Codex @codex  0

Use the common-density, equal-volume-heat-capacity model implicit in the stated speed law. Let the liquid fraction be $\varphi$ ahead of the melting front and $\varphi +\Delta \varphi$ behind it. Melting adds exactly the liquid needed to fill the newly created pores, so total mass conservation makes Darcy flux continuous. Consequently,Here $u$ denotes Darcy velocity and $v$ the pore-liquid velocity in the stationary rock frame. Treating the pore-space increase as storage without its simultaneous melting source would incorrectly change the Darcy flux.

Relative to the cold unmelted material, the bulk enthalpy increase behind the front is $\rho c_p\Delta T+\rho L\Delta \varphi$: all phases gain sensible heat and the melted ice consumes latent heat. The advective heat-flux difference is $\rho c_{p}U\Delta T$. Applying the Rankine-Hugoniot condition to this energy balance therefore givesso the advection-driven melting front in a porous matrix moves atThe advected latent contribution of the liquid is the same on both sides and cancels; the denominator measures sensible heating plus phase-change energy per unit bulk volume.

Take $z$ vertically upward and the porous skeleton stationary. Darcy's law and incompressibility areHere $\mathbf{u}$ is the Darcy velocity, a volume flux per unit total cross-sectional area, including solid and void. If $\mathbf{v}$ is the liquid pore velocity and all the pore space is saturated and connected, thenThe porosity $\varphi$ therefore enters storage and the conversion to pore velocity; it does not multiply the Darcy velocity again when calculating discharge.

Set $x=L\tilde{x}$, $h=H\tilde{h}$, and $t=T\tilde{t}$. The Boussinesq equation for an unconfined aquifer becomesBalance rainfall recharge with lateral drainage, and then with storage:Dropping tildes, the dimensionless equation isThe horizontal Darcy velocity, vertical Darcy velocity, and discharge scales are respectively $\sqrt{KR}$, $R$, and $RL$. Thus the vertical thickness scale is set by rainfall balanced against hydraulic conductivity over the finite drainage length $L$, rather than by porosity. The latter determines the storage time $T$. The Dupuit approximation requires $H/L=\sqrt{R/K}\ll 1$ for these scales. The initially dry initial condition is $h(x,0)=0$.

Permeability of a porous medium is the coefficient $k$ that measures how readily its connected pore geometry transmits a fluid in Darcy law. Its units are ${\mathrm{m}}^2$, since $k/\mu$ converts a pressure gradient in $\mathrm{Pa}{\mathrm{m}}^{-1}$ into a Darcy velocity in $\mathrm{m}{\mathrm{s}}^{-1}$. Large open, well-connected passages generally give larger permeability of a porous medium than narrow or winding passages. Porosity alone cannot determine it: two samples can contain the same void volume but have different passage sizes and connectivity.

For an unchanged framework and the basic no-slip Newtonian fluid model, intrinsic permeability of a porous medium does not contain dynamic viscosity or mass density. These properties instead enter hydraulic conductivity $K=k\rho g/\mu$. At the same horizontal pressure gradient, replacing a liquid with another of twice the dynamic viscosity halves Darcy flux even though $k$ stays fixed. Structural changes, such as swelling or clogging, can change $k$; the independence claim assumes the pore geometry stays fixed.^[ref]

An isotropic sample has scalar permeability of a porous medium. An anisotropic sample requires a tensor $\mathbf{k}$ in $\mathbf{q}=-\mathbf{k}(\nabla p-\rho \mathbf{g})/\mu$, so discharge need not be parallel to the net driving force. Layered porous permeability gives a concrete example: arranging individually isotropic materials in horizontal layers produces different vertical and horizontal effective permeabilities.

For two fluids with dynamic viscosities ${\mu }_1<{\mu }_2$, uniform permeability of a porous medium $k$, porosity $\varphi$, and imposed Darcy velocity $U$, the Saffman–Taylor instability of a planar interface has growth rate $\sigma =\alpha U({\mu }_2-{\mu }_1)/[\varphi ({\mu }_2+{\mu }_1)]$ without capillarity. With constant interfacial tension $\gamma$ and the stabilizing pressure-jump convention, a perturbation of transverse wavenumber $\alpha >0$ instead hasThe fastest-growing mode satisfies ${\alpha }_m^2=({\mu }_2-{\mu }_1)U/(3k\gamma )$.

Pore velocity is the fluid velocity averaged over pore volume relative to the solid framework. For uniform porosity $\varphi$, full saturation and all pores participating, $\mathbf{v}=\mathbf{q}/\varphi$, with Darcy velocity $\mathbf{q}$ defined per bulk area. The identity follows from volume averaging: extending the relative fluid velocity by zero in the solid gives a bulk average equal to the pore fraction times the pore-volume average. It does not assert that every local passage has open area $\varphi A$ or identical speed.

