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



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)



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



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.



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.



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.



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.



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.



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.



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.



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.



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).



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.



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


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))′.



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.



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 .


Slightly compressible two phase flow

Slightly compressible two phase flow


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)


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.


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)


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.


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.



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.



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.




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 .




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.



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,



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.



References

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pages 12–26, January 2014.

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Acad. Sci. Paris, Ser. I 347 (2009) 249-254.

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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.

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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.

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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.



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