Equations: Artificial lift

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

Depth of the static fluid level

H_{\text{st}} = H - \frac{\bar p_R - p_{\text{ann}}}{\rho g}

SI; pressures are gauge (above atmospheric)

Holds when: The well is shut in and the bottomhole pressure has equalised with the reservoir pressure. Below the level there is liquid of constant density with no gas. The weight of the gas column above the level is small. Depths are true vertical depths. In a deviated well, a level measured along the hole is converted using the directional survey (the record of the well's path).

Artificial lift: when it is needed and how to choose the method →

Pressure rise produced by the pump

\begin{aligned} \Delta p_{\text{pump}} &= p_{\text{out}} - p_{\text{in}},\\ p_{\text{out}} &= p_{wh} + \rho g H_{\text{pump}} + \Delta p_{\text{fr}},\\ p_{\text{in}} &= p_{wf} - \rho g\,(H - H_{\text{pump}}) \end{aligned}

SI (any consistent system of units)

Holds when: The liquid contains no free gas and has the same density in the tubing and below the pump. If there is gas in the tubing, the column is lighter, and its weight is calculated with multiphase flow correlations. Depths are true vertical depths. Friction is calculated along the length of the tubing. The formula also holds for a pump set below the perforated interval. Then $H - H_{\text{pump}} < 0$ , and the intake pressure is higher than the bottomhole pressure.

Artificial lift: when it is needed and how to choose the method →

Hydraulic power and power consumption

N = q\,\Delta p_{\text{pump}},\qquad N_{\text{el}} = \frac{N}{\eta}

Holds when: The flow is steady. The flow rate is the volume of liquid at intake conditions. Free gas, if it passes through the pump, is compressed, and for it the formula gives only an upper bound (a value that is too high). The system efficiency includes the losses in the motor, the transmission (rods, cable), the pump and the tubing.

Artificial lift: when it is needed and how to choose the method →

Free gas fraction at the pump intake

\varphi_{\text{g}} = \frac{q_o\,(R - R_s)\,B_g}{q_o\,(R - R_s)\,B_g + q_o B_o + q_w B_w}

any consistent units; pressures in $B_g$ are absolute, temperatures are in kelvins

Holds when: All the gas that is not dissolved in the oil at the intake pressure goes to the intake together with the liquid. If part of it is vented into the annulus, put only the remaining part into the formula. We ignore the solubility of gas in water.

Artificial lift: when it is needed and how to choose the method →

Bottomhole pressure from the dynamic fluid level

p_{wf} \approx p_{\text{ann}} + \rho_{\text{liq}}\,g\,(H - H_{\text{dyn}})

Holds when: The weight of the gas column above the level is small. At a high annulus pressure it is already a few per cent, and then it is added. The liquid in the annulus contains no gas and has a known density.

Artificial lift: when it is needed and how to choose the method →

The limiting number of strokes set by the free fall of the rods

n_{\text{lim}} = \frac{30}{\pi}\sqrt{\frac{2g}{S}} \approx \frac{42.3}{\sqrt{S}}

$S$ in metres, $g$ in m/s², $n$ in strokes per minute

Holds when: the polished rod moves in a harmonic (sine-wave) motion: the greatest acceleration is $\omega^2 S/2$ , where $\omega = \pi n/30$ is the angular velocity of the crank in radians per second the rods fall with the acceleration $g$ , with no liquid and no friction. In liquid they fall more slowly, because buoyancy reduces their weight and friction resists the motion. So the formula gives an upper limit, not a working value.

Sucker-rod pumping →

Theoretical displacement of a sucker rod pump

q_{\text{th}} = 1440\,A_{\text{pl}}\,S\,n, \qquad A_{\text{pl}} = \frac{\pi D^2}{4}

metres, square metres, strokes per minute; the result is in m³/d

Holds when: the plunger stroke is equal to the stroke of the polished rod, the barrel fills completely with liquid, and there are no leaks; the volumetric efficiency allows for the departures from this in the practice of the CIS countries, the stroke of the polished rod is put into the formula. In the API method the volume is calculated from the plunger stroke, and the loss of stroke is not part of the volumetric efficiency. When comparing coefficients from different sources, this must be checked.

Sucker-rod pumping →

Pump volumetric efficiency and its components

\alpha_{\text{p}} = \frac{q}{q_{\text{th}}} = \alpha_{\text{s}}\,\alpha_{\text{f}}\,\alpha_{\text{leak}}\,\alpha_{\text{sh}}

dimensionless fractions

Holds when: the factors are treated as independent; this is a convenient approximation for finding the main cause of the losses, not an exact law the delivery was measured in steady operation, not just after a start-up

Sucker-rod pumping →

Static loads on the polished rod

\begin{aligned} P_{\uparrow} &= P'_{\text{rod}} + P_{\text{liq}}, \qquad P_{\downarrow} = P'_{\text{rod}},\\ P'_{\text{rod}} &= m_{\text{rod}}\,g\,L\,k_{\text{b}}, \qquad P_{\text{liq}} = (p_{\text{out}} - p_{\text{in}})\,A_{\text{pl}} \end{aligned}

SI; both pressures gauge or both absolute, because only their difference enters the formula

Holds when: very slow motion: no inertia, friction or vibration the well is vertical; in a deviated well the weight of the rods is calculated from the vertical setting depth, not from the length of the string, and the friction of the rods against the tubing there increases the load on the upstroke and reduces it on the downstroke the rod string has one diameter; for a tapered string the weights of the tapers are added the fluid load is calculated on the full plunger area precisely because the weight of the rods is taken in liquid. If the weight of the rods is taken in air, the discharge pressure is multiplied by the plunger area less the cross-section of the rods, and the intake pressure still by the full area. The result is practically the same (the derivation is in the advanced section) the force with which the wellhead pressure pushes the rod out of the stuffing box is not included; it is usually a fraction of a kilonewton

