Depth-Dependent Dispersion Coefficient for Modeling of ... - CiteSeerX

Depth-Dependent Dispersion Coefficient for Modeling of Vertical Solute Exchange in a Lake Bed under Surface Waves Qin Qian1; Jeffrey J. Clark2; Vaughan R. Voller3; and Heinz G. Stefan4 Abstract: Variable pressure at the sediment/water interface due to surface water waves can drive advective flows into or out of the lake bed, thereby enhancing solute transfer between lake water and pore water in the lake bed. To quantify this advective transfer, the two-dimensional 共2D兲 advection-dispersion equation in a lake bed has been solved with spatially and temporally variable pressure at the bed surface. This problem scales with two dimensionless parameters: a “dimensionless wave speed” 共W兲 and a “relative dispersivity” 共␭兲. Solutions of the 2D problem were used to determine a depth-dependent “vertically enhanced dispersion coefficient” 共DE兲 that can be used in a 1D pore-water quality model which in turn can be easily coupled with a lake water quality model. Results of this study include a relationship between DE and the depth below the bed surface for W ⬎ 50 and ␭ 艋 0.1. The computational results are compared and validated against a set of laboratory measurements. An application shows that surface waves may increase the sediment oxygen uptake rate in a lake by two orders of magnitude. DOI: 10.1061/共ASCE兲0733-9429共2009兲135:3共187兲 CE Database subject headings: Dispersion; Lakes; Mass transfer; Hydraulic models; Sediment; Solutes; Surface wave; Water quality.

Introduction Water motion in a lake can have profound consequences for its ecosystem. Periodic surface waves induced by wind are among the important water movements in lakes 共Horne and Goldman 1994兲. Wind induced surface waves cause water pressure fluctuations at the lake bed which, in turn, affect pore pressures and can cause flow into and out of a porous lake bed. Similarly small bed forms over which a benthic current is flowing, induce pressure variations along the sediment surface as described by Huettel and Rusch 共2000兲 for small mound and ripples on shelf sediments, and by Elliot and Brooks 共1997兲 for dunes in streams. The pressure-driven advection of water into and out of the pore system of a lake bed can enhance the transfer of a dissolved substance 共solute兲 significantly. Wave-induced advective flow in permeable sediments has been described by Huettel and Webster 共2001, Chapter 7兲. Studies of pore-water advection have been conducted in marine environments. Shum and Sundby 共1996兲 and 1 Ph.D. Candidate, St. Anthony Falls Laboratory, Dept. of Civil Engineering, Univ. of Minnesota, Minneapolis, MN 55414; presently, Assistant Professor, Dept. of Civil Engineering, Lamar Univ., Beaumont, TX 77710 共corresponding author兲. E-mail: [email protected] 2 Associate Professor, Geology Dept., Lawrence Univ., Appleton, WI 54912. 3 Professor, St. Anthony Falls Laboratory, Dept. of Civil Engineering, Univ. of Minnesota, Minneapolis, MN 55414. 4 Professor, St. Anthony Falls Laboratory, Dept. of Civil Engineering, Univ. of Minnesota, Minneapolis, MN 55414. Note. Discussion open until August 1, 2009. Separate discussions must be submitted for individual papers. The manuscript for this paper was submitted for review and possible publication on October 21, 2007; approved on August 11, 2008. This paper is part of the Journal of Hydraulic Engineering, Vol. 135, No. 3, March 1, 2009. ©ASCE, ISSN 0733-9429/2009/3-187–197/$25.00.

Jahnke et al. 共2005兲 estimated the effect of advection on organic matter processing in continental shelf sediments. Van Rees et al. 共1996兲 indicated that, in addition to particle size, benthic organisms also affect the solute transport in lake sediments. Huettel and Rusch 共2000兲 found that the advective transport, associated with small mounds and ripples commonly found on shelf sediments, increased penetration depth of unicellular algae into sandy sediment up to a factor of seven and mass flux up to a factor of nine compared to a smooth sediment surface. Experiments on an intertidal sand flat 共Rusch and Huettel 2000兲 demonstrated that advective particle transport into permeable sediments depends on sediment permeability and particle size. Surface gravity waves can increase fluid exchange between sandy sediment and overlying shallow water 50-fold, relative to exchange by molecular diffusion 共Precht and Huettel 2003兲. Precht et al. 共2004兲 studied the effects of advective pore-water exchange driven by shallow water waves on the oxygen distribution in permeable natural sediment in a wave tank. The strong and relatively recent evidence of advective interaction between the surface water and the pore water, especially if the lake sediment is highly permeable, calls into question the often used model that molecular diffusion is the dominant solute transport process at the sediment/water interface 共Glud et al. 1996兲. The objective of this paper is to quantify the advective pore-water flow due to a progressive 共moving兲 sinusoidal pressure wave on a lake bed and its effect on solute transport in lacustrine sediments. The analysis will be 2D, but a 1D vertical, depthdependent “enhanced dispersion coefficient 共DE兲,” which accounts for both advection and hydrodynamic dispersion tensors in the permeable lake bed, will be related quantitatively to pressure wave amplitude 共a兲, wavelength 共L兲, wave speed 共c兲, bed hydraulic conductivity 共K兲, and depth below the lake bed/water interface 共y兲. The enhanced dispersion coefficient 共DE兲 can be used in a 1D 共vertical兲 dispersion equation which can also incorporate chemiJOURNAL OF HYDRAULIC ENGINEERING © ASCE / MARCH 2009 / 187

Downloaded 17 Feb 2010 to 128.101.188.45. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright

analytical solution of the Laplace equation. With this information we solve the 2D transient, advection-dispersion equation 共ADE兲 describing the transport of a conservative solute in the sediment bed. Initial conditions are a constant solute concentration C = C0 in the water and not solute, i.e., C = 0, in the bed. To obtain the 1D vertical, depth-dependent “enhanced dispersion coefficient 共DE兲, the calculated 2D concentration distribution is averaged horizontally at each time step, and the resulting 1D averaged concentration profile versus depth is matched to a solution of an unsteady vertical dispersion equation. A depth-variable enhanced dispersion coefficient 共DE兲 is thus determined by inverse modeling. Without loss of generality the enhanced dispersion coefficient is determined for a conservative 共nonreactive兲 solute. A similar enhanced dispersion coefficient has been obtained for transport into a stream bed under standing waves 共Qian et al. 2007b, 2008兲. The lake problem, studied here, has some key differences: the velocity field changes with time and the governing dimensionless parameters are different.

Governing Equations Two-Dimensional Solute Transfer Equation The 2D solute transport equation for a nonreactive solute in a porous lake bed is expressed as 共Zheng and Bennett 1995兲

冉 冊

⳵C ⳵C ⳵C ⳵共uC兲 ⳵共vC兲 ⳵ =− − + + Dxy Dxx ⳵x ⳵y ⳵t ⳵x ⳵y ⳵x + Fig. 1. Schematic sketch of sediment bed with boundary and initial conditions for solute concentrations

cal and microbiological kinetics in the lake bed. This equation can provide a more realistic transport link between lake water quality and sedimentary kinetics. In our study the lake bed is assumed to be homogeneous and isotropic below the lake bed/water interface. An extension to heterogeneous and anisotropic porous media is feasible but not included.

