United States
Environmental Protection
Agency
Off ice of Solid Waste
and Emergency Response
Washington DC 20460
July 1986
Solid Wane
Criteria for Identifying
Areas of Vulnerable
Hydrogeology Under the
Resource Conservation
and Recovery Act

Appendix B

Ground-Water
Flow Net/Flow Line
Construction
and Analysis
Interim Final

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                                   OSWER POLICY DIRECTIVE NO

                                 9472 •  00- 2A
 GUIDANCE CRITERIA FOR IDENTIFYING
  AREAS OF VULNERABLE HYDROGEOLOGY
             APPENDIX B
  GROUND-WATER FLOW NET/FLOW LINE
       CONSTRUCTION ANALYSIS
       Office of Solid Waste
     Waste Management Division
U.S. Environmental Protection Agency
         401 M Street, S.W.
      Washington, D.C.   20460
             July 1986

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                                                           QSWER POLICY DIRECTIVE NO.

                                                         9472 • 00-2A
                                   DISCLAIMER
     This Revised Final Report was furnished to the Environmental  Protection
Agency by the GCA Corporation, GCA/Technology Division,  Bedford, Massachusetts
01730 and the Battelle Project Management Division, Office of Hazardous Waste
Management, Richland, Washington 99532,  an fulfillment of Contract
No. 68-01-6871, Work Assignment No. 28.   The opinions, findings, and
conclusions expressed are those of the authors and not necessarily those  of
the Environmental Protection Agency or the. cooperating agencies.   Mention of
company or product names is not to be considered as an endorsement by  the
Environmental Protection Agency.

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                            ABSTRACT
   This  Technical Resource Document (TRD) is an appendix to
the Guidance Criteria for Identifying Areas of Vulnerable
Hydrogeology, and discusses the procedures necessary to construct
ground-water flow nets.  It was developed to assist EPA permit
writers and permit applicants in evaluating the suitability of
locations selected for hazardous waste disposal facilities.
The focus of this manual is on the construction
of vertical flow nets.
   The document discusses the step-by-step construction of flow
nets and is designed for persons with a limited background in
hydrology.  The manual is divided into five sections.  Section
1 is an introduction which discusses the background of location
guidance development and presents important definitions.
Ground-water flow theory is descussed in Section 2.  Graphical
construction of flow nets is discussed in Sections 3 and 4
through the use of several practical examples.  Section 5 discusses
mathematical techniques used to construct flow.nets.
   Given the complexity of geohydrologic systems, the reader is
cautioned that the proper development of a flow net is more complex
than it may first appear.  The reference list at the end of the
manual should be used as an aid to better understanding flow
nets and their construction.

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                                    CONTENTS
Abstract	     ill
Figures	     vi
Tables	     ix

     1.   Introduction 	       1

               Background	       1
               Organization of document	       2
               Definition of terms 	       3

     2.   Flow Net Theory and Hydrologic Considerations	       7

               General 	       7
               Flow net theory	       9
               Time of travel (TOT). .  .	      11
               Hydrologic considerations 	 	      11

     3.   Flow Net Construction	•.- ....      15

               Basic rules	      15
               Steps in flow net construction	      16
               Example of a homogeneous isotropic flow system	      18
               Second example of homogeneous isotropic flow system ...      20
               Construction of flow nets in heterogeneous, isotropic
                 systems	,	      27
               Construction of flow nets in homogeneous, anisotropic
                 systems	      43

     4.   Construction of Flow Nets in Special Settings	      53

               Example of • ground water mound over a leaking lagoon .  .      53
               Example of complex hydrogeology in an arid region ....      60
               Construction of flow nets for fractured bedrock
                 environments	      68
               Construction of flow nets in seepage face conditions. .  .      74
               Flow net construction for free-surface flow conditions.  .      77
               Situations in which flow nets are difficult or not
                 appropriate	      78
                                     iv

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                              CONTENTS (continued)






     5.   Mathematical Construction of Flow Nets	      83



Bibliography 	      ?g




Appendix




     A.   Glossary	     ._2

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                                                       OSWER POLICY DIRECTIVE NO.

                                                     9472 -  00- 2A
                                    FIGURES


Number                                                                    page

1.1       Measurement of subsurface water pressure 	     4

2.1       Graphical representation of ground water system types	     8

2.2       Types of boundary conditions 	    13

3.1       Hypothetical flow net	    18
                                           *

3.2       Introductory example -.Homogeneous isotropic flow system with
            no vertical gradient	•	    19

3.3       Location map of refinery sice and monitor wells. .  .  	    23

3.4       Potentiometric surface showing flow direction. ........    24

3.5       Cross section through the site between wells I and IV.'....    25

3.6       Flow net for the refinery	    26

3.7       Reconstructed potentiometric surface for refinery example.  .  .    28

3.8       Deflection of flow lines across materials of different
            hydraulic conductivity 	    30

3.9       Deflection of flow lines and illustration of the
            tangent law	    31

3.10      Cross section of heterogeneous isotropic ground water system
            with slurry wall	    33

3.11      Flow net developed to analyze Che effect of slurry wall. ...    34

3.12      Well location map for example of heterogeneous isotropic
            ground water system	    35
                                                              •
3.13a     Distribution of heads in silt layer	.-  .  .    36

3.13b     Interpolation and plotting of equipotential lines	    38

3.13c     Development of flow net	    39

                                      vi

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                              FIGURES  (continued)


Number                                                                    Pagg

3.13d     Flow net for siic and sand layers	    40

3.13e     Selection of flow line for use in TOT calculation	    42

3.14      Shrinking of anisotropic flow net	    45

3.15      Elongation of flow net to original dimensions	    45

3-16a     Transformed section using available well data	    47

3.16b     Contouring to determine placement of equipotential lines  ...    49

3>16c     Flow net construction in the silt layer	    50

3.16d     Flow net for silt layer returned to original dimensions.  ...    51

4.1       Well location and water table contour map Cor sand/gravel
            underlain by clay	    54

4.2a      Construction of tie lines to allow contouring to determine
            equipotential lines	    56

4.2b      Approximate equipotential map based on contouring of data
            points	*	    57

4.2c      Flow net for mound example 	 ............    58

4.3       Well location map	    61

4.4a      Cross section showing hydraulic head data. ... 	    62

4.4b      Construction of tie lines for interpolation of equipotential
            head values	    65

4.4c      Approximate equipotential map	    66

4.4d      Approximate flow net	    67

4.5       Well location map and ground surface elevations for fractured
            bedrock example	    70

4.6       Flow net for fractured bedrock example	    72

4.7       Cross section for seepage face	    75

4.8       Flow net for seepage face	    76
                                     vli

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                              FIGURES  (continued)
Number                                                                    Page

4.9       Diagram showing x and h(x) tor Dupuit-Forchheimer
            calculation	   79

4.10      Cross section for free-surface flow	   80

4.11      Dupuit-Forchheimer solution scheme for free-surface flow ...   81
                                    vlii

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                                     TABLES


Number                                                                    Page

 3-1      Data for Refinery Monitor Wells	    22

 4-1      Horizontal Hydraulic Conductivity Data from Slug Tests  and
            Pump Tests	-	    63

 4-2      Water Level and Stratigraphy for Monitor Wells  in Fractured
            Bedrock Example	    71

 5-1      Available Hand-Held Calculator Programs for Ground Water Flow
            and Transport	    85

 5-2      Available Microcomputer Programs for Ground Water Flow
            and Transport	  .    92
                                      ix

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                                   SECTION 1
                                  INTRODUCTION

BACKGROUND
   The EPA Office of Solid Waste is issuing a series of guidance documents
designed to encourage safe and proper siting of hazardous waste management
facilities subject to regulation under RCRA.  This document on flow net con-
struction and analysis is an appendix to the Guidance Criteria for Identifying
Areas of Vulnerable Hydrogeology.  The scope of this program is introduced below.
   Under Section 202 of the Hazardous and Solid Waste Amendments of 1984
(Minimum Technology Requirements), the Environmentatl Protection Agency (EPA)
is required to develop regulations that account for improvements in land
disposal system technology and address the location of hazardous waste
management facilities.  With regard to facility location, the regulations must
specify criteria for the acceptable location of new and existing treatment,
storage, or disposal facilities, as necessary to protect human health and the
environment.  Further, EPA is required to publish guidance criteria
identifying areas of vulnerable hydrogeology within 18 months after the
enactment of the Amendments.
   To address these requirements, the U.S. EPA Office of Solid Waste (OSW)
is developing guidance manuals to assist permit writers and permit applicants
in assessing the acceptability of physical locations for hazardous waste
treatment, storage, and disposal facilities.  EPA has developed RCRA site
selection criteria on the basis of ground-water vulnerability, as defined by
the estimated time of travel (TOT) of ground water at the site in question.
For all land disposal facilities and for land-based storage facilities,
current policy defines ground water to be vulnerable if the calculated TOT
along a 100 ft flow path (TOTioo) is less than on the order of 100 years.

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For  land-based storage  facilities,  current policy establishes ground water as vul-


nerable if TOT10o is  estimated  to be  less  than the time necessary to detect a release


and  implement  corrective action.


   This report, on the  use of flow  nets  for determining ground-water flow  direction


and  time of  travel (TOT), documents analytical procedures  that are essential tools


in assessing ground-water vulnerability.   If sufficient hydrologic data are available,


a flow net can be constructed to determine the flow path offering the least resistance


to ground-water flow  and, thus, highest  flow velocity;  TOT can then  be calculated


along that path.   This  use of flow  nets provides  a conservative approach to assessing


ground-water vulnerability beneath  a hazardous waste management facility.


   This Technical Resource Document (TRD)  provides RCRA permit applicants and permit


writers with guidelines for constructing vertical groundr-water flow  nets as a means


of estimating  TOT; thus, the construction  of horizontal flow  nets is not discussed


to a large extent in  this manual.   Introductory examples presented in Section 3


assume horizontal flow  conditions to illustrate flow net construction procedures and
                                                                     •*

concepts.  Practical  examples that  follow  assume  more realistic ground-water


conditions (e.g.,  vertical gradients, heterogeneity, anisotropy)  and illustrate the


usefulness of  flow nets under these circumstances.  Flow nets can be constructed


using either graphical  or mathematical techniques.  Graphical techniques are the


simplest and most commonly used and receive primary emphasis.


   The use of  flow nets has been applied in other areas of geotechnical


engineering  and is an established procedure for asssessing seepage through earth


fill dams.   For application to  subsurface  conditions at RCRA  facilities, the


concepts are the  same but the data  needs are quite different.  These special


considerations are explained in forthcoming sections of this  document.



ORGANIZATION OF DOCUMENT


     The manual is divided into five sections.  The remainder of  Section 1


presents technical definitions  that should be  understood before proceeding


through the  manual.   In development of this guidance manual,  it was  assumed


                                           2

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that the user has a limited knowledge of the basic concepts of geology  and
hydrology.  Section 2 provides a general introduction to flow net theory as  a
means of understanding flow net construction concepts.   Graphical construction
of vertical flow nets for different hydrologic systems  is discussed in
Section 3 and includes examples to demonstrate flow net construction
techniques.  Section 4 illustrates the construction of  flow nets in special
hydrologic settings, also by example.  Mathematical methods used to construct
flow nets are reviewed in Section 5.  It is recommended that the user of this
manual attempts to work through the examples provided in Sections 3 and 4 to
practice methods of flow net construction.

DEFINITION OF TERMS

    Several useful definitions are described below, in  alphabetical order,  to
assist in enabling a thorough understanding of subsequent sections of this
guidance document.  These terms are commonly used in the field of ground water
hydrology and are of importance in the application of flow nets.  A more
complete glossary is provided in Appendix A.
    Aquifer is defined as a geologic formation, group of formations, or part
of a formation capable of yielding a significant amount of ground water to
wells or springs (EPA, 1984c).  The uppermost aquifer is the aquifer nearest
to the ground surface.
                             -• ^
    Discharge velocity or velocity (v) is calculated based on the quantity of
water that percolates per unit time across a unit area of a section oriented
at right angles to the flow lines.  For laminar flow conditions (smooth,
uniform flow), velocity is defined by Darcy's Law (see  Section 2) as v  = ki,
where k is the coefficient of hydraulic conductivity and i is the hydraulic
gradient.  The units of velocity are commonly cm/sec or ft/day.  Seepage
velocity  (see below) is equal to the discharge velocity divided by the
effective porosity.
    Effective porosity is the volumetric percentage of the total volume of a
given mass of soil or rock that consists of interconnecting pore spaces
through which flow can occur.
    Elevation head (z) is the elevation difference between the point of
interest and the measurement datum point (see Figure 1.1).

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  piezometer
aqu ifer
                      T
    pressure  head
point of
interest
                I
T
2 =  elevation  head
                           H  =  hydraulic  head
                           -datum
      Figure 1.1.   Measurement of subsurface water pressure.

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    Equipotential lines are one of two sets of curves that make up a flow
net.  These lines are perpendicular to the flow and pass through points of
equal head.  They are representative of the head or driving force for ground
water flow.
    Equipotential space is the space between adjacent equipotential lines and
represents the incremental drop in head through that distance.
    Flow lines comprise the other set of curves that form a flow net.  Flow
lines represent the path that particles of water follow in passing through
subsurface materials.
    Flow path is the space between two flow lines and is sometimes referred to
as a flow tube or a flow channel.
    Ground water exists below the earth's surface in saturated and unsaturated
formations.  The water table is the division between the unsaturated zone and
the saturated zone.  It is the point in the vertical dimension where the
pressure head is equal to atmospheric pressure.  In confined aquifers
(artesian aquifers), the pressure head is greater than atmospheric and the
potentiometric surface extends above the confining layer.  If a well were
placed through such a confining layer, the ground water would rise to the
level of the potentiometric surface.
    Hydraulic conductivity is an expression of a material's ability to
transmit water.  The coefficient of hydraulic conductivity (k)  is dependent on
the properties of  the  flowing liquid.  It  is equivalent to the volumetric rate
of  flow of water through a cross-sectional area under a unit hydraulic
gradient.  The units of the coefficient of hydraulic conductivity are length
divided by time  (L/T), normally expressed  as cm/sec or  ft/day.
    Hydraulic gradient (i) is the change  in hydraulic head per  unit  length  in
the direction of flow.  For example, if the hydraulic head drops  1 m over a
100 m distance,  the gradient is 0.01 m/m.  The hydraulic gradient has both
horizontal and vertical components.  Nested piezometers must be used to
determine  the vertical component of the hydraulic gradient.
    Hydraulic head or  total head (H) is the sum of the  pressure head (4"), the
                                            2
elevation head  (z), and  the velocity head (v /2g) at the measuring point of
interest.  For  ground water flow, the velocity head term is generally
neglected so  that H = tp  + z (see Figure 1.1).  The hydraulic or total head is
the value measured in a  well or piezometer.

