Ensemble Score Filter for Data Assimilation of Two-Phase Flow Models in Porous Media

TL;DR

The paper presents Ensemble Score Filter (EnSF), a training-free data assimilation method for two-phase flow in porous media, leveraging score-based diffusion to approximate filtering distributions without neural-network training and employing an analytical update to mitigate degeneracy in high dimensions. The model combines a mixed FEM forward solver for pressure, velocity, and saturation with a diffusion-model-based sampler that evolves ensembles in pseudo-time to match partial, noisy observations. Key contributions include a Monte Carlo score estimator, a prediction-update framework within the Bayesian filter, and extensive numerical experiments showing EnSF outperforming LETKF under permeability uncertainties and incomplete fracture information. The approach offers scalable, computationally efficient state estimation for complex subsurface flows with significant parameter uncertainty and sparse data, enabling improved reservoir and geo-energy decision-making.

Abstract

Numerical modeling and simulation of two-phase flow in porous media is challenging due to the uncertainties in key parameters, such as permeability. To address these challenges, we propose a computational framework by utilizing the novel Ensemble Score Filter (EnSF) to enhance the accuracy of state estimation for two-phase flow systems in porous media. The forward simulation of the two-phase flow model is implemented using a mixed finite element method, which ensures accurate approximation of the pressure, the velocity, and the saturation. The EnSF leverages score-based diffusion models to approximate filtering distributions efficiently, avoiding the computational expense of neural network-based methods. By incorporating a closed-form score approximation and an analytical update mechanism, the EnSF overcomes degeneracy issues and handles high-dimensional nonlinear filtering with minimal computational overhead. Numerical experiments demonstrate the capabilities of EnSF in scenarios with uncertain permeability and incomplete observational data.

Figures (14)

• Figure 1: Permeability function
• Figure 2: Initial conditions of the saturation. a) illustrates the initial condition for the reference saturation $\hat{s}_h(t=0) = 0$ and b) presents the perturb initial condition for solving $s_h$ in the forward model $\bar{f}$.
• Figure 3: The comparison between the reference saturation, estimated saturation without observation, and the estimated saturation with EnSF at $t = 0.4$. a) the reference saturation $(\hat{s}_h)$ is simulated using $k({\mathbf{x}})$ and the true initial condition, without any perturbation. b)-c) Both estimated saturations are computed from noisy initial saturation-- b) one without observation, and c) the other using the EnSF with 100% observation of the saturation data.
• Figure 4: Estimated saturation ($\tilde{s}_h$) with EnSF based on partial observations of the saturation data.
• Figure 5: Error for partial observations of the saturation data.