Severe slug flow-induced nonlinear dynamic behavior of a flexible catenary riser Free

Flexible risers are widely employed in offshore oil and gas transportation, particularly in deep waters where the risers are typically arranged in free-hanging catenary configuration due to the simplest form and the least expensive.¹ The dynamic response of such catenary flexible risers is usually excited by either the external current or the internal oil–gas mixture flow or both,² as a result of the fluid–structure interaction. In the past few decades, considerable knowledge of the vortex-induced vibration (VIV) caused by the flow past slender cylinders has been gained, including the space–time varying dynamic behavior, spatial mode competition, temporal mode switching, standing-traveling wave propagation, mode transition, and vortex shedding features.^3–8 In contrast, the reported literature regarding the internal gas–liquid two-phase flow-induced vibration (FIV) is quite sparse for curved flexible pipelines. Although the underlying physics of single-phase FIV has been demonstrated as the chaotic effect excited by a sufficiently large flow velocity,⁹ the mechanism of two-phase FIV is fairly different due to the oscillating flow characteristics. Among the variety of gas–liquid flow regimes, slug flow consisting of hydrodynamic slug flow and severe slug flow (SS) is a typical and problematic one because of the space–time varying fraction and density and the fluctuation of pressure.^10–12 Theoretical and numerical investigations commonly assumed that the hydrodynamic slug flow is stable with a constant slug length and migration velocity.^13–16 In practice, however, even hydrodynamic slug flow is conveyed unstably with variable unit length.¹⁷ It has been experimentally confirmed that a variation of slug length and frequency contributes to a mode-switching phenomenon over time for the FIV of a flexible catenary riser.¹⁸ In comparison with the hydrodynamic slug flow, severe slug flow (SS) is characterized by the formation and cyclical production of long liquid slug and fast gas blowdown with typical consequences of the flooding of downstream pipeline and production facilities and an overall decrease in system productivity.^19,20 Severe slug flow is thus an undesirable flow-assurance issue.

Since the first observation of severe slugging by Yocum in 1973,²¹ a large amount of research has been conducted on the formation and characteristics of severe slug flow in stationary pipes. Schmidt et al.^22–24 investigated the air–kerosene mixture in an inclined-pipeline–vertical-riser system both experimentally and theoretically. They defined the severe slugging region I (SSI) and severe slugging region II (SSII) and divided the process into four stages: liquid fall back, slug formation (SF), slug movement into separator, and gas blowout (GB). The liquid slug length in SSI could be extended as long as the riser height, while it is less than a riser height in SSII due to the gas penetration. Luo et al.²⁵ experimentally examined the severe slugging flow characteristics in a wider range of flow velocities and inclined angles. Apart from SSI and SSII, another kind of severe slugging termed SSIII was identified with shorter aerated liquid slugs. Moreover, the severe slugging was also observed when the inclination angle of upstream straight pipeline is 0°, which is demonstrated by Wang and Guo²⁶ and Malekzadeh et al.²⁷ Wang and Guo²⁶ attributed this phenomenon to the influence of liquid viscosity. To highlight the difference of severe slugging, Malekzadeh et al.^27–29 divided SSI into five stages: (1) blockage of the riser base, (2) slug growth, (3) liquid production, (4) fast liquid production, and (5) gas blowdown. The liquid production stage is absent in SSII in spite of the other four stages same as SSI. SSIII presents the transient slugs, aerated slug growth, and fast aerated liquid production stages before the gas blowdown one. In addition, Malekzadeh et al.²⁷ illustrated a dual-frequency severe slugging with the higher frequency related to the classical severe slugging cycle and the lower one associated with the gradual cyclic transition of the system between the two metastable states. Based on the reported literature, the major characteristics of severe slugging in vertical riser can be drawn as follows.^10,11,19,30 For SSI, the length of the pure liquid slug is equal to or even more than a riser height and a liquid slug production stage exists before the blowout stage. For SSII, the length of the liquid slug is shorter than that of SSI with the liquid slug slightly aerated. The liquid slug containing more gas bubbles is the shortest in SSIII. As the submarine risers are usually deployed in a catenary configuration, Wordsworth et al.³¹ experimentally investigated the behavior of severe slugging in the inclined-pipeline–catenary-riser system. It was reported that the liquid slug in SSI is greater or equal to the hydrostatic head of the riser while the liquid slug in SSII is shorter than one riser length with intermittent gas penetration. Continuous gas penetration is observed at the riser bottom in SSIII. These flow characteristics are validated numerically and theoretically by Balino et al.^32,33 and Ehinmowo et al.³⁴ 

