An aid to making subsurface hydrology predictive

Especially as concerns unsaturated media, the current state of subsurface modeling is to use standard partial differential equations with constitutive relationships chosen to minimize interference with smooth numerical recipes. Boundary conditions are handled logically, but heterogeneity is not. The constitutive relationships contain parameters, which have no known relationship to geology or soil conditions. Simple predictions of the hydraulic conductivity in (nominally) homogeneous media using these relationships typically fail by orders of magnitude, so in heterogeneous applications researchers conduct sensitivity studies to see, which parameter has been misunderstood, and make unjustifiable inferences regarding the relationships between the medium and the parameter. Alternatively, one can calculate the constitutive relationships from first principles. Here I show how to obtain expressions for the unsaturated hydraulic conductivity, the air permeability, and the electrical conductivity. Comparison with experiment demonstrates not only that the theory is predictive, but, also that the model (truncated random fractals) is sound. The difficulty with these expressions is that they are based on percolation theory. Since near saturation the air phase becomes discontinuous, and for dry soils the water phase does, the physics of these phase transitions represents potential trouble for numerical modeling. But to put the demands of the numerical modeling first is what I would call "upside down" science.