Equations: Casing and cementing

The equations of the “Casing and cementing” lessons in “Drilling engineering”, with their conditions and units, and what each symbol means in them.

Inside diameter and mass per metre of a casing pipe

d = D - 2t, \qquad m = \rho_{\text{steel}}\,\pi\,(D - t)\,t

Holds when: The cross-section area of the wall, $\frac{\pi}{4}(D^2 - d^2)$ , equals $\pi(D - t)t$ . This is the mass of a plain pipe. The nominal weight in a catalogue is a little greater, because it includes the couplings or the upset (thickened) ends. These are nominal sizes. Real sizes differ within the tolerances of the standard.

Casing: purpose, types, properties and standards →

Internal yield pressure (the Barlow formula)

p_{\text{iy}} = 0.875\,\frac{2\,Y_p\,t}{D}

any consistent units: the pressure and the yield strength in the same units (MPa or psi), $t$ and $D$ in the same units

Holds when: The pipe is thin-walled, and the stress is taken as uniform across the wall (the derivation is in the advanced section). The pressure outside is zero. In a well, the pipe feels the difference between the internal and the external pressure. This is a rating of the pipe body. The pressure that the connection holds may be lower.

Casing: purpose, types, properties and standards →

Tensile strength of the pipe body

F_y = Y_p\,\frac{\pi}{4}\left(D^2 - d^2\right)

SI; it is convenient to take $Y_p$ in MPa and the diameters in mm: then $F_y$ comes out in newtons

Holds when: The load is axial, with no bending or pressure. The strength of the connection is checked separately.

Casing: purpose, types, properties and standards →

Collapse pressure: yield and elastic loss of stability

p_Y = 2\,Y_p\,\frac{D/t - 1}{(D/t)^2}, \qquad p_E = \frac{2E}{1 - \nu^2}\cdot\frac{1}{(D/t)\,(D/t - 1)^2}

any consistent units of pressure and modulus

Holds when: A long, perfectly round pipe with a uniform wall, zero pressure inside and no axial load. The standard reduces the theoretical $p_E$ by about 29% to allow for ovality and uneven wall thickness. Between the two limits it applies empirical formulas for “plastic” and “transition” collapse. The boundaries between these ranges depend on the grade.

Casing: purpose, types, properties and standards →

Weight of a metre of string when the fluids inside and outside differ

w = g\left[m\left(1 - \frac{\rho_{\text{out}}}{\rho_{\text{steel}}}\right) + (\rho_{\text{in}} - \rho_{\text{out}})\,A_{\text{in}}\right]

Holds when: A vertical string and fluids at rest. This is the same formula as in the lesson on drill string design (the effective force), where the volume of steel of the pipe with couplings is taken as $m/\rho_{\text{steel}}$ . With the same fluids inside and outside, the second term disappears, and we get $w = m\,g\,k_{\text{b}}$ with the buoyancy factor $k_{\text{b}} = 1 - \rho_m/\rho_{\text{steel}}$ . In an inclined hole the hook load also depends on the friction against the wall.

Wellheads and running casing →

Drop of the level in the annulus when the float valve fails

y = L_{\text{empty}}\,\frac{A_{\text{in}}}{A_{\text{in}} + A_{\text{ann}}}, \qquad \Delta p = \rho_m\,g\,y

SI; in metric units $\Delta p\,[\text{MPa}] = 0.00981\,\rho_m\,[\text{g/cm}^3]\,y\,[\text{m}]$

Holds when: The mud in the string and in the annulus has the same density, and the well is not being topped up at this time. The levels equalise completely. The areas are constant along the unfilled part and along the depth of the drop.

Wellheads and running casing →

Hook load of a string with fluids of different densities

W = W_m - g\,A_{\text{out}}\sum_j (\rho_{\text{out }j} - \rho_m)\,h_j + g\,A_{\text{in}}\sum_k (\rho_{\text{in }k} - \rho_m)\,h_k

SI; it is convenient to calculate $\sum(\rho - \rho_m)h$ in g/cm³·m and to multiply by 1000

Holds when: A vertical string and fluids at rest. When mud circulates, the friction of the flow against the pipe adds to the weight. The areas are the same along the length. For a string of sections with different thickness, $A_{\text{out}}$ hardly changes, and $A_{\text{in}}$ is taken for the section where the column stands.

Wellheads and running casing →

External and internal excess pressure

p_{\text{ne}}(z) = p_{\text{out}}(z) - p_{\text{in}}(z), \qquad p_{\text{ni}}(z) = p_{\text{in}}(z) - p_{\text{out}}(z)

any consistent units of pressure

Holds when: Both pressures are taken at the same depth and at the same moment. The profiles inside and outside must be compatible, that is, they must exist together in one operation. The pressures are gauge pressures: atmospheric pressure acts on both sides of the wall and cancels. The pressure of a column of fluid is calculated from the vertical depth, and the length of the pipe sections from the length along the hole.

Casing design →

Depth of the mud level in the string during lost circulation

L = h_{\text{loss}}\left(1 - \frac{\rho_{\text{loss}}}{\rho_m}\right)

any units of length; the densities in the same units

Holds when: The well is open at the wellhead, and the mud is at rest and has the same density throughout. The formation takes mud while the pressure of the column above it is higher than the pressure in the formation: $\rho_m (h_{\text{loss}} - L) = \rho_{\text{loss}}\,h_{\text{loss}}$ . While the level falls, the pressure on the formations above it decreases, and they may start to flow. Then the string will hold gas, not an empty space.

