NANJING INSTITUTE OF GEOPHYSICAL PROSPECTING AND INSTITUTE OF PHYSICS PUBLISHING
JOURNAL OF GEOPHYSICS AND ENGINEERING
J. Geophys. Eng. 2 (2005) 90–96
doi:10.1088/1742-2132/2/2/002
Determination of formation temperature from bottom-hole temperature logs—a generalized Horner method I M Kutasov1 and L V Eppelbaum2 1
Pajarito Enterprises, 3 Jemez Lane, Los Alamos, New Mexico 87544, USA Department of Geophysics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv 69978, Tel Aviv, Israel 2
E-mail:
[email protected]
Received 2 March 2005 Accepted for publication 14 April 2005 Published 9 May 2005 Online at stacks.iop.org/JGE/2/90 Abstract A new technique has been developed for the determination of the formation temperature from bottom-hole temperature logs. The adjusted circulation time concept, and a semi-analytical equation for the dimensionless temperature at the wall of an infinite long cylindrical source with a constant heat flow rate, is used to obtain the working formula. It is shown that the transient shut-in temperature is a function of the mud circulation and shut-in time, formation temperature, thermal diffusivity of formations and well radius. The sensitivity of the predicted values of formation temperature to the thermal diffusivity is shown. Two examples of calculations are presented. Keywords: borehole, formation temperature, Horner method, adjusted circulation time
Nomenclature cp r rw R T Ti Tc Ts Tw Tr X t tc tc∗ q Q
heat capacity of formations radial coordinate radius of the borehole relative accuracy, per cent temperature initial (undisturbed) formation temperature circulating mud temperature wellbore temperature after shut-in wall temperature radial temperature distribution parameter (32) time circulating time at a given depth adjusted circulating time heat flow rate cumulative heat flow
Greek symbols χ
thermal diffusivity of formations
1742-2132/05/020090+07$30.00
λ t ρ
thermal conductivity of formations shut-in time density of formations
Subscripts D
dimensionless
Introduction The determination of physical properties of reservoir fluids, calculation of hydrocarbon volumes (estimation of oil and gas formation volume factors, gas solubility), predictions of the gas hydrate prone zones, well log interpretation, determination of heat flow density and evaluation of geothermal energy resources require knowledge of the undisturbed formation temperature. In most cases bottom-hole temperature surveys are mainly used to determine the temperature of the earth’s interior. The drilling process, however, greatly alters the temperature of formation immediately surrounding the well. The temperature change is affected by the duration of drilling fluid circulation,
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90
Determination of formation temperature from bottom-hole temperature
the temperature difference between the reservoir and the drilling fluid, the well radius, the thermal diffusivity of the reservoir and the drilling technology used. Given these factors, the exact determination of formation temperature at any depth requires a certain length of time in which the well is not in operation. In theory, this shut-in time is infinitely long to reach the original condition. There is, however, a practical limit to the time required for the difference in temperature between the well wall and surrounding reservoir to become vanishingly small. The objective of this paper is to suggest a new approach in utilizing bottom-hole temperature logs in deep wells and to present a working formula for determining the undisturbed formation temperature. For this reason, here we do not conduct a review and analysis of relevant publications. We will discuss only the Horner method, which is often used in processing field data. Earlier we used the condition of material balance to describe the pressure build-up for wells produced at constant bottom-hole pressure (Kutasov 1989). The buildup pressure equation was derived on the basis of an initial condition approximating the pressure profile in the wellbore and in the reservoir at the time of shut-in. It was shown that a modified Horner method could be used to estimate the initial reservoir pressure and formation permeability. For many cases, the shut-in time (the time since end of mud circulation, t) or the duration of the mud circulation period (tc ) cannot be estimated (with a sufficient accuracy) from well reports. In these cases methods of correcting the bottom-hole temperature data for the uncertainty in determining values of t and tc can be utilized (Majorowicz et al 1990, Waples and Ramly 2001, Waples et al 2004). In this paper we will consider only bottom-hole temperature logs. This means that the thermal disturbance of formations (near the well’s bottom) is caused by short drilling time and, mainly, by one (prior to logging) continuous drilling fluid circulation period. The duration of this period is usually 3–24 h. It is known that the same differential diffusivity equation describes the transient flow of incompressible fluid in porous media and heat conduction in solids. As a result, a correspondence exists between the following parameters: volumetric flow rate, pressure gradient, mobility (formation permeability and viscosity ratio), hydraulic diffusivity coefficient; and heat flow rate, temperature gradient, thermal conductivity and thermal diffusivity. Thus, the same analytical solutions of the diffusivity equation (at corresponding initial and boundary conditions) can be utilized for the determination of the above-mentioned parameters. In this study we will use a similar technique (Kutasov 1989) for determining undisturbed (initial) formation temperature from bottom-hole temperature logs. As will be shown below, by introducing the adjusted circulation time concept, a new method of determining static formation temperature can be developed.
