Enhancing the Parameterization of Reservoir Properties for Data Assimilation Using Deep VAE-GAN

Ensemble smoothers are the most successful and efficient techniques currently available for history matching. However, because these methods rely on Gaussian assumptions, their performance is severely degraded when the prior geology is described in terms of complex facies distributions (non-Gaussian). In this way, for these methods, we need to apply efficient parameterization techniques. Currently, the most efficient methods for performing parameterization are deep learning models. However, given the variety of existing deep learning models, studies have not identified which is most suitable for use with ensemble-based methods, although some important models had already been evaluated. Based on a recent literature review, the most promising models selected were VAE-GAN, Latent Diffusion, and StyleGAN models. As a novel aspect of this work, data assimilation with the second generation of StyleGAN (StyleGAN2) model was performed using the latent z-space and intermediate w-space, separately. They were applied in two 2D case studies: one categorical (three facies) and the other continuous. The results demonstrated that all three models are highly efficient, with the StyleGAN2 model standing out for generating samples with geological realism and achieving excellent data matching in the cases studied. Our findings show that performing data assimilation with StyleGAN2 using the intermediate space (w-space) yielded better results than the traditional application in the latent space (z-space). This is due to the fact that ESMDA uses linear updates and the w-space is much more linear and disentangled than the highly entangled z-space, thereby ensuring that the updated vectors remain close to realistic geological patterns. These results were validated using main geostatistical and history matching metrics.

We propose an efficient retraining strategy for a parameterized Reduced Order Model (ROM) that attains accuracy comparable to full retraining while requiring only a fraction of the computational time and relying solely on sparse observations of the full system. The architecture employs an encode-process-decode structure: a Variational Autoencoder (VAE) to perform dimensionality reduction, and a transformer network to evolve the latent states and model the dynamics. The ROM is parameterized by an external control variable, the Reynolds number in the Navier-Stokes setting, with the transformer exploiting attention mechanisms to capture both temporal dependencies and parameter effects. The probabilistic VAE enables stochastic sampling of trajectory ensembles, providing predictive means and uncertainty quantification through the first two moments. After initial training on a limited set of dynamical regimes, the model is adapted to out-of-sample parameter regions using only sparse data. Its probabilistic formulation naturally supports ensemble generation, which we employ within an ensemble Kalman filtering framework to assimilate data and reconstruct full-state trajectories from minimal observations. We further show that, for the dynamical system considered, the dominant source of error in out-of-sample forecasts stems from distortions of the latent manifold rather than changes in the latent dynamics. Consequently, retraining can be limited to the autoencoder, allowing for a lightweight, computationally efficient adaptation procedure with very sparse fine-tuning data.

Data assimilation (DA) in subsurface flow entails calibrating model parameters to match observed data, typically at wells, while preserving geological realism. Latent diffusion models (LDMs) provide efficient mappings from high-dimensional geological model space to a low-dimensional latent variable, reducing the dimensionality of the inverse problem while maintaining plausibility in posterior geomodels. However, the high nonlinearity in the LDM mapping may degrade the performance of Kalman-gain-based ensemble updates. We present a systematic comparison of DA algorithms applied to large-scale 3D channelized geomodels with hierarchical geological uncertainty. We compare model-space and latent-space DA using the ensemble smoother with multiple data assimilation (ESMDA), and demonstrate a key trade-off: model-space updates achieve significant uncertainty reduction but produce geologically unrealistic posterior models, while latent-space updates preserve realism but exhibit limited uncertainty reduction. Motivated by this, we explore rigorous Markov chain Monte Carlo (MCMC) and Sequential Monte Carlo (SMC) algorithms in the 3D-LDM latent space. To accommodate their high computational demands, we develop a fast surrogate flow model that approximates well-rate responses. MCMC and SMC are evaluated against ESMDA across three synthetic test cases, with DA performed in the LDM latent space. All models maintain geological realism due to the LDM parameterization. MCMC and SMC are consistent with one another and achieve lower data mismatch and more uncertainty reduction than latent-space ESMDA. Our overall results demonstrate that ensemble Kalman methods may provide overestimated posterior uncertainty with highly nonlinear parameterizations, while rigorous Monte Carlo sampling, enabled by fast surrogate models, can provide a more reliable alternative.