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Appropriate Boundary Condition for Dupuit-Boussinesq Theory on the Steady Groundwater Flow in an Unconfined Sloping Aquifer With Uniform Recharge

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The application of Dupuit‐Boussinesq theory on the two‐dimensional steady groundwater flow in an unconfined sloping aquifer with uniform recharge is theoretically investigated. The examination of conditions at the downstream and upstream is performed. The method of phase plane is used to geometrically interpret the characteristics of the non‐linear theory by various initial conditions. For phase portrait analysis, three configurations for aquifers are considered. In each type of aquifer three conditions of recharges as well as downstream groundwater tables are also considered for comprehensive analysis. Based on the phase portrait analysis, one explicit formula in terms of known parameters and downstream boundary condition is derived to easily and straightforwardly determine an appropriate upstream boundary condition. The derived explicit formula can facilitate correct modelling using Boussinesq theory.
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Water Resources Research
Appropriate Boundary Condition for Dupuit-Boussinesq
Theory on the Steady Groundwater Flow in an
Unconfined Sloping Aquifer With
Uniform Recharge
Ying-Hsin Wu1, Takahiro Sayama1, and Eiichi Nakakita1
1Disaster Prevention Research Institute, Kyoto University, Uji, Kyoto, Japan
Abstract The application of Dupuit-Boussinesq theory on the two-dimensional steady groundwater
flow in an unconfined sloping aquifer with uniform recharge is theoretically investigated. The examination
of conditions at the downstream and upstream is performed. The method of phase plane is used to
geometrically interpret the characteristics of the nonlinear theory by various initial conditions. For phase
portrait analysis, three configurations for aquifers are considered. In each type of aquifer three conditions
of recharges as well as downstream groundwater tables are also considered for comprehensive analysis.
Based on the phase portrait analysis, one explicit formula in terms of known parameters and downstream
boundary condition is derived to easily and straightforwardly determine an appropriate upstream boundary
condition. The derived explicit formula can facilitate correct modeling using Boussinesq theory.
Plain Language Summary Groundwater is an important component affecting the human life and
ecosystem on aspects of drinking water supply, food production, and many environmental processes. As
is stored underground, groundwater distribution is always a puzzle to find out. In a hillslope groundwater
table information can be measured at a certain location by monitoring techniques, but information uphill
nearby is not directly accessible. For possibly knowing the unknown groundwater table, one of conventional
ways is computation on sophisticated groundwater theories with some field parameters. However, as
requiring high-level knowledge in computation as well as groundwater theories, correct calculation is
not easily achievable without expertise in both. To overcome this difficulty, with careful mathematical
investigation, we propose a simple formula to judge whether a steady groundwater table exists at the
upside of a sloping and soil-mantled hillslope in terms of parameters of hillslope inclination, rain rate,
and hydrological property of hillslope soil. This formula can intuitively facilitate correct assessing of
groundwater table.
1. Introduction
Modeling gravity-driven flow in a porous media has been intensely studied for several decades in many
aspects, for example, coastal engineering, geotechnical engineering, industrial process, and hydrology (e.g.,
Brutsaert, 1994; Huppert, 2006; Lambe & Whitman, 1979; Liu & Wen, 1997). In the hydrology field, ground-
water flow in an unconfined aquifer is of great importance on water resource management. Particularly, the
flow process in a soil layer on hillslope can influence slope stability and further induce mass wasting and land-
slide hazards. As more hazards were triggered by extreme rainfall on slopeland recently, understanding the
response of an unconfined sloping aquifer on intense rainfall is certainly important for accurate assessment
of hillslope stability.
To model groundwater flow in an unconfined sloping aquifer, one of conventional theories is the hydraulic
groundwater theory(Brutsaert, 2005), or called Dupuit-Forchheimer theory or Dupuit-Boussinesq theory in
some literatures(e.g., Bear, 1972; Guérin et al., 2014; Harr, 1990). The main feature of this theory is to assume
shallow flow motion of groundwater flow in an aquifer. The shallow flow obeys hydrostatic pressure. Thus,
the theory can be simplified by reducing the dependence of vertical coordinate, and infiltration process is
excluded for consideration. Although the theory itself becomes simplified, the simplification alters the the-
ory into a nonlinear one. For problems in a horizontal aquifer or without any source, the simplified theory
RESEARCH ARTICLE
10.1029/2018WR023070
Key Points:
• The nonlinear characteristics of
groundwater flow in an unconfined
Boussinesq aquifer are analyzed by
using the method of phase plane
• Appropriate boundary conditions
at the downstream and upstream
boundaries are examined
• An explicit formula to determine
the appropriate upstream boundary
condition is derived
Correspondence to:
Y.- H . W u,
yhwu@hmd.dpri.kyoto-u.ac.jp
Citation:
Wu,Y.-H.,Sayama,T.,&Nakakita,E.
(2018). Appropriate boundary
condition for Dupuit-Boussinesq
theory on the steady groundwater
flow in an unconfined sloping
aquifer with uniform recharge.
Water Resources Research,54.
https://doi.org/10.1029/2018WR023070
Received 5 APR 2018
Accepted 2 AUG 2018
Accepted article online 13 AUG 2018
©2018. American Geophysical Union.
All Rights Reserved.
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Water Resources Research 10.1029/2018WR023070
can be easily solved without any analytic difficulty(e.g., Bear, 1972; Brutsaert, 2005; Guérin et al., 2014; Harr,
1990; Lister, 1992; Polubarinova-Kochina, 1962; Troch et al., 2003; 2004; Vella & Huppert, 2006). However, for
problems in a steeper hillslope with rainfall recharge, the gravitational effect and the source term become
important in the theory, and to our knowledge, only one analytic solution in an implicit form has been pro-
posed (Henderson & Wooding, 1964; Schmid & Luthin, 1964) and discussed in literature(e.g., Basha & Maalouf,
2005; Beven, 1981; Chapman, 2005). Hence, the method of linearization has been widely used for finding
solutions for decades(e.g., Brutsaert, 1994; Verhoest & Troch, 2000). Recently, the transient groundwater flow
in a steep Boussinesq aquifer was numerically and analytically investigated using the full set of Boussinesq
theory (Bartlett & Porporato, 2018; Bogaart et al., 2013; Stagnitti et al., 2004) and then was applied to reces-
sion analysis (Bogaart et al., 2013; Dralle et al., 2014; Hogarth et al., 2014; Rupp & Selker, 2006). This explains
the importance of hydraulic groundwater theory on hillslope hydrology. A comprehensive review of the
application of Boussinesq theory has been summarized in great detail (Troch et al., 2013).
