This paper presents the problem of oil filtration in a fragmented non-homogeneous porous medium layer, its mathematical model, and software for graphical visualization of numerical results for process analysis.

This study introduces a mathematical framework for analyzing unsteady gas filtration in porous media with pressure-dependent porosity variations. The physical process is formulated as a nonlinear parabolic boundary value problem that captures the coupled interaction between pressure evolution and porosity changes during gas production. To solve the equation, a numerical strategy is developed by integrating the Alternating Direction Implicit (ADI) scheme with quasi-linearization iterations, employing finite difference discretization on a two-dimensional spatial grid. Extensive computational experiments are performed to investigate the influence of key reservoir parameters—including porosity coefficient, permeability, gas viscosity, and well production rate—on the spatiotemporal behavior of pressure and porosity during long-term extraction. The results indicate significant porosity variations near the wellbore driven by local pressure depletion, reflecting strong sensitivity of the system to formation properties. The validated numerical model provides valuable quantitative insights for optimizing reservoir management and improving production forecasting in gas field development. Overall, the proposed methodology serves as a practical tool for oil and gas engineers to assess long-term reservoir performance under diverse operational conditions and to design efficient extraction strategies that incorporate pressure-dependent formation property changes.

В данной статье представлена усовершенствованная математическая модель, описывающая фильтрационные процессы в двухслойных динамически взаимосвязанных нефтяных пластах с учетом зависимости гидродинамических параметров от изменения пластового давления. Модель позволяет более точно анализировать физические процессы, происходящие в недрах нефтяных месторождений. Для численного решения задачи предложен алгоритм на основе метода дифференциальной прогонки, обеспечивающий устойчивость и эффективность расчетов. Выполнены вычислительные эксперименты, демонстрирующие снижение коэффициентов проницаемости и пористости вблизи скважин при уменьшении давления. Использование параллельных вычислений позволяет значительно повысить скорость моделирования. Представленные графики наглядно иллюстрируют динамику изменения параметров фильтрации в пространстве и времени.

A mathematical framework is proposed for studying the process of gas displacement in multilayer systems within a gas reservoir, and a model problem is solved. In the considered problem, the boundaries of the section are unknown, and the boundary conditions are rather general; therefore, it is not possible to directly apply all the previously published methods discussed in the article. Since it is impossible to simultaneously capture both boundaries at a network node, the coefficient-smoothing method is applied. Numerical simulations have shown that the most efficient gas extraction occurs when the initial boundaries of the section are equidistant from the production well. A computational experiment has also demonstrated that, with increasing flow rate, the influence of differences in layer permeability becomes more pronounced. Furthermore, when the boundaries of the section are equidistant from the production well, the residual gas content after water breakthrough is higher compared to cases with unequal initial interface positions.