Question 4 of 7: Gas-well buildup – pseudopressure analysis
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
EGBC National Exam — Petroleum Engineering, 2015-May. 3 hours, closed book. This paper's own cover page reads “98-Pet-B5, Well Testing,” not Reservoir Mechanics, and every question below is pressure-transient/well-test analysis. NOTES item 4/5 state that five (5) questions constitute a complete exam and only the first five as they appear are marked; all seven questions on the paper are solved in full below. Four of the seven questions (Q3–Q6) are chart-reading questions built around semilog/log-log plots; where a printed data table exists (Q3, Q6) it was used directly, and every value read from a chart with no table (Q4, Q5) was read from the printed figure and is flagged check where it feeds a boxed result.
Reference texts: Lee, J., Well Testing, SPE Textbook Series Vol. 1 (diffusivity equation, radial flow, wellbore storage); Earlougher, R.C., Advances in Well Test Analysis, SPE Monograph Vol. 5 (Horner analysis, superposition in time, sealing faults, two-rate tests); Bourdet, D., Well Test Analysis: The Use of Advanced Interpretation Models, Elsevier (double-porosity model, hydraulically fractured wells); Warren, J.E. & Root, P.J., “The Behavior of Naturally Fractured Reservoirs,” SPE Journal, 1963; Cinco-Ley, H. & Samaniego, F., “Transient Pressure Analysis for Fractured Wells,” JPT, 1981 (infinite-conductivity vertical fracture linear flow).
Find. Approximate end of wellbore storage $\Delta t^*$, formation permeability $k$, and skin factor $S'$.
Approach. Two charts are supplied: a log-log plot of pseudopressure change vs. $\Delta t$, whose departure from the early unit-slope trend marks the end of wellbore storage, and a semilog Horner-ratio plot whose late-time straight line gives the slope $m$ (hence $k$) and, extrapolated to $\Delta t=1$ hr, the $p_{p,1hr}$ needed for skin. Both charts were digitized directly from the printed figure (Check throughout this question).
Fig. 3 – Log-log diagnostic (pseudopressure change vs. Δt, digitized). Early data track the unit-slope wellbore-storage line closely to about Δt≈0.04–0.06 hr, after which the curve visibly bends toward the flatter infinite-acting trend.
Fig. 4 – Semilog Horner-ratio plot (digitized), with the least-squares straight line fitted through the late-time (small-ratio) infinite-acting-radial-flow points.
End of wellbore storage. On the log-log plot (Fig. 3) the early points from $\Delta t\approx0.002$ to $\Delta t\approx0.045$ hr fall almost exactly on a unit-slope (45°) line – a ten-fold increase in $\Delta t$ produces a ten-fold increase in pseudopressure change – and the curve visibly departs from that trend shortly after,
$$\boxed{\Delta t^{*}\approx 0.06\ \text{hr}\quad(\text{check: chart-read})}$$
Semilog slope and permeability. A least-squares fit through the digitized late-time (small Horner-ratio) points on Fig. 4 gives
$$m\approx 3.77\times10^{7}\ \text{psia}^2/\text{cP per cycle}\quad(\text{check: digitized}),\qquad p_p^{*}\ (\text{at ratio}=1)\approx5.37\times10^{8}\ \text{psia}^2/\text{cP}$$
Using the gas-well buildup slope relation from the formula sheet, with $T=108.9+459.67\approx568.6\,{}^\circ\text{R}$,
$$k=\frac{1637\,q_gT}{mh}=\frac{1637(8000)(568.6)}{(3.77\times10^{7})(10)}$$
$$\boxed{k\approx 19.8\ \text{mD}}$$
Pseudopressure at $\Delta t=1$ hr. The Horner ratio at $\Delta t=1$ hr is $(t_p+1)/1=73$. Reading this point off the fitted straight line of Fig. 4,
$$p_{p,1hr}\approx p_p^{*}-m\log_{10}(73)\approx 4.67\times10^{8}\ \text{psia}^2/\text{cP}$$
Skin factor. Substituting into the formula sheet's gas-well buildup skin relation,
$$S'=1.151\left[\frac{p_{p,1hr}-p_{p,wf}}{m}-\log_{10}\!\left(\frac{k}{\phi\mu_gc_tr_w^2}\right)+3.23\right]$$
$$S'=1.151\left[\frac{4.67\times10^{8}-1.77\times10^{8}}{3.77\times10^{7}}-\log_{10}\!\left(\frac{19.8}{(0.106)(0.02)(2.238\times10^{-4})(0.46)^2}\right)+3.23\right]$$
$$\boxed{S'\approx 3.0}$$
