Nonlinear simulations of miscible viscous fingering with gradient stresses in porous media

Long-time behavior of the nonlinear simulations of miscible viscous fingering (VF), which arises during the displacement of a high viscous fluid by a lesser viscous one in a porous media, has been investigated in the presence of gradient stresses. Such non-conventional stresses appear in a miscible fluid system having steep concentration, density or temperature gradient. Experiments show that these gradient stresses, also called the Korteweg stresses, cause an effective interfacial tension (EIT) and act for the stabilization of the system against the growth of the fingers. Such fluid flow systems have been modeled by coupling the Darcy-Korteweg equation with the convection-diffusion equation for the evolution of the solute concentration. These equations are solved simultaneously using a highly accurate Fourier spectral method. Investigations have been carried out for classical single interface VF instability and it has been shown that the Korteweg stress stabilizes the downstream fingers more than the upstream fingers. The propagation of the non-zero axial velocity field, which is spread away from the fingertips, explains this effect. Korteweg stress seems to act against the broadening of the fingertip that resists the splitting of an isolated finger. The growth rate of the unstable modes at an early time of the nonlinear simulations is obtained for various flow parameters and the results obtained are found to be qualitatively in good agreement with the linear stability results available in the literature.