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This is to certify that the
thesis entitled

Bog - watershed Relationships Utilizing
Electric Analog MOdeling

presented by

John Egan Sander

has been accepted towards fulfillment
of the requirements for

PhoDo degree in Ge010gy

 

July 28, 1971

ABSTRACT

BOG-WATERSHED RELATIONSHIPS UTILIZING
ELECTRIC ANALOG MODELING

BY

John Egan Sander

Little is known about the role of bogs in the
hydrologic cycle although they comprise many millions of
acres within the United States and form the headwaters
of many major rivers. Current studies are principally
concerned with bogs that have been drained or otherwise
(listurbed. As a result, much remains to be learned about
'the hydrology of bogs in their natural state. This study
.is directed toward the development of a methodology for
'the study of wetlands and its application to a peat bog
i1: its natural state to supplement existing knowledge of
bog'hydrology.

A small undisturbed bog upland-watershed unit
Located.in northern Minnesota was investigated using
seismic refraction, gravity, and earth resistivity methods
to supplement geologic and hydrologic data. This infor-
mation was used to estimate watershed boundaries, trans—

missibilities, and storage values for the bog and the

John Egan Sander

watershed within which it occurs. Seismic refraction data

was the single most important geOphysical method and was
used to determine saturated thicknesses and transmissi-

bilities in the upland aquifer when combined with avail-

able pereability data. Earth resistivity measurements

served as a very limited aid in blind zone considerations

of saturated thickness obtained from seismic refraction

data. Gravity supplied no information directly perti-

nent to this study.
An electrical model of the bog and its watershed
was constructed from available hydrologic and geologic
data. Experimentation with this model, modified as a
result of comparing modeling results with historical
data, indicate many significant hydrologic characteris-
tics of the bog and its surrounding watershed.

Some of

the more significant conclusions reached from this

experimentation are: (l) the bog-watershed unit covers

a much larger area than local topography would suggest;
(2) the bog impedes flow of water through the aquifer
because of its generally low transmissibility and thus
raises the water table 'in the vicinity of the bog tending
to maintain the vertical development of the bog; (3) the
upper permeable layers of the bog have very high trans-
missibilities suggesting that flow in this region may not
obey Darcy's law; and (4) most of the surface discharge

from the bog is derived from horizontal inflow from the

John Egan Sander

upland watershed and from the underlying aquifer as
upwelling through openings within the bog. Short term
variations in surface discharge from the bog, however,
almost entirely reflect fluctuations in precipitation and

evapotranSpiration within the bog.

BOG-WATERSHED RELATIONSHIPS UTILIZING
ELECTRIC ANALOG MODELING

BY

John Egan Sander

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Geology

1971

To My Father

ii

ACKNOWLEDGMENT S

This study was aided by many individuals. W. J.
Hinze of Michigan State University initially coordinated
the project and made many helpful contributions both
during the course of the study and in the preparation of
the final manuscript. H. H. Bennett of Michigan State
University aided in data interpretation and in the prepara-
tion of the final manuscript. C. E. Prouty and J. w. Trow,
also of Michigan State University, made valuable sugges-
tions pertinent to the preparation of the final manuscript.

The study was undertaken with the assistance of
the U.S. Forest Service, and the contributions of all
participating Forest Service personnel are acknowledged.
In particular, D. H. Urie, whose contributions covered a
wide range, gave very generously of his time and effort,
and C. F. Hawkinson helped significantly both in the
collection of data and through his general knowledge of

the location of the study.

iii

TABLE OF CONTENTS

Chapter
I. INTRODUCTION . . . . . . . . . .
Statement of Purpose. . . . . .
Previous Studies Related to this
Investigation . . . . . . . .
II. DESCRIPTION OF THE AREA . . . . . .
General Description of the Marcell
Experimental Forest Region . .
Description of the S- 3 Bog-Watershed Unit.
Geological Setting of the Marcell
Experimental Forest Region . . . .
Glacial History of the Experimental
Forest. 0 O O O O O O O O 0
III. GEOPHYSICAL METHODS. . . . . . . .
Seismic Refraction Methods. . . . .
General Considerations . . . . .
The Hidden Layer Problem. . . . .
The Blind Zone Problem . . . . .
Seismic Results. . . . . . . .
Earth Resistivity Methods . . . . .
Gravity Methods . . . . . . . .
IV. COLLECTION AND INTEGRATION OF GEOPHYSICAL,

GEOLOGICAL, AND HYDROLOGIC INFORMATION .
Hydrologic Data . . . . . . . .

Upland Ground Water Wells . . . .
Instrumentation Within Bog S-3. . .
Temperature and Precipitation . . .
Bog Transmissibility and Storage .
Pumping Test Determination of Aquifer
Transmissibility and Storage. . .

iv

Page

11
16
19
22
22
27
29
35
37
43
49
49
49
50
52
53

54

 

 

 

 

 

Chapter

V.

VI.

Surface Geological Data . . . . . . .

Surface Topography . .
Surface Soil Examination . . . . . .

Subsurface Geological Data . . . . . .

Geophysical Data. . . . . . . . .
Bog Well Coring Data . . . . . . .
EXploratory Well Data . . . . . . .

Integration of Hydrologic and Geological
Information . . . . . . . . . .

DevelOpment of Watershed Limits. . . .
Development of the Aquifer Transmissi-
bility Map 0 O O O O O O O O 0

THE PHYSICAL ANALOG MODEL . . . . . . .

The Finite- Difference Approximation to

the Continuous System . . . . .
Development and Construction of the

Analog Model . . . . . . . . . .

Scale Factors . . . . . . . .
Use of Two Different Grid Sizes. . . .
The Current Input Representing
Precipitation Minus Evapotranspiration.
Initial and Boundary Conditions. . . .
Modeling the Bog-Aquifer Boundary . . .
Modeling Weir Outflow and Bog Upwelling .
Accuracy in Model Construction . . . .

RESULTS OF ELECTRIC ANALOG MODELING. . . .

Introduction--Verification of the Model. .
Modeled Recharge . . . . . . . . .

Recharge from the Outlying Regional
Watershed and an Estimation of the
Total Watershed Area. . . . . . .

Recharge from Precipitation in the

Upland Area. . . . .
Changing Upland Transmissibility

and Storage. . . . . . . .
Bog Recharge from Direct Precipitation .
Modeling of Storms . . . .

Bog Recharge from Upland Ground Water. .

Page
56

56
57

57
57
60
62
64
64
66

72

73
79

82
89

89
91
94
95
98

99

99
100

100
102

111
117
123
134

Chapter Page
Modeled Discharge . . . . . . . . . 136
Surface Discharge from the Bog . . . . 136
Discharge of Ground Water Flowing
Beneath the Bog . . . . . . . . 143

Experimentation with the Model. . . . . 144
Conclusions from Modeling . . . . . . 150

VII. CONCLUSIONS AND SUGGESTIONS FOR FURTHER
STUDY. 0 O O O I O O O O I O O O 154

Conclusions . . . . . . . . . . . 154
Suggestions for Further Study . . . . . 156

LIST OF REFERENCES. . . . . . . . . . . . 159
APPENDICES
Appendix

I. Basic Theory and Equations for the Analog
MOdel I O C I O O O O O O O O O 164

II. Theory of Geophysical Methods. . . . . . 186

III. F¢RTRAN Digital Computer Programs . . . . 195
IV. Seismic Refraction and Electrical
Resistivity Profiles . . . . . . . . 199

vi

LIST OF TABLES

Table Page

2-1. Glacial Events in the Marcell Experimental
Forest Area . . . . . . . . . . . 18

3-1. Average Value and Range of Values for
Compressional Wave Velocities Correlated

to Each Rock Type . . . . . . . . . 28

5-1. Analagous Properties Between the Hydrologic
and Electrical Systems . . . . . . . 73
III-1. Program B LAYER . . . . . . . . . . 196
III-2. Program RH¢. . . . . . . . . . . . 198

vii

Figure

2-10
2-20

2-3.

3-7.
3-8.
3-9.
3-10.
3-11.

3-12 0

LIST OF FIGURES

Map of Marcell Experimental Forest . . .
Surface Drainage Course for B09 S-3. . .

Schematic Block Diagram of the Surface
Geology in Marcell Experimental Forest .

Comparison of Seismic Profile 3 with
Drilling Data from Upland Well 305 . .

Comparison of Resistivity Profile 3 with
Drilling Data from Upland Well 305 . .

Time-Distance Graph of a Reversed Profile
Illustrating Two Layers with a Dipping
Interface. . . . . . . . . . .

Seismic Refraction Interpretation Without
Blind Zone Considerations . . . . .

Seismic Refraction Interpretation with
Blind Zone Considerations . . . . .

Overlay Method for Finding the Intercept
Times of a Horizontal Blind Zone Layer .

Profile 9 Apparent Resistivity . . . .
Profile 15 Apparent Resistivity . . . .
Profile 17 Apparent Resistivity . . . .
Residual Bouguer Gravity Anomaly Map . .

Bedrock Surface Topography Based on
Seismic Refraction Data . . . . . .

Bouguer Gravity Anomaly Map . . . . .

viii

Page

15

20

21

25

30

31

36
40
41
42

45

46

47

Figure Page

4-1. Location of Bog Wells and Adjacent Upland

Ground Water Wells . . . . . . . . 51
4-2. Seismic Refraction Profile Panel Diagram . 59
4-3. Is0pach Map of Bog S-3 . . . . . . . 61
4-4. Log of Upland Well 305 . . . . . . . 63
4-5. Aquifer Transmissibility Map . . . . . 69
4-6. Cross Section Through Bog S-3 and Adjacent

Upland . . . . . . . . . . . . 70
4-7. Location of Cross Section Through Central

Part of the Area of Study. . . . . . 71
5-1. Representation of a Continuous Conductive

Sheet of Resistivity p by a Network of

Discrete Resistors . . . . . . . . 76
5-2. Representation of Continuous Dielectric

Material by a Discrete Capacitor . . . 78
5-3. The Electrical Analog Model with

Peripheral Electrical Equipment. . . . 80
5-40 The RC Portion Of the MOdEl. o o o o o 81
5-5. Resistances Representing Transmissibility

for the Outer Part of the Modeled

Aquifer. . . . . . . . . . . . 84
5-6. Resistances Representing Transmissibility

for the Inner Part of the Modeled

Aquifer. . . . . . . . . . . . 85
5-7. Capacitors Representing Aquifer Storage . . 86
5-8. Resistances Representing Bog Transmissi-

bility O O O O O O O O O O O O 87
5-9. Capacitors Representing Bog Storage . . . 88
5-10. The Bog-Aquifer Interface. . . . . . . 96

ix

Figure
5—11 0
5—12.

6-1 a

6-10 0

6-11.

6-13.

6-14.

Page
Northwest Border of Bog S-3 . . . . . . 97
Southeast Border of Bog S-3 . . . . . . 97

Schematic Diagram of the Relative Areas
Represented by the Resistance-Capacitance
Section of the Model and by the Constant
Current Input . . . . . . . . . . 103

Measured and Modeled Upland Water Table
Elevations April 1, 1968-March 31, 1969 . 105

Measured and Modeled Upland Water Table
Elevations April 1, 1969-March 31, 1970 . 106

Measured and Modeled Upland Water Table
Elevations April 1, 1970-March 31, 1971 . 107

Upland Ground Water Recharge April 1,
1968-MarCh 31' 1969 o o o o o o o o 108

Upland Ground Water Recharge April 1,
1969-MarCh 31’ 1970 o o o o o o o o 109

Upland Ground Water Recharge April 1,
1970-October 31, 1970 . . . . . . . 110

Water Table Elevations at Upland Wells
April 1969—April 1970 . . . . . . . 113

Net Precipitation and Water Table Elevation
in Bog 8-3 April 1, 1968-March 31, 1969 . 119

Net Precipitation and Water Table Elevation
in Bog 8-3 April 1, 1969-March 31, 1970 . 120

Net Precipitation and Water Table Elevation
in Bog 8-3 April 1, 1970-March 31, 1971 . 121

Storm Rainfall Intensity and Measured and
Modeled Bog Surface Discharge August
28-29, 1969. o o o o o o o I o o 126

Measured and Modeled Water Table Elevation
Change at Bog Well 1 as a Result of the
Storm of August 28-29, 1969 . . . . . 127

Storm Rainfall Intensity and Measured and
Modeled Bog Surface Discharge September
4-5—6, 1969. o o o o o o o o o o 129

Figure
6-15.

6-18.

6-19.

6-20.

6-21.

6-23.

11.1.

II-ZO

II-3.

Measured and Modeled Water Table Elevation
Change at Bog Well 1 as a Result of the
Storm of September 4-5-6, 1969 . . . .

Storm Rainfall Intensity and Measured and

Modeled Bog Surface Discharge October 4-5,

1969. O O O O O O O I I O O 0
Measured and Modeled Water Table Elevation
Change at Bog Well 1 as a Result of the

Storm of October 4-5, 1969 . . . . .

Schematic Diagram of Ground Water Flow in
the Region of Bog S-3 . . . . . . .

Measured and Modeled Surface Discharge of
Bog 5-3 April 1, l968-March 31, 1969 . .

Measured and Modeled Surface Discharge of
Bog 8-3 April 1, 1969-March 31, 1970 . .

Measured and Modeled Surface Discharge of
Bog 5-3 April 1, 1970-March 31, 1971 . .

Modeled Ground Water Recharge and Water
Level Elevations in the Upland Area for
a Hypothetical High Precipitation Year .
Modeled Net Precipitation, Water Table
Elevations, and Surface Discharge of Bog
5-3 for a Hypothetical High Precipitation
. Year 0 O O O O O O O O O O O O
The Hydrologic and Electrical Systems . .

Horizontal Flow Through a, An Elemental
Volume of the Aquifer . . . . . . .

Snell's Law . . . . . . . . . . .

Time-Distance Graph for a Two-Layer Case
with a Horizontal Interface . . . . .

Colliner Resistivity Electrodes . . . .

xi

Page

130

131

132

138

139

141

142

146

148

169

183

189

189

191

Figure Page
II-4. Two-Layer Apparent Resistivity . . . . 191

IV-l. Location of Seismic and Resistivity
Profiles . . . . . . . . . . 199

IV-2. Seismic Refraction and Electrical
Resistivity Profiles. . . . . . . 200

xii

CHAPTER I

INTRODUCTION

Statement of Purpose

 

Wetlands comprise many millions of acres within
the northern lake states and form the headwaters of
many major rivers, yet relatively little is known about
the role played by bogs in the hydrologic cycle. Cur-
rent studies are mainly concerned with bogs that have
been altered by drainage or otherwise disturbed. Thus
a great deal remains to be learned about the hydrology
of bogs in their natural state. To this end, data have
been collected in recent years in several bog-watershed
complexes in the Marcell Experimental Forest located in
north-central Minnesota. These studies have yielded
information on the storage coefficient and permeability
of peats and have accumulated a store of climatic and
water budget data on bog-basin complexes in the region.
However, little is known about the role of the bog in
the ground water system. This investigation is directed
toward the deve10pment and testing of a methodology for

determining this role. In its approach, it supplements

existing data with information derived from geophysical
methods to formulate geologic-hydrologic parameters for
the region. This information is then used in the construc-
tion of an electric analog model of the bog and its
ground water system. More specifically, the objectives
are the development of techniques using geophysical
methods for the delineation of a bog and its watershed
and the development of hydrologic modeling methods for
a bog-watershed unit based on a knowledge of precipita-
tion, evapotranspiration, and the hydrogeologic proper-
ties of the hog and the basin in which it occurs. The
model is verified as a predictive tool by comparing its
input-output results with hydrologic data collected in
the area. The model also serves as a means of determin-
ing previously unrevealed geological conditions and
parameters of the hydrologic system. This is accomplished
by noting alterations of the initially prOposed model
that may be necessary to produce modeling results in
agreement with actual hydrologic data. The modeling
approach is used as a method of interpreting, verify-
ing, and making predictions from available geOphysical,
geologic, and hydrologic data.

Previous Studies Related to this
Investigation

 

There have been a number of studies applying

geophysical methods to ground water exploration, and

some examples are cited here. Watkins and Spieker (1965)
used detailed seismic refraction surveys to define the
extent and configuration of aquifers consisting of buried
Pleistocene valleys filled with gravel and sand. Zohdy
(1965) had some success in experimenting with selected
resistivity methods as a means of exploration for perme-
able strata. Joiner and others (1967) used seismic
methods along with tOpographic, geologic, and hydrologic
data to locate favorable sites for drilling. Seismic
methods were used to calculate depths to bedrock, with
resistivity methods employed to confirm these depths and
to locate gravel zones in buried stream channels.
Spangler and Libby (1967) used gravitational methods to
obtain a regional picture of thickness of unconsolidated
deposits as a basis for more detailed seismic work to
follow. Page (1969) worked with electrical methods as a
means for investigating geologic and hydrologic conditions.

There has been little work done on the direct
application of geophysical techniques to the study of
bogs. One known example is supplied by the Michigan
State Highway Department, Geophysical Unit (1963). They
employ a modified seismic refraction technique to
delineate bog thickness.

A variety of studies have been conducted on bogs
in northern Europe and Russia Asia. The Russians have

published several articles on wetland research, and a

few examples are noted here. N. I. P'yavchenko (1958)
gives an extensive descriptive summary of peat bogs of
the Russian forest-steppe. P. K. Vorobe'v (1963) investi-
gated the water yield prOperties of some swamps in
western Siberia and made comparisons between the various
types of materials comprising the swamps. S. M. Novikov
(1964) deve10ped a method for computing changes in bog
water table levels based on solar energy and precipi-
tation values. K. E. Ivanov (1965) developed a series
of equations for computations of the water balance of
swamps. L. G. Bavina (1967) describes a method developed
for calculating evaporation from bogs and discusses the
construction of new regional maps of bog evapotranspiration.
There has been relatively little study of wet-
lands in the United States. In the northern Great Lakes
region, B. K. Soper (1919), an early investigator, dis-
cussed the origin, classification, distribution, and uses
of peat deposits in Minnesota. Subsequent investigations
of wetlands in Minnesota were undertaken by other workers.
Particularly significant is the study made by Heinselman
(1963). The establishment around 1960 of several bog-
watershed units for study in Marcell EXperimental
Forest in northern Minnesota has resulted in an in-
creased amount of specific information both of bogs in
the region and on bog hydrology in general. R. R. Bay

(1961, 1963, 1966, 1967, 1969) in a series of publications

con-i

 

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discussed the results of several experiments conducted

at the Experimental Forest, and D. H. Boelter (1964, 1965,
1966) has done significant work on the hydrologic proper-
ties of peats in the Experimental Forest area.

Many articles on the application of analog model-
ing methods to ground water problems have appeared in
recent years covering a wide range of situations. How-
ever, no examples have been reported on the application

of analog modeling methods to problems in bog hydrology.

CHAPTER II

DESCRIPTION OF THE AREA

General Description of the Marcell Experimental
Forest Region

 

Marcell Experimental Forest of the North Central
Forest Experiment Station, Forest Service, U.S.D.A. is
located in the north-central region of Minnesota and
consists of two sections, a north and a south unit
(Figure 2-1). The south unit lies entirely within the
Mississippi Headwaters Watershed and contains Bog 8—3,
the specific bog investigated in this study. The con-
tinental divide passes through the north unit with north-
ward drainage to the Rainy River. Topographically, the
region consists of rolling plains and low hills with
lakes and peat bogs eSpecially predominant in depres-
sions in the hilly regions. The hills located in the
south unit are flat-topped and slope gently toward the
southeast. They are steep-sided and are separated by
irregular valleys and depressions with a maximum relief
of about 70 feet. The drainage pattern is poorly devel-
oped with many lakes and bogs having no surface drainage.

The climate is cool consisting of moderate sum-

mers and cold winters. The mean annual temperature in

6

 

 

 

 

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Grand Rapids, approximately 25 miles distant, is 39°F.
The mean for January is 7.8°F. and for June is 62°F.
(Bay, 1967). The average annual precipitation for the
Mississippi Headwaters Watershed is 25.33 inches and of
this amount, approximately 20 inches is lost through
evapotranspiration (Oakes and Bidwell, 1968). Storms
are common and often contribute one or more inches of

precipitation within a few hours.

Description of the 8-3 Bog-Watershed Unit

 

Bog S-3, located in a "T"-shaped ice-block
depression, is roughly 48 acres in size. The watershed
aquifer consists mainly of deep permeable glacial sands.
Approximately 300 acres surrounding the bog have been
intensively investigated by water table wells and seis-
mic studies without isolating a local watershed asso-
ciated with the bog. Regional studies indicate that the
bog lies within an extensive regional watershed that has
not been precisely defined. Groundwater flow diagrams
and surface water drainage patterns indicate that the
watershed continues to the northwest for several miles
and empties southeast of the bog through a swamp-lake
network into the Prairie River (Figure 2-2).

