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Effect of channel size on solute dispersion and hyporheic exchange in rivers 47

Effect of channel size on solute dispersion and hyporheic exchange in rivers

Zhi-Qiang Deng1, Hoon-Shin Jung, and Bhuban GhimireDepartment of Civil and Environmental Engineering, Louisiana State University,

Baton Rouge, LA 70803-6405, USA1 Corresponding author. E-mail address: [email protected];

Tel.: +1 225 578 6850; Fax: +1 225 578 8652

Abstract: The effect of channel size on residence time distributions (RTDs) of solute in rivers is investigated in this paper using tracer test data and the Variable Residence Time (VART) model. Specifically, the investigation focuses on the influence of shear disper-sion and hyporheic exchange on the shape of solute RTD, and how these two transport processes prevail in larger and smaller streams, respectively, leading to distinct tails of RTD. Simulation results show that (1) RTDs are dispersion-dependent and thereby channel-size (scale) dependent. RTDs in large rivers are controlled by strong instream advection-dispersion processes and thus exhibit lognormal distributions. The influence of hyporheic exchange on RTDs increases with decreasing channel size and thus de-creasing longitudinal dispersion coefficient. Small streams with negligible dispersion coefficient may display various types of RTD from upward curving patterns to a straight line (power-law distributions) and further to downward curving lognormal distribu-tions when plotted in log-log coordinates. Moderate-sized rivers are transitional in terms of RTDs and commonly exhibit lognormal and power-law RTDs; (2) The incor-poration of water and solute losses/gains in the VART model can improve simulation results and make parameter values more reasonable; (3) The ratio of time to peak con-centration to the minimum mean residence time is equal to the recovery ratio of tracer. The relation provides a simple method for determining the minimum mean residence time; and (4) The VART model is able to reproduce various RTDs observed in rivers with 3 – 4 fitting parameters while no user-specified RTD functions are needed.

Keywords: Channel size; hyporheic exchange; longitudinal dispersion; residence time distribution; VART model.

1 IntroductionHyporheic exchange is the process through which surface stream water and subsurface ground water exchange solute (nutrients, contaminants, and dissolved oxygen) and energy across the sediment-water interface. The ex-

48 Z.-Q. Deng, H.-S. Jung, and B. Ghimire

change controls nutrient uptake and retention in streams by increasing both residence time and the contact of nutrients with biogeochemically ac-tive surfaces [14,20]. The exchange can attenuate pollutants in contaminated streams [16,34]. The exchange also determines the thermal regime of chan-nel bed sediments [7,25] and the abundance of microbial and invertebrate communities in hyporheic zones [4,22]. Over the past decades, extensive investigations have been conducted to understand and simulate processes underlying in-channel dead zones and hyporheic exchange-induced long-tailed residence time distributions (RTDs) of solute in streams and rivers [3,5,6,13,17,19,23, 26,29,30,36,37]. The investigations have found that tem-poral concentration distributions of solute observed in streams can be de-scribed using exponential or power-law RTDs [17,18,19, 26] or lognormal RTDs [6,37]. The reported RTDs are generally based on tracer experiments performed in small streams. Alexander et al. [1] found that nitrogen-loss rates in streams decline rapidly with increasing stream depth. This means that channel size has a significant effect on mass losses in streams. However, it is not clear whether channel size and mass losses affect RTDs. In fact, the influence of river channel size on RTDs is rarely studied. The primary goal of this paper is to analyze the effect of channel size on the shape of the RTDs. This goal can be achieved by using field tracer test data collected from rivers and the Variable Residence Time (VART) model developed by Deng and Jung [12]. The US Geological Survey conducted nationwide dye tests in 1960s and the early 1970s on fifty-one river reaches [25], ranging from about 300 m – 300 km and delivering flows from about 0.85 – 6820 m3/s including hyporheic flow and associated mass losses/gains. The field tracer experiments resulted in excellent data for evaluating effects of channel size and hyporheic exchange on longitudinal transport of solute in rivers. The effect of channel size can be analyzed by comparing simu-lated RTDs for rivers with distinct flow depths.

2 Variable residence time (VART) modelA Variable Residence Time (VART) model was developed by Deng and Jung [12] for simulating longitudinal transport of solute in streams with uniform flow (constant discharge). A major advantage of the VART model is that it allows for generation of multiple types of solute RTDs observed in streams while no user-specified RTD functions are required. The VART model consists of the following equations for non-uniform flow:


∂C + U

∂C = KS

∂2C +

Aadv + Adif l (CS – C) + qL (CL – C) (1a) ∂ ∂x ∂x2 A TV A

∂CS = 1

(C – CS) + qh (Ch –CS) (1b) ∂t TV Aadv + Adif

Adif = 4 p DS tS (1c)

