A General Surrogate Model for CO₂ Flooding Dynamic Prediction Based on Dimensionless Features and Implicit Time

Abstract

In the development of CO₂ flooding reservoirs, complex multicomponent phase interactions and strong nonlinear flow behaviors cause the computational cost of traditional numerical simulations to increase significantly as the simulation period extends. This computational burden severely limits the efficiency of large-scale scheme optimization. To address this challenge, this paper proposes a general surrogate model based on the Transformer architecture, designed to achieve efficient and high-precision prediction of cumulative oil production and the CO₂ storage capacity. To tackle the generalization difficulty caused by differences in reservoir geological scales, a dimensionless feature system based on Pore Volume (PV) normalization was constructed. By converting absolute injection and production parameters into dimensionless numbers relative to the reservoir pore volume, this method effectively decouples the constraints of specific reservoir geometric scales, thereby establishing the physical universality of the model across different blocks. On this basis, by integrating an implicit time parameterization strategy with a Bayesian adaptive optimization algorithm, the model substitutes absolute physical time with cumulative fluid injection volume. This transformation converts discrete time-stepping prediction into a continuous mapping between development dynamics and the degree of injection. Validation experiments using 280,000 samples demonstrate that the model achieves R ² values of 0.9986 and 0.9968 for oil production and CO₂ storage predictions, respectively. In terms of computational efficiency, unlike numerical simulations, where computational costs grow cumulatively over time, this model exhibits remarkable time independence. Taking a typical 5 year full lifecycle prediction task as an example, under the current computing configuration, the proposed surrogate model achieves a speedup of nearly 60 times compared with numerical simulations of equivalent precision. This characteristic, coupled with its excellent generalizability, effectively mitigates the computational bottleneck of long-cycle simulations, providing robust technical support for rapid assessment in CCUS-EOR development.

1. Introduction

Carbon dioxide enhanced oil recovery (CO₂-EOR) technology has become a critical technical pathway for the efficient development of low-permeability reservoirs and the green transition of the energy industry, owing to its dual benefits of increasing crude oil production and facilitating the geological sequestration of greenhouse gases. , The scientific formulation and dynamic regulation of CO₂-EOR development schemes rely heavily on precise predictions of reservoir responses, which typically necessitate sensitivity analyses and parameter optimization across massive geological realizations, complex well patterns, and variable injection strategies. However, traditional numerical simulation methods are constrained by the strong nonlinear characteristics of multicomponent flow and complex interphase flow mechanisms inherent in CO₂ flooding, making single calculations computationally expensive. Consequently, traditional simulation becomes impractical when facing massive scheme optimization tasks, severely hindering the real-time formulation and dynamic adjustment of development strategies. − With the rapid development of artificial intelligence, data-driven surrogate models have emerged as research hotspots in this field, serving as efficient alternatives to traditional high-precision numerical simulations. ,

Although existing surrogate models demonstrate significant potential in alleviating computational pressure and improving prediction efficiency, current research still faces several significant limitations when dealing with complex multiphase CO₂-EOR systems:

First, constrained by the one-reservoir-one-model modeling paradigm, achieving cross-block application remains difficult. Since numerical models for different reservoirs vary in geological characteristics and grid dimensions, the input–output dimensions of prediction models cannot be uniformly adapted. − Furthermore, when modeling a single reservoir, geological parameters are often treated as fixed values rather than feature inputs, which further deprives the model of the ability to identify and generalize across different geological conditions.

Second, parameter representation based on absolute dimensions obscures the physical essence of flow in porous media, limiting the model’s cross-scale transferability. From the perspective of porous media fluid mechanics, although geological scales and injection-production intensities vary significantly across different reservoirs, their displacement processes often follow similar physical evolution laws within a dimensionless space. However, existing studies mostly establish prediction models using absolute dimensional parameters (e.g., absolute injection volume). While this approach can accurately characterize the production dynamics of a specific single reservoir, it inevitably couples physical displacement characteristics with the geometric scale of the reservoir. This coupling masks the intrinsic commonalities of fluid flow under different operating conditions, leading the model to fit numerical relationships specific to a certain scale rather than the underlying physical essence, thereby limiting its universality and transferability under cross-scale and variable geological conditions.

Finally, the prediction of continuous production dynamics is still in its infancy. Existing research is largely confined to a discrete mapping mode where a single injection-production schedule corresponds to a single training sample, and a time-consuming numerical simulation is used solely to extract output at a specific moment as a label. This approach not only results in high sample generation costs but also leads to the massive waste of time-series evolution information embedded within numerical simulations. The lack of diversified decomposition and utilization of single simulation processes restricts the effective expansion of training data scale and hinders the model’s ability to perform flexible, continuous predictions at any production moment, thus constraining the efficiency and accuracy of dynamic assessment.

Addressing these challenges, this study aims to construct a CO₂ flooding intelligent surrogate model that integrates cross-block universality, temporal continuity, and physical consistency. The specific technical pathways are as follows:

First, a dimensionless feature system based on PV normalization is constructed to break the constraints of geometric scale and achieve model universality. Taking a single well group as the object, this study constructs a dimensionless feature system based on PV normalization by comprehensively considering well pattern structures, static geological parameters, and injection-production parameters. By converting parameters with absolute dimensions, such as absolute injection volume and cumulative oil production into dimensionless numbers relative to the reservoir pore volume, the constraints of specific reservoir geometric scales (e.g., well group area, thickness) are effectively decoupled. This strategy eliminates differences in absolute physical dimensions, enabling the model to learn physical displacement laws rather than specific numerical values, thereby fundamentally establishing the model’s generalization and transferability across reservoirs of different geological scales.

Second, an implicit time parameterization strategy is proposed to achieve a leap from discrete point prediction to continuous dynamic process modeling. Unlike traditional methods that predict a future fixed moment (e.g., the 10th year), this strategy fully considers real-time changes in complex operating conditions, such as injection rates and alternating cycles, implicitly embedding the time dimension into the physical representation of cumulative injection volume. Based on the physical fact that cumulative injection volume is the integral of injection rate over the time domain, this study utilizes cumulative fluid injection to replace absolute physical time, thereby establishing a continuous mapping between development dynamics and the degree of injection. This transformation achieves a transition from discrete time-stepping prediction to continuous prediction at any moment, providing a flexible technical path for the real-time tracking and fine-tuning of reservoir development schemes.

Third, the Transformer architecture and Bayesian adaptive optimization algorithm are integrated to break through the modeling accuracy bottleneck of nonlinear multiphase flow coupling. Given the complexity of the CO₂ displacement process, the self-attention mechanism of the Transformer is utilized to precisely capture global long-range dependencies between geological attributes and dynamic operations, solving the problem of distortion in traditional models under extreme geological conditions. Simultaneously, Bayesian optimization is introduced to automatically search for the optimal network architecture for different prediction tasks (oil production and storage), ensuring high-fidelity simulation of complex physical fields while maintaining physical consistency.

