Available online at www.sciencedirect.com
Advances in Water Resources 31 (2008) 56–73
www.elsevier.com/locate/advwatres
Numerical modeling of two-phase flow in heterogeneous
permeable media with different capillarity pressures
Hussein Hoteit
a
a,1
, Abbas Firoozabadi
a,b,*
Reservoir Engineering Research Institute, Palo Alto, CA, USA
b
Yale University, New Haven, CT, USA
Received 8 February 2007; received in revised form 15 June 2007; accepted 21 June 2007
Available online 5 July 2007
Abstract
Contrast in capillary pressure of heterogeneous permeable media can have a significant effect on the flow path in two-phase immiscible
flow. Very little work has appeared on the subject of capillary heterogeneity despite the fact that in certain cases it may be as important as
permeability heterogeneity. The discontinuity in saturation as a result of capillary continuity, and in some cases capillary discontinuity
may arise from contrast in capillary pressure functions in heterogeneous permeable media leading to complications in numerical modeling. There are also other challenges for accurate numerical modeling due to distorted unstructured grids because of the grid orientation
and numerical dispersion effects. Limited attempts have been made in the literature to assess the accuracy of fluid flow modeling in heterogeneous permeable media with capillarity heterogeneity. The basic mixed finite element (MFE) framework is a superior method for
accurate flux calculation in heterogeneous media in comparison to the conventional finite difference and finite volume approaches. However, a deficiency in the MFE from the direct use of fractional flow formulation has been recognized lately in application to flow in permeable media with capillary heterogeneity. In this work, we propose a new consistent formulation in 3D in which the total velocity is
expressed in terms of the wetting-phase potential gradient and the capillary potential gradient. In our formulation, the coefficient of the
wetting potential gradient is in terms of the total mobility which is smoother than the wetting mobility. We combine the MFE and discontinuous Galerkin (DG) methods to solve the pressure equation and the saturation equation, respectively. Our numerical model is
verified with 1D analytical solutions in homogeneous and heterogeneous media. We also present 2D examples to demonstrate the significance of capillary heterogeneity in flow, and a 3D example to demonstrate the negligible effect of distorted meshes on the numerical
solution in our proposed algorithm.
2007 Elsevier Ltd. All rights reserved.
Keywords: Two-phase flow; Water injection; Heterogeneous media; Capillary pressure; Mixed finite element method; Discontinuous Galerkin; Slope
limiter
1. Introduction
Subsurface fluid flow problems are of importance in
many disciplines including hydrology and petroleum reservoir engineering. In such flows, the process of displacement
is mainly affected by the properties of the permeable medium, and fluids in single-phase both in homogeneous and in
*
Corresponding author. Address: Yale University, New Haven, CT,
USA.
E-mail address: [email protected] (A. Firoozabadi).
1
Present address: ConocoPhillips Co., USA.
0309-1708/$ - see front matter 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.advwatres.2007.06.006
heterogeneous media. In two-phase immiscible flows, the
interactions between the permeable medium and the fluids
also affect the path of flowing fluids. Such interactions are
defined by relative permeability and capillary pressure.
While in a homogeneous medium, the effect of capillary
pressure can be neglected, this may not be the case in heterogeneous permeable media. The contrast in capillary
pressure functions of different media may become a key
factor in fluid flow. In single-phase flow and in two-phase
flow with negligible capillary pressure, fluids generally
flow through regions with high effective permeabilities. In
heterogeneous media with capillary pressure heterogeneity,
H. Hoteit, A. Firoozabadi / Advances in Water Resources 31 (2008) 56–73
capillary pressure forces may completely change the path
of the flowing fluids. In heterogeneous petroleum reservoirs, depending on the recovery process, capillarity may
result in improved recovery or poor recovery performance.
In the imbibition process, capillarity may significantly
enhance the cross-flow in stratified reservoirs and therefore
improve the recovery efficiency of oil from tight permeable
media [1]. In the gravity drainage process, capillary forces
have an opposite effect resulting in low recovery efficiency
[2,3]. A large body of literature is devoted to the modeling
of two-phase flow in heterogeneous media without capillarity [4–9]. However, despite the fact that heterogeneity in
capillary pressure may have a significant effect on flow,
only a handful of authors have studied the problem. Yortsos and Chang [10] studied analytically the capillary effect
in steady-state flow in 1D heterogeneous media. They
assumed a sharp but continuous transition of permeability
to connect different permeable media of constant permeabilities. Van Duijn and De Neef [11] provided a semi-analytical solution for time-dependent countercurrent flow in
1D heterogeneous media with one discontinuity in the permeability, and studied the effect of the threshold capillary
pressure on oil trapping. They introduced an interface condition to account for the entry pressure and saturation discontinuity at the interface of different rock types. Niessnera
et al. [12] and Reichenberger et al. [13] described the van
Duijn/De Neef interface condition in the context of a control-volume finite element method (box-scheme). Beveridge
et al. [14] examined the sensitivity of the production rate to
the capillary forces. Correa and Firoozabadi [15] studied
the effect of capillary heterogeneity on two-phase gas-oil
drainage in 1D vertical flow. The authors demonstrated
that due to capillary pressure contrast, the gravity drainage
may become unstable. Without capillary pressure contrast,
the drainage is always stable.
A large number of numerical methods have been developed to model two-phase flow in heterogeneous media. In
the commercial simulators for multiphase flow, the finite
difference (FD) and finite volume (FV) methods are the
general framework for numerical simulation for the study
of fluid flow in very large problems [16–18]. The conventional FD method, however, is strongly influenced by the
mesh quality and orientation, which makes the method
unattractive for unstructured gridding [19,20]. Recently,
there have been attempts to improve the accuracy of the
FD and cell-centered FV methods on unstructured gridding by using multi-point flux approximation techniques
[21–25]. However, such techniques have not been demonstrated to be of value for heterogeneous media with contrast in capillary pressure. The vertex-centered FV may
be superior to the cell-centered FV method [26–28]. However, the former requires additional treatments to model
the flow in control volumes containing different permeable
media [29]. On the other hand, methods based on Galerkin
finite elements may not be robust in heterogeneous media.
Helmig and Huber [30] compared three Galerkin-type discretization methods. They showed that the standard and
57
the Petrov–Galerkin methods may produce unphysical
solutions in heterogeneous media and severe mesh restrictions are required for the stability of the fully upwind
Galerkin method in multidimensional space.
Accurate approximation of the flow-lines and flux, and
low mesh dependency are desirable features for successful
numerical schemes for modeling two-phase flow in heterogeneous media. Such an accurate approximation can be
achieved by the mixed finite element (MFE) method [31–
33], which is superior to the control volume FE and Galerkin FE methods in approximating the velocity field in heterogeneous media [34,35]. Various authors have explored
the MFE method for single-phase flow problems [36–40].
