Torque and drag decide whether a string can be run, rotated and pulled. They set the hook load against the rig's rating, the surface torque against the tool joints' make-up torque, and how much weight a sliding motor can put on the bit. This article works through the method on a 6⅛″ lateral: the equations, a hand calculation you can check, and what the broomstick plot says when the measured stands start to drift.
What torque and drag are
A string lying in a deviated hole presses on the wall. That contact force, multiplied by the friction factor, resists motion. Moving the pipe along the hole turns it into drag: extra hook load when you pick up, less when you slack off. Rotating turns it into torque: the friction acts around the pipe, at the radius of whatever touches the wall, usually the tool joints.
The contact force has two sources. In a straight inclined section it is the pipe's own buoyed weight, pressing on the low side. In a curve it is also the axial force: tension pulls the pipe against the inside of a build, compression pushes it against the outside. That second source is why the same dogleg costs more drag at 100 klbf of tension than at 20.
Rotating also changes drag. When the pipe turns, almost all of the friction acts around it rather than along it, so the rotating hook load sits close to the string's free weight. The gap between pick-up and rotating, and between rotating and slack-off, is the drag.
The soft-string model
The standard calculation treats the string as a cable that follows the hole's axis, with no bending stiffness. It splits the string into short elements, starts at the bit with the bit's load, and steps up to surface, adding each element's weight and friction. For an element of length Δs with buoyed weight w per unit length:
N = √[ (F ΔA sin Ī)² + (F ΔI + w Δs sin Ī)² ]
Ftop = Fbottom + w Δs cos Ī ± μ N
Ttop = Tbottom + μ N r
- N
- contact force on the element
- F
- axial force, tension positive
- Ī, ΔI, ΔA
- average inclination, and the change in inclination and azimuth across the element (radians)
- μ
- friction factor: + for pick-up, − for slack-off; when rotating, μN goes into torque instead
- r
- contact radius, half the tool joint's outside diameter
This is the method published by Johancsik, Friesen and Dawson in 1984 (SPE 11380), and it is still the default in planning because it is fast and, with calibrated friction, accurate for the drill pipe. Its assumption breaks down where the string is stiff relative to the hole: a heavy BHA in a high dogleg, a bent motor housing, or a stabilizer that holds the pipe off the wall. There a stiff-string model, a beam that can bend and touch the wall at discrete points, gives the real contact forces. DrillIQ runs the soft-string model by default and a finite-element stiff-string model over the BHA and the 300 m above it when you choose it.
The example well
The example is a horizontal oil producer in a carbonate reservoir, set in conditions typical of North Oman: a shallow, normally pressured reservoir at 1,500 m TVD, a water-based drill-in fluid and a long 6⅛″ lateral. The well is vertical to 900 m, built at 4°/30 m to a 22.4° tangent, then built again to land at 90° at 1,759 m MD and 1,500 m TVD. The 6⅛″ lateral runs to 2,759.2 m MD. The lateral's largest dogleg is 3.0°/30 m, where the well was steered back up after drifting towards the oil–water contact.
| Hole | 9⅝″ casing, a 7″ liner from 1,540 m to its shoe at 1,760 m, then 6⅛″ open hole to 2,759.2 m |
|---|---|
| String | 2,660 m of 4″ drill pipe (15.9 lb/ft adjusted, NC40 tool joints 5¼″ OD), 99 m BHA: PDC bit, 4¾″ motor, MWD/LWD, non-magnetic collar, jar, 6 joints of 4″ HWDP |
| Mud | drill-in fluid, 9.35 ppg, 260 gpm |
| Friction | 0.20 in casing, 0.25 in open hole |
| Drilling | 11 klbf WOB rotating (8 klbf sliding), 70 rpm, 2 kft·lbf at the bit; block 22 klbf |
| Limits | hook load 750 klbf, top drive 35 kft·lbf, NC40 make-up torque 16,000 ft·lbf |
A hand calculation for one stand
Before trusting any model, check its size by hand. Take one stand of 4″ drill pipe lying flat in the lateral.
- Buoyancy. Steel is 65.5 ppg, so the buoyancy factor in 9.35 ppg mud is 1 − 9.35 / 65.5 = 0.857. The pipe's buoyed weight is 15.9 × 0.857 = 13.6 lb/ft.
- Weight of a stand. A stand is 29.0 m, or 95.1 ft, so it weighs 13.6 × 95.1 ≈ 1,297 lbf.
- Contact force. At 90° there is no curvature and no axial weight component, so the whole weight presses on the low side: N = 1,297 lbf.
- Drag. At μ = 0.25 each stand adds 0.25 × 1,297 ≈ 324 lbf to pick-up and takes the same off slack-off.
