Numerical simulation of immiscible two-phase flow in porous media

Abstract

Nonlinear evolution of viscous and gravitational instability in two-phase immiscible displacements is analyzed with a high-accuracy numerical method. We compare our results with linear stability theory and find good agreement at small times. The fundamental physical mechanisms of finger evolution and interaction are described in terms of the competing viscous, capillary, and gravitational forces. For the parameter range considered, immiscible viscous fingers are found to undergo considerably weak interaction as compared to miscible fingers. The wave number of nonlinear fingers decreases rapidly due to the shielding mechanism and scales uniformly as t−1 at long times. We have observed that even a small amount of density contrast can eliminate viscous fingers. The dominant feature for these flows is the gravity tongue, which develops a “ridge instability” when capillary and gravity effects are of similar magnitude.

Influence of Relative Permeability on the Stability Characteristics of Immiscible Flow in Porous Media

“…The aspect ratio A = 2. Validation of the numerical method, reported by Riaz and Tchelepi (2005), shows good agreement between the linear stability analysis and numerical simulations for early times. Figure 9(a) shows the concentration contours at t = 0.1 where a large number of fingers are generated, consistent with the linear stability analysis.…”

“…At a later time of t = 0.2 shown in Figure 9(b) a smaller number of large amplitude fingers appear. These larger fingers increase their width by absorbing the smaller ones through shielding and merging mechanisms (Riaz and Tchelepi, 2005). The vorticity field at t = 0.2 is shown in Figure 9(c).…”

“…An important consideration is the effect of the functional form of the relative permeability curves on the nonlinear evolution of viscous instability. It would also be interesting to determine to what extent the basic mechanisms of finger interaction and propagation described by Riaz and Tchelepi (2005) are influenced by the specific form of the relative permeability functions.…”

“…The transport equation is solved explicitly in physical space by a 4th order Runga-Kutta method (Fletcher, 1991) with 6th order compact finite difference discretization of spatial derivatives (Lele, 1992). (See Riaz and Tchelepi (2005) for details on the numerical method).…”

Adaptive rational spectral methods for the linear stability analysis of nonlinear fourth-order problems

“…The rich variety of invasion patterns has been extensively characterized experimentally, and through simulations at the pore scale [67,23,55,85,56,68,58,43,69]. The classical macroscopic (continuum) models of multiphase flow in porous media, based on phase balance equations, generalizations of Darcy's law, and constitutive relationships for relative permeability and capillary pressure, are unable to explain these flow patterns in immiscible flow [38,79,65,1]. New macroscopic theories of flow of mixtures in porous media will emerge in the next few years, and most likely the mathematical structure of these new models will resemble that of models of phase transitions and surface growth [24].…”