Deep learning models of viscous fingering based on Koopman dynamics of dense embeddings
Phys. Rev. Fluids 10, 094502 – Published 29 September, 2025
DOI: https://doi.org/10.1103/knp4-cd89
Abstract
Mixing and spreading of fluids during viscous fingering in the presence of a continuous injection source boundary is an open question in energy and environment applications such as injection into oil reservoirs and groundwater remediation. Direct numerical simulation (DNS) of viscous fingering is computationally prohibitive at high viscosity contrasts (e.g., a displaced-to-displacing fluid viscosity ratio greater than 50) due to tip-splitting, merging, and channeling mechanisms that rapidly generate and eliminate large gradients in the concentration field. Recent advances in deep learning offer promise in overcoming the limitations of DNS. However, this promise has not been realized due to the formidable challenge posed by the multiscale dynamics of viscous fingering. Here, we present a novel deep learning framework that combines spatial embedding to extract the fingering mechanisms at multiple scales with Koopman-based temporal dynamics to learn the evolution of fingering-driven mixing. We demonstrate the framework using two models: a baseline autoencoder coupled with a transformer model (BAE-GPT2), which was recently proposed for the turbulent flow problem, and a new densely connected convolutional network autoencoder coupled with a long-short-term memory model (DAE-LSTM). Our DAE-LSTM model outperforms the BAE-GPT2 model in predicting concentration variance and mean scalar dissipation rate in the domain and the concentration breakthrough profile at the outlet. We derive the evolution equation for the degree of mixing, defined in terms of the concentration variance, in the context of continuous injection of the less viscous fluid—a problem that has remained unsolved until now due to challenges related to source/sink boundaries. Finally, we demonstrate that the DAE-LSTM model can forecast local-scale features, such as the number and size of fingers, as well as global-scale metrics, such as the degree and rate of mixing, for fluid-fluid displacement processes with high viscosity contrasts—a goal that has eluded many DNS methods.
