# Thermodynamic Modeling of Mineral Scaling in High-Temperature and High-Pressure Aqueous Environments

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{−3}. This failure occurs due to the model’s reliance on an empirical form of the Born equation which is unable to capture the trends observed in these high temperature, low density regimes. However, new models based on molecular solvent-solute interactions offer a pathway to overcome some of the deficiencies currently limiting high-temperature and high-pressure mineral scale predictions. Examples of the most common scale prediction methods are presented, and their advantages and disadvantages are discussed.

## 1. Introduction

_{sat}) and whether a particular fluid is above or below that limit. As such, knowing b

_{sat}is paramount to effective mineral scale control. To better predict b

_{sat}, the modeling of how it changes with system conditions has evolved from simple polynomials [4,5] to sophisticated thermodynamic programs [6,7,8,9] containing computational algorithms and thermodynamic databases [10,11,12]. This progression was a necessary response to the increasing complexity and diversity of how different mineral solubility limits change with the composition, temperature and pressure of fluids observed in applications which now range from low-temperature, low-pressure (LTLP) to high-temperature, high-pressure (HTHP) conditions [2,3]. Though complex, it is vital that thermodynamic models reliably capture these trends because system properties (temperature, pressure, etc.) vary dramatically within a system which can promote scaling in different regions (see Figure 1).

_{sp}) to assess a fluids tendency for scale formation [14,15]. This tendency is quantified with what is known as the scale saturation ratio (SR),

_{sp}for a given mineral [14,15].

## 2. Empirical Fitting Methods

^{2}value). These models either fit an equilibrium constant for a mineral precipitation reaction (solubility product) or a solubility limit concentration itself [4,5,22,23,24]. The choice of solubility product over solubility limit was often based on the desired use of the final fitting. Earlier works often focused on obtaining the solubility products, since these were not known for many mineral reactions, whereas later works focused on the solubility limits since these values are often used to validate new thermodynamic models for predicting mineral precipitation conditions.

_{sat}= a

_{0}+ a

_{1}t + a

_{2}t

^{2}+ a

_{3}t

^{3}+ a

_{4}t

^{4}+ a

_{5}t

^{5}+ a

_{6}t

^{6}

_{0}–a

_{6}are empirical constants used to fit the experimental data. As such, this empirical model provides reliable mineral solubility limits for the water-mineral system over a given temperature range. The work by Krumgalz demonstrates both the impact of temperature on the solubility of common minerals, and how they can vary dramatically between minerals for the same set of conditions (see Figure 3).

^{2}of 0.98 or better, with only three of the six empirical constants, but the number of constants required typically increased with the temperature range. Three to six terms are required for simple systems over small temperature ranges, it becomes increasingly difficult to accommodate additional dependences such as composition and pressure changes without a model being inundated by empirical constants.

_{sp}) of calcium carbonate minerals in different seawater salinities for a range of temperatures [5,22]. Their goal was to model both the temperature and salinity dependences of calcium carbonate. The resulting model to capture temperature and salinity trends are as follows:

_{10}K

_{sp}= a

_{0}+ a

_{1}T + a

_{2}T

^{−1}+ a

_{3}log

_{10}(T) + (b

_{0}+ b

_{1}T + b

_{2}T

^{−1})S

^{0.5}+ c

_{0}S + d

_{0}S

^{1.5}

_{0}–a

_{3}, b

_{0}–b

_{2}, c

_{0}and d

_{0}are all the empirical constants needed to model the dependence of the solubility product on salinity (S), (from 5 to 44 ppt), and temperature (T), (from 278.15 to 313.15 K). Note that K

_{sp}can be converted to b

_{sat}through the following expression:

_{sat}= K

_{sp}

^{0.5}

## 3. Solubility Product Methods

_{sp}and the ${\mathsf{\Delta}}_{\mathrm{r}}{G}^{0}$:

^{−1}K

^{−1}) and T is the thermodynamic temperature (in K). Despite the apparent simplicity of this approach, the underlying equations used to determine the necessary Gibbs energy data can be quite involved but they are able to capture an extraordinary range of temperatures and pressures.

