A posteriori error estimates, stopping criteria, and adaptivity for multiphase compositional Darcy flows in porous media

Transcription

1 A posteriori error estimates, stopping criteria, and adaptivity for multiphase compositional Darcy flows in porous media D. A. Di Pietro, M. Vohraĺık, and S. Yousef Université Montpellier 2 Marseille, 1 October, 2013

2 Outline 1 Model and discretization 2 A posteriori error estimates and adaptive resolution 3 Numerical results

3 In a nutshell I In the context of petroleum reservoir engineering, numerical simulation is used to predict oil production and to plan exploitation Modelling the flow of several fluids through a porous medium leads to highly nonlinear and computationally expensive problems Goal significantly reduce simulation time Idea use a posteriori error estimators to make smart online choices

4 In a nutshell II Fix M 0 and τ 0. Set t 0 0, n 0 and set the initial solution X 0 hτ. while t n t F do Set n n + 1, M n M n 1, τ n τ n 1. repeat { Equilibration of spatial and temporal errors } Set k 0 and X n,0 hτ X n 1 hτ. repeat { Newton iterations } k k + 1 and i 0. Set X n,k,0 hτ Set up the linear system repeat { Algebraic iterations } := X n,k 1 hτ. Set i = i + 1 and perform one iteration of the algebraic solver Compute ηsp n,k,i, η n,k,i tm, ηn,k,i lin, η n,k,i alg until stopping criterion until stopping criterion Adapt the time step τ n until time-space equilibration Set X n hτ X n,k,i hτ, and t n t n 1 + τ n. end while

5 In a nutshell III fixed thresholds adaptive criteria iterations iterations Figure: Cumulated linear solver iteration as a function of stopping criteria

6 The compositional Darcy model I Let two sets of phases P and components C be given We define for all p P and all c C the relevant subsets C p := {c C; c is present in p}, P c := {p P; c is present in p} H 2 O CO 2 N 2 C 7 H 16 w h o Figure: Example of a two-phase, three-component flux

7 The compositional Darcy model II Formulation inspired by [Coats, 1980, Eymard et al., 2012] The unknowns of the model are P P X := (S p ) p P = S (C p,c ) p P,c Cp (C p ) p P We define for all p P the phase pressure as P p (P, S) = P + P cp (S) The (average) phase velocity is given by Darcy s law v p = Λ ( P p + ρ p g z)

8 The compositional Darcy model III Let Ω denote the space domain and t F > 0 the simulation time Conservation of the quantity of matter: For all c C, t l c + Φ c = q c in Ω (0, t F ) with q c source piecewise constant in space-time and l c := φ ζ p S p C p,c, Φ c := {Φ p,c := ν p C p,c v p }, p P c p P c Initial and boundary conditions l c (0) = lc 0 in Ω, Φ c n Ω = 0 on Ω (0, t F )

9 The compositional Darcy model IV Saturation of the pore volume S p = 1 p P Partition of the matter into components C p,c = 1 p P c C p Thermodynamic equilibrium relations close the system

10 A classical finite volume scheme I We consider a popular fully implicity finite volume discretization The numerical fluxes are based on phase-upwind and two-point fluxes On phase-upwind cf., e.g., [Brenier and Jaffré, 1991]

11 A classical finite volume scheme II We consider a partition (t n ) 0 n N of (0, t F ) with t n = n τ i τ i > 0 1 i n i=1 We denote by (M n ) 0 n N a sequence of meshes of Ω with M n = {M} The discrete unknowns are, for all M M n, P n XM n := (X M n ) M M n, XM n := M S n M (C n p,m ) p P

12 A classical finite volume scheme III For each phase, the discrete phase pressure is given by P n p,m (P n M, S n M ) = P n M + P cp (S n M ) M M n The discrete phase velocity is given by, for all σ M L, F p,m,σ (X n M) := σ α M α L α M +α L [ P n p,m P n p,l + ρ n p,σg (z M z L ) ], while F p,m,σ (XM n ) = 0 if σ Ω

