# Research on the Inflow Performance of the Plunger Lift in the Shale Gas Horizontal Well

Authors: Changqing Ye; Hanbing Tang; Daogang Cai; Nanxing Xie; Qingrong Wang; Jianyi Liu; Lu Jiang
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Shale gas wells in the Changning block of Weiyuan, Sichuan, China, experience water production problems throughout their life cycles. In actual field production, plunger lift is typically used to discharge liquid from a shale gas well. However, it is difficult to test the formation inflow performance relationship curve (IPR) of a plunger lift well. Hence, data on the formulation and optimization of plunger lift well systems are lacking. Based on actual measured data related to casing pressure changes at the wellhead and the gas-liquid distribution characteristics in a wellbore during the shut-in phase, this paper establishes a method for calculating the formation IPR curve of a shale gas plunger lift well by calculating changes in wellbore gas volume continuously. This method has a fast calculation speed. The gas-water interface distribution can be obtained in 3–5 iterations, and the inflow dynamic curve can be calculated in less than 5 seconds. This method tested the IPR curves of gas wells in the Changning block. The IPR curves of plunger lift wells showed concave characteristics contrary to the conventional upward convex curves. This is because the bottom hole flowing pressure changes caused variations in the water saturation of the near-wellbore zone. Consequently, the productivity equation of the plunger lift well is established by introducing the variation coefficient of water saturation expressed by bottom hole flow pressure. This equation fits with the measured IPR curve with an averageR2 of approximately 0.9989.


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## Body

## 1. Introduction

Shale gas reservoirs have low porosity and permeability characteristics [1]. Because of this, they often use large-scale hydraulic fracturing for gas production [2, 3]. Coupled with the water production of the shale formation itself, shale gas wells generally experience water production problems throughout their life cycles [4–6]. In the early stage of gas production, the formation energy is sufficient for bringing out the water from the gas well. However, in the later stage, the formation energy is insufficient; hence, it cannot be used to bring out the gas from the wellbore, causing the gas well to accumulate fluid [7–9] and affecting wells’ gas productivity.As of October 2020, the daily production of shale gas in the Changning block of the Changning–Weiyuan National Shale Gas demonstration zone reached 2026 × 104 m3, ranking first in the daily gas production among shale gas fields in China [10]. However, shale gas wells in the Changning block generally suffer from fluid accumulation [11]. In the late stage of shale gas production, drainage gas recovery technology discharges bottom hole fluid from gas wells. This reduces the bottom hole flowing pressure, increases production pressure differences, and improves gas well productivity. Presently, the drainage gas production technology of shale gas wells mainly includes plunger lift, foam drainage, and gas injection lift. Among these, the plunger lift is widely used in the field because of its simple operating process, as well as low maintenance and operational costs [12–14].Plunger lift runs a cylindrical short rod (plunger) into the tubing of a shale gas production well. While opening the well, the plunger and upper liquid slug are lifted from the well’s bottom to the wellhead using the casing gas energy and formation supply. During the shut-in phase, the plunger falls to the well’s bottom, and the bottom liquid forms a water slug on the upper part of the plunger. These processes reciprocate the opening and closing of the well, forming a plunger lift cycle that lifts the bottom hole fluid accumulated in the shale gas well.Optimization of the plunger operation system has always been a key issue in the plunger lift process. Foss and Gaul (1965) [15] first proposed a static model of plunger motion based on the notion that the plunger rises at a constant speed. In addition, they introduced the concepts of minimum, maximum, and average casing pressures to design the plunger lift process. Based on these concepts, Hacksma (1972) [16] proposed a method for predicting the lift characteristics of the plunger using the plunger characteristic table and formation IPR curve. This static model can optimize the period of the plunger lift well, but it is impossible to obtain the plunger operation information and simulate the movement of the plunger in the wellbore accurately. Subsequently, researchers proposed a dynamic model for solving this problem. First such model proposed by Lea (1982) [17] regards instantaneous velocity, acceleration, and casing pressure as functions of plunger position. A set of equations obtained through force analysis of the plunger determined the plunger’s position, speed, acceleration, constantly changing casing pressure, and other working parameters. Marcano and Marcano (1994) [18] combined the laws of the conservation of momentum and conservation of mass to develop a complete dynamic model of the plunger’s up-and-down strokes. In this model, the Vogel’s equation describes the fluid inflow, and combining it with the experimental data derives quantitatively the channeling flow of the gas under the plunger and leakage of the upper liquid section of the plunger. Using exponential equations to describe gas inflow and linear equations to describe water inflow, Gasbarri and Wiggins (2001) [19] designed a plunger motion model combined with surface pipelines. In addition, previous studies predicted plunger’s movement by analyzing wellbore acoustic signals, casing pressure changes [20], plunger movement in the wellbore based on smart plunger wellbore parameter testing [21], and by dynamically simulating plunger movement using data mining [22]. Regardless of the method or model applied, gas well productivity equations based on a steady flow describe the formation inflow characteristics [23–25]. For the normal gas wells, scholars have established complete fine descriptions [26] and numerical simulation [27] methods to predict the formation inflow performance curve. However, shale gas wells in the Sichuan Basin in China generally pursue low-cost development, and most of the gas wells lack detailed geological data. It is difficult to complete geological modeling. However, the characteristics of the multiperiod and short production time of plunger lift wells have not been analyzed to date. To the best our knowledge, no reports investigating the supply characteristics linked to the formation of the wellbore during plunger lift exist presently.The formation IPR curve is the basis of all the work of well production [28, 29]. In addition, it is of great significance for simulating plunger motion, optimizing plunger lift well process systems, and their daily maintenance. Based on plunger lift technology and cost constraints, accurate testing of the bottom hole flowing pressure and gas production is difficult. In addition, the wellbore flow in the production stage of the plunger lift is unstable, and the plunger separates the space above and below; thus, the wellhead production is not equal to the amount of gas flowing into the wellbore from formation. Consequently, the formation IPR curve of the plunger lift well is difficult to determine.To accurately obtain the formation IPR curve of the plunger lift well, this paper proposes a method to determine the formation inflow performance based on the wellhead casing pressure data tested during the shut-in phase of the plunger lift. This method uses data related to changes induced in the wellhead casing pressure during the shut-in phase to calculate the gas volume flowing into the casing from the formation, thereby obtaining the formation IPR curve. Presently, the Engineering Technology Research Institute of the PetroChina Southwest Oil and Gas Field Company adopts an intelligent management platform to monitor production information related to the draining of gas-producing wells in the Changning block in real time, thereby realizing automatic well-opening and closing control. Furthermore, process implementation monitoring and the cloud storage of daily production data facilitate in easily obtaining the wellhead pressure change data of plunger lift wells required by this method. This method has significant importance for accurately understanding the formation inflow characteristics of plunger lift wells, the simulation of plunger movement in the wellbore, and the formulation and optimization of plunger lift system.