The aforementioned literature mainly focused on the hydrodynamic characteristics of the gas–liquid flow in risers with relatively large diameters (d ≥ 0.054 m) without taking the surface tension into account. The Eötvös number (Eo = gr2Δρ/cσ, where g is the acceleration of gravity, r is the riser radius, Δρ is the density difference of the internal flow, σ is the surface tension coefficient, c is a constant, and c = 1 for gas–liquid flow³⁵) was employed by Bretherton³⁵ and Liu et al.^36,37 to identify the capillary effect. Bretherton³⁵ reported that the critical value for the transition from macro- to micro-scale riser is about Eo = 0.84. Liu et al.³⁶ pointed out that the critical value is around 1 and the interfacial tension dominated the internal flow when Eo < 1. Barnea et al.³⁸ experimentally investigated the flow pattern transition in small diameter tubes (d = 4–12 mm) and found the deviation in flow map increases with decreasing the riser diameter.³⁹ Chinnov and Kabov⁴⁰ identifies the flow characteristics in risers of different diameters. It was reported that the surface tension does not affect the internal two-phase flow pattern when the inner diameter is larger than 12 mm, and a capillary force emerges when the inner diameter ranges from 5 to 12 mm and the capillary effect has a significant impact on the identification of flow patterns when the inner diameter is less than 5 mm. A capillary number was proposed by them to identify the effect of riser scale,

where ξ denotes the slope angle with respect to the vertical. When d > 5l_σ, the riser is called large-scale riser with the absence of capillary forces. The gravity-capillary risers (0.5l_σ < d < 5l_σ) and capillary-gravity risers (0.1l_σ < d < 0.5l_σ) are impacted by both capillary and gravitational forces with the former one being dominated by the gravitational force and the latter one being dominated by the capillary force. When d < 0.1l_σ, the influence of gravity is tiny. It demonstrated that the flow map in small-scale tube is different from that in large-scale one.^41–47 

A couple of questions arise as to whether the flow map in an oscillating small-scale riser is significantly altered and how about the associated dynamic response of such a curved flexible riser is especially when a severe slug flow occurs. The limited literature regarding the severe slug flow-induced vibration (SSIV) so far is about the dynamic response of rigid straight riser that presents negligible amplitude.^48,49

In this Letter, we experimentally investigate the SSIV of a flexible catenary riser with the aim of uncovering the underlying physics of severe slug flow-flexible pipe interaction and the relationship between the spatial-temporal features of dynamic response and the process of severe slug flow.

A dimensional analysis is conducted prior to the experiment setup. The maximum response amplitude of the flexible riser is assumed to be related to the riser diameter d, gravitational acceleration g, liquid surficial velocity v_sl, gas surficial velocity v_sg, fluid pressure p, gas density ρ_g, liquid density ρ_l, gas viscosity μ_g, and liquid viscosity μ_l. Taking d, v_sl, and ρ_l as three elementary quantities, eight normalized parameters are obtained via Buckingham's Pi theorem, including A_z,max/d, Fr_l = v_sl²/gd, v_sg/v_sl, We_l = ρ_ldv_sl²/σ, Eu_l = p/ρ_lv_sl2, ρ_g/ρ_l, Re_g = ρ_gdv_sg/μ_g, and Re_l = ρ_ldv_sl/μ_l. The responsibility of a riser in practice is lifting oil and gas mixture from seabed wellhead to water-surface platform. The gravitational force is the main impact force during the lifting process and hence the Froude number (Fr_l) is selected as the dimensionless similarity number. The model parameters listed in Table I are calculated according to the operating parameters of a riser in field (length of 40 m, inner diameter of 203 mm, and mixture transporting velocity ranges from 1.3 m/s to 9.6 m/s).

The employed air–water test loop consists of the water tank, water pump, air pump, buffer tank, valves, flowmeters, T-type mixer, funnel, and the flexible riser model, as depicted in Fig. 1. An electromagnetic air pump is used to deliver air at the standard room conditions (101 325 Pa and 293.15 K), and the stable air flow is created downstream the buffer tank with flow rate measured and regulated by a floater flowmeter and the associated stainless steel ball-type valve, respectively. A water pump submerged in a water tank is adopted to supply water at the standard conditions, and the associated flow rate is measured and regulated by a liquid turbine-type flowmeter and a needle valve, respectively. The water in the tank is dyed black for clearly distinguishing the liquid slugs from the gas bubbles. After that, the air and water flows are mixed through a T-type mixer with the same diameter as the upstream and downstream pipes. Before the entrance to the riser model, the gas–liquid mixture then flows through a horizontal tube with a length of 2.0 m to get fully development. Note that the inclined angle of upstream straight pipe is 0° for this pipe–riser system. The riser model of aspect ratio 200 is deployed as the catenary configuration in a vertical plane with the horizontal span (l₀) and height (h) of 0.989 and 0.6 m, respectively, and only tensioned by its weight. With the definition of the coordinate origin at the riser base, the initial position of the catenary riser is expressed as

where x and z are the horizontal and vertical coordinates, respectively.

