A 2-D Hybrid particle tracking/eulerian-lagrangian model for oil spill problems

Transcription

1 Indian Journal of Geo-Marine Sciences Vol. 42(1), February 2013, pp A 2-D Hybrid particle tracking/eulerian-lagrangian model for oil spill problems A. Attari Moghaddam A * & B. Dabir Chemical Engineering Department, Amirkabir University of Technology, Tehran, Iran * [ a.attari.m@aut.ac.ir] Received 27 June 2011; revised 20 October 2011 Present study consists a 2-D Random Walk Particle Tracking (RWPT) approach combined with Eulerian Lagrangian concentration model (ELM) to the problem of forecasting the dispersion of oil in open waters. For each oil spill processes the most trustful method that has presented so far is used. Application of the model in the Caspian Sea for a hypothetical spill is investigated. Results strongly confirm the use of a RWPT for the near field and ELM for the far field when modeling dispersion of oil in coastal waters. [Keywords: Oil spill, Particle Tracking, Eulerian-Lagrangian, Coastal Waters] Introduction Eulerian and Lagrangian models are widely used in coastal dispersion problems especially oil spill problems. One of the shortcomings of Eulerian Models (EM) is that they can t determine the concentration profile on scales less than the coordinate scale of grid 1. The second drawback of EMs is that they can just predict turbulent diffusion phenomena of oil spill by Fickian approach. While, field observations have confirmed that the movement of pollutants on the sea surface is non-fickian, with the dimension of the diffusing cloud is proportional non-linearly with time 2,3. Whereas Lagrangian memory of the element has a significant role in forecasting the path of next steps 4. Another limitation of EMs is the computational effort is spread evenly across the domain and this will increase the computational cost. According to the defects of EM, recently Lagrangian Models (LM) have been widely used in modeling oil spill problems 5-6,7 A shortcoming of LMs is that, even utilizing powerful computers, the element number must be limited because the storing and tracking of the history of each element needs huge memory and computational time. When such methods are used to generate concentration profile of oil, numerical errors appear during the conversion from the particle location to the concentration as the last step of RWPT in a fixed computational grid system. Thus, it is suggested that RWPT be used only in regions where it is completely essential. In the present study a combined model has been developed 8,9-10 in which, near regions of steep coordinate gradient, oil is considered as a large group of elements, each of which has a mass value. Also a non-fickian fractional Brownian motion (fbm) has been considered in this area. For the far field an ELM using 6-point method of characteristics has been applied. Application of the model for a hypothetical oil spill from Azerbaijan oil fields is investigated. Materials and Methods Oil Spill Processes The most important processes which affect the fate and spreading of an oil slick at sea are spreading, advection, turbulent diffusion, evaporation, vertical dispersion and emulsification. They are comprehended with different degrees of certainty and can be explained by mathematical models to some extent based and validated on experimental results. The character of the different processes will be considered and a complete algorithm is suggested to forecast their combined effect on the fate and spreading of oil spill at sea. Spreading is the horizontal growth of an oil slick as a result of mechanical forces such as inertia, gravity, viscous, and interfacial tension. Even though the conventional spreading equations presented by Fay 11 shape the fundamental of the most common spreading equations today, it is extensively identified that oil spreading cannot be completely described by these algorithms. Lehr et al. 12 intended an elongated ellipse