As mentioned previously, only numerical methods are practical for solving problems in a steep and uncon-
fined sloping aquifer using the nonlinear Boussinesq theory. When using numerical methods, correct bound-
ary conditions are certainly demanded for successful iteration algorithms. Similar issues for overland flow
have also been investigated numerically and experimentally (e.g., de Lima & Torfs, 1990; Morris, 1979; Van der
Molen et al., 1995) for some applications (e.g., Hu et al., 2014). In the overland flow modeling, velocity and
flow depth are two different physical variables, so zero volumetric discharge at the upper boundary can be
directly imposed as zero velocity without any attention on flow depth. However, in the subsurface modeling,
as seepage velocity utilizing Darcy’s law depends on groundwater table, the zero volumetric discharge can-
not be simply imposed as zero velocity, as is briefly explained in the following. In a two-dimensional sloping
aquifer having a flat bottom, with the Dupuit assumption and Darcy’s seepage, the volumetric discharge qin
the slope coordinates reads (Brutsaert, 2005)
q=−k0𝜂cos 𝛼d𝜂
dx+sin 𝛼,(1)
where 𝜂is the groundwater table, k0is the hydraulic conductivity, and 𝛼is the aquifer inclination. With
( (1)) the zero volumetric discharge at the upstream boundary yields two possible conditions that either
groundwater table or seepage velocity is 0, that is,
𝜂=0or d𝜂
dx=−tan 𝛼, (2)
as will be examined in detail in the following sections. For a correct numerical solution, the two possible con-
ditions cannot be simultaneously imposed at the same boundary. It requires some way to distinguish the
appropriate condition for a given problem. In literature, different upstream boundary conditions have been
used in different problems. Widely accepted byphysical intuition, the constant gradient of groundwater table
is usually imposed at the upstream boundary for analysis of long-time aquifer response (e.g., Hogarth et al.,
2014; Rupp & Selker, 2006; Troch et al., 2013). In some other literatures, the full equation of (1) is directly
imposed as the upstream boundary condition with linearization (e.g., Brutsaert, 1994; Dralle et al., 2014; Ver-
hoest & Troch, 2000). Particularly, without any linearization of (1), Henderson and Wooding (1964) proposed
a steady state solution as well as an implicit formula to judge the upstream boundary condition. So far, an
explicit formula to determine a proper upstream boundary condition has not been discussed yet. Therefore,
we aim to investigate this issue for finding out a new explicit and efficient way to determine the appropri-
ate boundary conditions for the Boussinesq theory. For the sake of simplicity, only the steady problem is
considered in this study.
Since the main purpose of the present study is to investigate the applicability of an upstream boundary condi-
tion, any presumption cannot be made at the upstream boundary. Under this restriction, the method of phase
plane (Jordan & Smith, 1999; Strogatz, 2015), mainly for analysis of nonlinear dynamical systems, is one of the
practical tools for analysis. The major advantage of this method is to be able to geometrically interpret the
characteristics of any given nonlinear system regarding various parameters and boundary conditions without
actually solving the system. In our problem, starting from a representative point imposed due to the down-
stream boundary condition, the method of line integration can be applied to demonstrate the evolution of a
trajectory in the phase diagram, then the end location of the trajectory can reveal the characteristics of the
upstream boundary. Hence, we can validate upstream boundary conditions and obtain some information to
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Figure 1. Sketch of our problem and definition of all dimensional variables. (x,z)is the slope coordinates, 𝜂is the
groundwater table, Iis the uniform rainfall recharge rate, 𝛼is the inclination, and Dis the groundwater table at the
downstream. Land Hare the characteristic length and height of the aquifer. x
u=Land x
d= 0 denote the upstream and
downstream locations. (a) and (b) illustrate the two conditions in (13) due to zero volumetric discharge.
judge the proper one. Up to now, the phase plane method has been widely applied in other scientific, engi-
neering, and financial fields, but not in groundwater Boussinesq theory yet. We would also like to elucidate
the applicability of the phase plane method on analyzing hydrological processes.
The content of this paper is as follows. The formulation is described in section 2. Downstream and upstream
conditions are examined in section 3. Phase portrait analysis is explained in section 4. Then, one explicit for-
mula is derived to distinguish an appropriate upstream boundary condition in section 5. Finally, conclusions
are summarized in section 6.
2. Formulation
2.1. Governing Equations
It is widely accepted that the soil layer mantling a steep and vegetated hillslope is usually thin in comparison
to the slope length. Groundwater table in the thin soil layer can be reasonably regarded to be shallow and
without large variation in the depth direction. Figure 1 illustrates the sketch and definition of our problem.
We adopt the slope coordinate system, of which the xaxis is aligned along and zaxis is perpendicularly and
upwardly directed from the surface of slope bottom. We consider a general hillslope having the characteris-
tic length L=(10)(m) and the characteristic depth H=(1)(m), where ()denotes the big O symbol
(Strogatz, 2015). Hence, we assume that
H
L=𝜖≪1.
The shallow flow assumption above gives a hydrostatic pressure distribution and the reduction of zdepen-
dence in the governing equations. As groundwater response to a rainfall event is rapid in a thin soil layer,
infiltration process is assumed to be neglected. Using Darcy’s seepage law and shallow flow assumption, the
steady groundwater flow with rainfall recharge can be approximately modeled by the Dupuit-Boussinesq
theory (Brutsaert, 2005):
d
dx𝜂d𝜂
dx+tan 𝛼d𝜂
dx=−I
k0
,(3)
where 𝜂is the groundwater table (m), k0is the hydraulic conductivity (m/s) ranging from 101to 105for
general hillslopes, 𝛼is the slope inclination (rad), and Iis the rain recharge rate (m/s), which generally ranges
from 10to 200 mm/hr to represent from a slight to an extremely intense rainfall. Equation (3) represents the
balance between mass influx/outflux and rainfall recharge. The depth-averaging specific flux, or called Darcy’s
seepage velocity, reads
u=−k0cos 𝛼d𝜂
dx+sin 𝛼,(4)
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where the minus sign ensures the rightward discharge is in the positive xdirection. With (4) the volumetric
discharge can be expressed as
q=𝜂u=−k0𝜂cos 𝛼d𝜂
dx+sin 𝛼.
Due to (3) two boundary conditions are required for a solution. A constant groundwater table is imposed at
the downstream boundary, as
𝜂=D=constant,at x=0.
Zero volumetric discharge is imposed at the upstream boundary, as below:
q=−k0𝜂cos 𝛼d𝜂
dx+sin 𝛼=0,at x=L.(5)
No simplification is made in the upstream boundary conditions. Besides, attention shall be paid to the two
possible conditions in (5). To our knowledge, as the governing equation (3) is nonlinear, even if an implicit
analytic solution exists, only numerical methods with an iteration algorithm are available for finding solutions.
However, the two conditions in (5) constitute two different boundary value problems. To clarify, we would
like to investigate the appropriate boundary condition for solving the nonlinear Dupuit-Boussinesq theory
without any simplification.
2.2. Normalization
All normalized variables, without primes, are assumed to be
x=x
Land 𝜂=𝜂
H.(6)
Using the normalized variables above, the governing equation (3) becomes
d
dx𝜂d𝜂
dx+𝛽d𝜂
dx=−𝛾, (7)
where
𝛽=Ltan 𝛼
Hand 𝛾=IL2
k0H2.(8)
Depending on the shallowness and inclination of an aquifer, 𝛽is called groundwater hillslope flow number.
A higher 𝛽represents the flow in a shallower or steeper aquifer. On the other hand, 𝛾denotes the ratio of
external rain rate to the hydraulic conductivity, and it represents the storage capability of an aquifer under
rainfall recharge. A higher 𝛾means a higher rainfall on a lower permeable aquifer. Hence, with (6) and (8) the
normalized Darcy’s velocity becomes
u=−𝜆d𝜂
dx+𝛽,
where one more normalized parameter is
𝜆=k0Hcos 𝛼
L,
reflecting the relation among shallowness, inclination, and aquifer permeability. Then, the normalized volu-
metric discharge reads
q=q
H2k0cos 𝛼L=−𝜂d𝜂
dx +𝛽.
