The wavelength of Darcy-scale immiscible viscous fingering

Immiscible viscous fingering in porous media continues to attract significant interest. The concept has been originally derived for Hele–Shaw geometries and then applied to (multiphase) immiscible displacement in Darcy scale porous media. In the Hele–Shaw geometry, the finger wavelength arises from a competition between the viscous instability and a capillary restoring force associated with the Laplace pressure of the curved interfaces between displacing and displaced phases. The well-known derivation by Saffmann and Taylor predicts that the dominant wavelength scales with the square root of interfacial tension and permeability over viscosity and velocity. However, for Darcy scale viscous fingering a linear scaling with interfacial tension is observed, consistent with the less well-known long-wavelength instability by King and Dunayevsky and Yortsos and Hickernell. In their linear stability analysis “mixing” within the capillary dispersion zone is assumed to moderate the mobility contrast at the displacement front, which controls the finger wavelength. By using fine-gridded Darcy scale numerical simulations, we find that the capillary dispersion zone has approximately the same width as the finger wavelength and both scale linearly with interfacial tension. This confirms that capillary dispersion is indeed the mechanism controlling finger wavelength, leading to a linear and not a square-root scaling. By using approximations for the width of the capillary dispersion zone, a very much simplified semi-analytical derivation provides a simple closed-form prediction of the finger wavelength. It naturally gives the correct scaling with interfacial tension, permeability, and injection velocity, and it predicts the finger wavelength within a factor of 2.3 or better for underground storage scenarios in the context of carbon capture and storage. Quantitative mismatches mainly originate from uncertainty in the choice of shock-front vs end point reference saturation to evaluate mobility contrast and capillary dispersion.