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EPA 600/R-14/273
September 2014
A for
M.S. Beljin & Associates, Cincinnati, OH
Subcontractor to CSS~Dynamac Corporation, Erlanger, KY
R.
U.S. EPA/National Risk Management Research Laboratory/
Ground Water and Ecosystems Restoration Division, Ada, OK
D.
U.S. EPA/National Risk Management Research Laboratory/
Ground Water and Ecosystems Restoration Division, Ada, OK
Office of Research and Development
National Risk Management Research Laboratory I Ground Water and Ecosystems Restoration Division
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The U.S. Environmental Protection Agency through its Office of Research and Development funded and managed
the research described herein under contract EP-W-12-026 to CSS-Dynamac Corporation. Mention of trade names,
products, or services does not convey, and should not be interpreted as conveying, official EPA approval, endorsement,
or recommendation.
All research projects making conclusions or recommendations based on environmentally related measurements and
information and funded by the Environmental Protection Agency arc required to comply with the requirements of the
Agency Quality Assurance Program. This project was conducted under an approved Quality Assurance Project Plan.
This document has been reviewed in accordance with U.S. Environmental Protection Agency policy and approved
for publication.
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Notice ii
List of Figures v
Abstract vi
1.0 Introduction 1
2.0 Description of the Spreadsheet Tool 3
2.1 Input Data 4
2.2 Results 5
2.3 Hydraulic Head and Gradient Plot 8
2.4 Hydraulic Gradient Plot 8
2.5 Groundwater Velocity Plot 8
2.6 Vector Plot 8
2.7 User Instructions 8
3.0 Application of the Three-point .Method of Analysis 9
3.1 Hydrogeologic Considerations 10
3.2 Data Collection Tools and Techniques 12
3.3 Assessment of Measurement Uncertainty in Hydraulic Gradient Analysis 14
3.3.1 Measurement Devices 14
3.3.2 Measurement Procedures 14
3.3.3 Reference Elevations 15
3.3.4 Monitoring Network Design 15
4.0 Example Applications 17
4.1 Application 1: Characterization of Nearby Extraction Well Effects 17
4.1.1 Background and Setting 17
4.1.2 Characterization Objective 17
4.1.3 Network Design and Data Acquisition 17
4.1.4 Results and Conclusions 17
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4.2 Application 2: Characterization of Groundwater/Surface Water Interactions 18
4.2.1 Background and Setting 18
4.2.2 Characterization Objective 18
4.2.3 Network Design and Data Acquisition 18
4.2.4 Results and Conclusions 18
4.3 Application 3: Spatial and Temporal Characterization of Groundwater Flow 19
4.3.1 Background and Setting 19
4.3.2 Characterization Objective 19
4.3.3 Network Design and Data Acquisition 19
4.3.4 Results and Conclusions 19
4.4 Application 4: Demonstration of Anisotropy Effects on Groundwater Flow 21
4.4.1 Problem Description 21
4.4.2 Example Data Set 21
4.4.3 Results and Conclusions 21
5.0 Summary 22
6.0 References 24
Appendix A: Theoretical Development Al
Appendix B: Comparison of 3PE Results with Published Problems Bl
Appendix C: 3PE Workbook Cl
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List of
Figure 1. Potential effects of horizontal anisotropy on groundwater flow 3
Figure 2. Spreadsheet screen capture depicting input data cells 4
Figure 3. Spreadsheet screen capture depicting the most significant results of the calculations 6
Figure 4. Spreadsheet screen capture depicting vector orientation and three-point triangle results ...7
Figure 5. Calculation of hydraulic gradient vector using hydraulic head data from three locations ..9
Figure 6. Example potenliometric surface map interpreted from hydraulic head data 9
Figure 7. Example misapplication of three-point analy sis of hy draulic gradient 10
Figure 8. Groundwater elevations in adjacent wells screened at different depths 11
Figure 9. Example of sensor drift during field deployment of a pressure transducer 13
Figure 10. Influence of pumping on groundwater flow near a contaminant plume 17
Figure 11. Hydraulic heads and the resulting hydraulic gradient magnitude and direction 18
Figure 12. Net groundwater flow direction near pond 19
Figure 13. Hydraulic heads and the resulting hydraulic gradient magnitude and direction 19
Figure 14. Hydraulic head in monitoring wells and local precipitation 20
Figure 15. Hydraulic heads and the resulting hydraulic gradient magnitude and direction 20
Figure 16. Monitoring network and the resulting average hydraulic gradient directions 20
Figure 17. Effects of horizontal anistropy on groundwater flow directions 21
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Evaluation of hydraulic gradients and the associated groundwater flow rates and directions is a fundamental aspect of
hydrogeologic characterization. Many methods, ranging in complexity from simple three-point solution techniques to
complex numerical models of groundwater flow, are available for hydraulic gradient estimation. In many situations
where the water table or other potentiometric surface can be represented as a plane, three-point estimation methods
will provide a quick and cost-effective means for estimating hydraulic gradients, particularly for initial evaluation
purposes.
The three-point solution method is well suited for implementation in a spreadsheet format. 3PE is an interactive
spreadsheet developed in Microsoft Excelฎ for estimation of horizontal hydraulic gradients and groundwater velocities.
It is particularly well suited for analyzing transient hydrologic conditions, allowing rapid visualization of hydraulic
gradient and groundwater velocity vectors. Applications include groundwater remediation performance assessments,
hydrologic conceptualization of groundwater/surface water interactions, and general characterization of site-specific
hydrology. Site-specific investigation objectives, supported by an analysis of measurement uncertainty, provide
the framework for determining the most appropriate measurement strategies and monitoring network designs for
estimating hydraulic gradients.
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1.0
Introduction
Evaluation of groundwater flow directions and rates
is a fundamental component of most hydrogeologic
investigations. At sites with groundwater contamination,
information concerning spatial and temporal
groundwater flow patterns is used to identify potential
receptors, design remediation systems, and evaluate the
effectiveness of existing systems. The determination
of how the groundwater flow field changes in response
to natural or man-made stresses often drives the
development of the site-specific conceptual model
in ecological as well as groundwater contamination
investigations.
Hydraulic gradients and. therefore, groundwater flow
direction and magnitude may vary with time in response
to factors such as precipitation, pumping, irrigation.
or interactions with surface water bodies. During
groundwater remediation, hydraulic stresses such as
remedial fluid injection and/or groundwater extraction
can result in significant changes to the groundwater flow
field. Physical modifications to aquifer structure such
as installation of permeable reactive barriers or low
permeability barriers may also significantly alter the
groundwater flow field. Characterization of the aquifer
response to such changes is a challenging aspect of the
evaluation of remedial effectiveness.
In simple terms, groundwater flow is controlled by
the hydraulic conductivity of the aquifer materials and
hydraulic gradients. Hydraulic gradient is the change
in hydraulic head per unit of distance measured in the
direction of the maximum rate of decrease in hydraulic
head, and represents the slope of the water table or
other potentiometric surface. The hydraulic gradient,
a vector having both a direction and magnitude, is
the driving force for groundwater flow within an
aquifer. Groundwater seepage velocity within an
aquifer is often estimated using a form of Darcy 's Law,
estimated hydraulic gradients, and estimates of hydraulic
conductivity and effective porosity.
Typical tools for calculating hydraulic gradients range
from simple graphical solutions (e.g.. Heath, 1983;
Fetter, 1981) using hydraulic head (i.e., groundwater
elevation) measurements from three locations forming
a triangle (i.e., three-point problem) to complex three-
dimensional models of groundwater flow. In situations
where key areas within Hie groundwater flow field can be
represented by a plane or a series of planes, calculation
of hydraulic gradients using a mathematical three-point
solution method offers a relatively simple and rapid
means for evaluating spatial and temporal changes
in the magnitude and direction of groundwater flow.
Groundwater flow can often be adequately represented
by a plane in portions of the aquifer separating areas of
aquifer recharge from discharge areas. Depending on
site-specific hydrology, areas with planar groundwater
flow often occur at the site to sub-regional scales and
can be indicated on water table or other potentiometric
surface maps as areas where equipotential lines are
straight and evenly spaced.
In many situations, three-point solution methods
can be particularly useful when combined with data
acquisition using data logging pressure transducers for
automated measurement of groundwater elevations. The
spreadsheet 3PE is a tool developed using an extension
of the three-point problem approach to allow rapid
calculation of the horizontal direction and magnitude
of groundwater flow in horizontally isotropic and
anisotropic settings. The 3PE workbook represents
an extension of previously available spreadsheet tools
such as those developed by Kelly and Bogardi (1989)
and Devlin (2003) and on-line calculators such as the
Three Point Gradient calculator (USEPA, 2014). The
3PE spreadsheet was developed in Microsoft Excelฎ
workbook fonnat using a rigorous mathematical
approach based on vector analysis, matrix algebra, and
trigonometric functions. Required user inputs include
well coordinates defining a triangle of wells (i.e., a
three-point problem); estimates of hydraulic conductivity
and effective porosity of the aquifer materials, if
groundwater velocity is to be calculated; the direction
in which the hydraulic conductivity is greatest, if the
aquifer has horizontal anisotropy; and the date/time
and hydraulic head values for each of the three wells.
3PE calculates hydraulic gradients and groundwater
velocities for each set of hydraulic head measurements.
The spreadsheet format is ideally suited as both a
teaching tool (e.g., for examining the effects of changes
in input parameters such as hydraulic conductivity) and
for rapid visualization and evaluation of seasonal and
other temporal fluctuations in groundwater flow rates
and directions. This tool is anticipated to be used by
project personnel with technical backgrounds, especially
by hydrogeologists, to evaluate the site-specific spatial
and temporal changes in hydraulic gradients and
associated groundwater velocities.
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Potential applications for three-point analysis methods
and the 3PE workbook include:
* Evaluation of the hydraulic impact of groundwater
remediation systems.
Conceptualization of potential groundwater/surface
water interactions.
* Rapid visualization of spatial and temporal
patterns in hydraulic gradients and groundwater
flow directions.
Enhanced groundwater flow model calibration
through improved characterization of the range of
hydrologic conditions.
* Improved conceptualization of site hydrology.
The theoretical development and derivation of the
mathematical equations solved in 3PE are provided in
Appendix A. Comparisons of the results of hydraulic
gradient and groundwater velocity calculations using
3PE with results from published problems and other
published calculators are presented in Appendix B. The
3PE workbook with an example data set is provided in
Appendix C.
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2.0
Description of the Spreadsheet Tool
The 3PE spreadsheet was developed using standard
functions built into Microsoft Excelฎ. There are no
macros or user-defined functions in the spreadsheet.
increasing portability. It is designed to estimate the
hydraulic gradient vector (the direction and magnitude)
from the coordinates of three wells and hydraulic head
measurements. Data can be entered manually or copied
from pressure transducer or other files. In the case
where data from more than one triangle of wells are
available, a separate workbook for each triangle of wells
would be used.
The tool allows calculation of horizontal hydraulic
gradient and groundwater velocity vectors in settings
where horizontal hydraulic conductivity is either
isotropic or anisotropic. A geologic formation is
isotropic if the properties of the medium (e.g., hydraulic
conductivity) are independent of direction. If the
properties vary with the direction of the measurement.
the medium is anisotropic. In anisotropic aquifers.
specially designed aquifer tests are used to estimate
the direction and magnitude of the axes of maximum
and minimum hydraulic conductivity. In some
cases, a heterogeneous aquifer may be effectively
represented using anisotropic conditions when it can
be conceptualized as a homogeneous, anisotropic
aquifer. Examples include a fractured aquifer that
can be represented as an equivalent porous medium
with higher hydraulic conductivity in the direction of
dominant fracture orientation and some coal bed aquifers
(e.g., Stoner, 1981; Kern and Dobson, 1998). In the
case of a horizontally isotropic aquifer, the groundwater
flow direction coincides with the hydraulic gradient.
However, in an anisotropic aquifer, the groundwater
flow direction is not necessarily parallel to the hydraulic
gradient vector (Figure 1). The groundwater flow
direction will depend on the orientation and magnitudes
of the maximum and minimum hydraulic conductivity
and the orientation of the hydraulic gradient.
Potentiometnc Surface
Tracer
Distribution
Direction of
Hydraulic
Gradient
Figure 1. In an anisotropic setting, the direction of the hydraulic gradient may be different from the direction of
groundwater flow. The direction in which hydraulic conductivity is highest (Kmax) is illustrated by the axis labeled
K*. The axis labeled Ky illustrates the direction in which hydraulic conductivity is lowest (&,,). For this example, the
magnitude ofKmax is much greater than &,,. The resulting groundwater flow direction is evidenced by the elongated
distribution of the dissolved tracer compound. This direction is essentially parallel to K* and significantly different
from the hydraulic gradient direction.
