Equations: Well completions

The equations of the “Well completions” lessons in “Oil and gas production”, with their conditions and units, and what each symbol means in them.

Kill fluid density

\rho_{\text{kill}} = \frac{k_{\text{m}}\,\bar p_R}{g\,h}

Holds when: The well is filled with a uniform fluid from the bottom to the wellhead, and the wellhead is open. The reservoir pressure is the current one, from the latest measurements. Too large a margin is harmful: the higher the overbalance (the amount by which the pressure in the well exceeds the reservoir pressure), the more fluid will be lost to the reservoir.

Well completion: opening the reservoir and the tubing string →

Pressure force on the seal assembly in a packer

\Delta F_{\text{p}} = \left(A_{\text{p}} - A_{\text{i}}\right)\Delta p_{\text{i}} - \left(A_{\text{p}} - A_{\text{o}}\right)\Delta p_{\text{o}}

any consistent units: MPa and mm² give newtons, psi and square inches give pounds-force (lbf)

Holds when: The tubing is open to the reservoir: the pressure in the tubing at the packer equals the pressure below the packer. The areas are calculated by the diameter of the seals, not of the couplings. If the packer bore is narrower than the outside diameter of the tubing, $A_{\text{p}} - A_{\text{o}} < 0$ , and the pressure in the annulus also pushes the tubing upwards.

Completion string equipment →

Ballooning: the force that would hold the tubing against a change in length

F_{\text{b}} = 2\nu\left(\Delta\bar p_{\text{i}}\,A_{\text{i}} - \Delta\bar p_{\text{o}}\,A_{\text{o}}\right)

any consistent units (as in the previous formula)

Holds when: Elastic deformation (the tubing springs back when the load goes) of tubing of a single wall thickness. With fluids of the same density and without friction, the pressure change is the same along the whole length and equals the change at the wellhead.

Completion string equipment →

Temperature: the force on latched tubing and the lengthening of free tubing

F_{\text{t}} = E\,\alpha\,A_s\,\left(-\Delta\bar T\right), \qquad \Delta L_{\text{t}} = \alpha\,L\,\Delta\bar T

MPa, mm² and °C give the force in newtons; lengths in metres

Holds when: Elastic steel, tubing of a single size. Heating ( $\Delta\bar T > 0$ ) lengthens free tubing and compresses latched tubing.

Completion string equipment →

Free and latched seals

\Delta L = \frac{\left(\Delta F_{\text{p}} + F_{\text{b}} + F_{\text{t}}\right)L}{E\,A_s}

SI (forces in newtons, $E$ in pascals, $A_s$ in square metres); it is convenient to take $E$ in MPa and $A_s$ in mm² — their product is also in newtons

Holds when: Vertical tubing of a single size; seal friction is not taken into account. Helical buckling of free tubing is added separately (advanced section).

Completion string equipment →

Control line pressure and the maximum setting depth of a safety valve

p_{\text{c}} \ge p_{\text{i}}(H) + p_{\text{open}} - \rho_{\text{c}}\,g\,H, \qquad H_{\max} = \frac{p_{\text{close}}}{\rho_{\text{c}}\,g}

SI (pressures in pascals, $\rho_{\text{c}}$ in kg/m³, $H$ in metres)

Holds when: The valve has a piston. The tubing pressure acts on it from below, and the line pressure acts on it from above. Friction and areas are already included in the data-sheet opening and closing pressures. $H_{\max}$ has no safety margin. With the line bled off and zero pressure in the tubing, the column of fluid in the line must not exceed the closing pressure. The manufacturer and the design add a margin. For greater depths we use valves with a balance line or a gas chamber. The column of fluid in the line has almost no effect on them.

Completion string equipment →

Shortening of free tubing due to helical buckling

F_{\text{f}} = A_{\text{p}}\,(p_{\text{i}} - p_{\text{o}}), \qquad n = \frac{F_{\text{f}}}{w}, \qquad \Delta L_{\text{buck}} = \frac{r^2 F_{\text{f}}^2}{8\,E\,I\,w}

Holds when: The well is vertical, the tubing is of a single size, the seals are free, and there is no friction against the casing. The formula holds while $n \le L$ . If the whole string is buckled, we need a different formula. With latched tubing, the tension that the packer transmits reduces the fictitious force, and there may be no buckling.

