1. Direct and iterative sparse linear solvers applied to groundwater flow simulations Matrix Analysis and Applications October 2007

2. Jocelyne Erhel INRIA Rennes Jean-Raynald de Dreuzy CNRS, Geosciences Rennes Anthony Beaudoin LMPG, Le Havre Partly funded by Grid’5000 french project

3. From Barlebo et al. (2004) Dispersion Flow Injection of tracer Tracer evolution during one year (Made, Mississippi) Heterogeneous permeability Physical context: groundwater flow Flow governed by the heterogeneous permeability Solute transport by advection and dispersion

4. Head Numerical modelling strategy Numerical Stochastic models Simulation results Physical model natural system Simulation of flow and solute transport Characterization of heterogeneity Model validation

5. Uncertainty Quantification methods Spatial heterogeneity Stochastic models of flow and solute transport -random velocity field -random solute transfer time and dispersivity Lack of observationsPorous geological media Solute dispersionHorizontal velocity Permeability field

6. HYDROLAB: parallel software for hydrogeoloy Numerical methods Physical models Porous Media Solvers PDE solvers ODE solvers Linear solvers Particle tracker Utilitaries Input / Output Visualization Results structures Parameters structures Parallel and grid tools Geometry PARALLEL-BASED SCIENTIFIC PLATFORM HYDROLAB Open source libraries Boost, FFTW, CGal, MPI, Hypre, Sundials, OpenGL, Xerces-C UQ methods Monte-Carlo Fracture Networks Fractured- Porous Media Object-oriented and modular with C++ Parallel algorithms with MPI Efficient numerical libraries

7.  Saturated medium: one water phase  Constant density: no saltwater  Constant porosity and constant viscosity  Linear equations  Steady-state flow or transient flow  Inert transport: no coupling with chemistry  No coupling between flow and transport  No coupling with heat equations  No coupling with mechanical equations  Classical boundary conditions  Classical initial conditions Physical equations Flow equations: Darcy law and mass conservation Transport equations: advection and dispersion

8. Flow and transport equations Fixed head and C=0 Fixed head and  C/  n=0 Nul flux and  C/  n = 0 injection Advection-dispersion equations Boundary conditions Initial condition Flow equations

9. Monte-Carlo simulations For j=1,…,M Compute V j using a finite volume method generate permeability field K j using a regular mesh End For

10. Discrete flow numerical model Linear system Ax=b b: boundary conditions and source term A is a sparse matrix : NZ coefficients Matrix-Vector product : O(NZ) opérations Regular 2D mesh : N=n 2 and NZ=5N Regular 3D mesh : N= n 3 and NZ=7N Need for parallel sparse linear solvers

11. Accuracy: condition number and variance Estimation with MUMPS solver Cond(A) in O(exp(  2 )) But in theory, cond(A) in O(Kmax/Kmin) thus in O(exp(  ))

12. Accuracy: condition number and scaling Estimation with Matlab without scaling and with scaling Scaled condition number in O(exp(  As expected

13. Componentwise condition number  Matlab condition number residualScaled condition number Componentwise condition number Solution error 16.2085e+0043.0868e-0151.6121e+0042.4745e+0036.7876e-015 21.1660e+0065.7614e-0156.8597e+0044.6892e+0031.8597e-014 33.4221e+0076.9331e-0151.7549e+0051.2636e+0048.4842e-015 42.1117e+0098.8442e-0155.0661e+0054.2924e+0045.2058e-013 52.0372e+0112.1596e-0141.6503e+0061.6898e+0054.7836e-014 Componentwise condition number estimated by || |A -1 | |A| |x| + |A -1 | b || 1 / ||x|| 1 Solution error means ||x-x s || / ||x s || n=64

14. Accuracy: condition number and system size Estimation with MUMPS for  Cond(A) in O(N) as expected Condition number not too large for  · 3 and for N up to 16 millions

15. Sparse direct linear solver UMFPACK multifrontal solver Robust to variance  but CPU time in O(N 1.5 ) As expected

16. Preconditioned Conjugate Gradient PCG with IC(0) slightly sensitive to variance  But very sensitive to size N Need for a multilevel preconditioner

17. Geometric multigrid HYPRE Solver SMG Linear CPU time in O(N) but sensitivity to variance As expected

18. Algebraic multigrid HYPRE Solver AMG Robust to variance and linear CPU time in O(N) As expected Less efficient than SMG for small variance

19. Algebraic multigrid with 3D domains Robust to variance and CPU time in O(N) Same properties as in 2D

20. Parallel computing facilities Numerical model Clusters at Inria Rennes Grid’5000 project 67.1 millions of unknowns in 3 minutes with 32 processors

21. Parallel performances with 2D domains Parallel CPU time in O(N) SMG more efficient than AMG for small  AMG much more efficient than SMG for large 

22. Longitudinal dispersion Transversal dispersion Macro-dispersion analysis Each curve represents 100 simulations on domains with 67.1 millions of unknowns

23. Conclusion Summary Efficient and accurate algebraic multigrid solver for groundwater flow in heterogeneous porous media Good performances with clusters Macro-dispersion analysis in 2D domains Current and Future work 3D heterogeneous porous media Subdomain method with Aitken-Schwarz acceleration Transient flow in 2D and 3D porous media Grid computing and parametric simulations UQ methods