UPSCALING IN SUBSURFACE TRANSPORT USING CLUSTER STATISTICS OF PERCOLATION

Transport/flow problems in soils have been treated in random resistor network representations (RRN's). Two lines of argument can be used to justify such a representation. Solute transport at the port-space leve l may probably be treated using a system of linear, first-order differ ential equations describing inter-pore probability fluxes. This equati on is equivalent to a random impedance network representation. Alterna tively, Darcy's law with spatially variable hydraulic conductivity is equivalent to an RRN. Darcy's law for the hydraulic conductivity is ap plicable at sufficiently low pressure 'head' in saturated soils, but o nly for steady-state flow in unsaturated soils. The result given here will have two contributions, one of which is universal to any linear c onductance problem, i.e., requires only the applicability of Darcy's ( or Ohm's) law. The second contribution depends on the actual distribut ion of linear conductances appropriate. Although nonlinear effects in RRN's (including changes in resistance values resulting from current, analogous to changes in matric potential resulting from how) have been treated within the framework of percolation theory, the theoretical d evelopment lags the corresponding development of the linear theory, wh ich is, in principle, on a solid foundation. In practice, calculations of the nonlinear conductivity in relatively (compared with soils) wel l characterized solid-state systems such as amorphous or impure semico nductors, do not agree with each other or with experiment. In semicond uctors, however, experiments do at least appear consistent with each o ther. In the limit of infinite system size the transport properties of a sufficiently inhomogeneous medium are best calculated through appli cation of 'critical rate' analysis with the system resistivity related to the critical (percolating) resistance value, R-c. Here well-known cluster statistics of percolation theory are used to derive the variab ility, W (R, x) in the smallest maximal resistance, R, of a path spann ing a volume x(3) as well as to find the dependence of the mean value of the conductivity, (sigma(x)). The functional form of the cluster st atistics is a product of a power of cluster size, and a scaling functi on, either exponential or Gaussian, but which, in either case, cuts of f cluster sizes at a finite value for any maximal resistance other tha n R-c. Either form leads to a maximum in W (R, x) at R = R-c. When the exponential form of the cluster statistics is used, and when individu al resistors are exponential functions of random variables las in stoc hastic treatments of the unsaturated zone by the McLaughlin group [see Graham and MacLaughlin (1991), or the series of papers by Yeh et nl. (1985, 1995), etc.], or as is known for hopping conduction in condense d matter physics), then W(R,x) has a power law decay in R/R-c, (or R-c /R), the power being an increasing function of x. If the statistics of the individual resistors are given by power law functions of random v ariables las in Poiseiulle's Law), then an exponential decay in R for W (R, x) is obtained with decay constant an increasing function of x. Use, instead, of the Gaussian cluster statistics alters the case of po wer law decay in R to an approximate power, with the value of the powe r a function of both R and x.