Géraldine Pichot

Influence of fracture scale heterogeneity on the flow properties of three‐dimensional discrete fracture networks (DFN)

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 2 106 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 Lc. 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 a2, 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. Citation: de Dreuzy, J.-R., Y. Meheust, and G. Pichot (2012), Influence of fracture scale heterogeneity

A Generalized Mixed Hybrid Mortar Method for Solving Flow in Stochastic Discrete Fracture Networks

The simulation of flow in fractured media requires handling both a large number of fractures and a complex interconnecting network of these fractures. Networks considered in this paper are three-dimensional domains made up of two-dimensional fractures intersecting each other and randomly generated. Due to the stochastic generation of fractures, intersections can be highly intricate. The numerical method must generate a mesh and define a discrete problem for any discrete fracture network (DFN). A first approach [Erhel, de Dreuzy, and Poirriez, SIAM J. Sci. Comput., 31 (2009), pp. 2688-2705] is to generate a conforming mesh and to apply a mixed hybrid finite element method. However, the resulting linear system becomes very large when the network contains many fractures. Hence a second approach [Pichot, Erhel, and de Dreuzy, Appl. Anal., 89 (2010), pp. 1629-1643] is to generate a nonconforming mesh, using an independent mesh generation for each fracture. Then a Mortar technique applied to the mixed hybrid finite element method deals with the nonmatching grids. When intersections do not cross or overlap, pairwise Mortar relations for each intersection are efficient [Pichot, Erhel, and de Dreuzy, 2010]. But for most random networks, discretized intersections involve more than two fractures. In this paper, we design a new method generalizing the previous one and that is applicable for stochastic networks. The main idea is to combine pairwise Mortar relations with additional relations for the overlapping part. This method still ensures the continuity of fluxes and heads and still yields a symmetric positive definite linear system. Numerical experiments show the efficiency of the method applied to complex stochastic fracture networks. We also study numerical convergence when reducing the mesh step. This method makes it easy to perform mesh optimization and appears to be a very promising tool to simulate flow in multiscale fracture networks.

Synthetic benchmark for modeling flow in 3D fractured media

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.

Simulating diffusion processes in discontinuous media

In this article, we propose new Monte Carlo techniques for moving a diffusive particle in a discontinuous media. In this framework, we characterize the stochastic process that governs the positions of the particle. The key tool is the reduction of the process to a Skew Brownian motion (SBM). In a zone where the coefficients are locally constant on each side of the discontinuity, the new position of the particle after a constant time step is sampled from the exact distribution of the SBM process at the considered time. To do so, we propose two different but equivalent algorithms: a two-steps simulation with a stop at the discontinuity and a one-step direct simulation of the SBM dynamic. Some benchmark tests illustrate their effectiveness.

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

with the local permeability distribution variance s 2 is nonnegligible but remains small. It is equal to 0.09 for sparse networks and 0.14 for dense networks representing 4.5% and 7%, respectively, of the full possible range of w values. Power averaging is not strictly valid but gives an estimate of upscaling at a few percent. Here w depends slightly on the local permeability distribution shape beyond its variance but mostly on the morphological network structures. Most of the morphological control on upscaling for on‐lattice structures is local and topological and can be explained by the dependence on the average number of neighbor by points (effective coordination number) within the flowing structure (backbone).

Domain Decomposition Methods in Science and Engineering XXI

This volume contains a selection of papers presented at the 21st international conference on domain decomposition methods in science and engineering held in Rennes, France, June 25-29, 2012. Domain decomposition is an active and interdisciplinary research discipline, focusing on the development, analysis and implementation of numerical methods for massively parallel computers. Domain decomposition methods are among the most efficient solvers for large scale applications in science and engineering. They are based on a solid theoretical foundation and shown to be scalable for many important applications. Domain decomposition techniques can also naturally take into account multiscale phenomena. This book contains the most recent results in this important field of research, both mathematically and algorithmically and allows the reader to get an overview of this exciting branch of numerical analysis and scientific computing.

Simulating Diffusion Processes in Discontinuous Media: Benchmark Tests

Abstract We present several benchmark tests for Monte Carlo methods simulating diffusion in one-dimensional discontinuous media. These benchmark tests aim at studying the potential bias of the schemes and their impact on the estimation of micro- or macroscopic quantities (repartition of masses, fluxes, mean residence time, …). These benchmark tests are backed by a statistical analysis to filter out the bias from the unavoidable Monte Carlo error. We apply them on four different algorithms. The results of the numerical tests give a valuable insight into the fine behavior of these schemes, as well as rules to choose between them.

