Abstract
Inaccessible pore volume, also known as dead pore space, is used when simulating enhanced oil recovery by polymer injection. We show that a widely used model for inaccessible pore volume can lead to an ill-posed problem, resulting in unphysical results. By considering shock solutions of the one-dimensional problem, we derive a necessary condition that an inaccessible pore volume model must fulfill in order to obtain well-posed equations. In this derivation, we use the Rankine–Hugoniot jump condition as a selection criterion for acceptable solutions. There are other possible criteria for the one-dimensional problem, in particular δ
δ
-shock solutions, which we also briefly describe, but these are challenging and impractical to use. Based on a heuristic understanding of relative permeability, we subsequently derive two modified models for inaccessible pore volume. The first model follows directly from the modeling assumptions, but it has limited applicability. If the inaccessible pore volume is larger than the irreducible water saturation, then the equations are ill-posed for convex relative permeabilities. A second model is derived by relaxing the assumption of inaccessibility, allowing a limited fraction of the polymer to enter the smallest pores. This second model fulfills our necessary condition for well-posedness for all values of the inaccessible pore volume and any choice of relative permeabilities. Through one- and two-dimensional numerical examples, the different models for inaccessible pore volume are compared. For our second suggested model, the polymer concentration is observed to stay below the maximum injected value, which is not the case for the conventional model. This enables a more stable implementation of the highly nonlinear system, and a reduction in the number of nonlinear iterations is also observed in some cases. As this suggested model is straightforward to implement into existing reservoir simulators and can be used for a wide range of polymer models, it serves as a possible alternative to the conventional model.
δ
-shock solutions, which we also briefly describe, but these are challenging and impractical to use. Based on a heuristic understanding of relative permeability, we subsequently derive two modified models for inaccessible pore volume. The first model follows directly from the modeling assumptions, but it has limited applicability. If the inaccessible pore volume is larger than the irreducible water saturation, then the equations are ill-posed for convex relative permeabilities. A second model is derived by relaxing the assumption of inaccessibility, allowing a limited fraction of the polymer to enter the smallest pores. This second model fulfills our necessary condition for well-posedness for all values of the inaccessible pore volume and any choice of relative permeabilities. Through one- and two-dimensional numerical examples, the different models for inaccessible pore volume are compared. For our second suggested model, the polymer concentration is observed to stay below the maximum injected value, which is not the case for the conventional model. This enables a more stable implementation of the highly nonlinear system, and a reduction in the number of nonlinear iterations is also observed in some cases. As this suggested model is straightforward to implement into existing reservoir simulators and can be used for a wide range of polymer models, it serves as a possible alternative to the conventional model.