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Article

A Preliminary Study on a Pumping Well Capturing

Groundwater in an Unconﬁned Aquifer with

Mountain-Front Recharge from Segmental Inﬂow

Haixiang Li 1and Xu-sheng Wang 1,2,*

1Ministry of Education Key Laboratory of Groundwater Circulation and Environmental Evolution, China

University of Geosciences, Beijing 100083, China; lihaixiangcugb@foxmail.com

2

Beijing Key Laboratory of Water Resources & Environmental Engineering, China University of Geosciences,

Beijing 100083, China

*Correspondence: wxsh@cugb.edu.cn; Tel.: +86-010-82322008

Received: 30 May 2019; Accepted: 11 June 2019; Published: 14 June 2019

Abstract:

Unconﬁned aquifers beneath piedmont pluvial fans are widely distributed in front of

mountains and proper for water supply with pumping wells. However, the catchment zone and

capture zones of a pumping well in such an unconﬁned aquifer is not well known. We develop a

preliminary simpliﬁed model where groundwater ﬂows between a segmental inﬂow boundary and a

discharge boundary of constant head. The catchment zone is delineated from numerical simulation

via MODFLOW and MODPATH. Results are expressed with dimensionless variables and lumped

parameters to show general behaviors. Sensitive analyses indicate that there are 4 types of the

catchment zone according to diﬀerent connections to the boundaries. The shape of the catchment

zone is quantitatively analyzed with typical shape factors. Capture zones with respect to special

travel times are identiﬁed from travel time distribution in the catchment zone. The modeling results

can be applied in the design of water supply wells and delineation of protection zones at a site with

similar hydrogeological conditions.

Keywords: water table; piedmont pluvial fans; catchment zone; travel time; numerical modeling

1. Introduction

The capture zone of a pumping well is the aquifer volume from which groundwater ﬂows to

the well after a limited travel time. In particular, the catchment zone refers to the maximum capture

zone without the limitation of the travel time [

1

]. The delineation of capture zones is important for

management of groundwater resources, such as to deﬁne wellhead protection zones or to optimize

pump-and-treat systems in projects of groundwater pollution remediation.

The characteristics of capture zones can be studied with numerical or analytical methods. In several

simpliﬁed aquifer conditions, the catchment zone of a pumping well has been soundly investigated

with analytical models. For a pumping well in an inﬁnite conﬁned aquifer with uniform

regional ﬂow

,

analytical solutions of potential and stream function were derived and used for determining the

catchment zone [

2

–

5

]. The method was also extended for unconﬁned conditions [

6

]. These analytical

models could not be used directly for conditions where the aquifer is limited by speciﬁc boundaries.

It was

found with analytical models of a near-stream well [

7

,

8

] that the shape of the catchment zone

is highly dependent on the well-stream distance and pumping rate. Intaraprasong and Zhan [

9

]

proposed an analytical model in which a pumping well is located in a conﬁned aquifer between

two streams along inﬁnite parallel lines that were treated as constant head boundaries. The eﬀect of

wedge–shape boundaries [

10

] and groundwater divide [

1

,

11

] were also investigated with analytical

Water 2019,11, 1243; doi:10.3390/w11061243 www.mdpi.com/journal/water

Water 2019,11, 1243 2 of 18

approaches.

In most

of the analytical models, the hydraulic head can be obtained directly from

closed-form solutions, however, boundary streamlines of the catchment zone have to be delineated

with indirect or numerical methods due to the complexity of stream functions.

In practice, capture zones are generally delineated with respect to a travel time that ranges from

tens of days to tens of years. For idealized radial ﬂow around a pumping well, simpliﬁed analytical

solution [

12

] and semi-analytical solutions [

13

,

14

] of the travel time have been proposed. For complex

conditions, a widely used approach is to perform a numerical modeling of groundwater ﬂow and then

determine capture zones with the particle tracking method [15,16].

The capture zone of a pumping well in a piedmont pluvial fan is of interest to hydrogeologists

because unconﬁned aquifers of pluvial deposits are common in front of mountains. A typical example

is the Qaidam Basin [

17

], China, where the climate is extremely dry but groundwater gains inﬂow

recharge from the surrounding mountains, as shown in Figure 1a. The general hydrogeological

conditions in the basin are shown in Figure 1b. Streams bring surface water from the mountains

toward the pluvial fans. At the mountain front, the water table in the aquifer is deep and so the streams

leak signiﬁcantly. Streams dry up along a short extending length and all surface water becomes the

supplement of groundwater. It produces a concentrate groundwater inﬂow to the pluvial fan that

has been known as mountain front recharge [

18

]. At the lower edge of the pluvial fan, water table

becomes shallow and discharges to springs and wetlands in the plain area (Figure 1b). The places

of spring form a groundwater discharge line where ﬁne sands and clayey sediments are the major

deposits in comparison with the gravel to coarse sands in the pluvial fan. A pumping well in the

pluvial fan aquifer would capture a portion of the concentrated inﬂow and the catchment zone should

be controlled by several key factors. However, the general features of this kind of capture zones were

not well known in the literature because the non-uniform ﬂow and the special boundary conditions

are diﬀerent from that in the previous models. For example, in the analytical model of Intaraprasong

and Zhan [

9

], the lateral groundwater ﬂow begins from a recharge river with inﬁnite length, which

could not describe the divergent ﬂow driven by the segmental inﬂow from the mountains.

In this study we develop a simpliﬁed model for the catchment and capture zones of a well

pumping groundwater with the mountain front recharge from the segmental inﬂow. Unconﬁned

groundwater ﬂow in this situation is numerically solved with MODFLOW. Particle tracking method is

used to delineate the time-related capture zone and the time-independent catchment zone. The general

shape of the catchment zone and distribution of the travel time are investigated. Sensitivity analyses

were performed to check the impacts of several lumped parameters.

Water 2019,11, 1243 3 of 18

Water 2019, 11, x FOR PEER REVIEW 3 of 19

Figure 1. Characteristics of pluvial fans in the Qaidam Basin: (a) A satellite photo of a pluvial fan,

with delineations of the groundwater inflow segment, discharge zone and potential catchment zone

of a pumping well; (b) A typical profile map of the general hydrogeological conditions (modified

from [17]).

In this study we develop a simplified model for the catchment and capture zones of a well

pumping groundwater with the mountain front recharge from the segmental inflow. Unconfined

groundwater flow in this situation is numerically solved with MODFLOW. Particle tracking method

is used to delineate the time-related capture zone and the time-independent catchment zone. The

general shape of the catchment zone and distribution of the travel time are investigated. Sensitivity

analyses were performed to check the impacts of several lumped parameters.

2. Conceptual and Mathematical Models of Groundwater Flow

2.1. Conceptual Model with Simplifications

The simplified conceptual model proposed in this study is shown in Figure 2. We assume that

the pluvial fans developed in front of mountains and distribute along a straightforward mountain-

Figure 1.

Characteristics of pluvial fans in the Qaidam Basin: (

a

) A satellite photo of a pluvial fan,

with delineations

of the groundwater inﬂow segment, discharge zone and potential catchment zone

of a pumping well; (

b

) A typical proﬁle map of the general hydrogeological conditions (modiﬁed

from [17]).

2. Conceptual and Mathematical Models of Groundwater Flow

2.1. Conceptual Model with Simpliﬁcations

The simpliﬁed conceptual model proposed in this study is shown in Figure 2. We assume that the

pluvial fans developed in front of mountains and distribute along a straightforward mountain-rim

line in a relatively uniform pattern (Figure 2a). The discharge boundary between pluvial fans and

the plain area is simpliﬁed as a line parallel to the mountain-rim with a constant hydraulic head (the

water table reaches the ground surface). Lateral inﬂow of groundwater in pluvial fans is segmentally

distributed at mountain passes. The recharge rate (Q

R

) across each segment is assumed to be equal

and steady.

In the

natural state, groundwater divides are developed between diﬀerent pluvial fans.

Water 2019,11, 1243 4 of 18

In considering of the symmetry, a block of double pluvial fans is selected as an interested study area

(Figure 2b) where the groundwater ﬂow along Divide-2 could be aﬀected by the pumping well.

Water 2019, 11, x FOR PEER REVIEW 5 of 19

Figure 2. Conceptual model of the unconfined aquifer in a pluvial fan: (a) uniform distribution of

pluvial fans in the plan view; (b) a representative area with a pumping well in the gray zone; (c) the

profile between P1 and P2.