- Torque. The tool joints touch the wall at 5.25 / 2 = 2.625 in, or 0.219 ft, so each stand adds 0.25 × 1,297 × 0.219 ≈ 71 ft·lbf when rotating.
- The whole lateral. From the liner shoe to TD is 999 m, 34.5 stands: about 11.2 klbf of drag each way and 2.45 kft·lbf of torque.
DrillIQ's engine, run on the full string from the shoe to TD, puts the rise in pick-up at 10.7 klbf and the rise in off-bottom torque at 2.54 kft·lbf. The hand figure gets both within 5 %. Slack-off drops further in the engine, by 16.9 klbf, because the hand sum leaves out the heavier BHA and the extra contact in the curves above the lateral, which grow as the lateral's drag loads them. That is the job of the hand check: if the model disagrees with it by a factor of two, look for a wrong unit, a missing buoyancy factor or a survey error before you look at friction.
Reading the broomstick plot
The broomstick plots hook load against depth for pick-up, slack-off and rotating, once for each friction factor in a range. The lines spread out like the bristles of a broom as the lateral gets longer. The measured hook loads at each connection go on top as points. Where the points fall among the bristles is the hole's friction; where they leave them is a problem.
- Pick-up
- Slack-off
- Rotating off bottom
- Measured, every stand
- Thin lines: open-hole friction 0.15 (innermost) to 0.35 (outermost) · thick: 0.25
Model: DrillIQ's soft-string engine on the example lateral, cased-hole friction held at 0.20, pumping 260 gpm. Measured: the hook load at each connection, every 29 m stand from 1,789 m to TD.
Show the data
| Bit depth | Pick-up | Slack-off | Rotating |
|---|---|---|---|
| 1,760 | 95.5 | 80.7 | 88.2 |
| 1,780 | 95.5 | 80.1 | 87.9 |
| 1,800 | 95.6 | 79.5 | 87.8 |
| 1,820 | 95.8 | 79.0 | 87.6 |
| 1,840 | 96.0 | 78.6 | 87.5 |
| 1,860 | 96.2 | 78.2 | 87.5 |
| 1,880 | 96.4 | 77.9 | 87.5 |
| 1,900 | 96.6 | 77.6 | 87.5 |
| 1,920 | 96.9 | 77.3 | 87.5 |
| 1,940 | 97.1 | 77.0 | 87.5 |
| 1,960 | 97.3 | 76.7 | 87.5 |
| 1,980 | 97.5 | 76.4 | 87.4 |
| 2,000 | 97.7 | 76.1 | 87.4 |
| 2,020 | 97.9 | 75.8 | 87.4 |
| 2,040 | 98.1 | 75.5 | 87.3 |
| 2,060 | 98.3 | 75.2 | 87.3 |
| 2,080 | 98.5 | 74.9 | 87.3 |
| 2,100 | 98.8 | 74.6 | 87.3 |
| 2,120 | 99.0 | 74.3 | 87.3 |
| 2,140 | 99.2 | 74.0 | 87.3 |
| 2,160 | 99.4 | 73.7 | 87.2 |
| 2,180 | 99.7 | 73.4 | 87.3 |
| 2,200 | 100.0 | 73.2 | 87.3 |
| 2,220 | 100.3 | 73.0 | 87.4 |
| 2,240 | 100.6 | 72.8 | 87.5 |
| 2,260 | 101.0 | 72.7 | 87.6 |
| 2,280 | 101.2 | 72.4 | 87.6 |
| 2,300 | 101.4 | 72.1 | 87.5 |
| 2,320 | 101.5 | 71.6 | 87.4 |
| 2,340 | 101.5 | 71.0 | 87.2 |
| 2,360 | 101.5 | 70.4 | 87.0 |
| 2,380 | 101.5 | 69.9 | 86.8 |
| 2,400 | 101.7 | 69.5 | 86.7 |
| 2,420 | 101.9 | 69.2 | 86.7 |
| 2,440 | 102.2 | 68.9 | 86.7 |
| 2,460 | 102.5 | 68.7 | 86.8 |
| 2,480 | 102.8 | 68.5 | 86.8 |
| 2,500 | 103.1 | 68.2 | 86.9 |
| 2,520 | 103.4 | 67.9 | 86.9 |
| 2,540 | 103.6 | 67.6 | 86.8 |
| 2,560 | 103.8 | 67.2 | 86.8 |
| 2,580 | 104.0 | 66.9 | 86.8 |
| 2,600 | 104.3 | 66.6 | 86.8 |
| 2,620 | 104.5 | 66.3 | 86.8 |
| 2,640 | 104.7 | 65.9 | 86.8 |
| 2,660 | 105.0 | 65.6 | 86.7 |
| 2,680 | 105.2 | 65.2 | 86.7 |
| 2,700 | 105.4 | 64.9 | 86.7 |
| 2,720 | 105.7 | 64.6 | 86.7 |
| 2,740 | 105.9 | 64.2 | 86.6 |
| 2,759 | 106.1 | 63.9 | 86.6 |
On this lateral the measured stands sit on the 0.25 lines for pick-up, slack-off and rotating alike, as they should in a clean hole. Two stands do not. At 2,195 m pick-up reads 2.9 klbf over the 0.25 line, and at 2,224 m 11.4 klbf over, while slack-off and rotating at the same depths stay on their lines.
