Equations: Drill string and bits

SI

Holds when: The pipe is plain and cylindrical. Grooves, spiral flutes and threads reduce the mass a little. To get the weight in mud, multiply by the buoyancy factor $1 - \rho_m/\rho_{\text{steel}}$ (it allows for the lift of the mud), as for casing.

MPa × mm² = N

Holds when: The load is axial and uniform over the section; there is no bending, torsion or pressure. Nominal dimensions of a new pipe; for a worn one a reduced outside diameter is used. This is the yield strength, not failure: above it the pipe stretches permanently. The working load is taken with a margin.

MPa, mm; the torque in N·mm (divided by $10^6$, in kN·m)

Holds when: The torsion is pure. The shear stress grows linearly from the axis to the outer surface, and yielding begins at the surface. Tension is not taken into account. Under tension the allowable torque is smaller (see the advanced section). This is a property of the pipe body. The tool joint is checked separately for its own torque.

any consistent ones: $F$ and $F_y$ in the same units of force

Holds when: The tension is uniform over the section; there is no bending or pressure. The von Mises criterion for the outer surface, where the shear stress is greatest. The pipe body; the tool joint is checked separately.

SI

Holds when: The pipe body between the tool joints does not touch the wall of the hole; the tool joints lie on the wall. The curvature of the hole is constant over the length of the pipe. Without tension the factor $KL/\tanh(KL)$ equals one, and the pipe simply follows the curvature of the hole.

SI. The densities can be in any units, as long as both use the same ones.

Holds when: The mud inside and outside the pipe has the same density. If the densities differ (the string is run empty, or there is a weighted slug, a small batch of heavy mud, inside), we calculate the weight by the general formula in the advanced section. The adjusted mass of a drill pipe is noticeably greater than the nominal one. Take a 127 mm 19.50 lb/ft pipe. Its nominal mass is 29.0 kg/m, the mass of the plain body is 26.7 kg/m, and with NC50 tool joints it is about 34 kg/m. Weight in air differs from mass by the factor $g$. One tonne-force (tf) is 9.807 kN.

SI

Holds when: The compressed part is in a straight section of the hole. The assembly is of one size. For several elements, subtract the weight of the lower element first, then of the next, and so on. Friction against the walls is not taken into account. With rotation, it hardly changes the axial force. Without rotation, part of the weight on bit goes into friction. The calculation uses the effective force, so the pressure of the fluid does not enter it (advanced section).

SI

Holds when: Drill collars of one size provide the weight on bit. If there are several sizes, find the length of the upper ones from the weight that remains after the lower ones. In a vertical hole $\cos\theta = 1$. For an inclined hole, see the next section, where HWDP can also provide the weight.

SI

Holds when: The section is straight, with constant inclination. The pipe lies on the low side under its own weight. The formula is used for inclinations of a few degrees and more. In a vertical hole there is no support, and the formula does not apply. The clearance is taken for the pipe body. The tool joints and the central upset of HWDP reduce the clearance and raise the critical load, so the estimate is on the safe side. The helical mode begins at a force roughly 1.4–2.8 times greater, depending on the solution. Rotation, irregularities and curvature of the hole change the picture. So we keep the compression below $F_{\text{cr}}$ with a margin.

Any units of force, as long as all quantities use the same units.

Holds when: The tension is static. The margin covers jerks and shocks. Torsion (twisting) is not taken into account. With a torque acting at the same time, the allowable tension is smaller (torsion under tension). The tensile strength of the tool joints is usually greater than that of the pipe body. Check it in the catalogue.

$D$ and $L_s$ must be in the same units.

Holds when: The slips and bushings are in good order and grip the pipe evenly along its whole length. Worn slips, an uneven grip, and a shock when the pipe is set in the slips cause crushing sooner. $\mu = 0.08$ is an assumed value. With greater friction $K$ is smaller, so a calculation with this value is on the safe side. The condition of the check: the weight of the string under the slips $P$, multiplied by $S_h/S_t$, must be no more than the same allowable load $P_a$.

SI

Holds when: The well is vertical. In a deviated well, use the force when pulling out, found by the soft-string model, instead of the weight. The first condition is the margin of overpull, and the second is crushing in the slips.

SI

Holds when: The section is straight. The pipe lies on the low side, and only its own weight presses it, $w\sin\theta$ per metre. The friction is Coulomb friction: the friction force is proportional to the contact force. When the string is rotated and moved at the same time, friction is shared between the axial and circumferential directions. The model gives only the extreme cases. Bending stiffness is not accounted for. For a stiff BHA in a curved hole, the model underestimates the contact forces.

SI. The angle is in radians.

Holds when: The build section is planar (it lies in one plane) and has constant dogleg severity. The azimuth (the compass direction of the hole) does not change. The tension presses the pipe against the high side harder than the weight presses it against the low side, so $F\kappa > w\sin\theta$ along the whole section. The integral of the contact force, $\bar F\Delta\theta - wA$, is taken with the mean force. Long sections are divided into parts. The derivation is in the advanced section.

SI

Holds when: The pipe is vertical, in a fluid at rest. With equal pressures inside and outside, $F_{\text{eff}} = F_{\text{true}} + pA_s$, where $A_s$ is the wall area. Then the weight of a metre is the usual weight in mud, $m g k_{\text{b}}$. The weight formula is written for a pipe of constant cross-section. For a pipe with tool joints and upsets, we take the difference $A_{\text{out}} - A_{\text{in}}$ in it as the mean along the length, that is $m/\rho_{\text{steel}}$. If the fluid inside is lighter (the string is run empty) or heavier (a weighted slug), the string on this section is lighter or heavier. The hook load shows this.

