Modelling the vertical distribution of radionuclides in soil. Part 1: the convection–dispersion equation revisited

Journal of Environmental Radioactivity 73 (2004) 127–150 www.elsevier.com/locate/jenvrad

Modelling the vertical distribution of radionuclides in soil. Part 1: the convection– dispersion equation revisited P. Bossew a,
, G. Kirchner b,1 a

Institute of Physics and Biophysics, University of Salzburg, Hellbrunner Strasse 34, A-5020 Salzburg, Austria b Department of Physics, University of Bremen, D-28336 Bremen, Germany Received 21 March 2003; received in revised form 29 July 2003; accepted 19 August 2003

Abstract The convective–dispersive transport and linear sorption model is discussed for the vertical migration of radionuclides in soil. An alternative procedure of solving the corresponding system of partial differential equations is presented as well as the special solution for the pulse-like fallout initial condition. Idealizations and simplifications of the model and properties of the solution are discussed. The model is fitted to a set of 528 measured radionuclide soil profiles and the resulting model parameters, apparent convection velocity v and apparent dispersion constant D, are evaluated statistically. Typical orders of magnitude of the velocities and the diffusion constants of Chernobyl-134Cs are 0.3 cm/year and 0.3 cm2/year, respectively. The mobilities of the radionuclides are ranked as 137Cs (global fallout) < 134Cs < 106Ru, 125Sb. Significant regional differences (related to different soils and geological properties below ground) of v and D exist. These analyses also indicate that v and D are not mere fitting parameters, but can be given a real physical interpretation. While in most cases, the convection–dispersion equation (CDE) model produces good descriptions for nearsurface soil layers, potentially important limitations are its failure to describe ‘‘young’’ profiles shortly after fallout. # 2003 Elsevier Ltd. All rights reserved. Keywords: Vertical migration of radionuclides in soil; Convection–dispersion model




Corresponding author. Tel.: +43-662-80-44-5702. E-mail address: peter.bossew@reflex.at (P. Bossew). 1 Present address: Federal Office for Radiation Protection, Ko¨penicker Allee 120-130, D-10318 Berlin, Germany. 0265-931X/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.jenvrad.2003.08.006

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1. Introduction For long, the vertical distribution of fallout radionuclides has been the subject of scientific investigations. The reason of the importance to understand and predict vertical migration is its potential radiological impact: slow migration results in the radionuclide being available to plant uptake, and gives rise to external doses for a long time. On the other hand, by fast downward migration, the radionuclide can enter the groundwater table quickly. Understanding the processes involved and modelling the migration is a necessary prerequisite for predicting the long-term behaviour. Two classes of migration models can be distinguished. The models of the first class use a number of compartments associated to soil layers which are defined by preset criteria, and calculate radionuclide fluxes between, and corresponding residence times within the compartments, without making assumptions on the migration mechanisms. Recently, however, it was pointed out (Kirchner, 1998) that this model implicitly assumes purely convective flow with dispersion characteristics defined by the number and sizes of the compartments. The models of the second class, which we call analytical models, result in an analytical formula for the concentration C(x,t), of a radionuclide in depth x below soil surface and after migration time t. Basically, there are two types of analytical models. The first type makes assumptions on physico-chemical mechanisms which govern the migration of radionuclides in soil and their interaction with soil particles (e.g. Frissel and Poelstra, 1967; Bear, 1972; Selim and Mansell, 1976; Bunzl, 1978; van Genuchten and Cleary, 1983; McKinley and Alexander, 1992; Roth, 1996). An analytical model sui generis has been proposed by Kirchner (1998) based on assumptions about statistical distributions of soil properties. The second type tries to describe empirical profiles mathematically rather than explaining them. Much used models are the exponential, CðxÞ  expðaxÞ, and Gaussian models, CðxÞ  expðax  x2 =bÞ, sometimes accounting for time dependence by, e.g. a ¼ aðtÞ, b ¼ bðtÞ (e.g. Konshin, 1992; Velasco et al., 1993; Toso and Velasco, 2001). Due to their purely empirical basis, this type of models should not be used for predictions. This paper is devoted to a particular analytical model of the first type, assuming physico-chemical processes to control the vertical migration of radionuclides in soil. These are convection and diffusion as transport mechanisms and sorption as interaction mechanism of radionuclides in liquid and solid phase. By diffusion, we understand an aggregation of dispersive processes which are of diffusion type in a mathematical sense, i.e. resulting from a random walk (Brownian motion, Einsteinian diffusion). For a good introduction into soil physics, see e.g. Roth (1996). In Section 2, the model will be presented in more detail. Its limitations will be discussed as well as a straight-forward way to solve the associated equation for a specific boundary condition. The procedure is shortly presented because it is somewhat different of that commonly given in the literature (e.g. Bunzl, 1978). Some properties of the solution will also be discussed. Section 3 presents the

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measurements to which the model will be fitted; the actual results of the fitting procedure, their statistical evaluation and discussion is the subject of Section 4. Whereas part 1 of the series discusses the solution of the simplest (also called standard) version of the convection–dispersion equation (CDE) model, future parts will be devoted to discussing the impact of various initial and boundary conditions on the model’s solution, to discuss model generalizations and to compare the CDE model to alternative modelling approaches.

