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Fractal Behaviors of Hydraulic Head and Surface Runoff of the Nested Groundwater Flow Systems in Response to Rainfall Fluctuations

Abstract and Figures

The responses of the nested groundwater flow systems (NGFS) to rainfall fluctuations and the impacts of the NGFS on surface runoff are explored using a fully coupled variably saturated groundwatersurface water model in conjunction with spectral analysis. Numerical experiments are designed to investigate the fractal behaviors of hydraulic head and surface runoff in different scenarios. The results show that the inlets, outlets, and flow patterns of the sub-systems of NGFS can change with rainfall. The scaling-exponent, defined as the slope of the power spectra of the fluctuations of hydraulic head or runoff, is used to depict the fractal behaviors. The scaling-exponent of the hydraulic head is highly variable within the unsaturated zone and local flow systems but remains constant within the intermediate and regional systems. Whether the scaling-exponent of surface runoff exhibits a linear relation with the watershed size depends on the proportion of baseflow in the runoff.
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1. Introduction
Groundwater is fundamental to connecting surface water, unsaturated zone, and climate, transporting chemicals
and energy, and sustaining the ecosystems (Tóth,1999). Groundwater flow in a river watershed is usually in
the form of nested groundwater flow systems (NGFS; Gleeson & Manning,2008; Wörman etal.,2006; Zlotnik
etal.,2011). Studies on NGFS can be traced back to the 1960s. When the water table consists of several crests and
troughs, the groundwater circulation occurs in the form of hierarchical NGFS (Tóth,1963). Since the 1960s, hy-
drogeologists have frequently assumed that the water table is a subdued replica of the ground topography (Carde-
nas,2007; Jiang etal.,2009, 2010; Zhang etal.,2020). Nevertheless, the water table is subject to fluctuations
from different time scales, for example, diurnal, seasonal, inter-annual, or even long-term climatic fluctuations.
In this case, a steady-state solution of the flow fields does not suffice.
Some researchers have examined the response of the NGFS to harmonic periodically oscillating water table
(Vandenberg, 1980; Zhao etal., 2018). They found that flow fields and recharge and discharge rates change
regularly with this periodical upper boundary. However, these studies on NGFS under transient conditions are
based on the assumptions that the upper boundary is a harmonic periodically oscillating water table, and that
the processes in the unsaturated zone exert little influence on the saturated groundwater flow. In fact, the water
table and flow patterns are significantly controlled by the ratio of rainfall recharge rate to hydraulic conductivity
Abstract The responses of the nested groundwater flow systems (NGFS) to rainfall fluctuations and the
impacts of the NGFS on surface runoff are explored using a fully coupled variably saturated groundwater-
surface water model in conjunction with spectral analysis. Numerical experiments are designed to investigate
the fractal behaviors of hydraulic head and surface runoff in different scenarios. The results show that the inlets,
outlets, and flow patterns of the sub-systems of NGFS can change with rainfall. The scaling-exponent, defined
as the slope of the power spectra of the fluctuations of hydraulic head or runoff, is used to depict the fractal
behaviors. The scaling-exponent of the hydraulic head is highly variable within the unsaturated zone and local
flow systems but remains constant within the intermediate and regional systems. Whether the scaling-exponent
of surface runoff exhibits a linear relation with the watershed size depends on the proportion of baseflow in the
runoff.
Plain Language Summary This study investigates the responses of the unsaturated zone,
groundwater flow systems, and surface runoff to rainfall fluctuations. The results reveal that the inlets and
outlets of the sub-systems can change with rainfall fluctuations. This has strong implications for understanding
the sources and destinations of solutes in an aquifer because these sub-systems are hidden channels of water
flow and also conveyors for chemicals or pollutants. The method to separate the sub-systems under steady-state
conditions is not applicable under transient conditions. Our study finds that the vertical distributions of the
fractal behaviors of the hydraulic heads can be used to separate sub-systems of the groundwater flow systems
under transient conditions. The proportion of the baseflow in the surface runoff can determine whether the
fractal behaviors of the runoff are linearly related to the watershed size. This can explain the controversy in
previous studies that the fractal behavior of river runoff can have either positive linear relation or no relation
with its watershed size. The spectral analysis is proven to be a useful way of analyzing the hydraulic features of
groundwater and surface runoff under transient conditions.
