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Velocity measurement of water flow within gravel layer with electrolyte tracer method under virtual boundary condition
Journal of Hydrology 527 (2015) 387–393
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Journal of Hydrology journal homepage: www.elsevier.com/locate/jhydrol
Velocity measurement of water flow within gravel layer with electrolyte tracer method under virtual boundary condition Xiaonan Shi a,⇑, Tingwu Lei b, Fan Zhang a, Yan Yan b a b
Key Laboratory of Tibetan Environment Changes and Land Surface Processes, Institute of Tibetan Plateau Research, Chinese Academy of Sciences, Beijing 100101, PR China College of Water Resources and Civil Engineering, China Agricultural University, Beijing 100083, PR China
a r t i c l e
i n f o
Article history: Received 30 June 2014 Received in revised form 20 March 2015 Accepted 6 May 2015 Available online 12 May 2015 This manuscript was handled by Konstantine P. Georgakakos, Editor-in-Chief, with the assistance of Ioannis K. Tsanis, Associate Editor Keywords: Flow velocity measurement Gravel layer Electrolyte tracer Model simulation
s u m m a r y The measurement velocity of water flow along gravel-rich hill slope is of great interest for hydrological simulation and prediction in alpine watersheds. The alpine terrain is often covered by rock debris of different origin, size, and type, here summarized as ‘gravel’ for the sake of simplicity. To do this, water flow velocity within a gravel layer was measured using the electrolyte tracer method under virtual boundary conditions, an improved method over pulse electrolyte tracer method. Models employing two virtual boundary conditions, a sine function and a normal distribution function, are suggested as preferable for determining flow velocity by simulating the transport processes of electrolytes along a gravel-covered flume at different flow rates (3, 6, and 12 L min1) and slope gradients (4°, 8°, and 12°). Flow velocities as well as determination coefficients and mean square errors were calculated at five measurement locations (0.3, 0.6, 0.9, 1.2, and 1.5 m) using model simulation. The feasibility and accuracy of velocity measurements for water flow within gravel layers are justified to the extent that the two models fit well with the experimental data, the velocity values predicted by the two models are almost identical and, finally, the relations between velocity and discharge and between velocity and slope gradient are rationally explained. The estimated velocity results also agree well with those previously measured by commonly-used methods such as the pulse boundary method and the conventional dye tracer method. Our results show that the virtual boundary conditions method, when used to measure water flow velocity within a gravel layer, predicts the flow velocity more rationally than the pulse boundary and dye tracer methods, and with greater precision. This study suggests that this method is not only readily applicable for obtaining water flow velocity measurements within gravel layers, but also for understanding hydrodynamics in typical watershed. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction Shallow water flow within a gravel layer (where the water surface is lower than the surface of the gravel layer), a common natural phenomenon, is a critical component of the watershed hydrological process. The velocity of water flow within a gravel layer is a fundamental parameter for process-based hydrology and soil erosion models (Abrahams et al., 1996; Poesen, 1998; Simanton et al., 1994; Takken et al., 2005). Accurate measurement of water flow velocity within a gravel layer is therefore necessary for reliable hydrological simulation and prediction, especially for high-alpine watersheds where rich gravel surface is one of typical
⇑ Corresponding author. E-mail address: [email protected] (X. Shi). http://dx.doi.org/10.1016/j.jhydrol.2015.05.008 0022-1694/Ó 2015 Elsevier B.V. All rights reserved.
landforms (De Jong et al., 2005; Kumar et al., 2011). Gravels of different shapes and sizes cause complex water flow processes (Comiti et al., 2007; Cerda, 2001), which need to be carefully distinguished from one another. An effective method for measuring flow velocities within gravel layers is thus a critical requirement. The traditional volumetric method uses a fine probe with scale to determine flow depth, whereby average velocity is determined by dividing flow rate by the product of the flow width and depth. The method is relatively simple and effective for regularly-shaped flow channels (Abrahams and Parsons, 1991). Comparatively, the tracer method seems to be a good option for velocity measurement of flows in which the flow depth and the cross-sectional area of the water flow path are impossible or not as easily determined. Commonly-used tracers include dyes of various colors, electrolytes, radioactive isotopes, fluorescent particles, magnetized materials, and heat, etc. (Abrahams et al., 1986, 1996;
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Lei and Nearing, 2000; Abril and Abdel-Aal, 2000; Ventura et al., 2001, 2002; Dunkerley, 2003; Richardson et al., 2004; Tauro et al. 2012a,b). The electrolyte tracer modeling method under pulse boundary conditions, referred to as the Pulse B.C. method (Lei et al., 2005) was proposed and validated for the measurement of water flow velocities on soil-formed hill-slopes with no correction required. The electrolyte tracer modeling method is based on the mathematic-physical model of solute transportation in a steady water flow. By fitting this solution model to the experimentally-measured solute transport process data, the flow velocity, as a model parameter, is then determined. The model is sensitive to boundary conditions, especially for high flow velocities and short distances (Lei et al., 2010; Shi et al., 2012). One of the critical factors in applying the method is the theoretical definition of the boundary condition as influenced by the physical injection of electrolytes into water flow. The electrolyte tracer modeling method has gradually been improved with three steps termed the Pulse B.C. method (Lei et al., 2005), the Measured B.C. method (Lei et al., 2010) and the Virtual B.C. method (Shi et al., 2012). The Pulse B.C. method (Lei et al., 2005) employs a pulse function to approximate the input boundary signal. The assumption of the pulse function makes it possible to produce an analytic solution to solute transport in water flow. This forms an improved foundation for measurement of flow velocity. The measured velocity becomes more accurate as the distance of electrolyte transport increases. This conflicts with the ideal/desired short distance measurement criterion, to be beneficial to make it possible to develop portable devices. To address this, the Measured B.C. method (Lei et al., 2010) was used to improve the solution to the partial differential equation of the solute transport in water flow, by measuring the practical electrolyte input signal with a sensor located as close to the electrolyte injector as possible. However, the boundary is still not the actual boundary but the signal at a position close to the electrolyte injector. Further modification to the method was proposed and termed the Virtual B.C. method (Shi et al., 2012). The boundary is imaginary on one side but real, reflecting the boundary’s form, on the other side. The shape or formation of the boundary was defined or approximated by a sine or normal distribution function, as determined by the mechanics of the boundary input device. Boundary conditions were determined simultaneously with the flow velocity through model simulation. Using the series of methods described, velocity measurement precision was significantly improved. Moreover, this series of methods has been validated by the volumetric floating object and dye tracer methods identified in previous studies (Lei et al., 2005, 2010; Shi et al. 2012). In addition, the two boundary conditions as defined by sine function and normal distribution functions were found to be equally applicable and accurate. Lei et al. (2013) have attempted to use the primary version of the Pulse B.C. method to measure flow velocity within a gravel layer. The results are encouragingly feasible even though the method was validated with lower measurement precision over a short distance. The purpose of this study is to employ the updated version of the Virtual B.C. method to improve the velocity measurement of water flow within a gravel layer. The method is tested in laboratory experiments. The specific issues dealt with in this study are: (1) the Virtual B.C. method is employed to simulate the experimentally-observed electrolyte transport process in gravel layer water flow so as to calculate flow velocities with both the sine function and normal distribution functions as boundary conditions; (2) The estimated flow velocity is evaluated through the coefficient of determination and the mean square error, the consistency of the results achieved by using the sine and normal distribution models, the relationship between velocity and discharge/slope degree, and further improvement over the Pulse B.C. and dye tracer methods.
