Response of a gravity current in a porous medium to a perturbation of the permeability field

One of the challenges with modelling subsurface flows is the uncertainty in measurements of geological properties, mostly due to limited resolution in observation methods. Many subsurface flows can be modelled as a gravity current, which, for uniform material properties and power-law injection rate, has a well-characterised similarity solution. The similarity solution forms a dynamical attractor that is typically approached rapidly from a host of initial conditions. Here, we consider the impact of transverse variations to the permeability field by performing a perturbation analysis of the self-similar spreading. This treats the response as perturbations to the self-similar flow. We restrict our focus to permeability fields that vary laterally, or across the flow, starting with the simple case of a sinusoidal perturbation to a uniform permeability. At early times, the height and nose position of the current are determined by the local permeability, and deviations to the height and nose grow at the same rate as the mean, and proportional to the amplitude, of the permeability variation. The transition between the early and late time regimes is governed by the wavelength of the permeability. At late times, lateral spreading between high and low permeability streaks is dominant; the height deviations decay, and the nose deviations approach a steady state. The magnitudes of both depend on the product of the wavelength and amplitude of the permeability. The single mode sets the groundwork for examining more complex, multimodal permeabilities, which are more representative of real geological structures.