Fully implicit finite difference pseudo-spectral method for simulating high mobility-ratio miscible displacements

A finite difference–pseudo-spectral (FD–PS) algorithm is developed to simulate the viscous fingering instability in high mobility-ratio (MR) miscible displacements. This novel algorithm uses the fully implicit alternating-direction implicit (ADI) method combined with a Hartley based pseudo-spectral method to solve the Poisson equation involving the streamfunction and the vorticity. In addition, under-relaxation in the iterative evaluation of the streamfunction is adopted. The new code allowed to model successfully the viscous fingering instability for mobility-ratios as high as 1800, and new non-linear viscous fingering mechanisms are discovered. A systematic analysis of the effects of the MR, the Peclet number and the aspect ratio on the finger growth is conducted. It is found that the growth of the interfacial instability accelerates with increase in the MR and Peclet number. At larger values of these parameters the increased stiffness of the corresponding numerical problem caused significant increase in the computational time as it required finer grids and smaller time steps to capture the fine structures of the viscous fingers.