**3. Formulation of the model considering non-equilibrium mass exchange**

The pollutant exchange between the main stream and underlying subsoil is nonequilibrium in nature. It can be seen most of the mountainous streams where the water with pollutant re-enters the stream in an slower phase. Simulation of non-equilibrium exchange processes along with advection and dispersion is not a simple case due the complexity of the processes of exchange (Cameron and Klute, 1977). Numerous investigators (Bencala and Walters, 1983; Runkel and Broshears, 1991; Runkel and Chapra, 1993; Czernuszenko and Rowinski, 1997; Runkel, 1998; Worman et al., 2002) have studied exchange of the pollutant between main stream and porous soil media. The concentrationtime profile of pollutant transport in such case is influenced significantly by the mass exchange. Cameron and Klute, 1977; Bajracharya and Barry, 1992; 1993; 1995 have illustrated that the pollutant exchange in the form of adsorption processes flatten more the concentration-time profile. Thus an exact pollutant transport simulation is important to correctly ascertain the assimilation capacity of streams. Consider a conceptualized hybrid model which incorporate non-equilibrium exchange of pollutant and which is expressed mathematically as follows

$$\frac{d\mathbf{C}\_s(\mathbf{x},t)}{dt} = R\_D \left[ \mathbf{C}(\mathbf{x},t) - \mathbf{C}\_s(\mathbf{x},t) \right] \tag{13}$$

where, *RD*…proportionality constant (per min), *Cs(x, t)…*concentration of pollutant adsorbed in mg/L, *C(x, t)…*concentration of pollutant in the water column in mg/L, *t*…the time in min.

For a steady state flow condition, performing the mass balance in a control volume within plug flow cell of hybrid model, one can get partial differential equation which governs hyporheic exchange coupled pollutant transport which will be same as eq. (2). Then Laplace transform has been used to solve it by combining eq. (13) and effluent concentration from plug flow zone is given by

$$\mathcal{C}\left(\mathbf{x},t\right) = \mathbb{C}\_{P}\left(a\boldsymbol{u},t\right) = \mathbb{C}\_{R}\exp\left(\boldsymbol{\gamma}\right)\left[\mathcal{U}\left(t-a\right) + \sqrt{\eta}\int\_{0}^{t} \mathcal{U}\left(\boldsymbol{\tau}-a\right)e^{-\mathcal{R}\_{0}\left(\boldsymbol{\tau}-a\right)}\frac{1}{\sqrt{\boldsymbol{\tau}-a}}I\_{1}\left(2\sqrt{\boldsymbol{\eta}\left(\boldsymbol{\tau}-a\right)}\right)d\boldsymbol{\tau}\right] \tag{14}$$

where 1 *DR <sup>x</sup> W DP B <sup>u</sup> <sup>e</sup> A* φ γ ⎛ ⎞ <sup>=</sup> − − ⎜ ⎟ ⎝ ⎠ ; 1 *DR <sup>x</sup> WDR PBD <sup>u</sup> <sup>e</sup> A* φ η⎛ ⎞ <sup>=</sup> ⎜ ⎟ <sup>−</sup> ⎝ ⎠, U(t - α)…step function which

is zero for t < α and it is 1 for t ≥ α, so eq. (14) is valid for t ≥ α; α…residence time of plug flow cell, which is x/u. As the eq. (14) is valid for t ≥ α,.it can be considered that the pollutant concentration is zero for t < α

Effluent of plug flow zone enters to the first well mixed cell, where it gets mixed before entering into the second well mixed cell. During these transports through mixed cells too, mass exchange activities follow the non-equilibrium type. Consider a unit step input, *CR* and perform the mass balance in the first thoroughly mixed zone which can be expressed as

$$\frac{d\mathbb{C}\_{M1}}{dt} = \frac{\mathbb{C}\_{R}}{T\_{1}} \frac{\mathcal{U}(t-a)}{T\_{1}} - \frac{\mathbb{C}\_{M1}}{T\_{1}} - \frac{\phi \text{ }\mathcal{W}\_{\text{P}} \, D}{A} \frac{d\mathbb{C}\_{s}}{dt} \tag{15}$$

Simulation of Stream Pollutant Transport with

depicted by the hybrid model with retardation.