Finally, with (6) the downstream boundary condition becomes
𝜂=D
H=𝜂0,at x=0.(9)
Attention is paid here that the conventional boundary condition of zero groundwater table at the down-
stream (e.g., Brutsaert, 2005), D=0, is not considered as it gives the trivial solution of no discharge. This
condition can also be explained by the phase portraits in the following section. At the upstream boundary,
the zero-volumetric-discharge boundary condition reads
q=−𝜂d𝜂
dx+𝛽=0,at x=1.(10)
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2.3. Volumetric Discharge Distribution
By integrating (7) with respect to xand applying the zero discharge (10) at the upstream boundary, the
volumetric discharge can be obtained as
q=−𝜂d𝜂
dx+𝛽=𝛾(x1),(11)
which is the same as the one in Basha and Maalouf (2005). The discharge distribution is linearly proportional
to 𝛾and has a maximum leftward discharge, 𝛾, at the downstream boundary. So far, no analytic solution
in an explicit form is available for (11). Although an implicit solution of the steady groundwater table profile,
𝜂, has been proposed using (11) (Henderson & Wooding, 1964), numerical methods are still demanded for
calculating the implicit solution.
3. Examination of Two Boundary Conditions
Despite nonlinearity and not having an explicitly analytic solution, (11) can still provide some information at
the upstream and downstream boundaries, as are discussed separately in the following.
3.1. Downstream Condition
At the downstream boundary, with the downstream boundary condition (9) and some algebra, the volumetric
discharge gives a gradient of groundwater table
d𝜂
dx=𝛾
𝜂0
𝛽at x=0,(12)
of which the minimum gradient is 𝛽when 𝛾=0. Beyond the objective of the present paper,the case of 𝛾=0
is excluded hereafter. The equation above simply gives a gradient of groundwater table in terms of 𝛽, 𝛾, and
𝜂0. By comparing the two terms in the right-hand side of (12), in the dimensional form, if total rainfall amount
on an aquifer is greater than drainage discharge at the downstream, IL>k0Dtan 𝛼, a convex groundwater
table exists near the downstream boundary. On the other hand, the groundwater table near the downstream
boundary is horizontal or concave if total rainfall amount is less than or equal to the drainage discharge, IL
k0Dtan 𝛼. Besides, if 𝜂00, the gradient of groundwater table approaches positive infinity. Therefore, no
groundwater at the downstream cannot give a reasonable solution.
3.2. Upstream Condition
At the upstream boundary, the zero-discharge condition (10) can only provide the ambiguous conditions that
either the groundwater table is zero, or the groundwater table gradient is 𝛽, or both are true, as below:
𝜂=0or d𝜂
dx=−𝛽, at x=1.(13)
Herein, the governing equation (7) is used for analyzing conditions at the upstream.
First, we examined the condition of a constant groundwater table gradient of 𝛽at x=1by substituting it
into (7) to obtain
𝜂d2𝜂
dx2=−𝛾, at x=1.
For physical significance, the equation above indicates that a finite and convex groundwater table must exist
at the upstream boundary:
𝜂>0and d2𝜂
dx2<0,at x=1.
However, the resultant equation above obviously contradicts the zero-groundwater-table condition, 𝜂=0.
On the other hand, we instead examined the zero-groundwater-table condition, 𝜂=0at x=1, by applying
it to (7) to obtain a quadratic equation of groundwater table gradient, as below:
d𝜂
dx2
+𝛽d𝜂
dx=−𝛾, at x=1.
Hence, we obtained two negative groundwater table gradients at the upstream boundary:
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1
2𝛽+𝛽24𝛾and 1
2𝛽𝛽24𝛾,(14)
where the discriminant for a real groundwater table gradient reads
Δ=𝛽24𝛾0.
The validity of the two gradients in (14) is investigated in the following sections. For physical significance in
our problem, the parameter bounds read
0<𝛾𝛽2
4.(15)
In (14) special attention is paid here that groundwater table gradients do not equal to 𝛽. This also testifies
again that both conditions in (13) cannot be true simultaneously. Then, recovering the dimensions of (15)
with some manipulations, we obtained a relation among rainfall intensity, permeability, and inclination of the
aquifer, for a real-valued groundwater table gradient, as below:
0<Ik0tan2𝛼
4.(16)
Taking some examples for using (16), in an aquifer inclining at 𝛼=45and consisting of well-sorted sand and
gravel k0103m/s, the maximum rain rate for a real-valued groundwater table gradient at the upstream is
2.5 ×104m/s, namely, 900 mm/hr, which represents an impossible extreme rainfall. If the same slope consists
of fine sand k0105m/s, the maximum rain rate becomes 9 mm/hr, which is a moderate rainfall on average.
Equation (16) provides parameter bounds for a real solution using Boussinesq theory.
We discover that if it holds a constant groundwater table gradient of 𝛽, there must be a finite and a convex
profile of groundwater table at the upstream. Otherwise, if the zero-groundwater-table boundary condition is
applied, the groundwater table gradient must not equal 𝛽. This result argues that only one condition in (13)
is valid for a given problem. However, at this point it is lacking of enough information to judge which condition
is appropriate at the upstream boundary. Only the groundwater table and its gradient at the downstream
boundary are known beforehand.
4. Phase Portrait Analysis
In our problem, the information of groundwater table and its gradient at the downstream boundary are
already known, but the appropriate upstream boundary condition is not. Herein, instead of actually solving
the nonlinear differential equation of our problem, we shall utilize the phase plane method to investigate the
condition at the upstream boundary by tracing a path in the phase plane starting from given downstream
groundwater table 𝜂0, hillslope flow number 𝛽, and auqifer storage capacity 𝛾. Consider a thin aquifer having
HL=0.1for a natural soil-mantled hillslope. Three configurations of inclination and shallowness for aquifers
are considered, including 𝛽=Ltan 𝛼H=10, 1, and 0.1, three different rainfall forcings 𝛾’s satisfying the
bounds of (16), and three downstream groundwater table 𝜂0s are considered.
4.1. Phase Plane System
The phase portrait analysis focuses on geometrically interpreting the relation between groundwater table
and its gradient. One new variable for groundwater table gradient is defined as
𝜁d𝜂
dx=(𝜂, 𝜁),(17)
and, by rearranging (7), the curvature of groundwater table, or 𝜁gradient, reads
d𝜁
dx=−
1
𝜂(𝜁2+𝛽𝜁 +𝛾)=(𝜂, 𝜁),(18)
where a singularity exists when 𝜂=0as d𝜁
dxapproaches negative infinity. As being a nontrivial solution, the
case of zero groundwater table is excluded for discussion. Herein, we only consider the condition of 𝜂>0for
physical significance and investigate the relation between 𝜂and 𝜁in the (𝜂,𝜁 )phase plane. With (17) and (18),
in the phase plane, the vector field can be expressed as (d𝜂
dx,d𝜁
dx)=(,), and the tangent function reads
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d𝜁
d𝜂=(𝜂, 𝜁)
(𝜂, 𝜁)=−
𝜁2+𝛽𝜁 +𝛾
𝜁𝜂 .(19)
With 𝛽and 𝛾satisfying (15), (19) is used for visually interpreting the relation between groundwater table and
its gradient in the phase plane.
