Space-Time Domain Decomposition Methods for Transport Problems in Porous Media

Transcription

1 1 / 49 Space-Time Domain Decomposition Methods for Transport Problems in Porous Media Thi-Thao-Phuong Hoang, Elyes Ahmed, Jérôme Jaffré, Caroline Japhet, Michel Kern, Jean Roberts INRIA Paris-Rocquencourt Maison de la Simulation October 16, 2015 Enit Lamsin, Tunis

2 2 / 49 OUTLINE 1 Introduction 2 Pure diffusion problems Multi-domain mixed formulations Nonconforming discretizations in time 3 Advection-diffusion problems Operator splitting 4 Extension to two-phase flow 5 Extension to reduced fracture models

3 3 / 49 Outline Introduction 1 Introduction 2 Pure diffusion problems Multi-domain mixed formulations Nonconforming discretizations in time 3 Advection-diffusion problems Operator splitting 4 Extension to two-phase flow 5 Extension to reduced fracture models

4 4 / 49 Introduction Objective: to formulate numerical methods for flow and transport in heterogeneous porous media Examples of heterogeneous media:

5 4 / 49 Introduction Objective: to formulate numerical methods for flow and transport in heterogeneous porous media Examples of heterogeneous media: porous media around underground nuclear waste deposit sites

6 5 / 49 Introduction Heterogeneities mean difficulties for simulation Deep underground repository (High-level waste) A repository 2km 2km

7 5 / 49 Introduction Heterogeneities mean difficulties for simulation Deep underground repository (High-level waste) Different materials strong heterogeneity, different time scales. Large differences in spatial scales. Long-term computations. A repository 2km 2km

8 6 / 49 Introduction Objective: to formulate numerical methods for flow and transport in heterogeneous porous media Examples of heterogeneous media: porous media around underground nuclear waste deposit sites porous media with fractures

9 7 / 49 Introduction Difficulty for modeling flow in media with fractures A problem requiring multi-scale modelling

10 Introduction Difficulty for modeling flow in media with fractures A problem requiring multi-scale modelling Fractures represent heterogeneities in porous media Usually of much higher permeability than surrounding medium May be of much lower permeability so that they act as a barrier 7 / 49

11 Introduction Difficulty for modeling flow in media with fractures A problem requiring multi-scale modelling Fractures represent heterogeneities in porous media Usually of much higher permeability than surrounding medium May be of much lower permeability so that they act as a barrier Fracture width much smaller than any reasonable parameter of spatial discretization. 7 / 49

12 Introduction Difficulty for modeling flow in media with fractures A problem requiring multi-scale modelling Fractures represent heterogeneities in porous media Usually of much higher permeability than surrounding medium May be of much lower permeability so that they act as a barrier Fracture width much smaller than any reasonable parameter of spatial discretization. Different types of models for flow in fractures double continuum models. discrete fracture networks (DFN s) (no exchange with surrounding matrix rock) reduced fracture models (with exchange with matrix rock) 7 / 49

13 8 / 49 Introduction Objective here: to formulate methods for subdomain time-stepping More specifically: develop and compare two different space-time (global in time) domain decomposition methods for the linear transport problem in mixed formulation. extend these methods to the case of a domain with a discrete fracture extend these method to two phase flow models, with discontinuous capillary pressure (in progress)

14 9 / 49 Introduction Domain decomposition (DD) methods Domain decomposition in space t y x

15 9 / 49 Introduction Domain decomposition (DD) methods Domain decomposition in space t y x

16 Introduction 9 / 49 Domain decomposition (DD) methods Domain decomposition in space t y x Discretize in time and apply DD algorithm at each time step: Solve stationary problems in the subdomains Exchange information through the interface Use the same time step on the whole domain.

17 Introduction 9 / 49 Domain decomposition (DD) methods Domain decomposition in space Space-time domain decomposition t t y y x x Discretize in time and apply DD algorithm at each time step: Solve stationary problems in the subdomains Exchange information through the interface Use the same time step on the whole domain.

18 Introduction 9 / 49 Domain decomposition (DD) methods Domain decomposition in space Space-time domain decomposition t t y y x x Discretize in time and apply DD algorithm at each time step: Solve stationary problems in the subdomains Exchange information through the interface Use the same time step on the whole domain. Solve time-dependent problems in the subdomains Exchange information through the space-time interface Enable local discretizations both in space and in time local time stepping

19 Introduction 9 / 49 Domain decomposition (DD) methods Domain decomposition in space t Space-time DD with Time windows t y y T1 x 0 x Discretize in time and apply DD algorithm at each time step: Solve stationary problems in the subdomains Exchange information through the interface Use the same time step on the whole domain.

20 Introduction 9 / 49 Domain decomposition (DD) methods Domain decomposition in space t Space-time DD with Time windows t y y T2 x T1 0 x Discretize in time and apply DD algorithm at each time step: Solve stationary problems in the subdomains Exchange information through the interface Use the same time step on the whole domain.

21 Introduction 9 / 49 Domain decomposition (DD) methods Domain decomposition in space t Space-time DD with Time windows t y T3 y T2 x T1 0 x Discretize in time and apply DD algorithm at each time step: Solve stationary problems in the subdomains Exchange information through the interface Use the same time step on the whole domain.

22 Introduction 9 / 49 Domain decomposition (DD) methods Domain decomposition in space t Space-time DD with Time windows t y T y T2 T1 x 0 x Discretize in time and apply DD algorithm at each time step: Solve stationary problems in the subdomains Exchange information through the interface Use the same time step on the whole domain. Perform few iterations per window Use different space-time grids in each window Use the solution in the previous window to calculate a good initial guess on the interface.