_{T}) relative to its enthalpy at a reference temperature (H

_{298}) [28]:

_{T}− H

_{298}= A + BT + CT

^{2}+ DT

^{−1}+ ET

^{1/2}+ FT

^{3}

^{−3}[8]. This density limitation excludes a sizable range of conditions of interest to petroleum and natural gas extraction [29] and some other important applications such as high-enthalpy supercritical geothermal technology [30] and supercritical water gasification of biomass [31] (see Figure 6).

_{1}–a

_{4}, c

_{1}–c

_{2}and ω are the seven empirical constants specific to each species. In Equation (10) $\mathsf{\Theta}$ = 228 K and $\mathsf{\Psi}$ = 2600 bar are constant values within the model. Ω

_{Pr,Tr}is the conventional born coefficient, ε is the relative permittivity of the solvent, and Y

_{Pr,Tr}is a constant based on of the relative permittivity of the solvent at 25 °C and 1 bar.

_{r}= 298.15 K and P

_{r}= 1 bar, a

_{1}and c

_{1}are empirical parameters responsible for all short range interactions and ${G}_{T,P}^{k}$ and ${S}_{Tr,Pr}^{k}$ are the contributions of specific molecular statistical interactions that account for hard sphere interactions [40] and electrostatic interactions [41,42,43]. Additionally, the standard state contributions are included to ${G}_{T,P}^{k}$ and ${S}_{Tr,Pr}^{k}$ [44]. Within these expressions empirical diameters of the solute (σ

_{i}) and solvent (σ

_{w}) and dipole moment of a molecule/ion pair (p

_{j}) are the three parameters that are fit for an ionic species with or without a dipole [6].

_{j}

^{HS}, is determined as [40]:

^{−1}mol

^{−1}, η = πN

_{A}ρσ

_{w}

^{3}/6, ρ is the molecular density, D = σ

_{i}/σ

_{w}, N

_{A}is the Avogadro number = 6.0221 10

^{23}mol

^{−1}, β = 1/(kT) where k is the Boltzmann constant = 1.3806 10

^{−23}J K

^{−1}. The mean spherical approximation (MSA) for an ion-dipole interaction was determined as [42]:

_{i}is the charge number of the ionic species, e is the elementary charge = 1.602 10

^{−19}C, ε is the permittivity of the pure solvent. ε is related to β

_{6}and β

_{3}through the well-known Wertheim equation given as:

_{2}is a parameter of the MSA theory. The dipole–dipole electrostatic term, G

_{j}

^{DD}, is calculated as [43]:

_{s}is the molar mass of the solvent = 18.015 10

^{−3}kg/mol and b

^{0}= 1 mol/kg is the standard molality. While this approach has been more successful than the HKF model, it has only been applied to a few dozen species and not the hundreds currently covered by the SUPCRT database.

^{−3}or prohibit calculations entirely [8,10]. As such, this approach to mineral scale modeling is only valid if (i) the reaction of interest is known, (ii) the database models are valid for the conditions of interest and (iii) the species of interest have available empirical constants for the conditions of interest. For these reasons, the multi-phase, multi-component systems, encountered in the applications mentioned earlier, require computational software beyond the basic solubility product methods for predicting the solubility limit of scaling minerals.

## 4. Speciation Model Methods

_{2}partial pressure on calcium carbonate scaling are well-known [9,15,45,46,47], and require more complex methods to predict the extent of scaling [9,48,49]. Likewise, the advent of ion-pairs at elevated temperatures and pressures [26,50,51,52], complicate the identification of the correct solubility product limiting mineral solubility.

