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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 q′in

the slope coordinates reads (Brutsaert, 2005)

q′=−k0𝜂′cos 𝛼d𝜂′

dx′+sin 𝛼,(1)

where 𝜂′is 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, 𝜂′is the

groundwater table, I′is 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 x′axis is aligned along and z′axis 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 z′depen-

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

dx′𝜂′d𝜂′

dx′+tan 𝛼d𝜂′

dx′=−I′

k0

,(3)

where 𝜂′is the groundwater table (m), k0is the hydraulic conductivity (m/s) ranging from 10−1to 10−5for

general hillslopes, 𝛼is the slope inclination (rad), and I′is 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 x′direction. With (4) the volumetric

discharge can be expressed as

q′=𝜂′u′=−k0𝜂′cos 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′=−k0𝜂′cos 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=I′L2

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 𝛼∕L=−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(x−1),(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, I′L>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, I′L≤

k0Dtan 𝛼. Besides, if 𝜂0→0, 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

2−𝛽+𝛽2−4𝛾and 1

2−𝛽−𝛽2−4𝛾,(14)

where the discriminant for a real groundwater table gradient reads

Δ=𝛽2−4𝛾≥0.

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<𝛾≤𝛽2

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<I′≤k0tan2𝛼

4.(16)

Taking some examples for using (16), in an aquifer inclining at 𝛼=45∘and consisting of well-sorted sand and

gravel k0≈10−3m/s, the maximum rain rate for a real-valued groundwater table gradient at the upstream is

2.5 ×10−4m/s, namely, 900 mm/hr, which represents an impossible extreme rainfall. If the same slope consists

of ﬁne sand k0≈10−5m/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

H∕L=0.1for a natural soil-mantled hillslope. Three conﬁgurations of inclination and shallowness for aquifers

are considered, including 𝛽=Ltan 𝛼∕H=10, 1, and 0.1, three diﬀerent rainfall forcings 𝛾’s satisfying the

bounds of (16), and three downstream groundwater table 𝜂0’s 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

𝜁≡d𝜂

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

2−𝛽+𝛽2−4𝛾,and 𝜁3=1

2−𝛽−𝛽2−4𝛾,(20)

with the discriminant (15) for a real-valued 𝜁. All three isoclines are horizontal lines. Upon (15) a maximum

𝛾=𝛽2∕4yields 𝜁2=𝜁3=−𝛽∕2. 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 𝜁2≤𝜁<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=1…5, 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 𝛽’s, 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<𝛾≤25. 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=𝛾∕𝜂0−𝛽,(21)

where i=1…5;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 p6’s 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<𝛾≤25.0, 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=𝛾∕𝜂0−𝛽,(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=1…5;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<𝛾≤0.0025. Here we consider three 𝛾=0.0005, 0.0015, and 0.0025. The

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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=𝛾∕𝜂0−𝛽,(23)

where i=1…5;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 𝜁≤𝜁3travel in the zone of 𝜁≤𝜁3and rapidly approach neg-

ative inﬁnity of 𝜁. Additionally, it must hold that 𝜁3≥−𝛽according 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 3a–3c 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 −𝛽≤𝜁0<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

2𝛽−𝛽2−4𝛾,(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 𝜂0≤0.5, 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(x−1).(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=1−x,(A2)

and a new dependent variable,

Q=𝜂

1−x=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

Q2−𝛽Q+𝛾=−

dX

X,(A6)

with the discriminant

Δ=𝛽2−4𝛾≥0

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=𝛽2−4𝛾=0(A6) can be further manipulated to

dm

m−𝛽

2d(m−1)=−

dX

X,(A7)

where m=Q−𝛽∕2. Integrating (A7) once and taking the exponential function to the resultant equation gives

C1

X=Q−𝛽

2exp 𝛽

𝛽−2Q,

where exp (⋅)denotes the exponential function and C1is an integration constant

C1=𝜂0−𝛽

2exp 𝛽

𝛽−2𝜂0

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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(1−x)=𝜂0−𝛽

2exp 𝛽

𝛽−2𝜂0

−𝛽(1−x)

𝛽(1−x)−2𝜂.(A8)

Substituting x=1into (A8) results in the groundwater table at the upstream boundary:

𝜂(x=1)=𝜂0−𝛽

2exp 𝛽

𝛽−2𝜂0

−E0,(A9)

where

E0=lim

x→1

𝛽(1−x)

𝛽(1−x)−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∕(𝛽−2𝜂0)in (A9) is always positive.

A2. Solution for 𝚫>0

Using partial fraction decomposition, (A6) becomes

Δ−𝛽

2ΔdQ

Q−(𝛽−Δ)∕2

+Δ+𝛽

2ΔdQ

Q−(𝛽+Δ)∕2

=−

dX

X,(A11)

with Δ=𝛽2−4𝛾>0. Integrating (A11) once and taking the exponential function with some algebraic

rearrangements gives

X

C2

=Q−(𝛽−Δ)∕2

(𝛽−Δ)∕2Δ

Q−(𝛽+Δ)∕2

(𝛽+Δ)∕2Δ

,(A12)

where

C2=𝜂0−(𝛽+Δ)∕2

(𝛽+Δ)∕2Δ

𝜂0−(𝛽−Δ)∕2

(𝛽−Δ)∕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

=1−𝜅−𝜂∕X(1−𝜅)∕2𝜅

1+𝜅−𝜂∕X(1+𝜅)∕2𝜅,(A14)

where 𝜅=1−𝜆=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<𝜂∕X<2.

However, as 𝜂∕Xis 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

2𝛽−Δ=1

2𝛽+Δ−Δ

(A12) can be rewritten as

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XQ−1

2𝛽−Δ=C2

1+Δ

Q−(𝛽+Δ)∕2

(𝛽+Δ)∕2Δ

,(A15)

and replacing Q=𝜂∕Xand X=(1−x)by original variables with some manipulations yields

𝜂−1

2(1−x)𝛽−Δ=C2

1+2(1−x)Δ

2𝜂−(1−x)(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−(𝛽−Δ)∕2

(𝛽−Δ)∕2Δ

.

Therefore, the two suﬃcient conditions for 𝜂(x=1)>0read

𝜂0−1

2𝛽−𝛽2−4𝛾>0and 𝜂0−1

2𝛽+𝛽2−4𝛾>0.

The intersection of the two conditions above reads

𝜂0>1

2𝛽+𝛽2−4𝛾.(A17)

Obviously, after taking reciprocals of both sides of (A17) with some manipulations, we can obtain the

inequality

𝛾

𝜂0

−1

2𝛽−𝛽2−4𝛾<0.(A18)

for ensuring a ﬁnite groundwater table at the upstream boundary provided that 𝛽2−4𝛾>0. Moreover, apply-

ing 𝛽2−4𝛾=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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