£EPA
United States
Environmental Protection
Agency
Robert S. Kerr Environmental
Research Laboratory
Ada, OK 74820
EPA/600/2-90/002
January 1990
Research and Development
A New Approach and
Methodologies for
Characterizing the
Hydrogeologic
Properties of Aquifers
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A NEW APPROACH AND METHODOLOGIES
FOR CHARACTERIZING THE HYDROGEOLOGIC
PROPERTIES OF AQUIFERS
by
FRED J. MOLZ; OKTAY GUVEN; JOEL G. MELVILLE
Civil Engineering Department
Auburn University, AL 36849
With Contributions By
IRAJ JAVANDEL
Earth Sciences Division
Lawrence Berkeley Laboratory
Berkeley, CA 94720
and
ALFRED E. HESS; FREDERICK L. PAJLLET
United States Geological Survey
Denver Federal Center
Denver, CO 80225
CR-813647
.,. ^
Project Officer
Lowell E. Leach
Robert S. Kerr Environmental Research Laboratory
Ada, OK 74820
ROBERT S. KERR ENVIRONMENTAL RESEARCH LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
ADA, OK 74820
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DISCLAIMER
The information in this document has been funded wholly or in part by the United
States Environmental Protection Agency under assistance agreement number CR-813647 to the
Board of Trustees of Auburn University, Auburn, Alabama, subject to the Agency's peer and
administrative review, and it has been approved for publication as an EPA document. Mention
of trade names or commercial products does not constitute endorsement or recommendation for
11
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FOREWORD
EPA is charged by Congress to protect the Nation's land, air and water systems.
Under a mandate of national environmental laws focused on air and water quality, solid waste
management and the control of toxic substances, pesticides, noise and radiation, the Agency
strives to formulate and implement actions which lead to a compatible balance between human
activities and the ability of natural systems to support and nurture life.
The Robert S. Kerr Environmental Research Laboratory is the Agency's center of
expertise for investigation of the soil and subsurface environment. Personnel at the laboratory
are responsible for management of research programs to: (a) determine the fate, transport and
transformation rates of pollutants in the soil, the unsaturated and the saturated zones of the
subsurface environment; (b) define the processes to be used in characterizing the soil and
subsurface environment as a receptor of pollutants; (c) develop techniques for predicting the
effect of pollutants on ground water, soil, and indigenous organisms; and (d) define and
demonstrate the applicability and limitations of using natural processes, indigenous to the soil
and subsurface environment, for the protection of this resource.
This report presents techniques for field measurements of the vertical distribution of
hydraulic conductivity in contaminated ground water at Superfund and other sites. These field
techniques allow fully three-dimensional characterization of aquifer properties which can be
used in advection-dominated transport models to significantly enhance our ability to
understand and predict contaminant transport, reaction and degradation in the field.
Clinton W. Hall
Director
Robert S. Kerr Environmental
Research Laboratory
111
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CONTENTS
Page
Preface viil
Abstract xi
Figures xil
Tables ixx
Abbreviations and Symbols xx
Acknowledgements xxiii
Executive Summary xxiv
I. Why a New Approach is Needed
1-1. Introduction 1
1-2. What's Wrong with Existing Procedures? 1
1-3. What's the New Approach? 7
II. The Impeller Meter Method for Measuring Hydraulic
Conductivity Distributions
II-1. Introduction 11
II-2. Performance and Analysis of Impeller Meter
Tests 11
II-3. Measurement of Hydraulic Conductivity at
Different Scales 22
II-4. Summary and Conclusions Concerning Impeller
Meter Applications 36
III. Multilevel Slug Tests for Measuring Hydraulic
Conductivity Distributions
III-l. Introduction 38
III-2. Performance of Multilevel Slug Tests 38
III-3. Analysis of Multilevel Data 41
III-4. Results, Discussion and Conclusions 53
IV. Characterizing Flow Paths and Permeability Distributions
in Fractured Rock Aquifers
IV-1. Introduction 61
IV-2. The U.S. Geological Survey's Thermal
Flowmeter 62
IV-3. Conclusions 71
V. Definition and Measurement of Hydraulic Conductivity
in the Vertical Direction
V-l. Introduction 75
V-2. Single-Well Tests 77
V-3. Tests with Two or More Wells 98
V-4. Conclusions 118
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VI. Field Methods for the Measurement of Effective Porosity,
Hydraulic Head and Storativity
VI-1. Introduction 119
VI-2. Field Measurement of Effective Porosity 119
VI-3. Definition of Hydraulic Head 124
VI-4. Field Measurement of Hydraulic Head 131
VI-5. Storage. Coefficient Definition and
Measurement . 132
VII. Mathematical Modeling of Advection-Dominated
Transport in Aquifers
VII-1. Introduction 135
VII-2. Governing Equations of Flow and Transport 135
VII-3. Numerical Solutions of Governing Equations .... 136
VII-4. Examples of Advection-Dominated Modeling .... 140
References 157
Appendices
AI. Overview and Evaluation of Methods for Determining
the Distribution of Horizontal Hydraulic Conductivity
in the Vertical Direction
AI-1. Introduction 173
AI-2. Straddle Packer Tests 174
AI-3. Particle Size Methods 176
AI-4. Empirical Relationships Between Electrical
and Hydraulic Conductivity 176
AI-5. Measurements Based on Natural Flow
Through a Well 177
AI-6. Single Well Electrical Tracer (SWET) Test 180
AI-7. Borehole Flowmeter Tests 183
AI-8. The Role of Geophysical Logging 183
AIL Analysis of Partially Penetrating Slug Tests
Considering Radial and Vertical Flow and
Anisotropy
AIM. Introduction 184
AII-2. Mathematical Model Development 184
AII-3. Model Solution and Parametric Study 186
AII-4. Numerical Example 187
AIII. The Physical Processes of Advection and Hydrodynamic
Dispersion
AIII-1. Introduction . 198
AIII-2. The Mechanisms of Dispersion 198
VI
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AIV. Table of Contents for the Proceedings Entitled
"New Field Techniques for Quantifying the
Physical and Chemical Properties of Heterogeneous
Aquifers"
AIV-1. Geology-Intensive Approaches to Property
Measurement 201
AIV-2. Property Measurement Using Borehole
Geophysics and Log Analysis 201
AIV-3. Measurement of Vertically-Averaged
Aquifer Properties 202
AIV-4. Tracer Techniques and Flow Regime
Characterization 202
AIV-5. Regulatory Problems and Multi-Level
Measurement of Aquifer Properties 203
AIV-6. Measurement of Chemical and Biochemical
Aquifer Properties 205
AIV-7. Measurement in the Unsaturated Zone 205
vil
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PREFACE
The decade of the eighties has seen the performance of several of the largest field
hydrologic transport and dispersion experiments ever attempted. Included in this number are
one or more studies performed at Base Borden, Ontario (University of Waterloo; Stanford
University), Cape Cod, Massachusetts (U.S. Geological Survey; Massachusetts Institute of
Technology), Chalk River, Ontario (Chalk River Nuclear Laboratories), Columbus, Mississippi
(Tennessee Valley Authority; Massachusetts Institute of Technology), Mobile, Alabama (Auburn
University), and Riverside, California (University of California at Riverside). Among other
things, all of these experiments were able to gather three-dimensional information, on a
relatively large scale, relating to parameters in the saturated zone or the unsaturated zone that
control transport and dispersion.
It is the present authors' opinion that one of the more important conclusions supported
by the studies mentioned above is that contaminant migration in both the saturated and
unsaturated zones is commonly more of an advection-based than a hydrodynamic dispersion-
based phenomenon. To understand what is going on in the subsurface, information is needed
primarily about the hydraulic conductivity distribution not aquifer dispersivity. This is
particularly true when one is attempting to understand or represent chemical and microbial
processes that are sensitive to the degree of irreversible mixing that occurs among the
components of a plume as it migrates downward or laterally away from its source. In aquifers,
it is clearly not possible to represent these processes accurately in most situations with a model
based on a two-dimensional (vertically-averaged) transmissivity field and large longitudinal and
transverse dispersion coefficients. In the unsaturated zone, such processes are not represented
well by a single downward flow velocity and a longitudinal dispersion coefficient, even when the
downward flow is driven by a long-term, constant irrigation rate.
It is our conclusion that the overall results of the large field transport and dispersion
experiments of the eighties, of which the present study is a member, have signaled an end to the
brief era of the dispersion-dominated transport model and the resulting preoccupation of
hydrologists with measurement, estimation, calibration, or otherwise divining a value for
longitudinal dispersivity, whether based on deterministic or stochastic methodology. It appears
likely that future models and regulations will increasingly be based on information about
groundwater or soil water flow patterns, flow rates and flow distributions, that are measured
directly or calculated based upon Darcy's law and a field-measured hydraulic conductivity
distribution. It may be that another type of model will come into use in unsaturated zone
problems because of the virtual impossibility of measuring accurate unsaturated hydraulic
conductivity and water retention distributions in the field using technology that is presently
available.
This report presents and summarizes most of the information and experience gained
from six years of field experimental and theoretical studies by Auburn University that were
funded by the U.S. Environmental Protection Agency through the Robert S. Kerr Laboratory.
In one way or another, most of this work dealt with the understanding and measurement of
hydraulic conductivity distributions in the field, with all measurements made in the saturated
zone. However, the title of the present report, "A New Approach and Methodologies for
Characterizing the Hydrogeologic Properties of Aquifers", implies more than the measurement
of horizontal hydraulic conductivity as a function of vertical position in a granular aquifer. The
additional information needed to make this report a more complete and useful document,
viii
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especially for individuals working in the field, was kindly supplied by colleagues at the Lawrence
Berkeley Laboratory and the U.S. Geological Survey in Denver.
The report itself is divided into seven chapters, four appendices and a list of references.
Chapter I, entitled "Why a New Approach is Needed", describes what the authors see as the
basis for the next logical step to be taken in the quest for improved hydrogeologic property
measurement in the field. An attempt is made also to place the needed changes within their
proper historical context. The next three chapters entitled "The Borehole Flowmeter Method
for Measuring Hydraulic Conductivity Distributions"; "Multi-Level Slug Tests for Measuring
Hydraulic Conductivity Distributions"; and "Characterizing Flow Paths and Permeability
Distributions in Fractured Rock Aquifers" are devoted to the better devices and techniques for
measuring vertical distributions of horizontal hydraulic conductivity in both granular and
fractured rock aquifers. These three chapters along with their two related and supporting
appendices ("Overview and Evaluation of Methods for Determining the Distribution of
Horizontal Hydraulic Conductivity in the Vertical Dimension", and "Analysis of Partially
Penetrating Slug Tests Considering Radial and Vertical Flow and Anisotropy") form the core of
this report. Most of the existing and emerging methodologies associated with our suggested new
approach are described in these sections of the report.
The final three chapters, entitled "Measurement of Hydraulic Conductivity in the
Vertical Direction", "Field Measurement of Effective Porosity, Storativity and Head", and
"Mathematical Modeling of Advection-Dominated Transport in Aquifers", make the overall
report more complete by defining and describing the measurement of related hydrogeologic
properties, and how these properties may be incorporated in mathematical models of advection-
dominated transport. Whenever applicable, we tried to remain consistent with the overall
theme of measuring vertically distributed rather than vertically-averaged quantities.
For those individuals not comfortable with the concepts of hydrodynamic dispersion or
for those wishing a review, we have included a third appendix entitled "The Physical Processes
of Advection and Hydrodynamic Dispersion".
Most individuals attempting to deal with subsurface contamination problems in the field
are well aware that more information is needed than that resulting from the measurement of
hydrogeologic properties as they are defined herein. Measurement of chemical and biochemical
subsurface properties, input from geologists, geophysicists, biologists and other scientists,
measurement of subsurface geometry, and information concerning the interplay of field
measurements and regulations are all important, but beyond the scope of this report. In order
to compensate for this shortcoming, a national conference entitled "New Field Techniques for
Quantifying the Physical and Chemical Properties of Heterogeneous Aquifers" was convened in
Dallas, Texas on March 20-23, 1989. Similar to this report, the conference was motivated by the
need to enhance field measurement capabilities if, as a nation, we are to solve the many site-
specific problems being addressed by the Superfund and other programs. The meeting provided
a much needed forum for professionals from government regulatory agencies, universities, and
private industry to discuss, describe or display the best and most applicable techniques or
equipment for measuring aquifer properties that have an important influence on contaminant
migration. The conference featured a broad spectrum of invited and submitted papers and
displays dealing with the most important topics facing ground water scientists, engineers and
consultants in this field of inquiry. Approximately 50 papers were presented and the 883 page
proceedings is available from the Water Resources Research Institute, 202 Hargis Hall, Auburn
University, AL 36849 at a moderate cost. The proceedings is intended to serve as a broad-
based supplement to this report, and the Table of Contents is included herein as Appendix IV.
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Given the current level of understanding concerning contaminant migration in porous
media, it was necessary for the authors of this report, and also for the participants in the
aforementioned conference, to attempt to identify practical and useful measurement techniques
and equipment while recognizing that we are working within a framework of basic
understanding that is far from perfect. This is a classical example of a situation that requires
innovative engineering solutions. Within this context, it is hoped that the work described in the
present report will serve as part of the basis for the "next step" in field measurements that must
be taken if we are to improve significantly our ability to characterize, evaluate and reclaim
contaminated aquifers.
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ABSTRACT
In the authors' opinion, the ability of hydrologists to perform field measurements of
aquifer hydraulic properties must be enhanced if we are to improve significantly our capacity to
solve ground water contamination problems at Superfund and other sites. Therefore, the
primary purpose of this report is to provide motivation and new methodology for measuring
K(z), the distribution of horizontal hydraulic conductivity in the vertical direction in the vicinity
of a test well. Measurements in nearby wells can then be used to build up three-dimensional
distributions. For completeness, and to enhance the usefulness of this report as a field manual,
existing methodology for the measurement of effective porosity, vertical hydraulic conductivity,
storativity and hydraulic head, are presented also. It is argued that dispersion-dominated
models, particularly two-dimensional, vertically-averaged (areal) models, have been pushed
about as far as they can go, and that two-dimensional vertical profile or fully three-dimensional
advection-dominated transport models are necessary if we are to increase significantly our
ability to understand and predict contaminant transport, reaction, and degradation in the field.
Such models require the measurement of hydraulic conductivity distributions, K(z), rather than
vertically averaged values in the form of transmissivities.
Three devices for measuring K(z) distributions (the impeller flowmeter, the heat-pulse
flowmeter, and a multi-level slug test apparatus) are described in detail, along with application
and data reduction procedures. Results of the various methods are compared with each other
and with the results of tracer studies performed previously. The flowmeter approach emerged
as the best candidate for routine K(z) measurements. Impeller meters are now available
commercially, and the more sensitive flowmeters (heat pulse and electromagnetic) are expected
to be available in the near future.
Three-dimensional transport models tend to be advection-dominated rather than
dispersion-dominated, and most of the standard finite-difference and finite-element algorithms
produce excessive amounts of numerical dispersion when applied to advection-dominated
models. Therefore this report closes by providing an introductory review of some newer
numerical methods that by tracking the flow produce a minimum of numerical dispersion.
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FIGURES
Number Page
1-1 Diagram illustrating the concept of a fully-penetrating pumping
test to determine K, the vertically = averaged hydraulic conductivity.
Conventional pumping tests, however, do not use the pumping well as
an observation well for measuring the drawdowns 3
1-2 Drawing providing a more realistic illustration of how water actually
moves in an aquifer. Variation of horizontal groundwater velocity
with vertical position is called differential advection. It is an
example of a shearing type of flow or simply shear flow 4
1-3 Schematic plot showing what happens when one assumes that
contaminants spread predominantly by a diffusion-like process
called hydrodynamic dispersion and attempts to measure dispersivity.
The dispersivity value depends on the size (scale) of the
measurement process 6
1-4 Schematic diagram showing the inherent lack of vertical contaminant
concentration structure that results from the solution of vertically-
averaged transport models 8
1-5 Plots of dimensionless horizontal hydraulic conductivity versus
elevation in three mutually perpendicular directions from well 12.
The basic data were obtained from single-well tracer tests 9
1-6 Dimensionless horizontal hydraulic conductivity distributions
based on impeller meter readings taken at the various
measurement intervals indicated on the figure 10
II-1 Subsurface hydrologic system at the Mobile site 12
II-2 Apparatus and geometry associated with a borehole
flowmeter test 13
II-3 Assumed layered geometry within which impeller meter data
are collected and analyzed. (Q(z) is discharge measured at
elevation z) 15
II-4 Details of well construction and screen types in wells E7 and A5 . . . 17
II-5 Hydraulic conductivity distributions calculated from flowmeter data
using two different methods 23
II-6 Comparison of hydraulic conductivity distributions for well E7 based
on tracer test data and impeller meter data 24
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II-7 Plan view of the field site where small-scale pumping tests were
performed. The numbers next to the dots are well designations, while
the values in parentheses are the average hydraulic conductivities
(in/day) assigned to the vicinity of each pumping well. Each arrow
represents a test and points from the observation well to the pumping
well. Wells with more than one arrow pointing toward them were
assigned average values 25
II-8 Results of small-scale pumping tests (m/day) wherein the pumping
wells were used as observation wells 27
II-9 Dimensionless horizontal hydraulic conductivity distributions based on
impeller meter readings taken at the various measurement intervals
indicated on the figure 28
11-10 Dimensionless horizontal hydraulic conductivity distributions based on
impeller meter readings taken at the various measurement intervals
indicated on the figure 29
11-11 Dimensionless horizontal hydraulic conductivity distributions based on
impeller meter readings taken at the various measurement intervals
indicated on the figure 30
11-12 Dimensionless horizontal hydraulic conductivity distributions based on
impeller readings taken at the various measurement intervals indicated
on the figure 31
11-13 Dimensionless horizontal hydraulic conductivity distributions based on
impeller meter readings taken at the various measurement intervals
indicated on the figure 32
11-14 Dimensionless horizontal hydraulic conductivity distributions based on
impeller meter readings taken at the various measurement intervals
indicated on the figure 33
11-15 Dimensionless horizontal hydraulic conductivity distributions based on
impeller meter readings taken at the various measurement intervals
indicated on the figure 34
11-16 Dimensionless hydraulic conductivity distributions at five-foot
intervals in well E7 taken 30 min., 60 min. and 120 min. after the
start of pumping. The results show good repeatability of the impeller
meter method 35
III-l Schematic diagram of the apparatus for performing a multi-level slug
test 39
III-2 Plan view of part of the well field at the Mobile site 40
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III-3 Multilevel slug test data from well E6. B=log(yi/y2)(t2-t1) =
magnitude of the slope of the log y(t)
response 42
III-4 Plot showing the reproducibility of data collected at well E6 43
III-5 Plots showing the influence of well development at two elevations
in well E6 44
III-6 Example type curves from Cooper, Bredehoeft and Papadopulos
(1967) 46
III-7 Slug test data from different elevations in well E6 47
III-8 K(z) profile at well E6 based on the radial transient analysis
, (Cooper et al., 1967) 50
III-9 Dimensionless flow parameter for consideration of vertical flow and
anisotropy. These curves apply for D/H > 2, i.e. effects of the far
boundary are not considered. For test interpretation when D/H <2,
the same curves can be applied if H is redefined as the distance
from the lower confining layer to the top of the straddle packer.
(Widdowson et al., 1989) 52
III-10 K(z) profiles for the three different methods of analysis at
well E6 54
III-ll K(z) profiles for the three different methods of analysis at
well E7 55
III-12 K(z) profiles for the three different methods of analysis at
well E3 56
III-13 Dimensionless hydraulic conductivity profiles at well E6 57
III-14 Dimensionless hydraulic conductivity profiles at well E7 58
III-15 Dimensionless hydraulic conductivity profiles at well E3 59
IV-1 The U.S. Geological Survey's slow-velocity-sensitive thermal
flowmeter (modified from Hess, 1986) 63
IV-2 The U.S. Geological Survey's thermal flowmeter with inflated flow-
concentrating packer (modified from Hess, 1988) 64
IV-3 Example of a thermal flowmeter calibration in a 6-inch (15.2 cm)
diameter calibration column 65
IV-4 Acoustic-televiewer, caliper, single-point-resistance, and flowmeter
logs for borehole DH-14 in northeastern Illinois 67
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IV-5 Acoustic-televiewer and caliper logs for selected intervals in a
borehole in southeastern New York 69
IV-6 Profile of vertical flow in a borehole in southeastern New York,
illustrating downflow with and without drawdown in the upper
fracture zone 70
IV-7 Distribution of fracture permeability in boreholes URL14 and
URL15 in southeastern Manitoba determined from acoustic-waveform
and other geophysical logs; fracture permeability is expressed as
the aperture of a single planar fracture capable of transmitting
an equivalent volume of flow 72
IV-8 Distribution of vertical flow measured in boreholes URL14 and
URL15 in southeastern Manitoba superimposed on the projection
of fracture planes identified using the acoustic
televiewer 73
V-l Downhole equipment arrangements for vertical well tests; A) single
interval test, and B) multiple interval test with sliding sleeve (after
Burns, 1969) 78
V-2 Aquifer with a partially penetrating well 80
V-3 A typical pulse test response in the lower perforated interval,
(modified from Hirasaki, 1974) 87
V-4 Sketch of the Hirasaki's test configuration 88
V-5 Dimensionless response time for pulse test; A) for semi-infinite
case, B) for a finite thickness layer with an impermeable lower
boundary, and C) for a finite thickness layer and constant head
at the lower boundary, (modified from Hirasaki,
1974) 90
V-6 Possible arrangements for conducting pressurized test (a) in
unconsolidated formations and (b) in consolidated formations,
(after Bredehoeft and Papadopulos, 1980) 92
V-7 Type curves of the function F(a,/J) against the product parameter
aft, (after, Bredehoeft and Papadopulos, 1980) 95
V-8 Arrangement of the borehole instrumentation as suggested by
Neuzil (1982) 96
V-9 Leaky aquifer with a constant head boundary at the top of the
aquitard 102
V-10 Leaky aquifer type curves based on r/B approach
(USER, 1977) 105
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V-ll Type curves of the function H(u,/J) against 1/u, for various values
of 0 (after Lohman, 1972) 110
V-12 A suggested arrangement for conducting a ratio-method
test 113
V-13 The variation of s'/s with tD' for a semi-infinite aquitard
(modified from Witherspoon et al. 1967) 115
V-14 Schematic diagram of two aquifer system 116
VI-1 Sketch of a sonic tool, showing ray paths for transmitter receiver
sets (modified from Kokesh et al., 1965) 121
VI-2 Schematic diagram showing two observation wells, one open in the
top fresh-water aquifer and the other screened in the lower saline
aquifer 127
VI-3 Heads in groundwater of variable density, (A) point-water head,
(B) fresh-water head, and (C) environmental head (modified from
Lusczynski, 1961) 129
VII-1 Vertical cross-sectional diagram showing single-well test
geometry 141
VII-2 Geometry and flow field for a two-well tracer test 143
VII-3 Hydraulic conductivity profile measured by Pickens and Grisak
(1981) and used in the calculations by Giiven et al.
(1985) 144
VII-4 Comparison of SWADM results with field data for the flow-weighted
concentration from an observation well one meter from the injection-
withdrawal well. From Guven et al. (1985) 146
VII-5 Comparisons of SWADM results with field data for the flow-weighted
concentration from an observation well two meters from the injection-
withdrawal well. From Guven et al. (1985) 147
VII-6 Comparison of SWADM results with field data for the concentration
leaving the injection-withdrawal well. From Giiven et al.
(1985) 148
VII-7 Results of various simulations of the first two-well tracer
test conducted at the Mobile site. No "calibration" or curve-
fitting of any type was used. From Molz et al.
(1986a) 150
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VII-8 Calculated tracer concentration versus time in the withdrawal well
based on an assumed homogeneous, isotropic aquifer with no local
dispersion (circles) shown together with the results of the first two-
well test (full line) at the Mobile site 151
VTI-9 Measured and predicted Br concentration versus time in the
withdrawal well during the second two-well tracer test at the
Mobile site. From Molz et al. (1988) 152
VII-10 Schematic diagram of the hypothetical domain, velocity distribution
and boundary conditions used in generic model simulations. O =
oxygen concentration; S = substrate concentration. From Molz and
Widdowson (1988) 154
VII-11 Simulated oxygen concentration versus depth below water table for
two different values of vertical dispersivity and an average
horizontal seepage velocity of 10 cm/day. From Molz and
Widdowson (1988) 155
VII-12 Plot of oxygen concentration versus depth below water table
(After Barker et al., 1987). From Molz and Widdowson
(1988) 156
AI-1 Details of an inflatable straddle packer design 175
AI-2 Schematic diagram illustrating a natural flow field in the vicinity
of a well 178
AI-3 Geometry and instrumentation associated with the dialysis cell
method for measurement of Darcy velocity 179
AI-4 Apparatus and geometry associated with the SWET
test 181
AI-5 Apparatus and geometry associated with a borehole flowmeter
test 182
AII-1 Diagram illustrating the geometry within which a partially
penetrating slug test is analyzed. Diagram (A) is for the
confined case and diagram (B) is for the unconfined
case 185
AII-2 Plots of dimensionless discharge, P = Q/2flKLy, for the
isotropic, confined aquifer problem as a function of L/rw
and H/L 188
AII-3 Plots of dimensionless discharge, P = Q/27rKLy, for the
isotropic, unconfined aquifer problem as a function of
L/rw and H/L 189
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AII-4 Multilevel slug test data from well E6,
B = log(y 1/y2)/(t2/t1) = magnitude of the slope of the
log(y(t)) response 196
AIII-3 Part (A) shows a hypothetical velocity distribution and an
initial distribution of tracer while part (B) shows how the
tracer would be dispersed by the moving groundwater at
several different scales. Three common mechanisms of pore
scale dispersion (velocity variation with a pore (a); flow
path tortuosity ($), and molecular diffusion due to concentration
differences (7) are illustrated also 199
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TABLES
Number
II-1 Impeller meter (discrete mode) and differential head data obtained
in Wells E7, and A5 at the Mobile Site. (z=depth, CPM=counts per
minute, and AH=head difference between static and dynamic
conditions) • • 19
II-2 Well screen discharge as a function of vertical position in Wells
E7 and A5 at the Mobile Site. (z=depth, Q=discharge rate in well
screen) • • • 20
II-3 Hydraulic conductivity distributions inferred from impeller meter
data using two different approaches described herein. Depth z is in
ft. and K(z) is in ft./min.) 21
III-l Multilevel slug test data and calculated hydraulic conductivities
at Well E6 (6/31/87 data) 48
V-l Values of G(ZD,ZD') 84
AIM Dimensionless discharge, P, as a function of H/L and L/rw for the
confined case with K/K, = 1.0 • • 190
AII-2 Dimensionless discharge, P, as a function of H/L and L/rw for the
confined case with K/K, = 0,2 ... 191
AII-3 Dimensionless discharge, P, as a function of H/L and L/rw for the
confined case with K/K, - 0.1 192
AII-4 Dimensionless discharge, P, as a function of H/L and L/rw for the
unconfined case with K/K^ = 1.0 193
AII-5 Dimensionless discharge, P, as a function of H/L and L/rw for the
unconfined case with K/fC, = 0.2 194
AII-6 Dimensionless discharge, P, as a function of H/L and L/rw for the
unconfined case with K/K^, = 0.1 195
XIX
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LIST OF ABBREVIATIONS AND SYMBOLS
Abbreviations
Br Bromide
CPM counts per minute
EFLOW computer code name
ELM Eulerian-Lagrangian Methods
EPA Environmental Protection Agency
IGWMC International Ground Water Modeling Center
NWWA National Water Well Association
OTA Office of Technology Assessment
SWADM Single Well Advection-Dispersion Model
TWAM Two-Well Advection Model
USGS United States Geological Survey
Symbols.(Chapters 1. 2. 3. 4. 7. and appendices)*
A screen area per unit length, (L)
Ac open cross-sectional area of casing, (L2)
B slope of semi-log plot, (T1)
B,b aquifer thickness, (L)
C solute concentration, (M/L3)
C0 Courant number, (-)
D aquifer thickness, (L)
D dispersivity matrk, (L2/T)
E , molecular diffusion coefficient, (L2/T)
DTO mechanical dispersion coefficient, (L2/T)
DM longitudinal component of dispersion matrk, (L2/T)
H distance from confining layer to straddle packer, (L)
h hydraulic head, (L)
h0 initial head, (L)
i counting index, (-)
K hydraulic conductivity, (L/T)
Ky hydraulic conductivity in radial direction, (L/T)
Kj, hydraulic conductivity in vertical direction, (L/T)
K hydraulic conductivity matrix, (L/T)
K vertically-averaged hydraulic conductivity, (L/T)
L length, (L)
Ois/Oi6 Oxygen-18/Oxygen-16 isotope ratio, (-)
P dimensionless flow parameter, (-)
Pe Peclet number, (-)
Q discharge rate, (L3/T)
QP pumping rate, (L3/T)
q Darcy velocity, (L/T)
Generalized symbols for the dimensions of length, time and mass will be L, T, and M
respectively. The symbol (-) indicates a dimensionless quantity.
-------
r radius, (L)
rc casing radius, (L)
Re radius of influence, (L)
Rf retardation factor, (-)
r plunger radius, (L)
rw well radius, (L)
S storage coefficient, (-)
Si volumetric sink of water, (L3/L3/T)
S0 volumetric source of water, (L3/L3/T)
Ss specific storage, (L"1)
T transmissivity, (Lr/T)
t time, (T)
U Darcy velocity, (L/T)
V pore or seepage velocity vector, (L/T)
Vr radial seepage velocity, (L/T)
x,y horizontal coordinates, (L)
y(t) head change in slug test, (L)
y0 initial head change, (L)
Z,z vertical coordinates, (L)
a type curve parameter for slug test, (-)
ar radial dispersivity, (L"1)
av vertical dispersivity, (L"1)
A prefix symbol indicating "change in", (-)
V gradient operator, (-)
7T 3.14159, (-)
6 porosity, (-)
Symbols (Chapters 5 and 6)
a half the distance between recharge and discharge walls in the tracer test,
(L)
B leakage factor, (L)
b,b' thickness of an aquifer and aquitard, respectively, (L)
C concentration of the tracer, (ML"3)
C0 input concentration of the tracer, (ML"3)
cw compressibility of water, (LT2M"1)
D dispersion constant or dispersivity, (L)
-Ei(-u) well function, (-)
Ew bulk modulus of elasticity of water, (ML^T2)
erfc(x) complementary error function, (-)
g acceleration of gravity vector, (LT2)
h hydraulic head, (L)
K,K' hydraulic conductivities of an aquifer and aquitard, respectively, (LT1)
K^Kj components of hydraulic conductivity in radial and vertical directions,
respectively, (LT1)
k intrinsic permeability, (L2)
1 depth of penetration, (L)
M,M' thickness of an aquifer and aquitard, respectively, (L)
P Peclet number, (-)
PD dimensionless pressure, (-)
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Q rate of pumping, (L^T1)
q pumping rate per unit aquifer thickness, (L2!"1)
r radial distance from a pumping well, (L)
rs radius of well in the tested interval, (L)
S,S' storage coefficient of an aquifer and aquitard, respectively, (-)
SS,S'S specific storage of an aquifer and aquitard respectively, (L'1)
s,s' drawdowns in an aquifer and aquitard, respectively, (L)
SD dimensionless drawdown, (-)
T time for a water particle to travel along a particular streamline between
two wells, (T)
t time, (T)
tD dimensionless time, (-)
V vector of apparent or Darcy's velocity, (LT1)
Vf sonic wave velocity of fluid filling the pores, (LT1)
W(u,r/B) well function of leaky aquifers, (-)
x,y,z coordinate system, (L)
a effective porosity, (-)
7 specific weight of water, (ML'2T2)
f hydraulic potential, (L2T*)
0 porosity, (-)
p density of fluid, (ML"3)
pb bulk density, (ML"3)
V gradient operator, (L"1)
li dynamic viscosity, (ML^T1)
0 stream function, (-)
XXLL
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ACKNOWLEDGEMENTS
This work was supported in part by the U.S. Environmental Protection Agency (contract
CR813647) through the Robert S. Kerr Environmental Research Laboratory. At Auburn
University, the project was administered by the Water Resources Research Institute (Joseph F.
Judkins, Director). The authors would like to acknowledge gratefully the support of both
institutions. Thanks are due also to the U.S. Geological Survey and the Lawrence Berkeley
Laboratory for indirect support.
The Alabama Power Company made their land available for the field experiments
performed by Auburn University and along with Southern Company Services, provided well
drilling services and materials free of charge. This help had a major positive impact on the
outcome of the project and is acknowledged with thanks.
Many individuals whose names do not appear on this report made valuable contributions
over the past four years. Those that come immediately to mind are Roger Morin of the USGS
Denver Federal Center for providing his help and expertise with early impeller meter
measurements, Ken Taylor and Joel Hayworth from the University of Nevada Desert Research
Institute for help with various geophysical methods, Post-Doctoral Fellow Mark Widdowson of
Auburn University (now an Assistant Professor at the University of South Carolina) for his
work on the analysis of multi-level slug tests, AU Field Engineers Clay Cardone and Jerry
Bowman for field work of various types, Daniel Ronen and Mordekai Magaritz of the Weizman
Institute, Rehovot, Israel for performing and analyzing the dialysis cell experiments for
measuring Darcy velocity, Lowell Leach of the R. S. Kerr Laboratory, U.S. Environmental
Protection Agency, for help and encouragement throughout the project, and last but not least,
AU Technical Secretary Kathy Seibel for her tireless typing of so many documents including this
report.
xxm
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EXECUTIVE SUMMARY
Introduction
The present writers are of the opinion that field measurement capability must increase if
we are to improve significantly our ability to handle ground water contamination problems
associated with Superfund and other sites. To this end, the single most important parameter
concerning contaminant migration is the hydraulic conductivity distribution. If one can not
predict where the water goes, how can one expect to predict the movement of a contaminant
that is carried by the water? Most conventional flow analyses are based on fully-penetrating
pumping tests to get a transmissivity field and large longitudinal dispersion coefficients to
account for contaminant spreading in the direction of flow. We call such models dispersion-
dominated. In the authors' opinion, the time has arrived to develop and apply aquifer tests for
determining the horizontal hydraulic conductivity as a function of vertical position (K(z)) within
a well or borehole. When this is done at a number of locations in the horizontal plane, the
resulting data can serve as a basis for developing two-dimensional vertical cross-section, quasi
three-dimensional or fully three-dimensional flow and transport models that do not require
large, scale-dependent, dispersion coefficients.
Shown in Figure 1-6 are dimensionless K(z) distributions obtained at four different scales
of measurement in a single well using an impeller meter (Hufschmied, 1983; Morin et al., 1988a;
Rehfeldt et al., 1988; Molz et al., 1989a). It is apparent that as the measurement interval varies
from 10 ft (3.05 m) to 1 ft (0.305 m), the apparent variability of the hydraulic conductivity
increases. This is the type of information that is lost when fully-penetrating pumping tests are
used to obtain vertically-averaged hydraulic conductivities.
There are several techniques for making vertically-distributed measurements, including
tracer tests, flowmeter tests, dilution tests and multi-level slug tests, that are described in this
report. Such measurements should serve as the basis for an improved understanding and
conceptualization of subsurface transport pathways, and may also allow the application of a new
generation of contaminant transport models that are advection-dominated and largely free of
the problems associated with large, scale-dependent, dispersion coefficients. All of this taken
together constitutes what the authors are advocating as the new approach to characterizing the
hydrogeologic properties of aquifers.
Selected Methodology
After studying a number of methodologies for measuring K(z) distributions, two
techniques, the flowmeter method and the multi-level slug test, emerged as the most practical
methodologies for obtaining K(z) information, and were, therefore, studied in detail. Of the
two, the flowmeter was more responsive, less sensitive to near-well disturbances due to drilling,
and easier to apply. As illustrated in Figure II-2, a flowmeter test may be viewed as a natural
generalization of a standard fully penetrating pumping test. In the latter application, only the
steady pumping rate, QP, is measured, whereas the flow rate distribution along the borehole or
well screen, Q(z), as well as QP is recorded during a flowmeter test.
Various types of flowmeters based on heat pulse or impeller technology have been
devised for measuring Q(z), and a few groundwater applications have been described in the
literature (Hada, 1977; Keys and Sullivan, 1978; Schimschal, 1981; Hufschmied, 1983; Hess,
1986; Morin et al., 1988a; Rehfeldt et al., 1988; Molz et al., 1989a,b). The most low-flow-
sensitive types of meters are based on heat-pulse, electromagnetic, or tracer-release technology
xxLv
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(Keys and MacCary, 1971; Hess, 1986), but to the authors' knowledge such instruments are not
presently available commercially, although several are nearing commercial availability. Impeller
meters (commonly called spinners) have been used for several decades in the petroleum
industry, and a few such instruments suitable for groundwater applications are now available for
purchase (Further information available upon request). A meter of this type was applied at the
Mobile site to produce the kind of hydraulic conductivity data shown in Figure 1-6.
Impeller Meter Tests
Once the necessary equipment is obtained, impeller meter tests can be a relatively quick
and convenient method for obtaining information about the vertical variation of horizontal
hydraulic conductivity in an aquifer. The idea and methodology behind the impeller meter test
are illustrated in Figure 11-2. One first runs a caliper log to ascertain that the screen diameter
is known and constant. If it is not constant, the variations must be taken into account when
calculating discharge. A small pump is placed in a well and operated at a constant flow rate,
QP. After pseudo steady-state behavior is obtained, the flowmeter, which when calibrated
measures vertical flow within the screen, is lowered to near the bottom of the well, and a
measurement of discharge rate is obtained in terms of impeller-generated electrical pulses over
a selected period of time. The meter is then raised a few feet, another reading taken, raised
another few feet~and so on. As illustrated in the lower portion of Figure II-2, the result is a
series of data points giving vertical discharge, Q, within the well screen as a function of vertical
position z. Just above the top of the screen the meter reading should be equal to QP, the
steady pumping rate that is measured independently on the surface with a water meter. The
procedure may be repeated several times to ascertain that readings are stable.
While Figure II-2 applies explicitly to a confined aquifer, which was the type of aquifer
studied at the Mobile site, application to an unconfined aquifer is similar. Most impeller meters
are capable of measuring upward or downward flow, so if the selected pumping rate, QP, causes
excessive drawdown, one can employ an injection procedure as an alternative. In either case,
there will be unavoidable errors near the water table due to the deviation from horizontal flow.
It is desirable in unconfined aquifers to keep QP as small as possible, consistent with the stall
velocity of the meter. Thus more sensitive meters will have an advantage.
The basic analysis procedure for flowmeter data is quite straightforward. One assumes
that the aquifer is composed of a series of n horizontal layers and takes the difference between
two successive meter readings, which yields the net flow, AQ;, entering the screen segment
between the elevations where the readings were taken, which is assumed to bound layer
i(i= l,2,...,n). One then employs the Cooper-Jacob (1946) formula for horizontal flow to a well
or an alternative procedure to obtain K(z) values.
Heat-Pulse Flowmeter Tests
Application of impeller meter technology will be limited at many sites due to the
presence of low permeability materials that will preclude the pumping of test wells at a rate
sufficient to operate an impeller meter. Another type of flowmeter that is in the prototype
stage, the heat-pulse flowmeter, can be used as an alternative to an impeller meter in virtually
any application, and it has the advantage of greater sensitivity. Spinner flowmeters measure a
minimum velocity ranging from about 3 to 10 ft/min (1 to 3 m/min), which limits their
usefulness in many boreholes having slower water movement. Flow volumes of as much as 4
gal/min (15 L/min) may go undetected in a 4-in (10 cm) diameter borehole when flow is
measured with a spinner flowmeter, and much larger volumes may go undetected in larger
xxv
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diameter holes. Heat-pulse flowmeters are particularly useful for application to fractured rock
aquifers where flows are often small and contaminant transport pathways difficult to visualize.
