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 unstabledisplacements. In both homogeneous and heterogeneous porous media, gravitysegregation is much more effective in 3D than in 2D computations. Whether flowis 2D or 3D, the presence of correlated heterogeneities lowers the range ofviscous/gravity ratio over which the transition from gravity- tofingering-dominated flow occurs. Introduction Viscous fingering has long been recognized as a factor inmiscible-gas-injection processes, and attempts to describe mathematically theinstability that triggers fingering and the subsequent growth of fingers havebeen the subject of much research over the last 3 decades.¹ Whilethe onset of fingering can be studied by linear stability theory, it is nowclear that the subsequent development of fingers must be examined byhigh-resolution numerical simulation. Several investigators have describednumerical schemes that reproduce both qualitative and quantitative features offinger growth with reasonable accuracy^2-5; however, thoseinvestigations have considered only 2D displacements. Advances in computing power have only recently made investigation ofunstable displacements in three dimensions possible. Two studies usedhigh-resolution numerical simulations in modeling unstable 3D flow. Zimmermanand Homsy⁶ simulated growth of viscous fingers in homogeneous porousmedia in the absence of buoyancy forces under conditions of isotropicdispersion using a spectral technique and found that transversely averagedconcentration profiles were similar in 2D and 3D displacements. Christie etal.⁷ compared recovery curves for several 2D and 3D simulationsin their study of the effects of varying water-alternating-gas ratios. Usingrelatively 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 threedimensions. 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 examinewhen 2D simulations reproduce the behavior of 3D flow and, more importantly,when they do not. First, we consider 2D and 3D fingering in homogeneous mediaand compare displacements with and without gravity segregation. Then we examine2D and 3D displacements in selected heterogeneous media, again with and withoutgravity segregation. The computations show conclusively that some situationsexist in which 2D simulations reproduce 3D behavior well and others for whichthey do not. Mathematical Model A hybrid finite-difference/particle-tracking technique described in moredetail elsewhere^4,10,11 was used to simulate unstable miscibledisplacements in 3D porous media. This technique has been shown to reproduce,qualitatively and quantitatively, experimental observations of fingeringbehavior in homogeneous and heterogeneous porous media.^4,10-12 Themathematical model used is based on the assumptions that (1) the porous mediumand fluids are incompressible; (2) fluids are first-contact miscible but havediffering densities and viscosities; (3) no volume change occurs on mixing, andviscosities 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 isgiven by Darcy's law; and (5) the dispersion tensor is anisotropic andvelocity-dependent. In the simulations discussed later, the domain is a parallelepiped with thez axis aligned with gravity. The length, L, width, w, andheight, h, refer to the dimensions along the x, y, and zaxes, respectively. A constant total injection rate was imposed at the inletboundary, which was taken to be the entire upstream face (i.e., x=0). Atthe outlet boundary (x=L), a constant flow potential was assumed.No-flow boundaries were imposed on the remaining sides. Thus, mean flow wasalways in the x direction. For each timestep in our hybrid technique, thepressure equation is solved with an explicit finite-difference scheme on apoint -distributed grid. Particles are then moved from their current locationsaccording to the interpolated velocity field obtained from the pressuredistribution. At the end of each explicit particle movement, the position ofthat particle is perturbed in the longitudinal and transverse directions by anamount scaled to reflect the levels of longitudinal and transverse dispersion.Thus, the numerical technique includes scaled perturbations at eachtimestep,¹⁰ and artificial triggering of fingers with concentrationor permeability variations is not required. The 2D and 3D simulations presented here are for a velocity-dependent localdispersion tensor with a strong anisotropy,a_L/a_T=30. For the 3D flows, thedispersivities in the two principal directions transverse to the local velocityvector were always equal. The longitudinal and transverse dispersivities werechosen to give Peclet numbers of L/a_L=505 andw/a_T =h/a_T=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 inthe x and z directions in all 2D simulations. The number ofparticles 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 massivelyparallel, single-instruction, multiple-data machine (MasPar). The code waswritten with a low-level programming...