2.2. Mathematical Model and Lumped Parameters

According to the assumptions in the conceptual model, the mathematical model of the two-

dimensional groundwater flow in the study area can be given as

()( )

0

ww w

hh

Kh h Qxxyy

xx yy

δδ

∂∂ ∂∂

+−−−=

∂∂ ∂∂

,

3, 0Bx B yL−<< < <

(1)

()

,,

c

hxy h=

3, Bx ByL−≤≤ =

(2)

or 3 , 0xB B yL=− ≤ ≤

(3)

0

2

R

hQ

Kh q

yD

∂==

∂

,

, or, 2 , 0xD x BDy≤−≤=

(4)

0

h

y

∂=

∂

,

, or, 2 , 0DxB Dx BBy<< <− < =

(5)

where K is the hydraulic conductivity [LT−

1

], h is the relative groundwater level [L], L is the distance

between the upper and lower boundaries [L], B and D refer to the half-widths of a fan block and an

inflow segment [L], respectively, along the x-direction, h

c

is the groundwater level on the discharge

boundary [L], q

0

refers to the flow rate across per-unit width of the inflow segment [L

2

T−

1

], Q

R

is the

total inflow from one segment [L

3

T−

1

], Q

w

is the pumping rate of the well [L

3

T−

1

], δ is the Dirac delta

function [L−

1

], x

w

and y

w

are the coordinates of the well [L].

Figure 2.

Conceptual model of the unconﬁned aquifer in a pluvial fan: (

a

) uniform distribution of

pluvial fans in the plan view; (

b

) a representative area with a pumping well in the gray zone; (

c

) the

proﬁle between P1 and P2.

Base rocks in the mountain area are treated as impervious media in comparison with the aquifers

in ﬂuvial fans, whereas each segment of mountain passes forms an inﬂow boundary with uniform

ﬂow rate across a per-length of the segment. A fully penetrating pumping well is located in the zone

between central-lines of the two fans, abstracting groundwater from the aquifer at the constant rate

of Qw.

The additional assumptions of the aquifer and groundwater ﬂow are speciﬁed to:

(1) The bottom of the unconﬁned aquifer is ﬂat and impervious (Figure 2c);

(2) The aquifer media beneath the fan is homogenous and isotropic;

(3) Vertical groundwater ﬂow is ignorable;

(4) Groundwater ﬂow is in steady state and satisﬁes the Darcy law;

(5) No vertical inﬁltration recharge and evapotranspiration loss of groundwater in the study zone;

(6) The radius of the well is suﬃciently small so that it can be treated as a point sink for the

horizontal ﬂow in the aquifer;

(7) The inﬂow rate on the upstream boundary and the hydraulic head on the downstream boundary

are not inﬂuenced by the well, i.e., the place of the discharge line (Figure 1b) is not inﬂuenced by

groundwater pumping;

(8) Divide-1 and Divide-3 are not inﬂuenced by the well, i.e., they are no-ﬂow boundaries as that

in the natural state.

Water 2019,11, 1243 5 of 18

Limitations of the model with these assumptions are discussed on Section 6.

2.2. Mathematical Model and Lumped Parameters

According to the assumptions in the conceptual model, the mathematical model of the

two-dimensional groundwater ﬂow in the study area can be given as

K∂

∂xh∂h

∂x+∂

∂yh∂h

∂y−Qwδ(x−xw)δ(y−yw)=0

−B<x<3B, 0 <y<L,

(1)

h(x,y)=hc,−B≤x≤3B,y=L(2)

x=−Bor 3B, 0 ≤y≤L(3)

Kh ∂h

∂y=q0=QR

2D,|x|≤D, or, |x−2B|≤D,y=0 (4)

∂h

∂y=0, D<|x|<B, or, D<|x−2B|<B,y=0 (5)

where Kis the hydraulic conductivity [LT

−1

], his the relative groundwater level [L], Lis the distance

between the upper and lower boundaries [L], Band Drefer to the half-widths of a fan block and an

inﬂow segment [L], respectively, along the x-direction, h

c

is the groundwater level on the discharge

boundary [L], q

0

refers to the ﬂow rate across per-unit width of the inﬂow segment [L

2

T

−1

], Q

R

is the

total inﬂow from one segment [L

3

T

−1

], Q

w

is the pumping rate of the well [L

3

T

−1

],

δ

is the Dirac delta

function [L−1], xwand yware the coordinates of the well [L].

At the site shown in Figure 1a, the hydrogeological conditions have not been surveyed in detail,

however, we can schematically characterize the shape of the pluvial fan and hydrological conditions as

that in Table 1. The h

c

value is approximately determined as the eﬀective thickness of the gravel to

coarse sands limited by the clayey sediments (Figure 1b). The Q

R

value is speciﬁed from streamﬂow

data of rivers in the eastern Qaidam Basin [

17

]. The Q

w

in fact is the total pumping rate of several

wells that allocated in a relatively small area around the position of (x

w

,y

w

). Pumping wells were

penetrated into the aquifer at a maximum depth around 100 m but can be plausibly considered as

fully penetrating wells in this study. In the modeling investigation, we will check the eﬀect of the well

position by replace (xw,yw) in Table 1with other values that are limited by Band L(Figure 2b).

Table 1. Characteristics of the site in Figure 1a.

B(km) D(km) L(km) hc(m)

6–11 2–4 8–11 100–160

QR(×104m3/d) q0(m2/d) K(m/d) Porosity

28–36 35–90 30–90 0.25–0.31

Qw(×104m3/d) xwyw

6.0–10.0 1.8 4.7

Introducing the following dimensionless variables

xD=x

B,yD=y

B,xwD =xw

B,ywD =yw

B,hD=h

hc, (6)

and lumped dimensionless parameters,

α=D

B,β=L

B,γ=QR

αKh2

c

,λ=Qw

QR

(7)

Water 2019,11, 1243 6 of 18

the mathematical model can be rewritten as

∂

∂xDhD∂hD

∂xD+∂

∂yDhD∂hD

∂yD−αγλδ(xD−xwD)δ(yD−ywD)=0,

−1<xD<3, 0 <yD< β (8)

hD(xD,yD)=1, −1≤xD≤3, yD=β(9)

∂hD

∂xD

=0, xD=−1 or 3, 0 ≤yD≤β(10)

hD

∂hD

∂yD

=γ

2,|xD|≤αor |xD−2|≤α,yD=0 (11)

∂hD

∂yD

=0, α < |xD|<1 or α < |xD−2|<1 , yD=0, (12)

Equations (8)–(12) yield a dimensionless model, indicating that groundwater ﬂow is generally

controlled by the four lumped parameters. The theoretical ranges of the four dimensionless parameters

are: 0 <

α

<1,

β

>0,

γ

>0, 0

≤λ≤

1. At the site of Figure 1a, the highly possible ranges are: 0.2 <

α

<

0.7, 0.7 <β<1.8, 0.2 <γ<5.4, 0.1 <λ<0.4.

In this study, we obtain solutions of the original physical-based model in Equations (1)–(4) with a

numerical method (Section 3.1). This is implemented by ﬁxing B=10 km, K=50 m/d, and h

c

=100 m

and changing the values of D,L,Q

R

, and Q

w

, as well as the x

w

and y

w

values. Then the results are

expressed with dimensionless variables in Equation (6) and lumped parameters in Equation (7) to

represent more general behaviors. Consequently, the result of the spatial distribution of groundwater

level, h(x, y), is expressed as h

D

(x

D

,y

D

) to indicate a solution of Equation (8). Dimensionless results are

useful for analyzing other sites with a diﬀerent size and/or diﬀerent physical parameters.

3. Numerical Methods

3.1. Numerical Solution of Groundwater Flow

The catchment zone of a pumping well has been well investigated by the analytical method.

This approach

has an advantage to obtain closed-form equations of the critical streamlines but limited

in a few of simple conditions. In addition, most of the analytical models are false to provide closed-form

equations for the travel time. For the mathematical model in Section 2.2, it is very diﬃcult to derive

an analytical solution of h(x,y), even more diﬃcult in obtaining analytical formulas of the catchment

zone and the travel time. Alternatively, we can suﬃciently obtain the numerical solution with eﬀective

simulation tools. In this study, the mathematical model is numerically solved with MODFLOW [

19

,

20

],

a widely adopted simulation tool for groundwater ﬂow modeling. A single-layer cell-centered grid is

generated to prepare data for the numerical model based on the ﬁnite-diﬀerence method.