That pattern matters. Higher friction along the whole open hole would move every pick-up to the right and every slack-off to the left together, and the points would ride a higher bristle. A jump in pick-up alone, at two neighbouring stands, points to something local: here a thin shale streak caving into the hole. The response is local too: back-ream the interval, circulate it clean, and watch the shakers for cavings, rather than adding lubricant to the whole system.
Friction factors also tell you how sensitive the plan is. On this lateral, 0.1 more open-hole friction adds 5.5 klbf to pick-up at TD and takes 6.7 klbf off slack-off. With pick-up at 106 klbf against a 750 klbf rig, there is a large margin. The more useful limit on a lateral like this is how much weight reaches the bit while sliding.
Torque against the connection limit
- Off bottom, rotating
- On bottom, drilling
- Measured, every stand
- Thin lines: open-hole friction 0.15 (left) to 0.35 (right) · thick: 0.25
Model and measured as in Figure 1; on bottom with 11 klbf on the bit, 70 rpm and 2 kft·lbf of bit torque.
Show the data
| Bit depth | Off bottom | On bottom |
|---|---|---|
| 1,760 | 1.29 | 3.65 |
| 1,780 | 1.36 | 3.72 |
| 1,800 | 1.43 | 3.79 |
| 1,820 | 1.49 | 3.86 |
| 1,840 | 1.55 | 3.92 |
| 1,860 | 1.61 | 3.98 |
| 1,880 | 1.66 | 4.02 |
| 1,900 | 1.71 | 4.07 |
| 1,920 | 1.75 | 4.12 |
| 1,940 | 1.80 | 4.17 |
| 1,960 | 1.85 | 4.22 |
| 1,980 | 1.90 | 4.28 |
| 2,000 | 1.95 | 4.33 |
| 2,020 | 2.00 | 4.38 |
| 2,040 | 2.05 | 4.42 |
| 2,060 | 2.10 | 4.47 |
| 2,080 | 2.15 | 4.52 |
| 2,100 | 2.20 | 4.57 |
| 2,120 | 2.25 | 4.62 |
| 2,140 | 2.30 | 4.67 |
| 2,160 | 2.35 | 4.72 |
| 2,180 | 2.40 | 4.76 |
| 2,200 | 2.44 | 4.79 |
| 2,220 | 2.49 | 4.83 |
| 2,240 | 2.54 | 4.87 |
| 2,260 | 2.58 | 4.92 |
| 2,280 | 2.63 | 4.98 |
| 2,300 | 2.68 | 5.05 |
| 2,320 | 2.73 | 5.12 |
| 2,340 | 2.79 | 5.20 |
| 2,360 | 2.85 | 5.26 |
| 2,380 | 2.90 | 5.31 |
| 2,400 | 2.95 | 5.35 |
| 2,420 | 3.00 | 5.39 |
| 2,440 | 3.05 | 5.43 |
| 2,460 | 3.09 | 5.47 |
| 2,480 | 3.14 | 5.52 |
| 2,500 | 3.19 | 5.57 |
| 2,520 | 3.24 | 5.62 |
| 2,540 | 3.29 | 5.67 |
| 2,560 | 3.34 | 5.72 |
| 2,580 | 3.39 | 5.76 |
| 2,600 | 3.44 | 5.81 |
| 2,620 | 3.49 | 5.86 |
| 2,640 | 3.54 | 5.91 |
| 2,660 | 3.58 | 5.96 |
| 2,680 | 3.63 | 6.01 |
| 2,700 | 3.68 | 6.06 |
| 2,720 | 3.73 | 6.11 |
| 2,740 | 3.78 | 6.16 |
| 2,759 | 3.83 | 6.21 |
Off-bottom torque rises from 1.3 kft·lbf at the liner shoe to 3.8 kft·lbf at TD, almost all of it from the lateral's contact. On bottom, the bit's 2 kft·lbf and the BHA's compression under weight on bit add about 2.4 kft·lbf, for 6.2 kft·lbf at TD. That is 39 % of the NC40 tool joints' 16,000 ft·lbf make-up torque and 18 % of the top drive's rating.
The make-up torque is the limit that matters in a lateral. Drill with surface torque above it and the tool joints keep screwing up downhole, which can damage the shoulders. A torque trend that climbs faster than the model, stand after stand, is often the first sign of a cuttings bed building on the low side, well before the hook loads show it.