Chavent et al. [41–43] extended the MFE method to twophase flow with a single capillary pressure function by
using the fractional flow formulation, where the governing
equations are written in terms of the total velocity and the
global pressure. The global pressure is defined in terms of
the arithmetic pressure of the wetting and non-wetting
phases and an integral with the integrand containing fractional flow function and the derivative of the capillary pressure with respect to the wetting phase saturation. The
gradient of the global pressure was then evaluated to be
a function of the fractional flow of the wetting phase,
and the gradient of the capillary pressure. Such a formulation based on the fractional flow formulation is sound as
long as a single capillary pressure function is used for the
entire permeable domain [42–46]. Since the MFE method
requires the primary variable and its derivative (flux) to
be continuous at the gridblock interfaces, the fractural flow
formulation with the global pressure as primary variable
becomes deficient for different capillary pressure functions
because of the discontinuity in global pressure at the interface [47] from saturation discontinuity. Nayagum et al. [48]
used the average pressure instead of the global pressure to
solve the inconsistency problem. Their method is demonstrated only in 1D space. Other formulations have also
been used with the MFE method [49,50], but none of the
proposed MFE formulations has been demonstrated for
two-phase flow in multidimensional, heterogeneous media
with different capillary pressure functions. In this work,
we provide a MFE formulation in 2D and 3D heterogeneous media that overcomes the drawbacks of the fractional flow formulation. Instead of the global pressure
variable that can be discontinuous in heterogeneous media,
we use the wetting-phase pressure as a primary variable
which is always continuous as long as none of the phases
is immobile. Our proposed formulation can correctly
describe discontinuities in saturation due to different capillary pressure functions as well as discontinuities in capillary
pressure at the interface of regions with threshold capillary
pressures (that is, entry pressures).
In the conventional MFE method for elliptic and parabolic equations, one calculates simultaneously the cell-pressures and the fluxes across the numerical block interfaces.
The resulting linear system is relatively large and indefinite.
In this work, we use the hybridized MFE method that
58
H. Hoteit, A. Firoozabadi / Advances in Water Resources 31 (2008) 56–73
provides a symmetric, positive definite linear system with
the face-pressures (traces of the pressure) as primary
unknowns. The hybridized MFE is algebraically equivalent
to the conventional MFE but it is more efficient [33,37,51].
In addition to the flux calculation accuracy, another
concern is the approximation of the mobilities at the interface of the numerical blocks. First-order methods, such as
the conventional FD and FV methods, with a single-point
upstream weighting technique, may produce significant
numerical dispersion in the vicinity of sharp fronts in the
saturation [52,53]. Higher-order FD methods such as
the two-point upstream weighting, total variation diminishing (TVD), and the essentially non-oscillatory (ENO)
approaches [54–60] reduce the numerical dispersion. In
structured meshes in multidimensional space, these methods can be implemented by using a directional splitting
technique so that the 1D formulation can be applied in
each directional space [61]. Higher-order FV methods have
been applied for different physical problems, such as conservative laws, advection and Navier–Stokes equations,
on unstructured meshes in 2D and 3D [62–67]. There are,
however, few attempts in the literature to implement
higher-order FV methods for two-phase flow in heterogeneous media on unstructured meshes in multidimensional
space.
For our problem, the discontinuous Galerkin (DG)
method is attractive because of its flexibility in describing
unstructured domains by using higher-order approximation functions. Its implementation is in line with saturation
discontinuity. It also conserves mass locally at the element
level.
The DG method was first implemented for nonlinear
scalar conservative laws by Chavent and Salzano [68],
who found that a very restrictive time step is required to
keep the scheme stable. Chavent and Cockburn [69]
improved the stability of the method by using a slope limiter following the work of van Leer [70]. Cockburn and Shu
introduced the Runge–Kutta Discontinuous Galerkin
(RKDG) method [71,72], which is an extension of the
DG method with higher-order temporal schemes. The
DG method has also been implemented for elliptic and parabolic equations [73–76] and for incompressible two-phase
flow in porous media [77–80], where the method is used to
approximate both the pressure and saturation equations.
In this work, however, we use the DG method with a second-order Runge–Kutta temporal scheme to approximate
only the saturation equation. The pressure equation is
approximated by the MFE method, as previously mentioned. A multidimensional slope limiter [42] is used to prevent the DG scheme from developing spurious oscillations
in the saturation.
In the past, the MFE and DG methods have been used
for different applications in single-phase [53,81–83] and
two-phase flow with a single capillary pressure function
in heterogeneous media [43]. In this work, we advance
the applicability of the combined MFE-DG method in heterogeneous media on unstructured gridding, where we
show how to account for the discontinuity in saturation
from different capillary pressure functions. The central
theme of this work is the displacement of the non-wetting
phase by the wetting phase. In the displacement of a wetting phase by the a non-wetting phase, the issue of the
threshold capillary (entry) pressure rises. For the sake of
completeness, we include the numerical modeling of the
threshold capillary pressure in 1D in this work. In a future
work, we will present examples of its significance in multidimensional flow.
This paper is organized as follows. First, we review the
governing equations of two-phase, incompressible fluid
flow in porous media. We then propose a new formulation
for the MFE method and present the approximations of
the velocity and volumetric balance equations. The DG
method is then used for the saturation equation, where
we approximate the saturation by using first-order polynomials on general elements. We also provide several numerical examples in one and multidimensional space in
heterogeneous media.
2. Mathematical formulation
The flow of two incompressible and immiscible fluids in
porous media is described by the saturation equation and
the Darcy law of the wetting and the non-wetting fluid
phases. The saturation equation of phase a is given by:
/
oS a
þ r ðva Þ ¼ F a ;
ot
a ¼ n; w;
ð1Þ
where / is the porosity of the medium, the subscripts n and
w denote the non-wetting and wetting phases, respectively,
Sa, F a , and va are the saturation, the external volumetric
flow rate, and the volumetric velocity of phase a,
respectively.
The velocity va is described by the Darcy law as follows:
va ¼ k ra
Kðrpa þ qa grzÞ;
la
a ¼ n; w:
ð2Þ
In Eq. (2), K is the absolute permeability tensor, pa, qa, kra,
and la are the pressure, density, relative permeability, and
viscosity of phase a, respectively, g is the gravity acceleration, and z is the depth. The saturations of the phases are
constrained by:
S n þ S w ¼ 1;
ð3Þ
and the two pressures are related by the capillary pressure
(pc) function:
pc ðS w Þ ¼ pn pw :
ð4Þ
As previously mentioned, the fractional flow formulation
used by several authors [42–46] becomes inconsistent when
implemented in the MFE method because of the discontinuity of the global pressure in heterogeneous media. In this
section, we present a new formulation that avoids the
drawback of the fractional flow implementation. We define
the flow potential, Ua, of phase a as follows:
H. Hoteit, A. Firoozabadi / Advances in Water Resources 31 (2008) 56–73
Ua ¼ pa þ qa gz;
a ¼ n; w:
ð5Þ
Replacing Eq. (5) in Eq. (4), we get the following expression of the capillary potential, Uc [84]:
Uc ¼ Un Uw ¼ pc þ ðqn qw Þgz:
ð6Þ
The total velocity vt is then written in terms of two velocity
variables va and vc, as follows:
vt ¼ vn þ vw ¼ kt KrUw kn KrUc ¼ va þ vc :
|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}
va
ð7Þ
kw
ðkt KrUw Þ ¼ fw va :
kt
ð8Þ
The expression of the balance of both phases in terms of va
and vc can be obtained by adding the saturation equations
given in Eq. (1) and using Eqs. (3) and (7). The governing
equations are, therefore, the total volumetric balance equation and the saturation equation of the wetting phase expressed in terms of va and fw from Eq. (8), that is,
r ðva þ vc Þ ¼ F n þ F w ;
oS w
/
þ r ðfw va Þ ¼ F w :
ot
ð9Þ
ð10Þ
The system of Eqs. (9) and (10) are subject to appropriate
initial and boundary conditions to describe the initial saturations, boundary pressures, and external flow rates. Let
C = CD [ CN be the boundary of the computational domain X, where CD and CN are non-overlapping boundaries
corresponding to Dirichlet and Neumann boundary conditions. The volumetric balance equation (9) is subject to the
following boundary conditions:
pw ðor pn Þ ¼ pD on CD ;
ðva þ vc Þ n ¼ qN on CN ;
S w ðor S n Þ ¼ S N on CN ;
3.1. Discretization of the velocity equation
The hybridized MFE method is based on the Raviart–
Thomas space with different orders of approximations
[33]. In this work, we use the lowest order Raviart–Thomas
space (RT0), where the degrees of freedom are the cellpotential average, the face-potential average, and the fluxes
across the faces of each cell. The RT0 basis functions, wK,E,
are defined in Appendix A. The velocity variable va,K over a
mesh element K can be determined from the flux variables
qa,K,E across each element-face E (see Appendix A), that is,
X
qa;K;E wK;E :
ð13Þ
va;K ¼
E2oK
By inverting the permeability tensor KK, the velocity va defined in Eq. (7) can be written as:
e 1 va;K ¼ rUw ;
K
K
ð14Þ
e 1 ¼ 1=kt;K K 1 . Note that kt,K is strictly positive.