The bog's water table surface is continuous with
the regional ground water table. This situation is con-
trasted with the "perched" type of bog also found in the

region. The latter type of bog is formed in clay surfaced

 

 

Wetland Area Downstream
tronIBog 5-3
is Stippled

Sane

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Figure 2-2

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Surface Drainage Course for Bog S-3

 

 

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10

depressions that lie above the regional water table in
which local surface runoff is trapped. Thus, the water
surface of the perched type of bog lies above the regional
water table. The continuity of 8—3 bog water levels with
the regional water table results in annual water table
fluctuations that are smaller in 5-3 than in the perched
type bog (Bay, 1967). Also, water flowing through S-3
contains nutrients leached from a large area of upland
soils. This causes an increase in the variety and growth
rate of the vegetation in Bog 8-3 as contrasted with the
sparce vegetation found in the perched type bog. The
bog's thickness varies with location and the maximum
recorded value from exploratory drilling is 28.5 feet.
Peat samples taken by workers in bog 8—3 show some aqua-
tic peat at lower depths with much of the upper layers
consisting of woody peat. This indicates that the bog
originated as a vegetation filled lake that was subse-
quently forested and has remained so for at least seve-
ral centuries (Bay, 1967).

The vegetation in the 8-3 bog-watershed complex
is representative for this type of region. The upland
area is largely forested with quaking aspen (Populus

tremuloides) being the predominant tree type. The aspen

 

is intermixed with a supporting variety of other deciduous
and coniferous species. The vegetation within bog 8-3

is relatively diverse with the principal bog tree being

11

black spruce (Picea mariana). Smaller amounts of birch,

 

fir, cedar, and larch are also present along with thick-
ets of high shrubs. Ground plants consist of ferns,
grasses, and mosses.

Geological Setting of the Marcell Experimental
Forest Region

 

Earlier geological studies relating to the Mar-
cell Experimental Forest have been restricted to inves-
tigations that are regional and of a reconnaissance
nature. These studies supply no direct information on
the surface and subsurface geology of the experimental
forest. However, it is possible, using both general
considerations and specific information from this study
to present a general framework of the geological struc-
ture and history of the region.

Little is known about the bedrock formations of
the area other than it has an obvious complex Precambrian
history. The surface geology and topography is the
result of recent glacial activity.

Seismic refraction studies support regional
findings by previous geologists that the bedrock is
igneous or metamorphic. Regional geologic maps indicate
the bedrock is Ely Greenstone of Precambrian age. Seis-
mic and gravity data indicate that the bedrock surface
is irregular and suggest that bedrock topography consists

of low hills and shallow valleys similar to the exposed

 

 

ODpA.

Univ-
5

0
III'.
\

"o.

'“a.

"A .

VII.

 

I
u
-

 

N.

12

areas of the Precambrian shield in northeastern Minnesota.
The data suggest that the maximum relief of this bedrock
surface is of the order of 130 feet. Although bedrock
topography influences glacial activity and the resulting
surface features on a regional basis, it is unlikely

that local bedrock topography is reflected in local
surficial glacial features.

It is possible that pre-Wisconsinan drift covers
much of the area, but if so it lies deeply buried beneath
younger drift and little is known of its composition and
extent. This drift would likely be a thin veneer lying
directly on top of the Precambrian surface. The Marcell
area was influenced by two major glacial events during
Wisconsinan time. In earlier Wisconsinan time, perhaps
14,000 years ago during the Cary substage, two lobes
from large glacial masses to the north had converged on
this region. To the northwest, the Wadena Lobe moved
inward from southern Manitoba to be met by the south-
westerly traversing Rainy Lobe which emanated from the
Patrician ice center north of Lake Superior (Wright and
Ruhe, 1965). This location could thus have been alter-
nately covered by Wadena and Rainy Lobe ice so that the
drift would reflect contributions from both lobes.

The second and final event occurred during the
Mankato substage about 12,000 years ago. At this time,

the large St. Louis Sublobe moved in from the northwest

Jul

.
I“

'11;

 

‘vH

-..

'1.

 

'e

I.“

I.“

 

 

 

 

 

13

following approximately the route of the earlier Wadena
Lobe. At this time its progress was not blocked by ice
to the east and so the lobe was able to move southeast-
wardly, nearly as far as Duluth (Wright and Ruhe, 1965).
Contributions from this lobe are represented by the upper
veneer of drift in the Marcell region.

The overlying glacial drift has, according to
seismic refraction data, an average thickness of 228 feet.
Seismic data on individual thicknesses range from 118 feet
to 389 feet. Drilling and seismic refraction data together
with surface geologic investigations indicate the drift
consists of several units. A thick layer of compacted
till consisting of sand and gravel in a clay matrix rests
directly on top of the bedrock. Only one exploratory
well was drilled into this layer and it did not reach
bedrock. Samples from this well show the compacted till
to be mixed with significant amounts of sand, and the till
is interpreted as ground moraine. Although seismic
refraction data do not indicate the universal presence of
this layer, its continuous extent throughout the region
is geologically plausible and can be reasonably inferred.

A thick layer of sand and gravel overlies this
clay. This sand is medium to coarse grained and well
sorted having a thickness from around 20 to over 100 feet.
A relatively thin layer is often found at the top of the

drift having thickness varying from a foot to over nine

14

feet. In some locations, this layer consists of sand with
gravel and small boulders, and in other locations it is
nearly pure clay. Although this upper layer of till is
not found everywhere in the region, it is prevalent

enough to be considered as a mapable unit.

The surface topography is an almost continuous
series of steep-sided flat-topped hills indented with
many kettle-like depressions which vary from nearly cir-
cular to very irregular in shape. Some of these depres-
sions display no surface drainage, and many of the larger
ones contain lakes or swamps. The tOp of the hills is
smooth and flat, and their elevations are nearly the same
resulting in a level skyline. This skyline has a gradual
upward slope toward the northwest for about five miles
where it meets the Marcell Hills, a prominent northeast-
southwest trending moraine. A schematic diagram of the
geologic structure in the experimental forest region is
given in Figure 2-3.

This information suggests that the basal clay
layer is overlain by a sandy outwash plain with numerous
ice block depressions. This plain is covered by a veneer
of ground moraine. In short, it represents an overridden
pitted outwash plain resting on ground moraine till. The
general upward slope of tOpography toward the northwest
indicates that the outwash was formed from a glacial lobe

coming from the northwest.

15

 

 

 

SURFACE TILL

 

OUTWASH

KETTLE DEPRESSION

 

 

 

 

 

 

 

 

BROWN SAND AND GRAVEL

 

 

COMPACTED TILL

 

 

 

 

 

 

 

 

 

 

CRYSTALLINE BEDROCK

lllllll

 

Figure 2—3

Schematic Block Diagram of the Surface
Geology in Marcell Experimental Forest

 

 

16
Glacial History of the Experimental
Forest

The glacial history in this region is undoubtedly
complex and little detailed work has thus far been done.
It is possible, nevertheless, to postulate some general
conclusions.

Most of the basal clay till was probably deposited
in early Wisconsinan time, perhaps on a thin veneer of
pre-Wisconsinan drift. As mentioned before, the Marcell
area is in the region of convergence of both the Rainy
and Wadena Lobes. Samples from the lower sand and the
underlying compacted till obtained from an exploratory
well drilled near bog 8-3 are grey colored and contain
carbonate rock fragments. These samples contrast with
the carbonate free brownish drift found near the surface.
The grey calcareous drift indicates a glacial source
from Mesozoic sediments to the east. With this evi-
dence, the most reasonable assumption is that glacial
drift has been supplied to the region from both the
Rainy and Wadena Lobes.

The upper layer of outwash sand was formed at a
later date and appears to have been formed by a glacial
lobe that moved in from the northwest. This would have
to be the St. Louis Lobe of the Mankato substage of
Wisconsinan time. This lobe must have remained stationary
at the site of the present Marcell Hills moraine and sup-

plied outwash to the southeast. The presence of ice-block

l7

depressions indicate that the lobe was receding. There
is no evidence that the Marcell Hills were being formed
by this lobe contemporaneous with the formation of the
outwash plain, and it is possible that the moraine was
formed during an earlier stage in the development of

the St. Louis Lobe. If the Marcell Hills are contem—
poraneous with the outwash plain, then this moraine must
be recessional and not terminal.

The thin layer of drift above the outwash sand
suggests a final southeasterly re-advance and deposition
of ground moraine by the St. Louis Lobe. If the ice
block depressions in the outwash plain had remained ice
filled during this re-advance, then the ice could have
travelled over this flat, gently sloping surface without
altering its tOpography.

A chronological chart outlining the geologic
history of the Marcell Experimental Forest region is

presented in Table 2-1.

18

Table 2-1 Glacial Events in the Marcell Experimental
Forest Area

 

 

ELAPSED
cgggégL gaggéigE OCCURRENCE TIME SINCE
OCCURENCE
WISCONSINAN MANKATO ADVANCE OF =12,ooo
ST. LOUIS YEARS
SUBLOBE
WISCONSINAN CARY ADVANCE AND =14,000
CONVERGENCE YEARS?
OF RAINY AND
WADENA LOBES
PRE-WISCONSINAN DEPOSITION >40,ooo

OF A THIN LAYER YEARS
OF DRIFT ON
CRYSTALLINE

BEDROCK?

 

CHAPTER III

GEOPHYSICAL METHODS

The high expense of detailed exploratory drilling
for the purpose of defining the subsurface hydrologic
characteristics made direct subsurface investigation
impractical. Therefore, surface geophysical methods were
used as an alternate approach. Geophysical methods are
most effective in providing geological information where
it is possible to calibrate geophysical data with direct
geologic control. Seismic refraction, earth resistivity,
and gravity methods were the geophysical methods used
in this study, and it was possible to calibrate seismic
and resistivity data with geological data at the site of
an exploratory well (upland well 305) in the experimental
forest. Comparisons between geological and geophysical
data at this site are shown in Figures 3-1 and 3-2.

Seismic refraction data provided the bulk of the
information for geological interpretation. Resistivity
data were employed as a very limited aid in suggesting
the presence of a clay layer undetected by the seismic

method. Gravity data were used with the aim of obtaining

19

20

 

 

 

 

 

 

 

 

 

 

UPLAND
DEPTH DEPTH WELL 305
(>r o'-
vpa :355 "I...
. . MIXED SAND
2° - 20 AND
vp- 5I70 "me GRAVEL
4O '- / ‘0 "
60 P 60 -
80 I' 30 L
fut
IOO -
Vp' 6446 "1...
l20 -
I4£Ib
ISO -
ISO -
200 B Vp' l4883 "Inc
feet

HORIZONTAL DISTANCE

9 . I900
feet

Figure 3-1

Comparison of Seismic Profile 3 with
Drilling Data from Upland Well 305

 

 

21

 

Apparent Resistivity, p xIO‘ ohm-cm UPLAND
WELL305

I 23 45 67 8
5'00"“. I I I I I I I
Spoflng

20-

 

MIXED
SAND
AND

40 _ BRAVE

60-

 

 

80 -
IOOr
I20?
I40 -
I60-
I80-
200 -
220)-
240 -

260 -

 

280 L
toot
Figure 3-2

Comparison of Resistivity Profile 3 with
Drilling Data from Upland Well 305

 

 

 

22

information on bedrock tOpography, but these results were

inconclusive.

Seismic Refraction Methods

 

General Considerations

Seismic refraction proved to be the most valuable
geophysical method employed in this study for determining
subsurface geology. The results of refraction data are in
the form of compressional wave velocities for the subsur-
face layers and depths to layer interfaces. The velocity
data can be translated into lithologic information where
geologic control exists.

The seismic apparatus used for this study was an
Electro Tech Model ER-75-12 Porta Seis recorder and a
1100 foot cable with leads for twelve Model EVS-4B 7 1/2
cycle geophones along the cable. The Porta Seis recorder
is constructed to record the first arrival compressional
waves generated when an explosive charge is detonated in
the ground. The cable was placed linearly along the
ground, and dynamite charges were successively detonated
at each end. The Porta Seis recorded the travel time
required for the first arrival pulse to reach each geo-
phone relative to the time of shot detonation. This
information was then plotted as a series of points on a
graph of time (ordinate) versus distance (abscissa).

These points form a series of line segments whose slope

 

.
we
8.

 

,l..'

0.1

 

.,.,
.
a...‘

3“

‘-

s
'-

 

I.“

 

23

decreases with distance from the origin to give the stand-

ard type time-distance graph of refraction seismology.

The data on this graph could then be processed to obtain

layer velocities and depths to velocity discontinuities
within the near surface. These data were then translated
with correlative geological information into geological
and hydrological information about the subsurface. A
brief discussion of basic seismic refraction theory
aPpears in Appendix II.

The theory for seismic refraction calculations in

a layered medium is based essentially on four assumptions
(EWeing, Woollard, and Vine, 1939): (1) Each layer is
bouruded top and bottom by planar interfaces and transmit
SeiSmic waves at a constant velocity. (2) Seismic waves
are :refracted at interfaces according to Snell's law.
(3) Interchanging shot and recording points does not
alt£ir seismic wave travel time. (4) The seismic velocities
ofthe layers increase with depth. Of these assumptions,
thfii first and to a lesser extent the fourth were most open
‘33 question in this study. Glacial deposits characteris-
tiCally change lithology within short distances, and inter-
faces in the drift may be irregular. In addition, situ-
ations may occur where velocities decrease with depth.

In Spite of these difficulties, however, the seismic

rEfraction method was employed with considerable success.

 

24

The above four assumptions serve as the basis

for the develOpment of elaborate computational procedures.
In the case of two layers containing a horizontal inter—
face, computations are relatively simple. The computa-
tions become increasingly involved as the number of hori-
zontal layers is increased. In addition, further complexi-
ties arise where the interfaces are inclined. Figure 3-3
shows a time-distance graph for two layers with a sloping
interface. It is necessary, in this situation, to obtain
a reverse profile for correct results. A reverse profile
Consists of recording the first arrivals at geophones
alcuig the spread from individual explosions detonated at
either end. Established computational procedures using
the=data from the reversed time-distance profile give the
inteerface depths and true velocities for each of the
diFuping layers. Many situations occurred in this study
Where four layers with dipping interfaces were encountered.
Thfi! large number of computations and chance of cumulative
e11‘1‘0r made it advisable to seek a digital computer solu-
'tiom for the seismic data, and, to this end, a four layer
Computational program was employed for the solution of
shallow seismic refraction problems up to four layers

(Johnson, 1964).

The results of refraction computations are

expressed in the form of a vertical profile along the

line of the seismic cable. The profile shows the seismic

25

p—zu

OODMHODEH mcflmmflo m cuflB muwhmq 039
mcflumuumsHHH maamonm cmmuo>mm m mo cacao mocmpmfloumefle

mum musmflm

woz<hm_o

 

I
I
I
#L
b-Sm

UN) I—Iu

 

 

 

26

compressional wave velocities for each layer and the depth
and inclination of interfaces between these layers along
the plane of the profile. As mentioned earlier in the
chapter, calibration of the seismic method was made
between a geologic profile based on well data from the
exploratory well (upland well 305) and a seismic profile
(profile 3) taken directly adjacent to it (Figure 3-1).
Calibration was accomplished in two ways. First, a com-
parison was made between depths to lithologic interfaces
hi the well profile and depths to velocity interfaces on
Una seismic profile. Second, compressional wave velocities
were: correlated with corresponding lithologic units in
the 'two profiles. These comparisons resulted in the
following correlation between lithologies and compres-

Sional wave velocities at the well site:

Unsaturated sand
and fine gravel . . . . 1,355 ft/sec

Saturated sand
and fine gravel . . . . 5,170 ft/Sec

Saturated compacted

till consisting of

sand and gravel in

a clay matrix . . . . 6,446 ft/sec

Igneous or meta-
morphic bedrock . . . . 14,883 ft/sec

'rhis correlation served as a basis for interpreting lith-
ologies from compressional wave velocity data in the rest

of the area of this study. The average and the range of

my

Iv

R
.JD

 

27

velocity values obtained from reversed profiles are corre-

lated with each lithology in Table 3-1.

The Hidden Layer Problem

 

The seismic procedures thus far considered assume
an increase in the velocity of each layer with depth. The
case where velocity decreases with depth at an interface
is called a velocity reversal. If a velocity reversal
occurs, the low velocity layer directly below the interface
will not be detected, and observed refraction data from
belrnn this interface will be in error.

A small portion of the surface watershed of bog S-3
conSiisted of a surface layer of nearly pure unsaturated

Clai' of undetermined thickness. From Table 3-1, the aver-
age ‘value of compressional waves in this medium is 1,722
ftZ/Sec while the average value of compressional waves in
unsaturated sand and fine gravel is 1,418 ft/sec. Thus,
if this surficial clay overlies unsaturated sand, a
Velocity reversal exists. This reversal would obscure
the unsaturated sand and give anomalous depths for lower
interfaces. It was not necessary to consider this prob-
lenh however, due to the fact that this thick surficial
clay occupied only about one per cent of the area of the
bog‘s surface watershed. Thus, any error introduced would

have a negligible effect on modeling computations.

28

Table 3-1 Average Value and Range of Values for Com-
pressional Wave Velocities Correlated

to Each Rock Type

 

 

Average
veloc1ty of Range of Number of
Rock type compre551ona1 veloc1t1es
samples
wave ft/sec
ft/sec

Unsaturated sand 1,850
and 1,418 to 19

fine gravel 1:050

1,870
Unsaturated clay 1 , 72 2 to 3

1,545

Saturated sand 5 , 780
and 5,228 to 20

fine gravel 3,935

Séturated compacted 8 , 291
tlldl consisting of 6,960 to 19

Sand and gravel in 6 , 122

a c lay matrix

I‘I-Ineous or 22 , 425
metamorphic 15 , 704 to 28

edrock 10 , 618

\

29

The Blind Zone Problem
At an interface between two strata recognized on

a seismic refraction time-distance graph where velocity
increases with depth, it is possible that a third layer

of intermediate velocity may exist undetected directly
above this interface. That is to say, the refracted first
arrivals from the intermediate layer will arrive slightly
later than those from the deeper layer. These later
arrivals cannot be detected in seismic refraction as the
inStruments in this case are designed to detect only the
first arrival compressional pulses. The blind zone refers
to tflne maximum thickness that this undetected layer may
haVEB, and this zone is a complex function of the velocity
and (depth of the deeper layer and the relative velocities
0f tdne three layers (Soske, 1959; Hawkins and Maggs, 1961).

The blind zone problem was of Special interest in

thifis study because of the possibility that a saturated
layer of impervious compacted till of significant thick-
neSS could exist undetected between bedrock and the sand
acluifer. As a result, the aquifer's thickness and cor-
reSponding transmissibility would be less than that indi—
cated on the refraction time-distance graph. As an example,
Figure 3-4 shows a time-distance graph suggesting a hori-
zOntal three layer case. Here a lOO-foot thick layer of
Saturated sand is found to overly bedrock at a depth of

200 feet. Figure 3-5 shows the first arrivals from a

3O

 

 

0.20 - T
‘-—---—‘——-—I— I”’ |
I“ I703 sec ,v” ' I500
o. I 5 x 5000
Time KJZTI 00¢
' 0. l 0 -
'" __I_ (0)
Seconds I500
0.05 '-
I I l l g
ICC 200 300 400 500

Distance in feet

 

. :lInsaiurated Sand 3 2 . _' ' . _ .‘ '
' . ': Isooftmc 2 '. '. .j- - lam."
MI " ' 1' " 2;.,'-,'.'. (b)

. Saturated Sand ”5 “Ref . I51I:%
‘ei'5°°° "4:; as: 1:": ' .. f'fia'm feet

 

(a) Time-distance graph of a three layer case that
does not consider a blind zone layer possibility.
(b) Profile showing interpretation of the graph in (a).

Figure 3-4

Seismic Refraction Interpretation Without
Blind Zone Considerations

31

A ’
BLIND ZONE L YER /

I 0F LESSER

BLIND ZONE
6500 THICKNE} /65—|;LAYER WITH

- AX. THICKNESS
I/‘/ i OFSBFEET
o 20 - /' -/
. I/- T
l/

4.. ’/' _l.._

’ ”,r ' I5000
0.! 5 r- —/ 5W

.I426see.
Time
in OJ 0 — (0)
Seconds |
Isoo
0.05 ..
l 4 1 1 J

 

 

I 00 200 300 400 500
Distance in feet

v7

- Unsaturated Sand.- _' _- :O
I50“0"I/sec . .. _. . 2 . I Oteet

 

“'> 2 1“» c 4Ine bfd I: I-> :M‘ >
"L‘4"A’L‘ ,:0 ON. f'. TOO 75:47‘01‘q:
"'<v<‘fl‘v‘v:< 7'604300 I..c.:L"v v<L7<viv

(ii) Time-distance graph of Figure 3-4 taking into consider-
ation a 6,500 ft/sec blind zone layer.

Profile showing the interpretation of the graph in (a)
for a blind zone layer of maximum thickness.

(13)

Figure 3-5

Seismic Refraction Interpretation
with Blind Zone Considerations

 

 

;.‘O*fl
.bUVV

 

'F‘y
w‘.

Mr.

.“P
‘I‘
di.‘

l
“A“

id.

e...
'4

e.._

‘3.“

Op‘

e.“

~\

I.

‘w

'4

'r

32

blind zone layer of impervious till as a dashed line seg-
ment with slope 1/6500. This line segment could never be
detected on the basis of first arrivals from the blind
zone layer for it is coincident with or occurs after the
refracted first arrivals from other layers. The segment
for a layer of maximum thickness is shown as tangent to
the time-distance plot with an intercept time of 0.1425
Seconds, and a higher intercept time for this line segment
would have indicated a lesser thickness. Calculations
indicate the maximum thickness in this case to be 68 feet.
The thickness of the saturated sand has been decreased
by 50 feet and the depth to bedrock has been increased by
18 feet. All bedrock depths recorded in this study were
those from seismic calculations that did not consider
the blind zone problem, as any change in bedrock depth
brought about by the presence of a blind zone layer had
no direct bearing on this study.
The blind zone layer can exist at any interface
Where there is an increase in seismic velocity with depth.
This study, however, considered only the situation of a
Clay layer existing in the particular blind zone at the
Saturated sand-bedrock interface. Consideration of blind
z°ne layers above any of the other interfaces would have
produced results that were geologically unlikely or would

have had no direct bearing on the results of this study.

