The Prediction of the Valve Opening Required for Slugging Control in Offshore Pipeline Risers Based on Empirical Closures and Valve Characteristics

Abstract

Topside choking is a common way to eliminate severe slugging flow in pipeline riser systems in offshore oil and gas fields. However, a lack of fundamentals in two-phase flow gives rise to difficulty in the model selection of valves and the effective control of the valves. In this study, the prediction of the valve opening required for slugging control based on measurable parameters is investigated experimentally and theoretically. It is found that the resistance coefficient factor of the valve is almost the same for pipeline risers and simple vertical pipes when severe slugging is eliminated. Therefore, fluid parameters can be approximated by the conditions of a simple vertical pipe. The target of control is to achieve dual-frequency fluctuation, and it is quantitatively converted to the pressure drop of the valve. Based on these two empirical enclosures, the valve opening can be worked out by using the gas fraction model and the theoretical model of valve flow resistance. The non-slip model is found to be better than the drift-flux model in the final prediction of the optimal valve opening. An explicit model for the calculation of the optimal resistance factor and the corresponding valve opening is established, making it more convenient to select the valve in the design stage of offshore oil and gas exploitation. The average absolute error of the proposed model is +0.01%, which is smaller than the simulation performed by OLGA 7.0 software (+4.91% before tuning and +0.08% after tuning). A field case with a flexible S-shape riser proves the good applicability of the model (with a deviation smaller than ±2%). The applications of the prediction model in the model selection of the valve and uncertain factors in the operation are also discussed.

1. Introduction

In offshore oil and gas exploitation, construction difficulties and high cost are encountered in the application of single-phase pipelines with underwater gas–liquid separators. Therefore, gas–liquid mixture pipelines are much more common in the transportation of oil and gas from seabed wells to onshore separation equipment [1], and they are the lifeline of offshore energy transportation [2]. In the mixed flow process of gas–liquid, two-phase flow, different flow patterns appear due to the different flow velocity and compressibility of each phase. Different flow patterns sometimes have different effects on pipeline risers and related equipment [3]. A hazardous two-phase flow regime and its control are among the key issues of flow assurance for gas–liquid mixture transporting pipelines in offshore oil and gas fields. The typical appearance of the hazardous two-phase flow regime is severe slugging. This flow regime occurs at small to moderate flow rates as a result of pipeline riser geometry and seabed terrain [4]. Severe slugging is a cyclic flow condition, which is commonly divided into four stages: liquid slug growth, liquid slug production, gas penetration and blowout, and liquid fallback and formation of liquid blockage [5,6,7,8], as shown in Figure 1. This flow regime has occurred in oil fields in the North Sea [9,10], Bohai Sea [11,12], and South China Sea [13]. The hazard of this flow regime lies in many aspects of operation processes, which include kinetic force on fittings and vessels, pressure cycling, control instability, and inadequate phase separation [14,15]. As the gas flow at the riser top is intermittent, gas turbine generators on the offshore platform will be shut down frequently, and the switch to diesel generators will result in greater cost and more pollutants [16]. On the other hand, the blowout stage is the fast outflow of gas and liquid at the riser top, which may not only result in separator overflow [17] but also cause pressure-fluctuation-induced vibration of the platform structure [18]. Therefore, this hazardous flow regime must be eliminated or controlled so as to ensure the operational safety of downstream facilities on the platform.

It is widely agreed that severe slugging cannot be completely avoided through the optimization of pipeline design [19]. So, measures should be taken to control (mitigate) and eliminate the flow regime, which we call slugging control. The most widely applied control method is topside choking [20]. A regulating valve is installed at the connection pipe between the riser top and the separator. By adjusting its opening, the peak value of the transient flow rate during the blowout stage is reduced. Meanwhile, the flow resistance across the valve fluctuates with the alternate passing of gas and liquid slugs. The fluctuation, in turn, enhances the entrance of gas (and suppresses liquid blockage) at the riser bottom as a result of the fluctuating, pressure-driven force.

According to laboratory studies [21,22], dynamic control can achieve better results, but it is less applicable at present, not only due to the difficulties in parameter setting and tuning of the dynamic control model, but also because the dynamic control model works only within a limited interval around the optimal valve opening, while there is no practical prediction method. In conventional studies of slugging control [23], the flow resistance factor is an adjustable parameter in the flow model that is not mapped to the opening. For the conventional proportional–integral (PI) control model, the performance depends on the relation between the process variable and the manipulated variable. If the relation is non-linear, integral control would result in sharp tuning of valve opening and significant overshoot of pressure (which may exceed the design pressure). If integral control is abandoned, the tuning is rather slow, and it would take tens of slugging cycles for elimination, as in the case presented in [13]. If the optimal opening can be estimated, the tuning can be more effective without causing safety problems. Then, the questions become clear—how to select the model of regulating valve, and how to find the optimal opening of the valve. This study focuses on the second issue, which also helps to find the working range of the dynamic model in the first issue.

