Wave Loads Computation for Offshore Floating Hose Based on Partially Immersed Cylinder Model of Improved Morison Formula

Send Orders for Reprints to [email protected] 130 The Open Petroleum Engineering Journal, 2015, 8, 130-137 Open Access Wave Loads Computation for Offshore Floating Hose Based on Partially Immersed Cylinder Model of Improved Morison Formula

Shi-fu Zhang1,*, Chang Chen2, Qi-xin Zhang2, Dong-mei Zhang1 and Fan Zhang3

1National Engineering Research Center for Disaster & Emergency Relief Equipment, Logistic Engineering University, ChongQing, 401311, China; 2Deptartment of Petroleum Supply Engineering, Logistic Engineering University, ChongQing, 401311, China; 3Deptartment of Mechanic and Electric Engineering, Logistic Engineering University, ChongQing, 401311, China

Abstract: Aimed at wave load computation of floating hose, the paper analyzes the morphologic and mechanical characteristics of offshore hose by establishing the partially immersed cylender model, and points out that the results of existing Morison equation to calculate the wave loads of floating hose is not precise enough. Consequently, the improved Morison equation has been put forward based on its principle. Classical series offshore pipeline has been taken as example which applied in the water area of different depth. The wave loads of pipeline by using the improved Morison equation and compared the calculation results with the existing Morison equation. Calculations for wave loads on pipelines in different depth were accomplished and compared by the improved Morison equation and the existing Morison equation. Results show that the improved Morison equation optimizes the accuracy of the computation of wave load on floating hose. Thus it is more suitable for analyzing the effects of wave loads on floating hose and useful for mechanic analysis of offshore pipeline. Keywords: Method improvement, morison equation, offshore floating hose, wave load.

1. INTRODUCTION that are the weighted and the ordinary least-squares proce- dures. Error analysis showed that the ordinary least-squares The offshore floating hose is widely used in port and provided better accuracy and, therefore, was used to evaluate ocean engineering due to its light weight, high tensile the ratio between inertia and drag forces for a range of strength and ability to achieve rapid deployment and with- Keulegan-Carpenter and Reynolds numbers. Two small- drawal. As an important part of FPSO system, floating hose scale field experiments on the effectiveness of Morison's are used to deliver the crude oil which extracted from off- equation have been carried out by Paolo Boccotti [3]. The shore field to shuttle tankers. Moreover, it has played an im- agreement between Morison’s equation and the observation portant role in the implementation of water-oil replenishment data is valuable which proved the method presented in an at sea for The Navy warships. Whether in the state of trans- earlier paper [4] to be suitable for field experiments and is porting oil or not, the hose are always floating on the sea recommended for future work. Yang, Wanli [5] presented under buoyancy for its light weight. Therefore, the floating the expanded Morison equation, which can afford hydrody- hose are vulnerable to ocean current and waves. namic pressure caused by inner water and outer water simul- Morison's equation (Morison et al. 1950) is used for es- taneously. The practical application of the expanded Morison timating the hydrodynamic forces on the relatively slender equation has been explored through analyzing a continuous members of offshore structures and is perhaps the most rigid-framed deep water bridge. The results demonstrate that widely quoted and used equation in offshore engineering. it is an approximate, convenient and efficient way to esti- Julian Wolfram [1] has proposed an alternative approach to mate the hydrodynamic pressure caused by both inner and linearization which yields corresponding linear forms that outer water under earthquakes. produce unbiased estimates for the outputs from nonlinear Many researches have been conducted the wave loads equations. The linearization factors for Morison's equation acting on offshore structures and the offshore structures are have been found for the cases of expected fatigue damage classified as large or small scale one according to the influ- and expected extreme environmental load. The Morison ence of wave movement on them. At present, offshore iso- equation inertia and drag coefficients were estimated by lated pile [6-8] and submarine oil pipeline [9-11] are main Julca Avila [2] with two parameter identification methods subjects of wave forces effects on small-size offshore struc-

tures. Researches aimed at the calculation of wave load of floating hose are relatively rare and Morison formula [12-14] which includes the inertia term (depending on wave acceleration) and the drag term (depending on square velocity)

1874-8341/15 2015 Bentham Open Wave Loads Computation for Offshore Floating Hose The Open Petroleum Engineering Journal, 2015, Volume 8 131 directly brings about inaccuracy. Thus, it is necessary to im- Shown as Fig. (2), only part of the hose is immersed in sea prove the Morison equation under the condition of the shape and the height below the sea level is less than the diameter of and loading features of hose floating on the sea. Improving the pipe. The relations are as follows. the accuracy of calculation of wave loads on hose is of vital A = 1 h importance for better analysis of the effects of marine hose 1 on wave loads and safe use of floating hose. A 1 D 2 = (2) A < A 2. MORISON EQUATION 1 2

