Equations: Directional drilling and measurements

metres (or feet), the same for all the quantities

Holds when: Both heights are measured from the same sea level (the same height system). The rotary table elevation equals the elevation of the ground plus the height of the rotary table above the ground; at sea it is the height of the rotary table above sea level.

metres; the angle in degrees (the arctan function in spreadsheets and programming languages returns radians: multiply by $180/\pi$)

Holds when: The arctangent gives an angle from −90° to +90°, so we check the quadrant from the signs: for $\Delta N < 0$ we add 180° to the result, and for $\Delta N > 0$ and $\Delta E < 0$ we add 360°. For $\Delta N = 0$ the azimuth is 90° (the target is to the east) or 270° (to the west). The azimuth is measured from the same north as the coordinate grid (see the box below).

$i$ in degrees per the reference length; $R$ in the same units of length as $l_0$

Holds when: The rate is constant and the azimuth does not change: the section is an arc of a circle in the vertical plane. The factor $180/\pi$ appears because the rate is given in degrees, while the length of an arc is the radius times the angle in radians (1 rad $= 180/\pi \approx 57.3^\circ$).

lengths in the same units; angles in degrees (if the arctangent and arcsine return radians, multiply the result by $180/\pi$)

Holds when: The profile lies in one vertical plane, which passes through the wellhead and the target; the azimuth is constant. The upper section is vertical. The build goes at a constant rate from zero to $\theta$, and then the well is straight right up to the target. The solution exists if the distance from the centre of the arc to the target is not less than the radius. Otherwise the target lies inside the build circle, and we need a higher rate or a shallower kick-off. The formula is written for angles up to 90°.

lengths in metres; $\theta$ in the formula for the length of the arc in degrees ($\pi\theta/180$ is the same angle in radians)

Holds when: The same as for the angle. The length of the tangent section is a leg of the right triangle O–E–T. A check: $L_{\text{tan}} = (H_{\text{T}} - H_{\text{E}})/\cos\theta = (A_{\text{T}} - A_{\text{E}})/\sin\theta$.

lengths in the same units; angles in degrees

Holds when: The profile is in one vertical plane: a build from zero to $\theta$, a tangent section, a drop from $\theta$ to zero, and then a vertical section to the target. The profile can be built only if the distance between the centres of the arcs is not less than the sum of the radii, $D \ge R_1 + R_2$. Otherwise the arcs do not have enough room. If the angle is dropped not to zero but to a given angle of entry into the formation, the calculation is based on the same condition of tangency, but the formulas are longer.

lengths in the same units; the trigonometric functions of the angle $\theta$ in the units that the calculator needs

Holds when: A flat profile: a vertical, an arc of a circle, and a tangent to it. The second equality is obtained by multiplying the first by $\cos\theta$ and by the denominator.

SI

Holds when: There are no leaks and no friction: the power of the flow $\Delta p\,Q$ turns entirely into the mechanical power $M\omega$, where $\omega = 2\pi n$. A real motor turns more slowly, because part of the mud leaks through the seal between the rotor and the stator, the more so the higher the pressure drop (the volumetric efficiency). It also gives a smaller torque because of friction (the mechanical efficiency). Real values are taken from the characteristic on the data sheet. The motor works within the range of flow rate and pressure drop of its data sheet.

SI; with the speed in rpm the power is $N_{\text{sh}} = 2\pi n M/60$

Holds when: The flow rate and the density of the mud are constant. The linear dependence of the torque on the speed follows from the turbine equation for blades with fixed angles. In real turbodrills it is close to linear in the working range. When the flow rate changes (with the same blades), the runaway speed changes in proportion to $Q$, the stall torque as $Q^2$, and the power as $Q^3$.

$\delta$ and $i$ in degrees; $L$ and $l_0$ in the same units of length

Holds when: The bit and both stabilisers touch the walls and lie on one circle; the stabilisers are full gauge. All the tilt is concentrated at the lower stabiliser. If the bend is above it, the effective angle is a little smaller than $\delta$. The angles are small. The formula follows from the fact that the angle between two chords of a circle equals half the sum of the arcs on which they rest (the derivation is in the advanced section). The clearances, the washing out of the hole in soft rock, the weight on the bit, the hardness and bedding of the rocks, and the ability of the bit to cut the wall are not taken into account. So this is an upper, geometric estimate.

rates in the same units, lengths in metres

Holds when: In rotation the assembly drills straight. If it has a tendency of its own, $i_{\text{rot}}$ (a build is positive, a drop negative), then $f = (i_{\text{plan}} - i_{\text{rot}})/(i_{\text{slide}} - i_{\text{rot}})$. During sliding the toolface is held in one position. The rate is averaged over a stand or two. Within the interval the hole is stepped.

angles in degrees

Holds when: The dogleg is small (a few degrees) and the toolface is held in one position all the time. The hole is not close to vertical. At an inclination of a few degrees the formula for the azimuth becomes crude, and in a vertical hole it loses its meaning (the exact formulas are in the advanced section). In the inverse problem the quadrant of the angle $\omega$ is chosen by the signs of the numerator and the denominator, as when calculating the azimuth. A turn to the left is a negative angle (or 270–360°), and a drop is more than 90° in absolute value.

