🔬 condensed matter

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

This paper investigates the physical mechanisms and modelling requirements for accurately simulating the onset and evolution of immiscible viscous fingering in porous media at high viscosity ratios, demonstrating that matching experimental finger scales and saturation patterns necessitates a small initial unstable wavelength, the inclusion of small-scale channelling effects to disrupt trailing stability, and a weakly oil-wet capillary pressure function to capture bypassed oil.

Original authors: Paulo L. K. Caetano Chang, Kundan Kumar, Arne Skauge, Kenneth S. Sorbie

Published 2026-08-24

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CC BY 4.0

Original authors: Paulo L. K. Caetano Chang, Kundan Kumar, Arne Skauge, Kenneth S. Sorbie

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a sponge, but one made of rock, filled with thick, sticky oil. Now, imagine trying to push water through it to push the oil out. In a perfect, uniform world, the water would move forward as a smooth, even wall, sweeping the oil ahead of it like a broom. But nature is rarely perfect. In the real underground world of oil reservoirs, fluids have different thicknesses, and the rock itself is never perfectly uniform. When a thin fluid like water tries to push a much thicker fluid like oil, the interface between them becomes unstable. Instead of a smooth wall, the water breaks through in jagged, finger-like paths, leaving vast pockets of oil untouched behind. This phenomenon, known as viscous fingering, is a major headache for engineers trying to extract oil or manage groundwater. It is also a puzzle for scientists trying to predict how fluids move through the complex, hidden maze of porous rock.

A team of researchers set out to solve a specific piece of this puzzle. They wanted to understand exactly how these fingers form, grow, and merge, and why standard computer models often fail to predict the messy reality seen in experiments. Their focus was on a specific experiment where water was injected into a slab of Bentheimer sandstone to displace oil that was two thousand times thicker than the water. In this extreme scenario, the water should theoretically race through the rock in a chaotic spray of fingers, leaving most of the oil stranded. However, when scientists ran computer simulations of this process, the results were too neat. The fingers in the simulation grew too wide, and a strange, stable zone of untouched rock often appeared right behind the fingers, a feature that simply did not exist in the real experiment. The researchers needed to figure out what physical ingredients were missing from their digital models to make them behave like the real rock.

The team began by looking closely at the early moments of the real experiment, using X-ray images to see how the fingers first appeared. They measured the spacing between these initial fingers and found them to be quite narrow. When they ran their standard simulations, the fingers were much too wide. Through careful analysis, they discovered a counterintuitive rule: to get the simulation to produce the correct, narrow fingers, the computer model had to be set up to create instability at a much smaller scale than the final fingers themselves. It is as if the model needed to start with a chaotic jumble of tiny, unstable ripples. As the simulation progressed, these tiny ripples would naturally merge and shield one another, coalescing into the larger, stable fingers seen in the experiment. If the model started with ripples that were already the size of the final fingers, the result was a simulation that was too smooth and too slow to match reality.

But matching the finger size was only half the battle. The real experiment showed no calm, stable zone trailing behind the fingers; the fingers stretched all the way back to the injection point. In contrast, the simulations kept producing this calm zone. The researchers realized that the rock in the experiment, while appearing uniform to the naked eye, must contain tiny, hidden variations in its structure. They introduced small-scale variations in the rock's ability to let fluid pass through, known as permeability. These tiny variations acted like a disruptor, breaking up the calm zone and forcing the fingers to twist and split, just as they did in the real rock. However, they had to be careful; if the variations were too strong, the water would get trapped in a speckled pattern of high and low saturation, which was not seen in the experiment. The key was finding a "Goldilocks" level of small-scale roughness—enough to break the calm zone, but not enough to create a speckled mess.

The final piece of the puzzle involved the oil that remained trapped in the rock after the water had finished its work. In the experiment, a significant amount of oil was bypassed, left behind in large pockets even after a long time. The researchers found that this happened because of a subtle interaction between the rock's wettability and its hidden variations. The rock in the experiment was slightly oil-wet, meaning the oil preferred to stick to the rock surfaces. When combined with the small-scale variations in permeability, this created a situation where the water was pushed away from the low-permeability zones, leaving the oil trapped there. By including this specific type of rock behavior in their model, the researchers were finally able to reproduce the large pockets of bypassed oil seen in the experiment.

The result is a new set of guidelines for modeling these complex fluid movements. The researchers showed that to accurately predict how oil and water interact in the ground, one cannot simply use average values for the rock or the fluids. Instead, the model must account for the fact that the smallest instabilities grow into the largest fingers, that tiny hidden variations in the rock prevent the flow from becoming too orderly, and that the way fluids stick to the rock surface can trap oil even in seemingly uniform stone. By combining these factors, the team created a simulation that matched the real-world experiment almost perfectly, capturing the chaotic beauty of the fingers, the lack of a calm trailing zone, and the stubborn pockets of oil left behind. This work provides a clearer path for understanding how fluids move through the Earth's crust, offering better tools for managing resources and predicting the behavior of underground reservoirs.

