- Email: [email protected]
The effect of spatially varying velocity field on the transport of radioactivity in a porous medium
Journal of Environmental Radioactivity 162-163 (2016) 285e288
Contents lists available at ScienceDirect
Journal of Environmental Radioactivity journal homepage: www.elsevier.com/locate/jenvrad
Short communication
The effect of spatially varying velocity field on the transport of radioactivity in a porous medium Soubhadra Sen*, C.V. Srinivas, R. Baskaran, B. Venkatraman Radiological Safety Division, Indira Gandhi Centre for Atomic Research, Kalpakkam, 603 102, India
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 February 2016 Received in revised form 30 May 2016 Accepted 31 May 2016
In the event of an accidental leak of the immobilized nuclear waste from an underground repository, it may come in contact of the flow of underground water and start migrating. Depending on the nature of the geological medium, the flow velocity of water may vary spatially. Here, we report a numerical study on the migration of radioactivity due to a space dependent flow field. For a detailed analysis, seven different types of velocity profiles are considered and the corresponding concentrations are compared. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Radioactive waste Radioactivity migration Spatially varying velocity field Dufort-Frankel scheme
1. Introduction The increasing demand of energy has forced the human civilization to look for a nuclear route of power generation. During the operation of a nuclear reactor, fissile atoms undergo fission causing the generation of fission products along with the release of energy. In the back end of a fuel cycle, the radioactive waste is isolated, then classified according to the activity level (low, intermediate and high) and subsequently disposed off following the safety norms. The low and intermediate category of wastes are immobilized and then stored in near earth surface repositories. On the other hand, high level wastes are vitrified in a glass matrix, then sealed in a tank called canister which is subsequently buried deep inside the earth surface (inside a rocky medium) to isolate the same (Wattal, 2013). In the event of a leak from the underground storage facility, the waste may come in contact with the flow of underground water. In such a situation, the radionuclides would migrate within the geological medium due to the combined effect of advection and diffusion. This is a serious issue related to environmental and radiological safety and thus requires an in-depth study. There are two ways to model the migration mechanism. Fig. 1 shows a schematic picture of a rock where solid porous blocks sit within the network of tiny fractures through which the laminar
* Corresponding author. E-mail addresses: [email protected], [email protected] (S. Sen). http://dx.doi.org/10.1016/j.jenvrad.2016.05.036 0265-931X/© 2016 Elsevier Ltd. All rights reserved.
flow of water occurs. Thus the concentration of the contaminant at a given point (as a function of time) can be estimated following a random walk approach (Williams, 1992; Giacobbo and Patelli, 2008; Sen and Mohankumar, 2013). One the other hand, this problem can also be addressed by considering a neat geometry of a rock (Fig. 2) where an infinite array of identical parallel fractures are separated by porous blocks of same width (Sudicky and Frind, 1982; Chen and Li, 1997; Mohankumar, 2007; Sen and Mohankumar, 2011, 2012, 2014; Sen et al., 2015). In all these works, it is assumed that the flow of water is constant in all the fractures and this in turn requires that the concentration of the dissolved species is same at a given horizontal distance along all the fractures (Fig. 2). This is clearly a very gross assumption to simplify the computation. From Fig. 1, one can easily notice that the geometry of a rock demands a spatial distribution of flow velocity. This point can be illustrated in a more detailed way by Fig. 3 where velocities at the points A, B, C, D, E, F and G cannot be the same. The key concept of this work is to point out that the transport of a dissolved contaminant depends largely on the space dependent flow field of water. To mimic the actual flow, one has to generate the velocity profile very accurately. In a bit simplified approach, we can estimate velocities at some nodal points (like A, B, C, D, E, F and G in Fig. 3) and fit into a polynomial to approximate the flow profile. This fitted profile can be utilized to estimate the transport through a selected path. This approach may raise another important question that the concentration of the dissolved radionuclide should also change from fracture to fracture at a given horizontal distance
286
S. Sen et al. / Journal of Environmental Radioactivity 162-163 (2016) 285e288
Fig. 3. Schematic picture of water flow through a rock medium.
