A quantitative study on accumulation of age mass around stagnation points in nested flow systems

Authors

  • Xiao-Wei Jiang,

    Corresponding author
    1. Key Laboratory of Groundwater Circulation and Evolution, China University of Geosciences, Ministry of Education,Beijing,China
    2. School of Water Resources and Environment, China University of Geosciences,Beijing,China
      Corresponding author: X.-W. Jiang, Key Laboratory of Groundwater Circulation and Evolution, China University of Geosciences, Ministry of Education, Beijing 100083, China. (jxw@cugb.edu.cn)
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  • Li Wan,

    1. Key Laboratory of Groundwater Circulation and Evolution, China University of Geosciences, Ministry of Education,Beijing,China
    2. School of Water Resources and Environment, China University of Geosciences,Beijing,China
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  • Shemin Ge,

    1. Department of Geological Sciences, University of Colorado Boulder,Boulder, Colorado,USA
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  • Guo-Liang Cao,

    1. Key Laboratory of Groundwater Circulation and Evolution, China University of Geosciences, Ministry of Education,Beijing,China
    2. School of Water Resources and Environment, China University of Geosciences,Beijing,China
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  • Guang-Cai Hou,

    1. Xi'an Center of Geological Survey, China Geological Survey,Xi'an,China
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  • Fu-Sheng Hu,

    1. Key Laboratory of Groundwater Circulation and Evolution, China University of Geosciences, Ministry of Education,Beijing,China
    2. School of Water Resources and Environment, China University of Geosciences,Beijing,China
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  • Xu-Sheng Wang,

    1. Key Laboratory of Groundwater Circulation and Evolution, China University of Geosciences, Ministry of Education,Beijing,China
    2. School of Water Resources and Environment, China University of Geosciences,Beijing,China
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  • Hailong Li,

    1. Key Laboratory of Groundwater Circulation and Evolution, China University of Geosciences, Ministry of Education,Beijing,China
    2. School of Water Resources and Environment, China University of Geosciences,Beijing,China
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  • Si-Hai Liang

    1. Key Laboratory of Groundwater Circulation and Evolution, China University of Geosciences, Ministry of Education,Beijing,China
    2. School of Water Resources and Environment, China University of Geosciences,Beijing,China
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Corresponding author: X.-W. Jiang, Key Laboratory of Groundwater Circulation and Evolution, China University of Geosciences, Ministry of Education, Beijing 100083, China. (jxw@cugb.edu.cn)

Abstract

[1] The stagnant zones in nested flow systems have been assumed to be critical to accumulation of transported matter, such as metallic ions and hydrocarbons in drainage basins. However, little quantitative research has been devoted to prove this assumption. In this paper, the transport of age mass is used as an example to demonstrate that transported matter could accumulate around stagnation points. The spatial distribution of model age is analyzed in a series of drainage basins of different depths. We found that groundwater age has a local or regional maximum value around each stagnation point, which proves the accumulation of age mass. In basins where local, intermediate and regional flow systems are all well developed, the regional maximum groundwater age occurs at the regional stagnation point below the basin valley. This can be attributed to the long travel distances of regional flow systems as well as stagnancy of the water. However, when local flow systems dominate, the maximum groundwater age in the basin can be located around the local stagnation points due to stagnancy, which are far away from the basin valley. A case study is presented to illustrate groundwater flow and age in the Ordos Plateau, northwestern China. The accumulation of age mass around stagnation points is confirmed by tracer age determined by 14C dating in two boreholes and simulated age near local stagnation points under different dispersivities. The results will help shed light on the relationship between groundwater flow and distributions of groundwater age, hydrochemistry, mineral resources, and hydrocarbons in drainage basins.

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.

Figure 1.

Schematics defining three types of stagnation points (SP): (a) A regional convergent SP formed by convergence of two flow systems; (b) A regional divergent SP formed by divergence of two flow systems; (c) A local SP formed by convergence and divergence of four flow systems.

[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 classicTóth [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.

2. Mathematical Models

[7] In studies on groundwater flow systems, it is usually assumed that: (1) a 2-D cross-section of a basin is representative of a three-dimensional basin when it is taken parallel to the direction of dip of the water table slope; and (2) the average water table over many years is constant and can be accepted as a specified head boundary [Tóth, 1963; Freeze and Witherspoon, 1967; Jiang et al., 2011]. Consequently, we use the steady state groundwater flow equation to obtain the flow field:

display math

where K is the hydraulic conductivity tensor, his the hydraulic head. The bottom of the basin is usually impervious. The two sides of the cross-section, which correspond to the basin valley and the basin divide, are also considered as no-flow boundaries.