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

Problem Statement
The simulation of viscous fingering in porous media is critical for understanding displacement processes in petroleum engineering and hydrogeology. While extensive theoretical and numerical research exists, particularly regarding miscible systems, modeling immiscible viscous fingering at the Darcy scale remains challenging. Previous studies, such as those by Salmo et al., successfully modeled finger evolution up to breakthrough using random correlated permeability fields and specific relative permeability functions. However, these models failed to capture significant "bypassed oil" regions observed in late-stage experiments (after 1+ pore volumes injected) and often produced a stable, unperturbed "rarefaction zone" trailing the fingers—a feature not seen in the specific high-viscosity-ratio experiments (µo/µw = 2000) conducted by Skauge et al. on Bentheimer sandstone slabs. The core problem addressed is the development of a physics-guided numerical model capable of reproducing both the early-stage finger scales and the late-stage saturation patterns, including bypassed oil, without relying on unrealistic heterogeneity magnitudes.

Methodology
The authors developed a numerical simulation framework using the commercial black-oil simulator IMEX, focusing on the E2000 experiment (water displacing 2000 mPa·s oil in a 30×30×2.55 cm Bentheimer slab). The methodology integrates three primary components:

Linear Stability Analysis (LSA): The authors utilized LSA to characterize the onset of instability. They analyzed experimental X-ray images to determine the dominant wavenumber ( $\hat{\nu} \approx 0.6 \text{ cm}^{-1}$ ) of the fingers. Crucially, they established that to match experimental finger scales in the nonlinear regime (after shielding and merging), the most unstable wavenumber ( $\nu_{max}$ ) predicted by LSA must be significantly larger (approximately 10 times) than the observed dominant wavenumber.
Flow Function Selection: Two sets of Relative Permeability (RP) and Capillary Pressure ( $P_c$ ) functions were tested. The selected functions (RP2 and Pc2) were chosen to maximize instability (high $\nu_{max}$ ) while maintaining consistency with production data (water cut and recovery). The $P_c$ function was modeled as weakly oil-wet, utilizing a Leverett J-function with a low ratio of derivative to magnitude ( $|J'_c|/|J_c|$ ) to enhance capillary heterogeneity effects.
Multiscale Permeability Modeling: To replicate the experimental observations, the authors constructed a multiscale permeability field:
- Small-scale heterogeneity: Random Gaussian simulations with short correlation lengths were used to seed finger formation and disrupt the stable rarefaction zone trailing the fingers (preventing the "stable tail" common in homogeneous simulations).
- Large-scale heterogeneity: A larger-scale correlated field, conditioned on X-ray scan data to align high/low permeability zones with observed saturation patterns, was superimposed to drive the formation of bypassed oil regions.
- Capillary Heterogeneity: The model incorporated a $P_c$ function dependent on both saturation and permeability, allowing capillary forces to trap the wetting phase (oil) in low-permeability regions.

Key Contributions and Results
The study successfully constructed a numerical model that matches the E2000 experiment's saturation fields and production data. Key findings include:

Wavenumber Disparity: The most unstable wavelength at the onset of instability must be several times smaller than the desired final finger width. This allows for the nonlinear processes of shielding and merging to reduce the wavenumber to the experimentally observed scale.
Disruption of the Rarefaction Zone: The absence of a stable trailing zone in the experiment is attributed to small-scale permeability heterogeneity. Without this, numerical simulations typically show a stable region behind the fingers; introducing small-scale variations disrupts this zone and promotes finger tip splitting.
Mechanism of Oil Bypass: The model demonstrates that bypassed oil in a relatively homogeneous medium can be explained by the combination of a weakly oil-wet capillary pressure function and moderate large-scale permeability heterogeneity. The capillary heterogeneity term acts as a barrier, preventing the invading water phase from entering low-permeability regions where the displaced oil (wetting phase) remains trapped.
Quantitative Match: The final model reproduced the breakthrough time, water cut, oil recovery, and the spatial distribution of saturation (verified via histograms) at various stages, including the late-stage bypassed oil regions.
Sensitivity to Wettability: Simulations without capillary heterogeneity or with a water-wet $P_c$ function failed to reproduce the bypassed oil, confirming that the specific interplay between wettability and permeability variations is essential for this phenomenon.

Significance and Claims
The paper claims to provide a set of "physics-guided constraints" and modeling guidelines for capturing immiscible viscous fingering dynamics. The authors assert that while a unique predictive model for such complex systems remains elusive, their approach offers a robust template for matching high mobility-contrast multiphase displacements.

The significance of the work lies in its ability to reconcile continuum-scale simulations with experimental observations of bypassed oil without invoking extreme heterogeneity. The authors argue that capillary heterogeneity effects, often considered negligible at the field scale, play a critical role in trapping the wetting phase at the macroscopic scale, thereby influencing overall displacement efficiency. By demonstrating that a weakly oil-wet system with modest permeability contrast can reproduce complex bypass patterns, the study offers a refined understanding of the physical mechanisms governing unstable displacements in porous media.

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