Reconstructing porous media using generative flow networks

Highlights

• Extension of digital rock physics workflow to include porous media generation.

• Deep learning generative flow models applied to 3D grain/pore structure.

• Rapid generation of large images of sandstone at pore scale.

• Minkowski functional evaluation of pore volumes created.

• Statistic evaluation of original and synthetic porous media generated.

Abstract

One area of intense scientific interest for the study of sandstones, carbonates, and shale at the pore scale is the use of limited image and petrophysical data to generate multiple realizations of a rock’s pore structure. Such images aid efforts to quantify uncertainty in petrophysical properties, including porosity–permeability transforms. We develop and evaluate a deep learning-based method to synthesize porous media volumes using a so-called generative flow model trained on x-ray microscope images of rock texture and pore structure. These models are optimized on a log-likelihood objective and they synthesize large and realistic three-dimensional images. We demonstrate the rapid generation of sandstone image volumes that display realism as gauged by quantitative comparison of topological features using Minkowski functionals of porosity, specific surface area, and the Euler–Poincaré characteristic (i.e., pore connectivity). We also evaluate the single-phase permeability using Navier–Stokes and lattice Boltzmann methods and show that transport properties of the generated samples match measured trends.

Introduction

Understanding fluid flow in porous media at the micro-scale is relevant to many fields, such as oil and gas recovery, geothermal energy, geological CO₂ storage, and battery energy storage. Porous medium properties, such as porosity and permeability, are often calculated from laboratory measurements or direct imaging of the microstructure (Walsh and Frangos, 1968, Ketcham and Carlson, 2001, Josh et al., 2012, Callow et al., 2020). Due to acquisition times and experimental costs, however, it is difficult to evaluate the variability of these properties across multiple macroscopic samples especially as sample permeability decreases. Instead, researchers often use statistical methods to reconstruct porous media based on either two-point or multi-point statistics (Okabe and Blunt, 2007, Wang et al., 2018). Reconstruction using these methods often requires prior knowledge about the pore- and throat-size distributions and can be time-consuming to generate multiple realizations of the same rock sample.

Various methods are available to reconstruct porous media, such as simulated annealing and multi-point statistical (MPS) methods. Simulated annealing can incorporate multiple correlation functions and statistical properties during reconstruction (Yeong and Torquato, 1998, Manwart et al., 2000, Pant and Berkeley, 2016). The method has been used to generate 2D and 3D Fontainebleau and Berea sandstone structures but can take hours to days to create large volumes (Manwart et al., 2000, Čapek et al., 2009, Pant and Berkeley, 2016). Extensions of the simulated annealing technique have also been used to generate multi-scale heterogeneous materials from multiple data sources, as well as generating 3D binary volumes from limited, multimodal data (Chen et al., 2016, Li et al., 2018). MPS techniques have been used to reconstruct large-scale reservoir systems, as well as micro-scale porous media, but also are slow to scale to generate multiple, large volumes (Caers, 2001, Okabe and Blunt, 2004, Okabe and Blunt, 2007, Bai and Tahmasebi, 2020). Pattern-based MPS methods have allowed for faster reconstruction of 3D networks using a cross-correlation based simulation method (CCSIM), but MPS techniques still struggle to capture long-range connectivity (Tahmasebi et al., 2012, Tahmasebi et al., 2014, Tahmasebi et al., 2017). Other recent methods for reconstructing three-dimensional porous volumes include texture synthesis and gradient-based methods as well (Liu and Shapiro, 2015, Fullwood et al., 2008).

Given the multi-scale nature of many rock systems, an ideal reconstruction method is one that recreates different length scales accurately and rapidly from a few datasets and is equally applicable to conventional and unconventional porous media. Recently, advances in machine learning, specifically in generative modeling, have focused on learning from limited datasets and generalizing these learned features for various applications (Goodfellow et al., 2014, Dinh et al., 2015, Zhu et al., 2017).

The major contribution of this work is the application of generative flow models to 3D volumes of porous media exhibiting grains, pore throats, and pore bodies. The advantage of this method is that the training is done on 2D grayscale images. The training time and model size are reduced, thereby improving scalability of the algorithm for training on larger image sizes. We generate 3D grayscale images via a latent space interpolation using a modifiable Gaussian distribution. The volumes generated are shown to be as useful as the original datasets for computing porosity, permeability, mercury injection capillary pressure, and other rock properties. We verify the accuracy of the generated images by calculating morphological parameters and comparing them against the real rock dataset. Accordingly, this paper proceeds by proposing implementation of generative flow models, metrics for evaluating the accuracy of model results, computation of permeability from 3D images, discussing results, and then conclusions.