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A POTENTIAL-FLOW REACTOR MODEL FOR INITIAL
DESIGN OF PULSE-PUMPED GROUNDWATER
REMEDIATION SYSTEMS

presented by

CRAIG MICHAEL TENNEY

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of the requirements for the

MS. degree in Chemical Engineering

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6/01 c:/CIRC/DateDue.pes—p. 15

A POTENTIAL-FLOW REACTOR MODEL FOR INITIAL DESIGN OF PULSE-
PUMPED GROUNDWATER REMEDIATION SYSTEMS

By

Craig Michael Tenney

A THESIS

Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of
MASTER OF SCIENCE

Department of Chemical Engineering and Materials Science

2004

ABSTRACT

A POTENTIAL-FLOW REACTOR MODEL FOR INITIAL DESIGN OF PULSE-
PUMPED GROUNDWATER REMEDIATION SYSTEMS

By

Craig Michael Tenney

A computational model is presented for the optimization of pulsed pumping
systems for efﬁcient in situ remediation of groundwater contaminants. In the pulsed
pumping mode of operation, periodic rather than continuous pumping is used. During the
pump-off or trapping phase, natural gradient ﬂow transports contaminated groundwater
into a treatment zone surrounding a line of injection and extraction wells that transect the
contaminant plume. Prior to breakthrough of the contaminated water from the treatment
zone, the wells are activated and the pump-on or treatment phase ensues, wherein
extracted water is augmented to stimulate pollutant degradation and recirculated for a
sufﬁcient period of time to achieve mandated levels of contaminant removal. An
important design consideration in pulsed pumping groundwater remediation systems is
the pumping schedule adopted to best minimize operational costs for the well grid while
still satisfying treatment requirements. Using an analytic two—dimensional potential ﬂow
model, optimal pumping ﬁ'equencies and pumping event durations have been investigated
for a set of model aquifer-well systems with different well spacings and well-line lengths
and varying aquifer physical properties. The ﬂow model serves as a computationally
efﬁcient tool for initial design and selection of the pumping regimen best suited for

pulsed pumping operation for a particular well conﬁguration and extraction rate.

To the Unknown. May there always be something new to ponder.

iii

ACKNOWLEDGMENTS

Thanks in particular go to Christian Lastoskie for much helpful support provided
during the course of this research. The advice and information provided at various stages
by Michael Dybas, David Hyndman, and David Wiggert is also very much appreciated.
Support for this project was provided by the Michigan Department of Environmental
Quality under contract Y403 86 and the National Science Foundation under award CTS-

9733086.

iv

PREFACE

Chapters 1 and 2 are excerpted from two separate manuscripts. Although Chapter
2 is a logical extension of Chapter 1, each chapter was written to stand alone, and either
chapter may be read independently of the other without loss of understanding or

completeness.

TABLE OF CONTENTS

 

LIST OF TABLES ............................................................................................................ vii

LIST OF FIGURES ......................................................................................................... viii

CHAPTER 1

A Reactor Model for Trap-and-Treat Groundwater Remediation ...................................... 1
Nomenclature .................................................................................................................. 1
Introduction ..................................................................................................................... 3
Methodology ................................................................................................................... 5
Results ........................................................................................................................... 12
Discussion ..................................................................................................................... 17
Conclusion .................................................................................................................... 22
Tables ............................................................................................................................ 23
F iglres ........................................................................................................................... 24

CHAPTER 2

Pulsed Pumping Process Optimization Using a Potential Flow Model ............................ 31
Nomenclature ................................................................................................................ 31
Introduction ................................................................................................................... 33
Methodology ................................................................................................................. 36
Results ........................................................................................................................... 41
Discussion ..................................................................................................................... 44
Conclusion .................................................................................................................... 5 1
F igﬂs ........................................................................................................................... 53

REFERENCES ................................................................................................................. 62

vi

LIST OF TABLES

Table 1-1: Varied ﬁnite difference model parameters. ..................................................... 23

Table 1-2: Average measured hydraulic conductivity at 15-well site (Hyndman et al.,
2000) ......................................................................................................................... 23

vii

LIST OF FIGURES

Figure 1-1. (a) Injection, extraction, and recirculation zones for a two-well system. (b)
Conceptual reactor network model for an arbitrary number of wells. The dashed box
encloses subsurface ﬂow within a single aquifer layer. Additional layers operate in
parallel with the layer shown. ................................................................................... 24

Figure 1-2. Streamlines and groundwater velocity contours for a single conﬁned layer
with Q,‘ = 4.8. The natural groundwater ﬂow gradient is in the positive y-direction.
(a) Two-well system with extraction well on the left and injection well on the right.
(b) Staggered three-well system with two extraction wells and one injection well.. 25

Figure 1-3. Dimensionless residence time distribution E" = EL/ Vreported in terms of
dimensionless time t" = tV/L for (a) two-well and (b) three-well systems with Qe“ =
9.6 .............................................................................................................................. 26

Figure 1-4. Extraction zone concentration histories from FD simulation (solid squares)
and ﬁ'om the reactor model (open squares) in a conﬁned aquifer with two wells. (a)
Conservative solute in a homogeneous aquifer with fully-screened wells. (b)
Conservative solute in a heterogeneous aquifer with fully-screened wells. (0)
Degradable solute in a homogeneous aquifer with fully-screened wells. (d)
Conservative solute in a homogeneous aquifer with partially-screened wells. In part
(d), the open squares and open diamonds show the reactor model results for 100%
and 75% efﬁciency, respectively. ............................................................................. 27

Figure 1-5. Measured and predicted tracer concentration histories for a 15-well ﬁeld
system. The experimental tracer measurements are shown as the solid squares; the
reactor model results for 100% and 75% efﬁciency are given by the open squares
and Open diamonds, respectively. ............................................................................. 28

Figure 1-6. Extraction zone solute concentration histories obtained for variation of key
parameters. (a) Three-well system with Q. = 2.0 m3/hr and k = 0.20 (diamonds), 2.0
(squares), and 20. hr'1 (triangles); (b) Two-well (diamonds), three-well (squares) and
ﬁve-well (triangles) systems with Q, = 2.0 m3/hr and k = 2.0 hr'l; (c) Three-well
system with k = 2.0 hr'1 and Q, = 8.0 (diamonds), 2.0 (squares) and 0.50 m3/hr
(triangles). ................................................................................................................. 29

Figure 1-7. (a) Effective recirculation zone width and (b) maximum allowable pump-off
time as a function of pump-on time for trap-and-treat operation in a conﬁned two-
layer heterogeneous aquifer. The conductivities of the low- and hi gh-conductivity
layers are 0.075 and 0.15 m/hr, respectively. ........................................................... 30

Figure 2-1. Model aquifer structure and well geometry used for analysis of pulsed
pumping. The geometric parameters for well length, well stagger, and aquifer layer

viii

thickness are shown for a hypothetical ﬁve-well, three-layer system with vertical
heterogeneity, as indicated by the layer-dependent speciﬁc discharge. ................... 53

Figure 2-2. Streamlines obtained from potential ﬂow model for an unstaggered, ﬁve-well
system in a single conﬁned aquifer layer with Q*=Q/( VHL)=30 and an equal
distribution of ﬂow between all injection and extraction wells. If natural gradient
groundwater ﬂow is assumed to be in the positive y—direction, three wells are
extracting groundwater and two wells are injecting. The recirculation zone is
highlighted. ............................................................................................................... 54

Figure 2-3. Allowable pump—off time to prevent contaminant breakthrough between
pumping events for an arbitrary pulsed pumping system operating in a
heterogeneous aquifer consisting of a low, medium, and high conductivity (K) layer.
The allowable pump-off time is shown on the right axis as a function of the duration
of each pumping event for the low (squares), intermediate (x’s) and high (triangles)
conductivity layers considered separately, and for the entire aquifer (solid line). The
overall fraction of time the pumps are activated (dashed line) is shown on the left
axis. ........................................................................................................................... 55

Figure 2-4. Dimensionless allowable pump-off time T“ versus dimensionless pump-on
time t“ for single-layer systems. All combinations of N = 5, 7, 9, 11, 13, 15, 21, 31,
45, 67, 99 wells and dimensionless total pumping rates Q“ = Q/( VHL) = 12, 20, 30,
50, 80, 120, 200, 300 are represented (scattered +’s). T"=7TV(N-1)/(PRL)].
t‘=t[V(N-1)/(PRL)][6Q*(N—1)/(1tN)][1 .948exp(-0.694N)+0.0001 08 1N+0.75 7]. Note
that 7" could equivalently be referred to as the recirculation zone width in units of
L/(N-l). The data were ﬁt to the equation T"=f(t*)=0.5404(t*-1)0.5292-0.0766(t"‘-
1) via least-squares regression (solid diamonds). The minimum of the t*/(t*+7"')
curve (solid line) corresponds to the point of operation for maximally efficient
pumping. Actual fractions of time spent with pumps on may be calculated from
t*/(t*+T“) and the deﬁnitions of t“ and 7“. .............................................................. 56

Figure 2-5. Total pumped volume required to ﬂush a 15m deep x 15m long x 0.3m wide
bioremediation zone versus the number of wells in the system. Values predicted by
the potential ﬂow model (connected diamonds) are compared with values calculated
using a MODF LOW ﬁnite-difference groundwater ﬂow model (solid squares,
Hyndman et a1. 2000). Values of V=0.356 m/week, P=0.3, and R=1 were applied to
the potential ﬂow model based upon the published aquifer characteristics .............. 57

Figure 2-6. Pump-on times that minimize total pumping requirements for pulsed
pumping systems operated in heterogeneous aquifers. Dimensionless pump-on time
t“ is plotted against the superﬁcial velocity ratio V“ = me/ Km for systems with
various dimensionless total pumping rates Q*=Q/( VHL). Systems with N = 7, 9, 11,
13, 15, 21, 31, 45, 67, and 99 wells are represented. t*=t[1/ V*][V,,.a,,(N-
1)/(PRL)] [6Q* (N- 1 )/(1tN)] [1 .948exp(-0.694N)+0.0001 08 1N+0.75 7]. ..................... 58

Figure 2-7. Pump-off times that minimize total pumping requirements for pulsed
pumping systems operated in heterogeneous aquifers. Dimensionless pump-off time

ix

T“ is plotted against dimensionless total pumping rate Q“ for systems with various
degrees of vertical heterogeneity, as measured by the superﬁcial velocity ratio V“.
Systems with N = 7, 9, 11, 13, 15, 21, 31, 45, 67, and 99 wells are represented.
T'=II V(N—1)/(PRL)]. ................................................................................................ 59