Completion string equipment →

Zone rates at a common bottomhole pressure

q_i = J_i\,(\bar p_{R,i} - p_{wf}),\qquad p_{wf} = \frac{\sum_i J_i\,\bar p_{R,i} - q}{\sum_i J_i},\qquad q = \sum_i q_i

metric oilfield units (any consistent system will do)

Holds when: The inflow of each zone is linear: single-phase (only liquid flows) and above the bubble point pressure (Inflow performance relationship (IPR)). For two-phase inflow we use the IPR curves of the zones (curves of rate against bottomhole pressure) instead of straight lines, and we find the common pressure by a numerical calculation. The pressures are referred to one depth: $\bar p_{R,i} = p_i - \rho g\,(z_i - z_0)$ , where $p_i$ is the pressure at the depth $z_i$ of the zone, $z_0$ is the datum depth, and $\rho$ is the density of the fluid in the casing between them. Pressure losses in the wellbore between the zones are small. The productivity indices do not change during the period considered.

Multiple-zone completions: commingled and segregated production →

Pressure in a shut-in well and crossflow through the wellbore

p_{ws} = \frac{\sum_i J_i\,\bar p_{R,i}}{\sum_i J_i},\qquad q_x = \sum_{\bar p_{R,i} < p_{ws}} J_i\,(p_{ws} - \bar p_{R,i})

metric oilfield units

Holds when: The well is closed at the wellhead, but the zones are connected through the wellbore. The injectivity (how easily a zone takes in fluid) of the zone that takes the crossflow is taken as equal to its productivity. If water or gas, with a different viscosity, goes into it, the injectivity will be different too. The estimate uses the productivity indices at constant average zone pressures. In the first hours after shut-in, while the pressure near the wellbore builds up, the crossflow is different. During a long shut-in the zone pressures converge and the crossflow dies out.

Multiple-zone completions: commingled and segregated production →

Share of a zone in a mixture from a tracer property

f_1 = \frac{C_{\text{mix}} - C_2}{C_1 - C_2},\qquad \delta f \approx \frac{\delta C}{\lvert C_1 - C_2\rvert}

any, the same for all three values of the property

Holds when: There are two zones, and the property mixes linearly. The oil samples of the zones are representative, and they have not changed since they were taken. The error of the share includes only the error of the mixture sample. The errors of the zone samples add to it.

Multiple-zone completions: commingled and segregated production →

The force with which the pressure pushes the line out of the well

F_{\text{ej}} = p_{wh}\,A_{\text{line}}, \qquad A_{\text{line}} = \frac{\pi d^2}{4}

SI; it is convenient to take the pressure in MPa and the area in mm², and their product is in newtons

Holds when: The pressure above the seal is atmospheric. Friction in the stuffing box or the grease head is not included. A weight margin covers it. The force does not depend on the depth of the tool string. It acts all the time that the line passes through the seal, but it matters most at the start of the run, when there is as yet no weight of line in the well.

Wireline operations →

Weight indicator reading with the string hanging freely

T = \left(W_{\text{tool}} + w\,L\right)\left(1 - \frac{\rho_{\text{L}}}{\rho_{\text{steel}}}\right) - p_{wh}\,A_{\text{line}}

SI; N, N/m, m, MPa and mm² are convenient

Holds when: The wellbore is vertical, and the tool string is stationary or moving uniformly. Friction is not included. On the way up it increases the reading, on the way down it decreases it, and in a deviated wellbore it can exceed the weight of the tool string. The liquid density is the same along the whole wellbore. In gas the buoyancy factor is close to one. For cable the factor is approximate, because a cable is lighter than steel of the same volume.

Wireline operations →

Margin of overpull at the tool

F_{\text{pull}} = k_{\text{all}}\,F_{\text{br}} - T

SI (any consistent units of force)

Holds when: The load is static. Jar blows create short-lived forces greater than the static ones. The tension is greatest at the wellhead, where the line carries both itself and the tool string. So overpulled wire usually breaks at the wellhead, and almost the whole line falls into the tubing.

Wireline operations →

Elongation of the line under a change of tension and heating

\Delta L = \frac{\Delta F\,L}{E\,A_{\text{line}}} + \alpha\,L\,\Delta\bar T

Holds when: The wire is solid and the wellbore is vertical. For braided line and cable the elongation under load is greater, and we take it from the data sheet. The stretch coefficients of slickline in the data sheets are also somewhat greater than a calculation from the modulus of steel gives. $\Delta F$ is the same along the whole length $L$ . This happens when the tool or the line is held and is pulled from the wellhead. Friction along the wellbore reduces it with depth. The effect of pressure on the elongation is not included.