The remainder of this paper is organized as follows: Section systematically reviews the research status of surrogate models in the petroleum engineering field; Section details the construction method of the general dynamic prediction framework for CO₂ flooding, including data generation strategies and network architecture design; Section presents the model’s prediction performance, generalization ability, and hyperparameter optimization analysis through multidimensional experiments; and Section summarizes the work and outlines future research directions.

2. Related Work

The landscape of surrogate modeling is undergoing a profound transformation, driven by the rapid evolution of machine learning algorithms that are pushing the field beyond the confines of traditional statistical regression toward deep learning paradigms. This shift represents more than a mere technical update; it reflects a collective effort within the research community to bypass the heavy computational burdens of numerical simulation by establishing instantaneous, data-driven mappings. Such techniques have successfully permeated the core workflows of reservoir engineering, spanning from history matching to the intricate optimization of injection-production schedules, demonstrating their potential to accelerate decision-making. However, the applicability of these models varies significantly depending on the physical complexity of the fluid system involved. To identify where the true technological bottlenecks lie, it is essential to scrutinize the divergent progress in two distinct contexts: the relatively established domain of water flooding and the fluid-mechanically chaotic environment of CO₂-EOR.

2.1. Current Status of Surrogate Models for Water Flooding

In the domain of water flooding, the computational intensity of a high-fidelity numerical simulation has driven a continuous search for faster alternatives. Initial breakthroughs were largely achieved by compressing the physical state space, with the aim of accelerating computation without sacrificing essential flow dynamics. Jin et al. for instance, demonstrated that integrating deep autoencoders into an embedded control (E2C) framework could boost simulation efficiency by nearly 3 orders of magnitude. This focus on dimensional reduction was further advanced by Zhang et al. whose deep reduced-order modeling algorithm proved that complex fluid behaviors could still be captured effectively even after significant compression of the computational grid.

Moving beyond spatial reconstruction, accurate forecasting of nonlinear production sequences requires models capable of handling temporal dependencies. Deep learning architectures, particularly the convolutional neural network (CNN) and recurrent neural network (RNN), have shown distinct advantages here over traditional shallow methods. By employing an AlexNet based multi-input architecture, Wang et al. achieved substantial accuracy improvements of 17.5% over BPNN and 20.8% over SVM specifically in scenarios where operational parameters fluctuate dynamically. Liu et al. took a different approach to address the inherent nonstationarity of production data; their hybrid model, which couples Ensemble Empirical Mode Decomposition (EEMD) with LSTMs, effectively isolates and learns multiscale evolutionary trends through Intrinsic Mode Functions.

Despite these gains in speed and accuracy, purely data-driven models often face criticism regarding their physical consistency and their ability to generalize across varying geological conditions. To bridge this gap, recent scholarship has increasingly turned toward embedding physical principles directly into the network architecture. Zhong et al. introduced a specialized surrogate for field-level rate prediction that significantly eases the burden of history matching, while Mo et al. utilized a deep Convolutional Encoder-Decoder to navigate the uncertainties inherent in heterogeneous multiphase flow. Perhaps the most rigorous attempt to enforce physical validity comes from Wang et al. who proposed the Theory-Guided neural network (TgNN) by incorporating governing equations into the loss function, a concept Yan et al. extended to simulate flow in 3D heterogeneous media. These diverse approaches confirm that while surrogates are highly effective for subsurface fluid migration, their robustness is markedly improved when data-driven flexibility is anchored by physical constraints.

2.2. Current Status of Surrogate Models for CO₂ Flooding

Unlike the relatively straightforward mechanics of water flooding, CO₂ flooding involves intricate multiphase behavior and compositional mass transfer. This heightened physical complexity imposes stringent demands on the nonlinear mapping capabilities of the surrogate models. Initial efforts in this domain predominantly utilized machine learning algorithms to establish direct functional mappings between inputs and outputs. Ahmadi et al. for instance, leveraged Least Squares Support Vector Machines (LS-SVM) combined with response surface methodology to provide rapid estimations of ultimate recovery in stratified reservoirs.Addressing key physical parameters critical for miscible displacement, Yang et al. developed an improved XGBoost model optimized by Particle Swarm Optimization (PSO) to predict minimum miscibility pressure (MMP). By incorporating critical temperature as a new variable and utilizing SHAP analysis for interpretability, their model outperformed traditional methods with an R² of 0.9845, demonstrating the potential of interpretable machine learning in capturing specific fluid phase behaviors. Davoodi et al. extended this approach by coupling Multi-Layer Perceptrons (MLPNN) with intelligent optimizers like Gray Wolf (GWO) and PSO, attempting to untangle the synergistic relationship between CO₂ storage and oil recovery in unconventional settings. Even for operationally complex scenarios such as Water-Alternating-Gas (CO₂-WAG), Li et al. demonstrated that Random Forest algorithms could establish robust nonlinear correlations, slashing prediction times to mere seconds while maintaining minimal error margins. Davoodi et al. recently pioneered a Committee Machine Learning Gaussian Process Regression (CML-GPR) model that integrates Random Forest, MELM, and GRNN. By leveraging the complementary strengths of these learners on a massive data set of over 21,000 simulations, their ensemble approach significantly improved predictive accuracy for both CO₂ storage mass and cumulative oil production, marking a breakthrough in precise, large-scale reservoir performance forecasting.

As deep learning has matured, the research focus has shifted toward more sophisticated architectures capable of handling multiobjective optimization and spatial reconstruction. Meng et al. developed a framework incorporating geomechanical risks, utilizing LSTM and Gaussian-SVR surrogates to achieve Pareto optimality between recovery factors and storage rates in high-water-cut reservoirs, all while managing fault slip risks. In the more challenging realm of spatial field prediction, Meng et al. introduced a Res U-Net-based model enhanced by a Structural Similarity (SSIM) loss function, successfully achieving high-fidelity reconstruction of the spatiotemporal evolution of pressure and saturation fields in heterogeneous low-permeability reservoirs. Addressing the computational burden of compositional simulation, Dong et al. proposed a hybrid deep learning surrogate integrating VQ-VAE with ConvLSTM. By projecting 3D reservoir properties into a 2D latent space via an autoencoder, their model accurately captures the dynamic characteristics of both production and distinct CO₂ storage forms. This approach demonstrated high-fidelity attribute restoration (correlation >0.98) and reduced computational costs by 2 orders of magnitude, proving the feasibility of deep learning in handling complex storage mechanisms. Other notable contributions include You et al. use of Gaussian-SVR for WAG injection scheme optimization and Tang et al. application of Graph Neural Networks (GNN) to handle global well control parameters on unstructured grids. These advancements confirm that data-driven methods have evolved sufficiently to serve as viable alternatives to traditional simulators, even for highly nonlinear CO₂-EOR problems.