TV = {Tmin for t ≤ Tmin (Tmin > 0) (1d) t for t ≥ Tmin

tS = {0 for t ≤ Tmin (1e) t – Tmin for t ≥ Tmin

where C = solute concentration [M/L3] in main channel; CS = solute con-centration [M/L3] in storage zones; Ch = solute concentration [M/L3] in the hyporheic exchange induced water gains and Ch = CS for water losses; CL = solute concentration [M/L3] in lateral inflows; U = cross-sectionally averaged flow velocity [L/T] in x [L] direction; t = time [T] and t = 0 cor-responds to the instant when tracer or solute injection starts in case of a Dirichlet type upper boundary condition for continuous injection. In case of multiple tracer injections, t = 0 corresponds to the initial release instant for the first injection; Tmin = minimum mean residence time [T] for solute to travel through the advection-dominated storage zone; tS = time [T] since the solute release from storage zones to the main stream; TV = actual vary-ing residence time [T] of solute. It can be seen from Eq. (1d) that solute residence time TV in transient storage zones is simply the time t since tracer injection when t ≥ Tmin; A = cross-sectional flow area of main channel [L2] (A = Q/U where Q = flow discharge); Aadv = area [L2] of advection-domi-nated transient storage zone; Adif = area [L2] of effective diffusion-domi-nated transient storage zone; qh = subsurface hyporheic exchange-induced water gain/loss rate per unit channel length [L2/T]; qL = surface lateral in-flow/outflow rate per unit channel length [L2/T]; KS = longitudinal Fickian dispersion coefficient excluding the transient storage effect [L2/T]. The pa-rameter KS is a function of flow velocity, depth, and channel width defining channel size and KS value increases with increasing channel size generally [8,11]; DS = effective diffusion coefficient [L2/T] in a hyporheic zone and it varies in a wide range from typical positive values of 1.0×10–5 m2/s – 1.0×10–10 m2/s [12,28] to zero and further to negative values [12]. Negative


diffusion or dispersion is characterized by the contraction of solute con-centration distributions and typically occurs when the net diffusive mass flux is smaller than the net advective mass flux in a reversing flow [24,35,38]. In case of hyporheic exchange, a net downward/forward mass flux occurs across the sediment water interface (SWI) due to the positive diffusion dur-ing the rising phase of instream solute concentration. It should be noted that the effective diffusion is not a pure diffusion process. Instead, it is the pore-scale dispersion that is primarily generated by differential advection of solute along advective hyporheic flow paths of varying lengths in the deep sediment layer [12]. A net upward/backward mass flux may occur during the receding phase of instream solute concentration when the hy-porheic flow-induced upward advective mass flux is greater than the down-ward diffusive mass flux, leading to the negative effective diffusion charac-terized by the contraction of transient storage zone area AS (= Aadv + Adif); l = dimensionless parameter [-] representing the ratio of hyporheic inflow rate to outflow rate, l > 1 for hyporheic exchange-induced water gains and l < 1 for water losses. If significant mass gains or losses are caused by lateral inflows or outflows, according to the mass conservation principle the concentration differential (CL – C) in Eq. (1a) can be replaced with (M – 1)C, where M is the recovery ratio of tracer/mass [27], and the last term in Eq. (1b) can be dropped. Likewise, if significant mass gains or loss-es are caused by hyporheic inflows or outflows, the concentration differen-tial (Ch – CS) in Eq. (1b) can be replaced with (M – 1)CS and the last term in Eq. (1a) can be removed. The flow velocity U, channel cross-section area A, and lateral inflow/outflow rate qL are commonly known or calculable for a given stream reach. If there is no lateral inflow or outflow (qL = 0), the parameter l can be calculated using l = (QD - QU + q)/q where QD and QU are stream flow rates (m3/s) at downstream and upstream sampling stations in a stream reach, respectively, and q is the flow rate from the stream to the storage zone. If no observed data is available for q, the parameter l can be roughly estimated using the relationship l ≈ QD/QU by assuming q = QU. Exclud-ing the lateral inflow/outflow rate qL, the parameter qh can be estimated using the relationship qh = (QD - QU)/L where L is the length (m) of the stream reach. If a water loss (l < 1) occurs, there is no need to determine qh because the second term on the right-hand side of Eq. (1b) disappears (Ch = CS). Therefore, there are at most four parameters (KS, DS, Aadv, and Tmin) to estimate in the VART model. Specifically, the four parameters (KS, DS/A, Aadv/A, and Tmin) can be determined using tracer test data and the frac-tional Laplace transform-based parameter estimation method proposed by


Deng et al. [10] or other parameter optimization methods [33]. There are only three parameters (KS, DS/A, and Aadv/A) to estimate for small and moderate-sized streams and some large rivers due to the new finding of a simple method for calculation of the parameter Tmin. A split-operator method used in the numerical solution of the VART model is provided in the Appendix.

3 Residence time distributions in large riversMost existing models for stream solute transport are tested using tracer experiment data collected from small streams. To understand the effect of channel size on RTDs and to evaluate the performance of VART model in simulating solute transport in large rivers, data of tracer injection experi-ments conducted in the Mississippi River, Red River, and Bayou Bar-tholomew are obtained from the USGS report by Nordin and Sabol [27]. The Mississippi River is one of the largest rivers in US and in the world. A dye test was conducted in the Mississippi River on August 7, 1968. Four sampling sites were located at 54.72 km (Crystal City, Missouri), 96.56 km (Genevieve, Missouri), 117.48 km (Chester, Illinois), and 294.51 km (Cairo, Illinois) downstream of the dye injection site. The reported flow discharge along the four river reaches was 6824.4 m3/s, meaning that l = 1, qL = 0, and qh = 0 (based on model calibration). Flow depths at the four sites were 9.24m, not available, 8.90 m, and 7.34 m. Recovery ratios of tracer at the four sites were 0.972, 0.843, 1.113, and 0.589, respectively. It should be noted that the recovery ratio values are directly taken from the USGS re-port [27] in which the recovery ratios were determined based on the mass conservation principle. The ratios are employed to represent solute losses (M < 1) or gains (M > 1) in the VART model. In order to identify the effect of water and solute losses/gains on RTDs, numerical simulations are per-formed under two cases where water and solute losses/gains are omitted in Case O and included in Case I. Estimates of VART parameters, used in producing the lognormal RTDs for the four reaches shown in Figure 1, are listed in Table 1 under the two (O and I) cases. Figure 1 shows that the type of lognormal RTDs was maintained along the 294.51 km long river reach. Table 1 indicates that the inclusion of solute losses in reaches 1, 2, and 4 and the solute gain in reach 3 has no effect on root mean square errors (RMSEs) because qL = 0 and qh = 0 make the last terms in Eqs. (1a) and (1b) dis-appear. It means that the solute losses/gains in the Mississippi River were


