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**A parallel scientific software for heterogeneous hydrogeoloy**

Conference on Parallel CFD Antalya, Turkey May 2007

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**Damien Tromeur-Dervout**

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 Grid’5000 french project

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**Physical context: groundwater flow**

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

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**Physical context: groundwater flow**

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

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**Numerical modelling strategy**

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

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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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**Hydrolab platform: physical models**

Porous Media Physical equations Fracture Networks Fractured Porous

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**Permeability field in porous media**

Simple 2D or 3D geometry stochastic permeability field finitely or infinitely correlated finitely correlated medium Multifractal D2=1.7 Multifractal D2=1.4

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**Well test Interpretation**

Generalized flux equation 10 h 100 h D=1 dw=2 D=2 dw=2 Cristalline aquifer of Ploemeur (Brittany, France) D=1.5 dw=2.8

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**Macro-dispersion analysis**

Longitudinal dispersion Transversal dispersion

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**Natural Fractured Media**

) l ( 0.1 1 10 100 -7 -6 -5 -4 -3 -2 -1 2 n probability density n ( l )~ l -2.7 Fracture length l Fractures exist at any scale with no correlation Fracture length is a parameter of heterogeneity 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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**Flow equations: Darcy law and mass conservation **

Physical equations Flow equations: Darcy law and mass conservation Transport equations: advection and dispersion 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

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**Hydrolab platform: numerical methods**

PDE solvers EDO/DAE solvers Linear solvers Particle tracker Multilevel methods Monte-Carlo method UQ methods Parametric simulations

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**Steady-state flow equations**

Darcy law and mass conservation Boundary conditions 2D porous medium 3D fracture network Nul flux Given head Nul flux Given Head Given Head Nul flux

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**Uncertainty Quantification methods**

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

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**Monte-Carlo simulations**

For j=1,…,M sample permeability field Kj sample network Ωj compute vj

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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**

©INRIA/Photo Jim Wallace Clusters at Irisa Grid’5000 project Numerical model Funded by French Government and Brittany council 16,8 millions of unknowns in 100 seconds with 16 processors

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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=n2 and NZ=5N Regular 3D mesh : N= n3 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**

Sparse direct linear solvers 2D heterogeneous porous medium memory size and CPU time with PSPASES Theory : NZ(L) = O(N logN) Theory : Time = O(N1.5) variance = 1, number of processors = 2

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**2D heterogeneous porous medium CPU time with HYPRE/AMG**

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

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**Flow computation in 2D porous medium**

Finitely correlated permeability field Impact of permeability variance matrix order N = 106 matrix order N = PSPASES and BoomerAMG independent of variance BoomerAMG faster than PSPASES with 4 processors

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**parallel sparse linear solvers**

2D heterogeneous porous medium Direct and multigrid solvers Parallel CPU time matrix order N = 106 matrix order N = variance = 9

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**Advection-dispersion equation Boundary conditions Initial condition**

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

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**Solute transport : particle tracker**

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

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**Solute transport in 2D heterogeneous porous media**

Permeability Particles Horizontal velocity Vertical velocity =1 and Pe=1 =3 and Pe=100

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**particle tracker : convergence analysis**

Pure advection Advection-diffusion with Pe=100 Longitudinal dispersion Transversal dispersion Np = 1000 is a good trade-off between efficiency and convergence

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**particle tracker : impact of diffusion and heterogeneity**

Grid size 8192*8192 with 32 processors More efficient in pure advection case Slightly more efficient with moderately heterogeneous media

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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**

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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