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Estimates of the magnitude of the groundwater velocity
vector (i.e., rate of groundwater flow) can also be
calculated by 3PE using Darcy's Law if estimates of
the hydraulic conductivity and the effective porosity
of the aquifer materials are provided. In general, such
estimates should be considered preliminary and only
local in nature since geologic settings often cannot be
adequately represented using a single value of hydraulic
conductivity.
The spreadsheet can be used to evaluate the gradient
and velocity using a single triangle with one or many
sets of hydraulic head measurements. The spreadsheet
calculates and records the coordinates of the starting and
ending points of each hydraulic gradient and velocity
vector. The vectors are plotted within the workbook to
allow rapid inspection. The length of the vector in the
plots represents its magnitude. This allows the user to
visually inspect not only the direction of the hydraulic
gradient or groundwater flow but, also, to observe
changes in the magnitudes of the vectors with time. In
that sense, the spreadsheet is ideal for analyzing transient
hydrologic conditions. The spreadsheet results can also
be copied to other software for production of customized
graphics such as rose diagrams displaying the
distribution of hydraulic gradients or groundwater flow
directions, histograms for evaluation of vector directions
or magnitudes, and for overlays on site-specific base
maps or other geographical information systems outputs.
2.1 Input Data
All cells in the 3PE spreadsheet for user input of data
are shaded green. Cells that are shaded blue present
significant results of the hydraulic gradient and velocity
calculations. Any consistent system of length and time
units can be used in 3PE (e.g., feet and days, meters and
days). However, the same set of units must be used in
all inputs.
Header Information
Space is provided at the top of the 3PE spreadsheet for
user input of project-specific text information such as the
site name, location, and measurement dates (Figure 2).
Any text may be entered in these cells.
Scale Factors for
Vector Plots
User input cells are shaded gra^n.
Principal Hydraulic
Conductivity Components
Figure 2. Screen capture from the 3PE spreadsheet depicting data to be input by user (green cells). Blue cells are
significant results of the calculations.
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Well and Coordinates
The spreadsheet requires the name and location in Mo-
dimensional Cartesian coordinates of each well forming
the triangle of the three-point problem: Well Name, X
Coordinate, and Y Coordinate (Figure 2). Any consistent
set of length units may be used for the well coordinates.
Principal Hydraulic Conductivity Components and
Effective Porosity
For conditions of horizontal anisotropy, the user
must enter the maximum (&,ซ) and minimum (Kmm)
horizontal hydraulic conductivity values (Figure
2) and the orientation of the axis of maximum
hydraulic conductivity (Kmax) in degrees measured
clockwise from North. Although calculated values
for the orientations of the hydraulic gradient and
groundwater velocity vectors can range from 0 to
360 degrees, the orientation of the axis of maximum
horizontal hydraulic conductivity must be input as a
value between 0 and 180 degrees. This direction and
representative values for the magnitude of the maximum
and minimum horizontal hydraulic conductivity will
generally be obtained using specialized hydraulic tests
(e.g., Hantush, 1966; Way and McKee, 1982; Neuman et
al., 1984; Maslia and Randolf, 1987). It is assumed that
Kmax is orthogonal to &.
In the more common case of an aquifer that is
represented as isotropic in the horizontal plane, hydraulic
conductivity is uniform (i.e., Kmax = Kmin). Thus, the
same value will be entered for both Kmm and Kmm and
the orientation is ignored in the 3PE calculations. The
length units used in specifying hydraulic conductivity
must be consistent with those used in specifying well
coordinates and hydraulic head.
Effective porosity is entered as a fraction between zero
and one. Effective porosity is the percentage of the total
mass of aquifer material occupied by interconnected
pores. It is often estimated from site-specific data related
to the composition of the aquifer material or the specific
yield of the aquifer. Note that a value other than
zero should be entered for effective porosity even if
hydraulic conductivity data are not available and
calculation of groundwater velocity is not required.
Entering an effective porosity of zero will result in
division by zero in the calculations and will render both
Vector Inspector and Vector Plot unusable.
Date/Time and Hydraulic
The date/time data are entered in "MM/DD/YY
HR:MN" format (Figure 2). Hydraulic head data should
be entered in the same length units as used for the well
coordinates (e.g., feet, meters). Data may be entered
manually or copied from electronic files using the "Paste
Special..." and "Values" options. The number of rows
of hydraulic head data for which hydraulic gradient and
groundwater velocity will be calculated is only limited
by the number of rows allowed in Excelฎ. Example
rows consisting of input data followed by calculations
and results arc provided in the 3PE spreadsheet. The
example input data in each row may be overwritten by
the user. Following input of the hydraulic head data,
the user should copy the calculation cells (i.e., 3PE
spreadsheet columns E through A A) from one of the
example rows and paste into the sheet (using the "Paste"
option) as many times as needed based on the number of
hydraulic head measurements to be solved as three-point
problems for the given triangle of wells.
2.2
The 3PE spreadsheet calculates and displays many useful
parameters in addition to the magnitude and orientations
of the hydraulic gradient and groundwater velocity
vectors. These include data describing the triangle used
in the three-point problem, vector components, and
simple statistics describing the hydraulic head input data
and the magnitudes of the calculated hydraulic gradient
and groundwater velocity vectors. In this workbook, the
symbol "i" is used to denote hydraulic gradient and the
symbol "V" is used to denote groundwater velocity.
The results displayed in the spreadsheet are described
below.
Statistics
The spreadsheet calculates the maximum, minimum, and
average hydraulic head and the range of the hydraulic
head data for each well. It also calculates the maximum,
minimum, and average magnitudes of the hydraulic
gradient and groundwater velocity vectors (Figure 3).
Vector Inspector
The hydraulic conductivity and velocity7 vectors for
a selected row of hydraulic head data are displayed
in the "Vector Inspector" chart as blue and red
arrows, respectively (Figure 3). The starting and
ending coordinates of each vector are provided
in the cells labeled "PLOTTED HYDRAULIC
GRADIENT ARROW COORDINATES" and
"PLOTTED GROUNDWATER VELOCITY'ARROW
COORDINATES". For a quick view of the vectors
for a particular Date/Time, the user enters the "Vector
Inspector Row of Interest" (i.e., the spreadsheet row
number where the Date/Time of interest is located). The
range of rows available for viewing is shown below the
Row of Interest cell. The "Vector Inspector" displays
the triangle and the vectors. The scale factors for vectors
in the "Vector Inspector" are functions of the magnitude
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of the vectors and the size of the problem grid. A
recommended scale factor is provided. However, the
user should adjust the scale factors to their satisfaction.
Large problem grid sizes (e.g., large x and y coordinate
values) may require large scale factors (e.g., > 10,000).
Proper scaling will allow the workbook to display the
vectors in both the "Vector Inspector" and as a separate
plot under the workbook tab labeled "Vector Plot".
Number of Measurements
These cells calculate the total number of hydraulic
head measurements (Figure 3) entered for each of the
wells. The number of measurements should be the same
for each well. If not, the user will receive the prompt:
WARNING: Please delete incomplete rows.
Number of Vectors within Each Compass Quadrant
The numbers of hydraulic gradient vectors (i-vectors)
and groundwater velocity vectors (F-vectors) indicating
directions of groundwater movement in each of the four
compass quadrants are displayed (Figure 4).
Triangle Information
Based on the coordinates of the triangle vertices, 3PE
calculates the following information (Figure 4):
The coordinates of the triangle centroid used as the
starting point for the vectors.
The area of the triangle.
The length of the triangle sides.
Angles of the triangle.
Triangle Plot Coordinates
The starting and ending coordinates of the line segments
(the triangle sides) define the three wells used in the
calculations (Figure 4). This information may be used in
other plotting software for presentation purposes.
Statistics Describing the
Hydraulic Head,
Hydraulic Gradient, &
Groundwater Velocity Data
Vector
Inspector
Number of Date/Time &
Hydraulic Head Data Entries
Direction/Magnitude of
the Hydraulic Gradient &
Groundwater Velocity Vectors &
the Angle Between the Vectors
Figure 3. Screen capture from the 3PE spreadsheet depicting results of the hydraulic gradient and groundwater
velocity vector calculations and the Vector Inspector feature. Cells for user input are shaded green. Blue cells are the
most significant results of the calculations.
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Hydraulic Conductivity Components
The principal horizontal hydraulic conductivity
components (i.e., the maximum horizontal hydraulic
conductivity (Kmax) and the minimum horizontal
hydraulic conductivity (&,,ซ)) and the orientation of Kmax
measured in degrees clockwise from North are used to
calculate the components of the hydraulic conductivity
tensor (i.e., K^,Kyy for anisotropic conditions and KJcy=Kyx
for isotropic conditions) (Figure 4). The rotation angle
(8) is calculated as the angle between the x-axis and the
Kma* axis (See Appendix A.).
Individual Results for Each Three-Point Problem
The 3PE spreadsheet calculates and displays the
following results for each row of hydraulic head input:
Hydraulic Gradient Magnitude (Column E) -
Magnitude of hydraulic gradient (L/L).
Hydraulic Gradient Direction (Column F) -
Orientation of hydraulic gradient in degrees
clockwise from North (positive y-axis).
Groundwater Velocity Magnitude (Column G) -
Magnitude of groundwater velocity vector (L/T).
Groundwater Velocity Direction (Column H) -
Orientation of groundwater velocity vector in
degrees clockwise from North (positive y-axis).
Angle Between Vectors (Column I) - Difference
between orientations of hydraulic gradient (i) and
groundwater velocity (V) vectors in degrees.
A, B, C (Columns K-M) - Constants of the equation
defining a plane solved to obtain hydraulic gradient.
ix (Column N) - Hydraulic gradient component in x
direction (L/L).
i (Column O) - Hydraulic gradient component in y
direction (L/L).
i-Quadrant (Column P) - One of four regions of the
two-dimensional Cartesian system, bounded by
two half-axes. In clockwise order, the quadrants are
1 (azimuth from 0 to 90 deg); 2 (azimuth from 90 to
180 deg); 3 (azimuth from 180 to 270 deg); and 4
(azimuth from 270 to 360 deg).
Vx (Column Q) - Groundwater velocity component
in the x direction (L/T).
Number of Vectors in
Each Compass Quadrant
Components of Hydraulic
Conductivity Tensor
Information Regarding
the Well Triangle &
Plotting Coordinates
Plotting Coordinates of Vectors
Calculated for Each Data Set
Figure 4. Screen capture from the 3PE spreadsheet depicting results describing the orientations of the vectors, the
triangle of wells used to calculate hydraulic gradients, the components of the horizontal hydraulic conductivity tensor,
and the vector plotting coordinates calculated for each data set.
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V (Column R) - Groundwater velocity component
in the y direction (L/T).
V-Quadrant (Column S) - One of four regions of
the two-dimensional Cartesian system, bounded
by two half-axes. In clockwise order, the quadrants
are 1 (azimuth from 0 to 90 deg); 2 (azimuth from
90 to 180 deg); 3 (azimuth from 180 to 270 deg);
and 4 (azimuth from 270 to 360 deg).
Hydraulic Gradient Arrow Plot Coordinates
x_start (Column T) - Starting x coordinate for
hydraulic gradient arrow plot.
* Hydraulic Gradient Arrow Plot Coordinates
y_start (Column U) - Starting y coordinate for
hydraulic gradient arrow plot.
* Hydraulic Gradient Arrow Plot Coordinates
x_end (Column V) - Ending x coordinate for
hydraulic gradient arrow plot.
Hydraulic Gradient Arrow Plot Coordinates
y_end (Column W) - Ending y coordinate for
hydraulic gradient arrow plot.
Groundwater Velocity Arrow Plot Coordinates
x__start (Column X) - Starting x coordinate for
groundwater velocity arrow plot.
Groundwater Velocity Arrow Plot Coordinates
y_start (Column Y) - Starting y coordinate for
groundwater velocity arrow plot.
Groundwater Velocity Arrow Plot Coordinates
x_end (Column Z) - Ending x coordinate for
groundwater velocity arrow plot.
Groundwater Velocity Arrow Plot Coordinates
y_end (Column AA) - Ending y coordinate for
groundwater velocity arrow plot.
