Groundwater flow model as a singularly perturbed diffusion equation

In formulating mathematical models of unconfined groundwater flow, the nature of the transition zone between saturated and dry aquifer material has been handled in a variety of ways. Although, the well known Richards' equation (Richards, 1931) describes the transition region well, in practice, it is difficult to use Richards' equation. One can obtain a simpler formulation assuming a free surface, but, in many practical situations such a formulation ceases to be valid. A useful intermediate approach is to apply a capillary correction term to the free surface formulation (Parlange and Brutsaert, 1987). For sudden drawdown, the situation can be expressed in the form of an initial boundary value problem as follows,

and

u(x, t) represents the height of the water table,  is the initial value of u and  is the drawdown height . The coefficients ,  and  are physical constants with  and . The term represents the correction for capillary with the magnitude of  indicating the amount of correction.  is proportional to the average suction required to extract the water held in the soil by capillarity. There will be no capillary correction when, but as the capillary effects become more important than gravity,  will take a non-zero value. It should be pointed out that the diffusion equation which results from (1) when , should not be viewed as a proper limiting case of the equation with . When , (1) belongs to the class of parabolic equations which have strong smoothing properties. However, when , one obtains a pseudoparabolic equation that preserves the regularity or singularity of initial data for all time. This means that the inclusion of capillary correction term represents a singular perturbation of the basic diffusion equation. Therefore, practitioners need to take care in applying such a model to realistic problems. We intend to provide more insight into this singularly perturbed equation by analyzing it theoretically and computationally.