Figure 28. Minimum required ﬁaction of time with pumps on for pulsed pumping
systems operated in heterogeneous aquifers. The fraction of pump-on time resulting
ﬁom use of the most efﬁcient pulsed pumping schedule is plotted against
dimensionless total pumping rate Q" for systems with various degrees of vertical
heterogeneity, as measured by the superﬁcial velocity ratio V". Systems with N = 3,
5, 7, 9, 11, 13, 15, 21, 31, 45, 67, and 99 wells are represented. .............................. 60

Figure 2-9. Minimum possible average dimensionless total pumping rate for pulsed
' pumping systems operated in heterogeneous aquifers. The average dimensionless
total pumping rate resulting from use of the most efﬁcient pulsed pumping schedule
is plotted against dimensionless total pumping rate Q“ for systems with various
degrees of vertical heterogeneity, as measured by the superﬁcial velocity ratio V*.
Average total pumping rate is the product of the fraction of pump—on time and the
total pumping rate. Systems with N= 3, 5, 7, 9, 11, 13, 15, 21, 31, 45, 67, and 99
wells are represented. ................................................................................................ 61

CHAPTER 1

A Reactor Model for Trap-and-Treat Groundwater Remediation

Nomenclature

a

EWD‘JDG

a»

CI: are ”U 2 l“

N

spatial coordinate of point source/sink
concentration

well borehole or injection/extraction zone diameter
exit-age residence time distribution
feed-normalized residence time distribution
aquifer layer thickness

degradation rate constant

aquifer layer conductivity

well line length

number of wells

porosity

volumetric flowrate

retardation factor

well line stagger (offset)

time

ﬂuid velocity

natural gradient speciﬁc discharge

complex potential function

spatial coordinate perpendicular to natural gradient

(m)
(mol/m3)
(m)
an“)

(m)
0H")
(In/hr)

(m/hr)
(m/hr)
(mZ/hr)

(m)

y spatial coordinate in the direction of natural gradient
2 complex spatial coordinate
Greek
'1’ stream function
r residence time
4) velocity potential
Subscripts
b bypass stream
c captured stream
d discharge stream
e extracted stream
1' injected stream
j layer index
k well index
m midpoint of timestep
r recirculated stream

(m)
(m)

(ml/hr)
(hr)

(mun)

Introduction

Trap-and-treat remediation refers to the use of periodic rather than continuous
pumping for the capture and degradation of groundwater contaminants. During the
pump-off phase of the trap-and-treat approach, contaminants are transported by natural
gradient groundwater ﬂow into an adsorption zone proximate to a transect of alternating
injection and extraction wells. Contaminant adsorbs onto aquifer solids in this region
until near saturation is attained, whereupon the well pumps are activated to establish
recirculation between adjacent injection and extraction wells. The recirculation zone is
then ﬂushed with augmented water to stimulate chemical or biological degradation of the
adsorbed contaminant. Once the zone has been cleansed of contaminant, the pumps are
shut off, allowing fresh contaminant to adsorb onto the aquifer solids from the next parcel
of groundwater that enters the treatment zone by natural gradient ﬂow. The alternating
sequence of pump-oﬁr (contaminant adsorption) and pump-on (contaminant degradation)
events is continued for the duration of the treatment project.

A hybrid scheme involving bioaugmentation and trap-and-treat operation has
been developed for the in-situ bioremediation of Schoolcraft Plume A, a carbon
tetrachloride plume in an unconﬁned aquifer in southwest Michigan (Dybas et al., 1998).
Bioaugrnentation of Plume A with Pseudomonas stutzeri strain KC, a non-native
microorganism, enables transformation of carbon tetrachloride into carbon dioxide and
other nonvolatile organic compounds without production of chloroform (Criddle et al,
1990; Dybas et al., 1995). By contrast, biostimulation of indigenous Plume A microﬂora
results in substantial chloroform production (Mayotte et al., 1996). Trap-and-treat

operation was implemented for a ﬁfteen-well plume transect at the Schoolcraﬁ site, with

weekly pumping events interrupting passive adsorption of carbon tetrachloride from the
groundwater onto the aquifer solids in the treatment zone. Over a 1550-day ﬁeld test
(Dybas er a1. , 2002), more than 96% of the adsorbed carbon tetrachloride was
transformed with minimal production of chloroform. A well-designed trap-and-treat
system may allow signiﬁcant cost savings relative to continuous pump-and-treat
remediation systems, which typically have relatively high maintenance costs and require
pumping and disposal of large volumes of water (Dybas, personal communication).

To achieve maximum effectiveness from trap-and-treat remediation, the
placement of injection/extraction wells and the pumping schedule must be judiciously
chosen. In this paper, we present a reactor model that has been developed for use as a
computationally efficient trap-and-treat design optimization tool. This model is primarily
intended for use in the early stages of system design, when detailed aquifer properties are
not likely known, to rapidly evaluate a large range of potential system conﬁgurations.
For comparison with trap-and—treat systems, the model can also predict the transient and
steady state behavior of continuously-pumped remediation systems, which are
fundamentally equivalent to pulsed systems with infrnite pump-on times. The reactor
model is largely analytic and is thus resource efﬁcient in comparison to numerical model
packages such as MODF LOW (McDonald and Harbaugh, 1983) and MT3D (Zheng,

1992).

Methodology

The aquifer is divided into laterally homogeneous layers with speciﬁc hydraulic
conductivities. The reactor model partitions each aquifer layer into injection, extraction,
and recirculation zones, as depicted in Figure 1-l(a) for a two-well system. The injection
and extraction well casings are enclosed by circular regions of diameter D equal to the
well borehole diameter. Ideal radial plug ﬂow is assumed in the injection and extraction
zones. The recirculation zone is defrned as the region between adjacent wells through
which pumped water ﬂows from the injection zone to the extraction zone. Plug ﬂow
does not occur in this region, but rather a distribution of ﬂuid velocities and residence
times is presumed.

Each of the three zones within an aquifer layer is modeled as a separate chemical
reactor, as indicated in Figure 1-1(b) for a single layer aquifer. For multiple layer
aquifers, with physical properties that vary across layers, vertical dispersion is neglected,
such that the recirculation zones of the layers operate in parallel and ﬂow and reaction in
each layer occur independently of the other layers. The outﬂows from the recirculation
zones combine and mix completely within a single extraction zone that vertically spans
all layers. Similarly, the injection zone is completely mixed, so that the injected ﬂuid
composition is the same in all layers.

For systems of more than two wells, there are multiple injection, extraction, and
recirculation zones within each aquifer layer. For computational efﬁciency these zones
are mathematically combined into single injection, extraction, and recirculation reactor
units that collectively represent the total ﬂow and reaction occurring within a given layer.

This simpliﬁcation is possible without loss of generality in the ﬁnal results because

reaction and ﬂow in the injection, extraction, and recirculation zones are modeled
according to residence time distributions rather than spatial location.

Assuming the diameter of the well casing is negligible relative to the borehole
diameter, the solute residence time r for radial plug ﬂow in the injection zone and the
extraction zone is

r = nDZPRH/(8Qg) (1)
where Q/H is the volumetric extraction rate per unit layer thickness for a given well; P is
the porosity; and R is the retardation factor for the solute in question assuming
equilibrium linear adsorption.

The boundaries of the recirculation zone within a given layer and the ﬂuid
residence time distribution within this zone depend upon the well conﬁguration and
spacing and the pumping extraction rate Q,. For very low values of Q, the extraction
rate is insufﬁcient to establish recirculation between adjacent wells, and the bypass
ﬂowrate Q1, of incoming ﬂuid not captured by the extraction well is nonzero. For larger
extraction rates that represent typical pumping conditions, all incoming groundwater is
captured (Q, = 0), and the recirculation zone volume and its associated residence time
distribution (RTD) are calculated using an analytic two-dimensional potential ﬂow model
(Bird et al., 1960; Columbini, 1999).

The potential ﬂow model assumes steady-state, continuous, incompressible,
inviscid, irrotational, two-dimensional ﬂow occurs within homogeneous, isotropic layers
of a conﬁned aquifer. The natural gradient groundwater ﬂow or speciﬁc discharge V is in
the positive direction along the y-coordinate. A set of N injection/extraction wells are

evenly spaced along a line of length L in the x-coordinate, with perpendicular stagger S

between consecutive wells. Sample conﬁgurations for two- and three-well systems are
shown in Figure 1-2. All wells are assumed to be fully screened ideal sources or sinks.

Continuity and irrotational ﬂow require that

 

 

6v‘+ﬂ’-=0 (2)
51: 5y

6“: 5Vy_

ay—Tch—‘O (3)

where vJr and vy are the rectangular components of the ﬂuid velocity. A solution may be
obtained in terms of the stream function SP and velocity potential Q, which are combined

into the complex potential w(z), where z = x + i y:

W(Z) = <D(x.y) + i WOW) (4)
0'"? 6(1)

vx = “a: = -3 (5)
W M

V}, = +3— : —E (6)

For a single layer, two-well system with an injection rate +Q/H at z = +L/2 and
extraction rate —Q,/H at z = —L/2 superimposed on the speciﬁc discharge, the complex

potential reported in terms of the dimensionless pumping rate Q,“ = Qe/(HL IO is

elk *_12 .
w*(z*) = Qzﬂ ln(:*+1;2)+rz* (7)

 

 

where 2* = z/L and w* = w/(L V). For a single layer, N-well system with even-numbered
extraction wells staggered a distance S in the +y direction behind odd-numbered injection

wells along a line of length L, the complex potential is

 

N at:
w*(z*) = Z Qzek ln(z*—ak *)+iz* (8)
k=l ”

 

 

 

 

 

V t
int[(ge+1)/2]’ kOdd
Qek*=< Q... (9)
e
—. , keven
L rnt[N/2]
1 -1 S*
i(-—+k )+i—, kodd
(Int) 2 N-l 2 (10)
" (1.1-1) .5: k
i 2 N—l 12, even

 

where int[x] is the truncated integer value of x and S” = S/L. Equation (9) stipulates that
the total volumetric ﬂowrate Q; is distributed equally across the extraction wells and
injection wells; e.g. for a ﬁve-well system with S" = 1, groundwater is extracted at a rate
—Qe*/2 from wells 2 and 4 at coordinates 02* = (—1/4, —i/2) and a4* = (+1/4, —i/2), and
injected at a rate +Qe*/3 from wells 1, 3 and 5 at coordinates a1* = (—1/2, +i/2),
a3* = (0, +i/2), and 05* = (+l/2, +i/2).

