A new empirical method based on log-transformation regressions for the estimation of static formation temperatures of geothermal, petroleum and permafrost boreholes

Previous research works in the worldwide literature have demonstrated that the reliable estimation of stabilized formation temperatures (SFTs) in geothermal and petroleum systems has relevance in the evaluation of geoenergy reserves (Santoyo et al 2000, Olea-González et al 2007, Kutasov and Eppelbaum 2011). Moreover, in climatic change studies, the determination of SFT in permafrost boreholes also exhibits scientific relevance to elucidating temperature changes on the Earth, specifically those related to surface temperature increases (Lachenbruch and Marshall 1986, Harris and Chapman 1997, Huang et al 2000).

The estimation of SFT from transient bottomhole temperature (BHT) measurements provides the opportunity to determine the virgin temperatures of the surrounding formation-rock during the thermal recovery processes of the borehole. For this reason, it is conceived of as a valuable tool in terms of planning, exploration, evaluation and developing geothermal, petroleum and permafrost projects (Verma et al 2006a, Bodri and Cermak 2007, Kutasov and Eppelbaum 2010).

In this context, numerical simulators and simplified analytical methods (based on heat transfer models) were initially developed for the thermal study of petroleum boreholes under drilling and completion conditions, and later extended to both the geothermal industry and to the study of the thermal history of permafrost zones (e.g. Harris and Chapman 1997, Davis et al 2011). Numerous simulators have been specifically developed for the determination of the complete thermal history of a drilled borehole and the surrounding rock-formation under a wide variety of heat transfer models and assumptions (Pioneer works: e.g. Raymond 1969, Keller et al 1973, Beirute 1991 and recent works: e.g. Olea-González et al 2007, 2008, Olea-González and García-Gutiérrez 2008, Espinosa-Paredes et al 2009, Porkhial et al 2015, Yang et al 2015).

On the other hand, a large number of analytical methods have been reported in the same literature for the estimation of the SFT in geothermal, petroleum, and permafrost boreholes (e.g. the pioneer works by Dowdle and Cobb 1975, Leblanc et al 1981; and recent contributions such as those reported by Bassam et al 2010, Wong-Loya et al 2012, 2015; among others). In the context of practical analytical tools, several works have criticized the validity of some analytical methods for calculating SFT in these geosystems (Luheshi 1983, Drury 1984, Shen and Beck 1986, Deming 1989, Andaverde et al 2005, Espinoza-Ojeda et al 2011). These studies have demonstrated that some simplified analytical methods were derived under unrealistic heat transfer assumptions, and with a wrong use of linear regression models in the simplified solutions to estimate the SFT. Significant errors have been actually reported when these analytical methods were used to estimate the SFT using synthetic BHT data sets, where the true formation temperature (TFT) is known with accuracy (e.g. Drury 1984, Andaverde et al 2005, Verma et al 2006a). In addition, large discrepancies have also been reported among the SFT estimates predicted by several analytical methods using the same BHT data sets (e.g. Santoyo et al 2000, Espinoza-Ojeda et al 2011, Wong-Loya et al 2012).

These conceptual (physical and statistical) problems explain why some analytical methods systematically show a tendency either to underestimate (e.g. the Horner-plot method: Dowdle and Cobb 1975) or overestimate (e.g. the spherical–radial method: Ascencio et al 1994) the SFT. However, in spite of these problems, some of these analytical methods are still being used in the geothermal and petroleum industry, probably for their simplicity in the calculation of SFT (e.g. Espinosa-Paredes and García-Gutiérrez 2003, Kutasov and Eppelbaum 2005, Goutorbe et al 2007, Pasquale et al 2008, Kutasov and Eppelbaum 2010, Eppelbaum and Kutasov 2011, Vaz de Medeiros Rangel 2014, Sulastri and Andriany 2015).

To overcome the conceptual problems detected with the use of previous analytical methods, and the necessity of additional information (e.g. the thermophysical and transport properties of drilling materials: mud, cement, rock-formation, etc), which is rarely available in drilling logging, a new empirical method to estimate SFT by using only transient BHT data sets logged from geothermal, petroleum, and permafrost boreholes has been developed. The new method performs analysis of transient BHT measurements (logged during borehole shut-in conditions) by applying an innovative mathematical methodology based on logarithmic transformation regressions. The SFT is estimated after reproducing the typical transient asymptotic behavior of BHT measurements (also called thermal recovery or the shut-in process) by assuming that the formation has reached a thermal equilibrium state at infinite time. The aim of this paper is to describe the mathematical basis of the new empirical method, the numerical algorithm used, and some application examples for demonstrating the effective and reliable prediction task of the SFT.

The main goal of this work is to reproduce the thermal recovery behavior of geothermal, petroleum and permafrost boreholes from the analysis of transient BHT data logged during their drilling and completion operations. For these purposes, borehole thermal recovery (also named shut-in) occurs after either cooling or heating the surrounding rock-formation by the drilling fluid circulation in the case of geothermal–petroleum or permafrost systems, respectively. After the thermal disturbance caused by the drilling process, rock-formation temperatures begin to increase in geothermal or petroleum systems, whereas in permafrost systems, a reverse tendency is observed in the borehole temperature profiles. The typical behavior pattern obtained under thermal recovery conditions generally exhibits an asymptotic curve, which can be observed by plotting the BHT versus shut-in time data (e.g. figures 2–14). The same process can also be observed in the estimation of the initial reservoir pressure of petroleum and geothermal boreholes, which commonly exhibit a similar recovery process at infinity time (e.g. Horner 1951, Fertl and Timko 1972, Grant et al 1983, Cao and Lerche 1990, Barelli et al 1994, Aragón et al 1999, Stevens 2000).

The thermal recovery observed during borehole drilling operations has been extensively studied by heat transfer models (conductive and convective), which have been applied to derive simplified analytical methods that are commonly used to reproduce the asymptotic tendency between BHT and shut-in time data, and hence, to determine the SFT in geothermal and petroleum boreholes.

Radial, cylindrical and spherical conductive models have been used to derive simplified analytical methods (e.g. Manetti 1973, Ascencio et al 1994), whereas radial and cylindrical convective models have also been employed to obtain other analytical tools (e.g. Hasan and Kabir 1994).

Other heat transfer models have been additionally oriented to studying the asymptotic behavior of BHT and shut-in time data (e.g. Middleton 1979, 1982, Barelli and Palama 1981, Leblanc et al 1981, Lee 1982, Luheshi 1983, Jones et al 1984, Ribeiro and Hamza 1986, Shen and Beck 1986, Cao et al 1988a, 1988b, Beirute 1991, Waples and Ramly 1995, 2001, Waples and Pederson 2004, Waples et al 2004a, 2004b).

For analyzing thermal recovery in a reliable way, the number of BHT data available for performing these calculations has been one of the major concerns of these tasks. In relation to this, and as a result of the high cost of well logging operations, most of the BHT measurements logged in petroleum and geothermal boreholes are usually not numerous, and commonly available in the interval of 3–10 measurements (e.g. Pasquale et al 2008). Temperature logging is usually carried out under short shut-in times (<30 h). Longer transient BHT data are sometimes collected in exploration exceptional cases (<200 h). In the case of permafrost systems (where most boreholes are drilled for research purposes), transient BHT measurements are commonly logged at long shut-in times (from a hundred to a thousand days), but with the same problem of having a limited number of BHT measurements.

On the other hand, the asymptotic curve pattern associated with the thermal recovery process may be alternatively reproduced by means of mathematical functions such as logarithmical, exponential or polynomial models (e.g. Espinoza-Ojeda 2011, Wong-Loya et al 2012, 2015). The use of regression models based on a logarithmic transformation of variables (either x or y) for reproducing the asymptotic tendencies among them has been previously reported in the literature (e.g. Baskerville 1972, Howarth and Earle 1979). More recently, Verma and Quiroz-Ruiz (2008), Verma (2009, 2015), and Verma and Agrawal (2011) published some works which proposed the use of logarithmic transformation in polynomial regression to obtain more precise critical values to be used in discordance and significance (F- and Student's t-) statistical tests. This methodology was proposed because the critical values also exhibit asymptotic patterns that can be accurately reproduced by using a logarithmic transformation model as the most suitable regression tool both to interpolate and to extrapolate such data.

