Mixed-dimensional model of poroelasticity Jan Bˇ rezina and Jan Stebel Institute of New Technologies and Applied Informatics Faculty of Mechatronics, Informatics and Interdisciplinary Studies Technical University of Liberec Studentsk´ a 1402/2, 461 17 Liberec, Czech Republic {jan.brezina,jan.stebel}@tul.cz July 28, 2022 Abstract The paper provides derivation and analysis of a poroelasticity model in a domain with fracture represented by a codimension-one manifold. The system of saturated flow and linear elasticity both in the matrix domain and in the fracture coupled through appropriate interface condi- tion is obtained from a continuum description by integration and semi- discretization in the normal direction of the fracture. The existence and uniqueness of a weak solution are proved with the help of a fixed-point argument. The analysis is complemented by a numerical example. Keywords: Biot’s poroelasticity, discrete fracture-matrix model, fixed-stress splitting. 2010 Mathematics Subject Classification: 35M33, 35Q86, 74S05, 74F10. 1 Introduction The hydromechanical (HM) interaction of fluid flow and rock mechanics plays a significant role in many important applications such as geothermal power utilization, nuclear waste deposition, or CO 2 storage [24]. The linear poroelastic model introduced by Biot [3], as well as some of its more recent non-linear extensions [24, 17] are formulated for a continuum description of fractured rock. The fracture network consists of fractures ranging from grain size cracks of a sub- milimeter scale to fault zones of a kilometer-scale [4]. The small-scale fractures can be described in terms of an equivalent continuum [22, 25]. However, the very high aspect ratio (size over aperture) of the fractures leads to large gradients of the principal unknowns, typically flow velocity and displacement, that can not be resolved by classical discretization methods (FVM, FEM). On the other hand, the discrete fracture network (DFN) approach [11, 15] that describes individual fractures as codimension-one manifolds is unable to treat a vast number of small- scale fractures. Various hybrid approaches are used to combine continuum and 1
Mixed-dimensional model of poroelasticity
May 29, 2023
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