A Precise Prediction Method for Subsurface Temperatures Based on the Rock Resistivity–Temperature Coupling Model

Abstract

The accuracy of deep temperature predictions is critical to the precision of geothermal resource exploration, assessment, and the effectiveness of their development and utilization. However, the existing methods encounter significant challenges in predicting the distribution characteristics of deep temperature fields with both efficiency and accuracy. Many of these methods rely on empirical formulas to approximate the relationship between geophysical parameters and temperature. Unfortunately, such approximations often introduce substantial errors, undermining the reliability and precision of the predictions. We present an advanced prediction methodology for deep temperature fields based on the rock resistivity–temperature coupling model (RRTCM). By converting the fixed parameters in the empirical formulas to variables dependent on the formation depth, we establish a dynamic model that correlates rock resistivity with temperature on the basis of limited constrained borehole data. We then input the 2D magnetotelluric inversion results into the model, and the subsurface temperature distribution can be predicted indirectly with high precision on the basis of the resistivity–temperature coupling relationship. We validated this method in the Xiong’an New Area, China, and the determination coefficient ($R^2$) of maximum temperature prediction reached 98.88%. The sensitivity analysis indicates that the prediction accuracy is positively correlated with the number and depth of the constrained boreholes and negatively correlated with the sampling interval of the well logging data. This method robustly supports geothermal resource development and enhances the understanding of geothermal field formation mechanisms.

1. Introduction

Geothermal energy is a clean, efficient, and stable form of green energy, and the precision of geothermal resource exploration directly affects the development and utilization of such resources [1]. Accurate prediction of the subsurface temperature is crucial in enhancing the efficiency and precision of geothermal resource exploration approaches, providing an essential reference and guidance for the development of geothermal energy techniques [2,3]. Consequently, the accurate prediction of subsurface temperatures has become an important research topic. The variation in rock resistivity is closely related to temperature changes, offering a theoretical basis for estimating temperature on the basis of the rock resistivity [4,5,6,7]. However, owing to the influence of complex factors such as lithology, rock formation structure, and geological background, the coupling relationship between rock resistivity and temperature is uncertain and complex [8,9,10]. Currently, subsurface temperature calculation methods primarily employ empirical formulas characterizing the rock resistivity–temperature relationship [11,12,13,14]. While such methods do not require complex calculations and allow quick estimation of deep temperatures, the two-dimensional distribution of the subsurface temperature cannot be calculated accurately; thus, such methods are useful only in local qualitative temperature analyses. Additionally, artificial neural networks (ANNs) have been widely applied to analyze correlations between rock temperature and resistivity and predict subsurface temperatures [2,15,16]. ANNs can be used to obtain accurate subsurface temperature predictions when sufficient borehole samples are available [17,18]. However, the prediction accuracy of ANNs depends on the depth of the training boreholes, and the use of shallow borehole training samples results in poor temperature prediction outcomes [19]. In recent years, Huang et al. [20,21] optimized empirical formulas characterizing the relationship between rock resistivity and temperature to predict deep subsurface temperature fields. This method can be used to obtain accurate subsurface temperature predictions with a limited number of constrained boreholes. However, in Huang et al. [20], the constrained borehole data were segmented, leading to significant errors when establishing temperature–resistivity coupling models for different strata. Furthermore, the key parameters of the two selected empirical formulas are different, complicating the calculation steps. Additionally, the lack of correlation between the two empirical formulas affects the accuracy of the temperature predictions. To address these issues, we optimized two correlated empirical formulas to predict subsurface temperatures precisely. The subsurface resistivity results were obtained on the basis of the magnetotelluric (MT) data, which serve as the basis for predicting deep stratigraphic temperature fields [22]. Initially, borehole temperature–resistivity data pairs from the study area or adjacent areas were normalized. Then, we calculated the temperature compensation coefficient (TCC), the resistivity compensation coefficient (RCC), and the temperature–resistivity correction coefficient (TRCC) at different depths with the temperature–resistivity coupling model. ANNs were used to derive the relationships between these three coefficients and depth. Moreover, we performed 2D MT inversion for the study area and normalized the obtained resistivity inversion results. Finally, combining the resistivity–temperature coupling model and the normalized resistivity from the MT inversion process, we precisely predicted the 2D subsurface temperature fields in the study area. Our proposed method, the rock resistivity–temperature coupling model (RRTCM), can accurately convert the macroscopic resistivity characteristics of a subsurface medium into an intuitive temperature field distribution, offering advantages such as deep prediction depth, wide range, and high accuracy.

2. Materials and Methods

2.1. Introduction to Empirical Formulas

The physical properties of rocks are significantly affected by temperature, with their resistivity showing a complex temperature dependence during freezing and thawing under different humidity conditions. The proposed geothermal field prediction method is primarily based on two sets of empirical formulas introduced by Campbell [23] and Keller and Frischknecht [24], as referenced in Eppelbaum et al. [25,26]. These empirical formulas, derived from extensive laboratory experiments, summarize the mapping relationship between rock resistivity and rock temperature, providing a theoretical foundation and empirical support for geothermal field predictions. In our proposed geothermal prediction method, RRTCM, we primarily utilize Formula (2). The structure of Formula (2) is relatively straightforward, allowing for seamless integration with well logging data. This enables the construction of a set of equations that can be used to calculate the model parameters at various depths.

$$R_T=R_0{e}^{-\alpha \left(T-T_0\right)}$$

(1)

$$R_T={\displaystyle \frac{R_0}{1+\alpha (T-T_0)}}$$

(2)

where $R_T$ denotes the resistivity of the rock at temperature T, $R_0$ denotes the resistivity of the rock at temperature $T_0$, and $\alpha$ denotes the temperature coefficient of the rock at temperature T.

