**1. Introduction**

Stream channel irregularities, meandering and hyporheic zones, is commonly seen in many riparian streams. These irregular channel geometries often influence the pollutant transport. The stagnation or dead zones are the pockets of stagnant water or water having very low velocities, which trap pollutants and release them at later time to mainstream flow at a rate that depends upon the concentration gradient of the pollutants between the two domains. The stagnation zones are formed near the concave banks of the stream and behind the irregular sand dunes formed on the bed of the stream. The stagnation zones may also be formed due to irregular stream boundaries and also due to localized channel expansions (Bencala & Walters, 1983). The hyporheic zone is a transition zone between terrestrial and aquatic ecosystems and is regarded as an ecologically important ecotone (Boulton et al., 1998; Edwards, 1998). The term hyporheic is derived from Greek language – hypo, meaning under or beneath, and rheos, meaning a stream (Smith, 2005). A number of definitions for the hyporheic zone exist (Triska et al., 1989; Valett et al., 1997; White, 1993), however, the most common connotations are: it is the zone below and adjacent to a streambed in which water from the open channel gets exchanged with interstitial water in the bed sediments; it is the zone around a stream in which fauna characteristic of the hyporheic zones are distributed and live; it is the zone in which groundwater and surface water mix (Smith, 2005). The physical process, of hyporheic exchanges, as described by many investigators (Stanford and Ward, 1988; Stanford and Ward, 1993; Triska et al., 1989; Valett et al., 1997; Brunke and Gonser, 1997, White, 1993) suggest that significant amounts of water are exchanged between the channel and saturated sediments surrounding the channel. Such exchanges have the potential to cause large changes in stream water chemistry and retard the transport of pollutants. The rates of biogeochemical processes and the types of processes governed by flow hydraulics may be fundamentally different. When the groundwater component is negligible, this is possible if there is a fine silt or clay formation underneath the hyporheic zone, the exchanges between the mainstream flow and the hyporheic zone for such condition shall be similar to the stagnation or dead zone processes, i.e., the stream water that enters the subsurface eventually re-enters the stream at some point downstream. The pollutant transport processes for such circumstances can be regarded as hyporheic exchange. Fig. 1 represents a stream with hyporheic zone.

Simulation of Stream Pollutant Transport with

Stream

Hyporheic zone

m2/s.

*CR Q*

Let V0

by, V0

<sup>∗</sup> , V1

<sup>∗</sup> , and V2

<sup>∗</sup> = (φ Ap D), V1

would respectively be: T1R and T2R.

Hyporheic Exchange for Water Resources Management 147

response of injected pollutant concentration, CR at the exit of Δx of the mainstream due to the affects of the hyporheic exchange using a hybrid model. The hybrid model has three compartments all connected in series; first compartment represents the plug flow cell of residence time, α; whereas second and third compartment, represent respectively two well mixed cells of unequal residence times, T1 and T2. The hybrid model simulates advectiondispersion transport of pollutant in regular channel for steady and uniform flow conditions when the size of the basic process unit, Δx, is equal to or more than 4 DL / u (Ghosh et al., 2008) where u…mean flow velocity in m/s, and DL…longitudinal dispersion coefficient in

The arrangement of the conceptualized hybrid model and the hyporheic zone has been shown schematically in Fig. 2. It is assumed that the hyporheic zones along streambed and banks below and around the plug flow cell and two well mixed cells are in equal proportion to their ratios of: cross-sectional areas, volumes of water and also mass of solute exchanges between the hyporheic zones and the mainstream water. Let V0, V1, and V2 be the volumes of the plug flow cell, and two well mixed cells in the mainstream, respectively. For stream flow Q, the residence time of pollutant in the plug flow and two well mixed cells of the main

well mixed cells of the mainstream water columns respectively. These volumes can be given

interface areas (in m2) of the hyporheic zone and the mainstream flow respectively for the plug flow and the two well mixed cells, D is the depth (in m) of the hyporheic zone below and around the mainstream, and φ is the porosity of the bed materials. When this zone represents a dead or stagnant pocket with only storage of water, φ = 1, and if there is no soil pores and no water storage, φ = 0. If the hyporheic zone is extended all along the wetted surfaces, the Ap, AM1, and AM2, in such cases, are represented respectively by: Ap = Wp (α u); AM1 = Wp (T1 u); and AM2 = Wp (T2 u), in which Wp is the wetted perimeter at the interface of the mainstream and the hyporheic zone and u is the mainstream flow velocity. If the ratios of the volume of water in the hyporheic zones to the three volumes in the mainstream are in proportion and constant, the ratios *VVVVVV* <sup>001122</sup> /// ∗∗∗ = = are also constant and defined as, say F. The total residence time of pollutant in the plug flow cell would thus be: (*V V QV VV Q* 0 0 0 00 ) / 1 / 1F R ( ) ( ) ∗ ∗ + = + =α + =α , where R…the retardation factor, and F…proportionality constant. Similarly, the total residence times in the two well mixed cells