13 A classical finite volume scheme IV Discrete conservation of the quantity of matter: For all c C, with M n t l c,m + σ E i,n M l n c,m = φ F c,m,σ (X n M) = M q n c,m p P c ζ p (P n p,m, C n p,m )S n p,m C n p,c,m M M n and molar component flux F c,m,σ (XM) n := { } F p,c,m,σ (XM) n := νpc n F p,c,mp p,m,σ (XM) n p P c Closure laws are enforced cell-wise

14 Outline 1 Model and discretization 2 A posteriori error estimates and adaptive resolution 3 Numerical results

15 Essential bibliography A posteriori estimates for model unsteady nonlinear problems [Eriksson and Johnson, 1995, Verfürth, 1998a, Verfürth, 1998b] A posteriori estimates for degenerate parabolic problems [Nochetto et al., 2000, Ohlberger, 2001] [Di Pietro et al., 2013b, Di Pietro et al., 2013a] Adaptive mesh refinement in reservoir simulation [Heinemann, 1983, Ewing et al., 1989] and many more [Mamaghani et al., 2011] (SAGD with #C = 2, #P = 3) Smart online choices [Jiránek et al., 2010] (stopping criteria) [Ern and Vohraĺık, 2013] (inexact Newton) [Di Pietro et al., 2013b] (stopping criteria + parameter selection)

16 Fully computable upper bound I Assumption (Weak solution) There exists a weak solution X such that For all p P, P p (P, S) X := L 2 (H 1 (Ω)) For all c C, l c Y := H 1 (L 2 (Ω)) and Φ c [L 2 (L 2 (Ω))] d The following equality holds for all ϕ X and all c C: tf 0 {( t l c, ϕ)(t) (Φ c, ϕ)(t)} dt = tf The initial condition and the closure equations hold 0 (q c, ϕ)(t)dt We equip the space X with the norm { N ϕ X := n=1 I n M M n ϕ 2 X,M dt } 1/2, ϕ 2 X,M := εh 2 M ϕ 2 M + ϕ 2 M

17 Fully computable upper bound II { 1/2 { N := N 2} c + c C Dual norm of the residual N c := tf sup ϕ X, ϕ X =1 0 p P N p 2 } 1/2 { ( tl c tl c,hτ, ϕ)(t) ( Φ c Φ c,hτ, ϕ ) (t) } dt with Φ c,hτ = p P c ν p(p p,hτ, S h,τ, C p,hτ )C p,c,hτ v p(p p,hτ, C p,hτ ) Nonconformity in X: { tf } 1/2 N p := inf Ψ p,c(p p,hτ )(t) Ψ p,c(ϕ p)(t) 2 dt ϕ p X c C 0 p with Ψ p,c(ϕ) := ν p(p p,hτ, S hτ, C p,hτ )C p,c,hτ Λ ϕ

18 Fully computable upper bound III Theorem (Fully computable upper bound) The following guaranteed upper bounds hold: { N } 1/2 ( N c η n R,M,c + ηf,m,c(t) n )2 dt c C, n=1 I n M M n { N } 1/2 ( N p η n NC,M,p,c (t) )2 dt p P, n=1 I n M M n c C p with estimators given by, for all c C and all M M n, ηr,m,c n := C P,M h M qc,h n t n l c,hτ Θ n c,h M, ηf,m,c(t) n := Θ n c,h Φ c,hτ (t) M, ηnc,m,p,c(t) n := Ψ p,c(p p,hτ )(t) Ψ p,c(p p,hτ )(t) M p P c. where, for all c C, Θ n c,h RTN(M n ) s.t. (qc,h n t n l c,hτ Θ n c,h, 1) M = 0 M M n.