where K
K
K
The MFE variational formulation is obtained by multiplying Eq. (14) by the RT0 basis functions, wK,E, using Eq.
(A.4), and integrating by parts over K, that is,
Z Z Z
Z Z
Uw r wK;E Uw wK;E nK;E
Z ZK Z
Z Z oK
1
1
¼
Uw Uw
jKj
jEj
K
E
|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl
ffl} |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}
e 1 va;K ¼
wK;E K
K
K
Z Z Z
Uw;K
Kw;K;E
ð11Þ
¼ Uw;K Kw;K;E ; E 2 oK:
ð12Þ
In the above equation, Uw,K and Kw,K,E are the wettingphase potential averages on cell K and face E, respectively.
By replacing the expression of va,K given in Eq. (13) in the
left-hand side of Eq. (15), one obtains:
X
qa;K;E0 AK;E;E0 ¼ Uw;K Kw;K;E ; E 2 oK;
ð16Þ
and the saturation equation (10) is subject to:
S w ¼ S 0 in X;
of edges(faces) in each cell, and NE be the number of edges
(faces) in the mesh not belonging to CD.
We use an implicit-pressure-explicit-saturation (IMPES)
approach, where the pressure equation and the saturation
equation are solved sequentially by the MFE and the DG
methods, respectively. In the MFE formulation, the velocity equation (7) and the volumetric balance equation (9) are
discretized individually. The procedure is described in three
steps as follows.
vc
In the above equation, ka = kra/la is the mobility of phase
a. The velocity variable va has the same driving force as the
wetting-phase velocity but with a smoother mobility kt,
that is, kt = kn + kw, than the wetting phase mobility.
The wetting-phase velocity, in terms of va and the wetting-phase fractional function, fw = kw/kt, reads as:
vw ¼
59
where n denotes the outward unit normal, qN and pD are
the imposed volumetric injection rate and pressure at CN
and CD, respectively, S0 is the initial saturation, and SN
is the boundary saturation of the injected fluid at CN.
3. Approximation of the flux
We consider a spatial discretization of the domain X
consisting of triangles or quadrilaterals in 2D, and tetrahedrons, prisms, or hexahedrons in 3D. The mesh cells are
denoted by K, and the cell edges(faces) by E. Let NK be
the total number of cells in the mesh, Ne be the numbers
ð15Þ
E0 2oK
RRR
e 1 wK;E0 .
where AK;E;E0 ¼
w K
K
K K;E
With simple manipulations (see Appendix B), one gets
from Eq. (16) an explicit expression of the flux qa,K,E in
terms of the cell-average potential and all the face-average
potentials in K,
X
qa;K;E ¼ aK;E Uw;K bK;E;E0 Kw;K;E0 ; E 2 oK;
ð17Þ
E0 2oK
where aK,E and bK,E (defined in Appendix B) are constants,
independent of the potential and flux variables.
60
H. Hoteit, A. Firoozabadi / Advances in Water Resources 31 (2008) 56–73
Eq. (17) is a key in the hybridized MFE method that
provides two local expressions of the flux at the interface
between two adjacent mesh elements. The continuity of
the flux and potential at the inter-element boundaries are
imposed by:
qa;K;E þ qa;K 0 ;E ¼ 0;
0
E ¼ oK \ oK ;
ð18Þ
therefore, the Schur complement matrix is readily computed. The resulting linear system is SPD whose primary
unknowns are the face-potentials Kw on the grid edges, that
is,
ðM RT D1 RÞKw ¼ RT D1 F þ V
ð27Þ
The definitions of the matrices in the above equation are
provided in Appendix B.
The preconditioned conjugate gradient (PCG) solver with
the Eisenstat diagonal preconditioner [85] is found to be
efficient for Eq. (27). The Eisenstat preconditioner, which
is also known as Eisenstat’s trick, is based on the incomplete Cholesky factorization with neglected off-diagonal entries. The preconditioned system for each CG iteration is
computed almost with the computational cost per CG iteration as the unpreconditioned system as only one matrixvector product is added. The cell-potential Uw and the flux
qa are, then, locally computed from Eqs. (24) and (13),
respectively.
3.2. Discretization of the volumetric balance equation
3.3. Approximation of the capillary flux
The volumetric balance given in Eq. (9) is integrated
locally over K by using the divergence theorem, that is,
Z Z
Z Z Z
ðva;K noK þ vc;K noK Þ ¼
ðF n þ F w Þ;
ð21Þ
In this section, we present the calculation of the degenerate flux (from capillary pressure and gravity effects), qc,
which is appeared in FK in Eq. (23). We should emphasize
that in our terminology, the capillary flux includes both the
effect of capillary pressure and gravity. The cell capillary
potentials Uc are calculated using the cell saturations from
the previous time step and the capillary pressure function.
We note that our method is locally conservative since the
fluxes are always continuous at the gridblock boundary
even when explicit time scheme is used to calculate the capillary potential.
For known Uc, different techniques can be used to
calculate the flux. One may use two-point and multi-point
flux-approximation techniques on orthogonal and nonorthogonal grids, respectively. In this work, we calculate
the capillary flux by the MFE method. Unlike the
common multi-point flux-approximation (MPFA) methods
[23], which in 1D becomes two-point approximation, the
MFE method relates the flux across each cell face to the
potential variables in the entire domain. This makes it superior to the MPFA and other alternative methods for heterogeneous media with capillary pressure heterogeneities.
The velocity variable vc given in Eq. (7) is discretized by
the MFE method similarly to discretization of va in Eq.
(15). The capillary flux can then be expressed in terms of
the cell capillary potential Uc and the face capillary potential Kc as follows:
X
^K;E;E0 Kc;K;E0 ; E 2 oK;
b
ð28Þ
qc;K;E ¼ ^aK;E Uc;K Kw;K;E ¼ Kw;K 0 ;E ¼ Kw;E ;
0
E ¼ oK \ oK :
ð19Þ
From Eqs. (18) and (17), one can readily eliminate the flux
variables and construct an algebraic system with unknowns
as the cell-potential, Uw, and face-potentials, Kw. The system in matrix form is given by:
RT Uw þ MKw ¼ V :
oK
ð20Þ
K
Similarly to the velocity expression of va,K in Eq. (13), the
velocity vc,K can be expressed in terms of the flux variables,
qc,K,E, and the RT0 basis function as follows:
X
vc;K ¼
qc;K;E wE :
ð22Þ
E2oK
Replacing Eqs. (13) and (22) in Eq. (21) and using the
properties of the RT0 functions given in Eq. (A.4), one gets:
X
qa;K;E ¼ F K ;
ð23Þ
E2oK
RRR
P
where F K ¼ E2oK qc;K;E þ
ðF n þ F w Þ.
