Finite-difference analysis of plane-parallel and plane-radial flows in the elastic mode of liquid and gas filtration

The theory of filtration of liquids and gases through porous media has historically been used to solve a large number of applied problems - from the movement of groundwater to the regularities of consolidation of biological tissues. This work examines two basic problems of the physics of oil and gas reservoirs - plane-parallel and plane-radial flows of liquid and ideal gas. Darcy's linear law is used as the equation of fluid motion. The constitutive equations for the skeleton include a term that takes account of the effect of fluid pressure on its deformation. In turn, the constitutive equations for the fluid take account of the influence of the skeleton on the compressibility of the fluid. In this way, a coupled problem is formulated for the elastic regime of fluid filtration. The model is verified based on analytical solutions, as well as numerical solutions obtained by other authors. It is shown that the obtained numerical solutions, namely the distributions of pore pressure, filtration rates and mass flow rates, coincide with high accuracy with analytical solutions. Additionally, fluid filtration through a quasi-isotropic medium is considered. It is shown that the presence of a layer with reduced permeability does not lead to nonlinearity in the velocity distribution for liquids and in the product of density and velocity for gas, but their values decrease. The pressure distribution profile, on the contrary, changes abruptly when moving from layer to layer. Formulas are proposed for determining the effective permeability of a medium based on numerical simulation data of a plane-parallel flow. The results obtained can be used in calculating the operating parameters of oil and gas fields.