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Mathematical model of oil displacement by water in a plane channel

Unstable displacement of immiscible liquids in a plane channel is a topical research in both theoretical and practical applications. In this paper, we consider a plane channel filled with an incompressible fluid. Over time, another fluid is injected into the channel. The fluids are immiscible. The paper builds a mathematical model of the process of oil displacement by water in a plane channel, which allows further numerical studies and comparison of the results with the obtained experimental data using the example of the Hele-Show cell. The mathematical model for a multiphase, multicomponent flow consists of the Navier-Stokes equations, the equations of conservation of mass, momentum and energy. Modern methods for modeling the dynamics of "viscous fingers“ are based mainly on numerical methods for solving systems of differential equations using the pressure gradient, viscosity and capillary forces as parameters. The influence of these parameters must be determined experimentally. To solve the problem, a quasi-hydrodynamic approach is used, based on the addition of a certain small parameter and allowing one to describe stable schemes with central differences. The complexity of solving such problems lies in the size of the considered models, which in practice have a wide range of applications from micro-scale to orders of one centimeter. A comprehensive study will allow us to evaluate and analyze the entire process as a whole, as well as to establish flow parameters to improve the efficiency of displacement and increase oil recovery, since in the numerical modeling of the process it is easier to create many independent experiments with the same initial data, in contrast to the experimental study.