Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol22.Issue6.1999:

Analysis of continuous and pulsed pumping of a

phreatic aquifer

H. Zhang

a,*

_{, D.A. Baray}

b

& G.C. Hocking

c a

Department of Civil Engineering National University of Singapore, Singapore 119260, Singapore b

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 (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In a phreatic aquifer, fresh water is withdrawn by pumping from a recovery well.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 The pumping e-ciency depends on the method by which the ¯uid is withdrawn.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 pum-ping 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 re-lationship between the pumping frequency and the ¯uid response. A Hele-Shaw model is set up for experimental veri®cation of the analytical and numerical so-lutions 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.

1 NOMENCLATURE

Ó1999 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0309-1708/99/$ ± see front matter

PII: S 0 3 0 9 - 1 7 0 8 ( 9 8 ) 0 0 0 3 8 - 4

B thickness of capillary fringe [L]

BEM Boundary Element Method

bm width of Hele-Shaw cell [L]

H aquifer depth [L]

hs vertical sink position [L]

hp vertical position of point P [L]

K hydraulic conductivity [LTÿ1_]

Km hydraulic conductivity of Hele-Shaw cell [LTÿ1]

k permeability [L2_]

n local coordinate in the normal direction on the

boundary [L]

p ¯uid pressure [MLÿ1Tÿ2]

Q pumping rate [L2_Tÿ1_]

Qc critical pumping rate [L2Tÿ1]

qc rate of local mass transfer across the free surface

[LTÿ1_]

qx qy discharge velocity in the (x, y) direction [LTÿ1]

t time [T]

x horizontal coordinate [L]

y vertical coordinate [L]

a weight factor in the ®nite-di€erence scheme

e e€ective porosity

g free surface elevation [L]

h volumetric moisture content

i angle between free surface and horizontal [Rad]

l ldynamic viscosity of ¯uid [MLÿ1Tÿ1]

q density of ¯uid [MLÿ3]

*

Corresponding author.

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 re-moval 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

as-sumed 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 ar-rangement, there is a critical pumping rate, the value of which was determined for a range of well-screen sepa-rations.

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 Garder16used

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 pre-dicted critical rates which are too low. Muskat and

Wycko€17 considered the problem of water coning

to-wards a vertical well. They calculated the potential function in the oil zone assuming horizontal radial ¯ow and neglecting the presence of water coning. Their cal-culated 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

Hocking23used a Boundary Integral Equation Method

to solve numerically the pumping problem in an axi-symmetric 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 ap-proximation to determine the shape of the rising inter-face. 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's9work, 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 en-tering the recovery well. The Boundary Integral Equa-tion 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 deter-mine 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 re-peats.

2.2 Pulsed pumping strategy: experiments

Some experimental work on the pulsed pumping

strat-egy has been reported. Wisniewski21 used

two-dimen-sional rectangular box models to investigate

simultaneous ¯ow of water and a denser ¯uid in an

r surface tension between air and glycerol

inter-face [MTÿ2]

/ potential head [L]

uncon®ned aquifer. It was shown that cyclic recovery at a higher ¯ow rate for a ®xed time may be more pro-ductive. But, in the long term, steady continuous pum-ping 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 two-dimensional 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

ex-ample, Khanet al.8modelled steady state ¯ow with

re-plenishment 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 pum-ping and numerical solutions for high ¯ow rate, pulsed pumping.

2.3 Scope of the study

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. Liet 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

pre-dicted 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.

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

boundary is considered. The physical plane is shown in

Fig. 1. A layer of water with depth H occupies a

ho-mogeneous and isotropic porous medium of constant

permeability, k, above a bottom boundary of

imper-meable rock. A line sink is located at a distancehsabove

the bottom boundary, and produces a total ¯ux Qper

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 two-dimensional ¯ow can be expressed as:

qxˆ ÿK o_/

o_x; qyˆ ÿK o_/

o_y; …1†

with the hydraulic conductivity Kˆqgk=l, where q is

the ¯uid density,lis the dynamic viscosity andgis the

magnitude of gravitational acceleration. The

pie-zometric head is /ˆp=qg‡y, where p is ¯uid

pres-sure. In the formulation of uncon®ned groundwater ¯ow problems, the potential within the domain must satisfy2:

r2

/ˆ0: …2†

UsingH as the characteristic length scale andK as the

¯ow rate scale, the following dimensionless variables can be de®ned:

whereeis the e€ective porosity andg is the position of

the free surface. We de®ne the non-dimensional vari-ables with an asterisk. The critical ¯ow rate is denoted

Q_c.