Casing design →

Pressure in a closed well filled with gas

p_{\text{shoe}} = \min\!\left[\,p_p - \rho_{\text{g}}\,g\,(h - h_{\text{shoe}});\ p_f\,\right], \qquad p_{\text{wh}} = p_{\text{shoe}} - \rho_{\text{g}}\,g\,h_{\text{shoe}}

SI; the pressures are gauge pressures

Holds when: The well is closed and full of gas from the formation to the wellhead. This is the worst case. When a small pocket of gas is circulated out, the pressures are much lower (Well kill methods). The gas density is taken as constant along the well, and taken with a margin on the low side: the lighter the gas, the higher the pressure at the wellhead. For this calculation the fracture pressure is taken with a margin on the high side: an underestimated value here underestimates the load. If there is a rock in the open hole that is weaker than the rock under the shoe, it sets the limit.

Casing design →

Strength conditions of a casing pipe

p_{\text{col}} \ge n_{\text{col}}\,p_{\text{ne}}, \qquad p_{\text{iy}} \ge n_{\text{bur}}\,p_{\text{ni}}, \qquad F_y \ge n_{\text{ten}}\,F

any consistent units for the rating and the load

Holds when: We check the conditions in every cross-section of the string, for every load case. The ratings already include the minimum yield strength (the stress at which the steel starts to stretch permanently) and the tolerance on the wall thickness, but not wear, corrosion or axial load. The values of the factors are set by regulatory documents and company standards.

Casing design →

Axial load from the weight of the string in mud

F(z) = g\,k_{\text{b}}\sum_i m_i\,l_i, \qquad k_{\text{b}} = 1 - \frac{\rho_m}{\rho_{\text{steel}}}

SI; the force in newtons, 1 MN = 10⁶ N ≈ 102 tonnes-force ≈ 224.8 thousand pounds-force

Holds when: The string hangs freely, with mud of the same density inside and outside. The hole is vertical. In an inclined hole, we take the projection of the length on the vertical instead of the length, and friction against the wall adds a load when the crew pulls the string or moves it up and down. This is the weight of the string in the fluid. The true axial force in the pipe body below the wellhead is a little smaller (see the advanced section), so the estimate is on the safe side.

Casing design →

Additional axial loads: bending, shock, pressure test

\sigma_{\text{bend}} = \frac{E\,D}{2R}, \qquad F_{\text{shock}} = \rho_{\text{steel}}\,c\,\Delta v\,A_s, \qquad F_{\text{test}} = p_{\text{test}}\,A_{\text{in}}

Holds when: Bending: the pipe follows the axis of the hole. We add the force $\sigma_{\text{bend}} A_s$ to the axial load as the equivalent of bending, and we check the sections with sharp curvature. Shock: an instant stop and an elastic wave in a long string. This is an upper estimate. With smooth work the force is much smaller. Pressure test: the bottom of the string is closed with a plug, and the pressure acts on the whole area of the inside cross-section.

Casing design →

Yield strength reduced for axial tension

Y_{pa} = Y_p\left[\sqrt{1 - 0.75\left(\frac{\sigma_a}{Y_p}\right)^2} - 0.5\,\frac{\sigma_a}{Y_p}\right]

any consistent units of stress

Holds when: The von Mises yield condition (a rule for when steel yields under combined stresses) for the hoop and axial stresses. The radial stress is not included (the derivation is in the advanced section). The formula is used for tension. The favourable effect of compression is not taken as a margin. Bending is not included in $\sigma_a$ . Internal pressure, if there is any, is taken into account separately.

Casing design →

Collapse pressure in the four regimes

\begin{aligned} p_Y &= 2\,Y_p\,\frac{D/t - 1}{(D/t)^2} && \text{— yield regime} \\ p_P &= Y_p\left(\frac{A}{D/t} - B\right) - C && \text{— plastic} \\ p_T &= Y_p\left(\frac{F}{D/t} - G\right) && \text{— transition} \\ p_E &= \frac{46.95\cdot 10^{6}}{(D/t)\,(D/t - 1)^2} && \text{— elastic} \end{aligned}

the formulas of the standard are written in psi: convert the yield strength to psi (1 MPa = 145.04 psi), and convert the result back to MPa

Holds when: We choose the regime by $D/t$ : yield up to $(D/t)_{YP}$ , plastic up to $(D/t)_{PT}$ , transition up to $(D/t)_{TE}$ , and elastic above. The boundaries are given below. The plastic-regime formulas are empirical, from tests of pipes. The numerator of the elastic formula is the theoretical value reduced by about 29% to allow for imperfections of shape. The pipe has no axial load and no internal pressure. We allow for tension by putting $Y_{pa}$ in place of $Y_p$ .