Nevertheless, a critical gap persists in the current body of knowledge. While computational efficiency has been significantly improved for specific reservoir scenarios, most existing studies remain confined to static mapping paradigms linking geological parameters to production metrics at fixed points in time rather than capturing the continuous, dynamic evolution of the entire lifecycle. More importantly, there is a notable absence of generalized solutions capable of scaling across different reservoir sizes or performing dynamic forecasting based on complete historical data. Addressing these limitations, this study seeks to forge a new path for generalizable surrogate modeling. By constructing a physics-based dimensionless feature system and integrating attention mechanisms, we aim to enable the continuous, dynamic simulation of complex CO₂-EOR processes across varying scales.

3. Methodology

The dynamic prediction framework for CO₂ flooding reservoir development proposed in this article is composed of four distinct steps:

Step 1: Data set Construction. An implicit time parametrization strategy is adopted to reconstruct the time dimension using cumulative injection volume, thereby establishing the mapping relationship between critical geological-engineering parameters and continuous dynamic responses.

Step 2: Data Preprocessing. A dimensionless feature system based on pore volume normalization is established. This approach eliminates the influence of reservoir geometric scale discrepancies on data distribution, thereby enhancing the physical consistency and convergence stability of the model training.

Step 3: Deep Learning. A Transformer prediction model incorporating a self-attention mechanism is constructed. Furthermore, a Bayesian optimization algorithm is introduced to perform adaptive optimization of the network architecture and hyperparameters, ensuring the precise capture of the nonlinear coupling relationships among parameters.

Step 4: Dynamic Prediction and Evaluation. The trained model is utilized to predict production dynamics under various injection-production schemes. The model’s performance advantages in terms of prediction accuracy, robustness, and continuous curve reconstruction are validated through comparative analysis with traditional methods such as XGBoost and Random forest.

3.1. Data Set Construction

3.1.1. Parameter Space Definition and Sampling Design

CO₂-EOR is not merely a fluid injection process but a complex domain governed by the intricate coupling of geological conditions and engineering interventions. Designing a data set that faithfully captures the physics of subsurface flow while reflecting real-world development scenarios requires more than a random selection of variables; it demands a synthesis of extensive literature and practical engineering judgment to identify the controlling factors. Consequently, we categorized the primary inputs into two distinct yet interacting groups: static geological parameters and dynamic operational parameters.

At the foundational level, static geological parameters characterize the inherent physical properties and spatial structure of the reservoir, forming the stage upon which fluid flow occurs. Key parameters selected include permeability, porosity, reservoir thickness, well pattern area, and initial formation pressure. These variables define the storage capacity, flow potential, and initial energy state of the porous media, serving as the internal determinants that ultimately constrain both oil recovery and carbon storage potential.

Superimposed upon this geological framework are the dynamic operational parameters and the external variables representing human intervention to reshape the subsurface flow field and induce phase behavior changes. Recognizing that the vast majority of CO₂-EOR projects are brownfield developments initiated only after prolonged water flooding, a realistic model must account for this historical legacy rather than assuming a “clean slate.” Accordingly, we divided the lifecycle into a water flooding stage and a CO₂ flooding stage, constructing a two-stage dynamic parameter system (Figure ).

1.

Two-stage division of dynamic development parameters.

The water flooding stage is defined by the duration of injection, the injection-production ratio, and the daily liquid production rate. These metrics are critical because the production history determines the reservoir’s state variables, specifically water cut and pressure distribution at the precise moment of conversion to gas injection. These states effectively become the initial boundary conditions for the subsequent CO₂ phase. Following this, the CO₂ flooding stage introduces parameters such as CO₂ injection rate, water injection rate (for WAG), annual CO₂ injection duration, and liquid production rate. These variables directly modulate injection intensity, interphase interactions, and the stability of the displacement front, acting as the core levers for optimizing recovery and storage efficiency.

To ensure the data set encompasses sufficient geological heterogeneity and diverse operational conditions, we anchored the boundary values of these controlling parameters in the geological characteristics and field practices of typical blocks in the Shengli Oilfield (Table ). This sampling design not only guarantees statistical breadth and diversity but also ensures that every data point adheres to the objective laws of reservoir engineering and remains practically feasible.

1. Summary of Value Ranges for Reservoir Sampling Parameters.

parameter category	feature	description	units	range
static geological
	permeability	characterizes the rock’s capacity to transmit fluids through its pore space	mD	0.01–1000
	porosity	the proportion of effective pore volume within the rock	fraction	0.1–0.35
	well pattern area	the planar area controlled by a single well group	km2	0.02–0.25
	reservoir thickness	the effective vertical thickness of the oil-bearing formation	m	1–50
	initial formation pressure	the initial pressure of the reservoir formation	bar	150–375
	well pattern	includes five-spot pattern and nine-spot pattern	fraction	5,9
water flooding stage
	development duration	the duration of the water flooding development phase	year	0–15
	injection-production ratio	the ratio of total injection volume to total liquid production volume	fraction	0.8–1.2
	water injection rate	the rate at which water is injected	m3/day	1–100
CO2 flooding stage
	CO2 injection rate	the rate at which CO2 is injected	m3/day	100–50000
	water injection rate	the rate at which water is injected (during WAG cycles)	m3/day	1–100
	annual CO2 injection duration	the number of months of continuous CO2 injection per year	month	4,6,8,9,12
	liquid production rate	the rate of liquid fluid production per single well	m3/day	1–40

3.1.2. Reservoir Model Establishment

To ensure the numerical simulation results possess both engineering representativeness and geological authenticity, a conceptual model was constructed based on actual geological and fluid parameters from a typical block in the Shengli Oilfield. The model employs an orthogonal Cartesian grid system for spatial discretization, with grid dimensions set to 31 × 31 × 3 (totaling 2883 grid cells). Targeting common well patterns in CO₂ flooding, two typical configurations the five-spot and nine-spot patterns were designed (Figure ). All wells were configured as fully penetrating and fully perforated to simulate realistic completion conditions. The development scheme strictly follows a composite mode of long-term water flooding, followed by CO₂ injection conversion. The total simulation duration was set to 30 years, aiming to comprehensively capture fluid migration and dynamic response characteristics throughout the entire process from water flooding to the CO₂ flooding stage.

2.

Well layouts (a): five-spot pattern; (b): nine-spot pattern.

Accurate description of fluid phase behavior is the core of the CO₂ flooding simulation. Based on PVT experimental data from the target block, this study employed the lumped parameter method to classify crude oil components into nine key pseudocomponents (CO₂, C1+, C2+, C5+, C8+, C10+, C12+, C17+, and C21+). The molar fractions of these components are 0.0045, 0.2470, 0.0832, 0.1499, 0.1275, 0.1124, 0.1209, 0.0853, and 0.0692, respectively. This refined component classification scheme precisely characterizes the phase equilibrium relationships, dissolution-swelling mechanisms, and interfacial tension variations between CO₂ and hydrocarbon components of varying carbon numbers, laying a solid petrophysical foundation for multiphase mass transfer simulation during gas flooding. ,

Thermodynamic calculations utilized the Peng–Robinson (PR) Equation of State (EOS) to construct a three-phase equilibrium model, ensuring an accurate solution of component phase evolution under high-pressure conditions. Based on this theoretical framework, a comprehensive database containing input parameters and two key response variables namely, CO₂ storage amount and oil production was established. Here, the CO₂ storage amount is defined as the quantity of carbon dioxide effectively trapped and permanently sequestered by the reservoir after injection operations, rather than simply the total injected volume. This indicator is of significant importance for evaluating carbon sequestration efficiency and is directly related to subsurface carbon capture and storage (CCS).

