Contribution to the formulation of the theory of groundwater flow through an elastic deformable aquifer

Abstract

The equation of motion (generalized Darcy's law) of a slightly compressible viscous liquid in an elastic porous medium is derived by means of the theory of interacting continua. The motion is expressed in terms of the macroscopic velocity of the fluid relative to the porous solid. The constitutive equation for the force due to interaction between the solids and the fluid is formulated and discussed. It is suggested that only the tangential (frictional) portion of this force is important in describing the flow of groundwater through a confined aquifer. The tangential part of the interacting force contributes mostly to distortion of the solid whereas the normal portion contributes mostly to the elastic volumetric deformation. In the formulation of the motion of the solid component, it is suggested that only the normal portion of the interacting force is important in defining the storage in an aquifer due to the deformation of the medium. The system of equations, composed of the equation of motion of each component, must be solved simultaneously because fluid pressure and aquifer dilatation are coupled. Uncoupling may be accomplished in several simplified cases: deformation of a porous medium in the vertical direction only; deformation in the radial (or horizontal) direction only, uniform over the thickness of the aquifer; leaky aquifer with b B < 0.3 ( b, the thickness of the aquifer; B, the leakage factor). The specific storage may be defined only in the cases where a simple relationship between fluid pressure and dilatation exists. Since this relationship is not unique, there is no unique definition of the specific storage. If the local fluid inertia is not negligible, the governing differential equation is the telegrapher's equation. The solution to a typical boundary-value problem yielded the following conclusion. The pressure disturbance travels through an elastic porous medium with a finite velocity. The more compressible the aquifer the slower the change in pressure that travels through the porous medium and the higher time lag that appears.