23 10 / 49 Outline Pure diffusion problems 1 Introduction 2 Pure diffusion problems Multi-domain mixed formulations Nonconforming discretizations in time 3 Advection-diffusion problems Operator splitting 4 Extension to two-phase flow 5 Extension to reduced fracture models

24 11 / 49 Model problem Pure diffusion problems Multi-domain mixed formulations Transport of a contaminant in a porous medium under the effect of diffusion, written in mixed form: L(c, r) := φ t c + div r = f in Ω (0, T ), M(c, r) := D 1 r + c = 0 in Ω (0, T ), c = 0 on Ω (0, T ), c(, 0) = c 0 in Ω, c concentration of a contaminant dissolved in a fluid, r diffusive flux. φ porosity; D symmetric, positive definite, time-independent diffusion tensor.

25 11 / 49 Model problem Pure diffusion problems Multi-domain mixed formulations Transport of a contaminant in a porous medium under the effect of diffusion, written in mixed form: L(c, r) := φ t c + div r = f in Ω (0, T ), M(c, r) := D 1 r + c = 0 in Ω (0, T ), c = 0 on Ω (0, T ), c(, 0) = c 0 in Ω, c concentration of a contaminant dissolved in a fluid, r diffusive flux. φ porosity; D symmetric, positive definite, time-independent diffusion tensor. Existence and uniqueness If D L (Ω), f L 2 (0, T ; L 2 (Ω)) and c 0 H0 1 (Ω) then problem above has a unique weak solution ( ) (c, r) H 1 (0, T ; L 2 (Ω)) L 2 (0, T ; H(div, Ω)) L (0, T ; L 2 (Ω)). Moreover, if D W 1, (Ω), f H 1 (0, T ; L 2 (Ω)) and c 0 H 2 (Ω) H0 1 (Ω), then ( ) (c, r) W 1, (0, T ; L 2 (Ω)) L (0, T ; H(div, Ω)) H 1 (0, T ; L 2 (Ω)). Proof. Galerkin s method and a priori estimates.

26 Pure diffusion problems Multi-domain mixed formulations Multi-domain problem t T y Γ Ω 1 Ω 2 x 0 Equivalent multi-domain problem: L(c 1, r 1 ) = f, on Ω 1 (0, T ), M(c 1, r 1 ) = 0, on Ω 1 (0, T ), c 1 (, 0) = c 0, in Ω 1, L(c 2, r 2 ) = f, on Ω 2 (0, T ), M(c 2, r 2 ) = 0, on Ω 2 (0, T ), c 2 (, 0) = c 0, in Ω 2, 12 / 49

27 Pure diffusion problems Multi-domain mixed formulations Multi-domain problem t T y Γ Ω 1 Ω 2 x 0 Equivalent multi-domain problem: L(c 1, r 1 ) = f, on Ω 1 (0, T ), M(c 1, r 1 ) = 0, on Ω 1 (0, T ), c 1 (, 0) = c 0, in Ω 1, L(c 2, r 2 ) = f, on Ω 2 (0, T ), M(c 2, r 2 ) = 0, on Ω 2 (0, T ), c 2 (, 0) = c 0, in Ω 2, together with the transmission conditions on the space-time interface c 1 = c 2 r 1 n 1 + r 2 n 2 = 0 on Γ (0, T ). 12 / 49

28 13 / 49 An overview Pure diffusion problems Multi-domain mixed formulations Different (equivalent) transmission conditions (TCs)

29 13 / 49 An overview Pure diffusion problems Multi-domain mixed formulations Different (equivalent) transmission conditions (TCs) GTP Schur Physical TCs + N-N preconditioner GTO Schwarz More general TCs with optimized parameters accelerate the convergence rate. Robin TCs Ventcell TCs

30 13 / 49 An overview Pure diffusion problems Multi-domain mixed formulations Different (equivalent) transmission conditions (TCs) GTP Schur Physical TCs + N-N preconditioner GTO Schwarz More general TCs with optimized parameters accelerate the convergence rate. Robin TCs Ventcell TCs Substructuring technique: Space-time interface problem

31 13 / 49 An overview Pure diffusion problems Multi-domain mixed formulations Different (equivalent) transmission conditions (TCs) GTP Schur Physical TCs + N-N preconditioner GTO Schwarz More general TCs with optimized parameters accelerate the convergence rate. Robin TCs Ventcell TCs Substructuring technique: Space-time interface problem Iterative solvers (GMRES, Richardson iteration)

32 T.T.P. Hoang, J. Jaffré, C. Japhet, M.K, and J. E. Roberts. Space-time domain decomposition methods for diffusion problems in mixed formulations. SIAM J. Numer. Anal., 51(6): , / 49 An overview Pure diffusion problems Multi-domain mixed formulations Different (equivalent) transmission conditions (TCs) GTP Schur Physical TCs + N-N preconditioner GTO Schwarz More general TCs with optimized parameters accelerate the convergence rate. Robin TCs Ventcell TCs Substructuring technique: Space-time interface problem Iterative solvers (GMRES, Richardson iteration)

33 14 / 49 Pure diffusion problems Multi-domain mixed formulations Time-dependent Steklov-Poincaré operator Dirichlet-to-Neumann operators, for i = 1, 2: where (c i, r i ), i = 1, 2, is the solution of S DtN i : (λ, f, c 0 ) (r i n i ) Γ, L(c i, r i ) = f, on Ω i (0, T ), M(c i, r i ) = 0, on Ω i (0, T ), c i = λ, on Γ (0, T ), c i (, 0) = c 0, in Ω i.

34 14 / 49 Pure diffusion problems Multi-domain mixed formulations Time-dependent Steklov-Poincaré operator Dirichlet-to-Neumann operators, for i = 1, 2: where (c i, r i ), i = 1, 2, is the solution of S DtN i : (λ, f, c 0 ) (r i n i ) Γ, L(c i, r i ) = f, on Ω i (0, T ), M(c i, r i ) = 0, on Ω i (0, T ), c i = λ, on Γ (0, T ), c i (, 0) = c 0, in Ω i. Space-time interface problem: S1 DtN (λ, f, c 0 ) + S2 DtN (λ, f, c 0 ) = 0, 2 2 Si DtN (λ, 0, 0) = Si DtN (0, f, c 0 ), i=1 i=1 Sλ = χ, on Γ (0, T ).

