A vertical dispersion model for solute exchange induced by underflow and periodic hyporheic flow in a stream gravel bed

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1 WATER RESOURCES RESEARCH, VO. 44,, doi: /2007wr006366, 2008 A vertical dispersion model for solute exchange induced by underflow and periodic hyporheic flow in a stream gravel bed Qin Qian, 1 Vaughan R. Voller, 1 and Heinz G. Stefan 1 Received 22 July 2007; revised 4 January 2008; accepted 15 May 2008; published 31 July [1] Interstitial flows in stream gravel beds are driven by stream slope and controlled by hydraulic conductivity (underflows) or induced by pressure differentials on the streambed surface (hyporheic flows). They enhance solute exchange between surface water and a streambed. To study the solute transport in a stream gravel bed, a 2-D transient advection dispersion mass transfer model was formulated. The velocity field includes an underflow and a spatially periodic hyporheic flow, e.g., due to standing surface waves or bed forms. Two dimensionless scaling parameters emerged: R measures the relative strength of hyporheic flow to underflow in the streambed and l is the ratio of dispersivity of the gravel bed to the pressure wavelength along the streambed. In the analysis of mass transfer of nonconservative substances into a streambed, an explicit 2-D analysis of interstitial flow is often undesirable. Therefore the numerical solutions for the 2-D concentration fields under periodic boundary conditions were reduced to 1-D vertical concentration profiles (by streamwise averaging of the solute concentrations). The profiles were matched to the solution of an unsteady vertical 1-D dispersion equation that introduces a depth variable enhanced dispersion coefficient D E (y) that lumps all forms of interstitial advective and dispersive transport in the streambed. Functions D E (y) were determined for many combinations of independent parameters by inverse modeling, and the dependence of D E (y) on the dimensionless parameters R and l was determined from these results. Analytical relationships for D E (y, R, l) have been proposed and validated against available experimental data. Knowledge of D E (y) allows the estimation of solute/mass transfer rates in streambeds under wavy boundary conditions, without explicit analysis of the interstitial flow. This is a distinct advantage for applications in stream water quality and/or pore water quality studies. Citation: Qian, Q., V. R. Voller, and H. G. Stefan (2008), A vertical dispersion model for solute exchange induced by underflow and periodic hyporheic flow in a stream gravel bed, Water Resour. Res., 44,, doi: /2007wr Introduction [2] Solute exchange between water in a stream and its gravel bed is a fundamental process that can affect water quality and stream ecology. For example, eggs of salmonids deposited in the gravel bed need dissolved oxygen from the overlying water to survive and to hatch [Bell, 1986]. On the other hand, if dissolved organic carbon (DOC) diffuses into the gravel bed, the pore water quality in the gravel bed can deteriorate [Hill, 2000]. The streambed layer in which surface water mixes with pore water is called the hyporheic zone. It can provide habitat for aquatic macroinvertebrates [Stanford and Ward, 1993] and juvenile fish [Vadas, 2003]. Hyporheic exchange is also important for nutrient cycling [Edwards, 1998] and biogeochemical processes [Fernald et al., 2001]. High hydraulic conductivity of gravel beds and steep slopes of streams increase the underflow in the hyporheic zone [Harvey and Wagner, 2000]. In 1 Department of Civil Engineering, St. Anthony Falls aboratory, University of Minnesota, Minneapolis, Minnesota, USA. Copyright 2008 by the American Geophysical Union /08/2007WR addition to underflow, periodic hydraulic features, e.g., standing surface waves [Tinkler, 1997] or bed forms [Elliott and Brooks, 1997; Elliott, 1990] can lead to a periodic pressure distribution along the gravel bed surface. This pressure distribution can induce a hyporheic flow in and out of the streambed [Thibodeaux and Boyle, 1987; Savant et al., 1987]. Both underflow and hyporheic flow affect the solute transfer process in a streambed significantly. [3] Different approaches have been proposed to model the solute exchange in the hyporheic zone. Richardson and Parr [1988] applied a modified 1-D Fickian model to the mass transfer induced by water flowing over a flat bed. The transient storage model (TSM) describes a linear first-order mass transfer process using a lumped exchange coefficient [Bencala and Walters, 1983; Young and Wallis, 1986, 1993; Worman et al., 1998; Runkel, 1998]. Worman [2000] matched the TSM and diffusive models with field data from tracer experiments. The TSM also predicted the hyporheic exchange of reactive solutes in streams [Jonsson et al., 2003]. Marion et al. [2003] and Zaramella et al. [2003] applied the TSM model to the prediction of hyporheic exchange induced by stationary bed forms. Hyporheic exchange is driven by multiple transport processes that 1of17

2 QIAN ET A.: DISPERSION MODE FOR SOUTE EXCHANGE Figure 1. Sketch of a stream reach showing the gravel bed and a periodic pressure wave distribution of amplitude (a) and wavelength () on the gravel bed surface. operate at different spatial scales [Packman and Bencala, 2000]. The effect of bed forms on the exchange process at a sediment-water interface was analyzed by Elliott and Brooks [1997] through laboratory experiments and by Cardenas and Wilson [2006, 2007] through simulation. A physical-chemical model was developed to predict the solute and colloid exchange in a streambed with stationary bed forms [Packman et al., 2000] and with moving bed forms [Packman and Brooks, 2001]. Bed form-induced advection and turbulence due to bed roughness enhance the hyporheic exchange [Packman et al., 2004]. Hyporheic exchange can be limited spatially by the structural heterogeneity of a streambed [Salehin et al., 2004]. Ren and Packman [2004] built a process-based model to represent the coupling of physical transport and chemical reactions. Marion and Zaramella [2005] applied the diffusive model to stationary bed forms experiments and indicated that the diffusion model can be used only in a narrow range of