Analysis of Model Error for a Continuum-Fracture Model of Porous Media Flow

Březina, Jan; Stebel, Jan

doi:10.1007/978-3-319-40361-8_11

Cited by 7 publications

(10 citation statements)

References 6 publications

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“…A comparison of our new reduced models to the Darcy flow models proposed in [19] is given in Section 6, where we also recover and extend the convergence results from [22]. In Section 7, we derive error estimates for reduced order models for anisotropic diffusion problems, which are sharper than the estimates given in [8]. From the exact and reduced order coupling conditions in Fourier space, we can infer the error of the traces of the matrix Fourier coefficients at the interfaces.…”

Section: Introductionmentioning

confidence: 65%

“…In [22], the authors use a suitable modification of the solution inside the fracture, in order to control its gradient and to show convergence for isotropic Darcy flow and a fracture of high resistivity. In [8], the authors also use a modification of the fracture solution, and derive error estimates for anisotropic Darcy flow. In [20], a rescaling of the fracture normal coordinate is used, which yields a rescaled model on a geometry independent of the fracture width.…”

Section: Introductionmentioning

confidence: 99%

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Modeling and Analysis of the Coupling in Discrete Fracture Matrix Models

Gander

Hennicker

Masson³

2021

SIAM J. Numer. Anal.

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This paper deals with the derivation and analysis of reduced order elliptic PDE models on fractured domains. We use a Fourier analysis to obtain coupling conditions between subdomains, when the fracture is represented as a hypersurface embedded in the surrounded rock matrix. We compare our results to prominent examples from the literature, for diffusive models. In a second step, we present error estimates for the reduced order models in terms of the fracture width. For the proofs, we rely on a combination of Fourier analysis, asymptotic expansions and functional analysis. Finally, we study the behaviour of the error of the reduced order solutions on numerical test cases, when the fracture width tends to zero.

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Section: Introductionmentioning

confidence: 65%

Section: Introductionmentioning

confidence: 99%

Modeling and Analysis of the Coupling in Discrete Fracture Matrix Models

Gander

Hennicker

Masson³

2021

SIAM J. Numer. Anal.

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“…Nevertheless, this type of DFM models has been developed for flow and transport, as well as mechanics and poroelasticity, and the explicit representation is particularly useful when the fractures deform. Simulation models that incorporate DFM principles include DuMuX [25], CSMP [26], MOOSE-FALCON [27,28], OpenGeoSys [29], and Flow123d [30].…”

Section: Introductionmentioning

confidence: 99%

“…By now, this type of DFM models have been developed for flow and transport, as well as mechanics and poroelasticity. Simulation models that incorporate DFM principles include DuMuX [26], CSMP [27], MOOSE-FALCON [28], [29], OpenGeoSys [30] and Flow123d [31].…”

Section: Introductionmentioning

confidence: 99%

PorePy: an open-source software for simulation of multiphysics processes in fractured porous media

et al. 2020

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Development of models and dedicated numerical methods for dynamics in fractured rocks is an active research field, with research moving towards increasingly advanced process couplings and complex fracture networks. The inclusion of coupled processes in simulation models is challenged by the high aspect ratio of the fractures, the complex geometry of fracture networks, and the crucial impact of processes that completely change characteristics on the fracture-rock interface. This paper provides a general discussion of design principles for introducing fractures in simulators, and defines a framework for integrated modeling, discretization, and computer implementation. The framework is implemented in the open-source simulation software PorePy, which can serve as a flexible prototyping tool for multiphysics problems in fractured rocks. Based on a representation of the fractures and their intersections as lower-dimensional objects, we discuss data structures for mixed-dimensional grids, formulation of multiphysics problems, and discretizations that utilize existing software. We further present a Python implementation of these concepts in the PorePy open-source software tool, which is aimed at coupled simulation of flow and transport in three-dimensional fractured reservoirs as well as deformation of fractures and the reservoir in general. We present validation by benchmarks for flow, poroelasticity, and fracture deformation in porous media. The flexibility of the framework is then illustrated by simulations of non-linearly coupled flow and transport and of injection-driven deformation of fractures. All results can be reproduced by openly available simulation scripts.

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“…The accuracy of the reduced models as compared with the full equi‐dimensional model has been analyzed by several authors. Březina and Stebel 27 derived error estimates for the weak solution of the reduced model in comparison to the weak solution of the full model for anisotropic Darcy flow. Independently of the current article, Gander et al 12 recently used a combination of Fourier analysis, asymptotic expansions, and functional analysis to obtain error estimates for the reduced models of anistropic diffusion in terms of the fracture width.…”

Section: Introductionmentioning

confidence: 99%

Modeling and discretization of flow in porous media with thin, full‐tensor permeability inclusions

Starnoni

Berre

Keilegavlen

et al. 2021

Numerical Meth Engineering

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When modeling fluid flow in fractured reservoirs, it is common to represent the fractures as lower‐dimensional inclusions embedded in the host medium. Existing discretizations of flow in porous media with thin inclusions assume that the principal directions of the inclusion permeability tensor are aligned with the inclusion orientation. While this modeling assumption works well with tensile fractures, it may fail in the context of faults, where the damage zone surrounding the main slip surface may introduce anisotropy that is not aligned with the main fault orientation. In this article, we introduce a generalized dimensional reduced model which preserves full‐tensor permeability effects also in the out‐of‐plane direction of the inclusion. The governing equations of flow for the lower‐dimensional objects are obtained through vertical averaging. We present a framework for discretization of the resulting mixed‐dimensional problem, aimed at easy adaptation of existing simulation tools. We give numerical examples that show the failure of existing formulations when applied to anisotropic faulted porous media, and go on to show the convergence of our method in both two‐dimensional and three‐dimensional.

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Analysis of Model Error for a Continuum-Fracture Model of Porous Media Flow

Cited by 7 publications

References 6 publications

Modeling and Analysis of the Coupling in Discrete Fracture Matrix Models

Modeling and Analysis of the Coupling in Discrete Fracture Matrix Models

PorePy: an open-source software for simulation of multiphysics processes in fractured porous media

Modeling and discretization of flow in porous media with thin, full‐tensor permeability inclusions

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