[8] In this study, we only care whether age mass could accumulate around stagnation points, so we calculate the steady state model age. Steady state groundwater flow field is used as input to obtain steady state age distribution. The equation for steady state age mass transport is [Goode, 1996]

display math

where τ is the model age of groundwater, θ is effective porosity, u is the pore velocity vector, Dis the dispersion coefficient tensor. For 2-D cross-section models,u = [ux, uz], and D = inline image, where D* is the molecular diffusion coefficient. To calculate each element of D, we define u = (ux2 + uz2)1/2, thus Dxx = inline image, Dxz = Dzx = inline image, and Dzz = inline image, where αL and αT are longitudinal and transverse dispersivities.

[9] At the top boundary of the basin, recharge zones obtained from flow simulation have zero age mass with τ = 0, while discharge zones have zero dispersive age mass flux with inline image = 0. All other boundaries have zero age mass flux with inline image = 0. The governing equations of groundwater flow and age mass transport are solved using the finite element method via COMSOL Multiphysics [COMSOL AB, 2008; Li et al., 2009].

3. A Synthetic Study

[10] Distributions of chemistry and age of groundwater are closely related to the velocity field. It is universally acknowledged that heterogeneity of hydraulic conductivity would greatly influence the distribution of velocity. In fact, the regional topography-driven flow alone is enough to create extremely heterogeneous velocity field. To exclude the influence of heterogeneity on the velocity field, we first use the classicTóth [1963] model to examine the relationship between stagnation points and groundwater age.

3.1. Basin Geometry and Parameters

[11] The geometry of the cross-sections of drainage basins is characterized by undulating water table on the top boundary as we used inJiang et al. [2009, 2010b]:

display math

where z0 is the elevation of the valley bottom, β is the angle of the regional slope, a and λ are the amplitude and wavelength of the local undulation of water table, and x is the horizontal distance from the basin valley. We assume that the basin bottom is flat and impervious.

[12] For the geometry of the basin cross-section defined inequation (3), we use z0 = 1000 m, tanβ = 0.02, a = 15 m, λ = 1500 m, and x ranges from 0 to 6000 m. We assume that the drainage basin has a hydraulic conductivity of K = 1 m d−1 and a porosity of θ = 0.3. In the calculation for the elements of D, we assume the effective molecular diffusion coefficient D* = 1.16 × 10−9 m2 s−1, the longitudinal dispersivity αL = 6 m, and the transverse dispersivity αT = 0.6 m. The longitudinal dispersivity and transverse dispersivity are assumed to be constant in space, which is typical in studies on regional flow and transport modeling [Zheng and Bennett, 2002].

3.2. Characteristics of Age Around Stagnation Points

[13] When the basin depth at the basin valley is 1000 m, three orders of flow systems, namely, local, intermediate and regional flow systems are all well developed (Figure 2a). There are one regional convergent stagnation point (SP 5), one regional divergent stagnation point (SP 6) and four local stagnation points (SP 1 through SP 4). Groundwater is increasingly older from the recharge zone to the discharge zone in each flow system. Near the end of an intermediate or regional flow system, due to the differences in travel distance, there is an abrupt change in groundwater age from basin top to basin bottom, i.e., the color representing logarithm of age changes from blue and yellow to red. To identify the characteristics of groundwater age around the local stagnation points, we show the distributions of groundwater age and flow systems around two of the four local stagnation points, SP 2 and SP 3 (Figure 3). Near each stagnation point, there is a closed line of contour, indicating trap of age mass around local stagnation points.

Figure 2.

The distributions of groundwater flow systems, stagnation points and groundwater age in synthetic drainage basins with different basin depths (Z). Solid lines are pathlines. (a) Z = 1000 m; (b) Z = 550 m; (c) Z = 500 m; (d) Z = 450 m; (e) Z = 400 m.

Figure 3.

Distributions of groundwater flow systems and age around local stagnation points. (a) SP 2; (b) SP 3. (The black lines are contours of groundwater age, the red lines are pathlines, and the arrows represent flow direction.)

[14] We also use some “boreholes” penetrating stagnation points to show the variations in groundwater age with depth (Figure 4). For the four local stagnation points, SP 1 through SP 4, age profiles show that groundwater ages have their local maximum values near stagnation points. For the two regional stagnation points, SP 5 and SP 6, groundwater ages have maximum values at the stagnation points. Moreover, groundwater at SP 5 is the oldest in the entire basin.

Figure 4.