Wireline operations →

Productivity index of a horizontal well (the standard model corrected for anisotropy)

\begin{gathered} J = \frac{q}{p_e - p_w} = \frac{2\pi k_h h}{\mu B\,R}\\ R = \ln\frac{a + \sqrt{a^2 - (L/2)^2}}{L/2} + \frac{I h}{L}\left[\ln\frac{I h}{r_w(I + 1)} + S\right]\\ a = \frac{L}{2}\sqrt{0.5 + \sqrt{0.25 + \left(\frac{2r_{eh}}{L}\right)^4}},\qquad I = \sqrt{\frac{k_h}{k_v}}\end{gathered}

SI; $\ln$ is the natural logarithm

Holds when: The inflow is steady-state (it does not change with time), at constant pressure at the boundary of a circular area, which is replaced by an ellipse of the same area. The formation is homogeneous, with $k_h$ the same in all horizontal directions. The wellbore is in the middle of the thickness, parallel to the top. The pressure along the wellbore is the same (there is no friction), and the flow is single-phase. $I h < L$ and $L/2 < 0.9\,r_{eh}$ . If these conditions are not met, we need numerical models.

Horizontal wells →

Equivalent skin and effective wellbore radius of a horizontal well

S_{eq} = R - \ln\frac{r_{eh}}{r_w},\qquad r_w' = r_w\,e^{-S_{eq}} = r_{eh}\,e^{-R}

dimensionless; radii in the same units

Holds when: The same drainage area and the same boundary conditions as in the productivity model.

Horizontal wells →

Inflow and friction along a horizontal wellbore

\frac{\dd q}{\dd x} = \frac{J}{L}\,\bigl(p_e - p(x)\bigr),\qquad -\frac{\dd p}{\dd x} = \lambda\,\frac{\rho\,v^2}{2d},\quad v = \frac{qB}{\pi d^2/4}

Holds when: The oil is single-phase, and the specific productivity is the same along the whole length. This underestimates the inflow to the ends of the wellbore. Friction is the same as in a pipe with no inflow through the wall (inflow through the wall changes it slightly). We neglect the acceleration of the flow. The system is solved numerically. We specify the drawdown at the heel and choose the drawdown at the toe so that the pressures agree.

Horizontal wells →

Inflow of a section through an inflow control device

p_e - p(x) = \frac{q'}{J/L} + c\,q'^2,\qquad \Delta p_{\text{ICD}} = c\,q'^2 \approx \frac{\rho\,v_n^2}{2C_d^2}

Holds when: Packers divide the annulus into sections, and all the inflow of a section passes through its device. The flow is single-phase. The pressure drop of nozzle-type devices is almost independent of viscosity, while that of channel-type devices depends on it appreciably. We choose the constant so that the pressure drop across the devices at the design rate is comparable with the drawdown on the formation, or greater.

Horizontal wells →

Critical water-free rate of a crest at vertical equilibrium

q_c = \frac{k_h\,\Delta\rho\,g\,\bigl[h^2 - (h - z_w)^2\bigr]\,L}{\mu B\,y_e}

SI (any consistent system of units)

Holds when: There is vertical equilibrium: oil flows to the wellbore horizontally from both sides through the layer above the crest. This is a plane problem, and the ends of the wellbore are not taken into account. The water is stationary, and the contact stays in place at the distance $y_e$ . The formation is isotropic. Right at the wellbore the assumption is crude, but it hardly affects the critical rate. Exact solutions of this plane free-boundary problem (a flow problem with a moving surface) for $y_e \gg h$ practically coincide with the formula. An approximation built from the pressure field without allowing for the crest (see below) gives several times more.

Horizontal wells →

A wellbore passing right through a rectangular block (pseudo-steady state)

J = \frac{q}{\bar p - p_w} = \frac{2\pi k_h h}{\mu B\,R_b},\qquad R_b = \frac{\pi y_e}{6L} + \frac{I h}{L}\left[\ln\frac{I h}{\pi r_w(I + 1)\sin(\pi z_w/h)} + S\right]

SI; $\ln$ is the natural logarithm

Holds when: The block is closed, and the regime is pseudo-steady state. The wellbore passes right through the block, so the flow is plane. The wellbore is at least $I h$ from the lateral boundaries of the block. Then the approximation of the shape factor is accurate to hundredths. If the wellbore is shorter than the block, we add the pseudo-skin of partial penetration of the block. Programs calculate such rectangular-block models.

