An Omni-Temporal Theory for Hydrodynamic Dispersion and Reaction in Porous Media
A frequency-based omni-temporal dispersion theory is developed to capture the transient interplay between diffusion, advection, and reaction during solute transport through porous media. Unlike classical asymptotic dispersion theories, which commonly rely on long-time approximation, the proposed framework simultaneously captures both fast and slow components of dispersion. The theory is formulated by volume averaging the Fourier-transformed pore-scale advection-diffusion equation, yielding four frequency-dependent upscaled transport coefficients for a periodic unit cell: a dispersion tensor, an advection-suppression transfer function, a spectral Sherwood number, and a reactivity-bias vector. These coefficients act as transfer functions that relate microscopic driving forces to corresponding effective fluxes in the frequency domain, enabling prediction of transient transport dynamics in the time domain through inverse Fourier transformation. The utility of the proposed framework is demonstrated by deriving analytical expressions for the transfer functions in Poiseuille flow between parallel plates and through circular tubes, and subsequently using them within a Fast Fourier Transform framework to obtain breakthrough curves. For fast solute pulses between inactive parallel plates, the proposed theory produces breakthrough curves in close agreement with direct numerical simulations, whereas conventional asymptotic theory overpredicts propagation rates by orders of magnitude. Finally, the framework is applied to reactive and non-reactive porous media consisting of periodic arrays of square rods under cross flow, demonstrating the generality and versatility of the proposed omni-temporal theory.