2 ATTARI & DABIR: A 2-D HYBRID PARTICLE TRACKING MODEL 43 along the direction of the wind to give an explanation for the detected non-symmetrical spreading of oil slicks as in equations 1 and 2: Δ Δ (1) Δ Δ Δ (2) where Q is the length of the minor axis, R is the length of the major axis, is the water density, is the oil density, V 0 is the initial volume of spill, W is the wind speed and t is the time. In order to make the notation given here simpler, the X axis is taken as adjusted in the direction of the wind. The slick is regarded as a shell of similar concentric ellipses with major and minor axes r and q, the boundary of which is that ellipse with R and Q the lengths of its major and minor axes, respectively. It is supposed that the slick elongates uniformly to some extent as the ratio of the lengths of the major (and minor) axes of an interior ellipse and the boundary ellipse is a constant over time. If the coordinates of the particle relative to the principal axes of the ellipse are (X,Y), we write cos, sin. Then the particle is displaced outwards with the same elliptical angle, that is 10 : cos (3) sin (4) Advection is a physical process dominated by winds, currents, and waves. The position of each particle at time i+1 due to advection in x and y directions can be computed by equations 5 and 6 respectively: α u u (5) α v v (6) where u is the surface water current velocity, which can be obtained from a 3-D hydrodynamic model; u is the wind velocity at 10 m above the water surface, whose deflection angels vary between 0 o and 25 o to the right side in the northern hemisphere to account for the rotation of the earth; is the current factor, usually selected as 1.0; and α is the wind drift factor, usually set at 0.03 T. T is the transformation matrix, which allows for introduction of the deviation angle 13 : cos (7) cos Where 40 8 is the deflection angle (degrees, clockwise in Northern Hemisphere). As well as the conventional displacements due to the mean current, oil droplets undergo a random walk process due to the turbulent diffusion. Stochastic velocity of drifter can be correlated with the time scale and the diffusion coefficient. Sanderson and Booth 4 observed that the tracking of satellite-tracked ocean surface drifters may be explained as fractional Brownian motion (fbm) with non-fickian scaling characteristics. The standard deviation σ of a diffusing cloud of fbm particles follows the following relationship 4 : σ 2Dt H (8) Where D is the fractal diffusion coefficient and H is the Hurst exponent. By analyzing the tracking of satellite tracked ocean surface drifters, the Hurst exponent H takes on an average value of about Equation 9 for estimating the movement of drifters due to fbm has been used: (9) Where BH(ti) is the ith discrete approximation to the fbm at time ti. M is limited memory used in the approximation of the fbm, and R(i) are random steps discretely sampled from a Gaussian probability distribution. The main advantage of fbm particle tracking model is that the Hurst exponent flexibly dominates the size of the diffusion cloud, while the particle cloud of regular Brownian motion is limited to diffuse in proportion to the time since release rose to the power of 0.5. Evaporation is a critical procedure that changes oil mass balance and physicochemical characteristics during the first hours of an oil spill. The rate of evaporation is related to the physicochemical characteristics of the oil as well as sea water temperatures, winds and other processes such as spreading and emulsification. Pseudo-component 14 and Analytical are two approaches for simulating the evaporation process of oil spilled on free surfaces.

3 44 INDIAN J. MAR. SCI., VOL. 42, NO. 1, FEBRUARY 2013 Here the analytical approach because it s more reliable results has been used. The most common relationship to forecast losses due to evaporation is equation 10 presents as below 15 : 1 ln 1 (10) In which is fraction evaporated, V is initial spill volume, C is constant dependent on temperature, A is area of the spill (thick part), R is gas constant, T is environmental constant, v is molar volume, t is time, K m is mass transfer coefficient which is proportional to U 0.78 (U is wind velocity), P 0 is liquid vapor pressure given by In P 0 =constant (1 T 0 /T) and T 0 is boiling point temperature. One of the most important problems in the oil spill modeling is a simple and yet truthful quantitative explanation of process of oil droplet formation, size distribution and dynamics. Oil dispersion near the surface can be governed by a lot of factors, of which the wave breaking phenomenon is the most important and very complicated. Following Mackay et al. 16 it is supposed that large droplets supply the oil resurfacing, and droplets smaller than some threshold radius, will permanently stay in the water column. Balancing the breaking wave energy and the droplet buoyancy, Tkalich and Chan 17 derived relationships for the oil mass vertical exchange between the slick and the dispersion layer, leading to equations 11 and 12: K O Λρ O hz C (11) C K Λρ O hz C (12) Where C em is oil concentration in dispersion layer, and h is thickness of oil slick. Other parameters used in these equations are computed as in 18. Emulsification is the process of the formation of water-in-oil emulsions that change the properties and characteristics of oil to a huge quantity. Mackay et al. 16 developed an equation to calculate the water uptake as a result of emulsification as follows: ln1 2/5/1 K t (13) Where W is the water content fraction of the emulsion; t is time; K 1 is a rate constant for water incorporation; K 2 is a rate constant dependent on coalescing tendency; and K 3 is a rate constant for water incorporation= W 2 in which Wind speed, W, is given in km/h. Many models used the above equation for modeling emulsification. Nakata 19 found that Mackay's relationship matched with their data for K 1 =0.926, K 2 =1.363, and K 3 =4.1/day. Oil Spill Model In the oil spill model presented here, Random Walk Particle Tracking (RWPT) model is used in the regions near the source point for tracking of oil particles in the coastal environment. Also, 2-dimensional Eulerian-Lagrangian (EL) model is used in the far field in order to achieve the most accurate results. These two numerical methods were merged by forming a hybrid method in which RWPT model was coupled with the EL model by projecting the number of particles into an Eulerian control volume. Random Walk Particle Tracking Method (RWPT) The particle tracking approach needs a powerful computing equipment to track each particle in a collection of particles spreading over the study area. The basis of the advective Lagrangian particletracking trajectory method is the following vector relationship:, (14) Where x i is the i th particle coordinate and V a the advective velocity due to wind and current at the particle coordinate. The hydrodynamic model determines the advective velocities at discrete times, t n for n= 0, 1, 2,..., and stores them in a binary file to save memory and reduce access times. The method used to solve the system of ordinary differential equation 14 is the 4 th Rung-Kutta. This method involves four velocity evaluations and the resulting equation for the position of every particle at time n+1 is as follows 6 : 2 2 (15) Where a i, b i, c i and d i are velocity evaluations. The total particle displacement at time step i is the summation of the advective component, diffusive component (i.e., the increment of fbm) and mechanical spreading over the time interval: (16) (17)