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    Piezometer is a field apparatus consisting of a standpipe with a porous
tip (to keep soil out and let water in), which is used to measure subsurface
water pressure.  It operates by converting pressure head to a readily
measurable elevation head.  The level to which water rises in the piezometer
tube represents the hydraulic head as referenced to the selected datum point.
    Potentiometric surface is the plane that describes the level to which
water will rise in a series of piezometers drilled into a confined aquifer
with horizontal flow.  Freeze and Cherry (1979) point out that a
potentiometric surface is basically "a map of hydraulic contours on a
two-dimensional horizontal cross section taken through the three-dimensional
hydraulic head patterns that exist in the subsurface in [the area of
concern].  If there are vertical components of flow, as there usually are,
calculations and interpretations based on this type of potentiometric surface
can be grossly misleading."
    Pressure head (40 is the elevation that ground water rises above the point
of interest (see Figure 1.1).
    Seepage velocity (v ) is defined as the average velocity at which water
percolates through the pores of a porous material and is equal to the
discharge velocity divided by the effective porosity of the material (v  =
                                                                       s
v/n ).  The units of seepage velocity are commonly cm/sec or ft/day.

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                                   SECTION 2
                  FLOW  NET THEORY AND HYDROLOGIC CONSIDERATIONS

GENERAL

    A ground water system can be represented by a  three-dimensional set  of
equipotential surfaces and orthogonal flow lines.   If a plan view or a
two-dimensional cross section is drawn to represent this system,  the resultant
equipotential lines and flow lines constitute a flow net.   A flow net can be
used to determine the distribution of heads, velocity distribution, flow paths
and flow rates, and the general flow pattern in a  ground water system
(McWhorter and Sunada, 1977).
    Four basic types of ground water systems exist based on the distribution
of hydraulic conductivity:

    •    homogeneous and isotropic;
    •    homogeneous and anisotropic;
    •    heterogeneous and isotropic; and
    •    heterogeneous and anisotropic.

Materials are homogeneous if the hydraulic conductivity does not vary
spatially, whereas materials are heterogeneous if hydraulic conductivities do
vary spatially.   If the hydraulic conductivity is independent of the direction
of measurement at a point in a geologic formation, the formation is isotropic
at that point.  If the hydraulic conductivity varies with the direction of
measurement at a  point (for example, when the vertical hydraulic conductivity
is different than the horizontal conductivity), the formation is anisotropic
at that point.  Figure 2.1 is a graphical representation of the four types of

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  Homogeneous, Isotropic
             Kz
 1 - *
                u
                             Homogeneous, Anisotropic
Kzl
t

L




  Heterogeneous,  Isotropic
"
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systems, where the hydraulic conductivity (horizontal and vertical)  is
represented in vector form and shown at two different locations within each
aquifer.

FLOW NET THEORY

    A flow net is a two-dimensional model of the ground water system which, as
mentioned above, identifies ground water flow directions and can be  used  to
calculate ground water flow rates.  Flow nets can also be used to identify
suitable locations for monitoring wells, as well as the screened interval of
the wells.  The conceptual ground water flow model of a site can be  tested .
using a flow net.  A flow net can be constructed for a site to represent  the
conceptual flow model.  The model can be tested by installing additional
piezometers at selected locations and comparing the actual head values at
these locations with those predicted by the flow net.
    The total quantity of water that flows through a given mass of geologic
material is equal to the sum of the quantities of water through each flow path
in the flow net.  A fundamental rule is that each flow path in a flow net must
transmit the same quantity of water.  Therefore, the total flow,* Q,  is equal
to the flow quantity in each flow path, q, multiplied by the total number of
flow paths, F.  Similarly, the total head loss, H, experienced in traversing
through one flow path of the entire flow net, is equal to the head loss
experienced in passing through any equipotential space multiplied by the
number of equipotential spaces, N.
    Flow, q, through any path in a flow net is defined by Darcy's Law:

                                    q  = k  i  A
where k is the coefficient of hydraulic conductivity (cm/sec or ft/day),
      i is the hydraulic gradient (dimensionless), and
      A is the cross sectional area through which flow occurs
        (sq. cm or sq. .ft).

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Till.- c.i I cu l.ittvl v.-ilin; nl <\ is for ;i unit widtli in the third dimension
orthogonal to the cross section.  Thus, the units of q are m /sec per meter
of width.  The total flow through a single flow path is found by multiplying q
by the width of interest.
    In a flow net consisting of squares of dimension s x  1 with head loss, h,
through a single equipotential space, Darcy's Law reduces to:

                                  q-^-S

Given that the flow net is comprised of rectilinear spaces that approximate
squares, s = 1 and:

                                     q  = k h

From preceeding discussions, it is known  that:
                                   "'•I
so that:
Knowing that the flow, q, through any square is described by:
where F  is  the  total number of  flow paths, it is demonstrated that the total
flow, Q,  is  calculated as  follows:

                                    Q - kH £

Accordingly,  the  total quantity of water  that will seep through a unit width
of a given  subsurface unit can  be found by constructing a  flow net for the
cross section and multiplying its hydraulic conductivity by  the total head
                                      10

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difference and the ratio of the number of flow paths to the number of
equipotential spaces.  Again, it should be noted that Q for the full width of
interest is determined by multiplying the quantity calculated above by width.

TIME OF TRAVEL (TOT)

    Upon construction of a flow net, a conservative determination of time of
travel can be calculated along the flow path considered to offer the least
resistance to ground water flow and, thus, highest flow velocity.  The flow
path of least resistance can be identified by inspection once a flow net  is
constructed for the site in question.  This application of flow nets provides
a conservative approach to assessing ground water vulnerability beneath a
hazardous waste management site.

HYDROLOGIC CONSIDERATIONS

    To enable proper construction of a flow net, certain hydrologic parameters
of the ground water system must be known, including:

    •    vertical and horizontal head distribution;
    •    vertical and horizontal hydraulic conductivity of the saturated zone;
    •    thickness of saturated layers; and
    •    boundary conditions.

Head Distribution

    Piezometers are used to determine the distribution of head throughout the
area of interest.  To be valid, head measurements must be time equivalent;
that is, all piezometric measurements must be made coincidentally or all
measurements must be made for the same ground water conditions.  Piezometers
should be spatially distributed and placed at varying depths to determine the
existence and magnitude of vertical gradients.  If vertical flow components
exist, the flow direction cannot be derived simply based on inspection of the
potentiometric surface in two dimensions.  A three-dimensional representation
                                     11

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of the potentiometric surface would be required to interpret the flow
direction.  Ground water will flow, however, from areas of high hydraulic head
to areas of low hydraulic head.

Hydraulic Conductivity

    Hydraulic conductivity is a measure of a material's ability to transmit
water.  Generally, clayey materials have low hydraulic conductivities,  whereas
sands and gravels have high conductivities.  Several laboratory and field
methods can be used to determine the saturated and unsaturated hydraulic
conductivity of soils including tracer tests, auger-hole tests and pumping
tests of wells (Todd, 1980; EPA, 19846).  Methods used to determine hydraulic
conductivity above and below the water table are often classified by the range
of hydraulic conductivity and the medium being tested.

Aquifer Thickness

    The thickness of an aquifer can be determined by evaluation of geologic
logs or by geophysical techniques.  Geologic logs from boreholes'may show
changes in lithology (the characteristics of the geologic material) indicating
the relative hydraulic conductivity of materials.  Various geophysical
techniques, both downhole and surface, have been documented in many textbooks
and EPA guidance manuals (1983a and 19835) and can be used to determine the
thickness of geologic units.

Boundary Conditions

    The boundary conditions of the area of investigation must be known to
properly construct a flow net.  The boundary conditions are used as the
boundaries of the flow net.  The three general types of boundaries are:
(1) impermeable boundaries; (2) constant-head boundaries; and (3) water table
boundaries (Freeze-and Cherry, 1979).  Ground water will not flow across an
impermeable boundary; it flows parallel to these boundaries.  Unfractured
granite is an example of an impermeable boundary (Figure 2.2a).  A boundary
where the hydraulic head is constant is termed a constant head boundary.

                                      12

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              a.
                          Impermeable  Layer
                          F1ow Line
                          Equipotential  Line
Impermeable (No Flow)  Boundary Condition
               Flow
               Lines
                            Equipotent!al  Line
                  b.   Constant-Head Boundary Condition
                             Water Table
                                       Flow Line
Equipotential Line
           c.  Water-Table Boundary Condition in Recharge Area
               Figure 2.2.  Types of Boundary Conditions.
                            Source:  Freeze and Cherry (1979).
                                   13

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Ground water flow at a constant-head boundary is perpendicular to the
boundary.  Examples of constant-head boundaries are lakes,  streams,  and ponds
(Figure 2.2b).  The water table boundary is the upper boundary of an
unconfined aquifer, and is a line of known and variable head.   Flow  can be at
any angle in relation to the water table due to recharge and the regional
ground water gradient (Figure 2.2c).  The boundary conditions  of an  aquifer
can be determined after a review of the hydrogeologic data  for a site.
    Guidance for determining the above parameters is provided  in:

    •    Permit Writer's Guidance Manual for Subpart F,  Ground Water
         Protection (EPA, 1983b);
    •    Permit Applicant's Guidance Manual for Hazardous Waste Land
         Treatment, Storage, and Disposal Facilities (EPA,  1984c);
    •    Method 9100, Saturated Hydraulic Conductivity,  Saturated Leachate
         Conductivity, and Intrinsic Permeability Methods (EPA, 1984b);
    •    Ground Water Monitoring Assessment Programs at Interim Status
         Facilities (EPA, 1982); and      '•
    •    Soil Properties, Classification, and Hydraulic Conductivity Testing
         (EPA, 1984d).

    After assessing the hydrologic parameters of the ground water system at
the site of concern, construction of the flow net and subsequent determination
of ground water flow direction and time of travel can proceed.
                                      14

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                                   SECTION 3

                             FLOW NET CONSTRUCTION


BASIC RULES


    The simplest ground water system is  one which is homogeneous  and
isotropic.  This type of geologic medium serves  as a simple basis for
describing the basic rules of flow net  construction, despite the  fact  that
homogeneous, isotropic media rarely occur in nature.  Regardless  of the type
of medium, the basic rules must be applied and necessary modifications are

made throughout the procedure to account for heterogenity or anisotropic
conditions.
    The fundamental rules and properties of flow nets are summarized below:


1.  Flow  lines and equipotential lines intersect at 90° angles;

2.  The geometric figures formed by the intersection of flow lines and
    equipotential lines must approximate squares;

3.  Equipotential lines must meet impermeable boundaries at right angles
    (impermeable boundaries are flow lines);

4.  Equipotential lines must be parallel to constant-head boundaries
    (constant-head boundaries are equipotential lines);

5.  The head difference (h) between any pair of equipotential lines is
    constant throughout the flow net;

6.  Each  flow path in a flow net must transmit the same quantity of water (q);
    and

7.  At any point in the flow net, the spacing of adjacent lines is inversely
    proportional to the hydraulic gradient (i) and the seepage velocity (vs).

Procedures that can be used for flow net construction include:

1.  Trial sketching

2.  Mathematical solution

                                     15

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3.  System modeling
A.  Electrical analogy

    Considering the specialized knowledge required for use of the latter
three, trial sketching is generally considered the best method for the novice
and is often the practice of choice by those who are knowledgeable of the
other methods.  Flow net sketching can be sufficiently accurate, if conducted
according to the basic rules outlined above with patience and a certain degree
of intuition that will develop with practice.  The precision associated with
flow net sketching is likely comparable to that associated with defining the
hydraulic conductivity of the media of concern.

STEPS IN FLOW NET CONSTRUCTION

    A relatively small number of flow lines are necessary to adequately
characterize flow conditions at the site in question.  The use of three to
five flow lines will generally be sufficient.  With this in mind,, the
following steps should be used to construct a flow net:
    1.   Draw a cross section, on a convenient scale, of the geologic mass of
         concern in the direction of flow.
    2.   Identify all points of known hydraulic head arad draw tie lines
         between them by traversing the shortest possible distances and
         avoiding crossing of lines.
    3.   Use the tie lines constructed in Step 2 to interpolate other
         hydraulic head values for the purpose of sketching equipotential
         contour lines.  Clearly, the accuracy of this interpolation procedure
         will depend upon the number and location of points of known hydraulic
         head.
    4.   Establish two boundary flow lines based on geologic information.
    5.   Using a trial-and-error method, sketch intermediate flow and
         equipotential lines, making sure that right angles and squares are
         formed.
    6.   Continue to sketch these lines until inconsistencies start to develop
         in the shape of the resulting flow net.  For example, angles other
         than right angles, or rectangles rather than squares are considered
         inconsistencies.

                                     16

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    7.    Make successive trials until the flow net is consistent throughout.
         Each inconsistency noted will indicate the direction and magnitude of
         change for the next trial.
    It is reiterated that only a few lines should be used in constructing the
net.  Further, all transitions that exist in the net should be smooth and the
size of the spaces should change gradually.
    As the flow net is constructed, the head distribution should be checked.
One should expect a greater head loss occurring in materials with low
hydraulic conductivity than in materials with high hydraulic conductivity.
Flow lines tend to follow or parallel zones of contact between materials that
have differences in hydraulic conductivity of 100 or more.  Flow nets drawn
for materials with a difference in hydraulic conductivity of a factor of 100
will look the same if the ratio of conductivities is 10   to 10   or
10   to 10  .  However, variations will be evident in the quantity of flow
and the TOT.  Directional differences in hydraulic conductivity within the
same geologic layer (i.e., anisotropy) are also of importance in flow net
construction.
    At most facilities, vertical and horizontal head data are obtained from
well and piezometer measurements, and from free surfaces such as springs,
lagoons, ponds and swamps.  Often, piezometers or wells have long, open
screened sections or have slotting below the water table to measure fluid
pressure.  The open interval on a piezometer should be as short as possible
with the midpoint of the interval being the measuring point.
    Wells with long screened sections can be used to obtain approximate
piezometer readings if the midpoint of the open interval is used.  The head
measured in such a well is the integrated average of all the different heads
over the entire length of the open interval.  In this instance, it is
important to note that if vertical gradients are present, the measured head
can be a function of the screened length of the well.  This must be considered
when piezometric data are collected from such wells and are interpreted for
the purpose of establishing hydraulic head conditions.
    A simple case of two-dimensional seepage is shown in Figure 3-1 to
demonstrate the basic characteristics of a flow net.  The flow lines are
parallel to each other.  In reality, there are a large number of flow lines
through a given cross section because each particle of water will follow its
                                      17

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own flow line.  For practical applications, however, a smaller number of flow
lin^s are drawn for problem resolution.  Generally, only four or five flow
lines are necessary to adequately describe flow conditions at the site.  Flow
lines are spaced so that the quantity of water flowing in each flow path is
Che sane.

