Physics-Constrained Deep Learning for Permeability Field Data Assimilation: A U-Net-Based Surrogate Enhanced with Attention Mechanisms

Accurate reconstruction of subsurface permeability fields is critical for reliable reservoir forecasting and decision-making. However, conventional numerical simulators are computationally expensive, and standard data-driven approaches often fail to capture the strong heterogeneity present in real reservoirs. This study proposes a physics-constrained deep learning framework for efficient permeability field data assimilation, integrating an attention-enhanced U-Net surrogate model with partial differential equation (PDE) constraints and evolutionary optimization. Permeability fields are generated via sequential Gaussian simulation (SGS), while a Differential Evolution (DE) algorithm is employed to assimilate them using sparse observations from five wells, including permeability and bottom-hole flow rate measurements. The U-Net architecture facilitates hierarchical feature extraction, and attention modules dynamically emphasize spatial dependencies between permeability and pressure distributions. In addition, the integration of PDE constraints ensures that the learned pressure field adheres to governing flow physics. Additionally, an adaptive learning rate strategy is introduced to improve convergence efficiency and model robustness. Benchmark comparisons indicate that the proposed PDE-constraint surrogate model improves pressure prediction accuracy by 87.88% relative to baseline architectures without attention and physics constraints. The end-to-end framework reduces computational cost to 8% of conventional simulators, while maintaining relative pressure prediction errors below 3% across diverse geological scenarios. Results highlight the model’s capacity to capture fine-scale heterogeneities, enhance physical fidelity, and accelerate the data assimilation process, offering a scalable solution for high-resolution reservoir characterization.