ZHANG ET AL.
© 2022. American Geophysical Union.
All Rights Reserved.
Fractal Behaviors of Hydraulic Head and Surface Runoff of
the Nested Groundwater Flow Systems in Response to Rainfall
Fluctuations
Xiaolang Zhang1,2 , Hailong Li2, Jiu Jimmy Jiao1 , Xin Luo1 , Xingxing Kuang2 , Rong Mao1, and
Wenli Hu1
1Department of Earth Sciences, The University of Hong Kong, Hong Kong, China, 2School of Environmental Science and
Engineering, Southern University of Science and Technology, Shenzhen, China
Key Points:
Fractal behaviors of hydraulic
head and surface runoff of nested
groundwater flow systems are
explored
The fractal behavior of the hydraulic
head can be used to separate the sub-
systems of groundwater flow systems
under transient conditions
The proportion of baseflow in surface
runoff can determine the relation
between fractal behavior of the surface
runoff and watershed size
Supporting Information:
Supporting Information may be found in
the online version of this article.
Correspondence to:
J. J. Jiao,
jjiao@hku.hk
Citation:
Zhang, X., Li, H., Jiao, J. J., Luo, X.,
Kuang, X., Mao, R., & Hu, W. (2022).
Fractal behaviors of hydraulic head and
surface runoff of the nested groundwater
flow systems in response to rainfall
fluctuations. Geophysical Research
Letters, 49, e2021GL093784. https://doi.
org/10.1029/2021GL093784
Received 19 APR 2021
Accepted 2 JAN 2022
Author Contributions:
Conceptualization: Xiaolang Zhang,
Rong Mao, Wenli Hu
Investigation: Xiaolang Zhang, Jiu
Jimmy Jiao, Xingxing Kuang
Methodology: Xiaolang Zhang, Rong
Mao, Wenli Hu
Supervision: Hailong Li, Jiu Jimmy Jiao,
Xin Luo
Validation: Xiaolang Zhang, Xingxing
Kuang
Writing – original draft: Xiaolang
Zhang
Writing – review & editing: Hailong Li,
Jiu Jimmy Jiao, Xin Luo
10.1029/2021GL093784
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(Liang etal.,2013), and the ratio is regulated by multi-timescale climate fluctuations. The thickness, moisture
content, and hydraulic conductivity of the unsaturated zone also change with rainfall fluctuations (Gleeson &
Manning,2008; Haitjema & Mitchell-Bruker,2005). The unsaturated zone can work as an additional stagnation
zone and water particles would undergo a long time lag before reaching the water table (Kollet & Maxwell,2008).
To the best of our knowledge, there is no systematic study on quantifying the temporal hydraulic changes of the
NGFS in response to rainfall fluctuations and the impacts of the NGFS on the surface runoff.
The temporal fluctuations of hydrological variables, including soil water content, hydraulic head, and river run-
off are usually correlated (Bellin etal.,2015; Cardenas,2008; Ji etal.,2016; Kantelhardt etal.,2006; Kirchner
& Neal, 2013; Zhang and Schilling, 2004). The spatial connectivity of fractal behaviors of the hydrological
variables is often employed to describe the complexities in hydrological processes (Hirpa etal.,2010; Rakhshan-
dehroo & Mehrab Amiri,2012). The slope of the power spectra of a hydrological variable has been used as an
indicator or measurement of stationarity of a time series (Davis etal.,1994). If the slope of the spectra is less than
1, the time series is stationary fractional Gaussian noise (fGn); if the slope is higher than 1, it is nonstationary
fractional Brownian motion (fBm; Davis etal.,1994). The previous study has noticed that the value of the slope
increases when water travels from rainfall to recharge, and then to discharge in a groundwater flow system (Yang
etal.,2017). However, the power spectra have rarely been employed to examine the responses of the water con-
tinuum of unsaturated zone-NGFS-surface runoff to rainfall fluctuations.