35 14 / 49 Pure diffusion problems Multi-domain mixed formulations Time-dependent Steklov-Poincaré operator Dirichlet-to-Neumann operators, for i = 1, 2: where (c i, r i ), i = 1, 2, is the solution of S DtN i : (λ, f, c 0 ) (r i n i ) Γ, L(c i, r i ) = f, on Ω i (0, T ), M(c i, r i ) = 0, on Ω i (0, T ), c i = λ, on Γ (0, T ), c i (, 0) = c 0, in Ω i. Space-time interface problem: S1 DtN (λ, f, c 0 ) + S2 DtN (λ, f, c 0 ) = 0, 2 2 Si DtN (λ, 0, 0) = Si DtN (0, f, c 0 ), i=1 i=1 Sλ = χ, on Γ (0, T ). Neumann-Neumann preconditioner with weights: ( ) σ 1 S1 NtD + σ 2 S2 NtD Sλ = χ, on Γ (0, T ), where σ i : Γ (0, T ) [0, 1] such that σ 1 + σ 2 = 1.

36 15 / 49 Pure diffusion problems Multi-domain mixed formulations GTO Schwarz: Robin transmission conditions Equivalent Robin TCs on Γ (0, T ): for α 1, α 2 > 0 r 1 n 1 + α 1 c 1 = r 2 n 1 + α 1 c 2, r 2 n 2 + α 2 c 2 = r 1 n 2 + α 2 c 1,

37 15 / 49 Pure diffusion problems Multi-domain mixed formulations GTO Schwarz: Robin transmission conditions Equivalent Robin TCs on Γ (0, T ): for α 1, α 2 > 0 r 1 n 1 + α 1 c 1 = r 2 n 1 + α 1 c 2, r 2 n 2 + α 2 c 2 = r 1 n 2 + α 2 c 1, Robin-to-Robin operators, for i = 1, 2 and j = 3 i: where (c i, r i ), i = 1, 2, is the solution of S RtR i : (ξ i, f, c 0 ) ( r i n j + α j c i ) Γ, L(c i, r i ) = f, on Ω i (0, T ), M(c i, r i ) = 0, on Ω i (0, T ), r i n i + α i c i = ξ i, on Γ (0, T ), c i (, 0) = c 0, in Ω i.

38 15 / 49 Pure diffusion problems Multi-domain mixed formulations GTO Schwarz: Robin transmission conditions Equivalent Robin TCs on Γ (0, T ): for α 1, α 2 > 0 r 1 n 1 + α 1 c 1 = r 2 n 1 + α 1 c 2, r 2 n 2 + α 2 c 2 = r 1 n 2 + α 2 c 1, Robin-to-Robin operators, for i = 1, 2 and j = 3 i: where (c i, r i ), i = 1, 2, is the solution of S RtR i : (ξ i, f, c 0 ) ( r i n j + α j c i ) Γ, L(c i, r i ) = f, on Ω i (0, T ), M(c i, r i ) = 0, on Ω i (0, T ), r i n i + α i c i = ξ i, on Γ (0, T ), c i (, 0) = c 0, in Ω i. Space-time interface problem with two Lagrange multipliers: or equivalently, ξ 1 ξ 2 = S2 RtR (ξ 2, f, c 0 ), = S1 RtR (ξ 1, f, c 0 ), on Γ (0, T ), S R ( ξ1 ξ 2 ) = χ R, on Γ (0, T ).

39 16 / 49 Pure diffusion problems OSWR algorithm with Robin TCs Multi-domain mixed formulations OSWR iterative algorithm: at the k th iteration, for i = 1, 2 and j = (3 i) φ i t ci k + div r k i = f in Ω i (0, T ) D 1 i r k i + ci k = 0 in Ω i (0, T ) r k i n i + α i ci k ( = r k 1 j n i + α i c k 1 j on Γ (0, T ), ) for given initial guess g i = r 0 i n i + α i ci 0, i = 1, 2.

40 16 / 49 Pure diffusion problems OSWR algorithm with Robin TCs Multi-domain mixed formulations OSWR iterative algorithm: at the k th iteration, for i = 1, 2 and j = (3 i) φ i t ci k + div r k i = f in Ω i (0, T ) D 1 i r k i + ci k = 0 in Ω i (0, T ) r k i n i + α i ci k ( = r k 1 j n i + α i c k 1 j on Γ (0, T ), ) for given initial guess g i = r 0 i n i + α i ci 0, i = 1, 2. Theorem (Convergence of OSWR algorithm in mixed formulation) ( ) If the algorithm above is initialized by (g i ) H 1 0, T ; L 2 (Γ) for i = 1, 2, then ( ) ( ) a sequence of iterates ci k, r k i H 1 0, T ; L 2 (Ω i ) L 2 (0, T ; H (div, Ω i )) is well-defined 2 i=1 ( ) ci k c Ωi H1 (0,T ;L 2 (Ω i )) + rk i r Ωi 2 k L 2 0. (0,T ;H(div,Ω i )) where H (div, Ω i ) := { } v H (div, Ω i ) : (v n i ) Γ L 2 (Γ). Remark. The proof is carried out for the multiple subdomain case.