Manual traversing is an awkward method that can only be applied in a laboratory setting to determine the reference optimal opening. This is because the response of the flow to control is slow in an offshore pipeline with tens of km or more than 100 km. An operator is impatient when spending tens of days seeking the optimal value, and the manual attempt to find the optimal opening was unsuccessful in an offshore oil field in China (hereinafter called Oil Field W) [13]. Based on certain rules, manual traversing can be replaced by an automatic scheme [24,25]. For Oil Field W, it took four days for the absolute change of 7% in valve opening [13]. However, the setting of the manipulated variable and the parameters of the automatic control model require inexpressible experience. For an automatic method without knowledge of pipe flow, the search for the optimal opening value cannot be sped up tremendously. Theoretically, if the flow rate (or daily production) could be metered, and the fluid properties and the factory parameters of the regulating valve were known, then the optimal opening could be determined. Unfortunately, there is no published model within the knowledge of the authors. The closest models are conventional flow regime transition criteria [8,26,27], which use the resistance factor as a known parameter to predict whether the flow regime is severe slugging. In other words, the calculation of the resistance factor is implicit and requires iteration or trials. In these models, the condition for the occurrence of the blowout stage was analyzed and regarded as the criterion. However, the volume of fluids in the riser makes the outflow condition more complex than the assumption of local flow patterns in the models, and the Initiation of a blowout is not always linked to the necessity of activating valve control. As Pedersen et al. [28] pointed out, an unstable flow regime is not always an unstable system. Andreolli et al. [29] attempted to use the discharge coefficient of a choke device, or C_d, to derive a stability criterion. C_d was set based on experience. The relation between the valve opening and flow regime transition is deeply implicit, and the result was qualitative.

There are two major problems to be solved before the prediction model can be established. First, what condition is the optimal condition (i.e., the target of control), and second, what are the values of some macroscopic flow parameters (pressure or pressure drop) at the optimal condition? In published works, whether severe slugging was eliminated or controlled was judged by observing the pressure fluctuation amplitude. If the amplitude was smaller than a given threshold, then severe slugging was regarded as eliminated. However, the threshold was not related to flow appearance or mechanism.

The obstacle In finding the relation between the parameters and the optimal condition lies in the two-phase flow model. Jansen et al. [26], Malekzadeh et al. [8], and Fadairo et al. [27] could predict a choke coefficient for eliminating severe slugging; however, the coefficient is NOT non-dimensional, which can not be mapped to the factory parameters of the regulating valve. Thus, the optimal opening remains unknown. There are two types of mathematical descriptions for gas–liquid two-phase flow in a pipeline riser: one simple model consists of four separated sub-models (corresponding to four stages of severe slugging); another model is based on governing equations of two-fluid flow (like OLGA software, which will be discussed later, and the codes developed by China University of Petroleum; see [30]). However, neither of them can directly solve for the optimal valve opening, and the settings require experience [29]. It is less effective with more effort, as an average-level designer or operator can only tune the diameter and C_d in the default orifice model in OLGA software, while the model cannot describe all valve types, which will be discussed later. A designer or operator has to try a few input opening values before the optimal value can be found, which is estimated to take several days. Moreover, the accuracy of prediction depends on the simulated flow regime or the accuracy of the two-phase flow model itself. If the simulated regime is not severe slugging while the actual condition is, then the opening can never be worked out. Therefore, it is necessary to establish an explicit model to predict the optimal opening with the information in the specification file of the valve and with fewer requirements for experience.

This study combines experiments and theoretical modeling. Choke experiments with different valve openings at different superficial velocities are performed (Section 2). The target of control is determined by the continuity of outflow (Section 3.1). Based on the comparison between the pipeline riser and a simple vertical pipe, the simplification of the two-phase flow model is discussed, and the magnitude of macroscopic flow parameters at the optimal condition is determined (Section 3.2). Then, a prediction model based on factory parameters of the regulating valve is established (Section 3.3). The predicted results are compared with experimental results and those predicted using OLGA software simulation (Section 4.1). Finally, the model is validated by an industrial case, and how to select the model of the regulating valve is briefly discussed (Section 4.2).