冉

⳵C ⳵C ⳵ Dyx + Dyy ⳵x ⳵y ⳵y

共1兲

where u and v = seepage 共local兲 velocity components in the wind direction 共x兲 and depth 共y兲 direction, respectively, and Dxx, Dxy, Dyx, and Dyy = components of the 2D dispersion coefficient tensor which have been given by 共Zheng and Bennett 1995兲 as Dxx = ␣L

uu vv + ␣T + Dem 兩V兩 兩V兩

Dxy = Dyx = 共␣L − ␣T兲 Problem Definition and Methodology Consider a lake bed 共Fig. 1兲 that is isotropic and homogeneous with a hydraulic conductivity 共K兲 and a porosity 共␧兲. At the lake bed surface, a series of periodic 共cosine兲 pressure waves of wavelength 共L兲, wave amplitude 共a兲, wave height 共H = 2a兲, and wave speed 共c兲 are induced by surface water waves. The relationship between the periodic pressure wave and the surface water wave will be discussed and quantified in “Range of Independent Parameter Values.” The pressure wave will drive a flow into and out of the sediment bed. Most likely, the pressure-induced flow near the lake bed surface is turbulent, but the quantification of this interaction is complex 共Thibodeaux and Boyle 1987兲. Therefore, Darcy flow within the sediment bed has been assumed as a starting point 共Elliot and Brooks 1997兲. To capture any wave effect we consider a vertical section through a lake—in the direction of the wind that creates the surface waves—which in turn generate pressure waves at the lake bed. We model the flow field and the transport of solute of concentration, C, in a 2D clean lake bed section by the advection flow. The transient velocity field in the lake bed is obtained from an

冊

Dyy = ␣L

vu 兩V兩

共2兲

uu vv + ␣T + Dem 兩V兩 兩V兩

In Eqs. 共2兲, ␣L and ␣T = longitudinal and transverse dispersivity, properties of the porous medium describing dispersive transport in the direction of flow, and normal to the flow, respectively; 兩V兩 = 冑u2 + v2 = magnitude of the seepage velocity; and Dem = effective molecular diffusion coefficient. Eq. 共1兲 is solved in a domain of depth 2L and length L, because the flow field is periodic with wavelength L, and cannot penetrate into the porous medium beyond a depth of 2L. The boundary conditions are as follows: 1. Along the lakebed/water interface C共x , 0 , t兲 = C0 at all times; 2. On the bottom of the domain 共y = 2L兲, the velocity is negligible, and does not significantly affect the results. A far-field “no flux with zero gradient” condition is imposed, i.e. 兩⳵C / ⳵y兩y=2L = 0; and 3. On the vertical boundaries at x = 0, and x = L, identical boundary conditions, C共0 , y , t兲 = C共L , y , t兲, are imposed, because

188 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / MARCH 2009

Downloaded 17 Feb 2010 to 128.101.188.45. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright

the force is exerted by periodic surface waves of wavelength L, and the response can be expected to be periodic as well. To obtain the local velocity components u共x , y , t兲 and v共x , y , t兲 in Eq 共1兲, a Laplace equation is solved for a periodic pressure head distribution h共x , y , t兲 along the sediment bed surface 共y = 0兲, representative of a wave-induced flow 关Eq. 共3兲兴 h共x,0,t兲 = hmean + a cos

冓

2␲共x − ct兲 L

冔

冊 冉 冊 冊 冉 冊

2␲共x − ct兲 2␲y vDarcy共x,y兲 2␲Ka = cos exp − ␧ L␧ L L

共4兲

In order to identify the dimensionless groups that govern the dispersion process in the lake bed, Eq. 共1兲 is made dimensionless. The following reference variables are proposed: C0 for concentrations in the lake bed, pressure wavelength 共L兲 for distances 共x and y兲, and Vref = Ka / L␧ for velocities in the lake bed. The dimensionless variables for concentrations 共C*兲, distances 共x* and y *兲, velocities 共u* and v*兲, and time 共t*兲 can then be defined as C , C0

x x* = , L

y y* = , L

u* =

u , Vref

v* =

v , Vref

t* =

L Vref 共5兲

The velocity field can then be written in dimensionless form as u* = 2␲ sin共2␲共x* − W兲兲exp共− 2␲y *兲 v* = 2␲ cos 2␲共x* − W兲exp共− 2␲y *兲

共6兲

where the dimensionless wave speed W is defined as W=

c␧L c = Vref Ka

D*xy =

Dyy v *v * ␣ T u *u * = + Vref␣L 兩V*兩 ␣L 兩V*兩

共8兲

冉 冊

Dyx ␣ T v *u * Dem Dxy = = 1− + Vref␣L Vref␣L ␣L 兩V*兩 Vref␣L

In Eq. 共8兲 two other dimensionless parameters appear: one is the longitudinal to transverse dispersivity ratio 共␣T / ␣L兲 that characterizes the porous medium and the other is Dem / 共Vref␣L兲 = Dem / Dref = 1 / P which is an inverse Péclet number that characterizes the relative importance of effective molecular diffusion to advective dispersion in the porous sediment bed. With the normalized variables from Eqs. 共5兲 and 共8兲, the mass transport Eq. 共1兲 can be expressed as

⳵C* ⳵t*

=−

冋 冉 冊册

⳵共u*C*兲 ⳵共v*C*兲 ⳵ ⳵C ⳵C + D*xy − + D* ⳵x* ⳵y* ⳵x* xx ⳵x ⳵y

冉

⳵ ⳵C ⳵C + D*yx D* ⳵y * yy ⳵y ⳵x

冊

␣L L

共9兲

In Eq. 共9兲, a single dimensionless length scale parameter ␭ = ␣L / L measures the longitudinal dispersivity to pressure wavelength. It is defined as “relative dispersivity” and characterizes the solute dispersion transfer process in the lake bed. One-Dimensional Vertical Solute Transfer Equation

Normalization of 2D Solute Transfer Equation

C* =

D*yy =

+

uDarcy共x,y兲 2␲Ka 2␲共x − ct兲 2␲y u共x,y兲 = = sin exp − ␧ L␧ L L v共x,y兲 =

Dxx u *u * ␣ T v *v * Dem = + + Vref␣L 兩V*兩 ␣L 兩V*兩 Vref␣L

共3兲

where hmean = mean piezometric head in the bed. There is no implicit restriction to small amplitude waves. Eq 共3兲 gives the pressure distribution at the sediment/water interface in response to a periodic surface wave, although the representation is better for small amplitude waves than steeper waves. Horizontal head gradients are zero at x = 0, L, 2L, etc. They are also zero at great depth, i.e., a far-field condition of 兩⳵h / ⳵y兩y=2L = 0 is imposed. The Laplace equation for the velocity field is obtained by combining the continuity equation with the Darcy relationship for flow in a porous medium. The Darcy velocity field for the wave-induced flow is obtained analytically 共Elliot and Brooks 1997; Ren and Packman 2004兲. Since the seepage velocity components used in Eq. 共1兲 can be obtained by dividing the Darcy velocity components by the porosity 共␧兲 of the bed the 2D field of seepage velocities can be expressed as

冉 冉

D*xx =

共7兲

and characterizes the wave speed relative to the advective flow velocities induced in the lake bed by the pressure wave. The dispersion coefficient tensor in Eq. 共2兲 also needs to be normalized. As a reference dispersion coefficient we have chosen Dref = Vref␣L. Because of their dependence on local velocity components u and v, the normalized dispersion coefficients will be locally variable

