Solubility Measurements and Predictions of Gypsum, Anhydrite, and Calcite Over Wide Ranges of Temperature, Pressure, and Ionic Strength with Mixed Electrolytes

Abstract

Today’s oil and gas production from deep reservoirs permits exploitation of more oil and gas reserves but increases risks due to conditions of high temperature and high pressure. Predicting mineral solubility under such extreme conditions is critical for mitigating scaling risks, a common and costly problem. Solubility predictions use solubility products and activity coefficients, commonly from Pitzer theory virial coefficients. However, inaccurate activity coefficients and solubility data have limited accurate mineral solubility predictions and applications of the Pitzer theory. This study measured gypsum solubility under its stable phase conditions up to 1400 bar; it also confirmed the anhydrite solubility reported in the literature. Using a novel method, the virial coefficients for Ca²⁺ and ${\text{SO}}_{4}^{2 - }$ (i.e., $\beta_{{{\text{CaSO}}_{4} }}^{(0)} ,\beta_{{{\text{CaSO}}_{4} }}^{(2)} ,C_{{{\text{CaSO}}_{4} }}^{\phi }$) were calculated over wide ranges of temperature and pressure (0–250 °C and 1–1400 bar). The determination of this set of virial coefficients widely extends the applicable temperature and pressure ranges of the Pitzer theory in Ca²⁺ and SO ²⁻₄ systems. These coefficients can be applied to improve the prediction of calcite solubility in the presence of high concentrations of Ca²⁺ and SO ²⁻₄ ions. These new virial coefficients can also be used to predict the solubilities of gypsum and anhydrite accurately. Moreover, based on the derived $\beta_{{{\text{CaSO}}_{4} }}^{(2)}$ values in this study, the association constants of ${\text{CaSO}}_{4}^{\left( 0 \right)}$ at 1 bar and 25 °C can be estimated by $K_{\text{assoc}} = - 2\beta_{{{\text{CaSO}}_{4} }}^{(2)}$. These values match very well with those reported in the literature based on other methods.

Appendices

Appendix 1: Fitted Parameters

See Table 8.

Table 8 Parameters fitted for $\beta_{{{\text{CaSO}}_{4} }}^{(0)} ,\beta_{{{\text{CaSO}}_{4} }}^{(2)} ,C_{{{\text{CaSO}}_{4} }}^{\phi }$

Appendix 2: The Pitzer Theory

Equation (3) shows the relationship among the solubility product (K _sp), the ion activity coefficients (γ), and the ion concentrations (molality, in mol/kg H₂O). For a specific mineral, its solubility product is only dependent on temperature and pressure, and independent of the solution compositions. Many mineral solubility products have been reported in the literature. The ion activity coefficients are as functions of temperature, pressure, and solution compositions. If the ion activity coefficients can be accurately predicted over wide ranges of temperature, pressure, and ion compositions, as has been done in this study, the ion concentrations (or mineral solubilities) can be predicted.

Based on the equation of excess Gibbs free energy, the activity coefficient of different species (γ, used in Eq. 3) and osmotic coefficient of water (ϕ) can be calculated with its differentiation in terms of mole amount (n) or weight of water (w _w ), respectively:

$$\begin{aligned} (\phi - 1) & = - \left( {\sum\limits_{i} {m_{i} } } \right)^{ - 1} \left( {\frac{{\partial (G^{\text{ex}} /RT)}}{{\partial w_{w} }}} \right)_{{T,P,n_{i} }} \quad a_{w} = \exp \left( { - \phi M_{w} \sum\limits_{i} {m_{i} } } \right) \\ \ln \gamma_{i} & = \left( {\frac{{\partial (G^{\text{ex}} /RT)}}{{\partial n_{i} }}} \right)_{{T,P,w_{w} ,n_{j \ne i} }} \\ \end{aligned}$$

The excess Gibbs free energy of the system can be represented with the Pitzer theory with the equations shown below (Christov and Moller 2004; Holmes et al. 1994; Li and Duan 2007; Moller 1988; Monnin 1999; Pitzer et al. 1984). It represents the non-ideality of the aqueous system consisting the long-range electrostatic interactions and the short-range interactions. The short-range interactions are represented by the virial coefficients as functions of temperature and pressure.

