Viscous fingering takes place when the viscous forces of a displacing phase has greater momentum than that of the displaced phase. Viscous fingering is an extremely important phenomenon in many applications of enhanced oil recovery, underground liquid waste disposal, and geothermal energy production. While the onset and propagation of viscous fingers during liquid-liquid displacement is considered to be of severe engineering consequences, little has been done to mathematically model the onset and propagation of a viscous finger. Viscous finger under double diffusive conditions is even scarcer. In this paper, two-dimensional non-linear double diffusive convection in a multi-porous cavity is considered, both numerically and experimentally. The Darcy equation, including Brinkman term to account for the viscous effects, is used as the momentum equation. The model consists of two rectangular cavities filled with a porous medium. The smaller cavity is located at the top of the larger one. The larger cavity is filled initially with glycerin while the smaller one contains fresh water. At the initial time, the fresh water is injected with either constant flow rate (numerical) or constant hydrostatic head (experiments) and the viscous fingering formation is studied in details. The momentum, solutal, energy and continuity equations are solved numerically using the finite element technique. This transient problem is solved to study the thermal displacement, the isothermal displacement and the microgravity displacement of glycerin by water to understand the onset and the propagation of viscous fingering. For each case, the variation of the distance between the tip of the finger with time is studied in details. The effects of aspect ratio and displacement velocity are studied, both in the context of onset and propagation of viscous fingers. Experimentally, an ingenious method is developed for visualizing 2-D flow in a porous medium. A carbonate formation is used as the porous medium. A chemical dye is used to delineate the propagating front of a viscous finger. Initial series of experiments are conducted under isothermal conditions.
The study of heat and mass transfer in porous media has a large number of applications in the areas of environmental geothermal, secondary and tertiary oil recovery, fixed bed regeneration in chemical processing, hydrology and filtration [1-8]. Viscous fingering generally refers to the onset and evolution of instabilities that occur in the displacement of fluids in porous bed. In most cases, this mechanism of instability is linked to the viscosity variation between the displaced and the displacing fluids. A fingering pattern may evolve when a less viscous fluid (higher mobility) penetrates a more viscous (lower mobility) fluid during a displacement process. This can occur for both miscible (no capillary force) and immiscible cases. A similar instability can occur when a more dense fluid displaces vertically down a less dense fluid in a porous medium or even a Hele-Shaw cell [1]. Also, fingering can be onset when a low viscosity Newtonian fluid, injected from a source, penetrates a Hele-Shaw cell (or a porous medium) filled with a miscible (or immiscible) non-Newtonian fluid.
