Bayes estimation: A novel approach to derivation of internally consistent thermodynamic data for minerals,their uncertainties,and correlations. Part I: Theory

Computation of phase diagrams in mineral systems and quantitative geothermobarometry thrive on the availability and accuracy of internally consistent thermodynamic datasets for minerals. The prevailing two methodologies applied to derive them, mathematical programming (MAP) and least squares regression (REG), have their very specific advantages and deficiencies which are to some extent complementary. Bayes estimation (BE), the novel technique proposed here for obtaining internally consistent thermodynamic databases, can combine the advantages of both MAP and REG but avoid their drawbacks. It optimally uses the information on thermochemical, thermophysical, and volumetric properties of phases and experimental reaction reverals to refine the thermodynamic data and returns their uncertainties and correlations. Therefore, BE emerges as the method of choice. The theoretical background of BE, and its relation to MAP and REG, is explained. Although BE is conceptually simple, it can be computationally demanding. Fortunately, modern computer technology and new stochastic methods such as Gibbs sampling help surmount those difficulties. The basic ideas behind these methods are explored and recommendations for their use are made using the Al₂SiO₅ unary as an example. The potential of BE and its future perspective for application to multicomponent-multiphase systems appear very promising. For the convenience of readers not interested in the mathematical details of BE, an illustrative example is given in the Appendix to promote an intuitive understanding of what BE is all about.     

An internally consistent thermodynamic dataset with uncertainties and correlations: 2. Data and results

Abstract This, the second of two papers, represents the application of a least squares approach, discussed in the previous paper, to the generation of an internally consistent thermodynamic dataset involving 60 reactions among 43 phases, in the system K₂O–Na₂O–CaO–MgO–Al₂O₃–SiO₂–H₂O–CO₂. We make the assumption that all the thermodynamic data, with the exception of enthalpies of formation of the phases, are well known, and solve for an internally consistent set of enthalpies which reproduces the 60, experimentally determined, phase equilibrium reactions. An important difference between our dataset and that of previous alternatives in the literature is that we are able to determine the uncertainties on, and correlations between, the enthalpies of formation for all phases in the set, and hence are able to apply simple error propagation techniques to determine the uncertainties in any phase equilibrium calculations performed using this dataset. Selection of reactions, for geothermometry and geobarometry, may be more readily made by choosing equilibria with small uncertainties in their thermodynamics. Our data are in reasonably close agreement with the high temperature molten oxide calorimetry results on silicate minerals where available, a fact which lends a degree of confidence to the results.     

An internally consistent thermodynamic dataset with uncertainties and correlations: 1. Methods and a worked example

Abstract This, the first two papers, sets out the philosophy and methods of determining an internally consistent thermodynamic dataset for minerals using the least squares method. The applicability of the least squares method is discussed, and it is applied to a small set of experimental equilibria in the system Na₂O–Al₂O₃–SiO₂–H₂O. The importance is stressed of defining not only the enthalpies of formation of minerals, but also the uncertainties and the correlations among them. The system which has been used as an illustration for this paper serves as a visual guide to the method, as it is small enough to represent graphically in two dimensions. In the paper which follows, we extend the method to a system of 60 equations (experimentally determined equilibria) involving 34 unknowns (enthalpies of formation of mineral end-members).     

The Bayesian approach to an internally consistent thermodynamic database: theory, database, and generation of phase diagrams

An internally consistent thermodynamic dataset has been derived for 148 endmember phases (145 solids and 3 fluids) comprising the elements Li, Na, K, Be, Mg, Ca, Ti, Cr, Mn, Fe, Zn, Al, Si, C, H, and O. This has been achieved by simultaneous treatment of phase property (like standard enthalpy of formation, standard entropy, molar heat capacity, molar volume, thermal expansivity, bulk modulus etc.) and reaction reversal data by the Bayesian method. The theory underlying the approach, and the computational methods involved, are briefly outlined. (For the benefit of readers unfamiliar with inference statistics, the basic concepts of the Bayes method are also presented in such a way that they can be grasped intuitively.) Although not yet addressed, this method can be extended to refine the thermodynamic mixing properties of crystalline solutions. The sources of the input data, culled from the literature, are summarized in the Appendix. The resulting database is succinctly documented in this paper. It includes the enthalpies of formation and entropies, their uncertainties, and the correlation among them. The database allows calculation of P-T, T-X _CO2, P-X _CO2, and T-f _O2 sections, with error propagation into the computed phase diagrams on a routine basis. A user-friendly computer program has been written to generate such phase diagrams. It is public domain software. The software and the thermodynamic database (which includes a complete documentation of the thermodynamic data above and beyond those listed (Table 2, here) may be downloaded from the web site http://homepage.ruhr-uni-bochum.de/niranjan.chatterjee/Index.htm. Examples of computed phase diagrams are given to illustrate the quality of the data and the capabilities of the software. Received: 11 March 1998 / Accepted: 11 June 1998     