2.2. General Scheme of the RRTCM

The general optimization approach for the empirical formula involves several key steps. Based on temperature–resistivity logging data, we first determine the temperature compensation coefficient (TCC $t^{\prime }$), the resistivity compensation coefficient (RCC $r^{\prime }$), and the temperature–resistivity correction coefficient (TRCC ${\alpha }^{\prime }$) for each measurement point. We then establish corresponding relationships between these three coefficients and depth and finally substitute the normalized apparent resistivity, along with the corresponding coordinates, into the optimized empirical formula to obtain an accurate 2D distribution of the underground temperature field. The specific optimization scheme for the empirical formulas will be detailed in the following sections (Figure 1).

Step 1: The temperature–resistivity logging data pairs for the M boreholes in the study area are obtained, and the resistivity of each borehole is normalized to obtain M sets of normalized resistivity–temperature datasets.

Step 2: On the basis of Equations (1) and (2) and the normalized resistivity–temperature datasets, the temperature compensation coefficient (TCC $t^{\prime }$), the resistivity compensation coefficient (RCC $r^{\prime }$), and the temperature–resistivity correction coefficient (TRCC ${\alpha }^{\prime }$) between each measurement point are derived.

Step 3: The target dataset is constructed on the basis of the obtained $t^{\prime }$, $r^{\prime }$, and ${\alpha }^{\prime }$ values, along with their corresponding depths. In addition, an ANN is constructed to perform a regression analysis based on this target dataset. The relationships between the changes in $t^{\prime }$, $r^{\prime }$, and ${\alpha }^{\prime }$ with depth, denoted as $T_0\left(z\right)$, $R_0\left(z\right)$, and ${\alpha }^{\prime }\left(z\right)$, respectively, are then analyzed.

Step 4: A 2D inversion of the MT data in the study area is performed, assuming that the inversion grid profile has i rows and j columns, to obtain the apparent resistivity $R_{i}nv(x,z)$ at the subsurface spatial location $P(x,z)$. The apparent resistivity profile is divided into j parts on the basis of the columns, and these j parts are normalized to obtain the normalized apparent resistivities $R_{NORMinv}(x,z)$ for each column, where $x=1,2,3,\dots ,i$ and $z=1,2,3,\dots ,j$.

Step 5: Based on the $T_0\left(z\right)$, $R_0\left(z\right)$, and ${\alpha }^{\prime }\left(z\right)$ obtained in Step 3 and the normalized apparent resistivity $R_{NORMinv}(x,z)$ obtained in Step 4, the predicted temperature $T(x,z)$ at each node $P(x,z)$ is determined by substituting the above values into Equation (1), and finally, the distribution of the subsurface temperature field in the study area is obtained.

In Step 1, the normalized resistivity data for the M boreholes are calculated as follows:

$$R_{NORM}\left(i\right)={\displaystyle \frac{R_{log}(i,j)}{max\left[R_{log}\left(i\right)\right]}}$$

(3)

where $R_{log}(i,j)$ denotes the resistivity logging data of the jth measurement point within the ith borehole, $R_{log}\left(i\right)$ denotes all resistivity logging data of the ith borehole, $max\left[R_{log}\left(i\right)\right]$ denotes the maximum resistivity value for the ith borehole, and $R_{NORM}\left(i\right)$ denotes the normalized resistivity data of the ith borehole.

In Step 2, the temperature compensation coefficient (TCC $t^{\prime }$), the resistivity compensation coefficient (RCC $r^{\prime }$), and the temperature–resistivity correction coefficient (TRCC ${\alpha }^{\prime }$) are calculated as follows.

First, Equation (2) is transformed into the following form:

$$T={\displaystyle \frac{R_0}{\alpha }}{\displaystyle \frac{1}{R_T}}+T_0-{\displaystyle \frac{1}{\alpha }}$$

(4)

Let the slope and intercept of the above equation be a and b, respectively, which are calculated as follows:

$$\left\{\begin{array}{c}{\displaystyle \frac{R_0}{\alpha }}=a\hfill \\ T_0-{\displaystyle \frac{1}{\alpha }}=b\hfill \end{array}\right.$$

(5)

Therefore, Equation (2) can be expressed as a linear equation for the temperature T and the inverse resistivity ${\displaystyle \frac{1}{R_T}}$, as follows:

$$T=a{\displaystyle \frac{1}{R_T}}+b$$

(6)

The normalized resistivity and temperature data for n rock pairs are substituted into Equation (6) to obtain the system shown in Equation (7):

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{t}_1=a_1{\displaystyle \frac{1}{r_1}}+b_1\hfill \\ t_2=a_1{\displaystyle \frac{1}{r_2}}+b_1\hfill \end{array}\right.\to \left\{\begin{array}{c}{t}_2=a_2{\displaystyle \frac{1}{r_2}}+b_2\hfill \\ t_3=a_2{\displaystyle \frac{1}{r_3}}+b_2\hfill \end{array}\right.\cdots \left\{\begin{array}{c}{t}_{n-1}=a_{n-1}{\displaystyle \frac{1}{r_{n-1}}}+b_{n-1}\hfill \\ t_n=a_{n-1}{\displaystyle \frac{1}{r_n}}+b_{n-1}\hfill \end{array}\right.\end{array}$$

