A Closed-Form Expression for the Maximum Inclination Angle of Build-and-Hold Directional Well Trajectories: A Unified Formulation Valid for Both r1 > X3 and X3 > r1

Abstract

The maximum (hold) inclination angle is the controlling design variable of a build-and-hold directional well, yet textbook solutions give two different expressions depending on whether the build radius exceeds the horizontal departure of the target, and the second expression is sensitive to quadrant errors. This study derives a single closed-form expression valid for both cases from the tangent geometry of the build arc, θ = 2 tan−1[X3/((D3 − D1) + S)], where S = [(D3 − D1)2 + X32 − 2r1X3]1/2 is the hold length. The same quantity yields a feasibility test and the minimum build rate needed to reach a given target. The expression reproduced three published worked examples to within 0.007° and closed the trajectory to within 5 × 10−12 ft in 193,120 randomly generated feasible designs. Parametric analysis showed that the maximum inclination rises by about 0.73° (r1 > X3) and 0.56° (X3 > r1) per 100 ft of horizontal departure, falls from 53.7° to 18.5° as target depth increases from 5,000 to 10,000 ft, increases from 16.9° to 48.3° as the kick-off point is deepened from 500 to 6,000 ft, and depends only weakly on build rate (20.8° to 18.7° for 1 to 5°/100 ft). A direct-difference simplification, θ = sin−1(X3/A), explored in preliminary work was found to be inexact, with errors from −2.5° to +15.7° and no solution in a region where the exact solution exists; it is not recommended. The closed form gives well planners a transparent, branch-free calculation and a geometric kernel for trajectory-optimisation workflows.