For an N-well system in an M-layer heterogeneous aquifer, the dimensionless
complex potential wf‘ = w/(L V1) in layer j is the same as that given by equations (8-10),
except that the dimensionless pumping rate in equation (9) is

Qe*____Q_§__

- M (11)
LZIHjVj
I:

where H, and V} are the respective thickness and speciﬁc discharge of the 1"“ layer. The

velocity ﬁeld v,(z) of layer j is obtained from the derivative of the complex potential:

 

dw- N *

J Qek 1 .

. .-_-——=V. V- 12
”‘2’ dz 1.7.322: mam“! ‘ )

where the x- and y-components of the velocity are given as the real and imaginary

portions of equation (12), respectively.

Figure 1-2(a) shows the ﬂow streamlines and groundwater velocity contours
calculated for a two-well system with Q; = 4.8. Figure 1-2(b) shows the results for a
staggered three-well system with the same pumping rate. Each well has a corresponding
stagnation point, indicated by the dark contours of Figures 2(a) and (b). The difference in
stream function values calculated at the stagnation points of consecutive wells is used to
determine the groundwater capture rate Q and the recirculation ﬂowrate Q, = Q, - Q for
each well (or well pair) in the trap-and-treat system. A similar approach has been used to
delineate well capture zones in two—dimensional groundwater ﬂow models (Bakker et al. ,
1996).

The recirculation zone is modeled as a segregated ﬂow reactor (Levenspiel, 1999;
Fogler, 1999) in which a set of plug ﬂow reactors (PFRs) with a distribution of residence
times operate in parallel. The residence time distribution (RTD) of the segregated ﬂow
reactor is determined from the ﬂuid streamlines in the recirculation zone obtained for a
given well geometry and pumping rate.

The PFR reactor network shown in Figure 1-1 (b) permits analytic solution of the
contaminant concentration versus time at any point within the network once the initial
conditions (i.e. Ce, C), and C, in Figure l-1(b) at t=0), aquifer physical properties, kinetic
parameters, and well geometry and pumping rates are speciﬁed. The solute concentration
C, of incoming captured groundwater is assumed constant within a given layer.
Equilibrium linear adsorption and ﬁrst order reaction are also assumed, with a

degradation rate constant k speciﬁc to each reactor unit.

For ﬁrst-order reaction, the contaminant concentration C ,(t) exiting the radial PF R
that represents the injection zones is obtained from the concentration Ce(t) entering this
zone as:

C,(t) = Ce(t—r) exp(—kr) (13)
where ris given by equation (1). Similarly, for the extraction zone:
Ce(t) = [(Qc/Qe)Cc + (Qr/Qe)Cr(t-T) ] eXIX-kt) (14)

The outlet concentration from the recirculation zone is given by the convolution integral
C, (t) = f E(r)C,-(t — r)exp(—kr)dr (15)

where the residence time distribution E( r) is calculated using a second-order numerical
particle-tracking algorithm. This is the only non-analytic calculation performed in
solving the reactor model. A selected number of particles are uniformly distributed
around the circumference of the injection well. The trajectory of each particle is tracked
forward in time until it either reaches an extraction well or it moves beyond a speciﬁed
distance (i.e. along a streamline outside of the recirculation zone; these particles are
discarded in the RTD analysis). The particle location at the end of the next timestep is
calculated ﬁom its current position and the analytic velocity ﬁeld as
z"‘(t + At) = 2*(t) + [v,,,*/|v,,,*| ] At (16)

vm“ = V*( 2* + ‘/2 [ V*(Z*)/IV*(Z*)| 1 At) (17)
where At is the time increment and the dimensionless velocity v*(z*) = —v,./ V + iv,/ Vat
the dimensionless spatial coordinate 2*. Figure 1-3 shows the particle travel time
distributions for two- and three-well systems with Q; = 9.6 in which the effects of

porosity and retardation are neglected. The travel time distribution is multiplied by the

10

porosity and retardation factor to yield E( z) for the recirculation zone. The integral of
equation (15) is then discretized to obtain the exit concentration from the recirculation
zone.

Once initial conditions are speciﬁed, the time-dependent concentrations C,, C,,
and C, are solved using E(z) for the speciﬁed well geometry and pumping rate by
combining equations (1 3)-(15) into one equation with t as the only unknown. The
extraction well concentration C, is typically of interest to compare against groundwater
ﬁeld measurements, whereas the discharge concentration

Cd = (Qc/Qd)Cl + (Qb/Qd)Cc (13)

where Q; = Q, + Q, is relevant in regard to regulatory compliance.

11

R_es_ul_ts

Transport tests with conservative tracers and reactive solutes were simulated
using the reactor model and the results were compared with calculations from a three-
dimensional ﬁnite difference (FD) MODFLOW/MT3D model using a 10 m x 10 m x 6 m
grid with a grid spacing of 0.1 m in the lateral (x- and y-) coordinates. Advection within
the FD model is simulated using a fourth-order Runge-Kutta hybrid method of
characteristics solution algorithm. The simulated trap-and-treat system consists of one-
meter diameter reactive (inj ection/extraction) zones around two wells spaced two meters
apart within a six-meter thick conﬁned aquifer that is assumed to be initially free of
solute. Transient solute concentrations are calculated for a hypothetical ten-hour
pumping interval with constant injection concentration C,- = 100 ppm. Results for single-
layer and three-layer aquifers are compared for the model systems listed in Table 1-1.
The following invariant parameter values were used in the ﬁnite difference simulations:
porosity = 0.3, dispersivity = 0.1 m, head gradient = 0.01, and leakance between layers =
0.01 hr”.

Figure l-4(a) shows the extraction well concentration C, predicted for a
conservative tracer in a homogeneous, single-layer conﬁned aquifer. Tracer extraction
begins after approximately two hours in the FD model, and after 3.5 hours in the reactor
model. The difference is due to longitudinal dispersion, which is accounted for in the FD
model but neglected in the reactor model. After approximately four hours, the extracted
tracer concentrations predicted by the reactor model are very close to those of the FD
model. The inability of the reactor model to simulate early, low concentration,

dispersion-induced breakthrough of solute is not expected to limit the model’s utility

12

because pulse-pumped systems are not generally expected to use such relatively short
pump-on times. Simulations with three-layer, homogeneous FD and reactor models
yielded similar results as those shown in Figure 14(a) and are not presented here.

Figure 1-4(b) shows FD and reactor model simulation results for a conservative
tracer in a three-layer, heterogeneous conﬁned aquifer in which the top and bottom
aquifer layers have a lower conductivity than the middle layer. The combined pumping
rate Q, for the three layers matches the pumping rate for the case shown in Figure 1-4(a).
The tracer breakthrough times for the FD and reactor models agree more closely for the
heterogeneous aquifer of Figure 1-4(b) than for the homogeneous aquifer of Figure 1-
4(a). It appears that neglect of longitudinal dispersion in the reactor model becomes less
signiﬁcant as heterogeneity is added to the system and dispersion effects are spread
across several layers. The extraction well concentration curves are again in good
agreement at later times.

Figure 1-4(c) shows FD and reactor model simulation results for a reactive solute
in a single-layer conﬁned aquifer with fully-screened wells. This case is identical to that
of Figure 1-4(a), except that solute degradation with ﬁrst-order kinetics occurs within the
aquifer. It is again observed that the reactor model predicts a longer breakthrough time
than the FD model, due to neglect of dispersion, whereas agreement between the models
at later times is quite good.

Figure 1-4(d) shows a comparison of the FD and reactor models for a
conservative tracer in a conﬁned aquifer with partially-screened wells. The FD model is
comprised of three layers, with a well screen present only in the middle layer, so that

groundwater can only be injected and extracted from this layer and not the others. The

13

reactor model results of Figure 1-4(d), meanwhile, are reported only for the layer in
which the well screen is present. The solute concentration curves predicted by the FD
and reactor models in Figure 1-4(d) differ signiﬁcantly. As in the previous cases, the two
models predict different tracer breakthrough times due to the effect of dispersion. More
signiﬁcant is that the reactor model overestimates the extraction well concentration at
longer pumping times. There are two reasons for this. First, solute arrival is delayed
because some ﬂuid streamlines in the three-dimensional FD ﬂow model are able to travel
through the unscreened layers, which results in residence times longer than predicted by
the two-dimensional analytic solution used in the reactor model. Second, the addition of
unscreened layers above and below the recirculation zone FD model allows for greater
solute loss due to dispersion from the recirculation zone. The reactor model curve in
Figure 1-4(d) labeled 100% efﬁciency assumes, as in Figures 4(a-c), that no solute is lost
ﬁ'om the recirculation zone. The reactor model curve labeled 75% efﬁciency in Figure 1-
4(d) is obtained if one assumes that 25% of the solute entering the recirculation zone is
permanently lost to dispersion before it reaches the extraction well. With this empirical
adjustment, the reactor model predictions can be brought into close agreement with the
FD model results for times out to ten hours. Although not shown, a 90% efﬁciency factor
provides a better ﬁt when steady state is achieved near 100 hours, suggesting that only
10% of the solute is actually lost to dispersion and providing an estimate of the degree to
which the remaining solute took a longer-than-predicted average path. Further study is
needed to determine the extent to which model predictions vary as actual system
properties depart ﬁ'om those assumed in the model. Recalling that this model is primarily

intended to rapidly and efﬁciently evaluate a large number of potential operational

14

conﬁgurations during the early stages of system design when detailed aquifer properties
may not be known, application of an empirical adj ustrnent based upon prior experience to
account for non-idealities should still provide conceptually useful results.

The FD and reactor model calculation times on a 400 MHz Intel Celeron
processor are shown for ten hours of simulated pumping for each case in Table 1-1. The
reactor model possesses a signiﬁcant computational advantage relative to the FD model,
which enables more rapid investigation of well geometries or changes in aquifer physical
properties.

The aquifer at the Schoolcraft Plume A site is composed primarily of sand and
gravel glacioﬂuvial sediments with signiﬁcant vertical heterogeneity. A regional layer of
clay exists approximately 27 meters below ground surface (bgs), and the water table is
approximately 4.5 meters bgs (Hyndman et al. , 2000). A trap-and-treat system of ﬁfteen
wells arranged in a line ﬁfteen meters long with no stagger was installed and screened
from 9.1 to 24.4 meters bgs. Table 1-2 lists the average hydraulic conductivities for
three-meter thick layers from 9 to 27 meters bgs as measured from well cores collected
during installation of the system. A hydraulic head gradient of 0.001 1, based on regional
measurements, was used to calculate speciﬁc discharge for each layer.