The aim of this study is to construct a two-dimensional variably saturated groundwater/surface water flow model
to investigate the fractal behaviors of hydraulic head and surface runoff of the NGFS in response to rainfall fluc-
tuations. Rainfall fluctuations are assigned as the driving force of the transient conditions in the fully coupled
system. We explored the temporal hydraulic changes of the NGFS in response to rainfall fluctuations and the im-
pacts of the NGFS dynamics on the surface runoff. Spectral analysis was utilized to depict the fractal behaviors of
the fluctuations of hydraulic head and surface runoff. Sensitivity analysis is carried out to examine other possible
causative factors on the fractal behaviors of hydraulic head and surface runoff. We further investigated the effects
of aquifer properties and climate fluctuations on the filtering processes of rainfall signals.
2. Methodology
2.1. Surface Topography
The topography of the ground surface consists of local topographic undulations and regional slope (see Figure
S1 in the Supporting InformationS1). The local topographic undulations of 11 para-curves can be defined and
generated using Equation1, and corresponding parameters are shown in Table S1 in Supporting InformationS1.
()=0+()2++
(1)
where x ranges from 0 to 6,727.11m, Z0 equals 500m; β is the regional topographic slope and equals 0.002; Be-
cause the local topographic undulation consists of 11 para-curves, a, b, and c are used to define these para-curves.
The values of a, b, and c of each para-curve are shown in Table S1 in Supporting InformationS1. The above
model setup is for Case 1.
There are six mountains (M1 to M6) and six valleys (Figure1b). These valleys are all potential groundwater dis-
charge zones and six streams (S1 to M6) can be developed at the bottom of the valleys. The summed horizontal
distances between the stream to adjacent mountains are defined as the capture zone length. There is no increasing
or decreasing trend of the capture zone length along the x-axis.
2.2. Fully Coupled Groundwater-Surface Water Modeling
Numerical modeling is conducted using HydroGeoSphere (HGS; Brunner & Simmons,2012). HGS solves the
Richards equation for subsurface flow and wave approximations of the Saint Venant equation for surface flow
simultaneously, in one fully integrated globally implicit matrix. A critical depth boundary is applied at the six
valleys (Goderniaux etal.,2009). The model domain is horizontally discretized at a 13.45m resolution. Vertical-
ly, the mesh has 100 layers for subsurface flow and one nodal sheet at the top for surface flow. For the uppermost
200 m of the subsurface where local groundwater flow changes instantaneously and variably saturated flow
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exists, vertical node spacing is set to be 2.53m. The underlying subsurface layers gradually increase in thickness.
Overall, the integrated surface-subsurface finite element mesh consists of 50,000 elements and 50,601 nodes in
the xz plane. A 40-year time series of monthly rainfall (June 1974 to December 2014) in Hong Kong is applied in
the warm-up period until a natural dynamic state is obtained. After that, the hydraulic head distribution and sur-
face water depth at the end of the warm-up period are used as the initial conditions of the running period. Case 1
is the base case and Cases 2–8 are designed to explore the effects of local and regional topographic gradients, hy-
draulic conductivity, specific storage, rainfall rate, and evapotranspiration on the NGFS dynamics (see Table1).
Actual evapotranspiration is computed using the model suggested by Kristensen and Jensen(1975) based on the
potential evapotranspiration in Hong Kong. In each case, the above-mentioned warm-up and running period are
performed. The simulated flow patterns are presented in Figure2 and S3-S9 in Supporting InformationS1, at
which the x-axis is normalized to the domain length and the z-axis to the domain depth.
2.3. Spectral Analysis of the Time Series Data
In Case 1, a total of evenly distributed 6,700 points in the subsurface domain are used to monitor the variations
of hydraulic head in response to rainfall. Four vertical profiles below (green dashed lines in Figure1b) are used
to monitor the variations of the hydraulic head among Cases 1–8. At the same time, the runoff in all six streams
is monitored and normalized to the average rainfall rate (Figure S10 in Supporting Information S1). Spectral
analysis of the monthly hydraulic head and surface runoff is carried out to determine the temporal scaling of the
two variables. The power spectra are calculated using the conventional Fast Fourier Transformation, which de-
composes the time series data into multi-components (Figures1c and1d). The spectrum (S)-frequency (f) relation
of each time series of the hydraulic head or surface runoff is displayed in the log-log scale. The slope of the S-f re-
lation is defined as the scaling-exponent (n). During the hydrological processes, components of different frequen-
Figure 1. (a and b) A conceptual model for the problem. The green dashed lines in Figure1b indicate 4 monitoring profiles. (c) Rainfall fluctuation and its spectral; (d)
Fluctuations of hydraulic head in different depth (at x/L=0.1) and their spectral of the Case 1.
