Interaction of Viscous Fingering, Permeability Heterogeneity, and Gravity Segregation in Three Dimensions

Abstract

Summary Two- and three-dimensional computations with a hybridfinite-difference/particle-tracking technique are compared for unstable displacements. In both homogeneous and heterogeneous porous media, gravitysegregation is much more effective in 3D than in 2D computations. Whether flow is 2D or 3D, the presence of correlated heterogeneities lowers the range of viscous/gravity ratio over which the transition from gravity- to fingering-dominated flow occurs. Introduction Viscous fingering has long been recognized as a factor inmiscible-gas-injection processes, and attempts to describe mathematically the instability that triggers fingering and the subsequent growth of fingers have been the subject of much research over the last 3 decades.1 While the onset of fingering can be studied by linear stability theory, it is now clear that the subsequent development of fingers must be examined by high-resolution numerical simulation. Several investigators have described numerical schemes that reproduce both qualitative and quantitative features of finger growth with reasonable accuracy2–5; however, those investigations have considered only 2D displacements. Advances in computing power have only recently made investigation of unstable displacements in three dimensions possible. Two studies used high-resolution numerical simulations in modeling unstable 3D flow. Zimmerman and Homsy6 simulated growth of viscous fingers in homogeneous porous media in the absence of buoyancy forces under conditions of isotropic dispersion using a spectral technique and found that transversely averaged concentration profiles were similar in 2D and 3D displacements. Christie et al.7 compared recovery curves for several 2D and 3D simulations in their study of the effects of varying water-alternating-gas ratios. Using relatively fine-grid simulations (60×30×30), they examined homogeneous porousmedia and a case where a distribution of shales was present. Strong gravitysegregation forces were always present in their calculations. They did notinvestigate the transition from gravity- to viscous-dominated flow in three dimensions. Chang et al.8 and Mohanty and Johnson9performed additional 3D studies with lower grid resolution. In this paper, we report results of high-resolution simulations that examine when 2D simulations reproduce the behavior of 3D flow and, more importantly, when they do not. First, we consider 2D and 3D fingering in homogeneous media and compare displacements with and without gravity segregation. Then we examine2D and 3D displacements in selected heterogeneous media, again with and without gravity segregation. The computations show conclusively that some situations exist in which 2D simulations reproduce 3D behavior well and others for which they do not. Mathematical Model A hybrid finite-difference/particle-tracking technique described in more detail elsewhere4,10,11 was used to simulate unstable miscible displacements in 3D porous media. This technique has been shown to reproduce, qualitatively and quantitatively, experimental observations of fingering behavior in homogeneous and heterogeneous porous media.4,10-12 The mathematical model used is based on the assumptions that (1) the porous medium and fluids are incompressible; (2) fluids are first-contact miscible but have differing densities and viscosities; (3) no volume change occurs on mixing, and viscosities of mixtures can be represented by a quarter-power blending rule;(4) flow takes place in two or three dimensions, and the local flow velocity is given by Darcy's law; and (5) the dispersion tensor is anisotropic and velocity-dependent. In the simulations discussed later, the domain is a parallelepiped with thez axis aligned with gravity. The length, L, width, w, and height, h, refer to the dimensions along the x, y, and zaxes, respectively. A constant total injection rate was imposed at the inlet boundary, which was taken to be the entire upstream face (i.e., x=0). At the outlet boundary (x=L), a constant flow potential was assumed. No-flow boundaries were imposed on the remaining sides. Thus, mean flow was always in the x direction. For each timestep in our hybrid technique, the pressure equation is solved with an explicit finite-difference scheme on a point -distributed grid. Particles are then moved from their current locations according to the interpolated velocity field obtained from the pressure distribution. At the end of each explicit particle movement, the position of that particle is perturbed in the longitudinal and transverse directions by an amount scaled to reflect the levels of longitudinal and transverse dispersion. Thus, the numerical technique includes scaled perturbations at each timestep,10 and artificial triggering of fingers with concentration or permeability variations is not required. The 2D and 3D simulations presented here are for a velocity-dependent local dispersion tensor with a strong anisotropy, aL/aT=30. For the 3D flows, the dispersivities in the two principal directions transverse to the local velocity vector were always equal. The longitudinal and transverse dispersivities were chosen to give Peclet numbers of L/aL=505 andw/aT =h/aT=3,750. The 3D simulations were performed with 128×64×64 gridblocks in the x, y, and z directions, respectively; 128×64 gridblocks were used in the x and z directions in all 2D simulations. The number of particles taken to represent a unit concentration in a gridblock was 64 and 100for 3D and 2D calculations, respectively. The simulation code used here was written specifically for a massively parallel, single-instruction, multiple-data machine (MasPar). The code was written with a low-level programming language, and the algorithms were designed to achieve optimum performance for the hybridfinite-difference/particle-tracking approach. The computational grid on which the pressure solution was obtained was mapped to processors. Each particle was mapped to a processor as well. Particle-grid communications were streamlined by use of segmentation and alignment algorithms. Typical simulation time for the3D displacements presented here was ˜5 hours of CPU time. p. 266–271