41 17 / 49 Pure diffusion problems Multi-domain mixed formulations GTO Schwarz: Ventcell transmission conditions With sufficient regularity equivalent Ventcell transmission conditions

42 17 / 49 Pure diffusion problems Multi-domain mixed formulations GTO Schwarz: Ventcell transmission conditions With sufficient regularity equivalent Ventcell transmission conditions In primal form: on Γ (0, T ): r 1 n 1 + α 1 c 1 + = r 2 n 1 + α 1 c 2 + r 2 n 2 + α 2 c 2 + = r 1 n 2 + α 2 c 1 +

43 Pure diffusion problems Multi-domain mixed formulations 17 / 49 GTO Schwarz: Ventcell transmission conditions With sufficient regularity equivalent Ventcell transmission conditions In primal form: on Γ (0, T ): ( r 1 n 1 + α 1 c 1 + β 1 φ2 t c 1 + div τ ( D 2,Γ τ c 1 ) ) = r 2 n 1 + α 1 c 2 + ( β 1 φ2 t c 2 + div τ ( D 2,Γ τ c 2 ) ), ( r 2 n 2 + α 2 c 2 + β 2 φ1 t c 2 + div τ ( D 1,Γ τ c 2 ) ) = r 1 n 2 + α 2 c 1 + ( β 2 φ1 t c 1 + div τ ( D 1,Γ τ c 1 ) ). α i, β i : positive constants to be optimized to accelerate convergence rate.

44 17 / 49 Pure diffusion problems Multi-domain mixed formulations GTO Schwarz: Ventcell transmission conditions With sufficient regularity equivalent Ventcell transmission conditions In primal form: on Γ (0, T ): r 1 n 1 + α 1 c 1 + β 1 ( φ2 t c 1 + div τ ( D 2,Γ τ c 1 ) ) = r 2 n 1 + α 1 c 2 + r 2 n 2 + α 2 c 2 + β 2 ( φ1 t c 2 + div τ ( D 1,Γ τ c 2 ) ) = r 1 n 2 + α 2 c 1 + β 1 ( φ2 t c 2 + div τ ( D 2,Γ τ c 2 ) ), β 2 ( φ1 t c 1 + div τ ( D 1,Γ τ c 1 ) ). α i, β i : positive constants to be optimized to accelerate convergence rate. In mixed form: introduce Lagrange multipliers on the interface, c i,γ and r Γ,i, for i = 1, 2, ( ) r i n i + α i c i,γ + β i φj t c i,γ + div τ r Γ,i = r j n i + α i c j,γ + ( ( )) β i φ j t c j,γ + div τ D j,γ D 1 i,γ r Γ,j, D 1 j,γ r Γ,i + τ c i,γ = 0. c i,γ : pressure trace on Γ. r Γ,i := D j,γ τ c i,γ : NOT the tangential trace of r i on Γ (0, T ).

45 18 / 49 Pure diffusion problems Nonconforming discretizations in time Nonconforming discretizations in time t T t 1 n t 2 m 0 Ω 1 Ω 2 x

46 18 / 49 Pure diffusion problems Nonconforming discretizations in time Nonconforming discretizations in time t T t 1 n t 2 m 0 Ω 1 Ω 2 x Time discretization: non-conforming time grids T 1, T 2 ; discontinuous Galerkin with piecewise polynomials of degree 0. Projections: Π ji is an L 2 projection from piecewise constant functions on T i onto piecewise constant functions on T j. Ex: (Π 21 (λ 1 )) m = 1 M 1 Jm 2 λ 1, for m = 1,, M 2. n=1 J 1 n J 2 m

47 19 / 49 Pure diffusion problems Nonconforming discretizations in time Semi-discrete transmission conditions with nonconforming time grids ( ) For GTP Schur method: take λ = λ 1,, λ M 1 piecewise constant on ( ) Jn 1 = t1 n, t n+1 1, for n = 0,, M 1 1. Continuity of concentration: c 1 = Π 11 (λ) and c 2 = Π 21 (λ). Conservation of the flux over each time subinterval (Π 11 (r 1 n 1 ) + Π 12 (r 2 n 2 )) dt = 0, for n = 0,, M 1 1. J 1 n Γ

48 19 / 49 Pure diffusion problems Nonconforming discretizations in time Semi-discrete transmission conditions with nonconforming time grids ( ) For GTP Schur method: take λ = λ 1,, λ M 1 piecewise constant on ( ) Jn 1 = t1 n, t n+1 1, for n = 0,, M 1 1. Continuity of concentration: c 1 = Π 11 (λ) and c 2 = Π 21 (λ). Conservation of the flux over each time subinterval (Π 11 (r 1 n 1 ) + Π 12 (r 2 n 2 )) dt = 0, for n = 0,, M 1 1. J 1 n Γ For GTO Schwarz method: conservation of the two Robin (Ventcell) conditions across the interface over each time subinterval [( r 1 n 1 + α 1 c 1 ) Π 12 ( r 2 n 1 + α 1 c 2 )] dt = 0, n = 0,, M 1 1, J 1 n J 2 m Γ Γ [( r 2 n 2 + α 2 c 2 ) Π 21 ( r 1 n 2 + α 2 c 1 )] dt = 0, m = 0,, M 2 1. Convergence of semi-discrete, nonconforming in time, OSWR algorithm

49 20 / 49 Outline Advection-diffusion problems 1 Introduction 2 Pure diffusion problems Multi-domain mixed formulations Nonconforming discretizations in time 3 Advection-diffusion problems Operator splitting 4 Extension to two-phase flow 5 Extension to reduced fracture models

50 21 / 49 Advection-diffusion problems Operator splitting Extension to advection-diffusion problems Linear advection-diffusion equation: Operator splitting φ tc + div (uc) + div r = f in Ω (0, T ), c + D 1 r = 0 in Ω (0, T ), c = 0 on Ω (0, T ), c(, 0) = c 0 in Ω. Advection eq.: explicit Euler + upwind, cell-centered finite volumes. Diffusion eq.: implicit Euler + mixed finite elements. CFL condition: sub-time steps for the advection. T = N 1 t 1 = N 2 t 2 t T L 1 t 1,a = t 1 t 2 = L 2 t 2,a 0 Ω 1 Ω 2 x