caused by some other mechanisms that are not included in the VART model. The RMSE is commonly employed as a metric for evaluating the goodness of fit of model simulations to measured data [2]. The Red River is a tributary of the Mississippi River. A dye test in the Red River was conducted on April 7, 1971. Four sampling sites were located at 5.74 km (Grand Ecore, Louisiana (LA)), 75.64 km (Colfax, LA), 132.77 km (Alexandria, LA), and 193.12 km (St. HWY 115, Moncla, LA) down-stream of the dye injection site, respectively. Flow discharges at the four sites were 230.2 m3/s, 245.2 m3/s, 249.5 m3/s, and 249.5 m3/s, respectively. Hyporheic exchange-induced water gains in reaches 1 – 4 were 0, 2.15× 10–4 m3/s.m, 7.44×10–5 m3/s.m, and 0, respectively. No lateral inflows/out-

Table 1. Parameter values used in Fig. 1 for the Mississippi River.

Case Reach U Ks Aadv/A Ds/A Tmin l M qh qL RMSE

(m/s) (m2/s) (1/s) (hours) (m2/s) (m2/s)

O 1 1.45 80 0.15 -3.26E-7 4.0 1 1 0.0 0.0 0.28 2 1.50 100 0.30 -5.68E-7 6.0 1 1 0.0 0.0 0.25 3 1.30 250 0.55 -8.09E-7 3.0 1 1 0.0 0.0 0.31 4 1.58 80 0.05 0.0 3.5 1 1 0.0 0.0 0.35

I 1 1.45 80 0.15 -3.26E-7 4.0 1 0.972 0.0 0.0 0.28 2 1.50 100 0.30 -5.68E-7 6.0 1 0.843 0.0 0.0 0.25 3 1.30 250 0.55 -8.09E-7 3.0 1 1.113 0.0 0.0 0.31 4 1.58 80 0.05 0.0 3.5 1 0.589 0.0 0.0 0.35

Figure 1. RWT (Rhodamine WT) concentration dis-tributions observed (circles) on August 7, 1968 in four sam-pling stations in the Mississippi River and simulated (lines) using the VART model for an instantaneous dye addition. The curves RTD-LO and RTD-LI are produced with the parameter values in Case O and Case I shown in Table 1, respectively.


flows were reported. Flow depths at the four sites were 4.82 m, not availa-ble, not available, and 1.62 m, respectively. Recovery ratios of tracer at the four sites were 0.741, 0.740, 0.695, and 0.587, respectively. Values of param-eters used in producing the lognormal RTDs (RTD-L) for the four reaches shown in Figure 2 are listed in Table 2 for the two cases. Figure 2 shows that the type of lognormal RTDs was maintained along the 193.12 km long river reach. Table 2 indicates that the incorporation of solute losses in the reaches and hyporheic flow-induced water gains in reaches 2 and 3 into the VART model reduces RMSEs (especially for the last reach) and thus im-proves the fitting of simulated RTDs to the observed ones. Bayous are commonly found in the south and southeastern United States

Table 2. Parameters values used in Fig. 2 for the Red River.


(m/s) (m2/s) (1/s) (hours) (m2/s) (m2/s)

O 1 0.64 40.0 0.15 -7.24E-7 0.34 1 1 0.0 0.0 0.24 2 0.64 40.0 0.15 -4.83E-7 6.10 1 1 0.0 0.0 0.25 3 0.62 60.0 0.16 -1.45E-7 8.60 1 1 0.0 0.0 0.18 4 0.52 60.0 0.05 -2.41E-7 13.62 1 1 0.0 0.0 0.29

I 1 0.64 40 0.15 -7.24E-7 0.34 1 0.741 0.0 0.0 0.24 2 0.64 40 0.15 -4.83E-7 6.1 1.07 0.740 2.15E-4 0.0 0.25 3 0.62 60 0.16 -1.45E-7 8.6 1.02 0.695 7.44E-5 0.0 0.17 4 0.52 60 0.08 -1.21E-7 13.62 1.00 0.587 0.0 0.0 0.24

Figure 2. RWT con-centration distribu-tions observed (cir-cles) on April 7, 1971 in four sampling reaches in series along the Red River and simulated (lines) using the VART model for an instan-taneous dye addi-tion.


and characterized by fine-grained substrates, low gradient, low flow veloc-ity, high TDS (total dissolved solids) and turbidity. Bayou Bartholomew is the largest bayou in the world and home to majestic cypress trees and ex-tensive amounts of wildlife and fish species, making it one of the most spe-cies-rich streams in North America. Substratum is predominantly silt and clay. The principal source of coarse substratum is large woody debris and gravel or cobble-sized particles from road and bridge construction. Bayou Bartholomew, flanked by wet bottomland forest, meanders through exten-sive croplands. An instantaneous RWT injection was performed on June 25, 1971 in the Bayou Bartholomew. Four sampling sites were established at Jones, Green Grove, Beekman, and Mouth, Louisiana, USA. The four sampling sites were located at 3.22 km, 25.75 km, 59.54 km, and 117.48 km downstream of the dye injection site, respectively. Flow discharges at the four sites were 4.1 m3/s, 4.8 m3/s, 6.5 m3/s, and 8.1 m3/s, respectively. Groundwater discharges into reaches 1 – 4 were 0, 3.14×10–5 m3/s.m, 5.03×10–5 m3/s.m, and 2.74×10–5 m3/s.m, respectively. No lateral inflows/outflows were reported. Flow depths at the four sites were 1.18 m, not available, 0.73 m, and 2.07 m, respectively. Recovery ratios of tracer at the four sites were 0.811, 0.842, 0.844, and 1.404, respectively. The flow veloc-ity in each river reach is calculated using the distance from the injection site and the time to peak concentration at each sampling site. Other four pa-rameters (KS, DS/A, Aadv/A, and Tmin) are determined using the method