Each cell

in the grid is square-shape in plane with the size of

∆

x=50 m. The aquifer type is speciﬁed to

“unconﬁned” in constructing the MODFLOW model. The inﬂow boundary segment with a constant

ﬂow rate is equivalently implemented by using the WELL package [

19

]. A virtual well with the

injecting rate of q

0∆

xis settled in each cell along this boundary, which is an equivalent implement of

Equation (4). The pumping rate Qwis speciﬁed to a well-block including the place of (xw,yw).

The PCG2 package [

21

] is chosen from the solver packages of MODFLOW to solve the

ﬁnite-diﬀerence equations. Criteria for the convergence are speciﬁed to: head change is

0.001 m;

the residual

of cell-by-cell ﬂow is 0.01m

3

/d. The grid resolution and the criteria have been checked

to obtain accurate modeling results without excessive computational cost. The dimensionless result,

hD(xD,yD), are obtained from the MODFLOW output, h(x, y), by using Equation (6).

Water 2019,11, 1243 7 of 18

3.2. Particle Tracking Method for Streamlines and Travel Time

We use MODPATH to observe the ﬂow-net and delineate the capture zone from the simulation

results of the MODFLOW model. It was developed on the basis of a semi-analytical particle tracking

method [

22

]. The new version, MODPATH version 6, improves the performance and output for

post-processing [

23

]. The pathline and step-by-step travel time of a particle can be extracted from the

MODPATH output. The catchment zone of a pumping well is delineated through a group of pathlines

that suﬃciently link the source on the boundaries and the well. The capture zone is a part within

the catchment zone where a particle can move to the well along a pathline in a period that is less

than a speciﬁed travel-time. In practice, 100 particles are placed on all sides of a well-cell and then

the pathlines of them are determined by backward tracking. Shafer [

15

] pointed out that 100 to 300

particles are generally suﬃcient to encompass the entire capture zone of a pumping well. We check the

use of diﬀerent numbers of particles and ﬁnd that 100 particles are enough in this studied model.

In this study, we attempt to obtain general results of the travel time with respect to the dimensionless

model in Equations (8)–(12), by introducing a dimensionless travel time. For a particle moving forward

along a streamline, the velocity is determined by the Darcy law as follows:

dl

dt =−K

φ

dh

dl =−Khc

φB

dhD

dlD

(13)

where lis the travel length along the streamline [L], dl is a small increase in travel length [L] with

respect to a small increase in the real travel time dt [T],

ϕ

is the eﬀective porosity [-], l

D

refers to the

dimensionless travel length (=l/B). The travel time is an integral of that:

t=Zl

0

1

(dl/dt)dl =BZlD

0

1

(dl/dt)dlD(14)

Substituting Equation (13) into Equation (14), we have:

t=−φB2

KhcZlD

0

1

(dhD/dlD)dlD(15)

To present the results in a general manner, the dimensionless travel time is deﬁned as

tD=−ZlD

0

1

(dhD/dlD)dlD=Khc

φB2t(16)

Accordingly, the MODPATH output can be also transformed into dimensionless results.

4. Results of the Catchment Zone

4.1. General Shape and Classiﬁcation

The characteristics of groundwater ﬂow ﬁeld in the modeling area and the shape of the

catchment zone

, as indicated by the simulation results, are highly dependent on the well location.

Figures 3–6show typical results for speciﬁed parameters:

α

=0.4,

β

=1,

γ

=1, and

λ

=0.2. When the

well is not far away from the Y-axis, the disturbance of the natural divergent ﬂow would be signiﬁcantly

limited in the zone between Divide-1 and Divide-2, as in the case of Figures 3and 4. Otherwise,

the catchment

zone could extend to the neighboring fan. We classiﬁed the shape of the catchment zone

into four types according to the hydraulic connection between the well and boundaries:

Type-I, the pumping well captures water from only one of the two inﬂow segments that is closer

to the well, as shown in Figure 3b;

Type-II, the pumping well captures water from the constant head discharge boundary and one of

the inﬂow segments that is closer to the well, as shown in Figure 4b;

Water 2019,11, 1243 8 of 18

Type-III, the pumping well captures water from both of the two inﬂow segments without

abstracting water from the constant head boundary, as shown in Figure 5b;

Type-IV, the pumping well captures water from both of the two inﬂow segments and the constant

head boundary, as shown in Figure 6b.

Water 2019, 11, x FOR PEER REVIEW 8 of 19

00

11

(/) (/)

D

ll

D

tdlBdl

dl dt dl dt

==

(14)

Substituting Equation (13) into Equation (14), we have:

2

0

1

(/)

D

l

D

cDD

B

tdl

Kh dh dl

φ

=−

(15)

To present the results in a general manner, the dimensionless travel time is defined as

2

0

1

(/)

D

lc

DD

DD

Kh

tdlt

dh dl B

φ

=− =

(16)

Accordingly, the MODPATH output can be also transformed into dimensionless results.

4. Results of the Catchment Zone

4.1. General Shape and Classification

The characteristics of groundwater flow field in the modeling area and the shape of the

catchment zone, as indicated by the simulation results, are highly dependent on the well location.

Figures 3–6 show typical results for specified parameters: α = 0.4, β = 1, γ = 1, and λ = 0.2. When the

well is not far away from the Y-axis, the disturbance of the natural divergent flow would be

significantly limited in the zone between Divide-1 and Divide-2, as in the case of Figures 3 and 4.

Otherwise, the catchment zone could extend to the neighboring fan. We classified the shape of the

catchment zone into four types according to the hydraulic connection between the well and

boundaries:

Type-I, the pumping well captures water from only one of the two inflow segments that is closer

to the well, as shown in Figure 3b;

Type-II, the pumping well captures water from the constant head discharge boundary and one

of the inflow segments that is closer to the well, as shown in Figure 4b;

Type-III, the pumping well captures water from both of the two inflow segments without

abstracting water from the constant head boundary, as shown in Figure 5b;

Type-IV, the pumping well captures water from both of the two inflow segments and the

constant head boundary, as shown in Figure 6b.

Figure 3. The flow net (a) and catchment zone (b) of type-I.

Figure 4. The flow net (a) and catchment zone (b) of type-II.

Figure 3. The ﬂow net (a) and catchment zone (b) of type-I.

Water 2019, 11, x FOR PEER REVIEW 8 of 19

00

11

(/) (/)

D

ll

D

tdlBdl

dl dt dl dt

==

(14)

Substituting Equation (13) into Equation (14), we have:

2

0

1

(/)

D

l

D

cDD

B

tdl

Kh dh dl

φ

=−

(15)

To present the results in a general manner, the dimensionless travel time is defined as

2

0

1

(/)

D

lc

DD

DD

Kh

tdlt

dh dl B

φ

=− =

(16)

Accordingly, the MODPATH output can be also transformed into dimensionless results.

4. Results of the Catchment Zone

4.1. General Shape and Classification

The characteristics of groundwater flow field in the modeling area and the shape of the

catchment zone, as indicated by the simulation results, are highly dependent on the well location.

Figures 3–6 show typical results for specified parameters: α = 0.4, β = 1, γ = 1, and λ = 0.2. When the

well is not far away from the Y-axis, the disturbance of the natural divergent flow would be

significantly limited in the zone between Divide-1 and Divide-2, as in the case of Figures 3 and 4.

Otherwise, the catchment zone could extend to the neighboring fan. We classified the shape of the

catchment zone into four types according to the hydraulic connection between the well and

boundaries:

Type-I, the pumping well captures water from only one of the two inflow segments that is closer

to the well, as shown in Figure 3b;

Type-II, the pumping well captures water from the constant head discharge boundary and one

of the inflow segments that is closer to the well, as shown in Figure 4b;

Type-III, the pumping well captures water from both of the two inflow segments without

abstracting water from the constant head boundary, as shown in Figure 5b;

Type-IV, the pumping well captures water from both of the two inflow segments and the

constant head boundary, as shown in Figure 6b.

Figure 3. The flow net (a) and catchment zone (b) of type-I.

Figure 4. The flow net (a) and catchment zone (b) of type-II.

Figure 4. The ﬂow net (a) and catchment zone (b) of type-II.

Water 2019, 11, x FOR PEER REVIEW 9 of 19

Figure 5. The flow net (a) and catchment zone (b) of type-III.

