| Abstract |
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 pore-space level may probably be treated using a system of linear, first-order differential equations describing inter-pore probability fluxes. This equation is equivalent to a random impedance network representation. Alternatively, Darcy’s law with spatially variable hydraulic conductivity is equivalent to an RRN. Darcy’s law for the hydraulic conductivity is applicable at sufficiently low pressure ‘head’ in saturated soils, but only for steady-state flow in unsaturated soils. The result given here will have two contributions, one of which is universal to any linear conductance problem, i.e., requires only the applicability of Darcy’s (or Ohm’s) law. The second contribution depends on the actual distribution 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 flow) have been treated within the framework of percolation theory, the theoretical development lags the corresponding development of the linear theory, which is, in principle, on a solid foundation. In practice, calculations of the nonlinear conductivity in relatively (compared with soils) well characterized solid-state systems such as amorphous or impure semiconductors, do not agree with each other or with experiment. In semiconductors, however, experiments do at least appear consistent with each other.In the limit of infinite system size the transport properties of a sufficiently inhomogeneous medium are best calculated through application 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 variability, W (R,x) in the smallest maximal resistance, R of a path spanning a volume x$^{3}$ as well as to find the dependence of the mean value of the conductivity, 〈σ(x)〉. The functional form of the cluster statistics is a product of a power of cluster size, and a scaling function, either exponential or Gaussian, but which, in either case, cuts off cluster sizes at a finite value for any maximal resistance other than 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 individual resistors are exponential functions of random variables (as in stochastic treatments of the unsaturated zone by the McLaughlin group [see Graham and MacLaughlin (1991), or the series of papers by Yeh et al. (1985, 1995), etc.], or as is known for hopping conduction in condensed 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 variables (as 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 power law decay in R to an approximate power, with the value of the power a function of both R and x. |