Morison equation, a semiempirical formula based on Where A1 represents the projected area of floating pipe per unit length in the vertical direction of the flow, A repre- flow theory, has been widely used in the calculation of wave 2 loads of small-scale marine structure whose ratio of radius sents the projected area of vertical cylinders per unit length and length is less than 0.2. The theory assumes that the in the vertical direction of the flow. small-scale marine structure has no significant effect on the wave motion, and that the effects of waves on the structure is mainly composed of viscous effect and the added mass effect. In case of a cylinder upright on the sea suffering the waves from the positive direction of axis x in which the in- tersection of seafloor and axis of cylinder is taken as the origin of XOZ coordinate system. Morison holds that the wave- induced force which offshore structures are subjected to the Fig. (1). The schematic diagram of offshore floating pipelines. inertia term (depending on wave acceleration) and the drag term (depending on square velocity). The derivation of the formula on the calculation of wave loads is presented as follow just taking one supposition as the premise that the drag force caused by wave movement and the one due to the uni- directional steady flow has the same mechanism.

fH = fD + fI 1 u = C Au u + 1+ C V x (1) 2 D x x ()m 0 t 1 D2 u = C Du u + C x 2 D x x M 4 t Where f is the horizontal wave forces acting on certain H Fig. (2). The sectional view of offshore floating pipelines. height of cylinder, f is the horizontal drag force, f is the D I horizontal inertial force, C is the drag coefficient, V is the Flow inertia force is caused by the disturbance of fluid D 0 volume of displacement of unit height, A is projection area pressure distribution due to the existence of structure. The volume of displaced fluid and pressure distribution without of unit height vertical to the wave motion direction, ux is the velocity in horizontal direction of wave particle, C is the structure are key factors of flow inertia force. As is shown in m added mass coefficient, C is the mass coefficient, and D diagram 1, the volume of displaced fluid of offshore floating M is the diameter of the cylinder. pipe is less than that of vertical cylinders per unit length, and relationship can be described as follows.

3. THE IMPROVED MORISON EQUATION V 1 S 1 = 1 Slender cylinder is often used as basic structure of ocean D2 V 1 engineering and the deducing of its Morison equation are 0 = (3) also based on the cylinder model. The diameter of floating 4 V < V hose is much smaller than the wavelength of the incident 1 0 wave. Thus, it is assumed that the floating hose has no significant influence on wave movement and the effect of Where V represents the volume of displaced fluid of 1 waves on the structure is mainly composed of viscous effect offshore floating pipe per unit length, S represents the pro- 1 and the added mass effect. Offshore floating hose is flat with jected area of per unit floating pipe in the vertical direction nothing in it but turn into cylindrical shape in oil-transferring of center axis of pipe, V represents the volume of displace- 0 state and always floating in the sea due to its light material ment of per unit vertical cylinders. According to the analysis and low density conveyance medium, as shown in Fig. (1). of key factors closely related to wave loads, a conclusion can The formation and change of the drag force is closely related be drawn that using the existing Morison formula to calcu- with the boundary layer formed on the surface of the column late the wave load of floating hose is inaccurate. Thus, it is body. In other words, it is closely related with the projected necessary to improve the Morison equation aiming at the area of the column per unit length in the vertical direction. calculation of wave load of floating hose. 132 The Open Petroleum Engineering Journal, 2015, Volume 8 Zhang et al.

Where h represents the height of immersed pipe below ' u the sea level, represents half of the corresponding central f = C V x I M 1 1 t angle of the bow-shaped above sea level. (7) ux Centre of mass is taken as the origin of XOZ coordinate = CM 1S1 system and the wave, ocean current, wind, gravity and buoy- t ancy loads are applied to the partially submerged pipe. Stress The shaded part area in Fig. (2) can be described in the analysis of floating pipe has been represented in the Fig. (3). other way as follows.