All these previous applications motivated the use of different regression models based on a logarithmic transformation of the independent variable (i.e. shut-in time) for a better fitting process of transient BHT measurements, and from this, to reproduce more accurately the typical thermal recovery behavior observed in drilled boreholes (geothermal, petroleum or permafrost). The fitting capability of these mathematical functions enabled the new empirical method to be developed as an innovated and improved analytical method. With this new regression tool the asymptotic behavior of BHT measurements with shut-in times was successfully reproduced, and finally used to determine the SFT with precision and accuracy.

The new empirical method proposes the application of a logarithmic transformation (Tln) in a regression model (GRM) that is either linear or polynomial, which can be represented by the following generalized equation:

$$\begin{eqnarray}&&y(x)\,=\,\underset{i-0}{\overset{m}{\sum}}\,{{a}_{i}}{{x}^{i}}\end{eqnarray} \tag{ 1 }$$

where y represents the dependent variable (BHT), x the independent variable (Δt or shut-in time), and a_i the coefficients of the regression model. As an initial step, three different logarithmic transformations (Tln) should be applied to the independent variables of the GRM to obtain three new improved independent variables (i.e. single (Tln1), double (Tln2), and triple (Tln3) functions which determine the new values of the independent variable: (ln Δt); ln(ln Δt); and ln(ln(ln Δt)), respectively). The main hypothesis of the new empirical method assumes that by applying linear or polynomial regression models to the BHT and the improved shut-in time data, the SFT will be estimated at infinite time conditions. Based upon the number of BHT data measurements, initially, it proposed the application of the GRM maximum order (different regression models) as the applicability criterion of the new empirical method. Based upon the fit results and the calculated residuals, the best regression model was selected to determine the SFT. These SFT estimates were compared with those SFT values predicted by some of the analytical methods commonly used in the geothermal and petroleum industry.

To derive the new empirical analytical method, a statistical–numerical methodology was proposed. The work methodology is schematically represented through the schematic flow diagram of figure 1. Basically the methodology consists of the following numerical and statistical tasks:

• (1)  
  To create a worldwide database from the BHT and shut-in time measurement logs of drilled boreholes. The database specifically consists of transient BHT data sets logged from 16 geothermal, 8 petroleum, and 3 permafrost boreholes, and 3 synthetic data sets where the TFT is known with accuracy.
• (2)  
  To apply the logarithmic transformation (Tln) functions (Tln1, Tln2, and Tln3) to the shut-in time data associated with each BHT data set, which will define the new improved value of the independent variable (equation (1)).
• (3)  
  To apply the applicability criterion to the maximum order of the GRM, which is defined as m  ⩽  (n  −  1)/2, where m and n are the maximum order of the regression model to be used, and the total number of BHT data, respectively. This first applicability criterion also allows the maximum order of the polynomial model (in the GRM) based on the total number of data sets to be evaluated. The nomenclature to determine the polynomial models to be used in the GRM (equation (1)) was as follows: linear (L); quadratic (Q); cubic (C); fourth (FO); fifth (FI); sixth (SI); seventh (SE); and eighth (EI).
• (4)  
  To calculate the GRM coefficients of equation (1) by using the regression numerical routines included in the commercial software STATISTICA (StatSoft 2003).
• (5)  
  To select the best GRM model that allows the reproduction of the BHT data with reliability, and hence, the estimation of the SFT at infinite recovery time (Δt  →  ∞) by using the following statistical criteria of evaluation:

  • Coefficient of determination (R²). From a statistical point of view, the parameter R² was used to evaluate the quality of the regression in terms of the variation between two variables (x and y) to be correlated in any regression model. This parameter is calculated for each regression model using the complete BHT data set, and a numerical value that approaches the unit will be expected when the regression is statistically acceptable (R²  ≈  1: Bevington and Robinson 2003).
  • Sum of the normalized squared residuals (RSSn). The different regression models applied to the relationship BHT–improved shut-in time (modified by the logarithmically transformed function (Tln)) were also evaluated through the estimation of the well-known statistical parameter RSSn. The best fit model will be given by the smaller RSSn numerical values. The RSSn was estimated by means of the equation:
    $$\begin{eqnarray}&&\text{RSSn}=\frac{{\sum}_{i=1}^{n}{{\left(\text{BH}{{\text{T}}_{i}}-{{T}_{i}}\right)}^{2}}}{n}\end{eqnarray} \tag{ 2 }$$
    where BHT_i and T_i are the measured and calculated (predicted by the new empirical method) transient temperatures, respectively.
  • Absolute extrapolation (Ext-Abs). A dimensionless statistical parameter that enables the accuracy of each regression model to be evaluated through the extrapolation of the last temperature measured (BHT_n) of the data set. The results of each GRM (with Tln) were compared to determine the best BHT_n predictor through the calculation of a dimensionless absolute difference between the measured BHT_n and the predicted T_n by the GRM (with Tln). In this case, the evaluation criterion establishes the small numerical value of the parameter Ext-Abs belongs to the best predictor of BHT_n. The parameter Ext-Abs was calculated as:
    $$\begin{eqnarray}&&\text{Ext-Abs}=|\frac{\text{BH}{{\text{T}}_{n}}-{{T}_{n}}}{\text{BH}{{\text{T}}_{n}}}|\end{eqnarray} \tag{ 3 }$$
  • Deviation percentage (%Dev). The dimensionless parameter %Dev is a combination of conditions used to calculate the SFT estimates and their associated uncertainty. All the different regression models (GRM with Tln) obtained in each BHT data set were simulated numerically at long times (Δt  →  ∞). In each analyzed data set, the BHT_n was used as reference data to determine the SFT. The dimensionless parameter %Dev was determined by means of the following equation:
    $$\begin{eqnarray}&&\%\text{Dev}=~|\frac{{{T}_{i+1}}-{{T}_{i}}}{{{T}_{i+1}}}|\times 100\end{eqnarray} \tag{ 4 }$$
    For the analysis of geothermal and petroleum cases, and according to the temperature magnitude, %Dev  ⩽  0.01 was used as a strict convergence criterion for estimating the SFT through the numerical simulation of each GRM (Tln) at infinite time. This means that in the case of using the value 0.01, if the numerical simulation does not accomplish the complementary conditions, the next value (0.001) will be used, and so on. The complementary condition to be accomplished in geothermal and petroleum application cases is that the SFT calculated at infinite time by the new method must be greater than the last BHT measured data (SFT(Tln)  >  BHT_n). For the analysis of permafrost boreholes, a value of %Dev  ⩽  0.001 was defined according to the BHT and long Δt data, which acts as a complementary condition for the estimation of the SFT using an inverse condition: SFT(Tln)  <  BHT_n. The general complementary condition to the systems under analysis was the simulation time (▵t_n+i) and in order to obtain the SFT by these means, the new method simulation time (Tln(Δt_n+i)) must be greater than the BHT_n shut-in time (▵t_n) measurements (Tln(▵t_n+i)  >  ▵t_n).

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For the determination of the SFT which corresponds to the BTH data sets under evaluation, the four statistical evaluation criteria were together applied both to decide the best GRM to fit the analyzed BHT and to reproduce the thermal recovery behavior of each data set. The results of the statistical evaluation parameters (R², RSSn, Ext-Abs and %Dev) were reported with the greatest possible number of digits to facilitate the final analysis of the selection. However, the SFT estimated by means of this numerical procedure was reported with the number of significant digits imposed by the uncertainty associated with the new method.

• (6)  
  For the accuracy analysis of the new empirical method, three synthetic data sets (CJON, SHBE and CLAH), were used because in these series the 'true' SFT (TFT) was known. TFT was therefore used as the reference parameter of accuracy evaluation. With this in mind, the next accuracy parameters were additionally calculated: (a) the error percentage (%Error) existing between the SFT calculated by the new method and the TFT reported for the synthetic data; and (b) the significant differences between the calculated SFT and the TFT by applying the statistical tests of F- and Student's t-.
• (7)  
  Two different types of comparative statistical analyses were finally carried out between the SFT values calculated by the new empirical method and those SFT estimates inferred from eight alternative analytical methods (already reported in the literature) using their approximated or simplified solutions with ordinary least-square (OLS) and quadratic regression (QR) models. These comparative analyses were performed by the following statistical procedures:

  • (7.1)  
    A comparative analysis of the SFT estimates between the values inferred from other analytical methods and those predicted from the new empirical method. For geothermal and petroleum borehole cases, the SFT calculated by the new method was always higher than the last BHT_n, which avoids an underestimation of the SFT, whereas for permafrost applications, the SFT estimated must always be lower than the last BHT_n; and
  • (7.2)  
    A comparative analysis between the SFT estimates predicted by the new empirical method and the mean values calculated from the SFT values inferred by the eight analytical methods. For the calculation of the mean SFT value and the associated standard deviation, a statistical normalization process was performed by using the computer software DODESYS, which uses discordant tests of univariate data to ensure Gaussian distributions of the SFT estimates (Verma et al 2008). The mean SFT estimated in each data set was finally compared with the predicted value by the new method using the F- and Student's t-statistical tests.

