E-proceedings of the 36th IAHR World Congress 28 June – 3 July, 2015, The Hague, the Netherlands

GROUNDWATER FLOW AND TRANSPORT SIMULATION FOR A CONFINED AQUIFER USING ANALYTIC ELEMENT METHOD AND RANDOM WALK PARTICLE TRACKING. 1

PARTHA MAJUMDER , T.I. ELDHO 1

2

Research Scholar, Department of Civil Engineering, IIT Bombay, [email protected] 2 Professor, Department of Civil Engineering, IIT Bombay, [email protected]

ABSTRACT In this paper, the groundwater flow and transport processes in a confined aquifer have been simulated using analytic element method (AEM) and random walk particle tracking (RWPT). The implementation of coupled AEM-RWPT to simulate solute transport processes is the first time in the field of groundwater flow and transport process. The AEM solves the groundwater flow equation by superposition of the analytic solution of simple hydrogeological features like, well, river, lake, recharge, etc. One of the main advantages of AEM is that, it generates continuous head and velocity, thus very suitable for particle tracking process, unlike finite difference (FD) or finite element (FE) based model where velocity needs to be interpolated from the nodal values. First using AEM, analytic elements are developed to represent pumping well, constant head boundary, no flow boundary, river, recharge etc. Thereafter, using AEM the flow processes of a hypothetical confined aquifer has been simulated. It is assumed that, there is an instantaneous injection of certain pollutant mass in the aquifer. The contaminant mass is represented by a large number of particles and solute transport process have been simulated by random walk particle tracking technique. The RWPT moves each particle by using the velocity field obtained from the AEM based flow simulation model to simulate advection and adds a random displacement to simulate dispersion. RWPT solves the transport equation directly and thus it is virtually free from numerical dispersion and artificial oscillations. Keywords: Analytic element method (AEM), Random walk particle tracking (RWPT). 1.

INTRODUCTION:

To simulate groundwater flow and solute transport processes, Eulerian numerical techniques like finite difference (FD) or finite element (FE) are the most widely used techniques ( Singh, 2014). However, some shortcomings have been noticed while simulating groundwater and solute transport processes by using FD/FE based numerical techniques. For example, using FD/FE it is difficult to obtain accurate head distribution near pumping wells. Also, there are difficulties to represent stream or river using FD/FE. As an alternative, the analytic element method (AEM) developed by Otto Strack in late seventies offers some advantages over FD/FE based numerical techniques. Using AEM it is possible to represent pumping well locations by their precise coordinates (Matott et al., 2006). Also AEM generates continuous head or velocity throughout the domain and give a more accurate water budget for the region. Using AEM, it is possible to model a large geographic area at high resolution with less computational time compared to FD/FE based technique (Bandilla et al., 2007). Eulerian transport models also often plagued by artificial oscillations or numerical dispersion, especially for advection dominated problems. To deal these problems a higher grid resolution and smaller time steps often necessary, resulting in long computational times (Liu et al., 2004, Salamon et al., 2006 a, b). Alternative to simulate solute transport processes we can use the Lagrangian approach. In particular the random walk particle tracking (RWPT) treats the solute mass via a large number of particles. It moves each particle through the porous medium using the velocity field obtained from the solution of the flow equation to simulate advection and adds a random displacement to simulate dispersion. This approach avoids solving the transport equation directly and therefore is virtually free from numerical dispersion and artificial oscillations. In RWPT, global mass conservation is automatically satisfied as particles are used to represent pollutant mass (Salamon et al., 2006 a, b). Also, it has been observed that the FD/FE based flow simulation model calculates groundwater head/velocity values only at the nodal points. To compute velocity at the particle’s position during particle movement, velocity needs to be interpolated from its nodal values in FD/FDM based model. But AEM based flow simulation model generates continuous head and velocity over the entire aquifer domain. So no velocity interpolation is required for particle tracking process if AEM is used to simulate groundwater flow processes. So AEM can be a better choice than FDM/FEM for particle tracking process. In the present study, analytic element method (AEM) is used to simulate groundwater flow process for a confined aquifer. Thereafter, using velocity information, the transport processes have been simulated using random walk particle tracking (RWPT). In this paper, coupled AEM-RWPT model is used for the first time to simulate solute transport process. 2. METHODOLOGY: Two important assumptions have been made to simplify the groundwater flow equation. (a) It is assumed that groundwater flow is horizontal and flow velocity does not vary with aquifer depth and is also known as Dupuit-Forchheimer assumption. (b) Flow in the aquifer is in steady state. Thus, there are negligible/no changes in aquifer storage with the time. Aquifer with a continuous pumping policy for a long period can be assumed as under steady state condition (Landmeyer, 1994). Considering the above two assumptions, the simplified governing equation for regional steady groundwater flow can be written as (Haitjema, 1995).

