Abstract
Bentonite clays are proven to be attractive as buffer and backfill material in highlevel nuclear waste repositories around the world. A quick estimation of swelling pressures of the compacted bentonites for different clay–water–electrolyte interactions is essential in the design of buffer and backfill materials. The theoretical studies on the swelling behavior of bentonites are based on diffuse double layer (DDL) theory. To establish theoretical relationship between void ratio and swelling pressure (e versus P), evaluation of elliptic integral and inverse analysis are unavoidable. In this paper, a novel procedure is presented to establish theoretical relationship of e versus P based on the Gouy–Chapman method. The proposed procedure establishes a unique relationship between electric potentials of interacting and noninteracting diffuse clay–water–electrolyte systems. A procedure is, thus, proposed to deduce the relation between swelling pressures and void ratio from the established relation between electric potentials. This approach is simple and alleviates the need for elliptic integral evaluation and also the inverse analysis. Further, application of the proposed approach to estimate swelling pressures of four compacted bentonites, for example, MX 80, Febex, Montigel and Kunigel V1, at different dry densities, shows that the method is very simple and predicts solutions with very good accuracy. Moreover, the proposed procedure provides continuous distributions of e versus P and thus it is computationally efficient when compared with the existing techniques.
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References
Bharat TV (2004) Simplified methods of evaluation of diffuse double layer parameters employed in geotechnical engineering. M. Sc. Dissertation, Indian Institute of Science, India
Bharat TV, Sivapullaiah PV, Allam MM (2008) Novel algorithm for estimation of swell pressure of fine grained soils based on diffuse double layer (DDL) theory. In: The 12th international conference of international association for computer methods and advances in geomechanics (IACMAG), 1–6 October, Goa, India
Bolt GH (1956) Physicochemical analysis of the compressibility of pure clays. Géotech 6(2):86–93
Bucher F, Spiegel U (1984) Quelldruck von hochverdichteten bentoniten. Nagra Technischer Bericht. 8418
Butcher F, MüllerVonmoos M (1989) Bentonite as a containment barrier for the disposal of highly radioactive waste. Appl Clay Sci 4(2):157–177
Chapman LA (1913) Contribution to the theory of electrocapillarity. Phil Mag Ser 6, 25(148):475–481
ENRESA (2000) FEBEX project—full scale engineered barriers experiments for a deep geological respiratory for high level radioactive waste in crystalline host rock. Final report, Publicación téechnica 4/2002, Empresa Nacional de Residuos Radiactivos SA (ENRESA), Madrid, Spain
ENRESA (2002) Thermohydromechanical characterisation of a bentonite from Cabo de Gata—a study applied to the use of bentonite as sealing material in high level radioactive waste repositories. Publicación téechnica 4/2002, Empresa Nacional de Residuos Radiactivos SA (ENRESA), Madrid, Spain
Gouy G (1910) Sur la constitution de la charge electrique a la surface d’un electrolyte. J Phys Theor Appl Ser 4(9):457–468 (in French)
Gratchev I, Towhata I (2011) Compressibility of natural soils subjected to longterm acidic contamination. Environ Earth Sci 64:193–200
Ishikawa H, Amemiya K, Yusa Y, Sasaki N (1990) Comparison of fundamental properties of Japanese bentonite as buffer materials for waste disposal. In: VC Farmer, Y Tardy (eds) Proceedings of the 9th international clay conference, Strasbourg, 28 Aug–2 Sept 1989. Université de Strasbourg, Strasbourg, France, pp 107–115
Japan Nuclear Cycle Development Institute (1999) H12: project to establish the scientific and technical basis for HLW disposal in Japan: supporting report 2 (respiratory design and engineering Technology). Japan Nuclear Cycle Development Institute, Tokyo