For the purposes of the comparative analyses (7.1 and 7.2), the following simplified analytical methods were used for the determination of SFT:

• (i)  
  the conductive radial heat source or Brennand method (BM: Brennand 1984);
• (ii)  
  the conductive–convective cylindrical heat source or Hasan–Kabir method (HKM: Hasan and Kabir 1994);
• (iii)  
  the constant linear heat source or Horner-plot method, (HM: Timko and Fertl 1972, Dowdle and Cobb 1975);
• (iv)  
  the generalized Horner or Kutasov–Eppelbaum method (KEM: Kutasov and Eppelbaum 2005);
• (v)  
  the conductive cylindrical heat source or Leblanc method (LM: Leblanc et al 1981);
• (vi)  
  the rectangular heat source or Leblanc–Middleton method (LMM: proposed by Middleton 1979 and improved by Leblanc et al 1982);
• (vii)  
  the conductive cylindrical heat source or Manetti method (MM: Manetti 1973); and
• (viii)  
  the spherical–radial heat flow method (SRM: Ascencio et al 1994).

Instead of providing details on the theory of each method in the present work, the reader is referred to the original literature source of each model or method for other physical, mathematical or assumption details.

4.1. Creation of a world database

4.2. Logarithmic transformation

4.3. Applicability criterion

4.4. Calculation of GRM coefficients

4.5. Selection of the 'best' GRM to reproduce transient temperature data (BHT) and calculation of SFT at infinite time (Δt  →  ∞)

Upon obtaining the regression equations (GRM with Tln) for each data set (equation (1)), we proceeded to evaluate them statistically through the application of the evaluation criteria, which will define the 'best' regression model that will be used to determine the SFT of the boreholes under study. With these evaluation purposes and using the same CH-A₉ data set (as an example) and a rigorous value of %Dev  ⩽  0.01 for the numerical simulation, the statistical parameters R², RSSn, Ext-Abs and %Dev were calculated (table 2).

Table 2. Statistical evaluation parameters R², RSSn, and Ext-Abs and the SFT estimated for the regression models (GRM (Tln)) of the geothermal data set CH-A₉ (using as rigorous value %Dev  ⩽  0.01 for the numerical simulation).

Model	R2	RSSn	Ext-Abs	SFT (°C)
L (Tln1)	0.995 663	0.981 68	0.001 733	156.462 92
Q (Tln1)	0.997 336	0.603 01	0.029 612	164.049 63
L (Tln2)	0.956 859	9.764 196	0.044 806	140.838 84
Q (Tln2)	0.998 68	0.298 777	0.012 476	156.518 24
L (Tln3)	0.877 831	27.650 918	0.077 915	138.960 73
Q (Tln3)	0.995 55	1.007 078	0.010 67	148.035 45

From a general view, it was observed that the quadratic models with Tln {Q(Tln1); Q(Tln2); and Q(Tln3)} have systematically much better values of R² and lower residuals (RSSn) than the linear models. In this way, the quadratic models fulfill enough of the two first two statistical criteria established by the parameters R² and RSSn, which is confirmed by observing the thermal behaviors plotted in figures 2(A), (C) and (E).

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An additional evaluation parameter was also calculated by applying the third criterion (Ext-Abs) to all the regression models. With these purposes, the value of BHT_n was determined by using the obtained regression equations {L(Tln1) and Q(Tln1); L(Tln2) and Q(Tln2); L(Tln3) and Q(Tln3)}. The resulting calculations were also shown in the plots of figures 2(B), (D) and (F).

The last selection criterion (referred to as deviation percentage: %Dev) was used as a dimensionless parameter for the determination of the SFT by using a numerical simulation of the thermal recovery patterns predicted for each GRM (with Tln). With these purposes, the last BHT measurement logged (e.g. for the CH-A₉ data set: BHT_n  =  138 °C at a shut-in time of 190.5 h) was used both to assure that the GRM (with Tln) does not underestimate the SFT with regard to the last logged BHT_n value and to calculate the SFT. For instance, given the magnitude of the geothermal borehole temperatures (logged for the CH-A₉), %Dev  ⩽  0.01 was used as the convergence criterion of the numerical simulation for predicting the SFT (i.e. the numerical prediction must fulfill (SFT_n+1  −  SFT_n)/SFT_n+1  ⩽  0.01).

For this particular case of analysis (CH-A₉), it was observed, in general, that in accordance with the increase of Tln level (from Tln1 to Tln3) in L regression models, the fitting quality of the BHT data decreases, whereas for the Q regression models a reverse behavior is obtained. However, as an exception case of the L regression model, it was observed that the predicted behavior by the regression model L(Tln1) provides a much better estimation of the extrapolated value of BHT_n in comparison with the other L or Q regression models (figure 2(B)). This result, even though seeming to contradict the previous evaluation results (see table 2), makes difficult the selection of the 'best' regression or fitting model. Therefore, a solid criterion of selection requires the largest number of evaluation criteria to come out for deciding the best regression model to finally determine the SFT.

With the integration of all these fundamental criteria and after analyzing all the results obtained for the thermal recovery behavior of the geothermal borehole CH-A₉ (table 2), it was determined that the 'best' regression model was the model Q(Tln2) because this model fulfilled, mostly, the evaluation criteria. Thus, the application of the model Q(Tln2) at infinite times allows the SFT of CH-A₉ to be determined as 157  ±  2 °C.

4.5.1. Application of the new method to analyze the thermal recovery process of geothermal boreholes.

In a similar way to the previously described numerical methodology, the geothermal borehole data sets were systematically analyzed for reproducing the thermal recovery processes and the determination of their SFT (CH-A₄, CH-A₁₁, MXCO₁, MXCO₂, ITAL, PHIL, JAPN, R #9-1, CB-1, SGIL, ROUX, KELLEY and GT-2). With these purposes, the methodology uses the applicability criterion reported in table 1.

To save space in the manuscript, a summary of the results obtained from the application of this numerical methodology is presented. Table 3 summarizes the numerical results obtained for the data sets CH-A₄ (948 m), CH-A₁₁ (2298 m), MXCO₁, MXCO₂, ITAL, PHIL, JAPN (700 m) and R #9-1 (1518 m). As we can observe in this table, the 'best' regression models that describe the thermal histories of the boreholes CH-A₄, CH-A₁₁, MXCO₁, MXCO₂, ITAL, PHIL, JAPN and R #9-1 were Q(Tln3), Q(Tln2), Q(Tln3), C(Tln1), SI(Tln3), C(Tln2), FO(Tln2) and C(Tln3), respectively.

Table 3. A summary of the numerical results obtained for the application of the new method to the geothermal data sets CH-A₄, CH-A₁₁, MXCO₁, MXCO₂, ITAL, PHIL, JAPN and R #9-1 (using as rigorous value %Dev  ⩽  0.01 for the SFT numerical prediction).

