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A Thermodynamic Model for the Solubility Prediction of Barite, Calcite, Gypsum, and Anhydrite, and the Association Constant Estimation of CaSO(0) 4 Ion Pair up to 250 °C and 22000 psi Zhaoyi Dai,* Amy Kan, Fangfu Zhang, and Mason Tomson Department of Civil and Environmental Engineering, Rice University, Houston, Texas 77005, United States S Supporting Information *

ABSTRACT: Mineral solubility predictions are critical for estimating scaling risks at conditions of high temperature, pressure, and ionic strength (IS) in mixed electrolytes which occur in various industrial processes. On the basis of the Pitzer theory, this study establishes a thermodynamic model to predict the solubilities and scaling risks of barite, calcite, gypsum, and anhydrite under these extreme conditions. This study combines the related equilibrium constants, virial coeﬃcients, and the solubilities of these minerals from 0 °C to 250 °C, from 14.7 psi to 22000 psi, with up to 6 mol NaCl/kg H2O with or without mixed electrolytes to determine the temperature and pressure dependences of the following virial coeﬃcients for Ca2+−SO42−, Ba2+− SO42−, Ba2+−Cl−, Na+−SO42−, and Ca2+−Br− binary interactions used in the Pitzer theory. Using these virial coeﬃcients, the solubilities of these four minerals can be accurately predicted with no apparent temperature, pressure, or IS bias in solutions under the extreme conditions. The association constants (Kassoc) for CaSO4(0) ion pairs calculated from the β(2) CaSO4 values derived in the model match well with experimental measurements at 25 °C and 1 atm, and show similar temperature and pressure dependence with those calculated from the Fuoss ion pair association theory and those listed in SOLMINEQ. 88.

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INTRODUCTION Mineral scale formation is a signiﬁcant problem in various industrial processes, such as the onshore and oﬀshore oil and gas production,1−5 geothermal energy production,6−8 water treatment with osmosis membrane,9,10 and sugar reﬁnery.11 In these processes, barite, calcite, gypsum, and anhydrite are among the most commonly observed scale minerals. High temperature (above 150 °C or 302 °F), pressure (above 14 500 psi or 1000 bar), and high concentrations of mixed electrolytes (above 250 000 mg/L total dissolved solids (TDS) or 4 molal NaCl) are often encountered in these industrial processes.1,12 A better prediction of scaling risks needs better understanding of the thermodynamic properties of the scale minerals under the conditions that might occur.3,13−15 However, the experimental data at such extreme conditions are rarely reported in the literature and the model predictions are rarely veriﬁed, especially when approaching the extreme conditions of temperature, pressure, and TDS at the same time.13,14,16−25 Furthermore, the applicability of the scaling formation models is usually restricted to a certain kind of mineral within a certain range of conditions due to the complex ion interactions in mixed electrolytes and their undetermined temperature or pressure dependences.22,26,27 Thus, it is important to develop a thermodynamic model to predict the solubility and scaling risks of a variety of minerals with mixed electrolytes over a wide range conditions. © XXXX American Chemical Society

The equilibrium constant of a chemical reaction is the ratio of products activity to the reactant activity when the reaction has reached equilibrium. Generally, the equilibrium constants can be expressed as the combination of a temperature-dependent function and a pressure-dependent function as follows: log10 K = log10 KT (T ) + log10 KP(P , T )

(1)

Speciﬁcally, the equilibrium constant of the solid dissolution reaction (i.e., the solubility products) is the product of the free lattice ion activities and represents the mineral solubility.28 Much research has been done to predict the thermodynamic equilibrium constants for the reactions involved in the solids precipitation or dissolution at temperatures up to 600 °C and pressures up to 5000 bar.20,21,25,29−35 Most of these equilibrium constants have been applied successfully in diﬀerent studies and will be applied in this research.14,15,27,32,36 In addition to these thermodynamic equilibrium constants (e.g., solubility product of each mineral, dissociation constant of carbonic acid), an electrolyte model to accurately predict the activity coeﬃcients (γ) over wide ranges of temperature, pressure, and IS with mixed electrolytes is required to calculate the ion activity in a solution. Received: September 24, 2014 Accepted: December 8, 2014

A

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Figure 1. Sketch of solubility measurement apparatus.65