2. Transport model 2.1. Model conceptualisation The model is based on the diffusive–convective transport Eq. (1), the continuity or conservation Eq. (2) and the interaction (sorption) Eq. (3). The basic, simplified model which will be dealt with here, looks as following (for the more general case see Appendix A): Jðx; tÞ ¼ D0

@CL ðx; tÞ þ v0 CL ðx; tÞ @x

ð1Þ

@Cðx; tÞ @Jðx; tÞ ¼  kCðx; tÞ @t @x

ð2Þ

CS ðx; tÞ ¼ kd CL ðx; tÞ

ð3Þ

ðlinear equilibrium sorptionÞ

In the following, the usual abbreviations f 0 :¼ @f =@x and f_ :¼ @f =@t will be used. In (1)–(3), the following notations have been used: x: depth in soil (cm); x ¼ 0: soil surface; t: migration time (years); CL;S(x,t): concentration in the liquid/sorbed phase, Bq/cm3; C(x,t): total volumetric concentration, Cðx; tÞ ¼ CS þ wCL ; w: water content, cm3 water/cm3 soil; J(x,t): flux, Bq/cm2a; D0 : dispersion constant which comprises molecular diffusion and hydrodynamic dispersion (a tensor-valued function of x in the general case) of the radionuclide in soil solution, cm2/a; v0 : interstitial water flow velocity (a vector-valued function of x in the general case), cm/a; kd: partition coefficient (dimensionless), as simplest case of the general sorption function f. Normally, the gravimetric definition of CS is used, Bq/g soil, which introduces the soil density into the equation and provides the commonly known dimension m2/kg to kd; k: radioactive decay constant, s1, representing a particular sink qðx; tÞ ¼ kCðx; tÞ. Here, C(x,t) is always the so-called residence concentration, Cðx; tÞ :¼ d3 Mðx; tÞ=dx3 , with M(x,t), the activity of a radionuclide in depth x at time t. This has to be distinguished from the flux concentration Cf ðx; tÞ :¼ Jðx; tÞ=Jw (Jw, water flux), resulting from sampling the solute over a time interval, rather than a space interval as with C(x,t) (see, e.g. Roth, 1996: p. 71f). The concentration modes are related through the conservation equation and, in general, differ in magnitude.

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Substitution of the sorption Eq. (3) yields a pair of linear first-order partial differential equations, which will be the base of further discussion: C_ ¼ J 0 ; J ¼ DC 0 þ vC

ð4Þ

with D ¼ D0 =Rd , v ¼ v0 =Rd , Rd ¼ w þ kd ¼ retardation factor. D and v are called effective or apparent dispersion coefficient and convection velocity, respectively. They also take into account soil porosity and tortuosity (Bunzl, 1978; Smith et al., 1995). Traditionally, the pair is merged into the well-known, one variable (C) secondorder convection–diffusion equation by forming the x-derive of the transport Eq. (1) and substitution into (2): C_  DC 00 þ vC 0 ¼ 0

ð5Þ

However, it has been pointed out (Kreft and Zuber, 1978; Parker and van Genuchten, 1984) that Eq. (5) is conceptually ambiguous since it describes the transport of two physically different concentration modes. Both the flux-averaged concentration Cf(x,t), or break-through curve, of a solute (measured e.g. in the outflow of a soil column) and the resident concentration C(x,t) (measured, e.g. by soil coring) satisfy Eq. (5), although these two types of concentrations differ not only conceptually, but in general also in magnitude. For a detailed discussion of these two concentration modes and the ambiguity of the classical CDE with regard to them, the reader is referred to Jury and Roth (1990). In contrast to this ambiguity of the classical convection–dispersion, the pair of first-order equations given in Eq. (4) is unequivocally formulated in terms of the resident concentration. In the following, a solution is presented which is based on the first order, two variable (C, J) pair of Eq. (4). 2.2. Simplifications of the model This model is of course a simplification whose validity must be thoroughly tested by checking its ability to describe vertical radionuclide depth distributions and their evolution with time, and to result in parameter values which reflect the physico-chemical properties of radionuclide transport in soils. In our opinion, the most important simplifications are: 1. No horizontal component of the transport is being considered, i.e. the model is purely one-dimensional in space. This seems to be realistic in most cases in that the vertical component is probably the dominant one. However, certain conditions may give rise to a substantial horizontal transport component, such as a gravitational force along a slope. A horizontal concentration gradient due to strong local variability of fallout concentrations (which has recently been observed, e.g. Lettner et al., 2000) will produce a horizontal diffusive flux. Macroscopic soil inhomogeneities such as stones and macropores, and spatial soil anisotropy will also cause a deviation of the flux from being purely vertical.

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2. The model parameters (v and D) are being considered constant over the soil column. For layered or vertically anisotropic soils, this is a simplification, but the ‘‘mean’’ parameters over the considered column depth, which result from the model fitting procedure, may still be typical for a soil/radionuclide system. 3. The assumption of linear sorption equilibrium may be an oversimplification. The linear model can hardly be justified theoretically; in contrary, a more realistic first-order model leads to a Langmuir sorption isotherm (Roth, 1996; for the Kd concept as an approximation to the Langmuir isotherm: Gutierrez and Fuentes, 1993). There is ample evidence that sorption of Sr and Cs depends non-linearly on its concentration in soil (e.g. Torstenfeld et al., 1982; Smith, 1990; Kirchner et al., 1993; Hsu et al., 1994; Kirchner et al., 1996). However, no analytical solution of the resulting non-linear CDE is available. 4. The anticipation of sorption equilibrium is probably justified for long-term migration periods, as equilibrium is established quickly (within hours to days: e.g. Thiry et al., 1994; Kirchner et al., 1996, or within minutes: Bunzl and Schimmack, 1991; Hsu et al., 1994). However, a fast non-equilibrium migration which is effective within a few hours after deposition of the Chernobyl fallout has been observed to produce initial depth distributions different from those predicted by equilibrium models (Giani et al., 1987; Schimmack et al., 1989; Do¨rr and Mu¨nnich, 1991; Anisimov et al., 1991). 5. In the model, only two radionuclide containing phases have been considered: the mobile soil solution (represented by CL) and the reversibly sorbed phase (CS). For caesium however, the experimental results (Comans and Hockley, 1992; Shand et al., 1994; Andersson and Roed, 1994; Fawaris and Johanson, 1995) indicate that some fraction may become almost irreversibly fixed. Mathematically this corresponds to a sink term in the continuity Eq. (2). 6. If initially a significant fraction of fallout is present in the form of hot particles or fuel fragments, which are eroded in soil only slowly, the CDE is modified by a source term Q(x,t) in Eq. (2). This has been found by Krouglov et al. (1998) and Arkhipov et al. (2001) in soils near Chernobyl. However, the fraction of hot particle-bound fallout was very low in Central Europe and mainly restricted to so-called type A-particles (essentially condensed ruthenium). 7. The parameters v and D are considered constant over time. There is no reason to state a time-dependence of v0 and D0 , but this can be different for the apparent quantities v and D, via the sorption constant kd (which may change with time). This could result in a slow fixation process different from the one discussed above (5). 8. The parameter D combines two different physical processes, molecular diffusion and hydrodynamic dispersion of the solute, into a single constant. It has been pointed out (Jury and Sposito, 1985) that this dispersion model is specific inasmuch as it shows a linear growth of the variance of a particle’s travel time (and hence, migration rate) with transport distance. In real soils showing spatially varying hydrodynamic and sorption properties, the dispersion of radionuclides may increase with the square of the transport time (Kirchner, 1998). This is related to the structure of the flux Eq. (1), which assumes a linear coupling of

