Simulation of nonlinear viscous fingering in miscible displacement

Abstract

The nonlinear behavior of viscous fingering in miscible displacements is studied. A Fourier spectral method is used as the basic scheme for numerical simulation. 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. There are two parameters, the Peclet number (Pe) and mobility ratio (M), that determine the stability characteristics. The result shows that at short times, both the growth rate and the wavelength of fingers are in good agreement with predictions from our previous linear stability theory. However, as the 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 the transverse dispersion. It is shown that once a finger becomes large enough, the concentration gradient of its front becomes steep as a result of stretching caused by the cross-flow, in turn causing the tip of the finger to become unstable and split. The splitting phenomenon in miscible displacement is studied by the authors for the first time. A study of the averaged one-dimensional axial concentration profile is also presented, which indicates that the mixing length grows linearly in time, and that effective one-dimensional models cannot describe the nonlinear fingering.

Investigation of concentration‐dependent diffusion on frontal instabilities and mass transfer in homogeneous porous media

“…The assumption of the constant diffusion coefficient D adopted by previous researchers [9,29,26] significantly simplifies the problem and reduces the nonlinearity of the convection-diffusion equation (Equation (6)). Such an assumption may be somewhat valid in the case where the concentration of the solvent in the oil phase is small.…”

“…In addition, it is worth noting that, although the apparent Pe is fixed in computation, the effective Pe may vary significantly depending on the spatial and temporal change of local velocities and the CDDC in the miscible displacement process. [15] Following previous studies, [9,26] the relation between concentration and viscosity is given by Equation (7). This exponential variation can closely characterize the "quarter-power mixing rule" adopted by petroleum engineers.…”

“…[3] Since then, researchers have studied the various aspects of such instability experimentally, analytically, and numerically. [4][5][6][7][8][9][10][11] Factors such as the injection rate, mobility ratio, heterogeneity, and geometry on the VF dynamics, breakthrough time, and oil recovery have been intensively studied. In these studies and in most previous investigations, the diffusion coefficient is assumed to be constant in the whole reservoir.…”

Dynamics of a Highly Viscous Circular Blob in Homogeneous Porous Media

“…Similarly, the semi-implicit time discretization schemes are found to show convergence only for some parameters. The Fourier pseudo-spectral method used by various researchers [9,13] to capture non-linear VF, is an example of the semi-implicit time discretization. This pseudo-spectral method converges only for R ≤ 3.…”

“…Its wide applicabilities, e.g., enhanced oil recovery [1], CO 2 sequestration [2], chromatographic separation [3,4], contaminant transport in aquifers [5], mixing in low-Reynolds number flow [6], intrinsic characteristics of nonlinear dynamics and pattern formation have fascinated active theoretical, numerical [6][7][8][9][10][11][12][13], and experimental [14] researchers for more than half a decade. Both rectilinear and radial displacements have been investigated rigorously and have their own significance that can be investigated independently as well as comparatively.…”