Fig. 1. Schematic picture of a rock.
Fig. 2. Waste matrix and fracture geometry in parallel fracture model.
from the source. To avoid further complicacy, in our present work we stick to the assumption that the concentration remains same at a given horizontal distance in all the fractures. Below we list out the two most important assumptions of this work: The flow velocity of water through a rock has a space dependent profile. The concentration of the dissolved contaminant is same in all the fractures at a given horizontal distance from the source at a given instant of time. In reality, all the transport related parameters (density of fractures, diffusivity, width of the fracture and other parameters) may vary spatially. As a sequel of our earlier works (Sen and
Mohankumar, 2011, 2012, 2013; Sen et al., 2015), we consider the transport to be dominated by advection and so the spatial variation of the diffusion coefficient is ignored. Moreover, for simplification other parameters are assumed to be constant. To highlight the importance of considering a proper field, we consider a basic 1-D model along with some simple space dependent velocity profiles. For a real life problem, we need to take some samples of the given medium and estimate the flow rates at some given spatial locations along a selected path for a given inflow at the source end. Now by fitting into a polynomial, the collected data can be used to generate a space dependent velocity profile along the path. It is important to note that the flow velocity may change from path to path. In such a case, the velocity of water in each path is needed to be estimated and this is not practically possible. In the parallel fracture model, the waste matrix is considered to be infinite which is also not a very correct assumption (Fig. 2). So for a waste matrix of finite size, flow velocities along the dominant paths (at the depth where the canister is buried) along a particular direction are enough for the simulation (for example horizontal direction in Fig. 2). Now along all such paths, we can experimentally measure the velocities at some nodal points and obtain a fitted function for the flow velocity. If the flow related characteristics of the medium (density of fractures, diffusivity, width of the fracture and other parameters) do not change drastically from region to region (a general assumption for a numerical simulation), the flow velocity profile would also remain nearly the same along all the paths. In such a case, an average velocity function can be used for all the paths and one may opt to use the parallel fracture modal to simulate the transport of radioactivity through the medium under consideration in a more accurate way. On the other hand, if the flow rate changes drastically from path to path or region to region, the space dependent velocity function for each path has to be considered separately. In this work, the limitation of the assumption of a constant flow velocity through all the paths is pointed out by considering a few theoretical velocity profiles. The main objective of this study is to estimate the migration of activity in a porous medium under the influence of a spatially varying velocity field. 2. Radioactivity migration in a porous medium We consider a simple one dimensional advection-diffusion equation to model the activity transport as given below.
vC 1 v 1 v vC þ ðvCÞ D þ lC ¼ 0 ; x 0 ; t 0 vt R vx R vx vx
(1)
Here, C(x, t) is the concentration of the active species, v is the flow velocity of water, D is the diffusion coefficient, R is the
S. Sen et al. / Journal of Environmental Radioactivity 162-163 (2016) 285e288
287
Fig. 4. Plot of velocity as a function of distance.
retardation factor and l is the decay constant of the radionuclide. One may note that we are ignoring the variation in the retardation coefficient R. As this is a safety related analysis where an upper bound estimation is important, we set R ¼ 1.0 which corresponds to the maximum transport (Sen et al., 2015). The present problem deals with the effect of variable transport coefficients on the activity migration where the overall mechanism is dominated by advection and diffusion plays a minor role and so the variation in the diffusion coefficient (D) is ignored. With this assumption, Eq. (1) takes the following form.
vC 1 v D v2 C þ ðvCÞ þ lC ¼ 0 ; x 0 ; t 0 vt R vx R vx2
(2)
Now, the point x ¼ 0 separates the source and the geological medium. By assuming a source of constant strength (C0), we write the boundary condition for the source end in the following way.
D
vC þ vCð0; tÞ ¼ vC0 vx x¼0
(3)
On the other hand, concentration must become zero as x tends to infinity. This can be represented mathematically in the following manner.