Moreover, it is worth noticing that in the solutions containing 2−2 electrolytes (e.g., Ba−SO4, Ca−SO4, Ca−CO3) the possible existence of corresponding ion pairs can signiﬁcantly impact the free ion activities. Many experimental methods (e.g., spectroscopy and conductivity measurements) have been developed to determine the ion pair association constants (Kassoc); however, these methods are hard to conduct under high temperatures and high pressures.51−53 The existence of a distinct species of ion pairs has been conﬁrmed by spectroscopic or ultrasonic data.54,55 The strongest 2−2 ion pair association is believed to occur at an intermediate range of IS from 0.03 M to 0.1 M.42,44 The temperature increase will lead to a decrease in dielectric constant of water and an increasing ion pairs formation.20,56,57 For example, Møller20 suggested that the CaSO4(0) (aq) ion pairs cannot be ignored at higher temperatures. Generally, there are two ways to include the ion pairs in the aqueous system. One way is to treat the ion pairs as a distinct species, and the stability constants are expressed explicitly.23,58 Some theoretical models have also been applied to calculate the Kassoc based on D-H theory and molecular dynamics.57,59−63 The other way is to implicitly treat the ion pair association as a stronger binary ion interaction. Because of the ambiguity of quantitatively setting a standard (e.g., the distance between the two ions) to diﬀerentiate ion pairs, ion interactions (e.g., binary interactions in Pitzer theory), and strong bonds in solids, Pitzer theory incorporated the ion pair associations implicitly into the binary ion interaction term β(2), with β(2) = −Kassoc/2.44 The β(2) term is controlled by the IS dependent multiplier (i.e., g(x) = 2[1 − (1 + x)e−x]/x2, with x = 12(IS)1/2) which makes the β(2) term negligible above 0.1 M. In this way, the association constants can be determined on the basis of the determination of virial coeﬃcients. It is noteworthy that such a method requires the support of the solubility data of corresponding minerals (e.g., barite for BaSO(0) 4 , gypsum and anhydrite for CaSO4(0)) under the intermediate IS range (approximately from 0.03 M to 0.1 M) where the ion pair association is stronger. Besides, when the aqueous solution is in equilibrium with mineral solids, the activity of ion pair equals the product of the solubility product of the solid and the association constants of ion pairs (i.e., aion pair = Ksp Kassoc), and is deﬁned as the intrinsic solubility.64 It indicates that the concentrations and (0) the fractions of BaSO(0) 4 and CaCO3 are much less than those of (0) CaSO4 because of their low solubility product, and thus the (0) association constants for BaSO(0) 4 and CaCO3 are harder to be accurately determined through solubility. Moreover, other virial coeﬃcients (i.e., β(0), β(1), Cϕ) for the same ion−ion interaction should be determined accurately so that β(2) will not be confused with other ion interactions. However, previously β(2) was usually omitted or used as the adjustable constants and thus cannot

The activity coeﬃcients of aqueous ions have been estimated by a variety of methods including the Debye−Huckel (DH) equation, Davis equation, TJ equation, Robinson-Stokes equation, Guggenheim equations, speciﬁc ion interaction (SIT) theory, and Pitzer theory.37−42 The ﬁrst three models mainly consider long-range ion−ion interactions, but cannot be applied to solutions with high IS and mixed electrolytes when short-range interactions and solvation impacts play a more important role. The Robinson-Stokes equation accounts for the ion−solvent interactions, but no short-range interactions. On the basis of DH theory, Guggenheim equations, and SIT theory, Pitzer and his co-workers developed the Pitzer theory by including not only long-range interactions and binary ion−ion interactions, but also ternary interactions including neutral compounds in aqueous solutions.40,43−45 The ion speciﬁc interactions among mixed electrolytes are represented by the virial coeﬃcients with temperature and pressure dependence and can be applied to any solutions. Such property makes Pitzer theory widely applicable in the industrial processes mentioned above with mixed electrolytes under complex conditions.44,46 In the past decades, starting from Pitzer’s analysis on Na−Cl interaction, the research on the Pitzer theory has been mostly focused on analyzing more ion interactions and expanding the temperature and pressure ranges.15,20,41,46−49 One possible way to extend the temperature and pressure ranges is based on the measurements of volumetric (e.g., density, compressibility) and calorimetric (e.g., enthalpy, heat capacity) properties of the components or the solution.22,46,47,50 However, for insoluble salts (e.g., CaSO4, CaCO3) the density changes due to mixing especially at high pressures are too small to be measured accurately. For example, the anhydrite solubility at 1000 bar and 200 °C in 2 m NaCl calculated by Monnin47 is about 30% more than that measured by Blount and Dickson.16 Another way to extend the temperature and pressure ranges is based on the use of the solubility measurements of minerals to calculate activity coeﬃcients with reliable solubility products. For example, Harvie, Møller, and their collaborators19,20,26 extended the temperature ranges for the H-Na-K-Ca-OH-Cl-SO4-HSO4-H2O system based on the solubility measurement. However, the pressure dependences of the virial coeﬃcients are often not fully considered and some important ions (e.g., Ba2+, CO32−) cannot be accurately incorporated in the model. As another example, Shi et al.27 found that the discrepancies in the solubility predictions of calcite with mixed electrolytes can be mainly attributed to the inaccuracies of the virial coeﬃcients for Ca2+ and SO42− interactions. Therefore, a more systematic estimation of virial coeﬃcients that is compatible for all main ion interactions and over wide ranges of temperature and pressure is needed. B

