## 1. Introduction

[2] Periodic undulations of the water table create gravity-driven, hierarchically nested flow systems, i.e., local, intermediate and regional flow systems in drainage basins [*Tóth*, 1963]. Even in homogeneous and isotropic basins, the distribution of groundwater velocity in such nested flow systems is extremely heterogeneous and stagnant zones with low velocity could develop due to convergence and/or divergence of groundwater flow systems. Mathematically, stagnant zones are associated with stagnation points. The dynamics of groundwater around stagnation points has been studied numerically by *Anderson and Munter* [1981], *Winter* [1976, 1978, 1999] and *Winter and Pfannkuch* [1984]in studies on surface water-groundwater interaction. The stagnation points had also been found to be useful in defining capture zones [e.g.,*Tosco et al.*, 2008]. Based on analytical solutions of hydraulic head and stream function, *Jiang et al.* [2011] studied the dynamics of groundwater around stagnation points in nested flow systems and found that stagnation points could be divided into three types. A regional convergent stagnation point, which is caused by convergence of two flow systems, is located at the basin bottom (Figure 1a); a regional divergent stagnation point, which is caused by divergence of two flow systems, is also located at the basin bottom (Figure 1b); a local stagnation point, is located inside the basin and is caused by divergence and convergence of four flow systems (Figure 1c). “Local” is due to the fact that at least one of these four flow systems belongs to a local flow system.

[3] Based on qualitative analysis, *Tóth* [1980, 1999] proposed that transported matter such as metallic ions and hydrocarbons could accumulate in stagnant zones. The accumulation of metallic ions or petroleum in stagnant zones at the discharge end of a basin where two regional flow systems converge or a regional flow system ascends, i.e., around the regional convergent stagnation points as shown in Figure 1a, has been reported by several researchers [*Baskov*, 1987; *Garven*, 1985; *Garven and Freeze*, 1984; *Sanford*, 1994; *Tóth*, 1980, 1988]. However, there has been little, if any, quantitative research on the accumulation of transported matter in stagnant zones around regional divergent stagnation points or local stagnation points. A quantitative understanding on whether transported matter could accumulate around stagnation points would allow direct application of the theory of regional groundwater flow to explorations of mineral resources and hydrocarbons.

[4] The age of groundwater, as an intrinsic property of groundwater, is an important factor in explaining the hydrochemistry and vegetation type on a regional scale [*Batelaan et al.*, 2003; *Freeze and Cherry*, 1979]. Groundwater age can be calculated mathematically by solving the transport equation or determined chemically by measuring tracers in groundwater [*Bethke and Johnson*, 2008; *Phillips and Castro*, 2003]. In this paper, the former is called model age, and the latter tracer age. Traditionally, model age is calculated using the piston-flow model.*Goode* [1996]defined the concept of “age mass” as the product of water mass and its age. Groundwater age is an intensive quantity of groundwater, which means that it is mass-independent and hence not additive. Age mass, on the contrary, is an extensive quantity which depends on the total mass of the system, and is additive. Therefore, age mass can be considered as transported matter which could accumulate in groundwater. By defining age mass,*Goode* [1996]developed the advection-dispersion equation for age mass transport, which accounted for the mixing of groundwater due to hydrodynamic dispersion. According to this definition, the age of a groundwater sample is the average age of all the water molecules in the sample for the length of time each molecule has spent in the subsurface. This new age mass transport approach is changing the field of groundwater age dating [*Bethke and Johnson*, 2008]. By comparing the model age distribution of a cross-section in the Carrizo aquifer calculated by three different approaches, i.e., the piston-flow approach, the tracer transport approach, and the age mass transport approach,*Castro and Goblet* [2005] found that when groundwater velocity is extremely heterogeneous within a single aquifer, the results obtained by the age mass transport approach yield the most consistent ages. Therefore, the age mass transport approach is the most suitable method to obtain model age in nested flow systems, whose velocity is extremely heterogeneous.

[5] The emphasis of this study is to identify the characteristics of age distribution around stagnation points where transported matter is expected to accumulate. According to *Tóth*'s [1980, 1999]study, the low velocities in stagnant zones are expected to result in old groundwater, i.e., accumulation of age mass would lead to the existence of older model ages around stagnation points and also older tracer ages around stagnation points in a real basin. Note that if the piston-flow model is employed, the age at a stagnation point would be infinitely large, which is obviously unrealistic.

[6] In this paper, we first numerically obtain model age distribution in a series of two-dimensional (2-D) synthetic drainage basins using the classic*Tóth* [1963] model and discuss the characteristics of age distribution around stagnation points. We then use the Ordos Plateau as an example to show the distributions of groundwater flow systems and age around stagnation points.