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

lo-cated at y ˆg…x;t†, where the following conditions

In eqn (6),n is the outward normal of the free surface

andiis de®ned as the angle formed by the free surface

with respect to the horizontal,

cosiˆ 1‡ og

The other boundary conditions are as follows:

/ˆ1; at xˆ x_l;

o_/

o_n ˆ0; at y

_ˆ0:

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 so-lution 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

dif-ference analog of eqn (6) can be written as11

/m‡1 ˆ/mÿ Dt

in which m de®nes the time step and a is a weighting

factor, which is taken as 1

2 in this study. We use the

Boundary Element Method11 (BEM) to solve the ¯ow

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; …8†

in which/ns is the non-singular portion and/

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 …0:7ÿ0:8†Q_c 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 su-percritical rate until the free surface drops below a certain

height,h

pˆh1, when pumping is stopped. The free

sur-face 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 ath_pˆh₁ to avoid air breaking through into

the well, then start another pumping cycle whenh_pˆh₂,

whereh₂lies in the fast rebound region in Fig. 2(b), thus

Fig. 2.Supercritical pumping usingQ_ˆ3Q

c. (a) Elevation change of point P. Pumping stops athpˆ0:2357. (b) Rate of elevation

the non-pumping time,T_off , is reduced as much as

pos-sible. For pulsed pumping, we seth₁to be at or near the

value ofh

pthat would result for steady pumping at the

design rate. For example, taking the design rate as

Q_ˆ0:75Q

c, we calculatehpfor this rate ashpˆ0:503.

Thus, we seth

1ˆ0:5. 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

period, T_on, reduces and the rebound period, T_off ,

in-creases as time proceeds, as shown in Fig. 4(a). If the

pumping frequency is kept ®xed,h₁andh₂will decrease,

and the free surface will move downwards and eventu-ally break through into the pump withdrawal location.

It is found, for a given value ofh_p, 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

pulsed pumping. Clearly, the initial supercritical pum-ping rate is a dominant feature in Fig. 4(b). Over ex-tremely long time periods, a subcritical steady pumping rate can be preferable. For example, consider an aquifer

with a depthHˆ10 m, permeabilitykˆ10ÿ8 _cm2 _and

porosity eˆ0:45. The water properties are qˆ1:0 g/

cm3 _{and lˆ1:0 cp. Therefore, from eqn (3) and}

ex-trapolation 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 se-lected 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.

3.3.2 Capillary e€ects

In a porous medium, the moisture content varies grad-ually from dry to wet through a zone of partially satu-rated 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 Fig. 3.The point P, rebounding betweenh

1ˆ0:5 andh2ˆ0:6,

for the high ¯ow rate pulsed pumping scheme. (a) Q_ˆ2Q

c; xl ˆ10, (b) Qˆ2Qc; xl ˆ20. The pump is

on for periodsT

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

offotherwise.

Fig. 4.High ¯ow rate pulsed pumping,h

1ˆ0:5 andh2ˆ0:6,

for Q_ˆ2Q

c. (a) Duration of the pumping cycles. (b)

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 kine-matic boundary condition of the free surface can be

expressed as10,

eog

whereqcis the rate of local mass transfer across the free

surface, non-dimensionalised as q

c ˆqc=K. From

con-servation of mass, we have18

Z 1

where h is the volumetric water content in the

unsatu-rated zone. The mass ¯ux, q_c, is determined from an

approximate solution of the unsaturated ¯ow equa-tion18,

where B is the thickness of the capillary fringe. Using

eqns (11) and (12), eqn (10) can be rewritten as10,

This equation can be non-dimensionalised to

o_/ (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_{ˆB=H, KˆKT=H and t}

1ˆt=T, where T is the

pumping period, leading to:

o_/

From eqn (15), we can assess the e€ect of capillarity for

pulsed pumping by comparingKandB, i.e.,KTandB.

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. How-ever, it is important for high pumping frequencies or a large capillary fringe. Physically, this re¯ects the be-haviour 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_ˆ3Q

c, h1ˆ0:5, h2ˆ0:7 and

B_{ˆ0:1. Fig. 5(a) and (b) show a comparison of the}

frequency of the pumping cycles for ¯ow with and

without the capillarity. It is clear that bothT_on andT_off

are shortened due to capillarity. The rate of the eleva-tion change of the free surface is increased. This be-haviour is consistent with the above interpretation that

Fig. 5.Comparison of ¯ow with (dashes,B_{ˆ0:1) and}

with-out (line,B_{ˆ0) capillarity, for supercritical ¯ow rate pulsed}

pumping, Q_ˆ3Q

c. (a) The position of point P of the free

surface, rebounding between h

1ˆ0:05 andh2ˆ0:7, (b)

the phreatic surface is able to ¯uctuate more rapidly with a capillary fringe present. Although capillary e€ects

reduceT

onandToff , they do not in¯uence the long term

productivity, as shown in Fig. 5(c).

