The physical reservoir simulations were performed using CMG-STARS, a robust numerical simulator capable of solving nonlinear systems governing multiphase, multicomponent fluid flow and heat transport in multidimensional porous media. To simplify numerical modeling, several reasonable assumptions were adopted: (1) chemical reactions within the reservoir were neglected; (2) mechanical effects were not considered; and (3) fluid flow was assumed to follow Darcy’s law.

In this study, the geothermal system involved a mixture of CO₂ and water as working fluids. Fluid flow through the reservoir matrix was described by a mass conservation equation based on Darcy’s law, expressed as follows: (1) φ ∂ ρ f ∂ t − ∇ • ρ f k μ f ∇ p + ρ f g ∇ z = − ρ f S ∂ p ∂ t − Q f where, φ denotes the reservoir porosity; ρ f is the density of the working fluid; t represents time; k is the reservoir permeability; μ f indicates fluid viscosity; p denotes reservoir pressure; g represents gravitational acceleration; and z indicates the vertical direction; S is the specific storage of the rock matrix; and Q_f describes the mass source/sink term associated with fluid injection or production within the rock matrix. Similarly, fluid flow within fractures is described by the mass conservation equation and Darcy’s law as shown in Eq. (2): (2) d f φ f ∂ ρ f ∂ t − ∇ T • d f ρ f k f μ f ∇ T p + ρ f g ∇ z = − d f ρ f S f ∂ p ∂ t + d f Q f where, d_f represents fracture aperture, and ∇ T denotes the gradient operator restricted to the fracture tangential plane; φ f is the fracture porosity; k_f and S_f are the fracture permeability and specific storage of the fracture, respectively.

In this study, the reservoir is assumed to be in local thermal equilibrium, implying instantaneous thermal equilibrium between the fluid and rock matrix. Thus, heat transfer within the porous medium can be described using a single conservation equation, as given in Eq. (3): (3) ρ C p e f f ∂ T ∂ t + ρ f C p , f u • ∇ T − ∇ λ e f f ∇ T = − Q f , E where, T represents temperature; C p , f denotes the heat exchange between the rock matrix and fracture; Q f , E represents the effective volumetric heat capacity; and λ e f f denotes the effective thermal conductivity; The effective volumetric heat capacity ρ C p e f f and effective thermal conductivity are calculated based on the volumetric averaging model, incorporating the thermal properties of both reservoir rock and the working fluid, as shown in Eqs. (4) and (5). (4) ρ C p e f f = ( 1 − φ ) ρ s C p , s + φ ρ f C p , f (5) λ e f f = ( 1 − φ ) λ s + φ λ e f f where, ρ s represents the density of the solid phase in the reservoir; C p , s denotes the specific heat capacity of the solid; λ s is the thermal conductivity of the solid phase. Similarly, the energy conservation equation for fractures within the porous medium can be expressed as Eq. (6): (6) d f ρ C p e f f ∂ T ∂ t + d f ρ f C p , f u f • ∇ T T − ∇ T • d f λ e f f ∇ T = d f Q f , E the parameters u and u_f in Eqs. (3) and (6) represent the Darcy velocities of the working fluid in the reservoir matrix and fractures, respectively.

CO₂-water EGS numerical simulations typically require significant computational resources due to the complex calculations involved in phase equilibrium and heat transfer, resulting in low computational efficiency. Mapping input data, such as permeability and porosity fields, to reservoir production data using traditional numerical simulators is both computationally expensive and time-consuming.

In this equation, H, W, and D denote the three-dimensional spatial grid resolutions in the spatial domain. Traditional numerical simulation approaches, when applied to such three-dimensional grid structures, commonly face challenges related to computational inefficiency due to inherent nonlinearities and spatiotemporal constraints. To overcome these limitations, this study proposes a multimodal fusion Transformer neural network model that significantly simplifies this procedure. The proposed model provides an efficient approximation method to predict geothermal production data, thereby considerably reducing computational costs. In this section, we detail the construction of the proposed neural network architecture.

