Analysis of Non-equilibrium Capillary Pressure-Saturation Relation using Direct Numerical Simulations with Volume-Of-Fluid (VOF) Method

In traditional two-phase flow models of porous media, capillary pressure (P$_{\mathrm{c}})$ and saturation (S$^{\mathrm{w}})$ are hysteretically related, i.e. different P$_{\mathrm{c}}$-S$^{\mathrm{w}}$ curves are obtained for drainage and imbibition. Extended two-phase flow theories hypothesize that inclusion of specific interfacial area (a$^{\mathrm{wn}})$ will result in a unique relation between capillary pressure, saturation and interfacial area (P$_{\mathrm{c}}$--S$^{\mathrm{w}}$--a$^{\mathrm{wn}})$. Several studies have confirmed the reduction of hysteresis in the P$_{\mathrm{c}}$--S$^{\mathrm{w}}$--a$^{\mathrm{wn}}$ relation under quasi-static conditions. However, the uniqueness of the P$_{\mathrm{c}}$--S$^{\mathrm{w}}$--a$^{\mathrm{wn}}$ relation under transient conditions is not clear. We investigate role of specific interfacial area under dynamic conditions using pore-scale direct numerical simulations (DNS) on two micromodels with volume-of-fluid (VOF) method. From the DNS data, Pc-S$^{\mathrm{w}}$ curves are estimated for drainage and imbibition; validity of different macroscopic capillary pressure definitions is evaluated. The quasi-static and dynamic P$_{\mathrm{c}}$--S$^{\mathrm{w}}$--a$^{\mathrm{wn}}$ relations are examined for uniqueness.