Deviation of linear relation between streaming potential and pore fluid pressure difference in granular material at relatively high Reynolds numbers

We conducted streaming potential measurements on the packing of glass beads, and investigated the deviation of streaming potential from the Helmholtz-Smoluchowski (H-S) equation. The H-S equation was originally derived on the assumption of laminar flows. Studies using a capillary have shown that the H-S equation is valid for turbulent flows in so far as the viscous sublayer is thicker than the electrical double layer and the entrance effect is negligible. Although the streaming potential in porous media has been reported to deviate from the H-S equation for turbulent flows, its mechanism is still poorly understood. We measured the fluid flux and the streaming potential as a function of the pore fluid pressure difference. The fluid flux begins to deviate from Darcy's law at Reynolds number >3, and the streaming potential begins to deviate from the linear relation at larger Reynolds numbers. When the flow is fast, the fluid inertia separates the boundary layer from the solid surface and induces the counter flows. The fluid in the counter-flow region is separated from the circulating fluid, and ions there cannot contribute to the convection current. We think that this results in a lower streaming potential than expected from the H-S equation.