Horizontal directional drilling

Estimating pullback force

Four things resist the pull: friction against the borehole wall, the capstan effect at every bend, drag from the drilling fluid, and the weight the pipe gains or loses changing depth. This is how each one is calculated, with a published worked example checked line by line and a calculator that reproduces it.

Start with buoyant weight, not weight

Nothing in the borehole is held down by what the pipe weighs in air. It is held against the crown or the invert by its net buoyant weight — the drilling fluid it displaces, minus its own weight and the weight of its contents:

w_b = 12 · (π/4) · OD² · γ_f · ρ_w − w_pipe − w_contents w_b = net upward buoyant force (lb/ft) · OD = pipe outside diameter (in) · γ_f = specific gravity of the drilling slurry · ρ_w = density of water, 0.0361 lb/in³ · 12 converts pounds per inch of length to pounds per foot · w_pipe, w_contents = weight of the pipe and of whatever is inside it (lb/ft)

Drilling slurry is heavy. ASTM F1962 recommends taking its specific gravity as 1.5; the PRCI steel example uses 1.44, which is 12 pounds per gallon. Either way the pipe floats. A 24 in DR 11 PE4710 pipe weighs 61.5 lb/ft in air and, empty in a 1.5 specific-gravity slurry, pushes upward at 232.4 lb/ft — nearly four times its own weight, pinning it to the crown of the hole. Even a 12.75 in × 0.25 in steel pipe comes out at 46.4 lb/ft net upward in 12 ppg mud, per the worked example in Huey, Hair and McLeod (1996).

This is why, as the PPI Handbook of PE Pipe puts it, “most major pullbacks are done wet” — the pipe is filled with water as it passes the break-over point. Run the arithmetic on that 24 in pipe: its bore is 19.636 in, which holds 131.2 lb of water per foot, so the net buoyant force falls from 232.4 to 101.2 lb/ft. That is a 56 per cent cut in every below-ground friction term on the page, for the price of a hose.

The four components

1. Frictional drag

Coulomb friction against the borehole wall or the ground, driven by the buoyant weight:

F = μ · w_b · L

Published coefficients: the PPI HDD chapter gives 0.25 between pipe and slurry and 0.40 between pipe and ground, and its worked example uses 0.4 for the pipe dragged over the ground before it enters the hole and 0.25 inside the lubricated bore. Slavin and Scholl take 0.1 above ground from ASTM F1962, for a steel line on low-friction rollers. The PRCI steel example uses 0.3 between pipe and borehole surface.

2. The capstan effect

A rope around a bollard holds far more than the friction of its contact area suggests, because the tension itself presses the rope into the surface. The same applies to a pipe in a curve: tension generates a normal force, the normal force generates friction, and the friction raises the tension. It compounds multiplicatively:

F_c = e^μθ · (μ · w_b · L) θ = angle of bend, in radians — not degrees

At μ = 0.25 a 10° bend multiplies tension by 1.045, a 15° bend by 1.068, a 90° bend by 1.481. On dry ground at μ = 0.40 that 90° bend costs 87 per cent. Note what the exponent does not contain: radius. The capstan penalty depends only on the total angle turned, so a long gentle curve and a tight one through the same angle cost the same here — the radius shows up instead in bending stress and in whether the pipe hangs up.

3. Fluidic drag

The pipe is being dragged through viscous slurry. ASTM F1962 treats it as a pressure acting on the annular area:

ΔT = P · (π/8) · (D_hole² − D²) P = hydrokinetic pressure (psi) · D_hole = borehole diameter (in) · D = pipe outside diameter (in)

PPI gives P as 30 to 60 kPa (4 to 8 psi); Slavin and Scholl report ASTM F1962 using 10 psi, and PPI’s own worked example uses 10 psi as well. The borehole is normally reamed to 1.2 to 1.5 times the carrier pipe diameter, so for a 24 in pipe in a 36 in hole this gives 2,830 lb at 10 psi, or 1,130 to 2,260 lb across PPI’s range. Small. And it is added to the local tension at each point, not accumulated along the pull.