Sucker-rod pumping →

Acceleration factor and loads with inertia

\begin{aligned} K_{\text{a}} &= \frac{\omega^2 S}{2g} = \frac{S\,n^2}{1790},\\ P_{\max} &\approx P'_{\text{rod}} + P_{\text{liq}} + K_{\text{a}}\,P_{\text{rod}}, \qquad P_{\min} \approx P'_{\text{rod}} - K_{\text{a}}\,P_{\text{rod}} \end{aligned}

$S$ in metres, $n$ in strokes per minute; the loads in any identical units

Holds when: harmonic motion of the polished rod; in an ordinary pumping unit the acceleration at the bottom is larger than the harmonic one, and at the top it is smaller the rods are rigid and move as one piece; elastic vibrations, the inertia of the liquid and friction are not included the estimate is good for small depths and small numbers of strokes. For design, engineers use methods based on solving the wave equation for the rod string (for example API RP 11L, now designated TR 11L)

Sucker-rod pumping →

Stretch of the rods and tubing, and the plunger stroke

\lambda_{\text{rod}} = \frac{P_{\text{liq}}\,L}{E\,A_{\text{rod}}}, \qquad \lambda_{\text{tbg}} = \frac{P_{\text{liq}}\,L}{E\,A_{\text{tbg}}}, \qquad S_{\text{pl}} = S - \lambda_{\text{rod}} - \lambda_{\text{tbg}}

Holds when: the rod string has one diameter; for a tapered string the stretches of the tapers are added: $\lambda_{\text{rod}} = (P_{\text{liq}}/E)\sum L_i/A_i$ the tubing is not anchored; if the bottom of the tubing is fixed with an anchor, $\lambda_{\text{tbg}} = 0$ statics: at large depths and numbers of strokes the inertia of the rods adds to the plunger stroke (advanced section) the rods are steel; for fibreglass rods the elastic modulus is several times smaller, and the stretch is correspondingly larger

Sucker-rod pumping →

The depth of the level from the echometer record

H_{\text{dyn}} = \frac{v\,t}{2}, \qquad v = \frac{2\,N_{\text{c}}\,l_{\text{tbg}}}{t_{\text{c}}}

metres, seconds, metres per second

Holds when: the speed of sound is the same over the whole height of the gas column. In fact it depends on the composition, the pressure and the temperature of the gas, and it is found from the upper part of the wellbore if the echoes from the collars cannot be read, the speed is taken from the composition and pressure of the gas, or from the echo of a reference marker (a reflector fixed on the tubing at a known depth) the depth is measured along the wellbore; in a deviated well it is converted to vertical depth with the directional survey

Sucker-rod pumping →

The clearance volume: the delay of the intake and the condition for gas lock

x_{\text{open}} = l_{\text{cl}}\left(\frac{p_{\text{out}}}{p_{\text{in}}} - 1\right), \qquad \text{gas lock when}\quad 1 + \frac{S_{\text{pl}}}{l_{\text{cl}}} \le \frac{p_{\text{out}}}{p_{\text{in}}}

any consistent units; the pressures are absolute

Holds when: the clearance volume is completely filled with gas. This is the worst case: liquid that stays in it reduces the volume of gas and the delay an ideal gas, with isothermal (constant-temperature) compression and expansion. When the gas is compressed fast, it heats up and the pressure grows faster, so the estimate is on the safe side the valves are tight, and the pressure losses in them are not included

Sucker-rod pumping →

The force balance of the rod string with the plunger

P = P_{\text{rod}} + p_{\text{out}}\,(A_{\text{pl}} - A_{\text{rod}}) - p_{\text{bar}}\,A_{\text{pl}}

SI; gauge pressures

Holds when: statics; rods of one cross-section; the pressure of the liquid on the side surface of the rods gives no vertical force the pressure losses in the valves are not included

Sucker-rod pumping →

The counterbalance effect and the power at the polished rod

C = P'_{\text{rod}} + \frac{P_{\text{liq}}}{2}, \qquad N_{\text{rod}} = \frac{P_{\text{liq}}\,S_{\text{pl}}\,n}{60}

SI; $n$ in strokes per minute

Holds when: an ideal static dynamometer card, with no inertia, friction or gas in the barrel. The real power is found from the area of the recorded dynamometer card in practice the balance is checked by comparing the largest motor current on the upstroke with that on the downstroke

Sucker-rod pumping →

An estimate of the added plunger stroke from inertia

\lambda_{\text{ov}} \approx \frac{\rho_{\text{steel}}\,g\,L^2}{E}\,K_{\text{a}}, \qquad S_{\text{pl}} = S - \lambda_{\text{rod}} - \lambda_{\text{tbg}} + \lambda_{\text{ov}}

Holds when: a steel string of constant cross-section; harmonic motion of the polished rod the acceleration is the same along the whole length of the string, so there are no wave effects. This is an estimate of the order of magnitude, not a design formula a bar of length $L$ under its own weight stretches by $\rho_{\text{steel}} g L^2/(2E)$ . Inertia adds the fraction $K_{\text{a}}$ of this stretch at the bottom of the stroke and takes away the same amount at the top, so the total change is doubled

Sucker-rod pumping →

The pressure rise, the power and the efficiency of a stage

\Delta p_{\text{stg}} = \rho g h, \qquad N_{\text{stg}} = \rho g\,q_{\text{in}}\,h, \qquad \eta = \frac{N_{\text{stg}}}{N_{\text{shaft,stg}}}

Holds when: a low-viscosity liquid (its viscosity is close to that of water) with no free gas. The manufacturer measures the curves on water, and viscosity and gas reduce the head and the efficiency the head and the efficiency are taken from the performance curve at the rate and the speed at which the stage works

