Equations: Gas lift

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

Bottomhole pressure of a gas-lift well

p_{wf} = p_{wh} + \Delta p_{\text{above}}\!\left(q,\ \text{GLR}_{\text{res}} + \frac{q_{\text{g}}}{q}\right) + \Delta p_{\text{below}}(q)

metric oilfield units; pressures are absolute, gas volumes are at standard conditions

Holds when: Steady state; all the injected gas enters the tubing through the operating valve. Gas from the reservoir is added to the injected gas wherever it is free. Below the injection point the pressure is usually above the bubble point (the pressure at which gas starts to come out of the oil), so the gas is dissolved in the oil. The pressure differences are calculated with correlations or models of multiphase flow (Multiphase flow in tubing). The examples in this lesson use a simple teaching drift-flux model (a model in which the gas drifts upward through the liquid at its own speed).

Gas lift: principles and choice of operating conditions →

Gas pressure in the annulus at a given depth

p_{\text{g}}(H) = p_{\text{g,wh}}\,\exp\!\left(\frac{M g H}{\bar Z R \bar T}\right)

SI; pressures are absolute

Holds when: The gas in the annulus is almost still. Its velocity is low, so friction is negligible. The temperature and $Z$ are taken as averages over the column. $Z$ depends on the average pressure, which is itself being found, so the calculation is repeated two or three times. The same gas fills the whole column. The formula is the same as for any column of gas (column of gas).

Gas lift: principles and choice of operating conditions →

Condition for injecting gas at a given depth

p_{\text{g}}(H_{\text{inj}}) - \Delta p_{\text{valve}} \ge p_{\text{t}}(H_{\text{inj}})

any consistent units; the pressures are all absolute or all gauge

Holds when: Steady state; gas enters the tubing only through the operating valve. The pressure drop across the valve depends on the gas rate and on the size of the valve port (its opening). How it is calculated is explained in the lesson Unloading a gas lift well: valves and mandrels.

Gas lift: principles and choice of operating conditions →

Condition for the technical optimum

\left(\frac{\partial \Delta p_{\text{weight}}}{\partial q_{\text{g}}}\right)_{q} + \left(\frac{\partial \Delta p_{\text{fr}}}{\partial q_{\text{g}}}\right)_{q} = 0

any consistent units

Holds when: The acceleration of the mixture is not taken into account. The column below the valve does not depend on the gas rate. The flow rate is at its maximum when extra gas stops lowering the pressure that is needed to lift this flow rate (the derivation is in the advanced section).

Gas lift: principles and choice of operating conditions →

Economic optimum of the gas rate

\frac{\dd q_o}{\dd q_{\text{g}}} = \frac{c_{\text{g}}}{v_o}

any consistent units: the ratio of the costs is in the same units as the slope of the curve

Holds when: The oil price and the cost of gas do not depend on the volumes. The gas supply of the system is not limited. If it is limited, the gas is shared between wells (next section). The limits of the surface facilities on water and gas are not taken into account. They move the optimum to the left.

Gas lift: principles and choice of operating conditions →

The best sharing of gas between wells

\frac{\dd q_{o,1}}{\dd q_{\text{g},1}} = \frac{\dd q_{o,2}}{\dd q_{\text{g},2}} = \dots = \beta, \qquad \sum_i q_{\text{g},i} = Q_{\text{g}}

any consistent units

Holds when: The curves of the wells bulge upward (they are concave): each next portion of gas gives less than the previous one. The wells do not affect each other: the pressure in the gathering system does not change with the way the gas is shared. A well whose marginal return is below $\beta$ even at a small gas rate gets no gas. If a well needs a minimum gas rate to start, the simple rule breaks down. We then compare the options with and without that well (below).

Gas lift: principles and choice of operating conditions →

Power of ideal isothermal compression of gas

N_{\text{iso}} = p_{\text{sc}}\,q_{\text{g}}\,\ln\frac{p_2}{p_1}

SI; pressures are absolute

Holds when: The gas is ideal and the compression is isothermal at 20 °C (an infinite number of stages with cooling), with no losses. This is a lower limit for real compressors. If the gas is compressed at a temperature $T$ , the power is higher by a factor of $T/(293\ \text{K})$ . The compressibility of a real gas ( $Z < 1$ ) reduces it by 5–10%. At 40–50 °C these two corrections almost cancel each other. A real multistage compressor with its drive uses more. In the example below it is assumed to use $1/0.65$ times as much.