1

E-proceedings of the 36th IAHR World Congress, the Netherlands 28 June – 3 July, 2015, The Hague, the Netherlands

(k xx ) ( k yy ) N x x y y Where, kxx , k yy

[1]

are values of hydraulic conductivity along x, y axes (L/T); is the potentiometric head (L);

N is the

volumetric flux per unit volume representing sources and/or sinks of water, with N 0 in case of pumping, and N 0 in case of recharge (T-1). The governing equation of solute transport for a conservative solute in porous media can be expressed as (Bear, 1979) C [2] .( DC ) .(vC ) t Where, C is the solute concentration (ML-3); v is the pore water velocity vector (M/L); t is the time (T); D is the hydrodynamic dispersion coefficient tensor (L2T-1). The groundwater flow and transport equation is related by Darcy’s equation (Bear, 1979) as [3] v k x The dispersion tensor ( D ) can be expressed as (Bear, 1979), vi v j Dij T | v | ij ( L T ) Dijm [4] |v|

1 ij 0

for i for i

j j

[5]

ij is the Kronecker delta; L is the longitudinal dispersion symbol; T is the transverse dispersion length; vi is the

component of mean pore velocity in the ith direction; Dijm is the tensor of molecular diffusion coefficient. Eq. [1] has been solved using analytic element method (AEM). Thereafter, using the flow velocity obtained by AEM, Eq. [2] has been solved random walk particle tracking. The analytic element method and the random walk particle tracking technique have been briefly described below. 2.1

Analytic element method (AEM):

Analytic element method (AEM) was developed by Otto Strack at the University of Minnesota (Strack and Haitjema, 1981a, 1981b). It is an alternative numerical method to simulate groundwater flow processes, which is based upon superposition of the exact analytical solution of hydrogeological features like wells, river, drain, lake, infiltration, inhomogeneity domain etc.

Figure 1. Permeability and discharge potential inside and outside of a rectangular domain .

Let us consider an aquifer domain with M numbers of head specified line sinks, N numbers of line doublets to represent inhomogeneity, P numbers of wells and a rectangular areal sink to represent infiltration. So the unknowns are M number of line sink strength, N number of jump in discharge potential across the doublet and an aquifer constant. Thus there is M+ N+ 1 unknown. At the control point of head specified line sink (generally, middle point) and at the reference point head value is known. Using the known head value, discharge potential ( ) is calculated using Eq. [6a, 6b] 1 [6a] kH kH 2 if H 2 1 [6b] k 2 if H 2 Thereafter M 1 number equations are generated at the middle point of head-specified line sink and at the reference point, by equating the discharge potential with the contribution from all the analytic elements present in the aquifer domain. M

N

P

j 1

j 1

j 1

i j , LS j , LD Areal _ Sink j ,Well C

where, i 1, 2,.....M 1

[7]

2

E-proceedings of the 36th IAHR World Congress 28 June – 3 July, 2015, The Hague, the Netherlands

Where, j , LS is the discharge potential contribution from j th Line sink; j , LD is the discharge potential contribution from j th Line doublet; Areal _ Sink is the discharge potential contribution from areal sink ; j ,Well is the discharge potential contribution from jth well and C is the aquifer constant. Another set of equations have been generated across the boundary of inhomogeneity domain by satisfying continuity of pressure and continuity of flow (Strack, 1989). As shown in Figure.1, if permeability and discharge potential inside and outside the rectangular domain is ,k and ,k respectively, then the relationship which satisfies the continuity of pressure can be written as (Strack, 1989), [8a] k k Using Eq. (4a) Jump in the potential across the boundary of inhomogeneity can be expressed as, k k k sn Where, n 1, 2,3......N [8b] k k Eq. [8] is used to generate N number of equation across the N doublets. So there are, M+ N+1 equations for M+ N+ 1 unknowns. Solution of these M+N+1 equations gives line sink strength at the control points, the aquifer constant and jump in potential across the doublets. The mathematics of the analytic elements is not presented here. Interested readers can find, detailed derivations of the analytic elements (i.e., well, Line sink, polygonal areal sink, line doublet, higher order analytic elements etc.) in Strack and Haitjema (1981 a, b), Strack (1989), Haitjema (1995), Fitts (1985), Janković (1997) and, Janković & Barnes (1999). 2.2

Random walk particle tracking (RWPT):

The random walk particle tracking technique solves the transport equation by utilizing the similarity of it with the Langevin equation (Risken, 1996).The RWPT uses particles to represent pollutant mass. It moves each particle through the porous medium using the velocity field obtained from the solution of the flow equation to simulate advection and adds a random displacement to simulate dispersion. The particle’s position in an unsteady non-uniform can be expressed as (Srinivasan et al. (2010)) X (t t ) X (t ) ( A [ X (t )] t ) ( B [ X (t )] (t ) t )

[9]

Here, X (t ) X 1 (t ), X 2 (t ) is the particle location at time t; t is the time step, A is the drift term and B is a second order T

tensor defining the strength of dispersion and [1, 2 ]T is Gaussian white noise with zero mean and unit variance. The drift vector ‘A’ and the second order tensor ‘B’ are related to the coefficient in Eq. (2) by Kinzelbach (1986, 1988) [10] A u D 1 T [11] D BB 2 uy ux 2( L | u | Dm ) 2(T | u | Dm ) | u | | u | B [12] uy ux u D u D 2( | | ) 2( | | ) L m T m |u| | u | Where, u is the velocity vector (L/T); u x and u y are the component of velocity vector u in the principal direction (L/T);