33

Consideration of the blind zone problem does not

in itself indicate the actual existence of an additional

layer. Supplemental data is necessary to determine

this, and in this respect, two methods were used. First,

seismic refraction profiles disclosed a layer of com-

pacted till overlying bedrock in a large part of the

bog's upland surface watershed. It is geologically palusi-

ble to infer the existence of this layer as a blind zone

layer in adjacent areas where it does not appear on a

refraction profile. Second, visual comparisons of graphs

of apparent resistivity versus electrode Spacing were made

between areas where the compacted till existed and areas
where it could be inferred as a blind zone layer. Simi-
larities brought out in comparing these graphs suggested
the existence of a blind zone layer of compacted till.

We have thus far dealt only with the existence of

a horizontal blind zone. layer above a horizontal inter-

face, and all known methods for blind zone layer computa-

tion limit themselves to this situation. However, the

ac3tua1 interfaces encountered in this study were inclined.
Therefore, it was necessary to develop a method for
determining the minimum depth to the upper surface of an

a‘ssumed horizontal blind zone layer for seismic profiles

showing inclined interfaces. The complexity of calcu-

la"tions increases for dipping interfaces, suggesting the

Ge‘IelOpment of a computer based method for the solution

34

of this type of problem. Again, the concern was only in

determining the minimum depth to the top of the blind

zone layer and thus the minimum thickness of the sand

aquifer.
In develOping the computational procedure, the

assumption was made that the upper surface of the blind
zone layer was horizontal and that the true velocity of
this layer was 6,500 ft/sec--the approximate mean velocity
for saturated, compacted sand and gravel in a clay matrix.

The first step was to calculate the apparent

Velocity of the blind zone layer. This was accomplished

bY means of a computer program developed from the basic

seismic program mentioned earlier. This program is listed

in Appendix III. The slopes of the reciprocals of these

apparent velocities were then plotted on a transparent
Overlay at the same scale as the observed time-distance

graph. The relative placement of these two line segments

Was guided by assumption (3) of the four basic assump-

tions. That is, interchanging shot and recording points

does not alter the travel time of the refracted seismic

wave. Thus, the maximum time for each reciprocal velocity

line segment should be the same, and their respective
0
'upper" time intercepts were accordingly placed at the

Se~l'rle time on the overlay. The overlay was next placed

over the time-distance graph with the respective time axes

of the overlay directly superimposed on those of the

35

initial graph. This overlay was then moved downward,

keeping the axes parallel, until one of the two line
segments of the overlay became tangential with its cor-

responding time-distance graph at the point of inter-

section of the l/V2 and l/V3 segments. The two line

segments of the overlay were then extended to their

respective time axes on the observed time-distance graph

to give the intercept times for the blind zone layer

(Figure 3-6) . With the intercept times and apparent

Velocities now known, the depth to the upper surface of
th£a blind zone layer could then be computed considering
the blind zone layer as the bottom layer in a three layer

CElse. The depth computed in this manner represents the

minimum depth or maximum thickness of the blind zone

laYer .

msmic Results

The results of the seismic refraction studies

are tabulated in Appendix IV as a series of profiles

listed by number. All profiles are included with the

eRception of profiles eight and ten. Profile eight is

located in the north unit of the Experimental Forest

about five miles from bog S—3 and supplies no informa-

tion directly pertinent to the study. Profile ten,

although located near bog 8-3, was very shallow and

s‘~1£>plied questionable information. A map of the south

36

 

 

 

 

 

14%

RECIPROCAL OF APPARENT
VELOCITY OF HORIZONTAL/

BLIND ZONE LAYER,” /

 

 

av ’ ’ ” ’ /
’ POINT OF
INTERcEPT :, ‘ ’/ /
.n ME , TANG Y
\ mn’acspr
\‘ \ / TIME
\Nt/ ’/
/ //’/\\ l
,/ x \S~.
/, / I' \ -—
/ / \VZD
‘\
\
\
\
_._ 2‘.
Va ‘

Figure 3-6

Overlay Method for Finding the Intercept
Times of a Horizontal Blind Zone Layer

n.‘ ‘U
Hub

.w-b

0 ‘ vu'

 

L...
P!
hay.
F

‘ e

 

-
Ce

egv‘

e.‘V

 

g

37

unit of the Experimental Forest is included showing the

numbered location of each seismic profile. In profiles

where the existence of a blind zone layer is considered,

its position is indicated on the profile by a horizontal

dashed line .

Earth Resistivity Methods

Earth resistivity measurements were made at twenty

0f the seismic profile sites. Resistivity measurements

Utilized the Wenner configuration and an instrument manu—

fflatured by W.G. Keck and Associates. Direct current was

employed and spacing between collinear electrodes varied

between a minimum of from 2 1/2 to 20 feet with spacings

increasing to 280 feet. An explanation of the Wenner and

Lee Partitioning configurations used in this study and a
1”l’l‘ief discussion of the elementary theory of the method

are given in Appendix II.
The purpose of earth resistivity interpretative

procedures is to determine the resistivity and depth of
as'Sumed horizontal layers within the earth. A variety of

theoretical and empirical methods have been developed to

achieve this purpose, and five of these methods were

tesated in this study. They met with very limited success.

Matching data of the study with theoretical curves
(MQOney and Wetzel, 1956) gave unrealistic results as did
a Itlethod deve10ped by Sanker-Narayan and Ramanujachary

(1967). A method of graphing summed apparent resistivities

.i‘s-

... e

d‘v

 

«A-

rot.

ll)

 

Jud

‘Io'u

‘u

 

 

\.l

\f‘

‘1

.‘ ‘
‘.e ‘F

38

(Moore, 1950) also gave inconclusive results. Conse-

quently, interpretation was limited to qualitative site
interpretations of apparent resistivity versus electrode
Spacing and qualitative use of the method of layer

resistivity interpretation developed by Barnes (1952) .

These results are shown as profiles in Appendix IV. A

computer program for determining apparent resistivity,

summed apparent resistivity, and Barnes layer resistivity

is given in Appendix III.
The region of study is made up of essentially five

different soil or rock types. These are unsaturated sand

and gravel, saturated sand and gravel, unsaturated clay,

saturated compacted till consisting of sand and gravel in

a Clay matrix, and igneous or metamorphic bedrock. In

order to interpret resistivity data from a geologic view-

point, it is necessary to relate electrical resistivity
There are a variety of factors that

Values to lithologies.
Two of

affect resistivity values for different rock types.

the most significant factors in this study are the effects

of water and clay. The overlying unsaturated sands and

gravels and the crystalline bedrock would be expected to

have the highest resistivity. The saturated sands in the

water table zone would have a somewhat lower resistivity

due to the presence of ions in the water. Saturated clays

would be expected to have the lowest resistivities due to
their ion exchange capacities. These considerations aided

i . . . . .
1'1 the reSist1v1ty interpretations of the study.

e-

we-

 

 

I

39

Graphs of apparent resistivity and Barnes layer

resistivities plotted against electrode spacing can be

classified into three general types. The first type is

shown schematically in Figure 3-7 and is representative

of resistivity profiles 6 and 9. This first type of

curve was obtained when the resistivity spread was

located near the bog. Here the low values of apparent

resistivity and of layer resistivity were caused by the

Electrolytes in the bog water acting in effect as a short

circuit between the electrodes. The second type of curve,

which is indicated schematically in Figure 3-8 and is

representative of resistivity profiles 12, 15, and 29.

This curve indicates an increase in apparent and layered

resistivity with increased electrode spacing. The geo-

logical interpretation of this type of graph is unclear,
but seismic results associated with this type shows that
the saturated zone is generally higher in velocity and

thus perhaps is related to a decreasing porosity in the

s aturated layer .
The third and most commonly observed resistivity

ellI‘ve, which is represented in Figure 3—9, showed low
val-lies of apparent and layer resistivities at small
e1ectrode spacings with an increase in these values at
intermediate spacings. This increase was followed by a
drop in resistivity values when Spacings became large.

Resistivity values on this type of curve were consistently

4O

Apparent Resistivity, pXIO‘ohm-cm
4 8 l2 IS

I I I I

 

O
O
h 1

I;
O
I

I60)-

200 _

EIOCII'OUO Spacing in Feet

240 -

 

280b

Figure 3-7

Profile 9
Apparent Resistivity

41

4
Apparent Resistivity, pXIO ohm-cm
4 8 l2 I6

I I

 

4o_

80-

200 -

Electrode Spacing in Feet
3

240»

 

280b

Figure 3-8

Profile 15
Apparent Resistivity

42

4
Apparent ResistivitprIO ohm-cm

 

 

4 8 I2 I6
40 -
‘2; 80 -
.2
.5
0 I20-
I:
’8
a ,
a) l60+
fl
'0
2
13 200 -
.2
m
240 ..
zeoL
Figure 3-9
Profile 17

Apparent Resistivity

43

higher where resistivity spreads were located over thick
unsaturated sands (profiles 5, 7, 11, 13, 16, 17, 20, 21,
22, and 23) than where the profiles were located closer
to the water table (profiles 3, 14, 18, 25, and 27).
The source of the low resistivity values at small
electrode spacings is possibly due to a higher electrolyte
content in the near-surface soil water. Comparison of
these graphs with associated seismic profiles suggests
that the higher intermediate depth resistivities can be
associated with clean sands and gravels of the water-
bearing aquifer. The drop in resistivity at larger
selectrode Spacings can be associated with an underlying
layer of compacted till in three seismic profiles
(profiles 3, l6, and 17). This correlation suggests
tflnat seismic profiles showing no layer of compacted till
overlying bedrock but having a resistivity profile of
t1'lis latter type could have compacted till within the

blind zone at the glacial drift-bedrock interface.

Gravity Methods
Reduced gravity data can be used to determine
tirends in bedrock topography where a relatively constant
density contrast exists between bedrock and the over-
Jflying earth materials. A gravity survey, consisting of
1373 gravity stations, was conducted in the region of
S5tudy utilizing a Worden gravimeter. The observed gravity

Iaeadings were converted to gravity units and corrected

44

for instrument drift, differences in station latitude,
and differences in station site elevation and interven-
ing mass. A correction was also made for the effect of
the regional trend on gravity patterns in the area of
study. These corrected gravity results are shown in a
residual Bouguer gravity anomaly map (Figure 3-10) . This map
shows the region of the survey within the area of the
south unit of the Experimental Forest. A simple Bouguer
map with no correction for regional trend is shown in
Figure 3-12. This map covers the entire area of the
gravity survey.

The area of study consists of low density glacial
drift overlying higher density crystalline bedrock. If
we make the assumption that the density of the drift
and that of the bedrock are constant throughout the area,
tZhe bedrock topography should be reflected in gravity
aI'lomalies recorded at the surface. That is, where the
bedrock surface is at a higher elevation, corrected
gravity readings will be greater due to the local increase
in the amount of mass. Similarly, the local lack of mass »
oVer a bedrock valley will be reflected in lower corrected
gravity readings. Thus bedrock hills will stand out on
a residual Bouguer gravity map as positive gravity anoma-
lies, and bedrock valleys will be associated with negative
gravity anomalies. The residual Bouguer gravity map can

be used in a qualitative sense as a tOpographic map of

45

 

 

 

      

EXPLANATION
E]
cum eec‘neu
emu can”

poem ITEM '0.‘ MIMI

 

Figure 3—10

Residual Bouguer Gravity Anomaly Map

 

46

 

 

 

 
 
  

E XPL ANAT ION
E] EEJ
caveman accrue
an acne:

CW” QTEI‘L ' 50 FEET

 

Figure 3-11

Bedrock Surface Topography Based on
Seismic Refraction Data

 

47

 

/ 5.0 N

 
 
 
 
  
  

T.59N.
T58 N.

R. 25 W. R.24W.

 

EXPLANATION

CONTOUR
INTERVAL
0.4 HILLIOALS

C3 amwnvsnwmu

SCALE
:2
O 0.25 0.6 "6
MILES

 

I
I
I
I
I
I
I
\
\

I
'e‘
I
I
I
I.
t
I
t
I
I
I
t
I
\
\

 

 

Figure 3-12 Bouguer Gravity Anomaly Map

 

138

I
"A I
:flv-
Iiod

4’
l \
4

env-

 

I

l
I...

v'y‘

ta]
'VI

9v

“A
‘H

48

the bedrock surface. However, this map also reflects

variations in bedrock and glacial drift density, and,
therefore, must be interpreted with considerable caution.
A comparison of the residual Bouguer map with a
map of bedrock topography based on seismic refraction
data (Figure 3-11) suggests a rough correlation between

a negative gravity anomaly and an east-west bedrock

valley that traverses beneath bog S-3. However, this

correlation is not strong, and there is in general little
similarity between the two maps.

It is significant to note that the bedrock
‘topography does not enter into the subsurface hydrology
<>f the area because the compacted till overlying the bed-

Jsock is effectively an impervious boundary to the flow

<>f ground water in the sand aquifer. Furthermore, the

topography of the upper surface of compacted till appears

'Uuarelated to the bedrock topography.

CHAPTER IV

COLLECTION AND INTEGRATION OF GEOPHYSICAL,

GEOLOGICAL, AND HYDROLOGIC INFORMATION

This chapter outlines the significant geological

and hydrological information acquired and utilized in

this study. It also includes a discussion of the geo-

logical interpretation of geophysical data and the inte—

gration of this with all geohydrologic data.

Hydrologic Data
A large amount of hydrologic information had
already been collected prior to this study, and this
information was supplemented by additional hydrologic

information taken during the study.

2% Ground Water Wells

Five upland ground water wells were installed

‘9‘ . .
‘1";th the upland watershed prior to the study (upland
‘9
$1.13 301, 302, 303, 304, and G-5), and monthly readings

at
t these wells supplied data on water table fluctuations.

49

F
t
e

 

~V

.0!

‘A_‘

50

Five additional upland ground water wells were installed

during this study (upland wells 305, 306, 307, 308, and

309). With the exception of well 305, these upland wells

are partially penetrating and penetrate the aquifer to

a depth of only a few feet below the water table. Well

305 is a fully penetrating well and, as such, extends to

the bottom of the aquifer. Well 305 is a continuously

recording well, and the other wells supply data through

monthly readings. The locations of these wells as well

as adjacent ground water wells within the area are shown

in Figures 2-1 and 4-1. Water level observations in

these wells provide the basic data for the construction

Of a regional water table map that shows general trends

of ground water flow.

Mmentation Within Bcg S—3

Bog S-3 contains eleven ground water wells, and

data from four of them (bog wells 1, 6, 9, and 11) were

“36d for analog modeling (Figure 4-1). Bog well 1 is a

continuously recording well while data from the remain-
ing ten wells are in the form of water level readings
taken at intervals of about six weeks.
Early in the study, a drainage ditch was con-
structed at the outlet of bog 8-3 to collect the outflow
of Water and thus make it possible to measure the sur-

:6
ace outflow from the bog. A weir and continuous water

51

 

 

 

 

 

I
I
D
I
MT E -- Leer or noon an eoe 3-3
I
I

 

 

Figure 4-1

Location of Bog Wells and Adjacent
Upland Ground Water Wells

ZOO-I

 

52

level recording gauge were installed for the purpose of

making these measurements. The gauge recorded the level

of the water at the weir, and a calibration was made

between the rate of flow of water through the weir and

water height at the weir. This calibration was accom-

plished by United States Forest Service personnel by

using a current meter to record the rate of flow through

the weir for various water heights. The calibration is

Presently between weir outflow and the water level in

bog well 1, the continuously recording well. The

accuracy of this calibration is unknown, but the relation-

Ship is clear.

T. emEerature and Precipitation

Precipitation was measured by a non-recording

rain gauge situated close to bog S-3, and data was taken

on cumulative precipitation at approximately weekly

intervals. In addition, precipitation as rainfall was

Ineasured at a continuously recording rain gauge situated

approximately one-half mile from bog S-3. Measurements

of Continuous precipitation were of value in this study

in the modeling of storms. Temperature was also recorded

on a continual basis. The method used was to take the

a"erage of the highest and lowest temperatures that

o .
ccunred during the interval between measurements.

Temperature data served as the basis for calcu-

l -
a“ti-on of evapotranspiration by the Thornthwaite method

53

(Thornthwaite, 1957). Potential evapotranSpiration

values were computed for the bog, as there is essentially

no depletion of soil moisture. Actual evapotranspiration

was calculated for the upland area utilizing a correction

for soil moisture depletion given by Thornthwaite.

Bog Transmissibility and Storage
Hydrologically, the surface peat of bog 8-3 is
highly permeable and has a high storage coefficient.

This upper peat grades rapidly downward through peats of

decreasing storage capacity and permeability. The

decrease in storage capacity and permeability is suffi-
Ciently great that, for the purpose of this study, the
Peat may be thought of as consisting of two layers.
These are: (1) an upper layer that is highly permeable
With high storage, and (2) a lower layer that is
essentially impermeable with very low storage. D. H.
Boelter (personal communication) estimates the thickness
of the upper layer for bog 8-3 to be 6 to 8 inches with
a permeability of s x 10'2 cm/sec (1.06 x 103 gal/day/ftz)
and a storage coefficient of 0.52. Lower horizon data
from two different sites in bog 8-3 show permeability
values of 6.0 X 10"4 cm/sec (12.73 gal/day/ft3) and
l - O X 10-5 cm/sec (0.212 gal/day/ftz) and storage

coefficients of 0.20 and 0.15 respectively.

uni!” —MI “‘

54

Lumping Test Determination of Aquifer
'I‘gansmissibility and Storage

The nonequilibrium pump test (Theis, 1935) was
employed to determine aquifer transmissibility (T) and
The pumping test was performed at well 305

storage (S) .

located 100 feet northwest of bog S-3. Ground water flow

in this region is directly toward the bog. The well is

45 feet deep and fully penetrating with a five-foot

screen at its lower end. Pumping was accomplished by

means of a gasoline-powered pump whose rate was periodi-

cally adjusted to maintain a constant discharge. This

discharge was measured by noting the time required to
fill a container of known volume, and a pumping rate was

maintained at values very close to an average of 56.6

gallons/minute. The pMped water was drained into the

hog, and it is unlikely that any significant amount of
this water re-entered the aquifer as recharge in the

vieinity of pumping. Two satellite wells were placed

at distances of 20 and 60 feet to the northwest of the
Well to measure drawdown during the test.

Theoretical methods of analysis for pumping tests

re(ll-lire conditions that are not fully realized in actual

s:'-"l‘-I.‘tations. Nevertheless the Theis method is widely

e . .
InE31oyed and gives reasonable results. The conditions

0
f this pmnping test were as follows:

1. Water table measurements at well 305

indicate that the water table level was

55

remaining constant immediately prior to
the test, and it can be assumed that
there was no recharge into or leakage
from the aquifer into the aquiclude of
compacted till.

2. Sampling suggests that the aquifer is
homogeneous and extends from the pumping
site for a considerable distance in
terms of the thickness of the aquifer.

3. The variation in saturated thickness due
to pumping was relatively small so that the
transmissibility remained essentially
constant.

4. It is assumed that most of the water in
storage was released rapidly during the
pumping test. In the unconfined situation,
a time lag in the release of water from
storage may lead to a deficient storage

coefficient.

The test had a duration of seventeen hours and
tWenty-five minutes (1045 minutes). Downdraw was
measured by a tape graduated to the hundredth of an inch
with measurements being estimated to the nearest thou-

Esiilitith of an inch. Maximum drawdown at the 20-foot well
‘T’ET‘B 0.453 feet and at the 60-foot well, 0.386 feet. The

E3 .
£3Illts were plotted, and values of S and T were obtained

56

by the Theis method. Data from the satellite well 20
3

feet distant indicated a transmissibility of 82.76 X 10

gal/day/foot and a storage coefficient of 0.07 at the
60-foot well, results indicated a transmissibility of

72.2 X 103 gal/day/foot and a storage coefficient of 0.05.

Permeability is calculated by dividing transmissibility

by saturated aquifer thickness (45 feet); thus values
3

for the 20- and 60-foot wells are respectively 1.84 X 10

EiaIL/day/ft2 and 1.61 X 103 gal/day/ftz. The storage

Coefficients of 0.07 and 0.05 were assumed to be somewhat

low due to the relatively short duration of the test, and

an arbitrary value of 0.1 was assigned for storage. This

Value was used for initial modeling.

These values for permeability and storage are
cor“parable with published data from aquifers of similar

lithology, but the storage coefficient is slightly low.

Surface Geological Data

It was possible to make inferences on the hydrology

and geology of the region of study by the use of surface

geologic data. Included in this category is sampling of

Surface soils and tOpographic data.

SEliEEEace Tepography
Local and regional topographic information was

a“Ogliired in three different ways. It was possible,

t
hrough working in the area, to gradually obtain a

57

knowledge of regional and local tOpographic patterns by

direct observation. This direct knowledge was supple-

mented by an examination of stereo pairs of aerial photo-
graphs and of detailed topographic maps. Topography
determines surface drainage patterns, and this knowledge

served as the basis for a first approximation in delimit—

ing the surface watershed for bog S-3.