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Description Features of the flow patterns
Separation depth (z/Z0, [-]) between
segments 1 and 2 in the vertical
distributions of the scaling-exponent of the
hydraulic head
Ratio of base
flow [-] in the
stream runoff
Capture zone length
versus flow rates
Capture zone length
versus scaling-
exponent of stream
runoff
Case1 Case 1 is the base case, whose topography
is defined using Equation1. K=1m/d,
Ss=0.001
Sub-systems change with rainfall 0.46 0.39 R2=0.31, p=0.55 R2=0.09, p=0.86
Case2 Regional topographic gradient is 0.5 times
of Case 1.
Sub-systems change with rainfall 0.4 0.38 R2=0.63, p=0.18 R2=0.01, p=0.98
Case3 Local topographic gradient is 0.8 times of
Case 1.
Sub-systems change with rainfall 0.48 0.48 R2=0.25, p=0.63 R2=0.11, p=0.83
Case4 K is reduced to 0.001m/d. Sub-systems are unchanged 0.58 0.97 R2=1.0, p<0.0001 R2=0.69, p=0.13
Case5 Ss is reduced to 0.0001. Sub-systems change with rainfall 0.62 0.41 R2=0.25, p=0.63 R2=0.08, p=0.88
Case6 Rainfall is reduced to 0.5 times of Case 1 Sub-systems change with rainfall 0.58 0.21 R2=−0.08,
p=0.89
R2=0.40, p=0.44
Case7 Rainfall is reduced to 0.05 times of Case 1 One regional flow system, one
perennial, and one seasonally
occurring local flow system
0.96 1.00 n.a. n.a.
Case8 Evapotranspiration is considered Sub-systems change with rainfall 0.47 0.30 R2=0.66, p=0.05 R2=0.04, p=0.72
Note. R2 is the coefficient of determination. p is the P-value. n.a. means not available.
Table 1
Description, Results, and Correlation Analysis of Cases 1–8
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cies of a spectrum will be successively filtered and removed, which ultimately leads to the change of n. Generally,
the high frequency components with low amplitude are easy to be removed, leading to the increasing of n; the
low frequency components with large amplitude are difficult to be removed, and if removed, n will decrease. The
spatial distribution of n in Case 1 is shown in detail in the contour map (Figure3a). The impacts of capture zone
size on surface runoff and the fractal behaviors of surface runoff are examined via regression analysis (Table1).
3. Results and Discussion
3.1. Evolution of the Sub-Systems and Internal Stagnation Points
Figures2a–2d show the evolutions of the flow patterns of Case 1 with rainfall. Stagnation point (SP), defined
as the point where groundwater velocity equals zero, is used to separate the sub-systems of the NGFS (Jiang
etal.,2011). Under steady-state conditions, intermediate flow systems are defined as those whose flow lines orig-
inate from the crests, extend over one or more crests and troughs, and then discharge to the troughs, similar to those
occurring in June and December (Figures2a and 2c). Nevertheless, this definition is not appliable for transient
conditions because intermediate flow systems may originate from the place where storage water is released, or
Figure 2. (a–d) Seasonal changes of the flow patterns of Case 1 in response to rainfall fluctuations. Panels (e and f) are the
insets of (a) and (d), respectively. Darcy velocity magnitude Log(V) (|V|, m/day) contours (in color bands), streamlines (black
solid lines), velocity direction (black arrows with uniform length), and transient SP (in yellow circle).