Casing design →

Coefficients of the collapse pressure formulas

\begin{aligned} A &= 2.8762 + 0.10679\cdot 10^{-5}\,Y_p + 0.21301\cdot 10^{-10}\,Y_p^2 - 0.53132\cdot 10^{-16}\,Y_p^3 \\ B &= 0.026233 + 0.50609\cdot 10^{-6}\,Y_p \\ C &= -465.93 + 0.030867\,Y_p - 0.10483\cdot 10^{-7}\,Y_p^2 + 0.36989\cdot 10^{-13}\,Y_p^3 \\ F &= \frac{46.95\cdot 10^{6}\,s^3}{Y_p\,(s - B/A)\,(1 - s)^2}, \qquad s = \frac{3\,B/A}{2 + B/A}, \qquad G = \frac{F\,B}{A} \end{aligned}

yield strength in psi

Holds when: The coefficients are valid only with $Y_p$ in psi. The boundaries of the regimes: $(D/t)_{PT} = \dfrac{Y_p\,(A - F)}{C + Y_p\,(B - G)}$ , $(D/t)_{TE} = \dfrac{2 + B/A}{3\,B/A}$ . The boundary $(D/t)_{YP}$ is the point where the yield formula and the plastic formula give the same value.

Casing design →

True axial force in a string of one wall thickness

F_a(z) = g\,m\,(L - z) - p_L\,A_s

Holds when: A vertical string of one wall thickness with an open bottom, and fluid of the same density inside and outside. If the bottom is closed by a float valve and the pressures inside and outside differ, the force on the end face is $p_{\text{out}} A_{\text{out}} - p_{\text{in}} A_{\text{in}}$ , where $A_{\text{out}}$ and $A_{\text{in}}$ are the areas of the outer and inner cross-sections. In a string of sections with different walls, the pressure also acts on the steps at the changes.

Casing design →

Von Mises equivalent stress on the inside surface of a pipe

\sigma_{\text{eq}} = \sqrt{\sigma_a^2 + \sigma_t^2 + \sigma_r^2 - \sigma_a\sigma_t - \sigma_t\sigma_r - \sigma_r\sigma_a}, \qquad \sigma_r = -p_{\text{in}}, \qquad \sigma_t = \frac{p_{\text{in}}\,(r_{\text{in}}^2 + r_{\text{out}}^2) - 2\,p_{\text{out}}\,r_{\text{out}}^2}{r_{\text{out}}^2 - r_{\text{in}}^2}

any consistent units of stress and pressure; radii in the same units

Holds when: An elastic pipe with a uniform wall. Under pressure, the highest equivalent stress is on the inside surface. Tension is positive. Shear stresses from torsion are not included. The condition describes the start of yield, not loss of stability: the collapse of thin-walled pipes is checked separately.

Casing design →

Change of the axial force in a fixed string

\Delta F = -E\,\alpha\,\Delta T\,A_s + 2\nu\left(A_{\text{in}}\,\Delta p_{\text{in}} - A_{\text{out}}\,\Delta p_{\text{out}}\right)

Holds when: The part is fixed at both ends and stays straight (no buckling). The changes of temperature and pressures are averaged along the length of the free part. The steel is elastic.

Casing design →

Density of a cement slurry from absolute volumes

\rho_{\text{sl}} = \frac{\sum_i m_i}{\sum_i m_i/\rho_i}

kg and kg/L: the volumes come out in litres, and the density in kg/L = g/cm³

Holds when: The volumes of the components add up. Dissolving and chemical shrinkage on mixing are not included. Additives that dissolve in water are counted by their own density. The error is small, because the amounts are small. There is no entrained air in the slurry. In foaming, the density is lower than the calculated one, so the crew checks it on the rig with a pressurised mud balance.

Oilwell cements and additives →

Slurry yield and consumption of materials

Y = \frac{\sum_i m_i/\rho_i}{m_{\text{cem}}}, \qquad M_{\text{cem}} = \frac{V}{Y}

masses in tonnes, densities in t/m³ (numerically equal to g/cm³)

Holds when: The same as for the density. A margin for losses on mixing and for the inaccuracy of the hole volume is added separately.

Oilwell cements and additives →

Fluid loss index of a cement slurry

V_{\text{API}} = 2\,V_t\,\sqrt{\frac{30}{t}}

mL and minutes

Holds when: The filtrate grows in proportion to $\sqrt t$ , as in filtration through a growing cake (filtration theory). When the sample dehydrates in minutes, this is only a conventional index for comparing recipes.

Oilwell cements and additives →

Standoff ratio of a string

\text{SO} = \frac{w_{\min}}{(D_{\text{hole}} - d)/2}\cdot 100\ \%

any consistent units of length

Holds when: The cross-section of the hole is round. In a washout (a place where the hole has been washed wider than the bit made it), $D_{\text{hole}}$ is its diameter. 100% means that the string is exactly in the middle, and 0% means that it touches the wall.

Primary cementing →

Ratio of the speeds on the narrow and the wide side of an eccentric annulus

\frac{v_{\text{nar}}}{v_{\text{wid}}} = \left(\frac{\text{SO}}{2 - \text{SO}}\right)^{(n+1)/n}

dimensionless

Holds when: Each part of the circumference of the annulus is treated as a narrow flat slot with its own width, and the pressure gradient along the length is the same for all parts (the narrow-slot approximation). The flow is laminar, and the fluid is a power-law fluid with no yield stress. In a mud with a yield stress, the flow on the narrow side can stop altogether. The width of the slot on the narrow side is proportional to $\text{SO}$ , and on the wide side to $2 - \text{SO}$ (as fractions of the nominal gap).