Vertical profiles of groundwater age through (a) Local stagnation points, and (b) Regional stagnation points, for the case shown in Figure 2a.

[15] Based on the contours of groundwater age around local stagnation points shown in Figure 3 and the maximum values of groundwater age in the profiles shown in Figure 4, we can conclude that older water can be found not only around regional convergent stagnation points (SP 5), but also around regional divergent stagnation points (SP 6) and local stagnation points (SP 1 through SP 4), i.e., age mass could accumulate around each type of stagnation points.

3.3. Effect of Depth of Local Stagnation Points on Age Distribution

[16] According to Tóth [2009], the influencing factors of flow pattern in basins can be grouped into water table configuration, basin depth, heterogeneity and anisotropy. Jiang et al. [2010b]studied the influence of depth-decaying hydraulic conductivity and porosity on model age in a unit basin and aTóth basin with the same basin depth. Here, we use drainage basins with different depths to discuss the relationship between the distributions of groundwater flow systems, depths of local stagnation points and groundwater age. Deeper local stagnation points have been suggested in settings where the amplitude of local undulation is high, the amplitude of regional undulation is low, the hydraulic conductivity decays rapidly with depth, or the anisotropy ratio (the ratio of horizontal hydraulic conductivity to vertical hydraulic conductivity) is small [Jiang et al., 2011; Tóth, 1963; Wang et al., 2011]. Therefore, although only the effect of basin depth is considered, the results of this study can be applied to other situations. For example, the flow pattern in a homogeneous basin with a large depth is similar to the flow pattern in a homogeneous basin with a small depth and a large anisotropy ratio.

[17] The distributions of groundwater flow systems, stagnation points, and model age for basins with different depths are shown in Figure 2. As the basin depth decreases, there is less room for intermediate and regional flow systems to develop, and the local stagnation points might reach the basin bottom. When the basin depth reduces to 500 m, SP 1 reaches the basin bottom and splits into two new stagnation points (SP 1 caused by convergence of two flow systems and SP 1′ caused by divergence of two flow systems as shown in Figure 2c). In this case, the regional flow cannot reach the basin valley. When the basin depth reduces to 400 m, SP 4 also reaches the basin bottom and splits into two new stagnation points (SP 4 and SP 4′ as shown in Figure 2e). In this case, groundwater recharged at the divide can only discharge locally.

[18] Accompanying with the changes in relative depths of local stagnation points, the location of maximum age in the basins, where transported matter is most likely to accumulate, also changes, although remains to happen at the basin bottom. When local, intermediate and regional flow systems are well developed (Cases a and b in Figure 2), the stagnation point below the basin valley (SP 5) has the maximum age. In Case c, when SP 1 reaches the basin bottom, groundwater at the new SP 1 (convergence) has the maximum age. When the basin depth reduces from 500 m (Case c) to 450 m (Case d), although the development of flow systems differs little (SP 4 does not reach basin bottom yet), the location of maximum age shifts from SP 1 to the basin bottom below SP 4. When the basin depth reduces to 400 m (Case e), SP 4 reaches the basin bottom, and groundwater at the new SP 4 (convergence) has the maximum age. We can infer that old groundwater could exist not only near the basin valley, but also other parts of a basin below the topographic lows where stagnation points exist. In other words, a long travel distance is not the only factor producing maximum groundwater age, and stagnancy within short travel distance could also result in maximum groundwater age. This finding is valuable for interpreting groundwater age distribution, groundwater chemistry and potential sites of concentrating minerals or petroleum.

4. A Field Study in the Ordos Plateau

[19] Groundwater age can be determined chemically by the concentration of tracers. A combination of tracer age and model age would be convincing to prove the accumulation of age mass around stagnation points. Here, we use a cross-section in the Ordos Plateau as a typical site to examine the relationship between groundwater flow systems, stagnation points and age distribution.

[20] The Ordos Basin, the second largest sedimentary basin in northwestern China, is abundant in fossil fuel and mineral resources. Unfortunately, the economic development of the Ordos Basin is restricted by limited water resources due to its arid to semiarid climate. Since 1999, a project “Groundwater Investigation in the Ordos Basin” was launched by the China Geological Survey [Hou et al., 2008a]. Results of this project indicate that groundwater flow is topographically controlled in the northern part of the Ordos Basin, which is also called the Ordos Plateau [Hou et al., 2008b]. Moreover, stable isotope analysis shows that groundwater in the Ordos Plateau is of meteoric origin [Yin et al., 2010; Yin et al., 2011].

