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A Preliminary Study on a Pumping Well Capturing Groundwater in an Unconfined Aquifer with Mountain-Front Recharge from Segmental Inflow

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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.
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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 dierent 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 eect 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 dierent 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 dierent 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 aected 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 suciently 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
xhh
x+
yhh
yQwδ(xxw)δ(yyw)=0
B<x<3B, 0 <y<L,
(1)
h(x,y)=hc,Bx3B,y=L(2)
x=Bor 3B, 0 yL(3)
Kh h
y=q0=QR
2D,|x|D, or, |x2B|D,y=0 (4)
h
y=0, D<|x|<B, or, D<|x2B|<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 [L1], 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 eective 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 eect 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
xDhDhD
xD+
yDhDhD
yDαγλδ(xDxwD)δ(yDywD)=0,
1<xD<3, 0 <yD< β (8)
hD(xD,yD)=1, 1xD3, yD=β(9)
hD
xD
=0, xD=1 or 3, 0 yDβ(10)
hD
hD
yD
=γ
2,|xD|αor |xD2|α,yD=0 (11)
hD
yD
=0, α < |xD|<1 or α < |xD2|<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 dierent size and/or dierent 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 dicult to derive
an analytical solution of h(x,y), even more dicult in obtaining analytical formulas of the catchment
zone and the travel time. Alternatively, we can suciently obtain the numerical solution with eective
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-dierence 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-dierence 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 suciently 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 sucient to encompass the entire capture zone of a pumping well. We check the
use of dierent 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 eective 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 36show 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
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 7.
General well-location zones with respect to dierent 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 dierent 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 dierent
λ
values; (
b
) Curves of w/dversus
α
with
dierent
β
values; (
c
) Curves of w/dversus y
wD
with dierent
α
values; (
d
) Curves of w/dversus y
wD
with dierent
λ
values; (
e
) Curves of r/dversus
α
with dierent
λ
values; (
f
) Curves of r/dversus y
wD
with dierent 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 eective
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 eect 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 suciently 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 dierent
λ
values; (
b
) Curves of w/(d
1
+d
2
) versus
α
with dierent
β
values; (
c
) Curves of w/(d
1
+d
2
) versus y
wD
with dierent
α
values; (
d
) Curves of
w/(d
1
+d
2
) versus y
wD
with dierent
λ
values; (
e
) Curves of r/(d
1
+d
2
) versus
α
with dierent
λ
values;
(f) Curves of r/(d1+d2) versus ywD with dierent 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
eect 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
ywR
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
VyQw
2πh(ywy)Qw
2πhc(ywy),(ywy)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
tD4παλ
γ
R2
d2,R0 (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
. Dierent 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 dierent 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 36. 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 eective 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 dierent 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 36, due to
dierent 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
(CC BY) license (http://creativecommons.org/licenses/by/4.0/).
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Article
Full-text available
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MODPATH is a particle-tracking post-processing program designed to work with MODFLOW, the USGS finite-difference groundwater flow model. MODPATH version 7 is the fourth major release since its original publication.
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An analytical solution is presented for delineating the catchment zone of a well located near a groundwater divide in a recharged aquifer. The analytical solution is derived for an infinitely wide, homogeneous, isotropic and uniformly thick semi-confined aquifer. It is shown that the shape and extent of the well catchment zone only depend on the distance to the groundwater divide and on the ratio of the well pumping and groundwater recharge rates. The catchment zone of a pumping well extends beyond the groundwater divide, such that the total area affected at the other side of the groundwater divide equals one half of the well-catchment area, and consists of a part from which groundwater is captured by the well and another part from which groundwater is flowing to the other side but bypassing the well. The amount of groundwater transfer involved equals half of the pumping rate. In case of an injection well, the catchment zone comprises the whole aquifer when the well is sufficiently close to the groundwater divide, and otherwise consist of an infinite but laterally bounded zone in the direction away from the groundwater divide. In both cases, the aquifer at the other side of the groundwater divide gains half of the well injection rate either directly by flow from the injection well or indirectly by capturing groundwater recharge from the other side of the groundwater divide.
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A capture zone of pumping well(s) in an aquifer with regional groundwater flow has been widely used to help design a pump-and-treat system for groundwater containment and remediation. Based on the theorem of potential and the principle of superposition, an analytical solution is derived in this study to calculate the capture zone for two arbitrarily located wells that pump water from a confined aquifer at an equal flow rate. The general two-well capture zone is composed of two sub-capture zones, one for each well. The shapes of the two sub-capture zones and their relative locations change with variations of the orientation and the separation of the two wells. The solution can be useful in verifying available numerical codes, as well as in solving practical field problems.