Primary cementing →

Volume of cement slurry

V_{\text{sl}} = \sum_i A_{\text{ann }i}\,L_i\,(1 + x_i) + V_{\text{rat}} + V_{\text{st}}, \qquad A_{\text{ann}} = \frac{\pi}{4}\left(D^2 - d^2\right)

Holds when: The volumes of the slurry do not change during pumping (we neglect compressibility and shrinkage). The margin covers the inaccuracy of the hole volume. If the hole turns out smaller, the cement will rise higher than planned.

Primary cementing →

Displacement volume

V_{\text{disp}} = \sum_j \frac{\pi}{4}\,d_{\text{in }j}^2\,L_j + V_{\text{pip}}

Holds when: Nominal inside diameters. The real volume differs by fractions of a per cent, so the crew expects the bump near the calculated volume. Pumps are counted by strokes with a volumetric efficiency, or by a flow meter. The compression of the mud and the string under pressure adds a little volume.

Primary cementing →

Imbalance of the columns and the pump pressure

\Delta p_{\text{U}} = 0.00981\left(\sum \rho_{\text{ann}} h_{\text{ann}} - \sum \rho_{\text{str}} h_{\text{str}}\right), \qquad p_{\text{pump}} = \Delta p_{\text{U}} + \Delta p_{\text{fr.ann}} + \Delta p_{\text{fr.str}}

metric: MPa, g/cm³, m; 0.00981 MPa is the pressure of a column of 1 m of fluid with a density of 1 g/cm³

Holds when: The heights are vertical. The columns are counted from the wellhead to the depth of the shoe. If $p_{\text{pump}}$ comes out negative by the formula, there is no pressure at the pumps: the fluid in the string goes down faster than it is pumped (free fall, the next section).

Primary cementing →

Flow rate out of the shoe during free fall

p_{\text{str}}(z_{\text{top}}) - \Delta p_{\text{fr.str}}(Q) = p_{\text{ann}} + \Delta p_{\text{fr.ann}}(Q), \qquad \frac{\dd V_{\text{void}}}{\dd t} = Q - Q_{\text{pump}}

metric

Holds when: The pressure in the void is taken as atmospheric. In reality there is a partial vacuum there, which holds the fall back a little: the void is slightly smaller than calculated. We ignore the inertia of the fluids: at each moment the flow rate is the one at which the heads balance. The fluid being pumped lies at once on top of the column. The rathole is filled with mud and does not take part in the flow. The volume of the head and the piping (0.2 m³) is not included: it only shifts the displacement by 0.2 m³.

Primary cementing →

Attenuation of the casing wave

\alpha = \frac{20}{\Delta s}\,\log_{10}\frac{A_1}{A_2}, \qquad \Delta\alpha = \frac{20}{L_{\text{eff}}}\,\log_{10}\frac{A_{\text{free}}}{A}

dB/m; amplitudes in the same units; 1 dB/ft = 3.28 dB/m

Holds when: The amplitude of the casing wave decays exponentially: on every metre of path it falls by the same factor. Both receivers see the same casing wave, and the losses on the stretches of path through the mud are equal for them. So these losses cancel in the ratio $A_1/A_2$ . In the second formula the reference is free casing in the same well: the same diameter and wall thickness of the pipe, the same mud and the same tool.

Remedial cementing, cement plugs and cement evaluation →

Bond index

\text{BI} = \frac{\Delta\alpha}{\Delta\alpha_{100}} = \frac{\log_{10}(A_{\text{free}}/A)}{\log_{10}(A_{\text{free}}/A_{100})}

dimensionless; amplitudes in mV

Holds when: The reference $A_{100}$ is taken in the same casing and in the same cement. A lightweight set cement weakens the wave less even with full bond, so the lead slurry needs its own reference. The index averages over the whole circumference: a channel over part of the circumference and a uniformly weak cement give the same BI. The travel time is constant, and arrivals through the rock do not fall into the window where the amplitude is measured (otherwise the readings are false; see the next section).

Remedial cementing, cement plugs and cement evaluation →

Wellhead pressure limit in a squeeze without fracturing

p_{\text{wh}}^{\max} = p_f - \Delta p_{\text{margin}} - 0.00981\sum_i \rho_i h_i

MPa; 0.00981 MPa is the pressure of a 1 m column of fluid with a density of 1 g/cm³

Holds when: The calculation is for rest. While we pump down the pipe, friction reduces the pressure at the bottom for the same wellhead pressure. So during pumping the pressure at the bottom stays below the limit, and in the pauses it just reaches it. The fracture pressure is taken from tests of this formation or a neighbouring one, if there are any, and not from a forecast. The heights are vertical: in a deviated well the columns are counted by the true vertical depth.