S1 =Ssector + Striangle

2-2 2 h r 2 S r a cos r sector = = 2 r (8) 2 S h r r 2 h r triangle = () ()

D r = 2 The height of immersed pipe below the sea level can be calculated by combining two equations and the flow drag force applying on the floating hose could be described as follows. Fig. (3). The stress analysis of offshore floating pipelines. ' 1 fD = CD 1 A1ux ux This is an extreme case where the loads of current, wave 2 (9) and wind has the same direction, resulting in the maximum 1 = C hu u horizontal load on the pipe. According to the force balance 2 D 1 x x principle, the following formula could be formed. To sum up, the improved Morison equation which FB =G1+G2 adapted for the analysis for offshore floating hose can be (4) F =F +F +F described as follows. P wind wave current f ' = f ' + f ' Where F represents the buoyancy loads of pipe, F H D I B P represents the anchor force in the horizontal direction, G 1 u 1 C A u u C V x (10) represents the weight of floating hose, G represents the = D 1 1 x x + M 1 1 2 2 t weight of oil transporting in the hose. The specific forms of 1 u each load are shown in formula (5). = C hu u + C S x 2 D 1 x x M 1 1 t

Where u represents the horizontal velocity of wave F = S L x B 1 1 1 ux particle at the center of the hose, represents the horizon- 2 2 t D D tal acceleration of wave particle at the center of the hose. G = - d L (5) 1 2 2 1 1 2 2 4. WAVE THEORY D G = d L Airy wave theory, Stokes wave theory, solitary wave 2 2 1 1 3 theory and conical wave theory are used to calculate the wave forces acting on a offshore structure based on different Where S represents the projected area of per unit float- 1 situations frequently. Zhu Yan-rong [15] presented the appli- ing pipe in the vertical direction of center axis of pipe, L 1 cation conditions of different wave theory as follow. represents the unit length, 1 represents the density of sea- water, 2 represents the density of floating hose, 3 repre- Choosing the suitable drag and mass coefficient is crucial sents the density of oil transporting in the hose, D repre- to calculate wave load of offshore structure by using the Mori- sents the outer diameter of floating hose, d1 represents the son equation. Many domestic experts and scholars have con- thickness of floating hose. ducted a lot of works on optimization of those parameters and some recommended values are listed in Table 2. Usually, it is 2 2 2 D D D essential to carry on the hydrodynamic experiment in water so d L + - d L 1 1 3 1 1 2 as to determine the value of specific coefficient [16]. 2 2 2 S = (6) 1 L 1 1 5. EXAMPLES Thus, the flow inertia force acting on the floating hose Floating hose is widely used in exploitation and transpor- could be described as follows. tation of offshore oil field. The article calculated the wave Wave Loads Computation for Offshore Floating Hose The Open Petroleum Engineering Journal, 2015, Volume 8 133 loads of widely used offshore pipe in the fourth sea level by C 1.2 D = using Morison equation and improved Morison equation (12) respectively. The parameters of pipeline and level 4 waves C = 2 M are listed in Tables 3 and 4. 5.1. Airy Wave Theory The height h and the projected area S1 can be obtained by taking above parameters into the formula (6) and (8).The Taking depth equal 6 as an example, the Airy theory is more result is listed in formula (11). suitable for wave loads calculation of offshore hose than others. The velocity and acceleration of wave partials in the horizontal h = 95.1mm direction should satisfy the condition of formula (13). h (11) = 0.834 Hgk cosh kz()+ d D u ()cos kx t x = = () 2 x 2 cosh kd S = 0.0091m () (13) 1 u 2 cosh kz+ d x 2 H ()() Hence, 83.4% of floating pipeline is immersed in the sea. a = = sin kx t x 2 () According to the applicable conditions of wave theory pre- t T sinh kd () sented in Table 1, the Airy theory should be chosen when the water is more than 4 meters deep and the Stokes theory will Where ux represents the horizontal velocity of wave par- be more suitable when it is between 2 and 4 meters deep. tial, ax represents the horizontal acceleration of wave par- Code of Hydrology for Sea Harbour pressed in 1998[17] has tial, represents the velocity potential of wave partial, k represents wave number which presented in formula (14). settled the value of CD and CM in China sea area.

2 Table 1. The applicable conditions of wave theory. k = (14) T The velocity and acceleration of wave partials in the ver- Condition Wave Theory tical direction should meet the condition of formula (15).

d / L 0.2, H / L 0.2 Airy wave theory Hgk sinh kz()+ d u ()sin kx t z = = () z 2 cosh kd 0.1 < d / L < 0.2, H / L 0.2 Stokes wave theory () (15) u 2 sinh kz+ d Conical wave theory z 2 H ()() 0.04 0.05 < d / L < 0.1 a = = cos kx t z 2 () Solitary wave theory t T sinh kd ()

Table 2. The recommended values of C and C . D M

Wave Theory CD CM Comments References

1.0 0.95 Mean values for ocean wave data on 13-24 in cylinders. Wiegel et al. (1957) Linear theory Recommended design values based on statistical analysis Agerschou and Edens 1.0-1.4 2.0 of published data. (1965)

Keulegan and carpenter Stokes 3rd order 1.34 1.6 Mean values for oscillatory flow for 2-3 in cylinders. (1958)

Recommended values based on statistical analysis of Agerschou and Edens Stokes 5th order 0.8-1.0 2.0 published data. (1965)

Table 3. Parameters of floating pipeline.