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the flux J and the concentration gradient, J ¼ DCL0 which is equivalent to a Gaussian random walk (i.e. Brownian motion), where the mean square displacement of a particle is proportional to the travelling time, hrðtÞ2 i  t. This socalled Fickian or normal diffusion can be generalized to hrðtÞ2 i  ta , a 6¼ 1, corresponding to non-linear relations J ¼ DðCÞ CL0 . Finally, boundary and initial conditions must be defined. As boundary conditions, a half-infinite space-time, x; t 2 ½0; 1Þ is assumed. The solution shall be finite, from which it follows Cðx ! 1; tÞ ! 0. The initial conditions are Cðx; 0Þ ¼: C 0 ðxÞ and Jð0; tÞ ¼: J 0 ðtÞ. The simplest case, which will be dealt with in the following, is C0 ðxÞ  0

and

J0 ðtÞ ¼ J0 dðtÞ;

ð6Þ

a pulse-like input at t ¼ 0 with deposition density J0 (Bq/cm2). 2.3. Solution A convenient way to solve Eq. (4) is to use the double Laplace transform technique. The main steps of the calculation are shown in Appendix B, for more details see Bossew (1997). For a general input function (flux at the surface) J(0,t), the solution is given as   rffiffiffiffiffi  ðt 1 v vx=D v s x 0 0 ks ðxvsÞ2 =ð4DsÞ pffiffiffiffiffiffiffiffi e e Cðx;tÞ ¼ dt Jð0;t Þe  erfc þ pffiffiffiffiffiffi 2D 2 D 2 Ds pDt 0 ð7Þ where s :¼ t  t0 . pffiffiffiffiffiffiffiffiffi The argument of the erfc-term can also be written as ðvs þ xÞ= 4Ds. For a pulse-like input J0 at t ¼ 0, Jð0; tÞ ¼ J 0 dðtÞ, we get the well-known solution (e.g. Bunzl, 1978; Roth, 1996; Ivanov et al., 1997; Kirchner, 1998), ( !) rffiffiffiffiffi 1 v vx=D v t x kt ðxvtÞ2 =ð4DtÞ pffiffiffiffiffiffiffiffi e Cðx; tÞ ¼ J0 e e  erfc þ pffiffiffiffiffiffi ð8Þ 2D 2 D 2 Dt pDt The variable transform vx=ð4DÞ ! x, v2 t=ð4DÞ ! t (x, t now dimensionless) yields the normalized form C(x,t), (  ) 2 pffiffi 1 eðx =tÞt x 2x 2x pffiffiffi pffiffi cðx; tÞ ¼ e t þ pffiffi  e erfc p t t ! z2 pffiffi e x ¼ e4x  erfcðzÞ ; with z :¼ t þ pffiffi ð80 Þ pt t Fig. 1 shows the function C(x,t) for four normalized migration times t ¼ 0:1; 0:3; 0:5; 1.

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Fig. 1. Normalized profiles of radionuclides in soil for four different normalized migration times.

Frequently, the solution 1 2 Cðx; tÞ ¼ C0 pffiffiffiffiffiffiffiffi eðxvtÞ =4Dt 2 pDt

ð9Þ

of the CDE (Eq. (4) or (5)) has been used to model the depth distribution of radionuclides in soils after a single deposition event (e.g. Kirchner and Baumgartner, 1992; Konshin, 1992; Szerbin et al., 1999). However, Eq. (9) is not a solution of the CDE with the initial and boundary conditions (6), although for t2D=v2 , it becomes a good approximation (Roth, 1996: p. 78; for typical values D ¼ 0:1 cm2 =a, v ¼ 0:1 cm=a, this is t20 years). In general, Eq. (9) underpredicts solute concentrations in soil, since it represents the solution of the CDE for an infinite space with tracer injected at t ¼ 0 moving in both directions (Kreft and Zuber, 1978). 2.4. Some properties of the solution (a) First, Eq. (8) shows an interesting symmetry: with the transformation t ! ct, D ! D=c, v ! v=c the solution is formally invariant, Cðx; ct; v=c; D=cÞ ¼ Cðx; t; v; DÞ. By setting c ¼ Rd (retardation factor), it follows immediately that Eq. (8) is equally valid for any solute showing sorption according to Eq. (3). (b) An incausality in the solution may be noted: for any, also very small t > 0, Cðx; tÞ > 0 for any, also very large x, which implies an infinitely fast propagation of some solute fraction. This is an effect of the boundary conditions; in order pffiffiffiffiffiffiffiffi to be causal, an additional condition, such as Cðx; tÞ ¼ 0 for x > ca: v0 t þ 2Dt must be introduced. For modelling, the long-term migration of solutes in soil, however, this incausality can be ignored.

