Modeling Fractures and Barriers as Interfaces for Flow in Porous Media

Martin, Vincent; Jaffré, Jérôme; Roberts, Jean E.

doi:10.1137/s1064827503429363

Cited by 485 publications

(657 citation statements)

References 11 publications

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“…As already observed (without convergence analysis) in [15] in the case of matching meshes, this implies exchanging data on the "immersed" interface Γ and building suitable Steklov-Poincaré operators acting on interface variables.…”

Section: Iterative Solution Of the Coupled Problemmentioning

confidence: 92%

“…This requires the derivation of a reduced model for the fracture flow, for which we refer to [1,2,12,15]. Following the cited references, we assume that the flow within the fracture is governed by a reduced Darcy's equation.…”

Section: Formulation Of the Problemmentioning

confidence: 99%

“…In [15], the well posedness of such coupled problem has been proved for 1 2 < ξ ≤ 1. We will show in the next section that this assumption is needed also in our formulation.…”

Section: Formulation Of the Problemmentioning

confidence: 99%

“…(iii) We observe that sinceŜ is positive definite, it is invertible; the iterative method (5.11) can be recast in the following preconditioned Richardson form, 15) where the linear operator R :=R + S is preconditioned byR =Ŝ −1 . Note that R andR are both symmetric positive definite, andR −1 R = I + T .…”

Section: As a Consequence P (K)mentioning

confidence: 99%

“…In the relevant case of fractures whose thickness is very small compared to the characteristic length of the domain, a reduced model can be set up in which the fracture is represented as an immersed interface. The development and the analysis of such models for the single phase flow have been extensively addressed in [1,12,15], where the fracture flow equations and the proper interface conditions across the fracture have been identified and mixed finite element schemes for the coupled porous medium flow/fracture flow have been proposed.…”

Section: Introductionmentioning

confidence: 99%

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A mixed finite element method for Darcy flow in fractured porous media with non-matching grids

2011

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Abstract.We consider an incompressible flow problem in a N -dimensional fractured porous domain (Darcy's problem). The fracture is represented by a (N −1)-dimensional interface, exchanging fluid with the surrounding media. In this paper we consider the lowest-order (RT0, P0) Raviart-Thomas mixed finite element method for the approximation of the coupled Darcy's flows in the porous media and within the fracture, with independent meshes for the respective domains. This is achieved thanks to an enrichment with discontinuous basis functions on triangles crossed by the fracture and a weak imposition of interface conditions. First, we study the stability and convergence properties of the resulting numerical scheme in the uncoupled case, when the known solution of the fracture problem provides an immersed boundary condition. We detail the implementation issues and discuss the algebraic properties of the associated linear system. Next, we focus on the coupled problem and propose an iterative porous domain/fracture domain iterative method to solve for fluid flow in both the porous media and the fracture and compare the results with those of a traditional monolithic approach. Numerical results are provided confirming convergence rates and algebraic properties predicted by the theory. In particular, we discuss preconditioning and equilibration techniques to make the condition number of the discrete problem independent of the position of the immersed interface. Finally, two and three dimensional simulations of Darcy's flow in different configurations (highly and poorly permeable fracture) are analyzed and discussed.Mathematics Subject Classification. 76S05, 35Q86, 65L60.

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Section: Iterative Solution Of the Coupled Problemmentioning

confidence: 92%

Section: Formulation Of the Problemmentioning

confidence: 99%

“…In [15], the well posedness of such coupled problem has been proved for 1 2 < ξ ≤ 1. We will show in the next section that this assumption is needed also in our formulation.…”

Section: Formulation Of the Problemmentioning

confidence: 99%

Section: As a Consequence P (K)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A mixed finite element method for Darcy flow in fractured porous media with non-matching grids

2011

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Physics-based preconditioners for flow in fractured porous media

2014

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Discrete fracture models are an attractive alternative to upscaled models for flow in fractured media, as they provide a more accurate representation of the flow characteristics. A major challenge in discrete fracture simulation is to overcome the large computational cost associated with resolving the individual fractures in large-scale simulations. In this work, two characteristics of the fractured porous media are utilized to construct efficient preconditioners for the discretized flow equations. First, the preconditioners are tailored to the fracture geometry and presumed flow properties so that the dominant features are well represented there. This assures good scalability of the preconditioners in terms of problem size and permeability contrast. For fracture dominated problems, numerical examples show that such geometric preconditioners are comparable or preferable when compared to state-of-the-art algebraic multigrid preconditioners. The robustness of the physics-based preconditioner for less favorable fracture conditions is further demonstrated by a systematic degradation of the fracture hierarchy. Second, the preconditioners are physics preserving in the sense that conservative fluxes can be computed even for an inexact pressure solutions. This facilitates a scheme where accuracy in the linear solver can be traded for efficiency by terminating the iterative solvers based on error estimates, and without sacrificing basic physical modeling principles. With the combination of these two properties a novel preconditioner is obtained which bridges the gap between multiscale approximations and iterative linear solvers.

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An upscaling procedure for fractured reservoirs with embedded grids

Fumagalli

Pasquale²,

Zonca

et al. 2016

Water Resources Research

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Upscaling of geological models for reservoir simulation is an active and important area of research. In particular, we are interested in reservoirs where the rock matrix exhibits an intricate network of fractures, which usually acts as a preferential path to the flow. Accounting for fractures' contribution in the simulation of a reservoir is of paramount importance. Here we have focused on obtaining effective parameters (e.g., transmissibility) on a 3-D computational grid on the reservoir scale, which account for the presence, at a finer spatial scale, of fractures and a network of fractures. We have essentially followed the idea illustrated in Karimi-Fard et al. (2006), yet this work has some notable aspects of innovation in the way the procedure has been implemented, and in its capability to consider rather general corner-point grids, like the ones normally used in reservoir simulations in the industry, and complex and realistic fracture networks, possibly not fully connected inside the coarse cells. In particular, novel contribution is the employment of an Embedded Discrete Fracture Model (EDFM) for computing fracture-fracture and matrix-fracture transmissibilities, with a remarkable gain in speedup. The output is in the form of transmissibility that, although obtained by considering single-phase flow, can be used for coarse-scale multiphase reservoir simulations, also via industrial software, such as Eclipse, Intersect, or GPRS. The results demonstrate the effectiveness and computational efficiency of the numerical procedure which is now ready for further testing and industrialization.

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Modeling Fractures and Barriers as Interfaces for Flow in Porous Media

Cited by 485 publications

References 11 publications

A mixed finite element method for Darcy flow in fractured porous media with non-matching grids

A mixed finite element method for Darcy flow in fractured porous media with non-matching grids

Physics-based preconditioners for flow in fractured porous media

An upscaling procedure for fractured reservoirs with embedded grids

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