Bundle-of-tubes model for calculating dynamic effects in the capillary-pressure-saturation relationship

Traditional two-phase flow models use an algebraic relationship between capillary pressure and saturation. This relationship is based on measurements made under static conditions. However, this static relationship is then used to model dynamic conditions, and evidence suggests that the assumption of equilibrium between capillary pressure and saturation may not be be justified. Extended capillary pressure-saturation relationships have been proposed that include an additional term accounting for dynamic effects. In the present work we study some of the underlying pore-scale physical mechanisms that give rise to this so-called dynamic effect. The study is carried out with the aid of a simple bundle-of-tubes model wherein the pore space of a porous medium is represented by a set of parallel tubes. We perform virtual two-phase flow experiments in which a wetting fluid is displaced by a non-wetting fluid. The dynamics of fluid-fluid interfaces are taken into account. From these experiments, we extract information about the overall system dynamics, and determine coefficients that are relevant to the dynamic capillary pressure description. We find dynamic coefficients in the range of 10²-10³ kg m^-1 s^-1, which is in the lower range of experimental observations. We then analyze certain behavior of the system in terms of dimensionless groups, and we observe scale dependency in the dynamic coefficient. Based on these results, we then speculate about possible scale effects and the significance of the dynamic term.