Figure 3. RWT con-centration distribu-tions observed (cir-cles) on June 25, 1971 in four sampling reaches in series along the Bayou Bar-tholomew and simu-lated (lines) using the VART model for an instantaneous dye addition.


presented by Deng et al. [10]. Estimated parameter values and RMSEs of the simulated RTDs shown in Figure 3 are listed in Table 3 for the two cases. The table indicates that the incorporation of solute losses in reaches 1 – 3 and the solute gain in reach 4 as well as hyporheic flow-induced water gains in reaches 2 – 4 into the VART model significantly reduces RMSEs (especially in the last reach) and thus improves the fitting of simulated RTDs to the observed ones. Silty and clayey streams like the Bayou Bartholomew are generally excluded from studies on transient storage effect including hyporheic ex-change that is commonly assumed to occur in sandy and gravel streams. Hulbert et al. [21] studied micrographs of fine-grained sediments from Louisiana bayous and found that the sediment-water interface is character-ized by great porosity and long (deep) pore-fluid pathways. It was also suggested that exchange between the pore fluid and the overlying water column would be relatively unhindered and the permeability of the undis-turbed interface would be relatively high. However, little is actually known about the transient storage effect of solute in bayous. Figures 1 – 3 clearly indicate that all the large rivers exhibit lognormal RTDs although the bed sediment (primarily silt and clay) in the Bayou Bartholomew is significantly different from those (primarily sand and gravel) in the Red and Mississippi Rivers. It means that the shape of tracer concentration distribution in large rivers is not controlled by sediment properties and hyporheic exchange. Figures 1 – 3 and Tables 1 – 3 also dem-onstrate that the VART model is able to reproduce the tracer concentration distributions observed in large rivers with a reasonable accuracy (RMSE). Simulated RTD-L distributions fit observed tracer concentration distribu-tions very well for all stations in the three large rivers except the last station

Table 3. Parameter values used in Fig. 3 for the Bayou Bartholomew.


(m/s) (m2/s) (1/s) (hours) (m2/s) (m2/s)

O 1 0.152 3 0.15 -7.58E-7 1.54 1 1 0 0 0.46 2 0.115 5 0.1 -2.35E-7 10.7 1 1 0 0 0.14 3 0.155 10 0.22 -1.61E-7 10.7 1 1 0 0 0.25 4 0.15 50 0.13 -2.68E-8 11 1 1 0 0 0.55

I 1 0.152 3 0.15 -7.58E-7 1.54 1 0.811 0 0 0.46 2 0.115 5 0.1 -2.35E-7 10.7 1.17 0.842 3.14E-5 0 0.13 3 0.155 10 0.22 -1.61E-7 10.7 1.35 0.844 5.03E-5 0 0.23 4 0.15 50 0.25 -7.16E-8 11 1.24 1.404 2.74E-5 0 0.49


in the Mississippi River and the Bayou Bartholomew. It can be seen from Figure 3 and Table 3 that a significant mass gain (M = 1.404) occurred in the last reach of the Bayou Bartholomew, causing a significant underestimate of tracer concentration in the falling limb of the BTC. It is not clear what caused the abnormal mass gain. It appears that large rivers are capable of maintaining a single type of RTDs over a long distance.

4 Residence time distributions in moderate-sized riversIn order to understand solute transport dynamics in moderate-sized rivers and the performance of VART model in such rivers, data of tracer experi-ments conducted in the Tickfau River and Tangipahoa River, Louisiana (LA), USA, are gathered from the USGS report by Nordin and Sabol [27]. The two rivers are located in the Lake Pontchartrain River Basin. The Tangipahoa River begins as an upland stream in Mississippi and flows southeastward for 127 km from the Mississippi-Louisiana state line through Tangipahoa Parish into Lake Pontchartrain. As it makes its way southward, it flows through rolling hills where it has a sand and gravel substrate. South of Highway 190 characteristics of the river change to those of a lowland stream where flat land levels off, substratum is silt and clay, and the water becomes sluggish, curved (meandering), and often muddy. A dye test in the Tangipahoa River was conducted on September 15, 1969. Four of seven sampling sites were located at 8.21 km (Kentwood, LA), 41.52 km (Amite, LA), 70.97 km (Natalbany, LA), and 93.98 km (Pon-chatula, LA) downstream of the dye injection site, respectively. Flow dis-charges at the four sites were 3.5 m3/s, 6.9 m3/s, 8.6 m3/s, and 10.8 m3/s, respectively. Flow depths at the four sites were 0.49 m, not available, 0.46 m, and 0.76 m, respectively. Average flow depth in the reaches was 0.52 m. Recovery ratios of tracer at the four sites were 1.023, 0.802, 0.741, and 0.696, respectively. The remaining three sites are not included here because they were very close to the last three sites and were thus not representative. Lateral (tributary) inflow rates of reaches 1 – 4 were 0, 1.05×10–4 m3/s.m, 0, and 9.72×10–5 m3/s.m, respectively. The discharge gain in stream reach 3 was attributed to the hyporheic flow-induced water gain at the rate of 5.67×10–5 m3/s.m because no tributaries joined the reach. Estimated para-meter values and RMSEs of the RTDs shown in Figure 4 are listed in Table 4 for the two cases. Table 4 indicates that the incorporation of solute losses in reaches 2 – 4 and the solute gain in reach 1 as well as water gains in