(7)

Solving the above system of $n-1$ quadratic equations yields the values of a and b between the two measurement points:

$$\{a_1,a_2,a_3,...,a_{n-1}\},\{b_1,b_2,b_3,...,b_{n-1}\}$$

(8)

With this method, the data are divided into $n-1$ segments, reducing the errors caused by data stratification; furthermore, assuming that the a and b values derived for each system of equations are the values corresponding to the midpoint between the two measurement points, the depths corresponding to coefficients a and b can be derived as follows:

$$\left\{{\displaystyle \frac{d_1+d_2}{2}},{\displaystyle \frac{d_2+d_3}{2}},\cdots ,{\displaystyle \frac{d_{n-1}+d_n}{2}}\right\}\to \left\{z_1,z_2,\cdots ,z_{n-1}\right\}$$

(9)

Additionally, Equation (1), which expresses the relationship between rock resistivity and temperature, can be transformed as

$$T=pln\left(R_T\right)+q$$

(10)

Here, we have

$$\left\{\begin{array}{c}p=-{\displaystyle \frac{1}{\alpha }}\hfill \\ q={\displaystyle \frac{1}{\alpha }}ln\left(R_0\right)+T_0\hfill \end{array}\right.$$

(11)

Similar to Equation (7), the measured normalized rock resistivity and temperature data can be substituted into Equation (11) to obtain the following system, as shown in Equation (12):

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{t}_1=p_{1}ln\left(r_1\right)+q_1\hfill \\ t_2=p_{1}ln\left(r_2\right)+q_1\hfill \end{array}\right.\to \left\{\begin{array}{c}{t}_2=p_{2}ln\left(r_2\right)+q_2\hfill \\ t_3=p_{2}ln\left(r_3\right)+q_2\hfill \end{array}\right.\cdots \left\{\begin{array}{c}{t}_{n-1}=p_{n-1}ln\left(r_{n-1}\right)+q_{n-1}\hfill \\ t_n=p_{n-1}ln\left(r_n\right)+q_{n-1}\hfill \end{array}\right.\end{array}$$

(12)

The above system of $n-1$ quadratic equations can be solved to determine the p and q values for the two measurement points:

$$\left\{p_1,p_2,p_3,\cdots ,p_{n-1}\right\},\left\{q_1,q_2,q_3,\cdots ,q_{n-1}\right\}$$

(13)

The TCC $t^{\prime }$ and RCC $r^{\prime }$ are also related as follows:

$$t^{\prime }=pln\left(r^{\prime }\right)+q$$

(14)

Combining Equations (5), (8), (13) and (14), we can find the $t^{\prime }$, $r^{\prime }$, and ${\alpha }^{\prime }$ values corresponding to the depth of each measurement point in the first borehole.

Following the above methodology, the $t^{\prime }$, $r^{\prime }$, and ${\alpha }^{\prime }$ values for the M boreholes can be obtained, yielding the following dataset:

$$\left\{(z_1,t_1^{\prime }),(z_2,t_2^{\prime }),(z_3,t_3^{\prime }),\cdots ,(z_{m-M},t_{m-M}^{\prime })\right\}$$

$$\left\{(z_1,r_1^{\prime }),(z_2,r_2^{\prime }),(z_3,r_3^{\prime }),\cdots ,(z_{m-M},r_{m-M}^{\prime })\right\}$$

$$\left\{(z_1,{\alpha }_1^{\prime }),(z_2,{\alpha }_2^{\prime }),(z_3,{\alpha }_3^{\prime }),\cdots ,(z_{m-M},{\alpha }_{m-M}^{\prime })\right\}$$

where m denotes the number of temperature and normalized resistivity data pairs.

In Step 3, to accurately determine the depth-dependent variations in $t^{\prime }$, $r^{\prime }$, and ${\alpha }^{\prime }$, we employed an ANN for regression analysis of these coefficients. The ANN consists of an input layer with one neuron, two hidden layers with 10 and 20 neurons, and an output layer with one neuron. The learning rate was set to 0.01. The dataset was randomly divided into training and test sets at an 8:2 ratio. Training was terminated once the fitting error of the training set fell below 10%. The test set was then input into the trained regression model, resulting in a test set fitting error of 8.2%.

The apparent resistivity data from the MT inversion are normalized in Step 4 with the following normalization equation:

$$R_{NORMinv}(x,z)={\displaystyle \frac{R_{inv}(x,z)}{max\left[R_{inv}\left(x\right)\right]}}$$

(15)

where $R_{inv}(x,z)$ denotes the apparent resistivity value at position $P(x,z)$, x denotes the horizontal position, and z denotes the depth. $R_{NORMinv}(x,z)$ denotes the normalized apparent resistivity value at position $P(x,z)$; $R_{inv}\left(x\right)$ denotes the apparent resistivity value at all depths at x; and $max\left[R_{inv}\left(x\right)\right]$ denotes the maximum value in $R_{inv}\left(x\right)$.

