D e t e r m i n a t i o n o f S u r f a c e H e a t F l o w 

i n Mazesta ( U S S R ) (*> 

H . A . L U B I M O V A , L . M . L U S O V A , F . Y . F I R S O V , 

G . N . S T A R I K O V A , A . P . S H U S H P A N O V ( * * ) 

Ricevuto il 7 dicembre 1960 

A great number of temperature measurements in deep bore-holes 
are made on the territory of the Soviet Union. The results of these mea-
surements are partly systematized in the work " Problems of Geother-
mie " t1). However they can be hardly used for the calculations of 
heat flow because of the absence of the corresponding determinations 
of the thermal conductivity of rock cores taken directly from these bore-
holes. In the present paper the temperature gradient and the thermal 
conductivity of samples of rocks for the same place are determined and 
by means of these data the value of the thermal flow is estimated. 

For temperature measurements the electric resistance thermometer 
was used t h a t is designed by Dergunov I. D. and improved for the con-
ditions of work in deep bore-holes (up to 4-5 km) at rather high pressures 
and temperatures. The electrical resistance was measured by means of 
the compensation methods by a special potentiometer and a high sensi-
tive mirror galvanometer. Four bringing wires are used for doing away 
with the erratic currents and the influence of the wires. As a result of 
it the accuracy of the temperature measurements is about 0.01°C. Fig. 1 
represents the exterior of the apparatus and the thermometer. The 
thermometer is a hollow cylinder on the surface of which a copper wire 
is winded bifllarily, whose resistance is measured. When the cylinder is 
buried into a bore-hole it is washed by a solution from outside and inside. 
Together with the application of ftorplast isolation this decreases the ther-
mal inertia of the thermometer down to 1.5 sec. For the sake of strength 
annular justs are made on the cylinder. The thermometer stands the 
pressure up to 600 a t m and the temperature up to 200°C. 

(*) P a p e r r e a d a t t h e Helsinky Assembly of t h e I.U.G.G., 1960. 
(**) I n s t i t u t e of E a r t h ' s Physics A c a d e m y of Science U S S R ) . 



1 5 8 L U B I M O V A , L U S O V A , F I R S O V , S T A R I K O V A , S I I U S H P A N O V 

l a t h e process of the determinations of t h e t e m p e r a t u r e t h e requi-
rements of t h e bore-holes measurements' methodics were strictly obser-
ved; the measurements were made by a point method in bore-holes t h a t 
were in rest for 8 to 10 months. 

As a total about 20 bore-holes with the depths f r o m 100 to 2600 m 
were investigated. Here the d a t a for the 3 bore-holes Mazesta-Hosta 

(Caucasus) are given. Temperatures were measured twice in it in 1955 a n d 
1957 and yielded t h e same results. As to its geology the region Mazesta 
refers to a great tectonic depression submerged below t h e level of t h e 
Black sea in its South-Western p a r t a n d extended in t h e direction of 
t h e general striking of t h e Caucasus for t h e length of 60 to 65 k m and 
amounting its m a x i m u m width of 20 to 25 k m . The complete formation 
of t h e depression and t h e submergence of its p a r t below t h e sea level are 
d a t e d to t h e end of the Tertiary and the beginning of t h e Q u a t e r n a r y (2). 
Pig. 4, represents t h e lithological and stratigraphical cross-section of t h e 
bore-hole 3T. 

The t h e r m a l constants of rock cores were determined b y t h e im-
pulse sound method. I n a core sample two holes are drilled a t a distance 
r from each other. One is for an electric heater and t h e other for ther-

Pig. 1 - T h e e x t e r i o r of t h e A p p a r a t u s a n d t h e r m o m e t e r for t h e m e a s u r e -
m e n t of t e m p e r a t u r e . 



D E T E R M I N A T I O N O F S U R F A C E I I E A T F L O W I N O L D M A Z E S T A 1 5 9 

mopair. The exterior of the a p p a r a t u s is represented in fig. 2. A current 
impulse is transmitted through the heater during 3 to 4 sec which creates 
a practically instantaneous source of heat since the time of the thermal 
impulse propagation from t h e source to the observations point is of the 
order of a hundred seconds. The movement of the light spot of a mirror 