reaches 2 and 4 due to lateral inflows and in reach 3 due to hyporheic ex-change into the VART model markedly reduces RMSEs (especially in reaches 2 and 4), and thus improves the fitting of simulated RTDs to the observed ones. The Tickfau (Tickfaw) River originates in Southern Mississippi, USA and flows southeastward from the Mississippi-Louisiana state line through St. Helena and Livingston Parishes, Louisiana and eventually empties into Lake Maurepas. The scenic portion of the stream, approximately 110 kil-ometers long, flows southward through flat, alluvial bottomland with seep-age (hyporheic flow) from ground water aquifers sustaining the stream flow. Substratum variation is similar to that of the Tangipahoa River. A dye

Table 4. Parameter values used in Fig. 4 for the Tangipahoa River.


(m/s) (m2/s) (1/s) (hours) (m2/s) (m2/s)

O 1 0.165 3 0.2 -4.29E-7 2.44 1 1 0 0 0.44 2 0.29 20 0.2 -2.43E-7 7.4 1 1 0 0 0.36 3 0.32 8 0.5 -1.46E-5 9.4 1 1 0 0 0.42 4 0.29 8 0.3 -1.72E-5 11.5 1 1 0 0 0.29

I 1 0.165 3 0.2 -4.29E-7 2.44 1 1.023 0 0 0.44 2 0.29 20 0.2 -2.43E-7 7.4 1 0.802 0 1.05E-4 0.29 3 0.32 8 0.45 -1.46E-6 9.4 1.24 0.741 5.67E-5 0 0.4 4 0.29 8 0.3 -1.72E-5 11.5 1 0.696 0.0 9.72E-5 0.22

Figure 4. RWT con-centration distribu-tions observed (cir-cles) on September 15, 1969 in four sampling reaches in series along the Tangipahoa River and simulated (lines) using the VART model for an instantaneous dye addition.


test in the Tickfau River was conducted on October 8, 1968 within the sce-nic portion. Four sampling sites were located at 6.44 km (Montpellier at Highway 16, LA), 22.53 km (Camp above Starns Bridge, LA), 38.62 km (Holden, LA), and 49.89 km (Springville, LA) downstream of the dye in-jection site, respectively. Flow discharges at the four sites were 2.0 m3/s, 2.2 m3/s, 1.9 m3/s, and 2.9 m3/s, respectively. Groundwater discharges into reaches 1 – 4 were 0, 1.23×10–5 m3/s.m, –2.29×10–5 m3/s.m, and 9.3×10–5 m3/s.m, respectively. No lateral inflows/outflows were reported. Flow depths at the four sites were 0.43 m, 0.80 m, 0.54 m, and 1.04 m, respec-tively. Average flow depth in the reaches was 0.70 m that was greater than

Table 5. Parameter values used in Fig. 5 for the Tickfau River.


(m/s) (m2/s) (1/s) (hours) (m2/s) (m2/s)

O 1 0.17 3 0.1 -4.07E-7 3 1 1 0.0 0.0 0.26 2 0.105 3 0.1 -9.17E-8 9.1 1 1 0.0 0.0 0.32 3 0.135 2 0.1 -1.09E-7 16.1 1 1 0.0 0.0 0.23 4 0.17 10 0.2 -5.36E-7 10.2 1 1 0.0 0.0 0.26

I 1 0.17 3 0.1 -4.07E-7 3 1 0.829 0.0 0.0 0.26 2 0.104 3 0.1 -9.17E-8 9.1 1.10 0.764 1.23E-5 0.0 0.23 3 0.135 2 0.15 -1.45E-7 16.1 0.84 0.560 -2.3E-5 0.0 0.22 4 0.177 6 0.18 -5.36E-7 10.2 1.56 0.781 9.3E-5 0.0 0.25

Figure 5a. RWT concentration dis-tributions observed (circles) on October 8, 1968 in four sam-pling reaches in se-ries along the Tick-fau River and RTD−L RTDs (lines) simulated using the VART model for an in-stantaneous dye addition.


that in the Tangipahoa River (0.52 m). Re-covery ratios of tracer at the four sites were 0.829, 0.764, 0.560, and 0.781, respectively. Estimates of parameters and RMSEs of the simulated RTDs shown in Figure 5a are listed in Table 5 for the two cases. The table indicates that the incorporation of solute losses in the four reaches and hyporheic flow-induced water gains in reaches 2 and 4 as well as the water loss in reach 3 into the VART model significantly reduces RMSEs (especially in the second reach) and thus improves the fitting of simulated RTDs to the observed ones. Figures 4 and 5 show that moderate-sized rivers may display either log-normal or power-law RTDs. Most reaches of the Tickfau River exhibit typical lognormal RTDs (RTD-L), as seen in Figure 5a. The first (upper) reach of the river can also be fitted using a power-law (RTD-P) distribution with RMSE = 0.37, as shown in Figure 5b. Likewise, the four reaches of the Tangipahoa River may also be fitted using power-law RTDs although the simulated RTDs shown in Figure 4 and Table 4 belong to RTD-L. In fact, the solute concentration distributions of the Tangipahoa River (with a mean water depth of 0.52 m) are closer to power-law distributions than those of the Tickfau River (with a mean water depth of 0.70 m). Residence time distributions in small streams were analyzed and reported by Deng and Jung [12]. The study showed that small streams may exhibit a wide variety of RTD types from upward curving patterns to a straight line (power-law distributions) and further to downward curving lognormal dis-tributions when plotted in log-log coordinates.

