Spatiotemporal dynamics of surfactant-driven secondary invasion in Gaussian pore networks.

Capillarity-dominated two-phase displacement in porous media can continue beyond the initial invasion-percolation (IP) breakthrough when surfactants progressively modify interfacial properties and reopen pathways previously sealed by capillary barriers. We study this post-breakthrough secondary invasion, in which adsorption-driven reductions in interfacial tension and wettability shifts lower the capillary-pressure entry thresholds of yet-uninvaded throats, enabling further displacement under a fixed inlet pressure. To capture this mechanism, we develop a time-dependent pore-network framework that couples quasistatic IP to a reduced-order transport-adsorption module: local fluxes follow Poiseuille flow on the invaded cluster, interfacial adsorption is described by a Langmuir isotherm, and wettability evolution is represented via a phenomenological relation. Network heterogeneity is prescribed by Gaussian throat-size distributions, with the variance controlling structural disorder. The resulting invasion trajectories are sigmoidal and well described by Gaussian cumulative statistics, indicating that surfactant mass-transfer kinetics and network variance primarily rescale invasion timescales while preserving the overall functional form. Overall, the framework links interfacial conditioning to time-varying capillary-pressure thresholds and clarifies how surfactant-mediated processes govern secondary, post-breakthrough dynamics in heterogeneous porous systems.