51 Advection-diffusion problems Discrete interface problems Operator splitting GTP Schur method: S h ( λa λ ) = χ h, on Γ (0, T ). = Generalized Neumann-Neumann preconditioner GTO Schwarz method with Robin TCs: λ a S R,h ξ 1 = χ R,h, on Γ (0, T ). ξ 2 = Optimized Robin parameters for the diffusion eq. only fully implicit scheme. Remark. λ a Λ N L h while λ, ξ 1, ξ 2 Λ N h. T.T.P. Hoang, J. Jaffré, C. Japhet, M.K., and J. E. Roberts. Proc. Mamern 2015, in preparation. 22 / 49

52 Advection-diffusion problems Operator splitting Test case 2: A near-field simulation (project PAMINA ) Performance Assessment Methodologies IN Application to Guide the Development of the Safety Case Parameters of the simulation Material Permeability (m.s 1 ) Porosity Diffusion (m 2. s 1 ) Host rock EDZ Vitrified waste m 0.6m 100m m / 49

53 Advection-diffusion problems Operator splitting Advection field: Darcy flow div u = 0 in Ω, u = K p in Ω. BCs: Homogeneous Neumann at x = 0 and x = 10, Dirichlet conditions with p = 100 Pa at y = 0 and p = 0 at y = / 49

54 25 / 49 Advection-diffusion problems Operator splitting Transport problem: time windows and decomposition Ω 7 Ω 8 Ω 9 Ω 4 Ω 5 Ω 6 Ω 1 Ω 2 Ω 3 Final time: T f = s ( years) 200 time windows with size T = 10 9 s. Decomposition into 9 subdomains. Nonconforming time grids: Diffusion step: t i = T /500, i = 5, t i = T /100, i 5. calculation and its decomposition. Diffusion-dominated: Pe L t a,i = t i. Non-uniform mesh in space: uniform mesh in the repository (10 by 10), then progressively coarser with a factor of 1.05.

55 26 / 49 Advection-diffusion problems Evolution of concentration field Operator splitting 100 years 5000 years years years In the repository In the host rock

56 27 / 49 Advection-diffusion problems Operator splitting Performance of one time window Log of error in c No Precond. Schur Precond. Schur Opt. Schwarz Error in c Error in r Number of subdomain solves Convergence with GMRES t Error in c with nonconforming time grids t Error in r with nonconforming time grids. Time grid 1: t i = T /500, i Time grid 2: t 5 = T /500, t i = T /100, i 5 Time grid 3: t 5 = T /100, t i = T /500, i 5 Time grid 4: t i = T /100, i

57 A subsurface waste storage simulation Zone Hydraulic conductivity Porosity Molecular diffusion K (m/year) φ d m (m 2 /year) terrain dalleprotec/dalleobtur voile remplissage conteneur1/conteneur dechet1/dechet radier drainant dalleobtur dalleprotec terrain voile remplissage conteneur2 conteneur1 dechet1 9.6 m dechet2 drainant forme radier drain 12 m

58 Advection-diffusion problems Operator splitting 29 / 49 Darcy flow div u = 0 in Ω, u = K h in Ω. BCs: Homogeneous Neumann at x = 0 and x = 12m, Dirichlet conditions with h = 9.998m at y = 0 and h = 10m at y = 9.6m Hydraulic head field

59 30 / 49 Advection-diffusion problems Operator splitting Concentration field after 500 years

60 31 / 49 Outline Extension to two-phase flow 1 Introduction 2 Pure diffusion problems Multi-domain mixed formulations Nonconforming discretizations in time 3 Advection-diffusion problems Operator splitting 4 Extension to two-phase flow 5 Extension to reduced fracture models

61 Extension to two-phase flow 32 / 49 Model problem: Two phase immiscible flow Mathematical model t (ωρ α S α ) + div (ρ α u α ) = q α mass conservation u α = k rα µ α K ( p α ρ α g) Darcy s law S n + S w = 1 p n p w = π(s w ) capillary pressure Phase α = w water, n gas or oil. π(s w ) increasing function on [0, 1] (extend coninuously to R). ω porosity S α phase saturation u α phase velocity k rα relative permeability K permeability p α : phase pressure ρ α phase density µ α viscosity

62 33 / 49 Simplified model Extension to two-phase flow Enchery et al. (06), Cances (08), Brenner et al. (13), no gravity 1 Global pressure (Chavent) P g (S) = p w + 2 Kirchhoff transformation : φ(s) = S 0 K S 0 k rn (u)/µ n π (u) du, k rn(u) k rw (u) µ n + µ w k rn (u)k rw (u) µ n k rw (u) + µ w k rn (u) π (u) du.

63 33 / 49 Simplified model Extension to two-phase flow Enchery et al. (06), Cances (08), Brenner et al. (13), no gravity 1 Global pressure (Chavent) P g (S) = p w + 2 Kirchhoff transformation : φ(s) = S 0 K S 0 k rn (u)/µ n π (u) du, k rn(u) k rw (u) µ n + µ w k rn (u)k rw (u) µ n k rw (u) + µ w k rn (u) π (u) du. µ w k rn (S) Transformed system : f (S) = µ w k rn (s) + µ n k rw (S), λ(s) = k rn(s) + k rw(s). µ n µ w { ω t S + div (f (S)q T ) φ(s) = 0 in Ω [0, T ] div q T = 0, q T = K λ(s) grad P g

