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Journal of Environmental Radioactivity 135 (2014) 100e107

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Journal of Environmental Radioactivity

journal homepage: www.elsevier .com/locate / jenvrad

Diffusion and decay chain of radioisotopes in stagnant water insaturated porous media

Juan Guzmán a,*, Jose Alvarez-Ramirez b, Rafael Escarela-Pérez a, Raúl Alejandro Vargas c

aDepartamento de Energía, Universidad Autónoma Metropolitana-Azcapotzalco, Av. San Pablo 180 Col. Reynosa Tamaulipas, México, D.F. 02200, MexicobDivisión de Ciencias Básicas e Ingeniería, Universidad Autónoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186 Col. Vicentina, México, D.F. 09340,Mexicoc Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional, Av. Instituto Politécnico Nacional s/n, U.P. Adolfo López Mateos, Col. San PedroZacatenco, México, D.F. 07738, Mexico

a r t i c l e i n f o

Article history:Received 29 November 2013Received in revised form21 March 2014Accepted 21 March 2014Available online

Keywords:RadioisotopeDispersionPorousMediumDiffusionDecay

Abbreviations: IR, injective reservoir; DR, diffusisorption and decay equation; DD, diffusion and decay* Corresponding author. Tel.: þ52 55 53189047; fax

E-mail address: doctor_guzman@hotmail.com (J. G

http://dx.doi.org/10.1016/j.jenvrad.2014.03.0150265-931X/� 2014 Elsevier Ltd. All rights reserved.

a b s t r a c t

The analysis of the diffusion of radioisotopes in stagnant water in saturated porous media is important tovalidate the performance of barrier systems used in radioactive repositories. In this work a methodologyis developed to determine the radioisotope concentration in a two-reservoir configuration: a saturatedporous medium with stagnant water is surrounded by two reservoirs. The concentrations are obtainedfor all the radioisotopes of the decay chain using the concept of overvalued concentration. A method-ology, based on the variable separation method, is proposed for the solution of the transport equation.The novelty of the proposed methodology involves the factorization of the overvalued concentration intwo factors: one that describes the diffusion without decay and another one that describes the decaywithout diffusion. It is possible with the proposed methodology to determine the required time to obtainequal injective and diffusive concentrations in reservoirs. In fact, this time is inversely proportional to thediffusion coefficient. In addition, the proposed methodology allows finding the required time to get alinear and constant space distribution of the concentration in porous mediums. This time is inverselyproportional to the diffusion coefficient. In order to validate the proposed methodology, the distributionsin the radioisotope concentrations are compared with other experimental and numerical works.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

The barrier systems used in radioactive repositories prevent therelease of radioactive substances to the ground surface. In order toavoid this release, the radioactive substances are vitrified andoverpacked for a certain period of time. However, this containmentof substances is impossible for all time. In order to minimize thepossible liberation of substances, a low permeability material isallocated between the overpack and the host rock so that water isstagnant in this material. This means that the release of substancesis limited by diffusion rather than advection in this material(Malekifarsani and Skachek, 2009). Therefore, it is important toanalyze the diffusion and decay of substances dissolved in stagnantwater in the two-reservoir configuration. The configuration

ve reservoir; DSD, diffusion,equation.: þ52 55 53947378.uzmán).

consists of a specimen (saturated porous medium) surrounded bytwo reservoirs: a reservoir consists of a high concentration of ra-dioisotopes (called injective reservoir, IR) and other is free of ra-dioisotopes (called diffusive reservoir, DR).

Several studies have examined the diffusion of substances inwater, which is stagnant in saturated porous media. In the works ofChen et al. (2012), Aldaba et al. (2010), García-Gutiérrez et al. (2001)and Shackelford (1991), the two-reservoir configuration is analyzedto determine the diffusion coefficients. Eriksen et al. (1999), Lü andAhl (2005), Lü and Viljanen (2002), Tits et al. (2003), Yamaguchiand Nakayama (1998) determine diffusion coefficients for situa-tions where the concentration of radioisotopes in IR is constant.While, Bharat et al. (2009), Lake and Rowe (2004), and Moridis(1999) find diffusion coefficients for situations where the concen-tration in the IR is variable.

In this work, a methodology is developed to determine theradioisotope concentration in water, which is stagnant in saturatedporous media for a two-reservoir configuration. The concentrationsare obtained for all the radioisotopes of the decay chain using theconcept of overvalued concentration. A methodology based, on the

Fig. 1. Radioisotopes intervening in the decay chain.