Gas lift: principles and choice of operating conditions →

Slope of the performance curve of a gas-lift well

\frac{\dd q}{\dd q_{\text{g}}} = -\,\frac{\partial p_{\text{outflow}}/\partial q_{\text{g}}}{\partial p_{\text{outflow}}/\partial q + 1/J}

any consistent units

Holds when: The IPR is a straight line, and the operating point is stable, so the denominator is positive (stability of the operating point).

Gas lift: principles and choice of operating conditions →

Depth to which the kick-off pressure pushes out the kill fluid

p_{\text{ko}}\exp\!\left(\frac{0.03418\,\gamma_{\text{g}}\,H}{\bar Z\,\bar T}\right) = p_{wh} + \rho_{\text{kill}}\,g\,H

metric oilfield units; pressures are absolute

Holds when: The left side is the pressure of the column of gas in the annulus, as in the lesson on gas lift. The right side is the pressure of the liquid in the tubing, which is full up to the wellhead. The balance is static. During displacement, flow friction adds to the pressure in the tubing, and the level stops a little higher still. There is a check valve (a one-way valve) at the bottom of the tubing, so the liquid does not go into the reservoir. Otherwise part of it would go into the reservoir, and the reservoir would first return that same liquid.

Unloading a gas lift well: valves and mandrels →

Gas pressure that opens the valve

p_{\text{dome}}A_{\text{b}} = p_{\text{g}}\,(A_{\text{b}} - A_{\text{p}}) + p_{\text{t}}A_{\text{p}}\quad\Rightarrow\quad p_{\text{g,open}} = \frac{p_{\text{dome}} - R\,p_{\text{t}}}{1 - R},\qquad R = \frac{A_{\text{p}}}{A_{\text{b}}}

any consistent units; the pressures are all absolute or all gauge, and the equation does not change, because the areas on each side add up to $A_{\text{b}}$

Holds when: The valve has no spring. The elasticity of the bellows itself is ignored. The pressures are uniform over the areas on which they act.

Unloading a gas lift well: valves and mandrels →

Closing pressure, and the difference between the opening and closing pressures

p_{\text{g,close}} = p_{\text{dome}},\qquad \Delta p_{\text{spread}} = p_{\text{g,open}} - p_{\text{g,close}} = k_{\text{t}}\,(p_{\text{dome}} - p_{\text{t}}),\qquad k_{\text{t}} = \frac{R}{1 - R}

any consistent units

Holds when: When the ball has moved away from the seat, the gas pressure acts on the whole area of the bellows, and the valve stays open as long as $p_{\text{g}} > p_{\text{dome}}$ . The valve has no spring. In reality the elasticity of the bellows shifts the closing pressure, as explained below.

Unloading a gas lift well: valves and mandrels →

Tubing pressure that opens a valve operated by the tubing pressure

p_{\text{t,open}} = \frac{p_{\text{dome}} - R\,p_{\text{g}}}{1 - R}

any consistent units; the pressures are all absolute or all gauge

Holds when: The valve has no spring. The tubing pressure acts on the bellows from outside, and the gas pressure acts on the seat.

Unloading a gas lift well: valves and mandrels →

Test-rack opening pressure

p_{\text{TRO}} = \frac{p_{\text{dome}}^{\text{rack}}}{1 - R}

gauge pressures, as shown by the gauges of the rack

Holds when: On the tubing side there is atmospheric pressure, that is, $p_{\text{t}} = 0$ on the gauge. The equation is a special case of the force balance.

Unloading a gas lift well: valves and mandrels →

Temperature correction for the nitrogen in the dome

C_T = \frac{p_{\text{dome}}^{\text{rack}}}{p_{\text{dome}}} = \frac{1}{1 + 0.00215\,(T_{^{\circ}\text{F}} - 60)}

temperature in degrees Fahrenheit; pressures are gauge, as in the accepted field method

Holds when: It is a straight-line fit to measurements on nitrogen at the constant volume of the dome. The tables of factors for setting valves are made from it. It is accurate enough at a nitrogen pressure of about 4 MPa. At higher pressures real nitrogen raises its pressure more when heated, and the correction underestimates it (advanced section). If part of the dome is filled with silicone fluid, it also expands on heating and adds pressure.