Electric submersible pumps (ESP) →

The required head of the pump

H_{\text{req}} = \frac{p_{\text{out}} - p_{\text{in}}}{\rho g} = H_{\text{dyn}} + \frac{p_{wh} - p_{\text{ann}} + \Delta p_{\text{fr}}}{\rho g}

SI; the pressures are all gauge or all absolute

Holds when: the liquid has no free gas and the same density in the tubing and in the annulus below the level. Gas in the tubing makes the column lighter. Then the discharge pressure is found from multiphase flow, and the required head is smaller the depths are vertical, and the friction is found from the length of the tubing: $\Delta p_{\text{fr}} = \lambda\,\dfrac{L}{d}\,\dfrac{\rho v^2}{2}$ , as in the lesson on nodal analysis the weight of the gas column in the annulus above the level is small

Electric submersible pumps (ESP) →

The number of stages and the shaft power of the pump

z_{\text{stg}} = \frac{H_{\text{req}}}{h(q_{\text{in}})}, \qquad N_{\text{shaft}} = \frac{\rho g\,q_{\text{in}}\,H_{\text{pump}}}{\eta(q_{\text{in}})}

Holds when: all stages are identical, the liquid has a low viscosity, and the share of gas is small. We allow for viscosity and gas with corrections or with extra stages to the power of the pump we add what the gas separator and the protector use, usually a few per cent

Electric submersible pumps (ESP) →

The affinity laws of a centrifugal pump when the speed changes

\frac{q_2}{q_1} = \frac{n_2}{n_1}, \qquad \frac{h_2}{h_1} = \left(\frac{n_2}{n_1}\right)^{2}, \qquad \frac{N_2}{N_1} = \left(\frac{n_2}{n_1}\right)^{3}, \qquad \eta_2 = \eta_1

the ratios are dimensionless; the quantities with the subscripts 1 and 2 are in the same units

Holds when: one and the same stage, a low-viscosity liquid with no gas. When the frequency is lowered a lot (below about 30–35 Hz, when the rated frequency is 50 Hz), the efficiency falls and the laws hold less well the laws link the corresponding points of two curves, not the operating points in the well. We find the operating point at a new frequency as the crossing of the new pump curve with the well curve

Electric submersible pumps (ESP) →

The share of free gas at the pump inlet, allowing for separation

\varphi_{\text{g}} = \frac{(1 - E_{\text{sep}})\,q_o\,(R - R_s)\,B_g}{(1 - E_{\text{sep}})\,q_o\,(R - R_s)\,B_g + q_o B_o + q_w B_w}

any consistent units; the pressures in $B_g$ are absolute, and the temperatures are in kelvins

Holds when: the gas that is not dissolved in the oil at the intake pressure comes to the intake together with the liquid; the solubility of gas in water is not included the share $E_{\text{sep}}$ is an estimate: it depends on the rate, the size of the bubbles, the viscosity and the design of the separator. At $E_{\text{sep}} = 0$ the formula is the same as the formula of the lesson on choosing a method

Electric submersible pumps (ESP) →

The velocity of the liquid along the submersible motor

v_{\text{cool}} = \frac{4\,q_{\text{in}}}{\pi\left(D_{\text{cas}}^2 - D_{\text{mot}}^2\right)}

Holds when: all the liquid that goes to the intake passes the motor: the motor is below the intake, and the inflow comes from below free gas cools badly, so the velocity is estimated for the liquid

Electric submersible pumps (ESP) →

The motor current and the losses in the cable

I = \frac{N_{\text{shaft}}}{\sqrt{3}\,U\,\eta_{\text{mot}}\cos\varphi}, \qquad \Delta N_{\text{cab}} = 3 I^2 R_{\text{cond}}, \qquad R_{\text{cond}} = \rho_{\text{el}}\,\frac{L_{\text{cab}}}{S}\,\bigl[1 + \alpha\,(t - 20)\bigr]

SI; the conductor cross-section in mm², the resistivity in Ω·mm²/m

Holds when: a balanced three-phase load; the motor works near its rated mode, and its efficiency and power factor are the nameplate values only the resistance of the conductors is included. The drop of the line voltage is estimated as $\Delta U \approx \sqrt{3}\,I R_{\text{cond}}\cos\varphi$ , and the inductive reactance of the cable increases it a little

Electric submersible pumps (ESP) →

The theoretical head of an impeller (the Euler equation)

H_{\text{th}} = \frac{u_2\,c_{u2} - u_1\,c_{u1}}{g}

Holds when: an ideal impeller: the flow follows the vanes exactly, and there are no losses. The real head is lower because of the finite number of vanes, friction and leaks with an inlet without swirl ( $c_{u1} = 0$ ), what is left is $H_{\text{th}} = u_2 c_{u2}/g$

Electric submersible pumps (ESP) →

The pressure balance of a hydraulic piston pump

p_{\text{pf}} - p_{\text{sp}} = (p_{\text{out}} - p_{\text{in}})\,\frac{P}{E} + F

any consistent units of pressure

Holds when: work per cycle: the engine receives $(p_{\text{pf}} - p_{\text{sp}})V_{\text{e}}$ , the pump end gives $(p_{\text{out}} - p_{\text{in}})V_{\text{p}}$ , and the difference is friction. $P/E = V_{\text{p}}/V_{\text{e}}$ (the derivation is in the advanced section) leaks are included separately, in the flow rates; the inertia of the masses and the pressure oscillations during the stroke are neglected $F$ grows with the number of strokes and with the viscosity of the liquid; it is taken from the manufacturer's data