K
The calculation of FK will be discussed later. The
unknowns qa,K,E are then eliminated from Eq. (23) by using
the flux expression defined in Eq. (17).
X
aK Uw;K aK;E Kw;K;E ¼ F K ;
ð24Þ
E2oK
P
where aK ¼ E2oK aK;E . In matrix form, Eq. (24), over all
mesh elements reads as:
DUw RKw ¼ F ;
ð25Þ
where D ¼ ½aK N K ;N K is a diagonal matrix and F ¼ ½F K N K .
The system of Eqs. (20) and (25) can be written together
as:
F
D R
Uw
¼
:
ð26Þ
V
R M
Kw
The potential variables Uw and Kw can be calculated simultaneously by solving the linear system in Eq. (26), which is
symmetric-positive definite (SPD) [51]. However, this approach is not efficient because of relatively large number
of unknowns. The matrix D is invertible and diagonal,
E0 2oK
A detailed formulation is provided in Appendix C. To link
the elements together, we impose the continuity of the flux
and the capillary potential at the inter-element boundaries
as follows:
qc;K;E þ qc;K 0 ;E ¼ 0;
E ¼ K \ K 0;
Kc;K;E ¼ Kc;K 0 ;E ¼ Kc;E ;
ð29Þ
0
E ¼K \K :
ð30Þ
H. Hoteit, A. Firoozabadi / Advances in Water Resources 31 (2008) 56–73
There are cases where the capillary pressure can be discontinuous and consequently Eq. (30) does not hold. The capillary pressure discontinuity issue is discussed in section
Numerical examples. Similarly to Eq. (20), we get the following linear system whose primary unknowns are the face
capillary potential Kc:
b Kc ¼ Vb R
b T Uc :
M
ð31Þ
b , R,
b and the vector Vb have similar strucThe matrices M
b in Eq.
tures as those defined in Eq. (20). The matrix M
(31) is SPD. The linear system is efficiently solved by the
PCG iterative solver. For known Uc and Kc, the capillary
flux is locally calculated from Eq. (28).
4. Approximation of the saturation
The saturation equation given in Eq. (10) is discretized by the discontinuous Galerkin (DG) method. Features of this method include the local conservation of
mass at the element level and the flexibility for complex
geometries by using unstructured griddings with higherorder approximations. The DG method better approximates sharp fronts in saturation than the first-order
methods. It produces less numerical dispersion and is
free from spurious oscillation when a suitable slope limiter is used.
We consider the same partition of the domain as previously described in the MFE formulation. Let nv be the
number of vertices in each cell K. The DG method is
described in two steps as follows.
4.2. Temporal approximation
The formulation in Eq. (33) leads to a system of ordinary differential equations of order nv over each element K.
After inverting the local mass matrix, which corresponds to
the integrals on the left-hand side of Eq. (33), one gets a
system in the following compact form:
dS w;K
¼ AðfK ; foK Þ;
dt
e nþ1=2 for known
1. Compute an intermediate saturation S
w;K
n
S w;K ,
e nþ1=2 ¼ S n þ Dt AðfK ðS n Þ; foK ðS n ÞÞ;
S
w;K
w;K
w;K
w;oK
2
In this step, the face fractional flow functions are calculated locally in K.
e nþ1 for known S n and S
e nþ1=2 ,
2. Compute S
w;K
w;K
w;K
nþ1=2
nþ1=2
e nþ1 ¼ S n þ DtAðfK ð e
S w;K Þ; foK ð e
S w;oK ÞÞ:
S
w;K
w;K
3. Reconstruct the updated saturations by applying the
slope limiter operator, L,
nþ1
e nþ1 Þ
S w;K
¼ Lð S
w;K
The wetting-phase saturation Sw,K in each cell K is
sought in a discontinuous finite element space with firstorder approximation polynomials. Then, Sw,K is expressed
over K as follows:
S w;K ðx; tÞ ¼
S w;K;j ðtÞuK;j ðxÞ
ð32Þ
j¼1
where, uK,j is a first-order shape function, and Sw,K,j is the
saturation at node j.
In the DG formulation, we multiply Eq. (10) by the
shape functions and integrate by parts, that is,
Z Z Z
nv
X
dS w;K;j
j¼1
dt
f~ w;oK;j
Z Z
K
oK
/uK;i uK;j ¼
ð34Þ
where Sw,K is a vector of dimension nv containing the nodal
unknowns Sw,K,j, and A represents the components of the
right-hand side in Eq. (33) multiplied by the inverse of the
mass matrix.
An explicit second-order Runge–Kutta scheme [83] is
used to approximate the time operator in Eq. (34). A slope
limiter procedure is applied to stabilize the method. The
computation procedure is illustrated by the following steps:
4.1. Spatial approximation
nv
X
61
nv
X
j¼1
uK;i uK;j va nÞ þ
ðfw;K;j
Z Z Z
Z Z Z
uK;j va ruK;i
K
uK;i F w;K
K
ð33Þ
where fw,K,j is the wetting-phase fractional flow function
at node j and f^ w;oK;j is the upstream value of the fractional
function at j defined from the direction of the velocity
field va.
The details of the slope limiter are provided in Appendix
D.
5. Numerical examples
We first verify our numerical model with known analytical solutions in 1D space. In Example 1, we solve the
Buckely–Leverett problem [86] in a homogenous medium
with different fluid properties and zero capillary pressure.
In Example 2, we compare our numerical solutions to
semi-analytical solutions of Van Duijn and De Neef problem [11] in a heterogeneous medium with different capillary
pressure functions. The objective of the third example is to
show the effect of capillary pressure in heterogeneous medium. In the last example, we show the robustness of the
MFE-DG method in 3D space on meshes of low quality.
All computations are performed on an Intel/Centrino
1.83 GHz PC with 512 MB of RAM.
5.1. Example 1: Buckely–Leverett problem
We consider a 1D horizontal homogeneous domain of
length 300 m, initially saturated with oil (non-wetting
62
H. Hoteit, A. Firoozabadi / Advances in Water Resources 31 (2008) 56–73
phase). Water (wetting phase) is injected with a constant
flow rate at one end to displace oil to the other end. The
pressure is kept constant at the production end and the
capillary pressure is neglected. The relevant data for this
problem are provided in Table 1. We use the conventional
FD (first-order) method [16,87] and our MFE-DG method
to numerically solve this problem with different relative
permeabilities and water to oil viscosity ratios, and compare the results to the analytical solutions.
The first-order FD and FV methods are standard
options in all industrial reservoir simulators while the
higher-order FD and FV methods are not currently available due to the computational cost of the problem and
the multidimensional features for multiphase flow. There
is, however, an interest in using higher-order FD (or FV)
methods in streamline approaches, where the method is
applied in 1D space [88,89]. To the best of our knowledge,
there is no claim in the literature of developing an efficient
and robust higher-order FV method in heterogeneous multidimensional space. Moreover, none of the industrial reservoir simulators supports higher-order FV methods in
multidimensional space. These simulators may support
two-point upstream weighting techniques to reduce numerical dispersion.