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NaCl.15,27 The solubilities of barite and calcite were also measured in synthetic brine with mixed electrolytes which represented the maximum concentrations of typical constituents (95% CI) in the oil and gas ﬁeld.75 The solubility data for calcite and barite are listed in Supporting Information, Tables 3, 5, and 7, and the compositions of synthetic brine water are listed in Supporting Information, Tables 4 and 6, respectively.

represent the ion pair association constants. For example, along with constant β(0) = 0.15, β(1) = 3, and Cϕ = 0, Monnin23 derived the association constant to be 20.4 at 25 °C and 1 bar (β(2) CaSO4 = −10.2) which is far less than 102.30 from experimental observations. Therefore, in this research, the CaSO4(0) ion pair association constants will be calculated over wide temperature and pressure ranges based on the derived β(2) CaSO4in Pitzer theory as mentioned above. Considering the diﬃculties mentioned above in scaling control, this study aims to (1) develop an overall model to predict the solubility of barite, calcite, gypsum, and anhydrite under extreme conditions of high temperature, high pressure, and high TDS with mixed electrolytes; (2) determine the virial coeﬃcients for main ion interactions with temperature and pressure dependences; and (3) predict the ion pair association constants for CaSO4(0) at high temperatures and pressures. To achieve these goals, this research will adopt the solubilities of barite, calcite, gypsum, and anhydrite, solubility products, and previously determined virial coeﬃcients for certain ion interactions (see discussion below) to derive the unknown virial coeﬃcients over a wide ranges of temperature and pressure, based on which the solubilities of these minerals and association constants of CaSO4(0) can be predicted.

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MODEL AND EQUATIONS In the mineral-solution systems, the saturation index (SI), the logarithm of the ratio of ion activity product to the solubility product, is a function of the related stability constants (solubility products of minerals, dissociation constants of carbonic acid), solubility of minerals (molality of ions), and activity coeﬃcients of each compound. It represents the saturation status of the system. In the solubility measurement experiments, if the bulk solutions have reached equilibrium with the solids, then the SIs should be 0 theoretically. If the SI is bigger than 0, the solution is oversaturated, and vice versa. Thus, the properties of these systems should follow the equations below: BaSO4 ⇄ Ba 2 + + SO24 − , SI = log10(mBa 2+γBa 2+mSO24−γSO2−/K sp,barite) = 0

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4

CO2 + H 2O ⇄ H 2CO3 ⇄ H+ + HCO−3 , a H+mHCO−3 γHCO− 3 K1 = a H2OmCO2(aq)γCO (aq)

EXPERIMENTS In our previous papers, Shi et al.15,27 had designed a novel ﬂowthrough apparatus (the sketch is replotted in Figure 1) for mineral solubility measurement. The near saturation feeding solution will ﬂow through the column packed with mineral solids and reach equilibrium at the conditions over wide ranges of temperature (i.e., 0 °C to 250 °C) and pressure (i.e., 14.7 psi to 24,000 psi) which are stably provided by the oven and the high pressure syringe pump. The retention time is chosen when longer retention time will not result in higher outﬂow concentration to make sure the equilibrium has been reached. The details of the apparatus and measurement procedures for diﬀerent systems can be found in the previous papers.15,27,65,66 Anhydrite (CaSO4) and gypsum (CaSO4·2H2O) are the two stable phases of calcium sulfates and will transit between each other or other phases (e.g., hemihydrate, CaSO4·0.5H2O) depending on the supersaturation state, temperature, pressure, and water activity.67,68 Under most conditions, the dominant and stable phase is believed to be gypsum under 40 °C, and anhydrite over 130 °C.68−73 This research will only adopt solubility data of gypsum and anhydrite at such temperature ranges. The gypsum solubility were measured at temperatures of (0, 25, and 40) °C, pressures of (14.7, 600, 7000, 16 000, and 20 000) psi with NaCl concentrations of (0, 1, 2, and 4) mol/kg H2O (Supporting Information Table 1). Anhydrite solubility from 100 °C to 250 °C has been systematically analyzed up to 14 500 psi in 0 to 6.2 m NaCl by Blount and Dickson.16,74 Anhydrite solubility is measured at three diﬀerent conditions identical to those measured by Blount and Dickson16 which are 150 °C and 19 400 psi, 200 °C and 19 000 psi, and 250 °C and 17 900 psi, respectively, all in 1.9 mol NaCl/kg H2O background. The results match well with the data reported in previous papers which are listed in Supporting Information Table 2. The anhydrite solubility data from Blount and Dickson16 listed in Supporting Information Table 1 will be applied in this research. In our previous papers, we have also measured the solubility of barite and calcite over wide temperature and pressure ranges (i.e., 0 °C to 250 °C and 14.7 psi to 22 000 psi) in 0.1 m to 6 m

