Back-Pressure Test and Absolute Open Flow: 24-Pet-B2 Oil and Gas Evaluation and Economics, May 2016, Q7

Question 7: Back-Pressure Test and Absolute Open Flow (20 marks)

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.

Flow rate (MSCFD)	$p_{wf}$ (psia)	$\bar p^2-p_{wf}^2$ (psia$^2$)
965	337.6	10,822
2,500	310.5	27,681
3,390	291.9	39,918
4,318	270.5	51,999

Average reservoir pressure $\bar p=352.4$ psia.

Find. Deliverability equation $q=C(\bar p^2-p_{wf}^2)^n$ and the absolute open flow (AOF) at $p_{wf}=14.7$ psia.

Approach. Fit the empirical deliverability equation by linear regression of $\log q$ vs. $\log(\bar p^2-p_{wf}^2)$ across all four test points, then extrapolate to atmospheric flowing pressure for the AOF.

Linearize and regress. Taking logs of $q=C(\bar p^2-p_{wf}^2)^n$ gives $\log q=\log C+n\log(\bar p^2-p_{wf}^2)$, linear in $\log(\Delta p^2)$. Least-squares fit through the four points: $$\boxed{n=0.934}, \qquad \boxed{C=0.175\ \text{MSCFD/(psia}^2)^{0.934}}$$
Check the fit. The fitted curve reproduces the four test points to within a few percent (e.g. at $\Delta p^2=10{,}822$: $q_{pred}=970$ MSCFD vs. 965 measured), confirming a stabilized single-line deliverability relation rather than a curved (non-Darcy-dominated at low rate) trend.
Absolute open flow. AOF is the rate at $p_{wf}=14.7$ psia (atmospheric): $$q_{AOF}=C\left(\bar p^2-14.7^2\right)^n=0.175\left(352.4^2-14.7^2\right)^{0.934}$$ $$\boxed{q_{AOF}=9{,}980\ \text{MSCFD}=9.98\ \text{MMSCFD}}$$

Log-log deliverability plot: the four measured points (black) fall on the fitted $q=C(\Delta p^2)^n$ line, extrapolated to $p_{wf}=14.7$ psia for the AOF (red).

Quantity	Result
Deliverability exponent, $n$	0.934
Deliverability coefficient, $C$	0.175 MSCFD/(psia$^2$)$^{0.934}$
Absolute open flow (AOF)	9,980 MSCFD (9.98 MMSCFD)

Topic: Gas well deliverability from a multi-point back-pressure (four-point flow-after-flow) test.

Key relations: $q=C(\bar p^2-p_{wf}^2)^n$; log-log linearization $\log q=\log C+n\log\Delta p^2$; $0.5\le n\le 1.0$ physically (turbulent-to-laminar range).

Why this works: The empirical back-pressure (LIT) equation captures both laminar Darcy flow ($n=1$) and fully turbulent non-Darcy flow ($n=0.5$) as limiting cases of the same power law, so a single log-log regression across the tested rate range characterizes the well without needing to separately model turbulence.

Common pitfall: Extrapolating the AOF far beyond the tested rate range without checking $n$ falls in the physically expected 0.5–1.0 band ($n=0.934$ here is close to the pure-laminar limit, plausible for this well’s relatively low rates) — a fitted $n$ outside that range would signal a data or fitting problem, not a real well.

Source: Lee & Wattenbarger, Gas Reservoir Engineering, Ch. 5 (deliverability testing); Katz et al., Handbook of Natural Gas Engineering, Ch. 7.

Question 7 of 7: Back-Pressure Test and Absolute Open Flow

Question 7: Back-Pressure Test and Absolute Open Flow (20 marks)