eISSN 2658–5782

DOI 10.21662/mfs

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2026. Vol. 21. Issue 1, Pp. 11–17
URL: https://multiphasesystems.online/mfs2026.1.003,en
DOI: https://doi.org/10.21662/mfs2026.1.003
Mathematical Modelling of Oil and Water Flow in a Porous Channel with Variable Cross Section
M.A. Zagorovskii1,2 🖂
1LLC “RN – Geology Research Development” PJSC Rosneft Oil Company, Tyumen, Russia
2University of Tyumen, Tyumen, Russia

Abstract

The paper describes process of transformation of a jet flow in a capillary channel with variable cross section using the numerical solution of Navier–Stokes equations. The channel emulates an element of the pore space of a reservoir rock and consists of two pores and a pore neck with a cross-sectional area ratio of 4/1. The influence of the surface tension between the phases and the ratio of the dynamic viscosities of the phases on the occurrence and frequency of the formation of a clear flow is studied. The Volume of Fluid method implemented in the open software package OpenFOAM is used for numerical simulation of a flow with an interfacial boundary. The stages of flow evolution have been studied. Presented simulation results, which demonstrate the formation of a train flow of oil and water. It is shown that the breakup of a stream of a non-wetting phase at the junction of two pores is caused by the action of surface tension forces from the wall layer of the wetting phase. It has been established that at the moment before the jet breakup, a counter current of the wetting phase occurs into the pore neck through areas of the cross-section not covered by displacement and leads to flow instability. Based on the conducted computational experiments, graphs of the dependence of the breakup frequency of the jet flow on the interfacial tension for various phase viscosity ratios have been compiled. The established dependence qualitatively explains the effect of the influence of surface tension and viscosity ratio on the relative phase permeability functions, based on the change in the intensity of the train flow in a porous medium. The conducted direct modelling used to evaluate effect of the capillary number on the amplitude parameter of the empirical interfacial interaction function, which used in calculating the relative phase permeabilites of oil and water based on a simplified cluster hydraulic model.

Received: 5.02.2026
Accepted: 2.03.2026
Published: 31.03.2026
Citation

Zagorovskii MA. Mathematical Modelling of Oil and Water Flow in a Porous Channel with Variable Cross Section. Multiphase Systems. 2026;21(1):11–17 (in Russian).

Keywords

hydrodynamic simulation;
digital rock analysis;
porous space;
two phase flow;
train flow;
OpenFOAM

Article outline

Mathematical modeling of fluid dynamic in the pore space of hydrocarbon reservoir rocks is relevant tool for determining the functions of relative phase permeability (RPP) and analysis the influence of various factors on the nature of multiphase flow and the efficiency of oil extraction. Existing methods for calculating fluid flow at the pore level form the basis of the Digital Core technology. Despite the development of computer technic, direct modeling methods based on the Navier-Stokes equations or their generalizations taking into account interphase interaction do not allow for the fast and massive calculation of RPP functions for a representative volumes due to high resource intensity. Interphase interaction leads to the formation of various flow regimes of oil and water, which can be classified as jet flow and train flow patterns. Taking into account the existing limitations of using direct modeling, it is advisable to use it to parameterize less resource-intensive methods for calculating the filtration properties of porous media.

The purpose of the study is analysis the mechanism of a train flow formation in a capillary channel of variable cross-section, as an element of a porous medium, and qualitatively assessment the frequency of formation depending on the coefficient of surface tension between phases and the ratio of viscosities. The Volume of Fluid method in the OpenFOAM used to model the flow with an interphase boundary. It is assumed that the phases (oil, water) are immiscible, incompressible, Newtonian. There is no mass transfer or chemical reactions between the phases. Dynamic viscosities and densities are constant within each of the phases. Water is the wetting phase. The flow process is isothermal. The shape of the channel cross-section is square, the ratio of the cross-sectional areas of the pore and the channel is 4/1. A uniform computational grid is used, the optimal detailization is estimated by the convergence analysis.

The article describes the stages of the flow evolution. Presented the calculation results, which demonstrate the formation of a train flow of oil and water. It is shown that a split of a jet of a non-wetting phase at the junction of two pores is caused by the action of surface tension forces from the wall layer of the wetting phase and the occurrence of conter-current into the pore neck. As a result of computational experiments, the positive dependence of the jet breakup frequency on the interfacial tension and the viscosity ratio has been established. Received results qualitatively explain the effect of interfacial tension and viscosity ratio on the functions of relative phase permeability, based on changes in the intensity of a train flow in a porous medium. The use of direct modeling to determine the dependences of flow characteristics on various properties of fluids and reservoir rock makes it possible to parameterize the empirical interphase interaction function used in the Shabarov-Stepanov-Zagorovsky express method for effective calculation of relative phase permeability in the oil-water system.

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