Introduction

Reliable storage systems are essential for long-term energy security, and energy storage is a key factor in economic growth and sustainability. The best geological storage media for storing energy resources such as compressed air, natural gas, oil, hydrogen, nuclear waste, and even renewable energy is salt rock1,2,3,4. Salt rock is especially well-suited for these uses due to its special mechanical and physical characteristics, such as low porosity, low permeability, and the capacity to mend when injured. These qualities make salt formations suitable for long-term energy storage, especially when paired with advantageous rheological behavior5,6,7.

Nevertheless, despite its benefits, creep deformation is a major problem with salt rock formations. A time-dependent phenomenon known as creep occurs when rock gradually deforms under continuous tension or strain, frequently producing significant changes over long periods. Even under relatively moderate stress, salt rock is especially vulnerable to substantial creep deformations because of its high ductility and non-linear deformation properties4,8,9. According to6 and10 this deformation may impair the salt caverns’ structural integrity and decrease their storage capacity, increasing the risk of leakage or collapse. For long-term storage sites to be stable and safe, it is crucial to comprehend and forecast the creep behavior of salt rock4,11,12. The creep behavior of salt rock is unique and can be divided into three stages: primary creep, in which the strain rate falls as the material adapts to the load; secondary creep, which is a steady-state stage with a constant strain rate; and tertiary creep, which accelerates until the material fails. Samples of high-quality salt rock with uniform surface finish and dimensions were created for this investigation. These samples were put in a triaxial testing device and exposed to confining pressure and axial load under controlled conditions. High-precision sensors were used to record creep data of axial strain, and tests were conducted for up to 22 days at greater pressures to catch only the primary creep stages. A crucial metric for evaluating the geomechanical stability of salt deposits and the structures that are connected to them, like storage caverns and boreholes, is axial strain, which is the ratio of the length change to the initial length under stress. Since distinct salt facies and stress conditions at varying depths must be taken into account during drilling and excavation, it is especially crucial to accurately predict axial strain. The processes causing salt rock creep have been the subject of numerous investigations, and several models have been created to forecast how the phenomenon will evolve under various stress and temperature scenarios13,14.

Recent developments, particularly for mining and energy storage applications, have greatly expanded our knowledge of subsurface deformation mechanisms and monitoring methods15. Developed a constitutive model for compressive damage that integrates compaction effects and linear energy dissipation laws for brittle geomaterials like coal. This model helps create safer geotechnical designs by improving predictions of mechanical reactions and fracture propagation under loads16. Investigated the permeability progression and creep behavior of coal pillar dams employed in subterranean water reservoirs around the same period. Their results demonstrated the crucial relationship between fluid migration and mechanical stress, highlighting the necessity of considering hydraulic and mechanical reactions when developing underground infrastructure. According to Yin et al.17, combining deep learning methods with image processing has enhanced environmental monitoring in mines. Their algorithms made it possible to classify dust pollution automatically and in real time, greatly improving compliance tracking and danger identification.

Although18, highlighted how the mineral richness and composition of clay affect soft rocks’ rheological and mechanical behavior. By tackling anisotropy and heterogeneity, their work helped to make constitutive modeling more realistic. Furthermore, they used acoustic emission (AE) techniques to provide early warning indications of instability by dynamically monitoring the breakdown of soft rocks saturated with water. Based on this19, created a model that looks at the elastic energy released in rock masses over time, offering a framework for figuring out the best excavation speeds. This effort aimed to reduce the chance of failure and avoid unexpected energy accumulation during mining and tunneling operations. To further improve failure prediction20, found that strain localization and gradual deformation occur before large-scale collapse in brittle materials. Their laboratory-scale tests support the necessity of constant monitoring in geomechanical designs.

Recently21, showed that long short-term memory (LSTM) neural networks are better at predicting slope stability than conventional models like random forests and support vector machines. These deep learning models provided more dependable tools for managing geohazards by enhancing temporal sensitivity and sequence-based forecasting.

In order to forecast the long-term behavior of salt rocks, Hou et al.1 and Zhao et al.6 performed uniaxial cyclic loading creep tests and put forth a novel constitutive model. They looked into how three cyclic conditions—average stress, half-amplitude, and cyclic period—affect the way salt rocks creep. Furthermore22 ,created a nonlinear creep damage constitutive model for argillaceous salt rock using uniaxial creep experiments, which consider the accelerated creep phase as a combined loading and creep process. Test results show that the model, which applies to mudstone, high-purity salt rock, and mudstone-salt mixtures, performs better than conventional models for rapid creep.

This offers crucial theoretical backing for interbedded energy storage in salt caverns to operate safely. Power law, the logarithmic model, Maxwell’s spring-dashpot model, and other conventional or analytical models have provided important insights into how salt rock deforms under continuous stress6,23. Although these traditional models have improved our knowledge of salt rock creep, they frequently fail to accurately predict long-term behavior. Promising substitutes for forecasting the creep response of salt rock are provided by recent developments in artificial intelligence (AI) and machine learning. The intricacies of rock deformation over a long period can be captured by sophisticated deep neural network (DNN) models, such as Neural Basis Expansion Analysis for Time Series (N-BEATS), Temporal Convolution Networks (TCN), Recurrent Neural Networks (RNN), and Transformers (TF). Following a resampling operation, these models can produce more accurate predictions of future deformation trends than traditional methods by training on experimentally acquired recorded creep data of axial strain responses with non-uniform sampling obtained from CSIRO’s Geomechanics and Geophysics Lab for various confining stages at room temperature conditions24,25.

In order to improve on earlier studies, this study compares the effectiveness of sophisticated deep-learning algorithms with conventional analytical models in forecasting the creep reaction of salt rock. In particular, we concentrate on the Frome rock salt formation, which is Devonian in age and may be used for long-term energy storage. We evaluate these models’ relative predictive accuracy for creep behavior by examining axial strain data. Statistical metrics like the R2 score and root mean squared error (RMSE), mean average error (MAE), mean average percentage error (MAPE in %), and symmetric mean average percentage error (SMAPE in %) will also be used to compare the predictions of DNN models with those of traditional models to assess their dependability.

