Mathematical analysis of compressible multifluid flows in...
Transcript of Mathematical analysis of compressible multifluid flows in...
Mathematical analysis of compressible multifluid flows in porous media
Mazen SAAD
Ecole Centrale de NantesLaboratoire de Mathematiques Jean Leray
In collaboration with C. Galusinski
–
MOMAS, Nice 5, 6, 7 october 2015
1 m-fluid flow : compressible immiscible, bounded densities
2 Slightly compressible two phase flowModelThe one dimensional modelDimension N ≥ 2
Mazen Saad (ECN) Multifluid flows MOMAS, Nice 5, 6, 7 october 2015 2 / 33
m-fluid flow : compressible immiscible, bounded densities
MULTIFLUID MODELbounded densities
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m-fluid flow : compressible immiscible, bounded densities
Multifluid compressible and immiscible flow
The formulation describing the immiscible displacement of m- compressible fluids is given bythe mass conservation of each phase.Mass conservation of each phase : for i = 1,m
φ∂t(ρi (pi )si ) + div(ρi (pi )Vi ) + ρi (pi )si fP = ρi (pi )s?i fI (1)
φ : Porositysi : Saturation of the i phasepi : Pressure of the i phaseρi : Density of the i phasefP : Production ratefI : Injection rate at given saturation s?i .
Saturations:m∑i=1
si = 1, si ≥ 0. (2)
Mazen Saad (ECN) Multifluid flows MOMAS, Nice 5, 6, 7 october 2015 4 / 33
m-fluid flow : compressible immiscible, bounded densities
Multifluid compressible and immiscible flow
Darcy’s law for velocities Vi
Vi = −KMi (∇pi − ρi (pi )g), i = 1 · · ·m
K(x) : permeability tensor
Mobility (Mi ) : mobility of the i phase
si → Mi (si , sj ) is increasing, and Mi (si = 0, sj ) = 0
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
0,25
0,5
0,75
1
Ml (sl ) mobility of liquid phase
Mg (sl ) mobility of gas phase
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
0,25
0,5
0,75
1
Total mobility: M = Ml + Mg ≥ m0
The mobility of each phase vanishes in the region where the phase is missing.Degenerate problem
Mazen Saad (ECN) Multifluid flows MOMAS, Nice 5, 6, 7 october 2015 5 / 33
m-fluid flow : compressible immiscible, bounded densities
Densities bounded
ρi = ρi (pi )
ρi ∈ C1(R,R+) increasing and bounded
0 < ρm ≤ ρi (pi ) ≤ ρM
Balance
m + 1 equations and 2m unknowns (si , pi ).To close the system, we introduce m − 1 capillary pressures.
Mazen Saad (ECN) Multifluid flows MOMAS, Nice 5, 6, 7 october 2015 6 / 33
m-fluid flow : compressible immiscible, bounded densities
Capillary pressures
Denotes = (s2, s3, · · · , sm)
Define m − 1 capillary pressure :
pcj (s) = p1 − pj ; pour j = 2,m. (3)
No existence results for the case (m > 2).
Objective :
Assumptions on data to ensure existence.
Mazen Saad (ECN) Multifluid flows MOMAS, Nice 5, 6, 7 october 2015 7 / 33
m-fluid flow : compressible immiscible, bounded densities
Assumptions
We assume that the map Pc : Rm−1 → Rm−1 such that Pc (s) = (pcj (s))j=2,m derives from apotential : There exists a function F : Rm−1 → R such that
Assumption (H1)
Pc (s) = F ′(s) ⇐⇒ pcj (s) =∂F (s)
∂sj, j = 2,m
Remark
If pcj (s) = pcj (sj ), the assumption (H1) is verified.
Mazen Saad (ECN) Multifluid flows MOMAS, Nice 5, 6, 7 october 2015 8 / 33
m-fluid flow : compressible immiscible, bounded densities
Notion of global pressure for m-phases
How to construct the global pressure?Each pressure pi is a variation of p denoted gi (s)
pi = p + gi (s), i = 1 · · ·m. (4)
Existence of functions gi? We want
V =m∑i=1
Vi = −m∑i=1
KMi (s)∇pi
= −Km∑i=1
Mi (s)∇p − Km∑i=1
Mi (s)∇gi (s)
︸ ︷︷ ︸=0
= −KM(s)∇p
Assumption (H2)
m∑i=1
Mi (s)∇gi (s) = 0.