A new method to calculate end-member thermodynamic properties of minerals from their constituent polyhedra II: heat capacity, compressibility and thermal expansion

Thermodynamic calculations in petrology are generally performed at pressures and temperatures beyond the standard state conditions. Accurate prediction of mineral equilibria therefore requires knowledge of the heat capacity, thermal expansion and compressibility for the minerals involved. Unfortunately, such data are not always available. In this contribution we present a data set to estimate the heat capacity, thermal expansion and compressibility of mineral end‐members from their constituent polyhedra, based on the premise that the thermodynamic properties of minerals can be described by a linear combination of the fractional properties of their constituents. As such, only the crystallography of the phase of interest needs to be known. This approach is especially powerful for hypothetical mineral end‐members and for minerals, for which the experimental determination of their thermodynamic properties is difficult. The data set consists of the properties for 35 polyhedra in the system K–Na–Ca–Li–Be–Mg–Mn–Fe–Co–Ni–Zn–Al–Ti–Si–H, determined by multiple linear regression analysis on a data set of 111 published end‐member thermodynamic properties. The large number of polyhedra determined allows calculation of a much larger variety of phases than was previously possible, and the choice of constituents together with the large number of thermodynamic input data results in estimates with associated uncertainty of generally <5%. The quality of the data appears to be sufficiently accurate for thermodynamic modelling as demonstrated by modelling the stability of margarite in the CASH system and the position of the talc–staurolite–chloritoid–pyrope absent invariant point in the KMASH system. In both cases, our results overlap within error with published equivalents.     

An improved and extended internally consistent thermodynamic dataset for phases of petrological interest,involving a new equation of state for solids

The thermodynamic properties of 254 end‐members, including 210 mineral end‐members, 18 silicate liquid end‐members and 26 aqueous fluid species are presented in a revised and updated internally consistent thermodynamic data set. The PVT properties of the data set phases are now based on a modified Tait equation of state (EOS) for the solids and the Pitzer & Sterner (1995) equation for gaseous components. Thermal expansion and compressibility are linked within the modified Tait EOS (TEOS) by a thermal pressure formulation using an Einstein temperature to model the temperature dependence of both the thermal expansion and bulk modulus in a consistent way. The new EOS has led to improved fitting of the phase equilibrium experiments. Many new end‐members have been added, including several deep mantle phases and, for the first time, sulphur‐bearing minerals. Silicate liquid end‐members are in good agreement with both phase equilibrium experiments and measured heat of melting. The new dataset considerably enhances the capabilities for thermodynamic calculation on rocks, melts and aqueous fluids under crustal to deep mantle conditions. Implementations are already available in thermocalc to take advantage of the new data set and its methodologies, as illustrated by example calculations on sapphirine‐bearing equilibria, sulphur‐bearing equilibria and calculations to 300 kbar and 2000 °C to extend to lower mantle conditions.     