The 1D vertical solute transfer equation for a lake bed is

冉 冊

¯ ¯ ⳵C ⳵C ⳵ = DE ⳵y ⳵t ⳵ y

共10兲

¯ = horizontally averaged concentration. Eq. 共10兲 allows where C for a much easier description of the solute exchange between surface water and pore water in the lake bed than Eq. 共9兲. Advection and hydrodynamic dispersion in the lake bed pore system are lumped into the enhanced dispersion coefficient DE共y兲, and an explicit description of the velocity field in the lake bed 关Eq. 共6兲兴 is no longer required. Eq. 共10兲 is a pure dispersion equation. The effect of the advection terms in Eq. 共9兲—in addition to the dispersion terms— therefore has to be captured in DE of Eq. 共10兲. The vertical velocity terms in Eq. 共6兲 vanish when averaged horizontally or over time, but their effect on mass transport in Eq. 共10兲 is captured in DE. Periodicity and the nonuniformity in the vertical velocity contribute strongly to DE, but horizontal advection and dispersion contribute as well. The interaction of vertical and horizontal advection with horizontal and vertical dispersion in Eq. 共9兲 is highly nonlinear. An interpretation of DE in terms of any single one of these processes is therefore not possible, although the periodic vertical velocity is thought to provide the major 共“pumping”兲 effect. DE can be obtained by rearranging and integrating Eq. 共10兲 to give Eq. 共11兲. Formally, DE is a function of both depth 共y兲 and time 共t兲, but only the asymptotic steady-state solution is of interest, as will be discussed later JOURNAL OF HYDRAULIC ENGINEERING © ASCE / MARCH 2009 / 189

Downloaded 17 Feb 2010 to 128.101.188.45. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright

DE共y兲 = −

冕

2L

y

¯ ⳵C 共␰,t兲d␰ ⳵t

冏 冏 ¯ ⳵C ⳵y

DE = 共11兲

冉

¯ −C ¯ C j−1 j+1 2⌬y

冊 兺冉 −1 2N

j=1

冊

¯ n+1 − C ¯ n−1 C j j ⌬y 2⌬t

共12兲

where y

The enhanced dispersion coefficient 共DE兲 can also be normalized with the reference dispersion coefficient Dref = Vref␣L to give D*E = ␧DE / Ka␭. The mass or material flux across the bed/water interface is ¯ / ⳵y兩 evaluated as J = 兩 − DE⳵C y=0 for incorporation in a surface water quality model. This flux 共J兲 has equal to the mass flux in the water above the sediment. The vertical enhanced dispersion coefficient 共DE兲 at y = 0 represents a mass-transfer coefficient relating the mass flux across the sediment-water interface and the concentration in the sediment phase 共pore water兲. The masstransfer coefficient was previously related in the literature 共Elliot and Brooks 1997; Higashino and Stefan 2005兲 to the Reynolds number of the flow and the Schmidt number of the transferred solute. In our study the value of DE is controlled by the steepness 共a / L兲 of the wave and the relative dispersivity 共␭ = ␣L / L兲, the hydraulic conductivity 共K兲, and the porosity of the lake sediment 共␧兲. Sediment particle size has a major influence, but solute characteristics do not significantly affect the process. This is not surprising because the analysis demonstrated that molecular diffusion, which is in the Schmidt number, makes a very small contribution to the solute transport when pressure wave induced flow is present in the lake bed. 共The influence of turbulence near the lake bed was not investigated.兲 Overall, the mass transfer coefficient Dwater in the water phase is related to DE in the sedi¯ / dy兩 兲 / 共兩dC ¯ water / dy兩 兲. ment phase by Dwater = DE共兩dC y=0 y=0

N

¯ =1 C Ci,j j N i=1

兺

and i and j indicate the grid number in x and y directions, respectively, and n indicates time step.

Range of Independent Parameter Values Four independent dimensionless parameters have emerged from the normalization of the governing equations: W = c␧L / Ka, ␣T / ␣L, ␭ = ␣L / L, and P = Vref␣L / Dem. These parameters are related to pressure wave 共c , a , a / L兲 and to porous media characteristics 共␣T , ␣L,Dem兲. Range of Wave Characteristics „c, a, and a / L… If the pressure wave on the sediment bed surface is induced by a progressive surface wave, the pressure wave speed 共c兲 and the pressure wavelength 共L兲 are the same as the surface wave speed 共cw兲 and the surface wavelength 共Lw兲. The substitutions c = cw and L = Lw can be made. The wave speed 共cw兲 of a progressive surface wave in a lake depends on both wavelength and water depth of the lake. For a small amplitude wave, cw can be estimated from 共White 1975兲

Numerical Solutions Two-Dimensional Solute Concentration Field The lake sediment bed is discretized within an N ⫻ 2N square control volume and Eq. 共9兲 is solved numerically using a finite volume approach 共Patankar 1980兲. N = 200 is the largest grid size number within one wavelength 共L兲. The concentrations are stored at the nodes 共located at the centers of the control volume兲 and flow velocity components are stored at the face midpoints. The choice of the numerical method used to solve the advection-dispersion Eq. 共9兲 is based on extensive numerical investigations focused on accuracy, grid dependence, and efficiency. Details and key findings of these investigations are reported in Qian et al. 共2007a,b兲. A third-order QUICK scheme 共Leonard 1979兲 written in terms of nodal-point values of the advected variable has been applied 共Qian et al. 2006兲 to reduce the numerical dissipation. The physically based 共phib兲 upwind flux limiter 共Qian et al. 2007a兲 is adopted in the advection term to diminish the numerical oscillations. A central difference formulation was used for the dispersion term. Eq. 共9兲 is solved using an explicit time stepping scheme to improve efficiency 共Qian et al. 2007a兲. Enhanced Vertical Dispersion Coefficient „DE… in Lake Bed Using central differences to discretize the averaged concentration values for time and space for Eq. 共11兲, the 1D enhanced dispersion coefficient 共DE兲 can be obtained in the N ⫻ 2N grid domain as

共13兲

c2w =

冉 冊

2␲h gLw tanh 2␲ Lw

共14兲

Typical wavelengths on lakes are in the range 0 ⬍ Lw ⬍ 100 m and typical water depths in the range 0 ⬍ h ⬍ 100 m. Small amplitude wave speeds are limited by lake depth. In deep water, wave speeds depend only on wavelength. A discussion and values of large amplitude waves speeds can be found in the oceanographic literature, 共Wiegel 1964; White 1975; USACE 1984; Mei et al. 2005兲. Dimensionless wave speeds 共W兲 have been calculated for different sediments and cover a range from about 50 to 5 ⫻ 106. Numerical simulations of concentration fields for W = 50, 5,000, and 500,000 predicted that 1D 共horizontally兲 averaged concentration profiles do not change over this range of wave speeds if other parameters are kept constant 共Fig. 2兲. It is concluded that wave speed does not affect the dispersion into the lake sediment bed because the wave-induced flow in the pore spaces is much slower than the wave celerity. The surface wave amplitude 共aw兲 is approximately half the wave height 共HW兲, i.e., Hw = 2aw. Wave climate on a lake can be derived from long-term wave recordings. If such measurements are not available, wave height can be estimated from wind speed, fetch, and/or wind duration 共Bretschneider 1959; Wiegel 1964; USACE 1984兲. To make an estimate of the wave characteristics on a lake, records of wind speed and direction need to be consulted. The maximum possible wave height 共Hw兲 for a progressive surface is given by the condition for the breaking of a wave

190 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / MARCH 2009

Downloaded 17 Feb 2010 to 128.101.188.45. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright

¯ *=C ¯ /C 兲 Fig. 2. Profile of horizontally averaged concentration 共C 0 with depth 共y * = y / L兲 for wave speeds W = c␧L / 共Ka兲 = 50– 500,000

冉 冊 Hw Lw

= 0.142 tanh max

冉 冊 2␲h Lw

共15a兲

for a Stokes wave 共Michell 1893兲

冉 冊 Hw h

共15b兲

= 0.78 max

for a solitary wave 共McCowan 1891兲. From Eqs. 共15a兲 and 共15b兲 one can derive a limiting wave steepness Hw / Lw ⬍ 0.08 or aw / Lw ⬍ 0.04. The amplitude 共a兲 of the pressure wave at the lake bed is equal to the product of the surface wave amplitude 共aw兲 and a water depth dependent function 共Kinsman 1965兲

冉

a = aw cosh

2␲ 共hmean − y兲 L

冊冒冉

cosh

¯ *=C ¯ /C 兲 Fig. 3. Profile of horizontally averaged concentration 共C 0 * with depth 共y = y / L兲 for relative dispersivity 共␭ = ␣L / L兲 from 0.001 to 0.1

冊

2␲ hmean L

terms in Eq. 共8兲. This indicates, as expected, that molecular diffusion makes a very small contribution to the solute transport when pressure wave induced flow is present in the lake bed. The inverse Péclet number can therefore be ignored when the results are presented. In summary, it appears that of the original four dimensionless parameters for the solute transport problem in a lake sediment bed, only one, the relative dispersivity ␭ = ␣L / L = dg / L, is a significant variable. The normalized wave speed 共W兲 in Eq. 共7兲, and the inverse Péclet number in Eq. 共8兲 cause insignificant changes in the results, and the dispersivity ratio ␣T / ␣L = 1 / 3 is more or less a physical constant.

共16兲

Therefore, a = aw can be used in a shallow lake; in deep water the pressure wave is transmitted with exponential decay. Since the amplitude a 艋 aw, the pressure wave induced by the progressive surface wave always has a wave steepness a / L ⬍ aw / L = 0.04. In addition, a / L ⬎ 0.001 is a reasonable lower limit for the pressure wave. With a sediment porosity in the range 0.24艋 ␧ 艋 0.61 共Zheng and Bennett 1995兲, the range of the parameter a / 共L␧兲 is therefore on the order of 0.002–0.2. Range of Sediment Characteristics „K , ␣T , ␣L , and Dem… The sediment size in a lake bed can range from fine silt to coarse gravel. The associated hydraulic conductivity is in the range 0.0001 cm/ s 艋 K 艋 10 cm/ s. The pore scale longitudinal dispersivity 共␣L兲 can be approximated by the sediment particle size 共dg兲 共Zheng and Bennett 1995兲 which can range from silt 共dg = 0.01 mm兲 to gravel 共dg = 10 mm兲. By comparison, a typical surface wavelength in a lake can be on the order of meters. With 0.00001⬍ ␣L ⬍ 0.01 m and 0.01⬍ L ⬍ 10 m, the estimated range of ␭ in a sediment bed can be from 10−6 to 1. Benekos 共2005兲 reported that ␣T / ␣L is usually 1 / 2 – 1 / 3, and ␣T / ␣L = 1 / 3 has been used in numerical simulations of macroscopic dispersion 共Zheng and Bennett 1995兲. Concentrations calculated with ␣T=␣L / 2 and ␣T = ␣L / 3 are very similar. Therefore, ␣T = ␣L / 3 has been uniformly applied throughout this study. In addition, the inverse Péclet number in Eq. 共8兲 was estimated for different ␭ values and Dem = 0.5⫻ 10−9 m2 / s 共typical of dissolved oxygen diffusion兲. It was found to contribute less than 0.1% of the other

Results 1D Concentration Fields Since the concentration field does not vary significantly with wave speed 共W兲, the wave speed was fixed at W = 5,000. Calculations were made for a matrix of a / L and ␭ values. Values of a / L = 0.01, 0.04, and 0.1, and for each a / L, values of ␭ = 0.0001, 0.005, 0.001, 0.01, 0.05, and 0.1 were chosen. The simulated 1D concentration profiles for ␭ = 0.0001, 0.005, and 0.001 are identical for the same a / L value. In other words, advective transport is dominant only when ␭ ⬍ 0.001. On the other hand, the simulated 1D concentration profiles for ␭ = 0.1 are the same as those for zero advection or pure dispersion. The solute transport for ␭ ⬎ 0.1 is dispersion dominated, and advection can be ignored. Therefore, 0.001⬍ ␭ ⬍ 0.1 is a range of practical interest. Fig. 3 shows 1D concentration profiles for ␭ = 0.001, 0.01, 0.05, and 0.1. Vertically Enhanced Dispersion Coefficient „DE… The vertically enhanced dispersion coefficient 共DE兲 in the lake bed was obtained as a function of depth from Eq. 共13兲. A normalized enhanced dispersion coefficient 共D*E = DE␧ / 共␭Ka兲兲 varies with depth y * = y / L below the bed/water interface. Fig. 4 shows the normalized dispersion coefficient D*E profile in the lake bed for ␭ = 0.01 and a / L = 0.01, 0.04 and 0.1. The D*E profile shown in Fig. 5 is for a / L = 0.1 and ␭ = 0.001, 0.01, and 0.1. Figs. 4 and 5 predict that the normalized dispersion coefficient D*E does not JOURNAL OF HYDRAULIC ENGINEERING © ASCE / MARCH 2009 / 191

Downloaded 17 Feb 2010 to 128.101.188.45. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright

Fig. 4. Profile of normalized enhanced vertical dispersion coefficient 共D*E = DE␧ / 共␭Ka兲兲 with depth 共y * = y / L兲 for relative dispersivity 共␭兲 = 0.01 and wave steepnesses 共a / L兲 = 0.01, 0.04, and 0.1

vary much with a / L- and ␭ values because the transfer by advection relative to hydrodynamic dispersion does not change much. The concentration predictions were obtained starting with C共x , y , 0兲 = 0 in the sediment bed and C = C0 in the overlying water. The DE共y兲 profiles extracted from the solutions therefore varied initially with time, but tended toward a steady-state distribution after a short initial development time. Similar results have been found in gravel stream beds with wave induced hyporheic flow 共Qian et al. 2008兲. We retained only the time-independent 共quasi-steady state兲 D*E distribution since the transient is the consequence of an arbitrary initial condition. Such a time requirement was also found for longitudinal dispersion in pipes and rivers 共Taylor 1953; 1954; Fischer et al. 1979兲. Development time is related to the size of the flow field, i.e., the time that is required for the solute to penetrate several grain lengths into the porous medium. If we choose arbitrarily four grain diameters, i.e., y = 4dg, a time scale can be estimated from Eq. 共10兲 as td = 4d2g / DE. This time would typically be short compared to the entire solute penetration process. For dg = 0.3 mm and DE = 10−7 m2 / s 共used in the application to DO diffusion in Application: Dissolved Oxygen Uptake by Lake Bed兲, t = 3.6 s can be estimated. For practical applications we propose a fitted equation for the normalized enhanced dispersion coefficient D*E as a function of depth 共Figs. 5 and 6兲. The relationship is

Fig. 5. Profile of normalized enhanced vertical dispersion coefficient 共D*E = DE␧ / 共␭Ka兲兲 with depth 共y * = y / L兲 for wave steepness 共a / L兲 = 0.1 and relative dispersivities 共␭兲 = 0.001, 0.01, and 0.1

