A Fractal Model for Effective Excess Charge Density in Variably Saturated Fractured Rocks

Abstract

Estimating hydraulic properties and monitoring water flow in fractured rocks using self-potential observations essentially relies on our ability to model streaming potential. One of the most promising approaches for modeling electrokinetic couplings is based on the macroscopic quantification of the excess charge which is effectively transported by pore water flow. In this study, we derive a fractal model to predict the effective excess charge density for fully and partially water saturated fractured media. Fractures are conceptualized as parallel plates with a fractal pattern described by the Sierpinski carpet. From the calculation of the excess charge in a single fracture and a flux averaging upscaling procedure, we obtain closed-form expressions for the effective excess charge density. This new analytical model explicitly depends on the fracture water ionic concentration, interface properties, water saturation, porosity and permeability. Model predictions under saturated conditions are compared to published data measured in laboratory during hydraulic fracturing. The model development also shows the independence of the excess charge from the pore shapes: when expressed in terms of hydrological parameters, one can find an expression identical to recently published models for sedimentary porous media using capillary tubes. These results extend the validity of the existing models to fractured rocks and highlight the importance of hydraulic parameters for an accurate modeling of electrokinetic couplings.