Abstract
We estimate the crystallisation pressure of gypsum quantitatively, with reference to the geological context of the Gypsum Keuper formation. The formation contains sulphatic claystones which have the property of swelling in the presence of water and have caused substantial structural damage to the linings of several tunnels in Switzerland and Germany. The swelling of these rocks is attributed to the transformation of anhydrite into gypsum, which occurs via the dissolution of anhydrite in pore water and the precipitation of gypsum from the solution. This simultaneous dissolution–precipitation process happens because the solubility of gypsum is lower than that of anhydrite under the conditions prevailing after tunnelling, and it does not cease until all of the anhydrite has been transformed. The elementary mechanism behind the development of the macroscopically observed swelling pressure is the growth of gypsum crystals inside the rock matrix: If a crystal is in contact with a supersaturated solution, but its growth is prevented by the surrounding matrix, it then exerts a socalled crystallisation pressure upon the pore walls. In the present paper, the crystallisation pressure is calculated by means of a thermodynamic model that takes coherent account of all relevant parameters, including the chemical composition of the pore water and pore size. Variations in these parameters lead to a very wide range of crystallisation pressures (from zero to several tens of megapascals). By using the results of mercury intrusion porosimetry and chemical analyses of samples from three Swiss tunnels, however, we show that the range of predicted values can be reduced significantly with the help of standard, projectspecific investigations.
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Abbreviations
 c:

Concentration
 c _{i} :

Concentration of ion i
 \( c_{{{\text{Ca}}^{{ 2 { + }}} }} \) :

Concentration of calcium ions
 c _{eq,A} :

Anhydrite equilibrium concentration
 \( c_{{{\text{eq}},{\text{G}}}}^{0} \) :

Gypsum equilibrium concentration at standard state
 \( c_{{{\text{SO}}_{4}^{2  } }} \) :

Concentration of sulphate ions
 K :

Ion activity product
 \( K_{{{\text{eq}},{\text{A}}}}^{0} \) :

Equilibrium solubility product of anhydrite at standard state
 \( K_{{{\text{eq}},{\text{G}}}}^{0} \) :

Equilibrium solubility product of gypsum at standard state
 \( K_{\text{sp}} \) :

Solubility product in Flückiger’s (1994) model
 \( K_{\text{sp}}^{0} \) :

Solubility product at standard state in Flückiger’s (1994) model
 n :

Pore percentage
 p _{G} :

Crystallisation pressure
 p _{G1} :

Crystallisation pressure at state 1
 p _{G2} :

Crystallisation pressure at state 2
 R :

Universal gas constant
 r _{G} :

Radius of gypsum particles
 r _{p} :

Pore radius
 \( S_{\text{A}}^{0} \) :

Molar entropy of anhydrite at standard state
 \( S_{\text{G}}^{0} \) :

Molar entropy of gypsum at standard state
 \( S_{\text{W}}^{0} \) :

Molar entropy of water at standard state
 T:

Temperature
 T _{ 0 } :

Temperature at standard state
 \( V_{\text{A}}^{0} \) :

Molar volume of anhydrite at standard state
 \( V_{\text{G}}^{0} \) :

Molar volume of gypsum at standard state
 α _{W} :

Water activity
 α _{ i } :

Activity of ion i
 \( \gamma_{{{\text{Ca}}^{2 + } }} \) :

Activity coefficient of calcium ions
 γ _{G} :

Surface free energy of the gypsum–water interface
 γ _{ i } :

Activity coefficient of ion i
 \( \gamma_{{{\text{SO}}_{4}^{2  } }} \) :

Activity coefficient of sulphate ions
 ΔG _{ r } :

Free energy of the transformation of anhydrite to gypsum in Flückiger’s (1994) model
 Δ_{ r,A} G ^{0} :

Standard Gibbs energy of anhydrite dissolution
 Δ_{ r,A} S ^{0} :

Standard entropy of anhydrite dissolution
 Δ_{ r,A} V ^{0} :