Remedial cementing, cement plugs and cement evaluation →

Balanced plug: heights and volumes

h_{\text{cem}} = \frac{V_{\text{cem}}}{A_{\text{ann}} + A_{\text{pipe}}}, \qquad V_{\text{sp2}} = V_{\text{sp1}}\,\frac{A_{\text{pipe}}}{A_{\text{ann}}}, \qquad V_{\text{disp}} = A_{\text{pipe}}\,(H - h_{\text{cem}} - h_{\text{sp}}), \qquad L_{\text{plug}} = \frac{V_{\text{cem}}}{A_{\text{hole}}}

SI: m, m², m³

Holds when: The fluids do not mix. The same fluids stand at the same height in the pipe and in the annulus. Then the pressures at the end of the pipe are equal, and there is no flow. The cross-section of the hole is constant (taken from the caliper log, a log that measures the diameter of the hole). The tool joints are ignored. They reduce the annulus slightly, so with the pipe in the hole the cement stands a little higher than calculated. The deviation of the hole is ignored. The pipe drains as it is pulled out, and all the cement stays in the wellbore.

Remedial cementing, cement plugs and cement evaluation →

Amplitude of an off-centre tool

\frac{A_e}{A} = J_0\!\left(2\pi f\,\tau\right), \qquad \tau = 2e\,\Delta t_m\cos\theta_c, \qquad \sin\theta_c = \frac{\Delta t_{\text{cas}}}{\Delta t_m}

SI; $\tau$ in the formula $2\pi f\tau$ is in seconds

Holds when: The delay round the circumference varies as $\tau\cos\varphi$ (the displacement is small compared with the radius of the casing), and the arrivals from all sides are equally strong. The signal is narrow-band: a sine wave of frequency $f$ . For a short pulse the weakening is of the same order. The ray approximation is used, as in the formula for the travel time of the head wave.

Remedial cementing, cement plugs and cement evaluation →

Decay of the ringing of the casing wall

D = 20\,\log_{10}\left|R_{\text{in}}\,R_{\text{out}}\right|, \qquad R = \frac{Z_2 - Z_1}{Z_2 + Z_1}, \qquad f_0 = \frac{v_{\text{steel}}}{2\,t}

SI; impedance is density times speed

Holds when: A plane wave at normal incidence. The attenuation in the steel itself and the spreading of the beam are ignored. They make the decay faster behind any material, so real times are shorter than calculated, and the tool finds the difference between materials by calibration. Behind the casing there is a layer that is thick compared with the wavelength. A thin layer or a microannulus changes the picture. Real processing fits a model to the whole record, with the transducer and the mud taken into account. The formula explains the principle.

