29 A journal of the AMERICAN Journal of Engineering, Mechanics and Architecture www. grnjournal.us AMERICAN Journal of Engineering, Mechanics and Architecture Volume 3, Issue 7, 2025 ISSN (E): 2993-2637 Solving a One-Dimensional Oil-Gas Two-Phase Flow Problem in Porous Media Using Numerical Methods Usarqulova Feruza Yusufjon kizi Karshi State Technical University E-mail: feruzausarqulova@gmail.com Abstract Modeling of oil and gas reservoirs on a computational platform requires a precise mathematical model of the system, which describes the fluid flow in porous media based on fundamental physical laws governing their movement within the reservoir. Currently, well-established mathematical models exist that describe the multiphase filtration processes of fluids in porous media. Using the filtration theory equations, this work derives a system of equations that describes the transient filtration processes of oil and gas in a reservoir. The model provides a theoretical basis for simulating complex multiphase flows, which are critical for optimizing reservoir management and production strategies. Keywords: Oil-gas reservoir, porous medium, multiphase filtration, mathematical modeling, fluid flow, transient filtration, filtration theory, numerical simulation. Introduction The efficient development and management of oil and gas reservoirs heavily depend on a thorough understanding of fluid flow behavior within porous geological formations. Due to the complex nature of multiphase fluid interactions in porous media, accurate mathematical modeling becomes essential for predicting reservoir performance and optimizing extraction processes. Computational simulation of these processes requires robust mathematical frameworks that reflect the physical laws governing fluid dynamics, phase interactions, and filtration mechanisms in reservoir rocks. Over the past decades, numerous mathematical models have been proposed to describe multiphase filtration in porous media, contributing significantly to enhanced reservoir characterization and production forecasting. This study focuses on deriving and analyzing a system of governing equations based on filtration theory to simulate the transient multiphase flow of oil and gas in porous reservoirs. The presented model aims to improve the predictive capabilities for reservoir behavior under varying operational conditions, thereby contributing to more efficient resource management. Simulation of oil or gas formations on computational devices necessitates a comprehensive mathematical model of the system, which accurately reflects the behavior of fluids in porous media as determined by fundamental laws governing their flow within the formation. To date, mathematical models describing multiphase fluid filtration in porous environments are well documented. Pioneering contributions to these models have been provided by experts including N.N. Veregin, V.N. Nikolaevskiy, V.M. Shestakov, E.S. Zakirov, B.B. Lapuk, F.B. Abutaliyev, mailto:feruzausarqulova@gmail.com 30 A journal of the AMERICAN Journal of Engineering, Mechanics and Architecture www. grnjournal.us D.F. Fayzullaev, R. Sadullaev, among others. Utilizing the governing equations of filtration theory, we obtain a set of equations that depict the non-stationary filtration phenomena of oil and gas in reservoirs. ( ) ( ) 1 , 1 ( ) , , 1. o o o o g go s o o g g o s g g g s o g i g o cog g o P z m S x x x t PP z z R x x x x x x m R S m S R q q t P P P S S                    − = −                        − + −                    = − + + +     − = + = (1) Here, (l= 0, g)— conductivity of the l-phase; K, r,— relative permeability for the l reservoir -phase; k— absolute permeability; m— porosity of the; μl — viscosity of the l-phase; ρl— density of the l-phase; Rs— oil solubility in gas; z— distance from a certain plane; q, l — volumetric flow rate withdrawn by the well for the l-phase; γ, l — specific weight of the l-phase. For the sake of convenience in notation, we assume that... ( ) ( ) == =−=− n i gig n i oio qxqqxq ii 11 .,  To close the system of equations, the following initial conditions are specified:     == == )()0,(),()0,( ),()0,(),()0,( xSxSxSxS xPxPxPxP H gg H oo H gg H oo (2) and