L is the longitudinal dispersion length (L); T is the transverse dispersion length (L). After moving the particles using RWPT, the whole domain is discretized on a two dimensional rectangular grid. If N number of particles are used to represent a pollutant mass of ‘ M ’ amount, the concentration c ( x , y , t t ) can be obtained by the counting number of particles over the region x, y at the time t t as (Kinzelbach, 1990)

c( x, y, t t ) (

nM ) N x y H n p

[13]

Where, c ( x , y , t t ) is the particles concentration at the mid-point ( x, y ) of the rectangular grid; n is the number of particles over the rectangular grid; x y is the area of the rectangular grid, H is the thickness and n p is the porosity of the medium. Besides the above-mentioned computational efficiency and the absence of numerical dispersion, random walk particle transport offers some further advantages. Based on the simplicity of the explicit equations it can be easily implanted over any type of flow model and due to the use of particles as discrete mass parcels, global mass conservation is automatically satisfied. 3. MODELING PROCEDURE (AEM-RWPT): Using the AEM and RWPT model discussed above, a coupled model called AEM-RWPT has been developed. The modeling procedure includes the following steps. 3

E-proceedings of the 36th IAHR World Congress, the Netherlands 28 June – 3 July, 2015, The Hague, the Netherlands

(i)

Define aquifer properties (base elevation, aquifer thickness, hydraulic conductivity, porosity, groundwater recharge, reference point location and reference point head etc.)

(ii)

Define well locations and associated pumping rates, line sink to represent constant head boundary and line doublet to represent no flow boundary.

(iii)

Generate equations at the mid points of line sink and at the reference point using Eq. [7].

(iv)

Represent no flow boundary using rectangular polygon of hydraulic conductivity of zero. Place one line doublet for each side of rectangular polygon.

(v)

Generate equations across the line doublets using Eq. [8].

(vi)

Solve the sets of equations and obtain line sink strength and jumps in potential across the line doublets.

(vii)

Compute discharge potential using Eq. [7] and determine head by inverting Eq. [6].

(viii)

Compute the Darcy velocity at the particles position’s using Eq. [3]. Thereafter compute pore water velocity, by dividing Darcy velocity by the porosity of the medium.

(ix)

Introduce an instantaneous tracer injection of mass ‘ M ’ amount. Represent the pollutant mass via large number of particles say N .

(x)

Move each of the particle using Eq.[9] for a certain duration using small time step.

(xi)

Compute concentration distribution using Eq.[13]. The AEM-RWPT model have been tested and validated by solving a two dimensional homogeneous aquifer problem taken from Park et al. (2008) and comparing its result with the analytical solution.

4. RESULT AND DISCUSSION: The AEM-RWPT method described above is applied to a two dimensional hypothetical confined aquifer (1000 m× 1000 m) as shown in Figure. 2. All the aquifer properties are listed in Table 1. First, the flow process of the hypothetical confined aquifer have been simulated by using AEM. Continuous velocity field obtained using AEM is compared against numerical path line code MODPATH (Figure.3) and found to be in good agreement.

Figure 2. A hypothetical confined aquifer

Table 1. Aquifer properties Aquifer properties

Values

Hydraulic conductivity (K) Porosity (n)

5 m/day 0.3

Aquifer Thickness(H) Horizontal anisotropy

25 m 1

Mass of the pollutant source

1 kg

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E-proceedings of the 36th IAHR World Congress 28 June – 3 July, 2015, The Hague, the Netherlands

Longitudinal dispersivity

40 m

Transverse dispersivity Effective molecular diffusion coefficient

20 m 0

W1 (Pumping)

5000 m /day

W2 (Pumping)

5000 m /day

3 3

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E-proceedings of the 36th IAHR World Congress, the Netherlands 28 June – 3 July, 2015, The Hague, the Netherlands

Figure.3. Path lines obtained using (a) AEM-Forward particle tracking (b) MODFLOW-MODPATH

Thereafter, both the pumping wells have been removed and the hypothetical problem have been simulated again using AEM. It is assumed that, an impulse injection of 1 kg of some pollutant have occurred over a rectangular area (50 m×50 m) at time t=0. The pollutant mass have been represented by particles and solute transport processes have been simulated using RWPT for a period of ten years. It is assumed that the time step is 1 day. By considering only 100 particles, AEM-RWPT model has been simulated and particle’s path line is shown in Figure.4. Assuming a circular observation point of radius (50 m) (Figure.2), breakthrough curves have been generated for different number of particles (Figure.5). It has been observed that for less number of particles, there is random fluctuations in computed concentration. However, the breakthrough curves improves with an increase in particle numbers. Also contour plot of contaminants at various time period is shown in Figure.6. From the Figure 6, it has been observed that, the tracer is moving from left to right with increase in time, and also tracer is expanding with time.

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E-proceedings of the 36th IAHR World Congress 28 June – 3 July, 2015, The Hague, the Netherlands

Fig.ure 4. Particles tracked for a period of ten years.