Mathematical models The determination of static formation temperatures from well logs requires knowledge of the temperature disturbance produced by circulating drilling mud.
To determine the temperature distribution T(r, t) in formations we will consider three mathematical models to describe the thermal effect of the circulating drilling fluid. Constant bore-face temperature The results of field and analytical investigations have shown that in many cases the temperature of the circulating fluid at a given depth can be assumed constant during drilling or production (Lachenbruch and Brewer 1959, Ramey 1962, Edwardson et al 1962, Jaeger 1961, Kutasov et al 1966, Raymond 1969). In this case it is necessary to obtain a solution of the diffusivity equation for the following boundary and initial conditions: rw r < ∞ T (r, 0) = Ti (1) T (rw , t) = Tw T (∞, t) = Ti t > 0 where Tw is the wall temperature. It is known that in this case the diffusivity equation has a solution in complex integral form (Jaeger 1956, Carslaw and Jaeger 1959). Jaeger (1956) presented results of a numerical solution for the dimensionless temperature TD (rD , tD ) with values of rD ranging from 1.1 to 100 and tD ranging from 0.001 to 1000. The dimensionless temperature TD , dimensionless distance rD and dimensionless time tD are T (r, t) − Ti r χt rD = tD = 2 TD (rD , tD ) = (2) Tw − Ti rw rw where χ is the thermal diffusivity of formations. Lachenbruch and Brewer (1959) have shown that the wellbore shut-in temperature mainly depends on the amount of thermal energy transferred to (or from) formations during drilling. For this reason we present below formulae which allow us to calculate the heat flow rate (q) and cumulative heat flow (Q) from the wellbore per unit of length: q = 2πλ(Tw − Ti )qD (tD )
(3)
where λ is the thermal conductivity of formations and qD is the dimensionless heat flow rate. Analytical expressions for the function qD = f (tD ) are available only for asymptotic cases or for large values of tD . The dimensionless flow rate was first calculated and presented in a tabulated form by Jacob and Lohman (1952). Sengul (1983) computed values of qD for a wider range of tD and with more table entries. We have found (Kutasov 1987) that for any values of dimensionless production time a semi-theoretical (4) can be used to forecast the dimensionless heat flow rate: 1 qD = (4) √ ln(1 + D tD ) 2 π 1 . (5) b= √ d= D=d+√ 2 tD + b 2 π −π The cumulative heat flow from (or into) the wellbore per unit of length is given by Q = 2πρcp rw2 (Tw − Ti )QD (tD )
(6)
where cp is the heat capacity of formations and QD (tD ) is the dimensionless cumulative heat flow (Kutasov 1987). 91
I M Kutasov and L V Eppelbaum
we obtain
Cylindrical source with a constant heat flow rate In this case the transient temperature Tw is a function of time, thermal conductivity and volumetric heat capacity of formations. Analytical expression for the function Tw is available only for large values of the dimensionless time (tD ). To determine the temperature Tw it is necessary to obtain the solution of the diffusivity equation under the following boundary and initial conditions: T (t = 0, r) = Ti
∂T r ∂r
rw
q =− 2πλ
rw r < ∞
(7)
T (t, r → ∞) → Ti
t > 0. (8)
It is well-known that in this case the diffusivity equation has a solution in complex integral form (Van Everdingen and Hurst 1949, Carslaw and Jaeger 1959). Chatas (Lee 1982) tabulated this integral for r = rw over a wide range of values of tD . For the wall transient temperature we obtained the following semi-analytical equation (Kutasov 2003): √ 1 q Tw = T (t, rw ) = Ti + ln 1 + c − tD √ 2πλ a + tD a = 2.701 0505
c = 1.498 6055.