2. Electrolyte tracer with Virtual B.C. method Electrolyte transport in a steady water flow is quantified by the convection and dispersion equation (CDE) for the 1-D solute transport as:
@ ðhwC Þ @ ðhwuC Þ @ @C hwDH þ ¼ @t @x @x @x
ð1Þ
where h is the depth of the water flow (m); w is the width of the flow (m); C is the electrolyte concentration (kg m3), proportional to the electrical conductivity of the solution; x is the coordinate down the slope (m); u is the flow velocity (m s1); t is time (s); and DH is the hydrodynamic dispersion coefficient (m2 s1). For the controlled flume experiments without rainfall and infiltration, the flow rate is constant and the flow velocity varies little, so that:
Q 0 ¼ hwu ¼ constant
ð2Þ
u ¼ constant where Q0 is the flow rate (m3 s1) leading to hw = constant. Therefore, Eq. (2) combined with Eq. (1) yields:
@C @C @ @C DH þu ¼ @t @x @x @x
ð3Þ
In order to solve the partial differentiation equation given in Eq. (3), some specific initial and boundary conditions based upon the actual situation encountered in each experiment are required. The upper and lower boundary conditions, as well as initial conditions, for Eq. (3) are given as:
Cðx ¼ 0; tÞ ¼ f ðtÞ
ð4Þ
Cðx ¼ 1; tÞ ¼ 0
ð5Þ
Cðx; t ¼ 0Þ ¼ 0
ð6Þ
Solving Eq. (3) with the initial and boundary conditions produces:
Cðx; tÞ ¼
Z
t
0
2ðt sÞ
x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f ðsÞds pDH ðt sÞ
ð7Þ
The upper boundary condition specified as f(t) cannot be captured through physical measurement. It is rendered as a pulse assumption in the Pulse B.C. model, as an approximate measurement made by a sensor close to the electrolyte injector in the Measured B.C. method, and as a virtual simulation in the Virtual B.C. method. For the last two methods, the injected electrolyte tracer is considered to follow a similar format to an output signal and can be described as both a sine and normal distribution function thus: ( A sin 2Bpt þ D 1 2Dp B 6 t 6 32 2Dp B f ðtÞ=C 0 ¼ ð8aÞ 0 Other
( f ðtÞ=C 0 ¼
h i 2 A exp ðtDÞ tP0 2 2B 0
ð8bÞ
Other
where C0 is a normalized coefficient of a dimensionless boundary R1 condition so that its integral with respect to time 0 Cðx; tÞdt is equal to 1. It is determined by the integral of the observed electrolyte transport curve with respect to time; A, B and D are parameters used to specify boundary condition functions. As for the Measured B.C. method, the function f(t) is determined by fitting it with the electrolyte transport process as registered by the first sensor positioned close to the electrolyte injector. After establishing the boundary conditions, the model solution
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(Eq. (7)) was fitted with experimental data logged by other sensors located at given distances in order to calculate the flow velocity values. In the Virtual B.C. method, the boundary conditions as well as the flow velocity were determined simultaneously through fitting the model with the experimental data. When the boundary conditions are normalized, i.e., the integral of the boundary curves is equal to 1, the coefficient D can be determined by A and B. Four unknown parameters therefore remain in the model solution (Eqs. (7), (8a) and (8b)), i.e. u and DH and the boundary condition parameters A and B. To determine these parameters, the solution model is used to fit the experimentally-obtained electrolyte transport process using the Least Square Method. Model deduction and parameterization details were as originally introduced by Shi et al. (2012). 3. Experimental materials and data analysis 3.1. Experimental materials The experiment flume in which the electrolyte is transported is 4 m long by 15 cm wide and 50 cm high. The experimental system is shown in Fig. 1. The bottom of the flume was filled to a depth of 5 cm with gravels, as a model substrate for different types rock debris found in alpine terrain, comprising a uniform mixture of two grain diameters: 2 cm and 4 cm. Electrical conductivity (EC) sensors, located at different distances from the electrolyte injector, were inserted into the gravel layer. The discharge rate was manually controlled from a tank. During each experiment, a water flow of designed rate was introduced into the flume from its upper end. When steady-state flow was established, a saturated KCl solution was injected into the water flow within the gravel layer at the upper end of the flume. The solute transport processes were logged into a computer at five data points per second. The experimental suites involved a combination of three flow rates (3, 6, and 12 L min1), three slope gradients (4°, 8° and 12°), and five measurement positions from the inlet (0.3, 0.6, 0.9, 1.2 and 1.5 m), with 3 replicates. 3.2. Data analysis The electrolyte contained in Part A (Fig. 1) was auto-released into water flow by computer (Part D in Fig. 1) running specially-designed software. The quantity of electrolyte injected during each injection may vary slightly and the tail-ends of observed downstream electrolyte concentration curves may not have returned to zero. Therefore, prior to model simulation, the observed data were pretreated by smoothing and zeroing the tails, and then dividing by the integral area of the curve to produce a
389
normalization of the injected electrolyte. The normalized data were firstly fitted with the solutions specified by Eq. (6) to obtain boundary condition parameters and an average flow velocity. After that, the velocities at different distances were estimated by fitting with the observed data registered by different EC sensors. In addition, the coefficient of determination (R2) and mean square error (MSE) between observation and fitted simulation were calculated for each treatment. 4. Results and discussion 4.1. Performance of the models Both solutions, deduced from both sine and normal distribution boundary conditions were employed to fit the experimentally-measured data at different locations under different experimental conditions. Fig. 2 shows the measured electrolyte transport processes (dots) at five locations (0.3, 0.6, 0.9, 1.2 and 1.5 m) and the fitted curves by the normal distribution and sine models (smooth curve) for each experimental condition. The estimated normal distribution and sine boundary conditions are plotted in this figure and given as ‘‘Boundary input’’; the corresponding boundary condition parameters are listed in Table 1. R2 and MSE, as well as velocity, are listed in Table 1. As shown in Fig. 2, the solutions under both virtual boundary conditions as the normal distribution and sine models fit the observed data very well at the five measurement positions, and for different experimental conditions. The fitted curves follow the peaks as well as the rising and the falling limbs of the experimentally-observed curves. The peaks of the curves decrease and the breakthrough times of the curves increase with distance under the same experimental conditions, due to longer transport distance increases the degree of electrolyte diffusion and dispersion. Simulation precision is reflected by R2 and MSE, as listed in Table 2. The R2 obtained at all the positions and for all the experimental treatments is >0.9. This indicates close agreement between the model simulations and the observed electrolyte transport processes. All the MSE values are <3.4 ⁄ 105, the smaller value exhibiting closer correlation between the model simulation and the measured data. These results convincingly support the conclusion that the modeling method using an electrolyte tracer may be used for water flow velocity measurement within a gravel layer. Further, the velocities calculated by both the normal distribution and sine models are compared (Fig. 3) to demonstrate the agreement of the two models for the determination of flow velocities. All the data points in Fig. 3 are close to the 1:1 line, indicating that the measured velocities within the gravel layer, as obtained using both methods, are almost identical for different slopes and flow rates. Moreover, results from T-test analysis further demonstrate that there is no significant difference between the measured velocities using both models. These results demonstrate that both models, as well as the parameter estimation procedures, are valid. It is thus feasible to determine velocity using the suggested mathematical models in tandem with the outlined experimental procedures. 4.2. Analysis of factors impacting on velocity
Fig. 1. The experimental system: (A) the electrolyte injector; (B) the electric conductivity sensor; (C) is the water flume; (D) the data logger and electrolyte injection auto-controller.
Flow velocities within a gravel layer are predicted for different positions and hydraulic conditions using model simulations. Due to the lack of effective and widely-acceptable methods for the precise determination of shallow water flow, the errors in measured velocities cannot be directly estimated (Labaky et al., 2007; Lei et al., 2010). An analysis of the relations between estimated
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X. Shi et al. / Journal of Hydrology 527 (2015) 387–393 .4
Q=3 L min-1 S=4°
.3
Observed data Simulated by Sine model Sine B.C. Simulated by Normal model Normal B.C.
.2 X=0.3m
.1
0.6m
0.9m
1.2m 1.5m
0.0
-1
Q=6 L min S=4°
.3
.2
.1
20
40
60
80
100
0
20
40
Time (s)
80
-1
.2
.1
0.0
0
60
80
Q=6 L min S=8°
.3
-1
.2
.1
100
20
40
60
80
.2
.1
0.0 40
60
80
20
40
60
80
100
Time (s) .4
Q=6 L min S=12°
.3
-1
.2
.1
0.0
100
-1
.1
0
Normalized concentration
Normalized concentration
Q=3 L min S=12°
100
.2
100
.4
.3
80
Q=12 L min S=8°
.3
Time (s)
-1
60
0.0
0
.4
20
40
.4
Time (s)
0
20
Time (s)
0.0 40
.1
100
Normalized concentration
Q=3 L min S=8°
.3
Normalized concentration
Normalized concentration
60
.4
20
.2
Time (s)
.4
0
-1
Q=12 L min S=4°
.3
0.0
0.0 0
Normalized concentration
Normalized concentration
.4
Normalized concentration
Normalized concentration
.4
Q=12 L min-1 S=12°
.3
.2
.1
0.0
0
40
20
Time (s)
60
80
100
0
Time (s)
20
40
60
80
100
Time (s)
Fig. 2. A comparison of measured and fitted electrolyte transport processes.
