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Advances in Water Resources Vol. 22, No. 6, pp 623±632, 1999
Ó 1999 Elsevier Science Ltd
Printed in Great Britain. All rights reserved
0309-1708/99/$ ± see front matter

Analysis of continuous and pulsed pumping of a
phreatic aquifer
H. Zhanga,*, D.A. Barayb & G.C. Hockingc
a
Department of Civil Engineering National University of Singapore, Singapore 119260, Singapore
Department of Civil and Environmental Engineering University of Edinburgh, Edinburgh EH9 3JN, UK
c
Department of Mathematics and Statistics Murdoch University, Murdoch, WA 6150, Australia

b

(Received 23 March 1998; revised 7 August 1998; accepted 19 August 1998)

In a phreatic aquifer, fresh water is withdrawn by pumping from a recovery well.

As is the case here, the interfacial surface (air/water) is typically assumed to be a
sharp boundary between the regions occupied by each ¯uid. The pumping eciency depends on the method by which the ¯uid is withdrawn. We consider the
eciency of both continuous and pulsed pumping. The maximum steady pumping rate, above which the undesired ¯uid will break through into the well, is
de®ned as critical pumping rate. This critical rate can be determined analytically
using an existing solution based on the hodograph method, while a Boundary
Element Method is applied to examine a high ¯ow rate, pulsed pumping strategy
in an attempt to achieve more rapid withdrawal. A modi®ed kinematic interface
condition, which incorporates the e€ect of capillarity, is used to simulate the ¯uid
response of pumping. It is found that capillarity in¯uences signi®cantly the relationship between the pumping frequency and the ¯uid response. A Hele-Shaw
model is set up for experimental veri®cation of the analytical and numerical solutions in steady and unsteady cases for pumping of a phreatic aquifer. When
capillarity is included in the numerical model, close agreement is found in the
computed and observed phreatic surfaces. The same model without capillarity
predicts the magnitude of the free surface ¯uctuation induced by the pulsed
pumping, although the phase of the ¯uctuation is incorrect. Ó 1999 Elsevier
Science Limited. All rights reserved
Key words: pulsed pumping, Hele-Shaw model, capillary fringe, free surface.

p
Q
Qc

qc

1 NOMENCLATURE
B
BEM
bm
H
hs
hp
K
Km
k
n

*

thickness of capillary fringe [L]
Boundary Element Method
width of Hele-Shaw cell [L]
aquifer depth [L]

vertical sink position [L]
vertical position of point P [L]
hydraulic conductivity [LT ÿ1 ]
hydraulic conductivity of Hele-Shaw cell [LT ÿ1 ]
permeability [L2 ]
local coordinate in the normal direction on the
boundary [L]

q x qy
t
x
y
a
e
g
h
i
l
q

Corresponding author.
623

¯uid pressure [MLÿ1 T ÿ2 ]
pumping rate [L2 T ÿ1 ]
critical pumping rate [L2 T ÿ1 ]
rate of local mass transfer across the free surface
[LT ÿ1 ]
discharge velocity in the (x, y) direction [LT ÿ1 ]
time [T ]
horizontal coordinate [L]
vertical coordinate [L]
weight factor in the ®nite-di€erence scheme
e€ective porosity
free surface elevation [L]
volumetric moisture content
angle between free surface and horizontal [Rad]
l dynamic viscosity of ¯uid [MLÿ1 T ÿ1 ]
density of ¯uid [MLÿ3 ]

624
r
/


H. Zhang et al.
surface tension between air and glycerol interface [MT ÿ2 ]
potential head [L]
superscript indicating a dimensionless variable

2 INTRODUCTION
In an aquifer or oil reservoir, ¯uid is withdrawn by
pumping from a well. When ¯uid is withdrawn from a
layered system of immiscible ¯uids, the withdrawn ¯uid
will come from the layer surrounding the point of removal until the critical ¯ow rate is reached. For a system
in which the interface is sharp, at the critical rate the
interface is drawn into a cusp shape. Above the critical
rate, the ¯uid from the adjacent layer will break through
into the well. Breakthrough is undesirable where water
enters an oil recovery well or, in coastal regions, where