In most of the situations, the pumping well does not significantly change the divergent flow at

the regional scale, as shown in Figures 3a, 4a, 5a, and 6a, but locally reshapes the streamlines to

produce a concentrated flow around the well. This local influence limited the size of the catchment

zone in an area with the length along the X-axis that is smaller than 2B. The width of the type-I and

type-II catchment zones in the X-direction could be even smaller than 2D, as shown in Figures 3b and

4b. The maximum length of the catchment zone in the Y-direction will reach to L in the type-II and

type-IV cases, as shown in Figures 4b and 6b.

Figure 6. The flow net (a) and catchment zone (b) of type-IV.

The relationship between the well location and the type of the catchment zone can be shown

with a map of well-location zones (Figure 7). For example, when the pumping well is located in the

II zone, a type-II catchment zone will be generated. The well-location zones symmetrically distribute

on sides of the Y-axis and Divide-2 due to the symmetry of the conceptual model. Therefore, we can

focus the investigation on the distribution of well-location zones in a typical area between the Y-axis

and Divide-2.

Figure 7. General well-location zones with respect to different types (from I to IV) of the catchment

zone.

Distribution of the well location zone depends on the control parameters, α, β, γ, and λ, that

defined in Equation (7). In particular, Figure 8 shows the change in the well-location zones with the

increasing λ value. An increase in the λ value indicates a relative increase in the well pumping rate

(Q

w

), which will reduce the area of the I zone and enlarge the area of the IV zone. When the λ value

is small, like that shown in Figure 8a, the II and IV zones will be limited in small areas in the vicinity

of the discharge boundary. While if the λ value is close or equal to 1.0, the I zone will significantly

shrink to a narrow small area that encloses a short part of the Y-axis, as shown in Figure 8d. The IV

zone extends to the X-axis in this situation, indicating that the constant head boundary could be

connected with the catchment zone for a well pumped heavy even it is very close to the X-axis.

Figure 5. The ﬂow net (a) and catchment zone (b) of type-III.

Water 2019, 11, x FOR PEER REVIEW 9 of 19

Figure 5. The flow net (a) and catchment zone (b) of type-III.

In most of the situations, the pumping well does not significantly change the divergent flow at

the regional scale, as shown in Figures 3a, 4a, 5a, and 6a, but locally reshapes the streamlines to

produce a concentrated flow around the well. This local influence limited the size of the catchment

zone in an area with the length along the X-axis that is smaller than 2B. The width of the type-I and

type-II catchment zones in the X-direction could be even smaller than 2D, as shown in Figures 3b and

4b. The maximum length of the catchment zone in the Y-direction will reach to L in the type-II and

type-IV cases, as shown in Figures 4b and 6b.

Figure 6. The flow net (a) and catchment zone (b) of type-IV.

The relationship between the well location and the type of the catchment zone can be shown

with a map of well-location zones (Figure 7). For example, when the pumping well is located in the

II zone, a type-II catchment zone will be generated. The well-location zones symmetrically distribute

on sides of the Y-axis and Divide-2 due to the symmetry of the conceptual model. Therefore, we can

focus the investigation on the distribution of well-location zones in a typical area between the Y-axis

and Divide-2.

Figure 7. General well-location zones with respect to different types (from I to IV) of the catchment

zone.

Distribution of the well location zone depends on the control parameters, α, β, γ, and λ, that

defined in Equation (7). In particular, Figure 8 shows the change in the well-location zones with the

increasing λ value. An increase in the λ value indicates a relative increase in the well pumping rate

(Q

w

), which will reduce the area of the I zone and enlarge the area of the IV zone. When the λ value

is small, like that shown in Figure 8a, the II and IV zones will be limited in small areas in the vicinity

of the discharge boundary. While if the λ value is close or equal to 1.0, the I zone will significantly

shrink to a narrow small area that encloses a short part of the Y-axis, as shown in Figure 8d. The IV

zone extends to the X-axis in this situation, indicating that the constant head boundary could be

connected with the catchment zone for a well pumped heavy even it is very close to the X-axis.

Figure 6. The ﬂow net (a) and catchment zone (b) of type-IV.

In most of the situations, the pumping well does not signiﬁcantly change the divergent ﬂow at the

regional scale, as shown in Figures 3a, 4a, 5a and 6a, but locally reshapes the streamlines to produce

a concentrated ﬂow around the well. This local inﬂuence limited the size of the catchment zone in

an area with the length along the X-axis that is smaller than 2B. The width of the type-I and type-II

catchment zones in the X-direction could be even smaller than 2D, as shown in Figures 3b and 4b.

The maximum

length of the catchment zone in the Y-direction will reach to Lin the type-II and type-IV

cases, as shown in Figures 4b and 6b.

The relationship between the well location and the type of the catchment zone can be shown

with a map of well-location zones (Figure 7). For example, when the pumping well is located in the

II zone

, a type-II catchment zone will be generated. The well-location zones symmetrically distribute

on sides of the Y-axis and Divide-2 due to the symmetry of the conceptual model. Therefore, we can

Water 2019,11, 1243 9 of 18

focus the investigation on the distribution of well-location zones in a typical area between the Y-axis

and Divide-2.

Water 2019, 11, x FOR PEER REVIEW 9 of 19

Figure 5. The flow net (a) and catchment zone (b) of type-III.

In most of the situations, the pumping well does not significantly change the divergent flow at

the regional scale, as shown in Figures 3a, 4a, 5a, and 6a, but locally reshapes the streamlines to

produce a concentrated flow around the well. This local influence limited the size of the catchment

zone in an area with the length along the X-axis that is smaller than 2B. The width of the type-I and

type-II catchment zones in the X-direction could be even smaller than 2D, as shown in Figures 3b and

4b. The maximum length of the catchment zone in the Y-direction will reach to L in the type-II and

type-IV cases, as shown in Figures 4b and 6b.

Figure 6. The flow net (a) and catchment zone (b) of type-IV.

The relationship between the well location and the type of the catchment zone can be shown

with a map of well-location zones (Figure 7). For example, when the pumping well is located in the

II zone, a type-II catchment zone will be generated. The well-location zones symmetrically distribute

on sides of the Y-axis and Divide-2 due to the symmetry of the conceptual model. Therefore, we can

and Divide-2.

Figure 7. General well-location zones with respect to different types (from I to IV) of the catchment

zone.

Distribution of the well location zone depends on the control parameters, α, β, γ, and λ, that

defined in Equation (7). In particular, Figure 8 shows the change in the well-location zones with the

increasing λ value. An increase in the λ value indicates a relative increase in the well pumping rate

(Q

w

), which will reduce the area of the I zone and enlarge the area of the IV zone. When the λ value

is small, like that shown in Figure 8a, the II and IV zones will be limited in small areas in the vicinity

of the discharge boundary. While if the λ value is close or equal to 1.0, the I zone will significantly

shrink to a narrow small area that encloses a short part of the Y-axis, as shown in Figure 8d. The IV

zone extends to the X-axis in this situation, indicating that the constant head boundary could be

connected with the catchment zone for a well pumped heavy even it is very close to the X-axis.

Figure 7.

General well-location zones with respect to diﬀerent types (from I to IV) of the catchment zone.

Distribution of the well location zone depends on the control parameters,

α

,

β

,

γ

, and

λ

,

that deﬁned

in Equation (7). In particular, Figure 8shows the change in the well-location zones with the increasing

λ

value. An increase in the

λ

value indicates a relative increase in the well pumping rate (Q

w

), which

will reduce the area of the I zone and enlarge the area of the IV zone. When the

λ

value is small,

like that shown in Figure 8a, the II and IV zones will be limited in small areas in the vicinity of the

discharge boundary. While if the

λ

value is close or equal to 1.0, the I zone will signiﬁcantly shrink to a

narrow small area that encloses a short part of the Y-axis, as shown in Figure 8d. The IV zone extends

to the X-axis in this situation, indicating that the constant head boundary could be connected with the

catchment zone for a well pumped heavy even it is very close to the X-axis.

Water 2019, 11, x FOR PEER REVIEW 10 of 19

Figure 8. Change in well-location zones with different values of λ when the other parameters are

specified as α = 0.4, β = 1 and γ = 1: (a) λ = 0.1; (b); λ = 0.3; (c): λ = 0.5; (d): λ = 1.0.

