Heat loss over an incremental length of a well (two-phase flow)

Input(s)

$T_{s}$: Temperature in the Well (Saturation Temperature) $\left({ }^{\circ} \mathrm{F}\right)$

$T_{e}$: Undisturbed Formation Temperature $\left({ }^{\circ} \mathrm{F}\right)$

y: Distance from the Bottom of the Well (ft)

$k$: Thermal Conductivity of Earth $\left(=33.6 \mathrm{BTU} /\left(\mathrm{ft} \mathrm{d}^{\circ} \mathrm{F}\right)\right)$

$f(t)$: Dimensionless Time Function that Represents the Transient Heat Transfer to the formation (dimensionless)

Output(s)

$d q:$ Heat Loss over an Incremental Length of the Wellbore (BTU/h)

Formula(s)

$$ \mathrm{dq}=\frac{2 \pi \mathrm{k}\left(\mathrm{T}_{\mathrm{s}}-\mathrm{T}_{\mathrm{e}}\right)}{\mathrm{f}(\mathrm{t})} \mathrm{dy} $$

Reference(s)

Ramey Jr, H. J. (1981). Reservoir Engineering Assessment of Geothermal Systems. Department of Petroleum Engineering, Stanford University. Page: 6.12.