Table 1 Boundary condition parameters. Slope (°)
Discharge (L min1)
Normal model
Sine model
A
B
D
A
B
D
4
3 6 12
0.184 0.181 0.171
2.169 2.202 2.331
4.016 4.427 5.784
0.187 0.226 0.157
16.783 13.902 19.952
0.040 0.437 0.146
8
3 6 12
0.278 0.304 0.153
1.435 1.311 2.601
2.590 2.525 3.801
0.307 0.316 0.172
10.250 9.931 19.026
0.078 0.080 0.387
12
3 6 12
0.329 0.352 0.269
1.212 1.134 1.484
2.193 2.211 2.116
0.336 0.355 0.282
9.363 8.856 11.715
0.089 0.003 0.449
velocities and measurement distances, slope gradients and flow rate was required to provide proof of relevance to a physical system (Fig. 4). As shown in Fig. 4, the mean velocities measured at different locations vary very little for any specified slope gradient and flow rate. The results agree with the theoretical assumption that flow velocity is a constant along the measured distance. Flow velocities within a gravel layer increase with slope gradient since the water depth declines as slope gradient increases. Moreover, this fits with the known accelerative function of gravity. By comparison, a considerably weaker dependence of velocity on discharge was noticed. Especially for lower slope gradients, the flow rate does not cause significant differences in flow velocities. According to volumetric
method theory, flow velocity is determined by dividing flow rate by water depth and width of wetting section. If the width of wetting section within a gravel layer is invariable, the velocity consequently depends upon both the flow rate and water depth in the gravel layer. Based on the experimental results, therefore, it could be deduced that with the increase in flow rate, water depth increases simultaneously; and consequently the velocities showed no significant differences under the three flow rates in this study. The effects of experimental factors on velocity estimation and modeling precision for both methods were analyzed. Table 3 lists the correlation coefficients of velocity and R2 with other variables using Kendall’s Tau Nonparametric Correlations analysis. The results show that both the velocity values and the corresponding R2 and MSE values of flow within gravel layers are significantly correlated with flume slope gradients. Nonetheless, velocity presents non-sensitivity to the modeling methods, discharges and/or measurement distances. Correlation analysis of R2 with the other variables in Table 3 shows that the prediction precision of flow velocities within gravel layers using both models is stable and unaffected by measurement distances and flow rates under designed experimental conditions.
4.2.1. Comparison of results with the Pulse B.C. and Dye Tracer methods The Virtual B.C. method is an updated version based on the Pulse B.C. method. Comparisons of the velocities measured by
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X. Shi et al. / Journal of Hydrology 527 (2015) 387–393 Table 2 The determination coefficients (R2), mean square error (MSE), and flow velocity (u). R2
Experimental treatments Discharge (L min
1
)
Measurement distance (m)
3
6
12
3
6
12
3
6
12
Velocity by Normal model (m/s)
0.08
MSE (⁄104 )
Sine model
Normal model
Sine model
Normal model
Sine model
Normal model
0.3 0.6 0.9 1.2 1.5 0.3 0.6 0.9 1.2 1.5 0.3 0.6 0.9 1.2 1.5
0.978 0.986 0.985 0.971 0.968 0.989 0.984 0.981 0.965 0.971 0.986 0.987 0.991 0.994 0.988
S = 4° 0.977 0.986 0.985 0.971 0.967 0.980 0.979 0.978 0.965 0.971 0.987 0.987 0.991 0.993 0.987
0.137 0.051 0.045 0.059 0.054 0.041 0.050 0.051 0.066 0.053 0.049 0.039 0.016 0.009 0.012
0.142 0.053 0.046 0.060 0.054 0.076 0.065 0.059 0.066 0.054 0.043 0.039 0.017 0.009 0.013
0.036 0.032 0.032 0.031 0.031 0.030 0.031 0.032 0.032 0.033 0.032 0.033 0.031 0.032 0.033
0.037 0.032 0.033 0.031 0.031 0.031 0.031 0.032 0.032 0.033 0.033 0.034 0.032 0.033 0.034
0.3 0.6 0.9 1.2 1.5 0.3 0.6 0.9 1.2 1.5 0.3 0.6 0.9 1.2 1.5
0.990 0.991 0.990 0.983 0.964 0.986 0.989 0.990 0.989 0.983 0.961 0.970 0.962 0.970 0.961
S = 8° 0.986 0.989 0.989 0.982 0.966 0.983 0.988 0.989 0.988 0.984 0.951 0.961 0.957 0.965 0.961
0.089 0.062 0.058 0.081 0.120 0.143 0.077 0.064 0.057 0.074 0.212 0.129 0.147 0.106 0.120
0.123 0.075 0.068 0.081 0.111 0.174 0.086 0.073 0.062 0.070 0.265 0.163 0.167 0.123 0.120
0.053 0.051 0.053 0.052 0.050 0.057 0.053 0.054 0.053 0.053 0.048 0.048 0.052 0.052 0.053
0.054 0.052 0.053 0.052 0.050 0.057 0.053 0.054 0.053 0.053 0.054 0.051 0.054 0.053 0.054
0.3 0.6 0.9 1.2 1.5 0.3 0.6 0.9 1.2 1.5 0.3 0.6 0.9 1.2 1.5
0.984 0.990 0.990 0.984 0.984 0.987 0.990 0.992 0.992 0.989 0.979 0.990 0.990 0.985 0.972
S = 12° 0.982 0.988 0.989 0.985 0.986 0.985 0.988 0.991 0.991 0.990 0.974 0.985 0.987 0.985 0.980
0.194 0.093 0.080 0.097 0.075 0.187 0.110 0.072 0.060 0.070 0.273 0.097 0.082 0.095 0.144
0.221 0.106 0.086 0.093 0.066 0.213 0.136 0.090 0.066 0.066 0.338 0.150 0.111 0.098 0.103
0.066 0.064 0.065 0.065 0.062 0.072 0.069 0.070 0.068 0.068 0.068 0.066 0.068 0.067 0.068
0.067 0.065 0.066 0.066 0.063 0.074 0.069 0.070 0.068 0.068 0.070 0.067 0.069 0.067 0.068
1:1
0.07
0.06
0.05
0.04
S=4° S=8°
0.03
S=12° 0.02
0.02
0.03
u (m s1)
0.04 0.05 0.06 0.07 Velocity by Sine model (m/s)
0.08
Fig. 3. A comparison of the velocities computed by sine and normal distribution models.
these two methods (Fig. 5) as well as by the dye tracer method (Fig. 6) are presented. The movement of dye within the gravel flow cannot be visually tracked and, therefore, only one velocity value along the entire slope length could be determined, by recording the time duration for the dye to travel from its injection until its first appearance in the outflow. There is a significant correlation between the velocities measured by the Pulse B.C. and Virtual B.C. methods for all experimental conditions and distances (Fig. 5). The velocities predicted by the Virtual B.C. method are about 10% higher than those determined by the Pulse B.C. method but about 20% lower than those obtained using the dye tracer method (Fig. 6). As a result, the Pulse B.C. method produced velocities about 30% lower than those rendered by the dye tracer method. The velocity measured by the dye tracer method is leading edge velocity, which is the fastest of the dye tracer and needs to be converted into average velocity with a correction coefficient (Dunkerley, 2001; Myers, 2002). The correction coefficient, as reported in most studies (Dunkerley, 2001; Emmett, 1970; Luk and Merz, 1992; Rouhipour et al., 1999; Pan
X. Shi et al. / Journal of Hydrology 527 (2015) 387–393
0.08
Q=3;S=4
0.07
Q=3;S=8 Q=3;S=12
0.06
Q=6;S=4 Q=6;S=8
0.05
Q=6;S=12
0.04
Q=12;S=4 Q=12;S=8
0.03
Q=12;S=12
0.02 0
0.3
0.6
0.9
1.2
1.5
1.8
Distance (m)
Velocity by Sine model (m/s)
0.10
(a)
0.08
Velocity by other method (m/s)
Velocity by Normal model (m/s)
392
0.08
1:1
Dye method: y = 1.227x R² = 0.858 Pulse B.C.: y = 1.118x R² = 0.901
0.06
0.04
Pulse B.C.
0.02
Dye method
Q=3;S=4 Q=3;S=8
0.07
Q=3;S=12
0.06
Q=6;S=4 Q=6;S=8
0.05
Q=6;S=12
0.04
Q=12;S=4 Q=12;S=8
0.03
Q=12;S=12
0.00 0.00
0.02
0.04
0.06
0.08
0.10
Velocity by Virtual B.C. (m/s) Fig. 6. A comparison of velocity measured by the Virtual B.C., Pulse B.C. and dye tracer methods for different slopes and flow rates. Note: the average velocities at five measurement positions obtained using the Virtual B.C. are compared with those by the Pulse B.C. and the dye tracer method.