Plug Flow Cell 1st mixed Cell 2nd

stream are α = V0 / Q, T1 = V1 / Q and T2 = V2 / Q respectively.

<sup>∗</sup> = (φ AM1 D) and V2

Fig. 2. Conceptual hybrid model with hyporheic zone

*CP CM1 CM2* 

<sup>∗</sup> be the volumes of the hyporheic zones below the plug flow and two

mixed Cell

<sup>∗</sup> = (φ AM2 D); where Ap, AM1, and AM2…the

*Q*

Fig. 1. Representations of a stream with underlying hyporheic zone

In the hyporheic zone, a fraction of stream water containing pollutants is temporarily detained in small eddies and stagnant water that are stationary relative to the faster moving water near the center of the stream. In addition, significant portions of flow may move through the coarse gravel of the streambed and porous areas within the stream banks (Runkel, 2000). The pollutants detained in the hyporheic zone as buffer is released back slowly to the mainstream when the concentration gradient reverses. The travel time for pollutants carried through these porous areas may be substantially longer than that for pollutants travelling within the water column. Streams with intact hyporheic zone provide more temporary storage space and residence time for water with pollutant than streams without them (Bencala & Walters, 1983, Berndtsson, 1990; Castro & Hornberger, 1991; Harvey et al., 1996; Runkel et al., 1996). The hyporheic exchange thus retards the transport of pollutants. The rate of exchange of pollutants between two domains may vary from constituent to constituent. The phenomena, that trap pollutants from the mainstream and hold into the buffer at the beginning and release them back to the mainstream water at later time, shall generate C-t profile representing characteristics of delayed transport, shifted time to peak, reduced peak concentration and also a long tail. To simulate the temporal variations of pollutant concentrations, a hybrid model which appends the retardation component with advection-dispersion pollutant transport has been conceptualized in this study. In this study, the hybrid model has been formulated for both equilibrium and non-equilibrium exchange of pollutant mass between main stream and underlying soil media. Efficacy of model was tested with the results of well known Advection Dispersion Equation and with field data collected from River Brahmani, India.

### **2. Formulation of the model considering equilibrium mass exchange**

Let us consider a straight stream reach of length, Δx, having irregular porous geomorphology. This irregular porous geomorphology is comprised of uniform formations of stagnation or dead or hyporheic zones along streambed and banks. These zones represent the hyporheic exchange, and are hydraulically connected with the mainstream flow. Further, there are exchanges of pollutants with the flow through the interface between the mainstream flow and the hyporheic zones. The stream reach has a steady flow rate, Q and initial concentration of pollutants Ci both in the mainstream as well as in the hyporheic zone. Let a steady state concentration of pollutant, CR, be applied at the inlet boundary of the stream at a time, t0. It is assumed that the river reach be composed of series of equal size hybrid units. It is required to derive the model that can simulate the

In the hyporheic zone, a fraction of stream water containing pollutants is temporarily detained in small eddies and stagnant water that are stationary relative to the faster moving water near the center of the stream. In addition, significant portions of flow may move through the coarse gravel of the streambed and porous areas within the stream banks (Runkel, 2000). The pollutants detained in the hyporheic zone as buffer is released back slowly to the mainstream when the concentration gradient reverses. The travel time for pollutants carried through these porous areas may be substantially longer than that for pollutants travelling within the water column. Streams with intact hyporheic zone provide more temporary storage space and residence time for water with pollutant than streams without them (Bencala & Walters, 1983, Berndtsson, 1990; Castro & Hornberger, 1991; Harvey et al., 1996; Runkel et al., 1996). The hyporheic exchange thus retards the transport of pollutants. The rate of exchange of pollutants between two domains may vary from constituent to constituent. The phenomena, that trap pollutants from the mainstream and hold into the buffer at the beginning and release them back to the mainstream water at later time, shall generate C-t profile representing characteristics of delayed transport, shifted time to peak, reduced peak concentration and also a long tail. To simulate the temporal variations of pollutant concentrations, a hybrid model which appends the retardation component with advection-dispersion pollutant transport has been conceptualized in this study. In this study, the hybrid model has been formulated for both equilibrium and non-equilibrium exchange of pollutant mass between main stream and underlying soil media. Efficacy of model was tested with the results of well known Advection Dispersion Equation and with

Hyporheic Zone Stream

Main Stream

Hyporheic zone

 (a) (b) Fig. 1. Representations of a stream with underlying hyporheic zone

field data collected from River Brahmani, India.

