Introduction

Background of study

Natural gas can be simply defined as a naturally occurring mixture of light hydrocarbons consisting mainly of methane but also contains varying amounts of heavier alkanes and other impurities such as carbon dioxide, hydrogen sulphide, helium, nitrogen and water. Natural gas exists in reservoir in different thermodynamic states; as dry gas, wet gas and as retrograde condensate. Although natural gas consumption in Nigeria is quite small compared to some countries such as the US, natural gas production is faced with variety of challenges one of which is the issue of liquid loading. Liquid loading, in a nutshell can be defined as the accumulation of liquid or water in the wellbore due to insufficient gas velocity to evacuate the liquid phase which is co-produced with the gas resulting in reduction or complete cessation of gas production. These usually occur due to depletion of the reservoir pressure and at this stage the gas produced is unable to lift the liquid produced alongside it, to the surface and at such causing an accumulation of the liquid in the wellbore, and overtime as the liquid accumulates it builds up a high hydrostatic pressure in the wellbore thereby preventing further production of gas. Liquid loading leads to the premature death of a gas well which becomes a financial loss to the operating company.

The liquid content in the gas come from a variety of sources either in liquid form or in vapour form depending on the prevailing well conditions, some sources are; free water present in the formation produced alongside the gas, water condensate and hydrocarbon condensate which enter the well as vapour but while travelling up the tubing condenses, in a case where the well completion is an open one, water or other liquid can flow from other zones into the well bore, in a case where there is an aquifer below the gas zone.

Liquid loading is not always easy to predict and recognize because a thorough diagnostic analysis of the well data needs to be done but accurate prediction of the problem is vitally important for taking timely measures to solve the problem. There are some symptoms also which can help predict a gas well which is under attack such as the onset of liquid slugs at the surface of the well, increasing difference between the tubing and casing pressures with time, sharp changes in gradient on a flowing pressure survey and sharp drops in production decline curve.

In a bid to predict the onset of liquid loading early which has been identified as the key step in salvaging the liquid loading problem, various investigators over the years have come up with models and methods to predict liquid loading in gas wells, the pioneer been Turner et al. (1969) subsequently other predictions came up most based on the foundation principle of Turner et al. (1969). Some of these models were better, some worse, but the various models used for the predictions had their merits and demerits. The Turner model and other models used in predicting liquid loading are included in subsequent sections.

After detection of liquid accumulation in the well, few solutions have been discovered to solve the problem of liquid loading in gas wells over the years and they have proved really helpful. These include foaming of the liquid water which can enable the gas to lift water from the well, the use of reduced tubing diameter and the heating the wellbore to the prevent further condensation, the use of plunger lift which is an intermittent artificial lift method which uses the energy of the gas reservoir to produce the liquid collected at the bottom of the hole, the use of gas lift method which involves injection of gas into the well from another source in other to increase gas production the use of the beam pumping which is a method commonly used in the oil industry to solve liquid loading in wells loaded with liquid hydrocarbon.

Statement of problem

From the above definition of liquid loading, it is clear that it is problematic to gas wells, ranging from decreasing the gas well production rate to completely preventing gas production from the well. However, if liquid loading can be predicted accurately and on time a suitable solution can be put together to combat it. This prediction in question is the primary aim of this study and research, due to the fact that the predicting methods at various points and times show various discrepancies, coming up or developing a better prediction mechanism will go a long way in saving maybe hundreds of petroleum gas wells which would have shut down prematurely due to liquid loading.

Aim of the study

This study aims at developing a new model for predicting critical velocity rate for gas wells building on the premise of Turner et al. (1969) and Li et al. (2001). The model developed would then be incorporated into an Ms-Excel-based program.

Significance of study

As was earlier stated, that early detection of liquid loading in gas wells helps the production team prepare a suitable technique and method to tackle the problem at its early stage before it aggravates. On the completion of this study, software which when provided with the necessary data has the ability to predict liquid loading status of a given well will be developed.

Materials and methods

This research work was done on a three-phase scale:

  1. 1.

    Gathering of the various prediction model, evaluating their errors and choosing the better one.

