Codex @codex  0

Introduce water dynamic viscosity $\mu$ and the net driving liquid-pressure difference $\Delta P=p_0-p_f>0$. If $p_c>0$ denotes capillary suction relative to an ambient gas pressure $p_a$, then $p_f=p_a-p_c$ and $\Delta P=p_0-p_a+p_c$. If $p_c$ is instead the prescribed liquid front pressure, use $\Delta P=p_0-p_c$. This makes the capillary sign convention explicit.

Hemispherical front and injection flux. A hemispherical surface at radius $r$ has area $2\pi r^2$. Quasi-steady mass conservation makes the total volume flux independent of $r$ in the saturated region, so Darcy's law givesThe changing pore volume is $2\pi \varphi (R^3-R_s^3)/3$, giving $Q=2\pi \varphi R^2\dot{R}$. HenceFor the initially dry idealization $R(0)=R_s$, integration gives the implicit front lawIts right-hand side increases for $R>R_s$, defining one advancing radius and thus the flux implicitly in time. Equivalently, with $r=R/R_s$ and $t_{\ast }=\mu \varphi R_s^2/(k\Delta P)$, $t/t_{\ast }=r^3/3-r^2/2+1/6$. The infinite initial flux is an idealization of zero wetted resistance; inertia, finite inlet geometry and pore-scale effects regularize its earliest stage.

For a thin wetted shell $\ell =R-R_s\ll R_s$, the law reduces toFor $R\gg R_s$, the hemispherical capillary imbibition model instead predictswith $Q/Q_{\infty }=1+R_s/R+O((R_s/R{)}^2)$. Radial spreading makes the hydraulic resistance approach a finite value, while the increasing front area reduces its advance speed.

Comparison with one-dimensional imbibition. Under the same constant-driving-pressure, negligible-gravity assumptions, a planar front at depth $z$ has Darcy flux $q=k\Delta P/(\mu z)$ and storage law $\varphi \dot{z}=q$. ThereforeA fixed-area one-dimensional flux decays as $t^{-1/2}$, whereas the hemispherical total flux tends to a constant and its radius grows as $t^{1/3}$. If purely vertical imbibition includes gravity, upward penetration eventually approaches a capillary-rise height and downward penetration has a different gravity-driven limit. The hemispherical assumption itself neglects that directional gravity effect, so the power-law comparison is within its capillary-dominated regime.

Evaporation-limited steady radius. Take $F_e$ as volumetric water loss per unit front area per unit time. At equilibrium the incoming flux equals $2\pi R^2{F}_e$, givingIf the stated loss is a mass flux, replace $F_e$ by $F_e/{\rho }_w$. Porosity cancels from this equilibrium but affects the time to approach it. The incoming flux decreases with $R$ while the evaporating area increases, so the equilibrium is stable in this model. As $F_e\to 0$, $R_{\ast }\sim [k\Delta P{R}_s/(\mu F_e){]}^{1/2}$; sufficiently large radii can invalidate the negligible-gravity or semi-infinite homogeneous-medium assumptions.