The flow equations are Poiseuille 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 also use a surface mesh
generation, with the software BLSURF.

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.

- J.-R. de Dreuzy and G. Pichot and B. Poirriez and J. Erhel

Synthetic benchmark for modeling flow in 3D fractured media

Computers and Geosciences, 2013, Volume 50, 59-71 - G. Pichot and J. Erhel and J.-R. de Dreuzy

A generalized mixed hybrid mortar method for solving flow in stochastic discrete fracture networks

SIAM Journal on scientific computing, 2012, Volume 34, B86-B105 - J.-R. de Dreuzy and Y. Meheust and G. Pichot

Influence of fracture scale heterogeneity on the flow properties of three-dimensional Discrete Fracture Networks

Journal of Geophysical Research, 2012, Volume 117, B11207 - G. Pichot and J. Erhel and J.-R. de Dreuzy

A mixed hybrid Mortar method for solving flow in discrete fracture networks

Applicable Analysis, 2010, Volume 89, 1629-1643 - J.-R. de Dreuzy and P. de Boiry and G. Pichot and P. Davy

Use of power-averaging for quantifying the influence of structure organization on permeability upscaling in on-lattice networks under mean parallel flow

Water Resources Research, 2010, Volume 46, 11 pages. - G. Pichot, J.-R. de
Dreuzy, J. Erhel and P. Davy

Flow in multi-scale fracture networks: numerical optimization by use of a Mortar-like method

*Proceedings of MAMERN'2009*,**2009**, Volume 2, 761-766 - J. Erhel and J.-R. de
Dreuzy and B. Poirriez

flow simulations in three-dimensional Discrete Fracture Networks

*SIAM Journal on Scientific Computing,***2009,**Volume 31, 2688-2705 - Beaudoin, A.; de Dreuzy, J.; Erhel, J. & Mustapha,
H.

Joubert, G.; Nagel, W.; Peters, F.; Plata, O.; Tirado, P. & Zapata, E.*(ed.)*

Parallel Simulations of Underground Flow in Porous and Fractured Media

*Parallel Computing: Current and Future Issues of High-End Computing, NIC,***2006***, 33*, 391-398. - J-R. de Dreuzy and J. Erhel.

Efficient algorithms for the determination of the connected fracture network and the solution to the steady-state flow equation in fracture networks.

*Computers and Geosciences*,**2003**, Volume 29, pp. 107-111 .

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.

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 on DFN and Domain Decomposition methods continued in
the Sage team (participant J.
Erhel) with the Ph-D of B. Poirriez, in
collaboration with J-R. de Dreuzy, defended in 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 Rennes.

The research on surface mesh generation was undertaken during the
post-doc of T. Dufaud,
started in January 2012 and ended in June 2013, in collaboration with
Geosciences Rennes, and the teams GAMMA3 and POMDAPI at Inria
Rocquencourt.

The research continues in the Sage team (participants J. Erhel, G.
Pichot), still in collaboration with Geosciences Rennes, POMDAPI and
GAMMA3 teams.

This work was first supported by the ACI Grid with the project Hydrogrid in 2002-2005.

The Ph-D of H. Mustapha was funded by the government, in complement of the Hydrogrid project.

Then work was supported by the ANR CIS with the project MICAS in 2008-2011.

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 work on benchmarking and mesh generation was funded by Inria as a collaborative action, with the project GeoFrac in 2012-2013.The post-doc of T. Dufaud was funded with the GeoFrac project.

Numerical experiments are undertaken with clusters from Grid'5000 consortium and computers from IDRIS Center.