Given. A nodal-analysis problem: reservoir inflow (IPR) must be matched against the tubing's vertical lift performance (VLP) to reach the required wellhead pressure.
Productivity index, $J$ (liquid basis)
0.5 STBL/day/psi
Static reservoir pressure, $\bar P_R$
2800 psi
Bubble-point pressure, $P_b$
3000 psi
Tubing length, ID
4000 ft, 2.441 in
Required wellhead pressure, $P_{wh}$
160 psi
Natural producing GLR
100 SCF/STBL
Water cut, $f_w$
50%
Oil gravity / gas gravity
35°API / 0.65
Water specific gravity
1.07
Average flowing temperature
150°F
Find. (a) Whether 400 STBO/day flows naturally to 160 psi at the wellhead; (b) the gas injection rate needed to reach 500 STBO/day.
Check: with no gradient-curve chart supplied, the tubing pressure traverse is computed with a homogeneous, no-slip (liquid+gas mixed, no relative velocity) black-oil mixture: liquid volume from $f_w$ (oil and water both taken near-incompressible, $B_o\approx B_w\approx1$ since neither is given), free-gas volume from the real-gas law with $z$ from a Standing pseudo-critical fit and an explicit (Papay-type) correlation, marched in small increments so the local gas density updates with pressure. Friction is neglected (no roughness/velocity data given) — this is a hydrostatic-dominated approximation appropriate to the data provided, not a full mechanistic (Hagedorn–Brown/Poettmann–Carpenter) correlation. The reservoir's straight-line $J$ is used directly (as the question's own "productivity index" framing implies), consistent with Question 1 of this same paper.
Approach. (a) Get $P_{wf}$ from the straight-line IPR at the total liquid rate, then march the no-slip mixture gradient up the tubing from that $P_{wf}$ and compare the resulting wellhead pressure with the 160 psi requirement. (b) Repeat at the higher rate, then bisect on the total (natural + injected) GLR until the traverse lands exactly on 160 psi at surface; the injected gas is the difference from the natural GLR.
Part (a) — inflow at 400 STBO/day. With $f_w=0.5$, total liquid rate $q_L=400/(1-0.5)=800$ STBL/day. IPR: $P_{wf}=\bar P_R-q_L/J=2800-800/0.5=2800-1600$. $\boxed{P_{wf}=1200\ \text{psi}}$.
Part (a) — VLP check. Marching the no-slip mixture gradient upward from $P_{wf}=1200$ psi over 4000 ft at the natural GLR of 100 SCF/STBL gives a computed wellhead pressure of only $\boxed{P_{wh,computed}\approx138\ \text{psi}}$ — short of the 160 psi required. $\boxed{\text{Answer (a): No, the well will NOT flow naturally at 400 STBO/day}}$ — it falls just short and needs artificial-lift assistance.
Part (b) — inflow at 500 STBO/day. $q_L=500/(1-0.5)=1000$ STBL/day; $P_{wf}=2800-1000/0.5=2800-2000$. $\boxed{P_{wf}=800\ \text{psi}}$ (even less BHP available than in part a, since the higher rate pulls $P_{wf}$ down further).
Part (b) — required total GLR. Bisecting the total GLR (natural + injected) so the marched traverse from $P_{wf}=800$ psi over 4000 ft lands exactly on $P_{wh}=160$ psi gives $\boxed{\text{GLR}_{total}=229.5\ \text{SCF/STBL}}$.
Part (b) — injection rate. Injected GLR $=229.5-100=129.5$ SCF/STBL; at $q_L=1000$ STBL/day, $Q_{inj}=129.5\times1000$. $\boxed{Q_{inj}\approx1.295\times10^5\ \text{SCF/day}\ (0.130\ \text{MMscf/day})}$, injected at the tubing intake to lighten the column enough to reach the required 500 STBO/day.
Fig. 4 — Oil-rate IPR (from the given straight-line J, 50% water cut) versus the 400 and 500 STBO/day targets; natural flow (286 STB/day computed via nodal VLP check) falls short of 400, motivating gas-lift assistance for part (b).