boundary conditions of the form 0,00 =   =   == Lx l x l x P x P (3) or ),(),(0 tPPtPP H lLxl H lxl == == (4) Where L - is the boundary of the filtration domain. In the filtration mathematical model of the oil-gas system, the capillary pressure between oil and gas is determined empirically as a function of gas saturation. Let us assume that oil is incompressible, i.e., ρoil= constant, while gas is compressible, with its density expressed as a function of pressure through the equation of state, i.e., 31 A journal of the AMERICAN Journal of Engineering, Mechanics and Architecture www. grnjournal.us g g P RTZ  = (5) where R - the universal gas constant, T - the temperature Z - the gas compressibility factor. The problem (1)– (5) was solved using a combined finite difference and iterative method. Computational experiments were conducted for various values of reservoir permeability coefficients, oil and gas viscosities, as well as well flow rates under symmetrical configuration. The redistribution of pressure and saturation fields of oil and gas in the reservoir over time was investigated.For determining the relative phase permeabilities, the following dependencies based on experimental results obtained by UZBEKNEFTEGAZ for the oil-gas system were used. 𝐾𝑜 = 0.839379 𝑆𝑔 3 + 1.12471 𝑆𝑔 2 − 1.0396 𝑆𝑔 + 0.182166, 𝐾𝑔 = −3.27135 𝑆𝑔 3 + 7.73761 𝑆𝑔 2 − 6.25468 𝑆𝑔 + 1.73322, The gas solubility in oil is expressed as: 𝑅𝑠 = 11.3 + 0.75 𝑃𝑜; The reservoir is assumed to be horizontal, and the effect of gravitational forces is considered negligible. In all calculations, the following values for reservoir parameters and boundary conditions were used: 𝐿𝑥 = 104𝑚; 𝑚 = 0.1; 𝐻 = 20𝑚; 𝐾𝐻 = 0.1дарси; 𝑃𝐻 = 300атм. ; 𝑃𝑜 = 0.87 г см3; 𝑃𝑔⁄ = 0.82 г см3; 𝑃𝑤⁄ = г см3; ⁄ R = 8.31 Дж (моль К); ⁄ Т=273 К; 𝑃𝑙 0 = 300; 𝑆𝑔 0 = 0.8; 𝑆𝑜 0 = 0.2. Based on the conducted computational experiments, the influence of parameter variations on the distribution of phase pressures and... was identified. Fig. 1. Distribution of gas and oil pressure in the reservoir and the corresponding saturations at 0.1, 4 , 0.01o gK сП сП = = = . Fig. 2. Distribution of gas and oil pressure in the reservoir and the corresponding saturations at 0.2, 4 , 0.01o gK сП сП = = = . Saturations, as well as pressure drops at the wells. 32 A journal of the AMERICAN Journal of Engineering, Mechanics and Architecture www. grnjournal.us Analysis of the results showed that these parameters have a significant impact on the distribution of oil and gas pressures within the reservoir and on the phase saturations. Conclusion The conducted numerical simulations confirm the effectiveness of the proposed mathematical model and computational algorithm for describing the unsteady, multiphase filtration processes occurring during the simultaneous flow of oil and gas in porous media. The results demonstrate that variations in key reservoir parameters such as permeability, phase viscosities, and well production rates significantly affect the distribution of pressures and saturations within the reservoir. Furthermore, the model allows for the analysis of pressure drops at production wells and dynamic changes in fluid distribution over time. These findings validate the applicability of the developed approach for evaluating filtration behavior in oil-gas systems and can be effectively utilized in the design, simulation, and optimization of oil and gas field development projects. LIST OF REFERENCES 1. Zakirov, S.N., Lapuk, B.B. Design and Development of Gas Fields. Nedra Publishing, Moscow, 1974, 376 pages. 2. Abutaliyev, F.B., Khadjibayev, N.N., Izmailov, I.I., Umarov, U. Application of Numerical Methods and Computers in Hydrogeology. Tashkent: “Fan” Publishing House, 1976, 160 pages. 3. Zakirov, S.N., Lapuk, B.B. Design and Development of Gas Fields. Nedra Publishing House, Moscow, 1974. 4. Neymatov, A., Nazirova, E.Sh. Numerical Modeling of Gas Filtration in a Porous Medium. International Academic Bulletin, 2016, No. 1(13), pp. 52–56. 5. Ravshanov, N., Nazirova, E. Numerical Simulation of Filtration Processes of Strongly Polluted Oil in a Porous Medium. Ponte, 2018, Vol. 74, No. 11/1, pp. 107–116. 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