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pffiffiffiffiffiffiffiffi (c) For small arguments ðvs þ xÞ= 4Dt (e.g. small migration times t), the erfcterm becomes negligible and the solution can be approximated with 1 2 Cðx; tÞ ¼ C0 pffiffiffiffiffiffiffiffi eðxvtÞ =ð4DtÞ pDt

ð10Þ

(identical with (9) apart from the factor 2 in the denominator of (9)). ‘‘Young’’ depth distributions have often been found (e.g. Miller et al., 1990; Poiarkov et al., 1995; Isaksson and Erlandsson, 1995) to be reasonably well described by an exponential decrease with depth, i.e. Cðx; tÞ ¼ Cðx ¼ 0Þ expðbtÞ

ð11Þ

where b is a purely empirical parameter, the inverse of which is often called relaxation length. Unfortunately, to our knowledge, no comparative assessment of Eqs. (10) and (11) is available. An overall better performance of Eq. (11) (for ‘‘young’’ profiles) could point to deficiencies of Eq. (10) and thus of the convection–dispersion model. (d) The locations of the maximum (mode) and the half value depth (median, i.e. the depth above and below which half of the inventory reside) cannot be given as analytical formulae. However, asymptotically with large t, both become equal to x0 ¼ vt. This follows from the first moment (mean), ð 1 1 xCðx; tÞdx; MðtÞ :¼ J0 0 for which holds    s pffiffiffi t!1 _ ðtÞ ¼ v 1 þ 1 p1ffiffiffi epffiffiffi  erfc s M ! v; 2 p s with s, normalized migration time, v2t/4D. Asymptotically, mean, maximum and half value depth must move with same speed due to the asymptotically Gaussian shape of the function C(x,t). Similarly, for the variance, ð 1 1 V ðtÞ :¼ ðx  MðtÞÞ2 Cðx; tÞdx; J0 0 we find that asymptotically V ðtÞ ¼ 2Dt for t ! 1. It can be shown that the position of the maximum, xm, can be better approximated as   0:8 xm ðtÞ  vt 1 þ 1:6 þ v2 t=D (for large t: xm ! vt, as stated above, for small t: xm ! 0 as it should be).

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The half value depth can be approximated as x1=2

D  vt þ v

1

0:801 ð0:0083 þ ðv2 tÞ=ð4DÞÞ

0:0443

135

! :

(e) A useful generalization is (for details see Appendix C) that asymptotically for large t, the solution C(x,t) (Eq. (7)) no longer depends on the deposition history J0(x,t), if only the input has ended. Therefore depth distributions originating from any, including the weapons fallout deposition history, can be approximated by Eq. (8) if only sufficient time has passed after the input ceased. The end of weapons fallout deposition can essentially be set to the early 1970s. As an approximation, the 1 January 1965 will be defined as input date for global fallout, since 1964 was the year of maximal fallout.

3. Depth distribution data 3.1. Materials and methods In the course of various radio-ecological survey and modelling projects in Austria since 1987, more than 500 soil profiles have been taken and investigated for concentrations of gamma emitting radionuclides. Most soil profiles were taken with a spade as soil ‘‘cubes’’ with an area of typically 18  18 cm2 and a depth of 10–20 cm. Some samples were taken as soil cores with a coring device. The cubes or cores were transferred to the laboratory and sliced into horizontal layers (thickness below 1 cm up to 2 cm), the dimensions as well as wet and dry weights of which were determined individually. Therefore for all profiles bulk density and water content profiles are available (although not needed for evaluations presented here). The sampling locations can roughly be attributed to six regions of different landscape type and geology which include: 1. 2. 3. 4. 5. 6.

Northern Austrian granite region promontory hill-type areas (mainly tertiary sediments) flatlands (mainly quarternary sediments) Limestone Alps Central Alps (granite, gneiss) Koralpe (gneiss).

Due to the large number of soil profiles taken, soil and physical and chemical analyses of the individual samples could not be performed. Assigning the sample locations to one of the regions listed above, therefore could introduce some error by neglecting small scale variabilities of geology and soil types.

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The majority of the resulting samples was measured with the HPGe system (Canberra) of the Austrian Institute of Applied Ecology, Vienna, the rest with the HPGe system (Ortec) of the university of Salzburg. The 137Cs fractions originating from global and Chernobyl fallout, respectively, were identified if 134Cs (only from Chernobyl) was present using the known 137/134Cs ratio, 1:764  0:088; referred to 1 May 1986 (Mu¨ck, 1988). Furthermore, to correctly determine 134Cs (and thus 137 Cs/b) summation effects in gamma spectrometry were taken into account. 3.2. Statistical analyses The data are stored in ASCII files which can be read by a home-made program which fits the CDE model to them according to the following least-square scheme: 2 Xð xi2 v2 ¼ Cðx; tÞdx  yðiÞðxi2  xi1 Þ ¼ min i

