This section of the report is a brief synopsis of the project's scientific aims and accomplishments. The discussion in this section is intended for a technical audience, but it does not assume that readers are specialists in mathematical modeling. The Appendix to this report, summarized in Section 2, consists of published scientific articles that describe the results of the project for specialists.
The original objective of the project was to develop numerical techniques for modeling groundwater contaminant flows in the presence of sharp fronts in contaminant concentration. Such fronts occur and persist in contaminant flows in which the spreading attributable to hydrodynamic dispersion is small compared with advective transport along the groundwater velocity field. This "advection-dominated" transport regime is well documented in the water resources literature.
Steep concentration fronts in advection-dominated flows pose severe problems for most standard numerical models. Such models usually rely on approximation schemes in which one treats the real continuous aquifer as a discrete network of cells or nodes, called a grid. In each cell, the model assumes that concentrations, velocities, and other cell variables vary in a simple fashion. For example, these quantities may be constant over each cell. When a steep front is present, many small cells are needed in the vicinity of the front to produce accurate approximations of the local variations in contaminant concentration. Since the cost of running a model increases with the number of cells used, it is useful to be able to use small cells-that is, to refine the grid—only in the small regions near the fronts, where improved resolution is needed.
Installing this capability in acutal computer codes is a challenging task. Since contaminant fronts move, the regions of refined grid must move adaptively as well. Mathematically, moving a zone of locally refined grid changes the algebraic relationships among the cell variables in a complicated manner that one cannot predict in advance of running the model. In contrast with the case when a single, coarse grid is adequate, grids having moving zones of local refinement require innovative algorithmic structures if they are to be computationally efficient. The purpose of this work has been to develop such structures.
1.2 Related applications
Adaptive local grid refinement has applications in a wide array of fluid-dynamic settings. In the field of groundwater contamination, adaptive local grid refinement is useful in a variety of problems beside the problem of passive solute transport. Of special interest are multiphase flows, such as air-water flows in the vadose zone or flows involving nonaqueous-phase liquids (NAPLs), where steep fronts or even shocks in phase saturations commonly arise.
1.3 Summary of accomplishments
Early in the project, considerable effort focused on adaptive gridding techniques for contaminant transport in one space dimension. We devised a finite-element collocation scheme that is quite effective in that setting and that is readily amenable to implementation on parallel-processing computers (Allen and Curran, 1989). However, that scheme does not readily extend to problems in higher dimensions.
We also investigated a class of methods for two-dimensional problems using highly parallelizable, alternating-direction collocation schemes (Curran and Allen, 1989 and 1990; Allen and Khosravani, 1990; Khosravani, 1989; Li, 1990). As part of this effort, we collaborated with researchers at the University of Vermont, sponsoring a week-long visit to Wyoming that culminated in the development of a parallelizable alternating-direction scheme suitable for tensor hydrodynamic dispersion (Guarnaccia and Finder, 1989).
To implement grid refinement in these two-dimensional codes, we revisited the one-dimensional case, devising a scheme that extends readily to the alternating-direction setting (Curran Allen, in preparation). The actual implementation of this technique is the subject of a Ph.D. dissertation in August, 1990 (Curran, 1990).
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