On a three-layer Hele-Shaw model of enhanced oil recovery with a linear viscous profile

TL;DR

This paper develops an exact analytical solution for a stability eigenvalue problem in a three-layer Hele-Shaw model with a linear viscous profile, using a transformation to Kummer's equation, and explores its asymptotic limits.

Contribution

It introduces a novel approach to solve a non-standard eigenvalue problem in Hele-Shaw flow by transforming it into Kummer's equation and analyzing its solutions.

Findings

Exact solution constructed via Kummer's equation

Dispersion relation derived from existence criteria

Solution converges to known limits in asymptotic cases

Abstract

We present a non-standard eigenvalue problem that arises in the linear stability of a three-layer Hele-Shaw model of enhanced oil recovery. A nonlinear transformation is introduced which allows reformulation of the non-standard eigenvalue problem as a boundary value problem for Kummer's equation when the viscous profile of the middle layer is linear. Using the existing body of works on Kummer's equation, we construct an exact solution of the eigenvalue problem and provide the dispersion relation implicitly through the existence criterion for the non-trivial solution. We also discuss the convergence of the series solution. It is shown that this solution reduces to the physically relevant solutions in two asymptotic limits: (i) when the linear viscous profile approaches a constant viscous profile; or (ii) when the length of the middle layer approaches zero.