Correct mathematical models of joint filtration of two immiscible viscous liquids

TL;DR

This paper emphasizes the importance of precise mathematical models for joint filtration of two immiscible viscous liquids, crucial for oil displacement simulations, and discusses microscopic modeling and homogenization techniques.

Contribution

It highlights the need for exact microscopic models over macroscopic ones and discusses homogenization methods for practical numerical implementation.

Findings

Existing macroscopic models lack microscopic detail.

Homogenization methods can bridge microscopic models and large-scale simulations.

Precise modeling is vital for efficient oil displacement simulation.

Abstract

Mathematical models of joint filtration of liquids are the main part of mathema-tical models of oil displacement by suspension. Since mining is a very important and urgent economic task, exact modeling of joint filtration of two different fluids is also an urgent economic task. For example, mathematical models of oil displacement by suspension are needed to create a hydrodynamic simulator of oil by suspension. All the existing simulators are based on the macroscopic Buckley-Leverett model, which does not distinguish between the free boundary separating liquids, and the details of liquid interaction. All these fundamental processes occur at a microscopic level corresponding to the average size of pores, while all proposed macroscopic models operate on completely different orders of magnitude and do not distinguish between free boundaries or the characteristics of fluid interactions and are simply a set of axioms. Exact modelling involves describing the process using equations from classical Newtonian mechanics of continuous media at the microscopic level (average size of tens of micrometers) followed by homogenization. The only obstacle to using such models is that, in areas measuring hundreds of metres, any numerical implementation would take years. A solution to this problem (the homogenisation method) was proposed in the papers of J. Keller and E. S\'anchez-Palencia.