Hydraulic pumps: piston and jet →

The delivery, the number of strokes and the flow rate of the power fluid

q = \alpha_{\text{p}}\,1440\,V_{\text{p}}\,n, \qquad q_{\text{pf}} = \frac{1440\,V_{\text{e}}\,n}{\alpha_{\text{e}}} = \frac{q}{\alpha_{\text{p}}\,\alpha_{\text{e}}\,(P/E)}

metric oilfield units; 1440 is the number of minutes in a day

Holds when: a double-acting unit: $V_{\text{p}}$ and $V_{\text{e}}$ are the volumes over a full cycle, up and down there is no free gas (gas that is not dissolved in the liquid) at the intake; with gas the volumetric efficiency falls, as with a sucker rod pump (Sucker-rod pumping)

Hydraulic pumps: piston and jet →

The dimensionless characteristic of a jet pump

\begin{aligned} H &= \frac{B - S}{1 + K_{\text{n}} - B},\\ B &= 2R + \frac{2M^2R^2}{1 - R} - (1 + K_{\text{td}})(1 + M)^2R^2,\\ S &= (1 + K_{\text{s}})\,\frac{M^2R^2}{(1 - R)^2} \end{aligned}

dimensionless quantities

Holds when: one-dimensional flow of an incompressible liquid; the densities of the power fluid and of the production are the same the mixing chamber is cylindrical, and the mixing in it is complete; the velocities at the intake and at the exit of the diffuser are small there is no free gas at the intake; if there is gas, its volume is added to the production rate, and the curve is only approximate the loss coefficients are found by tests; here the jet leaves the nozzle at the pressure at the inlet of the mixing chamber. In common field methods, the pressure drop in the nozzle is counted from the intake pressure. Then the denominator contains $B - S$ , and at the same $M$ the pressure ratio comes out lower (see the advanced section).

Hydraulic pumps: piston and jet →

The flow rate of the power fluid through the nozzle

q_{\text{pf}} = A_{\text{n}}\sqrt{\frac{2\,(p_{\text{pf}} - p_{\text{in}})}{\rho\,(1 + K_{\text{n}} - S)}}

Holds when: the velocity before the nozzle is small compared with the velocity of the jet $S$ is usually a few hundredths, and in estimates it is dropped. The flow rate then comes out slightly smaller

Hydraulic pumps: piston and jet →

The largest production rate of a jet pump without cavitation

q_{\text{cav}} = A_{\text{s}}\sqrt{\frac{2\,(p_{\text{in}} - p_{\text{v}})}{\rho\,(1 + K_{\text{s}})}}, \qquad A_{\text{s}} = A_{\text{t}} - A_{\text{n}}

SI; the pressures are absolute

Holds when: a one-dimensional estimate. In fact cavitation begins earlier, because in the boundary layer of the jet the pressure is lower than the average. So we leave a margin in the production rate there is no free gas. If the production contains gas, it expands at the inlet of the chamber and limits the production rate in a similar way

Hydraulic pumps: piston and jet →

The displacement per revolution and the theoretical delivery of a progressing cavity pump

V = 4\,e\,d\,t_{\text{s}}, \qquad q_{\text{th}} = 1440\,V n

metres and cubic metres; $n$ in rpm, $q_{\text{th}}$ in m³/d (1440 is the number of minutes in a day)

Holds when: a single-lobe rotor in a two-lobe stator is the most common geometry; for pumps with more lobes the displacement is found in a different way the cavities are completely filled with liquid, and there is no slippage; the volumetric efficiency allows for the departures from this

Progressing cavity pumps and hybrid systems →

The actual delivery and the volumetric efficiency of a progressing cavity pump

q = q_{\text{th}} - s\,\Delta p_{\text{pump}}, \qquad \eta_{\text{v}} = \frac{q}{q_{\text{th}}} = 1 - \frac{s\,\Delta p_{\text{pump}}}{1440\,V n}

m³/d, MPa, rpm

Holds when: the slippage depends linearly on the pressure drop. This is an approximation within the rated head. Near the limiting pressure drop, and in a worn pump, the slippage grows faster the coefficient $s$ is found on a test stand for a given liquid, temperature and interference. It falls as the viscosity and the interference grow there is no free gas in the liquid: gas takes up part of the cavities and reduces the delivery of liquid still more

Progressing cavity pumps and hybrid systems →

Laminar flow in the ring-shaped gap between the rods and the tubing

\frac{\Delta p_{\text{fr}}}{L} = \frac{8\,\mu\,Q}{\pi\left[r_2^4 - r_1^4 - \dfrac{(r_2^2 - r_1^2)^2}{\ln(r_2/r_1)}\right]}

Holds when: laminar flow of a Newtonian liquid (one whose viscosity does not depend on the flow rate): the Reynolds number $Re = \rho v\,(d_2 - d_1)/\mu$ (the number that tells smooth flow from turbulent flow) is less than about 2000 the rods are coaxial with the tubing. For the axial flow of a Newtonian liquid their rotation has no effect. Rods pressed against the tubing wall reduce the losses, and couplings and rod guides increase them, so the formula gives an estimate the viscosity is constant along the length; in fact the oil cools towards the wellhead and thickens

Progressing cavity pumps and hybrid systems →

The torque and the power at the polished rod

M = M_{\text{h}} + M_{\text{fr}} + M_{\text{rod}}, \qquad M_{\text{h}} = \frac{V\,\Delta p_{\text{pump}}}{2\pi}, \qquad N = M\,\omega, \qquad \omega = \frac{\pi n}{30}

Holds when: the hydraulic torque is found from the theoretical displacement: the pressure drop acts on the rotor whatever amount of liquid slips back the viscous torque of a smooth rod, coaxial with the tubing, in a Newtonian liquid: $M_{\text{rod}} = \dfrac{4\pi\mu\omega\,r_1^2 r_2^2}{r_2^2 - r_1^2}\,L$ (the symbols are as in the formula for friction in the ring-shaped gap). Couplings, rod guides and the pressing of the rods against the tubing in a curved wellbore increase it the friction of the rods against the tubing in deviated sections is not included