No equilibrium point exists in this system because letting =0and =0simultaneously cannot give any
real-valued pair of (𝜂, 𝜁). By letting =0or =0we can obtain three horizontal isoclines defined as below:
𝜁1=0,𝜁
2=1
2𝛽+𝛽24𝛾,and 𝜁3=1
2𝛽𝛽24𝛾,(20)
with the discriminant (15) for a real-valued 𝜁. All three isoclines are horizontal lines. Upon (15) a maximum
𝛾=𝛽24yields 𝜁2=𝜁3=−𝛽2. The geometrical property of each isocline is briefly explained in the following.
On the isocline of 𝜁=𝜁1=0, all vector direction along the line is vertically downward as d𝜁
dx=−
𝛾
𝜂<0.
So zero-gradient and negative curvature of groundwater table give a local maximum of groundwater table.
Then, on the other two isoclines of 𝜁=𝜁2or 𝜁=𝜁3, vector direction everywhere is horizontally leftward as
vertical component is always 0, d𝜁
dx=0. Both the two isoclines give a negative gradient and a zero curvature
of groundwater table. In particular, the groundwater table gradient on 𝜁=𝜁3is steeper than on 𝜁=𝜁2.Any
path on either the two isoclines can only stay on the line and approach 𝜂=0by a constant gradient of 𝜁2or
𝜁3. This indicates a linear groundwater table distribution if the downstream groundwater table gradient is 𝜁2
or 𝜁3. Moreover, 𝜁1is not a boundary to separate different behaviors of our nonlinear system; only 𝜁2and 𝜁3
are separatrices in the phase plane.
In different zones separated by 𝜁2or 𝜁3in (20), trajectories can reflect different behaviors of groundwater table
𝜂and its gradient 𝜁. In the zone of 𝜁>𝜁2, the direction of any phase path is downward as a negative vertical
component d𝜁
dx<0always holds. The horizontal direction is rightward provided that 𝜁>0then turns into
leftward as 𝜁2𝜁<0. In this zone, any path reflects a concave groundwater table having a maximum value.
Then, in the zone of 𝜁3<𝜁<𝜁
2, the horizontal and vertical components of any vector are always negative and
positive, respectively. So all phase paths go leftward and upward and rapidly converge to 𝜁2as 𝜂approaches 0.
In this zone, any groundwater table corresponding to any phase path has a decreasing distribution. A special
case exists for the separatrix of 𝜁=𝜁3. Any phase path on 𝜁=𝜁3can only stay on it and go leftward to
approach (0,𝜁
3), and the corresponding groundwater table has a linear distribution with a constant gradient
of 𝜁3in the whole aquifer. Finally, in the zone of 𝜁<𝜁
3, horizontal and vertical components of any vector are
both negative. So all paths diverge from 𝜁=𝜁3and rapidly approach negative infinity of 𝜁axis as 𝜂approaches
0 very closely. To sum up, phase paths approach 𝜁2in the zone of 𝜁>𝜁3, and negative infinity in the zone of
𝜁<𝜁
3. All the features can be recognized from the figures introduced in the next three subsections.
For phase portrait analysis, one should assign some representative points of specific interest in the phase
plane. In our analysis all representative points pare connected with the downstream conditions by letting
p=(𝜂0,𝜁
0). Hence, one phase path starting from a given representative point pcan be obtained by integrat-
ing from 0 to 1 with respect to xusing (17) and (18). According to section 3.1, once a downstream groundwater
table 𝜂0is given, the downstream groundwater table gradient 𝜁0can be determined by (12). Then, the rep-
resentative point for a true solution can be imposed under any given 𝜂0and is defined as p6in the following
analysis. However, for the sake of clear illustration of phase plane features, we assigned five more representa-
tive points piwhere i=15, covering possible parameter ranges of practical interest. On every trajectory,
the end location represents the upstream boundary. Therefore, the information of 𝜂and 𝜁at all these tra-
jectory ends can be used to validate the appropriate upstream boundary condition. To emphasize again,
none of presumption is made for upstream boundary because representative points are only assigned due to
downstream conditions.
For representative points of interest, three downstream groundwater tables are given as 𝜂0=1.0, 0.5, and 0.1.
As the valid range of 𝛾changes by different 𝛽s, all representative points for the phase portraits of 𝛽=10,1,
and 0.1 are defined separately in the following sections. Each case of 𝛽has nine phase portraits in total.
4.2. An Aquifer of 𝜷=10
A steeper and/or shallower aquifer is considered herein by assuming 𝛽=10. According to (15), the parameter
of 𝛽=10 gives the valid range of 0<𝛾25. Here we consider three 𝛾=5, 10, and 25. The representative
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Figure 2. (a– i) Phase portraits of 𝛽=10 under 𝛾=5, 15, and 25 and 𝜂0=1.0, 0.5, and 0.1. The black dotted lines are 𝜂=0 and the separatrices of 𝜁2and 𝜁3in
(20). The red dotted lines denote 𝜁=−𝛽. Colored solid circles and triangles denote the downstream and upstream locations, respectively. The trajectories of
𝜁0<𝜁
3go to negative infinity, but the others first converge to and then diverge from (0,𝜁
2). Each purple trajectory p6denotes the solution under the given
𝜁0,𝛾,and𝛽=10.
points are defined as pi=𝜂0,𝜁
0=−18 +6i,
p6=𝜂0,𝜁
0=𝛾𝜂0𝛽,(21)
where i=15;𝜂0=1.0, 0.5, and 0.1. Figure 2 illustrates all trajectories change rapidly in the nine phase
portraits. Only the trajectory of p6in each portrait reaches (0,𝜂
2). Trajectories with 𝜁0<𝛽go to negative
infinity, and the rest of trajectories converge to but then diverge from (0,𝜁
2). As a result, all p6s trajectory ends
reveal that the only upstream boundary condition is zero-groundwater table. Additionally, only the trajecto-
ries of p6in Figures 2a and 2d show the concave groundwater tables as the rainfall recharge is 𝛾=5.0; the
other solutions of groundwater table have convex profiles as 𝜁0>0. Particularly, no matter what groundwater
table is imposed at the downstream, all convex groundwater tables of 𝛾=15 and 25 exceed the top aquifer
surface as their maximums are greater than 1.0. All these cases mean that drainage occurs on the aquifer sur-
face, and the resultant profiles are mathematically valid in the phase plane, but out of physical significance.
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Figure 3. (a– i) Phase portraits of 𝛽=1 under 𝛾=0.05, 0.15, and 0.25 and 𝜂0=1.0, 0.5, and 0.1. The black dotted lines are 𝜂=0 and the separatrices of 𝜁2and
𝜁3in (20). The red dotted lines denote 𝜁=−𝛽. Colored solid circles and triangles denote the downstream and upstream locations, respectively. The trajectories
of 𝜁0<𝜁
3go to negative infinity, the ones of 𝜁0>𝜁3approach (0,𝜁
2), and the others satisfying 𝛽<𝜁
0<𝜁
3converge to (0,𝛽). Each purple trajectory p6,
which a linear curve with a constant slope gradient of 𝜁2=𝜁3, denotes the solution under the given 𝜁0,𝛾,and𝛽=1.