(9)
Then
2πλ(Tw − Ti ) . q
TwD (tD ) = ln 1 + c −
1 √ a + tD
r2 Ei − 4tDD . TrD (rD , tD ) = Ei − 4t1D
(16)
In this section we will show that by using the adjusted circulation time concept (Kutasov 1987, 1989) a well with a constant borehole wall temperature can be substituted by a cylindrical source with a constant heat flow rate. Let us assume that at a given depth the fluid circulation started at the moment of time t = 0 and stopped at t = tc . The corresponding values of the dimensionless heat flow rates (4) are qD (t = 0) = ∞
(17)
QD (tc ) = QD .
(18)
If the assumption is made that during the adjusted (equivalent) circulation period the heat flow rate is constant and equal to, q(t = tc ) = 2πλ(Tw − Ti )qD (tcD )
(11)
Values of TwD calculated from (11) and results of a numerical solution (‘exact’ solution) by Chatas (Lee 1982) were compared (Kutasov 2003). The agreement between values of TD calculated by these two methods was very good. For this reason the principle of superposition can be used without any limitations.
qD (t = tc ) = qD
and the values of the dimensionless cumulative heat flow are QD (0) = 0
(10) √ tD .
(15)
Adjusted circulation time
Let us introduce the dimensionless wall temperature TwD (tD ) =
1 1 TwD (tD ) = − Ei − 2 4tD
tcD =
χ tc rw2
(19)
then from (3) and (6) (using the condition of thermal energy balance) we obtain an equation for the adjusted circulation time. Q = 2πρcp rw2 (Tw − Ti )QD (tD ) = 2πλ(Tw − Ti )qD (tcD ) · tc∗
(20)
or Well as a linear source It is clear from physical considerations that, for large values of dimensionless time, the solutions for cylindrical and linear sources should converge. To develop the solution for a linear source, the boundary condition expressed by (8) should be replaced by the condition ∂T q lim r t >0 (12) =− r→0 ∂r 2πλ and the well-known solution for an infinitely long linear source with a constant heat flux rate in an infinite-acting medium is (Carslaw and Jaeger 1959) r2 q Tr (r, t) = Ti − Ei − (13) 4πλ 4χ t where Ei(–x) is the exponential integral. Introducing the dimensionless radial temperature TrD (rD , tD ) = 92
T (r, t) − Ti Tw − Ti
tD∗ =
or
∗ QD = qD tcD .
(21)
The values of qD and QD are presented in the literature (Van Everdingen and Hurst 1949, Jacob and Lohman 1952, Edwardson et al 1962, Sengul 1983). Using these parameters we obtained (Kutasov 1987) ∗ tcD = GtcD
1 G = 1 + 1+AF F = [ln(1 + tcD )]n
G= (14)
χ tc∗ λtc∗ QD = = 2 2 rw ρcp rw qD
(22)
tcD 10 n = 2/3 A = 7/8
√ ln tcD − exp(−0.236 tcD ) ln tcD − 1
tcD > 10.
(23)
(24)
The correlation coefficient G(tcD ) varies in the narrow limits: G(0) = 2 and G(∞) = 1 (figure 1).
Determination of formation temperature from bottom-hole temperature 2
G
440
Fluid temperature, oF
,
1.8
1.6
measured model
400
360
3.45 hrs going in hole shut-in
circulation at 109 gpm
320
280
0
4
8
12
16
20
Time, hours
1.4
tD 0
2
4
6
8
10
Figure 1. Correlation coefficient versus dimensionless time.
Figure 3. Circulating mud temperature at 23 669 ft (7214 m)—Mississippi well (Wooley et al 1984). Courtesy of the Society of Petroleum Engineers. Table 1. Dimensionless radial temperature TrD (rD , tD ) for a well with constant bore-face temperature, first line—(25), second line—numerical solution (Jaeger 1956).
