On the effect of background voidage on compressive solitary waves in compacting media

The effect of background voidage on the necessary conditions for the existence of compressive solitary wave solutions in the two-phase fluid flow of a medium compacting under gravity is investigated. It is assumed that K=K 0 phi n (1- phi ) -p and xi +4/3 eta =( xi +4/3 eta ) 0 phi -m (1- phi )q where K is the permeability of the medium, xi +4/3 eta is the effective viscosity of the solid matrix and phi is the voidage. It is shown that for compressive solitary wave solutions to exist, which satisfy certain boundary conditions, it is necessary that the background voidage phi 0 and the exponent n lie in two regions of the ( phi 0 , n)-plane when 0<or=p<1, and that this reduces to one region when p>or=1. Necessary conditions on the exponent m are also derived. Solitary wave solutions for specific values of n, m, p, q and phi 0 are obtained numerically and compared.