Dmm dmm g/cm3 g/cm3 g/cm3 () () 1 () 2 () 3 () 114 6 1.025 1.25 0.83

Table 4. Parameters of level 4 waves.

Lm Ts s-1 cm/s Hm () () () () () 20 3.57 1.93 5.6 1.25

134 The Open Petroleum Engineering Journal, 2015, Volume 8 Zhang et al.

For the calculation of wave load by Morison equation, pa- Thus, the wave load using Stokes two order wave theory rameters of level 4 waves in Table 4 are used to calculate the may follow the example of Airy theory. Parameters of level velocity and acceleration of wave partials in formula (13) and 4 waves are used to calculate the velocity and acceleration of formula (15) at the beginning. After that, the wave load could wave partials in formula (16) and formula (17). Thus, the be worked out by submitting the above result along with the wave load could be calculated by submitting those results external diameter and cross section area of the pipe to formula along with the height h and the projected area S1 to formula (1). For the calculation of wave load with Improved Morison (1) and (10). Extreme values of wave loads through the use equation, the velocity and acceleration of wave partials still of Stokes theory are listed in Table 6. need to be figured out. Afterwards, the wave load could be Wave load variations are shown in Fig. (6) and Fig. (7) calculated by submitting those results along with the height by using the Matlab program. h and the projected area S1 to formula (10) by using the Mat- lab program. Extreme value of wave loads by using the Mori- son theory and Improved Morison theory are listed in Table 5. 5.3. Analysis Wave load variations are shown in Fig. (4) and Fig. (5) It can be seen from Fig. (4) that the time-domain wave- by using the Matlab program. form based on improved Morison equation is similar to that of Morison equation and the value of wave loads based on 5.2. Stokes Two Order Wave Theory improved Morison equation is smaller than those based on Taking depth equal 3 as an example, the Stokes theory Morison equation. This is not a coincidence. The regulation can be found in Fig. (5) to Fig. (7). Wave loads of offshore should be chosen for wave loads calculation of offshore hose hose in case of depth equal 6 and depth equal 3 are listed in in this situation. The velocity and acceleration of wave par- Table 7. tials in the horizontal direction should satisfy the condition of formula (16). Where F represents the wave loads based on Morison 0 equation, F represents the wave loads based on Improved 1 H cosh()kz()+ d Morison equation, F represents the difference between the u = = cos kx t 2 x () value of F and F . x T sinh()kd 0 1 F = F F (18) 3 H H cosh() 2kz()+ d 2 0 1 + cos 2 kx t 4 () P represents the ratio of F to F . This index shows the 4 T L sinh kd 2 0 () (16) various degrees between two methods.

u 2 cosh kz+ d x H ()() a = = 2 sin kx t F x 2 () 2 t T sinh kd P = 100% (19) () F0 2 H H cosh() 2kz()+ d +3 sin 2 kx t 2 4 () Some characteristic value could be found through the sta- T L sinh ()kd tistical analysis of ratio P in horizontal and vertical direction with different wave theories shown in Table 8. The velocity and acceleration of wave partials in the vertical direction should satisfy the relationship as formula (17). Where P represents the ratio P referring to horizontal 1 wave load in Airy theory, P represents the ratio P referring 2 H sinh()kz()+ d to vertical wave load in Airy theory, P represents the ra- u sin kx t 3 z = = () z T sinh kd tio P referring to horizontal wave load in Stokes two order () wave theory, P represents the relative P referring to vertical 4 3 H H sinh() 2kz()+ d wave load in Stokes two order wave theory. + sin 2 kx t 4 () 4 T L sinh ()kd (17) The maximal value of ratio P is 23% and the mean value of it is upright to 15% indicating that it is not precisely u 2 sinh kz+ d z H ()() a = = 2 cos kx t enough by using the existing Morison equation to calculate z 2 () t T sinh kd () the wave loads of floating hose and the improved Morison equation is more suitable for this situation. The difference H 2 H sinh() 2kz()+ d 3 cos 2 kx t between the value of F and F can be visualized in Fig. (8) 2 4 () 0 1 T L sinh ()kd and Fig. (9) with the use of Matlab program.

