- Email: [email protected]
A three-dimensional stationary model of the thermal and thermoelastic fields of the Caucasus
191
Tec~o~~~, 227 (1993) 191-203 Elsevier Science Publishers B.V., Amsterdam
A three-dimensional stationary model of the thermal and thermoelastic fields of the Caucasus M.A. Alexidze, G.E. Gugunava, D.K.. Kiria and T.L. Chelidze ~~~~u~eof ~op~~~~c~,Academy of Sciences of Georgia, Rwkdze
Str 1,380093 7bilisi, Georgia
(Received May 23,1989; revised version accepted February 10, 1993)
ABSTRACT
Three-dimensional geothermal and thermoelastic models of the Caucasus have been produced. Application of the three-dimensional approach to regions as complex as the Caucasus has been justified. The model calculations were made excluding the surface heat flow and could therefore be used to test the validity of the model. The theoretical heat flow agreed well with the experimental model over the Black and Caspian Sea areas and over the whole Rioni-Kura depression. As was expected areas of young vdcanismpresented exceptions. Charts of temperature and thennoelastic displacement were plotted for the “granitic”, Conrad and Mohorovicic discontinuities.
The Earth’s thermal field is largely responsible for determining the main structurai features of the Earth and the character of processes within the Earth. Observational data for interpretation are available only from a thin surface layer, and this poses an extremely challen~ng geophysical problem, which is usually solved, by means of analytical extension of the surface heat flow to depth by a one-, or at most, a two-dimensional approximation (Liubimova, 1968, 1986; Gordienko et al., 1982; Cermak, 1983). Disregarding the three-dimensional problem whenever a complex erogenic area is concerned may lead to considerable errors. In addition, when values of the Earth’s surface heat flow are used as the starting point, we can hardly ever achieve direct geothermal assessment of the validity of a model, and we are thus eventually forced to use various indirect data from other geophysical methods such as seismology and ge~lec~ici~. 0040.1951/93/$06.00
The heterogeneity of the temperature field and geology calls for the formulation of three-dimensional equations to calculate thermoelastic stresses as the latter may constitute a vital geodynamic factor.
A thermal model of the Caucasus
A new numerical approach to the construction of a three-dimensional model for regions of complex geology has already been suggested by Alexidze et al. (1985) and has been applied to the Caucasus and the adjacent areas of the Black Sea and Caspian Sea. In addition, Hurtig et al. (1983) have advocated a similar approach. Our study area is approximated by a rectangular parallelepiped 210 km. deep, 2400 km long (northsouth and 1000 km wide east-west (Fig. 1). The depth H, to the sedimentary (i = l), “granitic” (i = 2) and “basaltic” (i = 3) layers were defined,
0 1993 - Elsevier Science Publishers B.V. All rights reserved
M.A. ALEXID%t
u
A’
c A Fig. 1. The parallelepiped used to approximate under investigation.
the region
(Fig, 2a-c) and a stationary inhomogeneous equation of heat conduction was solved:
where: a = the thermal diffusivity, aj = Ai/Cipi; hi = thermal conductivity, ci = heat capacity; pi = density; and T = temperature under the following boundary and conjugation conditions: TI ABCD
= To
l+T ax
=0
AA’B’B and DD’C’C’
aT
=0
G AA'D'D and BB’C’C
(4 Non-linear differential equations with partial derivatives were solved by the method of finite differences (Wasow and Forsythe, 1970). In addition to a stationary temperature distribution, some further a~umptions were necessary to caIulate the model, namely: (1) the lithospheric model corresponds to the seismo-gravitational model of the Caucasus (Bal-
I;?’ Al
avadze et al., 1979; Adamia et al., 1980) (Fig. 2a-cl; (2) the heat sources are: a horizon at a depth of 210 km with a constant temperature of 1,6OO”C, which is almost consistent with Pollack and Chapman’s and Anderson and Sammis’ geotherms (Anderson and Sammis, 1970, Pollack and Chapman, 19771, and crustal and mantle radioactive elements as well. Eieat generation values used were from Liubimova (1968); density and thermal characteristics were derived from laboratory observations (Shengelaya, 1968). Heat capacity was assumed to be L’= (0.19 + low4 T) cat/kg *"C.Allthese data are summarized in Table 1. It is thus apparent that, using this approach, the boundary conditions were defined without taking into account any field data about the surface heat flow. Therefore, surface heat flow data could be used quite efficiently to test the chosen model’s reality and the degree of influence of non-stationary processes (neovolcanism and sedimentation). All geographical details referred to in this paper are shown in Figure 2d. In calculating the model versions, an attempt was made to account for the dependence of A on temperature. This increased the difference between the experimental and model data; while if A was assumed to be constant, the difference decreased. The problem is that temperature, T, and pressure, P, have to be taken into account simultaneously, because they actually compensate each other’s action at high PT values (Lebedev et al., 1986) (Fig. 3). The model’s vahdity was tested by comparing the calculated heat flow over the Earth’s surface with field data (Kutas et al., 1978/79; Buachidze et al., 1980; Gordienko et al., 1982) (Fig. 4). The results of the comparison between the model and the field data are plotted in Figure 5 as a map of the difference between the theoretical, qm, and observed, qe, heat flow, Aq = qe -4,. It is evident that, over the entire area, except for the Greater and Lesser Caucasus and the district of Stavropol, the Aq value never exceeds 13 mW/m’, which is the value of the heat flow estimation error, allowing for possible distortion effects; hence, the observed flow appeared to be
MODEL OF THERMAL AND THERMOELEASTK
FIELDS OF THE CAUCASUS
close to a stationary value of q throughout most of the region. It was expected that qe would exceed the calculated values in the Greater and Lesser Caucasus and Stavropol areas due to voIcanic activity not taken into account by this model. However, it was interesting to observe good agreement between model and experimenta data for areas with a high rate of sedimentation; namely, the
193
Black Sea and Caspian Sea areas, since, according to many authors, a non-stationary sedimentation effect must give rise to a significant heat flow anomaly. In many reports observed heat flow has been assumed ‘a prriwi’ to be reduced by sedimentation; it was therefore considered necessary to make corrections for non-stationarity. Model calculations, however, proved the heat flow to be nearly stationary in sea areas with a high rate of
Fig. 2. Depth of crystalline basement (a), Conrad (b), and Moho (cl discontinuties for Caucasus and adjoining regions (Balavadse et al., 1979). (d) Map of geographic and tectonic elements cited in this paper. * = Neogene to Quaternary volcanoes; l-l = Volgograd-Nachichevan seismic profile, Java-Javakheti plateau; DB = Dzirula block; Rioni Bas = Rioni basin; Tse = Tsebelda; Kir = Kirovabad (Gyanja); St = Stepanavan, E = Elbrus; K = Kazbeck.
