ISSN 1069-3513, Izvestiya, Physics of the Solid Earth, 2016, Vol. 52, No. 6, pp. 836–843. © Pleiades Publishing, Ltd., 2016. Original Russian Text © H.H. Guliyev, Kh.B. Aghayev, G.H. Hasanova, 2016, published in Fizika Zemli, 2016, No. 6, pp. 63–70.
Determining the Elastic Moduli of the Third Order for Sedimentary Rocks Based on Well-Logging Data1 H. H. Guliyev*, Kh. B. Aghayev, and G. H. Hasanova Institute of Geology and Geophysics, Azerbaijan National Academy of Sciences, H. Javid ave. 119, Baku, AZ 1000 Azerbaijan *e-mail:
[email protected] Received February 21, 2015
Abstract—The method of determining the values of elastic moduli of the third order of sedimentary series of rocks is developed based on a complex of well-logging data. The calculation algorithm is based on the theory of elastic wave propagation in deformed solids, seismic-geological modeling, and the cluster analysis of well data on artificial neural networks. The elastic moduli of sedimentary series of rocks of oil- and gas-bearing structures of the Gunashli of the South Caspian Basin (SCB) are determined using the offered method. The numerical values of Poisson’s ratio are also determined for these rocks. The inaccuracies of the results obtained within the simplified theories are revealed in this example without considering the current geodynamic changes. Keywords: stressed medium, elastic moduli of the third order, well logging data, neural network, cluster DOI: 10.1134/S1069351316050062
INTRODUCTION There is a need to apply the nonlinear theory of deformation in solving various problems of theoretical and applied geophysics, internal structure, and dynamics of the Earth. In this regard, it is important to have quantitative information on the nonlinear physical-mechanical properties of rocks building up the studied geological medium. Various simplifying assumptions are usually used in model studies. For example, the existence of elastic potential which allows modeling the dependences between tensor components of stress and deformation is assumed in the problems of geomechanics (Biot, 1965; Guz, 1986). In general, as a first step, nonlinear elastic deformations are considered for the simplicity and development of theoretical approaches of the study. The data on linear (elastic moduli of the second order) and nonlinear (elastic moduli higher than the second order) physical-mechanical properties of rocks are used in the simplest case to study nonlinear deformation processes. These studies are well known for standard materials in mechanics of the continuum media (Biot, 1965; Wesolowski, 1978; Guz, 1986; Kuliev and Dzhabbarov, 1998). It succeeds in describing the observed nonlinear effects in relatively stiff compressible materials by involving elastic potentials that contain the third strain tensor invariant, along with the first and second invariants in its displays. The elastic moduli of 1 The article was translated by the authors.
the third order are used to describe nonlinear physical and mechanical properties of the material in the relevant equations and correlations in such cases. The use of the elastic potential of the Murnaghan type (Biot, 1965; Grinfel’d and Movchan, 1975; Guz, 1986; Kuliev, 2000; Aleksandrov et al., 2001; Vyzhva et al., 2005; Guliyev and Shirinov, 2006) is sufficiently tested within the isotropic approximations in the model problems. In this case, three new elastic moduli of the third order a, b, and c are also introduced in the theoretical constructions along with two elastic moduli of the second-order (Lame’s constants λ, μ). There are known experimental methods (Guz, 1986) for their determination in the case of standard materials (metals, plastics, glass, etc.). These methods are based on the results of the study of propagation of ultrasonic elastic waves in prestressed media. The theory of the propagation of elastic waves and the theory of nonlinear elastic waves are well developed and have found quite a wide application field in such media (Zarembo and Krasilnikov, 1966; Kostrov and Nikitin, 1969; Engelbrecht and Nigul, 1981; Nigul, 1981; Wesolowski, 1978; Guz, 1986; Kuliev and Dzhabbarov, 1998; Kuliev, 2000). These theories are also introduced in the studies of various problems of geology and geophysics in recent decades (Greenfield and Movchan, 1975; Bakulin and Protosenya, 1982; Nikitin, 1983; Yin and Rasolofosaon, 1994; Abasov et al., 2000; Aleksandrov et al., 2001; Prodayvoda et al., 2004; Guliyev and Shirinov, 2006; Guliyev, 2010). Approaches are also intensively
