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\bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \mathbf{u}_\alpha \bigg) \ \nabla T
= \frac{\delta E_H}{ \delta V \delta t} |
and
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its discrete computational scheme will be:
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\bigg( \sum_{a = \{w,o,g \}} \rho_\alpha^{k-1} \ c_{p \alpha}^{k-1} \ q_\alpha^{k-1} \bigg) T^{k-1} - \bigg( \sum_{a = \{w,o,g \}} \rho_\alpha^k \ c_{p \alpha}^k \ q_\alpha^k \bigg) T^k
= \sum_{a = \{w,o,g \}} \rho_\alpha^k \ c_{p \alpha}^k \ (q_\alpha^{k-1} - q_\alpha^k) \, (T_r^k + \epsilon_\alpha^k \delta P )p^k ) |
where
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| body | \delta p^k = p_e^k - p_{wf}^k |
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is drawdown, – formation pressure in th grid layer, – bottom-hole pressure across th grid layer, – reservoir temperature of th grid layer.The
axis is pointing downward along hole with th grid layer sitting above the th grid layer.
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The wellbore fluid velocity can be expressed thorugh the volumetric flow profile and tubing/casing cross-section area as:| LaTeX Math Block |
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| u_\alpha = \frac{q_\alpha}{\pi r_f^2} |
so that | LaTeX Math Block |
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| \bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \mathbf{u}_\alpha \bigg) \ \nabla T
= \frac{\delta E_H}{ \delta V \delta t} |
|
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