C enter for S ubsurface M odeling Comparison of Numerical Schemes for Accurate Long-term Simulation of Contaminant Migration Krzysztof Banaś 1,2, Steve Bryant 2 1 ICM, Cracow University of Technology 2 TICAM, The University of Texas at Austin

C enter for S ubsurface M odeling CSM Researchers Mary F. Wheeler (DIRECTOR) Todd Arbogast Steven Bryant Clint Dawson Rick Dean Eleanor Jenkins Phu Luong Victor Parr Malgorzata Peszynska Béatrice Rivière John Wheeler + several visitors + 10 graduate students : 5 PhD, 1 MS completed

C enter for S ubsurface M odeling Overview Experience with ParSSimExperience with ParSSim –First order Godunov –Higher order Godunov –Characteristics-mixed method Experience with DG research codesExperience with DG research codes

C enter for S ubsurface M odeling ParSSim Parallel Subsurface Simulator Multicomponent –logically rectangular 3D –operator splitting General biogeochemistry –interior point minimization of free energy –explicit integration of kinetics ODEs Scalable Parallel –domain decomposition (MPI) –SP2, cluster of PCs, T3E, Workstations –dynamic load balancing 1 flowing phase, N stationary phases

C enter for S ubsurface M odeling ParSSim solution scheme: operator splitting Solve flow equation Solve transport equations –Advect –React rate-limited reactions, mass transfer thermodynamic equilibrium –Diffuse –Update composition-dependent viscosity, permeability

C enter for S ubsurface M odeling ParSSim Flow Calculation Single phase Darcy flow Logically rectangular, cell-centered finite difference, implicit Glowinski-Wheeler domain decomposition

C enter for S ubsurface M odeling ParSSim Transport Linear sorption Biogeo- chemistry Radionuclide decay Injection/ extraction wells Advection step: solve

C enter for S ubsurface M odeling –Explicit characteristics-mixed method* Introduce total concentration T i : Resulting PDE: Solve by characteristic tracking: Extract advected concentrations: ParSSim Transport: advection step (1) *Arbogast and Wheeler, SIAM J. Numer. Anal. 32 (1995)

C enter for S ubsurface M odeling –Higher order Godunov* –Solve directly for advected concentrations –Formally 2 nd order, improved by postprocessing step –First order Godunov (no postprocessing) *Dawson, SIAM J. Numer. Anal. 30 (1993) ParSSim Transport: advection step (2)

C enter for S ubsurface M odeling ParSSim Transport: reaction step React the advected concentrations Equilibrium reactions handled by free energy minimization –Radionuclide decay Solve the PDE… By explicit integration –Biogeochemistry Solve the PDE… By explicit integration

C enter for S ubsurface M odeling ParSSim Transport: diffusion step Diffuse/disperse the reacted concentrations Solve the PDE… Implicitly by

C enter for S ubsurface M odeling Transport scheme comparisons Couplex1 Transport scheme benchmarks* –Moving hill –Curvilinear flow –Mixed waste *

C enter for S ubsurface M odeling Couplex1 Test Case

C enter for S ubsurface M odeling Couplex1 Simulation 129 I plume, Higher Order Godunov solution

C enter for S ubsurface M odeling Couplex1 Simulation 129 I plume, Higher Order Godunov solution

C enter for S ubsurface M odeling Couplex1 Variant Decrease clay layer diffusion coefficient 1000 times 129 I plume, Higher Order Godunov solution

C enter for S ubsurface M odeling Couplex1 Simulation 129 I plume, character- istics- mixed method solution

C enter for S ubsurface M odeling Couplex1 Simulation 129 I plume, Characteristics Mixed Method solution

C enter for S ubsurface M odeling Couplex1 Variant Decrease clay layer diffusion coefficient 1000 times 129 I plume, Characteristics Mixed Method solution Note “hot spot” at clay-limestone boundary

C enter for S ubsurface M odeling Moving Hill (Benchmark 6) Conservative tracer x 50  50 grid v x = v y N Pe(grid) = 10 2 N Cr = 0.1 y