saline water enters a freshwater well. We examine, as our
prototype problem, the withdrawal of water from a
phreatic aquifer. Our aim is to predict conditions under
which water can be withdrawn most eciently from a
single well, such that breakthrough of air into the well
does not occur.
2.1 Literature review
Steady withdrawal of one of a pair of immiscible ¯uids,
or a single ¯uid with a free surface, has been studied
using both the hodograph and numerical methods. Most
earlier research was based on subcritical and critical ¯ow
rates for steady conditions3,5,12,15,24. For example, Bear
and Dagan3 used the hodograph method to ®nd the
shape of the interface and the coning height for the case
of a sink on a horizontal, impermeable plane. Zhang
and Hocking22 employed a model in which it was assumed that the ¯owing layer is con®ned below by an
impermeable boundary. A nonlinear integral equation
solution for this model was solved numerically. With
this model, the critical ¯ow rate can be calculated for
any sink location. MacDonald and Kitanidis13 used

both linear stability theory and a Boundary Element
Method to model the ¯ow in a recirculation well, a
con®guration used in groundwater remediation
schemes. Numerical simulations show that, for this arrangement, there is a critical pumping rate, the value of
which was determined for a range of well-screen separations.
Axisymmetric, sink-like ¯ow problems cannot be
solved using conformal mapping methods. Previously,
various approximations were made to solve these
problems. For pumping of oil, Meyer and Garder16 used
Dupuit's well-discharge formula to derive a relation for
the critical rate which takes into account the presence of
the cone. They obtained a theoretical ¯ow maximum as
a function of the depth of penetration of the well below
the top of the oil layer (assumed to overlie a water layer)

and the thickness of the oil zone. However, they predicted critical rates which are too low. Muskat and
Wycko€17 considered the problem of water coning towards a vertical well. They calculated the potential
function in the oil zone assuming horizontal radial ¯ow
and neglecting the presence of water coning. Their calculated critical rates are about 20\% too high. In other
studies, such as Blake and Kucera4, a small perturbation

method and a Boundary Integral Method were applied,
assuming an approximate form for the well suction
pressure in an uncon®ned oil zone. Recently, Zhang and
Hocking23 used a Boundary Integral Equation Method
to solve numerically the pumping problem in an axisymmetric geometry.
Dagan and Bear5 considered withdrawal of fresh
water by shallow wells operating a short distance above
the salt water interface in a coastal aquifer. They applied
a small perturbation method based on a linearized approximation to determine the shape of the rising interface. Their approximations are valid until the crest of
the upconing interface advanced a third of the initial
distance between the interface and the sink. The analysis
was veri®ed by means of experiments in a sandbox
model.
In Lennon's9 work, a time-dependent problem with a
moving interface was considered. A denser ¯uid is
withdrawn through a recovery well from the lower layer
of a two-layer system, while a second well is drilled and
screened in the upper layer (the less dense ¯uid) to pull
the interface upward, so that the rebound time (time for
the interface to recover after pumping has ceased) is

reduced. Simultaneous pumping of water tends to cause
the interface to move upward, allowing the dense ¯uid
to be recovered at an increased rate without water entering the recovery well. The Boundary Integral Equation method was used to quantify the response of the
dense ¯uid near the recovery well.
As indicated above, the critical steady rate occurs
when the interface separating the two immiscible ¯uids
reaches the withdrawal point. For steady withdrawal,
the ¯ow rate clearly cannot exceed the critical rate if
breakthrough is to be avoided. A supercritical ¯ow rate,
pulsed pumping strategy is employed below to determine whether a more rapid withdrawal can be achieved.
In this strategy, when pumping begins in a phreatic
aquifer, the interface is drawn down rapidly, but before
the air breaks through into the well, pumping is stopped.
The interface is then allowed to rebound back towards
its initial position for a certain time. Then the cycle repeats.
2.2 Pulsed pumping strategy: experiments
Some experimental work on the pulsed pumping strategy has been reported. Wisniewski21 used two-dimensional rectangular box models to investigate
simultaneous ¯ow of water and a denser ¯uid in an

