Abstract In this paper, we develop a simplified formulation of the hydrocarbon system used for the petroleum reservoirs simulation.This system is a simplified model which describes a two-phase flow (oil and gas) with a mass transfer in a porous medium, that leads to the fluid compressibility.this kind of flow is modelled by a system of parabolic degenerated non linear convection-diffusion equations.Under certrain hypothesis, such as validity of Darcy&apos;s law, incompressibility of the porous medium, compressibility of the fluids, mass transfer between the oil and the gas and negligible gravity, the global pressure is formulated.This formulation allows the establishment of theoretical results on the existence and uniquness of the solution. Mathematics Subject Classification: 76D07, 76M2

F Z Nouri

S Gasmi

CiteSeerX

Applied Mathematical Sciences, Vol. 7, 2013, no. 42, 2055 - 2064
HIKARI Ltd, www.m-hikari.com
A Study of a Bi-Phasic Flow Problem
in Porous Media
S. Gasmi and F. Z. Nouri
Mathematical Modeling and Numerical Simulation Laboratory
Faculty of Sciences, University Badji Mokhtar
BP 12, Annaba, Ageria
gasmi.souad@yahoo.fr, tassili.nan09@gmail.com
Copyright c© 2013 S. Gasmi and F. Z. Nouri. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use, distribu-
tion, and reproduction in any medium, provided the original work is properly cited.
Abstract
In this paper, we develop a simplified formulation of the hydrocarbon sys-
tem used for the petroleum reservoirs simulation.This system is a simplified
model which describes a two-phase flow (oil and gas) with a mass transfer in
a porous medium, that leads to the fluid compressibility.this kind of flow is
modelled by a system of parabolic degenerated non linear convection-diffusion
equations.Under certrain hypothesis, such as validity of Darcy’s law, incom-
pressibility of the porous medium, compressibility of the fluids, mass transfer
between the oil and the gas and negligible gravity, the global pressure is for-
mulated.This formulation allows the establishment of theoretical results on the
existence and uniquness of the solution.
Mathematics Subject Classification: 76D07, 76M25
Keywords: Compressible Fluids, Porous Medium, Multiphasic Flows
1 Introduction
The mathematical study of the multiphasic flows in porous medium has at-
tracted attention to many researchers in oil industry for exploitation of oil or
gas field, in water resource management and many environment problems.
The hydrocarbon system is a simplified model which describes a two-phase
2056 S. Gasmi and F. Z. Nouri
flow (oil and gas) with a mass transfer in a porous medium, leading to the
fluid compressibility.The global pressure formulation of Guy Chavent [3] will
be introduced, this kind of flow (hydrocarbon system) is modeled by a system
of parabolic degenerated non linear convection-diffusion equations. This for-
mulation allows the establishment of theoretical results of the existence and
the uniquness of the solution.Based on different studies, for example, in [1]
the authors studied the case of two phase incompressible fluids without mass
transfer,[3] was devoted to analyze the one phase flow and [8] the black-oil
model.Furthermore a study of different numerical schemes have been consid-
ered by many authors; among others we can refer the reader to [4] who pur-
posed a finite element scheme, [5] who used a finite difference method while
[6] a higher order Godunov scheme and the authors in [9] have proposed a fi-
