Dependence of the conductivity of a porous medium on electrolyte conductivity

For an arbitrary geometry of insulating, but charged, objects immersed in an electrolyte for which diffusion currents are important, the mathematical problem of the dc electrical conductivity can be mapped onto that of an ordinary conduction problem without diffusion currents but with a conductive surface layer. As a result, using variational arguments we can prove two general theorems which hold irrespective of the geometry of the porous medium: (a) At high salinities, so that the conductivity of the pore fluid, ${\ensuremath{\sigma}}_{f}$, is large, the conductivity of the system as a whole, ${\ensuremath{\sigma}}_{\mathrm{eff}}$, is a linear function of ${\ensuremath{\sigma}}_{f}$, with a slope of 1/F and with an offset proportional to 1/\ensuremath{\Lambda}. (b) For lower values of salinity, ${\ensuremath{\sigma}}_{\mathrm{eff}}$ as a function of ${\ensuremath{\sigma}}_{f}$ is convex-up as long as the conductivity within the double-layer region is independent of the salinity of the pore fluid. The parameters F and \ensuremath{\Lambda} introduced previously [D. L. Johnson, J. Koplik, and L. M. Schwartz, Phys. Rev. Lett. 57, 2564 (1986); D. L. Johnson, J. Koplik, and R. Dashen, J. Fluid Mech. 176, 379 (1987)] are hereby shown to be relevant to the electrolyte problem. An illustration of an ordered suspension is given to show how to implement these ideas.