Environmental dispersivity in free-water-surface-effect dominated wetland: multi-scale analysis

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1 Front. Environ. Sci. Engin. China DOI /s RESEARCH ARTICLE Environmental dispersivity in free-water-surface-effect dominated wetland: multi-scale analysis Zi WU 1, Zhi LI 1, Li ZENG 1, Ling SHAO 1, Hansong TANG 2, Qing YANG 1, Guoqian CHEN ( ) 1 1 State Key Laboratory of Turbulence and Complex Systems, College of Engineering, Peking University, Beijing , China 2 Department of Civil Engineering, City University of New York City College, New York, NY 10031, USA Higher Education Press and Springer-Verlag Berlin Heidelberg 2011 Abstract Distinct from the case with width-dominated shallow wetland flows, the longitudinal evolution of contaminant concentration in the most-typical pattern of wetland as dominated by free-water-surface-effect is characterized by a multi-scale analysis in the present study. An environmental dispersion model for the evolution of the mean concentration is deduced as an extension of Taylor's classical formulation by Mei s multi-scale analysis. Corresponding environmental dispersivity is found identical to that determined by the method of concentration moments. Keywords free-surface wetland, environmental dispersion, multi-scale analysis 1 Introduction First introduced by G.I. Taylor in 1953 in his well-known research on soluble matter in solvent flowing slowly though a long and thin tube [1], dispersion refers to the process that solutes spread longitudinally as a result of uneven velocity and diffusion in the cross-section. This process occurs in a variety of environmental flows including that in open channels, rivers, estuaries, and wetlands [2 7]. Remarkable endeavors have been made to study the environmental dispersion in wetlands. Based on the theory of random walk of fluid particles, Nepf [8] devised a model to investigate the lateral spread of contaminant in the wetlands. As an improvement, the lateral spread with more factors including the flow velocity, drag coefficient, stem diameter, and characteristic distance between stems are considered in alternative models explored by Nepf and Serra [9,10] on the basis of the mechanism of turbulent and Received October 15, 2010; accepted January 15, gqchen@pku.edu.cn mechanical diffusion. Lightbody and Nepf [11,12] presented a formula to estimate the longitudinal dispersivity in flows through wetlands dominated by emergent vegetation, and the longitudinal dispersivity for salt mashes vegetated with Spartina alterniflora was estimated from experimental data of frontal area, stem diameter, and diffusivity. In view of the possible increase of dispersion due to the existence of dead-zones in wetlands that contain submerged canopies, Murphy et al. [13] analyzed the effect of relative submergence, i.e., the ratio of water-depth to height of canopy, on the longitudinal dispersivity by using a two-zone model. Nepf et al. [14] presented a two-box model to describe longitudinal dispersion in flow through wetlands with submerged vegetation, in which the renewal of water bodies are dominated by longitudinal advection and vertical exchanges. Though these models take into account the environmental dispersion contributed by the heterogeneity of velocity to some extent, the lateral or vertical variation of superficial velocity has not been considered. Analytical solutions for environmental dispersion in wetlands have been rigorously obtained by Chen and his fellows. For wetlands of distinct flow patterns, they respectively derived the flow profiles for the depth, width, depth-and-width dominated wetlands, and for a two-zone wetland as well [6,15 18], using fluid dynamics for porous media [19,20]. Based on an environmental dispersion model for the evolution of the mean concentration as an extension of Taylor s classical analysis on dispersion and Aris s method of concentration moments, analytical solutions for the environmental dispersivity have then been obtained for the specified flow velocity profiles [6,15 18]. The contaminant transport in the depth-dominated wetland is characteristic of multiple time scales. For an analytical solution, the multi-scale analysis presented by Mei for dispersion in the single phase fluid flow [21,22] is very illustrative. In the field of wetland science, Wu et al.

