Improved permeability bounds for highly polydisperse materials

The influence of polydispersity on the flow of fluid through a medium composed of solid spherical inclusions of many different sizes is studied. Focus is on the case of a medium composed of impenetrable, i.e., nonoverlapping inclusions, which is a good model for flow, e.g., through sand, and also on the case of randomly overlapping inclusions, which provides a well-studied model for flow through porous media. Bounds for the permeability are derived using trial functions of the kind developed by Doi [J. Phys. Soc. Jpn. 40, 507 (1976)] and by Weissberg and Prager [Phys. Fluids 5, 1390 (1962); 13, 2958 (1970)]. It is shown that the bounds for permeability are relatively insensitive to dispersivity of the inclusions or grains composing a material when they are plotted against the proper scaling variable. The porosity is a good scaling variable for materials composed of randomly overlapping inclusions. The mean number of inclusions or grains ρ〈a〉 is shown to be a good scaling variable for permeability bounds on systems of nonoverlapping inclusions. It is shown that the low-density limits of the standard Doi bounds and Weissberg–Prager bounds fail to reproduce the radius-averaged solution of Stokes for flow around a single spherical grain. Also, for extremely disperse media these bounds may become trivial. The improved bounds correct these difficulties.