4.1. An Overview of the Study Area

[21] The Ordos Plateau extends 360 km from north to south while about 210 ∼ 260 km from west to east, covering an area of 81,000 km2. Precipitation and evaporation are unevenly distributed in the study area (Figure 5). Precipitation decreases from 420 mm yr−1 in the southeast to 160 mm yr−1 in the northwest. The potential evaporation is extremely high, increasing from 2000 mm yr−1 in the southeast to 3200 mm yr−1 in the northwest. Precipitation mostly occurs as rainfall from July to September. The mean monthly temperature can be as high as 20.7°C in July but as low as −4.6°C in January, with a mean annual temperature of 6.5°C [Yin et al., 2010; Yin et al., 2011].

Figure 5.

The location of the study area and the spatial distributions of precipitation and evaporation in the Ordos Plateau (modified from Yin et al. [2011]).

[22] The Plateau has an undulating topography. The major basin divides are the Baiyu Mountain at the southern margin, with elevations ranging from 1500 to 1800 m, and the Sishi Ridge, which strikes roughly northeast, with elevations of 1400 ∼ 1500 m (Figure 5). The eastern, western and northern margins, at elevations of 1100 ∼ 1200 m, constitute the basin valleys. There are numerous local topographic lows and highs. As a result of groundwater discharge as well as surface runoff, lakes develop in some local lows. There are three major rivers in the area: the Molin River, the Dosit River and the Wuding River (Figure 5), all of which are supplied mainly by groundwater.

[23] In the Ordos Plateau, the basement consists of sandstones of Jurassic age (J), which is low in porosity and permeability. Overlying the basement is the major sandstone aquifer system of Cretaceous age, which comprises three groups: in ascending order, the Luohe Group (K1l), the Huanhe Group (K1h), and the Luohandong Group (K1lh). The Cretaceous sandstones, which are poorly consolidated, can be considered as a continuum type of porous medium. The sandstone aquifer system is overlain locally by Tertiary (E) mudstones and extensively by unconsolidated Quaternary (Q) sediments.

4.2. Geological and Hydrogeological Setting of the Typical Cross-Section

[24] Based on the equipotential map of the Ordos plateau, we selected a 2-D cross-section (A-A′ inFigure 5) for analysis of groundwater flow. The cross-section is generally parallel to the direction of groundwater flow with some adjustment to include as many research boreholes as possible. It spans about 240 km and has elevations ranging between 1200 and 1450 m. The Sishi Ridge at the middle of the cross-section has the highest elevation and is the main recharge area.

[25] Figure 6shows the stratigraphy and lithology of the cross-section. The basement geometry shows that the basin is an asymmetric syncline. At the core of the syncline, near borehole B2, the sandstone aquifer system has its maximum thickness of 950 m. The main sandstone aquifer system thins eastward and pinches out at the end of the basin.

Figure 6.

The geological cross-section of A-A′.

[26] The aquifer system of the cross-section consists mainly of K1h and K1l sandstones. The K1h sandstone was deposited in a fluvial environment. The eastern part of the K1l sandstone was deposited in an eolian environment while the western part was deposited in a fluvial environment. There are numerous sporadic clay lenses in the K1h and K1lsandstones. In the west end of the cross-section, the K1lhsandstone and Tertiary mudstones are locally distributed. Most part of the cross-section is overlain by thin Quaternary sediments.

[27] The K1l sandstone, with a thickness of about 200 m, is one of the two most important aquifers. Based on porosity measurements of rock samples from boreholes B2 and B7 (Figure 6), the mean porosity of fluvial sandstone is 21.1%. Eolian sandstone rock samples from borehole B15 were measured to have a mean porosity of 27.8%. Therefore, the porosity of the eolian sandstone differs greatly from that of the fluvial sandstone.

[28] The K1h sandstone is the other major aquifer. Its thickness reaches 750 m at the core of the syncline near B2. At the western edge, the thickness of K1h sandstone decreases to 600 m. At borehole B15, its thickness decreases to 260 m. At the eastern edge, the K1h sandstone finally pinches out. Based on porosity measurements of rock samples from B2, B7 and B15, the porosity of fluvial sandstone ranges between 16.5% and 24.5%.

[29] In borehole B2, pumping tests were conducted at three different depths using single well pumping tests. The results of pumping tests give hydraulic conductivity values of 0.38 m d−1 of the upper section of K1h sandstone and 0.18 m d−1 of the lower section of K1h sandstone. In contrast, the hydraulic conductivity of the K1l sandstone is 0.45 m d−1. This indicates that K1l sandstone has a larger hydraulic conductivity than K1h sandstone, and inside the K1hsandstone, the deep part has a smaller hydraulic conductivity than the shallow part. This implies a depth-dependent hydraulic conductivity structure in the K1h sandstone.