In this example, we assume the same viscosity for oil
and water phases for one case and change the viscosity
ratio for other cases, and use linear and nonlinear relative permeability functions. The relative permeabilities
are given by:
k rw ¼ S me ;
m
k rn ¼ ð1 S e Þ ;
ð35Þ
where m = 1 for linear relative permeabilities, and Se is the
normalized saturation defined as:
Se ¼
S w S rw
:
1 S rw S rn
ð36Þ
Srw and Srn are the residual saturations for the wetting and
non-wetting phases, respectively. Other relevant data are
provided in Table 1.
In Fig. 1a, the analytical solution and the FD and MFEDG solutions at different times are plotted. In this simple
case of a 1D homogeneous medium, the MFE-DG method
Table 1
Relevant data for Example 1
Domain dimensions
Rock properties
300 m · 1 m · 1 m
/ = 0.2, k = 1 md
Fluid properties
lw (cP)/ln (cP) = 1/1, 2/1, 2/3
qw = qn = 1000 kg/m3
Relative permeabilities
Capillary pressure
Residual saturations
Injection rate
Mesh size
Linear, quadratic (Eq. (35))
Neglected
Srw = 0, Srn = 0.2
5 · 104 PV/day
80 cells
shows less numerical dispersion than the FD solution as
expected; both solutions are obtained on a uniform mesh
of 80 cells. Very fine gridding is needed by the FD method
to match the MFE-DG solution [40]. The superiority in our
method comes from using higher-order (linear) approximations with the DG method, while piecewise constant
approximations are used with the FD method.
There are some special conditions where the FD
method results in low numerical dispersion. If the displacing fluid is more viscous than the fluid being displaced and linear relative permeabilities are used, the
analytical solution has one shock similar to the previous
case. The numerical dispersion is, however, low, as
shown in Fig. 1b.
In Figs. 1c and d, we plot the FD and MFE-DG solutions and compare the results with the analytical solutions
for lw/ln = 2/3 in both cases. In Fig. 1d, we use quadratic
relative permeabilities given in Eq. (35) with m = 2. In all
cases (Fig. 1a–d), the MFE-DG method shows a good
approximation of the solution. The convergence behavior
of our method on different refinements and different conditions is shown on Fig. 2. Results show low numerical dispersion with the MFE. The FD method has much higher
numerical dispersions (results not shown), as expected.
The experimental order of convergence from the
L1-error (1-norm) of the MFE-DG method and the conventional first-order FD method applied to the Buckley–
Leverett problem in Figs. 2a and b are provided in Tables
2 and 3, respectively. The average order of convergence of
the MFE-DG method is 0.78 and 1.18 for the problems in
Figs. 2a and 2b, respectively. We believe that there are two
reasons why the MFE-DG method does not attain a second order of convergence; the slope limiter that reduces
the order of convergence near shocks, and the regularity
of the problems. The method shows higher order of convergence in the second problem (Fig. 3), which is more regular
(no discontinuity in the solution) than the problem in
Fig. 2a. There is, however, merit in using the MFE-DG
method when compared to the conventional FD method
whose average order of convergence is less than 0.55 for
the studied cases.
5.2. Example 2: Van Duijn–De Neef problem
This example considers two-phase flow in a 1D horizontal domain of length 200 m which is composed of two permeable media of equal lengths and different permeabilities.
Both ends of the domain are closed. The left-hand side
(Part 1) and the right-hand side (Part 2) are initially saturated with the wetting fluid and the non-wetting fluid,
respectively. Because of the contrast in capillary pressure
at the interface, a redistribution of the fluids occurs from
countercurrent displacement. The total velocity vt is equal
to zero. Van Duijn and De Neef [11] provide a method
to construct a similarity solution of this countercurrent
problem with one discontinuity in the permeability. They
used the Brooks–Corey model [90] and the Leverett
H. Hoteit, A. Firoozabadi / Advances in Water Resources 31 (2008) 56–73
63
Fig. 1. Solution of the Buckley–Leverett problem with different relative permeabilities and viscosity ratios by the MFE-DG and FD methods, NK = 80:
Example 1. (a) Linear relative permeabilities: lw/ln = 1. (b) Linear relative permeabilities: lw/ln = 2. (c) Linear relative permeabilities: lw/ln = 2/3.
(d). Quadratic relative permeabilities: lw/ln = 2/3.
J-function [91] to describe the relative permeabilities and
capillary pressures, that is,
pc ¼ pt S e1=2 ;
k rw ¼ S 4e ;
2
k rn ¼ ð1 S e Þ ð1 S 2e Þ;
ð37Þ
where pt is the threshold capillary pressure assumed to be
proportional to (//k)1/2, and k is the absolute permeability (scalar value). Other relevant data are provided in
Table 4.
Fig. 2. Solution of the Buckley–Leverett problem with various refinements in the MFE-DG method: Example 1. (a) Linear relative permeabilities: lw/
ln = 1. (b) Linear relative permeabilities: lw/ln = 2/3.
64
H. Hoteit, A. Firoozabadi / Advances in Water Resources 31 (2008) 56–73
Table 2
Convergence order of the MFE-DG method and the conventional FD
method applied to the Buckley–Leverett problem as given in Fig. 2a
(Example 1)
MFE-DG method
FD method
Gridblocks
L1-error
20
40
80
160
0.016
0.010
0.006
0.003
Order
L1-error
Order
0.69
0.74
0.92
0.040
0.030
0.022
0.015
0.41
0.41
0.55
Table 3
Convergence order of the MFE-DG method and the conventional FD
method applied to the Buckley–Leverett problem as given in Fig. 2b
(Example 1)
MFE-DG method
FD method
Gridblocks
L1-error
20
40
80
160
0.011
0.005
0.002
0.001
Order
L1-error
Order
1.18
1.27
1.10
0.035
0.025
0.017
0.011
0.48
0.55
0.62
The authors neglected the hysteresis in the capillary
pressures and relative permeabilities. They used the same
capillary pressure functions for the imbibition and drainage processes. However, the problem is of interest for the
purpose of verifying our numerical model. Let (kl, kr) and
(pt,l, pt,r) be the relative permeabilities and the threshold
pressures in Part 1 (left) and Part 2 (right), respectively.
In this example, we compare the MFE-DG solutions to
semi-analytical solution for different permeability distributions (Cases A–D), where we get a continuity in capillary pressure in Cases A–C and discontinuity in Case
D.
Case A: We consider the same properties for the two
media, that is, kl/kr = 1 and pt,l/pt,r = 1. The
problem becomes similar to the countercurrent
displacement in a homogeneous medium by
McWhorter and Sunada [92], and Kashchiev
and Firoozabadi [93]. The boundary conditions
are, however, different. The MFE-DG solution
and the semi-analytical solutions at different
times are plotted in Fig. 3a. In this case, the cap-
illary pressure and the saturation are both continuous in the domain. The MFE-DG solution
correctly matches the semi-analytical solution
on a uniform mesh of 100 cells (gridding is similar for Case A and the other three cases). The
domain is discretized in such a way that the
interface between the two media coincides with
the numerical cells.
