Question 1 of 12: Resistivity of a Log and Invasion Profiles

Question 1: Resistivity of a Log and Invasion Profiles (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.

(a) Factors affecting the resistivity reading

A resistivity log records the true (or apparent) resistivity of the rock-fluid system in front of the tool, which is governed by Archie's relation $R_t = a\,R_w / (\phi^m S_w^n)$. Four independent factors control the reading:

• Porosity, φ. Resistivity is inversely proportional to φ^m — a higher-porosity rock carries more conductive brine per unit volume and reads a lower resistivity for the same saturation.
• Water saturation, S_w. Hydrocarbons are non-conductive; as S_w falls (S_h rises) the current path through the brine becomes more tortuous and R_t rises sharply (R_t ∝ S_w⁻ⁿ).
• Formation-water resistivity, R_w. R_w itself depends on brine salinity and formation temperature (R₂=R₁(T₁+6.77)/(T₂+6.77)); a fresher or cooler water reads a higher R_w and hence a higher R_t for the same rock and S_w.
• Rock texture / cementation (tortuosity factor a, cementation exponent m). Grain size, sorting, and degree of cementation change how tortuous the pore-brine current path is; a poorly-sorted, well-cemented rock has a higher m and reads a higher resistivity than a clean, well-sorted sand of the same φ. (Mud-filtrate invasion near the wellbore is a fifth, geometry-driven factor — developed in part (b).)

(b) Invasion profiles — transition vs. annulus

When water-base mud filtrate (resistivity R_mf) invades a permeable bed, it displaces the native fluids radially away from the wellbore. How the resistivity decays from the flushed zone (R_xo) out to the undisturbed formation (R_t) falls into recognizable invasion "profiles," read directly off a resistivity-vs-radius plot such as the one a suite of tools with different depths of investigation (microlog, shallow laterolog/SFL, deep induction/IL) effectively samples.

i) Transition zone. The mud filtrate and the native (oil + connate water) fluids mix progressively as the invasion front advances, so there is no sharp fluid boundary — the resistivity curve decreases (or increases, if R_mf < R_w) monotonically and smoothly from R_xo to R_t across the invaded annulus, di to dj. This is the profile assumed by most three-resistivity (SFL/ILM/ILD-type) interpretation charts because it produces a well-behaved, single-valued function of radius.

ii) Annulus zone. In an oil- or gas-bearing bed, filtrate first displaces the more mobile connate water ahead of the oil bank, so a ring of filtrate-flushed, near-water-saturated rock briefly exists just ahead of the true invasion front, before that ring itself mixes back with the annular mud filtrate/native-water transition farther out. This produces a local resistivity bump (the "annulus effect") between the flushed zone and the undisturbed formation: resistivity rises to a local maximum then falls again to R_t. The annulus effect is transient (it dissipates over days to weeks) and is a diagnostic that the bed is genuinely hydrocarbon-bearing, not simply water-wet.