timescales. [4] Previous studies examined how stream and streambed characteristics affect the residence time of solute transfer and the solute mass exchange rate, rather than the dispersion process which includes advection and hydrodynamic dispersion in the porous media. The main objective of this article is to simulate advective-dispersive solute transport induced by hyporheic flow and underflow in a stream gravel bed, and to derive the rate of vertical solute transfer into the streambed. More specifically we are interested in determining, by 2-D simulation, how a vertical enhanced dispersion coefficient (D E ) in a stream gravel bed depends on standing pressure wave distribution, the gravel bed characteristics, and the depth below the bed surface. One of the applications of this information is in the sedimentary module of surface water quality models. Knowledge of the depth-dependent, vertical enhanced dispersion coefficient D E (y) allows the analysis of conservative, as well as nonconservative solute (mass) transfer between a streambed and the water flowing above it, without an analysis of the flow through the pore space. We will not include turbulence penetration from the stream and the associated mass exchange [Zhou and Mendoza, 1993; Nagaoka and Ohgaki, 1990] affecting a thin bed layer at the sediment-water interface. [5] This analysis will evaluate solute/mass flux in the hyporheic zone of a gravel bed stream by a rigorous 2-D advective-dispersive solute transport model with periodic pressure as a boundary condition. It will parameterize the results, and translate them into a vertical enhanced dispersion coefficient that provides an equivalent solute/mass flux and is parameterized by the properties of the porous sediment bed (grain size, porosity and hydraulic conductivity) and the period pressure distribution on the sediment surface (wave amplitude, wavelength). The 2-D analysis illustrates the role of different transport mechanisms, while the simplification has potential for analyzing the large-scale nonconservative solute transport (water quality) in streams and rivers. Relative to previous studies, the advectivedispersive flow analysis in the sediment bed will be more rigorous, and the vertical dispersion coefficient will be parameterized with few assumptions based on this analysis. 2. Problem Definition [6] Consider a stream gravel bed that is isotropic and homogeneous with a hydraulic conductivity (K), a slope (S) and a porosity (e). At the bed surface there is a stationary 2of17

3 QIAN ET A.: DISPERSION MODE FOR SOUTE EXCHANGE pressure distribution driven by either surface standing waves or bed forms (Figure 1). On the basis of the previous work with sand dunes, it can be argued that without too much loss of generality the pressure distribution can be modeled as a cosine or sine function of amplitude (a) and length () of a pressure wave [Elliott and Brooks, 1997; Cardenas and Wilson, 2006, 2007]. This pressure (head) distribution on the gravel bed surface will drive a hyporheic flow into and out of the gravel bed, which can combine with the slope driven underflow in streamwise direction. Near the bed surface this combined flow can be turbulent, and Darcy s law may not apply [Thibodeaux and Boyle, 1987; Savant et al., 1987]. Inertial effects or turbulence flow through a pore system are not as easy to quantify as purely viscous flow [Barr, 2001; Packman et al., 2004]. Therefore, Darcy flow has been used as a starting point for this model [Elliott and Brooks, 1997; Cardenas and Wilson, 2006, 2007]. [7] We model the flow and solute transfer in a 2-D streamwise section of the gravel bed. We solve the 2-D flow field analytically, and the transient solute transport advection-dispersion equation (ADE) in the streambed, numerically. The initial condition is a concentration C = C 0 in the stream and a concentration C = 0 in the gravel bed. At each time step, the 2-D concentration distribution will be averaged horizontally. The resulting concentration profiles will be matched to the solution of an unsteady 1-D vertical dispersion equation. A depth variable enhanced dispersion coefficient (D E ) will thus be determined by inverse modeling. Without loss of generality the enhanced dispersion coefficient will be determined for a conservative (nonreactive) solute. 3. Governing Equations 3.1. Two-Dimensional Solute Transfer Equation in a Gravel Bed [8] The 2-D mass transport equation for a nonreactive solute in a porous streambed has been written as [Zheng and Bennett, uc ð ðvcþ þ D þ D yy : In equation (1), C is the concentration of solute, u and v are the seepage (local) velocity components in the downstream (x) and depth (y) directions, respectively, and D xx, D xy, D yx, and D yy are the components of the 2-D dispersion coefficient tensor which were given by Zheng and Bennett [1995] as uu D xx ¼ a jvj þ a vv T jvj þ D em; D xy ¼ D yx ¼ ða a T Þ vu jvj þ D em; vv D yy ¼ a jvj þ a T uu jvj þ D em: In equations (2) a is the longitudinal dispersivity, a property of the porous medium describing dispersive ð2þ transport in the direction of flow; a T is the transverse dispersivity, a similar property of the porous medium describing dispersive transport pnormal to the direction of flow. The absolute value V = ffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 þ v 2 is the magnitude of the seepage velocity. D em is the effective molecular diffusion coefficient. [9] Equation (1) is solved in a domain of depth 2 and length 10 (Figure 2). The boundary conditions are as follows: (1) along the sediment water interface the concentration C is constant, i.e., C (x,0,t)=c 0 at all times; (2) on the bottom of the domain (y = 2), a no flux with zero gradient condition is imposed as /@y y=2 =0;thismakesthe vertical advective and dispersive fluxes at the lower boundary of the domain equal to zero; and (3) on the upstream and downstream boundaries of the domain (x = 0, and x =10)the periodic condition C(0, y, t) =C(10, y, t) is imposed by equating the fluxes, i.e., both the concentration and the horizontal gradients on these boundaries are equated. [10] By combing Darcy s law with the continuity equation for a flow in a porous medium, a aplace equation is obtained and analytically solved for a given cosine pressure head distribution, h(x, 