Capillary imbibition ($\varphi \dot{z}=k\Delta P/(\mu z)$)

A wetting liquid penetrates pore space under a pressure difference that includes capillary suction. With a planar front, fixed pressure drop and negligible gravity/inertia, Darcy's law and pore-volume storage give $z\propto t^{1/2}$. The front pressure must be defined relative to the gas phase consistently; adding a positive suction to the inlet pressure is different from prescribing a positive liquid front pressure.

Radial capillary imbibition into a half-space has hemispherical flow area $2\pi r^2$. Its radial hydraulic resistance is proportional to $1/R_s-1/R$, so its inlet volume flux approaches $2\pi k\Delta P{R}_s/\mu$. The growing front area gives late-time $R\propto t^{1/3}$, unlike the planar $t^{1/2}$ law. Both results assume capillary-dominated geometry with negligible directional gravity effects.

When a hemispherical wetted front loses liquid at fixed volumetric flux per unit area $F_e$, equilibrium balances inlet flow with $2\pi R^2{F}_e$. The displayed radius exceeds $R_s$ and is stable because inlet flow decreases while evaporating area increases with radius. Porosity affects the adjustment time but cancels from the equilibrium radius.