Dual Virtual Element Method for Discrete Fractures Networks

Fumagalli, Alessio; Keilegavlen, Eirik

doi:10.1137/16m1098231

Cited by 55 publications

(58 citation statements)

References 36 publications

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“…We consider the fractures as (approximated by) planar objects, following e.g. [37,20,32,2,27]. The extension to curved fractures is possible, but the analysis and the presentation will increase in complexity, see [28,35].…”

Section: Mathematical Modelmentioning

confidence: 99%

Conforming, non-conforming and non-matching discretization couplings in discrete fracture network simulations

Fumagalli

Keilegavlen

Scialò

2019

Journal of Computational Physics

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Simulations of fluid flow in naturally fractured rocks have implications for several subsurface applications, including energy storage and extraction, and waste storage. We are interested in flow in discrete fracture networks, which explicitly represent flow in fracture surfaces, but ignore the impact of the surrounding host rock. Fracture networks, generated from observations or stochastic simulations, will contain intersections of arbitrary length, and intersection lines can further cross, forming a highly complex geometry. As the flow exchange between fractures, thus in the network, takes place in these intersections, an adequate representation of the geometry is critical for simulation accuracy. In practice, the intersection dynamics must be handled by a combination of the simulation grid, which may or may not resolve the intersection lines, and the numerical methods applied on the grid. In this work, we review different classes of numerical approaches proposed in recent years, covering both methods that conform to the grid, and non-matching cases. Specific methods considered herein include finite element, mixed and virtual finite elements and control volume methods. We expose our methods to an extensive set of test cases, ranging from artificial geometries designed to test difficult configurations, to a network extruded from a real fracture outcrop. The main outcome is guidances for choice of simulation models and numerical discretization with a trade off on the computational cost and solution accuracy.

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Section: Mathematical Modelmentioning

confidence: 99%

Conforming, non-conforming and non-matching discretization couplings in discrete fracture network simulations

Fumagalli

Keilegavlen

Scialò

2019

Journal of Computational Physics

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“…as well as exploring the fracture and barrier cases and comparing in a cheap way various non-linear solvers to (1.1)-(1.3). Crucially, the present approach can naturally be integrated into discrete fracture networks (DFNs) models [38,39,20,16], which in contrast to discrete fracture models (DFMs), do not consider the flow in the surrounding sub-domains, but handle both a large number of fractures and a complex interconnecting network of these fractures. For the presenting setting, we allow for the discretization of (1.1)-(1.3) by different numerical methods applied separately in the surrounding sub-domains and in the fracture.…”

Section: A)mentioning

confidence: 99%

“…The above problem can be seen as a DFNs system on the set of fractures, and as a domain decomposition problem between the 1-dimensional fractures γ i,j , 1 ≤ i < j ≤ 3, cf. [38,39,20] for more details.…”

Section: Domain Decomposition Formulationmentioning

confidence: 99%

A multiscale flux basis for mortar mixed discretizations of reduced Darcy–Forchheimer fracture models

Ahmed

Fumagalli

Budiša

2019

Computer Methods in Applied Mechanics and Engineering

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In this paper, a multiscale flux basis algorithm is developed to efficiently solve a flow problem in fractured porous media. Here, we take into account a mixed-dimensional setting of the discrete fracture matrix model, where the fracture network is represented as lower-dimensional object. We assume the linear Darcy model in the rock matrix and the non-linear Forchheimer model in the fractures. In our formulation, we are able to reformulate the matrix-fracture problem to only the fracture network problem and, therefore, significantly reduce the computational cost. The resulting problem is then a non-linear interface problem that can be solved using a fixed-point or Newton-Krylov methods, which in each iteration require several solves of Robin problems in the surrounding rock matrices. To achieve this, the flux exchange (a linear Robin-to-Neumann co-dimensional mapping) between the porous medium and the fracture network is done offline by pre-computing a multiscale flux basis that consists of the flux response from each degree of freedom on the fracture network. This delivers a conserve for the basis that handles the solutions in the rock matrices for each degree of freedom in the fractures pressure space. Then, any Robin sub-domain problems are replaced by linear combinations of the multiscale flux basis during the interface iteration. The proposed approach is, thus, agnostic to the physical model in the fracture network. Numerical experiments demonstrate the computational gains of pre-computing the flux exchange between the porous medium and the fracture network against standard non-linear domain decomposition approaches.

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“…For a discussion on more general compatibility conditions at the intersections, we refer the reader to [21,22,26,47]. In order to define the weak formulation of problem (9)-(11), we introduce the spaces for the velocity unknowns [3]…”

Section: The Case Of Multiple Intersecting Fracturesmentioning

confidence: 99%

“…The so-called extended finite element methods (XFEM), which permit to mesh the entire domain independently of the fractures, are described in [17,21] and references therein. In addition, further discretization schemes have been proposed for handling general elements and distorted grids, namely: mimetic finite difference methods [6,23], discontinuous Galerkin methods [5], virtual element methods [10,26], hybrid high-order methods [16], or multipoint flux approximation methods [1,45].…”

Section: Introductionmentioning

confidence: 99%

Mixed-Dimensional Geometric Multigrid Methods for Single-Phase Flow in Fractured Porous Media

Arrarás¹,

Gaspar²,

Portero³

et al. 2019

SIAM J. Sci. Comput.

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This paper deals with the efficient numerical solution of single-phase flow problems in fractured porous media. A monolithic multigrid method is proposed for solving twodimensional arbitrary fracture networks with vertical and/or horizontal possibly intersecting fractures. The key point is to combine two-dimensional multigrid components (smoother and inter-grid transfer operators) in the porous matrix with their one-dimensional counterparts within the fractures, giving rise to a mixed-dimensional multigrid method. This combination seems to be optimal since it provides an algorithm whose convergence matches the multigrid convergence factor for solving the Darcy problem. Several numerical experiments are presented to demonstrate the robustness of the monolithic mixeddimensional multigrid method with respect to the permeability of the fractures, the grid size and the number of fractures in the network. * Francisco J.

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Dual Virtual Element Method for Discrete Fractures Networks

Cited by 55 publications

References 36 publications

Conforming, non-conforming and non-matching discretization couplings in discrete fracture network simulations

Conforming, non-conforming and non-matching discretization couplings in discrete fracture network simulations

A multiscale flux basis for mortar mixed discretizations of reduced Darcy–Forchheimer fracture models

Mixed-Dimensional Geometric Multigrid Methods for Single-Phase Flow in Fractured Porous Media

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