2 2 Front. Environ. Sci. Engin. China [17] firstly adopted this analysis procedure to analyze the dispersion problem for the contaminant in wetland flows. However, the kind of wetland in their study is dominated by the effect of width, which means the dispersion process is determined by the shear flow confined between solid boundaries. This width-dominated wetland is most likely encountered in the case of shallow water wetland or the subsurface constructed wetland. For the more typical wetland naturally existed, a deep wetland flow with a free water surface can be very different in characteristics of contaminant dispersion due to the obviously diverse flow details. And dispersion process in the depth-dominated wetland flow can be of more importance. For the case of mass transport in fully developed flow through a depth-dominated wetland, presented in this work is a multi-scale analysis for the longitudinal evolution of the vertical mean concentration under environmental dispersion. The specific objectives of this paper are: 1) to give the governing equations for mass transfer in wetlands adopted in our research; 2) to present a longitudinal transport model for environmental dispersion; 3) to present a multi-scale analysis for the determination of the dominant variation of contaminant concentration; and 4) to determine and analyze the expression for the environmental dispersivity for the wetland flow. where U is the velocity ½m$s 1 Š, t time ½sŠ, φ porosity ½dimensionlessŠ, κ tortuosity ½dimensionlessŠ to account for the spatial structure of porous media, C concentration ½kg$m 3 Š, l concentration diffusivity ½m 2 $s 1 Š, and K concentration dispersivity tensor ½m 2 $s 1 Š. Like diffusivity for concentration transport valid for the description of the single phase clear water flow at the microscopic passage scale, microscopic dispersivities for concentration is properties valid for the description of superficial flow at the phase average scale out of the operation of phase average to smear out the discontinuity between the two phases of the ambient water and the solid structure. The equation for concentration transport is out of a combination of an advective-diffusive equation and a microscopic dispersion law. Consider solute transport in a fully-developed unidirectional flow through a depth-dominated wetland of depth H in a Cartesian coordinate system, with longitudinal x -axis aligned with the flow direction, vertical z- axis, and origin set at the bed wall, as shown in Figure Formulation for concentration transport Once there is an instantaneous release into a depthdominated wetland flow of soluble matter at the origin by the initial time, the evolution behavior of the solute cloud is characterized with two stages on the whole. Initially, there is a large longitudinal gradient of vertical averaged concentration, and the vertical concentration difference caused by the variance of longitudinal velocity in the vertical direction can t be balanced by the vertical diffusion generated by the vertical concentration gradient. The vertically averaged concentration thus forms a skewed longitudinal distribution. As time goes by, the longitudinal gradient of average concentration decreases, and finally the vertical concentration difference can be balanced by the vertical diffusion, and the longitudinal distribution of the vertical mean concentration tends to a normal distribution. Under such circumstances, the centroid of the solute moves at the vertical averaged velocity of the solvent, and it disperses longitudinally by a virtual diffusion coefficient [2,23]. The latter stage of the evolution is referred to as environmental dispersion. For a typical wetland