Analysis of continuous and pulsed pumping of a phreatic aquifer

uncon®ned aquifer. It was shown that cyclic recovery at
a higher ¯ow rate for a ®xed time may be more productive. But, in the long term, steady continuous pumping was found to be more e€ective than cyclic
pumping. No theoretical con®rmation of this behaviour
has been reported.
The Hele-Shaw model is a well known device for twodimensional ground water investigations. It was ®rst
developed by Hele-Shaw6,7 for studying the potential
¯ow patterns around variously shaped bodies. Since
then it has been used extensively by many investigators
for investigating groundwater ¯ow problems. For example, Khan et al.8 modelled steady state ¯ow with replenishment to horizontal tube drains in a two-layered
soil using a Hele-Shaw cell. Hele-Shaw cell experimental
investigations were carried out by Ram and Chauhan20
to model an unsteady rising water table pro®le in an
aquifer lying over a mildly sloping impervious bed in
response to constant replenishment. In our study, a
vertical Hele-Shaw analog is used to model the pumping
problem in a two-dimensional phreatic aquifer and
verify analytical solutions for steady, continuous pumping and numerical solutions for high ¯ow rate, pulsed
pumping.
2.3 Scope of the study

boundary is considered. The physical plane is shown in
Fig. 1. A layer of water with depth H occupies a homogeneous and isotropic porous medium of constant
permeability, k, above a bottom boundary of impermeable rock. A line sink is located at a distance hs above
the bottom boundary, and produces a total ¯ux Q per
unit time. A constant potential boundary at horizontal
distance, xl , from the sink is assumed. P is the lowest
point of the free surface, and is located at …0; hp †.
Darcy's law is valid, so the discharge velocity for twodimensional ¯ow can be expressed as:
o/
o/
; qy ˆ ÿK
;
…1†
ox
oy
with the hydraulic conductivity K ˆ qgk=l, where q is
the ¯uid density, l is the dynamic viscosity and g is the
magnitude of gravitational acceleration. The piezometric head is / ˆ p=qg ‡ y, where p is ¯uid pressure. In the formulation of uncon®ned groundwater
¯ow problems, the potential within the domain must
satisfy2:
qx ˆ ÿK

…2†
r2 / ˆ 0:
Using H as the characteristic length scale and K as the
¯ow rate scale, the following dimensionless variables can
be de®ned:
t ˆ

The presence of the capillary fringe has been shown to
a€ect beach water table ¯uctuations for high frequency
forcing at the shore line. In some respects, the e€ects
of pulsed pumping are similar to the response of a
coastal aquifer to wave-induced boundary ¯uctuations.
Li et al.10 derived their capillarity correction following
Parlange and Brutsaert18, who derived an approximate
boundary condition for capillary e€ects and applied it to
the Boussinesq model of uncon®ned aquifer ¯ow. A
similar approach was used by Barry et al.1, who predicted the behaviour of a phreatic aquifer subjected to
boundary forcing. Below, we use a modi®ed kinematic
boundary condition which incorporates the e€ects of
capillarity to simulate the ¯uid response in the vicinity of
the well for high frequency, pulsed pumping.
In the present paper, we examine steady withdrawal
and pulsed pumping of a phreatic aquifer. We describe
®rst the problem formulation. Then, both previously
derived analytical solutions24 for steady pumping, and
numerical solutions (applied for pulsed pumping), are
discussed. Finally, Hele-Shaw experiments are carried
out to verify the theoretical predictions.

tK
;
eH

Q ˆ

Q
;
KH

…3†

…/; g; x; y†
;
…4†
H
where e is the e€ective porosity and g is the position of
the free surface. We de®ne the non-dimensional variables with an asterisk. The critical ¯ow rate is denoted
Qc .
The free surface is typically assumed to be a sharp
boundary between saturated and dry material, i.e.,
capillarity is ignored. The free surface boundary is located at y ˆ g…x; t†, where the following conditions
hold11:
…/ ; g ; x ; y  † ˆ

/ ˆ g …x; t†;


…5†


@/
1 o/
ˆÿ
:
@t
cos i on

3 THEORETICAL ANALYSIS
3.1 Problem formulation
Fluid withdrawn through a line sink from a layered ¯uid
in a porous medium vertically con®ned by a solid

625

Fig. 1. Aquifer con®guration.

…6†

626

H. Zhang et al.

In eqn (6), n is the outward normal of the free surface
and i is de®ned as the angle formed by the free surface
with respect to the horizontal,
"
  2 #ÿ1=2
og
:
cos i ˆ 1 ‡
ox
The other boundary conditions are as follows:
/ ˆ 1;
o/
ˆ 0;
on

at

problem (2) to (7). The location of the free surface at
any time step can be calculated using eqns (5) and (7) in
the BEM solution. The well is represented by a sink.
This singular point is included in the solution by the use
of superposition,
/ ˆ /ns ‡ /s ;
in which
by

x ˆ xl ;

