The capillary pressure curves from upscaling interfacial and unsaturated flows in porous layers with vertical heterogeneity

We provide the capillary pressure curves $p_c\left(\overline{s}\right)$ as a function of the effective saturation $\overline{s}$ based on the theoretical framework of upscaling unsaturated flows in vertically heterogeneous porous layers proposed recently in [1]. Based on the assumption of vertical gravitational-capillary equilibrium, the saturation distribution and profile shape of the invading fluid can be obtained by solving a nonlinear integral-differential equation. The capillary pressure curves $p_c\left(\overline{s}\right)$ can then be constructed by systematically varying the injection rate. Together with the relative permeability curves ${\overline{k}}_{rn}\left(\overline{s}\right)$ that are already obtained in [1], one can now provide quick estimates on the overall behaviours of interfacial and unsaturated flows in vertically-heterogeneous porous layers.