2.3 Hydraulic and Plot
On the Hydraulic Head and Gradient Plot tab, the
workbook contains a plot of the direction of the
hydraulic gradient vector and the hydraulic head data
for each well as a function of the Date/Time input
in each data row in the 3PE spreadsheet. The plot
may be edited, as needed, and exported in portable
document format (PDF) for importation into slide show
presentation or other software, if desired. The plot
provides a rapid means of identifying significant changes
in the direction of the hydraulic gradient through time.
2.4 Hydraulic Plot
On the Hydraulic Gradient Plot tab, the workbook
contains a plot of the direction and magnitude of the
hydraulic gradient vector as a function of the Date/Time
input in each data row in the 3PE spreadsheet. The plot
may also be edited and exported for presentation. The
plot provides another tool for the rapid visualization of
any temporal trends in the hydraulic gradient that may
be seasonal or related to other hydrologic changes at the
site.
2.5 Groundwater Velocity Plot
In similar fashion, the direction and magnitude of
the groundwater velocity vector is also plotted on a
workbook tab.
2.6 Plot
A plot of the hydraulic gradient and groundwater
velocity vectors displayed in the "Vector Inspector"
section of the 3PE spreadsheet is included on the Vector
Plot tab to allow the user to more easily manipulate
the plotting functions and export the plot, if desired.
"Vector Plot" can be enhanced by adjusting the axes,
re-positioning the well labels, changing the chart title,
etc. Once the chart options are selected, a number of
charts can be generated by selecting a different "Vector
Inspector Row of Interest" in Row 5, Column H of the
3PE worksheet (i.e., Date/Time).
2.7 User Instructions
A simplified version of the instructions for use of 3PE is
provided on the first workbook tab.
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3.0
Application of the Three-Point Method of Analysis
The hydraulic gradient is usually estimated using
synchronous groundwater elevation measurements from
wells and piezometers (Figure 5). Estimates of the
direction and magnitude of the hydraulic gradient in a
given part of the aquifer may then be used with estimates
of the hydraulic conductivity and the effective porosity
to characterize the direction and rate of groundwater
flow (i.e., groundwater seepage velocity) using a form
of Darcy's Law. More extensive discussions of the
fundamental concepts of groundwater flow are available
from a variety of textbooks (e.g., Freeze and Cherry.
1979; Domenico and Schwartz, 1990; Fetter, 1988) and
other publications (e.g., USEPA, 1990).
Well A /
(100 m msl)
Well B o-o
(100m msl) /oo 20.0 40.0, so.o so.o loojfsjjo.o iuu>
/
-^0.0 J
Hydraulic Gradient
Veclor
Figure 5. Calculation of hydraulic gradient vector
using hydraulic head data from three locations (Wells
A, B, and C). The hydraulic gradient vector, calculated
graphically using the methods of Heath (1983) in this
case, is perpendicular to contours of equal hydraulic
head (equipotential lines) and is the direction of
groundwater flow in an isotropic setting.
One of the basic tools for evaluating hydraulic gradients
on a site-wide scale is the potentiometric surface map
(Figure 6). A potentiometric surface represents the
level to which water would rise in wells (Bates and
Jackson, 1987). It is produced either manually, or using
specialized software to plot and contour groundwater
elevation measurements that are representative of a
single hydrostratigraphic unit and obtained within a
limited timeframe. Following the convention used
in much of the hydrogeological literature (e.g., Bates
and Jackson, 1987; Fetter, 1988), the water table is
considered to be a particular potentiometric surface for
purposes of the discussions in this document. Assuming
a horizontally isotropic aquifer without a significant
component of vertical groundwater movement.
groundwater flows in directions perpendicular to
potentiometric surface contours (i.e., equipotential lines)
in the direction of decreasing hydraulic head
(i.e., direction of the hydraulic gradient vector).
Pumping Well
Water Table
March 7, 2014
Contour Interval
1 m
Figure 6. Example potentiometric surface map
interpreted from hydraulic head data measured in
multiple wells (black triangles). In general, groundwater
flows from areas where it enters the aquifer to areas
where it leaves the aquifer. Blue arrows depict
groundwater flow directions indicated by hydraulic
gradients. Green circles and ellipses highlight examples
of areas where the water table is sufficiently planar and
wells exist for analysis of hydraulic gradients using
three-point methods.
Examination of the potentiometric surface often
reveals basic characteristics of the groundwater flow
field that control groundwater flow and the transport
of any dissolved constituents (Figure 6). In general.
groundwater flows from recharge zones where water
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enters an aquifer, such as locations with significant
infiltration from precipitation or surface water, to
discharge zones where water leaves the aquifer.
such as pumping wells, seeps, springs, and surface
water bodies. Recharge zones may be reflected on
potentiometric surface maps as areas with higher
groundwater elevations and increased hydraulic gradient
(i.e., smaller spacing between equipotential lines). In
similar fashion, groundwater discharge zones may be
indicated by significant depression of the potentiometric
surface associated with groundwater discharge (e.g..
near extraction wells or discharge to surface water).
Examination of potentiometric surfaces obtained at
different points in time, such as wet and dry seasons, or
when local water supply wells are pumping at different
rates, can illustrate temporal fluctuations in hydraulic
gradients due to changes in hydrologic conditions.
Where sufficient hydraulic head data are available to
define the potentiometric surface, portions of the aquifer
where groundwater flow is planar will be reflected
as areas where the equipotential lines are straight.
rather than curved, and have a relatively uniform
spacing (Figure 6). In areas where groundwater flow
is planar, the hydraulic gradient can be estimated using
groundwater elevations from a minimum of three
wells that form a triangle. Calculation of the hydraulic
gradient using hydraulic heads measured in three wells
has been referenced in the literature by terms such as
the "three-point problem" or "three-point method".
The problem may be solved graphically (Figure 5)
(Heath, 1983) or using more advanced mathematical
techniques (e.g., Finder et. al, 1981; Cole and Silliman.
1996; Silliman and Frost, 1998; Devlin and McElwee,
2007). Although calculation of hydraulic gradient using
more than three wells can decrease uncertainty in some
situations, such as in areas where the hydraulic
gradient is low, the benefits of including more than three
monitoring points are often small in comparison with
other methods for reducing uncertainty (e.g., increasing
well spacing). In practice, the three-point method is
often a useful and cost effective tool, particularly for
initial investigations (Devlin and McElwee, 2007).
3.1 Hydrogeologic Considerations
As with every evaluation technique, there are
assumptions that should be met before the three-point
method is applied. Some of these assumptions are
specific to the three-point method while others are also
applicable to tools commonly used for estimation of
hydraulic gradients (e.g., potentiometric surface maps).
Key assumptions specific to the three-point method and
use of the 3PE spreadsheet are:
The water table or other potentiometric
surface within the triangle of wells can be
represented as a plane (i.e., the curvature of the
potentiometric surface is relatively small).
Conditions under which the potentiometric surface
is unlikely to be planar include situations where
there are point sources for water movement into or
out of the aquifer located within the triangle. For
example, surface water bodies often serve as either
recharge zones or discharge zones for the aquifer.
Similarly, active pumping wells screened within
or hydraulically communicating with the aquifer of
interest are obvious sources of discharge. The
effects of improperly applying three-point
analysis or other simple methods to estimate
hydraulic gradients in such areas can be dramatic
(Figure 7). In this example, an active pumping
well was included within the triangle of monitoring
wells used to estimate groundwater flow directions.
Figure 7. Misapplication of a three-point analysis
of groundwater flow direction surrounding an
actively pumping well. The contours of the
potentiometric surface reflect the non-planar
cone of depression associated with the pumping
well. Monitoring wells used to calculate the
hydraulic gradient by the three-point method
are labeled A, B, and C. The red arrow depicts
the groundwater flow direction calculated using
three-point analysis and the blue arrows depict
the actual groundwater flow direction. In this
case, improper application of the three-point
analysis results in a calculated flow direction that
is opposite to the actual direction.
-------
The curvature of the potentiometric surface is
significant in this area indicating it is not planar.
The red arrow depicts the groundwater flow
direction calculated using three-point
analysis and the blue arrows depict the actual
groundwater flow direction. In this case,
improper application of the three-point analysis
results in a calculated flow direction that is opposite
to the actual direction. Other less obvious
situations, such as areas with enhanced infiltration
or leaking water conveyance systems, can result in
significant curvature of the potentiometric surface.
It is recommended that such features be considered
during the choice of wells used for estimation of
hydraulic gradients by the three-point method.
Where sufficient data are available, potentiometric
surface maps should be used to aid in determining
areas of a site where planar groundwater flow
conditions likely exist.
The aquifer can be treated as homogeneous
within the triangular area used for the three-
point analysis.
Geologic heterogeneity that results in
significant changes in hydraulic conductivity can
result in significant changes in hydraulic
gradients. Such conditions also make specification
of representative values for hydraulic
conductivity and effective porosity subject
to increased uncertainty. If this occurs within
the area of the three-point problem, it is likely
that the resulting hydraulic gradient and
groundwater velocity vectors will not accurately
represent actual site conditions. Evaluations
of geologic logs and estimates of hydraulic
conductivity obtained from the wells defining the
three-point problem as well as consideration
of the geologic setting can be used to evaluate
the degree of heterogeneity that may exist.
Evaluation of potentiometric surfaces may also
reveal areas with changes in hydraulic gradients
that may be related to heterogeneity. In general,
effects due to heterogeneity may decrease as
the size of the three-point triangle increases relative
to the scale of the heterogeneity (Cole and Silliman,
1996; McKenna and Wahi, 2006). However,
increasing the size of the triangle must be balanced
by the requirement that groundwater flow within
the triangle be planar in order to apply three-point
analysis methods.
Key assumptions related to hydrogeology and well
construction that are applicable to most simple tools used
to estimate horizontal hydraulic gradients, including the
three-point method, are:
The wells are screened solely in the same aquifer
or hydrostratigraphic unit.
The distribution of hydraulic head and, therefore,
groundwater flow directions and rates may be
very different within different aquifers.
Consider the situation (Figure 8) where one well
(Well A) is screened at the water table, another
well (Well B) is screened within a deeper,
confined aquifer, and the third well (Well C)
is screened across the confining unit separating
the two aquifers. For this example, the hydraulic
head in the upper aquifer is higher than the head
in the lower aquifer. The water level in Well A is
higher than the water level in Well B and the water
level measured in the well screened across the
confining unit (Well C) is influenced by both
aquifers but is representative of neither aquifer.
Monitoring Wells
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Figure 8. Groundwater elevations in three
adjacent wells (A, B, and C) screened at different
depths. In this case, Well A is screened across
the water table in an unconfined aquifer, Well
B is screened in a deeper confined aquifer, and
Well C is screened in both aquifers. Differences
in hydraulic head in the two aquifers result
in significant differences in the groundwater
elevations measured in the wells.
-------
Three-point analysis methods, such as implemented
in the 3PE spreadsheet, would not be appropriate
for use with this well network. Use of data from
wells screened within different aquifers or across
confining units can result in estimated groundwater
flow directions and rates that are completely
unrepresentative of actual conditions. As a general
practice, the wells should be of similar screen
length and position within the aquifer. In
addition, the use of wells with long screens
should be avoided to minimize the potential for
screening zones with different hydraulic heads in
order to provide data most representative of aquifer
conditions. In situations where hydraulic gradients
are to be calculated for a confined aquifer, lack of
effective borehole seals can result in vertical
water flow outside the well casing and water-
level measurements that are not representative of
hydraulic head in the screened portion of the
aquifer. Information concerning local geology
and aquifer structure, as well as construction logs
for the wells to be used to estimate hydraulic
gradients, should be evaluated to insure all wells are
properly constructed and screened solely within the
aquifer or hydrostratigraphic unit of interest.
Groundwater elevation measurements are
synchronous.
At many sites, potentiometric surfaces and
hydraulic gradients change with time due to changes
in hydrologic conditions. These changes can be due
to such things as aquifer recharge from precipitation,
changes in pumping rates of nearby water supply
wells, or changes in nearby surface water elevations.
The frequency of such fluctuations should be
considered during planning of field activities to
insure water levels are measured synchronously.
Groundwater and surface water elevations used to
estimate hydraulic gradients should be measured
within a timeframe such that they are all
representative of the same point in time (i.e., no
significant changes occur during the measurement
period). For example, if several days were required
to conduct manual measurements of groundwater
elevations at a site and rainfall resulted in significant
increases in water levels during the measurement
period, hydraulic gradients estimated using this
data set may not be representative of site conditions.
