Characterization of hydrothermal alteration along geothermal wells using unsupervised machine-learning analysis of X-ray powder diffraction data

The geology of the Hachimantai geothermal field comprises a Quaternary formation that overlies a Tertiary formation (Kimbara 1985). The Tertiary formation consists of dacitic and andesitic tuff, and the Quaternary formation is approximately 450 m thick (Fig. 2b) (Kimbara 1985). The tonalite, which is encountered at depths of approximately 1500–1800 m, intrudes beneath the Tertiary formation (Fig. 2b) and is considered a heat source for the geothermal field. The metamorphism of the Tertiary wall rocks around the tonalite includes extensive development of biotite, cordierite, talc, magnetite, ilmenite, and other metamorphic minerals. The QI values in this study were obtained from cuttings along two neighboring wells called N19-HA-1 and N19-HA-2 that were drilled down to approximately 1750 and 1600 m, respectively. The QI values were reported in NEDO (2007, 2008). The ground facilities of the two wells are at almost the same position, but N19-HA-1 was drilled in an NW direction, whereas N19-HA-2 was drilled toward the NNW (NEDO 2007, 2008). Evidence of high-temperature fluid circulation can be found in the zonation of altered minerals around wells N19-HA-1 and N19-HA-2. These can be sub-divided into the Silica-Rutile, Silica-Anatase, and Tridymite-Cristobalite zones (around well N19-HA-1). Temperature logs mainly show patterns of conduction, with maximum temperatures of approximately 300 °C at the bottoms of the wells (NEDO 2007, 2008).

The rock cuttings (e.g., Fig. 3) were first crushed manually using a pestle and then disaggregated using a planetary mill (NEDO 2007). XRD analyses were subsequently performed on basally and randomly oriented samples using a diffractometer with CuK \({\alpha }_{1}\) radiation at 40 kV and 25 mA. The randomly oriented samples were scanned between \(2\)–\({62}^{^\circ }2\theta\) at a scan speed of \({2}^{^\circ }2\theta /\mathrm{min}\). Minerals were identified based on the International Centre for Diffraction Data (ICDD) PDF-2 database.

The QI of a mineral is defined as follows (Hayashi 1979):where \({I}_{m}\) is the peak intensity of the XRD value of a mineral and \({I}_{q}\) is the peak intensity of the XRD value of pure silica. The peak intensity of XRD is influenced not only by the amount of minerals but also by the crystallinity and preferential growth of the crystal structure. We thus used the normalized values (see Sect. 3) for machine-learning classification and did not take the absolute QI value into account.

Table 1 presents the 24 mineral types for which QI values were calculated: four clay minerals (smectite, chlorite, sericite, kaolinite), two zeolite minerals (laumontite, wairakite), two silica minerals (tridymite, cristobalite), seven silicate minerals (clinopyroxene, epidote, prehnite, anthophyllite, biotite, cordierite, talc), five oxide minerals (magnetite, ilmenite, hematite, anatase, rutile), one sulfide mineral (marcasite), two sulfate minerals (anhydrite, alunite), and one carbonate mineral (calcite) (NEDO 2007, 2008). These minerals are not components of the host rock and are likely to have formed as a result of hydrothermal alteration. A total of 43 QI values were obtained based on XRD of cuttings from N19-HA-1, and a total of 45 QI values were obtained from N19-HA-2 (NEDO 2008). The average depth spacing of the values was approximately 38 m. Depth profiles based on the QI showed that each mineral had its own characteristic depth range (Fig. 4). We confirmed that the standard deviations of QI values were in the range of 0.01–0.17, as calculated from three measurements using different samples at the same depth. Such low values indicate the validity of the QI values.

We compared and evaluated three classification methods—K-means clustering, the GMM, and AC—using synthetic QI data. In this synthetic dataset, we assumed four different mineral distributions along a well down to 1000 m with a depth spacing of 10 m. Mineral distributions exhibited a Gaussian shape across the depth range, with mean \({\mu }_{i}\) (\(i=1,\cdots 4\)), standard deviation \({\sigma }_{i}\) (\(i=1,\cdots 4\)), and maximum amplitude \({Amp}_{i}\) (\(i=1,\cdots 4\)), as shown in Table 2. Gaussian noise with a mean of 0 and a standard deviation of 0.25 was added to each data point, to simulate the fluctuation in QI value. The depth profile of the synthetic data indicates that each mineral is distributed within a characteristic depth interval, and some depth intervals contain multiple minerals (Fig. 5a). We assumed temperature increases with depth at a constant rate of 0.2 degrees C/m.