2. Experiment

2.1. Setup of Experimental Loop

The sketch of the experimental loop is shown in Figure 2. Although the loop itself is an oil–water–gas three-phase system, only water and air sources are displayed in Figure 2 since oil was not used in the experiments. The test section consists of a horizontal pipeline, a downward inclined pipe with an inclination angle of 5° to horizontal, and a vertical riser. The lengths of the three sections are 114 m, 20.4 m, and 16.3 m, respectively, and the inner diameter of the whole test section is 50 mm. Most of the test section is made of stainless steel, except for a transparent section of 2.75 m in the middle of the riser. The fluids, namely, air and tap water, are metered separately before mixing and entering the pipeline. A gas–liquid separator is installed at the highest position of the experimental facility. The riser top and the entrance of the separator are connected by an electric ball valve, with a nominal diameter of 50 mm. The deviation of the feedback opening from the set opening is ±0.5%. A cylinder with an inner diameter of 0.5 m is installed right below the separator to measure the accumulated liquid outflow, and the outflow rate can therefore be derived. An injection hole is located at the riser elbow, which is 40 cm above the lowest position of the elbow. The pressure and differential pressure transducers that are involved in this study are also marked in Figure 2. For detailed information on transducers and flow meters, see

Table 1. Installation and accuracy of transducers and flow meters.

Transmitter	Manufacturer	Transmitter Model	Range	Accuracy
Orifice Flowmeter (L)	YOKOGAWA	EJA-115	0–1.35 m3·h−1	±1% F.S.
Orifice Flowmeter (M)	YOKOGAWA	EJA-115	0–0.22 m3·h−1	±1% F.S.
Orifice Flowmeter (S)	YOKOGAWA	EJA-115	0–0.027 m3·h−1	±1% F.S.
Electromagnetic Flowmeter (L)	YOKOGAWA	AE204MG	0–42.5 m3·h−1	±0.5% F.S.
Electromagnetic Flowmeter (S)	YOKOGAWA	AE115MG	0–6.36 m3·h−1	±0.5% F.S.
Mass Flowmeter (L)	Siemens	Mass 6000	0–80 kg·h−1	±0.1% F.S.
Mass Flowmeter (S)	Emerson	CMF 010	0–24 kg·h−1	±0.1% F.S.
Pressure (P5/pb)	Micro	MPM480	0–500 kPa	±0.25% F.S.
Differential Pressure (DP5/ΔpR)	Rosemount	3051CD3	−20–200 kPa	±0.15% F.S.
Differential Pressure (DP11/Δpv)	Rosemount	3051CD3	0–240 kPa	±0.15% F.S.
Differential Pressure (DP12)	Rosemount	3051CD3	0–240 kPa	±0.15% F.S.

. More details of the experimental facility are available in [31], which presents the dimensions of the horizontal section.

2.2. Experimental Condition

The experiments were performed at room temperature. The pressure of the water supply was maintained at 1.2 Mpa (absolute) by regulating the inlet valve of the metering section and the valve on the return line. The pressure of the air supply was 0.9 Mpa (absolute).

The experimental work consists of choking experiments for pipeline risers and for a simple vertical pipe. Under the former condition, air was injected at the mixer, and the flow regime in the pipeline risers was severe slugging (Figure 3); under the latter condition, air was injected at the riser bottom, i.e., only the riser was in a state of two-phase flow. The opening of the regulating valve ranged from 15% to 100%. Nine cases of choking experiments for pipeline risers and the corresponding cases for the simple vertical pipe were conducted initially (see [31]), and another eleven cases for pipeline risers were performed to expand the dataset for validation. For all experimental cases, the valve opening was adjusted in a decreasing manner. The superficial velocities of the experiments were as follows:

Liquid superficial velocity, u_SL (m/s): 0.10, 0.25, 0.45, and 0.60;

Gas superficial velocity, u_SG0 (m/s): 0.10, 0.25, 0.45, 0.60, and 1.00.

2.3. Flow Coefficient

The flow coefficient and the type of inherent flow characteristic of the regulating valve were unavailable because the nameplate corroded after more than ten years. Instead, a calibration test using single-phase water was performed, as shown in Figure 4. For the opening between 10% and 35%, the flow coefficient, K_v, showed a log-linear relation with the valve opening, Z. For Z > 35%, K_v varied significantly with Z. A log-linear regression was performed for Z = 10–35%, and the regression line was applied for prediction. For Z > 35%, the K_v values at u_SL = 0.80 m/s were applied.