100
1

95
1
Flow
90
1
	 ^*-
Direction
85 80
1 1

75
1
Pressure
y
*
70
11
\ \
1 1
1 '
i !
i i
Equipotential '

1
1
FLOW PATH )
1
!
•
•



tFlow Lines
_
Equipotential

                                                       I
                                         I
                 k-10"4m/sec
                     F»3 (number of flow paths)
                     N-6 (number of equipotential spaces)
Figure 3.1.  Hypothetical flow net.
EXAMPLE OF A HOMOGENEOUS ISOTROPIC FLOW SYSTEM
    Figure 3.2a shows a cross section of a homogeneous, isotropic system with
no vertical hydraulic gradients to introduce flow net analysis procedures.
The cross section is drawn parallel to the direction of flow.  The water level
elevation is 102 m in Well 1 and 100 m in Well 2.  The aquifer consists of
fine sand with a hydraulic conductivity of 10~  m/sec.  Because this is an
idealized hydraulic system, the top and bottom of the aquifer are considered
impermeable (no-flow) boundaries and represent flow lines.  These flow lines
form a single flow path, which is sufficient in this simple case to construct
a flow net.  For this example, it is assumed that flow is horizontal, so that
vertical equipotential lines can be drawn at Wells 1 and 2.  Intermediate
equipotential lines are drawn by equally dividing the space between Wells 1
and 2 into squares (Figure 3.2b).  Once the flow net has been constructed, the
flow rate can be calculated using the equation:

                                     18

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                            Flow Direction
                  Well  1
                                                                  Well 2
                                    Water-Table  Surface
                         -   "..*"".   • Aquifer    . .  ^   ' .  '    " ,
Figure 3.2a.
                  Impermeable Boundary
Cross section of a homogeneous,  isotropic
aquifer; Parallel to the flow direction.
                             Flow  Direction
                                                        / /  /ss
                                                             100 m
                   Well 1
         F  =  1
         N  =  4
                      Equipotential Lines
                                            Well 2

=


1
LO
3
•^

/ \ '
/ N
/ ' si 2
r "*l °
1 '
Aquifer
V
o
0
/
/
/

Flow
^Lines
^
                     /////////
Figure 3.2b.   Construction of  a  simple  flow net for the       100 m
              system shown in  Figure  3.2a; Squares are
              formed by the equipotential lines and flow lines.
        F = 5
        N = 20
                             Flow Direction
                   Well  1
                                        Equipotential Lines

                                             //Well 2

                                 1  rr
               i
        I  I  I  I   I  I  I  I  '  '.  i  I   I  I  I  I  I   I  Kl I
i   i  i  i   1  1  i   i
                              .
                           A.qu.if
                                                i  i  i  i   i  i  i  1  1  1
                        i   i  i  i   i  i   i  i  i  i  i  i  i  i   i  i   i  i  i
 Figure 3.2c.   A more detailed  flow net for the system shown in Figure 3.2a.
 Figure 3.2.
Introductory Example - Homogeneous  Isotropic  flow system with
no vertical gradient.

                      19

-------
                                     k F H
                                 Q = -F-
where  Q = flow rate
       k = coefficient of hydraulic conductivity = 10~5 ra/sec
       F » number of flow paths = 1
       H • total head drop = 2 m
       N * number of equipotential spaces = 4
Using the flow net constructed for this problem,

                     Q = (10"3m/3ec)(l)(2m)

                     Q = 5 x 10   m /sec per meter of width

It is important to note that N is the number of equipotential spaces in one
flow path rather than the number of equipotential lines; N is one less than
the number of equipotential lines.
    The flow net in Figure 3.2b is the simplest one that can be drawn to
represent the system.  A more detailed flow net (Figure 3.2c) can be
constructed for this system, but the calculated flow rate is the same.  From
the flow net in Figure 3.2c, there are 5 flow paths and 20 equipotential
spaces. Thus, the calculated flow rate is:

                           . _ (10"3 m/sec)(5)(2m)
                           W            20
                     Q = 5 x 10~6 m^/sec per meter of width
and is equal to the value calculated from the flow net in Figure 3.2b.

SECOND EXAMPLE OF HOMOGENEOUS ISOTROPIC FLOW SYSTEM

    Another example of a homogeneous isotropic ground water system is shown in
Figure 3.3.  The facility is a landfarm at a petroleum refinery.  The
locations of four monitor wells are shown in Figure 3.3.  Well I is located
upgradient of the landfarm disposal area and Wells II, III, and IV are located

                                      20

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downgradient.   Water level data from nested piezometers indicate that ground
water flow at  the site is primarily horizontal.  Drilling logs indicate that
this site lies on interbedded layers of unconsolidated clay, silt, sand, and
gravel.  The silty clay unit ranges in thickness from 14 m to more than 27 m.
Beneath this unit is a layer of fine-to-medium sand with a thickness ranging
from 1 m to more than 10 m.  Underlying the sand unit is an impermeable till.
All wells are screened in the sand unit.
    Table 3.1 shows water level elevation data collected at the site on
July 19, 1983.  Plotting and contouring of-these data indicates that ground
water flow is to the northwest (Figure 3.A).   The data are contoured by
interpolating between the known hydraulic head elevations.  Using data from
Wells I and IV (Table 3.1), a cross section can be drawn parallel to the
direction of flow, southeast to northwest (Figure 3.5).
    A flow net (Figure 3.6) can be constructed to determine the rate of ground
water flow beneath the site.  Because the aquifer is relatively thin, the
entire flow system between Wells I and IV can be drawn as one flow path, with
the upper flow line representing the water table surface and the lower flow
line representing the impermeable till.  Equipotential lines can then be drawn
between these flow lines to form squares.  The closer spacing of equipotential
lines near Well I results from the change in aquifer thickness and indicates a
steeper hydraulic gradient near Well I.
    No hydrologic tests (i.e., slug or pump tests) were conducted at the
monitoring wells, so an average hydraulic conductivity for fine-to-medium
grained sand (10   m/sec) was selected from the literature (Freeze and
Cherry, 1979) for use in calculating the flow rate.  The flow rate is
calculated from the equation:
                               (10"4 m/sec)(l)(1.77m)
                           Q "            95
                               —6  3
                   Q = 1.9 x 10   m /sec per meter of width

Figure 3.6b shows an expanded portion of the flow net shown in Figure 3.6a.
The flow rate for this section is calculated below:
                                     21

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             TABLE 3.1.  DATA FOR REFINERY MONITOR WELLS
Monitor
well No.
I
II
III
IV
Ground
surface
elevation3
32.3
19.8
19.5
22.6
Water
level
elevation3
3.81
2.16
2.15
2.04
Elevation of
bottom of
sand unit3
3.1
1.2
2.7
-2.4
Elevation of
top of
sand unit3
4.0
5.9
12.2
7.3
aAll elevations are in meters, vs.  mean sea level.




Note:  Wells I and IV are approximately 223 m apart,
                               22

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   '/
V

•'//
                                               •Ugnt
                     ' • • ori* w; '      \
                                       V
                                        *\ !
	v'    . ,  -V\
V"» •"••• "O     '-    >\
 f -_'.-»\i     .   --^\
                        -  :••-•••-o ^-•-
                         • ••• m  • •    ~ Jl   *   '
                                                       •>••
ftf

1
       Borrow
       Figure 3.3.  Location map of  Refinery Site and Monitor Wells.
                                 23

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Downgradient
                                                IN
                                                          Hazardous Waste
                                                          Land Application
                                                          Sites
                                              Upgradlent
           100 m
Equipotential Lines
(In Meters)
Flow Lines

Wells with Well  Numbers
   Figure 3.4.  Potentiometric surface showing flow direction.
                                24

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                Well
              SE
                                                Water  Level

                                                                                                    Sand Unit
                     Screened Interval
                                                                            Screened Interval
N)
                                                                                    30 m
                                                                                              No  Vertical  Exaggeration
                            Figure 3.5.   Cross  section through the  site between wells I  and IV.

-------
A)
                                 3.0
                                  No Vertical  Exaggeration
                                                                       Flow Lines
                                                                i ' i i i t i i KK,i  ill i  i ! j

                                                                  Equipotential Lines
Well  IV  h;?.04

       —^  Direction of Flow
                                                                             30
        NU
            h-- Head

           Values in  Meters
           Above Sea  Levpl
           Well  I  h=3.81
   B)
                                                 Equipotential Lines
                                                                                     Direction of Flow
                                                                                             Values  in Meters
                                                                                             Above Sea Level
                                       Figure 3.6.   Flow net  for  the  refinery.

-------
                           0 = k F H
                           Q    -N

                               (10"4 m/sec)(l)(0.24 m)
                           4 ~            15

                   Q = 1.6 x 10   ID /sec per meter of width

The flow rate resulting from this calculation is virtually the same as  the
flow rate calculated using the entire length of the flow net.
    The flow net in Figure 3.6a indicates that the potentiometric surface in
Figure 3.4 is incorrect;  the contour lines should be more closely spaced near
Well I.  This can be corrected by inspection of the flow net.   Because  there
are 95 head drops (N) representing 1.77 m of total change in head (H),
                                                                  *
26.8 head drops represent 0.5 m change in head.  With this information  the
location of the 2.5, 3.0, and 3.5 m equipotential lines can be approximated
(see Figure 3.6a) and used to adjust the potentiometric surface (Figure 3.7).
    These introductory examples indicate the basic concepts of flow net
construction.  The remaining discussions and examples are practical in  nature
and show the influence of heterogeneity and anisotropy and the importance of
evaluating the vertical component of hydraulic gradient.  Procedures for
estimating time of travel are also documented.

CONSTRUCTION OF FLOW NETS IN HETEROGENEOUS, ISOTROPIC SYSTEMS

    Heterogeneous, isotropic ground water systems usually consist of two  or
more layers of materials with different lithologies and different hydraulic
conductivity.  This heterogeneity may result from vertical layering, sloping
strata, fault zones, igneous injection, or the existence of man-made
structures such as slurry walls.  Ground water flow in heterogeneous,
isotropic systems is controlled by the hydraulic conductivity of the layers,
as well as by boundaries within the system.
    The rules for construction of flow nets for heterogeneous, isotropic
systems are the same as for homogeneous, isotropic systems, except that the
"tangent law" (see below) must be satisfied at geologic boundaries.  If
squares are created in one portion of a formation, squares must be created
                                    27

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                                                                  i
100 M
     	Equipotential Lines from Flow Net
	  	 Equipctential Lines from Interpolation
             Flow  Lines
             Wells
Figure 3.7.  Reconstructed potentiometric surface for refinery example.
                                   28

-------
throughout that formation and throughout other formations that have the same
hydraulic conductivity.  Rectangles will be created in associated formations
that have different hydraulic conductivities (Freeze and Cherry,  1979).
    When a flow line crosses from a material of one hydraulic conductivity to
another, the flow line is deflected and the flow velocity changes.   Flow lines
tend to be parallel to the zone of contact between materials in the medium
with higher hydraulic conductivity, and perpendicular to contacts between
materials in the medium with lower hydraulic conductivity (Figure 3.8).  Flow
paths will be narrower in layers with high conductivity because less area is
necessary to conduct the same volume of water.  In media of. lower
conductivity, flow paths will be wider in order to conduct the same volume of
flow (Cedergren, 1977).
                    
-------
                   For both cases, k, >
Figure 3.8.
Deflection of flew lines across materials of
different hydraulic conductivity.
Source:  Freeze and Cherry (1979).
                            30

-------
3.9o
                                                           3 9b
                         SAND
                                                                           SILT
        Figure 3.9.   Deflection of flow lines and illustration of the tangent  law.

-------
    A simple example ot a heterogeneous,  isotropic ground water system is
shown in Figure 3.ID.  The conductivity of the slurry wall is a factor of 20
less than that of the sand.  For ground water to flow from the sand through
the slurry wall, the equipotential spaces of the flow net must compress in the
slurry wall section to l/20th of the equipotential space in the higher
conductivity sand layer.  Figure 3.11 shows that there are five equipotential
spaces (or five equal units of head loss) between points A and B and C and D,
or a total of ten equipotential drops for flow through the sand zones.   Flow
through the slurry wall from B to C consumes 20 equipotential drops because
there is one equipotential drop for each  equipotential space.  Thus,  the total
number of equipotential drops is 30 and the head loss through any
equipotential space is l/30th of the head loss through the system.   In
addition, flow through the slurry wall results in a head loss equal to two
thirds of the head loss through the system.  Most importantly, if piezometers
were installed in the soils along this cross section, the hydraulic head
profile or potentioroetric surface would closely follow A'-B'-C'-D', shown in
Figure 3.11.

Example of a Heterogeneous Isotropic Ground Water System

    The hazardous waste facility shown in Figure 3.12 is located over a
heterogeneous aquifer in a recharge area.  The site geology is composed of two
layers of material.  The upper layer is a 10 m thick layer of silt with a
hydraulic conductivity of  10~  m/sec (k.).  The lower layer is a 5 m thick
                                       •*•         _c
layer of sand with a hydraulic conductivity of 10   m/sec (k.).  The head
distribution  for  the system of water table wells and piezometers at the site
can be constructed by drawing tie lines between the measuring points, shown in
Figure 3.13a, making sure  that the tie lines do not cross each other.  The tie
lines can be used to interpolate equipotential head values of interest.  For
example, the distance along tie line 1 (Well P-8 to Well P-9) on Figure 3.13a
is 18 m and the head drop  is 1 m.  In order to contour the flow net at 1/2
meter head loss intervals, it is necessary to determine where the 10 m
equipotential contour intersects the tie line.  This is determined from the
ratio:
                                     32

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LJ
U)
                                 k«IO  cm/sec
                                                                     Proposed location of slurry
                                                                     cutoff walI
                                                        j     I ^— Keyed  into clay
                                                        i	1
                                                         62'
                                                  from lagoon to stream
          Figure 3.10.   Cross section of Heterogeneous Isotropic  ground water  system with slurry  wall

-------
                 ,	  The elevation  to which water would  rise
                      if piezometers were  installed.
                         B'
                                                                  20 Equipotential Drops
                                                                     in slurry walI
                                                                      Equipotential Drops
                                      Each  of  the  four  rectangles within  the
                                      slurry wall  includes 5 equipotential  spaces
Figure 3.11.  Flow net developed to analyze  the  effect  of  slurry wall

-------
                                                       Waste Management
                                                       Area
                                                  Well P-8
        -o*
Figure 3.12.
Well location map for example  of heterogeneous isotropic
ground water system.

                      35

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u>
6_
           WELL P-8
                10.5
          10m
                                  SILT
                                                                          .6
                                                                    k = 10  m/sec


1

9.5 8.5
_EL n
* SAND
5m
1
i

'-' rriNTArr
k = 10 m/sec
*

- MEASURED HEAD AT THE MEASURING POINT


  LOCATION OF OPEN SCREENS AND PIEZOMETER
                                                                                6m
                                                                        Scale
      MEASURING POINT
                              Figure 3.13a.   Distribution of heads in silt layer.

-------
where X is Che distance from the intersection to the measuring point and is
equal to 9 m.  Thus, the 10 m contour intersects the tie line at a distance of
9 m from Well P-8 and is located by measuring along tie line 1 (see
Figure 3.13b).  Each equipotential contour of interest is located along the
tie line in the same fashion.  Upon completion,  all points of equal head (from
the top of the layer to the base of the layer) are connected to form the
equipotential lines.  The relationship of equipotential lines to tie lines  is
shown in Figure 3.13b.
    Flow lines are added at right angles to the  equipotential lines so that
squares are formed (see Figure 3.13c).  At the contact of the silt layer with
the sand layer, the flow lines form an acute angle, o,.  This angle is used
in the tangent law to determine the angle of deflection, o •

                                 k      tan o
                                        tan a
From Figure 3.13d, o ± is 65°.  Therefore,
and tan Q£ = 0.21
        o2 = tan "! 0.21
        °  - 12.1°
                           ,«~6  .        tan o „
                           10   m/sec  = _ 2
                           10"5 m/sec    tan 65
    After determining this angle, the flow lines through the sand layer can be
drawn, as shown in Figure 3.13d.
    The dimensions of the rectangles in the sand layer can be determined by
considering the ratio of hydraulic conductivities.  If c is the length of the
rectangle and d is the width:
                                     37

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00
                                 EQUIPOTENTIAL LINES
                                                                                          TOP
                                                                                  WATER TABLE
                                                                                    P-IO
                                                                                      5m
                                                                                          SILT

                                                                                          = '°~6m/sec
CONTACT


SAND

k  =IO~5m/sec
                   Figure 3.13b.  Interpolation and  plotting of equipotential  lines.