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developed using nonlinear deformation processes in problems of processing and interpreting the seismic data of field and logging studies (Vyzhva, 2005; Guliyev and Aghayev, 2011). Methods of the experimental studies of the nonlinear elastic properties of the materials developed to study samples from standard materials are also applied to study samples of igneous and metamorphic rocks (Bakulin and Protosenya, 1982; Aleksandrov et al., 2001; Prodayvoda et al., 2004; Guliyev and Shirinov, 2006). There are only a few publications dedicated to these problems regarding the sedimentary rocks in the scientific literature (Yin and Rasolofosaon, 1994). It is necessary to determine the influence of current geodynamic processes (neotectonic activity, earthquakes, abnormal formation pressures, exploitation of hydrocarbon wells in the depletion drive, etc.) on the seismic data which requires solving nonlinear problems. Besides, core samples from the deeper interior of the Earth are not always available for direct experimental studies. All this makes it necessary to develop approaches to determine various physical and mechanical properties of rocks based on well, seismic, and seismological data. The method of determining the numerical values of the elastic moduli of the third order for a sedimentary series of rocks is proposed in this paper. A complex of well logging data is used as the source material. Such choice is natural for developing a methodological approach, as well logging data that are most reliable and qualitative; moreover, there are quietly tested standard methods for their interpretation (Darwin and Singer, 1987; Itenberg, 1987). Artificial neural networks (ANN) are used to split the well-logging data into clusters (Van der Baan and Jutten, 2000; Poulton, 2002; Uden et al., 2003; Aminzade and de Groot, 2006; Babkin, 2010; Chashkov and Valery, 2011; Sun et al., 2011). MATERIALS AND METHODS Well-logging data are used as the source material on the types of logs and thin-layered one-dimensional (1D) models of the physical properties of the medium built based on them (Gogonenkov, 1987; Guliyev and Aghayev, 2011): acoustic in pressure and shear waves; dense; pressure; lithologic (distribution of the lithology of rocks on the well section as codes of lithology). Well-logging data can also be used on oil saturation, porosity, etc. The method of the calculation algorithm is based on the theory of the propagation of elastic waves in deformed solid bodies, seismological modeling, and cluster analysis. Software NEUROXL (http://neuroxl.com/…) is applied for clusterization of well-logging data and ANN. One-dimensional models of well-logging data and elastic moduli are constructed using GEOPRESS the IZVESTIYA, PHYSICS OF THE SOLID EARTH
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(Guliyev and Aghayev, 2011) software package which is intended for modeling the physical parameters of the medium in the classical and nonclassical approaches of deformation theory. THEORETICAL BASIS OF THE METHOD The kinematic correlations of the theory of propagation of elastic waves in deformed solids are used. Various variations are considered within this theory, depending on the degree of deformations (Biot, 1965; Guz, 1986). It is assumed that the medium is subject to deformation of arbitrary size before the generation of a wave field in it. The deformation processes are modeled using the elastic potentials, including the potential of the Murnaghan type. Forms of elastic vibrations are modeled with and without changing the distance between the points of the medium in the predeformation. The accumulated experience of different variations of this theory shows that the most reliable results are obtained in cases of using the theory of large initial deformations and considering changes of distances in the forms of the elastic vibrations. The equations were obtained in the case of using the the elastic potential of the Murnaghan type for the basic kinematic characteristics of the wave field in the following variation of the theory (Guz, 1986):