Ultimately, this research seeks to enhance the accuracy and reliability of geomechanical models, contributing to the safe and efficient design of salt rock storage systems.

Material and methods

Study area & multi-stage triaxial test (MSTL)

For this study, the core samples were taken from the onshore FRDDH001 vertical well at the top edge of an interpreted salt diapir on the Jugurra Terrace, near the Fitzroy crossing in the Canning Basin, north of Western Australia26. This well intersects over 600 m of early Paleozoic rock salt, evaporites, and dolomitic breccia24,25.

This study used multi-stage triaxial (MSTL) tests on core samples extracted from salt-rock facies at around 1100 m, with varying confining pressures. Creep monitoring was done at each pressure stage under a constant axial load value, with the deviatoric stress set to 75% of the estimated yield stress. Radial strain was minimal and ignored due to its high sensitivity to temperature fluctuations and low signal-to-noise ratio. In contrast, though temperature fluctuations affected the axial strain, it showed a clear trend due to its high signal-to-noise ratio24,25,27. A typical four-stage triaxial test on a rock salt sample involves loading and unloading stages at four different confining pressures. The third stage matches the native in situ condition of the rock, usually at 25 MPa for clean salt facies. The first and second stages occur at lower pressures than those estimated for the recovery depth, while the fourth stage is at a higher pressure. These stages are performed at an axial strain rate of 10–6 s−1 with pressure changes between stages at a rate of 0.5 MPa/min24,25. Figure 1 shows axial creep strain and temperature fluctuations recorded at each stage of the MSTL test for salt rock. The creep rate trend changes after 0 to 10 days, depending on the applied stress and rock salt facies. The slope of the creep curve at the end of a test can serve as a proxy for estimating the steady-state creep rate. The exponential (tertiary) creep and failure stage was not reached in the four tests conducted on rock salt samples at room temperature and under deviatoric stress, approximately 75% of the short-term yield stress at the given confining pressure.

Fig. 1
figure 1

The MSTL creep data of axial strain and temperature sensor responses over time, whereas axial strain (milli-strain) and temperature (°C) with varying time (days) at various confining pressure levels such as for (a) 5MPa, (b) 15MPa, (c) 25MPa, and (d) 35MPa, respectively.

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A detailed summary of the dataset, including its duration and frequency statistics, is provided in Table 1.

Table 1 Time series statistics including training and test sets.
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Data collection and preparation

High-quality salt rock facies samples, prepared with consistent dimensions, were used in this study. These samples were placed in an MSTL testing apparatus, where they were subjected to controlled axial loads and confining pressures in a room-level experimental setup. Figure 2 shows how temperature variations (beyond diurnal) at about 300 h affect (bump) the series 3 data set at 25 MPa. However, after testing statistical techniques like Granger causality and Augmented Dickey–Fuller (ADF), we first ignored temperature as a component in this study. Particularly at 25 MPa, there is no discernible correlation between temperature and axial strain. The only discernible pattern was that axial strain appeared to decrease as temperatures grew. We implemented a univariate approach, with the main elements of this study being axial strain and time (index). Future multivariate time series analysis could explore the possibility of adding temperature as an extra variable to enhance forecasting and pattern recognition.

Fig. 2
figure 2

Four-time series creep datasets for axial strain responses over time with different time stamp lengths (hours) at various pressure levels.

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The raw axial strain data from MSTL tests underwent meticulous preparation to ensure data integrity. Inconsistencies and outliers were removed, and the datasets featured non-uniform time sampling (in milliseconds) at various constant confining stages4,8,9,26. A resampling operation was performed to streamline the data, converting high-frequency data into uniform hourly intervals. This reduced data volume and computational load while preserving significant trends and patterns. Normalization techniques were also applied to standardize the datasets, ensuring consistency across all data points.

The performance of predictive models was rigorously evaluated using validation metrics, allowing the identification of best-fitting parameters to ensure accurate predictions.

Predictive modeling architecture

In this study, we applied DNN and statistical baseline models to forecast the time-dependent deformation of salt rock, focusing on the axial strain response under varying pressures and temperatures. The resampled time series data, taken at fixed one-hour intervals, incorporated four time series datasets of axial strain responses over time. Before applying DNN and traditional forecasting models, we performed Seasonal-Trend decomposition (STL) using LOESS to separate the data into its original, trend, seasonal, and residual components28,29,30. Key statistical tests, such as Granger causality, ADF, and wavelet coherence plots (WCF), were applied31,32,33. The Granger test analyzed the influence of temperature on axial strain over time, while the ADF test examined the data’s stationarity.

We utilized various DNN models for forecasting, including N-BEATS, Temporal Convolutional Networks, Transformers, and Recurrent Neural Networks. Statistical models such as Exponential Smoothing (ES), Trigonometric Box-Cox, ARMA (autoregressive moving average), trend and seasonal components (TBATS), automatic autoregression integrated moving average (AutoARIMA), and Theta were employed34,35,36,37,38. Analytical models—power law, Kelvin, Burger, logarithmic, and Maxwell spring dashpot—were used to complement the deep learning forecasts for predicting axial strain over extended periods. Model performance was evaluated using key metrics such as RMSE, MAE, MAPE (%), and SMAPE (%)1,39. The results were further validated using Multiple Comparisons with the Best (MCB) tests based on mean rank comparisons39.

This comprehensive workflow, involving STL, causality, statistical tests, and rigorous model evaluation, ensured robust and reliable forecasting. Our findings provide valuable contributions to geoscience applications with statistically validated and effective models for real-world forecasting. Our proposed workflow summarizing these steps is presented in Fig. 3.

Fig. 3
figure 3

Proposed workflow for predictive modeling of salt rock’s creep deformation.