R=1.0

0

m for α = 1.7 min, T1 = 2.0 min, T2 = 6.3 min

0.2

0.4

Concentration (mg/L)

0.6

0.8

1

Hyporheic Exchange for Water Resources Management 153

comparison and it can be noted that the C-t profiles represented by the hybrid model with retardation shows the characteristics of delayed pollutant transport as expected. The impulse response, presented in fig. 5, represents the following; the rising limb occurred at a later time with reduction in magnitudes, the time to peak shifted, the peak concentration reduced, and the recession limb extended for a long time to that of the distributions

R=1.25

0 10 20 30 40 50

Time (min)

Fig. 4. Unit step responses of the hybrid model at the end of one hybrid unit of size ∆x = 200

The above data are adopted again and the C-t profiles of the hybrid model for a unit impulse input have been generated for n = 3, 6, and 11 using eqn (11) and shown in fig. 6 and it can be noted from the figure that as the pollutants move from the near field to the far

Equilibrium exchange

Eq. (15) can be solved analytically and effluent from the first well mixed cell will enter second well mixed cell. Thus similar mass balance equation can be formulated. Successive convolution numerical integration can be used by combining effluent concentrations of plug flow cell and well mixed cells to get effluent concentration at the end of first hybrid cell as follows

$$\mathcal{K}\left(n\Delta t\right) = \mathbb{C}\_{2}\left(n\Delta t\right) = \sum\_{\gamma=1}^{n} \left\langle \mathbb{C}\_{1}\left(\gamma\Delta t\right) - \mathbb{C}\_{1}\left(\left(\gamma - 1\right)\Delta t\right) \right\rangle \,\delta\_{M2} \left[\left(n - \gamma + 1\right), \Delta t\right] \tag{16}$$

where 1 1 ( ) { } () ( ) ( ) ( ) 1 1 1 , *n C nt C t C t n t PP M* γ Δ γΔ γ Δδ γ Δ = = − − ⎡ −+ ⎤ ∑ <sup>⎣</sup> <sup>⎦</sup> ; *δM1* & *δM2*…ramp kernel

co-efficients.

Eq. (16) is unit step response [K(.)] of first hybrid unit and one can get unit impulse response [k(.)] by differentiating Eq. (16). The output (Eq. 16) of pollutant from preceding hybrid unit forms the input to the succeeding hybrid unit, thus the response of the nth hybrid unit, n ≥ 2 for steady flow condition can be obtained using convolution technique which will be similar to eq. (11) and is given by

$$\mathbf{C}\{n\Delta\mathbf{x},t\} = \bigcap\_{0}^{t} \mathbf{C}\{(n-1)\Delta\mathbf{x},\tau\} \; \begin{array}{l} \mathbf{k}\{(n,T\_1,T\_2,R\_\mathcal{D},t-\tau)\} \; d\tau \\ \end{array} \tag{17}$$

where C[(n-1)Δx, t)]…effluent concentration from the preceding unit, k( )…unit impulse response of a single hybrid unit.

### **4. Simulation of pollutant transport**

### **4.1 Verification of model considering equilibrium exchange using synthetic data**

The hybrid model simulates the advection-dispersion governed pollutant transport in a regular channel under uniform flow conditions when the size of the basic process unit, Δx, is equal to or more than 4 DL/u. This means that the response of the hybrid model corresponding to that Δx for a specific u and DL, should be identical to that of the response of the Advection Dispersion Equation (ADE) model for that u and DL at the downstream distance, x = Δx.

Let the parameters,α T1, and T2 of the hybrid model for pollutant transport in a stream be known from the ADE model for a given value of u and DL satisfying Peclet number, Pe ≥4. Let the value of the parameters of the mainstream flow in a stream having hyporheic zones along streambed and banks, be: α = 1.70 min, T1 = 2.0 min and T2 = 6.30 min corresponding to Δx = 200 m, u = 20 m/min and DL = 1000 m2/min. The retardation factor, R is assumed to be 1.25. Using the above data, the unit step responses and the unit impulse responses of the hybrid model are generated at the end of the 1st hybrid unit applying eqns (9) and (10), respectively, and they are shown in figs 4 and 5.