## 5. Conclusions

^{−3}) regions. This issue stems from inherent flaws in the empirical models used to build these databases. Though models of new standard thermodynamic functions have proved to be successful, they still cover a very small number of species needed by the growing number of industrial applications in this space. Therefore, future efforts are needed to recreate these self-consistent databases which have underpinned so much recent growth in aqueous-based mineral scale modeling.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Brown, M. Full Scale Attack. Review
**1998**, 30, 30–32. [Google Scholar] - Kamal, M.S.; Hussein, I.; Mahmoud, M.; Sultan, A.S.; Saad, M.A.S. Oilfield Scale Formation and Chemical Removal: A Review. J. Pet. Sci. Eng.
**2018**, 171, 127–139. [Google Scholar] [CrossRef] - Olajire, A.A. A Review of Oilfield Scale Management Technology for Oil and Gas Production. J. Pet. Sci. Eng.
**2015**, 135, 723–737. [Google Scholar] [CrossRef] - Krumgalz, B.S. Temperature Dependence of Mineral Solubility in Water. Part 3. Alkaline and Alkaline Earth Sulfates. J. Phys. Chem. Ref. Data
**2018**, 47, 023101. [Google Scholar] [CrossRef] - Mucci, A. The Solubility of Calcite and Aragonite in Seawater at Various Salinities, Temperatures, and One Atmosphere Total Pressure. Am. J. Sci.
**1983**, 283, 780–799. [Google Scholar] [CrossRef] - Lvov, S.N.; Hall, D.M.; Bandura, A.V.; Gamwo, I.K. A Semi-Empirical Molecular Statistical Thermodynamic Model for Calculating Standard Molar Gibbs Energies of Aqueous Species above and below the Critical Point of Water. J. Mol. Liq.
**2018**, 270, 62–73. [Google Scholar] [CrossRef] - Helgeson, H.C. Thermodynamics of Hydrothermal Systems at Elevated Temperatures and Pressures. Am. J. Sci.
**1969**, 267, 729–804. [Google Scholar] [CrossRef] - Johnson, J.W.; Oelkers, E.H.; Helgeson, H.C. SUPCRT92: A Software Package for Calculating the Standard Molal Thermodynamic Properties of Minerals, Gases, Aqueous Species, and Reactions from 1 to 5000 Bars and 0 to 1000 °C. Comput. Geosci.
**1992**, 18, 899–947. [Google Scholar] [CrossRef] - Kan, A.T.; Dai, J.Z.; Deng, G.; Ruan, G.; Li, W.; Harouaka, K.; Lu, Y.T.; Wang, X.; Zhao, Y.; Tomson, M.B. Recent Advances in Scale Prediction, Approach, and Limitations. SPE J.
**2019**, 24, 2209–2220. [Google Scholar] [CrossRef] - Shvarov, Y.V. HCh: New Potentialities for the Thermodynamic Simulation of Geochemical Systems Offered by Windows. Geochem. Int.
**2008**, 46, 834–839. [Google Scholar] [CrossRef] - OLI Systems Inc. Selecting AQ, MSE or MSE-SRK for Your Scaling Calculation; OLI Systems Inc.: Parsippany, NJ, USA, 2017. [Google Scholar]
- Wang, P.; Springer, R.D.; Anderko, A.; Young, R.D. Modeling Phase Equilibria and Speciation in Mixed-Solvent Electrolyte Systems. Fluid Phase Equilib.
**2004**, 222–223, 11–17. [Google Scholar] [CrossRef] - FQE Chemicals. Pipe contaminated with Barium Sulfate and NORM scale. Available online: https://fqechemicals.com/contaminants/barium-sulfate-scale/ (accessed on 10 August 2022).
- Greg Anderson Rock-Water Systems. In Thermodynamics of Natural Systems; Cambridge University Press: Cambridge, UK, 2005; pp. 473–497.
- Zhang, P.; Kan, A.T.; Tomson, M.B. Oil Field Mineral Scale Control. In Mineral Scales and Deposits; Elsevier: Amsterdam, The Netherlands, 2015; pp. 603–617. ISBN 9780444627520. [Google Scholar]
- Kan, A.T.; Tomson, M.B. Scale Prediction for Oil and Gas Production. SPE J.