Mazen Saad (ECN) Multifluid flows MOMAS, Nice 5, 6, 7 october 2015 9 / 33
m-fluid flow : compressible immiscible, bounded densities
Notion of global pressure for m-phases
Construction of deviations gi (s)
p1 = p + g1(s)pi = p + gi (s)
=⇒ gi (s) = g1(s)− pci (s), i = 2 · · ·m
It suffices to construct g1 to deduce gi for i = 2,m.
The assumption (H2) is equivalent to
Total differential condition (H2)
There exists g1 : Rm−1 → R such that
∇g1(s) =1
M(s)
m∑j=2
Mj (s)∇pcj (s).
which ensures the existence of global pressure.
Mazen Saad (ECN) Multifluid flows MOMAS, Nice 5, 6, 7 october 2015 10 / 33
m-fluid flow : compressible immiscible, bounded densities
Integrant Factor
Assume that√
Mj (s)∇gj derives from a gradient of a function :
Integrant factor (H3)
There exists Aj : Rm−1 → R (j = 2 · · ·m) such that
∇Aj (s) =√
Mj (s)∇gj (s).
Denote A(s) = (Ai )i and suppose A−1 is θ-Holder function with 0 < θ < 1.
It can be viewed as Kirchhoff transform for degenerate multiphase flow.
Mazen Saad (ECN) Multifluid flows MOMAS, Nice 5, 6, 7 october 2015 11 / 33
m-fluid flow : compressible immiscible, bounded densities
Assumptions for the two phase flow (m=2)
For m=2, the three assumptions are fulfilled.
In fact, we have s = s2
(H1) F ′(s2) = pc2(s2).
(H2) g ′1(s2) = M2(s2)M(s2) p
′12(s2)
(H3) A′2(s2) =√
M2(s2)g ′2(s2) =√M2(s2)M1(s1)p′12(s2)
by a simple integration, we get F , g1, A2.
Mazen Saad (ECN) Multifluid flows MOMAS, Nice 5, 6, 7 october 2015 12 / 33
m-fluid flow : compressible immiscible, bounded densities
Assumptions for the three phase flow (m=3)
We have s = (s2, s3), pc (s) = (pc2(s), pc3(s))
(H1) pc2(s) = ∂F (s)∂s2
and pc3(s) = ∂F (s)∂s3
.
The function F is defined iff
∂pc2
∂s3(s) =
∂pc3
∂s2(s)
This condition is satisfied for pci (s) = pci (si ) , i = 2, 3.
(H2) Denote λi (s) = Mi (s)/M(s) (i = 1, 2), g1 satisfies∂g1∂s2
(s) = λ2(s) ∂pc2∂s2
(s) + λ3(s) ∂pc3∂s2
(s)∂g1∂s3
(s) = λ2(s) ∂pc3∂s2
(s) + λ3(s) ∂pc3∂s3
(s)
the function g1 is well defined iff ∂2g1∂s3∂s2
(s) = ∂2g1∂s2∂s3
(s).
Total differential condition of Chavent and Jaffre
∂λ2
∂s3
∂pc2
∂s2+∂λ3
∂s3
∂pc3
∂s2=∂λ2
∂s2
∂pc2
∂s3+∂λ3
∂s2
∂pc3
∂s3.
Mazen Saad (ECN) Multifluid flows MOMAS, Nice 5, 6, 7 october 2015 13 / 33
m-fluid flow : compressible immiscible, bounded densities
Assumptions for the three phase flow (m=3)
Algorithm of Chavent and Jaffre.For given a pair of capillary pressures
∂pc2
∂s3(s) =
∂pc3
∂s2(s),
we search the three mobilities solution of the following system :
∂λ2
∂s3
∂pc2
∂s2+∂λ3
∂s3
∂pc3
∂s2=∂λ2
∂s2
∂pc2
∂s3+∂λ3
∂s2
∂pc3
∂s3,
∂2A2
∂s3∂s2(s) =
∂2A2
∂s2∂s3(s),
∂2A3
∂s3∂s2(s) =
∂2A3
∂s2∂s3(s).