Horizontal wells →

Laterals in one formation: steady-state inflow (summation of segments)

\frac{2\pi k_h h\,(p_e - p_w)}{\mu B} = \sum_j q_j\,G_{ij} + q_i\,\frac{I h}{\Delta}\left[\ln\frac{I h}{r_w(I + 1)} + S\right],\qquad G_{ij} = \frac{1}{\Delta}\int_{\Delta_j}\ln\frac{r_{eh}}{\rho_{ij}(s)}\,\dd s

SI; $\ln$ is the natural logarithm

Holds when: We write the equation for the middle of each segment. All segments are at the same pressure $p_w$ , so there is no friction in the wellbores. We take the constant-pressure boundary $p_e$ at a distance $r_{eh}$ into account only roughly, as in the standard model. For laterals diverging from the centre of the area at equal angles, the exact solution for a circle (method of images) gives the same gains to within 0.5%, although the laterals reach half the radius. The vertical convergence to the wellbore does not overlap between different laterals. The laterals are farther from one another than $I h$ , except near the junction, where the model slightly overestimates the gain. That is why the gain curve starts at 15°.

Multilateral wells →

Symbols

Symbol	Meaning and context
$\Delta L$	shortening of free tubing (negative means lengthening); this is how far the seals will move in the packer bore, mCompletion string equipment elongation of the line in the well beyond what the counter showed, mWireline operations
$\Delta F$	change of tension along the whole length $L$ after the line has passed the counter — for example, an overpull beyond the weight, NWireline operations
$\bar p$	average pressure in the block, PaHorizontal wells
$\delta C,\ \delta f$	error in determining the property of the mixture and the resulting error of the share, units of the property, —Multiple-zone completions: commingled and segregated production
$a$	semi-major axis of the drainage ellipse with its foci at the ends of the wellbore and the area of a circle of radius $r_{eh}$ , mHorizontal wells
$A_{\text{i}},\ A_{\text{o}}$	areas by the inside and outside diameters of the tubing, mm²Completion string equipment
$A_{\text{line}}$	cross-sectional area of the wire, m²Wireline operations
$A_{\text{line}},\ d$	cross-sectional area and diameter of the line; for braided line and cable, the outer diameter, m², mWireline operations
$A_{\text{p}}$	area of the polished bore of the packer (the smooth bore in which the seals sit), taken by the diameter of the seals, mm²Completion string equipment
$A_s$	cross-sectional area of the tubing wall, mm²Completion string equipment
$\Delta L_{\text{buck}}$	shortening of the tubing due to buckling, mCompletion string equipment
$c$	constant of the devices, reduced to a metre of wellbore, Pa·s²/m⁴Horizontal wells
$C_{\text{mix}},\ C_1,\ C_2$	values of the property in the mixture and in the oil of zones 1 and 2, units of the propertyMultiple-zone completions: commingled and segregated production
$E$	Young's modulus of steel (how stiff it is), 207,000, MPaCompletion string equipment Young's modulus of steel (how stiff it is), 207,000 MPa = 2.07·10¹¹ Pa, PaWireline operations
$E\,A_s$	axial stiffness of the tubing: Young's modulus times wall area, NCompletion string equipment
$f_1$	share of zone 1 in the mixture — by mass if the property is given per unit of mass (like sulfur content), or by volume, —Multiple-zone completions: commingled and segregated production
$F_{\text{b}}$	ballooning force: positive when the tubing tends to shorten, NCompletion string equipment
$F_{\text{br}}$	breaking load of the line from the data sheet, NWireline operations
$F_{\text{ej}}$	ejection force, NWireline operations
$F_{\text{f}}$	fictitious force at the packer; buckling occurs if it is positive, NCompletion string equipment
$F_{\text{pull}}$	the largest static force that can be applied to a stuck tool in addition to its weight, NWireline operations
$F_{\text{t}}$	thermal force: positive (tension in latched tubing) on cooling, NCompletion string equipment
$g$	acceleration due to gravity, m/s²Well completion: opening the reservoir and the tubing string
$G_{ij}$	contribution of segment $j$ to the pressure drop at the middle of segment $i$ , —Multilateral wells
$h$	vertical depth of the reservoir (measured straight down), mWell completion: opening the reservoir and the tubing string
$H$	vertical setting depth of the valve, mCompletion string equipment
$h,\ r_w$	thickness of the formation and radius of the wellbore, mHorizontal wells
