Convergence of a Finite Volume Scheme and Numerical Simulations for - PowerPoint PPT Presentation

Convergence of a Finite Volume Scheme and Numerical Simulations for Water-Gas Flow in Porous Media M. Afif and B. Amaziane IPRA–LMA de Pau CNRS-UMR 5142 brahim.amaziane@univ-pau.fr Journ´ ee MoMaS : Mod´ elisation des Ecoulements Diphasiques Liquide-Gaz en Milieux Poreux : Cas Tests et R´ esultats IHP Paris, 23 Septembre 2010 M. Afif and B. Amaziane Journ´ ee MoMaS 23 Septembre 2010

PLAN 1 Mathematical model 2 Finite Volume Scheme 3 L ∞ and BV estimates 4 Numerical results of the BOBG test problem 5 References M. Afif and B. Amaziane Journ´ ee MoMaS 23 Septembre 2010

Mathematical model We consider the flow of two immiscible compressible fluids (w=water and g=gas) in a porous medium I =] a , b [= � 2 m =1 I m . Gas pressure - water saturation formulation   0 ≤ S ( x , t ) ≤ 1 in I × ]0 , T [,  � � � �   = Q w Φ ∂ t S + ∂ x f w ( S ) q − ∂ x K ∂ x α ( S ) in I × ]0 , T [, ρ � � � � (1) = Q g  Φ ∂ t P (1 − S ) − ∂ x P λ g ( S ) K ∂ x P in I × ]0 , T [,   σ g � �  q = − λ ( S ) K ∂ x P + β ( S ) in I × ]0 , T [, Φ( x ) is the porosity, K ( x ) absolute permeability, Q ν the source term of phase ν = w , g , kr ν ( S ) the ν -phase relative permeability, λ ν ( S ) = kr ν ( S ) where µ ν the ν -phase viscosity, µ ν λ ( S ) = λ w ( S ) + λ g ( S ) is the total mobility. M. Afif and B. Amaziane Journ´ ee MoMaS 23 Septembre 2010

Finite Volume Scheme Let { t 0 , ..., t N } be a partition of J = [0 , T ]; ∆ t n = t n +1 − t n ; � � N x � � Let x i i =0 be a partition of I and x i + 1 2 = x i +1 + x i / 2; Vertex-Centred control volumes I i := [ x i − 1 2 ; x i + 1 2 ]; h i = | I i | I 1 × J I 2 × J S 1 , n +1 i ∆ t n S n S n i − 1 i + 1 2 2 S 1 , n S 2 , n i i x i − 1 x i x i + 1 2 2 h i Figure: The control volume I i × J n at the interface for i = i 0 . M. Afif and B. Amaziane Journ´ ee MoMaS 23 Septembre 2010

Finite Volume Scheme Saturation equation � � � ∆ t n K i + j S n +1 α m ( S n +1 ) − α m ( S n +1 + i +2 j ) (2) i i Φ i h i ∆ x i + j − 1 j = ± 1 / 2 2 � � + � � ∆ t n � � + − f m � + ∆ t n = S n f m w ( S n 2 jq n w ( S n − 2 jq n Φ i ρ Q n − i ) i +2 j ) i i + j i + j w , i Φ i h i j = ± 1 / 2 Pressure equation � i +2 j ) 2 � λ m � g ( S n +1 ) i + j K i + j ) + ∆ t n ) 2 − ( P n +1 P n +1 S n +1 ( P n +1 (1 − i i i Φ i h i 2∆ x i + j − 1 j = ± 1 / 2 2 i ) + ∆ t n P n i (1 − S n Q n = (3) g , i Φ i σ g With continuity of the capillary pressure at the interface � � − 1 � � � S m =2 P 2 P 1 S m =1 = ) i 0 c c i 0 M. Afif and B. Amaziane Journ´ ee MoMaS 23 Septembre 2010

Finite Volume Scheme �� �� � � K i + 1 q n ∆ x i λ m ( S n ) i + 1 P n i − P n β m ( S n i ) − β m ( S n 2 := 2 + i +1 ) i + 1 i +1 2 Matrix form � S n +1 �� A n � S n +1 F n = � S n +1 , V n +1 �� B n � V n +1 G n = where for all n = 0 , ..., N − 1 S n := ( S n V n := ( v n i ) N x i := P n i (1 − S n i )) N x and i =0 i =0 � S n +1 �� � S n +1 , V n +1 �� A n � � � N x B n � � � N x A n B n := := and ij ij i , j =0 i , j =0 F n := ( F n G n := ( G n i ) N x i ) N x i =0 . and i =0 M. Afif and B. Amaziane Journ´ ee MoMaS 23 Septembre 2010