4. Depth change

For a pipe that floats — w_b upward, the usual case — dragging it down to depth costs w_b·H of extra tension, and on the rise to the far side buoyancy gives the same amount back. For a pipe that sinks, the signs reverse. These are the w_b·H terms in the equations below, where H is the depth of the bore.

The ASTM F1962 equations

The path is split into four horizontal spans: L1 is the pipe still lying on the surface, L2 the horizontal distance to reach depth, L3 the run at depth, L4 the rise to the exit. The load is calculated at four stages as the pull-nose reaches each point. The angles are named from the pipe’s side: α is where the product pipe goes into the ground, which is the end the drill came out of, and β is where it comes out at the rig.

T_A = exp(ν_aα) · [ ν_a w_a (L1 + L2 + L3 + L4) ] T_B = exp(ν_bα) · [ T_A + ν_b|w_b|L2 + w_bH − ν_a w_a L2 exp(ν_aα) ] T_C = T_B + ν_b|w_b|L3 − exp(ν_bα) · [ ν_a w_a L3 exp(ν_aα) ] T_D = exp(ν_bβ) · [ T_C + ν_b|w_b|L4 − w_bH − exp(ν_bα)(ν_a w_a L4 exp(ν_aα)) ] w_a = weight of the empty pipe above ground (lb/ft) · w_b = net buoyant weight below ground, upward positive (lb/ft) · ν_a, ν_b = friction coefficients above and below ground · α, β = pipe entry and exit angles in radians · H = depth of bore (ft)

These are the equations as Slavin and Scholl quote them from ASTM F1962. The PPI chapter gives the same form in its Figure 4, written without the absolute-value bars, which makes no difference for a pipe that floats. The chapter’s appendix prints some exponents differently — ν_b in place of ν_a in the last terms of T_B and T_D, and α in place of β at the start of T_D — but its published results are reproduced exactly by the form above, not by the printed one.

Worked, with every number

This is the river crossing example from PPI Handbook of PE Pipe Chapter 12. A 24 in OD PE4710 pipe, 2.182 in wall, 870 ft crossing, 35 ft deep, 10° entry, 15° exit, 100 ft still on the ground at the end of the pull.

w_a, pipe in air	61.54 lb/ft
w_b, net buoyant, empty	232.41 lb/ft upward
ν_a / ν_b	0.4 / 0.25
L1 / L2 / L3 / L4	100 / 401.07 / 201.55 / 267.38 ft
α / β	0.175 / 0.262 radians
T_A	25,610 lb
T_B	48,530 lb
T_C	54,680 lb
T_D	58,410 lb
ΔT, 10 psi in a 36 in hole	2,830 lb
Wall stress at D, (T_D + ΔT) ÷ 149.6 in²	409 psi

PPI adds the hydrokinetic increment at each stage and divides by the wall cross-section of 149.6 in². At D that is 409 psi against a 12-hour safe pull stress of 1,150 psi — 36 per cent of it. The allowable load that implies is 1,150 × 149.6 = 172,000 lb. (The example prints 164,000 lb for that figure, which its own inputs do not give; its header also says DR 12, while its wall thickness and every result are for DR 11.) The hydrokinetic increment is under 5 per cent of T_D — consistent with Slavin and Scholl’s note that this term is usually low compared with the tension estimates.

Two features of the result are worth reading. T_A, the load before the pipe has entered the ground at all, is 25,610 lb — 44 per cent of the final figure. 93 per cent of it is drag on 970 ft of pipe across the ground at μ = 0.4; the rest is the capstan factor over the entry bend. And the jump from T_A to T_B is the entry curve, where buoyancy, capstan and depth all land at once.

Run it on your own crossing

The calculator runs the equations above. Its defaults are the PPI example, so it opens on the published answer; change any input and every stage recalculates.

