= Thermal front in a porous medium
{title2=$V_T=\Gamma U,\quad\Gamma=C_f/C_{\rm eff}$}
= Porous thermal front
{synonym}
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 $C_{\rm eff}=\phi C_f+(1-\phi)C_s$, while the advected heat flux is $C_fUT$ with $U$ the <Darcy velocity>. Neglecting conduction gives $C_{\rm eff}T_t+C_fUT_x=0$, so the thermal front speed is $\Gamma U$ with $\Gamma=C_f/C_{\rm eff}$. The material <pore velocity> is $U/\phi$, so thermal retardation relative to that speed is $\Gamma\phi$. Temperature-dependent viscosity couples thermal-front perturbations back to the <Darcy flux>.
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