The Certified Reduced-Basis Method for Darcy Flows in Porous Media

Transcription

1 The Certified Reduced-Basis Method for Darcy Flows in Porous Media S. Boyaval (1), G. Enchéry (2), R. Sanchez (2), Q. H. Tran (2) (1) Laboratoire Saint-Venant (ENPC - EDF R&D - CEREMA) & Matherials (INRIA), (2) IFP Energies nouvelles 1 / 21 The Certified Reduced-Basis Method for Darcy Flows in Porous Media 07/11/2016

2 Reservoir simulation in petroleum industry Uses : well placement, optimization of production scenarios, History-matching, Sensitivity analysis, Quantification of uncertainties. Difficulties : Long simulation times (grid s size, time iterations...), Simulations should be rerun for different input parameters. A significant part of the simulation time is spent in the resolution of the pressure equation. Aim of this study : build a reduced solution for this problem, Among all possible methods : Reduced Basis (RB). 2 / 21 The Certified Reduced-Basis Method for Darcy Flows in Porous Media 07/11/2016

3 Outline Two-phase flow model Problem statement Parameterization Finite volume discretization Reduced model for the pressure equation Variational RB formulation A posteriori error Results First numerical results Reduction of the computational complexity Conclusions 3 / 21 The Certified Reduced-Basis Method for Darcy Flows in Porous Media 07/11/2016

4 Problem statement Equations in Ω R 2 where φ ts + div (f w (S)v) = g, (1) div (v) = f, (2) v + λ T (S)K P = 0, (3) λ T (S) = λ w (S) + λ o(1 S), λ α(s) = krα(s), α {w, o}, µ α ( ) 2 S Sw,i kr w (S) =, 1 S w,i S o,r ( ) 2 1 S So,r kr o(s) =, 1 S w,i S o,r f w (S) = λw (S) λ T (S). Slip boundary conditions v n Ω = 0. (4) Initial condition S(, 0) = 0.2 (5) 4 / 21 The Certified Reduced-Basis Method for Darcy Flows in Porous Media 07/11/2016

5 Parameterization Goal : assess numerically the potential of the enhanced oil recovery by polymer flooding. Variations of µ w due to operational conditions µ := µ w, µ P = [1, 30]. 5 / 21 The Certified Reduced-Basis Method for Darcy Flows in Porous Media 07/11/2016

6 IMPIMS scheme and finite volume discretization IMPIMS scheme for the time discretization φ Sn+1 µ Sµ n ( ) + div f w (Sµ n+1, µ)vµ n+1 t = g n+1 µ, (6) div (v n+1 µ ) = f, (7) v n+1 µ = λ T (S n µ, µ)k P n+1 µ. (8) A finite volume method for the space discretization (TPFA). For the pressure, one needs to solve a N N linear system : where (A µp n+1 µ ) K = A µp n+1 µ = f, a n σ (µ)(pn+1 µ,k Pn+1 µ,l ), σ=k L aσ(µ) n = harmonic mean of λ T (Sµ, n µ)k on the edge σ, f K = K f. 6 / 21 The Certified Reduced-Basis Method for Darcy Flows in Porous Media 07/11/2016

7 Homogeneous case Figure: Water saturations for different parameter values at T = 1, 000 days 7 / 21 The Certified Reduced-Basis Method for Darcy Flows in Porous Media 07/11/2016

8 SPE10 case 1 (layer 85) Figure: Water saturations for different parameter values at T = 1, 000 days 1. Tenth SPE comparative solution project : a comparison of upscaling techniques, Christie and Blunt (2001), SPE Reserv. Eval. Eng., 4(4) : / 21 The Certified Reduced-Basis Method for Darcy Flows in Porous Media 07/11/2016

9 Reduced model for the pressure equation The pressure equation reads div v µ = f, in Ω, v µ = a(µ) P µ, in Ω, v µ n Ω = 0, on Ω, P µ = 0, Ω where a(µ) = λ T (S µ, µ)k. (9) Difficulties : equation (9) is coupled with a transport equation for S µ. (9) is discretized with a cell-centered finite volume scheme. a(µ) does not fulfill the affine parameter dependence assumption. Idea 1. Let us suppose S µ (and so a(µ)) known for all µ P. 9 / 21 The Certified Reduced-Basis Method for Darcy Flows in Porous Media 07/11/2016

10 Discrete Galerkin Framework Idea 2. Express our finite volume discretization as a variational formulation. Let Q N be the space of piecewise-constant L2-functions subject to zero-mean condition, i.e., 1 K P K = 0. N Given µ P, consider the problem K where C N (P N µ, q; µ) = Find P N µ Q N such that C N (Pµ N, q; µ) = L f (q), q Q N, a ( ) σ(µ) Pµ,K N Pµ,L)( N qk q L and Lf (q) = f K q K. σ=k L K (10) The space Q N is equipped with the energy norm p 2 N, µ = C N (p, p; µ). (11) 10 / 21 The Certified Reduced-Basis Method for Darcy Flows in Porous Media 07/11/2016