/ns

is the non-singular portion and

…8†
/s

is given

q
Q
ln x2 ‡ …y  ÿ hs †2 :
…9†
2p
Thus, the object is to solve for /ns in the BEM, and then
add /s for the complete solution. The details of the
BEM have been described elsewhere11,19. Note that, for
the present simulations, the sink is assumed to lie on the
bottom impermeable boundary, i.e., at location …0; 0† in
Fig. 1. We ®nd, not surprisingly, that, although the
predicted critical ¯ow rate produces a stable cone just
above the well, the free surface is very close to the
withdrawal point. In practice, any perturbations in the
¯ow rate or local variations in hydraulic conductivity
would allow air to break through into the sink. Thus,
the range …0:7 ÿ 0:8†Qc is selected as the maximum
pumping rate in practice, so that the free surface will be
stable, and breakthrough will not occur when the system
is subject to perturbations. We will refer to the selected
rate as the ``design'' pumping rate.
During pulsed pumping, water is withdrawn at a supercritical rate until the free surface drops below a certain
height, hp ˆ h1 , when pumping is stopped. The free surface is allowed to rebound prior to the restarting of
pumping. Fig. 2(a) shows that a relatively long time is
taken to rebound. The free surface rebounds rapidly at the
beginning, but the recovery rate reduces with time, with
the reduction in head gradient, as shown in Fig. 2(b).
In order to improve pumping eciency, we cease
pumping at hp ˆ h1 to avoid air breaking through into
the well, then start another pumping cycle when hp ˆ h2 ,
where h2 lies in the fast rebound region in Fig. 2(b), thus
/s ˆ

at y  ˆ 0:

3.2 Analytical solution
Zhang et al.24 used the hodograph method to solve
above problem for steady cases at critical ¯ow rates. The
cusp shape of the interface can be calculated analytically
for all locations of the sink and the solid boundaries in
the case of steady, continuous pumping. The results of
the hodograph method establish a relationship between
locations of the sink, the solid boundary and the value
of the critical ¯ow rate. In our study, the analytical solution is to be veri®ed with the experimental data from a
Hele-Shaw model.
3.3 Numerical analysis
3.3.1 Saturated ¯ow (capillarity ignored)
In groundwater aquifer, we assume a sharp air/water
interface, i.e., ignoring capillary e€ects. The ®nite difference analog of eqn (6) can be written as11
" 
  m #
 m‡1

m‡1
m
Dt
o/
o/
a
/ ˆ / ÿ
‡ …1 ÿ a†
;
cos i
on
on
…7†
in which m de®nes the time step and a is a weighting
factor, which is taken as 12 in this study. We use the
Boundary Element Method11 (BEM) to solve the ¯ow

Fig. 2. Supercritical pumping using Q ˆ 3Qc . (a) Elevation change of point P. Pumping stops at hp ˆ 0:2357. (b) Rate of elevation
change of point P.

Analysis of continuous and pulsed pumping of a phreatic aquifer

627


the non-pumping time, Toff
, is reduced as much as possible. For pulsed pumping, we set h1 to be at or near the
value of hp that would result for steady pumping at the
design rate. For example, taking the design rate as
Q ˆ 0:75Qc , we calculate hp for this rate as hp ˆ 0:503.
Thus, we set h1 ˆ 0:5. This is the case for all examples in
this section.
Fig. 3 illustrates the ¯uid response in the vicinity of
the well with this scheme. It is found that the pumping


, reduces and the rebound period, Toff
, inperiod, Ton
creases as time proceeds, as shown in Fig. 4(a). If the
pumping frequency is kept ®xed, h1 and h2 will decrease,
and the free surface will move downwards and eventually break through into the pump withdrawal location.
It is found, for a given value of hp , that the free surface
shape in the pumping period is di€erent from that in the
rebound period, since the pressure distributions on the
free surface during pumping and rebound periods vary.
Fig. 4(b) shows that much higher productivity can be
achieved for a ®nite period using supercritical rate

Fig. 4. High ¯ow rate pulsed pumping, h1 ˆ 0:5 and h2 ˆ 0:6,
for Q ˆ 2Qc . (a) Duration of the pumping cycles. (b) Comparison of productivity for di€erent pumping schemes.

pulsed pumping. Clearly, the initial supercritical pumping rate is a dominant feature in Fig. 4(b). Over extremely long time periods, a subcritical steady pumping
rate can be preferable. For example, consider an aquifer
with a depth H ˆ 10 m, permeability k ˆ 10ÿ8 cm2 and
porosity e ˆ 0:45. The water properties are q ˆ 1:0 g/
cm3 and l ˆ 1:0 cp. Therefore, from eqn (3) and extrapolation of the curves in Fig. 4, we ®nd that, for at
least 301 d, the pulsed pumping has higher productivity.
The supercritical (pumping) ¯ow rate should be selected carefully. If the pumping rate is too high, the air
will have more chance to breakthrough into the well as
the free surface is moving down with increasing rapidity,
given that minor delays in shutting down the pump are
possible.