  2. 2.

    Discover the merit and demerit of the chosen model and improve on it.

  3. 3.

    Finally build it into an excel platform for easy usage.

Predicting liquid loading

Liquid load-up in gas wells is not always obvious; therefore, a thorough diagnostic analysis of well data needs to be carried out to adequately predict the rate at which liquids will accumulate in the well. Although this subject has been studied extensively but the results from previous investigators and the most commonly applied model in the industry still has a high degree of inaccuracy, especially in predicting the minimum gas flow rate required to prevent liquid loading into the well bore (Adesina et al. 2013).

Turner’s model

For the removal of gas well liquids, Turner et al. (1969) proposed two physical models: (1) the continuous film model (2) and entrained drop movement model.

The continuous film model

Turner et al. (1969) were of the opinion that since liquid film accumulation on the walls of a conduit during two-phase gas/liquid flow is inevitable due to the impingement of entrained liquid drops and the condensation of vapours. He also suggested that the annular liquid film must keep moving upwards along the conduit walls to keep a gas well from loading up, and in the same vain the minimum gas flow rate necessary to accomplish this is of primary importance in the prediction of liquid loading. The analysis technique used involves describing the profile of the velocity of a liquid film moving upward on the inside of a tube. The major shortcoming of this model was its inability to clearly define between the adequate and inadequate rates when it was analysed with field data.

Entrained drop movement model (spherical shape droplet model)

The existence of liquid drops in the gas stream presents a different problem in fluid mechanics, namely, that of determining the minimum rate of gas flow that will lift the drops out of the well (Turner et al. 1969). In a bid to do this, Turner et al. proposed a model to calculate the minimum gas flow velocity necessary to remove liquid drops from a gas well which is based on the sole principle of a freely falling particle in a fluid medium (by a transformation of coordinates, a drop of liquid being transported by a moving gas stream becomes a free falling particle and the same general equations apply as shown in Fig. 1). The minimum gas flow velocity necessary to remove liquid drop is given by the following equation:

Fig. 1
figure 1

Entrained droplet movement (Turner et al. 1969)

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$${V_{{\text{crit-Tunadjusted}}}}=1.593~\frac{{{\sigma ^{1/4}}{{({\rho _{\text{L}}} - ~{\rho _{\text{g}}})}^{1/4}}}}{{\rho _{{\text{g}}}^{{1/2}}}}.$$
(1)

Turner et al. (1969) discovered that the drop model prediction in most cases was too low and he blamed this on the values of critical weber number and drag coefficient used in the development of the model and also on the fact that the mathematical development predicts stagnation velocity, which must be exceeded by some finite quantity to guarantee removal of the largest droplets. Analysis of the Turner’s data reveals that the total contribution of these factors requires an upward adjustment of approximately 20%. Mapping Turner’s calculated model against actual test data (as shown in Fig. 2) reveals large discrepancies at lower flowrates; hence adjusted droplet model is given by the following equation to accommodate these discrepancies:

Fig. 2
figure 2

Turner et al. model-calculated minimum flow rates mapped against the test flow rates (Guo et al. 2006)

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$${V_{{\text{crit-T}}}}=1.92~\frac{{{\sigma ^{1/4}}{{({\rho _{\text{L}}} - {\rho _{\text{g}}})}^{1/4}}}}{{\rho _{{\text{g}}}^{{1/2}}}},$$
(2)
$${q_{{\text{crit-T}}}}=~\frac{{3060P{V_{{\text{crit-T}}}}A}}{{Tz}}.$$
(3)

Coleman model

Coleman et al. (1991) made use of the Turner model but validated with field data of lower reservoir and wellhead flowing pressure all below approximately 500 psia. Coleman et al. (1991) discovered that a better prediction could be achieved without a 20% upward adjustment to fit field data with the following expressions:

$${V_{{\text{crit-C}}}}=1.593{\left[ {\frac{{\sigma \left( {{\rho _{\text{L}}} - {\rho _{\text{g}}}} \right)}}{{\rho _{{\text{g}}}^{2}}}} \right]^{\frac{1}{4}}},$$
(4)
$${q_{{\text{crit-C}}}}=~\frac{{3060P{V_{{\text{crit-C}}}}A}}{{Tz}}.$$
(5)

Li’s model (flat-shaped droplet model)

Li et al. (2001) in their research posited that Turner and Coleman’s models did not consider deformation of the free falling liquid droplet in a gas medium, and furthermore, contended that as a liquid droplet is entrained in a high velocity gas stream, a pressure difference exists between the fore and aft portions of the droplet leading to its deformation and its shape changes from spherical to convex bean with unequal sides (flat) as shown in Fig. 3.