5 Discussion A comparison among the parameter values listed in Tables 1 – 5 shows that the three VART parameters Aadv/A, DS/A, and Tmin commonly vary in the ranges of 0.1 – 0.5, 1.0×10–7 – 9.0×10–7 s–1, and 0.3 – 16.0 hours, respec-tively, in large and moderate-sized rivers and also in small streams [12]. It appears that the parameter values/ranges are independent of channel size because there are no significant variation trends in the values of the three

Figure 5b. RWT concentration distribu-tions observed (circles) on October 8, 1968 in the upper reach of the Tickfau River and the RTD − P and RTD − E distributions (lines).


parameters that govern solute ex-change between surface stream water and subsurface sediment pore water. Among the four VART parameters KS, Aadv/A, DS/A, and Tmin, the pa-rameter that is most sensitive to changes in channel size is the longitu-dinal dispersion coefficient KS. The parameter KS varies over several or-ders of magnitude in the ranges of 0.1 – 1.0 m2/s in small streams [12], 1.0 – 10.0 m2/s in moderate-sized rivers, and 10.0 – 250.0 m2/s in large rivers, as shown in Tables 1 – 5 and in [8]. The formula for estimation of the longitu-

dinal dispersion coefficient presented by Deng et al. [8] shows that param-eter KS is proportional to flow depth and U2. Obviously, river size signifi-cantly affects longitudinal dispersion coefficient KS. In order to better un-derstand the effect of the variation in parameter KS on RTDs, two extreme but commonly-used scenarios are discussed as follows. In the first scenario it is assumed that the transient storage term including lateral inflows/outflows is negligible as compared to the longitudinal dis-persion term in Eq. (1). This is a fundamental assumption made in the clas-sical advection-dispersion theory for streams [8,15]. Then, Eq. (1) reduces to the conventional advection and dispersion equation that has the follow-ing analytical solution for an instantaneous slug injection of tracer:

C = W

exp ((x – Ut)2) (2) A √ 4p KSt 4KSt

where W is the mass of tracer injected. Figure 6 shows the solute concentra-tion distributions obtained from Eq. (2) by fixing values of parameters W, A, x, and U and changing the dispersion coefficient KS. The curves shown in Figure 6 are typical lognormal distributions. It means that a lognormal concentration distribution can be generated by the conventional advection and dispersion processes in streams without transient storage zones. This scenario may occur in large rivers where the longitudinal dispersion coef-ficient becomes large and advection and dispersion processes dominate sol-ute transport. In the second scenario it is assumed that the dispersion term is negligible

Figure 6. Lognormal concentration distribu-tions produced by advection and dispersion processes under three different longitudinal dispersion coefficients.


as compared to the transient storage term in Eq. (1). This assumption is often used for small streams [32,33]. Figure 7 shows the five types of VART series RTDs proposed by Deng and Jung [12] and simulated using Eq. (1) without the longitudinal dispersion term (KS = 0) by changing the para-meter DS/A and fixing other parameters. The VART series distributions, when plotted in log-log coordinates, switch from the upward curving VART+U and VART0U to a straight line (power-law: RTD-P) and further to the downward curving lognormal distributions (RTD-L) when the pa-rameter DS/A decreases from positive to zero and further to negative val-ues, as shown in Table 6 and Figure 7. The VART+U RTDs commonly occur in small streams [12]. Figure 7 clearly indicates that the transient storage term in the VART model, namely the way in which transient stor-age is parameterized in the model, can generate different tail behaviors of the RTDs due to the advection and effective diffusion (differential advec-tion) processes in the two types of hyporheic zone: (1) an advection-domi-nated upper transient storage zone and (2) an effective diffusion-dominated lower transient storage zone. It means that in theory (VART model) small streams with a negligible longitudinal dispersion coefficient are capable to produce various RTDs due to the hyporheic exchange. It should be noted that the lognormal, power-law, and exponential RTDs are not specific to the VART model. Instead, these RTDs can also be generated using oth-er models such as STAMMT-L [17] or CTRW [3]. The simulation results show that solute RTDs are dispersion-dependent and thereby scale-dependent. RTDs in

Table 6. Parameter values used in the VART series distributions shown in Fig. 7.

Parameter U (m/sec) KS (m2/s) Aadv/A DS/A (1/s) Tmin (hrs)

VART+U 0.37 0.0 0.12 +2.46E-8 4VART0U 0.37 0.0 0.12 0 4RTD-E 0.37 0.0 0.12 0 4RTD-P 0.37 0.0 0.12 -4.91E-8 4RTD-L 0.37 0.0 0.12 -3.69E-7 4

Figure 7. Five types of VART series RTDs (VART+U, VART0U, RTD-E, RTD-P, and RTD-L) produced with the VART model under a zero dispersion coefficient (KS = 0).


