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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 ${\alpha }_m^2=({\mu }_2-{\mu }_1)U/(3k\gamma )$.

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.

Porous-media flow is fluid motion through the interconnected voids of a porous medium, such as a sand bed, a rock or a filter. Individual passages can be tiny and tortuous, yet their combined discharge can be predicted without resolving every passage. Darcy law supplies this bulk description when viscous resistance dominates pore-scale inertia; porosity measures how much space stores fluid, while permeability of a porous medium measures how readily that space transmits it.

The basic fully saturated model assumes one Newtonian fluid fills the connected pore space of a stationary solid framework, with incompressible flow, a small pore-scale Reynolds number, and averaging volumes large compared with individual pores but small compared with macroscopic variations. These are assumptions of the model developed below, rather than a claim that Darcy law describes every porous-media flow. Two-phase porous-media flow and poroelastic aquifers require additional constitutive and storage information.

For a first calculation, see Darcy flow through a homogeneous slab: a pressure drop determines the discharge, but porosity determines the corresponding pore velocity. Layered porous permeability then shows how geometry alone can make the bulk response anisotropic.

A porous medium is a solid framework containing connected voids in which a fluid can reside and flow. Its porosity is the void fraction; its permeability of a porous medium measures the ease of pressure-driven flow through the connected pores. The averaged Darcy flux is measured per total cross-sectional area, whereas the pore velocity is that flux divided by porosity. Not every material with voids has connected permeable pores.

Suppose a porous layer has thickness $h(p)$ and permeability of a porous medium $k(p)$, and the pressure drop $p$ acts across its thickness. Darcy's law gives the drainage volume fluxIf $k/h=K{p}^{-\beta }$, then $w=(K/\mu )p^{1-\beta }$.

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.

Reactive infiltration instability is the positive feedback in which reaction or melting increases permeability of a porous medium, focuses more reacting fluid into that region, and further increases permeability. Finite reaction-zone thickness, transport, and pore geometry can change the short-wave behavior; Interacting length scales in the reactive-infiltration instability analyzes transport and reaction length scales in a dissolution model.

A dimensionless reduction of a porous formation's intrinsic permeability of a porous medium for one fluid phase in multiphase flow. Its effective phase permeability is $K{k}_{ri}$, and its phase mobility is $k_{ri}/{\mu }_i$. It depends on saturation and pore-scale phase configuration; it is not the magnetic use of relative permeability.

With permeability of a porous medium $k(z)=k_0(1+\beta z)$, the Dupuit approximation givesHere $\varphi$ is porosity and $R$ is recharge. The low-height limit has mobility proportional to $h$, whereas large heights have mobility proportional to $h^2$. This changes both the forced filling similarity for power-law diffusion and the separable draining profile for power-law diffusion.