Relationship between streaming potential and water saturation during drainage and imbibition in sandstones

The rock pore space in many subsurface settings is saturated with water and one or more immiscible fluid phases; examples include NAPLs in contaminated aquifers, supercritical CO₂ during sequestration in deep saline aquifers, the vadose zone and hydrocarbon reservoirs. To interpret spontaneous potential measurements for groundwater flow and hydraulic properties in these settings requires an understanding of the saturation dependence of the streaming potential. Vinogradov and Jackson [2011] reported measurements of the streaming potential during drainage and, for the first time, imbibition in two different sandstone plugs saturated with water and undecane. However, they reported effective values of the streaming potential coupling coefficient (C) at partial saturation (S_w), because S_w in the plugs was not uniform. The aim of this study is to determine the true value of C as a function of S_w in both samples. We use a three-step approach in which hydraulic and electrical parameters are determined using numerical simulation and Nelder-Mead simplex unconstrained optimisation or active-set constrained optimisation. In the first step, we determine the relative permeability and capillary pressure, assuming these are simple exponential functions of S_w (Corey-type) and using an objective function which is a weighted average of the measured (i) pressure drop across the plug, (ii) total fluid flow rate and (iii) water flow rate. In the second, we determine the saturation dependence of the electrical conductivity, assuming Archie's Law and using the measured conductivity of the plug as the objective function. In the final step, we determine the saturation dependence of the streaming potential, using the measured streaming potential across the plug as the objective function. We obtain a good match between simulated and measured values of C, and find that it (i) exhibits hysteresis, (ii) can vary non-monotonically with saturation, (iii) is non-zero when undecane flows at the irreducible water saturation, and (iv) can exceed the value observed at S_w = 1. We constrain the excess charge transported by the flow of water, given by Q_w(S_w)=(C(S_w)σ(S_w)μ_w)/(k(S_w) (1) where σ is the saturated rock electrical conductivity, k_w is the water permeability and σ_w is the viscosity) to increase monotonically with S_w during drainage, and decrease monotonically during imbibition, but it is not given by 1/S_w as suggested in previous studies. Instead, the variation of Q_w with S_w depends on the pore-level distribution of the fluids, which is controlled by rock texture and wettability and is different for the two sandstone samples investigated. Moreover, C(S_w) can exhibit complex, non-monotonic behaviour, depending upon the (monotonic) saturation dependence of the other parameters in (1). A reasonable model for Q_w is Q_w(S_w)=pS_w^-q (2) where p and q are determined from experimental data, obtained for a given sample/wettability/fluids/displacement. A similar approach is used to apply Archie's Law for electrical conductivity. Thus, the saturation dependence of the streaming potential is best determined by fitting models for k_w, σ and Q_w to experimental data.