ESSP for Ideal and Nonideal Shale Membranes: 24-Pet-B1 Natural Gas Engineering, December 2015, Q10

Question 10: ESSP for Ideal and Nonideal Shale Membranes (10 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.

Quantity	Value
Formation temperature, T	200°F
Mud-filtrate resistivity (at F.T.), R_mf	0.5 Ω·m
Formation-water resistivity (at F.T.), R_w	0.1 Ω·m
Shale resistivity (nonideal case, at F.T.), R_sh	2 Ω·m

Find. ESSP for (a) a perfect (ideal) shale membrane, and (b) a nonideal membrane with the given R_sh.

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(R_mf/R_w) law; the reduced SP is read from the departure-curve chart of R_mf/R_w vs. ESP parameterized by (R_sh/R_mf)@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) Nonideal membrane — chart parameters. $\frac{R_{mf}}{R_w}=\frac{0.5}{0.1}=5.0\qquad \left(\frac{R_{sh}}{R_{mf}}\right)_{@F.T.}=\frac{2.0}{0.5}=4.0$
(b) Read ESSP off the departure-curve chart. Entering the chart at R_mf/R_w=5.0 on the curve labelled (R_sh/R_mf)=4: $E_{SSP,\text{nonideal}}\approx\boxed{-30\text{ mV}}$ — 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.

Case	ESSP
(a) Perfect (ideal) shale membrane	−61.4 mV
(b) Nonideal membrane, R_sh/R_mf=4	≈ −30 mV

Check: part (b)'s value is read from the departure chart on the Attachment page, whose E_SP axis is labelled only every 20 mV. The reading was taken by tracing the printed curves against the chart's own axes (E_SP linear, 0 to −120 mV; R_mf/R_w logarithmic, 1 to 30): the (R_sh/R_mf)=4 curve crosses R_mf/R_w=5.0 at −29.5 mV, and the adjacent (R_sh/R_mf)=6 curve crosses the same ordinate at −58 mV, so the two curves are far enough apart that the reading cannot be confused between them. Quote it as −30 mV with about ±5 mV of chart precision. 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.

Topic: Static SP (SSP) for ideal vs. nonideal (leaky) shale membranes.

Key relations: $$K=61.3+0.133T$$ ; $E_{SSP}=-K\log_{10}[(R_{mf})_{eq}/(R_w)_{eq}]$ (ideal); departure chart of R_mf/R_w vs. ESP parameterized by (R_sh/R_mf)@F.T. (nonideal).

Why this works: The simple K·log formula assumes the adjacent shale is a PERFECT cation-selective membrane; when the shale is not much more resistive than the mud filtrate, ionic current leaks through it non-selectively, and the resulting membrane EMF is a function of the RATIO R_sh/R_mf as well as R_mf/R_w — which is exactly the two-parameter chart used in part (b).

Common pitfall: Applying the ideal K·log(R_mf/R_w) formula unconditionally whenever R_sh is given, instead of recognizing that a LOW R_sh/R_mf ratio signals a nonideal membrane requiring the departure chart, not the closed-form equation.

Source: Bassiouni, Ch. 5 (SP for nonideal shale membranes); Schlumberger, Log Interpretation Principles, Ch. 4.

Question 10 of 10: ESSP for Ideal and Nonideal Shale Membranes

Question 10: ESSP for Ideal and Nonideal Shale Membranes (10 marks)