Theoretical and numerical analyses of pore-fluid flow patterns around and within inclined large cracks and faults

Abstract

Large cracks and faults play an important and diverse role in controlling pore-fluid flow patterns in pore-fluid-saturated porous rocks. Examples of fault-related fluid flow include (1) geological structural controls on groundwater flow and contaminant transport, (2) formation and localization of some valuable mineral deposits around and within fault zones, (3) impacts of heterogeneous fault zone hydraulic properties on the formation and location of petroleum reservoirs through fault sealing, compartmentalization and variability in pore-fluid flow pathways and (4) induced rupture and failure processes by the interaction between material deformation and pore-fluid flow around faults through cycles of brittle deformation and seismicity. In this paper, theoretical and numerical methods are used to investigate pore-fluid flow patterns around and within large cracks and faults in pore-fluid-saturated porous rocks. Although considerable numerical investigations have been carried out on the flow pattern around large cracks and faults, a full set of exact analytical solutions in the case of a fault having any finite but non-zero permeability is not yet available due to mathematical difficulties in solving the related partial differential equations in a conventional Cartesian coordinate system. For the purpose of deriving a full set of exact analytical solutions, large cracks and faults are treated as elliptic inclusions, in which both the aspect ratio of an elliptic crack and the permeability ratio of the crack to its surrounding rock are used to represent the hydrodynamic property contrast of the elliptic crack. It is assumed that the pore-fluid flow in the far field away from the elliptic crack is uniform and that the long axis of the elliptic crack is inclined to the inflow direction in the undisturbed far field. Under these assumptions, a full set of analytical solutions has been derived for the pore-fluid velocity, stream function and excess pore-fluid pressure around and within an inclined elliptic crack in pore-fluid-saturated porous rocks. To overcome mathematical difficulties in describing and satisfying interface conditions between the elliptic crack and its surrounding rocks, analytical solutions are firstly derived in an elliptical coordinate system, and then explicitly expressed, through rigorous inverse mappings, in the conventional Cartesian coordinate system using elementary functions. This allows the present analytical solutions not only to be used to produce theoretical understanding of the pore-fluid flow pattern around and within a large crack, but also to be used as valuable benchmark solutions for validating any numerical methods. After a finite element computational model is validated by comparing the numerical solutions with the present analytical ones, it is used to investigate pore-fluid flow patterns around and within two parallel inclined cracks in pore-fluid-saturated porous rocks. Some conclusions in relation to the effects of inclined large cracks on pore-fluid flow patterns are made through both the theoretical and the numerical analyses.