Deep-learning-based surrogate models provide an efficient complement to numerical simulations for subsurface flow problems such as CO$_2$ geological storage. Accurately capturing the impact of faults on CO$_2$ plume migration remains a challenge for many existing deep learning surrogate models based on Convolutional Neural Networks (CNNs) or Neural Operators. We address this challenge with a graph-based neural model leveraging recent developments in the field of Graph Neural Networks (GNNs). Our model combines graph-based convolution Long-Short-Term-Memory (GConvLSTM) with a one-step GNN model, MeshGraphNet (MGN), to operate on complex unstructured meshes and limit temporal error accumulation. We demonstrate that our approach can accurately predict the temporal evolution of gas saturation and pore pressure in a synthetic reservoir with impermeable faults. Our results exhibit a better accuracy and a reduced temporal error accumulation compared to the standard MGN model. We also show the excellent generalizability of our algorithm to mesh configurations, boundary conditions, and heterogeneous permeability fields not included in the training set. This work highlights the potential of GNN-based methods to accurately and rapidly model subsurface flow with complex faults and fractures.

From a numerical perspective, resolving the wide range of spatiotemporal scales within such physical systems is challenging since extremely small spatial and temporal numerical We propose MeshfreeFlowNet, a novel deep learningbased stencils would be required. In order to alleviate the super-resolution framework to generate continuous computational burden of fully resolving such a wide range (grid-free) spatiotemporal solutions from the low-resolution of spatial and temporal scales, multiscale computational approaches inputs. While being computationally efficient, MeshfreeFlowNet have been developed. For instance, in the subsurface accurately recovers the fine-scale quantities flow problem, the main idea of the multiscale approach of interest. MeshfreeFlowNet allows for: (i) the output is to build a set of operators that map between the unknowns to be sampled at all spatiotemporal resolutions, (ii) a set associated with the computational cells in a fine-grid and the of Partial Differential Equation (PDE) constraints to be imposed, unknowns on a coarser grid. The operators are computed and (iii) training on fixed-size inputs on arbitrarily numerically by solving localized flow problems. The multiscale sized spatiotemporal domains owing to its fully convolutional basis functions have subgrid-scale resolutions, ensuring encoder.