0), of amplitude (a) and wavelength () along the streambed surface (y =0): hx; ð 0Þ ¼ h mean þ xs þ a cos 2px ; ð3þ where (h mean ) is the mean stream water depth. The mean head gradient in the horizontal direction is the stream slope (S), which by virtue of equation (3) is explicitly encountered at x = 0,, Head gradients in the vertical direction are zero at great depth, i.e., a far-field condition y=2 = 0 is imposed. The Darcy velocity field that is driven by the heads specified in equation (3) can be obtained by solving the aplace equation analytically [Elliott and Brooks, 1997; Ren and Packman, 2004]. An exact solution shown in equation (4a) requires image sources to account for upper (y = 0) and lower (y = d bed ) boundary conditions: u Darcy ðx; yþ ¼ 2pKa v Darcy ðx; yþ ¼ 2pKa sin 2px tanh 2pd bed cosh 2py þ sinh 2py þ KS cos 2px tanh 2pd bed sinh 2py þ cosh 2py : ð4aþ With the assumptions d bed =2 and tanh(4p) = 1, equation (4a) can be rewritten as u Darcy ðx; yþ ¼ 2pKa sin 2px exp 2py þ KS v Darcy ðx; yþ ¼ 2pKa cos 2px exp 2py ð4bþ : Equation (4b) is a very good approximation (without image terms) because at the upper boundary (y = 0) the velocity components are exact, and at the lower boundary (y =2) they are accurate to within 10 6 times the value at the upper 3of17

4 QIAN ET A.: DISPERSION MODE FOR SOUTE EXCHANGE Figure 2. Sketch of the computational domain (length 10, depth 2), the boundary conditions, and the concentration field for pressure wavelength 5 and 6. boundary. Because the focus of this study is on the dispersivity process close to the sediment surface and its relationship to the physical variables K, a, and, we can use Equation (4b). In general, even very small velocity differences at the bottom of the impermeable boundary (y = d bed ) do have an effect on the long tails of the residence time distribution in the entire sediment layer. [11] By dividing equation (4b) by the porosity (e) ofthe streambed, the seepage velocity components used in equation (1) can be obtained: ux; ð yþ ¼ u Darcyðx; yþ ¼ 2pKa sin 2px exp 2py þ KS e e e vx; ð y Þ ¼ v Darcyðx; yþ e ¼ 2pKa e cos 2px exp 2py : ð4cþ 3.2. Normalization of the 2-D Solute Transfer Equation in a Gravel Bed [12] It is desirable to normalize the solute transfer equation (1), in order to see the dimensionless groupings/ numbers that exert physical control over the dispersion process in the streambed. The proposed reference variables are: C ref = C o for concentrations in the streambed, periodic pressure wavelength () for distances (x and y), and V ref = Ka/e for velocities in the streambed. With these reference variables the dimensionless concentrations (C*), dimensionless distances (x* and y*), and dimensionless velocities (u*and v*) can be defined: C* ¼ C ; x* ¼ x C 0 ; y* ¼ y ; u* ¼ u ; v* ¼ v : V ref V ref Normalized concentrations (C*) in the streambed will be in the range 0 < C* < 1. Using the dimensionless variables, the velocity flow field can be written in dimensionless form as u* ¼ 2psinð2px* Þexpð 2py* Þþ 1 R v* ¼ 2pcosð2px* Þexpð 2py* Þ: ð5þ ð6þ 4of17

5 QIAN ET A.: DISPERSION MODE FOR SOUTE EXCHANGE [13] In equation (6), a dimensionless parameter R ¼ a S is used to capture the relative role of hyporheic flow velocity to underflow velocity. R is named steepness ratio because it is the pressure wave steepness (a/) divided by the stream steepness (S). [14] To normalize the equation (1), the dispersion coefficient tensor in equation (2) needs to be normalized in addition to the variables in equation (5). As a reference dispersion coefficient we have chosen D ref = V ref. Because of the dependence on local velocity components u and v,the normalized dispersion coefficients will be locally variable. Normalized local dispersion coefficients are D xx * ¼ D xx V ref ¼ a u*u* jv * j þ a T v*v* a jv* j þ D em V ref a D yy * ¼ D yy V ref ¼ a v*v* jv * j þ a T u*u* a jv* j þ D em D xy * ¼ D xy V ref ¼ D yx V ref ¼ a 1 a T a V ref a v*u* ; ; jv* j þ D em V ref a In equation (8) three other dimensionless parameters appear. The first is the dimensionless length scale parameter l = a / measures the longitudinal dispersivity to the wavelength of periodic pressure distribution. It characterizes the solute dispersion transfer process in the streambed. The second is the longitudinal to transverse dispersivity ratio (a T /a ) that characterizes the porous medium, i.e., streambed. The last one is an inverse Peclet number 1/Pe = D em /(V ref a ) that characterizes effective molecular diffusion to dispersivity in the porous streambed. [15] With the reference distance () and the reference velocity (V ref ), the normalized time can be defined as t* = /V ref. Combing the normalized variables from equations (5) and equation (8), equation (1) can be normalized as u*c* ð ðv*c* D* * þ D* * þ D* * þ D* * yx : 3.3. One-Dimensional Vertical Solute Transfer Equation [16] The transient advection and hydrodynamic dispersion inside a stream gravel bed can be represented by a single, depth-dependent, enhanced dispersion coefficient D E (y, t) in a one-dimensional vertical dispersion ¼ D ; where C is horizontally averaged concentration such that Cðy; tþ ¼ 1 Z =2 =2 : ð7þ ð8þ ð10þ Cx; ð y; tþdx: ð11þ This formulation allows for a much easier description of the interaction between stream water and pore water in the gravel bed than equation (1). The complexity of advection and dispersion in the gravel bed pore system is lumped into the enhanced dispersion coefficient (D E ). Equation (10) can be rearranged and integrated to obtain D E (y): D E ðyþ ¼ R 2 ðx; tþdx ; ð12þ y where x is a dummy variable. [17] Note that the mass or material flux across the bedwater interface can be calculated easily from equation (10) as J = D E /@y y=0 for incorporation in a surface water quality model. 