On the Theory of Gas Lift

V. A. Shvab

Oil production by means of gas lift is, in essence, the most important method for operating deep and high-yield wells. In addition, the principle of operation of gas lift is the basis of two methods of well operation—flowing and compressor operation. The wide spread of these methods in oil-producing regions is the reason why the problem of gas-lift operation has become a problem of great practical importance.

The field of application of gas lift is not limited to the oil-producing industry and is of considerable importance for lifting water by means of compressor wells when aquifer horizons occur at great depth. This method of lifting water from great depths has become fairly widespread. Thus, for example, one may point to several water-lift compressor stations providing water supply on the railways of the USSR.

Gas lift was first invented by the Russian engineers Shukhov and Bari. D. I. Mendeleev wrote about the invention of this method as early as 1886. The wide application of gas lift in the oil industry took place in the Baku region in 1897. Despite the long history of gas lift and its great practical importance in the oil industry, despite the considerable work carried out for the purpose of experimental and theoretical study of the gas-lift problem, up to the present time, from the theoretical point of view, the gas-lift problem remains far from solved.

In the domestic literature, the operation of gas lift is considered by Melikov, Shishchenko and Baklakov, by the AzNII brigade, by Krylov, and, finally, especially thoroughly, by Prof. Bogdasarov [1]. The indicated series of works gives an idea of the development of gas-lift theory. The essence of gas-lift action gradually appears more and more clearly. Errors in the general physical principles of an operating gas lift are gradually eliminated. However, a common feature of all theoretical works carried out is the hypothetical character of the laws of motion of a two-phase liquid in a gas lift. The absence of general principles concerning the motion of a mixture of gas and liquid, as a basis in considering gas-lift theory, led the authors of the works to considerable diversity, and at times directly to an erroneous conception of the features of two-phase flow. Very different initial positions on this question naturally led to considerable diversity also in the results of the work. The absence of satisfactory agreement with experimental investigation of gas lift compelled corrections to theoretical calculations. In view of the limited scope of the present article, there is no possibility of dwelling on the theories of gas lift available in the literature, and we can only refer to the detailed review made in the book by Prof. Bogdasarov.

The main purpose of the present work is the construction of a theory of gas lift from new positions. These positions are the new results of the theoretical study of two-phase flow, substantiated by many-

numerous experimental data concerning the structure of the flow, its hydraulic resistance, the acting head, etc.

Here we shall not dwell on the generally known physical principles of gas-lift operation (see the diagram in Fig. 1), but, as far as possible, shall disclose the fundamental features connected with the motion of a two-phase flow, on whose theoretical foundations the theory of gas lift proper may also be constructed.

On the basis of experimental investigation of the structure of a two-phase flow in pipes, in the case of their vertical arrangement, it has been established by a number of authors [2], [3], [4] that, for steady flow, the so-called “bubble” or “slug” regime of motion of a two-phase flow is characteristic. This regime is characterized by large bubbles occupying a considerable part of the pipe cross-section. However, a layer of liquid is always preserved between the bubble and the pipe wall. The bubbles alternate with a liquid slug.

Fig. 1

In his work S. I. Kosterin [4] makes an attempt to systematize the structure of the flow, from which it follows that, beginning with the very smallest mixture velocities and up to values of practical significance, a “slug” regime of mixture motion takes place, which in this case has a number of stages: a) quiet slug flow, b) slug flow with foaming behind the bubble, c) slug flow with foaming along the entire interface. At relatively significant velocities of motion, slug flow arises with slugs made of a gas-liquid emulsion and, finally, a more or less uniform emulsion flow for relatively small contents of the gaseous component. For a large volumetric content of the gas component, slug flow is maintained at considerably higher velocities, and from S. I. Kosterin’s data it does not follow that this flow develops into an emulsion flow. At large gas contents, as is obvious, what is characteristic is not an emulsion flow, but a core flow of gas carrying small droplets of liquid, while a moving liquid film is preserved on the pipe walls, having translational motion upward; however, at local thickenings of this film, liquid can break off downward, which is again picked up by the gas and, as already indicated above, transported by the latter in the form of small droplets suspended in the gas stream.