$$\begin{aligned} \frac{{G^{\text{ex}} }}{{RT \cdot w_{w} }} = & - \frac{{4 \cdot IS \cdot A^{\phi } }}{b}\ln \left( {1 + b \cdot I^{1/2} } \right) + 2\sum\limits_{c = 1}^{{N_{c} }} {\sum\limits_{a = 1}^{{N_{a} }} {m_{c} m_{a} \left( {B_{ca} + \left( {\sum\limits_{c}^{{N_{c} }} {m_{c} z_{c} } } \right)C_{ca} } \right)} } \\ & + \sum\limits_{c = 1}^{{N_{c} - 1}} {\sum\limits_{{c^{\prime } = c + 1}}^{{N_{c} }} {m_{c} m_{{c^{\prime }}} \left( {2\varPhi_{{cc^{\prime }}} + \sum\limits_{a = 1}^{{N_{a} }} {m_{a} \psi_{{cc^{\prime }a}} } } \right)} } + \sum\limits_{a = 1}^{{N_{a} - 1}} {\sum\limits_{{a^{\prime } = a + 1}}^{{N_{a} }} {m_{a} m_{{a^{\prime }}} \left( {2\varPhi_{{cc^{\prime }}} + \sum\limits_{c = 1}^{{N_{c} }} {m_{c} \psi_{{aa^{\prime }c}} } } \right)} } \\ & + 2\sum\limits_{n = 1}^{{N_{n} }} {\sum\limits_{c = 1}^{{N_{c} }} {m_{n} m_{c} \lambda_{nc} } } + 2\sum\limits_{n = 1}^{{N_{n} }} {\sum\limits_{a = 1}^{{N_{a} }} {m_{n} m_{a} \lambda_{na} } } + \sum\limits_{n = 1}^{{N_{n} }} {\sum\limits_{c = 1}^{{N_{c} }} {\sum\limits_{a = 1}^{{N_{a} }} {m_{n} m_{c} m_{a} \xi_{nca} } } } \\ \end{aligned}$$

$$A^{\phi } = \frac{1}{3}\left[ {\frac{{2\pi A_{v} \rho_{w} }}{1000}} \right]^{1/2} \left[ {\frac{{e^{2} }}{DkT}} \right]^{3/2}$$

$$B_{ca}^{{}} = \beta_{ca}^{(0)} + \beta_{ca}^{(1)} g(\alpha_{ca} \sqrt I ) + \beta_{ca}^{(2)} g(12\sqrt I )$$

$$C_{ca}^{{}} = \frac{{C_{ca}^{\phi } }}{{2\sqrt {|z_{c} z_{a} |} }}$$

where w _w is the weight of water; m is the molality of species in mol/kg H₂O; R is the gas constant; k is the Boltzmann constant; T is temperature in Kelvin; ρ _w is the water density; D is the dielectric constant of water; e is the unit electron charge; A _v is the Avogadro constant; z denotes the charge of an individual species; I is the ionic strength; b value is arbitrarily chosen as 1.2 (kg/mol)^1/2 for all electrolytes; $A^{\phi }$ is the Debye–Huckel limiting slope representing the long-range electrostatic interactions; $B\left( {\beta^{(0)} ,\beta^{(1)} ,\beta^{(2)} } \right),C\left( {C^{\phi } } \right),\varPhi ,\psi ,\lambda ,\xi$ are the short-range ion interaction terms represented with virial coefficients in the context; c, a, n in the subscripts represent the cation, anion, and neutral species, respectively; function g(x) = $2 \cdot [1 - (1 + x)e^{ - x} ]/x^{2}$ is the ionic strength-dependent function; α equals 2.0 (kg/mol)^1/2 for 1–1, 1–2, 2–1 ion interactions, 1.4 (kg/mol)^1/2 for 2–2 ion interactions (Pitzer 1995).

The virial coefficients adopted in this context represent the temperature and pressure dependence of the short-range interactions and will not change with the compositions. Thus, in different systems under different conditions, these virial coefficients are identical for their corresponding ion interactions. For each unknown short-range interaction, a set of parameters in Eq. (6) will be fitted to represent their temperature and pressure dependences based on the solubility data and dissolution equilibrium constants.