The Resistivity Profile

From Three Beds to a Whole Well

Sections 2.3 and 2.4 followed the filtrate into three half feet of Ogbon-1 and weighed what three tools read of each. A log is every half foot, and the same three tools draw three curves down the whole well, so what a reader meets is not a profile but how the curves sit against one another. Figure 6.4 draws them. A half foot in which the well has a permeability, 386 ft of the 600 ft logged, is invaded; a shale is not, because the filtrate cannot enter it. The virgin rock is the well's own deep resistivity, the flushed zone follows Section 2.3 ($S_{xo} = S_w^{0.2}$, the empirical relation of PetroWiki's Well log interpretation, and the well's filtrate of 0.40 ohm.m), and the tools are Section 2.4's: induction tools with depths of investigation of 10, 30 and 90 in, each reading a conductivity sum weighted by Doll's geometrical factor (Doll, 1949).

Where the Curves Lie Together

In a shale nothing invades, so the flushed zone and the virgin rock are the same rock and the three curves lie on one line: at 5830 ft all three read 2.37 ohm.m. In a permeable bed they part. That is the first use of the three curves, and it needs no calculation: curves that lie together mean no permeable bed, curves that part mean one the filtrate has entered. With a front 24 in across and a transition a quarter of the radius wide, the default, the shallow curve reads below the deep curve over 214.5 ft of the 386 ft of permeable rock and above it over 171.5 ft, and nowhere equal to it: no permeable half foot of Ogbon-1 reads the same on both.

Which Way They Part

Which side the shallow curve falls on is the question of Section 2.3 asked down the well. The flushed zone reads $R_{xo}/R_t = (R_{mf}/R_w)\,S_w^{1.6}$ times the virgin rock, which is one at a water saturation of 23.7 %. Ogbon-1 draws that line through its oil. The upper oil sand (72 ft, 19 % water) and the shaly oil sand (72 ft, 20 % water) lie under it, so in them the shallow curve reads under the deep. At 5890 ft it reads 16.8 ohm.m against 20.8 ohm.m. The oil of the main sand lies over it, from 27 % water at its top, so there the shallow curve reads above the deep. At 6150 ft it reads 8.57 ohm.m against 7.24 ohm.m. All 164 ft of the main sand part that way, oil and water leg alike, and so do 7.5 ft of the limestone, where the well's small bed-to-bed variation lifts a few half feet over the line. The direction of the separation therefore sorts this well's main sand from its other beds, and not its oil from its water: the shallow curve reads 1.18 times the deep in the oil at 6150 ft and 2.06 times in the water leg at 6230 ft. What tells the two apart is the size of the separation and, above all, the resistivity itself.

A Ramp Is Not a Step

Section 2.4 gave the invasion a sharp front. PetroWiki says that Rt correction is "usually performed assuming a simple three-parameter, step-profile invasion model", and of that model that "This is clearly not realistic, but it allows a complex problem to be solved relatively simply" (PetroWiki, Formation resistivity determination). Figure 2.3 gave the front a transition, centred on it, and Figure 6.4 lets you set its width, from a step to a transition as wide as the front's radius. How much a tool notices depends on the contrast. In the water leg with a front 60 in across, where the flushed zone is 10 times the virgin rock, widening the transition from a step to the widest lifts the medium tool from 1.10 ohm.m to 1.19 ohm.m, 8 % higher, and the three tools no longer agree where the front is: read as a step, the shallow tool's reading is a front 66.5 in across, the medium's 68.0 in and the deep's 71.1 in, where the front is 60 in. In the upper oil sand at the same front, with a flushed zone only 31 % under the virgin rock, no reading moves by more than 1.2 %. A transition changes how a contrast is spread, so it matters in proportion to the contrast.

The figure leaves out the annulus. PetroWiki writes that "Annulus formation is a common phenomenon", caused by "the sweeping of conductive ions" out of the formation, and it advises: "Before performing an inversion for invasion parameters, be sure the cause of curve separation is actually invasion." It lists other causes of separation: tool-response effects such as shoulder effect, and formation effects such as dipping beds and drilling-induced fractures. In this figure the only cause of separation is the invasion put into the model; on a real log, separation is a flag to be checked, not a proof.

What the Deep Curve Costs

The deep curve is the closest of the three to the virgin resistivity, but closest is not equal. At 5890 ft with the default front it reads 20.8 ohm.m, 0.9 % under the true 20.9 ohm.m. Put it into Archie's equation as $R_t$ (Archie, 1942), with Ogbon-1's own porosity and shale volume so that only the resistivity is in question, and the whole well gives 277.5 ft of net pay against the true 277.0 ft and a hydrocarbon column of 38.2 ft against 38.1 ft: a small cost, because the deep tool takes only 2 % of its signal from the flushed zone.

Push the front out and widen the transition and the cost shows. With a front 120 in across and the widest transition, the deep curve at 5890 ft reads 18.4 ohm.m, 12.3 % under, and at 6230 ft in the water leg 0.941 ohm.m against a true 0.606 ohm.m, 55 % over. Archie then reads 80 % water in a sand that is all water, and the net pay is 284.0 ft, 7.0 ft more than the truth, all of it added at the base of the pay, which moves from 6190.5 to 6197.5 ft. The base of the pay is where the water saturation sits near the 50 % cutoff, so a deep curve that is too high there tips half feet over it. In the oil sands the error runs the other way, the deep curve under the virgin rock and Archie reading too much water (20.1 % against 18.9 % at 5890 ft), but their saturations are far from the cutoff and no pay is lost: 0.0 ft. Section 6.5 uses all three readings to correct the deep one.

References

• Archie, G. E. (1942). The electrical resistivity log as an aid in determining some reservoir characteristics. Transactions of the AIME. SPE-942054-G.
• Doll, H. G. (1949). Introduction to induction logging and application to logging of wells drilled with oil base mud. Journal of Petroleum Technology 1(6), 148-162. SPE-949148-G.
• PetroWiki. Formation resistivity determination. Society of Petroleum Engineers, petrowiki.spe.org/Formation_resistivity_determination.
• PetroWiki. Well log interpretation. Society of Petroleum Engineers, petrowiki.spe.org/Well_log_interpretation.