Table 5. Extreme value of wave loads with the usage of Airy theory.

The Horizontal Wave Loads The Vertical Wave Loads Technique Max Value Mix Value Max Value Mix Value

Morison theory 89.46 -89.2 82.90 -83.0

Improved Morison theory 75.35 -75.2 75.55 -75.6

The unit of values in above table is N/m. Wave Loads Computation for Offshore Floating Hose The Open Petroleum Engineering Journal, 2015, Volume 8 135

Fig. (4). The comparison of horizontal wave force between the use Fig. (5). The comparison of vertical wave force between the use of of the improved Morison equation and existing Morison equation the improved Morison equation and existing Morison equation in in airy theory. airy theory.

Table 6. Extreme value of wave loads through the use of Stokes theory.

The Horizontal Wave Loads The Vertical Wave Loads Technique Max Value Mix Value Max Value Mix Value

Morison theory 257.48 -88.95 123.01 -124.42

Improved Morison theory 215.59 -74.76 103.39 -104.94

The unit of values in above table is N/m.

Fig. (6). The comparison of horizontal wave force between the use Fig. (7). The comparison of vertical wave force between the use of of the improved Morison equation and existing Morison equation in the improved Morison equation and existing Morison equation in in Stokes theory. Stokes theory.

The main reason for this phenomenon is that Morison ameter of the hose by default. However, the hoses are always equation is based on vertical cylinder model and supposes floating on the sea under buoyancy whether they are in the the infinitesimal section is completely immersed in water. state of transporting oil or not and only part of them are im- Thus, the displacement volume is considered as cylinder mersed in the sea. The displacement volume and the pro- volume and the projected area of per unit length in the verti- jected area of per unit length in the vertical direction of the cal direction of the flow are considered as the external di- flow is determined by the mechanical equilibrium in the 136 The Open Petroleum Engineering Journal, 2015, Volume 8 Zhang et al.

Table 7. The wave load of floating hose in different depth.

The Horizontal Load (N/m) The Vertical Loads (N/m) Wave Theory t(s) F0 F1 F2 P,% F0 F1 F2 P,%

0.00 84.21 70.25 13.96 16.58 -40.71 -36.29 -4.42 10.85 0.30 41.36 33.27 8.09 19.56 -54.89 -49.40 -5.49 10.01 0.90 -42.64 -38.01 -4.63 10.85 -77.18 -70.62 -6.56 8.50 1.50 -85.24 -72.28 -12.96 15.20 17.72 15.38 2.34 13.21 2.10 -38.43 -30.78 -7.66 19.93 56.15 50.56 5.59 9.95 Airy theory 2.70 42.73 38.09 4.64 10.86 76.00 69.57 6.44 8.47 3.30 86.09 72.93 13.15 15.28 -19.93 -17.39 -2.54 12.75 3.90 35.47 28.25 7.22 20.36 -57.44 -51.75 -5.68 9.90 4.50 -42.90 -38.23 -4.67 10.89 -74.71 -68.41 -6.30 8.44 5.00 -80.33 -68.45 -11.89 14.80 6.42 5.10 1.32 20.49 0.00 249.75 208.35 41.41 16.58 -68.20 -60.80 -7.40 10.85 1.00 -72.73 -63.18 -9.56 13.14 -29.47 -22.68 -6.79 23.04 2.00 -85.42 -71.22 -14.20 16.62 19.45 17.25 2.20 11.32 3.00 101.48 88.82 12.66 12.47 102.33 84.77 17.55 17.16 4.00 19.84 12.73 7.11 35.85 -120.51 -102.31 -18.19 15.10 Stokes theory 5.00 -88.54 -74.15 -14.39 16.25 18.37 16.72 1.65 8.98 6.00 -26.56 -20.55 -6.01 22.64 70.70 60.98 9.72 13.75 7.00 253.20 212.77 40.42 15.97 -49.81 -45.20 -4.61 9.25 8.00 -64.86 -57.46 -7.40 11.41 -66.67 -54.18 -12.49 18.73 9.00 -86.31 -72.02 -14.30 16.56 13.97 12.45 1.52 10.91 10.00 71.51 63.65 7.86 10.99 122.13 102.41 19.72 16.15

Table 8. Some characteristic value of the ratio P.

Parameter P1 P2 P3 P4

Max value 20.36 20.49 22.64 23.04 Average value 15.43 11.26 15.26 14.01

Fig. (8). The variation of wave load difference based on Airy the- Fig. (9). The variation of wave load difference based on Stokes ory. theory.

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