How a Gz bias reaches inclination, dip and azimuth

An MWD tool measures gravity with three accelerometers. Two are cross-axial. The third, the axial accelerometer or Gz, lies along the tool. In a straight hole it reads G·cos(inc): about 1 g with the tool vertical, and close to zero in a horizontal lateral.

A Gz bias is a constant offset on that reading. It is an error in one sensor, but it reaches the survey in several ways. It changes the calculated inclination at every station. It changes the calculated magnetic dip. Through both, it can change the azimuth. A total-gravity check cannot see it in a lateral.

This article follows a bias of b milli-g (mg) through each calculation. The worked examples use 2 mg because the effects are then easy to see. That is a large bias: the generic ISCWSA MWD error model allows 0.4 mg at 1σ for this term. To first order every effect scales with the bias, so 1 mg gives half of each value.

Inclination

To first order:

Δinc ≈ −(b / G) · sin(inc)

Here, Δinc is in radians, and b and G use the same units. A positive bias makes the axial reading too large, so the inclination reads low. The highside does not change, because it uses only the cross-axial readings. The result does not depend on toolface.

The effect is largest in a horizontal hole. There, 2 mg makes the inclination read 0.115° low at every station. The calculated hole points slightly down, so its TVD increases by 2.0 m for each 1,000 m of lateral, or 6.0 m over 3,000 m. On the published ISCWSA test well #1, 8,000 m long and horizontal for its last 2,600 m, the calculated TD is 10.8 m deeper.

The bias is the same at every station, so the error builds up along the well in one direction, like an uncorrected sag error, and does not average out like random noise. A negative bias gives the same values with the opposite sign.

A scale-factor error s on the same sensor behaves differently:

Δinc ≈ −s · sin(inc) · cos(inc)

It is zero in vertical and horizontal hole and largest at 45°. In a lateral, the bias matters; the scale-factor error mainly affects the build. See bias vs scale factor error for the general difference.

Why a gravity check misses it

The calculated total gravity changes by about:

ΔGtotal ≈ b · cos(inc)

TALLENTECH's default total-gravity tolerance is ±3 mg. A 2 mg bias passes that check at every inclination, and above 60° it uses no more than a third of it. In a lateral, where the inclination error is largest, the check cannot see the bias at all. A vertical station sees the full bias, but adds the axial scale-factor error and any error in the gravity reference to it. Only check shots at 0° and 180° separate the bias from them.

Check shots at 0° and 180°

With the tool vertical at 0° inclination, the axial accelerometer reads about +1 g. Turned over to 180°, it reads about −1 g:

r₀ = G·(1 + s) + b

r₁₈₀ = −G·(1 + s) + b

Half the sum of the two readings gives the bias, b. Half the difference gives G·(1 + s), and from it the scale-factor error. The bias needs no gravity reference: both shots see the same gravity, so it cancels. The scale-factor error does need one.

That reference must be local gravity in the units the tool was calibrated in. Gravity changes from place to place. Normal gravity is about 9.780 m/s² at the equator and 9.832 m/s² at the poles, a difference of about 5 mg, and it falls by about 0.3 mg for each 1,000 m of elevation. A tool calibrated on a bench in one place therefore reads slightly high or low at a rig in another. That change scales all three accelerometers equally, so it does not change inclination. Comparing a vertical shot with exactly 1 g mixes it into the result, so the axial error can look larger or smaller than it is.

Dip

The magnetic dip calculation uses the accelerometers to find the horizontal plane. A Gz bias tilts that plane, so to first order:

Δdip ≈ (b / G) · sin(inc) · cos(Am)

Here, Δdip is in radians, and Am is the magnetic azimuth. A horizontal hole heading magnetic north reads the dip 0.115° high for 2 mg. Heading south, it reads 0.115° low. Heading east or west, the dip does not change. Even the largest value is inside TALLENTECH's default dip tolerance of ±0.15°, but it uses about three quarters of it.

Long collar azimuth

The long collar (six-axis) azimuth also uses the accelerometers to rotate the magnetometer readings into the horizontal plane. The tilted plane moves the azimuth:

ΔAz ≈ (b / G) · tan(dip) · sin(inc) · sin(Am)

Here, ΔAz is in radians. This is the geometry of the ISCWSA weighting function for the axial accelerometer bias (ABZ). The effect is largest heading magnetic east or west and zero heading north or south. For 2 mg in a horizontal hole heading magnetic east or west, the azimuth error is 0.35° at a dip of 72° and 0.20° at 60°. A higher dip, as at high magnetic latitudes, makes it larger.

The dip error and the azimuth error have opposite patterns. The dip check sees the bias best on the headings where the long collar azimuth is least affected, and not at all where the azimuth is most affected.

ISCWSA test well #1 heads about 79° magnetic. On that well, the same 2 mg bias moves the long collar TD about 33 m sideways as well as 10.8 m down.

Figure 1 shows one survey station. Move the sliders to change the bias, the inclination, the dip and the azimuth. The calculations are the long and short collar methods that TALLENTECH's processing and MSA engine use.

Short collar azimuth

A short collar calculation does not use the axial magnetometer. TALLENTECH's method finds the azimuth at which the reference field, projected on the tool axis, agrees with the two cross-axial magnetometer readings. It still takes inclination and highside from the accelerometers, so a Gz bias reaches it through the inclination.

