold. As before,the initial water saturation must be specified (see (7)). If we again set

v = d∇s = kλλωλop′c∇s = kλλωλo∇pc, (59)

then, the boundary condition (58) is equivalent to

v · n = d∂s

∂n= −kλλωλo(ρω − ρo)g

∂z

∂n. (60)

We can derive a conservation relation associated with the transport part of the saturation equationas follows. First, note that

∫

Ω

(u · ∇λω + k(ρω − ρo)g∇z · ∇(λλωλo)

)dx =

−∫

Ω

λωq dx−∫

Ω

λλωλo(ρω − ρo)g∇k · ∇z dx+

∫

∂Ω

kλλωλo(ρω − ρo)g∂z

∂ndσ.

Thus, the conservation relation corresponding to (32) is

∫ tn,κ

tn,κ−1

∫

Ω

(φ∂s

∂t+ [λ′ωu+ k(λλωλo)

′(ρω − ρo)g∇z] · ∇s)dx = (61)

∫

Ω

φs(x, tn,κ) dx −∫

Ω

φs(x, tn,κ−1) dx−∫ tn,κ

tn,κ−1

∫

Ω

(λωq + λλωλo(ρω − ρo)g∇k · ∇z

)dx dt+

∫ tn,κ

tn,κ−1

∫

∂Ω

kλλωλo(ρw − ρo)g∂z

∂ndσ dt.

9.1 Pressure equation

Let us consider the discrete equations arising from the application of the same Raviart-Thomas ele-ments as in the gravity-free case, with the earlier notation being retained. Then, if uβ denotes theflux across the β-interface of an element, the pressure equation is again

∑

β

uβ = qh.

The equation for uβ in terms of the global pressure p, the Lagrange multipliers `gβ and `sβ now reads(c.f. (24))

uβ =2

hkλβ

(p− `gβ +

h

2g(λωρω + λoρo)(`sβ)∇z · nβ

).

The same consistency relations (41) hold here, and it follows that

`gβ =kp+ kβ pβ

k + kβ+h

2gk − kβk + kβ

(λωρω + λoρo)(`sβ)∇z · nβ ,

uβ =keff,βλβ

h

(p− pβ + gh(λωρω + λoρo)(`sβ)∇z · nβ

).

Thus, the pressure equation becomes

∑

β−

keff,βλβh

(p− pβ + gh(λωρω + λoρo)(`sβ)∇z · nβ

)= qh,

15


where summing on β− means that interfaces contained in ∂Ω are omitted in the summation. Notethat each gravity term above appears with the opposite sign in the equation on the β ′-element; hence,the sum of all gravity terms over all elements vanishes, with the consequence that these terms, whichare functions of sm and thus known at the time of the pm-calculation, can be taken to the right-handside of the equation to have the effect of modifying the external flow without affecting the requirementthat it be zero-sum:

∑

β−

keff,βλβh

(p− pβ) = qh− g∑

β−

keff,βλβ(λωρω + λoρo)(`sβ)∇z · nβ .

Thus, with this modification of the right-hand side, we have reduced the pressure problem with gravityincluded to the same problem as in the gravity-free case, and exactly the same iterative procedurecan be used to evaluate pm.

9.2 Saturation equation

Now, let us turn to the saturation calculation. Redefine b (see (8)) to be

b(x, s, u) = λ′ω(s)u+ k(x)(λλωλo)′(s)(ρω − ρo)g∇z,

and rewrite the saturation equation in the form

ψ∂s

∂τ+∇ · (kλλωλo∇pc) + λλωλo(ρω − ρo)g∇k · ∇z = (1− λω)q+.

The transport stage of the approximation of s will again consist of the MMOC microstep followedby an analogous modification to obtain the MMOCAA, followed at time tn+1 by a diffusive step. Letus outline the procedure.

Before we define the transport and diffusive steps for the saturation calculation, let us reexaminethe consistency relations (49). Since the term involving ∇k · ∇z is zero-order, the relation vβ +vβ′ = 0 remains valid, along with the continuity of the saturation as expressed through the Lagrangemultipliers. Thus, (50)-(52) are also correct. We will need to evaluate the Lagrange multiplier `sβ onthe boundary. This can be done as follows. It is convenient to write vβ (on the boundary only) interms of the capillary pressure function (see (59)):

vβ = − 2

hk(λλωλo)(`sβ)(pc(s)− pc(`sβ)). (62)

Then, combining (62) and the boundary condition (60), we see that

pc(`sβ) = pc(s)−h

2g(ρω − ρo)

∂z

∂nβ,

and the boundary value for the saturation (Lagrange multiplier) can be found simply by inverting thecapillary pressure function:

`sβ = p−1c

(pc(s)−

h

2g(ρω − ρo)

∂z

∂nβ

)on ∂Ω. (63)

It should be noted that later, when we move boundary vβ terms to the right-hand side of the algebraicequations in the iteration procedure, these boundary vβ ’s should be evaluated using the formula (62)with the `’s being evaluated with `sβ from (63).