3. Derivation of Optimal Valve Opening

3.1. Target of Control

Figure 5 shows the trend of pressure and differential pressure with the gradual turning down of the regulating valve. Four stages can be divided. When the opening was still large, cyclic flow cut-off was still observed, as indicated by the cyclic zero value of the pressure drop across the valve, Δp_v. With the decrease in opening, the duration of the flow cut-off reduced and reached zero. With a further decrease in opening, dual-frequency fluctuation appeared. Dual frequency is a condition where the long cycles of pressure fluctuation with large fluctuation still exist but are superimposed by short cycles with small fluctuation. Under this condition, the blowout stage is mitigated and turns into the fast outflow of a series of slugs, and the outflow covers the whole cycle of slugging. Finally, the low-frequency component disappeared, and the flow became stable with small pressure fluctuations. Dual-frequency fluctuation and stable flow were observed as slug flow with variable speed.

The time-averaged pressure at the riser bottom increased monotonically with the decrease in opening. However, the maximum pressure did not show a monotonous relationship with opening. It first increased, then decreased, and finally rose again with the decrease in opening. The flow regime at the time the maximum pressure rose again was dual-frequency fluctuation. Since the minimum value of Δp_v was above zero, the maximum value could be regarded as no greater than twice its average value, as validated by 16 experimental cases listed in

Table 2. Peak value of liquid outflow rate when dual-frequency fluctuation appears.

No.	uSL (m/s)	uSG0 (m/s)	Peak Value of uSL at Outlet (m/s)	Ratio (Peak Over Average)
1	0.10	0.10	0.176	1.76
2	0.10	0.25	0.188	1.88
3	0.10	0.60	0.154	1.54
4	0.10	1.00	0.171	1.71
5	0.25	0.10	0.355	1.42
6	0.25	0.25	0.398	1.59
7	0.25	0.60	0.387	1.55
8	0.25	1.00	0.447	1.79
9	0.45	0.10	0.564	1.25
10	0.45	0.25	0.591	1.31
11	0.45	0.60	0.713	1.58
12	0.45	1.00	0.601	1.34
13	0.60	0.10	0.775	1.29
14	0.60	0.25	0.750	1.25
15	0.60	0.60	0.772	1.29
16	0.60	1.00	0.719	1.20

(DP12 was unavailable for the remaining 4 cases). The outflow rate is smoothed by a 10-second window, as transient values are of no significance. The condition with the ratio of peak to average smaller than 2 was regarded as safe. Therefore, the target of control was set as dual-frequency fluctuation.

3.2. Macroscopic Flow Parameters at Optimal Condition

According to Figure 5, the average pressure under the condition of dual-frequency fluctuation was considered roughly the same as the maximum pressure under the condition of Z = 100%, or p_s + ρ_LgH. This approximation is not precise, as indicated by Figure 6; however, it is still acceptable, which will be discussed in Section 4.1. Then, an empirical closure could be established:

$$p_{\mathrm{b}}=p_{\mathrm{s}}+{\rho }_{\mathrm{L}}gH$$

(1)

In other words,

$$\Delta p_g+\Delta p_{\mathrm{v}}={\rho }_{\mathrm{L}}gH(1-{\alpha }_{\mathrm{R}})+{\rho }_{\mathrm{G}}gH{\alpha }_{\mathrm{R}}+\Delta p_{\mathrm{v}}={\rho }_{\mathrm{L}}gH$$

(2)

Rewrite Equation (2), and we derive

$$\Delta p_{\mathrm{v}}=({\rho }_{\mathrm{L}}-{\rho }_{\mathrm{G}})gH{\alpha }_{\mathrm{R}}$$

(3)

where

$${\alpha }_{\mathrm{R}}={\displaystyle \frac{{\alpha }_{\mathrm{b}}+{\alpha }_{\mathrm{t}}}{2}}$$

(4)

The relation between pressure drop and flow velocity is quadratic, which is expressed by

$$\Delta p_{\mathrm{v}}={\displaystyle \frac{f}{2}}{\rho }_{\mathrm{mix},\mathrm{t}}{u}_{\mathrm{mix},\mathrm{t}}^2={\displaystyle \frac{f}{2}}\left[{\rho }_{\mathrm{L}}(1-{\alpha }_{\mathrm{t}})+{\rho }_{\mathrm{G},\mathrm{t}}{\alpha }_{\mathrm{t}}\right]{\left(u_{\mathrm{SL}}+u_{\mathrm{SG},\mathrm{t}}\right)}^2$$

(5)

The next question is how to calculate the resistant factor, f. The average of f for each case was calculated based on the time averaged superficial velocity upstream of the valve and the time-average Δp_v recorded in the experiment. It is interesting to find that the resistant factor for pipeline risers became approximately the same as that for a simple vertical pipe at the same opening after dual-frequency fluctuation appeared, or f ratio ≈ 1, as shown in Figure 7 and

Table 3. Valve opening values when certain phenomena appear.

