Modelling the onset and evolution of immiscible viscous fingering in porous media

Technical Summary: Modelling the Onset and Evolution of Immiscible Viscous Fingering in Porous Media

Problem Statement
The simulation of immiscible viscous fingering in porous media is critical for understanding displacement processes in petroleum engineering and hydrogeology. While significant theoretical and numerical work exists, matching continuum-scale simulations to experimental data remains challenging. Specifically, previous models (e.g., Salmo et al.) successfully reproduced early-stage finger patterns and production curves for high viscosity ratios (e.g., $\mu_o/\mu_w = 2000$) but failed to capture two key late-stage experimental observations: (1) the absence of a stable, rarefaction zone trailing the fingers, and (2) the presence of significant bypassed oil regions after breakthrough. This paper addresses the physical mechanisms and modelling requirements necessary to replicate these specific features observed in the E2000 experiment (water displacing 2000 mPa·s oil in a Bentheimer sandstone slab).

Methodology
The authors developed a physics-guided numerical modelling approach using the commercial black-oil simulator IMEX. The methodology integrates three primary components:

1. Linear Stability Analysis (LSA): Used to characterize the onset of instability. The authors analyzed experimental X-ray images from the E2000 experiment to determine the dominant wavenumber ($\hat{\nu}$) of the fingers at early stages. They utilized LSA to select relative permeability (RP) and capillary pressure ($P_c$) functions that yield a most unstable wavenumber ($\nu_{max}$) significantly larger than $\hat{\nu}$. This accounts for the nonlinear effects of shielding and merging, which reduce the wavenumber as fingers thicken and coalesce.
2. Flow Function Selection: Two sets of RP functions (RP1 and RP2) and two $P_c$ functions (Pc1 and Pc2) were tested. The selected functions (RP2-Pc2) were chosen to maximize instability while maintaining consistency with experimental production data (water cut and oil recovery). The $P_c$ function was specifically selected to represent a weakly oil-wet system with a low ratio of derivative to magnitude ($|J'_c|/|J_c|$), enhancing capillary-heterogeneity effects.
3. Multiscale Permeability Heterogeneity: To replicate the experimental saturation fields, the authors constructed a multiscale permeability field. This field combines:
   • Small-scale heterogeneity: Randomly correlated fields to disrupt the stable rarefaction zone trailing the fingers and induce tip splitting.
   • Large-scale heterogeneity: Conditioned via an "image guide" derived from experimental X-ray scans to steer finger development and create preferential pathways.
   • Capillary Heterogeneity: The $P_c$ function was made dependent on permeability ($P_c \propto \tau^{-1/2}$), introducing a capillary-heterogeneity term that acts as a barrier to the invading phase in low-permeability regions when the displaced phase is wetting.

Key Results
The proposed model successfully matched the E2000 experimental data across multiple metrics:

• Finger Scales: By selecting flow functions where $\nu_{max} \approx 6 \text{ cm}^{-1}$ (roughly 10 times the experimental dominant wavenumber of $\approx 0.6 \text{ cm}^{-1}$), the model reproduced the correct finger width after accounting for nonlinear merging and shielding.
• Absence of Rarefaction Zone: The inclusion of small-scale permeability heterogeneity successfully disrupted the stable trailing zone commonly seen in homogeneous simulations, allowing fingers to extend to the inlet, consistent with the experiment.
• Bypassed Oil: The combination of a weakly oil-wet $P_c$ function and large-scale permeability heterogeneity (with a modest contrast of $K_{max}/K_{min} \approx 3.5$) successfully reproduced the bypassed oil regions observed at late stages (2+ PVI).
• Production Data: The model accurately matched the pressure drop, effluent water cut, and oil recovery curves.
• Saturation Distribution: The model captured the broadening of the saturation histogram, showing high saturation in high-permeability regions and low saturation (trapped oil) in low-permeability regions, a feature absent in models with fixed or zero capillary pressure.

Significance and Claims
The paper claims to provide a set of physics-guided constraints for modelling immiscible viscous fingering, moving beyond simple history matching to capture the underlying physical mechanisms. The authors emphasize that:

• LSA as a Guide: Linear stability analysis is a necessary guide for selecting flow functions; the initial most unstable wavelength must be several times smaller than the desired final finger width to account for nonlinear evolution.
• Role of Heterogeneity: Small-scale heterogeneity is required to disrupt the rarefaction zone, while large-scale heterogeneity (even with modest contrast) is necessary to create bypassed oil regions, particularly when coupled with capillary-heterogeneity effects in oil-wet systems.
• Capillary-Heterogeneity Effects: The study demonstrates that capillary heterogeneity, often neglected in field-scale models, plays a critical role in trapping the wetting phase (oil) in low-permeability regions, significantly impacting displacement efficiency even in relatively homogeneous media.

The authors conclude that while a unique predictive model for such complex systems remains elusive, their methodology offers a robust template for matching high mobility-contrast multiphase displacements by explicitly addressing the interplay between flow functions, multiscale heterogeneity, and capillary forces.