The topography of this region is mainly the result
015 glacial processes, so that topographic patterns could

be interpreted geologically in terms of glacial forms

and modes of origin.

fllr face Soil Examination

Soil information was obtained from casual observa-
tion of the surface soil and from materials obtained in

Shallow seismic shot holes less than ten feet in depth.

Subsurface Geological Data

Subsurface geological information from the area
0f study was in the form of geophysical data and explOra-

tory well data .

%bysica1 Data

A panel diagram (Figure 4-2), showing compres-
SiOnal wave velocities and interfaces, is the basic means
for presenting and relating the seismic profiles. This
pa~Ile1 diagram, as first constructed, did not include

b .
J~J~nd zone layers at the lowest (crystalline bedrock)

58

Figure 4—2

Seismic Refraction Profile
Panel Diagram

59

 

 

 

 

 

 

 

.‘II/ J \\T‘§ \
3/ ‘
. I/
I :‘I!
All ,‘l‘
:y 1!:
,‘O‘
I
)I
am; urea-cum /
met /
um um
cue: mm nevus
\
\‘.

SCALE I—

 

   

vnncn 1‘

III —
e an ace tee "IV

'I‘

re ,b-}.'I¢‘DVIL

hand—e

- - - —
O ‘00 em bit-9 IIIV

 

/

 

 

 

60

interface with a velocity corresponding to compacted till.

It did, however, clearly show the areal extent and thick—

ness of the compacted till where it appeared on seismic

profiles, and this compacted till was then postulated

to be present in the adjacent areas as a blind zone layer.

Resistivity profiles provided supporting evidence for

the blind zone layer in these adjacent areas, but it is

emphasized that this evidence is not conclusive.

Gravity data appeared in final form as a residual

Bouguer map. If it is assumed that the densities of
Crystalline bedrock and glacial drift remain constant
throughout the area, then this map will reflect trends in
bedrock topography; and the map can be used to check bed-
reek tOpography determined from seismic refraction pro-
fj-Z|.es. Comparison of data from the two sources shows
only limited correlation suggesting that the density
contrast between the bedrock and glacial drift is not

constant.

B\°S¥Well Coring Data

A series of bog water table wells were installed
in S-3 by United States Forest Service personnel prior to
this study, and drilling of these wells was extended
through the bottom of the peat to determine bog thickness.
This information is shown as contoured iSOpach map of
the bog in Figure 4-3. The bog is also represented in

three dimensional form in the panel diagram of Figure 4—2.

61

mum wen we on: rerOmH

m-e deemed

 

 

 

huquOOOOnOONOO. 0 00.

w 440%

9—3 4...: h4
8. no agxgh
g 802,100.. g
C“: 4.5. 8.

graiaaxw

hum... n u J<>¢wbz. 505.200

 

 

62

For simplicity in diagramming, the bog is shown with a
level surface and a constant thickness of 28.5 feet, the
maximum depth recorded from coring. The panel diagram
shows that the upland aquifer is thicker than the bog and
continues beneath it. Thus ground water can flow beneath

the bog as well as through it.

Exploratory Well Data

 

A deep exploratory well (upland well 305 whose
location is shown in Figure 4-1) gave direct subsurface
information on the aquifer and underlying compacted till
aquiclude. This information was of value in correlating
rock type with seismic velocity, thus permitting a geo-
logic interpretation of the seismic panel diagram. This
well, a driven well, was drilled near bog S-3, and its
casing penetrated to a depth of 72 feet. The casing was
subsequently raised to a depth of 46 feet and then instru—
mented to become a fully penetrating well that continuously
recorded the ground water level. Well samples were taken
at regular intervals during the drilling, and a log was
maintained describing the collected samples (Figure 4-4).
Examination of these samples showed that the penetrated
section of drift was composed of three distinct layers:

(1) an upper layer 35 feet thick consisting of brown sand
with isolated fine gravel pebbles with the upper parts
of this layer containing some interstitial clay, (2) an

underlying 12-foot thick layer of grey sand with some

63

 

 

DEPTH FEET WELL LOG
UPLAND WELL 305

SURFACE ELEVATION
I370 FEET

WATER TABLE
AT n FOOT DEPTH
SAND, BROWN

SAND, BROWN
WITH PEBBLES

' SAND, GRAY WITH PEBBLES
AND LIMESTONE
FRAGMENTS

COMPACTED TILL, GRAY

CONSISTING OF CLAY,

SAND, PEBBLES,

AND LIMESTONE
FRAGMENTS

 

 

Figure 4-4

Log of Upland Well 305

 

 

64

gravel, and small fragments of carbonate rock, and (3) a
deeper grey compacted till consisting of clay, sand,
pebbles, and small fragments of carbonate rock. The well

bottomed on this compacted till.

Several shallower wells were drilled in the
region to supply data on ground water levels for this
study. These wells penetrated to only a few feet below
the water table, and samples from them consisted mainly
of brown sand mixed with small pebbles. The location
of all ground water wells in the area of study is shown
in Figure 2-1.

Intggration of Hydrologic and
Geological Information

The collected hydrologic and geologic informa—
tion was combined to provide the geohydrologic setting
of the area to serve as a basis for construction of the
analog model. This integration may be separated into
essentially two phases. The first phase was concerned
with developing the effective limits of the bog's upland
watershed, and the second phase was directed toward the

develOpment of an aquifer transmissibility map.

Development of Watershed Limits

It was evident from the start of this project
that available data contained very little information
on the boundary of a watershed draining into bog S—3.

An estimation of watershed limits was obtained by

Aka
a:

R‘

C.

.I

 

l-i

nil
RN“

Po-

“5

no I

In

 

ILL

65

searching for divides by direct examination of measured
water—table elevations and by looking for topographic
divides with the assumption that water table divides would
follow topographic divides. This information was supple-
mented by construction of a flow line diagram to delineate
the watershed boundaries of hog S-3. These flow lines were
developed from a contour map of water table levels deter-
mined from a set of roughly synchronous well measurements
supplemented by surface topography and geophysical measure-
ments. These methods were considered satisfactory for
determining the southwest and northeast limits of the
watershed. The northwest boundary of the watershed,
however, remained in doubt. A series of exploratory wells
were drilled at increasing distances to the northwest.

They failed to indicate a local ground water divide sug-
gesting that local surface water divides are not reflected
at the water table. The ground water divide appears to

be located a considerable distance to the northwest, per-
haps as far out as the continental divide approximately
five miles away. It was advisable to at least estimate,

as a first approximation, the effective northwest limits
of the watershed. The most distant well in this direction,
upland well 309, is in the vicinity of Burrows Lake

(Figure 2-1) whose water table appeared to be higher than
the regional water table, suggesting that the lake was
perched on a clay layer. The effective northwest limit

for the watershed was placed between well 309 and the

66

southeast boundary of the lake. This choice was based

on the assumption that surficial clay beneath the lake
would at least partially divert incoming ground water
away from the bog. These limits for the watershed became
the limits for building and initially testing the analog
model.

Qevelopment of the Aquifer
Transmi§sibility Map

 

It was necessary to prepare a map showing trans-
missibility values within the watershed to serve as a
basis for initial construction of the analog model. There
is a considerable amount of uncertainty in the construc-
tion of such a map, for transmissibility has been
directly measured at only one location in the watershed.
This measurement was made by means of the pumping test
conducted in upland well 305. An attempt was made to
determine transmissibility values throughout the rest of
the watershed by projecting transmissibilities outward
from this known value by means of geophysical data.
Seismic refraction data aided by resistivity findings
proved to be the best method for this purpose.

The thickness of the saturated layer was first
determined from seismic profiling. Saturated layer
thicknesses at both ends of a profile were averaged and
the value placed at the location of the spread's center.

This, then, represented the saturated thickness in the

'11.

1
CE;

67

vicinity of each spread. In regions where seismic and
resistivity data suggested a clay blind zone layer,
the thickness of the saturated layer was reduced by an
amount equal to the calculated maximum thickness of
this blind zone layer.

The geological nature of the aquifer and drilling
results both indicate that the aquifer is relatively
homogeneous. Accordingly, the upland aquifer was assumed

3 gal/day/ftz,

to have a constant permeability of 1.771 X 10
the average of the results of the pumping test at well 305.
The transmiSsibility at various locations could then be

calculated by multiplying 1.771 X 103

by the aquifer's
saturated thickness, as seasonal variations in the water
table are small enough to assume that saturated thickness
remain constant. A simple analytical method devised by
Theis was used to subdivide the area into units of constant
transmissibility. This method divides the watershed into
a series of polygons, each enclosing a location where
transmissibility has been determined. These polygons were
constructed by erecting perpendicular bisectors to line
segments joining adjacent known transmissibility sites.
Each polygon contained one central point of measured
transmissibility, and any point within the polygon is

closer to this central point than to any other location

of measured transmissibility.

“O
NH

in

CE!

68

This information served as the basis for the
construction of the transmissibility map which appears
in Figure 4-5. A cross sectional diagram through the
central part of this area is shown in Figures 4-6 and its

location in the area is shown in Figure 4-7.

69

 

 

 

TRANSMISSIBILITIES
FOR THE MODELLED PORTION OF THE
AOUIFER IN GALLONS/DAY/FOOT

--L.TUMA.A ——muu.q

/
SCALE
I:
m 0 m m
MT

  
 

 

 

 

 

 

 

 

I ”5.040
I
| s
I 2 I
II 2
I T
“0.7 5
.0 O
N
IIOJO
‘
I
/
\ ‘il
I, ‘ : II"
II I I /
I I
: \ , , /
I. \\ '27.” [I /
.3.23° l’l .’.2’° \‘--‘ ’1’ I. /
’ 9:0,..‘s I, /
\\ (I _—- ’I,” -._"a— ~’
\ " ‘~\ I 1”
\ \_r
\ /
\ /
\ m:
\‘\ /
‘~ I /
Figure 4-5

Aquifer Transmissibility Map

70

 

 

50-

I00 '

 

 

3r
,.,

I I I I I l
ISOO 3000

   
  

ih’yfi'fi -~. ,. n.4, ,
firt$$ #6345?!" "p
{:6 Ain‘t: .
Kit-‘25

 

 

‘ CRYSTALLINEBEDROOK,

 

Figure 4-6

Cress Section Through Bog S-3
and Adjacent Upland

 

71

 

 

 

 

ZOO-I

 

 

no one ,
'll? . --Leeroreaeuuu :3 Ice e-s

 

 

 

 

Figure 4-7

Location of Cross Section Through Central
Part of the Area of Study

 

CHAPTER.V

THE PHYSICAL ANALOG MODEL

The basic theory of analog modeling, presented in
Appendix I, provides the foundation for the construction
of the analog model employed in this study. In the
modeling, there is an analogy between prOperties in both
systems (Table 5-1). More specifically, these analogous
properties are related by constants determined by model
construction. There is an exact analogy between the
hydrologic system and the electrical system when the
model is continuous. That is, the model is composed of
solid resistive material that has a continuous capacitance.
It is much easier, however, to construct a model of dis-
crete resistors and capacitors as an approximation to
the continuous model, and this was done in this study.

This chapter describes the analog model together
with the methods and assumptions employed in its con-

struction.

72

73

Table 5-1 Analagous Properties Between the Hydrologic
and Electrical Systems

 

Voltage I Water Table Elevation
or Piezometric Pressure
Quantity of .
. ‘——————q>
Electrical Charge Quantlty of Water

Flow Rate of Water

 

<————-—e> -
Current (Recharge or Discharge)
Time in Tim 'n D s
H
Seconds e 1 ay
TransmiSSibility
Capacitance .¢———————> Storage

 

The Finite-Difference Approximation
to the Continuous System

 

 

The finite-difference approximation replaces the
continuously distributed system with a network of discrete
elements in a manner such that the characteristics of
the two systems are essentially the same. The advantage
of employing a finite-difference electrical model in
this study is the reduction of material costs, the greater
ease of construction, and the simplification of any
later modifications that might be required in the model.
The methods used in this study are based on the discus-
sion of finite-difference approximations by Karplus
(1958). The substitution of a finite-difference approxi-
mation for the continuously distributed analog introduces

errors from this approximation process. However, by

74

making the spacing between discrete elements sufficiently
small in relation to the entire system, errors from the
approximation are reduced to the point where they may be
disregarded.

The finite difference model may be visualized
through a hydraulic analogy suggested by Karplus. We
first consider a continuous fluid flow field within a
thin sheet of porous medium. If this medium is replaced
by a grid of small tubules of properly chosen diameter
interconnected at right angles, it will act as a finite-
difference approximation to the continuous medium. Fluid
can be made to flow between any two points in the model
by adjusting the pressure distribution, but actual flow
can occur only in the X or Y directions. In the elec-
trical analog, the conductive sheet of resistivity p is
replaced by a grid network of discrete resistors in such
a manner that the resistance between any two points in
this network is essentially the same as that between
analogous points in the conductive sheet. Thus, identi-
cal voltage distributions in the continuous and finite
difference systems will cause analogous current flow
between corresponding points in both situations. How-
ever, while current can flow in any direction in the
continuous system, it can only flow in the X and Y
directions in the finite-difference approximation. That

is, each resistor represents the resistance of the

75

continuous sheet in a specific direction. The generalized
formula for calculation of the resistance of a discrete
resistor representing the resistance in the z direction

of a rectangular section of continuous material of length

i, width w, and resistivity p is

R2 = 3,}. (5-1)

The model used in this study simulates current
flow in the X and Y directions for a continuously dis-
tributed conductor of resistivity p. In this case,

equation (5-1) assumes the form

x AY (5-2)

for the resistance Rx of the resistor simulating the
resistance in the X direction of an element of length AX

and width AY. The equation

R = Pl—— (5-3)

gives the value of the resistor simulating the resistivity

in the Y direction of an element of length AY and width

AX. These equations define the values of resistors

within and at the boundary of a network (see Figure 5-1).
The collected data of this study indicate that

transmissibility varies within the upland aquifer, and

most likely these variations are not discrete but rather

“Y 'P AY I + -.

M I «:0:\I (a)

 

 

so‘j (b)
1 $3er

(a) Within the network
(b) At the network boundary

Figure 5-1
Representation of a Continuous Conductive Sheet

of Resistivity p by a Network of
Discrete Resistors

77

gradational. In Chapter 4 the development of the initial
transmissibility.map was discussed. A simplifying assump-
tion is made there that instead of a continuous distribu-
tion of transmissibility existing in the upland aquifer,
it is composed of a mosaic of interconnected segments,
each with a constant representative transmissibility.

This approximation is reflected in the analog model as

an analogous mosiac of resistive networks with the value
of the resistivity in each segment of the mosiac deter-
mined by Equation (I-11) in Appendix I.

Discrete capacitors in the finite difference
model simulate the capacitance of a continuously dis-
tributed dielectric. The value of the capacitor is
determined by the area of continuous dielectric that the

capacitor simulates according to the formula

K
c = 7.48 aZS —3 (5—4)
K2

where a2 is the number of square feet in the hydrologic
system whose storage (S) the discrete capacitor represents.
S is defined in Figure I-1, and K2 and K3 are defined in
equations I-7 and I-8 of Appendix I. The model was built

2

so that one inch of the model represents 1.5 X 10 feet

in the bog-watershed system. From this it follows that

a2 equals 2.25 X 104

times the number of square inches of
continuously distributed dielectric that the capacitor

represents (Figure 5-2).

78

 

(a) A discrete capacitor representing a square ele-
ment of continuous dielectric one node spacing
on a side.

(b) A discrete capacitor representing a square ele-
ment of continuous dielectric two node spacings
on a side. Everything else being equal, its
capacitance would be four times that of the
discrete capacitor in (b).

Figure 5-2

Representation of Continuous Dielectric
Material by a Discrete Capacitor

79

The continuous precipitation minus evapotranspira-
tion (P - Et) input was modeled with a series of discrete
current-conducting elements entering selected nodes, with
input nodes representing a specific area of input in the
continuously distributed analog. The amount of current
entering any particular node was determined by the rela-

tion
i = ——-a (5-5)

where the current i is in amperes and the recharge Q is

in gallons/day/ftz. K3 is defined by equation (I-8) of
Appendix I. Again a2 equals 2.25 X 104 times the number
of square inches of area on the surface of the continuous
model that the discrete input represents. The expression
a2 Q then represents the intensity of rainfall in gallons/
day over the analogous area in feet2 in the bog-watershed
system.

Develgpment and Construction of the
Analog ModeI

 

The analog model was constructed on a framework
of masonite pegboard mounted on a portable stand. This
pegboard supported a regular array of resistors and
capacitors with input current leads to selected nodes
simulating the input of (P - Et). Photographs of the

model appear in Figures 5-3 and 5-4.

 

80

 

Figure 5-3

The Electrical Analog Model with Peripheral
Electrical Equipment

81

(a)

5'.
v“. x}.

”716 'h‘b ’.

a

x

 

1.34» ,:

 

Resistance Components of the Electrical

Analog Model

w

 

 

Raised Area

1 Analog Model.
fer Underlying the Bog

ica

i

Represents Aqu

Capacitors of the Electr

Figure 5-4

The RC Portion of the Model

82

Scale Factors

The choice of values for resistors and capacitors
in the model was determined by the values assigned to the
four scale factors in equations (I-6) through (I-9) of
Appendix I. The values chosen for these constants was
influenced by several considerations. It was advisable
to use resistors and capacitors that were commercially
available at a reasonable price, and it was also necessary
to represent both the low transmissibility of the bog and
the high transmissibility of the sand aquifer within the
same model. In addition, it was essential to use voltages
and current that available instrumentation was capable
of handling. Values thus chosen for initial modeling
were determined in a manner similar to that used by

Walton and Prickett (1963) and are as follows:

13

K1 = 9.027 X 10 gal/coulomb (5-6)
K2 = 4.000 ft/Volt (5‘7)
K3 = 2.473 x 1010 gal/day/amp (5-8)
K = 3.650 x lo3 days/sec. (5-9)

The relationship between resistance and transmissibility

(T) is then represented by

83

6.183_X 109

R= T

 

(5-10)

where T is defined in Figure I—l of Appendix I. The
relationship between capacitance and storage is expressed

as

c a 3.314 x 10"10 aZS. (5-11)

The storage coefficient for the upper layer of the bog was
taken as 0.52, and S for the upland aquifer was taken as

0.1. Thus for the bog

c = 1.723 x 10'13 a2, (5-12)
and for the upland aquifer
c = 3.314 x 10‘14 a2. (5-13)

Aquifer transmissibilities varied from 6.375 x 104
gal/day/ft to limited areas that were considered to be
essentially impermeable. Bog transmissibility was taken

as 6.183 X 102 gal/day/ft. From these values, the

3

resistances used in the model ranged from 39 X 10 ohms

to 22 x 106

12

ohms while capacitor values ranged from 1,000

12 farads. Schematic dia-

x 10' farads to 15,000 x 10'
grams of the resistor and capacitor network comprising

the model are given in Figures 5-5 through 5—9.

84

 

 
  

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89

Use of Two Different Grid Sizes

 

The diagrams of the analog model show that two
different mesh sizes were used in the modeling procedure.
There was less concern for detailed relationships in the
outer regions of the watershed. Thus, the resistor grid
size in this region is twice as large as in the interior
portion of the model. Methods for varying grid size are
found in Karplus (1958). Capacitance grid size was also
varied, and larger capacitors were used to represent
larger segments at the modeled aquifer's periphery and
other selected regions where there was less concern for
detail. The size of the capacitor is determined by the
value of a2 assigned to equations 5-12 and 5-13.

The Current Input Representing Pre-
cipitation Minus EvapotranSpiration

 

 

The bog-watershed system represented by the model
is of small areal extent. Thus it is reasonable to assume
that the rainfall pattern throughout the region is repre-
sented by data from the rain gage located approximately
in the center of the watershed. Evapotranspiration
values within the bog, however, differ from those in
the upland region. If we consider (P - Et) as a single
value, then we find two separate annual patterns for
(P - Et), one for the hog and a second for the surround—
ing upland. In addition to these, there are other com-

plicating factors that give different precipitation input

9O

patterns for the bog and upland aquifer. More specifi-
cally, all precipitation falling on the bog enters it
directly, but there is a time lag in the upland region
between when precipitation enters the ground and when it
reaches the water table. Further, it is important to
consider the possibility of surface flow and interflow
in the upland area. For purposes of modeling, it was
assumed that these separate input patterns for upland and
bog were constant over their respective areas at any given
time. This is a good assumption for the bog, but is less
valid for the upland region, where variations in the
thickness and permeability in the glacial drift above the
water table affect the amount of water reaching the water
table at different locations. Lack of specific data makes
it not feasible to incorporate this effect in the model-
ing.

The two input patterns were incorporated into
the model by having two separate current input sources,
one representing (P - Et) for the bog and the other
representing (P - Et) for the upland watershed. With the
numerical evaluation of K3 in equation (5-8), (P - Et) is
now represented by current input into the model according
to the relation

2

i = a s9 10. (5—14)
2.473 x 10

 

91

Ideally the current input would be continuous over the
entire model as the (P — Et) input is continuous over the
bog-upland aquifer watershed region. In the actual model,
the current enters at discrete points as an approximation
to a continuously distributed input with the amount of
current entering at a given input proportional to the
area this input represents. The process of varying cur-
rent input with time was achieved through an arbitrary
function generator that was constructed Specifically to
vary voltage with time. This generator is described in

a publication by the Tektronix Corporation (1965). To
insure a reasonably uniform current input into the model,
a twenty-two megohm resistor was placed between each cur-
rent input and the arbitrary function generator which
served as the current source. Two of these generators
were constructed with one representing the (P - Et) input
for the hog and the other representing this input for the

upland watershed.