Progressing cavity pumps and hybrid systems →

The twist of the rod string and the stored energy

\theta = \frac{M L}{G J_p}, \qquad J_p = \frac{\pi d_{\text{rod}}^4}{32}, \qquad W = \frac{M\,\theta}{2}

Holds when: solid rods of one diameter; couplings and upset ends (the thickened ends of the rods) reduce the twist a little the torsion is elastic: the shear stresses in the rods are below the yield strength (the stress at which steel starts to deform permanently)

Progressing cavity pumps and hybrid systems →

Symbols

Symbol	Meaning and context
$1 + S_{\text{pl}}/l_{\text{cl}}$	compression ratio of the pump: the ratio of the largest volume under the plunger to the smallest, —Sucker-rod pumping
$A_{\text{n}}$	exit area of the nozzle, m²Hydraulic pumps: piston and jet
$A_{\text{pl}}$	cross-sectional area of the plunger, m²Sucker-rod pumping plunger area: the full area, without subtracting the cross-section of the rods, m²Sucker-rod pumping
$A_{\text{pl}},\ A_{\text{rod}}$	areas of the plunger and of the cross-section of the rod, m²Sucker-rod pumping
$A_{\text{rod}}$	cross-sectional area of the body of a rod, m²Sucker-rod pumping
$A_{\text{s}}$	area of the ring-shaped suction inlet of the production into the mixing chamber, m²Hydraulic pumps: piston and jet
$A_{\text{tbg}}$	cross-sectional area of the metal of the tubing (the wall), m²Sucker-rod pumping
$B$	pressure rise from the inlet of the mixing chamber to the discharge, in fractions of the velocity head of the jet $\rho v_{\text{n}}^2/2$ , —Hydraulic pumps: piston and jet
$B_g$	gas formation volume factor at intake conditions: $B_g = \dfrac{p_{\text{sc}}}{p}\cdot\dfrac{T}{T_{\text{sc}}}\,Z$ , m³/m³Artificial lift: when it is needed and how to choose the method formation volume factor of the gas at the intake conditions: $B_g = \dfrac{p_{\text{sc}}}{p}\cdot\dfrac{T}{T_{\text{sc}}}\,Z$ (the pressures are absolute, the temperatures are in kelvins, and Z is the gas compressibility factor, a correction for a gas that is not ideal), m³/m³Electric submersible pumps (ESP)
$B_o,\ B_w$	oil and water formation volume factors at intake conditions, m³/m³Artificial lift: when it is needed and how to choose the method formation volume factors of oil and water at the intake conditions, m³/m³Electric submersible pumps (ESP)
$C$	counterbalance effect: the force of the counterweights, converted into a force at the polished rod, NSucker-rod pumping
$c_{u1},\ c_{u2}$	circumferential components of the absolute velocity of the liquid at the inlet and at the outlet, m/sElectric submersible pumps (ESP)
$\Delta N_{\text{cab}}$	power loss in the three conductors of the cable, WElectric submersible pumps (ESP)
$d$	diameter of the cross-section of the rotor, equal to the width of the slot of the stator, mProgressing cavity pumps and hybrid systems
$D$	plunger diameter, mSucker-rod pumping
$D_{\text{cas}}$	inside diameter of the casing (or of the shroud), mElectric submersible pumps (ESP)
$D_{\text{mot}}$	outside diameter of the motor, mElectric submersible pumps (ESP)
$d_{\text{rod}}$	diameter of the body of the rod, mProgressing cavity pumps and hybrid systems
$e$	eccentricity: the offset of the axis of the rotor from the axis of the stator, mProgressing cavity pumps and hybrid systems
$E$	Young's modulus of steel (a measure of how stiff it is), about 207,000 MPa, PaSucker-rod pumping Young's modulus of steel, PaSucker-rod pumping
$E_{\text{sep}}$	the share of free gas that goes into the annulus, by natural separation at the intake and through the gas separator, —Electric submersible pumps (ESP)
$F$	friction losses in the unit, expressed as an extra pressure drop of the power fluid, MPaHydraulic pumps: piston and jet
$\Delta p_{\text{fr}}$	friction losses in the tubing, PaArtificial lift: when it is needed and how to choose the method pressure loss from friction in the tubing, from the pump to the wellhead, PaElectric submersible pumps (ESP) pressure loss from friction over the length $L$ , PaProgressing cavity pumps and hybrid systems
$g$	acceleration due to gravity, 9.81 m/s², m/s²Artificial lift: when it is needed and how to choose the method acceleration of free fall, 9.81 m/s², m/s²Sucker-rod pumping acceleration of free fall, m/s²Electric submersible pumps (ESP)
$G$	shear modulus of steel (its resistance to twisting), about $8\cdot 10^{10}$ , PaProgressing cavity pumps and hybrid systems
$h$	head of the stage at the rate $q_{\text{in}}$ , from the performance curve, mElectric submersible pumps (ESP)
$H$	true vertical depth of the middle of the perforated interval, mArtificial lift: when it is needed and how to choose the method pressure ratio $(p_{\text{out}} - p_{\text{in}})/(p_{\text{pf}} - p_{\text{out}})$ , —Hydraulic pumps: piston and jet
$h_1,\ h_2$	heads of the stage at these points, mElectric submersible pumps (ESP)
$H_{\text{dyn}}$	depth of the liquid level in the annulus, measured from the wellhead, mSucker-rod pumping vertical depth of the dynamic fluid level in the annulus, mElectric submersible pumps (ESP)
$H_{\text{pump}}$	head of the pump, $z_{\text{stg}}\,h$ ; for an exact choice it is equal to the required head, mElectric submersible pumps (ESP)
$H_{\text{pump}},\ H$	true vertical depths of the pump setting and of the middle of the perforated interval, mArtificial lift: when it is needed and how to choose the method
$H_{\text{req}}$	the head that the pump must create at the design rate, mElectric submersible pumps (ESP)
