Numerical Modeling of Methane Hydrate Evolution

Transcription

1 Numerical Modeling of Methane Hydrate Evolution Nathan L. Gibson Joint work with F. P. Medina, M. Peszynska, R. E. Showalter Department of Mathematics SIAM Annual Meeting 2013 Friday, July 12 This work was partially supported by NSF DMS Hybrid Modeling in porous media. N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 1 / 39

2 Outline 1 Introduction N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 2 / 39

3 Outline 1 Introduction 2 Model Development Conservation of mass Solubility constraints N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 2 / 39

4 Outline 1 Introduction 2 Model Development Conservation of mass Solubility constraints 3 Abstract Evolution Equation Examples Analysis N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 2 / 39

5 Outline 1 Introduction 2 Model Development Conservation of mass Solubility constraints 3 Abstract Evolution Equation Examples Analysis 4 Numerical aspects Fully discrete scheme Semismooth Newton solver N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 2 / 39

6 Outline 1 Introduction 2 Model Development Conservation of mass Solubility constraints 3 Abstract Evolution Equation Examples Analysis 4 Numerical aspects Fully discrete scheme Semismooth Newton solver 5 Some experiments N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 2 / 39

7 Outline 1 Introduction 2 Model Development Conservation of mass Solubility constraints 3 Abstract Evolution Equation Examples Analysis 4 Numerical aspects Fully discrete scheme Semismooth Newton solver 5 Some experiments 6 Conclusions and future work N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 2 / 39

8 Introduction Motivation Methane hydrates are an ice-like substance containing methane molecules trapped in a lattice of water molecules. N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 3 / 39

9 Introduction Motivation Methane hydrates are an ice-like substance containing methane molecules trapped in a lattice of water molecules. We will consider a simplified model of evolution of methane hydrates in the hydrate zone of the sea-bed. N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 3 / 39

10 Introduction Motivation Methane hydrates are an ice-like substance containing methane molecules trapped in a lattice of water molecules. We will consider a simplified model of evolution of methane hydrates in the hydrate zone of the sea-bed. Components: CH 4 H2 O [Images from DOE-NETL] N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 3 / 39

11 Model Development Conservation of mass Conservation of mass for CH 4 component Let Ω R 3 be a bounded region of points x Ω. Parameters: (assumed given) φ 0 porosity K 0 permeability f M external source of CH 4 N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 4 / 39

12 Model Development Conservation of mass Conservation of mass for CH 4 component Let Ω R 3 be a bounded region of points x Ω. Parameters: (assumed given) φ 0 porosity K 0 permeability f M external source of CH 4 Assumptions: Pressure P(x) given by hydrostatic gradients Temperature T (x) given by geothermal gradients High pressure and low temperature imply Hydrate zone: only liquid and hydrate phases present N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 4 / 39

13 Model Development Conservation of mass Conservation of mass for CH 4 component (cont.) Unknowns: S l, S h saturations (void fractions), with S h = 1 S l χ lm, χ lw, χ ls, with χ lm + χ lw + χ ls,= 1 mass fractions of methane, water and salt in liquid phase N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 5 / 39

14 Model Development Conservation of mass Conservation of mass for CH 4 component (cont.) Unknowns: Note: S l, S h saturations (void fractions), with S h = 1 S l χ lm, χ lw, χ ls, with χ lm + χ lw + χ ls,= 1 mass fractions of methane, water and salt in liquid phase Salinity, χ ls, assumed known and fixed to that of seawater χ hm, χ hw mass fractions in hydrate phase (assumed known) N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 5 / 39

15 Model Development Conservation of mass Conservation of mass for CH 4 component (cont.) Unknowns: Note: S l, S h saturations (void fractions), with S h = 1 S l χ lm, χ lw, χ ls, with χ lm + χ lw + χ ls,= 1 mass fractions of methane, water and salt in liquid phase Salinity, χ ls, assumed known and fixed to that of seawater χ hm, χ hw mass fractions in hydrate phase (assumed known) Conservation of mass for CH 4 component t (φ 0S l ρ l χ lm + φ 0 S h ρ h χ hm ) (D lm ρ l χ lm ) = f M (1) where densities ρ l, ρ h and diffusion D lm are assumed constant. N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 5 / 39

16 Model Development Conservation of mass Unified notation We redefine parameters and redefine variables R := ρ hχ hm, f := f M, D 0 := D lm, ρ l ρ l φ 0 φ 0 S := S l, v := χ lm, u := Sv + R(1 S), so that (1) becomes u t (D 0 v) = f, (2) where we may further scale the problem so that D 0 = 1. N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 6 / 39

