Transport of fluid and electric current through a single fracture

Abstract

One expects the hydraulic and electrical conductivities of rock to be related, since there is an analogy between the differential equations describing each process. Fractures in rock are commonly described by the parallel plate model, where the fracture surfaces are smooth and parallel with a separation or aperture of d. For this model the hydraulic conductivity is proportional to d3, whereas the electrical conductivity is proportional to d. Deviations from the parallel plate model are expected, since real fracture surfaces are rough and in partial contact. Computer simulations of fluid flow and conduction of electricity were performed on simulated fractures composed of rough surfaces generated with a fractal algorithm. The finite difference method was used to calculate the volume flow rate and electric current. This solution was used in the parallel plate model to get an effective hydraulic aperture dh and electric aperture de. The electric and hydraulic apertures are nearly the same when the surfaces are widely separated. However, de is always smaller than dh, with the difference increasing as the fracture closes. Additionally, the local directions of fluid flow and electric current are not the same. Thus contrary to the common assumption, the actual path length of a fluid particle as measured by the tortuosity is different for each process. The results of these simulations are consistent with the “equivalent channel model,” which shows that by introducing the tortuosity of the fluid flow or electric current paths, one can relate the microscopic physics of the transport properties to the macroscopic behaviors described by Darcy's and Ohm's laws.