## 2. Plunger Lift Cycle

Three stages of conventional plunger lift wells include the shut-in, well-opening, and afterflow stages (Figure1) [30, 31].Figure 1
Operating stages of a plunger lift well: (a) the shut-in stage, (b) the well-opening stage, and (c) the afterflow stage.
(a)(b)(c)Here, the shut-in stage is adopted as the beginning of a plunger gas lift well cycle. During the shut-in pressure recovery stage, the plunger rests on the seat, and the formation-produced water collects above the plunger. In the well-opening stage, the plunger in the wellbore moves from the bottom of the well to the wellhead under the action of differential pressure. In the afterflow stage, the plunger reaches the wellhead collection device, and all gas and liquid are produced at the upper part of the plunger. In this stage, the wellhead valve is kept open and production takes place continuously. Figure2 shows the pressure and output change curves at each stage.Figure 2
Pressure cycle and production curve of a plunger lift well (Gupta et al., 2017).In the shut-in stage, the plunger in the wellbore does not affect changes in the wellbore pressure. The gas and water produced in the formation enter the tube and case, respectively, causing the casing pressure to rise gradually. When the tubing and casing pressures rise to a certain value, the well is opened for production, and the wellhead tubing and casing pressure decreases. When the upper liquid section of the plunger reaches the wellhead, the tubing pressure may fluctuate slightly. When the upper liquid section of the plunger is completely produced, and after the plunger reaches the wellhead-fishing device, the well is kept open for production. Next, in the afterflow stage, the tubing and casing pressures continue decreasing until the well is closed, followed by the beginning of the subsequent production cycle.In current field production, most plunger gas lift wells have not used downhole pressure gauges to test the bottom hole flowing pressure, and the wellhead production does not fully reflect the formation inflow dynamics. The formation inflow dynamics reverse by changing the wellhead pressure. However, during the plunger’s ascending period, it divides the wellbore into two pressure systems. At this point, calculation of the wellbore gas volume and gas-liquid distribution and accurate determination of formation inflow performance is difficult. In the shut-in stage, the gas-liquid moves slowly in the wellbore and the pressure changes steadily. The formation inflow performance can be reversed through wellbore pressure changes at this stage, which can be used as the formation IPR curve of the well.