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       WELL P-8
TIELINE
                                                                                  _ TOP GRADE
  10


  8


.6
at
f 4


  2

  0
                                              WELL P-9
                                        WATER TABLE
                                      WELL P-10
                                       Mi8.5m
                                               SILT

                                           k, ='0'6m/sec
                                         7.5m
                            EQUIPOTENTIAL LINES
                         y
FLOW LINES
                                               CONTACT
                                                                                    SAND
                                                                                            m/sec
                           Figure 3.13c.  Development  of  flow net.

-------
                          Wosie  Monogemenl  Areo
.c-
o
                                                                                             TOP GRADE
                                                                                      WATER TABLE

                                                                                  WELL P-IO

                                                                                         .5m
                                    EOUIPOTENTIALc.-/7.

                                        1  INES        °
                                                                                              SILT
                           Figure 3.13d.   Flow net for silt and sand layers.

-------
                                 c
                                 d
Thus, by calculating the ratio of conductivities, the proportional
relationship between rectangle length and width can be determined, as follows:

                       c/d = k£/k1 = 1 x 10"5/1 x 10"6

Therefore, the width, d, of the rectangles in the sand is equal to l/10th of
the length, c.  By measuring the distance between adjoining flow lines, the
distance, d, is approximately 1.7 m while the distance, c, is approximately
17.0 m, as shown in Figure 3.13d.
    The flow net (Figure 3.13d) can now be used to determine the appropriate
flow path for TOT determination.  The flow path exhibiting the lowest gradient
and highest velocity is selected to determine TOT,Q0.  In this case, the
selected flow path is located at the downgradient edge of the WMA through the
silt.  The flow path shown in Figure 3.13e does not correspond to any of the
flow lines that were used to construct the flow net, but there are'an infinite
number of flow lines and equipotential lines that could have been .used to
construct the flow net.
    TOT.-- is based on the seepage velocity in each layer and the length of
the flow line in that layer.  The seepage velocity is:
                                        ki
                                  v  =  —
                                   s    n
The effective porosity  (n )  is given as 0.2 for both  layers  in this
example.  The hydraulic gradient,  i, for each layer is determined by
measurement  from  the  flow net, with the following results:
                      -  ...    ,     ,  ,     0.137
                      silt   L

                      i     - f* - I-!*- - 0.059
                      sand   L     17 m
Therefore,  the  seepage  velocity, v  , for each layer  is:
                                      41

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            Wosle Monogemenl  Areo
WELL P-8
                                                                       	. TOP GRADE
                                                                   .— WATER  TABLE
                                                                   WELL P-IO
                                                                              SILT
                        »-EOUIPOTENT|AL
                             LINES
MEASURED 100 ft.-
(30.5m) LENGTH
ALONG FLOW LINE
          Figure 3.13e.  Selection of flow line for use  in TOT calculation.

-------
           / • i  \   * w   »»• /    10~  m/sac (0.059)   , n     -6   .      .. _,   ,,
         v (sand) = 	—	 = 3.0 x 10   m/sec = 0.26 m/day
          s                 u * z

The flow path of highest velocity includes a sector of 10.5 ra in the silt and
20 m in the sand.  TOT1QO is calculated as follows:

             TOT(silt)  = L/vs(silt)  =  10.5  m/0.06 m/day  =  175  days
               TOT(sand) = L/vs(sand) = 20 ra/0.26 m/day = 77 days
                    TOT100 =  TOT(silt)  +  TOT(sand) = 252  days

According to current policy, the site would be considered vulnerable because
the calculated ground water TOT along a 100 ft flow path is less than
100 years.
    The volumetric flow  rate through this system can be calculated using the
flow net constructed for the silt.  It is best to calculate the flow rate for
one square of the flow net and multiply this value by the total number of flow
paths in the system.  The flow rate through one flow path is:
                              O - k F H
                              Q -- -—
                                  (IP"6 m/sec)(l)(lm)
                     Q = 1 x 10   m /sec per meter of width

Because there are approximately three flow paths in this system, the total
flow across the flow net is 3 x 10~  m /sec per meter of width.
CONSTRUCTION OF FLOW NETS IN HOMOGENEOUS, ANISOTROPIC SYSTEMS

    The previous example assumed that hydraulic conductivity within the flow
system was the same in all directions at any given point, but in nature many
                                     A3

-------
geologic formations .ire anisotropic.  In some sediments, such as clays and
silts, anisotropy is due to particles being flat instead of spherical.  These
particles are usually deposited with the flat side down, producing sheet-like
beds.  These beds decrease vertical hydraulic conductivity and result in
relatively higher horizontal hydraulic conductivity.  Anisotropy can also
result from lenses or pockets of material of different hydraulic conductivity
within a matrix.  A medium which is predominantly clay but includes sand
stringers will have a higher horizontal conductivity than clay without the
sand stringers.
    To construct flow nets for anisotropic media, it is necessary to shrink
the dimensions of the cross section in the direction of the higher hydraulic
conductivity, or expand the dimension of the cross section in the direction of
the lower hydraulic conductivity.  If the horizontal hydraulic conductivity,
kh, is greater than the vertical hydraulic conductivity, k , then the
reconstructed section can be reduced to a narrower horizontal dimension or
expanded to increase the vertical dimension.  If the reverse is true
(ky > k. ), the section could be lengthened horizontally or reduced
vertically (Cedergren, 1977).  Referring to Figure 3.14a, the expression
N/k /k. is multiplied by the horizontal.dimension (x) to transform the
cross section (Figure 3.146).  To transform the cross section vertically, one
would multiply the vertical dimension by the expression, s/k./k .  After
the cross section has been transformed, the flow net may be drawn as if it
were  in isotropic media.  The dimension which has not been transformed remains
the same.
    After the flow net has been constructed, it may be desirable to view  it in
its original dimensions.  This is done by dividing the dimension that was
transformed by the expression used  in the transformation, »Jk /k,  or
vk./k .  When the cross section has been returned to its original
dimensions, the  flow net can then be reconstructed using the same number  of
flow  lines and equipotential lines.  Transforming the flow net in this way
will  not change  the ratio of F to N and will not change any of the
calculations made on the transformed section.  The new flow net should be
composed of rectangles elongated in the direction of high hydraulic
conductivity.  Note that the intersections between flow lines and
                                     44

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a)
b)
             Figure 3.14.   Shrinking of Anisotropic  flow net.
    k,=
    X1=X/T74=0.5X
                          a)
                  b)
    Figure 3.15.  Elongation of flow net to original dimensions.

                  Source: Freeze and Cherry  (1979).

                                 45

-------
equipotential lines will not necessarily be at right angles in this
reconstructed view.  Figure 3.15 demonstrates the procedure of returning a
flow net to its original dimension, using a value of horizontal hydraulic
conductivity which is four times greater than the value of vertical hydraulic
conductivity.  Figure 3.15a is the transformed flow net and Figure 3.15b is
the flow net in the original scale.
    After the flow net is transformed, it must be double checked to make sure
that it appears to be reasonable.  The rectangles should be elongated in the
direction of greatest hydraulic conductivity.  The ratio of F to N will remain
the same and the expression Jky kh will be used as the effective hydraulic
conductivity in the transformed section.

Example of a Homogeneous Anisotropic Flow System

    The geology and orientation of the site are the same as that shown earlier
in Figure 3.13a with the same head distribution but different hydraulic
conductivitites.  For this example, only the upper silt layer is considered.
The silt layer has sediments which are anisotropic, and the vertical hydraulic
conductivity is l/10th of the horizontal conductivity.
    The hydraulic properties of the silt layer are:

                                 kv =  10   m/sec

                                 k  =  10"6  m/sec
                                  h

                                 ng =  0.2
be reduced to 31.6 pecent of the original, and/or both the horizontal and
vertical scales could be increased or decreased so that the ratio of these
dimensions is 0.316.  In this example, the horizontal scale is 1 in. » 6 m and
the vertical scale is 1 in. = 1.896 ra (note that 1.896/6 = 0.316).
Figure 3.16a shows the transformed geologic section.
                                      46

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WELL P-8
     ,10.5
V/T
*-
V>
O
0) VjJ
n
K>
<^>

SILT

-

-
tl9'5 R8'5 H7"5
                                                                     SAND
           Figure 3.16a.  Transformed section using available well data.

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    Flow net construction proceeds as if the sediments are isotropic, as shown
in Figures 3.166 and 3.16c.  Figure 3.16c shows the portion of the flow net
that could originate under the waste management area.  This occupies
approximately one-fourth of a flow path.  The upper boundary of this area is
used as the flow path to determine TOT,0f).
    To determine TOT in the silt, the completed flow net is transformed back
to its original dimensions (Figure 13.16d).  The hydraulic gradient is
determined from this transformed section and is equal to 0.09.  The equivalent
hydraulic conductivity of the silt layer is required to determine the seepage
velocity fhrough it.  This value is determined by:
            ke =Nkv kh = "(1 x 10")(1 x l°") " 3-16 x 10~7 m/sec.

The seepage velocity is:

      keX   (3.16 x 10"? m/sec)(0.09)   , •_   ,.-7   ,       , 0,   , n-2   ,.
 v  = 	 = 	     	 = 1.42 x 10   m/sec or 1.23 x 10   m/day
  S    fl              U • ^

The associated TOT.QO is 2480 days or 6.8 years.  Thus, the estimated TOT
has increased by two orders of magnitude as conditions have changed from
isotropic to anisotropic, with vertical conductivity an order of magnitude
less than horizontal conductivity.

Flow Net Construction in Heterogeneous. Anisotropic Systems

    Construction of flow nets in heterogeneous, anisotropic systems follows
the previously discussed rationale for flow net construction in homogeneous,
anisotropic systems with the additional factor that the ratio between vertical
hydraulic conductivity and horizontal hydraulic conductivity in the uppermost
layer of the system is assumed to be representative of the directional ratio
of conductivities in the lower layers.  This assumption is based on the.
likelihood that, in most sedimentary material, the horizontal hydraulic
conductivity is expected to be larger than the vertical hydraulic
conductivity.  In terms of practicality, if the assumption is made that the
                                      48

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                               WELL P-8
vO
                 TIE LINES BETWEEN
                 DAT*  POINTS-
                   5m -r
                                9 Si
                         .I.  ..I
                         I 2 3  4 5 4
               ( 0, = 316 0H)

                 SOLE

              VERT, l" : 1.896m
              HORIZ l" < 6m
                                                       WELL  P-9
                                                            9.9
WELL P-IO
       83
                                                EQUIPOTENTIAL  LINES
                                                                                               SILT
                                                                                               CONTACT
                                                                                               SAND
                                 O
                                 O
                                 f
                                                                                                                   3
                                                                                                                   o
                     Figure 3.16b.  Contouring to determine placement of equipotential  lines.

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                           •«
                                             .WATER TABLE
/
  /
            .   EOUIPOTENTIAL
           /       LINES
      /
     /
   /  FLOW LINES
 I 23456

    M
                                                             SAND
           Figure 3.16c.   Flow net construction in the silt layer.

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Wojie Monogemeni
     Areo
                                                        SAND
     3    6

  SCALE
 Figure 3.16d.   Flow net for silt layer returned  to  original dimensions.

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ratio of horizontal to vertical hydraulic conductivity is different from that
of the overlying sediments, then separate flow nets would have to be
constructed for each layer by using information from the upper layer as  the
starting point for construction of the lower layer.  Additionally, the
dimensions of this new flow net would have to be adjusted to the conductivity
ratio in the next layer and so on throughout the system.  The latter approach
should be followed, however, if geologic data indicate that such conditions
occur in reality and the 100 foot flow line of concern passes into such  media.
                                    52

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                                    SECTION 4
                  CONSTRUCTION OF  FLOW NETS IN SPECIAL SETTINGS

    This section provides examples for Che construction  of flow nets  and  the
calculation of TOT...- for special geologic settings.   The settings  include  a
ground water mound over a leaking lagoon, complex hydrogeology in an  arid
region, fractured bedrock environments,  seepage  face  flow conditions,  and free
surface flow conditions.

EXAMPLE OF A GROUND WATER MOUND OVER A LEAKING LAGOON

    The waste management unit of concern is an existing  lagoon which  was  used
for disposal of waste water from a metal plating and  finishing plant.   The
lagoon was considered to be an evaporation lagoon by  the owner because it did
not overflow and did not require pumping.  Consequently, the owner  elected  not
to monitor the lagoon.  During site closure,  however, the possibility of
subsurface leaking was noted.
    Figure 4.1 provides a plan view of the site, and  the location of  the
ground water investigation well and piezometers.  Site geology consists of
40 to 60 feet of glacial outwash sand overlying  a layer  of glacio-lucustrin
silt/clay of unknown depth.  The outwash sands are believed to be part of a
delta formed from an ice-dammed lake.  The presence of the clay has been
determined from one short boring and shallow seismic  refraction work.   No
piezometers have been installed in the clay.
    Based on review of the Part B permit application, it is evident that  the
applicant considers the ground water in the sand to be perched above  the  clay
and asserts that ground water flow is basically  horizontal.  The applicant
also believes that there is no basis for the placement of monitoring  wells  in
the clay or any need to investigate below the clay sediments.  Additionally,
the applicant stated that by drilling into or through the clay, connections
may be created between the upper sand layer and  deeper aquifers.

                                     53

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                  1
\
 \
  \
   I
    es
                                                       re






















1
1
1
/ • 78
/ /TV
/ //
/ /I





1
I
1
/
/

1 1 86
76 1

0 •


00

LAGOON



^

}
\
8(

|

V
1




I
1
\
1
\



W«B B-2

J
\ \
\ V" '
\ ^x
\

/


A
80
>
««ia_v

1
I
                                                  «
                                                  \
                                                                         ff
                                                                        /
                                                           77
                                 Seal* 1" = 20*
      Figure  4.1.
Well location and water  table  contour map for
sand/gravel underlain by clay.
                                    54

-------
    The intent ol drawing the vertical flow net is to investigate the claims
made by the applicant, determine the vulnerability of the site, and to
determine the need for additional work.
    Figure 4.2a presents a cross section through the subsurface below the
waste management area indicating the measuring points determined for each well
and piezometer.  The tie lines between each data point are shown.  Several
points should be noted about the cross section.  First, the bottom of the
lagoon forms a fixed head boundary.  The head at this boundary is the
elevation of .the liquid in the lagoon which is 90 feet.  The bottom and the
sides of the lagoon below the water level represent an equipotential line.
Flow lines that emerge from the lagoon will be at right angles to the bottom
of the lagoon. Second, any equipotential line that intersects the water table
does so at its characteristic hydraulic head.  Third, because the clay
sediments have a lower hydraulic conductivity than the overlying sand, the
flow lines entering the clay will bend toward an imaginary line normal to the
contact between the two sediments.
    Contouring the head data gives an approximate equipotential map as shown
in Figure 4.2b.  Five-foot contour intervals were selected for the
equipotential drops.  This map gives some idea of the final shape of the flow
net but cannot, at this point, be used to determine the quantity of flow or
TOTioo-
    Flow lines are now added to the equipotential map.  The flow lines and the
equipotential lines are shifted as needed to form squares.  Depending on the
complexity of the head distribution, 3 to 20 iterations can be expected.  The
equipotential lines cannot be moved in any way that would violate the known
data points at each well or piezometer location.
    Figure 4.2c is the final flow net.  The right hand side of the figure,
where the mound flow merges with the natural ground water, has not been
completed because of the lack of .data in this area.  One large square has been
subdivided to provide the necessary detail in shaping the squares in this
area.  By doing so, the 75 foot contour enters the bottom of the same layer
even though data are not available in this area.
    The flow net indicates that:
                                     55

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               SAND
-00



-80



-70



 60



-60



-40



-30



-20



-10



-0
k =  10   em/aac

n«r .2
                                                                      QROUNO SURFACE
                                                           76.6
CLAY

k r 10 "6 cm/sec

na = -2
         r
         0
    20
 I
40
 I
60
 1
80
100
120
 I
140
160
180
                                              Scato 1" r 20*
             Figure A.2a.  Construction of tie lines  to allow contouring  to determine
                           equipotential lines.