Fig. 6. Comparison of enhanced vertical dispersion coefficient profile 共DE共y兲兲 for fine sand lake bed with molecular diffusion coefficient 共Dm兲

D*E = 5共exp−6.15y/L兲 ␭Ka * D DE = ␧ E

for

0.001 艋 ␭ 艋 0.1 共17兲

for

0.001 艋 ␭ 艋 0.1

共17兲

Eq. 共17兲 is recommended for all practical normalized wave speeds 共W ⬎ 50兲. The value of D*E for ␭ = 0.001 also applies to ␭ ⬍ 0.001, and the values of D*E for ␭ = 0.1 is applicable for ␭ ⬎ 0.1. Eq. 共17兲 indicates that the enhanced dispersion coefficient DE increases with hydraulic conductivity 共K兲, wave steepness 共a / L兲, and longitude dispersivity 共␣L兲, and that it decreases with the sediment porosity 共␧兲. Sediment particle size 共dg兲 has a major effect on D*E: the longitudinal dispersivity 共␣L兲 is approximately equal to the sediment particle size 共dg兲, the hydraulic conductivity 共K兲 is proportional to 共dg兲2 共Harleman et al. 1963兲, and the porosity is somewhat independent of the grain size. Therefore, DE can increase eight times if d50 doubles. Eq. 共17兲 fits the numerical 2D simulation results very well. The maximum root-mean-square error 共RMSE兲 of D*E between y * = 0 and y * = 1 is less than 0.0003 for all wave steepnesses 共a / L兲 and relative dispersivities 共␭兲 investigated. Magnitude of DE „cm2 / s… The enhancement of vertical dispersion in a lake sediment bed by pressure waves needs to be compared to molecular diffusion and hydrodynamic dispersion coefficients “Range of Independent Parameter Values.” The maximum DE in Eq. 共17兲 is equal to 5共a / L兲␣LK / ␧. In we gave typical ranges of the parameter values that control DE. The range of a / L␧ in a lake is on the order of 0.02–0.2, while ␣L is equal to d50 which ranges from silt to gravel 共0.00001⬍ ␣L ⬍ 0.01 m兲, and K increases with d50 共Zaslavsky and Ravina 1965兲 共0.0001 cm/ s 艋 K 艋 10 cm/ s兲. For comparison the hydrodynamic dispersion coefficients 共DL兲 for dissolved oxygen 共DO兲 are computed from K, sediment grain size d50, and the molecular coefficient of DO: D = 10−5 cm2 / s based on the study by Pfannkuch 共1963兲, in Fig. 10.4.1 of Bear 共1972兲. The enhancement for the silt sediment is not obvious. Table 1 shows a comparison for fine sand to gravel sediments and indicates that the maximum enhanced dispersion coefficient 共DE兲 can be ten times larger than the hydrodynamic dispersion coefficient and 105 times larger than the molecular diffusion coefficient. Fig. 6 has been prepared as an example. In Fig. 6, the enhanced vertical dispersion coefficient DE共y兲 for a sandy lake bed, i.e., sediment with dg = 0.1 cm, K = 0.1 cm/ s, and ␧ = 0.38, and the

192 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / MARCH 2009

Downloaded 17 Feb 2010 to 128.101.188.45. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright

Table 1. Variation of Enhanced Dispersion Coefficient 共DE兲 with Hydraulic Conductivity 共K兲 and Grain Size 共d兲 of Sediment

K共cm/ s兲 d共cm兲 DE共cm2 / s兲 DL共cm2 / s兲 DE / DL DE / D

Fine sand

Sand

Coarse sand

Gravel

0.01 0.0125 10−5 10−4 10−5 10−5 1 10 1 10

0.1 0.0395 10−4 10−3 10−4 10−4 1 10 10 100

1 0.125 10−2 10−1 −2 10 10−2 1 10 104 103

10 0.395 10−1 10−1 1 104

1 10−1 10 104

molecular diffusion coefficient of DO D = 10−5 cm2 / s are plotted against depth 共y兲. DE共y兲 was calculated from Eq. 共17兲 for a very small wave amplitude a = 1 cm and a wavelength L = 1 m 共a / L = 0.01 and ␭ = 0.001兲. The enhancement of vertical dispersion by the progressive pressure waves is still 10–100-fold compared to the molecular value, and reaches down to 70 cm below the sediment surface. Penetration Depth „dp… The normalized enhanced dispersion coefficient 共D*E兲 tends to be constant below a certain depth 共Figs. 5 and 6兲. The distance from the sediment surface to the depth, where D*E is nearly constant, is defined as the penetration depth 共d p兲. It measures the depth in the lake bed down to which advection due to pressure waves is important. Because D*E approaches a constant value asymptotically with depth, we defined it as the distance from the lake bed surface to the depth where D*E共y兲 = 0.01 exp共−6.15y/L兲. With this definition Eq. 共17兲 gives the penetration depth d p = 0.75 L

共18兲

Our definition of penetration depth is based on the magnitude of DE relative to its value at the sediment-water interface. Because it is a relative measure, the properties of the porous medium— assumed to be homogeneous—do not influence the penetration depth. Only the horizontal length scale 共wavelength L兲 is found to be of importance, and the penetration depth is proportional to this length L. None of the dimensionless groups that control the solute penetration problem seems important.

Validation Experimental Data A set of experimental laboratory data collected in a recirculating flume at the St. Anthony Falls Laboratory, University of Minnesota, was used to validate the proposed model 关Eq. 共17兲兴. No other suitable laboratory or field data could be located. The flume was 9.1 m long, 0.51 m wide, 0.7 m deep, and had a 7.0 m long test section filled with a 0.2 m deep layer of well sorted, commercially available pea gravel 共Fig. 7兲. The test section was confined between impermeable boundaries at the upstream and downstream end such that solute could enter the bed only through the water-sediment interface. The gravel had a median particle diameter dg = 7.2 mm and a bulk porosity ␧ = 0.4; the hydraulic conductivity was measured to be K = 18 cm/ s. The top of the bed was flat, i.e., there were no bed forms and no slope, but small scale topography did vary locally due to individual grains and grain clusters. In these experiments the water depth was h = 0.145 m, the flow velocity was 0.233 m / s, the flow rate was 17.23 L / s,

Fig. 7. Diagram of recirculating flume. Periodic surface waves generated by plunger at head of flume were recorded with ultrasonic transducer located near middle of test section. Conductivity probe near sediment bed in middle of flume measured salt 共NaCl兲 concentration in recirculating water.