17 Model Development Solubility constraints Still need S l, χ lm F (x) provided by solubility constraints. First, we assume that the maximum solubility constraint χ max (x) is given. lm N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 7 / 39

18 Model Development Solubility constraints Still need S l, χ lm F (x) provided by solubility constraints. First, we assume that the maximum solubility constraint χ max (x) is given. lm Then the solubility satisfies a nonlinear complementarity constraint (NCC) Solubility constraint χ lm χlm max, S l = 1, χ lm = χlm max, S l 1, (χlm max χ lm)(1 S l ) = 0. (3) N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 7 / 39

19 Model Development Solubility constraints Still need S l, χ lm F (x) provided by solubility constraints. First, we assume that the maximum solubility constraint χ max (x) is given. lm Then the solubility satisfies a nonlinear complementarity constraint (NCC) Solubility constraint χ lm χlm max, S l = 1, χ lm = χlm max, S l 1, (χlm max χ lm)(1 S l ) = 0. (3) In our unified notation, we let v = χ max, so that lm 1 S v F (x) v v,s F (x; ) := [0,v (x)] {1} {v (x)} (0,1] Or equivalently, v α MH (x;u) := (u v (x)) + v (x), u R. (4) N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 7 / 39

20 Abstract Evolution Equation We consider the initial boundary-value problem Abstract Evolution Equation u v =f, t v α(u) on Ω (0,T ) v =0 on Ω (0,T ) u(,0) =u 0 ( ) on Ω. where α is maximal montone, or in the case of a measureable family {α(x; ) : x Ω}, each is a maximal monotone relation. N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 8 / 39

21 Abstract Evolution Equation Examples Examples: α(u) = α S, α MH, α E, α }{{ W, α } PM ; Note : β(v) = α 1 Toy models α S α E Stefan free-boundary problem: α S (u) = u + (u 1) + Elbow Showalter [1984] α MH v v R α W 1 Methane hydrate (our problem): α MH (x; ) = (u v (x)) + v (x), u R Porous medium equation: α = α PM (u) = u u m 1 (m > 1 slow diffusion, Woble 0 < m < 1 fast diffusion) N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 9 / 39

22 Abstract Evolution Equation Examples Examples: α(u) = α S, α MH, α E, α }{{ W, α } PM ; Note : β(v) = α 1 Toy models α S α E Stefan free-boundary problem: α S (u) = u + (u 1) + Elbow Showalter [1984] α MH v v R α W 1 Methane hydrate (our problem): α MH (x; ) = (u v (x)) + v (x), u R Porous medium equation: α = α PM (u) = u u m 1 (m > 1 slow diffusion, Woble 0 < m < 1 fast diffusion) N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 9 / 39

23 Abstract Evolution Equation Analysis Summary of theoretical results for α MH For the single graph case, we represent the non-linearity as a subgradient, and prove a useful comparison principle, which allows to extend the graph of β = α 1 to one which is affine bounded. Optimal regularity results follow. Properties of u,v are the same as those for the Stefan problem. N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 10 / 39

24 Abstract Evolution Equation Analysis Summary of theoretical results for α MH For the single graph case, we represent the non-linearity as a subgradient, and prove a useful comparison principle, which allows to extend the graph of β = α 1 to one which is affine bounded. Optimal regularity results follow. Properties of u,v are the same as those for the Stefan problem. We extend existing theory for porous medium equation to cover the case of a measureable family of graphs in order to show well-posedness. Based on a normal convex integrand construction. Details in Gibson, Medina, Peszynska, and Showalter [2013] N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 10 / 39

25 Numerical aspects Fully discrete scheme FE formulation for α = α MH (x, ) First, we apply fully implicit time stepping. Let V h V be the finite element space of continuous piecewise linears on triangulation of Ω. Find vh n V h at t n (n > 0) (uh n,ψ) + τ( v h n, ψ) = (un 1 h,ψ) u n h β(v h n) (uh 0,ψ) := (u 0,ψ), ψ V h N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 11 / 39

