Spatio-temporal dynamics of surfactant driven secondary invasion in Gaussian pore networks

TL;DR

This work addresses capillary-dominated two-phase displacement where surfactant adsorption lowers interfacial tension and alters wettability, enabling secondary invasion beyond IP breakthrough. It introduces a reduced-order framework that tightly couples a pore-network model with a surfactant transport module and an adsorption model to update capillary thresholds at fixed inlet pressure, producing adsorption-driven, time-dependent invasion. The main finding is that the invaded fraction follows a sigmoidal, Gaussian-CDF–like trajectory whose time axis is governed by a mass-transfer timescale; heterogeneity, encoded as Gaussian throat-size variance, stretches or compresses this timescale by exposing more reactive interfaces and promoting early activation of high-conductance throats. The framework offers a physically interpretable lens for surfactant-assisted displacement in heterogeneous porous media, with practical implications for enhanced oil recovery and subsurface remediation, and points to future work incorporating ganglion dynamics and richer adsorption kinetics.

Abstract

Capillarity-dominated two-phase displacement in porous media often continues beyond the initial invasion-percolation (IP) breakthrough, as surfactants alter interfacial properties and reopen pathways once sealed by capillary forces. This study examines such secondary invasion, where adsorption-driven reductions in interfacial tension and contact-angle shifts lower entry thresholds in yet uninvaded throats, enabling further displacement at a fixed inlet pressure. To capture this process, we employ a time-dependent pore-network framework that couples IP with a reduced-order transport-adsorption module. Local fluxes are governed by Poiseuille flow, interfacial adsorption follows a Langmuir isotherm, and wettability evolution is modeled through a calibrated phenomenological relation. Heterogeneity is prescribed by Gaussian throat-size distributions whose variance controls structural disorder. The resulting invasion trajectories are sigmoidal, consistent with Gaussian cumulative statistics, indicating that surfactant mass-transfer kinetics and network variance primarily rescale invasion timescales while preserving the overall functional form. The framework thus connects interfacial conditioning to time-varying capillary thresholds and reveals how surfactant-mediated processes govern post-breakthrough dynamics in heterogeneous porous systems.

Figures (10)

• Figure 1: Top-view of illustrative $22\times22$ micropillar networks showing baseline invasion-percolation (primary invasion) under constant inlet pressure, before surfactant effects. Throat radii are sampled from a Gaussian distribution with fixed mean and ensemble-specific variance. Open-boundary conditions are used; the inlet is fixed at the top-left and the outlet at the bottom-right (boundary arrows shown). Blue bonds delineate the first inlet-outlet connected path. The two panels are independent realizations with identical statistics, yielding distinct percolation paths that highlight the stochasticity of the pore-size heterogeneity. In the left panel, the green marker indicates a clustered region connected to the top boundary i.e., the clustered fluid can escape through the top. The trapped ganglia are apparent even as the interfacial tension is reduced, owing to the pore heterogeneity and local pressure drop. Axes indicate orientation; the view is along $+z$ (top view).
• Figure 2: Workflow linking primary invasion and hydrodynamics (PNM), surfactant transport (STM), and adsorption (SAM) to resolve surfactant-driven secondary invasion.
• Figure 3: Representative invaded fraction trajectories under open and restricted boundaries conditions, $\mathrm{f_t}$, together with their corresponding Gaussian CDF (error-function) fits as a function of time scaled with the median mass-transfer timescale. Results are obtained from an ensemble of 279 realizations generated from a PDF with a prescribed mean pore throat size of $50.0\,[\mu\text{m}]$ and variance of $4.94\,[\mu\text{m}^2]$. The specific network illustrated here has a mean pore throat size of $49.95\,[\mu\text{m}]$ and a variance of $5.14\,[\mu\text{m}^2]$. The temporal behavior reflects progressive surfactant adsorption at the oil-water interface (bulk concentration $\mathrm{C_B}=2\,\text{mM}$, surfactant S1), which drives successive invasion events. The invasion reaches different long-time limits under the two boundary conditions: complete displacement for the open case ($\mathrm{f_{\infty}}=1.0$) and partial saturation for the restricted case ($\mathrm{f_{\infty}}=0.936$). The representative fits capture the dynamics with high accuracy ($\mathrm{R^2}=0.996,\,0.997$), as further discussed in Section \ref{['analysis']}. Since the invasion dynamics are qualitatively similar, subsequent analysis in this paper focuses on the open boundary condition.
• Figure 4: (a) Normalized mass transfer coefficient for the ensemble, $\mathrm{\tilde{k}}$, (b) normalized interface size of the ensemble, $\mathrm{\tilde{r}_I}$, (c) normalized surface-area-to-volume ratio of the ensemble, $\widetilde{\dfrac{\mathrm{S}}{\mathrm{V}}}$, and (d) normalized mass transfer timescale, ${\tilde{\tau}}$, calculated for the S1 surfactant-driven invasion dynamics with different prescribed ensemble variance, $\overline{\mathrm{\sigma^2}}$, as indicated by the x-axis. The subscripts 'all' and 'sec' in the legend denote the ratio for all pore throat sizes and interface sizes at the onset of secondary invasion, respectively. At fixed inlet pressure, the surfactant lowers effective capillary thresholds ($\sigma\cos\alpha$), so larger $\overline{\mathrm{\sigma^2}}$ promotes earlier activation of high-conductance throats and faster connectivity of the invading pathway. As a result, $\mathrm{\tilde{k}}$ increases with $\overline{\mathrm{\sigma^2}}$, while the interface sizes decreases, yielding a higher $\widetilde{\dfrac{\mathrm{S}}{\mathrm{V}}}$-especially at the onset of secondary invasion-which enhances interfacial exchange. The combined effect is a systematic reduction in the mass transfer timescale. Normalization definition: for any quantity $q$ plotted in panels (a-d), $\tilde{q} \equiv \overline{q} / \overline{q}|_{\overline{\mathrm{\sigma^2}}=\overline{\mathrm{\sigma^2}}_{\min}}$, i.e., each value is divided by its value in the baseline case with the smallest prescribed ensemble variance$\overline{\mathrm{\sigma^2}}_{\min}$ (here, $4.953~\mu\mathrm{m}^2$).
• Figure 5: (a-b) Mean velocities of the invaded paths, $\mathrm{\overline{V}_{ij}}$ as a function of time scaled with the respective median mass-transfer timescale during S1 surfactant-driven invasion. Each panel shows two independent realizations from ensembles with mean variances, $\overline{\mathrm{\sigma^2}}=\{4.953,\,322.338\}\,\mu\mathrm{m}^2$. The values rise to a peak and then decay as capillary entry events rapidly accelerate newly opened pathways and, once the backbone forms, viscous redistribution and geometric bottlenecks progressively reduce the local driving. The surfactant lowers the effective capillary thresholds (via $\sigma\cos\alpha$), sustaining invasion beyond the first breakthrough and imprinting the transient peak in $\mathrm{\overline{V}_{ij}}$, (c) Normalized full-sweep time, $\mathrm{\tilde{t}}$, versus $\overline{\mathrm{\sigma^2}}$; the red line is a linear fit with the displayed equation, showing that larger variance shortens the full-sweep time.