0.02 0
0.3
0.6
0.9
1.2
1.51.8
Distance (m)
(b) Fig. 4. A comparison of the velocities measured at five locations for different slope and discharge conditions (a) by normal distribution model and (b) by sine model.
Table 3 Correlation coefficients of velocity and determination coefficient with other variables using Kendall’s Tau Nonparametric Correlations analysis.
a b
Method
Slope
Discharge
Distance
Velocity
Correlation coefficient Sig. (2-tailed) N
.064 .460 90
.823b .000 90
.072 .384 90
.027 .730 90
R2
Correlation coefficient Sig. (2-tailed) N
.101 .248 90
.163a .048 90
.052 .529 90
.079 .313 90
Correlation is significant at the 0.05 level (2-tailed). Correlation is significant at the 0.01 level (2-tailed).
Velocity by Virtual B.C. (m/s)
0.09 0.08
Sine: y = 1.101x R² = 0.910
0.07
Normal: y = 1.118x R² = 0.901
1:1
4.3. Discussions for further application in field study
0.06 0.05 0.04 0.03 0.02
Sine Model 0.01 0.00 0.00
Normal Model 0.01
0.02
0.03
and Shangguan, 2006), lies within the range of 0.365–0.825. Values are of the lowest in laminar flow, rise steeply through the transitional range, and continue rising slowly in turbulent flow (Li and Abrahams, 1997). In theory, the velocity measured by the electrolyte tracer method is the average velocity. Therefore, the correction coefficient is estimated 0.73 for the Pulse B.C. method and 0.81 for the Virtual B.C. method. The coefficient for turbulent flow on hill-slopes approaches 0.8 (Emmett, 1970). Comparatively, for water flow within a gravel layer, the corrected coefficient is theoretically higher than that for hill-slopes under the same flow rate conditions, since the velocity distribution along the depth is relatively uniform due to the gravel layer’s baffle effect. Based on this discussion, the value of 0.81 derived from the Virtual B.C. method seems more rational compared with the value of 0.73 obtained using the Pulse B.C. method. But as for how the results approaching to the true velocity, it still needs more tests in the further. R2 and MSE as a measure of how well a modeling curve fits observational data was compared for both the Virtual B.C. and Pulse B.C. methods. The average R2 value of 0.981 computed by Virtual B.C. is higher than that of 0.967 obtained using the Pulse B.C. method. Further comparison of MSE shows that the Virtual B.C. method produces rather lower MSE on the order of 105 m2/s than the Pulse B.C. method on 103 m2/s under various experimental conditions. These results indicate that the precision of velocity measurement using the Virtual B.C. method is relatively better than that obtained by the Pulse B.C. method.
0.04
0.05
0.06
0.07
0.08
0.09
Velocity by Pulse B.C. (m/s) Fig. 5. A comparison of velocity measured by the Virtual B.C. and Pulse B.C. methods for every experimental condition and measurement position.
The improved method, Virtual B.C. method, is satisfactorily sound for flow velocity measurement within a gravel layer under the condition of laboratory experiments. This study is an encouraging case to support further application of the method to field research. During practical application in the field, one sensor was deemed sufficient for velocity measurement. The input signal was released into the water flow using an electrolyte injector on the upper stream hill-slope, and the output signal was registered by a sensor that had been pre-inserted into the soil/gravel layer, angled at a given distance in a downstream direction. Using the output signal and the specified distance, average velocity within the given distance could be determined, based upon the modeling methods outlined. The distance between the injector and the
X. Shi et al. / Journal of Hydrology 527 (2015) 387–393
sensor was adjustable, but short-distance measurement is preferred for portable device development. On the one hand, short distance means small size and easy to operate. The flow velocity at a given point could be better estimated by the average of velocity as measured by the sensor at short distance. On the other hand, short-distance measurement could reduce electrolyte attenuation to a level which may not be sassily sensed. It is very important for shallow water flow along complex underlying surface in field application. Overall, since the method is sensitive to boundary conditions especially for short distances, the improvement of the electrolyte tracer method under virtual boundary condition will be considerable and hence meaningful for practical application. Moreover, these methods should be applied further to investigate the impact on flow velocity of factors such as sediment, infiltration, organic detritus and rainfall. 5. Conclusions Flow velocity within a gravel layer is difficult to measure accurately, due to flow resistance and the uncertainty of the saturated cross-section in any given gravel structure. The application advantage of an electrolyte tracer to measure velocity within a gravel layer is that it travels synchronously with the water flow and therefore reliably reflects the states and processes of that flow. In this study, the electrolyte tracer method with Virtual B.C. was employed to estimate flow velocity within a gravel layer for different hydraulic conditions as specified by slope gradients and flow rates. The Virtual B.C. method, namely the sine and normal distribution models, were defined as dependent upon the function formats of given virtual boundary conditions. Results show that both the sine and normal distribution models simulate the electrolyte transport process very well, with high coefficients of determination (R2). The velocity values estimated by the two models are identical, indicating that both model simulation and parameter estimation procedures are reasonable. The correlation of flow velocity with impact factors was analyzed; flow velocity was found to be significantly correlative with slope degree but non-sensitive to measurement distances and the specific flow rates in the study. Compared to results obtained using the Pulse B.C. and dye tracer methods, the Virtual B.C. method used fits more closely to the observed curves and predicts flow velocity with a greater precision. The velocity values determined by the Virtual B.C. method are about 10% higher than those obtained using the Pulse B.C. method, but about 20% lower than those by the dye tracer method. In summary, the electrolyte tracer method under virtual boundary condition can be reliably applied to measure flow velocity within a gravel layer. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 41230746 and 41190082), and the ‘‘Strategic Priority Research Program (B)’’ of the Chinese Academy of Sciences (Grant No. XDB03030305). References Abrahams, A.D., Parsons, A.J., 1991. Relation between sediment yield and gradient on debris-covered hillslopes, walnut-gulch, Arizona. Geol. Soc. Am. Bull 103 (8), 1109–1113.
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Abrahams, A.D., Parsons, A.J., Luk, S.H., 1986. Field measurement of the velocity of overland flow using dye tracing. Earth Surf. Proc. Land. 11, 653–657. Abrahams, A.D., Li, G., Parsons, A.J., 1996. Rill hydraulics on a semiarid hillslope, southern Arizona. Earth Surf. Process. Landforms 21 (1), 35–47. Abril, J.M., Abdel-Aal, M.M., 2000. A modelling study on hydrodynamics and pollutant dispersion in the Suez Canal. Ecol. Model. 128, 1–17. Cerda, A., 2001. Effects of rock fragment cover on soil infiltration, interrill runoff and erosion. Eur. J. Soil Sci. 52, 59–68. Comiti, F., Mao, L., Wilcox, A., Wohl, E.E., Lenzi, M.A., 2007. Field-derived relationships for flow velocity and resistance in high-gradient streams. J. Hydrol. 340, 48–62. http://dx.doi.org/10.1016/j.jhydrol.2007.03.021. De Jong, C., Ergenzinger, P., Borufka, M., et al., 2005. Geomorphological zoning: an improvement to coupling alpine hydrology and meteorlogy. In: De Jong, C., Collins, D., Ranzi, R. (Eds.), Climate and Hydrology in Mountain Areas. John Wiley & Sons, Ltd, pp. 247–260. Dunkerley, D., 2001. Estimating the mean speed of laminar overland flow using dye injection uncertainty on rough surfaces. Earth Surf. Process. Landforms 26, 363– 374. Dunkerley, D.L., 2003. An optical tachometer for short-path measurement of flow speeds in shallow overland flows, improved alternative to dye timing. Earth Surf. Process. Landforms 28, 777–786. Emmett, W.W., 1970. The hydraulics of overland flow on hillslopes. US Geological Professional Paper, 662-A, A-1A-68. Kumar, A., Chawla, A., Rajkumar, S., 2011. Characterization of Solang valley watershed in western Himalaya for bio-resource conservation using remote sensing techniques. Environ. Monit. Assess. 179, 469–478. http://dx.doi.org/ 10.1007/s10661-010-1749-0. Labaky, W., Devlin, J.F., Gillham, R.W., 2007. A probe for measuring groundwater velocity at the centimeter scale. Environ. Sci. Technol. 41 (24), 8453–8458. Lei, T., Nearing, M.A., 2000. Flume experiments of determining rill hydraulic characteristic erosion and rill patterns. J. Hydraul. Eng. 11, 49–54 (in China). Lei, T., Xia, W., Zhao, J., 2005. Method for measuring velocity of shallow water flow for soil erosion with an electrolyte tracer. J. Hydrol. 301, 139–145. Lei, T., Chuo, R., Zhao, J., Shi, X., Liu, L., 2010. An improved method for shallow water flow velocity measurement with practical electrolyte inputs. J. Hydrol. 390, 45– 56. Lei, T., Yan, Y., Shi, X., Chuo, R., Zhao, J., 2013. Measuring velocity of water flow within a gravel layer using an electrolyte tracer method with a Pulse Boundary Model. J. Hydrol. 500, 37–44. http://dx.doi.org/10.1016/j.jhydrol.2013.07.025. Li, G., Abrahams, A.D., 1997. Effect of saltating sediment load on the determination of the mean velocity of overland flow. Water Resour. Res. 33, 341–347. Luk, S.H., Merz, W., 1992. Use of the salt tracing technique to determine the velocity of overland flow. Soil Technol. 5, 289–301. Myers, T.G., 2002. Modelling laminar sheet flow over rough surfaces. Water Resour. Res. 38 (11), 1230. Pan, C., Shangguan, Z., 2006. Runoff hydraulic characteristics and sediment generation in sloped grassplots under simulated rainfall conditions. J. Hydrol. 331, 178–185. Poesen, J.W., 1998. Variation of rock fragment cover and size along semiarid hillslopes: a case-study from southeast Spain. Geomorphology 23, 323–335. Richardson, S.D., Willson, C.S., Rusch, K.A., 2004. Use of rhodamine water tracer in the marsh land upwelling system. Ground Water 42 (5), 678–688. Rouhipour, H., Rose, C.W., Yu, B., et al., 1999. Roughness coefficients and velocity estimation in well-inundated sheet and rilled overland flow without strongly eroding bed forms. Earth Surf. Process. Landforms 24, 233–245. Shi, X., Zhang, F., Lei, T., Chuo, R., Zhou, S., Yan, Y., 2012. Measuring shallow water flow velocity with virtual boundary condition signal in the electrolyte tracer method. J. Hydrol. 452–453, 172–179. http://dx.doi.org/10.1016/ j.jhydrol.2012.05.046. Simanton, J.R., Renard, K.G., Christiaensen, C.M., Lane, L.J., 1994. Spatial distribution of surface rock fragments along catenas in semi-arid Arizona and Nevada. Catena 23, 29–42. Takken, I., Govers, G., Jetten, V., et al., 2005. The influence of both process descriptions and runoff patterns on predictions from a spatially distributed soil erosion model. Earth Surf. Process. Landforms 30, 213–229. Tauro, F., Grimaldi, S., Petroselli, A., Porfiri, M., 2012a. Fluorescent particle tracers in surface hydrology: a proof of concept in a natural stream. Water Resour. Res. 48 (6). http://dx.doi.org/10.1029/2011WR011610. Tauro, F., Grimaldi, S., Petroselli, A., Rulli, M.C., Porfiri, M., 2012b. Fluorescent particle tracer for surface flow measurements: a proof of concept in a seminatural hillslope. Hydrol. Earth Syst. Sci. 16, 2973–2983. http://dx.doi.org/ 10.5194/hess-16-2973-2012. 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Journal of Hydrology journal homepage: www.elsevier.com/locate/jhydrol