During an initial tracer study, groundwater was extracted from odd-numbered
wells at a total rate of 9.1 m3/hr and injected into evenvnumbered wells for a period of
ﬁve hours. Sodium bromide was added continuously to the injected groundwater to bring
the average injected bromide concentration to 16 ppm. Bromide concentration was
measured at each extraction well at thirty-minute intervals. Figure 1-5 shows the average

extracted bromide concentration during the ﬁve-hour tracer test. The error bars denote

15

one standard deviation of the concentrations measured across the eight extraction wells.
Variations in extracted concentration among the individual wells at any given time are
primarily due to a combination of heterogeneity between the wells and variations in well
spacing. A ﬁve-layer aquifer reactor model was developed using the physical property
data of Table 1-2 and assuming an aquifer porosity of 0.3. The predictions of the reactor
model are compared to the experimental tracer data in Figure 1-5. As in Figure 1-4(d),
for 100% recirculation efficiency the reactor model overstates the extracted bromide
concentration, but at 7 5% recirculation efﬁciency a good ﬁt to the experimental data is
obtained. The difference between the model predictions and ﬁeld tracer measurements is
primarily attributed to the assumption of fully-screened wells in the reactor model.
Comparison of the reactor model and FD predictions indicates that the reactor
model yields quantitative predictions of the extracted solute concentration over
intermediate to long timescales, provided that the assumption of full well screening in a
conﬁned aquifer is met. When the reactor model is applied to partially-screened well
systems in conﬁned or unconﬁned aquifers, the solute breakthrough curves are
qualitatively similar to FD model results and experimental data, but a scaling factor that

accounts for the effect of partial-screening is required to achieve quantitative agreement.

16

Discussion

The effect of the reaction rate constant, well density, and pumping rate on the
extraction well contaminant concentration is shown in Figures 6(a), 6(b) and 6(0)
respectively. The results reported in Figures 6(a-c) are for a two-meter line of
unstaggered wells in a single conﬁned layer with C, = 100 ppm; V= 0.001 m/hr", P = 0.3;
H = 6 m; and D = 0.30 m. Degradation is assumed to occur only in the injection and
extraction zones, and retardation due to adsorption is neglected. For systems with an odd
number of wells, the end wells are operated as extraction wells. The initial contaminant
concentration in all zones is 100 ppm.

Figure 1~6(a) shows the sensitivity of the extracted solute concentration to
changes in the ﬁrst order degradation rate constant for a three-well system with Q, = 2.0
m3/ht. For k = 2.0 hr", approximately 50% of the solute is degraded with each pass
through the reactor network. This value of the rate constant is comparable to that
measured for CT degradation by strain KC (Dybas et al., 1995). As might be expected,
changing the rate constant by an order of magnitude in Figure 1-6(a) signiﬁcantly alters
both the steady-state extraction well solute concentration and the rate of approach to the
steady-state condition.

Figure l-6(b) shows how the extracted groundwater solute concentration changes
as the number of wells is varied while the well-line length is held constant. Increasing
the well density increases the groundwater retention time within the injection and
extraction zones, because the total volumetric ﬂow is divided among a greater number of
wells. The higher well density also increases the fraction Q/Q, of injected groundwater

that is recirculated. The higher recirculation ratio leads to longer overall retention times

17

and lower steady-state concentrations. The increased degradation efficiency afforded by
a high well density is achieved, however, at the expense of a higher well installation cost
per unit well-line length.

Figure l-6(c) shows the effect of varying the pumping rate on the extracted solute
concentration for a three-well system with a degradation rate constant of 2.0 hr".
Increasing the groundwater extraction rate has two competing effects. The recirculation
ratio Q/Q, increases as the pumping rate increases, resulting in longer overall retention
times within the system. As the extraction rate increases, however, the ﬂuid residence
time in the injection and extraction zones decreases, so that the amount of degradation
per ﬂuid pass decreases at high pumping rates. For the three-well system of Figure 1-
6(c), the effect of the shorter reactive zone residence time negates the beneﬁt of increased
recirculation, so that high pumping rates result in low degradation efﬁciency (i.e. C,
values). Furthermore, since operating costs increase as the total pumping burden
increases, there is an economic disincentive to operating at very high extraction rates.

For very low pumping rates, much lower than those shown in Figure 1-6(c), the
bypass ﬂowrate Q1, becomes nonzero, and the overall degradation efﬁciency decreases
due to the uncaptured groundwater fraction. Thus, there is an optimum intermediate
pumping rate, dependent on the number and conﬁguration of wells, that maximizes
overall degradation of the contaminant.

In order to minimize total pumping costs, the pumping schedule for trap-and-tr'eat
operation should provide the largest possible ratio of pump—off time to total pumped
groundwater volume. The principal trap-and-treat operating constraint is that the

breakthrough of contaminant in the recirculation zone during the pump-off phase is

18

prohibited. Speciﬁc trap-and-treat remediation systems may have additional constraints;
for example, the microorganisms in an in situ bioremediation system may require
scheduled delivery of nutrients by pumping.

An estimate of the allowable pump-off time, as a function of the pump-on
treatment time, can be obtained from the recirculation zone RTD in each aquifer layer.
Assuming that uncontaminated and/or nutrient enriched water is continuously injected
during the pump-on phase, the fraction of the recirculation zone volume in each layer that
is fully swept by injected water is equal to the value of the feed-normalized RTD, F (t),

for pumping time t

F(t) = £E(r)dr (19)

This is equivalent to calculating the volume traversed by streamlines in the recirculation
zone that arrive at the extraction well within the interval t. Dividing the swept volume by
the well spacing and aquifer layer thickness yields an estimate of the swept recirculation
zone width. The allowable pump-off time for a given layer is the time required for a
parcel of groundwater traveling at the natural gradient velocity to cross this width.

Figure 1-7 shows the recirculation zone width and maximum pump-off time for a
heterogeneous, two-layer, two-well aquifer, assuming negligible dispersion, chemical
reaction, and adsorption. For continuous pumping, all aquifer layers have the same
recirculation zone width, irrespective of their conductivities; but for intermittent
pumping, the recirculation zone width increases more rapidly with pump-on time for the
higher conductivity layer, as shown in Figure 1-7(a). The maximum time interval
between pumping events that avoids solute breakthrough in each layer is shown in Figure

1-7(b). For short pump-on time, breakthrough during the pump-off phase occurs ﬁrst in

19

the lower conductivity layer, because the swept width of the recirculation zone in this
layer is narrower than in the higher conductivity layer. In fact, for pumping durations of
less than 7.5 hours, the width of the swept region is zero in the lower conductivity layer,
because there is insufﬁcient time for injected groundwater to reach the extraction well.

For long pump-on times (e. g. r= 15 hours), the allowable pump—off time becomes
greater for the lower conductivity layer. Although the swept width of the recirculation
zone is always larger in the high conductivity layer, natural gradient ﬂow is sufﬁciently
slower in the lower conductivity layer so that it takes longer for a parcel of groundwater
to cross the swept width of this layer. The crossover point, at which both layers have the
same breakthrough time, occurs at a pump—on interval of eleven hours for the particular
two-layer aquifer investigated in Figure 1-7.

If no contaminant is to be allowed to exit the swept portion of the recirculation
zone during the pump-off phase, then the trap-and-treat system must be operated such
that the interval between pumping events does not exceed the shortest of the allowable
pump-off times for each aquifer layer; e. g. in Figure l-7(b), a pump-off or "trapping"
interval of 18 days for each pump-on or “treatment” period of 20 hours. Under this
constraint, maximum trap-and-treat efﬁciency is achieved by operating at the point of
crossover in the pump-off curves; i.e. in Figure 1-7(b), by adopting a schedule of eleven
hours of pumping once every twelve days. Pump-on times shorter than this duration are
disadvantageous because rapid breakthrough in the swept zone of the lower conductivity
layer compels more frequent pumping. Pump-on times longer than the optimum eleven-
hour duration, meanwhile, are also disadvantageous because the beneﬁt of the longer

pump—off time interval is negated by a disproportionately larger increase in the total

20

pumping burden incurred due to the longer pump-on time. For well conﬁgurations and
for aquifers with physical properties that are different from the two-well, two-layer case
considered in Figure 1-7, the preceding analysis will of course yield a different preferred
pumping scheme. However, the same process optimization considerations will apply.

If solute adsorption is signiﬁcant during both phases of operation, the swept
recirculation zone width increases more slowly over time, and the allowable pump-off
time increases. The curves of Figure l-7(a) will then shift rightward, and those of Figure
1-7(b) rightward and upward, by an amount proportional to the retardation factor. In
Figure 1-7(b), it was assumed that no reaction occurs between pumping events, but if
solute degradation does occur during the pump-off phase, the maximum pump-off time
again increases and the curves of Figure l-7(b) shift upward. Also, the results shown in
Figure 1-6 indicate that for certain conditions, the pumps must remain activated for a
period of time before the injected water becomes sufﬁciently "clean", i.e. the discharge
concentration attains its desired or steady-state concentration. In such cases, a longer
pumping time is required to effectively sweep the recirculation zone, and the curves of
Figures 7(a) and 7(b) shift rightward by an amount equal to the period of time needed to

achieve the desired discharge concentration.

21

Conclusion

The two-dimensional ﬂow, recycle reactor model presented in this work is
intended as a tool for rapid initial design and analysis of trap-and-treat remediation
systems. The model predicts transient and steady-state solute concentrations for different
conﬁgurations of intermittently pumped ﬁeld treatment systems. Comparisons with
three-dimensional FD model calculations demonstrate that the reactor model accurately
predicts solute transport in systems with ﬁrst-order reaction, fully-screened wells, and
heterogeneous layers when dispersion is not signiﬁcant. The reactor model predictions
also compare reasonably well with tracer breakthrough measurements from ﬁeld
experiments conducted at the Schoolcraft Plume A site. Extension of the reactor model
to include multiple solutes and phases, and to incorporate more complicated kinetics and
adsorption equilibria, is possible at the expense of computational speed. The
semianalytic reactor model is computationally efﬁcient for optimization of well spacing
and geometry, pump extraction rate, and pumping schedule for trap-and-treat operations.
For process modeling of aquifers with signiﬁcant lateral dispersion or vertical ﬂow, a
fully three-dimensional approach using FD or other numerical methods is recommended.

Additionally, the reactor model provides a simple framework for thinking about
relationships between remediation system behavior and general changes in aquifer
properties, well conﬁguration, and pumping rate and schedule. Future research will
illustrate some of these relationships, which might not be readily apparent when working
with conventional ﬁnite difference groundwater ﬂow models, and provide dimensionless

correlations for use in initial design studies.

