Detailed Simulation of Unstable Processes in Miscible Flooding

Abstract

Summary. This paper describes applications of linear and nonlinear simulations to unstable miscible flooding. The first section describes a method of calculating linear growth of unstable modes by use of finite differences in the direction of flow and Fourier decomposition perpendicular to flow. This work extends the previous long- and short-wavelength analytic results to cover the whole wave-number range. Results obtained are used to help validate a two-dimensional (2D) code to study nonlinear evolution of viscous fingers and to identify likely fingering regimes in the computed solution by identifying the range of wave numbers that dominates the linear growth behavior. The second section describes the numerical scheme for calculations on a fine grid of the nonlinear development of an instability. Results are presented on calculations of nonlinear growth at several mobility ratios and levels of diffusion. Comparisons with Blackwell's experimental data are presented. Good agreement is obtained. suggesting that the physical processes governing fingering are being correctly modeled. Introduction Miscible displacement at an unfavorable mobility ratio is known to be unstable to viscous fingering of the solvent into the oil. There have been many attempts to characterize this phenomenon through laboratory experiments, empirical methods. and direct simulation. Blackwell et al. studied experimentally the factors determining fingering behavior in miscible displacement at adverse mobility ratios. Their studies investigated the effects of viscosity ratio, system size, and heterogeneities on the production characteristics of horizontal and vertical linear floods. Habermann carried out experiments with five-spot geometries to examine the effects of miscible slug size on recovery from a heterogeneous system in the presence of fingering. Further linear experiments by Handy were reported by Dougherty. Empirical methods have been developed to fit experimental data. Kovat's K-factor method allows for viscosity ratio and heterogeneity effects by calculating an effective mixture viscosity that is then used in a simple fractional-flow formula. The parameters in the effective viscosity equation were determined by calibrating the model against experiments. Todd and Longstaff developed an approach for modeling fingering effects in a three-phase miscible simulator that uses a mixing parameter for viscosities and densities. A more recent approach by Fayers developed the concept of a "fingering function" to predict the flow behavior. Fayers was able to achieve good fits to the experimental data reported by Blackwell by using a consistent set of parameters in his model. To match Handy's data, however, it was necessary to adjust one of the parameters. This change was attributed to the length-to-width ratio of Handy's experiment. Detailed simulation has also been used to study viscous fingering, but the calculations generally have been on coarse meshes. An early study of fingering in a miscible system was undertaken by Peaceman and Rachford. They used a 40 × 20 grid and initiated fingers with a 2.5 or 5 % permeability fluctuation. More recently, Farinerg used moving-point techniques to study fingering in linear geometry and compared his results with an analytic solution for the single-finger case obtained by Jacquard and Seguier Another recent study was reported by Glimm et al., who used an interface-tracking technique to investigate the effects of converging and diverging geometry (near injection and production wells). Two techniques were used to initiate fingers-random initial conditions or permeability variations-and similar results were obtain from each. This paper addresses the following two questions.In an unstable displacement, what is the range of wavelengths that will dominate the linear growth?What is the nonlinear evolution of these instabilities'? The first section extends previous work on linear stability theory, applicable to the long and short wavelength limits, to cover the whole range. This provides a toot to calculate growth rate at any value of wave number. Linear theory is limited to small-amplitude behavior of fingering; therefore, other techniques must be used to investigate the large-amplitude regime. The second section describes the use of direct simulation to investigate the nonlinear regime of viscous fingering. The major difference between work described in the second section and previous work is the use of simulation on grids fine enough to resolve effects of both diffusion and fingering. This allows investigation of the nonlinear de-velopment of viscous fingering and effects caused by boundarie(e.g., length/width in a core). Calculation of Growth Rates by Use of Linear Stability Theory Linear stability theory has been used in several studies to predict growth rates for both miscible and immiscible displacement. In particular, two recent papers described stability analyses for miscible displacement in the presence of gravity and diffusion. The purpose of this section is first to discuss these analyses and to show that the results predicted are valid only for the short-wavelength region and second to describe a method of calculating linear growth rates that is valid for all wavelengths. Analytic Calculations of Growth Rates. To derive the results in this section. we assume 2D horizontal miscible flow. The governing, equations can then be written ..........................................(1) and ..........................................(2) If we perform a linear stability analysis for a Fourier mode that varies as ei (wt+ z) (where u) is the growth rate, a is the wave number. and z is a general coordinate perpendicular to the unperturbed flow), we can obtain analytic expressions for the growth rate as a function of wave number in both long- and short-wave length limits. ..........................................(3) SPERE P. 514^