Well-posedness of the Single-cell Transport Problem for Two-phase Flow with Polymer

A sequential splitting of the pressure and transport equations applied to a compressible two-phase flow with polymer leads to a considerable speed-up of the simulation, see [1]. To avoid excessive limitation on the size of the time step, the transport equations are solved implicitly. By using an iterative transport solver, in which the transport equations are solved cell-by-cell from upstream we can further decrease the computation time significantly. Such approach requires a robust single-cell transport solver. The single-cell problem consists of computing the saturation and the polymer concentration in a cell, given the total flux, the saturation, and the polymer concentration in the neighbouring cells. We derive a splitting and a discretization of the mass-conservation equations for which the single-cell problem is always well defined - for any time step size. We are now able to handle the compressible case, which requires a careful choice of the pressure equation and the segregation case, which requires to use a mixed upwind/downwind evaluation of the polymer concentration in the computation of the numerical flux.

[1] Lie K.A, Nilsen H., Rasmussen A, Raynaud X. Fast simulation of polymer injection in heavy-oil reservoirs based on topological sorting and sequential splitting SPE J. 2014