mtroleity, C

Accepted 14 October 2013Available online 7 November 2013

Keywords:Oil spillSubmarine pipeline

to investigate the process of oil spill from submarine pipeline to free surface. Effects of oil density, oil

into marine environment from the leak, causing extensive damage

making the risk relatively high. Given the frequent occurrence of oilspills, it has been a matter of constant concern from the viewpointsof environmental and social disasters.

current is particularly necessary. Besides, laying oil containmentso depends on theeading. Therefore,rsal could providement boom and

two-dimensionalmodeling system)om damaged sub-

marine pipeline, the migration of oil ow along the depth directionbecomes an important issue. An effective attempt has been madeby Li et al. [7] to observe the oil spill under the action of current andwave. However, the velocity of current in their study was uniform,which does not match with the actual shear velocity distributionunder sea surface. And the actual hydrostatic pressure distributionwas not used in their modeling. Moreover, the crucial parameter,the maximum horizontal migration distance of oil, was notconsidered in their research.

* Corresponding author. State Key Laboratory of Oil and Gas Reservoir Geologyand Exploitation, Southwest Petroleum University, Chengdu, Sichuan 610500,China. Tel.: 86 28 83032206.

Contents lists availab

Ener

journal homepage: www.els

Energy 64 (2014) 887e899E-mail address: ticky863@126.com (H. Zhu).to marine life, human health, and natural resources [1]. As theaccidental oil leakages in Gulf of Mexico, about twenty thousandbarrels of oil were released into the sea everyday, resulting inecological devastation in adjacent sea area. Similar oil spill acci-dents occurred at platform B and C of the Penglai 19-3 oileldlocated in Bohai Sea [2]. About 700 barrels of oil and 2500 barrels ofmineral oil-based drilling mud were released. Moreover, Bohai Seais a semi-enclosed shallow sea with average water depth of 18 m,

boom, as a basic way to control oil dispersal, alrising velocity of oil droplets and the trend of spran exact prediction of oil spill process and dispeuseful information for setting up oil containreducing the damage of future oil spills.

Most previous studies focused on the surfaceoil spill [4e6], in which ROMS (regional oceanis employed widely. However, for oil leakage frWashout and perforation failures are usually present in oilsubmarine pipelines due to corrosion or ow erosion. Then oil spills

required [3]. By this time, the information about the rise process ofoil droplets and the dispersal path of oil spill under the action ofVOF (volume of uid)CFD (computational uid dynamics)

1. Introduction0360-5442/$ e see front matter 2013 Elsevier Ltd.http://dx.doi.org/10.1016/j.energy.2013.10.037leaking rate, leak size and water velocity on the oil spill process are examined. High density, slow leaking,small leak size or fast current brings about long time for oil reaching the maximum horizontal migratedistance when it reaches surface. And this maximum horizontal migrate distance increases with theincrease of leak size or water velocity, while increases with the decrease of leaking rate. Then, thedimensionless time required for oil droplets which have the longest horizontal migrate distance whenthey reach the sea surface and the dimensionless longest horizontal distance the droplets migrate whenthey reach the sea surface are analyzed and the tting formulas are obtained. Only the formula for thedimensionless longest horizontal distance versus dimensionless density meets the polynomial, other veformulas meet the natural logarithm distribution. Using the formulas we can obtain when and where tosee oil reaching the sea surface, and conduct rapid response. Finally, the maximum horizontal migrationdistance of oil at certain time is predicted, and a forecasting model is proposed. The using methods oftting formulas and the forecasting model are shown in the paper by examples. These calculated resultsprovide useful guidance to place the oil containment boom.

2013 Elsevier Ltd. All rights reserved.

Once accidental oil leakages occur, a quick and adequateresponse in order to reduce the environmental consequences isReceived in revised form10 October 2013Article history:Received 19 June 2013

The objective of the present paper is to study the oil ows from damaged submarine pipelines withdifferent leak sizes. CFD (computational uid dynamic) simulations with FLUENT software are carried outA CFD (computational uid dynamic) sidamaged submarine pipeline

Hongjun Zhu a,b,*, Pengzhi Lin b, Qian Pan a

a State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Peb State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan Univers

a r t i c l e i n f o a b s t r a c tAll rights reserved.ulation for oil leakage from

um University, Chengdu, Sichuan 610500, Chinahengdu, Sichuan 610065, China

le at ScienceDirect

gy

evier .com/locate/energy

gy 6Without the effect of wave or current, a nearly vertical ascentof spilled oil droplets would present, and the diffusion radius of oillm in the surface is just a function of time. Where to place the oilcontainment boom can be easily obtained in this condition.However, there are waves and currents in ocean all the time.Under the action of wave and current, the trajectory of spilled oilruns off the straight line, and oil lm in the surface spreadsrapidly downstream. Therefore, the maximum horizontal migra-tion distance of oil is a key parameter to place the oil containmentboom.