Remedial cementing, cement plugs and cement evaluation →

Symbols

Symbol	Meaning and context
$\Delta p$	fall of the pressure on the formations of the open hole, PaWellheads and running casing
$\Delta F$	change of the axial force; positive is tension, NCasing design
$\Delta T$	average change of temperature of the free part, °CCasing design
$\Delta s$	distance between the receivers, mRemedial cementing, cement plugs and cement evaluation
$0.875$	allowance for the tolerance: the wall may be 12.5% thinner than nominal, —Casing: purpose, types, properties and standards
$2$	the conversion from the area of the cell (about 22.6 cm²) to the standard one, twice as large, —Oilwell cements and additives
$A_1,\ A_2$	amplitudes of the casing wave at the near and at the far receiver, mVRemedial cementing, cement plugs and cement evaluation
$A_{100}$	amplitude in that interval: the lowest steady amplitude of this cement, mVRemedial cementing, cement plugs and cement evaluation
$A_{\text{ann}},\ A_{\text{pipe}},\ A_{\text{hole}}$	cross-section areas of the annulus (the hole minus the outside diameter of the pipe), of the channel inside the pipe, and of the hole without the pipe, m²Remedial cementing, cement plugs and cement evaluation
$A_{\text{ann }i},\ L_i$	area of the annulus and the length of the $i$ -th interval (in a cased interval, $D$ is the inside diameter of the previous string), m², mPrimary cementing
$A_e,\ A$	amplitude of the casing wave with an off-centre tool and with a centred tool, mVRemedial cementing, cement plugs and cement evaluation
$A_{\text{free}},\ A$	amplitude at the near receiver in free casing of the same well and in the interval under study, mVRemedial cementing, cement plugs and cement evaluation
$A_{\text{in}}$	area of the inside cross-section of the pipe, m²Wellheads and running casing
$A_{\text{in}},\ A_{\text{ann}}$	area of the inside cross-section of the string and area of the annulus at the wellhead, m²Wellheads and running casing
$A_{\text{out}}$	outside area that displaces fluid: $A_{\text{in}} + m/\rho_{\text{steel}}$ , allowing for the steel of the couplings, m²Wellheads and running casing
$A_s$	area of the wall cross-section, m²Casing design
$A_s,\ A_{\text{in}}$	area of the cross-section of the pipe wall and area of its inside cross-section, m²Casing design
$A_s,\ A_{\text{in}},\ A_{\text{out}}$	area of the wall, and areas of the inside and outside cross-sections, m²Casing design
$A,\ B,\ C,\ F,\ G$	coefficients that depend on the yield strength (see below), C in psi, the others dimensionlessCasing design
$\text{BI}$	bond index: 0 is free casing, 1 is full bond, —Remedial cementing, cement plugs and cement evaluation
$c$	speed of the longitudinal wave along the pipe (the speed of sound along it, as in a rod), $\sqrt{E/\rho_{\text{steel}}}$ , about 5100 m/s; in massive steel the wave is faster, about 5900 m/s, m/sCasing design
$d$	inside diameter (nominal), mCasing: purpose, types, properties and standards
$D$	outside diameter, mCasing: purpose, types, properties and standards outside diameter of the pipe, mCasing design change of the amplitude of the ringing over one round trip across the wall and back, dBRemedial cementing, cement plugs and cement evaluation
$D_{\text{hole}},\ d$	diameter of the hole and outside diameter of the string; $(D_{\text{hole}} - d)/2$ is the gap with perfect centring (annular clearance), mmPrimary cementing
$d_{\text{in }j},\ L_j$	inside diameter and length of the $j$ -th section of the string from the wellhead to the landing ring, mPrimary cementing
$D,\ d$	outside and inside diameters, mCasing: purpose, types, properties and standards
$D/t$	ratio of the outside diameter to the wall thickness, —Casing design
$e$	displacement of the tool from the axis of the casing, mRemedial cementing, cement plugs and cement evaluation
$E$	Young's modulus of steel, about 207,000 MPa, PaCasing design
$E,\ \alpha,\ \nu$	Young's modulus (about 207,000 MPa), the coefficient of linear expansion (about $12\cdot 10^{-6}$ 1/°C) and Poisson's ratio (0.3) of steel, Pa, 1/°C, —Casing design
$E,\ \nu$	Young's modulus (for steel about 207,000 MPa) and Poisson's ratio (about 0.3), MPa, —Casing: purpose, types, properties and standards
$f$	frequency of the signal, HzRemedial cementing, cement plugs and cement evaluation
$f_0$	frequency of the ringing: the thickness resonance of the wall, HzRemedial cementing, cement plugs and cement evaluation
$F_a(z)$	axial force in the pipe body at the depth $z$ ; positive is tension, NCasing design
$F_{\text{shock}}$	additional force when the speed of the string changes suddenly by $\Delta v$ : landing on the slips, a jerk, moving the string up and down, NCasing design
$F_{\text{test}}$	additional tension when the string is pressure-tested with the pressure $p_{\text{test}}$ , when the top plug has landed on the float collar, NCasing design
$F_y$	the tensile load at which the stress in the pipe body reaches the yield strength (pipe body yield strength), NCasing: purpose, types, properties and standards tensile strength of the pipe body; for a connection, its tensile strength (the pull-out load, at which the thread starts to pull out of the coupling), MNCasing design
$F(z)$	tensile force in the cross-section at the depth $z$ , NCasing design
$\Delta p_{\text{fr.ann}},\ \Delta p_{\text{fr.str}}$	friction losses in the annulus and in the string at the given rate, MPaPrimary cementing
$\Delta p_{\text{fr.str}},\ \Delta p_{\text{fr.ann}}$	friction losses in the string and in the annulus at the rate $Q$ , MPaPrimary cementing
$g$	acceleration due to gravity, 9.81, m/s²Wellheads and running casing acceleration due to gravity, m/s²Casing design acceleration due to gravity, 9.81 m/s², m/s²Casing design
$h_{\text{cem}}$	height of the cement while the pipe is in the hole; it is the same in the pipe and in the annulus, mRemedial cementing, cement plugs and cement evaluation
$h_{\text{loss}}$	true vertical depth of the loss zone, mCasing design
