Paper detail

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

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.