Development and application of a new random walk model to simulate the transport of degradablepollutants*

Lin Zhang,Li-na Chen, Jian-yin Zhou,Jia-sheng Wang,Qi-hong Yang,Long-xi Han

1.Key Laboratory of Integrated Regulation and Resource Development on Shallow Lake of Ministry Education, Hohai University, Nanjing 210098,China

2.College of Environment, Hohai University, Nanjing 210098,China

3.Changjiang River Scientific Research Institute,Wuhan 430010,China

4.College of Agricultural Science and Engineering, Hohai University, Nanjing 210098,China

5.Key Laboratory of National Health and Family Planning Commission on Parasitic Disease Control and Prevention,Wuxi 214064,China

6.Jiangsu Provincial Key Laboratory on Parasite and Vector Control Technology,Jiangsu Institute of Parasitic Diseases,Wuxi 214064,China

Abstract:To simulate the pollutant transport with self-purification in inland waters,the widely used random walk model (RWM)is modified to include a source term for the degradation and to consider the impact of land boundaries.The source term for the degradation is derived from the assumption of the first-order reaction kinetics.Parameters for the new model are determined by a comparison to the analytical results.The proposed model is then applied to simulate and analyzethe transport of a test pollutant and its spatial distribution in a large reservoir in northeast China.Reasonable results are obtained,and the effects of the runoff,the flow structure,and the wind on the pollutant transport are analyzed.The results may help the risk assessment and the management of the water pollution in inland waters.

Key words:Degradable pollutants,inland water,Lagrangian method,random walk model

Introduction

After discharge,pollutants can mix with the water in the environment via advection,diffusion,and dispersion.Numerical models based on Eulerian or Lagrangian methods are widely used to simulate the pollutant transport in free-surface flows[1-2].With the Eulerian approach,the movement of a group of particles is considered by describing what happens in a fixed region in space.This approach is adopted by most of the researches of the pollutant transport,and the well-known advection–diffusion equations are normally used to describe the motion of substances in the water,which are based on a Eulerian framework[3].The equations are solved using analytical or numerical methods with the polluted water as the study object.The variations in the pollutant concentrations over time in each fixed region in the water body can then be obtained[4-5].By simply taking the tangent Eulerian endpoint as the starting point in each time step of the calculation,the error increases with the forward iterations,so Eulerian schemes often suffer from large errors.

In the Lagrangian approach,also known as the particle tracking method,the pollutants in the water are generalized as individual particles with a certain mass[6]and their trajectories in the flow field are tracked to acquire the spatial position of each pollutant particle at each moment[7-8].Then the spatial distribution of the pollutant concentrations at each time step is obtained according to the position and the number of the particles[9].Previous studies have shown that among Lagrangian-type methods,the random walk model (RWM)is one of the most effective[10-11].With the RWMs,the problem of“man-made diffusion”in the Eulerian methods is avoided,which allows the better simulation of the convective diffusion of materials in the water flows[12-13].The RWM is widely used for studies of the atmospheric environment since it wasproposed by Ghisalberti and Nepf[14].More recently,RWMs were used in the studies of the transport of contaminants,including soluble and floating substances,in both wide and shallow water bodies[15].For instance,Wang and Wang[16]designed an RWM to simulate the pollutant transport in the Yangtze River estuary.Durgut and Reed[17]studied the spreading of the spilled oil on the sea surface using random walk methods.Charles and colleagues[18]carried out the simulation of the pollutant transport in a shallow water using the RWMs.RWMs have also been utilized in studies related with the mixing in the flows through the submerged vegetation and the environmental management in coastal zones.Liang and Wu[19]applied an RWM to study the longitudinal dispersion processes in the open channel flows in the presence of submerged vegetation.Lv and co-workers[20]modeled the water exchange using the random walk particle tracking method to assess the impact of the sewage discharge on the coastal environment and to optimize theoutfalls.

RWMs are suitable for simulating the pollutant particles that are randomly transported by the turbulent flow at the water surface.However,in the existing RWMs for the surface water,the effect of the pollutant degradation is not taken into account[21-22].It is desirable to find a source term for the degradation in the RWMs.In this study,a widely used RWM is modified for the pollutant transport in free-surface flows by including a source term for the degradation based on the assumption of the first-order reaction kinetics[23].The new RWM is then used to simulate the pollutant transport with self-purification in an inland water.The parameters of the new RWM(e.g.,the diffusion and degradation coefficients)and those of a traditional Eulerian model are compared to determine a solution for the new RWM.The Dahuofang Reservoir is taken as a study case and the transport of the particles of a degradable pollutant in the reservoir flow is simulated by coupling the RWM with a two-dimensional hydrodynamic model.The proposed RWM is particularly suitable for simulating the degradable pollutants in narrow water bodies with complex configurations and in rivers and reservoirs with complicated flows.