## 3. Calculating the Inflow Performance Relationship Curve

The wellhead gas production of conventional wells equals the formation supply because of stable production. Thus, conventional gas wells obtain the formation IPR curve from the wellhead gas production during stable gas well production and measure the bottom hole flowing pressure. In plunger lift wells, the plunger separates the upper and lower spaces of the wellbore. These changes in the pressure difference between the upper and lower plunger, causing inequality in the gas production at the wellhead and formation supply. Consequently, conventional methods cannot be used to test the formation IPR curve of plunger lift wells. This section establishes a test method for deriving the formation IPR curve of a plunger lift well using wellhead tubing and casing pressure data obtained for the shut-in stage and wellbore gas-liquid interface distribution. The gas-liquid flow velocity during the shut-in stage is slower than that during the well-opening stage, so the gas-liquid two-phase flow effect is ignored in this method. Assuming that the annulus liquid level is 0, the gas-liquid interface distribution can be calculated according to the tubing and casing pressure difference at timet, and then the method introduced in Figure 3 is used to iteratively calculate the gas-water interface distribution at time t + 1. Due to the frequent opening and closing of plunger gas lift wells, the formation pressure can be considered unchanged in a short time and equal to the formation pressure measured at time t. This method uses the wellhead casing pressure data and the wellhead gas production to back-calculate the gas production of the formation, and the water production is calculated using a linear equation. The basic data used in this method was easily obtained, and additional testing equipment related to the operation of a gas well was not required. Thus, this method has a potentially broad range of applications.Figure 3
Flowchart for calculating wellbore liquid-level distribution.Continuous changes in the gas content of a wellbore: Based on the principle of equal pressure at the connecting interface of the tube and casing and as water is an incompressible fluid, the annulus liquid is assumed to be transferred to the tubing when the pressure in the annulus is too high. Contrarily, when the pressure in the tubing is too high, the liquid in the tubing is assumed to be transferred to the annulus. The fluid involved in the transfer was only liquid; the gas directly entered the annulus and tubing pipes according to the proportion and did not participate in the transfer until the liquid level in the annulus dropped to the tubing-casing joint. When the liquid level in the annulus dropped to the tubing-casing joint, i.e., when there was no liquid in the annulus, the pressure change in the tubing and casing directly reflected the distribution of gas produced in the formation.Based on the mass balance, the total gas volume in the tubing and casing is converted to ground conditions as follows:(1)Vcg=Ac−AtoHs−Hctwρgρg−airVtg=AtHs−Htwρgρg−air.Then, the gas supply of the formation from timet to time t + 1 can be written as follows:(2)qsct=qscct+qscttqsctt=Vtgt+1−VtgtΔtqscct=Vcgt+1−VcgtΔt.The above formula is adopted to calculate the gas production at each moment, and the corresponding bottom hole pressure at each moment is as follows:(3)pwft=pctwut+ρwgHx+ρwgHctwt.During the shut-in stage, the formation water enters the tubing and casing, which reduces the space occupied by gas. This affects the pressure of the tubing and the gas volume in the tubing and casing. Thus, the distribution of the gas-liquid interface in the wellbore at different shut-in times must be considered, and the formation water production at each time must be calculated.Distribution of the gas-liquid interface during the shut-in stage: To calculate the gas volume in the wellbore at a specific time, it is necessary to calculate the volume occupied by the gas in tubing and casing, i.e., the distribution of the gas-water interface in the wellbore at this time. Figure4 shows the distribution of the gas-liquid interface in the wellbore during the shut-in stage.Figure 4
Distribution of the gas-water interface in the tubing and casing.Regardless of the gas-liquid distribution in the wellbore, the pressures at the joint of the tubing and casing must be equal, i.e.,(4)pctwu+Hctwρwg=ptwu+Htwρwg.Since the wellbore gas flows slowly during the shut-in stage, the pressure on the surface of the liquid in the tubing and casing can be calculated using the static pressure of the gas column as follows [32]:(5)pctwu=pcexp0.03148γgHctgTct¯Zct¯ptwu=ptexp0.03148γgHtgTt¯Zt¯.The water production stage is given as follows:(6)VLt=qLΔt86400,where qL is described by a linear formula [33], i.e.,(7)qL=JLp¯R−pwf.Considering the influx of the liquid phase into the formation, from timet to time t + 1, the following unified material balance equation is adopted for calculating the liquid volume in the annulus and tubing:(8)HctwtAc−Ato+HtwtAt+VL=Hctwt+1Ac−Ato+Htwt+1At.Since the gas-water interface distribution is related to the water production stage in each stage, the interface distribution can only be calculated time by time using an iterative method. Figure3 shows the calculation flowchart.The specific calculation steps for calculating wellbore liquid-level distribution are as follows:Step 1: assume the height of the annulus liquid level at timet + 1 as Htw_t0t+1=Htwt; thus, Htgt+1=Hs−Htw_t0t.Step 2: calculate the formation water producted (VLt) at time t according to equation (6).Step 3: assume the annulus liquid level at timet as Hctwt. Assume the tubing liquid level Htwt combined with the hypothetical casing liquid level at t + 1 as Htw_t0t+1. The water produced at time t is obtained by substituting into equation (8). The height of the annulus liquid level at this time is Hctwt+1; hence, Hctgt+1=Hs−Hctwt+1,Htgt+1=Hs−Htwt+1.Step 4: calculate the pressure on the liquid surface of the annulus (pctwut+1) and the pressure on the liquid surface of the tubing (ptwut+1), on the basis of the wellhead tubing and casing pressures, i.e., Hctgt+1 and Htgt+1, respectively.Step 5: substituteHctwt+1, pctwut+1, and ptwut+1 into equation (4) to calculate Htw_tt+1.Step 6: set iteration precision toa=Htw_tt+1−Htw_t0t+1; if a >1e−6, repeat Steps 1–3, and if a ≤1e−6, quit the loop.Using the above iteration, the distribution of the gas-liquid interface in the wellbore at each time could be iteratively calculated using the casing pressure, annulus, and tubing liquid-level distribution at the initial shut-in time. Generally, the calculation can be completed in 3–5 iterations.