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 90




 80





 70




•60




•60




-40





-30




-20




-10
                      ^C±ST
                                                       l
                             76
               70
                         —„_.     .
                                 70.6

                                r*T
86
           \
                 \
            \
                                           80
                           80       I

                                    80
                                  \
    \
     \
                     \
               \
                \
                                       \    •78.6
                                     • 78.6
           —r
            40
                             120
        ~r
         20
80
80
         100
                                     Seal*: 1" : 20
140
180
180
    Figure 4.2b.  Approximate equipotential map based on contouring of data points.

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oo
 -80



 -70



 -60



 -60



 -40



 •30



 -20



 -10



--0
                   SAND
                       _o
                   k:  10  cm/aoc
GROUND SURFACE


— WATER TABLE       85
                                                    1
PARTIAL SQUARES
                                                                                                85
                                  Figure 4.2c.  Flow net for mound  example.

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         approximately 3.5 flow paths emerge from the lagoon,
         the influence of the lagoon extends to the base of the sand layer and
         probably through the clay layer, and
         the majority of the flow from the lagoon enters the clay strata and
         only about one-half to one flow path remains in the sand at the left
         side of the flow net.
By considering that 3.5 flow paths-leave the base of the lagoon and by
calculating the flow through one square, the total leakage from the lagoon can
be determined.  Using i = 0.56, A = 8 ft2,  and k = 2.83 ft/day; flow through
one square of unit thickness is:
               q = kiA =  (2.83)(0.56)(8) =  12.7 cubic  feet /day.
Leakage from the lagoon bottom through all flow paths in the cross section is:
                     Q = (3.5M12.7) = 44.5 cubic feet/day.
The calculated Q applies to a 1-foot thickness of lagoon bottom.  Considering
the lagoon length of 700 feet, the total flow from the lagoon is:
                Q(lagoon) » (700)(44.5) = 31,200 cubic feet/day.
    TOT    is determined by locating the flow line in the sand layer that
represents the highest ground water velocity.  A flow line near the water
table was selected.  Knowing that the effective porosity is 0.2, the seepage
velocity and TOT     along this flow line are calculated as:

                       v    ki   (2.83H0.56)   _ . . .  .
                        8 " *	672	 = ?*9 £t/day
                                    59

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                      TOTioo - 7- = 77?
    The tangent law indicates that the ground water flow will be nearly
vertical, directed downward through the clay layer.  The shape of the flow net
in the clay strata will be rectangles with the length of the rectangle
100 times the width (47 feet to 0.47).  This follows from the ratio,
    In assessing the applicant's claim that any leakage from the lagoon would
stay in the sand, the flow net indicates that the majority of the lagoon
leakage enters the clay layer.

EXAMPLE OF COMPLEX HYDROGEOLOGY IN -AN ARID REGION

    The following example points out typical problems that may be encountered
in drawing flow nets for cases with complex geology and hydrology.
Additionally, the example shows that in complex areas where construction of a
complete flow net is not possible, TOT can still be estimated.
    The cross section of the proposed site shows a 50 to 60 ft thick gravel
deposit above a 10 to 20 ft thick vetric tuff layer which overlies a 150 to
200 ft thick water worked tuffaceous layer which overlies a series of basalt
flows with interflow deposits.  The first continuous saturated zone lies in
the water worked tuff, 150 to 170 ft below the ground surface.  The thickness
of the saturated zone varies from 0 to over 100 ft depending upon the
topography and the amount of fracturing in the underlying basalt flow.
    Figure 4.3 indicates the locations of the wells that were installed during
site investigations, and shows the proposed point of compliance wells.   The
location of the cross section selected for flow net analysis is also shown.
Figure 4.4a indicates the location of the midpoints of the wells and
piezometers, all of which are in the tuff deposit or the underlying basalt.
Table 4.1 provides horizontal hydraulic conductivity data based on slug tests
and packer tests conducted in wells along the cross section.
    Pump tests were conducted to measure the vertical hydraulic conductivity
of the tuff and basalt, indicating an average of about 2.7 x 10   cm/sec in
each medium or about one to two orders of magnitude below the horizontal
                                    60

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r
T
X
                                                                                                 LEGEND
                     _	

                     931.2 = Ground Elevation
                             (II)
            PROPOSED POINT OF
            COMPLIANCE AROUND
            EACH UNIT
SECTION SELECTED FOR

      FLOW NET ANALYSIS
                                                                                700.5
                                       Figure  4.3.   Well location map.

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           •1000
           •060
           -900
N>
           •860

            W1
            dry

           -600
2  CC    H   CCC
   .6
   • 796      *»796.7
           • 814.7
                              VV       QQ       ZZ
                                      S.T84.3
                                                                     • DRY
                                                          '•766.7
           -760-
YV
                                                          WATER LADEN TUFF
                                                                                          «743.9
                                                                                          • 729 9
                                                                                          • 724 2
           -700
           -660
           • 600
                                                                                                     ^688 9
                    1"=60'
1"= 600'
                                    626.9

                                          BASALT         708.6*

  Figure  4.4a.  Cross section showing hydraulic head data.

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TABLE 4.1.  HORIZONTAL HYDRAULIC CONDUCTIVITY DATA FROM SLUG TESTS AND
            PUMP TESTS
 Well I.D.
Geologic unit
      k-cra/sec
    QQ
    YY

    VV

     R

    MM
Basalt
Tuff

Tuff

Basalt (top 30')
Basalt (bottom)

Basalt

Basalt

Tuff

Tuff
Basalt
2.99x10*5 (average from
2.31x10-6  slug tests)

7.52xlO-6

2.64xlO~4
6.34xlO-7

2.11x10-^

9.61x10-5 to ixio-8
7.73x10-5
2. 64xlO-4 to 4xlO-6
                                 63

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hydraulic conduct Lvit ies given in Table 4.1.  Based on Che conductivity
difference of 100:1, the horizontal axis of Figure 4.4 has been reduced by a
factor of 10:1 (e.g., Vkh/ky).  The flow net is now drawn as if the
subsurface materials were isotropic.
    The system of tie lines drawn between data points is shown in Figure 4.4b.
The tie lines can now be used for interpolation without consideration of site
layering.  A 10 foot contour interval is considered reasonable for this
system.  After interpolating to determine the location of intermediate
equipotential lines, an approximate equipotential map can be drawn as shown in
Figure 4.4c.  Upon completion, the geologic layering is superimposed onto the
map.
    The approximate equipotential map indicates that the gradients in the tuff
are generally less than those in the basalt.  In the basalt, however, there
are zones of very low gradients and very steep gradients.  These variations
reflect che changes in hydraulic conductivity noted in Table 4.1.
    The equipotential lines in the tuff layer have unusual shapes.  These
lines are probably artifacts of the contouring process because some lines
would require discharge of ground water upward to the unsaturated sediments,
which is not possible.  In Figure 4.4d, many of these unusual equipotential
lines have been smoothed, such that the orientation of the equipotential lines
is consistent with the downward gradients present in the ground water mound
between wells MM and R.
    The next step in the flow net construction process for'this site is to add
flow lines, as shown in Figure 4.4d.  In a geologic setting where the
hydraulic conductivity varies over a wide range within the same layer, it is
impossible to determine where a boundary is crossed and the accompanying
degree of flow line deflection.  In such cases, it is generally useful to
inspect the flow in a limited portion of the whole system.  Flow lines will
continue to cross the equipotential lines at right angles because the section
is already transformed.  To construct the flow net, a number of representative
flow lines can be placed on the equipotential map.  The majority of the flow
will be in areas other than the low hydraulic conductivity areas of basalt
with the highest gradients.  The equipotential map indicates that there are two
                                      64

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  805.6
                    796.7
   SCALE
\"-.20'
        I": 20'
                                                    766.7
                             713
                                                                                          743.9
                                                                                                 98
                                                                                                           688 98
                                                                                                           672.4
                                                                                                           672.1
                                                                                          708.56
        Figure A.4b.  Construction of tie lines  for interpolation of equipotential head values.

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 805.6
                  796.7
SCALE
                                                   766.7
     t"--20'
                                                                                        743.9
                           713
                                                                                                          688 98
                                                                                                          672 4
                                                              626.96
                                                                                                          672.1
                              Figure  4.Ac.   Approximate  equipotentlal map.

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80S.6
                                                BOUNDARIES
                                         OF THE PROPOSED FACILITY
                                          .WATERTABLE
                                                                          743
627.3 •  \    y
                                                                                        672
                                                                         •708.6
                                                  ASSUME 100 =
                                                                  H
                             Figure A.Ad.   Approximate flow net,

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zones in the basalt that have significantly higher hydraulic conductivities
than the medium as a whole.  These areas probably represent zones of
fracturing in the basalt.
    TOTjgy can be determined by selecting the flow path that represents the
highest ground water velocity.  This path will be the zone that has tha lowest
hydraulic gradient and could be represented by the flow path that passes near
well W and well Q in the tuff.  By measuring along this flow line in the
basalt, the hydraulic gradient is determined to be 0.025 (10/400).  Because
there are no hydraulic conductivity data for the tuff at well W (the area of
lowest gradient), the highest hydraulic conductivity for the tuff, as
documented in Table 4.1, will be assumed.
    Geophysical logging results indicate that the porosity of the tuff is
                                                                    t
about 40 percent.  However, the storage coefficient ranges from 0.01 to 0.1
based on pump test data.  Because the saturated zone in the tuff is
unconfined, it is reasonable to assume that the effective porosity, n , is
approximately equivalent to the storage coefficient determined in the pump
test.
    The seepage velocity and TOT10Q for the site can be determined as
follows:
     _ lei = (1.89 x 10"4 cm/sec)(0.25) _ (0.535 ft/day) (0.025)
   s " IT             OIo.oi

                        100 ft     100 ft      ,, , j
               TOTinn = 	 =   ,.    . — = 74.7 days
                  100     v      1.34 ft/day          '
CONSTRUCTION OF FLOW NETS FOR FRACTURED BEDROCK ENVIRONMENTS

    In order to construct a flow net for a fractured bedrock ground water
system, it roust be assumed that the system hydrology behaves similar to
granular porous media.  This would be the case if the fracture density of the
system was high, but variations in fracture spacing may cause the material to
exhibit heterogeneity or anistropy.  Therefore, it is necessary to evaluate
the fracture density of the system before constructing the flow net to ensure
                                     68

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that the assumption of porous media flow conditions is valid for the system.
If tl>e fracture density of a system is low,  it may be necessary to attempt to
analyze the flow in terms of individual fractures.  However, in systems where
the fractures are large and turbulent flow conditions exist, Darcy's Law is
not valid.  For these sytems, a method other than flow net construction must
be used to calculate ground water flow rates (Bear, 1972).
    An example of a hazardous wate management site overlying fractured bedrock
is shown in Figure 4.5.  The ground water system is comprised of a permeable
basalt flow overlying fractured basalt which overlies impermeable basalt.   The
permeable basalt layer has a measured hydraulic conductivity of 10   m/sec
at well 5, based on pump tests.  The fractured basalt has a measured
conductivity of 10~  m/sec.  For the purpose of demonstration,  porous media
laminar flow is assumed.  The basalts are assumed to be isotropic.  Ground
water flows through the aquifer in a northerly direction as determined from
measurements at several wells in the area.  Using well data from Table 4.2 and
the map in Figure 4.5, the flow through the  aquifer can be calculated by
following the rationale noted below.
    A cross section is constructed based on  the map and well data, using a
scale of 1 in = 25 m.  An approximate equipotential map of the head data and  a
flow net for the permeable basalt are drawn  using the water level elevations
indicated in Table 4.2.
    The approximate equipotential map indicates a relatively low gradient in
the top of the upper basalt flow layer.  A zone of higher hydraulic
conductivity at the top of basalt flows is common.  The noted low gradient
zone probably corresponds to a higher conductivity zone.  Because the low
hydraulic conductivity, high gradient zone in the center of the basalt flow
controls the flow in the system, the flow net analysis focuses on this region.
    The squares shown in Figure 4.6 represent a head drop of 30 m between
equipotential lines with 25 m wide flow paths.  The hydraulic conductivity
varies between the upper zone and the center zone by a tactor of about 30 to  1
(3 x 10~  m/sec vs. 1 x 10~  m/sec).  Because flow in the basalt is
approximately perpendicular to the contact between the two zones, no
deflection of the flow lines occurs.  However, the tangent law is used to
calculate the angle of flow line deflection, o.t as ground water flows into
the fractured basalt:
                                    69

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                                             Ground-Surface Elevation Contours
                                             in Feet Above Mean Sea  Level
Figure  4.5.  Well  location map  and ground-surface elevations
              for fractured bedrock example.
                                70

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    TABLE 4.2.   WATER LEVEL AND STRATIGRAPHY FOR MONITOR WELLS IN FRACTURED
                BEDROCK EXAMPLE
Well Data
Ground Surface Elevation
Elevation of First Encountered Water During
Drilling
Water-Level Elevation in Wells (Screened
portion is shown in Figure 4.6)
Fractured Basalt Layer Elevation (Top)
Fractured Basalt Layer Elevation (Bottom)
Well No.
5A (ra)a
1,431
1,412 „
1,410
7
1,374
1,364
Well No.
5B (m)
1,431
1,412
1,380
1,374
1,364
Well
No. 6 (m)
1,432
1,413
1,410
1,374
1,366
All elevations in meters AHSL.

aA well is shallow; B well is deep.

"Assumed equal to the water-table surface because the aquifer is unconfined.
                                      71

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   Well  #5A and B
Waste Management Area
Well *6
(5A)1410 '
(5B)1330
                                      Water-Table Surface

                                      Unfractured Basalt
                                      ki  = 10'7 m/sec
                                                                  1410(hydraulic head)
                                                                   Fractured  Basalt
                                                                   k?  =  10'5  m/sec
                              Scale  1  in.  =  25  m
                                 i   Screened  Interval of Well
                  Figure 4.6.  Flow net for fractured bedrock example.

