Heat flux density ($\mathbf{q}$)

Heat flux density is a vector $\mathbf{q}$ describing the rate of heat transfer per unit area, measured in $\mathrm{W}{\mathrm{m}}^{-2}$. Through a planar face of area $A$ with uniform normal component $q$, the signed rate is $\dot{Q}=Aq$ in $\mathrm{W}$; positive $q$ means heat travels in the chosen positive normal direction. Fourier's law determines the conductive contribution. In a bar with a constant cross-sectional area, steady thermal conduction without internal sources or side losses makes both $q$ and $\dot{Q}$ constant along the bar. Constant $\dot{Q}$ need not mean constant $q$ when the cross-sectional area varies.

A heat flow probe combines subsurface temperature measurements with an estimate of thermal conductivity to infer conductive heat flux. Multiple sensor depths help separate a stable geothermal gradient from surface-temperature transients and installation disturbance.

The sign depends on the coordinate convention. Let $z$ increase downward and let $k>0$ be the vertical thermal conductivity. Fourier's law gives downward heat flux $q_z=-kdT/dz$. The upward-positive heat flux is thereforeA subsurface temperature that increases with depth thus corresponds to positive upward conductive loss. The underlying gradient law is the same as in an ordinary conducting slab. ^[ref]

For example, a measured gradient of $25\mathrm{K}\mathrm{k}{\mathrm{m}}^{-1}$ and $k=2\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$ imply $q_{\mathrm{up}}=0.050\mathrm{W}{\mathrm{m}}^{-2}$. This inference requires representative thermal conductivity and an appropriate conduction model. Conductive heat flux can differ from total heat transport where fluid circulation also carries substantial energy.