An innovative application of deep learning in multiscale modeling of subsurface fluid flow: Reconstructing the basis functions of the mixed GMsFEM

In multiscale modeling of subsurface fluid flow in heterogeneous porous media, standard polynomial basis functions are replaced by multiscale basis functions. For instance, to produce such functions in the mixed Generalized Multiscale Finite Element Method (mixed GMsFEM), a number of Partial Differential Equations (PDEs) must be solved, which requires a considerable overhead. Thus, it makes sense to replace PDE solvers with data-driven methods, given their great capabilities and general acceptance in the recent decades. Convolutional Neural Networks (CNNs) automatically perform feature engineering, and they also need fewer parameters via defining two-dimensional convolutional filters without reducing the quality of models. This is why four distinct CNN models were developed to predict four different multiscale basis functions for the mixed GMsFEM in the present study. These models were applied to 249,375 samples, with the permeability field as the only input. The statistical results indicate that the AMSGrad optimization algorithm with a coefficient of determination (R2) of 0.8434-0.9165 and Mean Squared Error (MSE) of 0.0078-0.0206 performs slightly better than Adam with an R2 of 0.8328-0.9049 and MSE of 0.0109-0.0261. Graphically, all models precisely follow the observed trend in each coarse block. This work could contribute to the distribution of pressure and velocity in the development of oil/ gas fields. Looking at this work as an image (matrix)-to-image (matrix) regression problem, the constructed datadriven-based models may have applications beyond reservoir engineering, such as hydrogeology and rock mechanics.

Automated summary

This study investigates multiscale modeling by using data-driven methods to replace standard polynomial basis functions. Four distinct CNN models were developed to predict multiscale basis functions in mixed GMsFEM. The statistical results indicate that the AMSGrad optimization algorithm performs slightly better than Adam.