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.

翻译：地质参数化是指利用少量潜变量及其到网格属性（如孔隙度和渗透率）的映射关系来表征地质模型。参数化在数据同化（历史拟合）中具有重要作用，它能在减少待确定变量数量的同时保持地质真实性。扩散模型是一类新兴的生成式深度学习方法，在图像生成任务中已被证明优于以往的方法（如生成对抗网络）。扩散模型通过训练实现"去噪"，从而能够从随机噪声场中生成新的地质实现。本研究采用的潜变量扩散模型通过低维潜变量实现降维。本文构建的模型包含用于降维的变分自编码器和用于去噪过程的U-net网络。我们的应用针对条件化的二维三岩相（河道-天然堤-泥岩）体系。研究表明，潜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。通过评估涉及空间特征和流动响应统计量的定量指标，观察到扩散生成模型与参考实现之间总体吻合。本文进行了稳定性测试以评估参数化方法的平滑性。随后将潜扩散模型应用于基于集合的数据同化。研究考虑了两种合成"真实"模型。两种案例均实现了显著的不确定性降低、后验P$_{10}$-P$_{90}$预测区间对观测数据的有效包络，以及一致性良好的后验地质模型。