In Step 5, the temperature $T(x,z)$ at position $P(x,z)$ is calculated as

$$T(x,z)=T_0\left(z\right)+{\displaystyle \frac{1}{\alpha \left(z\right)}}ln\left({\displaystyle \frac{R_0\left(z\right)}{R_{NORMin\nu }(x,z)}}\right)$$

(16)

3. Prediction of Subsurface Temperatures in the Xiong’an New Area, China

3.1. Geological Setting

The geothermal geological structure of the Xiong’an New Area, located in the central part of the Jizhong depression of the Bohai Bay basin [27,28], is complex [29,30]. The structural units of the Xiong’an New Area include the Rongcheng uplift, the southern part of the Niutuozhen uplift, the southern part of the Niubei slope, and the Baiyangdian-Dahezhen depression [31]. These structural units not only determine the distribution of geothermal resources but also influence the formation and evolution of the geothermal system [32] (Figure 2a). The geothermal field of the Xiong’an New Area is controlled by multiple faults, with the primary fault structures including the Niunan fault, Niudong fault, Daxing fault, Xushui fault, and Rongcheng fault [33] (Figure 2b). Among them, the northeast-trending Xushui-Niunan fault separates the two uplifts (Rongcheng uplift and Niutuozhen uplift) from the Gaoyang low uplift [34]. These faults serve as efficient conduits for heat and fluid transfer, playing crucial roles in the formation of the geothermal system in the Xiong’an New Area [35]. Fault activity enhances the degree of rock fragmentation, increases the number of migration pathways for thermal fluids, and thereby promotes the accumulation of geothermal resources [36]. Additionally, fault structures impact groundwater circulation, allowing deep thermal water to migrate to shallower depths, further enhancing the development potential of the geothermal field [37,38]. Overall, the complex geothermal and geological structure of the Xiong’an New Area, as well as its fault structure, are crucial in the formation, distribution, and development of geothermal resources.

3.2. Results and Analysis

On the basis of data from the four boreholes (D16, D17, D32, and D35) near three MT survey lines in the Xiong’an New Area, the $t^{\prime }$, $r^{\prime }$, and ${\alpha }^{\prime }$ values between all pairs of temperature measurement points in the four boreholes were calculated separately. After excluding outliers, we conducted a regression analysis of the three key coefficients with the corresponding depths with an ANN. The ANN was trained on data from boreholes with maximum depths of approximately 4000 m, and the relationships between the three key parameters and depth were obtained (Figure 3). The dashed lines in Figure 3 represent the ANN’s extrapolation beyond the training data range (depths > 4000 m), which should be treated with appropriate caution as they lack direct validation from borehole measurements. The flattening of the curves below 4000m is an artifact of the ANN’s extrapolation behavior rather than a physical observation. While the ANN’s predictions at these depths provide a reasonable continuation of the observed trends, their accuracy cannot be directly verified without additional deep borehole data. This limitation should be considered when interpreting the temperature predictions at depths greater than 4000 m.

In a previous study, Huang et al. [20] (Figure 4) calculated the skin depths of the three MT profiles and estimated the regional strike and dimensionality of the MT data prior to 2D inversion. The MT data used in this study provided reliable apparent resistivity information down to depths of approximately 10 km, which was more than sufficient for our temperature predictions targeting depths up to 5 km. This verification supports the feasibility of performing 2D inversion based on the three MT profiles. The processed MT data were inverted using the nonlinear conjugate gradient algorithm developed by Rodi and Mackie [42], with the optimal regularization parameter determined through the L-curve method [43]. To enhance the reliability and resolution of the subsurface resistivity structure, we implemented a joint inversion of TE and TM modes, which exhibit complementary sensitivities to different aspects of the subsurface structure. The TM mode (electric field parallel to strike) shows greater sensitivity to lateral resistivity boundaries and provides superior resolution of vertical conductivity contrasts [44]. However, its sensitivity to the depth range of subsurface bodies is relatively limited. In contrast, the TE mode (electric field perpendicular to strike) offers better constraints on the depth of conductive structures but is more susceptible to three-dimensional effects [45].

Profile A: Based on the resistivity distribution characteristics of Profile A, strata less than 1 km deep, which mainly include the Quaternary and Neogene formations, generally have resistivity values less than 10 $\Omega ·$m. The area above the black dashed line in Profile A corresponds to the Jixian (Jxw) system strata, ranging from 2 km to 3 km in depth, with resistivity values generally less than 100 $\Omega ·$m. Between measuring points 3 and 10 in Profile A, there is an uplift below the black dashed line at depths of 2 km to 5 km, with resistivity values generally greater than 200 $\Omega ·$m, corresponding to the older Changcheng (Chg) system strata. The Jxw thermal reservoir is related to the Tertiary and Quaternary sandstone–mudstone strata, which have poor thermal conductivity and serve as excellent insulation layers, and the thermal reservoir of Jxw exhibits significant thickness and well-developed fractures and cavities. Additionally, on the basis of the resistivity distribution characteristics and geological data, the Rongxi fault and Xushui south fault are inferred to exist on both sides of the uplift between measuring points 3 and 10, providing good channels for the upward migration of deep thermal water.

Profile B: Above the black dashed line in Profile B, there is a low-resistivity body with a thickness of 3 km to 4 km, typically exhibiting resistivity values less than 40 $\Omega ·$m. This feature reflects the electrical characteristics of the Cenozoic strata (Kz). Below the black dashed line, the high-resistivity body corresponds to the Proterozoic basement rock (Pt). Between MT measurement points 15 and 42 and at depths of 3 km below these points, Profile B displays significant lateral discontinuities. This suggests that the region is affected by the Niunan Fault, which also serves as a conduit for the upward migration of deep geothermal water.

