The problem of fluid flow in a fractured reservoir has been a topic of interest to researchers since considerable amount of time. A coupled solid-fluid numerical procedure has always been a requirement in the petroleum and geotechnical industry. Porous media is modelled using Biot's theory using Darcy law for fluid flow. Naturally discontinuous porous rock mass constitutes two phases: fluid flow through pore spaces in rock matrix and fluid flow through fractures, joints, faults, etc. Because rock and discontinuities exhibit differential fluid storage capacity and fluid conductivity, modelling such dual porosity system poses mathematical as well as numerical challenges. Most of the existing coupled formulations of fluid rock interaction flow behavior are pressure based and this results in unsymmetrical stiffness matrices for absorbing viscous boundary condition in dynamic analysis. Hence, a displacement based extended finite element formulation is developed and presented in this paper for dual porous media having multiple intersecting discontinuities like joints, faults for symmetrical representation and for future application of infinite boundary conditions. This paper will mainly highlight on the formulation derivation based on rock displacement (u), joint displacement (a) and displacement of water in pore spaces (w) with its numerical implementation. A numerical example is illustrated in the later section for the efficacy of the procedure.
Rock mass in general is a composite of discrete fracture networks, faults planes, shear zones, fluid in the rock pores, fluid in the discontinuities and rock skeleton with different material properties. Understanding such complex phenomenon under different time varying stress loading conditions is vital for designing a stable underground structure. Classical one-dimensional consolidation theory of porous media was first proposed by Terzaghi in 1943 considering effective stress concept. Following the work done by Terzaghi, Biot (1941, 1955, 1956, 1957) thereby proposed a three-dimensional consolidation theory for porous media under static and dynamic loading conditions; which accounted for Darcy fluid flow in the rock matrix. A complete review on the historical development of the theory of porous media can be found in de Boer, 1996. Presence of fluid in the rock mass obviously causes difference in stress regime as compared to the conventional estimation (Theory of Elasticity) considering dry rock. Essentially nature of fluid flow in fractures is different than that in rock matrix showcasing two different permeability. Hence, such media is often referred to as Dual Porosity media (Barenblatt et al., 1960). Many researchers (Warren and Root, 1963; Aifantis, 1977; 1979; Ghafouri and Lewis, 1996; Bai et al., 1994; 1999) came up with analytical solutions for dual porosity media considering rock matrix and fracture network separately. Interaction between the two phases is achieved via a transfer function which is essentially fluid influx and out flux from the fracture to rock and vice versa. Complete closed form solution for such complex phenomenon is almost an impossible task considering huge variability in rock physics. Advent of numerical procedures have helped to solve approximately any kind of complicated physical problems. A brief review on the FEM procedures in porous media can be found in Khoei, 2015.