ρ C lx2 1 = λ + 2μ +
(
)
1 ⎡2 λ + μ σ 0 ( ) 11 3K 0μ ⎣
0 0 − λ σ 22 + σ 33 ⎤⎦ ( 2λ + 4μ + 2b + c ) ⎡λ + 2μ σ0 0 2 + 2 ( a + b ) αα + σ11 ; ρ C Sx =μ+⎢ 2 3K 0 ⎣ 3K 0 0 0 0 0 ⎤⎛ c ⎞ + b σ αα + σ 0 ; (1) × σ11 + σ 22 − 2λ σ 33 1 + 11 ⎥ ⎜ 4μ ⎟ 3K 0 3K 0 ⎠ ⎦⎝ ⎡λ + 2μ 0 2 0 0 ⎤ ρ C Sx3 = μ + ⎢ σ11 + σ 33 − 2λ σ 22 ⎥ 3K 0 ⎣ 3K 0 ⎦ 0 σ ⎛ ⎞ 0 × ⎜1 + c ⎟ + b αα + σ11; K 0 = λ + 2 μ, K μ 3 4 3 ⎝ ⎠ 0
(
)
(
)
where С lx1 is the velocity of the expansion wave (pressure wave); С Sx2 is the velocity of the shear wave polarized in the plane x10 x 2; С Sx3 is the velocity of the shear wave polarized in the plane x10 x3; ρ is the medium’s density; λ and μ are Lame’s elastic constants (of the second order); a, b, and c are the elastic moduli of the third order; and σ ij0 is the component of the stress tensor in the initial deformed state. It is known that for unstressed isotropic media 2
2
ρ C l0 = λ + 2μ, C S0 = ρμ,
(2)
where C l0,C S0 are the velocities of pressure and shear waves in a medium without initial stress. No. 6
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In the case of small elastic initial deformations, Eqs. (1) take the form (Guz, 1986)
ρ C lx2 1 = λ + 2μ + −λ
(
σ 022
+
0 σ 33
1 ⎡2 λ + μ σ 0 ( ) 11 3K 0μ ⎣
)⎤⎦ (λ + 2μ + 2b + c ) + 2 (a + b)
⎡2 ( λ + μ ) 0 σ 0αα 0 2 + σ11 =μ+⎢ σ11 ; ρ C Sx 2 3K 0 ⎣ 3K 0 ⎡λ + 2μ 0 0 ⎤ 0 (3) − λ σ 022 + σ 33 + c ⎢ σ11 + σ 22 ⎥ 3K 0 ⎦ 4μ ⎣ 3K 0 0 ⎡2 ( λ + μ ) σ αα 0 ⎤ 0 2 − 2λ σ 33 + + σ11 =μ+⎢ ; ρ C Sx b 3 ⎥ 3K 0 3K 0 ⎦ ⎣ 3K 0 ⎡λ + 2μ 0 0 0 ⎤ × σ11 − λ σ 22 + σ 33 ⎥ + c ⎢ 4 μ 3K 0 ⎦ ⎣ 3K 0 0 σ 0 0 0 ⎤ 0 × σ11 + σ 33 − 2λ σ 22 ⎥ + b αα + σ11. 3K 0 3 K ⎦ 0 It follows from the reduced equations that it is necessary to have information on velocities С lx1, C Sx2 , and C Sx3; the normal components of the initial stress tensor ×
(
)
(
(
(
)
)
)
σ ii0 (i = 1,2,3); and the medium’s density ρ and elastic moduli λ and μ to determine three elastic moduli of the third order a, b, and c. In the case of using large initial deformations, we apply Eqs. (1) and (2) and when using the theory of small elastic initial deformations, we apply Eqs. (2) and (3). The initial components of the stress tensor are determined by solving the boundary value problems of elasticity in the given form of the load (pressure), and they are well known (Guz, 1986). These known solutions are quite simple in the case of homogeneous initial deformations for isotropic media. The experimentally constructed dependences between the velocity of propagation of elastic waves and pressure indicate that the magnitude of the velocity of the pressure wave increases nonlinearly due to a rise in pressure (Guliyev and Shirinov, 2006). Such dependences allow using Eqs. (1) and (3) to determine the elastic moduli of the third order in cases of large and small initial deformation states accordingly. The method of determining the indicated moduli is implemented in the following sequence. Creation of 1D Model of Medium The thin-layered 1D models of the physical properties of medium are built based on the well-logging data (obtained by certain discrete intervals of measurements on the wellbore) according to the commonly accepted standard method (Darwin and Singer, 1987; Itenberg, 1987). Usually, the measurement interval on the borehole is a tenths of a meter, and there are several thousand measurements. In order to clarify the following reasoning, we note that a single
layer of the medium corresponds to each interval of discretization in 1D models.