Profile C: The resistivity distribution characteristics of Profile C indicate that the resistivity of the strata above the black dashed line is generally less than 50 $\Omega ·$m, corresponding to the Kz formation. The strata below the black dashed line exhibit higher resistivity values, corresponding to the Pt formation. The lateral electrical characteristics of Profile C show good continuity, with no significant electrical anomalies indicating the presence of faults, and the strata are relatively flat. Therefore, it is inferred that the primary heat sources for Profile C and nearby regions originate from deep crustal heat conduction.

The temperature field models for the three profiles in the Xiong’an New Area were calculated according to the relationships of $t^{\prime }$, $r^{\prime }$, and ${\alpha }^{\prime }$ with depth, which were then substituted into Equation (2) and combined with the 2D MT inversion results from the three survey lines in the Xiong’an New Area. We then compute temperature prediction plots for each of the three profile lines (Figure 5).

Profile A: Based on the temperature distribution, for depths less than 2 km ($z\le 2$ km), the temperature prediction shows higher resolution, indicating substantial temperature variations. Since the MT inversion resolves more resistivity variability in the shallower regions of the profile, the temperature prediction derived from the MT results will also show higher resolution. In contrast, for deeper formations, particularly below 4000 m where we rely on ANN extrapolation rather than direct borehole data, the resolution of the electrical structure decreases, producing more smeared gradual changes, which reduces temperature prediction accuracy, making it more difficult to capture detailed temperature variations. The confidence in our predictions decreases with depth, especially in regions where we depend on the extrapolated relationships between the key parameters and depth.

Profile B: On the basis of the results of the three electromagnetic inversion resistivity profiles and the corresponding temperature field distribution characteristics along the survey lines, local low-resistivity areas correspond to local high-temperature zones, whereas local high-resistivity areas correspond to local low-temperature zones. For example, in Profile A, there is a local high-resistivity body at depths of 3–5 km below points 5–8, which corresponds to a local low-temperature area in the temperature profile. Similarly, in Profile B, the areas below points 1–9, 21–34, and 60–65 at depths of 3–5 km are all local high-resistivity regions, which correspond to local low-temperature areas. Overall, the resistivity distribution characteristics and the temperature field distribution results exhibit an inverse relationship: areas with high resistivity values correspond to regions with low temperature values; conversely, areas with low resistivity values correspond to regions with high temperature values. This phenomenon is consistent with the relationship between rock resistivity and temperature: the resistivity of a rock is significantly affected by the presence and salinity of pore fluids. High resistivity at lower temperatures can generally be attributed to reduced ion mobility in the colder fluid, while higher temperatures tend to increase ionic conduction in the fluid, resulting in lower resistivity. The mineral composition of the rock also affects resistivity. Certain minerals, such as silicate minerals, exhibit higher resistivity at low temperatures, while hydrous minerals or minerals with conductive properties decrease resistivity as temperature increases [46].

Profile C: Fault zones at deeper positions typically correspond to localized low-temperature areas (characterized by concave temperature fields), whereas the shallower parts of the fault zones correspond to localized high-temperature areas (characterized by convex temperature fields). In Profile A, the temperatures in deep regions between the Rongxi fault and the Xushuinan fault, as well as in the areas on either side of the fault zones, tend to decrease, whereas the temperatures in shallow regions tend to increase. In Profile B, the temperature field characteristics at fault zone F3 and the Niunan fault are consistent with this pattern. This conclusion aligns with the thermal convergence model in the theory of heat conduction, further indicating that these faults are important conduits for heat and water transfer in the geothermal system of the Xiong’an New Area.

Furthermore, a comprehensive analysis of the horizontal variations in the temperature fields across the three survey lines reveals that the horizontal temperature change trend in Profile C is more gradual than those in Profiles A and B. This difference in temperature variation patterns can be attributed to several factors: (1) The limited resistivity variations in the MT data at depth naturally result in smoother temperature predictions, as our model primarily relies on resistivity variations to predict temperature changes; (2) The observed temperature variations in Profiles A and B are more pronounced, particularly in regions where fault structures are inferred from the resistivity model; (3) While significant thermal variations are observed in the shallow sections (0–3 km) of all profiles, the interpretation of these variations in terms of heat transfer mechanisms (thermal conduction vs. fault-related convection) would require additional geological and geophysical evidence beyond our current dataset.

3.3. Method Comparison

To analyze the accuracy of the temperature predictions for the individual boreholes, we used the goodness-of-fit formula to calculate the prediction accuracy for the individual boreholes:

$$RSS=\sum _k^n{\left(T_{pred}-{\overline{T}}_{true}\right)}^2$$

(17)

$$SST=\sum _k^n{\left(T_{pred}-{\overline{T}}_{true}\right)}^2+\sum _k^n{\left(T_{true}-T_{pred}\right)}^2$$

(18)

$$R^2={\displaystyle \frac{RSS}{SST}}$$

(19)

where $R^2$ denotes the coefficient of determination, $T_{pred}$ denotes the predicted temperature value for a given borehole, $T_{true}$ denotes the measured temperature value for a given borehole, and ${\overline{T}}_{true}$ denotes the average of the measured temperatures for a given borehole. $R^2$ ranges between 0 and 1. As the $R^2$ value approaches 1, the temperature prediction for the borehole tends to approach the measured temperature, indicating a greater accuracy for the temperature prediction.