Initial and Boundary Conditions

The analog model, as discussed in Appendix I, re-
flects a second order differential equation and as such
requires one initial value and one boundary value condition
for a unique solution. These conditions are difficult to
prescribe for the model due to a lack of detailed knowledge
of the regional drainage pattern and of the limits of the

effective watershed of the bog. An attempt was made

92

to approximate these conditions by grounding the periphery
of the portion of the aquifer under the bog which repre-
sents the outflow area. Each point of the remaining
periphery of the modeled aquifer was left unconnected as
an Open circuit. The hydraulic analogy to these conditions
is to imagine the watershed perimeter corresponding to

the ungrounded portion of the model's perimeter as having
a cement-filled trench dug around it that extends to bed—
rock. Thus the only incoming water would be that supplied
by precipitation within the watershed. If this perimeter
actually coincides with a water table divide, then the
insertion of this trench would have no effect on flow
patterns or water table levels in the region. The model
was then used to test the validity of these watershed
limits by inserting a current pulse equivalent to an
annual (P - Et). Voltage measurements indicated that the
model was reflecting a water table gradient much smaller
than that actually observed in the bog-watershed system.
In other words, much more water is flowing through this
watershed area than just that being supplied by rainfall
within the area included in this first approximation. The
limits of the actual watershed then is much greater than
those determined by the methods described in Chapter 4.

It is likely that water flowing through the bog area

comes from as far as the continental divide approximately

five miles away. This observation necessitated changing

93

the model's boundary conditions. A constant voltage was
applied at the northwestern and northern part of the
model to simulate ground water inflow into the region and
to give a voltage gradient that was more in accord with
the observed water table gradient. This voltage input
should actually oscillate slightly to correspond to
fluctuations in inflowing water due to annual variations
in precipitation in the outer areas of the watershed.
These fluctuations, small in comparison with the total
volume of inflowing water, reflected seasonal variations
of the water table within the modeled region of the bog-
watershed system. It is postulated that the damping
effect of slow infiltration of water through the unsatu-
rated zone and the slow release from storage of water
from the outer areas of the watershed would greatly
diminish these fluctuations. They would accordingly have
an insignificant effect on seasonal water table fluctua-
tions from direct precipitation within the modeled section
of the watershed. Therefore, a constant boundary current
input is considered to be a sufficiently accurate means
for determining initial and boundary conditions within
the model over short periods (i.e., one or two years).
The boundary conditions around the model thus remain
constant. The initial conditions represent the water
table levels existing when there is no (P — Et) current

input into the model. This condition is most closely

94

realized in the bog-watershed system in late March or
early April when water tables are at their annual low
levels. This time of year accordingly represents a good

initial time for modeling sequences.

Modeling the Bog-Aquifer Boundary

Data and observations at bog S-3 indicate that
it overlies a layer of sand aquifer varying from approxi—
mately 20 to 80 feet in thickness, and most of the ground
water entering the bog region passes through the highly
transmissible sand aquifer directly below S-3. Since
there is continuous outflow from the bog and the bog has
a much smaller seasonal variation in bog water levels
than diSplayed in the "perched" type of bog, the assump-
tion was made that there is a continuous flow into the
bog. The amount of water entering the bog was assumed
for modeling purposes to be limited only by the bog's
transmissibility. This situation was reflected in the
model by having the RC network representing the bog
placed in parallel with an underlying RC network repre-
senting the sand aquifer layer below the bog. This under-
lying network is continuous with the main portion of the
sand aquifer network and is connected to the bog's net-
work on three sides. Most of the electric current passes
through the underlying aquifer network with as much current
flowing through the bog network as its resistance will

permit.

95

A cross-section of the model at the bog-aquifer
interface and a cross-section of the simplified hydrologic
bog-aquifer interface that it represents is presented in
Figure 5-10. It will be noted in the hydrologic analogy
that water enters the swamp at the same piezometric pres-
sure as it enters the underlying aquifer. Similarly in
the model, current leaving an aquifer node at the aquifer-
bog interface enters both the bog and underlying aquifer
at the same voltage. Figure 5-11 is a schematic diagram
of the actual type of bog-aquifer interface according to
available data.

These diagrams represent the interface where water
enters the bog. At the "lower" or southeast end of the
bog, there is a much higher clay content in the upper
drift. This clay tends to block ground-water outflow from
the upper permeable layer of the swamp channeling it into
the outlet weir as surface discharge. A diagram of the
bog interface in this region is given in Figure 5-12.
Modeling Weir Outflow and
Bog Upwelling

Weir outflow was measured in the model by connect-
ing a low ohmage resistor between the modeled swamp outlet
and ground. Current was measured by noting the voltage
drop across this resistor and then converting it to out-
flow by equation 5—8.

Ground water upwelling in the bog has been observed

and verified by piezometric measurements in one location

96

Modeled Bog—Aquifer Interface

AQUIFER RESISTOR BOG RESISTOR
\ NETWORK ( NETWORK

    

‘

AOUIFER RESISTOR
NETWORK

Hydrologic Analogy of Bog-Aquifer Interface

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Figure 5-10

The Bog—Aquifer Interface

97

 

Figure 5-11

Northwest Border of Bog S-3

 

Figure 5-12

Southeast Border of Bog S-3

\
\
\
.x.‘

x\}\\\\\\\\

98

of the bog near its outlet. This is the only observed
bog upwelling, but it is likely that there are other
undetected areas. This upwelling was represented in the
model by a 200 kilohm resistor connected between the bog
and the underlying aquifer in approximately the same loca-
tion as the observed upwelling. Current through this
resistor was converted to upwelling recharge by equation
5-8. In final modeling experiments, this current repre-
sented about one-fourth the upland contribution to bog

discharge.

Accuracy in Model Construction

The relationships and equations expressed in 5-6
through 5-14 of this chapter and 6-1 through 6-6 of Chap-
ter VI have constants containing up to five significant
figures. This does not imply that the model represents
this degree of accuracy as data on which construction is
based is limited and has many uncertainties. Further,
electrical components that were readily available for
model construction have specified values with tolerances
up to i 20 percent, and in many cases these Specified
values are not exactly equivalent to values prescribed by
the equations. The model is quite adequate for a study of
this nature, however, and precision components of exact
specifications would not significantly enhance modeling

results.

CHAPTER VI

RESULTS OF ELECTRIC ANALOG MODELING

Introduction--Verification of
the Model

 

The electric analog model, whose construction was
discussed in the previous chapter, was assembled on the
basis of aquifer characteristics of the upland region
determined from geological and geophysical studies and on
data from the bog collected by U.S. Forest Service per-
sonnel. These data are quite limited and subject to errors.
Accordingly, the next step in the modeling procedure was
to verify the model. Verification was accomplished by
comparing modeled responses to known inputs with actual
measurements. Model responses consist of upland and bog
water table levels and their fluctuations and of bog sur-
face discharge rates. Model inputs, which simulate
recharge, are in the form of three current inputs. These
are: (l) the steady-state current representing recharge
from the regional watershed, (2) a time variable current
pulse that provides recharge from precipitation on the

immediate portion of the watershed represented by an RC

99

100

circuit, and (3) a second time variable current pulse
that represents (P - Et) within the bog. The effects of
these inputs on model response can be observed separately
or in combination, and this feature was an aid in con-
struction of some of the figures in this chapter.
Comparisons between modeled and measured responses
can be used to verify the model and alternatively to
suggest model modifications thus testing the first approxi-
mation of the hydrologic characteristics of the region.
As the model's capability to reproduce the hydrologic
history of the region increases through modifications, it
becomes an increasingly better predictive instrument for
determining the effects of various components of the
hydrologic cycle. Modeling covers the periods from April
l968--March 1969, April l969--March 1970, and April 1970--
March 1971. In addition, modeling covers three separate
storms which occurred in 1969. The use of the verified
analog model as a predictive tool is illustrated with a
study of the effects of a hypothetical year of high pre-

cipitation.

Modeled Recharge

 

Recharge from the Outlying Regional
Watershed and an Estimation of the
Total Watershed Area
The upland area represented by the RC network of

the model is only a small portion of the total area from

101

which ground water flows into the bog region. Recharge
from this outlying area is simulated by constant current
supplied to selected locations at the outer edge of the
upland RC network. This current is adjusted to give
modeled voltage gradients in the upland RC network that
correlate with the water table gradient based on early
spring upland water table measurements. Available field
measurements show very little difference in early spring
upland water levels from one year to the next.

An estimation was made of the size of the watershed
region that lies outside the area of the model as repre-
sented by the constant current which is supplied to
fixed external points in the model (Figure 5-5). The
location of three current leads in the northwest corner
of the model is based on ground water flow path information
derived from drilling data. The single current input in
the northern part of the model is used to bring modeled
water levels for upland well 303 up to reasonable agree—
ment with known values. This steady-state current was
converted into its equivalent in recharge by the relation-
ship defined by relationship 6-3, and the value obtained

was 103.4 x 107 gal/year or 1.43 X 106

gal/day. This
inflowing ground water comes from precipitation in the
outer region. Model studies and the use of field measure-

ments of precipitation adjusted by Thornthwaite (1957)

values of evapotranspiration indicate that infiltrating

102

recharge from precipitation in the upland area amounts to
about ten inches per year. Ten inches is equivalent to
27.2 X 104 gallons of recharge per acre. Thus a value

of 3,800 acres was obtained for the watershed region out-
side the model. Adding the 348 acres of modeled watershed
and bog, the total bog-upland watershed area is roughly
4,150 acres with bog S-3 comprising 48 acres of this
amount (Figure 6-1).

Recharge from Precipitation in
the Upland Area

A current pulse simulating recharge from precipi-
tation is next superimposed on the constant current
within the RC network of the upland portion of the model.
This pulse is adjusted so that it produces modeled upland
water table fluctuations that match measured fluctuations
at upland well 306. At this well site, there is an unsatu-
rated zone of about sixty feet above the water table.

The current pulse, determined in this manner, produces
modeled water table fluctuations that are in reasonable
agreement with measured values at the other well sites
(upland wells 302, 303, 305, 307, and 309) (Figure 4-1).
The current pulse is converted into its recharge equiva-
lent and compared with recharge calculated to be available
from measured precipitation in the area. Values for
recharge from measured precipitation are determined by

subtracting evapotranspiration estimates from measured

103

 

Upland Area

Represented by

the Constant Current
Input. The Specific
Shape of this Area

is not known.

,’ Upland Arg ‘

I, Represented
by the RC
\\ Model

I

 

1 Mile

Figure 6-1

Schematic Diagram of the Relative Areas Represented
by the Resistance-Capacitance Section of the
Model and by the Constant Current Input

 

 

 

104

precipitation using a method of evapotranspiration esti-
mation developed by Thornthwaite (1957). The effect of
deficiencies in root zone soil moisture in the upland
area are taken into account in the calculations. Thorn-
thwaite calculations are based on average effects of
temperature, type of vegetation, available soil moisture,
and amount of sunlight. Therefore, in this specific case,
the calculations are subject to a certain amount of error.
They are assumed, however, to be sufficiently accurate to
give meaningful estimates of evapotranspiration. Figures
6-2, 6-3, and 6-4 show measured water table fluctuations
at selected well sites (upland wells 302, 303, 305, 306,
307, and 309) and modeled fluctuations for the same sites.
Figures 6-5,-6-6, and 6-7 show modeled patterns for re-
charge at the water table and estimates of recharge avail-
able at the surface. Modeled recharge at the water table
is both delayed and spread out which reflects the infil-
tration rate and depth of the saturated zone. Most of the
potentially available recharge enters the ground in April,
but modeling results show that recharge arrives at the
water table in increasing amounts with maximum rates
occurring in May or June. If there are no more months of
heavy precipitation, recharge rates gradually decline
throughout the reaminder of the year and until the follow-
ing spring. It would be expected that the thicker the

unsaturated zone, the longer would be the time required

 

ELEVATION
FEET

. a~u . ‘————l _—_ o o x _ —_—f
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/ *9
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' fl
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Am MAY Jo» JUL Mac n! oer NOV 0:: JAN no RM

105

 

 

 

 

 

 

 

 

 

 

 

 

 

EXPLANATION

 

 

 

UPLA n we“ 30
metal; awn: “It!
lawman

3’]

arm” we“. %
0 ma 1'
IVAN”:

 

 

 

Figure 6-2

MEASURED AND NOOELEO UPLAND WATER TAaLE ELEVATIONS
APRlL l, l908 - MARCH 3|, l969

106

IT-"'___——_'—_T

“JV‘T’OM
“IT
I,“ - -

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

l9“ -'

[”4 "

 

' - r v v V V ' i V ‘7 fl ‘
”0 APR MAY Jun an. an up oer an 01c «mu m M

EXPLANATION

 

E309

 

 

 

UPLAND WELL '9’
MODELS WATER ME
EL 'ATION

 

30,

 

 

 

mun mu 399
mm Imam
ELEH‘TION

 

 

 

Figure 6-3

MEASURED AND MOOELED UPLAND WER TABLE ELEVATIONs
APRIL I, l989 - MARCH 3|, mo

 

107

 

 

 

”VAT"
FEET
I!“ - -
I!“ M x
— ’0 -
"-——_. _ J’o‘
- -./.,_——--—-—... I. , ".2.
I!” - _,
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m an an m m Ir oer M n: M "A MR

 

EXPLA NAT ION

 

m
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vat AID “ILL 39’
Non I no MAT" Tau
ELIVATION

 

m
_......f.

UPLAND WELL 309
MEASURED M1“ TABLE
ELEVATION

 

 

 

 

Figure 6-4

KAOURED AND MODELS!) m HTER TADLE ELEVATION:
APRIL I, Inc - NARCH 3|, l97l

 

108

 

MODELED RECHARGE AT THE WATER TABLE

TOTAL
5.7a meals

 

 

 

 

 

 

 

 

 

 

'APA MAT m'am Au ur'ocr Mw uc'm m MAIT‘

PRECIPITATION AVAILABLE FOR GROUND
134 ‘WATER RECHARGE CALCULATED BY ‘

 

Xi: THE THORNTHWAITE METHOD
\ *“amu. -

 

I4. ‘D INCHES

 

2.75

 

 

“3 (b)

mun U
Ann 8

//‘\ / ... -

 

 

 

 

 

 

 

J.
1

 

APR m an IIDL'Au'm Der nv‘m'm'm MRI

Figure 6-5

UPLAND GROUND WATER RECHARCE
APRIL l,l968 - MARCH 3|,l969

 

 

 

109

 

 

MODELED RECHARGE AT THE WATER TABLE

TOTAL

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

unfl ".38 INCH“

2.. — (a)

f

l- _

O T MN MW JUL A06 55 DC? AM one our ' m ‘ MAR '-

W“ PRECIPITATION AVAILABLE FOR GROUND

5' ,5, WATER RECHARGE CALCULATED BY ‘

«3 THE THORNTHWAITE METHOD
1- ‘ -
6- -
TOTAL.

5_ \ Io.16 INCH“ _

4- LSNOIIFALL _

3' ‘ (b)
2- _

I- ..

O- ..

 

 

 

M! m'm'Au'up ocr'm'ou'w'uu'm‘

Figure 6-6

UPLAND CROUND WATER RECHARCE
APRIL l,l969 - MARCH 3|,l970

 

 

110

 

 

MODELED RECHARGE AT THE WATER TABLE

TOTAL
5. 7B INC N‘S

2' ' (a)

 

 

 

 

 

 

 

 

 

 

j

APR MAY «on M Au‘up 'ur a» ‘7“ In. 'un'm

PRECIPITATION AVAILABLE FOR GROUND

 

,, m WATER RECHARGE CALCULATED BY -
KQI THE THORNTHWAITE AETHOD

TOTAL
I SAID-mu. 7.02 INCHES -

 

- (b)

 

 

 

 

 

u: 'JAAI'ru 'IIIAR‘

Figure 6-7

UPLAND GROUND WATER RECHARGE
APRIL I, I970 - OCTOBER 3|, I970

 

 

111

for recharge to infiltrate to the water table. Water
table measurements at upland wells for the year 1969
support this fact. The water table at upland wells 302
and 305 is about ten feet below the surface while the
water table at upland wells 303, 306, 307, and 309 is at
fifty to sixty feet below the surface. Water levels at
upland wells 302 and 305 show a sharp response to April
recharge and peak around the end of May. Water levels
in the remaining wells where a thicker unsaturated zone
is present rise more slowly and do not peak until July
(Figure 6-8).
Changing Upland Transmissibility
and Storage

The current pulse which produced correlative
modeled and measured water table fluctuations was con-
verted into its recharge equivalent and compared with
calculated available recharge. Comparison of modeled
recharge with values calculated to be potentially avail-
able at the surface indicated that only a portion of
potential recharge was actually reaching the water table.
In the interval from April 1969--October 1970, modeled
recharge was about 50 per cent of calculated available
recharge, and the value was only 19 per cent for April
1968--March 1969. The lower percentage in this latter
case can be accounted for by the fact that evapotrans-

piration strongly exceeded precipitation in the previous

112

onma kumdlmmma HHHQ<
maamz panama um maowum>mam UHQOB Hmumz

mIm mnsmflm

 

 

 

IIIII

 

 

 

114

year, 1967. This resulted in a sizeable soil moisture
deficiency at the end of the year that had to be made up
by Spring rain and snowmelt before any water was avail-
able for recharge. It is possible that this value too
would have been around 50 per cent had no soil moisture
deficiency occurred. If Thornthwaite estimates of
recharge are correct, then these results lead to two pos—
sible alternatives. The first alternative is that the
precipitation available for recharge is partially diverted
by surface drainage and interflow toward the bog. The
unsaturated zone in the upland watershed area consists
almost entirely of sands intermixed with gravel, and it is
more reasonable to assume that all infiltrating precipi-
tation percolates vertically down to the water table as
recharge.

The second alternative is that estimates of aqui-
fer storage or aquifer transmissibility and storage are
too low. This conclusion seems more reasonable in view
of the limited data on specific yield and area permeability.
0f the two aquifer parameters, storage has the greatest
likelihood of being incorrect and too low, and the model
was modified so that it represented an aquifer whose
storage coefficient is 0.2. This value is reasonable and
has been cited in the literature (Bedinger, Reed, Wells,
and Swafford, 1970). As such, storage for this aquifer

was now twice as great while transmissibility was

115

unchanged. Modeling with these constants did not suf-
ficiently raise the modeled recharge results to match the
observed values. Modeling with these constants requires
a more rapidly infiltrating early Spring recharge to
account for spring water table rises. Better results
were obtained when the model was modified so that it
represented an aquifer in which both transmissibility
and storage were everywhere doubled. Transmissibilities
were originally obtained using the permeability determined
from a pumping test and multiplying it by aquifer thick-
ness, and this permeability value may be somewhat low.
In addition, aquifer thickness was based on a maximum
thickness of the underlying confining layer which is
regarded as the blind zone layer in seismic calculations.
The thickness of the blind zone layer may be considerably
less than this maximum and thus the actual transmissibility
could be appreciably greater than originally estimated.
To simplify the modification process, the bog
was also represented as having twice the transmissibility
that initial estimates indicated. Doubling modeled bog
transmissibility does not affect the reliability of the
model since original estimates of transmissibility are
uncertain due to the dubious application of Darcy's Law
to bog peats. Selected capacitors in Figure 5-9 were
removed, however, so that modeled bog storage remained

essentially the same. With the modification, modeled

116

recharge is approximately equal to potentially available
recharge. The modified model is now defined by the

following constants:

 

Kl = 18.054 x 1013 gallons/coulomb (6-1)
K2 = 4.000 ft/volt (6—2)
K3 = 4.946 x 1010 gallons/day/coulomb (6-3)
K4 = 3.650 X 103 days/second (6-4)
and from this it follows that
12 266 x 109
c = 6.628 x 1010 a2 s. (6—6)

Upland modeling results with the model modified in this
manner are the same as those obtained with the previously
assumed constants with one exception; the modeled ground
water recharge values for each month are twice as great
with a resultant doubling of total recharge. Modeled
recharge results in Figures 6-5, 6-6, and 6-7 Show these
new recharge patterns after altering the model. These
figures Show that for the first year of modeling (Figure
6-5), modeled recharge is now somewhat less than half

estimated available precipitation. In the second two

117

years of modeling, values for recharge and available
precipitation are roughly equivalent.

Modeled upland recharge patterns in these figures
were obtained with both local surface recharge and steady
state inflow operating together. Subsequent evidence indi-
cates that modeling recharge patterns alone allows a more
accurate determination of the base voltage corresponding
to early April low water levels. Modeling in this manner
gives values for local surface recharge that are roughly
10 to 30 per cent lower. Thus it is possible that the con-
stants in relationships 6-1 through 6-6 should be adjusted
for upland transmissibility and storage coefficients as
much as two and one-half times greater than the initial
values used in modeling.