nite volume method.All these schemes seem to suffer from a lack of numerical
stability.It will be fruitfull if one can investigate numerical algorithms based
on stable methods, see for example [10] and |11].
Here, we are concerned by compressible fluids and our aim is to seek conditions
to prove the existence and uniqueness of the developed model’s solution..
2 Mathematical model
Let us consider a bounded connected open domain Ω of Rd with d = 2 or 3,
describing the porous medium (the reservoir), with a Lipchitz boundary Γ,
and let t be the time variable t ∈ [0, T [, T ≺ ∞.
We consider compressible flows, with constant dynamic viscosities and where
the gravity effect is neglected.Under these hypotheses, the Darcy’s law com-
bined with mass’s conservation equations for each of the component leads to
the following system of partial differential equations of parabolic convection-
diffusion type
φ(x)
∂
∂t
(ρoω
h
oSo) + div(ρoω
h
oUo) = 0 (1)
φ(x)
∂
∂t
(ρgSg + ρoω
l
oSo) + div(ρgUg + ρoω
l
oUo) = 0 (2)
Ui = −K(x)kri
μi
∇Pi (3)
where Si, Ui, Pi, ρi, μi, kri represent, the saturation, the velocity,the pressure,
the density, the viscosity and the relative permeability, of the phase i = o, g,
respectively, the parameters φ and K are the porosity and the absolute per-
meability of the medium and ωco, c = h, l is the massic fraction of component
A bi-phasic flow problem in porous media 2057
c, denoted by h for the heavy component and by l for the light component in
the oil phase.
We suppose that it is a saturated regime and is expressed by
So + Sg = 1 (4)
The capillary pressure is given by
Pg − Po = Pc(So) = pc(So)pcM (5)
where
pcM = sup |Pc(So)| and 0 ≤ pc(So) ≤ 1 (6)
We define the mobility of each phase by
λi =
kri
μi
, i = o, g (7)
and the total mobility λ by
λ = λo + λg (8)
For simplicity convenience, we set
ρho = ρoω
h
o , ρ = ρg + ρo, b = ρgλg + ρoλo, and d = ρg − ρo (9)
2.1 Reduced Saturation
Let us define by Si,m , the residual saturation of the fluid, i = o, g,; we write
Si,m ≤ Si, i = o, g (10)
and by Si,M ,the maximum saturation of the fluid, i = o, g, such that
Sg,M = 1 − So,m and So,M = 1 − Sg,m (11)
Si,m ≤ Si ≤ Si,M , i = o, g (12)
This leads to the reduced saturation S set as
S =
So − So,m
1 − Sg,m − So,m (13)
0 ≤ S ≤ 1 (14)
2058 S. Gasmi and F. Z. Nouri
2.2 Global pressure
If S = 0, equation (13) disappears.This is one of the main reasons for which
the terminology of the“global pressure” was introduced and written as
P =
1
2
(Pg + Po) + γ (S) (15)
with
γ (S) =
1
2
S∫
So,m
(
λg − λo
λ
)
p′c (ξ) pcMdξ (16)
The total velocity is given by the sum of the velocities of the two components
U = Ug + Uo (17)
and we write
γ(S) =
1
2
S∫
0
α (ξ) dξ (18)
where
α (S) =
(
λg (S) − λo (S)
λ (S)
)
p′c (S) pcM (19)
is the capillary diffusion.
2.3 Boundary and initial conditions
The system must be completed with boundary and initial conditions.We sup-
pose that the reservoir’s boundary is not permeable, we write
U.η = 0, on Γ × (0, T ) (20)
α(S)∇S = 0, on Γ × (0, T ) (21)
The parameter η denotes the normal vector.
The initial conditions for the saturation and the pressure are given by
S (x, 0) = S0 (x) in Ω (22)
P (x, 0) = P 0 (x) in Ω (23)
A bi-phasic flow problem in porous media 2059
Therefore, we write system (1 − 3) as
Φ(x)
∂
∂t
(
ρhoS
) − div (K(x)ρhoλo(S)∇P ) + div (K(x)ρhoα(S)∇S) = f1 (24)
Φ(x)
∂
∂t
(ρS) − div(K(x)b(S, P )∇P ) + div (K(x)d (P )α(S)∇S) = f2 (25)