The associated parameters are summarized in Table I. The Eötvös number (Eo = gr2Δρ/cσ) aforementioned in the Introduction in fact is a combined dimensionless parameter of Fr, We, and ρ_g/ρ_l, taking both gravitational and capillary forces into account. For the present riser model, Eo = 1.2 > 1 and 0.5 l_σmin < d < 5l_σmin (l_σmin = 3.05 mm), indicating that the gas–liquid mixture flow is dominated by the gravitational force despite the participation of capillary force. It further confirms the applicable of Fr as the similarity number.

As shown in Fig. 1, two pressure transducers are mounted on the riser base and top, respectively, to monitor the pressure changes over the riser with a sampling frequency of 100 Hz. Following the riser top outlet, the air is discharged to the atmospheric condition through a funnel-shaped air–water separator, whereas the water is returned to the water tank by the gravity effect.

The riser model is evenly marked with 29 black rings with the width of 6 mm and the center-to-center spacing of 40 mm along the span so that the non-intrusive measurement with high-speed cameras is employed to capture the vibration displacements of 29 markers as well as the severe slug flow characteristics.^{2,5,18,50–59} Note that the markers 1^#–29^# are counted from the riser top to its base. The in-plane (XOZ plane) and out-of-plane (Y direction) dynamic responses are simultaneously recorded by one camera placed beside the riser and the other camera arranged underneath the riser, respectively, with a sampling frequency of 100 Hz. The pixel calibration is conducted by connecting the width of each marker in the recorded images with its actual size of 6 mm. The displacement of each marker is extracted by comparing its central location from those of the adjacent images, and the length of each liquid slug is identified by scaling the pixels occupied by the blacked liquid slug to the actual size. Such non-intrusive measurement as well as image-processing method is proven to be reliable.^{2,5,18,50–59} The natural frequencies and damping ratios of the riser model in both empty and water-filled cases, as listed in Table I, are obtained from free decay tests performed by recording the free attenuation response with an initial imposed displacement on the riser.

A series of experiments has been conducted to examine the alteration of flow characteristics and the associated dynamic behavior of the catenary flexible riser excited by a gas–liquid two-phase flow with a variable gas superficial velocity. The observed severe slugging falls in SSII in terms of the cyclical three stages and aerated liquid slugs. In addition, another flow regime is observed with long liquid slugs and oscillating short slugs alternately occurring in the riser, termed transition flow. The flow characteristics of SSII and the resultant dynamic response of riser are discussed first as a typical example.

Figure 2 depicts the time histories of the monitored pressure at the riser base and the associated snapshots at thirteen moments during a fluctuation cycle. It is clearly seen that the pressure (P_b) varies cyclically over time, illustrating the periodic process of SSII.^11,12,19,20 One cycle of the evolution of SSII as well as the associated vibration response of the flexible riser recorded by the high-speed camera beside the riser are depicted in Fig. 3 (Multimedia view), in accordance with the time history of riser base pressure recorded by the pressure sensor and the time series of vibration displacements of two representative markers (9^# and 22^# markers, corresponding to the locations of the peak amplitudes in the x- and z-directions, respectively). From t₁ to t₈, P_b almost linearly increases with time, exhibiting an approximately constant growth rate $Ṗb$⁠. It is attributed to the accumulation of liquid starting from the riser base and the continuously elongated liquid slug length in the riser. At t₈, the accumulated liquid reaches the riser top and hence P_b is close to its maximum. This stage is thus termed slug formation (SF), at which most of the gas is blocked by the liquid slug and compressed in the upstream pipeline. Nevertheless, small gas bubbles are observed in the long liquid slug that possesses the same length as the riser, indicating the occurrence of gas penetration.¹² It is different from the severe slugging 1 (SS1) that is characterized by a pure liquid slug longer than the riser length.³³ The upstream horizontal pipeline (inclined angle of θ = 0°) and the vibration response of flexible riser possibly contribute to this difference. After t₈, the long-aerated liquid slug slowly moves out from the riser top to the separator while the blocked gas is continuously compressed until the appearance of the maximum P_b. Then P_b begins to decrease in accordance with a sudden drop of $Ṗb$⁠, suggesting that the pressure of upstream compressed gas surpasses the hydrostatic pressure of the liquid slug in the riser and hence gas starts to enter the riser.^34,49,60 From t₉ to t₁₂, the continuous shortening of liquid slug left in the riser results in the reduction of P_b and the rapid expansion and blowout of gas. As a result, the drop rate of P_b grows over time, leading to a shorter time for this gas blowout (GB) stage in comparison with the SF stage. Following the long liquid slug, some small slugs in different sizes emerge in the riser at t₁₁, indicating that some liquid is carried out with the expanded gas from the upstream. At t₁₂, the two small liquid slugs with lengths of 5.2d and 12d merge into a new slug of length 10d, implying the influence of riser's dynamic response on the internal slug migration. Additionally, some liquid falls into the riser base in the form of liquid film, contributing to the reduced length of merged slug and the almost completely gas-filled riser at t₁₃.⁴⁹ Therefore, this stage is named as transition (T), presenting a relatively small fluctuation in P_b and a near-zero $Ṗb$⁠. After that, the liquid accumulates at the riser base again and the cycle repeats. Nonetheless, the fluctuation amplitude of P_b slightly varies in different cycles, signifying the variation of the long liquid slug length. It is associated with the vibration response of the flexible riser.

