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A quantitative study on accumulation of age mass around

stagnation points in nested flow systems

Xiao-Wei Jiang,

1,2

Li Wan,

1,2

Shemin Ge,

3

Guo-Liang Cao,

1,2

Guang-Cai Hou,

4

Fu-Sheng Hu,

1,2

Xu-Sheng Wang,

1,2

Hailong Li,

1,2

and Si-Hai Liang

1,2

Received 6 June 2012; revised 26 September 2012; accepted 26 September 2012; published 6 December 2012.

[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

14

C 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.

Citation: Jiang, X.-W., L. Wan, S. Ge, G.-L. Cao, G.-C. Hou, F.-S. Hu, X.-S. Wang, H. Li, and S.-H. Liang (2012), A quantitative

study on accumulation of age mass around stagnation points in nested flow systems, Water Resour. Res., 48, W12502, doi:10.1029/

2012WR012509.

1. Introduction

[2] Periodic undulations of the water table create gravity-

driven, hierarchically nested ﬂow systems, i.e., local, inter-

mediate and regional ﬂow systems in drainage basins

[To

´

th, 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 diver-

gence 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, To

´

th [1980, 1999] pro-

posed that transported matter such as metallic ions and

hydrocarbons could accumulate in stagnant zones. The accu-

mulation 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,

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.

3

Department of Geological Sciences, University of Colorado Boulder,

Boulder, Colorado, USA.

4

Xi’an Center of Geological Survey, China Geological Survey, Xi’an,

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)

©2012. American Geophysical Union. All Rights Reserved.

0043-1397/12/2012WR012509

W12502 1of14

WATER RESOURCES RESEARCH, VOL. 48, W12502, doi:10.1029/2012WR012509, 2012

has been reported by several researchers [Baskov,1987;

Garven, 1985; Garven and Freeze, 1984; Sanford,1994;

To

´

th, 1980, 1988]. However, there has been little, if any,

quantitative research on the accumulation of transported mat-

ter in stagnant zones around regional divergent stagnation

points or local stagnation points. A quantitative understand-

ing 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 hydro-

chemistry 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 calcu-

lated 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 ground-

water, 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 trans-

ported 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 disper-

sion. 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 chang-

ing 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 trans-

port 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 charac-

teristics of age distribution around stagnation points where

transported matter is expected to accumulate. According to

To

´

th’s [1980, 1999] study, the low velocities in stagnant

zones are expected to result in old groundwater , i.e., accu-

mulation 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 stagna-

tion 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 To

´

th [1963] model and

discuss the characteristics of age distribution around stag-

nation 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 represen-

tative 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 [To

´

th, 1963; Freeze

and Witherspoon, 1967 ; Jiang et al., 2011]. Consequently,

we use the steady state groundwater ﬂow equation to obtain

the ﬂow ﬁeld:

rðKrhÞ¼0; (1)

where K is the hydraulic conductivity tensor, h is the hy-

draulic head. The bottom of the basin is usually impervi-

ous. The two sides of the cross-section, which correspond

to the basin valley and the basin divide, are also considered

as no-ﬂow boundaries.

Figure 1. Schematics deﬁning three types of stagnation

points (SP): (a) A regional convergent SP formed by conver-

gence 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.

W12502 JIANG ET AL.: AGE DISTRIBUTION AROUND STAGNATION POINTS W12502

2of14

[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]

rðDr ÞrðuÞþ ¼ 0; (2)

where is the model age of groundwater , is effective po-

rosity, u is the pore velocity vector, D is the dispersion

coefﬁcient tensor. For 2-D cross-section models, u ¼

[u

x

, u

z

], and D ¼

D

þ D

xx

D

þ D

xz

D

þ D

zx

D

þ D

zz

, where D

is the

molecular diffusion coefﬁcient. To calculate each element

of D, we deﬁne u ¼ (u

x

2

þ u

z

2

)

1/2

, thus D

xx

¼

L

u

2

x

u

þ

T

u

2

z

u

,

D

xz

¼ D

zx

¼ð

L

T

Þ

u

x

u

z

u

, and D

zz

¼

L

u

2

z

u

þ

T

u

2

x

u

, where

L

and

T

are longitudinal and transverse dispersivities.

