A Comparison of Mixed Hybrid Finite Element and
Finite Volume Method for Elliptic Problems
In Porous Media Flow
A. Elakkad
(1)
(1)*
, A.Elkhalfi
(2)
,N. Guessous
(3)
Laboratoire Génie Mécanique - Faculté des
Sciences et Techniques
B.P. 2202 – Route d'Imouzzer – FES Maroc
(2)
finite element method; Two-phase
simulation.
Governing equations
The pressure equation is given by:
Laboratoire Génie Mécanique - Faculté des
Sciences et Techniques
B.P. 2202 – Route d'Imouzzer – FES Maroc
(3)
Ecole normale Supérieure de Fès Maroc
*
Corresponding author:
We consider the saturation equation in its
simplest form (neglecting gravity and
[email protected]
capillary forces):
ABSTRACT
This paper is concerned with numerical
methods for the modelling of flow and
transport
in
porous
media.
The
sw + so = 1
diffusion equation is discretized by the
mixed hybrid finite element method [2,
3]. The saturation equation is solved by
a finite volume method [4]. We start
For the phase saturations
The source term for the saturation equation
becomes:
with incompressible single-phase flow
and move step-by-step to the black-oil
model for compressible two phase
Where
flow.
Numerical results are presented to see
the performance of the method [1, 5],
and
seems
to
be
interesting
f (S ) =
S2
S 2 + (1 − S 2 )
λ = λ0 + λw
by
comparing them with other recent
results.
Key Words: Saturation equation;
S0r : Irreducible oil saturation.
finite volume method; mixed hybrid
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9ième Congrès de Mécanique, FS Semlalia, Marrakech
4
Fig.1.The left plot show pressure contours
for a homogeneous quarter five-spot, and
the right plot the velocities at cell centre.
The pressure and velocity field are
symmetric.
S wc : Connate water saturation.
φ : Rock porosity
µ0 and µ w : viscosities
Numerical simulations
Velocities
1
0.8
We want to consider real two dimensional
0.6
0.4
0.2
flow problem in porous media. We first
0
-0.2
-0.4
want to look at the so called the quarter-
-0.6
-0.8
-1
five spot problems [1, 5]. We assume that
the reservoir is initially filled with pure oil.
Fig.3. Streamlines for a heterogeneous
quarter five-spot.
To produce the oil in the upper-right
corner, we inject water in the lower left.
We assume unit porosity, unit viscosities
for both phases, and set S wc = S0r =0. The
2
reservoir is the unit square [0,1] , with an
injector at (0, 0), a producer at (1, 1), and
Saturation t=0.14
Saturation t=0.42
0.9
0.9
0.9
0.9
0.8
0.8
0.8
0.8
0.7
0.7
0.7
0.7
0.6
0.6
0.6
0.6
0.5
0.5
0.5
0.5
0.4
0.4
0.4
0.4
0.3
0.3
0.3
0.3
0.2
0.2
0.2
0.2
0.1
0.1
0.1
no-flow boundary conditions. The pressure
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.1
0.1
0.9
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
equation is discretized by the mixed hybrid
finite element method. The saturation
equation is solved by a finite volume
method.
Velocities
Pressure
20
Saturation t=0.56
Saturation t=0.70
0.9
0.9
0.9
0.9
0.8
0.8
0.8
0.8
0.7
0.7
0.7
0.7
0.6
0.6
0.6
0.6
0.5
0.5
0.5
0.5
0.4
0.4
0.4
0.4
0.3
0.3
0.3
0.3
0.2
0.2
0.2
0.2
0.1
0.1
0.1
-0.5
18
16
-1
14
-1.5
12
-2
0.1
10
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.1
0.1
0.9
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
-2.5
8
6
-3
4
-3.5
2
5
10
15
20
Pressure
log10(K)
20
20
18
1
18
-0.5
16
16
14
0.5
-1
14
decreasing pressure gradient, the pressure
decays from the injector in the lower-left to
the producer in the upper-right corner.
12
12
-1.5
10
0
10
8
8
-2
6
6
-0.5
-2.5
4
4
2
2
-1
2
4
6
8
10
12
14
16
18
20
-3
5
10
15
20
Fig.2.The left shows logarithm of the
permeability for heterogeneous quarter
five-spot and the right show pressure
contours for heterogeneous quarter fivespot. As particles flow in directions of
Fig.4. Saturation profiles for
homogeneous quarter-five spot.
The
water
monotonically
saturation
toward
is
the
the
increasing
injector,
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9ième Congrès de Mécanique, FS Semlalia, Marrakech
5
meaning that more oil is gradually
displaced as more water is injected.
[1] J. E. Aarnes, T. Gimse and K.-A.
Table 1. CPU times and Saturation error
for the MHFE with various coarse meshes.
8 ×8
CPU MHFE 2.3280s
ε (S )
4.1387e-007
References
Lie. An introduction to the numerics
of flow in porous media using Matlab.
In "Geometrical Modeling, Numerical
20 × 20
245.4210s
30 × 30
Simulation, and Optimization: Industrial
2.5301e+003
3.4244e-005
2.5835e-006
Mathematics at SINTEF",
ε ( S ) : Saturation error.
Eds., G. Hasle, K.-A. Lie, and E.
Quak, Springer Verlag, pp. 265-306,
Conclusion
We were interested in this work in the
numeric solution for two dimensional
partial differential equations modelling
(or arising from) a fluid flow and
transport phenomena. The diffusion
equation is discredited by the mixed
hybrid finite element method. The
saturation equation is solved by a finite
volume method.
The pressure and velocity field are
symmetric about both diagonals for the
homogeneous field.
The water saturation is increasing
monotonically toward the injector for
the homogeneous field, meaning that
more oil is gradually displaced as more
water is injected. Numerical results are
presented to see the performance of
the method , and seems to be
interesting by comparing them with
other recent results.
(2007).
[2]
G.Chavent
Mathematical
and
Models
J.
Jaffré,
and
Finite
Elements for Reservoir Simulation,
Elsevier
Science
Publishers
B.V,
Netherlands, 1986.
[3]
H.Hoteit,
Philippe
and
J.Erhel
.R.Mosé,
B.
ph.Ackerer.Numerical
reability for mixed method applied to
flow
problem
in
porous
media.
Computational Geosciences 6: 161194, 2002.
[4] Robert Eymard ., Gallouet T.,
Herbin R., Finite volume methods, in
Ciarlet P.G., Lions J.L. eds., Handbook
of Numerical Analysis, vol. 7, pp. 7131020,
Elsevier
Science
B.V.,
Amsterdam, 1997.
[5] D. Burkle and M. Ohlberger,.
Adaptive finite volume methods for
displacement
problems
in
porous
media, Comput. Vis. Sci. 5, No.2.}
2002.
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9ième Congrès de Mécanique, FS Semlalia, Marrakech
6