4.2. Dependency of Shape Factors on Controls

The shape of a catchment zone can be characterized by several geometric elements. For the type-

I and type-III catchment zones with a well on the Y-axis or Divide-2, typical shape factors are defined

as that shown in Figure 9. The type-I catchment zone in Figure 9a collects water from a single inflow

segment where the source head width is d [L]. In fact, the d value is determined by the necessary

contribution of recharge from the source head to the well, which can be expressed as

w

0

==2

Q

dB

q

αλ

(17)

A line across the well center and parallel to the X-axis within the catchement zone characterizes

the size of the catchment zone near the well, which has a length of w [L]. The ratio w/d is a

dimensionless shape factor indicating the transectional expansion of the catchment zone along the

flow path from the source head to the well. The distance from the well center to the stagnation point

is defined as r [L]. The ratio r/d is another dimensionless shape factor indicating the expansion of the

catchment zone from the well to the natural downstream area. The type-III catchment zone in Figure

9b collects water from two inflow segments on the sides of Divide-2. The source head width along

the left segment is d

1

whereas the source head width along the right segment is d

2

. The total length of

the source head is defined as d

1

+ d

2

, which satisfies the following equation:

12

+=2dd B

αλ

(18)

The size of the catchment zone at the well, w, and the distance of the stagnation point, r, are

defined in Figure 9b. In this situation, the dimensionless shape factors are defined by w/(d

1

+ d

2

) and

r/(d

1

+ d

2

).

Figure 8.

Change in well-location zones with diﬀerent values of

λ

when the other parameters are

speciﬁed as α=0.4, β=1 and γ=1: (a)λ=0.1; (b); λ=0.3; (c): λ=0.5; (d): λ=1.0.

4.2. Dependency of Shape Factors on Controls

The shape of a catchment zone can be characterized by several geometric elements. For the type-I

and type-III catchment zones with a well on the Y-axis or Divide-2, typical shape factors are deﬁned as

that shown in Figure 9. The type-I catchment zone in Figure 9a collects water from a single inﬂow

Water 2019,11, 1243 10 of 18

segment where the source head width is d[L]. In fact, the dvalue is determined by the necessary

contribution of recharge from the source head to the well, which can be expressed as

d=Qw

q0

=2Bαλ (17)

Water 2019, 11, x FOR PEER REVIEW 11 of 19

Figure 9. Shape factors of a symmetrical catchment zone: (a) type-I; (b) type-III.

Shape factors of the catchment zone are controlled by several parameters, such as α, β, γ, and λ,

and also dependent on the well location. In Figure 10, the relationship between shape factors and key

parameters are illustrated from results of the sensitivity analysis for the Type-I zone. As shown in

Figure 10a,b, w/d increases between 0.5 and 2.5, nonlinearly with the decreasing α value between 0.2

and 1.0. Since the α value refers to the relative width of an inflow segment, this relationship indicates

that a shorter inflow segment will lead to a larger transectional expansion of the catchment zone

when groundwater flow toward the well. An increase in the λ value (relative pumping rate) will also

increase w/d, as shown in Figure 10a. The w/d value seems to be not sensitive to the change in the β

value (relative distance between the inflow and discharge boundaries), as indicated in Figure 10b,

even the β value has a slight negative influence. The impacts of α and λ on w/d are also exhibited with

the curves in Figure 10c,d, respectively, for the relationship between w/d and the well-location

represented by y

wD

. It can be seen that w/d is positively correlated with the y

wD

value, almost in a linear

manner. The r/d value is generally less than 1.0 and also increases with the decreasing α value as

shown in Figure 10e, indicating that the distance from the well to the stagnation point is generally

less than the width of source head. Similar to the relationship between w/d and y

wD

, r/d increases with

the increasing y

wD

value, as shown in Figure 10f, however, the relationship becomes nonlinear when

the λ value is large.

Figure 9. Shape factors of a symmetrical catchment zone: (a) type-I; (b) type-III.

A line across the well center and parallel to the X-axis within the catchement zone characterizes the

size of the catchment zone near the well, which has a length of w[L]. The ratio w/dis a dimensionless

shape factor indicating the transectional expansion of the catchment zone along the ﬂow path from the

source head to the well. The distance from the well center to the stagnation point is deﬁned as r[L].

The ratio r/dis another dimensionless shape factor indicating the expansion of the catchment zone

from the well to the natural downstream area. The type-III catchment zone in Figure 9b collects water

from two inﬂow segments on the sides of Divide-2. The source head width along the left segment is d

1

whereas the source head width along the right segment is d

2

. The total length of the source head is

deﬁned as d1+d2, which satisﬁes the following equation:

d1+d2=2Bαλ (18)

The size of the catchment zone at the well, w, and the distance of the stagnation point, r,

are deﬁned

in Figure 9b. In this situation, the dimensionless shape factors are deﬁned by w/(d

1

+d

2

) and r/(d

1

+d

2

).

Shape factors of the catchment zone are controlled by several parameters, such as

α

,

β

,

γ

, and

λ

,

and also dependent on the well location. In Figure 10, the relationship between shape factors and key

parameters are illustrated from results of the sensitivity analysis for the Type-I zone. As shown in

Figure 10a,b, w/dincreases between 0.5 and 2.5, nonlinearly with the decreasing

α

value between 0.2

and 1.0. Since the

α

value refers to the relative width of an inﬂow segment, this relationship indicates

that a shorter inﬂow segment will lead to a larger transectional expansion of the catchment zone when

groundwater ﬂow toward the well. An increase in the

λ

value (relative pumping rate) will also increase

w/d, as shown in Figure 10a. The w/dvalue seems to be not sensitive to the change in the

β

value

(relative distance between the inﬂow and discharge boundaries), as indicated in Figure 10b, even the

β

value has a slight negative inﬂuence. The impacts of

α

and

λ

on w/dare also exhibited with the curves

in Figure 10c,d, respectively, for the relationship between w/dand the well-location represented by y

wD

.

It can be seen that w/dis positively correlated with the y

wD

value, almost in a linear manner. The r/d

value is generally less than 1.0 and also increases with the decreasing

α

value as shown in Figure 10e,

indicating that the distance from the well to the stagnation point is generally less than the width of

source head. Similar to the relationship between w/dand y

wD

,r/dincreases with the increasing y

wD

value, as shown in Figure 10f, however, the relationship becomes nonlinear when the

λ

value is large.

Water 2019,11, 1243 11 of 18

Water 2019, 11, x FOR PEER REVIEW 12 of 19

Figure 10. Dependency of dimensionless shape factors on control parameters for x

wD

= 0 in the type-I

catchment zone: (a) Curves of w/d versus

α

with different

λ

values; (b) Curves of w/d versus

α

with

different

β

values; (c) Curves of w/d versus y

wD

with different

α

values; (d) Curves of w/d versus y

wD

with different

λ

values; (e) Curves of r/d versus

α

with different

λ

values; (f) Curves of r/d versus y

wD

with different

λ

values.

The impacts of control parameters and the well location on shape factors of the type-III

catchment zone are shown in Figure 11. The w/(d

1

+ d

2

) value could be significantly larger than 1.0,

indicating that the width of the catchment zone at the well center could be significantly larger than

the effective width of the source head. Both Figure 11a,b show that w/(d

1

+ d

2

) increases with the

decreasing α value in a nonlinear manner. The w/(d

1

+ d

2

) value also increases with the increasing λ

value as shown in Figure 11a, however, it is not sensitive to the change in the β value as shown in

Figure 11b. The relationship between w/(d

1

+ d

2

) and y

wD

is negative and nonlinear, as is clearly shown

in Figure 11c,d. In particular, Figure 11c indicates that a smaller α value leads to a larger range of

w/(d

1

+ d

2

) with respect to the same range of y

wD

. This effect seems can be also leaded by the change

in the λ value, as shown in Figure 11d, whereas the impact is not significant. Figure 11e exhibits the

negative nonlinear relationship between r/(d

1

+ d

2

) and

α

, where r/(d

1

+ d

2

) is less than 1.0 in most of

the situations. The relationship between r/(d

1

+ d

2

) and y

wD

is a bit complex as shown in Figure 11f

where r/(d

1

+ d

2

) does not simply increase with the decreasing y

wD

value but they could have a positive

relationship when y

wD

is high, especially for situations of a large λ value. In particular, r/(d

1

+ d

2

) is

not sensitive to the change in the λ value when y

wD

is sufficiently small (less than 0.4).

Figure 10.