Fig. 3
figure 3

a Flat-shaped droplet model. b Shape of entrained droplet moving in high velocity gas (Li et al. 2001)

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Spherical liquid droplets have a smaller efficient area and need a higher terminal velocity and critical rate to lift them to the surface. However, flat-shaped droplets have a more efficient area and are easier to be carried to the wellhead. This lead to the formulation of the following expression:

$${V_{{\text{crit-L}}}}=\frac{{0.7241{\sigma ^{1/4}}{{({\rho _{\text{l}}} - {\rho _{\text{g}}})}^{1/4}}}}{{\rho _{{\text{g}}}^{{1/2}}}},$$
(6)
$${q_{{\text{crit-L}}}}=~\frac{{3060P{V_{{\text{crit-L}}}}A}}{{Tz}}.$$
(7)

Nosseir’s model

Nosseir et al. (2000) focused his study mainly on the impact of flow regime and changes in flow conditions on gas well loading. Their work was similar to that of Turner but the difference between the both was that Turner did not consider the effect of flow regime on the drag coefficient (used in the derivation stage), and thereby making use of the same drag coefficient (0.44) for laminar, transient and turbulent flows. Nosseir derived the critical flow equations by assuming drag coefficient value of 0.44 for Reynolds number (Re) 2 × 105 to 104 and for Re value greater than 106 he took the drag coefficient value to be 0.2. Representation of the critical velocity equation by Nosseir’s model is summarized as:

$${V_{{\text{crit-N}}}}=1.938{\left[ {\frac{{\sigma ({\rho _{\text{l}}} - {\rho _{\text{g}}})}}{{\rho _{{\text{g}}}^{2}}}} \right]^{1/4}},$$
(8)
$${q_{{\text{crit-N}}}}=~\frac{{3060P{V_{{\text{crit-N}}}}A}}{{Tz}}.$$
(9)

Disk-shaped droplet model

Some authors consider that the four dimensionless numbers, Reynolds, Mortan, Eotvos, and Weber (especially the first two parameters), are always used to characterize the shape of bubbles or droplets moving in a surrounding fluid or continuous phase (Hinze 1955; Youngren and Acrivos 1976; Acharya et al. 1977). Wang and Liu (2007) were of the opinion that for typical oilfield condition, the Reynolds number ranges from 104 to 106, and the Morton number for low viscosity liquid in gas wells is possibly between 10−10 and 10−12. For the given ranges of the dimensionless numbers, they found that most of the entrained droplets in gas well are disk shaped. Similar to flat-shaped droplets proposed by Li et al. (2001), the disk-shaped droplets have a more efficient area and are more easily carried to the wellhead. For the disk-shaped droplets, the drag coefficient Cd is close 1.17. Wang’s expressions can be summarized as follows:

$${V_{{\text{crit-W}}}}=0.5213{\left[ {\frac{{\sigma ({\rho _{\text{l}}} - {\rho _{\text{g}}})}}{{\rho _{{\text{g}}}^{2}}}} \right]^{1/4}},$$
(10)
$${q_{{\text{crit-W}}}}=~\frac{{3060P{V_{{\text{crit-W}}}}A}}{{Tz}}.$$
(11)

Guohua and Shunli's model

In this work, Guohua and Shunli (2012) carried out reviews on various models spotting their merit and shortcomings but paid much attention to the Turner’s model (spherical shaped droplet model) and Li’s model (flat shaped droplet model).

