Topological Gradient‐Based Identification of Inhomogeneities in Variable‐Order Time‐Fractional Subdiffusion Models for Shale Gas Production

ABSTRACT This work addresses an inverse problem for a time‐fractional diffusion equation with spatially varying coefficients. The objective is to identify an unknown inhomogeneous region within a bounded domain using boundary measurements. The inhomogeneity induces localized perturbations in both the permeability and the memory‐effect weight functions, significantly influencing the diffusion dynamics. To reconstruct this region, the inverse problem is reformulated as an optimization problem. An asymptotic expansion of the perturbed solution is derived, allowing for the computation of the topological gradient of a suitable cost functional. This gradient underpins a non‐iterative reconstruction algorithm requiring only a single forward simulation. The proposed approach is validated through several numerical experiments, demonstrating its accuracy and efficiency in recovering the shape and location of the inhomogeneity from boundary data.