On the solubilities of anhydrite and gypsum

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1 On the solubilities of anhydrite and gypsum Serafeimidis Konstantinos and Anagnostou Georg ETH Zurich Switzerland Abstract. Claystones containing finely distributed anhydrite swell due to water uptake. Experience shows that the swelling may cause considerable damage and lead to high repair costs. During tunnel excavation a change in the field conditions occurs causing the dissolution of anhydrite into its ions and the subsequent precipitation of gypsum crystals. The formation of the latter is (together with water uptake by the clay minerals) the cause of the swelling process. Identifying the conditions under which anhydrite or gypsum is the stable phase is crucial for understanding the process. The current paper presents a thermodynamic model for the individual assessment of anhydrite and gypsum solubilities by additionally considering the partial contact stress between the minerals. The model is verified with experimental and theoretical results from the literature. 1 Introduction The distinguishing characteristic of swelling rocks is their volume increase when coming into contact with water. Anhydritic claystones of the Gypsum Keuper formation are particularly problematic. Their swelling has resulted in damage leading to high repair costs in numerous tunnels (see [1]). It occurs because of the transformation of anhydrite into gypsum crystals via the solution phase when the anhydrite solubility (i.e. its equilibrium concentration) in the field is higher than that of the gypsum. In general the solubility of a mineral depends on the temperature the pressure and the presence of foreign ions in the solution. The chemical reactions of gypsum and anhydrite are SO 2H 2O SO 2H2O and SO SO (1) respectively. The system SO H 2 O with and without salts has been examined by several authors both theoretically and experimentally (see [2]). Nonetheless most of the studies are outdated and many of them do not even provide all of the necessary information. Moreover their results often appear controversial. In the present study we therefore develop a rigorous thermodynamic model for predict-

2 2 ing the solubilities of anhydrite and gypsum by additionally considering the partial contact stress between the minerals a feature which has so far not been factored in by existing studies. The model predictions are compared with experimental and theoretical results from the literature. The present paper supplements recent research by the Authors into the kinetics of the reactions (1) and the simultaneous dissolution of anhydrite and precipitation of gypsum in a closed system ([3] []). 2 Thermodynamic derivations 2.1 Gypsum The direction of a chemical reaction depends on the difference in the Gibbs free energy G between the products and the reactants. Any transformation in a system takes place in order to minimize G i.e. a chemical reaction will occur spontaneously from a high G state to a low G state. The Gibbs Free Energy G change in a phase is expressed as follows: dg dn V dp SdT (2) i i i i where S is the entropy and T the temperature of the system. The subscript i corresponds to the different constituents of the system while repeated use of i indicates summation over all constituents. For the gypsum reaction i = ++ SO -- H 2 O or SO 2H 2 O. The term n i is the number of moles of constituent i (considered here to be unity) and i is the chemical potential. The latter determines how G changes due to chemical transformations at constant temperature and pressure and is given by i = i +RTlna i where a i i and R denote the chemical potential at standard state (atmospheric pressure p = and T = K) the activity of component i and the universal gas constant respectively. The activity of solids and thus of gypsum is unity. The activity of the ions is equal to i c i /c where c i is the concentration of the ions c is the concentration at standard state (c = 1 mol/l) and i 1 is the activity coefficient. The latter is a function of the concentrations of all ions present in the solution (including foreign ions) and accounts for interactions among them. For an extremely dilute solution i equals unity which leads to i = c i. Several theoretical approaches have been proposed in the literature for determining i. For dilute solutions the most common equations are the extended Debye Hückel and the Davies equations while for highly saline waters which is not the case in the present study the more complicated Pitzer equation is used. For details of the different methods the reader is referred to Merkel and Planer- Friedrich [5]. Finally the activity a W of water depends on the existence of foreign

3 3 ions and clay minerals in the system both of which tend to reduce it. For dilute solutions and in the absence of clay minerals a W = 1. The term V i dp i in Equation (2) with V i denoting the molar volumes of the constituents i accounts for the different pressures p i that might be exerted on the constituents. In a porous medium the pore pressure p is exerted on all constituents i while the solid phase (gypsum) is additionally subjected to the contact pressure among the solid grains. The contact pressure is related to the effective stress ʹ and can be taken equal to ʹ/(1--p/(1-where and denote the total stress andthe porosity respectively. The following equation then applies: 2. (3) V dp V dp V dp V dp V d p i i SO H 2O SO 2 H2O Since the molar volumes and entropies remain practically constant the difference in the Gibbs free energy G between the products and reactants for the arbitrary temperature T and pressures p i can be calculated by integrating Equation (2): G G G G G RT Q dp d dt () p T rg rg rg rg rg ln G T p T p p T p T where rg G G 2 f G SO f G G is the difference in the T p f HO 2 f SO2 HO 2 formation Gibbs free energies f G i between products and reactants at standard state (values for f G i can readily be found in the literature see Table 1) and Q G is 2 the activity product ratio (Q G c SO c SO a w ). The three integrals on the right hand side of Equation () read as follows (cf. [6]): p p rg G dp p rg V p G rg d VSO 2 H O 2 T T G dt S T T T rg rg (5) where V V V 2 V V rg HO 2 SO2HO 2 and S r G S S 2 S S H2O SO2H 2O are SO the differences in the molar volumes and entropies respectively between products and reactants. At equilibrium rg G and Q G then becomes equal to the equilibrium or T p solubility constant KG T p. From Equations () and (5) we obtain: 1 G exp r G r G T p T p SO2 H2O r G K G p V V S T T RT. (6) SO

