Conditional diffusion inversion for Darcy flows in heterogeneous porous media with random parameters

Diffusion models have recently become powerful tools for solving inverse problems in scientific and engineering applications. Many existing approaches treat pre-trained diffusion models as priors and employ additional sampling or correction strategies to incorporate observations. In this work, we consider a Bayesian inverse problem arising in subsurface flow, where the goal is to reconstruct spatially heterogeneous permeability fields from noisy flow rate observations governed by transient Darcy flow. We propose a two-stage conditional diffusion framework that explicitly separates the denoising of observations and the inversion of the quantity of interest. The first stage involves training a supervised conditional diffusion model to learn the mapping from noise-free simulated flow rates to the corresponding permeability fields. The second stage employs an unsupervised diffusion model to recover noise-free flow rates from noisy observations. This two-stage structure enables robust posterior sampling by producing permeability fields that match the observed flow rates. Numerical experiments on a synthetic reservoir model demonstrate the effectiveness of the proposed method in terms of reconstruction accuracy, uncertainty quantification, and computational efficiency.