Such meters may be used to located productive fracture zones and to characterize apparent
hydraulic conductivity distributions.
The urgent need for a reliable, slow-velocity flowmeter prompted the U.S. Geological
Survey to develop a small-diameter, sensitive, thermal flowmeter. This meter has
interchangeable flow-sensors, 1.63 and 2.5 in. (4.1 and 6.4 cm) in diameters, and has flow
sensitivity from 0.1 to 20 ft/min (0.03 to 6.1 m/min) in boreholes with diameters that range
from 2 to 5 in. (5 to 12.5 cm). Vertical discharge in a borehole is measured with the thermal
flowmeter by noting the time-of-travel of the heat pulse and determining water volume flow
from calibration charts developed in the laboratory using a tube with a diameter similar to that
of the borehole under investigation (Hess, 1986).
The basic measurement principle of the USGS meter is to create a thin horizontal disc
of heated water within the well screen at a known time and a known distance from two
thermocouple heat sensors, one above and one below the heating element. One then assumes
that the heat moves with the upward or downward water flow and records the time required for
the temperature peak to arrive at one of the heat sensors. The apparent velocity is then given
by the known travel distance divided by the recorded travel time. Thermal buoyancy effects are
eliminated by raising the water temperature by only a small fraction of a centigrade degree.
The geometry associated with the flowmeter is shown in Figure IV-2.
The USGS heat-pulse meter has been applied to the granular aquifer at the Mobile site
and several fracture flow systems. In the present report the authors describe applications to
fractured dolomite in northeastern Illinois, fractured Gneiss in southeastern New York, and a
granitic fracture zone on the Canadian shield in Manitoba. In these applications, supplemental
information was obtained from acoustic-televiewer logs, temperature logs, and caliper logs.
Information similar to that shown in Figure IV-8 was obtained. The case studies illustrate
potential application of the thermal flowmeter in the interpretation of slow flow in fractured
aquifers. The relative ease and simplicity of thermal-flowmeter measurements permits
reconnaissance of naturally occurring flows priors to hydraulic testing, and identification of
transient pumping effects, which may occur during logging. In making thermal flowmeter
measurements, one needs to take advantage of those flows that occur under natural hydraulic-
head conditions as well as the flows that are induced by pumping or injection. However,
thermal-flowmeter measurements interfere with attempts to control borehole conditions during
testing, because the flowmeter and wire-line prevent isolation of individual zones with packers.
In spite of this limitation, the simplicity and rapidity of thermal-flowmeter measurements
constitute a valuable means by which to eliminate many possible fracture interconnections and
identify contaminant plume pathways during planning for much more time consuming packer
and solute studies. The thermal flowmeter is especially useful at sites similar to the site in
northeastern Illinois, where boreholes are intersected by permeable horizontal fractures or
bedding planes. Under these conditions, naturally occurring hydraulic-head differences between
individual fracture zones are decreased greatly by the presence of open boreholes at the study
site. These hydraulic-head differences could only have been studied by the expensive and time
consuming process of closing off all connections between fracture zones in all of the boreholes
with packers. The simple and direct measurements of vertical flows being caused by these
hydraulic-head differences obtained with the thermal flowmeter provided information pertaining
to the relative size and vertical extent of naturally occurring hydraulic-head differences in a few
hours of measurements. Additional improvement of the thermal-flowmeter/packer system and
refinement of techniques for flowmeter interpretation may decrease greatly the time and effort
xxvi
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required to characterize fractured-rock aquifers by means of conventional hydraulic testing.
While the case studies described in the present report did not all involve contaminated
groundwater, the potential application to plume migration problems and sampling well screen
locating is obvious. The relationship of flowmeter measurements to more standard tests such as
caliper and televiewer logs is indicated also. Hopefully, thermal flowmeters and other
sensitive devices, such as the electromagnetic flowmeter being developed by the Tennessee
Valley Authority, will be available commercially in the near future.
Multilevel Slug Tests
The flowmeter testing procedure is generally superior to the multilevel slug test
approach, because the latter procedure depends on one's ability to isolate hydraulically a portion
of the test aquifer using a straddle packer. However, if reasonable isolation can be achieved,
which was the case at the Mobile test site, then the multi-level slug test is a viable procedure for
measuring K(z). All equipment needed for such testing is available commercially, and the test
procedure has the added advantage of not requiring any water to be injected into or withdrawn
from the test well if a water displacement technique is used to cause a sudden head change.
The testing apparatus used in the applications reported herein is illustrated schematically
in Figure III-l for the aquifer geometry at the Mobile site. Two inflatable packers separated by
a length of perforated, galvanized steel pipe comprised the straddle packer assembly. Aquifer
test length defined by the straddle packer was L=3.63 ft (1.1 m). A larger packer, referred to
as the reservoir packer, was attached to the straddle packer with 2" (2.54 cm) Triloc PVC pipe,
creating a unit of fixed length of approximately 100 ft (30.5 m) which could be moved with the
attached cable to desired positions in the well. When inflated, the straddle packer isolated the
desired test region of the aquifer and the reservoir packer isolated a reservoir in the 6" (15.2
cm) casing above the multilevel slug test unit and below the potentiometric surface of the
confined aquifer.
In a typical test, water was displaced in the reservoir above the packer. The head
increase then induced a flow down through the central core of the reservoir packer and down
the Triloc pipe to the straddle packer assembly. In the assembly, water flowed from the
perforated pipe, through the slotted well screen, and into the test region of the aquifer.
The multilevel unit described was used for slug tests by inserting the plunger, displacing
a volume of water in the reservoir and then recording the depth variation, y=y(t), relative to the
initial potentiometric surface (falling head test). Plunger withdrawal was used to generate a
rising head test. Head measurements were made with a manually operated digital recorder
(Level Head model LH10, with a 10 psig pressure transducer, In Situ, Inc.).
Three methods of analysis have been applied to the collected slug test data. In the first
and second methods, it is assumed that the flow from the test section is horizontal and radially
symmetric about the axis of the well. In the first method the quasi-steady state assumption is
made. In the second, a transient analysis is applied. In the third analysis, also quasi-steady
state, the possibility of vertical flow and anisotropy are considered using curves generated by a
finite element model.
Typical results of a series of tests at different elevations are shown for well E6 in Figure
III-3. The data shown are from plunger insertion tests where a sudden reservoir depth increase
to approximately y0=3 ft was imposed. The depth variation, y=y(t), which is nearly an
xxvii
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exponential decay, is a result of flow into the aquifer test section adjacent to the straddle
packer. The different slopes of the straight line approximations (least squares fits) are due to
the variability of the hydraulic conductivity in the aquifer at the different test section elevations.
Tests repeated at a given elevation were generally reproducible, with the maximum difference in
slopes of the straight line fits to the data being less than 11%. From this data it is easy to
calculate hydraulic conductivity distributions as described in the report.
Modeling of Advection-Dominated Flows
There are a variety of ways in which vertically-distributed hydraulic conductivity
distributions can be used to understand and assess problems involving contaminated ground
water. A significant amount of insight will be obtained simply by observing and discussing the
implications of such information on patterns of contaminant migration. However, use of the
vertically-distributed data in three-dimensional mathematical models will be a common
procedure for developing quantitative assessments of a variety of possible activities such as
evaluation of site remediation plans. Thus it is worthwhile to devote part of this report to a
discussion of the relationship between vertically-distributed hydraulic conductivity data and
mathematical modeling.
As pointed out previously, once one moves from the use of vertically-averaged aquifer
properties in two-dimensional mathematical models to the use of vertically-distributed aquifer
properties in three-dimensional models, the nature of the physical process represented by the
model changes dramatically. In many situations, the model changes from one being largely
dominated by dispersion (low Peclet number flows) to one largely dominated by advection (high
Peclet number flows). Unfortunately, most of the standard finite-difference and finite-element
algorithms for solving mathematical models do not work well when applied to advection-
dominated flows, especially those that involve chemical or microbial reactions. The necessary
evolution from dispersion-dominated to advection-dominated numerical algorithms for solving
the flow and transport equations is far from trivial, so it is important to call attention to some of
the newer numerical methods, particularly those that produce a minimum of numerical
dispersion when used to solve the transport equation.
A complete numerical analysis of contaminant migration in the subsurface usually
involves solution of the ground water flow equation and the transport (advection/dispersion)
equation. The latter equation is the more complicated due primarily to the existence of the
advective transport term which gives the transport equation a hyperbolic character and makes
its solution subject to numerical dispersion. In general, the techniques for solving such an
equation can be grouped into three classes; namely, Eulerian, Lagrangian, and Eulerian-
Lagrangian methods. Eulerian methpds are more suited to dispersion-dominated systems while
Lagrangian methods are most suited to advection dominated systems. Eulerian-Lagrangian
methods have been introduced to deal efficiently and accurately with situations in which both
advection and dispersion are important.
The Eulerian methods are based on the discretization of the transport equation on a
numerical solution grid that is fixed in space, and all of the terms of the equation, including the
advective transport term, are discretized together and the resulting algebraic equations are
solved simultaneously in one solution step. As discussed by Cady and Neuman (1987), while
Eulerian methods are fairly straightforward and generally perform well when dispersion
dominates the problem and the concentration distribution is relatively smooth, they are usually
constrained to small local grid Peclet numbers.
Methods which are based on solutions of the transport equation on a moving grid, or
xxyiii
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grids, defined by the advection field, or methods which do not rely on a direct solution of the
Eulerian transport equation but which are based on an analysis of the transport, deformation
and transformation of identified material volumes, surfaces, lines or particles by tracking their
motion in the flow field are generally called Lagrangian methods (Cady and Neuman, 1987). In
the present report, we reserve this term only for methods which are based on tracking alone and
win consider the moving-grid methods which have been called Lagrangian before by some
authors simply as special cases of the Eulerian-Lagrangian methods.
Reviews of Eulerian-Lagrangian methods (ELM) have also been presented recently by
Cady and Neuman (1987). These methods combine the advantageous aspects of the Lagrangian
and the Eulerian methods by treating the advective transport using a Lagrangian approach and
the dispersive transport and chemical reactions using an Eulerian approach. According to how
the advective transport is taken into account, these methods can be generally grouped into three
classes; one class makes use of particle tracking and relates the concentration at a grid node to
the solute mass associated with each particle and the particle density around that node, while
the second class treats concentration directly as a primary variable throughout the calculations
without resorting to the use of any particles, and the third class consists of models in which the
first and second approaches are used together in an adaptive manner depending on the
steepness of the concentration gradients.
It may be useful to point out that Eulerian-Lagrangian methods have been developed
extensively for the numerical modeling of complex three-dimensional industrial and
environmental flows and for the solution of various fluid mechanics problems particularly over
the last decade (Orin and Boris, 1987). These methods are now becoming popular also in the
area of subsurface contaminant migration modeling.
Availability of Computer Codes
Several well documented computer codes for three-dimensional flow and solute transport
modeling as well as parameter identification and uncertainty analysis are available in the public
domain. These codes have been developed by universities, various government agencies and
government supported laboratories. There are also several proprietary codes developed by
private consulting firms and research organizations such as the Electric Power Research
Institute. Many of these codes have been listed in the recent monographs by Javandel et al.
(1984) and van der Heijde et al. (1985). In this regard, the International Ground Water
Modeling Center (IGWMC) serves as an information, education, and research center for
groundwater modeling with offices in Indianapolis, Indiana (IGWMC, Holcomb Research
Institute, Butler University, 4600 Sunset Avenue, Indianapolis, Indiana 46208) and Delft, the
Netherlands. IGWMC operates as a clearinghouse for groundwater modeling codes and
organizes an annual series of short courses on the use of various codes. Similar specialized
short courses are also organized by various universities as well as professional organizations
such as the National Water Well Association (NWWA, 6375 Riverside Dr., Dublin, Ohio 43017).
The aforementioned references and organizations may be consulted for the availability of
various modeling codes.
Supplemental Information
This report is devoted mainly to a presentation of the information and experience gained
from six years of field experimental and theoretical studies by Auburn University that were
funded by the U.S. Environmental Protection Agency through the Robert S. Kerr Laboratory.
In one way or another, most of this work dealt with the understanding and measurement of
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hydraulic conductivity distributions in the field, with all measurements made in the saturated
zone. However, the title of the present report, "A New Approach and Methodologies for
Characterizing the Hydrogeologic Properties of Aquifers", implies more than the measurement
of horizontal hydraulic conductivity as a function of vertical position in a granular aquifer. The
additional information, including methodology for measuring specific storage, porosity, hydraulic
head, and hydraulic conductivity in the vertical direction, was kindly supplied by a colleague at
the Lawrence Berkeley Laboratory and is included in the report in two separate chapters.
Most individuals attempting to deal with subsurface contamination problems in the field
are well aware that raore information is needed than that resulting from the measurement of
hydrogeologic properties as they are defined herein. Measurement of chemical and biochemical
subsurface properties, input from geologists, geophysicists, biologists and other scientists,
measurement of subsurface geometry, and information concerning the interplay of field
measurements and regulation are all important, but beyond the scope of this report. In order to
compensate for this shortcoming, a national conference entitled "New Field Techniques for
Quantifying the Physical and Chemical Properties of Heterogeneous Aquifers" was convened in
Dallas, Texas on March 20-23, 1989. Similar to this report, the conference was motivated by the
need to enhance field measurement capabilities if, as a nation, we are to solve the many site-
specific problems being addressed by the Superfund and other programs. The meeting provided
a much needed forum for professionals from government regulatory agencies, universities, and
private industry to discuss, describe or display the best and most applicable techniques or
equipment for measuring aquifer properties that have an important influence on contaminant
migration. The conference featured a broad spectrum of invited and submitted papers and
displays dealing with the most important topics facing ground water scientists, engineers and
consultants in this field of inquiry. Approximately 50 papers were presented and the 883 page
proceedings is available from the Water Resources Research Institute, 202 Hargis Hall, Auburn
University, AL 36849 at a moderate cost. The proceedings is intended to serve as a broad-
based supplement to this report.
Given the current level of understanding concerning contaminant migration in porous
media, it was necessary for the authors of this report, and also for the participants in the
aforementioned conference, to attempt to identify practical and useful measurement techniques
and equipment while recognizing the fact that we are working within a framework of basic
understanding that is far from perfect. This is a classical example of a situation that requires
innovative engineering solutions. Within this context, it is hoped that the work described in the
present report will serve as part of the basis for the "next step" in field measurements that must
be taken if we are to improve significantly out ability to characterize, evaluate and reclaim
contaminated aquifers.
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CHAPTER I
WHY A NEW APPROACH IS NEEDED
1-1 Introduction
In a recent evaluation of the Superfund program, the Office of Technology Assessment
(OTA) asked the question "Are we cleaning up the mess or messing up the cleanup?" (U.S.
Congress, OTA, 1988). Their study suggested that in a significant number of cases the
reclamation technologies selected did not offer a permanent solution to the problem at hand.
Many surface and subsurface natural processes contribute to the difficulty in selecting the best
remediation report for each Superfund site, and no one technical paper can deal in depth with
all of them. However, significant concern at many sites relates to the diverse problems caused
by polluted groundwater. Groundwater is often the primary medium through which
contaminants are transported off site. Uncertainties concerning how fast, how far, in what
amounts and in what direction contaminants are expected to move can contribute greatly to
potential remediation, liability and monitoring costs.
The OTA suggests several reasons for its perceived problems with the Superfund
program including "too much flexibility, lack of central management control, and too much
preoccupation with site uniqueness." However, the fact that many of the very best existing tools
for evaluating groundwater contamination problems are not adequate for the problems at hand
is not emphasized. If the necessary tools are simply not available, no amount of management
wizardry will solve the problem.
It has never been easy and still is not easy to obtain good physical and chemical
information about the subsurface. An immense amount of basic research will be necessary for
years to come before our understanding approaches a satisfactory level. We have been
relatively successful in the modeling/computational area, which has resulted computer models
and other computational tools being developed far beyond our ability to measure the physical
and chemical parameters upon which they are based. Thus, it is not easy for individuals
working directly with Superfund problems to obtain the direct, specific and seemingly
straightforward information that is needed. In many cases the techniques for obtaining such
information either do not exist or are not developed sufficiently to be useful.
The present writers are of the opinion that field measurement capability must increase if
we are to improve significantly our ability to handle groundwater problems associated with
Superfund and other sites. Fortunately, it appears that such an increase is about to take place.
Therefore, the purpose of this introductory chapter is to describe what we see as the basis for
the next logical step to be taken in the quest for improved hydrogeological property
measurement procedures. We also wish to place the motivation for this change within its
proper historical context.
1-2 What's Wrong With Existing Procedures?
Most hydrologists will agree that the single most important parameter concerning
contaminant migration in groundwater is the hydraulic conductivity. If one can't predict where
the water goes, how can one expect to predict the movement of a contaminant that is carried by
the water? Recent studies have emphasized the importance of understanding the pattern of
groundwater movement, because in most situations contaminant migration is advection
-------
dominated (Molz et al., 1983; Osiensky et al., 1984; Fogg, 1986; Giiven et al., 1986; Molz et al,
1986a, 1986b; Ronen et al., 1987a, 1987b; Giiven and Molz, 1988; Molz and Widdowson, 1988;
Molz et al., 1988; Lehr, 1988).
As illustrated in Figure 1-1, the conventional field measurement of hydraulic conductivity
is almost always based on some variant of a fully penetrating pumping test, which results in a
vertically-averaged value for hydraulic conductivity (K). This parameter is an excellent index to
the water supply capability of an aquifer; certainly not surprising because the basics of
subsurface hydrology were developed during the 20th century in response to water supply
problems. Conceptual and computational tools based on vertically-averaged aquifer parameters,
such as transmissivity and storativity, work well in water supply situations because no one cares
where, from within an aquifer, the water comes. Everyone is happy as long as there is an
abundant and dependable supply.
This is not the situation, however, when one is dealing with groundwater quality
problems. In most situations, the essence of such problems is to determine from where the
contaminated water came and where it is going. As illustrated in the paper by Osiensky et al.
(1984) and also in Figure 1-2, practically all natural aquifers are highly heterogeneous. On the
scale of most contaminant plumes, there are usually irregular but non-random high hydraulic
conductivity pathways along which contaminated water moves preferentially. Therefore, plumes
often assume complicated shapes with certain portions moving more rapidly than other portions
as illustrated schematically in Figure 1-2. At least some understanding of this type of
groundwater motion is required in order to determine where a contaminant is coming from and
where it is going.
1-2.1 First Attempt to Adapt Water "Quantity" Tools to Water "Quality" Problems.
Beginning about 25 years ago, the concern of various citizens and federal regulatory
agencies began to include, to a greater and greater extent, groundwater quality problems. The
natural "first effort" of groundwater hydrologists to deal with such problems was to adapt the
existing water quantity tools to water quality problems. The most straightforward way to
accomplish this was to neglect the vertical hydraulic structure of aquifers and assume that
contaminants spread by a diffusion-like process called hydrodynamic dispersion. (For a
definition and brief discussion of hydrodynamic dispersion see Appendix III.) This resulted in
what was assumed to be a new aquifer property called full aquifer dispersivity. In principle,
dispersivity was supposed to be a tensorial quantity, but as a practical matter subsurface
hydrologists were satisfied to determine the principal longitudinal component which seemed to
dominate the field dispersion process. The principal transverse component was then estimated
as some fraction of the longitudinal component. Both components were assumed to apply over
the full-aquifer thickness, so in principle fully penetrating pumping tests and tracer tests would
be adequate to support the theory. The two-dimensional lateral-flow and transport models that
most commonly resulted from these concepts were called areal dispersion or areal transport
models.
To apply the areal dispersion models one would ideally determine the
vertically-averaged hydraulic conductivity distribution K(x,y) ((x,y) = horizontal, principal
Cartesian coordinates) using a series of fully penetrating pumping tests. Then the longitudinal
and transverse components of full-aquifer dispersivity could be determined from fully-
penetrating tracer tests. It was much more common, however, to use the principal dispersivity
components as calibration parameters for fitting the overall concentration distribution coming
out of the model to some measured distribution in the field. When areal dispersion models were
-------
Pumping well
0
/ / / /./ /
Confining bed.
////////////V////
Confined
-*-.
aquifer /
•• *••••*•
s///////////////////
///////////
'Confining bed
Figure 1-1. Diagram illustrating the concept of a fully-penetrating pumping test to determine
K, the vertically-averaged hydraulic conductivity. Conventional pumping tests,
however, do not use the pumping well as an observation well for measuring the
drawdowns.
-------
(A)
Hypothetical
Velocity
Distribution
Tracer at
V
vmin
max
(B)
Tracer Distribution
at Time>0
Pore-
Scale Dispersion
Multiple of
Figure 1-2. Drawing providing a more realistic illustration of how water actually moves in an
aquifer. Variation of horizontal groundwater velocity with vertical position is
called differential advection. It is an example of a shearing type of flow or
simply shear flow.
-------
first developed, and for some time thereafter, this was reasonable because even fully penetrating
pumping test data were scarce, and tracer tests are tedious, time consuming, expensive and
therefore seldom performed.
Regardless of the manner in which dispersivity values were measured, calibrated,
estimated, or otherwise divined, it gradually became apparent that something was seriously
wrong. By the early seventies, various investigators were questioning the meaning of the huge
longitudinal dispersivities that were resulting from field tracer tests and plume calibrations
(Cherry et al., 1975; Anderson, 1979). It also became apparent that field-measured dispersivity
values were scale-dependent (Robson, 1974; Anderson, 1979, 1984). This means that the value
one obtains for longitudinal dispersivity will increase as the size of the experiment used to
measure dispersivity is increased. Thus when one plots many values of longitudinal dispersivity
versus the scales of the experiments or natural processes used to obtain those values, one
produces a scatter diagram similar to that shown schematically in Figure 1-3. (For plots of
actual data see Anderson, 1984 and Gelhar, 1986.)
The controversy and confusion resulting from attempts to rationalize full aquifer
dispersion, which still have not abated completely, led the EPA Kerr Laboratory to fund a
combined experimental/theoretical study of contaminant dispersion in groundwater. The results
of this study (Molz et al., 1983; Guven et al., 1984, 1985, 1986; Giiven and Molz, 1986;
Huyakorn et al., 1986; Molz et al., 1985, 1986a, 1986b, 1988) led to the following working
conclusions relating to the concept of longitudinal dispersivity.
A. Local longitudinal hydrodynamic dispersion plays a relatively unimportant
role in the transport of contaminants in most aquifers. As illustrated in
Figure 1-2, differential advection (shear flow) in the horizontal direction is
much more important in determining the overall extent and structure of
most contaminant plumes.
B. The concept of full-aquifer dispersivity used in vertically-averaged (areal)
models will not be applicable over distances of interest in most
contamination problems. If one has no choice but to apply a full-aquifer
dispersion concept, the resulting dispersivity will not represent a physical
property of the aquifer. Instead, it will be a scale-dependent quantity that
will depend on the size and type of experiment used for its supposed
measurement.
C. Because of conclusion B, it makes no sense to perform tracer tests or any
other tests aimed at measuring full-aquifer dispersivity. If an areal model
is used, the modeler will end up adjusting the dispersivity during the
calibration process anyway, independent of the measured value.
Other theoretical studies and field experiments performed more recently support the
above conclusions and indicate further that local dispersivities, especially transverse
dispersivities, are very small in natural systems. (Moltyaner, 1987; Taylor and Howard, 1987,
1988; Giiven and Molz, 1988; Molz and Widdowson, 1988; Moltyaner and Killey, 1988a,b).
One does not have to be trained technically in order to grasp the idea that full aquifer
dispersion is a concept that has very limited physical meaning. One merely has to realize that
the use of the concept in a stratified aquifer flow system is equivalent to attempting to represent
a transport process analogous to that shown in Figure 1-2 with a model having the inherent
-------
1
>
>
CO
cr
UJ
CL
CO
Q
•
5ITUDINAL
\^s
0
-1
io4
•.
IO3
2
10^
10.
1.0
.10
.01
-
Upper
Boundary -x •
^ir a • •
«^* • •
^r • •
/« c
• • ji
* * * ^^"^
" /* • ^
/'• v^Y
" . . y*^ xLower
• *> Boundary
• /
- . '/
. i
' i
1 10 I02 IO3 IO4 IO5
SCALE (m)
Figure 1-3. Schematic plot showing what happens when one assumes that contaminants
spread predominantly by a diffusion-like process called hydrodynamic dispersion
and attempts to measure dispersivity. The dispersivity value depends on the size
(scale) of the measurement process.
-------
limitations illustrated in Figure 1-4. Obviously, the horizontal velocity variations that are
actually present will cause the plume to elongate dramatically in the horizontal direction. The
larger the value of Vmax - Vmin the larger the degree of spreading at a given time. To assume, as
the areal dispersion model does, that this spreading is proportional to the concentration
gradients illustrated in Figure 1-4 requires an increasing value for the constant of proportionality
(dispersion coefficient) to make up for the decreasing value of the concentration gradients
calculated by the vertically-averaged model.
3-3 What's The New Approach?
In the authors' opinion, the time has arrived to develop and apply aquifer tests for
determining the horizontal hydraulic conductivity as a function of vertical position (K(z)) within
a well or borehole. When this is done at a number of locations in the horizontal plane, the
resulting data can serve as a basis for developing two-dimensional vertical cross-section, quasi
three-dimensional or fully three-dimensional flow and transport models that do not require
large, scale-dependent dispersion coefficients. There are several methodologies for obtaining
K(z) distributions, and a major purpose of this report is to describe the methodologies
associated with the various field techniques, emphasizing those that are most promising. Shown
in Figure 1-5 are dimensionless K(z) distributions in three mutually perpendicular directions
based on results of single-well tracer tests (Molz et al., 1988). These distributions show that
K(z) varies with direction as well as with vertical position. Shown in Figure 1-6 are
dimensionless K(z) distributions obtained at five different scales of measurement in a single well
using an impeller meter (Hufschmied, 1983; Morin et al., 1988a; Rehfeldt et al., 1988; Molz et
al., 1989a). These differ from the tracer test results because at each measurement elevation, the
K value is averaged over the 360° polar angle, so that directional information is lost. It is
apparent that as the measurement interval varies from 10 ft (3.05 m) to 1 ft (0.3 m), the
apparent variability of the hydraulic conductivity increases, and there is every reason
to expect that it would increase further if the measurement interval were decreased below 1 ft
(0.3 m). This is the type of information that is lost when fully-penetrating pumping tests are
used to obtain vertically-averaged hydraulic conductivities. Of course, it is unlikely that
information as detailed as the 1 ft. data will be needed for practical purposes.
There are several techniques for making such measurements including tracer tests,
impeller meter tests, dilution tests and multi-level slug tests that are described in this report.
Such measurements should serve as the basis for an improved understanding and
conceptualization of subsurface transport processes, and may also allow the application of a new
generation of contaminant transport models that are advection-dominated and largely free of
the problems associated with large, scale-dependent, dispersion coefficients. All of this taken
together constitutes what the authors are advocating as the new approach to characterizing the
hydrogeologic properties of aquifers.
-------
(A)
Tracer at
time=0
c
o
c
0)
o
c
o
o
Tracer at later
times t, and t2>0
(B)
•Gradient (slope
ot curve)
0
Displacement
Figure 1-4. Schematic diagram showing the inherent lack of vertical contaminant
concentration structure that results from the solution of vertically-averaged
transport models.
-------
1
0)
(A
O
00
1
o-
0)
o
c
o
"o
9)
LJ
22
20
18
16
14
12
10
8
6
4
2
0
.
\ (12— E3)
\
B
; \
/
*NV
\.
^^^
• ^s^
S** i i
1. 2.
K/K
22
20
18
16
14
12
10
8
6
4
2
0
L ??
^\ (I2-~E6) 20
- / 18
- ^ .6
- ^x-"* 14
- \ 12
- / 10
/ 8
\ 6
*^>. 4
x'^ 2
X* I i
1. 2. 0
^
>f (12— E7)
- \
- \
/
/
^'
\
*^^^.
^'^^
f**^
S i i i
1. 2. 3.
K/K " K/K
Figure 1-5. Plots of dimensionless horizontal hydraulic conductivity versus elevation in three
mutually perpendicular directions from well 12. The basic data were obtained
from single-well tracer tests.
-------
WELL E8 IMPELLER METER
(I1 INTERVAL) -
130
140
150
2 160
N 170
ISO
190
200
0
130
140
150
160
170
180
190
200
i I i I
(51 INTERVAL)
j i i i
130
140
150
160
170
180
190
2.0 4.0
K/K
6.0
200
I I 1 I
(3' INTERVAL) -
0.0 2.0 _ 4.0
K/K
i I I I i
(10' INTERVAL)
i i i i
6.0
0.0 2.0 4.0 6.0
K/K
Figure 1-6. Dimensionless horizontal hydraulic conductivity distributions based on impeller
meter readings taken at the various measurement intervals indicated on the
figure.
10
-------
CHAPTER II
THE IMPELLER METER METHOD FOR MEASURING
HYDRAULIC CONDUCTIVITY DISTRIBUTIONS
II-1 Introduction
As outlined in Appendix I, one the best existing methodologies for obtaining vertically
distributed hydraulic conductivity information is the borehole impeller meter test. Such a test
may be viewed as a natural generalization of a standard fully penetrating pumping test. In the
latter application, only the steady pumping rate, QP, is measured, whereas the flow rate
distribution along the borehole or well screen, Q(z), as well as QP is recorded during an
impeller meter test.
Various types of flowmeters based on heat-pulse or impeller technology have been
devised for measuring Q(z), and a few groundwater applications have been described in the
literature (Hada, 1977; Keys and Sullivan, 1978; Schimschal, 1981; Hufschmied, 1983; Hess,
1986; Morin et al., 1988a; Rehfeldt et al, 1988; Molz et al., 1989a,b). The most low-flow-
sensitive types of meters are based on heat-pulse, electromagnetic or tracer-release technology
(Keys and MacCary, 1971; Hess, 1986), but to the authors' knowledge such instruments are not
presently available commercially, although several are nearing commercial availability. Impeller
meters (commonly called spinners) have been used for several decades in the petroleum
industry, and a few such instruments suitable for groundwater applications are now available for
purchase (Further information available upon request.). A meter of this type was calibrated,
applied, and the resulting data analyzed by Hufschmied (1983). A similar meter was applied in
the field and analyzed further by Rehfeldt et al. (1988). This latter study is probably the most
detailed to date regarding the analysis of assumptions one makes when using the borehole
impeller meter to measure formation hydraulic conductivity as a function of vertical position.
The main purpose of this chapter is to describe the application of an impeller meter to
measure K(z), at various locations in the horizontal plane, at a field site near Mobile, Alabama.
Identical or similar methodology should be applicable at many sites. The project site is located
in a soil borrow area at the Barry Steam Plant of the Alabama Power Company, about 32 km
(20 mi) north of Mobile, Alabama. The surface zone is composed of a low-terrace formation of
Quaternary age consisting of interbedded sands and clays deposited along the western edge of
the Mobile River. These sand and clay deposits extend to a depth of approximately 61 m (200
ft) where the contact between the Tertiary and Quaternary deposits is located. The water table
is about 3 m (9.84 ft) below the land surface. Below the contact, Miocene deposits are found
that consist of undifferentiated sands, silty clays and thin-bedded limestones extending to an
approximate depth of 305 m (1000 ft). The study formation is a sandy confined aquifer
approximately 21.3 m (70 ft) thick which rests on the Tertiary-Quaternary contact as illustrated
in Figure II-1.
II-2 Performance and Analysis of Impeller Meter Tests
II-2.1 Background Information
The idea and methodology behind the impeller meter test are illustrated in Figure II-2.
One first runs a caliper log to ascertain that the screen diameter is known and constant. If it is
not constant, the variations must be taken into account when calculating discharge. A small
pump is placed in a well and operated at a constant flow rate; QP. After pseudo steady-state
11
-------
*'"
K\*
\*
v
•V
hO ^
Supply^
U/«l I 1......V
Wei
XSI^IMZ^!^
v^
\
Injection
Well~\
\~\- A-~gaKkra3g!
\l
• •« 11\ I t'll'CTVV \1 \1\ \\ X' V"V \*
«
v\\\\\\\\\\\.
-------
OP-
PUMP
CAP ROCK
(QsDISCHARGE RATE)
BOREHOLE FLOW
METER •
ELEVATION=2
TO LOGGER (Q)
SURFACE
CASING
SCREEN
DATA
Q
Figure 11-2. Apparatus and geometry associated with a borehole flowmeter test.
13
-------
behavior is obtained, the flowmeter, which when calibrated measures vertical flow within the
screen, is lowered to near the bottom of the well, and a measurement of discharge rate is
obtained in terms of impeller-generated electrical pulses over a selected period of time. The
meter is then raised a few feet, another reading taken, raised another few feet—and so on. As
illustrated in the lower portion of Figure 11-2, the result is a series of data points giving vertical
discharge, Q, within the well screen as a function of vertical position z. Just above the top of
the screen the meter reading should be equal to QP, the steady pumping rate that is measured
independently on the surface with a water meter. The procedure may be repeated several times
to ascertain that readings are stable.
While Figure II-2 applies explicitly to a confined aquifer, which was the type of aquifer
studied at the Mobile site, application to an unconfined aquifer is similar. Most impeller meters
are capable of measuring upward or downward flow, so if the selected pumping rate, QP, causes
excessive drawdown, one can employ an injection procedure as an alternative. In either case,
there will be unavoidable errors near the water table due to the deviation from horizontal flow.
It is desirable in unconfined aquifers to keep QP as small as possible consistent with the stall
velocity of the meter. Thus more sensitive meters will have an advantage.
As shown in Figure II-3, the basic data analysis procedure is quite easy. One assumes
that the aquifer is composed of a series of n horizontal layers and takes the difference between
two successive meter readings, which yields the net flow, AQj, entering the screen segment
between the elevations where the readings were taken, which is assumed to bound layer i(i =
1,2,. ..,n). One then employs the Cooper- Jacob [1946] formula for horizontal flow to a well from
a layer, i, of thickness Azj, given by
} (II-l)
where AH; = drawdown in ith layer, AQj = flow from ith layer into the well, Kj = horizontal
hydraulic conductivity of the ith layer, AZ; = ith layer thickness, rw = effective well radius, t =
time since pumping started, and Si = storage coefficient for the ith layer. Solving equation (II-
1) for the Kj outside of the log term yields
g.= AQJ M1-5 /K.W'] (H-2)
^ 27TAHJAZJ r7J S;
which can be solved iteratively to obtain a value for Kj. Further details may be found in Morin
et al. [1988a] or Rehfeldt et al. [1988].
A convenient alternative method for obtaining a K distribution is based on the study of
flow in a layered, stratified aquifer by Javandel and Witherspoon [1969]. Their work showed
that in idealized, layered aquifers, flow at the well bore radius, rw, rapidly becomes horizontal
even for relatively large permeability contrasts between layers. As the writers point out, under
such conditions the radial gradients along the well bore are constant and uniform, and flow into
the well from a given layer is proportional to the transmissivity of that layer, that is:
AQj = aAZjKj (H-3)
where a is a constant of proportionality. This condition occurs_when the
dimensionless time tD = Kt/S/,, is > 100. (In this expression K is the average horizontal
aquifer hydraulic conductivity defined as 2KjAZj/b, where b is aquifer thickness, Ss is the aquifer
specific storage, t is time since pumping started and rw is well bore radius.)
14
-------
SURFACE-p
n^-
n-l
n-2-
i —
3 —
2-
i
0__
V
/ / /
•M w
V
/ / /
M* «^
., ^
/ / /
i
'
i
i
i
i
_ 1
i
i
_ — I
i
' (^WELL
' ' ' ' ' OlZj+iWFROM METER)
; f
/ i ' i
/X ^1 1 A=SCREEN AREA
1 — ' DARCY i | PER UNIT
i ^ VEL°fITY 1 D H LEN6TH= D
SCREEN '"',["*' ~*T~D='DIAMETER
[^ SEGMENT"^ i 1
, — _^_ | i
7 £_£)/ ////////////// /T 7' / wi^j/'ii-nuM MC. i tru
C— MEASUREMENT INTERVALS— >
V|
Figure II-3. Assumed layered geometry within which impeller meter data are collected and
analyzed. (Q(z) is discharge measured at elevation z.)
15
-------
To solve for a, sum the AQj over the aquifer thickness, to get
j = QP = a § AZjKj (II-4)
Multiplying the right-hand side of equation (II.4) by b/b and solving for a yields
a = (II-5)
bK
Finally, substituting for a in equation (II.4) and solving for Kj/K gives
; i = 1, 2, ... n (II-6)
K QP/b
To obtain equation (II-6) it was assumed that AQj and QP do not change with time (i.e. pseudo-
steady-state conditions apply). This will occur when r^jS/^Tt < 0.01, where S and T are aquifer
storage coefficient and transmissivity, respectively. Thus, from th_e basic data it is quite easy to
get a plot of K/K versus elevation. If one then has the value of K from a fully penetrating
pumping test, one can easily obtain dimensional values for K. At the Mobile site, the
theoretical time for tD > 100 is a fraction of a second^ and after about 3 min. of pumping,
pseudo-steady-state conditions are reached. The K/K approach has practical appeal because
one does not have to know values for rw or S(, which are impossible to specify precisely. Also,
multiplicative errors in flowmeter readings are cancelled out, and the meter does not have to be
calibrated. All that is needed is a linear response. However, a fully penetrating pumping test or
slug test must be performed along with each flowmeter test.
While the basic data analysis involved in the flowmeter method is quite simple, care
must be taken to come as close as possible to meeting all assumptions and measuring only the
actual flow caused by the pumping (Rehfeldt et al., 1988). For example, if there is ambient flow
in the well, this must be measured prior to any pumping so that the initial flow condition is
known. Alternatively, a two step pumping procedure can be used (Rehfeldt et al., 1988). In
addition, the basic data analysis procedure assumes horizontal flow in the aquifer and that
measured head loss in the well is due only to water flow through the undisturbed formation.
However, there are screen losses and head losses within the well. In general, hydraulic losses
can be minimized by pumping at the lowest rate consistent with the stall velocity of the impeller
meter. For a much more detailed discussion of well head losses and their possible correction
see Rehfeldt et al. (1988). Local deviations from horizontal flow will exist in most aquifers, but
the effects should be of second order compared to those of the average flow field as long as the
measurement intervals are not too small. As the Az; get smaller, one can expect errors due to
deviations from horizontal flow to get larger. Among other things, such errors will lead to poor
repeatability of flowmeter readings obtained from multiple tests performed in the same wefl.
II.2.2 Example Application at the Mobile Site
Testing began at the Mobile site with a mild redevelopment and cleaning of the test well
screens (Fig. II-4) with air. Ambient flow measurements were then made using a heat-pulse
flow meter developed by the U.S. Geological Survey, which has a measurement range of 0.1 to
20 ft./min (0.03 to 6.1 m/min.) (Hess, 1986), which is about 10 times more sensitive than any
impeller meter. Even at this sensitivity, however, no ambient vertical flow within the screen
could be detected. This is consistent with the study aquifer being relatively permeable and well
16
-------
Figure
Pockers
(AS)
Casing
Slotted
***»
„
and A5.