Under the action of current with shear velocity distribution,the length of time for oil to reach sea surface, and the distance foroil moving downstream when it reaches the surface are the twokey parameters to guide quick response, including the laying of oilcontainment boom. Especially for submarine pipelines near pro-duction platforms or shore, once oil spill is observed by ROV(remote operated vehicle) or acoustic detection, a quick rescueresponse can be implemented in a few minutes, even in 1 min. Inthis case, the process of oil spill from submarine pipeline to freesurface should be concerned to nd when and where the spilledoil droplets can reach the free surface. While for submarinepipelines far away from platforms and shore, the rescue shipsusually take hours to reach the accident location. Thus, themaximum horizontal migration distance of oil at certain time, thehours for rescue ships arrival after leakage started, is the majorconcern. For the process of oil spill consisting of two processes,rising process and drifting process, the maximum horizontalmigration distance is still related to the longest horizontal dis-tance the droplets migrate when they reach the sea surface. Afterthe oil droplets arriving at surface, the horizontal migration dis-tance is a function of surface water velocity. So the process of oilspill from submarine pipeline to free surface is still needed to besolved rstly.

Numerical simulation can provide detailed information on thehydrodynamics of oil ow, which is not easily obtained by physicalexperiments [8e10]. Therefore, in this work, CFD (computationaluid dynamic) model coupling with VOF (volume of uid) methodhas been used to investigate the process of oil spill from submarinepipeline to free surface. The actual shear velocity distribution ofcurrent and the actual hydrostatic pressure distribution areconsidered in this study. Detailed oil droplet and sea-surface in-formation could be obtained by the VOF model. By conducting aseries of numerical simulations, effects of oil density, oil leakingrate, leak size and water velocity on the oil spill process areexamined. Then, the dimensionless time required for oil dropletswhich have the longest horizontal migrate distance when theyreach the sea surface and the dimensionless longest horizontaldistance the droplets migrate when they reach the sea surface areanalyzed and the tting formulas are obtained. Using the formulaswe can obtain when and where to see oil reaching the sea surface,and conduct rapid response. Finally, the maximum horizontalmigration distance of oil at certain time is predicted, and a fore-casting model is proposed. The results provide useful guidance toplace the oil containment boom.

The remaining part of this paper is organized as follows. InSection 2, the description of mathematical models and simulationmethod are provided; Section 3 presents the simulated results anddiscussion; Section 4 is the concluding remarks.

2. Simulation method

2.1. Governing equations

Oil, water and air are treated as incompressible ows. And at

H. Zhu et al. / Ener888the interface of uids, no phase change and no-slip between uidswhere subscript a, o and w represent air, oil and water, respectively.In this study, Reynold number ranges from 181.25 to 4531.25. So

in condition of high Reynold number, realizable k- turbulencemodel [16e20] is employed to close the ow governing equationsand describe the turbulent properties:

vrk vrkui v"

m mtvk# Gk Gb r (9)Fo VoVc (4)

where Fo and Fw are oil and water fractional function, respectively,Vc, Vo and Vw represent volume of a cell, volume of oil inside the celland volume of water inside the cell, respectively.

And the two-dimensional transport equations for the fractionalfunctions are given by:

vFwvt

vuFwvx

vvFwvy

0 (5)

vFovt

vuFovx

vvFovy

0 (6)

Then, the density and viscosity can be expressed in followingequations:

r 1 Fw Fora Fwrw Foro (7)

y 1 Fw Foya Fwyw Foyo (8)equation for the volume fraction of uid. In this study, volumeof uid functions Fw and Fo are introduced to dene the waterregion and the oil region, respectively. The physical meaning ofthe F function is the fractional volume of a cell occupied by theliquid phase [14,15]. For example, a unit vale of Fw correspondsto a cell full of water, while a zero value indicates that the cellcontains no water. The fraction functions Fw and Fo are describedas follows:

Fw VwVc (3)are assumed. The ows of uids are governed by the RANS (Rey-nolds-Averaged-Navier-Stokes) equations, including continuity andmomentum equations written as follows [11e13]:

vuivxi

0 (1)

vuivt

vuiujvxj

1r

vpvxi

yV2ui vu0iu

0j

vxj gi (2)

where ui represents instantaneous velocity component in i direc-tion, for example u and v are velocity in x and y direction, respec-tively, while u0i is uctuation velocity component in i direction, xi isspace coordinate in i direction, gi is gravitational acceleration in idirection, t is time, p is pressure and r and y are density and kine-matic viscosity, respectively.