Partial melting of metagreywackes, Part II. Compositions of minerals and melts

A series of experiments on the fluid-absent melting of a quartz-rich aluminous metagreywacke has been carried out. In this paper, we report the chemical composition of the phases present in the experimental charges as determined by electron microprobe. This analytical work includes biotite, plagioclase, orthopyroxene, garnet, cordierite, hercynite, staurolite, gedrite, oxide, and glass, over the range 100–1000 MPa, 780–1025 °C. Biotites are Na- and Mg-rich, with Ti contents increasing with temperature. The compositions of plagioclase range from An₁₇ to An₃₅, with a significant orthoclase component, and are always different from the starting minerals. At high temperature, plagioclase crystals correspond to ternary feldspars with Or contents in the range 11–20 mol%. Garnets are almandine pyrope grossular spessartine solid solutions, with a regular and significant increase of the grossular content with pressure. All glasses are silicic (SiO₂ = 67.6–74.4 wt%), peraluminous, and leucocratic (FeO + MgO = 0.9–2.9 wt%), with a bulk composition close to that of peraluminous leucogranites, even for degrees of melting as high as 60 vol.%. With increasing pressure, SiO₂ contents decrease while K₂O increases. At any pressure, the melt compositions are more potassic than the water-saturated granitic minima. The H₂O contents estimated by mass balance are in the range 2.5–5.6 wt%. These values are higher than those predicted by thermodynamic models. Modal compositions were estimated by mass balance calculations and by image processing of the SEM photographs. The positions of the 20 to 70% isotects (curves of equal proportion of melt) have been located in the pressure-temperature space between 100 MPa and 1000 MPa. With increasing pressure, the isotects shift toward lower temperature between 100 and 200 MPa, then bend back toward higher temperature. The melting interval increases with pressure; the difference in temperature between the 20% and the 70% isotects is 40 °C at 100 MPa, and 150 °C at 800 MPa. The position of the isotects is interpreted in terms of both the solubility of water in the melt and the nature of the reactions involved in the melting process. A comparison with other partial melting experiments suggests that pelites are the most fertile source rocks above 800 MPa. The difference in fertility between pelites and greywackes decreases with decreasing pressure. A review of the glass compositions obtained in experimental studies demonstrates that partial melting of fertile rock types in the crust (greywackes, pelites, or orthogneisses) produces only peraluminous leucogranites. More mafic granitic compositions such as the various types of calk-alkaline rocks, or mafic S-type rocks, have never been obtained during partial melting experiments. Thus, only peraluminous leucogranites may correspond to liquids directly formed by partial melting of metasediments. Other types of granites involve other components or processes, such as restite unmixing from the source region, and/or interaction with mafic mantle-derived materials. Received: 11 July 1995 / Accepted: 27 February 1997     

A new method to calculate end-member thermodynamic properties of minerals from their constituent polyhedra I: enthalpy, entropy and molar volume

The thermodynamic properties of silicate minerals can be described as a linear combination of the fractional properties of their constituent polyhedra. In contrast, given the thermodynamic properties of these polyhedra, the thermodynamic properties of minerals can be estimated, where only the crystallography of the mineral needs to be known. Such estimates are especially powerful for hypothetical mineral end‐members or for minerals where experimental determination of their thermodynamic properties is difficult. In this contribution the fractional enthalpy, entropy and molar volume for 35 polyhedra have been determined using weighted multiple linear regression analysis on a data set of published mineral thermodynamic properties. The large number of polyhedra determined, allows calculation of a much larger variety of phases than was previously possible and the larger set of minerals used provides more confident fractional properties. The OH‐bearing minerals have been described by partial and total hydroxide coordinated components, which gives better results than previous models and precludes the need of a S–V term to improve estimates of entropy. However, the fractional thermodynamic properties only give adequate results for silicate minerals and double oxides, and should therefore not be used to estimate the properties of other minerals. The thermodynamic properties of ‘new’ minerals are calculated from a linear stoichiometric combination of their constituent polyhedra, resulting in estimates generally with associated uncertainty of <5%. The quality of such data appears to be of sufficient accuracy for thermodynamic modelling as shown for meta‐bauxites from the Alps and the Aegean, where the effect of Zn on the P–T stability of staurolite can be both qualitatively and quantitatively reproduced.     