Fig. 3. The point P, rebounding between h1 ˆ 0:5 and h2 ˆ 0:6,
for the high ¯ow rate pulsed pumping scheme.
(a) Q ˆ 2Qc ; xl ˆ 10, (b) Q ˆ 2Qc ; xl ˆ 20. The pump is

when hp decreases, and is o€ for periods
on for periods Ton

otherwise.
Toff

3.3.2 Capillary e€ects
In a porous medium, the moisture content varies gradually from dry to wet through a zone of partially saturated soil, called the capillary fringe. At steady state,
there is, of course, no change in the capillary fringe.
However, for unsteady ¯ow, the location of the phreatic
surface (where water pressure is atmospheric) varies
with time. If the capillary fringe is considered separately

628

H. Zhang et al.

from the fully saturated portion of the aquifer, then it
acts as either a source or sink from which the saturated
zone can gain or lose water. That is, this zone acts as a
temporary source/sink located at the free surface. This
e€ect can be accounted for approximately. The kinematic boundary condition of the free surface can be
expressed as10,
og
K o/
…10†
ˆÿ
ÿ qc ;
ot
cosi on
where qc is the rate of local mass transfer across the free
surface, non-dimensionalised as qc ˆ qc =K. From conservation of mass, we have18
Z 1
oh
og
dy ˆ e ‡ qc ;
…11†
ot
ot
g
where h is the volumetric water content in the unsaturated zone. The mass ¯ux, qc , is determined from an
approximate solution of the unsaturated ¯ow equation18,
Z 1

B o
oh
qc ˆ ÿ
dy ;
…12†
K ot
ot
g
where B is the thickness of the capillary fringe. Using
eqns (11) and (12), eqn (10) can be rewritten as10,


o/
K o/
B o o/
:
…13†
ˆÿ
ÿ
e
ot
cos i on cos i ot on
This equation can be non-dimensionalised to


o/
1 o/
B
o o/
:
…14†
ˆÿ
ÿ
ot
cos i on e cos i ot on
eqn (14) is the same as eqn (6), but with an extra term
(the second term on the right-hand side) representing
local mass transfer across the free surface as a result of
local pressure gradient changes10. Both mechanisms
contribute to the elevation change of the free surface.
We can also non-dimensionalise eqn (10) using
B ˆ B=H , K  ˆ KT =H and t1 ˆ t=T , where T is the
pumping period, leading to:


o/
K  o/
B
o o/
:
…15†
ÿ
ˆÿ
ot1
e cos i on e cos i ot1 on
e

From eqn (15), we can assess the e€ect of capillarity for
pulsed pumping by comparing K  and B , i.e., KT and B.
Obviously, the importance of the second term depends
on the pulsed pumping frequencies and the thickness of
the capillary fringe; this term is negligible for low pulsed
pumping frequencies or a small capillary fringe. However, it is important for high pumping frequencies or a
large capillary fringe. Physically, this re¯ects the behaviour of the capillary fringe as it responds to the
pulsed pumping. That is, the capillary fringe under high
frequency pumping is not able to self adjust to an
equilibrium state as the free surface rapidly changes
position, although pressure changes can be readily
propagated1,10. Consequently, the aquifer loses water to
the fringe during pumping, and gains water when the
pump is o€, this loss or gain of water being due to

¯uctuations of the phreatic surface within the capillary
fringe, rather than ¯ow of water to or from the free
surface.
The ®nite di€erence analog of eqn (14) is used in the
BEM solution described earlier. Supercritical, pulsed
pumping is simulated for the capillarity e€ects model
using the values: Q ˆ 3Qc , h1 ˆ 0:5, h2 ˆ 0:7 and
B ˆ 0:1. Fig. 5(a) and (b) show a comparison of the
frequency of the pumping cycles for ¯ow with and


and Toff
without the capillarity. It is clear that both Ton
are shortened due to capillarity. The rate of the elevation change of the free surface is increased. This behaviour is consistent with the above interpretation that

Fig. 5. Comparison of ¯ow with (dashes, B ˆ 0:1) and without (line, B ˆ 0) capillarity, for supercritical ¯ow rate pulsed
pumping, Q ˆ 3Qc . (a) The position of point P of the free
surface, rebounding between h1 ˆ 0:05 and h2 ˆ 0:7, (b) frequency of the pumping cycles, (c) cumulative withdrawal.

