To evaluate the variability of multiphase flow properties of porous media at the pore scale, it is necessary to acquire a number of representative samples of the void-solid structure. While modern x-ray computer tomography has made it possible to extract three-dimensional images of the pore space, assessment of the variability in the inherent material properties is often experimentally not feasible. We present a method to reconstruct the solid-void structure of porous media by applying a generative neural network that allows an implicit description of the probability distribution represented by three-dimensional image data sets. We show, by using an adversarial learning approach for neural networks, that this method of unsupervised learning is able to generate representative samples of porous media that honor their statistics. We successfully compare measures of pore morphology, such as the Euler characteristic, two-point statistics, and directional single-phase permeability of synthetic realizations with the calculated properties of a bead pack, Berea sandstone, and Ketton limestone. Results show that generative adversarial networks can be used to reconstruct high-resolution three-dimensional images of porous media at different scales that are representative of the morphology of the images used to train the neural network. The fully convolutional nature of the trained neural network allows the generation of large samples while maintaining computational efficiency. Compared to classical stochastic methods of image reconstruction, the implicit representation of the learned data distribution can be stored and reused to generate multiple realizations of the pore structure very rapidly.

We present an application of deep generative models in the context of partial-differential equation (PDE) constrained inverse problems. We combine a generative adversarial network (GAN) representing an a priori model that creates subsurface geological structures and their petrophysical properties, with the numerical solution of the PDE governing the propagation of acoustic waves within the earth's interior. We perform Bayesian inversion using an approximate Metropolis-adjusted Langevin algorithm (MALA) to sample from the posterior given seismic observations. Gradients with respect to the model parameters governing the forward problem are obtained by solving the adjoint of the acoustic wave-equation. Gradients of the mismatch with respect to the latent variables are obtained by leveraging the differentiable nature of the deep neural network used to represent the generative model. We show that approximate MALA sampling allows efficient Bayesian inversion of model parameters obtained from a prior represented by a deep generative model, obtaining a diverse set of realizations that reflect the observed seismic response.

The generation of multiphase porous electrode microstructures is a critical step in the optimisation of electrochemical energy storage devices. This work implements a deep convolutional generative adversarial network (DC-GAN) for generating realistic nphase microstructural data. The same network architecture is successfully applied to two very different three-phase microstructures: A lithium-ion battery cathode and a solid oxide fuel cell anode. A comparison between the real and synthetic data is performed in terms of the morphological properties (volume fraction, specific surface area, triple-phase boundary) and transport properties (relative diffusivity), as well as the two-point correlation function. The results show excellent agreement between datasets and they are also visually indistinguishable. By modifying the input to the generator, we show that it is possible to generate microstructure with periodic boundaries in all three directions. This has the potential to significantly reduce the simulated volume required to be considered "representative" and therefore massively reduce the computational cost of the electrochemical simulations necessary to predict the performance of a particular microstructure during optimisation.

Stochastic image reconstruction is a key part of modern digital rock physics and material analysis that aims to create representative samples of microstructures for upsampling, upscaling and uncertainty quantification. We present new results of a method of three-dimensional stochastic image reconstruction based on generative adversarial neural networks (GANs). GANs are a family of unsupervised learning methods that require no a priori inference of the probability distribution associated with the training data. Thanks to the use of two convolutional neural networks, the discriminator and the generator, in the training phase, and only the generator in the simulation phase, GANs allow the sampling of large and realistic volumetric images. We apply a GAN-based workflow of training and image generation to an oolitic Ketton limestone micro-CT unsegmented gray-level dataset. Minkowski functionals calculated as a function of the segmentation threshold are compared between simulated and acquired images. Flow simulations are run on the segmented images, and effective permeability and velocity distributions of simulated flow are also compared. Results show that GANs allow a fast and accurate reconstruction of the evaluated image dataset. We discuss the performance of GANs in relation to other simulation techniques and stress the benefits resulting from the use of convolutional neural networks . We address a number of challenges involved in GANs, in particular the representation of the probability distribution associated with the training data.

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