= Solution
Assume the dense current lies on the bottom, $\Delta\rho>0$, and its height is small compared with the confined aquifer depth $w$. This permits the <hydrostatic approximation> and neglect of leading-order resistance from the return flow of the original fluid. The dense-layer pressure anomaly on the basal seal is $\Delta\rho gh$. The along-layer <Darcy flux>, integrated over the current thickness, and the downward leakage flux are respectively
$$
q_x=-\frac{k\Delta\rho g}{\mu}h h_x,
\qquad q_\ell=\frac{\lambda\Delta\rho g}{\mu b}h.
$$
<Volume conservation> requires $\phi h_t+\partial_xq_x=-q_\ell$. Thus the <porous gravity current> with <hydrostatic leakage through a basal seal> obeys
$$
\boxed{h_t=D(hh_x)_x-\Omega h,
\quad D=\frac{k\Delta\rho g}{\phi\mu},
\quad\Omega=\frac{\lambda\Delta\rho g}{\phi\mu b}.}
$$
Here <permeability of a porous medium> has units of area, and $\Omega$ has units of inverse time. The original PDF instead prints the sink coefficient $\lambda k\Delta\rho g/(\mu b)$. With $\lambda$ defined there as a permeability, that coefficient has units of area per time and cannot multiply $h$ to give $h_t$. It also omits the division of leakage by <porosity>. This is a genuine source error, not a <Darcy law> convention. The remaining parts solve the stated linear-drainage equation for a constant $\Omega>0$; the physically consistent coefficient is the one above. Formally treating the printed coefficient as an independently prescribed inverse-time constant leaves those mathematical solutions unchanged.
Back to article page