Mazen Saad (ECN) Multifluid flows MOMAS, Nice 5, 6, 7 october 2015 14 / 33
m-fluid flow : compressible immiscible, bounded densities
Model
Search (pi ) , i = 1 · · ·m solution of the system
φ∂t(ρi (pi )si )− div(Kρi (pi )Mi (s)∇pi )
+ div(Kρ2i (pi )Mi (s)g) + ρi (pi )si fP = ρi (pi )s
I1fI , i = 1, 2
Boundary conditions ∂Ω = Γinj ∪ Γimps1(t, x) = 1, sj (t, x) = 0, i = 2,m; p1(t, x) = 0 on Γinj
Vj · n = 0 on Γimp , j = 1 · · ·m.
Initial conditionspj (0, x) = p0
j (x); sj (0, x) = s0j (x) in Ω j = 1 · · ·m
m∑j=1
s0j = 1, s0
j ≥ 0.
Mazen Saad (ECN) Multifluid flows MOMAS, Nice 5, 6, 7 october 2015 15 / 33
m-fluid flow : compressible immiscible, bounded densities
Energy estimates
φ∂t(ρi (pi )si )− div(Kρi (pi )Mi (s)∇pi ) + · · ·︸︷︷︸lower order terms
= 0
Multiplying by ri (pi ) =∫ pi
01
ρi (z)dz and integrating
m∑i=1
∫Ω∂t(ρi (pi )si
)ri (pi ) dx︸ ︷︷ ︸
=
∫Ω∂tE
+m∑i=1
∫Ω
KMi (s)∇pi · ∇pi dx + · · · = 0
Evolution term can be rearranged under the assumption (H1) to be
∂tE =m∑i=1
∂t(ρi (pi )si )ri (pi ).
where
E =m∑i=1
siHi (pi )− F (s),
The function Hi (pi ) := ρi (pi )ri (pi )− pi , satisfies
Hi (0) = 0, Hi (pi ) ≥ 0, |Hi (pi )| ≤ c|pi | for all pi .
The function F is bounded.Mazen Saad (ECN) Multifluid flows MOMAS, Nice 5, 6, 7 october 2015 16 / 33
m-fluid flow : compressible immiscible, bounded densities
Energy estimates
Estimate on the velocities
∫QT
m∑i=1
Mi (s)|∇pi |2 dxdt < +∞.
No control on ∇pi since Mi vanishes.
Estimate on the evolutive term :
φ∂t(ρi (pi )si )) is bounded in (L2(H1))′.
Mazen Saad (ECN) Multifluid flows MOMAS, Nice 5, 6, 7 october 2015 17 / 33
m-fluid flow : compressible immiscible, bounded densities
Energy estimates
Estimate on the velocities
∫QT
m∑i=1
Mi (s)|∇pi |2 dxdt < +∞.
No control on ∇pi since Mi vanishes.
Estimate on the evolutive term :
φ∂t(ρi (pi )si )) is bounded in (L2(H1))′.
Mazen Saad (ECN) Multifluid flows MOMAS, Nice 5, 6, 7 october 2015 17 / 33
m-fluid flow : compressible immiscible, bounded densities
Energy estimates
Estimate on the global pressure. Using assumption (H2), we have
m∑i=1
∫QT
Mi (s)∇pi · ∇pi =
∫QT
M(s)∇p · ∇p +m∑i=1
∫QT
Mi (s)∇gi · ∇gi .
Then∇p is bounded in L2(QT )√
Mi (s)∇gi is bounded in L2(QT )
Assumption (H3) leads to
m∑i=2
∫QT
Mi (s)∇gi · ∇gi dx =m∑i=2
∫QT
∇Ai (s) · ∇Ai (s) dx ,
and
Ai (s) is bounded in L2(H1) for all i = 2,m.
Mazen Saad (ECN) Multifluid flows MOMAS, Nice 5, 6, 7 october 2015 18 / 33
m-fluid flow : compressible immiscible, bounded densities
Energy estimates
Estimate on the global pressure. Using assumption (H2), we have
m∑i=1
∫QT
Mi (s)∇pi · ∇pi =
∫QT
M(s)∇p · ∇p +m∑i=1
∫QT
Mi (s)∇gi · ∇gi .