Finite Volume Scheme A n and B n are the sparse matrix with non nulls entries: � 1 + ∆ t n α ′ m � S n +1 � K i + j A n ii : = i + j , Φ i h i ∆ x i + j − 1 j = ± 1 / 2 2 − ∆ t n α ′ m � S n +1 � K i + j A n j = ± 1 / i , i +2 j : = i + j , 2 Φ i h i ∆ x i + j − 1 2 � 1 + ∆ t n � � S n +1 �� K i + j v n +1 � B n λ m ii : = , g i + j Φ i h i ∆ x i + j − 1 j = ± 1 / 2 2 − ∆ t n � � S n +1 �� K i + j B n v n +1 � λ m j = ± 1 / i , i +2 j : = , 2 , g i + j Φ i h i ∆ x i + j − 1 2 � 1 ˘ ˘ where for γ m ( S ) := − λ m g ( s ) ds , S � ˘ ˘ � � λ m g ( S i ) if S i = S i +2 j � λ m S i + j := γ m ( S i +2 j ) − γ m ( S i ) g otherwise . S i +2 j − S i M. Afif and B. Amaziane Journ´ ee MoMaS 23 Septembre 2010

L ∞ and weak BV estimates (A0) ρ, σ g , µ ν ( ν = w , g ), are positive constants. (A1) Φ ∈ L ∞ ( I ) such that, 0 < Φ − ≤ Φ( x ) ≤ Φ + ≤ 1 a.e. in I . (A2) 0 < K − ≤ K ( x ) ≤ K + < + ∞ a.e. in I . (A3) S 0 , K ∂ x α ( S 0 ) ∈ L ∞ ( I ) ∩ BV ( I ), 0 ≤ S 0 ( x ) ≤ 1 − ε . (A4) P 0 , K ∂ x P 0 ∈ L ∞ ( I ) ∩ BV ( I ), 0 < P 0 − ≤ P 0 ≤ P 0 + < + ∞ . (A5) λ ν , � λ g ∈ C 1 ([0 , 1]; R + ) such that ∀ s ∈ ]0 , 1[, λ ν ( s ) > 0 and � λ g ( s ) ≥ � λ − g > 0. (A6) λ ∈ C 1 ([0 , 1]; R + ) such that ∀ s ∈ [0 , 1], λ ( s ) ≥ λ − > 0. (A7) f w , ˘ f w ∈ C 1 ([0 , 1]; R + ) such that f w (0) = 0 and ∀ s ∈ ]0 , 1[, f ′ w ( s ) > 0. (A8) α, β ∈ C 1 ([0 , 1]; R + ) such that α ′ , β ′ (0) = 0 and ∀ s ∈ ]0 , 1[, α ′ , β ′ ( s ) > 0. (A9) Q ν ∈ L ∞ ( I × J ) ∩ BV ( I × J ), ∂ t Q ν ∈ L ∞ ( I × J ) and Q ν ( x , t ) ≥ 0 a.e. in I × J . M. Afif and B. Amaziane Journ´ ee MoMaS 23 Septembre 2010

L ∞ and weak BV estimates CFL condition � � 2∆ t n ˘ s f ′ � q n � ∞ ≤ 1 , sup w ( s ) + sup f w ( s ) (4) h φ − s and for i n ∈ { 0 , ..., N x } / S n +1 = max i S n +1 , we assume that i n i � q n � Q n w , i n = 0 and ∂ i n ≥ 0 . (5) � � � f w ( S ) where : ∂ i f := 1 if S � = 0 and ˘ f m 2 − f m f w ( S ) := S i + 1 i − 1 f ′ h i w ( S ) otherwise . 2 Proposition 1. Under the assumptions (A0)–(A9) , the CFL condition (4) and (5), the scheme (2)–(3) is L ∞ stable. Furthermore the following discrete maximum principle holds: for all i = 1 , ..., N x we have 0 ≤ S n +1 ≤ 1. i M. Afif and B. Amaziane Journ´ ee MoMaS 23 Septembre 2010

L ∞ and weak BV estimates Proposition 2. Under the assumptions (A0)–(A9) , the CFL condition (4) and (5), the scheme (2)–(3) is BV stable in space for all n = 1 , ..., N , furthermore we have the L 1 continuity in time. Theorem 1. Under the assumptions (A0)–(A9) , the CFL condition (4) and (5), the approximate solution ( S h , P h ) given by the scheme (2)–(3) converge in L 1 (Ω T ) to ( S , P ) a weak solution of (1) as H and ∆ t go to zero. M. Afif and B. Amaziane Journ´ ee MoMaS 23 Septembre 2010