22

Tables

Table 1-1: Varied ﬁnite difference model parameters.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Model Figure Layer H K Q, Computational time (s)
(m) (m/hr) (m /hr) FD model Reactor model
Homogeneous 1-4a 1 6 0.1 2 450 9
1 2 0.075 0.5
Heterogeneous l-4b 2 2 0.15 1 2670 10
3 2 0.07 5 0.5
Reactive 1 -40 1 6 0. l 2 420 9
k = 0.2 hr"
1 2 0.1 -
Partial Screen l-4d 2 2 0.1 0.67 1580 9
3 2 0.1 -

 

Table 1-2: Average measured hydraulic conductivity at 15-well site (Hyndman et al.,

 

 

 

 

 

 

 

2000)
Depth (m) 9-12 12-15 15-18 18-21 21-24 24-27
Number of measurements 41 35 44 33 41 26
Average (cm/s) 0.012 0.012 0.027 0.023 0.046 0.057
Standard Deviation 0.006 0.003 0.011 0.013 0.024 0.022
Coefficient of Variation (%) 53 25 4O 54 53 38

 

 

 

 

 

 

23

 

 

Figu_res

 

Q,

Injection
Zone ' -- irculaon

 

Injection Extraction
radial PFR (radial PFR

 

 

 

 

 

 

~L Zone

 

 

 

 

 

(a) (b)

Figure 1-1. (a) Injection, extraction, and recirculation zones for a two-well system. (b)
Conceptual reactor network model for an arbitrary number of wells. The dashed box
encloses subsurface ﬂow within a single aquifer layer. Additional layers operate in
parallel with the layer shown.

24

   

(a) (b)

Figure 1-2. Streamlines and groundwater velocity contours for a single conﬁned layer
with Q,‘ = 4.8. The natural groundwater ﬂow gradient is in the positive y-direction. (a)
Two-well system with extraction well on the left and injection well on the right. (b)
Staggered three-well system with two extraction wells and one injection well.

25

 

 

 

 

 

 

' I l I
2 wells
_ 8* = 0
0.5 " ‘
0 m r1 r1 al-t _ _ 1 _
0 0.5 1 1.5
Tiavel Tine (mrltiples of IN)

(a)

2

0.5

 

 

 

 

 

 

3 wells
3" = 0.5
0 0.5 1 1.5
Travel Tine (mrltiples of UV)

(b)

Figure 1-3. Dimensionless residence time distribution E" = EL/ V reported in terms of
dimensionless time t“ = tV/L for (a) two-well and (b) three-well systems with Q,* = 9.6.

26

2

 

 

 

 

 

NM NM

 

 

 

 

 

 

Figure 1-4. Extraction zone concentration histories from FD simulation (solid squares)
and ﬁom the reactor model (open squares) in a conﬁned aquifer with two wells. (a)
Conservative solute in a homogeneous aquifer with fully-screened wells. (b)
Conservative solute in a heterogeneous aquifer with fully-screened wells. (c) Degradable
solute in a homogeneous aquifer with fully-screened wells. ((1) Conservative solute in a
homogeneous aquifer with partially-screened wells. In part (d), the open squares and
open diamonds show the reactor model results for 100% and 75% efﬁciency,
respectively.

27

 

C. (ppm)

 

 

 

 

 

Figure 1-5. Measured and predicted tracer concentration histories for a 15-well ﬁeld
system. The experimental tracer measurements are shown as the solid squares; the
reactor model results for 100% and 75% efﬁciency are given by the open squares and
open diamonds, respectively.

28

 

100

C. (ppm)
3 a s
C. lpprn)

3

 

 

 

 

 

 

 

 

Figure 1-6. Extraction zone solute concentration histories obtained for variation of key
parameters. (a) Three-well system with Q, = 2.0 m3/hr and k = 0.20 (diamonds), 2.0
(squares), and 20. hr'1 (triangles); (b) Two-well (diamonds), three-well (squares) and
ﬁve-well (triangles) systems with Q, = 2.0 m3/hr and k = 2.0 hr"; (c) Three-well system
with k = 2.0 hr“ and Q, = 8.0 (diamonds), 2.0 (squares) and 0.50 m3/hr (triangles).

29

 

 

 

 

 

 

 

 

 

 

4 30 I I I
a 3 -
g a
., a:
g E

2 ' E—
2 a
a 1-

0 l J

0 5 IO 15 20 0 5 10 15 20
Pimp-0n Time (hours) Pimp-On Time (hurts)
(a) (b)

Figure 1-7. (a) Effective recirculation zone width and (b) maximum allowable pump—off
time as a function of pump-on time for trap-and-treat operation in a conﬁned two-layer
heterogeneous aquifer. The conductivities of the low- and hi gh-conductivity layers are
0.075 and 0.15 m/hr, respectively.

30

CHAPTER 2

Pulsed Pumping Process Optimization Using a Potential Flow Model

 

Nomenclature

a spatial coordinate of point source/ sink

B maximum possible recirculation zone volume

E exit-age residence time distribution

F feed-normalized residence time distribution

H aquifer layer thickness

K hydraulic conductivity

L well line length

M number of aquifer layers

N number of wells

P aquifer porosity

Q total pumping rate

R retardation factor for equilibrium linear sorption
S well line stagger (offset)

t pump-on time

T pump-off time

v ﬂuid velocity

V natural gradient speciﬁc discharge

w complex potential function

x spatial coordinate perpendicular to natural gradient

31

(m/hr)
(m)
(-)

(-)
(at/hr)
(-)

(m)
(hr)
(hr)
(tn/hr)
(m/hr)
(mz/hr)

(m)

y spatial coordinate in the direction of natural gradient

2 complex spatial coordinate
Subscripts

j layer index

k well index

m midpoint of timestep
max layer with highest conductivity

min layer with lowest conductivity

32

(m)
(m)

Introduction

Pulsed pumping or trap-and-treat remediation refers to the use of periodic rather
than continuous pumping for capture and degradation of groundwater pollutants. In the
pulsed system considered here, during the pump-off period, contaminants are transported
by natural gradient groundwater ﬂow into the region of the aquifer surrounding a transect
of alternating injection and extraction wells. After a speciﬁed period of time has elapsed
to permit intrusion of contaminated groundwater into this region, the well pumps are
activated and a recirculation zone is established between adjacent injection and extraction
wells along the well line. The extracted groundwater can be disposed of or treated for
removal of contaminants and the aquifer recharged with clean water that is free of
contaminants. Alternatively, the extracted groundwater can be augmented with
chemicals or nutrients that stimulate abiotic or biological degradation of pollutants and
then reinj ected into the recirculation zone for in situ contaminant removal. Once the
recirculation zone has been cleansed of contaminant, the pumps are shut off, allowing the
next volume of contaminated groundwater to enter into the treatment zone by natural
gradient ﬂow. The alternating pump-off and pump-on modes of operation are continued
sequentially for the duration of the project.

Pulsed pumping has been investigated, with varying levels of success, as a lower
cost alternative to continuous pumping for the treatment of LNAPLs (Kim and Lee, 2002;
Voudrias and Yeh, 1994), DNAPLs (Gerhard et al., 1998; Whelan et al., 1994) and
VOCs (Mackay et al., 2000). A hybrid scheme involving bioaugmentation and pulsed
pumping has been developed for the in situ bioremediation of Schoolcraft Plume A, a

carbon tetrachloride plume in near a village in southwest Michigan (Dybas et al. , 1998).

33

Bioaugrnentation of Plume A with Pseudomonas stutzeri strain KC, a non-native
microorganism, enabled transformation of more than 98% of the carbon tetrachloride in
the groundwater into carbon dioxide and forrnate without production of chloroform over
a 640-day monitoring period (Criddle et al, 1990; Dybas et al., 1995; Dybas et al., 2002).
Pulsed pumping was implemented for the ﬁfteen-well transect at the Plume A site, with a
weekly ﬁve to ten hour pumping event for nutrient addition interrupting the slow (one
meter per week) passage of the carbon tetrachloride-contaminated groundwater through
the treatment zone and stimulating biodegradation of the chlorinated solvent by strain KC
colonies in the aquifer solids. A substantial reduction of operating costs were realized in
comparison to the projected expense for continuous pump-and-treat remediation of this
site, which would have otherwise required pumping and disposal of large volumes of
contaminated groundwater.

To achieve maximum effectiveness ﬁ'orn the use of pulsed pumping in in situ
remediation, the placement of injection/extraction wells and the pumping schedule must
be sensibly chosen. Various mathematical models, based upon ﬁnite difference (Borden
and Kao, 1992; Gerhard et al., 2001), ﬁnite element (Armstrong et al., 1994; Schulenberg
and Reeves, 2002), Green’s function (Harvey et al., 1994) and boundary element (Zhang
et al. , 1999) methods, have been put forth to evaluate pulsed pumping performance and
economy relative to continuous pumping. Recently, a reactor model was developed for
use as a computationally efﬁcient pulsed pumping design and modeling tool (Tenney et
al. , submitted). This reactor model is semi-analytic and thus resource efﬁcient in
comparison to MODFLOW (McDonald and Harbaugh, 1983) and MT3D (Zheng, 1992)

numerical simulation methods. Comparison of the reactor model with ﬁnite difference

34

simulations has established that the model accurately predicts solute transport in systems
with ﬁrst-order reaction, fully-screened wells, and laterally homogeneous layers when
dispersion is not large. The reactor model predictions also compare reasonably well with
tracer breakthrough measurements from ﬁeld experiments conducted at the Schoolcraft
Plume A site. The semi-analytic reactor model is computationally efﬁcient for
optimization of the well conﬁguration and groundwater extraction rate for pulsed
pumping, and the effects of well density and pumping rate on degradation efﬁciency and
operating costs were considered for several treatment grids (T enney et al. , submitted).
An interesting ﬁnding of this work is that there is an optimum intermediate value of the
extraction rate, dependent on the number and conﬁguration of wells, that maximizes
contaminant degradation per unit of pumped groundwater volume. Preliminary
calculations also suggested that an optimum schedule for pulsed pumping can be
determined from the well geometry and aquifer hydrological proﬁle. In this paper, we
expand upon this latter concept and describe how potential ﬂow and particle tracking
algorithms can be combined into a predictive model for optimization of pumping

schedules for pulsed pumping remediation systems.

