Porous gravity current with background flow ($\varphi h_t+(Uh-K_{b}h{h}_x{)}_x=0$)

A shallow buoyant porous gravity current in a deep aquifer with uniform imposed Darcy velocity $U$ obeys $\varphi h_t+(Uh-K_{b}h{h}_x{)}_x=0$ away from sources, where $K_b=k\Delta \rho g/\mu$. Here $k$ is permeability of a porous medium, $\varphi$ is porosity, and $\mu$ is dynamic viscosity. The pore-fluid drift speed is $u=U/\varphi$, and the gravity diffusion coefficient multiplying $(h{h}_x{)}_x$ is $a=K_b/\varphi$. The sharp-interface model neglects capillary trapping and dissolution.

An instantaneous point release of actual volume $V$ per unit width in a porous gravity current with background flow has the Barenblatt solution $h=[R(t{)}^2-(x-ut{)}^2{]}_{+}/(6at)$, where $R^3=9aVt/(2\varphi )$. Its centre translates at pore speed $u=U/\varphi$ and its radius grows as $t^{1/3}$. The conserved physical volume is $\varphi \int hdx=V$.

Constant actual volume input $Q$ into a porous gravity current with background flow gives early symmetric scales $h\sim [(Q/\varphi {)}^{2}t/a{]}^{1/3}$ and $x\sim [a(Q/\varphi )t^2{]}^{1/3}$. For $U>0$, drift dominates after $t_c=\varphi K_{b}Q/U^3$. The late interior depth is $Q/U$ and the steady upstream edge is $x_{-}=-K_{b}Q/U^2$, with upstream linear profile $h=U(x-x_{-})/K_b$. Upstream volume flux is zero because background drift balances gravity spreading.

The downstream edge of a continuously fed porous gravity current with background flow has a transition width proportional to $\sqrt{a{h}_{\mathrm{int}}t}$ around drift position $ut$. Its similarity solution satisfies $(FF\prime )\prime +\eta F\prime /2=0$, $F(-\infty )=1$, and $F({\eta }_N)=0$. A finite front has $F\sim ({\eta }_N/2)({\eta }_N-\eta )$. A clipped linear ramp is a useful mass-preserving approximation, but not an exact solution of this nonlinear transition equation.