## 4. Method for Calculating Wellbore Gas Parameters

The distribution of gas density and theZ-factor in the wellbore is affected by the wellbore pressure distribution. Concurrently, when calculating the wellbore pressure distribution, it is also necessary to estimate the distribution of gas density and deviating factors in the wellbore. Thus, only an iterative method can be used to calculate the pressure, gas density, and deviation factors in the wellbore.The density andZ-factor were calculated using a modified PR equation [34], and the average pressure is as follows:(9)pcavg=pc+pctwu2ptavg=pt+ptwu2.The calculation block diagram showing the wellbore gas parameters according to an average pressure is shown in Figure5.Figure 5
Method for calculating the wellbore gas parameters.The steps for calculating the wellbore gas parameters are as follows:Step 1: assume the initial pressure on annulus liquid level value aspctwu_0 = pc and calculate the average pressure (pcavg_0) according to equation (9)Step 2: calculate the average deviation factorZ0¯ according to the average pressure and temperatureStep 3: calculatepctwu_i by substituting Z0¯ into equation (5)Step 4: set the iteration precisiona = pctwu_0 − pctwu_i; if a >1e−6, set pctwu_0 = pctwu_i to repeat Steps 1–3, and if a ≤1e−6, set pctwu = pctwu_iStep 5: substitutepctwu and pc into equation (9) to calculate the average pressure and then can calculate density and deviation by average pressure and temperatureGenerally, the calculation can be completed in 2–4 iterations.


## 5. Case Analysis

### 5.1. Basic Situation of Typical Wells

Plunger lift wells account for 57.41% of the total drainage gas production process wells in the Changning block. In daily production management, the intelligent management platform of the gas production process (developed by the Engineering Technology Research Institute of the PetroChina Southwest Oil and Gas Field Company) is used for automatic production data acquisition, maintaining good switching control, and obtaining the daily production data and basic well information related to plunger gas lift wells in real time. Wells Z201, Z202, and Z203 were adopted as examples, and Table1 lists the basic information of the three wells.Table 1
Basic information of typical plunger gas lift wells.
Well numberWell typeTool typeWell deep (m)Plunger length (m)Outer diameter of the plunger (mm)Inner diameter of the tubing (mm)Outer diameter of the casing (mm)A target measure depth (m)A target vertical depth (m)B target measure depth (m)B target vertical depth (m)Z201Down-dipFishbone plunger4874.927048.250.6139.73867362551673516Z202Up-dipFishbone plunger516727048.250.6138.74275379351673929Z203Down-dipPop-up plunger30433805250.6139.73867362551673515Figure6 shows the daily production data of the three wells (Z201, Z202, and Z203) obtained on September 20, 2020.Figure 6
Typical daily production data of three wells: (a) Z201, (b) Z202, and (c) Z203.
(a)(b)(c)Figure6 shows that the current production of the three wells is stable with no abnormalities. The average pressure difference between tubing and casing during the shut-in period for wells Z201 and Z202 was 1.09 and 1.25 MPa, respectively, and there was fluid accumulation in the wellbore. When the shut-in period of Z203 commenced, the casing pressure almost equated to the tube pressure, and no fluid accumulated in the wellbore. Because of the influence of wellbore fluid accumulation, the gas production from wells Z201 (3.34 × 104 m3/d) and Z202 (1.83 × 104 m3/d) in the afterflow stage was lower than for well Z203 (4.15 × 104 m3/d).


### 5.2. Data Smoothing

Presently, the wellhead tube and case pressure data derived by field tracking tests are typically tested once every 30 s. However, based on the degree of accuracy and stability of the test instrument, certain fluctuations exist in the data. Although it is not obvious from the numerical values of the tube and case pressures, the fluctuation phenomenon can clearly be observed by converting it into the pressure derivative. Figure7 shows the tested tubing and casing pressure curve obtained from well Z203. Figure 8 shows the pressure derivative curve.Figure 7
Production curve of well Z203.Figure 8
Pressure derivative curve of (a) the tubing pressure and (b) the casing pressure.
(a)(b)The pressure data in Figure7 show that the pressure curve rises continuously and smoothly in the shut-in stage. However, the pressure derivative curve of Figure 8 shows the pressure derivative as a series of points, indicating that the tested pressure points are not continuous and smooth. Thus, it is necessary to use the following function to fit and smooth the tested pressure data.This function is an empirical formula based on the characteristics of tubing and casing pressure. Test data were applied to fitn1, n2, n3, and n4 to calculate the smoothed data. Figure 9 shows the smoothed pressure curve.(10)y=n1+n2x+n3x0.5+n4exp−x.Figure 9
Comparison of pressure curves before and after smoothing for well Z203.Table2 lists the parameters used for smoothing for well Z203 and correlation coefficient (R2) before and after smoothing. The correlation coefficient was close to 1, and the degree of fit was very good.Table 2
Parameters and determination coefficients used in the data processing.
n1n2n3n4R2Tubing pressure8.2254−0.00040.05470.20460.9972Casing pressure10.01280.00010.00570.04710.9991Figure10 shows a comparison of the pressure derivative curve related to pressure before and after smoothing of well Z203. The pressure derivative was transformed from scattered points with specific rules into a smooth curve.Figure 10
Comparison of the pressure derivative curves before and after smoothing for well Z203.
(a)(b)


