Source-linked AI summary
Reconstruction of three-dimensional porous media using generative adversarial neural networks
Lukas Mosser, Olivier Dubrule, Martin J. Blunt
TL;DR
The paper addresses the difficulty of obtaining enough representative three-dimensional porous-media samples to assess pore-scale property variability. It uses GANs to learn an implicit distribution from segmented image datasets and generate synthetic realizations. These realizations capture characteristic statistical, morphological, and physical behavior while supporting efficient generation of large samples, under stated stationarity and training-data assumptions.
Problem
Obtaining enough representative three-dimensional pore-structure samples to assess variability in pore-scale flow properties is often experimentally infeasible.
Method
The paper trains generative adversarial neural networks on three-dimensional segmented images to learn an implicit representation of the porous-media image distribution.
Results
GAN-generated images capture characteristic statistical, morphological, and physical behavior of porous media, including two-point measures and effective permeability.
Takeaways & Limitations
Stored learned models can rapidly generate multiple large porous-media realizations for assessing material-property variability.
Takeaways & Limitations
The study treats training images as stationary and assumes a representative elementary volume; GAN training is also highly unstable and requires hyperparameter search.
Abstract
from arXiv · showhide
To evaluate the variability of multi-phase 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 novel 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 datasets. 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 GANs 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.
I. INTRODUCTION
The paper addresses the difficulty of acquiring enough representative three-dimensional porous-media images to assess pore-scale property variability. It proposes GAN-based stochastic reconstruction as a reusable, efficient alternative for generating synthetic samples from learned image distributions.
- Pore-scale flow processes depend on micron- and sub-micron-scale pore structures in sedimentary rocks.
- Micro-CT imaging requires specialized equipment, and CCS studies typically acquire only one image of the investigated rock type.
- Estimating variability across larger porous volumes would require numerous physical experiments, which high-resolution imaging costs and time often make infeasible.
- Existing reconstruction approaches include Boolean and process-based grain models, probabilistic spatial models, MPS, and simulated annealing.
- The study investigates GANs for three-dimensional porous-media textures by comparing synthetic statistical, morphological, and directional permeability properties with training images.
- The method can generate samples exceeding training-image dimensions, reuse stored trained parameters, and apply to volumetric images from other three-dimensional microscopy instruments.
II. GENERATIVE ADVERSARIAL NETWORKS
GANs learn an implicit image-distribution representation through adversarial training between a generator and discriminator. The approach enables fast parallel sampling but requires substantial effort to stabilize training.
- GANs learn implicit probability-density representations and support parallel generation of high-dimensional volumetric images.
- The generator maps random latent variables into images, while the discriminator classifies training-data images as real and generated images as fake.
- Training alternates discriminator and generator updates so generated samples increasingly fool the discriminator.
- The generator objective seeks a generated distribution pg(x) close to the real distribution pdata(x).
- The study notes that GAN training is highly unstable and requires many trial-and-error runs to identify hyperparameters supporting stable training.
III. METHODOLOGY
The evaluation treats porous-media images as stationary and assumes a representative elementary volume when assessing simulation quality.
- Simulation quality is evaluated against criteria defined from the training image datasets.
- The methodology assumes stationarity and the existence of a representative elementary volume for all images.
A. Evaluation Criteria
The evaluation combines two-point statistics, specific surface area, average chord length, and morphology-sensitive comparisons to assess reconstructed porous structures. These measures are computed with attention to anisotropy and compared against the original training images.
- 1. Two-Point Statistics: S2(r) is the probability that two points separated by lag vector r both lie in the pore phase, with S2(0)=φ and S2(r) approaching φ^2 at large r.
- 1. Two-Point Statistics: Because porous media may be anisotropic, S2(r) is computed along three Cartesian directions and as a radial average.
- 1. Two-Point Statistics: For Boolean models, S2(r) decays exponentially, whereas hard-sphere packings show dampened oscillations that reveal structural differences.
- 1. Two-Point Statistics: The specific surface area SV is related to the derivative of S2(r) at the origin for isotropic three-dimensional porous media.
- 1. Two-Point Statistics: Average chord length is obtained from the intersection of the slope of S2(r) at the origin with the x-axis.