[30] To examine the hydraulic conductivity structure of whole aquifer system, we use 170 hydraulic conductivity measurements in the K1h sandstone and 70 data in the K1l sandstone throughout the Ordos Plateau. These hydraulic conductivity values are obtained on the basis of pumping test at many wells, whose depths of test sections are different. Statistical analysis shows that hydraulic conductivity of the K1l sandstone has no significant correlation with depth, while mean hydraulic conductivity of the K1h sandstone decreases with depth (Figure 7). We use the exponential decay model [Jiang et al., 2010a; Jiang et al., 2009] to fit the relationship between mean hydraulic conductivity and depth. The decay exponent, A, is found to be 0.0022 m−1 (Figure 7).

Figure 7.

The change in mean hydraulic conductivity with depth in the K1h sandstone.

[31] There is a permeable fault near the western end of the cross-section. West of the fault are formations of Jurassic and older age that constitute an impervious boundary of the aquifer system. The Hekou Reservoir near the eastern end is the lowest discharge zone in the eastern part of the aquifer system, representing a no-flow boundary beneath the Hekou Reservoir. On the basis of water level, hydrogeochemistry and isotope data, it has been shown that groundwater flow is topographically controlled with three orders of flow systems (Figure 8) [Hou et al., 2008b; Yin et al., 2010; Yin et al., 2011]. Groundwater recharged around the Sishi Ridge could discharge locally via local flow systems, to the Dosit River via an intermediate flow system, and to the west end and the Hekou Reservoir near the east end via regional flow systems.

Figure 8.

The conceptual model of groundwater flow in cross-section A-A′ (modified fromHou et al. [2008b], with permission of Wiley).

4.3. Numerical Model

[32] In analyses of groundwater flow systems, available data are often insufficient to accurately determine the elevation of water table. It is, therefore, customary to assume that the long-term average water table over years conforms to the topography and groundwater flow is in steady state [Ophori and Tóth, 1989; Sykes et al., 2009; Tóth, 1963]. The 2-D steady state flow assumption has also been commonly used in regional-scale groundwater flow and transport studies such asBethke et al. [1999], Castro et al. [1998], Castro and Goblet [2003]. Because of limited historical data, we assume that groundwater flow is in steady state in our study area. The steady state groundwater flow assumption is an acceptable first step toward a quantitative understanding of stagnation zone development in the study area.

[33] A numerical model of steady state groundwater flow and age of the cross-section is developed. The model area is bounded by the water table on the top, the top of Jurassic sandstone at the bottom, the boundary between the fault and Jurassic and older formations at the west, and a water divide at the east (Figure 9). This implies that only saturated groundwater flow is considered. In the current study area, although the climate is semiarid, due to the differences in hydraulic conductivity between the Cretaceous sandstones and the Quaternary sediments, groundwater exists in the Quaternary sediments and the water table lies over the top of the Cretaceous sandstones. For the most part of the cross-section where the Quaternary sediments are thin, we use the top of the Cretaceous sandstones to represent the water table. For the east part where Quaternary sediments are thick, we use interpolation to obtain the water table based on limited water table measurements.

Figure 9.

The hydrostratigraphy of the model area and boundary conditions for groundwater flow.

[34] The model area includes five hydrostratigraphic units: the fault, the K1l fluvial sandstone, the K1l eolian sandstone, the K1h fluvial sandstone, and the Quaternary sediments (Figure 9). Note that the small K1lhsandstone and Tertiary mudstones near the western end of the cross-section, are lumped into the K1h fluvial sandstone.

[35] The 2-D steady state groundwater flow equation and steady state age mass transport equation are solved using the finite element method via COMSOL Multiphysics. The model area is discretized into triangles, with a maximum element size of 20 m. The mesh has 899,340 nodes, and 1,771,770 elements. For the groundwater flow model, the top boundary has specified heads and three other boundaries are impervious (Figure 9). For the age mass transport model, the no-flow boundaries for groundwater flow have zero age mass flux, the recharge zones on the specified head boundary have zero age mass, and the discharge zones on the specified head boundary have zero dispersive age mass flux.

