**Dogleg severity** (DLS) is essentially what oilfield engineers call curvature.

Oil wells are not simply vertical holes in the ground. The boreholes curve around underground, even if the intent is to drill a perfectly straight hole. With horizontal drilling the curvature can be substantial.

Dogleg severity is calculated by measuring **inclination** and **azimuth** every 100 feet along a well. Inclination is the angle the hole makes with the vertical axis, like the angle φ in spherical coordinates, 90° minus the latitude. Azimuth is the angle of the projection of the well to a horizontal plane, like the θ or longitude angle in spherical coordinates. So if a well is nearly vertical, the inclination angles will be small. If a hole were shaped like a corkscrew, the inclination would be constant while the azimuth makes multiple rotations.

When I said 100 feet along the well, that is 100 feet of arc length. If a hole were perfectly vertical, this would be a change of 100 vertical feet, but generally it is less. If a segment were perfectly horizontal, it would be a change of 100 horizontal feet with no change in vertical depth.

Dogleg severity models a section of an oil well as a piece of a big circle. We assume our two inclination and azimuth measurements taken at two points along this circle, separated by an arc of 100 feet. If this arc makes an angle ψ, then the length of the arc is

ρ ψ = 100 ft

where ρ is the radius of the circle. Engineers call the angle ψ the **dog leg angle**, the angle of the sector between two measurements. Mathematicians call 1/ρ the curvature, so curvature is proportional to DLS.

To calculate curvature or DLS you have to calculate ψ from the two inclination and azimuth readings, (θ_{1}, φ_{1}) and (θ_{2}, φ_{2}). This is exactly the same as calculating distance from longitude and latitude. Instead of longitude and latitude on the earth, imagine a large sphere tangent to the wellbore at the two points where the measurements were taken. To put it another way, we imagine this section of the well as being a piece of a great circle on this sphere.

The only difference is that instead of the radius ρ being fixed and solving for distance, as in the longitude and latitude problem, our distance is fixed at 100 ft and we want to calculate ρ, or equivalently calculate ψ. From these notes we have

ψ = cos^{−1}(cos φ_{1} cos φ_{2} + sin φ_{1} sin φ_{2} cos(θ_{1} − θ_{2})).

So, for example, suppose we had inclination 4° and azimuth 30° at one point, and inclination 7° and azimuth 40° at the next measurement, 100 ft of arc length away. Then

ψ = cos^{−1}(cos 4° cos 7° + sin 4° sin 7° (cos40° − 30°)) = 3.138°

This says ρ = 1826 feet, i.e. our section of the well is curved like a circle of radius 1826 feet.

We could compute this in Python as follows.

from numpy import sin, cos, arccos, deg2rad, rad2deg def dls(inc1, az1, inc2, az2): ph1 = deg2rad(inc1) ph2 = deg2rad(inc2) th1 = deg2rad(az1) th2 = deg2rad(az2) return rad2deg( arccos( cos(ph1)*cos(ph2) + sin(ph1)*sin(ph2)*cos(th1-th2)) )

Here we assume input and output are in degrees, but internally we do calculations in radians.

If we call `dls(4, 30, 7, 40)`

we get back 3.138°.

If Hewlett-Packard promotional materials from the 70s & 80s were accurate, there was a package you could get that would assign that formula to one of the keys on your HP65 or HP41. They made calculators look rugged and macho to young nerdy me. :–)