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                                                        OSWER POLICY DIRECTIVE NO.
                                                      9472  •  00-2A
                                 k     tan C

where:
                                    al = 85'
then:
                        i    m~7  /        tan o_
                        1  x 10   m/sec          2
                        i    in~5  /        tan 85°
                        1  x 10   m/sec
                                   02 = 6.5°

    Construction ot the flow net is continued into the fractured  basalt  as
shown in Figure 4.6.  Distance between equipotential lines in the fractured
basalt is 100 times greater than that in the unfractured basalt  based on the
difference in conductivity of a factor of 100.
    The total flow through the system can be calculated using the flow net  in
the upper basalt.  The flow through one square is multiplied by  the number  of
flow paths in the upper basalt layer to estimate the flow through the system.
In this example:

                F = 5                            (from flow net)
                k = 1 x 10~  m/sec               (from data)
                H = 30 m                         (from data)
                N = 1                            (from flow net)

Total flow for the cross section of unit width is calculated below:

                     n   (1 x 10"7 m/sec)(5)(30 m)
                     Q =	

                     Q = 1.5 x 10~5 m3/sec

    TOT100 is calculated based on a hydraulic gradient of approximately  1.2,
as determined from Figure 4.6 or the data presented in the text.   The
applicant has indicated an effective porosity for the unfractured basalt
of 0.1.
                                      73

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    The seepage velocity is:
                      ki   1 x 10"7(1.2)   ,  ,   .„-&  ,
                 vs = -r -- on -- l-2 x 10   m/sec
                              or 3.4 x 10-1 ft/day
    The associated TOT.-- is:
                       TOT          °f
                          100   0.34 ft/day

CONSTRUCTION OF FLOW NETS IN SEEPAGE FACE CONDITIONS

    A seepage face develops when a saturated-unsaturated flow system exists at
a free outflow boundary, such as a stream bank.  In this case, the ground
water flow will leave the system across the- seepage face.  The intersection of
the water table and the ground surface at the stream bank defines the upper
boundary of the seepage face.  Freeze and Cherry (1979) provide an indepth
explanation of this phenomenon and ramifications for drawing flow nets.
    Consider the case of a landfill located high on the bank of a river.   In
this example, an irrigation ditch upgradient from the landfill introduces
water which seeps through the landfill.  After the water passes through the
landfill, it flows through a layer of sand and silt that forms the top portion
of the river bank.  The water table and river bank intersect at a point,  the
elevation of which measures 315 m (the top of the seepage face).  Impermeable
bedrock is located below the sand and silt and acts as a boundary to ground
water flow.
    A monitoring well installed 78 m downgradient from the irrigation ditch
shows that the ground water level is at the ground surface during the summer
months when the irrigation ditch is in operation.  Through use of the cross
section shown in Figure 4. 7, ground water discharge into the river can be
calculated.  The flow net is drawn, as presented in Figure 4.8.  The
irrigation ditch is a constant-head boundary, the underlying bedrock is
impermeable, and the water flows out of the seepage face.  The top exit point
                                      74

-------
   Monitor Well
322 m
304 m
          Landfill
          k=10~5 m/sec
                            k=10~5 m/sec
                     Bedrock
322 m

315 m  Elevation  of
       Top of Seepage
       Face
                                                                    304 m
                           Scale  1 in.  = 15 m
 k286 m   River
                        Figure 4.7.   Cross section for seepage face,

-------
          Monitor Well
Constant Head
Boundary
            Impermeable
            Boundary
Sand and Silt Embankment
           Top Exit Point
                                                                                River
                                  Scale  1  in. =  15 m
                              Figure 4.8.   Flow net for seepage face.

-------
(see Figure 4.8) is unknown unless solved for by trial and error or
.ipprox imat(?'l from observations in the field, tli*> complicating factor in
solving this type of example.
    The flow through a unit width represented by the cross section  is
determined from:
where, in this example:
                    F = 3                         (from  flow net)
                    k = 1 x 10-5 ra/Sec            (from  data)
                    H = 18 ra                      (from  flow net)
                    N = 11                        (from  flow net)
Therefore,

                             (1 x 10"5 m/sec)(3)(18 m)
                        Q ~              11
                    Q = 4.9 x  iCT5 ra3/sec per meter of width

FLOW NET CONSTRUCTION FOR FREE-SURFACE FLOW CONDITIONS

    If the water  table itself  approximates a  flow line,  no  vertical  gradients
exist and this unusual condition is  referred  to  as free-surface  flow.   There
are two methods that can be used to  calculate flow for free-surface  flow
conditions.  The  first method  is to  construct a  flow  net and calculate  the
flow as previously discussed.  However,  the position  of  the entire free
surface may  not be known.  To  solve  the  flow  problem  in  this case, the
Dupuit-Forchheiraer theory of  free-surface flow  is used.   This  theory is based
on two assumptions:
     1.    Flow  lines  are  assumed  to  be  horizontal  and  equipotential  lines  are
          assumed  to  be vertical;  and
                                     77

-------
    2.   The hydraulic gradient is assumed to he equal to the slope of the
         f rre surf-'jcc ;ind to bo invarinnt with depth.
    With these assumptions, an empirical approximation can be used to
calculate flow, as follows:
where h(x) is the elevation of the water table above the datum used for the
flow system at x, and the gradient dh/dx is the slope of the free surface,
Ah/Ax at x (Figure 4.9).  In theory, this equation is representative of a free
surface that forms a parabola.  Calculations using the Dupuit-Forchheimer
theory produce the most accurate results when the slope of the free surface is
small and when the depth of the concpnfined aquifer is shallow (Freeze and
Cherry, 1979).
    Figure 4.10 shows a subsurface cross section in the direction of ground
water flow at a site where free-surface flow occurs.  The figure is vertically
exaggerated by a factor of 100.  Based on Figure 4.10, a Dupuit-Forchheimer
flow net can be constructed.  One flow path can be used and each equipotential
line represents the same drop in head (Figure 4.11).
    The flow rate is calculated using the Dupuit-Forschheimer solution at
point x,, where h(x) = 3.9 m.  The quantity dh/dx, at x,, is measured from
the figure, resulting in dh = 0.6 m and dx = 115 m.  Therefore, dh/dx =
5.2 x 10~  ra/ra.  Using the equation:

                  Q =   k h(x)|£

                 Q = (1 x 10~5 m/sec)(3.9 m)(5.2 x 10"3 m/m)(l m)
                 Q = 2 x 10   ra /sec

SITUATIONS IN WHICH FLOW NETS ARE DIFFICULT OR NOT APPROPRIATE

    There are situations in which the construction and use of flow  nets  is
difficult or impossible.  These situations occur when there are scaling
problems, when the geology is complex, when there is a lack of
                                     78

-------
Water Table Surface
     Figure 4.9.
Diagram showing x and h(x) for
Dupuit-Forchheimer calculation,
                           79

-------
                    7.5
00
o
                   6.0
                   4.S
                   3.0
                   1.5
                   0.0
Well  10
h=3.9 m
                                                                        Well 14
                                                     Water Table Surface  h=5-l m
                       Sand/Silt
                       k =  10"5m/sec
                                                        Impermeable Clav
                                                 3UC
                       450         600

                            Meters.
                                                                                   750
                                                                                                     Well 17
                                                                                                     h=5.7 m
                                                                                               900
                                                                                                          -Constant  Head
                                                                                                           Boundary
                                  Figure 4.10.  Cross section  for free-surface flow.

-------
oo
                     7.5
                     6.0
                  t.
                  a>
                     4.5
                     3.0
                     1.5-
                    0.0
                                      Well  10
dhC
Well 7
                                    XL
                                      y MX)
                                  / / / /  /  / /
                                                                                      / /
                                                                                               Well 17
                           -H	1	»	

                            0          150        300         450        600        750        900




                                                         Meters
                     Figure 4.11.  Dupuit-Forchhelmef  solution scheme  for' free-surface  flow.

-------
three-dimensional hydrologic data for Che ground water system, and when ground
water flow conditions do not conform to the principles expressed by Darcy's
Law.
    Scaling problems occur when the aquifer and/or geologic layers associated
with a particular ground water system are thin in relation to the length of
the flow net.  If a flow net is constructed for this situation, the flow net
will be made up of squares that are too small to work with unless the scale is
exaggerated.
    Lack of three-dimensional hydrologic data or hydrologically equivalent
data for a ground water flow system makes proper flow net construction
impossible.  Hydrologic testing at various depths within an aquifer and
determination of the vertical hydraulic conductivity and vertical gradient in
the aquifer are essential to provide reliable interpretive results.  These
data must be available before a flow net can be constructed.
    There are two types of ground water systems in which the principles
expressed by Darcy's Law do not apply.  The first is a system in which ground
water flows through materials with low hydraulic conductivities under
extremely low gradients (Freeze and Cherry, 1979) and the second is a system
in which a large amount of flow passes through materials with very high
hydraulic conductivities.  Darcy's Law expresses linear relationships and
requires that flow is laminar (flow in which stream lines remain distinct from
one another).  In a system with high hydraulic conductivity, flow is often
turbulent when it has a high velocity.  Turbulent flow is characteristic of
karstic limestone and dolomite, cavernous volcanics, and fractured rock
systems.  Construction of flow nets for areas of turbulent flow would not
provide accurate results.
                                      82

-------
                                    SECTION  5
                     MATHEMATICAL CONSTRUCTION OF FLOW NETS

    Two-dimensional, steady-state, boundary-value problems can be solved
mathematically, as well as by use of flow nets.  Freeze and Cherry (1979)
discuss the theory and mathematics of the problem.  It is clear that the time
and effort required to determine a mathematical solution is greater than the
effort to graphically construct a flow net.  Leliavsky points out that,
"... the analytical method, although rigorously precise, is not universally
applicable", because the number of known functions on which it depends is
limited.  However, except in a few elementary cases, the analytical method
lies beyond the raathemetics of practicing design offices" (in Cedergren,
1977).  Given the complexity of developing  flow nets using mathematical
techniques, they are recommended only for experienced technicians, and should
be used only if several flow net iterations would be necessary to calibrate
parameters or test  hypotheses.
    The basic equation  for saturated, steady-state, confined flow through
isotropic and homogeneous porous media (Laplace equation) in two dimensions  is:

                               2      2
The  solution  of  this equation  for  given  boundary conditions  is presented  in
Appendix  III  of  Freeze and  Cherry  (1979).  This equation describes a
homogeneous,  isotropic medium,  but  it  can  be modified  for  heterogeneous and
anisotropic materials.   This equation  is solved iteratively  to calculate  the
head at various  points along a  flow net.   A graphically constructed flow  net
is a solution of the Laplace equation.  A  complete discussion of mathematical
solutions for flow in one,  two,  and three  dimensions  is included in Freeze and
Cherry  (1979) and Bear (1972).

                                      83

-------
    With the advent of powerful calculators  and personal computers that are
"user Iriendly" and relatively inexpensive,  a number of simpler ways to
determine direction and velocity ot flow have been developed.   The data
collected to construct a flow net can be used to calculate flow direction and
velocity.
    Calculators and micro-computers greatly  reduce the amount  of work that
must be performed by the user, and thereby increase the speed  of problem
solving.  They can perform a large number of repetitive calculations in
minutes and present the output in numerical  or graphical form.   They also
eliminate the need for tables or graphs that are required for  hydraulic
computations.  Integration schemes that require complicated calculations can
be solved quickly and easily.
    Calculators and micro-computers also feature peripheral devices that aid
in calculation and presentation of hydrologic data.  The data  and the programs
to solve the mathematical expressions for flow problems can be  recorded on
magnetic cards, tapes, or disks.  The programs and data can be  stored until
such time as they are needed, and then can be readily loaded onto the
computational system.  Printers, plotters, and the software required to run
them are also available.  These devices can  be used to plot, contour, display,
and present analyses in a way that can be easily understood.
    The largest number of available programs are for well hydraulics
problems.  These programs can be useful to permit writers and  applicants in
situations where the effects of pumping wells in the vicinity  of a site must
be determined.  Table 5.1 lists several of these programs and  some of their
characteristics.
    Several simple contaminant transport programs possess the  capability to
track contaminated water and predict time of travel.  The draft TRD on
Leachate Plume Migration (Pettyjohn, et al., 1982) discusses mathematical
solutions for calculation of plume movement.  That document also includes a
TI-59 hand-held calculator program for calculating plume migration.  Table 5.2
lists several other programs for contaminant transport and some of their
capabilities.
    No attempt has been made to discuss the  hydraulic or transport codes for
hand-held calculators or micro-computers in  this document.  These techniques
are not difficult, but they must be diligently pursued if they  are to become a
                                    84

-------
                       TABLE 5.1.   AVAILABLE HAND-HELD CALCULATOR PROGRAMS  FOR GROUND WATER FLOW
                                   AND TRANSPORT (BROWN,  1983)
oc
Ground Water Systems Characterized with Well Data
Program Title
General Aquifer Analysis
for Nonsteady Thels Condi
tlons
Multiple Well. Variable
Pumping Rate Probluns
Constant or Variable
Pumping (Injection) Rate,
Single or Multiple Fully
Penetrating Wells
Constant or Variable
Pumping (Injection) Rate.
Single or Multiple Fully
Penetrating Wells
Dewaterlng Well Design
Thels Condition Well Field
Point Sink Aquifer Model
Nonsteady State Nonleaky
Artesian-Single Produc-
tion Well
AQMODl (4)
Nonsteady State Nonleaky
Artesian-Part ially Pene-
trating Wells
Aqulft
Type
C
C
C
C
C
C
C
C
C
C
r Charactei
Properties
H.I
H.I
H.I
H.I
H.I
H.I
H.I
H.I
H.I
• H.I
ristics
Extent
IN
IN
IN
IN
IN
IN
IN
IN
IN
IN
Well Cc
Number
24
1
"
'
?4
S7
SO
'
60
1
»nf Iguratlon
Penetration
FP
FP
FP
FP
FP
FP
FP
FP
FP
PP •
Time-
frame
TV
TV
TV
TV
TV
TV
TV
TV
TV
TV
Program
Output
0
0
0
D
0
D
0
0
0
0
Calculator
Type
TI-59
HP-29C
TI-59
HP-97
TI-59
HP-41
HP-41
TI-59
HP-41
TI-59
Reference
Sandberg et al . 1981
Prlckett and Vorhees I9fll
Picking 1979
Warner and Vow 1979
Rayner 1981
Koch and Associates (1)
IGWHC (?)
Ulrich (3)
Walton 1983 a & b
Rayner 1983
Walton 1983 alb
                                                                                       (Continued)

-------
                                          TABLE 5.1 (continued)
Program Title
Constant Pump (119
(Injection) Rate. Fully
Confined Aquifer. Parti-
ally Penetrating Well
Radial Flow to a Constant
Drawdown Hemisphere
Analysis of Source or Sink
Flow Rates with Drawdown
as a Given
Nonsteady Discharge of a
Flowing Wei 1
Anlsotroplc Confined
Aquifers
oo
cr- Jacob Leaky Artesian
Steady-State
Steady State leaky Arte-
sian - Single Production
Uell
Nonsteady State leaky
Artesian - Single
Production Uell
Leaky Aquifer Drawdown
Constant Pumping
(Injection) Rate. Single
Aquif
Type
C



C

C


C

C

L

L


L


L
I

er Charade
Properties
H.I



H.I

H.I


H.I

H.A

H.I

H.I


H.I


H.I
H.I

rlstics
Extent
IN



IN

IN


IN

IN

IN

IN


IN


IN
IN

Uell C(
Number
1



1

7


1

1

25

1


1


1
1

>nf Igurat ion
Penetrat ion
PP



PP

FP


FP

PP

FP

FP


FP


FP
FP

Tlroe-
frarae
TV



TV

TV


TV

TV

SS

SS


TV


TV
TV

Program
Output
D '



