Joint State-Parameter Estimation for the Reduced Fracture Model via the United Filter

TL;DR

The paper addresses online joint state and parameter estimation for flow and transport in fractured porous media using reduced fracture models. It introduces the United Filter, which alternates state estimation via the Ensemble Score Filter with parameter estimation via the Direct Filter, iterating on the forward map $oldsymbol{igPhi}(X_n, oldsymbol{ heta})$ and observations. The method demonstrates superior accuracy and robustness over AugEnKF across three test cases, including anisotropic and advection-dominated regimes, under sparse and noisy data. This provides a scalable PDE-constrained inverse solver for fractured networks with multi-scale features and model uncertainties.

Abstract

In this paper, we introduce an effective United Filter method for jointly estimating the solution state and physical parameters in flow and transport problems within fractured porous media. Fluid flow and transport in fractured porous media are critical in subsurface hydrology, geophysics, and reservoir geomechanics. Reduced fracture models, which represent fractures as lower-dimensional interfaces, enable efficient multi-scale simulations. However, reduced fracture models also face accuracy challenges due to modeling errors and uncertainties in physical parameters such as permeability and fracture geometry. To address these challenges, we propose a United Filter method, which integrates the Ensemble Score Filter (EnSF) for state estimation with the Direct Filter for parameter estimation. EnSF, based on a score-based diffusion model framework, produces ensemble representations of the state distribution without deep learning. Meanwhile, the Direct Filter, a recursive Bayesian inference method, estimates parameters directly from state observations. The United Filter combines these methods iteratively: EnSF estimates are used to refine parameter values, which are then fed back to improve state estimation. Numerical experiments demonstrate that the United Filter method surpasses the state-of-the-art Augmented Ensemble Kalman Filter, delivering more accurate state and parameter estimation for reduced fracture models. This framework also provides a robust and efficient solution for PDE-constrained inverse problems with uncertainties and sparse observations.

Figures (42)

• Figure 1: The domain $\Omega$ with the fracture $\Omega_{f}$ (left) and the fracture-interface $\gamma$ (right).
• Figure 2: [Test Case 1] (Left) Geometry and boundary conditions of the test case. (Right) Example of an uniform triangular mesh for spatial discretization.
• Figure 3: [Test Case 1] The initial condition for the 2D equation. (Left) The true initial condition. (Right) The initial condition for the state dynamics in data assimilation.
• Figure 4: [Test Case 1] The initial condition for the 1D equations. The straight blue line represents the true initial condition. The red curve depicts the initial condition used in the United Filter method for data assimilation.
• Figure 5: [Test Case 1] Heat map illustrating the accuracy of the United Filter's pressure state estimation. (First) Reference pressure field at final time $T$. (Second) Estimated pressure field state with $100\%$ direct observation. (Third) With $75\%$ direct observation. (Fourth) With $50\%$ mixed observation.

Theorems & Definitions (4)

• Remark 1
• Remark 2
• Remark 3
• Remark 4