(3)

2

aH m +

HCO−3 ⇄ H+ + CO32 − ,

K2 =

CO32 −

(2)

γCO2− 3

mHCO−3 γHCO− 3

(4)

⎛ m 2+γ 2+(m −γ −)2 K ⎞ Ca Ca HCO3 HCO3 2 ⎟=0 = log10⎜⎜ ⎟ m K K γ a ⎝ H2O CO2(aq) CO2(aq) 1 sp,calcite ⎠

(5)

CaCO3 ⇄ Ca 2 + + CO32 − , ⎛ mCa 2+γCa 2+mCO2−γCO2− ⎞ 3 3 ⎟⎟ SI = log10⎜⎜ K sp,calcite ⎝ ⎠

CaSO4 ·nH 2O ⇄ Ca 2 + + SO24 − + nH 2O, ⎛ mCa 2+γ 2+mSO2−γ 2−(a H O)n ⎞ Ca 4 SO4 2 ⎟=0 SI = log10⎜⎜ ⎟ K sp,gypsum/anhydrite ⎝ ⎠

(6)

where m and γ stands for molality and activity coeﬃcient of each species, respectively; aH2O is the activity of water; Ksp represents solubility product of each mineral; K1 and K2 are the dissociation constants of carbonic acid. The molality of each species can be measured or calculated in the experiments, and the stability constants can be calculated according to diﬀerent theories as stated below. The activity coeﬃcients can be expressed in the formulas of Pitzer theory with virial coeﬃcients to be determined. Li and Duan25 have studied the dissociation constants of carbonic acid which have been successfully applied in various researches.15,25,27,36 The solubility product of calcite is given by Plummer et al.35 based on experimental data.21,35 The solubility products of barite, gypsum, and anhydrite are based on the HKF theory by applying the estimations of heat capacity, enthalpy, and entropy for the temperature dependence up to 350 °C, and the C

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partial volume and compressibility (i.e., ΔV̅ ϕr and Δκϕr ) of the reaction for the pressure dependence.13,20,21,29,32−34,76 The temperature dependent parts for the four minerals and pressure dependent parts for barite and calcite ﬁtted and listed in SOLMINEQ. 88 will be adopted in this research (Table 1). In

Table 3. Sources of Virial Coeﬃcients Used in This Study

Table 1. Sources of the Stability Constants Applied in This Research stability constants

temperature-dependent part

K1, K2 of H2CO3 barite Ksp

Duan and Li (2007)

36

Duan and Li (2007)

Decock and Carrol (1982),34 Helgeson (1985)76 Plummer et al. (1976)35

calcite Ksp

Decock and Carrol (1982),34 Helgeson (1985)76 M?ller (1988),20 Decock and Carrol (1982)34

gypsum Ksp anhydrite Ksp

pressure-dependent part

∑ qi ·TC i ,

ln KT , P = ln KT , P0 −

RT

(P − P0) +

(1) φ β(0) NaCl, βNaCl, CNaCl θNaCa, θClHCO3, ψHCaCl, ψNaClHCO3

Pitzer, Peiper et al. 198448 Harvie, Möller et al. 198458

θHNa, ψNaCaCl β(1) CaSO4

He and Morse 199322 Kaasa 199814

(1) β(1) NaSO4, βBaSO4

He 199256

with full consideration of temperature dependence will be adopted in this research (also listed in Table 3). For the β(2) terms, only β(2) CaSO4 will be considered since the solubility data within the intermediate IS range (e.g., about 0.03 M to 0.1 M) is only available for calcium sulfates in our measurements and literature over a wide temperature and pressure ranges. Outside of the intermediate IS range, the 2−2 ion pair association is too weak to be examined. Besides, the contributions of β(2) terms to the binary ion interactions in Pitzer theory can be ignored at high IS (over 0.1 M) when the IS-dependent multiplier quickly diminishes with IS. Therefore, the virial coeﬃcients to be ﬁtted (2) ϕ (0) (0) ϕ (0) ϕ include β(0) CaSO4, βCaSO4, CCaSO4, βBaSO4, βBaCl, CBaCl, βNaSO4, CNaSO4, (0) βCaBr. The temperature and pressure dependence of these virial coeﬃcients are expressed as the semiempirical function of which the similar structures have been widely applied:15,22,48 z PZ(T , P) = z1 + 2 + z 3T + z4 ln T + z5T 2 T z ⎛ ⎞ + ⎜z6 + 6 + z8T + z 9 ln T + z10T 2⎟P ⎝ ⎠ T z12 ⎛ ⎞ + ⎜z11 + + z13T + z14 ln T + z15T 2⎟P 2 ⎝ ⎠ T