629

Analysis of continuous and pulsed pumping of a phreatic aquifer
the phreatic surface is able to ¯uctuate more rapidly
with a capillary fringe present. Although capillary e€ects


and Toff
, they do not in¯uence the long term
reduce Ton
productivity, as shown in Fig. 5(c).

Comparing these with Darcy's law (1), it is obvious that
the hydraulic conductivity of the space between the
plates can be written as,
qgb2m
:
12l
Continuity leads to the satisfaction of Laplace's equation as well. Therefore, the groundwater ¯ow in a
phreatic aquifer can be modeled using Hele-Shaw cell
experiments. Because the viscosity of glycerol varies
signi®cantly with temperature and concentration, it was
measured after each experiment using a Nrheology International RI:2:M viscometer. A video camera was
used to record the movement of the ¯uid. The MIH
IMAGE package was used to capture the images for
subsequent analysis.
Km ˆ

4 HELE-SHAW MODELLING
4.1 Description of the apparatus
Hele-Shaw experiments were carried out to examine the
validity of the theoretical solutions derived above.
The experimental model consisted of two parallel
Perspex plates (1.0 cm thick) oriented vertically. The
plates were kept apart at a ®xed distance, bm ˆ 2 mm, by a
network of spacers, as shown in Fig. 6. The dimensions of
the Hele-Shaw cell were 100  50 cm. Constant head
tanks were set up on both sides of the cell. The inside
dimensions of the tanks were 4:9  10:0  50:0 cm. A
viscous liquid, Glycerol, was allowed to ¯ow in the narrow space between the plates. There were 10 holes (with
di€erent diameters) drilled on the centre line of one plate.
The holes can be connected to a peristaltic pump to model
di€erent positions of a line sink (since the vertical HeleShaw cell represents a two-dimensional ¯ow domain).
The volumetric ¯ow rate Qpump can be converted to a
two-dimensional line sink ¯ow rate using Q ˆ Qpump =bm .
For ¯ow between vertical parallel plates, the speci®c
discharge between the plates can be described as2:


qgb2m o
p
;
…16†
y‡
qx ˆ ÿ
12l ox
qg


qgb2m o
p
:
…17†
y‡
qy ˆ ÿ
qg
12l oy

Fig. 6. Hele-Shaw model for investigation of pumping in a
phreatic aquifer.

4.2 Critical rate continuous pumping
For continuous pumping at the critical ¯ow rate, several
cases for di€erent heights of aquifer, sink positions and
¯uid viscosity were investigated. The parameters of each
experiment are listed in Table 1.
All parameters were non-dimensionalised for both
analytical and experimental analysis. In case 1, some dye
was mixed with the Glycerol, and the viscosity was
greatly reduced. Fig. 7 shows the comparison of the
theoretical and experimental results for case 1. In the
experiments, the sink has a ®nite width, e.g., hs ˆ 0:05
for case 1. In the theoretical model, the sink dimension is
in®nitely small. Figs. 7 and 8 show a comparison using
the bottom of the hole as the sink location, and the top

Fig. 7. Comparison of theoretical model predictions and experimental data on the free surface position for critical rate
continuous pumping (case 1 in Table 1).

Table 1. Parameters for di€erent cases of continuous pumping
Parameters

l (cp)

q (g/ml)

Q (ml/min)

hs (cm)

H (cm)

Km (m/s)

Case 1
Case 2
Case 3

111.28
1025.5
1025.5

1.205
1.255
1.255

6.5
10.0
7.5

0.22
0.22
5.11

4.0
10.5
10.5

0.0354
0.00399
0.00399

630

H. Zhang et al.

Fig. 8. Comparison of free surface positions obtained by the
analytical model and experiment for critical continuous pumping with di€erent sink positions: hs ˆ 0:02 and hs ˆ 0:486.
Circles are experimental data for case 2 (Table 1) and triangles
are for case 3.
Table 2. Comparison of ¯ow rates obtained analytically and
from the experimental model