4 Under the simplifying assumption that the calcium and sulphate ion concentrations are permanently equal during the chemical reaction K c a where G T p eq G W c c SO c eqg is the solubility of gypsum and SO the mean activity coefficient which in the present study is calculated according to the Davies equations. In conclusion the solubility of gypsum at arbitrary temperatures and pressures is calculated by the following equation: 1. (7) ceq G aw KG T p Table 1. Formation Gibbs free energies molar volumes and entropies after Anderson [6] with exception the value of VSO which is after Millero [7] Parameter f SO Value [kj/mol] G G f SO 2H2O G f HO 2 f G Parameter Value [cm 3 /mol] V 5.9 V SO SO 2H2O H2O 7.3 V 18 Parameter Value [J/mol/K] S 16.7 S SO SO2 H2O H2O 19.1 S V -18. S f GSO V SO S 2.1 SO 2.2 Anhydrite The solubility of anhydrite can be calculated with some minor modifications analogously to Section 2.1. Equation () becomes for the anhydrite reaction G G G G G RT Q dp d dt (8) p T ra ra ra ra ra ln A T p T p p T p T where G G G G. The three integrals on the right hand ra T p f SO f f SO side of Equation (8) read as follows: p p ra G dp p ra V p ra G d VSO T T G dt S T T T ra ra (9)

follows: V SO S r A T T V SO r (1) 2 2 c eq A. c eq A 1 K A T p.

5 5 where V V V and tion (6) the solubility constant K A T p ra V SOS SO 1 exp RTT ra S S SSO. Analogously A to Equa- S ra G p T p ra and assuming as before that the ion concentrations are equal K A T p The solubility of anhydrite at any temperature and pressure conditions c then reads as follows: V SO S r A T T V SO r (1) 2 2 c eq A. c eq A 1 K A T p. (11) 3 Model verification The model of Section 2 was testedd by comparing the modell predictions with experimental data and theoretical predictions obtained by different authors.. Figure 1a depicts the equilibrium concentration of anhydrite at different temperatures un- the der atmospheric pressure (p = ) and zero partial stress ( = ). In general results of the proposed model agree well with the literature data. d A greater devia- is tion is observed at low temperatures where the only available data however that of Kontrec et al. [8]. Similarly Figure 1b shows the equilibrium concentration of gypsum at atmospheric pressure = and a W = 1. The model m predictions are good in this case as well. Fig. 1. Model verificationn with literaturee data at atmospheric pressure: ( a) anhydrite andd (b) gyp- sum solubilities as a function of temperature

6 6 Fig. 2. Gypsum solubility at atmospheric pressure as a function of NaCl-molality at (a)) T = 15 C and (b) T = 3 C A further verification of the model is shown in Figure 2 which depicts thee solubil- ex- ity of gypsum as a function of NaCl-molality at different temperatures. In most isting investigations the molalityy of NaCl varies between and 6 mol/ /kg H 2 O which is a common range for saline deposits. However according to water analy- floor sis data from Belchentunnel (see [9]) which has experienced repeated heaves and swelling induced damage the NaCl-molality appears to be relatively low in the Gypsum Keuper (<..25 mol/kg H 2 O). Therefore the verification is restricted to low molalities of NaCl (there is experimental data at such loww molali- is ties only for gypsum in Marshall and Slusher [1] where the Davies equation still applicable (cf. Section 2.1). The results of the present model once again fit well with the existing experimental data. Stable phase We study the pressure and temperature conditions under which gypsum and anhy- wa- drite co-exist i.e. have the same equilibrium concentration in respect of pure ter without foreign ions or interactions with clay minerals (aa W = 1). The pressure- of temperature condition can be derived easily by setting the solubility constant gypsum (Eq. 6) to that of anhydrite (Eq. 1). Figure 3 shows this condition under the present model ass well as under MacDonald [11] and Marsal [12]. Thee dashed lines assume that there is no contact stress between the mineral grains ( = ) i.e. pore pressure alone acts uponn the solids (Marsal [12] studied this casee alone). The solid lines assume that the solid phase is subject to ann additional stress (cf. Section 2). In a similar way to MacDonald [11] we considerr lithostatic and hydro- of static total stress and a pore pressure conditions under a total unit weight 2 kn/m 3 and a water unit weight of 1 kn/m 3 respectively. Assuming g that the porosity is very low (i.e. ) = 1.p.

