| The nonlinear behavior of unstable viscous fingering in miscible displacements in porous media is studied by numerical simulation. A Fourier spectral method is used as the basic numerical scheme. In its simplest formulation, the problem can be reduced to two algebraic equations for flow quantities and a first order ordinary differential equation in time for the concentration of the solute responsible for the variation in viscosity. For displacements in homogeneous media, results show that small disturbances grow exponentially at short times. However, as time goes on, the nonlinear behavior of fingers becomes important. There are always a few dominant fingers that spread and shield the growth of other fingers. The spreading and shielding effects are caused by a spanwise secondary instability, and are aided by transverse disperson. It is observed that once a finger becomes large enough, the concentration gradient of its front becomes steep due to stretching as a result of the cross-flow, in turn causing the tip of the finger to become unstable and split. The splitting phenomenon is studied and its mechanism explained for the first time. A study of the 1-dimensional axial concentration profile averaged across the span of the flow is also presented, which indicates that the mixing length grows linearly in time, and that effective 1-dimensional models cannot describe nonlinear fingering.; For displacements in a non-homogeneous system in which permeability variations are present, the heterogeneity can be characterized by the auto-correlation function and the variance of the permeability fluctuations. Results indicate that for a neutral displacement (M = 1), the effective 1-dimensional dispersion becomes Fickian at large times. For the unstable cases (M {dollar}>{dollar} 1), the instability is enhanced by the heterogeneity of the medium, and becomes most unstable if the correlation length of the permeability fluctuations is comparable to the preferred wavelength of the fingers for the homogeneous case. The 1-dimensional averaged results indicate that the mixing length is strongly affected by the mobility ratio and the magnitude of the permeability fluctuations, as expected. |
