ЗАДАЧА О НЕЛИНЕЙНОМ ФИЛЬТРАЦИОННОМ ПОЛЕ ДАВЛЕНИЯ В СРЕДЕ СО СЛАБОСЖИМАЕМЫМ СКЕЛЕТОМ

Abstract

The solution of the problem of the pressure field during filtration of a compressible fluid in a porous medium with an incompressible skeleton in the presence of high-amplitude disturbances is presented. The equation describing pressure changes during field development takes into account the compressibility of the fluid and is presented in a nonlinear form. The known dependences of the density of the filtered medium on the pressure are approximated by a linear function. The movement is assumed to be one-dimensional and horizontal. The porosity, density and permeability of the porous medium skeleton, as well as the viscosity of the filtered medium are considered constant. The solution of the problem is found using an asymptotic expansion by a formal parameter added in the problem as a factor to the compressibility of the fluid. An approximate analytical solution of the nonlinear problem of the filtration pressure field in the zero and first approximations is found. The zero and first coefficients are represented by solutions of quasi-stationary equations, in which time is included as a parameter through the dimensions of the perturbation zone determined by the law of conservation of mass. It is established that taking into account the nonlinearity leads to a decrease in the size of the zone of disturbances of the pressure field. An approach to determining the upper boundary of the perturbation zone in nonlinear problems of this kind, which is based on the use of conservation laws, is proposed. It is shown that the special case of the zero approximation coincides with the solution of the linear problem obtained by the method of changing successive stationary states. The expressions found expand the possibilities of studying high-amplitude filtration processes, and the proposed approach removes the limitations of classical approaches associated with neglecting the dependence of fluid density on pressure in the divergent term of the continuity equation. The method used makes it possible to construct analytical expressions for decomposition coefficients orders of magnitude higher than the first, in addition, it creates the possibility of studying the contribution of nonlinearity caused by the dependence of permeability and viscosity on pressure.