$I$	second moment of area of the tubing cross-section (how well it resists bending), $\pi(D^4 - d^4)/64$ , m⁴Completion string equipment anisotropy index, —Horizontal wells
$\Delta p_{\text{i}},\ \Delta p_{\text{o}}$	changes in pressure at the packer in the tubing (which is also the pressure below the packer) and in the annulus, MPaCompletion string equipment
$I,\ h,\ r_w,\ S$	anisotropy index, thickness, wellbore radius, skin — as in the standard model, —, m, m, —Multilateral wells
$J$	productivity index in terms of the rate at standard conditions, m³/(s·Pa)Horizontal wells
$J_i$	productivity index of the zone, referred to the datum depth, m³/(d·MPa)Multiple-zone completions: commingled and segregated production
$J_i,\ \bar p_{R,i}$	productivity index and reservoir pressure of the zones, as above, m³/(d·MPa), MPaMultiple-zone completions: commingled and segregated production
$J/L$	specific productivity per metre of wellbore (from the standard model), m³/(s·Pa·m)Horizontal wells
$k_{\text{all}}$	allowable fraction of the breaking load; it is set by company rules, taking into account bending on the sheaves, wear and blows, —Wireline operations
$k_h,\ h,\ I,\ L,\ r_w,\ S,\ \mu,\ B$	as in the standard model, m², m, —, m, m, —, Pa·s, m³/m³Horizontal wells
$k_h,\ \Delta\rho,\ g,\ h,\ L,\ \mu,\ B$	horizontal permeability, density difference between water and oil, acceleration due to gravity, oil thickness, wellbore length, viscosity and formation volume factor, m², kg/m³, m/s², m, m, Pa·s, m³/m³Horizontal wells
$k_h,\ \mu,\ B$	horizontal permeability, viscosity and formation volume factor of the oil, m², Pa·s, m³/m³Multilateral wells
$k_h,\ k_v$	horizontal and vertical permeability, m²Horizontal wells
$k_{\text{m}}$	margin factor over the reservoir pressure, for example 1.05; it is set by the rules and the work plan, —Well completion: opening the reservoir and the tubing string
$L$	length of the tubing from the hanger to the packer, mCompletion string equipment length of line on which $\Delta F$ acts: to the tool or to the point where the line is held, mWireline operations length of the horizontal section that opens the formation, mHorizontal wells
$L,\ \Delta L_{\text{t}}$	length of the tubing to the packer and its change for free tubing, mCompletion string equipment
$n$	length of the buckled section above the packer (up to the neutral point, where the buckling force falls to zero), mCompletion string equipment
$\Delta F_{\text{p}}$	change in the force with which pressure pushes the tubing upwards: it shortens free tubing or pulls a latched assembly upwards, NCompletion string equipment
$p_{\text{c}}$	pressure in the control line at the surface (gauge), PaCompletion string equipment
$p_{\text{close}}$	closing pressure from the data sheet: the excess of the line pressure over the tubing pressure at which the spring begins to close the valve, PaCompletion string equipment
$p_e,\ p(x),\ J/L$	as above, Pa, Pa, m³/(s·Pa·m)Horizontal wells
$p_e,\ p_w$	pressure at the boundary of the area and in the wellbores, PaMultilateral wells
$p_{\text{i}},\ p_{\text{o}}$	pressures in the tubing and in the annulus at the packer, PaCompletion string equipment
$p_{\text{i}}(H)$	pressure in the tubing at the depth of the valve; the highest is in the shut-in well, PaCompletion string equipment
$p_{\text{open}}$	full-opening pressure from the data sheet: by how much the pressure in the line at the valve must exceed the pressure in the tubing, PaCompletion string equipment
$p_{wf}$	bottomhole pressure at the datum depth, MPaMultiple-zone completions: commingled and segregated production
$p_{wh}$	wellhead pressure, that is, the pressure in the lubricator (gauge), PaWireline operations
$p_{wh}A_{\text{line}}$	ejection force, NWireline operations
$p_{ws}$	pressure in the wellbore of the shut-in well at the datum depth, MPaMultiple-zone completions: commingled and segregated production
$p(x),\ p_e$	pressure in the wellbore and at the boundary, PaHorizontal wells
$q$	total rate of the well, m³/dMultiple-zone completions: commingled and segregated production
$q_c$	critical water-free rate at standard conditions, m³/sHorizontal wells
$q_i$	rate of zone $i$ at standard conditions (fixed reference pressure and temperature); negative means the zone is taking fluid, m³/dMultiple-zone completions: commingled and segregated production
