TABLES SIMPLIFY DETERMINING TEMPERATURE AROUND A SHUT-IN WELL

I.M. Kutasov Consultant Los Angeles To increase the accuracy of electric log interpretation and calculations for well cementing, tables were constructed to simplify obtaining a formation temperature around a shut-in well bore. Knowledge of temperature distribution around the well bore as a function of the circulation time, shut-in time, and the radial distance is very important for estimating the electrical resistance of formation water. Accurate temperatures will improve the quantitative

July 26, 1993

5 min read

I.M. Kutasov
Consultant
Los Angeles

To increase the accuracy of electric log interpretation and calculations for well cementing, tables were constructed to simplify obtaining a formation temperature around a shut-in well bore.

Knowledge of temperature distribution around the well bore as a function of the circulation time, shut-in time, and the radial distance is very important for estimating the electrical resistance of formation water. Accurate temperatures will improve the quantitative interpretation of electric logs.

Also, the temperature distribution around a shut-in well significantly affects cement thickening time, rheological properties, compressive strength development, and set time.

METHOD

Field and analytical investigations have shown that the circulating fluid temperature at a given depth can be assumed constant during drilling, injection, or production." This fact was used to obtain an approximate analytical solution that describes the temperature distribution in the formation around the well bore during drilling or production.2

For circulating periods, tables have been constructed that allow predicting radial temperature and the volumetric average temperature of the formation at various distances from the well bore.3 The following equation, Equation 1, was obtained using the principle of superposition. TrD is the dimensionless radial temperature of the formation during the shut-in period.

Trd =

Ei (- x1) - Ei (- x2)

_______________________ (1)

Ei (- xo)

TrD =

Tr - Ts

_______ = F (rD, tD, tsD) (2)

Tc - Ts

x1 =

rD2 rD2

____________; x2 = ____

4(taD + tsD) 4tsD

rD2

xo = ____; taD = GtD

r taD

at c ats

tD = ___; tsD = ___;

rw2 rw2

rD = __

where:

Tr = Radial temperature of formation, F.

Ts = Static (undisturbed) temperature of formation at a given depth, F.

Tc = Circulating temperature at a given depth, F.

tc = Circulating time, hr

ts = Shut-in time, hr

tD = Dimensionless circulating time

taD = Adjusted dimensionless circulation time

tsD = Dimensionless shut-in time

a = Thermal diffusivity of formation, sq ft/hr

r = Radial distance at well axis r = 0, ft

rw = Radius of the well (bit), ft

rD = Dimensionless radial distance

Ei(-x) = Exponential integral, a tabulated function

For tD 1,000

In tD

G = ________ (3)

In tD-1

For tD

A good estimate for thermal diffusivity of the sedimentary rocks is a = 0.04 sq ft/hr.5

A computer program was used to calculate functions TrD = F(rD, tD, tsD) at various dimensionless circulation and shut-in times, and dimensionless radial distances. The results are given in Tables 2-6.

EXAMPLE

A well drilled to 12,490 ft in Webb County, Tex., had a static formation temperature of Ts = 306 F. and circulating temperature of Tc = 251 F.6

If we assume that after 50 hr of mud circulation the well was shut-in for 100 hr and after that an electrical log was run near the bottom of the hole, the following steps are needed to calculate the radial temperature.

Assume a radius of investigation of r = 22 in., bit diameter of 2rw = 8.75 in., and formation thermal diffusivity of a = 0.04 sq ft/hr.

Step 1: Compute the dimensionless circulation time, dimensionless shut-in time, and dimensionless radial distance:

tD =

50 x 0.04 x (2 x 12)2

____________________

(8.75)2

= 15.0

tsD =

100 x 0.04 x (2 x 12)2

______________________

(8.75)2

= 30.0

rD =

22.0 x 2

________ = 5.0

8.75

Step 2: From Table 5 estimate the function TrD and from Equation 2 determine the value of Tr. For a tD = 15.0, tsD = 30.0, and rD = 5.0, TrD = 0.115, and therefore, Tr = 0.115 x (251 - 306) + 306 = 300 F.

Step 3: From Table 2 estimate the function TrD at r = rw. as 0.134. Therefore, Tr = 0.134 x (251 - 306) + 306 = 299 F.

Thus, the radial temperature of the formation is 299 F. at the well bore face and 300 F. at r = 22 in. The temperatures are therefore closer to the static temperature of 306 F. rather than to the circulating temperature of 251 F.

REFERENCES

Kritikos, W.P., and Kutasov, I.M., "Two-Point Method for Determination of Undisturbed Reservoir temperature," SPE Formation Evaluation, March 1988, pp. 222-26.
Kutasov, I.M., "Dimensionless Temperature, cumulative Heat Flow and Heat Flow Rate for a Well With a Constant Bore-face Temperature," Geothermics, Vol. 16, 1987, pp. 467-72.
Kutasov, I.M., "Tables can be used to determine formation temperature away from the well bore," OGJ, Jan. 16, 1989, pp. 62-63.
Taylor, A.E., "Temperatures and Heat Flow in a System of Cylindrical Symmetry Including a Phase Boundary," Geothermal Series, No. 7, 1978, p. 43.
Ramey, H.J. Jr., "Wellbore Heat Transmission," JPT, April 1962, pp. 427-35.
Venditto, J.J., and George, C.R., "Better Wellbore Temperature Data Equals Better Cement Jobs," World Oil, February 1984, pp. 47-50.