In these figures, the C-t profiles generated by the hybrid model corresponding to the same value of α, T1 and T2 without the retardation component, i.e., for R = 1, are also shown for

Eq. (15) can be solved analytically and effluent from the first well mixed cell will enter second well mixed cell. Thus similar mass balance equation can be formulated. Successive convolution numerical integration can be used by combining effluent concentrations of plug flow cell and well mixed cells to get effluent concentration at the end of first hybrid cell as

( ) ( ) { } () ( ) ( ) ( ) 2 11 2

 γΔ

> Δδ

*Knt C n t C t C t n t <sup>M</sup>*

Eq. (16) is unit step response [K(.)] of first hybrid unit and one can get unit impulse response [k(.)] by differentiating Eq. (16). The output (Eq. 16) of pollutant from preceding hybrid unit forms the input to the succeeding hybrid unit, thus the response of the nth hybrid unit, n ≥ 2 for steady flow condition can be obtained using convolution technique which will be similar

( , ) {( 1) , } ( , , , , )

Δτ

**4.1 Verification of model considering equilibrium exchange using synthetic data** 

*Cn xt C n x k T T R t d*

where C[(n-1)Δx, t)]…effluent concentration from the preceding unit, k( )…unit impulse

The hybrid model simulates the advection-dispersion governed pollutant transport in a regular channel under uniform flow conditions when the size of the basic process unit, Δx, is equal to or more than 4 DL/u. This means that the response of the hybrid model corresponding to that Δx for a specific u and DL, should be identical to that of the response of the Advection Dispersion Equation (ADE) model for that u and DL at the downstream

Let the parameters,α T1, and T2 of the hybrid model for pollutant transport in a stream be known from the ADE model for a given value of u and DL satisfying Peclet number, Pe ≥4. Let the value of the parameters of the mainstream flow in a stream having hyporheic zones along streambed and banks, be: α = 1.70 min, T1 = 2.0 min and T2 = 6.30 min corresponding to Δx = 200 m, u = 20 m/min and DL = 1000 m2/min. The retardation factor, R is assumed to be 1.25. Using the above data, the unit step responses and the unit impulse responses of the hybrid model are generated at the end of the 1st hybrid unit applying eqns (9) and (10),

In these figures, the C-t profiles generated by the hybrid model corresponding to the same value of α, T1 and T2 without the retardation component, i.e., for R = 1, are also shown for

 α

 γ

1 1 ,

 Δ δ

 γ

1 2

*<sup>D</sup>* −

τ τ

∫ (17)

= − − ⎡ −+ ⎤ ∑ <sup>⎣</sup> <sup>⎦</sup> ; *δM1* & *δM2*…ramp kernel

<sup>=</sup> = − − ⎡ −+ ⎤ ∑ <sup>⎣</sup> <sup>⎦</sup> (16)

1 1 ,

 Δ  γ  Δ

1

*C nt C t C t n t PP M*

 γ

=

γ

0

*t*

= −

where 1 1 ( ) { } () ( ) ( ) ( )

Δ

 γΔ

Δ

*n*

follows

co-efficients.

Δ

1

=

*n*

γ

Δ

to eq. (11) and is given by

response of a single hybrid unit.

distance, x = Δx.

**4. Simulation of pollutant transport** 

respectively, and they are shown in figs 4 and 5.

comparison and it can be noted that the C-t profiles represented by the hybrid model with retardation shows the characteristics of delayed pollutant transport as expected. The impulse response, presented in fig. 5, represents the following; the rising limb occurred at a later time with reduction in magnitudes, the time to peak shifted, the peak concentration reduced, and the recession limb extended for a long time to that of the distributions depicted by the hybrid model with retardation.

Fig. 4. Unit step responses of the hybrid model at the end of one hybrid unit of size ∆x = 200 m for α = 1.7 min, T1 = 2.0 min, T2 = 6.3 min

The above data are adopted again and the C-t profiles of the hybrid model for a unit impulse input have been generated for n = 3, 6, and 11 using eqn (11) and shown in fig. 6 and it can be noted from the figure that as the pollutants move from the near field to the far

Simulation of Stream Pollutant Transport with

n=3

0

6) and eleventh (n = 11) hybrid units.

using field data collected from a river.