xi1

where the index i denotes the soil layers, xi1 and xi2 are the lower and the upper depths of the soil layer i, respectively, and y(i) is the measured volumetric concentration (residence concentration, see Section 2.1) of soil layer i. An iterative procedure was used for fitting, which was stopped when the change of v2 was below a preset threshold. Parameters adjusted were v, D and J0. The migration time t was calculated as the interval between deposition (1 May 1986 for Chernobyl and 1 January 1965 for atomic weapons fallout, see 2.3.e) and sampling date. Fig. 2 shows an example of a CDE solution fitted to an empirical profile. 4. Results and discussion 4.1. General results Since the inventories due to Chernobyl fallout are relatively high in some parts of Austria (Bossew et al., 2001), ‘‘minor’’ nuclides from Chernobyl fallout including 106Ru, 144Ce and 110mAg were often detectable until the late 1980s and early 1990s; later only 134,137Cs and sometimes 125Sb could be determined. Other long lived gamma radionuclides, like 145,155Eu, 94gNb and 60Co which are present in Chernobyl fallout in minute traces can only be measured in very few cases, so that no statistical evaluation was possible. 241Am (mainly from global fallout) can be detected quite frequently but as being produced by the decay of 241Pu its distribution in soil follows a more complex input function than modelled here. Therefore it will not be discussed in the following. Of the 528 soil profiles which were investigated, only fitting results producing parameters v and D > 0 were used for statistical evaluation, i.e. those cases, for which the fitting algorithm could detect a parameter value greater than zero. The result v ¼ 0 effectively means that the resolution of the observed profile (thickness of the layers) was not fine enough to indicate a displacement of the maximum from the surface, and therefore the fitting algorithm yielded v ¼ 0. There is however no reason to believe that the velocity is strictly zero in a real situation. Therefore such

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137

Fig. 2. Observed and fitted profile.

cases (mostly profiles taken in 1987–1989 in soils with low v, i.e. small displacement  vt) were excluded from further statistical evaluation. For 134 Cs, e.g. this results in 519 values for D and 329 for v, respectively. The summary statistics of the fitted parameters is shown in Table 1 for all regions and each region separately. For the velocity v, the actual data were used (Table 1) since they approximately follow a normal distribution (Fig. 3) whereas for the diffusion coefficients values of log(D) are given (Table 1) since this parameter can be approximated by a lognormal distribution. Typical values are 0.1–0.5 cm/a for the velocity v, and 0.05–0.5 cm2/a for the diffusion constant D, respectively. 4.2. Differences between regions In Table 1, it can be observed that in most cases the coefficient of variation of the migration velocities within a region is smaller than that of the total dataset. This indicates that the regions in fact represent distinct environments with regard to the migration behaviour of the radionuclides studied. A visual inspection of the frequency distributions confirms this result, as in most cases, the frequency distribution associated to any single region appears to be more regular than that of the total dataset; e.g. see Fig. 3 (top). One notable exception is the migration velocity of 134Cs in region 5 (central Alps) with a very high variability (102%). The associated, quite irregularly looking frequency distribution is shown in Fig. 3 (bottom).

Cs/b

137

(a) Parameter v; unit of AM, SD and Med: cm/a 1 n 48 AM  SD 0:170  0:075 CV% 30 Med 0.153 2 n 34 AM  SD 0:126  0:071 CV% 57 Med 0.128 3 n 10 AM  SD 0:191  0:047 CV% 40 Med 0.170 4 n 19 AM  SD 0:118  0:047 CV% 40 Med 0.116 5 n 140 AM  SD 0:086  0:041 CV% 47 Med 0.085 6 n 55 AM  SD 0:106  0:055 CV% 52 Med 0.095 All n 308 AM  SD 0:113  0:063 CV% 56 Med 0.102

Region 55 0:31  0:21 66 0.27 57 0:49  0:26 52 0.50 20 0:43  0:21 49 0.41 19 0:27  0:23 83 0.23 133 0:21  0:22 104 0.15 44 0:27  0:20 74 0.24 329 0:30  0:24 81 0.24

Cs

134

50 0:58  0:46 79 0.46 18 0:51  0:29 58 0.45 3 0:461  0:091 20 0.486 13 0:39  0:18 46 0.41 55 0:35  0:31 76 0.35 34 0:37  0:25 66 0.37 173 0:46  0:35 75 0.41

Ru

106

Table 1 Summary statistics of parameters v and D of different nuclides, classified for regions 1–6

3 0:81  0:84 103 0.68

8 0:66  0:66 99 0.40

2 0:34  0:18 54 0.34 4 0:94  0:47 50 0.84

17 0:50  0:42 84 0.42

2 0:26  0:03 11 0.26

Ce

144

9 0:42  0:34 80 0.28 53 0:39  0:27 69 0.29 32 0:32  0:20 61 0.29 147 0:42  0:27 66 0.36

Ag

9 0:32  0:22 66 0.34

110m

36 0:46  0:27 89 0.048 15 0:63  0:31 48 0.68

Sb

125

138 P. Bossew, G. Kirchner / J. Environ. Radioactivity 73 (2004) 127–150

Table 1 (continued )

Cs/b

137 2

Cs

134

63 0.49
/3.4 0.79 24 0.82
/3.6 1.00 5 0.56
/3.3 0.90 19 0.51
/4.8 0.68 105 0.26
/2.7 0.24 54 0.50
/3.6 0.61 271 0.45
/3.7 0.41

Ru

106

52 0.80
/3.3 1.05 17 0.64
/3.7 0.54 2 2.65
/1.5 2.65 14 0.41
/3.7 0.58 87 0.26
/3.0 0.26 50 0.43
/3.5 0.44 223 0.43
/3.5 0.41

Sb

125

4 0.76
/10 0.85

10 0.52
/6.4 0.57

2 0.18
/1.5 0.18 10 0.31
/3.5 0.19

27 0.52
/3.6 0.44

Ce

144

3 1.54
/2.0 1.12

Ag

13 0.72
/3.1 0.78

110m

n, number of samples; AM, arithmetic mean; SD, standard deviation; GM, geometric mean ¼ expðAMflogðxÞgÞ; GSD, geometric standard deviation ¼ expðSDflogðxÞgÞ; CV% ¼ 100  SD=AM ¼ coefficient of variation; Med, median; 137Cs/b, 137Cs from global fallout (bomb-137Cs).

