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Convergence of a Finite Volume Scheme and Numerical Simulations for Water-Gas Flow in Porous Media M. Afif and B. Amaziane IPRALMA de Pau CNRS-UMR 5142 brahim.amaziane@univ-pau.fr Journ ee MoMaS : Mod elisation des Ecoulements

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SLIDE 1

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

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SLIDE 2

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

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SLIDE 3

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 Im.

Gas pressure - water saturation formulation          0 ≤ S(x, t) ≤ 1 in I×]0, T[, Φ∂tS + ∂x

  • fw(S)q
  • − ∂x
  • K∂xα(S)
  • = Qw

ρ

in I×]0, T[, Φ∂t

  • P(1 − S)
  • − ∂x
  • Pλg(S)K∂xP
  • = Qg

σg

in I×]0, T[, q = −λ(S)K∂x

  • P + β(S)
  • in I×]0, T[,

(1) Φ(x) is the porosity, K(x) absolute permeability, Qν the source term of phase ν = w, g, krν(S) the ν-phase relative permeability, λν(S) = krν(S)

µν

where µν the ν-phase viscosity, λ(S) = λw(S) + λg(S) is the total mobility.

  • M. Afif and B. Amaziane

Journ´ ee MoMaS 23 Septembre 2010

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SLIDE 4

Finite Volume Scheme

Let {t0, ..., tN} be a partition of J = [0, T]; ∆tn = tn+1 − tn; Let

  • xi

Nx

i=0 be a partition of I and xi+ 1

2 =

  • xi+1 + xi
  • /2;

Vertex-Centred control volumes Ii := [xi− 1

2 ; xi+ 1 2]; hi = |Ii|

I 1 × J I 2 × J xi xi− 1

2

xi+ 1

2

hi S1,n

i

S2,n

i

S1,n+1

i

Sn

i− 1

2

Sn

i+ 1

2

∆tn

Figure: The control volume Ii × Jn at the interface for i = i0.

  • M. Afif and B. Amaziane

Journ´ ee MoMaS 23 Septembre 2010

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SLIDE 5

Finite Volume Scheme

Saturation equation

Sn+1

i

+ ∆tn Φihi

  • j=±1

/

2

  • αm(Sn+1

i

) − αm(Sn+1

i+2j)

  • Ki+j

∆xi+j− 1

2

(2) = Sn

i

− ∆tn Φihi

  • j=±1

/

2

  • f m

w (Sn i )

  • 2jqn

i+j

+ − f m

w (Sn i+2j)

  • − 2jqn

i+j

+ + ∆tn Φiρ Qn

w,i

Pressure equation

Pn+1

i

(1 − Sn+1

i

) + ∆tn Φihi

  • j=±1

/

2

  • (Pn+1

i

)2 − (Pn+1

i+2j)2λm g (Sn+1)i+jKi+j

2∆xi+j− 1

2

= Pn

i (1 − Sn i ) + ∆tn

Φiσg Qn

g,i

(3) With continuity of the capillary pressure at the interface Sm=2

i0

=

  • P2

c

−1 P1

c

  • Sm=1

i0

)

  • M. Afif and B. Amaziane

Journ´ ee MoMaS 23 Septembre 2010

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Finite Volume Scheme

qn

i+ 1

2 :=

Ki+ 1

2

∆xi λm(Sn)i+ 1

2

  • Pn

i − Pn i+1

  • +
  • βm(Sn

i ) − βm(Sn i+1)

  • Matrix form
  • An

Sn+1 Sn+1 = Fn

  • Bn

Sn+1, Vn+1 Vn+1 = Gn where for all n = 0, ..., N − 1 Sn := (Sn

i )Nx i=0

and Vn := (v n

i := Pn i (1 − Sn i ))Nx i=0

  • An

Sn+1 :=

  • An

ij

Nx

i,j=0

and

  • Bn

Sn+1, Vn+1 :=

  • Bn

ij

Nx

i,j=0

Fn := (F n

i )Nx i=0

and Gn := (G n

i )Nx i=0.