#### 5.2.1. Measuring the Formation Inflow Performance Relationship Curve

The curve of shale plunger gas lift wells in the Changning block was calculated using the IPR curve method for plunger lift wells proposed in this paper. Taking wells Z201, Z202, and Z203 as examples, Figure7 shows the single-cycle production data of well Z203, and Figure 11 shows the single-cycle production data of wells Z201 and Z202. All three wells normally produced shale gas plunger lift wells.Figure 11
Single-cycle production data of wells Z201 and Z202: (a) the production curve for well Z201 and (b) the production curve for well Z202.
(a)(b)Figure12 shows the inflow performance curve calculated after data preprocessing of the three wells.Figure 12
Inflow performance curve of wells Z201, Z202, and Z203.The measured curves of the three wells show that the inflow performance curve of the plunger gas lift well was not a conventional convex curve. Neither an exponential productivity equation nor a binomial productivity equation fitted the IPR curve of the plunger gas lift well. Thus, a productivity equation suitable for a plunger gas lift well is proposed in this paper.


#### 5.2.2. Productivity Equation considering the Change in Water Saturation

There are two obvious production differences between plunger lift wells and conventional wells. (1) Frequently, wells are switched on and off, and the production time of a single cycle is short. (2) In a single production cycle, the water saturation near the well changes significantly. Researchers have not previously studied the productivity equation of plunger gas lift wells on the basis of their production characteristics.Currently, when establishing a plunger motion model, researchers typically use the exponential productivity equation to describe the formation inflow; however, the exponential productivity equation is obtained on the basis of the single-phase gas flow law of stable production. For water-producing gas wells, the influence of water saturation must be introduced.In a conventional plunger motion model of plunger lift wells, existing research typically applied a single-phase exponential productivity equation directly to describe the gas production of a gas well [35].(11)qsc=CpR¯2−pwf2n.For plunger lift wells, in this paper, the water saturation change factor (γ) is introduced to express the influence of water saturation near the well zone on formation inflow. Because water saturation is related to bottom hole flowing pressure, γ can be expressed as follows:(12)γ=e1/c1+c2lnpwf+c3pwfc4.Combined with exponential productivity equation andγ, the formation IPR curve of a water-producing gas well is as follows:(13)qsc=Cpr¯2−pwf2nγ=Cpr¯2−pwf2ne1/c1+c2lnpwf+c3pwfc4.The above formula is a two-phase correction of the conventional exponential equation including the conventional exponential productivity equation, and the correction term ofγ, in which the unknown parameters can be obtained by fitting the actual curve. This paper uses 1stOpt software to calculate, input the formula and original data, and select the genetic algorithm to automatically fit the parameters.


#### 5.2.3. A Typical Well Productivity Curve

Figure13 shows the comparison between the fitting results of the productivity equation considering the change factors of water saturation proposed in this paper and the measured data. Table 3 shows the relevant applied parameters.Figure 13
Inflow performance curves of wells Z201, Z202, and Z203.Table 3
Parameters and determination coefficients that were used in the data processing.
Cnc1c2c3c4R2Z2018.5240.6701.281−0.5050.945−0.0840.9982Z2022.6680.9611.125−0.4660.098−0.0980.9991Z2031.7400.885−1.0530.411−1.2070.0450.9994The formation inflow performance curve of the plunger gas lift well differed from the inflow performance curve calculated using the conventional exponential productivity equation. This was because, compared with conventional gas wells with long-term stable production, a plunger lift well needs periodic well-switching, which makes the single production time of the plunger lift well shorter. The well control area in the single cycle of the plunger lift well is near the bottom of the well, and the formation inflow gas is mainly affected by water saturation in the near-well zone. With a change in gas well-bottom hole pressure, the water saturation of the near-wellbore zone changes significantly, as does the gas permeability and gas flow into the formation. Consequently, the calculated IPR curve showed that gas production changed rapidly with a change in bottom hole pressure, and the productivity curve indicated a concave characteristic.Figure14 shows the change curve related to the water saturation changes in wells Z201, Z202, and Z203 and the bottom hole pressure. When the bottom hole flowing pressure was small, the water saturation near the well zone was also small, and the water saturation changed quicker. When the bottom hole flowing pressure was large, the near-well zone was almost entirely occupied by water, and the water saturation changed slowly.Figure 14
Change curve for the water saturation change factor with bottom hole flowing pressure.