IF

IF


IF

D

0

D


D


D
0

Calculator
Type
TI-S9



TI-59

TI-59


TI-59

TI-59
HP-41
TI-59

TI-59


TI-59


HP-41
TI-59

Reference
Warner and Vow I980b



Koch and Associates (1)

Sandberg et al. 1981
Prlckett and Vorhees 1981

Koch and Associates (1)

Parr et al . 1983

T.A. Prlckett and
Associates (5)
Ualton 1983 a » b


Walton 1983 a ft b


Ulrlch (3)
Uarner 1 Tow I980a

Fully Penetrating Uell.
Sent (confined Aquifer
                                                                                 (Continued)

-------
                                               TABLE 5.1 (continued)


Progna Title
Hantush "Well Function"
Nonsteady State Two
Mutually Leaky Artesian
Aquifers - Single Pro-
duction Well
Steady Radial Ground-Water
flow In a Finite leaky
Aquifer
Successive Steady States -
Constant Head Points -
Uncon fined Aquifer
Aquifer Characteristics Well Configuration

Type
I
I



I


WT



Properties
H.I
H.I



H.I


H.I



Extent
IN
IN



B


IN



Number
1
1



1


7



Penetration
FP
FP



fP


FP



Time-
frame
TV
TV



SS


TV



Program
Output
0
0



0


if .0



Calculator
Type
HP-41
TI-S9



HP-4J


TI-59




Reference
IGWHC(Z)
Walton 1983 a ft b



IGUHC(Z)


Koch and Associates (1)


CO
                                                                                      (Continued)

-------
TABLE 5.1 (continued)
Regions with Subsurface Drains
Program Title
Steady-State Draw-
down Around Fi-
nite Line Sinks
Successive Steady
States - Constant
Head Finite Line
Sinks - Compute
Drawdowns
Finite Line Sinks
for Nonsteady
Conditions
oo Line Sink Aquifer
°° Model
Study of Steady-
State Flow to
Finite Line
Sources or Sinks
with Drawdown as
the Given
Successive Steady
States - One
Dimensional In-
flow to a Line
Successive Steady
States - Constant
Head Finite Line
Sinks - Compute
Inflows
Type
C


C




C


C

C





C



C



Aquifer
Properties
H.I


H,l




H.I


H.I

H.I





H.I



H.I



Charact
Extent
IN


IN




IN


IN

IN





IN



IN



eristics
Dimensional It
X-Y


X-Y




X-Y


x-r

x-r





X



X-f



Ora
Numbe
10


10




IS


15

6





1



6



in ConfHura
Penetratlo
FP


FP




FP


FP

FP





FP



FP



tlon
1
Length
F


F




F


F

F





IN



F



Time-
frame
SS


SS




TV


TV

SS





SS



SS



Program
Output
0


0




0


D

IF





IN



IF



Calcu-
lator
T.pe
TI-59


TI-59




Tl-59


HP-41

TI-59





TI-59



TI-59



Reference
Sandberg et al . 1981
Prickett and Vorhees
1981
Koch and Associates (1 •




Sandberg et al . 1981
Prickett and Vorhees
1981
Ulrich (3)

Sandberg et al . 1981
Prickett and Vorhees
1981



Koch and Associates (!'



Koch and Associates (1;



                                            (Continued)

-------
TABLE 5.1 (continued)
Program Title
One Dimensional ,
Nonsteady Flow
to » Constant
Drawdown, Infi-
nite Line Sink
or Source
One Dimensional ,
Honsteady Flow
to an Increasing
Drawdown, Infi-
nite Line Sink
or Source
Boussinesq Solution
One Dimensional ,
2 Nonsteady Flow to
a Constant Draw-
down, Infinite
Line Sink or
Source with
Recharge
One Dimensional
Non-Steady Ground
Water Flow (3)
T»oe
C





C





UT
WT






WT


Aquifer
Properties
H.I





H.I





H.I
H.I






H.I


Charact
Extent
IN





IN





8
IN






IN


erlltitS
Dimensional It)
X





X





X
X






X


Ora
i Number
i





1





1
1






1


in Configure
Penetration
fP





FP





FP
FP






FP


tlon
Length
IN





IN





IN
IN






IN


Time-
frame
TV





TV





TV
TV






TV


Program
Output
IF.D





IF.D





IF.D
IF.O






IN.O


Calcu-
lator
Vpe
TI-59





TI-59





TI-59
TI-59






HP-41


Reference
Koch and Associates (1)





Koch and Associates (1)





Koch and Associates (1)
Koch and Associates (1)






IGWMC (2)


                                        (Continued)

-------
TABLE 5.1 (continued)
Models Described by Type of Transport Process
Program Title
Advecttve Mass Transport
Thels Particle Mover
Streamlines and Travel
Times for Regional
Ground-Water Flow
affected by Sources
and Sinks
Advectlve Transport Model
Advectlon and Dispersion
Regional Flow
vo Ground Water Dispersion
o
Plume Management Model
Calculator Code for Evalu
ating Landfill Leachate
Plumes
Dissipation of a Concen-
trated Slug of Contami-
nant
Advectlon and Dispersion
from a Stream
Advectlon and Dispersion
from a Single Pumping
Well
Advectlon and Dispersion
from a Slnole Solute Well
Type
C
C
C
C
C
C
C
C
C
C
C
Aqu i f el-
Properties
H.I
H.I
H.I
H.I
H.I
H.I
H.I
H.I
M.I
H.I
H.I
Charact
Exten
IN
IN
IN
IN
IN
IN
IN
IN
IN
IN
IN
eristics
Dimensional it
X-Y
X-Y
X-Y
X-Y
X-Y
X-Y
X-Y
X-Y
X
R
R
transport
Processes
AD
AD
AD
AD. DS.
RD. DC
AD. OS.
RD. DG
AD. DS.
RD. DG
AD. DS.
RO. OG
AD. DS.
RD. DG
AD. DS
AD. OS
AD. DS
Timefram
TV
TV
TV
TV
TV
TV
TV
TV
TV
TV
TV
Program
Output
PL
PL
PL
CN
CN
CN
CN
CN
CN
CN
CN
Calculate
Type
TI-59
HP-41
HP-41
TI-59
TI-58/S9
TI-59
TI-59
TI-59
TI-59
TI-59
HP-41
Reference
Sandberg et al . 1981
Prlckett and Vorhees
1981
IGWHC (2)
Ulrich (3)
Walton 1983 a & b
Kelly 1982
Sandberg et al . 1981
Prtckett and Vorhees
1981
Pettyjohn et al.
T.A. Prickett and
Associates (5)
Walton 1983 a & b
Walton 1983 a & b
IGWMC (2)
                                            (Continued)

-------
                                                            TABLE 5.1  (continued)
                                                                  Pond Sources

Program Title
Analysis of Ground Mater
Hounding Beneath
Tailings Ponds
Circular Recharge Area
Circular Basin Recharge
Hound
Aquifer Characteristics
Type
WT
WT
UT
Properties
H.I
H.I
H.I
Extent
IN
IN
IN
Dimension*) Ity
R
R
R

Pond
Configuration
Cl
Cl
Cl

Time-
frame
TV
TV
TV

Program
Output
NH
HH
HH

Calculator
Type
TI-59
TI-59
HP-41

Reference
Sandherg et al . 1901
Prickett and Vorhees
1981
Mai ton 1983 a & b
Ulrich (3)
v£>
(I)   Programs  available as of October 1983 fron Koch and Associates. Denver,  Colorado
(2)   Programs  available as of August 1983 from the International Ground Water Modeling Center. Hoi comb Research
     Institute.  Butler University,  Indianapolis, Indiana
(3)   Programs  available as of August 1983 from James S. Ulrich and Associates, Berkeley. California
(4)   Programs  can also consider regional water level changes with time and the effects of a regional gradient
(5)   Programs  available as of July  1983 from Thomas A. Prickett and Associates. Inc., Urbana, Illinois
(6)   Program can consider four boundary conditions for drain:  constant head,  constant flux, linearly varying
     head  and  linearly varying flux
            LEGEND:   C  - Confined
                     L  - Leaky
                     WT - Water Table
                     H  - Homogeneous
                     I  - Isotropic
                     A  - AnIsotropic
                     IN - Infinite
                     B  - Bounded
                     X  - Longitudinal
                     Y  - Lateral
                     F  - Finite
                     R  - Radial
                               FP - Fully Penetrating
                               PP - Partially Penetrating
TV - Time Varying
SS - Steady State
D  - Drawdown
IF - Inflow
Cl - Ci rcular
HH - Hydraulic Head
CN - Concentration
PL - Particle Location
     wi th Time
AD - Advection
DS - Dispersion
RD - Retardation
DG - Degradat ion

-------
                           TABLE 5.2  AVAILABLE MICROCOMPUTER PROGRAMS FOR GROUND WATER FLOW
                                      AND TRANSPORT (BROWN, 1983)
CO
Ground Water Systems Characterized with Well Data
Program Title
General Aquifer
Analysis (THEIS)
THUELLS
GWFLOU (3)
Nonsteady State
Nonleaky Arte-
sian - Single
Production Well
Honsteady State
Nonleaky Arte-
sian - Partially
Penetrating Wells
Leaky Aquifer
Analysis (LEAKY)
Steady State Leaky
Artesian - Single
Production Well
Nonsteady State
Leaky Artesian -
Single Production
Well
Type
C
C
C.L
C
C
L
L
L
Aquifer
Properties
H.I
H.I
H.I
H.I
H.I
H.I
H.I
H.I
lharact
Extent
IN
IN
IN
IN
IN
IN
IN
IN
eristics
Dimensionality
X-Y
X-Y
X-Y
R
R
X-Y
R
R
Well C
Number
100
-
1
1
1
100
1
1
anf Iguration
Penetration
TP
FP
FP.PP
FP
.PP
FP
FP
FP
Tiaeframe
TV
TV
TV
TV
TV
TV
SS
TV
Program
Output
0
0
D
0
0
0
D
0
Computer
Type
TRS-80
Apple
IBM -PC
Os borne
Osborne
Sharp
PC 1500
TRS-BO
TRS-80
TRS-80
TRS-80
TRS-80
Reference
Koch and Associates (1)
IGWHC (2)
IGWHC (2)
Walton 1983 a & b
Walton 1983 a & b
Koch and Associates (1 )
Walton 1981 a i b
Walton 1983 a » b
                                                                                     (Continued)

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TABLE 5.2 (continued)


Program Title
Nonsteady State Two
Mutually Leaky
Artesian Aquifers -
Single Production
Well
Aquifer Characteristics

Type
I





'roper ties
H.I





Extent
IN





)lmenslonal Ity
R




Well Configuration

Number Penetration
1 ft






Timeframe
TV





Progran
Output
D





Computer
Type
TRS-80






Reference
Walton 1983 a S b




                                     (Continued)

-------
TABLE 5.2 (continued)
Models Described by Type of Transport Process
Program Title
Advectlon and Dis-
persion - Region-
al Flow
MAP PLUME



PLUME


PLUME

<£>
*" PLOSBNB

PLUME CROSS-
SECTION


RANDOM WALK






RANDOM WALK




Type
C


C



C


C


C

C



C.I.
UT





C.L.
HI



Aquifer I
1
Properties
H.I


H.I



H.I


H.I


H.I

H.I



H.I






H.I




!haract
Extent
IN


IN



IN


IN


IN

IN



IN






IN




frlstics
Dimensional ity
x-r


I-Y



x-v


x-r


x-v

x-z



x-»






x-v •




Transport
Processes
AD. OS.
NO. OG

AO. OS.
RD. DG


AO. OS.
RD. DG

AO. OS.
RO. DG

AD. DS.
RD, DG
AO. DS.
RO. DG


AO. OS.
RD. OG





. AO. OS.
RO, OG


i
Tlmeframe
TV


TV



TV


TV


TV

TV



TV






TV




PrograM
Output
CN


CN



CN


CN


CN

CN



CN






CN




Computer
Type
TRS-80


Apple
Kaypro II
Victor
Vector
Super -
brain
TRS-80
Sharp -
PC1SOO

Osborne

Apple
Kaypro II
Victor
Vector
Apple
Ka ypro 1 1
Victor
Vector
1RS-80
Sharp -
PC 1500
Super -
>rain
Is borne
Sharp -
PCI 500
Reference
Walton 1983 a * b


NCGWR d)



IGWMC (2)


NCGWR (If)


Voorhe.es (5)

NCGWR (d)



NCGWR (It)






IGWMC (?)




                                          (Continued)

-------
VO
                                                        TABLE 5.2  (continued)


Program Title
Aquifer Characteristics |

Type Properties
RUOSBNB C H.I

RWMY C.I. H.I
Wt

Extent
IN

IN

1 Transport
Dimensional lt>| Processes


Tlmeframe

Program
Output

Computer
Type

Reference
X-Y AO. OS. TV CN Osborne Voorhees (5)
RO. OG
X-Y AD. OS. TV CN Osborne Voorhees (5)
HI). OG
                  Advectlon and Dis-
                   persion from a
                   Stream

                  Advectlon and Dis-
                   persion from a
                   Single Pumping
                   Uell
H.I
H.I
IN
IN
AO. OS      TV     CN    TRS-80    Walton 1983 a & b
AO. OS      TV     CN    TRS-80    Walton 1983
                                                                                                           (Continued)

-------
                                                           TABLE 5.2  (continued)
                                                                Pond  Sources
Program Title
Circular Recharge
Area
Hounding
Type
HT
WT
Aquifer 1
Properties
H.I
H.I
lharact
Extent
IN
IN
eristics
Dimensional Ity
R
R
Pond
Configuration
Cl
Cl
Tine frame
TV
TV
Prograa
Output
HH
HH
Computer
Type
IRS -80
Apple
Kay pro II
Victor
Vector
Reference
Mai ton 1983 a & b
NCGWR (It)
\o
(1)   Programs available as of October 1983 from Koch and Associates,  Denver. Colorado

(2)   Programs available as of November 1983 from the International  Ground Water Modeling Center, Ho I comb Research
     Institute, Butler UniversIty, Indianapolis, Indiana

(3)   GWFLOW  Is a series of eight flow solutions, including  one  for  mounding estimation

(k)   Programs available as of October 1983 from the  National  Center for Ground Water Research, Oklahoma State
     University, St11(water, Oklahoma

(5)   Programs available as of November 1983  from Dr. Michael L. Voorhees of Warzyn Engineering,  Inc.,
     Madison, Wisconsin
                 LEGEND:  C  - Confined
                         L  - Leaky
                         H  - Homogeneous
                         I  - I sotropic
                         IN - Infinite
                         X  - Longitudinal
                         Y  - Lateral
                         R  - Radial
                         Z  - Vertical
                               fP - Fully Penetrating
                               PP - Partially  Penetrating
TV - Time Varying
SS - Steady State
D  - Drawdown
PL - Particle Location
     with Time
CN - Concentration
AD - Advectlon
OS - Dispersion
RD - Retardation
DG - Degradation

-------
working part of a permit applicant's or permit writer's skills.   Walton (1984)
discusses 35 codes for micro-computers.  Holcomb Research Institute (1983)
serves as a source of documentation for hand-held calculator codes.  Brown
(1983) also discusses hand-held calculator and micro-computer programs for
ground water flow and transport.
    Large computers are now commonly used to model hydrologic systems at
hazardous waste sites.  Researchers have found that flow codes can easily be
modified to calculate stream function and steady-state potentials.  The
results can then be plotted into a flow net (Christian, 1980).  Fogg and
Senger (1984) discuss the use of a finite-element code to draw flow nets for
hetergeneous, anisotropic media in the absence of internal sources or sinks.
                                     97

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                                 BIBLIOGRAPHY*
Bates, R. L.,  and J. A. Jackson (eds).   Glossary of Geology.   American
    Geological Institute.  Falls Church, VA.

