Computational Mechanics 23 (1999) 108±116 Ó Springer-Verlag 1999
Finite element simulation of nonlinear viscous fingering in miscible displacements with anisotropic dispersion and nonmonotonic viscosity profiles A. L. G. A. Coutinho, J. L. D. Alves
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Abstract In this work parallel ®nite element techniques for the simulation of nonlinear viscous ®ngering in miscible displacements are addressed. The governing equations are approximated in space by equal order elements. The resulting semi-discrete equations are approximated in time by a block-iterative predictor-multicorrector algorithm. Feedback control theory is used for time step selection. The linear systems of equations at each blockiteration are solved with parallel element-by-element iterative techniques. Numerical simulations in different physical situations, involving anisotropic dispersion and a nonmonotonic viscosity law are shown.
1 Introduction Oil recovery from a reservoir can be greatly improved by injection of a displacing material that is miscible with the oil. However, a mechanical instability, called viscous ®ngering, reduces the ef®ciency of this injection. A comprehensive review of the mechanics of viscous ®ngering in homogeneous porous media can be found in Homsy (1987). Many factors in¯uence the severity of viscous ®ngering. Waggoner et al. (1991) and Moissis et al. (1987) examined the effects on ®ngering of statistically generated heterogeneous permeability ®elds, while Moissis et al. (1993) studied the combined effects of gravity and heterogeneity. In a series of works, Zimmerman and Homsy (1991, 1992a, b) studied the effects of viscosity contrast, anisotropic dispersion and velocity dependence of dispersion, on both two and three-dimensional homogeneous problems. More recently Manickam and Homsy (1993, 1994, 1995) have studied the in¯uence of nonmonotonic viscosity pro®les and gravity effects. These works describe A. L. G. A. Coutinho, J. L. D. Alves Center for Parallel Computations, COPPE/Federal University of Rio de Janeiro, PO Box 68506, 21945-970 Rio de Janeiro, RJ, Brazil e-mail:
[email protected] and
[email protected] Correspondence to: A. L. G. A. Coutinho Computer time on the CRAY J90 was provided by the Center of Parallel Computing of the Graduate School of Engineering of the Federal University of Rio de Janeiro, Brazil. We are indebted to Mr. T. Bulh~oes from Silicon Graphics/CRAY Research Division, Brazil by his invaluable help. This work was partially supported by research grants 350145/93-8 and 350148/93-7 from CNPq, the Brazilian Research Council.
the ®nger interaction mechanisms of shielding, spreading, tip splitting, pairing, multiple coalescence and fading. When nonmonotonic viscosity pro®les and gravity were present, Manickam and Homsy (1994, 1995) reported the phenomena of reverse ®ngering, where the ®ngers spread farther in the backward than in the forward direction. Thus, nonmonotonicity offers a potential method to control the growth rate of the ®ngering zone. Water-alternating-gas (WAG) schemes are based on this concept, introducing spatial nonmonotonicities in the mobility pro®les, altering the composition of the injected ¯uid. Christie et al. (1991) have shown the efectiveness of WAG schemes using numerical simulations. According to Manickam and Homsy (1994), these effects are naturally achieved when viscosity varies nonmonotonically with the local concentration of the ¯uid mixture, as for the pair alcohol-water. Some related work on porous media can also be found in Saldanha da Gama and Martins-Costa (1997). Several different numerical methods were employed in these simulations. Finite differences (Christie et al., 1991, Waggoner et al., 1991), pseudo-spectral methods (Zimmerman and Homsy, 1991, 1992a, 1992b, Manickam and Homsy, 1994, 1995), and the modi®ed method of characteristics combined with mixed ®nite elements (Moissis et al., 1987, 1993). Finite differences are employed in most of commercial reservoir simulators, in spite of its dif®culties to handle complex geometries. Pseudo-spectral methods are very ef®cient and highly accurate for simulations in simple domains with periodic boundary conditions. Mangiavacchi et al. (1997) have shown that pseudo-spectral methods are very ef®cient in the simulation of the growth of viscous ®ngering in porous media on parallel machines such as the CRAY T3D and IBM SP2. The combination of a ®nite element modi®ed method of characteristics and a mixed ®nite element method generates a method with very little numerical dispersion, allowing the use of large time steps and good accuracy in the computation of velocities. However, this combined method involves different interpolation schemes for pressure, velocity and concentration. Recently we employed in the simulation of miscible displacements in random heterogeneous media a parallel ®nite element method (Coutinho and Alves, 1996) where all variables were approximated by equal order interpolations. In this method pressure is computed by Galerkin's method, and a global post-processing technique is used to evaluate velocities. The SUPG formulation with discontinuity-capturing is employed for spatial discretization of the concentration equation. The
resulting semi-discrete equations are approximated in time by a predictor-multicorrector algorithm with variable time stepping. In this work we show that our formulation is able to simulate the evolution of viscous ®ngering in different physical situations, achieving good parallel performance on the CRAY J90, a low-cost version of the third-generation parallel-vector CRAY Y-MP, with scalar performance enhancements. The remainder of this paper is organised as follows. In the next section we brie¯y review the mathematical model for miscible displacements. The section that follows presents the ®nite element equations and the computational strategies. Two numerical examples are shown in Section 4. The ®rst one is the simulation of the evolution of viscous ®ngering in a homogeneous media with anisotropic dispersion. In the second we consider a nonmonotonic viscosity pro®le. The paper ends with a summary of the main conclusions of this work.
2 Problem statement The governing equations for the miscible displacement of one incompressible ¯uid by another, in a porous medium X 2
