Bit nozzle selection - Optimized hydraulics for two and three jets

Input(s)

$N_{a}$: Jet Size for nozzle 1 (in.)

$N_{b}$: Jet Size for nozzle 2 (in.)

$N_{c}$: Jet Size for nozzle 3 (in.)

Ca: Circulation Rate (gpm)

$\mathrm{Cb}$: Circulation Rate (gpm)

Wm: Mud Weight (ppg)

Cpa: Circulating Pressure 1 (psi)

$\mathrm{Cpb}$: Circulating Pressure $2(\mathrm{psi})$

$P_{\max }$: Max surface pressure (psi)

$\mathrm{d}$: Dia of the nozzle chosen (in.)

$\mathrm{n}$: No. of nozzles (unit)

Output(s)

A: Nozzle Area (in. ${ }^{2}$ )

Pba: Bit Nozzle Pressure Loss 1 (psi)

Pbb: Bit Nozzle Pressure Loss 2 (psi)

$P_{c a}$: Total Pressure Loss for pump pressure 1 except bit nozzle (psi)

$P_{c b}$: Total Pressure Loss for pump pressure 2 except bit nozzle (psi)

M: Slope of Line (unitless)

$P_{\text {iopt }}$: Optimum Pressure for Impact Force (psi)

$P_{\text {hopt }}$: Optimum Pressure for hydraulic Horsepower (psi)

Pilb: Pressure loss at the bit due to impact force (psi)

Phlb: Pressure loss at the bit due to hydraulic horsepower (psi)

$Q_{\text {iopt }}$: Optimum flowrate for impact force (gpm)

$Q_{\text {hopt }}$: Optimum flowrate for hydraulic horsepower (gpm)

$A_{\text {in }}$: Area of nozzle for Impact force (in. ${ }^{2}$ )

$A_{h n}$: Area of nozzle used for hydraulic horsepower (in. ${ }^{2}$ )

$A_{n}$: Area of nozzle used (in. ${ }^{2}$ )

Formula(s)

$$ \begin{aligned} & \mathrm{A}=\frac{\mathrm{N}_{\mathrm{a}}^{2}+\mathrm{N}_{\mathrm{b}}^{2}+\mathrm{N}_{\mathrm{c}}^{2}}{1303.8} \\ & P b a=(C a)^{2} * \frac{W m}{10858 * A} \\ & P b b=(C b)^{2} * \frac{W m}{10858 * A} \\ & P_{c a}=C p a-P b a \\ & P_{c b}=C p b-P b b \\ & M=\frac{\log \left(\frac{P_{c a}}{P_{c b}}\right)}{\log \left(\frac{C a}{C b}\right)} \\ & P_{\text {iopt }}=2 * \frac{P_{\max }}{M+1} \\ & P_{\text {hopt }}=2 * \frac{P_{\max }}{P_{\max }+1} \\ & \text { Pilb }=\left(\frac{P_{\text {iopt }}}{P_{\text {max }}}\right)^{1+M} * C a \\ & \text { Phlb }=\left(\frac{P_{\text {hopt }}}{P_{\text {max }}}\right)^{1+M} * C a \\ & Q_{\text {iopt }}=P_{\text {max }}-P_{\text {iopt }} \\ & Q_{\text {hopt }}=P_{\text {max }}-P_{\text {hopt }} \\ & A_{\text {in }}=\left(Q_{\text {iopt }}^{2} * \frac{W m}{10858 * P_{\text {max }}}\right)^{0.5} \\ & A_{\text {hn }}=\left(Q_{\text {hopt }}^{2} * \frac{W m}{10858 * P_{\text {max }}}\right)^{0.5} \\ & A_{n}=d *\left(\frac{A}{n * 0.7854}\right)^{0.5} \end{aligned} $$

Reference(s)

Formulas and Calculations for Drilling, Production and Workover, Second Edition, Lapeyrouse, Page: 166.