Fig. 2 - P h o t o g r a p h of t h e a p p a r a t u s for t h e m e a s u r e m e n t of t h e t h e r m a l 
c o n s t a n t s of rocks 

galvanometer indicates t h e moment when the maximum temperature is 
reached a t a given point. The time interval tmax and t h e value of the 
m a x i m u m temperature 0m a x a t a given point of observation are deter-
mined and used for t h e determination of the thermal diffusivity lc 
and the thermal capacity eg simultaneously. These d a t a are applied for 
the calculation of the thermal conductivity A = Iccg. This method lias 
essential advantages over the stationary method, the so-called " divided 
b a r " method. I t does not require a through polishing of a core sample 
before the experiment, which greatly disturbs its n a t u r a l state. The 
preparation of the sample for the experiments consists only in boring of 
holes for t h e heater and t h e thermopair. The experiment takes some 



1 6 0 L U B I M O V A , L U S O V A , F I R S O V , S T A R I K O V A , S I I U S H P A N O V 

minutes. A pressing spring being a heater's spiral a t t h e same t u n e prov-
ides a close contact with the rock. This method permits to determine t h e 
heat conductivity both along and across the rocks striking and can be 
used for the determination of the thermal conductivity " in situ " . 

d i s t a n c e s r f r o m t h e source. 

The heater is a kind of a rod with the radius of 0,3 cm and t h e length 
2 1 = 5 cm. F o r calculation such a source can be considered neither point 
nor spherical. The formula for the t e m p e r a t u r e distribution a b o u t an 
infinitely long and thin source in an infinite medium was used 

where r is t h e distance between t h e source and t h e thermopair, Q is t h e 
generated heat for the unit length of t h e source. 



D E T E R M I N A T I O N OF S U R F A C E I I E A T F L O W I N O L D M A Z E S T A 1 6 1 

The values k and eg are determined from t h e conditions of t h e 
id 

extremum = 0 by t h e formulas 
o t 

r2 Q.e~ 1 
1 = it 5 « = 6 71V2 > [2] ^ 1 m a x " m a x • ' ' 

where i m a x is t h e time moment when 0 — 6max a t a given point. 
However t h e comparison of t h e experimental and theoretical curves 

0 calculated by means of formula [1] of the t e m p e r a t u r e variation with 
time (fig. 3) shows t h a t these curves are similar as to their p a t t e r n b u t 
tliey are somewhat shifted relative to the time axis a n d have different 
values of t h e t e m p e r a t u r e maximum. This difference does not disappear 
if to allow for the ultimate duration of the impulse At and the ultimate 
size of t h e sample. These effects account for the error not greater t h a n 1 
per cent if the conditions A t < 10 sec are observed and t h e diameter of 
t h e sample a t r < 3 sm. 

A more essential error is due to the assumption t h a t the heater is 
an infinite straight line. 

The integration of the function describing t h e temperature influence 
of t h e instantaneous point heat source along with the cylinder surfaces 
with t h e radius a and length 21 yields a formula 

131 

which gives the temperature distribution about a cylindrical instanta-
neous source of finite dimensions, where I„ (x) is t h e Bessel function of 
t h e zero order from an imaginary argument, erf is t h e integral of t h e errors. 
Hence t h e t h e r m a l diffusivitv k is determined graphically from t h e equa-

tion for = 0 : 
0 t 

ar _ ^ U f c w ) _ 1 . 6XP (~ ±kt) 
4 f c < m a x 2 A ^ a x y ( OT \ 2 J-' Tlkt erf / }_ ^ 

2fc<max/ \ 2 ]/ kt 

using the obtained k we calculate heat capacity 

* = t M Z G X P ( - l i a x ) e r f ( ^ f c ) [ 5 ] 



1 6 2 L U B I M O V A , L U S O V A , F I R S O V , S T A R I K O V A , S I I U S H P A N O V 

The error due to some loss of heat through t h e top surface and to 
non-absolute contacts of the heater and the thermopair with t h e rock 
can be allowed empirically. Let us consider the simplest case a n d cal-
culate the temperature curve using formula [1]. The differences of t h e 
experimental and theoretical curves of fig. 2 can be used for t h e deter-
mination of t h e correction coefficients which are inserted in formula [1] 

ciet up. czetaceous Cou/e z 
1 c S. \Tuz. s 

a, a va6anginLan -Bait em can 
1 c fcmest s-M S. €imes tone mi.croQ7on 

5k-
32-

•H 38SBBH«f 
5k-

32-
^ CO £ 

30-
28- - '05 
26- »» % ,. ... 0 26-

• 

0 
24-/ 1 . 