Cð∞; tÞ ¼ 0 ; t 0
(4)
The initial condition is given by
Cðx; 0Þ ¼ 0 ; x 0
(5)
Eq. (2) has to be solved along with Eqs. (3) and (4) for different flow velocity v. We use a set of space dependent functions for v
which are listed below.
iÞ v ¼ u0 ; iiÞ v ¼ u0 ð1 þ xÞ; iiiÞ v ¼ u0 1 þ x0:5 ; ivÞ v ¼ u0 1 þ x0:25 ; vÞ v ¼ u0 1 þ x0:25 ; viÞ v ¼ u0 1 þ x0:5 ; viiÞ v ¼ u0 1 þ x1
(6)
where u0 is a constant. For the numerical solution, a finite difference technique namely the Dufort-Frankel method has been implemented. The solution of a set of coupled one dimensional transport equations describing the migration of activity in a rock medium using this scheme has been reported in our earlier work (Sen and Mohankumar, 2011). Here, just by ignoring the transport through the perpendicular dimension, the earlier code has been modified and used for the present problem. In the next section, we briefly discuss the Dufort-Frankel scheme. 3. The Dufort-Frankel (DF) scheme This method is an explicit numerical technique which can provide second order accuracy in both the space and the time variables. Let us indicate the space variable by the index i and the time part by j. In the DF scheme, the derivatives are centered around the time jDt. The approximations of the derivatives using this scheme are listed below. j
jþ1
j1
vCi Ci Ci z vt 2Dt
Fig. 5. Plot of concentration (after 50 years) as a function of distance.
(7)
288
S. Sen et al. / Journal of Environmental Radioactivity 162-163 (2016) 285e288
j
j
j vCi Ciþ1 Ci1 z vx 2Dx j
v2 C i vx
z 2
j
jþ1
Ciþ1 Ci
(8) j1
Ci
j
þ Ci1
ðDxÞ2
(9)
It is very important to note that the accuracy of the solution depends critically on the estimation of the interface derivative ½vC=vxx¼0 . Mohankumar and Auerbach (2004) have shown that a 7-point approximation is necessary for an accurate estimation of the said term and we follow the same. 4. Results and discussion In this work, all the calculations are performed for a period of 50 years in double precision. We have used a grid of 0.2 m along with a time step of 0.001 year. All the calculations are re-performed with half of the grid size and half of the time step to ensure convergence. The set of parameters used in the calculations are listed below.
. D ¼ 1:0 m2 year ; R ¼ 1:0; u0 ¼ 1:0ðm=year Þ
(10)
In Fig. 4, we plot the velocity profile as a function of distance. Fig. 5 shows the corresponding concentration profiles (normalized with respect to the source) after a period of 50 years. An important point to be noted is that at the source end, velocity is taken as constant (u0) for all the velocity profiles. It can be easily noted from Fig. 5 that the constant velocity gives the maximum concentration up to a distance beyond which the curve falls rapidly. With the increase in the positive power of x in the velocity function, the concentrations at smaller distances come down systematically. It is also important to note that the contaminant reaches larger distances with this kind of velocity. On the other hand, for an increasing negative index of x in the velocity profile, the concentration increases systematically at the smaller distances and for a decreasing negative power, the contaminant reaches longer distances. So in a nutshell, for increasing positivity of the index of x, the concentration reduces at the lower end but the contaminant reaches longer distances and with decreasing positivity of the same (in other words with increasing negativity), the concentration profile slowly shifts to that of a constant velocity. The nature of the plots can be explained easily. Let us consider a velocity profile of the following kind.