SI

Holds when: The arc is planar. $F\kappa > w\sin\theta$, so the pipe lies on the high side. Bending stiffness is not accounted for.

metric: $R$ in m/h, $N$ in rpm, $\delta$ in mm

Holds when: The cutters of one track are evenly spaced round the circumference and remove equal shares of the penetration. The bit penetrates uniformly over the whole bottom, without rocking or vibration. The tracks of neighbouring cutters overlap, so the depth of cut varies across the width of a cutter. The formula gives the greatest depth of cut.

$\omega$ and $\Omega$ in the same units

Holds when: The cone is a single cone, its apex lies on the bit axis, and there is no axis offset. The bottom of the hole is flat and the bit does not rock.

MPa; 0.00981 is the pressure of a column of liquid with a density of 1 g/cm³ and a height of 1 m, in MPa

Holds when: a permeable rock with a mudcake: the whole pressure difference acts on the skeleton as an effective confining stress. In tight clays and salts the pore pressure under the bit changes as the rock breaks, and the simple formula works only as an estimate (see the advanced section); a Mohr–Coulomb straight line with constant $C_0$ and $\varphi$, as in the lesson on sand production. At high confining pressures the strength grows more slowly, and the formula overestimates it; when we drill with circulation, $\Delta p$ also includes the pressure lost to friction in the annulus (the space between the pipe and the wall of the hole). We express it as the equivalent circulating density (ECD): the mud density that would give the same pressure at the bottom as the moving mud does. The example takes the mud density, which slightly underestimates the confining pressure.

rpm × h × 60 min/h

Holds when: the rotary speed is constant during the run. If the parameters change, we add up the revolutions over the periods of time; the life of the bearing also depends on the load, the temperature and the mud. The number of revolutions is a simple measure of how much of it is used, but not the only one.

any identical units of length

Holds when: a cutter broken off flush with the body, or a lost cutter, counts as completely worn: 8; the code takes the averages, rounded to a whole number; the manufacturer often grinds the gauge cutters flat in advance. This cut is not wear.

mm; 1.5875 mm = 1/16 inch

Holds when: the cones have no play. In a bit with a broken-down bearing the gap depends on how the cone is pressed, and the measurement is unreliable; we measure at the gauge inserts or at the inserts of the back cone, whichever are closer to the nominal diameter.

m and h

Holds when: the trip time is charged to the run at the end of which the bit is pulled; some calculations add the time of connections, circulating and reaming to the denominator. When we compare runs, we count it in the same way for all of them.

any currency; time in hours, footage in metres

Holds when: the formula leaves out costs that do not depend on the bit or on time (casing, cement, mobilisation, which is moving the rig and crew to the site). It compares runs and does not estimate the cost of a well; the time of problems and repairs is not charged to the bit, unless the bit is to blame; the formula does not count risk (breakage, metal left on the bottom, an unsuccessful next run). We weigh it separately.

m, min and h

Holds when: this is an estimate for planning. In reality the speed of running in and pulling out changes along the hole. In open hole (the part not yet lined with casing), the pressure surges that the moving pipe causes limit the speed (running and pulling pipe), and in cased hole it is higher; the lesson's examples take $\tau_{\text{st}}$ = 2.5 min, $t_{\text{fix}}$ = 2.0 h and $\tau_{\text{conn}}$ = 5 min. These are assumptions, not norms.

SI. In practical units the advance in mm equals the weight in kN multiplied by $\sum L_i\cdot 10^6/(E A_i)$ ($L_i$ in m, $E$ in MPa, $A_i$ in mm²), and the rate in m/h is $3.6\,\Delta L/\Delta t$ ($\Delta L$ in mm, $\Delta t$ in s)

Holds when: a vertical or nearly vertical well: a change of weight is passed along the whole string without losses to friction against the walls; the top of the string is fixed: the stretch of the drilling line when the hook load rises is small compared with the stretch of the string; the cross-section is the pipe body. Tool joints and upset ends (the thickened ends of the pipes) are stiffer, so the compliance and the rate are overstated a little, by a few per cent; we assign the rate within a step to its mean weight. We take small steps, but such that each lasts at least tens of seconds, otherwise the error of timing is large.

SI (any consistent system; $N$ and $R$ in the same units of time)

Holds when: the formula takes the weight and the torque at the bit itself. The torque at the surface includes the friction of the string against the walls and overstates the energy; the energy of the mud jets is not counted: this is the mechanical energy of the bit; the values are averaged over a time during which the rock and the parameters did not change.

consistent units. In metric units $M = \mu G D/3000$ (torque in kN·m, weight in kN, diameter in mm), and the rotational part of the MSE is $160\,\mu N G/(D R)$ MPa ($N$ in rpm, $R$ in m/h)

Holds when: the weight is spread evenly over the face of the bit, so the mean radius at which the friction force acts is a third of the diameter; we find $\mu$ for a given bit and rock from torque measurements. It changes with wear, with balling and in another rock.

oilfield units of the original model

Holds when: the factors are independent: the influence of each does not depend on the others. This is a simplification; we fit the coefficients $a_1 \ldots a_8$ by regression (a statistical fit) to the data of offset wells for each unit of rock. The authors suggested reasonable limits, for example $a_5$ from 0.5 to 2.0, $a_6$ from 0.4 to 1.0, and $a_7$ from 0.3 to 1.5; the model was built for roller-cone bits. For PDC bits we use it with care, or use other models.

$, h, m

Holds when: the tripping time of a run is constant. In fact it grows with the depth of pulling, so the calculation tells us to pull a little too late; the rock in the interval is uniform, and the rate falls only because of wear; the risk of breakage and of losing the cutting structure is not counted. It makes us pull earlier.

m, h

Holds when: the rock, the weight and the rotary speed are constant; only the wear changes; $R_0$ and $\lambda$ are fitted to the record of the rate in similar runs. With a change of rock the curve is different.