Temperature, oF
TrD · 1000
12.177
tD /rD
1.1
1.2
1.5
2.0
3.0
5.0
7.0
10.0
2.0
912 924
834 854
642 677
418 458
172 194
22 22
1 1
0 0
5.0
934 940
875 886
726 746
543 568
310 332
97 101
26 24
2 2
10.0
945 949
896 903
771 784
614 631
404 422
180 188
77 77
18 16
20.0
953 956
912 916
804 813
668 681
481 497
266 277
148 153
59 57
30.0
957 959
919 922
820 827
694 705
520 534
314 325
194 201
93 94
50.0
961 963
926 929
837 843
723 731
564 574
370 381
253 260
144 146
14.965
Measured data 14.965 ft 12.177 ft predicted
Time, hours Figure 2. Comparison of measured and predicted circulating mud temperatures, Well 1 after Sump and Williams (1973).
Circulation period Field investigations have shown that the bottom-hole circulating (without penetration) fluid temperature after some stabilization time can be considered constant (figures 2 and 3). The solid curves in figure 2 present the calculated circulating mud temperatures (at a constant heat transfer coefficient) by using the Raymond (1969) model. We also found that the exponential integral can be used to describe the temperature field of formations around a well with a constant bore-face temperature (Kutasov 1987) r2 Ei − 4tD∗ T (r, t) − Ti cD . = (25) TrD (rD , tD ) = Tw − Ti Ei − 4t1∗ cD
In table 1 values of TrD (rD , tD ) calculated with equation (25) and results of a numerical solution are compared. The agreement between values of rD and TD (rD , tD ) calculated
by these two methods is seen to be good. It is easy to see that (16) and (25) are similar and identical at G → 1. Introducing the adjusted circulation time into (9) we obtain 1 q ln 1 + c − Tw = T (t, rw ) = Ti + GtD √ 2πλ a + GtD (26) where q = q(t = tc ).
Horner method The Horner method is widely used in petroleum reservoir engineering and in hydrogeological exploration to process the pressure-build-up test data for wells produced at a constant flow rate. From a simple semilog linear plot the initial reservoir pressure and formation permeability can be estimated. Using the similarity between the transient response of pressure and temperature build-up, it was suggested that the Horner 93
I M Kutasov and L V Eppelbaum
method be used for prediction of formation temperature from bottom-hole temperature surveys (Timko and Fertl 1972, Dowdle and Cobb 1975, Fertl and Wichmann 1977, Jorden and Campbell 1984). It is assumed that the wellbore can be considered as a linear source of heat. Santoyo et al (2000) performed an interesting thermal evolution study of the LV-3 well in the Tres Virgenes geothermal field, Mexico. Several series of temperature logs were run during LV-3 drilling and shut-in operations. The temperature build-up tests were limited to short shut-in times (up to 24 h). Static formation temperatures (SFT) were computed by five analytical methods (including the Horner plot), which are the most commonly used in the geothermal industry. The authors observed that the SFT predictions made by the use of the Horner method were always less than the temperatures provided by other methods. In the Horner method the thermal effect of drilling is approximated by a constant linear heat source. This energy source is in operation for some time tc and represents the time elapsed since the drill bit first reached the given depth. For a continuous drilling period the value of tc is identical to the duration of mud circulation at a given depth. The well-known expression for the borehole temperature is ((13) r = rw ) 1 q Ei − Tw (rw , tc ) − Ti = − . (27) 4πλ 4tcD Using the principle of superposition the following equation for shut-in temperature can be obtained: 1 q −Ei − Ts (rw , ts ) − Ti = 4πλ 4tcD + tsD 1 χ t + Ei − (28) tsD = 2 4tsD rw where t is the shut-in time. The logarithmic approximation of the exponential integral function (with a good accuracy) is valid for small arguments Ei(−x) = ln x + 0.577 22
x < 0.01.
