Reading True Rt

The Deep Reading Is Not Rt

Section 2.4 gave every resistivity tool a depth of investigation and showed that it is not a wall: invasion that is deep beside that depth pulls even the deep reading toward the flushed zone. This section takes the pull back out. Take the water leg of Ogbon-1 at 6230 ft with a front 96 in across. The virgin rock reads 0.606 ohm.m, the flushed zone 6.06 ohm.m, and the deep tool 0.788 ohm.m, 30.0 % above the truth. Archie's equation on that reading gives 88 % water in a rock that is all water, a wet sand that looks as if it holds oil, the same trap as taking the flushed reading for $R_t$ in Section 2.3. Figure 6.5 runs the three tools of Section 2.4, 10, 30 and 90 in, on the same three beds, and asks whether the other two readings can undo the pull.

Three Readings, Three Unknowns

A step profile has three unknowns: the flushed-zone resistivity $R_{xo}$, the radius of the front $r_i$ and the true resistivity $R_t$. In Doll's theory (Doll, 1949) a two-coil tool reads the conductivities of the rock around it, each weighted by the share of its signal that comes from there. If $G_j$ is the share of tool $j$'s signal from inside the front, its reading is

Divide by $R_t$ and what is left depends on the contrast $c = R_{xo}/R_t$ and the radius alone: $R_j/R_t = 1/(G_j/c + 1 - G_j)$. So the deep reading over the shallow and the medium over the deep are functions of $c$ and $r_i$ only, not of the bed. Plot one against the other for every contrast and every front and you have the mesh of plate (a). That is the tornado chart; PetroWiki describes it as "the solution offered to users of the dual induction logs", a "graphical solver called the tornado chart" (PetroWiki, Induction logging). A point that lands on the mesh names the contrast and the diameter, and then the deep reading gives the true resistivity:

The figure's default is the upper oil sand at 5890 ft with a front 60 in across, a step, and no error. The three tools read 15.4 ohm.m, 17.2 ohm.m and 19.9 ohm.m; the deep tool alone is 5.1 % below $R_t$. The two ratios are 1.29 and 0.86, the point lands at a contrast of 0.69 and a diameter of 60 in, and, since the deep tool takes 12 % of its signal from inside the 30 in radius, the formula returns 20.9 ohm.m, the true value. The fit is exact because the figure made these readings from a step with the same model it inverts: Ogbon-1 has only a deep resistivity log, so the shallow and medium readings are computed, and a chart that returns the truth here checks the arithmetic and nothing about the method.

Where the Chart Earns Its Keep

With a step and exact readings the chart gives back the true $R_t$ at every front from 12 to 120 in, in all three beds, while the deep tool does not. At 120 in the deep tool is 13.1 % below $R_t$ in the upper oil sand and 44.1 % above it in the water leg, where the flushed zone is 10 times as resistive as the virgin rock; the chart gives 20.9 ohm.m and 0.606 ohm.m. Plate (c) shows the gap opening as the front deepens. This is the case the tornado chart was made for.

What a Step Inversion Assumes

The chart is a forward model run backwards, and it is only as good as that model. It assumes a step, with nothing between the flushed zone and the virgin rock; it assumes the tools respond as Doll's theory says, which neglects skin effect and, in the words of PetroWiki, is valid only at very low conductivities; it leaves out the borehole and the beds above and below; and it assumes the readings are exact. The trouble is that it cannot check any of them. Three readings fix three unknowns exactly, so there is no spare equation to test the step with: a profile that is not a step still gives two ratios, and the ratios still name a contrast and a diameter. PetroWiki lists three invasion models for array-induction processing, step, ramp and annulus, and says that with the additional curves of an array the profile parameters can be solved for by inverse methods. Its example of the three-curve case is a simulated Phasor induction log of a gas reservoir in which "an annulus has developed": the deep reading falls well below $R_t$ and, with only three measurements, the three-parameter invasion model cannot return the correct $R_t$ (PetroWiki, Induction logging). The figure lets you break each assumption in turn.

An error in one reading. Put 3 % of error in the shallow reading in the water leg with a 60 in front. The chart's $R_t$ moves from the true 0.606 ohm.m to 0.608 ohm.m, 0.4 % above, but its contrast moves from 10 to 12, 16 % off. Even at 5 % in either direction its $R_t$ in the water leg is never more than 4.2 % off at any front it answers at. The resistivity is well determined; the contrast and the diameter are not determined nearly as well. Where the contrast is near one it is worse. In the upper oil sand the flushed zone reads only 0.69 of the virgin rock, the mesh folds back on itself, and a shallow reading 2.5 % high already puts the dot off the mesh at every front from 76 in up: no step reproduces the two ratios, and the chart returns nothing.

A ramp. Let the resistivity change log-linearly from the front out to twice its radius, a profile of the figure's own that is wider than the transition of Figure 2.3. In the main sand's oil with a front 60 in across, the deep tool reads 7.72 ohm.m, 7.2 % above the true 7.19 ohm.m. The chart's $R_t$ is 7.24 ohm.m, 0.6 % above, a good answer, but its diameter is 87 in: it is the step that reads like the ramp, not the front.

An annulus. Let a ring from the front to 1.5 times its radius hold 1.5 times the water saturation of the virgin rock, the water the filtrate pushed ahead of itself (the figure's own numbers, and an annulus needs oil to displace, so the water leg has none). In the main sand's oil the deep tool reads 6.50 ohm.m, 9.6 % below $R_t$, and the dot lands on the mesh at a contrast of 1.7 and 24 in. The chart returns 6.45 ohm.m, 10.4 % below the truth, which is further off than the deep tool; of the 70 fronts in that bed at which it answers at all, it is further off than the deep reading at 69. Nothing on the chart says the profile was not a step. In the upper oil sand the same annulus puts the dot off the mesh, which is a warning there, but not a test.

Crain (2005) put the same doubt in the title of a column, "Myth #1: Tornado charts correct for invasion", arguing that readings from invaded oil and gas zones often do not fall on the chart at all, so that no correction is made. The figure's off-the-mesh states are one way that happens. The figure's chart is its own forward model, not a service company's chart for its tools, so what it shows is what a step inversion does, not what any one tool's chart does. Where it lands on the mesh for a profile that is not a step it can even correct the wrong way: in the main sand's annulus it moves the deep reading from 6.50 ohm.m down to 6.45 ohm.m, when the truth, 7.19 ohm.m, lies above it.

Into Archie, With the Warning Attached

The chart's $R_t$, not the deep reading, goes into Archie's equation (Archie, 1942) with the well's porosity, $R_w$ of 0.04 ohm.m and $a$, $m$, $n$ of 1, 2 and 2. In the water leg with 96 in of invasion the deep reading gives 88 % water and the chart's $R_t$ gives 100 %, the truth. In the main sand's annulus the deep reading gives 31 % and the chart 31 %, where the well has 29 %: the correction did not help, and it looked as sure as when it did. Use the chart where the profile is plausibly a step and the contrast is large; compare it with the deep reading, and treat a disagreement as information about the profile, not about the rock. Section 6.6 puts the resistivity on the whole of Ogbon-1.

References

• Archie, G. E. (1942). The electrical resistivity log as an aid in determining some reservoir characteristics. Transactions of the AIME. SPE-942054-G.
• Crain, E. R. (2005). Myth #1: Tornado charts correct for invasion. Myth Interpretation, a series on common misconceptions. CWLS InSite, September 2005.
• 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. Induction logging. Society of Petroleum Engineers, petrowiki.spe.org/Induction_logging.