194
M.A. ALEXIDZE
NACidEVAti
4oa
450
Fig, 2 (~ntinued~.
TABLE 1 Values used in the model Parameters
A lo3 (Cal/cm. s. grad) A f.10’ W/fm . grad)) P Wcm3Xp b/m31 w (10’3 Cal. cm3. s) w lOi W/m3
Layers Sediments
“Granite”
“Basalt”
Mantle
3.4 (1.8-2.2) 13.28 (7.56-2.24) 2.4 2.3 9.66
6.88 28.9 2.7 3.0 12.6
7.0 29.4 2.9 1.3 5.45
10.0 42.0 3.3 0.03 0.126
A = values for marine sediments (given in brackets>, w = beat generation of the rock.
ET AL.
195
MODEL OF THERMAL AND THJZRMOELEASTIC FIELDS OF THE CAUCASUS
a, rd’m?s” 4l t ,f
'2 0 0 20
*5 O.i 85
Fig. 3. (a) Thermobaric dependence
,
fib*
0.2 455
0.3 PU
f5 9.4 255
H,wn P, Wa T,,‘C
a 20
a.4
0.Q 8.3 0.4p,G/h 85 455 St4 25s r,‘C
of temperature conductivity and (6) heat conductivity A under various PT conditions for charnokite (I) and granite (2) (Lebdev et al., 1986).
~d~entati~n and the corrections were, in fact, not necessary. This could be explained in two ways: (1) the sediments were old enough to allow the heat flow to reach a stationary state; (2) the sediments being very young, the heat exchange rate was higher than in a purely conductive mechanism. This concept has gained sound support lately: in appears that fluids and loose sediments undergo active convective movements in marine ~d~enta~ sequences (Ether-
idge et al., 1984; Kerrich et al., 1984) and the convection is certain to enhance greatly the establishment of a heat stability. We did not consider a third possibility, namely that the model is incorrect and that coincidence is fortuitous: such suspicions can be expressed for almost any geophysical model. After the model had been shown to conform with experimental results, isotherm charts were plotted for every 100°C to show the temperature dist~bution over the important geological discon-
Fig. 4. JZxperimentai heat Bow values over the Earth’s surface for the Caucasus and adjoining regions (in mW/m’). al., 1980; Gordienko et al., 1982).
(Buachidze et
M.A. ALEXID%L-
E-1’Al.
Fig. 5. Map showing the difference between the observed (q,f and theoretical (qm) heat flow (6q = qc - qmf in the Caucasus aad Black and Caspian Sea areas (mW/m2).
tinuities (crystalline basement, Conrad and Moho), and also heat flow distribution at these levels (Alexidze et al., 1985). In this way “cold” and “hot” areas (temperature provinces) could be recognized (Fig. 6). To confirm the validity of one- and two-dimensional geothermal appro~mations, it is particu-
larly interesting to examine the heat flow distribution over the Moho d~s~ntin~~ (Fig. 7). The maximum variation of qm over the discontinuity in a horizontal direction for the Caucasus as 20-25% in our model; using a one-dimensional model (Cermak, 1983) these variations reached 100%.
Fig. 6. Temperature distributjoo at the Moho ~~ntioui~
(“Cl.
MODEL OF THERMAL AND THERMOELEASTIC
FIELDS OF THE CAUCASUS
Fig. 7. Calculated heat flow (at the Moho) for the Caucasus, Black Sea and Caspian Sea areas (mW/m2).
This difference must have been a result of neglecting lateral com~nents of the heat flow in such complex structures as the Caucasus. The main geological features of the crust and mantle in the Caucasus; in particular, the features of the relief of the Moho, Conrad and crystalline basement discontinuities, could be traced at various depths down to 140 km. The relief of the isotherms is specified basically by the geological boundaries; as a rule, the uprise of the isotherms correspond to the thickest sedimentary sequences and the descent of the isotherms to the thinnest. All the isotherm charts clearly show two basic features: (1) the marine areas of ascending isotherms and (2) the Dzirula area between them of dipping isotherms. Two large intermediate areas (the Kura and Colchis Basins), with relatively high temperatures, also feature dipping isotherms. It is interesting to compare the behaviour of isotherms from this model along NakhichevanVolgograd geotraverse with the data of Gordienko et al. (1982). According to the traditional one-dimensional model, the isotherms must reflect anomalous heating beneath the Greater Caucasus; according to the present model, however, the isotherms dip here as a result of the absence of a sedimentary sequence. It is worthwhile noting that a dip in the isotherms beneath
I
300
4000
I
4330 TV
Fig, 8. The geotherms of Pollack and Chapman and Anderson and Sammis, and those constructed according to our calcuiations. i, 2, 3 = geotberms for this model for the Elbrns, Black and Caspian Seas, respectively; 4,s = Anderson and Sammis’ (1970) geotherms for dry and partially molten lherzolite mantle, 6, 8 = Pollack and Chapman’s (1977) oceanic geothetms for q = 40 and 60 mW/m’ respectively; 7, 9 = PoUack and Chapman’s (1977) continental geoterms for q = 40 and 30 mW/mz, respectively.