The VOF approach is based on the solution of one mo-mentum equation for the mixture of the phases, and one

4 (2014) 887e899vt vxi vxj sk vxj

gy 6Under the action of current, spilled oil spreading directlydownstream can reach the maximum horizontal migration dis-vrvt

vruivxi

vvxj

"m mt

s

v

vxj

# rC1S rC2

2

k yp C11 C3

kGb

(10)

where,

C1 max0:43;

h

h 5

(11)

h S k

(12)

S 2Sij$Sij1=2 (13)

Sij 12

vuivxj

vujvxi

!(14)

Gk ru0iu0jvujvxi

(15)

Gb gimtPrt

vr

rvxi(16)

mt rCmk2

(17)

where k and represent turbulent kinetic energy and turbulentkinetic energy dissipation rate per unit mass, respectively, h is therelative strain parameter, S is the strain rate, Gk and Gb representproduction term of turbulent kinetic energy due to the averagevelocity gradient and production term of turbulent kinetic energydue to lift, respectively, m ry is dynamic viscosity of uid, mt isturbulent viscosity, Prt is Prandtl number taken as 0.85, Cm, C1, C2and C3 are empirical model constants taken as 0.09, 1.44, 1.9 and0.9, respectively, and st and s are turbulent Prandtl numbers takenas 1.0 and 1.2, respectively.

2.2. Numerical method

FVM (nite volume method) is employed to discretize aboveequations. All the simulations are carried out using a commercialsoftware package FLUENT 14.0. In calculations, Patankars well-known SIMPLE algorithm [21], applied well in many similar simu-lation studies [22e24], is employed to solve the pressureevelocitycoupling to satisfy the conservation law of mass. In order to ensurethe accuracy of calculation, second-order upwind scheme andsecond-order central-differencing scheme are used for convectiveterms and diffusion terms, respectively. For convective terms,second-order upwind scheme can save CPU time [25], while fordiffusion terms second-order central-differencing scheme is a goodchoice [26]. The convergent criteria for all calculations are set asthat the residual in the control volume for each equation is smallerthan 104.

2.3. Computational domain and mesh

H. Zhu et al. / Enertance. Therefore, two-dimensional ow simulation is accurateenough to capture the maximum horizontal migration distance. Inaddition, three-dimensional simulation needs a higher CPU cost.Due to time limitations, 2D simulation is applied in this work.

Fig. 1 shows a sketch of the geometry and numerical grid forcomputational domain investigated in this study. The averagedepth of water (14.5m) in Kenli oileld located in Bohai Sea is takenas the model depth in order to facilitate the comparison. The wholecomputational domain is a rectangle with a length of 20 m and aheight of 15 m. The length of computational domain is largeenough, which is larger than the longest horizontal distance the oildroplets migrate when they reach the sea surface. Water occupiesthe lower regionwith height of 14.5m, while air occupies the upperregion.

In the computational domain, the damaged submarine pipewith the outer diameter (D) of 0.6 m, the most commondiameter of submarine pipe used in Bohai oileld, is located inthe sea bed, 1.8 m (3D) downstream of the inlet. There is aleakage hole on the top of pipe, opening upwards. The size ofthe leakage hole (d) is a variable ranging from 0.01 m to 0.05 mwith increment of 0.01 m, in order to examine the effect of leaksize.

GAMBIT 2.3 mesh-generator is employed to perform all geom-etry generation and meshing. As shown in Fig. 1, computationaldomain is divided into two blocks. The water occupying region isdiscretized with triangular cells, while the upper region is dis-cretized with quadrilateral cells. Progressive mesh is used to cap-ture the near-leak ow properties. A suitable grid density isreached by repeating computations until a satisfactory indepen-dent grid is found. At last, the number of grid cells used in calcu-lation is 9011.

2.4. Boundary conditions

A logarithmic velocity prole is adopted to meet the actual ownear seabed as vw vwmax{1 [1 y/(H D)]2}, where vwmax is themaximumvelocity (vwmax 0.1 m/s, the most commonly measuredmaximum ow rate in Bohai Sea surface) presenting at the freesurface,H is the height from leakage hole to the free surface, y is theindependent variable (0 y H D) and origin of coordinates islocated in the seabed. This velocity prole is dened for the leftinlet of computational domain. In order to nd the effect of watervelocity on the displacement of oil droplets, the maximum watervelocities are taken as 0.04 m/s and 0.07 m/s in comparingcases. For the right outlet, a linear static pressure prole isemployed to meet the actual hydrostatic pressure distribution aspout rg(H D y).