Sliding, buckling and lock-up
When a steerable motor slides, the string does not rotate, so the full axial friction acts against the weight you set down. The question is how much of what you slack off at surface reaches the bit.
The hook load less the 22 klbf block, with the string sliding at TD. Model: the same engine, soft-string, helical contact by Mitchell's r·F²/4EI.
Show the data
| WOB (klbf) | Left at the hook (klbf) | Helical buckling |
|---|---|---|
| 0.0 | 41.9 | none |
| 2.0 | 39.2 | none |
| 4.0 | 36.5 | none |
| 6.0 | 33.7 | 865–900 m |
| 8.0 | 30.7 | 795–1,245 m |
| 10.0 | 27.0 | 710–1,245 m |
| 12.0 | 23.1 | 620–1,245 m |
| 14.0 | 19.5 | 535–1,245 m |
| 16.0 | 15.6 | 445–1,245 m |
| 18.0 | 11.3 | 345–1,245 m |
| 20.0 | 6.4 | 235–1,245 m |
| 21.0 | 3.8 | 175–1,245 m |
| 22.0 | 0.9 | 105–1,245 m |
| 22.3 | 0.0 | lock-up |
With the bit just off bottom, the string hangs 41.9 klbf from the hook (the hook load less the 22 klbf block). Setting 8 klbf on the bit, the sliding weight used on this well, leaves 30.7 klbf at the hook. Push harder and the weight at the hook falls faster than the weight at the bit rises: at 20 klbf on the bit only 6.4 klbf is left, and at 22.3 klbf the string locks up. More slack-off at surface then puts no more weight on the bit.
The lateral is not what limits it. At 8 klbf the drill pipe from 795 m to 1,245 m, the vertical section and the 22° tangent, is in helical buckling: the compressed pipe coils inside the 9⅝″ casing, and the coils press on the wall with a force that grows with the square of the axial load. That contact eats the push. The build sections do not buckle, because curvature raises the buckling load several times, and the lateral only starts to buckle sinusoidally at about 30 klbf, beyond lock-up.
For this well the practical sliding limit is about 15–18 klbf, with margin to lock-up. If the next lateral is longer, the options are the usual ones: heavier pipe or HWDP in the vertical to resist buckling, an agitator to break static friction, or rotary steerable drilling, which keeps the string rotating.
Calibrating friction from measured stands
Friction factors are not material properties. They lump together mud lubricity, cuttings, tortuosity the survey misses and tool-joint contact, so they are calibrated, not looked up. The method is a least-squares fit: choose the friction that brings the model closest to the measured pick-up, slack-off and rotating hook loads, and separately the rotating friction that matches the off-bottom torque.
Use every stand and both trip directions. Each reading here carries about ±2.5 % of scatter, which is ±2.5 klbf on a 100 klbf pick-up, or about ±0.05 of friction from one stand alone. Fitted over all 35 stands, pick-up and slack-off together, the open hole comes back at 0.250 axial and 0.251 rotating, with the cased-hole 0.20 held fixed. The two tight stands barely move it, which is why the fit should be read beside the broomstick, not instead of it.
A checklist for the rig team
- Check the model by hand for one stand before relying on it.
- Calibrate friction from a run of stands in both directions, never from one reading.
- Plot every connection's pick-up, slack-off and rotating weight on the broomstick, at the same pump rate and speed each time.
- Read pick-up and slack-off together: both moving outwards means friction; one moving alone means a local problem.
- Track off-bottom torque against the model; a steady climb often comes before the hook loads change.
- Keep surface torque below the tool joints' make-up torque, with your company's margin.
- Before sliding, know the lock-up weight on bit and where the string buckles first.
How DrillIQ handles it
In DrillIQ the Torque & Drag module plans each section with soft-string and stiff-string models, checks buckling, lock-up, von Mises stress and drill-pipe fatigue, and builds the road map for the run. While drilling, it plots each connection's hook loads and torque on the broomstick from the rig's live data, calibrates friction from the measured stands, and flags points that leave their bristles. Everything above was worked out with the same engine.
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References
- Johancsik, C.A., Friesen, D.B. and Dawson, R. Torque and Drag in Directional Wells: Prediction and Measurement. SPE 11380, 1984.
- Dawson, R. and Paslay, P.R. Drillpipe Buckling in Inclined Holes. SPE 11167, 1984.
- Mitchell, R.F. Effects of Well Deviation on Helical Buckling. SPE 29462, 1997.
- Aadnøy, B.S. Modern Well Design, 2nd ed. CRC Press, 2010.
About this example. The well in this article is a representative scenario set in typical North Oman conditions and worked through with DrillIQ's engines. It is not a client's well, and its figures are not field results.