We need to discretize several expressions before we can define the transport and diffusive saturationsteps. First, consider the term containing ∇k ·∇z in the conservation equation (57). Since k is treatedas being constant on each element, the integrand is a distribution with support on the edges of theelements. We require some new notation to be able to describe the result simply. Let e denote

16


the set of edges of the partition into elements, and denote the (unambiguously determined) Lagrange

multiplier `sβ = ˜sβ′ on the edge e by `se. Then, use the approximation

∫ tn,κ

tn,κ−1

∫

Ω

λλωλo∇k · ∇z dx dt ≈ ∆tsth∑

e⊂Ω

(λλωλo)(E(∆tst)`se,n,κ)∇k(e) · ∇z,

where ∇k(e) is to be interpreted as the distributional gradient of k. Thus, we can redefine Qn,κ−1 tobe

Qn,κ−1 = h2∑

j

φj sj,n,κ−1 + ∆tsth2∑

j

λω(E(∆tst)sj,n,κ

)qj,n,κ + (64)

∆tsth(ρω − ρo)g∑

e

(λλωλo)(E(∆tst)`se,n,κ

)∇k(e) · ∇z −

∆tsth(ρw − ρo)g∑

e⊂∂Ω

k(e)(λλωλo)(E(∆tst)`se,n,κ

) ∂z∂n

;

`se is evaluated for boundary edges by (63) and on interior edges by (51). Finally, it is necessary totreat the discretization of the term ∇k · ∇z in the diffusive step of the saturation calculation. Thus,we want to evaluate (λλωλo∇k · ∇z, s) when s is the characteristic function of an element Ωj ; theproblem is that both k and s jump on the edges of Ωj . However, replacing the jumps by piecewiselinear changes over short intervals and taking the limit as the intervals go to zero indicates that theproper evaluation is given by

(λλωλo∇k · ∇z, s) =h

2

∑

β

(λλωλo)(E(∆tst)`sβ,n,κ

)(kβ − kβ)∇z · nβ

in the case that the domain lies in two-space. (The factor h should read hd−1 in the d-dimensionalcase.)

Now, to perform the transport microstep, redefine the δj,n,κ of (35) to be

δj,n,κ =∆tstφj

(λ′ω (E(∆tst)sn,κ(xj))E1(m, tn,κ)um(xj) + (65)

k(λλωλo)′ (E(∆tst)sn,κ(xj)) (ρω − ρo)g∇z

).

Then, xj,n,κ and sj,n,κ can be evaluated as before, with the exception that sn,κ−1 cannot be extendedoutside of Ω by simple reflection. Instead, the bilinear extrapolation should be defined by usingthe values of s at the centers of boundary elements and the boundary Lagrange multipliers justcomputed above. Then, the MMOC portion of the calculation is exactly as before, using the value ofδj,n,κ determined above in (65). Moreover, the alterations to preserve mass through the MMOCAAprocedure are unchanged, except that Qn,κ−1 is evaluated using (64).

The diffusive-stage equation (39) needs to be changed to

hφjsj,n+1 − sj,n+1

∆tsd+ vj,n+1,L + vj,n+1,R + vj,n+1,T + vj,n+1,B = (66)

1

∆tsdhSj,n+1 −

1

2(ρω − ρo)g

i2∑

k=1

∑

β

(λλωλo)(E(∆tst)`sβ,n,κ

)(kβ − kβ)∇z · nβ

to take gravity into account; here, Sj,n+1 has the same meaning as before:

Sj,n+1 = ∆tsth2i2∑

κ=1

(1− λω(E(∆tst)sj,n,κ)) q+j,n,κ.

17


The iterative procedure to find sn+1 is affected only by the movement of the gravity term in (66)to the right-hand side of the equation.

Thus, the modifications to the code for the discrete problem caused by taking gravity into accountare, for practical purposes, confined to a few lines required to evaluate the terms moved to the right-hand sides of the equations.

10 Numerical Experiments

We present the results of grid refinement studies for the numerical simulation of one- and two-dimensional fluid flows associated with four distinct flooding problems:

• One-dimensional flow in a 512 m long, homogeneous reservoir.

• Two-dimensional flow in a rectangular, heterogeneous reservoir (slab geometry) having 512 m× 128 m.

• Two-dimensional flow in a five spot pattern, homogeneous reservoir having 256 m × 256 m.

• Two-dimensional flow in a five spot pattern, heterogeneous reservoir having 256 m × 256 m.

In all simulations, the reservoir contains initially 79% oil and 21% water. Water is injected uni-formly into the reservoir at a constant rate of one pore-volume every five years. For the heterogeneousreservoir flow studies, we consider a scalar, heterogeneous absolute permeability field taken to be (thelog of) a realization of a (Gaussian) random fractal field (see [32, 33]) with moderately large hetero-geneity strength — standard deviation of permeability values equal to the mean permeability value.This fractal field is displayed in Figure ??. The data below are held fixed in all flow studies.