35

Methodology

The aquifer is divided into laterally homogeneous layers with different values of
natural gradient speciﬁc discharge V(Figure 2-1). N evenly spaced and fully screened
injection/extraction wells are arranged in a line of length L perpendicular to the direction
of natural gradient ﬂow with the injection wells placed up- or downstream of extraction
wells by a stagger distance S. (For the results described later in this study, S is uniformly
set to zero.) During the pump—on periods, recirculation zones are established in each
layer between adjacent injection and extraction wells (Figure 2-2). A recirculation zone
is formally deﬁned as the region between adjacent wells through which pumped water
ﬂows from the injection well to the extraction well. Because vertical ﬂow is neglected,
the recirculation zones in any given layer evolve independently of those in other layers.

The boundaries of a recirculation zone within a given layer depend upon the
number of wells and their spacing and the dimensionless total pumping rate
Q*=Q/( VHL), where Q is the total rate of groundwater extraction and H is the layer
thickness. As the pumping rate or well spacing increases, the maximum possible volume
of the recirculation zone likewise increases. Figure 2-2 shows the two-dimensional ﬂow
streamlines and resulting recirculation zones for any layer of a ﬁve-well system with
Q*=30 assuming an equal distribution of ﬂow among injection and extraction wells. In a
multi-layer system the value of Q“ is the same for all layers, eliminating the need to
perform ﬂow calculations for each layer.

The boundaries and volume of the recirculation zone are determined from an
analytic potential ﬂow model (Bird et al., 1960; Columbini, 1999) that assumes steady-

state, continuous, incompressible, inviscid, irrotational, two-dimensional ﬂow occurs

36

within the layers of a conﬁned aquifer. All injection and extraction wells are respectively
assumed to be ideal sources and sinks. A solution for the ﬂow ﬁeld may be obtained in
terms of the complex potential w(z), where z = x + iy and the natural groundwater
gradient is in the positive direction of the y spatial coordinate. For an N-well system in
an M-layer heterogeneous aquifer with even-numbered extraction wells staggered behind

odd-numbered injection wells, the dimensionless complex potential w“ = wj/(L V,-) is

 

 

 

 

NQk" .
w*(z*)= 22—1n(z*—ak*) +12* (1)
k=l 7’
T Q*
, , kodd
Q“: j int[(N :l)/ 2] (2)
-,——, keven
L rnt[N/2]
i[-%+;—‘_ll]+ 5—, kodd
“"E‘ 1 k—l 5* (3)
(——+ )—i——, keven
i 2 N—l 2
Q*=—MQ—— (4)
LZVJ'HJ

where int[x] is the truncated integer value of x, 2* = z/L, S“ = S/L, and H,- and V,- are the
respective thickness and speciﬁc discharge of the jth layer. Equation (2) states that the
total dimensionless pumping rate Q* is distributed equally across the extraction wells and
injection wells; e. g. for a three-well system with S“ = 1, groundwater is extracted at the
rate —Q* from well 2 at coordinate of = (0, —i/2), and injected at the rate +Q*/2 into

wells 1 and 3 at coordinates a1* = (-l/2, +i/2) and (13* = (+1/2, +i/2).

37

The velocity ﬁeld v,(z) of layer j may be obtained from the derivative of the

complex potential

dwj .
‘)j(z)=_= Vjﬂkzﬁgklk 27: (2* *—ak*)j+l} (5)

where the velocity components v, and v,. are given respectively as the real and imaginary

 

parts of equation (5).

As previously noted, pulsed pumping is characterized by long periods where the
remediation system is idle, punctuated by recurrent pumping events in which the
contaminated groundwater that has entered the treatment zone is purged and replaced
with clean or augmented water. An estimate of the allowable pump-off time for the
trapping phase, as a function of the pump-on time for the treatment phase, can be
obtained if the recirculation zone residence time distribution E(t) is known for each pair
of adjacent wells in an aquifer layer. Assuming that contaminated groundwater is
continuously extracted during the pump—on phase and replaced by clean injected water,
the ﬁaction of a recirculation zone volume that is fully swept by the injected water is
equal to the value of the feed-normalized residence time distribution F (t). For pumping

time t
F(t)= £E(Z’)d1’. (6)

For each recirculation zone E(r) is calculated using a second-order numerical particle-
tracking algorithm in which a predetermined number of particles are unifome
distributed around the circumference of the injection well. Using the ﬂow ﬁeld solution
obtained ﬁ'om equation (1), the trajectory of each particle is tracked forward in time until

it either reaches an extraction well or it moves a speciﬁed distance from its original

38

position (i.e. along a streamline directed outside of the recirculation zone). The latter
particles are discarded from the analysis of the recirculation zone residence time
distribution. The location of a particle at the end of the next timestep is calculated from
its current position and ﬁom the velocity ﬁeld of equation (5) as
z*(t + At) = z*(t) + [ v,,* / |v,,,*| ] At (7)

vm“ = V*( 2* + V2 I V*(Z*) / IV*(Z*)| ] A!) (3)
where At is the time increment and the dimensionless velocity v*(z*) = -—v,,/ V + ivy/ V at
the dimensionless spatial coordinate 2*.

Once the residence time distribution has been numerically determined, F(t) is
calculated by discretizing the integral of equation (6). The product of F (t) and the
maximum possible recirculation zone volume B is equivalent to the volume traversed by
those streamlines in the recirculation zone that arrive at the extraction well within the
pumping interval t. Dividing the volume of this swept zone by the well spacing, L/(N —
1), and the aquifer layer thickness yields an estimate of the swept recirculation zone
width. In the absence of signiﬁcant degradation or sorption, the allowable pump-off time
7}(t) for a given layer for pump-on time t is the time required for a parcel of groundwater
traveling at the natural gradient groundwater ﬂow velocity to cross the swept width of the
narrowest recirculation zone in this layer during the idle period of no pumping activity:

TJ(I)=(N—1)BF(t)/(LI£-Vj) (9)
In this manner, permissible pump-off times (i.e. those which avoid contaminant
breakthrough) can be estimated for separate layers in a heterogeneous aquifer, given a
speciﬁed well geometry, the aquifer physical properties, and the duration and rate of

groundwater extraction maintained during the pump-on period. A presentation of results

39

and observations using this method for a broad range of model aquifer/well systems now

follows.

40

3.2821192

Figure 2-3 shows the maximum pump-off time as a function of pump-on time for
an arbitrary pulsed pumping system operating in a three layer heterogeneous aquifer
consisting of a low, medium, and high conductivity (K) layer. While all aquifer layers in
the system have the same maximum possible steady state recirculation zone width, during
pulsed pumping operation the swept recirculation zone width increases most rapidly with
pump—on time for the highest conductivity layer, and least rapidly for the lowest
conductivity layer. This is reﬂected in the shapes of the pump-off time vs. pump-on time
curves 7)’(t) shown for the three layers in Figure 2-3. If the pulsed pumping timetable is
constrained so as to avoid breakthrough of contaminated water in all layers, then the
maximum allowable pump-off time for any given pump-on interval t is given by the
minimum of the T1, T2 and T 3 curves at that value of t. This maximum pump-off time to
prevent breakthrough in any layer is depicted by the solid line in Figure 2-3.

It can be seen that for pump-on intervals of less than 13.6 hours, the allowable
pump-off time is constrained by the breakthrough of contaminant in the lowest
conductivity layer. The reason for this is that the swept width of the recirculation zone in
this layer is narrower than in the higher conductivity layers. In fact, it is apparent ﬁ'om
Figure 2-3 that, in order for pulsed pumping to be carried out in this model aquifer, the
pumps must remain activated for at least twelve hours during each pump—on interval. For
pumping durations shorter than this, there is insufﬁcient time to fully sweep the
recirculation zone of the lowest conductivity layer; i.e. none of the water injected into this
layer reaches the extraction well within the allotted time period, reducing the allowable

pump—off zero.

41

Conversely, for pumping durations that exceed 13.6 hours, the allowable pump-
off time becomes constrained by the highest conductivity layer. Although the swept
width of the recirculation zone is always larger in this layer than in the other layers,
natural gradient ﬂow is also fastest in the highest conductivity layer, and so it takes less
time for a parcel of groundwater to cross the swept width of this layer compared with the
other layers. The crossover point, at which the most and least conductive aquifer layers
have the same breakthrough time, occurs for the pump-on interval of 13.6 hours for the
particular model aquifer considered in Figure 2-3. Signiﬁcantly, the fractional pump-on
time for pulsed pumping passes through a minimum at this pumping duration.

Maximum operational efﬁciency is achieved by adopting the pumping schedule
corresponding to the crossover point; i.e. activating the pumps for 13.6 hours, and then
idling the pumps for 2.2 days. Pump-on times shorter than 13.6 hours are
disadvantageous because rapid breakthrough in the swept zone of the lowest conductivity
layer compels more frequent pumping. Pump-on times longer than the optimum are also
disadvantageous because the potential beneﬁt of a longer pump-off time interval is
negated by an overall increase in the ﬁaction of time that the pumps are turned on.
Departures ﬁom the pumping schedule at the crossover point in either direction therefore
lead to increased operating costs due to the energy expended in groundwater extraction
and re-injection.

For well conﬁgurations or aquifer constructs that have different physical
properties than those considered in Figure 2-3, the preceding analysis will naturally yield
a different optimum pumping schedule. However, the same process optimization

considerations will apply. It is noteworthy that the properties of the aquifer layer with

42

intermediate conductivity had no bearing on the determination of the process optimum in
Figure 2-3; i.e. only the conditions in the highest and lowest conductivity layers were
signiﬁcant to the outcome. It can be shown that for multi-layered, heterogeneous
systems, the minimum ﬁaction of pump—on time will always occur where allowable
pump-off times for the least and most conductive layers become equal. (For single-layer
systems the minimum required fraction of pump-on time occurs when the slope of
allowable pump-off time versus pump-on time, dT/dt, is equal to T/t.) Consequently,
ﬁnding the optimal pumping scheme for pulsed pumping in an aquifer that has three or
more physically distinct vertical layers simpliﬁes to an analysis of contaminant
breakthrough in just two of the layers, those that have the highest and the lowest speciﬁc
discharge, and the pump-on vs. pump—off correlations may be expressed in terms of a
speciﬁc discharge ratio V“ = er/ V,,,,-,, for the most conductive and least conductive

layers.