(b) Parameter log D; unit of GM and exp(Med): cm /a 1 n 49 90 GM
/GSD 0.19
/3.3 0.77
/4.0 exp(Med) 0.18 0.46 2 n 36 71 GM
/GSD 0.13
/5.7 0.68
/3.9 exp(Med) 0.25 0.95 3 n 11 30 GM
/GSD 0.36
/2.8 0.72
/2.6 exp(Med) 0.36 1.50 4 n 20 27 GM
/GSD 0.059
/5.1 0.29
/2.9 exp(Med) 0.047 0.28 5 n 150 223 GM
/GSD 0.045
/3.2 0.17
/2.7 exp(Med) 0.043 0.15 6 n 60 75 GM
/GSD 0.16
/3.0 0.27
/2.8 exp(Med) 0.17 0.27 All n 328 519 GM
/GSD 0.087
/4.2 0.30
/3.4 exp(Med) 0.084 0.27

Region

P. Bossew, G. Kirchner / J. Environ. Radioactivity 73 (2004) 127–150 139

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P. Bossew, G. Kirchner / J. Environ. Radioactivity 73 (2004) 127–150

Fig. 3. Frequency distributions of parameter v, for all regions (left) and for region 5 (right).

134

Cs and

137

Cs/b (=137Cs from global fallout); values

This may indicate that this geographical classification is too general; it has in fact been observed that in Alpine environments the Cs migration behaviour can be very different depending of the type of meadow (Bossew et al., 2000). The hypothesis that the selected regions represent different migration conditions was tested with analysis of variance (ANOVA) technique, using the region as grouping parameter. For 137Cs/b (global fallout), 134Cs and 125Sb, the grouping is significant for both parameters v and D; for 106Ru, this holds for D only and for 110m Ag, for v only. Statistical details are summarized in Table 2. The significance could probably be improved with a more detailed definition of the regions. However, even on the basis of quite coarse grouping, these results clearly demonstrate that the fitted parameter values v, D do not only represent arbitrary values of free fit parameters, but reflect the physico-chemistry of radionuclide transport in the soil studied. As a consequence model and parameters derived in our study could be used for predictive modelling. 4.3. Differences between radionuclides Table 1 also demonstrate that in terms of v and D, the radionuclides show differing migration behaviour. For any pair of radionuclides, this has been quantified by

Ag

Ce

144

110m

Sb

125

Ru

106

Cs

134

Cs/b

137

Bartlett F Kruskal–Wallis Bartlett F Kruskal–Wallis Bartlett F Kruskal–Wallis Bartlett F Kruskal–Wallis Bartlett F Kruskal–Wallis Bartlett F Kruskal–Wallis

Test 2.1e7! (0) 3.5e12 0.57 0 0 4.5e5! (0.050) 0.18 0.21 0.0080 0.018 0.25 0.013 0.056 0.045! 0.45 0.59

v 6.7e3! (0) 0 6.8e3! (0) 0 0.029! (0) 2.8e7 0.74 0 9.0e6 0.57 0.085 0.11 0.16 0.36 0.32

log D

Table 2 ANOVA of parameters v and D of different nuclides for factors ‘‘region’’. 1st line: Bartlett test for homogeneities of variances: if pðBartlettÞ < 0:05 (denoted by !), then the condition for the F-test of homogenous variances is not fulfilled and the F-test therefore meaningless. Second line: F-test: if p < 0:05, then there is a significant difference of means. If pðBartlettÞ < 0:05, then p(F) cannot be used. Third line: Kruskal–Wallis test: parameter free, more robust. Bold values of p: significant (p < 0:05)

P. Bossew, G. Kirchner / J. Environ. Radioactivity 73 (2004) 127–150 141

b

a

GM GSD ratios which are significantly 6¼1b 1.31 12 0.51 18 0.63 10 2.85 4.1 0.65 3.1 0.62 3.3

+/+/> 

 

Sb

125

Significance of paired sample t-test/significance of Wilcoxon test/log(i)log(j)<>0. þþ p < 0:05; þ 0:05  p < 0:10;  p  0:10. GM, geometric mean; GSD, geometric standard deviation (see legend of Table 1).

Ratio (c) Comparison of some nuclide v(134Cs)/v(137Cs/B) v(134Cs)/v(106Ru) v(134Cs)/v(125Sb) D(134Cs)/D(137Cs/B) D(134Cs)/D(106Ru) D(134Cs)/D(125Sb)

134

  

a

Ru

106

(b) Tests for parameter D: paired sample test of logðDi Þ  logðDj Þa Cs ++/++/< 106 Ru ++/++/< ++/++/< 125 Sb ++/++/< ++/++/< 110m Ag ++/++/<  144 Ce ++/++/< 

Cs

134

/+/< +//> 

Cs/b

137

(a) Tests for parameter v: paired sample test of logðvi Þ  logðvj Þ 134 Cs +/++/< 106 Ru ++/++/< ++/++/< 125 Sb ++/++/< ++/++/< 110m Ag  /++/< 144 Ce ++/+/< 

j# i!