## 5.1. Basic Situation of Typical Wells

Plunger lift wells account for 57.41% of the total drainage gas production process wells in the Changning block. In daily production management, the intelligent management platform of the gas production process (developed by the Engineering Technology Research Institute of the PetroChina Southwest Oil and Gas Field Company) is used for automatic production data acquisition, maintaining good switching control, and obtaining the daily production data and basic well information related to plunger gas lift wells in real time. Wells Z201, Z202, and Z203 were adopted as examples, and Table1 lists the basic information of the three wells.Table 1
Basic information of typical plunger gas lift wells.
Well numberWell typeTool typeWell deep (m)Plunger length (m)Outer diameter of the plunger (mm)Inner diameter of the tubing (mm)Outer diameter of the casing (mm)A target measure depth (m)A target vertical depth (m)B target measure depth (m)B target vertical depth (m)Z201Down-dipFishbone plunger4874.927048.250.6139.73867362551673516Z202Up-dipFishbone plunger516727048.250.6138.74275379351673929Z203Down-dipPop-up plunger30433805250.6139.73867362551673515Figure6 shows the daily production data of the three wells (Z201, Z202, and Z203) obtained on September 20, 2020.Figure 6
Typical daily production data of three wells: (a) Z201, (b) Z202, and (c) Z203.
(a)(b)(c)Figure6 shows that the current production of the three wells is stable with no abnormalities. The average pressure difference between tubing and casing during the shut-in period for wells Z201 and Z202 was 1.09 and 1.25 MPa, respectively, and there was fluid accumulation in the wellbore. When the shut-in period of Z203 commenced, the casing pressure almost equated to the tube pressure, and no fluid accumulated in the wellbore. Because of the influence of wellbore fluid accumulation, the gas production from wells Z201 (3.34 × 104 m3/d) and Z202 (1.83 × 104 m3/d) in the afterflow stage was lower than for well Z203 (4.15 × 104 m3/d).


## 5.2. Data Smoothing

Presently, the wellhead tube and case pressure data derived by field tracking tests are typically tested once every 30 s. However, based on the degree of accuracy and stability of the test instrument, certain fluctuations exist in the data. Although it is not obvious from the numerical values of the tube and case pressures, the fluctuation phenomenon can clearly be observed by converting it into the pressure derivative. Figure7 shows the tested tubing and casing pressure curve obtained from well Z203. Figure 8 shows the pressure derivative curve.Figure 7
Production curve of well Z203.Figure 8
Pressure derivative curve of (a) the tubing pressure and (b) the casing pressure.
(a)(b)The pressure data in Figure7 show that the pressure curve rises continuously and smoothly in the shut-in stage. However, the pressure derivative curve of Figure 8 shows the pressure derivative as a series of points, indicating that the tested pressure points are not continuous and smooth. Thus, it is necessary to use the following function to fit and smooth the tested pressure data.This function is an empirical formula based on the characteristics of tubing and casing pressure. Test data were applied to fitn1, n2, n3, and n4 to calculate the smoothed data. Figure 9 shows the smoothed pressure curve.(10)y=n1+n2x+n3x0.5+n4exp−x.Figure 9
Comparison of pressure curves before and after smoothing for well Z203.Table2 lists the parameters used for smoothing for well Z203 and correlation coefficient (R2) before and after smoothing. The correlation coefficient was close to 1, and the degree of fit was very good.Table 2
Parameters and determination coefficients used in the data processing.
n1n2n3n4R2Tubing pressure8.2254−0.00040.05470.20460.9972Casing pressure10.01280.00010.00570.04710.9991Figure10 shows a comparison of the pressure derivative curve related to pressure before and after smoothing of well Z203. The pressure derivative was transformed from scattered points with specific rules into a smooth curve.Figure 10
Comparison of the pressure derivative curves before and after smoothing for well Z203.
(a)(b)


### 5.2.1. Measuring the Formation Inflow Performance Relationship Curve

The curve of shale plunger gas lift wells in the Changning block was calculated using the IPR curve method for plunger lift wells proposed in this paper. Taking wells Z201, Z202, and Z203 as examples, Figure7 shows the single-cycle production data of well Z203, and Figure 11 shows the single-cycle production data of wells Z201 and Z202. All three wells normally produced shale gas plunger lift wells.Figure 11
Single-cycle production data of wells Z201 and Z202: (a) the production curve for well Z201 and (b) the production curve for well Z202.
(a)(b)Figure12 shows the inflow performance curve calculated after data preprocessing of the three wells.Figure 12
Inflow performance curve of wells Z201, Z202, and Z203.The measured curves of the three wells show that the inflow performance curve of the plunger gas lift well was not a conventional convex curve. Neither an exponential productivity equation nor a binomial productivity equation fitted the IPR curve of the plunger gas lift well. Thus, a productivity equation suitable for a plunger gas lift well is proposed in this paper.


