Given. A Vogel IPR curve is given directly (so $q_{o,max}=1500$ STB/day needs no fitting); the well must lift that oil through 8000 ft of 2.441 in ID tubing to a fixed 400 psig at the wellhead, where a surface choke sits.
Vogel $q_{o,max}$ (given)
1500 STB/day
Average reservoir pressure, $\bar P_R$
4000 psig
Bubble-point pressure, $P_b$
4500 psi
Well depth / tubing ID
8000 ft / 2.441 in
GOR ($f_w=0$)
400 SCF/STB
Required wellhead pressure, $P_{wh}$
400 psig
Find. (a) The natural oil rate this well delivers to 400 psig at the wellhead; (b) the choke bean size currently installed.
Check: oil API gravity and gas gravity are not stated in this question's own data block. The same paper's Q4/Q5 both use 35°API oil and 0.65 gas gravity for otherwise-similar wells — those values are adopted here for the tubing-intake calculation and flagged as an engineering assumption, not a given. The no-slip mixture tubing-intake model (homogeneous liquid+gas, no relative velocity, hydrostatic-dominated, friction neglected) is the same one used for Questions 4 and 5 of this paper, for internal consistency.
Approach. March the no-slip mixture gradient UP the tubing from an assumed $P_{wf}$ and bisect on $P_{wf}$ until the traverse lands exactly on the required 400 psig at the wellhead; read the corresponding oil rate off the given Vogel curve; then size the choke for that rate and wellhead pressure with a standard critical-flow correlation.
Part (a) — tubing-intake pressure. Bisecting $P_{wf}$ so that marching the no-slip mixture gradient (GOR = 400 SCF/STB, 35°API, 0.65 gas gravity, 150°F assumed avg. flowing temperature) up 8000 ft lands on exactly 400 psig at surface gives $\boxed{P_{wf}=1844.1\ \text{psi}}$.
Part (a) — oil rate from the IPR. $R=P_{wf}/\bar P_R=1844.1/4000=0.4610$: $q_o=1500\times(1-0.2(0.4610)-0.8(0.4610)^2)=1500\times0.7381$. $\boxed{q_o=1106.6\ \text{STB/day}}$ — this is the operating point where the reservoir's IPR and the tubing's intake requirement intersect.
Part (b) — choke sizing (Gilbert critical-flow correlation). $P_1=435\,q_L\,\text{GLR}^{0.546}/d^{1.89}$ ($q_L$ in Mstb/day, GLR in scf/STB, $d$ in 64ths, $P_1$ upstream/wellhead in psig — the Ros-family form permitted by the question). With $q_L=1.1066$ Mstb/day, GLR$=400$: $d=\left(\dfrac{435\times400^{0.546}\times1.1066}{400}\right)^{1/1.89}$. $\boxed{d\approx6.2\ \text{64ths}\ (\approx3/32\ \text{in})}$ — the nearest standard bean is a 6/64 in choke.
Fig. 3 — Nodal analysis: the Vogel IPR intersected against the tubing-intake requirement at a fixed 400 psig wellhead pressure, giving the operating point (1107 STB/day, 1844 psi).