26 Numerical aspects Fully discrete scheme FE formulation for α = α MH (x, ) First, we apply fully implicit time stepping. Let V h V be the finite element space of continuous piecewise linears on triangulation of Ω. Find vh n V h at t n (n > 0) (uh n,ψ) + τ( v h n, ψ) = (un 1 h,ψ) u n h β(v h n) (uh 0,ψ) := (u 0,ψ), ψ V h Let M : mass matrix K : stiffness matrix v n h vn R M u n h un R M Mu n + τkv n = Mu n 1 N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 11 / 39

27 Numerical aspects Fully discrete scheme FE formulation for α = α MH (x, ) First, we apply fully implicit time stepping. Let V h V be the finite element space of continuous piecewise linears on triangulation of Ω. Find vh n V h at t n (n > 0) (uh n,ψ) + τ( v h n, ψ) = (un 1 h,ψ) u n h β(v h n) (uh 0,ψ) := (u 0,ψ), ψ V h Fully discrete scheme { u n + τa h v n = u n 1 v n j,un j β(x j; ) := β j ( ) where the constraint is applied point-wise. Let M : mass matrix K : stiffness matrix v n h vn R M u n h un R M Mu n + τkv n = Mu n 1 Mass-lumping A h = M 1 K allows N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 11 / 39

28 Numerical aspects Fully discrete scheme Lemma [N.Gibson, P. Medina, M. Peszynska, R.E. Showalter] For every n > 0 there is a unique solution v n R M of the discrete problem for β = β MH (x; ) it is the unique minimizer of the appropriate functional Ψ(v) for which the discrete problem is the Euler-Lagrange condition. N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 12 / 39

29 Numerical aspects Fully discrete scheme Lemma [N.Gibson, P. Medina, M. Peszynska, R.E. Showalter] For every n > 0 there is a unique solution v n R M of the discrete problem for β = β MH (x; ) it is the unique minimizer of the appropriate functional Ψ(v) for which the discrete problem is the Euler-Lagrange condition. Corollary The discrete scheme is uniquely solvable for each of β = β MH (x, ), β S, β E, β W. N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 12 / 39

30 Numerical aspects Fully discrete scheme Nonlinear algebraic subproblem Newton-type solvers have difficulties near singularties; may not be defined for multi-valued operators. N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 13 / 39

31 Numerical aspects Fully discrete scheme Nonlinear algebraic subproblem Newton-type solvers have difficulties near singularties; may not be defined for multi-valued operators. Relaxation solvers require a number of iterations proportional to the number of degrees of freedom. N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 13 / 39

32 Numerical aspects Fully discrete scheme Nonlinear algebraic subproblem Newton-type solvers have difficulties near singularties; may not be defined for multi-valued operators. Relaxation solvers require a number of iterations proportional to the number of degrees of freedom. In each, optimal convergence results are close to O(h) for v and O(h 1 2 ) for u in L 2 (Q). N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 13 / 39

33 Numerical aspects Fully discrete scheme Nonlinear algebraic subproblem Newton-type solvers have difficulties near singularties; may not be defined for multi-valued operators. Relaxation solvers require a number of iterations proportional to the number of degrees of freedom. In each, optimal convergence results are close to O(h) for v and O(h 1 2 ) for u in L 2 (Q). We propose a scheme which does not require regularization and can be applied when neither α nor β are functions. N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 13 / 39

34 Numerical aspects Fully discrete scheme Nonlinear algebraic subproblem Newton-type solvers have difficulties near singularties; may not be defined for multi-valued operators. Relaxation solvers require a number of iterations proportional to the number of degrees of freedom. In each, optimal convergence results are close to O(h) for v and O(h 1 2 ) for u in L 2 (Q). We propose a scheme which does not require regularization and can be applied when neither α nor β are functions. The method also applies when constraints are parameterized by x. N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 13 / 39

35 Numerical aspects NCC and Semismooth Newton solver Nonlinear Complementarity Problem (NCP) We represent as an NCP-function v,u β φ(u,v) = 0. N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 14 / 39

36 Numerical aspects NCC and Semismooth Newton solver Nonlinear Complementarity Problem (NCP) We represent as an NCP-function v,u β φ(u,v) = 0. For example, v,u β MH φ MH (u,v) := min(u v,v (x) v) = 0. N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 14 / 39

37 Numerical aspects NCC and Semismooth Newton solver Nonlinear Complementarity Problem (NCP) We represent as an NCP-function v,u β φ(u,v) = 0. For example, v,u β MH φ MH (u,v) := min(u v,v (x) v) = 0. Similarly, v,u β E φ E (u,v) := min(u,1 v) = 0, v,u β W φ W (u,v) := min(1 u,v) = 0, v,u β S φ S (u,v) := u v max(0,min(u,1)) = 0. N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 14 / 39

