Effect of No-Flow Boundaries on Viscous Fingering in Porous Media of Large Aspect Ratio Zhengming Yang, SPE, and Yanis C. Yortsos, SPE, U. of Southern California Summary We report on a boundary effect associated with the preferential propagation of viscous fingers along no-flow boundaries in mis- cible displacements in two-dimensional (2D) rectilinear geometries near conditions of transverse equilibrium (TE) (large values of the generalized aspect ratio R L ). It is shown that this effect intensifies with an increase in the mobility ratio, and diminishes with increased permeability disorder, and as the system departs from conditions of TE. The effect disappears altogether if periodic boundary condi- tions are used. We show that this effect results from the violation of the no-slip condition at no-flow boundaries, associated with Darcy’s law. It must be considered in simulations of viscous fingering in relatively homogeneous media, near conditions of TE, particularly when gravity segregation occurs. Introduction Viscous fingering in porous media is the evolution of unstable disturbances when a higher mobility fluid displaces another one of lower mobility. Fingering patterns in random media develop fol- lowing the various mechanisms of tip-splitting, shielding, fading, etc., discussed in detail in many previous studies (e.g., see Homsy, 1 or Araktingi and Orr 2 ). More generally, the displacement patterns are influenced by the fluid characteristics, including the possible nonmonotonicity in the viscosity profile, 3 and the heterogeneity of the medium, namely the variance and the correlation structure of the permeability field. 2,4,5 Viscous fingering is mitigated by small- scale dispersion (molecular and hydrodynamic at the pore and pore-network levels) and by large-scale heterogeneity effects (large permeability disorder in uncorrelated media and channeling in correlated media). The effect of the geometry on viscous fingering has also received some attention (Waggoner et al., 4 Sorbie et al., 5 Zimmerman and Homsy 6 ). Typically, simulations of viscous fingering for upscaling purposes are conducted in geometries with aspect ratio of 3 (e.g., Christie 7 ). Waggoner et al. 4 simulated displacements at conditions of vertical equilibrium 8,9 (for which a more appropriate terminol- ogy should be TE, to be subsequently used in this paper). This limit is reached when the generalized aspect ratio, R L  ( l / b)  k  x / k  y , is large, where l and b = length and width, respectively, k  = the average permeability, and subscripts x and y = directions parallel and transverse to the main flow direction, respectively. Sorbie et al. 5 studied the sensitivity of displacement patterns to this param- eter in heterogeneous reservoirs and showed that it significantly affects the delineation in the parameter space of the various dis- placement regimes (fingering, dispersion, channeling). In recent studies, 10, 11 we provided an asymptotic description of displacements in porous media, including viscous fingering, in the two limits when parameter R L is large or small, respectively. The case of large R L corresponds to conditions of TE. This regime is reached in long and narrow isotropic reservoirs, in reservoirs in which the permeability transverse to the applied pressure gradient substantially exceeds the permeability parallel to it, and in slim tubes (commonly used in laboratory studies of CO 2 displacements). It is a regime of intense transverse mixing. The case of small R L corresponds to the opposite regime of zero transverse mixing and is better known as the Dykstra-Parsons approximation. 12 Differ- ences and similarities between these two limits are discussed in detail in Ref. 11. In parallel, Yang 13 reported on the sensitivity of the viscous fingering patterns to R L by means of high resolution simulation (HRS). He observed that, for uncorrelated, weakly heterogeneous media and at conditions near TE, most of the viscous fingering ultimately occurs near the lateral, no-flow boundaries. Specifically, he found that narrow, single fingers originated at these boundaries and propagated faster than the fingers in the interior of the domain until a small permeability value was randomly encountered, at which point the fingers turned inwards. The intensity of this effect was found to depend on the viscosity ratio, M, and on the heter- ogeneity parameter, V DP , and to diminish away from the TE regime. Such an effect has not been reported previously. Simulations of viscous fingering are typically conducted at R L values of O(1), where this boundary effect does not appear. In their renormalization study of viscous fingering, Morris and Ball 14 alluded to the pos- sibility of an (unspecified) interference from no-flow boundaries at large aspect ratios, which forced them to restrict their investigation to smaller aspect ratios. They also speculated that use of periodic boundary conditions may eliminate this interference. Simulations of displacement at TE conditions in isotropic media (namely, at large aspect ratios) using no-flow boundaries have also been reported. 4,5 Waggoner et al. 4 investigated the effect of the heter- ogeneity (Dykstra-Parsons) index, V DP . Fig. 4 of their work for M = 10 (reprinted here as Fig. 1) shows a pronounced boundary effect for the case of small disorder, V DP = 0.01. However, the authors did not comment on its origin or significance. Sorbie et al. 5 reported on the effect of the aspect ratio on the displacement regimes. However, their results are for strong heterogeneity disorder ( V DP  0.5), which competes strongly with, and partly obscures, viscous fin- gering (including fingering at the boundaries). Under these condi- tions, there is no noticeable boundary interference, as can be also confirmed in Fig. 1 when V DP is large. Boundary effects were not reported in simulations with periodic boundary conditions. For example, Zimmerman and Homsy 6 studied the effect of large aspect ratio with spectral simulation methods, which require the use of periodic boundary conditions. They did not report any unusual boundary interference. The unusual aspects of the boundary effect noted in Ref. 13 led us to investigate its origin. In this paper, we report on a numerical study that examines its sensitivity to parameters, such as R L , M, and V DP and to the use of periodic boundaries. We show that this effect is not a numerical artifact, but arises as a consequence of the slip condition implied by the use of Darcy’s law along no-flow bound- aries. The paper is organized as follows. First, we proceed with a formulation of the problem for general R L , including the limit of TE ( R L =). Then, we report on the sensitivity of this effect to the various parameters and to the two different boundary conditions used. Next, we present an analysis in which the effect is attributed to the use of Darcy’s law near no-flow boundaries. Possible ramifications for simulations, including gravity segregation effects, are discussed. Mathematical Formulation We consider first-contact miscible displacement in a heterogeneous 2D porous medium of a rectilinear geometry, in which a lower viscosity fluid displaces a higher viscosity fluid at constant velocity q. The analysis to follow is also applicable to immiscible displace- Copyright 1998 Society of Petroleum Engineers This paper (SPE 51257) was revised for publication from paper SPE 38430. Original manuscript received for review 17 May 1996. Revised manuscript received 20 April 1998. Revised manuscript approved 22 May 1998. 285 SPE Journal, September 1998