From (28) and (29) we obtain the Horner equation tc q Ts (rw , ts ) = Ti + M ln 1 + M= . t 4πλ
(29)
(30)
Thus from a semilog plot we can obtain the undisturbed formation temperature and the parameter M. In many cases the dimensionless parameters tcD and tsD are small and (29) cannot be applied. In addition, as was shown by Lachenbruch and Brewer (1959), the heat source strength (at a given depth) while drilling might more realistically be considered as a decreasing function of time. It should also be taken into account that drilling records show that the mud is circulating for only a certain part of the time required to drill the well. The evaluation and limitations of the Horner technique are discussed in the literature (Dowdle and Cobb 1975, Drury 1984, Beck and Balling 1988). In table 2 the function TD∗ (tD ) = TD (tD ) (15) and the ‘exact’ solution of Chatas (Lee 1982) are compared. Thus we can conclude that at small values of tD (due to short drilling fluid circulation time and low values of thermal diffusivity of formations) the 94
Table 2. Comparison of the values of dimensionless wall temperaturea. tD
TDCh
∗ TwD
TDCh − TD∗
R (%)
0.4 0.8 1.0 1.4 2 4 6 8 10 15 20 30 40 50
0.5645 0.7387 0.8019 0.9160 1.0195 1.2750 1.4362 1.5557 1.6509 1.8294 1.9601 2.1470 2.2824 2.3884
0.2161 0.4378 0.5221 0.6582 0.8117 1.1285 1.3210 1.4598 1.5683 1.7669 1.9086 2.1093 2.2521 2.3630
0.3484 0.3009 0.2798 0.2578 0.2078 0.1465 0.1152 0.0959 0.0826 0.0625 0.0515 0.0377 0.0303 0.0254
61.71 40.73 34.89 28.14 20.38 11.49 8.02 6.17 5.01 3.42 2.63 1.76 1.33 1.06
∗ TDCh—‘exact’ solution, TwD —(15), R = (TDCh − TD∗ )/TDCh × 100 (%).
a
borehole cannot be considered as a linear heat source. Below we present a simple example. Let us assume that the well radius = 0.1 m, the thermal diffusivity = 0.0040 m2 h–1, the fluid circulation time = 5 h, and the shut-in time is 2 and 5 h. Then the value tD (t = 1 h) = (1 × 0.0040)/(0.1 × 0.1) = 0.4, and in (28) the corresponding values of dimensionless time are tD (t = 7 h) = 2.8; tD (t = 2 h) = 0.8; tD (t = 10 h) = 4.0; tD (t = 5 h) = 2.0. From table 2 follows that (28) and (29) cannot be used to process field data. We consider (30) only as an extrapolation formula.
The new equation Using (26) and the principle of superposition for a well as a cylindrical source with a constant heat flow rate q = q(tc ) which operates during the time t = Gtc and shut-in thereafter, we obtain a working formula for processing field data T (rw , ts ) = Ti + m ln X √ 1 + c − a+√Gt1 +t GtcD + tsD cD 1sD √ X= 1 + c − a+√tsD tsD
(31) (32)
q . (33) 2πλ The constants a and c were defined earlier (9). As can be seen from equation (31) the processing of field data (semilog linear log) is similar to that of the Horner method. For this reason we have given the name ‘generalized Horner method’ to the procedure just described for determining the static temperature of formations. It is easy to see that for large values of tcD (G → 1) and tsD we obtain the well-known Horner equation (30). To calculate the ratio X the thermal diffusivity of formations (χ ) should be determined with a reasonable accuracy. The effect of variation of this parameter on the accuracy of determining undisturbed formation temperature will be shown below. The value of χ = 0.04 ft2 h–1 = 0.0037 m2 h–1 was found to be a good estimate for sedimentary rocks (Ramey 1962). m=
Determination of formation temperature from bottom-hole temperature
Table 3. Shut-in temperatures and data used in example 1 (Schoeppel and Gilarranz 1966). ◦
No
ts (hr)
Ts ( F)
Ts ( C)
ts /tc
