The Use of Higher-Order Differencing Techniques in Reservoir Simulation
- I.J. Taggart | W.V. Pinczewski
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Engineering
- Publication Date
- August 1987
- Document Type
- Journal Paper
- 360 - 372
- 1987. Society of Petroleum Engineers
- 5.4.9 Miscible Methods, 5.3.2 Multiphase Flow, 5.5 Reservoir Simulation, 5.3.1 Flow in Porous Media, 5.2 Reservoir Fluid Dynamics
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Summary. Uniformly second- and third-order finite-difference schemes are developed for convection-dominated flows in porous media. The schemes are applied to immiscible and miscible displacements characterized by low levels of physical dispersion. The high-order schemes developed are shown to reduce significantly both dispersion of sharp fronts and sensitivity to grid orientation when compared with solutions obtained with the commonly used one-point upwind differencing schemes. Moreover, for explicit time-stepping simulators, such as those based on implicit-pressure explicit-saturation (IMPES) formulations, the improved performance of the high-order methods is obtained with only a modest increase in computational effort.
A number of important reservoir engineering problems have solutions characterized by low levels of physical dispersion. Consequently, these solutions frequently possess or develop sharp fronts or discontinuities as the displacement proceeds. Examples include high-velocity immiscible flows, EOR processes where small slugs of fluid are injected into the reservoir, thermal recovery processes, and miscible displacements where compositional effects are important. For these conditions, convective terms dominate in the governing transport equations, which take on a wavelike or hyperbolic nature. Low-order finite difference methods (one-point upstream differencing) have been the conventional means of solving these problems, because of their simplicity and computational economy. These methods are known to suffer from excessive numerical diffusion, however, and to exhibit severe grid orientation sensitivity. They are therefore inappropriate for the problems described and, if applied, produce solutions of unacceptable accuracy. A number of alternative solution techniques have been proposed to overcome the shortcomings of simple proposed to overcome the shortcomings of simple singlepoint upwind differencing. These include the method of characteristics, modified random choice methods, and various flux-updating schemes. Although the method of characteristics is capable of producing exact solutions for certain simple problems, its use as a general simulation tool is limited by the complexity of the computer codes required to apply the method to general three-dimensional multiphase-flow problems. Complexity also eliminates the random choice and flux-updating methods, both of which are based on the method of characteristics. Although variational methods (finite elements and related weighted residual schemes, such as collocation) have been shown to produce solutions that are more accurate and less prone to grid orientation sensitivity than those obtained with simple finite-difference schemes, the practical implementation of these methods to reservoir engineering problems is limited by their complexity and computational expense. The inherent simplicity and generality of finite-difference techniques make them attractive. This is reflected in the fact that the majority of commercial simulators use this form of approximation. Nine-point finite-difference schemes based on linear combinations of the conventionally differenced flow equations for parallel and diagonal grids have been shown to reduce grid orientation sensitivity considerably without incurring the computational expense of variational methods. These methods are known to produce physically unrealistic "bullet-like" displacement fronts, physically unrealistic "bullet-like" displacement fronts, however, and offer no significant improvements in truncation error over the simpler five-point schemes. The success of the variational methods in significantly reducing numerical dispersion and grid orientation sensitivity is largely a result of the higher-order approximations commonly used with these techniques. Higher-order finite-difference techniques have also been proposed, but these do not appear to share the favorable properties of the variational methods. Two-point upwind differencing schemes, which are second-order accurate in space but only first-order accurate in time, often produce grid orientation effects of magnitude similar to those resulting from one-point upwind schemes. They are limited further by more restrictive timestep limitations than the one-point methods.
In contrast, workers in gas dynamics who also solve hyperbolic transport equations numerically have experienced considerable success with higher-order difference schemes. The difference is that they use schemes that are usually second-order accurate in both space and time. For highly convective conditions, the solution to hyperbolic problems is a function of x/t, and in computations involving a fixed grid (Eulerian approach), it is necessary to consider both spatial and time approximations with equal importance.
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