[36] Porosity, hydraulic conductivity and dispersivity are primary input parameters for groundwater flow and age mass transport modeling. Porosity is mainly determined by porosity measurements of rock samples from B2, B7, and B15 and the ratio of sandstones to clays in the sandstone aquifer system. The distribution of porosity follows the division of the model area shown in Figure 9. When the clays are considered, the porosity of K1h fluvial sandstone, K1l fluvial sandstone and K1l eolian sandstone is 19.3%, 18.9%, and 27.8%, respectively. No measurements of porosity of the fault and the Quaternary sediments are available. The porosity of the fault is assumed to be larger than that of the K1h fluvial sandstone and K1l fluvial sandstone, while the porosity of the Quaternary sediments is assumed to be larger than that of the K1l eolian sandstone. The porosity of the fault and the Quaternary sediments is set to be 25% and 30%. Due to their small area compared with the Cretaceous sandstones, we believe that this uncertainty would not have a significant influence on the simulation results.

[37] For large-scale models, sandstones with clays can be characterized as an anisotropic medium, whose horizontal hydraulic conductivity (Kx) is mainly determined by the sandstones and the vertical hydraulic conductivity (Kz) by the clays. Although in situ measurements of transmissivity are available, they cannot be directly applied in the numerical model due to the existence of clays, which impose uncertainties on the effective thickness of the aquifer, thus the calculation of Kx. Here, the values of Kx and anisotropy ratio (Kx/Kz) are inversely determined. The value of Kx/Kz is assumed to be constant within each hydrostratigraphic unit. In the hydrostratigraphic unit of K1hsandstone, the depth-decaying hydraulic conductivity shown inFigure 7 is also applied.

[38] Defining dispersivity values for field-scale transport simulation is inherently difficult and controversial [Zheng and Bennett, 2002]. In Castro and Goblet's [2005] study on age mass transport and 14C transport models, the longitudinal dispersivity was assumed to be 125 m, which is around 1/1000 of the distance from the recharge zone to the discharge zone in the cross-section. In our cross-section, the distance from the divide to the basin valleys is around 100 km, so we assume the longitudinal dispersivity to be 100 m and the transverse dispersivity to be 10 m for calibration of the model. Sensitivity analysis of dispersivity on age distribution had also been conducted.

[39] The information used for model calibration includes the conceptual model of groundwater flow shown in Figure 8, the hydraulic head measured at different depths of B15 by packer test, and the isotopic age of 14C in the K1l sandstone part of B2 and B7. After model calibration by the trial and error method, the horizontal hydraulic conductivity, Kx, of K1l fluvial sandstone is found to be 1.4 m d−1, of K1l eolian sandstone is found to be 2.1 m d−1. The initial Kx (horizontal hydraulic conductivity at or near ground surface) of K1h fluvial sandstone was determined at 0.8 m d−1. The values of Kx/Kz of K1l fluvial sandstone, K1l eolian sandstone, and K1h fluvial sandstone are 700, 600 and 520, respectively. These anisotropy ratio values are in agreement with Bethke's [1989] and Deming's [2002] findings that anisotropy ratio could be of the order of 102 to 103 in large sedimentary basins. The selections of hydraulic conductivity and anisotropy ratio of the Quaternary sediments and fault are more arbitrary due to limited information available and their small area. Because they are more permeable than the sandstone aquifer system, the Kx of Quaternary sediments is set to be 10 m d−1 and that of the fault is set to be 4 m d−1. Because they are relatively unconsolidated, the Kx/Kz of Quaternary sediments and fault are set to be 50 and 100, respectively.

[40] The measured hydraulic head by packer test and the simulated hydraulic head at different depths of B15 are shown in Figure 10. The absolute values of the differences between measured and simulated hydraulic head range between 0.20 m and 1.51 m, demonstrating that the heterogeneity and anisotropy of hydraulic conductivity is well characterized. During calibration, we found that the rate of decrease in hydraulic head with depth in the K1h sandstone is sensitive to the anisotropy ratio, suggesting that the hydraulic head measured at different depths of a borehole could be useful in estimating this parameter.

Figure 10.

The measured and simulated hydraulic head at different depths of B15.

[41] Groundwater were sampled at different parts of boreholes B2 and B7, which were screened at different depths. The 14C ages, which were measured by the IAEA using the method of liquid scintillation counting, in the K1l sandstone part of B2 and B7 were used for model calibration. According to the conceptual model of groundwater flow shown in Figure 8, groundwater mainly flows horizontally in the K1l sandstone part of B2, and is near a regional divergent stagnation point in the K1l sandstone part of B7. The 14C ages in the K1l sandstone part of B2 and B7 are 21,400 years and 19,110 years, respectively (Figure 11c). The simulated age in the K1l sandstone part of B2 is around 21,500 year, and is very close to the measured age. B7 is located near the divide, where a stagnation point exists at the bottom. The simulated age around the stagnation point, SP 3, which is about 19,000 ∼ 24,000 years, is also in agreement with the measured age in the K1l sandstone, which equals 19,110 years.

