Daniel Zegarra Vasquez: Thursday, 15th May 2025 at 10:30
Abstract:
This talk is a rehearsal for my thesis defense that will take place on Tuesday, May 27th 2025.
This thesis focuses on single-phase flow in three-dimensional underground porous media, characterized by fractures, narrow discontinuities that are ubiquitous within the rock matrix. In this work, fractures are specifically considered as preferential flow paths and are modeled using the Discrete Fracture Matrix (DFM) approach. This model preserves the three-dimensional structure of the rock matrix while representing the fracture network as a codimension-one object. The flow is governed by coupled Darcy-type partial differential equations (PDEs), which describe exchanges between the rock and the fractures. While existing literature typically addresses DFMs with a few thousand fractures, this thesis deals with models involving up to several hundreds of thousands.
The objective of this work is to design, implement, and analyze a simulation method tailored to such DFMs. The main challenge lies in the efficient solution of the linear systems resulting from the discretization of the PDEs, which may involve several hundreds of millions of degrees of freedom (DOFs).
Chapter [1] addresses the mathematical and numerical analysis of the PDE system. It provides the functional framework of the mixed variational formulation and proves the existence and uniqueness of the solution. Mixed finite elements are used for the discretization, for which existence and uniqueness are also demonstrated. A priori error estimates are established as well. The equivalent mixed-hybrid formulation (MHFE) leads to a sparse, square, symmetric, and positive definite linear system. First-order numerical convergence of the MHFE scheme is validated through an academic test case with an analytical solution.
Chapter [2] introduces a methodology to assess the performance of linear solvers applied to the system arising from the MHFE discretization presented in Chapter [1]. The fractured networks studied are large-scale, involving several tens of thousands of fractures. As the number of fractures increases, the resulting linear systems become increasingly ill-conditioned, a phenomenon exacerbated by strong contrasts in hydraulic conductivity. Direct solvers are impractical due to their excessive memory consumption. Conjugate Gradient (CG), preconditioned with algebraic multigrid methods, reduces memory usage, but the most complex test cases require several days of computation and hundreds of thousands of iterations, sometimes without convergence. This highlights the need for a more robust preconditioner. Finally, the chapter shows that iterative methods require very strict tolerances, as otherwise the poor quality of the solution leads to non-physical results.
Chapter [3] addresses the initial challenge using a two-level Schwarz domain decomposition method, where the coarse level is built using the spectral GenEO method (“Generalized Eigenvalue problem on the Overlap”), implemented in the HPDDM library. GenEO guarantees a bound on the condition number of the preconditioned operator, and thus on the number of Krylov iterations. This chapter demonstrates the applicability of HPDDM GenEO to the flow problem under study. It describes, in the case of DFMs, a partitioning algorithm adapted to the 2D/3D nature of the domain and to the construction of the Neumann matrices required by GenEO. The fractured networks considered here involve several hundreds of thousands of fractures. The methodology from Chapter [2] is reused to assess CG and GMRES preconditioned with HPDDM GenEO, which significantly outperforms the previous approaches. Even with strong contrasts in hydraulic conductivity, the iterative method preconditioned with HPDDM GenEO converges in under 6 minutes and 51 iterations for the largest DFM case (697k fractures, 243M DOFs) using 6825 MPI processes.
[1] M. Kern, G. Pichot, and D. Zegarra Vasquez. Mathematical and numerical analysis of the mixed formulation of single phase flow in three dimensional fractured porous media. Preprint HAL. URL : https://inria.hal.science/hal-05029638 . 2025.
[2] M. Kern, G. Pichot, and D. Zegarra Vasquez. Performance of algebraic preconditioners for large-scale simulations of single-phase flow in three-dimensional fractured porous media. Preprint HAL. URL : https://inria.hal.science/hal-05029652 . 2025.
[3] P. Jolivet, M. Kern, F. Nataf, G. Pichot, and D. Zegarra Vasquez. Domain decomposition preconditioners for efficient parallel simulations of single-phase flow in three-dimensional fractured porous media with a very large number of fractures. Preprint HAL. URL : https://inria.hal.science/hal-05029676 . 2025.
