Self-similar waves in a three-dimensional porous medium in the presence of non-stationary singular source or absorption

Abstract

We study an important for applications a model of three-dimensional porous medium in the presence of non-stationary singular source or absorption. We obtained all invariant submodels of this model. These submodels are described by invariant solutions. We found that among the invariant solutions of rank 1, this model has only three types of essentially different (not connected by point transformations) self-similar waves. These waves include: a self-similar wave, propagating in a porous medium along one of the axes of coordinates, a plane self-similar circular wave and a self-similar spherically symmetric wave. We obtained containing arbitrary constant, integral equations describing these self-similar waves. We established the existence and uniqueness of self-similar waves for which at an initial instant of the time at a fixed point, or on a fixed circle, or on a fixed sphere, either a pressure and rate of its change, or a pressure and its gradient are given. The pressure distribution graphs, obtained as results of a numerical solution of these boundary value problem, are given. The obtained results can be used to study to describe the processes associated with a underground fluid or gas flow, with water filtration, with the engineering surveys in the construction of the buildings, and also with shale oil and gas production.