**2012**, 17, 362–378. [Google Scholar] [CrossRef] - Fan, C.; Kan, A.T.; Zhang, P.; Lu, H.; Work, S.; Yu, J.; Tomson, M.B. Scale Prediction and Inhibition for Oil and Gas Production at High Temperature/High Pressure. SPE J.
**2012**, 17, 379–392. [Google Scholar] [CrossRef] - Kan, A.T.; Dai, Z.; Tomson, M.B. The State of the Art in Scale Inhibitor Squeeze Treatment. Pet. Sci.
**2020**, 17, 1579–1601. [Google Scholar] [CrossRef] - Mpelwa, M.; Tang, S.F. State of the Art of Synthetic Threshold Scale Inhibitors for Mineral Scaling in the Petroleum Industry: A Review. Pet. Sci.
**2019**, 16, 830–849. [Google Scholar] [CrossRef] - Helgeson, H.C.; Kirkham, D.H.; Flowers, G.C. Theoretical Prediction of the Thermodynamic Behavior of Aqueous Electrolytes at High Pressures and Temperatures: Calculation of Activity Coefficients, Osmotic Coefficients, and Apparent Molal and Standard and Relative Partial Molal Properties to 600 °C And. Am. J. Sci.
**1981**, 281, 1249–1516. [Google Scholar] [CrossRef] - Plummer, N.; Busenberg, E. The Solubilities of Calcite, Aragonite and Vaterite in CO
_{2}-H_{2}O Solutions between 0 and 90 C, and an Evaluation of the Aqueous Model for the System CaCO_{3}-CO_{2}-H_{2}O. Geochim. Cosmochim. Acta**1981**, 46, 1011–1040. [Google Scholar] [CrossRef] - Krumgalz, B.S. Temperature Dependence of Mineral Solubility in Water. Part I. Alkaline and Alkaline Earth Chlorides. J. Phys. Chem. Ref. Data
**2017**, 46, 043101. [Google Scholar] [CrossRef] - Krumgalz, B.S. Temperature Dependence of Mineral Solubility in Water. Part 2. Alkaline and Alkaline Earth Bromides. J. Phys. Chem. Ref. Data
**2018**, 47, 013101. [Google Scholar] [CrossRef] - Marion, G.M.; Millero, F.J.; Feistel, R. Precipitation of Solid Phase Calcium Carbonates and Their Effect on Application of Seawater SA-T-P Models. Ocean Sci.
**2009**, 5, 285–291. [Google Scholar] [CrossRef] - Djamali, E.; Chapman, W.G.; Cox, K.R. A Systematic Investigation of the Thermodynamic Properties of Aqueous Barium Sulfate up to High Temperatures and High Pressures. J. Chem. Eng. Data
**2016**, 61, 3585–3594. [Google Scholar] [CrossRef] - Monnin, C. A Thermodynamic Model for the Solubility of Barite and Celestite in Electrolyte Solutions and Seawater to 200 °C and to 1 Kbar. Chem. Geol.
**1999**, 153, 187–209. [Google Scholar] [CrossRef] - Robie, R.; Hemingway, B.; Fisher, J. Thermodynamic Properties of Minerals and Related Substances at 298.15 K and 1 bar Pressure and at Higher Temperatures. US Geol. Surv. Bull.
**1984**, 1452. [Google Scholar] [CrossRef] - Phi, T.; Elgaddafi, R.; Al Ramadan, M.; Fahd, K.; Ahmed, R.; Teodoriu, C. Well Integrity Issues: Extreme High-Pressure High-Temperature Wells and Geothermal Wells a Review. In SPE Thermal Well Integrity and Design Symposium; OnePetro: Richardson, TX, USA, 2019. [Google Scholar] [CrossRef]
- Friðleifsson, G.; Elders, W.A.; Zierenberg, R.A.; Fowler, A.P.G.; Weisenberger, T.B.; Mesfin, K.G.; Sigurðsson, Ó.; Níelsson, S.; Einarsson, G.; Óskarsson, F.; et al. The Iceland Deep Drilling Project at Reykjanes: Drilling into the Root Zone of a Black Smoker Analog. J. Volcanol. Geotherm. Res.
**2020**, 391, 106435. [Google Scholar] [CrossRef] - Su, H.; Yan, M.; Wang, S. Recent Advances in Supercritical Water Gasification of Biowaste Catalyzed by Transition Metal-Based Catalysts for Hydrogen Production. Renew. Sustain. Energy Rev.
**2022**, 154, 111831. [Google Scholar] [CrossRef] - Shock, E.L.; Helgeson, H.C. Calculation of the Thermodynamic and Transport Properties of Aqueous Species at High Pressures and Temperatures: Correlation Algorithms for Ionic Species and Equation of State Predictions to 5 Kb and 1000 °C. Geochim. Cosmochim. Acta
**1988**, 52, 2009–2036. [Google Scholar] [CrossRef] - Haas, J.R.; Shock, E.L.; Sassani, D.C. Rare-Earth Elements in Hydrothermal Systems—Estimates of Standard Partial Molal Thermodynamic Properties of Aqueous Complexes of the Rare-Earth Elements at High-Pressures and Temperatures. Geochim. Cosmochim. Acta
**1995**, 59, 