and the water surface slope was 0.00049. Flow was generated by gravity, and a periodic wave generator 共plunger兲 at the inlet of the flume induced water surface fluctuations. These fluctuations had some three-dimensional characteristics; however, a progressive wave motion in the streamwise direction was clearly observed. Water surface fluctuations in the transverse direction were much less than in the streamwise direction and were ignored. Results from the 2D model applied to a vertical longitudinal section were used for comparison with the experimental data. Changes in water surface elevations with time were recorded at a station 3.2 m downstream of the start of the test section, using a Massa ultrasonic transducer. The wave amplitude 共aw兲 was calculated from the water surface elevation record by taking the standard deviation of the time series data. To estimate the wave frequency the power spectral density 共PSD兲 function was determined. The wavelength was estimated from the wave frequency and the water depth using Fig. 2-3 by White 共1975兲. A surface wave amplitude aw = 0.82 cm and a wavelength Lw = 114.3 cm were estimated from the experimental data. The pressure wave amplitude on the surface of the sediment bed was then calculated as a = aw exp共−2␲h / L兲 = 0.45aw = 0.37 cm 共Kinsman 1965兲. Solute transfer into the sediment bed was tracked by introducing a conservative tracer 共NaCl兲 into the recirculating stream. Changes in surface water conductivity were recorded at 30 s intervals at the outlet by a multiparameter sonde 共Hydrolab 4Ms兲. Electric conductance and solute concentration in the water were linearly correlated, as established by a calibration of the probe. Solute concentrations were therefore easily determined from the conductivity record over time. Validation Procedure The following steps were taken to compare the model 关Eq. 共17兲兴 of the 1D vertically enhanced dispersion coefficient 共DE兲 with the experimental data: 1. Estimate the dispersivity ␣L = dg = 0.72 cm, and calculate ␭ = ␣L / L = 0.0063. 2. With pressure wave amplitude a = 0.37 cm, hydraulic conductivity K = 18 cm/ s, porosity ␧ = 0.40, and relative dispersivity ␭ = 0.0063 find the enhanced dispersion coefficient DE共y兲 from Eq. 共17兲: DE共y兲⫽0.0000524 exp共−6.15 y / 1.143兲 =0.0000524 exp共−0.053806 y兲 m2 / s. 3. With this function for DE obtain stepwise 共⌬t = 0.001 s兲 numerical solutions of the finite-difference approximation of the 1D solute transport equation. JOURNAL OF HYDRAULIC ENGINEERING © ASCE / MARCH 2009 / 193

Downloaded 17 Feb 2010 to 128.101.188.45. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright

2.

Fig. 8. Comparison of calculated and measured normalized solute 共NaCl兲 concentrations 关C0共t兲 / C0共0兲兴 in recirculating water versus time. Difference 共bias兲 between experimental and numerical results during period from 700 to 2,000 s can be explained as effect of finite thickness of sediment bed.

冉 冊

4.

¯ ¯ ⳵C ⳵C ⳵ = DE ⳵y ⳵t ⳵ y

4.

共19兲

to track the vertical concentration profiles in the gravel bed with time. With this information calculate for each time step the mass 共solute兲 flux through the gravel bed surface into the gravel bed. This flux must equal the mass 共solute兲 loss from the overlying water. This flux balance can be written as

冏 冏

¯ dCWATER ⳵C d⬘ = DE dt ⳵y

5.

3.

共20兲 y=0

where d⬘ = 1.01 m = effective water depth above the sediment bed, calculated from the total recirculating water volume 共5.09 m3兲 per unit bed area 共4.64 m2兲. At each time step calculate the concentration in the water CWATER, by coupling an Euler solution of Eq. 共20兲 with the finite-difference results of the surface fluxes from Eq. 共19兲. The resulting simulated values of CWATER versus time can be compared with the experimental measurements in the flume.

Validation Results Fig. 8 gives the computed normalized concentrations in water 共C / C0兲 versus time and the data points from the experiments. C0 is the solute concentration in the recirculating water at the beginning of the experiment. Over the duration of the experiment C / C0 changed from 1.0 to about 0.932. The RMSE between the simulated normalized solute concentrations and the measured normalized concentrations 共C / C0兲 was found to be 0.00013 for a change from 1.0 to 0.932= 0.068. We conclude that the experimental and the computational results matched well, overall. Details are discussed in the next section. Discussion of Experimental Details Some features of the experimental data in Fig. 8 deserve additional discussion. 1. Solute concentrations in the water column above the sediments were uniform in vertical direction because the water in the flume was recirculated through a pump and well mixed before it entered the flume. The water in the experimental flume was turned over every 180 s, and the residence time of

5.

the water on the sediment bed was 30 s. Normalized concentrations in the experimental water column dropped from 1.0 to 0.94 over a period of about 1,000 s, and then to 0.93 over about 3,000 s. This change in boundary condition was simulated in the numerical analysis which solved the unsteady dispersion Eq. 共10兲. There is considerable variability in measured concentrations over a period of about 700 s after the onset of the experiment 共Fig. 8兲. The fluctuations occurred because the brine was introduced in the tailwater tank of the experimental flume. After four hydraulic residence times or about 700 s, the total water mass recirculating in the flume had achieved a uniform salinity. Development time is the time before a vertical concentration profile has developed such that Eq. 共17兲 can be applied. Development time was previously estimated as td = 4dg2 / DE. In the experiment with dg = 7.2 mm and DE = 0.00003 m2 / s development time is on the order of 10 s. This is short enough to be ignored compared to the duration of the experiment. The difference 共bias兲 between experimental and numerical results during the period from 700 to 2,000 s 共Fig. 8兲 can be explained as the effect of the finite thickness of the sediment bed. The experimental sediment bed is shallow 共0.20 m兲 compared to the predicted penetration depth 共d p = 0.75L = 0.85 m兲. Vertical velocity components in the sediment pore space are therefore more limited in the experiment than in the simulations. DE is consequently smaller in the experiment than predicted by Eq. 共18兲. Indeed, the solute depletion in the overlying water 共and hence slower solute penetration into the sediment兲 is slower in the experiment than simulated between 700 and 2,000 s. The bias is clearly noticeable although the difference is less than 1% of the initial solute concentration. Overall, this is not a serious issue. The duration of 3,000 s 共Fig. 8兲 was a reasonable choice to study solute dispersion at both a short and a long timescale. The measured and simulated solute concentrations after 3,000 s are identical at 0.932, and the rate of concentration change with time is virtually zero, indicating that the solute has fully penetrated to the bottom of the experimental sediment bed. The normalized solute concentration at the bottom of the sediment bed after 3,000 s can be calculated from Eq. 共10兲 and was found to be 0.9 for the experimental conditions.

Application: Dissolved Oxygen Uptake by Lake Bed A lake bed is a mix of pore water and solid particles. The constituents of the solid particles are typically inorganic and/or organic chemical species. Nutrients, oxidants, and a biological community including bacteria and macrofauna are common in lake sediment 共DiToro 2001兲. The exchange of solutes between lake water and lake sediments is an important component of the chemical/biological cycles in a lake. For example, organic matter in the lake sediment consumes DO resulting in DO depletion at the bottom of a domestic lake in late summer. This DO depletion may cause increased phosphorus release from the lake sediment, which in turn can stimulate undesirable phytoplankton blooms in a lake. Solute exchange with lake sediments also controls sulfate uptake by lake sediments, and the recycling of metals which can be toxic. The exchange rate of conservative or nonconservative solutes between lake water and lake bed can be estimated only if dispersion coefficients in the lake sediment are known in addition to kinetic rate coefficients.

194 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / MARCH 2009

Downloaded 17 Feb 2010 to 128.101.188.45. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright

With DE, the SOD is 10–30 times larger than that predicted with 0.5Dm, indicating that SOD predicted by using Dm can be erroneous when the SOD is high and wave effects are ignored.

Summary and Conclusions

Fig. 9. Sediment oxygen demand 共SOD兲 versus DO uptake rate coefficient 共k1兲 共day−1兲 in sediment bed of 4-m deep lake, calculated for pure molecular diffusion and with enhanced dispersion under surface waves of 0.4 m wave height and 10 m wavelength. Particle size 共dg兲 = 0.3 mm, porosity 共␧兲 = 0.38, hydraulic conductivity 共K兲 = 0.38 cm/ s.