64 33 / 49 Simplified model Extension to two-phase flow Enchery et al. (06), Cances (08), Brenner et al. (13), no gravity 1 Global pressure (Chavent) P g (S) = p w + 2 Kirchhoff transformation : φ(s) = S 0 K S 0 k rn (u)/µ n π (u) du, k rn(u) k rw (u) µ n + µ w k rn (u)k rw (u) µ n k rw (u) + µ w k rn (u) π (u) du. µ w k rn (S) Transformed system : f (S) = µ w k rn (s) + µ n k rw (S), λ(s) = k rn(s) + k rw(s). µ n µ w { ω t S + div (f (S)q T ) φ(s) = 0 in Ω [0, T ] div q T = 0, q T = K λ(s) grad P g Simplified system: neglect advection ω t S φ(s) = 0 in Ω [0, T ] Nonlinear (degenerate) diffusion equation

65 Extension to two-phase flow Discontinuous capillary pressure: transmission conditions Two subdomains Ω = Ω 1 Ω 2, Ω 1 Ω 2 =. Γ = Ω 1 Ω 2 P_c2(1) P_c1(1) Capillary pressure P_c P_c2(0) P_{c1}(0) 0 s_2 s_1 1 Saturation Transmission conditions on the interface Continuity of capillary pressure π 1 (S 1 ) = π 2 (S 2 ) on Γ Continuity of the flux φ 1 (S 1 ).n 1 = φ 2 (S 2 ).n 2 on Γ Chavent Jaffré (86), Enchéry et al. (06), Cances (08), Ern et al (10), Brenner et al. (13). 34 / 49

66 35 / 49 Extension to two-phase flow Non-linear Schwarz algorithm Robin transmission conditions φ 1 (S 1 ).n 1 + β 1 π 1 (S 1 ) = φ 2 (S 2 ).n 2 + β 1 π 2 (S 2 ) φ 2 (S 2 ).n 2 + β 2 π 2 (S 2 ) = φ 1 (S 1 ).n 1 + β 2 π 1 (S 1 ) Schwarz algorithm Given S 0 i, iterate for k = 0,... Solve for S k+1 i, i = 1, 2, j = 3 i ω t S k+1 i φ i (S k+1 i ) = 0 in Ω i [0, T ] φ i (S k+1 i ).n i + β i π i (S k+1 i ) = φ j (S k j ).n j + β i π j (S k j ) on Γ [0, T ], (β 1, β 2 ) are free parameters chosen to accelerate convergence Basic ingredient: subdomain solver with Robin bc.

67 36 / 49 Extension to two-phase flow Finite volume scheme (1) Extension to Robin bc of cell centered FV scheme by Enchéry et al. (06). Triangulation T, cells K T, boundary faces σ Γ. Unknowns : cell values (S K ) K T, boundary face values (S σ ) σ EΓ Notations: K L = edge between K and L, τ K L = average). m(k L) K K L (eg harmonic

68 37 / 49 Extension to two-phase flow Finite volume scheme (2) Interior equation m(k ) Sn+1 K S n K δt + L N (K ) + σ E Γ E K τ K,σ ( ) τ K L φ(s n+1 K ) φ(s n+1 L ) ( ) φ(s n+1 K ) φ(s n+1 σ ) = 0, K T. Robin BC for boundary faces ( ) τ K,σ φ(s n+1 K ) φ(s n+1 σ ) + βm(σ)π(s n+1 σ ) = g σ, σ E Γ Implemented with Matlab Reservoir Simulation Toolbox (K. A. Lie et al. (14)) Solver with automatics differentiation : no explicit computation of Jacobian

69 Extension to two-phase flow 10 2 alpha=0.009,beta=0.008 Numerical example Homogeneous medium, Ω 1 = (0, 100) 3, Ω 2 = (100, 200) (0, 100) 2. Mobilities λ 0 (S) = S, S [0, 1], Capillary pressure π(s) = 5S 2, S [0, 1] 10 0 alpha=8, beta=0.008 alpha=4,beta=4 alpha=0.008, beta= Convergence history for various parameters 38 / 49

70 39 / 49 Extension to two-phase flow Evolution of the concentration

71 40 / 49 Outline Extension to reduced fracture models 1 Introduction 2 Pure diffusion problems Multi-domain mixed formulations Nonconforming discretizations in time 3 Advection-diffusion problems Operator splitting 4 Extension to two-phase flow 5 Extension to reduced fracture models

72 t p i + div u i = q i in Ω i (0, T ), 41 / 49 and we denote by n i the outward unit normal vector field on Ω i. We use the convention Extension that to reduced for anyfracture scalar, models vector or tensor valued function φ defined on Ω, φ i denotes the A reduced model: δ interface-fracture Alboin-Jaffré-Roberts-Serres (2002) n 1 Martin-Jaffré-Roberts (2005) Knabner-Roberts (2014) γ 1 (Forchheimer γ 2 flow) Ω 1 Ω f Ω 2 In this work: assumenthat 2 D/δ large concentration continuity across the fracture In the subdomains n 1 Ω 1 Ω 2 Figure φ 1. Left: The domain Ω with the fracture Ω f. Right: The domain Ω with the i t c i + div r i = f i in Ω interface-fracture i (0, T ), γ. r i = D i c i in Ω i (0, T ), restriction of φ to Ω i, i = 1, c2, i f. We = 0rewrite problem on Ω (1) i as Ω the following (0, T ), transmission for i = problem: 1, 2, c i = c γ on γ (0, T ), c t p i + div u i = q i in Ω i (0, T ), i = 1, 2, f, i (, 0) = c 0,i in Ω i, u i = K i p i in Ω i (0, T ), i = 1, 2, f, p i = 0 on Ω i Ω (0, T ), i = 1, 2, f, p i = p f on γ i (0, T ), i = 1, 2, u i n i = u f n i on γ i (0, T ), i = 1, 2, p i (, 0) = p 0,i in Ω i, i = 1, 2, f. 2. Reduced fracture model In this section, the model where the fracture is treated as an interface γ between subdomains Ω 1 and Ω 2 [1,4] is considered. We use the notation τ and div τ for the tangential gradient and divergence, respectively. A reduced model consists of the equations in the subdomains n 2 γ (2)

