Stability Analysis of Miscible Displacement Processes

Abstract

Abstract This paper presents a linear perturbation analysis of viscosity- and gravity-induced instability in miscible displacement processes. The analysis takes into account the time-dependent behavior of the basic flow solution. It also considers the velocity dependence in the hydrodynamic dispersion expressions. By solving analytically the perturbation equations, the necessary and sufficient conditions for instability in miscible displacement processes are derived. These criteria are related to the well-known Dumore's velocity and to a critical wavelength for finger growth. Furthermore, they are calculated at the most unstable location in the solvent-oil mixing zone. The equations derived then relate quantitatively major process variables to the onset of instability in a miscible displacement process. Examples are presented to illustrate the applications of the criteria derived. It is shown that as the displacement process proceeds, the injection rate can be gradually increased. Furthermore, for dipping and stratified reservoirs, and for laboratory core experiments, a threshold time can be reached, after which the process is said to be in a state of unconditional stability. The transverse interlayer mixing is an important variable in smearing the fingers. From a qualitative analysis on the stability of horizontal displacement processes, it is revealed that transverse dispersion is more effective in smearing the fingers than longitudinal dispersion. However, the existence of a dispersed longitudinal mixing zone is necessary for the stabilizing effect of the transverse dispersion to be operative. Oscillatory instability can occur in a miscible displacement process. It is attributable solely to the existence of a longitudinal wave. The existence of a longitudinal perturbation can further stabilize the displacement process. For horizontal displacement processes with an adverse mobility ratio and in the absence of a gravitational stabilizing force, transverse dispersion alone is incapable of smearing the fingers for typical reservoir length scales. On the other hand, in the case of laboratory core experiments, the small transverse dimension of the core allows the dispersion mechanism to effectively damp the growth of the fingers, resulting in a stable displacement.