Unloading a gas lift well: valves and mandrels →

Greatest depth of the next unloading valve

p_{\text{t,des}}(H_n) + \rho_{\text{kill}}\,g\,(H_{n+1} - H_n) = p_{\text{g}}\big(H_{n+1};\ p_{\text{g,wh},n+1}\big) - \Delta p_{\text{margin}},\qquad p_{\text{g,wh},n} = p_{\text{ko}} - (n - 1)\,\Delta p_{\text{step}}

metric oilfield units; pressures are absolute

Holds when: The well is vertical. (In a deviated well, true vertical depths are converted into measured depths along the hole for the mandrels.) Below the working valve, the annulus and the tubing hold kill fluid, and the level in the tubing stays at this valve. The friction of the liquid while it is pushed out is not included, and the margin covers it. The tubing pressure at the working valve is taken from the design line, and enough gas passes through the valve for this mode.

Gas lift design →

Spacing to the next valve with constant gradients

H_{n+1} - H_n = \frac{p_{\text{g,wh},n+1} + g_{\text{g}}H_n - \Delta p_{\text{margin}} - p_{\text{t,des}}(H_n)}{\rho_{\text{kill}}\,g - g_{\text{g}}}

metric oilfield units; pressures are absolute

Holds when: The gas gradient is constant. It is taken from the formula for a column of gas at the average pressure over the interval, so the error is small. For the first valve $H_0 = 0$ and $p_{\text{t,des}}(0) = p_{wh}$ .

Gas lift design →

The stock of gas in the annulus and the rate of change of the pressure in it

V_{\text{sc}} = V_{\text{ann}}\,\frac{\bar p}{p_{\text{sc}}}\,\frac{T_{\text{sc}}}{\bar T}\,\frac{1}{\bar Z},\qquad \frac{\dd\bar p}{\dd t} \approx \bar p\,\frac{q_{\text{in}} - q_{\text{out}}}{V_{\text{sc}}}

any consistent units; pressures are absolute

Holds when: The temperature and $\bar Z$ hardly change during a swing. The pressure in the annulus changes almost equally over the whole height. The second equality follows from the first: with constant $V_{\text{ann}}$ , $\bar T$ and $\bar Z$ , the pressure is proportional to the amount of gas.

