Analytical and numerical modeling of the work of a perfect well in anisotropic homogeneous soil formation

The extraction of oil and gas reservoirs in complex geological structures is constantly increasing. Thus, it is obvious that the reliable mathematical models governing such porous media must be of interest. The problem is concerned with the work of a well in anisotropic soil formation in the case of a random smooth (piecewise-smooth) external boundary of reservoir. The permeability of the soil is described by an asymmetric second-rank tensor. To find a solution to the problem is difficult because of the complicated form of the basic equation. The solution can be simplified if the equation is transformed into the Laplace equation. To do this, we turn on the auxiliary plane using homeomorphic (affine) transformation. In a special case when the external boundary of reservoir of the auxiliary plane takes the form of a circle, we arrive at a closed-form (analytical) solution. These analytic solutions are useful in testing numerical codes. In the general case, the problem is reduced to a system of integral equations with Cauchy kernel and the integral relation. The system is solved by the method of discrete singularities. Convergence of the algorithm is studied. Convergence solutions are improved with an increase in the number of division points. The convergence solutions with a piecewise-smooth (square) external boundary of reservoir are slightly worse compared with the smooth (circular) ones. Soil anisotropy greatly affects the production rate and can strongly change the production rate compared to the rate of the well in an isotropic soil. Permeability tensor components located on the main diagonal have a major impact on the rate of flow in the anisotropic soil. This method can be used to solve different problems in filtering anisotropic porous medium.