Model	R2	RSSn	Ext-Abs	SFT (°C)
CH-A4 (n  =  6, BHTn  =  169 °C)
Q (Tln3)	0.996 722	2.956 543	0.001 296	194.916 32
L (Tln2)	0.964 099	32.376 074	0.053 473	181.4573
L (Tln3)	0.892 873	96.607 761	0.108 527	175.047 61
CH-A11 (n  =  6, BHTn  =  145 °C)
Q (Tln2)	0.996 878	0.785 206	0.027 293	172.059 23
Q (Tln3)	0.996 684	0.833 766	0.005 419	158.900 02
Q (Tln1)	0.988 733	2.833 226	0.058 883	191.930 27
MXCO1 (n  =  6, BHTn  =  247.7 °C)
Q (Tln3)	0.996 613	1.460 254	0.009 664	320.090 27
L (Tln3)	0.920 687	34.194 589	0.044 714	262.985 46
L (Tln2)	0.967 773	13.894 016	0.029 139	290.549 15
MXCO2 (n  =  7, BHTn  =  247.1 °C)
C (Tln1)	0.998 045	1.017 936	0.000 701	274.976 16
C (Tln3)	0.997 908	1.088 956	0.003 577	354.613 97
L (Tln3)	0.796 159	51.260 305	0.096 799	262.197 53
ITAL (n  =  39, BHTn  =  118.7 °C)
SI (Tln3)	0.999 832	0.008 434	0.002 586	119.832 27
SI (Tln2)	0.999 82	0.009 014	0.002 446	119.702 77
PHIL (n  =  14, BHTn  =  146 °C)
C (Tln2)	0.989 219	3.375 62	0.040 132	178.8161
L (Tln3)	0.910 849	24.476 029	0.079 313	158.406 08
L (Tln2)	0.892 521	33.652 404	0.071 256	183.4237
JAPN (n  =  10, BHTn  =  170.9 °C)
FO (Tln2)	0.999 962	0.019 106	0.014 319	176.437 47
FO (Tln1)	0.999 943	0.029 189	0.186 46	171.092 71
FO (Tln3)	0.999 889	0.056 454	0.048 056	201.353 17
R #9-1 (n  =  7, BHTn  =  170 °C)
C (Tln3)	0.997 299	0.242 136	0.017 264	196.104 66
C (Tln2)	0.997 245	0.247	0.018 191	381.457 47
Q (Tln3)	0.997 153	0.255 235	0.000 013	172.920 47

Table 4 summarizes the numerical results obtained for the analysis of the geothermal data sets CB-1, SGIL, ROUX, KELLEY and GT-2, where we can observe that the 'best' fix models that describe the thermal recovery of the boreholes CB-1 (994 m), CB-1 (1494 m), CB-1 (1987 m), CB-1 (2583 m), SGIL, ROUX (1518 m) and KELLEY (1035 m) were L(Tln1), L(Tln2), L(Tln2), L(Tln3), FO(Tln3), L(Tln3) and L(Tln3), respectively. These results were represented and validated with the plots shown in figures 3–7, according to the following group distribution: figure 3 (CH-A₄ (948 m), CH-A₁₁ (2298 m), and MXCO₁); figure 4 (MXCO₂, ITAL, and PHIL); figure 5 (JAPN (700 m) and R #9-1 (1518 m)); figure 6 (CB-1 (994 m), CB-1 (1494m), CB-1 (1987m), and CB-1 (2583 m)); and figure 7 (SGIL, ROUX (1518 m), KELLEY (1035 m) and GT-2 (1595 m)). These plots include the BHT measurements, the numerical simulation of BHT through the Tln method and the extrapolation analysis (Ext-Abs) using a graphical zoom around the data reference BHT_n (see figures 3(B), (D), (F), 4(B), (D), (F), 5(B), (D), 6(B), (F) and 7(B), (F)).

Table 4. A summary of the numerical results obtained for the application of the new method to the geothermal data sets CB-1, SGIL, ROUX and KELLEY (using as rigorous value %Dev  ⩽  0.01 for the SFT numerical prediction).

Model	R2	RSSn	Ext-Abs	SFT (°C)
CB-1 (994 m) (n  =  4, BHTn  =  52.3 °C)
L (Tln1)	0.979 109	0.022 249	0.002 719	57.565 91
L (Tln2)	0.965 923	0.036 292	0.001 377	54.488 601
L (Tln3)	0.936 624	0.067 496	0.005 306	53.219 369
CB-1 (1494 m) (n  =  3, BHTn  =  65.8 °C)
L (Tln2)	0.998 433	0.012 299		78.473 125
L (Tln3)	0.992 25	0.060 828		72.811 573
CB-1 (1987 m) (n  =  3, BHTn  =  90 °C)
L (Tln2)	0.979 413	0.279 573		98.063 551
L (Tln3)	0.944 743	0.750 396		93.341 348
CB-1 (2583 m) (n  =  4, BHTn  =  102.7 °C)
L (Tln3)	0.905 541	3.555 599	0.048 389	108.479 79
SGIL (n  =  12, BHTn  =  96.13 °C)
FO (Tln3)	0.998 917	0.005 393	0.002 418	101.803 52
C (Tln3)	0.998 816	0.005 896	0.002 224	99.8027
Q (Tln3)	0.998 78	0.006 073	0.001 518	100.063 15
ROUX (n  =  3, BHTn  =  155.56 °C)
L (Tln3)	0.990 872	0.368 253		179.869 65
L (Tln2)	0.982 425	0.709 079		199.781 08
L (Tln1)	0.972 76	1.098 997		252.543 85
KELLEY (n  =  3, BHTn  =  94.44 °C)
L (Tln3)	0.989 185	0.202 301		112.233 54
L (Tln1)	0.997 895	0.039 367		145.889 75
L (Tln2)	0.994 087	0.110 598		121.598 46
GT-2 (1595 m) (n  =  77, BHTn  =  123.817 °C)
SE (Tln3)	0.999 72	0.122 01	0.000 121	123.832 32
FI (Tln3)	0.999 563	0.190 541	0.000 068	124.212 13
SI (Tln3)	0.999 587	0.179 983	0.000 135	137.162 91

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Thirty-nine BHT measurements were logged for the ITAL data set; therefore, the maximum order (EI) was achieved for the GRM. However, only two GRM models satisfied the established conditions in the four evaluation criteria.

For the data set PHIL's case, it was possible to apply the GRM of the sixth degree; therefore, numerical results with a variety of models were obtained. In the case of the JAPN and R#9-1 data sets, it was found that the three 'best' fix models were consistently polynomial models.

On the other hand, the geothermal data sets CB-1, ROUX and KELLEY showed a few BHT measurements (n  ⩽  5). Thus, in these examples it was just possible to apply the linear models with Tln. In the corresponding data sets of CB-1 (1494 m), CB-1 (1987 m), ROUX (1518 m) and KELLEY (1035 m), the criterion Ext-Abs was not applied due to its application requirements. As a consequence, only the obtained numerical results of the criteria R², RSSn and %Dev were analyzed. This showed a limitation of the new method, because only linear models can be applied.

The data set GT-2 (1595 m) is an example that has not been commonly reported in international literature, due its numerous BHT measurements (n  =  77). Therefore, it becomes an ideal case for the application of the new method or for any thermal recovery analysis on geothermal boreholes.

From the results shown in table 4, it can be observed, almost consistently, the linear models prevail as the 'best' regression models, and consequently they have been suggested as the tools to estimate the SFT whereas some exceptions were found in the case of the KELLEY and GT-2 results where the polynomial models provide the 'best' regression models to obtain the SFT.

4.5.2. Application of the new method to analyze the thermal recovery process of synthetic datasets.

Table 5 summarizes the numerical results obtained for the synthetic data sets CJON, SHBE and CLAH, whereas the predicted thermal recovery processes are shown in figure 8. The CJON data set was selected because it deals with data that correspond to low BHT and relatively shorter recovery times (Δt  ⩽  1.5 h). This example was very useful to verify the efficiency of the new method applied to this kind of synthetic BHT data. On the other hand, the SHBE data set was selected as a good example of medium BHT measurements, whereas the BHT measurements of the CLAH data set enabled us to represent a typical case of a geothermal borehole with relatively high temperature.

Table 5. A summary of the numerical results obtained for the application of the new method to the synthetic data sets CJON, SHBE and CLAH (using as rigorous value %Dev  ⩽  0.01 for the SFT numerical prediction).

Model	R2	RSSn	Ext-Abs	SFT (°C)
CJON (n  =  12, BHTn  =  19.6 °C, TFT  =  20.25 °C)
C (Tln1)	0.999 158	0.003 433	0.038 091	19.662 649
L (Tln1)	0.888 882	0.453 271	0.099 774	22.695 986
SHBE (n  =  8, BHTn  =  75.5 °C, TFT  =  80.0 °C)
FO (Tln2)	0.999 98	0.000 757	0.000 505	78.176 02
FO (Tln1)	0.999 976	0.000 910	0.002 418	76.744 824
C (Tln1)	0.999 969	0.001 146	0.000 277	77.195 069
CLAH (n  =  15, BHTn  =  119.1 °C, TFT  =  120.0 °C)
SI (Tln2)	0.999 989	0.000 622	0.001 137	119.566 18
SI (Tln3)	0.999 981	0.000 668	0.001 163	119.519 52
FI (Tln3)	0.999 981	0.000 679	0.000 365	119.578 56

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From the numerical results reported in table 5, it can be observed in the case of CJON analysis that only two models fulfilled the requirements of the evaluation criteria, and the calculation of SFT was through the application of a polynomial regression (C) model (Tln1). On the other hand, it was also found for the SHBE and CLAH synthetic data sets that the fourth and sixth polynomial regression models with Tln2 were the best fitting models, respectively.