Figure 5. Breakthrough curves (a) Number of particles (1000) (b) Number of particles (10000)

Figure 6. Contour of contaminants in mg/L at various times due to an impulse tracer injection

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E-proceedings of the 36th IAHR World Congress, the Netherlands 28 June – 3 July, 2015, The Hague, the Netherlands

5.

CONCLUSION

In the present study, flow and transport processes of a hypothetical confined aquifer have been simulated by coupling analytic element method with random walk particle tracking. The proposed methodology (AEM-RWPT) has been applied hypothetical confined aquifer problem and performance is found to be satisfactory. One of the main advantage of AEM is that, it generates continuous head and velocity, which makes it very suitable for particle tracking method. As, AEM-RWPT model uses continuous velocity information, so there is no need to do tedious velocity interpolation and mass conservation unlike finite difference (FD) or finite element (FE) based model. FD or FE based models also often plagued by artificial oscillations or numerical dispersion, especially for advection dominated problems. But the proposed AEM-RWPT model eliminates artificial oscillation or numerical dispersion error. However, there are some limitations of both AEM and RWPT. To simulate transient flow process using AEM is still a challenge. And also for highly heterogeneous media and for anisotropic dispersion very large number of particles may require to reduce random fluctuations of commutated concentration. That may make the model computationally expensive. However by using projection function the computational time can be reduced significantly. References: Bandilla, K. W., Janković, I., & Rabideau, A. J. (2007). A new algorithm for analytic element modeling of large-scale groundwater flow. Advances in water resources, 30(3), 446-454 Bear, J. (1979). Groundwater hydraulics. McGraw, New York. Fitts, C. R. (1985). Modeling aquifier inhomogeneties with analytic elements (Master's thesis) Haitjema, H. M. (1995). Analytic element modeling of groundwater flow. Academic Press. Jacob, B. (1979). Hydraulics of groundwater. MacGraw-Hill, New-york, 567p. Janković, I. (1997) High-Order Analytic Elements in Modeling Groundwater Flow (Ph.D. Thesis) Janković, I., & Barnes, R. (1999). High-order line elements in modeling two-dimensional groundwater flow. Journal of Hydrology, 226(3), 211-223. Kinzelbach, W. (1986). Groundwater modelling: an introduction with sample programs in BASIC. Elsevier. Kinzelbach, W. (1988). The random walk method in pollutant transport simulation. In Groundwater flow and quality modelling (pp. 227-245). Springer Netherlands. Kinzelbach, W. (1990). Simulation of pollutant transport in groundwater with the random walk method. Groundwater Monitoring and Management, (173). Landmeyer, J. E. (1994). Description and application of capture zone delineation for a wellfield at Hilton Head Island, South Carolina. US Department of the Interior, US Geological Survey. Liu, G., Zheng, C., Gorelick, S.M., 2004. Limits of applicability of the advection–dispersion model in aquifers containing connected high-conductivity channels. Water Resources Research 40, W08308.doi:10.1029/2003 WR002735. Matott, L. S., Rabideau, A. J., & Craig, J. R. (2006). Pump-and-treat optimization using analytic element method flow models. Advances in Water Resources, 29(5), 760-775. Park, C. H., Beyer, C., Bauer, S., & Kolditz, O. (2008). An efficient method of random walk particle tracking: accuracy and resolution. Geosciences Journal,12(3), 285-297. Salamon, P., Fernàndez-Garcia, D., & Gómez-Hernández, J. J. (2006). A review and numerical assessment of the random walk particle tracking method. Journal of contaminant hydrology, 87(3), 277-305.

Salamon, P., Gómez-Hernández, J. J., & Fernandez-Garcia, D. (2006). On modeling contaminant transport in complex porous media using random walk particle tracking (Doctoral dissertation, Ph. D. Thesis, Instituto de Ingenieria del Agua y Medio Ambiente Universidad Politecnica de Valencia).

Srinivasan, G., Tartakovsky, D. M., Dentz, M., Viswanathan, H., Berkowitz, B., & Robinson, B. A. (2010). Random walk particle tracking simulations of non-Fickian transport in heterogeneous media. Journal of Computational Physics,229 (11), 4304-4314. Risken, H. (1985). The Fokker-Planck equation: methods of solution and applications. Springer-Vlg. Singh, A. (2014). Groundwater resources management through the applications of simulation modeling: A review. Science of The Total Environment, 499, 414-423. Strack, O. D., & Haitjema, H. M. (1981 a). Modeling double aquifer flow using a comprehensive potential and distributed singularities: 1. Solution for homogeneous permeability. Water Resources Research, 17(5), 1535-1549. Strack, O. D. L., & Haitjema, H. M. (1981 b). Modeling double aquifer flow using a comprehensive potential and distributed singularities: 2. Solution for inhomogeneous permeabilities. Water Resources Research, 17(5), 1551-1560. Strack, O. D. (1989). Groundwater mechanics. Prentice Hall. University of Minnesota.