Using experimental results from Awolusi (2005) and Wei et al. (2007) (both though did different experiments came out with the same conclusion), Guohua and Shunli (2012) discovered that the Turner’s model overestimated the probability of the occurrence of liquid loading in gas wells and Li’s model underestimated after a plot of critical gas flow rate was done against flow tubing pressure, for measured flow rates and calculated flow rates from Turner’s and Li’s model.

Guohua and Shunli (2012) came up with a dimensionless parameter of loss factor S which he introduced to his new model and using data from Coleman’s work. This loss factor S ranges from zero to unity (the larger it is, the closer the calculated results are to Turner’s model and if zero the new model matches Li’s model). Luan compared results from his work with Turner and Li and discovered that his was more accurate. Summary of Luan model given below:

$${V_{{\text{crit-S}}}}={V_{{\text{crit-L}}}}+0.83 \times ({V_{{\text{crit-T}}}} - {V_{{\text{crit-L}}}}),$$
(12)
$${q_{{\text{crit-S}}}}=~\frac{{3060P{V_{{\text{crit-S}}}}A}}{{Tz}},$$
(13)

Guo’s model

Guo et al. (2006) were of the opinion that although various investigators have suggested several methods of predicting liquid loading but the results from these methods often show discrepancies and are not easy to use due to the difficulties with prediction of the bottom hole pressure in multiphase flow.

Guo et al. (2006) formulated a closed form analytical equation for predicting the minimum gas flow rate for the continual removal of liquid from the gas well. He did this by substituting the Turner et al.’s entrained drop movement model (for predicting minimum gas velocity for the continual removal of liquid from the gas well) into the kinetic energy equation to obtain the minimum kinetic energy the gas must possess in other to keep the well unloaded. The final stage of his work was to apply the minimum kinetic energy equation to the four phase mist flow model in gas wells to obtain the equation for predicting the minimum gas flow rate which can be solved using numerical method (Newton Raphson) or the go seek function built in the MS Excel. Below is a summary of Guo’s model:

$$b\left[ {6.46 \times {{10}^{ - 13}}\frac{{{S_g}TQ_{{{\text{gm}}}}^{2}}}{{{A^2}{E_{{\text{km}}}}}} - {p_{{\text{hf}}}}} \right]+\frac{{1 - 2bm}}{2}\ln \left| {\frac{{{{\left[ {6.46 \times {{10}^{ - 13}}\frac{{{S_g}TQ_{{{\text{gm}}}}^{2}}}{{{A^2}{E_{{\text{km}}}}}}~+m} \right]}^2}+n}}{{{{\left( {{P_{{\text{hf}}}}+m} \right)}^2}+n}}} \right| - \frac{{m+~\frac{b}{c}~n~ - b{m^2}}}{{\sqrt n }}~\left[ {{{\tan }^{ - 1}}\left[ {\frac{{6.46~ \times {{10}^{ - 13}}\frac{{{S_g}TQ_{{{\text{gm}}}}^{2}}}{{{A^2}{E_{{\text{km}}}}}}+m}}{{\sqrt n }}} \right]~ - ~{{\tan }^{ - 1}}\left[ {\frac{{{P_{{\text{hf}}+m}}}}{{\sqrt n }}} \right]} \right]=a\left( {1+~{d^2}e} \right)L,$$
(14)
$$a=~\frac{{15.33{S_{\text{S}}}{Q_{\text{S}}}+86.07{S_{\text{w}}}{Q_{\text{w}}}+86.07{S_{\text{O}}}{Q_{\text{O}}}+0.01879{S_{\text{g}}}{Q_{\text{g}}}~}}{{T{Q_{\text{g}}}}}\cos (\theta ),$$
(14a)
$$b=n=~\frac{{{c^2}e}}{{{{(1+~{d^2}e)}^2}}}~\frac{{0.2456{Q_{\text{s}}}+1.379{Q_{\text{w}}}+1.379{Q_{\text{o}}}}}{{~T{Q_{\text{g}}}}},$$
(14b)
$$c=~\frac{{4.712~ \times {{10}^{ - 5}}T{Q_{\text{g}}}}}{A},$$
(14c)
$$d=~\frac{{{Q_{\text{s}}}+5.615\left( {{Q_{\text{w}}}+~{Q_{\text{O}}}} \right)}}{{86400A}},$$
(14d)
$$e=~\frac{f}{{2g{D_{\text{H}}}\cos (\theta )}},$$
(14e)
$$f=~{\left[ {\frac{1}{{1.74~ - 2\log \left[ {\frac{{2\varepsilon }}{{{D_{\text{H}}}}}} \right]}}} \right]^2},$$
(14f)
$$m=~\frac{{cde}}{{1+~{d^2}e}},$$
(14g)
$$n=~\frac{{{c^2}e}}{{{{(1+~{d^2}e)}^2}}}.$$
(14h)

