Constrained Transformer-Based Porous Media Generation to Spatial Distribution of Rock Properties

Abstract

Pore-scale modeling of rock images based on information in 3D micro-computed tomography data is crucial for studying complex subsurface processes such as CO2 and brine multiphase flow during Geologic Carbon Storage (GCS). While deep learning models can generate 3D rock microstructures that match static rock properties, they have two key limitations: they don't account for the spatial distribution of rock properties that can have an important influence on the flow and transport characteristics (such as permeability and relative permeability) of the rock and they generate structures below the representative elementary volume (REV) scale for those transport properties. Addressing these issues is crucial for building a consistent workflow between pore-scale analysis and field-scale modeling. To address these challenges, we propose a two-stage modeling framework that combines a Vector Quantized Variational Autoencoder (VQVAE) and a transformer model for spatial upscaling and arbitrary-size 3D porous media reconstruction in an autoregressive manner. The VQVAE first compresses and quantizes sub-volume training images into low-dimensional tokens, while we train a transformer to spatially assemble these tokens into larger images following specific spatial order. By employing a multi-token generation strategy, our approach preserves both sub-volume integrity and spatial relationships among these sub-image patches. We demonstrate the effectiveness of our multi-token transformer generation approach and validate it using real data from a test well, showcasing its potential to generate models for the porous media at the well scale using only a spatial porosity model. The interpolated representative porous media that reflect field-scale geological properties accurately model transport properties, including permeability and multiphase flow relative permeability of CO2 and brine.

Figures (16)

• Figure 1: The workflow of Vector Quantized Variational Autoencoder
• Figure 2: Workflow of the spatially assembled transformer. This process models spatial autoregression among patches $x$, associated with coordinates $(i,j,k)$ within a large porous medium $\mathcal{X}$. The transformer is trained on both the compressed spatial image token set $\mathcal{S}$ and corresponding rock properties, in this case porosity $\phi$. The transformer predicts the last patch token set $\mathcal{S}$ given all current and previous porosity information and all previous token sets $\mathcal{S}^{i=0,j=0,k=0}, ..., \mathcal{S}^{i=n,j=n,k=n-1}$, where n is the dimension of $\mathcal{X}$. The encoder $E$ was pretrained during the VQVAE workflow. Rock properties can be extracted using either an image processing tool to extract static properties like porosity or a pore network model to obtain flow-related variables. For simplification, we use the image processing tool PoreSpy 2019porespy to extract porosity.
• Figure 3: Scatter plot of original input porosity versus reproduced porosity in VQVAE model. The mean absolute error between the original porosity and reproduced porosity is $0.003$
• Figure 4: Scatter plot of original permeability versus reproduced permeability in VQVAE model. The mean absolute error between original permeability and reproduced permeability is $50$ md
• Figure 5: Histogram of original image patch porosity (left) versus transformer sampled image patch porosity (right)