73 t p i + div u i = q i in Ω i (0, T ), 41 / 49 and we denote by n i the outward unit normal vector field on Ω i. We use the convention Extension that to reduced for anyfracture scalar, models vector or tensor valued function φ defined on Ω, φ i denotes the A reduced model: δ interface-fracture Alboin-Jaffré-Roberts-Serres (2002) n 1 Martin-Jaffré-Roberts (2005) Knabner-Roberts (2014) γ 1 (Forchheimer γ 2 flow) Ω 1 Ω f Ω 2 In this work: assumenthat 2 D/δ large concentration continuity across the fracture In the subdomains n 1 Ω 1 Ω 2 Figure φ 1. Left: The domain Ω with the fracture Ω f. Right: The domain Ω with the i t c i + div r i = f i in Ω interface-fracture i (0, T ), γ. r i = D i c i in Ω i (0, T ), restriction of φ to Ω i, i = 1, c2, i f. We = 0rewrite problem on Ω (1) i as Ω the following (0, T ), transmission for i = problem: 1, 2, c i = c γ on γ (0, T ), and in the fracture c t p i + div u i = q in Ω (0, T ), i = 1, 2, f, i (, = c 0) u i 0,i i in = K i p i Ω i, i in Ω i (0, T ), i = 1, 2, f, p i = 0 on Ω i Ω (0, T ), i = 1, 2, f, p i = p f on γ i (0, T ), i = 1, 2, (2) u i n i i = 1, 2, φ γ t c γ + div p i (, τ 0) r γ ini γ = 1, (0, 2, f. T ), = u f n i on γ i (0, T ), = f γ p 0,i + ( ) r 1 nin 1 γ Ω i, + r 2 n 2 γ r γ = D γδ τ c γ in γ (0, T ), c γ 2. = Reduced 0 fracture modelon γ (0, T ), In this section, the model c γ(, where 0) the= fracture c 0,γ is treated as an interface γ between in γ. subdomains Ω 1 and Ω 2 [1,4] is considered. We use the notation τ and div τ for the tangential gradient and divergence, respectively. A reduced Communication model consists between of the equations the fracture in the and subdomains the rock matrix. n 2 γ

74 42 / 49 Extension to reduced fracture models Formulation as an interface problem (GTP Schur) The same (as with simple DD) Dirichlet-to-Neumann operators, for i = 1, 2: where (c i, r i ), i = 1, 2, is the solution of S DtN i : (λ, f, c 0 ) (r i n i ) Γ, L(c i, r i ) = f, in Ω i (0, T ), M(c i, r i ) = 0, in Ω i (0, T ), c i = λ, on γ (0, T ), c i (, 0) = c 0, in Ω i.

75 42 / 49 Extension to reduced fracture models Formulation as an interface problem (GTP Schur) The same (as with simple DD) Dirichlet-to-Neumann operators, for i = 1, 2: where (c i, r i ), i = 1, 2, is the solution of Different space-time interface problem: 2 i=1 S DtN i : (λ, f, c 0 ) (r i n i ) Γ, L(c i, r i ) = f, in Ω i (0, T ), M(c i, r i ) = 0, in Ω i (0, T ), c i = λ, on γ (0, T ), c i (, 0) = c 0, in Ω i. instead of 2 Si DtN (λ, 0, 0) = Si DtN (0, f, c 0 ), i=1 Sλ = χ, on γ (0, T ).

76 42 / 49 Extension to reduced fracture models Formulation as an interface problem (GTP Schur) The same (as with simple DD) Dirichlet-to-Neumann operators, for i = 1, 2: where (c i, r i ), i = 1, 2, is the solution of Different space-time interface problem: S DtN i : (λ, f, c 0 ) (r i n i ) Γ, L(c i, r i ) = f, in Ω i (0, T ), M(c i, r i ) = 0, in Ω i (0, T ), c i = λ, on γ (0, T ), c i (, 0) = c 0, in Ω i. L γ(λ, r γ) + Sλ = χ + f γ, in γ (0, T ), M γ(λ, r γ) = 0 in γ (0, T ), λ(, 0) = c 0,γ, in γ.

77 42 / 49 Extension to reduced fracture models Formulation as an interface problem (GTP Schur) The same (as with simple DD) Dirichlet-to-Neumann operators, for i = 1, 2: where (c i, r i ), i = 1, 2, is the solution of Different space-time interface problem: Two possible preconditionners: S DtN i : (λ, f, c 0 ) (r i n i ) Γ, L(c i, r i ) = f, in Ω i (0, T ), M(c i, r i ) = 0, in Ω i (0, T ), c i = λ, on γ (0, T ), c i (, 0) = c 0, in Ω i. L γ(λ, r γ) + Sλ = χ + f γ, in γ (0, T ), M γ(λ, r γ) = 0 in γ (0, T ), λ(, 0) = c 0,γ, in γ. a Neumann-Neumann preconditionner with weights

78 Extension to reduced fracture models Formulation as an interface problem (GTP Schur) The same (as with simple DD) Dirichlet-to-Neumann operators, for i = 1, 2: where (c i, r i ), i = 1, 2, is the solution of Different space-time interface problem: Two possible preconditionners: S DtN i : (λ, f, c 0 ) (r i n i ) Γ, L(c i, r i ) = f, in Ω i (0, T ), M(c i, r i ) = 0, in Ω i (0, T ), c i = λ, on γ (0, T ), c i (, 0) = c 0, in Ω i. L γ(λ, r γ) + Sλ = χ + f γ, in γ (0, T ), M γ(λ, r γ) = 0 in γ (0, T ), λ(, 0) = c 0,γ, in γ. a Neumann-Neumann preconditionner with weights a local preconditioner (coming from the observation that the interface problem is dominated by the 2nd order operator, Amir, MK, Martin, Robert, Arima 06) 42 / 49

