Uncertainty Quantification of Two-Phase Flow in Heterogeneous Porous Media

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1 Uncertainty Quantification of Two-Phase Flow in Heterogeneous Porous Media M.Köppel, C.Rohde Institute for Applied Analysis and Numerical Simulation Inria, Nov 15th, 2016

2 Porous Media Examples: sponge, subsurface, fuel cells, (human) tissues, etc pore scale REV solid matrix averaging wetting phase volume V W non-wetting phase volume V NW Parameters: averaged velocity (flux) permeability (proportionality constant in Darcys law) porosity (ratio of volumes) saturation (ratio of fluid phases) 1

3 Motivation Multi-phase flow in heterogeneous porous media permeability layers fractures Quantify uncertain system parameters, e.g. location of the interfaces Layer 1 Layer 2 Small stochastic dimension N Layer 3 2

4 Outline Fractional Flow Formulation Stochastic Galerkin Model Numerical Experiments Conclusion and Outlook 3

5 Outline Fractional Flow Formulation Stochastic Galerkin Model Numerical Experiments Conclusion and Outlook 4

6 Deterministic model Capillarity-free fractional flow formulation of two immiscible and incompressible fluid phases (D R 2, T > 0, D T := D (0, T )) Unknowns: saturation S, total velocity v, global pressure p v = Kλ(S) p, and div (v) = q, in D T, φs t + div (vf (x, S)) q = 0, in D T, S(, 0) = S 0, in D. Mixed hyperbolic-elliptic system coupled by v, λ(s) and f (x, S). 5

7 Fractional flux function with heterogeneity Test case: vertical injection in porous medium with heterogeneous lense f D 1 (1, 1) f 1 f 2 (0, 0) 0.6 D (-0.5, -0.5) 0.2 (-1, -1) S Non-linear flux function f (x, S, γ) := γ(x)f 1 (S) + (1 γ(x))f 2 (S), with γ(x) := { 1, x D 1, 0, x D 2. 6

8 Vertical injection with heterogeneous lense (deterministic) (a) (b) Saturation (a) and total velocity (b), Standing shock wave at the heterogeneity interface, (a): central-upwind FV scheme (Kurganov & Petrova, 2005). (b): mixed finite elements (Alberta-FEM, Schmidt & Siebert, 2005). 7

9 Fractional flux function with heterogeneity and uncertainty f D 1 (ξ) (1, 1) f 1 f D 2 (ξ) (-1, -1) S Let ξ(ω) := {ξ 1(ω 1),..., ξ N (ω N )} be a N-dimensional random vector on (Ω, P). The non-linear flux function reads f (x, S, γ, ξ) := γ(x, ξ)f 1 (S) + (1 γ(x, ξ))f 2 (S), with γ(x, ξ) := { 1, x D 1 (ξ), ξ [0, 1] N, 0, x D 2 (ξ), ξ [0, 1] N. 8

10 Outline Fractional Flow Formulation Stochastic Galerkin Model Numerical Experiments Conclusion and Outlook 9

11 Polynomial Chaos We set N = 1. Let ξ = ξ(ω) be a random variable on (Ω, P) and {ϕ p(ξ)} p N0 a family of polynomials satisfying ϕ p(ξ), ϕ q(ξ) L 2 (Ω) := ϕ p(ξ)ϕ q(ξ) dp(ω) = δ pq for p, q N 0. ω The random field S = S(x, t, ξ), (x, t) D T, with finite variance can be represented by S ( x, t, ξ ) = S p ( ) (x, t)ϕ p ξ, where p=0 S p := S p (x, t) := S, ϕ p L 2 (Ω) for p N 0, Expectation and variance are given by E [S] = S 0 and Var[S] = (S p ) 2. p=1 10

12 Stochastic Galerkin (SG) approach Exemplarily we consider the hyperbolic equation for given velocity: We test the equation with ϕ 0,..., ϕ No and obtain the system Ω ( ) φs t(ξ) + div (vf(s, γ, ξ)) q ϕ p(ξ) dp(ω) = 0 We replace S(x, t, ξ) by the truncated PC-expansion N o Π No [S] (x, t, ξ) := S p (x, t)ϕ p(ξ). p=0 for p = 0,..., N o. Introduced by R.G. Ghanem and P.D. Spanos (1989) (elliptic problem), R. Abgrall (2008), B. Després et al. (2008) (hyperbolic problem) 11

13 Stochastic Galerkin Orthogonality of ϕ p yields a truncated SG system for the deterministic coefficients S 0,..., S No φst 0 + div vf(π No [S]), ϕ 0 L 2 (Ω) q0 = 0,. φs No t + div vf(π No [S]), ϕ No L 2 (Ω) qno = 0. Hyperbolic (N o + 1)-dimensional system. Numerical challenges: Strongly coupled. System complexity increases w.r.t. polynomial order N o. Parallel computing requires synchronization in each time-step. Huge hyperbolic system and no structural information. 12

14 Hybrid stochastic Galerkin (HSG) approach Multi-element discretization of the stochastic space with 2 Nr R. Bürger, I. Kröker & C. Rohde (2013), (J. Trygoen, O. Le Maître & A. Ern (2012)) elements, Let ξ = ξ(ω) be a random variable. Assume ξ U(0, 1). Define ϕ Nr i,l (ξ) = 2Nr/2 ϕ i(2 Nr ξ l), i = 0,..., N o, l = 0,..., 2 Nr 1. Here ϕ i is the i-th Legendre polynomial. The polynomials ϕ Nr 0,0,..., ϕnr N o,2nr satisfy 1 ϕ Nr i,k, ϕnr j,l = δ ijδ kl. and their support is Supp(ϕ Nr i,k ) = [2 Nr k, 2 Nr (k + 1)]. 13