IYX
M.A.
the Greater Caucasus occurs even when the increased heat generation in the ‘granitic’ roots of the mountains is taken into account. We have also calculated the geotherms, as shown in Figure 8. Geotherms obtained for the most specific areas of Elbrus Sea (Liubimova, 1968) the Black Sea (Gordienko, 1975) and the Caspian Sea (Kutas, 1978) were compared with those of Pollack and Chapman and Anderson and Sammis (Anderson and Sammis, 1970; Pollack and Chapman, 1977). They appear to agree in the depth interval of 150-80 km. Geotherms over areas with a thick sedimentary sequence (* 20 km) yield anomalously high surface gradients, owing to the screening effect of the sediments. Geotherms below 150 km (Liubimova, 1968; Gordienko, 1975; Kutas, 1978) are different from these suggested by Pollack and Chapman (1977) and Anderson and Sammis (1970). A thermoelastic model of the Caucasus The thermal model calculated above allows us to evaluate thermoelastic displacement in the Caucasian lithosphere. This requires additional data on the elastic instants; namely Lame’s coefficients h, and II, a thermal expansion coefficient, LYand pressure, & arising whenever the body temperature increases by 1°C without the body expansion. /3 is related to the linear temperature expansion coefficient and compression modules as: p = 3Ka, where K = 3A, + 2~/3, K = the bulk compression modulus. The values of p were calculated for: Ly= (4 - 1.3 x 10-V)
c’1’ AL
TABLE 2 Lame’s coefficients used in the model Lame’s coefficient A,
i=l,2,3,4
10’~
p. 10’0
Sediments
“Granite”
“Basalt”
Mantle
24.832 16.224
26.75 32.375
41.295 41.063
71.108 70.587
In the first stage, we chose the simplest model of heat evolution in the region which is, presumably, the originally ‘cold’ nonhomogeneous structure of the Caucasus (Balavadze et al., 1979; Adamia et al., 1980) which was heated to temperatures defined by our stationary heat model. This is, of course, an oversimpli~~tion, nevertheless it yields quite acceptable results. When a thermal field is superimposed at each point of parallelepiped ABCD A’B’C’D’ a vector of thermoelastic displacements v’, with compowhere z is nents ul, u2, u3 (l-x,2-y,3-z, normal to the surface) arises. The displacements are calculated from a three-dimensional equation of the elasticity theory (Muskhelishvili, 1935):
a2tdl -+p i ax*
a2u,
-i”-
ay2
a*u, a2*
a2u, a2td2 s+axay+-
+(4?+& i =
+(A, =
a%, axa 1
T&T)
a*u, -++ ax*
x lo-s(“c)-l
Lame’s coefficients were taken from Table 2. In the present paper the thermoelastic effects are calculated for the purely elastic behaviour of material. Of course, a rigorous approach implies taking into account the relaxation process and this will be the object of our research in the future. Thus, the following calculations should be considered as the first crude approximation of the real the~~lastic evolution of the Caucasus.
ALEXIDZE
(3)
a2u,
a2u2 ay2 + a.$
-t-F)
a2u,
-+ hay
a2ti, -+ay*
au; ayaz
;(Sr)
(4)
Y
a*u, -+ ax*
a2u, ---+ay2
a2u, az* a2u2 -+g ayaz
= z&W
a*u,
(5)
MODEL OF THERMAL AND THERMOELEASTIC FIELDS OF THE CAUCASUS
and the following boundary plied:
conditions
~~lABGD=~~IABCD=~jIABCD=O ui = U2 = u1=uz
u3 1AA’B’B and DD’C’C
= u3 1AA’D’D
are ap-
= 0
(7)
and BB’C’C = 0
(8)
+ %)~*,B,c,D,=O
W
(6)
Equations (6-8) indicate that the sides &CD, AA’B’B, DD’C’C, AA’D’D and BB’C’C of parallelepiped ABCDA’B’C’D’ are fixed. Of course, these conditions are not adequate to describe the real the~oelasti~ displacements on the sides of the parallelepiped but in is only necessary to move inside the parallelepiped by 1 or 2 spacings of the grid (that is, by 50-100 km) and the approximation becomes reasonable. This is because the inner nodes of grid are not fixed and yield to thermoelastic forces. That is why we considered such a large volume. It can also be noted that changing the boundary conditions (when normal derivatives of the d~placement vector equals zero on the same sides) did not significantly change the distribution of displacements and stresses inside the parallelepiped: p( 2
199
(9)
au,
au3
au3
-+z
+25
aY
II
-_
0
A’B’C’D’
1
(11)
Equations (9-11) indicate that the upper surface of the area in question (the Earth’s surface) is free of stresses and this seems quite reasonable from a physical point of view. Equations (12-14) represent the conjugation conditions in the theory of elasticity (the equality of displacement vectors and normal stresses) for the ith-(i + 1)st layer interfaces:
au1
Iln
i
z-+ax )=pn+l(gl+y au3
(12)
pn(~+~)=~n+l(~+~) au, au,
4l
(
---g+-+-g
= A
au,
)
ay
n+l (
au, ---fax
ay
au3
+a.$5
au,
(13)
au, +=
+a&+$-
au3
(14)
i
It is evident that, in this approach, stresses and displacements caused by plate motions are neg-
Fig. 9. Vertical displacements over the “granite” surface (ml
200
M.A.
Fig. 10. Vertical dis~Iacements over the Conrad d~s~~tinuity (m).
Fig. 11. Vertical displacements over the Maho discontinuity (ml.
ALEXID%fi
E.T AI..
MODEL OF THERMAL AND THERMOELEASTIC
201
FIELDS OF THE CAUCASUS
lected and only the~~lastic effects are considered. The solution was obtained using the method of finite differences. The resulting systems of algebraic equations were solved by iterative methods of upper relaxation. The solution of the boundary value problem provided the displacement components used to find the stress components and the main stress. The values obtained for ui, u2, us were defined as the differences between the positions of a given point, in an “unheated” and a “heated” states. From the calculated results for the displacements of the Moho discontinuity, four major zones of anomalous vertical displacements (up to 1200 m> were recognized in the Caucasus and adjacent territories, with “background” displacements of approximately 500-800 m (150-200 m in marine areas>. These are the Elbrus-Stavropol uplift and the western, central and eastern uplifts, located along the southern Black Sea coast and its eastem extension (Figs. 9-11). Notice that the uplift gradients are not large for the Moho discontinuity (maximum values of gradiu, never exceed 2-6 m/km). All the four major zones of vertical displacement occur over the Conrad discontinuity. The
highest value of H reaches 1300 m in the central parts of two of them (Elbrus-Stavropol and the central uplift). In addition, the Conrad discontinuity shows local uplifts in the territory or Armenia and to the south (Fig. 13). The value of ug varies both in lateral (grad,+) and vertical (grad& directions. Close examination has shown that almost all the similar zones with large grad,+ have deep roots, at least as deep as the middle of the basaltic layer. It should be noted that the increasing dampening with depth of ug is typical only of thermoelastic components of displacement. Several zones with various thermoelastic stresses can be identified in the chart of horizontal displacements, uh: Uh =
(Uf + uy2
(Figs. 1%14), as for u3 (Figs. 9-111, these are: (1) areas of predominantly “northward” displacement, with a boundary between them running along the south of the Caspian Sea, the Greater Caucasus piedmonts, the Rioni Basin and the south of the Black Sea; (2) areas with high positive and negative uh gradients (alternating compression and extension zones), which outline the anomalous zones of the
Fig. 12. Horizontal displacements at the crystalline basement Cm).