Qc (experimental)
Qc (analytical)
Error (%)

Case 1

Case 2

Case 3

0.038
0.040
5.0

0.199
0.210
5.24

0.149
0.158
5.70

of the hole as the sink location for di€erent sink location
respectively. Clearly, using the top of the hole gives the
best comparison. The critical ¯ow rates Qc are also
calculated and listed in Table 2.
The analytical analysis of Zhang et al.24 indicates that
the solution is very sensitive to small perturbations of
parameters when Q is close to the critical value. Therefore, it is dicult to control the pump to reach the
critical value accurately, since any perturbations will
allow air to break through into the sink. In the critical
rate continuous pumping experiment the pumping rates
were by necessity slightly less than the critical values, as
shown in Table 2.
4.3 Pulsed pumping
Tests for pulsed pumping were also carried out. The
parameters for various cases are listed in Table 3.
For a ¯uid between two completely wetted vertical
plates where the plate separation, bm , is known, the
capillary fringe length can be calculated as14:

Fig. 9. Comparison of elevation change of point P for single
cycle, obtained by the numerical and experimental models
(case: Single in Table 3).

Bˆ

2r
;
bm …q2 ÿ q1 †g

…18†

in which r is the surface tension between the air/Glycerol interface, while q1 and q2 are the densities of air and
glycerol, respectively. In this model, taking r ˆ 63:4
dynes/cm, q2 ˆ 1:255 g/cm3 (q1 is negligible), B ˆ 0:5 cm
(B ˆ 0:06) can be calculated from eqn (17). The results
for a single cycle in which the pump stopped at hp ˆ 2:7
cm (hp ˆ 0:34) are shown in Fig. 9. The ®gure shows the
elevation change of point P from the experiment along
with the numerical predictions. The latter are both in
reasonably good agreement with the data, although the
case with B ˆ 0:06 appears more accurate. From the
single cycle case, the phase changes due to capillary effects are hard to identify. Therefore, pulsed pumping
cases over several cycles were examined. In the pulsed
pumping strategy, for example of case Pulsed 1 in Table 3, the ¯uid was pumped until the point P (Fig. 1)
reaches h1 ˆ 3:9 cm (hp ˆ 0:37), when the ¯uid was allowed to rebound, the next pumping cycle starting at
h2 ˆ 3:2 cm (hp ˆ 0:30). In Fig. 10 the experimental results are compared with the numerical solutions with
and without capillary e€ects. Fig. 10 shows very clearly
how the inclusion of capillary e€ects in the numerical
model has improved its accuracy. The amplitude of the
oscillations is the same for both cases, since it is ®xed by
the pulsed pumping strategy, but the phase is very different, and neglecting capillary e€ects leads to an error
in phase which increases over time.

Table 3. Parameters for di€erent cases of pulsed pumping
Parameters

l (cp)

q (g/ml)

Q (ml/min)

hs (cm)

H (cm)

Km (m/s)

Single
Pulsed 1
Pulsed 2

1025.5
1025.5
439.0

1.255
1.255
1.239

11.5
12.0
37.0

0.22
0.22
1.15

7.9
10.5
9.4

0.00399
0.00399
0.00922

Analysis of continuous and pulsed pumping of a phreatic aquifer

631

riod but, for long times, subcritical steady pumping is
more ecient. It was also shown that capillary e€ects
in¯uence signi®cantly the ¯uid response to the pumping.
A Hele-Shaw model was set up for experimental
veri®cation of the analytical and numerical solution in
steady and unsteady cases for pumping of a phreatic
aquifer. Close agreement was found in the computed
and simulated phreatic surfaces. For the pulsed pumping case, the in¯uence of capillarity was con®rmed.

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Fig. 10. Experimental results and theoretical predictions
showing the e€ect of the capillary fringe on the elevation
change of point P for pulsed pumping (case: Pulsed 1 in
Table 3).

Fig. 11. Comparison of elevation change of point P for pulsed
pumping, obtained by numerical (with capillarity) simulation
and experiment, (case: Pulsed 2 in Table 3).

Fig. 11 reinforces this fact by showing a separate case
in which there is excellent agreement in phase between
the numerical model with capillary e€ects included and
the experimental results.

5 CONCLUSIONS
In this paper, we examined the withdrawal of ¯uid by
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