Upscaling Capillary Pressure‐Saturation Curves in Heterogeneous Porous Media

Abstract

Numerical models of flow in the vadose zone require capillary pressure‐saturation relationships for the grid blocks discretizing the medium. However, the scale of these grid blocks is generally much larger than the sample scale at which capillary pressure‐saturation relationships are measured. This study investigates the upscaling of capillary curves under static conditions of capillary‐gravity equilibrium. The averaging regions are horizontal layers, parallel to the phreatic surface, as in one‐dimensional models of infiltration. At the point or sample scale it is assumed that all capillary curves can be normalized using the Leverett (1941∥ approach to a common dimensionless form described by the Corey‐Brooks model. Static water saturation at any point in the medium is then expressed as a parametric function of three regionalized variables, permeability, porosity, and irreducible water saturation. Water saturations at the layer scale are obtained by spatial averaging of point‐scale values. With certain assumptions it is shown that the nondimensional relationship between capillary pressure and saturation at the layer scale is the same as at the sample scale. In general, the nondimensional scaling of layer‐averaged capillary pressure requires exhaustive knowledge of the spatial distribution of the permeability‐porosity ratio within the averaging region. However, when permeability and porosity are jointly lognormal, this spatial distribution may be approximated by a lognormal model requiring only two parameters for each averaging region. The developments are illustrated using published field data from the Apache Leap tuff site. Exact and estimated upscaled capillary curves are compared in simulated coregionalized fields of permeability, porosity, and irreducible water saturations. Results show that the upscaled capillary pressure‐saturation curves accurately reproduce the vertical saturation profile obtained by exhaustive spatial averaging of point‐scale water saturations.