v ¼ u0 ð1 þ xh Þ
(11)
First, we set h to be positive. This means that with the increase of x, the velocity increases. As we have set a constant velocity (u0) at the source end, all the velocity profiles start from a common point and increases with the increasing values of x where the rate of increment depends on h (for higher values of h, the velocity increases in a faster way). These trends are visible in Fig. 4. One can note that if we apply these velocity profiles to Eq. (2), it would drive the contaminant to the larger distances resulting in a decrease in concentration at the smaller distances (Fig. 5). Now, let us consider h to be negative. For this kind of power of x, the velocity comes down systematically to the profile of the constant velocity. With the increasing negativity of the index of x, the curve reaches the constant velocity profile in a faster manner (Fig. 4). So for these velocity functions, with the increasing negativity of the power of x, the concentration curve systematically moves towards the corresponding curve of a constant velocity (Fig. 5). Now if the flow velocity of water in a particular geological medium follows a
particular profile and we try to simulate the migration considering some other profile (for example a space dependent velocity field is modeled with a constant velocity), it would lead to a gross error. Thus by considering a set space dependent velocity profiles, we try to point out the importance of considering a proper velocity field for the simulation of the transport of radioactivity in a porous medium. 5. Conclusion In this work, we have studied the effect of a spatially varying velocity field on the transport of a radioactive contaminant in a geological (rock) medium. The results show that a constant value of velocity causes maximum concentrations at smaller distances where as a spatially varying velocity function results in a decrease in the concentration at the lower distances but higher concentration at larger distances. So it points out the importance of choosing a proper velocity profile. The present study also highlights that a large difference in the simulated concentrations may arise if the spatial variation in the velocity is not considered properly. This point is demonstrated by considering a few simple theoretical velocity profiles. For example when the velocity follows a function like uo(1 þ x) in place of uo, it has been found that the simulated concentrations beyond a distance of 100 m (after 50 years) differ by more than one order. As the final remark, it is important to remember that the simulation of the underground transport of radioactivity is meant to address the safety related aspects in the event of an accidental release of radionuclides from an underground storage facility and so a conservative approach is always advisable. Thus for an upper bound estimation of the concentration of the contaminant at smaller distances (may be enough for a short term storage of wastes in near earth surface facilities), one may opt to use a constant velocity function. On the other hand, if we want to do the same at the larger distances (important for the storage facilities of high level waste), we need to use a proper velocity function. Thus in a nutshell, this study highlights the inadequacy of using a constant flow velocity in realistically representing the real world situation. Also it indicates a simple way to overcome the limitation. References Chen, C.T., Li, S.H., 1997. Radionuclide transport in fractured porous mediaanalytical solutions for a system of parallel fractures with a constant inlet flux. Waste Manage. 17 (1), 53e64. Giacobbo, F., Patelli, E., 2008. Monte Carlo simulation of radionuclide transport through fractured media. Ann. Nucl. Energy 35, 1732e1740. Mohankumar, N., Auerbach, S.M., 2004. On the use of higher-order formula for numerical derivatives in scientific computing. Comput. Phys. Commun. 161, 109e118. Mohankumar, N., 2007. On the numerical solution of radioactivity migration in a porous medium. Ann. Nucl. Energy 34, 222e227. Sudicky, E.A., Frind, E.O., 1982. Contaminant transport in fractured porous media: Analytical solutions for a system of parallel fractures. Water Resour. Res. 18 (6), 1634e1642. Sen, S., Mohankumar, N., 2011. A computational strategy for radioactivity migration in a porous medium. Ann. Nucl. Energy 38, 2470e2474. Sen, S., Mohankumar, N., 2012. A note on higher order numerical schemes for radioactivity migration. Ann. Nucl. Energy 49, 227e231. Sen, S., Mohankumar, N., 2013. A random walk based methodology for the realistic estimation of radioactivity migration in a porous medium. Ann. Nucl. Energy 60, 202e205. Sen, S., Mohankumar, N., 2014. Migpore, a code package for the estimation of migration of radioactive species in a porous medium. Comput. Phys. Commun. 185, 302e306. Sen, S., Srinivas, C.V., Baskaran, R., Venkatraman, B., 2015. Numerical simulation of the transport of a radionuclide chain in a rock medium. J. Environ. Radioact. 141, 115e122. Wattal, P.K., 2013. Indian programme on radioactive waste management. Sadhana 38 (5), 849e857. Williams, M.M.R., 1992. A new model for describing transport of radionuclides through fractured rock. Ann. Nucl. Energy 19, 791e824.