0.02

0.04

Concentration (mg/L)

0.06

0.08

0.1

Hyporheic Exchange for Water Resources Management 155

Eqilibrium mass exchange

n=1 Solid lines (R=1.25)

n=6

Dash lines (R=1.00)

n=11

0 40 80 120 160 200

Time (min)

Fig. 6. Unit impulse responses of the hybrid at the end of first (n = 1), third (n = 3), sixth (n =

**4.2 Verification of model considering non-equilibrium exchange using synthetic data**  Let the parameters, α T1, and T2 of the hybrid model are being 1.7 min, 2.0 min and 6.3 min respectively. Non-equilibrium mass exchange rate RD is assumed to be 0.25 per min which is a time reciprocal parameter. The C-t profiles of the hybrid model for a unit impulse input have been generated having parameters (α T1, T2 and RD) for n = 3, 6, and 11 using eq. (17) which is presented in fig. 7. In this study, the synthetic data was used for non-equilibrium exchange because of absence of field data to calculate mass exchange rate (RD). However, the section 4.3 explains the simulation of pollutant transport with equilibrium exchange

field the C-t distributions of the hybrid model get more and more attenuated, elongated and delayed in terms of occurrence of the rising limbs, the times to peak, and the peak concentrations. This means that the total residence time of the hybrid unit is increased by a factor of R for one unit as compared with the hybrid unit without mass exchange. As pollutant move down stream by n units, the residence time increased by a factor of "n times R". These characteristics of the C-t profiles for a steam with retardation component are in the expected lines. This may be due to more permanent loss of pollutant within the hyporheic zone or the water along with pollutant may take very long time to re-enter stream. The model is also capable of simulating non-equilibrium exchange which has been demonstrated in section 4.2 below with synthetic data.

Fig. 5. Unit impulse responses of the hybrid model at the end of one hybrid unit of size ∆x = 200 m for α = 1.7 min, T1 = 2.0 min, T2 = 6.3 min

field the C-t distributions of the hybrid model get more and more attenuated, elongated and delayed in terms of occurrence of the rising limbs, the times to peak, and the peak concentrations. This means that the total residence time of the hybrid unit is increased by a factor of R for one unit as compared with the hybrid unit without mass exchange. As pollutant move down stream by n units, the residence time increased by a factor of "n times R". These characteristics of the C-t profiles for a steam with retardation component are in the expected lines. This may be due to more permanent loss of pollutant within the hyporheic zone or the water along with pollutant may take very long time to re-enter stream. The model is also capable of simulating non-equilibrium exchange which has been

Equilibrium exchange R=1.0

R=1.25

0 10 20 30 40 50

Time (min)

Fig. 5. Unit impulse responses of the hybrid model at the end of one hybrid unit of size ∆x =

demonstrated in section 4.2 below with synthetic data.

0

200 m for α = 1.7 min, T1 = 2.0 min, T2 = 6.3 min

0.02

0.04

0.06

Concentration (mg/L)

0.08

0.1

Fig. 6. Unit impulse responses of the hybrid at the end of first (n = 1), third (n = 3), sixth (n = 6) and eleventh (n = 11) hybrid units.

### **4.2 Verification of model considering non-equilibrium exchange using synthetic data**

Let the parameters, α T1, and T2 of the hybrid model are being 1.7 min, 2.0 min and 6.3 min respectively. Non-equilibrium mass exchange rate RD is assumed to be 0.25 per min which is a time reciprocal parameter. The C-t profiles of the hybrid model for a unit impulse input have been generated having parameters (α T1, T2 and RD) for n = 3, 6, and 11 using eq. (17) which is presented in fig. 7. In this study, the synthetic data was used for non-equilibrium exchange because of absence of field data to calculate mass exchange rate (RD). However, the section 4.3 explains the simulation of pollutant transport with equilibrium exchange using field data collected from a river.

