The viscous froth model is used to predict rheological behaviour of a two-dimensional (2D) liquid-foam system. The model incorporates three physical phenomena: the viscous drag force, the pressure difference across foam films and the surface tension acting along them with curvature. In the so-called infinite staircase structure, the system does not undergo topological bubble neighbour-exchange transformations for any imposed driving back pressure. Bubbles then flow out of the channel of transport in the same order in which they entered it. By contrast, in a simple single bubble staircase or so-called lens system, topological transformations do occur for high enough imposed back pressures. The three-bubble case interpolates between the infinite staircase and simple staircase/lens. To determine at which driving pressures and at which velocities topological transformations might occur, and how the bubble areas influence their occurrence, steady-state propagating three-bubble solutions are obtained for a range of bubble sizes and imposed back pressures. As an imposed back pressure increases quasi-statically from equilibrium, complex dynamics are exhibited as the systems undergo either topological transformations, reach saddle-node bifurcation points, or asymptote to a geometrically invariant structure which ceases to change as the back pressure is further increased.

The two-dimensional (2D) viscous froth model was initially designed to capture the dynamic behavior of dry foams within a Hele–Shaw cell, characterized by two parallel covering plates separated by a small gap. However, due to its inherent 2D nature, this model does not explicitly account for the dimension across the gap. To address this limitation, we have opted for a three-dimensional (3D) version of the viscous froth model. In this 3D model, the dynamic effect is introduced through the motion of the surface Plateau borders, while the configurations of films in bulk are determined via surface energy minimization subject to specified bubble volumes and surface Plateau border locations. We use this 3D model to simulate the motion of a viscous froth lens within a straight channel. The steady states of the viscous froth lens in 3D are primarily influenced by the driving velocity-to-gap size ratio, particularly at relatively small values of this ratio. By contrast, as the ratio becomes relatively large, the gap size begins to play a significant role in influencing the behavior of the viscous froth lens in 3D. Differences are observed in the steady-state configurations of the viscous froth lens in 3D when compared to those in 2D. However, the behavior of the viscous froth lens in 2D can be better aligned with the results in 3D by treating the drag coefficient required in the 2D viscous froth model as a fitting parameter. A further quantitative analysis indicates that the drag coefficient needed in the 2D viscous froth model may not serve as a uniform parameter for the entire foam structure. Instead, it may depend on the specific location along the evolving foam films over time.

With a view towards modelling the foam improved oil recovery process, fractional flow theory is used to study the dynamics of a foam as it propagates in a porous medium that is initially filled with liquid. In particular, a case is studied whereby, at a certain time, the net pressure driving the foam is decreased below the hydrostatic pressure at depth, leading to a local change in the flow direction. This is known as flow reversal. In both forward and reverse flow, the boundary between foamed gas and liquid is found as a discontinuous jump in liquid saturation. Over a certain thickness in the neighbourhood of this discontinuity, foam is finely textured, and the mobility of foamed gas drops by orders of magnitude relative to either pure gas or pure liquid. In reverse flow, however, the foam mobility itself and also the thickness over which low mobilities apply might differ from the forward flow case. Fractional flow theory reveals that the thickness of the low mobility region, and hence the resistance to motion that it presents, increases directly proportional to the distance travelled. Previous studies recognised this, but assumed the thickness of this region to be just a small fraction of the distance travelled by the discontinuity. Here, however, we demonstrate that the extent of the low mobility region, in both forward and reverse flow, accounts for a considerable fraction of the distance travelled by the foam, despite what was assumed in previous works.

A mathematical model formulated as a system of Hamilton–Jacobi equations describes implicitly the propagation of a foam–liquid front in an oil reservoir, as the zero-level set of the solution variable. The conceptual model is based on the ‘pressure-driven growth’ model in Lagrangian coordinates. The Eulerian mathematical model is solved numerically, where the marching is done via a finite volume scheme with an upwind flux. Periodic reinitialization ensures a more accurate implicit representation of the front. The numerical level set contour values are initially formed to coincide with an early time asymptotic analytical solution of the pressure-driven growth model. Via the simulation of the Eulerian numerical model, numerical data are obtained from which graphical representations are generated for the location of the propagating front, the angle that the front normal makes with respect to the horizontal and the front curvature, all of which are compared with the Lagrangian model predictions. By making this comparison, it is possible to confirm the existence of a concavity in the front shape at small times, which physically corresponds to an abrupt reorientation of the front over a limited length scale.