4. Numerical Model 4.1. Two-Dimensional Solute Concentration Field [18] The streambed is discretized using a 10-wavelength (10 2) domain, and 10N 2N square control volume cells [Patankar, 1980]. N = 200 is the largest grid size number used for one pressure wavelength (). The solute transfer into and out of a single control volume cell has been specified by Qian et al. [2007a]. The concentrations are stored at the nodes (located at the control volume centers) and fluid velocity components are stored at the face midpoints. [19] The advection-dispersion equation (1) is solved using an explicit time stepping scheme. A third-order Quadratic Upstream Interpolation for Convective Kinematics (QUICK) [eonard, 1979] scheme written in terms of nodal point values of the advected variable has been applied [Qian et al., 2006] to decrease the numerical dissipation (smear). The physically based (phib) flux limiter [Qian et al., 2007b] is adopted to diminish the numerical dispersion. [20] Calculated concentrations near the center of the 10-wavelength (10 2) domain indicate a periodic solution, i.e., the predicted concentration field under length 5 is essentially identical to that under length 6 (Figure 2). To save computational time all subsequent calculations were made on the single length ( 2) domain, with periodic boundary conditions applied on the vertical sides (Figure 2) Enhanced Vertical Dispersion Coefficient (D E ) in Gravel Bed [21] To obtain the 1-D enhanced dispersion coefficient (D E ), the central differences of the averaged concentration values for time and space are applied to solve equation (12) for the N 2N grid: where D E ¼ C 1 j 1 C jþ1 X 2N 2Dy j¼1 C j ¼ 1 N X N C nþ1 j C i;j ; C n 1 j 2Dt! Dy; ð13þ ð14þ 5of17

6 QIAN ET A.: DISPERSION MODE FOR SOUTE EXCHANGE i and j stand for the grid number in x-direction and y-direction, respectively, and n stands for the time step 5. Range of Independent Parameters [22] Four independent dimensionless parameters R = a/(s), l = a /, a T /a and 1/Pe = D em /(V ref a ) have emerged from the normalization of the governing equations. To estimate the reasonable ranges of those parameters is important for us to analyze the problem Range of Pressure Distribution Parameters (a, ) and Steepness Ratio (R) [23] The periodic pressure distribution can be driven by bed forms or by stationary water surface waves. A sinusoidal pressure variation on the flat bed surface is a reasonable simulation for the sand dune bed forms [Elliott and Brooks, 1997]. An empirical expression for the dynamic pressure variation h m = a, caused by a bed form is [Elliott and Brooks, 1997] 8 H b =h 3=8 h m ¼ 0:28 U 2 >< H b =h 0:34 0:34 2g H b =h 3=2 >: H b =h 0:34; 0:34 ð15þ where H b is the height of the bed forms, h is the water depth, and U is the stream velocity. According to Elliott and Brooks [1997, Figure 2], a/ 0.1 can be estimated with the substitution of bed form wavelength b =. [24] Standing surface waves are a common feature in gravel bed streams because they are steep and have near critical flow conditions (Froude number 1.0) [Chanson, 1996]. The nature of standing waves has been observed in sediment-free, i.e., rock or gravel bed channels [Tinkler, 1997]. Standing waves also relate to mobile bed forms, e.g., antidunes for sand bed. Equations to estimate surface wave characteristics in a stream reach have been published. To obtain these relationships, surface wavelengths ( w ) and wave amplitudes (a w ) were measured at near-critical flow conditions in laboratory and field open channels, and empirical equations were fitted to the measurements. Equations based on fundamental concepts of hydrodynamics can also be found in the literature. Kennedy [1963] related the wavelength of stationary surface waves to Fr number and water depth (h) as w h ¼ 2pFr2 : ð16þ [25] The results are well connected with field observations in bedrock rivers [Tinkler, 1997]. The dimensionless wave amplitude (a w /h) of free surface undular flow in a stream is proportional to (1 Fr) according to Andersen [1978]: 8 a < w h ¼ : 0:656ð1 FrÞ for 0:7 Fr < 0:9 0:741ð1 FrÞ 1:028 for 1 < Fr 1:6 1:7: ð17þ Therefore, a w / w 0.08 can be estimated by taking the ratio of equation (17) to equation (16). This result agrees closely with the findings of Chanson [2000] for standing waves in a fixed bed channel. [26] Kinsman [1965] analyzed the relationship between the pressure distribution and the surface wave. In deep and intermediate water, the pressure steepness (a/) on the gravel bed surface is equal to the product of the surface wave steepness (a w / w ) and the depth-dependent function cosh((2p/)(h mean y))/cosh((2p/)h mean ). However, in shallow water, the pressure distribution, due to the local kinetic energy, cannot be ignored. Even if the pressure steepness cannot be connected to the surface wave steepness directly as in the deep water, a cosine pressure distribution is still a realistic description [Kinsman, 1965]. Chanson [1996] indicated the pressure steepness (a/) can be 20% less than a w / w at the wave crests and 20% larger than a w / w at the wave trough from the experiment data. Therefore, the range of a/ can be estimated as less than = < 0.1. Overall, the steepness (a/) of the pressure distribution wave due to bed forms and standing surface wave is very likely in the range from 0.01 to 0.1. Once the steepness (a/) is known, the steepness ratio R = a/(s) is easily determined. The streambed slope (S) of interest can be from to 0.1. Therefore, the steepness ratio (R) can be from 0.1 to However, to obtain R = 0.1 the wave steepness a/ must be on the order of 0.01 or less, i.e., a w / w 0.01 is a reasonable approximation. This represents a fairly flat wave with a weak pressure gradient on the bed. By dividing equation (17) by equation (16), a w / w is obtained as a function of Froude number. For R = 0.1 the Froude number estimated from a w / w 0.01 becomes negative, which is physically impossible. It is concluded that R = 0.1 is unrealistic or an unlikely combination of wave steepness and stream slope. Therefore, only R values from 1 to 1000 are of practical interest. [27] The numerically predicted solute concentrations for R = 1, 10, 100 and 1 at l = 0.01 in Figure 3 gave identical 2-D distribution plots when R 100, i.e., when the underflow can be ignored compared to the hyporheic flow. Hence, choosing R = 1, 10, and 100 in the model predictions covers the range of practical interest Range of Dispersivities (a, a T ) and Normalized ength Scale (l) [28] ongitudinal dispersivity (a ) in groundwater flow has been related to the length of the flow path at the field scale. Neuman [1990] presents an approximate scaling rule for longitudinal dispersivity in the empirical form: a = s ( s < 3500 m), where s is the length of the flow path. We cannot use this relationship because our scale is the pore scale. The pore-scale dispersivity measured in the laboratory is on the order of centimeters [Fetter, 1994]. The longitudinal dispersivity (a ) is in the rage from to 0.01 m [Zheng and Bennett, 1995] or to 0.01 m [Gelhar, 1993]. [29] To derive values for a and a T at the pore scale, we relate longitudinal dispersivity (a ) to the longitudinal dispersion coefficient (D ) derived experimentally for onedimensional flow and mass transport. The relationship between molecular diffusion and convective dispersion in one-dimensional flow was addressed by Saffman [1960] and Pfannkuch [1963]. A hydrodynamic longitudinal dispersion coefficient (D ) was measured experimentally and related to a particle Peclet number defined as Pe d = U d g /D m, 6of17