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E-proceedings of the 36th IAHR World Congress 28 June – 3 July, 2015, The Hague, the Netherlands

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E-proceedings of the 36th IAHR World Congress, the Netherlands 28 June – 3 July, 2015, The Hague, the Netherlands

10

GROUNDWATER FLOW AND TRANSPORT SIMULATION FOR A CONFINED AQUIFER USING ANALYTIC ELEMENT METHOD AND RANDOM WALK PARTICLE TRACKING. 1

PARTHA MAJUMDER , T.I. ELDHO 1

2

Research Scholar, Department of Civil Engineering, IIT Bombay, [email protected] 2 Professor, Department of Civil Engineering, IIT Bombay, [email protected]

ABSTRACT In this paper, the groundwater flow and transport processes in a confined aquifer have been simulated using analytic element method (AEM) and random walk particle tracking (RWPT). The implementation of coupled AEM-RWPT to simulate solute transport processes is the first time in the field of groundwater flow and transport process. The AEM solves the groundwater flow equation by superposition of the analytic solution of simple hydrogeological features like, well, river, lake, recharge, etc. One of the main advantages of AEM is that, it generates continuous head and velocity, thus very suitable for particle tracking process, unlike finite difference (FD) or finite element (FE) based model where velocity needs to be interpolated from the nodal values. First using AEM, analytic elements are developed to represent pumping well, constant head boundary, no flow boundary, river, recharge etc. Thereafter, using AEM the flow processes of a hypothetical confined aquifer has been simulated. It is assumed that, there is an instantaneous injection of certain pollutant mass in the aquifer. The contaminant mass is represented by a large number of particles and solute transport process have been simulated by random walk particle tracking technique. The RWPT moves each particle by using the velocity field obtained from the AEM based flow simulation model to simulate advection and adds a random displacement to simulate dispersion. RWPT solves the transport equation directly and thus it is virtually free from numerical dispersion and artificial oscillations. Keywords: Analytic element method (AEM), Random walk particle tracking (RWPT). 1.

INTRODUCTION:

To simulate groundwater flow and solute transport processes, Eulerian numerical techniques like finite difference (FD) or finite element (FE) are the most widely used techniques ( Singh, 2014). However, some shortcomings have been noticed while simulating groundwater and solute transport processes by using FD/FE based numerical techniques. For example, using FD/FE it is difficult to obtain accurate head distribution near pumping wells. Also, there are difficulties to represent stream or river using FD/FE. As an alternative, the analytic element method (AEM) developed by Otto Strack in late seventies offers some advantages over FD/FE based numerical techniques. Using AEM it is possible to represent pumping well locations by their precise coordinates (Matott et al., 2006). Also AEM generates continuous head or velocity throughout the domain and give a more accurate water budget for the region. Using AEM, it is possible to model a large geographic area at high resolution with less computational time compared to FD/FE based technique (Bandilla et al., 2007). Eulerian transport models also often plagued by artificial oscillations or numerical dispersion, especially for advection dominated problems. To deal these problems a higher grid resolution and smaller time steps often necessary, resulting in long computational times (Liu et al., 2004, Salamon et al., 2006 a, b). Alternative to simulate solute transport processes we can use the Lagrangian approach. In particular the random walk particle tracking (RWPT) treats the solute mass via a large number of particles. It moves each particle through the porous medium using the velocity field obtained from the solution of the flow equation to simulate advection and adds a random displacement to simulate dispersion. This approach avoids solving the transport equation directly and therefore is virtually free from numerical dispersion and artificial oscillations. In RWPT, global mass conservation is automatically satisfied as particles are used to represent pollutant mass (Salamon et al., 2006 a, b). Also, it has been observed that the FD/FE based flow simulation model calculates groundwater head/velocity values only at the nodal points. To compute velocity at the particle’s position during particle movement, velocity needs to be interpolated from its nodal values in FD/FDM based model. But AEM based flow simulation model generates continuous head and velocity over the entire aquifer domain. So no velocity interpolation is required for particle tracking process if AEM is used to simulate groundwater flow processes. So AEM can be a better choice than FDM/FEM for particle tracking process. In the present study, analytic element method (AEM) is used to simulate groundwater flow process for a confined aquifer. Thereafter, using velocity information, the transport processes have been simulated using random walk particle tracking (RWPT). In this paper, coupled AEM-RWPT model is used for the first time to simulate solute transport process. 2. METHODOLOGY: Two important assumptions have been made to simplify the groundwater flow equation. (a) It is assumed that groundwater flow is horizontal and flow velocity does not vary with aquifer depth and is also known as Dupuit-Forchheimer assumption. (b) Flow in the aquifer is in steady state. Thus, there are negligible/no changes in aquifer storage with the time. Aquifer with a continuous pumping policy for a long period can be assumed as under steady state condition (Landmeyer, 1994). Considering the above two assumptions, the simplified governing equation for regional steady groundwater flow can be written as (Haitjema, 1995).

