Tortuosity: a guide through the maze

Despite its widespread use in petrophysics, tortuosity remains a poorly understood concept. Tortuosity can have various meanings when used by physicists, engineers or geologists to describe different transport processes taking place in a porous material. Values for geometrical, electrical, diffusional and hydraulic tortuosity are in general different from one another. Electrical tortuosity is defined in terms of conductivity whereas hydraulic tortuosity is usually defined geometrically, and diffusional tortuosity is typically computed from temporal changes in concentration. Abetter approach may be to define tortuosity in terms of the underlying flux of material or electrical current with respect to the forces which drive this flow. Unsteady transport processes, including diffusion, can be described only by a population of tortuosities corresponding to the different how paths taken by particles traversing the medium. In measurements of steady how (e.g., those normally used to obtain resistivity or permeability), information about particle travel times is lost, and so the multiple values of tortuosity are homogenised. It can be shown that the maximum amount of information about pore structure is embedded in transport processes that combine advective and diffusive elements. Most existing formulations of tortuosity are model-dependent, and cannot be correlated with independently measurable pore-structure properties. Nevertheless, tortuosity underpins the rigorous relationships between transport processes in rocks, and ties them with the underlying geometry and topology of their pore spaces. Tortuosity can be redefined in terms of the energetic efficiency of a flow process. The efficiency is related to the rate of entropy dissipation (or isothermally, energy dissipation) with respect to a simple, non-tortuous model medium using the postulates of non-equilibrium thermodynamics. Through Onsager's reciprocity relation for coupled hows it is possible to inter-relate efficiency for pairs of transport processes, and so go some way towards unifying tortuosity measures. In this way we can approach the goal of predicting the Value of one transport parameter from measurements of another.