7 QIAN ET A.: DISPERSION MODE FOR SOUTE EXCHANGE Figure 3. Two-dimensional solute concentration fields for steepness ratios R = a/(s) = 1, 100, 10, 1, normalized dispersivity l = a / = 0.01, and normalized time t* = Figure 4. Two-dimensional solute concentration fields for normalized dispersivity l = , , 0.1 at R = 100 and normalized time t* = of17

8 QIAN ET A.: DISPERSION MODE FOR SOUTE EXCHANGE molecular diffusion, i.e., D /D m = D em /D m = 0.5. Hydrodynamic dispersion controls the mass transport process for Pe d 1. In this case, the experimentally determined hydrodynamic dispersion coefficient (D ) in equation (18) can be approximated as the dispersion coefficient tensor in x-direction (D xx ) by setting u = U = KS and v = 0, i.e., D xx = D. Therefore, combining equations (2) and (18) gives D xx ¼ a KS þ D em ¼ D ¼ Pe d ¼ KSd g ¼ a KS þ 0:5: ð19þ D m D m D m D m D m Equation (19) can be rewritten as a d g ¼ 1 0:5D m KSd g ¼ 1 1 2Pe d : ð20þ Figure 5. (top) One-dimensional (laterally averaged) concentration profile for R = 100, l = 0.01, and t* = 16; (middle) enlarged view of 1-D concentration profile and (bottom) its curvature very near the gravel bed surface (0 < y* < 0.1). where U is the seepage velocity in the longitudinal direction, d g is the mean particle size and D m is the molecular diffusion coefficient. Although Pfannkuch [1963] identifies five regimes, there are essentially two domains with different dispersive behavior for the line with most of the data points in Bear [1972, Figure ]: D D m ¼ 8 < 0:5 Pe d < 1 : Pe d Pe d 1: ð18þ [30] Equation (18) shows that the molecular diffusion (D m ) is dominant for Pe d < 1 and D is the effective The second term on the right-hand side of equation (20) becomes relatively small (<0.1) for Peclet numbers Pe d >5 and it can be dropped. Hence, a /d g = 1 can be obtained, i.e., the longitudinal dispersivity in a sediment bed is approximately equal to the particle size. The mean particle size for a gravel bed varies from millimeters to centimeters [Bear, 1972], and the range of a is therefore from approximately to 0.1 m. The pressure wavelength () which relates to a practical bed form or standing surface wave in a stream can be on the order of centimeters to meters [Kennedy, 1963], i.e., 0.1 < < 10 m. Therefore the range of l is from to 1 in a gravel bed stream. Concentration distributions for l = , 0.001, 0.01 and 0.1 at the same R number and time are plotted in Figure 4. With increasing l the concentration field is more uniform in both distance and depth. This suggests that the higher dispersivity enhances the solute transfer downward and sideways into the gravel bed as to be expected. [31] As has been suggested in other studies, we will keep the ratio of transverse dispersivity to longitudinal dispersivity (a T /a ) constant. Benekos [2005] reported that a T /a at the pore scale is usually 1/2 to 1/3 instead of 1/10, as previously reported [Bear, 1972]; a T = a /3 has also been used in numerical simulations of macroscopic dispersion [Zheng and Bennett, 1995]. With a d g, = w or b and a T /a 1/3 the dispersion coefficient tensor for a known velocity field can be calculated using equation (2) Range of Dispersivity Peclet Number (Pe) [32] The dispersivity Peclet number is defined as Pe = V ref a /D em in equation (8). The effective molecular diffusion coefficient (D em ) can be determined from the relationship D em = w D m, where w is an empirical coefficient with a typical value of 0.5, but can be as low as 0.01 [Freeze and Cherry, 1979]. A value of w can also be calculated from the porosity divided by the square of the tortuosity [Berner, 1971]. The molecular diffusion coefficient (D m ) depends on the chemical nature of the solute. A representative value, e.g., for dissolved oxygen in water is D m =10 9 m 2 /s. Substitute V ref = Ka/e, the corresponding Pe is equal to Kaa /(ewd m ). With typical values of K/e in a gravel bed, a typical range of a/ and a, D m =10 9 m 2 /s and w = 0.5, the value of Pe can be as great as Therefore, the 1/Pe = D em /V ref a becomes very small compared to the other terms in equation (8). This indicates, as expected, that molecular diffusion makes a very small contribution to the mass 8of17

9 QIAN ET A.: DISPERSION MODE FOR SOUTE EXCHANGE Figure 6. Normalized vertical enhanced dispersion coefficient (D E *) versus depth (y*) profile (a) at different normalized times (t*) for R = 100 and l = 0.01, (b) at l = and l = with R = 100, (c) for a/ = 0.01 and a/ = 0.1 with l = 0.1, and (d) for a T = a /2 and a T = a /3 with R = 100 and l = 0.1. transport when hyporheic flow is present in a sediment bed. The dispersivity Peclet number can therefore be ignored when the results will be presented. [33] In summary, of all the physical variables in the problem only the wave amplitude a, the particle size d g a, the wavelength and the stream slope S are significant. Their effect can be lumped into the two dimensionless parameters (R = a/s) and (l = a /). A fixed constant a T /a = 1/3 can be applied in the model as recommended in the macroscopic dispersion simulations. 