3.1.3. Database Generation

Based on the parameter space defined in Section , this study employed the Latin Hypercube Sampling (LHS) algorithm to generate 1000 statistically independent initial parameter combinations. The LHS algorithm ensures uniform coverage of the high-dimensional parameter space through a stratified sampling mechanism, effectively controlling computational costs while guaranteeing sample representativeness. For each parameter combination, a full-life-cycle numerical simulation (encompassing both water flooding and the CO₂ flooding stages) was executed to acquire complete data.

The traditional sample construction method is illustrated in Figure . In conventional modeling workflows, a complete numerical simulation run is often treated as a single training sample, with output labels typically restricted to oil production and CO₂ storage amounts at specific fixed moments. However, in oilfield development, particularly during the transition from water flooding to CO₂ flooding, engineering decisions frequently require predictions of oil production and storage at arbitrary time points, which is crucial for development plan optimization. The discretized labels of traditional sample construction significantly limit the temporal extrapolation capabilities of neural networks. For instance, if training samples only contain data for years 1, 5, and 10, the model struggles to accurately predict dynamic responses for years 2, 6, or even 11, thereby reducing its reliability in practical engineering applications.

3.

Traditional sample construction method.

To overcome these limitations, this paper proposes an implicit time parameterization modeling strategy. The core idea is to reconstruct water and gas injection volumes as cumulative injection volumes up to time t, thereby implicitly embedding the time dimension into the injection history. By transforming the reservoir dynamic evolution process from discrete time-point mapping to historical cumulative drive mapping, the model can predict production responses at any moment based on continuous injection history, fundamentally realizing the shift from discrete prediction to continuous temporal prediction. The corresponding sample construction process is shown in Figure : for each parameter combination, instead of generating a single sample, the entire CO₂ flooding process is sliced into different time steps to form multiple sample points. Specifically, considering that this study focuses on predicting dynamic responses during the CO₂ flooding stage, a targeted time-domain screening was implemented during sample construction, retaining only all time-step data after the initiation of CO₂ injection, thus, eliminating interference from redundant time nodes in the early water flooding stage on model training.

4.

Implicit-time sample construction strategy.

Based on the implicit time strategy, this study constructed a comprehensive input feature system comprising 14 dimensions, as shown in Table This system categorizes model inputs into three classes: static geological parameters, water flooding stage parameters, and CO₂ flooding stage parameters, thereby systematically integrating key control information regarding the reservoir from its initial state to the current development stage.

2. 14-Dimensional Input Features for the Reservoir Dynamic Prediction Model.

input category	variable name	unit	source
static geological parameters	permeability (perm)	mD	parameter sampling range (same as Table )
	porosity (por)	fraction
	well pattern area (WA)	km2
	reservoir thickness (RT)	m
	well pattern (WP)	fraction
water flooding stage	recovery factor (RF)	%	numerical simulation: water flooding cumulative oil/initial oil in place
	water cut (WC)	fraction	numerical simulation calculation
	formation pressure (PC)	fraction	simulation calculation: water flooding pressure/initial pressure
CO2 flooding stage	annual CO2 injection duration (WAG)	month	parameter sampling range (Table )
	cumulative gas injection (GIT)	m3	numerical simulation
	cumulative water injection (WIT)	m3	numerical simulation
	CO2 injection rate (GIR)	m3/day	parameter sampling range (same as Table )
	water injection rate (WIR)	m3/day
	liquid production rate (LPR)	m3/day

Static geological parameters are directly derived from the parametrized sampling space in Table and remain constant throughout the time series, providing fundamental constraints on the reservoir’s inherent properties.

Water flooding history state parameters are obtained through numerical simulation of the water flooding stage. The complete history of water flooding is compressed into three state variables, which reduces the input dimensionality while ensuring physical continuity between the two development stages.

Current stage dynamic parameters incorporate cumulative gas injection volume and cumulative water injection volume to represent the long-term effects of injection history; gas injection rate, water injection rate, and liquid production rate are taken directly from the sampling space to reflect current operational intensity and capture short-term dynamic responses; well pattern type and WAG scheme are included to distinguish displacement characteristics of different development modes.

Through the aforementioned screening and extraction, the 1000 numerical simulation cases generated a total of 287,414 time-series sample points. Each sample contains complete input features and corresponding target outputs (cumulative oil production and the CO₂ storage amount). This high-dimensional, high-fidelity database constitutes the foundational data set for the training and validation of the surrogate model.

3.2. Nondimensionalization and Normalization

In the data modeling process for reservoir numerical simulation, input features originate from diverse sources and possess distinct properties, universally exhibiting significant discrepancies in physical units and inconsistencies in the data distribution. Statistical analysis of the raw data (Figure .) reveals that while static geological parameters generated via Latin Hypercube Sampling (LHS) are distributed very uniformly within their value ranges, dynamic response indicators calculated by numerical simulation (e.g., cumulative oil production, cumulative injection volume) exhibit markedly unbalanced distribution characteristics due to the influence of complex subsurface flow processes. Furthermore, the numerical magnitudes of these values often differ by several orders of magnitude. To eliminate interference caused by varying physical magnitudes and enhance convergence stability during model training, this study constructed a hybrid preprocessing workflow integrating a physical dimensionless transformation and data standardization. This workflow aims to remove magnitude discrepancies while maximally preserving the intrinsic physical laws and relative proportional relationships within the data.

5.

Histograms of the original distributions of 16 features (14 inputs and 2 outputs).

To eliminate the physical constraints imposed by reservoir geometric scale differences (e.g., well pattern area, effective thickness) on the model’s generalization capability and to enable cross-scale transfer prediction, this study introduces a pore volume (PV) dimensionless mechanism. Specifically, all physical quantities involving the fluid volume (cumulative gas/water injection volumes, injection/production rates, and output variables) are divided by the total reservoir pore volume. This transformation converts absolute volumes into relative pore volume multiples, enabling the model to learn intrinsic flow laws independent of geometric boundaries

$$L^{*}=\frac{L}{\mathrm{P}\mathrm{V}}=\frac{L}{A\times h\times \phi }$$ 1

where L* is the dimensionless parameter, L is the original volume parameter, A is the well pattern area, h is the reservoir thickness, and φ is the porosity.

