High-Accuracy Implicit Finite-Difference Simulations of Homogeneous and Heterogeneous Miscible-Porous-Medium Flows
- Eckart Meiburg (U. of Southern California) | C.-Y. Chen (U. of Southern California)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- June 2000
- Document Type
- Journal Paper
- 129 - 137
- 2000. Society of Petroleum Engineers
- 5.5 Reservoir Simulation, 4.1.5 Processing Equipment, 5.3.1 Flow in Porous Media, 5.3.4 Integration of geomechanics in models, 4.1.2 Separation and Treating, 5.4.9 Miscible Methods, 5.4.1 Waterflooding, 5.7.2 Recovery Factors, 5.3.2 Multiphase Flow, 4.3.4 Scale, 6.5.2 Water use, produced water discharge and disposal
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A novel computational algorithm for the direct numerical simulation of miscible-porous-media flows is developed and applied that offers multiple benefits. It is based on the streamfunction-vorticity formulation of Darcy's law and resolves all physically relevant length scales including diffusion. By employing an implicit discretization based on compact finite differences, it combines ease of implementation and the ability to handle nontrivial geometries with the superior computational accuracy usually reserved for spectral methods. The formal accuracy is O(?x4,?t2), although diffusion terms are discretized with O(?x6). Test calculations are performed that provide a quantitative comparison with linear stability results for strong mobility contrasts in the practically relevant-quarter five-spot configuration. The excellent agreement, both with regard to the algebraic growth rate as well as the preferred wave number, demonstrates the very low levels of numerical diffusion, thereby eliminating grid orientation effects. The capabilities of the numerical method are furthermore demonstrated by means of representative calculations for both homogeneous and heterogeneous displacements. Among other findings, these simulations exhibit a minimal recovery efficiency for intermediate values of the correlation length of the permeability heterogeneities.
The continuous enhancement of simulation techniques for flows in porous media poses a longstanding and ongoing challenge. Accurate predictive capabilities are crucial for, among other areas, problems in hydrology and groundwater contamination, as well as enhanced oil recovery. Here, improved reservoir simulations will yield qualitative and quantitative information essential both for correctly anticipating oil reservoir performance, as well as for assessing the effectiveness of strategies for enhancing recovery. All of the above areas require the quantitatively accurate simulation of complex physical processes in complicated geometries. The main difficulty lies in the correct reproduction of the wide range of relevant length and time scales that typically characterize these problems. At one end of the spectrum, the size of the smallest scales is determined by the action of physical diffusion or dispersion mechanisms, while at the other end, reservoir size or well distances determine the large scales. An improved understanding at the fundamental level of how the governing physical mechanisms and interactions between them are affected, for example, by the mobility contrast or the level of heterogeneity, can potentially be obtained by direct numerical simulations that resolve the entire spectrum of length scales, without introducing significant levels of numerical diffusion. Consequently, especially for low levels of physical diffusion and dispersion, the accurate and detailed representation of the dominant mechanisms and their dynamical interactions remains a formidable challenge. The numerical approach to be discussed in the present paper aims at this area of direct numerical simulation.
Efforts in the field of porous media flow simulation date back at least to the early finite difference work by Peaceman and Rachford,1 as well as the mixed Lagrangian-Eulerian approach by Garder et al.2 Subsequently, a host of novel numerical approaches for the simulation of miscible displacements were developed. The research groups around Wheeler and Ewing played a very active role in developing new computational techniques for porous-media flows, cf. the early review by Russell and Wheeler.3 Douglas et al.4 employ a self-adaptive finite-element method, cf. also the analysis by Bell et al.5 Darlow et al.6 describe so-called mixed finite-element methods, which realize certain computational advantages, in particular for strongly heterogeneous media, by solving for pressure and velocity simultaneously. Ewing et al.7 analyze a modified method of characteristics for handling the concentration equation in miscible porous-media flows. Ewing et al.8 perform more-detailed finite-element simulations for anisotropic, heterogeneous, miscible flows in the quarter five-spot geometry, using finer grids up to 100×100. While these detailed simulations allow the authors to draw a variety of conclusions about the underlying physics, they nevertheless state a number of issues left unresolved, thereby emphasizing the need for further, higher-accuracy simulations. A comprehensive review of the above methods, as well as related work on Eulerian-Langrangian approaches9,10 is provided in Refs. 11 and 12, along with a discussion of these methods in light of recent developments in computer architectures, cf. also Ref. 13.
More conventional finite-difference discretizations have been used as well to obtain improved physical understanding,14-20 although alternate approaches such as particle tracking have also been pursued.21 These methods are typically of second-order overall accuracy, but often still suffer from significant levels of numerical dispersion. In terms of formal accuracy, an important milestone is represented by the work of Leventhal,22 who applied the fourth-order operator compact implicit family of finite-difference schemes23,24 to one-dimensional, two-phase immiscible waterflood problems, and demonstrated a significant reduction of the adverse effects of numerical diffusion.
A different line of research has aimed at modifying numerical approaches originally developed for compressible flows with shock discontinuities, e.g., Godunov-based approaches25,26 and true verticle depth (TVD) techniques.27,28 These methods typically are of second-order accuracy, although Liu et al.29 have recently developed a TVD scheme that discretizes first-order spatial derivatives with third-order accuracy. More recently, Batycky et al.30 utilize mapping of numerical solutions along streamlines in order to simulate first-contact miscible displacement. Their simulations, which take into account gravitational forces as well, exhibit vigorous viscous fingering. The method furthermore allows big improvements in efficiency, although in the absence of physical diffusion or dispersion, numerical diffusion sets the short-wave cutoff length scale and, hence, has a substantial effect on the results.
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