To validate the performance of the RRTCM, a comparative study was conducted against a well-established artificial neural network (ANN) method [17] and the CCMOT method proposed by Huang et al. [20]. Both methods were applied to predict temperature distributions in four validation boreholes (D16, D17, D32, and D35) under identical conditions. The ANN architecture was specifically designed for geothermal temperature prediction with the following:

• Input layer: 3 neurons (receiving spatial coordinates x, y and resistivity values);
• Hidden layers: Two layers with 20 and 15 neurons, respectively;
• Output layer: 1 neuron (producing predicted temperature values).

The ANN was rigorously trained until achieving a convergence criterion where the mean squared error (MSE) between predicted and actual temperatures fell below 0.1. This MSE threshold of 0.1 represents a balance between model accuracy and training efficiency. Following training, the model was used to predict temperatures by inputting the resistivity measurements and spatial coordinates from the validation boreholes. This standardized comparison framework allows for a direct assessment of both methods’ predictive capabilities.

The RRTCM demonstrated exceptional performance in predicting subsurface temperatures across all four validation boreholes. For boreholes D16, D17, D32, and D35, our method achieved remarkably high prediction accuracies of 97.85%, 98.88%, 96.49%, and 97.53%, respectively (Figure 6). In comparison, the conventional artificial neural network approach yielded significantly lower accuracies of 86.70%, 85.83%, 85.56%, and 77.52% for the same boreholes. The prediction results of CCMOT and ANNs are similar. The superiority of RRTCM is particularly evident in two aspects: first, it consistently maintained prediction accuracies above 90% for all boreholes, The fitting degree of the temperature prediction results of the D17 and D35 well loggings is as high as 98.88% and 97.53%, respectively. The main reason is that the temperature changes in the D17 and D35 are relatively gentle and change linearly with depth, and the amount of measured temperature data of these two well loggings is relatively small, and RRTCM can easily obtain a high prediction accuracy. Second, it showed substantial improvement over the neural network method, most notably for borehole D35, where RRTCM’s accuracy was 20.01 percentage points higher (97.53% versus 77.52%).

In addition, compared with ANNs and CCMOT, the RRTCM better reflects the observed temperature change trends. For example, for the D16 borehole, the increase in the temperature slows at a depth of approximately 1200 m, and the RRTCM prediction results accurately capture this inflection point. A comparison of the prediction results for boreholes D17, D32, and D35 reveals that the prediction accuracy of the ANN decreases with increasing depth, whereas the RRTCM still produces relatively accurate predictions at deeper temperatures.

4. Analysis of Factors Affecting the Temperature Prediction Accuracy

To predict deep formation temperatures with the RRTCM, three key factors are considered: (i) the sampling interval of the logging data for the constrained boreholes, $T_{gap}$; (ii) the number of constrained boreholes, $CN$, and their locations; and (iii) the maximum depth of the constrained boreholes, $DL$. In calculating the variation in the three key parameters with depth, $T_{gap}$, $CN$, and $DL$ affect the amount of data needed and the quality of the data used in the calculations. Therefore, to study the effectiveness and reliability of the RRTCM in predicting temperature distributions at substantial depths, the impacts of these three factors on the temperature prediction results must be analyzed.

4.1. Effect of the Sampling Interval of the Logging Data from the Constrained Boreholes

To study the effect of $T_{gap}$ on the temperature prediction accuracy, we selected five fixed boreholes, D12, D16, D17, D32, and D35, as constrained boreholes and set the maximum depth of the constrained boreholes, $DL$, to 4000 m. The minimum value of $T_{gap}$ was set to 0.15 m, and the maximum value was set to 2.1 m. Fourteen temperature predictions were made for each of the boreholes in incremental steps of 0.15 m. From the experiment we found that the temperature prediction accuracy is negatively correlated with $T_{gap}$ when $T_{gap}$ is less than 1.35 m and fluctuates with $T_{gap}$ when $T_{gap}$ is greater than 1.35 m (Figure 7). The highest temperature prediction accuracy was obtained for the four validation boreholes in the three profiles when $T_{gap}$ was 0.15 m. Among these boreholes, the best temperature prediction was obtained for D17 in Profile A. When $T_{gap}$ was 1.95 m, the lowest temperature prediction accuracy was obtained for D16 in Profile A and D35 in Profile B; when $T_{gap}$ was 1.8 m, the lowest prediction accuracy was obtained for D17 in Profile A and D32 in Profile C. Thus, as the sampling interval increases, the amount of data involved in the calculations for the constrained boreholes decreases, thus introducing some errors. However, when $T_{gap}$ is increased to a certain value, some anomalous measurement points in the logging data may be eliminated, thus improving the temperature prediction accuracy to some extent.

4.2. Effects of the Number of Constrained Boreholes CN and Their Locations

We conducted a comprehensive analysis of how temperature prediction accuracy is influenced by both the quantity and spatial distribution of constrained boreholes. In this analysis, we maintained constant values for $T_{gap}$ (0.15 m) and $DL$ (4000 m) while systematically varying the borehole configurations. The detailed combinations of constrained boreholes and their corresponding prediction accuracies reveal a clear positive correlation between the number of constrained boreholes and prediction accuracy, with optimal results achieved when using five boreholes (Figure 8 and Figure 9). Notably, when boreholes D12, D16, D17, D32, and D35 were selected as constraints (combination $C{5}_{06}$), the model achieved its highest prediction accuracy of 98.88% for borehole D17. This superior performance can be attributed to the increased number of temperature measurement points available for calculation, which enables more accurate regression analysis of the key parameters through ANNs. Additionally, our analysis of the results presented in

Table 1. The R² values of the predicted temperature profiles with different constrained boreholes.