The result depends strongly on the hole attitude:

• In a horizontal hole away from magnetic east–west, the short collar removes almost all of the Gz azimuth error. At a dip of 72° and 2 mg, it is 0.000° heading 45° and 0.001° heading 80°, against 0.25° and 0.35° for the long collar.
• As the inclination falls, the two errors come closer together. On northerly headings, the short collar error can be larger than the long collar error: by up to about 3% at a dip of 72°, 8% at 60° and 15% at 50°. On southerly headings, it is smaller.
• Near horizontal east–west, the short collar is sensitive to every input error. At exactly horizontal east–west, it has no solution.

Values are for a 2 mg bias at a dip of 72°.

On ISCWSA test well #1, the short collar moves TD about 17 m sideways, half the long collar's shift. Most of it comes from the build and the 60° tangent, where the two methods agree. The vertical error is the same, because both methods use the same inclination.

What multi-station analysis does with it

Multi-station analysis (MSA) fits sensor corrections so that each station's total gravity, total field and dip agree with their reference values. A Gz bias enters the gravity and dip comparisons. In a lateral, the gravity comparison cannot see it, which leaves the dip.

We ran TALLENTECH's MSA engine, with its production settings, on test runs with a known 2 mg Gz bias and no other sensor error: a build-and-lateral run and a lateral-only run, at several headings, at dips of 72° and 60°. Without check shots, the engine holds the Gz bias at zero. It also holds the reference dip and field-strength offsets fixed. The axial magnetometer bias (Bz bias) is free, as it must be to correct drillstring interference.

• Inclination is not corrected. It stays 0.115° low, and the run's TVD is 6 m too deep after about 3,000 m of lateral.
• The dip misfit moves into a false Bz bias. At horizontal, a Bz bias of (b / G)·Be / sin(dip) cancels the dip error: about 116 nT for 2 mg at a dip of 72° and a field of 55,000 nT. The engine took 70 to 110 nT of it. Small cross-axial magnetometer terms took part of the rest.
• The long collar azimuth improved, by accident. At oblique headings, the false Bz bias turns the azimuth back. Heading 45° magnetic, the error fell from 0.25° to 0.09° in the build-and-lateral run and to −0.02° in the lateral-only run. Heading east or west, there was no dip misfit, so MSA could not see the bias and the full 0.35° remained.
• The short collar azimuth got worse. At oblique headings, it was almost free of the bias in the lateral before MSA. After MSA, the false magnetometer corrections added 0.02° to 0.05°. With the dip offset released, or the Bz bias held, the short collar error grew to 0.11° to 0.19°.

So a Gz bias can add azimuth error through MSA, most clearly to a short collar azimuth. It can also hide inside a magnetometer correction that has nothing to do with it. By calculation, a false Bz bias of 100 nT found on a north–south section turns the long collar azimuth on an east–west section by about 0.3° at a dip of 72°. While the same Gz bias is present, that turn mostly cancels the Gz error. If the Gz bias is corrected later and the Bz correction is kept, or if the Bz correction is applied to another tool, the 0.3° becomes an error.

Holding the Gz bias at its true value removed all of these errors. Releasing it recovered the 2 mg in the build-and-lateral run, but only because the test tool had no axial scale-factor error. With a 1.1 mg bias and a 0.08% scale-factor error, the released bias came out at 1.9 mg: across a build, the fit reads the bias and the scale-factor error together. The correction then overshot the lateral inclination by 0.05°. In the lateral-only run, the released bias was poorly determined.

In the build-and-lateral runs, a wrong Gz bias left signs: the engine treated the gravity values at the vertical and build stations as outliers and raised warnings. In the lateral-only runs, nothing in the quality checks showed it.

The uncertainty model and the shift

The generic ISCWSA model treats the axial bias as an uncertainty of 0.4 mg at 1σ. On ISCWSA test well #1, that term alone is 2.2 m (1σ) in the vertical at TD, beside about 20 m for uncorrected BHA sag and 9.4 m for tool misalignment. Uncertainty terms add in quadrature, so removing this one changes the 22.7 m total by only about 0.1 m. An actual, uncorrected bias does not add in quadrature. It is a shift: about 5.4 m per mg on that well. Multi-station analysis compares the two.

What to check

• Take check shots at 0° and 180° inclination for each tool. Record both the bias and the scale-factor error, and apply both.
• Compare a vertical shot with local gravity in the tool's calibration units, not with exactly 1 g.
• Do not use total gravity to confirm the axial bias in a lateral.
• Treat a Bz bias that appears mainly on north–south sections, or that changes with heading, as a possible Gz bias.
• Do not release the Gz bias across a build unless the scale-factor error is known.
• For a short collar azimuth, review the magnetometer corrections that MSA applies. They can add the error that the short collar had avoided.

References

• Williamson, H.S. (2000). Accuracy Prediction for Directional Measurement While Drilling. SPE Drilling & Completion 15(4), SPE 67616. The ISCWSA error model and its test wells.
• ISCWSA (January 2023). Definition of the ISCWSA Error Model, Rev5.13. Weighting functions for the axial accelerometer bias (ABZ) and scale factor (ASZ).
• ISCWSA. Introduction to Wellbore Positioning, October 2017.