1 . r . 1 1 , 1 . 1 , 
-05 

200 W 600 800 M 
Fig. 4 - Lithological a n d s t r a t i g r a p h i o a l cross-section of 

t h e bore-hole 3T Old Mazesta. 

so t h a t these curves coincide within t h e given accuracy. F o r example 
the following empirical formulas can be taken 

_6_ =
 A 6 X P ( ~ ~ 4 k f ) _ JB2 _ Ae-i-Q 

Q ' 4 W ; °Q ~ 6maxnR* [ 6 ] 

where A(r), R(r) are obtained f r o m experience for the given conditions 
of an experiment and allow for all t h e possible deviations of the idealized 
scheme simultaneously. The comparison of the results obtained b y two 
methods f r o m formulas 4-5 and 6 give discrepancies not greater t h a n 



D E T E R M I N A T I O N O F S U R F A C E I I E A T F L O W I N O L D M A Z E S T A 1 6 3 

~ 5 per cent, which is a good selfcheck. An independent study of one 
and t h e same specimen of chalk b y our method and by the method of 
regular regime in another I n s t i t u t e gave a rather satisfactory coincidence 
of t h e results. 

Temperature measurements in the bore-liole 3 T Old Mazesta were 
m a d e to t h e depth of 2165 m and with t h e interval of 25 m The results 
are given in table 1. The core samples were selected in t h e near lying 
bore-hole 4 T located a t t h e distance of ~ 200 m. B o t h lithologically 
and stratigraphycally the selected core-samples correspond to rocks in 
t h e bore-hole 3 T in t h e range of 175 to 950 m. 

Three intervals with different t e m p e r a t u r e gradients answering t h e 
alternation of layers can be distinguished on the temperature curves 
(fig. 4) corresponding to these depths. The valued grad T and T0 in t h e 
formula 

T = T0 + D • grad T [7] 

is determined b y the least square method for each interval, where I ) 
is t h e depth, T is the temperature a t a given depth, T0 is the constant 
designating the value of temperature extrapolated to the level of the 
E a r t h ' s surface. 

Thus the temperature curve can be represented b y t h e following 
equations: 

a) in t h e range of 175 to 250 m 

T = 19,01 + 2,812 • 10~'D , grad T = 2,812 • 10 "4°C/cm = 28,12°(7/km 

b) in the range of 250 to 375 

T = 24,23 + 0,658 • 10' 4 D , grad T = 0,658 • 10-40C/cm = 6,58° G / km 

c) in the range of 375 to 950 

T = 22,10 + 1,264 • 10" 4 D , grad T = 1,264 • 10"40(7/om = 12,64°C/km 

The deviations of AT calculated from these formulas of the tem-
p e r a t u r e values from the observed ones are represented b y crosses in 
fig. 4. 

An increase of t h e gradient in the upper range a) m a y be connected 
with the influence of the relief and the other fa 'tors. Therefore we shal 
not t a k e into account this range of depths and calculate t h e thermal 
flow for t h e range of 250 to 950 m. 



1 6 4 L U B I M O V A , L U S O V A , F I R S O V , S T A R I K O V A , S I I U S H P A N O V 

The d e t e r m i n a t i o n of t h e average g r a d i e n t for t h e I I a n d I I I inter-
vals b y t h e least suquare m e t h o d gives f o r t h e range of 250 to 950 m 
T = 22,40 +1,224.10-4Z>, 

grad T = 12,24°C/km . [8] 

The deviation of AT2 observed t e m p e r a t u r e f r o m t h e calculated 
using formulas [8] is given in t a b l e 4, column 1. The s t a n d a r d error is 
0.11°C. 

Below t h e values of t h e h e a t c o n d u c t i v i t y of rock cores across t h e 
striking are represented. 

250-290 0,00725 
0,00653 

0,00689 Limestone of t h e T u r o n 

290-315 0,00517 
0,00594 

c p e f l H e e 0,00556 Marls of t h e senoman tufaceous s a n d s t o n e 
of t h e senoman 

315-328 0,00417 

328-375 0,00664 
0,00627 

c p e , a ; H e e 0,00646 Marls of t h e Alb a n d t h e A p t 

375-950 0,00677 
0,00666 
0,00737 
0,00658 

cpeflHee 0,00685 Limestone of t h e V a l a n g i n i a n - B a r r e m i a n 

T h e average v a l u e of t h e t h e r m a l c o n d u c t i v i t y in t h e r a n g e of 250 
to 950 m consisting of a great n u m b e r of layers was calculated b y m e a n s 
of t h e f o r m u l a 

^ DI — DK 70000 
Di A + l Dk_ 1000 57500 
Xi Xi - f 1 " h 6,89 • 1 0 - s 6,85 • 10~3 