J. Guzmán et al. / Journal of Environmental Radioactivity 135 (2014) 100e107 101

variable separation method, has been proposed for the solution ofthe radioisotope transport equation. This methodology proposes tofactorize the overvalued concentration in two terms: a factor thatdescribes the diffusion without decay and other factor that de-scribes the decay without diffusion.

It is considered that the radioisotopes can be absorbed by theporous medium (sorption process) and that these radioisotopes candecay. Henceforth, the transport of radioisotopes is examined bysolving the diffusion, sorption and decay equations (DSD equa-tions). It is considered that the sorption process follows a linearrelationship, so that DSD equations are reduced to equations onlyinvolving diffusion and decay (DD equations). In order to solve theDD equations, a separation of variables is employed. A decouplingof the DD equations is then achievable, leading to an equation thatdescribes the diffusion and other equation that incorporates decay.The resulting equations of the decay are then handled through theLaplace Integral Transform Technique (Arfken and Weber, 2005),which provides analytical expressions. The diffusion equation canbe analytically or numerically solved, depending on the complexityof the problem. In this work, a numerical solution is establishedusing the finite element method and a commercial softwarepackage (COMSOL, 2008). In order to validate the proposed meth-odology, our solution of the radioisotope concentration iscompared with the solution of Moridis (1999) and Chen et al.(2012). Moridis (1999)have modeled the radioisotope transportusing the Laplace transform technique for the decay of the parentradioisotope. The decay of the descendents of the parent has notbeen modeled. However, the diffusion and decay of the radioiso-tope and its first descendent are modeled in the work of Chen. Theeffect of the decay of other descendents was not considered. In thiswork, the transport of all descendents of the radioisotope is fullyconsidered.

Fig. 2. Scheme of the two reservoir method.

2. Preliminaries

Water is modeled by an incompressible fluid, which is stagnatedin a homogeneous saturated porous medium. Henceforth, theadvective part in the transport of radioisotopes is not considered.The transport of radioisotopes dissolved in water is produced bydiffusion. It is considered that the radioisotopes can be absorbed(sorption process) by the porous medium. Also, the radioisotopescan decay into other radioisotopes, as is illustrated in Fig. 1. In Fig. 1,the radioisotope labeled 1 is the first radioisotope in the decaychain and decays in the radioisotope labeled 2, which in turn de-cays in the radioisotope labeled 3, and so on. In Fig. 1, the decaychain is formed by N radioisotopes.

Using the mass balance (Bear and Cheng, 2010), the equationsthat describe the radioisotopes transport due to diffusion, sorptionand decay in a porous medium are obtained:

vC1ðx; tÞvt

¼ Dm1v2C1ðx; tÞ

v2x� rb

F

vS1vt

� rbF

l1S1 � l1C1ðx; tÞ; (1)

and

vCiðx; tÞvt

¼ Dmiv2Ciðx; tÞ

vx2� rb

F

vSivt

� liCiðx; tÞ �rbF

liSi

þ li�1Ci�1ðx; tÞ þrbF

li�1Si�1 i ¼ 2;.;N;(2)

where Ci(x,t) is the aqueous concentration per unit volume of waterof the radioisotope i in the decay chain formed by N radioisotopes,at the position x and time t; Ci�1 is the aqueous concentration offather radioisotope of i per unit volume of the water; li and li�1 arethe decay constant of the radioisotope i and its parent radioisotope,

respectively; Dmi is the molecular diffusion coefficient of theradioisotope i; Si is the absorbed mass of the radioisotope i in thesoil matrix per unit bulk dry mass of the porous medium; F is theporosity; rb is the bulk dry density of the porous medium (bulk soildensity).

Eqs. (1) and (2) can be recast into one equation given by:

vCiðx; tÞvt

¼ Dmiv2Ciðx; tÞ

vx2� rb

F

vSivt

� liCiðx; tÞ �rbF

liSi

þ li�1Ci�1ðx; tÞ þrbF

li�1Si�1 i ¼ 1;.;N;(3)

This equation is not consistent with Eq. (1), since it contains thefictitious radioisotope C0(x,t). As a result, constant l0 is defined asl0 ¼ 0, so as to make them consistent.

The first term on the left-hand side of Eq. (3) describes thediffusion of water in pores. The second term describes the ab-sorption processes of movable substances: sorption and precipita-tion. If a movable substance is precipitated in a porousmedium, thissubstance becomes immovable since porosity is small enough.Therefore, the precipitation effect is described by means of ab-sorption of movable substances. The remaining terms in Eq. (3)describes decay of the radioisotope and its parent, respectively.