Table 3 Comparison of parameters vi and Di associated to different nuclides (i)





110m

Ag

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P. Bossew, G. Kirchner / J. Environ. Radioactivity 73 (2004) 127–150

143

performing paired sample t-tests and Kruskal–Wallis tests of the logarithms of individual pairs of v (and D, respectively), including all soil samples for which parameter values of the two radionuclides are available (thus testing the ratios v(106Ru)/v(134Cs), etc., against the null hypothesis H0: ratio ¼ 1). Details of the statistical analyses are given in Table 3. Most interestingly, values of both mobility parameters increase (p < 0:05, one sided) in the order 137Cs/b < 134 Cs < 106Ru  125Sb. In addition, for v, 137Cs/b < 144Ce and for D, 137Cs/b < 110m Ag, 144Ce are significant. As the ratios of the migration velocities and dispersion coefficients of different radionuclides are approximately log-normally distributed (Fig. 4), Table 3 shows the geometrical means and standard deviations of some of these ratios. Apparently, both migration velocity v and dispersion D are in general higher for 106Ru and

Fig. 4. Frequency distributions of the ratios of parameters v and D associated to different nuclides.

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125

Sb than for 134Cs. This result is likely to reflect the well-known effective sorption of Cs by clay minerals which are almost ubiquitous in soil. 4.4. Differences between Chernobyl and weapons fallout radiocaesium

Of the results shown in Table 3, particular attention should be given to the differing mobilities of Chernobyl (134Cs) and weapons fallout (137Cs/b) radiocaesium, since being chemical identical their mobilities should be equal. Potential explanations for this observation are as follows. It cannot be excluded that v and D may show vertical gradients, i.e. are not constant over the soil column, either as a consequence of variabilities of the retardation factor R with depth, R ¼ RðxÞ, or of a slow sorption process, R ¼ RðtÞ, as discussed above. However, no evidence is available to us of the potential role of any of these processes in the soils studied here. Another possible explanation is related to a particular idealization of the migration model (see the discussion in Section 2.1, topic 4). An initial fast migration as a result of slow sorption kinetics or fast infiltration caused by heavy rainfall during the deposition phase, would produce a profile, which, interpreted with the standard CDE model, yields CDE parameters v and, in particular D, which are too high as they reflect the initial fast transport phase. Only later, when the CDE-type migration becomes the dominant contribution to the shape of the profile (since the hypothesized fast component is only effective for a short initial time period), the model fitting produces parameters which are ‘‘genuinely’’ related to the CDE mechanism. In fact, most of the profiles investigated here have been taken between 1987 and 1989, i.e. only allowing for a relatively short migration time of Chernobyl compared to global fallout caesium. It should be noted that in various parts of Austria, deposition of Chernobyl fallout radionuclides was associated with heavy rainfall. We plan to explore these alternatives in more detail both by experiments and modelling.

5. Conclusions The convection–dispersion model has been fitted to more than 500 depth distributions of 106Ru, 110mAg, 125Sb, 134Cs, 137Cs and 144Ce which have been measured in soils taken in various regions of Austria. This to our knowledge most extensive model testing performed so far demonstrates that despite of its many simplifications, the convection–dispersion model adequately represents radionuclide profiles in undisturbed soils. Our study did not include soils which show large variabilities of hydrological or sorption properties between layers (e.g. forest soils), since the simple CDE model is not expected to give reliable simulations for those soils. Statistical analyses showed that the values of the apparent migration velocity v and the dispersion coefficient D which resulted from the fitting procedure can be grouped with regard to geography (and thus roughly soil type and geological base)

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145

and that parameter variabilities are significantly reduced within groups. This result indicates the values we determined are not mere fitting parameters, but are related to the physico-chemistry of radionuclide transport in soils included in our study. Radionuclide mobilities increase in the sequence: humic Alpine soils < podsols < brown earths ¼ chernozems. A more detailed classification of soil types may even further reduce parameter variabilities within groups. In general, mobilities of radionuclides in the soils studied increase in the sequence 137Cs (weapons fallout) < 134Cs (Chernobyl) < 106Ru  125Sb. The differing mobilities of the two Cs fractions indicate that transport of radiocaesium in soils may be influenced by processes not adequately taken into account by the CDE model and the assumptions used in our simulations.

Acknowledgements Financial support of one of the authors (P. Bossew) by the Austrian Ministry of Science is gratefully acknowledged.

Appendix A General, three-dimensional case of the convection–dispersion model ~CL ðx; tÞ þ~ ~ J ðx; tÞ ¼ DðxÞr vðxÞCL ðx; tÞ

ðA:1Þ

C_ ðx; tÞ ¼ div~ J ðx; tÞ þ qðx; tÞ

ðA:2Þ

CL ðx; tÞ ¼ f ½CS ðx; tÞ:

ðA:3Þ

Appendix B In the following the main steps of solving the system will be presented. Intermediate steps which involve some simple, but a bit strenuous analysis will be omitted. The first step is to form the Laplace transforms of Eq. (4) with respect to x, Lx , resulting in ^ ðu; tÞ þ DCð0; tÞ þ vC ^ ðu; tÞ J^ ðu; tÞ ¼ D u C ^ ^ ðu; tÞ  uJ^ ðu; tÞ þ Jð0; tÞ ¼ C_ ðu; tÞ: kC The variables in Laplace space corresponding to x and t are denoted by u and s, respectively. The ^-sign denotes the Lx transformed functions,  refers to Lt . (For extensive tables of Laplace transforms, see, e.g. Fodor (1965) or Oberhettinger and Badii (1973). The technique will not be further discussed here.)

