Abstract
This paper reports an investigation of unstable fingering in two-fluid flow in a porous medium to determine if lambda the dimensionless finger width, is unique For a viscous finger A is the ratio of finger width to the distance between the tips of the two trailing fingers adjacent to the leading finger. For a gravity finger lambda is defined as the ratio of finger width, to "height" of the medium perpendicular to hulk flow. This work confirms previous experiments and existing theory that for viscous fingering lambda approaches a value of 0.5 with increasing ratio of viscous to interfacial force. However, for a given fluid pair and given, medium, this ratio can he increased only by increasing the, velocity. Experiments on gas liquid systems show that the asymptotic value of lambda with velocity is not always 0.5. Apparently, for gas-liquid systems, the influence of the interfacial force cannot always he eliminated by increasing the velocity. For such systems lambda is a function of fluid pair and media permeability. If the gravity force normal to the hulk permeability. If the gravity force normal to the hulk flow is active, it damps out the viscous fingers except for an underlying or overlying finger. The dimensionless width of this gravity finger strongly depends on velocity and height of the medium, as well as the fluid and media properties. The existing experiments and theories are reviewed and the gravity, stable, and viscous flow regimes are described in view of these experiments and theories. The existence of a gravity-dominated unstable regime, a gravity-viscous balanced stable regime, and a viscous-anminated regime was demonstrated experimentally by increasing flow velocity bin a rectangular glass head model. Asymptotic values of the dimensionless finger width were determined in various-sized Hele-Shaw models with gravity perpendicular and parallel to flow. The dimensionless perpendicular and parallel to flow. The dimensionless finger width lambda was determined as a function of applied force, flow resistance, and fluid properties. The results are interpreted dimensionally. Some comments are made concerning possible scaling and meaningful extensions of theory to describe these regimes in three-dimensional flow. Previous description of unstable two-fluid flow in porous media is mainly restricted to studies of viscous-dominated instability. The direction of this study is to provide data and understanding to consider the more realistic problem of predicting flow in three dimensions that may result in instabilities that are combinations of all, four flow regimes.
Introduction
The unstable flow of two fluids is characterized by interface changes between the fluids as a result of changes in relative forces. In a given porous medium and for a given fluid pair the gravity force dominates flow at low displacement velocities. As the velocity increases the viscous forces begin to affect flow significantly, and eventually there is a balance between effects of the gravity and viscous forces. As velocity increases further, the viscous force dominates flow. In the plane parallel to gravity, four flow regimes result as the velocity is increased: a gravity-induced stable flow regime; a gravity-dominated unstable flow regime; a stable regime resulting from a balance between gravity and viscous forces; and a viscous-induced unstable flow regime. The gravity-induced stable regime is represented schematically in Fig. 1a. This general flow pattern persists with the displacing fluid contacting all of persists with the displacing fluid contacting all of the in-place fluid until the interface becomes parallel to the bulk flow. At this velocity a gravity finger forms, and the interface, is unstable in that the length of the gravity finger grows and the fluid behind the nose of the finger is practically nonmobile because of the small pressure gradient along the finger. The gravity-dominated unstable flow is shown schematically in Fig. 1b. As the injection rate is increased, the gravity finger thickens, perhaps until it spans the medium creating a stable interface where all of the in-place, fluid is again mobile. This regime would, not occur in the absence of gravity. It occurs due to the counter effects of the gravity and viscous forces (Fig. 1c). As the velocity of the displacing fluid increases, the viscous forces dominate, and, the interface breaks into viscous fingers (Fig. 1d).
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Numerical Simulation of Flow over Stepped Spillways with varying Step-Angle
