# Interactions between Two Deformable Droplets in Tandem Fixed in a Gas Flow Field of a Gas Well
## 1 Introduction  
Gas-well liquid loading is one of the most serious problems encountered in gas production engineering, which can reduce the gas production rate and eventually kill the gas well due to low pressure or energy. It occurs as a result of liquid accumulating at the bottomhole. The controlling factor to prevent liquid accumulating in a wellbore is the gas velocity in the tubing. It is believed that there is a critical gas velocity in gas wells, below which liquid loading occurs and a static liquid column develops. To keep a well producing without a liquid-loading problem, the velocity of the gas stream must be greater than the critical gas velocity. The commonly used model for predicting the critical gas velocity is the entrained-droplet model proposed by [PERSON] et al. (1969) [1]. According to the model assumptions, a droplet will not fall downward and liquid loading will not occur if the gas velocity is great enough to keep the droplet suspended or to carry it upward. [PERSON]'s entrained-droplet model was derived from the force balance of suspending a maximally sized droplet in agas stream, by assuming that the liquid droplet would be spherical and using a critical Weber number value of 30 to estimate the maximum droplet size. For security considerations, they increased the critical gas rate by 20% to match the value measured in gas wells. [PERSON] et al. (1991) [2] found that the predicted critical gas rate without the increase could better match that of gas wells with a low pressure and a low liquid rate. Later, [PERSON] et al. (1997) [3], [PERSON] et al. (2001) [4], [PERSON] et al. (2010) [5], and [PERSON] et al. (2015) [6] considered the effect of the flow regime, droplet deformation, and droplet amount in the gas stream on the critical gas velocity, and presented various entrained-droplet models. The single-droplet models were proposed based on an isolated droplet, for which the influence of droplet interaction was not considered. For the gas wells with a higher liquid production rate, the number of droplets increases and the distance between the droplets decreases in the annular flow field of the gas wellbore. At such flow conditions, the accuracy of these entrained-droplet models based on single, isolated droplets decreases. Thus, it is necessary to accurately understand the interaction between droplets. There exist numerous droplets in annular flow with different diameters and relative positions, as shown in Figure 1a. The relative positions of the droplets can be defined using separating space L and angle \(\theta\), as shown in Figure 1b, where \(\theta\) is the angle between the gas stream and the line connecting the centers of two droplets. For tandem arrangements, \(\theta=90^{\circ}\), as shown in Figure 1c. For side-by-side arrangements, \(\theta=0^{\circ}\), as shown in Figure 1d. A review of the literature finds that the interaction between two rigid particles has acquired enough investigation, especially in the lower Reynolds number ranges ([PERSON] et al., 1993 [7]; [PERSON] et al., 2003 [8]; [PERSON] et al., 2003 [9]; [PERSON] et al., 2007 [10]; [PERSON] and [PERSON], 2007 [11]; [PERSON] et al., 2009 [12]; and [PERSON] et al. 2010 [13]). Because of the internal circulation and transient effects caused by droplet deformation or breakup, the interaction between deformable droplets is much more complex in comparison with rigid particles ([PERSON] et al., 2016 [14]; [PERSON] et al., 2019 [15]; [PERSON] et al., 2021 [16]). For tandem arrangements, [PERSON] (1990) [17] and [PERSON] (1993) [18] investigated the interaction between two vaporizing liquid-fuel droplets subject to uniform flows. Numerical simulations were carried out for the Reynolds numbers of Re = 50-200. The previous studies of the interactions between two droplets in tandem were limited to Reynolds numbers less than 200 [17; 18]. However, in a natural gas well with a high pressure, the Reynolds number of a droplet would increase to several thousand, but its dynamics are not known and have not been investigated.