Contents lists available at ScienceDirect
Journal of Environmental Radioactivity journal homepage: www.elsevier.com/locate/jenvrad
Short communication
The effect of spatially varying velocity field on the transport of radioactivity in a porous medium Soubhadra Sen*, C.V. Srinivas, R. Baskaran, B. Venkatraman Radiological Safety Division, Indira Gandhi Centre for Atomic Research, Kalpakkam, 603 102, India
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 February 2016 Received in revised form 30 May 2016 Accepted 31 May 2016
In the event of an accidental leak of the immobilized nuclear waste from an underground repository, it may come in contact of the flow of underground water and start migrating. Depending on the nature of the geological medium, the flow velocity of water may vary spatially. Here, we report a numerical study on the migration of radioactivity due to a space dependent flow field. For a detailed analysis, seven different types of velocity profiles are considered and the corresponding concentrations are compared. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Radioactive waste Radioactivity migration Spatially varying velocity field Dufort-Frankel scheme
1. Introduction The increasing demand of energy has forced the human civilization to look for a nuclear route of power generation. During the operation of a nuclear reactor, fissile atoms undergo fission causing the generation of fission products along with the release of energy. In the back end of a fuel cycle, the radioactive waste is isolated, then classified according to the activity level (low, intermediate and high) and subsequently disposed off following the safety norms. The low and intermediate category of wastes are immobilized and then stored in near earth surface repositories. On the other hand, high level wastes are vitrified in a glass matrix, then sealed in a tank called canister which is subsequently buried deep inside the earth surface (inside a rocky medium) to isolate the same (Wattal, 2013). In the event of a leak from the underground storage facility, the waste may come in contact with the flow of underground water. In such a situation, the radionuclides would migrate within the geological medium due to the combined effect of advection and diffusion. This is a serious issue related to environmental and radiological safety and thus requires an in-depth study. There are two ways to model the migration mechanism. Fig. 1 shows a schematic picture of a rock where solid porous blocks sit within the network of tiny fractures through which the laminar
* Corresponding author. E-mail addresses: [email protected], [email protected] (S. Sen). http://dx.doi.org/10.1016/j.jenvrad.2016.05.036 0265-931X/© 2016 Elsevier Ltd. All rights reserved.
flow of water occurs. Thus the concentration of the contaminant at a given point (as a function of time) can be estimated following a random walk approach (Williams, 1992; Giacobbo and Patelli, 2008; Sen and Mohankumar, 2013). One the other hand, this problem can also be addressed by considering a neat geometry of a rock (Fig. 2) where an infinite array of identical parallel fractures are separated by porous blocks of same width (Sudicky and Frind, 1982; Chen and Li, 1997; Mohankumar, 2007; Sen and Mohankumar, 2011, 2012, 2014; Sen et al., 2015). In all these works, it is assumed that the flow of water is constant in all the fractures and this in turn requires that the concentration of the dissolved species is same at a given horizontal distance along all the fractures (Fig. 2). This is clearly a very gross assumption to simplify the computation. From Fig. 1, one can easily notice that the geometry of a rock demands a spatial distribution of flow velocity. This point can be illustrated in a more detailed way by Fig. 3 where velocities at the points A, B, C, D, E, F and G cannot be the same. The key concept of this work is to point out that the transport of a dissolved contaminant depends largely on the space dependent flow field of water. To mimic the actual flow, one has to generate the velocity profile very accurately. In a bit simplified approach, we can estimate velocities at some nodal points (like A, B, C, D, E, F and G in Fig. 3) and fit into a polynomial to approximate the flow profile. This fitted profile can be utilized to estimate the transport through a selected path. This approach may raise another important question that the concentration of the dissolved radionuclide should also change from fracture to fracture at a given horizontal distance
286
S. Sen et al. / Journal of Environmental Radioactivity 162-163 (2016) 285e288
Fig. 3. Schematic picture of water flow through a rock medium.
