Every MWD inclination is the tilt of a sensor package bolted inside a drill collar — and that collar bends under its own weight between the stabilisers that hold it off the low side of the hole. The sensor faithfully reports the local slope of the bent steel, which is not the direction of the borehole. The difference is sag: typically +0.05° to +0.5°, systematically toward the low side, in every survey of every directional well ever drilled.
A near-bit stabiliser and a string stabiliser pin the collar at two points. Between them ~10–20 m of steel hangs unsupported. Gravity's transverse component pulls it toward the low side; the string takes a gentle catenary-like curve; the D&I sensor — sitting somewhere on that curve — reads the local tangent, tilted from the borehole axis by the slope of the deflection at that exact point.
Millimetres of deflection over a ten-metre span is a slope of a few hundredths of a degree — which is exactly the size of error that moves TVD by metres over a long lateral.
Sag is not an empirical fudge — it falls straight out of beam mechanics. Four ingredients:
1 — Buoyed weight. Steel in mud weighs less; the mud both inside and around the collar carries part of it.
2 — Transverse load. Only the component of weight across the hole bends the collar — nothing in a vertical well, full weight in a horizontal one.
3 — Stiffness. Bending resistance comes from the steel cross-section — outer diameter to the 4th power, minus the bore. A sonde suspended inside a collar does not stiffen it: the EI section is the collar wall, not the flow path.
4 — The beam equation. The deflection v(x) of the string under that load, pinned at every wall contact:
Why you cannot just scale by sin(INC). The wall is in the way. As inclination grows, the sagging collar lands on the low side of the hole and the wall starts carrying it — a one-sided (unilateral) contact that switches on and off depending on how far the string sags. That makes the problem non-linear: sag grows from vertical, then eases at high angle as more steel rests on the wall. Every station gets its own contact set, found iteratively.
A constant inclination bias δ (radians) in a lateral leaks into vertical depth with every metre drilled:
Sag is systematic — always the same sign for a given BHA and angle — so it never averages out. It accumulates. That is why a definitive survey without a sag correction is not definitive.
Everything else — meshing, contact search, the solve — is the computer's problem, in about a second per profile.