15 Hybrid stochastic Galerkin approach The projection of a random field S(x, t, ) L 2 (Ω) is defined by Π No,Nr [S] (x, t, ξ) := S Nr i,l (x, t) := 2 Nr 1 l=0 N o i=0 S(x, t, ), ϕ Nr S Nr i,l (x, t)ϕnr i,l (ξ), i,l. For N r, N o N 0 find S Nr i,l : D T R such that ( ) φ tπ No,Nr [S] (ξ) + div (vf(π No,Nr [S], γ, ξ)) q ϕ Nr i,l dp(ω) = 0 Ω for all (i, l), where i = 0,..., N o and l = 0,..., 2 Nr 1. We obtain a (N o + 1)2 Nr - dimensional system 14

16 Hybrid stochastic Galerkin approach Orthogonality of ϕ Nr i,l leads to the partially decoupled system with 2 Nr blocks, each of dimension (N o + 1) φ ts Nr 0,0 + div vf ( Π No,Nr [S], γ ), ϕ Nr 0,0 q N r 0,0 = 0,. φ ts Nr N + div o,0 vf ( Π No,Nr [S], γ ), ϕ Nr N o,0 q N r N o,0 = 0,. φ ts Nr + div vf ( Π No,Nr [S], γ ), ϕ Nr q Nr = 0, 0,2 Nr 1 0,2 Nr 1 0,2 Nr 1. φ ts Nr N o,2 Nr 1 + div vf ( Π No,Nr [S], γ ), ϕ Nr N o,2 Nr 1 q Nr N o,2 Nr 1 = 0. Better for parallel computing Reduction of polynomial order N o 15

17 Stochastic Mixed Finite Element Method (SMFEM) Consideration of the elliptic part K 1 λ 1 (S) v + p = 0 in D (0, T ), div (v) = q in D (0, T ). Basis functions of v and p based on lowest order Raviart-Thomas (RT0) elements and HSG expansion v(x, ξ) Π No,Nr [v h ] := 2 Nr 1 No l=0 i=0 p(x, ξ) Π No,Nr [p h ] := 2 Nr 1 l=0 No i=0 Ne j=1 vnr j,i,l (x)ϑj(x)ϕnr i,l (ξ), Nt j=1 pnr j,i,l (x)χj(x)ϕnr i,l (ξ). with N e number of edges, N t number of triangles, deterministic velocity and pressure basis functions ϑ j(x) resp. χ j(x). Related work of the mixed elliptic problem by L. Traverso, T.N. Phillips, Y. Yang (2014) O.G. Ernst et al. (2009) 16

18 Stochastic Mixed Finite Element Method This yields a matrix saddle-point problem [ ] [ ] ] ˆBT v [ p =, ˆB 0 p q with block matrices  and ˆB given by A 0,0 A 0, P 1  = , A P 1,0 A P 1, P 1 and B 0,0 0 ˆB = G B =..., 0 B P 1, P 1 with P := (N o + 1)2 Nr and diagonal, stochastic matrix G, 17

19 Stochastic Mixed Finite Element Method and block vectors v = v 0. v No2 Nr R Ne(No+1)2Nr, p = p 0. p No2 Nr R N t(n o+1)2 Nr, where each vector block is of size dim(v l ) = N e resp. dim(p l ) = N t for l = 0,..., N o2 Nr. Analogous for p and q. Remark: system matrix is not fully assembled degrees of freedom: (N t + N e)(n o + 1)2 Nr. appropriate preconditioning necessary 18

20 Outline Fractional Flow Formulation Stochastic Galerkin Model Numerical Experiments Conclusion and Outlook 19

21 Vertical injection with heterogeneous lense & randomness (a) (b) Expectation (a) and variance (b) of the saturation, N = 2 (stoch. dim), N r = 1, N o = 1, (hyp system is 12-dimensional), Smeared heterogeneity interface due to uncertainty. 20

22 Vertical injection with heterogeneous lense & randomness (a) (b) Expectation (a) and diag. of covariance (b) of the velocity [m/h], N = 2 (stoch. dim), N r = 1, N o = 1, (ellipt dim: 12 (N t + N e)), smooth transition at vertical boundaries of the lense due to uncertainty. 21

23 Vertical injection with heterogeneous lense & randomness (a) (b) Expectation (a) and variance (b) of the pressure [bar], N = 2 (stoch. dim), N r = 1, N o = 1, (ellipt dim: 12 (N t + N e)). 22

24 Model performance N r/#cores: 0 / 1 1 / 2 2 / 4 3 / 8 4 / 16 5 / 32 N o = 1, h N o = 2, h N o = 3, h N o = 1, N o = 2, N o = 3, N o = 1, % N o = 2, % N o = 3, % Table: Computation time (hours h), average iteration counts ( ) and time ratio of all elliptic solves to global solve (%) with preconditioner (N = 1). 23

25 Outline Fractional Flow Formulation Stochastic Galerkin Model Numerical Experiments Conclusion and Outlook 24

26 Conclusion and Outlook Conclusion First uncertainty quantification (UQ) for full (hyperbolic-elliptic) two-phase system in 2D based on SG. The HSG approach requires a higher dimensional system than SG, but this system is partially decoupled and needs lower polynomial order. Effective parallel computations. Outlook Apply HSG to fractured porous media Comparsion with other UQ techniques 25

27 Thank you for your attention! 26

28 Questions 27