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24-Pet-B5 Reservoir Mechanics · May 2016

Question 4 of 6: Buildup test – permeability, initial pressure, and skin (Horner)

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

EGBC National Exam — Petroleum Engineering, 2016-May. 3 hours, closed book, non-communicating calculator. This paper's own cover page reads “98-Pet-B5, Well Testing,” not Reservoir Mechanics — every question below is pressure-transient/well-test analysis. NOTES items 4/5 state that five (5) questions constitute a complete exam and only the first five as they appear are marked; all six questions on the paper are solved in full below. Three of the six questions (Q3, Q4, Q6) are chart-reading questions built around semilog/log-log plots with no printed data table for Q3/Q4; every value read from those charts is flagged check where it feeds a boxed result. Q6 ships a short printed data table for its early-time linear-flow fit; its flow-regime identification uses the full log-log pressure-change/derivative plot.

Reference texts: Lee, J., Well Testing, SPE Textbook Series Vol. 1 (diffusivity equation, radial flow, wellbore storage, superposition); Earlougher, R.C., Advances in Well Test Analysis, SPE Monograph Vol. 5 (Horner analysis, interference/pulse tests, reservoir-limit tests); Bourdet, D., Well Test Analysis: The Use of Advanced Interpretation Models, Elsevier (derivative diagnostic plot, flow-regime identification); Cinco-Ley, H. & Samaniego, F., “Transient Pressure Analysis for Fractured Wells,” JPT, 1981 (infinite-conductivity vertical fracture linear flow).

Question 4: Buildup test – permeability, initial pressure, and skin (Horner) (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.

Check: Horner ratios and shut-in pressures below were read from the printed Horner-plot chart (no printed data table for this question) — the chart's own $x$-axis runs in the standard reversed-log convention (ratio decreasing left to right).

Given. $q=480$ STBD, $N_p=20{,}000$ STB (so $t_p=N_p/q=41.7$ hr), $p_{wf}=2832$ psia, $h=20$ ft, $\phi=0.18$, $r_w=0.5$ ft, $B_o=1.2$ bbl/STB, $c_t=2.6\times10^{-5}\ \text{psi}^{-1}$, $\mu=1.5$ cP. Digitized Horner data (excerpted; full 16-point set spans ratio $=9780$ down to $48.7$):

Horner ratio vs. shut-in pressure (read from the printed chart)
$(t_p+\Delta t)/\Delta t$$p_{ws}$ (psia)
97802883
6723006
2383157
1373205
963237
643269
493293

Find. Permeability $k$, initial reservoir pressure $p_i$, and skin factor $S$.

Approach. Plot $p_{ws}$ vs. $\log_{10}[(t_p+\Delta t)/\Delta t]$ (Horner plot); regress the straight-line MTR (excluding the earliest, storage-distorted points), extrapolate to ratio $=1$ for $p^\ast\approx p_i$ (the shut-in period is short relative to the reservoir, so $p^\ast$ is taken directly as $p_i$), and read $p_{1hr}$ off the same line at $\Delta t=1$ hr for the skin equation.

1 10 100 1000 10000 100000 2650 2900 3150 3400 3650 Horner time ratio (tp+dt)/dt Shut-in pressure (psia) Q4 - Buildup Horner plot
Digitized Horner data (red, reversed-log x-axis) with the fitted MTR line (dashed); shaded band marks the regression window ratio $=238$ down to $48.7$.
  1. Fit the Horner straight line. Least-squares fit of $p_{ws}$ vs. $\log_{10}[(t_p+\Delta t)/\Delta t]$ over the 9 points from ratio $=238$ down to $48.7$ ($R^2=0.999$, excluding the earlier storage-affected points) gives $$m\approx 196.1\ \text{psi/cycle},\qquad p^\ast\ (\text{extrapolated to ratio}=1)\approx 3625\ \text{psia}$$
  2. Permeability. $$k=\frac{162.6q\mu B_o}{mh}=\frac{162.6(480)(1.5)(1.2)}{(196.1)(20)}$$ $$\boxed{k\approx 35.8\ \text{mD}}$$
  3. Initial reservoir pressure. With a producing time of only $t_p=41.7$ hr against a shut-in duration comparable to $t_p$, $p^\ast$ is taken directly as the (near-virgin) initial pressure: $$\boxed{p_i\approx 3625\ \text{psia}}$$
  4. Skin factor. Reading $p_{1hr}$ off the fitted line at Horner ratio $=t_p+1=42.7$, $$p_{1hr}=3625-196.1\log_{10}(42.7)\approx 3305\ \text{psia}$$ $$S=1.151\left[\frac{p_{1hr}-p_{wf}}{m}-\log_{10}\!\left(\frac{k}{\phi\mu c_tr_w^2}\right)+3.23\right]$$ $$S=1.151\left[\frac{3305-2832}{196.1}-\log_{10}\!\left(\frac{35.8}{(0.18)(1.5)(2.6\times10^{-5})(0.5)^2}\right)+3.23\right]$$ $$\boxed{S\approx -1.9}$$
ResultValue
Horner slope, $m$≈ 196 psi/cycle
Permeability, $k$≈ 35.8 mD
Initial pressure, $p_i$≈ 3625 psia
Skin factor, $S$≈ −1.9