The one-dimensional problem of elastic filtration of fluid through moving boundary is considered. The boundary conditions for invariant problem is introduced. The problem is reduced to overdetermine boundary problem for Veber equation. The exact solutions are obtained. For arbitrary invariant filtration law the relationship between overdetermine invariant boundary conditions is obtained.
asymptotics of solutions
Problem: get the relationship between the function of the moving boundary, pressure and flow rate for the elastic filtration of the fluid into the reservoir under various self-similar regimes.
Methods: a group analysis of a boundary value problem with a free boundary was used, nonlinear elastic filtration equation, methods for solving boundary value problems of hypergeometric equations in an unbounded domain.
In a study was determined:
1. The elastic regime of filtration is described by a nonlinear parabolic equation, which is linearized by a point change to the linear filtration equation.
2. Invariant solutions of the boundary value problem with a free boundary in an unbounded domain are found, and a relationship between the function of the moving boundary, pressure and flow rate under various self-similar regimes is obtained.
3. Exact solutions of invariant submodels given by the Weber equation for special regimes of filtration and approximate solutions for arbitrary self-similar regimes are obtained.