Pb、Sr和REE在矿物中的扩散补偿关系及其对扩散系数的预测

对现有实验扩散数据的检查发现，不仅Pb、Sr和REE元素在不同矿物中存在着扩散补偿关系，不同元素在同一矿物或同族矿物中也满足扩散补偿关系。阴离子孔隙度作为矿物内部离子堆积密度的一种量度，它与Pb和REE扩散活化能之间存在负的线性相关性，在固定温度下与Sr扩散系数（lnDT)之间呈正的线性相关，因而可以用于预测元素在矿物中的扩散系数。本文分别采用离子孔隙度法和双补偿法预测了Pb、Sr和REE在不同矿物中的扩散系数，这些结果与已有的实验数据在实验误差范围内是一致的，因此可以应用到与扩散有关的地球化学动力学研究中。     

Correction to: A novel methodological approach for land subsidence prediction through data assimilation techniques

Computational Geosciences - A Correction to this paper has been published: https://doi.org/10.1007/s10596-021-10079-6     

Chemical mass transfer in magmatic processes IV. A revised and internally consistent thermodynamic model for the interpolation and extrapolation of liquid-solid equilibria in magmatic systems at elevated temperatures and pressures

A revised regular solution-type thermodynamic model for twelve-component silicate liquids in the system SiO₂-TiO₂-Al₂O₃-Fe₂O₃-Cr₂O₃-FeO-MgO-CaO-Na₂O-K₂O-P₂O₅-H₂O is calibrated. The model is referenced to previously published standard state thermodynamic properties and is derived from a set of internally consistent thermodynamic models for solid solutions of the igneous rock forming minerals, including: (Mg,Fe²⁺,Ca)-olivines, (Na,Mg,Fe²⁺,Ca)^M2 (Mg,Fe²⁺, Ti, Fe³⁺, Al)^M1 (Fe³⁺, Al,Si)₂ ^TETO₆-pyroxenes, (Na,Ca,K)-feldspars, (Mg,Fe²⁺) (Fe³⁺, Al, Cr)₂O₄-(Mg,Fe²⁺)₂ TiO₄ spinels and (Fe²⁺, Mg, Mn²⁺)TiO₃-Fe₂O₃ rhombohedral oxides. The calibration utilizes over 2,500 experimentally determined compositions of silicate liquids coexisting at known temperatures, pressures and oxygen fugacities with apatite ±feldspar ±leucite ±olivine ±pyroxene ±quartz ±rhombohedral oxides ±spinel ±whitlockite ±water. The model is applicable to natural magmatic compositions (both hydrous and anhydrous), ranging from potash ankaratrites to rhyolites, over the temperature (T) range 900°–1700°C and pressures (P) up to 4 GPa. The model is implemented as a software package (MELTS) which may be used to simulate igneous processes such as (1) equilibrium or fractional crystallization, (2) isothermal, isenthalpic or isochoric assimilation, and (3) degassing of volatiles. Phase equilibria are predicted using the MELTS package by specifying bulk composition of the system and either (1) T and P, (2) enthalpy (H) and P, (3) entropy (S) and P, or (4) T and volume (V). Phase relations in systems open to oxygen are determined by directly specifying the f _{o 2} or the T-P-f _{o 2} (or equivalently H-P-f _{o 2}, S-P-f _{o 2}, T-V-f _{o 2}) evolution path. Calculations are performed by constrained minimization of the appropriate thermodynamic potential. Compositions and proportions of solids and liquids in the equilibrium assemblage are computed.     

Extension of lattice strain theory to mineral/mineral rare-earth element partitioning: An approach for assessing disequilibrium and developing internally consistent partition coefficients between olivine, orthopyroxene, clinopyroxene and basaltic melt

Olivine/melt and orthopyroxene/melt rare-earth element (REE) partition coefficients consistent with clinopyroxene/melt partition coefficients were determined indirectly from subsolidus partitioning between olivine, orthopyroxene, and clinopyroxene after suitable correction for temperature. Heavy- and middle-REE ratios for olivine/clinopyroxene and orthopyroxene/clinopyroxene pairs correlate negatively with effective cationic radius, whereas those for the light REEs correlate positively with cationic radius, generating a U-shaped pattern in apparent mineral/clinopyroxene partition coefficients versus cationic radius. Lattice strain models of partitioning modified for subsolidus conditions yield negative correlations of olivine/clinopyroxene and orthopyroxene/clinopyroxene with respect to cationic radii, predicting well the measured partitioning behaviors of the heavy and middle REEs but not that of the light REEs. The light-REE systematics cannot be explained with lattice strain theory and, instead, can be explained by disequilibrium enrichment of the light REEs in melt inclusions or on the rims of olivine and orthopyroxene. Realistic light-REE partition coefficients were thus extrapolated from the measured heavy- and middle-REE partition coefficients using the lattice strain model. Light REE olivine/melt and orthopyroxene/melt partition coefficients calculated in this manner are lower than most published values, but agree reasonably well with partitioning experiments using the most recent in situ analytical techniques (secondary-ionization mass spectrometry and laser ablation inductively coupled plasma mass spectrometry). These new olivine/melt and orthopyroxene/melt partition coefficients are useful for accurate modeling of the REE contents of clinopyroxene-poor to -free lithologies, such as harzburgitic residues of melting. Finally, the application of the lattice strain theory to subsolidus conditions represents a framework for assessing the degree of REE disequilibrium in a rock.     