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Time series decomposition & statistical tests

Observing and assessing seasonality, trends, and noise is possible when examining a time series’s temporal data40,41,42,43. Each element can be investigated separately to better comprehend the data. However, two more elements can be seen when examining the data as a whole: stationarity and autocorrelation. It is essential to comprehend these elements while choosing a time series model to get precise forecast outcomes. Seasonal-trend decomposition with LOESS is a reliable technique for breaking down time series, where LOESS is a nonlinear relation estimation approach. The STL method estimates the trend components—a flexible estimate of the underlying pattern in the data—using a locally weighted regression technique called LOESS. The time series data Y(t) at a specific time (t) is broken down by STL into three components: a trend component T(t), a seasonal component S(t), and a remainder or residual component R(t). The findings are displayed as follows in Eq. (1)39,43,44:

$$Y\left( t \right) = T\left( t \right) + S\left( t \right) + R\left( t \right),\; t = 1,2,3 \ldots N$$
(1)

where Y (t) is the actual time series dataset with the function of varying time t = 1,2,3…N.

ADF test, the Granger causality test, and WCP to analyze time series datasets comprehensively31,45. The ADF test ensures stationarity in the analyzed time series, which is crucial for reliable time series modeling. The Granger causality test identifies causal relationships between series, enhancing forecasting accuracy. Wavelet coherence plots visualize correlations across time and frequency domains, providing deeper insights into dynamic interactions. These methods offer a robust way to understand dependencies, improve model performance, and inform decision-making in various applications such as economics, finance, environmental, and geoscience studies, especially related to long-term storage of energy—wavelet coherence is particularly used to detect common time-localized oscillations in non-stationary signals. If one time series influences another, one may use the phase of the wavelet cross-spectrum to identify the relative lag within the time series datasets.

Time series forecasting (TSF)

A time series consists of chronologically ordered observations recorded at fixed intervals, with the forecasting problem focusing on predicting future values based on past observations (lags). Given historical data \(X = \left\{ {x_{1} , x_{2} ,x_{3} , \ldots x_{T} } \right\},\) and a desired forecasting horizon H, the goal is to predict subsequent values \(\left\{ {x_{T + 1} , x_{T + 2} ,x_{T + 3} , \ldots x_{T + H} } \right\}.\)

Being \(\hat{X} \left\{ {x_{1} ,x_{2} ,x_{3} , \ldots ,x_{T} } \right\}\) is the vector of predicted value, While minimizing prediction error (PE) is defined as34,35,38:

$$PE = \mathop \sum \limits_{i = 1}^{h = H} \left| {x_{T + i} - \hat{x}_{T + i} } \right|$$
(2)

Time series can be classified as univariate (single observation) or multivariate (multiple observations). This study focuses on univariate time series analysis of axial strain recorded over time.

In recent decades, machine and deep learning techniques have gained prominence for time series forecasting (TSF), serving as advanced alternatives to traditional statistical methods. TSF is crucial in various fields, including energy consumption, financial forecasting, and anomaly detection35. Effective TSF models must address trends, seasonal variations, and correlations among closely observed values.

Deep neural networks (DNNs) have emerged as powerful tools for TSF, capable of capturing complex non-linear relationships that traditional models cannot. This study evaluates four DNN models—N-BEATS, TCN, RNN, and Transformer—alongside four statistical baseline models—ES, TBATS, Auto ARIMA, and Theta—specifically for forecasting axial strain deformation in geoscience applications.

We aim to provide a reliable benchmark for future studies by comparing these models without requiring refined architectures tailored to specific problems. The proposed domain-independent approaches are intended to guide researchers in effectively tackling forecasting challenges in various contexts.

Global forecasting DNN models

The DNN models—N-BEATS, TCN, RNN, and Transformers offer unique advantages and challenges for modeling nonlinear time series. N-BEATS excels in decomposing time series data into basis functions, and TCNs leverage dilated causal convolutions for capturing temporal dependencies. RNNs maintain a hidden state for sequence modeling, and Transformers utilize self-attention mechanisms for parallel processing and capturing long-range dependencies. In this study, we considered the following deep learning methods for forecasting salt rock deformation description given below;

N-BEATS

N-BEATS32 is a deep neural network architecture designed to forecast time series data by decomposing it into basis functions. This approach enables the model to effectively learn and combine complex patterns for accurate predictions. The N-BEATS model is structured with multiple layers that handle trend and seasonality components, enhancing its performance across various time series datasets. Its interpretability allows time series modeling to generate better forecasts.

TCN

TCN15,29,46 utilizes dilated causal convolutions to efficiently capture temporal dependencies in sequences. Unlike RNNs, TCN relies on these convolutions to consider only past information, thus avoiding gradient explosion and vanishing issues. Due to their parallel processing capabilities and stable training dynamics, TCNs are particularly effective for modeling long-term dependencies and handling large datasets.

RNN

RNNs31,39 model is designed for sequence modeling, maintaining a hidden state that captures temporal dependencies across inputs. This characteristic allows RNNs to process sequences of varying lengths and is crucial for tasks like time series prediction. However, when dealing with long sequences, RNNs face challenges such as gradient vanishing and explosion.

TF

The Transformer model17 is renowned for its self-attention mechanism, which allows it to weigh distinct parts of an input sequence differently. This feature is beneficial for capturing long-range dependencies and handling complex temporal patterns in time series data. Although Transformers were initially developed for natural language processing, their ability to process sequences in parallel and capture global dependencies makes them valuable for time series forecasting. Various adaptations have been made to tailor Transformers for sequential data, improving their performance in this domain.

Statistical baseline models

ES

ES is a basic statistical technique used to smooth out time series data that often exhibits long-term and short-term variability. It applies uniform or exponential weights to past observations, making long-term variability more evident as short-term variability (noise) is removed. Unlike the simple moving average, this model assigns weights that decrease exponentially for older data points. This method is widely used for forecasting because it effectively captures short-term fluctuations and trends while smoothing noise and irregularities30,33.

TBATS

TBATS30,31,33 is a time series forecasting model that can handle intricate seasonal patterns, especially ones with several seasonalities. The data is particularly helpful when non-linear trends, high-frequency seasonality, and variable degrees of seasonality are present throughout the series. ARMA is used to model residuals, while Box-Cox transformations stabilize the variance. It incorporates multiple components such as trend, seasonality, and irregular patterns, whether yearly, weekly, daily, or hourly. It offers robust performance and incorporates automatic model selection and parameter estimation.

