Oil displacement by slug injection: a rigorous justification for the Jouguet principle heuristic

Abstract

In this paper we discuss a one-dimensional model for two-phase Enhanced Oil Recovery (EOR) floods, primarily for the polymer flood. We improve upon the method for the construction of semi-analytical solutions for the oil displacement by a water slug containing dissolved chemicals given in (Pires, Bedrikovetsky and Shapiro, 2006) and later generalized in (Apolin\'{a}rio, de Paula and Pires, 2020), (Apolin\'{a}rio and Pires, 2021). This method utilizes a transformation into the Lagrange coordinates that splits the equations and allows one to solve the chromatographic one-phase problem separately. The solution is then substituted into a scalar hyperbolic conservation law, which is solved using the method of characteristics. However, there is often a gap in the characteristics near the chemical shock front. It was posited to the authors that the Jouguet principle could be used to close that gap. However, no rigorous justification was given for this approach, and as such it remained a heuristic. We analyze the conditions for the appearance of the gap and its properties, and give a proper argumentation for the Jouguet heuristic and its applicability based on the Kru\v{z}kov-type uniqueness theorem for the conservation law system. Additionally, a second splitting technique within the Lagrange coordinates is developed that simplifies this analysis and the construction of characteristics. Keywords: Enhanced oil recovery, Polymer flooding, Slug injection, Conservation laws, Hyperbolic systems of partial differential equations