For continuous variables generated by numerical simulation (such as cumulative oil production and injection/production rates), this study further employs a shifted logarithmic transformation

$$X_{\log }=\log (a+X)$$ 2

where X is the input value to be processed, X _log is the value after logarithmic transformation, and a is a constant (shift factor) used to avoid taking the logarithm of zero while preserving the relative proportional relationships of the original data.

It is worth noting that water cut (WC) is inherently a parameter within the [0, 1] interval and directly reflects the water cut rising law and displacement degree during the development stage. Therefore, this study maintains the original values of WC without additional transformation to preserve its explicit physical significance and original temporal characteristics.

Finally, Min-Max normalization is uniformly applied to all features except WC, mapping each feature value to the [0, 1] interval. The calculation formula is as follows

$$X_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}=\frac{X-X_{\min }}{X_{\max }-X_{\min }}$$ 3

where X _norm is the normalized input value, and X _min and X _max are the minimum and maximum values of that feature in the training set, respectively.

After processing with the aforementioned strategies (Figure .), the features in each dimension effectively eliminated dimensional differences and exhibited quasi-normal distribution characteristics. This significantly enhanced the density and orthogonality of the feature space, providing an optimized numerical foundation for the subsequent model to capture high-dimensional nonlinear manifolds.

6.

Histograms of the 16 feature distributions after normalization.

3.3. Transformer Network Design for Dynamic Feature Extraction

Leveraging its robust capacity for deep nonlinear feature representation, deep learning has emerged as a core tool for constructing surrogate models of reservoir numerical simulation and accelerating production performance prediction. , Among various network architectures, the Transformer model, distinguished by the introduction of the self-attention mechanism, demonstrates significant advantages over traditional methods in capturing representations of reservoir dynamic evolution. Unlike CNN constrained by local receptive fields or RNN reliant on temporal recurrence, Transformer possesses parallel processing capabilities. It can explicitly model long-range dependencies and high-order correlations between arbitrary elements within an input sequence. This characteristic makes it particularly suitable for characterizing the strong nonlinear flow responses and coupling mechanisms existing between static geological properties and dynamic engineering parameters during the CO₂-EOR process.

This study constructs a surrogate model as illustrated in Figure , employing an Encoder-Decoder architecture to accommodate multiobjective prediction requirements. The model’s raw input consists of a 14-dimensional feature vector, encompassing static reservoir physical properties such as permeability and porosity as well as dynamic control variables like injection-production rates and the timing of displacement conversion.

7.

Basic architecture of the Transformer model containing self-attention mechanism.

Encoder (Figure ): The input vector is projected into a 64-dimensional latent space via a linear embedding layer to enhance the feature expressiveness. Subsequently, learnable positional encodings are superimposed to explicitly preserve temporal dependencies and the logical sequence of development stages. The encoder structure is composed of stacked homogeneous sublayers, each containing a feed-forward network, residual connections, and layer normalization, thereby achieving stable training of deep features and hierarchical abstraction of nonlinear mappings.

8.

Encoder architecture.

Self-Attention Mechanism (Figure ): This is the core component of the Transformer. This mechanism enables the adaptive aggregation of global information by dynamically calculating the correlation weights among various feature vectors in the input sequence. Compared with traditional structures based on fixed convolution kernels or temporal recurrence, the self-attention mechanism can establish direct connections between arbitrary features. This facilitates the efficient capture of long-range dependencies and potential coupling relationships between distinct geological properties and engineering parameters during CO₂ displacement.

9.

Self-attention mechanism.

Decoder (Figure ): The decoder is primarily responsible for mapping the highly dimensional latent features extracted by the encoder back into the physical target space. Given the need to simultaneously predict cumulative oil production and CO₂ storage two target variables with distinct physical meanings and significantly different magnitudes the model adopts a dual-branch independent output structure. Each branch is equipped with a prediction head based on a multilayer perceptron (MLP), realizing an end-to-end nonlinear mapping from high-dimensional features to cumulative oil production and CO₂ storage.

10.

Decoder architecture.

To clearly present the dimensional evolution laws during the model’s forward propagation, Table details the input and output tensor shapes of each key functional module. This structural design not only guarantees the learning independence of the two heterogeneous tasks (oil displacement and sequestration), effectively avoiding gradient interference between multitasks, but also ensures the model’s precise prediction for the dual objectives of enhanced oil recovery and carbon storage.

3. Tensor Shapes of Various Modules in the Transformer Model.

type	size	output
inputs	(batch,14)	(batch,14)
embedding layer	(batch,14)	(batch,64)
reshape	(batch,64)	(batch,1,64)
FFN	(batch,1,64)	(batch,1,64)
self-attention	(batch,1,64)	(batch,1,64)
squeeze	(batch,1,64)	(batch,64)
MLP layer 1	(batch,64)	(batch,64)
MLP layer 2	(batch,64)	(batch,32)
MLP layer 3	(batch,32)	(batch,1)
output	(batch,1)	(batch,1)

3.4. Continuous Production Curve Prediction

Based on the implicit time modeling strategy introduced in Section , the constructed surrogate model inherently possesses the capability to predict production dynamics at arbitrary discrete time points. Specifically, for a given set of input parameters, the model outputs cumulative oil production and cumulative CO₂ storage at a specific target moment. However, the evaluation of practical development plans requires complete continuous production performance curves across the time domain to support full-life-cycle analysis and decision-making for optimization. To achieve the transformation from discrete point predictions to continuous production curves, this section proposes a continuous production curve fitting method.

The execution workflow of this method is illustrated in Figure First, input sample sets corresponding to n different time points are constructed and fed into the surrogate model to obtain the corresponding discrete predicted values. Here, t ₁,t ₂,···,t _n represent the input samples corresponding to the n time steps. By sequentially inputting each sample into the surrogate model, the predicted oil production values (Oil_1,Oil_2,···, Oil_n) and CO₂ storage values (CO₂_1,CO₂_2,···,CO₂_n) are obtained. Subsequently, curve fitting is applied to convert these discrete predictions into continuous curves for cumulative oil production and cumulative CO₂ storage.

11.

Continuous production curve prediction and fitting workflow.

Considering that cumulative reservoir production follows the physical law of monotonic increase, an exponential growth model is adopted to fit the discrete prediction points

$${\mathrm{N̂}}_p(t)=a(1-{\mathrm{e}}^{-bt})+c,t\in [t_1,t_n]$$ 4

where ${\mathrm{N̂}}_p(t)$ represents the fitted cumulative oil production at time t parameter a represents the maximum reservoir reserves, b represents the production growth rate, and c represents the initial cumulative oil production.

Considering that the CO₂ geological sequestration process is influenced by factors such as injection rate regulation, reservoir pressure response, and injection-production schemes, the cumulative storage amount exhibits stepwise increasing characteristics accompanied by short-term fluctuations. To accurately capture the overall evolutionary trend of storage amount while suppressing noise, the moving average method is employed to fit the prediction sequence

$${\mathrm{V̅}}_{\mathrm{c}{\mathrm{o}}_2}(t)=\frac{1}{2k+1}\sum _{i=-k}^k{V}_{\mathrm{c}{\mathrm{o}}_2}(t+i),t\in [t_1,t_n]$$ 5

where ${\mathrm{V̅}}_{c{o}_2}(t)$ denotes the smoothed CO₂ storage volume after moving average processing; $V_{c{o}_2}(t+i)$ represents the original predicted value at time step t + i, k is the moving window width, and 2k+1 is the window radius.