  • M. Afif and B. Amaziane

Journ´ ee MoMaS 23 Septembre 2010

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SLIDE 7

Finite Volume Scheme

An and Bn are the sparse matrix with non nulls entries: An

ii :

= 1 + ∆tn Φihi

  • j=±1

/

2

Ki+j ∆xi+j− 1

2

α′m Sn+1

i+j,

An

i,i+2j :

= − ∆tn Φihi Ki+j ∆xi+j− 1

2

α′m Sn+1

i+j,

j = ±1/

2

Bn

ii :

= 1 + ∆tn Φihi

  • j=±1

/

2

  • v n+1

λm

g

  • Sn+1

i+j

Ki+j ∆xi+j− 1

2

, Bn

i,i+2j :

= − ∆tn Φihi

  • v n+1

λm

g

  • Sn+1

i+j

Ki+j ∆xi+j− 1

2

, j = ±1/

2,

where for γm(S) := − 1

S

˘ ˘ λm

g (s)ds,

  • λm

g

  • S
  • i+j :=

˘ ˘ λm

g (Si)

if Si = Si+2j

γm(Si+2j)−γm(Si) Si+2j−Si

  • therwise.
  • M. Afif and B. Amaziane

Journ´ ee MoMaS 23 Septembre 2010

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SLIDE 8

L∞ and weak BV estimates

(A0) ρ, σg, µν(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) S0, K∂xα(S0) ∈ L∞(I) ∩ BV (I), 0 ≤ S0(x) ≤ 1 − ε. (A4) P0, K∂xP0 ∈ L∞(I) ∩ BV (I), 0 < P0

− ≤ P0 ≤ P0 + < +∞.

(A5) λν, λg ∈ C 1([0, 1]; R+) such that ∀s ∈ ]0, 1[, λν(s) > 0 and λg(s) ≥ λ−

g > 0.

(A6) λ ∈ C 1([0, 1]; R+) such that ∀s ∈ [0, 1], λ(s) ≥ λ− > 0. (A7) fw, ˘ fw ∈ C 1([0, 1]; R+) such that fw(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) Qν ∈ L∞(I × J) ∩ BV (I × J), ∂tQν ∈ L∞(I × J) and Qν(x, t) ≥ 0 a.e. in I × J.

  • M. Afif and B. Amaziane

Journ´ ee MoMaS 23 Septembre 2010

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SLIDE 9

L∞ and weak BV estimates

CFL condition 2∆tn hφ−

  • sup

s f ′ w(s) + sup s

˘ fw(s)

  • qn∞ ≤ 1,

(4) and for in ∈ {0, ..., Nx} / Sn+1

in

= maxi Sn+1

i

, we assume that Qn

w,in = 0 and ∂in

  • qn

≥ 0. (5) where : ∂if := 1 hi

  • f m

i+ 1

2 − f m

i− 1

2

  • and ˘

fw(S) :=

  • fw(S)

S

if S = 0 f ′

w(S)

  • therwise.

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, ..., Nx we have 0 ≤ Sn+1

i

≤ 1.

  • M. Afif and B. Amaziane

Journ´ ee MoMaS 23 Septembre 2010

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SLIDE 10

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 L1 continuity in time. Theorem 1. Under the assumptions (A0)–(A9), the CFL condition (4) and (5), the approximate solution (Sh, Ph) given by the scheme (2)–(3) converge in L1(ΩT) 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

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SLIDE 11

Numerical results

Test case BOBG (C. Chavant, 2008) Pg = 105 Pa Pw = Pg − Pc(S) Pg = Pw = 105 Pa S = 0.77 S = 1.0 BO BG Capillary pressure: Sm(Pc) :=

  • 1 +
  • Pc

Am

  • 1

1−Bm −Bm

Relative permeability: krg(S) := (1 − S2)(1 − S

5 3) and

krm

w (S) :=

  • 1 + (S−Cm−1)Dm

Fm

−Em for m = 1 or 2. m Φm Km (m2) Am (Pa) Bm Cm Dm Em Fm 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

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SLIDE 12

Numerical results

Capilary pressure: Sm(Pc) :=

  • 1 +
  • Pc

Am

  • 1

1−Bm −Bm

Relative permeability: krg(S) := (1 − S2)(1 − S

5 3) and

krm

w (S) :=

  • 1 + (S−Cm−1)Dm

Fm

−Em for m = 1 or 2.

