Similarity solution for instabilities in double-phase flow through porous media

Abstract

This paper analytically discusses the phenomenon of instabilities (fingering) which arise frequently in displacement processes through porous media. The underlying assumptions of the investigation are that the two flowing phases are immiscible liquids with a large viscosity difference, the porous medium is homogeneous, and the instabilities are described by their statistical behavior. A mathematical solution of the nonlinear differential system governing fingering has been obtained by using the group-transformation technique of similarity analysis. Finally, an analytical expression for the average cross sectional area occupied by fingers has been derived, and the possibility of finger stabilization is shown under certain specific conditions.

Variational Iteration Method and Analytic Solution for Laplace Equation for Steady Groundwater Flow

“…Numerous researchers have discussed this phenomenon from various perspectives. For example, Klute [2] and Hank Bower [3] employ a finite difference method; Philips [4] uses a transformation of variable technique; Mehta [5] discussed a multiplescale method; Verma [1,6] has obtained Laplace transformation and similarity solution, and Sharma [7] discusses a variational approach. Bruce and Klute [8]; Gardner and Mayhugh [9]; Nielson and Bigger [10]; Rawlins and Gardner [11], Terwilliger [12], Van Vorts [13], and Rahme [14] have described the phenomenon of gravity drainage of liquids through porous media and supported their theoretical investigation by experimental results.…”

Mathematical Model of Instability Phenomenon with Magnetic Fluid through Homogeneous Porous Media

“…Verma and Rajput [14] discussed the phenomenon of instabilities that arises in the displacement process involving two immiscible phases through dipping porous media with ferrofluid under certain conditions. Verma and Mishra [12]discussed Similarity solution for instabilities in double-phase flow through porous media. The displacement problems of this type in the porous medium have gained much current importance and many authors have analysed these problems; for example, Scheidegger [9], Verma [13], Rosenweig [7], Mehta [5] has discussed the instability phenomenon with different viewpoint.…”

Mathematical Model and Solution for Fingering Phenomenon in Double Phase Flow through Homogeneous Porous Media

“…Mehta has used special relation with capillary pressure and he used singular perturbation technique to find its solution [ 3 ]. Verma and Mishra have discussed similarity solution for instability phenomenon in double phase flow through porous media [ 4 ]. Pradhan et al have discussed the solution of instability phenomenon by finite element method [ 5 ].…”