Fig. 1. Schematic picture of a rock.
Fig. 2. Waste matrix and fracture geometry in parallel fracture model.
from the source. To avoid further complicacy, in our present work we stick to the assumption that the concentration remains same at a given horizontal distance in all the fractures. Below we list out the two most important assumptions of this work: The flow velocity of water through a rock has a space dependent profile. The concentration of the dissolved contaminant is same in all the fractures at a given horizontal distance from the source at a given instant of time. In reality, all the transport related parameters (density of fractures, diffusivity, width of the fracture and other parameters) may vary spatially. As a sequel of our earlier works (Sen and
Mohankumar, 2011, 2012, 2013; Sen et al., 2015), we consider the transport to be dominated by advection and so the spatial variation of the diffusion coefficient is ignored. Moreover, for simplification other parameters are assumed to be constant. To highlight the importance of considering a proper field, we consider a basic 1-D model along with some simple space dependent velocity profiles. For a real life problem, we need to take some samples of the given medium and estimate the flow rates at some given spatial locations along a selected path for a given inflow at the source end. Now by fitting into a polynomial, the collected data can be used to generate a space dependent velocity profile along the path. It is important to note that the flow velocity may change from path to path. In such a case, the velocity of water in each path is needed to be estimated and this is not practically possible. In the parallel fracture model, the waste matrix is considered to be infinite which is also not a very correct assumption (Fig. 2). So for a waste matrix of finite size, flow velocities along the dominant paths (at the depth where the canister is buried) along a particular direction are enough for the simulation (for example horizontal direction in Fig. 2). Now along all such paths, we can experimentally measure the velocities at some nodal points and obtain a fitted function for the flow velocity. If the flow related characteristics of the medium (density of fractures, diffusivity, width of the fracture and other parameters) do not change drastically from region to region (a general assumption for a numerical simulation), the flow velocity profile would also remain nearly the same along all the paths. In such a case, an average velocity function can be used for all the paths and one may opt to use the parallel fracture modal to simulate the transport of radioactivity through the medium under consideration in a more accurate way. On the other hand, if the flow rate changes drastically from path to path or region to region, the space dependent velocity function for each path has to be considered separately. In this work, the limitation of the assumption of a constant flow velocity through all the paths is pointed out by considering a few theoretical velocity profiles. The main objective of this study is to estimate the migration of activity in a porous medium under the influence of a spatially varying velocity field. 2. Radioactivity migration in a porous medium We consider a simple one dimensional advection-diffusion equation to model the activity transport as given below.
vC 1 v 1 v vC þ ðvCÞ D þ lC ¼ 0 ; x 0 ; t 0 vt R vx R vx vx
(1)
Here, C(x, t) is the concentration of the active species, v is the flow velocity of water, D is the diffusion coefficient, R is the
S. Sen et al. / Journal of Environmental Radioactivity 162-163 (2016) 285e288
287
Fig. 4. Plot of velocity as a function of distance.
retardation factor and l is the decay constant of the radionuclide. One may note that we are ignoring the variation in the retardation coefficient R. As this is a safety related analysis where an upper bound estimation is important, we set R ¼ 1.0 which corresponds to the maximum transport (Sen et al., 2015). The present problem deals with the effect of variable transport coefficients on the activity migration where the overall mechanism is dominated by advection and diffusion plays a minor role and so the variation in the diffusion coefficient (D) is ignored. With this assumption, Eq. (1) takes the following form.
vC 1 v D v2 C þ ðvCÞ þ lC ¼ 0 ; x 0 ; t 0 vt R vx R vx2
(2)
Now, the point x ¼ 0 separates the source and the geological medium. By assuming a source of constant strength (C0), we write the boundary condition for the source end in the following way.
D
vC þ vCð0; tÞ ¼ vC0 vx x¼0
(3)
On the other hand, concentration must become zero as x tends to infinity. This can be represented mathematically in the following manner.