Auto ARIMA

AutoARIMA is a model class that predicts a time series’s future values based on its past lagged values (lags) and lagged errors. It is effective for non-seasonal time series exhibiting patterns, not random white noise. ARIMA is agile in adapting to various types of time series data and can be trained on smaller datasets. However, ARIMA may not handle data with large mean shifts and nonlinear time series. AutoARIMA is an automated approach to fitting ARIMA models to time series data. It automatically selects optimal parameters (p, d, q) based on statistical metrics like AIC (Akaike Information Criterion) and BIC (Bayesian Information Criterion), which are statistical measures used to evaluate and select models, simplifying the forecasting process. The model with the lowest AIC and BIC values is considered the best28,33,36.

Theta

In time series analysis, the Theta model36 is a forecasting technique that uses ES33,34. For future trend prediction, it combines level and trend data components. A straightforward linear model and a decomposition technique are used to create this potent time series forecasting technique. The Theta model begins by breaking down a given time series Y(t) into its component parts, such as the trend component, T(t), seasonal component, S(t), and residual component, R(t), as was previously described in Eq. (1). Using the Theta (θ) coefficient to alter the trend component is the main concept of the Theta model during the Theta Transformation. The Theta transformation modifies the trend to change the series’ curvature:

$$Y_{\theta } \left( t \right) = \theta \times T\left( t \right) + \left( {1 - \theta } \right) \times Y\left( t \right)$$
(3)

By strengthening or weakening the trend, this equation alters the original time series.

The series is detrended and deseasonalized when θ = 0, emphasizing the error and seasonal components. In this instance, it generates a series in which the trend is eliminated, leaving only the noise and seasonal components, like

$$Y_{0} \left( t \right) = Y\left( t \right) - T\left( t \right)$$
(4)

When θ = 2, the trend is overstated, which may make it easier to identify the long-term trend. In this instance, it results in a series that highlights a tendency, like

$$Y_{2} \left( t \right) = 2\cdot T\left( t \right) - S\left( t \right) - R\left( t \right)$$
(5)

A basic forecasting technique (often ES or linear extrapolation) is used for every θ-transformed series. These two Eqs. (4) and (5), where θ = 0 and 2, are combined to get the final forecast Y ̂(t + h), which is usually derived by averaging the forecasts from several θ-transformed series:

$$\hat{Y}\left( {t + h} \right) = \frac{1}{2}\left[ {\hat{Y}_{0} \left( {t + h} \right) + \hat{Y}_{2} \left( {t + h} \right)} \right]$$
(6)

where h is the forecasting horizon, and \({\widehat{Y}}_{\theta }(t+h)\) is the forecasted value for time t + h.

With this method, the advantages of both the accentuated trend line (θ = 2) and the deseasonalized trend line (θ = 0) are combined. It is a statistical baseline model that works well for time series forecasting because it strikes a fair mix between capturing the long-term trend and short-term seasonality. It is crucial to comprehend that the model generates the final forecast by transforming the trend and seasonal components in various ways. The Theta model is therefore a straightforward yet effective time series forecasting method36.

A crucial tool for producing accurate forecasts based on past data trends is offered by the four statistical baseline forecasting models mentioned above.

Analytical models

The viscoplastic behavior of materials under creep circumstances can be understood and predicted with the help of analytical models4,8,14,42,43. Several researchers can learn more about the properties of material deformation by fitting these models to experimental data, which improves the precision of long-term trajectory predictions. As earlier discussed, the creeping phenomenon as a function of time, stress, temperature, and material qualities is expressed using empirical or phenomenological equations, which are commonly used to predict creep responses over time. Because it captures the nonlinear relationship between stress and strain rate, the power law model is the most commonly used for understanding the nature of the viscoelastic response of creep. The creep deformation over time under a given stress is forecast by the various analytical models, such as the power law, extended Kelvin, Burger, or spring dashpot model.

In order to effectively depict viscoplastic effects, Burger’s model combines aspects of the Maxwell and Kelvin–Voigt models. These analytical models are appropriate for complex viscoelastic behavior since they take into consideration both instantaneous and time-dependent viscous reactions2,4,8. Time-dependent elastic and viscous trends are also captured by the Kelvin–Voigt model; these trends are especially pertinent for materials with delayed elastic responses, such as polymers and biological tissues. Furthermore, by simulating the time-dependent deformation under various confining pressure circumstances, the Maxwell model—which excels at modeling viscoplastic behavior—is distinguished by its spring-dashpot configuration1,2,4,10,42,43,44,45. The stimulation of creep processes is often facilitated by the combination of the logarithmic and power law models. The axial strain response over time, ϵ(t), as a function of time t, can be expressed via Kelvin, Burgers, Maxwell, Power Law, and Logarithmic analytical models. Table 2 summarizes how these models use empirical creep constants for long-term predictions. Despite being a simplified depiction of real-world creep, the power law model might not adequately account for the intricacies of various materials and circumstances. For salt rock deformation, the related creep data on axial strain responses, as shown in Table 2, were previously utilized to fit popular creep models7,23,45.

Table 2 A list of analytical models for creep responses for long-term forecasting.
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Model implementation and training process

We used deep learning models to predict axial strain time series data methodically (See Darts library developed with the Python programming language). To enable accurate performance evaluation, data preparation entailed splitting the dataset into training and testing sets (Fig. 4). The Number of Epochs, input, and output chunk lengths were two important hyperparameters that were tuned for the model’s performance. To enhance DNN convergence, we scaled each training time series to a range of 0 to 1 once the data was separated. Using a Multi-Input Multi-Output (MIMO) approach, which has been demonstrated to perform better than single-output techniques by preventing error accumulation and increasing computing efficiency, training examples were produced31,34. For example, Fig. 5 shows input–output pairings with a forecast horizon of 12 and a history length of 24 generated by a fixed-size sliding window. To determine the ideal training setup, we also experimented with different batch sizes, learning rates, and normalization strategies; the results are shown in Table 3.

Fig. 4
figure 4

Split data for training (70%) and validation test (30%) for axial strain response time series analysis at different pressure levels.