M.A. ALEXIDZE ET AI.
Fig. 13. Horizontal displacements at the Conrad discontinuity {III).
Anatolian Fault, Lesser Caucasus, Eibrus, Crimea and Dagestau. Thermoelastic stresses calculated by the present model can reach quite high values, of the
order of 10” we/cm2 (‘IO9Pa). Obviously, the thermal fields and thermaelastic stresses, must exert a large ixxfluenceon other geophysical fields and geudyuamic processes.
MODEL OF THERMAL AND THERMOELEASTIC FIELDS OF THE CAUCASUS
Conelusions The present investigation has yielded the following results: (1) A three-dimensional stationary heat flow model of the Earth’s crust in the Caucasus and Black Sea areas has been designed with assumptions that allowed us to exclude surface heat flow values from the calc~ations and, therefore, to use them in testing the model’s approximation to the real conditions. (2) The heat flow model correlates well with the experimental data recovered from the Black and Caspian Sea areas and the Rioni-Kura Depression (i.e., the heat flow is close to a stationary one), and differs, as expected, in the Greater and Lesser Caucasus, where young volcanism is present. This kind of correlation over the various regions indicates that, in addition to the conductive heat transfer, other forms of heat transfer (e.g., convection) must also be considered in the Black and Caspian Sea areas, where the conductive mechanism is not enough to attain the stationary state in young sediments. (3) A three-dimensional stationary thermoelastic model of the Caucasus, Black Sea and Caspian Sea areas has been calculated and horizontal and vertical ~m~nents of stresses and displacements have been evaluated. It has been shown that thermoelastic stresses reaching lo9 Pa induce Iarge vertical (over 1000 m) and horizontal (up to 400 m) displacements in geologically inhomogeneous media.
References
Adamia,Sh.A., Balavadze,B.K., loseliani, VS. and Shengeiaya, G.Sh., 1980. Results of an integrated geoiogical-geophysical interpretation of the Caucasian region. in: Crustal Structure of Central and Eastern Europe by Geophysical Data. Naukova Dumka, Kiev. Alex&e, MA., Buachidze, G.I., et al., 1985. A three-dimensional geothermal model of the Caucasus. In: Geophysical Fields and Deep Structure of Transcaucasia. Nauka, Moscow, pp. 123-133.
203
Anderson, D.L. and Sammis, G., 1970. Partial melting in the upper mantle. Phys. Earth Planet. Inter., 3: 41-50. Balavadze, B.K., Shengelaya, G.Sh. and Mindeii, P.Sh., 1979. A gravitational model of the earth crust for the Caucasus and Black and Caspian Sea areas. In: Gravitational Models of the Earth Crust and Upper Mantle. Naukova Dumka, Kiev. Buachidze, I.M., Buachidze, G.I., et al. 1980. Geothermal Conditions and Geothermal Waters of Georgia. Sabchota Sakartvelo, Tbilisi, 206 pp. Ccrmak, V., 1983. Crustal temperature and mantle heat flow in Europe. Tectonophysics, 83: 123-142. Cermak, V. and Rybach, L., (Editors), 1982. Heat Field of Europe. Mir, Moscow, 376 pp. Etheridge, M.S., Wall, V.G., Cos, and Vernon, R.H., 1984. High fluid pressures during regional metamo~hism and deformation: apptications for mass transport and deformation mechanism. J. Geophys. Res. Gordienko, V.V., 1975. Heat Anomalies of Geosynchnes. Naukova Dumka, Kiev, p. 144. Gordienko, V.V., Zavgorodnaia, G.V. and Jakobi, N.M., 1982. Continental Heat Flow. Naukova Dumka, Kiev, 184 pp. Hurtig, E., Rugenstern, B. and Stromeyer, 1983. Three-dimensional modelling of crustal temperature and Moho heat flow in Central Europe and adjacent areas. Abstr. IUG Assembly (Hamburg~, p. 476. Kerrich, T.E., La Tourend, and Wilhnore, L., 1984. Fluid participation in deep fault zones: evidence from geological, geochemical and ‘*O/ I60 relations. J. Geophys. Res., 89: 4331-4334. Kutas, R.N., 1978. Heat Flow Field and Thermal Model of the Earth Crust. Naukova Dumka, Kiev, 146 pp. Kutas, R.N., Liubimova, EA. and Smirnova, E.V., 1978/79. A heat flow map of European part of the USSR. Pure Appl. Geophys., 117. Lebedev, T.G., Korchin, V.A., Savenko, Y., Shapoval, V.I. and Shepel, S.I., 1986. Physical Properties of Mineral Matter in Thermobaric Conditions of Lithosphere. Naukova Dumka, Kiev, 198 pp. Liubimova, E.A., 1968. The Earth’s and Moon’s Thermics. Nauka, Moscow, 278 pp. Moiseenko, U.I., Smyslov, A.A., 1986. Temperatures of the Earth’s Interior. Leningrad, Nedra, 1’78pp. Muskhelish~ii, NJ., 1935, Some Basic Problems of the Mathematical Elasticity Theory. Acad. Sci. USSR, MoscowLeningrad. Pollack, H.N. and Chapman, D.S., 1977. On the regional variations of heat flow, geotherms, and lithospheric thickness. Tectonophysics, 38: 279-296. Shengelaya, GSh., 1968. Structure of Earth Crust in the West of the R. Kura Basin. Metsniereba, Tbilisi, 177 pp. Wasow, W. and Forsythe, G., 1970. Finite-difference Methods for Partial Differential Equation. John Wiley Sons, NYLondon.