43

Discussion

Figure 2-4 is a plot of dimensionless pump—off time versus dimensionless pump-
on time for a single layer with all combinations of N = 5, 7, 9, 11, 13, 15, 21, 31, 45, 67,
and 99 wells and dimensionless total pumping rates Q‘ = Q/( VHL) = 12, 20, 30, 50, 80,
120, 200, and 300. Pump—off time is shown in units of PRL/[V(N-1 )] where P is aquifer
porosity and R is a retardation factor accounting for sorption effects. A dimensionless
pumpooff time T" equal to unity corresponds to the time required to cross a recirculation
zone with a width equal to the well spacing, i.e. 7" could equivalently be referred to as
the recirculation zone width in units of L/(N-l). Pump-on time is shown in units of

PRL 7rN l

. . 10
V(N —l) 6Q*(N —1) 1.9486xp(—0.694N )+0.0001081N +0.757 ( )

 

The ﬁrst two terms in (10) represent the time required to travel directly between any pair
of adjacent (and unstaggered) injection and extraction wells in the absence of natural
gradient ﬂow and neglecting inﬂuences from other wells in the system. The third term in
(10) is an empirical correction to account for the inﬂuence of other wells in the system
upon the travel time between the adjacent wells. This empirical correction was used for
presentation of the data instead of the exact expression because calculating exact inter-
well travel times for systems with an arbitrary number of wells is very cumbersome.
Presented in the units described above, a dimensionless pump-on time t* equal to unity
roughly corresponds to the initial breakthrough of recirculation ﬂow between the adjacent
wells. Substituting the deﬁnition of Q* into (10), it can be seen that this breakthrough
time is proportional to [L/(N-l)]2, which is the square of the well separation distance, and
inversely proportional to Q/(2NH), which is roughly equal to the pumping rate per unit

layer thickness per well.

44

Using least-squares regression, the data was ﬁt to the equation
T" = 0.5404(t*-1)0.5292 - 0.0766(t“-1) (11)

Note that this equation is empirical and should not be extrapolated beyond its maximum
value at approximately t*=17. As shown in Figure 2-4 by the curve of t*/(t*+ T“), the
required fraction of pump-on time for a single-layer system reaches a minimum value
when the dimensionless pump—on time is approximately equal to 1.85. The
dimensionless pump-off time at that point is approximately 0.44, which corresponds to a
recirculation zone width equal to 44% of the distance between adjacent wells. Actual
fractions of time spent with pumps on may be calculated from t“/(t*+7“') and the
deﬁnitions of t* and 7*.

Although not shown in Figure 24, systems with three wells generally have longer
allowable pump-off times than systems with more wells because only in three-well
systems are all recirculation zones the same size and shape. Three well systems are
particularly efficient because the narrowest recirculation zone in a layer determines the
maximum purnp—off time and remaining recirculation zones might be considerably wider.
As stated earlier, total extraction and injection rates were assumed to be distributed
equally among extraction and injection wells, respectively. With an odd number of wells
greater than three under these pumping conditions, the recirculation zone between the
second and third well is always the narrowest. For systems with Q‘ greater than 10, the
width of this recirculation zone approaches a limiting value approximately equal to 110%
of the distance between adjacent wells. Systems with Q* less than 10 approach limiting
recirculation zone values less than this distance and so generally have shorter allowable

pump-off times. Cursory investigation suggests that overall pumping requirements for

45

pulsed operation could be reduced if individual well pumping rates were adjusted to
result in a more even distribution of recirculation ﬂow between various well pairs. For
example, the calculated pumping requirements for a single layer, 15-well system
operating at Q*=30 were reduced by approximately 12% by simply adjusting ﬂow rates
to result in the same ﬂuid velocity at the midpoints of the gaps between all pairs of
adjacent wells. Further study is required to determine whether this strategy would be
beneﬁcial in a ﬁeld setting.

If it is necessary to account for degradation or non-linear sorption during the
pump-off period, this more complicated behavior can be simulated by combining
equation (11) with a customized one-dimensional ﬂow model. In this case equation (11)
is used to calculate the recirculation zone width, and the one-dimensional ﬂow model is
used to calculate the allowable pump-off time by simulating contaminant transport across
the recirculation zone width. A similar adjustment could also be made to the pump-on
time if necessary, but it would be less straightforward because the effect would need to be
spread over a continuum of path lengths, which are not explicitly shown in equation (11).

Although equation (11) applies speciﬁcally to recirculation zone behavior in only
one layer, it can be used to predict the behavior of multi-layer systems. Recalling that
V*=V,,.,,/ Vm, if T’=f(t*) is the allowable pump-off time for the layer with the greatest
superﬁcial velocity, then V*f(t*/ V*) is the allowable pump-off time for the layer with the
lowest superﬁcial velocity. The allowable pump-off time for the entire system at any
given value of t* is then the minimum of the two values f(t"‘) and V‘f(t*/ V). Note that in
this case the superﬁcial velocity used to convert the actual pump-off and pump—on times

from their dimensionless forms is Vmax. Also note that the allowable pump-off time for

46

layers with intermediate values of conductivity can be similarly calculated but that these
layers have no inﬂuence on the allowable pump-off time for the entire system.
Predictions made by equation (11) were compared to published results (Hyndman
et al. , 2000) describing an optimization study conducted using the MODF LOW ﬁnite
difference groundwater ﬂow model to design a pulsed-pumping system intended to feed
microbes once per week in a 15m deep x 15m long x 0.3m wide bioremediation zone.
(Based upon the published aquifer characteristics, V=0.356 m/week, P=0.3, and R=1
were chosen to calculate t from (10) and equation (11) for various N and Q.) Figure 2-5
summarizes the results of this comparison by plotting the predicted total pumped volume
per pumping event versus number of wells for 7-, 9-, 11-, 13-, and 15-well systems.
(Total pumped volumes from Hyndman were calculated from values apparently
published on a per-well basis.) The agreement between the predictions ﬁom equation
(11) and the MODFLOW simulations is very good. As is commonly the case, signiﬁcant
information about aquifer heterogeneity was not available prior to installing the
remediation system in question, and so initial design simulations had to be conducted
using relatively simple aquifer models. This necessarily simpliﬁed approach was
continually validated as more data became available (Hyndman et al. , 2000). Note that
equation (11) inherently assumes that the total pumped volume per pumping event
remains constant regardless of the actual pumping rate chosen if the ﬁnal recirculation
zone width is speciﬁed. This is equivalent to stating that pump-on time varies inversely
with pumping rate for a given number of wells. In support of this assumption, it was

found using conventional groundwater models to simulate the system described above

47

that total pumped volumes varied by less than 3% while varying total pumping rates from
0.4 to 125 m3/hr (Hyndman et al., 2000).

The pulsed-pumping schedule (i.e. pump—on and pump-off time) that minimizes
total pumped volume for an N-well system operated at dimensionless total pumping rate
Q‘ and installed in a heterogeneous aquifer with superﬁcial velocity ratio V* was
identiﬁed for all combinations ofN= 3, 5, 7, 9, 11, 13, 15, 21, 31, 45, 67, 99; Q* = 2, 3,
5, 8, 12, 20, 30, 50, 80, 120, 200, 300; and V“ = 1, 2, 3, 5, 8, 12, 20, 30, 40. Results from
these calculations are presented in Figures 6-9.

Figure 2-6 is a plot of the dimensionless pump-on time 1* for most efficient
pumping versus the superﬁcial velocity ratio for 7- to 99-well systems operating at
various dimensionless total pumping rates. Calculations for systems with three or ﬁve
wells generally yield larger values of t*. The pump-on time for multi-layer systems is
shown in units of

V at: . PRL . 7W . 1 (12)
Vmax (N — l) 6Q * (N — 1) 1.948 exp(—0.694N) + 0.0001081N + 0.757

 

which reduces to (10) for single-layer systems. As previously seen in Figure 2-4, t*
approaches a value of approximately 1.85 for homogeneous aquifers, i.e. when V*=1.
Also note that the dimensionless pump-on time for any given Q* decreases to a limiting
value as V* increases. The value of V* at which t* becomes constant is equal to the value
of t* at which the recirculation zone width reaches its maximum limiting value in the
most conductive layer for that pumping rate. Although the dimensionless pump-on time
for most efﬁcient pumping decreases with increasing V*, the actual pump-on time t must

always increase with V*.

48

Figure 2-7 is a plot of the dimensionless pump-off time 7“ for most efficient
pumping versus the dimensionless total pumping rate for 7- to 99-well systems in
aquifers with various degrees of layered heterogeneity, as measured by V*. Calculations
for systems with three or ﬁve wells generally yield larger values of 7“. As expected,
pump-off times approach zero as Q* approaches unity, and continuous pumping becomes
necessary when pumping is no longer strong enough to establish recirculation zones
between all wells. Figure 2-7 also shows that the most efficient pump-off time does not
change signiﬁcantly as Q* is increased beyond a particular value for any given V*.
Similarly, pump-off time does not change signiﬁcantly as V* is increased beyond
approximately 12 for any given pumping rate. The combination of V* and Q" values
above which 7" becomes constant is again related to the point at which the recirculation
zone width reaches its maximum limiting value in the most conductive layer.

Figure 2—8 is a plot of the minimum possible ﬁaction of time spent with pumps
turned on versus dimensionless total pumping rate for 3- to 99-well systems in aquifers
with various degrees of layered heterogeneity. Figure 2-8 provides a quantitative
estimate of how required system operation varies from continuous pumping through
degrees of pulsed pumping as the total pumping rate is changed. Figure 2-8 also shows
that heterogeneity can have a signiﬁcant negative impact upon overall pumping
requirements. As seen previously in Figure 2-7, pumping becomes continuous as Q"
approaches unity and recirculation between wells becomes nonexistent. Because Figure
2-8 assumes retardation and degradation are negligible, for pulsed operation the actual

required fraction of time spent with pumps on will be lower than that shown in Figure 2-8

49

if retardation and/or degradation are signiﬁcant due to the increase in allowable pump—off
time.

Figure 2-9 is a plot of the minimum possible time-averaged dimensionless total
pumping rate versus dimensionless total pumping rate for 3- to 99-well systems in
aquifers with various degrees of layered heterogeneity. The average total pumping rate is
calculated by multiplying the ﬁaction of time spent with pumps on (ﬁom Figure 2-8) by
the total pumping rate. Figure 2-9 provides a quantitative estimate of the degree of
retardation and/or degradation required in order to provide pulsed-pumping with an
efﬁciency advantage relative to continuous pumping. For example, if retardation and
degradation are negligible during the pump—off phase (as is assumed in Figures 8 and 9),
pulsed operation will always increase pumping requirements relative to continuous
operation. This potential disadvantage for pulsed systems relative to continuous systems
becomes more signiﬁcant as heterogeneity increases. Because this model assumes only
two—dimensional ﬂow, actual ﬂow through low conductivity layers might be reduced
relative to model predictions for systems in which signiﬁcant vertical ﬂow between
layers is possible. This would result in an effective value of V* greater than the

calculated value of V*, worsening the case for pulsed operation.