Cluster Analysis Generally, the dependences between the parameters of a lithological composition and their petrophysical properties, including the elastic properties of rocks, are complex (Marmonshtein, 1985; Bayuk et al., 1988; Dortman, 1988; Chandong et al., 2006). Lithologically the same rocks significantly differ based on petrophysical properties due to differences in age, saturation, cementation, thermodynamic conditions of bedding, etc. or, in contrast, different rocks have similar values based on certain petrophysical properties. Such situations are observed in all complicatedly constructed geological media. The recognition of similar rocks based on lithology and petrophysics, the definition of the dependences between some of their physical properties based on the well-logging data, and the use of standard approaches do not always provide the required accuracy of the results. Approaches based on cluster analysis involving ANN have been introduced to eliminate such shortcomings in the geophysical surveys in recent years (Van der Baan and Jutten, 2000; Aminzade and de Groot, 2006). This approach is also applied in this paper. Based on the purpose of the set problems and according to Eqs. (1)–(3), we conduct the clusterization of the well-logging data on a group of parameters (С lx1, C Sx2 , C Sx3 , ρ, P) within each code of the lithology of rocks. Here σ ii0 = α i P (i = 1,2,3); P is the pressure parameter; α i are the coefficients determining the direction of the action of external loads, angles between them, and the coordinate axes, as well as the relationship of the values of these forces (which are defined in the solution of the problems on the initial stress state). For example, in the case of volumetric compression at infinity along the coordinate axes α1 = α 2 = α 3 = − 1. We test the optimal parameters of clusterization and use them to split the well-logging data into clusters. We choose quite a large number of clusters to provide the necessary level of data splitting and for the stability of the clusterization process. We determine the quality of clusterization from the analysis of the distribution character of the clusters in the depth of the well and on the dependences between the petrophysical properties of the rocks of the medium. These distributions and dependences should correspond to the seismological model built on well-logging data. Each cluster consists of those layers in the model whose rocks have quite similar values on the group of parameters (petrophysical properties) of clusterization. The criterion of quite close values is to maintain
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a certain range of variation between the values of petrophysical properties and all their possible relationship within the cluster. As the number of clusters increases, the value of this range decreases, which improves the result of the selection of the same rocks. At the same time, an excessive increase in the number of clusters may lead to the instability of the clusterization process. The number of clusters will depend on the required accuracy and number of parameters subjected to clusterization. Based on the requirements of the theory of propagation of elastic waves in stressed media, we enter the data on the rise of the pressure values starting from their minimum values within each cluster. As a result, we obtain a cluster type with fewer members instead of the model seismo-geological section usually consisting of several tens of thousands of layers. Calculation of the Elastic Moduli Further, we determine the elastic moduli of the third order for the entire series of rocks of the lithological code using Eqs. (1)–(3) successively. We calculate the elastic parameters within the lithological code of the rocks using the data of each cluster. We take their minimum values corresponding to the minimum value of pressure and related to the layers of the considered data cluster as the initial values of pressure, velocities of pressure and shear waves, and density of the rocks. The minimum values are determined from the acoustic logs in the borehole assuming that they do not belong to the stress state. Such an approximation is forced. Unfortunately, there is no data base of laboratory experiments for all the rocks in the considered specific geological sedimentary section. Such an approximate approach is justified by the fact that it uses only well-logging data as the source materials. RESULTS AND DISCUSSION The proposed method was applied to determine the elastic moduli of the 3rd order of a sedimentary series of rocks of oil- and gas-bearing structures of the Gunashli of the SCB. The logging data and codes of the lithology given in Fig. 1 were used to build up thin layered one-dimensional models of the medium. The lithology of rocks was divided into five codes, where 1 is anhydride; 2 is green and gray mudstone, shale, clay, compacted and clay; 3 is brown mudstone; 4 is sand; and 5 is siltstone and silt. The number of layers of the medium comprising the lithological codes of rocks is respectively 487, 3586, 17275, 4497, and 5110. The total number of layers is 30 955. The well-logging data were divided into ten clusters within each code of lithology. The results of the cluster analysis on the distribution of the data clusters relating to the third code of the lithology–brown mudstone—were given as an examIZVESTIYA, PHYSICS OF THE SOLID EARTH