4329–4350. [Google Scholar] [CrossRef] - Sverjensky, D.A.; Shock, E.L.; Helgeson, H.C. Prediction of the Thermodynamic Properties of Aqueous Metal Complexes to 1000 Degrees C and 5 Kb. Geochim. Cosmochim. Acta
**1997**, 61, 1359–1412. [Google Scholar] [CrossRef] - Shock, E.L.L.; Sassani, D.C.; Willis, M.; Sverjensky, D.A. Inorganic Species in Geologic Fluids: Correlations among Standard Molal Thermodynamic Properties of Aqueous Ions and Hydroxide Complexes. Geochim. Cosmochim. Acta
**1997**, 61, 907–950. [Google Scholar] [CrossRef] - Shock, E.L.; Helgeson, H.C. Calculation of the Thermodynamic and Transport Properties of Aqueous Species at High Pressures and Temperatures: Standard Partial Molal Properties of Organic Species. Geochim. Cosmochim. Acta
**1990**, 54, 915–945. [Google Scholar] [CrossRef] - Pokrovskii, V.A.; Helgeson, H.C. Calculation of the Standard Partial Molal Thermodynamic and Activity Coefficients of Aqueous KC1 at Temperatures and Pressures. Geochim. Cosmochim. Acta
**1997**, 61, 2175–2183. [Google Scholar] [CrossRef] - Hall, D.M.; Lvov, S.N.; Gamwo, I.K. Prediction of Barium Sulfate Deposition in Petroleum and Hydrothermal Systems. In Solid–Liquid Separation Technologies: Applications for Produced Water; CRC Press Taylor & Francis: Boca Raton, FL, USA, 2022. [Google Scholar]
- Morey, G.W.; Hesselgesser, J.M. The Solubility of Some Minerals in Superheated Steam at High Pressures. Econ. Geol.
**1951**, 46, 821–835. [Google Scholar] [CrossRef] - Mansoori, G.A.; Carnahan, N.F.; Starling, K.E.; Leland, T.W., Jr. Equilibrium Thermodynamic Properties of the Mixture of Hard Spheres. J. Chem. Phys.
**1971**, 54, 1523–1525. [Google Scholar] [CrossRef] - Triolo, R.; Grigera, J.; Blum, L. Simple Electrolytes in the Mean Spherical Approximation. J. Phys. Chem.
**1976**, 80, 1858–1861. [Google Scholar] [CrossRef] - Blum, L. Solution of a Model for the Solvent-Electrolyte Interactions in the Mean Spherical Approximation. J. Chem. Phys.
**1974**, 61, 2129. [Google Scholar] [CrossRef] - Bandura, A.V.; Holovko, M.F.; Lvov, S.N. The Chemical Potential of a Dipole in Dipolar Solvent at Infinite Dilution: Mean Spherical Approximation and Monte Carlo Simulation. J. Mol. Liq.
**2018**, 270, 52–61. [Google Scholar] [CrossRef] - Lvov, S.N.; Wood, R.H. Equation of State of Aqueous NaCl Solutions over a Wide Range of Temperatures, Pressures and Concentrations. Fluid Phase Equilib.
**1990**, 60, 273–287. [Google Scholar] [CrossRef] - Heberling, F.; Trainor, T.P.; Lützenkirchen, J.; Eng, P.; Denecke, M.A.; Bosbach, D. Structure and Reactivity of the Calcite-Water Interface. J. Colloid Interface Sci.
**2011**, 354, 843–857. [Google Scholar] [CrossRef] - García, A.V.; Thomsen, K.; Stenby, E.H. Prediction of Mineral Scale Formation in Geothermal and Oilfield Operations Using the Extended UNIQUAC Model. Part II. Carbonate-Scaling Minerals. Geothermics
**2006**, 35, 239–284. [Google Scholar] [CrossRef] - Li, J.; Duan, Z. A Thermodynamic Model for the Prediction of Phase Equilibria and Speciation in the H
_{2}O-CO_{2}-NaCl-CaCO_{3}-CaSO_{4}System from 0 to 250 °C, 1 to 1000 Bar with NaCl Concentrations up to Halite Saturation. Geochim. Cosmochim. Acta**2011**, 75, 4351–4376. [Google Scholar] [CrossRef] - Amiri, M.; Moghadasi, J. The Effect of Temperature on Calcium Carbonate Scale Formation in Iranian Oil Reservoirs Using OLI ScaleChem Software. Pet. Sci. Technol.