- 1. Two-Point Statistics: The study compares covariance S2(r), specific surface area SV, and related measures between GAN reconstructions and original training images.
2. Morphological Measures
The study evaluates porous-media morphology using Minkowski functionals and empirical covariance, linking these measures to storage, interface processes, connectivity, and pore-scale flow.
- Porosity φ is the void-volume fraction and measures the ability of a porous medium to store fluids.
- Specific surface area SV characterizes the void-solid interface and its inverse defines a characteristic pore size.
- For voxelized polyhedra, the Euler characteristic is computed from vertices, edges, faces, and objects using χ = V − E + F − O.The relationship supports efficient computation of Minkowski functionals on volumetric voxel domains.
- The Euler characteristic is used to characterize pore connectivity, which is critical for fluid flow.
- Reconstructions should closely match observed Minkowski functionals because these measures represent physical processes at the pore scale.Specific surface area and porosity are also compared with estimates from empirical non-centered covariance S2(r).
3. Single-Phase Permeability
Single-phase permeability is evaluated by solving slow, incompressible Stokes flow on connected pore domains, while a volumetric GAN architecture generates synthetic reconstructions.
- 3. Single-Phase Permeability: Single-phase permeability is evaluated by solving the Stokes equations for slow, incompressible flow with small inertial forces.
- 3. Single-Phase Permeability: The flow domain includes only pore space connected to the fluid inlet and outlet, defining an effective porosity relevant to transport.
- B. Neural Network Architecture: The reconstruction network uses fully convolutional volumetric generator and discriminator networks based on DCGAN.The generator upsamples latent vectors using volumetric transposed convolutions, while the discriminator uses volumetric convolutions.
- B. Neural Network Architecture: Gaussian input noise and one-sided label smoothing are used to stabilize GAN training and prevent mode collapse.
- B. Neural Network Architecture: Generated images are median-filtered and then segmented with Otsu’s method before comparison with binary training images.
IV. EXPERIMENTAL DATA
Three voxelized porous-media datasets—bead pack, Berea sandstone, and Ketton limestone—provide training material for three-dimensional reconstruction experiments.
- IV. EXPERIMENTAL DATA: All datasets are segmented into three-dimensional binary voxel images separating pore space from grain structure.Training databases are formed by extracting subvolumes from the voxelized images, including overlapping subsets.
- IV. EXPERIMENTAL DATA: Training-image sizes are selected using estimated grain size and must exceed the structuring element to match covariance and morphology.Images larger than 128^3 voxels were excluded because of computational limitations.
- IV. EXPERIMENTAL DATA: The bead pack contains equally sized ceramic spheres in disordered close packing, with 500^3 voxels at 3 µm resolution.Its individual spheres have diameter 50 voxels, and 1,727 training images of size 128^3 were extracted.
- IV. EXPERIMENTAL DATA: Berea sandstone is a fluvial, medium- to fine-grained sandstone with angular grains and a 400^3-voxel image at 3 µm resolution.Training images of size 64^3 were extracted at 16-voxel spacing, yielding 10,647 images.
3. Ketton
The beadpack GAN reproduces key covariance, permeability, and morphological statistics while generating realizations that approximate spherical grains and their contacts. It also exhibits small porosity error but biases surface area and connectivity.
- Three GANs were trained for 24 hours each, with convergence checked through manual inspection and intermediate covariance and Minkowski-functional evaluations.
- The GAN captures the beadpack’s strong covariance hole effect caused by spherical grains.
- Synthetic beadpack grains deviate from perfect spheres, increasing specific surface area and reducing connectivity relative to the training image.
- The GAN reproduces isotropic directional covariance without directional variation and agrees with the training image up to 128^3 voxels.
- The synthetic realizations show close agreement in permeability magnitude and variance, although effective porosity tends to be higher.
- The generated realizations reproduce circular-to-ellipsoidal grains and complex single-point grain contacts.
B. Berea
For Berea sandstone, GAN realizations reproduce radial and directional covariance structure, Minkowski-functional distributions, and the observed permeability range, variability, and trend.
- The radial covariance shows near-exponential decay and stabilizes at a lag distance of 30 voxels in both training and synthetic images.
