The migration of melt through the mantle of the Earth is governed by a third-order nonlinear partial differential equation for the voidage or volume fraction of melt. The partial differential equation depends on the permeability of the medium which is assumed to be a function of the voidage. It is shown that the partial differential equation admits, as well as translations in time and space, other Lie point symmetries provided the permeability is either a power law or an exponential law of the voidage or is a constant. A rarefactive solitary wave solution of the partial differential equation is derived in the form of a quadrature for the exponential law for the permeability.
N. Mindu. D. P. Mason. "Permeability Models for Magma Flow through the Earth's Mantle: A Lie Group Analysis." J. Appl. Math. 2013 (SI27) 1 - 8, 2013. https://doi.org/10.1155/2013/258528