Velocity measurement of water flow within gravel layer with electrolyte tracer method under virtual boundary condition Xiaonan Shi a,⇑, Tingwu Lei b, Fan Zhang a, Yan Yan b a b
Key Laboratory of Tibetan Environment Changes and Land Surface Processes, Institute of Tibetan Plateau Research, Chinese Academy of Sciences, Beijing 100101, PR China College of Water Resources and Civil Engineering, China Agricultural University, Beijing 100083, PR China
a r t i c l e
i n f o
Article history: Received 30 June 2014 Received in revised form 20 March 2015 Accepted 6 May 2015 Available online 12 May 2015 This manuscript was handled by Konstantine P. Georgakakos, Editor-in-Chief, with the assistance of Ioannis K. Tsanis, Associate Editor Keywords: Flow velocity measurement Gravel layer Electrolyte tracer Model simulation
s u m m a r y The measurement velocity of water flow along gravel-rich hill slope is of great interest for hydrological simulation and prediction in alpine watersheds. The alpine terrain is often covered by rock debris of different origin, size, and type, here summarized as ‘gravel’ for the sake of simplicity. To do this, water flow velocity within a gravel layer was measured using the electrolyte tracer method under virtual boundary conditions, an improved method over pulse electrolyte tracer method. Models employing two virtual boundary conditions, a sine function and a normal distribution function, are suggested as preferable for determining flow velocity by simulating the transport processes of electrolytes along a gravel-covered flume at different flow rates (3, 6, and 12 L min1) and slope gradients (4°, 8°, and 12°). Flow velocities as well as determination coefficients and mean square errors were calculated at five measurement locations (0.3, 0.6, 0.9, 1.2, and 1.5 m) using model simulation. The feasibility and accuracy of velocity measurements for water flow within gravel layers are justified to the extent that the two models fit well with the experimental data, the velocity values predicted by the two models are almost identical and, finally, the relations between velocity and discharge and between velocity and slope gradient are rationally explained. The estimated velocity results also agree well with those previously measured by commonly-used methods such as the pulse boundary method and the conventional dye tracer method. Our results show that the virtual boundary conditions method, when used to measure water flow velocity within a gravel layer, predicts the flow velocity more rationally than the pulse boundary and dye tracer methods, and with greater precision. This study suggests that this method is not only readily applicable for obtaining water flow velocity measurements within gravel layers, but also for understanding hydrodynamics in typical watershed. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction Shallow water flow within a gravel layer (where the water surface is lower than the surface of the gravel layer), a common natural phenomenon, is a critical component of the watershed hydrological process. The velocity of water flow within a gravel layer is a fundamental parameter for process-based hydrology and soil erosion models (Abrahams et al., 1996; Poesen, 1998; Simanton et al., 1994; Takken et al., 2005). Accurate measurement of water flow velocity within a gravel layer is therefore necessary for reliable hydrological simulation and prediction, especially for high-alpine watersheds where rich gravel surface is one of typical
⇑ Corresponding author. E-mail address: [email protected] (X. Shi). http://dx.doi.org/10.1016/j.jhydrol.2015.05.008 0022-1694/Ó 2015 Elsevier B.V. All rights reserved.
landforms (De Jong et al., 2005; Kumar et al., 2011). Gravels of different shapes and sizes cause complex water flow processes (Comiti et al., 2007; Cerda, 2001), which need to be carefully distinguished from one another. An effective method for measuring flow velocities within gravel layers is thus a critical requirement. The traditional volumetric method uses a fine probe with scale to determine flow depth, whereby average velocity is determined by dividing flow rate by the product of the flow width and depth. The method is relatively simple and effective for regularly-shaped flow channels (Abrahams and Parsons, 1991). Comparatively, the tracer method seems to be a good option for velocity measurement of flows in which the flow depth and the cross-sectional area of the water flow path are impossible or not as easily determined. Commonly-used tracers include dyes of various colors, electrolytes, radioactive isotopes, fluorescent particles, magnetized materials, and heat, etc. (Abrahams et al., 1986, 1996;
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Lei and Nearing, 2000; Abril and Abdel-Aal, 2000; Ventura et al., 2001, 2002; Dunkerley, 2003; Richardson et al., 2004; Tauro et al. 2012a,b). The electrolyte tracer modeling method under pulse boundary conditions, referred to as the Pulse B.C. method (Lei et al., 2005) was proposed and validated for the measurement of water flow velocities on soil-formed hill-slopes with no correction required. The electrolyte tracer modeling method is based on the mathematic-physical model of solute transportation in a steady water flow. By fitting this solution model to the experimentally-measured solute transport process data, the flow velocity, as a model parameter, is then determined. The model is sensitive to boundary conditions, especially for high flow velocities and short distances (Lei et al., 2010; Shi et al., 2012). One of the critical factors in applying the method is the theoretical definition of the boundary condition as influenced by the physical injection of electrolytes into water flow. The electrolyte tracer modeling method has gradually been improved with three steps termed the Pulse B.C. method (Lei et al., 2005), the Measured B.C. method (Lei et al., 2010) and the Virtual B.C. method (Shi et al., 2012). The Pulse B.C. method (Lei et al., 2005) employs a pulse function to approximate the input boundary signal. The assumption of the pulse function makes it possible to produce an analytic solution to solute transport in water flow. This forms an improved foundation for measurement of flow velocity. The measured velocity becomes more accurate as the distance of electrolyte transport increases. This conflicts with the ideal/desired short distance measurement criterion, to be beneficial to make it possible to develop portable devices. To address this, the Measured B.C. method (Lei et al., 2010) was used to improve the solution to the partial differential equation of the solute transport in water flow, by measuring the practical electrolyte input signal with a sensor located as close to the electrolyte injector as possible. However, the boundary is still not the actual boundary but the signal at a position close to the electrolyte injector. Further modification to the method was proposed and termed the Virtual B.C. method (Shi et al., 2012). The boundary is imaginary on one side but real, reflecting the boundary’s form, on the other side. The shape or formation of the boundary was defined or approximated by a sine or normal distribution function, as determined by the mechanics of the boundary input device. Boundary conditions were determined simultaneously with the flow velocity through model simulation. Using the series of methods described, velocity measurement precision was significantly improved. Moreover, this series of methods has been validated by the volumetric floating object and dye tracer methods identified in previous studies (Lei et al., 2005, 2010; Shi et al. 2012). In addition, the two boundary conditions as defined by sine function and normal distribution functions were found to be equally applicable and accurate. Lei et al. (2013) have attempted to use the primary version of the Pulse B.C. method to measure flow velocity within a gravel layer. The results are encouragingly feasible even though the method was validated with lower measurement precision over a short distance. The purpose of this study is to employ the updated version of the Virtual B.C. method to improve the velocity measurement of water flow within a gravel layer. The method is tested in laboratory experiments. The specific issues dealt with in this study are: (1) the Virtual B.C. method is employed to simulate the experimentally-observed electrolyte transport process in gravel layer water flow so as to calculate flow velocities with both the sine function and normal distribution functions as boundary conditions; (2) The estimated flow velocity is evaluated through the coefficient of determination and the mean square error, the consistency of the results achieved by using the sine and normal distribution models, the relationship between velocity and discharge/slope degree, and further improvement over the Pulse B.C. and dye tracer methods.