It is considered that the sorption process follows a linear rela-tionship. That is, Si ¼ KiCi, where Ki is the distribution coefficient ofradioisotope i. Therefore, Eq. (3) is given by:

vCiðx; tÞvt

¼ Div2Ciðx; tÞ

vx2� l0iCiðx; tÞ þ l0i�1Ci�1ðx; tÞ i ¼ 1;.;N;

(4)

where l’1 ¼ l1, l0i�1 ¼ ðRi�1=RiÞli�1 ðis2Þ, Ri¼ 1þ (rb/F)Ki is the

retardation factor of radioisotope i, Di is the apparent diffusioncoefficient of radioisotope i and given by:

Di ¼Dmi

Ri: (5)

In order to validate the proposed methodology, proven solu-tions, provided in the works of Moridis (1999) and Chen et al.(2012), have been employed to. In these works, the two-reservoirmethod is used, where the saturated porous medium is sur-rounded by: a reservoir containing a high radioisotope concentra-tion (called injective reservoir) and other reservoir that is initiallyfree of radioisotopes (called diffusive reservoir), as shown in Fig. 2.A one-dimensional geometry is considered, where variable x de-scribes position. In Fig. 2, Ci(x,t) represents the concentration of theradioisotope i in the porous medium at position x and time t; CIR,i(t)

J. Guzmán et al. / Journal of Environmental Radioactivity 135 (2014) 100e107102

is the aqueous concentration per unit volume of water of radio-isotope i in the injective reservoir (IR) at time t; CDR,i(t) is theaqueous concentration per unit volume of water of radioisotope i inthe diffusive reservoir (DR) at time t; L is the thickness of the porousmedium; VDR is the volume of DR; VIR is the volume of IR; x ¼ 0represents the interface between the porous medium and theinjective reservoir; x ¼ L represents the interface between theporous medium and the diffusive reservoir.

Initial conditions:It is initially assumed that the saturated porousmedium and the

diffusive reservoir is free of radioisotopes; while the injectivereservoir contains radioisotopes; that is,

CIR;iðt ¼ 0Þ ¼ Ci0 injective reservoirCiðx; t ¼ 0Þ ¼ 0 porous mediumCDR;iðt ¼ 0Þ ¼ 0 diffusive reservoir

(6)

where Ci0 is the initial concentration of the radioisotope i in IR.Using mass balance in the reservoirs (Moridis, 1999), the

boundary conditions are determined as:Boundary Conditions in IR (x ¼ 0):

CIR;1ðtÞ ¼ C1ðx ¼ 0; tÞ

VIRdCIR;1dt

¼ �l1VIRCIR;1 þ AFDm1vC1vx

����x¼0

(7)

and

CIR;iðtÞ ¼ Ciðx ¼ 0; tÞ i ¼ 2;.;N

VIRdCIR;idt

¼ �liVIRCIR;i þ li�1VIRCIR;i�1

þAFDmivCivx

����x¼0

i ¼ 2;.;N;

(8)

Boundary Conditions in DR (x ¼ L):

CDR;1ðtÞ ¼ C1ðx ¼ L; tÞ

VDRdCDR;1dt

¼ �l1VDRCDR;1 � AFDm1vC1vx

����x¼L

(9)

and

CDR;iðtÞ ¼ Ciðx ¼ L; tÞ i ¼ 2;.;N

VDRdCDR;idt

¼ �liVDRCDR;i þ li�1VDRCDR;i�1

�AFDmivCivx

����x¼L

i ¼ 2;.;N;

(10)

Using Eq. (5), the boundary conditions (Eqs. (7)e(10)) can bewritten as:

CIR;iðtÞ ¼ Ciðx ¼ 0; tÞ i ¼ 1;.;N

VIRdCIR;idt

¼ �liVIRCIR;i þ li�1VIRCIR;i�1 þAeiDivCivx

����x¼0

i ¼ 1;.;N

CDR;iðtÞ ¼ Ciðx ¼ L; tÞ i ¼ 1; :::;N

VDRdCDR;idt

¼ �liVDRCDR;i þ li�1VDRCDR;i�1

�AeiDivCivx

����x¼L

i ¼ 1;.;N;

(11)

where Aei ¼ AFRi.