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Fig. 5
figure 5

A moving window technique example that gathers input/output examples for model training. The models are given an instance in this example with a forecasting horizon of 12 values and an input history window of length 24.

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Table 3 Darts-based default hypertunning parameters for DNN models used in this study.
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For evaluating model performance and directing modifications, the test dataset was essential. The outcomes showed how deep learning models could capture intricate temporal patterns. In order to forecast axial strain responses, we used DNN models (N-BEATS, TCN, RNN, and Transformers), statistical baseline models (ES, TBATS, Auto ARIMA, and Theta), and analytical models (power law, Burger’s, Kelvin’s, logarithmic, and spring dashpot models).

To improve performance, grid search and cross-validation were used to adjust the hyperparameters of each model, which was customized for the time series data.

Model performance

Four statistical criteria were used to assess each model’s performance4,14,28,29;

$$RMSE = \sqrt {\frac{1}{n}\sum\nolimits_{i = 1}^{n} {\left( {y_{i} - \hat{y}_{i} } \right)^{2} ,} }$$
(7)
$$MAE = \frac{1}{n}\sum\nolimits_{i = 1}^{n} {\left| {y_{i} - \hat{y}_{i} } \right|,}$$
(8)
$$MAPE \left( {in \% } \right) = \frac{1}{n}\sum\nolimits_{i = 1}^{n} {\left| {\frac{{y_{i} - \hat{y}_{i} }}{{y_{i} }}} \right|} \times 100,$$
(9)
$$SMAPE \left( { in \% } \right) = \frac{1}{n}\sum\nolimits_{i = 1}^{n} {\left| {\frac{{y_{i} - \hat{y}_{i} }}{{\left( {{\raise0.7ex\hbox{${\left| {y_{i} } \right| + \left| {\widehat{{y_{i} }}} \right|}$} \!\mathord{\left/ {\vphantom {{\left| {y_{i} } \right| + \left| {\widehat{{y_{i} }}} \right|} 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right)}}} \right|} \times 100$$
(10)

where \({y}_{i}\) represents the actual value of axial strain and \({\widehat{y}}_{i}\) represents the forecast value of axial strain obtained from different forecast models, and n represents the length of the series. By definition, the lower the values of these metrics, the better the forecasting models.

The output of the evaluation metrics is summarized in Tables 5 and 6 later in the result section, which detail model performance across four different confining pressures. It is important to note that discrepancies between model outputs and actual values arise from two primary sources: sample error, which can be mitigated through effective outlier processing, and model error, which is linked to the model’s generalization ability, which can limit predictive accuracy. Forecast accuracy is a critical measure that reflects how well the model fits the actual data, underscoring the importance of refining models to improve their predictive capabilities.

Results

Preliminary data analysis

The STL decomposition of the axial strain dataset (Fig. 6a, b) exhibited minimal seasonal variation, with significant shifts in trend and residual components after the time data were converted to hourly intervals. At 5 MPa confining pressure, the residual component displayed pronounced fluctuations, suggesting an influence of pressure on the strain response.

Fig. 6
figure 6

An example of STL decomposition that decomposed data into original, trend, seasonal, and residual components for (a) frequency in seconds and (b) frequency in hours, respectively, at a pressure level of 5 MPa.

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Granger causality analysis (Fig. 7a, b) revealed no significant causal relationship between axial strain and temperature across most conditions. However, at 5 MPa, a p-value of 0.016574 confirmed a significant causal link between the variables. The ADF test (Fig. 7c) further validated the stationarity of the datasets, with all p-values falling below the 0.05 threshold, indicating the data’s suitability for reliable predictive modeling (Table 4). WCP (Fig. 7d) identified clear cyclic patterns and external correlations within the dataset, particularly in lower frequency bands. These coherence magnitudes, ranging between 0 and 1, pointed to strong correlations in specific frequency ranges, demonstrating external influences on axial strain behavior over time. Such findings highlight the dataset’s potential for accurate forecasting in future applications.

Fig. 7
figure 7

An example of a time series for axial strain data is a causality test with different statistical tools such as (a) granger test, (b) granger test with increases lags more data points below or lies at significant levels (p-value less than 0.05), (c) ADF test for stationarily or non-stationary, and (d) WCP respectively at confining pressure condition of 5 MPa.

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Table 4 Statistical analysis summary with STL and causality test for creep data of axial strain responses with varying time at various confining pressure level conditions.
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Outcome from DNN models

Figures 8 and 9 showcase the forecats performance of the N-BEATS model, validated against test samples to ensure accuracy. In comparison, Figs. 10 and 11 highlight the forecasting capabilities of the TCN model across specific time intervals under varying confining pressure conditions. In our analysis of DNN models for time series forecasting, N-BEATS emerged as the top performer, achieving the lowest RMSE, MAE, MAPE (%), and SMAPE (%) values across all evaluated metrics. This model effectively captured complex temporal dependencies, making it suitable for short to medium-term forecasts. Specifically, the average performance metrics for N-BEATS varied as follows: RMSE (0.33–0.95), MAE (0.28–0.96), MAPE (1.45–4.54%), and SMAPE (1.46–4.62%) at various confining pressure levels (5–35 MPa) as summarized in Table 5. In contrast, the TCN ranked second, with slightly higher average error metrics (RMSE: 0.54–1.35, MAE: 0.47–1.17, MAPE: 2.85–6.28, SMAPE: 2.88–6.03). While it outperformed other models in capturing longer-term trends, the Transformer model showed moderate performance, hindered by the limited dataset size. Although RNNs like LSTM and GRU outperformed baseline models, they did not match the accuracy of N-BEATS and TCN.

Fig. 8
figure 8

The proposed N-BEATS model forecasts for (a) 5MPa, (b) 15MPa, (c) 25MPa, and (d) 35MPa, respectively, MSTL creep data of axial strain data, whereas the axial strain is in milli-strain and the time stamp is in hours. The best prediction performance was achieved using evaluation metrics such as RMSE, MAE, MAPE (%), and SMAPE (%).