**2. Formulation of the model considering equilibrium mass exchange** 

Let us consider a straight stream reach of length, Δx, having irregular porous geomorphology. This irregular porous geomorphology is comprised of uniform formations of stagnation or dead or hyporheic zones along streambed and banks. These zones represent the hyporheic exchange, and are hydraulically connected with the mainstream flow. Further, there are exchanges of pollutants with the flow through the interface between the mainstream flow and the hyporheic zones. The stream reach has a steady flow rate, Q and initial concentration of pollutants Ci both in the mainstream as well as in the hyporheic zone. Let a steady state concentration of pollutant, CR, be applied at the inlet boundary of the stream at a time, t0. It is assumed that the river reach be composed of series of equal size hybrid units. It is required to derive the model that can simulate the response of injected pollutant concentration, CR at the exit of Δx of the mainstream due to the affects of the hyporheic exchange using a hybrid model. The hybrid model has three compartments all connected in series; first compartment represents the plug flow cell of residence time, α; whereas second and third compartment, represent respectively two well mixed cells of unequal residence times, T1 and T2. The hybrid model simulates advectiondispersion transport of pollutant in regular channel for steady and uniform flow conditions when the size of the basic process unit, Δx, is equal to or more than 4 DL / u (Ghosh et al., 2008) where u…mean flow velocity in m/s, and DL…longitudinal dispersion coefficient in m2/s.

Fig. 2. Conceptual hybrid model with hyporheic zone

The arrangement of the conceptualized hybrid model and the hyporheic zone has been shown schematically in Fig. 2. It is assumed that the hyporheic zones along streambed and banks below and around the plug flow cell and two well mixed cells are in equal proportion to their ratios of: cross-sectional areas, volumes of water and also mass of solute exchanges between the hyporheic zones and the mainstream water. Let V0, V1, and V2 be the volumes of the plug flow cell, and two well mixed cells in the mainstream, respectively. For stream flow Q, the residence time of pollutant in the plug flow and two well mixed cells of the main stream are α = V0 / Q, T1 = V1 / Q and T2 = V2 / Q respectively.

Let V0 <sup>∗</sup> , V1 <sup>∗</sup> , and V2 <sup>∗</sup> be the volumes of the hyporheic zones below the plug flow and two well mixed cells of the mainstream water columns respectively. These volumes can be given by, V0 <sup>∗</sup> = (φ Ap D), V1 <sup>∗</sup> = (φ AM1 D) and V2 <sup>∗</sup> = (φ AM2 D); where Ap, AM1, and AM2…the interface areas (in m2) of the hyporheic zone and the mainstream flow respectively for the plug flow and the two well mixed cells, D is the depth (in m) of the hyporheic zone below and around the mainstream, and φ is the porosity of the bed materials. When this zone represents a dead or stagnant pocket with only storage of water, φ = 1, and if there is no soil pores and no water storage, φ = 0. If the hyporheic zone is extended all along the wetted surfaces, the Ap, AM1, and AM2, in such cases, are represented respectively by: Ap = Wp (α u); AM1 = Wp (T1 u); and AM2 = Wp (T2 u), in which Wp is the wetted perimeter at the interface of the mainstream and the hyporheic zone and u is the mainstream flow velocity. If the ratios of the volume of water in the hyporheic zones to the three volumes in the mainstream are in proportion and constant, the ratios *VVVVVV* <sup>001122</sup> /// ∗∗∗ = = are also constant and defined as, say F. The total residence time of pollutant in the plug flow cell would thus be: (*V V QV VV Q* 0 0 0 00 ) / 1 / 1F R ( ) ( ) ∗ ∗ + = + =α + =α , where R…the retardation factor, and F…proportionality constant. Similarly, the total residence times in the two well mixed cells would respectively be: T1R and T2R.