In Bohai oil eld, due to corrosion or ow erosion, perforationsusually present in submarine pipelines. Only a few pipes havecracks on them due to mechanical damage. According to a lot offailure marine pipe tests, the perforation holes mainly show cir-cular shape or roughly circular shape. For roughly circular shape,we can use a circular instead of it with an equivalent diameter (thetwo have the same equal area). By measuring a large number ofperforation holes, the pore sizes mainly range from 0.006 m to0.08 m. For analyzing the effect of pore size, ve different leak di-ameters, 0.01 m, 0.02 m, 0.03 m, 0.04 m and 0.05 m, are chosen insimulations.

Oil leaking rate is related to leak size, hydrostatic pressure ofwater above the pipe, the pressure within the pipe and pressuredrop of oil owing through the leak. Therefore, perforations indifferent oil pipelines or in different locations at one oil pipelinehave the different oil leaking rates. In the Bohai Sea environment,oil leaking rate usually ranges from 0.1 m/s to 10 m/s. In order tofacilitate comparative analysis, we have selected ve rates rangingfrom1m/s to 5m/s to conduct simulations. Pressure inlet boundary

4 (2014) 887e899 889conditionwith value of 0 Pa is used for the three edges of air region.

H. Zhu et al. / Energy 64 (2014) 887e899890At the initial time, the lower area is lled with water and theupper area is lled with air. Still water surface is assumed at theinterface of the two regions. And the pressure in air region isdened as 0 Pa, meaning the atmospheric pressure.

In simulations, densities of air and water are seemed constant as1.225 kg/m3 and 1025 kg/m3, respectively. While the density of oil

Fig. 1. Sketch of the geometry and numerical grid for computational domain: (a) overallcomputational domain.

Table 1Simulation cases.

Case Oil density(kg/m3)

The maximumwater velocity(m/s)

Oil leakingrate (m/s)

1 780 0.1 22 810 0.1 23 840 0.1 24 870 0.1 25 900 0.1 26 930 0.1 27 960 0.1 28 870 0.1 19 870 0.1 310 870 0.1 411 870 0.1 512 870 0.1 213 870 0.1 214 870 0.1 215 870 0.1 216 870 0.04 217 870 0.07 2is a variable in different case ranging from 780 kg/m3 to 960 kg/m3

with increment of 30 kg/m3, in order to analyze the effect of oildensity. The viscosities of air, oil and water are dened as1.8 105 Pa$s, 1.003 103 Pa$s and 0.048 Pa$s, respectively.

The information of simulation cases is listed in Table 1, in whichoil density, oil leaking rate, diameter of leak and the maximum

view of the computational domain and boundary conditions; (b) grid distribution of

Diameter ofleak (m)

Volume uxof leaking oil(m3/s)

Flux multiple(comparingwith case 12)

0.05 0.003925 250.05 0.003925 250.05 0.003925 250.05 0.003925 250.05 0.003925 250.05 0.003925 250.05 0.003925 250.05 0.0019625 12.50.05 0.0058875 37.50.05 0.00785 500.05 0.0098125 62.50.01 0.000157 10.02 0.000628 40.03 0.001413 90.04 0.002512 160.05 0.003925 250.05 0.003925 25

gy 6H. Zhu et al. / Enerwater velocity are the four variables. For ow rate and leak size arerelated to each other, we have set the minimum volume ux occursin case 12 (oil leaking rate 2 m/s and leak size 0.01 m). Andvolume uxes in others cases are some times more than the min-imum volume ux, as listed in Table 1. Therefore, a wide range ofvolume ux of leaking oil has been considered in this paper.

3. Numerical results and discussion

3.1. Standard case

We adopt case 4 (as shown in Table 1) as the standard case. Fig. 2presents the volume fraction of water at different times, corre-sponding to the process of oil spill from submarine pipeline to freesurface. It is indicated 39 s are required for oil dropletswhich have thelongest horizontal migrate distance when they reach the sea surface.

As Fig. 2 shows, continuous oil ow stream presents justreleasing from the leakage hole with the height of about 3 m.However, it is tore apart at a certain depth by current. Under thejoint action of gravity, inertia force, buoyancy and shear stress, oilspill appears in the form of droplets or droplet groups inmost of thecomputational region. With the increase in rising height, oildroplets become more dispersed. At t 39 s, the horizontal span ofoil droplets is about 3.36 times longer than that at t 15 s.