Nodal analysis

Nodal analysis can be used to estimate the onset of liquid loading in gas wells with more accuracy since it considers the complete flow path of fluids from reservoir to wellhead (Nallaparaju et al. 2012). In nodal analysis, the system is divided into two subsystems at a certain location called nodal point. The first subsystem takes into account inflow from reservoir to nodal point (IPR), while the other subsystem considers outflow from the nodal point to the surface (TPR). The curves formed by this relation on the pressure–rate graph are called the inflow curve and the outflow curve, respectively. The point where these two curves intersect denotes the optimum operating point where pressure and flow rate are equal for both of the curves. Turner’s critical rate prediction in relation with the IPR and TPR curves can be used to predict liquid loading.

In a study carried out by Nallaparaju et al. 2012, we observe that Turner’s critical rate model was plotted with IPR and TPR for a Well with a given initial static reservoir pressure of 3281 psi and Tubing head pressure of 1500 psi as shown in Fig. 4 above. The point at which the Turner’s curve meets with the IPR curve represent the minimum flowrate to prevent liquid loading (10MMSCFD), whereas the optimum operating flowrate for the well is around 75MMSCFD. The farther the operating point from the minimum flowrate, the lower the tendency for liquid loading to occur.

Fig. 4
figure 4

IPR and TPR overlap with Turner (Nallaparaju et al. 2012)

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Zhou model

Turner et al. (1969) entrained droplet model is the most popular model in predicting liquid loading in gas wells (Zhou and Yuan 2009). However, there were still quite a few wells that could not be covered even after a 20% upward adjustment (Turner et al. 1969). By studying the droplet model and liquid film mechanisms, Zhou came up with a new model.

In his work, he studied the force balance on a single liquid droplet, which are the upward drag force, FD; the upward buoyant force, FB; and downward gravity force, FG, which for unloading to be possible and for the droplets to move upwards (FD + FB) must be greater than FG otherwise the droplet will accelerate downwards. The Turner et al. (1969) droplet model was based on the balance of these forces (FD+FB=FG).

Zhou argued that Turner et al. (1969) based his model on force balance on a single liquid droplet, but which in a gas stream, there are more than one as shown above (droplets A and B in Fig. 5). And further argued that if a turbulent flow existed in the gas stream, due to the irregularity of the flow droplets will encounter each other and coalesces will occur as in Fig. 6 below forming a droplet AB (A + B). This newly formed droplet in a situation the gas stream does not have enough energy to lift the droplet it starts falling and may scatter into smaller droplets 1, 2, 3 and the cycle goes on and on for similar larger droplets and these leads to an accumulation of liquid down hole.

Fig. 5
figure 5

Encountering two liquid droplets in turbulent gas stream

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Fig. 6
figure 6

Liquid loading when liquid droplet number reaches a threshold value (Zhou and Yuan 2009)

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Zhou and Yuan (2009) were of the opinion that if there are more liquid droplets in the gas stream, the chance of the process of liquid droplet encountering, coalescing, falling and scattering increases (Fig. 6). As the number of liquid droplets in a gas stream, called liquid droplet concentration increases to a threshold value β, the process of droplets encountering, coalescing, falling and scattering will continue and bring those liquid droplets down to the well bottom. The liquid droplet concentration in a gas stream is defined by

$${H_{\text{l}}}=\frac{{{\text{Liquid~}}\;{\text{superficial~}}\;{\text{velocity}}}}{{{\text{Gas~}}\;{\text{superficial}}\;{\text{~velocity+liquid~}}\;{\text{superficial}}\;{\text{~velocity}}}}=\frac{{{V_{{\text{sl}}}}}}{{{V_{{\text{sg}}}}+{V_{{\text{sl}}}}}}.$$
(15)

