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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.