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In case of dual-barrier single-string completion with fluid (stagnant or moving) filling in the annulus (see Fig. 1) the HTC is defined by the following equation:


LaTeX Math Block
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\frac{1}{ r_{ti} \, U} = \frac{1}{r_{ti} \, U_{ti}} + \frac{1}{r_{ti} \, U_t} +
 \frac{1}{d_{ann} \, U_{ann}} +
\frac{1}{r_{ci} \, U_c}  + \frac{1}{r_c \, U_{cem}}


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Fig. 1. 


where


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LaTeX Math Inline
bodyr_t

outer radius of tubing


LaTeX Math Inline
body--uriencoded--r_%7Bti%7D

inner radius of the tubing

LaTeX Math Inline
body--uriencoded--h_t = r_t - r_%7Bti%7D

tubing wall thickness

LaTeX Math Inline
bodyr_c

outer radius of casing

LaTeX Math Inline
body--uriencoded--r_%7Bci%7D

inner radius of the casing

LaTeX Math Inline
bodyh_c = r_c - r_i

casing wall thickness

LaTeX Math Inline
body\lambda_t

thermal conductivity of tubing material

LaTeX Math Inline
body\lambda

thermal conductivity of fluid moving through the tubing

LaTeX Math Inline
body--uriencoded--\lambda_%7Bann%7D = \lambda_a \cdot \epsilon_a

effective thermal conductivity of the annulus 

LaTeX Math Inline
body\epsilon_a

Natural Convection Heat Transfer Multiplier

LaTeX Math Inline
body\lambda_a

thermal conductivity of fluid in the annulus

LaTeX Math Inline
body--uriencoded--\displaystyle U_%7Bti%7D = \frac%7B\lambda%7D%7B2 \, r_%7Bti%7D%7D \, %7B\rm Nu%7D_%7Bti%7D

Pipe Flow Heat Transfer Coefficient

LaTeX Math Inline
body--uriencoded--\displaystyle U_t = \frac%7B\lambda_t%7D%7Br_%7Bti%7D \cdot \ln (r_t/r_%7Bti%7D)%7D

Tubing Wall Conductive Heat Transfer Coefficient

LaTeX Math Inline
body--uriencoded--\displaystyle U_%7Bann%7D = \frac%7B\lambda_%7Bann%7D%7D%7Bd_%7Bann%7D%7D \, %7B\rm Nu%7D_%7Bann%7D

Annular Flow Heat Transfer Coefficient

LaTeX Math Inline
body--uriencoded--\displaystyle U_c = \frac%7B\lambda_c%7D%7Br_%7Bci%7D \cdot \ln (r_c/r_%7Bci%7D)%7D

Casing Wall Conductive Heat Transfer Coefficient

LaTeX Math Inline
body--uriencoded--\displaystyle U_%7Bcem%7D = \frac%7B\lambda_%7Bcem%7D%7D%7Br_c \cdot \ln (r_w/r_c)%7D

Cement Conductive Heat Transfer Coefficient

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