Question 7 of 7: Deliverability Equation and Absolute Open Flow from a Back-Pressure Test

Question 7: Deliverability Equation and Absolute Open Flow from a Back-Pressure Test (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. $\bar p=3000$ psia; four stabilized ($q$, $p_{wf}$) test points (table above).

Find. The deliverability equation $q=C(\bar p^2-p_{wf}^2)^n$ and the AOF.

Approach. Compute $\Delta p^2=\bar p^2-p_{wf}^2$ for each point, linearize by taking logs, least-squares fit $n$ (slope) and $C$ (intercept) on the log-log plot, then evaluate the fitted equation at $p_{wf}\approx0$ for AOF.

1. Compute $\Delta p^2$ for each point. $\Delta p^2=\bar p^2-p_{wf}^2=3000^2-p_{wf}^2$:

2. Linearize and fit. $\log_{10}q=\log_{10}C+n\log_{10}\Delta p^2$. Least-squares regression of $\log_{10}q$ on $\log_{10}\Delta p^2$ across the four points gives $\boxed{n=0.750}$ and $\boxed{C=2.00\ \text{SCFD/psia}^{2n}}$ — the fit reproduces every test point to within 0.1 SCFD, confirming a clean four-point back-pressure line with a textbook-typical exponent ($0.5\le n\le1.0$).
3. Absolute open flow. AOF is the rate at $p_{wf}=14.7$ psia (atmospheric, the standard AOF reference): $q_{AOF}=C(\bar p^2-14.7^2)^n=2.00(3000^2-14.7^2)^{0.750}$: $\boxed{q_{AOF}=328{,}626\ \text{SCFD}=0.329\ \text{MMSCFD}}$.