Gas lift operation and optimisation →

Symbols

Symbol	Meaning and context
$\bar Z$	gas deviation factor (Z-factor) at the average pressure and the average temperature of the column, —Gas lift: principles and choice of operating conditions
$\bar T$	average absolute temperature of the gas in the annulus, KGas lift: principles and choice of operating conditions
$\bar p,\ \bar T,\ \bar Z$	average absolute pressure of the gas in the annulus, its average absolute temperature and its gas deviation factor, MPa, K, —Gas lift operation and optimisation
$(\cdot)_{q}$	derivative at a constant liquid flow rate, —Gas lift: principles and choice of operating conditions
$0.03418$	$0.02897\cdot 9.81/8.314$ : the molar mass of air (kg/mol) multiplied by $g$ and divided by the universal gas constant, K/mUnloading a gas lift well: valves and mandrels
$A_{\text{b}},\ A_{\text{p}}$	effective area of the bellows, and area of the seat along the line where the ball touches it, cm²Unloading a gas lift well: valves and mandrels
$\Delta p_{\text{above}}$	pressure difference in the tubing from the gas injection point up to the wellhead: the weight of the mixture and friction, found from a multiphase flow calculation (gas and liquid flowing together), MPaGas lift: principles and choice of operating conditions
$\Delta p_{\text{below}}$	pressure difference from the bottom to the injection point: the weight of the liquid and friction, MPaGas lift: principles and choice of operating conditions
$c_{\text{g}}$	cost of a thousand cubic metres of lift gas: compression, drying and maintenance, and for bought gas its price as well, US$/thousand m³Gas lift: principles and choice of operating conditions
$C_T$	correction factor from the temperature of the valve to the rack temperature of 60 °F, —Unloading a gas lift well: valves and mandrels
$\Delta p_{\text{fr}}$	the part of the pressure difference that comes from friction, MPaGas lift: principles and choice of operating conditions
$g$	acceleration due to gravity, 9.81 m/s², m/s²Gas lift: principles and choice of operating conditions acceleration due to gravity, 9.81 m/s². The product $\rho_{\text{kill}} g H$ is converted from pascals to megapascals, m/s²Unloading a gas lift well: valves and mandrels
$g_{\text{g}}$	pressure gradient of the gas in the annulus, taken as constant, MPa/mGas lift design
$\text{GLR}_{\text{res}}$	gas–liquid ratio of the fluid produced from the reservoir: the producing gas–oil ratio multiplied by the share of oil in the liquid, m³/m³Gas lift: principles and choice of operating conditions
$H$	true vertical depth, mGas lift: principles and choice of operating conditions true vertical depth of the liquid level in the annulus, mUnloading a gas lift well: valves and mandrels
$H_{\text{inj}}$	depth of gas injection, which is the depth of the operating valve, mGas lift: principles and choice of operating conditions
$H_n,\ H_{n+1}$	true vertical depths of the working valve and of the next one, mGas lift design
$J$	productivity index, m³/(d·MPa)Gas lift: principles and choice of operating conditions
$k_{\text{t}}$	production pressure effect factor (TEF): by how much the opening pressure falls when the tubing pressure rises by one unit, —Unloading a gas lift well: valves and mandrels
$M$	molar mass of the gas, $0.02897\,\gamma_{\text{g}}$ , where $\gamma_{\text{g}}$ is the specific gravity of the gas relative to air, kg/molGas lift: principles and choice of operating conditions
$\Delta p_{\text{margin}}$	margin: the amount by which the gas at the opening valve must exceed the pressure of the liquid, MPaGas lift design
$N_{\text{iso}}$	power of ideal isothermal compression, WGas lift: principles and choice of operating conditions
$\partial p_{\text{outflow}}/\partial q$	slope of the outflow curve at the given gas rate, MPa per m³/dGas lift: principles and choice of operating conditions
$\partial p_{\text{outflow}}/\partial q_{\text{g}}$	how much extra gas lowers the pressure needed to lift the same flow rate (negative up to the technical optimum), MPa per thousand m³/dGas lift: principles and choice of operating conditions
$p_1,\ p_2$	absolute pressures at the compressor inlet and outlet, PaGas lift: principles and choice of operating conditions
$p_{\text{dome}}$	nitrogen pressure in the dome at the temperature of the valve in the well, MPaUnloading a gas lift well: valves and mandrels nitrogen pressure at the temperature of the valve in the well, MPa (gauge)Unloading a gas lift well: valves and mandrels
$p_{\text{dome}},\ p_{\text{g}},\ R$	nitrogen pressure at the temperature of the valve, gas pressure at the valve, ratio of the seat area to the bellows area, MPa, MPa, —Unloading a gas lift well: valves and mandrels
$p_{\text{dome}}^{\text{rack}}$	nitrogen pressure in the dome at the temperature of the test rack, MPa (gauge)Unloading a gas lift well: valves and mandrels
$p_{\text{g}}$	gas pressure in the annulus at the depth of the valve, MPaUnloading a gas lift well: valves and mandrels
$p_{\text{g}}(H_{\text{inj}})$	gas pressure in the annulus at this depth, MPaGas lift: principles and choice of operating conditions
$p_{\text{g}}(H;\ p_{\text{g,wh}})$	gas pressure in the annulus at depth $H$ for a pressure $p_{\text{g,wh}}$ at the wellhead, from the formula for a column of gas (gas pressure in the annulus), MPaGas lift design
$p_{\text{g}}(H)$	gas pressure in the annulus at depth $H$ , absolute, PaGas lift: principles and choice of operating conditions