Simulation of Stream Pollutant Transport with

(Adopted from Muthukrishnavellaisamy K, 2007)

*reach between Tikira & Talcher)*

**Location Measured** 

Point 5 32.2 mg/L (Source: Muthukrishnavellaisamy K, 2007)

**Water Quality data** 

Fig. 8. Map showing study river reach and sampling points

**Channel geometry, flow characteristics and dispersion co-efficient** 

**Location** *Q* **(m3/s)** *U* **(m/s)** *A* **(m2)** *H* **(m)** *W* **(m)** *DL(m2/s)*  Before Tikira (Point 2) 195.98 0.83 218.12 3.21 67.91 430.74 After Tikira (Point 3) 239.72 0.92 257.99 3.43 75.10 490.05 Talcher 238.62 0.9 257.00 3.42 74.93 488.61

**Model parameters having U = 0.91 m/s and DL = 489.3 m2/s** *(these are average values for river* 

**Location Measured** 

**concentration** 

Cell size (∆x), m Pe α, min T1, min T2, min

3200 5.9 15.5 19.4 25.7

Point 1 (Q = 13.3 m3/s) 242.3 mg/L Point 2 (Q = 195.98 m3/s) 25 mg/L Point 3 38 mg/ L Point 4 36 mg/L

**concentration** 

Table 1. Flow and water quality data collected from the river Brahmani

Hyporheic Exchange for Water Resources Management 157

Fig. 7. Unit impulse responses of the hybrid at the end of first (n = 1), third (n = 3), sixth (n = 6) and eleventh (n = 11) hybrid units

### **4.3 Model verification using field data**

In order to verify the performance of the model, field data from the river Brahmani, India has been collected. A river reach from Rengali dam to Talcher is affected seriously by the waste water discharged by river Tikira, tributary of the main river. The Talcher Township is located 26km downstream of Tikira confluence. Fig. 8 shows the study area with locations of sampling points. Data collections from the field have been tabulated in Table 1.

If observed *C-t* profile is available, as an inverse problem model parameters can be estimated using optimization. In absence of observed *C-t* profile, model parameters can be obtained by relating with longitudinal dispersion co-efficient, *D*L, satisfying the condition of

Non-equilibrium mass exchange

Solid lines (RD=0.25 per min) n=1

Dash lines (RD=0.00 per min)

n=11

0 40 80 120 160 200

Time (min)

Fig. 7. Unit impulse responses of the hybrid at the end of first (n = 1), third (n = 3), sixth (n =

In order to verify the performance of the model, field data from the river Brahmani, India has been collected. A river reach from Rengali dam to Talcher is affected seriously by the waste water discharged by river Tikira, tributary of the main river. The Talcher Township is located 26km downstream of Tikira confluence. Fig. 8 shows the study area with locations of

If observed *C-t* profile is available, as an inverse problem model parameters can be estimated using optimization. In absence of observed *C-t* profile, model parameters can be obtained by relating with longitudinal dispersion co-efficient, *D*L, satisfying the condition of

sampling points. Data collections from the field have been tabulated in Table 1.

n=6

0

6) and eleventh (n = 11) hybrid units

**4.3 Model verification using field data** 

0.02

0.04

n=3

Concentration (mg/L)

0.06

0.08

0.1

(Adopted from Muthukrishnavellaisamy K, 2007)



(Source: Muthukrishnavellaisamy K, 2007)

Table 1. Flow and water quality data collected from the river Brahmani

Simulation of Stream Pollutant Transport with

**5. Conclusions** 

advection and dispersion processes.

model.

streams.

Hyporheic Exchange for Water Resources Management 159

One can choose any peclet number between 4 and 8 in order to match the response of hybrid model with the ADE model. Having u = 0.9 m/s and DL = 490 m2/s, the hybrid unit size (∆x) has been chosen as 3200 m by satisfying the condition of peclet number and the parameters of the hybrid model (α, T1 and T2) are approximately estimated as: α = 15 min, T1 = 19 min and T2 = 25 min. The reach length of 26 km (from Tikira confluence to Talcher) is covered with 8 hybrid units. By successive convolution, the pollutant concentrations at Talcher (26 km downstream of pollutant source) are predicted for step input. Having the above data collected from river Brahmani, using eqn 11, pulse response at 3.2km (1st hybrid unit), 16km (5th hybrid unit) and 26km (8th hybrid unit) have been simulated with different retardation factors (R = 1.0 & 2.1) and presented in fig. 9. The maximum pollutant concentration at various locations downstream of pollutant disposal point can be derived by numerical integration of pulse responses obtained using hybrid model for those downstream locations. The maximum pollutant concentration at Talcher is about 34.5 mg/L, which is very close to the measured value of 32.2 mg/L. It clearly demonstrates the influence of retardation process of pollutant transport. In order for complete verification of model, numerous data are needed towards calculating mass exchange rate constant (R).