Dependency of dimensionless shape factors on control parameters for x

wD

=0 in the type-I

catchment zone: (

a

) Curves of w/dversus

α

with diﬀerent

λ

values; (

b

) Curves of w/dversus

α

with

diﬀerent

β

values; (

c

) Curves of w/dversus y

wD

with diﬀerent

α

values; (

d

) Curves of w/dversus y

wD

with diﬀerent

λ

values; (

e

) Curves of r/dversus

α

with diﬀerent

λ

values; (

f

) Curves of r/dversus y

wD

with diﬀerent λvalues.

The impacts of control parameters and the well location on shape factors of the type-III catchment

zone are shown in Figure 11. The w/(d

1

+d

2

) value could be signiﬁcantly larger than 1.0, indicating

that the width of the catchment zone at the well center could be signiﬁcantly larger than the eﬀective

width of the source head. Both Figure 11a,b show that w/(d

1

+d

2

) increases with the decreasing

α

value in a nonlinear manner. The w/(d

1

+d

2

) value also increases with the increasing

λ

value as

shown in Figure 11a, however, it is not sensitive to the change in the

β

value as shown in Figure 11b.

The relationship

between w/(d

1

+d

2

) and y

wD

is negative and nonlinear, as is clearly shown in

Figure 11c,d. In particular, Figure 11c indicates that a smaller

α

value leads to a larger range of w/(d

1

+d

2

) with respect to the same range of y

wD

. This eﬀect seems can be also leaded by the change in

the

λ

value, as shown in Figure 11d, whereas the impact is not signiﬁcant. Figure 11e exhibits the

negative nonlinear relationship between r/(d

1

+d

2

) and

α

, where r/(d

1

+d

2

) is less than 1.0 in most of

the situations. The relationship between r/(d

1

+d

2

) and y

wD

is a bit complex as shown in Figure 11f

where r/(d

1

+d

2

) does not simply increase with the decreasing y

wD

value but they could have a positive

relationship when y

wD

is high, especially for situations of a large

λ

value. In particular, r/(d

1

+d

2

) is

not sensitive to the change in the λvalue when ywD is suﬃciently small (less than 0.4).

Water 2019,11, 1243 12 of 18

Water 2019, 11, x FOR PEER REVIEW 13 of 19

Figure 11. Dependency of dimensionless shape factors on control parameters for x

wD

= 1 in the type-

III catchment zone:(a) Curves of w/d versus

α

with different

λ

values; (b) Curves of w/(d

1

+ d

2

) versus

α

with different

β

values; (c) Curves of w/(d

1

+ d

2

) versus y

wD

with different

α

values; (d) Curves of

w/(d

1

+ d

2

) versus y

wD

with different

λ

values; (e) Curves of r/(d

1

+ d

2

) versus

α

with different

λ

values;

(f) Curves of r/(d

1

+ d

2

) versus y

wD

with different

λ

values.

5. Travel Time Analyses for Capture Zones

5.1. General Travel Time Distribution

Distributions of the dimensionless travel time around the pumping well are typically shown in

Figure 12 for the four types of the catchment zone. In general, the travel time increases with the

distance from the starting point to the well but the change patterns are not uniform. When control

parameters are fixed, the maximum travel time for particles on the source head increases when the

catchment zone changes from type-I to type-IV, due to the increase in the maximum length of

streamlines linking the source head and the well. A capture zone is enclosed by a contour of travel

time (t

D

= 1, 2, 3, etc.) in the map. In the type-I and type-II catchment zones shown in Figure 12a,b,

the capture zone could be significantly stretched to the upstream area along the middle line between

the sides of the catchment zone. In the type-III (Figure 12c) and type-IV (Figure 12d) catchment zones,

this stretch effect also exists for relatively small travel times, whereas the capture zone will be

stretched to the upstream area along the sides of the catchment zone for relatively large travel times,

because the double source heads are not in the middle. The capture zone will be preferentially

stretched toward a closer recharge boundary when the pumping well is not rightly located at Divide-

2.

Figure 11.

Dependency of dimensionless shape factors on control parameters for x

wD

=1 in the type-III

catchment zone:(

a

) Curves of w/dversus

α

with diﬀerent

λ

values; (

b

) Curves of w/(d

1

+d

2

) versus

α

with diﬀerent

β

values; (

c

) Curves of w/(d

1

+d

2

) versus y

wD

with diﬀerent

α

values; (

d

) Curves of

w/(d

1

+d

2

) versus y

wD

with diﬀerent

λ

values; (

e

) Curves of r/(d

1

+d

2

) versus

α

with diﬀerent

λ

values;

(f) Curves of r/(d1+d2) versus ywD with diﬀerent λvalues.

5. Travel Time Analyses for Capture Zones

5.1. General Travel Time Distribution

Distributions of the dimensionless travel time around the pumping well are typically shown in

Figure 12 for the four types of the catchment zone. In general, the travel time increases with the distance

from the starting point to the well but the change patterns are not uniform. When control parameters

are ﬁxed, the maximum travel time for particles on the source head increases when the catchment

zone changes from type-I to type-IV, due to the increase in the maximum length of streamlines linking

the source head and the well. A capture zone is enclosed by a contour of travel time (t

D

=1, 2, 3,

etc.) in

the map

. In the type-I and type-II catchment zones shown in Figure 12a,b, the capture zone

could be signiﬁcantly stretched to the upstream area along the middle line between the sides of the

catchment zone.

In the type-III (Figure 12c) and type-IV (Figure 12d) catchment zones, this stretch

eﬀect also exists for relatively small travel times, whereas the capture zone will be stretched to the

upstream area along the sides of the catchment zone for relatively large travel times, because the

double source heads are not in the middle. The capture zone will be preferentially stretched toward a

closer recharge boundary when the pumping well is not rightly located at Divide-2.

Water 2019,11, 1243 13 of 18

Water 2019, 11, x FOR PEER REVIEW 14 of 19

Figure 12. Travel time (t

D

, dimensionless) distribution in the catchment zone: (a) type-I; (b) type-II; (c)

type-III; (d) type-IV. Dashed lines are the contours of t

D

. Parameter values are specified in the model

as

α

= 0.4, β = 1, γ = 1, λ = 0.2.

5.2. The Relationship between the Travel Time and the Size Factor

The shape and area of a capture zone are not only dependent on the well location and control

parameters but also dependent on the specified travel time. To quantitatively analyze the controls of

the capture zone, shape factors are required for description. Without significant loss of general, we

investigate symmetrical capture zones in the type-I and type-III catchments with a size factor, R, that

defined in Figures 13a and 14a, respectively. The axis of symmetry for the type-I catchment zone is

the Y-axis, which is also shown in Figure 9a. For the type-III catchment zone, the axis of symmetry is

Divide-2, which is also shown in Figure 9b. The contour of a specified travel time, t

D

, and the

symmetry axis have an intersection point. R denotes the distance between this intersection point and

the well. It certainly increases with the increasing t

D

as a function whereas the function is controlled

by the well location and parameters.

0

5

10

15

2

0

01234

y

wD

=0.4

y

wD

=0.5

y

wD

=0.6

λ=0.3, γ=1,

β=1, α=0.4

0

5

10

15

20

01234

λ=0.1

λ=0.3

λ=0.5

α=0.4, γ=1,

β=1, y

wD

=0.6

0

5

10

15

20

01234

α=0.2

α=0.4

α=0.6

λ=0.3, γ=1,

β=1, y

wD

=0.6

Y

X

(a)

Wel l

R

Capture zone of t

D

d

(b)

(d)

R/d

R/d

γt

D

/(

α

λ)

γt

D

/(

α

λ)

(c)

R/d

γt

D

/(

α

λ)

Figure 12.

Travel time (t

D

, dimensionless) distribution in the catchment zone: (

a

) type-I; (

b

) type-II;

(c) type-III;

(

d

) type-IV. Dashed lines are the contours of t

D

. Parameter values are speciﬁed in the model

as α=0.4, β=1, γ=1, λ=0.2.

5.2. The Relationship between the Travel Time and the Size Factor

The shape and area of a capture zone are not only dependent on the well location and control

parameters but also dependent on the speciﬁed travel time. To quantitatively analyze the controls

of the capture zone, shape factors are required for description. Without signiﬁcant loss of general,

we investigate

symmetrical capture zones in the type-I and type-III catchments with a size factor, R,

that deﬁned in Figures 13a and 14a, respectively. The axis of symmetry for the type-I catchment zone is

the Y-axis, which is also shown in Figure 9a. For the type-III catchment zone, the axis of symmetry

is Divide-2, which is also shown in Figure 9b. The contour of a speciﬁed travel time, t

D

, and the

symmetry axis have an intersection point. Rdenotes the distance between this intersection point and

the well. It certainly increases with the increasing t

D

as a function whereas the function is controlled

by the well location and parameters.