For gas-lift operation, the “bubble,” or otherwise “slug,” flow structure is of primary importance. The emulsion flow structure, which appears at relatively very high velocities of mixture motion, does not arise in a gas lift, since the required velocities are practically unattainable. In gas lifts with a very low located reduced level of the lifted liquid, or, in the terminology of Prof. Bogdasarov, at

In gas lifts with a “small power effect,” a flow of a two-phase liquid arises, characterized by core flow of the gas owing to the extremely large volumetric gas content. In this case the liquid phase moves in the peripheral annular layer, and is also transported in the form of droplets by the gas stream.

In the present work, at the first stage, we confine ourselves to an analysis of gas-lift operation under the plug regime of motion of a two-phase liquid.

In further investigations of gas-lift operation we shall rely on the results of work on the theory of two-phase flow published by the author in 1948 and 1949 [5], [6]. The plug structure of the flow is characterized by the coefficient of relative volumetric gas content \(\xi\), within values of the latter from 0 to 0.95, where

\[ \xi=\frac{u_1}{u_1+u_2}, \]

\(u_1\) — the reduced gas velocity, equal to the volumetric gas flow rate referred to the cross-sectional area of the pipe;

\(u_2\) — the reduced liquid velocity, equal to the volumetric liquid flow rate referred to the cross-sectional area of the pipe.

The frictional resistance in the motion of a two-phase flow with this flow structure is determined by the relation

\[ dp_f=\left[1-\left(1-k^2\varphi^2 M^2\right)\frac{\xi}{k\varphi z}\right]\lambda\frac{\gamma_2 U_0^2}{2gd}\,dy . \tag{1} \]

For a gas–liquid mixture at pressures up to 50 ata, the preceding formula is simplified and becomes

\[ dp=(1-\xi)\lambda\frac{\gamma_2 U_0^2}{2gd}\,dy . \tag{2} \]

The mean specific weight of the gas–liquid mixture, occurring under the conditions of motion through the pipe (with allowance for the relative motion of gas and liquid), is determined by the relation

\[ \gamma_{\mathrm{cp}}=\gamma_2\left[1-\left(1-\frac{\gamma_1}{\gamma_2}\right)\frac{\xi}{k\varphi}\right]. \tag{3} \]

In the expressions given above the following notation has been adopted:

\(U_0\) — the mean discharge velocity of the two-phase flow, equal to the sum \(u_1+u_2\);

\(\gamma_1\) and \(\gamma_2\) — the specific weight of the gas and of the liquid;

\(k\) — the ratio of the maximum velocity of the velocity profile of a single-phase flow to the mean velocity in the section, \(\dfrac{U_m}{U_0}\);

\(z\) — the square of the ratio of the bubble diameter to the pipe diameter,

\[ z=1-\frac{k\varphi-1}{mk\varphi}, \]

where

\[ k=\frac{U_m}{U_0}=1+3.75\sqrt{\frac{\lambda}{8}}; \]

\[ \varphi=\frac{1}{1+\dfrac{0.975\,k-1}{0.24}\dfrac{\gamma_1}{\gamma_2}}; \]

\[ m=1-\frac{0.855}{1+0.87\sqrt{\frac{\gamma_2}{\gamma_1}}}; \]

\(M\) — a function determining the relation between the velocity of motion of the bubble and the motion of the liquid layer between the bubble and the wall;

\(\lambda\) — coefficient of frictional resistance in the region of the quadratic law of resistance, determined for rough pipes from known relations;

\(d\) — pipe diameter;

\(g\) — acceleration due to gravity.

The formulas determining the frictional resistance were tested on the basis of numerous experimental data. In [6] formulas determining the pressure drop due to frictional resistance, as well as the value of the mean specific weight of the mixture, were tested on the basis of a number of experimental data.

Armand’s work [2], recently published and devoted to the experimental determination of the mean specific weight of a two-phase mixture in motion through vertical pipes, excellently confirms the previously published theoretical conclusions concerning the mean specific weight (of a two-phase mixture), formula (3).

The differential equation of motion of a two-phase liquid in the lift pipe of a gas lift will be

\[ -dp_t=dp+dp_w+\gamma_{cp}\sin\alpha\,dy. \tag{4} \]

Here

\(p\) — total pressure drop;

\(p_f\) — pressure drop due to frictional resistance;

\(p_w\) — pressure drop due to acceleration of the flow;

\(y\) — coordinate coinciding with the direction of the vertical;

\(\alpha\) — angle between the axis of the gas-lift pipe and the horizontal in the inclined position of the gas lift.