Viscosity µω = .5 cP µo = 10 cPPorosity φ = .2Residual saturations sro = .15 srω = .2Relative permeability kro(s) = 1− (1− sro)−1s2functions krω(s) = (1− srω)−2(s− srω)2

Capillary pressure pc(s) = η((s− srω)−2 − ζ(1− s)−2)function s0 = 1− sro ζ = s2

ro(s0 − srω)−2

η = 3000 dynes / cm 2

Each set of simulations is performed using both the usual MMOC and the new MMOCAA timestepping procedures. The saturation level curves displayed refer to saturation values from 0.24 to0.82, with an increment of 0.04 in the saturation values between two consecutive level curves.

We discuss first the set of simulations for the one-dimensional, homogeneous reservoir flow problem.Figure ?? shows the effects of grid refinement in the numerical solution of this problem when theMMOCAA is used. The computed saturation profiles after 100, 200, and 300 days of simulation ongrids having 64, 128, 256, and 512 elements are shown.

Analogous results are shown in Figure ?? for the usual MMOC. For comparison, we have alsoplotted in this figure the results obtained with the MMOCAA on a grid with 512 elements. Note asystematic delay in the sharp front position computed with the MMOC, when compared to the frontposition computed with the MMOCAA. Such a delay, or inaccuracy in predicting the sharp frontposition, is a consequence of the inability of the MMOC to conserve mass, since the MMOCAA andthe MMOC solutions essentially coincide in the region away from the saturation front.

Mass balance results for the simulations reported in Figures ?? and ?? appear in Figure ??. Inthis figure we plot the relative error for the mass balance, measured relatively to the total amountof injected water, as a function of time. Note that the MMOCAA conserves mass exactly — therelative error is identically zero. In contrast, the relative error curves for the MMOC display someoscillation in the beginning of the simulation, and then converge to a value close to 5% after 300 daysof simulation.

18


In Figure ?? we show the relative error curves for the MMOC on two finer grids, having 1024and 2048 elements. Under this further grid refinement the mass balances produced by the MMOCimproved slightly, with the relative errors going down to 3% after 300 days of simulation.

We discuss now the set of simulations performed for the two-dimensional flow in a rectangular,heterogeneous reservoir. Figures ?? and ?? display the results of grid refinement studies using theMMOC and the MMOCAA, respectively. The grids used are finer in going from top to bottom, andhave 256 × 64 (top), 512 × 128 (middle) and 1024 × 256 (bottom) elements. Saturation level curvesare shown after 200 days of simulation for the three levels of spatial discretization. For each level ofdiscretization the fingers computed by the MMOCAA have advanced further into the reservoir thanthose computed by the MMOC, reflecting the conservation of mass by the MMOCAA in contrastto the steady loss of water mass by the MMOC.

Mass balance results are shown in Figure ??. As expected, the mass balance produced by theMMOC improves as the computational grid is refined, but even for the finest simulation grid usedthe relative error is not less than 8%. In contrast, as for the one-dimensional flow problem, theMMOCAA solution conserves mass exactly.

The overall delay of the sharp front of a simulation performed with the usual MMOC is furtherillustrated in Figure ?? where surface plots of the water saturation are shown as for a computationalgrid having 512 × 128 elements.

We discuss next the set of simulations performed for the two-dimensional flow problem in a fivespot pattern, homogeneous reservoir. Figure ?? shows the saturation level curves after 230 days ofsimulation obtained with both the MMOC and the MMOCAA for three levels of spatial discretiza-tion. The plots on the left column are the results with the usual MMOC and the ones on the rightthe results with the MMOCAA. The grids are finer in going from top to bottom and have 64 × 64(top), 128 × 128 (middle), and 256 × 256 (bottom) elements. Note, for each row (each grid), theconsiderable delay of the tips of the fingers in the MMOC solution (due to mass balance errors). Notealso that the breakthrough time predicted with the MMOC on a grid having 256 × 256 elementsis comparable to the prediction made by the MMOCAA on a computational grid having 64 × 64elements. A quantitative comparison between the two schemes is given in Figure ?? where the relativeerrors for the mass balances are displayed against the time. Note again that although the mass balancein the MMOC solution improves under grid refinement, even for the finest grid we considered therelative error is not less than 4%.

Finally, we discuss the simulations for the two-dimensional flow problem in a five-spot pattern,heterogeneous reservoir. The same remarks made above for Figures ?? and ?? hold for Figures ??and ??, respectively. It is important to note that, as expected, given a computational grid the relativemass balance errors produced by the MMOC procedure are higher in the heterogeneous simulations.This indicates clearly the importance of using the MMOCAA time stepping procedure in simulatingflows in heterogeneous formations. Figure ?? shows surface plots of saturation values for the finest(256 × 256) computational grid used in our simulations. It illustrates further the overall delay of thetip of the fingers produced by the usual MMOC.

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