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,bmˆ2 mm, by a

network of spacers, as shown in Fig. 6. The dimensions of

the Hele-Shaw cell were 10050 cm. Constant head

tanks were set up on both sides of the cell. The inside

dimensions of the tanks were 4:910:050:0 cm. A

viscous liquid, Glycerol, was allowed to ¯ow in the nar-row 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 Hele-Shaw cell represents a two-dimensional ¯ow domain).

The volumetric ¯ow rate Qpump can be converted to a

two-dimensional line sink ¯ow rate usingQˆQpump=bm.

For ¯ow between vertical parallel plates, the speci®c

discharge between the plates can be described as2:

qxˆ ÿ

Comparing these with Darcy's law (1), it is obvious that the hydraulic conductivity of the space between the plates can be written as,

Kmˆqgb

2

m

12l :

Continuity leads to the satisfaction of Laplace's equa-tion 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 In-ternational 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.

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.,h_s ˆ0:05

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

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 1 111.28 1.205 6.5 0.22 4.0 0.0354

Case 2 1025.5 1.255 10.0 0.22 10.5 0.00399

Case 3 1025.5 1.255 7.5 5.11 10.5 0.00399

Fig. 7. Comparison of theoretical model predictions and ex-perimental data on the free surface position for critical rate

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 Zhanget al.24indicates that

the solution is very sensitive to small perturbations of

parameters whenQis close to the critical value.

There-fore, 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:

Bˆ 2r

bm…q2ÿq1†g

; …18†

in whichr is the surface tension between the

air/Glyc-erol interface, whileq1andq2are the densities of air and

glycerol, respectively. In this model, taking rˆ63:4

dynes/cm,q2ˆ1:255 g/cm3(q1is negligible),Bˆ0:5 cm

(B_{ˆ0:06) can be calculated from eqn (17). The results}

for a single cycle in which the pump stopped athpˆ2:7

cm (h_pˆ0:34) 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 Bˆ0:06 appears more accurate. From the

single cycle case, the phase changes due to capillary ef-fects 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 Ta-ble 3, the ¯uid was pumped until the point P (Fig. 1)

reachesh1ˆ3:9 cm (hpˆ0:37), when the ¯uid was

al-lowed to rebound, the next pumping cycle starting at

h2ˆ3:2 cm (hpˆ0:30). In Fig. 10 the experimental

re-sults 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 dif-ferent, and neglecting capillary e€ects leads to an error in phase which increases over time.

Table 2. Comparison of ¯ow rates obtained analytically and from the experimental model

Case 1 Case 2 Case 3

Qc(experimental) 0.038 0.199 0.149

Qc(analytical) 0.040 0.210 0.158

Error (%) 5.0 5.24 5.70

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 1025.5 1.255 11.5 0.22 7.9 0.00399

Pulsed 1 1025.5 1.255 12.0 0.22 10.5 0.00399

Pulsed 2 439.0 1.239 37.0 1.15 9.4 0.00922

Fig. 9.Comparison of elevation change of point P for single cycle, obtained by the numerical and experimental models

(case: Single in Table 3).

Fig. 8. Comparison of free surface positions obtained by the analytical model and experiment for critical continuous pum-ping with di€erent sink positions: hs ˆ0:02 and hs ˆ0:486.

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 continuous and pulsed pumping from a recovery well in a phreatic aquifer. An existing analytical solution based on the hodograph method was used to determine the critical ¯ow rate for the steady continuous situation. A Boundary Element Method was applied to examine a high ¯ow rate, pulsed pumping strategy in an attempt to achieve a more rapid withdrawal. A modi®ed kinematic interface condition that incorporates the e€ects of cap-illarity is used to simulate the ¯uid response in the vi-cinity of the well. It was found that supercritical ¯ow rate pulsed pumping is more productive for a ®nite

pe-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 pum-ping case, the in¯uence of capillarity was con®rmed.

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2. Bear, J., Dynamics of Fluids in Porous Media. American Elsevier, New York, 1972.

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