  The self-potential (SP) method is a passive geophysical method that relies on
the measurement of naturally occurring electrical field. One of the
contributions to the SP signal is the streaming potential, which is of
particular interest in hydrogeophysics as it is directly related to both the
water flow and porous medium properties. The streaming current is generated by
the relative displacement of an excess of electrical charges located in the
electrical double layer surrounding the minerals of the porous media. In this
study, we develop a physically based analytical model to estimate the effective
excess charge density dragged by the water flow under partially saturated
conditions. The proposed model is based on the assumption that the porous media
can be represented by a bundle of tortuous capillary tubes with a fractal pore
size distribution. The excess charge that is effectively dragged by the water
flow is estimated using a flux averaging approach. Under these hypotheses, this
new model describes the effective excess charge density as a function of
saturation and relative permeability while also depending on the chemical and
interface properties, and on petrophysical parameters of the media. The
expression of the model has an analytical single closed-form which is
consistent with a previous model developed from a different approach. The
performance of the proposed model is then tested against previous models and
different sets of laboratory and field data from the literature. The
predictions of the proposed model fits fairly well the experimental data and
shows improvements to estimate the magnitude of the effective excess charge
density over the previous models. This new model proposes a simple and
efficient way to model the streaming current generation for partially saturated
porous media.
