This article focuses on development of software for calculating the temperatures and pressures of rock formation. The assessment of the physicochemical conditions of natural mineral formation is based on solving the inverse problem of convex programming - modeling mineral equilibria by minimizing the Gibbs free energy of a mineral system. The proposed algorithm includes the parametric minimization of the criterion function of deviation of the observed and calculated mineral paragenesis over the entire set of specified identifying parameters. The method was verified using the example of thermobarometric conditions for the formation of metamorphic rocks of the Okhotsk complex (North-East of Russia). In the conclusion, the main directions of using the obtained estimates in geological applications are outlined.
Determination of temperatures and pressures of formation of rocks in the thickness of the earth's crust is associated with laborious experimental or calculated geochemical studies. We propose a different approach to assessing the physicochemical conditions of natural mineral formation, as solving the inverse problem of convex programming modeling mineral equilibria by minimizing the Gibbs free energy of a mineral system using all available petrological information, including the chemical composition of the rock and the composition of minerals in a given mineral paragenesis. A correct assessment of the parameters helps to understanding the natural laws in the history of the mechanism of formation and transformation of minerals in different geological eras.
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The main criterion for finding the optimal values of temperature and pressure for the formation of a reference rock sample is to find the maximum approximation of the observed and calculated mineral paragenesis over the entire set of specified identifying parameters.
In the mathematical description, the problem can be attributed to a special class of
problems in the theory of pattern recognition - the method of comparison with the
prototype [
( − )2), ∈ , ∈ where A is a standard (real rock), B is a model calculation,
M is a set of parameters of the mineral composition, C is a set of data on the quantitative chemical composition of a rock.
The weighting factors wj allow to assign a certain weight for the parameter in proportion to the degree of importance of the feature in the classification problem (wj> 0). If such additional information is absent, all wj are taken equal to one. The search for the minimum of the criterion function is carried out by the golden ratio method. This method provides high reliability, reliability and efficiency in solving deterministic problems.
It is considered to use a combined measure of distance: weighted Euclidean distance (for parameters of mineral composition) and squared Euclidean distance (for rock composition). Typically, the second measure of distance is used when you want to give more value to objects that are more distant from each other. But in this case, since the parameters are determined in relative units in the interval (0, 1), the second measure represents a smaller contribution to the overall functional. Thus, the composition of minerals (parameters) make a more significant contribution to finding a solution than the value of the amount of minerals in the rock. This ratio is accepted due to the fact that determining the composition of minerals is the most reliable and verified information available to almost any petrologist. In the case of the chemical composition of the rock, a certain gap is in the calculations due to accuracy of silicate analysis, determination of the degree of oxidation, not always complete and objective assessment of the role of volatile components in the formation of the rock, and other factors that increase the degree of error in the initial data.
The problem of determining the equilibrium rock composition at fixed T, P and fluid composition is solved by minimizing the Gibbs energy of the system for the model calculation of the rock sample. The reduced potential of the Gibbs energy of the mineral system has the form [11]:
G( x) = ∑ (
g j∈S RT
j + ln aijd + lng j ) x j where gj is the standard isobaric-isothermal potential of the j-th component; a
ijd - the activity of the minal j in a mineral with ideal mixing, which allows to take into account the contributions of configuration factors to the ideal activity; γj is the activity coefficient; xj is the molar amount of the j-th minal; S - a set of indices j, denoting the components of the mineral system.
In the case of solid solutions in which the mixing of components occurs at one position of the crystal structure, the activity is equal to the molar fraction of the minal in the mineral phase aijd = x j / Xα , where Xα is the molar amount of the α phase of the solid solution.
From a practical point of view, it is necessary to take into account that the error in calculating the values of T and P can be associated with small variations in the criterion function in the case of systems with a sufficiently wide PT-range of stability of a certain paragenesis. Therefore, increased requirements are applied to the accuracy of the initial data on the composition of minerals. It can also be recommended to consider several mineral systems at once in one area of the metamorphic strata, represented by rocks with different compositions of coexisting minerals. 3
All calculations were carried out on the basis of rock samples from the Okhotsk metamorphic complex of the southwestern part of the Verkhoyansk-Chukotka fold region (North-East of Russia) [12]. The weight ratio fluid / rock (W / R) in the models was 0.005-0.0005.
Table 1 shows a comparison of modeling data with estimates derived from the
basis of known experimental and empirical classical thermobarometers [
There is a good convergence in pressure in samples (A-218, 138, A-234-1) for which there are well-developed experimental thermobarometers, since the differences in pressure estimates are on average no more than 1 kbar. Differences in temperature Parameters
T
Grt-Bt ther- 634
mometer [
P 6.7 6.1 T
P
T
660
estimates reach 50-700 ºС. Good convergence is observed between the Grt-Bt-Pl of
the Wu barometer readings [
The P-T estimates obtained in this work for the Okhotsk granulite complex are 5.2–7 kbar in pressure and 620–770 ºС in temperature (Table 1). On the whole, they are quite close to earlier estimates [13]. The probable error in finding the P-T estimates was (except for sample 441-g) 1-1.5 kbar in pressure and 200-500 ºC in temperature.
The P-T parameters of the Okhotsk complex metamorphism are shown on the Pattison diagram [14], where they form a small field (Fig. 2) corresponding to the boundary conditions of the amphibolite and granulite facies, which corresponds to all available data on the Okhotsk complex metamorphism regime. 4
The developed software is a rapid temperatures and pressures of rock formation assessment tool. It allows to significantly enhance the capabilities of classical geothermobarometry tools, which are rely on experimental studies or calculation methods based on the equations of a geothermobarometer or a fugometer, representing a set of chemical reactions of the formation of natural mineral associations. The obtained values can be used in the construction of models of the evolution of the geological environment, including ore deposit formation process. 5
The research was carried out within the state assignment of Ministry of Science and Higher Education of the Russian Federation № 0350-2019-0011. 11. Chudnenko, K.V.: Thermodynamic modeling in geochemistry: theory, algorithms, software, applications. Novosibirsk: Academic publishing house "Geo", 287 p. (2010) 12. Avchenko, O.V., Chudnenko, K.V., Vakh, A.S.: Analysis of mineral paragenesis by the
Gibbs energy minimization method. Moscow: GEOS, 252 p. (2018) 13. Avchenko, O.V.: Mineral equilibria in metamorphic rocks and problems of geobarothermometry. Moscow: Nauka, 184 p. (1990) 14. Pattison, D.R.M.: Petrogenetic significance of orthopyroxene-freegarnet + clinopyroxene+plagiclase ± quartz-bearing metabasites with respect to the amphibolite and granulite facies. Journ. Metamorph. Geol, V. 21, pp. 21-34. (2003)