Initial-boundary Problem for the Piezoconductivity Equation Describing Filtration in a Layered Oil Reservoir

On the basis of the complex use of the asymptotic, spectral and methods of the theory of generalized functions, a solution to the problem of the pressure field in a layered inhomogeneous oil and gas reservoir, limited in length, operated in a given depression mode, is found. The original problem, containing partial differential equations with variable coefficients, was transformed into a chain of coefficient problems. In order find a solution, the classical methods developed for equations with constant coefficients were used. Analytical expressions for the zero coefficient are constructed using the spectral method, and a universal procedure based on the theory of generalized functions is proposed to find the first and higher coefficients of the asymptotic expansion for the pressure fields and the filtration velocity. It is shown that in the zero approximation the pressure field in the reservoir does not depend on the transverse coordinate. This means that the flow in a non-uniform confined reservoir in the zero approximation is one-dimensional, and horizontal filtration flows are predominant. It is shown that in a flat two-dimensional filtration flow in an ideally exposed homogeneous bounded formation, interlayer crossflows are always absent, and in heterogeneous formations, transverse flows arise only if the pressure fields are unsteady. The found analytical expressions for the pressure field and filtration velocity can be widely used to study filtration processes in real oil and gas reservoirs.