2. Electrolyte tracer with Virtual B.C. method Electrolyte transport in a steady water flow is quantified by the convection and dispersion equation (CDE) for the 1-D solute transport as:
@ ðhwC Þ @ ðhwuC Þ @ @C hwDH þ ¼ @t @x @x @x
ð1Þ
where h is the depth of the water flow (m); w is the width of the flow (m); C is the electrolyte concentration (kg m3), proportional to the electrical conductivity of the solution; x is the coordinate down the slope (m); u is the flow velocity (m s1); t is time (s); and DH is the hydrodynamic dispersion coefficient (m2 s1). For the controlled flume experiments without rainfall and infiltration, the flow rate is constant and the flow velocity varies little, so that:
Q 0 ¼ hwu ¼ constant
ð2Þ
u ¼ constant where Q0 is the flow rate (m3 s1) leading to hw = constant. Therefore, Eq. (2) combined with Eq. (1) yields:
@C @C @ @C DH þu ¼ @t @x @x @x
ð3Þ
In order to solve the partial differentiation equation given in Eq. (3), some specific initial and boundary conditions based upon the actual situation encountered in each experiment are required. The upper and lower boundary conditions, as well as initial conditions, for Eq. (3) are given as:
Cðx ¼ 0; tÞ ¼ f ðtÞ
ð4Þ
Cðx ¼ 1; tÞ ¼ 0
ð5Þ
Cðx; t ¼ 0Þ ¼ 0
ð6Þ
Solving Eq. (3) with the initial and boundary conditions produces:
Cðx; tÞ ¼
Z
t
0
2ðt sÞ
x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f ðsÞds pDH ðt sÞ
ð7Þ
The upper boundary condition specified as f(t) cannot be captured through physical measurement. It is rendered as a pulse assumption in the Pulse B.C. model, as an approximate measurement made by a sensor close to the electrolyte injector in the Measured B.C. method, and as a virtual simulation in the Virtual B.C. method. For the last two methods, the injected electrolyte tracer is considered to follow a similar format to an output signal and can be described as both a sine and normal distribution function thus: ( A sin 2Bpt þ D 1 2Dp B 6 t 6 32 2Dp B f ðtÞ=C 0 ¼ ð8aÞ 0 Other
( f ðtÞ=C 0 ¼
h i 2 A exp ðtDÞ tP0 2 2B 0
ð8bÞ
Other
where C0 is a normalized coefficient of a dimensionless boundary R1 condition so that its integral with respect to time 0 Cðx; tÞdt is equal to 1. It is determined by the integral of the observed electrolyte transport curve with respect to time; A, B and D are parameters used to specify boundary condition functions. As for the Measured B.C. method, the function f(t) is determined by fitting it with the electrolyte transport process as registered by the first sensor positioned close to the electrolyte injector. After establishing the boundary conditions, the model solution
X. Shi et al. / Journal of Hydrology 527 (2015) 387–393
(Eq. (7)) was fitted with experimental data logged by other sensors located at given distances in order to calculate the flow velocity values. In the Virtual B.C. method, the boundary conditions as well as the flow velocity were determined simultaneously through fitting the model with the experimental data. When the boundary conditions are normalized, i.e., the integral of the boundary curves is equal to 1, the coefficient D can be determined by A and B. Four unknown parameters therefore remain in the model solution (Eqs. (7), (8a) and (8b)), i.e. u and DH and the boundary condition parameters A and B. To determine these parameters, the solution model is used to fit the experimentally-obtained electrolyte transport process using the Least Square Method. Model deduction and parameterization details were as originally introduced by Shi et al. (2012). 3. Experimental materials and data analysis 3.1. Experimental materials The experiment flume in which the electrolyte is transported is 4 m long by 15 cm wide and 50 cm high. The experimental system is shown in Fig. 1. The bottom of the flume was filled to a depth of 5 cm with gravels, as a model substrate for different types rock debris found in alpine terrain, comprising a uniform mixture of two grain diameters: 2 cm and 4 cm. Electrical conductivity (EC) sensors, located at different distances from the electrolyte injector, were inserted into the gravel layer. The discharge rate was manually controlled from a tank. During each experiment, a water flow of designed rate was introduced into the flume from its upper end. When steady-state flow was established, a saturated KCl solution was injected into the water flow within the gravel layer at the upper end of the flume. The solute transport processes were logged into a computer at five data points per second. The experimental suites involved a combination of three flow rates (3, 6, and 12 L min1), three slope gradients (4°, 8° and 12°), and five measurement positions from the inlet (0.3, 0.6, 0.9, 1.2 and 1.5 m), with 3 replicates. 3.2. Data analysis The electrolyte contained in Part A (Fig. 1) was auto-released into water flow by computer (Part D in Fig. 1) running specially-designed software. The quantity of electrolyte injected during each injection may vary slightly and the tail-ends of observed downstream electrolyte concentration curves may not have returned to zero. Therefore, prior to model simulation, the observed data were pretreated by smoothing and zeroing the tails, and then dividing by the integral area of the curve to produce a
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normalization of the injected electrolyte. The normalized data were firstly fitted with the solutions specified by Eq. (6) to obtain boundary condition parameters and an average flow velocity. After that, the velocities at different distances were estimated by fitting with the observed data registered by different EC sensors. In addition, the coefficient of determination (R2) and mean square error (MSE) between observation and fitted simulation were calculated for each treatment. 4. Results and discussion 4.1. Performance of the models Both solutions, deduced from both sine and normal distribution boundary conditions were employed to fit the experimentally-measured data at different locations under different experimental conditions. Fig. 2 shows the measured electrolyte transport processes (dots) at five locations (0.3, 0.6, 0.9, 1.2 and 1.5 m) and the fitted curves by the normal distribution and sine models (smooth curve) for each experimental condition. The estimated normal distribution and sine boundary conditions are plotted in this figure and given as ‘‘Boundary input’’; the corresponding boundary condition parameters are listed in Table 1. R2 and MSE, as well as velocity, are listed in Table 1. As shown in Fig. 2, the solutions under both virtual boundary conditions as the normal distribution and sine models fit the observed data very well at the five measurement positions, and for different experimental conditions. The fitted curves follow the peaks as well as the rising and the falling limbs of the experimentally-observed curves. The peaks of the curves decrease and the breakthrough times of the curves increase with distance under the same experimental conditions, due to longer transport distance increases the degree of electrolyte diffusion and dispersion. Simulation precision is reflected by R2 and MSE, as listed in Table 2. The R2 obtained at all the positions and for all the experimental treatments is >0.9. This indicates close agreement between the model simulations and the observed electrolyte transport processes. All the MSE values are <3.4 ⁄ 105, the smaller value exhibiting closer correlation between the model simulation and the measured data. These results convincingly support the conclusion that the modeling method using an electrolyte tracer may be used for water flow velocity measurement within a gravel layer. Further, the velocities calculated by both the normal distribution and sine models are compared (Fig. 3) to demonstrate the agreement of the two models for the determination of flow velocities. All the data points in Fig. 3 are close to the 1:1 line, indicating that the measured velocities within the gravel layer, as obtained using both methods, are almost identical for different slopes and flow rates. Moreover, results from T-test analysis further demonstrate that there is no significant difference between the measured velocities using both models. These results demonstrate that both models, as well as the parameter estimation procedures, are valid. It is thus feasible to determine velocity using the suggested mathematical models in tandem with the outlined experimental procedures. 4.2. Analysis of factors impacting on velocity
Fig. 1. The experimental system: (A) the electrolyte injector; (B) the electric conductivity sensor; (C) is the water flume; (D) the data logger and electrolyte injection auto-controller.