[

9] At the top boundary of the basin, recharge zones

obtained from ﬂow simulation have zero age mass with

¼ 0, while discharge zones have zero dispersive age mass

ﬂux with Dr ¼ 0. All other boundaries have zero age

mass ﬂux with u Dr ¼ 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 acknowl-

edged that heterogeneity of hydraulic conductivity would

greatly inﬂuence the distribution of velocity. In fact, the re-

gional topography-driven ﬂow alone is enough to create

extremely heterogeneous velocity ﬁeld. To exclude the inﬂu-

ence of heterogeneity on the velocity ﬁeld, we ﬁrst use the

classic To

´

th [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 in Jiang et al. [2009, 2010b]:

z

s

ðxÞ¼z

0

þ x tan þ

a

cos

sin

2x

cos

; (3)

where z

0

is the elevation of the valley bottom, is the angle

of the regional slope, a and are the amplitude and wave-

length 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 in

equation (3), we use z

0

¼ 1000 m, tan ¼ 0.02, a ¼ 15 m,

¼ 1500 m, and x ranges from 0 to 6000 m. We assume

that the drainage basin has a hydraulic conductivity of K ¼

1md

1

and a porosity of ¼ 0.3. In the calculation for

the elements of D, we assume the effective molecular diffu-

sion coefﬁcient D

¼ 1.16 10

9

m

2

s

1

, the longitudinal

dispersivity

L

¼ 6 m, and the transverse dispersivity

T

¼ 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 dis-

tance, there is an abrupt change in groundwater age from ba-

sin 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 stagna-

tion 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.

[

14] We also use some ‘‘boreholes’’ penetrating stagna-

tion 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. More-

over, groundwater at SP 5 is the oldest in the entire basin.

[

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 To

´

th [2009], the inﬂuencing factors of

ﬂow pattern in basins can be grouped into water table con-

ﬁguration, basin depth, heterogeneity and anisotropy . Jiang

et al. [2010b] studied the inﬂuence of depth-decaying hy-

draulic conductivity and porosity on model age in a unit ba-

sin and a To

´

th basin with the same basin depth. Here, we

use drainage basin s 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 condu ctivity to verti-

cal hydraulic conductivity) is small [Jiang et al., 2011;

To

´

th, 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

W12502 JIANG ET AL. : AGE DISTRIBUTION AROUND STAGNATION POINTS W12502

3of14

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, stag-

nation 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 sys-

tems 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

0

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

0

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 accu-

mulate, also changes, although remains to happen at the ba-

sin 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 max-

imum 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 bot-

tom 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 maxi-

mum age. We can infer that old groundwater could exist

not only near the basin valley, but also other parts of a ba-

sin 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 accumula-

tion of age mass around stagnation points. Here, we use a

cross-section in the Ordos Plateau as a typical site to exam-

ine 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

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 ¼ 1000 m;

(b) Z ¼ 550 m; (c) Z ¼ 500 m; (d) Z ¼ 450 m; (e) Z ¼ 400 m.

W12502 JIANG ET AL.: AGE DISTRIBUTION AROUND STAGNATION POINTS W12502

4of14

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 iso-

tope 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, cover-

ing an area of 81,000 km

2

. Precipitation and evaporation

are unevenly distributed in the study area (Figure 5). Pre-

cipitation decreases from 420 mm yr

1

in the southeast to

160 mm yr

1

in the northwest. The potential evaporation is

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.)

W12502 JIANG ET AL. : AGE DISTRIBUTION AROUND STAGNATION POINTS W12502

5of14

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 tem-

perature of 6.5

C[Yin et al., 2010 ; 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 ele-

vations of 1400 1500 m (Figur e 5). The eastern, western

and northern margins, at elevations of 1100 1200 m, con-

stitute the basin valleys. There are numerous local topo-

graphic 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.

Figure 4. Vertical proﬁles of groundwater age through (a) Local stagnation points, and (b) Regional

stagnation points, for the case shown in Figure 2a.

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]).