Then∇p is bounded in L2(QT )√
Mi (s)∇gi is bounded in L2(QT )
Assumption (H3) leads to
m∑i=2
∫QT
Mi (s)∇gi · ∇gi dx =m∑i=2
∫QT
∇Ai (s) · ∇Ai (s) dx ,
and
Ai (s) is bounded in L2(H1) for all i = 2,m.
Mazen Saad (ECN) Multifluid flows MOMAS, Nice 5, 6, 7 october 2015 18 / 33
m-fluid flow : compressible immiscible, bounded densities
Energy estimates
Estimate on the global pressure. Using assumption (H2), we have
m∑i=1
∫QT
Mi (s)∇pi · ∇pi =
∫QT
M(s)∇p · ∇p +m∑i=1
∫QT
Mi (s)∇gi · ∇gi .
Then∇p is bounded in L2(QT )√
Mi (s)∇gi is bounded in L2(QT )
Assumption (H3) leads to
m∑i=2
∫QT
Mi (s)∇gi · ∇gi dx =m∑i=2
∫QT
∇Ai (s) · ∇Ai (s) dx ,
and
Ai (s) is bounded in L2(H1) for all i = 2,m.
Mazen Saad (ECN) Multifluid flows MOMAS, Nice 5, 6, 7 october 2015 18 / 33
m-fluid flow : compressible immiscible, bounded densities
Main result when densities are bounded
Theorem
The couples (si , pi )i , i = 1 · · ·m, are a weak solution in the sense :
p ∈ L2(H1Γinj
), s ∈(L
2θ (W τ, 2
θ ))m−1
, τ < 1
A(s) ∈ (L2(H1Γinj
)m−1
pi ∈ L2(L2),
and satisfy for all ϕi ∈ C1([0,T ];H1Γinj
(Ω)) with ϕi (T ) = 0,
−∫QT
φρi (pi )si∂tϕi −∫
Ωφ(x)ρi (p
0i (x))s0
i (x)ϕi (0, x) dx
+
∫QT
Kρi (pi )Mi (s)∇p · ∇ϕi +
∫QT
Kρi (pi )√
Mi (s)∇Ai (s) · ∇ϕi
−∫QT
Kρ2i (pi )Mi (s)g · ∇ϕi +
∫QT
ρi (pi )si fPϕi =
∫QT
ρi (pi )sIi fI ϕi .
Mazen Saad (ECN) Multifluid flows MOMAS, Nice 5, 6, 7 october 2015 19 / 33
Slightly compressible two phase flow
Slightly compressible two phase flow
Mazen Saad (ECN) Multifluid flows MOMAS, Nice 5, 6, 7 october 2015 20 / 33
Slightly compressible two phase flow Model
Model : Slightly compressible
We consider the case of slightly compressible phases where the density of each phase is takenas an exponential law with small compressibility factor.Mass conservation of each phase:
φ∂t(ρi si ) + div(ρiVi ) = 0, i = 1, 2 (5)
The velocity of each fluid Vi
Vi = νiV − Kα(s1)∇si , i = 1, 2 (6)
The total velocityV = −KM(s1)∇p, (7)
and νi = Mi/M, and α(s1) = M(s1)ν1(s1)ν2(s1)p′c2(s1) ≥ 0.
Slightly compressible assumption
we assume that ρi = ρi (p) satisfies
dρi
dp(p) = ziρi (p), zi > 0. (8)
Mazen Saad (ECN) Multifluid flows MOMAS, Nice 5, 6, 7 october 2015 21 / 33
Slightly compressible two phase flow Model
dρi
dp(p) = ziρi (p), zi > 0.
Mass conservation of each phase can be written as: for i=1,2
φ∂tsi + φsizi∂tp + div(νiV) + νiziV · ∇p − div(Kα∇si )− Kαzi∇si · ∇p = 0. (9)
Pressure equation. Adding the two equations for i=1, 2
φ (z2 + (z1 − z2)s)︸ ︷︷ ︸d(s)>0
∂tp + div V︸ ︷︷ ︸dissipative
+ (z2 + (z1 − z2)ν(s))︸ ︷︷ ︸f (s)
V · ∇p︸ ︷︷ ︸≈|∇p|2
−Kα(s)(z1− z2)∇s ·∇p = 0.
Saturation equation. The equation for i = 1, is considered to be the saturation one
φ∂ts + φsz1∂tp + div(ν1(s)V) + ν1(s)z1V · ∇p − div(Kα(s)∇s)− Kα(s)z1∇s · ∇p = 0.