In this situation, it may be necessary to identify
smaller regions within the monitoring network to
allow manual water level measurements within
a time period suitable for application of three-
point problem analysis. As a general practice,
groundwater elevation measurements should
be obtained from all wells used to create
potentiometric surfaces or estimate hydraulic
gradients in as short a timeframe as practical.
Data from pressure transducers can be used to
characterize the frequency of significant temporal
changes and allow more informed planning of future
field activities.
Other important sources of transient behavior
or time lag effects that should be considered are
discussed by Post and Asmuth (2013). These
include the well-specific response time following
a change in groundwater pressure due to well
volume, screen length, and permeability of materials
adjacent to the well screen. This may be particularly
important in low permeability settings and where
fluctuations in water elevations are rapid.
such as near a beach where wave action results
in periodic changes in water elevations. Inadequate
well development may also result in similar time lag
effects.
The effects of time lags related to barometric
pressure changes should also be considered.
In an unconfined aquifer, significant delays in
the transmission of atmospheric pressure
changes to the water table may be observed if
the water table is deep or the vadose zone lias
a low permeability to air (Post and Asmuth,
2013). In such situations, correction of water-level
measurements for barometric pressure effects should
be evaluated and implemented if needed to reduce
uncertainty.
Groundwater flow is predominantly horizontal.
Hydraulic gradients estimated using the three-
point method or other simple tools generally
represent the horizontal component of the hydraulic
gradient. In a setting where hydraulic head within
the aquifer of interest changes with depth, there
will also be a vertical component to the hydraulic
gradient and, potentially, a significant vertical
component of groundwater flow. In settings where
a vertical hydraulic gradient exists and vertical
hydraulic conductivity is not significantly lower
(e.g., two orders of magnitude) than horizontal
hydraulic conductivity, a significant component
of vertical flow may exist and should be considered
prior to application of simple three-point analysis
methods. More complex analyses would be needed
to fully characterize the three-dimensional aspects
of groundwater flow in situations where the vertical
flow is not negligible.
3.2 Collection Tools and Techniques
Various tools are commonly used for the accurate
measurement of water level in wells. In general practice.
-------
measurements are made with respect to a surveyed
elevation which, for monitoring wells or piezometers.
is usually a reference point marked on the top of the
well casing. This allows measurements of the depth to
groundwater to be expressed as hydraulic head relative
to a common datum, which is required for calculation
of hydraulic gradients and most other hydrogeologic
analyses.
Submersible pressure transducers that automatically
compensate for temperature combined with data loggers
for storing the measurements have become more
portable, reliable, and affordable. In some situations.
the use of pressure transducers allows a cost-effective
way of collecting a large amount of hydraulic head data
with a pre-programmed time interval, ensuring that all
measurements are made synchronously. This makes
pressure transducers well suited for use in investigations
of temporal changes in hydraulic gradients that occur on
a time scale that is impracticable to fully characterize
using manual measurements of groundwater elevations.
Pressure transducers commonly used in groundwater
monitoring are either vented to the atmosphere by a
vented cable (gauged transducer), which means the
readings are automatically compensated for barometric
pressure, or unvented (absolute transducer), where
the pressure transducer measures the total pressure
(i.e., atmospheric pressure plus the pressure of the
water column above the transducer). When using an
unvented transducer, site-specific barometric pressure
must also be measured and subtracted from the total
(absolute) pressure reading to obtain the water pressure.
If groundwater elevation is independently measured
coincident with pressure transducer measurements, the
transducer data can then be converted to groundwater
elevations using methods described in the operations
manuals that accompany most instruments.
Pressure transducers are designed to be used under
specified pressure ranges. The accuracy and precision
of the transducer are often a function of the transducer
range. As the transducer range decreases, the
uncertainty in the measurement due to the accuracy
and precision of the transducer often decreases. For
example, the reported accuracy and precision of current
absolute pressure instruments with a pressure range
below approximately five meters of water is often +/- a
few millimeters of water. The accuracy and precision
of similar instruments with a much higher pressure
range (e.g., 100 meters of water) may be +/- a few
centimeters of water. Product specifications provided
by the manufacturer should be consulted to determine
the accuracy and precision of particular pressure
transducers. Previous site-specific monitoring data
should be consulted to estimate the range of water levels
that may be observed. This information will allow
the optimum pressure range for the transducer to be
specified.
The user should also consider transducer performance
in the areas of temperature compensation and pressure
sensor drift (Figure 9). Historically, transducer drift
(i.e., deterioration in sensor accuracy over time) and its
potential effects on hydrogeologic studies have been
noted (e.g., Rosenberry, 1990). Recently, Sorensenand
Butcher (2011) describe the results of independent tests
of pressure transducers used for water-level monitoring.
Pressure sensor drift was judged to be significant for
some of the tested transducers over the course of a
99-day field test. Drift was found to be linear in some
instances and nonlinear in others. In addition, their
results indicate that the degree to which the instruments
accurately compensated for changes in temperature
varied. Unfortunately, product specifications often do
not describe transducer performance with respect to
sensor drift or temperature compensation. Therefore.
assessment of these instrument characteristics by the
user may be needed in some situations, such as the long-
term monitoring of water levels where a high degree
of accuracy is required. In addition, periodic manual
measurements of water levels will generally be needed
during field deployment of either vented or unvented
transducers to confirm that transducer accuracy meets
project requirements and, potentially, correct the data for
sensor drift.
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Figure 9. Example of pressure sensor drift unrelated to
temperature compensation issues during extended field
deployment of a pressure transducer. The plot depicts the
difference between hydraulic head measured using a pressure
transducer installed in a monitoring well and hydraulic
head synchronously measured using an electric water-level
indicator in the same well (open circles). In this case.
instrument drift steadily increased throughout the deployment.
The blue line is a simple linear regression of the data.
-------
3.3 of
Uncertainty in Hydraulic Gradient Analysis
Measurement uncertainty can have a significant impact
on the accuracy of hydraulic gradients calculated
using three-point methods (Silliman and Mantz, 2000).
Awareness of the sources of measurement: uncertainty
or error and the magnitude of the uncertainty is often
essential to the evaluation of hydraulic gradients, design
of the monitoring network, and decisions concerning
instrumentation. Uncertainty in hydraulic head
measurements is introduced to varying degrees by each
part of the measurement process, including uncertainty
due to the accuracy/precision of the measurement device.
the measurement technique, and even the specification of
reference elevations.
3.3.1 Measurement Devices
Devices commonly used for the measurement of
water levels in wells include the steel tape, electric
water-level indicator, air line, float-activated recorder,
and submersible pressure transducers. Each tool has
advantages and disadvantages with respect to the
others. In addition, there is measurement uncertainty
inherent in the use of each device. In many instances,
the measurement uncertainty associated with the device
can be readily quantified. For many devices, the
accuracy is often specified by the manufacturer. For
example, most current electric water-level indicators are
marked to 0.01 ft or 1 mm, calibrated, and traceable to
national standards. Periodic comparison of a water-level
indicator with a reference tape can be used to ensure
calibration is maintained and documented.
Similarly, the accuracy and precision of pressure
transducers used to measure hydraulic head are generally
stated by the manufacturer and can be significantly
greater than the measurement uncertainty associated
with manual measurements of water level. In addition,
sensor drift and inadequate temperature compensation
can reduce accuracy under some field conditions,
such as long-term deployments and conditions where
temperatures vary significantly. When using unvented
(absolute) transducers, there will be measurement
uncertainty associated with both the submerged
transducer and the barometric pressure transducer or
other instrument used to measure barometric pressure
that will be cumulative and should be considered.
The choice of appropriate devices for measuring
hydraulic head at a given site should consider several
factors, including:
Study objectives (i.e., what question is to be
answered).
Measurement accuracy required to meet the
objectives.
* Measurement accuracy of the device and proposed
measurement technique.
* Advantages, disadvantages, and limitations of the
device.
3,3.2 Measurement Procedures
The thoughtful development of standard operating
procedures for obtaining the data used to calculate
hydraulic head and estimate hydraulic gradients is a key
element in controlling measurement uncertainty. The
procedures should identify and quantify7 the various
sources of uncertainty associated with the particular
data collection instruments and provide procedures to
control the uncertainty. For example, proper techniques
for obtaining manual water-level measurements in a
monitoring well using an electric water-level indicator
generally include:
* Measuring the depth to water with respect to a
surveyed reference mark with the required accuracy
(e.g., nearest 1 mm or nearest 0.01 ft).
ซ Recording of the values in a notebook that meets
quality assurance standards.
* Obtaining duplicate measurements in all or a
specified percentage of the wells.
* Insuring that the water level in the well is stable
following insertion of the monitoring probe,
particularly in wells with a small internal diameter
(e.g., less than 5 cm), in situations where the well
may be screened in aquifer materials with a low to
moderate hydraulic conductivity, and in wells that
may not be adequately vented to the atmosphere.
Procedures for testing the instrument battery and for
regular battery replacement.
* Site-specific procedures for decontaminating the
probe between wells.
* At sites where the water density is variable,
procedures for head corrections to account for
density effects may be needed (Post et al, 2007).
* At sites where light nonaqueous phase liquid
(LNAPL) may be present, procedures to
measure LNAJPL thickness and correct hydraulic
head measurements for LNAPL presence may be
needed (Newell et al., 1995).
Procedures for obtaining data using submersible pressure
transducers should include:
* Specification of transducer range. Consideration
should be given to the required measurement
accuracy as well as the likely range of water
levels that will be observed during the measurement
period. Ideally, the specified pressure range
-------
of the instrument should be large enough to allow
measurements of the highest and lowest
projected water levels but still provide data with the
required accuracy. When available, previous
data concerning the temporal patterns of
water-level fluctuations should be used to
determine both the appropriate transducer and
the elevation within the water column to suspend
the transducer to insure data are obtained over the
full range of water levels. Consideration should
also be given to placement at a depth such that
the transducer remains both submerged and within
its pressure range throughout deployment.
Procedures for suspension of the transducer in the
well to insure that the transducer remains at
a constant elevation throughout the investigation.
Consideration should be given to the point of
attachment on the well and use of a suspension
line that will not stretch or corrode (e.g., stainless
steel cable).
Procedures for measurement of barometric
pressures synchronous with the submerged
transducer readings if absolute pressure
(unvented) transducers are used.
Barometric pressure can be measured using a
specialized barometric pressure transducer installed
on-site. Alternatively, barometric pressure
data can be obtained from other measurement
devices such as an on-site or nearby barometer.
Consideration should be given to installation of a
backup measurement device for monitoring
barometric pressure, since it is a critical parameter
for correcting data from absolute pressure
transducers.
Procedures for programming data acquisition,
including synchronization of instrument clocks, and
for downloading and archiving the data.
In most situations, the well should be vented
to allow equilibration of the internal casing with
atmospheric pressure.
For vented transducers, inspection of the transducer
vent to ensure it is open to the atmosphere and
routine replacement of the desiccant capsule, if so
equipped.
Corrections for water density- due to temperature or
salinity where such corrections are warranted.
Periodic manual measurements of hydraulic
head. These data are needed to convert the pressure
measurements to groundwater elevations and to
evaluate instrument drift.
Routine checks of instrument calibration with
recalibration. as needed.
A succinct discussion of water-level measurements and
recommended procedures for use of various devices
(e.g., steel tape, electric indicator, air line, float-
activated recorder, submersible pressure transducers)
for measuring water levels in wells is available in
Cunningham and Schalk (2011).
3,3.3 Reference Elevations
The accuracy of the surveyed elevation of the reference
point used to calculate groundwater elevation should
always be specified (e.g., +/- 0.003 m or +/- 0.01 ft).
However, significant errors in the specified reference
elevations can occur, particularly at sites with a complex
characterization history, due to a variety of conditions,
including:
Wells installed in phases were sun-eyed using a
different reference datum.
* Transcription errors occurred and propagated.
Well is misidentified in the field.
* Top of well casing is uneven and has no marked
survey reference point.
Reference elevation has changed due to conditions
such as frost heaving.
* Survey used poor technique or lack of accuracy
control.
Errors in the specified reference elevations are often
more difficult to evaluate than uncertainties associated
with the measurement device or measurement technique.
Such errors are sometimes evidenced by obviously
abnormal or odd groundwater elevation measurements
that are not readily interpretable. In situations where
a high degree of measurement accuracy is required,
careful examination of survey history and resurvey of
the reference elevations at key wells may be warranted.