The root-mean-squared (RMS) response amplitudes of the flexible riser and the instantaneous spanwise profiles of the z-directional amplitude during the three SSII stages are displayed in Fig. 4. It is seen from Fig. 4(a) that the out-of-plane response (A_y,rms/d) is quite smaller than the in-plane response (A_x,rms/d and A_z,rms/d) and hence is negligible. The main reason is the SSII process occurs in the curvature plane of the riser and the exerted fluid forces are also along the two directions of the plane. Both the x- and z-directional profiles of root-mean-squared amplitudes exhibit a trough and two peaks, suggesting that the response is dominated by the fundamental model. Nevertheless, unlike the approximately symmetric spatial distribution of A_z,rms/d about the midspan, the upper part of the riser experiences significantly higher A_x,rms/d compared to the lower part, due to the incoming flow direction and the spanwise varying curvature along the catenary riser. The non-zero node of RMS amplitude and the variable intersection of response profiles illustrate the co-existence of multiple modes.¹⁰ It is demonstrated from the instantaneous profiles, which intermittently present the second-mode shape. As shown in Fig. 2, the transient liquid slugs distributed along the riser with variable lengths and locations mainly contribute to the co-occurrence of first and second modes. It is observed from Fig. 4(b) that the lower part of the riser moves upwards from the equilibrium position to the upmost location and then returns back to the equilibrium position in the SF stage, while the upper part of the riser moves in the opposite direction as a result of the curvature configuration. In contrast, the lower part of the riser moves downwards from the equilibrium position to the lowest location and then returns to the equilibrium position in the GB stage. Although the maximum amplitudes are equal in the two stages, the movement speed in the GB stage is obviously faster than that in the SF stage, which is closely related to the flow characteristics.

The mixed standing and traveling wave features are observed from the spatial-temporal response amplitudes with traveling wave component oriented preferentially from the lower part to the upper part, corresponding to the flow direction, as depicted in Fig. 4(c). It is noted that the appearance of the maximum amplitude does not correspond to the occurrence of the maximum P_b. Instead, the maximum amplitude occurs once in the SF stage and once in the GB stage. The time histories of the response amplitudes at two representative markers corresponding to the trough and one peak of both the x- and z-directional RMS amplitude profiles are also plotted in Fig. 4(c) in accordance with the wavelet time-varying frequency. It is seen that the cyclic variation of the response amplitude at the crest marker (9^# for x direction and 22^# for z direction) is more obvious than that at the trough marker (22^# for x direction and 17^# for z direction), indicating that the response at the former is dominated by the first mode while the mode competition occurs at the latter. It is also illustrated from the broader frequency band of the latter. Such spatial out-of-sync response characteristics coincide with those excited by hydrodynamic slug flow.^18,57