Oil droplets move downstream under the action of waterowing from the left boundary. However, this phenomenon is not

Fig. 2. The process of oil spill from submarine4 (2014) 887e899 891obvious near the seabed for the small water velocity. With thedecrease in water depth, the horizontal migrate distance has asignicant increase. The reason is that high-speed water has agreater horizontal shear stress exerting on spilled oil and transfersgreater kinetic energy to oil ow, resulting in a longer horizontalmigrate. Therefore, the shear distribution current plays an impor-tant role in oil migration. We should consider the actual watervelocity distribution to conduct a relative accurate prediction.

From the Fig. 2 we can see that the rst oil droplet reachingthe free surface is not the one which has the longest horizontalmigrate distance. This main reason is that oil droplets havedifferent sizes and the sizes are variable with time due to thedispersion and combining of oil droplets under the joint action ofgravity, inertia force, buoyancy and shear stress. In the process ofoil spill from the leak to free surface, the oil droplet which rea-ches the free surface rstly has larger buoyancy, and then thefurthest-migration droplet receives a greater shearing action ofcurrent. The time required for the rst oil droplet reaching thefree surface is less than 35 s. While the droplet which has thelongest horizontal migrate distance reaches the surface att 39 s. And the longest horizontal migrate distance is 16.7 m(18.5 m 1.8 m, the horizontal distance from the inlet to pipeleak). At this time, the horizontal and vertical displacement ratioof this droplet is 1.201.

Since the oil spill is subjected to the joint action of gravity,inertia force, buoyancy and shear stress, in the later stage, we have

pipeline to free surface at standard case.

Fig. 3. The process of oil spill from submarine pipeline to free surface at different oil densities: (a) ro 780 kg/m3; (b) ro 870 kg/m3; (c) ro 960 kg/m3.

H. Zhu et al. / Energy 64 (2014) 887e899892

gy 6H. Zhu et al. / Enerchanged the oil density, oil leaking rate, leaking size and watervelocity to explore their inuences.

3.2. Effect of oil density

In this section, to study the effect of oil density on the length oftime for oil to reach the sea-surface and the distance for oil movingdownstream, simulations are conducted by changing the oil densitywhile leaving other parameters same as those in the standard case.Fig. 3 illustrates the process of oil spill from submarine pipeline tofree surface at three different oil densities. It can be seen that thelarger the oil density, the longer the time required for oil to reachfree surface. For ro 960 kg/m3, the required time for themaximum horizontal migration is about 1.84 times as long as that

Fig. 4. The process of oil spill from submarine pipeline to free surface at di4 (2014) 887e899 893when oil density is 780 kg/m3. It is attributed to the increasinggravity of oil droplets. In the vertical direction, an oil droplet ismainly subject to the force of gravity and buoyancy. For twodroplets of the same size, the upward buoyant forces are the same,while the droplet of larger density has a larger gravity. Therefore,the nal vertical upward force is small for high density droplet,resulting in a slow rising rate.

At the same time (t 15 s), the maximum horizontal migrationdistance of light oil droplet (ro 780 kg/m3) is 7 m (8.8 m minus1.8 m), about one time longer than that with density of 960 kg/m3.Before the leaking oil reaching the sea-surface, the horizontalmigration of oil ow under the sea-surface can not be easilyobserved without monitoring instruments. In addition, oil contain-ment boom is laid oating on the sea-surface. So the maximum

fferent oil leaking rates: (a) vo 1 m/s; (b) vo 3 m/s; (c) vo 5 m/s.

gy 6H. Zhu et al. / Ener894horizontal migration distance when the oil droplet reaches the freesurface is a very vital parameter. This horizontal migration distancefor ro 960 kg/m3 is 17.1m (18.9m 1.8m), a little shorter than thatwhen oil density is 780 kg/m3 which is 18 m (19.8 m 1.8 m). The

Fig. 5. The process of oil spill from submarine pipeline to free surface at different w4 (2014) 887e899main cause of this result is that low density oil droplets rise fasterand enter into high-speed water zone earlier, leading to shearingaction of current acting on oil earlier. However, the difference in themaximum horizontal migration distance is little. Therefore, light oil

ater velocities: (a) vwmax 0.04 m/s; (b) vwmax 0.07 m/s; (c) vwmax 0.1 m/s.

Fig. 6. The process of oil spill from submarine pipeline to free surface at different oil leak sizes: (a) d 0.01 m; (b) d 0.03 m; (c) d 0.05 m.