3.3,4 Monitoring Network
As with most methodologies, measurement uncertainty-
should be considered in the design of the monitoring
network used for three-point analysis of hydraulic
gradients. The overall uncertainty in the measurement
is the accumulation of uncertainties related to the
various investigation-specific sources. Data quality
objectives providing a quantitative, as well as
qualitative, description of the data quality required to
support the proposed analyses and decisions should be
developed for each investigation (Post and Asmuth,
2013). Detailed discussion regarding the development
of data quality- objectives is provided in USEPA (2006).
Following specification of the required data quality.
measurement methods and a monitoring design capable
of obtaining the desired accuracy can be chosen and
measurement uncertainty quantified. Quantifying
-------
measurement uncertainty is particularly important in
a setting where the hydraulic gradient is low or the
distance between wells is small. In such situations,
the differences in hydraulic head among the wells may
be very small. Measurement uncertainty may be too
large to allow estimation of hydraulic gradients with the
required degree of accuracy. In which case, use of other
measurement methods or a larger well spacing may be
needed to reduce uncertainty.
With respect to design of monitoring networks for
hydraulic gradient analysis using the three-point method.
the following guidelines are provided by Devlin and
McElwee (2007) and McKenna and WaM (2006):
* The measured groundwater elevation differences
between wells should be much greater than the
expected measurement uncertainty to produce
reliable estimates of hydraulic gradients. This
implies that it may be necessary to increase the
spacing between measurement points in some
situations, particularly in settings where the
hydraulic gradient is low. The degree to which the
measured water-level difference should exceed the
measurement uncertainty will be site specific and
depend on the data quality7 requirements of the
investigation. For example, studies reported by
Devlin and McElwee (2007) at the Geohydrologic
Experimental and Monitoring Site, an alluvial
aquifer study site with a low hydraulic gradient
(approximately 0.0005), resulted in a site-
specific guideline that the monitored area should
be large enough that the head drop across the
area would be at least three times the expected
measurement uncertainty. This guideline
provided data of sufficient quality to meet the
particular study objectives. In similar fashion,
McKenna and Ward (2006) recommend the use
of the relative head measurement error (RHME)
in designing the monitoring network for use with
three-point solution methods. The RHME is defined
as measurement error normalized by the head
drop across the three-point estimator. The RHME
can be used as an objective means for evaluating the
minimum size of the three-point estimation triangle.
Distortion of the triangle formed by the three
monitoring points can result in increased
uncertainty. Ideally, the ratio of any two of the sides
of the triangle would be close to 1. The dimension
of the triangle perpendicular to the direction of
the hydraulic gradient can be increased in order
to reduce the uncertainty in the estimate of the
hydraulic gradient direction (Devlin and McElwee,
2007). McKenna and Walii (2006) found
that triangles with base-to-height ratios between
0.5 and 5.0 resulted in the most accurate
gradient estimates.
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4.0
Example Applications
Brief examples of the application of three-point solution
methods and the 3PE spreadsheet are discussed below.
4.1 Application #1: Characterization of
Nearby Extraction Well Effects
At some sites, groundwater flow and contaminant
transport or the effectiveness of a remediation system
may be affected by pumping of off-site wells. The
following example illustrates the use of the 3PE
spreadsheet in characterizing such effects.
4.1.1 Background and Setting
The site is a closed industrial facility situated in a
valley, overlying an unconsolidated aquifer composed
of alluvial fan and flood-plain deposits. Estimates
of hydraulic conductivity within the alluvial aquifer
range from less than 0.3 m/d to approximately 50
m/d. In general, groundwater flows in a north to
northwest direction from the site. Groundwater
contamination associated with the facility has migrated
off site, extending approximately 1.5 km north of
the facility boundary. Implementation of hydraulic
controls to contain the plume are currently being
considered. During the characterization process, it was
discovered that a high-capacity irrigation well located
approximately 3 km northeast of the site is active during
the agricultural growing season.
4.1.2 Characterization Objective
The investigation was designed to answer the following
question:
Does agricultural pumping affect groundwater flow
directions within the contaminant plume?
4.1.3 Network Design and Data Acquisition
For the initial investigation of the possible effects of
agricultural pumping on groundwater flow within the
plume, three existing wells screened at similar elevations
within the aquifer were instrumented with pressure
transducers to synchronously measure hydraulic heads
six times daily for approximately nine months (January
through September). The wells were located near
the downgradient margin of the plume. Groundwater
elevations were also measured monthly using an electric
water-level indicator to document the accuracy of
the transducer measurements and allow evaluation of
instrument drift.
4.1.4 Results and Conclusions
The magnitude and direction of the hydraulic gradient
were calculated using 3PE for each of the approximately
1,600 data sets obtained during the monitoring
period. The results indicate that the irrigation well
is, potentially, a major influence on groundwater flow
near the downgradient margin of the contaminant
plume (Figure 10). Prior to the onset of the irrigation
season, the average direction of the hydraulic gradient
and, therefore, groundwater flow was approximately
north. During this period, the average magnitude of
the hydraulic gradient was approximately 0.0009 m/m.
During the irrigation season, which began in mid-March.
the hydraulic gradient was oriented in a northeast
direction toward the irrigation well with an average
magnitude of 0.0035 m/m, which is approximately four
times larger than the magnitude of the hydraulic gradient
under non-pumping conditions.
Hydraulic Gradient Azimuth
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Well
Well A
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Figure 10. Influence of irrigation pumping on
groundwater flow at downgradient limits of a
contaminant plume. Hydraulic gradients were calculated
using the three-point method implemented in 3PE with
hydraulic head data obtained from wells A, B, and
C. The distribution of the vector directions during the
monitoring period is presented using a rose diagram.
The average hydraulic gradient vector during the
irrigation season (red vector) is significantly different
from the average vector while pumping is not occurring
(blue vector). The directions of the arrows indicate
the groundwater flow directions. The difference in the
magnitudes of the hydraulic gradients is indicated by the
relative lengths of the vector arrows.
-------
The direction and magnitude of the hydraulic gradient
change abruptly with the onset of pumping (Figure
11). The magnitude of the gradient also appears to
increase as the agricultural season progresses. Based on
these initial results, it was determined that agricultural
pumping may be a major influence on groundwater flow
and would require both additional characterization and
consideration during remedial design.
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Figure 11. Hydraulic heads measured using pressure
transducers in wells A, B, and C and the resulting
magnitude and direction (azimuth) of the hydraulic
gradient measured in degrees from North. When the
irrigation well is not pumping, the hydraulic gradient
is low and oriented approximately north. During the
agricultural season, the hydraulic gradient increases in
magnitude and the orientation shifts to the northeast
toward the irrigation well.
4.2 Application #2: Characterization of
Groundwater/Surface Water Interactions
Groundwater/surface water interactions can control
the transport and the environmental fate of dissolved
constituents at sites where surface water bodies are
hydraulically connected with the aquifer. The following
example illustrates the use of the 3PE spreadsheet in the
initial hydrologic characterization of these interactions.
4.2.1 Background and Setting
The site is a closed waste management unit (WMU)
situated adjacent to a pond. The site lies above an
unconsolidated alluvial aquifer that is approximately
30m thick. Estimates of average hydraulic conductivity
within the aquifer range from approximately 3 m/d
to over 15 m/d. In general, groundwater flows north
beneath the WMU and has elevated concentrations
of inorganic constituents. Based on an initial
potentiometric surface map, it appears that groundwater
on the eastern margin of the site may discharge to the
pond.
4.2.2 Characterization Objective
The investigation was designed to answer the following
question:
Do hydraulic gradients near the pond provide consistent
indications of groundwater flow toward the pond or do
they fluctuate through time?
4.2.3 Network Design and Data Acquisition
For the initial investigation of temporal trends in
hydraulic gradients, hydraulic head was monitored
in three existing wells near the portion of the pond
where groundwater discharge was indicated in the
potentiometric surface. Surface water elevation was
also monitored using a staff gauge installed in the pond.
Monitoring was performed for ten months to evaluate
potential seasonal differences in hydraulic gradients
using pressure transducers programmed to obtain a
synchronous data set twelve times each day and periodic
manual measurements of groundwater elevations to
confirm the accuracy of the pressure transducer data.
4.2.4 Results and Conclusions
The magnitude and direction of the hydraulic gradient
were calculated using 3PE for each of the data sets
obtained from the wells during the monitoring period
(Figure 12). In addition, precipitation data were obtained
from a nearby weather station to aid in conceptualizing
site conditions. The results indicate that the direction
of the hydraulic gradient was generally toward the pond
(Figure 13) with the exception of three very brief periods
when surface water elevations rose rapidly in response
to rainfall and temporary increases in the height of the
dam due to beaver activity. During these brief periods.
the flow direction near the pond reversed and the pond
provided recharge to the aquifer. The magnitude of the
hydraulic gradient appeared to display some degree of
seasonality with a decrease in magnitude during the late
summer and early fall that may be related to periods of
decreased aquifer recharge from precipitation in areas
upgradient of the pond.
This initial characterization of the hydrologic
setting near the pond provided the framework for
subsequent investigations of the transport of dissolved
constituents into the pond. Based on the results of
these investigations, remedial measures were ultimately
undertaken to mitigate further discharge of contaminated
groundwater.
-------
Pond
Waste
Management
Unit
Figure 12. Site map depicting triangle of wells used to
estimate hydraulic gradients. The average hydraulic
gradient vector (red arrow) indicated groundwater flow
was to the pond during the monitoring period.
0.03
2 ฃ0.02
1.001 -
o
180
0
67.0
!|
If.665
I
66.0 '
' i-
\;
" Well A
Well B
Well C
Jan
Apr
Jul
Ocl
Figure 13. Hydraulic heads measured using pressure
transducers in three wells and the resulting magnitude
and direction (azimuth) of the hydraulic gradient
measured in degrees from North. Precipitation data was
obtained from a nearby meteorological station.
4.3 Application #3: Spatial and Temporal
Characterization of Groundwater Flow
The determination of groundwater flow directions
is often one of the first steps in characterizing the
hydrogeology of a site. Potentiometric surface maps
can provide a general understanding of groundwater
flow directions at a given point in time. The following
example depicts the use of 3PE in conjunction with
software to visualize the average (net) groundwater flow
directions across a site.
4.3.1 Background and Setting
The site was the location of unregulated dumping
of industrial waste. The aquifer of interest consists
of glacial till and outwash deposits. Hydraulic
conductivity values range from 0.01 m/d to over 100
m/d. Groundwater generally flows in a northwesterly
direction before discharging to a stream.
4.3.2 Characterization Objective
The investigation was designed to answer the following
question:
What are the average groundwater flow directions in key
areas of the site?
4.3.3 Network Design and Data Acquisition
Ten wells screened in the shallow glacial aquifer were
chosen for this investigation. The wells were selected
to form a triangular network of eleven, three-point
problems, providing coverage across the central portion
of the site. The wells were instrumented with pressure
transducers which recorded changes in water levels
twelve times each day for a sixteen month period.
Manual water-level measurements were obtained five
times during the study using an electric water-level
indicator to document the accuracy of the pressure
transducers and allow evaluation of transducer drift.
4.3.4 Results and Conclusions
Groundwater elevations across the site varied up to
approximately 0.8 m during the study period in response
to precipitation events (Figure 14). The groundwater
flow directions remained relatively consistent and
the magnitude of the hydraulic gradient varied less
than a factor of two throughout the study, despite the
significant variations in water levels (Figure 15). Note
that the direction of the hydraulic gradient vector was
approximately north (i.e., 0 degrees) throughout the
monitoring period. The large shift in the numerical
value of the azimuth from approximately 0 degrees
to approximately 360 degrees merely reflects a small
shift in direction from slightly east of north to slightly
west of north. In this case, the net groundwater flow
direction was determined by averaging the calculated
hydraulic gradient direction data for each triangle in the
monitoring network. The average hydraulic gradient
directions and coordinates for each triangle were
overlaid on a site base map. The resulting figure
(Figure 16) illustrates the average directions of the
hydraulic gradients and, therefore, groundwater flow
during the sixteen month time period.
-------
Figure 14. Hydrographs of hydraulic head in
monitoring wells and precipitation measured at a local
meteorological station.
Figure 15. Hydraulic heads measured using pressure
transducers in three wells and the resulting magnitude
and direction (azimuth) of the hydraulic gradient
measured in degrees from North. Precipitation data were
obtained from a local meteorological station.
T18
J23
J27
Figure 16. Network of three-point problems used to calculate hydraulic gradients and the resulting average (net)
direction of the hydraulic gradient within each triangle (blue arrows). The monitoring wells (crosses), three-point
solution triangles, and output vectors from the 3PE spreadsheet were plotted on a georeferenced base map to enhance
conceptualization of groundwater flow directions across the site.
