Analytical estimation of solution parameters for elliptic equations with variable coefficients in fluid dynamics

Abstract

In this paper we consider multidimensional and two-dimensional elliptic partial differential equations of the second order with variable coefficients. They are often used in the oil and gas industry to calculate the inflow volume of liquids and gases from natural reservoirs to exciting structures. The analytical solution of that equations with variable values of the permeability coefficient is possible in rare cases, when the change in the formation permeability is expressed in the form of simple analytical dependencies over the area of fluid filtration. In this paper, we give estimated formulas for fluid inflow to structures under conditions of reservoir heterogeneity, which retain an analytical relationship with the permeability coefficient. The value of the proposed method lies in the ability to establish important qualitative regularities of an underground flow formation, based on simple formulas. The obtained expressions of the above theorems are consistent with the calculation formulas of underground fluid dynamics, and allows one to perform calculations of oil production in heterogeneous formations.