To illustrate its usefulness, the proposed enhanced dispersion coefficient model is applied to the estimation of sediment oxygen demand 共SOD兲 in a shallow lake. We included the effect of progressive waves on the surface of the shallow lake 共water depth h = 4 m兲. Wave amplitude aw = 0.2 m and wavelength Lw = 10 m were used. The wave speed was determined to be cw = 1 m / s. The lake bed had a particle diameter dg = 0.3 mm and a porosity ␧ = 0.38. A hydraulic conductivity K = 0.38 cm/ s 共Harleman et al. 1963兲 and longitudinal dispersivity ␣L = 0.3 mm were estimated from dg = 0.3 mm. We assumed an initial DO concentration C = C0 = 8 mg/ l at the water/bed interface, and C = 0 in the sediment bed. The molecular diffusivity of dissolved oxygen is D = 10−9 m2 / s. Microbial uptake of DO in the sediment pore system was previously modeled by Higashino and Stefan 共2005兲 using Michaelis–Menten and other kinetic relationships. To model the DO diffusion process in the sediment pore system, we used DE 关Eq. 共17兲兴 instead of Dm / 2 used by Higashino and Stefan 共2005兲. We proceeded step by step as outlined in “Validation Procedure”: 1. Dispersivity ␣L = dg = 0.3 mm, and relative dispersivity ␭ = ␣L / L = 0.3 ⫻ 10−4. 2. Enhanced vertical dispersion coefficient DE共m2 / s兲 = 10−7共exp−6.15共y/10兲兲 from Eq. 共17兲. 3. DO concentration profiles in the lake bed from a discretized numerical form of Eq. 共21兲 with a first-order kinetic rate coefficient k1 = 0.01– 0.5共L / day兲

冉 冊

⳵C ⳵C ⳵ = DE − k 1C ⳵y ⳵t ⳵ y

4.

共21兲

Boundary conditions for solving Eq. 共21兲 are: 共1兲 C共0 , t兲 = C0 = 8 mg/ L and 共2兲 no flux at y = 2 L in the sediment. DO flux across the lake bed surface 共SOD兲 SOD = DE

冏 冏 ⳵C ⳵y

共22兲

Progressive surface waves on a lake can produce pressure waves on a lake bed that force flow through the pores of the permeable sediment. This flow enhances solute transport from the overlying water into the lake bed significantly. It was shown that the enhanced vertical solute flux into and within a lake bed can be several orders of magnitude greater than the molecular flux. The enhancement was incorporated in a depth-dependent vertical dispersion coefficient that can be used in a 1D vertical solute transport equation for a lake sediment bed. The value of the enhanced vertical dispersion coefficient is controlled by the steepness 共a / L兲 of the wave, the relative dispersivity or sediment size to wavelength ratio 共␭ = ␣L / L兲, the hydraulic conductivity, and the porosity of the lake sediment. Sediment particle size has a major influence. A numerical simulation of the 2D unsteady solute transfer in a permeable porous medium under progressive pressure waves was made. The simulated 2D concentration field was horizontally averaged and used to calculate the value of the enhanced vertical dispersion coefficient 共DE兲 across a wide and meaningful range of ␭ and a / L values. Results were normalized using the pressure wave amplitude 共a兲, wavelength 共L兲, sediment hydraulic conductivity 共K兲, dispersivity 共␣L兲, and porosity 共␧兲 to arrive at D*E = ␧DE / 共Ka␭兲, where ␭ = ␣L / L. The maximum value for D*E is ⬃0.05, and D*E decays with normalized depth 共y * = y / L兲 below the lake bed. Normalized wave speed 共W = c␧L / Ka兲 does not affect D*E for the realistic range W ⬎ 50. For practical applications, a function D*E共y *兲 was fitted to the numerical results 关Eq. 共17兲兴. The equation was applied to large-scale experimental data with good agreement. The penetration depth of the wave effect is approximately 75% of the wavelength. A sample application of the enhanced dispersion coefficient to the estimation of sediment oxygen demand by a lake bed demonstrated SOD estimated with the enhanced dispersion coefficient can be one order of magnitude larger than that estimated from the molecular diffusion. This indicates that solely using the molecular diffusion or hydrodynamic dispersion coefficient to analyze the solute flux in lake sediments in water quality models when progressive surface waves are present may yield erroneous results. Eq. 共17兲 is proposed for the estimate of a more realistic enhanced vertical dispersion coefficient in the sediment pores when surface waves are present. It can be used in a 1D vertical dispersion equation to simulate the transport of conservative or nonconservative solutes in a lake sediment bed.

Acknowledgments

y=0

after the DO profiles obtained from Eq. 共21兲 have reached steady state. 5. SOD from Eqs. 共21兲 and 共22兲. Two sets of computations were made: one with the depth-dependent DE from step 2 above and the other with DE = 0.5Dm, substituted into Eqs. 共21兲 and 共22兲. A comparison of the SOD values for both cases is given in Fig. 9.

The writers were granted access to the computational resources of the Minnesota Supercomputing Institute for Digital Simulation and Advanced Computation at the University of Minnesota, and express their gratitude for this privilege. This work was partially supported by the STC program of the National Science Foundation via the National Center for Earth-Surface Dynamics under agreement No. EAR-0120914. Helpful comments by two anonyJOURNAL OF HYDRAULIC ENGINEERING © ASCE / MARCH 2009 / 195

Downloaded 17 Feb 2010 to 128.101.188.45. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright

mous reviewers and the associate editor are acknowledged. The editor recommended numerous editorial improvements. The writers thank everyone.

Notation The following symbols are used in this paper: a ⫽ pressure wave amplitude 共m兲; aw ⫽ wave amplitude 共m兲; C ⫽ solute concentration 共mg/L兲; C0 ⫽ concentration at stream bed/water interface 共mg/L兲; ¯ ⫽ concentration C共x , y兲 averaged in x-direction, equal C ¯ 共y兲 共mg/L兲; to C c ⫽ pressure wave speed 共m/s兲; cw ⫽ wave speed 共m/s兲; Dem ⫽ effective molecular diffusion coefficient in sediment pore system, 共m2 / s兲; Dm ⫽ molecular diffusion coefficient, 共m2 / s兲; Dxx , Dyy , Dxy , Dyx ⫽ components of two-dimensional dispersion coefficient tensor 共m2 / s兲; DE ⫽ enhanced dispersion coefficient 共m2 / s兲; Dwater ⫽ mass transfer coefficient 共m2 / s兲; d p ⫽ penetration depth 共m兲; dg ⫽ particle diameter 共mm兲; g ⫽ acceleration due to gravity, 共m / s2兲; h ⫽ velocity head or water depth 共m兲; H ⫽ wave height, equal to 2a 共m兲; Hw ⫽ surface water wave height 共m兲; K ⫽ hydraulic conductivity 共m/s兲; k1 ⫽ oxygen uptake coefficient in lake bed 共L/day兲; L ⫽ pressure wavelength 共m兲; Lw ⫽ surface wavelength 共m兲; N ⫽ maximum grid number; P ⫽ dispersivity Péclet number; t ⫽ time 共s兲; td ⫽ development time 共s兲; u , v ⫽ local 共seepage兲 velocity components 共m/s兲; 兩V兩 ⫽ absolute velocity value; equal to 冑共u2 + v2兲; Vref ⫽ reference velocity in stream bed 共porous medium兲 共m/s兲; W ⫽ dimensionless wave speed; equal to c␧L / Ka; x ⫽ downstream distance 共m兲; y ⫽ vertical distance from stream bed surface 共m兲; ␣L ⫽ longitudinal dispersivity 共m兲; ␣T ⫽ transverse dispersivity 共m兲; ␧ ⫽ porosity= volume of pore space/total volume; ␭ ⫽ relative dispersivity= ␣L / L; and * ⫽ normalized parameter.

