 Fixed-pressure planar Darcy displacement

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 $X^{\prime }=U/\varphi$. 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.