3. Proposed solution methodology

Eq. (5) shows that the apparent diffusion coefficient Di dependson the radioisotope i by means of the molecular diffusion coeffi-cient Dmi and the retardation factor Ri. Therefore, coefficient Di foreach radioisotope is not the same. The aim of this work is todevelop a methodology able to predict radioisotope concentrationsthat does not harm human health. If the overvalued concentration(which is greater than the actual concentration) of each radioiso-tope does not harm human health, then the actual concentrationwill not do it either. Therefore, the case, corresponding to over-valued concentrations, is analyzed here. These overvalued con-centrations are achieved if the molecular diffusion coefficient ofeach radioisotope i is equal to the highest value Dmax of all mo-lecular diffusion coefficients of the radioisotopes:

Dmax ¼ maxfDmi; i ¼ 1;.;Ng; (12)

Also, overvalued concentrations are achieved if the distributioncoefficient Ki of each radioisotope i is equal to the smallest valueKmin of all distribution coefficients of the radioisotopes: Kmin ¼min{Ki, i ¼ 1,.,N}. The equations describing the diffusion, sorptionand decay of the overvalued concentrations of each radioisotopewith its initial and boundary conditions are obtained by means ofEqs. (4), (6) and (11) with Dmi / Dmax and Ki / Kmin. Hence:

vCover iðx; tÞvt

¼ Dv2Cover iðx; tÞ

vx2� liCover iðx; tÞ

þ li�1Cover i�1ðx; tÞ; (13)

where, Cover i(x,t) is the overvalued concentration in the porousmedium at position x and time t of the radioisotope i, D¼ Dmax/Rminand Rmin ¼ 1þrb/FKmin.

Initial condition:

CIR;over iðt ¼ 0Þ ¼ Ci0 injective reservoirCover iðx; t ¼ 0Þ ¼ 0 porous mediumCDR;over iðt ¼ 0Þ ¼ 0 diffusive reservoir

(14)

where, CIR,over i(t) is the overvalued concentration in the injectivereservoir of the radioisotope i; and CDR,over i(t) is the overvaluedconcentration in the diffusive reservoir of the radioisotope i.

Boundary conditions:

CIR;over iðtÞ ¼ Cover iðx ¼ 0; tÞ

VIRdCIR;over i

dt¼ �liVIRCIR;over i þ li�1VIRCIR;over i�1

þAeDvCover i

vx

����x¼0

CDR;over iðtÞ ¼ Cover iðx ¼ L; tÞ

VDRdCDR;over i

dt¼ �liVDRCDR;over i þ li�1VDRCDR;over i�1

�AeDvCover i

vx

����;(15)

where Ae ¼ AFRmin and i ¼ 1,.,N. A method based on the separa-tion of variables is proposed here for the solution of the diffusion,sorption and decay equations with its initial and boundary

J. Guzmán et al. / Journal of Environmental Radioactivity 135 (2014) 100e107 103

conditions (Eqs. (13)e(15)). Let the overvalued concentrations bethe product of two functions:

CIR;over iðtÞ ¼ cIRðtÞniðtÞ; (16)

Cover iðx; tÞ ¼ cðx; tÞniðtÞ; (17)

and

CDR;over iðtÞ ¼ cDRðtÞniðtÞ: (18)

where c(x,t) is a function of the position x and time t and cDR(t) andcIR(t) are functions of time t. The functions: c(x,t), cDR(t) and cIR(t) donot depend on the radioisotope i. The function ni(t) is a function of tand depends on the radioisotope i.

It is shown in Appendix A that factor ni(t) and c factors (c(x,t),cDR(x,t) and cIR(x,t)) satisfy the following equations and conditions:

Factor ni(t):

dniðtÞdt

¼ li�1ni�1ðtÞ � liniðtÞ i ¼ 1;.;N (19)

Initial condition:

niðt ¼ 0Þ ¼ Ci0 (20)

C factors:

vc x; tð Þvt

¼ Dv2c x; tð Þ

vx2; (21)

Initial condition:

cIRðt ¼ 0Þ ¼ 1 injective reservoir; (22)

cðx; t ¼ 0Þ ¼ 0 porous medium; (23)

cDRðt ¼ 0Þ ¼ 0 diffusive reservoir: (24)

Boundary conditions:

cIRðtÞ ¼ cðx ¼ 0; tÞ; (25)

VIRdcIRðtÞdt

¼ AeDvcðx; tÞ

vx

����x¼0

; (26)

cDRðtÞ ¼ cðx ¼ L; tÞ; (27)

VDRdcDRðtÞ

dt¼ �AeD

vcðx; tÞvx

����x¼L

; (28)

Eq. (19) is similar to the equations that describe the decay ofradioisotopes without diffusion-sorption, while Eq. (21) is similarto the equation that describes the diffusion of substances in stag-nant water in a porous medium without decay. The factor ni(t) iscalled pure decay factor and the c factors (c(x,t), cDR(x,t) and cIR(x,t))are called pure diffusion factors.

