A Dual-Branch Coupled Fourier Neural Operator for High-Resolution Multi-Phase Flow Modeling in Porous Media

This paper investigates a physics-informed surrogate modeling framework for multi-phase flow in porous media based on the Fourier Neural Operator. Traditional numerical simulators, though accurate, suffer from severe computational bottlenecks due to fine-grid discretizations and the iterative solution of highly nonlinear partial differential equations. By parameterizing the kernel integral directly in Fourier space, the operator provides a discretization-invariant mapping between function spaces, enabling efficient spectral convolutions. We introduce a Dual-Branch Adaptive Fourier Neural Operator with a shared Fourier encoder and two decoders: a saturation branch that uses an inverse Fourier transform followed by a multilayer perceptron and a pressure branch that uses a convolutional decoder. Temporal information is injected via Time2Vec embeddings and a causal temporal transformer, conditioning each forward pass on step index and time step to maintain consistent dynamics across horizons. Physics-informed losses couple data fidelity with residuals from mass conservation and Darcy pressure, enforcing the governing constraints in Fourier space; truncated spectral kernels promote generalization across meshes without retraining. On SPE10-style heterogeneities, the model shifts the infinity-norm error mass into the ${10}^{-2}$ to ${10}^{-1}$ band during early transients and sustains lower errors during pseudo-steady state. In zero-shot three-dimensional coarse-to-fine upscaling from $30\times 110\times 5$ to $60\times 220\times 5$, it attains $R^2=0.90$, RMSE = $4.4\times {10}^{-2}$, and MAE = $3.2\times {10}^{-2}$, with more than 90% of voxels below five percent absolute error across five unseen layers, while the end-to-end pipeline runs about three times faster than a full-order fine-grid solve and preserves water-flood fronts and channel connectivity. Benchmarking against established baselines indicates a scalable, high-fidelity alternative for high-resolution multi-phase flow simulation in porous media.