flow, basic equation for mass transfer can be adopted generally at the phase average scale as [6,19,24] φ C t þr$ðucþ ¼r$ðκlφrCÞþκr$ðK$rCÞ, (1) Fig. 1 Sketch for a depth-dominated wetland flow For the depth-dominated wetland flow, Eq. (1) becomes C þ u C t φ x ¼ κ l þ K 2 C φ x 2 þ κ l þ K 2 C φ z 2 : (2) Consider a uniform and instantaneous release of contaminant with mass Q at the cross-section of x ¼ 0 at time t ¼ 0, the initial condition can be set as Cx,z,t ð Þj t¼0 ¼ QδðÞ x φh, (3) where δðxþ is the Dirac delta function. The non-penetration conditions at the bed wall of z ¼ 0 and free water surface of z ¼ H read as C z j z¼0 ¼ C z j z¼h ¼ 0: (4) Since the amount of released contaminant is finite, we have upstream and downstream boundary conditions as Cx,z,t ð Þj x¼1 ¼ 0: (5)

3 Zi WU et al. Environmental dispersivity in free-water-surface-effect dominated wetland 3 3 Multi-scale analysis for concentration transport Multi-scale analysis was initiated by Mei for the case with a single phase flow [21]. For the present case for a wetland flow, there are three time scales in the dispersion stage of the transport process, i.e., T 1 ¼ H 2 = κ l þ K as the φ diffusion time across the depth of the wetland H, T 2 ¼ L=u m as the convection time across the characteristic length L of the contaminant cloud (u m is the vertical mean velocity), which is significantly greater than the depth of H, and T 3 ¼ L 2 = κ l þ K as the diffusion time across φ the cloud length of L. With a small parameter of ε ¼ H=L << 1, the relations between the time scales can generally be expressed as T 1 : T 2 : T 3 ¼ 1 : 1 ε : 1 ε2, (6) by considering that the Péclet number is of order unity. With dimensionless parameters of Pe ¼ H u m, (7) φκ l þ K φ ψ ¼ u, ¼ z u m H, ¼ x κ l þ K L, τ ¼ φ H 2 t, (8) the governing equation and boundary conditions for concentration can be rewritten as C τ þ εpeψc ¼ ε2 2 C 2 þ 2 C 2, (9) C ¼0 ¼ C ¼1 ¼ 0: (10) Here we introduce the multiple time coordinates t 0 ¼ τ, t 1 ¼ ετ, t 2 ¼ ε 2 τ, (11) thus the variation of concentration C with dimensionless time τ can be expressed as C τ C t 0 þ ε C t 1 associated with a multiple scale expansions þ ε 2 C t 2, (12) Cð,,t 0,t 1,t 2 Þ¼C 0 ð,,t 0,t 1,t 2 ÞþεC 1 ð,,t 0,t 1,t 2 Þ þ ε 2 C 2 ð,,t 0,t 1,t 2 ÞþOðε 3 Þ, (13) Substituting Eq. (12) and Eq. (13) into Eq. (9) and Eq. (10), we have 2 C 0 t 0 2 þ ε þ C 1 2 C 1 t 1 t 0 2 þ Peψ C 0 þ ε 2 þ C 1 þ C 2 2 C 2 t 2 t 1 t 0 2 þ Peψ C 1 2 C 0 2 þ Oðε 3 Þ¼0, (14) ½C 0 þ εc 1 þ ε 2 C 2 þ Oðε 3 ÞŠ j ¼0 ¼ ½C 0 þ εc 1 þ ε 2 C 2 þ Oðε 3 ÞŠ j ¼1 ¼ 0: (15) According to Eq. (14) and Eq. (15), the perturbation problem for Oðε 0 Þ is obtained as t 0 ¼ 2 C 0 2, (16) j ¼0 ¼ j ¼1 ¼ 0: (17) To solve this partial differential equation, we resort to the integral transform technique. First we introduce a transform for the concentration as ec 0 ð,β n,t 0,t 1,t 2 Þ¼! 1 0 C 0ð,,t 0,t 1,t 2 Þcosðβ n Þd, (18) where β n ¼ nπðn ¼ 0,1,2:::Þ are the eigenvalues. Apply this transform to Eq. (16), we have d~c 0 ¼ β 2 dt n~c 0, (19) 0 and the solution to this ordinary differential equation is ec 0 ð,0,t 0,t 1,t 2 Þ¼f 0 ð,t 1,t 2 Þ, (20) ec 0 ð,β n,t 0,t 1,t 2 Þ¼f n ð,t 1,t 2 Þe β2 nt 0 ðn 0Þ, (21) where f n are functions with variables of, t 1, and t 2. Then the solution for C 0 is promptly found by an inverse transformation as C 0 ð,,t 0,t 1,t 2 Þ ¼ f 0 ð,t 1,t 2 Þþ2 X1 n¼1 f n ð,t 1,t 2 Þe β2 nt 0 cosðβ n Þ: (22) Evidently for long time evolution, C 0 is independent of and t 0 since the second term at the right hand side of Eq. (22) dies out quickly because of its exponential decay, thus we have