Simulation of Stream Pollutant Transport with

applied at the inlet boundary of the plug flow cell.

first one except residence time

cell of residence time, T2 is given by,

response function, K(t) is given by:

the Advection-dispersion equation.

is given by

*M2 R*

() ( ) 1

α

Hyporheic Exchange for Water Resources Management 149

( )

Eqn (6) is valid for t ≥ α R which gives the time varying concentration of pollutants at the exit of the first well mixed cell coupled with the retardation due to a unit step input, CR

The time varying outputs from the first well mixed cell form the inputs to the second well mixed cell. Alike first well mixed cell, pollutant before being exited from the cell exchanges to the adjoining hyporheic zone and release back to the mainstream water. Performing the mass balance over a time interval t to t + ∆t, one can get a differential equation similar to the

> ( ) 1

α

*t R R T*

2 2 2

<sup>⎡</sup> ( ) ( ) <sup>⎤</sup> <sup>⎢</sup> <sup>⎥</sup> <sup>⎢</sup> <sup>⎥</sup> ⎢⎣ ⎥⎦

*1 2 1 2*

1 2 1 2

α

*TT TT*

α

− − − − <sup>⎡</sup> <sup>⎤</sup> <sup>−</sup> <sup>⎢</sup> <sup>⎥</sup> <sup>=</sup> <sup>−</sup> <sup>−</sup> <sup>⎢</sup> <sup>⎥</sup> ⎢⎣ ⎥⎦

Eqns (9) and (10) are valid for t ≥ α R, and they respectively represent the unit step and the unit impulse response functions of a hybrid unit coupled with the retardation. If R = 1, eqns (9) and (10) respectively represent the unit step and the unit impulse response functions of

Let the stream reach downstream of a point source of pollution be composed of series of equal size hybrid units coupled with the hyporheic zone, each having linear dimension, Δx

− − − − <sup>⎡</sup> <sup>⎤</sup> <sup>⎢</sup> <sup>⎥</sup> =− − <sup>+</sup> <sup>⎢</sup> − − <sup>⎥</sup> ⎢⎣ ⎥⎦

The unit impulse response function, k (t) is the derivative of eqn (9) with respect to't' which

1

⎢ ⎥ − − ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ <sup>=</sup> <sup>−</sup> <sup>−</sup> (7)

*1 2 t -α R t-α <sup>R</sup> - - 1 2 R T R T*

( ) ( )

*t R t R*

1 2 1 2

( ) ( ) 1 2

 α

*tR tR*

*T -T T -T* (8)

 α

(9)

(10)

( )

( )

α

*C Ut R e*

*R*

2 2

*T T C =C U t- <sup>α</sup> R 1- e + e*

*T T R T R T Kt Ut R <sup>e</sup> <sup>e</sup>*

*Ut R RT RT k t e e*

( )

α

*RT T*

( ) ( )

1 2

*M M s*

<sup>−</sup> <sup>−</sup> ⎡ ⎤

*dC C dM dt T T V dt*

Solving Eqns (1) and (7), the concentration of pollutants at the end of the second well mixed

If CR = 1, eqn (8) represents the unit step response function. Designating K(t) as the unit step

1

( ) <sup>⎡</sup> <sup>⎤</sup> <sup>⎢</sup> <sup>⎥</sup> <sup>⎢</sup> <sup>⎥</sup> ⎢⎣ ⎥⎦ *1 t -αR - R T C =C U t- M1 R αR 1-e* (6)

It is also assumed that the retardation process of pollutants takes place in all the cells of the hybrid model due to the hyporheic exchange. In natural riparian rivers, the hyporheic exchange is a complex process which may follow non-equilibrium exchange between main stream and underlying stagnation zone. Decay of pollutant will take place both in main stream and stagnation zone, if the pollutant is of non-conservative type. It is worth trying with a conservative pollutant's equilibrium exchange between main stream channel and hyporheic zone. Hence, in this study retardation process is considered to be followed the linear equilibrium condition, which is expressed as:

$$\mathbf{C}\mathbf{s}(\mathbf{x},\mathbf{t}) = \mathbf{F}\mathbf{C}(\mathbf{x},\mathbf{t})\tag{1}$$

where Cs(x, t)…the concentration of pollutant which is trapped in the hyporheic zone in mg/L, F…the proportionality constant and C(x, t)…the concentration of pollutant in the main stream in mg/L.

