Tortuosity and Archie’s Law

Abstract

Despite the popularity of Archie’s Law, parameterizing bulk electrical conductivity as a power-law function of porosity seems to lack support from first principles. In this chapter, we renew the discussion on improving the way to upscale electrical current in porous media. We notice that in a solute diffusion problem (without advection), which is mathematically equivalent to the electrical current problem at the pore scale, bulk diffusivity is upscaled to be a linear function of porosity. The paradox of this upscaling problem results from the difficulty in quantifying the effects of the twisted and windings paths of transport in porous media, and from the ambiguous correlation between tortuosity and porosity. We argue that tortuosity is not well defined at the microscale; it is a macroscopic property. We show that the intuitive definition of tortuosity from an effective length is not a fruitful approach in terms of rigorous quantitative analysis, even for simple tubes with non-uniform cross-sectional area. Moreover, even though empirical relationships between tortuosity and porosity widely exist in the literature, our numerical study of electrical current in 3-D porous media demonstrates that tortuosity does not have to exhibit intrinsic correlation with porosity. Furthermore, we show that as a macroscopic parameter to capture the overall impediment that soil grains have on solute diffusion or electric current, the tortuosity is a tensorial property (i.e., directionally dependent), and cannot be predicted from porosity.