19 Linearization and algebraic resolution I Solving the discrete problem amounts to zeroing the residuals R n c,m ( ) ( X n ) l c,m X n M l n 1 M := c,m M + ( F τ n c,m,σ X n ) M M q n c,m = 0 σ E i,n M The Newton method generates a sequence (X n,k M ) k 0 by solving Rn c,m X n M M n M ( X n,k 1) ( M X n,k M X n,k 1 M ) ( + R n c,m X n,k 1) = 0 M

20 Linearization and algebraic resolution II The resulting linear system can be solved by an iterative linear solver The residuals at Newton iteration k and linear solver iteration i read R n,k,i c,m ( ) = M lc,m X n,k 1 M + L n,k,i c,m ln 1 c,m + τ n σ E i,n M F n,k,i c,m,σ M qn c,m, where F n,k,i c,m,σ is a linearized component flux

21 Linearization and algebraic resolution III Corollary (Time-localized a posteriori error estimate) For a given time step n, Newton iteration k, and linear iteration i we have N n c N n p { I n M M n { c C p I n M M n 1/2 ( η n,k,i 2dt} R,M,c + ηn,k,i F,M,c (t) + ηn,k,i NA,M,c) c C, ( η n,k,i NC,M,p,c (t)) 2 dt} 1/2 p P, where η n,k,i NA,M,c is related to the nonlinear accumulation term.

22 Distinguishing the error components I We decompose the component flux reconstruction Θ n,k,i c,h RTN(Mn ) as Θ n,k,i c,h := Θ n,k,i disc,c,h + Θn,k,i lin,c,h + Θn,k,i alg,c,h The discretization flux reconstruction Θ n,k,i disc,c,h RTN(Mn ) is s.t. (Θ n,k,i disc,c,h nm, 1)σ := F c,m,σ(x n,k,i M ) The linearization error flux reconstruction Θ n,k,i lin,c,h RTN(Mn ) is s.t. (Θ n,k,i c,m,σ lin,c,h nm, 1)σ = F n,k,i Fc,M,σ(X n,k,i M ) The algebraic error flux reconstruction Θ n,k,i alg,c,h RTN(Mn ) is s.t. (Θ n,k,i alg,c,h nm, 1) M := R n,k,i c,m

23 Distinguishing the error components II Space error estimator { η n,k,i ( sp,m,c (t) := ηn,k,i R,M,c + Θn,k,i disc,c,h Φn,k,i c,hτ (tn ) M + η n,k,i NC,M,p,c (t)) 2 p P c Time error estimator η n,k,i tm,m,c (t) := Φn,k,i c,hτ (tn ) Φ n,k,i c,hτ (t) M Linearization error estimator Algebraic error estimator η n,k,i lin,m,c := Θn,k,i lin,c,hm + η n,k,i NA,M,c η n,k,i alg,m,c := Θn,k,i alg,c,h M } 1/2

24 Distinguishing the error components III Corollary (Distinguishing the different error components) The following estimate holds: N n { with estimators given by c C ( η n,k,i sp,c ηsp,c {4 n,k,i := I n { η n,k,i tm,c := η n,k,i lin,c := η n,k,i alg,c := } 1/2 + η n,k,i tm,c + η n,k,i lin,c + ) 2 ηn,k,i alg,c, M M n 2 I n M M n { 2τ n { 2τ n M M n M M n ( η n,k,i sp,m,c (t)) 2 dt} 1/2, ( η n,k,i tm,m,c (t)) 2 dt} 1/2, ( η n,k,i lin,m,c) 2} 1/2, ( η n,k,i alg,m,c) 2} 1/2.