The factorization in the overvalued concentration can be ach-ieved because the diffusion effect induces a percent change in theconcentration. If the diffusion conditions are the same for all iso-topes of the chain, then the diffusion effect induces the samepercent change in all concentrations of the isotopes. Therefore, theconcentration of each isotope can be factorized in a diffusion factorand a decay factor.

3.1. Decay factor ni(t)

The solution of the function ni(t) is obtained by solving Eq. (19)along with its initial condition (20), and can be solved utilizing theLaplace Integral Transform technique (Arfken and Weber, 2005;Davies, 2000):

niðtÞ ¼Xi

l¼1

0@Cl0

li

Xi

j¼ l

aijle�ljt

1A (29)

here,

aijl ¼

Yin¼ l

ln

Yin ¼ lnsj

�ln � lj

� (30)

3.2. Diffusion c factors

Solution of the c functions (c, cDR and cIR) is obtained by solvingEq. (21) (diffusion equation) along with initial and boundary con-ditions (Eqs. (22)e(28)). This solution can be found using a greatamount of numerical methods such as the multi-compartmentmethod (Chen et al., 2012), the inverse Laplace transform(Moridis, 1999) and the finite element method (Fish andBelytschko, 2007). In this work, the finite element method (FEM)has been employed.

4. Model verification

In order to validate the methodology proposed here, the solu-tions of our approach are compared with solutions ofother methods: Moridis (1999) and Chen et al. (2012).Moridis methodology has been validated experimentally, whereasChen methodology has been numerically verified but its resultsalso agree well with results of other methodologies (which havebeen validated experimentally). The comparison of our results withthese two approaches validates our proposed methodology, mak-ing, in fact, an experimental validation. Further, the solutions of ourmethodology are compared with experimental data obtained byYamaguchi and Nakayama (1998).

4.1. Comparison with Chen et al. methodology

Chen et al. (2012) described the transport of radioisotopes usingthe dispersion-decay model. In the model of Chen, the decay of theparent radioisotope and its daughter is considered. However, thisapproach did not consider the decay of the daughter. Therefore, Eq.(4) was solved for the parent and daughter radioisotope (i¼ 1,2) butit was not solved for later generations of the daughter. The initialconditions that were considered are given by Eq. (6) with i ¼ 1,2,whereas, the boundary conditions are given by Eq. (11) with i ¼ 1,2.Chen used the multi-compartment method to divide the porousmedium in M compartments (Chen et al., 2012). In addition, Chenapplied the Laplace transform method to derive the radioisotopeconcentrations of the parent and its daughter.

The results of Chen work have been compared with the resultsof our proposed methodology using the data shown in Table 1:case B. The decay of later generations, other than the first genera-tion, has not been computed. Hence, the results of our proposed

J. Guzmán et al. / Journal of Environmental Radioactivity 135 (2014) 100e107104

methodology for the parent radioisotope and its first generation arethe only ones to be compared. Notice that Chen uses onecompartment in the multi-compartment method. In addition, thesame diffusion and distribution coefficients are chosen both forparent and daughter radioisotopes: D1 ¼ D2 and K1 ¼ K2. Thismeans thatD¼D1¼D2, Kmin¼ K1¼ K2, Ae¼ Ae1¼ Ae2, meaning thatthe overvalued concentration must coincide with the actual con-centration: Cover i ¼ Ci, CIR,over i ¼ CIR, i and CDR,over i ¼ CDR, i. Fig. 3shows the daughter radioisotope concentration in the diffusiveand injective reservoirs. It can be observed that the concentrationsof the parent radioisotope have the same curves, that is, CIR,1/n1 ¼ CIR,2/n2 and CDR,1/n1 ¼ CDR,2/n2. It is found that the solution ofthe proposed methodology coincides with the curves of Chen forboth the parent radioisotope and its first progeny. Thus, thesefindings confirm validity of our methodology.

Fig. 3. Comparison of the daughter radioisotope concentration.

4.2. Comparison with Moridis methodology

Moridis describes the transport of radioisotopes using theadvectionedispersion-decay model. Here, the decay of the parentradioisotope is not considered, that is, Eq. (4) was solved for the firstradioisotope (i ¼ 1) without considering the remainder of radio-isotopes. The initial condition used in Moridis model is given by Eq.(6) with i ¼ 1, whereas the boundary conditions are given by Eq.(11). Moridis obtained the solution of Eq. (4) in the Laplace domain.

