Passive pollutant transport in the Arabian Gulf

Passive pollutant transport in the Arabian Gulf R. W. Lardner

Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada H. M. Cekirge, A. H. AI-Rabeh and N. Gunay

Water Resources and Environment Division, Research Institute, Kin9 Fahd University o[ Petroleum and Minerals, Dhahran, Saudi Arabia A convection-diffusion model is formulated to simulate transport of passive or noninteracting pollutants in the Arabian Gulf. The pollutant is assumed to be a large ensemble of small discrete quantities. The elements of the ensemble are subject to transport by convection and diffusion. The convection is due to movement of the surrounding water body where current velocity is obtained from a three-dimensional hydrodynamic model of the Arabian Gulf. The diffusion of the pollutant is simulated by the Monte Carlo method. A case study of discharge of a pollutant in the Arabian Gulf is presented.

1. I N T R O D U C T I O N The adverse effects of the spilling of pollutants in water have focused attention on the behaviour of pollutants. Over the past few years a lot of research has been directed towards the development of mathematical models to describe the behaviour and fate of pollutants such as oil and oil products. Such research has primarily been motivated by the practical consideration that a successful model will be of greater value in predicting where to deploy containment and collection systems to mitigate the effects of the pollutant on the environment. Generally, the behaviour of spilled pollutants can be affected by physical, chemical, biological processes. These may include advection, spreading, evaporation, dissolution, emulsification, dispersion, auto-oxidation, biodegradation, and sinking/sedimentation ~. Transport processes have been studied extensively. Surface transport and, more recently, subsurface transport, have received attention. Recent studies utilize three-dimensional hydrodynamic models; for a survey on the topic 1. Models describing the evaporation, emulsification, and dispersion processes contain a high degree of empiricism as the mechanisms driving these processes are still not well understood~'2 In this paper, we will study the convection-diffusion processes in the Arabian Gulf. It is assumed that the pollutant is passive or noninteractive. The Monte Carlo simulation technique is applied to track the trajectories of pollutants which are assumed to consist of small discrete quantities or elements each of which is moving randomly, and independently. Thus, a Lagrangian approach is used to simulate convection of pollutant while the random walk technique is used to simulate the turbulent diffusion component. The technique offers the advantage of simplicitly and elimination of numerical instability which may be introduced if numerical techniques are used to solve the transport equations. The three-dimensional hydrodynamic model of the Arabian Gulf is used to compute the velocity of the transporting medium (see Refs 3-7). A discharge of a pollutant from a pipe at the bottom of the Arabian Gulf ~ Computational Mechanics Publications 1988

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Adv. Water Resources, 1988, Volume 11, December

near the Arabian coastline is simulated. The results are presented in Section 3. 2. ANALYSIS Assume that the pollutant is divided into a large set of discrete elements. These elements are convecting with the surrounding water body and diffused as a result of random processes. The convection element is simulated through a Lagrangian approach where the current velocities are obtained from the three-dimensional hydrodynamic model of the Arabian Gulf 3 ~. The brief formulation of the model will also be given in this paper. The dispersive velocity component is modelled by using the random walk method. Based on the studies of Ref. 2 and Okubo s, the vertical and horizontal diffusion coefficients are selected and used in the model. The governing equation of convection-diffusion is: ?p ~+V.Vp=V.{DVp)+S

(1)

where, p is the density of the pollutant, t is time, V is the convective velocity of the surrounding medium, V is the gradient operator, D is the dispersion tensor, and S is the sink or source term. Equation (1) can be solved by using a variety of numerical techniques, including finite differences and finite elements. However, in this study the Monte Carlo solution technique is adopted due to its simplicity and elimination of numerical instability that is associated with other numerical schemes. A significant factor in the transport of pollutants is the convective velocity field,

V=V[u(x,y,z,t), v(x,y,z,t), w = 0 ]

(2)

which is obtained from the three-dimensional hydrodynamic model of the Arabian Gulf. In equation (2), u, v, and w are the velocities in x, y, and z direction, respectively; x, y, and z are the Cartesian coordinates; z is the vertical direction, and t is time. As the vertical

Passive pollutant transport in the Arabian Gulf." R. W. Lardner et al. convection is small compared with the horizontal convection, we assume that the vertical velocity is zero, w - 0 . The pollutant elements convect with the velocity field, V. If we assume that there are n elements of pollutant in the medium, where the coordinates of the ith element at a given time are X °, yo, and Z °, then the coordinates of the ith element after Atp time step are given by,

Xi= X° +uAtp

(3)

Yi= yo + vAtv

(4)

Zi=Z °

(5)

where d~m5is the root mean square distance and D h is the horizontal diffusion coefficient. For any individual element, the diffusion step size, d, is generated randomly by, d = [R]~,:

o~
(7)

where [R]~ is the random number in the interval o to r, and R is an integer. The value of r is chosen so that d,ms must equal the mean square of all values of d. As the random number varies from 0 to 1, and using

[,i'~2~112,1,~

,8,

where Atp is the time step selected for observing the pollutant transport. The pollutant elements diffuse in the fluid medium• Diffusion should be considered as horizontal and vertical dispersion. An average element travels in time Atp (Ref. 9),

then, the distance that any element travels by diffusion is

d,-ms= x/4DhAt v

The diffusion component must be added to the horizontal

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The location q/" the spill in the coarser grid of the Arabian Gulf used in the three-dimensional hydrodynamic model

Adv. Water Resources, 1988, Volume II, December

159

Passive pollutant transport in the Arabian Gulf: R. W. Lardner et al. travel of the element as, X i= X°i + uAtp + dh cos 0

(1o)