Finally, table 6 shows the SFTs and their uncertainties calculated for each geothermal and synthetic data set, inferred by means of the GRM (with Tln). The BHT_n from each data set are included as additional data with the purpose of demonstrating that the ideal GRM (with Tln) does not underestimate the SFT.

Table 6. Summary of the obtained SFT estimations through the log-transformation (Tln) method using the geothermal and synthetic temperature data sets.

Data sets	BHTn (°C)	Model Tln	SFT (°C)
Geothermal boreholes:
CH-A4	169	L (Tln2)	181  ±  2
CH-A9	138	Q (Tln3)	148  ±  1
CH-A11	145	Q (Tln3)	159  ±  2
MXCO1	247.7	Q (Tln3)	320  ±  3
MXCO2	247.1	C (Tln1)	275  ±  3
ITAL	118.7	SI (Tln3)	120  ±  1
PHIL	146	L (Tln3)	158  ±  2
JAPN	170.9	FO (Tln2)	176  ±  2
R #9-1	170	Q (Tln3)	173  ±  2
CB-1 (994 m)	52.3	L (Tln1)	58  ±  1
CB-1 (1494 m)	65.8	L (Tln2)	79  ±  1
CB-1 (1987 m)	90	L (Tln2)	98  ±  1
CB-1 (2583 m)	102.7	L (Tln3)	109  ±  1
SGIL	96.13	FO (Tln3)	102  ±  1
ROUX	155.56	L (Tln3)	180  ±  2
KELLEY	94.44	L (Tln3)	112  ±  1
GT-2 (1595 m)	123.817	SE (Tln3)	124  ±  1

Synthetic data:
CJON	19.6	C (Tln1)	19.7  ±  0.2
SHBE	75.5	FO (Tln2)	78  ±  1
CLAH	119.1	SI (Tln2)	120  ±  1

4.5.3. Application of the new method to analyze the thermal recovery process of petroleum boreholes.

Table 7 presents the obtained numerical results for the petroleum data sets USAM (4900 m), COST, MALOOB, PPH-ABA and BECU (2700 m). The data group COST consists of three logged data sets at different depths and temperature ranges of thermal recovery. In the petroleum case MALOOB, the data sets MALOOB-456 and MALOOB-309D were used, while for the case PPH-ABA, the data set Franciacorta (3328 m) was used. Figures 9(A), (C) and (E) show the measured BHT data from COST (1420 m), COST (3710 m) and COST (4475 m), and the obtained results from the numerical simulations. Furthermore, the plots of figures 9(B), (D) and (F) correspond to the graphical representation of the Ext-Abs criterion numerical results. Finally, the BHT data from the petroleum boreholes USAM, MALOOB, PPH-ABA and BECU, as well as the BHT numerical reproduction by means of the GRM (Tln) models' computer simulation and the respective extrapolation analysis (Ext-Abs), are shown in figures 10 and 11.

Table 7. A summary of the numerical results obtained for the application of the new method to the petroleum data sets COST, USAM, MALOOB, PPH-ABA and BECU (using as rigorous value %Dev  ⩽  0.01 for the SFT numerical prediction).

Model	R2	RSSn	Ext-Abs	SFT (°C)
COST (1420 m) (n  =  6, BHTn  =  56.11 °C)
L (Tln3)	0.821 036	3.580 292	0.074 993	61.311 327
L (Tln2)	0.756 32	4.874 985	0.091 726	63.710 132
L (Tln1)	0.677 809	6.445 655	0.114 891	69.358 841
COST (3710 m) (n  =  5, BHTn  =  150 °C)
Q (Tln3)	0.984 304	2.423 206	0.032 93	165.724 55
Q (Tln2)	0.984 261	2.429 81	0.033 565	164.354 74
Q (Tln1)	0.983 694	2.517 377	0.034 948	153.060 21
COST (4475 m) (n  =  4, BHTn  =  174.44 °C)
L (Tln3)	0.956 214	0.830 434	0.004 879	188.1315
L (Tln2)	0.952 101	0.908 436	0.003 154	197.0859
L (Tln1)	0.945 707	1.029 703	0.004 879	220.199 27
USAM (n  =  14, BHTn  =  147.27 °C)
SI (Tln1)	0.999 999	0.000 004	0.000 003	149.204 34
SI (Tln2)	0.999 999	0.000 005	0.000 059	150.3086
SI (Tln3)	0.999 999	0.000 005	0.000 064	150.328 02
MALOOB—309D (n  =  7, BHTn  =  118 °C)
L (Tln3)	0.481 088	29.419 119	0.248 828	126.6137
L (Tln2)	0.470 75	30.005 256	0.2579	127.115 75
L (Tln1)	0.449 803	31.192 791	0.277 808	128.439 84
MALOOB—456 (n  =  9, BHTn  =  127 °C)
Q (Tln2)	0.948 886	22.285 757	14.168 513	139.768 06
Q (Tln3)	0.945 954	23.564 209	10.848 95	135.843 55
L (Tln2)	0.657 459	149.347 77	52.868 032	130.091 16
Franciacorta (3328 m) (n  =  4, BHTn  =  92 °C)
L (Tln1)	0.999 655	0.001 725	0.001 453	100.800 77
L (Tln2)	0.989 163	0.054 187	0.007 453	95.745 028
L (Tln3)	0.963 931	0.180 344	0.012 594	93.593 147
BECU (n  =  5, BHTn  =  72 °C)
L (Tln2)	0.962 76	0.297 923	0.016 895	86.050 726
L (Tln3)	0.931 861	0.545 114	0.023 003	78.950 547

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In a general way and in relation to the obtained numerical results from the petroleum boreholes COST (1420 m), COST (3710 m), COST (4475 m), MALOOB-456, MALOOB-309D, Franciacorta (3328 m) and BECU (2700 m), the best fix models were L(Tln3), Q(Tln3), L(Tln3), Q(Tln2), L(Tln3), L(Tln1) and L(Tln2), respectively. In the particular case of the data set USAM (table 7 and figure 10), it can be observed that there is practically no existence of numerical difference between the predictions carried out by the three 'best' fix models (SI(Tln1), SI(Tln2) and SI(Tln3)), as what happens in a lot of geothermal cases.

Table 8 shows a summary of the calculated SFTs and their respective uncertainties for each petroleum data set analyzed with the selected GRM (Tln) models.

Table 8. SFT estimations for the petroleum data sets, by means of the application of the Tln method.

Data sets	BHTn (°C)	Model Tln	SFT (°C)
Petroleum boreholes:
USAM (4900 m)	147.27	SI (Tln1)	149.2  ±  0.2
COST (1420 m)	56.11	L (Tln3)	61  ±  1
COST (3710 m)	150	Q (Tln3)	166  ±  2
COST (4475 m)	174.44	L (Tln3)	188  ±  2
MALOOB—456	127	Q (Tln2)	140  ±  1
MALOOB—309D	118	L (Tln3)	127  ±  1
Franciacorta (3328 m)	92	L (Tln1)	101  ±  1
BECU (2700 m)	72	L (Tln2)	86  ±  1

4.5.4. Application of the new method to the analysis of the thermal recovery process of permafrost boreholes.

In a similar way to the previously described numerical methodology, the permafrost borehole data sets were systematically analyzed for reproducing the thermal recovery processes and the determination of their SFT. Tables 9–11 and figures 12–14 show the numerical and graphical results for each data set of the permafrost boreholes REINDEER, MOKKA and P-RIVER, which received the same treatment of the last examples. The three 'best' fix models are listed, but for space limitations, only some of them are shown in figures 12(A), (C), (E), 13(A), (C), (E) and 14(A), (C), (E). The graphic amplification of the obtained numerical approximations of the GRM (Tln) models is shown in figures 12(B), (D), (F), 13(B), (D), (F) and 14(B), (D), (F). These permafrost exploration boreholes were selected to be analyzed due to their different ranges of 'cold' temperatures, around or less than 0 °C, and depths.

Table 9. A summary of the numerical results obtained for the application of the new method to the permafrost data borehole REINDEER (using as rigorous value %Dev  ⩽  0.001 for the SFT numerical prediction).