The results of our proposed methodology are now comparedwith the methodology of Moridis using the values of case B* shownin Table 1. The decay of the progeny of the radioisotope is notconsidered in the work of Moridis, which means that one radio-isotope in the decay chain is only considered, and therefore D ¼ D1,Kmin ¼ K1. This means that the overvalued concentration mustcoincide with the actual concentration: Cover ¼ C1. In Fig. 4, theconcentration of the radioisotope in the diffusive and injectivereservoirs is depicted. It can be observed that Moridis results matchthe solution of this work. These findings validate our proposedmethodology by providing equal results.

Fig. 4. Comparison of the radioisotope concentration.

4.3. Experimental comparison

Our proposed methodology is validated in this section with thedetermination of the diffusion coefficient using experimental data.Experimental data were obtained by Yamaguchi and Nakayama(1998) for the determination of the diffusion coefficient ofuranium-233 in Inada granite. The experimental setup of this workuses a two-reservoir configuration. A detailed description of thisexperimental setup is given in Yamaguchi and Nakayama (1998)and Yamaguchi et al. (1993). The data used in this work areshown in Table 2. The soil data can be found in Yamaguchi et al.(1997).

The concentration of the uranium radioisotope in the diffusivereservoir is measured to determine the coefficients: FDm1 and FR1.The time distribution of this concentration is shown in Fig. 5.Yamaguchi and Nakayama (1998) determined the diffusion anddistribution coefficients using the analytic solution of Crank (1975).

The diffusion and distribution coefficients were found using ourproposed model. A square least method has been employed to

Table 1Values used in this work.

L (cm) Ae (cm2) VIR (cm3) VDR (cm3)

Case B 0.2 1 10 10Case B*

a NC: not considered.

minimize the deviation between measurement and our numericalprediction. The obtained values for the coefficients are shown inTable 2. The diffusion and distribution coefficients obtained byYamaguchi and Nakayama (1998) can also be found in this table.The differences between the results of Yamaguchi and Nakayama(1998) and ours for the coefficient FDm1 are 1% and for the coeffi-cient FR1 is 27%, respectively. The difference for FR1 can be due tothe extrapolation of the linear portion of the diffusion curve usedby Yamaguchi and Nakayama. Comparison of the diffusion coeffi-cient obtained by Yamaguchi and Nakayama with ours, the validityof our proposed model proposed is attained. Also, Fig. 5 shows thatthe discrepancy between experimental measurement and our nu-merical prediction is quite acceptable, except at some small valuesof concentration.

D1

�cm2

day

�l1

�1

day

�D2

�cm2

day

�l2

�1

day

�C10, C20

1E-4 1E-3 1E-4 1E-4 1, 01E-4 1E-5 aNC NC 1, NC

Table 2Data used in the experimental determination of coefficients.

Geometric data

L (cm) A (cm2) VIR (cm3) VDR (cm3)

0.5 12.56 116 116

Subsoil data

r (g/cm3) F

2.6 0.007

Coefficients

FDm1

�cm2

day

�FR1

Yamaguchi and Nakayama 1.35e�4 0.266Our numerical prediction 1.34e�4 0.365

Fig. 5. Time distribution of concentration of uranium in diffusive reservoir.

J. Guzmán et al. / Journal of Environmental Radioactivity 135 (2014) 100e107 105

5. Results and discussion

It was observed in section 3 that the radioisotope concentrationis the product of two types of terms: factors describing the diffusionwithout decay (c terms: cIR, cDR, and c, called diffusion factors), andanother factor representing the decay without diffusion (ni, calleddecay factors). In this section is analyzed the effect of the decay,diffusion and absorption on the previous factors.

Fig. 6. Pure decay factors.

5.1. Effect of decay constants

The decay effect is described by the pure decay factor ni(t). In orderto simplify the analysis, the factors of the parent radioisotope and itsdaughter (n1,n2) are only considered. Similar analyses can be per-formed for the descendents of the daughter radioisotope by consid-ering the daughter radioisotope as a new parent radioisotope. Fig. 6illustrates a number of curves for n1 and n2 with different decay con-stants. The values for the case l1 > l2 (l1 ¼1.0 and l2 ¼ 0.5) and casel1< l2 (l1¼0.5 and l2¼1.0) have been considered. The units of decayconstants are 1/days and the initial concentrations are taken as:C10¼ 1,0, C20¼ 0. It is observed in both cases that the concentration ofparent nuclide n1 decays while the concentration of daughter nucliden2 is initially increased until it reaches a maximum value, decayingafterward. The case l1 > l2 leads to a sharp decay of n1, whereas n2continue to decay where n1z0. The case l1< l2 is characterized by aslowofn1andn2 continues todecayuntiln1zn2 (secularequilibrium).