7 Fig. 3. Pressure Temperature conditionn at phase equilibrium between anhydrite a and gypsum In both cases (dashedd and solid lines of Figure 3) the slopes are almost the same.

According to MacDonald [11] and Marsal [12] anhydrite and gypsum co-exist at atmospheric pressure at a temperature of T = and 22 C re- of spectively while the present model suggests a transition

The difference in the temperature mates is also a result of the different thermodynamic values assumed.

7 7 Fig. 3. Pressure Temperature conditionn at phase equilibrium between anhydrite a and gypsum In both cases (dashedd and solid lines of Figure 3) the slopes are almost the same. The minor differences can only be attributed to different thermodynamicc values that have been used. According to MacDonald [11] and Marsal [12] anhydrite and gypsum co-exist at atmospheric pressure at a temperature of T = and 22 C re- of spectively while the present model suggests a transition temperature T = 8.85 C. This value is in the e middle of the range of the transition tempera- esti- tures in the literature (2 6 C cf. [2]). The difference in the temperature mates is also a result of the different thermodynamic values assumed. 5 Conclusions A thermodynamic model for estimating anhydrite and gypsum solubility was de- stress. In addition the model takess account (via the water activity and thee ion ac- veloped which considers the effect of temperature pore pressure and effective tivity coefficient) of the presence of foreign ions as well as possible interactions with clay minerals. It may serve as a component of more complex models of the swelling process in anhydritic claystones.

8 8 REFERENCES [1] Amstad C. Kovári K.: Untertagbau in quellfähigem Fels. Schlussbericht Forschungsauftrag 52/9 des Bundesamts für Strassen ASTRA (21) [2] Freyer D. Voigt W.: Crystallization and phase stability of SO of and SO based salts. Monatshefte für Chemie (23) [3] Serafeimidis K. Anagnostou G.: On the kinetics of the chemical reactions underlying the swelling of anhydritic rocks. Eurock 212 Stockholm. (212) [] Serafeimidis K. Anagnostou G.: Simultaneous anhydrite dissolution and gypsum precipitation in a closed swelling rock system. ARMA 212 Chicago. In Press (212) [5] Merkel J.B. Planer-Friedrich B.: Groundwater Geochemistry 2nd edition. Springer (28) [6] Anderson G.M.: Thermodynamics of Natural systems. University of Toronto John Wiley and Sons Inc (1996) [7] Millero F.J.: The partial molar volume of electrolytes in aqueous solutions. Water and Aqueous Solution Wiley (1972) [8] Kontrec J. Kralj D. Brečević L.: Transformation of anhydrous calcium sulphate into calcium sulphate dihydrate in aqueous solutions. J. of Crystal Growth (22) [9] Wegmüller M.C.: Einflüsse des Bergwassers auf Tiefbau/Tunnelbau (21) [1] Marshall W.L. Slusher R.: Thermodynamics of calcium sulphate dihydrate in aqueous sodium chloride solutions -11 C. J. Phys. Chem (1966) [11] MacDonald G.J.F.: Anhydrite-gypsum equilibrium relations. Am. J. of Science (1953) [12] Marsal D.: Der Einfluss des Druckes auf das System SO H 2 O. Heidelberger Beiträge zur Mineralogie and Petrographie (1952) [13] Innorta G. Rabbi E. Tomadin L.: The gypsum-anhydrite equilibrium by solubility measurements. Geochim. Cosmochim. Acta (198) [1] Bock E.: On the solubility of anhydrous calcium sulfate and of gypsum in concentrated solutions of sodium chloride at 25 C 3 C C and 5 C. n. J. Chem (1961) [15] Posnjak E.: The system SO. Am. J. of Sc. 5th edition 35-A (1938) [16] Raju K. Atkinson G.: Thermodynamics of scale mineral solubilities. 3. lcium sulfate in aqueous NaCl. J. of Chem. and Engin. Data (199) [17] Blount C.W. Dickson F.W.: Gypsum-Anhydrite equilibria in systems SO H 2 O and CO 3 -NaCl-H 2 O. Am. Mineralogist (1973)

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