## 2 Development of Seepage Field Prediction Methods
### Deep Learning-Based Method for Seepage Field Prediction
The model converts the discretized residuals of the seepage control equations into a differential convolutional kernel structure and embeds them into the network loss function, which improves the physical consistency of the prediction results and the boundary processing capability. In the case of a typical reservoir with high WCP, PD-CNN shows good prediction ability in complex regions such as pressure degradation and drastic change of WC at the end of the recovery period, which is suitable for the tasks of real-time production monitoring and dynamic optimization deployment [2]. [PERSON] et al. (2023) [3] constructed a Deep Learning-based saturation field prediction model for the problem of oil saturation evolution in the high WC stage in complex fracture block reservoirs, combining sample pool optimization and U-Net structure extraction features to achieve dynamic modeling of three-dimensional time-series saturation fields. The study drove the model training by historical development data, and it reconstructed the evolution path of oil-rich zones by combining the data with a spatial interpolation algorithm, which could be used for identifying and dynamically tracking residual oil potential zones in high WCPs. Their experiments showed that the method can reconstruct high-resolution seepage response distribution maps under low well control conditions, providing technical support for reservoir digital twin modeling. [PERSON] and [PERSON] (2023) [15] proposed an intelligent modeling framework integrating numerical modeling and Deep Learning based on the coupled pressure-saturation control equation for simulating the evolution of oil-water two-phase flow in complex inhomogeneous reservoirs. The method was based on regression prediction of the coupled field evolution results by Deep Neural Networks, which realizes efficient modeling of the time-sequential changes of pressure and saturation fields, and which is especially suitable for intelligent assessment of injection and extraction responses during the optimization phase of regulation in high WCPs. The study emphasized the role of the physical-data-fusion strategy in improving the prediction accuracy and training stability of complex seepage systems. [PERSON] et al. (2024) proposed a multi-scale nonlinear seepage modeling method based on Graph Neural Networks (GNNs) for the problem of high-resolution seepage prediction at the Digital Core Scale (DCS) [16]. The method constructs the pore structure as a graph structure input, and it simulates the local velocity and saturation change process in a two-phase flow through the node information transfer mechanism. In the water-driven-dominated micro-replacement process, the model can accurately identify the micro-channel flow paths, residual oil retention areas, and non-homogeneous response characteristics in the high water-bearing stage, which provides a new numerical modeling path for the intelligent analysis of the micro-scale replacement mechanism in the high-water-bearing stage. [PERSON] et al. (2024) constructed an agent modeling framework based on Convolutional Spatio-temporal Networks (CSNs) for rapid prediction of production response during reservoir development, covering key variables such as pressure, production, and WC (2018). The method established a mapping of spatio-temporal dynamic relationships by extracting input features such as injection and production time series, well group locations, and geological parameters to realize rapid response prediction under complex flow conditions during high WCPs. In summary, Deep Learning methods have been widely explored in the prediction of seepage fields in oilfields with high WCP, especially in the modeling of key variables, such as pressure field, saturation field, WC evolution, and flow rate response, on which significant progress has been made.