To sum up, under any reasonable rainfall recharge, 0<𝛾25.0, a groundwater table in an aquifer of a
higher 𝛽can only be 0 at the upstream boundary no matter what the downstream groundwater table is. The
zero-groundwater-table condition is the only choice for upstream boundary condition.
4.3. An Aquifer of 𝜷=1
An aquifer having a moderate inclination and/or shallowness is considered herein by assuming 𝛽=1. Com-
paring to the steeper aquifer of 𝛽=10, the milder aquifer can be regarded as having either a milder inclination
or a greater depth. According to (15), 𝛽=1givesthe valid range of rainfall recharge parameter as 0<𝛾 0.25.
Here we consider three 𝛾=0.05, 0.15, and 0.25. The representative points are defined as
pi=𝜂0,𝜁
0=−3+i,
p6=𝜂0,𝜁
0=𝛾𝜂0𝛽,(22)
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Figure 4. (a– i) Phase portraits of 𝛽=0.1 under 𝛾=0.0005, 0.0015, and 0.0025 and 𝜂0=1.0, 0.5, and 0.1. The black dotted lines are 𝜂=0 and the separatrices
of 𝜁2and 𝜁3in (20). The red dotted lines denote 𝜁=−𝛽. Colored solid circles and triangles denote the downstream and upstream locations, respectively. The
trajectories of 𝜁0<𝜁
3go to negative infinity, the ones of 𝜁0>𝜁3approach (0,𝜁
2), and the others satisfying 𝛽<𝜁
0<𝜁
3converge to (0,𝛽). Each purple
trajectory p6denotes the solution under the given 𝜁0,𝛾,and𝛽=0.1.
where i=15;𝜂0=1.0, 0.5, and 0.1. Comparing with the case of 𝛽=10 aquifer in Figure 2, all trajec-
tories in Figure 3 have the same geometrical features but do not change rapidly. As a result, both of the
zero-groundwater-table and the constant groundwater table gradient of 𝛽appear at the upstream bound-
ary under certain parameters. The zero-groundwater-table upstream boundary condition appears in the
cases of shallower downstream groundwater tables of 𝜂0=0.1and 0.5, but the constant gradient of 𝛽
exists in the cases of deeper one of 𝜂0=1.0. It is intuitive that much groundwater exists in an aquifer, much
groundwater can be accumulated everywhere, and a finite groundwater table can appear at the upstream
boundary. This indicates each boundary condition in (13) can hold independently under certain parameters.
4.4. An Aquifer of 𝜷=0.1
Finally, a relatively mild and/or deeper aquifer is considered herein by assuming 𝛽=0.1. According to (15),
𝛽=0.1gives the valid range of 0<𝛾0.0025. Here we consider three 𝛾=0.0005, 0.0015, and 0.0025. The
WU ET AL. 10
Water Resources Research 10.1029/2018WR023070
Figure 5. Criteria for appropriate upstream boundary condition. Solid
black circles and triangles denote the cases of 𝛽=1.0 and 0.1. The blue,
red, and green lines are the criteria under different 𝜂0=1.0, 0.5, and 0.1
by using (24). The gray part is excluded for solutions as the black dashed
lines denote the lower bound for having real-valued solutions using (15).
representative points are defined as
pi=𝜂0,𝜁
0=−0.24 +0.08i,
p6=𝜂0,𝜁
0=𝛾𝜂0𝛽,(23)
where i=15;𝜂0=1.0, 0.5, and 0.1. Figure 4 illustrates that all tra-
jectories in the nine phase portraits have slow change. Only the constant
groundwater table gradient of 𝛽appears at the upstream boundary.
From all the nine cases together with the ones of 𝛽=1in Figures 3a–3c,
it can be readily observed that representative points of p6are all located
between 𝜁3and 𝛽when the trajectories approach 𝜁=−𝛽. This implies
the existence of the constant groundwater table gradient at the upstream
boundary and will be used in the next section.
5. Criterion for Appropriate Upstream
Boundary Condition
Section 3.2 states that the two upstream boundary conditions in (13) can-
not coexist for a true solution. Recalling from section 4.1, all phase paths
starting from any location of 𝜁>𝜁3always converge to (0,𝜁
2). Only the
paths from 𝜁𝜁3travel in the zone of 𝜁𝜁3and rapidly approach neg-
ative infinity of 𝜁. Additionally, it must hold that 𝜁3𝛽according to (20). To combine them, only the paths
starting from 𝛽<𝜁<𝜁
3depart from 𝜁=𝜁3and go downward in the phase plane. This is testified by our
analysis. According to Figures 3a3c and 4, the phase portraits reveal that the constant groundwater table
gradient of 𝛽exists at the upstream provided that the relating representative points are all located in the
range of 𝛽𝜁0<𝜁
3. Therefore, we can reasonably conclude the sufficient condition of a finite groundwa-
ter table existing at the upstream boundary shall be that the downstream groundwater table gradient must
be less than 𝜁3. So by applying the sufficient condition above with some algebraic manipulations, an explicit
criterion formula can be expressed as
𝛽,𝛾, 𝜂0=𝛾
𝜂0
1
2𝛽𝛽24𝛾,(24)
with (15) for real-valued solutions. When <0, the constant groundwater table gradient shall be imposed
at the upstream boundary, d𝜂
dx=−𝛽at x=1; otherwise, when ≥0, the zero-groundwater-table condition,
𝜂=0, must hold instead. An analytical verification can be referred in the appendix.
Figure 5 illustrates the criteria of the cases under different 𝛽, 𝛾, and 𝜂0for whether a groundwater table exists
or not at the upstream boundary. All cases of 𝛽=10 are excluded as only the zero-groundwater-table con-
dition exists at the upstream boundary. All thick color lines are obtained letting =0using (24). The cases
of 𝛽=1.0and 0.1 are denoted by black solid circles and triangles, respectively. As all black circles are located
below the criterion of 1.0=(𝜂0=1.0)in Figure 5, this means that an aquifer of 𝛽=1.0has a constant
groundwater table gradient of 𝛽at the upstream boundary if the downstream groundwater table is 𝜂0=1.0;
however, the other has zero-groundwater-table at the upstream if 𝜂00.5, as are shown in Figures 3d– 3i.
Finally, as shown in the inlet figure, the cases of 𝛽=0.1are all below the criteria of 0.1,0.5, and 1.0.Asa
result, all cases of 𝛽=0.1have the constant groundwater table gradient of 𝛽at the upstream boundary, as
are testified in Figure 4.
6. Concluding Remarks
The appropriate boundary conditions for Dupuit-Boussinesq theory on the steady flow in an unconfined
sloping aquifer have been comprehensively investigated in this study. Using the phase plane method, we
have explored the nonlinear features of the full Boussinesq theory without any simplification. Two parame-
ters representing inclination, shallowness, and rainfall recharge are defined for analysis. We have examined
all conditions at the downstream and upstream boundaries and obtained a parameter bound for true solu-
tions. In the phase portrait analysis, three aquifer flow number 𝛽, aquifer storage capacity 𝛾, and downstream
groundwater table 𝜂0are considered. The phase portraits show that the a steeper and/or shallower aquiferhas
a zero-groundwater-table if the downstream groundwater table is shallower; otherwise, a finite groundwa-
ter table with a constant gradient shall exist at the upstream boundary. Hence, an explicit criterion formula is
successfully derived to straightforwardly determine an appropriate upstream boundary condition. Any given
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problems can be numerically easily solved by adopting a correct upstream boundary condition. To conclude,
the major merit of the present work is to provide a simple and explicit formula in terms of known parameters
and downstream groundwater table for efficiently judging whether a groundwater table exists or not at the
upstream boundary in a thin and unconfined sloping aquifer.