Bog_Recharge from Direct
Precipitation

Bog recharge was modeled by means of a current
pulse simulating patterns of (P - Et). In contrast to the
upland region, precipitation falls directly into the bog
water and does not pass through a thick unsaturated zone
before reaching the water table. Thus, recharge can be
directly computed by subtracting theoretical estimates of
evapotranspiration from measured precipitation. The bog
is continually supplied with incoming ground water, and
annual water level fluctuations within the bog are within
a few tenths of a foot. Estimates of evapotranspiration,

in this case, are made with the assumption that there is

118

no soil moisture depletion affecting evapotranspiration
rates. When the bog was modeled as an RC network, initial
modeled bog water table fluctuations were several orders
of magnitude greater than actual measurements. Recharge
estimates are assumed to be essentially correct. Any
reasonable changes in bog storage would not sufficiently
affect the analog results. It was therefore assumed that
the bog's transmissibility is higher than anticipated,
especially during periods of high water tables. Low ohmage
shunt resistors were placed between current input nodes to
increase modeled transmissibility. Modeled bog fluctuations
were comparable to measured values when these resistors had
a value of about 2 kilohms resulting in an estimate of bog
transmissibility of 3 X 106 gal/day/ft. This value for
transmissibility is of the order of 5,000 times higher than
initial modeling estimates. The value of 3 X 106 must be
considered only an estimate because it is questionable that
surface flow in the bog can be represented by Darcy flow
through a porous medium or by a resistive electric analog.
Bog well fluctuations due to modeled precipitation
were approximately the same order of magnitude as well
measurements after raising modeled transmissibilities in
the bog. Modeling results over the period from April 1968
through March 1971 are Shown in Figures 6-9, 6-10, and

6-11. The greatest rise in bog levels occurs early in the

Spring. Spring release of this precipitation stored on

119

 

 

 

BOG PRECIPITATION MINUS

:: § EVAPOTRANSPIRATION CALCULATED

mm... BY THE THORNTHWAITE METHOD

:28 3
‘I:/ —J_I—_I “322.12%“; “I

 

 

 

 

 

 

 

 

 

 

 

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Figure-6-9

Net Precipitation and Water Table
Elevation in Bog S-3
April 1, 1968-March 31, 1969

 

 

120

 

 

menu BOG PRECIPITATION MINUS
8- EVAPOTRANSPIRATION CALCULATED

2‘ BY THE THORNTHWAITE METHOD
5- I.
4- Smut-Au. _
,_ _ (a)
z - Mouuo FMICIPITATION

"I... I VWTRMJPIRATION
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' AM ' M1 ' IW JUL MIG sEP ' on NOV on JAM "I "M
(LIVAVIDAI
"If

 

I350_3 ._ WATER TABLE ELEVATION -
é BOG WELL NO.I
E MODELING MEASUREMENTS
1350.2 -; ‘ _
E “\\ "/—
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I350" _; munch.“ -
[350.0 5. .

 

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E
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-
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WATER TABLE ELEVATION
BOG WELL NO.I
WELL MEASUREMENTS -

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1351.0

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[$56.8 fl 7 w r . a . —
A" III" Jul: Jul- Au up oer an n: W no .44

Figure 6-10

Net Precipitation and Water Table
Elevation in Bog S-3
April 1, 1969-March 31, 1970

 

 

121

 

 

BOG PRECIPITATION MINUS
EVAPOT RANSPIRAT ION CA LCUL AT ED
BY THE THORNTHWAITE METHOD

 
    
    

in“

AA-cmrAnw
”fizc’mnnumnmou

 

 

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":1

WATER TABLE ELEVATION -

 

[950.31
BOG WELL NO.I
i MODELING MEASUREMENTS

Ml. —: _
g \‘\._ ______

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. sum 87A?!
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SI
1

WATER TABLE ELEVATION
BOG WELL NO.I

m”; WELL MEASUREMENTs I -

lis‘loé ..

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: // \

mu é \-\/' ..

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Figure 6-11

Net Precipitation and Water Table
Elevation in Bog S-3
April 1, 1970-March 31, 1971

(a)

(b)

k)

 

 

122

the surface as snow causes the sharp rise in bog water
levels in early April. This rise peaks and then falls

off rapidly, essentially disappearing by the end of May.
As there is little evidence of surface flow from spring
melting in the upland areas, it is assumed that snowmelt
within the bog is the sole cause of this sharp early spring
rise in bog water levels. Generally during the summer
months, there is a net water loss from the bog as Thorn-
thwaite calculations indicate an excess of bog evapotrans-
piration over precipitation. Recharge is possible during
the fall because evapotranSpiration is often much lower
than precipitation. Any precipitation after the middle

of November can be considered as stored as snow to be
released the following spring.

It is expected that the net late summer and fall'
precipitation in the bog determined from Thornthwaite
measurements will cause an increase in bog water levels
and surface discharge. Actual measurements for the fall
of 1968, however, show bog levels (Figure 6-9 [c]) and
discharge rates (Figure 6-12 [b]) remaining essentially
constant. The reason for this is uncertain, but it is
possible that at this time the bog could store additional
water.

Modeled water levels indicate a total water level
drop of about one-half foot along the flow path from bog

’

well 1 to the outlet for bog surface discharge. The

123

outlet is at an elevation of about 1,350 feet. The
modeled water table at bog well 1 remains at a level

less than 1,350.5 feet whereas measured values at this
well are approximately 1,357 feet. Modeling assumes

Darcy or resistive flow, and the much higher observed
water table gradient implies a mechanism other than simple

Darcy flow.

Modeling of Storms

Modeling results suggest that any effect summer
bog precipitation and evapotranspiration patterns have on
upland water levels would be small and limited to the
upland area immediately adjacent to the bog. This can be
demonstrated in modeling by altering (P - Et) current
patterns for the bog and observing upland water level
behavior at the modeled well sites. The minor effect of
summer bog precipitation on upland wells results mainly
from the fact that ground water flow is from the upland
region into the bog. Thus, excluding local effects within
the very immediate area of the bog, it was possible to
model general upland behavior without taking the bog into
consideration. However, bog water table levels are
affected by conditions in the upland watershed. It was
necessary, in modeling total bog response over periods
in excess of one or two months, to incorporate both the
bog and upland watershed components. The exception to

this is in modeling the effect of storms where precipitation

124

is intense and occurs over a short interval. Upland
water table recharge from these storms probably does not
begin to reach the bog region until at least a month after
the storm. The arrival of this recharge is also spread
out in time so that it is difficult to determine the
effect on water table levels from any one storm. This is
in contrast with the sharp response of bog water table
levels and bog discharge to an individual storm. Thus,
with respect to modeling, we may consider the upland com-
ponent as constant, disregarding it altogether, and study
modeled bog response to storms by observing only the bog
component.

Three storms were modeled and the results are
Shown in the graphs of Figures 6-12 through 6-17. Each
storm consisted of a series of three separate storm com-
ponents that covered a period of from less than one to
slightly more than two days. Each individual storm was
represented by a rectangular current pulse calibrated
to the intensity and duration of the storm. Three pulse
generators,with each producing a separate pulse, were
employed to model the individual storms. When these
generators were combined on the same circuit, there was
a limitation on their voltage differentials. An example
is shown in Figure 6-14 (a). The second pulse could not
be maintained at fourteen volts when the first and third

pulses were at about two volts. Accordingly, these pulses

125

Figure 6-12

Storm Rainfall Intensity and Measured
and Modeled Bog Surface Discharge
August 28-29, 1969

126

 

 

 

 

 

 

 

 

 

"3"" RAINFALL INTENSITY
Ao- " -
Is- -
n- - (a)
- I
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o i E 3 #339
STORM
"”0"" MODELED INCREASE IN 806
'°‘ SURFACE DISCHARGE “
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127

 

 

MODELED WMTER TABLE ELEVNHON CHANGE

 

 

 

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OA- -
05- -
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0', - .-
0-2- -
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° 1 2 3 4 s . 7 32:1."
MEASURED WATER TABLE ELEVAHON CHANGE
WAT" TMLB
fig (0)
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0.2- '
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STORM

Figure 6-13

Measured and Modeled Water Table Elevation
Change at Bog Well 1 as a Result of
the Storm of August 28-29, 1969

 

 

128

Figure 6—14

Storm Rainfall Intensity and Measured and
Modeled Bog Surface Discharge
September 4-5-6, 1969

129

 

 

 

 

 

 

 

 

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orn- °
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' RAINFALL INTENSITY
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- [I ~V‘fitfi‘
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STORM
IIJ '
’ / ”m MODELED INCREASE IN 806
SURFACE DISCHARGE ‘
5‘ ’ (b)
a- o I 'z a ‘23:."3"
$190!

806 SURFACE DISCHARGE MEASUREMENTS
gal/49, m‘

I./ -

003:3 -

 

T T T

O

(c)

‘ DAYS m

START or

i 2 3
STORM RAINFALL INTENSITY AND MEASURED AND MODELED‘N'"

BOG SURFACE DISCHARGE SEPTEMBER 4-5-6,l969

 

 

130

 

 

MODELED WMTER TABLE ELEVAHON CHANGE

 

 

 

WATER TAR“
nu
PC"
0.0- _
DIS- -
a.- _ (a)
6.3- '-
Al- "
0.! - ..
° " a 3' 72 3 2 E . E1 “5%; gen
STORM
MEASURED WATER TABLE ELEVNHON CHANGE
“87%.”!
I!"
0.3 - _ (b)
0-2 - _
OJ - /\.. -
° ’ a i 2' s a 3 i it " 11332”
STORM

Figure 6-15

Measured and Modeled Water Table Elevation
Change at Bog Well 1 as a Result of the
Storm of September 4-5-6, 1969

 

 

131

 

 

 

 

 

 

 

mtg-,2, RAINFALL INTENSITY
IO- -
' F1 (3)
5- -
- manual
E [f (I 1:“.
°‘ ° ” ‘1 ‘ i ' 5 33:71?"
STORM
gAI/Joymo'
.- MODELED INCREASE IN 806 -
9 ' SURFACE DISCHARGE
a -
1..
6 -
s- - (b)
4-
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z-
1-
STORM
BOG SURFACE DISCHARGE MEASUREMENTS
gal XIO‘
F” (c)
I-I - ’r-\ ..
\\
9323 ' ° ' 3 ' '2 ' T mm
STORM

Figure 6-16

Storm Rainfall Intensity and Measured and
Modeled Bog Surface Discharge
October 4-5, 1969

 

 

132

 

 

MODELED WMIER TABLE ELEVNHON CHANGE

 

mm TAILE

RI“

“If

04 - -

0.2 - -

O-I - '-

O " r— f 1 fl # fl -

O 1 Z 3 4- 5 DAYS FROM

START OF
.T'ORM

MEASURED WATER TABLE ELEVAHON CHANGE

 

 

 

“TIN TABLE

RI“

"‘7
M - - (b)
o" - / .\ -

O - T Y 5 TI— fi—I "
O 1 DAYS FROM
3 4 5 START OF
STORM

Figure 6—17

Measured and Modeled Water Table Elevation
Change at Bog Well 1 as a Result of the
Storm of October 4-5, 1969

 

 

133

were raised and their duration shortened so that they
still delivered the same amount of charge. Modeling
response to a higher pulse of shorter duration would be
in the form of a higher voltage peak which would diminish
very rapidly as contrasted with the response to a lower
voltage pulse of longer duration. These instrumentally
necessary sharper pulses result in distinct peaks in
modeled bog discharge which are not observed in the
natural situation.

Bog water level data on the storm of August 28-
29, 1969, (Figures 6-12, 6-13) illustrate the effects
of the increase in the bog's storage and permeability as
the water table rises (Boelter, 1964, 1966). The modeled
water level response shows two prominent rises of roughly
the same magnitude (Figure 6-13 [a]). In contrast, well
measurements Show the latter rise to be smaller than the
initial rise (Figure 6-14 [b]). The modeled response
represents the case where storage and transmissibility
remain constant at all levels in the upper part of the
bog. A reasonable assumption is that the difference
between the actual and modeled response patterns is due
to an upward increase in bog storage with an accompanying
increase in bog permeability. There is some question as
to whether bog flow in levels very near to the surface
can be adequately described by Darcy's law, and accord-

ingly it is difficult to accurately establish any

134

relationship between bog transmissibility and water level
gradients in this zone.

The modeled volume of bog discharge from these
storms is roughly eight times greater than actual measure-
ments at the bog weir. The modeled volume of discharge
is considered reliable since precipitation from these
storms is not significantly affected by evapotranspiration.
Two explanations are proposed here to account for this
discrepancy. First, the actual weir measurements may be
low. Second, there may be a considerable amount of
discharge flowing out within the bog peats on either side
of the weir.

Bog Recharge from Upland
Ground Water

Adjustments in modeled bog transmissibility
resulted in a lowering of the modeled water table gradient
so that potential levels at the inter-nodes within the
bog were only slightly above ground potential. There is
actually a surface water drop of about eight feet along
the flow path through the bog. Modeling results suggest
that the transmissibility of the bog's upper layer is
much too high to support this type of gradient. These
findings suggest that the deeper, dense, essentially
impermeable layers of compact vegetation that constitute
the bulk of the bog's volume act as a dam, raising the

water table in the permeable sands around the bog. Under

135

these conditions, it is probable that ground water enters
the bog in two ways. The first is by direct flow into the
upper permeable layer and then through this layer down

the steep incline formed by the surface of the underlying
dense peats to the bog's outlet. Darcy's equation for

flow in porous media has only limited application in this
instance. This upper layer of permeable peat may be

broken up by a network of channels of very high transmis-
sibility so that flow through this layer is closely related
to a stream flow phenomenon.

The second way ground water enters the bog iS by
upward flow. Artesian conditions beneath the bog lead to
this upward flow. One very large vertical opening has
been discovered in bog S-3 and measurements here show the
upward piezometric gradient. Upward flow in this large
opening is perhaps supplemented by seepage of ground water
to the surface in several other small fissures in the bog.
Upward seepage or upwelling was modeled by connecting one
lead of a 200 kilohm resistor to a node located in the
model in approximately the same location as the large
vertical opening in the bog. The other lead of the resis-
tor was connected to a node directly beneath in the modeled
aquifer. The current through this resistor represents the
total upwelling within the bog. The value of the resistor
was chosen to give a modeled bog discharge that was in

reasonable agreement with measured values. Results from

136

this modeling show that upwelling accounts for about one-
fourth of bog recharge from upland ground water (Figure 6-
19 [c]). A schematic cross sectional diagram of the bog
and surrounding ground water system illustrating this
discussion is shown in Figure 6-18.

If upland inflow into the bog were constant
throughout the year, then changes in bog water levels
would be solely the result of precipitation and evapo-
transpiration within the bog. There are, however, cyclic
changes during the year in the rate of upland inflow which
also contribute to changes in bog water levels. Thus,
actual measured fluctuations in bog levels represent the
effect of changes in bog precipitation and evapotranSpira—
tion superimposed on the longer term fluctuations from
changes in upland inflow. Figures 6-9 (b), 6-10 (b), and
6-11 (b) Show modeled levels in bog well 1 for the three
years of modeling resulting from changes in bog precipita-
tion and evapotranspiration superimposed on changes in
upland inflow. These same graphs Show water levels if the
upland inflow rate remained constant at late winter values

throughout the year.

Modeled Discharge

Surface Discharge from the Bog
Modeled surface discharge from the bog is the

resultant of the combined effect of the upland watershed

137

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138

 

 

 

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139

 

 

 

 

 

 

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'”°‘ MODELED BOG DISCHARGE
I- -
TOTAL -
7 - OOO Dam-OE
‘5'- , \_./ T (a)
cammuwnu OP
’ ‘ 0m «mow TO ’
z _ BOO DISCHARGE _
I - -
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" {\ WEI R MEASUREMENT OF "
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1_ -
.-I \. -
,_ | \ -
" I \ ' (b)
3* I \ .. ‘
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2— . - . -.__-__-_._---_._--__-
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”w” MODELED EFFECT OF UPWELLING m“, ”mm“ a." ”u”
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mm“
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Figure 6-19

Measured and Modeled Surface Discharge of
Bog S-3 April 1, 1968-March 31, 1969

 

 

140

and bog components. The results of modeling bog dis-
charge are shown in Figures 6-19, 6-20, and 6—21. In view
of the comparative size of the areas, it would be expected
that the contribution of the upland watershed would pre-
dominate. Modeling results verify that the component of
hog discharge derived from ground water flowing into the
bog from the surrounding aquifer comprises the bulk of

bog outflow. This component is essentially constant
throughout the year. Variations in bog outflow are pri-
marily the result of precipitation and evapotranspiration
within the bog itself. The bog contributes to discharge
during periods of heavy precipitation. Thornthwaite
estimates of evapotranspiration in this study indicate
that evapotranspiration frequently exceeds precipitation
during the summer months. In this case, the bog would
have an opposite effect. It not only makes no contribution
to discharge but also actually reduces total discharge
through evapotranspiration losses of incoming ground water.
An example of this is given in Figure 6-19 (a). The total
discharge, which consists of combined upland inflow and
the effect of hog precipitation and evapotranspiration,
and the component of upland inflow alone are shown on the
same graph. High spring discharge is followed by a drop
in discharge during part of all of the summer. Bog dis—
charge rates during this period are less than upland

ground water recharge rates into the bog. From about the

141

 

 

MODELED BOG DISCHARGE

 

 

goody x IO‘
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WEIR MEASUREMENT OF BOG DISCHARGE
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0- ..

 

Figure 6-20

Measured and Modeled Surface Discharge of
Bog S—3 April 1, 1969-March 31, 1970

 

 

142

 

 

WMXM‘

MODE LED BOG DISCHARGE

1'0 TAL .0.

‘ _, DISCHARGE

 

4..

CONT RIDUT ION

3 - OF UPLAND IAIFLONI

TO .06 DIRCHAR“

I -

 

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APR HAT aw am. A» up per ' no u: JAM Pu MR

WEIR MEASUREMENT OF BOG DISCHARGE

Why ' '0’
3 _

1-

 

Figure 6-21

Measured and Modeled Surface Discharge of
Bog S-3 April 1, 1970-March 31, 1971

(a)

(b)

 

 

143

middle of November to March, bog surface discharge is only
that resulting from upland inflow.

Values for bog discharge observed in modeling are
greater than measured discharge. In this case, modeled
values are roughly twice measured values. Thornthwaite
values for (P - Et) in the upland and bog regions are
assumed to be accurate, thus, the same explanation is
offered for the higher modeled rates as was offered in
the case of storms. These are that (1) measured bog dis—
charge rates are lower than actual discharge through the
weir, and (2) a considerable portion of this discharge
passes through surface peats in the bog outlet on either
side of the weir so that only a part of the total discharge
is available for measurement.

The function generator modeling bog precipitation-
evaportranspiration patterns cannot supply a sufficiently
high or rapidly rising current pulse to model normal April
peaks in bog recharge from snowmelt and still model pat-
terns for the remainder of the year with any degree of
accuracy. As a result, modeled peaks for spring rises in
bog water levels are smaller than they should be and in
some instances occur in late May and early June.

Discharge of Ground Water
Flowing Beneath the Bog

 

 

Modeling calculations indicate that 103.4 X 107

gallons of water per year flow into the modeled portion

144

of the upland region from the outlying area. This recharge
is simulated by the steady-state current. Assuming ten
inches of recharge per year within the modeled portion of
the upland area, then recharge would add 8.2 X 107 gal/year
giving a total of 111.6 X 107 gal/year of upland recharge.
Modeling results indicate that of this amount, roughly

16 X 107 gal/year of inflowing ground water enters the bog
and leaves the area as bog surface discharge or evapotrans-
piration. The remaining 95.6 X 107 gal/year leaves the
basin as ground water flow beneath the bog. That is, about
86 per cent of the inflowing ground water leaves the

region by flowing out as underflow beneath the bog with

only 14 per cent entering the bog.

Experimentation with the Model

The model was utilized as a predictive instrument
in the modeling of a hypothetical wet year. The previous
years that had been modeled have moderate to sparse
recharge from precipitation with evapotranspiration gener-
ally exceeding precipitation for the bulk of the summer
period. It is therefore of interest to observe modeled
response to a wet year. The results of this portion of
modeling are shown in Figures 6-22 and 6-23. The constants
used in modeling are those defined in relationships 6—l
through 6-6. The most significant aspect of this hypo-
thetical year is that precipitation always exceeds evapo-

transpiration. If it is assumed that there is no upland

145

Figure 6—22

Modeled Ground Water Recharge and Water Level
Elevations in the Upland Area for a
Hypothetical High Precipitation
Year

146

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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7 - ..
6 ' PRECIPITATION AVAILABLE FOR "
5 GROUND WATER RECHARGE _
4 - -
(a)
3 - _
Z - _
TOTAL
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0 - An‘mv‘wu 'Ju'A-vs'sv ‘ocr mvlotc 'JAu'cu‘m' -
3 ‘ MODELED RECHARGE AT '-
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,5“ - WATER TABLE ELEVATION _
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1' 306
1362 - -
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147

Figure 6-23

Modeled Net Precipitation, Water Table Elevations,
and Surface Discharge of Bog S-3 for a
Hypothetical High Precipitation Year

148

 

 

 

6 - AVAILABLE BOG PRECIPITATION _

4 ..
/ (a)
3 - TOTAL IS INCHES -

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1 - ’K -

O - I V W

 

 

 

 

 

 

 

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goo/4.} xno‘ MODELED BOG SURFACE DISCHARGE
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149

soil moisture deficit carried over from the previous year
in the upland area, then upland evapotranspiration rates
would not be affected by any soil moisture deficiency and
available recharge from precipitation for the upland and
bog regions would be equivalent. The value for recharge
is taken as fifteen inches. The pattern for recharge
arriving at the water table was estimated from previous
modeled patterns based on known water table fluctuations.
Maximum recharge is assumed to be in June with rates then
diminishing until the beginning of the following spring.
All available recharge (fifteen inches) was modeled as
infiltrating directly to the water table.