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ple in Fig. 2. The complex behavior of the distribution of clusters at a depth justifies the use of cluster analysis. Complex nonlinear behavior of a change in the pressure wave velocity is observed in the depth interval 2900–3700 m. Layers of the medium are alternated on seven data clusters by various acoustic properties. It is seen from Fig. 2 that the straight-line trend of the change of values of the seismic wave velocities and density of the rocks is generally observed within each data cluster. Such behavior of the change of the indicated parameters is a major factor in choosing the number of clusters on each lithology of rocks. The certainty of determining the parameters of the nonlinearity of the rocks is provided for each cluster while fulfilling this condition and requirements on the stability of clusterization process. Similar results were obtained on all codes of the lithology of rocks. Figure 3 shows the numerical values of the elastic moduli of (a), (b), and (c) on lithological codes using Eqs. (1) and (2). Table 1 shows the numerical values of the elastic moduli of (a), (b), and (c) calculated by the data of Figs. 1 and 2. The numerical data specified in the numerators for the elastic moduli were calculated according to Eqs. (1), and in the denominators on Eqs. (3). They show that the values of the elastic moduli of the third order obtained within the theory of large initial deformations (Eq. (1)) and the second variant of the theory of small initial deformations (Eq. (3)) are differentiated between themselves. The influence of the deformation values on the results also differ for various codes of the lithology of the rocks. The received results can be used to solve various problems of geology and geophysics. Poisson’s ratio of different rocks is one of the main subjects when studying the problems of the internal structure and dynamics of the Earth, seismology, and seismic exploration. The numerical values of Poisson’s ratio for some rocks were calculated using the data in Table 1 and Eq. (9) (Kuliev, 2000) (Table 2). The results calculated on the equations of the linear theory of elasticity (i.e., in the media without initial deformations) were given in the numerators and in the denominators according to Table 1 in the case of applying correlations (1) and (2). It follows from the analysis of these results that the use of real data (such as the data shown in Fig. 1) in the simplified theory leads to significant errors, values of which are given in the third line of Table 2. At the time, the use of such an approach led to the wrong conclusion on the existence of a paradox of Poisson’s ratio (Puzyrev et al., 1985). This paradox is as follows. Poisson’s ratio is determined using the following equation in the applied studies of seismic exploration and seismology: No. 6
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(a) Velocity, km/s 0.2 1.2 2.2 3.2 0.8
4.2
(b) Density, g/cm3 1.8 2.0 2.2 2.4 2.6
(c) Pressure, GPa 0.01 0.04 0.06 0.09
(d) Lithology code 0 1 2 3 4 5
1.3
Depth, km
1.8
2.3
2.8
3.3 CSx2
Clx 1
3.8 Fig. 1. Charts of the velocity variations of pressure and shear waves (a), density of rocks (b), geostatic pressure,(c) and the codes of the lithology (d) of the rocks in depth.
which is obtained within the linear theory of the elasticity of isotropic homogeneous media and the linear theory of elastic wave propagation. The real data of the geological media С l and C S determined by various
for ν in some cases. At the same time, there is 0 < ν ≤ 0.5 for existing natural and artificial materials. This problem was studied in detail and it was concluded that there are no paradoxes (Kuliev, 2000). The received abnormalities are related with the violation of the restrictions adopted in the derivation of Eq. (4) relative to isotropism, homogeneity, elasticity, and the smallness of the deformations and Hooke’s law.
methods were substituted in Eq. (4) instead of С l0, C S0 SCB that resulted in negative or large positive values
These results indicate that processing and interpreting the data of geophysical studies should be based
2
ν=
С l0 − 2C S0
(
2
2
2 С l0 − C S0
2
)
(4)
,
Table 1 Elastic moduli of the 3rd order
Lithology code 1
2
3
4
5
− 509.692 − 510.689
− 11.144 − 11.235
− 19.920 − 19.710
− 26.462 − 26.446
− 67.123 − 67.472
−3
17.936 18.975
− 4.155 − 4.015
− 2.517 − 2.126
1.548 2.024
− 2.727 − 2.543
−3
2.557 2.602
− 0.129 − 0.271
− 0.045 − 0.071
0.126 0.105
0.059 0.025
−3
а × 10 , GPa
b × 10 , GPa c × 10 , GPa
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(a) 0.2 0.8
Velocity, km/s 1.2 2.2 3.2
841
(b)
(c) 3
Density, g/cm 1.8 2.0 2.2 2.4 2.6
4.2
1.3
Pressure, GPa 0.01 0.04 0.06 0.09 cluster 1 cluster 2 cluster 3 cluster 4 cluster 5 cluster 6 cluster 7 cluster 8 cluster 9 cluster 10
Depth, km
1.8
2.3
2.8
3.3
CSx2
Clx 1
3.8
Fig. 2. Charts of velocity variations of pressure, shear waves (a), density (b), and geostatic pressure (c) on clusters of rocks of mudstone depending on depth.
on the theories that properly take into account the influence of the current geodynamic changes on the studied processes. Similar results were also obtained for other elastic physical characteristics of the medium. However, they are not given here; i.e., the aim of this work is to determine the nonlinear elastic moduli.