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 as given

$$\frac{\partial \mathbb{C}(\mathbf{x},t)}{\partial t} + \mu \frac{\partial \mathbb{C}(\mathbf{x},t)}{\partial \mathbf{x}} = -\frac{\oint \mathcal{W}\_{\text{p}} \mathrm{D}}{A} \frac{\partial \mathbb{C}\_{s}(\mathbf{x},t)}{\partial t} \tag{2}$$

where A…the cross sectional area of any control volume in the plug flow cell of the mainstream water in m2.

Solving eqns (1) and (2), the concentration of pollutants at the end of the plug flow cell of residence time, α is given by,

$$\mathbf{C}\left(\mathbf{x},t\right) = \mathbf{C}\_P\left(a\mathbf{u},t\right) = \mathbf{C}\_R\mathbf{L}\left(t-a,R\right) \tag{3}$$

where, CP(αu, t)…concentration of pollutant at the end of plug flow cell in mg/L, U(\*)…the unit step function, and t…the time reckoned since injection of pollutants in min.

In the first well mixed cell, pollutants after travelling a distance of 'αu' through the plug flow cell enter and exchange to the adjoining hyporheic zone and release back to the mainstream water, before making an exit from it. The inputs to this cell are thus the outputs from the plug flow cell. The pollutants are thoroughly mixed in this cell.

Performing the mass balance in the first well mixed cell along with the hyporheic zone over a time interval t to t + ∆t, one can get governing differential equation as

$$\frac{d\mathbb{C}\_{M1}}{dt} = \frac{\mathbb{C}\_{R}}{T\_{1}} \frac{\mathbb{U}\left(t - \alpha R\right)}{T\_{1}} - \frac{\mathbb{C}\_{M1}}{T\_{1}} - \frac{1}{V\_{1}} \frac{dM\_{s}}{dt} \tag{4}$$

Replacing dMs/dt by derivative of concentration, eqn (4) transforms to:

$$\frac{d\mathbf{C}\_{M1}}{dt} = \frac{\mathbf{C}\_{R}}{T\_{1}} \frac{\mathbf{U}\{t - a\mathbf{R}\}}{T\_{1}} - \frac{\mathbf{C}\_{M1}}{T\_{1}} - \frac{\phi}{A} \frac{\mathcal{W}\_{\text{P}} \, D}{A} \frac{d\mathbf{C}\_{s}}{dt} \tag{5}$$

Solving eqns (1) and (5), the concentration of pollutants at the end of the first well mixed cell of residence time, T1 is given by,

It is also assumed that the retardation process of pollutants takes place in all the cells of the hybrid model due to the hyporheic exchange. In natural riparian rivers, the hyporheic exchange is a complex process which may follow non-equilibrium exchange between main stream and underlying stagnation zone. Decay of pollutant will take place both in main stream and stagnation zone, if the pollutant is of non-conservative type. It is worth trying with a conservative pollutant's equilibrium exchange between main stream channel and hyporheic zone. Hence, in this study retardation process is considered to be followed the

where Cs(x, t)…the concentration of pollutant which is trapped in the hyporheic zone in mg/L, F…the proportionality constant and C(x, t)…the concentration of pollutant in the

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

( ,, , ) ( ) *<sup>s</sup>* ( ) *C xt C xt W DP C xt*

φ

∂ ∂ ∂

+ =−

*t x At*

where A…the cross sectional area of any control volume in the plug flow cell of the

Solving eqns (1) and (2), the concentration of pollutants at the end of the plug flow cell of

*C xt C ut C U t R* ( , , ) = = − *P R* (α

where, CP(αu, t)…concentration of pollutant at the end of plug flow cell in mg/L, U(\*)…the

In the first well mixed cell, pollutants after travelling a distance of 'αu' through the plug flow cell enter and exchange to the adjoining hyporheic zone and release back to the mainstream water, before making an exit from it. The inputs to this cell are thus the outputs

Performing the mass balance in the first well mixed cell along with the hyporheic zone over

*dCM C Ut R <sup>R</sup> C dM M s* 1 *dt T T V dt* −α

1 1 *<sup>M</sup> C Ut R <sup>R</sup> M s W DP dC C dC dt T T A dt*

Solving eqns (1) and (5), the concentration of pollutants at the end of the first well mixed cell

1 11

φ

( ) 1 1

( ) 1 1

−α

unit step function, and t…the time reckoned since injection of pollutants in min.

