12 ème Congrès de Mécanique 21-24 Avril 2015 - Casablanca (Maroc)

COMPUTATIONAL MODELLING Materials and methods This study used numerical model predictions. Initially, OF TIDAL EFFECT ON bathymetric and sea-level data were calculated as a WASTEWATER DISPERSION IN preliminary assessment for the hydrodynamic characteristics of the bay. These data were used then to COASTAL BAY: CASE OF assessment a case of pollutant dispersion over idealized TANGIER’S BAY . tidal cycle. Sub-sections describe the study area and summarize the essential steps of mathematical and JEYAR M. 1, CHAABELASRI EM. 1,* , SALHI N. 1 numerical model formulations. a. Description of the system 1. LME, Faculté des Sciences, Université Mohammed Premier, 60000 Oujda, Maroc * Auteur correspondant (chaabelasri @gmail.com) The Bay of Tangier (Cf. FIG. 1) is a semi-enclosed shallow basin located on the west coast of the southern Abstract: Strait of Gibraltar. The bed topography is spatially quite This paper describes a numerical modelling of the process non-uniform, with the mean water depth increasing of coastal pollution by wastewater in the Bay of Tangier. progressively from 0 m at the shore to about 50 m at the The numerical model is based on Shallow Water interface with the greater Mediterranean Sea. The vertical equations used for the hydrodynamics and an equation of flow structure in the Bay of Tangier consists of an upper advection-diffusion for describes the wastewater layer of incoming cold fresh surface Atlantic water assimilated of a passive pollutant, then, the obtained overlying a deep current of outgoing warmer salt water model is resolved using a unstructured triangular finite [4]. The bay is heavily used by shipping, whose volume volumes method. The found results show that: (i) the tidal has grown since the recent development of a flows have a direct effects on direction of wastewater Mediterranean port at Tangier. Assessment of the dispersion that can used to control its evacuation out of environmental risk is therefore required, and the present bay (ii) with the numerical model used we can modelling numerical tool is being developed with this longer-term domains that have more complicated geometry and aim in mind. In this work, the domain area was bathymetry, and this, in a large time scale. discredited in unstructured meshes composed of triangles. Along the simulation time we use the same mesh Key words: Pollutant, Numerical modeling, Tanger’s composed by 4140 nodes and 7611 triangles. bay, Shallow water equation. 6000 4140 Nodes / 7611 Elements Introduction Q The wastewater discharged into the sea through bays or 4000 lagoons has particularly adverse impact on marine life. 100 75 [m] Z

50 The specialists have found that many of the chemicals and Y [m] 2000 industrial hamper the chemical reactions and cause the P 25 0 0 (1) 0 1000 (5) 1000 2000 death and sometimes the disappearance of certain species. 2000 3000 (4) Y [m 3000 4000 ] X [m] 0 (2) 4000 5000 The increase in water temperature and the concentration (3) 5000 6000 0 2000 4000 6000 of polluting substances assumes at the same time an X [m] increase in the consumption of oxygen, which is harmful FIG. 1 – Tanger’s bay. Left: mesh of unstructured to marine life. triangles (1, 2, 3, 4 and 5 the location connected rivers) . These studies of local situations have motivated Right: three-dimensional view bathymetry. mathematicians and computer scientists, since permitted levels of contamination have been largely surpassed in b. Hydrodynamic and pollutant advection- many different societies and regions. Models, although of diffusion models several different types, have been approximated by The 2D shallow water equations and the advection- feasible numerical methods for some years [1, 2, 3]. diffusion with source terms of bed slope and bed friction Several numerical methods developed for hyperbolic is adopted and may be written in tensor notation and in a systems of conservation laws were applied to the shallow compact conservative form as [5]: water equations that are used in this work to model the flows prevailing in the bay. In this order we describe a ∂()W +∂ (() FW − GW ()) = SW () finite volume shock-capturing scheme for the high- t xi resolution solution of the non-homogenous hyperbolic where: system of equations that represent shallow flow pollutant 0  transport processes. The method is high resolution and h      τ τ ∂Z aimed at analyzing the processes of direct discharge of W= hu , S =−−wi bi gh b  , i  ρ ρ ∂  wastewater on two-dimensional domains with   xi hC    complicated boundary and bed geometry, such as the Bay S  of Tangier. c

