In case of single-string well completion with flowing fluid in the annulus (see Fig. 3) the HTC is defined by the following equation:

\frac{1}{ d_{ti} \, U} = \frac{1}{d_{ti} \, U_{ti}} + \frac{1}{\lambda_t} \, \ln \frac{d_t}{d_{ti}} +
+ \frac{1}{\lambda_{a, \rm eff}} \ln \frac{d_{ci}}{d_t} +
\frac{1}{\lambda_c} \ln \frac{d_c}{d_{ci}} + \frac{1}{\lambda_{cem}} \ln \frac{d_w}{d_c}

where

	outer radius of tubing (with outer radius )
	inner diameter of the tubing (with inner radius )
	tubing wall thickness
	outer radius of casing (with outer radius )
	inner diameter of the casing (with inner radius )
	casing wall thickness
	thermal conductivity of tubing material
	thermal conductivity of fluid moving through the tubing
	effective thermal conductivity of the annulus
	Natural Convection Heat Transfer Multiplier
	thermal conductivity of fluid in the annulus
	heat transfer coefficient (HTC) between inner surface of tubing and moving fluid

In case the annulus is filled with stagnant fluid the annulus fluid convection will be natural and the Convection Heat Transfer Multiplier is a function of Rayleigh number .

In case the annulus fluid is moving the annulus fluid convection will be forced and the Convection Heat Transfer Multiplier can be approximated as:

See also