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A Note on the Theory of Temperature Logging

Interest in temperature logs has been renewed

recently. One of the main probiems of temperature

logs in injection wells is that of determining the

zones that are taking fluids. A great step toward

solving this problem has been reported in a recent

paper. 1

The purpose of this paper is to point out another

aim of temperature logging—namely, that of relating

the flow rate in water injection wells to some

characteristics of the temperature logs. It has been

statedz that a factor of 6:1 gives approximate

vaiues in converting A into B/D.

~~e .-.,..

,~~.u~ .fi,

which has been found empiric ally, may be explained

from theoretical considerations and because of this,

it may be estimated more accurately.

It has been showns that, for flow of a liquid

a’~w Tw -

T~

=0,......... 1

dz

A

where

~ =

WC[k

rlu~(~)l

. . . . .

(2)

2f771uk ‘ ““”

a quantity that is different from zero.

Eq. 1 can be written as

Te-Tw=A

. . . . . . . . . . . .

3

grad Tw ‘ “

,

which shows that A,

as is defined in Ref. 2, is

identical to A.

For injection down casing, the over-all heat

transfer coefficient, U, may be considered infinite.

Therefore,

4

A =

we/ t

— . . . . . .

2nk

. . . . . . . . .

(4)

Considering the wellbore as a linear point source,

or, if

A= FQ, . . . . . . . . . . . . . . “(6)

()

2

F.–~Ei – .... . . ... 7)

It has been observeds that surprisingly good

results are obtained by using the values

72 L n... IA. X,.4*.OF

~ = ,,. ” “LU, U-J-.. .

and

a = 0.96 sq ft/day

for different locations. Taking the values

~ = 350 lb/bbl, c = I Btu/lb-°F,

It should be noted that the lower curve of Fig. 1

of Ref. 3 does not agree, for low values of t,with

the solution

u

t =-~ Ei ’ ~ ’ , . . . . . . . . . 9

4at

corresponding to the constant heat flux line source

4

and for this reason the graph should be used with

caution.

Ea. 8 has been piotteri in Fig. 1

foi thi~~ Yri ’ues

of th; external radius r2’. It may be used to estimate

the rate of water injection down casing from the

shape of the injecting temperature log above the

zone of entry of fluids.

10,

I

I

\

I I I Ill I I I I 11111

.

0

+

“

*

L

-,

I

A-FQ

Q - ,NJECTION RATE

I

, ,

I

I

~= T.-T.

I

5

rod

:,, , 1

T.-RESEF

INJECTION

TIME

9AYS

FIG. 1 —

THE FACTOR F AS A FUNCTION OF TIME.

DECEMBER, 1969

.Ffl—.

876


2/3

Example: Fig.

2 shows typical temperature

(injecting and shut-in), microlog and spinner logs

of a well in the San Andres oil field. This is a Gulf

Coast field in which the productive formation is art

oolitic limestone of Jurassic age. Pay thickness is

about 70 m or 230 ft.

h Fig. 3 the measured geothermal gradient and

temperature log injecting

475

cu m/day, or 3,000 B/D

a~e shown.

The geothermal gradient for this area

was obtained as an average of five determinations

in

four “ells,

“ti5itig

c ectr~ca resistance,

continuous

thermometers

and point measured

temperature thermometers. The vaIue of the

geothermal gradient is 0.0314°C/m or 0.0 H32°F/ft.

Injecting and geothermal temperatures at 3,100-m

depth (10,17o ft), according to Fig. 3,

are

51.5° C

and 115.3°C, respectively. Measured injecting

temperature gradient turns out to be 0.00875 ”C/m.

Then,

Te -

Tw ~

115.3 - 51.5 %

grad

Tw

0.00875

OC/m

= 7,291m = 23,914ft

(Im = 3.28 ft).

Injection time, down 6

5/8-in.

casing, is about

1,140 days, then by Fig. 1, or by a straightforward

calculation

F =

8.6 ft/(B/D), and therefore:

~ = 29,914ft

= 2>775 B/D .

8.6 ft/(B/D)

I

I

, 00 Wmrn,

s’ ‘— –_ .-

319 ‘,. ., ..”,

~

—

‘“+.__.

-—..=.

3xc Ucrc”s

-— s, - — m ‘Y

,0

FIG. 2 —

TEMPERATURE, MICROLOG AND SPINNER

LOGS IN WELL A, SAN ANDRES FIELD.

