Question 7 of 7: Deliverability from a Back-Pressure Test

Question 7: Deliverability 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. $\bar p=3884$ psia (shut-in); four stabilized ($q$, $p_{wf}$) test points (table above).

Find. The deliverability equation $q=C(\bar p^2-p_{wf}^2)^n$ and $q$ at $p_{wf}=2500$ psia.

Approach. The empirical back-pressure equation linearizes on log-log axes: $\log q=\log C+n\log(\bar p^2-p_{wf}^2)$. Fit $n$ (line slope) and $C$ (intercept) from the four test points — equivalent to reading the straight line on the given log-log chart — then evaluate the fitted equation at the target $p_{wf}=2500$ psia.

1. Tabulate $\Delta p^2=\bar p^2-p_{wf}^2$.

2. Log-log regression (equivalent to the chart’s straight-line fit). Least-squares fit of $\log q$ vs. $\log\Delta p^2$ over the four points gives $\boxed{n=0.873}$ and $\boxed{C=4.079\times10^{-3}}$ (field units, $q$ in MSCFD, $\Delta p^2$ in psia$^2$) — i.e. $q=4.079\times10^{-3}(\bar p^2-p_{wf}^2)^{0.873}$, which reproduces all four test points to within about 1%.
3. Evaluate at $p_{wf}=2500$ psia. $\Delta p^2=3884^2-2500^2=8.835\times10^6\ \text{psia}^2$. $q=4.079\times10^{-3}(8.835\times10^6)^{0.873}$: $\boxed{q=4758\ \text{MSCFD}=4.76\ \text{MMSCFD}}$.