Zhou and Yuan (2009) concluded that if liquid droplet concentration is above the threshold value, then the new model can be used but below the threshold value Turners model is used. Summary of Zhou model:

$${V_{{\text{crit-Z}}}}={V_{{\text{crit-T}}}}=1.593\frac{{{{\left[ {\sigma ({\rho _l} - {\rho _{\text{g}}})} \right]}^{1/4}}}}{{\rho _{{\text{g}}}^{{1/2}}}}~{\text{for}}\quad ~{H_{l~}} \leqslant ~\beta ,$$
(16)
$${V_{{\text{crit-Z}}}}={V_{{\text{crit-T}}}}+~\ln \frac{{{H_{\text{l}}}}}{\beta }+\alpha ~\;{\text{for~}}\quad {H_{{\text{l~}}}}>\beta ,$$
(17)
$$Q{\left( {\frac{{Mcf}}{D}} \right)_{{\text{crit-Z}}}}=\frac{{3060P{V_{{\text{crit-Z}}}}A}}{{TZ}}.$$
(18)

Experimental works on liquid loading

A series of experiments have been designed by Awolusi (2005) and Wei et al. (2007) which were done separately. The primary aim of their work was to investigate and evaluate discrepancies in the previous works on critical gas velocities required to keep liquid from accumulating. Which resulted to the discovery that Turner model prediction was high compared to the actual observed critical rate, while Li’s model prediction was low compared to the actual observed critical rate as can be seen in Fig. 7 below.

Fig. 7
figure 7

Critical rate against flow tubing pressure (Awolusi 2005)

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Model development

As was earlier stated at the beginning of this chapter, we are aimed at developing a new model which can be used to calculate the critical rate and in turn predict the onset of liquid loading in gas wells.

This new model was developed by combining the Turner critical rate model and the Li’s critical rate model.

Theory

$${V_{{\text{crit-T}}}}=1.593{\left[ {\frac{{\sigma \left( {{\rho _{\text{l}}} - {\rho _{\text{g}}}} \right)}}{{\rho _{{\text{g}}}^{2}}}} \right]^{\frac{1}{4}}}.$$
(19)

Turner’s critical model given above was developed based on the assumption that the droplet is a sphere and remains spherical all through the entire wellbore,

$${V_{{\text{crit-L}}}}=\frac{{0.7241{\sigma ^{1/4}}{{({\rho _{\text{l}}} - {\rho _{\text{g}}})}^{1/4}}}}{{\rho _{{\text{g}}}^{{1/2}}}}.$$
(20)

On the other hand, Li’s model above was developed based on the assumption that the liquid droplets are flat in shape and remains same all through the wellbore. Having considered the two above droplet shape, in a case where the droplet is neither spherical nor flat as can be observed in Fig. 8 invariably the Turner’s model and the Li’s model would not be able to predict correctly the critical rate required to lift that droplet which is because the droplet deformation was not taken into account during the development of the models. As the droplet travel up the wellbore they tend to deform and can take different shapes at different points in the well. For simplicity sake, the only droplet shapes consider in this paper are droplet shapes ranging from the spherical shape to the flat shape (Fig. 9).

Fig. 8
figure 8

Droplet shapes

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Fig. 9
figure 9

User interface for LOADCALC Software

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In the new model, a deformation coefficient C is introduced to cater for the deformation of the liquid droplet along the wellbore and in turn be able to predict correctly the critical rate when the droplet varies from the spherical shape to the flat shape.