A study of Late Pleistocene loess deposits,South Canterbury,New Zealand: Part II. Paleosols and their stratigraphic implications

Examination of loess columns in coastal South Canterbury, New Zealand, and the recognition of paleosols with comparable morphology to surface soils allowed the division of the loess column into six members. The upper loess members 1–4 are grouped into a proposed Dashing Rocks Formation. This commonly overlies an erosion surface on Timaru Basalt, but in some situations is underlain by loess member 5 and mixed loess and weathered basalt member 6. The paleosol developed on loess member 5, on both morphological and chemical evidence, is indicative of a period of soil formation of longer duration or greater intensity than is indicated by the overlying paleosols or surface soils. Therefore members 5 and 6 are grouped into a separate formation.A radiocarbon chronology for loess members 1 and 2 of the Dashing Rocks Formation, suggests loess accumulation phases from 9900 to 11,800 and prior to 31,000 radiocarbon yr BP, followed by periods of soil formation and contemporary peat development in surface depressions.A hypothesis is presented suggesting that at least loess members 1 and 2 of the Dashing Rocks Formation accumulated during periods of glacial recession which in turn precipitated fluvial and eolian erosion, transport and redeposition of fluvioglacial deposits. The periods of soil development indicated by the paleosols were initiated during warm interstadial conditions and continued throughout the cooling of the following stadial. Such an interpretation has its parallels in the northern hemisphere but is in slight disagreement with previous glacial and fluvioglacial chronologies accepted in New Zealand.     

Optimization of the garnet-biotite geothermometer: Part II. Calibration equations and accuracy of the estimation

Calibrations of the garnet-biotite geothermometer are considered with allowance for the effect of the nonideality of the coexisting garnet and biotite upon K _D . The calibrations based on the model of an asymmetric subregular Prp-Alm-Grs-Sps solid solution with W _i−j ^Grt (after Ganguly et al. (1996) and Holdaway (2000) and taking into account the experimental data published by Perchuk et al. (1983)) are the most accurate. The same may be said about the equations additionally taking into consideration the model of the symmetric regular Phl-Ann-Ms-(Ti-Bt) solution with W _i−j ^Bt after Holdaway et al., (1997), Kaneko and Miano (2004), and the data published by Ferry and Spear (1978) and Perchuk et al. (1983). These equations well reproduce the temperature at which the Prp + Ann = Alm + Phl equilibrium is reached in experiments, and they are the least sensitive to the variation of the Ca and Mn contents in garnet and Al^VI and Ti in biotite. The uncertainty of the temperature interval is 30–46°C (2σ).     

A fully implicit and consistent finite element framework for modeling reservoir compaction with large deformation and nonlinear flow model. Part II: verification and numerical example

A fully implicit, fully coupled, and fully consistent finite element framework has been formulated in part I of this work for modeling reservoir compaction through linearizing coupled solid and flow field equations and constructing a local material integrator. In part II of this work, we focus on verification and performance analysis of our numerical formulation and computer implementation using several numerical examples. First, we design a cube problem in triaxial compression to verify our numerical formulation and computer code implementation especially for rock formation in compaction using cap plasticity models. The finite element prediction on stresses is compared with the analytical solution. The second problem we select is a strip footing problem popular in the geotechnical area where the evolution of soil consolidation degrees following the diffusion of pore pressure is the main interest. In this example, we demonstrate a good performance of the proposed numerical formulation on solving different shear and compaction-dominated deformation behaviors by varying the footing length. Importantly, an extremely sharp cap model based on real experimental data for Leda clays, a challenging cap model, is successfully applied in this footing problem. Our focus in this work is to model field reservoirs undergoing serious compaction. A reservoir with complex payzone geometries, multiple horizontal wells, and cap plasticity models with sharp cap surfaces has been successfully solved using our fully implicit formulation. The last example is to model a horizontal wellbore damage problem. Finally, the sensitivity of predicted subsidence to nonlinear flow model, cap hardening parameters, and Lode angles have been systemically investigated and documented in detail, which can provide a constructive guidance on how to successfully model field reservoir compaction problems with cap plasticity models.     