17
-------
confined in an area of low horizontal hydraulic gradient. If there had been a significant ambient
flow, the ambient flow at any measurement level z{ would have been subtracted from the
impeller meter reading at that level prior to data analysis.
The test well geometry shown in Figure II-2 is similar to that at the Mobile site, with the
4 in (10 cm) ID well screen (0.01 inch slotted plastic or plastic wire-wrap, see Fig. II-4)
extending from about 130 ft (39.6 m) to 200 ft (61 m) below the land surface. None of the
screens had a sand pack. To prepare for a test, the well screens were cleaned with air (clean
and open screens are important), and caliper and ambient pressure logs were run. Data
obtained from the caliper log were used to verify and compute the cross-sectional area of the
well, and the hydraulic-head distribution derived from the pressure log served as a calibration
reference for evaluating AHj produced by pumping. Subsequently, a pressure transducer and an
impeller meter with centralizer were lowered into the well, followed by a small submersible
pump capable of pumping about 60 gpm (227 liter /mm). The pump was started and allowed to
run for about an hour prior to taking pressure and impeller meter readings, more than adequate
time for drawdown to reach a pseudo-steady-state as defined by the Cooper-Jacob criterion
discussed in a following section. Data analysis showed that AHj varied only slightly over the
length of the screens.
An impeller meter can function in either a stationary or a trolling mode. In the
stationary mode, the meter is held at a series of set elevations and readings are taken in the
form of pulses per unit time with the aid of an electronic pulse counter. In the trolling mode,
the meter is raised or lowered at a constant known rate, and the reading reflects a superposition
of the trolling and water flow velocities. For fine-scale groundwater applications, the stationary
mode seems better suited; however, both methods of data acquisition were used during this
study.
Listed in Table II-1 are the basic impeller meter data obtained in wells E7, and A5,
along with the corresponding head difference between static and dynamic (pumping) conditions
derived from the pressure logs. In order to convert impeller meter readings into discharge, the
meter was calibrated in-situ by placing it in the top unslotted extension of the well screens (Fig.
II-4) and then pumping at three different rates which were measured independently at the
surface. In all cases the response was quite linear, and a straight line approximation was drawn
through the calibration points. For wells E7 and A5, the calibration equation was Q =
0.00428(CPM) in both cases, where Q is in ft3/min and CPM represents impeller "counts per
minute." Applying this equation to the data listed in Table II-1 resulted in the discharge profiles
presented in Table II-2.
II-2.3 Data Analysis
As discussed previously, we visualize two procedures for inferring a hydraulic
conductivity function, K(z), from impeller meter data. One approach involves the application of
equation (II-2) to each depth interval. This was done for wells A5 and E7 using a storage
coefficient St = 10"5-AZj, where the average specific storage of lO^ft"1 (3.05 x 10 m"1) was
determined from a pumping test performed previously (Parr et al., 1983). The results of this
computation are presented in Table II-3 as Kl(z), with depth values corresponding to the
midpoint (of the assumed layers) between impeller meter readings. Also shown in Table II-3 as
K2(z) are the results of applying equation (II-6) to each measurement^ interval in wells A5 and_
E7. To obtain these results, values of the dimensionless function Kj/K were calculated, where K
is average hydraulic conductivity. Knowing K from the analysis of a standard, fully penetrating
pumping test enables us to solve for each Kj. For a K of 0.121 ft/min (3.69 x 10"2m/min), a
18
-------
Table II-1. Impeller meter (discrete mode) and differential head data obtained in Wells E7,
and A5 at the Mobile Site. (z=depth, CPM=counts per minute, and AH=head
difference between static and dynamic conditions.)
z(ft)
130
135
140
145
150
155
160
165
170
175
180
185
190
Well #E7
CPM
1983
1933
1886
1764
1705
1607
1561
1468
1118
994
911
638
277
AH(ft)
1.218
1.202
1.189
1.177
1.166
1.157
1.149
1.143
1.139
1.138
1.138
1.138
1.138
z(ft)
132.5
137.5
142.5
147.5
152.5
157.5
162.5
167.5
172.5
177.5
182.5
187.5
190.0
WeU #A5
CPM
2024
1968
1885
1799
1652
1488
1362
1106
882
740
506
293
57
AH(ft)
1.210
1.201
1.170
1.147
1.136
1.132
1.132
1.132
1.138
1.156
1.173
1.186
1.193
19
-------
Table II-2. Well screen discharge as a function of vertical position in wells E7 and A5 at
the Mobile site. (z=depth, Q = discharge rate in well screen.)
z(ft)
130
135
140
145
150
155
160
165
170
175
180
185
190
Well #E7
Q(ft3/min)
8.49
8.27
8.07
7.55
7.30
6.88
6.68
6.28
4.79
4.25
3.90
2.73
1.19
z(ft)
132.5
137.5
142.5
147.5
152.5
157.5
162.5
167.5
172.5
177.5
182.5
187.5
190.00
Well #A5
Q(ft3/min)
8.66
8.42
8.07
7.70
7.07
6.37
5.83
4.73
3.77
3.17
2.17
1.25
0.244
20
-------
Table II-3. Hydraulic conductivity distributions inferred from impeller meter data using two
different approaches described herein. (Depth z is in ft. and K(z) is in ft./min.)
z
132.5
137.5
142.5
147.5
152.5
157.5
162.5
167.5
172.5
177.5
182.5
187.5
195.0
Well #E7
Kl(z)
0.050
0.046
0.128
0.059
0.104
0.048
0.100
0.405
0.139
0.088
0.315
0.421
0.154
K2(z)
0.042
0.038
0.100
0.049
0.083
0.040
0.080
0.299
0.109
0.071
0.236
0.310
0.120
z
135
140
145
150
155
160
165
170
175
180
185
189
195
Well #A5
Kl(z)
0.055
0.083
0.091
0.163
0.184
0.140
0.297
0.257
0.156
0.263
0.237
0.536
0.027
K2(z)
0.043
0.063
0.069
0.119
0.134
0.104
0.212
0.185
0.115
0.189
0.171
0.371
0.022
21
-------
value obtained from a pumping test performed in the vicinity of E7 and A5, corresponding
values of K2(z) are listed in Table II-3. These hydraulic conductivity profiles are plotted also in
Figure II-5,
33-2.4 Comparison of Impeller Meter Tests With Tracer Tests
Examination of Figure II-5 shows that the trends in the data are virtually identical for
wells A5 and E7. There is also fairly good agreement between the absolute (dimensional)
values calculated for the hydraulic conductivity.
It is of interest to compare the hydraulic conductivity distributions inferred from the
impeller meter data with those obtained previously using single well tracer tests (Molz et al.,
1988). These tests involved one fully penetrating tracer injection well and one multilevel
sampling/observation well located about 20 ft (6.1 m) away. A bromide tracer solution was
injected at a constant rate through the injection well, while water samples were collected
periodically from up to 14 different elevations of the multilevel sampling well. Analysis of the
samples for Br concentration allowed one to determine the travel time distribution between the
injection and sampling wells as a function of elevation. From this information it is possible to
infer a relative hydraulic conductivity distribution (Molz et al., 1988). There is no reason to
expect detailed agreement between the impeller meter results and the single-well tracer test
results because the latter data reflect an average hydraulic conductivity value inferred over a
travel distance of approximately 20 ft (6.1 m). However, as shown in Figure II-6, the agreement
is reasonably good, indicating that the overall trend in K(z) persists over the 20 ft (6.1 m) travel
distance of the tracer test (Molz et al., 1988).
Ibl Measurement of Hydraulic Conductivity at Difference Scales Using
Impeller Meter Tests and Pumping Tests
The main purpose of this section is to describe the application of impeller meter tests
and pumping tests so that the reader will develop an appreciation for the type and extent of
hydraulic information that can be assembled at a particular site. Once again, the site chosen for
this detailed application was the Mobile site. Vertical scale information was obtained using the
impeller meter, while fully penetrating pumping tests were employed for obtaining information
at various lateral scales. The testing procedures where those described in the previous sections.
The fully penetrating pumping tests were analyzed using the Cooper-Jacob Method (Freeze and
Cherry, 1970 (or almost any contemporary groundwater text)).
II-3.1 Results of Tests
Shown in Figure II-7 is a plan view of the Mobile study site where the pumping tests and
impeller meter tests were performed. The various wells are designated as 12, E6, A3, etc. The
number in parentheses next to each well designation is the vertically-averaged hydraulic
conductivity in meters per day, K(x,y), that resulted from one or more small-scale pumping tests
in which the designated well was the pumping well. Arrows indicate the pattern of the
observation well/pumping-well arrangement in Figure II-7. They point from the observation
well towards the pumping well. Each arrow represents a single test with a pumping rate of
about 0.22 m3/min (58 gpm). The repeatability of any one test was good with the drawdown
data falling within 5% of each other.
22
-------
N
130
140
150
160
170
180
190
WELL A5
200
0.0
130
140
150
s 160
N 170
180
190
0.2 0.4
K (ft/min)
J =!
WELL E7
0.6
200
0.0
0.2 0.4
K (ft/min)
0.6
Figure II-5. Hydraulic conductivity distributions calculated from flowmeter data using two
different methods.
23
-------
N
130
140
150
160
170
180
190
200
WELL E7
I ^ >^^^^^^^^^^^«^^»
—IMPELLER METER (Kl)
—TRACER TEST
0.0
O.I
0.2 0.3
K (ft/mln)
0.4 0.5
Figure II-6. Comparison of hydraulic conductivity distributions for well E7 based on tracer
test data and impeller meter data.
24
-------
*IO-
A5(53)
AK55.8)
A6(5I.8)
•
j
A7(53.9) A3(55.8)
12(54.9) E7(
*E8l53.3)
\-
(/>
o
-5-
E9(57) El 0(54.9)
A2(5I.5) A8(59.l) A4
E6(52.l)
-10-
A9(58.8)
I I I I
10 15 20 25
DISTANCE (M)-
30 35
40 45
Figure II-7. Plan view of the field site where small-scale pumping tests were performed. The
numbers next to the dots are well designations, while the values in parentheses
are the average hydraulic conductivities (m/day) assigned to the vicinity of each
pumping well. Each arrow represents a test and points from the observation well
to the pumping well. Wells with more than one arrow pointing toward them
were assigned average values.
25
-------
A series of pumping tests in which the pumped wells were used as observation wells
were performed also. Once again the pumping rate was approximately 0.22 m3/min (58 gpm).
The results of these tests are shown in Figure II-8.
K/K distributions based on impeller meter tests performed in wells E7, E8, E9, A6, A7,
A8, and A9 are shown in Figures II-9 through 11-15, respectively. Each figure was obtained with
the use of equation II-6 applied to impeller meter data from measurement intervals of 0.3 m (1
ft), 0.91 m (3 ft), 1.52 m (5 ft), and 3.108 m (10 ft), as illustrated in Figure II-3.
As with the fully penetrating pumping tests, repeatability of the impeller meter tests was
good. Evidence for this is shown in Figure 11-16 which documents the results of repeated
impeller meter tests in well E7.
II-3.2 Discussion of Results
The vertically-averaged hydraulic conductivity, K(x,y), shown in Figure II-7 seems to
imply that the study aquifer is fairly homogeneous. The mean value of hydraulic conductivity is
54.9 m/day with a standard deviation of only 2.4 m/day*. The mean value agrees well with the
result of a large-scale pumping test (53.4 m/day) performed previously using 12 as the pumping
well and pumping at the rate of 1.48 m3/min (390 gpm) (Parr et al, 1983).
As one would expect, the results shown in Figure II-8 are more variable because a
pumping test using the pumping well as an observation well will sample a smaller volume of the
aquifer. Here the mean value is only 3.5% smaller at 53.0 m/day, but the standard deviation
has increased by a factor of 4.75 to 11.4 m/day.
In the authors' judgement, no distinct pattern emerges from Figure II-7
or Figure II-8. It is probable that K(x,y) will show lateral trends over distances in excess of 38
m, which is the approximate distance between wells 12 and E10. However, over the lateral
distances representative of Figures II-7 and II-8 the K(x,y) variations appear more or less
random.
Given the generally layered nature of geologic deposits in a fluvial environment, one
would expect much more variability of horizontal hydraulic conductivity as a function of vertical
position, K(z), than of vertically-averaged horizontal hydraulic conductivity as a function of
lateral position, K(x,y). Examination of Figures II-9 through 11-15 shows this to be the case.
(Note that K(z) at any particular z is still averaged over the 360° polar
angle, so that the impeller meter test gives no information about lateral heterogeneity or
anisotropy around a given well.) Different degrees of heterogeneity are apparent at the various
measurement scales common to each figure. As the measurement scale varies from 10 ft (3.05
m) to 1 ft (0.3 m), the measured variation in hydraulic conductivity increases, and there is every
reason to expect that it would increase further if the measurement scale were decreased to 0.5
ft, 0.25 ft. etc. Obviously, this type of heterogeneity is not reflected in the results of fully
penetrating pumping tests.
Since the data are correlated, the standard deviation is not well defined in a statistical sense.
We are using it here just as a convenient measure of variation.
26
-------
+10-
+5-
1 E3
UJ t-
u
I-
w
Q
-5-
-10-
A6(32.0)
•
A5(50.7)
AK66.4)
A7(47.8)
•
A3(62.2)
•
12"^ ^7(66.5)
E6(37.5)
d H
H h
E9(50.2) EIO
A2(66.l)
A8(53.9) A4
A9(57.2)
10 15 20 2
DISTANCE (M)
30 35 40 45
Figure II-8. Results of small-scale pumping tests (m/day) wherein the pumping wells were
used as observation wells.
27
-------
WELL E7 IMPELLER METER
130
140
150
s= 160
««*—
N 170
180
190
200
0.0
I I
(5' INTERVAL)
130
140
150
160
170
180
190
200
2.0 _ 4.0 6.0
K/K
6.0
(10' INTERVAL)
0.0 2.0 4.0 6.0
K/K
Figure II-9. Dimensionless horizontal hydraulic conductivity distributions based on impeller
meter readings taken at the various measurement intervals indicated on the
figure.
28
-------
WELL E8 IMPELLER METER
130
140
150
160
170
180
190
200
130
140
150
160
170
180
190
200
i r i
(I1 INTERVAL) -
0.0 2.0 4.0
K/K
ii i i r
(5' INTERVAL)
I
0.0 2.0 4.0
K/K
6.0
6.0
130
140
150
160
170
180
190
200
130
140
150
160
170
180
190
200
i i r i r
(31 INTERVAL)
i i i i
I i I r
(10' INTERVAL)
0.0 2.0 4.0 6.0
K/K
0.0 2.0 4.0 6.0
K/K
Figure 11-10. Dimensionless horizontal hydraulic conductivity distributions based on impeller
meter readings taken at the various measurement intervals indicated on the
figure.
29
-------
WELL E9 IMPELLER METER
130
140
150
170
ISO
190
200
0.0
(51 INTERVAL)
2.0
4.0
6.0
130
140
150
~ 160
N 170
180
190
200
0.0
2,0 _ 4.0
K/R
I 1
(10' INTERVAL)
6.0
K/K.
2.0 4.O 6.0
K/K
Figure 11-11. Dimensionless horizontal hydraulic conductivity distributions based on impeller
meter readings taken at the various measurement intervals indicated on the
figure.
30
-------
WELL A6 IMPELLER METER
200
0.0
130
140
150
160
170
180
190
200
• i
(5' INTERVAL)
j i i i
130
140
150
160
170
180
190
200
130
140
150
170
180
190
0.0 2.0 4.0 6.0
K/K
200
0.0
(3' INTERVAL)
0.0 2.0 4.0
K/K
6.0
(10' INTERVAL)
2.0
4.0
6.0
K/K
Figure 11-12. Dimensionless horizontal hydraulic conductivity distributions based on impeller
meter readings taken at the various measurement intervals indicated on the
figure.
31
-------
WELL A7 IMPELLER METER
130
140
150
170
180
190
200
i I
(5' INTERVAL)
0.0 2.0 4.0
K/K
130
140
150
160
170
180
190
6.0
200
2.0 4.0 6.0
K/K
(10' INTERVAL)
0.0 2.0 _ 4.0
K/K
6.0
Figure 11-13 Dimensionless horizontal hydraulic conductivity distributions based on impeller
meter readings taken at the various measurement intervals indicated on the
figure.
32
-------
WELL A8 IMPELLER METER
130
140
150
= 160
N 170
180
190
200
130
140
150
= 160
N 170
180
190
200
0
(I1 INTERVAL)
130
140
150
= 160
N 170
180
190
200
0.0 2.0 4.0
K/K
6.0
(5' INTERVAL)
2.0 4.0
K/K
6.0
130
140
150
= 160
N 170
180
190
200
0
(3' INTERVAL)
0.0 2.0 4.0 6.0
K/K
(10' INTERVAL)
j i I I
2.0 _ 4.0
K/K
6.0
Figure 11-14. Dimensionless horizontal hydraulic conductivity distributions based on impeller
meter readings taken at the various measurement intervals indicated on the
figure.
33
-------
WELL A9 IMPELLER METER
200
0.0
130
140
150
160
170
180
190
200
i i i i
(3' INTERVAL)
(5' INTERVAL)
0.0 2.0 4.0
K/K
6.0
130
140
150
I6°
170
180
190
200
0.
(10' INTERVAL)
2.0 _ 4.O 6.0
K/R
Figure 11-15. Dimensionless horizontal hydraulic conductivity distributions based on impeller
meter readings taken at the various measurement intervals indicated on the
figure.
34
-------
WELL E7 IMPELLER METER
(5f DATA)
30 MIN. PUMPING
60 MIN. PUMPING
2.0 4.0 6.0
K/K
120 MIN. PUMPING
6.0
Figure 11-16. Dimensionless hydraulic conductivity distributions at five-foot intervals in well E7
taken 30 min., 60 min. and 120 min. after the start of pumping. The results show
good repeatability of the impeller meter method.
35
-------
As shown in Figure II-8, wells E7, E8 and E9 are located along a common line, which
runs approximately in an east-west direction. As the measurement scale gets larger, all three
wells begin to approximate something like a triangular distribution with a peak in the bottom
third of the aquifer. In the perpendicular direction, the same general behavior is demonstrated
by wells A7, A8, A9 and to a lesser extent by A6. Thus, with respect to the vertical coordinate,
the mean horizontal hydraulic conductivity is not constant with depth. There is a definite trend
toward a hydraulic conductivity peak in the bottom third of the aquifer, and a smaller, less
consistent peak in the top third. In the language of stochastic hydrology, one would say that the
K(z) distribution is nonstationary (Russo and Jury, 1987a,b).
II-4 Summary and Conclusions Concerning Impeller Meter Applications
Once the necessary equipment is obtained, impeller meter tests can be a relatively quick
and convenient method for obtaining information about the vertical variation of horizontal
hydraulic conductivity K(z) in an aquifer. This information can be used in a variety of ways
including the design of sampling/monitoring wells or pump and treat systems. It can also be
used as the basis for the development of three-dimensional flow and transport models which will
be far more realistic than their vertically-averaged forerunners. (Applications to fractured rock
hydrology are described in Chapter IV.)
Over the past several years at the Mobile site, a fairly large amount of hydraulic
conductivity data have been developed based in part on fully penetrating pumping tests, both
large and small scale, and impeller meter tests. As far as contaminant transport predictions are
concerned, the pumping tests alone are of limited use because by their nature they fail to show
the large amount of vertically-distributed heterogeneity that is apparent to varying degrees in
the impeller meter tests. This may be obvious, but merits emphasis because fully penetrating
tests and vertically-averaged properties are still the mainstay for dealing with contaminant
migration problems in the field. Obviously, this has to change. Vertically distributed
information is vital to successful remediation design and meaningful simulation of contaminant
transport in aquifers.
Although much less restrictive than the assumption of a vertically homogeneous aquifer,
the assumption of a layered, stratified aquifer in the vicinity of a test well is still limiting to
complete characterization of the unknown three-dimensional variations that actually exist. Thus
errors will be made in analyzing a given test, and discrepancies will arise when different types of
tests are compared, or even when the same test is analyzed using different methods. Our data
suggest that the best strategy for suppressing such errors or discrepancies may consist of using
an impeller meter to obtain a dimensionless K/K distribution and then a standard pumping test,
or a slug test, to compute K. Combining both types of information enables one to "fit" an
impeller meter test to a given aquifer and to obtain dimensional values for K(z). Shown in
Figure II-5 is the type of information that results when the two testing procedures are
combined.
In the flowmeter applications at Mobile, a different K(z)/K distribution was obtained at
every vertical scale of measurement at each of seven different wells. As one would expect, the
smaller the vertical scale of measurement the larger the degree of heterogeneity that becomes
apparent.
36
-------
In such a system that shows increasing heterogeneity at decreasing scales of
measurement, one is led to ask the question: "What scale of measurement is appropriate for a
given application?" Although there are some promising approaches in the area of geostatistics
for answering this question, the authors have left this question largely unanswered in a general
sense. However, based on our studies at the Mobile site, a rule of thumb would be to use
measurement intervals of about one tenth of the aquifer thickness [Molz et al., 1989b].
However, once the equipment is set up, one foot measurement intervals would be practical in
most aquifers. If at a later time less detailed data is desirable, one could use every other data
point, or every third data point, etc.
37
-------
CHAPTER III
MULTI-LEVEL SLUG TESTS FOR MEASURING
HYDRAULIC CONDUCTIVITY DISTRIBUTIONS
III-l Introduction
As discussed in Appendix I, the impeller meter test procedure is generally superior to
the multilevel slug test approach, because the latter procedure depends on one's ability to isolate
hydraulically a portion of the test aquifer using a straddle packer. However, if reasonable
isolation can be achieved, which was the case at the Mobile test site, then the multi-level slug
test is a viable procedure for measuring K(z). All equipment needed for such testing is
available commercially, and the test procedure has the added advantage of not requiring any
water to be injected into or withdrawn from the test well if a water displacement technique is
used to cause a sudden head change.
The testing apparatus used in the applications reported herein is illustrated schematically
in Figure III-l for the aquifer geometry at the Mobile site. Two inflatable packers separated by
a length of perforated, galvanized steel pipe comprised the straddle packer assembly. The test
length of aquifer defined by the straddle packer was L=3.63 ft. (1.1 m). A larger packer,
referred to as the reservoir packer, was attached to the straddle packer with 2" (5.08 cm) Triloc
PVC pipe, creating a unit of fixed length of approximately 100 ft (30.5 m) which could be moved
with the attached cable to desired positions in the well. When inflated, the straddle packer
isolated the desired test region of the aquifer and the reservoir packer isolated a reservoir in
the 6" (15.2 cm) casing above the multilevel slug test unit and below the potentiometric surface
of the confined aquifer.
An advantage of this unit design was that the length of 2 in (5.08 cm) connecting pipe
and other geometric factors contributing to frictional losses remained unchanged regardless of
packer elevation in the well. (This was possible at the Mobile site because the piezometric
surface was approximately 120 ft (36.6 m) above the top of the 70 ft (21.3 m) thick aquifer.)
For the straddle packers, the inflatable lengths were 24.5 in. (62.2 cm) (model 36, pneumatic
packer, Tigre Tierra, Inc.) and for the reservoir packer the inflatable length was 39.0 in. (99.1
cm) (model 610, pneumatic packer, Tigre Tierra, Inc.)
III-2 Performance of Multi-Level Slug Tests
Multilevel slug tests are described for three wells (E3,E6,E7) at the Mobile, Alabama
site shown in the plan view of Figure III-2. The wells, formerly used as multilevel tracer
sampling wells (Molz, et al. 1988), were constructed of 6 in (15.2 cm) PVC casing down 130 ft
(39.6 m) to the top of the medium sand aquifer. Fully slotted 4 in (10.2 cm) PVC pipe
extended an additional 70 ft (21.3 m) through the aquifer as shown in Figure III-l. Well E3 was
an exception, having 3 ft (0.91 m) slotted pipe sections separated by 7 ft (2.13 m) solid sections
through the aquifer.
In a typical test, water was displaced in the reservoir above the packer. The head
increase then induced a flow down through the central core of the reservoir packer and down
the TriLoc pipe to the straddle packer assembly. In the assembly, water flowed from the
perforated pipe, through the slotted well screen, and into the test region of the aquifer.
38
-------
depth recorder
cable
Figure III-l. Schematic diagram of the apparatus for performing a multi-level slug test.
39
-------
E1 E5
E2E3
-50 (ft)
-12 E7 E8 E9E10
i CD i i | O i i i | O i GI i i
150
EG
n injection wells
o multilevel observation/slug test wells
Figure III-2. Plan view of part of the well field at the Mobile site.
40
-------
The multilevel unit described was used for slug tests by inserting the plunger, displacing
a volume of water in the reservoir and then recording the depth variation, y=y(t), relative to the
initial potentiometric surface (falling head test). Plunger withdrawal was used to generate a
rising head test. Head measurements were made with a manually operated digital recorder
(Level Head model LH10, with a 10 psig pressure transducer, In Situ, Inc.).
Typical results of a series of tests at different elevations are shown for well E6 in Figure
III-3. The data shown are from plunger insertion tests where a sudden reservoir depth increase
to approximately y0=3 ft (0.91 m) was imposed. The depth variation, y=y(t), which is nearly an
exponential decay, is a result of flow into the aquifer test section adjacent to the straddle
packer. The different slopes of the straight line approximations (least squares fits) are due to
the variability of the hydraulic conductivity in the aquifer at the different test section elevations.
Tests repeated at a given elevation were generally reproducible, as shown in Figure III-4, with
the maximum difference in slopes of the straight line fits to the data being 10% or less.
An initial exception to the general rule of reproducible behavior was observed in well
E6. Shown in Figure III-5 are the results of tests at two elevations conducted on three different
days in well E6. For the tests of July 20, 1987, the well had been undisturbed for approximately
40 days. For the repeated tests on July 21, 1987, the well was developed by repeated air
injection and flushing prior to the slug testing. Noting the significant change, particularly for the
curves having larger slope (higher hydraulic conductivity), the tests were repeated on July 30,
1988 after more extensive development with an air jetting tool at the bottom of the well. With
this third set of data in close agreement with the second set, it was decided that the
development was sufficient. This behavior was not observed at the other test wells; however, all
tests were done after a small amount of well re-development. The well construction, originally
done for tracer observations (Molz, et al., 1986a, 1986b), was intended to minimally disturb the
aquifer close to the screen. Particularly after the passage of several months, some minor
redevelopment is important prior to hydraulic testing. It seems that clay and silt materials tend
to migrate into the well, coat the screen and often collect at the well bottom.
Multilevel slug testing will be meaningful only if the straddle packer system isolates
hydraulically the segment of the screen and the adjacent aquifer. Channels, which will negate
the packer seal, may be present in the well screen structure or can be caused by failure of the
formation or backfill material to fill the annulus between the screen and the borehole.
Similarly, any backfill material of greater permeability than the formation can allow flow to
bypass the packers rather than having outward flow into the test section. Additional pressure
monitoring above and below the straddle packer assembly may be desireable if nonisolation
problems are suspected (Taylor et al., 1989).
III-3 Analysis of Multilevel Data
Three methods of analysis have been applied to the collected data. In the first and
second methods, it is assumed that the flow from the test section is horizontal and radially
symmetric about the axis of the well. In the first method the quasi-steady state assumption is
made. In the second, a transient analysis is applied. In the third analysis, also quasi-steady
state, the possibility of vertical flow and anisotropy are considered using a finite element model.
41
-------
300.
time (sec)
log(y)=-
log(y)=-
|0g(y)=-
log(y)=-
log(y)«-
\og(y)—-
log(y)=-
log(y)=-
log(y)=-
log(y)=-
log(y)=-
•0.0280t+0.47
-0.0045t-t-0.49
-0.0020t+0.50
-0.0038t+0.49
•0.00411+0.51
-0.0034t+0.49
-0.0026t+0.47
-0.0046t+0.48
-0.0053t+0.48
-0.00711+0.47
-0.0092H-0.50
Z=11.2 ft
Z«17.2 ft
Z=23.2 ft
Z= 5.2 ft
Z=29.2 ft
Z=35.2 ft
Z=41.2 ft
Z=47.2 ft
Z=53.2 ft
Z=59.2 ft
Z=65.2 ft
Figure IQ-3. Multilevel slug test data from well E6. B=log(y1/y2)/(t2-t1) = magnitude of the
slope of the log y(t) response.
42
-------
I I I
Z=5.2 ft June 9. 1987
-^ 8=0.0060
8=0.0063
8=0.0060
50
100 150 200
time (sec)
250 300
Figure III-4. Plot showing the reproducibility of data collected at well E6.
43
-------
July 20
July 21
July 30
July 20
July 21
July 30
100 150 200
time (sec)
250 300
Figure III-5. Plots showing the influence of well development at two elevations in well E6.
44
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III-3.1 Quasi-Steady Radial Flow Assumption
A very common analysis of slug test data (Hvorslev, 1951) assumes that the radial
seepage is steady state (no aquifer storage effects) and that the head increase induced during
the test is zero at the radius of influence, r=re. If B is the magnitude of the slope of the log
y(t) response (Figure III-3), the hydraulic conductivity, based on this radial, quasi-steady analysis
is
K = 2.30B(rc2 - rp2)ln(re/rw)/2L (III-l)
where K = hydraulic conductivity, rc = casing radius, rp = plunger radius, B = magnitude of the
slope of log [y(t)j, re = radius of influence, rw = well screen effective radius, and L = separation
length of straddle packers. If the plunger is not used, or in the case of plunger withdrawal
(rising head test), then rp = 0 in equation (III-l). This method of interpretation is somewhat
arbitrary since re is not measured and is basically an adjustable parameter in the Hvorslev
(1951) method. Chirlin (1989) pointed out that re should be interpreted as a storage-effect
parameter. For this study, re = 60 ft (18.29 m) was chosen. This value of re results in K(z)
values similar to those calculated in the transient analysis which follows. This similarity of K(z)
values and dependence on re is consistent with Figuje 4 of the critique by Chirlin (1989). For
well E6, the slopes from Figure III-3 and the K(z) |alues calculated using equation (III-l) are
shown in Table III-l.
III-3.2 Radial. Transient Flow Assumption
With the radial flow assumption, the slug test interpretation is equivalent to the fully-
penetrating analysis of Cooper, et al. (1967) and th
3 extension of the analysis by Papadopulos et
al. (1973). In these analyses, the response of a finiie diameter well to an instantaneously
injected or withdrawn volume of water is considered, and a set of type curves, Figure III-6, is
presented which permit the matching of slug test data to determine the transmissivity of the
aquifer. The storage coefficient of the aquifer can, in theory, also be determined with this
matching method, but the accuracy is poor as discussed in the two original papers. In this
transient model, there is no need for a re parameter.
Previous full-aquifer pumping tests at the Mobile site (Parr et al., 1983) have indicated a
specific storage of 7.14 x 1(T (ft'1) (2.18 x 10"6 m). If it is assumed that the full-aquifer result
applies on the smaller scale of the multilevel tests described here, then the storage coefficient
for the test length of L = 3.63 ft (1.1 m) is S = 2.6 x 10'5.
For slug testing with the plunger in place the parameter defined by Cooper et al., (1967)
is modified to a = Srv//(rc2 - rp2). Thus, for the match point, (t,T) the hydraulic conductivity is
determined from
K = T(rc2 - rp2)/Lt. (III-2)
Slug test data taken at 11 elevations in well E6 as shown in Figure III-7 are plotted on a
log-linear scale. Determination of storage coefficients from these data, which has questionable
reliability using this method (Cooper et al. 1967), was not done since the value of S determined
from full aquifer tests was available. These points for the data and type curves are shown in
Table III-l. The match points were obtained by visual inspection in the attempt to best fit the
data between the a= 10^ and the a= 10"5 curves of Figure III-6. These bounds were obtained
based on our measured values for S. Match points for the curves of Figure III-7 are shown in
Table III-l, with the corresponding calculations for hydraulic conductivity at each elevation.
Repeated independent visual fits using this method provided reproducibility within
45
-------
icr2 icr1 10° 101
T (dimensionless time)
10a
Figure III-6. Example type curves from Cooper, Bredehoeft and Papadopulos (1967).
46
-------
Well E6 (7/31/87)
Figure III-7. Slug test data from different elevations in well E6.
47
-------
Table III-l. Multilevel Slug Test Data and Calculated Hydraulic Conductivities at Well E6 (6/31/87 data)
Elevation
Z(ft)
5.2
11.2
17.2
23.2
29.2
35.2
41.2
47.2
53.2
59.2
65.2
Slope (Figure III-3)
B (I/sec)
0.00383
0.0284
0.00453
0.00205
0.00411
0.00339
0.00261
0.00463
0.00525
0.00712
0.00918
Eq. (III-l)
K (ft/day)
34.8
258.2
41.1
18.6
37.4
30.8
23.7
42.1
47.7
64.7
83.4
T
1.0
1.0
3.0
1.0
1.0
9.2
1.0
2.9
3.0
9.0
5.0
Match Point
t(sec)
40
6
100
70
40
400
50
90
80
200
90
Eq. (III-2)
K (ft/day)
33.6
223.7
40.3
19.2
33.6
30.9
26.9
43.4
50.4
60.5
74.6
Eq. (III-8)
K (ft/day)
22.3
162.4
25.9
11.7
23.4
19.3
14.9
26.4
30.0
40.7
54.5
K = 57.6 K = 53.8 K = 36.4
-------
approximately 10% which is consistent with field data accuracy. Numerical fitting procedures
have also been applied but did not yield a significant change in the K(z) values calculated.
The variation of horizontal hydraulic conductivity with respect to elevation, K(z), is
shown in Figure III-8 for well E6. This K(z) profile is based on one slug test at each of the 11
elevations. The length of the aquifer isolated by the straddle packers, L=3.63 ft (1.1 m), is
shown in the figure. Short sampling lengths and/or more sampling elevations would produce
K(z) profiles of more detail.
III-3.3 Quasi-Steady State Radial and Vertical Flow Assumption
Dagan's (1978) partially-penetrating well test analysis considers the double packer
configuration for determining hydraulic conductivity in unconfined aquifers. The effects of
vertical as well as radial components to the aquifer flow are included in the analysis. However,
the analytical solution presented and the resulting dimensionless curves apply to an unconfined
aquifer and are valid only if the distance between the packers (L) is at least 50 times greater
than the well screen radius (rw). Braester and Thunvik (1984) simulated double packer tests by
solving an equation for axisymmetric flow in cylindrical coordinates by the Galerkin method and
were able to produce results consistent with those derived from Dagan's solution. While the
study considered the effects of anisotropy and inhomogeneity on interpretation of double packer
slug test data, a general solution technique to extract a value for hydraulic conductivity from
experimental data was not offered.
The analysis developed is an extension of Dagan's dimensionless-curve approach (1978)
using a method similar to Braester and Thunvik (1984). A finite element model (EFLOW,
licensed through the Electric Power Research Institute) was modified to simulate multilevel slug
tests for a variety of flow configurations and aquifer conditions. Using EFLOW simulation
results for a generic confined aquifer, a family of curves was developed and a procedure by
which hydraulic conductivity is determined was derived.
Mathematically, the aquifer head distribution is described by the equation,
r dr 3r dz2
Where r = radial coordinate, z = vertical coordinate (as shown in Figure III-2), h = head in the
aquifer, K = hydraulic conductivity in the horizontal, radial direction, and Kj = hydraulic
conductivity in the vertical direction. Equation (III-3) is independent of time but will be
subjected to a time-dependent boundary condition at r = rw. Thus the transient slug test is
approximated by a succession of steady states (quasi-steady state model). The boundary
conditions are:
3h/ x 3h / i-.x n f .
3-(r> °) = 3— (r' D) = 0. f°r rw < r < re
oz oz
— (rw, z) = 0, for 0 < z < (D-H) and (D-H + L) < z < D
or
and
h(re, z) = h0, for 0 < z < D
h(rw, z, t) = h0 + y(t), for (D - H) < z < (D - H + L)
49
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^straddle packer
separation
200 300 400
K(z) (ft/day)
500
Figure III-8. K(z) profile at well E6 based on the radial transient analysis (Cooper et al.,
1967).
50
-------
where y is the depth increase in the well casing reservoir, rw = the screen radius, D = the
aquifer thickness, h0 = the undisturbed head in the aquifer, and H = the vertical depth to the
lower packer (Figure III-2).
By conservation, the change of the reservoir head with respect to time, dy/dt, is related
to the volumetric flow rate into the aquifer, Q, by
- 7T(rc2 - rp2)dy/dt = Q (III-4)
For the succession of steady states and a given aquifer geometry, the flow into the aquifer at
any instant of time is proportional to y, i.e. because of the steady-state flow distribution at each
time instant, doubling y will double the flow into the aquifer. More precisely,
Q = 0y (III-5)
where /3 is the constant of proportionality. Combining equations (III-4) and (III-5) and
rearranging yields
(l/y)dy/dt = d(ln(y))/dt = - 0/0c(rc2 - rp2)) (III-6)
Since the right-hand side of (III-6) is constant, this shows that a plot of ln(y) versus t will yield
a straight line.
Through the use of Darcy's law the flow into the aquifer may also be written as
D-H+Lah(rw,z)
Q = 27TrwKj _ w 'dz (III-7)
D-H dr
One may now define a dimensionless flow parameter, P, given by
2flKLy Ly D-H dr
The parameter, P, depends only on the geometry of a particular slug test. From numerical
solutions of equation (III-3) for different geometries and using equation (III-8), Figure III-9 was
generated showing the dependence of P on H/L and L/rw. The anisotropy considered here was
= 0.1 (Parr et al., 1983), which is approximately that which applies at the Mobile site.
Once Figure III-9 is developed for a given anisotropy ratio, it may be used in
combination with a semi-log plot of slug test data to calculate the hydraulic conductivity. The
procedure is to enter Figure III-9 with the appropriate values of L, rw, and H and obtain the
corresponding number for P (call it Pn). Then using equation (III-4) one notes that
Q = 2*KLyPn = - 7T(rc2 - rp2)dy/dt (III-9)
Employing the fact that (l/y)dy/dt = d(ln(y))/dt and solving equation (III-9) for K yields
d(ln(y)) } , (rc2 - rp2) ( }
' V '
2LPn dt 2LP
where B is the slope of a semi-log plot (base 10 logs) of y versus t as shown in Figure III-3. An
example calculation of K using equation (III- 10) is shown in Appendix II.
The radius of influence for this study was chosen as re = 60 ft (18.3 m) to be consistent
with that used in method 1. For the geometries of the slug tests described here, the
dependence of K on re is not so strong as it is in equation (III-l). For example, if re is
51
-------
CO
o
II
QL
0.40
0.35
0.30
0.25
0.20
0.15
I I I I I I I I I I I I I I I I I 1 I I I I I I I I I I I
Kz/Kr = O.I
= 1.0
= 1.25
= 1.5
oH/L = 2
-o-H/L = 00
0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2
log (L/rw)
Figure III-9. Dimensionless flow parameter for consideration of vertical flow and anisotropy.
These curves apply for D/H > 2, i.e. effects of the far boundary are not
considered. For test interpretation when D/H <2, the same curves can be
applied if H is redefined as the distance from the lower confining layer to the top
of the straddle packer. (Widdowson et al., 1989).
52
-------
increased from 60 ft (10.3 m) to 600 ft (182.9 m), the increase in K is less than 5%. (For
equation (III-l) the increase in K would be by a factor of 2.3). The matter of K dependence on
re and Ss as studied by Chirlin (1989) deserves further investigation for the geometry of the
multilevel slug tests.