12 ème Congrès de Mécanique 21-24 Avril 2015 - Casablanca (Maroc)

  dij and xi is the intersection of the orthogonal hu i      0 gh 2   bisectors of the edge Γ ,meas (Γ ) is the length of the F=hu u +  , G =  0  i ij i j 2  ∂ ∂  edge. The source terms are balanced by means of a two-   C  hui C (hD ) dimensional implementation of the upwind scheme  ∂xi ∂ x  i i  proposed by Vazquez et al. [8] Integration of the source

term over the control volume Vi gives: where i and j are indices and the Einstein summation ∫ Sn ( W ) dV = convention is used, i.e. repeated indices mean a Vi summation over the space coordinates .h is water depth, − n Γ ∑ [[I AWn ( ,ij ) AWn ( , ij )]. s ( XXWWn i , jijij , , , )]. ij j∈ N( i ) t is time, Ui depth-averaged velocity,Zb is the bed elevation above a fixed horizontal datum,C is depth- Where I is the identity matrix, A( W , n ij ) is the Roe flux n averaged concentration,Di is the dispersion coefficient jacobien and s(, Xi X j ,, WWn i j ,) ij represent an i Γ in the - direction,Sc is the depth-averaged source term, approximation of the source term at the cell interfaceij . τ τ Finally, to obtain higher-order spatial accuracy, the fluxes bi, wi are respectively the bed shear stress and wind at each edge are calculated using a piecewise linear stress, given respectively by [6]: . function of the state variable W inside the control − 2 τ= ρ gnh2 6 u uu volume, for more details we refer the readers to works(3) [7] bi b i j j . τ=× ρ ρ + × −3 wi a(0.75 0.067ww jj) 10 www ijj Results and discussion in which n the Manning’s coefficient at the bed , ρ For numerical simulation of water circulation and b a wastewater dispersion in Tanger’s bay, we propose is the air density and ρ is the water density. scenario with tidal flow. The tidal flow was forced at In the context of finite volume, the integral of equation Atlantic ocean side boundaries through the specification (1) over a triangular element (see FIG. 1) is written as the of the surface tidal elevation. We have limited our sum of each edge contribution [5], numerical modeling study to the semidiurnal component, n+1 n ∆t r forcing the model with the M 2, S 2 and N 2 components:(4) = −FW(n , nd ) Γ Wi W i ∑ ∫ 3 Vi j∈ N( i ) Γij ht()= A cos(ω t + ψ ) ∆t r ∑ n n n +∑ ∫GW(,)n nd Γ + ∫ SW () n dV n=1 V ∈ i j N( i ) Γij Vi ω ψ n whereAn is the amplitude, n is the pulsation and n W i To evaluate the state ,an approximation is required of of the semi-diurnal component. the convective and diffusive flux terms at each cell edge The model was run for three days with tidal forcing in .The integral along the i− j edge of a control volume order to achieve a steady flow exchange. The steady n exchange obtained is characterized by an inflow and an of the normal flux FW( , n ) = Fn + Fn can be 1x 2 y outflow behavior. Indeed, this is evident in a cross section written, of simulated water height along the distance PQ at times n 1 t=T, t=T/4 and t=3T/4 shown in FIG. 2. (5) FWnd( ,)Γ = ((,) FWn + FWn (,)) ∫ i ij j ij Then, in third day, we start the wastewater injection, Γij 2 which is assimilated by a passive pollutant, through five 1 −RWn(,)(,)(,)( AWn LWn W − W ). L connected rivers (see FIG. 1). 2 ij ij ij j i ij

Spanish t=T + + + is the flux jacobian evaluated using Roe’s 50 ++ A( W , n ij ) t=T/4 + + t=3T/4 + + + + + 40 + + average state,R and L are the right and left eigenvector + + ++ ++ 30 ++ + matrice of A . + + + 20 + A four point finite volume interpolation is used to Tangier ( port ) + Water height [m] ++ + + + + + + + evaluate the diffusion flux through an inner edge Γ so 10 + + + ij + + + that: 1000 2000 3000 4000 5000 Distance along the width of the bay from the port entrance [m] ∂C C− C (6) ∇C. nd Γ= d Γ= meas ( Γ ) j i FIG. 2 – Cross section of simulated depth-averaged water ∫ ∫ ∂ ij height along the PQ section at three tidal times. Γij Γ ij n d ij