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Next, the second equation is inserted into the first after some rearrangement:   ^_ ^ ðu; tÞ  k þ Du  v  C ðu; tÞ þ Jð0; tÞ  DCð0; tÞ ¼ 0: C u u u As second step, performing Lt yields   ~ ^ ^ ~ ~ ~ ð0; sÞ ¼ 0 ^ ðu; sÞ  k þ Du  v  sC ðu; sÞ þ C ðu; 0Þ þ J ð0; sÞ  DC C u u u u and after some rearranging: ~ ^ ðu; sÞ ¼ C

Du2

~ ð0; sÞ ^ ðu; 0Þ þ J~ ð0; sÞ uDC C  :  vu  ðs þ kÞ Du2  vu  ðs þ kÞ

ðB:1Þ

The back-transform of Eq. (B.1) is the solution which we are interested in. It can be seen that, as expected, the decay term (k) leads to a contribution exp(kt) when L1 is performed, since L1 ðf ðs þ aÞÞ ¼ expðatÞ L1 ðf ðsÞÞ. It will therefore be t omitted in the following for simplicity. The back-transform of (A.1) is a bit lengthy. Firstly, we define rffiffiffiffiffi   v s ~ ð0; sÞ; F :¼ 1 C ^ ðu; 0Þ þ J~ ð0; sÞ ; ; B :¼ A :¼ ; E :¼ C 2D D D which yields ~ ^ C ðu; sÞ ¼

uE F  2 : 2 u  2Au  B2  2Au  B

ðB:2Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Next, we back-transform (B.2), L1 A2 þ B2 ¼ x , using the discriminant d :¼ dðsÞ, which, after rearranging, gives u2

ðAdÞx eðAþdÞx ~ ðx; sÞ ¼ e ðEðd  AÞ þ #x F Þ  ðEðd þ AÞ þ #x F Þ C 2d 2d

ðB:3Þ

where #x denotes convolution with respect to x. Now comes the only tricky point of the derivation. The second term of (B.3) has p p the structure ðexp sÞ= s, which makes it divergent in general. Therefore we need a condition to this term to be convergent; however, this is less arbitrary than it may appear, since we have one degree of freedom left, namely the one which is provided by the presence of three dependent initial conditions in (B.1) (C(0,t), C(x,0), J(0,t)) rather than two independent ones. For practical reasons, we decide for eliminating C(0,t), i.e. expressing it (or its Laplace transform) in terms of the other two conditions, see Eq. B.5 below. As a necessary condition for the convergence of the second term on the right side of Eq. (B.3), we choose eðAþdÞx ½Eðd þ AÞ þ #x F  ¼: gðx; sÞ with limðs ! 1Þ gðx; sÞ < 1

ðB:4Þ

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147

~ ð0; sÞ, F ¼ J~ ð0; sÞ=D. Since J(0,t) and C(x,0) are independent, the cases (i) E¼C Jð0; tÞ > 0 and Cðx; 0Þ ¼ 0 and (ii) Jð0; tÞ ¼ 0 and Cðx; 0Þ > 0 can be investigated separately. Since this paper restricts itself to the discussion of pure fallout situations (i), i.e. not the propagation of existing profiles (ii), we do not need to further discuss case (ii) (a planned forthcoming paper will be devoted to this task). With C(x,0) DD 0 and inserting for E and F, the condition (B.4) reads   ~ ð0; sÞðd þ AÞ þ J~ ð0; sÞ=D ¼ gðx; sÞ eðAþdÞx C and hence   1 ~ eðAþdÞx ~ ð0; sÞ ¼ 1 J ð0; sÞ þ Dgðx; sÞ C : D d þA d þA

ðB:5Þ

Since C˜(0,s) does not depend on x, as opposed to the second term in the last equation, which does depend on x (otherwise g(x,s) would have to be ~edx, which is in contradiction to the definition of g, Eq. (B.4), it follows gðx; sÞDD0. Inserting C˜(0,s) into (B.3) and rearranging a bit, we find ðAdÞx 1 ~ ~ C ðx; sÞ ¼ ð1=DÞðe =ðd þ AÞÞJ ð0; sÞ. Further back-transforming, Lt , is straight-forward if transform tables are used. Not exactly surprisingly, the result is: Cðx; tÞ ¼ Jð0; tÞ#t ekt eðv=ð2DÞÞ=ðxðvt=2ÞÞ ( !) rffiffiffiffiffi 1 v ðv=ð2DÞÞ=ðxðvt=2ÞÞ v t x x2 =ð4DtÞ e  pffiffiffiffiffiffiffiffi e  erfc þ pffiffiffiffiffiffi ; 2D 2 D 2 Dt pDt which after rearranging gives the result (7).

Appendix C Let C(x,t;J1) and C(x,t,J2) be the solutions related to different deposition histories J1(0,t) and J2(0,t). Then Cðx; t; J1 Þ  Cðx; t; J2 Þ

J0;1 ; J0;2

Ð1 where J0,1 and J0,2 are the respective total deposits, J0 ¼ 0 Jð0; tÞdt. Proof. The functions C(x,t;J) are of the form Cðx; t; JÞ ¼ Jð0; tÞ # cðx; tÞ. Furthermore, lim f ðtÞ ¼ limsf~ðsÞ;

t!1

s!0

f~ðsÞ :¼ ðLf ÞðsÞ; LðJð0; tÞ#cðx; tÞÞ ¼ J~ ð0; sÞ~cðx; sÞ

Therefore J~ 1 ð0; sÞ Cðx; t; J1 Þ sJ~ 1 ð0; sÞ ¼ lim ¼ lim ; ~ 2 ð0; sÞ s!0 J~ 2 ð0; sÞ t!1 Cðx; t; J2 Þ s!0 sJ ~  ðt J~ ð0; sÞ J ð0; sÞ ¼ lim L1 limJ~ ð0; sÞ ¼ lims ¼ lim Jð0; sÞds ¼ J0 ; t!1 t!1 0 s!0 s!0 s s lim

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hence lim

t!1

Cðx; t; J1 Þ J0;1 ¼ ; Cðx; t; J2 Þ J0;2

q:e:d:

Therefore two profiles which are due to two different deposition histories with same integral (J 0;1 ¼ J 0;2 ) are asymptotically equal.

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