Buildup test – permeability, skin, average pressure: 24-Pet-B5 Reservoir Mechanics, December 2015, Q4

Question 4: Buildup test – permeability, skin, average pressure (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. The 11-point Horner buildup table above, with $q$, $t_p$, $p_{wf}(\Delta t=0)$, $h$, $\phi$, $r_w$, $B_o$, $c_t$, $\mu_o$ as listed.

Find. $k$, skin factor $S$, and the average reservoir pressure.

Approach. Plot $p_{ws}$ vs $\log_{10}[(t_p+\Delta t)/\Delta t]$ (Horner plot); the early points (huge pressure rise over tiny $\Delta t$) are wellbore-storage distorted, so identify the true middle-time straight line from consecutive-point slope stability rather than fitting every point.

Dashed red line: Horner semilog straight line through the last 5 points ($\Delta t=0.53$ to 18.9 hr, $R^2=0.9998$). The first 6 points are wellbore-storage distorted and excluded from the fit.

Identify the MTR and fit the Horner slope. Consecutive-point slopes for the last 5 points cluster tightly ($-232$ to $-252$ psi/decade) while earlier points swing wildly ($-58$ to $-2014$); regression on the last 5 gives $R^2=0.9998$ with $$m=238.9\text{ psi/cycle},\qquad p^*=3702.2\text{ psia (intercept at Horner ratio}=1\text{)}$$
Permeability. $$k=\frac{162.6\,qB_o\mu}{mh}=\frac{162.6(800)(1.12)(0.9)}{238.9(50)}$$ $$\boxed{k\approx 11.0\text{ mD}}$$
Skin factor. At $\Delta t=1$ hr, Horner ratio $=(140+1)/1=141$; reading the fitted line gives $p_{1hr}=3188.7$ psia. $$S=1.151\left[\frac{p_{1hr}-p_{wf}}{m}-\log_{10}\left(\frac{k}{\phi\mu c_tr_w^2}\right)+3.23\right]$$ $$S=1.151\left[\frac{3188.7-400}{238.9}-\log_{10}\!\Big(\frac{11.0}{(0.15)(0.9)(5.8\times10^{-6})(0.3)^2}\Big)+3.23\right]$$ $$\boxed{S\approx 7.7\text{ (moderately damaged)}}$$
Average reservoir pressure. No drainage-shape/Dietz-shape-factor data is given to apply the full MBH correction, so – per this exam's own simplification – the Horner extrapolated pressure is taken directly as the average reservoir pressure estimate: $$\boxed{\bar{p}\approx p^*=3702\text{ psia}}$$

Quantity	Result
Semilog slope, $m$	238.9 psi/cycle
Permeability, $k$	11.0 mD
Skin factor, $S$	7.7
Average reservoir pressure, $\bar{p}$	3702 psia ($\approx p^*$)

Check: $\bar{p}\approx p^*$ is an accepted exam-level simplification only for a well centred in a roughly circular/square drainage area; a rigorous MBH (Matthews-Brons-Hazebroek) correction would need the drainage shape and $t_{pDA}$, neither of which is given here.

Topic. Horner buildup analysis, MTR identification, and $p^*$ as an average-pressure proxy.

Key relations. Horner ratio $(t_p+\Delta t)/\Delta t$; $m=162.6qB_o\mu/(kh)$; $S=1.151[(p_{1hr}-p_{wf})/m-\log_{10}(k/\phi\mu c_tr_w^2)+3.23]$.

Why this works. Superposition of the well's own production history plus the shut-in "negative twin" reduces a variable-rate history to a single semilog straight line in $\log[(t_p+\Delta t)/\Delta t]$; extrapolating that line to $\Delta t\to\infty$ (Horner ratio $\to1$) estimates the pressure the reservoir would reach given infinite shut-in time.

Common pitfall. Using the raw $p_{ws}$ value nearest $\Delta t=1$ hr instead of the value read off the fitted line at $\Delta t=1$ hr – the two differ whenever the nearest data point does not sit exactly at $\Delta t=1$ hr, and only the fitted-line value is consistent with the skin formula's derivation.

Source. Lee, Well Testing, ch.3–4 (Horner analysis); Earlougher, Advances in Well Test Analysis, ch.2, 4 (buildup superposition, average pressure).

Question 4 of 7: Buildup test – permeability, skin, average pressure

Question 4: Buildup test – permeability, skin, average pressure (20 marks)