Question 10 of 10: ESSP for Ideal and Nonideal Shale Membranes
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Examinations, 17-Pet-B1, Well Logging and Formation Evaluation — December 2019, 3 hours, closed book (Casio or Sharp approved calculators permitted), 10 questions, all marked.
Reference texts: Bassiouni, Theory, Measurement, and Interpretation of Well Logs (SPE Textbook Series Vol. 4); Asquith & Krygowski, Basic Well Log Analysis, 2nd ed. (AAPG); Ellis & Singer, Well Logging for Earth Scientists, 2nd ed.; Schlumberger, Log Interpretation Charts / Log Interpretation Principles and Applications.
Question 10: ESSP for Ideal and Nonideal Shale Membranes (10 marks)
Find. ESSP for (a) a perfect (ideal) shale membrane, and (b) a nonideal membrane with the given Rsh.
Approach. (a) Apply the attachment's ESSP formula directly, using K=61.3+0.133T. (b) An imperfect (leaky) shale membrane no longer follows the simple K·log(Rmf/Rw) law; the reduced SP is read from the departure-curve chart of Rmf/Rw vs. ESP parameterized by (Rsh/Rmf)@F.T.
Temperature coefficient K.$$K=61.3+0.133T=61.3+0.133(200)=61.3+26.6=\boxed{87.9}$$
(a) Ideal (perfect) membrane — direct formula.$$E_{SSP}=-K\log_{10}\!\left(\frac{R_{mf}}{R_w}\right)=-87.9\log_{10}\!\left(\frac{0.5}{0.1}\right)=-87.9\log_{10}(5)=-87.9(0.699)=\boxed{-61.4\text{ mV}}$$
(b) Read ESSP off the departure-curve chart. The chart is reproduced on source page 13: (Rmf/Rw) on a logarithmic ordinate from 1 to 30, ESP on a linear abscissa from 0 to −120 mV, and a family of curves labelled (Rsh/Rmf)@F.T. = 1, 2, 4, 6, 8, 10, 12, 15, 20. Entering at Rmf/Rw=5.0 and running across to the curve labelled 4:
$$E_{SSP,\text{nonideal}}\approx\boxed{-27\text{ mV}}$$
— a much smaller deflection than the ideal −61.4 mV, because a shale resistivity only 4× the mud filtrate is a comparatively leaky (nonideal) cationic membrane that develops only a fraction of the ideal electrochemical EMF. The chart is steep here: the neighbouring curve (Rsh/Rmf)=6 does not reach Rmf/Rw=5.0 until about −58 mV, so which curve is entered matters far more than the precision of the read. (The exam's own attachment page reproduces a similar departure chart with a worked example annotated at ESP=−50 mV giving (Rmf/Rw)eq≈4 — that is the chart's FORWARD use (an SP reading to Rmf/Rw for a water-resistivity determination) with different input parameters than this part, so it is not re-used directly here, but confirms the same chart family and the reasonableness of a chart-read answer in the −20 to −30 mV range.)
Case
ESSP
(a) Perfect (ideal) shale membrane
−61.4 mV
(b) Nonideal membrane, Rsh/Rmf=4
≈ −27 mV
Check: part (b) is a chart read from the printed attachment, so −27 mV carries about ±3 mV. It was read against the chart’s own axes (plot box 0 mV at the left border to −120 mV at the right; logarithmic ordinate with printed 2, 3, 4, 5, 8 and 10 gridlines), and cross-checked against two other points on the same curve family that can be read independently: the (Rsh/Rmf)=6 curve reaches Rmf/Rw=4.0 at −37 mV and 4.5 at about −49 mV, and the (Rsh/Rmf)=4 curve begins at Rmf/Rw=4.27 at the chart’s −10 mV left limit. The qualitative point — a shale only 4× more resistive than the mud filtrate develops well under half of the ideal SSP — is the intended lesson and is insensitive to that tolerance.