The deep learning surrogate model was trained for 200 epochs with a learning rate of 0.0001, taking approximately 530 s on an NVIDIA TITAN RTX. Table 2 presents the parameter settings for the deep learning surrogate model. The variation of root mean square error (RMSE) with epochs is shown in Fig. 5. The training set exhibits an average RMSE of 3%, with standard deviations ranging from 1.0% to 5.0%. Meanwhile, the testing set shows an average RMSE of 4.35%, with standard deviations between 1.57% and 7.13%. These results indicate that the model fits the training data well. Fig. 6 presents the parity plot of the parameters predicted by the forward surrogate model against the reference simulation results. The data points are closely distributed along the diagonal line, indicating that the surrogate model provides highly accurate predictions. This strong agreement confirms the reliability of the proposed surrogate in replacing time-consuming numerical simulations for subsequent optimization and analysis.

Fig. 7 compares the RMSE distributions of four methods on this dataset, where GRU, RNN, and LSTM are typical neural networks used for time series forecasting. In contrast, the average RMSE values for the GRU, RNN, and LSTM models are approximately 8.9%, 8.4%, and 8.1%, respectively, with a wider error distribution range. Fig. 8 presents the comparison between the predicted and actual values for CO₂ sequestration and geothermal energy production across different models. It is evident from the figure that, compared with GRU, RNN, and LSTM, the “Proposed” model exhibits data points that are more tightly clustered around the 45° reference line in both plots, indicating higher prediction accuracy and lower dispersion. In contrast, the scatter distributions of the other models are relatively farther from the reference line, suggesting the presence of systematic biases or random errors that result in larger discrepancies between the predictions and the true values. These results demonstrate that the model proposed in this study offers significant advantages in both prediction accuracy and stability.

By adjusting the number of layers in both the encoder and decoder of the Transformer model, we explored its impact on model performance. Four different configurations were employed (i.e., 2, 4, 6, and 8 layers for both the encoder and decoder). As shown in Fig. 9, the Transformer models with different depths exhibit varying RMSE distributions in the prediction task. Specifically, the 2-layer model has an average RMSE of approximately 5.2% with a standard deviation of about 3.2%; the 4-layer model sees a reduction to around 4.5% average RMSE and a standard deviation of approximately 2.4%; the 6-layer model further decreases the average RMSE to roughly 3.9% with a standard deviation of about 2.2%; however, when the number of layers increases to 8, the average RMSE slightly rebounds to approximately 3.4% with a standard deviation of around 2.0%. Fig. 10 illustrates that for models with different depths, the data points for both the total CO₂ sequestration (left panel) and total energy production (right panel) are distributed near the ideal 45° reference line, indicating an overall high prediction accuracy. In summary, these figures demonstrate that adjusting the number of layers affects the prediction accuracy and stability. An appropriate network depth enhances the consistency of predictions for CO₂ sequestration and energy output. This trend suggests that moderately increasing the network depth can effectively improve the model’s feature extraction capabilities and reduce prediction errors. However, when the network becomes too deep, training becomes more challenging and the risk of overfitting increases, which may lead to a decline in prediction accuracy. Therefore, in practical applications, it is essential to balance network depth and model complexity based on the scale of the task, data volume, and available computational resources to achieve optimal predictive performance.

We employed 5-fold cross-validation to assess the generalization performance of the proposed model. Specifically, the entire dataset was randomly partitioned into five non-overlapping subsets (200 data samples per subset). In each fold, one subset was designated as the validation set, while the remaining subsets formed the training set. For each fold, the model was independently initialized and trained for 100 epochs. As shown in Fig. 11, the RMSE distributions across the different folds (fold1–fold5) are generally similar, with both the mean and standard deviation showing no significant differences. This indicates that the model achieves relatively consistent prediction accuracy across the folds, with a concentrated error distribution and no evident outliers. Furthermore, the R² results (Fig. 12) reveal that the goodness-of-fit in each fold is consistently high with minimal variation. This suggests that under different data partitioning conditions, the model maintains stable explanatory power for the target variables and achieves robust fitting performance. In summary, the model’s performance during the 5-fold cross-validation is both stable and reliable, demonstrating strong generalization ability across different data splits.