large rivers are controlled primarily by instream advection-dispersion proc-esses and thus exhibit lognormal distributions (RTD−L). The influence of hyporheic exchange on RTDs increases with decreasing channel size and thus decreasing longitudinal dispersion coefficient. Small streams with neg-ligible dispersion coefficient may display various types of the VART series distributions, as shown in Figure 7. Moderate-sized rivers are transitional in terms of RTDs. Thus, moderate-sized rivers commonly exhibit lognor-mal (RTD−L) and power-law (RTD−P) RTDs, as shown in Figures 4 and 5. More investigations are definitely needed to provide quantitative criteria for describing the transition from one type of RTDs to another one. Any-way, there is no single scale-independent type of RTDs that can be univer-sally applied to any rivers and streams. In that sense, RTDs are channel size or spatial scale dependent. Eqs. (1a) – (1e) clearly shows that VART model is also a temporal scale-dependent model [11] because an actual variable residence time is employed in Eqs. (1a) and (1b). It should be noted that the channel size or spatial scale refers primarily to flow depth and channel width which affect dispersion coefficient and thereby the relative impor-tance of shear dispersion and hyporheic exchange. Tables 1 – 5 and Figures 1 – 5 demonstrate that losses/gains of water and solute affect RTDs. The incorporation of losses/gains of water and solute in the VART model can make parameter values more meaningful, improving simulation results. However, the influence of parameter l alone on simu-lated RTDs is small when the change in discharge is small (0.8 < l < 1.6). The negligible influence of l is attributed to the wide range of variation in the variable residence time TV in Eq. (1a). To reduce the number of fitting parameters, taking l ≈ 1 is accurate enough when no observed hyporheic flow data is available. The influence of variation in l may also be incorpo-rated into the parameter TV or Tmin. It is interesting to find that the ratio of the time to peak concentration (Tpeak) to the minimum mean residence time (Tmin), Tpeak/Tmin, is very close to the recovery ratio of tracer, Mrec/Minj,, i.e., Tpeak/Tmin ≈ Mrec/Minj, where Mrec is the tracer mass recovered at the end of a tracer experiment and Minj is the mass injected. Figure 8 includes both measured data and simulated results of tested river reaches including the small streams simulated by Deng and Jung [12]. The field data of Tpeak and recovery ratio (Mrec/Minj) are taken from the USGS report by Nordin and Sabol [27]. The minimum residence time (Tmin) shown in Figure 8 is estimated using the fractional Laplace trans-form-based parameter estimation method proposed by Deng et al. [10]. Figure 8 illustrates that all small and moderate-sized rivers have an almost


perfect correlation between the recov-ery ratio, Mrec/Minj, and the ratio of the time to peak to the minimum resi-dence time, Tpeak/Tmin. It means that the ratio, Tpeak/Tmin, represents mass loss (Tpeak/Tmin < 1) or gain (Tpeak/Tmin >1) or balance (Tmin/Tpeak = 1) during solute transport in a stream. The sig-nificance of this finding is that the minimum mean residence time Tmin involved in the VART model can be simply calculated using the relation Tpeak/Tmin = Mrec/Minj if the data of Tpeak and Mrec/Minj are available, pro-viding a simple method for determina-tion of the parameter Tmin. Actually, the finally adopted Tmin values, shown in Figures 2, 3 (first two reaches), 4, and 5 in this paper, and in Deng and Jung [12] were determined using this method as the Tmin values initially estimated using the optimization procedure [10] are very close to those determined from the relation Tpeak/Tmin = Mrec/Minj. The Tmin values shown in Figures 8 are determined using the optimization procedure. The values of parameter Tmin for both the case O and the case I in Tables 1 – 5 are determined using the rela-tion Tpeak/Tmin = Mrec/Minj. Therefore, Tmin values do not change when solute gains or losses are included. Consequently, the VART model contains only three fitting parameters Aadv/A, Ds/A, and KS for small and moderate-sized streams and some large rivers like the Red River, greatly simplifying the application of VART model. It is not very clear why the five reaches of two large rivers do not follow the relation Tpeak/Tmin = Mrec/Minj. A possible ex-planation for the deviation is that the solute losses/gains in the five reaches were not caused by the hyporheic exchange (controlling the relation Tpeak/Tmin = Mrec/Minj). In fact, qh = 0 in the four reaches of the Mississippi River. It appears from the Mississippi River case that the relation Tpeak/Tmin = Mrec/Minj may also be utilized to determine the contribution of hyporheic exchange to the attenuation of nutrients and contaminants in streams. More field obser-vation efforts are needed to confirm the findings. It should be pointed out that mass losses and gains generally occur in rivers. Solute losses or gains may re-duce or increase peak concentration, time to the peak concentration, and the minimum mean residence time [27,31,33], and thereby affect RTDs.

Figure 8. Correlation between Tpeak/Tmin and Mrec/Minj.


6 Conclusions The major findings of the paper can be summarized as follows:

(1) Instream advection and dispersion processes can produce only lognor-mal concentration distributions.

(2) The hyporheic exchange process described by the VART model with-out the dispersion and lateral inflow/outflow terms is able to generate a wide variety of RTD types from the upward curving VART+U to a straight line (RTD−P) and further to the downward curving RTD−L distributions when plotted in log-log coordinates.

(3) RTDs depend on both temporal and spatial scales. In terms of temporal scale, RTDs depend on an actual variable residence time. In terms of spatial scale, RTDs are channel-size dependent. Stream channel size af-fects the pattern of RTDs of solute through changing the relative con-tributions of the dispersion and hyporheic exchange terms to solute transport described in the VART model. Large rivers are dominated by instream advection and dispersion processes due to large longitudinal dispersion coefficient and tend to exhibit lognormal distributions. The influence of hyporheic exchange on RTDs increases with decreasing channel size. RTDs in small streams are affected more significantly by hyporheic exchange. Therefore, small streams may display various types of the VART series distributions. Moderate-sized rivers are transitional in terms of RTDs and can exhibit both lognormal and power-law RTDs. There is no single scale-independent type of RTDs that can be generally applied to any rivers and streams.