SI

Holds when: The free section of the collars is a beam on two supports under a uniform cross-hole load $w\sin\theta$; each support carries half of this load. The weight on the bit is not taken into account: by bending the collars it reduces the pendulum effect. The collars do not touch the low side between the bit and the stabiliser. If they do, the length of the pendulum is up to the point of contact, and raising the stabiliser higher no longer increases the force.

angles in degrees; the quadrant of $\Delta\varphi$ is chosen by the signs of the numerator and the denominator

Holds when: The hole over the interval is an arc of a circle in one plane (the toolface stays the same relative to this plane). The inverse problem, finding $\beta$ and $\omega$ from two directions, is solved in the same way: $\beta$ is the angle between the vectors $\mathbf{t}_1$ and $\mathbf{t}_2$, and $\omega$ is the direction of the projection of $\mathbf{t}_2$ on the plane of the vectors $\mathbf{h}$ and $\mathbf{r}$.

dimensionless components; angles in degrees

Holds when: The $N$ axis points north, the $E$ axis east and the $H$ axis vertically down. The azimuths of all the readings are measured from the same north. At $\theta = 0$ the first two components are zero, and the azimuth does not affect the result: a vertical well has no azimuth.

degrees; the result is brought to the range 0–360°

Holds when: The declination is taken from a geomagnetic model for the place and the date of the survey. It changes from year to year, and in some places noticeably. The grid convergence depends on the position of the point in the zone of the coordinate grid. In the northern hemisphere, east of the central meridian of the zone it is positive, and west of it negative. As a rough rule, $\gamma \approx \Delta\lambda\sin B$, where $\Delta\lambda$ is the difference of the longitudes of the point and of the central meridian, and $B$ is the latitude. The signs follow the definitions above. In programs and documents the conventions may be different, so the sign convention is always checked.

lengths in metres (or in feet, the same for all)

Holds when: The well over the interval is a straight segment with the direction of the lower reading. The upper reading is not used.

lengths in metres

Holds when: The well over the interval is two straight segments of equal length $\Delta L/2$ with the directions of the upper and of the lower readings.

lengths in metres; angles in degrees

Holds when: The well over the interval is a straight segment of length $\Delta L$ with the averaged direction. The azimuths are averaged along the shorter arc: for 350° and 10° the change is +20° and the average azimuth is 0°, not 180°. If one of the readings is almost vertical, its azimuth is random, and it cannot be averaged (an example is in the section on comparing the methods).

lengths in metres; the differences of angles in the denominators in radians

Holds when: The inclination changes linearly along the length of the well, and the azimuth linearly along the length of the horizontal projection. If $\theta_1 = \theta_2$, then $\Delta H = \Delta L\cos\theta$ and $\Delta A = \Delta L\sin\theta$. If $\varphi_1 = \varphi_2$, then $\Delta N = \Delta A\cos\varphi$ and $\Delta E = \Delta A\sin\varphi$. These are the limits of the formulas for a zero denominator. In space such a curve is not an arc of a circle but a helix (a spiral). For an almost vertical reading the azimuth is random, and the method “trustfully” uses it.

degrees

Holds when: The formula is the scalar product of two unit direction vectors. For small angles it is more accurate to use the equivalent formula with the sines of half-angles (the advanced section): the cosine is close to one, and significant digits are lost in rounding.

lengths in metres; $\beta$ in the factor in radians

Holds when: The well over the interval is an arc of a circle that touches the directions of both readings. The factor is greater than one and grows with the bend: for $\beta = 3^\circ$ it is 1.00023, and for $10^\circ$ it is 1.0025.

$\beta$ and $i$ in degrees; $\Delta L$ and $l_0$ in the same units of length

Holds when: The bend is taken as uniform over the interval: this is an average rate. Local bends inside the interval cannot be seen from two readings.

metres; angles in degrees

Holds when: The quadrant of the departure azimuth is chosen by the signs of $N$ and $E$, as for the azimuth of the target. All azimuths are from the one north in which the coordinates are given.

any units of angle, the same for all the quantities

Holds when: It follows from the formula with cosines by the substitution $1 - \cos x = 2\sin^2(x/2)$; it is equivalent to it for any angles.

the components are in units of $g$ or in m/s²: only their ratios enter the formulas

Holds when: The tool is at rest. When it moves, the accelerometers also feel the accelerations of the string and its vibration. The sensor axes coincide with the tool axes, and the tool axis with the axis of the hole. If the tool sits crooked in the hole, or the drill collar sags, the inclination is in error. The signs follow the convention used here. At $\omega = 0$ the $x$ axis points to the high side and the $y$ axis points to the right. Manufacturers orient the axes differently, and then the signs in the formulas change.