H. Zhu et al. / Energy 64 (2014) 887e899 895

can reach surface quickly, requiring short response times, while thelocation be laidwith oil containment boom to control oil dispersal isbasically the same for different-density oil ow.

3.3. Effect of oil leaking rate

Oil leaking rate is one of the key factors which impacts thediffusion of oil spill. Fig. 4 depicts the process of oil spill fromsubmarine pipeline to free surface at different oil leaking rates. Atsmall leaking rate (vo 1 m/s), 79 s is required for oil ow to reachthe maximum horizontal migrate distance when it reaches the freesurface, and the maximum horizontal migration distance is arrivedat 16.4 m (18.2 m minus 1.8 m). However, for higher leaking rate(vo 5 m/s), just 15 s is needed for oil to reach the maximumhorizontal migrate distance when it reaches the free surface. It canbe explained that high-speed leaking oil has more ascending ki-netic energy. From Fig. 4, we can also see more dispersed oildroplets present in computational domain at high leaking rate. Thereason is that the total amount of released oil is larger as the massrate of oil is larger (For incompressible uid, mass ow rate in-creases as the increase in velocity). Thus, in order to reduce theenvironmental consequences, a relatively fast response is requiredfor high-speed leaking oil, and an adequate response should be

wmax

The time required for oil ow to reach the maximum horizontalmigrate distance when it reaches the free surface is 36 s when themaximum water velocity is 0.04 m/s, 2 s earlier than that forvwmax 0.07 m/s and 3 s earlier than that for vwmax 0.1 m/s. Theroute of oil droplet is the square root of the sum of the squares onvertical depth and horizontal displacement. Therefore, the route ofoil droplet is relative small in vwmax 0.04 m/s due to the shorthorizontal displacement, resulting in the short rising time. There-fore, short response time is required for slow current, while oilcontainment boom should be laid a longer distance for fast current.

3.5. Effect of oil leak size

Fig. 6 shows the process of oil spill from submarine pipeline tofree surface at different oil leakage sizes. The results indicate thatthe effect of the diameter of leakage hole plays a signicant role inthe spread of oil spill. With increasing leakage size, the timerequired for oil to reach the maximum horizontal migrate distancewhen it reaches the free surface is shortened. As leakage size re-duces from 0.05 m to 0.01 m, the required time decreased by 23.53percents. It can be explained that at the same leaking rate, thebigger the diameter of leak, the larger the amount of released oiland the greater the upwardmomentum. Due to the largemass owrate, oil droplets released from the leak with d 0.05 m are easier

H. Zhu et al. / Energy 64 (2014) 887e899896that for vwmax 0.1 m/s.considered to solve a large number of oil spills.

3.4. Effect of water velocity

Current as a carrier plays a crucial role in the migration of the oilow. Therefore, in this section, we have changed the water velocityto nd its effect on oil spill. The processes of oil spill from subma-rine pipeline to free surface at different water velocities are shownin Fig. 5. The larger the water velocity, the more obvious the tra-jectory of oil ow skewed to the downstream. The reason is thathigh-speed water exerts more shear stress on oil droplets andtransfers more kinetic energy to oil droplets. The maximum hori-zontal migrate distance for vwmax 0.04 m/s is 14.2 m (16 mminus1.8 m), 0.9 m less than that for v 0.07m/s and 2.5 m less thanFig. 7. Dimensionless time required for oil droplets which have the longest horizontal migratto collision and have greater chance of gathering into large droplets,as shown in Fig. 6. Though the water velocities are the same, largeactive faces of big oil droplets lead to great shear stress. Under theaction of shear stress, the maximum horizontal migrate distance,16.7m (18.5mminus 1.8m), presents in the case of d 0.05m. Thisdistance is about 1.5 times than the maximum horizontal migratedistance for d 0.01 m. Therefore, big-hole leaks may lead to moreserious consequences.

3.6. Dimensionless analysis

As shown in Fig. 7, the dimensionless time required for oildroplets which have the longest horizontal migrate distance whenthey reach the sea surface is analyzed and the tting formulas areobtained. We can clearly see that the larger the density of oil, thee distance when they reach the sea surface (vot/H) versus ro/rw, 10vwmax/vo and 100 d/H.

slower the oil leaking, or the smaller the leak size, the longer thedimensionless oil spill time is. All the three parameters meet thenatural logarithm distributionwell, and the adjust R-squares are alllarger than 0.98. The three tting formulas are as follows:

votH

6:14146 106 exprorw$

10:07163

4:2567 (18)

votH

3:40596 exp 10vwmax

vo$

10:13146

5:68636 (19)

votH

3:41601 exp 100d

H$

10:12289

5:43198 (20)

where t is the time required for oil droplets which have the longesthorizontal migrate distance when they reach the sea surface.

