  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.