4 ATTARI & DABIR: A 2-D HYBRID PARTICLE TRACKING MODEL 45 Where ( ) is the position of each particle at time n, ( ) is displacement due to advection, ( ) is displacement due to fbm and ( ) is displacement due to spreading. In order to test the practicability of the RWPT model, we first applied it to a simple system with known analytical solutions. Under the assumption of uni-directional, uniform flow from left to right, the analytical solution for instantaneous release of mass M on a flat bed can be written as 20 : C M exp D D (18) The results presented in Figure 1, demonstrate that the first stage of the RWPT has good agreement with the analytical solution, but that after this stage, the PTM swerves a little away from the analytical solution due to the random characteristic of the particle displacements. These results recommend that PTM can be logical near the source point, where there are high concentration gradients. However PTM is not applicable in the regions from the source point where the ambient turbulent field governs. EL Model The ELM used here is based on the research of Gane 21. By applying an ELM it is possible to use large Courant numbers and this will definitely reduce the computational time. This characteristic of ELM is especially important in marine environment in which flows are governed by advection and therefore the Courant number is high. The governing equation of mass transport of oil in a 1-dimensional domain can be written as: u D S (19) Where is concentration of oil in oil slick, x is the coordinate, t is time and S is the amount of oil lost by physical or chemical processes. According to linearity of the above equation, splitting methods used to divide it into two equations called respectively pure advection and pure diffusion as: u 0 (20) S (21) Pure advection equation states that the concentration remains constant along characteristics line 22. As shown in figure 2, for calculating the node concentration (j, n), using known concentrations at time n-1, a characteristic line between the unknown node and the previous time is drawn. The characteristics line defined as below: (22) After obtaining position of characteristics line, a 6-point method of characteristics is used to interpolate the concentration of oil at foot of characteristics line. Model Validation Tkalich 18 has done a simulation of hypothetical oil spill in a simplified case. RWPT/EL model validation is done by comparing results for mentioned problem. Figure 3 shows the comparison and there is a complete agreement between the model and Tkalich results. Fig. 1 Compariso of random walk particle distribution (dashed lines) with analytical results (continuous lines) on flat bed.

