Evolution of convection in a layered porous medium

The effect of a series of thin, horizontal, low-permeability layers on convective motion from a distributed dense source along an upper boundary in an otherwise homogeneous, two-dimensional porous medium is considered. This set-up provides an idealised version of a relatively common form of heterogeneity in geological formations. The thickness and permeability of the thin layers are assumed to be small relative to the distance between them and the bulk permeability, respectively. As such, the layers can be parameterised by their impedance $\varOmega$ – a dimensionless ratio of the effective layer thickness and permeability – while the strength of convection is controlled by the dimensionless distance $H \gg 1$ between layers, which can also be interpreted as an effective Rayleigh number for the flow. The role of $\varOmega$ is explored with the aid of high-resolution numerical simulations, and simple analytical models are developed for the evolution of the mean concentration and the flux in the limits of small and large $\varOmega$. For intermediate values of $\varOmega$, the flow undergoes a transition from predominantly diffusive transfer across the layers to predominantly advective transfer, and the lateral scale of the flow can become very large. This transition is characterised and a simple model is developed.