The influence of the confining layer boundaries is nearly negligible for the tests
considered here. For the tests completed adjacent to the boundary, P = 0.26 is found from
Figure III-8. For all the other tests, P = 0.27. When P = 0.27 the values of K calculated from
equations (III- 10) (method 3) and (III-l) (method 1) are related by
K(1) / P ln(re/rw)
= 0.63 K^ (III-ll)
Thus, the K(z) profile shapes calculated using equation (111-10) of method 3 are the same as
those calculated using equation (III-l). Thus methods 1 and 3 give proportional but not
identical results.
III-4 Results. Discussion and Conclusions
Well E6 slug data, Figure III-3, and the corresponding hydraulic conductivity
calculations, Table III-l, are summarized in the K(z) profiles shown in Figure III- 10. Based on
similar field tests and analyses, K(z) profiles for wells E7 and E3 are shown in Figures III-ll
and III- 12. For all three wells, the K(z) values calculated using methods 1 and 2 are very
similar. Of course, as discussed earlier, the magnitude of K(z) for method 1 is strongly
dependent on the choice of re. The profiles for method 3 are 63% of those calculated using
method 1 (with small deviation for those slug tests conducted adjacent to the confining layer
boundary).
Although method 3 is under further investigation, it is probable that the magnitudes of
the K values based on this method are improved approximations of the aquifer values compared
with the K values based on the horizontal, radial flow models. For a slug test in the
configuration of Figure III-2, it is obvious that the seepage to or from the aquifer test section is
not horizontal only. The vertical component of seepage contributes to the slug test response,
y=y(t), and the resulting K values are smaller than those based on radial flow interpretations.
Method 3 also allows one to consider the effects of anisotropy.
The similarity of profile shapes for the different methods of analysis is apparent in
Figures 111-10, III-ll, and III- 12. If an average of K(z) is calculated for each profile
K = (l/D)JK(z)dz (111-12)
0
where the trapezoidal rule is applied to approximate the integral (the values of K are shown in
Figure 111-10, III-ll, and 111-12), then nondimensional profiles can be generated. These profiles
of K(z)/K are shown in Figure 111-13, 111-14, and 111-15. The collapse to nearly identical
profiles, regardless of the method of data analysis, is evident.
If an average hydraulic conductivity, K, it available from full aquifer pumping tests,
multilevel tests like those described here could be used to develop K(z)/K profiles. Then, the
full aquifer K could be applied to generate the dimensional K(z) profile. For the
characterization of contaminant transport, this method of obtaining K(z) profiles would appear
53
-------
N
method 2 K=53.8 ft/day
-e- method 1 R=57.6
-4- method 3 R=36.4
300 400 500 600
K(z) (ft/day)
Figure III-10. K(z) profiles for the three different methods of analysis at well E6.
54
-------
1 I I I I I I I
method 2 K=281.5 ft/day
method 1 K=309.0
method 3 K=194.4
100 200 300 400 500 600 700 800
K(z) (ft/day)
Figure III-11. K(z) profiles for the three different methods of analysis at well E7.
55
-------
o
» | i i i i |J i i i I i i i i
method 2 K=306.8 ft/day^
method 1 R=318.2 j
method 3 K=200.1 :
400 500 600
K(z) (ft/day)
Figure III-12. K(z) profiles for the three different methods of analysis at well E3.
56
-------
N
method 2 K=53.8 ft/day
method 1 R=57.6
method 3 K=36.4
o
K(z)/K
Figure III-13. Dimensionless hydraulic conductivity profiles at well E6.
57
-------
I I I I I I I I I I I I I I I
method 2 K=281.5 ft/day
method 1 K~309.0
method 3 K= 194.4
K(z)/K
Figure 111-14. Dimensionless hydraulic conductivity profiles at well E7.
58
-------
' ' method' 2 R«3oi.8 ft/dayl
method 1 R-318.2 j
method 3 R«200.1 I
tititiiilti
2 3
K(z)/K
Figure III-15. Dimensionless hydraulic conductivity profiles at well E3.
59
-------
practical until more confidence in the magnitudes from the multilevel tests can be developed.
This point is made also in Chapter I dealing with the impeller meter test.
The K(z) profiles at wells E7 and E3 (Figure III-ll and 111-12) are quite similar. A full-
aquifer pumping test at well E7, using observation of drawdown at E7, resulted in K = 218
ft/day (66.4 m/day) which supports the validity of the slug tests and analysis completed for well
E7. Full aquifer pumping test data at E3 are not available. A full aquifer pumping test at E6
resulted in K = 123 ft/day (37.5 m/day). This value of K does suggest a heterogeneity of low
permeability, but is not so small as the K values calculated from the multilevel slug tests at well
E6.
Several colleagues have expressed serious reservations about the reliability of slug test
data. We are sure that the slug tests performed in wells having slotted screens at the Mobile
site are reasonably accurate. However, we were unable to perform slug tests in wells having
wire-wrapped screens because of vertical water leakage in the screen structure that could not be
prevented with packers. As discussed in Braester and Thunvik (1984), partially-penetrating slug
tests are very sensitive to cylindrical annuli of high or low permeability surrounding a well.
Thus test wells must be prepared and developed very carefully if they are going to be used for
slug tests. Gravel or sand packs must never be used. Tests in unscreened boreholes are
questionable because the surface of the formation can become coated with low permeability
materials. It must be admitted that all of these considerations make multi-level slug testing
much more problematical than impeller meter testing. However, if the formation permeability
is sufficiently low, an impeller meter can not be used because of stall speed problems, and
multilevel slug testing may be a viable alternative.
60
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CHAPTER IV
CHARACTERIZING FLOW PATHS AND PERMEABILITY DISTRIBUTIONS
IN FRACTURED ROCK AQUIFERS*
IV-1 Introduction
In chapters II and III, the impeller meter test and the multi-level slug test were
described as a means for measuring hydraulic conductivity distributions in the vertical. This
chapter deals with the application of another type of borehole flowmeter called the heat-pulse
flowmeter. This meter can be used as an alternative to an impeller meter in virtually any
application, and it has the advantage of greater sensitivity. Increased sensitivity is needed near
the bottom of almost any test well where flow velocities are small. Spinner flowmeters measure
a minimum velocity ranging from about 3 to 10 ft/min (1 to 3 m/min), which limits their
usefulness in many boreholes having slower water movement. Flow volumes of as much as 4
gal/min (15 1/min) may go undetected in a 4-in (10 cm) diameter borehole when flow is
measured with a spinner flowmeter, and much larger volumes may go undetected in larger
diameter holes. However, impeller flowmeters are available commercially and to the authors'
knowledge, heat-pulse flowmeters of the type described herein are not.
Analysis of data obtained with a heat-pulse flowmeter in granular aquifers is identical to
that applied to impeller meter data. Therefore, the present chapter will be devoted to the
application of flowmeters, particularly heat-pulse flowmeters, to fractured rock aquifers. Such
meters may be used to locate productive fracture zones and to characterize apparent hydraulic
conductivity distributions. Because flow from or to individual fractures is often small,
flowmeters more sensitive than impeller meters are commonly needed.
Several thermal, flow-measuring techniques have been developed or considered for the
measurement of slow borehole flow, including a thermal flowmeter described by Chapman and
Robinson (1962) and an evaluation of hot-wire and hot-film anemometers in liquids by Morrow
and Kline (1971). Dudgeon et al. (1975) reported the development of a heat-pulse flowmeter
for boreholes that used a minimal-energy thermal pulse in a tag-trace, travel-time technique.
This flowmeter had accurate slow-flow resolution and was only 1.63 in. (41 mm) in diameter, so
it could be used in small-diameter boreholes. Although the other thermal flowmeters
considered have not proved to be practical in a borehole environment, the commercial version
of the Dudgeon-style, heat-pulse flowmeter was determined to be viable even though the
instrument lacked features important for borehole use, such as seals which could withstand
water pressures to at least 10,000 ft (3,048 m), insensitivity to changes in logging cable resistance
and to stray electrical currents which exist in the ground, and integral centralizer (Hess, 1982).
The basic measurement principle of the USGS Meter is to create a thin horizontal disc
of heated water within the well screen at a known time and a known distance from two
thermocouple heat sensors, one above and one below the heating element. One then assumes
that the heat moves with the upward or downward water flow and records the time required for
the temperature peak to arrive at one of the heat sensors. The flow velocity is then given by
Material in this chapter was prepared by Alfred E. Hess and Frederick L. Paillet under
sponsorship of the Water Resources Division, U.S. Geological Survey, at the Denver
Federal Center, Denver, CO 80225.
61
-------
the known travel distance divided by the recorded travel time. Thermal buoyancy effects are
eliminated by raising the water temperature only a small fraction of a centigrade degree.
This chapter describes three case studies where flow measurements were used to provide
a quick survey of aquifer hydraulic response in fractured rock. This information then was made
available for subsequent studies, which markedly decreased the time that would have been
required to complete aquifer characterization by means of conventional hydraulic tests and
tracer studies.
IV-2 The U.S. Geological Survey's-Thermal Flowmeter
The urgent need for a reliable, slow-velocity flowmeter prompted the U.S.G.S. to develop
a small-diameter, sensitive, thermal flowmeter that would operate to depths of 10,000 ft (3,048
m) or more using 16,000 ft (5,000 m) or longer lengths of conventional four-conductor logging
cable (Figure IV-1). The thermal flowmeter developed by the U.S.G.S. has interchangeable
flow-sensors, 1.63 and 2.5 in. (41 and 64 mm) in diameters, and has flow sensitivity from 0.1 to
20 ft/min (0.03 to 6.1 m/min) in boreholes with diameters that range from 2 to 5 in. (50 to 125
mm). Vertical velocity in a borehole is measured with the thermal flowmeter by noting the
time-of-travel of the heat pulse and determining water velocity (or volume flow) from
calibration charts developed in the laboratory using a tube with a diameter similar to that of the
borehole under investigation (Hess, 1986).
After the thermal flowmeter was tested at several sites, the U.S.G.S. developed an
inflatable, flow-concentrating packer to decrease the measurement uncertainties caused by
geothermally induced convection currents within the borehole fluid and to increase flow
sensitivity in larger diameter holes. The thermal flowmeter and packer have been integrated
into a single probe that operates on logging lines having four or more conductors (Figure IV-2).
The packer system requires only a single conductor (plus the cable armor). The packer also can
be used with other borehole probes, such as spinner flowmeters and pressure transducers, whose
function would be enhanced with the use of an easily controlled packer (Hess, 1988). The
thermal flowmeter, with or without packers, was used to measure natural or artificially induced
flow distributions, or both, in boreholes with diameters ranging from 3 to 10 in. (75 to 250 mm),
at temperatures from 6 to 60°C, and in a variety of lithologies including basalt, dolomite, gneiss,
granite, limestone, sandstone, and shale. When the packer is used to direct all borehole flow
through the measurement section, measured thermal travel-times correlate with borehole
discharge, rather than average vertical velocity. With the flow-concentrating packer inflated, the
thermal flowmeter measures borehole flows in the range of 0.02 to 2 gal/min (0.04 to 8 L/min).
A representative flow calibration chart for the thermal flowmeter, with separate curves for
operation when the concentrating packer is inflated, deflated, or not installed is shown in Figure
IV-3. The inverse of the time-of-travel is used on the calibration chart for ease and accuracy of
reading the calibration curves (Hess, 1982).
The thermal flowmeter was used initially to define naturally occurring flows in boreholes.
However, the flowmeter has been used for additional applications, such as locating fractures that
produce water during aquifer tests and identifying flows induced in adjacent boreholes during
such tests. The capability of rapid measurement provided by the thermal flowmeter means that
a few hours of flowmeter measurements have the potential for saving many days or weeks of
investigation using conventional packer and tracer techniques.
62
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METRE
r-l.O
-0.9
-0.8
-0.7
-0.6
-0.5
-0.3
-0.2
-0.1
"—0.0
ELECTRONICS
SECTION
BOWSPRJNG
,CENTRALIZERS
__LJ-FLO« SENSOR
ROW SENSOR
CH»«T n£co»ot»
0
O
oo
/
(
\
O
O
oo
no* ioc
Figure IV-1. The U.S. Geological Survey's slow-velocity-sensitive thermal flowmeter (modified
from Hess, 1986).
63
-------
?=£T
3.0
2.S
20
•1.5
-1.0
•0.5
LO.O <-o.o
METESS
1.0
-0.9
-0.8
-0.7
-0.6
-O.S
-0.4
-0.3
ELECTRONIC
SECTION
FLOW SENSOR
WITH INFLATED
PACKER
PACKER
PUMP
Figure IV-2. The U.S. Geological Survey's thermal flowmeter with inflated flow-concentrating
packer (modified from Hess, 1988).
64
-------
or
Ul
0. 1
Ul -1
s
-2 -
-4=T
INFLATED PACKER
-15
3-
II
S.S 2
5CC
a.
1-
-0.8
-0.6
i
-0.4 -0.2 0.0 0.2
INVERSE RESPONSE TIME, IN HERTZ
0.8
0.6
0.4
0.2
0.0
•0.2
-0.4
-0.6
-0.8
z
S
-------
IV-2.1 Case Study 1-Fractured Dolomite in Northeastern Illinois
Acoustic-televiewer, caliper, single-point-resistance and flowmeter logs were obtained in
a 210 ft (64 m) deep borehole in northeastern Illinois as part of a study of contaminant
migration in fractured dolomite (Figure IV-4). The acoustic-televiewer log is a magnetically
orientated, television-like image of the borehole wall, which is produced with a short-range
sonar probe (Zemanek and others, 1970). Irregularities in the borehole wall, such as fracture
and vugular openings absorb or scatter the incident acoustic energy, and result in dark features
on the recorded image. Such televiewer logs may be used to determine the strike and dip of
observed features (Paillet and others, 1985). The acoustic-televiewer and caliper logs for
borehole DH-14 indicate a number of nearly horizontal fractures that seem to be associated
with bedding-lanes. The largest of these fractures are designated A, B, C, and D in Figure IV-
4. The caliper log indicates that the major planar features on the televiewer log are large
fractures or solution openings associated with substantial borehole diameter enlargements. The
large but irregular features on the televiewer log between fractures B and C also are associated
with borehole enlargements, but these are interpreted as vugular cavities within the dolomite
rather than fractures. The single-point-resistance log indicates abrupt shifts in resistance, at
depths of about 130 and 185 ft (40 and 56 m). These shifts may reflect differences in the
dissolved-solids concentration of the water in the borehole.
The pattern of vertical flow determined by the flowmeter measurements indicated the
probable origin for the inferred water-quality contrasts in the borehole (Figure IV-4). The
flowmeter log indicated downflow, which probably was associated with naturally occurring
hydraulic-head differences, causing water to enter at the uppermost fracture, A, and exit at
fracture B. A much smaller flow of water with the same electrical conductivity and dissolved-
solids concentration continued down the borehole to fracture C. At this fracture, the downflow
increased and the water inflow apparently contained a greater concentration of dissolved solids,
-,/hich accounts for the shift to greater electrical conductivity. This increased downflow exited
the borehole at fracture D, where there was another, somewhat smaller shift in single-point-
resistance. Although not rigorously proven from the geophysical logs, the second shift in
resistance appears to be associated with the dissolved-solids concentration of the water entering
at fracture C.
Subsequent water sampling confirmed that there were differences in the dissolved-solids
concentration of the water at the different depths. Sample analysis indicated that the water
entering at fracture A had a dissolved-solids concentration of about 750 mg/L; the water
entering at fracture C had a dissolved-solids concentration of about 1,800 mg/L. In this
instance, the geophysical data, especially the thermal-flowmeter data, were useful in planning
subsequent packer testing of the aquifer and in interpreting the results of water-quality
measurements. Identification of substantial natural differences in background water quality was
useful in planning for modeling of conservative-solute transport. At the same time,
measurements of vertical-velocity distributions in the borehole provided useful indications of
hydraulic-head differences between different depth intervals. This information could not be
obtained from conventional water-level measurements without the time-consuming installations
of packers at multiple levels in all of the boreholes at the site.
IV-2.2 Case Study 2-Fractured Gneiss in Southeastern New York
Conventional geophysical and televiewer logs were obtained in a 400 ft (123 m) deep
borehole completed in fractured gneiss at a site with contaminated ground water in southeastern
New York. This borehole was located about 200 ft (70 m) from Lake Mahopac. After a night
66
-------
CALIPER LOG
FLOWMETER LOG
ACOUSTIC
TELEVIEWER LOG DIAMETER, IN INCHES
5 6 7 8 9 10
RELATIVE SINGLE-POINT
RESISTANCE LOG
100
UJ
< 120
-------
of recovery from the effects of pumping nearby wells, the water level in the borehole appeared
to be slightly higher than the lake level, but the lake level generally was higher than the water
level in the borehole during the day. The acoustic-televiewer log indicated that fractures
intersected almost every depth interval of this borehole. Brine-solution tracing had indicated
that there was downflow within the borehole, but the locations of the fractures providing entry
and exit points for the downflow were uncertain.
Acoustic-televiewer and caliper logs for selected intervals of the borehole are shown in
Figure IV-5. The caliper log indicates several borehole enlargements at point A just below the
bottom of the casing and other enlargements, B and C, near the bottom of the borehole. The
selected intervals of the televiewer log indicate the large number of major fractures that could
be entry and exit points for flow in the borehole.
The flowmeter logs obtained in the borehole indicated both the entry and exit points of
the downflow (Figure IV-6). With just a few hours of flowmeter measurements, we learned that
the entry points of the downflow were the uppermost fractures and most of the inflow was from
fracture A. Consistent differences in the downflow indicated that about 20 percent of the flow
exited at fracture B and that the rest of the downflow exited at fracture C.
The flowmeter measurements also indicated a series of transient fluctuations in
downward flow in the borehole. These fluctuations were attributed to the effects of pumping in
nearby water-supply wells. For example, the downward flow between fractures A and B was
determined to vary from a maximum of about 0.7 gal/min (2.7 L/min) to a minimum of 0.4
gal/min (1.5 L/min). These transient changes occurred during periods ranging from a few
minutes to an hour and were accompanied by changes in the water level in the borehole. The
transient-flow changes probably represented the effects of pumping in one or more nearby
water-supply wells on the local hydraulic-head differences between individual fractures
intersected by the borehole. The transient changes in downflow may have been related to a
pattern of changing hydraulic-head differences between shallow and deep fractures during
pumping of nearby wells. Flowmeter measurements could not be made quickly enough to
characterize the transient changes in the upper part of the borehole. However, several fractures
just below fracture A in Figure IV-5 may have contributed flow during periods when the
changes affected the flow distribution in the borehole. This preliminary conclusion provides
information about possible fracture interconnections within the vicinity of fracture A that can be
tested during future tests.
These results enabled the hydrologists studying the contamination problem to infer local
flow conditions in the aquifer. The results of the flowmeter measurements provide useful
information about the extent and characteristics of the hydraulic-head differences between the
upper and lower permeable-fracture zones and the extent of interconnection between individual
fracture sets within those two zones. Of special interest in this situation is the small proportion
of the many large fractures indicated by the caliper log that actually produced or accepted flow
under ambient hydraulic-head conditions. This conclusion agrees with conclusions reached in
several other recent studies of fractured-rock aquifers (Paillet et al., 1987; Paillet and Hess,
1987).
IV-2.3 Case Study 3-Water Movement In and Around a Fracture Zone on the Canadian Shield
in Manitoba
This case study describes the flow in interconnected fractures inferred for an isolated
fracture zone on the southeastern margin of the Canadian Shield in Manitoba, Canada. Two
68
-------
CALIPERLOG
DIAMETER, IN
INCHES
5678
ACOUSTIC so
TELEVIEWER
LOG
B
• •*•*•%
4
100 -
i200
t
K
5 250
Q.
£
/
300 -
350 -
400
- A
"i
I
-
B
C
J
>
V
- 100
15
DIAMETER. IN
CENTIMETERS
20
Figure IV-5. Acoustic-televiewer and caliper logs for selected intervals in a borehole in
southeastern New York.
69
-------
ACOUSTIC
TELEVIEWER
LOG
VERTICAL FLOW, IN
GALLONS PER MINUTE
10C
LU
QC
Cf>
O
§ 200
UJ
CD
LU
UJ
U.
X
o,
LU
D 300
400 LJ
1.0 -0.5 0.0 0
DO\
. WITHOUT
-
I
I
ft/NFLOW
. » !
• " DRAWDOWN
i
WITH DRAWDOWN
P
*~ t • ••
I i
1
UPFLOW
I
5
0
25
50
75
00
UJ
3
LL
cc
eo
Q
z
3
UJ
m
cr
UJ
UJ
z
Q.
UJ
Q
-3.0 -2.0 -1.0 0.0 1.0
VERTICAL FLOW, IN
LITERS PER MINUTE
Figure IV-6. Profile of vertical flow in a borehole in southeastern New York, illustrating
downflow with and without drawdown in the upper fracture zone.
70
-------
boreholes spaced about 425 ft (130 m) apart intersected a fracture zone at a depth of about 870
ft (265 m). The nominal depths on the logs are somewhat greater than actual vertical depths
because the boreholes had been angled deliberately from the vertical by about 20 degrees. The
boreholes intersected almost no fractures except for those associated with the major zone
(Figure IV-7). The results indicate substantial permeability in the main fracture zone and in
several sets of fractures that appear to splay off from the main zone.
Flowmeter tests in these boreholes indicated that each borehole produced water from
the vicinity of the fracture zone during pumping, but at markedly different rates. In borehole
URL14, a pumping rate of only 0.07 gal/min (0.25 L/min) maintained a drawdown of more
than 260 ft (80 m). In borehole URL15, a pumping rate of 5 gal/min (19 L/min) resulted in
only 5 ft (1.5 m) of drawdown. All water production in borehole URL14 came from a minor
fracture, far below the major fracture zone, whereas all of the water production in borehole
URL 15 came from the lower one-half of the major fracture zone.
The hydraulic connection between the two boreholes was investigated by measuring the
flow in borehole URL15 while borehole URL14 was pumped. Flow was determined to enter
borehole URL15 at the main fracture zone at a depth of 880 ft (270 m) and then flow downhole
and exit at an apparently minor fracture about 50 ft (15 m) below. Flow entered borehole
URL14 at a minor fracture about 130 ft (40 m) below the main fracture zone (Figure IV-8).
Measured outflow from borehole URL15 was equal to measured inflow to URL14 to within the
measurement uncertainty associated with the thermal flowmeter. Projection of the exit and
entry fractures with respect to the plane connecting the two boreholes indicates that there was
no direct planar connection between the exit point in borehole URL15 and the entry point in
borehole URL14. This analysis indicates that the hydraulic connection between the boreholes
occurred by means of irregular fracture intersections beneath the main fracture. Although the
major fracture zone was the primary producer when borehole URL15 was pumped, that zone
produced no inflow in borehole URL14 when it was pumped.
The apparent small size of the fracture conveying the flow between boreholes URL14
and URL15 raises questions about why fractures located away from the main fracture zone
should provide the only hydraulic connection between the two boreholes. Other geophysical logs
provided additional information pertaining to how this connection may have been achieved.
Local stress concentrations inferred from borehole-wall breakouts identified on acoustic-
televiewer logs, and later confirmed by hydraulic-fracturing stress measurements, may have
caused local rock mass dilatency, accounting for the permeable pathway below the main fracture
zone.
IV-3 Conclusions
These case studies illustrate potential application of the thermal flowmeter in the
interpretation of slow flow in fractured aquifers. The relative ease and simplicity of thermal-
flowmeter measurements permits reconnaissance of naturally occurring flows prior to hydraulic
testing, and identification of transient pumping effects, which may occur during logging. In
making thermal flowmeter measurements, one needs to take advantage of those flows that occur
under natural hydraulic-head conditions as well as the flows that are induced by pumping or
injection. However, thermal-flowmeter measurements interfere with attempts to control
borehole conditions during testing, because the flowmeter and wire-line prevent isolation of
individual zones with packers. In spite of this limitation, the simplicity and rapidity of thermal-
flowmeter measurements constitute a valuable means by which to eliminate many possible
fracture interconnections and identify contaminant plume pathways during planning for much
71
-------
SOUTH BOREHOLE
URL15
BOREHOLE
URL14
NORTH
ESTIMATED FRACTURE
APERATURE. IN
INCHESxIO2
i.o
800
850
LU
o
tn
Q
LU
CD
ti
LU
LL
950
1000
FRACTURE
ZONE
0.0 0.2 0.4
240
250
260
270
280
290
LU
O
LL.
cc
ID
CO
Q
1
i
LU
m
CO
LT
LU
ti
300
310
0.0 0.2 0.4
JJ320
ESTIMATED FRACTURE
APERTURE, IN
MILLIMETERS
Figure IV-7. Distribution of fracture permeability in boreholes URL14 and URL15 in
southeastern Manitoba determined from acoustic-waveform and other
geophysical logs; fracture permeability is expressed as the aperture of a single,
planar fracture capable of transmitting an equivalent volume of flow.
72
-------
URL14
NORTH
200
O
UJ
tr
§
9
u.
z
t
UJ
Q
PROJECTIONS
OF FRACTURES
1000 -
1100-
Figure IV-8. Distribution of vertical flow measured in boreholes URL14 and URL15 in
southeastern Manitoba superimposed on the projection of fracture planes
identified using the acoustic televiewer.
73
-------
more time consuming packer and solute studies. The thermal flowmeter is especially useful at
sites similar to the site in northeastern Illinois, where boreholes are intersected by permeable
horizontal fractures or bedding planes. Under these conditions, naturally occurring hydraulic-
head differences between individual fracture zones are decreased greatly by the presence of
open boreholes at the study site. These hydraulic-head differences could only have been studied
by the expensive and time consuming process of closing off all connections between fracture
zones in all of the boreholes with packers. The simple and direct measurements of vertical
flows being caused by these hydraulic-head differences obtained with the thermal flowmeter
provided information pertaining to the relative size and vertical extent of naturally occurring
hydraulic-head differences in a few hours of measurement. Additional improvement of the
thermal-flowmeter/packer system and refinement of techniques for flowmeter interpretation
may decrease greatly the time and effort required to characterize fractured-rock aquifers by
means of conventional hydraulic testing.
While the case studies described in this chapter did not all involve contaminated
groundwater, the potential application to plume migration problems and sampling well screen
locating is obvious. The relationship of flowmeter measurements to more standard tests such as
caliper and televiewer logs is indicated also. Hopefully, thermal flowmeters and other sensitive
devices, such as the electromagnetic flowmeter being developed by the Tennessee Valley
Authority, will be available commercially in the near future. Additional information is available
in the proceedings discussed in the Preface (New Field Techniques for Quantifying the Physical
and Chemical Properties of Heterogeneous Aquifers).
74
-------
CHAPTER V
DEFINITION AND MEASUREMENT OF HYDRAULIC CONDUCTIVITY
IN THE VERTICAL DIRECTION*
V-l Introduction
Chapters II and III dealt primarily with the measurement of hydraulic conductivity in the
horizontal direction, while chapter IV was devoted to various types of borehole flowmeter
measurements in fractured rock aquifers. Problems in the subsurface involving predominantly
horizontal flow are common. Nevertheless, situations will often arise wherein one will need a
value for hydraulic conductivity in the vertical direction. In general, a value will be needed
whenever one wishes to determine how quickly contaminated water may move from one aquifer
to another located at a different elevation.
One obvious method for obtaining vertical K values is to perform laboratory
permeability tests on cores (Freeze and Cherry, 1979). This technique is well developed and
applied often, but suffers from the usual problems of sample disturbance and non-representative
sampling. In this chapter techniques will be presented that when successful will result in a
vertical K value averaged over a volume of subsurface material much larger than that contained
in a core. Of course, each of the techniques discussed has its own limitations and is not
applicable in all situations. For each technique, such limitations are identified and discussed.
It is also convenient in this chapter to review some of the basic definitions and principles
of Darcian flow in porous media. The remainder of this introduction will be devoted to that
endeavor.
Hydraulic conductivity is the constant of proportionality in Darcy's law,
V = - K} (V-l)
dx
where
K = hydraulic conductivity.
V = Darcy's velocity.
x = distance.
h = hydraulic head.
_ = hydraulic gradient.
dx
Hydraulic conductivity, which is sometimes called the coefficient of permeability, has
been shown to be related to the fluid properties and the permeability of the porous medium by
the following formula (Hubbert, 1940):
Material in this chapter has been taken from the Lawrence Berkeley Laboratory Report
No. LBL-15050 entitled "Field Determination of the Hydrologic Properties and
Parameters that Control the Vertical Component of Groundwater Movement" by Iraj
Javandel, prepared under sponsorship of the U.S. Nuclear Regulatory Commission.
75
-------
K-
M
where
k = specific or intrinsic permeability of the porous medium.
p = density of fluid.
H = dynamic viscosity of fluid.
g = gravitational acceleration.
Intrinsic permeability k, which is a function of mean grain diameter, grain size
distribution, sphericity, and roundness of the grains, is a measure of the ability of the medium to
transfer fluids, independent of the density and viscosity of any particular fluid.
The hydraulic conductivity of geological materials varies from approximately 1 to 10"13
m/s. This is a very wide range of variation. There are very few physical parameters that take
on values over 13 orders of magnitude (Freeze and Cherry, 1979). Values of hydraulic
conductivity of a geological formation can vary in space. This property of the medium is called
heterogeneity. They can also show variations with the direction of measurement at any given
point. This property is called anisotropy and is quite common in sedimentary rocks. In such
rocks, hydraulic conductivity along the layers is sometimes several orders of magnitude larger
than across the layers. This property becomes especially important in layered formations where
some thin layers of very low permeability appear within highly permeable sediments.
Anisotropy is also quite common in fractured rocks where aperture and spacing of joints varies
with direction.
As a result, in an anisotropic medium, hydraulic conductivity in its general form may be
represented by a 3x3 symmetric matrix. The components of fluid velocity in an anisotropic
medium may then be written by the following equations:
v = K dh v dh ^ dh rv ?^
vx " rSa -5— " rSty-JTT " IVxz ?cr V " '
OX
_ ™- dh v dh v dh ,-v ^
y ~ " ^x ~s~ ~ ^y*— " ^z 5— v v"-
(V-4)
dx y
The values of K in the above equations are components of the hydraulic conductivity matrix (or
tensor). It has been shown that an appropriate selection of coordinate system enables one to
diagonalize a symmetric matrix. The necessary and sufficient condition that allows such a
transformation is that the principal directions of anisotropy coincide with the x, y, and z
coordinate axes. If the system allows such a simplification, then the three components of flow
velocity may be represented by the following equations
v* = -K,-^ (V-5)
dx
vy . -S, » (V-6)
v, = -K, -2 (V-7)
dz
76
-------
where K^ Ky and K,. are principal values of hydraulic conductivity which are now in the direction
of x,y, and z. Therefore, depending on the media, the vertical velocity of groundwater
movement may be given by one of the two equations, (V-4) or (V-7). Equation (V-7) indicates
that the vertical component of groundwater motion is controlled by ^ alone. In cases where
vertical velocity is given by equation (V-4), values of hydraulic conductivity in other directions
are also required.
In this section some of the conventional, as opposed to multilevel, methods for
determination of in situ hydraulic conductivity in geological materials will be discussed.
Emphasis will be placed on the methods that lead also to determination of vertical hydraulic
conductivity, which is the main motivation for this chapter. Some methods which have been
recently developed for finding horizontal hydraulic conductivity in tight formations will also be
examined.
In general these tests may be divided into two categories: those which are performed in
a single well and those whose execution requires more than one well.
V-2 Single-Well Tests
V-2.1 Burns' Single-Well Test
Burns (1969) proposed a method of estimating vertical permeability of rocks. Following
is a modification of that method.
A) Purpose
The purpose of this test is to find in-situ vertical permeability of geological
material in the vicinity of the test well. Average horizontal permeability may also be estimated
by this method.
B) Procedure
This test can be performed with several alternative arrangements of down-hole
equipment. Two useful arrangements proposed by Burns are illustrated in Fig. V-l. Here the
procedure for the more simple test (Fig. V-1A) is described. For further detail the reader is
referred to Burns (1969).
A well is drilled into the zone of interest. Assuming the well is cased, the
annulus between the casing and the formation should be tightly cemented to
prevent any sort of vertical flow. Arnold and Paap (1979) have presented a
method for monitoring water flow behind a well casing. If the process of
cementing fills up the voids in the vicinity of the well it may cause an artificial
reduction of permeability. Then the casing and cement should be perforated at
least at two different intervals separated from each other by a few feet.
A packer is placed between these two perforated intervals to seal the hydraulic
connection between them from inside the casing. Care should be taken that the
change of pressure on one side does not transmit through the packer.
Installation of two packers with about half a foot distance may achieve this goal.
77
-------
1
GROUT
-f:"l
ROCK
j A ^
A 1 **
WIRE LINE
PACKER-)}.
1 M T~l 1 D 1 M C*
IN 1 UblNb
'"'•""I
'"/J
rKLboUKcJ
GAUGE ;/;{
- ** * 1
; « v e
PERFORATIONS
*•',•
'.-':'
•V.
- -
„" 'Q
»•'
.'_;;
*»».
'•".
:?.
£
«
c
L
j
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s
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i
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-i
X
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^ s
•.••V"'>"
'.•••f':\' :->J
,-JWELL :,v
BASING .}>;"
'::'.)''/' V:"^
:- *'Vt >%^
iiiFLOW •$
=
;llHERE /::
- " -Vf *" > v"" '
• 7r >•,' * -*v
.'|y^ SLIDING
:-JPACKER SLEEVE
r-'/ADAiiMn ^>
./AKUUIMU r-4
''\TI IRIMr;
• -,U U DIIMO ,r r
^- L> ' ' A /
>}•>::. :;)
,t-;:
=-//-v -v:
.^OBSERVE
^THERE 'v!{
J^i S?
' - •
:-'.•".'•
r "
:-;;:
BS
-',
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f.:-
',:,-
r1"
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X
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AND
OBSERVE
U C"DC
HLKL
IV."
F
'<-•.'•
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OBSERVE
LJ f DC"
n LKL
< j -<
^ ^ *
T''"
',"'>'
,%V.'
f- «,
OBSERVE
HERE
B
XBL 826-837
Figure V-l Downhole equipment arrangements for vertical well tests; A) single interval test,
and B) multiple interval test with sliding sleeve (after Burns, 1969).
78
-------
One pressure transducer is installed on each side of the packer. The ambient
pressure trend is monitored by both of the transducers for some period before
injection. To facilitate test data interpretation, the values of ambient pressure
should either remain constant or change linearly during the trend monitoring
period.
Start injecting into or pumping from the upper perforated zone. The rate of flow
should remain constant during this period. Flow rate should be monitored very
accurately.
Production or injection may be stopped after the pressure change recorded in the
lower part reaches at least 10 times the sensitivity of the gauge.
Recording of pressure at both intervals should continue throughout the producing
or injection period and afterwards for a period equal to at least 20 percent of the
elapsed flow period.
Caution: extra packers may be used to minimize the effect of well bore storage.
C) Theory
The theory behind this method rests on the derivation of pressure changes due to
a finite-length vertical line source in a homogeneous, anisotropic infinite aquifer bounded
between two impermeable confining layers. The solution of this problem has been given by
Hantush (1957), and Nisle (1958). See also Hantush (1964).
According to Hantush the change of hydraulic head or drawdown s(r,z,t) in a piezometer
having a depth of penetration z and being at a distance r from a steadily discharging well (with
infinitesimal diameter) that is screened between the penetration depths d and 1 in the
anisotropic aquifer of Fig. V-2 is given by
s = (Q/4ffKrb){W(ur) + f} (V-8)
where
« nfll nfld nflz
f = [2b/fl(l-d)] 2 1/n [sin _ - sin _ ] • cos _ •
n=l b b b
• w{ur , l*L(?Lf} , and ur = ^ (V-9)
X r W K, vb ' ' r 4K,bt
The two functions, W(ur) and W{ur , & ("IL )2 }, have been tabulated and
given by Hantush (1964). The solution presented by Burns is based on the more complicated
form which Nisle (1958) presented.
Introducing the following dimensionless parameters
bKJ
SD= _Z_ andtD=
Q Sr
79
-------
Ground surface
K.
1
Piezometer
XXAAAAA^
i
'XXXXXXXX
r
Figure V-2. Aquifer with a partially penetrating well.
XBL 826-838
80
-------
one can compute families of type curves of SD versus tD for different
dimensionless parameters such as _ , _ , _, and _I .
b b b b
D) Analysis of Field Data
Plot observed values of pressure versus time on rectangular coordinates.
Draw the best straight line through the pressure response measurements during
the trend-monitoring period, and extend it to the end of the flow period.
The difference between the measured pressures and the original trend, Ap = YS,
is determined as a function of time since initiation of flow.
Knowing dimensionless parameters such as £.,_,_. and_^L , a
b b b b
family of type curves (log-log plot of SD vs tD) is prepared from equations (V-8)
through (V-10) ,for different values of __L.
KT
Variation of s versus time is plotted on another log-log paper with the same scale
as the type curve plots.
The observed plot is then compared with the type curves.
Keeping the axes of the two plots parallel, find the position that the observed
plot matches best with one of the type curves.
V
Read the value of ^ and pick up a point on the top paper and
KT
identify the corresponding point right beneath that on the other plot. Read the
coordinates of the two points i.e. s,t,sD and tD.
Calculate the value of K, from the definition of SD, and K^ from
the ratio of ^ .
K,
The value of S may now be computed from equation (V-10).
E) Multiple Tests in the Same Well
Several tests may be performed over different portions of a formation in the same well.
In this case two or more packers may be used to isolate the testing portions of the well.
Multiple tests can sometimes determine whether the response is characteristic of the formation
or is a result of behind-casing leaks arising from poor cementing.
81
-------
F) Uncertainties
This test relies heavily on the assumption that the cementing behind the casing is
not leaking. The existence of cement leaks behind the casing could result in
abnormally high vertical permeability measurements. Sufficiently large values of
leakage behind the casing could cause almost equal response at the transducers
in the flow and measurement zones.
If the well has skin damage or if discontinuous shale barriers are locally present
in the tested interval, then the calculated vertical permeability would be lower
than the actual regional value.
Within low permeability materials, if proper instrumentation is not utilized, the
period of time required to reach a stabilized pressure before beginning the test
might be long. In this case linear extrapolation of test pressure trends might lead
to errors.
The value of the hydraulic conductivity calculated by this method corresponds to
a small volume of rock located in the vicinity of the testing zone.
V-2.2 Prats' Single-Well Test
Prats (1970) proposed a method for estimating in-situ vertical permeability of geological
materials which we shall describe here. This test requires injection or production at a constant
rate from a short perforated interval and measurement of the pressure response at another
perforated interval that is isolated from the first by a packer.
The purpose of this test is estimating the in-situ vertical permeability of materials in the
vicinity of a well. The test procedure is essentially the same as for the previously discussed
Burns' test (1969), but probably less accurate.
A) Procedure
Consider a single well with a casing cemented to the rock.
Perforate two small intervals into the casing in the zone to be tested.
Set a packer in the casing between the two perforations.
Set one pressure transducer close to each perforation and monitor changes in
pressure with time. As was discussed in the previous test, to avoid transfer of
pressure through the packer, more than two packers may be used for separation
of the flow and measurement zones.
After pressure is almost stabilized, inject into the formation with a constant rate
Q for some period of time until a reasonable amount of pressure response is
picked up by the transducer at the other perforation zone. In order to minimize
the time required for pressure to stabilize, isolate the injection and observation
zones from the rest of the well.
