Detailed Validation of an Empirical Model for Viscous Fingering With Gravity Effects

Abstract

Summary. This paper extends to two-dimensional (2D) flows the derivation and validation of an empirical model for viscous fingering previously developed. Fine-scale numerical simulations are used to provide basic data for validating the approximations, and these fingering results are also checked against a range of experiments. The flow rate dependence of gravity segregation in vertical section experiments conducted by van der Poel is examined, where the broadly acceptable agreement of the empirical model is limited by some identified additional features. Introduction In an earlier paper a new approach to representing miscible fingering phenomena was described. The aim of that study was to derive an empirical fingering model that was based on physically plausible simplifying assumptions and that could be consistently plausible simplifying assumptions and that could be consistently extended to multidimensional displacement. The advantage of the new approach is that the use of parameters with an identifiable physical significance is intuitively more appealing than the physical significance is intuitively more appealing than the earlier Koval and Todd and Longstaff methods. The key assumption of the new model is that the flow may be described as the displacement of bypassed oil of unmodified composition by a second, displacing fluid. This displacing fluid, corresponding to the fingers, propagates as a continuously varying mixture of solvent and oil, having an oil-rich composition at the leading edge and a solvent-rich composition at the trailing edge of the fingering zone. A second assumption of the model is that the fractional width, A, occupied by the fingers grows away from the leading edge of the fingers (see Fig - 1). Fayers proposed that the finger growth could be expressed by (1) where a = initial fractional width at which fingers begin togrow, Cf= solvent concentration in fingers, and= growth rate exponent. However, other monotonic increasing functions may also be suitable. For homogeneous media, the reservoir is eventually completely swept, so we require that a+b= 1 as 1. For a heterogeneous reservoir, it is realistic to suppose that a + b is less than 1, so that the model brings with it a simple expression for the heterogeneous sweep efficiency through the parameter b. Thus each of the parameters in the model have a simple physical interpretation. Parameter values were determined by fitting a fractional flow solution of the model equations to the experimental line-drive recovery data obtained by Blackwell et al. Suitable choices were found to be (2a) and (2b) where M is the mobility ratio, . In this paper, we use the empirical model summarized above to tackle the following objectives.We examine in detail the approximations necessary to derive the empirical model by comparison with direct numerical solution of the miscible displacement equations. The detailed fingering behavior that occurs is analyzed to indicate which of the model assumptions have a sound physical basis and also to show when the assumptions cease to be valid.The empirical model is extended to 2D and used to obtain predictions of recovery in quarter five-spot patterns. The model predictions of recovery in quarter five-spot patterns. The model predictions are validated against the Lacey et al. and Habermann predictions are validated against the Lacey et al. and Habermann experiments.Displacements are studied where the flow is influenced by both viscous and buoyancy effects. We consider vertical downward displacement, analyzing the predictions of both direct simulation and the Fayers empirical model. Results are compared with the vertical flow experiments of Blackwell et al. and Dumore.In the final section of the paper, we examine 2D flow in vertical sections. Results from the empirical model are again compared with direct simulation, and with the vertical section experiments of van der Poel. Flow rates are identified for which gravity override is not the dominant physical process and viscous fingering may still occur. Validation of the Empirical Model for Linear Systems Much of the work of this paper rests on the use of very detailed solution of unstable fingering growth to confirm attributes of the empirical model. Highly detailed numerical simulations of the development of unstable viscous fingers have been reported by Christie and Bond, and a full description of the accurate numerical procedures used is given by Christie. The approach adopted is to procedures used is given by Christie. The approach adopted is to solve the pressure and concentration equations directly on very fine regular Cartesian grids. The concentration equation is solved explicitly with flux-corrected transport. The code that has been developed also allows detailed analysis of fingering effects in the presence of gravity. presence of gravity. Example results from a detailed simulation of Blackwell et al.'s experiment at an adverse viscosity ratio M= 86 are shown in Figs. 2A and 2B. These results require input of only the viscosity data, flow rate, and an appropriate estimate of the dispersion coefficient. Note the excellence of the match to the measured recovery profile (Fig. 2B). This level of agreement is obtained without the need to tune any of the input parameters, such as the physical dispersion coefficient. The details of the calculated physical dispersion coefficient. The details of the calculated spatial concentration and pressure distributions then provide quantitative results about the nature of fingering that cannot normally be obtained from experimental data. This detailed information is used as a source of data for validating the simplifying assumptions in the empirical model. SPERE P. 542