The integral of equation (4) can be obtained without any substantial simplifications, and this will be shown below. Striving, however, to simplify the method of calculating a gas lift, we shall use the usually adopted assumption that the influence of the pressure drop due to acceleration of the flow is insignificant. Then

\[ -dp=dp_f+\gamma_{cp}\sin\alpha\,dy. \tag{5} \]

Using the values of \(dp\) and \(\gamma_{cp}\) from expressions (2) and (3), we obtain

\[ -dp=(1-\xi)\lambda\,\gamma_2\frac{U_0^2}{2gd}\,dy +\gamma_2\left[1-\left(1-\frac{\gamma_1}{\gamma_2}\right)\frac{\xi}{k\varphi}\right]\sin\alpha\,dy. \tag{6} \]

The specific weight of the gas \(\gamma_1\) in any section of the gas lift can be determined from the condition of isothermal expansion

\[ \gamma_1=\gamma_{11}\frac{p}{p_{11}}, \tag{7} \]

where \(\gamma_{11}\) is the specific weight of the gas at the barometric pressure equal to \(p_{11}\). In the absence of back pressure at the outlet of the gas lift, the pressure \(p_{11}\) corresponds to the pressure in the outlet section of the gas lift. The volumetric gas flow rate.

\(V_1\) through any section of the gas lift under an isothermal process of gas expansion is determined by the condition

\[ V_1 = V_{11}\frac{p_{11}}{p}, \tag{8} \]

where \(V_{11}\) is the gas flow rate through the outlet section.

The reduced gas velocity, as a result of dividing (8) by the cross-sectional area of the pipe, will be

\[ u_1 = u_{11}\frac{p_{11}}{p}, \tag{9} \]

where \(u_{11}\) is the reduced velocity in the outlet section of the gas lift.

The coefficient of relative volumetric gas content

\[ \xi = \frac{1}{1+\dfrac{u_2}{u_{11}}} \tag{10} \]

on the basis of equality (9) will be

\[ \xi = \frac{1}{1+\dfrac{u_2}{u_{11}}\dfrac{p}{p_{11}}}. \tag{11} \]

The mean flow velocity of the two-phase mixture is equal to the sum of the reduced velocities of the gas and liquid, i.e.,

\[ U_0 = u_1 + u_2 \]

or

\[ U_0 = \frac{u_{11}p_{11}}{p}\left(1+\frac{u_2}{u_{11}}\frac{p}{p_{11}}\right). \tag{12} \]

Equation (6) can now be represented as

\[ -dp = \left(1-\frac{1}{1+\dfrac{u_2}{u_{11}}\dfrac{p}{p_{11}}}\right) \left(1+\frac{u_2}{u_{11}}\frac{p}{p_{11}}\right)^2 \frac{u_{11}^{2}p_{11}^{2}}{p^{2}} \frac{\lambda\gamma_2}{2gd}\,dy + \]

\[ +\gamma_2\sin\alpha \left[ 1- \left(1-\frac{\gamma_{11}p}{\gamma_2p_{11}}\right) \frac{1}{\left(1+\dfrac{u_2}{u_{11}}\dfrac{p}{p_{11}}\right)k\varphi} \right]dy \tag{13} \]

After separating the variables, after multiplying by

\[ \frac{p}{p_{11}}\left(1+\frac{u_2}{u_{11}}\frac{p}{p_{11}}\right), \]

we obtain

\[ \frac{ -\dfrac{p}{p_{11}} \left(1+\dfrac{u_2}{u_{11}}\dfrac{p}{p_{11}}\right) d\left(\dfrac{p}{p_{11}}\right) }{ \dfrac{u_2}{u_{11}}\dfrac{\lambda u_2^2}{2gd} \left(1+\dfrac{u_2}{u_{11}}\dfrac{p}{p_{11}}\right)^2 + \dfrac{p}{p_{11}} \left(1+\dfrac{u_2}{u_{11}}\dfrac{p}{p_{11}}\right)\sin\alpha - \left(1-\dfrac{\gamma_{11}}{\gamma_2}\dfrac{p}{p_{11}}\right) \dfrac{p}{p_{11}}\frac{1}{k\varphi}\sin\alpha } = \frac{\gamma_2}{p_{11}}\,dy. \tag{14} \]