= 6,64 • 10~3 ± o , 6 o • 10-3 . cm20C sec cm20C sec 



D E T E R M I N A T I O N OF S U R F A C E I I E A T FLOW IN OLD MAZESTA 1 6 5 

The average values of A for t h e I I and I I I intervals are equal to 

5,84 • 1 0 3 and 6,85 • 10"3 — ~ — . We obtain for the h e a t flow 
cm200 sec 

B = I grad T = 6,64 • 1(T3 • 1,22 • 1(T4 = 0,81 • 10"' — — ± 
cm2 sec 

cal 
± 0,07 • lO"6 

cm2 sec 

T a b l e No. 1 

^ D 
D e p t h T ATX AT, Z — • 1 0

5 

K at3 
m. °C °C °C cm

2 sec °C <>C 
cal 

1 7 5 2 3 , 8 2 — 0 , 1 1 
2 0 0 2 4 , 7 2 + 0 , 0 9 
2 2 5 2 5 , 4 5 + 0 , 1 1 
2 5 0 2 5 , 9 2 — 0 , 1 2 + 0 , 0 5 + 0 , 4 6 4 2 , 8 0 8 + 0 , 6 2 
2 7 5 2 6 , 0 0 — 0 , 0 4 + 0 , 2 3 4 7 , 0 8 9 + 0 , 3 5 
3 0 0 2 6 , 1 6 — 0 , 0 4 + 0 , 0 9 5 1 , 3 7 0 + 0 , 1 6 
3 2 5 2 6 , 4 1 + 0 , 0 4 + 0 , 0 3 5 5 , 6 5 1 + 0 , 0 5 
3 5 0 2 6 , 3 7 + 0 , 0 4 — 0 , 1 1 5 9 , 9 3 1 — 0 , 1 4 
3 7 5 2 6 , 3 8 — 0 , 0 2 — 0 , 1 6 — 0 , 3 1 6 4 , 2 1 2 — 0 . 3 8 
4 0 0 2 7 , 2 4 + 0 , 0 8 — 0 , 0 6 6 7 , 8 6 3 — 0 , 1 2 
4 2 5 2 7 , 6 1 + 0 , 1 4 + 0 , 0 1 7 1 , 5 1 2 — 0 , 0 5 
4 5 0 2 7 , 8 7 + 0 , 0 8 — 0 , 0 4 7 5 , 1 6 2 — 0 , 0 5 
4 7 5 2 8 , 2 0 + 0 , 1 0 - 0 , 0 1 7 8 , 8 1 2 — 0 , 0 6 
5 0 0 2 8 , 4 8 + 0 , 0 6 — 0 , 0 3 8 2 , 4 6 1 — 0 , 0 8 
5 2 5 2 8 , 8 2 + 0 , 0 9 0 8 6 , 1 1 1 — 0 , 0 4 
5 5 0 2 9 , 1 2 + 0 , 0 7 - 0 , 0 1 8 9 , 7 6 0 — 0 , 0 5 
5 7 5 2 9 , 3 5 - 0 , 0 2 — 0 , 0 9 9 3 , 4 1 0 — 0 , 1 2 
6 0 0 2 9 , 6 5 — 0 , 0 3 — 0 , 0 9 9 7 , 0 6 0 — 0 , 1 2 
6 2 5 2 9 , 8 0 — 0 , 2 0 — 0 , 2 5 1 0 0 , 7 0 9 — 0 , 2 7 
6 5 0 3 0 , 1 9 — 0 , 1 3 — 0 , 1 7 1 0 4 , 3 5 9 — 0 , 1 8 
6 7 5 3 0 , 4 9 — 0 , 1 4 — 0 , 1 7 1 0 8 , 0 0 8 — 0 , 1 8 
7 0 0 3 0 , 8 4 — 0 , 1 1 — 0 , 1 3 1 1 1 , 6 5 8 — 0 , 1 3 
7 2 5 3 1 , 2 0 — 0 , 0 6 — 0 , 0 7 1 1 5 , 3 0 8 — 0 , 0 7 
7 5 0 3 1 , 5 7 — 0 , 0 1 — 0 , 0 1 1 1 8 , 9 5 7 0 
7 7 5 3 1 , 9 0 0 + 0 , 0 1 1 2 2 , 6 0 7 + 0 , 0 3 
8 0 0 3 2 , 2 8 + 0 , 0 7 + 0 , 0 9 1 2 6 , 2 5 6 + 0 , 1 1 
8 2 5 3 2 , 6 1 + 0 , 0 8 + 0 , 1 1 1 2 9 , 9 0 5 + 0 , 1 4 
8 5 0 3 2 , 8 7 + 0 , 0 3 + 0 , 0 7 1 3 3 , 5 5 6 + 0 , 1 0 
8 7 5 3 3 , 2 3 + 0 , 0 7 + 0 , 1 2 1 3 7 , 2 0 5 + 0 , 1 5 
9 0 0 3 3 , 5 5 + 0 , 0 7 + 0 , 1 3 1 4 0 , 8 5 5 + 0 , 1 7 
9 2 5 3 3 , 8 5 + 0 , 0 6 + 0 , 1 3 1 4 4 , 5 0 4 + 0 , 1 7 
9 5 0 3 4 , 1 0 — 0 , 0 1 + 0 , 0 7 1 4 8 , 1 5 4 + 0 , 1 2 