1

E-proceedings of the 36th IAHR World Congress, the Netherlands 28 June – 3 July, 2015, The Hague, the Netherlands

(k xx ) ( k yy ) N x x y y Where, kxx , k yy

[1]

are values of hydraulic conductivity along x, y axes (L/T); is the potentiometric head (L);

N is the

volumetric flux per unit volume representing sources and/or sinks of water, with N 0 in case of pumping, and N 0 in case of recharge (T-1). The governing equation of solute transport for a conservative solute in porous media can be expressed as (Bear, 1979) C [2] .( DC ) .(vC ) t Where, C is the solute concentration (ML-3); v is the pore water velocity vector (M/L); t is the time (T); D is the hydrodynamic dispersion coefficient tensor (L2T-1). The groundwater flow and transport equation is related by Darcy’s equation (Bear, 1979) as [3] v k x The dispersion tensor ( D ) can be expressed as (Bear, 1979), vi v j Dij T | v | ij ( L T ) Dijm [4] |v|

1 ij 0

for i for i

j j

[5]

ij is the Kronecker delta; L is the longitudinal dispersion symbol; T is the transverse dispersion length; vi is the

component of mean pore velocity in the ith direction; Dijm is the tensor of molecular diffusion coefficient. Eq. [1] has been solved using analytic element method (AEM). Thereafter, using the flow velocity obtained by AEM, Eq. [2] has been solved random walk particle tracking. The analytic element method and the random walk particle tracking technique have been briefly described below. 2.1

Analytic element method (AEM):

Analytic element method (AEM) was developed by Otto Strack at the University of Minnesota (Strack and Haitjema, 1981a, 1981b). It is an alternative numerical method to simulate groundwater flow processes, which is based upon superposition of the exact analytical solution of hydrogeological features like wells, river, drain, lake, infiltration, inhomogeneity domain etc.

Figure 1. Permeability and discharge potential inside and outside of a rectangular domain .

Let us consider an aquifer domain with M numbers of head specified line sinks, N numbers of line doublets to represent inhomogeneity, P numbers of wells and a rectangular areal sink to represent infiltration. So the unknowns are M number of line sink strength, N number of jump in discharge potential across the doublet and an aquifer constant. Thus there is M+ N+ 1 unknown. At the control point of head specified line sink (generally, middle point) and at the reference point head value is known. Using the known head value, discharge potential ( ) is calculated using Eq. [6a, 6b] 1 [6a] kH kH 2 if H 2 1 [6b] k 2 if H 2 Thereafter M 1 number equations are generated at the middle point of head-specified line sink and at the reference point, by equating the discharge potential with the contribution from all the analytic elements present in the aquifer domain. M

N

P

j 1

j 1

j 1

i j , LS j , LD Areal _ Sink j ,Well C

where, i 1, 2,.....M 1

[7]

2

E-proceedings of the 36th IAHR World Congress 28 June – 3 July, 2015, The Hague, the Netherlands

Where, j , LS is the discharge potential contribution from j th Line sink; j , LD is the discharge potential contribution from j th Line doublet; Areal _ Sink is the discharge potential contribution from areal sink ; j ,Well is the discharge potential contribution from jth well and C is the aquifer constant. Another set of equations have been generated across the boundary of inhomogeneity domain by satisfying continuity of pressure and continuity of flow (Strack, 1989). As shown in Figure.1, if permeability and discharge potential inside and outside the rectangular domain is ,k and ,k respectively, then the relationship which satisfies the continuity of pressure can be written as (Strack, 1989), [8a] k k Using Eq. (4a) Jump in the potential across the boundary of inhomogeneity can be expressed as, k k k sn Where, n 1, 2,3......N [8b] k k Eq. [8] is used to generate N number of equation across the N doublets. So there are, M+ N+1 equations for M+ N+ 1 unknowns. Solution of these M+N+1 equations gives line sink strength at the control points, the aquifer constant and jump in potential across the doublets. The mathematics of the analytic elements is not presented here. Interested readers can find, detailed derivations of the analytic elements (i.e., well, Line sink, polygonal areal sink, line doublet, higher order analytic elements etc.) in Strack and Haitjema (1981 a, b), Strack (1989), Haitjema (1995), Fitts (1985), Janković (1997) and, Janković & Barnes (1999). 2.2

Random walk particle tracking (RWPT):

The random walk particle tracking technique solves the transport equation by utilizing the similarity of it with the Langevin equation (Risken, 1996).The RWPT uses particles to represent pollutant mass. It moves each particle through the porous medium using the velocity field obtained from the solution of the flow equation to simulate advection and adds a random displacement to simulate dispersion. The particle’s position in an unsteady non-uniform can be expressed as (Srinivasan et al. (2010)) X (t t ) X (t ) ( A [ X (t )] t ) ( B [ X (t )] (t ) t )

[9]

Here, X (t ) X 1 (t ), X 2 (t ) is the particle location at time t; t is the time step, A is the drift term and B is a second order T

tensor defining the strength of dispersion and [1, 2 ]T is Gaussian white noise with zero mean and unit variance. The drift vector ‘A’ and the second order tensor ‘B’ are related to the coefficient in Eq. (2) by Kinzelbach (1986, 1988) [10] A u D 1 T [11] D BB 2 uy ux 2( L | u | Dm ) 2(T | u | Dm ) | u | | u | B [12] uy ux u D u D 2( | | ) 2( | | ) L m T m |u| | u | Where, u is the velocity vector (L/T); u x and u y are the component of velocity vector u in the principal direction (L/T);