Stop injection and continue to monitor the change of pressure at both
transducers until the original ambient condition is almost reached.
82
-------
B) Theory
The supporting theory behind this method is based on the pressure response of a
confined homogeneous, anisotropic infinite aquifer due to a continuous point source. Thus, not
only is the well considered to be of zero radius, but the perforation length of both injection zone
and pressure measurement zone are also assumed to be vanishingly small. Based on these
assumptions, the pressure change at a point z due to the release of a constant rate of flow Q at
the point z', both located on the axis of the well, may be given by
ZD'-2n|
+« erfcf _ " ] erfcf _ < _ _ 1
n = -« |ZD - ZD' -2n| |ZD + Z,,' -2n|
where
erfc = complementary error function.
7 - Z
ZD - _
IT
D
b = aquifer thickness.
T - ^
Sb
Y = unit weight of fluid.
z = vertical distance of the point of measurement from the base of the aquifer.
z' = vertical distance of the point of injection from the base of the aquifer.
For large times, equation (V-ll) may be simplified to
AP = __ [^ .. + G(ZD, ZD')
.. ,
Sb |z-z I
where G(ZD, ZD') may be obtained from Table V-l.
C) Analysis of Field Data
Calculate pressure changes at the measuring interval with the same procedure
mentioned in the previous test.
Plot pressure changes at the measuring interval versus time on a semi-
logarithmic paper.
If the test was run long enough, the above curve should become a straight line at
large value of time. Measure the slope m of that portion as AP/cycle.
83
-------
Table V-1. Values of G(ZD,ZD')
oo
•is
ZD'
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.1
4.188
2.542
1.742
1.289
1.017
0.857
0.777
0.759
0.797
0.2
2.511
1.701
1.237
0.953
0.781
0.685
0.651
0.669
0.740
0.3
1.685
1.210
0.916
0.732
0.625
0.577
0.581
0.634
0.741
0.4
1.210
0.904
0.709
0.591
0.532
0.523
0.563
0.655
0.807
0.5
0.919
0.712
0.582
0.512
0.492
0.520
0.599
0.738
0.954
0.6
0.743
0.600
0.517
0.485
0.502
0.569
0.696
0.900
1.214
0.7
0.648
0.550
0.505
0.509
0.565
0.680
0.872
1.174
1.657
0.8
0.617
0.555
0.544
0.586
0.689
0.868
1.159
1.631
2.488
0.9
0.644
0.613
0.638
0.725
0.891
1.169
1.629
2.435
4.087
0.10
0.729
0.729
0.796
0.944
1.207
1.653
2.446
4.086
9.072
-------
Calculate the radial hydraulic conductivity
fromK, = i!2?
47fbm
Extrapolate the straight-line portion of the plot to a value of t = 1 hr, and read
the pressure change at that time. This pressure change will be denoted as AP(1).
Read the value of G(ZD,ZD') from Table V-l.
Determine Kj from the following formula:
AP(1)'
All dimensions in equation (V-13) are in SI units. Note that K^
can be calculated only when S is known.
D) Advantages and Limitations
The advantage of this method over the Burns' method is its simplicity in application. No
type curve is necessary and analysis may be carried out with a small calculator.
Major limitations are as follows:
The injection and measuring intervals must be short compared with the distance
between them, probably 10 percent or less.
If the distance between the injection (production) interval and the measuring
interval is relatively long and the net vertical permeability is low, the pressure
response may not be measured even in weeks. If this distance is relatively short,
then the assumptions of point recharge (discharge) and point measurement
become questionable.
The thickness of the aquifer and the coefficient of storage are assumed to be
known from other sources of information.
The method will probably produce representative results in sands containing shaly
streaks of limited extent, say not more than a few feet in length. But its
application is subject to question in the case of a reservoir with rather extended
lenses of shale which could have significant local but not regional effects on
vertical permeability.
The method is rather sensitive to variations in the mass rate of fluid injection
(production). The rate of flow is supposed to be constant.
The method can only give the horizontal and vertical hydraulic conductivity of the
materials immediately adjacent to .he well being tested.
V-2.3 Hirasaki's Single-Well Pulse Test
Hirasaki (1974) has proposed a pulse test technique for estimating in-situ vertical
permeability. The test consists of pumping or injecting a small interval of a well for a short
time, shutting in, and then measuring the time for the maximum pressure response to occur at
85
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another small interval of the well. This method has been used to estimate the vertical
permeability of a low-permeability zone in the Fahud field, Oman (Rijnders, 1973).
A) Purpose
The purpose of this technique is also to provide a simple means of estimating in-situ
vertical permeability of an aquifer in the vicinity of the testing location.
B) Procedure
Perforate the casing of the well over a short interval at the top of the aquifer just
beneath the confining layer.
Perforate another short interval at a distance z below the first interval.
Isolate these two intervals with a packer.
Pump water from or inject into the upper perforated interval for a short time
and measure pressure changes at the lower interval.
Stop pumping or injection and continue measuring pressure change at the lower
interval until the major part of the pulse test curve is obtained. Figure V-3
shows a typical curve which may be obtained from such a pulse test. Note that
the pumping or injection period should be short compared with the time required
to reach the maximum pressure response (e.g., less than 10 percent) in the lower
interval.
C) Theory
The theory of this technique rests on an approximation of the recovery equation for a
continuous point source in a homogeneous anisotropic medium. Consider a continuous point
source at z = 0 on the axis of the well, operating for a period t = tv as shown in Fig. V-4. The
pressure response of a semi-infinite medium (b is so large that the lower boundary is not
touched) to the source at the point z and at any time t>tj may be given by
PD=JL [ erfc (JEy- erfc
where
Z
b
. Si
Sb
86
-------
Pressure change
CIi
(X
CD
I
a.
I
/—v
3
a
CL
I
3
CD
X
03
9s
03
CO
CO
-------
XXXXXXXAA
FlOW
Packer
Measurement_±
point
Figure V-4. Sketch of the Hirasaki's test configuration.
XBL 826-840
88
-------
If tj is much less than t, then equation (V-14) may be approximated by
p . TJ_ exp - [ZDV4r]
JT r3/*
Equation (V-15) represents the pulse-pressure curve. The arrival time of the peak of this curve
may easily be obtained by setting its derivative equal to zero, which would give
T0= ZjL (V-16)
o
Substituting for T0 gives
K, = SbZP2 (V-17)
6t
o
As was mentioned above, equations (V-14) through (V-17) apply to a semi-infinite medium.
Two other cases have also been considered by this author. In one case the aquifer is considered
to be finite in thickness, which is treated by the introduction of a no flow condition at the lower
boundary. In the other case the lower boundary is assumed to remain at constant head. A
family of curves has been presented in Fig. V-5, which gives the variation of T0 versus ZD for all
three cases. It is interesting to note that the equation (V-17) holds for all three cases as long as
ZD < 0.6.
D) Analysis of Field Data
Plot the variation of AP versus time as measured at the lower interval. The same
precautions for measuring AP apply here as were discussed in previous methods.
If this curve shows a peak like that on Fig. V-3, then measure the time t0
corresponding to the maximum pressure response.
Modify the time t0 by subtracting half of the flow period tt
If the distance z between the upper and lower intervals is relatively short with
respect to the thickness of the aquifer (z<0.6b), then calculate ^ using equation
(V-17) by employing t0' instead of t0.
If the distance z is larger than 0.6b, then calculate ZD = z/b and determine the
type of lower boundary which best approximates field conditions.
Determine the value of T0 from the appropriate curve of Fig. V-5.
Calculate vertical hydraulic conductivity ^ from the following equation:
Here again it is assumed that the specific storage Ss is known from other
information.
89
-------
0.20
0.16 -
0.12 -
0.08 -
0.04 -
0.3
0.9
D
XBL 826-841
Figure V-5. Dimensionless response time for pulse test; A) for semi-infinite case, B) for a
finite thickness layer with an impermeable lower boundary, and C) for a finite
thickness layer and constant head at the lower boundary, (modified from
Hirasaki, 1974).
90
-------
E) Uncertainties
This test is based on the assumption that the period of injection or pumping is
almost negligible in comparison with the time to reach the maximum pressure
response.
Possible leaks behind the casing lead to erroneously high values of vertical
hydraulic conductivity.
The hydraulic conductivity measured by this method is representative of materials
very close to the well.
V-2.4 Bredehoeft-Papadopulos Single-Well Test
Bredehoeft and Papadopulos (1980) have proposed a method of measuring permeability
which is a modification of the conventional slug test. Although their method is designed for
measuring horizontal rather than vertical hydraulic conductivity, we shall discuss it here because
(1) the conventional methods for measuring vertical hydraulic conductivity in tight formations
are associated with uncertainties, and the value of horizontal hydraulic conductivity could give
an upper limit for the vertical component provided that major vertical fractures are absent, and
(2) as we saw before, in some cases in addition to the vertical value one also needs horizontal
hydraulic conductivity to evaluate the vertical component of fluid flow.
A) Purpose
The purpose of this test is to measure in-situ horizontal hydraulic conductivity of so
called "tight formations", such as tightly compacted clays, rock units in which fractures, if they
exist, are essentially closed or filled, or matrix rock between fractures.
B) Procedure
Figure V-6 depicts set-ups for the test in (a) an unconsolidated formation and (b) a
consolidated formation. Depending on the time elapsed since the well has been drilled, the
water level in the hole may or may not have stabilized to the ambient hydraulic head at the
interval to be tested. To start the test, the test system is filled with water and, after a period of
observing the water level for ambient conditions, the test interval is suddenly pressurized by
injecting an additional amount of water with a high pressure pump. The test interval is then
shut-in, and the head change H0 caused by the pressurization is allowed to decay. As water
slowly penetrates into the formation, H0 will drop. The variation of H0 with time is recorded.
C) Theory
Apart from the conventional initial-boundary value formulation which is generally
adopted for simple radial flow in a confined and infinitely long aquifer, one specific constraint
used in this development is as follows. The driving force governing the movement of water
from the well into the formation is the expansion that the water stored within the pressurized
system undergoes as the head, or the pressure within the system, declines. Thus, the rate at
which water flows from the well is equal to the rate of expansion. In a conventional slug test
the water flow into the formation comes directly from the volume of stored water in the system
under normal hydrostatic pressure. The solution for the modified slug test has been presented
in the form
91
-------
Valve
Valve
Pressure Gage
sat
Pressure
System Riled
with Watery
— ? — ?— ? — ? — ?
irvinnnnn.-ir:
• — — • _i
_?_?_?_?_?_
-Casing
Land Surf ace-
Initial Head
Open Hole
I -_-3
"i. -T: —_ H
_r
--_-_-_-_-_-_-_-_-_-i
h-
1-
t-:
••.r-
•1T.
• »—
•:^
. ^
"be Tested -----
--^-•ir-r-r-r-_-~
tv^riririririr;
~ ~- --- -Well Poinl.- ~_ ----- _- -
(a)
^ System Riled
with Water
-Pump
(b)
Figure V-6. Possible arrangements for conducting pressurized test (a) in unconsoh'dated
formations and (b) in consolidated formations, (after Bredehoeft and
Papadopulos, 1980).
92
-------
5- = F(a,/0 (V-20)
Wo
where H0 and H are values of head measurement in the hole at the time of shut-in and
following that with respect to the background head, respectively, a and ft are given by
a =
VwCwpwg
7rTt
(V-22)
wwpwg
where
rs = radius of well in the tested interval.
t = time.
S = storage coefficient of the tested interval.
Vw = volume of water within the pressurized section of the system.
Cw = compressibility of water.
pw = density of water.
T = transmissivity of the tested interval.
g = gravitational acceleration.
Tables of the function F(a,/f) for a large range of variation of a and ft are given by the
above authors as well as Cooper et al. [1967] and Papadopulos et al. [1973].
Major assumptions applied in development of this method are as follows:
Flow in the tested interval is radial, which will also imply that the flow at any
distance from the well is limited to the radial zone defined by the tested interval.
Hydraulic properties of the formation remain constant throughout the test.
The casing and the formation on the side of the borehole containing the water
are rigid and do not expand or contract during the test.
Before the system is pressurized, the water level in the well has come to a near
equilibrium condition with the aquifer.
D) Analysis of Field Data
Bredehoeft and Papadopulos have proposed two different techniques, one for a < 0.1
and the other for a > 0.1. If a < 0.1 the following steps should be taken.
Prepare a family of type curves, one for each a, of ¥(a,ft) against ft on
semilogarithmic paper. A table giving the value of F(a,/3) as a function of a and
ft is presented by Bredehoeft and Papadopulos (1980).
93
-------
Plot observed values of H/H0 versus time t on another semilog paper of the
same scale as the type curves.
Match the observed curve with one of the type curves keeping the ft and taxes
coincident and moving the plots horizontally.
Note the value of a of the matched type curve, and the values of /J and t from
the match point.
Calculate values of S and T from the definitions of a and ft given by equations
(V-21) and (V-22).
The above method is not suitable for a > 0.1. In this range of a, this method can only
give the product of transmissivity and storage coefficient, TS. This product may be calculated by
matching the field curve of H/H0 versus time t with a type curve family of F(a,/}) versus the
product a/8 (Fig. V-7).
E) Merits of the Method
As the authors have shown in one example, a conventional slug test in a formation with
hydraulic conductivity of K = 10"12 m/s may last more than one year whereas the modified slug
test method as discussed here may take only a few hours.
F) Uncertainties
The major assumption employed in this method is that "volumetric changes due
to expansion and contraction of other components of the system are negligible."
In other words, expansion of the pipes and contraction of the rock in the test
zone is negligible relative to that of water. This assumption may introduce large
errors into the calculation of hydraulic conductivity. Neuzil (1982) has referred
to a test in which the compressibility in the shut-in well was approximately six
times larger than the compressibility of water.
The other major assumption which was employed in this method was that before
the system was pressurized, either the water level in the well had come to a near
equilibrium condition with the aquifer or that the observed trend could be
extended throughout the test. Neuzil (1982) has pointed out that this assumption
may also lead to erroneous results. He argues that the pressure changes due to
nonequilibrium conditions before shut-in become much more rapid after the well
is pressurized. Neuzil (1982) has proposed the following modifications in the
setup and procedure for performing the test.
Modify the test equipment to that shown on Fig. V-8.
Fill the borehole with water and set two packers near each other.
Set up two pressure transducers as shown in the figure.
Close the valve, shutting in the test section, and monitor the pressures in both
sections until they are changing very slowly.
94
-------
10 10* 10> 10* 10»
Figure V-7. Type curves of the function F(cr,/J) against the product parameter aft, (after,
Bredehoeft and Papadopulos, 1980).
95
-------
rump~ — : — - —
Remotely
Operated *
Valve
t
^
s
.
t
0
^
r
r
\
-•
)'l
•
_
X
/
r
(«•
\
/
3
10 Kecoraers
— Ground surface
Water level
Packer
Transducer
Packer
Transducer
Figure V-8. Arrangement of the borehole instrumentation as suggested by Neuzil (1982).
96
-------
Open the valve, pressurize the test section by pumping in a known volume of
water, and reclose the valve.
Measure the net pressure decay (slug) by subtracting the decline due to transient
flow prior to the test from the measured total pressure.
Analyze data using the technique prepared by Bredehoeft and Papadopulos
(1980) as was mentioned before, except that the term for the compressibility of
water cw is replaced by the ratio c, defined as
c = (AV/v)/AP (V-23)
where v is the volume of the shut-in section, and Av is the volume of water added
to generate a pressure change of Ap. Neuzil (1982) indicates that a rise in
pressure measured by the transducer between the two packers may indicate
leakage upward from the test section. However, two other phenomena may
cause some rise of pressure in the middle section. One is increase of pressure
inside the formation adjacent to the test section, which may or may not be
significant. The other reason is the possibility of transfer of pressure by the
packer itself, from the test section to the middle section.
V-2.5 General Comments About Single-Well Tests
The following problems are inherent in all single well tests.
The hydraulic conductivity measured by these tests is only representative of a
small zone around the testing interval. A thin lens of very small permeability
located between injection and measuring zones could lead to an erroneously low
vertical hydraulic conductivity, even if it is only locally present. This problem
may be overcome by conducting several tests within the total thickness of a given
formation. However, the lateral variation of vertical hydraulic conductivity could
be another problem which requires either other types of testing or performance
of a number of single-well tests.
Because the horizontal permeability of sedimentary materials is usually much
larger than the vertical permeability, flow lines generated by either injection or
pumping in these tests are predominantly horizontal. Therefore, a long time may
be required to have significant pressure disturbances in measuring intervals
located vertically above or below the flow zone. A small pressure change
together with the possibility of leakage behind the casing due to poor cementing
will result in an increased degree of uncertainty in the credibility of these tests in
tight formations.
Measurement of change of pressure due to pumping or injection in single-well
tests is another source of uncertainty. This is because the test may often start
before the pressure at the measuring interval has stabilized. One way to handle
this problem is to minimize the volume of the measurement cavity in the well
with the help of extra packers. This will shorten the time required for pressure
stabilization.
97
-------
In a single-well test, injection is preferred over pumping unless the well will flow
without artificial lift (Earlougher, 1980). In a tight formation, indeed, injection is
the only feasible way to test.
The injection or pumping zone should be packed off to minimize well bore
storage.
V-3 Tests with Two or More Wells
Tests involving two or more wells measure the response of a much larger volume of
rocks than tests from a single well. Therefore, the value of hydraulic conductivity obtained from
multiple well tests is usually more representative of the large scale behavior of the formation.
The only problem with these tests is that they cannot be directly used within the formation of
interest, once the permeability of that formation becomes very low. Wells completed in very
low permeability materials are unable to produce fluid for the required test period. Fluid could
be injected in these wells; however, it may take years before any useful response can be
measured in observation wells at a distance of 5 to 10 m.
In the following discussions readers are assumed to be familiar with general pump test
design and operation. For more information on this subject readers are referred to Stallman
(1971).
V-3.1 Weeks' Method
Weeks (1964) proposed a method of calculating vertical hydraulic conductivity of higher
conductivity aquifers. A brief description of his method is given here.
A) Purpose
The purpose of this method is to determine in-situ vertical and horizontal hydraulic
conductivity of anisotropic aquifers.
B) Procedure
Consider a pumping well which is only partially penetrating an anisotropic
aquifer. The well is open to the aquifer over a length of (1-d), (see Fig. V-2).
Also consider one or more piezometers at distances ri( from the axis of the
pumping well, such that each rt is smaller than half of the thickness of the
aquifer.
Pump the well with a constant rate of discharge Q, for a period of time.
Measure water level variations in the piezometers and record these variations
against the time of measurement.
The solution for the drawdown around a partially penetrating well in an anisotropic
aquifer has been given by Hantush (1957, 1964).
98
-------
s = (Q/47TKrb){w(u) + f} (V-8)
where
*} (v-9)
This equation was presented in a slightly different context in section V-2.1.
Hantush (1961) has given another form for f which is valid at large values of time.
Weeks (1964) has modified this solution for anisotropic aquifers.
f = JL_ Z 1 (sin ! - sin1 )cosz. ^. 1/2] (V-24)
Tr(l-d) n=lnb b b b K,
where K,, is a modified Bessel function of the second kind and zero order. Equation (V-24) is
valid for large values of time when
ur < (JLr)2 *L (V-25)
r v b' 20K,
or t > Ji£ (V-26)
2K,
Let us introduce the following dimensionless terms:
SD = *s (V-27)
tD = -LL (V-28)
r*S
where T and S are transmissivity and storage coefficient of the aquifer, respectively.
Given the geometry of the system, one can calculate r/b, z/b, 1/b, and d/b. Assuming
different values for K^/Kp a family of type curves showing the variation of SD against tD can be
prepared for the above known dimensionless parameters.
One may have noticed that the methods proposed by Weeks and Burns are both based
on the same theory. Burns' method applies the theory to a single well, and Weeks' method
applies it to multiple wells. Saad (1967) and Weeks (1969) have proposed other methods for
calculating the ratio of horizontal to vertical permeability in aquifers. Both of those methods
are also based on the theory of the Partially Penetrating Wells which was discussed above.
D) Analysis of Field Data
Plot SD versus tD calculated from equation (V-8), (V-9), (V-27), and (V-28) for
the dimensionless parameters of the system and for different values of Kj/K,. on
log-log paper. Note that equation (V-24) is independent of time. Therefore, it is
much simpler to use equation (V-24) in place of equation (V-9) for those times
99
-------
i_o
when t > . This means that the order of magnitude of S and
j
KZ should be estimated in advance. An extensive table evaluating equation (V-8)
for the simple case of d=0 and 1^/1^=1 is given by Witherspoon et al. (1967),
which could be easily modified for the case of d?*0 and an anisotropic medium.
Plot values of drawdown versus time as measured by each piezometer on another
log-log paper with the same scale as the type curves.
Using the superposition technique, find the best match between the observed
data and one of the type curves.
When the best match is achieved read the ^/K,. corresponding to the type curve
and the coordinates of a match point on both graphs.
Calculate the radial hydraulic conductivity and the storage coefficient of the
aquifer from the following equations.
S = J±±_ (V-30)
tor2
where s, t, tD and SD are coordinates of the match point.
Calculate the vertical hydraulic conductivity of the aquifer from
V — ( ^^i\V f\T 11\
"T. ~ \—~)^r (.""^-U
V-3.2 Tests Based on the Theory of Leaky Aquifers
The term leaky aquifer generally refers to a system in which an aquifer is overlain
and/or underlain by much less permeable layers. Once the pressure in the aquifer drops while
being pumped, water from saturated less permeable layers lying above or below leaks into the
aquifer. Sometimes the amount of leakage is so great that its effect can be detected in the
aquifer being pumped. In this case the confining beds are called "aquitards" and the aquifer is
referred to as being 'leaky'. When the amount of leakage is so little that its effect cannot be
easily detected in the aquifer, then the confining beds are called "aquicludes" and the aquifer is
termed 'slightly leaky' (Neuman and Witherspoon, 1968).
Much work has been done on the theory of leaky aquifers. The first group of papers
appeared before 1960 (Jacob, 1946, Hantush and Jacob, 1955, Hantush, 1956) and were based
on the assumption that the storage capacity of the aquitard was negligible. Later, Hantush
(1960a,b) introduced a new solution for leaky aquifers in which he had considered the effect of
storage capacity on the confining bed. Neuman and Witherspoon (1969, 1972) evaluated the
significance of the assumptions applied in the earlier work and provided more generalized
solutions. A brief description of these methods will be given in the following sections.
One may ask what the relation is between leaky aquifers and the subject of field
determination of vertical hydraulic conductivity. Why should we study the leaky aquifer pump
100
-------
test techniques? As we shall see later, all of the leaky aquifer solutions which are discussed
here are based on the assumption that the flow in the less permeable layer, above or below an
aquifer, is essentially vertical. Therefore, application of these methods should give an overall
vertical hydraulic conductivity for the confining layer.
A) Hantush and Jacob Solution
Jacob (1946) developed a partial differential equation for a leaky aquifer and solved it
for a bounded reservoir. Hantush and Jacob (1955) solved the same problem for a radially
infinite aquifer. Because of its simplicity, in spite of the fact that in some cases it leads to
erroneous results, these solutions have been widely used by groundwater hydrologists.
Purpose
The purpose of this section is to evaluate the possibility of determining the vertical
hydraulic conductivity of the confining layer and discuss the assumptions and limitations
encompassing the method of approach.
Procedure
The procedure for conducting the test is similar to that for a standard pump test within
a simple aquifer. From such a test one obtains a table of observed drawdown in an observation
well or a piezometer against the time elapsed from the start of pumping.
Figure V-9 depicts the arrangement of the system to be studied. A semi-permeable
layer (aquitard) with a constant thickness of b' is overlying an aquifer with much higher
hydraulic conductivity. The aquitard is overlain by another highly permeable extensive aquifer.
The lower aquifer is being pumped with a constant rate of discharge Q. Hantush and Jacob
(1955) obtained an expression which gives the drawdown distribution in the pumped aquifer as a
function of time. Derivation of this solution was based on the following major assumptions: (1)
flow is essentially horizontal in the aquifer and vertical in the aquitard, (2) no drawdown is
permitted in the upper aquifer because of pumping in the lower aquifer, (3) leakage into the
pumped aquifer is proportional to the potential drop across the aquitard. This last assumption
is equivalent to assuming that the storage capacity of the confining bed is negligible, and all the
water leaking into the pumped aquifer comes directly from the upper aquifer; thus the aquitard
behaves only as a conduit between the two aquifers. The solution to this problem as given by
Hantush and Jacob (1955), sometimes referred to as the (r/B solution) is
where
u =
4tK
B = _—_, called the leakage factor.
JC
K, K' = hydraulic conductivity of the aquifer and aquitard, respectively.
101
-------
Q
Source layer
Aquitard
/
K
Aquifer
K
,/VWS<
'vxxxx
XBL 826-842
Figure V-9. Leaky aquifer with a constant head boundary at the top of the aquitard.
102
-------
Ss = specific storage.
s = drawdown in the aquifer.
b, b' = thickness of the aquifer and the aquitard, respectively.
W(u,r/B)=
u 4yB y
This last term is called the well function of leaky aquifers and has been extensively tabulated
(Hantush, 1956).
Analysis of Field Test Data
Several methods based on the r/B solution are conventionally used for interpretation of
leaky aquifer pump test data. Here we shall discuss two of these methods.
Walton's Type-Curve Method (1960^
Prepare a family of type curves by plotting on a log-log paper the values of the
function W(u,r/B) versus 1/u with r/B as the running parameter of the curves.
Note that the curve with r/B = 0 is the Theis curve.
Plot the drawdowns versus time as were recorded within an observation well
(after appropriate adjustments) on another log-log paper with the same scale as
that used for the type curves.
Follow the regular procedure for curve matching1 and read the appropriate value
of r/B by interpolating the position of the data curve among the type curves.
Also read the dual coordinates of the matching point, s,t,l/u, and W(u,r/B).
Calculate the hydraulic conductivity of the pumped aquifer from
K = Q W(u,r/B) (V-33)
4?rbs
Calculate the specific storage of the pumped aquifer from
SS=_J!L_ (V-34)
r'Cl/u)
Finally, calculate the vertical conductivity of the aquitard from
K' = 2*' (_L)2 . -(V-35)
r2 B
1 A unique fitting position is difficult to obtain unless sufficient data are available from the
period when the leakage effect is insignificant (Hantush, 1964).
103
-------
U.S.B.R. Method
U.S. Bureau of Reclamation (1977) has published a groundwater manual as a guide for
field personnel in groundwater investigation. Following is the method which that manual
suggests for interpretation of pump test data of a leaky aquifer. Fig. V-10 shows a family of
type curves prepared from Jacob's leaky aquifer solution (1946). As was discussed before,
Jacob's solution was developed for a radially bounded aquifer. However, in developing Fig. V-
10 the outer boundary was located at a sufficient distance that the effect of pumping never
reached it (Glover, Moody, and Tapp, 1960). This approach permits the curves to be used for
infinite aquifers. The steps to be used in applying the USER method are as follows:
Drawdown versus time from two or more observation wells (after appropriate
corrections) located at different radial distances r from the pumped well should
be plotted on a log-log paper with the same scale as Fig. V-10.
Superimpose the field curve with those of Fig. V-10.
After obtaining the best match read the dual coordinates of a match point (s,t, u
and rj), and the x value of the best fitting type curve. Interpolation may be
required to find the x value.
Calculate the hydraulic conductivity of the aquifer from
K - .. (V-36)
2wMs
Calculate the hydraulic conductivity of the aquitard from
K' = KMMY.L)2 (V-37)
r
Finally, calculate the storage coefficient of the aquifer from
S » El (V-38)
In the above equations M and M' indicate the thickness of the aquifer and the aquitard,
respectively. The ratio r/x is the leakage factor B used in the development of the theory. The
definitions of other terms are given in Fig. V-10.
The following is a quotation from the U.S.B.R. staff on the interpretation of leaky
aquifer pump test data from the Missouri river basin project (Glover, Moody, and Tapp, 1960,
p. 175).
"When drawdown data from well tests are compared with
drawdown curves computed for idealized conditions a lack of
perfect agreement is generally evident".
Other methods of analysis of field data based on r/B solution have been suggested by Hantush
(1964, p. 416-417), and Narasimhan (1968).
104
-------
I
7-i
_ CHART SHOWING IN DIMENSIONLESS
— FORM THE TIME-DRAWDOWN RELATIONS
_ FOR A WELL DISCHARGING AT A STEADY .
RATE FROM AH INFINITE AQUIFER J-
- OVERLAIN BY A BED OF LOW PERMEABILITY
I | | III I III -4-14
Figure V-10. Leaky aquifer type curves based on r/B approach (USER, 1977).
105
-------
Uncertainties
The problem of flow to a pumped well in a hydrologic system consisting of several
aquifers separated by less pervious aquitards or aquicludes is in fact three dimensional. A
rigorous approach to the solution of such a problem is analytically intractable. Therefore, it has
been customary to simplify the problem by assuming that flow is essentially horizontal in the
aquifers and vertical in the aquitards and aquicludes. The validity of this assumption which was
used in the derivation of the r/B solution, was evaluated by Neuman and Witherspoon (1969).
They noted that the errors introduced by this assumption are less than 5 percent provided that
the conductivities of the aquifers are more than three orders of magnitude greater than that of
the aquitards. These errors increase with time and decrease with radial distance from the
pumping well. One should note that the 5 percent error given by Neuman and Witherspoon
(1969) is the percentage difference between drawdowns calculated by the analytic solution based
on the above assumption and drawdowns obtained by a finite-element numerical analysis
without that assumption. The magnitude of the error which may result in the calculation of the
hydraulic properties of the confining layer is not known.
Another assumption used in the derivation of the r/B solution is that no water is
released from storage in the aquitard. Neuman and Witherspoon (1969) have found that this
assumption tends to result in overestimating the permeability of the aquifer and underestimating
the permeability of the aquitard.
An important uncertainty about the r/B solution is that it does not provide a means of
distinguishing whether the leaking bed lies above or below the aquifer being pumped. In the
case of leakage from both above and below the aquifer, this method does not provide a means
for determining conductivities of individual aquitards. This becomes particularly important
when one is looking for the hydraulic conductivity of a certain confining bed rather than that of
the aquifer itself.
When the hydraulic conductivity of the confining bed becomes so small that the ratio of
K'/K tends to zero, the drawdown distribution in the aquifer becomes essentially the same as
would be predicted by the Theis solution for an aquifer without leakage. As a result, techniques
based on observation in the aquifer alone fail to give the properties of the confining bed.
B) Hantush Modified Solution
In 1960 Hantush published another paper in which he introduced a new treatment of
leaky aquifers which overcame some of the difficulties of the r/B solution.
Purpose
The Hantush modified solution provides a more accurate approach to the evaluation of
the vertical hydraulic conductivity of less permeable layers which confine permeable aquifers.
Procedure
The test procedure again follows the same steps as a regular pump test. Needed for
interpretation is a record of drawdown versus time in one or more observation wells around a
pumping well.
106
-------
Theory
In this development, in addition to assigning a storage capacity to the confining aquitard,
Hantush (1960) solved the problem for two different cases: (1) an infinite horizontal aquifer
overlain by an aquitard whose upper boundary does not experience any change in drawdown,
and (2) the same situation but with an impermeable bed overlying the aquitard. Other
assumptions applied in the development of the r/B solution, including vertical flow in the
aquitard and horizontal flow in the aquifer, still hold. In this solution Hantush considered
leakage into the aquifer from both above and below. He presented the solutions for two ranges
of time t as indicated below.
Solutions for Small Values of Time
For t less than both b'S'/10K' and b"S"/10K", the solution for both cases is the same and
is given by
S = S— H(u,j3) (V-39)
where
H(u,0) = ^
(rX)/4
IK s?+ /K" JT
/jKbb'lT N/KbbII~S
u
4tbK
s = drawdown in the aquifer.
S",S' = storage coefficient of the lower and upper aquitards, respectively.
K",K' = hydraulic conductivity of the lower and upper aquitards,
respectively.
r = radial distance of the observation well from the pumped well.
b",b' = thickness of aquitards below and above the aquifer, respectively.
H(u,/f) has been extensively tabulated (Hantush, 1960b). A short table of H(u,/3) is also
available (Hantush, 1964).
Solution for Large Values of Time
Case 1.
In this case, t should be larger than both 5b'S'/K' and 5b"S"/K". The solution is
then given by
107
-------
s = _2_W(u6.,a) (V-40)
47TbK l 1 } ^ '
where
W(u61; a) = f 2L exp(.y - *.)
u5l y 4y
is the well function for leaky aquifers which is tabulated by Hantush (1956);
Me
a = r / —
N/ bb'.
S1 = 1 +
L K"
K ' bW
S' + S"
3S
The other terms are the same as defined before.
Case 2.
For t greater than both 10b'S'/K' and 10b"S"/K" the expression for drawdown in
the aquifer is
s = Q W(u62) (V-41)
4?rKb V 2' ^ '
where
m -y
W(u62) = J — dy is the well function
u62 y
X, - S> + S" 4. 1
2 S
At this point, before describing the method of interpreting the pump test data, the
applicability of the different operations given above will be reviewed. For large values of time,
equation (V-40) indicates that, even when one considers the storage capacity of the confining
bed, the r/B solution could be safely used for evaluation of the aquifer and aquitard, provided
5b'S'
that t > This solution may qualify at relatively small values of time
when the aquitard is thin, when it has a relatively high hydraulic conductivity and is
incompressible (i.e. very small S'). For example, if b' = 5m, K' = 2xlO"7 m/s, and S' = 2xlO"5,
then the r/B solution is applicable after 2500 seconds, or approximately 41 minutes after the
start of the test. In applying the simpler r/B solution, note that u should be replaced by
S'
u(l + ). Also, the aquifer above the aquitard should not show any drawdown during the test.
If the overlying aquifer does show some drawdown, then the r/B solution tends to
underestimate the hydraulic conductivity of the aquitard. On the other hand, if the confining
bed is relatively thick and elastic with low hydraulic conductivity, then the r/B solution is not
applicable. For example, if b' = 50m, K' = 5xlO"9 m/s and S' = 10"3, then the r/B solution is
only applicable after 5xl07 seconds, or approximately 1.5 years after the test has started.
Equation (V-41) suggests that when the confining bed is thin, relatively permeable, and
incompressible, and overlain by an impermeable layer which cannot supply water, the drawdown
108
-------
data in the aquifer will follow the Theis solution at relatively small values of time. In applying
S'
the Theis solution, note that u should be replaced by u(l + — ).
O
Equation (V-39) is the solution for small values of time. It can also be applied to
relatively large values of time when the aquitard is thick, relatively impermeable and
compressible. For example, if b' = 100m, K' = 10'9 m/s, and S' = 10 , then equation (V-39) is
applicable for 107 seconds or the first 115 days of the test. Note that within this range of time
the effect of pumping would not reach the upper boundary of the aquitard. Therefore, the
assumption of a constant head boundary there does not introduce any error. The above
discussion was made only with reference to the upper confining bed. In each case, however,
both the upper and lower beds must meet the same criteria for these simplifications to apply.
Analysis of the Field Data
Figure V-ll shows a family of type curves on a log-log plot of H(u,/J) versus 1/u which
can be used for the analysis of the Hantush modified solution.
Plot the variation of drawdown versus time on a log-log paper with the same
scale as that of the type curves.
Use the superposition method to find the best match between the observed plot
and the appropriate type curve.
Read the value of /J from the type curve which matches the observed plot, and
the dual coordinates H(u,/3), 1/u, t, and s of the match point.
Calculate the hydraulic conductivity of the aquifer from
K = Q H(u,/J) (V-42)
Calculate the storage coefficient of the aquifer from
S = 4tbKu (V-43)
Calculate \ from
_ 4/3
(V-44)
If we assumed that the lower layer is completely impermeable, then
K'S' * X2Kbb'S
If one can determine the magnitude of the storage coefficient of the aquitard S'
from other methods, then the hydraulic conductivity of the aquitard may be
obtained from
K, = A'Kbb'S
S'
109
-------
10 :
10
Figure V-ll. Type curves of the function H(u,j8) against 1/u, for various values of J3 (after
Lohman, 1972).
110
-------
Uncertainties
Except for very large values of ft, the type curves have shapes that are not too different
from the Theis curve. Thus, it is difficult to decide which of the type curves to use in matching
against field data. When b is very small, one may easily choose a ft which could be off by two
orders of magnitude. Since K'S' = (ft2) , an error in choosing ft would lead to a much
r2
larger error in the calculation of (K'S'). Thus, two orders of magnitude error in estimating ,6
would lead to four orders of magnitude error in (K'S').
In order to improve this problem, Weeks (1977) suggested that data from at least two
observation wells at different distances from the pumping well should be used. A composite
plot of the drawdown versus t/r2 is made on a log-log paper with the same scale as that of the
type curves. As a result, one should obtain two or more type curves each with different values
of ft proportional to the value of r. A unique match may then be obtained by adding the extra
constraint that r values for observation wells must fall on curves having proportional ft values
(Weeks, 1977). This method could somewhat improve the results but, when )3<0.01, type curves
with different values for ft are so close together that a unique match is still next to impossible.
Very often both layers above and below an aquifer constitute leakage to the aquifer. If
this is the case, one may not be able to find the properties of either of the confining layers. All
this method can give is the value of A (equation V-44), which is a parameter depending on the
properties of both confining layers and the aquifer. This method provides no means for
independently determining the properties of both confining layers.
Even when leakage comes only from one of the confining layers, this method gives the
product of the hydraulic conductivity and the storage coefficient of the aquitard. The value of
the storage coefficient for the aquitard must be found by some other means before one can
finally obtain the vertical hydraulic conductivity.
C) Witherspoon and Neuman Ratio Method
When the ratio of K'/K decreases, both r/B and ft, as defined in previous methods,
decrease and equations (V-32) and (V-39) will eventually reduce to the Theis solution.
Therefore, it is obvious that determining the hydraulic conductivity of a tight confining layer by
observations in the aquifer alone, if at all possible, is associated with a great many uncertainties.
Witherspoon et al. (1962) suggested a method of calculating the permeability of the caprock of
gas storage reservoirs which was based on using observations of drawdown in both the aquifer
and the overlying aquiclude. Later, Witherspoon and Neuman (1967) presented as
improvement over the previous method. This work, together with their more recent works
(Neuman and Witherspoon, 1972), will be discussed here.
Purpose
The purpose of this section is to describe a method of determining the vertical hydraulic
diffusivity of a low permeable layer overlying an aquifer.
Procedure
Complete a pumping well through the total thickness of the aquifer.
Ill
-------
Construct an observation well in the aquifer at a distance r from the axis of the
pumping well.
Establish at least three transducers at three different elevations within the
confining bed as shown in Fig. V-12. It is required that the radial distance of all
three transducers from the pumping well be the same as that of the observation
well. To avoid the effect of possible inhomogeneity of the media, it is preferred
to have all the transducers in the same well close to the observation well.
Start recording water levels in the observation well and values of pressure
measured by the transducers long before the start of the pumping test. It is very
important that the values of pressure measured by the transducers come to an
equilibrium condition before the beginning of the test.
Start producing from the pumping well with a constant rate Q. Pumping should
continue until at least half a meter of drawdown is observed by the middle
transducer in the aquiclude. Recently very accurate pressure-measurement
instruments have been introduced to the market which are able to measure
pressure changes equivalent to 1 cm or less of water. If such instruments are
available for use, then 10 cm of drawdown would be sufficient. Recording of
water level in the observation well and pressures measured by the transducers
should continue at least a few days after pumping has stopped.
