ALGORITHM FOR SOLVING THE PROBLEM OF GAS RESERVOIR FILTRATION WITH PRESSURE-DEPENDENT POROSITY

ABSTRACT

This paper presents the development of an algorithm based on numerical modeling for calculating key parameters of gas reservoir development, taking into account pressure-dependent variations in porosity. The study focuses on the formation of an input data array, the design of a user-friendly interface, and the structure of software modules. This approach enhances the reliability of calculations, enables clear visualization of numerical results in various graphical formats, and provides deeper insight into the filtration process.

АННОТАЦИЯ

В статье рассматривается разработка алгоритма на основе численного моделирования для расчёта ключевых параметров эксплуатации газового месторождения с учётом изменения пористости в зависимости от давления. Особое внимание уделено формированию информационного массива исходных данных, созданию удобного пользовательского интерфейса, а также проектированию и структуре программных модулей. Такой подход позволяет повысить надёжность вычислений, обеспечить наглядное представление результатов в различных графических формах и углублённо изучить процессы фильтрации

Keywords: algorithm, filtration, porous media, interface, gas, pressure.

Ключевые слова: алгоритм, фильтрация, пористые среды, интерфейс, газ, давление.

1. Introduction

The automation of boundary-value filtration problem solving in porous media, aimed at identifying the hydrodynamic characteristics of reservoirs, enables more reliable analyses and faster forecasting of filtration dynamics. Consequently, the development and effective implementation of automated systems for studying gas filtration in porous media are of particular importance.

The design of stable computational schemes, algorithms, and software packages for solving multidimensional gas filtration problems in porous media not only enhances the reliability of the obtained numerical results but also accelerates the overall computational process. Software solutions built on such algorithms improve the analysis and forecasting of key indicators in gas field development and ensure the achievement of consistent and dependable results. The inherent complexity of gas filtration processes in porous media necessitates the universality and flexibility of the computational algorithms and software modules being developed [1–6].

2. Materials and methods

The mathematical model describing gas filtration in porous media with pressure-dependent porosity is represented as follows:

             (1)

The initial and boundary conditions of the problem are as follows:

                                                     (2)

                                     (3)

                                  (4)

                                                             (5)

In addressing the boundary-value problem, the continuous domain was discretized into a computational grid and solved using a combination of the quasilinear approach, the finite difference method, and iterative techniques. Based on the developed numerical model, a computational algorithm along with a corresponding software implementation was designed [7, 8].

3. Computational algorithm  

Based on the numerical model developed above, the two-dimensional boundary-value problem of gas filtration in a porous medium is reduced to two finite-difference problems (in the x and y directions) using the alternating direction scheme, and is solved by the sweep method (tridiagonal matrix algorithm) [9, 10]. In each direction, i.e., within a half-time layer, the pressure function is computed iteratively.

The calculation is carried out in two stages at each time layer.

Stage 1. The first stage of the algorithm, aimed at calculating the main parameters of gas field development, involves determining the pressure function values at the time layer l + 0.5 in the x-direction. In this case, the sweep method in the direction of the variable x is used and the calculation is performed as follows:

- the value of the porosity coefficient for the new time layer is calculated;

- the   coefficients of the finite-difference equation (13) are determined;

-   – initial values of the coefficients of the sweep method, determined from the boundary conditions of the left part of the discrete filtration region, that is, from the first equation of the system;

-  and  the values of the coefficients of the sweep method are calculated;   

- the final value of the pressure function is determined by using the right boundary conditions of the discrete filtration region and the third equation of the system    

- the values of the pressure function  are calculated backwards using a simple sweep method.    

Stage 2. The second stage is performed in a similar way along the y-direction at the l + 1 time layer. The iterative process for the pressure function continues until the specified convergence condition is satisfied, if the condition is not met, further iterations of the stage repeated.

The first and second stage algorithms are repeated at each time layer, with each solution serving as the initial condition for the subsequent time step. The block diagram of the numerical solution algorithm is presented in Fig. 1.

4. Results and discussion

The proposed algorithm for solving the gas filtration problem with pressure-dependent porosity significantly accelerates computational experiments on key indicators in the study of non-stationary gas filtration processes in porous media. The use of this system enhances the efficiency of gas reservoir development and enables faster design and preparation of fields for production.

Figure 1. Block diagram of the algorithm for calculating key indicators in gas field development and the module of the sweep method

Conclusion

The developed algorithm and software enable comprehensive studies of both operating and newly commissioned gas wells. The system provides the capability to perform computational experiments in real time and to obtain reliable information about the reservoir. The program allows users to conduct various computational experiments on key indicators when solving gas filtration problems in porous media and, based on the results obtained, to carry out different analyses and forecasts, as well as to pursue scientific investigations.

Acknowledgment

This research was funded by the State Budget of the Republic of Uzbekistan.

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