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.