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A parallel scientific software for heterogeneous hydrogeoloy Conference on Parallel CFD Antalya, Turkey May 2007

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Jocelyne Erhel INRIA Rennes Jean-Raynald de Dreuzy Geosciences Rennes Anthony Beaudoin LMPG, Le Havre Etienne Bresciani INRIA Rennes Damien Tromeur-Dervout CDCSP, Lyon Partly funded by Grid5000 french project

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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

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Physical context: groundwater flow Spatial heterogeneity Stochastic models of flow and solute transport -random velocity field -random solute transfer time and dispersivity Lack of observations Porous geological media fractured geological media Flow in highly heterogeneous porous medium 3D Discrete Fracture Network

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

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Hydrolab scientific software Object-oriented and modular with C++ Parallel algorithms with MPI Efficient numerical libraries free software

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PHYSICAL MODELS Porous Media Fracture Networks Fractured Porous Hydrolab platform: physical models Physical equations

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Permeability field in porous media Simple 2D or 3D geometry stochastic permeability field finitely or infinitely correlated Multifractal D 2 =1.7 finitely correlated medium Multifractal D 2 =1.4

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D=1 d w =2 10 h 100 h D=2 d w =2 Well test Interpretation Generalized flux equation Cristalline aquifer of Ploemeur (Brittany, France) D=1.5 d w =2.8

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Longitudinal dispersion Transversal dispersion Macro-dispersion analysis

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Natural Fractured Media Fractures exist at any scale with no correlation Fracture length is a parameter of heterogeneity n(l)~l -2.7 probability density n ( l ) Fracture length l Site of Hornelen, Norway

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Discrete Fracture Networks with impervious matrix Stochastic computational domain length distribution has a great impact : power law n(l)=l -a 3 types of networks based on the moments of length distribution Existing mean No variation 2 < a < 3 Existing mean Existing variation No third moment 3 < a < 4 Existing mean Existing variation Existing third moment a > 4

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Output of simulations in 2D fracture networks : upscaling

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

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NUMERICAL METHODS PDE solvers EDO/DAE solvers Linear solvers Particle tracker Multilevel methods Monte-Carlo method UQ methods Parametric simulations Hydrolab platform: numerical methods

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Darcy law and mass conservation Darcy law and mass conservation Boundary conditions Boundary conditions Given head Nul flux 3D fracture network Given Head Nul flux 2D porous medium Steady-state flow equations

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Uncertainty Quantification methods Probabilistic framework Given statistics of the input data, compute statistics of the random solution stochastic permeability field Kstochastic network Ω stochastic flow equations stochastic velocity field n(l)=l -a

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Monte-Carlo simulations For j=1,…,M sample network Ω j compute v j sample permeability field K j

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Spatial discretization 2D heterogeneous porous medium Finite volume and regular grid 3D Discrete Fracture Network Mixed Finite Elements and non structured grid

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Meshing a 3D fracture network Direct mesh : poor quality or unfeasible Projection of the fracture network: feasible and good quality

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Mesh and flux computation in 3D fracture networks

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Parallel algorithms Domain decomposition

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Parallel computing facilities 16,8 millions of unknowns in 100 seconds with 16 processors ©INRIA/Photo Jim Wallace Numerical model Clusters at Irisa Grid5000 project Funded by French Government and Brittany council

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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 Direct linear solvers: fill-in in Cholesy factor Regular 2D mesh : N=n 2 and NZ=5N Regular 3D mesh : N= n 3 and NZ=7N Fracture Network : N and NZ depend on the geometry N = 8181 Intersections and 7 fractures Parallel sparse linear solvers

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2D heterogeneous porous medium memory size and CPU time with PSPASES Theory : NZ(L) = O(N logN)Theory : Time = O(N 1.5 ) variance = 1, number of processors = 2 Sparse direct linear solvers

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2D heterogeneous porous medium CPU time with HYPRE/AMG Linear complexity of BoomerAMG Sparse iterative linear solvers variance = 1, number of processors = 4 residual=10 -8

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Flow computation in 2D porous medium Finitely correlated permeability field Impact of permeability variance matrix order N = 10 6 PSPASES and BoomerAMG independent of variance BoomerAMG faster than PSPASES with 4 processors matrix order N = 16 10 6

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parallel sparse linear solvers 2D heterogeneous porous medium Direct and multigrid solvers Parallel CPU time variance = 9 matrix order N = 10 6 matrix order N = 4 10 6

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Solute transport Fixed head and C=0 Fixed head and C/ n=0 Nul flux and C/ n = 0 injection Advection-dispersion equation Boundary conditions Initial condition

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injection Solute transport : particle tracker Many independent particles Bilinear interpolation for V Homogeneous molecular diffusion D m Stochastic differential equation First-order explicit scheme Parallel particle tracker Bunch of independent particles Subdomain decomposition

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1 and Pe=1 3 and Pe=100 Solute transport in 2D heterogeneous porous media PermeabilityParticles Horizontal velocityVertical velocity

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particle tracker : convergence analysis N p = 1000 is a good trade-off between efficiency and convergence Longitudinal dispersion Transversal dispersion Pure advectionAdvection-diffusion with Pe=100

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Grid size 8192*8192 with 32 processors More efficient in pure advection case Slightly more efficient with moderately heterogeneous media particle tracker : impact of diffusion and heterogeneity

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particle tracker : parallel performances Grid size 4096*4096 Good speed-up in any configuration

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Current work and perspectives Current work Iterative linear solvers for 3D fracture networks 3D heterogeneous porous media Subdomain method with Aitken-Schwarz acceleration Transient flow in 2D and 3D porous media Grid computing and parametric simulations Future work Porous fractured media with rock Site modeling UQ methods

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