In the past decades, several EOR (Enhanced Oil Recovery) techniques have been developed to increase the amount of oil recoverable from a reservoir. Among these techniques, polymer flooding consists in the injection of a solution of water and polymer into the reservoir, and it is performed in order to lower the mobility of injected water. A peculiarity of a water-polymer flow through a porous medium is the velocity enhancement, or hydrodynamic acceleration, that affects the polymer molecules. Experimentally, polymers are observed to travel faster than inert chemical species. Therefore, when simulating numerically a polymer flooding, a velocity enhancement effect has to be incorporated into the governing equations to accurately predict oil recovery. The simple model that introduces a constant enhancement factor, usually implemented in commercial simulators, leads to an ill-posed problem, causing stability issues in the simulations, which further leads to a monotonicity loss in the solution: the polymer is predicted to pile-up at the water-oil interface. While accumulation of polymer at the water front is not necessarily unphysical, it should not be the result of a mathematical ill-posed problem. Hence, an alternative model, employing a saturation dependent factor, has been proposed in the work of Bartelds et al. (G.A. Bartelds, J. Bruining, and J. Molenaar. The modeling of velocity enhancement in polymer flooding. Transport in Porous Media, 26(1):75–88, 1997), and has been extended by Hilden et al. (S.T. Hilden, H.M. Nilsen, and X. Raynaud. Study of the well-posedness of models for the
inaccessible pore volume in polymer flooding. Transport in Porous Media, 114(1):65–86, 2016) to overcome some physical restrictions. In this thesis, these models are re-examined through a thorough analytical study in the one-dimensional case. An analytical solution, resulting in an acceleration of the polymer front, is computed for the well-posed model proposed by Bartelds, while the extended enhancement factor derived by Hilden is shown to lead again to an ill-posed problem. A study of the monotonicity of the numerical schemes reveals that accumulation of polymer at the water front is strictly related to ill-posedness.
Obtaining accurate numerical solutions is not an easy task. Commercial simulators adopt fully implicit schemes to solve the multi-phase flow, and transport of chemical agents is solved through the development and coupling of separate modules. In this case, the use of high-resolution methods for the transport of polymer is not compatible with the first order solver for the underlying flow.
To investigate the interaction of the enhancement model proposed by Bartelds with other physical phenomena, adsorption of polymer onto the reservoir rock is added to the governing equations. The problem maintains the well-posedness property, but because of a raise in equations complexity, only numerical results are presented. The model is then extended to the two-dimensional case and, relying on numerical solutions, similar conclusions as in the one-dimensional case are derived.