A Mortar BDD Method for Solving Flow in Stochastic Discrete Fracture Networks

In this paper, flow in Discrete Fracture Networks (DFN) is solved using a Mortar Mixed Hybrid Finite Element Method. To solve large linear systems derived from a nonconforming discretization of stochastic fractured networks, a Balancing Domain Decomposition is used. Tests on three stochastically generated DFN are proposed to show the ability of the iterative solver SIDNUR to solve the flow problem.

Random Walk Particle Tracking

The classical advection-dispersion equation of a conservative solute in porous media can be written as [1]

Monte Carlo simulations in media with interfaces

This note presents briefly some recent results about Monte Carlo simulations in media with interfaces.Spectrum-preserving two-scale decompositions with applications to numerical homogenization and eigensolversThe workshop has brought together experts in the broad field of partial differential equations with highly heterogeneous coefficients. Analysts and computational and applied mathematicians have shared results and ideas on a topic of considerable interest both from the theoretical and applied viewpoints. A characteristic feature of the workshop has been to encourage discussions on the theoretical as well as numerical challenges in the field, both from the point of view of deterministic as well as stochastic modeling of the heterogeneities.The workshop has brought together experts in the broad field of partial differential equations with highly heterogeneous coefficients. Analysts and computational and applied mathematicians have shared results and ideas on a topic of considerable interest both from the theoretical and applied viewpoints. A characteristic feature of the workshop has been to encourage discussions on the theoretical as well as numerical challenges in the field, both from the point of view of deterministic as well as stochastic modeling of the heterogeneities. Mathematics Subject Classification (2010): 35A15, 35A35, 35J60, 49S05, 65C05, 35B27, 74Q15, 65N99, 74Q20, 37A25. Introduction by the Organisers Multiscale problems are ubiquitous in modern science and engineering. Applications are many: materials science, underground flows, wave propagation, etc. Indeed, materials that seem to be a continuum are made up of atoms. Geological media have many scales ranging from very small (atomistic), to an order of magnitude larger (pore and crack sizes), and to even much larger sizes of tectonic plates. Composite materials that are critical for modern technology are typically multiscale (e.g., small filling particles of different sizes in a sample of much larger size or layers of various thickness in a laminate composite). Many, though not all, of the models involve random modeling. Mathematics evidently plays a major role in this area. It covers the whole spectrum from theory 802 Oberwolfach Report 14/2013 (with homogenization theory, variational calculus, etc) to scientific computing, via numerical analysis (with the development of adequate, multiscale, finite element methods and related approaches). Statistics-based methods are also employed. While modeling at each specific fundamental scale (e.g., molecular dynamics at atomistic scales, continuum PDEs at much larger scales) has been quite successful, the coupling between different scales is not well understood mathematically. The workshop has been an opportunity to make a state-of-the art review of the mathematical knowledge in a broad sense, to draw up a list of the challenges to overcome in the near future. It was a unique opportunity to bring together experts from all the areas involved. Approximately 45 scientists with very different backgrounds attended. Many delivered a talk, presenting their recent contributions in the light of their own perspectives. The quite compact schedule we had did not prevent many informal interactions to take place over coffee breaks, meals and lively evenings. On the theoretical side, we heard about recent theoretical achievements on quantitative rates of convergence in stochastic homogenization, on the Einstein relation, or on derivations of models for rough boundaries. Several contributions presented novel results on homogenization in the presence of large random potentials and on the derivation of macroscopic models to understand the energy density and phase information of waves propagating in highly heterogeneous media. Several discussions focused on the relatively uncharted territory of understanding the propagation of uncertainty from the coefficients in an equation to the solution of said equation. Many numerical analysis talks were devoted to novel numerical methods for approximating the solution space of PDEs with rough coefficients and possibly with non-separated scales. This generated intense discussion among the participants. Common features and differences between the approaches were discussed in terms of robustness, cost, assumptions made, generality, ease of implementation, mathematical rigor and optimality. While there is a variety of heuristic numerical schemes, the mathematical understanding of such problems remains relatively underdeveloped. The workshop helped fill that gap through such discussions. Techniques using polynomial chaos expansion were also a topic of choice. Variance reduction issues, and more general aspects of Monte-Carlo methods, were the topic of several talks. Applications as varied as transport processes through membranes, wave propagation phenomena, underground flows in porous media, nanotechnologies, colloid dynamics, optimal design of trussed materials, modeling of composite materials used in the aerospace industry, were addressed. Based on the feedback already received from participants, we consider the organization of such a workshop equally beneficial for scientists interested in theoretical issues, applied mathematicians developing numerical techniques, and mechanical engineers in contact with practical problems. The mixing of experts at deterministic and stochastic techniques was extremely rewarding for both camps. We hope other workshops in the same spirit will be organized in the near future. Theory and Numerics for Deterministic and Stochastic Homogenization 803 Workshop: Interplay of Theory and Numerics for Deterministic and Stochastic Homogenization