$h_{\text{sp}}$	height of the spacer: $V_{\text{sp1}}/A_{\text{ann}}$ , the same in the annulus and in the pipe, mRemedial cementing, cement plugs and cement evaluation
$h,\ h_{\text{shoe}}$	depths of the formation and of the shoe, vertical, mCasing design
$\Delta p_{\text{in}},\ \Delta p_{\text{out}}$	average change of pressure inside and outside along the free part, PaCasing design
$J_0$	Bessel function of the first kind of order zero, —Remedial cementing, cement plugs and cement evaluation
$k_{\text{b}}$	buoyancy factor, —Casing design
$L$	depth of the mud level in the string, measured from the wellhead, mCasing design depth of the shoe, mCasing design
$L_{\text{eff}}$	path of the wave along the casing to the near receiver: the 0.914 m spacing minus the shift caused by the two stretches of path through the mud, which gives about 0.9 m, mRemedial cementing, cement plugs and cement evaluation
$L_{\text{empty}}$	length of the unfilled upper part of the string, mWellheads and running casing
$l_i$	length of the part of the $i$ -th section that lies below the cross-section, mCasing design
$L_{\text{plug}}$	length of the plug after the pipe is pulled out, mRemedial cementing, cement plugs and cement evaluation
$m$	mass of a metre of plain pipe, without couplings, kg/mCasing: purpose, types, properties and standards nominal weight (mass per metre) of the pipe with couplings, kg/mWellheads and running casing mass of a metre of plain pipe (without couplings), kg/mCasing design
$m_{\text{cem}}$	mass of cement in the recipe, tOilwell cements and additives
$M_{\text{cem}}$	mass of cement for the job; we find the water and the additives by multiplying by their share of the cement mass, tOilwell cements and additives
$m_i$	nominal weight (mass per metre) of the pipes of the $i$ -th section, with couplings, kg/mCasing design mass of a component: cement, water, each additive, kgOilwell cements and additives
$\Delta t_m,\ \Delta t_{\text{cas}}$	interval transit times of the mud and of the casing wave, µs/mRemedial cementing, cement plugs and cement evaluation
$\Delta p_{\text{margin}}$	margin that the programme leaves below the fracture pressure, MPaRemedial cementing, cement plugs and cement evaluation
$n$	flow behaviour index of the power-law model of the fluid, —Primary cementing
$n_{\text{col}},\ n_{\text{bur}},\ n_{\text{ten}}$	design factors for collapse, internal pressure and tension, —Casing design
$p_{\text{ann}}$	the hydrostatic pressure of the columns in the annulus at the shoe, MPaPrimary cementing
$p_{\text{col}}$	collapse pressure of the pipe, MPaCasing design
$p_E$	external pressure of the elastic loss of stability of a perfectly round pipe, MPaCasing: purpose, types, properties and standards
$p_f$	fracture pressure of the rock under the shoe, PaCasing design fracture pressure at the depth of the holes, MPaRemedial cementing, cement plugs and cement evaluation
$p_{\text{in}},\ p_{\text{out}}$	pressure inside and outside the pipe, MPaCasing design
$p_{\text{iy}}$	internal yield pressure (burst): the internal gauge pressure at which the hoop stress in the wall (the stress around the circumference of the pipe) reaches the yield strength, MPaCasing: purpose, types, properties and standards internal yield pressure of the pipe, MPaCasing design
$p_L$	fluid pressure at the depth of the shoe, PaCasing design
$p_{\text{ne}}$	external excess pressure, the collapse load, MPaCasing design
$p_{\text{ne}},\ p_{\text{ni}},\ F$	external and internal excess pressure and axial force in the cross-section considered, MPa, MPa, MNCasing design
$p_{\text{ni}}$	internal excess pressure, the burst load, MPaCasing design
$p_{\text{out}},\ p_{\text{in}}$	pressure outside and inside the string at the depth $z$ , MPaCasing design
$p_p$	pore pressure of the highest-pressure formation of the next interval, at the depth $h$ , PaCasing design
$p_{\text{pump}}$	pump pressure (at the cementing head), MPaPrimary cementing
$p_{\text{shoe}}$	pressure inside the string at the depth of the shoe, PaCasing design
$p_{\text{str}}(z_{\text{top}})$	the hydrostatic pressure of the column in the string at the shoe, when its top has gone down to the depth $z_{\text{top}}$ (above it there is a void), MPaPrimary cementing
$p_{\text{wh}}$	pressure inside the string at the wellhead, PaCasing design
$p_{\text{wh}}^{\max}$	highest allowed pressure in the pipe at the wellhead, at rest, MPaRemedial cementing, cement plugs and cement evaluation
$p_Y$	external pressure at which the inner surface of the wall reaches yield, MPaCasing: purpose, types, properties and standards
$p_Y,\ p_P,\ p_T,\ p_E$	collapse pressure in the yield, plastic, transition and elastic regimes, psiCasing design
$Q,\ Q_{\text{pump}}$	the flow rate out of the shoe (which is also the returns at the wellhead) and the pump rate, m³/minPrimary cementing
$R$	radius of curvature of the hole, mCasing design
$r_{\text{in}},\ r_{\text{out}}$	inside and outside radii, mmCasing design
$R_{\text{in}},\ R_{\text{out}}$	reflection coefficients at the inner surface (steel to mud) and at the outer surface (steel to the material behind the casing), —Remedial cementing, cement plugs and cement evaluation
$s$	an auxiliary quantity, —Casing design
$\text{SO}$	standoff ratio, the ratio of the smallest gap to the nominal gap, %Primary cementing standoff, as a fraction, —Primary cementing
$t$	wall thickness (nominal), mCasing: purpose, types, properties and standards the time of the test: 30 min, or the moment when the sample dehydrated and gas started to go through the filter, minOilwell cements and additives
$t,\ D$	nominal wall thickness and outside diameter, mmCasing: purpose, types, properties and standards
$\Delta p_{\text{U}}$	excess of the annulus columns over the columns in the string at the depth of the shoe; negative means that the string is heavier, MPaPrimary cementing
$V$	the required volume of slurry (from the cementing design), m³Oilwell cements and additives
$V_{\text{API}}$	fluid loss index: the volume of filtrate for 30 min, reduced to the standard area, mL/30 minOilwell cements and additives