Flow velocities within a gravel layer are predicted for different positions and hydraulic conditions using model simulations. Due to the lack of effective and widely-acceptable methods for the precise determination of shallow water flow, the errors in measured velocities cannot be directly estimated (Labaky et al., 2007; Lei et al., 2010). An analysis of the relations between estimated
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Q=3 L min-1 S=4°
.3
Observed data Simulated by Sine model Sine B.C. Simulated by Normal model Normal B.C.
.2 X=0.3m
.1
0.6m
0.9m
1.2m 1.5m
0.0
-1
Q=6 L min S=4°
.3
.2
.1
20
40
60
80
100
0
20
40
Time (s)
80
-1
.2
.1
0.0
0
60
80
Q=6 L min S=8°
.3
-1
.2
.1
100
20
40
60
80
.2
.1
0.0 40
60
80
20
40
60
80
100
Time (s) .4
Q=6 L min S=12°
.3
-1
.2
.1
0.0
100
-1
.1
0
Normalized concentration
Normalized concentration
Q=3 L min S=12°
100
.2
100
.4
.3
80
Q=12 L min S=8°
.3
Time (s)
-1
60
0.0
0
.4
20
40
.4
Time (s)
0
20
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0.0 40
.1
100
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Q=3 L min S=8°
.3
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60
.4
20
.2
Time (s)
.4
0
-1
Q=12 L min S=4°
.3
0.0
0.0 0
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.4
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.4
Q=12 L min-1 S=12°
.3
.2
.1
0.0
0
40
20
Time (s)
60
80
100
0
Time (s)
20
40
60
80
100
Time (s)
Fig. 2. A comparison of measured and fitted electrolyte transport processes.
Table 1 Boundary condition parameters. Slope (°)
Discharge (L min1)
Normal model
Sine model
A
B
D
A
B
D
4
3 6 12
0.184 0.181 0.171
2.169 2.202 2.331
4.016 4.427 5.784
0.187 0.226 0.157
16.783 13.902 19.952
0.040 0.437 0.146
8
3 6 12
0.278 0.304 0.153
1.435 1.311 2.601
2.590 2.525 3.801
0.307 0.316 0.172
10.250 9.931 19.026
0.078 0.080 0.387
12
3 6 12
0.329 0.352 0.269
1.212 1.134 1.484
2.193 2.211 2.116
0.336 0.355 0.282
9.363 8.856 11.715
0.089 0.003 0.449
velocities and measurement distances, slope gradients and flow rate was required to provide proof of relevance to a physical system (Fig. 4). As shown in Fig. 4, the mean velocities measured at different locations vary very little for any specified slope gradient and flow rate. The results agree with the theoretical assumption that flow velocity is a constant along the measured distance. Flow velocities within a gravel layer increase with slope gradient since the water depth declines as slope gradient increases. Moreover, this fits with the known accelerative function of gravity. By comparison, a considerably weaker dependence of velocity on discharge was noticed. Especially for lower slope gradients, the flow rate does not cause significant differences in flow velocities. According to volumetric
method theory, flow velocity is determined by dividing flow rate by water depth and width of wetting section. If the width of wetting section within a gravel layer is invariable, the velocity consequently depends upon both the flow rate and water depth in the gravel layer. Based on the experimental results, therefore, it could be deduced that with the increase in flow rate, water depth increases simultaneously; and consequently the velocities showed no significant differences under the three flow rates in this study. The effects of experimental factors on velocity estimation and modeling precision for both methods were analyzed. Table 3 lists the correlation coefficients of velocity and R2 with other variables using Kendall’s Tau Nonparametric Correlations analysis. The results show that both the velocity values and the corresponding R2 and MSE values of flow within gravel layers are significantly correlated with flume slope gradients. Nonetheless, velocity presents non-sensitivity to the modeling methods, discharges and/or measurement distances. Correlation analysis of R2 with the other variables in Table 3 shows that the prediction precision of flow velocities within gravel layers using both models is stable and unaffected by measurement distances and flow rates under designed experimental conditions.
4.2.1. Comparison of results with the Pulse B.C. and Dye Tracer methods The Virtual B.C. method is an updated version based on the Pulse B.C. method. Comparisons of the velocities measured by
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X. Shi et al. / Journal of Hydrology 527 (2015) 387–393 Table 2 The determination coefficients (R2), mean square error (MSE), and flow velocity (u). R2
Experimental treatments Discharge (L min
1
)
Measurement distance (m)
3
6
12
3
6
12
3
6
12
Velocity by Normal model (m/s)
0.08
MSE (⁄104 )
Sine model
Normal model
Sine model
Normal model
Sine model
Normal model
0.3 0.6 0.9 1.2 1.5 0.3 0.6 0.9 1.2 1.5 0.3 0.6 0.9 1.2 1.5
0.978 0.986 0.985 0.971 0.968 0.989 0.984 0.981 0.965 0.971 0.986 0.987 0.991 0.994 0.988
S = 4° 0.977 0.986 0.985 0.971 0.967 0.980 0.979 0.978 0.965 0.971 0.987 0.987 0.991 0.993 0.987
0.137 0.051 0.045 0.059 0.054 0.041 0.050 0.051 0.066 0.053 0.049 0.039 0.016 0.009 0.012
0.142 0.053 0.046 0.060 0.054 0.076 0.065 0.059 0.066 0.054 0.043 0.039 0.017 0.009 0.013
0.036 0.032 0.032 0.031 0.031 0.030 0.031 0.032 0.032 0.033 0.032 0.033 0.031 0.032 0.033
0.037 0.032 0.033 0.031 0.031 0.031 0.031 0.032 0.032 0.033 0.033 0.034 0.032 0.033 0.034
0.3 0.6 0.9 1.2 1.5 0.3 0.6 0.9 1.2 1.5 0.3 0.6 0.9 1.2 1.5
0.990 0.991 0.990 0.983 0.964 0.986 0.989 0.990 0.989 0.983 0.961 0.970 0.962 0.970 0.961
S = 8° 0.986 0.989 0.989 0.982 0.966 0.983 0.988 0.989 0.988 0.984 0.951 0.961 0.957 0.965 0.961
0.089 0.062 0.058 0.081 0.120 0.143 0.077 0.064 0.057 0.074 0.212 0.129 0.147 0.106 0.120
0.123 0.075 0.068 0.081 0.111 0.174 0.086 0.073 0.062 0.070 0.265 0.163 0.167 0.123 0.120
0.053 0.051 0.053 0.052 0.050 0.057 0.053 0.054 0.053 0.053 0.048 0.048 0.052 0.052 0.053
0.054 0.052 0.053 0.052 0.050 0.057 0.053 0.054 0.053 0.053 0.054 0.051 0.054 0.053 0.054
0.3 0.6 0.9 1.2 1.5 0.3 0.6 0.9 1.2 1.5 0.3 0.6 0.9 1.2 1.5
0.984 0.990 0.990 0.984 0.984 0.987 0.990 0.992 0.992 0.989 0.979 0.990 0.990 0.985 0.972
S = 12° 0.982 0.988 0.989 0.985 0.986 0.985 0.988 0.991 0.991 0.990 0.974 0.985 0.987 0.985 0.980
0.194 0.093 0.080 0.097 0.075 0.187 0.110 0.072 0.060 0.070 0.273 0.097 0.082 0.095 0.144
0.221 0.106 0.086 0.093 0.066 0.213 0.136 0.090 0.066 0.066 0.338 0.150 0.111 0.098 0.103
0.066 0.064 0.065 0.065 0.062 0.072 0.069 0.070 0.068 0.068 0.068 0.066 0.068 0.067 0.068
0.067 0.065 0.066 0.066 0.063 0.074 0.069 0.070 0.068 0.068 0.070 0.067 0.069 0.067 0.068
1:1
0.07
0.06
0.05
0.04
S=4° S=8°
0.03
S=12° 0.02
0.02
0.03
u (m s1)
0.04 0.05 0.06 0.07 Velocity by Sine model (m/s)
0.08
Fig. 3. A comparison of the velocities computed by sine and normal distribution models.