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terminate at the location where the storage is increased. For example, in September, the intermediate flow sys-
tems replenish the aquifer storage (Figure2b); in March, water in the intermediate flow systems is released from
the aquifer storage (Figure2d). Actually, the inlets and outlets of the intermediate flow systems change seasonally
under transient conditions. Similar results can also occur in other cases (Table1). This phenomenon is important
because the intermediate flow systems are hidden channels for not only groundwater discharge (Robinson &
Love,2013) but also for chemical or pollutant migration.
Two new types of internal SPs are found under transient conditions. One is below the highest or lowest local flow
systems (L1 or L11) and the other is below the water table. In March, when the downward-moving local flow
system L11 below M6 meets the upward-moving intermediate flow system I5, the transient internal SP occurs
(see the yellow circle in Figures2e and2f). This SP, however, will disappear in other months. Similarly, in a rainy
month, when the upward-moving local flow system below S1 meets the downward-moving intermediate flow
Figure 3. (a and b) Spatial and vertical distributions of the n of the hydraulic head of Case 1, respectively; (c) Vertical
distributions of the n of hydraulic head in the 4 vertical profiles in Figure1b of Cases 1–8.
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system, the transient internal SP may also occur, but disappear in other seasons. Another type of transient SP
occurs because the thickness of the unsaturated zone increases from the streamside to the mountain top. In rainy
seasons, especially when heavy rain occurs, the streamside water table increases more rapidly than that under
the mountain top. Hence, the water level in the streamside is temporally higher than that below the mountain
top, leading to the groundwater flow from the streamside toward the watershed divide and the formation of this
type of transient SP. These SPs of transient and ephemeral nature are reported in this study for the first time.
This is because previous studies on NGFS mostly focused on the steady-state condition, without considering the
changes of hydraulic conductivity and storage in the unsaturated zone (Freeze & Witherspoon,1966,1967; Jiang
etal.,2009; Liang etal.,2013).
3.2. Fractal Behaviors of Hydraulic Head
Figure3a shows the spatial distribution of n of the hydraulic head in Case 1. As presented in the inset, n first
increases then decreases along the flow line in the shallow aquifer, indicating that n is modified during the
processes of recharge and discharge. Below the shallow part of the aquifer, n first increases downward and then
remains constant. Previous studies have found that the aquifer can work as a fractal filter and the n of hydraulic
heads decreases toward the stream in a sloping aquifer, as concluded by Zhang and Schilling(2004) and Zhang
and Li(2005). Their finding uncovered the changing trend of n adjacent to the streams. In this study, a more
complete picture of the spatial distribution of n in the NGFS is revealed.
The vertical distribution of n of hydraulic heads is shown in Figure3b. Generally, n displays two segments
vertically. Segment 1 is corresponding to the influence sphere of the variably saturated zone and the local flow
systems. The unsaturated zone has a damping effect on the movement of water particles, which can dissipate
the energy stored in the fluctuation and delay the time for the water particles to move down-gradient (Yang
etal., 2017). The moisture in the unsaturated zone either evaporates, or flows down-gradient, or is held in the
pores by capillary forces. During rainy seasons, the gravity flow is generated once the water content is sufficiently
high. After the rainy season, the gravity flow will still sustain for a while until the capillary forces are compa-
rable to the gravity. In such wetting and drying processes, the crests and troughs of the rainfall fluctuations are
smoothened and delayed in the unsaturated zone. The damping effect in local flow systems is relatively weaker
because the saturated zone has a higher hydraulic conductivity and thus a less significant damping effect than
the unsaturated zone. Hence, in Segment 1, n first increases rapidly with z/Z0, then increases relatively slowly.
The n of the rainfall fluctuation is 0.07, which means that the rainfall is an fGn signal. Segment 1 shows that
the unsaturated zone and local flow systems can transfer an fGn signal in rainfall to an fBm signal. The time lag
between the fluctuations of rainfall and the hydraulic head mainly depends on the hysteresis of the movements of
water particles during the wetting and drying processes. When the rainfall signals are propagated to the saturated
zones, the time lag is significantly shortened because the pressure perturbation is transmitted via the aquifer's
elastic skeleton and pore water.