The simulated results are presented in FIG. 3, and show the concentration contour of wastewater at three separate

12 ème Congrès de Mécanique 21-24 Avril 2015 - Casablanca (Maroc) instances from thirteenth days after injection. A simple [3] Z. Wu, G.Q. Chen, L. Zeng, Environmental dispersion inspection of this results reveals that the change of in a two-zone wetland, Ecological Modelling 222 (2011) velocity field direction during the time process according 456–474. to one period of the considered tides have a direct effect [4] M. Gonzalez, A. Sánchez-Arcilla, Un Modelo on direction of propagation of wastewater. The influence Numérico en Elementos Finitos para la Corriente of tidal currents appears at the right of the bay in terms of Inducida por la Marea. Aplicaciones al Estrecho de iso-concentrations are essentially moved, this reflects the Gibraltar, Revista Internacional de Métodos Numéricos fact that the wastewater mass is mainly moved by the tidal para Cálculo y Diseño en Ingenieria. 11 (1995) 383–400. flow. Finally, the results show that wastewater [5] E. M. Chaabelasri, A. Amahmouj, M. Jeyar, A. G. L. concentration decreased by diffusing effect due the tidal Borthwick, N. Salhi, and I. Elmahi, “Numerical Survey of waves. Contaminant Transport and Self-Cleansing of Water in Nador Lagoon, Morocco,” Modelling and Simulation in Engineering, vol. (2014), ID 179504 . doi:10.1155/2014/179504. C: 0.6 2.0 3.3 4.6 5.9 7.2 8.5 9.9 C: 0.6 2.0 3.3 4.6 5.9 7.2 8.5 9.9

4000 t=0 t=T/4 [6] F. Benkhaldoun, I. Elmahi and M. Seaid, Well- balanced finite volume schemes for pollutant transport by shallow water equations on unstructured meshes, Journal Y [m] 2000 of Computer Physics , 226 (2007) pp. 180–203. [7] F. Benkhaldoun, I. Elmahi and M. Seaid, Application of mesh-adaptation for pollutant transport by water flow, 0 Mathematics and Computers in Simulation 79 (2009) C: 0.6 2.0 3.3 4.6 5.9 7.2 8.5 9.9 C: 0.6 2.0 3.3 4.6 5.9 7.2 8.5 9.9

4000 t=3T/4 t=T/2 3415–3423. [8] Vazquez-Cendon ME. Improved treatment of source terms in upwind schemes for shallow water equations in Y [m] 2000 channels with irregular geometry. Journal of Computational Physics 1999; 148:497– 526.

0 0 2000 4000 6000 0 2000 4000 6000 Y [m] Y [m] FIG. 3 – Concentration contours of wastewater at four times measured during a typical tidal period.

Conclusion In this study, a numerical model is used for numerical analysis of process of coastal pollution by wastewater in the Bay of Tangier. The numerical model is based on a coupled Shallow Water and advection-diffusion equations for describing, respectively, the water flow and wastewater advection-dispersion. The pollutant concentration is captured and the flow field is resolved reasonably well. The features obtained illustrate the robustness of the numerical model used and its capability to run simulation of such area during moderated time scales. As a perspective of this work, various research topics can be considered. Indeed, it seems very interesting to calibrate the numerical model before starting a realistic modelling, also extend the model to a three-dimensional in order to take into account the real participation of vertical variation of the physical variables, this will be a challenge for next work.

References [1] Marchuk G.I., Mathematical Models in Environmental Problems, vol. 16, Studies in Mathematics and Its Applications. North-Holland, Amsterdam (1986). [2] Y. Montano-Ley , R. Peraza-Vizcarra, F. Páez-Osuna, The tidal hydrodynamics modeling of the Topolobampo coastal lagoon system and the implications for pollutant dispersion, Environmental Pollution 147 (2007) 282-290.