Drilling guides › Directional survey calculations: minimum curvature step by step
Directional survey calculations: minimum curvature step by step
A directional survey measures where the hole is pointing at a given depth along the hole. From a series of those measurements we calculate where the hole actually is: how deep vertically, and how far north and east of the surface location. Those positions decide whether the well hits its target, whether it stays clear of nearby wells, and what the true vertical depth is for every pressure calculation. Getting them right matters well beyond the directional driller's office.
The terms
| Term | Meaning |
|---|---|
| Measured depth (MD) | Length along the hole from the depth reference (usually the rotary table or drill floor). |
| Inclination (I) | Angle of the hole from vertical: 0° is vertical, 90° is horizontal. |
| Azimuth (A) | Direction of the hole in the horizontal plane, measured clockwise from north, 0° to 360°. State which north: true, grid or magnetic. |
| True vertical depth (TVD) | Vertical distance below the reference. Hydrostatic and formation pressures use TVD. |
| Northing, easting | Horizontal displacement north (negative = south) and east (negative = west) of the well's surface location. |
| Dogleg (DL) | Total change in hole direction between two stations, combining inclination and azimuth change. |
| Dogleg severity (DLS) | Dogleg per 100 ft of MD (or per 30 m in metric). |
A survey tool only gives MD, inclination and azimuth. Everything else is calculated, and the result depends on how we assume the hole bends between stations.
The dogleg
The dogleg is the angle between the two direction vectors of the hole at the upper and lower stations. It is found from:
The second term shows something that surprises many people: a change in azimuth counts for little in a near-vertical hole, because sin I is small, but counts fully in a horizontal one. A 6° turn at 10° inclination is a mild dogleg; the same turn at 60° is a severe one.
Dogleg severity matters because pipe running through a dogleg is bent, and under tension it is pressed against the wall. High DLS raises torque and drag, wears casing, causes key seats, and gives fatigue in rotating drill pipe. Casing and completion design often limit the DLS that can be accepted in parts of the well.
Minimum curvature method
The minimum curvature method assumes the hole between two stations is the smoothest possible curve: a circular arc lying in one plane, tangent to the measured direction at each station. It is the industry standard for final well positions. The steps are:
- Find the dogleg DL from the formula above, in radians for the next step.
- Find the ratio factor RF, which corrects the straight-line averages for the curvature of the arc.
- Calculate the changes in north, east and vertical depth, and add them to the upper station's position.
For normal survey intervals, RF is only a little more than 1. Its main purpose is to make the method consistent for any dogleg. A common mistake is to calculate DL in degrees and then use it in the RF formula, which needs radians. If DL is zero (a perfectly straight section), the formula would divide by zero; set RF to 1.
Worked example 1: two stations by minimum curvature
Upper station: MD 5,000 ft, I 10°, A 40°. Position TVD 4,962.4 ft, N 210.3 ft, E 168.7 ft. Lower station: MD 5,100 ft, I 14°, A 48°. ΔMD = 100 ft.
The hole went 100 ft along its length but only 97.8 ft deeper, and moved about 20.7 ft sideways in that interval.
Simpler methods for comparison
Older or quick methods treat the interval as a straight line.
- Tangential uses only the lower station's angles for the whole interval. It is simple but biased: in a building hole it puts the well too far out and too shallow. Errors accumulate quickly and it should not be used for anything that matters.
- Average angle uses the average inclination and the average azimuth along a straight line. Over short intervals and small doglegs it comes very close to minimum curvature, which makes it a good field check by hand. Take care averaging azimuths across north, for example 350° and 10°, whose average is 0°, not 180°.
| Method (same two stations) | ΔN (ft) | ΔE (ft) | ΔTVD (ft) |
|---|---|---|---|
| Minimum curvature | 14.75 | 14.58 | 97.80 |
| Average angle (I 12°, A 44°) | 14.96 | 14.44 | 97.81 |
| Tangential (I 14°, A 48°) | 16.19 | 17.98 | 97.03 |
Over one interval the tangential method is already 3.4 ft off to the east. Over a few hundred stations, that kind of bias can put a well far from where it really is.
Worked example 2: an azimuth change at high angle
Two stations 90 ft apart, both at 60° inclination. The azimuth turns from 120° to 126°.
Minimum curvature gives ΔN = −42.42 ft, ΔE = +65.32 ft and ΔTVD = 45.03 ft for this interval. The negative northing means the hole is heading south of east, as an azimuth between 90° and 180° should. A turn of only 6° is close to 6°/100 ft of dogleg severity here, which could already exceed a casing wear or completion limit.
Closure and vertical section
Two more numbers appear on every survey report. The closure is the straight-line horizontal distance from the surface location to the station, √(N² + E²), with its direction. The vertical section is the horizontal displacement projected onto a chosen plane, usually the direction from the surface location to the target: VS = closure × cos(target azimuth − closure azimuth). The directional driller steers to keep the well on the planned vertical section and TVD. For the lower station in example 1, the closure is √(225.1² + 183.3²) ≈ 290.3 ft towards 39.2°; with a target direction of 45°, the vertical section is about 288.8 ft.
Where survey errors come from
The calculation is exact for its assumptions, but the inputs are not. Magnetic tools are disturbed by nearby casing, steel in the BHA and the earth's changing field. Depth is measured on pipe that stretches and is corrected only roughly. Inclination sensors drift. Each survey therefore has an uncertainty, which grows with depth and is shown as an ellipse around the calculated position. That is why anti-collision work uses the uncertainty ellipses, and not only the calculated centre line, when deciding whether two wells are too close.
Metric notes
The formulas are the same with MD in metres; the displacements come out in metres. DLS is usually quoted per 30 m. Since 30 m is 98.43 ft, °/30 m = °/100 ft × 0.984.
Good practice
- Check every survey for quality: magnetic interference, tool face, gravity and magnetic field values against the expected ones, before it goes into the definitive file.
- Record which north reference, grid convergence and magnetic declination were used. Mixing them shifts the whole well.
- Recompute from the tie-in point when a survey is corrected, rather than editing one station.
- Use TVD from the survey file, not MD, for every mud weight, kill and pressure calculation.
- Watch the DLS trend, not only the position: a hole that reaches the target with severe doglegs can still cause problems later.