1.Materialsand methods

Thebasic principlein the RWMsistherelease of a large number of marked particles and the tracking of their movements in a flow field.A series of random displacements are applied to simulate the turbulent diffusion.By tracking their trajectories,we can calculate the overall temporal and spatial distributions of these particles and obtain a transport law for the pollutants.

The spatial position of a single particle at time( +1)n tΔis

where+1nX andnX denote the column vectors for the spatial position of the particle at times( +1)n tΔ and n tΔ ,respectively,CXΔ denotes the column vector for the variation in the spatial position caused by the flow convection;andDXΔ denotes the column vector for the variation in the spatial position caused by the turbulent diffusion in the water flow(also known asthe random walk distance).

1.1 Movement via flow advection

The particle movement due to the water flow convection is simulated using the deterministic method.Under normal circumstances,the difference of the densities between the general pollutants and the water is neglected,so the advection velocity of the particles is considered to be the same as the water velocity at each corresponding point.

After a duration tΔ ,the column vector for the variation in the spatial position of a particle due to the movement caused by the flow convection is

wherenu and+1nu are the water velocity at the spatial point for a particle at times n tΔ and( +1)n tΔ ,respectively.

1.2 Random walk movement of particles

The random walk of the particles leads to changes in the scale and the shape of the pollution cloud over time.The column vector for the random walk distance is

where a and b are two independent random variables obtained from a normal distribution with the mean of zero and the standard deviation of one,and Kxand Kyare the turbulent diffusion coefficients in the x and y directions,respectively.

1.3 Improvements for simulation of a degradable pollutant

wherejV is the water volume of the cell j and K is the self-purification coefficient for the pollutant.

1.4 Modification of trajectories at land boundaries

The flow in coastal waters is generally in the direction parallel to the shore,yet there may be certain angles between the flow velocity at coastal grid nodes and the coastline.As the particles migrate to coastal waters,the calculationsbased on the advection and the diffusion can yield some unreasonable results with the particles being deposited onto the shore.In this study,we use the shortest distance rule to correct the particle coordinates at the end of a deposition period to the grid node coordinates for the water located closest to the deposition point.

2.Determination of the model parameters

(1)Degradation coefficient:In the improved RWM,individual particles are considered,while the analytic model considers the whole pollutant group.The two models are essentially the same as they both conform to the basic assumption of the first-order reaction kinetics.Hence,the degradation coefficient K is set the same value in both models.(2)Turbulent diffusion coefficient:According to the assumption of the isotropic turbulence in a homogeneous flow field,the diffusion coefficients in the x and y directions are equal.Therefore,we only examine the relationship in the x direction between the diffusion coefficientsKxin the RWM and Exin the analytic model.

In the analytic model,the turbulent diffusion coefficient Exis determined using the Elder equationEx=αxhu*,where αxis an empirical parameter,h is the water depth and u*is the shear velocity.The diffusion coefficient K in the RWMs is not well studied.The empirical equation Kx=βxhu*is typically used for K using the same diffusion coefficient[24],where βxis an empirical coefficient.The relationship between the empirical coefficientsαxand βxmust be determined.Since βxis related to the diffusive movement of the microscopic particles,it is difficult to determine both experimentally and theoretically.There are few studies so far concerning βx,but there are various studies concerning αx.In this study,the concentration fields of the two models are well matched by adjusting Kx.Then the values for the diffusion coefficients( Ex, Kx)and the empirical coefficients(αx, βx)are analyzed to obtain the correlation between αxand βx.

Fig.1 (Color online)Relationship between the empirical coefficients xα and xβ for the two models to yield the same result

From Fig.1,αxislinearly correlated to βxas

Using Eq.(5),the empirical relationship for the diffusion coefficient Kxin the RWM under a uniform flow isobtained as

3.Numerical calculation

(1)Input the following data:the flow field,the particle location,and the residual particle mass at time tn.The hydrodynamic input data for the particle can be obtained from approaches such as the hydroacoustic measurements and the hydrodynamic modeling.In our case the data are obtained in a fixed grid using the hydrodynamic model DHI MIKE21/3.The input data are used to simulate the travel time and the concentration distributions for the particles.(2)The particle displacements driven by the advection and the diffusion within the time period tΔ are calculated from Eqs.(2),(3).(3)These calculated results and the particle location at timent are put into Eq.(1)to obtain the particle location at time+1nt .(4)The residual pollutant mass at time+1nt is obtained approximately according to Eq.(4),in which the source term for the degradation is included using the first-order reaction kinetics assumption.(5)Repeat steps(1)-(4)until the final time point is reached.The time step used in the numerical solution is set according to the information of the flow field and the pollutant properties,as well as our simulation experience.