Appendix A : Revisit of Conventional Steady State Solution
Here we shall revisit the conventional analytic solution (Henderson & Wooding, 1964) to derive an explicit
criterion formula to determine an appropriate upstream boundary condition. This criterion formula shall be
used to verify the one obtained by phase portrait analysis.
Rearranging the equation of volumetric discharge (11) results in
𝜂d𝜂
dx=−𝛽𝜂 𝛾(x1).(A1)
Equation (A1) can be regarded as an Abel equation of the second kind and can be implicitly expressed in a
parametric form (Polyanin & Zaitsev, 2002). Here we directly revisit the conventional solution (Henderson &
Wooding, 1964) instead of investigating the parametric solutions above.
For integration of (A1), inspired by Schmid and Luthin (1964), two new variables are defined, including the
first variable to transform the xcoordinate into a reverse one:
X=1x,(A2)
and a new dependent variable,
Q=𝜂
1x=𝜂
X.(A3)
With the two variables above the downstream boundary condition (9) gives
Q=𝜂0,at X=1.(A4)
With the chain rule and (A3) and (A2), we have
d𝜂
dx=−XdQ
dX+Q.(A5)
Hence, after some algebra (A1) results in
QdQ
Q2𝛽Q+𝛾=−
dX
X,(A6)
with the discriminant
Δ=𝛽24𝛾0
for ensuring a real-valued Q. Again, having been explained in section 3.2, this discriminant denotes the suf-
ficient condition for finding a real-valued solution. In what follows the solutions of Δ=0and Δ>0are
discussed separately.
A1. Solution for 𝚫=0
With Δ=𝛽24𝛾=0(A6) can be further manipulated to
dm
m𝛽
2d(m1)=−
dX
X,(A7)
where m=Q𝛽2. Integrating (A7) once and taking the exponential function to the resultant equation gives
C1
X=Q𝛽
2exp 𝛽
𝛽2Q,
where exp ()denotes the exponential function and C1is an integration constant
C1=𝜂0𝛽
2exp 𝛽
𝛽2𝜂0
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by applying (A4). Recovering Qand Xback into the original variables with some algebra, we obtain the
solution in an implicit form of 𝜂:
𝜂+𝛽
2(1x)=𝜂0𝛽
2exp 𝛽
𝛽2𝜂0
𝛽(1x)
𝛽(1x)−2𝜂.(A8)
Substituting x=1into (A8) results in the groundwater table at the upstream boundary:
𝜂(x=1)=𝜂0𝛽
2exp 𝛽
𝛽2𝜂0
E0,(A9)
where
E0=lim
x1
𝛽(1x)
𝛽(1x)−2𝜂=0,
provided that a finite groundwater table exists at the upstream boundary, 𝜂(x=1)>0. For ensuring a finite
groundwater table at the upstream, an inequality must hold
𝜂0>𝛽
2(A10)
as exp 𝛽∕(𝛽2𝜂0)in (A9) is always positive.
A2. Solution for 𝚫>0
Using partial fraction decomposition, (A6) becomes
Δ−𝛽
2ΔdQ
Q−(𝛽Δ)∕2
+Δ+𝛽
2ΔdQ
Q−(𝛽+Δ)∕2
=−
dX
X,(A11)
with Δ=𝛽24𝛾>0. Integrating (A11) once and taking the exponential function with some algebraic
rearrangements gives
X
C2
=Q−(𝛽Δ)∕2
(𝛽Δ)∕2Δ
Q−(𝛽+Δ)∕2
(𝛽+Δ)∕2Δ
,(A12)
where
C2=𝜂0−(𝛽+Δ)∕2
(𝛽+Δ)∕2Δ
𝜂0−(𝛽Δ)∕2
(𝛽Δ)∕2Δ
,(A13)
by applying (A4). To compare with the conventional solution(Henderson & Wooding, 1964), we replaced the
normalized parameters analogous to their definition by 𝛽=2and 𝛾=𝜆and rearranged the sign convention
to obtain
X
C2
=1𝜅𝜂X(1𝜅)2𝜅
1+𝜅𝜂X(1+𝜅)2𝜅,(A14)
where 𝜅=1𝜆=Δ∕2. (A14) exactly equals the conventional solution. Henderson and Wooding (1964)
proposed that a constant gradient of groundwater table exists at the upstream provided that 1+𝜅<𝜂X<2.
However, as 𝜂Xis unknown before having a solution, the inequality is of impractical use. Generally, numeri-
cal techniques utilizing an iterative algorithm is necessary to solve (A12) with an initial guess of solution. The
numerical solution is neither efficient to obtain nor lacking of an appropriate way to impose a reasonable ini-
tial guess. Hence, for easier and efficient use, it may demand an explicit formula in terms of known parameters
and the downstream boundary condition.
To find an explicit formula, another form of the solution is used here. With
1
2𝛽Δ=1
2𝛽+ΔΔ
(A12) can be rewritten as
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XQ1
2𝛽Δ=C2
1+Δ
Q−(𝛽+Δ)∕2
(𝛽+Δ)∕2Δ
,(A15)
and replacing Q=𝜂Xand X=(1x)by original variables with some manipulations yields
𝜂1
2(1x)𝛽Δ=C2
1+2(1x)Δ
2𝜂−(1x)(𝛽+Δ)
(𝛽+Δ)∕2Δ
,(A16)
0 where C2is expressed in (A13). Equation (A16) is an implicit solution of groundwater table 𝜂. With unknown 𝜂
we have no information about the profile of groundwater table.However, we only have to investigate whether
a finite groundwater table exists at the upstream boundary; that is, 𝜂(x=1)>0. Hence, substituting x=1
into (A15) simply yields
𝜂(x=1)=C2=𝜂0−(𝛽+Δ)∕2
(𝛽+Δ)∕2Δ
𝜂0−(𝛽Δ)∕2
(𝛽Δ)∕2Δ
.
Therefore, the two sufficient conditions for 𝜂(x=1)>0read
𝜂01
2𝛽𝛽24𝛾>0and 𝜂01
2𝛽+𝛽24𝛾>0.
The intersection of the two conditions above reads
𝜂0>1
2𝛽+𝛽24𝛾.(A17)
Obviously, after taking reciprocals of both sides of (A17) with some manipulations, we can obtain the
inequality
𝛾
𝜂0
1
2𝛽𝛽24𝛾<0.(A18)
for ensuring a finite groundwater table at the upstream boundary provided that 𝛽24𝛾>0. Moreover, apply-
ing 𝛽24𝛾=0to (A18) can also yield the same inequality of (A10). It is proven that (A18) is exactly equivalent
to (24) obtained by phase portrait analysis.
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Acknowled gments
E. N. and Y. H. W. would like to thank the
financial support of Japan Society for
the Promotion of Science (JSPS)
Grants-in-Aid for JSPS Research Fellow
(Grant 16F16378). Y. H. W. appreciates
JSPS FY2016 Postdoctoral Fellowship
for Overseas Researchers. The authors
are indebted to reviewers for the
valuable comments greatly improving
this paper and to Adrean Webb for
editing assistance. The paper is
theoretical, and no data are used.
