The Fractal Nature of Viscous Fingering in Porous Media

Abstract

SPE Members Abstract An experimental study was undertaken to investigate the fractal nature of viscous fingering. Unstable first-contact miscible displacements were performed in a two-dimensional areal model. Images of the fingering patterns were captured and analyzed with a microcomputer-based imaging workstation. Results show that the areal sweep efficiency of the unstable displacements follows a fractal scaling law with a fractal dimension between 1.9 and 2.0. These results have potential application in the mathematical modelling potential application in the mathematical modelling of unstable EOR displacements and in the scaling of laboratory displacements to field conditions. Introduction The problem of viscous fingering caused by hydrodynamic instability is of concern in many enhanced oil recovery (EOR) displacements. Not only does the phenomenon reduce the displacement efficiency, it also makes the displacement process difficult to model and therefore, difficult to predict. Because of its widespread occurrence in predict. Because of its widespread occurrence in many EOR processes, there is considerable incentive within the EOR community to characterize the phenomenon quantitatively. phenomenon quantitatively. A modern mathematical tool that has shown promise in the quantitative characterization of promise in the quantitative characterization of viscous fingering is fractal geometry. Popularized by Mandelbrot, fractal geometry is a tool which can be used to characterize certain irregular and fragmented objects. Studies have been reported in which special types of viscous fingering have been characterized with fractal geometry. Some of these studies have examined the highly ramified fingering patterns observed when a low viscosity fluid is injected into a very viscous non-Newtonian slurry. Other studies have characterized fingering patterns in Hele-Shaw models without porous media. Yet others have characterized porous media. Yet others have characterized viscous fingering in porous media but at unrealistically high mobility ratios. There is very little reported on the fractal characterization of the types of viscous fingering typically encountered in EOR displacements. These fingering patterns in Newtonian fluids have less ramified morphologies than their non-Newtonian counterparts. The fractal analysis of these types of viscous fingering was the objective of this study. THEORETICAL BASIS OF VISCOUS FINGERING FRACTALS Viscous fingering, which is a manifestation of a hydrodynamic instability, is an example of a chaotic motion often encountered in nonlinear dynamics. A choatic process is extremely sensitive to initial conditions. Consequently, it is difficult to reproduce it experimentally because of the inevitable imperfections that are present in any experimental setup. Therefore, if one were to repeat a viscous fingering experiment many times, the outcome of the displacement for each experiment would be somewhat different because of the underlying chaotic nature of the displacement. There is increasing evidence that chaos and fractals are intimately connected. Therefore, the use of fractal geometry to characterize viscous fingering has a sound if not completely enunciated theoretical basis. The concept of a fractal growth process in a 2-dimensional setting may easily be described as follows. Consider the displacement of one fluid by another in a quarter five-spot pattern at a mobility ratio of 1. Theoretical as well as experimental evidence indicates that at some instant of time, before breakthrough and before cusping to the producing well, the displacement front will be radial as shown in Fig-1. P. 225