Upscaling unsaturated flows in vertically heterogeneous porous layers

Characterising interfacial and unsaturated flows in heterogeneous porous layers is of both fundamental and practical interest. Under the assumption of vertical gravitational-capillary equilibrium, we present a theoretical model to describe one-dimensional flows in a porous layer with vertical variations in average pore size, porosity, intrinsic permeability, and capillary pressure jump between invading and displaced fluids. The model leads to asymptotic solutions for the saturation distribution and outer envelope of the invading fluid, and for the background pressure drop across the porous layer. Eight dimensionless parameters are recognised after appropriate non-dimensionalisation of the governing equations, the influence of which are demonstrated through a series of example calculations. In particular, four asymptotic regimes are identified, representing unconfined sharp-interface flows, confined sharp-interface flows, unconfined unsaturated flows, and confined unsaturated flows. Finally, in the context of flow upscaling, analytical solutions are derived for the effective relative permeability curves on the basis of exact solutions of the saturation field and interface shape, shedding light on the subtle influence of competition between injection/pumping and gravitational forces, wetting and capillary effects, viscosity contrast between the invading and displaced fluids, and vertical heterogeneity of the porous layer.