∇P.η = 0, α(S)∇S = 0, on Γ × (0, T ) (26)
S (x, 0) = S0 (x) , P (x, 0) = P 0 (x) in Ω (27)
Where
f1 = −φ(x)So,m ∂
∂t
(
ρho
)
and f2 = −φ(x) ∂
∂t
(
ρSo,m + ρg
)
3 Existence and uniqueness of the solution
Let Ω be a connected open set in Rd ( d = 2 or 3), with a Lipschiz boundary Γ,
to ensure the existence and the uniqueness of the weak solution, for our case,
we start by setting the following hypotheses:
1. K (x) ∈ L∞ (Ω) , such that
K− ≤ K (x) ≤ K+ a.e.in Ω
2. φ(x) ∈ L∞ (Ω) , such that:
0 ≺ φ− ≤ φ (x) ≤ φ+ a.e.in Ω
3. ρho(P ) ∈ L∞ (Ω, (0, T )) ∩ H1 (Ω, (0, T )) , such that:
ρho− ≤ ρho (P ) ≤ ρho+ a.e.in Ω × (0, T )
4. α(S) ∈ L∞ (Ω, (0, T )) , such that:
α− ≤ α (S) ≤ α+ a.e.in Ω × (0, T )
5. λo(S) ∈ L∞ (Ω, (0, T )) , such that:
λo− ≤ λo (S) ≤ λo+ a. e.in Ω × (0, T )
2060 S. Gasmi and F. Z. Nouri
6. d(P ) ∈ L∞ (Ω, (0, T )) , such that:
d− ≤ d (P ) ≤ d+ a.e.in Ω × (0, T )
7. b(S, P ) ∈ L∞ (Ω, (0, T )) , such that:
b− ≤ b (S, P ) ≤ b+ a.e.in Ω × (0, T )
8. So, P o ∈ L∞ (Ω, (0, T )) , such that:
0 ≤ S (x, t) ≤ 1 a.e.in Ω × (0, T )
9. ρg (P ) ∈ H1 (Ω, (0, T )) .
Note that a.e stands for almost everywhere.Then we introduce the following
functional spaces
H (div, Ω) =
{
v ∈ (L2 (Ω, (0, T )))d , div (v) ∈ L2 (Ω, (0, T )) , d = 2, 3} (28)
V (Ω) = {v ∈ H (div, Ω) , v.η = 0 on Γ} (29)
W (Ω) = {v ∈ V (Ω) , v (x, T ) = 0 in Ω}
The weak formulation of problem (24 − 27) is written as
(
Φ(x)ρhoS,
∂v
∂t
)
Ω
− (K(x)ρhoλo(S)∇P,∇v)Ω +
(
K(x)ρhoα(S)∇S,∇v
)
Ω
= (f1, v) (30)
(
Φ(x)ρS,
∂v
∂t
)
Ω
− (K(x)b(S, P )∇P,∇v)Ω +
(K(x)d (P )α(S)∇S,∇v)Ω = (f2, v) (31)
where (., .)Ω is the inner product defined on W (Ω).
Proposition 3.1
A bi-phasic flow problem in porous media 2061
Under assumption (1.−9.), problem (24 − 27) has a unique solution (S, P ) ∈
(V (Ω))2.
Proof
To prove this result, we adapt results in [1] and [3] to our model so that
system (24 − 25) can be written in a simplified form as:
Φ(x)b(S,P )
∂
∂t
(
ρhoS
) − b(S, P )∇. (K(x)ρhoλo(S)∇P ) +
+ b(S, P )∇. (K(x)ρhoα(S)∇S) = b(S, P ).f1 (32)
Φ(x)ρhoλo(S)
∂
∂t
(ρS) − ρhoλo(S)∇.(K(x)b(S, P )∇P ) +
+ρhoλo(S)∇. (K(x)d (P )α(S)∇S) = ρhoλo(S).f2 (33)
If we multiply equations (33) by v1 and (34) by v2 and integrate over Ω we get(
Φ(x)
∂
∂t
(
ρhoS
)
, v1
)
− (K(x)ρhoλo(S)∇P,∇v1) +
+
(
K(x)ρhoα(S)∇S,∇v1
)
= (f1, v1) (34)
(
Φ(x)
∂
∂t
(ρS) , v2
)
− (K(x)b(S, P )∇P,∇v2) +
+ (K(x)d (P )α(S)∇S,∇v2) = (f2, v2) (35)
where ∇v1 = b(S, P ) and ∇v2 = ρhoλo(S). Therefore∫
Ω
Φ(x)
(
∂
∂t
(
ρhoS
)
.v1 − ∂
∂t
(ρS) .v2
)
+
+
∫
Ω
K (x) ρhoα(S) (b(S, P ) − d (P )λo(S))∇S =
∫
Ω
f1v1 − f2v2 (36)
2062 S. Gasmi and F. Z. Nouri
We deduce that
Φ(x)
(
∂
∂t
(
ρhoS
)
.v1 − ∂
∂t
(ρS) .v2
)
+
+K (x) ρhoα(S) (b(S, P ) − d (P )λo(S))∇S = f1v1 − f2v2 a.e in Ω (37)
in the distribution’s sense.
Now, we introduce a new unknown V = K (x) ρhoα(S) (b(S, P ) − d (P )λo(S))∇S
to get the following two equations:
Φ(x)
(
∂
∂t
(
ρhoS
)
.v1 − ∂
∂t
(ρS) .v2
)
+ V = f1v1 − f2v2 (∗)
and
V = K (x) ρhoα(S) (b(S, P ) − d (P )λo(S))∇S (∗∗)
We first assume that the unknown V ∈ L2 (Ω, (0, T )) is known and we try to
find S ∈ V (Ω, (0, T )) from equation (∗∗) . We multiply equation (∗∗) by a test
fonction v ∈ V (Ω, (0, T )) and we integrate over Ω to get
a (s, v) = (V, v)
∇S = 0 on Γ
where
a (s, v) =
(
K (x) ρhoα(S) (b(S, P ) − d (P )λo(S))∇S, v
)
This problem has a unique solution s in V (Ω) So,if we define A ∈ £ (V, V ′) as
(AS, v) = a (s, v) (38)
we can write
AS = V a.e in (0, T ) (39)
A is an isomorphism from V into V ′ (according to the above conditions (1, 3 − 7)),