6. Main Results 6.1. Horizontally Averaged 1-D Concentration Profiles [34] From the 2-D concentration distributions (examples in Figures 3 and 4) we calculated laterally averaged 1-D concentration versus depth profiles using equation (14). From these profiles we then obtained the depth variable enhanced dispersion coefficient D E (y) as outlined by equation (13). The top plot in Figure 5 is an example of the 1-D concentration profile at dimensionless time t* = 16 for l = 0.01 and R = 100. Since the velocity decays exponentially with depth, it is reasonable to expect that the enhanced dispersion coefficient related to this plot will also decay exponentially. In the immediate vicinity of the sedimentwater interface, however, the depth averaged concentration profiles obtained from the numerical solutions have an interesting feature, which can lead to an increase of the calculated dispersion coefficient with depth. The middle and bottom plots in Figure 5 show a blowup of the concentration profile and its curvature (d 2 C/dy 2 ) in this region (0 < y* < 0.15). These plots indicate an inflection point, i.e., a change in sign of the curvature at about y* This observation is important in the sense that it is known that in matching a steady state concentration profile to a dispersion coefficient, a region of decreasing concentration gradients will coincide with an increasing value of the dispersion coefficient and vice versa. It is reasonable to expect that the 1-D enhanced dispersion coefficient D E (y) calculated from the profile in Figure 5 will show the same property, i.e., an increase immediately below the sediment-water interface until the inflection point is reached, and followed by an exponential decay Vertical Enhanced Dispersion Coefficient (D E ) Dependence of D E on Depth, R, and l [35] A distribution of the vertical enhanced dispersion coefficient (D E ) at different times and values of R and l, with depth in the gravel bed was obtained from equation (13). For cases of practical interest, i.e., when l < 0.1 and R > 1, values of (D E ) were several (1 3) orders of magnitude larger than the horizontally averaged downward hydrodynamics dispersion tensor (D yy ) in equation (2). Using a reference dispersion coefficient D ref = V ref, a normalized enhanced dispersion coefficient D E *=D E /(Ka/e) can be defined. 9of17

10 QIAN ET A.: DISPERSION MODE FOR SOUTE EXCHANGE Figure 7. Normalized vertical enhanced dispersion coefficient (D E *) versus depth (y*) profile for 1 < R < 100 and < l < 0.1 and t* > 8 (quasi steady state conditions). [36] The D E * value varies not only with values of R and l, but also with depth (y) below the gravel bed surface. Initially it also varies with time, but it reaches a quasi steady state distribution after some initial development time. The enhanced dispersion coefficient does not change with time physically; rather the initial discontinuous concentration profile with depth needs to become more gradual to establish the dispersion process. An initial establishment time also exists for longitudinal dispersion in pipes or open channels [Taylor, 1953, 1954; Fischer et al., 1979]. The time to reach quasi steady state was longer with lower l value and higher R number. Figure 6a shows the D E * value varying with normalized time t* forr = 100 and l = A normalized time t* = 5 was sufficient to establish quasi steady state conditions. The normalized enhanced dispersion coefficients (D E )forl = and are identical. Below l = the enhanced dispersion coefficient does not change any more as shown in Figure 6b. One can conclude that advective transport dominates over dispersion and the wave effect on the penetrative enhanced convection is purely by advection if l On the other hand, when l > 0.1 (very large grain size relative to the wavelength), the normalized enhanced dispersion coefficient (D E ) remains the same for a/ = 0.1 and 0.01 as shown in Figure 6c, which illustrates the loss of the effectiveness of wave steepness a/ in a very coarse gravel bed. This indicates the transfer process switches from the advection dominated to dispersion dominated. It reaches a point where the solute transfer is not enhanced by the wave induced hyporheic flow. Therefore, only l values from to 0.1 are of practical interest. In addition, transverse dispersivities of a T = a /2 and a T = a /3 were tested in our model, but the effect on the enhanced dispersion coefficient (D E ) was negligible as long as R 1 and l < 0.1. An example is given in Figure 6d. [37] The enhanced dispersion coefficient is given in normalized (dimensionless form) in Figure 7 as a function of the two key independent variables, R and l. (Figure 4 gives 2-D concentration profiles from which the enhanced dispersion coefficient is eventually derived). A single function D E *(y*) can be fitted to the D E * versus y* plot when y* is beyond (greater than) the inflection point, regardless of the l value when R >10. R >10 represent a regime dominated by hyporheic flow. For R 10, the values of D E * at a given depth become smaller as the R number is decreased, i.e., the hyporheic flow is weaker because the pressure distribution on the streambed has smaller amplitude. In that case a stronger underflow inhibits the vertical penetration of the hyporheic flow into the gravel bed Maximum of D E [38] Owing to the near bed surface inflection in the averaged concentration profile, there is, as previously noted, an increase in the values of D E * with depth to a maximum at the inflection point followed by an exponential decay thereafter. The normalized vertical depth (d MAX *) between the surface to the inflection point and the value of the maximum normalized enhanced dispersion coefficient (D EMAX *) are of interest. Their dependence on R and l is complex. For the same l value, the depth (d MAX *) from the bed surface to the D EMAX *, i.e., the inflection point, remains the same for R > 10, but is a little smaller for R = 1. Figure 7 also shows that D EMAX * varies with the l value and this variation is consistent with the inflection point. The 10 of 17