$q_j$	rate of the $j$ -th segment of the lateral at standard conditions, m³/sMultilateral wells
$q_x$	crossflow through the wellbore: the rate taken by the zones with a pressure below $p_{ws}$ ; the others give the same amount, m³/dMultiple-zone completions: commingled and segregated production
$q'$	inflow per metre of wellbore, m³/(s·m)Horizontal wells
$q(x)$	rate accumulated in the wellbore up to the point $x$ (standard conditions), m³/sHorizontal wells
$r$	radial clearance between the tubing and the casing, mCompletion string equipment
$\bar p_R$	current reservoir pressure at depth $h$ (gauge pressure, that is, measured above atmospheric pressure), PaWell completion: opening the reservoir and the tubing string
$R$	dimensionless resistance to inflow (the analogue of $\ln(r_e/r_w) + S$ of a vertical well), —Horizontal wells
$R_b$	resistance: linear inflow to the wellbore plus convergence in the vertical direction, —Horizontal wells
$r_{eh}$	radius of the circular drainage area of the horizontal well, mHorizontal wells radius of the drainage area of the well, mMultilateral wells
$r_w'$	radius of a vertical well with no skin that is equivalent to the horizontal one, mHorizontal wells
$R,\ r_{eh},\ r_w$	resistance to inflow, radius of the area and radius of the wellbore, —, m, mHorizontal wells
$\bar p_{R,i}$	average reservoir pressure of zone $i$ , referred to the datum depth, MPaMultiple-zone completions: commingled and segregated production
$S$	damage skin around the horizontal wellbore, —Horizontal wells
$S_{eq}$	skin of a vertical well of radius $r_w$ that is equivalent to the horizontal one, —Horizontal wells
$T$	tension in the line above the seal — the weight indicator reading, NWireline operations
$v_n,\ C_d$	velocity in the nozzle and discharge coefficient (for nozzle-type devices), m/s, —Horizontal wells
$w$	weight per metre of tubing in the fluid: steel plus the fluid inside minus the fluid displaced outside, N/mCompletion string equipment
$W_{\text{tool}}$	weight of the tool string in air, NWireline operations
$w,\ L$	weight of a metre of line in air and the length of line in the well, N/m, mWireline operations
$x$	distance along the wellbore from the toe to the heel, mHorizontal wells
$y_e$	distance from the wellbore to the line where the OWC stays in place (half the distance between neighbouring wellbores), mHorizontal wells full width of the block across the wellbore (in the crest formula $y_e$ is half the distance between wellbores); the wellbore is in the middle of the width, mHorizontal wells
$z_w$	height of the wellbore above the oil–water contact (OWC); for a wellbore at the top, $z_w = h$ , mHorizontal wells height of the wellbore above the base, mHorizontal wells
$\alpha$	coefficient of linear thermal expansion of steel (how much each unit of length grows per degree), $1.24\cdot10^{-5}$ , 1/°CCompletion string equipment coefficient of linear expansion of steel, $1.24\cdot 10^{-5}$ , 1/°CWireline operations
$\Delta\bar T$	change in the length-averaged temperature of the tubing, °CCompletion string equipment heating of the line, averaged along its length, relative to the temperature at the counter, °CWireline operations
$\Delta$	length of a segment, mMultilateral wells
$\Delta\bar p_{\text{i}},\ \Delta\bar p_{\text{o}}$	changes in pressure in the tubing and in the annulus, averaged over the length of the tubing, MPaCompletion string equipment
$\lambda$	Darcy friction factor (Haaland formula, Multiphase flow in tubing), —Horizontal wells
$\mu,\ B,\ p_e,\ p_w$	viscosity and formation volume factor of the oil, pressure at the boundary and in the wellbore, Pa·s, m³/m³, Pa, PaHorizontal wells
$\nu$	Poisson's ratio of steel, 0.3, —Completion string equipment
$\rho,\ v,\ d$	density of the oil in the wellbore, mean velocity, internal diameter of the liner, kg/m³, m/s, mHorizontal wells
$\rho_{\text{c}}$	density of the fluid in the control line, kg/m³Completion string equipment
$\rho_{ij}(s)$	distance from the middle of segment $i$ to the point $s$ of segment $j$ , mMultilateral wells
$\rho_{\text{kill}}$	density of the kill fluid, kg/m³Well completion: opening the reservoir and the tubing string
$\rho_{\text{L}},\ \rho_{\text{steel}}$	density of the liquid in the well and of steel (7850 kg/m³); their ratio gives the buoyancy (the lifting effect of the liquid), kg/m³Wireline operations