Torque & drag in directional wells
How tension, torque and side force are computed along the string — from Johancsik's soft-string model to 3D stiff-string with unilateral contact.
In a directional well the drillstring does not run free: it leans against the borehole wall along every curved and every inclined section. That contact produces friction, and friction eats part of the weight meant to reach the bit and part of the torque applied at surface. When the numbers do not add up, the string locks up short of target, weight on bit turns into compression that buckles the pipe, or rotary torque exceeds the connection limit.
Predicting these values before drilling is what separates a planned well from one discovered mid-operation. The calculation has been public for four decades — what varies is how rigorously the pipe-to-wall contact is treated.
The soft-string model (Johancsik)
The classical model treats the string as a flexible cable: it carries tension and torque but offers no bending stiffness. Each element between two survey stations is solved bottom-up, carrying forward the effect of the element below.
The normal force against the wall combines two contributions: the buoyed weight of the element projected sideways, and the effect of tension passing through a curve. The second one surprises most people on first reading — through a build section, tension presses the string against the HIGH side of the hole, opposing gravity. In a catenary profile the two cancel and contact tends to zero.
With the normal force known, the tension increment adds axial weight and subtracts (or adds, running in) friction; the torque increment is friction times pipe radius. Tension enters signed, and that is what makes all four quadrants — build or drop, tension or compression — fall out of the same expression.
Fn = √[(W·sinθ̄ − Ft·Δinc)² + (Ft·Δazi·sinθ̄)²] ΔFt = W·cosθ̄ ± μ·Fn ΔM = μ·Fn·r
Where soft-string breaks down, and 3D stiff-string
Zero bending stiffness is a fair assumption for drill pipe over long, gentle sections. It stops holding where stiffness matters: in a short, heavy BHA, across an abrupt build-rate transition, or when the string is in compression and close to buckling.
The stiff-string model solves the fourth-order static beam equations in both lateral planes, in a frame that follows the wellbore. Geometric torsion of the hole couples the build and walk planes, and pipe-to-wall contact is unilateral — the pipe pushes the wall, but the wall never pulls the pipe. That is a non-linear problem, solved by Newton-Raphson with continuation on contact stiffness.
The difference shows up exactly where the design is decided: concentrated loads at curvature transitions, which the soft-string model simply cannot see.
Effective tension and hydraulic coupling
One distinction is routinely missed: the tension that governs buckling is not the real wall stress. Effective tension discounts internal and external pressure acting over the corresponding areas.
The Hammerlindl and Lubinski fictitious buckling force, derived for tubing anchored in a packer, corresponds in a drillstring to effective tension itself. The gap between real tension and effective tension is precisely the piston term. Without coupling the hydraulic pressure profile, that term drops out and the buckling margin comes back optimistic in high-pressure wells.
Te = T_real + p_ext·A_ext − p_int·A_int
References
- SPE 11380Johancsik, C. A., Friesen, D. B., & Dawson, R. (1984). Torque and Drag in Directional Wells — Prediction and Measurement. JPT, 36(6), 987–992.
- SPE 15463Sheppard, M. C., Wick, C., & Burgess, T. (1987). Designing Well Paths to Reduce Drag and Torque. SPE Drilling Engineering, 2(4), 344–350.
- SPE/IADC 112623Mitchell, R. F. (2009). Drillstring Solutions Improve the Torque-Drag Model.
- IADC/SPE 98965Menand, S., Sellami, H., Tijani, M., Stab, O., Dupuis, D. C., & Simon, C. (2006). Advancements in 3D Drillstring Mechanics: From the Bit to the Topdrive.
- SPE 178Lubinski, A. (1962). Helical Buckling of Tubing Sealed in Packers. JPT, 14(6), 655–670.
- SPE 5143Hammerlindl, D. J. (1977). Movement, Forces, and Stresses Associated With Combination Tubing Strings Sealed in Packers. JPT, 29(2), 195–208.