Figure 5 further illustrates the relationship of riser vibration with the evolution of liquid slugs. It is seen that both the x- and z-directional forces exerted by the liquid slugs on the riser closely follow the distribution of the liquid inventory along the riser length, which is also depicted in Fig. 2. The greatest forces occur at the final moments of SF stage when the liquid fills the entire riser. Nevertheless, the twice peak amplitudes occur at t₃ in the SF stage and t₁₁ in the GB stage, respectively. The common feature is that the liquid slug length at the two moments is approximately equal to half of the riser length, although the locations are different. It is reasonable to believe that the violent response emerges when the fluid phase is seriously asymmetrically distributed in the riser. In contrast, the forces are dispersed along the riser due to the distributed liquid slugs, resulting in the weak response. The hydrostatic pressure (P_static) formed by the liquid inventory strictly follows the sum length of liquid slugs in the riser. The pressure drop between the riser bottom and top varies nearly parallel to P_static in the SF stage, and the difference between them indicates the pressure loss caused by the flow resistance. This difference is amplified in the GB stage as a result of the accelerated migration velocity of liquid slug. Additionally, the riser motion speed of the riser is also increased in the GB stage in comparison with the SF stage. Therefore, the dynamic response is more vigorous in the GB stage.

Figure 6 depicts the phase difference between the x- and z-directional responses in the curvature plane of the riser. It is seen that the phase difference varies spatially and temporally. For the majority part of the riser, the phase angle concentrates at 180°, presenting the antiphase variation in A_X/d and A_Z/d and the inclined linear oscillation and phase-averaged trajectories, where the oscillation trajectories are denoted by gray lines and the phase-averaged trajectories are highlighted in green lines overlying the former. For the middle part, the phase difference shifts to 90°, exhibiting the elliptic trajectories. The phase angle at the parts close to both ends of the riser switches between 0° and 360°, illustrating the in-phase characteristic. It is noted that the vibration frequencies in both x and z directions coincide with the fluctuation frequencies of the monitored pressure at the riser base. Such resonance between the dynamic response and the pressure fluctuation is associated with the SSII characteristics (as in Figs. 2–5).

The observed flow regimes including SSII and transition flow (TS) patterns in present small-scale riser are highlighted in the flow map drawn from the reported literature concerning the gas–liquid mixture in large-scale tubes (d = 26–75 mm), as depicted in Fig. 7, where the green lines denote the boundaries of different flow patterns. It is seen that the SSII and transition flow occurring in the present riser of d = 6 mm deviate from the reported flow partitions in large-scale tubes,²⁵ due possibly to the occurrence of surface tension that influences the flow pattern transition.^40,44–46

As displayed in Fig. 7, the mean pressure in riser base decreases with increasing the gas superficial velocity as a result of the growth in appearance of the high-frequent small-amplitude pressure fluctuation caused by oscillating liquid slugs. In contrast, the fluctuation amplitude of P_b increases with the increase in gas superficial velocity. Moreover, the fluctuation frequency grows accordingly, attributed to the appearance of oscillating liquid slugs with an accelerated migration speed. Nevertheless, the vibration response is weakened with increasing the gas superficial velocity, as shown in Figs. 7(d) and 7(e). It indicates that the contribution of longer liquid slugs is heavier than the oscillating shorter liquid slugs within the considered cases.

The current work pioneers the experimental investigation of severe slug flow-induced vibration (SSIV) of a small-scale flexible catenary riser, improving the understanding of SSIV and associated flow characteristics applicable to the offshore oil and gas applications. The major conclusions are drawn as follows.

1. Two flow regimes are identified in present experiments: SSII characterized with a cyclical process of three stages including slug formation, gas blowout and transition stages, and the transition flow from severe slugging to unstable oscillating featured by alternate appearance of long slugs and oscillating short slugs. These two kinds of flow patterns deviate from the reported flow zones in large-scale tubes, due possibly to the effect of surface tension in such a small-scale riser system.

2. The x- and z-directional forces exerted by liquid slugs on the riser are related to the spatial distribution of the liquid inventory along the length. The violent dynamic response occurs when the fluid phase is seriously asymmetrically distributed in the riser. Additionally, the spatial-temporal dynamic behavior of the flexible riser is closely connected with the flow regime, liquid slug length, and occurrence frequency. Compared to the response amplitude excited by SSII, the vibration amplitude is smaller in the transition flow pattern, suggesting that the contribution of longer liquid slugs on the dynamic behavior is greater than the oscillating shorter liquid slugs.

See the supplementary material for one cycle of the evolution of SSII and the associated dynamic response of the flexible riser those were recorded by the high-speed camera. Additionally, the time history of riser base pressure recorded by the pressure sensor and the time series of vibration displacements of two representative markers are also depicted in the video, presenting the feature of SSII as well as the resultant dynamic behavior of flexible riser.

The research work was supported by National Natural Science Foundation of China (No. 51979238) and State Key Laboratory of Hydraulic Engineering Simulation and Safety (Tianjin University) (No. HSSE-2005).

The data that support the findings of this study are available within the supplementary material.