Taking Eq. (18) for example, if the actual parameters are thesame as that used in this paper such as H 13.9 m andrw 1025 kg/m3, this equation can be simplied as:

t 4:2683 105exp0:014ro 29:584 (21)Then we can select an arbitrary oil density substituted into the

formula to calculate the required time. For example, if the oildensity is set as 850 kg/m3, the time would be calculated as 35.86 s.

10vwmax/vo and 100 d/H, they meet the natural logarithm distri-butionwell, and the adjust R-squares are all larger than 0.99. Whilefor ro/rw, polynomial is the most appropriate one. These threetting formulas are as follows:

LfD 194:556 ro

rw 107:6361

rorw

2 118:7261 (22)

LfD 19:738 exp

10vwmax

vo$

10:25236

30:4548 (23)

LfD 21:4234 exp

100d

H$

10:09199

31:1406 (24)

where Lf represents the maximum horizontal migrate distancewhen oil droplets reach the free surface.

Here we take Eq. (22) for example, if the actual parameters arethe same as that used in simulation, this equation can be simpliedas:

Lf 0:1169ro 6:48 105r2o 71:236 (25)Based on Eq. (25), if the oil density is set as 850 kg/m3, the

maximum horizontal migrate distance when oil reaches the freesurface can be calculated as 18.69 m. For different oil leaking rates,

3.7. Prediction of the maximum horizontal migration distance

H. Zhu et al. / Energy 64 (2014) 887e899 897Using the same approach, we can calculate the required times fordifferent oil leaking rates, water velocities and leak sizes based onthe Eq. (19) and Eq. (20).

Fig. 8 shows the dimensionless longest horizontal distance thedroplets migrate when they reach the sea surface. The dimen-sionless longest horizontal distance increases with the increase ofleak size, while increases with the decrease of oil leaking rate. Withthe increase in oil density, the dimensionless longest horizontaldistance decreases rstly and then increases. But the value does notchange signicantly. Adopting the same method as above, we haveobtained three tting formulas about the dimensionless longesthorizontal distance versus ro/rw, 10vwmax/vo and 100 d/H. ForFig. 8. The dimensionless longest horizontal distance the droplets migrate whIn most cases, the leak in submarine pipeline is far away fromplatforms and shore. Thus, a few hours would be taken for rescueships to reach the accident location. When the rescue ships arrive,how long the horizontal migration distance is the major concern. Inwater velocities or leak sizes, we can also predict the maximumhorizontal migrate distance by solving Eq. (23) and Eq. (24). Theabove tting equations may provide some useful information forgovernment and oil business to adopt rapid response in case oilspill occurs.en they reach the sea surface (Lf/D) versus ro/rw, 10vwmax/vo and 100 d/H.

order to prevent further oil spilling, the maximum horizontalmigration distance of oil at certain time should cover all oil drop-lets, then the location is a proper place to lay oil containment boom.

As shown in Fig. 9, after oil droplets reaching the free surface,they will drift downward with current. So the process of oil spillconsists of rising process and drifting process. The rising processhas been analyzed in detail in the above, and we can predict whenand where the spilled oil droplets reach the maximum horizontalmigration distance when they reach the free surface. For driftingprocess, the main motion of oil droplets is moving downstreamalong the free surface. In this process, water is acting as a carrier.Therefore, the horizontal displacement of oil at free surface is afunction of surface water velocity. Although the arrival of oil

appears in the form of droplets. Under the action of watershear stress, oil droplets become more dispersed with theincrease in rising height. The rst oil droplet reaching the freesurface is not the one which has the longest horizontalmigrate distance for the size of oil droplet varying with time.For standard case, 39 s is required for oil reaching the longesthorizontal migrate distance when it reaches the surface, andthe longest horizontal migrate distance is 16.7 m.

(2) Vertical upward force is small for high density oil, leading toslow rising rate of droplets. For ro 960 kg/m3, the requiredtime for the maximum horizontal migration is about 1.84times as long as that when oil density is 780 kg/m3. However,the difference in the maximum horizontal migration dis-tance is little for different-density oil ow. Due to high-speed

H. Zhu et al. / Energy 64 (2014) 887e899898droplets causes the slight undulation of sea surface, it has a littleeffect on the horizontal displacement of oil droplets. Even thoughits effect is considered, the undulation of sea surface just plays asobstacle for oil migration. Therefore, the following equation can beused to predict the maximum horizontal migration distance of oilat certain time:

L Lf atr tvwmax (26)

where L represents the maximum horizontal migrate distance of oilat certain time, tr is the certain time and a is a coefcient, which canbe taken as 1.0e1.2. A larger a corresponds to a longer migrationdistance, which can cover all oil droplets. Since the undulation ofsea surface and collision of oil droplets exert resistance on oilmigrate, the actual value of amay less than 1. However, we take a as1.0e1.2 to ensure the prediction more secure and reliable.