38 Numerical aspects NCC and Semismooth Newton solver Semismooth Newton solver Problem solved at every time step becomes { u + τa h v min(u j v j,v (x j ) v j ) = b = 0, j N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 15 / 39

39 Numerical aspects NCC and Semismooth Newton solver Semismooth Newton solver Problem solved at every time step becomes { u + τa h v min(u j v j,v (x j ) v j ) = b = 0, j It can be shown that each of φ MH,φ E,φ W,φ S is semi-smooth the Jacobian is never singular Semismooth Newton converges superlinearly for these NCC problems Ben Gharbia, Gilbert, and Jaffre [2011]; Ulbrich [2011] N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 15 / 39

40 Some experiments α = α E. Toy model (Showalter [1984]) Frame I N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 16 / 39

41 Some experiments α = α E. Toy model (Showalter [1984]) Frame II N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 17 / 39

42 Some experiments α = α E. Toy model (Showalter [1984]) Frame III N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 18 / 39

43 Some experiments α = α E. Toy model (Showalter [1984]) Frame IV N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 19 / 39

44 Some experiments α = α E. Toy model (Showalter [1984]) Frame V N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 20 / 39

45 Some experiments α = α E. Toy model (Showalter [1984]) Frame VI N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 21 / 39

46 Some experiments α = α E. Toy model (Showalter [1984]) N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 22 / 39

47 Some experiments Convergence in u and v for α E Using τ = h 10, h 100, h2 error rate error rate error rate 1/h 1/τ N it e u,2 r u,2 e v,2 r v,2 e q r q e e e e e e e e e e e e e e e e e e (with quasi-norm: n τ Ω u un h v v h n dx, Ebmeyer and Liu [2008]). Observed rates e u,2 O(h 1/2 ), e v,2 O(h), e q O(h) N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 23 / 39

48 Some experiments α = α MH, v max 1. No analytical solution. Frame I N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 24 / 39

49 Some experiments α = α MH, v max 1. No analytical solution. Frame II N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 25 / 39

50 Some experiments α = α MH, v max 1. No analytical solution. Frame III N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 26 / 39

51 Some experiments α = α MH, v max 1. No analytical solution. Frame IV N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 27 / 39

52 Some experiments α = α MH, v max 1. No analytical solution. Frame V N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 28 / 39

53 Some experiments α = α MH, v max 1. No analytical solution. Frame VI N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 29 / 39

54 Some experiments α = α MH, v max 1. No analytical solution. N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 30 / 39

55 Some experiments Convergence in u and v for α MH, v max 1. Using τ = h 10, h 100, h2 1/h 1/τ N it e u,2 r u,2 e v,2 r v,2 e q r q e e e e e e e e e e e e e e e e e e Observed rates e u,2 O(h 1/2 ), e v,2 O(h), e q O(h) N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 31 / 39

56 Some experiments α = α MH (x; ), v max(x) = (1 + x)/2 N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 32 / 39

57 Some experiments Convergence rates in u and v for α = α MH (x; ), v max(x) = (1 + x)/2 Using τ = h 10, h 100, h2 1/h 1/τ N it e u,2 r u,2 e v,2 r v,2 e q r q e e e e e e e e e e e e e e e e e e Observed rates e u,2 O(h 1/2 ), e v,2 O(h), e q O(h) N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 33 / 39

58 Some experiments α = α MH (x; ), v max(x) = (1 + 2x x 2 )/2 N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 34 / 39

59 Some experiments Convergence in u and v for α = α MH (x; ), v max(x) = (1 + 2x x 2 )/2 Using τ = h 10, h 100, h2 1/h 1/τ N it e u,2 r u,2 e v,2 r v,2 e q r q e e e e e e e e e e e e e e e e e e Observed rates e u,2 O(h 1/2 ), e v,2 O(h), e q O(h) N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 35 / 39

60 Some experiments Convergence in S for α = α MH Using τ = h 100 constant affine non-affine 1/h e S,2 r S,2 e S,2 r S,2 e S,2 r S, e e e e e e e e e Observed rates e S,2 O(h 1/2 ) (Similar to rates in u.) N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 36 / 39

61 Conclusions and future work Summary We described a solubility constrained methane hydrate model. N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 37 / 39

62 Conclusions and future work Summary We described a solubility constrained methane hydrate model. We reformulated it into an abstract evolution equation constrained by parameter-dependent familes of graphs. N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 37 / 39