Assuming

\[ \frac{p}{p_{11}}=q \quad \text{and} \quad \frac{\lambda u_2^2}{g d}=n \]

and grouping by powers of \(q\), we obtain\(^1\)

\[ -\frac{q\,dq+\dfrac{u_2}{u_{11}}q^2\,dq} {\left[\dfrac{u_2}{u_{11}}\left(\dfrac{n}{2}+\sin\alpha\right) +\dfrac{\gamma_{11}}{\gamma_2}\dfrac{1}{k\varphi}\sin\alpha\right]q^2 +\left[n+\left(1-\dfrac{1}{k\varphi}\right)\sin\alpha\right]q +\dfrac{n}{2}\dfrac{u_{11}}{u_2}} = \frac{\gamma_2}{p_{11}}\,dy . \tag{15} \]

We shall integrate the left-hand side of this equation within the limits from

\[ q_2=\frac{p_2}{p_{11}} \quad \text{to} \quad q_1=\frac{p_{10}}{p_{11}}, \]

having first changed the overall sign to the opposite one, and the right-hand side from \(y=0\) to \(y=H\), taking \(p_{10}\) as corresponding to the pressure at the gas-lift shoe, \(p_2\) as the pressure at the outlet cross section, and the value \(H\) as equal to the length of the gas lift. Thus,

\[ \int_{q_2}^{q_1} \frac{q\,dq+\dfrac{u_2}{u_{11}}q^2\,dq} {Aq^2+Bq+C} = \frac{\gamma_2 H}{p_{11}}, \tag{16} \]

where

\[ A=\left(\frac{n}{2}+\sin\alpha\right)\frac{u_2}{u_{11}} +\frac{\gamma_{11}}{\gamma_2}\frac{1}{k\varphi}\sin\alpha, \]

\[ B=n+\left(1-\frac{1}{k\varphi}\right)\sin\alpha, \]

\[ C=\frac{n}{2}\frac{u_{11}}{u_2}. \]

The integral of the equation will be the equality

\[ \frac{1}{A}\left\{ 1.15\left(1-\frac{u_2}{u_{11}}R\right) \log\frac{t_1^2+a^2}{t_2^2+a^2} -\frac{1}{a}\left[ \frac{u_2}{u_{11}}S+\frac{R}{2}\left(1-\frac{u_2}{u_{11}}R\right) \right] \left(\operatorname{arctg}\frac{t_1}{a} -\operatorname{arctg}\frac{t_2}{a}\right) +\frac{u_2}{u_{11}}(q_1-q_2) \right\} = \frac{\gamma_2 H}{p_{11}} \tag{17} \]

or

\[ \frac{1}{A}\left\{ 1.15\left(1-\frac{u_2}{u_{11}}R\right) \log\frac{t_1^2-a^2}{t_2^2-a^2} -\frac{1.15}{a}\left[ \frac{u_2}{u_{11}}S+\frac{R}{2}\left(1-\frac{u_2}{u_{11}}R\right) \right] \log\frac{(t_1-a)(t_2+a)}{(t_1+a)(t_2-a)} +\frac{u_2}{u_{11}}(q_1-q_2) \right\} = \frac{\gamma_2 H}{p_{11}}. \tag{18} \]

\(^1\) The coefficient \(\varphi\) is a weakly dependent function of \(\dfrac{\gamma_1}{\gamma_2}\) and, in absolute value, is close to unity; it is determined for the average conditions in the gas lift.

In these expressions

\[ R=\frac{B}{A} \]

\[ S=\frac{C}{A} \]

\[ a^2=S-\frac{R^2}{4} \]

\[ t_1=q_1+\frac{R}{2} \]

\[ t_2=q+\frac{R}{2} \]

\[ H \]
is the length of the gas lift;

\[ q_1=\frac{p_{10}}{p_{11}}; \]

\[ q_2=-\frac{p_2}{p_{11}}; \]

\(p_2\) is the pressure at the outlet from the gas lift, equal to \(p_{11}\), the back pressure created at the outlet from the gas lift.