W12502 JIANG ET AL.: AGE DISTRIBUTION AROUND STAGNATION POINTS W12502

6of14

[23] In the Ordos Plateau, the basement consists of sand-

stones of Jurassic age (J), wh ich is low in porosity and per-

meability. Overlying the basement is the major sandstone

aquifer system of Cretaceous age, which comprises three

groups: in ascending order, the Luohe Group (K

1

l), the

Huanhe Group (K

1

h), and the Luohandong Group (K

1

lh).

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 Terti-

ary (E) mudstones and extensively by unconsolidated Qua-

ternary (Q) sediments.

4.2. Geological and Hydrogeological Setting of the

Typical Cross-Section

[

24] Based on the equipotential map of the Ordos pla-

teau, we selected a 2-D cross-section (A-A

0

in Figure 5) for

analysis of groundwater ﬂow. The cross-section is gener-

ally parallel to the direction of groundwater ﬂow with some

adjustment to include as many research boreholes as possi-

ble. 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 6 shows 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 maxi-

mum thickness of 950 m. The main sandstone aquifer sys-

tem thins eastward and pinches out at the end of the basin.

[

26] The aquifer system of the cross-section consists

mainly of K

1

h and K

1

l sandstones. The K

1

h sandstone was

deposited in a ﬂuvial environment. The eastern part of the

K

1

l sandstone was deposited in an eolian environment

while the western part was deposited in a ﬂuvial environ-

ment. There are numerous sporadic clay lenses in the K

1

h

and K

1

l sandstones. In the west end of the cross-section,

the K

1

lh sandstone and Tertiary mudstones are locally dis-

tributed. Most part of the cross-section is overlain by thin

Quaternary sediments.

[

27] The K

1

l sandstone, with a thickness of about 200 m,

is one of the two most important aquifers. Based on poros-

ity 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%. There-

fore, the porosity of the eolian sandstone differs greatly

from that of the ﬂuvial sandstone.

[

28]TheK

1

h 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 K

1

h sandstone

decreases to 600 m. At borehole B15, its thickness decreases

to 260 m. At the eastern edge, the K

1

h sandstone ﬁnally

pinches out. Based on porosity measurements of rock sam-

ples 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 K

1

h sandstone and

0.18 m d

1

of the lower section of K

1

h sandstone. In con-

trast, the hydraulic conductivity of the K

1

l sandstone is

0.45 m d

1

. This indicates that K

1

l sandstone has a larger

hydraulic conductivity than K

1

h sandstone, and inside the

K

1

h sandstone, the deep part has a smaller hydraulic conduc-

tivity than the shallow part. This implies a depth-dependent

hydraulic conductivity structure in the K

1

h sandstone.

[

30] To examine the hydraulic conductivity structure of

whole aquifer system, we use 170 hydraulic conductivity

measurements in the K

1

h sandstone and 70 data in the K

1

l

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 differ-

ent. Statistical analysis shows that hydraulic conductivity

of the K

1

l sandstone has no signiﬁcant correlation with

depth, while mean hydraulic conductivity of the K

1

h sand-

stone decreases with depth (Figure 7). We use the exponen-

tial 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).

[

31] There is a permeable fault near the western end of

the cross-section. West of the fault are formations of Juras-

sic and older age that constitute an impervious boundary of

the aquifer system. The Hekou Reservoir near the eastern

Figure 6. The geological cross-section of A-A

0

.

W12502 JIANG ET AL. : AGE DISTRIBUTION AROUND STAGNATION POINTS W12502

7of14

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, hydro-

geochemistry 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.

4.3. Numerical Model

[

32] In analyses of groundwater ﬂow systems, available

data are often insufﬁcient to accurately determine the ele-

vation 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 To

´

th, 1989; Sykes et al., 2009 ; To

´

th, 1963].

The 2-D steady state ﬂow assumption has also been com-

monly used in regional-scale groundwater ﬂow and trans-

port studies such as Bethke et al. [1999], Castro et al.