3.5. Hyperparameter Optimization

The selection of hyperparameters has a significant impact on the prediction accuracy and generalization capability of deep learning models. Since hyperparameters cannot be directly learned from data via gradient descent algorithms, their configuration typically relies on researcher experience or random trial-and-error. However, in high-dimensional and complex parameter spaces, traditional manual tuning or grid search methods are not only computationally expensive but also struggle to guarantee convergence to the global optimal solution. To address this challenge, this study employs the Bayesian Optimization (BO) algorithm to establish an automated hyperparameter tuning workflow (Figure ), aiming to efficiently search for the optimal parameter combination with minimal iterations.

12.

Hyperparameter optimization workflow using Bayesian optimization.

Bayesian optimization is a global optimization strategy designed for objective functions that are difficult to evaluate. Unlike the blind nature of random searches, this method constructs a probabilistic surrogate model to approximate the mapping relationship between the objective function and hyperparameters. Its core philosophy is to intelligently guide the next search direction by balancing exploitation (leveraging known good areas) and exploration (investigating uncertain areas) based on historical evaluation results.

Specifically, Bayesian optimization assumes that the objective function f­(x) to be optimized follows a Gaussian Process (GP) prior distribution. With each evaluation of a hyperparameter configuration x _t , the observation y _t = f­(x _t )+ϵ (where ϵ represents noise) is used to update this prior to obtaining a posterior distribution, representing a progressive Bayesian inference of the objective function. On this basis, an acquisition function α|(D_1:t) is constructed to quantify the potential utility of candidate points. Common choices include Expected Improvement (EI) and the Upper Confidence Bound (UCB). The next set of hyperparameters to be evaluated is determined by maximizing the acquisition function

$$x_{t+1}={\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}x}_{\mathbf{x}\in \mathcal{X}}\alpha (\mathbf{x}|{\mathcal{D}}_{1:t})$$ 6

where ${\mathcal{D}}_{1:t}={\{({\mathbf{x}}_i,y_i)\}}_{i=1}^t$ represents the historical data set formed by the previous t evaluations, and x is the predefined hyperparameter search space.

4. Results

4.1. Model Performance Evaluation and Analysis

4.1.1. Data Set Partitioning

Rational data set partitioning is a critical step in defining the model’s generalization boundary and mitigating the risk of overfitting. Traditional methods based on fixed ratios often lack an empirical basis tailored to specific physical problems. Therefore, this study conducted a systematic evaluation experiment based on data size sensitivity, aiming to investigate the dependence of different prediction targets on training sample density, thereby determining the optimal data set partitioning scheme.

Specifically, three training set ratios (70%, 80%, and 90%) were established, while the validation set remained fixed at 10% of the total sample size across all experiments. To eliminate statistical bias caused by random data partitioning and network initialization, five independent random repeated trials were executed for each configuration scheme, with the average mean squared error (MSE) serving as the final evaluation metric.

Figure illustrates the evolution trend of the model prediction accuracy under different training set sizes. For cumulative oil production prediction, the model achieved optimal generalization performance with a 70% training set ratio; performance did not improve significantly with further increases in training samples. This indicates that the evolution of oil production is primarily driven by relatively stable pressure differentials and possesses strong physical regularity, meaning a smaller sample size is sufficient to cover its feature space.

13.

Comparison of model prediction performance under different training set ratios.(a) cumulative oil production; (b) CO₂ Storage.

Conversely, the prediction accuracy for cumulative CO₂ storage exhibited a monotonic increasing trend as the training set proportion increased, reaching its optimum at 90%. This phenomenon is attributed to the fact that the CO₂ sequestration process involves complex gas–water interactions. Its physical response is characterized by high nonlinearity and uncertainty, thus, requiring higher-density sample data to fully capture the underlying mapping mechanisms.

Based on the aforementioned empirical analysis, this study adopted a task-differentiated partitioning strategy for subsequent modeling: a 70% training set was employed for oil production prediction to balance computational efficiency with generalization capability, while a 90% training set was utilized for the CO₂ storage prediction to maximize the learning accuracy regarding complex nonlinear features. This differentiated configuration ensures the optimal allocation of limited data resources across different tasks.

4.1.2. Validation Case

To systematically evaluate the modeling capability of the proposed Transformer model in reservoir dynamic prediction tasks, this study conducted a comparative analysis against two typical machine learning models XGBoost and Random Forest under identical data set partitioning conditions. This ensures fairness and reproducibility of the evaluation results.

Figures and display the scatter density plots of predicted versus actual values for the three models on the test set for cumulative oil production (validation sample size N = 86,224) and CO₂ storage (validation sample size N = 28,741), respectively. The 45° diagonal line serves as the baseline for ideal fitting. Visual results indicate that the Transformer model exhibits optimal spatial concentration in both tasks: its prediction points are distributed tightly and symmetrically along the diagonal without significant systematic deviation. Even in high-value regions, it maintains good convergence, demonstrating strong global consistency and extrapolation stability. In contrast, XGBoost tends to underestimate results as the true value increases in both tasks, indicating modeling bias under extreme operating conditions. Although Random Forest shows a relatively high overall fit, significant deviations from the diagonal occur in medium-to-high-value regions, suggesting potential issues with local overfitting or a mismatch in nonlinear response modeling. These structural observations confirm that the Transformer possesses not only higher fitting accuracy but also stronger physical consistency and robustness in modeling complex dynamic behaviors.

14.

Prediction results of cumulative oil production for each model: (a) Transformer; (b) XGBoost; (c) Random forest.

15.

Prediction results of CO₂ storage for each model: (a) Transformer; (b) XGBoost; (c) Random forest.

Table summarizes the quantitative evaluation results of each model on the test set. For both cumulative oil production and CO₂ storage tasks, the proposed model significantly outperforms the comparison models in terms of the coefficient of determination (R²), root mean square error (RMSE), and mean absolute error (MAE), demonstrating consistent and robust high-precision prediction capabilities. XGBoost performed the weakest across all metrics, with the largest errors and insufficient stability. While Random forest achieved an R² close to that of the Transformer in CO₂ storage prediction, its RMSE and MAE remained relatively high, indicating that while it can capture trends, its control over error magnitude is limited.

