Geological parameterization entails the representation of a geomodel using a small set of latent variables and a mapping from these variables to grid-block properties such as porosity and permeability. Parameterization is useful for data assimilation (history matching), as it maintains geological realism while reducing the number of variables to be determined. Diffusion models are a new class of generative deep-learning procedures that have been shown to outperform previous methods, such as generative adversarial networks, for image generation tasks. Diffusion models are trained to "denoise", which enables them to generate new geological realizations from input fields characterized by random noise. Latent diffusion models, which are the specific variant considered in this study, provide dimension reduction through use of a low-dimensional latent variable. The model developed in this work includes a variational autoencoder for dimension reduction and a U-net for the denoising process. Our application involves conditional 2D three-facies (channel-levee-mud) systems. The latent diffusion model is shown to provide realizations that are visually consistent with samples from geomodeling software. Quantitative metrics involving spatial and flow-response statistics are evaluated, and general agreement between the diffusion-generated models and reference realizations is observed. Stability tests are performed to assess the smoothness of the parameterization method. The latent diffusion model is then used for ensemble-based data assimilation. Two synthetic "true" models are considered. Significant uncertainty reduction, posterior P$_{10}$-P$_{90}$ forecasts that generally bracket observed data, and consistent posterior geomodels, are achieved in both cases. PLEASE CITE AS: 10.1016/j.cageo.2024.105755 https://www.sciencedirect.com/science/article/pii/S0098300424002383 NOT WITH THE ARXIV VERSION

翻译：地质参数化是指利用少量隐变量及其到网格属性（如孔隙度和渗透率）的映射关系来表示地质模型。参数化在数据同化（历史拟合）中具有重要价值，它能在保持地质真实性的同时减少待确定变量的数量。扩散模型是一类新兴的生成式深度学习框架，在图像生成任务中已被证明优于生成对抗网络等传统方法。扩散模型通过训练实现"去噪"功能，从而能够从随机噪声场中生成新的地质实现。本研究采用的隐式扩散模型通过低维隐变量实现降维处理。本文构建的模型包含用于降维的变分自编码器以及用于去噪过程的U-net网络。该模型应用于二维三岩相（河道-堤岸-泥岩）条件建模。结果表明，隐式扩散模型生成的地质实现与地质建模软件样本在视觉上具有一致性。通过空间特征和流动响应统计量的定量指标评估，发现扩散模型生成结果与参考实现总体吻合。研究进行了稳定性测试以评估参数化方法的平滑性。随后将隐式扩散模型应用于基于集合的数据同化，针对两个合成"真实"模型案例开展实验。两种情形均实现了显著的不确定性降低，后验P$_{10}$-P$_{90}$预测区间基本覆盖观测数据，且获得了地质特征一致的后验模型。