On the crystallisation pressure of gypsum

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 so-called 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, project-specific investigations.

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):

$$ p_{\text{G}} = \frac{{\Delta G_{r} (T) - \Delta G_{r} (T_{0} )}}{{V_{\text{G}}^{0} }}. $$

(12)

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

$$ \Delta G_{r} (T) - \Delta G_{r} (T_{0} ) = - (T - T_{0} )\Delta_{{r,{\text{GA}}}} S^{0} , $$

(13)

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):

$$ \Delta_{{r,{\text{GA}}}} S^{0} = S_{\text{G}}^{0} - S_{\text{A}}^{0} - 2S_{\text{W}}^{0} , $$

(14)

Taking into account the values of the molar constants according to Table 1, Eqs. (12)–(14) lead to

$$ p_{\text{G}} = \frac{{\left( {T - T_{0} } \right)\left( {S_{\text{G}}^{0} - S_{\text{A}}^{0} - 2S_{\text{W}}^{0} } \right)}}{{V_{\text{G}}^{0} }} \cong 3.6\;{\text{MPa}}\;. $$

(15)

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:

$$ p_{G1} \; = \frac{RT}{{V_{\text{G}}^{0} }}\ln \frac{{K_{\text{eq,A}}^{0} }}{{K_{\text{eq,G}}^{0} }}, $$

(16)

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):

$$ RT\ln \frac{K}{{K_{\text{eq,G}}^{0} }} = V_{G}^{0} \;p_{\text{G2}} \; + \left( {T - T_{0} } \right)\,\Delta_{{r,{\text{G}}}} S^{0} , $$

(17)

where the standard entropy of gypsum dissolution Δ _r,G S ⁰ can be determined from the molar entropies:

$$ \Delta_{{r,{\text{G}}}} S^{0} = S_{{Ca^{2 + } }}^{0} + S_{{S{\text{O}}_{4}^{2 - } }}^{0} + 2S_{\text{W}}^{0} - S_{\text{G}}^{0} $$

(18)

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:

$$ RT\ln \frac{K}{{K_{\text{eq,A}}^{0} }} = \left( {T - T_{0} } \right)\,\Delta_{{r,{\text{A}}}} S^{0} , $$

(19)

where the standard entropy of anhydrite dissolution is

$$ \Delta_{{r,{\text{A}}}} S^{0} = S_{{{\text{Ca}}^{{ 2 { + }}} }}^{0} + S_{{{\text{SO}}_{ 4}^{{ 2 { - }}} }}^{0} - S_{\text{A}}^{0} $$

(20)

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:

$$ p_{G2} \; = p_{G1} + \left( {T - T_{0} } \right)\frac{{S_{G}^{0} - S_{A}^{0} - 2S_{W}^{0} }}{{V_{G}^{0} }}. $$

(21)

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.