**2012**, 30, 453–466. [Google Scholar] [CrossRef] - Hajirezaie, S.; Wu, X.; Soltanian, M.R.; Sakha, S. Numerical Simulation of Mineral Precipitation in Hydrocarbon Reservoirs and Wellbores. Fuel
**2019**, 238, 462–472. [Google Scholar] [CrossRef] - Ho, P.C.; Palmer, D.A. Ion Association of Dilute Aqueous Potassium Chloride and Potassium Hydroxide Solutions to 600 °C and 300 MPa Determined by Electrical Conductance Measurements. Geochim. Cosmochim. Acta
**1997**, 61, 3027–3040. [Google Scholar] [CrossRef] - Ho, P.C.; Bianchi, H.; Palmer, D.A.; Wood, R.H. Conductivity of Dilute Aqueous Electrolyte Solutions at High Temperatures and Pressures Using a Flow Cell. J. Solution Chem.
**2000**, 29, 217–235. [Google Scholar] [CrossRef] - Ho, P.; Palmer, D. Ion Association of Dilute Aqueous Sodium Hydroxide Solutions to 600 C and 300 MPa by Conductance Measurements. J. Solution Chem.
**1996**, 25, 711–729. [Google Scholar] [CrossRef] - Fritz, J.J. Chloride Complexes of CuCl in Aqueous Solution. J. Phys. Chem.
**1980**, 84, 2241–2246. [Google Scholar] [CrossRef] - Balashov, V.N.; Schatz, R.S.; Chalkova, E.; Akinfiev, N.N.; Fedkin, M.V.; Lvov, S.N. CuCl Electrolysis for Hydrogen Production in the Cu–Cl Thermochemical Cycle. J. Electrochem. Soc.
**2011**, 158, B266–B275. [Google Scholar] [CrossRef] - Lvov, S.N.; Akinfiev, N.N.; Bandura, A.V.; Sigon, F.; Perboni, G. Multisys: Computer Code for Calculating Multicomponent Equilibria in High-Temperature Subcritical and Supercritical Aqueous Systems. In Steam, Water, and Hydrothermal Systems: Physics and Chemistry Meeting the Needs of Industry; Tremaine, P.R., Hill, P.G., Irish, D.E., Palakrishnan, P.V., Eds.; NRC Press: Ottawa, ON, USA, 2000; pp. 866–873. [Google Scholar]
- Du, P.C.; Mansoori, G.A. Phase Equilibrium of Multicomponent Mixtures: Continuous Mixture Gibbs Free Energy Minimization and Phase Rule. Chem. Eng. Commun.
**1987**, 54, 139–148. [Google Scholar] [CrossRef] - Shobu, K.; Tabaru, T. Development of New Equilibrium Calculation Software: CaTCalc. Mater. Trans.
**2005**, 46, 1175–1179. [Google Scholar] [CrossRef] - Bethke, C.M. Geochemical Reaction Modelling: Concepts and Applications; Oxford University Press: New York, NY, USA, 1996. [Google Scholar]
- Bethke, C.M. The Geochemists Workbench Version 4.0: A User’s Guide; University of Illinois: Urbana, IL, USA, 2002. [Google Scholar]
- Cleverley, J.S.; Bastrakov, E.N. K2GWB: Utility for Generating Thermodynamic Data Files for The Geochemist’s Workbench® at 0–1000 °C and 1–5000 Bar from UT2K and the UNITHERM Database. Comput. Geosci.
**2005**, 31, 756–767. [Google Scholar] [CrossRef] - Shvarov, Y. A Suite of Programs, OptimA, OptimB, OptimC, and OptimS Compatible with the Unitherm Database, for Deriving the Thermodynamic Properties of Aqueous Species from Solubility, Potentiometry and Spectroscopy Measurements. Appl. Geochem.
**2015**, 55, 17–27. [Google Scholar] [CrossRef] - OLI Systems. OLI Studio Stream Analyzer User Guide. V 9.5; OLI Systems: Parsippany, NJ, USA, 2018. [Google Scholar]
- Lencka, M.M.; Springer, R.D.; Wang, P.; Anderko, A. Modeling Mineral Scaling in Oil and Gas Environments up to Ultra High Pressures and Temperatures: Paper No. C2018-10828. In CORROSION; OnePetro: Richardson, TX, USA, 2018; pp. 1–15. [Google Scholar]
- Shvarov, Y.V. A General Equilibrium Criterion for an Isobaric-Isothermal Model of a Chemical System. Geochem. Int.
**1981**, 18, 38–45. [Google Scholar] - Shvarov, Y.V. Algorithmization of the Numeric Equilibrium Modeling of Dynamic Geochemical Processes. Geochem. Int.
**1999**, 37, 571–576. [Google Scholar] - Hall, D.M.; Akinfiev, N.N.; LaRow, E.G.; Schatz, R.S.; Lvov, S.N. Thermodynamics and Efficiency of a CuCl(Aq)/HCl(Aq) Electrolyzer. Electrochim. Acta
**2014**, 143, 70–82. [Google Scholar] [CrossRef] - Koukkari, P.; Pajarre, R. Calculation of Constrained Equilibria by Gibbs Energy Minimization. Comput. Coupling Phase Diagr. Thermochem.
**2006**, 30, 18–26. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) A generic demonstration of how mineral solubility limits (b