4.Comparison

To test its performance,the new RWM is used in a case of uniform flow field with the velocity of 0.1 m/s and the water depth of 1 m,with an instantaneous release of 8 000 particles at time t =0.We assume that the turbulence is uniformly isotropic.The degradation coefficient isk =0.10 d-1and the time step isΔt =600 s .The initial particle location is set tox =0,at the middle of the cross-section.The RWM output is then compared with that obtained by the analytic model.Figure 2 shows the mass concentration contours obtained from the analytic solution and the RWM results at t =5 h,8 h.

As shown in Fig.2,there is a slight bias in the particle concentration distributions between the simulated results and the analytic solution,with the simulated results giving slightly higher or lower concentrations.Possible sources of the bias include the lack of reliable information and uncertainties arising from the solving process.Despite the bias,one sees a good agreement between the RWM simulation resultsand the analytic solution (Fig. 2).

5.Applicationsand discussions

Fig.2 Comparison of concentration contours for the two modelsat =5 h t ,=8 h t

Fig.3 (Color online)Study site (Dahuofang Reservoir)in Fushun City,Liaoning Province,China

The Dahuofang Reservoir (Fig.3)is chosen as the study case.The reservoir is located in the eastern mountainous area of Liaoning Province and controlsa basin area of 5 437 km2.The reservoir supplies water to 2.3×107people in seven cities in Liaoning Province,and is thus a key water source in China.The water quality in the reservoir directly affects not only the drinking water safety but also the industrial and agricultural usage.

Under the conditions of the steady outflow and inflow out of and into the reservoir,the wind-driven flow field is simulated for the annual average wind speed of 2.2 m /s-1and the dominant wind direction of 45°NE.With the hydrodynamics simulations for Dahuofang Reservoir,the surface flow field is obtained.It is assumed that the source mass is 130 kg,which is divided into 25 000 mass points,and that the pollutants are instantaneously discharged into the reservoir.The degradation coefficient k =0.03 d-1and the empirical parametersαx=αy=0.327.

The pollutant distributions over time and space(Figs.4,5)are obtained using the improved RWM.The spatial distribution of the pollutant group at different times reflects the water flow in Dahuofang Reservoir from the Hunhe River entrance to the dam,the shallow water close to the pollution source has significant nearshore characteristics.Therefore,after the pollutants flow into the reservoir,they move from the emission point to the southern bay of the reservoir along the northeast bank under the influence of the northeastward nearshore water flow.Owing to the random walk of individual pollutant particles,the shape of the polluted area changesduring the transport and its size increases.Since the emission point is located in the northern bay of the reservoir,where the flow velocity is relatively small,the subsequent movement of the pollutants is also slow.At =1h t ,the center of the pollutant group is located 11 m downstream the emission point.At that time,the pollutant group leaves the northern bay and the nearby flow velocity is large,so the migration of the pollutant group is correspondingly fast.At =14 h t , the center of the pollutant group is located 950 m downstream the emission point.Under the influence of the southwestward water flow,the pollutant group gradually moves to the southern bay of the reservoir.Because the water is relatively contained,one sees a counterclockwise circulation in the left flow field.As a result,the pollutant group is retained in the bay as a whole.In addition,the flow field relatively far from the left bank moves in a southeast direction.Therefore,some particles move towards the dam under the influence of this flow.

Fig.4 (Color online)Path followed by the pollutants

Fig.5 Distribution of pollutant concentration contoursat various times

6.Conclusion

In this paper,the widely used RWM for the pollutant transport in free-surface flows is modified to include a source term for degradation,based on the assumption of the first-order reaction kinetics.The new RWM is then applied to simulate the pollutant transport with self-purification in an inland water.The parameters of the new RWM (e.g.,the diffusion and the degradation coefficients)and those of a traditional Eulerian model are compared to determine a solution for the new RWM.The model is then applied to simulate a hypothetical event involving pollutant release in Dahuofang Reservoir.The results show that the pollutant particles generally move from the emission point to the dam.Owing to its specific hydrodynamic features,the pollutant particles linger in the reservoir bay.In addition,since the pollutant mass is located in places far from the left bank,they generally move towards the southeast.Under the influence of the reservoir flow,a portion of the particles move towards the dam.The proposed model provides a foundation with improved accuracy in predicting the water quality in narrow water bodies such as reservoirs,and can also help the water quality management for reservoirs.

AcknowledgementsThis work was supported by the Jiangsu Provincial Project of Invigorating Health Care through Science Technology and Education (Grant No.wk018-005),the Yangtze River Scientific Research Foundation of Public Welfare Institutions(Grant No.CKSF 2019511)and the Yangtze River Scientific Research Foundation of Public Welfare Institutions(Grant No.CKSF2019490).