CN	Constrained Borehole Combination	Constraint Data (Borehole)						Max (GOF)	Min (GOF)	Avg (GOF)
		D11	D12	D16	D17	D32	D35
1	C101	√						0.832	0.626	0.726
	C102		√					0.629	0.561	0.603
	C103			√				0.912	0.752	0.816
	C104				√			0.925	0.759	0.814
	C105					√		0.804	0.719	0.770
	C106						√	0.886	0.627	0.708
2	C201	√	√					0.962	0.697	0.809
	C202	√		√				0.931	0.717	0.806
	C203	√			√			0.937	0.750	0.810
	C204	√				√		0.924	0.768	0.858
	C205	√					√	0.942	0.651	0.758
	C206		√	√				0.936	0.754	0.855
	C207		√		√			0.921	0.760	0.819
	C208		√			√		0.934	0.726	0.814
	C209		√				√	0.928	0.809	0.887
	C210			√	√			0.908	0.758	0.813
	C211			√		√		0.941	0.711	0.874
	C212			√			√	0.955	0.647	0.841
	C213				√	√		0.952	0.812	0.874
	C214				√		√	0.996	0.804	0.870
	C215					√	√	0.948	0.765	0.866
3	C301	√	√	√				0.942	0.767	0.833
	C302	√	√		√			0.935	0.753	0.817
	C303	√	√			√		0.950	0.782	0.867
	C304	√	√				√	0.954	0.792	0.880
	C305	√		√	√			0.917	0.760	0.809
	C306	√		√		√		0.958	0.733	0.876
	C307	√		√			√	0.963	0.666	0.833
	C308	√			√	√		0.957	0.814	0.869
	C309	√			√		√	0.951	0.783	0.849
	C310	√				√	√	0.930	0.781	0.870
	C311		√	√	√			0.911	0.772	0.820
	C312		√	√		√		0.941	0.741	0.882
	C313		√	√			√	0.965	0.728	0.871
	C314		√		√	√		0.967	0.817	0.881
	C315		√		√		√	0.960	0.825	0.870
	C316		√			√	√	0.957	0.771	0.865
	C317			√	√	√		0.964	0.781	0.867
	C318			√	√		√	0.948	0.763	0.846
	C319			√		√	√	0.944	0.713	0.873
	C320				√	√	√	0.942	0.812	0.881
4	C401	√	√	√	√			0.919	0.774	0.817
	C402	√	√	√		√		0.959	0.762	0.886
	C403	√	√	√			√	0.975	0.738	0.863
	C404	√	√		√	√		0.940	0.822	0.869
	C405	√	√		√		√	0.988	0.805	0.869
	C406	√	√			√	√	0.924	0.800	0.880
	C407	√		√	√	√		0.980	0.784	0.865
	C408	√		√	√		√	0.974	0.764	0.846
	C409	√		√		√	√	0.953	0.805	0.895
	C410	√			√	√	√	0.984	0.813	0.886
	C411		√	√	√	√		0.976	0.827	0.883
	C412		√	√	√		√	0.973	0.825	0.873
	C413		√	√		√	√	0.945	0.803	0.897
	C414		√		√	√	√	0.982	0.820	0.895
	C415			√	√	√	√	0.985	0.855	0.912
5	C501	√	√	√	√	√		0.980	0.861	0.922
	C502	√	√	√	√		√	0.922	0.882	0.897
	C503	√	√	√		√	√	0.954	0.890	0.918
	C504	√	√		√	√	√	0.973	0.881	0.925
	C505	√		√	√	√	√	0.975	0.894	0.925
	C506		√	√	√	√	√	0.975	0.909	0.950

“√” indicates that the corresponding borehole was selected as a constrained borehole. Max (R²) represents the maximum temperature prediction accuracy for the four verification boreholes when this combination was used. Min (R²) represents the minimum temperature prediction accuracy for the four verification boreholes when this combination was used. Avg (R²) represents the average temperature prediction accuracy for the four verification boreholes when this combination was used.

and Figure 8 indicates that combinations including boreholes D11 and D12 generally yielded lower prediction accuracies. This phenomenon can be explained by examining the spatial relationships detailed in

Table 2. Distance from the constrained boreholes to the MT profiles [20].

Constrained Boreholes	Profile A (km)	Profile B (km)	Profile C (km)
D11	12.6	20.2	31.4
D12	4.5	11.5	22.7
D16	0.3	7.7	19.1
D17	0.2	9.1	20.7
D32	21.9	12.4	0.6
D35	11.4	0.9	12.9

, which shows that D11 and D12 are situated significantly farther from the MT survey lines compared to other constrained boreholes. This greater distance introduces discrepancies in geological background, rock structure, and electrical characteristics between these boreholes and the survey profiles, ultimately affecting the accuracy of temperature predictions.

4.3. Effect of the Maximum Depth of the Constrained Boreholes

We performed a detailed investigation into the relationship between maximum borehole depth ($DL$) and temperature prediction accuracy by selecting D12, D16, D17, D32, and D35 as constrained boreholes while maintaining $T_{gap}$ at 0.15 m. Starting from 1000 m, we incrementally increased $DL$ by 500 m intervals until reaching 4000 m, analyzing the prediction accuracy at each depth increment. The results demonstrate distinct patterns of depth-dependent accuracy across different boreholes (Figure 10). Most boreholes in our dataset have maximum depths around 3 km, with only D17 extending to 4 km. The flattening of prediction accuracy at 3 km for most boreholes is more reasonable than the continued improvement observed for D35. For boreholes D35 and D17, the prediction accuracy shows a positive correlation with $DL$ up to 3 km, though D17 exhibits a notable stabilization around 97% accuracy when $DL$ exceeds 3000 m. Similarly, boreholes D16 and D32 display a positive correlation between accuracy and $DL$ in the 0–3 km range. Beyond 3 km, due to the limited control of our data, the accuracy of the prediction results above 4000 m is highly dependent on the extrapolation accuracy of the artificial neural network for three key parameters.

4.4. Comprehensive Comparison of the Sensitivity Analysis Results

To comprehensively compare the influence of the aforementioned three parameters on the temperature prediction accuracy for the boreholes, we used the coefficient of variation as a sensitivity evaluation index.

$$C_v={\displaystyle \frac{SD}{MN}}$$

$$SD=\sqrt{{\displaystyle \frac{1}{N_{cv}}}\sum _{i=1}^{N_{cv}}{(y_i-MN)}^2}$$

where $C_v$ represents the coefficient of variation, $SD$ denotes the standard deviation of the dataset, $MN$ indicates the mean value of the observations, $y_i$ corresponds to individual data points, and $N_{cv}$ signifies the total number of samples in the dataset. A higher $C_v$ value for a given factor indicates its greater influence on the temperature prediction accuracy, thereby providing a quantitative measure of the factor’s relative importance in the prediction model.

Analysis of the $C_v$ values across the four validation boreholes reveals distinct patterns of parameter sensitivity (Figure 11). The number of constrained boreholes ($CN$) demonstrates the lowest influence on prediction accuracy, indicating that the RRTCM can maintain high accuracy even with fewer boreholes. This characteristic represents a significant advantage over traditional ANN methods, which typically require extensive training samples. However, our analysis also reveals that certain boreholes (D32, D35, and D16) are particularly important for maintaining prediction accuracy, suggesting they should be included in the constrained borehole set. The sensitivity analysis further shows that the sampling interval ($T_{gap}$) is most critical for boreholes D35 and D16, while the maximum borehole depth ($DL$) exhibits the strongest overall impact on prediction accuracy. Based on these findings, we recommend an optimized implementation strategy that (1) prioritizes drilling to the maximum possible depth, as this has the strongest influence on prediction accuracy; (2) includes boreholes D32, D35, and D16 in the constrained set; (3) uses smaller sampling intervals, particularly for boreholes D35 and D16; and (4) focuses on drilling fewer but deeper boreholes while maintaining small sampling intervals. This approach not only ensures high prediction accuracy but also provides a cost-effective solution for practical applications of the RRTCM method.

5. Conclusions

We employed an accurate prediction method for deep geothermal fields based on the rock resistivity–temperature coupling model (RRTCM) to obtain precise temperature predictions for the deep strata of the Xiong’an New Area. The predictions of the temperatures of four validation boreholes (D16, D17, D32, and D35) yielded accuracies of 90.93%, 98.88%, 96.49%, and 97.53%, respectively.

Our model significantly outperformed methods based on ANNs given the same number of constrained boreholes, with the improvements in the prediction accuracy ranging from 4.23% to 20.01%. When analyzing the influence of three parameters on the temperature prediction accuracy, we found that $DL$ had the greatest impact, followed by $T_{gap}$, whereas $CN$ had the smallest impact. This finding indicates that the RRTCM can achieve high accuracy in temperature field predictions even with a limited number of constrained boreholes. Additionally, positioning the constrained boreholes closer to the MT survey lines and reducing $T_{gap}$ can further increase the prediction accuracy.

The RRTCM can calculate key parameters in empirical formulas under different stratigraphic conditions, providing a more detailed characterization of the relationship between rock resistivity and temperature. While RRTCM demonstrates superior performance in this study, it is important to note that this method may not universally outperform other temperature prediction approaches under all geological conditions. The effectiveness of RRTCM relies heavily on several key factors:

• Quality of MT data: The accuracy of RRTCM predictions is directly dependent on the resolution and reliability of magnetotelluric inversion results. In areas with strong electromagnetic noise or complex 3D structures, the method’s performance may be compromised.
• Geological complexity: In regions with highly heterogeneous lithology or complex fluid systems, the relationship between resistivity and temperature might deviate significantly from the empirical formulas used in RRTCM, potentially leading to less accurate predictions. In fault zones, particularly those with extensive fracturing or significant displacement, the thermal conductivity and heat flow within the rocks may become heterogeneous, thereby rendering the empirical formula ineffective and preventing the RRTCM from providing accurate temperature predictions.
• Depth limitations: The prediction depth of RRTCM is constrained by the effective penetration depth of MT signals. For very deep temperature predictions (typically >10 km), other methods such as thermal modeling or integrated geophysical approaches might be more suitable.
• Data availability: While RRTCM can achieve good results with relatively few boreholes, it requires both high-quality MT data and temperature-resistivity logging data. In areas where such data are limited or unavailable, machine learning methods that can utilize other types of geophysical data might be more practical.

Therefore, the selection of appropriate temperature prediction methods should be based on careful consideration of local geological conditions, data availability, and specific research objectives. In some cases, a combination of different methods might provide the most robust results. Future work could focus on integrating multiple methods to overcome these limitations and further improve prediction accuracy.