In this Letter, we use a model fluid mechanics experiment to elucidate the impact of curvature heterogeneities on two-dimensional fields deriving from harmonic potential functions. This result is directly relevant to explain the smooth stationary structures in physical systems as diverse as curved liquid crystal and magnetic films, heat and Ohmic transport in wrinkled two-dimensional materials, and flows in confined channels. Combining microfluidic experiments and theory, we explain how curvature heterogeneities shape confined viscous flows. We show that isotropic bumps induce local distortions to Darcy's flows, whereas anisotropic curvature heterogeneities disturb them algebraically over system-spanning scales. Thanks to an electrostatic analogy, we gain insight into this singular geometric perturbation, and quantitatively explain it using both conformal mapping and numerical simulations. Altogether, our findings establish the robustness of our experimental observations and their broad relevance to all Laplacian problems beyond the specifics of our fluid mechanics experiment.