4 4 Front. Environ. Sci. Engin. China C 0 ð,,t 0,t 1,t 2 Þj t0 1 ¼ C 0 ð,t 1,t 2 Þ: (23) We define the depth-average for any variable f as hi¼ f! 1 f d, (24) 0 and apply the average operation to Eq. (16) to result in ¼ 0: (25) t 0 According to Eq. (14) and Eq. (15), the perturbation problem for Oðε 1 Þ is the following: t 1 þ C 1 t 0 þ Peψ ¼ 2 C 1 2, (26) C 1 j ¼0 ¼ C 1 j ¼1 ¼ 0: (27) It is readily seen from Eq. (11) that t 0 is a much larger time scale than t 1, and the second term at the left hand side of Eq. (26) can be neglected. Eq. (26) can be rewritten as t 1 þ Peψ ¼ 2 C 1 2 : (28) We apply the average operation to Eq. (28) and get t 1 þ Pehψi ¼ 0: (29) Multiplying a quantity of ε to each term of Eq. (29), we have ε t 1 þ εpehψi ¼ 0: (30) Subtracting Eq. (29) from Eq. (28) gives Peψ# ¼ 2 C 1 2, (31) where ψ# ¼ ψ hψi is the non-uniformity of the velocity. Since C 0 is independent of, we have C 1 ¼ Pe GðÞ: (32) Substituting Eq. (32) into Eq. (31) and Eq. (27) results in dgðþ d d 2 GðÞ d 2 ¼ ψ#, (33) j ¼0 ¼ dgðþ d j ¼1 ¼ 0, (34) then GðÞ can be calculated with specified ψ# to obtain the explicit expression for C 1. According to Eq. (14) and Eq. (15), the perturbation problem for Oðε 2 Þ is the following: t 2 þ C 1 t 1 þ C 2 t 0 þ Peψ C 1 ¼ 2 C 0 2 þ 2 C 2 2, (35) C 2 j ¼0 ¼ C 2 j ¼1 ¼ 0: (36) Since t 0 is a much larger time scale than t 1 and t 2, the third term at the left hand side of Eq. (35) is negligible, and Eq. (35) can be rewritten as t 2 þ C 1 t 1 þ Peψ C 1 ¼ 2 C 0 2 þ 2 C 2 2 : (37) According to Eq. (29) and Eq. (32), we have C 1 ¼ PeGðÞ ¼ Pe 2 hψigðþ 2 C 0 t 1 t 1 2, (38) and Eq. (37) can be rewritten as t 2 þ Pe 2 ψ#gðþ 2 C 0 2 ¼ 2 C 0 2 þ 2 C 2 2 : (39) Applying the depth-average operation to Eq. (39) and multiplying a quantity ε 2 at each term of this equation, we have ε 2 t 2 ¼ ε 2 ½1 Pe 2 hψ#gðþiš 2 C 0 2 : (40) Adding Eq. (40) and Eq. (30) to Eq. (25) results in t 0 þ ε t 1 þ ε 2 t 2 þ εpehψi ¼ ε 2 ½1 Pe 2 hψ#gðþiš 2 C 0 2, (41) and if we use the relation presented in Eq. (12) inversely, the expression can be expressed as τ þ HPehψi x ¼ H 2 ½1 Pe 2 hψ#gðþiš 2 C 0 x 2 : (42) With T ¼ τ, ¼ x=h Pehψiτ, Eq. (42) can be rewritten as T ¼½1 Pe2 hψ#gðþiš 2 C 0 2 : (43) 4 Environmental dispersion Taking the average operation for Eq. (9), with the aid of Eq. (10), gives hci τ þ εpe hψci ¼ ε 2 2 hci 2 : (44)

5 Zi WU et al. Environmental dispersivity in free-water-surface-effect dominated wetland 5 Substituting decompositions of C ¼ hciþ C# and ψ ¼ hψiþ ψ# into Eq. (44) gives hci τ þ εpehψi hci ¼ ε 2 2 hci 2 εpe h ψ #C# i : (45) To close the equation, the new term hψ#c# i is modeled as [2] hψ#c# hci i ¼ εd L, (46) where D L is the Taylor dispersivity. Analogous closure models have been extensively employed in dealing with environmental transport in rivers, estuaries, etc. For the mean concentration of solute in confined porous media, Hamdan et al. [25] even presented an extended high-order model, though with complexity beyond reach of practical applications for the time being. Similar operation of average to yield Taylor dispersion is found in studies of a packed bed associated with groundwater pollution and chemical engineering [7,26]. Substituting Eq. (46) into Eq. (45) results in hci τ þ εpehψi hci ¼ ε 2 ð1 þ PeD L Þ 2 hci 2, (47) and with T ¼ τ, ¼ x=h Pehψiτ, Eq. (47) can be rewritten as hci T ¼ hci D2 2, (48) where D ¼ 1 þ PeD L is the environmental dispersivity, and the transformation to the, T system allows us to view the mass transport from an observer moving at a speed of Pehψi. Considering that hci ¼ C 0 under long time evolution, comparison between Eq. (48) and Eq. (43) gives D ¼ 1 Pe 2 hψ#gðþi: (49) 5 Environmental dispersivity Zeng and Chen [6] have already obtained the analytical solution for velocity distribution in the