Multivariate generative modelling of subsurface properties with diffusion models

Accurate multivariate parametrization of subsurface properties is essential for subsurface characterization and inversion tasks. Deep generative models, such as variational autoencoders (VAEs) and generative adversarial networks (GANs), are known to efficiently parametrize complex facies patterns. Nonetheless, the inherent complexity of multivariate modeling poses significant limitations to their applicability when considering multiple subsurface properties simultaneously. Presently, diffusion models (DM) offer state-of-the-art performance and outperform GANs and VAEs in several tasks of image generation. In addition, training is much more stable compared to the training of GANs. In this work, we consider score-based DMs in multivariate geological modeling, specifically for the parametrization of categorical (facies) and continuous (acoustic impedance – I­_P) distributions, focusing on a synthetic scenario of sand channel bodies in a shale background. We benchmark modeling performance against results obtained by GAN and VAE networks previously proposed in literature for multivariate modeling. As for the GAN and VAE models, the DM was trained with a training dataset of 3000 samples, consisting of facies realizations and co-located I­_P geostatistical realizations. Overall, the trained DM shows significant improvements in modeling accuracy, for all evaluation metrics considered in this study, except for the sand-to-shale ratio, where the values are comparable to those of the GAN and VAE. In particular, the DM is 26% more accurate at reproducing the average (nonstationary) facies distribution and up to 90% more accurate at reproducing the I_P marginal distributions for both sand and shale classes. Higher accuracy is also found in the reproduction of the facies-to-I­_P joint distribution, whereas the spatial I­_P distributions generated by the DM honour the two-point statistics of the training samples. The iterative generative process in DMs generally makes these networks more computationally demanding than VAEs and GANs. However, we demonstrate that with appropriate network design and training parametrization, the DM can generate realizations with significantly fewer sampling iterations while maintaining accuracy comparable to these benchmarking networks. Finally, since the proposed DM parametrizes the joint prior probability density function with a Gaussian latent space, it is straightforward to perform inversion. In addition to improved modeling accuracy, the mapping between the latent and image representations preserves a better topology than that of GANs, overcoming the well-known limitation of the latter for inference tasks, particularly for gradient-based inversion.