Hydro-mechanical coupling in zero-thickness interface elements, formulation and applications in geomechanics

Abstract

Zero-thickness joint/interface elements of the Goodman type, have been advantageously used to solve many problems in solid mechanics involving material interfaces or discontinuities. Some years ago, the authors have also proposed a version of such element for flow/diffusion and hydro-mechanical (H-M) coupled problems, either geomechanical or multiphysics. Some advantages are for instance that fluid pressure discontinuities and localized flow lines may be represented on the same FE mesh used for the mechanical problem, as well as the influence of fluid pressure on mechanical stresses or, conversely, of crack openings on the flow redistribution (“cubic law”). In the paper, previous developments are briefly described, together with some new Geomechanical applications under development, particularly the application to the hydraulic fracture problems, which in the past have been studied mainly via analytical or semi-analytical formulations, or using mixed FE-FD approaches. 2 ZERO-THICKNESS INTERFACE ELEMENTS, CONSTITUTIVE LAWS Zero-thickness joint or interface elements are finite elements introduced between adjacent continuum elements, with the special feature that they have one less dimension than the standard continuum elements, that is, they are lines in 2D, or surfaces in 3D. The integration of these elements is done through a local orthogonal coordinate system defined on the interface line or surface. The interface constitutive behavior is formulated in terms of the jump of the main variable across the mid-plane of the interface, and the corresponding force-type conjugate variable. In the standard mechanical problem, those variables are the normal and tangential components of the relative displacements, and their counterpart stress tractions (Fig. 1a). 2.1 Elastoplastic constitutive law The standard interface constitutive model implemented in the code for rock mechanics purposes is a relatively general elastoplastic formulation, which is formulated in terms of normal and shear stress, and the corresponding normal and tangential relative displacements, and includes a hyperbolic failure surface, a range of hardening-softening and dilatancy laws, step-by-step numerical integration, etc. (Gens et al. 1995). However, a simplified version exists that, at the expense of some restrictive assumptions, becomes adequate for computationally efficient explicit integration (Gens et al. 1995). The main simplifying assumptions are perfect plasticity, no dilatancy, and a linear elastic relationship between the normal stress and the normal relative displacement in compression (zero normal stress in tension). The yield surface in the σσ − ττ plane, where ττ = �ττ1 + ττ2 is defined by: FF = ττ2 − tan2φ (σσ2 − 2aaσσ) = 0 (1) Due to the expression of the yield surface and the elastic relationship between the normal stress and the normal relative displacement, once the normal stress is known, the ratio between ττ1 and ττ2 is the only unknown in the integration of the constitutive law. The angle λ, which represents this ratio, can be obtained using the following equation: tan � 2 � = tan �0 2 �� ττ+�ττ 2+aa2tan2φφ ττ0+�ττ0+aa2tanφφ � −tt KKnn Δvv