Study of a numerical scheme for miscible two-phase flow in porous media. R. Eymard, Université Paris-Est

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1 Study of a numerical scheme for miscible two-phase flow in porous media R. Eymard, Université Paris-Est in collaboration with T. Gallouët and R. Herbin, Aix-Marseille Université C. Guichard, Université Paris 6 R. Masson, Université Nice V. Schleper, Universität Stuttgart october, 23

2 Water and gas flow in porous media Conservation equations Φ(x) t(s(q)ρ w ) div(λρ w k w (S(q)) p w ) = ρ w f w Φ(x) t(s(q)ρ w X (p g ) + ( S(q))ρ g ) div[λρ w k w (S(q))X (p g ) p w + Λρ g k g (S(q)) p g + Dρ w S(q) X (p g )] = ρ g f g with q = p w p g s(q) X (pg) X q pg Existence of continuous solution under various hypotheses : see F. Caro, B. Saad, M. Saad

3 Formal weak sense p g L 2 (, T ; H ()), p w L 2 (, T ; H ()), α = ρw ρ g, D = T + T Φ(x) ( ts(q)t w + t ( αs(q)x (pg ) + ( S(q)) ) T g ) dxdt ( kw (S(q)) p w T w + ( k w (S(q))αX (p g ) p w + k g (S(q)) p g ) Tg ) dxdt = T (f w T w + f g T g ) dxdt T w, T g L 2 (, T ; H ())

4 Space estimates : formal computation on continuous equations set Choose q S(q) := S (τ)τ dτ p X (p) := X (τ)τ dτ T w = p w α p g X (p)dp T g = p g remarkable property ts(q)t w + t ( αs(q)x (pg ) + ( S(q)) ) T g = t ( S(q) + αs(q) X (pg ) ) k w (S(q)) p w T w + ( k w (S(q))αX (p g ) p w + k g (S(q)) p g ) Tg = k w (S(q)) pw 2 + k g (S(q)) pg 2 thanks to p w ( p g X (p)dp ) + X (p g ) p w p g = need of a nonlinear function of p g as test function and Stampacchia s result (ν(u))(x) = ν (u(x)) u(x), for a.e. x, u H (), ν Lip(R)

5 Need of regularization for strong convergence of S(q) and X (p g ) s(q) S max S min q k w (S(q)) pw 2 + k g (S(q)) pg 2 C( pw 2 + pg 2 )

6 Time estimates : formal computation on continuous equations ( ) 2 Φ(x) ( ts(q)) ω dx C p w 2 L 2 () ω 2 L 2 () Φ t(αsx ) = div ( ) (αx + )k w (S) p w + k g (S) p g ( ) 2 Φ t(s(q)x (p g ))ω dx C( p w 2 L 2 () + p g 2 L 2 ()) ω 2 L 2 () bounds ts(q) L 2 (,T ;H C and ( Φ ()) t S(q)X (pg ) ) L 2 (,T ;H C Φ ()) with { v H := sup () Φ Φ(x)v(x)ω(x) dx strong convergence of S(q) and S(q)X (p g ) } ω H () with ω H () = use of regularization for strong convergence of X (p g ) X (p g ) = S(q)X (pg ) S(q)

7 Numerical Scheme : first idea for reproducing continuous estimates set D = (X D,, Π D, D) and look for p w (n+), p g (n+) X D, such that with T ( Φ(x) δ (n+ 2 ) D S(q)Π DT w + δ (n+ 2 ) ( D αs(q)x (pg ) + ( S(q)) ) ) Π DT g dxdt T ( + k w (S(Π Dq)) Dp w DT w + ( ) ) kw (S(Π Dq))αX (Π Dp g ) Dp w + k g (S(Π Dq)) Dp g DT g dxdt T = (f w Π DT w + f g Π DT g ) dxdt T w, T g X D, δ (n+ 2 ) S (n+) S (n) D S = Π D δ (n+ δt (n+ 2 ) D Main problem : reproduce the remarkable property 2 ) (SX ) = Π D S (n+) X (n+) S (n) X (n) ( p g Dp w D X (p)dp ) + X (Π Dp g ) Dp w Dp g = δt (n+ 2 ) unfortunately no hope on general meshes and gradient schemes except if X (p g ) = meaning no dissolution (extension of Richards Problem)

8 Case X (p g ) = : sufficient conditions for convergence (D m = (X Dm,, Π Dm, Dm )) m N Piecewise constant reconstruction Π D(g(u)) = g(π Du) and Π D(uv) = Π DuΠ Dv Π Dv L Coercivity C D = max 2 () v X D, \{} v D discrete Poincaré inequality C Dm remains bounded Consistency : S Dm (ϕ) ( ) ϕ H (), S D(ϕ) = min ΠDv ϕ L v X 2 () + Dv ϕ L 2 () d D, Limit-conformity : W Dm (ϕ) ϕ H div (), W D (ϕ) = max u X D, \{} u D ( D u ϕ + Π D u divϕ) dx Compactness For all bounded sequence u m X Dm,, m N, sequence Π Dm u m rel. compacte in L 2 () (implies coercivity)

9 Case X (p g ) = : convergence analysis estimates lead to existence thanks to topological degree method 2 Dp w, Dp g bounded implies Π Dp w,π Dp g bounded (coercivity) implies Π Dp w,π Dp g weakly converge to some p w, p g 3 Dp w, Dp g bounded implies control on space translate of Π Dp w, Π Dp g (compactness), implies control on space translate of S(p w p g ) 4 time control (piecewise const. rec.) on S(q) implies convergence in L 2 ( (, T )) of S(q) to some S 5 Minty s trick implies S = S( p w p g ) 6 Interpolation of regular test functions, consistency and limit conformity imply limits are solution of weak formulation Examples of gradient scheme?

