Preconditioning Techniques for the Numerical Solution of Flow in Fractured Porous Media
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Preconditioning Techniques for the Numerical Solution of Flow in Fractured Porous Media Paola F. Antonietti1 · Jacopo De Ponti1 · Luca Formaggia1
· Anna Scotti1
Received: 2 June 2019 / Revised: 9 November 2020 / Accepted: 15 November 2020 © The Author(s) 2020
Abstract This work deals with the efficient iterative solution of the system of equations stemming from mimetic finite difference discretization of a hybrid-dimensional mixed Darcy problem modeling flow in fractured porous media. We investigate the spectral properties of a mixed discrete formulation based on mimetic finite differences for flow in the bulk matrix and finite volumes for the fractures, and present an approximation of the factors in a set of approximate block factorization preconditioners that accelerates convergence of iterative solvers applied to the resulting discrete system. Numerical tests on significant three-dimensional cases have assessed the properties of the proposed preconditioners. Keywords Porous media flow · Fractured media · Preconditioners
Introduction The simulation of underground flows in fractured porous media is of great interest for a large number of geophysical applications, such as oil production, CO2 storage, and groundwater contamination and remediation. It is well-known that the presence of fractures and/or faults strongly influence subsurface flows. The major challenges from the numerical viewpoint are represented by (i) geometric complexity, and (ii) strong heterogeneity of materials at different space scales. While micro-fractures can be accounted for by means of homogeniza-
This work was partially funded by INdAM—GNCS. P.A. also acknowledges the financial support of PRIN research Grant No. 201744KLJL funded by MIUR.
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Luca Formaggia [email protected] Paola F. Antonietti [email protected] Jacopo De Ponti [email protected] Anna Scotti [email protected]
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MOX, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy 0123456789().: V,-vol
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Page 2 of 32
Journal of Scientific Computing
(2021) 86:2
tion/upscaling techniques, large fractures and faults play a more complex role, acting as paths or barriers for the flow, and therefore they have to be included in the model explicitly. Fractures are characterized by a small aperture compared to their typical length and the size of the domain, thus a widely employed approach consists in modeling them as (d − 1)-dimensional interfaces immersed in a d-dimensional porous medium (indicated in the following as the bulk). A reduced (d − 1)-dimensional problem is then solved on the surfaces representing the fractures, with physically-consistent coupling conditions accounting for the exchange of fluid between fractures and porous medium. From the computational viewpoint, this dimensionally-hybrid setting avoids the need for extremely fine grids to resolve the fracture’s scales. Assuming that the fractures are filled by a porous medium with its own porosity and permeability, Darcy’s law can be u
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