Mazen Saad (ECN) Multifluid flows MOMAS, Nice 5, 6, 7 october 2015 22 / 33
Slightly compressible two phase flow The one dimensional model
The one dimensional model
We set QT = Ω× (0,T ). We investigate the following nonlinear boundary value problem ofparabolic type in QT
γd(s)∂tp − ∂x (M(s)∂xp)− γβ(s)|∂xp|2 − γα(s)∂x s∂xp = 0, (10)
∂ts + γb(s)∂tp − ∂x (α(s)∂x s) + γk(s)|∂xp|2 + aγ(s)∂x s∂xp = 0. (11)
γ : compressibility factor. z1 = 2γ, z2 = γMixed boundary conditions
p(t, 0) = 1, p(t, 1) = 0,s(t, 0) = 0, α(s)∂x s(t, 1) = 0,
Main properties
d(s) ≥ d0 > 0, M(s) ≥ m0 > 0
b, k, f and aγ continuous on [0, 1]
Main assumption
α(s) ≥ α0 > 0 (Non degenerate case)
Mazen Saad (ECN) Multifluid flows MOMAS, Nice 5, 6, 7 october 2015 23 / 33
Slightly compressible two phase flow The one dimensional model
The one dimensional model
Denote S0 = u ∈ H1(0, 1), u(0) = 0. We search (p, s) ∈ P × S :
P = u ∈ L∞(H10 ) ∩ L2(H2), ∂tu ∈ L2(L2)
S = u ∈ L∞(S0) ∩ L2(H2), ∂tu ∈ L2(L2).
Theorem
Let (p0, s0) ∈ H10 (0, 1)× S0.
(Local existence) For every γ > 0, there exists a number T∗γ ∈]0,T ] such that thesystem has a strong solution on QT∗γ .
(Admissibility) Moreover, if 0 ≤ s0(x) ≤ 1 a.e in x ∈ [0, 1], then 0 ≤ s(t, x) ≤ 1 a.e inx ∈ [0, 1], for all t ∈ [0,T∗γ ].
(Uniquness) The system has a unique strong solution.
(Consistency) The solution of the system (pγ , sγ) converges to the solution of the usualincompressible model, obtained for γ equal to zero, as γ goes to zero.
We give explicitly the dependence of solutions on the compressibility factor γ.The proof is based on the choice of test functions : p, ∂tp, ∂xxp, s, ∂ts, ∂xx s, Sobolevinjection in dimension 1.No possibility to extend results to dimension N ≥ 2.
Mazen Saad (ECN) Multifluid flows MOMAS, Nice 5, 6, 7 october 2015 24 / 33
Slightly compressible two phase flow Dimension N ≥ 2
Dimension N ≥ 2
We consider the specific case :
The two fluids have the same compressibility factor : γ = z1 = z2
dρi
dp(p) = γρi (p), γ > 0
The quadratic term γV · ∇p ≈ γ‖V‖2 ≈ 0 is neglected.
Pressure equation :
φ (z2 + (z1 − z2)s)︸ ︷︷ ︸=γ
∂tp +div V + (z2 + (z1− z2)ν(s)) V · ∇p︸ ︷︷ ︸≈‖V‖2≈0
−Kα(s) (z1 − z2)︸ ︷︷ ︸=0
∇s · ∇p = 0.
The saturation equation :
φ∂ts + γφs∂tp + div(ν(s)V) + γ ν(s)V · ∇p︸ ︷︷ ︸≈γ‖V‖2≈0
− div(Kα(s)∇s)− γKα(s)∇s · ∇p = 0.
Mazen Saad (ECN) Multifluid flows MOMAS, Nice 5, 6, 7 october 2015 25 / 33
Slightly compressible two phase flow Dimension N ≥ 2
Modelling : Slightly compressible N ≥ 2
Pressure equation :γφ∂tp − div(KM(s)∇p) = fI − fp
Saturation equation :
φ∂ts − s div(V) + div(ν(s)V)− div(Kα(s)∇s)− γKα(s)∇s · ∇p = (1− s)fp
Pressure regularity: p ∈ L2(H1), V ∈ (L2(QT ))N .Degenerate dissipation in saturation: several cases
α(0) > 0, α(1) = 0
α(0) = 0, α(1) > 0
α(0) = α(1) = 0
Assume that : α(s)∇s := ∇β(s) ∈ (L2(QT ))N .
