Theoretical Analysis of Groundwater Flow Patterns Near Stagnation Points

Abstract

The importance of stagnation points for characterizing groundwater flow patterns has long been recognized. However, the possible streamline configurations near stagnation points under the constraints of Darcy's law have not been thoroughly investigated. We fill this gap by conducting a systematic analysis of groundwater flow patterns near stagnation points in two and three dimensions. The approach borrows ideas from dynamical systems theory, as often done in fluid mechanics. The most general form of Darcy's law, which applies to variable-density, compressible fluids flowing through heterogeneous, anisotropic, compressible porous media, is first considered. Under these conditions, there are no major restrictions on the possible flow patterns near stagnation points. The common types of stagnation points are thus minimums (spiral or nonspiral), maximums (spiral or nonspiral), and saddles (in three dimensions, these can be converging or diverging and spiral or nonspiral). The implications of dealing with more restrictive fluid and porous medium properties are then systematically investigated. In particular, important restrictions on the possible types of stagnation points exist when the flow is constant density or divergence free. The theoretical analysis is complemented by a series of examples of groundwater flow fields in which different types of stagnation point arise. The findings highlight key differences between two-dimensional and three-dimensional flows and provide new insights on the patterns of variable-density groundwater flow. The fundamental knowledge on groundwater flow patterns gained from this study will benefit both theoretical and applied studies of groundwater flow and solute transport. Plain Language Summary Groundwater flow patterns reveal how water flows and solutes are transported in the subsurface. Therefore, characterizing them allows for understanding the origin and fate of water and solutes in the subsurface. Such information is absolutely essential for developing appropriate water resources management strategies. This paper investigates stagnation points in groundwater flow fields, where the flux is zero. Although the flux at these points is zero, these points are extremely important to identify and analyze in order to characterize the flow patterns. The reason is that at stagnation points, groundwater flow can converge, diverge, or abruptly change its course. This is in contrast with other (nonzero flux) points, at which groundwater locally flows uniformly. This article provides for the first time a thorough investigation of the possible flow patterns near stagnation points in porous media in two and three dimensions. It thus advances fundamental knowledge on groundwater flow patterns that will benefit both theoretical and applied studies of groundwater flow and solute transport.