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WU ET AL. 15
... They further concluded that a steeper and/or shallower aquifer has a zero-groundwater table if the downstream groundwater table is shallower. Here we point out that the conceptual model adopted by Wu et al. (2018) not reasonable and the results are quite different from those with unsaturated flow considered in the real hillslope aquifer. Specifically, the water table at the upstream boundary could be positive or negative due to the lateral unsaturated flow effect, and hence, the seepage velocity must be set to zero to satisfy the zero discharge boundary condition at the upstream boundary. ...
... Clearly, a correct boundary condition is crucial for the numerical model. Wu et al. (2018) proposed two possible boundary conditions at the upstream boundary, namely, zero groundwater table or zero seepage velocity, which are given as ...
... Note that h in equations (1) and (2) equals the η ' used in Wu et al. (2018). Based on our recent findings, the no-flow boundary described in equation (1) is problematic, because h could be either positive or negative. ...
Article
Full-text available
By neglecting unsaturated flow, Wu et al. (2018, https://doi.org/10.1029/2018WR023070) claimed that the zero volumetric discharge at the upstream boundary results in two possible boundary conditions of either zero groundwater table or zero seepage velocity based on the original Dupuit‐Boussinesq theory. They further concluded that a steeper and/or shallower aquifer has a zero‐groundwater table if the downstream groundwater table is shallower. Here we point out that the conceptual model adopted by Wu et al. (2018) not reasonable and the results are quite different from those with unsaturated flow considered in the real hillslope aquifer. Specifically, the water table at the upstream boundary could be positive or negative due to the lateral unsaturated flow effect, and hence, the seepage velocity must be set to zero to satisfy the zero discharge boundary condition at the upstream boundary. In addition, the zero groundwater table cannot ensure a zero discharge at the upstream boundary. Furthermore, we argue that the discharge with respect to the groundwater table is linearly distributed along the slope distance only when both saturated and unsaturated flows are considered. The saturated flow discharge itself is not distributed linearly. Our comment highlights the importance of unsaturated flow in studying steady groundwater flow in an unconfined sloping aquifer with uniform recharge using the Dupuit‐Boussinesq theory.
... The authors would like to thank Kong and other colleagues for their interest in and a comment article (Kong et al., 2019) made on our research paper (Wu et al., 2018) investigating appropriate boundary conditions for the original Dupuit-Boussinesq theory for steady two-dimensional groundwater flow in an unconfined sloping aquifer with uniform rainfall recharge. We are delighted to read the comments (Kong et al., 2019) pointing out shortcomings of the classical Dupuit-Boussinesq theory for general sloping aquifers and providing possible directions for broadening the scope of analytical analysis on shallow groundwater flow in unconfined aquifers. ...
... We are delighted to read the comments (Kong et al., 2019) pointing out shortcomings of the classical Dupuit-Boussinesq theory for general sloping aquifers and providing possible directions for broadening the scope of analytical analysis on shallow groundwater flow in unconfined aquifers. We do agree with Kong and other colleagues that an unsaturated zone is of importance for general and natural aquifers that are not the investigation focus of our paper (Wu et al., 2018). Instead, our paper's main purpose is to demonstrate one important analytical issue of determination of an appropriate boundary condition, which has not drawn much attention and comprehensively discussed in the past. ...
... However, the original theory still can provide approximate solutions very close to ones obtained by a more complete formulation, so it is the method of choice in many investigations (Brutsaert, 2005) or for further developments of improvement (e.g., Hilberts & Troch, 2005, Kong et al., 2016, Troch et al., 2003. But, as is clearly pointed out in Wu et al. (2018), it is still lacking of an efficient way to determine appropriate boundary conditions when applying this original and classical theory. Based on its simplicity and relevance to hillslope hydrology analysis, the original theory is still worthy of our investigation focus as a foundation for further analytical development. ...
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The article aims to respond a comment made on our paper about appropriate boundary condition for the original Dupuit-Boussinesq theory for two-dimensional steady groundwater flow in an unconfined sloping aquifer with uniform rainfall recharge. To respond to the comments arguing the existence of lateral groundwater flows and negative groundwater table, clarifications are made for our analysis focusing on two-dimensional groundwater flow without considering lateral effects by using the original and classical approximate theory.
... Attention is paid here that two possibilities for zero discharge exist in (5). Setting up an appropriate upstream boundary condition can refer to our recent work 4) . An initial groundwater table is imposed as ...
... However, a technically-sound theoretical solution for the original transient problem has not been proposed yet, so short-time transient simulation could not be fully verified so far. But, as the capability of correct long-time simulation is also important for a numerical model, we shall verify our model with the steady-state solution 4) . ...
... The solution for the steady-state problem, (8) without the LHS term, has been revisited recently 4) . As the analytical steady state solution is nonlinear and implicit, it still demands an iterative root-finding method for an approximate solution. ...
Article
This study presents a new numerical model for transient shallow groundwater table in an unconfined sloping aquifer. The theory for groundwater table evolution is the hydraulic groundwater theory, or called Dupuit-Boussinesq theory. The relaxation approach is applied for numerically calculating the nonlinear advection-diffusion equation with the help of the Godunov based wave-propagation algorithm. The transient numerical model is verified with the steady-state solution. We performed two case studies considering variable rainfall patterns and variable hydraulic conductivity. The resultant hydrographs of outflow discharges and groundwater tables at the downstream outlet are obtained. The results verify that our transient model is practical for modeling motion of thin groundwater table in an unconfined sloping aquifer.
... M3 Dupuit-Boussinesq aquifer model, that is groundwater flow moves shallower in the subsoil and it follows hydrostatic pressure (Wu et al., 2018), 'and the simplifying assumptions of a porous, free, homogeneous, isotropic, aquifer with no capillary effect, that is limited by an impermeable horizontal layer at the level of the outlet that is considered to be localized' (Boussinesq, 1877) to give an exact solution of the diffusion equation (Dewandel et al., 2003). Finally, model T A B L E 1 Descriptive characteristics of the eight catchments including annual rainfall, location, catchment morphology and soil texture classes M3 is a modification of M1 presented by Pizarro et al. (2013) which implies a greater participation of the recession coefficient at the time of estimating Q(t), thus yielding α values significantly higher than those obtained by the remaining models. ...
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In central Chile, many communities rely on water obtained from small catchments in the coastal mountains. Water security for these communities is most vulnerable during the summer dry season and, from 2010 to 2017, rainfall during the dry season was between 20 and 40% below the long‐term average. The rate of decrease in stream flow after a rainfall event is a good measure of the risk of flow decreasing below a critical threshold. This risk of low flow can be quantified using a recession coefficient (α) that is the slope of an exponential decay function relating flow to time since rainfall. A mathematical model was used to estimate the recession coefficient (α) for 142 rainstorm events (64 in summer; 78 in winter) in eight monitored catchments between 2008 and 2017. These catchments all have a similar geology and extend from 35 to 39 degrees of latitude south in the coastal range of south‐central Chile. A hierarchical cluster analysis was used to test for differences between the mean value of α for different regions and forest types in winter and summer. The value of α did not differ (p<0.05) between catchments in winter. Some differences were observed during summer and these were attributed to morphological differences between catchments and, in the northernmost catchments, the effect of land cover (native forest and plantation). Moreover, α for catchments with native forest was similar to those with pine plantations, although there was no difference (p<0.05) between these and eucalyptus plantations. The recession constant is a well‐established method for understanding the effect of climate and disturbance on low flows and baseflows and can enhance local and regional analyses of hydrological processes. Understanding the recession of flow after rainfall in small headwater catchments, especially during summer, is vital for water resources management in areas where the establishment of plantations has occurred in a drying climate. This article is protected by copyright. All rights reserved.