Yi = yo + vAtp + dh sin 0

(11)

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where D~, is the vertical diffusion coefficient. The form in (13) covers the transport of the element upward and downward when the random number varies from 0 to 1, the vertical position of the element is, Z~=Z~ +dv/H

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where H is the total height of the water column. If the element moves out of the bottom or top boundaries, then the following will reflect the element at I

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the boundaries for as many times as required to ensure its position within the water body, Z , = [!Z*]- 2([IZ* l + 1]/2) I

(15)

where II is the absolute operation, Z* is the vertical coordinate of an element which moved out off the boundaries, and ( ) is the integer operator. Equations (10), ( 11 ), (14), and (I 5 ) define the location of the ith element after Atp time step. The number of the elements which arrived at every grid point is counted to find the vertical and horizontal distribution of the pollutant. This can be done easily by converting Xi, Yi, and Zi coordinates to the grid points and then by tallying elements at every grid point. By noting that Z~ varies between 0 and 1, ten grid points were selected in the vertical direction.

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Adv. Water Resources, 1988, Volume 11, December

3. APPLICATION A spill at the bottom of the sea at grid point (24,6), see Fig. 1, is used for an example. The spill rate is 100 units per second. A wind blows 135 degrees from north, and it increases linearly from 0 to 40 knots. It should be noted that the average speed of wind in twenty-four hours is 20

Passive pollutant transport in the Arabian Gulf: R. V¢~ Lardner et al. 1

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which varies from a = 0.0 at the b o t t o m to a = 1.0 at the surface. Fig. 4 presents the h o r i z o n t a l p o l l u t a n t d i s t r i b u t i o n in p o l l u t a n t unit per square meter in the whole water c o l u m n , i.e., from a = 0.0 to a = 1.0, after 24 hours. In F i g s 2, 3, and 4, S is the area of the mesh cell which equals 25 × 10 ° square meters.

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4. C O N C L U S I O N S

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A passive p o l l u t a n t t r a n s p o r t m o d e l for the A r a b i a n G u l f is studied in this paper. T h e m o d e l is reduced to a single e q u a t i o n with given p a r a m e t e r s for c o n v e c t i o n a n d diffusion. The solution is o b t a i n e d using the M o n t e C a r l o m e t h o d . T h e m o d e l f o r m u l a t e d in this p a p e r provides estimates of the t r a n s p o r t a t i o n of n o n i n t e r a c t i n g p o l l u t a n t s in the A r a b i a n Gulf.

26

ACKNOWLEDGEMENTS W e t h a n k the K F U P M Research Institute a n d the A r a b i a n A m e r i c a n O i l C o m p a n y for s u p p o r t of this research effort u n d e r K i n g F a h d University of P e t r o l e u m a n d M i n e r a l s Research I n s t i t u t e C o n t r a c t N o . 24079, a n d the A r a b i a n A m e r i c a n Oil C o m p a n y a n d the Saudi A r a b i a n M i n i s t r y of P e t r o l e u m a n d Minerals Resources for their a u t h o r i z a t i o n to publish this paper.

REFERENCES

I SO

51

Fi 9. 4. Horizontal pollutant distribution C, in pollutant units per square meter, after 24 hours in total water column: i.e., from a=O.O to a = 1.0: S =area of the 9rid cell and equals to 25 × 106 square meters knots. The h o r i z o n t a l a n d vertical diffusion c o n s t a n t s are chosen as Dh= 1000m2/sec and D~,=0.01 m2/sec (Refs 6 a n d 7). H o w e v e r , these values should be selected a c c o r d i n g to the type of p o l l u t a n t spilled in the sea. The area a r o u n d the grid point (24,6) is magnified to a grid of (41 × 21) with a mesh size of a p p r o x i m a t e l y 5 kilometers. The t r a n s p o r t of p o l l u t a n t is observed in this grid. Figs 2 a n d 3 present the h o r i z o n t a l d i s t r i b u t i o n of the p o l l u t a n t in p o l l u t a n t units per square meter at 24 hours after the spill on the surface (a = 1.0) a n d on the b o t t o m (a = 0.0) respectively, a is the vertical c o o r d i n a t e

1 Huang, J. C. A review of the state of the art of oil spill fate, behavior models, Proceedings of Oil Spill Conf., API. Washington DC, 1983 2 Lardner, R. W., Cekirge, H. M. and Gunay, N. Numerical solution of the two-dimensional tidal equations using the method of characteristics, Comput. Math. Applic., 1986, I2A(10t, 1065 1080 3 Cekirge, H. M.. Lardner, R. W. and Fraga. R. J. Adaption of the solution of the two-dimensional tidal equations using the method of characteristics to wind-induced currents and storm surges, Comput. Math. Applic., 1986, 12A(10), 1081-1090 4 Lardner, R, W. and Cekirge, H. M. An efficient three-dimensional algorithm for the computation of wind-driven circulation, Engng. Analysis, 1987, 4(2) 89-94 5 Lardner, R. W. and Cekirge, H. M. A new algorithm for threedimensional tidal and storm surge computations, Applied Mathematical Modelling, in press 6 Cekirge, H. M., AI-Rabeh, A, H. and Gunay. N. Implementation of the three-dimensional hydrodynamic model for the Arabian Gulf, (submitted for publication) 7 Okubo, A. Ocean diffusion diagrams, Deep Seu Research. 1971, 18, 789-802 8 Nihoul, J. Modelling of Marine Systems. Elsevier Pub. Com., Amsterdam, 1975 9 Ahlstrom, S. A Mathematical Model for Predicting the Transport of Oil Slicks in Marine Waters. Batelle Laboratories, Richland, Washington, 1975

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