Question 10 of 11: Shale index from GR vs. spectral log; apparent/true porosity in an oil-bearing limestone
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exams — December 2016. 98-Pet-B1, Well Logging and Formation Evaluation (every question is log-interpretation content, not gas-engineering material). 3-hour closed-book exam, 11 questions, all marked, calculators and attached graphs/formula sheet permitted.
Reference texts: Bassiouni, Theory, Measurement, and Interpretation of Well Logs (SPE Textbook Series Vol. 4); Asquith & Krygowski, Basic Well Log Analysis, 2nd ed.; Ellis & Singer, Well Logging for Earth Scientists, 2nd ed.; Schlumberger, Log Interpretation Charts.
Check: Q3, Q7, Q8, Q9(b), Q10 and Q11 are built on the paper's printed logs and attached charts. Values printed as annotations on the logs (Q11's SSP, PSP and GR labels) are used exactly as printed. Values read off a curve or a chart (Q9(b) and the Q8 chart check, Q10's track readings) are read from the printed figure and flagged inline with their precision. All arithmetic that follows is exact.
Question 10: Shale index from GR vs. spectral log; apparent/true porosity in an oil-bearing limestone (10 marks)
[Figure not reproduced: Source GR and Th-U-K spectral log with Zone A. See the official exam paper or the cited reference text.]
Fig. Q10 — The printed log (page 11). Left: total GR, 0–100 API. Right: Th 0–20 ppm, U 0–10 ppm and K 0–5%. Zone A is the bracketed interval below the 16,600 ft depth mark.
Given (read from the printed log).Check: track readings read from the printed log, about ±10%
Track
Rest of the logged section
Zone A
Total GR (0–100 API)
≈ 65–80 API; lowest spike ≈ 50 API
Runs off the right edge of the track; the wrapped backup trace peaks near 140 API
Thorium, Th (0–20 ppm)
≈ 10 ppm; range ≈ 6–16 ppm
≈ 5 ppm (dips to ≈ 3 ppm, the lowest on the log)
Uranium, U (0–10 ppm)
≈ 2 ppm
≈ 7 ppm, peaks near 10 ppm
Potassium, K (0–5%)
≈ 1–1.5%
≈ 1–1.5% (no distinct change)
Find. (a) Shale index of Zone A from total GR and from the spectral (thorium) response, and which is more representative. (b) Apparent and true porosity of Zone A as an oil-bearing limestone drilled with oil-based mud.
Approach. No pure clean bed or pure shale bed is isolated on this short section. The baselines are therefore taken from the section itself: the lowest reading is the clean line, and the highest reading of the ordinary background trace is the shale line. The linear index $I_{sh}=(X-X_{clean})/(X_{shale}-X_{clean})$ is then applied once with $X=$ GR and once with $X=$ Th.
Shale index from total GR. Clean line $GR_{clean}\approx50$ API (lowest reading on the section). Shale line $GR_{shale}\approx80$ API (highest reading of the background trace outside Zone A). Zone A $\approx140$ API:
$$I_{sh,GR}=\frac{GR_A-GR_{clean}}{GR_{shale}-GR_{clean}}=\frac{140-50}{80-50}=3.0$$
An index above 1 has no physical meaning as a shale fraction, so it is capped:
$$\boxed{I_{sh,GR}\ge 1\ \Rightarrow\ \text{Zone A reads as }100\%\text{ shale on total GR}}$$
Shale index from the spectral thorium response. Clean line $Th_{clean}\approx3$ ppm (lowest reading). Shale line $Th_{shale}\approx16$ ppm (highest reading). Zone A $\approx5$ ppm:
$$I_{sh,Th}=\frac{Th_A-Th_{clean}}{Th_{shale}-Th_{clean}}=\frac{5-3}{16-3}=\frac{2}{13}$$
$$\boxed{I_{sh,Th}\approx 0.15\ (15\%)}$$
Compare and explain (part a, 6 pts). The two responses point in opposite directions. Total GR makes Zone A the hottest bed on the section, while thorium makes it one of the cleanest. The uranium track explains the conflict. U jumps from about 2 ppm in the background to about 7 ppm (peaks near 10 ppm) in Zone A, while K is unchanged and Th falls. With the usual conversion of roughly 8 API per ppm of uranium, that 5–8 ppm excess is worth about 40–65 API, which accounts for most of Zone A's excess gamma ray. Uranium in a clean carbonate is typically carried by organic matter, phosphate or fracture-deposited minerals, not by clay. Total GR counts it anyway, so the GR index badly overstates shaliness here. Thorium (and potassium) are held in clay minerals, so the thorium-based spectral index, ≈ 15%, is the more representative shale index. Zone A is a relatively clean, uranium-rich bed, which is consistent with part (b) describing it as a limestone reservoir.
Part (b): apparent and true porosity with oil-based mud.Check: the printed figure has no density track, so, as the paper's Note 2 invites, a bulk density $\rho_b=2.35\ \text{g/cm}^3$ is assumed for Zone A. The limestone matrix is $\rho_{ma}=2.71\ \text{g/cm}^3$.
Apparent porosity is what the density log computes on its standard fresh-water scale ($\rho_f=1.0$):
$$\phi_{app}=\frac{\rho_{ma}-\rho_b}{\rho_{ma}-1.0}=\frac{2.71-2.35}{2.71-1.0}=\frac{0.36}{1.71}\qquad \boxed{\phi_{app}\approx 0.211\ (21.1\%)}$$
True porosity uses the density of the fluid actually in the flushed zone that the tool reads. That zone holds 65% oil-mud filtrate plus 35% residual oil. Both are oil, so taking the filtrate at the reservoir oil's density gives
$$\rho_{fl}=S_{xo}\,\rho_{mf}+S_{or}\,\rho_o=0.65(0.70)+0.35(0.70)=0.70\ \text{g/cm}^3$$
$$\phi_{true}=\frac{\rho_{ma}-\rho_b}{\rho_{ma}-\rho_{fl}}=\frac{2.71-2.35}{2.71-0.70}=\frac{0.36}{2.01}\qquad \boxed{\phi_{true}\approx 0.179\ (17.9\%)}$$
The fresh-water scale therefore overstates porosity by about 3 porosity units, because light oil occupies the pores. The 35% residual oil saturation sets the proportions of the flushed-zone fluid mix. It changes the answer only if the oil-mud filtrate is lighter or heavier than the reservoir oil: a diesel-type filtrate at 0.85 g/cm³ gives $\rho_{fl}=0.80$ and $\phi_{true}=18.8\%$.