Numerical results Test case BOBG (C. Chavant, 2008) BO BG S = 1 . 0 S = 0 . 77 P g = 10 5 Pa P g = P w = 10 5 Pa P w = P g − P c ( S ) � � � 1 − Bm � − B m 1 P c Capillary pressure: S m ( P c ) := 1 + A m 5 Relative permeability: kr g ( S ) := (1 − S 2 )(1 − S 3 ) and � � − E m for m = 1 or 2. 1 + ( S − Cm − 1) Dm kr m w ( S ) := F m K m ( m 2 ) m Φ m A m ( Pa ) B m C m D m E m F m 1 0.30 1.E-20 1.5E+6 0.060 16.67 1.880 0.5 4.0 2 0.05 1.E-19 1.0E+7 0.412 2.429 1.176 1.0 1.0 M. Afif and B. Amaziane Journ´ ee MoMaS 23 Septembre 2010

Numerical results � � � 1 − Bm � − B m 1 Capilary pressure: S m ( P c ) := P c 1 + A m 5 Relative permeability: kr g ( S ) := (1 − S 2 )(1 − S 3 ) and � � − E m for m = 1 or 2. 1 + ( S − Cm − 1) Dm kr m w ( S ) := F m 1 1E+09 0.9 9E+08 6E+06 kr w 1 (s) 0.8 8E+08 4E+06 2 (s) kr w 0.7 7E+08 2E+06 kr g (s) 0.6 6E+08 0 0.9 0.925 0.95 0.975 1 0.5 5E+08 0.4 4E+08 Pc 1 (s) 0.3 3E+08 2 (s) Pc 0.2 2E+08 0.1 1E+08 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Relative permeability (left) and Capillary pressure (right) M. Afif and B. Amaziane Journ´ ee MoMaS 23 Septembre 2010

Numerical results Total mobility: λ m ( S ) := kr m w ( S ) + kr g ( S ) µ w µ w kr m w ( S ) fractional flow: f m w ( S ) := µ w λ m ( S ) , for m = 1 or 2 1 2000 50000 2E-07 1500 0.8 40000 1000 1E-07 0.6 500 30000 0 0.75 0.8 0.85 0.9 0.95 1 0 0 0.1 0.2 0.3 0.4 0.4 1 (s) lambda 20000 1 (s) f w 2 (s) lambda 0.2 2 (s) f w 10000 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Total mobility λ ( S ) (left) and fractional flow f w (right) M. Afif and B. Amaziane Journ´ ee MoMaS 23 Septembre 2010

Numerical results BO BG S = 1 . 0 S = 0 . 77 P g = 10 5 Pa P g = P w = 10 5 Pa P w = P g − P c ( S ) 1 0.95 0.9 0.85 0.8 0.75 S w (t=0) S w (1.e5 s) 0.7 S w (1.e6 s) S w (1.e7 s) 0.65 S w (1 y) S w (10 y) 0.6 S w (1.e2 y) S w (5.e2 y) 0.55 S w (1.e3 y) 0.5 -0.5 -0.25 0 0.25 0.5 Figure: Water saturation S M. Afif and B. Amaziane Journ´ ee MoMaS 23 Septembre 2010

Numerical results BO BG S = 1 . 0 S = 0 . 77 P g = 10 5 Pa P g = P w = 10 5 Pa P w = P g − P c ( S ) 1 9E+07 0.95 8E+07 P c (t=0) P c (1.e5 s) 0.9 7E+07 P c (1.e6 s) P c (1.e7 s) 0.85 P c (1 y) 6E+07 P c (10 y) 0.8 P c (1.e2 y) 5E+07 P c (5.e2 y) 0.75 S w (t=0) P c (1.e3 y) 4E+07 S w (1.e5 s) 0.7 S w (1.e6 s) 3E+07 S w (1.e7 s) 0.65 S w (1 y) 2E+07 S w (10 y) 0.6 S w (1.e2 y) S w (5.e2 y) 1E+07 0.55 S w (1.e3 y) 0 0.5 -0.5 -0.25 0 0.25 0.5 -0.5 -0.25 0 0.25 0.5 Figure: Water saturation S Figure: Capillary pressure P c M. Afif and B. Amaziane Journ´ ee MoMaS 23 Septembre 2010