Pyrolysis of Transvaal kerogens. II. An evaluation of vacuum pyrolysis with polyethylene,polystyrene and their mixtures with minerals

A series of experiments have been conducted with polyethylene and polystyrene standards in an attempt to define the advantages and limitations of a vacuum pyrolysis—gas chromatography—mass spectrometry procedure for the characterization of kerogen and other macromolecular substances. Effects of variations in pyrolysis temperatures and times, sample sizes (weights) and thickness were evaluated together with the reproducibility of the nature and abundances of pyrolyzates. The effects of minerals (illite and quartz) admixed in the polymers were also considered with reference to the nature of the breakdown products. Optimal pyrolysis conditions, where primary pyrolyzates were sufficiently abundant and secondary products did not hinder characterization, were attained at 450°C and 60–90 min. The reproducibility of the nature and quantities of pyrolyzates was rather satisfactory at this temperature and pyrolysis time. However, relatively large samples of macromolecular matter, which is considerably volatile at this temperature, led to the synthesis of an abundant yield of secondary products, but sample thickness does not affect the nature of pyrolyzates. Admixed mineral matter affected the nature and relative abundances of the pyrolyzates but did not impede characterization of samples, as primary breakdown products were discernible. Macromolecular substances of limited volatility, heterogeneous chemical composition and containing intractable mineral matter, such as many kerogens, need to be pyrolyzed as relatively large samples. The vacuum procedure used in these studies may be to advantage, as compared with some other methods, to pyrolyze such samples. This method seems to be also suitable for the pyrolysis of volatile macromolecular matter, provided that small samples are employed.     

Experimental determination of the reaction paragonite=jadeite+kyanite+H2O,and internally consistent thermodynamic data for part of the system Na2O−Al2O3−SiO2−H2O,with applications to eclogites and blueschists

Hydrothermal reversal experiments have been performed on the upper pressure stability of paragonite in the temperature range 550–740 ° C. The reaction $$\begin{gathered} {\text{NaAl}}_{\text{3}} {\text{Si}}_{\text{3}} {\text{O}}_{{\text{1 0}}} ({\text{OH)}}_{\text{2}} \hfill \\ {\text{ paragonite}} \hfill \\ {\text{ = NaAlSi}}_{\text{2}} {\text{O}}_{\text{6}} + {\text{Al}}_{\text{2}} {\text{SiO}}_{\text{5}} + {\text{H}}_{\text{2}} {\text{O}} \hfill \\ {\text{ jadeite kyanite vapour}} \hfill \\ \end{gathered}$$ has been bracketed at 550 ° C, 600 ° C, 650 ° C, and 700 ° C, at pressures 24–26 kb, 24–25.5 kb, 24–25 kb, and 23–24.5 kb respectively. The reaction has a shallow negative slope (? 10 bar °C^?1) and is of geobarometric significance to the stability of the eclogite assemblage, omphacite+kyanite. The experimental brackets are thermodynamically consistent with the lower pressure reversals of Chatterjee (1970, 1972), and a set of thermodynamic data is presented which satisfies all the reversal brackets for six reactions in the system Na₂O-Al₂O₃-SiO₂-H₂O. The Modified Redlich Kwong equation for H₂O (Holloway, 1977) predicts fugacities which are too high to satisfy the reversals of this study. The P-T stabilities of important eclogite and blueschist assemblages involving omphacite, kyanite, lawsonite, Jadeite, albite, chloritoid, and almandine with paragonite have been calculated using thermodynamic data derived from this study.