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The nonlinear equation of Boussinesq [1877] is a foundational approach for studying groundwater flow through an unconfined aquifer, but solving the full nonlinear version of the Boussinesq equation remains a challenge. Here, we present an exact solution to the full nonlinear Boussinesq equation that not only applies to sloping aquifers but also accounts for source and sink terms such as bedrock seepage, an often significant flux in headwater catchments. This new solution captures the hysteretic relationship (a loop rating curve) between the groundwater flow rate and the water table height, which may be used to provide a more realistic representation of streamflow and groundwater dynamics in hillslopes. In addition, the solution provides an expression where the flow recession varies based on hillslope parameters such as bedrock slope, bedrock seepage, plant transpiration and other factors that vary across landscape types.
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Exact solutions have always played and still play an important role in properly understanding the qualitative features of many phenomena. The Handbook of Exact Solutions for Ordinary Differential Equations contains a collection of more than 5,000 ordinary differential equations and their solutions. Coverage focuses on two types of equations: those that are of interest to researchers but are difficult to integrate (Abel equations, Emden-Fowler equations, Painlev equations, etc.), and equations relevant to applications in heat and mass transfer, nonlinear mechanics, hydrodynamics, nonlinear oscillations, combustion, chemical engineering, and other related fields. The authors also pay special attention to equations containing arbitrary functions, and devote other sections to equations contain one or more arbitrary parameters that the reader can fix at will.
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We investigate the response of a laboratory aquifer submitted to artificial rainfall, with an emphasis on the early stage of a rain event. In this almost two-dimensional experiment, the infiltrating rainwater forms a groundwater reservoir which exits the aquifer through one side. The resulting outflow resembles a typical stream hydrograph: the water discharge increases rapidly during rainfall and decays slowly after the rain has stopped. The Dupuit-Boussinesq theory, based on Darcy's law and the shallow-water approximation, quantifies these two asymptotic regimes. At the early stage of a rainfall event, the discharge increases linearly with time, at a rate proportional to the rainfall rate to the power of 3/2. Long after the rain has stopped, it decreases as the squared inverse of time (Boussinesq, C. R. Acad. Sci., vol. 137, 1903, pp. 5-11). We compare these predictions with our experimental data.
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The linearized hillslope Boussinesq equation, introduced by Brutsaert[1994], describes the dynamics of saturated, subsurface flow from hillslopes with shallow, unconfined aquifers. In this paper, we use a new analytical technique to solve the linearized hillslope Boussinesq equation to predict water table dynamics and hillslope discharge to channels. The new solutions extend previous analytical treatments of the linearized hillslope Boussinsq equation to account for the impact of spatiotemporal heterogeneity in water table recharge. The results indicate that the spatial character of recharge may significantly alter both steady-state subsurface storage characteristics and the transient hillslope hydrologic response, depending strongly on similarity measures of controls on the subsurface flow dynamics. Additionally, we derive new analytical solutions for the linearized hillslope-storage Boussinesq equation and explore the interaction effects of recharge structure and hillslope morphology on water storage and baseflow recession characteristics. A theoretical recession analysis, for example, demonstrates that decreasing the relative amount of downslope recharge has a similar effect as increasing hillslope convergence. In general, the theory suggests that recharge heterogeneity can serve to diminish or enhance the hydrologic impacts of hillslope morphology.
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[1] Hillslope response to rainfall remains one of the central problems of catchment hydrology. Flow processes in a one-dimensional sloping aquifer can be described by Boussinesq's hydraulic groundwater theory. Most hillslopes, however, have complex three-dimensional shapes that are characterized by their plan shape, profile curvature of surface and bedrock, and the soil depth. Field studies and numerical simulation have shown that these attributes are the most significant topographic controls on subsurface flow and saturation along hillslopes. In this paper the Boussinesq equation is reformulated in terms of soil water storage rather than water table height. The continuity and Darcy equations formulated in terms of storage along the hillslope lead to the hillslope-storage Boussinesq (HSB) equation for subsurface flow. Solutions of the HSB equation account explicitly for plan shape of the hillslope by introducing the hillslope width function and for profile curvature through the bedrock slope angle and the hillslope soil depth function. We investigate the behavior of the HSB model for different hillslope types (uniform, convergent, and divergent) and different slope angles under free drainage conditions after partial initial saturation (drainage scenario) and under constant rainfall recharge conditions (recharge scenario). The HSB equation is solved by means of numerical integration of the partial differential equation. We find that convergent hillslopes drain much more slowly compared to divergent hillslopes. The accumulation of moisture storage near the outlet of convergent hillslopes results in bell-shaped hydrographs. In contrast, the fast draining divergent hillslopes produce highly peaked hydrographs. In order to investigate the relative importance of the different terms in the HSB equation, several simplified nonlinear and linearized versions are derived, for instance, by recognizing that the width function of a hillslope generally shows smooth transition along the flow direction or by introducing a fitting parameter to account for average storage along the hillslope. The dynamic response of these reduced versions of the HSB equation under free drainage conditions depend strongly on hillslope shape and bedrock slope angle. For flat slopes (of the order of 5%), only the simplified nonlinear HSB equation is able to capture the dynamics of subsurface flow along complex hillslopes. In contrast, for steep slopes (of the order of 30%), we see that all the reduced versions show very similar results compared to the full version. It can be concluded that the complex derivative terms of width with respect to flow distance play a less dominant role with increasing slope angle. Comparison with the hillslope-storage kinematic wave model of Troch et al. [2002] shows that the diffusive drainage terms of the HSB model become less important for the fast draining divergent hillslopes. These results have important implications for the use of simplified versions of the HSB equation in landscapes and for the development of appropriate analytical solutions for subsurface flow along complex hillslopes.
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Subsurface flow from a hillslope can be described by the hydraulic groundwater theory as formulated by the Boussinesq equation. Several attempts have been made to solve this partial differential equation, and exact solutions have been found for specific situations. In the case of a sloping aquifer, Brutsaert [1994] suggested linearizing the equation to calculate the unit response of the hillslope. In this paper we first apply the work of Brutsaert by assuming a constant recharge to the groundwater table. The solution describes the groundwater table levels and the outflow in function of time. Then, an analytical expression is derived for the steady state solution by allowing time to approach infinity. This steady state water table is used as an initial condition to derive another analytical solution of the Boussinesq equation. This can then be used in a quasi steady state approach to compute outflow under changing recharge conditions.
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An approximation is obtained for the recession of a sloping aquifer. The analytical approximation can provide a useful tool to analyze data and obtain physical properties of the aquifer. In contrast to the case of a horizontal aquifer, when plotting the time derivative of the flux vs. the flux on a log scale, the result shows that the flux derivative reaches a minimum value and that the curve can have a slope of unity as often observed. Illustration of the application of the analytical results to the Mahantango Creek data is also discussed.