No.	uSL (m/s)	uSG0 (m/s)	Set Z1 * %	Set Z2 * %	f Ratio * at min (Z1, Z2)
1	0.10	0.10	17	17	0.90
2	0.10	0.25	18	18	1.03
3	0.10	0.60	18	17	0.90
4	0.25	0.10	22	22	0.98
5	0.25	0.25	22	23	0.92
6	0.25	0.60	23	23	0.99
7	0.45	0.10	28	27	1.06
8	0.45	0.25	28	26	0.95
9	0.45	0.60	26	27	0.86
10	0.25	0.60	23	23	0.99

* Z₁ is the opening value below which the time-averaged resistance factor of the valve is approximately the same for pipeline risers and simple vertical pipes. Z₂ is the opening value below which the flow condition in the riser is dual-frequency fluctuation or stable flow. Deviation within ±2% is acceptable, as explained in Section 4.1. The “f ratio” refers to the time-averaged f for pipeline risers over that for simple vertical pipes.

. In Figure 7, the resistant factor, f, is larger for pipeline risers than for a simple vertical pipe before severe slugging is eliminated. The reasons can be explained as follows: before severe slugging is eliminated, severe blowout will cause large peak values of Δp_v. As flow resistance is approximately proportional to the square of flow velocity, the time-averaged Δp_v is significantly larger for pipeline risers than for a simple vertical pipe. While ρ_mix,t and u_mix,t change little for both pipeline risers and a simple vertical pipe, f is also larger for pipeline risers. With a smaller opening, the blowout becomes less severe, and the time average Δp_v becomes closer to the condition of a simple vertical pipe. Therefore, the deviation of Δp_v or f from those of a simple vertical pipe can act as an indicator of whether severe slugging is eliminated, and it is possible to calculate f based on the time-averaged flow parameters in a simple vertical pipe.

3.3. Optimal Valve Opening

For a simple vertical pipe, the gas fraction, α, is calculated by non-slip model (Equation (6)), although more accurate drift-flux models are available. The deviation between drift-flux models will be discussed in Section 4.1.

$$\alpha ={\displaystyle \frac{u_{\mathrm{SG}}}{u_{\mathrm{SG}}+u_{\mathrm{SL}}}}$$

(6)

Since u_SG is related to local pressure, α_t and Δp_v have to be solved in an iterative manner. After the flow parameters are solved, f can be worked out by Equations (6) and (7). With f calculated, the corresponding opening value, Z, can be determined. The relation between f and Z is connected by the flow coefficient, K_v. The definition of K_v is the flow rate of liquid (usually water, in m³/h) at the resistance of 1 × 10⁵ Pa, i.e.,:

$${\displaystyle \frac{f}{2}}{\rho }_{\mathrm{L}}{\left({\displaystyle \frac{K_{\mathrm{v}}}{3600A}}\right)}^2={10}^5$$

(7)

Rewrite Equation (7), and we derive

$$K_{\mathrm{v}}=900\pi d_{\mathrm{v}}^2\sqrt{{\displaystyle \frac{200000}{f{\rho }_{\mathrm{L}}}}}$$

(8)

Then, Z can be obtained by the relation between K_v and Z.

4. Validation

4.1. Laboratory Results and Discussion

The experimental results, i.e., the optimal valve opening listed in Table 2, were used to validate the model established in Section 3. The flowchart is shown in Figure 8. The final result of the model was predicted by the K_v-Z relation of the valve shown in Figure 3. The predicted optimal opening versus the reference value is plotted in Figure 9. The absolute deviation ranges from −1.73% to +4.91%, with the average of +0.60%. The cases where obvious positive deviation occurred were those with smaller gas fraction, where more pressure was required (the cases above the reference line in Figure 6). Evidently, the reason is that Δp_v was underpredicted. The deviation of most cases was within an acceptable range of ±2% due to the actual property of the actuator of common automatic valves, whose deviation of feedback opening from the set opening can be as large as ±2%. For negative deviation, our recent related study showed that no obvious “overchoking” (increase in pressure from the value at reference opening) was found; for the specific experimental apparatus, the increase was smaller than 10 kPa for most cases [32]. Although a few cases show larger deviation, the predicted opening could still act as the average opening for dynamic choking once the reliability of the frequent action of the valve could be overcome, since the opening interval within which dynamic choking works could be 10–20% [21,22].