11 QIAN ET A.: DISPERSION MODE FOR SOUTE EXCHANGE Figure 8. Profile of normalized vertical enhanced dispersion coefficient (D E *) versus depth (y*) for < l <0.1 and R = 10. D EMAX * is very near the gravel bed surface for l = 0.001, it moves downward when l = 0.01, appears to reach a maximum depth when l = 0.05, and moves back close to the gravel bed surface when l = 0.1. This behavior is shown in more detail in Figure 8. Table 1 lists the inflection point, i.e., (D EMAX *, d MAX *) for steepness ratios R = 1 and R 10 at different l values. The value of d MAX * has a maximum, while D EMAX * has a minimum at l = Empirical Equations for the Enhanced Dispersion Coefficient (D E ) [39] Overall, the enhanced dispersion coefficient decreases more or less exponentially with depth, except in the top region, and reaches a very low and constant value at a depth less than the pressure wavelength (Figures 6 and 7). For use in practical applications, e.g., water quality models, we shall fit an empirical equation D E *(y*, l, R) to the bulk of the numerical results in Figures 6 8. [40] Near the gravel bed surface, we will assume that the enhanced dispersion coefficient follows a straight line between the point (D E *(0), 0) on the gravel bed surface and the intersection point (D E *(d I *), d I *) on the exponential decay function line (Figure 8). The normalized depth from the bed surface to the intersection point (d I *) can be related to different R and l values. A relationship d I *(R, l) can be expressed in first approximation as for 0:001 l 0:05 8 0:0218 lnðlþþ0:1513 R ¼ 1 >< d I * 0:0326 lnðlþþ0:2208 R 10 ¼ 0:434 lnðrþð0:0108 lnðlþþ0:3721þ >: þ 0:0218 lnðlþþ0: R < 10 for l > 0:05; 8 0:086 R ¼ 1 >< d I * ¼ 0:123 R 10 >: 0:0161 lnðrþþ0:086 1 R < 10: ð21aþ Substituting y* =d I * into the exponential function D E *(y*) as defined in equation (21b), the corresponding D E *(d I *) can be found. When l is less than 0.01, the value D E *(d I *) is identical to the value D EMAX *(d MAX *) in Table 1. The point (D E *, d I *) moves down from the point (D EMAX *, d MAX *) for l great than Since D E *(0) does not change much with R, it can be expressed as a function of l only with an R 2 close to In summary, the enhanced dispersion coefficient can be approximated by equation (21b): D E ðy* Þ ¼ where 8 >< >: Ka e Ka e D E * d I * DE * ð0þ y* þ D d I * E * ð0þ exp Ay* þ KSd g þ 0:8Kal e er D E * d I * ¼ exp Ad* I þ Sdg a þ 0:8l R ; D E * ð0þ ¼ 5:4732l þ 0:0153; 8 < 6:15 R > 10 A ¼ : 9:9253R 0: R 10: for y* < d I * for y* d I *; ð21bþ The enhanced dispersion coefficient for y* d I * in equation (21b) includes three parts: The first term, (Ka/e)(exp Ay* ), is related to the hyporheic flow, the second term, KSd g /e, is connected to the underflow, and the third term, 0.8 Kal/eR, represents the dispersion tensor. The parameterization Ka/e was used by Elliott [1990] in an expression for a vertical dispersion coefficient D =(1/p)(3.5/4) 2 (Ka/e). Depth below the sediment bed was not included in this expression. [41] The maximum root mean square error (RMSE) of D E *(y*) obtained from equation (21b) has been determined for R =1,R = 10 and l = 0.001, 0.005, 0.01, 0.05 and 0.1. It was found to be less than for all cases. 7. Penetration Depth (d p ) [42] Figures 6 and 7 indicate that the enhanced dispersion coefficient diminishes with depth. The distance from the gravel bed surface to the normalized depth where D E *is nearly constant or zero, is the penetration depth (d p ). It Table 1. Maximum Normalized Enhanced Dispersion Coefficient (D EMAX *) and the Depth of Occurrence (d MAX *) for R = 1 and R 10 at Different l Values l R =1 R 10 D EMAX * d MAX * D EMAX * d MAX * of 17

12 QIAN ET A.: DISPERSION MODE FOR SOUTE EXCHANGE where 0 J disp ðyþ ¼ Z 0 J adv ðyþ ¼ 1 D dx þ Z 0 Z 0 1 D dx A vcdx ¼ JðyÞ J disp : The normalized total flux (J/(C 0 Ka/e)), the hydrodynamic dispersive flux (J disp /(C 0 Ka/e)), and the advective flux (J adv /(C 0 Ka/e)) are plotted versus depth for R = 1, 10 and 100 and l = 0.01 in Figure 10, and for l = 0.01, 0.05, 0.1 and R = 100 in Figure 11. The total normalized flux Figure 9. Dependence of hyporheic flow penetrations depth (d p *) on steepness ratio R = a/(s) and normalized dispersivity l = a /. measures the depth in the gravel bed over which advection due to hyporheic flow induced by periodic pressure variations at the gravel bed surface has an appreciable effect on mass transport. Because D E * approaches a constant value asymptotically with depth, an accurate value of the penetration depth is not easily obtained. To estimate the penetration depth (d p ) we defined it as the distance from the sediment bed surface to the depth where the term, (Ka/e)(exp Ay* ), becomes 1% of D E *(d I *). Hence d p can be found from equation (21b) by setting or e Adp ¼ 0:01e Ad I * d p ¼ 4:605 A þ d I * : ð22þ The normalized penetration depth can reach a value d p *= d p / = 0.75 when R 10 and l > For R between 1 and 10, the penetration depth decreases with R. The distribution of penetration depth (d p *) with R and l is shown in Figure Flux Analysis [43] The total vertical mass flux through the gravel bed surface is of particular interest, e.g., in water quality modeling of the overlying water. How much solute penetrates into the gravel bed can be calculated from the 1-D model equation J(y) = D E (y)/@y. The mass or material flux across the bed-water interface can be calculated as J = D E (0)/@y y=0. [44] From the results of the 2-D simulation model results, three solute flux components can be computed. The total vertical solute flux J(y) is the sum of two partial fluxes: one due to hydrodynamic dispersion (J disp ) and the other due to advection (J adv ): Z 2 JðyÞ ¼ ðx; tþdx ¼ J disp ðyþþj adv ðyþ; ð23þ Figure 10. Normalized solute flux (J*) versus normalized depth (y*) for R = 1, 10, 100 and l = of 17

13 QIAN ET A.: DISPERSION MODE FOR SOUTE EXCHANGE decreases monotonically with depth, whereas, the advective flux and hydrodynamic dispersive flux do not follow the same pattern. Quite interestingly the maximum advective flux and the minimum dispersive flux occur at the same depth near the streambed-water interface and this depth of the maximum advective flux in Figure 10 is consistent with the location of inflection point in the averaged concentration profiles of Figure 5. It is believed that the complex combination of advection and hydrodynamic dispersion causes the 1-D concentration profile to have an inflection point below the streambed-water interface. [45] In Figure 11, the transition from predominantly advective solute transport (l = 0.01) to predominantly dispersive solute transport (l = 0.1) is very evident. The D EMAX * occurs at the same depth as the maximum advective flux if advection is stronger than dispersion. On the other hand, the D EMAX * occurs at the same depth as