thus equation (∗∗) has a unique solution S (x, t) for t ∈ (0, T ) (from Lax-
Milgram theorem [7]).
Let us now study equation (∗): Φ(x) ( ∂
∂t
S1
)
+ V = f1v1 − f2v2 where S1 =(
ρhoS
)
.v1 − ∂∂t (ρS) .v2; from (39) and (40) we get
S = A−1V
A bi-phasic flow problem in porous media 2063
where A is an isomorphism.Note that at t = 0: V0 = AS0.If we multiply
equation (∗) by a test fonction u ∈ H1 (Ω) and integrate over Ω we get∫
Ω
∂
∂t
(
A−1V
)
.u +
(
1
φ (x)
V, u
)
= (f1v1 − f2v2.v)
∫
Ω
∂
∂t
A−1V.u +
(
1
φ (x)
V, u
)
= (f1v1 − f2v2.v)
If we define the norm ((., .)) on Ω by ((u, v)) =
∫
Ω
A−1 (u) .v, we have the
relation
((V, u)) +
(
1
φ (x)
V, u
)
= 0, V (0) = V0
This problem has a unique solution V in L2 (Ω) ∩ L∞ (Ω).
4 Conclusion
In this paper, we have introduced a simplified formulation of the Hydrocarbon
system where the unknowns are the reduced saturation of one of the fluids
and the global pressure.This formulation transforms a coupled degenerate non
linear parabolic system to a familly of elliptic equations. Hence we prove
a theorecal result for the existence and the uniquness of the solution of the
resulting system.This is proved by decoupling the equations and using twice
Lax-Milgram’s theorem [7], first to determine the saturation S assuming that
V
(
V = K(x)ρhoα(S) (b(S, P ) − d (P )λo(S))∇S
)
is known from equation (∗),
and then obtain V from equation (∗∗).
References
[1] M.Afif and B.Amaziane, On convergence of finite volume schemes for one-
dimensional two-phase flow in porous media, Journal of Computational
and Applied Mathematics, Vol.145 (2002), pp.31–48.
[2] F.Boyer, F.Hubert, Finite volume method for 2D linear and nonlinear
elliptic problems with discontinuities, Journal of Numerical Analysis,
(2010).
[3] G.Chavent and J.Jaffre, Mathematical Models And Finite Elements For
Reservoir Simulation, Single Phase, Multiphase and Multicomponent
Flows through Porous Media, Elsevier Science Publishers B.V., 1986.
2064 S. Gasmi and F. Z. Nouri
[4] Z.Chen, Finite element methods for the black oil model in petroleum
reservoirs, I.M.A.Preprint Series, Juillet 1994
[5] Z.Chen, Reservoir Simulation, Mathematical Techniques in oil Recovery,
Society for Industrial and Applied Mathematics, Philadelphia, 2007.
[6] E.M.Dicks, higher order Godunov black-oil simulations for compressible
flow in porous media, Thèse, Université de reading, Mars 1993.
[7] R.Dautray, J.L.Lions, Analyse mathématique et calcul numérique pour
les sciences et les techniques, masson, Paris 1987.
[8] G.Gagneux, A-M.Lefevere, M.Madaune-Tort, Analyse mathématique de
modèles variationnels en simulation pétrolière, Le cas du modèle black-
oil pseudo-compositionnel standart isotherme”, Revista Matematica de la
Universidad Complutense de Madrid,1989.
[9] A.Mahamane, O.Diallo, F.Benkhaldoun, Simulation numérique par vol-
umes finis d’un écoulement diphasique eau-pétrole en milieu poreux,
M.S.A.S., 2006, pp.47-54.
[10] M. Maouni and F.Z. Nouri, Numerical Results for flows in a Complex
geometry,Int.J.Applied Math.and Mech.5(4), 2009, pp.19-31.
[11] F. Saci, F.Z. Nouri and E.Canot, A spectral element method for the so-
lution of flows with obstacles, Int.J.Applied Math.and Mech. 8(18), 2012,
pp 79-99.
Received: January 7, 2013


A Study of a Bi-Phasic Flow Problem in Porous Media

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A Study of a Bi-Phasic Flow Problem in Porous Media

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