$B$ is in nT and $G$ is in $g$. The formula uses a ratio, so the unit of the field does not matter

Holds when: The magnetometers see only the field of the Earth: there is no steel nearby, or its effect has been allowed for. The quadrant follows from the signs: the numerator plays the role of a sine and the denominator the role of a cosine (like the atan2 function in computer programs). As $\theta \to 0$ the numerator and the denominator both go to zero, and the azimuth loses its meaning, as it does for a truly vertical well.

degrees; in the approximate form $\gamma$ and $\Delta\lambda$ are in the same unit of angle

Holds when: A transverse Mercator projection (Gauss–Krüger or UTM, the usual ways to draw the Earth on a flat map) and a spherical Earth. The flattening of the Earth changes $\gamma$ in a zone 6° wide by only fractions of an arc second (an arc second is 1/3600 of a degree). The approximate form differs from the exact one by thousandths of a degree within the zone. The central meridians of the 6° zones are 3, 9, 15, … °E. For Gauss–Krüger zone number $n$ the central meridian is $6n - 3$. In UTM the number of the same zone is 30 higher.

$\Delta B_z$ and $B_h$ are in the same units (nT); $\Delta\varphi$ is in radians

Holds when: The interference acts along the axis of the tool. The formula does not describe cross interference (from neighbouring casing strings, or from magnetised spots in the non-magnetic drill collar itself). The formula is a first-order approximation (it keeps only the main term). It holds when $\Delta B_z$ is much smaller than $B_h$. The error is systematic: it is the same in all surveys as long as the assembly and its magnetisation do not change.

SI: $P$ in Wb, $d$ in m, $\Delta B_z$ in T

Holds when: The poles are points and lie on the axis of the tool. The field of the far ends of the steel parts, which has the opposite sign, is not included, so the estimate is slightly too high. When the steel is magnetised by the field of the Earth, the lower end of the steel above the sensors and the upper end of the steel below them give a field in the same direction. Magnetisation from knocks, from magnetic particle inspection and from earlier jobs cannot be predicted in strength or sign. So we take the pole strengths from experience with surveys in the same kinds of assemblies.

$G$ in $g$, $B$ in nT

Holds when: The tool is at rest. The reference $B$ and $I$ come from a field model for the place and the date, or from local measurements. The tolerances come from the error model of the particular instrument. Different companies use different ones.

°/h; latitude in degrees

Holds when: The tool is at rest relative to the Earth. The latitude is the geographic latitude.

lengths in metres, angles in radians

Holds when: The well is vertical down to the kick-off point and below it runs in one vertical plane. The three errors are independent and constant along the whole well. The azimuth error turns the whole path, seen from above, around the wellhead. The inclination error turns the inclined part around the kick-off point. The depth error stretches the whole path from the rotary table. Random errors of single surveys, tilts, sag and interference are not included. A real error model has dozens of terms and rules for adding them from survey to survey.

metres; the result has no units

Holds when: The uncertainty regions come from the error models of both wells, for one probability level (often 2–3σ). Company standards have variants: some subtract the radii of the wells from the distance, and some add the sizes in other ways. The standards themselves set the limits of allowed values.

$\varepsilon$ and $\Omega$ are in the same unit of angular speed; the result is in radians

Holds when: The zero offsets of the two sensors are independent and equal in size. The inclination and the toolface are known exactly, and the tool is at rest.

SI

Holds when: The fluid is Newtonian. Drilling mud is not Newtonian, so we put an effective viscosity in place of $\mu$. The estimate is rough. The losses come only from viscous friction in the layer at the wall, which is thin compared with the radius of the pipe. Turbulence of the flow and free gas add losses, so the calculation more likely underestimates the attenuation. The pipes are the same along their whole length. There are no reflections from narrowings, the downhole motor, pumps and pulsation dampeners (devices that smooth out the pressure surges of the pumps). In practice such reflections change the signal strongly at some frequencies.

bits and seconds

Holds when: The quantity is quantised evenly over its whole range (it is split into equal steps). Data compression and error-correcting codes are not included. $R$ is the rate of useful data after decoding, not the frequency of the signal.

SI

Holds when: A uniform conducting medium and a plane wave. This estimate gives only the order of magnitude, that is, the right power of ten. In reality the current spreads from the sub along the string and through layers of different resistivity. The signal also weakens with distance where the skin effect is small, and casing and conducting mud add losses. The real range is set by the ratio of signal to noise at the receiver. Displacement currents (the currents through insulating material) are negligible. At frequencies of a few hertz this is true.

metres; the angle in degrees

Holds when: The boundary is a plane, and the well runs in one vertical plane. The angle is apparent, in the direction of the well. The true dip is larger if the well does not run along the dip. The depths and positions are for the point of the sensor, not the bit, and come from the calculation of the path.

SI

Holds when: The pipe is thin-walled and free to move in the axial direction. The gas is in small bubbles that are compressed isothermally (at constant temperature; the bulk modulus ≈ the pressure). In adiabatic compression (with no exchange of heat) the modulus is 1.2–1.4 times larger, and the speed is a little higher. The wavelength is much larger than the diameter of the pipe and the size of the bubbles.