Figure 11.

The distributions of hydraulic head, groundwater flow systems and age: (a) The contours of hydraulic head; (b) streamlines showing local (green lines), intermediate (red lines) and regional (blue lines) groundwater flow systems; (c) The distributions of groundwater age (gray lines are contour, red lines represent boreholes, and black dots on the boreholes represent age measurements).

[42] Groundwater sampled from the K1h sandstone part of B7 was measured to be 440 years (Figure 11c), however, the simulated age ranges between 0 and 9000 years from the top to the bottom of the K1h sandstone. It is hard to compare these two values because it is difficult to tell which depth groundwater had been sampled at. In B15, the measured age of groundwater sampled at eight different depths has a narrow variation ranging between 1450 years and 2240 years, while the simulated age at corresponding depths increases nonlinearly from 2000 years to 4000 years.

4.4. Simulation Results

[43] The distributions of hydraulic head, groundwater flow systems and groundwater age of the cross-section obtained from the calibrated model are shown inFigure 11. Hydraulic head is high around the Sishi Ridge, and has a general trend of decreasing toward west and east (Figure 11a). Around the Sishi Ridge, which is the regional recharge zone, hydraulic head ranges between 1370 and 1380 m. In the middle of the Sishi Ridge and the Hekou Reservoir, there is a zone with hydraulic head larger than 1370 m. This local high constitutes a local recharge zone. To the west of the Dosit River, there is also a local recharge zone, with hydraulic head larger than 1230 m.

[44] At the recharge zones, for example, around the Sishi Ridge, the shallow part of the aquifer system has a higher hydraulic head than the deep part. This indicates that groundwater mainly flows downward. At the discharge zones, for example, around the Dosit River, the shallow part of the aquifer system has a lower hydraulic head than the deep part, which indicates that groundwater flows upward. In the zone between the Sishi Ridge and the Dosit River, the contours of hydraulic head are almost vertical, which implies that horizontal flow dominates.

[45] As discussed by Anderson and Munter [1981], Tóth [1988], and Jiang et al. [2011], groundwater around stagnation points has the characteristics of potentiometric minimum. Based on the shape of contours of hydraulic head near the Dosit River, we can infer that there is a stagnation point below this zone.

[46] Streamlines help to identify the flowpath of groundwater and the distribution of groundwater flow systems (Figure 11b). The Sishi Ridge is the recharge zone of two regional flow systems, one intermediate flow system and one local flow system. Figure 11b also shows the location of two local stagnation points (SP 1 and SP 2) and one regional divergent stagnation points (SP 3). SP 1 west of the Dosit River divides four flow systems, including two local flow systems, one intermediate flow system and one regional flow system. SP 2 east of the Sishi Ridge also divides four flow systems, including three local flow systems and one regional flow system. Moreover, due to the differences in basin depth, the locations of the two stagnation points differ greatly, with SP 1 around the middle and SP 2 near the bottom of the aquifer system. SP 3 below the Sishi Ridge divides two regional flow systems.

[47] Due to factors as basin geometry (varying basin thickness) as well as heterogeneities and anisotropy of the medium, the distribution of the groundwater age pattern is more complex in the study area than that in the theoretical cases shown in Figure 2. West of the Sishi Ridge, local flow systems form in the shallow parts of the aquifer system, and intermediate and regional flow systems develop in the deep parts. Consequently, groundwater is generally older in the deep part than at shallow depths. Near the western end of the aquifer system, i.e., the ascending limb of regional flow system, groundwater can be as old as exceeding 40,000 years. At the ascending limb of intermediate flow system near the Dosit River, groundwater can be as old as around 20,000 years.

[48] East of the Sishi Ridge, the Hekou Reservoir is the lowest discharge region. Although groundwater can reach the Hekou Reservoir through a regional flow system, local flow systems dominate. Due to the large penetration depths of the two local flow systems over SP 2 in Figure 11b, groundwater has its maximum age (larger than 60,000 years but smaller than 120,000 years) around SP 2. This phenomenon is similar to the age distribution around SP 4 in Figure 2c. Sensitivity analysis of dispersivity shows that, smaller dispersivity would lead to an even greater maximum age, while larger dispersivity would result in much younger waters.