For a positive value of the coefficient

\[ a^2=S-\frac{R^2}{4}, \]

the solution of the gas-lift equation corresponds to equality (17), and for a negative value of this coefficient, to equality (18).

Equation (17) or (18) is the gas-lift equation, determining the fundamental relations among its characteristic parameters for gas contents within the range of the coefficient \(\xi\) from 0 to 0.95.

Let us now consider the gas-lift equation taking inertial forces into account. The pressure drop due to acceleration of the flow, restricting ourselves to an approximate expression in terms of mean velocities, will be

\[ dp_w=\rho U_0\,dU_0. \]

From the condition of continuity of motion it follows that

\[ \rho U_0=\mathrm{const}, \]

where

\[ \mathrm{const}=\frac{\gamma_2}{g}\left[u_2+\frac{\gamma_{11}}{\gamma_2}\frac{u_{11}}{k\varphi}\right]. \]

The basic gas-lift equation (4) in this case will be

\[ -dp=dp_f+\mathrm{const}\,dU_0+\gamma_{cp}\sin\alpha\,dy. \tag{19} \]

After transformations analogous to the preceding case, we obtain

\[ -\frac{ q\,dq+\frac{u_2}{u_{11}}\,dq-\frac{\mathrm{const}\,u_{11}}{p_{11}}\cdot\frac{dq}{q} -\frac{\mathrm{const}\,u_2}{p_{11}}\,dq }{ Aq^2+Bq+C } = \frac{\gamma_2}{p_{11}}\,dy. \]

The integral of this equation for a positive value of \(a^2\) will be

\[ \frac{1}{A}\left\{ 1.15\left(1-\frac{u_2}{u_{11}}R\right)\log\frac{t_1^2+a^2}{t_2^2+a^2} -\frac{1}{a}\left[ \frac{u_2}{u_{11}}S+\frac{R}{2}\left(1-\frac{u_2}{u_{11}}R\right) \right]\left(\operatorname{arctg}\frac{t_1}{a}-\right. \]

\[ -\operatorname{arctg}\frac{t_2}{a}\right) +\frac{u_2}{u_{11}}\left(q_1-q_2\right)\right\} +\frac{1}{A}\left\{1.15D\log\frac{t_1^2+a^2}{t_2^2+a^2}+\right. \]
\[ \left. +\frac{1}{a}\left[\varepsilon+F-\frac{R}{2}D\right] \left(\operatorname{arctg}\frac{t_1}{a}-\operatorname{arctg}\frac{t_2}{a}\right) +2.3N\log\frac{q_1}{q_2}\right\} =\frac{\gamma_2H}{p_{11}} \tag{20} \]

or, for a negative value of \(a^2\),

\[ \frac{1}{A}\left\{1.15\left(1-\frac{u_2}{u_{11}}R\right) \log\frac{t_1^2-a^2}{t_2^2-a^2} -\right. \]
\[ -\frac{1.15}{a}\left[ \frac{u_2}{u_{11}}S+\frac{R}{2}\left(1-\frac{u_2}{u_{11}}R\right) \right]\log \frac{(t_1-a)(t_2+a)}{(t_1+a)(t_2-a)} +\right. \]
\[ \left. +\frac{u_2}{u_{11}}\left(q_1-q_2\right)\right\} +\frac{1}{A}\left\{1.15D\log\frac{t_1^2-a^2}{t_2^2-a^2}+\right. \]
\[ \left. +\frac{1.15}{a}\left[\varepsilon+F-\frac{RD}{2}\right]\log \frac{(t_1-a)(t_2+a)}{(t_1+a)(t_2-a)} +2.3N\log\frac{q_1}{q_2}\right\} =\frac{\gamma_2H}{p_{11}}. \tag{21} \]