L is the longitudinal dispersion length (L); T is the transverse dispersion length (L). After moving the particles using RWPT, the whole domain is discretized on a two dimensional rectangular grid. If N number of particles are used to represent a pollutant mass of ‘ M ’ amount, the concentration c ( x , y , t t ) can be obtained by the counting number of particles over the region x, y at the time t t as (Kinzelbach, 1990)

c( x, y, t t ) (

nM ) N x y H n p

[13]

Where, c ( x , y , t t ) is the particles concentration at the mid-point ( x, y ) of the rectangular grid; n is the number of particles over the rectangular grid; x y is the area of the rectangular grid, H is the thickness and n p is the porosity of the medium. Besides the above-mentioned computational efficiency and the absence of numerical dispersion, random walk particle transport offers some further advantages. Based on the simplicity of the explicit equations it can be easily implanted over any type of flow model and due to the use of particles as discrete mass parcels, global mass conservation is automatically satisfied. 3. MODELING PROCEDURE (AEM-RWPT): Using the AEM and RWPT model discussed above, a coupled model called AEM-RWPT has been developed. The modeling procedure includes the following steps. 3

E-proceedings of the 36th IAHR World Congress, the Netherlands 28 June – 3 July, 2015, The Hague, the Netherlands

(i)

Define aquifer properties (base elevation, aquifer thickness, hydraulic conductivity, porosity, groundwater recharge, reference point location and reference point head etc.)

(ii)

Define well locations and associated pumping rates, line sink to represent constant head boundary and line doublet to represent no flow boundary.

(iii)

Generate equations at the mid points of line sink and at the reference point using Eq. [7].

(iv)

Represent no flow boundary using rectangular polygon of hydraulic conductivity of zero. Place one line doublet for each side of rectangular polygon.

(v)

Generate equations across the line doublets using Eq. [8].

(vi)

Solve the sets of equations and obtain line sink strength and jumps in potential across the line doublets.

(vii)

Compute discharge potential using Eq. [7] and determine head by inverting Eq. [6].

(viii)

Compute the Darcy velocity at the particles position’s using Eq. [3]. Thereafter compute pore water velocity, by dividing Darcy velocity by the porosity of the medium.

(ix)

Introduce an instantaneous tracer injection of mass ‘ M ’ amount. Represent the pollutant mass via large number of particles say N .

(x)

Move each of the particle using Eq.[9] for a certain duration using small time step.

(xi)

Compute concentration distribution using Eq.[13]. The AEM-RWPT model have been tested and validated by solving a two dimensional homogeneous aquifer problem taken from Park et al. (2008) and comparing its result with the analytical solution.

4. RESULT AND DISCUSSION: The AEM-RWPT method described above is applied to a two dimensional hypothetical confined aquifer (1000 m× 1000 m) as shown in Figure. 2. All the aquifer properties are listed in Table 1. First, the flow process of the hypothetical confined aquifer have been simulated by using AEM. Continuous velocity field obtained using AEM is compared against numerical path line code MODPATH (Figure.3) and found to be in good agreement.

Figure 2. A hypothetical confined aquifer

Table 1. Aquifer properties Aquifer properties

Values

Hydraulic conductivity (K) Porosity (n)

5 m/day 0.3

Aquifer Thickness(H) Horizontal anisotropy

25 m 1

Mass of the pollutant source

1 kg

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E-proceedings of the 36th IAHR World Congress 28 June – 3 July, 2015, The Hague, the Netherlands

Longitudinal dispersivity

40 m

Transverse dispersivity Effective molecular diffusion coefficient

20 m 0

W1 (Pumping)

5000 m /day

W2 (Pumping)

5000 m /day

3 3

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E-proceedings of the 36th IAHR World Congress, the Netherlands 28 June – 3 July, 2015, The Hague, the Netherlands

Figure.3. Path lines obtained using (a) AEM-Forward particle tracking (b) MODFLOW-MODPATH

Thereafter, both the pumping wells have been removed and the hypothetical problem have been simulated again using AEM. It is assumed that, an impulse injection of 1 kg of some pollutant have occurred over a rectangular area (50 m×50 m) at time t=0. The pollutant mass have been represented by particles and solute transport processes have been simulated using RWPT for a period of ten years. It is assumed that the time step is 1 day. By considering only 100 particles, AEM-RWPT model has been simulated and particle’s path line is shown in Figure.4. Assuming a circular observation point of radius (50 m) (Figure.2), breakthrough curves have been generated for different number of particles (Figure.5). It has been observed that for less number of particles, there is random fluctuations in computed concentration. However, the breakthrough curves improves with an increase in particle numbers. Also contour plot of contaminants at various time period is shown in Figure.6. From the Figure 6, it has been observed that, the tracer is moving from left to right with increase in time, and also tracer is expanding with time.

