Propagation, drainage and leakage of constant-flux input fluids in porous media over wavy topography

Wavy topography can exert a significant influence on gravity-driven flows in porous media. Building on the low-dimensional theoretical framework for a wavy topography of height f (x) = A1 - (x), where A is the amplitude and is the wavenumber of the topography, under small-slope conditions (A 1) Di et al. (2025 J. Fluid Mech. , vol. 1016, A16), we extend the framework to constant-flux injection while incorporating uniform drainage and localised leakage through low-permeability substrates. A key dimensionless topographic intensity, emerges as the ratio of the pressure gradient required to overcome topographic slopes to the characteristic viscous gradient driving the flow, thereby quantifying topographic resistance. Our results show that a larger topographic intensity retards current advancement, while drainage, governed by the drainage intensity, imposes an upper bound on propagation distance. Leakage proves highly sensitive to the along-slope position of fissured zones. Comparisons with a macroscopic sharp-interface flow model indicate that the low-dimensional model simplifies the two-phase dynamics in substrates via a Darcy’s sink term, yielding underestimates of propagation during drainage and leakage. Applied to the field of carbon dioxide sequestration, our low-dimensional model reveals how injection flux modulates the early-stage flow dynamics over wavy cap rocks, offering theoretical insights into sequestration performance.