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Fig. 9
figure 9

The proposed N-BEATS model validation between forecasts against test data for (a) 5MPa, (b) 15MPa, (c) 25MPa, and (d) 35MPa, respectively, MSTL creep data of axial strain data, whereas the axial strain is in milli-strain and the time stamp is in hours. The best prediction performance was achieved using evaluation metrics such as RMSE, MAE, MAPE (%), and SMAPE (%).

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Fig. 10
figure 10

The TCN model forecasts for (a) 5MPa, (b) 15MPa, (c) 25MPa, and (d) 35MPa, respectively, MSTL creep data of axial strain, whereas the axial strain is in milli-strain and the time stamp is in hours. The best prediction performance was achieved using evaluation metrics such as RMSE, MAE, MAPE (%), and SMAPE (%).

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Fig. 11
figure 11

The TCN model validation between forecasts against test data for (a) 5MPa, (b) 15MPa, (c) 25MPa, and (d) 35MPa, respectively, MSTL creep data of axial strain data, whereas the axial strain is in milli-strain and the time stamp is in hours. The best prediction performance was achieved using evaluation metrics such as RMSE, MAE, MAPE (%), and SMAPE (%).

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Table 5 Summary of performance metrics and adjustable training parameters of deep learning models for forecasting the axial strain response creep datasets at various confining stress levels (best highlighted in bold).
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The outcome from the statistical baseline models

The Theta model consistently excels in forecasting the axial strain responses. It achieved the lowest statistical error metrics across the datasets, with RMSE values ranging from 0.53 to 0.85, MAE from 0.47 to 0.80, MAPE from 2.18 to 7.31%, and SMAPE from 2.18 to 7.61% (Table 6). Notably, at a confining pressure of 5 MPa, the Theta model recorded with MAPE of 2.42%, SMAPE of 2.42%, RMSE of 0.55, and MAE of 0.48, as reflected in Fig. 12.

Table 6 Summary of statistical evaluation metrics for statistical baseline models, where bold highlighted for the best models.
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Fig. 12
figure 12

An example from statistical baseline models such as for (a) ES, (b) TBATS, (c) AutoARIMA, and (d) Theta, respectively, at a confining pressure level of 5MPa. In comparison, the theta model outperformed for minimal evaluation metrics error values.

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The ES model also demonstrated strong performance at specific confining pressure levels, achieving the lowest error metrics at 25 MPa and maintaining good accuracy at 35 MPa. The TBATS model effectively handled complex seasonality at 15 MPa but did not consistently outperform the Theta model. While AutoARIMA provided reliable forecasts, it was generally less effective than the Theta and deep learning models, as indicated by its higher evaluation metrics (Table 6). The results clearly illustrate that the Theta model is the most effective choice for accurate predictions across varying confining pressure levels.

Key outcome from analytical models

We employed notable analytical models to validate and support the deep learning models, including the Kelvin, Burgers, power law, spring-dashpot, and logarithmic models, with predicted values illustrated in Fig. 13 across various pressure levels. The power law model demonstrated the best fit with R2 (0.98), yielding creep parameters C (0.10 day⁻1) and k (0.40) (Tables 7 and 8). The Burgers model followed closely with fitting parameters A (0.10), B (0.022 × 1014 s⁻1), and k (2.64 day⁻1) and an R2 of 0.97. Evaluation metrics in Table 7 a revealed superior forecasting performance for the power law model, exhibiting RMSE (0.33), MAE (0.28), MAPE (1.45%), and SAMPE (1.46%), compared to the Burgers model, which had RMSE (0.52), MAE (0.44), MAPE (3.799%), and SAMPE (3.98%).

Fig. 13
figure 13

Analytical models for long-term prediction, such as Burger’s, Kelvin, spring-dashpot, power law, and logarithmic, for creeping results of axial strain (milli-strain) over time (hour), forecast trends from left to right at different confining pressure level conditions.

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Table 7 Performance metrics of the analytical models from training data (best results are highlighted).
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Table 8 Creep parameters obtained by fitting the best two considered analytical models, such as Power-law creep followed by Burger creep, recorded for the homogeneous rock salt facies (Temperature: 27 °C = 300K).
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Discussions

This study assesses the long-term creep deformation at ambient temperature under a range of confining pressures. Minimal causality between axial strain and temperature was confirmed by STL analysis, which broke down the creep datasets into trend, seasonal, and residual components. The data’s linearity was evaluated using significant causality tests, such as Granger and ADF (Table 4). For time-dependent deformation, DNN models—N-BEATS, TCN, RNN, and Transformers in particular—showed excellent predictive ability, with low RMSE, MAE, MAPE, and SMAPE values (Table 5). Traditional baseline models, such as AuToARIMA and ES, on the other hand, fared better under different conditions (Table 6). The incorporation of mathematical models strengthened the DNN results’ resilience. The study demonstrates how well sophisticated DNN models work with the Darts library to forecast time series. With preliminary data analysis, the investigation verified that the triaxial creep data is stationary, non-seasonal, and increases linearly with time, demonstrating the data’s dependability for upcoming projections. Axial strain and temperature are not causally related, emphasizing the need for more sophisticated modeling methods to produce reliable forecasts. These important forecasting model performances were discussed in detail as summarized below;

Forecasting performance of DNN models

The N-BEATS model forecasts both the trend and seasonal components by decomposing down time series data into trend, seasonal, and residual components. This allows it to achieve exceptional precision and interpretability. likewise, the TCN model forecasts the residual term, which allows for reliable long-term strain response forecasting outside of test datasets. Precise decision-making enabled both the N-BEATS and TCN models to achieve high forecast accuracy; these models’ respective RMSE values are 3.7441 for N-BEATS and 4.5044 for TCN, and their MAPE values are 2.50% and 2.001% (as seen in Table 5). Traditional statistical approaches, such as the Theta model and ES, by contrast, increased average RMSE by 20.18%, MAE by 17.5%, and MAPE by 15% under different confining pressures (as seen in Table 6). The TCN’s robustness for enormous time series datasets is increased by its architecture, which includes residual blocks and causal convolution. N-BEATS’s great accuracy and efficacy are further confirmed by its effective training. When it comes to forecasting, N-BEATS and TCN routinely perform better than other DNN models.