$p_{\text{g,close}}$	gas pressure at which the open valve closes, MPaUnloading a gas lift well: valves and mandrels
$p_{\text{g,wh}}$	gas pressure in the annulus at the wellhead, absolute, PaGas lift: principles and choice of operating conditions
$p_{\text{g,wh},n}$	gas pressure at the annulus wellhead at which valve $n$ opens, MPaGas lift design
$p_{\text{g,wh},n+1},\ \Delta p_{\text{margin}},\ p_{\text{t,des}}(H_n)$	as in the equation for the greatest depth of the next valve, MPaGas lift design
$p_{\text{ko}}$	kick-off pressure: the highest gas pressure at the annulus wellhead that the compressor or another source can supply, MPa (abs.)Unloading a gas lift well: valves and mandrels kick-off pressure: the gas pressure at the wellhead during unloading, MPaGas lift design
$p_{\text{sc}}$	standard pressure, 101,325 Pa, PaGas lift: principles and choice of operating conditions
$p_{\text{sc}},\ T_{\text{sc}}$	standard conditions: 0.101325 MPa and 293.15 K (20 °C), MPa, KGas lift operation and optimisation
$p_{\text{t}}$	pressure in the tubing at the depth of the valve, MPaUnloading a gas lift well: valves and mandrels
$p_{\text{t}}(H_{\text{inj}})$	pressure in the tubing at this depth in the working mode, MPaGas lift: principles and choice of operating conditions
$p_{\text{t,des}}(H_n)$	tubing pressure at depth $H_n$ on the design line; for the first valve, the tubing pressure at the wellhead $p_{wh}$ at $H_0 = 0$ , MPaGas lift design
$p_{\text{t,open}}$	tubing pressure at the valve at which it opens, MPaUnloading a gas lift well: valves and mandrels
$p_{\text{TRO}}$	gas pressure that opens the valve on the test rack, MPa (gauge)Unloading a gas lift well: valves and mandrels
$p_{wf}$	bottomhole pressure at the middle of the perforated interval, MPaGas lift: principles and choice of operating conditions
$p_{wh}$	pressure in the tubing at the wellhead (tubing head pressure), MPaGas lift: principles and choice of operating conditions pressure in the tubing at the wellhead (the back pressure of the flowline), MPa (abs.)Unloading a gas lift well: valves and mandrels
$q$	liquid flow rate, m³/dGas lift: principles and choice of operating conditions
$q_{\text{g}}$	rate of injected gas at standard conditions (a fixed reference pressure and temperature, so that gas volumes can be compared), m³/dGas lift: principles and choice of operating conditions rate of injected gas, thousand m³/dGas lift: principles and choice of operating conditions gas rate at standard conditions, m³/sGas lift: principles and choice of operating conditions
$Q_{\text{g}}$	all the gas that the system can supply, thousand m³/dGas lift: principles and choice of operating conditions
$q_{\text{g},i}$	gas rate given to well number $i$ , thousand m³/dGas lift: principles and choice of operating conditions
$q_{\text{in}},\ q_{\text{out}}$	gas rate into the annulus from the surface, and from the annulus into the tubing, at standard conditions, m³/minGas lift operation and optimisation
$q_o$	oil rate, $q\,(1 - \text{WC})$ , where $\text{WC}$ is the water cut (the share of water in the liquid), m³/dGas lift: principles and choice of operating conditions
$q_{o,i}$	oil rate of well number $i$ , m³/dGas lift: principles and choice of operating conditions
$R$	universal gas constant, 8.314 J/(mol·K), J/(mol·K)Gas lift: principles and choice of operating conditions ratio of the seat area to the bellows area; for unloading valves it is usually a few hundredths, —Unloading a gas lift well: valves and mandrels ratio of the seat area to the bellows area, —Unloading a gas lift well: valves and mandrels
$\Delta p_{\text{spread}}$	valve spread, the difference between the opening and closing pressures, MPaUnloading a gas lift well: valves and mandrels
$\Delta p_{\text{step}}$	pressure step: the amount by which each next valve opens at a lower pressure at the wellhead, MPaGas lift design
$T_{^{\circ}\text{F}}$	temperature of the valve in the well, °FUnloading a gas lift well: valves and mandrels
$V_{\text{ann}}$	volume of the annulus from the wellhead to the operating valve, m³Gas lift operation and optimisation
$v_o$	net value of a cubic metre of oil: the price less the payments taken from revenue and the variable costs, US$/m³Gas lift: principles and choice of operating conditions
$V_{\text{sc}}$	gas in the annulus, converted to standard conditions, m³Gas lift operation and optimisation
$\Delta p_{\text{valve}}$	pressure drop across the operating valve that is needed to pass the given gas rate, MPaGas lift: principles and choice of operating conditions
$\Delta p_{\text{weight}}$	the part of the pressure difference in the tubing above the valve that comes from the weight of the mixture, MPaGas lift: principles and choice of operating conditions
$\beta$	the common marginal return of gas: how much oil one more thousand cubic metres of gas would give, m³ of oil per thousand m³Gas lift: principles and choice of operating conditions
$\gamma_{\text{g}},\ \bar Z,\ \bar T$	specific gravity of the gas, and the average gas deviation factor and the average absolute temperature of the gas in the annulus, —, —, KUnloading a gas lift well: valves and mandrels
$\rho_{\text{kill}}$	density of the kill fluid, kg/m³Unloading a gas lift well: valves and mandrels density of the kill fluid; $\rho_{\text{kill}} g$ is converted to MPa/m with the factor $10^{-6}$ , kg/m³Gas lift design
$\rho_{\text{kill}}\,g$	gradient of the kill fluid, MPa/mGas lift design