5 46 INDIAN J. MAR. SCI., VOL. 42, NO. 1, FEBRUARY 2013 For the simulation, advection, diffusion, spreading and vertical dispersion are considered. As in Tkalich 18, a surface water current of 0.1 m/s from west to east and a wind velocity of 5 m/s from south to north is took into account. As is expected, the movement of slick is in direction of superposition of wind and current. The tail of slick is formed in direction of wind that is in agreement with Elliott 23 which states the shear diffusion caused by wind will result in tail-like elongation of slick. Application of model in Caspian Sea The Caspian Sea extends zonally from 46.6 E to 54.8 E and meridionally from 36.6 N to 47.0 N. Over 60% of the sea is shallower than 100 m. There are two relatively deep basins (about 600 and 800 m, respectively) in the central and southern parts of the sea. The Caspian Sea is an enclosed sea with major freshwater input from the Volga River, balanced by evaporation. There has been oil pollution issue in the Caspian Sea for many years, but the problem has Fig. 2 Characteristic line from time n to n-1. become specifically critical in recent years because of the development of new, offshore oil wells 24. For modeling, a hypothetical instantaneous spill at 40 N and 52 E is considered. The amount of oil released is m 3 and density of oil is 890 kg/m 3. Other constant parameters used in the model are presented in Table 1. The mean surface current extracted from hydrodynamic model is shown in Figure 4. Wind profile is considered as in Korotenko et al. 25. A clockwise wind direction from northwest to west with constant amount of 5 m/s is applied. Results For near field regions where there is steep concentration gradient ( 0.001), RWPT is used. Table 1 Parameters used for application of model in Caspian Sea Parameter Amount Oil density (kg/m 3 ) 890 Initial oil volume (m 3 ) Initial particles 2000 Water density (kg/m 3 ) 1000 Ambient temperature () 8 Oil spill location 40, 52 PTM time step (sec) 1000 ELM time step (sec) 100 Grid size (m) Modeling duration (day) 16 Fig. 3 Comparison of Tkalich results and PTM/ELM model (14.4hr and 28.8hr after spill).

6 ATTARI & DABIR: A 2-D HYBRID PARTICLE TRACKING MODEL 47 Fig. 5 Thickness counters of oil slick in 4 days time interval (greater than 2 mm, between 0.5 to 2 mm and less than 0.5 mm thickness). Fig. 4 Mean surface current of Caspian Sea. With going away from the source point, concentration gradient will reduce until the magnitude of that become Thereafter ELM will used to predict the oil slick trajectory. Results are shown in figure 5. As seen from this figure oil is transported from spill location to Iranian coastline. Also thickness counters of slick are presented in the figure. The simulations state that after 15 days the slick is located in Iranian shoreline border. These findings are in good agreement with Korotenko et al 26 results. Figure 6 shows the oil slick maximum thickness changes. As time is passed the oil is spread and therefore the maximum thickness of slick is reduced. The rate of reduction for initial hours of spill is much more than the final hours of simulation. Figure 7 shows the oil slick balance. Evaporation and vertical Fig. 6 Oil slick maximum thickness versus time after spill slick balance. dispersion are incorporated in reducing mass of oil while the effect of evaporation in mass reduction is very intense. For initial hours of spill the rate of change of mass evaporated is very high. The reason is that light components of oil that locate on the upper layer of slick evaporate in these hours. After time is passed the rate of evaporation is reduced because of the reduction of light fractions of oil in slick layer.

