A reduced model for Darcy's problem in networks of fractures
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 4, p. 1089-1116
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Subsurface flows are influenced by the presence of faults and large fractures which act as preferential paths or barriers for the flow. In literature models were proposed to handle fractures in a porous medium as objects of codimension 1. In this work we consider the case of a network of intersecting fractures, with the aim of deriving physically consistent and effective interface conditions to impose at the intersection between fractures. This new model accounts for the angle between fractures at the intersections and allows for jumps of pressure across intersections. This fact permits to describe the flow when fractures are characterized by different properties more accurately with respect to other models that impose pressure continuity. The main mathematical properties of the model, derived in the two-dimensional setting, are analyzed. As concerns the numerical discretization we allow the grids of the fractures to be independent, thus in general non-matching at the intersection, by means of the extended finite element method (XFEM). This increases the flexibility of the method in the case of complex geometries characterized by a high number of fractures.

DOI : https://doi.org/10.1051/m2an/2013132
Classification:  65N30,  76S05,  86A60
Keywords: reduced models, fractured porous media, XFEM
     author = {Formaggia, Luca and Fumagalli, Alessio and Scotti, Anna and Ruffo, Paolo},
     title = {A reduced model for Darcy's problem in networks of fractures},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {48},
     number = {4},
     year = {2014},
     pages = {1089-1116},
     doi = {10.1051/m2an/2013132},
     zbl = {1299.76254},
     mrnumber = {3264347},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2014__48_4_1089_0}
Formaggia, Luca; Fumagalli, Alessio; Scotti, Anna; Ruffo, Paolo. A reduced model for Darcy's problem in networks of fractures. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 4, pp. 1089-1116. doi : 10.1051/m2an/2013132. http://www.numdam.org/item/M2AN_2014__48_4_1089_0/

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