Forecasting performance of statistical models

The Theta model demonstrated superior performance based on the analysis of statistical baseline models—ES, Auto ARIMA, Theta, and TBATS. Across the board, lower error metrics, including RMSE, MAE, MAPE (%), and SMAPE (%), support this conclusion. The Theta model consistently minimized prediction errors, as evidenced by the lowest RMSE, and provided the most reliable forecasts, indicated by the smallest MAE. Furthermore, its robust handling of relative error, with the lowest MAPE, and its balanced performance in managing both overestimations and underestimations, reflected in the superior SMAPE, underscore its accuracy and reliability in forecasting axial strain and time over the given forecast horizon of 3. These findings establish the Theta model as the most suitable and preferred method among the models evaluated, particularly for time series forecasting tasks where precision and consistency are critical.

Comparative significance analysis and advantages of DNN over analytical models

Deep Neural Network models—particularly N-BEATS and Temporal Convolutional Networks—demonstrate significantly higher predictive performance metrics by approximately 15–20%, compared to traditional analytical models such as the Power Law and Burger models in predicting axial creep strain. This improvement is supported by superior evaluation metrics including R2, RMSE, MAE, MAPE, and SMAPE. While analytical models like the Power Law (R2 = 0.97–0.98, RMSE = 0.33–0.42) and Burger (R2 = 0.94–0.97, RMSE = 0.52) are effective in fitting static or short-term trends, they are limited in capturing the dynamic temporal dependencies inherent in long-term creep behavior. In contrast, DNN models dynamically learn from data through techniques such as window-shifting and data-splitting, enabling them to produce long-term forecasts with data-driven validation. This capability is crucial for accurately modeling non-linear and time-dependent phenomena. Analytical models are primarily curve-fitting approaches that describe the general trend of the data. Although the Power Law and Burger models performed relatively well among the analytical approaches, outperforming other models like Kelvin’s and spring-dashpot models, as shown in Tables 7 and 8, they still fall short in accurately representing complex temporal patterns. Their inability to model unseen future response limits their reliability for long-term prediction when observational data is unavailable.

On the other hand, DNN models such as N-BEATS and TCN are inherently more adaptive. They effectively capture intricate temporal dependencies and non-linear variations in the data, making them particularly suitable for dynamic systems. By segmenting datasets into training and testing windows, these models allow for controlled validation against actual observations of axial creep strain over extended periods. This approach not only enhances predictive precision but also ensures model reliability by tuning data lengths to maintain consistency and accuracy.

Thus, DNN models are far more capable than traditional analytical models for applications requiring precise long-term forecasting. Their dynamic architecture and ability to learn from time-dependent data make them the preferred choice for modeling real-world time-dependent deformation response in complex systems.

Statistical significance of the results: MCB test

We next compared the predictions produced by the N-BEATS and TCN models to those produced by other benchmark forecasters to ascertain the statistical significance of the predictions. RMSE, MAE, MAPE (%), and SMAPE (%) scores across different time series datasets and confining pressure level conditions at their corresponding critical distances were used to calculate the average ranks of the various methods to assess their relative performance using multiple comparisons with the best test.

According to the MCB test findings in Table 9 and Figs. 14, 15, 16, 17 and 18, the N-BEATS model performed better across the board in all measures at the confining pressure level of 5–35 MPa, consistently achieving the lowest average 1. For the best-fitted model, the reference value is the upper bound of the critical distance. Critical intervals for all benchmark forecasters (in terms of RMSE, MAE, MAPE, and SMAPE scores) were higher than the reference value for DNN models, suggesting noticeably poorer performance compared to N-BEATS. Other baseline forecasters had non-overlapping intervals, indicating their poorer performance, but N-BEATS and TCN showed a minor overlap for the SMAPE metric. The N-BEATS and TCN models regularly outperformed the others, as seen by the MCB plots, which also showed that the theta model had the lowest mean rank among statistical baseline models.

Table 9 Average rank derived from the MCB test for DNN and statistical baseline models corresponding to the performance metrics (the best one is highlighted).
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Fig. 14
figure 14

Visualization of multiple comparisons with the best test analysis with mean rank value across four performance metrics: (a) RMSE, (b) MAE, (c) MAPE in %, and (d) SMAPE in % at a confining pressure level of 5MPa, respectively. Each subplot presents the mean rank of different forecasting models, where lower ranks signify better performance. For instance, in subplot (a), ‘N-BEATS-1’ indicates that the N-BEATS model achieved the top rank for RMSE. This comprehensive visualization offers insights into model performance across various evaluation criteria, similar to others.

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Fig. 15
figure 15

Visualization of multiple comparisons with the best test analysis with mean rank value across four performance metrics: (a) RMSE, (b) MAE, (c) MAPE in %, and (d) SMAPE in % at 15MPa, respectively.

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Fig. 16
figure 16

Visualization of MCB test analysis with mean rank value across four performance metrics: (a) RMSE, (b) MAE, (c) MAPE in %, and (d) SMAPE in % at 25MPa, respectively.

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Fig. 17
figure 17

Visualization of MCB test analysis with mean rank across four performance metrics: (a) RMSE, (b) MAE, (c) MAPE in %, and (d) SMAPE in % at 35MPa, respectively.

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Fig. 18
figure 18

Another example of a box view for MCB analysis (left) with error plot (right) for DNN and statistical baseline models for metrics such as RMSE, MAE, MAPE (%), and SMAPE (%) at a confining pressure level of 5MPa, respectively. The mean/average rank value of the N-BEATS model is 1, which shows the best model in terms of accuracy among other models.

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Figure 18, a heatmap (left) and error analysis (right) for the MCB test in another statistical way is used to visualize and test for improved performance predictive models in terms of statistical evaluation error analysis metrics. This was also done for confining pressure at 5MPa and creep data of axial strain responses. To find the best model for each metric, such as RMSE, MAE, SMAPE, and SMAPE, look for the models with the reddest cell. The best model is indicated in red, and its rank number is displayed in bold black in the error plot, which displays the average rank of models over metrics with an 85% confidence interval length.