Standard volume of anhydrite dissolution
 Δ_{ r,GA} G ^{0} :

Standard Gibbs energy of the transformation of anhydrite to gypsum
 Δ_{ r,GA} S ^{0} :

Standard entropy of the transformation of anhydrite to gypsum
 Δ_{ r,GA} V ^{0} :

Standard volume of the transformation of anhydrite to gypsum
 Δ_{ r,G} G ^{0} :

Standard Gibbs energy of gypsum dissolution
 Δ_{ r,G} S ^{0} :

Standard entropy of gypsum dissolution
 Δ_{r,G} V ^{0} :

Standard volume of gypsum dissolution
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Acknowledgments
The authors appreciate the support of the Swiss National Science Foundation (SNF, Project Nr. 200021126717/1) and by the Swiss Federal Roads Office (FEDRO, Project Nr. FGU 2010007).
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Appendix: on crystallisation pressure after Flückiger et al. (1994)
Appendix: on crystallisation pressure after Flückiger et al. (1994)
From Eqs. (3)–(6), we obtain the following expression for crystallisation pressure according to Flückiger et al. (1994):
The nominator of the right side is equal to the change in the Gibbs energy ΔG _{ r } of the anhydrite hydration reaction due to a change in temperature from the standard temperature T _{0} to another temperature T. It is well known (cf., e.g. White (2005) that
where the standard entropy Δ_{ r,GA} S ^{0} of the anhydrite hydration reaction can be determined from the molar entropies of the reaction products (gypsum) and reactants (anhydrite and water):
Taking into account the values of the molar constants according to Table 1, Eqs. (12)–(14) lead to
This value is close to Flückiger et al. (1994) result (3.7 MPa). The difference is due to rounding errors and to the fact that Flückiger (1994) used Kelley et al.’s (1941) empirical equation rather than the molar constants of Anderson (1996).
In the following, we will show that the pressure according to Eq. (15) is equal to the increase in the crystallisation pressure of the gypsum that would occur if the solution was permanently saturated with respect to anhydrite; the anhydrite was under atmospheric pressure and the temperature was reduced from the standard temperature of T = T _{0} = 25 °C (hereafter referred to as “state 1”) to T = 20 °C (hereafter referred to as “state 2”). The decrease in temperature causes an increase in the crystallisation pressure of the gypsum because the solubility of anhydrite increases with decreasing temperature (cf., Freyer and Voigt 2003, for example) and, consequently, supersaturation with respect to gypsum is higher in state 2 than in state 1. At state 1, the crystallisation pressure of gypsum reads as follows:
where K ^{0}_{eq,A} and K ^{0}_{eq,G} denote the equilibrium solubility products of anhydrite and gypsum, respectively, at standard conditions. Eq. (16) follows directly from Eq. (2) considering that the solution is saturated with respect to anhydrite and, therefore, K = K ^{0}_{eq,A} . The crystallisation pressure p _{c2} at an arbitrary temperature T can be calculated from the following equation (cf., e.g. White 2005):
where the standard entropy of gypsum dissolution Δ_{ r,G} S ^{0} can be determined from the molar entropies:
If the solution is always saturated with respect to anhydrite and the anhydrite is under atmospheric pressure, then the solubility product K can be obtained (analogously to Eq. 17) from the following equation:
where the standard entropy of anhydrite dissolution is
Inserting K from Eq. (19) into Eq. (17) and taking account of Eqs. (14) (18) and (20) leads to the following expression for the crystallisation pressure of gypsum in state 2:
The last term on the right side of this equation is identical with Eq. (15). This means that the value determined by Flückiger et al. (1994) is equal to the change in the crystallisation pressure (p _{G2}–p _{G1}) that would occur if the temperature decreases from T _{ 0 } = 25 °C to T = 20 °C.
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Serafeimidis, K., Anagnostou, G. On the crystallisation pressure of gypsum. Environ Earth Sci 72, 4985–4994 (2014). https://doi.org/10.1007/s1266501433667
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DOI: https://doi.org/10.1007/s1266501433667