_{sat}) can vary with temperature for three different minerals (not to scale), (

**b**) an example of how temperature and pressure gradients can promote mineral scale formation (without considering the scaling kinetics), and (

**c**) an image capturing the possible consequences of mineral scale formation [13]. Figure 1c is reprinted with permission from [13]. Copyright 2022 FQE Chemicals.

**Figure 2.**Model evolution tree of mineral scale solubility models from the 1980s to today. The grey boxes are examples of the different techniques.

**Figure 3.**Calcium sulfate (

**a**) and barium sulfate (

**b**) empirical fits of mineral solubility limits in pure water as a function of temperature [4].

**Figure 5.**Comparison between experimental data (green) [4] and model predictions (red) using Equations (5)–(7). All calculations were performed assuming negligible solution non-ideality.

**Figure 6.**Temperature and pressure ranges (shown in red) where the HKF model has been observed to fail due to model limitations and insufficient experimental data.

**Figure 7.**Observed (●) solubility of quartz in supercritical water at t = 500 °C as a function of pressure [39] compared to NETL-PSU model [6] calculations (●) and HKF results which failed due to model limitations (○) [6]. The authors generated these plots assuming negligible activity coefficient contributions.

**Figure 8.**Comparison between HKF model (red) predictions to experimental data (green), (2700% deviations at 400 °C due to invalid empirical parameters and model limitations) [38]. The authors generated these plots assuming negligible activity coefficient contributions.

**Figure 10.**The effect of forming an ion pair on the solubility of barite at high temperatures [38]. The computation program predicts that mineral solubility limit is limited by the formation of ionic species at low temperatures and the ionic pair at high temperatures. The authors generated these plots assuming negligible activity coefficient contributions.

Method | Empirical Fits | Solubility Product | Speciation Models |
---|---|---|---|

Benefits | Easy to implement | Easy to implement | Predictive capabilities |

Predictive capabilities | Works with multiphase systems | ||

Provides solution compositions | |||

Limitations | No predictive capabilities | Requires a thermodynamic database | Requires a thermodynamic database |

Requires system-specific solubility data | Limited to the temperature and pressure range of the database | Limited to the temperature and pressure range of the database | |

Limited to the temperature and pressure range of the fit | Limited to simple systems |

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Hall, D.M.; Lvov, S.N.; Gamwo, I.K.
Thermodynamic Modeling of Mineral Scaling in High-Temperature and High-Pressure Aqueous Environments. *Liquids* **2022**, *2*, 303-317.
https://doi.org/10.3390/liquids2040018

**AMA Style**

Hall DM, Lvov SN, Gamwo IK.
Thermodynamic Modeling of Mineral Scaling in High-Temperature and High-Pressure Aqueous Environments. *Liquids*. 2022; 2(4):303-317.
https://doi.org/10.3390/liquids2040018

**Chicago/Turabian Style**

Hall, Derek M., Serguei N. Lvov, and Isaac K. Gamwo.
2022. "Thermodynamic Modeling of Mineral Scaling in High-Temperature and High-Pressure Aqueous Environments" *Liquids* 2, no. 4: 303-317.
https://doi.org/10.3390/liquids2040018