Change of pressure in wells opening up highly inhomogeneous porous collectors

We consider a generalized averaged model of the filtration of a weakly compressible liquid in a highly inhomogeneous medium. We construct an approximate analytical solution of the problem that describes the change in the pressure field around a well in a circular bed. We investigate the effect of the parameters of a highly inhomogeneous bed on the form of the well pressure stabilization curve.Notation c Euler constant - D diameter of a weakly permeable block - Ei symbol of the integral exponential function - h bed thickness - K(t), K_*(t) kernels of integral operators - K₀ symbol of the Macdonald function - k permeability - P pressure - Q constant discharge of the well - r radius - s complex variable - t time - t_* time of the relaxation of a weakly permeable block - U(r, t) reduced pressure - U(t) function taking into account the internal flow of the fluid from one medium to the other - fraction of the volume of the bed per block - ^* reduced coefficient of compressibility - characteristic scale of inhomogeneity - ratio of piezoconductivities of a block and a crack - Laplace operator - piezoconductivity - ratio of elastic amounts of weakly permeable blocks and systems of cracks - viscosity of the fluid - dimensionless time - integration variable. Indices: 1, weakly permeable blocks - 2 system of cracks - 0 initial value - w value in the well - e effective value - overbar Laplace transformInstitute for the Problems of Oil and Gas of the Russian Academy of Sciences, Moscow. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 68, No. 3, pp. 444–450, May–June, 1995.