63 Conclusions and future work Summary We described a solubility constrained methane hydrate model. We reformulated it into an abstract evolution equation constrained by parameter-dependent familes of graphs. One can extend monotone operator theory to case of measureable family of graphs to show well-posedness. N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 37 / 39

64 Conclusions and future work Summary We described a solubility constrained methane hydrate model. We reformulated it into an abstract evolution equation constrained by parameter-dependent familes of graphs. One can extend monotone operator theory to case of measureable family of graphs to show well-posedness. Regularity of solutions is the same as in the Stefan problem, at least in the single graph case. N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 37 / 39

65 Conclusions and future work Summary We described a solubility constrained methane hydrate model. We reformulated it into an abstract evolution equation constrained by parameter-dependent familes of graphs. One can extend monotone operator theory to case of measureable family of graphs to show well-posedness. Regularity of solutions is the same as in the Stefan problem, at least in the single graph case. We have proposed a numerical scheme which applies semismooth Newton to complementarity conditions. N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 37 / 39

66 Conclusions and future work Summary We described a solubility constrained methane hydrate model. We reformulated it into an abstract evolution equation constrained by parameter-dependent familes of graphs. One can extend monotone operator theory to case of measureable family of graphs to show well-posedness. Regularity of solutions is the same as in the Stefan problem, at least in the single graph case. We have proposed a numerical scheme which applies semismooth Newton to complementarity conditions. Semismooth Newton solver requires mesh independent iterations. N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 37 / 39

67 Conclusions and future work Summary We described a solubility constrained methane hydrate model. We reformulated it into an abstract evolution equation constrained by parameter-dependent familes of graphs. One can extend monotone operator theory to case of measureable family of graphs to show well-posedness. Regularity of solutions is the same as in the Stefan problem, at least in the single graph case. We have proposed a numerical scheme which applies semismooth Newton to complementarity conditions. Semismooth Newton solver requires mesh independent iterations. Convergence rates for examples agree with optimal results for the Stefan problem. N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 37 / 39

68 Conclusions and future work Summary We described a solubility constrained methane hydrate model. We reformulated it into an abstract evolution equation constrained by parameter-dependent familes of graphs. One can extend monotone operator theory to case of measureable family of graphs to show well-posedness. Regularity of solutions is the same as in the Stefan problem, at least in the single graph case. We have proposed a numerical scheme which applies semismooth Newton to complementarity conditions. Semismooth Newton solver requires mesh independent iterations. Convergence rates for examples agree with optimal results for the Stefan problem. Incidentally discovered a semismooth Newton method for the Stefan problem. N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 37 / 39

69 Conclusions and future work Future work Implementation and convergence studies for the gas zone. N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 38 / 39

70 Conclusions and future work Future work Implementation and convergence studies for the gas zone. Include salinity as unknown. N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 38 / 39

71 Conclusions and future work Future work Implementation and convergence studies for the gas zone. Include salinity as unknown. Semi-implicit time stepping. N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 38 / 39

72 Conclusions and future work References I. Ben Gharbia, J. C. Gilbert, and J. Jaffre. Nonlinear complementarity constraints for two-phase flow in porous media with gas phase appearance and disappearance. Numerical approximation of hysteresis problems, C. Ebmeyer and W. B. Liu. Finite element approximation of the fast diffusion and the porous medium equations. SIAM J. Numer. Anal., 46(5): , N. L. Gibson, F. P. Medina, M. Peszynska, and R. E. Showalter. Evolution of phase transitions in methane hydrate. To appear in Journal of Mathematical Analysis and Applications, R. H. Nochetto and C. Verdi. Approximation of degenerate parabolic problems using numerical integration. SIAM J. Numer. Anal., 25(4): , J. Rulla. Error analysis for implicit approximations to solutions to Cauchy problems. SIAM J. Numer. Anal., 33(1):68 87, R. E. Showalter. A singular quasilinear diffusion equation in L 1. J. Math. Soc. Japan, 36(2): , R. E. Showalter. Monotone operators in Banach space and nonlinear partial differential equations, volume 49 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, Michael Ulbrich. Semismooth Newton methods for variational inequalities and constrained optimization problems in function spaces, volume 11 of MOS-SIAM Series on Optimization. SIAM, Philadelphia, PA, N. L. Gibson (OSU) Methane Hydrate Evolution SIAM AN13 39 / 39