50

Conclusion

The two-dimensional potential ﬂow model and recirculation zone analysis
presented in this work is intended as a tool for initial design of in situ remediation
systems that employ pulsed pumping. The model provided predictions comparable to
those from a conventional ﬁnite-difference groundwater ﬂow model used during the
initial design and cost-analysis of a pulse pumped bioremediation system. A virtue of the
semi-analytic model is its speed of application for identifying the pulsed pumping
schedule best-suited for a given well conﬁguration and aquifer physical properties. For
selection of the pumping schedule, the principal constraint applied is that pumping costs
are to be minimized by selecting the largest possible ratio of pump-off time to total
pumped groundwater volume, subject to the requirement that contaminant breakthrough
in the swept portion of the recirculation zone during the pump—off period is prohibited.
Bioremediation systems may be additionally constrained by the feed requirements of the
in situ microbial population, and may therefore necessitate pulsed pumping of enriched
nutrient medium more frequently than would otherwise be adopted based on hydrologic
considerations alone.

The results from this model provide a lower bound to pulsed pumping
requirements, particularly for systems with a high degree of vertical heterogeneity. The
model predictions are based upon hydrologic considerations and do not take into account
dispersion or vertical ﬂow in the recirculation zone during the pump-on phase, which
may be signiﬁcant in some aquifers. If the actual system uses partially-screened wells,
ﬂow paths making up the upper and lower regions of the recirculation zone will tend to

be longer than predicted by this model. Similarly, since some degree of vertical ﬂow

51

between layers of differing conductivity will likely be present during pumping events,
actual ﬂow through the recirculation zone of low conductivity layers will likely be
reduced relative to that predicted by this model, and so a greater amount of pumping will
be required. In such cases, three-dimensional groundwater ﬂow models are
recommended for carrying out ﬁnal process optimization; or alternatively, a scale factor
for the recirculation zone RTD can be applied to mimic for the longer ﬂuid streamlines in

three—dimensional ﬂow.

52

 

’

 

 

—>
O ———.
V;

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2-1. Model aquifer structure and well geometry used for analysis of pulsed
pumping. The geometric parameters for well length, well stagger, and aquifer layer
thickness are shown for a hypothetical ﬁve-well, three-layer system with vertical
heterogeneity, as indicated by the layer-dependent speciﬁc discharge.

53

 

 

 

 

 

 

Figure 2-2. Streamlines obtained ﬁ‘om potential ﬂow model for an unstaggered, ﬁve-well
system in a single conﬁned aquifer layer with Q’=Q/(VHL)=30 and an equal distribution
of ﬂow between all injection and extraction wells. If natural gradient groundwater ﬂow
is assumed to be in the positive y-direction, three wells are extracting groundwater and
two wells are injecting. The recirculation zone is highlighted.

54

 

100% ~ .......................... .

Fraction Time wl Pumps On
Allowable Pump-Off Time, days

 

 

 

 

Pump-On Time, hours

Figure 2-3. Allowable pump-off time to prevent contaminant breakthrough between
pumping events for an arbitrary pulsed pumping system operating in a heterogeneous
aquifer consisting of a low, medium, and high conductivity (K) layer. The allowable
pump-off time is shown on the right axis as a ﬁrnction of the duration of each pumping
event for the low (squares), intermediate (x’s) and high (triangles) conductivity layers
considered separately, and for the entire aquifer (solid line). The overall fraction of time
the pumps are activated (dashed line) is shown on the left axis.

55

 

 

 

 

1.2 'l — — 4— ﬁ 1
t. g ..
l- tz .2 in ‘* ' ,
6 A _ h . ' t , + +
1'?" e
E 1 T (Wt. - ~'. *4“ . t ’ 4, T 4'
0- " v i J
1- ,, -+ 0.95
q 0.8 t, I
a 0 e , 0.9 i
v

a ‘
8 3-
E 0.4
.2
O ' 0.85
c .
0 0.2 »
E r J
-
o r

r i

o ‘ _ ._L f n + 1 i i + r if r i‘" I- i 1 Ti 1 + r + r i 1 Ir 0.8

 

 

 

 

0 1 2 3 4 5 s 1 a 0 10111213
Dimensionless Pump-On Time, t‘

Figure 2-4. Dimensionless allowable pump-off time 7““ versus dimensionless pump-on
time t“ for single-layer systems. All combinations of N = 5, 7, 9, ll, 13, 15, 21, 31, 45,
67, 99 wells and dimensionless total pumping rates Q* = Q/( VHL) = 12, 20, 30, 50, 80,
120, 200, 300 are represented (scattered +’s). T“=TI V(N-1)/(PRL)]. t*=t[ V(N-
1)/(PRL)][6Q*(N-1)/(1rN)][1.948exp(-0.694N)+0.0001081N+0.757]. Note that 7" could
equivalently be referred to as the recirculation zone width in units of L/(N -1). The data
were ﬁt to the equation P=f(t*)=0.5404(t*-1)0.5292-0.0766(t‘-1) via least-squares
regression (solid diamonds). The minimum of the t*/(t*+7"') curve (solid line)
corresponds to the point of operation for maximally efﬁcient pumping. Actual fractions
of time spent with pumps on may be calculated from t*/(t*+7") and the deﬁnitions of t*
and T".

56

l
' l
l

140+

 

 

 

0
E
.1-
5 0
> ! ‘
I“ l
i .
0 120 if
E i l
% L 0
E. 100 i
a I
9; 00 T \\\\\\ w T
s i a
,_ J
00 T i H _ Tia—T: +
7 9 11 13 15
Number of wells

Figure 2-5. Total pumped volume required to ﬂush a 15m deep x 15m long x 0.3m wide
bioremediation zone versus the number of wells in the system. Values predicted by the
potential ﬂow model (connected diamonds) are compared with values calculated using a
MODFLOW ﬁnite-difference groundwater ﬂow model (solid squares, Hyndman et al.
2000). Values of V=0.356 m/week, P=0.3, and R=1 were applied to the potential ﬂow
model based upon the published aquifer characteristics.

57

 

N

 

 

 

 

 

3... A
.. 1.8 K (
0 w.
E a \\
I'- ” it \
= 1 .6 '- ‘§\>
i i
E ..
3 1 4 - \ ' " ** '
n_ _ \\ N=7 to 99 wells
\ \
3 * \\ \ Direction of increasing number
.2 L \i of wells within each series:
\ \\
.9 1 2 '- \\\‘:\\‘~\-:~_ 2,; __________
0 . \ ~42E_:—~_—:;—~:—==—:—=::=-_:-=::: Q*=2
: __ a, 1 ____ .__ __ ___ .
0 _
_E_ _ .2......._,_._,,,__. o*=3
0 — _ Q*=5
1 b *=300
l l l l J l l 1 l1 1 lLlllJlllllllllllllllllLl
1 10 20 30 40

Superﬁcial Velocity Raﬂo, V*

Figure 2-6. Pump-on times that minimize total pumping requirements for pulsed
pumping systems operated in heterogeneous aquifers. Dimensionless pump-on time t* is
plotted against the superﬁcial velocity ratio V* = V,,.,,./ Km for systems with various
dimensionless total pumping rates Q"'=Q/( VHL). Systems with N = 7, 9, 11, 13, 15, 21,

31, 45, 67, and 99 wells are represented. t*=t[l/ V*][V,,.,,(N-1)/(PRL)][6Q*(N—

1)/(1rN)][1.948exp(-O.694N)+0.0001081N+0.757].

58

 

_L
N

 

 

 

 

 

 

 

L.

f_ * =7 1° 99 “"3 4,155; eesggg1¥§E§§§5E32§ae,nas; v*>12
.. r

0 1 _ ”
.§ .
P r—
a:
o. _
a. 0.8 P
E _
3 _
a _
g 0.6 _-
t:
.2 —
e 0.4 —
g _
D _

l-

J l l 1 I I I ll 1 J l 1 I L L l L L 1
0.2100 101 102

Dimensionless Total Pumping Rate, 0*

Figure 2-7. Pump-off times that minimize total pumping requirements for pulsed
pumping systems operated in heterogeneous aquifers. Dimensionless pump-off time T“
is plotted against dimensionless total pumping rate Q‘ for systems with various degrees
of vertical heterogeneity, as measured by the superﬁcial velocity ratio V*. Systems with
N = 7, 9, ll, 13, 15, 21, 31, 45, 67, and 99 wells are represented. P=7I V(N-1)/(PRL)].

59

 

   
  

*V*=20'l. N=3 to 99 wells
— V*'3 ,3 Direction of increasing number
- of wells within each series: i

   

.0
.h.
I T ﬁt I l

Fraction of Time with Pumps On
0
to

 

 

0 1 1 l 1 1 l 1 ll 4
10° 101
Dimensionless Total Pumping Rate, 0*

 

Figure 2-8. Minimum required fraction of time with pumps on for pulsed pumping
systems operated in heterogeneous aquifers. The fraction of pump-on time resulting from
use of the most efﬁcient pulsed pumping schedule is plotted against dimensionless total
pumping rate Q‘ for systems with various degrees of vertical heterogeneity, as measured
by the superﬁcial velocity ratio V*. Systems with N= 3, 5, 7, 9, 11, 13, 15, 21, 31, 45,
67, and 99 wells are represented.

60

Dimensionless Average Total Pumping Rate

ANN“

A

 

 

 

V*=40

- , v*=20

 

    
  

 

 

0 E
5 E-
0 2-
5 :-
0f-
r
5 l—
, ,,,,,, N=3t099 wells
Direction of increasing number ‘
of wells within each series:
I I I I I I I I1 I I I I J IIll l 1
10° 101 102

Dimensionless Total Pumping Rate, 0*

- v*=a

: v*=3

V*=1

Figure 2-9. Minimum possible average dimensionless total pumping rate for pulsed
pumping systems operated in heterogeneous aquifers. The average dimensionless total
pumping rate resulting from use of the most efficient pulsed pumping schedule is plotted
against dimensionless total pumping rate Q* for systems with various degrees of vertical
heterogeneity, as measured by the superﬁcial velocity ratio V*. Average total pumping
rate is the product of the fraction of pump-on time and the total pmnping rate. Systems
with N= 3, 5, 7, 9, 11, 13, 15, 21, 31, 45, 67, and 99 wells are represented.

61

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IIIIIIIIIIIIIIIIIIIII LIBRARIES

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