Data classification typically entails preprocessing of observed data. A common preprocessing method is to normalize both the mean and standard deviation of the data. The effectiveness of this method is well-known, but normalization of the mean value produces negative QI values that are not found in the original data. To assess the impact of this preprocessing strategy, we applied additional preprocessing to normalize only the standard deviation of the observed data and compared the classification results of these two preprocessing methods.

Suppose that the QI values are measured for \(N\) samples along a depth profile, and the data are represented as the matrix \(\overrightarrow{{\varvec{x}}}=\left(\overrightarrow{{{\varvec{x}}}_{1}},\cdots ,\overrightarrow{{{\varvec{x}}}_{{\varvec{n}}}},\cdots ,\overrightarrow{{{\varvec{x}}}_{{\varvec{N}}}}\right)\) with the dimension of \(N\times D\), where \(\overrightarrow{{{\varvec{x}}}_{{\varvec{n}}}}\) is a \(D\)-dimensional vector \(\overrightarrow{{{\varvec{x}}}_{{\varvec{n}}}}={\left({x}_{n,1},\cdots ,{x}_{n,d},\cdots ,{x}_{n,D}\right)}^{T}\) that contains the QI values of \(D\) different minerals. We now consider the classification of QI values from different samples into \(K\) classes. We assign a sample to the k^th class when the sample’s QI is at the minimum distance from the mean of the data in the k^th class \({\mu }_{k}\). This classification is attained by minimizing the following objective function \(J\):where \({r}_{n,k}\in \left\{\mathrm{0,1}\right\}\) is the coefficient that becomes 1 if \({x}_{n}\) is classified into the k^th cluster and otherwise becomes 0. \({r}_{n,k}\) and \({\mu }_{k}\) are generally unknown, and the classification problem can be defined as the estimation of optimal \({r}_{n,k}\) and \({\mu }_{k}\) that minimize the objective function. This procedure is called “K-means clustering” (MacQueen 1967).

The GMM is a type of unsupervised learning in Bayesian modeling that assumes Gaussian distribution of classes (McLachlan and Peel 2000). The Gaussian mixture is defined as follows:where \(N\left(\overrightarrow{{\varvec{x}}}|{\mu }_{k},{\Sigma }_{k}\right)\) is the Gaussian distribution of the k^th mixture component with a mean of \({\mu }_{k}\) and a mean and covariance matrix of \({\Sigma }_{k}\). \({\pi }_{k}\) is the mixing coefficient that sums to 1 (\(\sum_{k=1}^{K}{\pi }_{k}=1\)) such that the integral of the Gaussian mixture \(p(\overrightarrow{{{\varvec{x}}}_{{\varvec{n}}}})\) is 1. In terms of classifying data \(\overrightarrow{{\varvec{x}}}\), the GMM estimates the optimal \({\mu }_{k}\), \({\Sigma }_{k},\) and \({\pi }_{k}\) by fitting the Gaussian mixture and categorizes \(\overrightarrow{{\varvec{x}}}\) into \(K\) clusters. Therefore, the objective function to be minimized is as follows:where \(\widehat{{\mu }_{k}},\widehat{{\Sigma }_{k}}\), and \(\widehat{{\pi }_{k}}\) indicate the optimal values of the mean, covariance matrix, and mixing coefficient, respectively. Because Eq. (4) cannot be solved analytically, numerical optimization is generally employed. In this study, the expectation–maximization algorithm was used for numerical optimization (Dempster et al. 1977; McLachlan and Krishnan 1997).

The K-means classification method is theoretically interpreted as cases wherein the covariance of the Gaussian mixture becomes zero and the classification depends on the distance from the mean (Bishop, 2006). This theoretical implication can be understood based on the distinction whereby the GMM method estimates both the mean and covariance matrix of the k^th class, whereas the K-means algorithm estimates the mean only.

Furthermore, the GMM calculates the probability of the classification from the mean and covariance matrix, and the probability can be used to obtain the optimal number of classes based on an information criterion, such as the Akaike information criterion (AIC) (Akaike 1974) adopted here.where \(\overrightarrow{{\varvec{m}}}\) is the parameters to be estimated, and \(M\) is the number of parameters. AIC enables the comparison of models that include different numbers of classes, and the optimal number of classes has a minimum AIC value. The AIC values were obtained in cases where the number of classes ranged from 1 to 20.

AC classifies data by measuring the distances between data points (Lior and Maimon 2005). At the beginning of this algorithm, each data point forms its own class, leading to \(n\) single-object classes. Based on the measured distances, two classes with the shortest distance merge. The distances are measured again with the newly formed class, and a new class is created with the data points or clusters with the shortest distances. This procedure is repeated until the number of classes is identical to the a priori determined value or all distances are over a certain threshold. In this algorithm, the classification result is influenced by the choice of distance measure. In this study, Ward’s method was applied (Ward 1963), which defines the distance of two clusters \({d}_{{z}_{i},{z}_{j}}\) as follows:where \({z}_{i}\) indicates the i^th cluster and \({z}_{i}\cup {z}_{j}\) indicates a new cluster that is obtained after merging \({z}_{i}\) and \({z}_{j}\). \({x}_{{z}_{i}}\) and \({\mu }_{{z}_{i}}\) are the data points in cluster \({z}_{i}\) and the mean of the data in cluster \({z}_{i}\), respectively. Ward’s method minimizes the variance of data in a new cluster with respect to the variance of data in two existing clusters.

A decision tree is a type of unsupervised classification that uses a flowchart-like graph to represent each test of attributes to explain known classes (Breiman 2001). More specifically, heuristic criteria are applied to attributes as a plausible boundary of different classes, and the tests are applied continuously until classification is complete or the number of tests reaches a certain threshold. Heuristic criteria were selected based on the Gini coefficient \(G\) in this study, as follows:where \(K\) is the total number of classes and \(p(k)\) is the ratio of data that are categorized into a class. One of the major advantages of the decision tree is that the classification flow is explainable because the optimal heuristic criteria can be visualized via a flowchart. For the optimization of the decision tree, K-fold cross-validation was applied, which is a versatile and simple evaluation technique in machine learning. In K-fold cross-validation, the available dataset is divided into K subsets, with one subset used for validation and the other subsets used for training for supervised classification. Data in this study were divided into 10 non-overlapping subsets, resulting in tenfold cross-validation (James et al. 2014).

Figure 5b–g shows the classification results of three algorithms (i.e., K-means, the GMM, and AC), with each algorithm producing two sets of results representing the different preprocessing methods (i.e., normalization of the mean and standard deviation or normalization of the standard deviation). The boundaries of each class are depicted as horizontal lines in Fig. 5a. Based on the AIC values obtained from the GMM, six classes were determined to be optimal. Therefore, classification with six classes was performed for all three algorithms. Figure 5b–g shows that the classification results differed depending on the algorithm used, whereas the preprocessing strategies did not significantly influence the classification results. The reason for the variance in classification results among algorithms might have been each algorithm’s different mathematical background. For example, Euclidean distance was used in the K-means algorithm, whereas Eq. (6) was used as a distance measure in the AC algorithm. Because no universal answer to unsupervised classification exists, different mathematical backgrounds can yield classification results from different viewpoints. Thus, an integrated interpretation of the classification from the different unsupervised algorithms may facilitate a more detailed extraction of the features inherent in the data.

Of the three classification results, the result of GMM classification is relatively easy to interpret. The first class (A′ and A″) corresponds to the shallowest synthetic QI (a blue line in Fig. 5a), and the second class (B′ and B″) roughly corresponds to the depth overlapping the shallowest and next shallowest synthetic QI values (blue and green lines in Fig. 5a). The following classes produced by the GMM also correspond to the QIs of various mineral types. Notably, this GMM classification result was similar to that of K-means clustering, possibly because the two algorithms share a similar mathematical background. On the other hand, the AC classification result was unique. The first class (A‴) covered a broader depth range compared with those of the K-means (A′) and GMM (A″) classifications, which corresponds to the shallowest synthetic QI (a blue line in Fig. 5a), and the classes at subsequent depths were similar to those from either the K-means or GMM classification. For example, the depth ranges for the classes C‴ and C″ were approximately equal, and the depth ranges for the classes D‴, E‴, and F‴ were similar to those for D′, E′, and F′. Overall, we showed that the classification results of the three algorithms were consistent with the distribution of the synthetic QI values. Thus, in the following application to real data, we used all three algorithms and compared their results to obtain the features buried within the data.

For preprocessing, the standard deviation of the data was set to a value of 1 to normalize the data variation. The AIC values obtained from the GMM were larger for low numbers of classes, and the value decreased with an increase in the number of classes (Fig. 6). However, if the number of classes exceeds 7, the AIC values increase again (Fig. 6). Because low AIC values relate to a statistically optimal model, the class number with the minimum AIC value is optimal, which in this study corresponds to seven classes (Fig. 6). Because the optimal distribution of classes approximately followed depth, the classes were labeled A to G (shallow to deep) (Fig. 7). Single, double, and triple quotation marks attached to the class names in Fig. 7 indicate that the classes were obtained using the K-means, GMM, and AC algorithms, respectively.

Classification using K-means clustering and the GMM was similar. The shallow layer of N19-HA-1, down to approximately 400–500 m (in the Quaternary layer), was divided into the classes A′, B′, C′, and D′ (GMM: A″, B″, C″, and D″) (Fig. 7b and c). However, the same depth range in N19-HA-2 was categorized into the classes A′ and D′ (A″ and D″) (Fig. 7f and g). Depths from 450 m to 1300–1500 m were categorized as classes E′ and F′ (E″ and F″) (Fig. 7b, c, f and g). Most of the depth distribution for classes E′ and E″ was shallower than that for classes F′ and F″, and was found in the first, second, and third Tertiary layers (Fig. 1b). Most of the minerals in classes F′ and F″ are found in the fourth Tertiary layer. Classes G′ and G″ included the deepest minerals, with depths of > 1500 m in N19-HA-1 and 1400 m in N19-HA-2 (Fig. 7b, c, f and g). Notably, the classification result is consistent with the resistivity logs (Fig. 7a and e). For example, classes A′, B′, C′, and D′ (A″, B″, C″, and D″) correspond to a low resistivity zone (< 10^1.5 Ω∙m) in both N19-HA-1 and N19-HA-2, and the resistivities of classes E′ and E″ were approximately 10² Ω∙m (Fig. 7).

Based on classification using the AC algorithm, the shallow part of N19-HA-1, down to approximately 400–500 m, was grouped into three classes (A‴, B‴ and C‴ in Fig. 7d). Comparison of the classification results across AC, K-means clustering, and the GMM showed that class B‴ included classes B′, B″, C′, and C″ from the K-means and GMM classifications (Fig. 7b, c and d). On the other hand, the class D‴ from AC classification did not correspond to any of the classes of the K-means and GMM classifications (Fig. 7). Class E‴ of AC roughly corresponded to classes F′ (F″) and E′ (E″) of the K-means (GMM) classification, whereas class F‴ did not correspond to any classes of the K-mean and GMM classifications (Fig. 7). Class G‴ of AC was comparable with class G′ (G″) of the K-means (GMM) classification (Fig. 7). Therefore, compared with the K-means and GMM classification results, unique information obtained from the AC algorithm were found in classes D‴ and F‴.

Based on the analysis of the synthetic data in Sect. 3.1 and the agreement with K-means clustering, we relied on GMM classification and illustrated cross-plots of the cluster classification using the GMM (Fig. 8). Figure 8 shows that most samples in class A″ were associated with large QI values corresponding to smectite and a certain amount of anhydrite. The low resistivity of class A″ minerals may be attributable to smectite. Class B″ contained relatively high concentrations of cristobalite, marcasite, and alunite compared to class C″, which contained kaolinite in addition to smectite (Fig. 8). Class D″ contained smectite, but to a lesser extent than in class A″, and characteristically contained tridymite, clinopyroxene, and magnetite (Fig. 8). The clay minerals in class E″ were largely chlorite and small amounts of sericite (Fig. 8). In addition, the QI values of wairakite and carbonate in class E″ were larger than those in the other clusters. Epidote was only observed at depths below that of class E″ (Fig. 8). Class F″ was characterized by larger amounts of sericite than those found in the other clusters, although smaller amounts of chlorite and anhydrite were also present in class F″ (Fig. 8). Class G″ also contained a certain amount of chlorite and was characterized by the presence of epidote, prehnite, biotite, cordierite, talc, magnetite, and ilmenite (Fig. 8).