Figure 6-22 (c) shows modeled upland water level
responses which display an unusually high spring water
level rise. Peak water levels occur about a month after
the peak in recharge rates to the upland water table. The
bog recharge pattern and modeled bog responses are shown
in Figure 6-23. Figure 6-23 (b) shows modeled levels in
bog well 1 for this hypothetical wet year resulting from
bog (P - Et) superimposed on changes in upland inflow.

To illustrate the effect of variable upland inflow on the
hypothetical wet year, water table elevations are also
illustrated assuming a constant upland inflow rate equiva-
lent to late winter values.

Modeling of bog surface discharge shows a greater

volume of bog outflow for the hypothetical year. This is

150

mainly the result of increased upland inflow to the bog
area and the fact that there is no net removal of water

through bog evapotranspiration.

Conclusions from Modeling

Electrical analog modeling provides a useful method
for semi-quantitative analysis of hydrologic systems such
as the bog-upland watershed region of this study. The
model makes possible a solution when many variables are
operating simultaneously under complex initial and boundary
conditions. The parameters of the hydrologic system repre-
sented in this study are in general poorly defined. The
size of the upland area supplying ground water to the bog
can only be approximated. Permeability and storage values
for the upland aquifer are available from only one pumping
test. Bog permeability is based on limited data, and the
amount of upland ground water recharge was approximated
by a theoretical formula.

The complex interaction of these factors was
examined under various assumptions with the analog model,
and modeled data were compared with historical hydrologic
data. Where there was disagreement, the model was modified
until there was approximate agreement. Through these modi-
fications, estimates can be made of poorly known factors.
The estimates are not unique, however, for more than one
combination of factors may produce modeled data that are

in reasonable agreement with the limited amount of collected

151

data. The modeling results may be considered as semi-
quantitative in that they produce neither a unique nor

a completely accurate solution. The model nevertheless
provides a means for testing conceptual models of the
hydrologic characteristics which are based on geological
characteristics.

Results from the analog model show that the upland
watershed region supplying ground water inflow to bog S—3
is very large. The value of permeability and storage
coefficients determined for the portion of this watershed
represented by the RC circuit were high but within the
limits expected for the type of material encountered. The
bog itself has very little effect on ground water behavior
within the region. The bog's effect is confined to the
immediate region of the bog and is of two forms. First,
the blocking or damming effect of the bog impedes ground
water flow and raises water table levels in the immediate
vicinity of the bog thus slightly increasing the amount
of ground water in storage. Second, the bog depletes
local ground water supplies during the summer months
through evapotranspiration of inflowing ground water.

Runoff and evapotranspiration from bog S-3 affect
surface discharge from the watershed area. This discharge
consists of contributions from both the bog and regional
ground water inflow. The bog does not act in a manner

to insure an even flow of surface discharge throughout

152

the year. The bog immediately releases excess water dur-
ing periods of high precipitation instead of storing it

to be gradually released during dry periods. The bog
exhibits considerable evapotranSpiration during dry periods
in the growing season. As such, it not only fails to
release stored water, but actually removes water through
evapotranspiration. The bog thus exerts no regulative

or conserving effect on discharge, but rather acts in quite
the Opposite manner. Although the influence of bog S-3

on the regional ground water system is minor, the bog
drains directly into a much larger marsh area (Figure 2—1),
and the combined effect of the total system of bogs on
regional ground water could be proportionately much
greater. This system of bogs, which merges into lakes
downstream, eventually drains into the Prairie River
(Figure 2-2). It is likely that this total bog system
exerts an appreciable effect on surface discharge into the
river similar to that of bog S-3 at its outlet.

Modeling results suggest that the dense, decom-
posed, deeper layers of peat in bog S-3 have a damming
effect that raises the level of incoming ground water.

The bog likely had its origin as a vegetation filled
stream or pond. Successive additions of peat caused a
buildup in the lower dense layers with a resulting block-
age of incoming ground water. This blockage raised water

levels and thus allowed growth of new peat producing

153

plants at successively higher levels. This form of bog
accretion has continued until the bog surface and adjacent
ground water levels are at their present elevations.

This same mechanism has likely been an important factor in
bog development in other regions of Minnesota such as the
extensive bogs found in the eastern part of the bed of
former Lake Agassiz. There is evidence (Heinselman, 1963)
of the spreading of isolated peat bogs outward from the
shallow valleys of their sources to the extent that the
low divides have been covered and the whole region has-
become a continuous bog. Whether the larger bogs have

similar ground water-bog relationships is unknown.

CHAPTER VII

CONCLUSIONS AND SUGGESTIONS FOR

FURTHER STUDY

Conclusions

 

This study is concerned with the application of
a combination of investigative techniques designed to
determine the role of a small undisturbed ground water
peat bog in the hydrologic cycle. Geophysical methods
and exploratory drilling combined with hydrologic tests
and observations were used to determine the hydrologic
characteristics of the bog-watershed unit. These charac-
teristics served as the basis for construction of an
electric analog model which was verified and then used
as a predictive instrument. These methods were shown
to be successful in gathering information on the role of
the bog in the hydrology of this region. This particular
bog was small in size with respect to its ground water
watershed; thus its effect on the area's hydrology was
quite limited. A larger bog would have had a more pro-

nounced effect.

154

155

No one method used in this study was in itself
capable of providing all the information sufficient to
determine the role of the bog in its hydrologic system,
however, certain methods stand out as being particularly
effective. The analog model was the most important tool
in determining the role of the bog in this system. It
was also possible through modeling to augment available
data on the hydrologic and related geologic characteristics
of the system. This was accomplished by noting modifica-
tions necessary in the model to bring modeling results
into agreement with historical hydrologic data. Second
in importance were the seismic refraction studies and
supportive data from exploratory well drilling. The
applicability of seismic refraction in determining aquifer
thickness was restricted because of the possibility of a
blind zone layer of uncertain thickness. In this respect,
earth resistivity methods were of limited value in sug-
gesting the presence of this layer. Gravity methods
played no direct role in determining the hydrologic
characteristics of the area.

The final result in the development and applica-
tion of these techniques was the use of the analog model
to predict, and the specific application in this study
was the modeling of a hypothetical year of high precipi-

tation.

156

Suggestions for Further Study

 

There are several possibilities for the exten-
sion of geophysical exploration and modeling methods to
the further study of bogs. It is feasible to enlarge and
refine the present electrical analog model as additional
information is collected within the S-3 bog-watershed
area. Analog models may be applied to the perched bogs
in the Marcell Experimental Forest to the extent that
resistive flow is important in the hydrologic process.
Data indicate that the upper peats in the bogs steadily
increase in permeability toward the bog surface. This
upper region of hog S-3 was modeled as a single layer
whose transmissibility was roughly equivalent to the sum
of the transmissibilities of the incremental layers. It
may be possible to further refine the modeling of these
upper peat horizons by the use of Zener diodes or transis-
tors. These devices could be incorporated into the bog
circuitry so as to cause a decrease in resistance with an
increasing voltage gradient across the bog. This would
correspond to the increase in bog transmissibility noted
during periods of high bog flow. Introducing various
precipitation-evapotranspiration sequences into the
perched bog model and comparing known results with modeled
data should provide useful estimates of vertical drain-
age, evapotranspiration rates, transmissibilities, and

storage coefficients.

157

Some of the techniques of the study might be
applied to larger bog areas as, for example, the bogs in
the Lake Agassiz region. In these bogs, seismic refrac-
tion methods might be an effective means of determining
bog thickness. Much of the old lake bottom contains
clay, and saturated clays would have a significantly
higher compressional wave velocity than saturated peats.
As such, the interface between the hog and old lake
bottom would be mappable with the seismic refraction
method.

Analog modeling methods also appear to have
applicability in the study of these bogs. It is possible
to model the area of an entire large bog using resistance-
capacitance networks on a scale of around one inch to the
mile (and this network could contain current inputs for
combined precipitation and evapotranspiration). A paral-
lel network beneath that of the bog with interconnecting
resistors to the upper network would model upward verti-
cal seepage from underlying mineral soils if flow of this
nature was suspected. The initial precipitation-
evapotranspiration input could be constant and represent
the estimated average for a number of years. Modeled
flow patterns and water table gradients could be compared
with known data and modifications made in modeled re-
sistances where discrepancies were found. It is assumed

here that bog resistances would represent estimates of

158

average transmissibilities of vertical profiles of hog
material. It is likely that these averages would repre-
sent, in effect, only the near surface layers of bog
peats, as in all probability the deeper peats are compact
and essentially impermeable. Transistors or Zener diodes
might also be used here to model the upwardly increasing
permeability of the near surface peats. Once reasonable
steady-state simulation in the model has been achieved,
various precipitation-evapotranspiration pulses could be
introduced and modeled fluctuations of bog water levels
compared with known values. This procedure should supply
information on evapotranspiration rates in larger bogs
and possibly help to further refine estimates of the

bog's transmissibility and storage.

LI ST OF REFERENCES

LIST OF REFERENCES

Barnes, H. E., 1952, Soil investigation employing a new

Bavina,

Bay,

R.

method of layer value determinations: Bull.
Highway Research Board, v. 65, p. 26.

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, 1967, Ground water and vegetation in two peat

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415.

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25-Oct. 7, 1967.

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160

Bay, R. R., 1969, Runoff from small peatland watersheds:
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Bedinger, M. 8., Reed, J. E., Wells, C. J., and Swafford,
B. F., 1970, Methods and applications of electri-
cal simulation in groundwater studies in the lower
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and Oklahoma: U.S. Geol. Survey Water-Supply
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Boelter, D. H., and Blake, G. R., 1964, Importance of
volumetric expression of water contents of
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, 1964, Water storage characteristics of several
peats in situ: Soil Sci. Soc. Am. Proc., v. 28,
no. 3, p. 433-435.

, 1965, Hydraulic conductivity of peats: Soil
SCio, V010 100, I10. 4’ p. 227-231.

, 1966, Hydrologic characteristics of organic
soils in lakes states watersheds: Jour. of Soil
and Water Conserv., v. 21, no. 2, p. 50-53.

Dobrin, M. B., 1960, Introduction to geophysical prospect-
ing: New York, McGraw-Hill Book Co.

Ewing, M., Woolard, G. P., and Vine, A. C., 1939, Geophysi-
cal investigations in the emerged and submerged
Atlantic coastal plain: Part III: Barnegat Bay,
New Jersey, section: Bull., G. S. A., v. 50, p.
257-296.

Hawkins, L. V., and Maggs, D., 1961, Monograms for deter-
mining maximum errors and limiting conditions in
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problem: Geophysical Prospecting, v. Ix, p.
526-532.

Heinselman, M. L., 1963, Forest sites, bog processes,
and peatland types in the glacial lake Agassiz
region, Minnesota: Ecol. Monog., v. 33, p. 327-
374.

Ivanov, K. E., 1965, Fundamentals of the theory of swamp
morphology and hydromorphological relationships:
Soviet Hydrology: Selected Papers, no. 3, p. 224-
258.

Jakowsky, J. J., 1950, Exploration geophysics: Newport
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Johnson, R. B., 1964, IBM program for refraction seismic
problems: Dept. of Geology, Colorado State
University, Fort Collins, Colorado.

Joiner, T. J., Warman, J. C., and Scarbrough, W. L.,
1968, An evaluation of some geophysical methods
for water exploration in the piedmont area:
Ground Water, v. 6, no. 1, p. 19-28.

Karplus, W. J., 1958, Analog simulation: New York,
McGraw-Hill Book Co.

Michigan State Highway Department, 1963, "The application
of geophysics to highway engineering by the
Michigan State Highway Department.

Mooney, H. M., and Wetzel, N. W., 1956, The potentials
about a point electrode and apparent resistivity
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Moore, R. W., 1952, Geophysical methods adapted to high-
way engineering problems:, Geophysics, v. 17,
p. 505-530.

Novlkov, S. M., 1964, Computation of the water-level
regime of undrained upland swamps from metero-
logical data: Soviet Hydrology: Selected Papers,
no. 1, p. 1-22.

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of the Mississippi headwaters watershed north-
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Investigations Atlas HA-278.

Page, L. M., 1969, The use of the geoelectric method for
investigating geologic and hydrologic conditions
in Santa Clara County, California: Journal of
Hydrology: v. 7, p. 167-177.

P'Yavchenko, N. I., 1958, Peat bogs of the Russian forest—
steppe: Izdatel'stvo Akademii S.S.S.R. MOSKVA,
1958. Published for the U.S. Dept. of Agriculture
and the National Science Foundation, Washington,
D.C. by the Israel Program for Scientific Trans-
lations, Jerusalem 1964.

Sanker Narayan, P. V., and Ramanujachary, K. R., 1967,
An inverse slope method of determining absolute
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Univ. Minn. Geol. Surv. Bull., v. 16.

162

Soske, J. L., 1959, The blind zone problem in engineering
geophysics: GeOphysics, v. 24, p. 359-365.

Spangler, D. P., and Libby, F. J., 1968, An application
of the gravity survey method to watershed
hydrology: Ground Water, v. 6, no. 6, p. 21—26.

Tektronix, Inc., 1965, Operational amplifiers and their
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6-3.

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Trans. Amer. Geophysical Union, v. 16, p. 519-524.

Thornthwaite, C. W., and Mather, J. R., 1957, Instructions
and tables for computing potential evapotranspi-
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Technol. Lab. Climatol., Publications in Clima-
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Vorobe'v, P. K., 1963, Investigations of water yield of
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States, a review volume for the VII Congress of
the International Association for Quaternary
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29-41.

Walton, W. C., and Prickett, T. A., 1963, Hydrogeologic
electric analog computers: Hydraulics Div.
Jour., Am. Soc. Civil Engineers, v. 89, p. 67-91.

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p. 41-48.

General References

 

Brophy, J. J., 1966, Basic electronics for scientists:
New York, McGraw-Hill Book Co.

163

Heath, R. C., and Trainer, F. W., 1968, Introduction to
ground-water hydrology: New York, John Wiley
and Sons, Inc.

Kreyszig, E., 1967, Advanced engineering mathematics:
New York: John Wiley and Sons, Inc.

Todd, D. K., 1959, Ground water hydrology: New York:
John Wiley and Sons, Inc.

APPENDICES

APPENDIX I

BASIC THEORY AND EQUATIONS FOR THE

ANALOG MODEL

APPENDIX I

BASIC THEORY AND EQUATIONS FOR THE

ANALOG MODEL

An extensive treatment of the basic theory of
electric analog modeling and its applications has been
provided by Karplus (1958). Walton and Pickett (1963) and
other workers have put forth guidelines for the application
of these modeling principles to hydrologic situations, and
this Appendix supplements these works. It is necessary to
understand the theory of electric analog modeling to ade-
quately understand the assumptions, limitations, and
validity of its application to the hydrologic situation.
This first section deve10ps the formulae and processes
needed for analog modeling in this study, and the approach
in this development has been to combine intuitive under-
standing with rigorous theory. The development of analog
theory is followed by a second section establishing
specific hydrologic equations describing the bog-basin
groundwater system of this study.

The basic criterion we will use to establish a

modeling relationship between two systems is whether they

164

165

may be described by the same characteristic equation. The
equations for both systems must not only be of the same
general form, but each constant in one equation must be
numerically equal to the respective constant in the other.
As an example, if a rocket is accelerating according to

the formula
2
——— = 100 ft/sec ,

and if a racing car's acceleration may also be described by

93§-= 100 ft/secz,
dt
we may find out exactly how far the rocket has climbed at
the end of the 3 1/2 seconds from blast-off by simply
noting how far the racing car has traveled in its first
3 1/2 seconds. The racing car then may be used to model
the rocket. In the following paragraphs, we will examine
the feasibility of using a continuous electrical system
for exactly modeling a hydrologic system and then briefly
introduce the more readily constructed finite difference

electrical analog as a somewhat less exact.method of

modeling.

The Diffusion Equation

 

Let us first consider a linear partial differential

equation which describes a certain system of interest.

166

This equation is of the form

—+——=K—— (1-1)

It is necessary to specify its boundary condition (the
constant set of values h(X,Y) around the boundary of the
system) and the initial condition (the set of values
h(X,Y,O) throughout the interior of the system at time
t = 0). This form of equation is known as the diffusion
or heat flow equation and is extensively discussed in the
literature. It can be shown that this equation possesses
a unique solution determined by the value of the constant
K and the initial and boundary conditions. By a unique
solution, it is meant that h(X,Y,t) as determined by
equation I-l with its initial and boundary conditions
has only one value for each set of values of X, Y, and t.
It should also be noted that the solution h(X,Y,t) is
changed to K'h(X,Y,t) by changing the boundary condition
to K'h(X,Y) and the initial condition to K'h(X,Y,0) Where
K' is a particular constant. This can be readily veri-
fied by direct substitution into equation I-l.

As a first step in showing that an electrical
system can be used to model a hydrologic system, consider

two specific examples of the diffusion equation

a:
:3‘
c)
:3‘
o)

_ h _.
+ 2_K—E- (12)

167

and
2 2
3 V 3 V 3V
__ + = K' -— (1-3)
3x2 ayi at

with initial and boundary conditions numerically equivalent.
We may ask under what conditions will V(X,Y,t) be a solution
for equation I-2. That is, from previous considerations of
uniqueness, when will V(X,Y,t) = h(X,Y,t). Direct substi-
tution shows that this can only be true when K = K'.

Assume

V(X,Y,t) = h(X,Y,t)

substituting V(X,Y,z) into equation I-2 we have

32v + 32v = K. 33
3x2 M at
But from equation I-3
32v + 32v _ K av
"§ “‘7 ‘ 3?
ex ay
thus
42. av
K at ’ K at

168

From this the converse follows by the uniqueness theorem.
That is if K = K', then V(X,Y,t) = h(X,Y,t).

We may next construct a pair of two-dimensional
physical systems, A and B, as shown in Figure I-l. Both
systems are the same shape but in system A, x and Y are
measured in feet and t in days while system B is scaled
down prOportionately so that it has the same dimensions
but X and Y are measured in inches and t is measured in
seconds. It can be shown that these systems may be
described by the following equations. For system A
——— + ——— = 7.48 §_%% (1-4)
where, as we recall from above, x and Y are in feet and t

is in days. For system B

32 2

V
———-+ = pC —— (I-5)
3x2 ayi at

where x and y are in inches and t is in seconds. If we
make 7.48 S = C and l/T = p so that 7.48 S/T = pC and

prescribe identical boundary conditions

h(X,Y) V(X,Y)
and the initial conditions

h(XIYIO) V(X,Y,O)

169

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170

then from the above considerations, it follows that
h(X,Y,t) = V(X,Y,t)

Let us consider a specific example of what these
equations represent. System A is a sand aquifer in an
island which contains a very gently sloping horseshoe-
shaped watershed with a stream outlet for discharge at
its base. The water table at the periphery of the island
merges with the surrounding lake surface. This aquifer
can be represented to a high degree of accuracy in system
by a uniformly resistive sheet of continuous capacitance
that is grounded around its periphery. The stream outlet
in system B is represented by a grounded wire of very low
but measurable resistance. A contour map is available
for system A showing the water table at the beginning of
a 30-day dry period. We may adjust the voltage in system
B so that we can prepare a contour map of voltage values
identical to water table levels in the water table map and
then remove the voltage source so that system B is allowed
to discharge. We next make a voltage contour map at a
given number of seconds (say 4 1/4 seconds) after the
beginning of discharge. If this map were to be compared
with a piezometric map of system A at the same number of
days (4 1/4) after the onset of the 30-day dry period, we
would find the two maps identical. Further, Ohm's and

Darcy's laws describe the flow of water and electricity

171

in these two systems. The relationship in this instance
between S, T, p, and C, namely 7.48 S = C and l/T = p,
prescribe identical flow of voltage and current in the

two systems. That is, if we were to find that water flowed
at the rate of 5 gal/day/ft2 in a specific direction at a
point (X,Y) in the hydraulic system at say 5 days from a
reference time to’ we would find an analagous current flow
of 5 amperes/inch2 in the same direction at the corres—
ponding point (X,Y) in the model at 5 seconds from to' In
other words, system B, the electrical system, represents

a model of system A, the hydraulic system.