However, this chapter theoretically compares the model with limited field data.

A Hybrid model coupled with hyporheic exchange has been derived by incorporating a time delay factor termed as "retardation factor" with each of the three compartments in the hybrid model to simulate retardation governed pollutant transport in riparian streams or rivers. A linear equilibrium condition between the concentration of pollutants in the hyporheic zone and the mainstream water has been considered. The stagnation or dead or hyporheic zone retards the transport of downstream pollutants. The hybrid model is a fourparameter model representing three time parameters and one constant factor. Theoretical study on non-equilibrium exchange of pollutant has also been done to demonstrate the

The unit step response and the unit impulse response functions of the hybrid model have been simulated with synthetic data and limited field data. The characteristics of the concentration-time profiles generated by the hybrid model are comparable to the physical processes of pollutant transport governed by the advection-dispersion-retardation both in equilibrium and non-equilibrium exchanges in a natural stream. This present model can be used to obtain theoretically exact solutions and can be compared with results of ADE model considering with and without retardation of pollutant transport in a stream along with

Data regarding the influence of the hyporheic zone to pollutant trap in streams are rare due to the absence of simple techniques to get necessary parameters and complexity of the phenomenon. The pollutant exchange between the main channel and the hyporheic zone is very variable and estimation of exchange rate is mostly inaccurate due to channel irregularities and other complexities. In depth analysis and understanding about the hyporheic exchange will over-come the problem in collecting relevant data from natural

It can be concluded that the presented hybrid model for pollutant transport in streams affected by hyporheic exchange is a useful tool in predicting water quality status streams.

Peclet number, *P xu D e L* = Δ / which should be greater than or equals to 4 and less than 8 (Ghosh et al., 2004; Muthukrishnavellaisamy K, 2007; Ghosh et al., 2008) as follows:

$$\alpha = \frac{0.04 \text{ A} \text{x}^2}{D\_L} \tag{18}$$

$$T\_1 = \frac{0.05\,\text{A}\text{x}^2}{D\_L} \tag{19}$$

$$T\_2 = \frac{\Delta \mathbf{x}}{\mu} - \frac{0.09 \text{ } \Delta \mathbf{x}^2}{D\_L} \tag{20}$$

Fig. 9. Pulse responses of the hybrid at the end of 3.2 km, 16 km and 26 km for the data collected from river Brahmani

One can choose any peclet number between 4 and 8 in order to match the response of hybrid model with the ADE model. Having u = 0.9 m/s and DL = 490 m2/s, the hybrid unit size (∆x) has been chosen as 3200 m by satisfying the condition of peclet number and the parameters of the hybrid model (α, T1 and T2) are approximately estimated as: α = 15 min, T1 = 19 min and T2 = 25 min. The reach length of 26 km (from Tikira confluence to Talcher) is covered with 8 hybrid units. By successive convolution, the pollutant concentrations at Talcher (26 km downstream of pollutant source) are predicted for step input. Having the above data collected from river Brahmani, using eqn 11, pulse response at 3.2km (1st hybrid unit), 16km (5th hybrid unit) and 26km (8th hybrid unit) have been simulated with different retardation factors (R = 1.0 & 2.1) and presented in fig. 9. The maximum pollutant concentration at various locations downstream of pollutant disposal point can be derived by numerical integration of pulse responses obtained using hybrid model for those downstream locations. The maximum pollutant concentration at Talcher is about 34.5 mg/L, which is very close to the measured value of 32.2 mg/L. It clearly demonstrates the influence of retardation process of pollutant transport. In order for complete verification of model, numerous data are needed towards calculating mass exchange rate constant (R). However, this chapter theoretically compares the model with limited field data.