The optimal valve opening was also investigated using OLGA 7.0 software, an industry-standard tool for transient simulation of multiphase petroleum production developed by SPT Group. The geometry and nodes for simulation are presented in Figure 10. It should be noted that the default valve model in OLGA, “Orifice”, is unable to describe the specific valve. In the default orifice model, only a fixed orifice diameter and a fixed C_d value are set, and the flow characteristics are also fixed. As a result, different velocities require different C_ds to make the optimal valve opening consistent with experimental results, which is impossible for a single valve. For Cases 1–8 in Table 2, the tuned C_d ranges from 0.12 to 0.35 to obtain the experimentally determined optimal opening, while the default C_d is 0.84 in OLGA 7.0 software, and the recommended C_d by Al-Safran and Kelkar [33] was 0.75. Hence, the flow characteristics of the valve (Figure 4) have to be input manually. Since the software uses C_v, the flow coefficient in British units, the K_v-Z relation is replaced by the C_v-Z relation, where C_v = 1.156 K_v. The C_v-Z relation was input as a “Table” on “Library”, and the “Model” of the valve was set as “Hydrodynamic”. As displayed in Figure 9, the optimal valve opening for all cases was overpredicted. The deviation for most cases was above +2%, which is unacceptable. A detailed check on each case finds that the predicted flow regime for the case (u_SL = 0.60 m/s, u_SG0 = 0.10 m/s) at Z = 100% is stable, with continuous gas outflow at the riser top; the predicted flow regime for the case (u_SL = 0.60 m/s, u_SG0 = 0.25 m/s) at Z = 100% is almost stable. So, for the specific facility, the reason why OLGA software overpredicts the optimal opening lies in the deviation of flow regime prediction.

It is possible to tune the “table” of the C_v-Z relation to obtain better predicted results; see Figure 11. After the “table” is tuned, the average deviation is reduced from +4.91% to +0.08%, and the absolute error of each case is within ±2%. However, in the real application, there are NO test data for tuning the parameter before severe slugging occurs. So, it must follow the factory data or calibration data before its application.

For the above results, the gas fraction was calculated using the non-slip model. In fact, the gas fraction should be smaller because of the drift flow of gas. A widely used drift-flux model [34] was also used to calculate the gas fraction. However, the opening is seriously overpredicted, as shown in Figure 12. Two aspects may lead to these results. One is that the time-averaged gas fraction is larger for pipeline risers than for simple vertical pipes due to severe flow intermittency, as observed in the previous study [31]; so, Δp_v is underpredicted. In addition, smaller gas fractions result in a bigger average mixture density and, hence, a smaller resistant factor and a positive deviation of predicted opening. The other is that the drift flow of gas recedes with the decrease in the valve opening. Direct observation at the transparent section in the riser found that the slug flow at the choking condition was not at a steady velocity. Owing to the pressure fluctuation at the valve induced by the alternate passing of gas and liquid slugs, the velocity rose when gas passed the valve and dropped when liquid passed the valve. In other words, the slug flow “stepped” upwards at the choking condition. With liquid and gas velocities getting closer, the gas fraction should be larger. One more factor in the experimental data processing may also lead to better accuracy of the non-slip model. In this study, the experimental two-phase flow resistant factor was calculated by the mixture velocity, and the average fluid density used to calculate the resistant factor was calculated using the non-slip model. We have no other choice because the actual gas and liquid flow velocities were unmeasurable. We believe this is also an important reason why the non-slip model performs better.

However, even if the theoretically largest gas fraction is substituted into Equation (3), Δp_v is still underpredicted at smaller gas–liquid ratios (the cases above the reference line in Figure 6). The underprediction leads to a large deviation of Z because of the small absolute value of Δp_v. For correction, we suggest that α_R in Equation (3) should be substituted by α_t, and we derive

$$\Delta p_{\mathrm{v}}=({\rho }_{\mathrm{L}}-{\rho }_{\mathrm{G}})gH{\alpha }_{\mathrm{t}}$$

(9)

When Equation (9) is combined with Equations (5), (6), and (8), the positive deviation at a small gas–liquid ratio (GLR) decreases significantly, while the negative deviation at a large GLR shows only a slight increase, as demonstrated in Figure 12. The absolute deviation ranges from −1.93% to +2.87%, with a mean value of +0.01%. Therefore, correction Equation (9) should be applied.