### 5.2.2. Productivity Equation considering the Change in Water Saturation

There are two obvious production differences between plunger lift wells and conventional wells. (1) Frequently, wells are switched on and off, and the production time of a single cycle is short. (2) In a single production cycle, the water saturation near the well changes significantly. Researchers have not previously studied the productivity equation of plunger gas lift wells on the basis of their production characteristics.Currently, when establishing a plunger motion model, researchers typically use the exponential productivity equation to describe the formation inflow; however, the exponential productivity equation is obtained on the basis of the single-phase gas flow law of stable production. For water-producing gas wells, the influence of water saturation must be introduced.In a conventional plunger motion model of plunger lift wells, existing research typically applied a single-phase exponential productivity equation directly to describe the gas production of a gas well [35].(11)qsc=CpR¯2−pwf2n.For plunger lift wells, in this paper, the water saturation change factor (γ) is introduced to express the influence of water saturation near the well zone on formation inflow. Because water saturation is related to bottom hole flowing pressure, γ can be expressed as follows:(12)γ=e1/c1+c2lnpwf+c3pwfc4.Combined with exponential productivity equation andγ, the formation IPR curve of a water-producing gas well is as follows:(13)qsc=Cpr¯2−pwf2nγ=Cpr¯2−pwf2ne1/c1+c2lnpwf+c3pwfc4.The above formula is a two-phase correction of the conventional exponential equation including the conventional exponential productivity equation, and the correction term ofγ, in which the unknown parameters can be obtained by fitting the actual curve. This paper uses 1stOpt software to calculate, input the formula and original data, and select the genetic algorithm to automatically fit the parameters.


### 5.2.3. A Typical Well Productivity Curve

Figure13 shows the comparison between the fitting results of the productivity equation considering the change factors of water saturation proposed in this paper and the measured data. Table 3 shows the relevant applied parameters.Figure 13
Inflow performance curves of wells Z201, Z202, and Z203.Table 3
Parameters and determination coefficients that were used in the data processing.
Cnc1c2c3c4R2Z2018.5240.6701.281−0.5050.945−0.0840.9982Z2022.6680.9611.125−0.4660.098−0.0980.9991Z2031.7400.885−1.0530.411−1.2070.0450.9994The formation inflow performance curve of the plunger gas lift well differed from the inflow performance curve calculated using the conventional exponential productivity equation. This was because, compared with conventional gas wells with long-term stable production, a plunger lift well needs periodic well-switching, which makes the single production time of the plunger lift well shorter. The well control area in the single cycle of the plunger lift well is near the bottom of the well, and the formation inflow gas is mainly affected by water saturation in the near-well zone. With a change in gas well-bottom hole pressure, the water saturation of the near-wellbore zone changes significantly, as does the gas permeability and gas flow into the formation. Consequently, the calculated IPR curve showed that gas production changed rapidly with a change in bottom hole pressure, and the productivity curve indicated a concave characteristic.Figure14 shows the change curve related to the water saturation changes in wells Z201, Z202, and Z203 and the bottom hole pressure. When the bottom hole flowing pressure was small, the water saturation near the well zone was also small, and the water saturation changed quicker. When the bottom hole flowing pressure was large, the near-well zone was almost entirely occupied by water, and the water saturation changed slowly.Figure 14
Change curve for the water saturation change factor with bottom hole flowing pressure.


## 5.2.1. Measuring the Formation Inflow Performance Relationship Curve

The curve of shale plunger gas lift wells in the Changning block was calculated using the IPR curve method for plunger lift wells proposed in this paper. Taking wells Z201, Z202, and Z203 as examples, Figure7 shows the single-cycle production data of well Z203, and Figure 11 shows the single-cycle production data of wells Z201 and Z202. All three wells normally produced shale gas plunger lift wells.Figure 11
Single-cycle production data of wells Z201 and Z202: (a) the production curve for well Z201 and (b) the production curve for well Z202.
(a)(b)Figure12 shows the inflow performance curve calculated after data preprocessing of the three wells.Figure 12
Inflow performance curve of wells Z201, Z202, and Z203.The measured curves of the three wells show that the inflow performance curve of the plunger gas lift well was not a conventional convex curve. Neither an exponential productivity equation nor a binomial productivity equation fitted the IPR curve of the plunger gas lift well. Thus, a productivity equation suitable for a plunger gas lift well is proposed in this paper.


