Onset of viscous fingering for miscible liquid-liquid displacements in porous media

The displacement of one fluid by another miscible fluid in porous media is an important phenomenon that occurs in petroleum engineering, in groundwater movement, and in the chemical industry. This paper presents a recently developed stability criterion which applies to the most general miscible displacement. Under special conditions, different expressions for the onset of fingering given in the literature can be obtained from the universally applicable criterion. In particular, it is shown that the commonly used equation to predict the stable velocity ignores the effects of dispersion on viscous fingering.Nomenclature C Solvent concentration - Unperturbed solvent concentration - D_L Longitudinal dispersion coefficient [m²/s] - D_T Transverse dispersion coefficient [m²/s] - g Gravitational acceleration [m/s²] - I_sr Instability number - k Permeability [m²] - K Ratio of transverse to longitudinal dispersion coefficient - L Length of the porous medium [m] - L_x Width of the porous medium [m] - L_y Height of the porous medium [m] - M Mobility ratio - V Superficial velocity [m/s] - V_c Critical velocity [m/s] - V_s Velocity at the onset of instability [m/s] - µ Viscosity [Pa/s] - Unperturbed viscosity [Pa/s] - µ₀,µ_s Viscosities of oil and solvent, respectively [Pa/s] - Density [kg/m³] - ₀,_s Densities of oil and solvent, respectively [kg/m³] - Porosity - Dimensionless length