Tec~o~~~, 227 (1993) 191-203 Elsevier Science Publishers B.V., Amsterdam
A three-dimensional stationary model of the thermal and thermoelastic fields of the Caucasus M.A. Alexidze, G.E. Gugunava, D.K.. Kiria and T.L. Chelidze ~~~~u~eof ~op~~~~c~,Academy of Sciences of Georgia, Rwkdze
Str 1,380093 7bilisi, Georgia
(Received May 23,1989; revised version accepted February 10, 1993)
ABSTRACT
Three-dimensional geothermal and thermoelastic models of the Caucasus have been produced. Application of the three-dimensional approach to regions as complex as the Caucasus has been justified. The model calculations were made excluding the surface heat flow and could therefore be used to test the validity of the model. The theoretical heat flow agreed well with the experimental model over the Black and Caspian Sea areas and over the whole Rioni-Kura depression. As was expected areas of young vdcanismpresented exceptions. Charts of temperature and thennoelastic displacement were plotted for the “granitic”, Conrad and Mohorovicic discontinuities.
The Earth’s thermal field is largely responsible for determining the main structurai features of the Earth and the character of processes within the Earth. Observational data for interpretation are available only from a thin surface layer, and this poses an extremely challen~ng geophysical problem, which is usually solved, by means of analytical extension of the surface heat flow to depth by a one-, or at most, a two-dimensional approximation (Liubimova, 1968, 1986; Gordienko et al., 1982; Cermak, 1983). Disregarding the three-dimensional problem whenever a complex erogenic area is concerned may lead to considerable errors. In addition, when values of the Earth’s surface heat flow are used as the starting point, we can hardly ever achieve direct geothermal assessment of the validity of a model, and we are thus eventually forced to use various indirect data from other geophysical methods such as seismology and ge~lec~ici~. 0040.1951/93/$06.00
The heterogeneity of the temperature field and geology calls for the formulation of three-dimensional equations to calculate thermoelastic stresses as the latter may constitute a vital geodynamic factor.
A thermal model of the Caucasus
A new numerical approach to the construction of a three-dimensional model for regions of complex geology has already been suggested by Alexidze et al. (1985) and has been applied to the Caucasus and the adjacent areas of the Black Sea and Caspian Sea. In addition, Hurtig et al. (1983) have advocated a similar approach. Our study area is approximated by a rectangular parallelepiped 210 km. deep, 2400 km long (northsouth and 1000 km wide east-west (Fig. 1). The depth H, to the sedimentary (i = l), “granitic” (i = 2) and “basaltic” (i = 3) layers were defined,
0 1993 - Elsevier Science Publishers B.V. All rights reserved
M.A. ALEXID%t
u
A’
c A Fig. 1. The parallelepiped used to approximate under investigation.
the region
(Fig, 2a-c) and a stationary inhomogeneous equation of heat conduction was solved:
where: a = the thermal diffusivity, aj = Ai/Cipi; hi = thermal conductivity, ci = heat capacity; pi = density; and T = temperature under the following boundary and conjugation conditions: TI ABCD
= To
l+T ax
=0
AA’B’B and DD’C’C’
aT
=0
G AA'D'D and BB’C’C
(4 Non-linear differential equations with partial derivatives were solved by the method of finite differences (Wasow and Forsythe, 1970). In addition to a stationary temperature distribution, some further a~umptions were necessary to caIulate the model, namely: (1) the lithospheric model corresponds to the seismo-gravitational model of the Caucasus (Bal-
I;?’ Al
avadze et al., 1979; Adamia et al., 1980) (Fig. 2a-cl; (2) the heat sources are: a horizon at a depth of 210 km with a constant temperature of 1,6OO”C, which is almost consistent with Pollack and Chapman’s and Anderson and Sammis’ geotherms (Anderson and Sammis, 1970, Pollack and Chapman, 19771, and crustal and mantle radioactive elements as well. Eieat generation values used were from Liubimova (1968); density and thermal characteristics were derived from laboratory observations (Shengelaya, 1968). Heat capacity was assumed to be L’= (0.19 + low4 T) cat/kg *"C.Allthese data are summarized in Table 1. It is thus apparent that, using this approach, the boundary conditions were defined without taking into account any field data about the surface heat flow. Therefore, surface heat flow data could be used quite efficiently to test the chosen model’s reality and the degree of influence of non-stationary processes (neovolcanism and sedimentation). All geographical details referred to in this paper are shown in Figure 2d. In calculating the model versions, an attempt was made to account for the dependence of A on temperature. This increased the difference between the experimental and model data; while if A was assumed to be constant, the difference decreased. The problem is that temperature, T, and pressure, P, have to be taken into account simultaneously, because they actually compensate each other’s action at high PT values (Lebedev et al., 1986) (Fig. 3). The model’s vahdity was tested by comparing the calculated heat flow over the Earth’s surface with field data (Kutas et al., 1978/79; Buachidze et al., 1980; Gordienko et al., 1982) (Fig. 4). The results of the comparison between the model and the field data are plotted in Figure 5 as a map of the difference between the theoretical, qm, and observed, qe, heat flow, Aq = qe -4,. It is evident that, over the entire area, except for the Greater and Lesser Caucasus and the district of Stavropol, the Aq value never exceeds 13 mW/m’, which is the value of the heat flow estimation error, allowing for possible distortion effects; hence, the observed flow appeared to be
MODEL OF THERMAL AND THERMOELEASTK
FIELDS OF THE CAUCASUS
close to a stationary value of q throughout most of the region. It was expected that qe would exceed the calculated values in the Greater and Lesser Caucasus and Stavropol areas due to voIcanic activity not taken into account by this model. However, it was interesting to observe good agreement between model and experimenta data for areas with a high rate of sedimentation; namely, the
193
Black Sea and Caspian Sea areas, since, according to many authors, a non-stationary sedimentation effect must give rise to a significant heat flow anomaly. In many reports observed heat flow has been assumed ‘a prriwi’ to be reduced by sedimentation; it was therefore considered necessary to make corrections for non-stationarity. Model calculations, however, proved the heat flow to be nearly stationary in sea areas with a high rate of
Fig. 2. Depth of crystalline basement (a), Conrad (b), and Moho (cl discontinuties for Caucasus and adjoining regions (Balavadse et al., 1979). (d) Map of geographic and tectonic elements cited in this paper. * = Neogene to Quaternary volcanoes; l-l = Volgograd-Nachichevan seismic profile, Java-Javakheti plateau; DB = Dzirula block; Rioni Bas = Rioni basin; Tse = Tsebelda; Kir = Kirovabad (Gyanja); St = Stepanavan, E = Elbrus; K = Kazbeck.
