Electrohydrodynamic channeling effects in narrow fractures and pores

In low-permeability rock, fluid and mineral transport occur in pores and fracture apertures at the scale of micrometers and below. At this scale, the presence of surface charge, and a resultant electrical double layer, may considerably alter transport properties. However, due to the inherent nonlinearity of the governing equations, numerical and theoretical studies of the coupling between electric double layers and flow have mostly been limited to two-dimensional or axisymmetric geometries. Here, we present comprehensive three-dimensional simulations of electrohydrodynamic flow in an idealized fracture geometry consisting of a sinusoidally undulated bottom surface and a flat top surface. We investigate the effects of varying the amplitude and the Debye length (relative to the fracture aperture) and quantify their impact on flow channeling. The results indicate that channeling can be significantly increased in the plane of flow. Local flow in the narrow regions can be slowed down by up to $5\%$ compared to the same geometry without charge, for the highest amplitude considered. This indicates that electrohydrodynamics may have consequences for transport phenomena and surface growth in geophysical systems.

Figure 1

Schematic setup of the model system. The inlet, charged-surface, and outlet areas (see text) are indicated. The $x$ direction is periodic. Note that the dimensions are not to scale.

Figure 2

Comparison of 2D simulations of two channel lengths to theoretical predictions. The blue points, where the resolution $\delta x$ has a subscript $ln$ correspond to simulations with channel length $160R$, and the red points, with a subscript $sh$, to simulations with channel length $40R$. Both channels have a width of $6R$. The solid lines denote the analytical results from Eqs. (17) (a) and (20) (b). In both cases the center half of the channel has a surface charge. (a) Plot of the electric viscosity as a function of $\kappa a$. (b) The streaming potential in units of the thermal voltage.

Figure 3

Comparison of 3D versus 2D channel flow simulations. The 2D channel is the short channel described in Fig. 2, and the 3D channel has the same size in the streamwise and vertical dimensions, while the additional horizontal dimension is periodic with length $R$. The analytical predictions shown as solid lines are the same as in Fig. 2. (a) Electric viscosity as a function of $\kappa a$. (b) The streaming potential in units of thermal voltage as a function of $\kappa a$.

Figure 4

Comparison of electric viscosity and streaming potential in 3D channels with varying undulation amplitude. The channels, shown schematically in Fig. 1, have dimensions given in the text, and $A$ is given in the legend. The solid lines are theoretical predictions and the same as in Figs. 2 and 3. (a) The electric viscosity plotted as a function of $\kappa a$. (b) The streaming potential in units of thermal voltage plotted as a function of $\kappa a$.

Figure 5

Simulations in an undulated 3D channel without any electric effects included, i.e., $\kappa a=0$. (a) Flow rate in undulated channels as a function of undulation amplitude $A$, relative to the flat channel $A=0$. (b) The absolute asymmetry of the flow in the channel as a function of amplitude $A$. The linear dependence of ${\mathrm{\Theta }}_x$ on $A$ is in good agreement with the theoretical prediction ${\mathrm{\Theta }}_x^{\mathrm{theor}}$ derived in Appendix pp3.

Figure 6

Relative asymmetries are plotted as a function of $\kappa a$. (a) The relative asymmetry in the plane of flow, ${\theta }_x$. (b) The relative asymmetry normal to the plane of flow, ${\theta }_y$.

Figure 7

The flow field at $\kappa a=3$ divided by the flow field for the flow without any electric effects, probed as described in the text. (a) to (d) show increasing amplitude.