Slope-driven porous gravity current ($u=K\Delta \rho g\sin \theta /\mu$)

A buoyant phase beneath a sloping impermeable cap has a slope-driven Darcy velocity proportional to density deficit and slope. Far from thickness-adjustment regions, its uniform plateau speed is $K\Delta \rho g\sin \theta /\mu$. With thickness measured normal to the slope, buoyant cap overpressure is $\Delta \rho gh\cos \theta$. A complete depth model also includes the along-current gradient of thickness.

A buoyant current beneath an overlying low-permeability cap can enter it when current overpressure exceeds the capillary entry threshold. The plateau criterion compares $\Delta \rho gh\cos \theta$ to that threshold. Cold and warm portions can have different depths and mobilities, giving distinct onset injection rates. These onset criteria do not determine the depleted current after leakage begins.

A localized temperature adjustment separates regions with different fluid density, viscosity and Darcy speed. Mass conservation in the moving frame gives the displayed plateau-depth relation. A speed stated as $V_T=\Gamma u_c$ with $u_c$ a Darcy velocity has effective retardation parameter $\varphi \Gamma$. Setting both laboratory-frame mass fluxes equal neglects storage associated with passage of the temperature front.