New model development

$${\text{New}}\;{\text{~critical}}\;{\text{~velocity~}}\;{\text{model}}={\text{Li}}'{\text{s}}~\;{\text{critical}}~\;{\text{velocity}}~\;{\text{model}}+{\text{deformation~}}\;{\text{coefficient}} \times ({\text{Turner's}}~\;{\text{critical}}\;{\text{~velocity}}~\;{\text{model}} - {\text{Li's}}~\;{\text{critical}}~\;{\text{velocity}}\;{\text{~model}}),~$$
(21)
$${V_{{\text{crit-new}}}}={V_{{\text{crit-L}}}}+C \times ({V_{{\text{crit-T}}}} - {V_{{\text{crit-L}}}}).$$
(22)

The deformation coefficient used in this work is adapted from Kelbaliyev and Ceylan (2007). Detailed derivation of the deformation coefficient can be found in “Appendix 5”.

$$C=\frac{{{a_0}}}{{{b_0}}}=\frac{{R\left( {1 - {\lambda _{\text{v}}}We} \right)}}{{R\left( {1+{\raise0.7ex\hbox{${{\lambda _{\text{v}}}}$} \!\mathord{\left/ {\vphantom {{{\lambda _{\text{v}}}} 2}}\right.\kern-0pt}\!\lower0.7ex\hbox{$2$}}We} \right)}}.$$
(23)

Using the experimental data from Raymond and Rosant (2000), \({\lambda _{\text{v}}}\) is estimated as:

$${\lambda _{\text{v}}}=\frac{1}{{12}}\left( {1 - \frac{{3~We}}{{25~Re}}} \right).$$
(24)

Weber’s number can be obtained from

$$We=\frac{{\rho \times {v^2}l}}{\sigma },$$
(25)

whereas Reynolds number is obtained from

$$Re=\frac{{Q{D_{\text{H}}}}}{{vA}}.$$
(26)

Computing the values of Table 1 into Eqs. (35) and (36), the value of C was calculated to be:

Table 1 Summary of all assumptions used to prepare critical velocity rate for our new model
Full size table

C = 2.261921523

$${V_{{\text{crit-new}}}}={V_{{\text{crit-L}}}}+2.261921523 \times ({V_{{\text{crit-T}}}} - {V_{{\text{crit-L}}}}),$$
$${V_{{\text{crit-new}}}}=\frac{{0.7241{\sigma ^{1/4}}{{({\rho _{\text{l}}} - {\rho _{\text{g}}})}^{1/4}}}}{{\rho _{{\text{g}}}^{{1/2}}}}+2.261921523 \times \left[ {\frac{{1.593{\sigma ^{1/4}}{{({\rho _{\text{l}}} - {\rho _{\text{g}}})}^{1/4}}}}{{\rho _{{\text{g}}}^{{1/2}}}} - \frac{{0.7241{\sigma ^{1/4}}{{({\rho _{\text{l}}} - {\rho _{\text{g}}})}^{1/4}}}}{{\rho _{{\text{g}}}^{{1/2}}}}} \right],$$
$${q_{{\text{crit-new}}}}=~\frac{{3060P{V_{{\text{crit-new}}}}A}}{{Tz}}={\text{Critical}}\;{\text{~flow}}\;{\text{~rate}}{\text{.}}$$
(27)

The derivation of the Turner and Li models is shown in “Appendix 1”.

Software development

After the new model was developed, the new model and other related correlations were converted to computer codes and built into an excel platform leading to the development of a new software LOADCALC. The computer codes used for the development of LOADCALC are shown in appendix D.

LOADCALC features

LOADCALC software is colourful software and it is user friendly and consists of various sections;

  1. a.

    Wellbore data section

  2. b.

    Fluid properties section

  3. c.

    Introductory section

  4. d.

    Flow parameter section

  5. e.

    Output section

  6. f.

    Analysis section

Results

In other to be able to compute the critical rate using the new model and also compare with the critical rate from Turner and Li, the Turner et al. (1969) data on liquid loaded wells were utilised. Turner published important parameters which influence critical liquid loading rate calculation such as producing depth, wellhead pressures, liquid gas rate, fluid properties, tubing inner/outer diameter, casing diameter and also information on the well flow rate and status (loaded up, near load up and unloaded) at time of test was also included in the database. These data are shown in “Appendix 3”.

