Thermal front in a porous medium ($V_T=\Gamma U,\Gamma =C_f/C_{\mathrm{eff}}$)

A sharp advected temperature transition in a fluid-filled porous medium. If fluid and solid equilibrate locally in temperature, their effective heat capacity per total volume is $C_{\mathrm{eff}}=\varphi C_f+(1-\varphi )C_s$, while the advected heat flux is $C_{f}UT$ with $U$ the Darcy velocity. Neglecting conduction gives $C_{\mathrm{eff}}{T}_t+C_{f}U{T}_x=0$, so the thermal front speed is $\Gamma U$ with $\Gamma =C_f/C_{\mathrm{eff}}$. The material pore velocity is $U/\varphi$, so thermal retardation relative to that speed is $\Gamma \varphi$. Temperature-dependent viscosity couples thermal-front perturbations back to the Darcy flux.