Let us first discuss the theory which was developed for evaluating a slightly permeable
aquiclude. A review of more recent works from Neuman and Witherspoon will then follow.
Consider an aquifer of finite thickness overlain by a semi-infinite confining bed. When
the ratio of K'/K is sufficiently small, then under the influence of pumping the aquifer, the flow
in the confining bed is essentially vertical, and the drawdown in the aquifer can be closely
approximated by the Theis solution. The term semi-infinite has been used to indicate that the
aquiclude is so thick that the effect of pumping the aquifer does not reach the top of the
aquiclude. With the above assumptions in mind, Witherspoon and Neuman (1967) derived the
following expression which gives the drawdown in the aquiclude as a function of time t and
elevation z above the top of the aquifer.
s' =
J " - El [- to>y2 _ ] e'^dy (V-45)
27T3/2Kb l/4tD> tD(4tDy-l)
where
112
-------
PUMPING WELL
I
i
AQUICLUDE
/
K
i
j
AQUIFER t
K
i
D
f
5 :
2
J
r
k
1
/
b ,
i
t
'2.
b'/4
' t
yj
K
1
1
^
S-
1
ft
;lj
j
M
1
I
s
[ft
J«
Ji
|
|
|
y
£
|
•;.
V
1
"
— *- TU KtUUKU
i
\
\
\
\
. ^^TRANSDUCERS
/
/
/
^-OBSERVATION
•^^ WELL
WYYVW.
XXXXX>:
XBL 826-843
Figure V-12. A suggested arrangement for conducting a ratio-method test.
113
-------
z = vertical distance from the top of the aquifer
Ss, Ss' = specific storage of the aquifer and the aquiclude, respectively.
Equation (V-45) has been evaluated over a practical range for the two parameters tD and
tD'. Calculated values of s' and s'/s for different tD and tD' have been tabulated in Appendix G
of Witherspoon et al. (1967). Figure V-13 shows a family of curves presenting variation of s'/s
versus tD' for different values of tD.
A variation of the above problem, involving a finite thickness aquiclude, has also been
solved by Neuman (1966). In this derivation the hydraulic head was assumed to be constant at
the top of the aquiclude. This solution has been evaluated over a practical range of relevant
dimensionless parameters and the results are tabulated in Appendix H of Witherspoon et al.
(1967).
Later, Neuman and Witherspoon (1969a) developed a complete solution for the
distribution of drawdown in a system consisting of an aquitard separated by two aquifers as
shown on Fig. V-14. In each aquifer the solution depends on five dimensionless parameters,
and in the aquitard six dimensionless parameters are involved. Consequently, Neuman and
Witherspoon (1972) stated that "This large number of dimensionless parameters make it
practically impossible to construct a sufficient number of type curves to cover the entire
range of values necessary for field application." Hantush (1960) apparently had noticed this
problem before as he stated that "It should be remarked that rigorous solutions can be obtained
for the actual nonsteady three-dimensional flow in layered aquifers, as well as solutions for flow
systems in which the condition of vertical leakage is removed. These solutions, however, are
very difficult to evaluate numerically and are therefore not presented here."
As a result, in spite of the development of more sophisticated theories, because of the
difficulties which appear in the process of their application in the field, authorities seem to go
back and recommend the simpler approaches. For example, all the methods of analysis of the
leaky-aquifer pump tests described by Hantush (1964), appearing four years after he introduced
the modified theory (1960), are based on the r/B solution. Neuman and Witherspoon (1972),
too, stated that "We therefore decided to adopt the ratio method as a standard tool for
evaluating the properties of aquitards." This happened five years after their original
introduction of the ratio method (1967).
Analysis of the Field Data
Observe the pressure record of the transducer at the top of the confining bed. If
it shows any drawdown beyond the error limits of the system, note the time of
such observation and ignore all records of drawdown measured after that time.
Calculate the hydraulic conductivity K, and the specific storage Ss of the aquifer
using Hantush's modified solution and the drawdown record from the observation
well.
Plot the values of drawdown, measured both in the aquifer and the aquiclude, on
log-log paper and draw smooth curves through the data.
Select several arbitrary values of time t. All values should be smaller than the
time when drawdown was first noted at the top transducer.
114
-------
10'
IO
'2
10"
ICT
10
-2
10'
10
n
Jl
-r
AQUITARD K1 .[
1
;; AQUIFER K
10'
K't
10'
X8L835-I828
Figure V-13 The variation of s'/s with tD' for a semi-infinite aquitard (modified from
Witherspoon et al. 1967).
115
-------
//\\VA\y// — i
k
k
b
>2 K2
/ /
i Lr
1 i
k
•™
,
V
{,
Unpumped aquifer
Aquitard
>i K, Pumped aquifer
....i.j.,
XBL 828-2382
Figure V-14. Schematic diagram of two aquifer system.
116
-------
Calculate tD for each selected value of t from the following equation -•'
t,,- EL (V~46)
S/
At each value of time select representative values of s and s' from the time
drawdown plots.
Using the appropriate curve corresponding to each value of tD from Fig. V-13,
find tD' for each ratio of s'/s.
Calculate the vertical hydraulic diffusivity of the confining bed for each value of t
and z of a particular transducer from
* -$SL. (V~47)
For each value of z find the average value of K'/SS' calculated for different
selected times. The average value calculated for each z should represent the
diffusivity of that part of the confining layer between the top of the aquifer and
that particular elevation.
Advantages
As was noted before, if the aquifer received leakage from both above and below, then
r/B and ft methods, which relied on the measurement of drawdowns in the aquifer alone, failed
to lend themselves to calculation of the hydraulic conductivities of the confining layers. The
ratio method, on the other hand, can provide a means of calculating hydraulic difi'usivities of
both upper and lower confining beds.
Uncertainties
The ratio method can only lead to the calculation of the vertical diffusivity of the
confining beds. If one can calculate the specific storage by other means, then the
vertical hydraulic conductivity of those layers may be computed. Leahy (1976)
has used the following approach to overcome the above difficulty. He used
Hantush's (1960) $ solution to find the product of K' and Ss', and the
Witherspoon and Neuman (1967) ratio method to find the ratio of K'/Sg'. Then,
he calculated the value of K' from
K'- ((EL)-(K'-S'J)1/' (V-48)
^s
The method is based on the assumption that the hydraulic head remains constant
at the top of the confining bed. Depending on the thickness and the hydraulic
properties of the aquitard, this may or may not cause errors in the result. If the
aquitard is thin with a small storage coefficient, the transient effect may
completely penetrate it at relatively earl" stages of the pump test.
117
-------
Wolff (1970) reported that piezometers completed in the aquitard exhibit reverse
water-level fluctuations, in that water levels rise for some period of time after the
start of pumping from the aquifer. He relates these changes to radial and
vertical deformation of the aquifer and aquitard resulting from their
compressibility. Because the ratio method does not take such phenomena into
account, Weeks (1977) warns the investigators against application of this method.
This phenomenon has not been observed in other tests such as the ones reported
by Leahy (1977), and Neuman and Witherspoon (1972).
V-4 Conclusions
Vertical hydraulic conductivity of permeable formations can be easily obtained by the
analysis of appropriate aquifer pump tests. For less permeable formations, two general types of
field tests are available which could estimate vertical hydraulic conductivity. The first includes
methods based on single well tests in the low permeability formation itself, while the second
includes large scale multiple well pumping tests designed and interpreted based on the various
theories of leaky aquifer systems.
The problem inherent in the first type of tests is that the measured hydraulic
conductivity is normally only representative of a small zone around the testing interval. Hence,
again, a large number of testing wells is required to give a clear picture of the distribution of the
vertical hydraulic conductivity in the area of interest.
The most commonly used method among the second type of test is based on an early
leaky aquifer solution of Hantush (1956). This solution ignores the storativity of the confining
bed. Neuman and Witherspoon (1969) have noted that the application of this method tends to
overestimate the hydraulic conductivity of the aquifer and underestimate that of the confining
bed. When the confining layer is thin and relatively permeable and incompressible, however,
this method could give useful results.
Hantush's (1960) modified method and the "ratio method" of Neuman and Witherspoon
(1972) are two other techniques of the second type of tests which under certain circumstances
could be used for determination of (KSS) and (K/SJ, respectively. Unfortunately, neither of
these two methods can yield vertical hydraulic conductivity unless the specific storage of the low
permeability layer is independently identified (see Chapter VI). Furthermore, Hantush's
method is unable to separately distinguish the contribution of leakage from upper and lower
confining beds, thus introducing further difficulties in calculation of the vertical hydraulic
conductivity of the individual confining layers.
118
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CHAPTER VI
FIELD METHODS FOR THE MEASUREMENT OF EFFECTIVE POROSITY,
HYDRAULIC HEAD, AND STORATIVITY*
VI-1 Introduction
As the reader will have realized by now, most of this report is devoted to the
measurement of hydraulic conductivity distributions and a brief discussion of how such
distributions may be used in mathematical modeling. However, in a complete study of the
hydraulics of contaminated ground water it is necessary to measure other hydrogeologic
properties or parameters such as effective porosity, hydraulic head, and storativity. For this
reason, and in the interest of completeness, Chapter VI will be devoted to a review of the
definition and measurement of these three quantities.
While not nearly as important or variable as hydraulic conductivity, porosity of
geologic materials plays an important role in controlling groundwater travel time and transport
of hazardous substances between sources and the accessible environment. Therefore,
determination of a representative value of porosity for a given volume of geologic materials, or
a statistical distribution of this parameter, is of interest in predicting the transport of hazardous
waste substances in projects such as radioactive waste isolation and underground injection of
hazardous waste liquids. Another important area of such need is in projects related to
groundwater contamination and remediation. In this chapter, some of the conventional
methods used for the field measurement of porosity will be reviewed. The limitations and
uncertainties related to each method will be discussed.
Hydraulic head measurements are fundamental to determining rates and directions of
movement of contaminant plumes. Usually, such measurements are straightforward except for
the case where multiple fluids of different densities, such as salt water and fresh water, are
present. Therefore, an overview of hydraulic head measurement is presented with attempts to
clarify the measurement of head in fluids having non-negligible density differences.
When contaminant transport in time varying flow fields is of interest, one may have the
need to measure aquifer storativity or storage coefficient. Methods for accomplishing this are
reviewed briefly.
VI-2 Field Measurement of Effective Porosity
The porosity of a material is defined as the ratio of void space to total bulk volume.
Sometimes some of the voids are isolated and do not play a role in transmitting fluid. This is
the reason for introducing the concept of effective porosity, which is defined as the ratio of the
volume of connected pores to the bulk volume of the material. Porosity is a scalar property of
the rock, which means it is independent of direction.
Material in this chapter has been taken from the Lawrence Berkeley Laboratory Report
No. LBL-15050 entitled "Field Determination of the Hydrologic Properties and
Parameters that Control the Vertical Component of Groundwater Movement" by Iraj
Javandel, prepared under sponsorship of the U.S. Nuclear Regulatory Commission.
119
-------
There are several methods which are commonly used in the laboratory to measure the
porosity of a rock core sample. These techniques, including the direct method, mercury
injection, gas expansion, and imbibition have been fully discussed in an American Petroleum
Institute report (1960). Because laboratory techniques are not within the scope of this report,
we shall not discuss them further.
In the field, porosity may be obtained by several methods including well logging and
tracer tests. The following is a brief discussion of several of these techniques, such as Sonic,
Formation Density, and Neutron Logs, and tracer tests. There are various private companies
that perform porosity measurements as well as other types of geophysical measurement services.
VI-2.1 Sonic Log Method
A more detailed description of this method and additional references are given in
Schlumberger (1972). Generally, for a given rock, when porosity increases the sonic velocity
decreases. The Sonic Log is a recording of interval transit time (At) versus depth. The interval
transit time is the time required for a compressional sound wave to traverse through one foot of
formation. This transit time for a given formation is a function of its lithology and porosity.
The Sonic Log is therefore a useful means for obtaining porosity, provided the lithology is
known.
A) Theory and Procedure
A sonic tool, consisting of two transmitters and two pairs of receivers, is lowered into an
uncased well filled with drilling mud or other fluid, (See Figure VI- 1). A pulse is generated by
each of the two transmitters and the difference between the arrival times of the first wave at the
corresponding pair of receivers is measured. The At from the two sets of receivers are averaged
and recorded as a function of depth.
The wave generated by the transmitter will travel through different available media.
However, since the speed of the wave in the formation is generally larger than that in the
drilling fluid or the sonde itself, the wave which will first arrive at the receivers is the one which
has traveled through the formation very close to the wall of the hole. As we measure the
difference in travel time to the two receivers, the period of time corresponding to travel through
the drilling fluid is cancelled out. As a result, knowing the constant of the instrument, the
measured At can be adjusted to show the reciprocal of the velocity in the formation. At is
generally recorded in microsecond/foot (/isec/ft) and it varies between about 44 /isec/ft (for
dense, zero porosity dolomite) to about 190 /fsec/ft for pure water.
Wyllie et al. (1956, 1958) have proposed the following empirical formula for determining
the porosity 0 of a consolidated formation with uniformly distributed pores:
= Atlog ' Atma
Atf-Atma
where
Atlog = transit time reading on the Sonic Log, in /isec/ft
At^ = transit time for the rock matrix material (values for different rocks are
given in Table VI- 1)
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I
"
O
o
No
Upper transmitter
Receivers
Lower transmitter
Figure VI-1. Sketch of a sonic tool, showing ray paths for transmitter receiver sets (modified
from Kokesh et al., 1965).
121
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Atf = the inverse of the velocity of a Sonic Wave in the pore fluid (about 189
/isec/ft).
Table VI-1. Values of transit time for common rocks and casing (modified from
Schlumberger, 1972).
Rock Atma(/isec/ft)
Sandstones 51.0-55.5
Limestones 47.5
Dolomites 43.5
Anhydrite 50.0
Salt 67.0
Casing (iron) 57.0
To calculate porosity at a given depth, one should identify the type of rock from cores and/or
cuttings and determine the value of Atma from Table VI-1, or other sources. Atlog from the Sonic
Log is then measured for that particular depth. Equation VI-1 may then be used to calculate
porosity at the depth under consideration.
B) Uncertainties
The depth of penetration of the recorded wave is only a few inches from the borehole
wall. Thus, the value of porosity obtained by this method is limited to a very small zone around
the well.
According to Wyllie et al. (1956, 1958), the sound velocity in vuggy materials (materials
containing fractures or solution openings) depends mostly on the primary porosity. Therefore,
the sonic method tends to ignore secondary porosity such as fractures. The Sonic Logs in
comparison with the Density Logs and Neutron Logs could, however, give a measure or
secondary porosity.
The method is not suitable for determining effective porosity if there is a significant
volume of isolated pore space.
VI-2.2 Density Log Method
A downhole radioactive source, in contact with the borehole wall, emits medium-energy
gamma rays into the surrounding formation. After colliding with electrons in the formation, the
scattered gamma rays are counted by a detector placed at a fixed distance from the source. The
response to such a bombardment is determined essentially by the electron density of the
formation, which is a function of the true bulk density, pb. Therefore, the porosity of the
formation may be calculated if the densities of the rock matrix and the pore fluid are known.
A) Theory and Procedure
A formation density logging device, consisting of a source and one or two detectors
attached to a skid, is lowered into an uncased well filled with drilling mud or other fluid. The
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device is designed such that the source and detectors come in contact with the borehole wall.
The variation of bulk density against depth is recorded. These tools are usually calibrated to
indicate apparent bulk density. For some types of rock such as sandstone, limestone and
dolomite, in the saturated zone the apparent bulk density is essentially equal to the bulk density
itself. Other types of rock bulk density should be estimated from apparent values using
available graphs (Schlumberger, 1972). The porosity, 0, of the rock can be estimated from
Pma - Pt
where
pb = bulk density of the rock obtained from the log.
pt = the density of pore fluids close to the well.
pma = rock matrix density.
B) Uncertainties
This method determines total porosity. It does not differentiate between
connected and isolated pore spaces within the formation.
The presence of shale or clay in the formation introduces some errors into the
results.
-2.3 Neutron Log Method
This method can determine the amount of in situ liquid-filled porosity of a given
material. The technique is based essentially on a measurement of the amount of hydrogen
present in the formation. If the pore space of the rock is filled with water, and no other source
of hydrogen, such as the water in gypsum (CaSO4 + 2H2O), is present, then the response of this
test is a measure of porosity.
A) Theory and Procedure
There are at least three different kinds of Neutron Logs which are currently available.
Gamma Ray Neutron Tool (GNT), Sidewall Neutron Porosity (SNP), and Compensated
Neutron Log (CNL) use plutonium-beryUium or americium-beryllium as sources of neutrons
with initial energies of several million electron volts (Schlumberger, 1972). Here, we shall only
address the SNP method. Information about other tools and additional references on those
tools may be obtained from Schlumberger (1972).
In the SNP method, a neutron source and a detector are mounted on a skid which is
lowered into an uncased well, preferably without fluid and drilling mud. This tool is designed
such that it comes in contact with the borehole wall. The neutrons emitted by the source, after
penetrating the formation and colliding with the nuclei of the formation materials are received
by the detector. The response is measured against depth. A surface panel automatically makes
necessary corrections for salinity, temperature, and hole diameter variations, and records the
porosity directly. If the hole is filled with drilling mud, porosity values should be corrected for
the mud-cake thickness, using available charts (Schlumberger, 1972).
123
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Neutrons are electrically neutral particles, each with the mass of a hydrogen atom
(Tittman, 1956). The source on the tool continuously emits fast neutrons. These neutrons
collide with nuclei of the formation materials and lose some of their energy. The amount of
energy which a neutron loses in each collision depends on the relative mass of the nucleus
against which the neutron collides. Collision with a hydrogen nucleus causes the maximum
energy loss. Thus, the slow-down of neutrons depends largely on the amount of hydrogen in the
formation, which in turn is related to the amount of water in the formation. The SNP method
has the advantages that borehole effects are minimized (Schlumberger, 1972) and that most of
the corrections required are performed automatically in the panel.
B) Uncertainties
This method can measure effective porosity only if the isolated pores are free of
liquid, otherwise the method does not differentiate between connected and
isolated pores.
The tool responds to all the hydrogen atoms in the formation including those
chemically bound to formation materials.
In shaly formations the porosity derived from the neutron response will be
greater than the effective porosity.
The zone of influence of this method depends on the porosity of the formation,
but generally it is limited to a short distance from the wall of the hole.
VI-2.4 Tracer Techniques
There are several tracer methods for determination of aquifer properties such as
effective porosity and dispersivity. The literature is replete with descriptions of theory and
practice of tracer tests, see for example, Nir and Kirk (1982); Klett et al. (1981); Benson (1988);
Fried (1975); Hoehn and Roberts (1982); Malaszowski and Zuber (1985); Lenda and Zuber
(1970); Ivanovich and Smith (1978) and Giiven et al. (1986). Many types of radioactive and
nonradioactive tracers have been used. A list of some of the tracers which have been used in
groundwater studies has been given by Davis et al. (1980) and Thompson (1981).
Unfortunately, most of the analyses of tracer tests for porosity computation are based on
the assumption of a two-dimensional, vertically homogeneous aquifer. Thus unknown
variations of hydraulic conductivity will lead one to compute porosity values or variations that
may be erroneous. For this reason, the commonly used tracer based methods for measuring
effective porosity are not discussed herein.
VI-3 Definition of Hydraulic Head
An important parameter which controls the movement of groundwater is the hydraulic
gradient. Distribution of hydraulic head within a given hydrologic system is generally controlled
by the conditions at the boundaries of the system and the properties of the media.
The potential 0 of a given fluid at any point in space is generally defined as the
mechanical energy per unit mass of the fluid, which has three components
124
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where
g = gravitational acceleration.
z = elevation of the point above the chosen datum.
v = velocity of fluid.
P = pore water pressure at the point of interest.
p = density of fluid.
P0 = atmospheric pressure.
The potential 0 is the amount of work required to bring a unit mass of fluid from an
arbitrary standard state to the point under consideration. The standard state is usually
considered to be at elevation z = 0, velocity v=0, and pressure P = P0, atmospheric pressure. The
first term of the right hand side of Equation (VI-3) represents the work against gravity
necessary to bring a unit mass of fluid from the standard position to the elevation z. The
second term is the work required to increase the velocity of the unit mass from zero to V.
Finally, the third term is the work required to bring the pressure of the fluid from P0 to P.
For the case of flow through porous media, where velocity is generally very small, the
term v2/2 may be ignored with respect to the other terms. In the case of slightly compressible
fluids, where the dependence of p on pressure can be ignored in computing potential, the third
term may also be simplified and equation (VI-3) becomes
0 = gz + ?£? (VI-4)
P
One may note that in some cases, such as fracture flow close to a well or a shaft, where
fluid velocity is relatively large, the term v2/2 may not be so small as to be negligible.
If we refer to P-P0 as gauge pressure, then the expression for potential becomes
0 = gz + I_ (VI-5)
P
A term for potential which is commonly used in groundwater hydrology is hydraulic head which
is defined as energy per unit weight of the fluid. Therefore,
h = * = z +JL (VI-6)
g PS
For a homogeneous fluid with constant p, Darcy's law states that fluid flows from regions
of higher heads toward regions of lower heads, and that the flow velocity is proportional to the
gradient of the hydraulic head. However, if we have more than one type of fluid or p changes
from one aquifer to the other, which could occur because of changes of temperature or salt
concentration, then at each point in space one can define as many potentials as there are
densities (Hubbert, 1940). For example, if we have three different densities such as plt p2 and
125
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P3, then at any point in the space with elevation z and pore pressure P, regardless of which fluid
occupies that space, we can write
0! = gz + — (VI-7)
Pi
02 = gz + £_ (VI-8)
P2
03 = gz + :L_ (VI-9)
Pi
In this case, according to Hubbert (1940), motion of fluid i with the density p{ should be
solely studied by the distribution of its own potential { or head hj = QJg. Let us emphasize
that potential { based on the density p{ is defined everywhere in the space including the space
occupied by fluids of other densities. This concept is very important when we are investigating
flow between two aquifers of differing salinity separated by some low permeability layer.
To make this point clear let us consider the following example. A look at Fig VI-2
without attention to the quality of water of two aquifers makes one think that there is a drop of
hydraulic head from the freshwater aquifer downward to the saline water, and thus flow is
downward. However, the following calculations show that flow is actually occurring upward
from the saline aquifer towards the fresh-water aquifer.
The values of potential of fresh water at points A and B, top and bottom of the aquitard,
are
0f = 160g + 120/?fg = 280g (VMO)
A pf
h = lOOg + 178psg = 284.5g (VI-11)
B pf
It is now apparent that the fresh-water potential at point B is larger than that at A, thus causing
an upward flow from B to A. Assuming a linear variation of potential between A and B, salt-
water potential gradient between B and A is
= (278 - 275.77) =
6 5
60 60
The vertical velocity component is upward with the magnitude of
vz = -_b pW = - A (0.0385gpf)
Ms dz ns
In the above calculations the density of saline water ps is 1.036 g/cm3.
For the study of groundwater movement in a porous medium containing fresh, diffused,
and salt water, Lusczynski (1961) has introduced three different types of head at each point i
within the medium: fresh-water head hif, point-water head hip, and environmental-water head
hin, which are defined as follows.
126
-------
/A$y/>$//s
Fresh water aquifer
IDS =500ppm
Aquitard 2*
Saline water aquifer
IDS =50,OGOppm
i
I
— :
fe
AM%&\
H
1
1 .A |
JOm
-K .- 1
i
Z
—
—
; —
=
=
=
m
m
^
^
i — i
-
21
'8m
2
z=250m
z=!60m
z=IOOm
z=0m
Figure VI-2. Schematic diagram showing two observation wells, one open in the top fresh-
water aquifer and the other screened in the lower saline aquifer.
127
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Fresh-water head at point i, Fig. V1-3.B, is defined as the height of the water above the
datum in a well filled with fresh-water from point i to a level high enough to balance the existing
pressure at point i. Based on this definition, fresh-water head at point i may be written as
ha = Zj + — (VI- 13)
SPt
where
hif = fresh-water head at the point i,
Zj = elevation of point i,
P = gage pressure at point i.
pf = density of fresh-water.
As defined above, hit is the energy per unit weight of fresh water at the point i, as was defined
by Hubbert (1940).
Point-water head at point i, Fig. VI-3, in groundwater of variable density, is defined as
the water level above the datum in a well filled with water of the type found at point i to
balance the existing pressure at point i. From this definition one can write
hip = z, + JL (VI-14)
BPi
where
hip = point-water head at point i.
Pi = density of water at i.
Environmental-water head at a given point i, Fig. (VI-3), in groundwater of variable
density is defined as the fresh-water head reduced by an amount corresponding to the difference
of salt mass in fresh-water and the environmental water between point i and the top of the zone
of saturation. Environmental water between point i and the top of the zone of saturation is
herein defined to be the water of constant or variable density occurring in the environment
along a vertical line between point i and the top of the zone of saturation.
Based on the above definition, environmental-water head at point i may be written as
h^ =!_- (zr - Zi)A + Zf
PS Pt
or in terms of fresh-water head one can write
hin - htf - (zr - zs)(l - 1) (VM6)
Pf
where
hta = environmental head at point i.
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B
[51
TOP OF ZONE
OF SATURATION
FRESH
WATER
DIFFUSED^
WATER
•."•• SALT
;v. WATER
=w
hin
- u
11
DATUM
Figure VI-3. Heads in groundwater of variable density, (A) point-water head, (B) fresh-water
head, and (C) environmental head, (modified from Lusczynski, 1961).
129
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zr = vertical distance between datum and top of the zone of saturation.
pa = average density of water between point i and top of the zone of saturation.
Pa = — - f r P(z)dz (VI-17)
Zr - Zj Zj
Lusczynski (1961) states that "fresh-water heads define hydraulic gradients along a
horizontal. However, along a vertical environmental-water heads should be used to define the
hydraulic gradient". Although along a horizontal the Lusczynski and Hubbert theories match, in
a vertical direction their theories lead to different values of gradients. Based on the Lusczynski
approach, environmental head at point A of the example of Fig. (VI-2) is the same as the fresh-
water head which is 280 m. The value of environmental head at point B may be calculated from
equations (VI-1), (Vl-16), and (VI-17). Assuming that the aquitard is occupied by saline water,
the value of pa may be calculated from equation (VI-17) to be 1.0146pf. Substituting for pa and
hjf in equation (VI-16), one obtains the value of h^ at B to be 282.31m. Therefore, Lusczynski's
approach also gives the direction of flow from B to A. The magnitude of gradient is also the
same as that obtained by the Hubbert approach for saline water. If the concentration of water
in the aquitard is somewhere between fresh and saline water, then the environmental water
head at point B would be larger than 282.31m leading to a gradient different from those
obtained by the Hubbert approach.
Lusczynski (1961) has given the following formula for calculation of components of
velocity in the horizontal and vertical directions at point i
ft
ft dz
Some authors believe that in dealing with problems in which density is a function of
space, it is more convenient to formulate the groundwater flow equation in terms of pressure
rather than head, because pressure head (P/pg) is dependent on fluid density which in turn is
dependent on salt concentration (Anderson, 1979). In terms of pressure, Darcy's law at a point
i in a groundwater system may be written as (Scheideger, 1960):
v, = -M (VPj-pil) (VI-20)
where
Vj = Darcy's velocity vector.
VPj = gradient of pressure at point i.
ft = viscosity of fluid at point i.
[k] = permeability matrix.
Pi = density of fluid at point i.
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g = gravity vector.
Let us now examine the example in Fig. (VI-2) with the approach of equation (VI-20).
Values of pressure at points A and B are 120/o,g and 184.5pfg, respectively. If we assume a
linear variation of pressure between A and B, then the component of pressure gradient in the
vertical direction becomes
and
op
-Tl - Peg = (1.075-1.0365)gpf = 0.0385g/>f (VI-22)
oz
and the vertical velocity component at the point B is
vz = - JE (0.0385g/>f) (VI-23)
B /i
which is exactly the same magnitude as obtained from the Hubbert approach.
The above discussion was based on the gradient of hydraulic head alone. Other types of
gradient such as chemical, electrical, and thermal are also effective in moving fluid, even in the
absence of any hydraulic head gradient (Philip and de Vries, 1957; Casagrande, 1952). In
particular, for a problem such as the above example, where one is dealing with a big contrast of
concentration, a certain amount of solute moves from the higher concentration zone to the
lower one with water moving in the opposite direction. The law governing this type of motion is
called Pick's first law of diffusion which is
F = - D^£ (VI-24)
where
F = mass of solute passing through a unit area per unit time,.
D = diffusion coefficient.
C - concentration of solute.
Although the value of D is generally very small, over a long period of time this diffusion
process could cause a considerable amount of contaminant transport. Note that in the above
example the chemical gradient acts in the same direction as the hydraulic head gradient.
Clarification of one point seems to be in order here. A layer of compacted clay restricts
the passage of ions while allowing relatively unrestricted passage of neutral species (Freeze and
Cherry, 1979). Thus, salt may not easily move across a compacted clay layer while water may,
if, of course, the hydraulic gradient allows.
VI-4 Field Measurement of Hydraulic Head
Hydraulic head at a given point in a geological formation occupied by a fluid may be
measured both directly and indirectly. Hydraulic head may be measured directly by a pipe with
131
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one end open at the point of interest and the other end open to the atmosphere. This pipe is
generally referred to as a piezometer. The elevation of fluid in this pipe at equilibrium relative
to an arbitrary datum is the hydraulic head at the point of interest, where water is allowed to
enter the pipe. The end of the pipe which allows water to enter is usually equipped with a small
section of slotted pipe or a device called a well point. Hydraulic head may be obtained
indirectly by measuring the pore water pressure at any point with the help of a transducer.
Most commercially available transducers generate a voltage proportional to pressure which can
be converted to the actual pressure of the water at the point. The value of pressure and the
elevation of the point of measurement may be substituted into equation (VI-6) to give the
hydraulic head at the point of interest.
Within a single-layer aquifer where flow is essentially horizontal and equipotential lines
are vertical (hydraulic head remains constant with depth), water level in an observation well
which is screened along all or part of the thickness of the aquifer would give the value of
hydraulic head of the aquifer at the position of the well. If for some reason such as
stratification of the aquifer, proximity to the zones of recharge or discharge, or change in water
quality with depth, hydraulic head varies with depth, then the observation well can only give an
average value of head of the aquifer for the screened interval. This head may not be accurate
enough for a critical study of groundwater movement.
As we discussed above, an important parameter which one should always measure
together with hydraulic head is the density of fluid at the point of measurement. Density varies
with temperature and chemical properties. It is recommended that a water sample be taken
from the point of interest for chemical analysis. If the medium is occupied by freshwater, i.e.,
total dissolved solid (TDS) less than 1000 mg/1, one can ignore the effect of density variation,
and hydraulic head as defined by equation (VI-6) is adequate for calculation of the velocity
components of groundwater movement. If, however, TDS is very high and its magnitude
changes with position, then one must consider the density of water at each point where
hydraulic head is measured. At 75°F and atmospheric pressure, the relation between NaCl
water salinity and water density may be approximated by
p = 1 + .73C (VI-25)
where C is NaCl concentration is ppm x 10"6. A chart showing variation of water density with
temperature and pressure at different NaCl concentrations is given in page 47 of Schlumberger
(1972).
The above methods of hydraulic head measurement are only practical when the
formation is reasonably permeable such that height of water within the pipe comes to
equilibrium with the formation pressure at the point within a reasonably short period of time.
Measurement of hydraulic head in less permeable formations is quite involved. This is because
of the long period of time required for water pressure in the pipe to come to equilibrium with
the formation pressure. To overcome this difficulty one should pack off the test interval from
the rest of the hole to minimize the volume of water needed to be produced by the formation.
For further information about installation of piezometers in fine-textured soils and application
of inflatable straddle packers for hydrologic testing, readers are referred to Johnson (1965) and
Shuter and Pemberton (1978), respectively.
VI-5 Storage Coefficient Definition and Measurement
The storage coefficient or the storativity S of a saturated confined geological bed of
thickness b is defined as the volume of water that the bed releases from storage per unit surface
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area of the bed per unit decline in the component of hydraulic head normal to that surface.
This term has been commonly defined for aquifers (Freeze and Cherry, 1979, Hantush, 1964).
However, storativity has been generally used for aquitards and aquicludes as well. Some authors
have used the term storativity for both confined and unconfined aquifers (USER, 1977). A
particularly thorough discussion of storativity may be found in Narasimhan and Kanehiro (1980).
Note that in the above definition it is inherent that the hydraulic head is the same
through the total thickness of the bed. This may not be a valid assumption for cases in which
hydraulic head varies with elevation. Consequently, Hantush (1964) has used the term average
hydraulic head in the definition of the storage coefficient in order to overcome the above
problem. A more accurate term to use is the specific storage. The specific storage Ss of a
saturated confined geological bed is defined under a unit decline in hydraulic head. For cases
where the hydraulic head remains constant throughout the total thickness b of the bed, then the
following relation holds
S = bSs (VI-26)
The storativity and the specific storage are scalar parameters. They may be space
dependent, but they are independent of direction.
A decrease in hydraulic head leads to a decrease in fluid pressure and an increase in
effective stress. Therefore, the volume of water that is released from storage due to decreasing
the hydraulic head h is produced by two mechanisms: (1) the expansion of the water caused by
decreasing the pore water pressure, and (2) the compaction of the skeleton of the medium
caused by increasing the effective stress. The expansion of the water is controlled by its
compressibility /? and the compaction of the medium by the matrix compressibility a. Therefore,
it can be shown that the specific storage Ss is given by
Ss = pg(a + W} (VI-27)
where
p - density of the water
g = acceleration of gravity
$ = porosity of the medium.
Equations (VI-26) and (VI-27) indicate that Ss has the dimension of
(L)"1 and S is dimensionless.
VI-5.1 Methods of Measurement
In general, methods of in-situ measurement of the storage coefficient fall into two
categories: (1) methods which are based on well testing of aquifers, and (2) techniques which
rely on the change of barometric pressure and earth tides. In addition, some indirect methods
such as measurement of subsidence and consolidation have been used to obtain a rough
estimate of the storativity of the shallower unconsolidated materials.
Average storativity of an aquifer can be determined by the common pump test
technique. Some of these tests were discussed in Chapter V. Unfortunately, the suitability of
the current pump test techniques to determine the storativity of less pervious confining beds is
questionable. As we discussed before, the original leaky aquifer theory (r/B method) simply
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ignores the storage capacity of the aquitard. The more recent theories such as the B solution of
Hantush (1960) and the ratio method of Witherspoon and Neuman (1967) cannot single out the
storativity of the aquitards. The Hantush solution could at best give the product of the specific
storage and the hydraulic conductivity of the aquitard, and the ratio method yields the hydraulic
diffusivity of the aquitard. In fact, we noticed that calculation of hydraulic conductivity was
only possible if one could obtain the storativity from other sources.
Application of the B solution of Hantush combined with the ratio method of
Witherspoon and Neuman has been reported (Leahy, 1977) to yield a value for the storativity of
the aquitard. However, despite the fact that the B method cannot differentiate properties of the
two confining layers above and below the aquifer, the procedure used by Leahy is suitable for
cases where one is certain that leakage into the aquifer is from only one of the confining beds.
In regard to single well tests, the modified Burns' method as described in a previous
section should give a value for the storativity of the layer being tested. Recalling the limitations
of that test, the estimated value of storativity belongs to the materials very close to the well.
Other single well tests are either unable to give Ss or, if they do, the reliability of the calculated
value is questionable (Papadopolus et al., 1973).
Barometric efficiency of a well can also be used to find the average storage coefficient of
a confined aquifer (Jacob, 1940)
S = (VI-28)
EWB k '
where
S = storage coefficient.
0 = porosity.
b = aquifer thickness.
Ew = bulk modulus of elasticity of water.
B = barometric efficiency.
7 = specific weight of water.
Fluctuation of the water level in a well due to the earth tide has occasionally been used
to estimate the storativity of deep confined aquifers (Kanehiro and Narasimhan, 1980).
However, because of uncertainties in estimating input data this method is not commonly applied
in the field.
1.34
-------
CHAPTER VII
MATHEMATICAL MODELING OF ADVECTION-
DOMINATED TRANSPORT IN AQUIFERS
VII- 1 Introduction
There are a variety of ways in which vertically-distributed hydraulic conductivity
distributions can be used to understand and assess problems involving contaminated ground
water. A significant amount of insight will be obtained simply by observing and discussing the
implications of such information on patterns of contaminant migration. However, use of the
three-dimensional vertically-distributed data in three-dimensional mathematical models will be a
common procedure for developing quantitative assessments of a variety of possible activities
such as evaluation of site remediation plans. Thus it is worthwhile to devote a chapter to a
discussion of the relationship between vertically-distributed hydraulic conductivity data and
mathematical modeling.
As pointed out in Chapter I, once one moves from the use of vertically-averaged aquifer
properties in two-dimensional mathematical models to the use of vertically-distributed aquifer
properties in three-dimensional models, the nature of the physical process represented by the
model changes dramatically. In many situations, the model changes from one being largely
dominated by dispersion (low Peclet number flows) to one largely dominated by advection (high
Peclet number flows). Unfortunately, most of the standard finite-difference and finite-element
algorithms for solving mathematical models do not work well when applied to advection-
dominated flows, especially those that involve chemical or microbial reactions. The necessary
evolution from dispersion-dominated to advection-dominated numerical algorithms for solving
the flow and transport equations is far from trivial, so it is important to call attention to some of
the newer numerical methods, particularly those that produce a minimum of numerical
dispersion when used to solve the transport equation. Therefore, we will present a brief review
of numerical methods, emphasize the approaches that work best for advection-dominated
transport problems, and present several overviews of example applications developed by the
authors during the past several years.
VII-2 Governing Equations for Flow and Transport
In developing a model of contaminant migration in the subsurface, one usually must deal
with the flow equation, the transport equation or both. As developed in Jacob (1940), Bear
(1979) and numerous other sources, what we will refer to as the flow equation may be written as
where t is time, Ss is the specific storage, which depends on the specific weight and
compressibility of the water and compressibility of the porous matrix (see, e.g., Bear, 1979,
sections 5-2 and 5-11), h is hydraulic head, K is the hydraulic conductivity matrix, 2S0 is the sum
of volumetric sources of water and 2S; is the sum of volumetric sinks.
Equation (VII-1) may be solved, together with the appropriate initial and boundary
conditions, for the head, h, provided that the hydraulic conductivity K is known as a function of
position (x) in the medium and provided that information is available on the spatial distribution
of the specific storage Ss and the spatial and temporal variations of the source(s) and sink(s). It
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is assumed that the relevant fluid properties such as density and viscosity are all known and
nearly constant, independent of position or composition. Once the solution for the head is
obtained, the flow field can be obtained using Darcy's law, which is given by
v(x,t) = - K.Vh/6 (VII-2)
where^v is the pore velocity vector and 8 is the porosity.
If we are interested in the transport of a single species, and if it is assumed, for
simplicity, that the reaction terms can be described by a first-order decay expression with the
same decay rate constant A for both the solid phase and water phase components of the species,
and that the linear Freundlich isotherm is applicable, then the solute transport equation may be
written as
+ V.(9Cv) - V.(95 -VC) = - XRj8C + 2S0C0 -
where Rf is the retardation factor, C is solute concentration, D is the dispersivity maxtrix and C0
and C; are solute concentrations in the source(s) and sink(s) respectively. Given the flow field v
and the spatial distribution of the medium parameters such as porosity, retardation factor,
tortuosity and dispersivity and the other parameters such as the decay rate constant and
molecular diffusion coefficient, the solute transport equation can be solved, together with the
appropriate initial and boundary conditions, for the concentration C.
