Modeling the Influence of a Variable Permeability Inclusion on Free-Surface Flow in an Inclined Aquifer

The interaction of sub-surface, gravity-driven flows with inclusions of different permabilities are investigated theoretically using a model that exploits the relative shallowness of the motion. Numerically computed solutions for steady motion around cylindrical inclusions reveal a range of behaviours dependent on the ratio of the interior to exterior permeability and a dimensionless flow parameter that measures the far-field thickness to the product of the gradient of the slope down which the fluid flows and the width of the inclusion. When the inclusion is relatively narrow, the depth of the flow is little changed from its far-field value and the fluid is focussed into inclusions of higher permeability and deflected around those of lower permeability. However, if the inclusion is relatively wide then three qualitatively different regimes emerge, dependent on the ratio of permeabilities. When the interior and exterior permeabilities are similar, then negligible deviation of the flow occurs apart from within thin transition layers at the boundary of the inclusion. When the permeabilities differ significantly, the flow forms deep ponds at either the upstream or downstream boundary of the inclusion for relatively low or high permeability inclusion, respectively, which arise due to deflection or focussing. In each case, asymptotic relationships are derived between the depth of the flow and the parameters. Inclusions of differing cross-section are also analysed numerically and analytically to draw out the interplay between adjustment, deflection and focussing.