194
M.A. ALEXIDZE
NACidEVAti
4oa
450
Fig, 2 (~ntinued~.
TABLE 1 Values used in the model Parameters
A lo3 (Cal/cm. s. grad) A f.10’ W/fm . grad)) P Wcm3Xp b/m31 w (10’3 Cal. cm3. s) w lOi W/m3
Layers Sediments
“Granite”
“Basalt”
Mantle
3.4 (1.8-2.2) 13.28 (7.56-2.24) 2.4 2.3 9.66
6.88 28.9 2.7 3.0 12.6
7.0 29.4 2.9 1.3 5.45
10.0 42.0 3.3 0.03 0.126
A = values for marine sediments (given in brackets>, w = beat generation of the rock.
ET AL.
195
MODEL OF THERMAL AND THJZRMOELEASTIC FIELDS OF THE CAUCASUS
a, rd’m?s” 4l t ,f
'2 0 0 20
*5 O.i 85
Fig. 3. (a) Thermobaric dependence
,
fib*
0.2 455
0.3 PU
f5 9.4 255
H,wn P, Wa T,,‘C
a 20
a.4
0.Q 8.3 0.4p,G/h 85 455 St4 25s r,‘C
of temperature conductivity and (6) heat conductivity A under various PT conditions for charnokite (I) and granite (2) (Lebdev et al., 1986).
~d~entati~n and the corrections were, in fact, not necessary. This could be explained in two ways: (1) the sediments were old enough to allow the heat flow to reach a stationary state; (2) the sediments being very young, the heat exchange rate was higher than in a purely conductive mechanism. This concept has gained sound support lately: in appears that fluids and loose sediments undergo active convective movements in marine ~d~enta~ sequences (Ether-
idge et al., 1984; Kerrich et al., 1984) and the convection is certain to enhance greatly the establishment of a heat stability. We did not consider a third possibility, namely that the model is incorrect and that coincidence is fortuitous: such suspicions can be expressed for almost any geophysical model. After the model had been shown to conform with experimental results, isotherm charts were plotted for every 100°C to show the temperature dist~bution over the important geological discon-
Fig. 4. JZxperimentai heat Bow values over the Earth’s surface for the Caucasus and adjoining regions (in mW/m’). al., 1980; Gordienko et al., 1982).
(Buachidze et
M.A. ALEXID%L-
E-1’Al.
Fig. 5. Map showing the difference between the observed (q,f and theoretical (qm) heat flow (6q = qc - qmf in the Caucasus aad Black and Caspian Sea areas (mW/m2).
tinuities (crystalline basement, Conrad and Moho), and also heat flow distribution at these levels (Alexidze et al., 1985). In this way “cold” and “hot” areas (temperature provinces) could be recognized (Fig. 6). To confirm the validity of one- and two-dimensional geothermal appro~mations, it is particu-
larly interesting to examine the heat flow distribution over the Moho d~s~ntin~~ (Fig. 7). The maximum variation of qm over the discontinuity in a horizontal direction for the Caucasus as 20-25% in our model; using a one-dimensional model (Cermak, 1983) these variations reached 100%.
Fig. 6. Temperature distributjoo at the Moho ~~ntioui~
(“Cl.
MODEL OF THERMAL AND THERMOELEASTIC
FIELDS OF THE CAUCASUS
Fig. 7. Calculated heat flow (at the Moho) for the Caucasus, Black Sea and Caspian Sea areas (mW/m2).
This difference must have been a result of neglecting lateral com~nents of the heat flow in such complex structures as the Caucasus. The main geological features of the crust and mantle in the Caucasus; in particular, the features of the relief of the Moho, Conrad and crystalline basement discontinuities, could be traced at various depths down to 140 km. The relief of the isotherms is specified basically by the geological boundaries; as a rule, the uprise of the isotherms correspond to the thickest sedimentary sequences and the descent of the isotherms to the thinnest. All the isotherm charts clearly show two basic features: (1) the marine areas of ascending isotherms and (2) the Dzirula area between them of dipping isotherms. Two large intermediate areas (the Kura and Colchis Basins), with relatively high temperatures, also feature dipping isotherms. It is interesting to compare the behaviour of isotherms from this model along NakhichevanVolgograd geotraverse with the data of Gordienko et al. (1982). According to the traditional one-dimensional model, the isotherms must reflect anomalous heating beneath the Greater Caucasus; according to the present model, however, the isotherms dip here as a result of the absence of a sedimentary sequence. It is worthwhile noting that a dip in the isotherms beneath
I
300
4000
I
4330 TV
Fig, 8. The geotherms of Pollack and Chapman and Anderson and Sammis, and those constructed according to our calcuiations. i, 2, 3 = geotberms for this model for the Elbrns, Black and Caspian Seas, respectively; 4,s = Anderson and Sammis’ (1970) geotherms for dry and partially molten lherzolite mantle, 6, 8 = Pollack and Chapman’s (1977) oceanic geothetms for q = 40 and 60 mW/m’ respectively; 7, 9 = PoUack and Chapman’s (1977) continental geoterms for q = 40 and 30 mW/mz, respectively.