± 0 , 1 1 ± 0 , 1 8 



1 6 6 L U B I M O V A , L U S O V A , F I R S O V , S T A R I K O V A , S I I U S H P A N O V 

In accordance with a more precise method of the determination of 
the average heat flow through a multi-layered medium [4] we h a v e 

T T 
E = ,, ; [9] 

2 DijXi 

2 Di 
Values — — for each range of depths are given in table 1, 

Xi 
column 5. The corresponding deviations AT3 of the observed value T 
from the calculated one using formula [9] are given in the last column. 

The equation for T is 

T = 21,77 + 0,82 • 10-* £ D -
»=1 M 

cal cal 
H = 0,82 • 10"6 ± 0,05 • 10-6 ~ 

cm2 sec cm2 sec 

which is very close to t h e value obtained above. We obtained t h e values 
of t h e t e m p e r a t u r e s for t h e near located bore-holes 2 T New Mazesta 
and 2 T Hosta also and determined t h e heat flow on the base of t h e 
known lithological cross-sections and d a t a on X: 

1) for t h e bore hole 2 T New Mazesta 
T = 28,58 + 0,987 • 10~4D , grad T = 0,987 • 10"1 ; H = 0.68 • lO"6 

2) for t h e bore hole 2 T H o s t a 
T = 16,95 + 1,810 • 10~4D , grad T = 1,81 • 10"4 , H = 1,14 • 10"6 . 

Then t h e average value of heat flow for Mazesta-Hosta region 
C£Jjl 

is equal (0,88 ± 0,16) • 1 0 - 6 . I t is possible there is t h e coo-
cm2 sec 

ling effect of t h e near located Black Sea and of the Mazesta's springs. 

SUMMARY 

The results of the measurements of the temperature and the thermal 
conductivity of rocks for the three boreholes Mazesta's region are described. 
The special thermometer of the resistance ivas used. The method of the mea-
surements of the thermal conductivity and capacity based upon the principle 
of the instant linear heat source. Then calculated magnitude of heat floiv 
is equal 0.88 • 10~e cal/cm2/sec. It is likely that there is the cooling 
effect of the near located Blaclc Sea and of the Mazesta's springs. 



D E T E R M I N A T I O N OF S U R F A C E I I E A T F L O W I N O L D M A Z E S T A 1 6 7 

RIASSUNTO 

Sono descritti i risultati delle misure della temperatura e della condut-
tivita termica delle roece per i 3 fori praticati nella regione della Mazesta. 
E stato usato lo speciale termometro a resistenza. II metodo delle misure della 
eonduttivita e della capacita termica si basa sul principio della sorgente 
improvvisa lineare di colore. L'ampiezza calcolata delVemissione di colore 
e uguale a 0.88 • 10~6 cal/cm2/sec. 

R E F E R E N C E S 

(x) Problems of Geothermie. T r a n s a c t i o n s of tlie All-Union Congress of geo-
t h e r i n i c i n v e s t i g a t i o n s , Acad. Sci. U S S R . (1959). 

(2) MAKARENKO F . A., On the Genesis of the Sulpheretted Hydrogen Waters of 
Mazesta. T r a n s a c t i o n s of t h e L a b o r a t o r y of Hydrogeological P r o b l e m s 
Acad. Sci. U S S R , v. 2, (1949). 

(3) BENFIELD A. E., Terrestrial Heat Flow in Great Britain, P r o c . Roy. Soc. 
173, 428-450, ( 1 9 3 9 ) . . 

(4) BULLARD E . C., Heat Flow in South Africa. P r o c . Roy. Soc. A., 173, 
474-502, (1939).