Motion of an oil droplet through a capillary with charged surfaces

Abstract

A model developed by Wilmott et al. (J. Fluid Mech., vol. 841, 2018, pp. 310–350) for the advance of a charged oil droplet along a charged capillary pore is considered. The oil droplet is surrounded by an aqueous phase filling the pore, and the model considers a uniformly curved capillary static droplet front plus an aqueous thin film separating the body of the oil droplet from the capillary wall, with these two regions being joined by a transition region. The methodology follows a classical asymptotic approach proposed by Bretherton (J. Fluid Mech., vol. 10, 1961, pp. 166–188) but incorporates additional electro-osmotic effects (specifically an electro-osmotic disjoining tension) due to the charged surfaces. A number of dimensionless parameters control the model’s behaviour, of which the most important is denoted $\unicode[STIX]{x1D712}^{\prime }$ and represents the ratio between the ‘nominal’ thickness of the aqueous film (as determined neglecting any electrostatic effects) and the Debye length within the film, which is sensitive to ion concentrations and hence to salinity. When $\unicode[STIX]{x1D712}^{\prime }$ is large, electro-osmotic effects are screened and Bretherton’s classical results are recovered. However as $\unicode[STIX]{x1D712}^{\prime }$ decreases, electro-osmotic effects come into play and the film becomes much thicker than Bretherton’s prediction to ensure that screening effects are not altogether lost, and also there is a noticeable increase in the pressure needed to drive the droplet front along. These results apply with minor variations in the case of singly charged surfaces (charge on either oil or on the capillary wall), oil and wall surfaces with like charges, or oil and wall surfaces with opposite but unequal charges. However in the case of opposite and equal charges, the system’s behaviour changes dramatically. There is now a conjoining electro-osmotic pressure rather than a disjoining tension, the film becomes thinner than the analogous Bretherton film, and the pressure needed to drive the droplet front along decreases. Surprisingly in this case, for sufficiently small $\unicode[STIX]{x1D712}^{\prime }$, the work done by the conjoining pressure can exceed the work done against viscous dissipation, meaning the pressure required to drive the droplet front is not just smaller than in Bretherton’s predictions but also slightly less than would be estimated based on capillary forces alone. Although the main effect of reducing salinity is to increase Debye length and hence reduce $\unicode[STIX]{x1D712}^{\prime }$, salinity also affects surface charges. A situation is explored whereby reducing salinity affects charges, producing a switch from disjoining tensions to conjoining pressures and back again: this leads to a non-monotonic response in film thickness and pressure required to drive the droplet front along.

Viscous and electro-osmotic effects upon motion of an oil droplet through a capillary

“…with the composition of sea water) often perform less well in oil recovery terms than lower salinity floods (up to a couple of orders of magnitude less saline). The difference in behaviour between high and low salinity floods has been postulated to be due to the difference in electro-osmotic forces between these different situations (Lager et al 2008; Wilmott et al 2018; Grassia 2019). Although a full predictive model of how to optimise waterflood performance by manipulation of salinity has yet to be developed, it is clear that enhancing understanding of how electro-osmotic effects influence displacement of an oil droplet along a capillary represents a step towards a predictive model.…”

“…Extensions to the classical system of Bretherton (1961) to incorporate the effect of electrostatic potentials have been considered by a number of authors (Teletzke, Davis & Scriven 1987, 1988; Krechetnikov & Homsy 2005; Wilmott et al 2018; Grassia 2019). In a typical situation (Grassia 2019) in which the capillary wall and droplet have like charges, we see a large electro-osmotic repulsion in the thin film region (the repulsive force can scale with an inverse square law of distance) but no electro-osmotic effects in the capillary static region (since charges are screened in that latter region (Wright 2007) according to Debye–Hückel theory). Expressing this in terms of electro-osmotic disjoining tension (tension is the negative of pressure), implies (Wilmott et al 2018) a large (and positive) electro-osmotic tension for a thin film but a zero electro-osmotic tension in the capillary static region.…”

“…It follows that, whereas the capillary pressure gradient generates Poiseuille flows that make films thinner (as noted above), repulsive electro-osmotic interactions generate opposing Poiseuille flows tending to thicken them. In an extreme case, films thicken to the point in which capillary and electro-osmotic pressure gradients cancel each other out (Grassia 2019). A so-called augmented Young–Laplace state (Teletzke et al 1988) is then reached in which films are found to be much thicker than their classical counterparts from Bretherton (1961), and as a result viscous forces cease to play a role (shear rates and hence associated viscous forces only being significant in thin films).…”

“…The above arguments assume repulsive electro-static interactions below the charge screening distance. It is possible, however, to consider a system (Grassia 2019) in which the electro-static interactions below the charge screening distance instead are attractive conjoining forces: this is considered next.…”

“…This will increase the net rightward Poiseuille flow driven by these combined pressures, which will then offset even more of the leftward flow carried by the moving capillary wall. The result (Grassia 2019) is that, for attractive electro-osmotic interactions, the thin film will actually reduce in thickness compared to the classical case of Bretherton (1961), not increase in thickness, as happens for repulsive interactions.…”