IYX
M.A.
the Greater Caucasus occurs even when the increased heat generation in the ‘granitic’ roots of the mountains is taken into account. We have also calculated the geotherms, as shown in Figure 8. Geotherms obtained for the most specific areas of Elbrus Sea (Liubimova, 1968) the Black Sea (Gordienko, 1975) and the Caspian Sea (Kutas, 1978) were compared with those of Pollack and Chapman and Anderson and Sammis (Anderson and Sammis, 1970; Pollack and Chapman, 1977). They appear to agree in the depth interval of 150-80 km. Geotherms over areas with a thick sedimentary sequence (* 20 km) yield anomalously high surface gradients, owing to the screening effect of the sediments. Geotherms below 150 km (Liubimova, 1968; Gordienko, 1975; Kutas, 1978) are different from these suggested by Pollack and Chapman (1977) and Anderson and Sammis (1970). A thermoelastic model of the Caucasus The thermal model calculated above allows us to evaluate thermoelastic displacement in the Caucasian lithosphere. This requires additional data on the elastic instants; namely Lame’s coefficients h, and II, a thermal expansion coefficient, LYand pressure, & arising whenever the body temperature increases by 1°C without the body expansion. /3 is related to the linear temperature expansion coefficient and compression modules as: p = 3Ka, where K = 3A, + 2~/3, K = the bulk compression modulus. The values of p were calculated for: Ly= (4 - 1.3 x 10-V)
c’1’ AL
TABLE 2 Lame’s coefficients used in the model Lame’s coefficient A,
i=l,2,3,4
10’~
p. 10’0
Sediments
“Granite”
“Basalt”
Mantle
24.832 16.224
26.75 32.375
41.295 41.063
71.108 70.587
In the first stage, we chose the simplest model of heat evolution in the region which is, presumably, the originally ‘cold’ nonhomogeneous structure of the Caucasus (Balavadze et al., 1979; Adamia et al., 1980) which was heated to temperatures defined by our stationary heat model. This is, of course, an oversimpli~~tion, nevertheless it yields quite acceptable results. When a thermal field is superimposed at each point of parallelepiped ABCD A’B’C’D’ a vector of thermoelastic displacements v’, with compowhere z is nents ul, u2, u3 (l-x,2-y,3-z, normal to the surface) arises. The displacements are calculated from a three-dimensional equation of the elasticity theory (Muskhelishvili, 1935):
a2tdl -+p i ax*
a2u,
-i”-
ay2
a*u, a2*
a2u, a2td2 s+axay+-
+(4?+& i =
+(A, =
a%, axa 1
T&T)
a*u, -++ ax*
x lo-s(“c)-l
Lame’s coefficients were taken from Table 2. In the present paper the thermoelastic effects are calculated for the purely elastic behaviour of material. Of course, a rigorous approach implies taking into account the relaxation process and this will be the object of our research in the future. Thus, the following calculations should be considered as the first crude approximation of the real the~~lastic evolution of the Caucasus.
ALEXIDZE
(3)
a2u,
a2u2 ay2 + a.$
-t-F)
a2u,
-+ hay
a2ti, -+ay*
au; ayaz
;(Sr)
(4)
Y
a*u, -+ ax*
a2u, ---+ay2
a2u, az* a2u2 -+g ayaz
= z&W
a*u,
(5)
MODEL OF THERMAL AND THERMOELEASTIC FIELDS OF THE CAUCASUS
and the following boundary plied:
conditions
~~lABGD=~~IABCD=~jIABCD=O ui = U2 = u1=uz
u3 1AA’B’B and DD’C’C
= u3 1AA’D’D
are ap-
= 0
(7)
and BB’C’C = 0
(8)
+ %)~*,B,c,D,=O
W
(6)
Equations (6-8) indicate that the sides &CD, AA’B’B, DD’C’C, AA’D’D and BB’C’C of parallelepiped ABCDA’B’C’D’ are fixed. Of course, these conditions are not adequate to describe the real the~oelasti~ displacements on the sides of the parallelepiped but in is only necessary to move inside the parallelepiped by 1 or 2 spacings of the grid (that is, by 50-100 km) and the approximation becomes reasonable. This is because the inner nodes of grid are not fixed and yield to thermoelastic forces. That is why we considered such a large volume. It can also be noted that changing the boundary conditions (when normal derivatives of the d~placement vector equals zero on the same sides) did not significantly change the distribution of displacements and stresses inside the parallelepiped: p( 2
199
(9)
au,
au3
au3
-+z
+25
aY
II
-_
0
A’B’C’D’
1
(11)
Equations (9-11) indicate that the upper surface of the area in question (the Earth’s surface) is free of stresses and this seems quite reasonable from a physical point of view. Equations (12-14) represent the conjugation conditions in the theory of elasticity (the equality of displacement vectors and normal stresses) for the ith-(i + 1)st layer interfaces:
au1
Iln
i
z-+ax )=pn+l(gl+y au3
(12)
pn(~+~)=~n+l(~+~) au, au,
4l
(
---g+-+-g
= A
au,
)
ay
n+l (
au, ---fax
ay
au3
+a.$5
au,
(13)
au, +=
+a&+$-
au3
(14)
i
It is evident that, in this approach, stresses and displacements caused by plate motions are neg-
Fig. 9. Vertical displacements over the “granite” surface (ml
200
M.A.
Fig. 10. Vertical dis~Iacements over the Conrad d~s~~tinuity (m).
Fig. 11. Vertical displacements over the Maho discontinuity (ml.
ALEXID%fi
E.T AI..
MODEL OF THERMAL AND THERMOELEASTIC
201
FIELDS OF THE CAUCASUS
lected and only the~~lastic effects are considered. The solution was obtained using the method of finite differences. The resulting systems of algebraic equations were solved by iterative methods of upper relaxation. The solution of the boundary value problem provided the displacement components used to find the stress components and the main stress. The values obtained for ui, u2, us were defined as the differences between the positions of a given point, in an “unheated” and a “heated” states. From the calculated results for the displacements of the Moho discontinuity, four major zones of anomalous vertical displacements (up to 1200 m> were recognized in the Caucasus and adjacent territories, with “background” displacements of approximately 500-800 m (150-200 m in marine areas>. These are the Elbrus-Stavropol uplift and the western, central and eastern uplifts, located along the southern Black Sea coast and its eastem extension (Figs. 9-11). Notice that the uplift gradients are not large for the Moho discontinuity (maximum values of gradiu, never exceed 2-6 m/km). All the four major zones of vertical displacement occur over the Conrad discontinuity. The
highest value of H reaches 1300 m in the central parts of two of them (Elbrus-Stavropol and the central uplift). In addition, the Conrad discontinuity shows local uplifts in the territory or Armenia and to the south (Fig. 13). The value of ug varies both in lateral (grad,+) and vertical (grad& directions. Close examination has shown that almost all the similar zones with large grad,+ have deep roots, at least as deep as the middle of the basaltic layer. It should be noted that the increasing dampening with depth of ug is typical only of thermoelastic components of displacement. Several zones with various thermoelastic stresses can be identified in the chart of horizontal displacements, uh: Uh =
(Uf + uy2
(Figs. 1%14), as for u3 (Figs. 9-111, these are: (1) areas of predominantly “northward” displacement, with a boundary between them running along the south of the Caspian Sea, the Greater Caucasus piedmonts, the Rioni Basin and the south of the Black Sea; (2) areas with high positive and negative uh gradients (alternating compression and extension zones), which outline the anomalous zones of the
Fig. 12. Horizontal displacements at the crystalline basement Cm).
