Porous gravity current

A porous gravity current spreads because a density difference creates a lateral hydrostatic-pressure gradient. For a two-dimensional current of thickness $h$, density deficit $\Delta \rho$, permeability $k$, viscosity $\mu$, and porosity $\varphi$,

For a thin buoyant layer beneath an inclined impermeable roof, the hydrostatic approximation and Darcy's law give $u=kg\Delta \rho \sin \theta /\mu$ and the volume flux per unit width $F=uh-u\cot \theta h{h}_x$. The first term transports fluid upslope; the second spreads the layer through its hydrostatic pressure gradient. The porosity multiplies the stored fluid volume, so the advection speed is $u/\varphi$, not the Darcy velocity $u$.

With constant porosity $P$ and $X=x\tan \theta$, $Y=y\tan \theta$, a translating frame $\xi =X-T$, $\eta =Y$ has area Jacobian determinant $dxdy={\cot }^2\theta d\xi d\eta$. Actual fluid volume is $P\iint hdxdy$, proving the displayed normalization. Replacing the cotangent factor by a tangent factor is incompatible with these coordinates except at a slope of $\pi /4$.

For porosity $P{z}^{\alpha -1}$ and permeability of a porous medium $K{z}^{\beta -1}$ above an inclined impermeable plane, stored liquid volume per area is $P{h}^{\alpha }/\alpha$. The depth-integrated Darcy flux is $\rho gK{h}^{\beta }(\sin \theta -\cos \theta h_x,-\cos \theta h_y)/(\beta \mu )$. Conservation of mass yields the displayed equation with $X=x\tan \theta$, $Y=y\tan \theta$ and $T=t\alpha \rho gK\sin \theta \tan \theta /(\beta \mu P)$.

In a slender steady porous gravity current, the dominant balance is $(h^{\beta }{)}_X=(h^{\beta }{h}_Y{)}_Y$. If its width and height scales are $W,H$, fixed downslope volume flux gives $H^{\beta }W=\text{constant}$, and transverse spreading gives $W^2\sim HX$. Eliminating $H$ yields the displayed width exponent. The porosity exponent affects transient storage but not this steady balance.

With exponentially decreasing injection, the advective approximation $\varphi h_t+u{h}_x=0$ and inlet condition $uh(0,t)=Q{e}^{-t/\tau }$ give $h_0=Q/u$ and the displayed solution along characteristic curves. It applies behind the advective nose $X=ut/\varphi$. The relative size of the discarded nonlinear spreading term for this profile is $2\varphi h\cot \theta /(u\tau )$. Smallness of that quantity, and matching through a narrow leading layer, are necessary qualifications: elapsed time alone does not make the approximation uniform at the nose. In a translating frame the full equation remains a porous medium equation with coefficient $u\cot \theta /\varphi$.

A localized release of actual fluid volume $V$ per unit width on a reflecting half-line $x\ge 0$ has the Barenblatt solution $h=[L^2-x^2{]}_{+}/(6\lambda t)$, where $L^3=9\lambda Vt/\varphi$ and $\lambda =k\Delta \rho g/(\varphi \mu )$. The mass conservation normalization is $\varphi {\int }_0^{L}hdx=V$. Zero slope at the reflecting boundary gives zero Darcy flux. For symmetric spreading on the full line, the same formula has $L^3=9\lambda Vt/(2\varphi )$ when $V$ denotes the total released volume. A finite initial patch is not determined by volume alone; this exact source profile has a concentrated initial release.

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.

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.

A leaky porous gravity current loses flux through a localized fracture or a distributed permeable boundary. At a point fracture, mass conservation makes the horizontal Darcy flux discontinuous by exactly the leakage rate.

A dense porous gravity current of height $h$ places excess hydrostatic pressure $\Delta \rho gh$ on a basal seal. For seal permeability of a porous medium $\lambda$ and thickness $b$, Darcy law gives downward flux $\lambda \Delta \rho gh/(\mu b)$. Dividing the stored height-volume balance by porosity $\varphi$ gives a linear drainage term $-\Omega h$. The coefficient has inverse-time units and does not contain an extra factor of the aquifer permeability.

The porous medium equation $h_t=\lambda (h{h}_x{)}_x-\Gamma h$ with constant $\Gamma >0$ becomes $H_{\tau }=\lambda (H{H}_x{)}_x$ under the displayed transformation. Substitution gives $h_t+\Gamma h=e^{-\Gamma t}{\tau }^{\prime }{H}_{\tau }$ and $\lambda (h{h}_x{)}_x=\lambda e^{-2\Gamma t}(H{H}_x{)}_x$, so ${\tau }^{\prime }=e^{-\Gamma t}$. This transformation applies to every solution with appropriately transformed initial and boundary conditions, not just a similarity solution. A one-sided constant-volume porous gravity current therefore has remaining physical volume $V{e}^{-\Gamma t}$ and a limiting front $[9\lambda V/(\varphi \Gamma ){]}^{1/3}$. The finite transformed time $\tau \to 1/\Gamma$ means that arbitrary initial profiles need not reach the universal large-time source shape before drainage removes them.

For an impulsive leaky porous gravity current, write $s=(1-e^{-\Omega t})/\Omega$ and $h=(1-\Omega s)[R^2-x^2{]}_{+}/(6Ds)$, where $R=C{s}^{1/3}$. At a fixed lateral location, $h_t=0$ when $x^2=R^2(1+2\Omega s)/3$. The maximum invaded thickness is the envelope $x_{\ast }(s)=C{s}^{1/3}\sqrt{(1+2\Omega s)/3}$, $h_{\ast }(s)=C^2(1-\Omega s{)}^2/(9D{s}^{1/3})$. The source solution is singular at the initial line release; finite source size and aquifer depth limit its near-source validity.