11 The reduced problem reads Find Pµ N Q N such that C N (Pµ N, q; µ) = L f (q), q Q N. (12) The reduced basis method consists in solving a small N x N system A N (µ)p N µ = f N, (13) with N P N µ R N such that Pµ N = (P N µ) np µn, n=1 A N (µ) R N N, such that A N m,n(µ) = f N R N such that fn N = Pµ N n,k K T K a σ(µ)(pµ N m,k Pµ N m,l)(pµ N n,k Pµ N n,l), σ=k L f. 11 / 21 The Certified Reduced-Basis Method for Darcy Flows in Porous Media 07/11/2016

12 A posteriori error estimate first attempt Definition 1 (Residual and associated norm). Let p Q N. For all q Q N, define the residual R N (p; µ), q = C N (p, q; µ) L f (q), and the associated (discrete) norm R N (p; µ) N,,µ = sup q Q N R N (p; µ), q. q N, µ Proposition 1 (Energy estimate). with P N µ P N µ N, µ en N (µ) := α N (µ) = 1 R N (P N µ ; µ) α N (µ) C N (q, q; µ) inf. q Q N \{0} q N, µ N,,µ (14) 12 / 21 The Certified Reduced-Basis Method for Darcy Flows in Porous Media 07/11/2016

13 Practical computation of error bounds Computing the stability factor α N (µ) = 1, µ P Computing the norm of the residual R N (Pµ N ; µ) = (f A µp N P N µ) t A µ(f A µp N P N µ) 2 N,,µ = f t A µf 2f t A µa µp N P N µ + (P N P N µ) t A µp N P N µ Pros & cons 1. trivial computation of α N (µ) No affine parameter dependence assumption A µ depends on µ 13 / 21 The Certified Reduced-Basis Method for Darcy Flows in Porous Media 07/11/2016

14 A posteriori error estimate second attempt Definition 2 (Norm of the residual). Let p Q N, define a new dual norm as R N (p; µ) = sup N,,µ ref q Q N R N (p; µ), q. q N, µref Proposition 2 (Energy estimate). with P N µ P N µ N, µref en N (µ) := α N (µ) = 1 R N (P N µ ; µ) (15) α N (µ) N,,µ ref C N (q, q; µ) inf. q Q N \{0} q N, µref Pros & cons 2. A µ no longer depends on µ complex evaluation of α N (µ) 14 / 21 The Certified Reduced-Basis Method for Darcy Flows in Porous Media 07/11/2016

15 Numerical results Figure: Representative solution and pointwise error for two values of N (homogeneous case). 15 / 21 The Certified Reduced-Basis Method for Darcy Flows in Porous Media 07/11/2016

16 Figure: Representative solution and pointwise error for two values of N (SPE10 case). 16 / 21 The Certified Reduced-Basis Method for Darcy Flows in Porous Media 07/11/2016

17 Figure: A posteriori error bound. Comparison of the maximum and average exact error Pµ N Pµ N µ and the estimator en N (µ) (#Ξ train 300). 17 / 21 The Certified Reduced-Basis Method for Darcy Flows in Porous Media 07/11/2016

18 Reduction of the computational complexity To solve A N (µ)p N µ = f N, µ P, one needs to build A N whose coefficients are A N m,n(µ) = a σ(µ)(pµ N m,k Pµ N m,l)(pµ N n,k Pµ N n,l). Idea 3. σ=k L Replace a(µ) with a collateral affine expansion a M (µ) = Empirical Interpolation Method a to obtain A N,M m,n (µ) = M ( m=1 σ=k L M Θ m(µ)ζ m using the m=1 ) ζ m,σ(pµ N m,k Pµ N m,l)(pµ N n,k Pµ N n,l) Θ m(µ). (16) a. An empirical interpolation method : application to efficient reduced-basis discretization of partial differential equations, Barrault, Maday, Nguyen and Patera (2004), C. R. Acad. Sci. Paris, Ser. I, 339(9) : / 21 The Certified Reduced-Basis Method for Darcy Flows in Porous Media 07/11/2016

19 EIM-RB a posteriori error estimate Define the residual ( R M (Pµ N,M ; µ), q = C N P N,M µ, q; a M (µ) ) + L f (q), q Q N. Proposition 3. The following a posteriori error estimate holds P N µ where P N,M µ N,µref 1 α N (µ) ( RM (P N,M µ ; µ) N,,µ ref + γ(µ)δ a(µ) ) C N (Pµ N,M, q; 1 δ a(µ) = a(µ) a M (µ) L (Ω), γ(µ) = sup q Q N q N, µref ) (17) 19 / 21 The Certified Reduced-Basis Method for Darcy Flows in Porous Media 07/11/2016

20 Figure: Convergence of the collateral reduced-basis approximation for the homogeneous example. 20 / 21 The Certified Reduced-Basis Method for Darcy Flows in Porous Media 07/11/2016

21 Conclusions and future works Results Reduction of the pressure problem (elliptic) with non-affine dependence in the parameter. Certification of the RB model (a posteriori error estimate in energy norm). Efficient implementation of the RB method, i.e. the reduced problems can be constructed in complexity independent of N. Certification of the RB-EIM model. Future works Improvement of the efficiency of the a posteriori error estimate for the highly heterogenous case. Adapt this procedure to reduce the collection of all pressure equations. Consider more complex parameterizations (multi-dimensional parameter). Thank you for your attention Any questions? 21 / 21 The Certified Reduced-Basis Method for Darcy Flows in Porous Media 07/11/2016