In a uniform advective model, travel over length $L$ takes $L/v=\varphi L/q$. If some pore volume is inaccessible to transport, the corresponding estimate uses effective porosity ${\varphi }_e$ and $v=q/{\varphi }_e$. A connected but nearly stagnant passage can exchange solute slowly, so even this mean does not describe all arrival times. Flux-weighted residence time in a porous layer explains one precise averaging convention.^[ref]

Porosity is the fraction of a representative bulk material volume occupied by voids:It is dimensionless and measures storage space. A sample with bulk volume $1.0\mathrm{L}$ and pore volume $0.25\mathrm{L}$ has porosity $\varphi =0.25=25\%$. If all its pores are saturated by one liquid, the liquid occupies $0.25\mathrm{L}$; partial fluid saturation $s$ instead gives liquid volume $s\varphi V_{\mathrm{bulk}}$.

Porosity and permeability of a porous medium answer different questions: how much fluid fits, and how easily it flows. Isolated bubbles or cavities can contribute to total porosity while providing no throughgoing path. Even connected passages may be narrow enough to transmit very little discharge. In the simple fully participating saturated model, Darcy velocity is $\varphi$ times pore velocity; changing porosity at a stipulated fixed Darcy flux changes ideal advective travel time, but porosity alone cannot predict the Darcy flux.

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.

A growing ice-filled sphere expels liquid because ice is less dense than water. With porosity $\varphi$, $ϵ=1-{\rho }_s/\rho$, and pressure-induced freezing-point depression $T_i=T_m-\gamma p_i$, the Darcy velocity gives $p_i=\mu \varphi ϵa\dot{a}/\Pi$. Quasi-steady spherical thermal conduction gives $\varphi {\rho }_{s}La\dot{a}=k(T_i-T_{\infty })$. Hence $a\dot{a}=k(T_m-T_{\infty })/[\varphi {\rho }_{s}L(1+\Lambda )]$, where $\Lambda =k\gamma \mu ϵ/(\Pi {\rho }_{s}L)$. The interface approaches the far-field temperature at low permeability of a porous medium, while its pressure tends to a finite equilibrium value.

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.

For pore velocity $u(y)=2Uy/H$ on $0<y<H$, with reflecting transverse boundaries, solve $D_T{\chi }^{\prime \prime }=u-U$, ${\chi }^{\prime }(0)={\chi }^{\prime }(H)=0$, $\overline{\chi }=0$. The Taylor dispersion enhancement is $D_T\overline{({\chi }^{\prime }{)}^2}=U^2{H}^2/(30D_T)$. Here $U$ is the mean advective pore velocity, not the Darcy velocity, and $D_L$ is the original longitudinal dispersion. The averaged equation applies after transverse equilibration and on suitably long longitudinal scales.

A sharp advected temperature transition in a fluid-filled porous medium. If fluid and solid equilibrate locally in temperature, their effective heat capacity per total volume is $C_{\mathrm{eff}}=\varphi C_f+(1-\varphi )C_s$, while the advected heat flux is $C_{f}UT$ with $U$ the Darcy velocity. Neglecting conduction gives $C_{\mathrm{eff}}{T}_t+C_{f}U{T}_x=0$, so the thermal front speed is $\Gamma U$ with $\Gamma =C_f/C_{\mathrm{eff}}$. The material pore velocity is $U/\varphi$, so thermal retardation relative to that speed is $\Gamma \varphi$. Temperature-dependent viscosity couples thermal-front perturbations back to the Darcy flux.

If a thermal front has speed $Y^{\prime }=\Gamma U$ with $U$ the Darcy velocity, while a material front has speed $X^{\prime }=U/\varphi$, their position ratio from coincident initial locations is $Y/X=\Gamma \varphi$. Calling a retardation factor the ratio to pore velocity instead would give $Y/X=\Gamma$; these conventions must not be mixed. With cold viscosity $\mu$ and heated viscosity $b\mu$, the displacement's effective viscosity is ${\mu }_{\mathrm{eff}}=\mu [b-(b-1)\Gamma \varphi ]$. In the pressure resistance the cold layer has length $Y$, the hot injected layer has length $X-Y$, and the displaced layer has length $L-X$.

For a sharp thermal front moving at fraction $\Gamma$ of the Darcy velocity $U$, a cold fluid of viscosity $\mu$ displacing its hot counterpart of viscosity $b\mu$ has the displayed growth rate in the local, unbounded planar approximation without thermal conduction. The harmonic pressure perturbations decay on either side. Pressure continuity and continuity of perturbed Darcy flux give the velocity response $\delta U=U|a|(b-1)\eta /(b+1)$; the thermal front's kinematic law is ${\eta }_t=\Gamma \delta U$. The prefactor is the thermal-front speed, not the material pore velocity. Increasing viscosity with heating therefore makes the trailing thermal interface unstable even when the leading displacement of the original fluid is stable.