A review of porosity and Diffusion in Bentonite (pdf) (2.4 MB) - Posiva

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19More simply, the individual concentrations in DDL water can be calculated by averagingthe potential in DDL water, and calculating the concentration with Boltzmann’sequation: - ziFDDLcDDL, i ciexp(34) RTwhere ψ DDL is the averaged potential in DDL water. Appelo and Wersin (2007) haveshown that the averaged concentration gives an accurate approximation for the totalconcentration that follows from integrating the exponential concentration distribution inthe DDL. With Equation (34), the flux through DDL water can be calculated as shownin the next section.3.2 Which concentration gradient should be used?Many authors, cited before, distribute a high proportion of the mass of cations to DDLand/or interlayer water, and calculate the diffusive flux without realizing that the drivingforce for diffusion is not the concentration, but the concentration gradient (formally, thepotential gradient). Imagine two tubes, one filled with 1 M NaCl, the other with 0.01 MNaCl, while the concentration gradient is the same in both tubes. If the flux in the ensembleis weighed according to concentration, the flux would be 99 % located in thetube with the 1 M NaCl solution. But of course, the flux is the same in the two tubes.Actually, because the activity coefficient is somewhat smaller in 1 M solution, the fluxin that solution is slightly less.In free porewater and DDL water the molal scale is used for the activity of solutes(mol/kg H 2 O, divided by 1 M to arrive at a dimensionless number). The flux in freeporewater is, from combining Equations (20) and (29), assuming γ i = 1:Jfree,i free dci free D2 w,i(35) dxfreeAppelo and Wersin (2007) have shown that the potential gradient in free porewater alsoapplies to diffusion in DDL water. Thus the flux through DDL water follows fromEquations (14-15), (29) and (34):JDDL,iDDLcDDL,idci DDLD2 w,i(36) c dxDDLiFor exchangeable cations in interlayer water the molar- or equivalent-fraction is thecommonly used concentration scale. Both molar- and equivalent-fraction scales are wellestablished by experience and experiment, and explain typical aspects of ion exchangesuch as dilution and retardation in a complex mixture of cations (Appelo and Postma,2005). The choice between the equivalent and the molar scale is mostly a matter of convenience(Gaines and Thomas 1953). The equivalent scale has the advantage that thesum of the exchangeable cations equals the CEC (a more or less fixed number), and willbe used here.
20The equivalent fraction of an exchangeable cation is:β i = (eq i /kg) / CEC (37)Again, the flux in the interlayer follows from Equations (14-15) and (29):JIL,i cIL IL,i di ILD2 w,i(38) dxILiwhere c IL,i is the concentration in the interlayer (mol/m 3 ). For example, the Namontmorillonitediscussed before has 0.6 Na + ions in an interlayer volume of 2.84×10 -28m 3 for the 2-layer hydrate. Dividing by Avogadro’s number gives the concentrationc IL,Na = c IL,CEC = 3.51 kmol/m 3 . In general for exchangeable cations,c IL,i = β i (c IL,CEC ) / z i (39)and therefore: cIL,CECILdiJ IL,i ILD 2 w,iz(40)i dxIL 3.3 Example: fluxes of I - , Cs + and Sr 2+ in the 3 porosity domainsAs an example, the concentration gradients and fluxes can be calculated in freeporewater, DDL water and interlayer water in a column containing bentonite packed toa dry density of 1700 kg/m 3 . Steady state concentration differences are maintained at thecolumn ends (0 and 1 m) of the NaCl concentration (0.3 and 0.1 M), and trace concentrationsof I - , Cs + and Sr 2+ (3 and 1 μmol/L). Using the parameters derived fromMuurinen’s data (Table 1), the porosities are:I = 0.1 I = 0.3 Averageε IL = 0.299 - 0.0128×I 0.298 0.295 0.30ε DDL = 0.0363 / √I 0.115 0.066 0.09ε tot = 1 - 1700 / 2760 0.384 0.384 0.40 (adapted total of ε’s)ε free = ε tot - ε IL - ε DDL (-0.028) 0.023 0.01 (small positive number)For I NaCl = 0.1, the total of DDL and interlayer water is calculated to be larger than thetotal porosity. The free porosity would be negative, which is impossible, of course, anda small positive value (0.01) is assumed in the average. In the averaged numbers, interlayerwater constitutes 75 % of the total porosity.The total CEC = 1 eq/kg × 1.7 kg/L total volume = 1.7 eq/L. The CEC can be distributedover interlayer water and the sum of DDL- and free porewater in proportion withspecific surface areas (Table 1). Thus, the interlayer charge (c IL,CEC ) is (583 / (583 + 86)= 0.87) × 1.7 eq/L / (0.3 L interlayer water/L total volume) = 4.94 eq/L interlayer water.The DDL charge is 0.13 × 1.7 eq/L / (0.09 L DDL water/L total volume) = 2.43 eq/L.
Page 1 and 2: Working Report 2013-29A Review of P
Page 3 and 4: 2Katsaus bentoniitin huokoisuuteen
Page 5 and 6: 21 INTRODUCTIONBentonite is foresee
Page 7 and 8: 42 POROSITY IN BENTONITELaboratory
Page 9 and 10: 6t IL = 1.41×10 -9 - 4.9×10 -13
Page 11 and 12: 8data are few and not very precise
Page 13 and 14: 10Figure 4. (A): Anion-accessible p
Page 15 and 16: 123 DIFFUSION IN BENTONITE: INTRODU
Page 17 and 18: 14Dp,iDa,i (22)Rwhere R is the reta
Page 19 and 20: 16In Equation (26) it is assumed th
Page 21: 18Figure 7. The effective diffusion
Page 25 and 26: 22Table 3. Concentrations, concentr
Page 27 and 28: 24Table 4. Specific volume and rela
Page 29 and 30: 26Furthermore, if the flux of Cs +
Page 31 and 32: 286 ACKNOWLEDGMENTSComments of Urs
Page 33 and 34: 30Doherty, J., 1994. PEST, Model-in
Page 35 and 36: 32Muurinen, A., Karnland, O. and Le

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