Validity of the "thin" and "thick" double-layer assumptions to model streaming currents in porous media

Measurements of the streaming potential component of the spontaneous potential have been used to characterize groundwater flow and subsurface hydraulic properties in numerous studies. Streaming potentials in porous media arise from the electrical double layer which forms at solid-fluid interfaces. The solid surfaces typically become electrically charged, in which case an excess of counter-charge accumulates in the adjacent fluid. If the fluid is induced to flow by an external pressure gradient, then some of the excess charge within the diffuse part of the double layer is transported with the flow, giving rise to a streaming current. Divergence of the streaming current density establishes an electrical potential, termed the streaming potential. Within the diffuse layer, the Poisson-Boltzmann equation is typically used to describe the variation in electrical potential with distance from the solid surface. In many subsurface settings, it is reasonable to assume that the thickness of the diffuse layer is small compared to the pore radius. This is the so-called 'thin double layer assumption', which has been invoked by numerous authors to model streaming potentials in porous media. However, a number of recent papers have proposed a different approach, in which the thickness of the diffuse layer is assumed to be large compared to the pore radius. This is the so-called 'thick double layer assumption' in which the excess charge density within the pore is assumed to be constant and independent of distance from the solid surface. The advantage of both the 'thin' and 'thick' double layer assumptions is that calculation of the streaming current is greatly simplified. However, perhaps surprisingly, the conditions for which these assumptions are valid have not been determined quantitatively, yet they have a significant impact on the interpretation of streaming potential measurements in natural systems. We use a simple capillary tubes to model investigate the validity of the thin and thick double layer assumptions. We find that the thin double layer assumption is valid so long as the capillary radius is >200 times the thickness of the double layer, while the thick double layer assumption is valid so long as the capillary radius is >6 times smaller than the thickness of the double layer. At low surface charge density (<10mCm^-3) or high electrolyte concentration (>0.5M) the validity criteria are less stringent: the thin double layer assumption is valid so long as the capillary radius is >25 times the thickness of the double layer, while the thick double layer assumption is valid so long as the capillary radius is less than the thickness of the double layer. Our results suggest that the thin double layer assumption is valid in sandstones at low specific surface charge (<10mCm^3), but may not be valid in sandstones of moderate to small pore-throat size at higher surface charge, if the salt concentration in the groundwater is low (<0.001M). The thick double layer assumption is valid in mudstones at moderate salt concentration (<0.1M) and surface charge (<10mCm^3) but, at higher surface charge, it is likely to be valid only at very low salt concentration (<0.003M). Mudstones saturated with saline groundwater will often lie in the range where neither the thin nor thick double layer assumption is valid.