Model	R2	RSSn	Ext-Abs	SFT (°C)
REINDEER (18.3 m) (n  =  9, BHTn  =  −7.69 °C)
L (Tln3)	0.944 148	0.233 615	0.021 614	−9.117 11
L (Tln2)	0.929 968	0.292 926	0.005 69	−9.629 47
L (Tln1)	0.895 713	0.436 205	0.025 361	−10.931 07
REINDEER (48.8 m) (n  =  9, BHTn  =  −5.58 °C)
C (Tln3)	0.999 958	0.000 113	0.009 629	−5.664 63
C (Tln2)	0.999 934	0.000 176	0.012 824	−6.026 79
C (Tln1)	0.999 812	0.000 503	0.023 539	−12.4364
REINDEER (79.2 m) (n  =  10, BHTn  =  −5.14 °C)
Q (Tln3)	0.987 126	0.028 617	0.028 054	−5.249 55
L (Tln3)	0.925 19	0.166 289	0.149 964	−6.836 49
L (Tln2)	0.904 862	0.211 473	0.167 925	−7.175 05
REINDEER (140.2 m) (n  =  10, BHTn  =  −4.3 °C)
Q (Tln1)	0.958 276	0.073 738	0.117 861	−5.700 82
Q (Tln3)	0.957 635	0.074 87	0.129 196	−6.192 97
Q (Tln2)	0.956 732	0.076 466	0.12 805	−6.161 24
REINDEER (201.2 m) (n  =  10, BHTn  =  −3.11 °C)
Q (Tln1)	0.997 466	0.001 624	0.000 612	−3.112 16
Q (Tln2)	0.994 872	0.003 288	0.022 786	−3.188 08
Q (Tln3)	0.993 422	0.004 217	0.031 296	−3.258 95
REINDEER (292.6 m) (n  =  10, BHTn  =  −1.28 °C)
C (Tln2)	0.990 826	0.001 329	0.086 32	−1.455 95
C (Tln3)	0.989 779	0.001 481	0.093 912	−1.561 14
Q (Tln3)	0.983 646	0.002 369	0.117 03	−2.204 25
REINDEER (414.5 m) (n  =  10, BHTn  =  1.25 °C)
C (Tln2)	0.999 969	0.000 011	0.001 253	1.215 17
FO (Tln1)	0.999 973	0.000 009	0.016 733	1.194 65
C (Tln1)	0.999 967	0.000 011	0.011 554	−0.785 04
REINDEER (506 m) (n  =  10, BHTn  =  3.49 °C)
FO (Tln1)	0.999 971	0.000 009	0.000 119	3.472 45
C (Tln2)	0.999 967	0.000 01	0.001 036	3.475 19
REINDEER (597.4 m) (n  =  10, BHTn  =  5.97 °C)
C (Tln2)	0.999 853	0.000 038	0.002 746	5.9448
C (Tln3)	0.999 841	0.000 041	0.002 979	5.9529
Q (Tln3)	0.999 835	0.000 043	0.002 297	5.959 66

Table 10. A summary of the numerical results obtained for the application of the new method to the permafrost data borehole MOKKA (using as rigorous value %Dev  ⩽  0.001 for the SFT numerical prediction).

Model	R2	RSSn	Ext-Abs	SFT (°C)
MOKKA (15.2 m) (n  =  6, BHTn  =  −15.371 °C)
Q (Tln2)	0.998 955	0.015 957	0.006 378	−15.499 38
Q (Tln1)	0.998 944	0.016 138	0.014 344	−15.392 82
L (Tln3)	0.998 929	0.016 363	0.001 092	−15.378 63
MOKKA (30.5 m) (n  =  9, BHTn  =  −15.016 °C)
C (Tln3)	0.992 194	0.053 543	0.006 403	−15.116 95
C (Tln2)	0.992 187	0.053 593	0.006 250	−15.104 76
Q (Tln3)	0.992 186	0.053 596	0.002 711	−15.085 16
MOKKA (45.7 m) (n  =  9, BHTn  =  −14.629 °C)
C (Tln1)	0.998 453	0.006 363	0.005 049	−14.716 63
Q (Tln1)	0.998 269	0.007 118	0.007 963	−14.929 27
C (Tln2)	0.998 257	0.007 169	0.008 639	−14.883 71
MOKKA (61 m) (n  =  7, BHTn  =  −14.304 °C)
C (Tln2)	0.997 490	0.011 827	0.013 306	−14.386 77
Q (Tln1)	0.995 861	0.019 499	0.022 153	−14.558 68
Q (Tln2)	0.994 532	0.025 76	0.026 75	−14.829 09
MOKKA (76.2 m) (n  =  9, BHTn  =  −14.083 °C)
C (Tln2)	0.993 874	0.026 332	0.007 205	−14.126 19
C (Tln3)	0.993 392	0.028 402	0.011 565	−14.191 20
Q (Tln1)	0.992 369	0.032 802	0.018 473	−14.388 44
MOKKA (91.4 m) (n  =  8, BHTn  =  −13.808 °C)
Q (Tln1)	0.996 520	0.018 257	0.013 921	−13.970 35
Q (Tln2)	0.992 879	0.037 363	0.027 100	−14.046 11
Q (Tln3)	0.992 823	0.037 654	0.027 821	−14.083 75
MOKKA (106.7 m) (n  =  9, BHTn  =  −13.265 °C)
C (Tln2)	0.996 978	0.014 221	0.034 122	−13.298 17
C (Tln3)	0.996 577	0.016 105	0.011 845	−13.338 79
Q (Tln1)	0.994 549	0.025 648	0.023 884	−13.610 69
MOKKA (152.4 m) (n  =  7, BHTn  =  −11.277 °C)
C (Tln1)	0.996 042	0.011 100	0.003 741	−11.363 62
C (Tln2)	0.995 957	0.011 338	0.000 530	−11.292 62
C (Tln3)	0.995 937	0.011 394	0.001 889	−11.594 63
MOKKA (198.1 m) (n  =  8, BHTn  =  −9.32 °C)
Q (Tln1)	0.963 645	0.062 832	0.085 784	−9.847 98
Q (Tln3)	0.962 290	0.065 174	0.083 807	−10.054 34
Q (Tln2)	0.960 462	0.068 333	0.086 776	−10.069 75
MOKKA (320 m) (n  =  7, BHTn  =  −6.906 °C)
C (Tln3)	0.997 157	0.001 604	0.016 952	−7.499 04
C (Tln2)	0.996 862	0.001 771	0.019 815	−7.350 62
C (Tln1)	0.994 900	0.002 878	0.030 533	−6.989 53
MOKKA (441.9 m) (n  =  9, BHTn  =  −2.361 °C)
Q (Tln1)	0.943 364	0.012 715	0.019 642	−2.403 88
Q (Tln2)	0.941 163	0.013 210	0.000 115	−2.429 55
Q (Tln3)	0.940 259	0.013 412	0.006 898	−2.460 57

Table 11. A summary of the numerical results obtained for the application of the new method to the permafrost data borehole P-RIVER (using as rigorous value %Dev  ⩽  0.001 for the SFT numerical prediction).