Finally, it should be noted that the proposed model is independent of flow regime determination. The model should only be applied only when necessary and should therefore be used in conjunction with appropriate flow regime transition criteria. These criteria should be selected based on actual control requirements [27].

4.2. Field Case

Oil Field W consists of one FPSO and five wellhead platforms [13]. The pipelines connecting platform W8-3A to FPSO via platform W14-3A experienced severe slugging issues [35], with the flowpath configuration illustrated in Figure 13. Initially, operators tried manual tuning of the regulating valve with the “equal percentage” characteristic, and set the opening as 27%. Following implementation of an automatic model [24], the value opening was gradually increased over four days and eventually stabilized at 34% [13], which was determined to be the optimal setting at Z = 34%. The production data from this period were used to validate the current model. The known conditions, including pipeline geometry and measurable parameters, are summarized in

Table 4. Known conditions in Oil Field W used for validation.

Parameter	Value	Unit
H	138.9	m
dR	250.9	mm
ps,avg	560	kPa(a)
Twellhead	329.25	K
Tambient	303.15	K
QL	1985	m3/d
QG0	102,705	m3/d
ρL	850.7	kg/m3
ρG0	1.179	kg/m3
Δpv,avg	566	kPa
σ	0.0691	N·m
dv	250	mm
Cv,max	1000	gal·min−1·psi−1
valve flow characteristic	equal percentage	-

. Based on experimental work by Ye et al. [36], the flow pattern of the oil–water mixture was considered to be like an oil-in-water emulsion; consequently, the surface tension was assumed to be equivalent to that of water. The average temperature of the wellhead and ambient was taken as the fluid temperature. The rangeability of the valve, R, which is the ratio of K_v,max (or C_v,max) over K_v,min (or C_v,min), was not recorded on the nameplate; and it was assumed to be 50, which is a common value. Then, the ideal K_v-Z relation of this specific valve was obtained according to the definition of equal percentage characteristic:

$$Z={\displaystyle \frac{100}{\lg R}}\lg {\displaystyle \frac{K_{\mathrm{v}}}{K_{\mathrm{v},\min }}}=58.859191\lg {\displaystyle \frac{1.156K_{\mathrm{v}}}{C_{\mathrm{v},\min }}}$$

(10)

The diameter of the regulating valve is different from that of riser. Therefore, the target resistant factor was calculated based on the superficial velocities at the valve location, with the results presented in

Table 5. Calculated results for the case of Oil Field W.

Parameter	Value	Unit
αt (αv)	0.7965
αb	0.7673
αR	0.7819
Δpv	887.87	kPa
uSL,v	0.465	m/s
uSG,v	1.819	m/s
ρmix,v	185.52	kg/m3
f	1810
Kv	63.69	m3·h−1·(105 Pa)−1
Z	33.06% (Equation (9))

. Due to the flow characteristic of “equal percentage”, a large deviation of Δp_v does not translate to a significant deviation in Z. The predicted optimal opening of 33.06% shows satisfactory agreement with the reference value, falling within an absolute deviation of ±2%, which demonstrates both the accuracy of the model and its applicability to a flexible S-Shape riser. The case is of a large GLR, and Equation (3) gives a similar predicted opening of 33.31%. However, if Shi et al.’s model [34] is applied, Z is severely overpredicted (51.07%).

Under actual operating conditions, the flow characteristics may deviate from the factory specifications, and the calibrated rangeability often differs from ideal conditions. For instance, Chinese-manufactured values with equal percentage characteristics typically exhibit rangeability values between 30 and 50. As defined in the IEC standard 60534-2-4:2009 [37], the calibrated rangeability value is defined as the K_v at Z = 100% over K_v at Z = 5%. However, the same standard strictly limits deviations from ideal flow characteristics (log-linear relation for equal percentage characteristics) within the operating range of Z = 10–80%. So, it is still reasonable to adopt the ideal flow characteristics.

Table 6. Calculated results of Equation (9) for the case of Oil Field W concerning the deviation of valve flow characteristics.

Rangeability	Definition of Rangeability	Predicted Z/%
50	Kv(Z = 100%)/Kv(Z = 0%)	33.06%
40	Kv(Z = 100%)/Kv(Z = 0%)	29.01%
30	Kv(Z = 100%)/Kv(Z = 0%)	23.00%
50	Kv(Z = 100%)/Kv(Z = 5%)	36.40%
40	Kv(Z = 100%)/Kv(Z = 5%)	32.55%
30	Kv(Z = 100%)/Kv(Z = 5%)	26.85%

shows the results concerning the above factors.