25 A fully adaptive algorithm Fix M 0 and τ 0. Set t 0 0, n 0 and set the initial solution X 0 hτ. while t n t F do Set n n + 1, M n M n 1, τ n τ n 1. repeat { Equilibration of spatial and temporal errors } Set k 0 and X n,0 hτ X n 1 hτ. repeat { Newton iterations } k k + 1 and i 0. Set X n,k,0 hτ Set up the linear system repeat { Algebraic iterations } until until := X n,k 1 hτ. Set i = i + 1 and perform one iteration of the algebraic solver Compute ηsp n,k,i, η n,k,i tm, ηn,k,i lin, η n,k,i alg until Adapt the time step τ n Set X n hτ X n,k,i hτ, and t n t n 1 + τ n. end while

26 A fully adaptive algorithm Fix M 0 and τ 0. Set t 0 0, n 0 and set the initial solution X 0 hτ. while t n t F do Set n n + 1, M n M n 1, τ n τ n 1. repeat { Equilibration of spatial and temporal errors } Set k 0 and X n,0 hτ X n 1 hτ. repeat { Newton iterations } k k + 1 and i 0. Set X n,k,0 hτ Set up the linear system repeat { Algebraic iterations } until until := X n,k 1 hτ. Set i = i + 1 and perform one iteration of the algebraic solver Compute ηsp n,k,i, η n,k,i tm, ηn,k,i lin, η n,k,i alg until η n,k,i alg,c γ alg(ηsp,c n,k,i Adapt the time step τ n Set X n hτ X n,k,i hτ, and t n t n 1 + τ n. end while + η n,k,i tm,c + ηn,k,i lin,c ) c C

27 A fully adaptive algorithm Fix M 0 and τ 0. Set t 0 0, n 0 and set the initial solution X 0 hτ. while t n t F do Set n n + 1, M n M n 1, τ n τ n 1. repeat { Equilibration of spatial and temporal errors } Set k 0 and X n,0 hτ X n 1 hτ. repeat { Newton iterations } k k + 1 and i 0. Set X n,k,0 hτ Set up the linear system repeat { Algebraic iterations } := X n,k 1 hτ. Set i = i + 1 and perform one iteration of the algebraic solver Compute ηsp n,k,i, η n,k,i tm, ηn,k,i lin, η n,k,i alg until η n,k,i alg,c γ alg(ηsp,c n,k,i until η n,k,i lin,c γ lin(ηsp,c n,k,i + η n,k,i tm,c ) Adapt the time step τ n until Set X n hτ X n,k,i hτ, and t n t n 1 + τ n. end while + η n,k,i tm,c + ηn,k,i lin,c ) c C c C

28 A fully adaptive algorithm Fix M 0 and τ 0. Set t 0 0, n 0 and set the initial solution X 0 hτ. while t n t F do Set n n + 1, M n M n 1, τ n τ n 1. repeat { Equilibration of spatial and temporal errors } Set k 0 and X n,0 hτ X n 1 hτ. repeat { Newton iterations } k k + 1 and i 0. Set X n,k,0 hτ Set up the linear system repeat { Algebraic iterations } := X n,k 1 hτ. Set i = i + 1 and perform one iteration of the algebraic solver Compute ηsp n,k,i, η n,k,i tm, ηn,k,i lin, η n,k,i alg until η n,k,i alg,c γ alg(ηsp,c n,k,i until η n,k,i lin,c γ lin(ηsp,c n,k,i + η n,k,i tm,c ) Adapt the time step τ n until γ tmη n,k,i sp,c η n,k,i tm,c Γtmηn,k,i sp,c Set X n hτ X n,k,i hτ, and t n t n 1 + τ n. end while + η n,k,i tm,c + ηn,k,i lin,c ) c C c C c C

29 Adaptive stopping criteria total space time linearization algebraic adaptive stopping criterion adaptive stopping criterion classical stopping criterion GMRes iteration 10 6 classical stopping criterion Newton iteration Figure: Evolution of the error component estimators for a fixed mesh as a function of GMRes (left) and Newton iterations (right)

30 Outline 1 Model and discretization 2 A posteriori error estimates and adaptive resolution 3 Numerical results

31 A numerical example Production well Injection well Figure: Numerical example: injection of a mixture of CO 2 and N 2 into a reservoir initially saturated with C 7H 16

32 Homogeneous medium I Figure: Liquid saturation, classical (left) vs. adaptive (right) resolution at times s and s

33 Homogeneous medium II Oil production (m 3 ) Time (second) classical resolution adaptive resolution 10 8 Figure: Cumulated oil production, classical vs. adaptive resolution

34 Homogeneous medium III Newton iterations Time classical adaptive 10 8 Cumulated number of Newton iterations 1,500 1, classical adaptive 1758 iterations 853 iterations Time 10 8 Figure: Newton iterations at each time step (left) and cumulated number of Newton iterations as a function of time (right).