Model	R2	RSSn	Ext-Abs	SFT (°C)
P-RIVER (15.24 m) (n  =  9, BHTn  =  −9.369 °C)
C (Tln3)	0.977 025	0.163 278	0.295 452	−9.577 25
Q (Tln3)	0.957 816	0.299 796	0.222 731	−11.302 70
Q (Tln2)	0.957 187	0.304 265	0.301 706	−10.941 26
P-RIVER (30.48 m) (n  =  9, BHTn  =  −9.167 °C)
FO (Tln3)	0.989 711	0.078 599	0.385 402	−9.257 49
FO (Tln2)	0.988 417	0.088 491	0.424 543	−9.234 46
Q (Tln3)	0.942 530	0.439 038	1.296 310	−11.666 43
P-RIVER (45.72 m) (n  =  9, BHTn  =  −9.052 °C)
FO (Tln3)	0.988 655	0.090 420	76.752 338	−9.426 17
FO (Tln2)	0.986 400	0.108 393	254.789 827	−13.156 41
Q (Tln1)	0.946 489	0.426 489	20.487 645	−9.360 02
P-RIVER (60.96 m) (n  =  9, BHTn  =  −8.957 °C)
FO (Tln3)	0.991 309	0.061 889	45.148 756	−9.234 93
FO (Tln2)	0.989 991	0.071 274	131.669 369	−10.927 99
Q (Tln3)	0.954 231	0.325 926	5.310 914	−11.086 51
P-RIVER (91.44 m) (n  =  9, BHTn  =  −8.771 °C)
FO (Tln3)	0.988 684	0.087 261	53.101 663	−9.513 73
FO (Tln2)	0.986 072	0.107 401	169.442 480	−15.726 59
Q (Tln1)	0.931 838	0.525 597	25.027 252	−9.132 18
P-RIVER (152.4 m) (n  =  9, BHTn  =  −8.124 °C)
C (Tln3)	0.978 598	0.157 363	16.835 435	−8.703 13
Q (Tln3)	0.944 509	0.408 021	8.286 078	−11.536 75
Q (Tln1)	0.908 429	0.673 308	75.034 554	−10.257 37
P-RIVER (304.81 m) (n  =  5, BHTn  =  −5.462 °C)
Q (Tln2)	0.982 172	0.083 204	21.286 175	−8.537 79
Q (Tln3)	0.959 846	0.187 405	18.710 074	−7.019 35
L (Tln1)	0.959 032	0.191 202	6.580 069	−8.195 71
P-RIVER (335.28 m) (n  =  5, BHTn  =  −4.935 °C)
Q (Tln2)	0.979 947	0.080 370	20.546 817	−7.632 48
L (Tln1)	0.967 906	0.128 630	3.930 330	−7.546 67
Q (Tln3)	0.958 627	0.165 820	16.542 837	−6.321 09
P-RIVER (396.24 m) (n  =  9, BHTn  =  −4.039 °C)
FO (Tln3)	0.984 232	0.031 941	50.830 807	−4.241 41
C (Tln3)	0.967 932	0.064 958	8.825 567	−6.880 25
C (Tln2)	0.960 977	0.079 048	30.572 812	−7.626 70
P-RIVER (579.12 m) (n  =  9, BHTn  =  −0.778 °C)
FO (Tln3)	0.998 885	0.000 985	0.866 575	−1.435 53
FO (Tln2)	0.998 834	0.001 030	2.153 146	−1.191 09
C (Tln2)	0.998 718	0.001 132	0.083 099	−1.876 62
P-RIVER (609.6 m) (n  =  9, BHTn  =  −0.195 °C)
Q (Tln3)	0.909 995	0.090 411	2.007 245	−1.796 72
Q (Tln1)	0.878 840	0.121 707	11.443 311	−0.663 68
Q (Tln2)	0.865 085	0.135 524	4.941 936	−2.029 43
P-RIVER (640.08 m) (n  =  9, BHTn  =  0.761 °C)
Q (Tln1)	0.971 860	0.095 737	6.078 248	0.520 04
Q (Tln3)	0.964 014	0.122 429	2.139 353	−1.579 50
Q (Tln2)	0.952 984	0.159 955	4.881 918	−1.695 50
P-RIVER (670.56 m) (n  =  5, BHTn  =  1.664 °C)
Q (Tln2)	0.990 178	0.062 682	2.121 075	−0.018 68
Q (Tln3)	0.989 466	0.067 225	1.458 316	−0.235 47
L (Tln3)	0.975 693	0.155 114	0.597 726	1.142 11
P-RIVER (701.04 m) (n  =  8, BHTn  =  2.885 °C)
C (Tln3)	0.992 035	0.288 610	0.414 031	2.857 82
Q (Tln3)	0.975 785	0.109 671	1.822 725	−0.060 16
Q (Tln2)	0.975 421	0.111 319	3.046 635	1.190 55

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From table 9, only one linear model provided the best results in the first case (REINDEER 18.3 m), so this model was used to determine the SFT in the corresponding section. In the remaining sections (REINDEER) and the boreholes MOKKA (table 10) and P-RIVER (table 11), polynomial models were the 'best' fix models according to the evaluation criteria. In general, relevant discrepancies are not observed between the numerical predictions provided with each of the analyzed models (figures 12, plots (C) and (E); figures 13, plots (A), (C) and (E)). Graphic–numerical discrepancies in the simulations are easier to detect in figures 12(A) and 14(A), (C), (E). Nevertheless, in the amplified plots from figures 12–14(B), (D) and (F), respectively, the emphasized differences between the results of the Ext-Abs criterion to each model can be observed; however, some of these results are not 'visible' in the numerical results of tables 9–11.

As of these last results, finally the SFTs were determined to each depth from the exploration permafrost boreholes REINDEER, MOKKA and P-RIVER, which are summarized in table 12.

Table 12. Summary of the obtained SFT for the permafrost borehole data sets REINDEER, MOKKA, and P-RIVER, by the application of the Tln method.

Data sets	BHTn (°C)	Model Tln	SFT (°C)
Permafrost borehole:
REINDEER (18.3 m)	−7.69	L (Tln3)	−9.12  ±  0.01
REINDEER (48.8 m)	−5.58	C (Tln3)	−5.67  ±  0.01
REINDEER (79.2 m)	−5.14	Q (Tln3)	−5.25  ±  0.01
REINDEER (140.2 m)	−4.3	Q (Tln1)	−5.70  ±  0.01
REINDEER (201.2 m)	−3.11	Q (Tln1)	−3.112  ±  0.003
REINDEER (292.6 m)	−1.28	C (Tln2)	−1.456  ±  0.001
REINDEER (414.5 m)	1.25	C (Tln2)	1.215  ±  0.001
REINDEER (506 m)	3.49	FO (Tln1)	3.472  ±  0.003
REINDEER (597.4 m)	5.97	C (Tln2)	5.95  ±  0.01
MOKKA (15.2 m)	−15.371	Q (Tln2)	−15.50  ±  0.01
MOKKA (30.5 m)	−15.016	C (Tln3)	−15.117  ±  0.002
MOKKA (45.7 m)	−14.629	C (Tln1)	−14.717  ±  0.002
MOKKA (61 m)	−14.304	C (Tln2)	−14.387  ±  0.001
MOKKA (76.2 m)	−14.083	C (Tln2)	−14.126  ±  0.001
MOKKA (91.4 m)	−13.808	Q (Tln1)	−13.970  ±  0.001
MOKKA (106.7 m)	−13.265	C (Tln2)	−13.2982  ±  0.0001
MOKKA (152.4 m)	−11.277	C (Tln1)	−11.36  ±  0.01
MOKKA (198.1 m)	−9.32	Q (Tln1)	−9.85  ±  0.01
MOKKA (320 m)	−6.906	C (Tln3)	−7.50  ±  0.01
MOKKA (441.9 m)	−2.361	Q (Tln1)	−2.404  ±  0.001
P-RIVER (15.24 m)	−9.369	C (Tln3)	−9.58  ±  0.01
P-RIVER (30.48 m)	−9.167	FO (Tln3)	−9.2575  ±  0.0001
P-RIVER (45.72 m)	−9.052	FO (Tln3)	−9.43  ±  0.01
P-RIVER (60.96 m)	−8.957	FO (Tln3)	−9.2349  ±  0.0001
P-RIVER (91.44 m)	−8.771	FO (Tln3)	−9.51  ±  0.01
P-RIVER (152.4 m)	−8.124	C (Tln3)	−8.7031  ±  0.0001
P-RIVER (304.81 m)	−5.462	Q (Tln2)	−8.5378  ±  0.0001
P-RIVER (335.28 m)	−4.935	Q (Tln2)	−7.6325  ±  0.0001
P-RIVER (396.24 m)	−4.039	FO (Tln3)	−4.241  ±  0.004
P-RIVER (579.12 m)	−0.778	FO (Tln3)	−1.436  ±  0.001
P-RIVER (609.6 m)	−0.195	Q (Tln3)	−1.797  ±  0.002
P-RIVER (640.08 m)	0.761	Q (Tln1)	0.520  ±  0.001
P-RIVER (670.56 m)	1.664	Q (Tln2)	−0.018 68  ±  0.000 02
P-RIVER (701.04 m)	2.885	C (Tln3)	2.8578  ±  0.0003

After the analysis of all the cases of geothermal, petroleum and permafrost application, it is demonstrated that the coefficient of determination (R²), the residuals normalized (RSSn), and the dimensional parameters of extrapolation and simulation convergence (Ext-Abs and %Dev, respectively) are fundamental for the evaluation of the regression models and log-transformation (Tln) applied together to each data set. These parameters are statistical criteria used to define the best fix model that describes the behavior of the BHT and shut-in time. This process has allowed us to indicate that each evaluation criterion is complementary to the others, a requirement that guarantees the effectiveness and reliability of the choice of the best models that will represent the analyzed temperature data sets with the new method.

