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Table 2 Key wellbore equations

From: Modelling an unconventional closed-loop deep borehole heat exchanger (DBHE): sensitivity analysis on the Newberry volcanic setting

\(\frac{\partial M^{\kappa }}{\partial t} = q^{\kappa } + F^{\kappa }\) (7)
\(M^{\text{E}} = \rho_{\text{L}}S_{\text{L}} \left(U_{\text{L}}+\frac{u^{2}_{\text{L}}}{2}+gz\cos {\theta }\right) +\rho_{\text{G}}S_{\text{G}} \left(U_{\text{G}}+\frac{u^{2}_{\text{G}}}{2}+gz\cos {\theta }\right)\) (8)
\(F^{\text{E}} = -\lambda \frac{\partial T}{\partial z} - \frac{1}{\sigma }\frac{\partial }{\partial z} \left[\sigma \rho_{\text{L}}S_{\text{L}}u_{L} \left(h_{\text{L}}+\frac{u^{2}_{\text{L}}}{2} + gz\cos {\theta }\right) + \sigma \rho _{\text{G}}S_{\text{G}}u_{\text{G}} \left(h_{\text{G}}+\frac{u^{2}_{\text{G}}}{2} + gz\cos {\theta }\right)\right]\) (9)