-------
4.4 Application #4: Demonstration of
Anisotropy Effects on Groundwater Flow
The principal direction of groundwater flow in a
setting where the hydraulic conductivity varies with
horizontal direction is a function of the directions and
magnitudes of the minimum and maximum hydraulic
conductivity axes and the direction of the hydraulic
gradient. It is often instructive to illustrate the
potential effects of anisotropy on groundwater flow
assuming various hydraulic conductivity distributions
that may be encountered in a given geologic setting.
Such illustrations can aid in developing site-specific
conceptual models.
4.4.1 Problem Description
For this example, the 3PE spreadsheet was used to
calculate the principal direction of groundwater flow
assuming isotropic and anisotropic conditions. The
results were then compared to evaluate the potential
effects on a site-specific conceptual model. For these
calculations, data concerning horizontal anisotropy
characterized by Stoner (1981) in a coal bed aquifer
were used in conjunction with an assumed set of
hydraulic head data to aid in visualizing the potential
effects of this anisotropy on an arbitrary groundwater
flow field.
4.4.2 Example Data Set
The aquifer was assumed to have an average maximum
hydraulic conductivity of 0.65 m/d oriented 85 deg
clockwise from North. The average minimum hydraulic
conductivity is assumed to be 0.26 m/d and
perpendicular to the direction of the maximum hydraulic
conductivity. The following well configuration and
hydraulic head data were assumed for this illustration:
Well
Well A
Well B
Well C
X
Coordinate
(m)
722229
722179
722279
Y
Coordinate
(m)
156500
156400
156400
Hydraulic
Head
(m msl)
100.00
100.00
99.00
4.4.3 Results and Conclusions
Under the given conditions, the direction of the hydraulic
gradient, which would be the groundwater flow direction
in an isotropic setting, is significantly different from
the groundwater flow direction due to the moderate
anisotropy in horizontal hydraulic conductivity. The
groundwater flow direction shifts markedly toward
the direction of maximum hydraulic conductivity. In
this case, the horizontal hydraulic conductivity only
varied by a factor of 2.5 (i.e., KmaxIKmin = 2.5) in the
horizontal plane of the aquifer. This resulted in an 18
deg difference in the direction of groundwater flow from
the direction of the hydraulic gradient (Figure 17). If the
degree of anisotropy is increased to a factor of 10 (i.e..
Kmax/Kmm = 10), the difference between the directions of
the hydraulic gradient and groundwater flow increases
to 28 deg. This indicates the potential impact that
anisotropy may have on proper conceptualization of
groundwater flow under anisotropic conditions.
Well A
Direction of Maximum
Hydraulic Conductivity
Well B
WellC
Anisotropy Ratio = 10:1
Anisotropy Ratio = 2.5:1
Hydraulic Gradient
Direction
Figure 17. Effects of anisotropy
in horizontal hydraulic
conductivity on groundwater
flow directions. The blue arrow
(direction of the hydraulic
gradient) would be the direction
of groundwater flow in an
isotropic setting. In a situation
with horizontal anisotropy, the
direction of groundwater flow
shifts toward the direction of the
maximum hydraulic conductivity
as the ratio Kmax'.Kmin increases.
-------
5.0
Summary
In many situations, hydraulic head data, used in
hydrogcologic assessments will increasingly be obtained
using automated means, such as pressure transducers.
The large amount of the data recorded can only be
efficiently analyzed using a computer-based tool. The
3PE spreadsheet is a tool that can be used for solving the
classical three-point problem to determine the hydraulic
gradient for a single triangle or multiple triangles
of monitoring points. In addition to calculating the
horizontal hydraulic gradient magnitude and direction.
3PE calculates the groundwater velocity magnitude and
direction, if estimates of hydraulic conductivity and
effective porosity are provided.
Hydraulic gradient estimation using three-point
solution methods, such as 3PE, can be a powerful tool
in situations where the potentiometric surface can be
represented by a plane. This tool is particularly well
suited for rapid analyses of temporal trends in hydraulic
gradients and groundwater flow velocity using large
datasets. Potential applications for 3PE include:
Evaluation of the design and effectiveness of
groundwater remediation systems.
Groundwater remediation systems require detailed
knowledge of the ambient (natural) groundwater
flow field for system design and the changes
in the flow field during system operation.
Information regarding groundwater flow rates and
directions prior to system design and after remedy
implementation is a critical data need for
technologies using groundwater extraction or fluid
injection as well as more passive technologies such
as permeable reactive barriers. Similar information
is also needed to design robust monitoring networks.
Characterization of groundwater/surface water
interactions.
In situations where groundwater/surface water
exchange may occur, initial evaluations of
temporal trends in groundwater flow rates and
directions can significantly improve
conceptualization of the possible effects of such
exchanges on surface water and, potentially,
groundwater quality.
Rapid visualization of spatial patterns in
hydraulic gradients (groundwater
How directions).
At sites where key portions of the groundwater
flow field can be adequately represented by a
scries of three-point problems, 3PE offers a tool
for determining groundwater flow directions across
the site at individual points in time or as averages
(net flow directions) that can be readily plotted on
site maps for improved visualization.
* Enhanced groundwater flow model calibration.
Analyses of the range of hydraulic gradients and
groundwater flow directions for a number of
selected triangles in sensitive areas of the model can
be compared to simulated gradients and flow-
directions. This can provide additional confidence
in the applicability of the flow model under a wider
variety of hydrologic conditions.
* Improved conceptualization of site hydrology.
Changes in hydraulic gradient through time can
often be correlated with hydrologic changes such
as local pumping or irrigation schedules, variations
in aquifer recharge due to changing precipitation
patterns, and changes in nearby surface water
elevations. Knowledge of such correlations
often leads to an improved understanding of
the dominant controls on groundwater flow and
appropriate engineering methods for attaining
site-specific objectives.
The following general recommendations concerning the
application of 3PE and other simple tools for estimating
hydraulic gradients are provided:
1. Determine that the potentiometric surface in the
area of interest can be adequately described as a
planar surface or a series of planes.
2. Use hydraulic head data that are synchronously
obtained from wells and piezometers of
appropriate constructions and representative of
the same portion of the aquifer.
3. Develop clear monitoring objectives. Use
detailed definitions of the particular questions
to be answered by the investigation to properly
design the monitoring network and monitoring
frequency required to support the analyses.
4. Consider potential effects of site-specific
temporal variability on hydraulic
gradients. Evaluation of the possible frequency
of significant changes in hydraulic gradients
-------
is Hie basis for informed decisions regarding the
appropriate frequency for monitoring
hydraulic head.
Consider measurement uncertainty and its
effects on the attainment of investigation
objectives. Monitoring network designs,
measurement tools, and monitoring
procedures should be chosen based, in part, on
an analysis of the uncertainty that will be
inherent in the measurements. In some
situations, it may be useful to use 3PE
to examine the sensitivity of the hydraulic
gradient calculations to site-specific well
placement and potential measurement errors.
-------
6.0
References
Abriola, M.L., and G.F. Finder. 1982. Calculation of
velocity in three space dimensions from hydraulic head
measurements. Ground Water. 20(2):205-213.
Bates, R.L., and J.A. Jackson, editors. 1987. Glossary of
Geology^. American Geological Institute, Alexandria, VA.
788 pp.
Bear. J. 1972. Dynamics of Fluids in Porous Media.
Dover Publications, Mineola, NY. 764 pp.
Bear, J. 1979. Hydraulics of Groundwater. McGraw-
Hill New York,'NY. 569 pp.
Bear, J., and A. Verruijt. 1987. Modeling Groundwater
Flow and Pollution. D. Reidcl Publishing Company,
Boston, MA. 414pp.
Cole, B.E., and S.E. Silliman. 1996. Estimating the
horizontal gradient in heterogeneous, unconfined
aquifers: Comparison of three-point schemes. Ground
Water Monitoring & Remediation. 16:2(84-91).
Cunningham, W.L., and C.W. Schalk, compilers.
2011. Groundwater Technical Procedures of the U.S.
Geological Survey. Techniques and Methods 1-A1. U.S.
Geological Survey, Reston, VA. 151 pp.
Devlin, J.F. 2003 . A spreadsheet method of estimating
best-fit hydraulic gradients using head data from multiple
wells. Ground Water. 41(3):316-320.
Devlin, J.F., and C.D. McElwee. 2007. Effects of
measurement error on horizontal hydraulic gradient
estimates. Ground Water. 45(l):62-73.
Domenico, P.A., and F.W. Schwartz. 1990. Physical and
Chemical Hydrogeology. John Wiley & Sons, New York,
NY. 824 pp.
Fetter, C.W. 1981. Determination of the direction of
ground-water How. Ground Water Monitoring Review .
Fetter, C.W. \9%%. Applied Hydrogeology. Merrill
Publishing, Columbus, Ohio. 592 pp.
Ficncn, M.N. 2005. The three-point problem, vector
analysis and extension to the n-point problem. Journal of
Geoscience Education. 53(3):257-262.
Freeze. A., and J. Cherry. 1979. Groundwater. Prentice-
Hall. Englewood Cliffs. New Jersey. 604 pp.
Hantush, M.S. 1966. Analysis of data from pumping
tests in anisolropic aquifers. Journal of Geophysical
Research. 71(2):421-426.
Heath, R.C. 1983. Basic Ground-Water Hydrology.
Water-Supply Paper 2220. U.S. Geological Survey.
Reston, VA. 82 pp.
Kelly, W.E., and I. Bogardi. 1989. Flow directions with a
spreadsheet. Ground Water. 27(2):245-247.
Kern, J.W., and C.W. Dobson. 1998. Determination of
variances for maximum and minimum transmissivities
of anisotropic aquifers from multiple well pumping test
data. Ground Water. 36(3):457-464.
Maslia, M.L., andR.B. Randolph. 1987. Methods and
computer program documentation for determining
anisotropic transmissrvity tensor components of two-
dimensional ground-water flow. Water-Supply Paper
2308. U.S. Geological Survey, Reston, VA. 46 pp.
McKenna. S. A., and A. Wahi. 2006. Local hydraulic
gradient estimator analysis of long-term monitoring
networks. Ground Water. 44(5):723-731.
Newell, C.J., S.D. Acrce, R.R. Ross, and S.G.
Ruling. 1995. Light Nonaqueous Phase Liquids.
EPA/540/S-95/500. U.S. Environmental Protection
Agency, Ada, OK. 28 pp.
Neuman, S.P.. G.R. Walter, H.W. Bently. J.J. Ward,
andD.D. Gonzalez. 1984. Determination of horizontal
aquifer anisotropy with three wells. Ground Water.
22(l):66-72.
Finder, G.F., M. Celia, and W.G. Gray. 1981. Velocity
calculation from randomly located hydraulic heads.
Ground Water. 19(3):262-264.
Post. V.E.A. and J.R. von Asmuth. 2013. Review:
Hydraulic head measurements - new technologies,
classic pitfalls. Hydrogeology Journal. 21(4):737-750.
Post, V, H. Kooi, and C. Simmons. 2007. Using
hydraulic head measurements invariable-density ground
water flow analyses. Ground Water. 45(6):664-671.
Rosenbeny, D.O. 1990. Effect of sensor error on
interpretation of long-term water-level data. Ground
Water. 28(6):927-936.
Silliman, S.E., and C. Frost. 1998. Monitoring hydraulic
gradient using three-point estimator. Journal of
Environmental Engineering. 124(6):517-523.
Silliman, S.E., and G. Mantz. 2000. The effect of
measurement error on estimating the hydraulic gradient
-------
in three dimensions. Ground Water. 38(1): 114-120.
Sorenson, J.P.R., and A.S. Butcher. 2011. Water
level monitoring pressure transducers: A need for
industry-wide standards. Ground Water Monitoring &
Remediation. 31(4):56-62.
Stoner, J.D. 1981. Horizontal anisotropy determined
by pumping in two Powder River Basin coal aquifers,
Montana. Ground Water. 19(1):34-40.
Stack, O.D.L. 1989. Groundwater Mechanics. Prentice-
Hall. Englewood Cliffs, New Jersey. 732 pp.
USEPA. 1990. Handbook: Ground Water, Volume 1:
Ground Water and Contamination. EPA/625/6-90/016a.
U.S. Environmental Protection Agency, Washington,
DC. 144 pp.
USEPA. 2006. Guidance on Systematic Planning Using
the Data Quality Objectives Process: EPA QA/G-4.
EPA/240/B-06/201. U.S. Environmental Protection
Agency, Washington, DC. Ill pp.