References Bear, J. 共1972兲. Dynamics of fluids in porous media, American Elsevier, New York. Benekos, D. I. 共2005兲. “On the determination of transverse dispersivity: Experiments and simulations in a helix and a cochlea.” Ph.D. dissertation, Stanford Univ., Stanford, Calif. Bretschneider, C. L. 共1959兲. “Wave variability and wave spectra for wind generated gravity waves.” Technical Memo No. 118, U.S. Army Corps of Engineers, Beach Erosion Board, Washington, D.C. DiToro, D. M. 共2001兲. Sediment flux modeling, Wiley, New York. Elliot, A. H., and Brooks, N. H. 共1997兲. “Transfer of non-sorbing solutes

to a streambed with bed forms: Theory.” Water Resour. Res., 33共1兲, 123–136. Fischer, H. B., List, E. J. R., Koh, C. Y., Imberger, J., and Brooks, N. H. 共1979兲. Mixing in inland and coastal waters, Academic, New York. Glud, R. N., Forster, S., and Huettel, M. 共1996兲. “Influence of radical pressure gradients on solute exchange in stirred benthic chambers.” Mar. Ecol.: Prog. Ser., 141, 303–311. Harleman, D. R. E., Mehlhorn, P. F., and Rumer, R. R. 共1963兲. “Dispersion-permeability correlation in porous media.” J. Hydr. Div., 89共HY2兲, 67–85. Higashino, M., and Stefan, H. G. 共2005兲. “Oxygen demand by a stream bed of finite length.” J. Environ. Eng., 131共3兲, 350–358. Horne, A. J., and Goldman, C. R. 共1994兲. Limnology, McGraw-Hill, New York. Huettel, M., and Rusch, A. 共2000兲. “Transport and degradation of phytoplankton in permeable sediment.” Limnol. Oceanogr., 45共3兲, 534– 549. Huettel, M., and Webster, I. T. 共2001兲. Porewater flow in permeable sediments: Chapter 7 in the Benthic boundary layer, B. P. Boudreau and B. B. Jorgensen, eds., Oxford University Press, New York, 144– 179. Jahnke, R., Richards, M., Nelson, J., Robertson, C., Rao, A., and Jahnke, D. 共2005兲. “Organic matter remineralization and porewater exchange rates in permeable South Atlantic bight continental shelf stream beds.” Cont. Shelf Res., 25, 1433–1452. Kinsman, B. 共1965兲. Wind waves, Prentice-Hall, Englewood Cliffs, N.J. Leonard, B. P. 共1979兲. “A stable and accurate convective modeling procedure base on quadratic upstream interpolation.” Comput. Methods Appl. Mech. Eng., 19, 59–98. McCowan, J. 共1891兲. “On the solitary wave.” London, Edinburgh Dublin Philos. Mag. J. Sci., 32, 45–58 Mei, C. C., Stiassnie, M., and Yue, D. K.-P. 共2005兲. Theory and applications of ocean surface waves, World Scientific, Singapore. Michell, J. H. 共1893兲. “On the highest wave in water.” Philos. Mag., 36共5兲, 430–435. Patankar, S. V. 共1980兲. Numerical heat transfer and fluid flow, Hemisphere, New York. Pfannkuch, H. O. 共1963兲. “Contribution a l’etude de deplacement des fluides miscibles dans un milieu poreux.” Revue de l’Institut Francais du Petrole, 2共18兲, 215–270. Precht, E., Franke, U., Polerecky, L., and Huettel, M. 共2004兲. “Oxygen dynamics in permeable stream beds with wave-driven pore water exchange.” Limnol. Oceanogr., 49共3兲, 693–705. Precht, E., and Huettel, M. 共2003兲. “Advective pore-water exchange driven by surface gravity waves and its ecological implications.” Limnol. Oceanogr., 48共4兲, 1674–1684. Qian, Q., Stefan, H. G., and Voller, V. R. 共2007a兲. “A physically based flux limiter for QUICK calculations of advective scalar transport.” Int. J. Numer. Methods Fluids, 55, 899–915. Qian, Q., Voller, V. R., and Stefan, H. G. 共2006兲. “Benchmark of QUICK scheme for environmental application.” Research Rep. No. 2006/94, University of Minnesota Supercomputing Institute 共UMSI兲, Minneapolis. Qian, Q., Voller, V. R., and Stefan, H. G. 共2007b兲. “Modeling of solute transport into sub-aqueous sediments.” Appl. Math. Model., 31, 1461–1478. Qian, Q., Voller, V. R., and Stefan, H. G. 共2008兲. “A vertical dispersion model for solute exchange induced by underflow and periodic hyporheic flow in a stream gravel bed.” Water Resour. Res., in press. Ren, J., and Packman, A. I. 共2004兲. “Modeling of simultaneous exchange of colloids and sorbing contaminants between streams and streambeds.” Environ. Sci. Technol., 38, 2901–2911. Rusch, A., and Huettel, M. 共2000兲. “Advective particle transport into permeable sediments—Evidence from experiments in an intertidal sandflat.” Limnol. Oceanogr., 45共3兲, 525–533. Shum, K. T., and Sundby, B. 共1996兲. “Organic matter processing in continental shelf stream beds—The subtidal pump revisited.” Mar. Chem., 53, 81–87.

196 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / MARCH 2009

Downloaded 17 Feb 2010 to 128.101.188.45. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright

Taylor, G. I. 共1953兲. “Dispersion of soluble matter in solvent flow slowly through a tube.” Proc. R. Soc. London, Ser. A, 219, 186–203. Taylor, G. I. 共1954兲. “The dispersion of matter in turbulent flow through a pipe.” Proc. R. Soc. London, Ser. A, 223, 446–468. Thibodeaux, L. J., and Boyle, J. D. 共1987兲. “Bedform-generated convective transport in bottom sediment.” Nature (London), 325, 341–343. United States Army Corps of Engineers 共USACE兲. 共1984兲. Shore protection manual, U.S. Government Printing Office, Washington, D.C. Van Rees, K. C. J., Reddy, K. R., and Rao, P. S. C. 共1996兲. “Influence of benthic organisms on solute transport in lake sediments.” Hydrobiologia, 317, 31–40.

White, F. M. 共1975兲. Hydrodynamics of coastal areas, Introduction to ocean engineering, H. Schenck, Jr., ed., Chap. 2, McGraw-Hill, New York, 8–42. Wiegel, R. 共1964兲. Oceanographical engineering, Prentice-Hall, Englewood Cliffs, N.J. Zaslavsky, D., and Ravina, I. 共1965兲. “Measurement and evaluation of the hydraulic conductivity through moisture moment method.” Soil Sci., 100, 104–107. Zheng, C., and Bennett, G. D. 共1995兲. Applied contaminant transport modeling: Theory and practice, Van Nostrand Reinhold, New York.

JOURNAL OF HYDRAULIC ENGINEERING © ASCE / MARCH 2009 / 197

Downloaded 17 Feb 2010 to 128.101.188.45. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright