Research on undesirable events in offshore oil wells is not a single anomaly detection problem, but involves three levels: anomaly detection, event recognition, and early prediction. This paper defines these tasks under the 3W label structure. The 3W data are organized as multivariate time-series instances. Version-1.0.0 of the 3W Dataset contains 1984 CSV instances and 8 process variables, and includes missing, frozen, and outlier characteristics from real industrial data.

An undesirable event refers to an operating condition that causes an oil well to deviate from normal production and may lead to production loss, increased maintenance cost, or safety risk. In the 3W label system, normal operation is coded as 0, steady-state undesirable events are usually coded as 1 - 8, transition states are coded as 101 - 108, and unknown states are coded as NaN. Among them, 101 - 108 can be mapped to the transient precursors of the corresponding steady-state events 1 - 8 (Table 1).

Anomaly detection corresponds to a binary classification task: given a time window, the model determines whether it deviates from the normal production pattern. A window containing only valid label 0 is labeled normal; a window containing steady-state event labels 1 - 8 or transition labels 101-108 is labeled anomalous. Windows containing a large proportion of NaN values are not used for supervised loss, or they participate only in unsupervised representation learning through masking.

Event recognition corresponds to a multi-class classification task: under the condition that an anomaly has been detected, the model further determines which type of undesirable event is more likely. Specifically, steady-state labels are used as classes, while transition label 100+e is mapped to the same event e so that the transient segment and the steady-state segment remain semantically consistent.

Table 1. Abnormal state types in the 3W dataset [4].

Early prediction corresponds to an early-warning task: before the steady-state undesirable event begins, the model predicts event risk and type based on multivariate changes in the late normal stage or in the transition segment. Its core indicators are not ordinary window accuracy, but the lead time of the first valid alarm relative to the onset of the steady-state event, the early-detection rate, and the false-alarm rate.

From the perspective of traditional machine learning, the survey by Chandola et al. [11] and the review of time-series anomaly detection by Blázquez-García et al. [12] indicate that anomaly detection methods can be divided into four categories: statistical-boundary-based methods, density-estimation-based methods, distance-measurement-based methods, and normal-domain-learning methods. In offshore oil well scenarios, these methods have long occupied an important position. They require relatively low sample sizes and annotation quality, and they are also easier to transform into prediction rules in industrial systems.

The Local Outlier Factor proposed by Breunig et al. [13] identifies outliers based on local density differences. The Isolation Forest proposed by Liu et al. [14] amplifies the separability of abnormal points through random partitioning, whereas the one-class support vector machine proposed by Schölkopf et al. [15] realizes novelty detection by learning the support domain of normal samples. These models jointly form the baseline system for early 3W anomaly detection and also explain why Fernandes. [7] found that LOF and OCSVM remain competitive in multiple groups of experiments. However, the review of condition monitoring and prognostics by Jardine et al. [16] has pointed out that reliable diagnosis of complex industrial systems ultimately requires stronger time-series representation and higher-level anomaly detection.

Deep learning has promoted the transformation of industrial time-series anomaly detection from shallow discriminators to end-to-end representation learning. The review of time-series classification by Fawaz et al. [17] indicates that convolutional networks are better at extracting local patterns, recurrent networks are more suitable for modeling sequential dependence, and hybrid networks often show stronger adaptability to complex industrial data. Malhotra. [18] and Hundman. [19] demonstrate, respectively, from general industrial sequences and spacecraft telemetry scenarios, that LSTM-based prediction error or reconstruction error can effectively characterize abnormal behavior.

In unsupervised and semi-supervised directions, OmniAnomaly proposed by Su et al. [20], MAD-GAN proposed by Li et al. [21], DAGMM proposed by Zong et al. [22], and USAD proposed by Audibert et al. [23] have become representative methods for industrial multivariate anomaly detection. These methods improve the modeling capability of complex normal patterns from the perspectives of stochastic recurrent modeling, generative adversarial learning, joint density estimation, and dual-autoencoder adversarial training, respectively. However, they also face problems such as training instability and sensitivity to missing values.

In recent years, Transformers and graph neural networks have further expanded the modeling boundary of anomaly detection. The Transformer proposed by Vaswani et al. [24] provides a general attention framework for long-dependency modeling. Informer proposed by Zhou et al. [25] reduces the computational overhead of long-sequence processing through sparse attention. Anomaly Transformer proposed by Xu et al. [26] uses association discrepancy to characterize anomalous points. TranAD proposed by Tuli et al. [27] enhances the efficiency and stability of multivariate anomaly diagnosis. DCdetector proposed by Yang et al. [28] strengthens anomaly representation from the perspective of contrastive learning and the modeling of interactions among variables. Compared with models that encode only along the time dimension, such methods are more suitable for industrial scenarios because sensors often correspond to clear physical coupling relationships.

For offshore oil wells, this means that future high-performance models are unlikely to arise from the extreme deepening of a single paradigm. Instead, they are more likely to come from systematic designs that integrate missing awareness, dual-domain representation, graph-temporal fusion, and multi-task prediction.

This paper proposes SDG-Former (Missing-aware Multi-scale Time-frequency Graph Transformer) as a conceptual framework for predicting undesirable events in offshore oil wells. Its objective is to translate the data challenges of the 3W Dataset into verifiable model design hypotheses: the coexistence of long-term dependence and local abrupt changes, the coexistence of variable coupling and structural missingness, the coexistence of anomaly scarcity and event diversity, and the coexistence of engineering deployment requirements and deep-model complexity.

SDG-Former is constructed around the 3W Dataset and offshore oil well scenarios with four core modules: first, missing-aware input encoding based on sliding windows; second, dual-domain representation that preserves both time-domain evolution and frequency-domain disturbances; third, a graph-temporal Transformer backbone for learning variable associations and cross-scale dependencies; and fourth, multi-task output heads for anomaly detection, event recognition, and lead-time evaluation.

For the problem of inconsistent sensor availability in offshore oil wells, this paper does not simply impute all missing values as a single value. Instead, it constructs a value matrix X, a missing mask, and a time-interval input Delta X for each time step. The purpose is to enable the model to distinguish between a genuinely low observation and the temporary unavailability of a variable, thereby avoiding the mislearning of missingness patterns as stable discriminative features.

Considering further that the available sensor sets may differ significantly across wells and files, this paper first generates structural missingness based on the available sensor set, and then uses a lightweight router to determine which parameter subspace the sample should enter. Compared with training a complete model separately for each missingness pattern, this shared-backbone and missingness-adaptive strategy preserves structural differences, avoids the unlimited expansion of the model pool, and is more suitable for the large sample size and significant heterogeneity of the 3W data.

If modeling is performed only in the time domain, short-term oscillations, phase shifts, and multi-period disturbances may easily be compressed into superficially smooth trends. If modeling is performed only in the frequency domain, the order and duration structure of anomaly development may be ignored. Therefore, this paper adopts a dual-domain representation strategy. On the time-domain side, the window sequence is divided into several temporal patches to reduce the computational overhead of long-sequence attention. On the frequency-domain side, short-time Fourier transform (STFT) and three-band statistics are applied to each variable window to explicitly retain frequency clues related to typical anomalies such as mechanical oscillation, choke fluctuation, and flow instability.

Temporal patches emphasize the process of anomaly development, whereas frequency-band statistics emphasize the form of abnormal vibration. These two representations are highly complementary in industrial scenarios. In offshore oil wells, for example, severe slugging and flow instability often show stronger periodicity in the frequency domain, whereas choke restriction and rapid productivity decline rely more heavily on time-domain trends and variable linkage relationships. Therefore, dual-domain representation is closer to the detection logic of practical engineering than single-domain modeling. Nevertheless, dual-domain representation remains a design hypothesis to be validated by experiments, rather than a conclusion empirically verified in this paper.

At the backbone level, this paper adopts a three-stage structure consisting of graph structure learning, temporal self-attention, and cross-domain fusion. A learnable adjacency matrix and a graph initialized by physical priors are used to model the relationships among downhole pressure, upstream and downstream choke pressure, gas-lift flow rate, and related temperature variables. The graph structure is not directly fixed; instead, it is allowed to undergo lightweight adjustment during training so as to adapt to dynamic associations across different wells and operating conditions.

In the temporal dimension, a hybrid encoder combining sparse self-attention and local convolution is introduced. Sparse self-attention follows the efficient design of Informer [25] for long-sequence processing, while local convolution is used to preserve short-term abrupt changes and edge information. This avoids the over-smoothing of pure Transformer models in industrial short-window tasks and mitigates the insufficient long-distance dependency capture of pure convolutional models.

Finally, two gating units, namely sensor attention and frequency-band attention, are designed in the fusion layer. The former dynamically measures which variables are more important for the current judgment, while the latter distinguishes the frequency region in which the current anomaly is mainly expressed. Compared with Anomaly Transformer [26], which focuses on locating anomalies by association discrepancy, this design embeds interpretability into the model structure, thereby providing variable-level and frequency-band-level auxiliary evidence for subsequent engineering analysis.

To demonstrate that the core modules of SDG-Former are not simply superimposed, this paper further summarizes the basis for key design choices and the hypotheses to be verified from three perspectives: data characteristics, model structure, and task objectives. Specifically, the window length mainly serves the needs of transition process coverage and real-time early warning; the time-frequency dual-domain representation is used to simultaneously characterize trend changes and oscillation disturbances; the learnable adjacency structure is used to express the potential correlations between multivariable sensors; and the multi-task loss is used to unify the objectives of anomaly detection, event recognition, and early prediction(Table 2). The relevant design choices, their basis, and the hypotheses to be verified are shown in Table 2.

Table 2. Key SDG-former design choices, rationale, and hypotheses to be validated.

Continued

This paper uses the 3W Dataset [4] as the core benchmark. The dataset contains undesirable-event instances from naturally flowing offshore oil wells. It is multivariate, time-series-based, rare-anomaly-oriented, and clearly grounded in engineering scenarios. Therefore, it is suitable for anomaly detection, event classification, and early-warning research (Table 2). Existing studies based on the 3W Dataset indicate that its research value lies not only in the inclusion of multiple anomaly classes but also in its data characteristics: coexistence of missing and frozen values, significant class imbalance, and large differences in duration and dynamic characteristics among different instances (Table 3).

Table 3. Core features of the 3W dataset and their modeling implications.

Data processing is performed by sliding-window segmentation. For missing values, neighboring-time imputation is used when the missing proportion is low; if missingness shows a structural distribution, the missing mask is preferentially retained. The sliding-window length is set to 120 - 192 sampling points according to anomaly duration, and the step size is set to half of the window length in order to balance the number of samples and temporal resolution.

The main experiment uses instances with all labels 0 - 8. Ablation experiments may separately report results using only real instances and using real, simulated, and hand-drawn instances together (Table 3). All sliding windows must inherit their source file ID and event ID. Training, validation, and test splits should use file ID as the minimum grouping unit. If well IDs can be parsed from file names, well-level grouping should be preferred; otherwise, file-level GroupKFold should be adopted. Five-fold GroupKFold is recommended. All overlapping windows generated from the same file must appear in the same fold, and random splitting by window is prohibited.

Missing-value processing follows the rule of “short-gap imputation, long-gap masking, and structural-missingness preservation.” When a continuous missing segment lasts no more than 10 seconds, linear interpolation or forward/backward neighboring imputation within the same file is used. When a continuous missing segment lasts more than 10 seconds but no more than 60 seconds, the imputed value is used only as a numerical placeholder, while the missing mask M = 0 and the time interval are retained. When a continuous missing segment lasts more than 60 seconds, or when the missing proportion of a variable in a file exceeds 30%, it is no longer treated as an ordinary gap but as a structurally unavailable variable. In this case, only zero-filled numerical placeholders, the missing mask, and the time interval are used, and the value is not regarded as a real observation.

Frozen values need to be treated separately because freezing is not ordinary missingness but information distortion caused by a sensor holding a constant value for a period of time. It is recommended to detect consecutive repeated values for each variable: if a variable remains exactly the same for no fewer than 60 consecutive sampling points, or if the absolute value of the first-order difference after normalization remains continuously lower than 10⁻⁴, the interval is marked as a frozen segment. Frozen-segment values are not directly deleted. Instead, a frozen mask F is retained in the input, and the segment is down-weighted or excluded when calculating statistical features, STFT, and supervised loss. Normalization parameters must be fitted only on the training set within each cross-validation fold to avoid test leakage. For continuous variables such as pressure, temperature, and flow rate, robust standardization based on normal windows in the training set is recommended:

where median train and IQR train are estimated only from normal windows in the training set, and epsilon is used to avoid division by zero. For variables with natural ranges, such as valve opening, min-max scaling based on the training-set range or physical range can be used. For binary or multi-valued valve states, the original discrete meaning is retained and a state mask is added. The validation and test sets only apply the parameters fitted on the training set and do not re-estimate them.

Comparative models cover three categories. The first category consists of traditional machine learning baselines, including LOF [13], Isolation Forest [14], OCSVM [15], and Random Forest [5]. The second category consists of common deep learning baselines for industrial anomaly detection, including OmniAnomaly [20], MAD-GAN [21], DAGMM [22], USAD [23], and LSTM autoencoders [18][19]. The third category consists of recent graph-temporal and Transformer baselines, including Anomaly Transformer [26], TranAD [27] DCdetector [28], and recent improved graph-temporal models [29].

The evaluation metrics include Accuracy, Recall, F1, ROC-AUC, and PR-AUC. For rare-anomaly scenarios, PR-AUC often better reflects the model’s recognition quality for minority-class windows than ROC-AUC. For engineering early-warning tasks, the average early-detection ratio and the first detection time of the anomaly should also be added.

In terms of experimental settings and evaluation, the training stage adopts the AdamW optimizer, cosine-annealing learning rate, class reweighting, and early stopping. During inference, anomaly scores, variable attention, and frequency-band attention are retained for subsequent manual review and engineering interpretation. For edge-deployment scenarios, lower latency can be obtained by reducing the number of Transformer layers, shortening the patch length, and pruning the frequency-band branch.

Traditional methods usually perform well in two types of scenarios: cases where anomaly boundaries are clear and statistical features are obvious, and cases where sample sizes are limited and deep models are prone to overfitting. Random Forest [5], LOF [13], and Isolation Forest [14] have clear advantages in this regard because they have low training costs, are friendly to small samples, and are less sensitive to hyperparameters and optimization procedures.

However, for adverse-event prediction in offshore oil wells, the limitations of traditional methods are also clear. They have difficulty directly using the continuously evolving temporal structure in transition stages and often rely on window-level summary statistics. Their ability to express variable coupling is limited, and they may simplify multivariate linkage anomalies into several univariate shifts. When structural missingness and cross-well operating-condition changes exist, shallow models are more likely to mistake distribution habits in the training set for generalizable rules.

In comparison, the SDG-Former proposed in this paper has the following expected design advantages: it explicitly exposes weak precursors from both the time and frequency domains through dual-domain representation, thereby reducing reliance on manually designed statistics; it learns dynamic relationships among variables through graph-temporal fusion rather than treating each sensor as an independent channel; and it incorporates structural missingness into the model design through missingness routing instead of passively masking it during preprocessing.

Compared with deep learning models, this paper does not regard the reconstruction of normal patterns as the only objective. OmniAnomaly [20], MAD-GAN [21], DAGMM [22], and USAD [23] are representative models in industrial multivariate anomaly detection. However, such models usually rely on reconstruction error or energy functions to characterize anomalies. Therefore, when normal operating conditions themselves fluctuate substantially and normal patterns differ significantly across wells, threshold transfer becomes difficult.

Compared with graph-temporal and Transformer models, Anomaly Transformer [26], TranAD [27], DCdetector [28], and recent improved graph-temporal models have significantly improved multivariate anomaly detection performance. Nevertheless, most of them still assume fixed input dimensions and give insufficient consideration to changes in sensor availability. In addition, many models focus on detecting anomalous points, while unified modeling of event-type recognition and early-warning lead time remains insufficient.

For offshore oil well scenarios, the most interpretable result is often not a small improvement in a single AUC value, but whether the model can remain stable in cross-well testing, provide sufficiently early prediction before steady-state anomalies occur, and output key variables that are consistent with engineering knowledge (Table 4).

However, the proposed method also has limitations. Dual-domain representation and graph-temporal fusion increase implementation complexity and impose higher requirements on data preprocessing protocols. If the training data cover too few well types and operating conditions, the model may still be affected by distribution shift. Although multi-task learning improves task completeness, it may also introduce objective conflicts, requiring careful adjustment of loss weights and training strategies.

Table 4.Comparison of existing model routes and SDG-former design hypotheses.