USEPA. 201.4. On-Site: the On-line Site Assessment
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Assessment Calculation. U.S. Environmental Protection
Agency. http://www.epa.gOY.at.hens/learn2model/part-
two/onsi le/gradientS ns .htm 1
Way, S.C., and C.R. McKee. 1982. In-situ determination
of three-dimensional aquifer permeabilities. Ground
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-------
A
Theoretical
A.I Representing Wells in a Two-Dimensional Cartesian System A2
A.2 Hydraulic Gradient in 2-D Flow Field A4
A.3 Specific Discharge A6
A.4 Ground-water Velocity A13
Figures
Figure A-1. Representation of the wells in a two-dimensional Cartesian system as implemented in
the 3PE spreadsheet A2
Figure A-2. Definition of the quadrants and vector angles used in this derivation A5
Figure A-3. Representation of the rotation of the coordinate sy stem so that the principal directions
of anisotropy, K,,,m and K,,,m, correspond with the coordinate system x', y'. respectively A7
Figure A-4. Terminology used in calculation of the hydraulic conductivity ellipsoid A8
Figure A-5. Terminology used in this derivation of the direction of the groundwater velocity vector
in an anisotropic medium A13
-------
The solution of the three-point problem for calculations of hydraulic gradient and groundwater velocity vectors
under both isotropic and anisotropic conditions implemented in the Excelฎ workbook relies on numerical rather
than graphical methods. Mathematical conventions and notations used in the theoretical development differ slightly
from those used in the spreadsheet. The conventions implemented in the spreadsheet were developed based on the
hydrogeological concept of the representations of planes and angles. The derivations of the mathematical expressions
implemented in Excelฎ are provided below.
A.I Representing Wells in a Two-Dimensional Cartesian System
The Cartesian coordinates (Figure A-l) are unique and the points of a Cartesian plane can be identified with all
possible pairs of real numbers. For example, the locations of three points on the plane are fully defined with their
coordinates (i.e., (x^y,), (x2,y2 ),and (x3,y3)).
y 4
Node #3
23
d
13
Node #2
Node #1
WellC(x3,y3-)
WellB(_x2,y2)
(0,0)
Figure A-l. Representation of the wells in a two-dimensional Cartesian system as implemented in the 3PE spreadsheet.
Terminology used to designate the sides and vertices of the triangle (well locations) is depicted in the left triangle. The
well coordinate convention and location of the centroid of the triangle are depicted in the triangle on the right.
-------
The distance between the nodes (i.e., the lengths of the triangle sides) can be computed using the Cartesian version of
Pythagoras' theorem:
-i - y2)2
(A-la)
O2 -
(A-lb)
y3)2
(A-lc)
The centroidof the triangle (xc,yc) can be computed as
x2+
(A-2a)
+ ^2 +
(A-2b)
When the three sides of a triangle are known, the angles at the nodes (#1, #2, and #3) can be found
_,
Angle @#1 = cos ]
(A-3a)
Angle @#2 = cos
j Ad12)2+ (^23)2~ (d13)
1 - -
V 2d12^23
(A-3b)
= 180- Angle @#1 - Angle @#2
(A-3c)
The area of triangle, A, is computed as
= -det\x2
y3
(A-4)
-------
A.2 Hydraulic Gradient in 2-D Flow Field
The hydraulic gradient, i, is the change in hydraulic head per unit of distance in the direction in which the maximum
rate of decrease in head occurs (Heath, 1983). The direction of the slope of the water table or potentiometric surface is
important because in a homogeneous and isotropic aquifer it indicates the direction of the groundwater flow.
Over a relatively small area, the water table or potentiometric surface can often be approximated as a plane. The
general equation of a plane in the Cartesian coordinate system (x, y) is
h = Ax + By + C
(A-5)
where A, B, and C are constants of the plane equation.
Three points on the plane that form a triangle are sufficient to define the plane. That means that the hydraulic heads
in three wells located at the corners of a triangle are sufficient to define the local water table or potentiometric surface.
The slope of the plane (i.e., the hydraulic gradient of the water table or potentiometric surface) can be computed if the
equation of the plane is known. If the corners of the triangle are labeled 1, 2, and 3 and the hydraulic heads at the
corners of the triangle are h= h(x1,y1), h2= h(x2, y2), h = h(x3,y3) and are known, then the following equations
can be written
^ +C
(A-6a)
h2 = Ax2 + By2 + C
By3 +C
(A-6b)
(A-6c)
where A, B, and C are constants yet to be determined. The above equations (A-6a,b,c) can be written in the matrix
notation as
72
(A-7)
Solving for [A, B, C\
-i
(A-8)
-------
The hydraulic gradient, i = - grad h = -Vh, is a vector defined by its direction and magnitude. The components of the
vector are ix = - dhldx and iy=- dhldy. The physical interpretation of the constants A and B is that they are actually the
components of the hydraulic gradient in the x- and y- direction, ix and iy, respectively. Inspection of Equation (A-5)
reveals that the coefficient C is equal to the hydraulic head at the origin of the Cartesian system (x=y = 0).
The magnitude of the hydraulic gradient can now be computed from its components as
I =
2+ i 2
iy
(A-9)
For the purposes of this derivation, the direction of the hydraulic gradient is the angle (a) between the positive x-axis
and the hydraulic gradient (Figure A-2):
North
>'
in oar-,-)
I QU+.+)
Azimuth
Figure A-2. Definition of the quadrants and vector angles used in this derivation. Note
that the mathematical convention used to number the quadrants in the derivation proceeds
in a counter-clockwise direction from the positive x-axis. In the 3PE spreadsheet, the
familiar geological convention in which the quadrants are numbered in a clockwise
fashion starting at the positive y-axis (North) was implemented.
(A-10)
-------
The axes of the two-dimensional Cartesian system divide the plane into four regions called quadrants, each bounded
by two half-axes. These regions are denoted by Roman numerals: I (+,+), II (-,+), III (-,-), and IV (+,-). According
to the mathematical convention used in this appendix, the numbering goes counter-clockwise starting from the upper
right ("northeast") quadrant (Figure A-2). Note that the convention of naming quadrants in this section differs from
that used in the 3PE spreadsheet, and supporting documentation.
Equation (A-10) would yield the same angle for the case when ix< 0 and iy > 0 (the II quadrant or Q2 orientation) and
in the case when the signs are reversed (the IV quadrant or Q4 orientation), it is important to know in which quadrant
the hydraulic gradient components lie.
A.3 Specific Discharge
The specific discharge (Darcy's velocity), q, in a homogeneous isotropic medium can be expressed by the following
equation (Bear and Verruijt, 1987, p. 34):
q= Ki (A-ll)
where K is the hydraulic conductivity (L/T) and /' is the hydraulic gradient (L/L).
Three-dimensional groundwater flow in a homogeneous anisotropic medium can be expressed by the following set of
equations (Freeze and Cherry, 1979):
KXX- Kxy-KXZ (A.12a)
a/i
-------
In its two-dimensional form, Equations (A-12a,b) become
Qx KXX ,-! Kxy a
dh
~Kyxdx
dh
fy
dh
(A-13a)
(A-13b)
The directions in space corresponding to the maximum and minimum values of the hydraulic conductivity are
called the principal directions ofanisotropy. Let a coordinate system x', y' coincide with the principal directions of
anisotropy with^'= Km* oriented along the x' axis and Ky = Kmm along they' axis (Figure A-3). Equations (A-13a,b)
are then further reduced to their simplest form
^771(71
y 4
Figure A-3. Representation of the rotation of the coordinate system so that the principal
directions of anisotropy, Kmax and Kmm, correspond with the coordinate system x', y',
respectively.
dh
~T~7
(A-14a)
dh
~ ~ "min ~~,
(A-14b)
-------
The specific discharge q in the direction s is defined as
dh
(A-15)
If/3 is the angle between the specific discharge qs and the x'-axis (Figure A-4), then
Figure A-4. Terminology used in calculation of the hydraulic conductivity ellipsoid
including the major semi-axes oriented in the directions of the maximum and minimum
hydraulic conductivity. The hydraulic conductivity Ks in any direction of flow in an
anisotropic aquifer can be determined graphically or using Equation (A-20).
qxl
qyl
(A-16a)
(A-16b)
The chain rule applied to the derivatives dh/ds yields
dh dhdx' dhdy' dh
T~ = T~!~^ -- h T~7^T~ = T~?
3.9 dx ds dy ds dx
dh
TT sin/?
dy
(A-17)
-------
Substituting Equation (A-17) into (A-15) and simplifying yields
-f = cos/? + ^- sin/?
Substituting Equations (A-16a,b) into (A-17) would yield
s
cos/? H sin/?
V V If
"S *vmcur "
Finally.
1 cos2 /? sin2 /?
= + (A-20)
or since r2 = (jc')2 + (/)2 , then cos2/? = ^ / 2 and sin2/? -
rZ i"u --"ll P / 2
V 17 17
"<: "J71/7V "77117J
This equation is known as the hydraulic conductivity ellipsoid with major semi-axes -max and
(Figure A-4). The hydraulic conductivity Ks in any direction of flow in an anisotropic aquifer can be determined
graphically or using Equation (A-20) and solving it for Ks
=
Kmin
Because the coordinates of the points are often represented in a global coordinate system (e.g., UTM), which may or
may not coincide with the principal directions of anisotropy, Equations (A-14a,b) in a local coordinate system cannot
always be applied directly.
-------
When the x',y' coordinate system is rotated counterclockwise by some angle (9) to coincide with the x,y coordinate
system, then the relationship between the specific discharge in two coordinate systems is given as
(Jx = Ixi cos 9 ~
-------
By multiplying and grouping the similar terms
dh dh dh ,
qx = ~ Kmax cos2 0 - Kmax sin 9 cos 9 - Kmin sin2 6 (A-27a)
dh
~ sin 9 cos 8
dh dh , dh ,. _ x
Sil1 S + * Si" 9 CฐS ฐ
-Kmin cos
Simplifying,
qx = ~ ( Kmax cos2 0 + Kmin sin2 0) - ((Kmax - Kmin ) sin 8 cos 8) (A-28a)
dh . dh
qy=~ ((Kmax ~ Kmin ) sin 8 cos 8) - (Kmax sin2 8 + Kmin cos2 8 ) (A-28b)
A comparison of Equations (A-13a,b) and (A-28a,b) provides equations for computing Kv components (Strack, 1989;
p. 14):
Kxx - Kmax cos2 B + Kmin sin2 B (A-29a)
Kyy = Kmax sin2 6 + Kmin cos2 6 (A-29b)
~ Kmin) sin 8 COS 8 (A-29c)
Using the trigonometric identities,
2 sin2 9 = 1 - cos 26 (A-30a)
2 cos2 0 = 1 + cos 26> (A-30b)
sin 28 = 2 sin 8 cos 8 (A-30c)
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the hydraulic conductivities from Equations (A-29a,b,c) can be expressed as (Bear, 1979; p.73)
I 17 _ 17 17 / A O 1 \
"-min "-max Amin (A-jla)
., ,
Krr = + COS 26
*xx
,, Kmax + Kmin Kmax Kmin (A-31b)
Kyy = cos 26
,. ,,
Kxy = Kyx = 5 Sm2# (A-31c)
Equation (A-3 Ic) can be rearranged when the right-hand side of the equation is multiplied by cos 261 cos 26
., Kmax ~~ "mm ซ.,, ,, /A ,ON
Kxy = tan 26 cos 28 (A-32)
and solved for tan 26
y-v
tan 26 = ^ (A-33)
~ Kmin) COS 26
Subtracting Equation (A-3 Ib) from Equation (A-3 la) yields
is i/ _ (is is \ cos 26
JLA yy \. itiitA itaiij
Inspection of Equations (A-34) and (A-33) reveals the equation for computing the rotation angle 0 when the K
components are known (Bear, 1972; p. 140)
tan 26 = _xy (A-35)
KXX "
If the x', y' are principal directions (Figure A-3), the rotation angle 9 can be computed from Equation (A-35) and the
principal hydraulic conductivity components Kmaxw& Kmm are computed as (Bear, 1972; p. 141)
.J(fkz!fc)'
I i ^^ yy i i i/ i ,-ป ->^ \
y (A-36a)
+ ^yy _ If Kxx ~ Kyy\
2 ^l\ 2 )
2 (A-36b)
V 7
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A.4 Groundwater Velocity
Darcy 's law for groundwater flow in an isotropic medium can be expressed by the following equation (Freeze and
Cherry, 1979):
ne np
(A-37)
where Vis the groundwater velocity, q is the specific discharge, K is the hydraulic conductivity, ne is the effective
porosity, and / is the hydraulic gradient.