Helgeson (1978),13 Kharaka et al. (1989)21 Helgeson (1978),13 Kharaka et al. (1989)21 Atkinson and Mecik (1997)33 Atkinson and Mecik (1997)33

i = 0, 1, 2, 3 ΔVr̅ ϕ, T , P0

sources Christov and Möller 200426 He and Morse 199322

36

addition, Atkinson and Mecik33 ﬁtted both ΔVϕr,T,P0 and Δκϕr as a function of temperature (eq 7).14,33 The ﬁtted values of the q sets are listed in Table 2. The pressure-dependent part of gypsum and anhydrite will be calculated as eq 8 below. f (TC) =

virial coeﬃcients (1) φ β(0) CaCl, βCaCl, CCaCl, θHCa (0) (0) βNaHCO3, βNaHCO3,CφNaHCO3, λCO2Na, λCO2Cl

(7)

Δκr̅ ϕ (P − P0)2 2RT (8)

where P0 is the reference pressure (i.e., 1 atm below 100 °C and water vapor pressure above 1 atm). In this research, the temperature-dependent part from HKF theory and the pressuredependent part from Atkinson and Mecik will be incorporated to represent the solubility products of gypsum and anhydrite as a function of temperature and pressure. The equations of Pitzer theory for activity coeﬃcient calculations can be found in Pitzer’s book40 and related papers.43,44,48,15,18,20,22,23,36,46 In this study, the main binary ion interactions that should be considered include Na−Cl, Na−SO4, Na−HCO3, Ca−Cl, Ca−SO4, Ca−HCO3, Ca−Br, Ba−Cl, and Ba-SO4. Among them, some virial coeﬃcients (listed in Table 3) have been well established over wide temperature and pressure ranges. Shi et al.27 observed that Ca-HCO3 is less sensitive to the pressure changes. To take account for the SI deviation of calcite in mixed electrolytes, interactions of Ca−Br and Ca−SO4 instead of Ca−HCO3 are to be adjusted with temperature and pressure dependence in this research. Moreover, Pitzer et al.48 suggested that in NaCl solutions, β(0) and Cϕ terms are functions of both temperature and pressure, but independent of IS, whereas the β(1) term does not change with pressure much and is supposed to be only a function of temperature. Thus, most of the β(1) terms

(9)

Theoretically, the SI value of each solubility measurement should be 0 when reaching equilibrium. The sum of SI2 (i.e., ΣSI2 = Σ(SI − 0)2) for each solubility data in the solutions containing the four minerals are minimized to ﬁt the temperature and pressure-dependent parameters in eq 9 (i.e., z1 to z15) for each undetermined virial coeﬃcient. The genetic algorithm on MATLAB is used to optimize the objective function. The program is running on the super computer, DAVinCI, at Rice University.

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RESULTS AND DISCUSSION Solubility Predictions. The temperature- and pressuredependent parameters for each virial coeﬃcient are ﬁtted and listed in Table 4, Table 5, and Table 6. With these parameters, the values of each virial coeﬃcient under diﬀerent conditions can be calculated, and the activity coeﬃcient for each ion in these aqueous solutions can thus be predicted accurately. On the basis

Table 2. Parameters for the Temperature Dependence of ΔVϕr and Δκϕr for Gypsum and Anhydrite14,33 q2

q3

gypsum

ΔVϕrT,p0

−47.722

q0

0.22247

−1.8252·10−3

−2.1242·10−6

anhydrite

Δκϕr ΔVϕrT,p0

−17.83 −55.379

1.543·10−2 0.22423

1.601·10−3 −1.9922·10−3

−1.684·10−5 −2.0301·10−6

Δκϕr

−15.89

6.0247·10−3

−1.673·10−5

minerals

q1

0

ΔVϕrT,p0 in unit of cm3/mol, Δκϕr in unit of 103 cm3/(mole bar). D

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Table 4. Parameters for Virial Coeﬃcients Representing Ca−SO4 Interactions z1 z2 z3 z4 z5 z6 z7 z8 z9 z10 z11 z12 z13 z14 z15

β(0) CaSO4

β(2) CaSO4

CϕCaSO4

−1.23124210308582·1000 7.49671194585645·1002 −4.73763515840242·10−03 −2.60916455110925·10−01 9.19230233971759·10−06 −2.07071679632132·10−04 6.31787287140741·10−01 −1.70093829312035·10−06 −1.19779536757841·10−04 −1.83541477168724·10−09 −5.85524620503627·10−07 3.38280269359202·10−04 −1.19715195898137·10−09 −3.80583050754424·10−08 −3.91432244208833·10−12

4.25491942056627·1001 2.39473064409586·1004 −3.75264582714950·10−01 −1.34694736649945·1000 −8.32171829852575·10−04 3.70010720113461·10−02 −1.37447763082107·1001 2.05970175045264·10−04 6.70523676039265·10−03 3.86683716907682·10−10 −1.68872098058175·10−05 −1.39913318717239·10−02 1.30199555939164·10−07 −1.49805850283876·10−06 −2.06536262643717·10−10

