A reduced model for Darcy’s problem in networks of fractures^∗

¹ MOX - Dipartimento di Matematica “F. Brioschi” – Politecnico di Milano - via Bonardi 9, 20133 Milan, Italy.
luca.formaggia@polimi.it; anna.scotti@polimi.it
² IFP Energies nouvelles – 1 and 4, avenue de Bois-Prau, 92852 Rueil-Malmaison Cedex, France.
alessio.fumagalli1984@gmail.com
³ ENI Spa – Exploration and Production Division – 5° Palazzo Uffici, Room 4046 E, GEBA Dept. - via Emilia 1, San Donato Milanese, 20097 (MI), Italy.
paolo.ruffo@eni.com

Received: 12 February 2013
Revised: 18 September 2013

Abstract

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.