[1998], Castro and Goblet [2003]. Because of limited histor-

ical data, we assume that groundwater ﬂow is in steady state

in our study area. The steady state groundwater ﬂow assump-

tion is an acceptable ﬁrst step toward a quantitative under-

standing 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 satu-

rated groundwater ﬂow is considered. In the current study

area, although the climate is semiarid, due to the differen-

ces in hydraulic conductivity between the Cretaceous sand-

stones 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.

[

34] The model area includes ﬁve hydrostratigraphic

units: the fault, the K

1

l ﬂuvial sandstone, the K

1

l eolian

sandstone, the K

1

h ﬂuvial sandstone, and the Quaternary

Figure 8. The conceptual model of groundwater ﬂow in cross-section A-A

0

(modiﬁed from Hou et al.

[2008b], with permission of Wiley).

Figure 7. The change in mean hydraulic conductivity

with depth in the K

1

h sandstone.

W12502 JIANG ET AL.: AGE DISTRIBUTION AROUND STAGNATION POINTS W12502

8of14

sediments (Figure 9). Note that the small K

1

lh sandstone

and Tertiary mudstones near the western end of the cross-

section, are lumped into the K

1

h 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 bounda-

ries 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 sandst ones to clays in the sandstone

aquifer system. The distribution of porosity follows the di-

vision of the model area shown in Figure 9. When the clays

are considered, the porosity of K

1

h ﬂuvial sandstone, K

1

l

ﬂuvial sandstone and K

1

l eolian sandstone is 19.3%,

18.9%, and 27.8%, respectively. No measurements of po-

rosity of the fault and the Quaternary sediments are avail-

able. The porosity of the fault is assum ed to be larger than

that of the K

1

h ﬂuvial sandstone and K

1

l ﬂuvial sandstone,

while the porosity of the Quaternary sediments is assumed

to be larger than that of the K

1

l eolian sandstone. The po-

rosity 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 horizon-

tal hydraulic conductivity (K

x

) is mainly determined by the

sandstones and the vertical hydraulic conductivity (K

z

)by

the clays. Although in situ measurements of transmissivity

are available, they cannot be directly applied in the numeri-

cal model due to the existence of clays, which impose

uncertainties on the effective thickness of the aquifer, thu s

the calculation of K

x

. Here, the values of K

x

and anisotropy

ratio (K

x

/K

z

) are inversely determined. The value of K

x

/K

z

is assumed to be constant within each hydrostratigraphic

unit. In the hydrostratigraphic unit of K

1

h sandstone, the

depth-decaying hydraulic conductivity shown in Figure 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

14

C transport models, the longi-

tudinal 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 cali-

bration 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

14

C in the K

1

l sandstone

part of B2 and B7. After model calibration by the trial and

error method, the horizontal hydraulic conductivity, K

x

,of

K

1

l ﬂuvial sandstone is found to be 1.4 m d

1

,ofK

1

l eolian

sandstone is found to be 2.1 m d

1

. The initial K

x

(horizon-

tal hydraulic conductivity at or near ground surface) of K

1

h

ﬂuvial sandstone was determined at 0.8 m d

1

. The values

of K

x

/K

z

of K

1

l ﬂuvial sandstone, K

1

l eolian sandstone, and

K

1

h 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 anisot-

ropy ratio could be of the order of 10

2

to 10

3

in large sedi-

mentary 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

Figure 9. The hydrostratigraphy of the model area and boundary conditions for groundwater ﬂow.

W12502 JIANG ET AL. : AGE DISTRIBUTION AROUND STAGNATION POINTS W12502

9of14

sandstone aquifer system, the K

x

of Quaternary sediments

is set to be 10 m d

1

and that of the fault is set to be

4md

1

. Because they are relatively unconsolidated, the

K

x

/K

z

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 hetero-

geneity and anisotropy of hydraulic conductivity is well

characterized. During calibration, we found that the rate of

decrease in hydraulic head with depth in the K

1

h sandstone

is sensitive to the anisotropy ratio, suggesting that the hy-

draulic head measured at different depths of a borehole

could be useful in estimating this parameter.