Similar analyses are found if the decay chain contains three ormore radioisotopes. For example, if lk is the smallest decay constant

then the decay of any radioisotope i > k for t [ 0 is described bythis decay constant lk:

niyC10l1l2.lk.li

ðl1 � lkÞðl2 � lkÞ.ðlk�1 � lkÞðlkþ1 � lkÞ.ðli � lkÞe�lkt :

(31)

5.2. Effect of the diffusion coefficient

The results shown in Figs. 3 and 4 demonstrate that the pro-posed methodology can describe the diffusive effect. This effect isdescribed by the c pure diffusion factors: c(x,t) ¼ Cover i(x,t)/ni(t),cIR(x,t) ¼ CIR,over i(x,t)/ni(t), cDR(x,t) ¼ CDR,over i(x,t)/ni(t).

Fig. 7 shows the space distribution in porous medium of thediffusive factor c(x,t) at different times. The corresponding data areshown in Table 2. It is observed that the space distribution of c(x,t)tends to be uniform as time progresses. In turn, the gradient of c(x,t)(vc(x,t)/vx) become zero as time advances. The situation defined byvc(x,t)/vx z 0 is called equilibrium condition. This means that thediffusive flux of the overvalued concentration is practically zero(�Dmaxnivc(x,t)/vx z 0) in equilibrium condition. Also, the spacedistribution of the factor c(x,t) can be considered lineal at times tafter 0.1/0.007Dmax.

Fig. 8 shows the time distribution in injective and diffusivereservoirs of the diffusive factors cIR(t) and cDR(t). The geometricand subsoil data are shown in Table 2. It is observed that the factorcIR(t) decreases while the factor cDR(t) increases until these factorsare practically equal (equilibrium condition) in time t after 13/0.007Dmax. This means that the equilibrium condition is reachedsooner for higher values in Dmax. Note the time scale utilized t0/0.007Dmax. Therefore, higher values in Dmax correspond to fasterchanges in factors cIR(t) and cDR(t).

5.3. Effect of absorption

The effect of absorption is described by means of the distribu-tion coefficient Ki. In fact, this distribution coefficient is the slope ofthe isotherm absorption (Freeze and Cherry, 1979). Therefore,higher sorption effects correspond to higher distribution co-efficients. Fig. 8 shows the time distribution of cIR(t) and cDR(t) atdistinct minimum distribution coefficients Kmin: 0.0015, 0.099, and0.8 in cm3/g. The values of 0.099 and 0.0015 (cm3/g) correspond toexperimental values obtained by Yamaguchi and Nakayama (1998)for the UO2ðCO3Þ4�3 and UO2þ

2 , respectively. The value of 0.8 (cm3/g)is a theoretical value introduced for comparison analysis. It can beseen that the time distribution of cIR(t) and cDR(t) is not sensitive tothe experimental distribution coefficients: 0.099 and 0.0015. Also,cIR(t) and cDR(t) decrease little for the high theoretical distribution

Fig. 7. Space distribution of the factor c(x,t). t ¼ t0/0.007Dmax, Dmax in cm2/days, t indays.

Fig. 8. Factors cIR and cDR at different distribution coefficients. t ¼ t0/0.007 Dmax, Dmax

in cm2/days, t in days, Kmin in cm3/g.

J. Guzmán et al. / Journal of Environmental Radioactivity 135 (2014) 100e107106

coefficient: 0.8. The little sensitive can be due to that the length L issmall.

6. Conclusions

The aim of this work was to present a methodology based uponthe use of the method of separation of variables to simulate thetransport of radioisotopes in water which is stagnated in a tworeservoir configuration: a saturated porous medium that is sur-rounded by two reservoirs. This methodology is capable of ac-counting for the processes of diffusion, sorption and decay, whichoccurs during the transport of radioisotopes in stagnated water insaturated porous medium.

Solutions of the radioisotope concentration are obtained for allisotopes of the decay chain using the concept of overvalued con-centration. The overvalued radioisotope concentration is calculatedusing the following methods: variable separation, finite elementand Laplace transform. The proposed methodology factorizes theovervalued concentration in two factors: a factor that describes the

diffusion without decay and another one that describes the decaywithout diffusion.