from the plug flow cell. The pollutants are thoroughly mixed in this cell.

a time interval t to t + ∆t, one can get governing differential equation as

Replacing dMs/dt by derivative of concentration, eqn (4) transforms to:

Cs(x, t) = F C(x, t) (1)

<sup>∂</sup> ∂ ∂ (2)

 α

= −− (4)

= −− (5)

) ( ) (3)

linear equilibrium condition, which is expressed as:

hyporheic exchange coupled pollutant transport as given

*u*

main stream in mg/L.

mainstream water in m2.

residence time, α is given by,

of residence time, T1 is given by,

$$C\_{MI} = C\_R \, U \left( t \text{ - } aR \right) \left[ I \text{ - } e^{-\frac{\left( t \cdot aR \right)}{R \cdot T\_I}} \right] \tag{6}$$

Eqn (6) is valid for t ≥ α R which gives the time varying concentration of pollutants at the exit of the first well mixed cell coupled with the retardation due to a unit step input, CR applied at the inlet boundary of the plug flow cell.

The time varying outputs from the first well mixed cell form the inputs to the second well mixed cell. Alike first well mixed cell, pollutant before being exited from the cell exchanges to the adjoining hyporheic zone and release back to the mainstream water. Performing the mass balance over a time interval t to t + ∆t, one can get a differential equation similar to the first one except residence time

$$\frac{\mathbf{C}\_{R}}{dt} = \frac{\mathbf{C}\_{R} \cdot \mathbf{U} \left(t - \alpha R\right) \left[1 - e^{-\frac{\left(t - \alpha R\right)}{R\_{1}T\_{1}}}\right]}{T\_{2}}\tag{7}$$

$$\frac{\mathbf{d}C\_{M2}}{dt} = \frac{\mathbf{C}\_{M2}}{T\_{2}} - \frac{\mathbf{C}\_{M2}}{T\_{2}} - \frac{1}{V\_{2}} \frac{d\mathcal{M}\_{s}}{dt}\tag{7}$$

Solving Eqns (1) and (7), the concentration of pollutants at the end of the second well mixed cell of residence time, T2 is given by,

$$C\_{M,2} = C\_R U \left( t \cdot a \, R \right) \left[ I \cdot \frac{T\_I}{T\_I \cdot T\_2} e^{-\frac{\left( t \cdot a \, R \right)}{R \cdot T\_I}} + \frac{T\_2}{T\_I \cdot T\_2} e^{-\frac{\left( t \cdot a \, R \right)}{R \cdot T\_2}} \right] \tag{8}$$

If CR = 1, eqn (8) represents the unit step response function. Designating K(t) as the unit step response function, K(t) is given by:

$$K(t) = \mathcal{U}(t - \alpha \,\mathrm{R}) \left[ 1 - \frac{T\_1}{T\_1 - T\_2} \, e^{-\frac{\left(t - \alpha \,\mathrm{R}\right)}{\mathrm{R} \cdot T\_1}} + \frac{T\_2}{T\_1 - T\_2} e^{-\frac{\left(t - \alpha \,\mathrm{R}\right)}{\mathrm{R} \cdot T\_2}} \right] \tag{9}$$

The unit impulse response function, k (t) is the derivative of eqn (9) with respect to't' which is given by

$$k(t) = \frac{\mathcal{U}(t - \alpha \ R)}{\mathcal{R}\left(T\_1 - T\_2\right)} \left[ e^{-\frac{\left(t - \alpha \ R\right)}{\mathcal{R}\left(T\_1\right)}} - e^{-\frac{\left(t - \alpha \ R\right)}{\mathcal{R}\left(T\_2\right)}} \right] \tag{10}$$

Eqns (9) and (10) are valid for t ≥ α R, and they respectively represent the unit step and the unit impulse response functions of a hybrid unit coupled with the retardation. If R = 1, eqns (9) and (10) respectively represent the unit step and the unit impulse response functions of the Advection-dispersion equation.

Let the stream reach downstream of a point source of pollution be composed of series of equal size hybrid units coupled with the hyporheic zone, each having linear dimension, Δx

Simulation of Stream Pollutant Transport with

expressed mathematically as follows

plug flow zone is given by

α

where 1 *DR <sup>x</sup> W DP B <sup>u</sup> <sup>e</sup> A* φ

pollutant concentration is zero for t < α

γ

min.

