Modeling flow in porous media with fractures; Discrete fracture models with matrix-fracture exchange

Jaffré, Jérôme; Roberts, Jean E.

doi:10.1134/s1995423912020103

Cited by 19 publications

(9 citation statements)

References 9 publications

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“…Geosciences very often produce applications with huge domains and terrific geometrical complexities. Within this context, the Discrete Fracture Network (DFN) model was developed for modeling the flow in the geological fractured media [6][7][8][9] and is object of a very large numerical bibliography [10][11][12][13][14][15][16][17][18][19][20]. Due to the huge uncertainty in the definition of the underground fracture distribution, this model instantiates a fracture distribution by a stochastic procedure starting from probabilistic distributions of geometrical parameters: direction, dimension, aspect ratio; and from probabilistic distributions of thickness and other hydrogeological properties.…”

Section: Introductionmentioning

confidence: 99%

Orthogonal polynomials in badly shaped polygonal elements for the Virtual Element Method

Berrone

Borio

2017

Finite Elements in Analysis and Design

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In this paper we propose a modified construction for the polynomial basis on polygons used in the Virtual Element Method (VEM). This construction is alternative to the classical monomial basis used in the classical construction of the VEM and is designed in order to improve numerical stability. For badly shaped elements the construction of the projection matrices required for assembling the local coefficients of the linear system within the VEM discretization of Partial Differential Equations can result very ill conditioned. The proposed approach can be easily implemented within an existing VEM code in order to reduce the possible ill conditioning of the elemental projection matrices. Numerical results applied to an hydro-geological flow simulation that often produce very badly shaped elements show a clear improvement of the quality of the numerical solution, confirming the viability of the approach. The method can be conveniently combined with a classical implementation of the VEM and applied element-wise, thus requiring a rather moderate additional numerical cost.

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Section: Introductionmentioning

confidence: 99%

Orthogonal polynomials in badly shaped polygonal elements for the Virtual Element Method

Berrone

Borio

2017

Finite Elements in Analysis and Design

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“…In Hyman et al (2014) an approach with conforming finite elements is used. In Jaffré and Roberts (2012) a mixed finite elements approach is proposed, and in Karimi-Fard and Durlofsky (2014) a local adaptive upscaling method is proposed.…”

Section: Introductionmentioning

confidence: 99%

Machine learning for flux regression in discrete fracture networks

et al. 2021

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In several applications concerning underground flow simulations in fractured media, the fractured rock matrix is modeled by means of the Discrete Fracture Network (DFN) model. The fractures are typically described through stochastic parameters sampled from known distributions. In this framework, it is worth considering the application of suitable complexity reduction techniques, also in view of possible uncertainty quantification analyses or other applications requiring a fast approximation of the flow through the network. Herein, we propose the application of Neural Networks to flux regression problems in a DFN characterized by stochastic trasmissivities as an approach to predict fluxes.

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“…Such PDE systems are often classified into PDE systems with co-dimension 1, i.e., 3D-2D or 2D-1D coupled PDEs, and PDEs with co-dimension 2, i.e., 3D-1D or 2D-0D coupled problems. The numerical analysis and discretization of these types of problems have been investigated in several publications 6,17,20 . For the numerical treatment of interface problems with co-dimension 1, XFEM methods have proved to be very effective 10,32 .…”

Section: Introductionmentioning

confidence: 99%

Mathematical modeling, analysis and numerical approximation of second-order elliptic problems with inclusions

Köppl

Vidotto

Wohlmuth

et al. 2018

Math. Models Methods Appl. Sci.

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Many biological and geological systems can be modeled as porous media with small inclusions. Vascularized tissue, roots embedded in soil or fractured rocks are examples of such systems. In these applications, tissue, soil or rocks are considered to be porous media, while blood vessels, roots or fractures form small inclusions. To model flow processes in thin inclusions, one-dimensional (1D) models of Darcy- or Poiseuille type have been used, whereas Darcy-equations of higher dimension have been considered for the flow processes within the porous matrix. A coupling between flow in the porous matrix and the inclusions can be achieved by setting suitable source terms for the corresponding models, where the source term of the higher-dimensional model is concentrated on the center lines of the inclusions. In this paper, we investigate an alternative coupling scheme. Here, the source term lives on the boundary of the inclusions. By doing so, we lift the dimension by one and thus increase the regularity of the solution. We show that this model can be derived from a full-dimensional model and the occurring modeling errors are estimated. Furthermore, we prove the well-posedness of the variational formulation and discuss the convergence behavior of standard finite element methods with respect to this model. Our theoretical results are confirmed by numerical tests. Finally, we demonstrate how the new coupling concept can be used to simulate stationary flow through a capillary network embedded in a biological tissue.

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Modeling flow in porous media with fractures; Discrete fracture models with matrix-fracture exchange

Cited by 19 publications

References 9 publications

Orthogonal polynomials in badly shaped polygonal elements for the Virtual Element Method

Orthogonal polynomials in badly shaped polygonal elements for the Virtual Element Method

Machine learning for flux regression in discrete fracture networks

Mathematical modeling, analysis and numerical approximation of second-order elliptic problems with inclusions

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