Fixed-pressure planar Darcy displacement (${\mu }_{h}LX+\frac{1}{2}(\mu -{\mu }_h)X^2=k\Delta Pt/\varphi$)

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.

If a thermal front has speed $Y^{\prime }=\Gamma U$ with $U$ the Darcy velocity, while a material front has speed $X^{\prime }=U/\varphi$, their position ratio from coincident initial locations is $Y/X=\Gamma \varphi$. Calling a retardation factor the ratio to pore velocity instead would give $Y/X=\Gamma$; these conventions must not be mixed. With cold viscosity $\mu$ and heated viscosity $b\mu$, the displacement's effective viscosity is ${\mu }_{\mathrm{eff}}=\mu [b-(b-1)\Gamma \varphi ]$. In the pressure resistance the cold layer has length $Y$, the hot injected layer has length $X-Y$, and the displaced layer has length $L-X$.