[

41] Groundwater were sampled at different parts of

boreholes B2 and B7, which were screened at different

depths. The

14

C ages, which were measured by the IAEA

using the method of liquid scintillation counting, in the K

1

l

sandstone part of B2 and B7 were used for model calibra-

tion. According to the conceptual model of groundwater

ﬂow shown in Figure 8, groundwater mainly ﬂows horizon-

tally in the K

1

l sandstone part of B2, and is near a regional

divergent stagnation point in the K

1

l sandstone part of B7.

The

14

C ages in the K

1

l sandstone part of B2 and B7 are

21,400 years and 19,110 years, respectively (Figure 11c).

The simulated age in the K

1

l 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 K

1

l sandstone, which equals

19,110 years.

[

42] Groundwater sampled from the K

1

h 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 K

1

h 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 ran ging 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 in Figure 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 consti-

tutes 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], To

´

th

[1988], and Jiang et al. [2011], groundwater around stagna-

tion points has the characteristics of potentiometric mini-

mum. 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 ground-

water 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 re-

gional 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 re-

gional ﬂow system. SP 2 east of the Sishi Ridge also

divides four ﬂow systems, including three local ﬂow sys-

tems and one regional ﬂow system. Moreover, due to the

differences in basin depth, the locations of the two stagna-

tion 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 sys-

tem, 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

Figure 10. The measured and simulated hydraulic head

at different depths of B15.

W12502 JIANG ET AL.: AGE DISTRIBUTION AROUND STAGNATION POINTS W12502

10 of 14

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

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).

W12502 JIANG ET AL. : AGE DISTRIBUTION AROUND STAGNATION POINTS W12502

11 of 14

age, while larger dispersivity would result in much younger

waters.

4.5. Discussions

[

49] The characteristics of dynamics and age of ground-

water around the local stagnation point SP 1 in Figure 11b

are discussed below. This point is chosen because borehole

B2, where measurements of

14

C age are available, might be

within the zone of inﬂuence of SP 1. Figure 12 shows 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 pre-

cisely divide the four ﬂow systems.

[

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 K

1

h and K

1

l,

groundwater age decreases to a certain value.

[

51] In B2, the

14

C age in the lower part of K

1

h sand-

stone is measured to be 26,060 years, which is several thou-

sands years older than groundwater in the K

1

l sandstone

(21,400 years). In our calibrated model (

L

¼ 100 m), the

maximum age in the lower part of K

1

h sandstone is about

24,260 years, which is almost 3000 years older than the

simulated age in the K

1

l sandstone (Figure 13b). If a

smaller dispersivity

L

¼ 30 m is used, the maximum age

in the lower part of K

1

h sandstone is about 7000 years older

than the simulated age in the K

1

l sandstone. If a larger dis-

persivity

L

¼ 300 m is used, the maximum age in the

lower part of K

1

h sandstone is still about 2000 years older

than the simulated age in the K

1

l 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 stag-

nant zones around local stagnation points. If an extremely

large dispersivity is used, however, it is possible that accu-

mulation 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 700800 m.

5. Conclusions

[54] We analyzed groundwater ﬂow systems and ground-

water age in cross-sections of drainage basins of varying

depth. The characteristics of groundwater age around stagna-

tion points are emphasized. We ﬁnd that age mass can accu-

mulate around stagnation points. In basin s where local,

intermediate and regional ﬂow systems are all well developed,

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).

W12502 JIANG ET AL.: AGE DISTRIBUTION AROUND STAGNATION POINTS W12502

12 of 14

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 distan-

ces 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 meas-

ured in the K

1

l sandstone of two boreholes. Due to the dif-

ference 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 measured

14

C 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 ground-

water ﬂow, such as interpreting tracer age and hydrochemi-

cal patterns, and exploration of mineral deposits or

petroleum. In the future, more ﬁeld work is needed to fur-

ther demonstrate the accumulation of transported matter due

to stagnancy of groundwater and to obtain basin-scale val-

ues for dispersivity.

[57] Acknowledgments. This study was supported by China Geologi-

cal 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 Jo´zsef To´th as a reviewer for their valuable comments that

have signiﬁcantly enhanced the quality of this manuscript. The authors

also thank Associate Editor, Daniel Ferna`ndez-Garcia, and Editor, Graham

Sander, for their constructive suggestions.

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