Cð∞; tÞ ¼ 0 ; t 0
(4)
The initial condition is given by
Cðx; 0Þ ¼ 0 ; x 0
(5)
Eq. (2) has to be solved along with Eqs. (3) and (4) for different flow velocity v. We use a set of space dependent functions for v
which are listed below.
iÞ v ¼ u0 ; iiÞ v ¼ u0 ð1 þ xÞ; iiiÞ v ¼ u0 1 þ x0:5 ; ivÞ v ¼ u0 1 þ x0:25 ; vÞ v ¼ u0 1 þ x0:25 ; viÞ v ¼ u0 1 þ x0:5 ; viiÞ v ¼ u0 1 þ x1
(6)
where u0 is a constant. For the numerical solution, a finite difference technique namely the Dufort-Frankel method has been implemented. The solution of a set of coupled one dimensional transport equations describing the migration of activity in a rock medium using this scheme has been reported in our earlier work (Sen and Mohankumar, 2011). Here, just by ignoring the transport through the perpendicular dimension, the earlier code has been modified and used for the present problem. In the next section, we briefly discuss the Dufort-Frankel scheme. 3. The Dufort-Frankel (DF) scheme This method is an explicit numerical technique which can provide second order accuracy in both the space and the time variables. Let us indicate the space variable by the index i and the time part by j. In the DF scheme, the derivatives are centered around the time jDt. The approximations of the derivatives using this scheme are listed below. j
jþ1
j1
vCi Ci Ci z vt 2Dt
Fig. 5. Plot of concentration (after 50 years) as a function of distance.
(7)
288
S. Sen et al. / Journal of Environmental Radioactivity 162-163 (2016) 285e288
j
j
j vCi Ciþ1 Ci1 z vx 2Dx j
v2 C i vx
z 2
j
jþ1
Ciþ1 Ci
(8) j1
Ci
j
þ Ci1
ðDxÞ2
(9)
It is very important to note that the accuracy of the solution depends critically on the estimation of the interface derivative ½vC=vxx¼0 . Mohankumar and Auerbach (2004) have shown that a 7-point approximation is necessary for an accurate estimation of the said term and we follow the same. 4. Results and discussion In this work, all the calculations are performed for a period of 50 years in double precision. We have used a grid of 0.2 m along with a time step of 0.001 year. All the calculations are re-performed with half of the grid size and half of the time step to ensure convergence. The set of parameters used in the calculations are listed below.
. D ¼ 1:0 m2 year ; R ¼ 1:0; u0 ¼ 1:0ðm=year Þ
(10)
In Fig. 4, we plot the velocity profile as a function of distance. Fig. 5 shows the corresponding concentration profiles (normalized with respect to the source) after a period of 50 years. An important point to be noted is that at the source end, velocity is taken as constant (u0) for all the velocity profiles. It can be easily noted from Fig. 5 that the constant velocity gives the maximum concentration up to a distance beyond which the curve falls rapidly. With the increase in the positive power of x in the velocity function, the concentrations at smaller distances come down systematically. It is also important to note that the contaminant reaches larger distances with this kind of velocity. On the other hand, for an increasing negative index of x in the velocity profile, the concentration increases systematically at the smaller distances and for a decreasing negative power, the contaminant reaches longer distances. So in a nutshell, for increasing positivity of the index of x, the concentration reduces at the lower end but the contaminant reaches longer distances and with decreasing positivity of the same (in other words with increasing negativity), the concentration profile slowly shifts to that of a constant velocity. The nature of the plots can be explained easily. Let us consider a velocity profile of the following kind.