depth-dominated wetland as uðþ ¼ 1 α 2 1 cosh½αð 1ÞŠ cosh α, (50) where ¼ z=h is a dimensionless parameter, and sffiffiffiffiffiffiffiffiffiffiffiffi FH 2 α ¼, (51) L þ reflects the combined action of viscous stress, vegetation drag, and the lateral momentum microscopic dispersion; while F is shear factor, L vertical momentum microscopic dispersivity and dynamic viscosity. The vertical mean velocity of the wetland flow is found as u m ¼ α tanh α α 3 : (52) According to the definition of dimensionless velocity in Eq. (8), with aid of Eq. (50), the velocity non-uniformity can be calculated as αf1 cosh½αð 1ÞŠsechαg ψ#ðþ ¼ 1: (53) α tanh α Solving Eq. (33) with the help of Eq. (34) and Eq. (53), we have GðÞ ¼ 2α 2 C st coshα 2coshðα αþþαð 2 þ 2 2C st Þsinhα, 2αðαcosh α sinh αþ (54) where C st is a constant. Consequently, hψ#gðþi ¼ 12 þ 8α2 2ð6 þ α 2 Þcoshð2αÞþ9αsinhð2αÞ 12α 2 ð αcosh α þ sinh αþ 2 : For comparison we define D s ¼ D 1 (55) ¼ Pe 2 12 þ 8α 2 2ð6 þ α 2 Þcoshð2αÞþ9αsinhð2αÞ 12α 2 ð αcosh α þ sinh αþ 2, (56) which is found to be identical to the result presented in Zeng and Chen [6] and Zeng [23]. Figure 2 shows the variation of D s with Pe when α = 1.0, 3.0, 5.0, 7.0, 9.0, respectively. Figure 3 shows the variation of D s with α when Pe = 1.0, 3.0, 5.0, 7.0, 9.0, respectively. It can be found that D s increases with the increase of Pe, and decreases with the increase of α. These patterns are in accordance in trend with that have been found for the width-dominated wetland flow [17], though different in details. Evidently the dispersivity in the depth-dominated wetland flow is larger in magnitude, indicating a more intensive dispersion process. 6 Conclusions Distinct from the case with width-dominated shallow wetland flows, the longitudinal evolution of contaminant concentration in the most-typical pattern of wetland as

6 6 Front. Environ. Sci. Engin. China performed for the contaminant concentration, and the expression for the dominant one, which also known as the vertical mean concentration, is deduced for a diffusion-like dispersion process as observed in a moving coordinate with a speed equal to the product of the Péclet number and the vertical mean velocity. The environmental dispersivity is then determined by the comparison between this diffusion equation and the environmental dispersion model. The result is found identical to that determined by the method of concentration moments. Acknowledgements This work is supported by the National Natural Science Foundation Program of China (Grant No ). Help from Simon Skraatz is acknowledged. Fig. 2 Variation of D s with Pe for α = 1.0, 3.0, 5.0, 7.0, 9.0 Fig. 3 Variation of D s with α for Pe = 1.0, 3.0, 5.0, 7.0, 9.0 dominated by free-water-surface-effect is characterized by a multi-scale analysis in the present study. With essential implications for ecological risk assessment, the case of concentration transport in the depth-dominated wetland flow is formulated with the basic equation for concentration transport in porous media flows. With the rigorous extension of Taylor s classical analysis on dispersion in pure fluid flows, a basic environmental dispersion model is presented to model the transport of the contaminant in the depth-dominated wetland flow. Mei s multi-scale analysis for dispersion in the single phase fluid flow is elaborated and generalized for the case of depth-dominated wetland flow to find out the expression for the dominant variation of contaminant concentration. Three time-scales including the diffusion time across the depth of the wetland, the convection time across the contaminant cloud length, a length far exceeding the depth, and the diffusion time across the length of L; and their relations have been introduced to construct the multi-time coordinates. A similar multi-scale expansion has been References 1. Taylor G I. Dispersion of soluble matter in solvent flowing slowly through a tube. Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 1953, 219(1137): Fischer H B, List E J, Koh R C Y, Imberger J, Brooks N H. Mixing in Inland and Coastal Waters. New York: Academic, Elder J W. The dispersion of marked fluid in turbulent shear flow. Journal of Fluid Mechanics, 1959, 5(04): Holley E R, Harleman D R F, Fischer H B. Dispersion in homogeneous estuary flow. Journal of Hydraulic Engineering, 1970, 96: Zeng L, Chen G Q. Notes on modelling of environmental transport in wetland. Communications in Nonlinear Science and Numerical Simulation, 2009, 14(4): Zeng L, Chen G Q. Ecological degradation and hydraulic dispersion of contaminant in wetland. Ecological Modelling, 2011, 222(2): Chen G Q, Zeng L. Taylor dispersion in a packed tube. Communications in Nonlinear Science and Numerical Simulation, 2009, 14(5): Nepf H M, Sullivan J A, Zavistoski R A. A model for diffusion within emergent vegetation. Limnology and Oceanography, 1997, 42(8): Nepf H M. Drag, turbulence, and diffusion in flow through emergent vegetation. Water Resources Research, 1999, 35(2): Serra T, Fernando H J S, Rodríguez R V. Effects of emergent vegetation on lateral diffusion in wetlands. Water Research, 2004, 38(1): Lightbody A F, Nepf H M. Prediction of velocity profiles and longitudinal dispersion in emergent salt marsh vegetation. Limnology and Oceanography, 2006, 51(1): Lightbody A F, Nepf H M. Prediction of near-field shear dispersion in an emergent canopy wiht heterogeneous morphology. Environmental Fluid Mechanics, 2006, 6(5): Murphy E, Ghisalberti M, Nepf H M. Model and laboratory study of dispersion in flows with submerged vegetation. Water Resources Research, 2007, 43(5 W05438): W Nepf H M, Ghisalberti M, White B, Murphy E. Retention time and dispersion associated with submerged aquatic canopies. Water

7 Zi WU et al. Environmental dispersivity in free-water-surface-effect dominated wetland 7 Resources Research, 2007, 43(4 W04422): W Chen G Q, Zeng L, Wu Z. An ecological risk assessment model for a pulsed contaminant emission into a wetland channel flow. Ecological Modelling, 2010, 221(24): Zeng L, Chen G Q, Tang H S, Wu Z. Environmental dispersion in wetland flow. Communications in Nonlinear Science and Numerical Simulation, 2011, 16(1): Wu Z, Li Z, Chen G Q. Multi-scale analysis for environmental dispersion in wetland flow. Communications in Nonlinear Science and Numerical Simulation, 2011, 16(8): Wu Z, Chen G Q, Zeng L. Environmental dispersion in a two-zone wetland. Ecological Modelling, 2011, 222(3): Liu S, Masliyah J H. Dispersion in Porous Media. In: Vafai K, ed. Handbook of Porous Media. USA: CRC Press, 2005: Bear J. Dynamics of Fluids in Porous Media. New York: American Elsevier Pub, Mei C C. Dispersion of supension in a steady shear flow. Lecture notes in Fluid Dynamics, 2002, 1.63J/2.01J (MIT Opencourse) 22. Mei C C, Auriault J L, Ng C O. Some applications of the homogenization theory. Advances in Applied Mechanics, 1996, 32: Zeng L. Analytical Study on Environmental Dispersion in Wetland Flow. Dissertation for the Doctoral Degree. Beijing: Peking University, Rajagopal K R, Tao L. Mechanics of Mixtures. Singapore: World Scientific Publishing Co Pte Ltd, Hamdan E, Milthorpe J F, Lai J C S. An extended macroscopic model for solute dispersion in confined porous media. Chemical Engineering Journal, 2008, 137(3): Fried J J. Groundwater Pollution: Theory, Methodology, Modelling and Practical Rules. Amsterdam: Elsevier Scientific Publishing Company, 1975