−1.76772320113455·10−01 5.81629887084948·1002 −1.42138137539279·10−03 −1.18210241736728·10−01 5.93239717040146·10−06 −1.36220540397882·10−04 −3.51870462858910·10−01 1.61008388326011·10−06 4.33467129265007·10−05 1.77591567840277·10−09 −2.68618439387069·10−07 −2.14146437837877·10−04 2.64778512244175·10−09 3.76232892608941·10−08 2.20648208389312·10−12

Table 5. Parameters for Virial Coeﬃcients Representing Ba−SO4 and Ba−Cl Interactions β(0) BaSO4 z1 z2 z3 z4 z5 z6 z7 z8 z9 z10 z11 z12 z13 z14 z15

βϕBaCl

β(0) BaCl

−4.26330251270769·10 7.42382833736292·1003 3.04849752726983·10−01 −1.41723945559967·1001 1.33518609963378·10−03 −1.03555829753726·10−01 3.92654431330031·1001 −1.86934602374119·10−04 −1.53737016653037·10−02 2.54069658089713·10−07 −2.61017347580062·10−05 2.49758972711934·10−02 −8.26350437814859·10−08 −3.49783425209638·10−06 2.55795827827990·10−10 02

−01

8.59856204794662·10 −2.59927251898707·1002 −1.20930708314635·10−03 1.16099604287759·10−01 −2.56335261874473·10−07 7.66684936632549·10−04 −1.47914826039283·10−01 −1.23715710565886·10−06 7.10852570753079·10−05 −1.39810660572220·10−09 3.81774953486760·10−07 −1.36630572912662·10−04 −7.05400660573214·10−10 3.54950146112999·10−08 −1.88087221511704·10−13

2.87267795731851·10−01 7.76053203972202·1000 −1.65560042276988·10−03 5.74475887931065·10−03 1.71042441444030·10−06 1.26202145397791·10−04 −7.20883638599124·10−02 −8.23471123248952·10−07 1.65712376102864·10−05 1.40384049508077·10−09 1.87179334896813·10−07 1.45060566996736·10−05 −7.61906299467309·10−10 3.05828039359368·10−09 7.72357096028062·10−13

Table 6. Parameters for Virial Coeﬃcients Representing Na−SO4 and Ca−Br Interactions CϕNaSO4

β(0) NaSO4 z1 z2 z3 z4 z5 z6 z7 z8 z9 z10 z11 z12 z13 z14 z15

−01

2.94047000625712·10 −1.03208976081242·1002 −4.78076291256761·10−04 3.88849980209989·10−02 4.66052460396946·10−07 2.88921790846207·10−04 −2.78962580662594·10−02 −7.62320101281829·10−07 1.48313995121977·10−05 −4.55358823231623·10−10 6.21509889288744·10−08 −2.12609587508880·10−05 −1.38020398207072·10−10 7.10159088489194·10−11 9.88889955911576·10−13

β(0) CaBr −01

1.68758976332341·10 −3.81278402408175·1001 −7.27594471084674·10−04 2.01764726480784·10−02 4.22172705234184·10−07 1.85297267923483·10−04 −2.20092324044005·10−02 −3.50233495321443·10−07 9.89675129894031·10−06 1.30133707767905·10−10 −2.74790889995891·10−08 3.62454795826759·10−05 −4.71069172129012·10−10 −1.63293554236369·10−08 8.42701604200576·10−13

2.83982376071247·1000 −1.04321995787844·1003 8.39931524247816·10−03 6.97507618215969·10−01 −4.99360253648432·10−05 4.66694128657528·10−03 −1.30485985528794·1000 1.39848176702415·10−05 9.26730217736824·10−04 −4.45621831057912·10−08 1.49654610232772·10−07 −1.43221473761194·10−03 7.97450510957051·10−09 2.41944015734913·10−07 −1.14917791306241·10−11

packed solids. The average SI value for the 296 data points is 0.0061 with a standard of deviation 0.072. More than 85 % of the SI values are within the range of [−0.1, 0.1], which means most of the relative errors of the solubility predictions are within 12 % (i.e., |1−10±0.05|). Under the normal distribution assumption, the 95 % conﬁdential interval is about [−0.135, 0.147]. The ﬁtted

of the predicted activity coeﬃcients and related equilibrium constants, the SI values are calculated under various conditions and plotted in Figure 2 (listed in Supporting Information Tables 1, 3, 5, and 7). SI values represent the predicted scaling risks of these minerals under each condition and should be 0 theoretically since the solutions are in equilibrium with the E

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Figure 3. Association constants for CaSO(0) 4 ion pairs at inﬁnite dilution at diﬀerent temperatures and pressures derived from Pitzer theory (this study), Fuoss theory, and the data in SOLMINEQ. 88.

