Global Jacobian Mortar Algorithms for Multiphase Flow in Porous Media Ben Ganis* Collaborators: Kundan Kumar*, Gergina Pencheva*, Mary F. Wheeler*, Ivan Yotov** *Center for Subsurface Modeling, ICES, UT Austin **University of Pittsburgh ICES Seminar-Babuska Forum Series November 7, 2014 C enter for S ubsurface Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu M odeling
Multiscale Mortar Mixed FEM • Mortar finite elements are a domain decomposition technique to couple unknowns across: – Multiple Scales – Multiple Physics – Multiple Numerics – Multiple Processors Subdomains Ω k Interfaces Γ kl • Note that Domain Decomposition is not the same as “Data Decomposition”. • The “Global Jacobian” algorithms developed in this research seek to have the best of both worlds. C enter for S ubsurface Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu M odeling
Models used with mortars • Mortars have been used with: – 1,2,3 phase flows in porous media – CG, Mixed, DG methods – Linear elastic solid mechanics – Bricks, prisms, tetrahedra – Porescale network models • Example: Saturation field in two phase flow, with two subdomains. • Prior to this research, the solution algorithm for nonlinear problems relied on two Newton loops with a forward difference approximation. C enter for S ubsurface Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu M odeling
Selected ¡References ¡on ¡Mortars ¡ Single Phase Mortar Theory • Glowinski, R., and Wheeler, M.F. 1988. Domain decomposition and mixed finite element methods for elliptic problems. In 1 st international symposium on domain decomposition methods for PDEs . • Arbogast, T., Cowsar, L.C., Wheeler, M.F. and Yotov, I. 2000. Mixed finite element methods on nonmatching multiblock grids. SIAM Journal on Numerical Analysis 37 (4): 1295– 1315. • Arbogast, T., Pencheva, G., Wheeler, M.F., and Yotov, I. 2007. A multiscale mortar mixed finite element method. Multiscale Modeling & Simulation 6 (1): 319–346. Forward Difference (FD) Algorithms for Nonlinear problems • Peszynska, M., Wheeler, M.F., and Yotov, I. 2002. Mortar upscaling for multiphase flow in porous media. Computational Geosciences 6 (1): 73–100. • Yotov, I. 2001. A multilevel Newton–Krylov interface solver for multiphysics couplings of flow in porous media. Numerical Linear Algebra and Applications , 8 (8): 551–570. Global Jacobian (GJ) Algorithms for Nonlinear problems • Ganis, B., Juntunen, M., Pencheva, G., Wheeler, M.F., and Yotov, I. 2014. A global Jacobian method for mortar discretizations of nonlinear porous media flows. SIAM Journal on Scientific Computation 36 (2): A522–A542. • Ganis, B., Kumar, K., Pencheva, G., Wheeler, M.F., and Yotov, I. 2014. A global Jacobian method for mortar discretizations of a fully-implicit two-phase flow model. Multiscale Modeling & Simulation 12 (4): 1401–1423. • Ganis, B., Kumar, K., Pencheva, G., Wheeler, M.F., Yotov, I. A multiscale mortar method and two- stage preconditioner for multiphase flow using a global Jacobian approach. SPE 172990-MS. C enter for S ubsurface Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu M odeling
Outline 1. Multiscale, Multiphase Problem Setting 2. Fully-implicit two-phase model for flow in porous media 3. Global Jacobian algorithms – Schur complements – Interface unknowns – Upwinding scheme 4. Numerical results – Strongly Heterogeneous Case – Two Rock Type Case – Non-matching Geometry Case 5. Two-Stage Preconditioner and Parallel Results C enter for S ubsurface Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu M odeling
Problem Setting • Non-overlapping domain decomposition on spatial domain Use mixed finite elements on Use high-order mortars (Lagrange Subdomains Ω k Interfaces Γ kl structured subdomain grids multipliers) on non-matching interfaces • Application : Multiphase flow in porous media • Goal : Develop simple algorithms with parallel scalability • Key Idea : Global linearization • Capillarity, gravity, and compressibility. C enter for S ubsurface Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu M odeling