4.5. Discussions

[49] The characteristics of dynamics and age of groundwater around the local stagnation point SP 1 in Figure 11b are discussed below. This point is chosen because borehole B2, where measurements of 14C age are available, might be within the zone of influence of SP 1. Figure 12shows the distributions of groundwater age in the western part of the cross-section, as well as four contours of hydraulic head, two dividing streamlines and four schematic streamlines showing the flow direction. The four contours of hydraulic head of 1220.51 m and 1220.55 m, show the potentiometric minimum around SP 1. The two dividing streamlines precisely divide the four flow systems.

Figure 12.

Dynamics and age of groundwater around SP 1 (the gray lines around the SP are contours of hydraulic head and the red lines are dividing streamlines).

[50] In Figure 12, it is evident that groundwater below SP 1 is much older than groundwater above SP 1. We plot the vertical distribution of groundwater age through SP 1 under different dispersivities (Figure 13a). Under different longitudinal dispersivities ranging between 30 and 300 m, groundwater age has an abrupt increase near SP 1, and reaches a maximum value below SP 1. Beyond the zone of influence of the stagnation point, due to the heterogeneity caused by lithology difference between K1h and K1l, groundwater age decreases to a certain value.

Figure 13.

Vertical distributions of model age through (a) SP 1 and (b) in borehole B2 under different dispersivities.

[51] In B2, the 14C age in the lower part of K1h sandstone is measured to be 26,060 years, which is several thousands years older than groundwater in the K1l sandstone (21,400 years). In our calibrated model (αL = 100 m), the maximum age in the lower part of K1h sandstone is about 24,260 years, which is almost 3000 years older than the simulated age in the K1l sandstone (Figure 13b). If a smaller dispersivity αL = 30 m is used, the maximum age in the lower part of K1h sandstone is about 7000 years older than the simulated age in the K1l sandstone. If a larger dispersivity αL = 300 m is used, the maximum age in the lower part of K1h sandstone is still about 2000 years older than the simulated age in the K1l sandstone.

[52] Here, both tracer age and model age under different dispersivities demonstrate that within the zone of influence of a stagnation point, groundwater age can be higher than surrounding areas, i.e., age mass could accumulate in stagnant zones around local stagnation points. If an extremely large dispersivity is used, however, it is possible that accumulation of age mass might be negligible due to the high degree of mixing.

[53] The age data in the study area are limited at this stage, but available data support the model results. In the three available deep wells with age measurements, we found that two of them are located near stagnation points, i.e., B2 is located near a local stagnation point and B7 is located near a regional divergent stagnation point below the divide. Future efforts could be directed to collecting more age data in the area around the Dosit River (Figure 11c) at the elevation of 700∼800 m.

5. Conclusions

[54] We analyzed groundwater flow systems and groundwater age in cross-sections of drainage basins of varying depth. The characteristics of groundwater age around stagnation points are emphasized. We find that age mass can accumulate around stagnation points. In basins where local, intermediate and regional flow systems are all well developed, the maximum groundwater age is located at the stagnation point below basin valley. When regional flow is weak or absent, local stagnation points can be close enough to, or even reach, the basin bottom. In such cases maximum groundwater age can be located around local stagnation points, which are far away from the basin valley. Consequently, maximum groundwater age can be caused not only by long travel distances combined with stagnancy, but also by stagnancy with a short travel distance.

[55] A cross-section model of steady state groundwater flow and age in the Ordos Plateau was constructed. The model is calibrated using hydraulic head measurements from different depths of one borehole and tracer ages measured in the K1lsandstone of two boreholes. Due to the difference in basin depth, groundwater flow patterns west and east of the Sishi Ridge differs greatly. A relatively shallow sub-basin east of the Sishi Ridge results in dominantly local flow systems, while a relatively deep sub-basin west of the Sishi Ridge leads to well developed local, intermediate and regional flow systems. The measured14C age in borehole B2 and the model age at SP 1 and nearby areas under different dispersivities demonstrate that within their zones of influence, age mass could accumulate around local stagnation points.

[56] The results reported in this study are fundamental to the future applicability of the theory of regional groundwater flow, such as interpreting tracer age and hydrochemical patterns, and exploration of mineral deposits or petroleum. In the future, more field work is needed to further demonstrate the accumulation of transported matter due to stagnancy of groundwater and to obtain basin-scale values for dispersivity.

Acknowledgments

[57] This study was supported by China Geological Survey (grant 1212011121145), National Natural Science Foundation of China (grant 41202173), and the Fundamental Research Funds for the Central Universities of China. The authors acknowledge three anonymous reviewers and József Tóth as a reviewer for their valuable comments that have significantly enhanced the quality of this manuscript. The authors also thank Associate Editor, Daniel Fernàndez-Garcia, and Editor, Graham Sander, for their constructive suggestions.