Figure 2. Predicted SI values and distribution by the new thermodynamic model for all the data points adopted in this research.

temperatures from 0 to 350 °C based on the solubility measurements which are plotted in Figure 3.21,79 In their paper, the predicted value of Kassoc,CaSO(0) at 25 °C and 1 atm is 4 2.30 10 . This research calculates the stability constant at 25 °C and 1 atm to be 102.15 (141) based on the derived β(2) CaSO4 values, and the value matches well with the data by Bell and George, Martell and Smith, and Kalyanaraman. At higher pressures, the experimentally measured association constants of CaSO(0) 4 have not been reported in the literature. For comparisons with the ion pair association constant calculated from the virial coeﬃcients ﬁtted in this study, Fuoss theory for ion pair association constants is applied to calculate Kassoc for CaSO4(0) over the temperature and pressure ranges at inﬁnite dilution. The Fuoss theory is derived from the consideration of columbic interactions, statistical arguments, and the theory of diﬀusion.63 By incorporating the extended Debye−Huckel theory for activity coeﬃcients estimation (the third term in eq 10), it can be expressed as follows:57,59,60,63

results show a good applicability of this model in scale prediction and control under extreme conditions in diﬀerent industrial processes. In detail, for the calcium sulfates solubility or scaling risks prediction, only two anhydrite data points measured by Blount and Dickson16 (i.e., the sample ID is 80 at 184 °C,1450 psi, and 82 at 263 °C, 14504 psi) have the SI values larger than 0.15.16 For the calcite and barite scaling risks predictions in solutions with mixed electrolytes, all SI values are within [−0.10, 0.10] showing good prediction capability of this model in mixed electrolytes. For the scaling risk predictions of calcite in solutions with NaCl, only three data points (sample IDs 112, 120, and 139) have SI values out of the [−0.15, 0.15] range. They are measured under very high temperature (i.e., 250 °C) or very high pressure (i.e., 21 000 psi). The imperfect measurements under such extreme conditions might lead to such deviations. More data points are available for barite solubility measurements, and among them only 10 predicted SI values are out of the [−0.15, 0.15] range. Most of these deviated data points are measured at 0 or 200 °C and 500 psi. Especially, the only two data points with SI bigger than 0.20 (i.e., sample IDs 241 and 242) are measured at 200 °C, 500 psi with 0.1 m and 1 m NaCl in the solution, respectively. Such deviations might probably be due to random measurement errors. Mineral solubilities at such high temperature and pressure are extremely diﬃcult to measure experimentally. More research is needed to validate the solubility data over wide temperature and pressure ranges. Ion Pair Association Constant Predictions. In Pitzer theory, the ion pair association is treated as a strong ion−ion binary interaction. The ion pair association constant at inﬁnite dilution can be calculated by Kassoc = −2β(2). On the basis of the accurate solubility measurements of calcium sulfates (i.e., gypsum and anhydrite) in 0 m to 4 m NaCl solutions over wide ranges of temperature and pressure, the values of β(2) CaSO4 can be calculated using eq 9. The CaSO(0) ion pair association 4 constants Kassoc,CaSO(0) are calculated and plotted in Figure 3. 4 value was studied by diﬀerent Previously, the Kassoc,CaSO(0) 4 experimental methods at 1 atm. For example, Bell and George77 calculated Kassoc,CaSO(0) to be 102.30 at 25 °C and 1 atm based on 4 the solubility measurement of thallous iodate in diﬀerent salt solutions. Martell and Smith78 reviewed a lot of research and to be 102.30 to 102.31. Kalyanaraman et al. suggested Kassoc,CaSO(0) 4 (1973) calculated the association constants for CaSO4(0) at

K assoc =

⎡ −z z e 2 ⎤ 4000πNava3 M L ⎥ exp⎢ 3 4 πε ⎣ 0DkTa ⎦ ⎤ ⎡ z Mz Le 2κ ⎥ exp⎢ ⎣ 4πε0DkT (1 + κa) ⎦

κ2 =

2000e 2Nav IS ε0DkT

(10)

(11)

where a is the closest center-to-center approach distance of the ions in pair (usually set as 5·10−10 m); Nav is Avogadro constant; e is the elementary charge; k stands for the Boltzmann constant; ε0 is the dielectric constant of vacuum; D is the dielectric constant of solvent (water) which is in function of temperature and pressure;80 κ is the Debye−Huckel ion atmosphere parameter and will be zero when IS = 0; ZM and ZL are the charges of the cation and the anion, respectively. This equation is an approximation of charge speciﬁc ion pair association constants and shows the trend of the association constants change with temperature, pressure, and IS. The association constants for 2−2 ion pairs at certain temperatures and pressures are calculated and plotted in Figure 3. In detail, this model estimates the association constant for the 2−2 ion pairs to be 101.98 (95) which is a little smaller than the experimental results. It also shows that the association constants will increase with temperature and decrease with pressure. This trend can be F