Madsen FT (1998) Clay mineralogical investigations related to nuclear—wate disposal. Clay Min 33:109–129
Mitchell JK (1993) Fundamentals of soil behavior. Wiley, New York
MüllerVonmoos M, Kahr G (1982) Bereitstellung von bentonit fur laboruntersuchungen. Nagra Technischer Bericht. 8204
Overbeek JThG (1952) Electrochemistry of double layer and the interaction of colloidal particles. In: Kruyt HR (ed) Colloid science, vol 1. Elsevier, Amsterdam
Pusch R, Yong R (2006) Microstructure of smectite clays and engineering performance. Taylor and Fracis, New York
Schanz T, Tripathy S (2009) Swelling pressure of a divalentrich bentonite: diffusedouble layer theory revisited. Wat res res 45(5)
Sridharan A, Choudhury D (2002) Swelling pressure of sodium montmorillonites. Géotech 52(6):459–462
Sridharan A, Jayadeva MS (1982) Double layer theory and compressibility of clays. Geotech 32(2):133–144
Tripathy S, Sridharan A, Schanz T (2004) Swelling pressures of compacted bentonites from diffuse double layer theory. Can Geotech J 41:437–450
van Olphen H (1977) An introduction to clay colloid chemistry. Interscience, New York
Verwey EJW, Overbeek JTG (1948) Theory of the stability of lyophobic colloids. Elsevier, Amsterdam
Yong RN, Mohamed AMO (1992) A study of particle interaction energies in wetting of unsaturated expansive clays. Can Geotech J 29:1060–1070
Zheng L, Samper J, Montenegro L (2011) A coupled THC model of the FEBEX in situ test with bentonite swelling and chemical and thermal osmosis. J Contam Hydrol 126(1–2):45–60
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Appendix 1
Appendix 1
Calculation of swelling pressure from halfspace distance by the proposed approach:
Known
Specific surface, S = 800 m^{2}/g; base exchange capacity, B = 100 meq/100 g (P′ = 0.125 meq/100 m^{2}); n = 0.001 M; v = 1; ε = 78.54 T = 298 K; and d = 50 Å
Calculation

Step 1. Surface potential of single clay particle (Eq. 9),
$$ \phi_{0} = 0.1725\frac{T}{\nu }\sinh^{  1} \left( {\frac{{1256.81 \times P^{'} }}{{\sqrt {n\varepsilon T} }}} \right) = 214.554\,{\text{mV}} $$ 
Step 2. Scaled surface potential,
$$ z = \frac{{\nu F\varphi_{0} }}{RT} = \frac{{1 \times 9.6487 \times 10^{7} \times 0.214554}}{{8.314 \times 10^{3} \times 298}} = 8.3556 $$ 
Step 3. The dimensionless parameter, ξ, at x, which equals to the halfspace distance d, where the midplane potential is to be calculated
$$ \xi = \kappa \times x = \kappa \times d = 10^{6} \times 50 \times 10^{  8} = 0.5 $$ 
Step 4. Scaled potential at any distance from the particle surface,
$$ y = 2 \times \ln \left( {\frac{{\exp \left( \xi \right) + \tanh (\frac{z}{4})}}{{\exp \left( \xi \right)  \tanh (\frac{z}{4})}}} \right) = 2 \times \ln \left( {\frac{{\exp \left( {0.5} \right) + \tanh (\frac{8.3556}{4})}}{{\exp \left( {0.5} \right)  \tanh (\frac{8.3556}{4})}}} \right) = 2.6997 $$ 
Step 5. Electric potential of a single noninteracting plate at distance, x, which equals to halfspace distance d, where the midway potential is to be calculated,
$$ \phi_{x = d} = y\frac{RT}{\nu F} = 2.6997 \times \frac{{8.314 \times 10^{3} \times 298}}{{1 \times 9.6487 \times 10^{7} }} \times 1000 = 69.319\,{\text{mV}} $$ 
Step 6. The midplane potential is calculated using (12),
$$ \phi_{d} =  6.24 \times 10^{  4} \phi_{x = d}^{2} + 1.205\phi_{x = d} + 8.582 = { 89}. 1 1 3 {\text{ mV}} $$ 
Step 7. Scaled midplane potential
$$ u = \frac{{\nu F\phi_{d} }}{RT} = \frac{{1 \times 9.6487 \times 10^{7} \times 89.113}}{{8.314 \times 10^{3} \times 298}} \times 10^{  3} = 3.47 $$ 
Step 8. The swell pressure corresponding to the halfspace distance 50 Å is calculated using the Langmuir’s formula (5).
$$ P = 2nRT(\cosh u  1) = 74.77\,{\text{kPa}} $$
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Bharat, T.V., Sivapullaiah, P.V. & Allam, M.M. Novel procedure for the estimation of swelling pressures of compacted bentonites based on diffuse double layer theory. Environ Earth Sci 70, 303–314 (2013). https://doi.org/10.1007/s1266501221287
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DOI: https://doi.org/10.1007/s1266501221287