We develop and analyze a model for the interaction of a quasi-Newtonian free fluid with a poroelastic medium. The flow in the fluid region is described by the nonlinear Stokes equations and in the poroelastic medium by the nonlinear quasi-static Biot model. Equilibrium and kinematic conditions are imposed on the interface. We establish existence and uniqueness of a solution to the weak formulation and its semidiscrete continuous-in-time finite element approximation. We present error analysis, complemented by numerical experiments.

SUMMARYConstitutive laws for rock joints should be able to reproduce the fundamental mechanical behaviour of real joints, such as dilation under shear and strain softening due to surface asperity degradation. In this work, we extend the model of Plesha to include hydraulic behaviour. During shearing, the joint can experience dilation, leading to an initial increase in its permeability. Experiments have shown that the rate of increase of the permeability slows down as shearing proceeds, and, at later stages, the permeability could decrease again. The above behaviour is attributed to gouge production. The stress-strain relationship of the joint is formulated by appeal to classical theories of interface plasticity. It is shown that the parameters of the model can be estimated from the Barton-Bandis empirical coefficients; the Joint Roughness Coefficient (JRC) and the Joint Compresive strength (JSC). We further assume that gouge production is also related to the plastic work of the shear stresses, which enables the derivation of a relationship between the permeability of the joint and its mechanical aperture. The model is implemented in a finite element code (FRACON) developed by the authors for the simulation of the coupled thermal-hydraulic-mechanical behaviour of jointed rock masses. Typical laboratory experiments are simulated with the FRACON code in order to illustrate the trends predicted in the proposed model.

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