$V_{\text{cem}}$	volume of cement slurry for the plug, m³Remedial cementing, cement plugs and cement evaluation
$V_{\text{disp}}$	volume of displacement fluid up to the bump, m³Primary cementing
$V_{\text{disp}},\ H$	volume of displacement and depth of the end of the pipe (the bottom of the plug), m³, mRemedial cementing, cement plugs and cement evaluation
$v_{\text{nar}},\ v_{\text{wid}}$	average speed of the fluid in the narrowest and in the widest part of the annulus, m/sPrimary cementing
$V_{\text{pip}}$	volume of the head and the piping from the place where the top plug is released to the string, m³Primary cementing
$V_{\text{rat}},\ V_{\text{st}}$	volume of the rathole (the hole below the shoe) and of the shoe track, m³Primary cementing
$V_{\text{sl}}$	volume of cement slurry (one portion or all), m³Primary cementing
$V_{\text{sp1}},\ V_{\text{sp2}}$	volume of the spacer ahead of the cement (it goes into the annulus) and behind it (it stays in the pipe), m³Remedial cementing, cement plugs and cement evaluation
$v_{\text{steel}},\ t$	speed of the compressional wave in steel (about 5900 m/s) and the wall thickness, m/s, mRemedial cementing, cement plugs and cement evaluation
$V_t$	the volume of filtrate collected in the test by the time $t$ , mLOilwell cements and additives
$V_{\text{void}}$	the volume of the void at the top of the string, m³Primary cementing
$w$	effective weight of a metre of string: the contribution of one metre to what the weight indicator shows, N/mWellheads and running casing
$W$	hook load of the string (without the travelling block and the drive), NWellheads and running casing
$W_m$	weight of the string in mud, with mud inside and outside, NWellheads and running casing
$w_{\min}$	the smallest distance between the outer surface of the string and the wall of the hole (standoff), mmPrimary cementing
$x_i$	margin: zero for a cased interval, and for the open hole taken from the caliper log and the experience of the area, fractionPrimary cementing
$y$	drop of the level in the annulus; the level in the string rises to the same depth, mWellheads and running casing
$Y$	slurry yield, the volume of slurry per unit mass of cement, m³/tOilwell cements and additives
$Y_p$	minimum yield strength of the steel, MPaCasing: purpose, types, properties and standards minimum yield strength, PaCasing: purpose, types, properties and standards minimum yield strength of the steel (or the reduced $Y_{pa}$ ), psiCasing design yield strength, psiCasing design
$Y_p,\ D/t$	yield strength and the ratio of the outside diameter to the wall thickness, MPa, —Casing: purpose, types, properties and standards
$Y_{pa}$	reduced yield strength: the yield strength of a pipe that is also being pulled. It is lower than $Y_p$ , and we use it instead of $Y_p$ to calculate the collapse pressure of a pipe in tension, MPaCasing design
$z$	true vertical depth, mCasing design
$Z_1,\ Z_2$	acoustic impedances of steel and of the material on the other side of the surface, $10^6$ kg/(m²·s)Remedial cementing, cement plugs and cement evaluation
$\alpha$	attenuation of the casing wave over the stretch between the two receivers, dB/mRemedial cementing, cement plugs and cement evaluation
$\Delta\alpha$	attenuation that the cement added to the losses of the free pipe, found from one receiver, dB/mRemedial cementing, cement plugs and cement evaluation
$\Delta\alpha_{100}$	added attenuation in an interval of full bond of the same cement in the same casing, dB/mRemedial cementing, cement plugs and cement evaluation
$\rho_{\text{ann}},\ h_{\text{ann}}$	density and height of the columns of fluids in the annulus, g/cm³, mPrimary cementing
$\rho_{\text{g}}$	average density of the gas in the well, kg/m³Casing design
$\rho_i$	true (absolute) density of a component: cement about 3.15, water 1.00, silica flour about 2.65, g/cm³ (= kg/L)Oilwell cements and additives
$\rho_i,\ h_i$	densities of the fluids in the pipe and below the packer, down to the depth of the holes, and their vertical heights, g/cm³, mRemedial cementing, cement plugs and cement evaluation
$\rho_{\text{in }k},\ h_k$	density and height of the $k$ -th column inside the string, kg/m³, mWellheads and running casing
$\rho_{\text{loss}}$	equivalent mud weight of the pressure in the loss zone (the density of mud that would give the same pressure there), g/cm³Casing design
$\rho_m$	mud density, kg/m³Wellheads and running casing density of the drilling mud, kg/m³Wellheads and running casing density of the mud in the well, g/cm³Casing design
$\rho_m,\ \rho_{\text{steel}}$	density of the mud and of steel (7.85 g/cm³), g/cm³Casing design
$\rho_{\text{out}},\ \rho_{\text{in}}$	density of the fluid outside and inside (an empty string has $\rho_{\text{in}} = 0$ ), kg/m³Wellheads and running casing
$\rho_{\text{out }j},\ h_j$	density and height of the $j$ -th column of fluid behind the string, kg/m³, mWellheads and running casing
$\rho_{\text{sl}}$	density of the cement slurry, g/cm³Oilwell cements and additives
$\rho_{\text{steel}}$	density of steel, about 7850 kg/m³, kg/m³Casing: purpose, types, properties and standards density of steel, 7850 kg/m³, kg/m³Wellheads and running casing
$\rho_{\text{str}},\ h_{\text{str}}$	the same in the string, g/cm³, mPrimary cementing
$\sigma_a$	axial tensile stress: the axial force divided by the area of the pipe wall, MPaCasing design axial stress: the true axial force divided by the wall area, plus the bending stress, MPaCasing design
$\sigma_{\text{bend}}$	the greatest bending stress in the pipe wall in a curved hole: tension on the convex (outer) side of the bend, PaCasing design
$\sigma_{\text{eq}}$	equivalent stress; we compare it with the yield strength divided by the design factor (usually 1.25), MPaCasing design
$\sigma_t,\ \sigma_r$	hoop and radial stresses on the inside surface (the Lamé solution for a thick-walled pipe), MPaCasing design
$\tau$	how much earlier the wave arrives from the near side (and later from the far side), sRemedial cementing, cement plugs and cement evaluation