(4) The effect of water and solute losses/gains on RTDs is significant. Math-ematically, the incorporation of water and solute losses/gains into the VART model is to maintain the mass balance of tracer and improves simulation results, making parameter values more reasonable. Physi-cally, solute losses or gains can change peak concentration, time to the peak concentration, and the minimum mean residence time and thereby affect RTDs.

(5) The ratio of the time to peak to the minimum mean residence time in the VART model is equal to the recovery ratio of tracer. The relation provides a simple method for determination of the VART parameter Tmin. Consequently, the VART model contains only three fitting pa-rameters (Aadv/A, DS/A, and KS) for small and moderate-sized streams and some large rivers, greatly simplifying and enhancing the application of VART model.


(6) The VART model is able to reproduce essentially any type of solute RTDs observed in rivers and streams with 3 – 4 fitting parameters while no prescribed RTD functions are needed.

AcknowledgmentSupport for this research by the USGS and Louisiana Water Resources Re-search Institute is gratefully acknowledged. The writers thank anonymous reviewers and the editor for providing constructive comments that greatly improved this paper.

Appendix

Numerical solution procedure of VART modelA split-operator method can be utilized to split Eq. (1a) into a pure advec-tion equation and a dispersion equation with the transient storage and lat-eral inflow terms. The pure advection process in Eq. (1a) can be simulated using the following hyperbolic sub-equation:

∂C + U

∂C = 0 , t ∈ (tn, tn+1/2) (A1)

∂t ∂x

where n stands for the time step. Eq. (A1) can be solved using the Semi-Lagrangian approach [10]. The dispersion and transient storage-release processes in Eq. (1a) can be simulated using the following dispersion sub-equation:

∂C = KS

∂2C +

Rl (CS – C) +

qL (CL – C) , t ∈ (tn+1/2, tn+1) (A2) ∂t ∂x2 TV A

in which R = (Aadv + Adif)/A = the ratio of transient storage zone area (AS = Aadv + Adif) to main stream area (A). Eq. (A2) in conjunction with Eqs. (1b) – (1e) can be solved using the forward time scheme and the fully implicit F.3 central finite-difference scheme presented by Deng et al. [9]. Eq. (A2) can be discretized as follows:

C n + 1 – C n + 1/2 =

KS (C n + 1 – 2C n + 1 + C n + 1) +

i

Dt/2 i

(Dx)2 i + 1 i i – 1

eR (C n + 1 + C n + 1/2 – C n + 1 + C n + 1/2 ) + m (C n + 1 – C n + 1/2) (A3) Si 2 Si i 2 i i i


where e = l/TV, (M – 1)C = (CL – C), and µ = qL(M-1)/(2A) are introduced. Eq. (1b) is discretized as

C n + 1 – C n + 1/2 =

e (C n + 1 + C n + 1/2 – C n + 1 + C n + 1/2 ) + Si Dt/2 Si i 2 i Si 2 Si q (C n + 1 + C n + 1/2) (A4a)

Si Si

where (M – 1)CS = (Ch – CS) and θ = qh(M-1)/(2AS) are introduced. From Eq. (A4a), CSni + 1 can be expressed as

C n + 1 = (1 – eDt/4 + qDt/2) C n + 1/2 + (eDt/4) (C n + 1 + C n + 1/2)

(A4b) Si

Si

i

i

1 + eDt/4 – qDt/2

Substituting Eq. (A4b) into Eq. (A3) and rearranging Eq. (A3) so that all known quantities appear on the right-hand side and all unknown quantities appear on the left-hand side yields:

– aC in++11 + [1 + 2a + (R – b)φ – h]C in + 1 – aC in–+11 =

{1 – (R – b)φ + h}C in + 1/2 + [φR + b (1 – φ + w)]CSni+ 1/2 (A5a)

where the following definitions are utilized:

a = KSDt , b =

eRDt , φ =

eDt , h =

2(Dx)2 4(1 + eDt / 4 – qDt / 2) 4mDt

, w = qDt

(A5b) 2 2

The dimensionless parameters α, β, φ, η, and ω are either known or calcu-lable. As CSni + 1/2 is unknown, it is assumed that CSni + 1/2 = CSni and R = Rn+1/2. Then, all quantities appearing on the right-hand side of Eq. (A5a) are known. Eq. (A5a) can be further simplified by grouping terms as

WC in++11 + PC in + 1 + WC in–+11 = W n + 1/2 (A6a)

whereW = – a, P = 1 +2 a + (R – b)φ – h, and

W n + 1/2 = [1 – (R – b)φ + h]C in + 1/2 + [φR + b (1 – φ + w)]CSni+ 1/2 (A6b)

The left-hand side of Eq. (A6a) may be assembled into a tridiagonal matrix and solved to determine concentrations at the time level n+1 in the gridded computational domain and thereby RTDs of solute in a river. The concen-tration C in + 1/2 is computed from the solution of Eq. (A1). The assumptions CSni+ 1/2 = CSni and R = Rn + 1/2 will not affect the early-portion (including


the rising limb and the peak) of computed concentration BTCs when time t ≤ Tmin. The short portion of computed BTCs affected by the assumptions primarily lies between the early times and late times. Consequently, poten-tial computation errors caused by the assumptions are very limited.

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