- The GAN captures Berea’s directional covariance, including the small hole effect in z and near-exponential decay in x.
- Synthetic and training images have comparable distributions of porosity, specific surface area, and Euler characteristic.
- Synthetic permeability captures the training dataset’s values, variability, and trend, with a similar effective-porosity relationship.
- A 64^3 training-image size was sufficient to capture Berea’s long- and short-range correlations because its structuring elements were smaller.
C. Ketton
For Ketton limestone, GAN realizations reproduce oolitic covariance structure, morphology, connectivity, and permeability, including anisotropic and oscillatory directional behavior.
- The GAN captures Ketton’s pronounced covariance hole effect associated with ellipsoidal oolitic grains.
- Generated images reproduce Ketton’s oscillatory and anisotropic directional covariance behavior.
- Approximately 1% porosity error was achieved between GAN-generated images and the original Ketton dataset.
- Synthetic specific surface area agrees with training data on average, although its variance is higher.
- Synthetic permeability distributions overlap the training-image values, while Euler characteristic and permeability jointly match observed connectivity.
VI. DISCUSSION
The GAN framework reconstructs porous-media structures beyond acquired image sizes while learning morphology and physical properties from training data. Its efficiency supports large ensembles, but fidelity depends on suitable training-image scale, stationarity assumptions, and accurate porosity learning.
- Objectives: GANs generate stochastic porous-media reconstructions exceeding the size of acquired image datasets and support probabilistic treatment of pore-scale properties such as permeability.The method targets larger synthetic domains and repeated realizations without requiring numerous images of one rock type.
- Scale: High-resolution generated domains can bridge micro-CT trade-offs between representative sample size and pore-scale resolution for macro-scale models.The approach incorporates micro-scale features into larger porous domains.
- Training-image scale: Training images smaller than the average grain size produce artifacts and distorted shapes, while 128^3 voxel beadpack images better capture complete particle geometry.A 64^3 voxel beadpack model learned short-scale structure but failed to reproduce long-distance correlation.
- Training-image scale: Training images should exceed the average grain size, although no evaluated two-point or morphological measure reliably predicts the required training-image size.For Boolean-like models, covariance S2(r) stabilization can estimate structuring-element size; complex samples require another measure.
- Training-data suitability: The beadpack was the most challenging training image because GANs implicitly learned deviations from its exactly spherical grains, misrepresenting effective properties.The authors therefore suggest fit-for-purpose GAN use for training images with variable grain sizes and shapes.
- Assumptions: The evaluated datasets were treated as stationary, while modeling non-stationary single-rock domains would require many images of the same porous medium.Representative volumetric images may also be harder to obtain for carbonate samples with complex pore-grain structures.
- Property matching: Matching the training-image porosity distribution was the main GAN training challenge because porosity emerges from data rather than being explicitly specified.The passage notes that a three-percent porosity error would mismatch synthetic permeability, although the sentence is truncated before giving the mismatch magnitude.
- Computational efficiency: Once trained, the fully convolutional GAN generates images larger than the original samples efficiently, with computational cost increasing linearly with voxel count.Training requires substantial computational resources, whereas reconstruction requires little effort and low runtime.
VII. CONCLUSIONS
The study evaluates GAN reconstruction on beadpack, Berea sandstone, and Ketton limestone using statistical, morphological, and transport properties. Synthetic images capture characteristic porous-media behavior, while sampling is efficient and future work targets improved reconstruction quality.
- Evaluation: GANs trained on beadpack, Berea sandstone, and oolitic Ketton limestone were evaluated for stochastic porous-media image reconstruction.These three image datasets served as the training images.
- Results: Synthetic images captured characteristic statistical and physical behavior across the evaluated porous media using two-point statistics, morphology, and single-phase effective permeability.The evaluation covered statistical, morphological, and transport properties.
- Efficiency: Despite costly GAN training, sampling from the learned representation is highly efficient and the learned models are easily stored for reuse.This supports repeated generation of synthetic realizations after training.
- Future work: Future work includes evaluating generator-discriminator architectures, grayscale and multi-channel training images, and large multi-scale domains.These directions are intended to improve image-reconstruction quality.