$H_{\text{st}}$	true vertical depth of the static fluid level below the wellhead, mArtificial lift: when it is needed and how to choose the method
$H_{\text{th}}$	theoretical head of the impeller, mElectric submersible pumps (ESP)
$H,\ H_{\text{dyn}}$	true vertical depths of the middle of the perforated interval and of the dynamic fluid level, mArtificial lift: when it is needed and how to choose the method
$h(q_{\text{in}})$	head of one stage at the rate $q_{\text{in}}$ and at the frequency at which the pump will work, mElectric submersible pumps (ESP)
$I$	current in each conductor of the cable, AElectric submersible pumps (ESP)
$J_p$	polar moment of inertia of the cross-section of the rod (how the cross-section resists twisting), m⁴Progressing cavity pumps and hybrid systems
$K_{\text{a}}$	acceleration factor: the greatest acceleration of the polished rod as a fraction of $g$ , —Sucker-rod pumping acceleration factor, $S n^2/1790$ , —Sucker-rod pumping
$k_{\text{b}}$	buoyancy factor, $1 - \rho/\rho_{\text{steel}}$ ; $\rho$ is the density of the liquid in the tubing, and $\rho_{\text{steel}} = 7850$ kg/m³ is the density of steel, —Sucker-rod pumping
$K_{\text{n}},\ S$	loss coefficient in the nozzle, and the fall of the pressure of the production at the inlet of the chamber (from the characteristic), —Hydraulic pumps: piston and jet
$K_{\text{n}},\ K_{\text{s}},\ K_{\text{td}}$	loss coefficients in the nozzle, at the suction inlet of the production, and in the mixing chamber with the diffuser (found by experiment; in the lesson 0.03, 0 and 0.30), —Hydraulic pumps: piston and jet
$L$	length of the rod string (along the wellbore), mSucker-rod pumping length of the rod string and of the tubing down to the pump, mSucker-rod pumping length of the rod string, mSucker-rod pumping length of the tubing from the pump to the wellhead, mProgressing cavity pumps and hybrid systems
$L_{\text{cab}},\ S$	length of the cable and cross-section of the conductor, m, mm²Electric submersible pumps (ESP)
$l_{\text{cl}}$	height of the clearance volume: its volume divided by the plunger area, mSucker-rod pumping
$l_{\text{tbg}}$	average length of one tubing joint, mSucker-rod pumping
$\ln$	natural logarithm, —Progressing cavity pumps and hybrid systems
$M$	ratio of the flow rates of the production and of the power fluid, —Hydraulic pumps: piston and jet torque at the polished rod, N·mProgressing cavity pumps and hybrid systems the torque that is passed, N·mProgressing cavity pumps and hybrid systems
$M_{\text{fr}}$	friction torque of the rotor in the stator (from stand tests), N·mProgressing cavity pumps and hybrid systems
$M_{\text{h}}$	hydraulic torque, N·mProgressing cavity pumps and hybrid systems
$m_{\text{rod}}$	mass of one metre of rods with couplings, kg/mSucker-rod pumping
$M_{\text{rod}}$	viscous friction torque of the turning rods, N·mProgressing cavity pumps and hybrid systems
$n$	number of strokes (double strokes) per minute, min⁻¹Sucker-rod pumping number of strokes per minute, min⁻¹Sucker-rod pumping number of double strokes per minute, min⁻¹Hydraulic pumps: piston and jet rotational speed of the rotor, rpmProgressing cavity pumps and hybrid systems rotational speed, rpmProgressing cavity pumps and hybrid systems
$N$	hydraulic power — the useful work of lifting the liquid per unit of time, WArtificial lift: when it is needed and how to choose the method power at the polished rod, WProgressing cavity pumps and hybrid systems
$n_1,\ n_2$	rotational speeds. For a motor fed by a variable speed drive they are almost proportional to the frequency $f$ of the current, rpmElectric submersible pumps (ESP)
$N_1,\ N_2$	shaft powers, kWElectric submersible pumps (ESP)
$N_{\text{c}}$	number of collars whose echoes have been counted on the record, —Sucker-rod pumping
$N_{\text{el}}$	power drawn from the grid (or the equivalent of the energy expended), WArtificial lift: when it is needed and how to choose the method
$n_{\text{lim}}$	the number of strokes per minute at which the acceleration of the polished rod at the top is equal to the acceleration of free fall, min⁻¹Sucker-rod pumping
$N_{\text{rod}}$	power at the polished rod: the work per cycle (the area of the dynamometer card) multiplied by the number of cycles per second, WSucker-rod pumping
$N_{\text{shaft}}$	power at the pump shaft, WElectric submersible pumps (ESP) power at the motor shaft, WElectric submersible pumps (ESP)
$N_{\text{shaft,stg}}$	shaft power spent on the stage, WElectric submersible pumps (ESP)
$N_{\text{stg}}$	hydraulic (useful) power of the stage, WElectric submersible pumps (ESP)
$P$	load on the polished rod, NSucker-rod pumping
$P_{\uparrow},\ P_{\downarrow}$	load on the polished rod on the upstroke and on the downstroke, NSucker-rod pumping
$p_{\text{ann}}$	gas pressure in the annulus at the wellhead (gauge; zero if the annulus is open), PaArtificial lift: when it is needed and how to choose the method gas pressure in the annulus at the wellhead, PaArtificial lift: when it is needed and how to choose the method
$p_{\text{bar}}$	pressure in the barrel under the plunger: on the upstroke it is equal to the intake pressure, and on the downstroke it is equal to the discharge pressure, PaSucker-rod pumping
$p_{\text{in}}$	pump intake pressure, PaArtificial lift: when it is needed and how to choose the method pressure at the pump intake, MPaHydraulic pumps: piston and jet intake pressure, PaHydraulic pumps: piston and jet
$P_{\text{liq}}$	fluid load, NSucker-rod pumping
