Circulating Bottomhole Pressure, Fracture Safety and Bit Nozzle Sizing: 24-Pet-A4 Oil and Gas Well Drilling and Completion, May 2013, Q4

Question 4: Circulating Bottomhole Pressure, Fracture Safety and Bit Nozzle Sizing (equal value, part (c) 10 points)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Depth $12{,}000$ ft; mud weight $11$ ppg; $Q=600$ gpm; pump pressure $3{,}000$ psi; friction losses per table above; $3$ bit nozzles; fracture gradient $12.5$ ppg.

Find. (a) circulating bottomhole pressure; (b) whether it is safe against the fracture gradient; (c) the (equal) nozzle sizes in 32nds of an inch.

Approach. Circulating BHP adds only the ANNULAR friction losses (drill-collar and drill-pipe annulus) to the static hydrostatic column, since losses inside the string are spent before the fluid reaches bottom. Compare that to the fracture pressure for the safety check. For the nozzles, the bit pressure drop is whatever remains of the pump pressure after every other listed loss, then invert the standard bit-hydraulics equation for total nozzle area.

(a) Circulating bottomhole pressure. Static hydrostatic $=0.052(11)(12{,}000)=6{,}864$ psi. Add only the two annular friction terms (drill-collar annulus $250$ psi $+$ drill-pipe annulus $150$ psi): $P_{BH,circ}=6{,}864+250+150$, so $\boxed{P_{BH,circ}=7{,}264\ \text{psi}}$.
(b) Fracture safety check. Fracture pressure at $12{,}000$ ft $=0.052(12.5)(12{,}000)$, so $\boxed{P_{frac}=7{,}800\ \text{psi}}$. Since $P_{BH,circ}=7{,}264\ \text{psi} < P_{frac}=7{,}800\ \text{psi}$, there is a $536$ psi margin — $\boxed{\text{yes, safe to continue drilling}}$.
Bit pressure drop. Pump pressure is spent across every loss in the flow path including the bit: $\Delta P_{bit}=P_{pump}-(\Delta P_{surf}+\Delta P_{DP}+\Delta P_{DC}+\Delta P_{DCann}+\Delta P_{DPann})=3{,}000-(50+800+600+250+150)$, so $\boxed{\Delta P_{bit}=1{,}150\ \text{psi}}$.
(c) Total nozzle area. Using $\Delta P_{bit}=\dfrac{MW\,Q^2}{12{,}032\,C_d^2\,A_t^2}$ with $C_d=0.95$ (standard bit discharge coefficient): $A_t^2=\dfrac{11(600)^2}{12{,}032(0.95)^2(1{,}150)}=\dfrac{3{,}960{,}000}{12{,}487{,}700}=0.3171\ \text{in}^4$, so $\boxed{A_t=0.563\ \text{in}^2}$.
Per-nozzle diameter. With 3 equal nozzles, $a_n=A_t/3=0.1877\ \text{in}^2$, so $d_n=\sqrt{4a_n/\pi}=0.489\ \text{in}=15.6/32$ in, rounding to the nearest standard 32nd: $\boxed{d_n=16/32\ \text{in} = 1/2\ \text{in per nozzle}}$.

Check: nozzle sizing assumes a standard bit discharge coefficient $C_d=0.95$ (not given in the source data) and equal-size nozzles; rounding the computed 15.6/32 in up to the nearest stocked size (16/32 in) is the conservative choice (slightly larger total area, marginally lower actual $\Delta P_{bit}$ and pump pressure than the target).

Quantity	Value
(a) Circulating bottomhole pressure	7,264 psi
(b) Fracture pressure at 12,000 ft	7,800 psi — safe (536 psi margin)
Bit pressure drop, $\Delta P_{bit}$	1,150 psi
Total nozzle area, $A_t$	0.563 in$^2$
(c) Nozzle size (3 nozzles, equal)	16/32 in each (computed 15.6/32 in)

Topic: Circulating bottomhole pressure, formation-fracture safety, and bit-nozzle (hydraulics) sizing.

Key relations: $P_{BH,circ}=0.052\,MW\,D+\sum \Delta P_{annulus}$; $P_{frac}=0.052\,G_{frac}\,D$; $\Delta P_{bit}=\dfrac{MW\,Q^2}{12{,}032\,C_d^2\,A_t^2}$.

Why this works: bottomhole pressure while circulating equals the static hydrostatic column plus whatever additional resistance the returning fluid must overcome in the annulus (friction losses in the string itself are "used up" reaching the bit and never reappear at the bottom). The bit-nozzle equation follows from Bernoulli's equation applied across a sharp-edged orifice, with $C_d$ correcting for the vena-contracta and turbulent losses; because pump pressure is a fixed budget split across every loss in series, whatever is not accounted for elsewhere must be consumed at the bit.

Common pitfall: adding ALL the friction losses (including the in-string ones) into the circulating BHP calculation, or forgetting that the bit pressure drop is a byproduct of the OTHER losses (found by difference from pump pressure), not computed independently first.

Source: Bourgoyne et al., Applied Drilling Engineering, Ch. 4 (drilling hydraulics) & Ch. 5 (bit hydraulics, nozzle sizing and jet impact force), §4.2, §5.4.

Question 4 of 5: Circulating Bottomhole Pressure, Fracture Safety and Bit Nozzle Sizing

Question 4: Circulating Bottomhole Pressure, Fracture Safety and Bit Nozzle Sizing (equal value, part (c) 10 points)