Capturing the development of CO₂ front instability in a system with unstable fluid-fluid interfaces requires addressing the combined effects of inertial forces, viscous dissipation, and the inherent complexities of immiscible displacement. To achieve this using the conventional extension of Darcy’s law, here are key considerations and methodologies:

1. Extended Darcy's Law for Two-Phase Flow

Darcy's law can be adapted for multiphase flow by incorporating the relative permeability concept:

u_α=−k_rα*k/μ_α*∇P_α

where:

• u_α​: Darcy velocity of phase α,
• k: intrinsic permeability of the porous medium,
• k_rα​: relative permeability of phase α,
• μ_α​: viscosity of phase α,
• P_α​: pressure of phase α.

This provides a foundation for modeling immiscible displacement but requires modifications to address instability dynamics.

2. Relative Permeability Models

Relative permeability functions are crucial for predicting displacement patterns:

• Use hysteresis-aware relative permeability models to capture dynamic changes in fluid saturation and interface instability.
• Apply models incorporating capillary pressure (P_c=P_non-wetting-P_wetting​) and its dynamic effects due to interface movement.

3. Incorporating Instability Dynamics

CO₂ front instability often results from:

• Inertial forces: Introduce a Forchheimer term to extend Darcy's law, accounting for non-linear flow at higher Reynolds numbers: u_α=−k_rα*k/μ_α*∇P_α-β*u_α*2
• where β is the inertial resistance factor.
• Viscous dissipation: Include viscous effects in the energy balance equation to better capture the interplay of viscous and inertial forces.
• Interfacial tension effects: Use capillary number (Ca=μ*v/σ​) and mobility ratio to predict unstable patterns.

4. Numerical Simulation Framework

Develop a simulation that couples the following:

5. Multiphysics Coupling

• Gravity effects: Account for buoyancy-driven instability if the density difference between phases is significant.
• Heterogeneity: Incorporate spatial variations in permeability and porosity to reflect realistic pore invasion patterns.

6. Validation and Calibration

• Conduct experiments or high-fidelity simulations to observe intermittent pore invasion and validate the relative permeability and instability predictions.
• Use experimental data to fine-tune parameters such as relative permeability, k_rα​, and inertial resistance.

Outcome

This approach provides a predictive capability for immiscible displacement and instability by coupling extended Darcy's law with dynamic interface modeling and relative permeability. By calibrating against observed behaviors, it can offer insights into optimizing CO₂ injection processes and other multiphase flow scenarios in porous media.