Pipe and fluid

Pipe outside diameter (in) Dimension ratio, DR (OD ÷ minimum wall) Pipe material specific gravity (PPI example: 0.95) Drilling slurry specific gravity (F1962: 1.5) Pipe filled with water below ground (a “wet” pull)

Path

Depth of bore, H (ft) Pipe entry angle, α (degrees) Pipe exit angle, β (degrees) Crossing length, L2 + L3 + L4 (ft) Pipe still on the surface at the end, L1 (ft)

Friction, drag and allowable stress

Friction above ground, ν_a Friction in the bore, ν_b Hydrokinetic pressure (psi) Reamed hole ÷ pipe OD Safe pull stress (psi; PE4710, 12 h: 1,150)

Assumptions, all from the F1962 model as PPI applies it: entry and exit at the same elevation; each curve’s horizontal run is the small-angle 2H/α and 2H/β, so angles are limited to 30°; the pipe above ground is empty; no pipe bending stiffness, which suits PE and understates stiff pipe; an ideal, clean, fully reamed borehole. Wall stress uses the minimum wall, OD ÷ DR. PPI calls results from these formulas qualitative, for preliminary estimates only.

The other method gives a different answer

PRCI’s method (Huey, Hair and McLeod, 1996; ASCE MOP 108) was written for steel and differs in two ways. It models the pipe’s bending stiffness — the reaction forces needed to force a stiff pipe to follow the borehole curve, solved iteratively because tension depends on the reactions and the reactions depend on tension. And it treats fluid drag as a viscous stress distributed over the pipe’s whole outer surface: originally 0.05 psi, reduced to 0.025 psi on field results (Puckett, 2003).

That second change dominates. Take the same 24 in pipe over 870 ft, counting the crossing length as the length in the hole: the wetted surface is 787,000 in², and at 0.025 psi the drag term alone is 19,700 lb — seven times what the F1962 hydrokinetic equation gives at 10 psi.

Slavin and Scholl ran both methods on the same 1,500 ft steel crossing (12.75 in × 0.25 in, 100 ft deep, 20° entry and 14° exit, 1,000 ft and 1,200 ft radii). PRCI returned about 46,000 lb; ASTM F1962 returned about 27,000 lb. Stiffness accounted for 15 per cent of the PRCI figure; almost all the rest of the difference was fluid drag. The two are not calibrated to the same borehole quality: F1962 assumes a well-prepared bore, PRCI assumes a worse one.

How much to trust any of it

PPI states the position directly — the formulas are “for guidelines only”, the values obtained “should be considered only as qualitative values and used only for preliminary estimates”, and they assume an ideal borehole: a rigid tunnel, gradual curvature, no dog-legs, no collapses, near-complete cuttings removal and good slurry circulation.

The honest measure of the uncertainty is field data. Knight and co-workers reported controlled HDD experiments in what appeared to be almost identical conditions producing pull loads differing by almost two to one. No calculation method closes that gap, because the variable is the ground and the cleanliness of the hole.

Two operational notes follow from the physics rather than the formulas. Pullback runs at 1 to 2 ft per minute and should not stop except to break out rods: when motion stops, a buoyant pipe pressed against the crown squeezes the lubricating mud out of the contact, the mud decants, and PPI warns that the load needed to break it free can be very high — none of the equations on this page predict it. And the pull is applied through the reamer ahead of the pipe, so what reaches the pull-head is only part of the rig’s indicated force.

BorePlanner Coming soon plans the profile these equations run on — the depth, the entry and exit pitch, and the tightest radius along the path — on an Android phone at the jobsite. The pullback estimate itself is the calculator above, with every coefficient exposed so it can be re-run against the assumptions it actually depends on.

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Sources

Plastics Pipe Institute, Handbook of PE Pipe, Chapter 12 — equations 13 to 16, friction coefficients, pullback speed, worked river crossing.
Slavin and Scholl, “Which Method to Use When Estimating Maxi-HDD Installation Loads — ASTM F1962 or the PRCI Method?”, ASCE Pipelines 2014 — segment equations, slurry density, the 0.025 psi drag value, side-by-side comparison, Knight’s field results.
ASTM F1962; Huey, Hair and McLeod (1996); ASCE Manual of Practice 108 — as cited by the two documents above.