$P_{\max},\ P_{\min}$	estimates of the greatest and the smallest load on the polished rod in a cycle, NSucker-rod pumping
$p_{\text{out}}$	pump discharge pressure — what is needed to lift the liquid to the wellhead, PaArtificial lift: when it is needed and how to choose the method pressure above the plunger, that is, the pump discharge pressure, PaSucker-rod pumping pressure at the discharge of the pump end, MPaHydraulic pumps: piston and jet
$p_{\text{out}},\ p_{\text{in}}$	pressure at the pump discharge and at the pump intake, PaSucker-rod pumping pressure at the pump discharge and at the pump intake, both absolute, PaSucker-rod pumping pressures at the pump discharge and at the pump intake, PaElectric submersible pumps (ESP)
$p_{\text{pf}}$	pressure of the power fluid at the inlet of the engine, MPaHydraulic pumps: piston and jet
$p_{\text{pf}},\ p_{\text{in}}$	pressure of the power fluid before the nozzle, and the intake pressure, PaHydraulic pumps: piston and jet
$P_{\text{rod}}$	weight of the rods in air, $m_{\text{rod}}\,g\,L$ : inertia depends on the mass, and the buoyant force does not affect it, NSucker-rod pumping weight of the rods in air, NSucker-rod pumping
$p_{\text{sp}}$	pressure of the spent power fluid at the outlet of the engine, MPaHydraulic pumps: piston and jet
$p_{\text{v}}$	vapour pressure of the liquid at the temperature at the intake (absolute), PaHydraulic pumps: piston and jet
$p_{wf}$	bottomhole pressure at the middle of the perforated interval, PaArtificial lift: when it is needed and how to choose the method
$p_{wh}$	pressure at the wellhead in the tubing, PaElectric submersible pumps (ESP)
$p_{wh},\ p_{wf}$	tubing pressure at the wellhead and bottomhole pressure at the middle of the perforated interval, PaArtificial lift: when it is needed and how to choose the method
$P'_{\text{rod}}$	weight of the rod string in liquid, NSucker-rod pumping
$P'_{\text{rod}},\ P_{\text{liq}}$	weight of the rods in liquid and the fluid load, NSucker-rod pumping
$P/E$	ratio of the working volumes (effective areas) of the pump end and of the engine, —Hydraulic pumps: piston and jet
$\Delta p_{\text{pump}}$	pressure rise the pump must produce, PaArtificial lift: when it is needed and how to choose the method pressure rise of the pump, PaArtificial lift: when it is needed and how to choose the method pressure drop of the pump: the difference between the discharge and intake pressures, MPaProgressing cavity pumps and hybrid systems pressure drop of the pump, PaProgressing cavity pumps and hybrid systems
$q$	volumetric flow rate through the pump at intake conditions, m³/sArtificial lift: when it is needed and how to choose the method actual delivery of liquid, measured at the surface, m³/dSucker-rod pumping delivery of production at intake conditions, m³/dHydraulic pumps: piston and jet actual delivery at intake conditions, m³/dProgressing cavity pumps and hybrid systems
$Q$	flow rate of the liquid, m³/sProgressing cavity pumps and hybrid systems
$q_1,\ q_2$	rates at similar (corresponding) points of the curves, m³/dElectric submersible pumps (ESP)
$q_{\text{cav}}$	production rate at which the pressure at the inlet of the mixing chamber falls to the vapour pressure, m³/sHydraulic pumps: piston and jet
$q_{\text{in}}$	rate: the volume of liquid that passes through the pump, at the intake conditions, m³/sElectric submersible pumps (ESP) rate at the intake conditions: the oil and water rates multiplied by their formation volume factors, plus the free gas that has not been separated, m³/sElectric submersible pumps (ESP) rate at the intake conditions, m³/sElectric submersible pumps (ESP)
$q_o,\ q_w$	oil and water rates at standard conditions, m³/dArtificial lift: when it is needed and how to choose the method
$q_{\text{pf}}$	flow rate of the power fluid, m³/dHydraulic pumps: piston and jet
$q_{\text{th}}$	theoretical displacement: the volume that the plunger would displace in a day if its stroke were equal to the stroke of the polished rod, m³/dSucker-rod pumping theoretical delivery, m³/dProgressing cavity pumps and hybrid systems
$\bar p_R$	average reservoir pressure, PaArtificial lift: when it is needed and how to choose the method
$R$	producing gas–oil ratio, m³/m³Artificial lift: when it is needed and how to choose the method ratio of the areas of the nozzle and of the mixing chamber, —Hydraulic pumps: piston and jet
$r_1$	radius of the body of the rod, mProgressing cavity pumps and hybrid systems
$r_2$	inside radius of the tubing, mProgressing cavity pumps and hybrid systems
$R_{\text{cond}}$	resistance of one conductor at its average temperature, ΩElectric submersible pumps (ESP)
$R_s$	solution gas–oil ratio of the oil at intake pressure and temperature, m³/m³Artificial lift: when it is needed and how to choose the method solution gas–oil ratio of the oil at the pressure and the temperature of the intake, m³/m³Electric submersible pumps (ESP)
$s$	slippage coefficient: how much liquid flows back for each megapascal of pressure drop, m³/(d·MPa)Progressing cavity pumps and hybrid systems
$S$	length of the stroke of the polished rod, mSucker-rod pumping fall of the pressure of the production from the intake to the inlet of the mixing chamber, in the same fractions, —Hydraulic pumps: piston and jet
$S_{\text{pl}}$	length of the plunger stroke relative to the barrel, mSucker-rod pumping length of the plunger stroke, mSucker-rod pumping
$\Delta p_{\text{stg}}$	pressure rise in one stage, PaElectric submersible pumps (ESP)