Water 2019, 11, x FOR PEER REVIEW 14 of 19

Figure 12. Travel time (t

D

, dimensionless) distribution in the catchment zone: (a) type-I; (b) type-II; (c)

type-III; (d) type-IV. Dashed lines are the contours of t

D

. Parameter values are specified in the model

as

α

= 0.4, β = 1, γ = 1, λ = 0.2.

5.2. The Relationship between the Travel Time and the Size Factor

The shape and area of a capture zone are not only dependent on the well location and control

parameters but also dependent on the specified travel time. To quantitatively analyze the controls of

the capture zone, shape factors are required for description. Without significant loss of general, we

investigate symmetrical capture zones in the type-I and type-III catchments with a size factor, R, that

defined in Figures 13a and 14a, respectively. The axis of symmetry for the type-I catchment zone is

the Y-axis, which is also shown in Figure 9a. For the type-III catchment zone, the axis of symmetry is

Divide-2, which is also shown in Figure 9b. The contour of a specified travel time, t

D

, and the

symmetry axis have an intersection point. R denotes the distance between this intersection point and

the well. It certainly increases with the increasing t

D

as a function whereas the function is controlled

by the well location and parameters.

0

5

10

15

2

0

01234

y

wD

=0.4

y

wD

=0.5

y

wD

=0.6

λ=0.3, γ=1,

β=1, α=0.4

0

5

10

15

20

01234

λ=0.1

λ=0.3

λ=0.5

α=0.4, γ=1,

β=1, y

wD

=0.6

0

5

10

15

20

01234

α=0.2

α=0.4

α=0.6

λ=0.3, γ=1,

β=1, y

wD

=0.6

Y

X

(a)

Wel l

R

Capture zone of t

D

d

(b)

(d)

R/d

R/d

γt

D

/(

α

λ)

γt

D

/(

α

λ)

(c)

R/d

γt

D

/(

α

λ)

Figure 13.

The size factor of a capture zone (

a

) with limited travel time in a type-I catchment for a well

on the Y-axis and the curves of γtD/(αλ) versus R/dwith varying α(b), λ(c) and ywD (d).

Water 2019,11, 1243 14 of 18

Water 2019, 11, x FOR PEER REVIEW 16 of 19

that the bound of R/(d1 + d2) decrease in Figure 14b,c, respectively, with the increasing

α

and λ values.

However, an increase in the ywD value will enlarge the range of R and consequently increase the bound

of R/(d1 + d2), as shown in Figure 14d.

Figure 14. The size factor of a capture zone (a) with limited travel time in a Type-III catchment for a

well on Divide-2 and the curves of γtD/(

α

λ) versus R/d with varying

α

(b), λ (c) and ywD (d).

6. Discussions on the Application

The modeling results with dimensionless variables in this study represent general behaviors of

the catchment zone and capture zones of a pumping well with hydrogeological conditions that

illustrated in Figure 2. It can be applied in such conditions to determine the type and the shape of the

catchment zone for a well in water supply projects. In the example of Figure 1a with parameters in

Table 1, the pumping site is generally located in the type-I zone so that the shape of the catchment

zone is similar to that shown in Figure 3b. The width of the catchment zone on the source head, d,

can be estimated with Equation (16), which is 0.7–2.9 km. The w/d value is larger than but close to d.

General results in this study can be used to optimize the location of a pumping site and to

delineate protection zones for different security levels. Three major points should be concerned:

First of all, it is noticeable that the catchment zone would be one of the four types shown in

Figures 3–6. The type-I catchment has a single source head, whereas the others have two (type-II,

type-III) or three (type-IV) source heads. An increase in the number of source heads would increase

the risk of contamination on water supply if all the source heads could be probably influenced by

pollutions. It would also increase the cost of groundwater monitoring because all the source heads

should be monitored to achieve an effective response in time of pollution events. Thus, the type-I

catchment is recommended for safety. As indicated in Figures 7 and 8, the type of the catchment is

highly dependent on the location and pumping rate of the well. When the pumping rate is fixed and

less than QR (λ < 1.0), the catchment will be a type-I catchment if the position is far enough away the

discharge boundary and close enough to the Y-axis. When the position of the well is fixed, it would

be a type-I catchment if the pumping rate is not too high. In the example of Figure 1a, when Qw is

larger than 20 × 104 m3/d, the possibility of forming a type-III catchment is high, indicating that the

pumping site should be moved westward to maintain a type-I catchment zone.

Second, the shape and size of the catchment zone depend on well position, pumping rate, and

aquifer parameters with different sensitivities. To reduce the risk of contamination, a catchment of

0

20

40

60

80

100

01234

y

wD

=0.4

y

wD

=0.5

y

wD

=0.6

λ=0.3, γ=1,

β=1, α=0.4

0

20

40

60

80

100

01234

α=0.4, γ=1,

β=1, y

wD

=0.6

λ=0.1

λ=0.3

λ=0.5

0

20

40

60

80

100

01234

α=0.2

α=0.4

α=0.6

λ=0.3, γ=1,

β=1, y

wD

=0.6

(b)

(c) (d)

Capture zone of t

D

(a)

X

Y

d

1

d

2

RWel l

Divide-2

R/(d

1

+d

2

)

γt

D

/(

α

λ)

γt

D

/(

α

λ)

γt

D

/(

α

λ)

R/(d

1

+d

2

)

R/(d

1

+d

2

)

Figure 14.

The size factor of a capture zone (

a

) with limited travel time in a Type-III catchment for a

well on Divide-2 and the curves of γtD/(αλ) versus R/dwith varying α(b), λ(c) and ywD (d).

The travel time of a particle starting from a place in a catchment zone depends on the streamline

length from the place to the well, as well as the change in the ﬂow velocity along the streamline.

In particular

, the size factor, R, in Figure 13a is dependent on a streamline along the Y-axis oriented to

the well. According to the relationship formulas presented in Equation (16), the relationship between

Rand tDcan be expressed as

tD=Khc

φB2t=Khc

B2Zyw

yw−R

1

Vy(y)dy (19)

where tis the real travel time [T] and V

y

is the Darcy velocity [LT

−1

] in the Y-direction along the

Y-axis. In the vicinity of the pumping well, the ﬂow is almost in a radial form where the radial velocity

(oriented toward the well), Vr, depends on the radial distance from the well, r, as follows

Vr=Qw

2πrh (20)

where rcan be expressed as |y

w−

y|in the model. Accordingly, in the vicinity of the pumping well

along the Y-axis, Vycan be approximately estimated by

Vy≈Qw

2πh(yw−y)≈Qw

2πhc(yw−y),(yw−y)→0 (21)

where hroles as the saturated thickness of the unconﬁned aquifer and is assumed to be h

c

for small

drawdown condition. Substituting Equations (16) and (21) into Equation (19), we have

tD≈4παλ

γ

R2

d2,R→0 (22)

Water 2019,11, 1243 15 of 18

Equation (22) implies that in the vicinity of the well the dimensionless travel time depends on R/d

and

αλ

/

γ

. When Ris not small, signiﬁcant error could be arisen by Equation (22), whereas the general

relationship between Rand tDcan be checked by a formula as follows

γtD

αλ =FR

d(23)

where F() represents a function of R/dthat would be dependent on or independent on a few parameters.

For the type-III catchment zone, R/(d1+d2) is used in the F() function to replace R/d.

Figure 13b–d show curves of

γ

t

D

/(

αλ

) versus R/dfor capture zones in the type-I catchment.

As indicated

,

γ

t

D

/(

αλ

) is almost linearly dependent on R/d. An increase in the

α

value will cause a

decrease in the slope of the curve, as indicated in Figure 13b. The slope of curves also decreases with

the increasing

λ

value, as shown in Figure 13c, but the response is not signiﬁcant as that inﬂuenced

by the

α

value. Figure 13d shows that the curves are not sensitive to y

wD

even the increase in y

wD

could increase the slope. It is clearly indicated that the relationship between

γ

t

D

/(

αλ

) and R/d, i.e.,

the function

F(R/d), is almost independent on

λ

and y

wD

when R/dis less than 1.0 but still signiﬁcantly

depends on

α

. Diﬀerent curves of

γ

t

D

/(

αλ

) versus R/(d

1

+d

2

) are shown in Figure 14b–d for capture

zones in the type-III catchment. Unlike those shown in Figure 13, these curves are signiﬁcant nonlinear.

