Investigations on Viscous Fingering by Linear and Weakly Nonlinear Stability Analysis

Abstract

Summary. In this paper we examine the growth of unstable disturbances (viscous fingers) in a water/oil displacement process with a mobile interstitial water saturation. Both linear and weakly nonlinear methods of analysis that include continuously changing mobility and capillary effects are presented. The fingers are assumed to develop as a result of perturbations of a steady displacement. perturbations of a steady displacement. After a brief review of the linear stability results, we implement a weakly nonlinear stability analysis for flow processes near the onset of instability (finite wave-number cutoff). Our numerical results indicate that disturbances near the critical point of the linear theory become stabilized to an equilibrium value. which is a function of the mobility ratio and the capillary number. This behavior suggests supercritical stability, in contrast to the instability predictions of the linear theory. The results of the linear theory are subsequently used to investigate the mechanism of finger growth at the wavelength of the fastest-growing disturbance. Saturation contours and the water flow field are numerically computed as a function of time. The local mechanism for finger growth is shown to be the continuous transfer of water (oil) from oil (water)-rich to water (oil)-rich regions. Although the linear analysis predicts constant increase in the size of the regions, the weakly nonlinear theory shows that fingers eventually stabilize to a configuration of a fixed amplitude. Typical finger growth processes are numerically illustrated. Introduction Displacement processes in porous media frequently exhibit unstable behavior on the macroscopic scale (viscous fingering). For instance, in rectilinear flow geometries and under certain conditions, the various flow variables are not uniform in the direction lateral to the direction of the displacement. This nonuniformity is-the result of the continuous growth of unstable disturbances that arise from random heterogeneities and manifest the dynamic competition between viscous and capillary forces. Because of its considerable importance in the sweep efficiency, this process has been the subject of extensive past research. A fundamental understanding of unstable behavior principally requiresthe delineation of conditions describing the onset of in-stability, obtained by a linear stability analysis, andth determination of the subsequent growth of the unstable disturbances (fingers). In the case of immiscible displacement. linear stability studies were carried out by Chuoke et al., Raghavan and Marsden, and Peters and Flock based on Hele-Shaw flow conditions. Recently, Jerauld et al., Yortsos and Huang, and Huang et al. provided a different approach that includes certain previously provided a different approach that includes certain previously ignored two-phase flow features, such as relative permeability and capillary pressure effects. A limited number of investigations have addressed the question of the subsequent-to-inception growth of fingers. Notable are the early attempts by Outmans, Sheldon and Fayers, and Jacquard and Seguier, which provided solutions for Hele-Shaw-like flows. A different approach seeking to incorporate effects of heterogeneity and fingering into effective fractional flow curves was also pursued by van Meurs and van der Poel for immiscible pursued by van Meurs and van der Poel for immiscible displacement and Koval, Dougherty, Claridge, and Fayers for miscible displacement. The accurate prediction of the growth of viscous fingers is clearly a nontrivial problem. Preliminary insight can be obtained by investigating the early stages of growth, where linear analysis results are expected to be valid. Additional information, at the expense of more elaborate computations, can be furnished by studying the growth near the critical wavelength, where the process is marginally stable, by use of weakly nonlinear methods. In this paper, both linear and weakly nonlinear approaches are used. Our methodology is based on related previous work on this subject, thus the following pertain to immiscible water/oil displacement in the absence of gravity with an initial mobile water saturation. Standard saturation-dependent relative permeability and capillary pressure functions are used. Inclusion of gravity effects in inclined thin reservoirs can be obtained in a relatively direct way. Extension to processes with an initially immobile water saturation, however, does not necessarily follow from the above and needs further investigation. The paper is organized as follows: we first review and discuss recent results on linear stability. Next. a weakly nonlinear stability analysis is carried out. The growth of viscous fingers is examined numerically on the basis of the two methods. Linear Stability Results In a previous publication, the linear stability problem was solved numerically to yield the dependence of the dimensionless rate of growth, w, on the dimensionless wave number, or, of the disturbance and the process parameters. Typical numerical results showed that the values of w where the displacement is unstable (omega greater than 0) are generally small (on the order of 10–2). and they correspond to equally small values of sigma. Accordingly, we expect that reasonably accurate estimates in such regions can be obtained from an asymptotic analysis of the eigenvalue problem in the limit or sigma less than 1 (long-wave disturbances). Such an approach was recently carried out and is reported in detail elsewhere. The results obtained are briefly summarized below for the sake of completeness and as an introduction to the ensuing weakly nonlinear analysis. We use notation similar to that in Refs. 6 and 7 as described in Appendix A. In the limit sigma less than 1, the following asymptotic expansion has been obtained: (1)(2) SPERE P. 1268