=Paper=
{{Paper
|id=Vol-2677/paper10
|storemode=property
|title=Developing application for thermobarometric calculation in geology
|pdfUrl=https://ceur-ws.org/Vol-2677/paper10.pdf
|volume=Vol-2677
|authors=Konstantin V. Chudnenko,Oleg V. Avchenko,Anastasia K.Popova
|dblpUrl=https://dblp.org/rec/conf/itams/ChudnenkoAP20
}}
==Developing application for thermobarometric calculation in geology==
https://ceur-ws.org/Vol-2677/paper10.pdf
Developing application for thermobarometric calculation
in geology
Konstantin V. Chudnenko1 [0000-0002-1547-2188], Oleg V. Avchenko2, Anastasia K.
Popova3 [0000-0001-6209-678X]
1
Vinogradov Institute of Geochemistry SB RAS, Irkutsk, Russia
chud@igc.irk.ru
2
Far East Geological Institute FEB RAS, Vladivostok, Russia
3
Matrosov Institute for System Dynamics and Control Theory, SB RAS, Irkutsk, Russia
Abstract. This article focuses on development of software for calculating the
temperatures and pressures of rock formation. The assessment of the physico-
chemical 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 ob-
tained estimates in geological applications are outlined.
Keywords: Geothermobarometry, Inverse Problems, Convex Programming,
Parametric Minimization.
1 Introduction
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.
Copyright Β© 2020 for this paper by its authors. Use permitted under Creative Com-
mons License Attribution 4.0 International (CC BY 4.0).
2 Mathematical model
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 identify-
ing 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 [1-4], where a real rock sample, represented by a vector of basic parameters
and chemical composition (Fig. 1), is selected as a standard. Experimental (model)
rock samples obtained as a result of solutions for various temperature (T), pressure
(P) and fluid compositions represent a set of recognizable classes. The measure of
proximity between objects of classification and a given standard is established de-
pending on the selected metric distance between them, which largely determines by
the results of classification.
Fig. 1. Software
The identification of thermobaric conditions is carried out by solving the problem of
finding the minimum of the function f, which determines the proximity of the mod-
eled rock composition to the reference at different T and P from a given Tmin-Tmax and
Pmin-Pmax intervals:
ππ = ππππππ( οΏ½οΏ½ π€π€ππ (π΄π΄ππ β π΅π΅ππ )2 + οΏ½(π΄π΄ππ β π΅π΅ππ )2 ), ππ β ππ, ππ β πΆπΆ
ππ ππ
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 pro-
portion 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 determinis-
tic problems.
It is considered to use a combined measure of distance: weighted Euclidean dis-
tance (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 compo-
sition of minerals (parameters) make a more significant contribution to finding a solu-
tion 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 composi-
tion of the rock, a certain gap is in the calculations due to accuracy of silicate analy-
sis, determination of the degree of oxidation, not always complete and objective as-
sessment 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]:
gj
G ( x) = β ( + ln a idj + ln g j ) x j
j βS RT
where gj is the standard isobaric-isothermal potential of the j-th component;
a idj - 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;
Ξ³j 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 po-
sition of the crystal structure, the activity is equal to the molar fraction of the minal in
the mineral phase Ξ± idj = x j / X Ξ± , where XΞ± is the molar amount of the Ξ± 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 criteri-
on 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 consid-
er several mineral systems at once in one area of the metamorphic strata, represented
by rocks with different compositions of coexisting minerals.
3 Verification of the model
All calculations were carried out on the basis of rock samples from the Okhotsk met-
amorphic 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 ba-
sis of known experimental and empirical classical thermobarometers [5-7, 9-10] and
P-T estimates under the PET program [8].
Table 1. Comparison of estimates Π (kbar) and Π’(ΒΊΠ‘)
Sample Π-433 441-g 251-Π Π-218 138 Π-234-1 Π-208
Parameters T P T P T P T P T P T P T P
Grt-Bt ther- 634 - 660 - 640 - - - 583 - 635 660
mometer [5]
Grt-Opx
- - - - - - 700 - 700 - 750 - - -
thermometer
[6]
Grt-Opx-Pl-
Q barometer - - - - - - - 6.2 - 5.5 - 4.5 - -
[7]
PET [8] 680 6.7 - - 690 6.5 890 8.3 690 6.7 700 5.7 710 6.1
Grt-Bt ther-
680 - 720 - 690 - - - 690 - 700 - 676 -
mometer [9]
Grt-Bt-Pl
barometer - 6.1 - - - 7.0 - - - 7.0 - 4.8 - 5.1
[10]
This study 690 6.0 770 6.1 620 5.6 630 7.0 652 5.2 700 5.6 650 5.6
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
estimates reach 50-700 ΒΊΠ‘. Good convergence is observed between the Grt-Bt-Pl of
the Wu barometer readings [10] and our data. Only in one sample (sample 138) the
divergence in the pressure estimates exceed the error in determining the pressure us-
ing the Wu geobarometer [10]. But sample 138 is distinguished by an increased con-
tent of magnetite and increased oxidation of garnet and biotite, which, apparently,
introduces an error in the pressure estimate according to Wu [10]. Quite good conver-
gence (except for sample 218) is noted between our calculation and the PET program.
The P-T estimates obtained in this work for the Okhotsk granulite complex are
5.2β7 kbar in pressure and 620β770 ΒΊΠ‘ in temperature (Table 1). On the whole, they
are quite close to earlier estimates [13]. The probable error in finding the P-T esti-
mates was (except for sample 441-g) 1-1.5 kbar in pressure and 200-500 ΒΊC in tem-
perature.
Fig. 2. PβT field of the conditions of formation of metamorphic rocks of the Okhotsk complex
on the Pattison diagram [14]. A colon in front of mineral indices indicates that these minerals
may be missing. Minerals: Grt - garnet, Cpx - clinopyroxene, Opx - orthopyroxene, Pl - plagio-
clase, Amp - amphibole, Qz - quartz.
The P-T parameters of the Okhotsk complex metamorphism are shown on the Pat-
tison 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 Conclusions
The developed software is a rapid temperatures and pressures of rock formation as-
sessment tool. It allows to significantly enhance the capabilities of classical geother-
mobarometry 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 Acknowledgments
The research was carried out within the state assignment of Ministry of Science and
Higher Education of the Russian Federation β 0350-2019-0011.
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