Topographic viscous fingering: fluid-fluid displacement in a channel of non-uniform gap width

We consider the displacement of one fluid by a second immiscible fluid through a long, thin permeable channel whose thickness and permeability decrease away from the axis of the channel. We build a model that illustrates how the shape of the fluid-fluid interface evolves in time. We find that if the injected fluid is of the same viscosity as the original fluid, then the cross-channel variations in permeability and thickness tend to focus the flow along the centre of the channel. If the viscosity of the injected fluid is smaller than the original fluid, then this flow focusing intensifies, leading to very poor sweep of the original fluid in the system, with the injected fluid bypassing much of the channel. We also show that if the viscosity ratio of the injected fluid to the original fluid is sufficiently large, then a blunt nose may develop at the leading edge of the injected fluid, whereas the remainder of the fluid-fluid interface becomes stretched out along the edges of the channel. This leads to a much more efficient sweep of the original fluid from the channel. We generalize the model to illustrate how buoyancy forces and capillary pressure affect the evolution of the system and compare our model predictions with some simple laboratory experiments. This partial stabilization of a fluid interface in a channel of non-uniform width represents a generalization of the classical Saffman-Taylor instability, and our nonlinear solutions for the evolution of the interface highlight the importance of cross-channel variations in permeability and thickness in modelling flow in channelled reservoirs. This article is part of the themed issue 'Energy and the subsurface'.