The two systems just described had electrical and
hydraulic units that were numerically the same. That is,
they were related by scale factors equal to one. In
practice, it is nearly always more feasible to relate them
by scale factors other than one. The four fundamental

scale factors are defined aS‘

q(gallons) = KlQ*(coulombs) (I-6)
h(feet) = K2V(volts) (I-7)
Q(gallons/day)= K3i(amperes) (I-8)
td(days)' = K4ts(seconds) (I—9)

where the scale factors K1, K2, K3, and K4 are numbers that
we choose in the beginning. We are not completely free
to choose all the numbers at random, however, for some of

the scale factors within each set are related to one

another as described next. The current i in ampres

172

(remember that 1 stands for a number) may be written as

i coulombs
1 second

 

. From this we see that the following relation-

ship exists between the scale factors

Kl(i)gallons Kl(i)gallons

 

 

 

 

 

 

 

K = Q gallons/days = K4(1)daYS _ 1 coulombs
3 i coulombs/second i colombs - K4(l)days
1 second
1 second

= Kl gallons/coulomb

K4days/second

that is K3 is numerically equal to Kl/K4 or

= 1 (I-lO)

 

We next wish to see if altering our initial
electrical system by means of these scale factors results
in a new electrical system which may still be used to model
the hydraulic system. To do this we will first want to
develop a relationship between the initial electrical
system (system B) and the new electrical system. First
the new relationship between transmissibility and resis-
tivity may be determined from Ohm's law, Darcy's law, and
equations I-7 and I-8. Consider a voltage applied across
Opposite sides of one square inch of conducting sheet
and a piezometric gradient applied across opposite sides

of a square foot column of aquifer. From Ohm's law,

by Darcy's law,

o-JII—a
I
our

Thus,

0
ll
lu

alt-4

. (I-11)

7:

Finally, as a capacitor gains or loses stored charge in
accordance with the change in voltage applied to it, so
does a given section of the aquifer gain or lose stored
water in proportion to the change in hydraulic head. This
prOperty is expressed in the storage coefficient S. From
the definition of capacitance and equations I-6 and I-7
there results

= Q* coulombs _ q/K1 coulombs 9.K

_ = 2 coulombs
V volts h/K2 volts h

volt '

 

C

 

LEI

but by the definition of storage

ft3/ft2

ft

 

9 gallons = (7.48 gal/ft3)(S

h feet ) = 7.48 gal/ft.

Thus

N

C = 7.48 S (I-12)

N

174

The resistance-capacitance system can now be modi-
fied by equations I-6 through I-12 to give a new resis-
tivity 3 and a new capacitance C. Considerations similar
to those used in the development of equation I-4 show

that the equation for this system would be of the form

32v 32v _ — — @1— (1-13)
2 +' 2 ‘ p C at
ax ay

From equations I-11 and I-12 we know that

 

— — _ s

a v + a v = 7.48 §_gy
ax2 ari K4 T ats
0r
32v + 32v = 7 48 g av
ax2 3Y2 T 3(K4t;)

Where tS emphasizes that time is being measured in seconds.
If we consider K4tS as the time variable, then this
equation has the same value for its constant as does
equation I-4 which describes the hydraulic system. Thus
given identical boundary and initial conditions, both
systems will behave the same. However, an event taking
one day in the hydrologic system will take 1/K4 seconds

to occur in the electrical system.

175

This does not imply that we now make V represent
h as V = h, for our choice of the particular resistance
and capacitance from equations I-11 and I-12 was based
on the initial way we decided to relate h and V by means
of the scale factor K

namely h = KZV' The resistance

2:
and capacitance of this system will "remind" us of this
fact by the way they govern current flow. We must set
our initial and boundary conditions in the model by the
relationship V = h/KZ' otherwise there would be a dis-
proportionate amount of current flowing in the electrical

system and Q = K i would no longer hold. Remember that

3
changing the initial and boundary conditions by the
factor 1/K2 changes V throughout the model by l/Kz.

The differential equation describing the aquifer
system may be readily altered to include the effects of
recharge from precipitation minus evapotranspiration
(QP _ Et). The resulting equation, which although not
completely representative, approximates the unconfined

aquifer situation sufficiently closely and is written

as follows
(I-14)

where QP is given in gallons/day/ftz. The quantity

- Et

QP _ Et' in regions where EH1 unsaturated zone overlies

the water table, represents recharge from percolation

176

through the unsaturated zone. The resistor-capacitor
system may be altered to model the effect of precipitation
and evapotranspiration by the addition of a continuous
current input over the entire resistive surface. The

equation describing this modified system is
_ 33.! _ - _
— pC 3t p1 (I 15)

where i is measured in amperes/inchz. It is possible,
using a reasoning process similar to that developed
earlier in this section, to show that this electrical
system models the modified hydrologic system. Electrical
and hydrologic units used in equation I-13 are again re-
lated in these two new systems by the constants in
equations I-6 through I-9. In addition, the current

in equation I-15 is related to recharge by

i = Q—P—Z—ElE . (I-16)
K3
The model, as just described, was scaled such
that one inch in the model was equivalent to one foot in
the hydrologic system. If we were to construct a new
model scaled at one inch to two feet (or one-half inch
to one foot), then X and Y in the model would be measured
in units of one-half inch instead of one inch. It would

thus be necessary to construct this new model of material

with electrical properties such that a square in the model

177

one-half inch on a side would have the same resistance

and capacitance as one square inch of the initial model.

A two dimensional conducting sheet of length i and width w
having a resistivity of p ohms will have a resistance R

in ohms to current flow in the 2 direction given by the

formula

R=p£

a: . (1'17)

This equation shows that halving both i and w does not
change R. Thus, the properties of a two dimensional con-
ducting sheet are such that it is not necessary to change
the resistivity of the material for constructing the model
at this or any other new scale. The new material, hOwever,
would require a capacitance in farads per square inch four
times greater. In general, changing from a scaling of

one inch to one foot to a scaling in one inch to a feet
would require a change to a2 farads per square inch in

the capacitance of the modeling material. Equation I-12
may then be modified to provide for a scaling of one inch
in the model to a feet in the hydrologic system in the

form

C = 7.48 a S — , (I-18)

where C is the capacitance in farads per square inch of

the material used in model construction. Similarly, the

178

amount of current i flowing into one square inch of the
model when the model is scaled at one inch to a feet is

given by

Q _
i = a2 _P__E (1-19)

In practice, the actual hydrologic system has
completely changing prOperties which are at best only
partially known. An element of area in this system can be
approximated by an idealized hydraulic system with homo-
geneous properties. This idealized hydraulic system can
be closely modeled by a continuous resistive—capacitive
electrical model. The actual model, however, is more
practically constructed as a finite difference approxi-
mation to the continuous model. Here the continuous
resistive-capacitive system is replaced by regular arrays
of resistors and capacitors with the spatial dimensions
of this system the same as those of the continuous system.
Generally each discrete resistor represents a piece of
continuous material of length 1 and width w having a
resistance of R ohms. The value of this resistor is

calculated by equation I-17 as

where 2 is in the direction of current flow. Thus one

resistor represents the resistivity to current flow of

179

the continuous material only for a specific direction in
accordance with the vector area technique described by
Karplus.

In a similar manner, the capacitance of a dis—
crete capacitor representing the capacitance in a square
inch of the continuous model with a scaling of one inch
to a feet would be determined by equation I-18 as

c = 7.48 aZS 53 .
K1
A discrete capacitor representing, as an example, four
square inches of this continuous model would have a
capacitance four times larger.

The continuous current input representing QP _ Et
can be approximated by discrete current conducting wires
entering selected nodes with the appropriate voltage
applied to them. From equation I-19, the amount of
current flowing through a wire representing a continuous

input over one square inch of the model when the scaling

is one inch to a feet would be

In this section, we have established an idea of
what the modeling process is and how we may establish
and justify the use of an electrical model to simulate a

hydrologic system. In summary, water table elevationnwas

180

related to voltage by the diffusion equation. Four scale
factors were then introduced as a basis for relating all
quantities in the hydrologic and electrical systems. In
addition, Darcy's and Ohm's laws were introduced to relate
resistance to transmissibility, and the definition of capa-
citance related capacitance to storage. This development
was followed by a discussion of modifications to include
recharge from precipitation minus evapotranspiration, a
discussion of scaling the model, and a discussion of the
finite difference approximation to the continuous model.
Finally, this development was related to an unconfined

aquifer. It also can be applied to the confined situation.

Development of Hydrologic Formulas
for an Unconfined Aquifer

It is necessary to make certain assumptions in
develOping formulas for an unconfined aquifer which in
the physical situation are only approximately true. These
approximations are embodied in the following assumptions.

1. The water is incompressible.

2. Any changes in the aquifer's saturated thick-
ness is small enough that transmissibility
at any given location may be considered

constant.

3. The water budget for a vertical section of
aquifer is completely described by the

equation

181

(horizontal ground water flow entering the
section of aquifer) - (horizontal ground water
flow leaving the section) + (ground water
entering as precipitation - ground water leav-
ing as evapotranspiration) = (change in the

amount of ground water in storage).

Let us next express these three statements in a mathe-
matical format. Consider an elemental volume of the
aquifer of dimensions AX, AY, and b in feet as shown in
Figure I-2. Here b represents the vertical thickness of
the aquifer. At any given location in the aquifer, fluid
flow = T ah/BS where as is in the direction of maximum

head loss. In vector notation, this is eXpressed as
fluid flow = T GRAD h

This vector is taken at a point along the surface so that
it represents the average flow either into or out of the
elemental volume. Here T, the coefficient of transmissi-
bility, is in gallons/day/foot. Assumption (3) may now
be stated as

a) horizontal ground water entering the aquifer

in units of gallons/day/foot for an increment
of time At in days

TIGRAD h|(AX cos ¢2 + AY cos ¢1)At

but AX = AY
TIGRAD h|(Ax cos p2 + AX cos ¢1)At

182

TIGRAD hl(cos ¢2 + cos ¢1)AXAt

T(IGRAD hlcos ¢2* + IGRAD hlcos ¢1)AXAt

Q) 0)
xi:
+
Q) Q)
KID‘

a) a)

 

]AXAt

Similarly

b) horizontal ground water leaving the aquifer

TIGRAD hI(AX cos ¢3 + AY cos p4)At

TIGRAD h|(AY cos o3 + AY cos ¢4)At

TIGRAD h|(cos p3 + cos ¢4)AYAt

39. + 33 ]AXAt

I
*3

8

 

where the subscripts a) and b) indicate the faces of cur-
rent inflow and current outflow respectively.

It follows that (horizontal ground water flow
entering the section of aquifer) - (horizontal ground water
flow leaving the section) for the increment of time At may

now be expressed as

 

*IGRAD hlcos ¢2 is the scalar product of the
gradient vector with a unit vector in the direction of the
X axis. It may readily be shown that this is equivalent
to ah/ax. Similar reasoning applies to the develOpment
of ah/BY.

183

Y
5 Coordinate Axes AX IAY
AY

 

CURRENT FLOW
ITS DIRECTION
AND MAGNITUDE
IS REPRESENTED
BY - T GRAD h

 

 

,;---—---
’l

x’ D

’ x

 

 

 

CURRENT FLOW
VECTOR

 

 

 

 

FLOW
VECTOR

Figure I-2

Horizontal Flow Through a, An Elemental
Volume of the Aquifer

 

 

 

 

= 113% - 13- + 11% - p—g ]AXAt
a) b) a) b)
22 - 1. 2E _ i.
= T axa)Ax axb) AX + Ya)AY ayb) AY AXAt
Eh _ 12 22 - 3.1
= T axa)Ax xb) + a a)AY ayb) AXAYAt .

In the limit as AX and AY approach zero, this expression
becomes

2 2
= T i.% + i.% AXAYAt
ax 3y

 

c) Ground water entering as precipitation -
ground water leaving as evapotranspiration for
the time increment At in gallons/day/ft2

= (QPpt. gal/day/ft2 _ QEt gal/day/ft2)AXAYAt

and

d) change in the amount of ground water in
storage

0)

= 7.48 s 33 AXAYAt

where ah/Bt represents instantaneous change in water
table elevation in feet/day, and S, the storage coef-

ficient, is a dimensionless number.

185
In its entirety, (3) may be stated as

2 2
T[§—% + §—%}AXAYAt + (QP t - Q t)AXAYAt
3x aY p ‘ E

- 911
— 7.48 s at AXAYAt

_ § 32.- l
——— + -—— — 7.48 T T(Q

t Ppt.

Representing precipitation and evapotranspiration as one

quantity (QP _ Et) results in

(I-14)

NI
+
NI
II
\I
.2.
on
Film
0) Q)
r4:
I
Hill--l

QP - Et'

In the case where QP is zero, equation I-l4 reduces

— Et
to

2 2
a h a h _ 3 3h _
——5 + -—— — 7.48 ¥«—— . (I 4)

ax ay t

APPENDIX II

THEORY OF GEOPHYSICAL METHODS

APPENDIX II

THEORY OF GEOPHYSICAL METHODS

The basic theory of seismic refraction, earth
resistivity, and earth gravitational methods have been
extensively developed and are covered in detail by many
authors. It is beyond the purpose of this study to
attempt to review this theory in any detail. The intention
here is rather to present for the reader who is unfamiliar
with these methods a very basic introduction to the pro-

cesses involved.

Seismic Refraction Methods

 

The discovery that seismic waves in a layered
earth obey the same basic optical laws of refraction and
reflection as light waves led to simple and direct methods
for processing seismic data. Snell's law, which may be

stated as

-+———— — —— (II-l)

can be used to calculate the behavior of refracted

seismic waves at an interface. Referring to Figure II-l,

186

187

01 and 02 are the angles that the incident and refracted
rays make with a normal to the interface, and V1 and V2
are the respective velocities of the incident and re-

fracted waves with V being greater than V For this

2 1'

condition as O1 increases, O2 also increases to the point

where 62 = 90°. Snell's law then takes the form

V
l 1
V— (II-2)

2

91 = sin-
and the refracted wave travels at the higher velocity of
the lower medium.

These foregoing relationships serve as the basis
for the develOpment of formulas that give the depths to
interfaces and compressional wave velocities in a layered
medium from the first arrival compressional wave recorded
at the surface. Calculations are relatively simple for
two layers separated by a horizontal interface at depth Z
with compressional wave velocities V1 and V2 as defined
before. Figure II-l shows the seismic profile for this
two-layer case. The explosive is located at the point
X = 0, and a series of detectors are located along a
straight line extending out at increasing distances (X)
from the shot point. The explosive is detonated at time
T = 0, and the time of arrival (T) of the first compres-
sional pulse at each detector is recorded. This data can
be plotted on a graph of T vs. X as in Figure II-2. Near

the shot, the compressional wave in the upper layer arrives

188

first with a velocity Vl as indicated by the straight
line with slope l/VZ. Beyond a certain distance, the
critical distance (Xc),which is a function of V1, V2, and
Z, waves from the deeper layer arrive first with a
velocity of V2. These arrivals appear on the graph as

a second line with slope l/Vz. It can be shown that this

line is represented by the formula

 

2 2
22ml .. v0

T = —— + . (II-3)
V1 V1V0

 

Ti' the intercept time, is the value of T on this line
when X = 0, and a value for Ti can be obtained graphi-
cally by extending this line back to the vertical axis
(Figure II—l). With X = 0 and a value for Ti Obtained,

equation II-3 may be rewritten to solve for Z in the form

 

 

T.
z = 7% l 0 (II-4)

These readily deve10ped formulas are derived and
discussed by many authors. More complex extensions of the
reasoning leading to these formulas produce solutions for
depths and velocities in cases where multiple horizontal
or multiple inclined interfaces are encountered (Dobrin,

1960; Jakosky, 1950; Ewing, Woolard, and Vine, 1939).

189

 

 

INTERFACE

J

---J-—-

  
 

 

 

Figure II-l

Snell's Law

 

 

 

 

 

 

Figure II-2

Time-Distance Graph for a Two—Layer Case
with a Horizontal Interface

 

 

 

190

Earth Resistivitngethods

 

The resistivity p of any material is defined as
the electrical resistance across Opposite faces of a unit
cube of this material. If the resistance is measured in
ohms and the units of length are in centimeters, then the
resistivity is given in ohm—cm. A rectangular block of
conducting material with two Opposite faces of area 52

separated by a length I would display a resistance of

R = -—- (II-5)

to a voltage applied across these two faces. Conversely
the resistivity of this material may also be computed from
equation II-5 if its resistance and dimensions are known.
A knowledge of resistivity in the underlying earth
can furnish information on its composition and structure.
A method frequently employed to determine earth resistivity
is to introduce current into the ground by means of two
electrodes placed in the ground with a voltage established
across them. Two additional electrodes are used to re—
cord the resulting potential drOp between selected points
on the surface of the ground. These four electrodes are
arranged along a straight line as shown in Figure II-3.
It can be shown (Jakowsky, 1950; Dobrin, 1960) that the
resistivity of the underlying earth may be calculated from

this electrode configuration by the formula

191

 

 

 

 

 

 

 

 

 

 

 

Zfi077777IIII71/1I/I/I/I/I/II/I/I/IIIII
L—————q§ «on——1h-——9

 

Figure II-3

Colliner Resistivity Electrodes

 

 

 

APPARENT RESISTIVITY
PXIO3 OHM ' CM

 

 

 

O . I DO ZOO SURFACE
0 q f
SPAcmG . 60' P08 200:: l030HM-CM
FEET A L
I00 3 P. sloaxlo3 arm-cm
200 T
: Figure II—4
- Two-Layer Apparent Resistivity
300 T

 

 

 

 

192

 

(II-6)

where r r and r4 are distances between the

1' 2' r3'
electrodes, V is the voltage between the inner electrodes,
and i is the current passing between the outer electrodes.
In the particular instance where the four electrodes are
spaced collinearly an equal distance apart and the volt-

age electrodes are placed between the current electrodes;

equation II—6 reduces to

p = Zna (II—7)

El
1
where”H'is the spacing between any two adjacent electrodes.
This particular electrode arrangement, known as the Wenner
configuration, was employed in this study. A variation of
the Wenner configuration is called the Lee Partitioning
method. In this arrangement an extra electrode is placed
in the center of the spread. Potential differences are
measured between the original potential electrodes as well
as between the center electrode and the original potential
electrodes. This configuration can be used to check the
homogeneity of resistivity values measured along the line

of electrode placement for a given a" Spacing.
If resistivity is constant throughout the earth,
then equation II-7 will give the same value for p regard-

less of the "a" spacing. When earth resistivity is not

193

constant but, as an example varies with depth, then the
value obtained for p by equation II-7 represents what is
called an effective or apparent resistivity. This value
will vary with the "a" spacing. When the spacing is small,
the value of p represents the near surface resistivity.
Increasing the "a" spacing results in values of p that
represent a combination of the surface and increasingly
deeper layers. With increasing electrode separation, the
effect of the surface resistivity will diminish. Thus by
varying the "a" spacings, we can study the variation of
resistivity with depth. An example of this is shown in
Figure II-4. The apparent resistivity has been plotted
against "a" spacing for a specific resistivity spread on
the left, and an interpretation of this graph is given on
the right. As the "a" spacing is increased along a
resistivity spread, the apparent resistivity of an in—
creasingly deeper section of material is measured. It is
assumed that this increment in depth is the same as the
increment in 9a" spacing. As deeper layers of earth are
penetrated, the effect of the resistivity of these layers
on the measured apparent resistivity is obscured by the
increasing volume of overlying earth. The Barnes layer
method (Barnes, 1952) tends to minimize the masking effect
of the overlying layers. That is for any two successively
increasing values of "a" spacing, the Barnes' layer
equation mathematically strips away the overlying material

and measures the apparent resistivity of the layer of

194

material added by this increment. This equation is of

the form

1 (II-8)

 

where Aa is the increment in "a" spacing in feet, pL is

the apparent resistivity of the incremental layer, and Rn

Rh_l are the apparent resistivities measured for electrode

spacings of a and a + Aa respectively with these resis-

tivities being measured in ohm-centimeters.

APPENDIX III

F¢RTRAN DIGITAL COMPUTER PROGRAMS

APPENDIX III

F¢RTRAN DIGITAL COMPUTER PROGRAMS

Program B LAYER

 

Program B LAYER (Table III-l) computes apparent
seismic refraction compressional wave velocities for a
horizontal bottom layer in a three-layer case when the
interface between the first and second layers is inclined.
The program will handle as many successive profiles as
there is data for. The nomenclature and conventions in
the program are those used by Johnson (1964). Program
inputs are an identifying number for the profile (PRFLNO):
the velocity of the surface layer (V1), the true velocity
of the second layer (V2T), the true velocity of the bottom
layer (V3T), the critical angle at the inclined interface
(ANGI12), and the dip of the inclined interface (ANG¢12).
ANG¢12 is positive if the "A" end of the refraction pro—
file (arbitrarily chosen in the beginning) is the deep
end and negative if it is the shallow end. The apparent
velocities are given in the output as V3A and V3B and
refer to the velocities that would be obtained in shooting

from the "A" and "B" ends of the profile respectively.

195

 

196

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197

Blind zone layer computations in this study used the value

of 6500 for V3T.

Program RH¢

 

Inputs for program RH¢ are direct ratios of V/i

utilizing a Wenner configuration at a spacings from 20
to 280 feet. Measurements of V/i are read into a dimen-
sioned array (R(1), R(2),...,R(15)) starting at the 20

foot "a' spacing and incrementing to 280 feet. The pro-
gram will handle as many successive profiles as there is
data for. The program computes apparent resistivity
(SIMPLE RHO), summed apparent resistivity (CUMULATIVE RH¢),
and Barnes layer resistivities (RH¢ 20, RH¢ 40, ...,

RH¢ 280).

198

 

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APPENDIX IV

SEISMIC REFRACTION AND ELECTRICAL

RESISTIVITY PROFILES

 

 

EXPLANATION

 

27

\

Profile 27

 

 

 

 

 

eh:

Section Corners .,
T 5%

 

 

 

Numbers With an
Asterisk Indicate
onIy 3 Seismic Profile

13
SCALE
Feet 18
W

 

Figure IV-l

Location of Seismic and Resistivity
Profiles

199

 

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