Here

\[ A=\left(\frac{n}{2}+\sin\alpha\right)\frac{u_2}{u_{11}} +\frac{\gamma_{11}}{\gamma_2}\frac{1}{k\varphi}\sin\alpha; \qquad q_1=\frac{p_{10}}{p_{11}}; \]

\[ B=n+\left(1-\frac{1}{k\varphi}\right)\sin\alpha; \qquad q_2=\frac{p_2}{p_{11}}; \]

\[ C=\frac{n}{2}\frac{u_{11}}{u_2}; \qquad D=-\frac{\mathrm{const.}\,u_{11}}{S\cdot p_{11}}; \]

\[ R=\frac{B}{A}; \qquad \varepsilon=\frac{\mathrm{const.}\,u_2}{p_{11}}; \]

\[ S=\frac{C}{A}; \qquad F=-\frac{R}{S}\frac{\mathrm{const.}\,u_{11}}{p_{11}}; \]

\[ a^2=S-\frac{R^2}{4}; \qquad N=-D. \]

\[ t_1=q_1+\frac{R}{2}; \]

\[ t_2=q_2+\frac{R}{2}; \]

The solution obtained, which takes into account the influence of the inertial forces of the flow, is of rather fundamental significance, since for gas lift this influence is relatively small, while it considerably complicates the solution.

Thus, in practice it is recommended to calculate gas lift by equation (17) or (18)—for a negative value of the coefficient that ensures the correct calculation of gas lift when the relative volumetric gas content \(\xi\) lies within the range from 0 to 0.95.

Equation (17) is an implicit function of dimensionless criteria of the form

\[ \frac{H\gamma_2}{p_{11}} = f\left( \frac{\lambda u_2^2}{g d}, \frac{u_{11}}{u_{11}+u_2}, \frac{p_2}{p_{11}}, \frac{p_{10}}{p_{11}}, \sin\alpha \right). \tag{22} \]

For vertical gas lifts we obtain a simpler dependence

\[ \frac{H\gamma_2}{p_{11}} = f\left( \frac{\lambda u_2^2}{g d}, \frac{u_{11}}{u_{11}+u_2}, \frac{p_{10}}{p_{11}} \right), \tag{23} \]

where it is also assumed that \(p_2 \cong p_{11}\).

For a broad practical solution of the problem of calculating a gas lift, a series of graphs should be constructed in accordance with the scheme (Fig. 2).

Particular attention should be given to determining the pressure \(p_{10}\) at the base of the gas lift. This pressure will be equal to the natural pressure at the shoe of the gas lift, created by the pressure of the column of liquid and gas in the formation, with allowance for the pressure drop due to the local resistances at the inlet and the pressure drop equal to the kinetic energy of the mixture at the base of the gas lift. At the outlet from the gas lift the pressure \(p_2\) is equal to atmospheric pressure \(p_{11}\) plus the back pressure created beyond the outlet section of the gas lift. In the special case when the back pressure is relatively small, \(p_2=p_{11}\).

Fig. 2

For the purpose of experimentally verifying the gas-lift calculation obtained, the author made use of the very detailed experiments of Prof. Bogdasarov, carried out on a special laboratory gas-lift installation at AzNII. The calculation results were compared with experiments for which \(\xi\) did not exceed the value 0.95. In the calculation, the local resistances at the shoe of the gas lift indicated by Prof. Bogdasarov and the back pressure formed at the outlet were taken into account.

From the experimental data the criteria were determined

\[ \frac{\lambda u_2}{g d}, \qquad \xi=\frac{u_{11}}{u_{11}+u_2}, \qquad \frac{p_{10}}{p_{11}} \ \text{and}\ \frac{p_2}{p_{11}}, \]

which were then substituted into equation (17), making it possible to determine the height of the gas lift, which was compared with the true height of the gas lift, equal to \(19.3\ \text{m}\). Some results of these calculations are given in the table.

TABLE

References

1. Bogdasarov, V. G.—Theory, Calculation, and Practice of Gas Lift. Gostoptekhizdat, 1947.
2. Armand, A. A.—Izv. VTI, No. 2, 1950.
3. Kosterin, S. I.—Izv. Academy of Sciences of the USSR, Division of Technical Sciences, December, 1949.
4. Prandtl, L.—Hydromechanics, Moscow, 1949, pp. 395 and 396.
5. Shvab, V. A.—Hydraulics of two-phase flow in horizontal and vertical pipes. Transactions of TEMIIT, Vol. XIV, 1948.
6. Shvab, V. A.—Theoretical foundations of the steady motion of two-phase flow. Transactions of TEMIIT, Vol. XV, 1949.