CONCLUSIONS Based on the geophysical data from wells, the method of calculating the elastic moduli of the 3rd order for sedimentary rocks was proposed. The method allows studying the cases of small and large initial deformations. It was applied to study the nonlinear physical and mechanical properties of some
Table 2 Lithology code Poisson’s ratio
ν Error, %
1
2
3
4
5
0.4155 0.4159
0.4200 0.475
0.349 0.3201
0.3013 0.236
0.3400 0.375
0.1
13.1
–8.28
–21.67
10.29
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−70 0.8
(a) a × 10−3, GPa −50 −30 −10
−30
(b) b × 10−3, GPa −10 10
30
(c) c × 10−3, GPaW −2.2 −1.2 −0.2 0.8 1.8
1.3
Depth, km
1.8
2.3
2.8
3.3
3.8
Fig. 3. Charts of variations of values of elastic moduli “a” (a), “b” (b), and “c” (c) on clusters of rocks of mudstone depending on depth.
sedimentary rocks of the SCB. The obtained specific numerical results proved its effectiveness. This method can be improved and used to study field seismic and seismological source data with appropriate corrections. ACKNOWLEDGMENTS This work was supported by the Science Foundation of SOCAR, grant project no. 18 (2013–2015), and the research program of the Azerbaijan National Academy of Sciences (ANAS) “On the Complex of Theoretical and Experimental Studies of Interdisciplinary Problems of Geomechanics” approved by resolution no. 5/3 on February 11, 2015 (2015–2017) of the Presidium of ANAS. REFERENCES Abasov, M.T., Kuliev, G.G., and Jevanshir, R.D., The Model of lithospheric evolution, Herald Russ. Acad. Sci., 2000, vol. 70, no. 1, pp. 66–71. Aleksandrov, K.S., Prodaivoda, G.T., and Maslov, B.P., A method ofor determining nonlinear elastic constants of rocks, Dokl. Earth Sci., 2001, vol. 380, no. 7, pp. 827–829.
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DETERMINING THE ELASTIC MODULI Darwin, V.E. and Singer, J.M., Well Logging for Earth Scientists, 2nd ed., Dordrecht: Elsevier, 2007. Dortman, N.B., Fizicheskie svoistva gornykh porod i poleznykh iskopaemykh (petrofizika): Spravochnik geofizika (Physical Properties of Rocks and Minerals (Petrophysics): A Geophysicist’s Reference Book), 2nd ed., Moscow: Nedra, 1984. Engel’brekht, Yu.K. and Nigul, U.K., Nelineinye volny deformatsii (Nonlinear Strain Waves), Moscow: Nauka, 1981. Gogonenkov, G.N., Izuchenie detal’nogo stroeniya osadochnykh tolshch seismorazvedkoi (Studying the Detailed Structure of Sedimentary Strata by Seismic Prospecting), Moscow: Nedra, 1987. Grinfel’d, M.A. and Movchan, A.A., Implications of preliminary deformation for the propagation of elastic waves, Izv. Akad. Nauk SSSR, Fiz. Zemli, 1975, no. 8, pp. 29–35. Guliyev, H.H., A new theoretical conception concerning the tectonic processes of the Earth, New Concepts Global Tecton. Newsl., 2010, vol. 56, pp. 50–74. Guliyev, H.H. and Aghayev, Kh.B., Determination of physical-mechanical parameters of sedimentary cover rocks on the base of seismic, borehole data and the theory of stressed medium elastic waves, Geofiz. Zh., 2011, vol. 33, no. 6, pp. 126–135. Guliyev, H.H. and Shirinov, N.M., The definition of elastic moduli of the third order in the stressed nonlinear isotropic media, Izv. Nats. Akad. Nauk Azerb., Ser. Nauk o Zemle, 2006, no. 3, pp. 31–35. Guz, A.N., Uprugie volny v telakh s nachal’nymi napryazheniyami. T. 2. Zakonomernosti rasprostraneniya (Elastic Waves in the Pre-Stressed Bodies. Vol. 2: Regularities of Propagation), Kiev: Naukova dumka, 1986. Itenberg, S.S., Interpretatsiya rezul’tatov geofizicheskikh issledovanii skvazhin: Ucheb. posobie dlya vuzov (Interpretation of Geophysical Logs: Student Manual), 2nd ed., Moscow: Nedra, 1987. Kostrov, B.V. and Nikitin, L.V., The propagation of seismic waves in prestressed elastic medium, General Assembly Int. Union Geodesy Geophys., 1969, vol. 15, Part 1, p. 98. Kuliev, G.G., Determination of the Poisson’s ratio in stressed media, Dokl. Earth Sci., 2000, vol. 370, no. 1, pp. 206–209. Kuliev, G.G. and Dzhabbarov, M.D., On elastic waves propagation in stressed anisotropic media, Izv. Akad. Nauk Azerb., Ser. Nauk Zemle, 1998, no. 2, pp. 102–112.
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No. 6
2016
