Section: New Results

Models and simulations for flow in porous fractured media

This work is done in collaboration with J.-R. de Dreuzy, from Geosciences Rennes (who is on leave until 2013 at UPC, Barcelona, Spain, see 8.2.1 ). It is done in the context of the GEOFRAC project (see 8.1.5 ), the H2OGUILDE project (see 8.1.4 ), the HEMERA project (see 8.1.2 ), and the Joint Laboratory for Petascale Computing (see 8.3.7 ). Computations are partly done with GENCI supercomputers (see 8.1.6 ), using the platform H2OLab (see 5.1 ) and the software GWNUM, GWUTIL, MPFRAC (see 5.3 , 5.2 , 5.4 ).

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

Participant : Géraldine Pichot.

This work is published in a journal [21] .

While permeability scaling of fractured media has been so far studied independently at the fracture- and network- scales, we propose a numerical analysis of the combined effect of fracture-scale heterogeneities and the network-scale topology. The analysis is based on $2\times {10}^6$ discrete fracture network (DFNs) simulations performed with highly robust numerical methods. Fracture local apertures are distributed according to a truncated Gaussian law, and exhibit self-affine spatial correlations up to a cutoff scale $L_c$. Network structures range widely over sparse and dense systems of short, long or widely-distributed fracture sizes and display a large variety of fracture interconnections, flow bottlenecks and dead-ends. At the fracture scale, accounting for aperture heterogeneities leads to a reduction of the equivalent fracture transmissivity of up to a factor of 6 as compared to the parallel plate of identical mean aperture. At the network scale, a significant coupling is observed in most cases between flow heterogeneities at the fracture and at the network scale. The upscaling from the fracture to the network scale modifies the impact of fracture roughness on the measured permeability. This change can be quantified by the measure ${\alpha }_2$, which is analogous to the more classical power-averaging exponent used with heterogeneous porous media, and whose magnitude results from the competition of two effects: (i) the permeability is enhanced by the highly transmissive zones within the fractures that can bridge fracture intersections within a fracture plane; (ii) it is reduced by the closed and low transmissive areas that break up connectivity and flow paths.

Synthetic benchmark for modeling flow in 3D fractured media

Participants : Jocelyne Erhel, Géraldine Pichot.

This work is published in a journal [22] .

Intensity and localization of flows in fractured media have promoted the development of a large range of different modeling approaches including Discrete Fracture Networks, pipe networks and equivalent continuous media. While benchmarked usually within site studies, we propose an alternative numerical benchmark based on highly-resolved Discrete Fracture Networks (DFNs) and on a stochastic approach. Test cases are built on fractures of different lengths, orientations, aspect ratios and hydraulic apertures, issuing the broad ranges of topological structures and hydraulic properties classically observed. We present 18 DFN cases, with 10 random simulations by case. These 180 DFN structures are provided and fully documented. They display a representative variety of the configurations that challenge the numerical methods at the different stages of discretization, mesh generation and system solving. Using a previously assessed mixed hybrid finite element method (Erhel et al., 2009a), we systematically provide reference flow and head solutions. Because CPU and memory requirements stem mainly from system solving, we study direct and iterative sparse linear solvers. We show that the most cpu-time efficient method is a direct multifrontal method for small systems, while conjugate gradient preconditioned by algebraic multrigrid is more relevant at larger sizes. Available results can be used further as references for building up alternative numerical and physical models in both directions of improving accuracy and efficiency.

Robust numerical methods for solving flow in stochastic fracture networks

Participants : Jocelyne Erhel, Géraldine Pichot.

This work is published in a journal [20] and was presented at a conference (plenary talk) [33] .

Working with random domains requires the development of specific and robust numerical methods to be able to solve physical phenomena whatever the generated geometries. Hydrogeology is a typical area of application where one has to face uncertainty about the geometry and the properties of the domain since the available information on the underground media is local, gathered through in-situ experiments with outcrops and wells. From measurements, statistical laws are derived that allow the generation of natural-like random media. The focus of this talk will concern flow in discrete fracture networks. The parameters governing the fractures lengths, shapes, orientations, positions as well as their hydraulic conductivity are stochastic. Our objective is to design robust numerical methods to solve Poiseuille's flow in large and heterogeneous stochastic fracture networks. The first part deals with the meshing strategies required to obtain a good quality mesh for any generated networks. The second part is devoted to numerical techniques to solve the flow equations. A Mortar-like method to deal with nonconforming meshes at the fracture intersections is presented as well as a Schur complement approach to solve the linear system of interest in parallel.

Deflation and Neumann-Neumann Preconditionner for Schur Domain Decomposition Method

Participants : Jocelyne Erhel, Géraldine Pichot.

This work was presented at a conference [34] . A paper is in preparation.

We study a domain decomposition method, which takes advantages from both the direct method and the Preconditioned Conjugate Gradient (PCG). This Schur method reduces the global problem to an interface problem, with a natural domain decomposition based on fractures or fracture packs. We propose an original approach for optimizing the algorithm and a global preconditioning of deflation type. Since the Schur complement $S$ is spd, we apply PCG to solve the linear system $Sx=b$. We use the classical Neumann-Neumann (NN) preconditioner. To gain in efficiency, we use only one Cholesky factorization of the subdomain matrices for the preconditionning and the conjugate gradient steps. We also define a coarse space, based on the subdomain definition, to apply a deflation preconditioner. We do a theoretical complexity study of our algorithm. We use this study, with the numerical data, to compute experimental complexity. We compare the results between several combination for the preconditioner. Then, we confront our results with existing solvers.

Flow in complex 3D geological fractured porous media

Participants : Thomas Dufaud, Jocelyne Erhel, Géraldine Pichot.

This work was presented at a conference [24] .

This communication focuses on numerical techniques to compute flow in complex 3D geological fractured porous media, where water can flow both in the rock matrix and in the fractures. This study is an extension of the models designed in the teams SAGE and POMDAPI. The numerical model deals with steady-state flow for single phase and incompressible fluid. In the rock matrix, the flow is governed by Darcy’s law, while the flow in the fractures is governed by Poiseuille’s law. For both, the law of mass conservation is verified. In a first part, we present the model. Then we propose a test case and its discretization considering a Mixed Hybrid Finite Element Method.