4. Comparison of Model Performance Metrics.

	method	metrics
forecasting		R 2	RMSE	MAE
cumulative oil productin	Transformer	0.9986	1541.63	1149.48
	XGBoost	0.8592	15152.35	10631.63
	Random forest	0.7992	18094.49	12836.31
CO2 storage	Transformer	0.9968	3.96e+06	1.01e+06
	XGBoost	0.9873	7.77e+06	5.04e+06
	Random forest	0.9952	4.76e+06	1.94e+06

To further analyze the accuracy distribution characteristics of the prediction results, Figure illustrates the proportion of samples within four relative error intervals (<5%, 5–10%, 10–20%, > 20%) for each model. The results show that the Transformer has the highest proportion in the high-precision interval (<5%), reflecting the high concentration and low uncertainty of its predictions. XGBoost has the highest proportion in the large-error interval (>20%), suggesting a higher risk of prediction failure. Random forest shows a decent distribution in the medium-error range (5–10%) for the CO₂ storage task, but its proportion of tail errors remains significantly higher than that of the Transformer, particularly showing poor stability in oil production prediction. This error distribution pattern further corroborates that the proposed model possesses superior risk control capabilities and decision-making reliability for engineering applications.

16.

Comparison of model error distributions: (a) cumulative oil production model; (b) CO₂ storage model.

It is worth noting that as the prediction time span extends (i.e., as corresponding cumulative values increase), the prediction accuracy of all benchmark models decays to varying degrees due to the superimposed effects of changing geological conditions and dynamic injection-production strategies. However, the performance decay rate of the Transformer model is the most gradual. This phenomenon profoundly reflects the core advantage of the self-attention mechanism within the Transformer architecture: unlike XGBoost and Random Forest, which rely on local partitioning of the feature space, the Transformer can explicitly capture long-range dependencies in a time series. The model effectively decouples and memorizes the cumulative impact of historical states on the current moment, thereby overcoming the challenge of error accumulation in long-term predictions.

In summary, the proposed framework comprehensively outperforms traditional machine learning models in four dimensions: goodness of fit, error control, extrapolation stability, and error distribution concentration. This validates its algorithmic superiority and potential for engineering applications in complex reservoir dynamics and carbon sequestration simulations.

It is noteworthy that as the prediction time horizon extends, the prediction accuracy of all models exhibits a decline; This phenomenon is primarily attributed to the cumulative error propagation inherent in recursive multistep prediction, where minor deviations in early steps amplify over time. However, it is noteworthy that the proposed Transformer model mitigates this decay trend more effectively than the baseline RNN and LSTM models. Unlike RNN-based architectures that rely on sequential processing and are prone to information loss over long sequences, the Transformer’s self-attention mechanism allows the model to capture global temporal dependencies directly, thereby maintaining higher stability and robustness over extended prediction horizons. However, the Transformer model demonstrates the most gradual rate of performance decay. This phenomenon is likely attributed to the superimposed effects of multiple factors in long-term forecasting such as variations in geological conditions and adjustments in production strategies, which amplify the complexity and uncertainty of the prediction task.

4.1.3. Extreme Cases

To further evaluate the model’s generalization capability and prediction stability at the boundaries of the training data distribution, this study designed an extreme case experiment. Input parameters were set to the minimum and maximum values appearing in the training set to simulate extreme conditions that are rare but possible in actual reservoir development. On this basis, a continuous dynamic prediction spanning 60 time steps (corresponding to a 5 year development cycle) was conducted to generate time-series curves for cumulative oil production and CO₂ storage, validating the reliability of the model’s response to extreme initial conditions during long-term extrapolation.

Figures and illustrate the comparison of prediction results among the Transformer model, Random forest, and XGBoost under different feature extreme value conditions. It is evident that when input parameters are at the edges of the distribution, the prediction curves of traditional machine learning models deviate significantly from actual values, exhibiting marked distortion, especially in the middle and late stages of prediction. This reflects their insufficient extrapolation capability in data-sparse regions. In contrast, the Transformer model maintains an evolutionary trajectory highly consistent with actual values throughout the entire 5 year prediction domain. Whether driven by minimum or maximum values, the prediction curves for oil production and CO₂ storage remain smooth, reasonable, and within controllable error limits. This demonstrates a reasonable prediction for extreme situations and effectively addresses complex prediction problems in the field of CO₂-EOR reservoirs.

17.

Comparison of model prediction performance under maximum input conditions: (a) Transformer; (b) XGBoost; (c) Random forest.

18.

Comparison of model prediction performance under minimum input conditions: (a) Transformer; (b) XGBoost; (c) Random forest.

It is worth noting that, in addition to superior prediction accuracy and generalization performance, the proposed model demonstrates a fundamental distinction in computational efficiency mechanisms compared to traditional numerical simulation. As illustrated in Figure , the traditional numerical simulation is constrained by the iterative time-stepping solution mechanism, where the computational cost inevitably exhibits a cumulative growth trend as the simulation duration extends.

19.

Time consumption comparison between numerical simulation and transformer surrogate model.

In contrast, benefiting from the Implicit Time Parameterization strategy described earlier, the proposed model utilizes the cumulative injection volume to establish a direct continuous mapping of development dynamics, thereby eliminating the recursive dependence on discrete time steps. This mechanism effectively decouples the computational cost from the prediction time horizon: under the current experimental hardware configuration, regardless of whether predicting short- or long-term sequences, the model is essentially performing a single direct inference for a specific physical state. Its single inference time is 1.13 s; in comparison, the traditional numerical simulation requires 68 s for the equivalent prediction task, implying that the proposed model achieves a computational efficiency improvement of approximately 60 times. To ensure the reproducibility of these computational results, it is noted that all experiments were conducted on a workstation equipped with an Intel Core i5-8400 CPU and an NVIDIA GeForce GTX 1060 GPU, implemented using the PyTorch 2.3.1 deep learning framework. This time-independent high-efficiency characteristic effectively overcomes the time barrier of long-period simulations, making large-scale multidimensional parameter optimization and real-time rolling decision-making feasible.

These results indicate that the Transformer not only can accurately model development dynamics under typical operating conditions but also possesses the capability for stable extrapolation in extreme scenarios not fully covered by data. Simultaneously, its architecture based on implicit time modeling naturally supports continuous prediction output at arbitrary moments, avoiding the dependence of traditional models on fixed time nodes. Therefore, when facing complex reservoir systems with strong geological heterogeneity and variable operating conditions, the Transformer exhibits stronger adaptability and engineering credibility, proving to be a robust surrogate modeling tool.

4.2. Hyperparameter Optimization Results and Model Configuration

Based on the Bayesian optimization framework established in Section , this study conducted systematic hyperparameter tuning separately for the cumulative oil production and CO₂ storage prediction tasks. Considering the significant differences in dynamic evolutionary characteristics and data distributions between the two physical processes (as detailed in Section ), this study adopted an independent search strategy. Within a search space covering seven key parameters including model dimension, number of network layers, and feed-forward network size, optimal topological configurations adapted to the specific physical characteristics of each task were sought independently. The specific search ranges and the finally determined optimal hyperparameter combinations are detailed in Table .

