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24-Pet-A6 Well Logging and Formation Evaluation · May 2015

Question 7 of 8: Pressure Buildup Test — Afterflow, Permeability, Skin

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

Notes on this paper

98-Pet-A6 — Reservoir Mechanics · National Exams, May 2015 · 3 hours, closed book, Casio/Sharp approved calculator only · eight problems set (candidates answer Problems 1 and 2 plus any three of the remaining six per the exam's own instructions; all eight are solved in full below as a complete study resource), all questions equal value.

Reference texts: Craft, B.C. & Hawkins, M.F., Applied Petroleum Reservoir Engineering, 3rd ed. (material balance, decline curves, transient well testing, permeability averaging); Ahmed, T., Reservoir Engineering Handbook, 5th ed. (material balance, decline-curve analysis, pressure buildup, PVT correlations); Golan, M. & Whitson, C.H., Well Performance, 2nd ed. (reserves methods, water/gas influx); Lyons, W.C. (ed.), Standard Handbook of Petroleum and Natural Gas Engineering, 3rd ed.

Check: this paper's own title page reads “98-PET-A6: Reservoir Mechanics”, not “Well Logging and Formation Evaluation” — the subject heading it is listed under does not match its content. Every problem below is answered as the paper actually printed it (material balance, decline-curve analysis, pressure-transient testing and permeability averaging — classic Reservoir Mechanics/Fundamental Reservoir Engineering topics), not well-logging.

Problem 7: Pressure Buildup Test — Afterflow, Permeability, Skin (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. Rising-liquid-level buildup data above; $q=988$ STB/d, $B=1.126$ rbbl/STB, $\mu=0.55$ cp, $h=7$ ft, $\phi=0.14$, $c_t=16\times10^{-6}$ psi$^{-1}$, $r_w=0.5$ ft, cumulative production 12,173 STB.

Find. (1) end of afterflow distortion; (2) $k$; (3) skin $S$; (4) $\Delta p_{skin}$.

Approach. A rising liquid level is classic wellbore-storage (afterflow); estimate the storage coefficient $C$ from the annular geometry, use the standard storage-duration criterion together with the semilog data's own departure from the early curved trend to bracket when afterflow ends, fit the late (infinite-acting) points on a Horner plot for $m$ and $p_{1hr}$, then get $k$, $S$ and $\Delta p_{skin}$ from the standard buildup formulas.

  1. Wellbore-storage coefficient. For a rising liquid level, $C=\dfrac{144\,A_{wb}}{5.615\,\rho}=\dfrac{144(0.0218)}{5.615(54.8)}$, so $\boxed{C=0.0102\ \text{bbl/psi}}$.
  2. Pseudo-producing time. $t_p=\dfrac{24\,N_p}{q}=\dfrac{24(12{,}173)}{988}$, giving $t_p=295.7$ hr, used to form the Horner ratio $(t_p+\Delta t)/\Delta t$.
  3. Horner-plot regression. Fitting $p_{ws}$ vs. $\log_{10}[(t_p+\Delta t)/\Delta t]$ over the 10 latest points ($\Delta t\geq9.86$ h) gives slope magnitude $\boxed{|m|=489.3\ \text{psi/cycle}}$ ($R^2=0.987$) and intercept $p_{1hr}=3567$ psia at $\Delta t=1$ hr (Fig. 5); the 6 earlier points sit visibly below this trend — the signature of wellbore-storage/afterflow distortion.
  4. (1) End of afterflow. Using the provisional $k$ and $S$ from steps 5–6 in the storage-duration criterion $\Delta t_{wbs}\gtrsim\dfrac{170{,}000\,C\,e^{0.14S}}{kh/\mu}=\dfrac{170{,}000(0.0102)e^{0.14(1.26)}}{(29.0)(7)/0.55}$ gives $\boxed{\Delta t_{wbs}\approx5.6\ \text{hr}}$, comfortably below the $\Delta t=9.86$ hr cutoff actually used for the straight-line fit — consistent margin, confirming afterflow has died out well before the fitted points begin.
  5. (2) Permeability. $|m|=\dfrac{162.6\,qB\mu}{kh}\ \Rightarrow\ k=\dfrac{162.6(988)(1.126)(0.55)}{7(489.3)}$, so $\boxed{k=29.0\ \text{md}}$.
  6. (3) Skin factor. $S=1.151\left[\dfrac{p_{1hr}-p_{wf}}{|m|}-\log_{10}\!\left(\dfrac{k}{\phi\mu c_tr_w^2}\right)+3.23\right]$, with $p_{wf}=p_{ws}(\Delta t=0)=709$ psia. $\dfrac{3567-709}{489.3}=5.841$; $\phi\mu c_tr_w^2=0.14(0.55)(16\times10^{-6})(0.25)=3.08\times10^{-7}$, so $\log_{10}(29.0/3.08\times10^{-7})=7.975$. $S=1.151[5.841-7.975+3.23]$, giving $\boxed{S=1.26}$ — a very small, near-undamaged skin, consistent with the question's own statement that no significant formation damage is expected.
  7. (4) Pressure drop across the altered zone. $\Delta p_{skin}=0.87\,|m|\,S=0.87(489.3)(1.26)$, so $\boxed{\Delta p_{skin}=537\ \text{psi}}$.
1000 1750 2500 3250 4000 1 10 100 p₁hr ≈ 3567 psia Δt, hr (log scale) p_ws, psia m = -489 psi/cycle grey = afterflow-affected (Δt<9.86 hr)
Fig. 5 (Horner/MDH semilog): p_ws vs. log Δt. Grey points (Δt<9.86 hr) sit above the infinite-acting straight line (afterflow-distorted); the 10 later points define the IARF trend used for m and p₁hr.
Check: $r_e=1320$ ft is given for a possible drainage-area average-pressure (Dietz/MBH) step beyond the buildup's stated primary objective; since the question asks only for afterflow time, $k$, $S$ and $\Delta p_{skin}$, $r_e$ is not needed further here.
QuantityValue
(1) Time afterflow ceased distorting data≈5.6 hr (theoretical); IARF fit begins by 9.86 hr
(2) Formation permeability, $k$29.0 md
(3) Skin factor, $S$1.26
(4) Pressure drop across altered zone, $\Delta p_{skin}$537 psi