This chapter discusses how a data-driven machine learning approach can reproduce key aspects of the physical behavior of multiphase flows in complex geological formations. We propose an end-to-end graph neural surrogate tailored to CO$_2$ plume migration forecasting in geological storage. The method is evaluated on the SPE11A benchmark, a well-known industry test case designed to assess CO$_2$ storage scenarios and characterized by sharp gas-water interfaces, strong advective transport, and rapid convective mixing with fingering development. The benchmark is reformulated as a graph in which nodes represent computational cells and edges encode transmissibility-based interactions enriched with geometric attributes. Directional transport arising from grid geometry, permeability contrasts, and geological heterogeneity is captured through an anisotropic message-passing mechanism, where interaction weights are computed via geometry-conditioned edge embeddings, biasing message aggregation toward physically relevant transport directions. Temporal evolution is modeled in latent space using an autoregressive residual formulation trained with multi-step supervision. The proposed model produces competitive forecasts of gas saturation and liquid-phase density, which are key indicators for CO$_2$ storage monitoring, with cumulative errors that remain moderate over extended forecasting horizons.

翻译：本章探讨了数据驱动的机器学习方法如何再现复杂地质构造中多相流物理行为的核心特征。我们提出了一种端到端的图神经代理模型，专门用于地质封存中CO₂羽流迁移预测。该方法在SPE11A基准测试上进行了评估——该基准是评估CO₂封存场景的知名工业测试案例，其特征包括尖锐的气水界面、强对流传输以及伴随指进现象的快速对流传质。我们将该基准重新表述为图结构，其中节点代表计算单元，边编码基于传导率的相互作用，并辅以几何属性增强。通过各向异性消息传递机制捕获由网格几何、渗透率差异及地质非均质性引起的定向传输，该机制中相互作用权重通过几何条件化的边嵌入进行计算，使消息聚合偏向物理相关的传输方向。时间演化采用隐空间自回归残差公式进行建模，并通过多步监督训练。所提模型对气体饱和度及液相密度（CO₂封存监测的关键指标）产生了具有竞争力的预测结果，且在扩展预测时域内累积误差保持适度水平。