Darcy velocity

Porous-media flow

{c}
{title2=$Q=Ak\Delta p/(\mu L)$}

Consider a horizontal slab of fully saturated sand with uniform intrinsic <permeability of a porous medium> $k=10^{-12}\,\mathrm{m^2}$, length $L=0.50\,\mathrm m$, total cross-sectional area $A=0.020\,\mathrm{m^2}$, and <porosity> $\phi=0.25$. Use one <Newtonian fluid> with <mass density> $\rho=1000\,\mathrm{kg\,m^{-3}}$ and <dynamic viscosity> $\mu=10^{-3}\,\mathrm{Pa\,s}$. Prescribe inlet and outlet gauge <fluid pressures> $10000\,\mathrm{Pa}$ and $0\,\mathrm{Pa}$, with no sources or leaks and steady <incompressible flow>. The two ends have the same elevation, so <gravity> has no component along the slab. <Mass conservation> makes the axial <Darcy flux> constant, and <Darcy law> then gives a linear <fluid pressure> profile:
$$
p(x)=10000\left(1-\frac{x}{L}\right)\mathrm{Pa},\qquad \frac{dp}{dx}=-20000\,\mathrm{Pa\,m^{-1}},\qquad q=-\frac{10^{-12}}{10^{-3}}(-20000)=2.0\times10^{-5}\,\mathrm{m\,s^{-1}}.
$$
The measured <volume flux> and mean <pore velocity> are
$$
Q=Aq=4.0\times10^{-7}\,\mathrm{m^3\,s^{-1}}=0.40\,\mathrm{mL\,s^{-1}},\qquad v=\frac q\phi=8.0\times10^{-5}\,\mathrm{m\,s^{-1}}.
$$
The slab's total volume is $AL=0.010\,\mathrm{m^3}$, of which $\phi AL=0.0025\,\mathrm{m^3}=2.5\,\mathrm L$ is pore volume. In the ideal uniform advective model, a nonreacting tracer has transit time $L/v=6250\,\mathrm s\simeq1.74\,\mathrm h$, consistent with pore volume divided by <volume flux>, $\phi AL/Q$. The incorrect substitution of <Darcy velocity> for <pore velocity> would give $25000\,\mathrm s$, four times too long. Real passages have different path lengths and local speeds; these numbers describe a bulk mean rather than every particle's journey.

As an independent check, <gravitational acceleration> magnitude $g=9.81\,\mathrm{m\,s^{-2}}$ gives <hydraulic conductivity> $K=9.81\times10^{-6}\,\mathrm{m\,s^{-1}}$, <hydraulic head> drop $\Delta h=10000/(1000\times9.81)\,\mathrm m\simeq1.01937\,\mathrm m$, and $q=K\Delta h/L=2.0\times10^{-5}\,\mathrm{m\,s^{-1}}$ again. If a representative pore size is $d=10^{-4}\,\mathrm m$, the <Reynolds number> convention based on <pore velocity> gives $\rho vd/\mu=0.008$, supporting neglect of pore inertia. This size is an illustrative extra measurement, not something determined by <porosity> and intrinsic <permeability of a porous medium> alone.

\Image[/darcy-flux-and-pore-velocity.png]
{title=The same discharge gives different Darcy and pore velocities}

Regenerate the diagram by running `python ../darcy-flux-and-pore-velocity.py` from `_media`; it writes its PNG into the current directory.


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 Incoming links: Darcy flow through a homogeneous slab