4.3. Discussion on Application

The proposed model provides valuable support for regulating valve selection. The selection process should begin by calculating the minimum K_v,min or C_v,min at the lowest estimated production. The corresponding opening value should be maintained at 10% or greater; otherwise, the operating condition may exceed the calibration range. Z = 10% is the second smallest calibrated opening suggested by IEC Standard 60534-2-4:2009 [37]. The smallest is Z = 5%, but the deviation of ±2% of the feedback opening should also be considered. Although the calculated K_v,min or C_v,min is independent of d_v, the actual K_v,min or C_v,min is dependent on d_v due to the structure and manufacturing technique of the valve, or more exactly, its closure member. Generally, K_v,min or C_v,min is larger for bigger valves. Then, we choose an initial d_v of the rangeability, R, and the maximum K_v,max or C_v,max can be determined. If this value is smaller than K_v or C_v corresponding to the largest production rate within the occurrence of severe slugging, a valve with larger R must be selected; if not possible, another alternate valve with larger K_v,max has to be laid in a parallel branch pipe. According to the discussion above, d_v, R, and K_v,max are constrained by each other. The selection of the valve may also be subjected to other restrictions like the operation of the gas–liquid separator or slug catcher and the dead band, as well as the deviation of the feedback value from the set opening of the valve. Hence, the optimization model for valve selection is worthy of study in future work.

Based on the experiments and field data, we believe that the model works within the pressure range (separator pressure) from atmospheric to 1 MPa. We also consider that the pressure range can be extended. For bigger p_s, the slugging itself is less severe, as indicated by the reduced velocity range of occurrence on the flow regime map [38]. With a less severe blowout, the difference of average Δp_v between pipeline risers and a simple vertical pipe is smaller, resulting in a bigger optimal opening, which is consistent with the experimental observation [38]. The temperature range of common fields is several dozen degrees centigrade, which does not differ much from experimental conditions and should be within the application range. As for the gas–liquid ratio, its application range is related to the actual necessity of control, as mentioned in Section 4.1.

In the field configuration, solid particles in the products may attach to the valve core, reducing the actual cross-sectional area of the flow channel in the valve, and the actual optimal opening should be larger as a result of the smaller actual K_v. However, the deviation will not shake the fundamentals of the model, which is that the two-phase flow resistance factor is the same for pipeline risers and simple vertical pipes. As for the valve type, it is not a direct parameter. Each type should provide the characteristics of the K_v-Z relation. Once K_v-Z is determined, the influence of valve type only lies in the aspect of the precision of travel control, which is out of the scope of this study.

Another uncertainty of application lies in the complex pipe geometry from the riser top to the separator. On the deck of an FPSO, the pipe from the single point mooring to the separator includes a series of upward, horizontal, and downward sections and may exhibit a concave geometry. According to the OLGA simulation, the optimal opening varies with the valve position on the pipe. If the valve is installed on the concave part of the pipe, the actual optimal opening should be smaller than the predicted result. Future work can focus on the modification of the model under these conditions.

5. Conclusions

In this study, a prediction model of valve opening required for slugging control in offshore pipeline risers is proposed. The model is based on macroscopic phenomena of slugging elimination and the flow characteristics of the regulating valve. Two empirical closures are added based on experimental observation. The proposed model is validated by laboratory and field cases. The major findings and contributions of the study are as follows:

• The target of slugging control can be set as the occurrence of dual-frequency fluctuation. At the time, the sum of the time-averaged total pressure drop along the riser and that across the valve is approximately the same as the gravity pressure drop of single-phase water in the riser without choking. So, the conservation of total pressure drop can be assumed.
• The resistance factor of the valve is approximately the same as slug flow in a simple vertical pipe at the time severe slugging is eliminated, indicating the feasibility of calculating the resistance factor by a two-phase flow model in a simple vertical pipe.
• Because of severe flow intermittency in pipeline risers and weaker gas drift flow, and in order to be consistent with the processing of measured data, the non-slip assumption is recommended in the calculation of the gas fraction in the riser.
• The average absolute error of the proposed model is +0.01%, which is smaller than the simulation performed by OLGA software (+4.91% before tuning and +0.08% after tuning). The validation of the field case also proves the good applicability of the model (with deviation smaller than ±2%).
• Future work can focus on the application of the model to the valve selection at the design stage and the modification of the model for different valve installation positions on a real platform at the operation stage.