10 Conforming finite elements with mass lumping N N finite element basis (ϕ i ) i=,...,n, with ϕ i H () X D, = R N Π Du = N u i ϕ i and Π Du = i= N u i χ Ki i= Du = N u i ϕ i = Π Du i= K i Then Π Du Π Du L 2 () h Π Du L 2 () d implies Poincaré inequality coercivity S D(ϕ) Ch D Ŵ D(ϕ) = Rellich theorem consistency limit conformity compactness

11 Mixed finite elements (φ i ) i=,...,n L 2 () characteristic functions of a mesh of V h H div () finite dimensional subspace { } X D, = u = (u,..., u N ) Π Du = N Du V h s.t. for all ψ V h, u i φ i ( ψ Du + Π ) Dudivψ dx = i= Then C D defined by L.B.B. constant Convergence of MFE on linear problems Approximation in H div () coercivity consistency limit conformity compactness

12 Vertex Approximate Gradient scheme X D = { discrete value u K at the cell center x K and u s at the vertex s } Barycentric cutting of a cell in tetrahedra x σ = s V σ CardV σ x s, u σ = s V σ CardV σ u s x s Local discrete gradient on each tetrahedra T T u = s V σ (u s u K )g s T x K x σ x s Piecewise constant gradient in L 2 () d Du = T u on each tetrahedra T Reconstruction operator Π Du(x) = u K on K VAG is vertex-centered : unknowns u K can be eliminated in the linear system

13 The Multi Point Flux Approximation O-scheme on cubes X D, values in K and σ, s Π Du(x) = u K, x K K,s u = (u σ,s u K ) n K,σ d K,σ σ E K,s Du(x) = K,s u, x V K,s x K s x σ

14 The Mixed Mimetic Hybrid Family : the example of the SUSHI scheme u X D, = R M Π Du(x) = u K, x K u σ = aσ K u K K K u = σ K (uσ u K ) n K,σ σ E K R K,σ u = u σ u K K u (x σ x K ) 3 K,σ u = K u + R K,σ u n K,σ d K,σ Du(x) = K,σ u, x D K,σ C D C P S D(ϕ) Ch D W D(ϕ) Ch D discrete func.anal. coercivity consistency limit conformity compactness x K x σ

15 Case X (p g ) > : second idea : TP - two-point flux approximation see R.E., T. Gallouët, C. Guichard, R. Herbin, R. Masson finite volume mesh with (x K, x L ) σ K,L div(λ u)dx = K λ u n K,σ ds σ E K σ σ E K F K,σ σ xk dσ σ EK EL dσ nk,σ xl Dσ L λ σ on D σ with D σ = σ dσ d Dσ K u K,σ = u L for σ E K E L, u K,σ = for σ F K,σ = λ σ σ d σ (u K,σ u K ), σ E K F K,σ + F L,σ = gradients on D σ σu = d u K,σ u K n K,σ d σ σv n K,σ = v K,σ v K d σ v K F K,σ = Π Eλ Du Dvdx = K M σ E K σ λ σ (u L u K )(v L v K ) + d σ σ E K E L Π Eλ Du Dudx = d σ E K,σ λ σ σ d σ u K v K Π Eλ Du 2 dx

16 Resulting scheme with T ( Φ(x) δ (n+ 2 ) D S(q)Π DT w + δ (n+ 2 ) ( D αs(q)x (pg ) + ( S(q)) ) ) Π DT g dxdt T ( + k w (S(Π Eq)) Dp w DT w + ( kw (S(Π Eq))αX (Π Ep g ) Dp w + k g (S(Π Eq)) ) ) Dp g DT g dxdt T = (f w Π DT w + f g Π DT g ) dxdt T w, T g X D, δ (n+ 2 ) S (n+) S (n) D S = Π D δ (n+ δt (n+ 2 ) D 2 ) (SX ) = Π D S (n+) X (n+) S (n) X (n) δt (n+ 2 ) Reproduction of the remarkable property with definition of X (Π Ep g ) Dp w ( p g D X (p)dp ) + X (Π Ep g ) Dp w Dp g = pg,k ensured for σ E K E L under the condition pg,l X (p)dp X (p)dp = pg,l p g,k X (p)dp = X (p g,σ)(p g,l p g,k )

17 Convergence analysis Mimicks continuous estimates provide existence by topological degree method Pass to the limit in Π Eλ Du Dϕ dx λ ū ϕdx weak convergence of Du by conservativity strong convergence of Π Eλ and of Dϕ by consistency

18 Numerical example D injection of gas on the left in pure water Φ(x) =.2 Λ(x) = 2 ρ w = 3 kg m 3 ρ g = kg m 3 f w (t, x) = f g (t, x) = D = S(q) = (( ) n ) m with n = 2, m = /3, p = 5 Pa max{ q,} p + X (p g ) = min{c pp g, X } with X =. k w (S) = max{ S 2 µ w, 5 } with µ w = 3 ( S)2 k g (S) = max{, 5 } with µ g = 5. µ g p g (, x) = 5 Pa p g (t, ) = 3 5 Pa p w (t, ) = 5 Pa p w (, x) = 5 Pa p g (t, L) = 5 Pa p w (t, L) = 5 Pa.

19 Results c p = 3 7 L = 3 m S(q). X(p g ) x x 5 p g.5 5 x.2 5 x.5 x 5 p w x X does not attain its maximum value

20 Results c p = 3.5 7, L = 4. S(q). X(p g ) x x 5 p g.5 5 x.2 5 x.5 x 5 p w x X does attain its maximum value

21 Conclusion Convergence of gradient schemes if X (p g ) = Not clear that possible to relax TP if X (p g ) > Open problem : relax regularization of S(q) with TP scheme (difficult)