When R/dis less than 1.0, the curves are close to each other and are not sensitive to the changes in

α

,

λ

and y

wD

. Otherwise the curves are signiﬁcantly inﬂuenced by these parameters. Note that the

intersection of the X-axis and Divide-2 is a stagnation point for the Type-III catchment, which limits

the value of R/(d

1

+d

2

) but leads to an inﬁnite t

D

value because of the tiny V

y

value in the vicinity of

the stagnation point. As a result, the curves are approximately vertical to the horizontal axis when

R/(d

1

+d

2

) is close to the maximum value. According to Equation (18), both

α

and

λ

increase d

1

+d

2

so

that the bound of R/(d

1

+d

2

) decrease in Figure 14b,c, respectively, with the increasing

α

and

λ

values.

However, an increase in the y

wD

value will enlarge the range of Rand consequently increase the bound

of R/(d1+d2), as shown in Figure 14d.

6. Discussions on the Application

The modeling results with dimensionless variables in this study represent general behaviors

of the catchment zone and capture zones of a pumping well with hydrogeological conditions that

illustrated in Figure 2. It can be applied in such conditions to determine the type and the shape of the

catchment zone for a well in water supply projects. In the example of Figure 1a with parameters in

Table 1,

the pumping

site is generally located in the type-I zone so that the shape of the catchment

zone is similar to that shown in Figure 3b. The width of the catchment zone on the source head, d,

can be estimated with Equation (16), which is 0.7–2.9 km. The w/dvalue is larger than but close to d.

General results in this study can be used to optimize the location of a pumping site and to delineate

protection zones for diﬀerent security levels. Three major points should be concerned:

First of all, it is noticeable that the catchment zone would be one of the four types shown in

Figures 3–6. The type-I catchment has a single source head, whereas the others have two (type-II,

type-III) or three (type-IV) source heads. An increase in the number of source heads would increase

the risk of contamination on water supply if all the source heads could be probably inﬂuenced by

pollutions. It would also increase the cost of groundwater monitoring because all the source heads

should be monitored to achieve an eﬀective response in time of pollution events. Thus, the type-I

catchment is recommended for safety. As indicated in Figures 7and 8, the type of the catchment is

highly dependent on the location and pumping rate of the well. When the pumping rate is ﬁxed and

less than Q

R

(

λ

<1.0), the catchment will be a type-I catchment if the position is far enough away the

discharge boundary and close enough to the Y-axis. When the position of the well is ﬁxed, it would be

a type-I catchment if the pumping rate is not too high. In the example of Figure 1a, when Q

w

is larger

than 20

×

10

4

m

3

/d, the possibility of forming a type-III catchment is high, indicating that the pumping

site should be moved westward to maintain a type-I catchment zone.

Water 2019,11, 1243 16 of 18

Second, the shape and size of the catchment zone depend on well position, pumping rate,

and aquifer

parameters with diﬀerent sensitivities. To reduce the risk of contamination, a catchment of

small wand ris recommended. For a well with ﬁxed pumping rate, the dvalue of a type-I catchment

and d

1

+d

2

value of a type-II catchment are ﬁxed according to Equations (17) and (18), respectively.

Thus, the changes in the shape factors, w/dand r/d, for the type-I catchment, or w/(d

1

+d

2

) and

r/(d1+d2)

for the type-III catchment, can be considered to check the catchment size. The relationships between

these shape factors and the parameters are shown in Figures 9and 10. Note that in the type-I catchment

the w/dvalue increases with y

wD

, whereas in the type-III catchment the w/(d

1

+d

2

) value decreases

with ywD. The r/dand r/(d1+d2) values could be nonlinearly dependent on ywD.

The third point is the dependency of the size factor, R, on the well location and parameters for

capture zones with required maximum travel times. In practice, the maximum travel time is hundreds

of days to high security levels and several of years to low security levels. A small R/dor

R/(d1+d2)

value is recommended for the same security level to reduce the protection area. As indicated in

Figure 13,R/dincreases with the dimensionless travel time,

γ

t

D

/(

αλ

), almost following a linear manner

in the type-I catchment. However, R/(d

1

+d

2

) nonlinearly increases with

γ

t

D

/(

αλ

) in the type-III

catchment with a limitation that positively depends on y

wD

, as shown in Figure 14. In application,

one should notices the conversion between the dimensionless travel time t

D

and the real travel time, t,

according to Equation (16).

Limitations of the model should be also paid attention to in application. One assumption in the

model is the equal length of each inﬂow segment and inﬂow rate. This will be not satisﬁed so far

if the mountain-front recharge of groundwater is signiﬁcantly heterogeneous. However, the results

of type-I catchment are still useful because it is dominated by a single inﬂow segment. The model

would be invalid when the pumping rate is higher than double of the recharge rate from an inﬂow

segment, because in this situation Divide-3 (Figure 2) and other divides far away the well could be

inﬂuenced. The constant head discharge boundary is a simpliﬁcation and needs to be checked in

reality, especially for a pumping well located in the II and IV zones, which may signiﬁcantly inﬂuence

the discharge boundary. A time-dependent h

c

would exist on a natural discharge boundary because of

the unsteady groundwater ﬂow. When the seasonal ﬂuctuation of groundwater level in the natural

state (generally less than 2 m at the site in Figure 1a) is signiﬁcantly less than h

c

(>100 m at the site in

Figure 1), the mean annual state could be adopted in analyses to represent a steady state ﬂow. Vertical

recharge to or discharge from water table should be also checked. At the site in Figure 1, the climate is

extremely dry (mean annual precipitation is less than 50 mm, whereas mean potential evaporation is

higher than 1500 mm) so that inﬁltration recharge could be ignored. Depth of water table in the pluvial

fan area is generally larger than 10 m, which does not yield a signiﬁcant loss of groundwater from

evapotranspiration. The model in this study is false if there is a river ﬂowing across the whole study area

and providing persistent leakage recharge to groundwater. More comprehensive models are required

to analyze capture zones for a complex pumping site, however, with more uncertain parameters.

7. Conclusions

A conceptual model of the unconﬁned aquifer beneath ﬂuvial fans in front of mountains is

developed in this study, where the aquifer is limited by a segmental inﬂow boundary and a discharge

boundary of constant head. The capture zone of a pumping well in such an unconﬁned aquifer

is investigated with a numerical approach using MODFLOW and MODPATH. Shape factors are

introduced to quantitatively analyze the catchment zone. Travel time distribution in the catchment

zone is identiﬁed to delineate the capture zones with respect to special travel times. Results are

transformed into dimensionless variables and parameters to observe general behaviors.

The characteristics of the catchment zone and capture zones for a pumping well in such an aquifer

can be summarized as:

Water 2019,11, 1243 17 of 18

(1) The catchment zone has 4 types of shape, in terms of Type-I to type-IV in Figures 3–6, due to

diﬀerent connections with the boundaries, depending on the location of the pumping well and several

control parameters.

(2) The dimensionless width of the catchment zone in the vicinity of the well (related to the source

head width) decreases with the relative length of inﬂow segments but increases with the relative

pumping rate, as indicated in Figures 10 and 11;

(3) The dimensionless size of capture zones (related to the source head width) increases with the

relative travel time almost in a linear manner when the source is contributed by a single inﬂow segment,

as shown in Figure 13. A signiﬁcant nonlinear relationship exists when the source is contributed by

double inﬂow segments, as shown in Figure 14.

Simpliﬁcations in the model also bring limitations of the results in application.

Author Contributions:

Conceptualization, H.L. and X.-s.W.; methodology, H.L.; software, H.L.; validation, H.L.

and X.-s.W.; formal analysis, H.L.; data curation, H.L.; Writing—Original Draft preparation, H.L.; Writing—Review

and Editing, X.-s.W.; supervision, X.-s.W.; project administration, X.-s.W.; funding acquisition, X.-s.W.

Funding:

This research was funded by the Fundamental Research Funds for the Central Universities (2652018191),

the Foundation Research Project of Qinghai Province (No. 2018-ZJ-740) and the National Natural Science

Foundation of China (No. 41772249).

Conﬂicts of Interest: The authors declare no conﬂict of interest.

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2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access

article distributed under the terms and conditions of the Creative Commons Attribution

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