In an anisotropic medium, the groundwater velocity components are computed as (Finder, et al., 1981)
1 dh dh
x ~ \. xx -. ' ^xy a,.)
(A-38a)
1 dh dh
T/ / TS I Jf "
y ~ ~ n/ yx dx yy dy
(A-38b)
The magnitude of the groundwater velocity is computed as:
V=
V
(A-39)
Using the conventions of this derivation, the direction of the groundwater velocity vector is the angle (ij) between the
positive x-axis and the hydraulic gradient (Figure A-5):
equipotential linen
Figure A-5. Terminology
used in this derivation of the
direction of the groundwater
velocity vector in an anisotropic
medium. The hydraulic
gradient vector is the blue
arrow labeled i and the
groundwater velocity vector is
the red arrow labeled V.
-------
r? = tan 1 j y/v j (A-40)
/ X
In an anisotropic aquifer, the groundwater velocity and the hydraulic gradient vectors are not collinear (Figure A-5).
The angle between the vectors (<$) can be computed by subtracting the /' direction from the V direction; however, a
better approach is to use the dot product or the inner product because the angle between the vectors is always the inner
angle, which is less than 180 degrees (Fienen, 2005)
S =
The above derivation provides mathematical expressions for hydraulic gradient and groundwater velocity that can be
computed using standard functions available in commercial spreadsheet software.
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B
Comparison of 3PE Results with Published Problems
B.I Validation of Hydraulic Gradient Computation B2
Heath (1983; pp. 10-11) B2
Finder etal. (1981) B3
Siiiiman and Frost (1998) B4
On-Site: Hie On-line Site Assessment Tool - Three Point Gradient Calculator (USEPA, 2014).B4
Gradicnt.XLS (Devlin, 2003) B5
B.2 Validation of Velocity Computations B6
Bear, I and A. Vemiijt (1987) Problem 2.7 (p. 387) B6
Bear, land A. Verruijt (1987) Problem 2.9 (p. 387) B7
Bear, J. and A. Vemiijt (1987) Problem 2.11 (p. 387) B8
Abriola and Finder (1982) B9
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Results from the numerical computations using the 3PE spreadsheet were verified through comparison with results
of published problems available in the literature, and with results obtained from other published hydraulic gradient
calculators.
B.I Validation of Hydraulic Gradient Computation
Heath (1983; pp. 10-11)
This is a classic problem used to explain the graphical method of solving a three-point problem. The information
about the hydraulic heads is given: Well 1 (26.26 m), Well 2 (26.20 m), and Well 3 (26.07 m). The coordinates of the
wells were not provided; the distances between the wells are given: dn = 165.0 m; d23 = 150.0 m, and dn = 215.0 m.
The computed hydraulic gradient is (0.13 m)/(133 m) or 0.000977.
Because the well coordinates were not provided in Heath (1983), the triangle position with respect to North is not
uniquely defined. For purposes of this comparison, Well 1 was placed at the origin of the coordinate system and Well
2 on the x-axis. The coordinates of Well 3 were then computed using trigonometric functions.
Well Name
Welll
Well 2
Wells
X Coordinate
(m)
0.00
165.00
154.39
Y Coordinate
(m)
0.00
0.00
149.62
The computations of the length of the triangle sides were checked:
Triangle Centroid
Triangle Area
Distance #1 - #2
Distance #2 - #3
Distance #1 - #3
106.46
12,344.03
165.00
150.00
215.00
49.87
(LA2)
(L)
(L)
(L)
Angle of Triangle (degrees) @
Welll
Well 2
WellS
44.10
85.95
49.95
The magnitude of the hydraulic gradient computed by 3PE is 0.000966, which is equivalent to the published value
considering that the Heath (1983) solution was determined graphically and, therefore, subject to the increased
uncertainty associated with graphical solutions.
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Finder etal. (1981)
The coordinates and the hydraulic heads are given. The hydraulic heads are: Well 101 (11.00 m), Well 104 (12.00 m),
and Well 103 (10.00 m) and the well coordinates are:
Well Name
MW-101
MW-104
MW-103
X Coordinate
M
0.00
1.00
0.00
Y Coordinate
(m)
0.00
1.00
2.00
The hydraulic conductivity components K** and Ky? are both 2 m/d and the effective porosity is 0.25. The published
results are as follows: the velocity components F* and Vy are -12 m/d and 4 m/d, respectively; the hydraulic gradient
components i* and iy are -1.5 and 0.5, respectively.
The 3PE results are identical to the published results. 3PE also computed the orientation of the velocity vector as 288
degrees.
> Hydraulic Gradient Direction
> Groundwater Velocity Direction
MW-103
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Silliman and Frost (1998)
The hydraulic heads and well coordinates are given. The hydraulic heads in Well 1, Well 2, and Well 3 are 132.37 m.
131.86 m, and 132.01 m, respectively. The well coordinates are as follows:
Well Name
Welll
Well 2
WellS
X Coordinate
(m)
534.12
439.43
422.13
Y Coordinate
(m)
134.37
236.34
162.33
The magnitude of the gradient is 0.003667 (m/m) and the orientation is reported to be "approximately 45 degrees
below the positive x-axis" (Silliman and Frost, 1998, p. 518). The magnitude of the gradient computed by 3PEhas
the same value. However, the orientation of the hydraulic gradient vector computed by 3PE is 316 degrees in the
clockwise direction from North, which is different from the reported orientation. The orientation computed by 3PE
was verified using a graphical solution. It appears that the published orientation is reported incorrectly, and should
read "approximately 45 degrees above the negative x-axis".
250.0
200.0
I
>-
150.0
100.0
Well 2
> Hydraulic Gradient Direction
Welll
Well 3
400.0
450.0
X(m]
500.0
550.0
On-Site: the On-line Site Assessment Tool - Three Point Gradient Calculator (USEPA, 2014)
The output from the 3PE spreadsheet was compared with the results of a sample problem computed using the on-line
Three Point Gradient calculator (USEPA, 2014). For this example, the well coordinates and hydraulic heads presented
in Silliman and Frost (1998) and discussed above were used as input into the on-line calculator. The magnitude of the
gradient computed using the on-line calculator is 0.003666 and the orientation is 316 degrees from North, which is the
same as computed using the 3PE spreadsheet.
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GradienLXLS (Devlin, 2003)
The output from the 3PE spreadsheet was also compared with the results of two sample problems computed using
the spreadsheet Gradient.XLS (Devlin, 2003). For the first problem, the well coordinates and hydraulic heads are as
follows:
Well Name
Well A
WellB
WellC
X Coordinate
(m)
0.00
80.00
0.00
Y Coordinate
M
0.00
0.00
80.00
Hydraulic Head
(m)
100.0
101.0
102.0
The magnitude of the hydraulic gradientvectorcomputedusingGradient.XLS is 0.02795 with an orientation of-116.6
degrees off the x-axis or an azimuth of 206.6 degrees from North. Identical results were obtained using 3PE.
100.0
80.0
60.0
40.0
20.0
0.0
> Hydraulic Gradient Direction
0.0 20.0
40.0 60.0
X(m)
80.0
100.0
For the second problem, the well coordinates were the same as in the first comparison but the hydraulic head data
were:
Well Name
Well A
WellB
WellC
Hydraulic Head
(m)
101.0
100.0
102.0
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The magnitude of the hydraulic gradient vector computed using Gradient.XLS is 0.01768 with an orientation of -45.0
degrees off the x-axis or an azimuth of 135.0 degrees from North. Identical results were again obtained using 3PE.
100.0
80.0
60.0 -
40.0
20.0
-> Hydraulic Gradient Direction
20.0
40.0 60.0
X(m)
80.0
100.0
B.2 Validation of Velocity Computations
Bear, J. and A. Verruijt (1987) Problem 2.7 (p. 387)
This problem is used to validate the computation of the direction of the groundwater velocity vector in an isotropic
aquifer.
Given the piezometric heads in three observation wells located in a homogeneous confined aquifer of constant
transmissivity T = 5,000 m2/d (Well A (0,0, h = 10.0 m), Well B (0, 300, h = 8.4 m) and Well C (200, 0,
h = 12.5 m)), determine the direction of discharge through the aquifer.
Well Name
Well A
WellB
WellC
X Coordinate
(m)
0.00
0.00
200.00
Y Coordinate
(m)
0.00
300.00
0.00
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ANSWER: The direction of discharge through the aquifer is -23 degrees from the negative x-axis.
300.0
250.0
wellB
> Ground water Velocity Direction
-50.0 0.0 50.0 100.0 150.0 200.0 250.0
The orientation of the groundwater flow velocity vector calculated by 3PE is 293 degrees (23 degrees clockwise from
the negative x-axis), which is identical to the published answer.
Bear, J. and A. Verruijt (1987) Problem 2.9 (p. 387)
The following problem is used to validate the computation of the orientation of the groundwater velocity vector in an
anisotropic aquifer.
Repeat Problem 2.7 when the aquifer is anisotropic with K& = 30 m/d, Kyy = 10 m/d, and K^ = Kyi = 8 m/d.
The aquifer thickness is 50 m.
ANSWER: The discharge direction is -8 degrees from the negative x-axis.
In order to use 3PE in the solution of this problem, the principal hydraulic conductivity components and the angle of
rotation must be calculated for input into the spreadsheet. The angle of rotation is computed using Equation A-35 and
the principal hydraulic conductivity components are computed with Equations A-36a,b:
Principal Hydraulic Conductivity Components
Kmax -
Kmin =
Orientation of Kmax -
9 =
32.81
7.19
70.67
19.33
(L/T)
(L/T)
(degrees from N)
(degrees from X)
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The orientation of the groundwater velocity vector calculated by 3PE is 262 deg clockwise from North (or 8 degrees
measured counter-clockwise from the negative x-axis), which is identical to the vector orientation given in the
published problem.
350.0
300.0 Well B
-> Hydraulic Gradient Direction
-> Groundwater Velocity Direction
-100.0 -50.0 0.0 50.0
X(m)
100.0 150,0 200.0
250.0
Bear, J. and A. Verruijt (1987) Problem 2.11 (p. 387)
The following problem is used to validate the computation of the angle between the groundwater velocity vector and
the hydraulic gradient vector in an anisotropic aquifer.
Let K*=36 m/d and Ky = 16 m/d be the principal values of hydraulic conductivity in an anisotropic aquifer,
in the x andy directions, respectively, in two-dimensional flow. The hydraulic gradient is 0.004 in a direction
making an angle of 30 degrees with the positive x-axis. Determine the angle between the vectors.
ANSWER: 15.6 degrees.
Because the well coordinates and the hydraulic heads were not given, a simple layout of the wells was assumed and
the hydraulic heads were computed based on the given hydraulic gradient.
Well Name
Well A
Well B
WellC
X Coordinate
(m)
0.00
866.00
866.00
Y Coordinate
(m)
0.00
0.00
500.00
Hydraulic Head
(m)
104.0
101.0
100.0
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The direction of the hydraulic gradient calculated by 3PE is 60 degrees (30 degrees counter-clockwise from the
positive x-axis); and the direction for the velocity vector is 76 degrees (14 degrees counter clockwise from the positive
x-axis). The difference between the directions of the hydraulic gradient vector and velocity vector is 16 degrees, in
close agreement with the published result.
1,000.0
800.0
600.0
400.0
200.0
-> Hydraulic Gradient Direction
-> Groundwater Velocity Direction
200.0
400.0
X(m)
600.0
WellC
800.0
1,000.0
Abriola and Finder (1982)
The following problem is used to validate the computation of the magnitude of velocity vector components in an
isotropic aquifer.
The horizontal velocity components are calculated for wells with coordinates and hydraulic heads given as:
Well 1 (-10, 10, 20.0), Well 2 (0, 0, 20.0) and Well 4 (0, 10, 22.8). Also given are values for porosity (0.25) and
Kxx = Kyy = 2.
The published results are V* = -2.24 and Vy = -2.24.
The 3PE results are identical to the published results (Vx = -2.24 and Vy = -2.24).
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Appendix C
3PE Workbook
Excelฎ file available for download at http://www.epa.gov/nrmrl/gwerd/publications.html
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United States
Environmental Protection
Agency
PRESORTED STANDARD
POSTAGES FEES PAID
EPA
PERMIT NO. G-35
Office of Research and Development (8101R)
Washington, DC 20460
Official Business
Penalty for Private Use
$300
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