Abstract
Geological parameterization procedures entail the mapping of a high-dimensional geomodel to a low-dimensional latent variable. These parameterizations can be very useful for history matching because the number of variables to be calibrated is greatly reduced, and the mapping can be constructed such that geological realism is automatically preserved. In this work, a parameterization method based on generative latent diffusion models (LDMs) is developed for three-dimensional channel-levee-mud systems. Geomodels with variable geological metaparameters (also referred to as hyperparameters or scenario parameters), specifically mud fraction, channel orientation, and channel width, are considered. A perceptual loss term is included during training to improve geological realism. For any set of scenario parameters, an (essentially) infinite number of realizations can be generated, so our LDM parameterizes over a very wide model space. New realizations constructed using the LDM procedure are shown to closely resemble reference geomodels, both visually and in terms of one- and two-point spatial statistics. Flow response distributions, for a specified set of injection and production wells, are also shown to be in close agreement between the two sets of models. The parameterization method is applied for ensemble-based history matching, with model updates performed in the LDM latent space, for cases involving uncertain scenario parameters. For three synthetic true models, we observe clear uncertainty reduction in both production forecasts and geological scenario parameters. The overall method is additionally shown to provide posterior geomodels consistent with the synthetic true model in each case.
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Acknowledgements
We are grateful to the Stanford Doerr School of Sustainability and to the industrial affiliates of the Stanford Smart Fields Consortium for financial support. We also thank the SDSS Center for Computation for providing the computational resources used in this work.
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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. The second author is on the editorial board of this journal.
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The source codes and datasets used in this work are available for download at https://github.com/guidodf09/ldm_3d_geomodel. The codes are based on the implementations provided in the https://github.com/huggingface/diffusers and https://github.com/Project-MONAI/GenerativeModels repositories. The Python libraries diffusers and monai are used in our implementation.
Appendices
Appendix A 3D-LDM Algorithms
VAE training
U-net training
Inference (generation)
Appendix B 3D-LDM Model Architecture
Tables 3 and 4 define the VAE and U-net architectures used in our 3D-LDM implementation.
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Di Federico, G., Durlofsky, L.J. Three-Dimensional Latent Diffusion Models for Parameterizing and History Matching Facies Systems Under Hierarchical Uncertainty. Math Geosci (2025). https://doi.org/10.1007/s11004-025-10245-x
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DOI: https://doi.org/10.1007/s11004-025-10245-x