Biased-diffusion calculations of electrical transport in inhomogeneous continuum systems

Simulations in which random walkers move in the presence of a weak uniform bias are shown to provide an efficient method for estimating the electrical properties of disordered continuum systems. This approach is particularly well suited to the description of transport in systems with macroscopic inhomogeneities, e.g., layering normal to the vertical (z) axis. In general, the horizontal conductivity ${\sigma }_{\perp }$(z) in such systems is not subject to the same constraints as in more familiar homogeneous disordered systems. Using biased diffusion, we evaluate ${\sigma }_{\perp }$(z) in granular packings with alternating large and small grain layers and in consolidated networks with (roughly) grain-size horizontal cracks. In both systems the two-dimensional porosity φ(z) varies rapidly with z. Although, in principle, ${\sigma }_{\perp }$(z) may be a nonlocal function of φ(z), it is interesting that our calculations indicate that the conductivity is well described by the simple relation ${\sigma }_{\perp }$(z)∼[φ(z${\left)\right]}^{3/2}$. .AE