Statistical Hydraulic Conductivity Models and Scaling of Capillary Phenomena in Porous Media

The 1956 Miller and Miller scaling theory predicts that the capillary pressure h (θ) and conductivity K (θ) at liquid content θ in a given member of a family of similar media are related to the corresponding functions h o (θ) and K o (θ) in a “reference” member by h (θ) = (1/α) h o (θ) and K (θ) = α 2 K o (θ), where α is a scale factor related to pore size. However, in field soils, hydraulic properties are better characterized by the set of scaling relations h ( S ) = (1/α h ) h o ( S ) and K ( S ) = (α K ) 2 K o ( S ), where S is the liquid‐filled fraction of the effective porosity, and α h and α k are scale factors that are generally different from each other. Both results are shown to be theoretically possible through dimensional inspection of statistical hydraulic conductivity models. The models predict the scale factor relation α K = (α h )(α P ) n , where α P is the effective porosity ratio P/P o and n is a model‐dependent constant. According to this result, α h = α K if α P = 1 as in Miller and Miller theory, but α h ≠ α K whenever α P ≠ 1.