Each substance in the environment, including radioactive isotopes, interacts with groundwater and geological formations (soils and rocks). Transport in the pores of rocks and soils occurs via the migration of water-soluble materials. The migration in porous solid media is influenced both by hydrological processes and by the inter­action between the soluble substances and the geological formations. The migration of a substance in a porous solid medium is influenced by the flowing medium (typ­ically groundwater in geological formations), the chemical species of the migration substance, and its sorption properties. Thus, the migration is affected by the follow­ing factors:

• Advection: the migration of soluble components with flowing medium.

• The mixing of solutions in macropores of the solid medium, which is due to the different flowing rates of solutions in the pores with different sizes.

• Diffusion of dissolved components in the liquid phase.

In addition, the interactions of solid matrices and dissolved substances are the following:

• Adsorption and ion exchange

• Precipitation

• Structural modification and destruction of materials.

In the case of the migration of nonsorbing substances, the transport of the dis­solved substances is determined by the first three processes, namely advection, mixing, and diffusion. This means that the dissolved components move with water (e. g., chloride ions in geological formations). The flux of flow is described by dif­ferent migration equations. A frequently used migration equation is:

Sc Sc

J0 = -[0Dh + 0Deff ] 7е + v0c = —0D 7е 1 qc (9.36)

ox ox

where J0 is the flowing rate of water (flux), 0 is the humidity of medium, Dh is the hydrodynamic dispersion coefficient, Deff is the effective diffusion coefficient, c is the concentration of flowing material at place x, v0 = q is the volume flowing in a unit time, and v is the linear flowing rate.

If the solid medium reacts with the dissolved components (e. g., by adsorption or ion exchange), their flowing rate decreases:

where p is the density of the matrix and a is the sorbed amount.

When the dissolved components are adsorbed on the solid matrix or they take place in ion exchange reactions, or precipitate, their migration rates can decrease significantly. The degree of decrease is determined by the chemical spe­cies of the given substance under chemical conditions characteristic of the solutions in geological formations (groundwater).

As discussed in Section 9.3.2.1.1, the migration equations (including Eqs. (9.36) and (9.37)) cannot be solved generally; only partial solutions can be obtained in certain initial and boundary conditions. In addition, the sorption has to be included, e. g., by a sorption isotherm equation that describes the relation between the con­centration of the dissolved components in the solid and solution phases. As an example, the Langmuir adsorption isotherm is mentioned:

where m is the mass of the adsorbent (g), w is the pore volume saturated with water (dm3), y is the ratio of the substance dissolved in the pore water, x is the ratio of the substance sorbed on the solid phase, k is the distribution coefficient (k = x/y), Ce is the equilibrium concentration of the solution, a is the adsorbed quantity, z is the number of the active sites of sorption, and K is the parameter that is characteris­tic of the sorption energy.

Neglecting the advection, Eq. (9.37) is simplified as follows:

By substituting the adsorbed quantity (a) from Eq. (9.38), we obtain:

8C w 8C @2C

1 k = D 8t m 8t 8×2

and from here,

The quantity of the value in brackets can be interpreted as migration coefficient (Dm):

Equation (9.42) is equivalent to Fick’s second law (Eq. (9.32)), but the interpreta­tion of Dm is slightly different. A similar mathematical procedure can be applied for ion exchange too.

Figure 9.7 Migration cell for the study of radionuclide transport. Source: Reprinted from Nagy and Konya (2005), with permission from Elsevier.

The denominator of Eq. (9.41) is called the “retardation factor,” which is the ratio of the migration coefficients of a nonsorbing substance (e. g., chloride or the migrating medium, water, itself) and a sorbing substance.

As discussed in Section 7.3, geological repositories play an important role in the safe storage of nuclear waste. The migration rate of the radioactive isotopes, both in the engineering barrier system and in the surrounding geological formation, is a sig­nificant factor that must be considered. The migration rate of the radionuclides stud­ied in laboratory model experiments using a migration cell is shown in Figure 9.7.

In the migration cell, the sample (a bentonite clay layer in Figure 9.7) is located in the middle of the donor and receptor half-cells. The solution of the studied radio­active isotopes is filled into the donor half-cell and is permitted to migrate through the sample in the middle. As discussed in Section 9.3.2.1.1, there are two possibili­ties to determine the diffusion coefficient: the first is that solution samples are taken at different times from the receptor cell and the concentration of the migrat­ing substances is determined as a function of time (in this case, x = constant). The other possibility is that the rock sample is cut into thin layers after a given time (t = constant), and the concentration of the migrating substances is determined as a function of distance.

The solution of Fick’s second law for this migration cell is as follows, assuming the boundary condition (C = C0, x = 0, t > 0) and the initial condition (C = 0, x > 0, t = 0):

where erfc (z) = 1 — erf(z), where z is the fraction behind the erfc function in Eq. (9.43).

The two possibilities to determine the diffusion coefficient are shown in Figures 9.8 and 9.9.

These figures show the migration studies of chloride ions labeled by the 36Cl isotope and carrier-free 137Cs+ ions in bentonite clay. The diffusion coefficient of a chloride ion provides the maximum migration rate in the bentonite clay because the chloride ion is not sorbed in the clay. However, cesium ions are fairly easily sorbed by cation exchange in bentonite clay. As a result, the diffusion coefficient in this case is about two orders of magnitude lower than in the case of chloride ion. This illustrates that the sorption process in clay plays an important role in the isola­tion of radioactive ions from the environment.

8000 -|

7000

6000

5000 О

4000

II

3000 2000 1000 0

0.0E+00 5.0E+05 1.0E+06 1.5E+06 2.0E+06 2.5E+06 3.0E+06
t (s)

Figure 9.8 Migration of 36Cl_ ions bentonite clay: intensity proportional to the concentration versus time. Diffusion coefficient calculated by Eq. (9.43) is 7.76 X 10_12m2/s.