An Efficient Hierarchical Preconditioner-Learner Architecture for Reconstructing Multi-scale Basis Functions of High-dimensional Subsurface Fluid Flow

Abstract

Modeling subsurface fluid flow in porous media is crucial for applications such as oil and gas exploration. However, the inherent heterogeneity and multi-scale characteristics of these systems pose significant challenges in accurately reconstructing fluid flow behaviors. To address this issue, we proposed Fourier Preconditioner-based Hierarchical Multiscale Net (FP-HMsNet), an efficient hierarchical preconditioner-learner architecture that combines Fourier Neural Operators (FNO) with multi-scale neural networks to reconstruct multi-scale basis functions of high-dimensional subsurface fluid flow. Using a dataset comprising 102,757 training samples, 34,252 validation samples, and 34,254 test samples, we ensured the reliability and generalization capability of the model. Experimental results showed that FP-HMsNet achieved an MSE of 0.0036, an MAE of 0.0375, and an R2 of 0.9716 on the testing set, significantly outperforming existing models and demonstrating exceptional accuracy and generalization ability. Additionally, robustness tests revealed that the model maintained stability under various levels of noise interference. Ablation studies confirmed the critical contribution of the preconditioner and multi-scale pathways to the model's performance. Compared to current models, FP-HMsNet not only achieved lower errors and higher accuracy but also demonstrated faster convergence and improved computational efficiency, establishing itself as the state-of-the-art (SOTA) approach. This model offers a novel method for efficient and accurate subsurface fluid flow modeling, with promising potential for more complex real-world applications.

Figures (9)

• Figure 1: A permeability field of fractured porous medium with $K_m$ of 1 millidarcy and $K_f$ of 1000 millidarcy. The number of fractures is 15. The fine grid in blue refers to the matrix, and those in yellow refer to the fractures. We assign coordinates to the coarse grid system from bottom to top and from left to right. The coarse grid in the lower left corner is No.1 coarse grid, and so on.
• Figure 2: Architecture of Preconditioner-Sovler Structure. (A) Fourier Transform-based Preconditioner. Spectral denotes spectral convolution. (B) Hierarchical Neural Operator. (C) Structure of a single spectral conv layer in (A).
• Figure 3: Reconstruction results for coarse grid system. The data used for reconstruction comes from the training set, validation set, and testing set (each with two examples). The left image shows the reconstruction results based on four models (the main model, ablation 1, ablation 2, and ablation 3).
• Figure 4: Error distribution of our proposed method and ablation studies.
• Figure 5: Histogram plot of the evaluation results (Training set, Validation set, Testing set and total dataset.