Learning multi-phase flow and transport in fractured porous media with graph neural networks

The EDFM framework introduces fracture cells with pore volumes magnitudes of order smaller than the matrix cells when discretizing stochastically generated fracture networks. This severely affects the convergence performance of nonlinear solvers and leads to low computational efficiency. Moreover, numerical simulations must be run over many realizations of fracture networks for uncertainty quantification. Considering the unstructured topology of the computation grid resulting from EDFM discretization, we propose to learn the complex multiphase flow and transport in fractured porous media with a graph neural network. The computational grids are first converted to a graph structure by treating each grid cell as a node and forming an edge between a pair of nodes whenever the two corresponding cells are connected in the mesh. The static grid properties (pore volume, depth, etc.) and the dynamic reservoir states (pressure and saturation) are then used to build feature vectors for both the nodes and edges. An encoder-processor-decoder architecture is adopted to build the graph neural network. The encoder encodes the raw node and edge features into latent embeddings, which are then passed through a series of massage-passing processors that process the node and edge embeddings based on the node connectivity. Finally, the decoder decodes the processed node embeddings into change of pressure and saturations that are used to update the pressure and saturation at the next time step. To train the model, one hundred realizations of discrete fracture networks are generated stochastically and a two phase, oil-water, flow problem is simulated using the EDFM method. In the first stage of training, the simulation data is organized into one-step rollout dataset as training data. The input data passes through the pressure model and saturation model independently, and the mean square error between model predictions and ground truth labels of both models are combined together into a single loss function. To limit the error accumulation during multi-step rollout, in the second stage of training, the pressure and saturation models are rolled forward in time multiple steps for any given input and the loss at each prediction step is calculated and summed as the final loss function. The trained graph neural network models show good generalization capability for unseen realizations of fracture networks when used both for one-step rollout and multi-step rollout, demonstrating the potential of graph neural models as invaluable tools for geoscience applications that involve optimization and uncertainty quantification of multiphase flow in fractured reservoirs.