7 48 INDIAN J. MAR. SCI., VOL. 42, NO. 1, FEBRUARY 2013 Fig. 7 Oil slick balance. Discussion A 2-dimensional hybrid oil spill model has been developed to predict the trajectory of surface oil slicks, the mass balance of oil spill, and the oil particle concentration distribution. In modeling the diffusion process, a discrete method is used to generate fractional Brownian motion. The application of the fbm technique causes achieving a more truthful prediction of oil slick tracking than regular Fickian modeling. To preserve the conservation law of material transport for advection-dominated coastal regions such as Caspian Sea, we used ELM scheme. However, to reduce numerical errors related to concentration models in regions with high concentration gradient, we used a RWPT scheme for near field only. Because of RWPT application in the far field involves much excessive computation and can give unreal results, it was applied only in the near field. Simple model tests demonstrated that the critical value of concentration gradient for switching from RWPT to ELM is Simulations of the spreading of oil spill using the proposed schemes showed trends in reasonable agreement with Tkalich 18 results. Also with applying the model in Caspian Sea, results prove there is good agreement with Korotenko et al. 26 results. Thus, the proposed hybrid method can be applied in coastal regions with satisfactory results. References 1. Yapa PD, Shen HT, Wang D, Angammana K. An Integrated Computer Model For Simulating Oil Spills In The Upper St. Lawrence River. J. Great Lakes Res (2): Okubo A. Oceanic diffusion diagrams. Deep-Sea Research. 1971;18: Osborne AR, Kirwan AD, Provenzale A, Bergamasco L. Fractal drifter trajectories in the Kuroshio extension. Tellus. 1989;41A: Sanderson BG, Booth DA. The fractal dimension of drifter trajectories and estimates of horizontal eddy-diffusivity. Tellus. 1991;43A: Badri MA, Azimian AR. Oil spill model based on the kelvin wave theory and artificial wind field for the Persian Gulf. Indian Journal of Marine Sciences. 2010; 39 (2): GARCIA-MARTINEZ R, FLORES-TOVAR H. Computer Modeling of Oil Spill Trajectories With a High Accuracy Method. Spill Science & Technology Bulletin. 1999; 5 (5/6): LONIN SA. Lagrangian Model for Oil Spill Diffusion at Sea. Spill Science & Technology Bulletin. 1999; 5 (5/6): Neuman SP. Adaptive Eulerian Lagrangian finite element method for advection dispersion. International Journal of Numerical Methods in Engineering. 1984;20: Zhang XY. Ocean outfall modeling interfacing near and far field models with particle tracking method, MIT; Guo WJ, Wang YX. A numerical oil spill model based on a hybrid method. Marine Pollution Bulletin. 2009;58: Fay JA. Physical processes in the spreading of oil on a water surface. Paper presented at: Proc., Joint Conf. on Prevention and Control of Oil Spills1971; Washington D. C. 12. Lehr WJ, Cekirge HM, Fraga RJ, Belen MS. Empirical studies of the spreading of oil spills. Oil and Petrochemical Pollution. 1984;2: Samuels WB, Huang NE, Amstutz DE. An oil spill trajectory analysis model with a variable wind deflection angle. Ocean Engineering. 1982;9: Habibi S, Torabi AM, Bidokhti AA. A numerical model for the prediction of movement of gas condensate from spill accidents in the Assalouyeh marine region, Persian Gulf, Iran. Indian Journal of Marine Sciences. 2008; 37 (3): Stiver W, Mackay D. Evaporation rate of spills of hydrocarbons and petroleum mixtures. Environment Science & Technology. 1984;18: Mackay D, Buist I, Mascarenhas R, Paterson S. Oil Spill Processes and Models. Ottawa Tkalich P, Chan ES. Vertical mixing of oil droplets by breaking waves. Marine Pollution Bulletin. 2002; 44 (11): Tkalich P. A CFD solution of oil spill problems. Environmental Modelling & Software. 2006;21: Nakata K. Oil spill fate model developed in Japan and future plan. Journal of Advanced Marine Technology Conference Kang YH. Eulerian and Lagrangian Approaches to Simulate Solute Transport in a Rectangular Harbour, University of Bradford; Gane S. Solution of the advection equation using finite difference schemes and the method of characteristics, University of Glasgow; Dimou K. 3-D HYBRID EULERIAN-LAGRANGIAN / PARTICLE TRACKING MODEL FOR SIMULATING MASS TRANSPORT IN COASTAL WATER BODIES: MASSACHUSETTS INSTITUTE OF TECHNOLOGY Elliott AJ, Hurford N, Penn CJ. Shear diffusion and the spreading of oil slicks. Marine Pollution Bulletin. 1986;17: Mamaev V. The Caspian Sea: European Environment Agency; 2002.

8 ATTARI & DABIR: A 2-D HYBRID PARTICLE TRACKING MODEL KOROTENKO KA, MAMEDOV RM, MOOERS CNK. Prediction of the Transport and Dispersal of Oil in the South Caspian Sea Resulting from Blowouts. Environmental Fluid Mechanics. 2002;1: Korotenko KA, MAMEDOV RM, MOOERS CNK. Prediction of the Dispersal of Oil Transport in the Caspian Sea Resulting from a Continuous Release. Spill Science & Technology Bulletin. 2000;6(5/6):