Case B: We use the less permeable medium in Part 1. The
permeability and threshold
pffiffiffi pressure ratios are kl/
kr = 1/2 and pt;l =pt;r ¼ 2, respectively. Because
of the difference in capillary pressure functions,
there is a discontinuity in saturation at the interface between the two parts. Let I denote the interface between the two media located at the middle
of the domain (x = 100 m), and Sw,l and Sw,r be
the left- and right-hand sides of the wetting phase
saturation at I. The capillary pressure is always
continuous across the heterogeneity interface
except when one phase is immobile [3,11,30].
The continuity of capillary pressures at I is
expressed by:
pcI ðS w;l Þ ¼ pcI ðS w;r Þ:
ð38Þ
A discontinuity in the capillary pressure occurs
when for a given Sw,l, there is no feasible value
of Sw,r that satisfies the continuity condition in
Eq. (38). The upper capillary pressure function
in Fig. 4a corresponds to the fine medium (Part
1) and the lower function corresponds to the
coarse medium (Part 2). When flow occurs, the
wetting phase saturation Sw,l decreases and Sw,r
increases. Following the upper capillary pressure
curve, pc,l, in Fig. 4a in the decreasing direction of
Sw,l, there always exists a right-hand side saturation, Sw,r, that satisfies Eq. (38). The capillary
pressure is, therefore, continuous. Fig. 3b shows
good agreement between the MFE-DG solution
and the semi-analytical solution of the wettingphase saturation at different times.
Case C: In this case, unlike the previous one, we use a
more permeable medium in Part 1. The permeability and threshold capillarypffiffipressure
ratios
ffi
are kl/kr = 2 and pt;l =pt;r ¼ 1= 2, respectively.
Van Duijn et al. [3,11] showed that there is a
Table 4
Relevant data for Example 2
Domain dimensions
200 m · 1 m · 1 m
Rock properties
/ = 0.25, kl (d)/kr (d) = 85.8/85.8,
42.9/85.8, 85.8/42.9, 85.8/21.45
Fluid properties
Relative permeabilities
Capillary pressure
Residual saturations
Injection rate
Mesh size
lw = ln = 1 cP, qw = qn = 1000 kg/m3
Brooks–Corey model Eq. (37))
Laverett model (Eq. (37)), pt,l (bar)/pt,r (d)=0.1/0.1, 0.141/0.1, 0.1/0.141, 0.1/0.2
Srw = 0, Srn = 0
0
100 cells
H. Hoteit, A. Firoozabadi / Advances in Water Resources 31 (2008) 56–73
65
Fig. 3. Solution of the Van Duijn–De Neef problem at different times with
pressures by the MFE-DG method,
pffiffiffi different permeabilities andpcapillary
ffiffiffi
NK = 100: Example 2. (a) kl/kr = 1; pt,l/pt,r = 1. (b) k l =k r ¼ 1=2; pt;l =pt;r ¼ 2. (c) k l =k r ¼ 2; pt;l =pt;r ¼ 1= 2. (d) kl/kr = 4; pt,l/pt,r = 1/2.
critical saturation S w , with pc;l ðS w Þ ¼ pc;r ð1Þ, such
that Eq. (38) is satisfied if Sw,l (t > 0) is less than
S w (see Fig. 4b). The authors provided a method
to calculate Sw,l(t > 0), which depends on the
threshold capillary pressure ratios and the type
of the capillary pressure functions. In this case,
S w;l < S w ; as a result there is continuity in capillary pressure. Fig. 3c shows the saturation profiles
at different times versus the domain length. The
MFE-DG solution is in agreement with the
semi-analytical solutions without imposing any
additional conditions other than the continuity
of the flux and capillarity pressure given in Eqs.
(29) and (30).
Case D: We increase the permeability and capillary pressure contrast between the two media (kl/kr = 4
and pt,l/pt,r = 1/2). The left-hand saturation is
greater than S w ¼ 1=4 (see Fig. 4c); therefore,
Fig. 4. Capillary-pressurepfunctions
in the left-p(Part
1) and right-hand (Part 2) of the media vs. the wetting-phase saturation for different rock properties:
ffiffiffi
ffiffiffi
Example 2. (a) pt;l =pt;r ¼ 2. (b) pt;l =pt;r ¼ 1= 2. (c) pt,l/pt,r = 2.
66
H. Hoteit, A. Firoozabadi / Advances in Water Resources 31 (2008) 56–73
continuity in the capillary pressure cannot be
established. At time t > 0, we get a jump in the
right-hand saturation to one, that is Sw,r = 1
[3,11]. In this case, the continuity condition of
the capillary pressure in our numerical method
given in Eq. (30) is not valid. We relaxed the continuity constraint by allowing two different values
of the capillary pressure at I. The system can be
closed from the knowledge of the right-hand saturation, that is, pc,r = pc,r(1) = pt,r. The jumps in
the saturations at different times appear in
Fig. 3d. The MFE-DG solution can describe correctly the left-hand jump in saturation; our calculated results are in agreement with the semianalytical solutions.
5.3. Example 3: effect of capillarity on flow in heterogeneous
media
In this example, we show the significance of capillary
pressure contrast in heterogeneous media. We consider a
2D horizontal domain with two different configurations
for the permeability distribution. In the first configuration
(Example 3a), the domain is composed of layers of alternate permeabilities (1 md and 100 md), as shown in
Fig. 5a. Water (wetting phase) is uniformly injected across
the left-hand side of the layered domain, which is initially
saturated with oil (non-wetting phase). The production is
across the opposite right-hand side. The injection rate in
pore volume (PV) is 0.11 PV/year. The capillary pressuresaturation function is given by:
Fig. 5. Two-dimensional domains with heterogeneous permeabilities: Example 3. (a) Layered heterogeneities. (b) Random heterogeneities.
Table 5
Relevant data for Examples 3a and 3b
Example 3a
Example 3b
Domain dimensions
Rock properties
500 m · 270 m · 1 m
/ = 0.2, k = 1, 100 md
500 m · 270 m · 1 m
/ = 0.2, k = 0.1–100 md
Fluid properties
lw = 1 cP, ln = 0.45 cP,
qw = 1000 kg/m3, qn = 660kg/m3
As in Example 3a
Relative permeabilities
Capillary pressure
Residual saturations
Injection rate
Mesh size
Quadratic (Eq. (35))
Bc = 5, 50 bar (Eq. (39))
Srw = 0, Srn = 0
0.11 PV/year
4500 rectangles
As in Example 3a
Bc = 1–33 bar (Eq. (39))
As in Example 3a
0.06 PV/year
3158 triangles
270
Width (m)
180
150
120
90
60
30
100
200
300
Length (m)
400
500
Sw(fraction)
240
0.90
0.85
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
210
Width (m)
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
210
0
0
270
Sw(fraction)
240
180
150
120
90
60
30
0
0
100
200
300
Length (m)
400
500
Fig. 6. Wetting-phase saturation profiles at 0.5 PVI with zero and nonzero capillary pressure: Example 3a. (a) Zero capillary pressure. (b) Nonzero
capillary pressure.
H. Hoteit, A. Firoozabadi / Advances in Water Resources 31 (2008) 56–73
270
Width (m)
180
150
120
90
60
30
0
100
200
300
Length (m)
400
500
Sw (fraction)
240
0.90
0.85
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
210
Width (m)
0.90
0.85
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
210
0
270
Sw (fraction)
240
67
180
150
120
90
60
30
0
0
100
200
300
Length (m)
400
500
Fig. 7. Wetting-phase saturation profiles at 0.5 PVI with homogeneous and heterogeneous capillary pressure functions: Example 3b. (a) Homogeneous
capillary pressure. (b) Heterogenous capillary pressure.
pc ðS e Þ ¼ Bc log S e ;
ð39Þ
where the capillary
pressure parameter Bc is inversely propffiffiffi
portional to k .
