A surrogate model for the prediction of permeabilities and flow through porous media: a machine learning approach based on stochastic Brownian motion

Abstract

In this contribution we propose a data-driven surrogate model for the prediction of permeabilities and laminar flow through two-dimensional random micro-heterogeneous materials; here Darcy’s law is used. The philosophy of the proposed scheme is to provide a large number of training sets through a numerically “cheap” (stochastic) model instead of using an “expensive” (FEM) one. In order to achieve an efficient computational tool for the generation of the database (up to 10 3 and much more realizations), needed for the training of the neural networks, we apply a stochastic model based on the Brownian motion. An efficient algebraic algorithm compared to a classical Monte Carlo approach is based on the evaluation of stochastic transition matrices. For the encoding of the microstructure and the optimization of the surrogate model, we compare two architectures, the so-called UResNet model and the Fourier Convolutional Neural Network (FCNN). Here we analyze two FCNNs, one based on the discrete cosine transformation and one based on the complex-valued discrete Fourier transformation. Finally, we compare the flux fields and the permeabilities for independent microstructures (not used in the training set) with results from the FE 2 method, a numerical homogenization scheme, in order to demonstrate the efficiency of the proposed surrogate model.