Segment 2 is corresponding to the influence sphere of the intermediate and regional flow systems. The hydraulic
head in the intermediate and regional flow systems are subjected to both the local and regional pressure pertur-
bations. Therefore, when the local pressure perturbations are transmitted from the local flow systems to the in-
fluence spheres of the intermediate and regional flow systems, the energy of water particles is averaged with that
of ambient particles, which are more inertial. At this segment, the fluctuations of the hydraulic head are further
filtered and smoothened, and n remains almost constant with z/Z0. This suggests that the vertical distributions of
n of the hydraulic head can be used to separate the local flow systems from the intermediate and regional ones.
The filter processes of rainfall signals in the NGFS not only depend on the aquifer properties, for example, hy-
draulic conductivity and specific storage but also on the rainfall fluctuations. Fluctuations of the hydraulic head
along four vertical profiles (see Figure1b for locations of the profiles) are observed in all 8 cases. As shown in
Figure3c, Cases 1–8 present three different types of signal filter processes due to damping effects to different
degrees. For the first type, in Cases 1, 2, 5, 6, and 8, the high frequency components are greatly removed by the
aquifer while the low frequency counterparts are only slightly removed. Thus, n first monotonically increases
downward and then remains constant in these cases. For the second type, in Cases 3 and 4, after the high frequen-
cy components are greatly removed in the shallow part of the aquifer, the low frequency counterparts are also
greatly removed when the pressure perturbations are transmitted further downward. Therefore, n first increases,
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then decreases downward, and finally remains constant in Cases 3 and 4. However, this transition depth at which
low frequency components start to be significantly removed is different in these two cases. The transition depth is
smaller in Case 4 than in Case 3, indicating that the aquifer setting in Case 4 results in a stronger damping effect.
For the third type, the amplitude of the rainfall fluctuations in Case 7 is significantly reduced due to the fact that
the rainfall is 0.05 times of Case 1. At the very beginning of rainfall infiltration, both low and high frequency
components of the rainfall signal are removed by the aquifer. Hence n first decreases downward and then remains
constant in Cases 3 and 4.
The vertical distribution of n of the hydraulic head may be used to separate the sub-systems of the NGFS. It
works well when using SP to separate the sub-systems of the NGFS under steady-state conditions. The fact that
the NGFS change significantly with rainfall fluctuations in most cases makes it hard to separate the sub-systems
using changing SPs under transient conditions. The vertical distribution of n of the hydraulic head is suggested to
separate the local flow systems from the intermediate and regional ones under transient conditions. Here Case 4
is used to validate this method because its flow patterns are nearly unchanged with rainfall. The deepest locations
of local flow systems in Figure S5 in Supporting InformationS1 are consistent with the transition depth between
Segments 1 and 2 in Figure3c. Case 4 confirms the validity of using vertical distributions of n to separate the
local flow systems from the intermediate and regional ones. If the fluctuations of hydraulic heads in shallow and
deep wells are monitored, which is fairly common in major aquifers worldwide, fractal behaviors of these water
level data can be used to explore the nature and characteristics of NFGS in a basin.
3.3. Fractal Behaviors of the Surface Runoff
Fluctuations of the stream runoff in the six valleys of Cases 1–8 are monitored (Figure S10 in Supporting In-
formation S1) and decomposed to multi-components via Fourier Transformation (Figure S11 in Supporting
InformationS1). Stream runoff consists of stormflow and baseflow. The results show that the percentage of base-
flow in the stream runoff is different in these cases (Table1). The proportion of baseflow in Case 5 is the largest
with a value of 0.97 and the smallest in Case 8 with a value of 0.21. Because the hydraulic conductivity in the un-
saturated zone can change over several orders of magnitude (van Genuchten,1980), the existence of the unsaturated
zone can reduce the rainfall infiltration to the groundwater. Though the saturated hydraulic conductivity in Case 4
is the smallest among these cases, rainfall is infiltrated rapidly due to the almost fully saturated subsurface domain.