No control of ∇s.How to control the terms : s div(V), γKα(s)∇s · ∇p.
Mazen Saad (ECN) Multifluid flows MOMAS, Nice 5, 6, 7 october 2015 26 / 33
Slightly compressible two phase flow Dimension N ≥ 2
Weighted weak solution : Slightly compressible N ≥ 2
Φ∂ts − s div(V) + div(ν(s)V)− div(K∇β(s))− γK∇β(s) · ∇p = (1− s)fp
Suppose
∇p, V et ∇β(s) are bounded in L2(QT ).
• consider a regular test function ψ,
−∫
Ωs div(V)ψ =
∫ΩψV · ∇s︸︷︷︸
/∈L2
+
∫ΩsV · ∇ψ, (no sense)
• The following formulation has a ”sense” if we consider κ(s)ψ as test function
−∫
Ωs div(V)κ(s)ψ =
∫ΩψV · ∇(s κ(s))︸ ︷︷ ︸
∈L2
+
∫Ωsκ(s)V · ∇ψ,
where κ(s) is a function to be chosen. However, the dissipatif term has ”no sense”,
−∫
Ωdiv(K∇β(sη))κ(sη)ψ =
∫Ω
K∇β(sη) · ∇κ(sη)ψ︸ ︷︷ ︸convergence?
+
∫Ω
K∇β(sη) · ∇ψκ(sη).
=⇒ Notion of weighted degenerate weak solutions.
Mazen Saad (ECN) Multifluid flows MOMAS, Nice 5, 6, 7 october 2015 27 / 33
Slightly compressible two phase flow Dimension N ≥ 2
Weighted weak solution : Slightly compressible N ≥ 2
φ∂ts − s div(V) + div(ν(s)V)− div(Kα(s)∇s)− γKα(s)∇s · ∇p = (1− s)fP
Assume : α(1) > 0 et α(s) ≈ sr avec r > 1
Consider test function κ(s) = sr−1 and denote h(s) =∫ s
0 κ(y)dy , we obtain
d
dt
∫Ωφh(s)dx + (r − 1)
∫Ωα(s)sr−2︸ ︷︷ ︸≈s2r−2
|∇s|2dx = −r∫
Ωsr−1∇s · V dx
+ (r − 1)
∫Ων(s)sr−2︸ ︷︷ ︸≈sr−1
V · ∇s + γ
∫Ωα(s)sr−1∇p · ∇s +
∫Ω
(1− s)sr−1fp .
Then,
‖sr−1∇s‖(L2(QT ))N ≤ C .
Mazen Saad (ECN) Multifluid flows MOMAS, Nice 5, 6, 7 october 2015 28 / 33
Slightly compressible two phase flow Dimension N ≥ 2
Weighted weak solution : Slightly compressible N ≥ 2
Nondegenerate Problem : let η > 0
φ∂tsη−sη div(Vη)+div(ν(sη)Vη)−div(Kα(sη)∇sη)−η∆sη−γKα(sη)∇sη ·∇pη = (1−sη)fP
Lemma
The solutions of the saturation equation) satisfy
(i) 0 ≤ sη(t, x) ≤ 1, a.e. (t, x) in QT .
(ii) (sr−1η ∇sη)η , (s
r−22
η α12 (sη)∇sη)η and (α(sη)∇sη)η are bounded in (L2(QT ))N .
(iii) (h(sη))η is bounded in L∞(0,T ; L1(Ω)).
(iv) (η12 s
r−22
η ∇sη)η is bounded in (L2(QT ))N .
(v) (φ(x)∂th(sη))η is bounded in L1(0,T ; (W 1,q(Ω))′) for q > N.
(vi) (h(sη))η and (sη)η are relatively compact in L2(QT ).
Lemma
The sequence ((s3r+2η α(sη)∇sη))η is a Cauchy sequence in measure.