the maximum dispersive flux. Figure 11. Normalized solute flux (J*) versus normalized depth (y*) for l = 0.01, 0.05, 0.1 and R = Validation 9.1. Experimental Data [46] To validate the foregoing model (equation (21b)) of enhanced vertical dispersion (D E ), experimental data illustrating the dispersion of a conservative tracer in a stream/ gravel bed are required. We used a data set measured by Packman et al. [2004]. In Packman et al. s experiment, bed forms created pressure gradients along the bed as described by Elliott and Brooks [1997]. Those experiments were conducted in a recirculating flume, 2.5 m long and 0.2 m wide. The channel was 0.5 m deep and the gravel bed is d bed = 0.19 m deep. The slope of the flume was from to , and water depths varied from m to m. The gravel bed had a mean particle diameter d g = 6 mm, the mean bulk porosity of the clean sediment was e = 0.38, and the hydraulic conductivity was K = 0.15 m/s. The data for three of the experiments labeled 7, 8, and 9 are listed in Table 2, where U is the mean flow velocity in the flume, h is the flume water depth, S is the flume slope, H b is the bed form height (trough to crest), and b is the bed form wavelength. A pressure variation (amplitude) (h m ) was calculated from equation (15). To match these conditions to the model characteristics, the substitutions a = h m, and = b are made. The calculated steepness ratio R = a/s is also given in Table 2. R values ranged from 0.8 to 5 in the experiments. Analysis of the experiments also required a value for the total volume of recirculating water (V R = d 0 A), where A is the surface area of the flume bed, and d 0 is defined as an effective stream depth [Packman et al., 2004]. The value d 0 can be easily obtained from a solute mass balance at the beginning and at the end of the experiment, since the solute mass is conservative. One obtains d 0 = ed bed C steady /(C 0 C steady ), where C 0 is the initial solute concentration, C steady is the final steady state concentration in the recirculating flume (obtained directly from Packman et al. s measurements) Validation Procedure [47] The following steps were taken to validate the model of the 1-D vertical enhanced dispersion coefficient (D E ) with the experimental data. [48] 1. Estimate the dispersivity a = d g = 0.6 cm, and calculate l = a / = [49] 2. Find d I * (l, R) from equation (21a) for the intersection point. [50] 3. Calculate the enhanced dispersion coefficient function D E (y) from equation (21b). Table 2. Flow Conditions in Packman et al. s [2004] Experiments Run Number U (cm/s) h (cm) Slope H b (cm) b (cm) h m (cm) R of 17

14 QIAN ET A.: DISPERSION MODE FOR SOUTE EXCHANGE Figure 12. (top) Comparison of solute concentrations in the water above the gravel bed in runs 7, 8, and 9 of Packman et al. s [2004] experiments with simulation results using the 1-D depth variable D E (y) from equation (21b); (middle) experimental data (run 7) and simulation results using Packman et al. s D eff-best and D E (y) from equation (21b); and (bottom) experimental data (run 8) and simulation results using only the exponential function in equation (21b) or the complete equation (21b) to compute D E. [51] 4. Use a finite difference approximation ¼ D ð24þ to track the concentration in the gravel bed with time and depth. [52] 5. At the gravel bed surface, the mass flux into the gravel bed must be equal to the mass flux out of the overlying water. This flux balance can be written as dc WATER d 0 ¼ D E : ð25þ y¼0 [53] 6. At each time step calculate the concentration in the water C WATER, by coupling an Euler solution of equation (25) with the finite difference result of equation (24). The resulting simulated values of C WATER can be compared with the experimental measurements Validation Results [54] We compared our model results with the data from runs 7, 8, and 9 (read from Figures 2 and 3) of Packman et al. [2004]. The top plot of Figure 12 gives the computed normalized concentrations (C/C 0 ) versus time and the data points from the experiments. The root mean square error (RMSE) was estimated for each run and ranged from to The experimental and the computational results are matched well, although run 8 has a slower decline in concentrations than the experimental data. In the model, any coupling between the flow in the streambed and the surface flow in the stream was ignored. However, turbulent momentum transfer from the surface water flow into the porous and permeable sediment may provide an additional enhancement of vertical solute transport resulting in increased vertical diffusion near the sediment surface. This becomes more significant for the lower-pressure heads h m and may explain the slower decline in model concentrations for run 8. [55] For performance comparison, the results in Figure 12 also show predictions made with simpler forms for the enhanced dispersion coefficient. For example, a constant effective diffusion coefficient D eff was estimated by Packman et al. [2004] for the entire gravel bed. For run 7 (middle plot of Figure 12) a single, constant, effective value D eff = cm 2 /s, which gives the best fit to the experimental data, was used to plot a second predictive line. The prediction is very good only at the beginning of the experiment. When the nonmonotonic near surface variations are neglected, so that D E (y*) is solely modeled by the exponential decay in equation (21b), the solute flux into the gravel bed is overpredicted as to be expected. The prediction is shown for comparison with the data from run 8 in the bottom plot of Figure 12. The need to account for the nonmonotonic component of D E (y*) is illustrated by this example. [56] The simulated dimensionless solute exchange m f * versus exchange timescale t f * can also be tested against data. Packman et al. [2004] give the definitions m f *=(d 0 / ed bed )(1 C water /C 0 )/(C water /C 0 ), and t f * = (2pK/) (tanh(2p/d bed )/d bed ) h m t. Figure 13 shows the measured dimensionless solute exchange in runs 7, 8 and 9 along with simulation results using the 1-D depth variable enhanced dispersion coefficient. The agreement is obvious. 10. Summary and Conclusions [57] Interstitial flow through the pores of a stream gravel bed enhances the solute transport from the stream water into 14 of 17