The controls of capture zone size on the stream runoff and fractal behaviors of stream runoff are also different
among these cases. The capture zone length and flow rates exhibit a highly significant positive relation in Case 4,
a less significant positive relation in Case 8, and no apparent relation in other cases. Thus, the capture zone size
not always has a significant impact on the surface runoff. These numerical results are consistent with Shaman
etal.’s(2004) data-driven conclusion that capture zone size is not the sole controlling factor and many other fac-
tors can influence the stream runoff, such as topography, evapotranspiration, and rainfall. The intermediate and
regional flow systems play an important role in transfer the water from one basin with higher elevation to another
basin with lower elevation. These inter-basin flows will weaken the topographic control on the stream runoff. In
Cases 4 and 6, the capture zone length and n of stream runoff present a weak positive relation. Nevertheless, n of
the stream runoff in other cases has no identifiable relation with the capture zone length. If baseflow and storm
flow are comparable, the relation between the capture zone length and n of stream runoff will be very weak.
When baseflow is much smaller or greater than storm flow, the relation between the capture zone length and n
of the runoff is strengthened.
The fractal behaviors of river runoff are often used to capture the properties of runoff fluctuations because these
data are directly related to flood or drought frequency (Dey & Mujumdar,2018; Hirpa et al., 2010; Mudel-
see,2007). When examining the relation of the fractal behaviors of the river runoff with drainage area, however,
some researchers reported that the fractal behaviors depend on the watershed size (Hirpa etal.,2010), but some
others failed to observe this dependence (Tessier etal.,1996). An increasing n of the stream runoff with the wa-
tershed size indicates that the runoff fluctuations from larger watershed sizes are more persistent. This means that
high-intensity runoff is more likely to occur in a large watershed (Hirpa etal.,2010). On the one hand, the signal
in the stream runoff comes from rainfall and baseflow. The latter is damped by the unsaturated zone and aquifer
systems, leading to the out-sync and interference of the two signals. In streams that are solely dominated by base-
flow or stormflow, the fractal behaviors of stream runoff are likely to depend on the watershed size. On the other
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hand, a larger watershed can also increase the rainfall contribution to the intermediate flow, which is ultimately
discharged to the low-lying streams. Local and intermediate flow systems are characterized by different ground-
water residence times (Jiang etal.,2012). The two asynchronous signals in the baseflow can also interfere with
each other. Thus, when investigating the relation of the fractal behaviors of the river runoff with the watershed
size, the groundwater flow patterns should be considered.
4. Summary
Using a fully coupled model, this study explores the responses of the water continuum of unsaturated zone-ground-
water-surface runoff to rainfall fluctuations. The results show that the sub-systems of the NGFS change season-
ally in most cases. Spectral analysis on the variations of hydraulic head and surface runoff is carried out to ex-
tract the dynamic features of NGFS and surface runoff in response to rainfall fluctuations. The spectrum results
demonstrate that the NGFS plays an important role in transferring an fGn signal in rainfall to an fBm signal in
groundwater and surface runoff. The vertical distributions of the fractal behaviors of the hydraulic head can be
utilized to separate the sub-systems of the NGFS. When the surface water is mainly recharged by either base flow
or storm flow, the fractal behaviors of the surface runoff is likely correlated with the watershed sizes.
Data Availability Statement
The input files and output results of the numerical models are available online (http://doi.org/10.5281/zenodo.
4672136).
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Acknowledgments
This study was supported by grants from
the National Natural Science Foundation
of China (Grant Nos. 41572208 and
91747204) and the Research Grants
Council of Hong Kong (HKU 17304815
and 17307620). The editor and anony-
mous reviewers are thanked for their fruit-
ful comments on the original manuscript.
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... A reason for the deviation might be monthly to weekly fluctuations in river height of the surrounding streams (e.g., the Rauhe Ebrach, south of Birkach, see supporting information for a hydrograph), the short term lateral inflow or highly fractured karstic materials which is not captured by the theoretical spectrum (Equation 8). Furthermore, the shape of the spectrum can be influenced by the superposition of local and intermediate flow systems, wetting and drying in partly saturated shallow aquifers (Zhang et al., 2022) as well as other hydrogeological processes. ...
... How large the representative volume of the aquifer characterized by the spectral analysis finally is, depends on the hydrogeological setting and its heterogeneity. Besides that, also topography, nested flow systems, and vertical hydraulic gradients will change the shape of the power spectra of hydraulic heads of different depths (Zhang et al., 2022). Consequently, the assumed Dupuit model may no longer be appropriate. ...
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