Question 3 of 6: Drawdown test – permeability, skin, and end of wellbore storage

Question 3: Drawdown test – permeability, skin, and end of wellbore storage (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. $q=550$ STBD, $h=20$ ft, $B_o=1.45$ bbl/STB, $\phi=0.23$, $c_t=6\times10^{-5}\ \text{psi}^{-1}$, $\mu=0.42$ cP, $r_w=0.5$ ft, $p_i=5665$ psia. Drawdown data read from the printed chart (excerpted; full 23-point set spans $t=0.13$ to $726$ hr):

Find. (a) Permeability $k$; (b) Skin factor $S$; (c) End of wellbore storage; (d) Acidizing recommendation.

Approach. Plot $p_{wf}$ vs. $\log t$ (MDH plot); the log-log $\Delta p$-vs-$t$ plot flattens out of its early steep decline by $t\approx0.5$–1 hr, marking the approach of the end of wellbore storage, while a regression search across the full semilog data set (excluding early storage-affected and any late boundary-affected points) finds the true infinite-acting radial-flow straight line spanning $t=4.2$ to $245$ hr ($R^2=0.999$) — comfortably more than a log cycle past the storage-affected region, the standard buffer. That line gives $m$ and the extrapolated $p_{1hr}$ for the permeability and skin equations.

1. Fit the middle-time-region (MTR) straight line. Least-squares fit of $p_{wf}$ vs. $\log_{10}t$ over $t=4.2$ to $245$ hr (12 points, $R^2=0.999$) gives slope $$m\approx 115.1\ \text{psi/cycle},\qquad p_{1hr}\approx 5600\ \text{psia (extrapolated on the fitted line)}$$
2. (a) Permeability. $$k=\frac{162.6q\mu B_o}{mh}=\frac{162.6(550)(0.42)(1.45)}{(115.1)(20)}$$ $$\boxed{k\approx 23.7\ \text{mD}}$$
3. (b) Skin factor. Using the drawdown skin formula with $p_i=5665$ psia and the fitted $p_{1hr}$, $$S=1.151\left[\frac{p_i-p_{1hr}}{m}-\log_{10}\!\left(\frac{k}{\phi\mu c_tr_w^2}\right)+3.23\right]$$ $$S=1.151\left[\frac{5665-5600}{115.1}-\log_{10}\!\left(\frac{23.7}{(0.23)(0.42)(6\times10^{-5})(0.5)^2}\right)+3.23\right]$$ $$\boxed{S\approx -3.9}$$
4. (c) End of wellbore storage. The log-log $\Delta p$-vs-$t$ plot shows a steep early decline (consistent with unit-slope storage domination) that has already flattened toward the eventual radial-flow trend by $t\approx0.5$–1 hr; the semilog straight line is not reliably established until $t\approx4$ hr (about a log cycle later, the standard buffer used to avoid residual storage distortion in the fitted slope). Check: chart-read estimate. $$\boxed{t_{WBS,end}\approx 1\ \text{hr}}$$
5. (d) Acidizing recommendation. The computed skin is strongly negative ($S\approx-3.9$), indicating a completion that already performs better than an undamaged (zero-skin) well — not a damaged one. $$\boxed{\text{No, acidizing is not recommended: }S<0\text{ shows no formation damage.}}$$