Filtration Pressure Field at High-Amplitude Perturbations
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Journal of Engineering Physics and Thermophysics, Vol. 93, No. 6, November, 2020
FILTRATION PRESSURE FIELD AT HIGH-AMPLITUDE PERTURBATIONS A. I. Filippov, A. A. Koval′skii, and O. V. Akhmetova
UDC 532.685
A study has been made of the pressure field in filtration of a weakly compressible fluid in a homogeneous isotropic compressible medium at high-amplitude perturbations. The equation used in formulating the problem is nonlinear: the densities of a porous medium and of a saturating fluid are assumed to be dependent on the function sought. Use is made of the fact that the dependences of the densities of the fluid and the skeleton material on pressure are approximated with a high degree of accuracy by the linear function. Consideration is given to one-dimensional plane horizontal filtration. The porosity and permeability of a porous medium, and also the viscosity of a filtering medium, are considered to be constant. A solution to the problem has been found with asymptotic expansions where a cofactor of the compressibility factor of the liquid acts as a small parameter. Analytical expressions have been found for the zero and first expansion factors. It has been shown that the zero expansion factor may be used to investigate the evolution of the pressure field of an incompressible liquid, whereas the expression for the first factor contains information on the contribution of nonlinearity due to the fluid′s compressibility. Values of the zero and first residual terms have been determined. It has been proved that the zero and first residual terms contain terms of only higher orders as far as the asymptoticexpansion parameter is concerned, i.e., the corresponding requirement of asymptoticity of expressions of the zero and first factors is met. On the basis of a computational experiment, the regularities of the dependence of the contribution of the nonlinearity under study on time and on the spatial coordinate have been established. Keywords: nonlinear equation, filtration, pressure, asymptotic expansion. Introduction. Under today′s conditions of development of oil and gas fields, new objects are frequently put into operation due to noncommercial strata with difficult-to-extract hydrocarbon reserves [1]. Oil and gas extraction under such conditions requires that high-amplitude pressure differences be used in the face zone [2–5]. Therefore, the development of extraction technologies [3, 6, 7] makes it necessary to solve a number of nonlinear problems of filtration theory, in which account is taken of the dependence of physical parameters of the stratum on pressure. Since finding solutions to the problems containing nonlinear equations is of significance both for the development of optimum engineering solutions and in terms of general science, the employed mathematical methods are being continuously improved. Here we present a novel approach to solving problems of this kind which is based on asymptotic expansions. In the paper, in contrast to the classical piezoconductivity equation [8, 9], the constraint that in the diver
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