Directional trajectory: minimum curvature and well planning
How a survey becomes a position in space, how a well path is designed to a target, and why measured azimuth needs correcting before any of it.
A survey tool measures three numbers per station: measured depth, inclination and azimuth. None of them is a position. Turning that sequence of angles into north, east and vertical coordinates is the calculation everything else rests on — anti-collision, torque and drag, hydraulics, open-hole volume.
Get that conversion wrong and everything downstream is wrong with it, silently: the trajectory still looks perfectly plausible on the plot. It is worth knowing where the numbers come from.
Minimum curvature: the arc between two stations
The naive approach treats the interval between two stations as a straight line along the average direction. That underestimates displacement whenever the path curves, and the error grows with dogleg.
Minimum curvature treats the interval as a circular arc in space whose subtended angle is the dogleg itself. The correction enters as a ratio factor multiplying the balanced tangential result — for small dogleg it tends to 1 and both methods agree; for large dogleg it is what saves the calculation.
It has been the industry standard for decades, and it is the core the whole application depends on.
RF = tan(β/2) / (β/2) ΔN = (ΔMD/2)·(sinI₁·cosA₁ + sinI₂·cosA₂)·RF ΔE = (ΔMD/2)·(sinI₁·sinA₁ + sinI₂·sinA₂)·RF ΔTVD = (ΔMD/2)·(cosI₁ + cosI₂)·RF
Designing the path to target
The inverse problem is the interesting one: given a target, what geometry reaches it? The classical profiles — build-hold (slant), build-hold-drop (S), aggressive J — have known solutions in the vertical plane, and their closure equations are solved by 2D Newton-Raphson.
Out of plane, things change. Build-turn, optimum alignment between targets at different azimuths and curve-hold-curve have no useful closed form. The way through is numerical shooting: guess the free parameters, reconstruct the whole trajectory by minimum curvature, measure the residual to target, and correct by damped Newton with a finite-difference Jacobian.
The payoff of that choice is consistency: the residual the solver drives to zero is the closure of the same reconstruction the rest of the system uses, never a parallel formula that could disagree with it.
Before the maths: correcting azimuth
A magnetic tool measures azimuth relative to magnetic north, which is neither true north nor map grid north. These are two distinct corrections and both are mandatory.
Magnetic declination — the angle between magnetic and true north — comes from a global geomagnetic model (IGRF or WMM), varies with position and with time, and exceeds twenty degrees in some regions. Grid convergence is the angle between the local meridian and UTM grid north; it depends on how far you sit from the zone's central meridian.
Skip either one and the entire well shifts laterally, in proportion to departure. In an anti-collision calculation, that is the difference between a safe offset well and an intersection.
γ ≈ (λ − λ₀) · sin(φ)
References
- SPE 84246Sawaryn, S. J., & Thorogood, J. L. (2005). A Compendium of Directional Calculations Based on the Minimum Curvature Method. SPE Drilling & Completion, 20(1), 24–36.
- Inglis, T. A. (1987). Directional Drilling. Springer / Petroleum Engineering and Development Studies, vol. 2.
- Bourgoyne, A. T., Millheim, K. K., Chenevert, M. E., & Young, F. S. (1986). Applied Drilling Engineering. SPE Textbook Series, vol. 2.
- Mitchell, R. F., & Miska, S. Z. (2011). Fundamentals of Drilling Engineering. SPE Textbook Series, vol. 12.
- Alken, P., et al. (2021). International Geomagnetic Reference Field: the thirteenth generation. Earth, Planets and Space, 73(49).