Silicon Dale

Drillhole De-Surveying

I was astonished to receive an email a few days ago from Datamine's central software development office, asking how their own de-survey algorithm worked. Astonished for two reasons.

First, that nobody there seems to know (and therefore they can't answer clients who asked that perfectly reasonable question) - despite their having used the algorithm in 'new' Earthworks software which they are selling. How can a software vendor sell a supposedly new product, or indeed even an old one, without knowing how a vital part of it actually works? (How much else within their systems is a mystery to them, I wonder ?)

Second, that they came to me for this information. Although I certainly do know (and will explain in this article - especially for the benefit of those Datamine users who may wish to know), it surprises me that Datamine think I have any inclination to help them out of a hole, considering the circumstances of my departure from the company seven years ago.

De-surveying of drillhole data is an arcane and specialised art. The basic problem is that nobody actually knows the exact three dimensional co-ordinates of any point along any drillhole - except for its collar location. Too many assumptions have been made in the past and are still made routinely - for example that holes which are drilled vertically actually remain vertical for their entire length.

The 'remedy' to this problem is to survey holes at intervals along their length. These surveys provide directions (dip and azimuth) at known downhole depths. What they emphatically do not provide is a set of absolute XYZ co-ordinates along the hole. It is necessary to carry out some mathematical processing to obtain such co-ordinates. There are modern downhole survey technologies, such as the Reflex MAXIBOR system, which provide much more closely spaced measurements than the old photographic and other methods. But even these provide only orientation data, not co-ordinates.

The simplest method of determining co-ordinates down hole is the straight-line method. In this method, the hole direction at each survey point is projected to the depth of the next survey point. This has the advantage of being very easy to compute. Co-ordinates at each survey point are computed directly by simple trigonometry from the position and orientation of the previous survey point. Unfortunately, this method leads to large errors which increase with depth through ignoring the fact that a drillhole is in fact continuously curved, and does not bend abruptly and only at points where survey readings happen to have been taken. Also there is a systematic bias in the interpreted co-ordinates, in that each surveyed direction is taken to apply for a length of hole below - but not above - the position of the measurement.

A centred straight-line method avoids the bias problem, by assigning a given direction to a length of hole both above and below each measurement, half-way to the next higher or lower measurement. Unfortunately this method still does not account for the real curvature of the hole.

A curved-hole method was developed in the 1970s for Rio Tinto's suite of mining software. In this method, the co-ordinates at any downhole depth were computed by separately interpolating dip and azimuth of the hole between adjacent survey points. Effectively, what this meant was that the horizontal co-ordinates were obtained by drawing a circular arc from one survey point to the next using azimuth data, and the vertical co-ordinate by drawing a circular arc from one survey point to the next using dip data. Unfortunately this method suffered from serious disadvantages. Imagine two adjacent survey points, dipping 70 due east at depth 100 metres and 70 due west at depth 200 metres. The old Rio Tinto method would interpret the behaviour of the hole between these two points as follows: the dip would remain the same 70 at all depths from 100m to 200m. The azimuth would change continuously from 90 at 100m to 270 at 200m though whether it would change clockwise or anticlockwise is not defined in this pathological example. In either case, this section of the hole would be interpreted as corkscrew-like. This is in fact one of a number of pathological cases where this particular method gives ambiguous or even nonsensical results.

In 1982-3, while developing Datamine, I was familiar with the Rio Tinto code, and was aware of the problems with their de-survey algorithm. It became clear that the problems arose through misunderstanding of the simple 3D geometry. In fact, the simplest curved-hole interpretation should use spherical arcs rather than circular. The survey measurements are in fact unit vectors in 3D space: dip and azimuth cannot be treated independently. Between any two orientations, for a given, known, downhole length, there is exactly one spherical arc between them. Knowing the co-ordinates of the first point, the co-ordinates of the second are uniquely fixed, as are the positions of every point between. Furthermore, since the spherical arc is tangential to the orientation at each survey point, curvature along the entire hole is guaranteed to be continuous, with no sharp angles as in any straight-line method. There is just one pathological case - where the directions at successive survey points are exactly opposite (e.g. 70 downwards to the east at one point, and 70 upwards to the west at the next) where a spherical arc cannot be defined. However, this case is so implausible that it can be discounted as a real problem. This algorithm has been used in Datamine from the earliest days, and found to be reliable almost all the time. There have been occasional problems resulting from 'special cases' of the trigonometric functions that are used in the computation, but these were steadily patched over the years: the principle of this method has consistently been found to be reliable.

There was a 'new' Rio Tinto de-survey method, developed in the mid 1980s by Mark Howson and Ed Sides. This new method took the concept of continuity of curvature a step further and used piecewise spline functions to interpolate between adjacent survey points. It required each hole to have at least four survey points, and guaranteed continuity of first and second derivatives of the direction over the entire surveyed length. However, this method can be less stable than the spherical-arc method, in situations where, for example, survey points are irregularly spaced, and it cannot be used at all when there are only two or three survey points. Various comparative studies also showed that the actual discrepancies between the spline and spherical-arc methods were relatively small - certainly much smaller than the discrepancies between either and the straight-line methods.

Whichever downhole survey method is used, it is necessary to be aware that it offers only an interpretation - a model - of the real drillhole. No-one imagines that any hole actually follows a piecewise spherical or piecewise spline curve any more than it follows a set of straight lines. It is merely more likely that a curved interpretation is a bit closer to the unknown reality. And the close that you can get, the better your deposit model. It is very easy in a 200-metre hole to have a 10-metre uncertainty in the bottom-of-hole position. This can lead to errors of many thousands of tonnes of ore.

Stephen Henley
Matlock, England

Copyright © 2000 Stephen Henley
Drillhole De-Surveying, Earth Science Computer Applications, v.16, no.2, p.3-5