The relative permeabilities are quadratic function of
water saturation. Other relevant parameters are provided
in Table 5. In Fig. 6, we compare the calculated wettingphase saturation with and without capillary pressure at
0.5 pore volume injection (PVI). In Fig. 6a, with zero capillary pressure, the injected water flows faster in the more
permeable layers, as expected. In Fig. 6b, where we take
the capillary pressure into account, the flow in the more
permeable layers slows down because of the cross-flow
between the layers owing to the contrast in capillary pressure. A two-phase flow occurs in the transverse directions
of the adjacent layers in a very narrow region similar to
the one observed in fractured media [94]. In this example,
the capillary pressure is continuous because the threshold
capillary pressure is zero, as described in Eq. (39).
In the second configuration, we use a random distribution of permeabilities in the domain, as shown in Fig. 5b.
Water is injected at one corner to displace oil to the opposite corner with a constant injection rate of 0.06 PV/year.
The capillary pressure and the relative permeability models
are the same as in Example 3a (Table 5). In Fig. 7a, we
show the water saturation profile at PV = 0.5 by considering a single capillary pressure function for the whole
domain. The capillary pressure parameter Bc is computed
using an arithmetic average permeability equal to 0.5 md.
The water saturation profile presented in Fig. 7b is calculated at PV = 0.5 by considering different capillary pressure
functions corresponding to different permeabilities. There
is a significant effect of the capillary pressure contrast that
leads to a less diffusive solution and, therefore, a more efficient recovery. Note that unlike in homogeneous media,
where capillary pressure results in a diffusive front, in heterogeneous media with capillary pressure contrast the front
may become less diffusive. We like to emphasize that when
a single capillary pressure function is assigned to a heterogeneous media, there is very little effect on flow on the type
of problems discussed in this work.
The computational time for all cases in this example are
given in Table 6. The increase in CPU time when the capillary pressure is taken into account is due to the increase in
nonlinearity of the problem that affects the size of time
step.
5.4. Example 4: effect of mesh on the MFE method
The purpose of this last example is to show the robustness of the MFE method on unstructured meshes with low
quality. We consider a 3D tilted domain with dimension
(200 m · 400 m · 200 m). The injection and production
wells are located at the coordinates (0, 0, 0), and (200 m,
400 m, 200 m), respectively. The injection rate is 0.0625
PV/day. The rock and fluid properties are provided in
Table 7. We consider different mesh types made of hexahedrons and prisms and examine the effect of mesh quality on
the recovery and the saturation distribution. In Figs. 8a
and 8c, we discretize the 3D domain into meshes made of
Table 6
Computational CPU time for the problem in Example 3 that correspond
to Figs. 5a and b and 6a and b
Time(s)
Fig. 5a
Fig. 5b
Fig. 6a
Fig. 6b
105
152
141
169
Table 7
Relevant data for Example 4
Domain dimensions
Rock properties
Fluid properties
Relative permeabilities
Capillary pressure
Residual saturations
Injection rate
Mesh size
200 m · 400 m · 200 m
/ = 0.2, k=1 md
As in Example 3a
Cubic (Eq. (35))
Neglected
Srw = 0.1, Srn = 0.1
0.0625 PV/day
4500 hexahedrons, 4760 prisms
68
H. Hoteit, A. Firoozabadi / Advances in Water Resources 31 (2008) 56–73
Z
Z
Y
X
X
300
300
200
400
100
Z (m)
Z (m)
200
300
0
0
100
Y
400
100
300
0
0
200
X 100
(m
)
)
(m
200
X 100
(m
)
200 0
100
Y
)
(m
200 0
Z
Z
Y
X
Y
X
300
300
200
200
400
100
300
0
0
200
X 100
(m
)
100
Y
)
(m
200 0
Z (m)
Z (m)
Y
400
100
300
0
0
200
X 100
(m
)
100
Y
)
(m
200 0
Fig. 8. Meshes with different elements for a 3D tilted domain: Example 4. (a) Uniform hexahedrons. (b) Non-uniform hexahedrons. (c) Uniform prisms.
(d) Non-uniform prisms.
4500 uniform hexahedrons (parallelepipeds) and 4760 uniform prisms (have parallel opposite faces). In Figs. 8b and
8d, the nodes in the hexahedron- and prism-meshes are
randomly perturbed while the cell faces are kept coplanar.
The water saturation contours on the four meshes at
PV = 0.5 are shown in Fig. 9. The MFE method shows
minor mesh dependency on the distorted meshes.
Fig. 10a presents oil recovery versus the pore volume injection from the four meshes. In a magnified plot in Fig. 10b,
there is a minor discrepancy in the recovery curves
obtained on different meshes.
The accuracy depends on the integration formula used
to approximate the mass (elementary) matrix. In this example, we approximate the integrations for the mass (elementary) matrix by using 6- and 8-point Gaussian methods for
uniform prisms and hexahedrals, and 15- and 27-point
methods for nonuniform prisms and hexahedrals, respectively. The CPU time for all cases in this example are given
in Table 8.
6. Conclusions
A consistent numerical model for the flow of two
incompressible and immiscible fluids in heterogeneous permeable media with distinct capillary pressures is presented. The MFE and DG methods are combined to
approximate the pressure and saturation equations. We
introduce a formulation for the MFE method that overcomes the deficiencies of the fractional flow formulation
in heterogeneous media. Our proposal can correctly
describe the discontinuity in the saturation from the difference in capillary pressure functions and the discontinuity
in capillary pressure from the threshold capillary pressure.
We present numerical examples to demonstrate the significance of capillary contrast in heterogeneous media. The
MFE method has also the advantage in modeling of
unstructured grids with low grid dependency. The numerical results show that the MFE-DG method has better
shock capturing features and less numerical dispersion
H. Hoteit, A. Firoozabadi / Advances in Water Resources 31 (2008) 56–73
Z
69
Z
Y
X
X
Sw
Sw
(fraction)
(fraction)
0.85
0.80
300
0.85
0.80
300
0.72
400
100
0.50
0.40
300
0.20
0
0
0.72
200
0.70
Z (m)
Z (m)
200
X 100
(m
)
Y
)
(m
0.70
400
100
0.50
0.40
300
0.20
0
0
200
100
200
X 100
(m
)
200 0
100
Y
)
(m
200 0
Z
Z
Y
X
Sw
Sw
(fraction)
(fraction)
0.85
0.80
0.85
0.80
300
0.70
400
100
0.50
0.40
300
0.72
Z (m)
0.72
200
200
400
100
300
0.20
0
0
X 100
(m
)
100
)
(m
0.50
0.40
200
X 100
(m
)
200 0
0.70
0.20
0
0
200
Y
Y
X
300
Z (m)
Y
100
Y
)
(m
200 0
Fig. 9. Wetting-phase saturation profiles at 0.5 PVI with different meshes: Example 4. (a) Uniform hexahedrons. (b) Non-uniform hexahedrons. (c)
Uniform prisms. (d) Non-uniform prisms.
Fig. 10. Recovery of the non-wetting phase vs. PV injection with different gridings in the MFE-DG method: Example 4.
Table 8
Computational CPU time for the problem in Example 4 that correspond
to Figs. 9a–d
Time (s)
Fig. 9a
9b
Fig. 9c
Fig. 9d
523
575
757
797
than the first-order FD method. In a forthcoming work,
we will demonstrate major advantages of the combined
MFE-DG method in fractured media for immiscible
two-phase flow.
