On the Validity of the “Thin” and “Thick” Double-Layer Assumptions When Calculating Streaming Currents in Porous Media

We find that the thin double layer assumption, in which the thickness of the electrical diffuse layer is assumed small compared to the radius of curvature of a pore or throat, is valid in a capillary tubes model so long as the capillary radius is >200 times the double layer thickness, while the thick double layer assumption, in which the diffuse layer is assumed to extend across the entire pore or throat, is valid so long as the capillary radius is >6 times smaller than the double layer thickness. At low surface charge density (<10？ ) or high electrolyte concentration (>0.5？M) the validity criteria are less stringent. Our results suggest that the thin double layer assumption is valid in sandstones at low specific surface charge (<10？ ), but may not be valid in sandstones of moderate- to small pore-throat size at higher surface charge if the brine concentration is low (<0.001？M). The thick double layer assumption is likely to be valid in mudstones at low brine concentration (<0.1？M) and surface charge (<10？ ), but at higher surface charge, it is likely to be valid only at low brine concentration (<0.003？M). Consequently, neither assumption may be valid in mudstones saturated with natural brines. 1. Introduction Streaming potentials in porous materials arise from the electrical double layer which forms at solid-fluid interfaces (e.g., [1]). The solid surfaces typically become electrically charged, in which case an excess of countercharge accumulates in the adjacent fluid, in an arrangement called the electrical double layer. The double layer comprises an inner compact (Stern) layer and an outer diffuse (Gouy-Chapman) layer. Most of the countercharge typically resides within the Stern layer; however, if the fluid is induced to flow by an external pressure gradient, then some of the excess charge within the diffuse 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 (e.g., [2–4]). 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 cylindrical coordinates and assuming a symmetric, monovalent electrolyte, the Poisson-Boltzmann equation is given in dimensionless form by [5] where the dimensionless electrical potential is , and dimensionless radial position is (Figure 1) (see Table 1 for the nomenclature). The electrical potential is denoted is temperature, k is Boltzmann’s constant, is the electron charge, is a