Conformal prediction for N-BEATS and TCN models

After analyzing the MCB test and mean ranks to facilitate the best DNN models for forecasting for time series datasets at different confining pressure level conditions, we introduced two best DNN models for time series datasets with superior evaluation metrics that promise to forecast, we conducted the conformal prediction with 95% prediction intervals between actual and model predicted. The N-BEATS overperformed, followed by the TCN model, with evaluation metrics such as RMSE, MAE, MAPE, and SMAPE (Table 4) in terms of accuracy compared to other DNN models. That is why we focus on CP with the two best proposed DNN models in this case for analysis, shown in Figs. 19 and 20, respectively. CP provides a probabilistic way of generating prediction intervals, which is crucial for industrial applications where point forecasts are seldom needed39.

Fig. 19
figure 19

The proposed N-BEATS was utilized for conformal prediction between precedent and actual data in 95% prediction intervals for (a) 5MPa, (b) 15MPa, (c) 25MPa, and (d) 35MPa, respectively. The horizontal axis corresponds to the index (time in hours), and the vertical axis to axial strain values (milli-strain × 10–3).

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Fig. 20
figure 20

The TCN model followed by the N-BEATS was utilized for conformal prediction between precedent against actual data in 95% prediction intervals for (a) 5MPa, (b) 15MPa, (c) 25MPa, and (d) 35MPa, respectively. The horizontal axis corresponds to the index (time in hours), and the vertical axis corresponds to axial strain values (milli-strain × 10–3).

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Model forecasts: N-BEATS and TCN

The Darts Python library has revolutionized time series forecasting with its user-friendly interface and diverse range of models, from classical approaches to cutting-edge deep neural networks. This library simplifies the forecasting process by employing familiar methods such as .fit() and .predict(), akin to scikit-learn, enabling seamless model training and forecasting. One of Darts’ standout features is its robust support for univariate and multivariate time series, accommodating potentially large datasets with multiple series. Moreover, the library facilitates backtesting, model combination, and integration of external data, enhancing the accuracy and flexibility of predictions and forecasting. Notably, Darts includes ML-based models capable of probabilistic forecasting, providing valuable insights into forecast uncertainty. Models like N-BEATS and TCN have demonstrated remarkable performance beyond sample datasets.

In our evaluations, the N-BEATS model excelled in short-term forecasting, efficiently capturing patterns in time series data with shorter durations. Conversely, the TCN model proved superior in forecasting over longer durations, demonstrating its effectiveness in capturing complex trends and patterns. Figure 21 illustrates the N-BEATS model successfully forecasting 100-h samples beyond the set, highlighting its robust performance in the short term. Meanwhile, Fig. 22 showcases the TCN model accurately forecasting up to 200-h samples, underscoring its capability in longer-term forecasts.

Fig. 21
figure 21

The N-BEATS model forecasts beyond the test sample for various confining pressure conditions, such as for (a) 5 MPa, (b) 15 MPa, (c) 25 MPa, and (d) 35 MPa, respectively. The N-BEATS model forecasts that the next value is realistic, limited to the length of the test duration.

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Fig. 22
figure 22

The TCN model forecasts beyond the test sample for various confining pressure conditions, such as for (a) 5MPa, (b) 15MPa, (c) 25MPa, and (d) 35MPa, respectively. The TCN model forecasts the next value to be realistic, limited to the length of the test duration.

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Implications and limitations

The study underscores the limitations of traditional causality tests and analytical models in forecasting axial strain deformation response over extended periods. Specifically, DNN models—such as N-BEATS and TCN—demonstrate significant advantages in capturing complex temporal dependencies and patterns, yielding more accurate forecasts in dynamic systems. Their applicability extends beyond energy storage to other critical fields, including weather forecasting and geoscience, where precise forecasting is essential.

While DNN models show strong forecasting performance, they also have certain limitations, such as high complexity and substantial computational resource requirements, particularly for GPU acceleration. Additionally, they may struggle with data spikes, leading to prediction inaccuracies, especially when data is limited. Nevertheless, DNNs have key advantages in forecasting creep deformation, as they capture temporal dependencies and allow for trend validation over both short and long-term scenarios. In contrast, analytical models typically fit data trends over time but cannot validate forecasts without additional future data, which limits their reliability for long-term trend validation. Unlike DNNs, analytical models do not require a defined forecast window length; they simply fit data over time to obtain a trend or pattern. For example, using a five-day dataset of axial strain creep deformation, an analytical model might provide a trend over an extended period. However, without future data, these models cannot validate or challenge the forecasted trend, posing a significant drawback compared to DNNs. In DNN models, however, a time-shifting procedure for input and output windows can be applied, allowing for the division of creep datasets into training and testing sets. This enables model validation against actual recorded data, supporting both forecasting and validation of axial strain deformation trends. Consequently, DNN models play a pivotal role in addressing the limitations of traditional methods for long-term creep deformation forecasting.

Future work could enhance DNN forecasting accuracy by incorporating temperature fluctuations and other engineering features, further improving the model performance of rock formation’s creep response under in situ conditions. Overall, these findings support a shift towards advanced DNN models for more robust and accurate long-term forecasting.

Conclusions

This study highlights the exceptional performance of deep neural network (DNN) models in forecasting long-term axial strain under varying confining pressures at room temperature.

In this study, some important outcomes of this paper are as below;

  • STL decomposition and causality tests revealed no significant relationship between strain and temperature, with minimal seasonality observed. The stationary p-value (< 0.05) and identifiable patterns in creep deformation suggest the necessity of advanced modeling.

  • The N-BEATS and TCN models led in predictive accuracy with notably low on-average RMSE (3.7441, 4.5044) and MAPE in % (2.50, 2.001), respectively, outperforming other DNNs (RNN, Transformer) and statistical models (e.g., ES, Theta), demonstrating high precision in time series forecasting.

  • N-BEATS and TCN showed a 15–20% performance metrics improvement over traditional models, capturing intricate temporal patterns and dependencies, which analytical models, limited to linear trends, could not achieve.

  • The study establishes a foundation for applying DNNs to time-dependent deformation modeling. Future work could expand these approaches to other forecasting challenges in the field of geophysics, enhancing the understanding of complex patterns in similar domains.