Submodels of 2-d model of the motion of fluid or gas in a porous medium with an external non-stationary source or absorption

Abstract

The motion of fluid or gas in a porous medium is the main object of investigation of underground hydromechanics. The model describing this motion was called “Porous medium”.In our paper, we study a general 2-d model of the motion of fluid or gas in porous medium with an external non-stationary source or absorption. As the main research method, we use the method of group (symmetry) analysis, which is one of the most effective ways to obtain information about the properties of solutions to differential equations that define a physical model. A group classification of the differential equation defining studied physical model is carried out. As a result, all basic models with different symmetry properties are obtained. For the basic model, which admits the widest group a classification of all its invariant submodels is performed and all invariant submodels, defined by invariant solutions of rank 1 are studied. All solutions found have physical meaning.The set of explicitly found solutions forms a 47-parametric family of exact solutions. Each solution in this family depends on an arbitrary function defining an external non-stationary source or absorption. For each solution, for the specific values of the parameters on which it depends and for this specific function, we obtained a graph of the pressure distribution.Other solutions are used to describe the movement of a liquid or gas if either the pressure and its rate of change, or the pressure and one of its gradient components, are given at an initial time at a fixed point. The set of these solutions depends on 13 parameters and a function defining an external non-stationary source or absorption. For specific values of this function and specific values of the parameters on which each solution depends, the existence and uniqueness of a solution to the corresponding boundary value problem under some additional conditions has been established. Each such problem is solved numerically. The results of these decisions we reflected in the graphs.The significance of obtained solutions lies in the fact that we have obtained a database consisting of two multiparameter families of invariant solutions of rank 1 of the equation describing the motion of fluid or gas in a porous medium in the presence of an external non-stationary source or absorption. Every solution depends on an empirically determined function. These solutions describe specific physical processes. They can also be used as test solutions in numerical calculations related to the study of the motion of fluid or gas in a porous medium during fluids filtering, engineering surveys in the construction of building structures and in the extraction of shale oil or gas.