Mechanical energy balance for wellbore fluids

Input(s)

$$p_{1}, p_{2}$$: Pressure at positions 1 and 2 (\mathrm{psi})\$

$$\rho_{f}$$: Fluid Density (lbm/ft3)

$$g_{c}$$: Gravitational System Conversion Constant $$\left(32.17(\mathrm{lbm} / \mathrm{ft}) /\left(\mathrm{lbf} / \mathrm{s}^{2}\right)\right)$$

$$g$$: Acceleration of Gravity $$\left(\mathrm{ft} / \mathrm{s}^{2}\right.$$ )

$$Z_{1}, Z_{2}$$: Fluid elevation at Positions 1 and $$2(\mathrm{ft})$$

$$v_{1}, v_{2}$$: Fluid velocity at Positions 1 and $$2(\mathrm{ft} / \mathrm{s})$$

$$E_{l}$$: Irreversible Energy Loss between Positions 1 and 2 (ft lbf/lbm)

Output(s)

$$W$$: Work done by the Fluid while in Flow (ft lbf/lbm)

Formula(s)

$W=-\left[\int_{P_{1}}^{P_{2}} \frac{d p}{\rho_{f}}+\frac{g}{g_{c}}\left(Z_{2}-Z_{1}\right)+\frac{\rho_{f}\left(v_{2}^{2}-v_{1}^{2}\right)}{2 g_{c}}+E_{l}\right]$

Reference(s)

Watson, D., Brittenham, T., & Moore, P. L. (2003). Advanced Well Control (Vol. 10). Society of Petroleum Engineers, Page: 8.

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