High-Resolution Simulation of Unstable Flows in Porous Media

Abstract

Summary This paper describes accurate numerical methods used to solve equations governing two-phase, three-component flow on a fine grid to obtain predictions of fingering behavior for both miscible and immiscible flows. The reasons for the choice of finite-difference scheme are discussed. Results are presented that show almost complete stabilization of fingering for illustrative line-drive and quarter five-spot configurations when water and solvent are injected simultaneously at the optimum water-alternating-gas (WAG) ratio. Introduction Viscous fingering is an instability that occurs when oil is displaced by a less viscous (more mobile) fluid, such as water or a miscible solvent. Homsy1 recently reviewed viscous fingering in homogeous porous media. Experimental studies of factors governing fingering behavior inmiscible displacements have been reported by several authors. Blackwell et al.2 investigated the effects of viscosity ratio and system size and orientation on the production characteristics of miscible linear flood, and their experiments provide a test of the ability of direct simulation to model detailed fingering behavior. Further linear experiments carried out by Handy were reported by Dougherty,3 and Habermann4 carried out experiments on miscible fingering in a quarter five-spot. Empirical models5–7 that account for the effects of fingering in reservoir simulators have been developed by fitting to experimental data. Fayers and Newley8 compare emipirical models and direct simulation. It is important to obtain solutions that are representative of the physical processes being modeled rather than the underlying grid. The first direct simulation of fingering in miscible flow was presented by Peaceman and Rachford.9 They obtained several viscous fingers initiated by permeability fluctuations on a 40×20 grid. Since their work, several authors10–12 have shown calculations of fingering for miscible flow, but they have geneally been limited by grid resolution. King et al.13 showed calculations of fingering in immiscible flow, which they compare with front-tracking approaches described by Glimm et al.14 Christie and Bond15 presented detailed simulations of unstable miscible flow on an extremely fine (130×130) two-dimensional (2D) finite-difference grid. Input data were taken directly from Blackwell et al.'s experiments, and fingers were triggered by a random perturbation to the initial concentration. Good agreement was obtained between simulation and experiment for mobility ratios of 5 and 86. The purposes of this paper are to describe the accurate numerical methods used by Christie and Bond for detailed simulation of miscible viscous fingering and to extend the work to coupled miscible and immiscible flows. This extension allows us to perform calculations when water and solvent flow simultaneously and thus to calculate directly the stabilizing effects of a WAG scheme. Theoretical Aspects of Model When diffusive and dispersive effects are ignored, the equations describing fingering allow infinite growth rates for small wave-lengths for both miscible and immiscible displacements. In a numerical solution, short wavelength modes are damped out by the mesh, and the solution obtained depends on the grid size no matter how much the grid is refined. When physical dispersion and diffusion terms are included, short wavelength modes are damped out and it becomes possible to obtain solutions that converge under grid refinement. Mathematical Formulation. The processes we wish to examine are the following: fingering in miscible and immiscible systems for different geometries; stabilization of fingering through the effects of gravity, diffusion, and capillary pressure; and stabilization of fingering through a WAG process. To obtain the highest resolution of fingering behavior, we use a simplified physical description and make the following assumptions:two-phase flow (aqueous and oleic phases),three components (water, oil, and solvent),all components incompressible,oil and solvent first-contact miscible,ideal mixing for oil and solvent densities, andone-fourth-power mixing rule for viscosities. The conservation equations describing the flow are written according to the total velocity formulation described by Peaceman.16 The equations are reproduced in dimensionless form below. Pressure Equation. Equation 1 where p=average pressure, (po+pw)/2; Pc=capillary pressure, po-pw; g=(normalized) gravity term, (gx, gy); ?o=oil-phase mobility, kro(Sw)/µo(C); ?w =water-phase mobility, krw(Sw)/µw; and ?(C)=oleic-phase specific gravity, [C?s+(1-C)?o]/?w. Total Velocity. Equations 2 and 3 Conservation of Water. Equations 4–6 Conservation of Solvent Equation 7 where D=dispersion tensor. For all the cases studied, we used the simplified formEquation 8 Numerical Solution of Equations. To otain solutions that are convergent under grid refinement, we need to resolve the physical dispersion and dimension terms, which are small compared with convective terms even on a laboratory scale. The resolution of these low values of dispersion requires at least 100 gridblocks in each direction, which effectively limited us to 2D models in the present study. Several factors influenced the choice of finite-difference algorithins used.Higher-order accuracy schemes are used in conjunction with a flux-limiting algorithm for both the saturation and concentration equations to obtain high resolution near steep gradients.Explicit time integration is chosen to advance the equations because the accuracy condition for tracking fronts is of the same order as the explicit stability limit. 17