Hyporheic Exchange for Water Resources Management 151

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

> ( ) () () , , , *<sup>s</sup> D s*

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

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

( ) ( ) () ( ) ( ) ( ) <sup>1</sup> ( ) ( ) 0

*A*

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

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

− − <sup>⎡</sup> <sup>⎤</sup> = = <sup>⎢</sup> −+ − <sup>−</sup> <sup>⎥</sup> <sup>⎢</sup> <sup>−</sup> <sup>⎥</sup> <sup>⎣</sup> <sup>⎦</sup> <sup>∫</sup> (14)

 η τα

, , exp 2 *<sup>D</sup> <sup>t</sup> <sup>R</sup> C xt C ut C U t U e P R <sup>d</sup>*

> φ η

( ) 1 1

−α

1 1 *<sup>M</sup> C Ut <sup>R</sup> M s W DP dC C dC dt T T A dt*

φ

 α

⎛ ⎞ <sup>=</sup> − − ⎜ ⎟ ⎝ ⎠ ; 1 *DR <sup>x</sup> WDR PBD <sup>u</sup> <sup>e</sup>*

γ

*R C xt C xt*

*dt* <sup>=</sup> ⎡− ⎤ <sup>⎣</sup> <sup>⎦</sup> (13)

1

τ α

⎛ ⎞ <sup>=</sup> ⎜ ⎟ <sup>−</sup> ⎝ ⎠, U(t - α)…step function which

= −− (15)

Ι

 ητ α τ

τ α

*dC x t*

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

and consisting of a plug flow cell, and two unequal well mixed cells. The stream reach has identical features all along downstream; i.e., mainstream flow and hyporheic or dead zones. The exchange of pollutant takes place in all the hybrid units. Assuming that output 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, as:

$$\mathbf{C}\{n\Delta\mathbf{x},t\} = \mathop{\rm C}\limits\_{\mathbf{0}}^{t}\{\mathbf{x} - \mathbf{1}\}\Delta\mathbf{x},\tau\}\ \;\;k\{\mathbf{a},T\_{1},T\_{2},\mathbf{R},t-\tau\}\,d\tau\tag{11}$$

where C((n-1)Δx,τ)…input of nth hybrid unit in mg/L, k(α,T1,T2, R, τ )…the unit impulse response (in mg/L. min) as given by eqn (10).

The retardation factor, R can be calculated as given

$$R = \left(1 + \frac{\phi w\_p D.F}{A}\right) \tag{12}$$

Eqn (12) is a function of ratio between areas of the hyporheic zone and the mainstream flow, porosity of bed materials. The retardation coefficients have been estimated with different porosity values using eqn (12) by keeping all other parameters constant as shown in Fig. 3.

Knowing parameters α, T1, and T2 of the hybrid unit, and using estimated value of the retardation factor, R, one can predict concentration profiles at multiple distance downstream from source at {n Δx}, n = 1,2,3,…. in a stream having steady flow and homogeneous reach conditions making use of eqn (11).

Fig. 3. Variation of retardation factor due to porosity

and consisting of a plug flow cell, and two unequal well mixed cells. The stream reach has identical features all along downstream; i.e., mainstream flow and hyporheic or dead zones. The exchange of pollutant takes place in all the hybrid units. Assuming that output 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

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

Δτ

*Cn xt C n x k T T Rt d*

where C((n-1)Δx,τ)…input of nth hybrid unit in mg/L, k(α,T1,T2, R, τ )…the unit impulse

. <sup>1</sup> *w DF <sup>p</sup> <sup>R</sup>*

= + ⎜ ⎟ ⎝ ⎠

Eqn (12) is a function of ratio between areas of the hyporheic zone and the mainstream flow, porosity of bed materials. The retardation coefficients have been estimated with different porosity values using eqn (12) by keeping all other parameters constant as shown in Fig. 3. Knowing parameters α, T1, and T2 of the hybrid unit, and using estimated value of the retardation factor, R, one can predict concentration profiles at multiple distance downstream from source at {n Δx}, n = 1,2,3,…. in a stream having steady flow and homogeneous reach

*A* ⎛ ⎞ φ

0

*t*

= −

Δ

response (in mg/L. min) as given by eqn (10).

conditions making use of eqn (11).

The retardation factor, R can be calculated as given

Fig. 3. Variation of retardation factor due to porosity

1 2

 τ τ

(12)

<sup>−</sup> ∫ (11)

 α

convolution technique, as:
