= Fixed-pressure planar Darcy displacement
{title2=$\mu_hLX+\tfrac12(\mu-\mu_h)X^2=k\Delta P\,t/\phi$}
For two incompressible fluids separated by a planar material front $X(t)$, the <Darcy flux> is uniform and the pressure drop sums the resistances of the two layers: $U=k\Delta P/[\mu X+\mu_h(L-X)]$. The front moves at the <pore velocity> $X'=U/\phi$. Integrating from $X(0)=0$ yields the displayed relation. The formula holds up to breakthrough $X=L$, with either sign of the viscosity difference; equal viscosities give constant speed.
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