Bear, J.  1975.  Dynamics of Fluids in  Porous Media.  American Elsevier
    Publishing Co., Inc.  New York, NY.

Bennett, R. R. 1962.  Flow Net Analysis.  In: Ferris,  J.  G.,  D.  B.  Knowles,
    R. H. Brown, and R. W. Stallman; Theory of Aquifer Tests.  Geological
    Survey Water Supply Paper 1536-E.  U.S. Government Printing Office.
    Washington, DC.

Brown, S. M. 1983.  Simplified Methods  for the Evaluation of  Subsurface  and
    Waste Control Remedial Action Technologies at Uncontrolled Hazardous Waste
    Sites.  Draft, Anderson-Nichols and Co.,  Inc.  Palo Alto, CA.

Bureau of National Affairs.  1984.  Envirbnmental Protection  Agency
    Regulations for Federally Administered Hazardous Waste Permit  Programs.
    Environment Reporter - Federal Regulations.  Washington,  DC.

Bureau of National Affairs.  1983.  Environmental Protection  Agency
    Regulations for Owners and Operators of Permitted  Hazardus Waste
    Facilities.  Environment Reporter - Federal Regulations.   Washington,  DC.

Casagrande, A.  1937.  Seepage Through  Dams.   New England Water Works
    Association.  Vol. 51, No. 2.

Cedergren, H.  R.  1977.  Seepage, Drainage, and Flow Nets. John Wiley and
    Sons.  New York, NY.

Christian, J.  T.  1980.  Flow Nets by the Finite Element  Method.  Ground Water.
    Vol. 18, No. 2.

Davis, S. N. and R. J. M. DeWeist.  1966.  Hydrogeology.   John Wiley and Sons.
    New York,  NY.

Fetter, Jr., C. W.  1980.  Applied Hydrogeology.  Charles E.  Merrill Publishing
    Co.  Columbus, OH.

Fogg, G. E. and R. K. Senger.  1984. Automatic Generation of Flow Nets  with
    Conventional Ground Water Modeling  Algorithms.  In:  Proceedings of  the
    Practical Applications of Ground Water Models.
*Site specific references have been omitted to maintain confidentiality.

                                     98

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Freeze, R. A. and J. A. Cheery.  1979.  Ground Water.   Prentice-Hall, Inc.
    Englewood Clitfs, NJ.

Holcomb Research Institute.  1983.   HP-41C Program Package.   International
    Ground Water Modeling Center, Butler University.  Indianapolis, IN.

Kelly, W. E.  1982.  Field Reports  - Ground Water Dispersion Calculations with
    a Programmable Calculator.  Ground Water.  Vol. 20, No.  6.

Lohman, S. W.  1979.  Ground Water  Hydraulics.  Geological Survey Professional
    Paper 708.  U.S. Government Printing Office.  Washington, DC.

McWhorter, D. B. and D. K. Sunada.   1977.  Ground-Water Hydrology and
    Hydraulics.  Water Resources Publications.  Fort Collins, CO.

Oberlander, P. L., and R. W. Nelson.  1984.  An Idealized Ground-Water  Flow
    and Chemical Transport Model (S-PATH).  Ground Water.  Vol.  22, No. 4.

Parr, A. D., J. G. Melville, and F. J. Molz.  1983.  HP41C and TI59 Programs
    for Anisotropic Confined Aquifers.  Ground Water.   Vol.  21,  No. 2.

Pettyjohn, W. A., D. C. Kent, T. A. Prickett, H. E. LeGrand, and F. Witz.
    1982.  Methods for the Prediction of Leachate Plume Migration and Mixing.
    Draft EPA Technical Resource Document.

Picking, L. W.  1979.  Field Reports - Programming a Pocket  Calculator  for
    Solving Multiple Well, Variable Pumping Rates Problems.   Ground Water.
    Vol. 18, No. 2.

Prickett, T. A., and M. L. Vorhees.  1981.  Selected Hand-Held Calculator Codes
    for the Evaluation of Cumulative Strip-Mining Impacts on Ground Water
    Resources.  Prepared for the Office of Surface Mining, Region V, Denver,
    CO.  Thomas A. Prickett and Associates, Inc.  Urbana, IL.

Rayner, F. A.  1983.  Discussion of Programmable Hand Calculator Programs for
    Pumping and Injection Wells: I  - Constant or Variable Pumping (injection)
    Rate, Single or Multiple Fully  Penetrating Wells.   Ground Water. Vol.  19,
    No. 1.

Sandberg, R. R., R. B. Scheibach, D. Koch and T. A. Prickett.  1981. Selected
    Hand-Held Calculator Codes for  the Evaluation of the Probable Cumulative
    Hydrologic Impacts of Mining.  Report H-D3004/030-81-1029F.   Prepared for
    the Office of Surface Mining, Region V, Denver CO., by Hittman Associates,
    Inc.

Terzaghi, K. and K. B. Peck.  1967.  Soil Mechanics in Engineering Practice.
    Second Edition.  John Wiley and Sons.  New York, NY.

Todd, D. K.  1980.  Ground Water Hydrology.  John Wiley and  Sons.  New  York,
    NY.
                                    99

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U.S. Department of Agriculture.  1979.   The Mechanics of Seepage Analyses.
    Soil Mechanics Note No. 7.

U.S. Environmental Protection Agency.  1984a.   Ground-Water Protection
    Strategy.  Office of Ground Water Protection.   Washington,  DC.

U.S. Environmental Protection Agency.  1984b.   Method 9100.  Saturated
    Hydraulic Conductivity, Saturated Leachate Conductivity and Intrinsic
    Permeability Methods.  Washington,  DC.

U.S. Environmental Protection Agency.  1984c.   Permit Applicants' Guidance
    Manual for Hazardous Waste Land Treatment, Storage and Disposal
    Facilities.  Final Draft.  EPA 530  SW-84-004.    Washington, DC.

U.S. Environmental Protection Agency.  1984d.   Soil Properties,
    Classification, and Hydraulic Conductivity Testing.   Draft.  SW-925.
    Washington, DC.

U.S. Environmental Protection Agency.  1983a.   Permit Writers'  Guidance Manual.
    Draft.  Washington, DC.
    i
U.S. Environmental Protection Agency.  1983b.   Permit Writers'  Guidance Manual
    for Subpart F, Ground-Water Protection. Washington, DC.

U.S. Environmental Protection Agency.  1982.  Ground-Water Monitoring
    Assessment Program at Interim Status Facilities.  Draft SW-954.
    Washington, DC.

U.S. Environmental Protection Agency.  1980.  Procedures Manual for  Ground
    Water Monitoring at Solid Waste Disposal Facilities.  SW-611.
    Washington, DC.

Walton, W. C.  1984a.  Handbook of Analytical  Ground Water Models.
    International Ground Water Modeling Center,  Hoicomb  Research Institute,
    Butler University.  Indianapolis, IN.

Walton, W. C.  1984b.  35 Basic Ground  Water Model  Programs for Desktop
    Micro-Computers.  International Ground  Water Modeling Center, Hoicomb
    Research Institute, Butler University.   Indianapolis, IN.

Walton, W. C.  1983a.  Handbook of Analytical  Ground Water Models.   Distributed
    at the short course "Practical Analysis of Well Hydraulics  and Aquifer
    Pollution."  Holcomb Research Institute, Butler University.  Indianapolis,
    'IN.  April 11-15.

Walton, W. C.  1983b.  Handbook of Analytical  Ground Water Codes for Radio
    Shack TRS-80 Pocket Computer and Texas  Instruments TI-59 Hand-Held
    Programmable Calculator.  Distributed at the short course "Practical
    Analysis of Well Hydraulics and Aquifer Pollution."   International Ground
    Water Modeling Center, Holcomb Research Institute, Butler University.
    Indianapolis, IN.  April 11-15.
                                    100

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Warner, D. L. and M. G. Yow.  1980a.   "Programmable Hand Calculator Programs
    for Pumping and Injection Wells:  III - Constant Pumping (Injection) Rate,
    Fully Confined Aquifer, Partially Penetrating Well.  Ground Water.   Vol.
    18, No. 5.

Warner, D. L. and M. G. Yow.  1980b.   "Programmable Hand Calculator Programs
    for Pumping and Injection Wells:  II - Constant Pumping (Injection)  Rate,
    Single Fully Penetrating Well, Semiconfined Aquifer."  Ground Water.  Vol,
    18, No. 5.

Warner, D. L. and M. G. Yow.  1979.  "Programmable Hand Calculator Programs
    for Pumping (Injection) Rate, or  Single or Multiple Fully Penetrating
    Wells."  Ground Water.  Vol. 17,  No. 6.
                                   101

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                                   APPENDIX A

                                   GLOSSARY*
AnisoCropic—Descriptive of a geologic medium Chat has hydraulic properties
(i.e., hydraulic conductivity) that vary with direction.

Aquiclude—Defined as a saturated geologic unit that is incapable of
transmitting significant quantities of water under ordinary hydraulic
gradients (Freeze and Cheery, 1979).

Aquifer—A geologic formation, group of formations, or part of  a formation
capable of yielding a significant amount of ground water  to wells or  springs
(EPA, 1984b).

Aquitard—Defined as a stratigraphic sequence of geologic beds  that may  be
permeable enough to transmit water in quantities that are significant  in the
study of regional water flow, but may not'have a permeability sufficient to
allow completion of production wells within them (Freeze  and Cherry,  1979).

Artesian—Descriptive of ground water that is under sufficient  hydrostatic
pressure to rise above the aquifer containing it.

Confined Aquifer—An aquifer bounded above and below by impermeable beds; an
aquifer containing confined ground water.

Constant-Head Boundary—A boundary along which the hydraulic head is
constant.  An equipotential line.  Flow lines must meet at a constant-head
boundary at right angles, and adjacent equipotential lines must be parallel to
the boundary (Freeze and Cherry, 1979).

Discharge Velocity—The quantity of water that percolates per unit time  across
a unit area of a section oriented at right angles to the  flow lines.   For
laminar flow conditions, it is defined by Darcy's Law as  the coefficient of
hydraulic conductivity multiplied by the hydraulic gradient (Terzaghi  and
Peck, 1967).

Effective Porosity—The volumetric percentage of the total volume of  a given
mass of soil or rock that consists of interconnecting pores spaces through
which flow can occur.
*A11 definitions are after Bates and Jackson, 1980,  unless otherwise noted.
                                    102

-------
Equipotential Line—A contour line on the potentiometric surface;  a line along
which the hydraulic head of ground water in an aquifer is the same.  Flow is
perpendicular to these lines in the direction of decreasing ground water
potential.

Flow Line—A line that is indicative of the direction of ground water flow,
and is perpendicular to equipotential lines (Freeze and Cherry, 1979).

Flow Net— In the study of ground water phenomena, a graph of flow lines and
equipotential lines that represents two-dimensional flow through porous media.

Flow Path—^the area between two adjacent flow lines (Freeze and Cherry, 1979).

Ground Water—Subsurface water that exists in saturated and unsaturated
formations.

Head—(a) The elevation to which water rises at a given point as a result of
reservoir pressure,  (b) Water-level elevation in a well, or elevation  to
which the water of a flowing artesian aquifer will rise in a piezometer.

Heterogeneous—Descriptive of a ground water formation where hydraulic
properties vary spatially (Freeze and Cherry, 1979).

Homogeneous—Descriptive of a ground water formation where hydraulic
properties are the same at every point (Freeze and Cherry, 1979).

Hydraulic Conductivity—The volumetric rate of flow of water per unit time
through a unit cross-sectional area, under a unit hydraulic gradient, at the
prevailing temperature.

Hydraulic Conductivity Boundary—The boundary between two materials having
different hydraulic conductivities (Freeze and Cherry, 1979).

Hydraulic Gradient— The rate of change of total head per unit distance in the
direction of flow at a given point.

Hydrostratigraphic Unit—A body of rock having considerable lateral extent and
composing a geologic framework for a reasonably distinct hydrologic system.

Hydrostatic Pressure—The pressure exerted by water at any given point  in a
body of water at rest.  The pressure is generally due to the weight of  the
water at  higher levels in the zone of saturation.

Isotropic—Descriptive of the condition in which hydraulic properties of an
aquifer are the same in all directions (Fetter, 1980).

Laminar Flow—Smooth, uniform water  flow  in which the stream lines remain
distinct  from one another, as distinguished from turbulent flow.

Lithology—The description of rocks  on the basis of such characteristics as
color, mineralogic composition, and  grain size.
                                     103

-------
Permeability—The capacity of a porous rock, sediment, or soil to transmit
fluid.  It is a measure of the relative ease of fluid flow under a hydraulic
gradient.

Piezometer—A nonpumping well, generally of a small diameter,  which is used to
measure the elevation of ground water or potentiometric surface.  A piezometer
generally has a short well screen isolated in the aquifer of interest (Fetter,
1980).

Porosity—The percentage of the bulk volume of a rock or soil  that is occupied
by pore spaces, whether isolated or connected.

Potentiometric Surface—An imaginary surface representing the  areal head  of
ground water as defined by the level to which the water rises  in a group  of
wells.  The water table is an example of a potentiometric surface.

Pressure Head—The distance between the point of measurement (e.g., bottom of
a well) and the water level in a well (Freeze and Cherry, 1979).

Seepage Velocity—Defined as the average velocity at which water percolates
through porous material and is equal to the discharge velocity divided by'the
effective porosity of the material (Terzaghi and Peck, 1967).

Stratification—The formation, accumulation, or deposition of  material in
layers; specifically, the arrangement or disposition of sedimentary rocks.

Tie Line—In the context of flow net analysis, this is a straight line drawn
between points of equal pressure head, as measured by a piezometer or a well,
on a flow net diagram.  It is used to interpolate the magnitude-of otherwise
unknown hydraulic head values at selected locations.

Turbulent Flow—Water flow in which flow lines cross and intermix at random,
as opposed to laminar flow.

Unconfined Aquifer—An aquifer having a water table as the upper boundary; an
aquifer containing unconfined ground water.

Uppermost Aquifer—The geologic formation nearest to the natural ground
surface that is an aquifer, as well as lower aquifers that are hydraulically
interconnected with this aquifer (EPA, 1984b).

Water Table—The surface of a body of unconfined ground water  at which the
pressure is equal to that of the atmosphere.

Zone of Aeration—A subsurface zone containing water at pressure less than
atmospheric, including water held by capillary forces.  This zone is limited
above by the land surface and below by the water table.

Zone of Saturation—A subsurface zone in which all of the pore spaces are
filled with water under pressure greater than that of the atmosphere.
Although the zone may contain sporadic gas-filled pore spaces  or pore spaces
filled with a fluid other than water, it is still considered saturated.
                                   104

-------