VII-3 Numerical Solutions of Governing Equations
Several books are available in the hydrology literature which are devoted to the study of
numerical solutions of subsurface flow and transport equations (e.g., Remson et al, 1971;
Huyakorn and Finder, 1983; Bear and Verruijt, 1987). A concise history of the applications of
numerical methods to the solution of subsurface flow and transport problems can be found in
the work by Huyakorn and Finder (1983). Recently, a number of comprehensive reviews and
state-of-the-art reports have been presented on the subject (Odeh, 1987; Allen, 1987; Cady and
Neuman, 1987; Abriola, 1987; Finder, 1988a,b).
A major step in the numerical solution of either the flow or the transport equation is the
discretization of the solution domain and the governing equation. This may be accomplished in
various ways including the finite difference method, the integrated finite difference method and
the finite element method (see, e.g. Huyakorn and Finder, 1983; Narasimhan and Witherspoon,
1976). The advantages and disadvantages of each of these methods are discussed briefly by
Finder (1988a). The discretization process leads to a set of algebraic equations in terms of the
nodal values of head, h, or the concentration, C, over a numerical solution grid. These
equations are linear in the case of the flow equation (VII- 1) and the transport equation (VII-
3). However, in general, in the case of unsaturated flow or multiphase flow arid transport
equations, the discretization results in a set of nonlinear equations which are often subjected to
an intermediate linearization step to obtain a set of linear algebraic equations in terms of the
independent variables (see, e.g., Odeh, 1987). Alternatively, an iterative nonlinear equation
solver may be used (Remson et al., 1971).
The solution of the linear algebraic equations may be accomplished using either direct
elimination or iterative methods (Odeh, 1987; Finder, 1988a; Schmid and Braess, 1988; Lapidus
and Finder, 1982). While the direct elimination procedure is more reliable and the method of
choice for systems involving less than a few hundred equations, iterative methods are preferred
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in large three-dimensional flow and transport problems, as they require less computer storage
and time (Odeh, 1987; Finder, 1988a). Several iterative methods in current use are discussed
briefly by Odeh (1987) and Allen (1987). These include the strongly implicit procedure, SIP, the
successive over relaxation method, SOR, and the conjugate gradient method with matrix
preconditioning, PCG. Another iterative method is the multigrid, MG, method which has been
described by Brandt (1977) and originally suggested by Federenko (1961). The multigrid
method appears to be very suitable for the solution of large equation systems associated with
three-dimensional flow and transport modeling problems in heterogeneous media (see, e.g.,
Schmid and Braess, 1988; Cole and Foote, 1987,1988). A comparison of various solution
methods for some two-dimensional groundwater flow problems, including the direct Choleski
decomposition method and the iterative PCG and MG methods have been presented recently by
Schmid and Braess (1988). More detailed discussions of various matrix solution methods and
additional references can be found in the recent work by Oran and Boris (1987).
VII-3.1 Flow Equation Solution
The solution of the flow equation (VII-1) together with the initial and boundary
conditions is fairly straightforward (see, e.g., Huyakorn and Pinder, 1983). However, problems
arise with respect to computer storage and time limitations when large scale three-dimensional
flow problems in heterogeneous media are considered which require the solution of a large
system of equations. As discussed briefly above, these problems may be alleviated greatly by the
use of the multigrid iterative method, although multigrid solution methods are not yet widely
used in three-dimensional subsurface flow applications. At the present time, most three-
dimensional flow codes seem to favor the strongly implicit procedure (McDonald and Harbaugh,
1984; Ababou, Oelhar and McLaughlin, 1988), while the applications of the multigrid technique
have been limited mostly to two-dimensional problems (Cole and Foote, 1987; Schmid and
Braess, 1988). Applications of the MG methods to three-dimensional flow problems are just
beginning to be made (e.g., Cole and Foote, 1988).
The use of the finite difference or the integrated finite difference methods with simple
rectangular grids appear to be the most practical approach at present to three-dimensional flow
problems in heterogeneous media (McDonald and Harbaugh, 1984; Ababou et al., 1988). It
may be useful to mention also a versatile finite element approach, particularly suited to
fractured media, described by Kiraly (1988) and Hufschmied (1989), which can incorporate one-
dimensional and two-dimensional elements together with three-dimensional elements for the
purpose of large scale groundwater flow modeling in highly heterogeneous media.
VII-3.2 Transport Equation Solution
The solution of the transport equation (VII-3) is more complicated than the solution of
the flow equation. This is primarily due to the existence of the advective transport term (second
term on the left-hand side) in (VII-3) which gives the transport equation a hyperbolic character.
Reviews of various techniques for the solution of the transport equation have been presented by
Allen (1987), Cady and Neuman (1987), Farmer (1987), Hauguel (1985), and Baptista, Adams
and Stolzenbach (1985). In general, these techniques can be grouped into three classes; namely,
Eulerian, Lagrangian, and Eulerian-Lagrangian methods. Eulerian methods are more suited to
dispersion dominated systems while Lagrangian methods are most suited to advection
dominated systems. Eulerian-Lagrangian methods have been introduced to deal efficiently and
accurately with situations in which both advection and dispersion are important.
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A) Eulerian Methods
The Eulerian methods are based on the discretization of the transport equation on a
numerical solution grid that is fixed in space, and all of the terms of the equation, including the
advective transport term, are discretized together and the resulting algebraic equations are
solved simultaneously in one solution step. As discussed by Cady and Neuman (1987), while
Eulerian methods are fairly straightforward and generally perform well when dispersion
dominates the problem and the concentration distribution is relatively smooth, they are usually
constrained to small local grid Peclet numbers (Pe=vAs/Dss, where v is the magnitude of the
seepage velocity, As is the grid spacing along the flow direction, and Dss is the longitudinal
component of the dispersion matrix along the flow direction), and to small grid Courant
numbers (Co=vr At/As, where At is the time step and vr is the retarded seepage velocity =
v/Rf). Daus and Frind (1985) describe an alternating-direction Galerkin finite element method
for two-dimensional domains which is constrained to Pe < 2 and Co < 1. Burnett and Frind
(1987a,b) present a similar method for three-dimensional domains. Frind, Sudicky, and
Schaellenberg (1988) apply this method to study the detailed evolution of contaminant plumes in
heterogeneous media. Another example of an Eulerian method applied to a three-dimensional
domain is the curvilinear finite element method of Huyakorn et al. (1986a,b).
B) Lagrangian Methods
Methods which are based on solutions of the transport equation on a moving grid, or
grids, defined by the advection field, or methods which do not rely on a direct solution of the
Eulerian transport equation (VII-3) but which are based on an analysis of the transport,
deformation and transformation of identified material volumes, surfaces, lines or particles by
tracking their motion in the flow field are generally called Lagrangian methods (Cady and
Neuman, 1987; Oran and Boris, 1987). Here, we will reserve this term only for methods which
are based on tracking alone and will consider the moving-grid methods which have been called
Lagrangian before by some authors simply as special cases of the Eulerian-Lagrangian methods.
It appears that the first applications of the Lagrangian method in groundwater solute
transport problems are the immiscible front movement analyses of Muskat (1937). These
analyses and other studies of immiscible front movements are described by Bear (1979). This
approach is most suited to situations where advection is the dominant process and the effects of
hydrodynamic dispersion and chemical reactions are negligible. Nelson (1978a,b,c,d) and Nelson
et al. (1987) have used this approach to calculate arrival time distributions to evaluate the
environmental consequences of groundwater contamination in various geological settings.
Giiven et al. (1986) have developed a simplified Lagrangian analysis of two-well tracer tests in
stratified aquifers, neglecting the effects of hydrodynamic dispersion, and Molz, Giiven and
Melville (1983) have used a similar Lagrangian approach in their analysis of some of the single-
well tracer test results of Pickens and Grisak (1981a,b).
If sorption can be approximated as an equilibrium process, the effects of sorption can be
easily taken into account in a Lagrangian framework by advecting the tracked particles or the
fronts with the retarded velocity vr. First-order decay or degradation can also be simulated
fairly easily in a number of ways (Prickett, Naymik, and Lonnquist, 1981; Kinzelbach, 1988).
The effects of hydrodynamic dispersion can be incorporated in a Lagrangian framework
by introducing a large number of particles into the flow field, each with a specified quantity of
solute mass associated with it, and adding a random-walk component to the motion of each
particle as the particles are advected in the flow field. This method has been popularized by the
138
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random-walk solute transport model of Prickett et al. (1981). Improvements to the basic
random-walk model, to take into account the effects of nonzero spatial derivatives of the
dispersion coefficients in flow fields with spatially variable dispersion coefficients, are discussed
by Kinzelbach (1988), Uffink (1988) and Ackerer (1988). A discussion of the advantages and
disadvantages of Lagrangian methods can be found in the work by Kinzelbach (1988).
Lagrangian methods are described and discussed further in the recent book by Bear and
Verruijt (1987).
C) Eulerian-Lagrangian Methods
Reviews of Eulerian-Lagrangian methods (ELM) have been presented recently by Cady
and Neuman (1987) and others (e.g., Hauguel, 1985; Baptista, Adams and Stolzenbach, 1985).
These methods combine the advantageous aspects of the Lagrangian and the Eulerian methods
by treating the advective transport using a Lagrangian approach and the dispersive transport
and chemical reactions using an Eulerian approach. According to how the advective transport is
taken into account, these methods can be generally grouped into three classes; one class makes
use of particle tracking and relates the concentration at a grid node to the solute mass
associated with each particle and the particle density around that node, while the second class
treats concentration directly as a primary variable throughout the calculations without resorting
to the use of any particles, and the third class consists of models in which the first and second
approaches are used together in an adaptive manner depending on the steepness of the
concentration gradients. The first type of method, which was originally suggested by Garder,
Peaceman and Pozzi (1964) according to Cady and Neuman (1987) and also applied by Konikow
and Bredehoeft (1978), is often called the method of characteristics (MOC). This type of
method which makes use of particles had been denoted as ELM/P by Baptista et al. (1985).
The second type of method has been called the "single-step reverse particle tracking" method by
Neuman (1984) and the "modified method of characteristics" by others (e.g., Ewing, Russell and
Wheeler, 1984). Since the concentration, C, is used as the primary variable throughout the
calculations without resorting to the use of any particles, Baptista et al. (1985) have denoted this
method as ELM/C. The third method, developed by Neuman (1984), and Cady and Neuman
(1987), is called the adaptive Eulerian-Lagrangian method; in keeping with the notation of
Baptista et al. (1985), we may denote this method as ELM/A.
The ELM/C method has been used by a number of investigators including Baptista,
Adams and Stolzenbach (1984), Cheng, Casulli and Milford (1984), Holly and Usseglio-Polatera
(1984), Wang, Cofer-Shavica and Fatt (1988), Sorek (1988), Wheeler et al. (1987) and others
(see, e.g., Cady and Neuman, 1987; Baptista, Adams and Stolzenbach, 1985; Hauguel, 1985).
A variation of the ELM, other than ELM/P, ELM/C or ELM/A methods, has been
described by Doughty et al. (1982). The method of Doughty et al. (1982) has been applied by
Guven et al. (1985) to the analysis of a single-well tracer test in a stratified aquifer and by Molz
et al. (1986a) to a coupled transport problem involving biodegradation.
It may be useful to point out that Eulerian-Lagrangian methods have been developed
extensively for the numerical modeling of complex three-dimensional industrial and
environmental flows and for the solution of various fluid mechanics problems particularly over
the last decade (see, e.g., Oran and Boris, 1987; Hauguel, 1985). These methods are now
becoming popular also in the area of subsurface contaminant migration modeling.
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VII-3.3 Available Computer Codes
Several well documented computer codes for three-dimensional flow and solute transport
modeling as well as parameter identification and uncertainty analysis are available in the public
domain. These codes have been developed by universities, various government agencies and
government supported laboratories. There are also several proprietary codes developed by
private consulting firms and research organizations such as the Electric Power Research
Institute. Many of these codes have been listed in the recent monographs by Javandel et al.
(1984) and van der Heijde et al. (1985). In this regard, the International Ground Water
Modeling Center (IGWMC) serves as an information, education, and research center for
groundwater modeling with offices in Indianapolis, Indiana (IGWMC, Holcomb Research
Institute, Butler University, 4600 Sunset Avenue, Indianapolis, Indiana 46208) and Delft, the
Netherlands. IGWMC operates as a clearinghouse for groundwater modeling codes and
organizes and annual series of short courses on the use of various codes. Similar specialized
short courses are also organized by various universities as well as professional organizations
such as the National Water Well Association (NWWA, 6375 Riverside Dr., Dublin, Ohio 43017).
The aforementioned references and organizations may be consulted for the availability of
various modeling codes.
VII-4 Examples of Advection-Dominated Modeling
Much of this report is devoted to field methods for obtaining the vertical distribution of
horizontal hydraulic conductivity. Such information obtained at an array of observation wells
can serve, among other things, as a basis for the development of fully three-dimensional models
or quasi-three-dimensional models of transport processes in aquifers. In order to represent the
transport process in a manner that is realistic physically, such models must be advection-
dominated and solved in such a way that significant amounts of numerical dispersion is not
introduced.
The problem of developing numerical schemes that allow for the preservation of large
gradients on grids of practical size has been with us ever since the development of modern
numerical models. Such large persisting gradients can occur at the boundaries of contaminant
plumes, at interfaces of fresh and salt water, and in various multiphase flows in oil reservoirs.
Techniques that allow one to preserve large gradients in advection-dominated flows may be
termed "front-tracking methods", and were discussed previously in this chapter. There are now
a variety of such techniques that fall into the classification of Lagrangian methods or Eulerian-
Lagrangian methods (Cady and Neuman, 1987). The remainder of this chapter will review and
elaborate on some of the recent models and modeling applications of the Lagrangian or
Eulerian-Lagrangian type. Specific applications will involve concentration measurements made
during forced-gradient tracer experiments and measurements made in aquifer fluids supporting
microbial activity.
VII-4.1 Performance of Single-Well and Two-Well Tracer Experiments
The most common types of forced gradient tracer experiments are single-well ^
experiments and two-well experiments. In the past, both types have been performed in Canada
(Pickens and Grisak, 1981) and in the United States (Molz et al., 1985, 1986b). Shown in
Figure VIM is a typical configuration for a single-well experiment. The term "single-well"
represents the fact that only one pumping well is required in order to perform the experiment.
As detailed in Giiven et al. (1985), an observation well with multilevel samplers is required in
order to obtain concentration versus time data at selected locations. One or more such
140
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FLOW RATE
DISTRIBUTION
IN AQUIFER
TRACER
IN
TO SAMPLE
COLLECTION
f
/////////////
TRACER
. INJECTION
-*2-~ WELL • •
. OBSERVATION
' .'• WELL • ; 7^_
•. ' WITH • - '• .
• MULTILEVEL
, • SAMPLERS
BASIC CONCEPT OF SINGLE-WELL
TRACER TEST
Figure VII-1. Vertical Cross-sectional diagram showing single-well test geometry.
141
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observation/sampling wells may be used in any particular tracer study. Actual performance of
the experiment involves the injection of water having a known concentration of tracer in a well
which is fully penetrating (Fig. VII-1). After some time, the flow may be reversed and the
tracer-labeled water removed from the same well.
A typical configuration and flow pattern for a two-well tracer experiment is illustrated in
Fig. VII-2. Here there are two pumping wells because the experiment involves the simultaneous
operation of an injection well and a withdrawal well, both of which are fully penetrating. Water
is pumped into the injection well at a steady flow rate and is removed from the withdrawal well,
usually at the same rate, although two-well tests have been performed in which the flow rates in
the two pumping wells were not equal. A conservative tracer of known concentration is added
at the injection well for a period of time and the concentration of tracer in the water leaving the
withdrawal well is measured and recorded as a function of time to give a concentration versus
time breakthrough curve. Multilevel sampling wells may be placed at various locations in the
flow field (Molz et al., 1986b, 1988).
VII-4.2 Advection-Dominated Modeling of a Single- Well Tracer Experiment
A single-well tracer experiment was performed by Pickens and Grisak (198 la) in which
K(z) was measured, and concentration vs. time breakthrough curves were recorded in several
multilevel sampling wells. An Eulerian-Lagrangian numerical model called SWADM was
developed by Giiven et al. (1985) to simulate several of the experiments. This model took into
account depth-dependent advection in the radial direction and local hydrodynamic dispersion in
the vertical and radial directions (Giiven et al. 1985). The model is based on the equation given
by
+ vr = _ r
at or r or or
where r is the radial coordinate, C = C(r,z,t) is the tracer concentration, vr = vr(r,z) is the
radial seepage velocity, Dr = Dd + ar|vr| is the radial (longitudinal) dispersion coefficient, Dz =
Dd + Ov|vr| is the vertical dispersion coefficient, Dd is the effective molecular diffusion
coefficient, and ar and Oy are the radial and vertical local dispersivities.
The detailed single-well tracer dispersion experiment of interest was performed in a
shallow unconfined aquifer. A volume of 95.6 cubic meters of tracer-labeled water was injected
into an 8.2 m thick aquifer at a rate of 3.2 m3/hr for a period of 30 hours and then withdrawn
at the same rate. Withdrawal began immediately at the end of injection. Multi-level samplers
were located in the aquifer at distances of 1, 2, 3, 4 and 6 m from the injection-withdrawal well.
From the relative tracer arrival times at different elevations in the observation wells, a
dimensipnless radial hydraulic conductivity distribution in the vertical (K/K) was calculated,
where K is vertically averaged K. Additionally, Pickens and Grisak (198 la) determined that the
effective porosity was 0.38 and found the values of local longitudinal dispersivity at each _
sampling point to be fairly constant with an average magnitude of about 0.007 m. The K/K
distribution inferred from the breakthrough data at the observation well at a distance of 1 m
from the injection-withdrawal well in test SW1 was used in the SWADM simulation. This
profile is shown in Figure VII-3.
142
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Injection
Well
Multilevel
Observation
Well
t
Withdrawal
Well
\ \ \ \\\\ \\ \ \ \ \ V
• .' I • .
• «-_ I I ^—»
. i ! ' • • .
.'.-. :ji
i !•
Aquifer
• • j!.'- • -U- • • • •• •'•
• /J ! ' * U- 'L^J Li • • '•
* '"".'• i I • ' ' • ' M * ' ' I I * ' ' ' '
''////////7V/ ///////// ////////) ///
//'
Figure VII-2. Geometry and flow field for a two-well tracer test.
143
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depth
from
upper
confining
layer
(meters)
*
0
1 .0
2.0
3.0
5 Q
6.0
7.0
8.0
o
r= I m
I .0
K/K
2.0
Figure VII-3. Hydraulic conductivity profile measured by Pickens and Grisak (1981) and used
in the calculations by Giiven et al. (1985).
144
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In Figures VII-4 and VII-5, the flow-weighted concentration breakthrough curves from
observation wells located 1 and 2 m from the injection-withdrawal well respectively (Pickens and
Grisak, 198 Ib) are shown along with the corresponding breakthrough curves calculated by
SWADM. The experimental concentration versus time data measured at the injection-
withdrawal well are shown in Figure VII-6 along with the results of the SWADM simulation
using the unsteady input concentrations. The early part of the experimental data seems to show
a large amount of scatter, however, this part of the curve is closely modeled by SWADM using
the actual unsteady injection concentration. The later part of the breakthrough curve is
underestimated by SWADM, and this is discussed further in Giiven et al. (1985). In obtaining
the results shown in Figures VII-4 and VII-5, and VII-6, no "model calibration" of any type was
performed. Only parameter values measured by Pickens and Grisak (198 la) were utilized. The
resulting curves represent very accurate simulations which indicate an advection-dominated
dispersion process with local dispersivities approaching those measured in the laboratory. As
also discussed in more detail by Molz et al. (1983) and Giiven et al (1984), it is clear that if a
full-aquifer dispersivity were calculated from these data it would not represent a physical
property of the aquifer.
VII-4.3 Advection-Dominated Modeling of Two-Well Tracer Experiments
Both fully Lagrangian and fully Eulerian models have been developed for simulating two-
well tracer experiments. Giiven et al. (1986) describe application of a Lagrangian-based analysis
of the two-well tracer experiment performed by Pickens and Grisak (1981a). Theory and
experiment matched well, indicating once again the advection domination of the transport
process. Both Eulerian and Lagrangian models were used to simulate a two-well tracer
experiment performed by Molz et al. (1986b,c). The comparison of the two numerical
approaches is interesting and is described below.
Application of the two modeling approaches was done separately and independently.
Under contract to Auburn University, Geotrans, Inc. developed a three-dimensional Eulerian
advection-dispersion model that took advantage of the particular geometry of the experiment
(Huyakorn et al., 1986a,b). The aquifer was divided vertically into 12 layers of varying
thicknesses and flow between the injection and production wells was assumed to be stratified,
steady and horizontal within each layer. The advection pattern for such a situation is well
known, so the Darcy velocity, U, could be calculated at any particular point within the 12-layer
system (Huyakorn et al., 1986a,b). Given the known velocity distribution, the advection-
dispersion equation was solved using a finite element approach (Huyakorn et al., 1986b) with
the governing equation written in three-dimensional curvilinear streamline coordinates (s,n,z)
where s and n are the horizontal coordinate along and normal to a local streamline, and z is the
vertical coordinate. This type of coordinate system minimizes numerical dispersion problems.
The transformed advection-dispersion equation is given by
_L L (h2Ds^l) +_L !_ (\D?£-) +1 (Dza.£) -H-fS -^ = 0 (VII-5)
h2dsV2 sas' h^dn ' ndn dz^ zdz Q ds dt V '
where Ds, Dn and Dz are principal components of the hydrodynamic dispersion matrix in the
longitudinal, transverse and vertical directions, respectively, and hj and h2 are the scale factors
of the curvilinear coordinate system (Huyakorn et al., 1986a).
The second model used to simulate the two-well experiment is called the two-well
advection model (TWAM) (Giiven et al., 1986). A Lagrangian solution method is used in this
model based on the travel times of tracer along various flow lines from one well to the other.
In this model, it is assumed that the aquifer is horizontal, confined, of constant thickness and
145
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I .00-
0.75-
0.50-
0.25-
0.00-
•EXPERIMENTAL DATA
[PICKENS AND GRISAK, I98lb]
NUMERICAL RESULT
= a = 0.007m
0
15 20 25
30
TIME (HOURS)
Figure VII-4. Comparison of SWADM results with field data for the flow-weighted
concentration from an observation well one meter from the injection-withdrawal
well. From Giiven et al. (1985).
146
-------
.00-
0.75-
0.50-
0.25-
0.00-1
0
• EXPERIMENTAL DATA
[PICKENS AND GRISAK, I98lb]
NUMERICAL RESULT
Q=a= 0.007m
10 15 20
TIME (HOURS)
25
30
Figure VII-5. Comparisons of SWADM results with field data for the flow-weighted
concentration from an observation well two meters from the injection-withdrawal
well. From Guven et al. (1985).
147
-------
1.0-
0.8-
0.6-
0.4-
0.2-
0.0-
= 3.2mVhr
= 8.2m
V, =9 5.6m3
©EXPERIMENTAL
DATA
[PICKENS AND GRISAK,"
I 98 la]
-NUMERICAL
RESULT
ar=avB0.007m
o o
0.0 0.4 0.8 1.2 1.6
VOL. WITHDRAWN / VOL. INJECTED
Figure VII-6. Comparison of SWADM results with field data for the concentration leaving the
injection-withdrawal well. From Giiven et al. (1985).
148
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porosity, and perfectly stratified in the vicinity of the test weJls. TWAM takes into account the
depth-dependent advection in the horizontal planes, but neglects completely any local
hydrodynamic dispersion. Thus any simulations resulting from application of TWAM will yield
dispersed concentration distributions based solely on differential advection.
Shown in Figure VII-7 are the results of the three-dimensional dispersion simulation and
the advection simulation using the model called TWAM (Gilven et al, 1986). Both models do a
good job of predicting the recovery concentrations during the two-well test, although the
Lagrangian model ran in a small fraction of the time required by the Eulerian approach. The
result of the Lagrangian analysis showed that local hydrodynamic dispersion played a very minor
role in determining the time distribution of tracer concentration in the withdrawal well. The
entire experiment, which involved estimated travel distances over individual flow paths ranging
from 38.3 m to about 90 m in the most permeable layer, was highly advection-dominated.
If one assumed a homogeneous aquifer during the performance of the two-well
experiment being analyzed, the tracer concentration versus time curve shown in Figure VII-8
would be predicted. One could make the theoretica.1 curve fit the data well by using a scale-
dependent longitudinal dispersivity that was about 4 m at the 38 m separation of the injection
and production wells ((Huyakorn et al, 1986b). However, the previous results show that this
traditional approach is purely curve-fitting.
During the summer of 1985 a second two-well tracer experiment was performed at the
Mobile site in a different portion of the study aquifer (Molz et al., 1988), The same Lagrangian
model (TWAM), but with a different hydraulic conductivity distribution, was used to simulate
the tracer recovery concentration in the withdrawal well. The results shown in Figure VII-9 are
again good. This decreases the probability that the relatively good comparison between theory
and experiment at the Mobile site was simply fortuitous.
VII-4.4 Advection-Dominated Modeling of JSubsurface TrjLnsporLPrQcggses
that Incorporate Chemical .and,Micrgbiai Kinetics
Contaminant transport models that incorporate chemical or microbial reaction terms are,
in many common situations, very sensitive to hydrodynamic dispersion, particularly vertical
transverse dispersion (Molz and Widdowson, 1988). This steins from the fact that for such
reactions to occur, it is necessary for a number of dissolved substances to come into intimate
physical contact. Thus anything in a mathematical model that causes artificial mixing, like
dispersion coefficients that are unrealisticaliy large, will cause the model to overpredict the rates
and distribution of possible chemical and microbial reactions.
In order to study such processes with a mathematical model relatively free from
numerical dispersion, Molz et al. (1986a) developed an Eulerian-Lagrangian model for
simulating microbial growth dynamics coupled to nutrient and oxygen transport in porous media.
This model was based on the split-operator approach of Holly and Preissman (1977), and the
microbial kinetics employed by Benefield and Molz (1983). A two-dimensional version of the
model was used to explore the transport implications of large chemical concentration gradients
that have been measured recently in natural aquifers by several research groups (Barker, Patrick
and Major, 1987; Ronen et al., 1987a,b; Smith et al., 1987). The actual equations used have
been described in detail previously (Molz et al., 1986a; Widdowson et ai., 1988), and values
selected for the various parameters may be found in Molz and Widdowson (1988). The overall
model consists of two advection-dispersion-adsorption-consumption equations describing
transport of substrate (organic carbon and energy source) and oxygen at the Darcy scale, two
149
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f
CONCENTRATION (MG/L)
^ a S?
VD GO ^>
f^f-i *"*• r*.
ON CB ^
P> ' Si
8
S,l
5'S
I'I
8,1
93 W
[13 ^
CB
o a.
3 P
O
D
r-o
O!
(Jl
b
-^i
Ot
—
p
b
—
ro
01
—
Ol
b
— N
H O
o! b
w
ro
U1
ro
OI
b
-------
o
I-
ir
H
LU
O
O
O
50h
45
40
35
30
25
20
I 5
10
O
o
o
o
o
o
o
o
o
40 80 120 160 ZOO 240 280 32O 360 400 440 480 520 560 600 64O 68O 720 760
TIME (hrs.)
Figure VII-8. Calculated tracer concentration versus time in the withdrawal well based on an
assumed homogeneous, isotropic aquifer with no local dispersion (circles) shown
together with the results of the first two-well test (full line) at the Mobile site.
151
-------
A Predicted
•• Measured Data
40 80 120 160 200 240 280 320 360 400 440 480 520 560 600 640 680 720 760
Time (hours)
Figure VII-9. Measured and predicted Br concentration versus time in the withdrawal well
during the second two-well tracer test at the Mobile site. From Molz et al.
(1988).
152
-------
algebraic chemical kinetic equations describing uptake of substrate and oxygen by microbes, and
one additional kinetic equation describing growth or decay of the particle-bound microbial
population. The flow direction is assumed to be horizontal, with flow magnitude varying only
with vertical position. Diffusion/dispersion, however, is fully two dimensional.
Shown in Figure VII-10 is the hypothetical domain and velocity distribution that was
selected as the basis for the generic modeling effort. Flow in a vertical cross section of the
aquifer is assumed to be horizontal and perfectly stratified. However, both oxygen and substrate
are allowed to move in the horizontal and vertical directions. Boundary conditions were
selected to be compatible with a contaminant source in the upper left-hand region of the
domain. Oxygen but not substrate (contaminant) was allowed to cross the water table.
Shown in Figure VII-11 is simulated oxygen concentration versus depth for two different
values of vertical dispersivity. The profiles correspond to a travel distance of 62 m at an
average horizontal seepage velocity of 10 cm/day. Indicated on the figure is the maximum value
of oxygen concentration gradient. It is evident that to obtain a concentration gradient and
overall oxygen distribution roughly similar to that measured by Barker et al. (1987), and shown
in Figure VII-12, it is necessary to use a vertical transverse dispersivity of about 0.1 cm. This
shows simultaneously the numerical capability of Eulerian-Lagrangian models and the necessity
for using such models if one is to simulate accurately chemical/microbial processes that occur
commonly in porous media.
153
-------
Figure VII-10. Schematic diagram of the hypothetical domain, velocity distribution and boundary
conditions used in generic model simulations. O - oxygen concentration; S =
substrate concentration. From Molz and Widdowson (1988).
154
-------
OXYGEN CONCENTRATION (mg/f)
-0 2 ^7 4 6 8 re 10
^T^^^^~ I
GRADIENT=0.066 mg/J?/cm
ot=0.l cm
AVERAGE SEEPAGE
VELOCITY = IOcm/day
TRAVEL DIST.
I61-
Figure VII-11. Simulated oxygen concentration versus depth below water table for two different
values of vertical dispersivity and an average horizontal seepage velocity of 10
cm/day. From Molz and Widdowson (1988).
155
-------
E
JH i±
_Q I"
O
H
X_
(D
"5
_o
(D
DQ
0_
-------
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APPENDIX I
OVERVIEW AND EVALUATION OF METHODS FOR DETERMINING
THE DISTRIBUTION OF HORIZONTAL HYDRAULIC
CONDUCTIVITY IN THE VERTICAL DIMENSION
AI-1 Introduction
This appendix will start with an overview of several techniques for measuring K(z), the
vertical distribution of horizontal hydraulic conductivity. It is based largely on a paper by Taylor
et al. (1989) published in Ground Water. Therefore, much of the credit for the overview portion
of Appendix I should be given to the major authors (K. Taylor, S. Wheatcraft, J. Hess and J.
Hayworth) of the Taylor et al. (1989) paper. This reference should be consulted for a more in-
depth evaluation of those techniques not emphasized in this report.
As discussed in more detail in Chapter I of this report, application of the advection-based
modeling approach requires measurement of hydraulic conductivity distributions. This has been
done previously using forced-gradient tracer techniques (Molz et al., 1988, 1989b), but in the
authors' opinion, tracer technology is expensive and time-consuming and can only be justified in
granular aquifers for exceptional situations. Therefore, tracer methodology will not be discussed
further. The remainder of this appendix will be devoted to tests designed to be performed in
boreholes. We will assume that the holes are screened wells because most past applications have
been in granular aquifers. Undisturbed cores are also difficult to obtain in such formations and are
of limited use.
An important general consideration when making measurements of hydraulic conductivity
is the volume over which the measurement is averaged. This volume spans the range of a few
tenths of a liter for core studies to hundreds or thousands of cubic meters for hydraulic testing of
aquifers. Depending on the intended use of the hydraulic conductivity data, the volume over which
the measurement is made may be significant. When the measured volume is too large, important
small scale features may be ignored. When the measured volume is small, there is a tendency to
undersample, which can also result in missing significant features. The exact definition of large or
small depends on the local variability of the hydraulic properties and the intended application of
the data.
Another important consideration is that most natural formations have a strong horizontal
to vertical anisotropy ratio with respect to hydraulic conductivity. Anisotropy ratios on the order
of 10:1 or more are common [Freeze and Cherry, 1979]. In such situations, measurements of
hydraulic conductivity made in one direction are of limited use when modeling fluid movement that
is occurring in another direction. When hydraulic conductivity is treated as a scalar or a
diagonalized matrix, which for convenience it usually is, it is important to be sure that the fluid
movement that is being modeled is consistent with the directions in which the scalar values of
hydraulic conductivity are measured.
All borehole methods sense the properties of the formation immediately surrounding the
well. The distance into the formation away from the well for which the measurement is
representative is referred to as the radius of investigation. Depending on the method, the radius
of investigation can range from about 0.05 to 5 m. It is important to ensure that this zone is not
disturbed significantly by the presence of the borehole or drilling effects such as drilling mud forced
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into the formation. Morin et al. (1988b) discuss the effects of various drilling methods on the
development of the disturbed zone.
AL2 Straddle Packer Tests
One of the most common methods for determination of the vertical distribution of average
horizontal hydraulic conductivity is to perform hydraulic testing over short intervals of a borehole,
The section of the borehole that is of interest is commonly isolated from the rest of the borehole
by a straddle packer (Fig. AI-1). There are several variations of straddle packer tests. For
example, one can pump at a constant rate from or into the packed-off section while measuring
head, or inject at a constant head while measuring flow. A common method is to change the head
suddenly in the packed-off section by adding or displacing a volume of water, and then monitoring
the return to equilibrium by recording head vs time. This latter procedure may be called a multi-
level slug test. In any case, these methods are accurate only if the packer is effective in
hydraulically isolating the segment of the borehole. If channels exist around the well screen, it will
not be possible to obtain a good seal with the packer. Channels may be present in the structure
of a well screen or can be caused by the failure of the formation or backfill material to fill the
annulus between the casing and the borehole wall. A similar problem may occur if a gravel pack
is installed that has a greater hydraulic conductivity than the formation. In this case, fluid will
preferentially flow along the well, bypassing the packer, instead of flowing radially toward, or
outward, from the well as is desired. Ideally, the well is constructed with short screened intervals
that are isolated from one another by grouting between the intervals. However, this type of
construction is expensive.
If leakage around the packers exist, results obtained with a straddle packer test will indicate
a hydraulic conductivity that is erroneously high. To test if this is occurring, it is desirable to
monitor the head in the zones that are hydraulically isolated from the packed off interval.
Typically, this would require a pressure transducer above and below the packed off interval.
However, if the transmissivity of the segments of the well that are not packed-off is larger than the
transmissivity of the segment of the well that is packed-off, leakage of fluid around the straddle
packer will not cause a detectable change in head outside the packed off interval. To identify this
problem, it is necessary to install a second set of packers (Fig. AI-1). The hydraulic head in the
segments of the well that are between the two sets of packers will now be sensitive to leakage
around the first set of packers that are isolating the segment that is being tested.
If the hydraulic head in the segments between the two sets of packers is influenced
significantly by the hydraulic testing in the packed-off interval, it is then known that the straddle
packer is not isolating the segment of the well and, hence, the results are not valid. If this situation
occurs, it is usually not possible to correct and the straddle packer method cannot be used. Because
of the need for four packers and three transducers, it can be a cumbersome arrangement to operate
in the field. Nevertheless, based on comparisons with other test results, the straddle packer
technique worked well at the Mobile site and, therefore, was selected for detailed study.
The straddle packer method can be used to measure hydraulic conductivity over well
segments that range from centimeters to hundreds of meters in length. However, the data must
be analyzed carefully for small test intervals because the flow can have significant vertical
components (Dagan, 1978; Melville et al., 1989). The calculated hydraulic conductivity reflects that
of the formation material within 25 to 35 well radii for a typical 2 in (5 cm) well (Braester and
Thunvik, 1984).
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PRESSURE
B
Figure AI-1. Details of an inflatable straddle packer design.
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AI-3 Particle Size Methods
In a material consisting of unconsolidated particles, the hydraulic conductivity is controlled,
in part, by the size and distribution of the pores. In an effort to quantify this, Fair and Hatch
(1933) and Masch and Denny (1966) have developed analytical approaches to estimate hydraulic
conductivity from a description of the formation grains. The model proposed by Fair and Hatch
requires that the distribution of grain sizes be known. The model proposed by Masch and Denny
requires that the mean and standard deviation of the grain size be known.
Both of these methods suffer from several of the fundamental problems listed below.
1. Samples must be collected during drilling. This is not always done, hence for many
if not most existing wells, these methods cannot be used.
2. To determine the grain-size statistics required for the analysis, the formation must
be sieved. Obviously, features such as small scale layering, compaction, and sorting
are destroyed by this process. If these features exist, which is usually the case, then
the material which is evaluated will not be representative of the formation.
3. Sampling bias may be introduced by the sampling method. The sampling method
may be unable to collect large material such as gravel or may not adequately sample
fine particles.
4. The methods are limited to clean formations with sand-size particles (greater than
0.06 mm). Formations that have silt or clay-size material cannot be accurately
analyzed with these methods.
Because of these problems, grain-size analysis methods have a restricted application and
should be considered as approximate. They are unlikely to be suitable for characterization of
aquifers for detailed contaminant transport modeling.
AI-4 Empirical Relationships Between Electrical and Hydraulic Conductivity
The electrical conductivity of a porous medium is a measure of the ability of the medium
to conduct electrical current. In natural formations, electrical conduction occurs along two paths.
The first path is by ionic conduction through the pore fluid. This is controlled by the electrical
conductivity of the pore fluid, the volume of the pore fluid, and the manner in which the pores are
connected. The size of individual pores does not influence the electrical conductivity of the fluid.
The second path for electrical conduction is along the surface of the formation matrix. This path
is a function of the type and distribution of the matrix mineralogy, particularly clay minerals. In
clay-free formations, with a constant pore fluid electrical conductivity, the formation electrical
conductivity is usually found to be a function of porosity. Archie's rule is a frequently used
relationship relating electrical conductivity and porosity in clay-free formations (Keller and
Frischknecht, 1966).
The hydraulic conductivity of a porous medium is a function of the size of the pores and
the manner in which they are connected. Two formations with the same pore fluid that have the
same pore volume (porosity), but have different size pores, will have the same electrical conductivity
and porosity, but different hydraulic conductivities. Hence, in general, there is not a unique
relationship between electrical and hydraulic properties. The situation is further complicated when
anisotropic effects are considered, because the axis of anisotropy for electrical and hydraulic
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conductivity may not coincide. The presence of clays will further complicate any relationship
between electrical and hydraulic conductivity.
Despite these problems, there exist numerous examples in the literature of empirical
relationships that have been developed between electrical and hydraulic properties (Mazac et al.,
1985; Kwader, 1985; Huntley, 1986; Urish, 1981). These relationships were developed in clay-free
formations where electrical conduction by the matrix was not a significant factor. It is also
necessary for the formation to have a relationship between porosity and hydraulic conductivity and
to have a pore fluid of constant and known electrical conductivity. Depending on the formation and
the methods used to measure the properties, both positive and negative correlations between the
two properties have been observed. These empirical relationships are only applicable over limited
areas of a specific formation. Such restrictions, and the need to measure the hydraulic
conductivities at numerous locations first to define the relationship, severely limit the utility of this
approach. However, if a relationship can be defined, the electrical measurements can be made
rapidly and a large number of hydraulic conductivity determinations can be made with little
additional effort.
