Total heat loss of a geothermal well

Input(s)

$T_{o}$: Inflowing Fluid Temperature $\left({ }^{\circ} \mathrm{F}\right)$

w: Mass Flow Rate $(\mathrm{lb} / \mathrm{h})$

$c$: Thermal Heat Capacity of the Fluid $\left(\mathrm{BTU} / \mathrm{lbm}^{\circ} \mathrm{F}\right)$

a: Geothermal Gradient $\left({ }^{\circ} \mathrm{F} / \mathrm{ft}\right)$

A: Diffusion Depth (ft)

$H$: Total Well Depth $(\mathrm{ft})$

$b$: Surface Temperature $\left({ }^{\circ} \mathrm{F}\right)$

Output(s)

q: Total Heat Loss of a Geothermal Well (BTU/h)

Formula(s)

$$ q=-w c\left[a H-\left(T_{o}+a A-b\right)\left(1-e^{\frac{-H}{A}}\right)\right] $$

Reference(s)

Horne, R. N., \& Shinohara, K. (1979). Wellbore Heat Loss in Production and Injection Wells. Journal of Petroleum Technology, 31(01), Page: 117.