Towards weakly hyperbolic models for inaccessible pore volume in polymer flooding
Résumé
In the simulations of enhanced oil recovery by polymer flooding, it is widely acknowledged that the inaccessible pore volume (IPV) must be properly accounted for. The effect of IPV is to make the interstitial velocity of polymer molecules higher than that of water molecules. This acceleration is traditionally modeled by a velocity enhancement factor. The difficulty, however, lies in finding an adequate closure law for this factor that preserves well-posedness of the overall flow model. Previous works by Bartelds et al. (Transp. Porous Med. 26, 75-88, 1997) and Hilden et al. (Transp. Porous Med. 114, 65-86, 2016) have demonstrated that illposedness occurs for the popular constant enhancement factor and proposed some alternative physically motivated IPV laws that partially addressed the question of stability. This paper is aimed at bringing new mathematical insights into the design of IPV laws that ensure weak hyperbolicity for a Buckley-Leverett type two-phase polymer flow model. To begin with, we show that by switching to Lagrangian coordinates, it is possible to derive several practical and meaningful sufficient conditions for local weak hyperbolicity. Next, special solutions to some of these sufficient conditions are shown to coincide with well-known IPV laws in the literature and then investigated more thoroughly. Finally, by patching together these piecewise sufficient conditions, we are in a position to develop original IPV laws that guarantee global weak hyperbolicity for the flow model, at least under mild restrictions.
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