Assumptions

All assumptions are summarized in Table 1 below;

Calculated parameters

Due to the large amount of database, the calculations for the below parameters are done using an excel sheet.

Parameters

  1. a.

    Gas density

  2. b.

    Production area

  3. c.

    Compressibility factor

The results for the calculated parameters, formulas, correlations used in calculation are all stated and outlined in “Appendix 2”.

Critical velocity results

Using the production data from the 105 gas wells, assumed parameters and calculated parameter, an excel sheet was used to compute the critical rate for.

  • The new critical velocity model.

  • The Turner critical velocity model.

  • The Li’s critical velocity model.

The results from the different critical velocity models are tabulated in “Appendix 2”.

When the measured flow rate from the different gas wells was plotted against the critical flow rates from the different models in the above table, the following figures were obtained, Figs. 10, 11, 12.

Fig. 10
figure 10

Crossplot of test flowrate against critical flowrate using Li’s model

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Fig. 11
figure 11

Crossplot of test flowrate against critical flowrate using Turner model

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Fig. 12
figure 12

Crossplot of test flowrate against critical flowrate using New model

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Discussion

The above figures are constructed in such a way that if a well’s actual test flow rate equals its critical flow rate for liquid removal, the point will plot on the diagonal, so therefore for critical velocity model to be correct, the wells that are tested at conditions near load-up should plot near this diagonal. Wells that unload easily during a test should plot above the diagonal and those that do not unload should plot below the line. The ability of a given analytical model to achieve this data separation is a measure of its validity.

As can be seen from Figs. 10, 11 and 12 above the most data separation is achieved by the new model in Fig. 12 and in turn provides a better prediction.

Li’s model as can be seen in Fig. 10 was not able to separate the loaded wells from the unloaded wells, almost all the points plotted above the diagonal line, suggesting that all the wells are unloaded which is not correct compared to the well status obtained from the different wells during test.

Turner model as can be seen in Fig. 11 predicted better than Li but majority of the loaded wells still plotted in the unloaded region above the diagonal line.

From Fig. 12, it can be observed that the new model prediction was quite better than Turner and Li due to the fact that most of the loaded wells plotted in the loaded region, most of the unloaded wells plotted in the unloaded region and the near load up wells plotted close to the diagonal.

Software test

To test the efficiency of the LOADCALC software, the well data from a loaded well, an unloaded well and a well which is near load up was collected and feed in the software.

The summary of the production data from the three different wells is shown in Table 2.

Table 2 Summary of the production data from the three different wells
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Loaded up test

See Fig. 13.

Fig. 13
figure 13

LOADCALC software interface showing loaded up test

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4.3 Near load up test

See Fig. 14.

Fig. 14
figure 14

LOADCALC software interface showing near load up test

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Unloaded test

See Fig. 15.

Fig. 15
figure 15

LOADCALC software interface showing unloaded test

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As can be observed from the test figures above the results from the software matches with the test result in Table 2.

Conclusion

This paper presents a new empirical model that calculates the critical velocity and flowrate for low-pressure gas wells by taking into account droplet deformation coefficient. The new model is derived from a modification of Turner’s model and Li’s model. The Turner’s critical model above was developed based on the assumption that the droplet is a sphere and remains spherical all through the entire wellbore On the other hand, Li’s model above was developed based on the assumption the liquid droplets are flat in shape and remains same all through the wellbore. By applying the deformation and elasticity theory, a dimensionless coefficient, C was obtained as a function of the gas Weber number and Reynolds number assuming the surface area of a particle increases after deformation, while the volume remains constant.

Comparative analysis of well test results obtained from 105 gas wells between the new model, Turner’s spherical model and Li’s flat model reveal that droplet deformation plays an important role in the accuracy of prediction of critical flowrate for liquid loading. Error analysis carried out reveals that the new model predicted the onset of liquid loading in gas wells with the least error of 20%, whereas Turner’s model had 26% error and Li, 35% error.

Finally, the recommended applicable range of the new model is when the wellhead pressure is less than 500 psia and the liquid/gas ratios are in the range of 1–130 bbl/MMscf, which is suggested by Turner et al. (1969) to ensure a mist flow in gas wells.