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A stabilized Lagrange multiplier finite-element method for flow in porous media with fractures

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Abstract

In this work we introduce a stabilized, numerical method for a multidimensional, discrete-fracture model (DFM) for single-phase Darcy flow in fractured porous media. In the model, introduced in an earlier work, flow in the \((n-1)\)-dimensional fracture domain is coupled with that in the n-dimensional bulk or matrix domain by the use of Lagrange multipliers. Thus the model permits a finite element discretization in which the meshes in the fracture and matrix domains are independent so that irregular meshing and in particular the generation of small elements can be avoided. In this paper we introduce in the numerical formulation, which is a saddle-point problem based on a primal, variational formulation for flow in the matrix domain and in the fracture system, a weakly consistent stabilizing term which penalizes discontinuities in the Lagrange multipliers. For this penalized scheme we show stability and prove convergence. With numerical experiments we analyze the performance of the method for various choices of the penalization parameter and compare with other numerical DFM’s.

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Acknowledgements

The authors would like to thank the German Research Foundation (DFG) for financial support of the project within the Cluster of Excellence in Simulation Technology (EXC 310/2) at the University of Stuttgart, and the two referees whose comments helped improving this paper.

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Authors and Affiliations

  1. Universtität Stuttgart, IANS, Pfaffenwaldring 57, 70569, Stuttgart, Germany

    Markus Köppel

  2. Université de Technologie de Compiègne (UTC), LMAC, Rue du docteur Schweitzer CS 60319, 60203, Compiègne Cedex, France

    Vincent Martin

  3. INRIA Paris, 2 Rue Simone Iff, 75589, Paris, France

    Jean E. Roberts

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  1. Markus Köppel
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  2. Vincent Martin
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  3. Jean E. Roberts
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Correspondence to Markus Köppel.

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Köppel, M., Martin, V. & Roberts, J.E. A stabilized Lagrange multiplier finite-element method for flow in porous media with fractures. Int J Geomath 10, 7 (2019). https://doi.org/10.1007/s13137-019-0117-7

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