1 2 3 4 5 6 7 8 9 10 11 12
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0
179.50 187.80 191.92 195.37 198.13 200.20 201.58 202.27 202.96 203.65 204.34 205.03
81.94 86.56 88.84 90.76 92.29 93.44 94.21 94.59 94.98 95.36 95.74 96.13
0.333 0.667 1.000 1.333 1.667 2.000 2.333 2.667 3.000 3.333 3.667 4.000
Formation temperature (214 oF)
200
Mud temperature, oF
◦
220
180
160
Examples As will be shown by the following example, it is difficult to determine the accuracy of the Horner method in predicting undisturbed formation temperatures. Example 1. Basic data used in the example (Schoeppel and Gilarranz 1966) are shown in table 3. The example applies to a borehole to 10 000 ft (3050 m). The geothermal gradient is 1.4 ◦ F/100 ft (2.55 ◦ C/100 m) and the bottomhole circulating temperature is determined to be 145 oF (62.8 ◦ C). The undisturbed formation temperature is 214 oF (101.1 oC), the well radius is 0.329 ft (0.10 m), and the formation diffusivity is 0.0431 ft2 h–1 (0.0040 m2 h–1). Figure 4 shows the computed temperature-time relation (Schoeppel and Gilarranz 1966). We used (30) and (31) and a computer linear regression program for input data processing. The predicted values of Ti are presented in table 4. The accuracy of the Ti prediction in this example depends on the duration of the shut-in period. For example, if the shut-in period is 3 h, Ti = 101.11–95.70 = 5.41 (◦ C).
Circulation mud temperature (145 oF) 140 0
8
16
Total time, hours Figure 4. Shut-in temperatures for example 1 as given by Schoeppel and Gilarranz (1966).
At the same time the temperature deviations (R) from the Horner plot are small (table 4). Now let us assume that the thermal diffusivity of the formation is determined with an accuracy of ±20%, then for the last combination (12 points) we obtain (after (31)): Ti = Ti (χ = 0.0048 m2 h–1)–Ti (χ = 0.0040 m2 h–1) = 101.26–101.46 = −0.20 (◦ C); Ti = Ti(χ = 0.0032 m2 h–1)–Ti(χ = 0.0040 m2 h–1) = 101.71–101.46 = 0.25 (◦ C). Thus the effect of variation of thermal diffusivity of formations on the value of Ti can be estimated.
Table 4. Predicted formation temperature for example 1. (31)
(30)
Combination
Ti (◦ C)
−m (◦ C)
R (%)
Ti (◦ C)
−M (◦ C)
R (%)
1–3 1–4 1–5 1–6 1–7 1–8 1–9 1–10 1–11 1–12
101.04 101.19 101.51 101.77 101.88 101.80 101.68 101.58 101.50 101.46
37.89 38.24 38.99 39.61 39.89 39.68 39.37 39.08 38.86 38.74
0.1 0.1 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2
95.70 96.31 96.94 97.45 97.82 98.01 98.13 98.24 98.34 98.45
9.94 10.45 11.01 11.49 11.84 12.03 12.16 12.28 12.38 12.51
0.0 0.2 0.4 0.5 0.6 0.6 0.6 0.5 0.5 0.5
Table 5. Predicted formation temperature for example 2. (31) ◦
◦
(30)
ts (hr)
◦
◦
Ts ( C)
Tf ( C)
−m ( C)
R (%)
Tf ( C)
−M (◦ C)
R (%)
14.3 22.3 29.3
83.9 90.0 94.4
111.32
84.64
0.4
107.26
38.68
0.5
95
I M Kutasov and L V Eppelbaum
Example 2. This example is from Kelley Hot Springs geothermal reservoir, Moduc County of California, where the depth
is 1035 m (3395 ft) (Roux et al 1980). The parameter χ rw2 = 0.27 h–1 and tc = 12 h. The results of temperature measurements and predicted formation temperatures are presented in table 5.
Conclusions A new method of determination of formation temperature from bottom-hole temperature logs is developed. It is assumed that the circulating mud temperature is constant. A semi-analytical equation for the transient bore-face temperature during shut-in is presented. At large values of shut-in and mud circulation dimensionless time the suggested equation transforms to the Horner formula (plot).
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