Inversion of thermal flow in porous media using conditional Karhunen–Loéve expansion

Inverse problems associated with thermal flow in porous media are typically ill-posed due to strong parameter heterogeneity, limited observations, and measurement noise. Efficient parameterizations and robust optimization strategies are therefore essential to ensure stable and computationally feasible reconstructions. In this work, we formulate the inversion of permeability fields as a PDE-constrained optimization problem, where space-dependent and partially observed parameters are represented through a conditional Karhunen–Loève expansion (cKLE). The coefficients of the cKLE are treated as decision variables and are estimated by minimizing a nonlinear least-squares objective function that measures the mismatch between simulated and observed pressure and temperature data. To solve the resulting inverse problem, we employ deterministic optimization schemes based on the Levenberg–Marquardt (LM) method and Singular Value Decomposition (SVD). The use of cKLE enables an effective reduction of the parameter space while preserving the dominant spatial variability, significantly improving numerical stability and computational efficiency. The proposed methodology is evaluated using a field-scale thermal flow model based on Layer 10 of the SPE 10 benchmark, considering hot water injection under fixed-temperature boundary conditions. Numerical experiments with noise-free and noisy synthetic observations demonstrate that both LM and SVD schemes accurately reconstruct heterogeneous permeability fields, with robust performance under measurement uncertainty. The results highlight the effectiveness of combining dimensionality reduction with deterministic solvers for large-scale inverse problems in porous media.