35 Homogeneous medium IV GMRes iterations classical adaptive Time/Newton step 10 8 Cumulated number of GMRes iterations classical adaptive iterations 5110 iterations Time 10 8 Figure: GMRes iterations for each Newton iteration (left) and cumulated as a function of time (right)

36 Heterogeneous medium I Figure: Liquid saturation, classical (left) and adaptive (right) resolutions at times s, s, and s (heterogeneous medium)

37 Heterogeneous medium II Cumulated number of Newton iterations 2,000 1,500 1, classical adaptive 1975 iterations 995 iterations Time 10 8 Cumulated number of GMRes iterations classical adaptive 5907 iterations iterations Time 10 8 Figure: Cumulated Newton (left) and GMRes (right) iterations as a function of time (right) (heterogeneous medium)

38 References I Brenier, Y. and Jaffré, J. (1991). Upstream differencing for multiphase flow in reservoir simulation. SIAM J. Numer. Anal., 28: Coats, K. H. (1980). An equation of state compositional model. Society of Petroleum Engineers, 20(5): Di Pietro, D. A., Flauraud, E., Vohraĺık, M., and Yousef, S. (2013a). A posteriori error estimates, stopping criteria, and adaptivity for multiphase compositional Darcy flows in porous media. Submitted. Preprint hal Di Pietro, D. A., Vohraĺık, M., and Yousef, S. (2013b). Adaptive regularization, linearization, and discretization and a posteriori error control for the two-phase Stefan problem. Math. Comp. Accepted for publication. Preprint Eriksson, K. and Johnson, C. (1995). Adaptive finite element methods for parabolic problems. IV. Nonlinear problems. SIAM J. Numer. Anal., 32(6): Ern, A. and Vohraĺık, M. (2013). Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs. SIAM J. Sci. Comput. DOI / Ewing, R. E., Boyett, B. A., Babu, D. K., and Heinemann, R. F. (1989). Efficient use of locally refined grids for multiphase reservoir simulation. Society of Petroleum Engineers. Eymard, R., Guichard, C., Herbin, R., and Masson, R. (2012). Vertex-centred discretization of multiphase compositional darcy flows on general meshes. Comput. Geosci., 16:

39 References II Heinemann, Z. E. (1983). Using local grid refinement in a multiple-application reservoir simulator. Society of Petroleum Engineers. Jiránek, P., Strakoš, Z., and Vohraĺık, M. (2010). A posteriori error estimates including algebraic error and stopping criteria for iterative solvers. SIAM J. Sci. Comput., 32(3): Mamaghani, M., Enchéry, G., and Chainais-Hillairet, C. (2011). Development of a refinement criterion for adaptive mesh refinement in steam-assisted gravity drainage simulation. Comput. Geosci., 15(1): Nochetto, R. H., Schmidt, A., and Verdi, C. (2000). A posteriori error estimation and adaptivity for degenerate parabolic problems. Math. Comp., 69(229):1 24. Ohlberger, M. (2001). A posteriori error estimate for finite volume approximations to singularly perturbed nonlinear convection diffusion equations. Numer. Math., 87(4): Verfürth, R. (1998a). A posteriori error estimates for nonlinear problems. L r (0, T ; L ρ (Ω))-error estimates for finite element discretizations of parabolic equations. Math. Comp., 67(224): Verfürth, R. (1998b). A posteriori error estimates for nonlinear problems: L r (0, T ; W 1,ρ (Ω))-error estimates for finite element discretizations of parabolic equations. Numer. Methods Partial Differential Equations, 14(4):