In an enlightening article published in 2015 in Physical Review Letters, Holtzman and Segre studied the effect of the contact angle (wettability) in immiscible two-phase flow in porous media in the unstable case of a less viscous invading fluid displacing a more viscous one. The title of the article “Wettability stabilizes fluid invasion into porous media via nonlocal, cooperative pore filling”, indicates the main conclusion of the work.
They carried out numerical experiments in a two-dimensional disordered medium constructed with an array of solid cylinders of varying diameter set within a large rectangular area. A degree of disorder was introduced by randomly selecting the cylinder’s diameter from a uniform distribution. The array of cylinders consisted of repeated triangular lattices of fixed spacing between vertexes (the cylinders’ centres). The fluid dynamics modelling at the pore scale took into account three types of meniscus instabilities, namely, bursting (Haines jumps), occurring when the curvature reaches a critical value; touching, when a meniscus intersects a third solid surface, and overlapping, when adjacent menisci encounter. The latter is essential in creating a compact, smooth, displacing front as it promotes non-local, cooperative pore filling. They used this porous medium model to simulate the experiments of Trojer, Szulczewski, and Juanes (2015). The radial fluid motion was produced by injecting air at a certain rate, in a small region of the domain initially saturated with a solution of water and glycerol. The simulations stopped when air reached one of the outer boundaries.
In various applications, it is of interest to have a compact moving front in immiscible two-fluid motion as this generates the more efficient displacement mechanism. This is desirable, for instance, in oil recovery processes.
An important contribution of the study by Holtzman and Segre is the introduction of two novel dimensionless parameters, namely, a modified capillary number, obtained by altering the classical capillary number (ratio of viscous to capillary forces) with a function of the contact angle and a dimensionless geometric ratio, and the so-called “cooperative number”, given as a function of these two parameters and the angle between two adjacent menisci. This number is a measured of the likelihood of filling the pores by menisci overlaps. In their simulations, the angle between two adjacent menisci was fixed to 120o.
By computing these two parameters for the same data used in the numerical study, Holtzman and Segre correlated their values with the salient features of the interfacial morphology from the simulations. These predictions followed the invasion regimes observed in the experiments referred to above. For modified capillary numbers greater than approximately 4 x 10-3, simulations showed viscous fingering, characterized by long and thin fingers, regardless of the magnitude of the cooperative number. On the other hand, for a modified capillary number smaller than 4 x 10-3 and positive cooperative numbers, indicating invasion by a wetting fluid, simulations predicted compact displacement, characterized by short and thick “fingers”. For the remaining sector, for which the cooperative number is negative (non-wetting invasion), results showed thinner, adjacent fingers within fractal, long interfaces, typical of the capillary fingering pattern. Therefore, their results showed that wetting invasion (imbibition) favors the formation of a compact front for low capillary numbers.
Finally, it is noteworthy that more recently, in 2016, Zhao, MacMinn, and Juanes, conducted a series of experiments by radially displacing a very viscous silicone oil in a porous system using water while systematically varying the wettability of the medium. They found a trend similar to the one described by Holtzman and Segre in that increasing the wettability of the invading fluid promotes cooperative pore filling and hence a more compact front displacement. Nonetheless, Zhao et al. surprisingly found that for a strongly wetting invading fluid, this trend is dramatically reversed and the displacement becomes markedly less efficient. They noted that this sudden change, not reported previously, is caused by the dominant presence of corner flow, for which the invading fluid moves without filling the pore bodies, over cooperative pore filling.