Perturbation analysis of the equation for the transport of dissolved solids through porous media: II. Basic non-linear problem

Abstract

Singular perturbation methods are used to treat analytically the problem of steady one-dimensional flow in a porous column, assuming non-linear exchange equilibrium and concentration-dependent diffusivity, when the Péclet number is large. The breakdown of the original problem into a non-linear outer problem, which is hyperbolic and of first order, and an inner problem which is equivalent to a set of ordinary differential equations, greatly facilitates analysis. The development and movement of concentration discontinuites (jumps) within the flow is examined via the outer expansion, the first term of which neglects diffusion entirely. The next term gives a first estimate of diffusion effects. It is found that material diffusing towards a discontinuity increases its speed, as would be expected. In the neighborhood of a discontinuity, a state of quasi-equilibrium exists between diffusion and convective effects due to the relative velocity between the characteristics and the discontinuity. A quasi-steady diffusion zone of thickness O (ε) exists, in contrast to the zone, O ( ε 1 2 t 1 2 ), which forms in the linear case. The first term of the inner expansion is matched both ahead of and behind the jump with the aid of only one arbitrary function of integration; the second function remains undetermined. At the next order, this function is evaluated as a shift of origin of the inner expansion to match the displacement of the jump due to diffusion. However, at this next order of matching, a new undetermined function appears. Extensions of the method to more general cases of non-linear, non-equilibrium exchange, and when source terms are present, are discussed briefly.