An analytical study on stagnation points in nested flow systems in basins with depth‐decaying hydraulic conductivity

Abstract

[1] The existence of stagnation points in nested flow systems is relevant to a range of geologic processes. There has been no analytical study on the characteristics and locations of stagnation points in nested flow systems. We derived analytical solutions for hydraulic head and stream function in basins with isotropic and depth-decaying hydraulic conductivity. The solutions of hydraulic head and stream function are used to identify the positions of stagnation points and discuss the dynamics of groundwater around the stagnation points. Three types of stagnation points are identified by analytical and graphical means. For stagnation points on the basin bottom below the valley, only two regional flow systems converge from opposite directions. For stagnation points on the basin bottom below the regional high, only two regional flow systems part toward opposite directions. In contrast, for stagnation points under counterdirectional local flow systems, flow systems converging from and parting toward opposite directions coexist, and these stagnation points move deeper as the water table configuration becomes more rugged and the decay exponent of hydraulic conductivity increases. Moreover, the dividing streamlines around stagnation points under counterdirectional local flow systems are used to divide the local, intermediate, and regional flow systems accurately, from which the penetration depths of local and intermediate flow systems are precisely determined. A clear understanding of the location of stagnation points is critical for characterizing the pattern of hierarchically nested flow systems and has potential implication in studying solute and mineral concentration distributions in drainage basins.

Partitioning a regional groundwater flow system into shallow local and deep regional flow compartments

“…Analytic and numerical solutions have been derived by assuming that the water table is equal to the topography [e.g., Cardenas, 2007;Craig, 2008;Dahl et al, 2007;Freeze and Witherspoon, 1967;Jiang et al, 2011;Winter, 1978;Zijl, 1999]. These studies have shown the complexity of the nested flow structures, the importance of location of the discharge zones, and the sensitivity of the deep flows given specific water table surfaces.…”

A new analytical solution of topography‐driven flow in a drainage basin with depth‐dependent anisotropy of permeability

“…[5] In this study, similar to the work by Tóth [1963], Jiang et al [2011], and Zlotnik et al [2011], a steady state Darcian flow in a cross section is investigated. The flow is assumed to be limited below a flat line boundary.…”

“…[7] The analytical solutions of topography driven flow in drainage basins for two special cases have been obtained: the case for k x = k z and a = b is given by Jiang et al [2011], and the case for k x ≠ k z and a = b is given by Zlotnik et al [2011]. In this study, a new analytical solution is presented to satisfy more general conditions.…”

“…It has been found with numerical solutions that the patterns of the groundwater flow, solute transport, and residence time distributions are sensitive to the depth-decaying behavior of hydraulic conductivity in gravity-driven drainage basins [Jiang et al, 2009;Cardenas and Jiang, 2010]. As a further development, Jiang et al [2011] derived analytical solutions of groundwater flow in isotropic basins with exponentially depth-decaying permeability. In these investigations, only scalar values of permeability are available.…”

“…[4] To improve the integrity of the theory of regional groundwater flow, following previous works [Tóth, 1963;Zijl, 1999;Jiang et al, 2011;Zlotnik et al, 2011], a new analytical solution for topography-driven flow is developed by considering the systematic trend of permeability as well as the trend of anisotropy with depth. The depth-dependent anisotropy is highlighted in this study.…”