A high‐resolution local RBF collocation method for steady‐state poroelasticity and hydromechanical damage analysis

Abstract

SummaryIn this work, we describe a meshless numerical method based on local collocation with RBFs for the solution of the poroelasticity equation. The RBF finite collocation approach forms a series of overlapping nodal stencils, over which an RBF collocation is performed. These local collocation systems enforce the governing PDE operator throughout their interior, with the intersystem communication occurring via the collocation of field variables at the stencil periphery. The method does not rely on a generalised finite differencing approach, whereby the governing partial differential operator is reconstructed at the global level to drive the solution of the PDE. Instead, the PDE governing and boundary operators are enforced directly within the local RBF collocation systems, and the sparse global assembly is formed by reconstructing the value of the field variables at the centrepoint of the local stencils. In this way, the solution of the PDE is driven entirely by the local RBF collocation, and the method more closely resembles the approach of the full‐domain RBF collocation method. By formulating the problem in this fashion, high rates of convergence may be attained without the computational cost and numerical ill‐conditioning issues that are associated with the full‐domain RBF collocation approach.An analytical solution is formulated for a 2D poroelastic fluid injection scenario and is used to verify the proposed implementation of the method. Highly accurate solutions are produced, and convergence rates in excess of sixth order are observed for each field variable (i.e. pressure and displacement) and field‐variable derivative (i.e. pressure gradients and stresses). The stress and displacement fields resulting from the solution of the poroelasticity equation are then used to describe the formation and propagation of microfractures and microfissures, which may form in the presence of large shear strain, in terms of a continuous damage variable which modifies the mechanical and hydraulic properties of the porous medium. The formation of such hydromechanical damage, and the resulting increase in hydraulic conductivity, is investigated for a pressurised injection into sandstone. Copyright © 2014 John Wiley & Sons, Ltd.