The estimation is considered satisfactory for

secondary

recovery

project calculations since the

difference between measured and calculated

rates

of injection is 7 percent. Similar results were

obtained for a number of wells.in the San Andr6s

and in other oil fields. Should the value of

F

be

takenas 5,

he value of Q would have been 3,875

B/D, or a difference of 29 percent of the measured

rates of injection. What is important is that the

previously

empirically determined value of the

factor Fcan

be predicted by the transient heat flow

theory and that Fcan

be”more ~~~iii~i~l~

calculated.

Calculation of F from Eq. 8 can be made readily

by means,of Fig. G.8 of Ref. 5 or, for long times of

injection, by

means

of the well known approximation:

Ei -x)=lnx+O.3772.

. . . . . . .. (10)

For injection down tubing, according to Eqs. 2

and 6, the factor

F

is given by

F = 1.658

1 + p/(t)

. . . . . . . . . .

(11)

P

wherein

B=+ .. . . . . . . . . . . .

..-(lz)

In Ref. 3, /(t) is given as a function of the

injection time, the casing external radius and @ as

a parameter. For times longer than a week, all

calculated values of /(t) under different imposed

boundary conditions are coincident.

From Eq. 1 it may be seen that if

Tw = Te,

grad

Tw

is zero. In other words, if the geothermal

gradient line eventually crosses the injecting

\

‘\\

I

L ,

\

-d ---- . --- . . -

. \ -—

,\

_.._-

& & &

To. 32C5 0

io w & 100 1;0 120 l-m

TEMPERATuRE- “C

FIG. 3

— TEMPERATURE LOG OF WELL A, SAN

ANDR ES F IE LD.

S76

sOCIETY OF PETROLEUM ENGINEERs J OURNAL


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temperature profile, the tangent to this curve at the

point of interception must be vertical. This property

may be used t o better delineate the temperature

profile when enough temperature readings are not

available.

A=

c=

Ei =

F=

/ t) =

k=

Q=

71 =

NOMENCLATURE

function defined by Eq. 2, ft

specific heat of water, Btu/lb-°F

transcendental mathematical function

proportionality factor defined by Eq. 6, ft/

(B/D)

transient heat conduction time function

thermal conductivity of earth, Btu/day-ft-°F

volume injection rate, B/D

inside radius of tubing, ft

r2 ’ =

outside radius of casing, ft

Tw =

temperature of water in the well

Te =

geothermal temperature

t = time of injection, days

U =

over-all heat transfer coefficient, Btu/day-sq

ft-°F

W =

weight injection rate, lb/day

. . ....4--C. (,

z = aepdi b~b >U..ak. , ..

a = thermal diffusivity of earth, sq ft/day

A=;a; ;;, ft

p = specific gravity of water, lb/bbl

1 .

2 .

3 .

4 .

5 .

REFERENCES

Cocanower,

R. D. ,

Morris, B. P. and Dillingham, M.:

“Comtmterized Temperature Decay — An As se t t o

Tem pe r a t u r e Loggin g”, J . Pe t . Tech. Aug., 1969 )

933-941 .

Bir d , J . M.:

s

lInte~re@tiOn

of Tem pe ra t u re Logs in

Wat er an d Gas In jec t ion We lls an d Gaa Produ c in g

Wells”,

Dr i l l . a nd Pr od . Pr ac ., API 1954 ) 187 .

Ram ey, H. J . , J r . :

c~Wellbore Hest Transmiss ion” ,

]. Pet . T ech . p ril, 1 96 2) 4 27 -4 35 .

MOSS, J . T. an d Wh it e , P. D.: “How t o Ca lcu la t e

Tem per a t u re Pro file s in s Ws t e r In je c t ion We ll”, Oil

Gas J . March 9 , 1 95 9 ) Vol. 5 7 , 1 74 .

Ma t t h ews , C. S. an d Ru sse ll, D. G.:

Pf essu r e B ui l du p

a nd F l ow T est s i n W el l s , Mon ogr aph Se r ie s , c ie t y of

Pe t roleu m En gin ee r a , Da lla s , Text 19 67 ) Vol. 1 , 1 63 .

ANTONIO ROMERO-JUAREZ

Petr61eos Mexicanos

Mexico City, Mexico

DECEMBER. 1 9 6 9

S7 7

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