The equations governing transient flow of groundwater have the same form as those governing transient flow of heat in solids. The equations that are usually applied to solve problems of transient groundwater flow in confined systems are a simplified form of the equations that govern poro-elasticity. The equations that govern transient flow in unconfined aquifers are a linearized form of the nonlinear equations that apply to such flow, obtained using the Dupuit-Forchheimer approximation. A large body of solutions to the equations exist, and many are found in Carslaw and Jaeger [1959].

Transient Shallow Confined Flow

Transient effects in aquifer systems come about when boundary values or infiltration rates vary with time. A common example is a well that is switched on at some time; on starting the pump, the heads and pressures in the aquifer system change gradually until, for all practical purposes, steady-state conditions are reached. In a confined aquifer, the transient effects are caused by the compression of the grain skeleton as a result of decreasing pressures; if both the aquifer material and the fluid were incompressible, steady-state conditions would be reached instantaneously.

The problem of transient flow in a confined aquifer is a coupled one; the deformation of the grain skeleton is coupled to the groundwater flow. The problem is very complex, as the constitutive equations for soil are highly non-linear, even under dry conditions, and the coupling with the groundwater flow increases the complexity further still. Biot [1941] formulated the coupled problem mathematically, approximating the grain skeleton as a linearly elastic material. Fortunately, the pressure changes due to groundwater flow are usually small compared with the overburden stresses in the grain skeleton, which allows approximations to be made that simplify the problem considerably.We formulate the problem in terms of two equations: the equation describing storage and a simplified equation for the deformation of the grain skeleton. All strains and stresses are taken as positive for contraction and compression, respectively.

The Storage Equation

The storage equation expresses that water may be stored in an elementary volume V of porous material due to both an increase of V and a decrease in the volume Vw of the water.