Algorithms for nonlinear mortar problems New Methods Prior Method • This algorithm uses local linearizations GJ Method GJS Method FD Method for subdomain and mortar unknowns separately. – Two nested Newton-Krylov loops Time Step Time Step Time Step – Outer loop formes a numerical Jacobian with a forward difference Nonlinear Global Nonlinear Global Nonlinear Interface Newton Step Newton Step FD-Newton Step – Requires delicate choice of four tolerances and difference parameter Linear Global Linear Interface Linear Interface GMRES Step GMRES Step – Challenging to precondition outer GMRES Step GMRES Linear Subdomain + Allows multiple physics and multiple Nonlinear Subdom. GMRES Step Newton Step time steps Linear Subdom. GMRES Step = convergence check = forward difference approximation used C enter for S ubsurface Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu M odeling
Algorithms for nonlinear mortar problems New Methods Prior Method GJ Method GJS Method FD Method Time Step Time Step Time Step Nonlinear Global Nonlinear Global Nonlinear Interface Newton Step Newton Step FD-Newton Step Linear Global Linear Interface Linear Interface GMRES Step GMRES Step GMRES Step Linear Subdomain Nonlinear Subdom. GMRES Step Newton Step Linear Subdom. GMRES Step = convergence check = forward difference approximation used C enter for S ubsurface Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu M odeling
Novelty of this work Ω 1 Ω 2 Ω 3 Γ 12 Γ 23 Proc1 Proc2 Ω 1 J ΘΘ � � J ΘΛ Ω 2 Ω 1 Ω 2 = Ω 3 J ΛΘ J ΛΛ Γ 23 Γ 12 Ω 3 Γ 12 Γ 23 Global linearization: • Augment linear systems to reuse codes. • Utilize existing preconditioners for multiscale models. • Simplify algorithms by having fewer nested iterations. • Demonstrate parallel scaling with strong nonlinearities. • Improve saturation with careful mobility upwinding. C enter for S ubsurface Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu M odeling
Parallel scaling, nonlinear single phase Homogeneous, Heterogeneous, No Preconditioning AMG+ILU Preconditioner 1 day GJ 1 hour GJS GJ 40 min FD 6 hours 20 min Wall clock time Wall clock time 1 hour 10 min 20 min 5 min 5 min 4 min 3 min 2 min 1 min 2 4 8 16 32 64 128 1 2 4 8 16 32 64 128 256 512 Processors (Subdomains) Processors (Subdomains) Strong scaling, O(10 6 ) elements [2] B. Ganis, M. Juntunen, G. Pencheva, M.F. Wheeler, I. Yotov. A global Jacobian method for mortar discretizations of nonlinear porous media flows. SIAM Journal on Scientific Computation , Vol. 36, No. 2, (2014) pp. A522-A542. C enter for S ubsurface Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu M odeling
Two Phase Model ∂ in Ω k ⇥ (0 , T ] Mass Balance: ∂ t ( φ s α ρ α ) + r · u α = q α in Ω k ⇥ (0 , T ] Auxiliary Velocity: u α = � K ( r p α � ρ α g ) e e u α = k r α ρ α in Ω k ⇥ (0 , T ] Darcy Law: e u α µ α p α = p α , 0 at Ω ⇥ { t = 0 } , Initial condition: u · n = 0 on ∂ Ω ⇥ (0 , T ] Boundary condition: p α = p Γ α ( λ 1 , λ 2 ) on Γ ⇥ (0 , T ] , Lagrange multiplier: α · n k + u l α · n l = 0 on Γ kl ⇥ (0 , T ] u k Flux continuity: s w + s o = 1 Saturation constraint: p c ( s w ) = p o � p w Capillary pressure: ρ α ( p α ) = ρ ref α e c α p α . Slightly compressible density: C enter for S ubsurface Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu M odeling
Finite element discretization e ( p o , n o ), Primary Unknowns: s ( e u o , e u w , u o , u w ), Phase Velocities: Lowest Order Raviart- s ( λ 1 , λ 2 ) Lagrange Multipliers: Thomas (RT0) mixed finite elements with mortars Velocity, Pressure, Mortar Spaces: mortar velocity N Ω N Ω M M V k W k V h = W h = h , h , pressure k =1 k =1 N Ω M M kl d M H = M M H . k =1 Time discretization: on 0 = t 0 < t 1 < · · · < t N T = T , with δ t n = t n � t n � 1 . T C enter for S ubsurface Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu M odeling
Fully discrete system Expanded multiscale mortar n α = ρ α s α Phase concentration: method for fully-implicit two- phase flow: m α = k r α ρ α Phase mobility: d µ α � � u k A k Ω k u k Ω k m α � α · v dx − α · v dx = 0 , α = � � � � N Ω � u k D k Ω k K − 1 � Ω k p k Γ kl p Γ α · v dx − α ∇ · v dx − Ω k ρ α g · v dx + α v · n d σ = 0 , α = l =1 ,l ̸ = k � � � φ n k α − φ n n − 1 B k α Ω k ∇ · u k w dx + α w dx − Ω k q α w dx = 0 , α = δ t Ω k � � � u k α · n k + u l Flux continuity equation H α = µ d σ = 0 . α · n l Γ kl C enter for S ubsurface Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu M odeling
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