After analyzing the results related to the best R², the lower residuals, and the most accurate numerical extrapolation and simulation (tables 2–5, 7 and 9–11), it is inferred systematically that the best fix models used together with the Tln of shut-in time were given by the achieved predictions with the polynomial models (e.g. Q(Tln1), Q(Tln2), Q(Tln3), ..., EI(Tln1), EI(Tln2), EI(Tln3)). On and after the analysis of 17 geothermal temperature data sets, 8 were described through linear models (40%), while the remaining percentage were better analyzed by polynomial models (60%). In the eight petroleum cases, five were better represented by linear models (63%), than by the difference obtained with polynomial models (37%). Finally, in the analysis of the permafrost borehole data, only 1 of the 34 data sets was described by a linear model (3%) and the 97% were obtained successfully by polynomial models. In this way, and by doing a global analysis, it was found that from the 62 analyzed cases of thermal recovery, only 17 were described by linear models (27%) whereas 45 cases were better represented with non-linear models.

Therefore, it is concluded that the processes of thermal recovery exhibited by most of the borehole and synthetic temperature data sets are better described by polynomial models. It was found that the larger the number of measurements, the higher the order of the polynomial that will describe this physical phenomenon. In fact, these observations are in agreement with some of the results shown in early works, which mention that the analysis of the thermal recovery process of drilled boreholes and consequently the most reliable estimation of SFT are achieved by means of the use of polynomial regression (quadratic) that must be applied in some analytical methods (e.g. Andaverde et al 2005 and Espinoza-Ojeda et al 2011).

Finally, an important observation was given in the numerical simulation of the fix model GRM (Tln) for the estimation of SFT. It was found that the application of %Dev  ⩽  0.01 (0.01; 0.001; ...), as a convergence criterion in the numerical simulation of the fix models, was very suitable and satisfactory for the cases of boreholes with temperatures higher than 100 °C (e.g. the synthetic data set CLAH). However, in the case of medium or low BHT (BHT  ⩽  100 °C; e.g. the synthetic data sets CJON and SHBE), the value was not enough; therefore, it suggests that we define this parameter with a low value due to the orders of magnitude from the borehole temperatures (e.g. %Dev  ⩽  0.001).

4.6. Accuracy analysis between the estimated SFTs

In this last section, an analysis of the synthetic temperature data was conducted where the SFT obtained by the log-transformation (Tln) method was compared with the TFT reported in the synthetic data. So, with this purpose, the synthetic data CJON, SHBE and CLAH were used; the obtained results with the new method were then compared with the estimated SFT by the analytical methods using their approximate solutions with OLS and QR models. In the accuracy evaluation, the following statistical tests were applied: (i) the error percentage (%Error) between the estimated SFT and the TFT; and (ii) the statistical tests F- and Student's t- for significance analysis between the calculated SFT and TFT. The accuracy analysis results are shown in figure 15. The comparison between the estimated SFT and TFT through the error percentage (%Error) is shown in figures 15(A)–(C), while the results of the F- and Student's t-tests are displayed in figures 15(D)–(F). In both figures, the new method results are represented by an empty triangle symbol, while the uncertainties are shown with error bars (see figures 15(D)–(F)). As a reference the calculated SFTs by the analytical methods are included.

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After analyzing the global results of these tests, it was found that only in the case of CLAH (see figures 15(C) and (F)) did the new method provide a most suitable estimation with a lower error percentage of 0.36%. The SFT estimated with the new method (SFT  =  120  ±  1 °C: SI(Tln2)) was compared with the reported TFT (120 °C) through the F- and Student's t-tests. The obtained results from this analysis do not show such significant differences that make the null hypothesis (H₀) acceptable. The deviation percentages found by the GRM (Tln) models (figures 15(A) and (B)) are inside of the  ±3% of the average deviation.

In the case of the data set CJON (TFT  =  20.25 °C; SFT  =  19.7  ±  0.2 °C: C(Tln1)), the new method provides predictions with the highest average error of 2.9%, when it is compared with the TFT (figure 15(A)), while with the SHBE (TFT  =  80 °C; SFT  =  78  ±  1 °C: FO(Tln2)), the comparison error with the TFT was 2.3% (figure 15(B)). Also, it can be observed that lower error percentages (between  −3.1% and 2.5%) were obtained for the data set CLAH (figure 15(C)), while the highest values (between  −10% and 8%) were reached for SHBE (figure 15(B)). Finally, the results of significance tests of the data sets CJON and SHBE applied to the SFT estimations and TFT (figures 15(D) and (E)) do not show significant differences with a 95% confidence analysis. These results are in total agreement with the results shown in figures 15(A)–(C).

4.7. Comparative analysis between the estimated SFTs

A new practical method based on logarithmic transformation regressions was successfully developed for the determination of the SFTs of geothermal, petroleum and permafrost boreholes. The new method involved the application of multiple linear and polynomial (from quadratic to eight-order) regression models to BHT and log-transformation (Tln) shut-in times. The best regression model was used for reproducing the asymptotic thermal recovery process of the boreholes with accuracy, and later used for the reliable determination of the SFT.

A geochemometric evaluation methodology was applied for demonstrating the efficiency and prediction capability of the new log-transformation method. Four statistical parameters (R², RSSn, Ext-Abs and %Dev) were successfully applied for the evaluation of the linear and non-linear regression models.

It was found that the temperature measurements and the shut-in times corrected by the log-transformation (Tln) were mostly better reproduced through non-linear regression models (e.g. Q(Tln1), Q(Tln2), Q(Tln3),..., EI(Tln1), EI(Tln2), EI(Tln3)). Nevertheless, in some particular cases where the thermal recovery shows quasi-linear tendencies, the linear regression models with Tln appeared as the best fitting tool.

All these assertions were supported based on the results of the 'best' coefficients to determine the regression models (R²), lower residuals (RSSn), numerical extrapolation (Ext-Abs) and more accurate simulations given by the evaluation parameter %Dev.

It is important to emphasize that in the geothermal or petroleum borehole analysis, the definition criterion of the convergence parameter for the new method was %Dev  ⩽  0.01 for temperature data sets above 100 °C, whereas for temperatures in the interval 10  ⩽  BHT (°C)  ⩽  100, a value of %Dev  ⩽  0.001 was adopted. In the case of permafrost systems, the parameter %Dev  ⩽  0.001.

As part of the new numerical methodology developed, it was also found that the availability of data sets with a high number of measurements (n  >  10) provides the possibility to obtain a major number of regression models (GRM (Tln)) for a much better analysis, which will provide the most reliable SFT estimations. It is important to note that the number and the quality of the measured data also play an important role for a more efficient application of the new method.

According to the accuracy analysis, it was demonstrated that the SFT predictions from the new developed method present average deviations less than 3%.

The difficulties that prevent an integral evaluation and a better selection of analytical and empirical methods are basically due to the following factors: (i) the limited number of BHT measurements gathered in the different applications (in order to be highly desirable there must be at least 30 measurements, a situation that can be viably reached with current measuring technology, such as optical or digital fiber); (ii) the limited number of synthetic data sets of different temperature ranges, whether high, medium or low, and knowledge of their TFT for a more accurate analysis; and (iii) reassessment of some of the physical—conceptual—and mathematical models of simplified analytical methods due to the presence of complex heat transfer phenomena that form part of these thermal disturbance processes.

Inside this context of difficulties and limitations, the Tln method offers several technical and practical advantages over the existing analytical methods, among which the following stand out: (i) knowledge of circulation time is not required, as it is a physical parameter that is very hard to determine with high precision and accuracy under field conditions; (ii) neither are the thermophysical and transport property data from the formation, cementation, mud and drilling tubing required; and (iii) the input data on the BHT and thermal recovery time (shut-in time) measurements are also required.

Finally, we can also conclude that the newly developed empirical method (Tln) for SFT calculation in geo-systems achieves the expected theoretical conditions and results, and that it makes for a practical, useful and effective tool to use in these kinds of tasks. Also it can have additional applications in the development and calibration of numerical simulators to reproduce the thermal histories of boreholes and/or the surrounding formations.

The first author wants to thank CONACyT (México) for the PhD, postdoctoral and the Chair-Program scholarships. The authors want to give special thanks to Dr S P Verma, for all the intellectual contributions and technical suggestions, which greatly helped them to improve this work. The authors are also grateful to the anonymous reviewers for their helpful and constructive comments on an earlier version of the paper. Finally, the second author also acknowledges the CONACyT fellowship program and the CIICAp-UAEM Institution for supporting his sabbatical year.