## 5.2.2. Productivity Equation considering the Change in Water Saturation

There are two obvious production differences between plunger lift wells and conventional wells. (1) Frequently, wells are switched on and off, and the production time of a single cycle is short. (2) In a single production cycle, the water saturation near the well changes significantly. Researchers have not previously studied the productivity equation of plunger gas lift wells on the basis of their production characteristics.Currently, when establishing a plunger motion model, researchers typically use the exponential productivity equation to describe the formation inflow; however, the exponential productivity equation is obtained on the basis of the single-phase gas flow law of stable production. For water-producing gas wells, the influence of water saturation must be introduced.In a conventional plunger motion model of plunger lift wells, existing research typically applied a single-phase exponential productivity equation directly to describe the gas production of a gas well [35].(11)qsc=CpR¯2−pwf2n.For plunger lift wells, in this paper, the water saturation change factor (γ) is introduced to express the influence of water saturation near the well zone on formation inflow. Because water saturation is related to bottom hole flowing pressure, γ can be expressed as follows:(12)γ=e1/c1+c2lnpwf+c3pwfc4.Combined with exponential productivity equation andγ, the formation IPR curve of a water-producing gas well is as follows:(13)qsc=Cpr¯2−pwf2nγ=Cpr¯2−pwf2ne1/c1+c2lnpwf+c3pwfc4.The above formula is a two-phase correction of the conventional exponential equation including the conventional exponential productivity equation, and the correction term ofγ, in which the unknown parameters can be obtained by fitting the actual curve. This paper uses 1stOpt software to calculate, input the formula and original data, and select the genetic algorithm to automatically fit the parameters.


## 5.2.3. A Typical Well Productivity Curve

Figure13 shows the comparison between the fitting results of the productivity equation considering the change factors of water saturation proposed in this paper and the measured data. Table 3 shows the relevant applied parameters.Figure 13
Inflow performance curves of wells Z201, Z202, and Z203.Table 3
Parameters and determination coefficients that were used in the data processing.
Cnc1c2c3c4R2Z2018.5240.6701.281−0.5050.945−0.0840.9982Z2022.6680.9611.125−0.4660.098−0.0980.9991Z2031.7400.885−1.0530.411−1.2070.0450.9994The formation inflow performance curve of the plunger gas lift well differed from the inflow performance curve calculated using the conventional exponential productivity equation. This was because, compared with conventional gas wells with long-term stable production, a plunger lift well needs periodic well-switching, which makes the single production time of the plunger lift well shorter. The well control area in the single cycle of the plunger lift well is near the bottom of the well, and the formation inflow gas is mainly affected by water saturation in the near-well zone. With a change in gas well-bottom hole pressure, the water saturation of the near-wellbore zone changes significantly, as does the gas permeability and gas flow into the formation. Consequently, the calculated IPR curve showed that gas production changed rapidly with a change in bottom hole pressure, and the productivity curve indicated a concave characteristic.Figure14 shows the change curve related to the water saturation changes in wells Z201, Z202, and Z203 and the bottom hole pressure. When the bottom hole flowing pressure was small, the water saturation near the well zone was also small, and the water saturation changed quicker. When the bottom hole flowing pressure was large, the near-well zone was almost entirely occupied by water, and the water saturation changed slowly.Figure 14
Change curve for the water saturation change factor with bottom hole flowing pressure.


## 6. Conclusions

This paper established a test method for deriving the IPR curve of a plunger lift well, using wellhead tubing and casing pressure data at the well’s shut-in time, focusing on the problem that the conventional inflow performance test method cannot be used to obtain the IPR curve of a single well in a plunger lift well; combined with wellbore gas-liquid distribution, a shale gas horizontal well in the Changning block that applied a plunger gas lift in Sichuan Province was tested, and the following conclusions were drawn:(1)
The inflow performance test method for a plunger lift well established in this paper calculated the formation supply to the wellbore at specific time intervals using wellhead tubing and casing pressure data to determine the inflow performance curve of a plunger lift well. This method did not require the use of permeability, a relative permeability curve, or other parameters. Accordingly, it is easy to use and has broad applicability.(2)
A smoothing method for field-measured tubing and casing pressure data were proposed, focusing on the problem related to the insufficient accuracy of field instruments-measured. After smoothing out abnormal test points, the pressure and its derivative could be depicted as smooth curves.(3)
By conducting a field test, this study found that the concave characteristics of the formation IPR curve in plunger lift wells differed from the convex characteristics of conventional gas wells. This was because, compared with conventional gas wells, the single production time of plunger gas lift wells of periodic switching wells is shorter, and gas wells were mainly affected by the water saturation near the well zone. Water saturation near the well zone changed significantly, resulting in large changes in gas-phase permeability and large changes in formation inflow gas.(4)
On the basis of the IPR curve characteristics of plunger lift wells, the water saturation change factor expressed by bottom hole flowing pressure was introduced to modify the conventional exponential equation; additionally, a productivity equation of plunger gas lift wells that considered the influence of water saturation near the well zone was proposed. The productivity equation proposed in this paper showed a good fit with the measured data, and the averageR2 was 0.9989.



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*Source: 4401128-2022-02-27.xml*