Lets take the standard case for example. If 2 h are needed forrescue ships to reach the accident location, themaximumhorizontalmigration distance of oil when rescue ships arrive is 876.02 m,which is calculated based on Eq. (26) where a is taken as 1.2.

4. Conclusions

An approach, for predicting the process of oil spill under theaction of current with shear velocity distribution, by nite volumesimulation combined with VOF method is proposed. Effects of oildensity, oil leaking rate, leak size and water velocity are examined.And tting formulas are obtained to predict when and where to seeoil reaching the sea surface. Finally, a forecasting model of themaximum horizontal migration distance of oil at certain time isproposed. These results provide useful guidance to place the oilcontainment boom. Based on our numerical results, we draw thefollowing conclusions:

(1) Continuous oil ow stream presents just releasing from theleakage holewith the height of about 3m. Then it is tore apartat a certain depth (for standard case it is 3 m) by current andFig. 9. The whole proleaking oil has more ascending kinetic energy, for vo 5 m/s,just 15 s is needed for oil to reach the maximum horizontalmigrate distance, while it is 79 s for vo 1 m/s. Since theshearing action times of water are different, the maximumhorizontal migrate distance for vo 1 m/s is 5.2 m longerthan that for vo 5 m/s. High-speed water exerts more shearstress on oil droplets and transfers more kinetic energy to oildroplets. Therefore, the longest horizontal migrate distancefor vwmax 0.04 m/s is 14.2 m, 2.5 m less than that forvwmax 0.1 m/s. Because the route of oil droplet is the squareroot of the sum of the squares on vertical depth and hori-zontal displacement, the route of oil droplet is relative smallin vwmax 0.04 m/s, resulting in the short rising time. Thebigger the diameter of leak, the larger the amount of releasedoil and the greater the upward momentum is. As leakage sizereduces from 0.05 m to 0.01 m, the required time decreasedby 23.53 percents. The maximum horizontal migrate dis-tance presenting in the case of d 0.05 m is about 1.5 timesthan that for d 0.01 m.

(3) The larger the density of oil, the slower the oil leaking, or thesmaller the leak size, the longer the dimensionless oil spilltime is. The three tting formulas of dimensionless timemeet the natural logarithm distribution well. The dimen-sionless longest horizontal distance increases with the in-crease of leak size, while increases with the decrease of oilleaking rate. With the increase in oil density, the dimen-sionless longest horizontal distance decreases rstly andthen increases. The two formulas for dimensionless rate anddimensionless leak size meet the natural logarithm distri-bution, while the formula for dimensionless density meetsthe polynomial. Using the formulas we can obtain when andwhere to see oil reaching the sea surface, and conduct rapidresponse. We have taken ro 850 kg/m3 for example toshow the using method of formulas. The calculated resultsare 35.86 s and 18.69 m.cess of oil spill.

(4) The process of oil spill consists of rising process and driftingprocess. For drifting process, the main motion of oil dropletsis moving downstream along the free surface, and water isacting as a carrier. A forecasting model, L Lf a (tr t)vwmax, is proposed to calculate the maximum horizontalmigration distance of oil at certain time. For standard case, itsvalue would be 876.02 m if 2 h are needed for rescue ships toreach the accident location.

Acknowledgments

Research work was supported by the special fund of ChinasCentral Government for the development of local colleges anduniversitiesdthe project of national rst-level discipline in Oil andGas Engineering (P019) and Key Project of Sichuan Provincial Ed-ucation Department (No: 12ZA189). Without the support, this workwould not have been possible.

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A CFD (computational fluid dynamic) simulation for oil leakage from damaged submarine pipeline1 Introduction2 Simulation method2.1 Governing equations2.2 Numerical method2.3 Computational domain and mesh2.4 Boundary conditions

3 Numerical results and discussion3.1 Standard case3.2 Effect of oil density3.3 Effect of oil leaking rate3.4 Effect of water velocity3.5 Effect of oil leak size3.6 Dimensionless analysis3.7 Prediction of the maximum horizontal migration distance

4 ConclusionsAcknowledgmentsReferences