these two methods (Fig. 5) as well as by the dye tracer method (Fig. 6) are presented. The movement of dye within the gravel flow cannot be visually tracked and, therefore, only one velocity value along the entire slope length could be determined, by recording the time duration for the dye to travel from its injection until its first appearance in the outflow. There is a significant correlation between the velocities measured by the Pulse B.C. and Virtual B.C. methods for all experimental conditions and distances (Fig. 5). The velocities predicted by the Virtual B.C. method are about 10% higher than those determined by the Pulse B.C. method but about 20% lower than those obtained using the dye tracer method (Fig. 6). As a result, the Pulse B.C. method produced velocities about 30% lower than those rendered by the dye tracer method. The velocity measured by the dye tracer method is leading edge velocity, which is the fastest of the dye tracer and needs to be converted into average velocity with a correction coefficient (Dunkerley, 2001; Myers, 2002). The correction coefficient, as reported in most studies (Dunkerley, 2001; Emmett, 1970; Luk and Merz, 1992; Rouhipour et al., 1999; Pan
X. Shi et al. / Journal of Hydrology 527 (2015) 387–393
0.08
Q=3;S=4
0.07
Q=3;S=8 Q=3;S=12
0.06
Q=6;S=4 Q=6;S=8
0.05
Q=6;S=12
0.04
Q=12;S=4 Q=12;S=8
0.03
Q=12;S=12
0.02 0
0.3
0.6
0.9
1.2
1.5
1.8
Distance (m)
Velocity by Sine model (m/s)
0.10
(a)
0.08
Velocity by other method (m/s)
Velocity by Normal model (m/s)
392
0.08
1:1
Dye method: y = 1.227x R² = 0.858 Pulse B.C.: y = 1.118x R² = 0.901
0.06
0.04
Pulse B.C.
0.02
Dye method
Q=3;S=4 Q=3;S=8
0.07
Q=3;S=12
0.06
Q=6;S=4 Q=6;S=8
0.05
Q=6;S=12
0.04
Q=12;S=4 Q=12;S=8
0.03
Q=12;S=12
0.00 0.00
0.02
0.04
0.06
0.08
0.10
Velocity by Virtual B.C. (m/s) Fig. 6. A comparison of velocity measured by the Virtual B.C., Pulse B.C. and dye tracer methods for different slopes and flow rates. Note: the average velocities at five measurement positions obtained using the Virtual B.C. are compared with those by the Pulse B.C. and the dye tracer method.
0.02 0
0.3
0.6
0.9
1.2
1.51.8
Distance (m)
(b) Fig. 4. A comparison of the velocities measured at five locations for different slope and discharge conditions (a) by normal distribution model and (b) by sine model.
Table 3 Correlation coefficients of velocity and determination coefficient with other variables using Kendall’s Tau Nonparametric Correlations analysis.
a b
Method
Slope
Discharge
Distance
Velocity
Correlation coefficient Sig. (2-tailed) N
.064 .460 90
.823b .000 90
.072 .384 90
.027 .730 90
R2
Correlation coefficient Sig. (2-tailed) N
.101 .248 90
.163a .048 90
.052 .529 90
.079 .313 90
Correlation is significant at the 0.05 level (2-tailed). Correlation is significant at the 0.01 level (2-tailed).
Velocity by Virtual B.C. (m/s)
0.09 0.08
Sine: y = 1.101x R² = 0.910
0.07
Normal: y = 1.118x R² = 0.901
1:1
4.3. Discussions for further application in field study
0.06 0.05 0.04 0.03 0.02
Sine Model 0.01 0.00 0.00
Normal Model 0.01
0.02
0.03
and Shangguan, 2006), lies within the range of 0.365–0.825. Values are of the lowest in laminar flow, rise steeply through the transitional range, and continue rising slowly in turbulent flow (Li and Abrahams, 1997). In theory, the velocity measured by the electrolyte tracer method is the average velocity. Therefore, the correction coefficient is estimated 0.73 for the Pulse B.C. method and 0.81 for the Virtual B.C. method. The coefficient for turbulent flow on hill-slopes approaches 0.8 (Emmett, 1970). Comparatively, for water flow within a gravel layer, the corrected coefficient is theoretically higher than that for hill-slopes under the same flow rate conditions, since the velocity distribution along the depth is relatively uniform due to the gravel layer’s baffle effect. Based on this discussion, the value of 0.81 derived from the Virtual B.C. method seems more rational compared with the value of 0.73 obtained using the Pulse B.C. method. But as for how the results approaching to the true velocity, it still needs more tests in the further. R2 and MSE as a measure of how well a modeling curve fits observational data was compared for both the Virtual B.C. and Pulse B.C. methods. The average R2 value of 0.981 computed by Virtual B.C. is higher than that of 0.967 obtained using the Pulse B.C. method. Further comparison of MSE shows that the Virtual B.C. method produces rather lower MSE on the order of 105 m2/s than the Pulse B.C. method on 103 m2/s under various experimental conditions. These results indicate that the precision of velocity measurement using the Virtual B.C. method is relatively better than that obtained by the Pulse B.C. method.
0.04
0.05
0.06
0.07
0.08
0.09
Velocity by Pulse B.C. (m/s) Fig. 5. A comparison of velocity measured by the Virtual B.C. and Pulse B.C. methods for every experimental condition and measurement position.
The improved method, Virtual B.C. method, is satisfactorily sound for flow velocity measurement within a gravel layer under the condition of laboratory experiments. This study is an encouraging case to support further application of the method to field research. During practical application in the field, one sensor was deemed sufficient for velocity measurement. The input signal was released into the water flow using an electrolyte injector on the upper stream hill-slope, and the output signal was registered by a sensor that had been pre-inserted into the soil/gravel layer, angled at a given distance in a downstream direction. Using the output signal and the specified distance, average velocity within the given distance could be determined, based upon the modeling methods outlined. The distance between the injector and the
X. Shi et al. / Journal of Hydrology 527 (2015) 387–393
sensor was adjustable, but short-distance measurement is preferred for portable device development. On the one hand, short distance means small size and easy to operate. The flow velocity at a given point could be better estimated by the average of velocity as measured by the sensor at short distance. On the other hand, short-distance measurement could reduce electrolyte attenuation to a level which may not be sassily sensed. It is very important for shallow water flow along complex underlying surface in field application. Overall, since the method is sensitive to boundary conditions especially for short distances, the improvement of the electrolyte tracer method under virtual boundary condition will be considerable and hence meaningful for practical application. Moreover, these methods should be applied further to investigate the impact on flow velocity of factors such as sediment, infiltration, organic detritus and rainfall. 5. Conclusions Flow velocity within a gravel layer is difficult to measure accurately, due to flow resistance and the uncertainty of the saturated cross-section in any given gravel structure. The application advantage of an electrolyte tracer to measure velocity within a gravel layer is that it travels synchronously with the water flow and therefore reliably reflects the states and processes of that flow. In this study, the electrolyte tracer method with Virtual B.C. was employed to estimate flow velocity within a gravel layer for different hydraulic conditions as specified by slope gradients and flow rates. The Virtual B.C. method, namely the sine and normal distribution models, were defined as dependent upon the function formats of given virtual boundary conditions. Results show that both the sine and normal distribution models simulate the electrolyte transport process very well, with high coefficients of determination (R2). The velocity values estimated by the two models are identical, indicating that both model simulation and parameter estimation procedures are reasonable. The correlation of flow velocity with impact factors was analyzed; flow velocity was found to be significantly correlative with slope degree but non-sensitive to measurement distances and the specific flow rates in the study. Compared to results obtained using the Pulse B.C. and dye tracer methods, the Virtual B.C. method used fits more closely to the observed curves and predicts flow velocity with a greater precision. The velocity values determined by the Virtual B.C. method are about 10% higher than those obtained using the Pulse B.C. method, but about 20% lower than those by the dye tracer method. In summary, the electrolyte tracer method under virtual boundary condition can be reliably applied to measure flow velocity within a gravel layer. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 41230746 and 41190082), and the ‘‘Strategic Priority Research Program (B)’’ of the Chinese Academy of Sciences (Grant No. XDB03030305). References Abrahams, A.D., Parsons, A.J., 1991. Relation between sediment yield and gradient on debris-covered hillslopes, walnut-gulch, Arizona. Geol. Soc. Am. Bull 103 (8), 1109–1113.
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