$t$	time from the pulse to the arrival of the echo from the level, sSucker-rod pumping average temperature of the conductor along the well, °CElectric submersible pumps (ESP)
$t_{\text{c}}$	arrival time of the echo from the last counted collar, sSucker-rod pumping
$t_{\text{s}}$	pitch of the stator, twice the pitch of the rotor, mProgressing cavity pumps and hybrid systems
$U$	line voltage at the motor, VElectric submersible pumps (ESP)
$u_1,\ u_2$	circumferential speeds of the impeller at the inlet and at the outlet, $u = \pi D n/60$ , m/sElectric submersible pumps (ESP)
$v$	average speed of sound in the gas of the annulus, m/sSucker-rod pumping
$V$	displacement per revolution: the volume displaced in one turn of the rotor, m³Progressing cavity pumps and hybrid systems displacement per revolution of the pump, m³Progressing cavity pumps and hybrid systems
$v_{\text{cool}}$	average velocity of the liquid in the annular gap along the motor, m/sElectric submersible pumps (ESP)
$V_{\text{p}},\ V_{\text{e}}$	volume displaced by the pump end and by the engine in one double stroke, m³Hydraulic pumps: piston and jet
$V,\ n$	displacement per revolution and rotational speed, m³, rpmProgressing cavity pumps and hybrid systems
$W$	elastic energy stored in the twisted string, JProgressing cavity pumps and hybrid systems
$x_{\text{open}}$	the distance that the plunger travels from the bottom of the stroke before the standing valve opens, mSucker-rod pumping
$z_{\text{stg}}$	number of stages; the result is rounded up, —Electric submersible pumps (ESP)
$\alpha$	temperature coefficient of the resistance of copper, 0.0039, 1/°CElectric submersible pumps (ESP)
$\alpha_{\text{e}}$	the share of the power fluid that does work in the engine; the rest is leaks past the piston and the spool valve, —Hydraulic pumps: piston and jet
$\alpha_{\text{f}}$	fillage: the share of the volume swept by the plunger that is occupied by liquid; the rest is free gas (gas that is not dissolved in the liquid), —Sucker-rod pumping
$\alpha_{\text{leak}}$	the share that is left after leaks through the plunger clearance, the valves and the tubing threads, —Sucker-rod pumping
$\alpha_{\text{p}}$	pump volumetric efficiency, —Sucker-rod pumping pump volumetric efficiency of the pump end: leaks and incomplete filling, —Hydraulic pumps: piston and jet
$\alpha_{\text{s}}$	the share of the stroke of the polished rod that reaches the plunger: $S_{\text{pl}}/S$ ; the rest is the elastic stretch of the rods and tubing, —Sucker-rod pumping
$\alpha_{\text{sh}}$	shrinkage, $1/B_{\text{liq}}$ : the liquid at the intake contains dissolved gas and takes up more volume than at the surface; $B_{\text{liq}}$ is the formation volume factor of the liquid at the intake conditions, —Sucker-rod pumping
$\eta$	system efficiency: the share of the energy supplied that goes into lifting the liquid, —Artificial lift: when it is needed and how to choose the method efficiency of the stage, which is also the efficiency of a pump made of identical stages, —Electric submersible pumps (ESP)
$\eta(q_{\text{in}})$	efficiency of the pump at this rate, —Electric submersible pumps (ESP)
$\eta_1,\ \eta_2$	efficiencies, —Electric submersible pumps (ESP)
$\eta_{\text{mot}},\ \cos\varphi$	efficiency and power factor of the motor (the power factor shows how much of the current does useful work), —Electric submersible pumps (ESP)
$\eta_{\text{v}}$	volumetric efficiency, the same concept as the pump volumetric efficiency of a sucker rod pump, —Progressing cavity pumps and hybrid systems
$\theta$	angle of twist of the string, radProgressing cavity pumps and hybrid systems
$\lambda_{\text{ov}}$	added plunger stroke due to the inertia of the rods, mSucker-rod pumping
$\lambda_{\text{rod}},\ \lambda_{\text{tbg}}$	elastic stretch of the rod string and of the tubing string under the fluid load, mSucker-rod pumping
$\mu$	dynamic viscosity of the liquid, Pa·sProgressing cavity pumps and hybrid systems
$\rho$	density of the liquid in the well, kg/m³Artificial lift: when it is needed and how to choose the method density of the liquid in the tubing and below the pump, kg/m³Artificial lift: when it is needed and how to choose the method density of the pumped liquid at the conditions in the pump, kg/m³Electric submersible pumps (ESP) density of the liquid in the tubing and in the annulus below the level, kg/m³Electric submersible pumps (ESP) density of the power fluid, kg/m³Hydraulic pumps: piston and jet
$\rho,\ g$	density of the pumped liquid at the conditions in the pump, and the acceleration of free fall, kg/m³, m/s²Electric submersible pumps (ESP)
$\rho,\ K_{\text{s}}$	density of the production and the loss coefficient at its suction inlet, kg/m³, —Hydraulic pumps: piston and jet
$\rho_{\text{el}}$	resistivity of copper at 20 °C, 0.0175, Ω·mm²/mElectric submersible pumps (ESP)
$\rho_{\text{liq}}$	average density of the liquid in the annulus below the level, kg/m³Artificial lift: when it is needed and how to choose the method
$\rho_{\text{steel}}$	density of steel, 7850 kg/m³, kg/m³Sucker-rod pumping
$\varphi_{\text{g}}$	volume fraction of free gas in the flow at the intake, —Artificial lift: when it is needed and how to choose the method volume fraction of free gas in the flow that enters the pump, —Electric submersible pumps (ESP)
$\omega$	angular velocity of the crank, $\pi n/30$ , rad/sSucker-rod pumping angular velocity, rad/sProgressing cavity pumps and hybrid systems