M.A. ALEXIDZE ET AI.
Fig. 13. Horizontal displacements at the Conrad discontinuity {III).
Anatolian Fault, Lesser Caucasus, Eibrus, Crimea and Dagestau. Thermoelastic stresses calculated by the present model can reach quite high values, of the
order of 10” we/cm2 (‘IO9Pa). Obviously, the thermal fields and thermaelastic stresses, must exert a large ixxfluenceon other geophysical fields and geudyuamic processes.
MODEL OF THERMAL AND THERMOELEASTIC FIELDS OF THE CAUCASUS
Conelusions The present investigation has yielded the following results: (1) A three-dimensional stationary heat flow model of the Earth’s crust in the Caucasus and Black Sea areas has been designed with assumptions that allowed us to exclude surface heat flow values from the calc~ations and, therefore, to use them in testing the model’s approximation to the real conditions. (2) The heat flow model correlates well with the experimental data recovered from the Black and Caspian Sea areas and the Rioni-Kura Depression (i.e., the heat flow is close to a stationary one), and differs, as expected, in the Greater and Lesser Caucasus, where young volcanism is present. This kind of correlation over the various regions indicates that, in addition to the conductive heat transfer, other forms of heat transfer (e.g., convection) must also be considered in the Black and Caspian Sea areas, where the conductive mechanism is not enough to attain the stationary state in young sediments. (3) A three-dimensional stationary thermoelastic model of the Caucasus, Black Sea and Caspian Sea areas has been calculated and horizontal and vertical ~m~nents of stresses and displacements have been evaluated. It has been shown that thermoelastic stresses reaching lo9 Pa induce Iarge vertical (over 1000 m) and horizontal (up to 400 m) displacements in geologically inhomogeneous media.
References
Adamia,Sh.A., Balavadze,B.K., loseliani, VS. and Shengeiaya, G.Sh., 1980. Results of an integrated geoiogical-geophysical interpretation of the Caucasian region. in: Crustal Structure of Central and Eastern Europe by Geophysical Data. Naukova Dumka, Kiev. Alex&e, MA., Buachidze, G.I., et al., 1985. A three-dimensional geothermal model of the Caucasus. In: Geophysical Fields and Deep Structure of Transcaucasia. Nauka, Moscow, pp. 123-133.
203
Anderson, D.L. and Sammis, G., 1970. Partial melting in the upper mantle. Phys. Earth Planet. Inter., 3: 41-50. Balavadze, B.K., Shengelaya, G.Sh. and Mindeii, P.Sh., 1979. A gravitational model of the earth crust for the Caucasus and Black and Caspian Sea areas. In: Gravitational Models of the Earth Crust and Upper Mantle. Naukova Dumka, Kiev. Buachidze, I.M., Buachidze, G.I., et al. 1980. Geothermal Conditions and Geothermal Waters of Georgia. Sabchota Sakartvelo, Tbilisi, 206 pp. Ccrmak, V., 1983. Crustal temperature and mantle heat flow in Europe. Tectonophysics, 83: 123-142. Cermak, V. and Rybach, L., (Editors), 1982. Heat Field of Europe. Mir, Moscow, 376 pp. Etheridge, M.S., Wall, V.G., Cos, and Vernon, R.H., 1984. High fluid pressures during regional metamo~hism and deformation: apptications for mass transport and deformation mechanism. J. Geophys. Res. Gordienko, V.V., 1975. Heat Anomalies of Geosynchnes. Naukova Dumka, Kiev, p. 144. Gordienko, V.V., Zavgorodnaia, G.V. and Jakobi, N.M., 1982. Continental Heat Flow. Naukova Dumka, Kiev, 184 pp. Hurtig, E., Rugenstern, B. and Stromeyer, 1983. Three-dimensional modelling of crustal temperature and Moho heat flow in Central Europe and adjacent areas. Abstr. IUG Assembly (Hamburg~, p. 476. Kerrich, T.E., La Tourend, and Wilhnore, L., 1984. Fluid participation in deep fault zones: evidence from geological, geochemical and ‘*O/ I60 relations. J. Geophys. Res., 89: 4331-4334. Kutas, R.N., 1978. Heat Flow Field and Thermal Model of the Earth Crust. Naukova Dumka, Kiev, 146 pp. Kutas, R.N., Liubimova, EA. and Smirnova, E.V., 1978/79. A heat flow map of European part of the USSR. Pure Appl. Geophys., 117. Lebedev, T.G., Korchin, V.A., Savenko, Y., Shapoval, V.I. and Shepel, S.I., 1986. Physical Properties of Mineral Matter in Thermobaric Conditions of Lithosphere. Naukova Dumka, Kiev, 198 pp. Liubimova, E.A., 1968. The Earth’s and Moon’s Thermics. Nauka, Moscow, 278 pp. Moiseenko, U.I., Smyslov, A.A., 1986. Temperatures of the Earth’s Interior. Leningrad, Nedra, 1’78pp. Muskhelish~ii, NJ., 1935, Some Basic Problems of the Mathematical Elasticity Theory. Acad. Sci. USSR, MoscowLeningrad. Pollack, H.N. and Chapman, D.S., 1977. On the regional variations of heat flow, geotherms, and lithospheric thickness. Tectonophysics, 38: 279-296. Shengelaya, GSh., 1968. Structure of Earth Crust in the West of the R. Kura Basin. Metsniereba, Tbilisi, 177 pp. Wasow, W. and Forsythe, G., 1970. Finite-difference Methods for Partial Differential Equation. John Wiley Sons, NYLondon.