5. Hyperparameter Search Ranges and Results.

hyperparameter	hyperparameter search range	optimal value
		cumulative oil production	cumulative CO2 storage
dimension of model	32, 64, 128, 256	256	64
number of attention heads	2, 4, 8, 16	2	2
number of transformer layers	1, 2, 3, 4	3	1
dimension of feed forward network	128, 256, 512, 1024	256	128
dropout rate	0.0, 0.05,0.1, 0.2,	0.05	0.1
learning rate	1e-2, 1e-3, 1e-4	1e-3	1e-4
batch size	16, 32, 64, 128	64	128

The optimization results revealed significant task-specific architectural preferences. For cumulative oil production prediction, the optimal model exhibited a deeper and wider structure (3 Transformer layers, 256-dimensional hidden state, 2 attention heads). This indicates that the process possesses high nonlinearity and long-term dependency characteristics, necessitating a stronger model capacity to accurately capture development dynamics. Conversely, the CO₂ storage prediction task favored a more lightweight architecture (1 Transformer layer, 64-dimensional hidden state, 2 attention heads). This reflects that its evolutionary trend is relatively smooth and easier to model; a shallower network structure suffices for efficient feature extraction while effectively avoiding the risk of overfitting caused by over parameterization.

Figure further validates the effectiveness of Bayesian optimization. In terms of prediction accuracy, the validation set loss for the oil production model decreased from an initial value of 0.0062 to 0.001574 after optimization. More significantly, the loss for the CO₂ storage model dropped substantially from 0.0084 to 0.00098, indicating that the optimized model’s ability to approximate physical laws was significantly enhanced. Regarding computational efficiency, the optimization strategy also yielded considerable benefits: the total training time for the oil production model was shortened from 1200 to 780 s, while that for the CO₂ storage model was reduced from 1800 to 810 s, effectively improving overall modeling efficiency.

20.

Comparison of model performance before and after optimization: (a) before optimization; (b) after optimization.

In summary, hyperparameter tuning based on Bayesian optimization not only significantly improved the prediction accuracy and training efficiency of the surrogate models but also, more importantly, revealed the intrinsic principle that physical characteristics dictate the network framework through a data-driven approach. This demonstrates the necessity of adaptive architectural adjustment for distinct reservoir engineering tasks.

4.3. Sensitivity Analysis of Input Values

This study employs the SHAP (SHapley Additive exPlanations) method to conduct a model interpretability analysis. To quantitatively evaluate the contribution of each input variable to the prediction results within the optimal Transformer model, SHAP, rooted in cooperative game theory, fairly attributes the predicted value of a single data point to each input feature, thereby revealing the driving effects of different parameters on the model output. The SHAP value quantifies the marginal contribution of each feature in a single prediction: a positive value indicates that the feature pushes the predicted output higher; a negative value indicates it lowers the output; and the absolute value reflects the magnitude of the impact. The SHAP scatter plot (Figures a and a) displays the contribution distribution of each feature across different samples and its relationship with feature values (Red = High, Blue = Low), thereby revealing the dependency patterns between features and predictions. By averaging the absolute SHAP values across all samples and plotting a summary bar chart (Figures b and b), we obtained a global feature importance ranking to identify key and secondary driving factors.

21.

SHAP features of the oil production prediction model.(a) SHAP Beeswarm plot; (b) SHAP feature importance bar.

22.

SHAP features of the CO₂ storage prediction model.(a) SHAP Beeswarm plot; (b) SHAP feature importance bar.

The SHAP feature importance analysis presented in Figures , and indicates that in both the oil production and CO₂ storage prediction models dynamic development parameters are the dominant factors, with Total Gas Injection (GIT) ranking first. Specifically, in the oil production model, the importance of injection-production related features such as GIT, Total Water Injection (WIT), and Liquid Production Rate (LPR) is significantly higher than that of geological properties. This demonstrates that under given geological conditions, the injection-production regime and dynamic development history are the primary determinants of oil displacement efficiency and storage capacity.

5. Conclusions

To address the challenges of time-consuming numerical simulations, difficulties in characterizing multiphysics coupling, and the limited generalization of traditional surrogate models during gas flooding development, this paper proposes a Transformer-based production dynamics surrogate model. This model integrates implicit time parameterization, physics-constrained nondimensionalization, and Bayesian adaptive optimization. By deeply mining the temporal dependencies and physical mechanisms within injection-production data, the model achieves high-fidelity, cross-scale predictions for cumulative oil production and CO₂ geological storage. The main conclusions are as follows:

• (1)Proposed an implicit time parameterization strategy to overcome the limitations of discrete prediction. By implicitly embedding physical time into cumulative injection features, the traditional reliance on fixed time steps is replaced by reconstructing the injection history. This mechanism transforms discrete point prediction into continuous process modeling based on cumulative driving effects, endowing the model with the capability to dynamically capture arbitrary injection-production stages and achieving continuous, smooth prediction of production curves.
• (2)Constructed a nondimensional input framework to facilitate cross-scale model transferability. A feature processing system based on pore volume (PV) normalization was established, effectively eliminating absolute differences in reservoir geometric scales (e.g., well spacing, thickness). This approach enables the model to learn physical flow laws rather than specific numerical mappings, significantly enhancing cross-block universality and resolving the reusability challenge of the “one reservoir, one model” paradigm associated with traditional surrogate models.
• (3)Validated the Transformer architecture’s advantage in capturing long-range dependencies, establishing a high-precision benchmark. The self-attention mechanism effectively addresses modeling challenges related to multiphase flow phase behavior changes and long-term historical dependencies. Experiments demonstrate that the Transformer architecture excels in capturing high-order nonlinear interactions (R ² > 0.99). Particularly under extreme geological parameter boundaries, it maintains superior physical consistency and robustness, overcoming the overfitting and distortion issues common in traditional machine learning models.
• (4)A Bayesian task-adaptive optimization approach was employed to elucidate physics-architecture mapping rules. Beyond automated hyperparameter tuning, this framework identified differentiated topological requirements: oil production forecasting necessitates deep and wide networks to capture strong nonlinearities, while storage prediction favors lightweight architectures to enhance generalization. Consequently, for a typical 5 year prediction task, this task-driven customization yields a 60-fold computational speedup over numerical simulations without compromising accuracy, offering an efficient technical pathway for multiobjective collaborative optimization.

However, the limitations of this study should be acknowledged. Specifically, since the model is trained on data generated from numerical simulations, strictly speaking, there may still be a gap when applying it to complex real-world field conditions involving high noise and extreme heterogeneity.

Future work can be expanded in three aspects: First, the seepage-stress-thermal multiphysics coupling mechanism is incorporated to enhance physical consistency under complex injection conditions. Second, real field data are introduced and transfer learning is combined to bridge the distribution gap between simulation and reality mentioned above, thereby strengthening field adaptability. Third, expanding into a multiobjective collaborative optimization platform that integrates recovery factor, carbon storage, and economic targets to build an integrated prediction optimization decision intelligent development system.

The authors declare no competing financial interest.