Numerical modeling of the performance of a well with an arbitrary piecewise-smooth external boundary in an anisotropic heterogeneous reservoir
DOI:
https://doi.org/10.7242/1999-6691/2022.15.2.17Keywords:
filtration theory, orthotropic reservoir, conductivity tensor, integral equation, production, rectangular reservoir, mathematical modelingAbstract
The extraction of oil and gas from productive reservoirs of complex geological structures is constantly increasing. Thus, it is obvious that reliable mathematical models able to manage such porous media are of particular interest. In this paper, a model of filtration flow to a well located in an anisotropic heterogeneous soil formation, with a piecewise-smooth external contour, is numerically studied. The permeability of soil, as a porous medium, is characterized by a second-rank tensor. Soil heterogeneity is modeled by a power function of one of the coordinates. We consider a particular case with a positive exponent. The solution of the stated boundary problem of well operation in a porous reservoir causes significant mathematical difficulties associated with the complicated form of the basic equation. The formulation of the problem is reduced to a canonical form, which greatly simplifies the solution. To do this, we turn on the auxiliary plane using homeomorphic (affine) transformation. The problem is reduced to a system that includes a Fredholm type integral equation and an integral relation. The results were obtained using the discrete singularity method. It was found that the algorithm convergence is slightly better under weak anisotropy. The influence of anisotropy and heterogeneity of the soil on the production in the case of the productive reservoir with a rectangular boundary was assessed. The anisotropy can greatly change the production rate compared to the rate of a well in an isotropic soil. Permeability tensor components are located on the main diagonal and have a major impact on the rate of the flow in the anisotropic and heterogeneous soil. This method can be used to solve different problems of fluid filtration in an anisotropic porous medium.
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