v ¼ u0 ð1 þ xh Þ
(11)
First, we set h to be positive. This means that with the increase of x, the velocity increases. As we have set a constant velocity (u0) at the source end, all the velocity profiles start from a common point and increases with the increasing values of x where the rate of increment depends on h (for higher values of h, the velocity increases in a faster way). These trends are visible in Fig. 4. One can note that if we apply these velocity profiles to Eq. (2), it would drive the contaminant to the larger distances resulting in a decrease in concentration at the smaller distances (Fig. 5). Now, let us consider h to be negative. For this kind of power of x, the velocity comes down systematically to the profile of the constant velocity. With the increasing negativity of the index of x, the curve reaches the constant velocity profile in a faster manner (Fig. 4). So for these velocity functions, with the increasing negativity of the power of x, the concentration curve systematically moves towards the corresponding curve of a constant velocity (Fig. 5). Now if the flow velocity of water in a particular geological medium follows a
particular profile and we try to simulate the migration considering some other profile (for example a space dependent velocity field is modeled with a constant velocity), it would lead to a gross error. Thus by considering a set space dependent velocity profiles, we try to point out the importance of considering a proper velocity field for the simulation of the transport of radioactivity in a porous medium. 5. Conclusion In this work, we have studied the effect of a spatially varying velocity field on the transport of a radioactive contaminant in a geological (rock) medium. The results show that a constant value of velocity causes maximum concentrations at smaller distances where as a spatially varying velocity function results in a decrease in the concentration at the lower distances but higher concentration at larger distances. So it points out the importance of choosing a proper velocity profile. The present study also highlights that a large difference in the simulated concentrations may arise if the spatial variation in the velocity is not considered properly. This point is demonstrated by considering a few simple theoretical velocity profiles. For example when the velocity follows a function like uo(1 þ x) in place of uo, it has been found that the simulated concentrations beyond a distance of 100 m (after 50 years) differ by more than one order. As the final remark, it is important to remember that the simulation of the underground transport of radioactivity is meant to address the safety related aspects in the event of an accidental release of radionuclides from an underground storage facility and so a conservative approach is always advisable. Thus for an upper bound estimation of the concentration of the contaminant at smaller distances (may be enough for a short term storage of wastes in near earth surface facilities), one may opt to use a constant velocity function. On the other hand, if we want to do the same at the larger distances (important for the storage facilities of high level waste), we need to use a proper velocity function. Thus in a nutshell, this study highlights the inadequacy of using a constant flow velocity in realistically representing the real world situation. Also it indicates a simple way to overcome the limitation. References Chen, C.T., Li, S.H., 1997. Radionuclide transport in fractured porous mediaanalytical solutions for a system of parallel fractures with a constant inlet flux. Waste Manage. 17 (1), 53e64. Giacobbo, F., Patelli, E., 2008. Monte Carlo simulation of radionuclide transport through fractured media. Ann. Nucl. Energy 35, 1732e1740. Mohankumar, N., Auerbach, S.M., 2004. On the use of higher-order formula for numerical derivatives in scientific computing. Comput. Phys. Commun. 161, 109e118. Mohankumar, N., 2007. On the numerical solution of radioactivity migration in a porous medium. Ann. Nucl. Energy 34, 222e227. Sudicky, E.A., Frind, E.O., 1982. Contaminant transport in fractured porous media: Analytical solutions for a system of parallel fractures. Water Resour. Res. 18 (6), 1634e1642. Sen, S., Mohankumar, N., 2011. A computational strategy for radioactivity migration in a porous medium. Ann. Nucl. Energy 38, 2470e2474. Sen, S., Mohankumar, N., 2012. A note on higher order numerical schemes for radioactivity migration. Ann. Nucl. Energy 49, 227e231. Sen, S., Mohankumar, N., 2013. A random walk based methodology for the realistic estimation of radioactivity migration in a porous medium. Ann. Nucl. Energy 60, 202e205. Sen, S., Mohankumar, N., 2014. Migpore, a code package for the estimation of migration of radioactive species in a porous medium. Comput. Phys. Commun. 185, 302e306. Sen, S., Srinivas, C.V., Baskaran, R., Venkatraman, B., 2015. Numerical simulation of the transport of a radionuclide chain in a rock medium. J. Environ. Radioact. 141, 115e122. Wattal, P.K., 2013. Indian programme on radioactive waste management. Sadhana 38 (5), 849e857. Williams, M.M.R., 1992. A new model for describing transport of radionuclides through fractured rock. Ann. Nucl. Energy 19, 791e824.