Method of numerical analysis of fluid flows in porous media under a cas-cade of hydraulic structures

Abstract

Fluid flows in porous media are widespread in nature and they often come to the need for consideration in the course of economic activity. The paper deals with the problem of the theory of stationary fluid flows in porous media in the ground under the construction of hydraulic structures under the assumption that Darcy’s law is fulfilled. The mathematical model of this problem is the elliptic equations for the stream function with boundary conditions of the second kind on sections of the boundary of the reservoir and boundary conditions of the first kind on sections of the boundary that are impermeable to liquid. At the same time, the formulation of the problem includes the unknown values of the total fluid flow rates under each of the hydraulic structures of the cascade, for the determination of which additional integral relations are formulated. For the numerical analysis of the problem, it is proposed to use the structural-variational method (the R-functions method), which will make it possible to fully take into account in the computational algorithm all the geometric and analytical information that is included in the formulation of the problem. In accordance with the principle of superposition from the original problem, a transition was made to a set of boundary-value problems with known boundary conditions. For each of these problems, according to the method of R-functions, the structures of the solution are constructed that accurately take into account all the boundary conditions, and the use of the Ritz variational method for approximation of the uncertain component is justified. After that, of the additional integral relations, the approximate values of the unknown flow rates of the fluid, and hence the approximate solution of the original problem, are found. A computational experiment was carried out for the case of a constant filtration coefficient in an area that has the form of the lower half of a ring with two semicircular burials located symmetrically. The proposed method of numerical analysis has shown its effectiveness in solving a test problem and can be used to solve applied problems. The advantages of the developed numerical method are the possibility of obtaining the solution of the boundary value problem in the form of a single analytical expression and the exact satisfaction of all the boundary conditions.