0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

krw

1(s)

krw

2(s)

krg(s)

0.2 0.4 0.6 0.8 1 1E+08 2E+08 3E+08 4E+08 5E+08 6E+08 7E+08 8E+08 9E+08 1E+09

Pc

1(s)

Pc

2(s) 0.9 0.925 0.95 0.975 1 2E+06 4E+06 6E+06

Relative permeability (left) and Capillary pressure (right)

  • M. Afif and B. Amaziane

Journ´ ee MoMaS 23 Septembre 2010

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SLIDE 13

Numerical results

Total mobility: λm(S) := krm

w (S)

µw

+ krg(S)

µw

fractional flow: f m

w (S) := krm

w (S)

µwλm(S), for m = 1 or 2

0.2 0.4 0.6 0.8 1 10000 20000 30000 40000 50000

lambda

1(s)

lambda

2(s) 0.75 0.8 0.85 0.9 0.95 1 500 1000 1500 2000 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

fw

1(s)

fw

2(s)

0.1 0.2 0.3 0.4 1E-07 2E-07

Total mobility λ(S) (left) and fractional flow fw (right)

  • M. Afif and B. Amaziane

Journ´ ee MoMaS 23 Septembre 2010

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SLIDE 14

Numerical results

Pg = 105 Pa Pw = Pg − Pc(S) Pg = Pw = 105 Pa S = 0.77 S = 1.0 BO BG

  • 0.5
  • 0.25

0.25 0.5 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Sw(t=0) Sw(1.e5 s) Sw(1.e6 s) Sw(1.e7 s) Sw(1 y) Sw(10 y) Sw(1.e2 y) Sw(5.e2 y) Sw(1.e3 y)

Figure: Water saturation S

  • M. Afif and B. Amaziane

Journ´ ee MoMaS 23 Septembre 2010

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SLIDE 15

Numerical results

Pg = 105 Pa Pw = Pg − Pc(S) Pg = Pw = 105 Pa S = 0.77 S = 1.0 BO BG

  • 0.5
  • 0.25

0.25 0.5 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Sw(t=0) Sw(1.e5 s) Sw(1.e6 s) Sw(1.e7 s) Sw(1 y) Sw(10 y) Sw(1.e2 y) Sw(5.e2 y) Sw(1.e3 y)

Figure: Water saturation S

  • 0.5
  • 0.25

0.25 0.5 1E+07 2E+07 3E+07 4E+07 5E+07 6E+07 7E+07 8E+07 9E+07 Pc(t=0) Pc(1.e5 s) Pc(1.e6 s) Pc(1.e7 s) Pc(1 y) Pc(10 y) Pc(1.e2 y) Pc(5.e2 y) Pc(1.e3 y)

Figure: Capillary pressure Pc

  • M. Afif and B. Amaziane

Journ´ ee MoMaS 23 Septembre 2010

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SLIDE 16

Numerical results

  • 0.5
  • 0.25

0.25 0.5 15000 30000 45000 60000 75000 90000 105000 120000 135000 150000 165000 180000 195000 Pg(t=0) Pg(1.e5 s) Pg(1.e6 s) Pg(1.e7 s) Pg(1 y) Pg(10 y) Pg(1.e2 y) Pg(5.e2 y) Pg(1.e3 y)

Figure: Gas pressure P

  • M. Afif and B. Amaziane

Journ´ ee MoMaS 23 Septembre 2010

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SLIDE 17

Numerical results

  • 0.5
  • 0.25

0.25 0.5 15000 30000 45000 60000 75000 90000 105000 120000 135000 150000 165000 180000 195000 Pg(t=0) Pg(1.e5 s) Pg(1.e6 s) Pg(1.e7 s) Pg(1 y) Pg(10 y) Pg(1.e2 y) Pg(5.e2 y) Pg(1.e3 y)

Figure: Gas pressure P

  • 0.5
  • 0.25

0.25 0.5

  • 9E+07
  • 8E+07
  • 7E+07
  • 6E+07
  • 5E+07
  • 4E+07
  • 3E+07
  • 2E+07
  • 1E+07

Pw(t=0) Pw(1.e5 s) Pw(1.e6 s) Pw(1.e7 s) Pw(1 y) Pw(10 y) Pw(1.e2 y) Pw(5.e2 y) Pw(1.e3 y)

Figure: Water pressure Pw

  • M. Afif and B. Amaziane

Journ´ ee MoMaS 23 Septembre 2010

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SLIDE 18

References

  • M. Afif and B. Amaziane, Convergence of a 1–D finite volume

scheme and numerical simulations for water-gas flow in porous

  • media. Submitted to Applied Numerical Mathematics (2010).
  • O. Angelini, C. Chavant, E. Ch´

enier, R. Eymard, S. Granet, Finite volume approximation of a diffusion-dissolution model and application to nuclear waste storage. Preprint submitted to Mathematics and Computers in Simulation, (2010).

  • M. Afif and B. Amaziane

Journ´ ee MoMaS 23 Septembre 2010