79 43 / 49 Extension to reduced fracture models Transmission conditions for a GTO Schwarz method Taking a linear combination of the transmission conditions for the GTP Schur method we obtain: r 1 n 1 + α 1 c 1,γ + φ γ t c i,γ + div τ r γ,1 = r 2 n 1 + α 1 c 2,γ + f γ r γ,1 = D γδ τ c 1,γ r 2 n 2 + α 2 c 2,γ + φ γ t c 2,γ + div τ r γ,2 = r 1 n 2 + α 2 c 1,γ + f γ r γ,2 = D γδ τ c 2,γ

80 Extension to reduced fracture models 44 / 49 Formulation as an interface problem (GTO Schwarz) We use Ventcell to Robin operators, for i = 1, 2: S VtR i : (θ i, f, c 0, f γ, c 0,γ ) ( r i n j + αc i ) Γ, where ( c i, r i, c i,γ, r γ,i ), i = 1, 2, is the solution of L(c i, r i ) = f, in Ω i (0, T ), M(c i, r i ) = 0, in Ω i (0, T ), r i n i + αc i,γ + φ γ t c i,γ + div τ r γ,i = θ i on γ (0, T ), r γ,i + D γδ τ c i,γ = 0, on γ (0, T ), c i (, 0) = c 0, in Ω i c i,γ (, 0) = c 0,γ, in γ.

81 44 / 49 Extension to reduced fracture models Formulation as an interface problem (GTO Schwarz) We use Ventcell to Robin operators, for i = 1, 2: S VtR i : (θ i, f, c 0, f γ, c 0,γ ) ( r i n j + αc i ) Γ, where ( c i, r i, c i,γ, r γ,i ), i = 1, 2, is the solution of L(c i, r i ) = f, in Ω i (0, T ), M(c i, r i ) = 0, in Ω i (0, T ), r i n i + αc i,γ + φ γ t c i,γ + div τ r γ,i = θ i on γ (0, T ), r γ,i + D γδ τ c i,γ = 0, on γ (0, T ), c i (, 0) = c 0, in Ω i c i,γ (, 0) = c 0,γ, in γ. Space-time interface problem: θ 1 θ 2 = S2 VtR (θ 2, f, c 0, f γ, c 0,γ ) + f γ, on γ (0, T ), = S1 VtR (θ 1, f, c 0, f γ, c 0,γ ) + f γ, on γ (0, T ).

82 45 / 49 Numerical results Extension to reduced fracture models 1 c=0 c=0 0 c= c=1 Geometry and boundary conditions. Isotropic coefficients: D i = 1, i = 1, 2, and D γ = 1/δ = 1000.

83 45 / 49 Numerical results Extension to reduced fracture models 1 c=0 c=0 0 c= c=1 Geometry and boundary conditions. Isotropic coefficients: D i = 1, i = 1, 2, and D γ = 1/δ = Zero source terms and initial condition. Spatial discretization: uniform rectangular mesh h = 1/100 mixed FE with the lowest-order Raviart-Thomas spaces. Time discretization (case 1): conforming grids t m = t γ = T /300 with T = 0.5.

84 46 / 49 Extension to reduced fracture models Snapshots of solution - concentration field c t= t t=t/4 t=t/2 t=t

85 47 / 49 Extension to reduced fracture models Snapshots of solution - diffusive flux r 1 1 t= t t=t/ t=t/ t=t

86 Extension to reduced fracture models Convergence - GMRES Number of Iterations Number of Iterations Figure 4. Convergence curves for the compressible flow: errors in p (on the lef TITLE TITLE WILL WILL BE SET BE BY SETTHE BY PUBLISHER THE PUBLISHER (on the right) - Method 1 with 13no preconditioner 13 (blue), Method 1 with local pr (green), Method 1 with Neumann-Neumann preconditioner (cyan) and Metho Error in p Error in p Error in u Error in u Error in u Number Number of Iterations of Iterations Number Number of Iterations of Iterations Figure 5. The errors (in logarithmic scale) after 10 Jacobi iterations for variou Figure Figure 4. Convergence 4. curves curves for the forcompressible the flow: flow: errors errors in p in (onp (onthe left) the Robin left) andparameter. in andu in uthe red star shows the optimized parameters computed by (on the (onright) the right) - Method - Method 1 with 1 with no preconditioner no (blue), (blue), Method Method 1 with 1 with local local preconditioner minimizing the continuous convergence factor. (green), (green), Method Method 1 with 1 with Neumann-Neumann preconditioner (cyan) (cyan) and Method and Method 2 (red). 2 (red). T.T.P. Hoang, J. Jaffré, 8 8 C. Japhet, M.K., and J. E. Roberts. Space-time Domain Error in u Error in u Anal., to appear, α L 2 concentration errors (c) L 2 flux errors (r) L 2 error versus α GT Schur with no preconditioner GTP Schur with local preconditioner GTP Schur with NN preconditioner GTO Schwarz method the number of iterations for different schemes are shown: errors in p (on the left) and in see that Method 2 is superior to the other schemes. For Method 1, the Neumann-Neumann a better acceleration than the local one, and both are better than without a precondition For Method 2, we vary Robin parameters and plot the logarithmic scale of the error iterations. In Figure 5, we see that the optimized Robin parameter (the red star), w numerically minimizing the convergence factor, are located close to those giving the sm same number of iterations. Decomposition and Mixed Formulation for reduced fracture models. SIAM J. Numer / 49

87 49 / 49 Extension to reduced fracture models Conclusions perspectives Space time DD method with Robi TC for diffusion and advection diffusion Extension to fractured media Convergence for GTP Schur (Gander et al. for homogeneous media) Convergence for fractured media Influence of Robin parameter β, find optimal parameter Study interface problem for non-linear case, Jacobi (SWR) vs Newton Extension to full two-phase model Convergence of Schwarz alg. for nonlinear case Large scale parallel solver (MdS)