C enter for S ubsurface M odeling ms/pcl/prob1/ashok1.htm

C enter for S ubsurface M odeling ms/pcl/prob1/ashok1.htm

C enter for S ubsurface M odeling Benchmark 6: First order Godunov

C enter for S ubsurface M odeling Benchmark 6: Higher order Godunov

C enter for S ubsurface M odeling Benchmark 6: Characteristics Mixed Method

C enter for S ubsurface M odeling Benchmark 6: Discontinuous Galerkin Method

C enter for S ubsurface M odeling Benchmark 6: Characteristics Mixed Method, N Cr =1

C enter for S ubsurface M odeling Benchmark 6: Results First order GodunovHigher order Godunov Characteristics-mixed methodDiscontinuous Galerkin

C enter for S ubsurface M odeling Benchmark 5 Description solute inlet effluent collection window 150 x 150 grid zero diffusion

C enter for S ubsurface M odeling Benchmark 5: CMM

C enter for S ubsurface M odeling Benchmark 5: DG

C enter for S ubsurface M odeling Benchmark 5: Tracer Effluent Analytical HOG CMM

C enter for S ubsurface M odeling Benchmark 5: DG results Results from current research code, courtesy K. Banas

C enter for S ubsurface M odeling Benchmark 5: Reactive Solute Effluent

C enter for S ubsurface M odeling Benchmark 5: Reactive Solute Effluent Analytical, half order HOG

C enter for S ubsurface M odeling Transport scheme comments Characteristics-mixed method (CMM) –Minimal numerical dispersion –No CFL constraint (except that arising from domain decomposition) –Good scaling in parallel –Computationally expensive Higher order Godunov –Locally mass conservative –CFL constraint –More numerical dispersion than CMM –Computationally inexpensive Discontinuous Galerkin –Even less numerical dispersion than CMM –Subject of current research

C enter for S ubsurface M odeling DG Methods Unstructured, non-matching grids Element refinements/de-refinements Local high order of approximations Arbitrary tensor coefficients and BC Locally conservative Error estimators (for hp adaptivity) Positive definiteness

C enter for S ubsurface M odeling 3D elements of “arbitrary” shape Tetrahedral Prismatic Hexahedral

C enter for S ubsurface M odeling Unstructured grids

C enter for S ubsurface M odeling Non-matching grids

C enter for S ubsurface M odeling Non-matching grids

C enter for S ubsurface M odeling Slope limiting

C enter for S ubsurface M odeling hp-adaptivity Order of approximation specified element-wise Isotropic element divisions Anisotropic element divisions

C enter for S ubsurface M odeling Example of 3D convection

C enter for S ubsurface M odeling Example of 3D convection

C enter for S ubsurface M odeling Example of 3D convection

C enter for S ubsurface M odeling Iterative linear equations solver Large, sparse, non-symmetric, positive definite systems of linear equations Single level GMRES as default solver Multi-level GMRES for elliptic problems Preconditioning by ILU, domain decomposition Implementation based on block Gauss-Seidel iterations Easy parallelization

C enter for S ubsurface M odeling Application to subsurface flows Single phase flow Miscible displacement Two-phase flow Interfaces with IPARS framework

C enter for S ubsurface M odeling Application to subsurface flow

C enter for S ubsurface M odeling Miscible Displacement Concentration Front

C enter for S ubsurface M odeling Miscible Displacement Concentration isocontour of solvent in unstable flow

C enter for S ubsurface M odeling Conclusions Accurate dispersion calculation critical to Couplex1 behavior Transport Schemes: ParSSim testbed –Lower order Godunov –Higher order Godunov –Characteristics Mixed Method –CMM, HOG as good or better than TVD Research continues on discontinuous Galerkin –Appears to be best yet

C enter for S ubsurface M odeling Contact ParSSim Todd Arbogast: Steve Bryant: IPARS Malgorzata Peszynska: DG Béatrice Rivière: Visit us at