70
H. Hoteit, A. Firoozabadi / Advances in Water Resources 31 (2008) 56–73
A velocity vector vK over K can be uniquely written in
terms of the fluxes qK,E across edge(face) E and the basis
functions given in Eqs. (A.1)–(A.3):
X
vK ¼
qK;E wE ;
ðA:5Þ
Appendix A. The Raviart–Thomas basis functions
The Raviart–Thomas (RT0) space defines the velocity
vector over each cell K in terms of the fluxes across the cell
faces E. The RT0 basis functions are available for all standard geometrical elements (see Fig. A.1). The basis functions for the hexahedral, prismatic, and tetrahedral
reference elements are given below in following equations,
respectively.
wEi ;i¼1;...;6
0 1
u
B C
: @ 0 A;
0
u1
1
B
@
0
C
A;
0
0 1
0
B C
@ 0 A;
0
wEi ;i¼1;...;5
0
0
0
where oK = {Ei; i = 1,. . .,Ne}.
Appendix B. Discretization of the velocity va
Eq. (15) can be written for all faces E in K in the matrix
form:
1
B
C
@ v 1 A;
0
0
AK Qa;K ¼ Uw;K e Kw;K ;
0
f1
1
0
1
u
u1
B
C B
C
: @ v 1 A; @ v A;
0
0
0
1
0
B
C
2@ 0 A :
f1
0
1
0
1
u
u1
B
C
B
C
wEi ;i¼1;...;4 : 2@ v 1 A; 2@ v A;
f
f
0
1
u
B
C
2 @ v A:
1
Qa;K ¼ Uw;K A1
K e AK KK :
ðA:1Þ
0 1
0
B C
2@ 0 A ;
0 1
u
B C
@ v A;
0
ðB:1Þ
whereAK = [AK,E,E 0 ]E,E 0 2oK; Qa,K = [qa,K,E]E2oK;
Kw,K =
[Kw,K,E]E2oK; e = [1]E2oK.
The matrix AK is symmetric and positive definite. By
inverting AK, Eq. (B.1) becomes:
1
B
C
@ 0 A:
f
0
0 1
0
B C
@ v A;
E2oK
ðB:2Þ
Eq. (17) can be obtained by expanding Eq. (B.1). The coefficients bK,E and aK,E in Eq. (17) and aK in Eq. (24) are defined by:
X
X
bK;E;E0 ¼ A1
aK;E ¼
bK;E;E0 ; aK ¼
aK;E
K;E;E0 ;
f
E0 2oK
In Eq. (20), R is an NK · NE rectangular matrix, M is an
NE · NE square matrix, and V is a vector of size NE that
describes the boundary conditions. The entities of R and
M are:
R ¼ ½RK;E N K ;N E ; RK;E ¼ aK;E E 2 oK;
X
M ¼ ½M E;E0 N E ;N E ; M E;E0 ¼
bK;E;E0 E 62 CD ;
ðB:3Þ
E;E0 3oK
X
V ¼ ½V E N E ; V E ¼
bK;E;E0 Kw;E0 :
ðA:2Þ
0 1
u
B C
2 @ v A;
f
E2oK
ðA:3Þ
E0 2oK\CD
f1
Appendix C. Discretization of the velocity vc
The basis functions are linearly independent and satisfy the
following properties:
r wE ¼
1
;
jKj
wE :nE0 ¼
1=jEj if E ¼ E0 ;
ðA:4Þ
if E 6¼ E0 ;
0
The velocity variables va and vc have similar forms (see
Eq. (7)). However, unlike the coefficient kt in va, the mobility coefficient kn in vc can be zero and so it cannot be
inverted as in Eq. (14). We multiply vc by the inverse matrix
of KK to obtain:
where |K| and |E| are the volume and the area of the cell K
and face E, respectively. In 2D, similar relations apply.
E5
w5
ζ
w3
E2
v
w2
w
E1 1
1
u
1
w4
1
w6
E6
0
E4
1
w1 E1
E2 w2
w
E4 4
E3
ðC:1Þ
ζ
ζ
1
K 1
K vc;K ¼ kn rUc :
E3
w3 E3
1
E2 w
2
v
w5
E5
w1 E1
u
0
1
1
w3
w4
v
E4
0
Fig. A.1. Raviart–Thomas basis functions on hexahedral, prismatic, and tetrahedral reference elements.
1
u
H. Hoteit, A. Firoozabadi / Advances in Water Resources 31 (2008) 56–73
Following a similar procedure used for Eqs. (15) and (16)
(see Appendix B), one gets:
!
X
X
1
1
^
b
b
0
qc;K;E ¼ kn;E
A
A
; E 2 oK;
K;E;E0 Uc;K K;E;E0 Kc;K;E
E0 2oK
E0 2oK
ðC:2Þ
RRR
1
b
w K wK;E0 .
where A K;E;E0 ¼
K K;E K
In the above equation, ^
kn;E is the non-wetting phase
mobility at the interface E of a cell K and the neighboring
cell K 0 (E = K \ K 0 ). The interface mobility is calculated
from data in the upstream cell, that is:
(
kn;K;E if qc;E P 0 ði:e:; effluxÞ;
^
kn;E ¼
ðC:3Þ
kn;K 0 ;E if qc;E < 0 ði:e:; influxÞ:
The flux qc,E in Eq. (C.3) is known from the previous time
step. Between homogeneous cells, the mean value weighting technique can also be used instead of the fully upstream
^K;E;E0 and ^aK;E in
technique in Eq. (C.3). The coefficients b
Eq. (28) are defined by:
X
^K;E;E0 ¼ kn;K;E A
b 1 0 ; ^
b 1 0 :
b
aK;E ¼ kn;K;E
A
K;E;E
K;E;E
E0 2oK
Appendix D. Slope limiter
We use the multidimensional slope limiter introduced by
Chavent and Jaffré [42]. It is formulated in such a way to
avoid local minima or maxima at the grid nodes. In each
cell K, the saturation variable at a vertex i should be within
the minimum and the maximum of the cell-average saturations of all neighboring elements. Let Ti be the set of all
cells having i as a vertex. We define the notation:
Z Z Z
1
S w;K ¼
S w;K ; fS w;K g; S w; min ¼ min S w; max
i
i
K2T i
jKj
K
¼ maxfS w;K g:
K2T i
e w;K Þ is the solution of the following leastThen, S w;K ¼ Lð S
squares problem:
8
e w;K k;
>
min kW S
>
>
W2Rnv
>
>
>
>
< with the linear constraints :
nv
P
ðD:1Þ
W ¼ n1v
W i ¼ S w;K ;
>
>
>
i¼1
>
>
>
>
: S w; min 6 W i 6 S w; max;i ; i ¼ 1; . . . ; nv :
i
In the minimization problem in Eq. (D.1), we seek the closest solution, Sw,K = W, to the initial distribution of saturae w;K , in K that keeps the same total material and is free
tion, S
from local minima and maxima at the nodes. The problem
can be solved efficiently by using an iterative procedure
[42,95] that requires at most 2nv iterations to converge.
We note that the Chavent-Jaffré slope limiter in [42] has
a tuning parameter a 2 [0,1] that controls the degree of
restriction of the slopes. The parameter a does not appear
in our definition in Eq. (D.1) as it is set to one.
71
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