The flow equations are Darcy and mass conservation in each fracture. The rock matrix surrounding the fractures is considered as impervious. Continuity of hydraulic head and flux is imposed at the intersections between fractures. Stochastic models generate a random geometry and a random permeability field.
In 2D fracture networks, each fracture is a line and intersections are points in general. We use a 1D finite volume method in each fracture.
In 3D fracture networks, the main difficulty is to define a
mesh. We first discretize the intersections and generate a mesh in each
fracture. The grids can match or not. We use a Mixed Hybrid Finite
Element method,
where unknowns are the hydraulic head in each cell and at each edge.
Thus continuity equations become straightforward in the discrete
formulation with matching grids. With non matching grids, we define a
Mortar like method, where the difficulty is to deal with intricate
geometries and partitions.
Discrete flow equations are a sparse linear system. Solvers
so far studied are direct methods, preconditioned Krylov methods and
multigrid methods. We have developed a new solver, based on domain decomposition, using the Schur complement, preconditioned by Neumann Neumann combined with deflation.
Parallel computations can be done in all steps, from data generation to velocity computation. They are based on a partition of the network into groups of fractures.
We are integrating the Mortar method and the Schur solver into the
software MPFRAC, of the platform H2OLab. We use parallelism, with MPI,
in order to run large scale simulations.
Now, we want also to study transient flow in Discrete Fracture Networks, where the rock matrix is no longer impervious. This objective is described in the project GeoFrac (ARC).
This topic started in the Sage team (participant J. Erhel), in collaboration with the UMR Geosciences at Rennes, during the Ph-D thesis of J-R. de Dreuzy, under the direction of P. Davy. The thesis was defended in 1999 and J-R. de Dreuzy was hired by CNRS in 2001, in the UMR Geosciences. The collaboration continued with the Ph-D of H. Mustapha, supervised by J-R. de Dreuzy and J. Erhel, and defended in 2005.
The research continues in the Sage team (participant J. Erhel) with the Ph-D of B. Poirriez, in collaboration with J-R. de Dreuzy, started in october 2007, and to be defended in December 2011. The research on non matching grids was undertaken during the post-doc of G. Pichot, started in February 2008 and ended in August 2009, at Geosciences.
The research on interactions between the matrix and the fractures is done in collaboration between the teams Sage, Estime, Gamma, and Geosciences Rennes.
This work was supported by the ACI Grid with the project Hydrogrid in 2002-2005 then supported by the ANR CIS with the project MICAS in 2008-2011.
The Ph-D of H. Mustapha was funded by the government, in complement of the Hydrogrid project.
The Ph-D of B. Poirriez was funded by the government in 2007-2010 then he was ATER at INSA in 2010-2011.
The post-doc of G. Pichot was funded by the ANR CIS with the project MICAS in 2008-2009.
The project GeoFrac is funded as a collaborative action by INRIA. Only travel expenses are funded.
Numerical experiments are undertaken with clusters from Grid'5000 consortium and computers from IDRIS Center.