If one radioisotope is considered in the decay chain, the over-valued concentration is equal to the actual concentration. Thismeansthat the actual concentration for the first isotope in the chain can befactorized into a decay factor and a diffusion factor. If the diffusionand distribution coefficients have basically same values, the actualconcentration of all radioisotopes in the chain can then be factorizedin these two factors. Also, it is observed that the concentration is notvery sensitive to experimental values of the distribution coefficients,if the dimensions of the porous medium are small.

The proposed methodology is important since it allows deter-mining a limiting value in the initial concentration, so that theactivity of the dispersion does not harm human health. Also, thedecay of overvalued activity is described by the smallest decayconstant for enough long times (secular equilibrium).

It is possible with our methodology to determine the requiredtime to obtain equal injective and diffusive concentrations in res-ervoirs. In fact, this time is inversely proportional to the diffusioncoefficient. In addition, our methodology allows finding therequired time to get a linear and constant space distribution of theconcentration in porous mediums. This time is inversely propor-tional to the diffusion coefficient.

Acknowledgment

This research was supported by the program: “Programa para elMejoramiento al Profesorado (PROMEP)”.

Appendix A

The demonstration that the ni and c terms satisfy Eqs. (19)e(28)is divided into two steps. In the first step is demonstrated that niand c satisfy Eqs. (19) and (21). In the second step is shown that niand c satisfy boundary and initial conditions: Eqs. (20), (22)e(28).

First stepSubstituting Eqs. (16)e(18) in Eq. (13) yields:

cdnidt

þ nivcvt

¼ niDv2cvx2

þ li�1ni�1c� linic i ¼ 1;.;N (32)

These equations are satisfied if the functions c(x,t) and ni(t)satisfy the following equations:

ni tð Þ vc x; tð Þvt

¼ ni tð ÞD v2c x; tð Þvx2

i ¼ 1;.;N; (33)

and

dniðtÞdt

cðx; tÞ ¼ li�1ni�1ðtÞcðx; tÞ � liniðtÞcðx; tÞ i ¼ 1;.;N:

(34)

Simplifying, these equations:

vc x; tð Þvt

¼ Dv2c x; tð Þ

vx2; (35)

and,

dniðtÞdt

¼ li�1ni�1ðtÞ � liniðtÞ i ¼ 1;.;N (36)

Second stepEqs. (16)e(18) are substituted in initial and boundary conditions

(Eqs. (14) and (15)):

J. Guzmán et al. / Journal of Environmental Radioactivity 135 (2014) 100e107 107

Initial condition:

cIRðt ¼ 0Þniðt ¼ 0Þ ¼ Ci0 injective reservoircðx; t ¼ 0Þniðt ¼ 0Þ ¼ 0 porous mediumcDRðt ¼ 0Þniðt ¼ 0Þ ¼ 0 diffusive reservoir

(37)

Boundary conditions:

cIRðtÞniðtÞ ¼ cðx ¼ 0; tÞniðtÞ

VIR

�dcIRðtÞ

dtniðtÞ þ cIRðtÞ

dniðtÞdt

�¼ li�1VIRcIRðtÞni�1ðtÞ

� liVIRcIRðtÞni�1ðtÞ þ AeDniðtÞvcðx; tÞ

vt

����x¼0

cDRðtÞniðtÞ ¼ cðx ¼ L; tÞniðtÞ

VDR

�dcDRðtÞ

dtniðtÞ þ cDRðtÞ

dniðtÞdt

�¼ li�1VDRcDRðtÞni�1ðtÞ

� liVDRcDRðtÞni�1ðtÞ þ AeDniðtÞvcðx; tÞ

vt

����x¼L

ð38Þ

The initial and boundary conditions (Eqs. (37) and (38)) can besatisfied if the functions c, cDR, cIR, and ni fulfill:

Initial condition:

niðt ¼ 0Þ ¼ Ci0 injective reservoircIRðt ¼ 0Þ ¼ 1 injective reservoircðx; t ¼ 0Þ ¼ 0 porous mediumcDRðt ¼ 0Þ ¼ 0 diffusive reservoir

(39)

Boundary conditions:

cIRðtÞ ¼ cðx ¼ 0; tÞ

VIRdcIRðtÞdt

¼ AeDvcðx; tÞ

vt

����x¼0

cDRðtÞ ¼ cðx ¼ L; tÞ

VDRdcDRðtÞ

dt¼ AeD

vcðx; tÞvt

����x¼L

(40)

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