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E-proceedings of the 36th IAHR World Congress 28 June – 3 July, 2015, The Hague, the Netherlands

Fig.ure 4. Particles tracked for a period of ten years.

Figure 5. Breakthrough curves (a) Number of particles (1000) (b) Number of particles (10000)

Figure 6. Contour of contaminants in mg/L at various times due to an impulse tracer injection

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E-proceedings of the 36th IAHR World Congress, the Netherlands 28 June – 3 July, 2015, The Hague, the Netherlands

5.

CONCLUSION

In the present study, flow and transport processes of a hypothetical confined aquifer have been simulated by coupling analytic element method with random walk particle tracking. The proposed methodology (AEM-RWPT) has been applied hypothetical confined aquifer problem and performance is found to be satisfactory. One of the main advantage of AEM is that, it generates continuous head and velocity, which makes it very suitable for particle tracking method. As, AEM-RWPT model uses continuous velocity information, so there is no need to do tedious velocity interpolation and mass conservation unlike finite difference (FD) or finite element (FE) based model. FD or FE based models also often plagued by artificial oscillations or numerical dispersion, especially for advection dominated problems. But the proposed AEM-RWPT model eliminates artificial oscillation or numerical dispersion error. However, there are some limitations of both AEM and RWPT. To simulate transient flow process using AEM is still a challenge. And also for highly heterogeneous media and for anisotropic dispersion very large number of particles may require to reduce random fluctuations of commutated concentration. That may make the model computationally expensive. However by using projection function the computational time can be reduced significantly. References: Bandilla, K. W., Janković, I., & Rabideau, A. J. (2007). A new algorithm for analytic element modeling of large-scale groundwater flow. Advances in water resources, 30(3), 446-454 Bear, J. (1979). Groundwater hydraulics. McGraw, New York. Fitts, C. R. (1985). Modeling aquifier inhomogeneties with analytic elements (Master's thesis) Haitjema, H. M. (1995). Analytic element modeling of groundwater flow. Academic Press. Jacob, B. (1979). Hydraulics of groundwater. MacGraw-Hill, New-york, 567p. Janković, I. (1997) High-Order Analytic Elements in Modeling Groundwater Flow (Ph.D. Thesis) Janković, I., & Barnes, R. (1999). High-order line elements in modeling two-dimensional groundwater flow. Journal of Hydrology, 226(3), 211-223. Kinzelbach, W. (1986). Groundwater modelling: an introduction with sample programs in BASIC. Elsevier. Kinzelbach, W. (1988). The random walk method in pollutant transport simulation. In Groundwater flow and quality modelling (pp. 227-245). Springer Netherlands. Kinzelbach, W. (1990). Simulation of pollutant transport in groundwater with the random walk method. Groundwater Monitoring and Management, (173). Landmeyer, J. E. (1994). Description and application of capture zone delineation for a wellfield at Hilton Head Island, South Carolina. US Department of the Interior, US Geological Survey. Liu, G., Zheng, C., Gorelick, S.M., 2004. Limits of applicability of the advection–dispersion model in aquifers containing connected high-conductivity channels. Water Resources Research 40, W08308.doi:10.1029/2003 WR002735. Matott, L. S., Rabideau, A. J., & Craig, J. R. (2006). Pump-and-treat optimization using analytic element method flow models. Advances in Water Resources, 29(5), 760-775. Park, C. H., Beyer, C., Bauer, S., & Kolditz, O. (2008). An efficient method of random walk particle tracking: accuracy and resolution. Geosciences Journal,12(3), 285-297. Salamon, P., Fernàndez-Garcia, D., & Gómez-Hernández, J. J. (2006). A review and numerical assessment of the random walk particle tracking method. Journal of contaminant hydrology, 87(3), 277-305.

Salamon, P., Gómez-Hernández, J. J., & Fernandez-Garcia, D. (2006). On modeling contaminant transport in complex porous media using random walk particle tracking (Doctoral dissertation, Ph. D. Thesis, Instituto de Ingenieria del Agua y Medio Ambiente Universidad Politecnica de Valencia).

Srinivasan, G., Tartakovsky, D. M., Dentz, M., Viswanathan, H., Berkowitz, B., & Robinson, B. A. (2010). Random walk particle tracking simulations of non-Fickian transport in heterogeneous media. Journal of Computational Physics,229 (11), 4304-4314. Risken, H. (1985). The Fokker-Planck equation: methods of solution and applications. Springer-Vlg. Singh, A. (2014). Groundwater resources management through the applications of simulation modeling: A review. Science of The Total Environment, 499, 414-423. Strack, O. D., & Haitjema, H. M. (1981 a). Modeling double aquifer flow using a comprehensive potential and distributed singularities: 1. Solution for homogeneous permeability. Water Resources Research, 17(5), 1535-1549. Strack, O. D. L., & Haitjema, H. M. (1981 b). Modeling double aquifer flow using a comprehensive potential and distributed singularities: 2. Solution for inhomogeneous permeabilities. Water Resources Research, 17(5), 1551-1560. Strack, O. D. (1989). Groundwater mechanics. Prentice Hall. University of Minnesota.

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