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temperature, pressure, and TDS, and to explore more robust temperature and pressure dependent functions for the virial coeﬃcients. Following the procedures of this research, the virial coeﬃcients for more ion speciﬁc interaction can be ﬁtted with wide ranges of temperature and pressure dependences. These virial coeﬃcients can improve the solubility prediction of related minerals under extreme conditions, and can give more thermodynamic information on the system. Finally, it will help handle the scale problem in many various disciplines.

partially attributed to the impacts of the dielectric constant of water which decreases with temperature and increases with pressure. When IS increases, the association constants will drop very quickly. As a rough example, at 25 °C and 1 atm, the association constant for the 2−2 ion pairs is 101.98 at IS = 0 M, 101.13 at IS = 0.1 M, and 100.08 at IS = 4 M. Figure 3 shows that at 1 atm or saturated vapor pressure, the three sets of data have similar increasing trend with temperature. At relatively higher temperatures, the data from SOLMINEQ. 88 shows stronger temperature dependence and larger values than that from Fuoss theory and Pitzer theory based on this research. Over the wide temperature and pressure ranges, the Kassoc,CaSO(0) 4 values from the Pitzer theory and the Fuoss theory show similar pressure dependence. Especially, at lower temperatures most data based on Pitzer theory and Fuoss theory has very good identity with each other. But at higher temperatures (i.e., 200 °C and 250 °C) the Fuoss theory gives larger estimation for the association constants, and there are two data points with deviation at 7000 psi and 16 000 psi under 0 °C. These deviations show that the β(2) CaSO4 term derived in this research underestimates the association of CaSO(0) 4 at these conditions which might be due to the imperfect solubility measurements at 0 °C and 250 °C or the incomplete temperature and pressure dependence in these two theories. It deserves notice that the equation structures of the two methods (i.e., eq 9 for the Pitzer theory versus eq 10 for the Fuoss theory) are totally diﬀerent from each other. On the basis of the derivation of virial coeﬃcients for Ca−SO4 interactions in this study, the Pitzer theory gives a good estimation of the Kassoc,CaSO(0) values over wide temperature and pressure ranges. 4 However, more research is still in need to clarify these controversies by giving more accurate solubilities, solubility products, and other related virial coeﬃcients.

■

ASSOCIATED CONTENT

S Supporting Information *

Original solubility data, solution compositions, and the calculated SI for diﬀerent minerals applied in this paper. This material is available free of charge via the Internet at http://pubs. acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*Email: [email protected] Funding

This work was ﬁnancially supported by Brine Chemistry Consortium companies of Rice University, including Baker Hughes, BCS, BWA, CARBO, Chevron, ConocoPhillips, Dow, Hess, Kemira, Kinder Morgan, Marathon, Nalco Champion/An Ecolab company, Occidental, Petrobras, Saudi Aramco, Schlumberger, Shell, SNF, Statoil, Total, and Weatherford. This work was supported in part by the Data Analysis and Visualization Cyberinfrastructure funded by NSF under Grant OCI-0959097. Notes

The authors declare no competing ﬁnancial interest.

■

■

CONCLUSION Previous papers of our group and others reported the solubility of barite, calcite, and calcium sulfates over wide ranges of temperature, pressure, and TDS. With these solubility data, the temperature- and pressure-dependence of the virial coeﬃcients for Ca-SO 4 , Ba−Cl, Na-SO 4 , Ca−Br interactions were developed. When these virial coeﬃcients are applied in the Pitzer ion interaction model, the saturation indices of barite, calcite, and calcium sulfates from 0 °C to 250 °C, 14.7 psi to 22000 psi, in 0 mol to 6 mol NaCl/kg H2O or even with mixed electrolytes can be predicted without apparent temperature, pressure, or IS bias. Furthermore, the association constant of CaSO(0) 4 ion pair was derived over wide ranges of temperature and pressure according to the β(2) CaSO4 values developed in this derived from β(2) study. The Kassoc,CaSO(0) CaSO4 matches well with that 4 from other experimental work at 1 atm and 25 °C and Kassoc,CaSO(0) 4 predicted at high temperature and pressure follows the same trend with that calculated using Fuoss ion pair association theory. The association constants prediction shows that the new model not only can beneﬁt the solubility prediction of minerals but can also be useful for research on the thermodynamic properties of ions, ion pairs, and solids in the aqueous system over wide ranges of temperature and pressure. This thermodynamic model will be incorporated in the software ScaleSoftPitzer developed by our group for scale prediction and control in the oil and gas industry. However, the inaccuracies of some data points in the solubility prediction and the ion pair association estimation show the necessity to gather more solubility data over wider ranges of

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