Mazen Saad (ECN) Multifluid flows MOMAS, Nice 5, 6, 7 october 2015 29 / 33
Slightly compressible two phase flow Dimension N ≥ 2
Weighted degenerate weak solution
Theorem
Let θ ≥ 6r + 6. There exists (p, s) such that
p ∈ L2(H1) ∩ L∞(L2), V ∈ (L2(QT ))N , φ∂tp ∈ L2((H1(Ω))′)
0 ≤ s(t, x) ≤ 1, a.e. in QT
κθ(s) = sr−1+θ , hθ(s) =∫ s
0βθ(y)dy ∈ L2(0,T ; H1(Ω))
α12 (s)β′
12 (s)∇s ∈ (L2(QT ))N ,
satisfying for all ψ ∈ L2(H1)
γ < φ∂tp, ψ > +
∫QT
KM(s)∇p.∇ψ dxdt =
∫QT
(fI − fP )ψ dxdt,
we define
F (s, p, χ) =−∫QT
φ(x)hθ(s)∂tχ dxdt −∫
Ω
φ(x)hθ(s0(x))χ(0, x)dx
+
∫QT
V · ∇(sκθ(s)χ) dxdt −∫QT
ν(s)V · ∇(κθ(s)χ) dxdt
+
∫QT
α(s)K∇s · ∇(κθ(s)χ) dxdt − γ∫QT
Kα(s)κθ(s)∇s · ∇pχ dxdt
−∫QT
(1− s)fPκθ(s)χ dxdt,
Mazen Saad (ECN) Multifluid flows MOMAS, Nice 5, 6, 7 october 2015 30 / 33
Slightly compressible two phase flow Dimension N ≥ 2
Weighted degenerate weak solution
Theorem
and F satisfying
F (s, p, χ) ≤ 0 for all χ ∈ C1([0,T [×Ω) with supp χ ⊂ [0,T [×Ω and χ ≥ 0 (12)
furthermore,
for all ε > 0, there exists Qε ⊂ QT ,meas(Qε) < ε, such that,
F (s, p, χ) = 0 for all χ ∈ C1([0,T [×Ω) with supp χ ⊂(
[0,T [×Ω)\Qε.
(13)
Formally, we have shown that
for θ ≥ 7r + 5, the saturation equation is satisfied
sθ(φ∂ts − s div(V) + div(ν(s)V)− div(Kα(s)∇s)− γKα(s)∇s · ∇p − (1− s)fp
)= 0,
in the sense of distribution except in small region Qε ⊂ (0,T )× Ω, mes(Qε) < ε.
Mazen Saad (ECN) Multifluid flows MOMAS, Nice 5, 6, 7 october 2015 31 / 33
Slightly compressible two phase flow Dimension N ≥ 2
References
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pages 12–26, January 2014.
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Acad. Sci. Paris, Ser. I 347 (2009) 249-254.
C. Galusinski, M. Saad, A nonlinear degenerate system modeling water-gas in reservoir flow, Discrete and continuous dynamical systems
series B, Vol. 9, Num. 2, pp. 281–308, March 2008.
C. Galusinski, M. Saad, Two compressible and immiscible fluids in porous media, J. Differential Equations 244 (2008), 1741-1783.
F. Caro, B. Saad, M. Saad Two-Component Two-Compressible Flow in Porous Media, Acta Applicandae Matehematicae, February 2012,
Volume 117, Issue 1, pp. 15–46.
Z. Khalil, M. Saad, Degenerate two-phase compressible immiscible flow in porous media : The case where the density of each phase depends
on its own pressure. Mathematics and computers in simulation, Vol. 81, issue 10, June 2011, pp. 2225-2233.
Z. Khalil, M. Saad, On a fully nonlinear degenerate parabolic system modelling immiscible gas-water displacement in porous media.
Nonlinear Analysis: Real World Applications , Vol. 12, issue 3, 2011, 1591-1615.
Z. Khalil, M. Saad, Solutions to a model for compressible immiscible two phase flow in porous media, EJDE, Vol. 2010(2010), No. 122, pp.
1-33.
F. Caro, B. Saad, M. Saad, Study of degenerate parabolic system modeling the hydrogen displacement in a nuclear waste repository,
Discrete and Continuous Dynamical Systems Series S, Volume 7, Number 2, April 2014, pp. 191–205.
Mazen Saad (ECN) Multifluid flows MOMAS, Nice 5, 6, 7 october 2015 32 / 33
Slightly compressible two phase flow Dimension N ≥ 2
Merci
Mazen Saad (ECN) Multifluid flows MOMAS, Nice 5, 6, 7 october 2015 33 / 33