A framework for coupling flow and deformation of the porous solid

TL;DR

This paper introduces a comprehensive mathematical framework for modeling coupled fluid flow and elastic deformation in porous solids, extending classical models and enabling simulation of complex geomechanical phenomena.

Contribution

It develops a thermodynamically consistent mixture theory model that extends Darcy's law to include elastic deformation effects and provides numerical algorithms for coupled simulations.

Findings

Model recovers results similar to Biot and Terzaghi theories

Capable of simulating consolidation and subsidence

Demonstrates numerical coupling algorithms effectiveness

Abstract

In this paper, we consider the flow of an incompressible fluid in a deformable porous solid. We present a mathematical model using the framework offered by the theory of interacting continua. In its most general form, this framework provides a mechanism for capturing multiphase flow, deformation, chemical reactions and thermal processes, as well as interactions between the various physics in a conveniently implemented fashion. To simplify the presentation of the framework, results are presented for a particular model than can be seen as an extension of Darcy's equation (which assumes that the porous solid is rigid) that takes into account elastic deformation of the porous solid. The model also considers the effect of deformation on porosity. We show that using this model one can recover identical results as in the framework proposed by Biot and Terzaghi. Some salient features of the framework are as follows: (a) It is a consistent mixture theory model, and adheres to the laws and principles of continuum thermodynamics, (b) the model is capable of simulating various important phenomena like consolidation and surface subsidence, and (c) the model is amenable to several extensions. We also present numerical coupling algorithms to obtain coupled flow-deformation response. Several representative numerical examples are presented to illustrate the capability of the mathematical model and the performance of the computational framework.