The radius of investigation of this method is dependent on the process that is used to
determine the electrical and hydraulic conductivities. The hydraulic conductivities are usually
measured with hydraulic testing and, hence, typically have a radius of investigation of several
meters. The radius of investigation of the electrical measurements is controlled by the
instrumentation used to make the measurement and should be comparable to the radius of
investigation of the hydraulic measurements.
AI-5 Measurements Based on Natural Flow Through a Well
There are several techniques whereby one can obtain information about the hydraulic
conductivity distribution surrounding a well by measuring the natural fluid velocity distribution
through the well. These methods work best when the fluid velocity is horizontal. The basic idea
of these methods is illustrated in Figure AI-2. They differ mainly in how the velocity measurement
is made within the packed-off section of the well. There are heat-pulse devices for making the
measurement (Melville et al., 1985) and various types of point-dilution approaches (Drost et al.,
1968; McLinn and Palmer, 1989; Taylor et al., 1989). In this latter approach, a tracer is injected
into the segment of the well of interest and must be kept well mixed. The tracer is removed from
the segment by diffusion and by advection of the fluid moving horizontally through the well.
Vertical fluid movement is blocked by packers. If the fluid velocity is high, the tracer concentration,
which must be recorded, will decrease more rapidly than if the fluid velocity is low. Because the
decay is exponential, the slope of the tracer decay curve on a semi-log plot is a function of the
horizontal fluid velocity.
A new type of point-dilution apparatus based on an arrangement of dialysis cells is
illustrated in Figure AI-3. Glass cylinders having selected types of semipermeable membranes as
their ends are mounted along a positioning rod. Each cell, which has a flexible rubber seal above
and below, is filled with water that is depleted in the isotope oxygen-18, i.e. the O18/O16 ratio is
different for the water within the cell compared to the natural groundwater (Alternatively, some
other tracer may be used.). The entire apparatus, which may contain something like 20 or more
dialysis cells spaced at equal intervals along the rod, is then lowered into the well and positioned
within the screen. Immediately after positioning, oxygen-18 begins to diffuse into each cell, and the
rate of diffusion depends on the groundwater flow velocity in the immediate cell vicinity. By
measuring O18/O16 ratios in each cell before and after a test period, the groundwater flow velocity
as a function of cell position can be calculated. The calculation procedure is only moderately
177
-------
Land
Groundwater
Flow
Surface
Flow Velocity
Meas
Here
Straddle
Packer system
Figure AI-2. Schematic diagram Illustrating a natural flow field in the vicinity of a well.
178
-------
To The
I Surface
Well Screen
Positioning Rod
Flexible Seals
Dialysis Cell
Semipermeable
Membrane
Darcy
Figure AI-3. Geometry and instrumentation associated with the dialysis cell method for
measurement of Darcy velocity.
179
-------
involved and is described in Ronen et al., 1986, who developed and applied the technique in an
unconfined aquifer. The method was also applied with apparent success at the Mobile site. Once
the cells and cell holders/isolators are available in large quantities, many measurements can be
made fairly efficiently.
As the title of this section implies, the natural flow methods result in a velocity
measurement not a hydraulic conductivity measurement. If one assumes that the head gradient is
predominantly in the horizontal direction, constant with depth and the formation has a constant
porosity, then K(z) will be proportional to the fluid velocity distribution v(z). An approach that
results in a more direct calculation of K is described by Taylor et al., 1989.
All of the natural flow methods are relatively difficult to apply and the resulting data
difficult to interpret. Due to a variety of factors, a complex flow pattern develops around a well
screen that is sensitive to near-hole disturbances. Some methods require that the packed-off section
be filled with glass beads, and it is difficult or impossible to achieve the same bead packing in all
the measurement sections.
AI-6 Single Well Electrical Tracer (SWET) Test
In the single well electrical tracer (SWET) method (Taylor et al., 1988), salt water is injected
under steady-state conditions into a well. While injection of the tracer continues, the radius of
invasion of the tracer is determined with a borehole induction tool (Figure AI-4). By repeatedly
measuring the depth of invasion at different times, the rate of invasion can be determined. The
hydraulic head, which is a measure of the driving force required to inject the fluid, is also noted.
The tracer will invade different intervals of the formation at different rates depending on the
hydraulic properties of each interval of the formation. This information can be used to calculate
a hydraulic conductivity log of the formation. Because multiple induction logs are run, the rate of
invasion can be determined at several different radii of invasion. Hence, the hydraulic conductivity
log of the formation can be calculated at several different radii of invasion. A porosity log can also
be calculated by using a model of formation electrical conductivity that accounts for variations in
matrix conductivity and porosity. The SWET test procedure was field-tested for the first time at
the Mobile site during the summer of 1987.
The hydraulic conductivities calculated by the SWET test are representative of the formation
for a radius around the well that is roughly equivalent to the radius of effect of the well. This
radius is larger than the radius of invasion because the rate of invasion is not only influenced by
the hydraulic properties of the portion of the formation that is invaded, but is also influenced by
the portion of the formation into which the displaced native fluid is forced. At the Mobile site, this
was on the order of 4 m. Because most wells have a disturbed zone around them, techniques that
have a shallow radius of investigation will be inaccurate; the SWET test minimizes these problems.
Another advantage of the SWET test is that the entire well is subjected to the same hydraulic head.
Straddle packer tests pressurize only a portion of the well and will be in error if there is leakage
around the packer.
A disadvantage of the SWET test is that the method requires the careful injection of a large
volume of electrolyte. At some locations this may not be allowed. However, in the author's opinion
further study of the SWET test is warranted.
180
-------
ELECTROMAGNETIC
CONDUCTIVITY
TEST
ELECTROLYTE INJECTION*^ — -
LINE "' •''
. —^
•
^
•
/"-
".
• Y
TO LOGC
*-LAND
CAP
.ER
SURFACE
ROCK
ELECTROLYTE
FRONT
HIGH PERMEABILITY
ZONES
CONDUCTIVITY
PROBE
Figure AI-4. Apparatus and geometry associated with the SWET test.
181
-------
PUMP
CAP ROCK
(Q=DISCHARGE RATE)
BOREHOLE FLOW
METER •
ELEVATION=Z
v
TO LOGGER (Q)
SURFACE
CASING
SCREEN
DATA
Q
Figure AI-5. Apparatus and geometry associated with a borehole flowmeter test.
182
-------
AI-7 Borehole Flowmeter Tests
The idea behind this test is illustrated in Figure AI-5. A small pump is placed in a well and
operated at a constant flow rate, Q. After something near steady-state behavior is obtained, the
flowmeter, which measures vertical flow, is lowered to near the bottom of the well and a reading
taken. The meter is then raised a few feet, another reading taken, raised another few feet - and
so on. As illustrated in the lower portion of Figure AI-5, the result is a series of data points giving
vertical discharge within the well screen as a function of position. Just above the top of the screen
the meter reading should be equal to Q, the steady pumping rate that is measured independently
on the surface.
The basic data analysis procedure is quite easy. One simply takes the difference between
two successive meter readings, which yields the net flow entering the screen segment between the
elevations where the readings were taken. This information may be analyzed in several ways to
obtain a K(z) value. Along with the multi-level slug test, the flowmeter test was selected for
detailed study at the Mobile site.
The flowmeter test suffers from the lack of ready availability of impeller meters designed
for water well applications. Also, other types of promising technologies for flowmeter applications,
such as heat-pulse (Hess and Paillet, 1989) and electromagnetic (Young and Waldrop, 1989)
techniques, are not fully developed. However, it does appear that some types of heat-pulse (Hess
and Paillet, 1989) and electromagnetic (Young and Waldrop, 1989) water well flow meters will be
available in the very near future.
AI-8 The Role of Geophysical Logging
The more traditional geophysical logging methods such as gamma logs, electric logs of
various types, nuclear logs, etc., can be used to help identify the overall stratigraphy and geological
setting of a site. They can also provide information of a general nature concerning hydraulic
conductivity distributions. Applicable techniques are reviewed by Taylor (1989), and an approach
to application and interpretation is provided by Paillet (1989) for groundwater investigations.
Detailed descriptions of methodology may be found in Keys and MacCary (1971), and a
bibliography of borehole geophysics as applied to groundwater hydrology has been developed by
Taylor and Dey (1985).
183
-------
APPENDIX II
ANALYSIS OF PARTIALLY PENETRATING SLUG TESTS CONSIDERING
RADIAL AND VERTICAL FLOW AND ANISOTROPY
AII-1 Introduction
The authors are aware of three approaches or techniques for analyzing partially
penetrating slug tests that allow for both radial and vertical flow in an aquifer that is assumed to
be locally homogeneous and isotropic for the purpose of test analysis (Boast and Kirkham, 1971;
Bouwer and Rice, 1976; Dagan, 1978). None of these approaches are entirely satisfactory,
especially for test sections that have relatively large diameter to length ratios (Melville et al.,
1989; Widdowson et al., 1989). Therefore, there is a need for a more general approach that is
reasonably accurate, free from limiting assumptions and easy to use. In addition, it is desirable
to have a procedure that includes the effect of anisotropy in the test aquifer, since this physical
phenomenon is so common.
The purpose of Appendix II is to present the details of a procedure for analyzing slug
test data that was developed recently at Auburn University and which considers radial and
vertical, anisotropic, axi-symmetric flow to or from the test interval. This procedure was used as
one of the 3 analysis techniques in Chapter III dealing with multi-level slug tests. It is based on
a finite element model called EFLOW that was licensed through the Electric Power Research
Institute and modified at Auburn University for partially penetrating slug test analysis.
AII-2 Mathematical Model Development
Diagrams of the two-dimensional geometry within which the mathematical model is
applied are shown in Figure AII-1. Diagram (A) applies specifically to the case of a confined
aquifer, while diagram (B) applies to the unconfined case. When analyzing a partially
penetrating slug test in an unconfined aquifer one assumes that the water table stays at a
constant elevation throughout the test (Dagan, 1978).
In a homogeneous, anisotropic aquifer, the equation governing transient, axi-symmetric
flow is given by
Ss = K( +) + K. (AIM)
3t dr2 r dr
where Ss is specific storage, h is hydraulic head, t is time, r is radial distance, z is vertical
distance, and K and KZ are hydraulic conductivities in the radial and vertical directions
respectively. The initial and boundary conditions for simulating a slug test within the geometry
of Figure AII-1 are:
LC.) h(r, z, 0) = h0 (AII-2)
B.C.) h(rw, z, t) =h0 - y(t), for (D-H) < z * (D-H+L) (AII-3)
^L(r, 0, t) = %L(r, D, t) = 0, for rw< r
-------
(A)
•« — *•
y(t)
h(t)-h«-
2r
x. \ \ \. s. x. \ \ s. \ \ \ \\
2
\ \ \
H
\\\\\\\\\\\
R
e
(B)
-yU) +
\ \
2
t
I
2
•
B
i
v
rc
(t)
I
rw
•I
m
z;
i *
j
L
r
\ ^
, >
H
i
v \ X \
^
D
\ \ \N\\\\
R.
J
^
\ \ \
k
ho
f
Figure AII-1. Diagram illustrating the geometry within which a partially penetrating slug test is
analyzed. Diagram (A) is for the confined case and diagram (B) is for the
unconfined case.
185
-------
— (rw, z, t ) = 0, for 0 < z <(D-H) and (D-H+L) < z
-------
Since the right-hand side of (AII-1 1) is constant, this shows that a plot of ln(y) versus t will yield
a straight line.
Through the use of Darcy's law the flow into the aquifer may also be written as
-H+L,,
^(r^dz (AII-12)
D-H ar
One may now define a dimensionless flow parameter, P, given by
-H+L-,
(An'13)
The parameter, P, depends only on the geometry of a particular slug test. From numerical
solutions of equation (AII-8) for different geometries and using equation (All- 13), Figures AII-
2 and AII-3 were generated for the confined and unconfined cases respectively, snowing the
dependence of P on H/L and L/rw for isotropic conditions. Dimensionless curves for K/I^ of
1, 0.2 and 0.1 are also presented in tabular form in Tables AII-1 through AII-6. From this data
it is possible for an individual to develop his own detailed figures, or interpolate directly from
the tables.
Once the various figures or tables are developed for a given anisotropy ratio, they may
be used in combination with a semi-log plot of slug test data to calculate the hydraulic
conductivity in the radial direction. For example, assume one is working with Fig. AII-2. The
procedure is to enter the figure with the appropriate values of H/L, and L/rw, and obtain the
corresponding number for P (call it Pn). Then using equation (AII-9) one notes that
Q = 27rKLyPn = - Ac(dy/dt) (AII-14)
Employing the fact that (l/y)dy/dt = d(ln(y))/dt and solving equation (AII-14) for K yields
K = -_s_ = .__c__ (2.3B) (All- 15)
2flLPn dt 2wLPnV ' V '
where B is the slope of a semi-log plot (base 10 logs) of y vs. t, with the y vs. t values obtained
from an actual slug test. B should always be considered a negative number regardless of
whether y is above or below the reference level during the slug test.
AII-4 Numerical Example
Multi-level slug test data from the Mobile site has been analyzed using the method
presented here (Melville et al., 1989). Data from eleven levels in a test well are shown in
Figure AII-4 along with straight line representations using linear regression. The following,
which applies specifically to the data centered at z~ 11.2 ft in Figure AII-4, is an outline of the
procedure by which the individual hydraulic conductivity values were calculated:
1. Obtain a measurement or estimate of aquifer anisotropy ratio.
= 6.7:1. (Parr et al., 1983)
187
-------
Q.
CN
"i i t i I i i i i I i i i i i i i i a I i i i i I i i i t\ i i i
).8 1.0 1.2 1.4 1.6 1.8 2.0 2.2
Log (L/rw)
Figure AII-2. Plots of dimensionless discharge, P = Q/27rKLy, for the isotropic, confined
aquifer problem as a function of L/rw and H/L.
188
-------
CL
CM
O
0.70
0.60
0.50
0.40
0.30
I I I I I I I I I I I I I I I I I I I
I I
Q 90 " ' " " ' ' i i i I i ii i i i i i
0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2
Log (L/rw )
Figure AII-3. Plots of dimensionless discharge, P = Q/2flKLy, for the isotropic, unconfined
aquifer problem as a function of L/rw and H/L.
189
-------
vD
O
Table AIM. Dimensionless discharge, P, as a function of H/L and L/rw for the confined case with
= 1.0.
H/L
1
1.25
1.5
2
4
8
16
L/rw = 8
.4117
.4448
.4617
.4805
.5029
.5155
.5243
12
.3610
.3905
.4045
.4219
.4370
.4463
.4526
18
.3196
.3456
.3570
.3725
.3829
.3898
.3945
24
.2964
.3202
.3303
.3402
.3519
.3576
.3610
36
.2675
.2882
.2965
.3045
.3140
.3183
.3207
48
.2497
.2685
.2757
.2828
.2908
.2945
.2964
72
.2293
.2455
.2515
.2575
.2645
.2674
.2687
96
.2147
.2295
.2348
.2400
.2459
.2484
.2496
-------
Table AII-2. Dimensionless discharge, P, as a function of H/L and L/rw for the confined case with
K/K, = 0.2.
H/L
1
1.25
1.5
2
4
8
16
L/rw = 8
.3205
.3428
.3533
.3660
.3771
.3837
.3878
12
.2874
.3076
.3165
.3279
.3360
.3411
.3442
18
.2597
.2778
.2852
.2950
.3013
.3053
.3076
24
.2434
.2601
.2667
.2741
.2806
.2840
.2858
36
.2230
.2377
.2434
.2487
.2551
.2577
.2589
48
.2102
.2238
.2288
.2336
.2392
.2415
.2424
72
.1955
.2078
.2124
.2168
.2215
.2232
.2237
96
.1847
.1957
.1997
.2034
.2076
.2092
.2096
-------
Table AII-3. Dimensionless discharge, P, as a function of H/L and L/rw for the confined case with
K/K, = 0.1.
vO
NJ
H/L
1
1.25
1.5
2
4
8
16
L/rw = 8
.2914
.3121
.3209
.3295
.3401
.3453
.3463
12
.2634
.2821
.2894
.2979
.3055
.3096
.3105
18
.2398
.2567
.2630
.2701
.2765
.2798
.2800
24
.2256
.2410
.2457
.2523
.2588
.2615
.2616
36
.2078
.2207
.2255
.2302
.2357
.2378
.2387
48
.1966
.2085
.2129
.2172
.2219
.2238
.2245
72
.1839
.1949
.1990
.2028
.2068
.2081
.2083
96
.1742
.1840
.1876
.1909
.1945
.1958
.1960
-------
Table AII-4. Dimensionless discharge, P, as a function of H/L and L/rw for the unconfined case with
K/K, = 1.0.
H/L
1.25
1.5
2
4
8
16
L/rw = 8
.6564
.6207
.5912
.5616
.5505
.5453
12
.5487
.5219
.4955
.4783
.4701
.4662
18
.4658
.4455
.4241
.4129
.4066
.4036
24
.4186
.4018
.3883
.3748
.3697
.3672
36
.3644
.3515
.3410
.3305
.3264
.3244
48
.3329
.3220
.3132
.3042
.3007
.2990
72
.2973
.2887
.2813
.2736
.2707
.2695
96
.2742
.2667
.2605
.2540
.2516
.2505
-------
"O
Table AII-5. Dimensionless discharge, P, as a function of H/L and L/rw for the unconfined case with
K/K, = 0.2.
H/L
1.25
1.5
2
4
8
16
L/rw = 8
.4528
.4351
.4201
.4047
.3988
.3960
12
.3944
.3802
.3683
.3564
.3517
.3494
18
.3469
.3356
.3256
.3166
.3128
.3110
24
.3187
.3090
.3018
.2926
.2894
.2879
36
.2853
.2774
.2708
.2639
.2612
.2601
48
.2651
.2582
.2524
.2463
.2441
.2431
72
.2423
.2362
.2311
.2259
.2242
.2238
96
.2258
.2206
.2162
.2117
.2102
.2097
-------
SO
Table AII-6. Dimensionless discharge, P, as a function of H/L and L/rw for the unconfined case with
K/K, = 0.1.
H/L
1.25
1.5
2
4
8
16
LAW = 8
.3960
.3824
.3724
.3587
.3540
.3517
12
.3498
.3386
.3292
.3195
.3157
.3139
18
.3114
.3023
.2946
.2867
.2835
.2821
24
.2883
.2804
.2737
.2667
.2640
.2628
36
.2605
.2539
.2482
.2424
.2402
.2393
48
.2434
.2376
.2326
.2274
.2255
.2248
72
.2237
.2185
.2141
.2098
.2085
.2083
96
.2096
.2051
.2012
.1974
.1962
.1960
-------
0.
100.
time (sec)
300.
-B-
log(y)=-
Iog(y)=-
log(y)=-
log(y)=-
log(y)=-
log(y)=-
log(y)=-
log(y)=-
iog(y)=-
|0g(y)=-
log(y)=-
-0.0280t+0.47
-0.0045t-fO.4-9
-0.0020t+0.50
-0.0038t+0.49
-0.00411+0.51
-0.0034t+0.49
-0.0026t+0.47
-0.0046t-fO.48
-0.0053t-fO.48
-0.007U+0.47
-0.0092t-fO.50
Z=11.2 ft
Z=17.2 ft
Z=23.2 ft
Z= 5.2 ft
Z=29.2 ft
Z=35.2 ft
Z=41.2 ft
Z=47.2 ft
Z=53.2 ft
Z=59.2 ft
Z=65.2 ft
Figure AII-4. Multilevel slug test data from well E6, B=log(y1/y2)/(t2-t1) = magnitude of the
slope of the log(y(t)) response.
196
-------
2. Calculate H/L and log(L/rw) from experimental geometry.
Aquifer thickness, D = 70 ft
Packer separation length, L = 3.63 ft
Distance (H) to closest boundary = 13.01 ft
Radius of screen = 0.167 ft
H/L= 3.58
L/rw = 21.8; loglo(L/rw) - 1.34
3. Select dimensionless discharge by interpolating between 1:5 and 1:10 anisotropy values in
Tables AII-2 and AII-3.
Pn = 0.277
4. Determine slope of semi-log data plot (Figure AII-4).
B = 0.028 sec'1
5. Calculate hydraulic conductivity from equation (All-15).
With Ac = 0.180 ft2, K = 0.00183 ft/sec = 158 ft/day
197
-------
APPENDIX III
THE PHYSICAL PROCESSES OF ADVECTION
AND HYDRODYNAMIC DISPERSION
AIII-1 Introduction
Most individuals have an intuitive understanding for what is meant by molecular
diffusion. Molecular diffusion of a solute (such as salt) is caused by the random thermal
motions (Brownian motion) of the water and solute molecules. In a solution undergoing perfect
laminar flow, the Brownian motion will cause an irreversible mixing of the solute and water
molecules, so that solute will slowly migrate from zones of higher concentrations to zones of
lower concentrations. If allowed to proceed for a sufficiently long time, the solute and water
molecules will ultimately become uniformly mixed. Molecular diffusion is a relatively simple
process to visualize and represent mathematically, even though the motions involved on the
molecular scale are very complex. This is because the scale of mixing due to Brownian motion
is very small compared to the size of a container or flow regime within which the fluid/solute
mixture is confined.
If one wanted to increase the rate of mixing in a fluid/solute mixture undergoing perfect
laminar flow, one could, in principle, cause some type of mixing process in the mixture as it
moved along. If this mixing process, whatever it might be, caused random mixing on a scale that
was larger than the diffusion scale but still relatively small compared to the size of the container
or flow regime, then to a good approximation it could be visualized and represented
mathematically as a diffusion-like process with a "diffusion" coefficient having a larger value
than the coefficient of molecular diffusion. In fact, given the appropriate conditions, the
combined mixing process could be represented as a diffusion-like process with a diffusion
coefficient given by Dd + Dm, where Dd is the coefficient of molecular diffusion and Dm could
be called the coefficient of mechanical dispersion in recognition of whatever mechanical process
caused the additional small-scale mixing.
When a solution moves in perfect laminar flow through a porous medium, the irregular
and tortuous structure of the pore space and the velocity gradients below the Darcy scale cause
a type of mixing on a scale of many pores that is small compared to the size of the problem of
interest which is defined by the overall Darcy flow field. This pore-scale mixing is analogous to
the hypothetical "mechanical dispersion" discussed in the previous paragraph and, in fact, is
called mechanical dispersion. In a porous medium, diffusion and mechanical dispersion take
place simultaneously, and the sum of the two processes at each point of the medium is called
local hydrodynamic dispersion. These concepts will be elaborated further in the next section.
Additional discussion may also be found in Bear (1979).
AIII-2 The Mechanisms of Dispersion
In order to improve our capability of modeling solute transport, it is important to
understand the major physical mechanisms which affect the evolution of an existing groundwater
contamination plume or the future course of an anticipated plume. The catch-all name given to
the spreading of a contaminant in groundwater is dispersion, a term which is familiar to almost
everyone. However, as illustrated in Figure AIII-1, many different phenomena contribute to the
dispersion process in aquifers. At the stage of development of the hypothetical tracer plume in
Figure AIII-1, the extent of horizontal spreading is determined mainly by the elapsed travel time
198
-------
(A)
Hypothetical
/Velocity
*v j Distribution
-Tracer at
Time=0
Tracer Distribution
at Time >0
Scale Dispersion""
Multiple of
Figure AIII-1. Part (A) shows a hypothetical velocity distribution and an initial distribution of
tracer while part (B) shows how the tracer would be dispersed by the moving
groundwater at several different scales. Three common mechanisms of pore
scale dispersion (velocity variation within a pore (a); flow path tortuosity (/J), and
molecular diffusion due to concentration differences (Y)) are illustrated also.
199
-------
and the difference between the maximum and minimum values of the horizontal advective
velocities. These velocity variations result primarily from the larger-scale variations of hydraulic
conductivity. This type of plume dispersion is due to the fact that the solute is simply carried
along with the moving ground water at different fluid velocities depending on position. Such
dispersion may be called advective dispersion or shear-flow dispersion. Considered alone,
advective dispersion causes spreading but no irreversible mixing of the tracer with water. Much
of this report is devoted to methods for measuring the hydraulic conductivity variations that give
rise to advective dispersion.
To get irreversible mixing, one must, as a minimum, consider the dilution within the
plume and along the plume boundaries that is caused by pore-scale mixing (the local
hydrodynamic dispersion discussed previously) due in part to molecular diffusion, velocity
variations within each pore, and the overall tortuosity of the flow path. Mixing due to local
hydrodynamic dispersion is relatively large at plume boundaries where concentration variations
with distance (gradients) are largest. It is evident, therefore, that advective dispersion and
hydrodynamic dispersion act synergistically to increase irreversible mixing. Advective dispersion
tends to increase the surface area of a plume where the concentration gradients are largest, and
this promotes increased mixing due to local hydrodynamic dispersion.
Some researchers have suggested that random, diffusion-like mixing on a scale much
larger than the pore-scale (the scale of local hydrodynamic dispersion) can be conceptualized.
Depending on the scale being considered, names for these hypothesized mixing processes
include field-scale dispersion, macro-dispersion and, when the mixing scale is across the entire
vertical dimension of an aquifer, full-aquifer dispersion. These concepts are somewhat
controversial at the present time, and the present authors are of the opinion that their practical
use in field hydrology is limited. Out viewpoint is presented in Chapter I, and further discussion
may be found in Giiven and Molz (1988).
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APPENDIX IV
TABLE OF CONTENTS FOR THE PROCEEDINGS ENTITLED
"NEW FIELD TECHNIQUES FOR QUANTIFYING THE
PHYSICAL AND CHEMICAL PROPERTIES OF
HETEROGENEOUS AQUIFERS"*
Session 1
Geology-Intensive Approaches to Property Measurements
Emergence of Geologic and Stochastic Approaches for Characterization of Heterogeneous Aquifers,
by Graham E. Fogg (The University of Texas at Austin, Bureau of Economic Geology, University
Station, Box X, Austin, TX 78713-7508).
Statistical Analysis of Hydraulic Conductivity Distributions: A Quantitative Geological Approach
by Fred M. Phillips, John L. Wilson and John M. Davis (Geoscience Department and Geophysical
Research Center, New Mexico Institute of Technology, Socorro, NM 87801).
Identification of Aquifer Interconnection and Continuity in a Coastal Plain Geologic Environment
by Matthew J. Gordon and Robert L. Powell (ENVIRON Corporation, 1000 Potomac St., NW,
Washington, DC 20007).
Methods for Evaluating Large-Scale Heterogeneity in Fine-Grained Glacial Sediment by William
W. Simpkins (Department of Geology and Geophysics, University of Wisconsin-Madison, Madison,
WI53706), Kenneth R. Bradbury (Wisconsin Geological and Natural History Survey, 3817 Mineral
Point Road, Madison, WI 53705), and David M. Mickelson (University of Wisconsin). x
Empirical Approaches for Estimating Flow and Transport Parameters by William A. Milne-Home
(Centre for Groundwater Management and Hydrogeology, University of New South Wales, P.O.
Box 1, Kensington, New South Wales 2033, Australia) and Franklin W. Schwartz (Department of
Geology and Mineralogy, The Ohio State University, Columbus, OH 43210).
Session 2
Property Measurement Using Borehole Geophysics and Log Analysis
A Generalized Approach to Geophysical Well Log Analysis and Interpretation in Hydrogeology by
Frederick L. Paillet, (U.S. Department of the Interior, U.S. Geological Survey, Box 25046, MS
403, Denver, CO 80225).
Review of Borehole Methods for Characterizing the Heterogeneity of Aquifer Hydraulic Properties
by Kendrick Taylor, (Desert Research Institute, Water Resources Center, P.O. Box 60220, Reno,
NV 89506-0220).
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A Direct Integral Method for the Analysis of Borehole Fluid Conductivity Logs to Determine
Fracture Inflow Parameters by Chin-Fu Tsang and Frank V. Hale (Earth Sciences Division,
Lawrence Berkeley Laboratory, 1 Cyclotron Road, University of California, Berkeley, CA 94720).
On the Field Determination of Effective Porosity by Iraj Javandel (Earth Sciences Division,
Lawrence Berkeley Laboratory, 1 Cyclotron Road, Berkeley, CA 94720).
Session 3
Measurement of Vertically-Averaged Aquifer Properties
Application of Interference Testing to Characterize Three Coal Seam Aquifers Near Birmingham,
Alabama by Robert A. Koenig (In-Situ, Inc., P.O. Box I, Laramie, WY 82070-0920).
Analysis of Early-Time Oscillatory Aquifer Response by Daniel A. Giffin and David S. Ward
(GeoTrans, Inc., 250 Exchange Place, Suite A, Herndon, VA 22070).
Delineating Geometry of Unconfined Aquifer Heterogeneities with Microgravity Surveys During
Aquifer Testing by Eileen P. Poeter (Department of Geological Engineering, Colorado School of
Mines, Golden, CO 80401).
A New and Comprehensive Method for the Characterization of Horizontally Anisotropic Aquifers
Through the Analysis of Data from Pumped Wells by Garry Grimestad (Hydralogic, P.O. Box 4722,
Missoula, MT 59806).
Conductive Slug Tracing as a Single-Well Test Technique for Heterogeneous and Fractured
Formations by Andrew Michalski (TRC Environmental Consultants, Inc., 18 Worlds Fair Drive,
Somerset, NJ 08873).
Session 4
Tracer Techniques and Flow Regime Characterization
Ground Water Flow Regime Characterization, Columbia Plateau Physiographic Province, Arlington,
North Central Oregon by Ching-Pi Wang (Washington Department of Ecology, 4350 150th Avenue
NE, Redmond, WA 98052-5301) and Stephen M. Testa (Engineering Enterprises, Inc., 21818 South
Wilmington Avenue, Long Beach, CA 90810).
A Downhole Column Technique for Field Measurement of Transport Parameters by Douglas R.
Champ and Grigory L. Moltyaner (Chalk River Nuclear Laboratories, Chalk River, Ontario,
Canada KOJ 1JO).
Application of Data-Logger/Pressure-Transmitter/Conductivity-Probe Instrumentation in Long-
Term Salt-Tracer Studies in a Fractured, Saturated Geologic Medium by J.A. Paschis, R.A. Koenig
and J.E. Benedik (In-Situ, Inc., P.O. Box I, Laramie, WY 82070).
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Quantifying the Nature and Degree of Heterogeneity Using Concepts of Non-linear Dynamics and
Fractals by Stephen W. Wheatcraft, Scott W. Tyler and Michael J. Nicholl (Desert Research
Institute, University of Nevada, P.O. Box 60220, Reno, NV 89506).
Session 5
Regulatory Problems and Multi-Level Measurement
of Aquifer Properties
Hydrochemical Monitoring and Hydrogeologic Characterization: Conflict and Resolution by Ronald
Schalla, Stuart P. Luttrell and Ronald M. Smith (Site Characterization and Assessment Section,
Geoscience Department, Battelle Pacific Northwest Labs, P.O. Box 999, Richland, WA 99352).
Ground-Water Monitoring at Hazardous Waste Facilities: Regulatory Changes that Consider Site-
Specific Subsurface Characteristics by Joseph M. Abe, James R. Brown and Vernon B. Myers
(Waste Management Division, Office of Solid Waste, U.S. Environmental Protection Agency, 401
M Street, SW, Washington, DC 20460).
Scale of Measurements, REV and Heterogeneities by Arthur W. Warrick and T-C. Jim Yeh (The
University of Arizona, Department of Soil and Water Science, 429 Shantz Building #38, Tucson,
AZ 85721).
Characterization of the Hydrogeologic Properties of Aquifers: The Next Step by Fred J. Molz,
Oktay Giiven and Joel G. Melville (Civil Engineering Department, 238 Harbert Engineering Center,
Auburn University, AL 36849-5337).
Application of the Borehole Flowmeter Method to Measure The Spatially Variable Hydraulic
Conductivity at the Macro-Dispersion Experiment (MADE) Site by Kenneth R. Rehfeldt (Illinois
Stale Water Survey, Ground-Water Section, 2204 Griffith Drive, Champaign, IL 61820-7495).
Characterizing Flow Paths and Permeability Distribution in Fractured Rock Aquifers Using a
Sensitive, Thermal Borehole Flowmeter by Alfred E. Hess and Frederick L. Paillet (U.S.
Department of Interior, U.S. Geological Survey, Box 25046, MS 403, Denver Federal Center,
Denver, CO 80225).
An Electromagnetic Borehole Flowmeter for Measuring Hydraulic Conductivity Variability by
Steven C. Young and William R. Waldrop (TVA Engineering Laboratory, P.O. Drawer E, Norris,
TN 37828).
Laboratory Testing and Comparison of Specific-Conductance and Electrical-Resistivity Borehole-
Dilution Devices by Eugene L. McLinn (Residuals Management Technology, Inc., Suite 124, 1406
East Washington Avenue, Madison, WI53705) and Carl D. Palmer (Department of Environmental
Science and Engineering, Oregon Graduate Center, 19600 von Neumann Drive, Beaverton, OR
97006-1999).
Use of a Borehole Flowmeter to Determine Spatial Heterogeneity of Hydraulic Conductivity and
Macrodispersion in a Sand and Gravel Aquifer Cape Cod, Massachusetts by Kathryn M. Hess (U.S.
Geological Survey, WRD, 28 Lord Road, Suite 200, Marlboro, MA 01752).
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Aquifer Parameter Estimation with the Aid of Radioactive Tracers by Grigory S. Moltyaner
(Atomic Energy of Canada Limited, Environmental Research Branch, Chalk River Nuclear
Laboratories, Chalk River, Ontario, Canada KOJ 1JO).
Measuring Hydraulic Conductivity in a Steep Appalachian Watershed by Gary C. Pasquarell and
Douglas G. Boyer (USDA/Agricultural Research Service, Appalachian Soil and Water Conservation
Research Laboratory, Box 867, Airport Road, Beckley, WVA 25802-0867).
Patterns of Grotrndwater Movement in Water Wells East Salt River Valley, Arizona by Errol L.
Montgomery, Andrew A. Messer and Ronald H. DeWitt (Errol L. Montgomery & Associates, Inc.,
1075 East Fort Lowell Road, Suite B, Tucson, AZ 85719).
A Direct-Reading Borehole Flowmeter by William B. Kerfoot and Lawrence Kiely (K-V
Associates, Inc., 281 Main Street, Falmouth, MA 02540).
An Analytic Solution Relating Wellbore and Formation Velocities with Application to Tracer
Dilution Problem by Pascal Bidaux (CNRS Laboratoire d'Hydrogeologie, Universite des Sciences
et Techniques, PL E. BataUlon, 34060 Montpellier Cedex, France) and Chin-Fu Tsang (Lawrence
Berkeley Laboratory, University of California, Berkeley, CA 94720).
An Advanced Technology for the In-Situ Measurements of Heterogeneous Aquifers by Julie M.
Smythe and Philip B. Bedient (Rice University, Department of Environmental Science and
Engineering, P.O. Box 1892, Houston, TX 77251), Rick A. Klopp (Terra Technologies, Houston,
TX 77099) and Chen Y. Chiang (Shell Development Company, Houston, TX 77001).
The Hydrogeological Properties of the Gulf Coast Upper Chicot Aquifer Near Port Arthur, Texas
by Andreas Haug, R. Harold Petrini, and Gerald E. Grisak (Intera Technologies, Inc., Suite 300,
6850 Austin Center Blvd., Austin, TX 78731) and Kim Klahsen (Chemical Waste Management, Inc.,
P. O. Box 2563, Port Arthur, TX 77640).
A Three-Dimensional Lagrangian Numerical Model of Two-Well Tracer Tests in Confined Aquifers
by Oktay Giiven, Fred J. Molz and Joel G. Melville (Department of Civil Engineering, Harbert
Engineering Center, Auburn University, AL 36849-5337).
Free Gasoline Thickness in Monitoring Wells Rekted to Ground Water Elevation Change by
William T. Hunt, Jeffrey W. Wiegand and John D. Trompeter (Alton Geosciences, Inc., 16510
Aston Street, Irvine, CA 92714).
Multilevel Slug Tests: Performance and Analysis by Joel G. Melville, Fred J. Molz and Oktay Giiven
(Department of Civil Engineering, Harbert Engineering Center Auburn University, AL 36849-
5337).
Quantitation of Aviation Fuel Contaminant Levels in Sandy Subsurface Core Material by Gas
Chromatography by Steve A. Vandegrift (NSI Technology Services Corporation, Ada, OK) and Don
H. Kampbell, (EPA Robert S. Kerr Environmental Laboratory, P.O. Box 1198, Ada, OK 74820).
Cone Penetrometer Tests and Hydropunch'™ Sampling: An Alternative to Monitoring Wells for
Plume Definition by Mark Smolley and Janet C. Kappmeyer (EMCON Associates, 1921 Ringwood
Avenue, San Jose, CA 95131).
U. S. GOVERNMENT PRINTING OFFICE 1930/748-159/00381
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Session 6
Measurement of Chemical and Biochemical Aquifer Properties
In-Situ Methods for Evaluating Retardation Factors and Biotransformation Parameters by Robert
W. Gillham (Waterloo Centre for Groundwater Research, University of Waterloo, Waterloo,
Ontario, Canada, N2L 3G1).
Subsurface Microbiota as Monitors of Contaminant Migration and Mitigation by David C. White
(Institute for Applied Microbiology, University of Tennessee, 10515 Research Drive, Suite 300,
Knoxville, TN 37932-2567) and John T. Wilson (R.S. Kerr Environmental Research Laboratory,
USEPA, P.O. Box 1198, Ada, OK 74820).
A Multi-Layer Sampler for Groundwater: New Field Technique to Study Chemical Processes and
Transport Phenomena in Aquifers by Daniel Ronen and Mordeckai Magaritz (Isotope Department,
The Weizmann Institute of Science, 76 100, Rehovot, Israel).
Measurement of Volatile and Aqueous Geochemical Properties in Shallow and Deep Groundwaters:
Two Innovative Sampling Devices by Barbara Sherwood Lollar and Shaun K. Frape, et al
(Department of Earth Sciences, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1).
Leaky Microcosms are Representative of BTX Biodegradation in the Borden Sand Aquifer by J.F.
Barker, G.C. Patrick, DJ. Berwanger and E.A. Sudicky (Waterloo Centre for Groundwater
Research, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1).
Geochemical Evaluation of Reduced Subsurface Environments by Dhanpat Rai and John M.
Zachara (Battelle Pacific Northwest Laboratories, P.O. Box 999, Richland, WA 99352).
Session 7
Measurement in the Unsaturated Zone
Vertical Profiles and Near Surface Traps for Field Measurement of Volatile Pollution in the
Subsurface Environment by David W. Ostendorf and Don H. Kampbell (Robert S. Kerr
Environmental Research Laboratory, USEPA, P.O. Box 1198, Ada, OK 74820).
Evaluation of Ceramic Vacuum Soil Solution Samplers in Sandy Soils Overlying Aquifers by
Nathaniel O. Bailey, Tammo S. Steenhuis and Jean Y. Parlange (Cornell University, Department
of Agricultural and Biological Engineering, Riley-Robb #30, Cornell, Ithaca, NY 14853).
Methods of Estimating Hydraulic and Transport Parameters for the Unsaturated Zone by Jack C.
Parker and Srikanta Mishra (Center for Environmental and Hazardous Materials Studies, Virginia
Polytechnic Institute and State University, 332 Smythe Hall, Blacksburg, VA 24061).
For a copy of the proceedings contact:
Water Resources Research Institute
202 Hargis Hall
Auburn University, AL 36849
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