Sim?E¥ P ROF ESS ION A£ PAPER
Numerical Simulation Analysis 

of the Interaction of 

Lakes and Ground \Vater

By THOMAS C. WINTER

GEOLOGICAL SURVEY PROFESSIONAL PAPER 1001

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1976
UNITED STATES DEPARTMENT OF THE INTERIOR 

THOMAS S. KLEPPE, Secretary

GEOLOGICAL SURVEY 

V. E. McKelvey, Director

First printing 1976 

Second printing 1977

Library of Congress Cataloging in Publication Data 

Winter, Thomas C.

Numerical simulation analysis of the interaction of lakes and ground water. 

(Geological Survey Professional Paper 1001) 

Bibliography: p.

1. Groundwater flow Mathematical models. 2. Groundwater flow Data processing. 3. Lakes Mathematical models. 4. LakesData processing. 5. Numerical analysis.

I. Title. II. Series: United States Geological Survey Professional Paper 1001. 

GB1001.72M35W56 551.4'82 76-608309

For sale by the Branch of Distribution, U.S. Geological Survey 

1200 South Eads Street, Arlington, VA 22202
CONTENTS

rage

Abstract.......................................................................................................................................... 1

Introduction......................................................................................... ......................................... 1

Background of the study, including previous investigations................................................ 1

Purpose and scope.................................................................................................................. 3

Acknowledgment.................................................................................................................... 3

Ground-water flow systems........................................................................................................... 3

Theoretical background......................................................................................................... 3

Models of hydrologic sections................................................................................................ 3

Digital models of ground-water flow near lakes .......................................................................... 5

Practical considerations and assumptions of this study........................................................ 5

Ground-water-flow diagrams................................................................................................. 7

One-lake system...................................................................................................................... 11

Multiple-lake system............................................................................................................... 29

Moderate regional slope and local relief........................................................................ 29

Low regional slope and local relief................................................................................ 31

Quantitative aspects of ground-water interchange with lakes.............................................. 42

Areal and temporal variations........................................................................................ 42

Implications for field studies of the interaction of lakes and ground water........................ 43

Summary........................................................................................................................................ 44

References cited.............................................................................................................................. 44

ILLUSTRATIONS

Page 

FIGURE 1. Sketch showing physical meaning of mathematical expressions for boundaries of hydrologic-section models........................... 4

2. Hydrologic section showing local, intermediate, and regional ground-water-flow systems determined from an analytical

solution to the ground-water-flow equation ............................................................................................................................ 6

3. Hydrologic section showing ground-water-flow systems determined from a numerical solution to the ground-water-flow

equation........................................................................................................................................................................^ 6

4. Hydrologic section showing a quasi-quantitative flow net of ground-water flow near lakes in a multiple-lake system that

does not contain aquifers.......................................................................................................................................................... 8

5. Hydrologic section showing a quasi-quantitative flow net of ground-water flow near lakes in a multiple-lake system that

contains aquifers........................................................................................................................................................................ 10

6. Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a multiple-lake system that

does not contain aquifers.......................................................................................................................................................... 12

7. Hydrologic section showing the dimensions of features included in the diagrams of the one-lake system.................................. 13

8. Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a one-lake system that does

not contain aquifers or lake sediments..................................................................................................................................... 14

9. Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a one-lake system that contains

lake sediments but does not contain aquifers........................................................................................................................... 16

10. Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a one-lake system that has a

low water-table mound on the downslope side of the lake and does not contain aquifers.................................................... 17

11. Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a one-lake system that has

low water-table mounds on both sides of the lake and does not contain aquifers ................................................................. 18

12. Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a one-lake system that has

an extensive aquifer of relatively low hydraulic conductivity at the base of the system......................................................... 19

13. Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a one-lake system that has

an extensive aquifer of relatively high hydraulic conductivity at the base of the system....................................................... 20

14. Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a one-lake system that has

an extensive aquifer of relatively low hydraulic conductivity in the middle of the system.................................................... 21

15. Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a one-lake system that has

an aquifer of limited extent upslope from the lake at the base of the system......................................................................... 22

16. Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a one-lake system that has

an aquifer of limited extent beneath the lake at the base of the system.................................................................................. 23
IV CONTENTS

Page

FIGURE 17. Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a one-lake system that has an

aquifer of limited extent downslope from the lake at the base of the system......................................................................... 24

18. Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a one-lake system that has

a deep lake and an extensive aquifer of relatively low hydraulic conductivity at the base of the system.............................. 25

19. Sketch showing key to table 1.......................................................................................................................................................... 27

20. Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a multiple-lake system that

has an extensive aquifer of relatively high hydraulic conductivity at the base of the system ................................................ 33

21. Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a multiple-lake system that

has water-table mounds of various height and aquifers of limited extent in the middle of the system................................. 34

22. Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a multiple-lake system that

has low water-table mounds and aquifers of limited extent at the base of the system........................................................... 35

23. Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a multiple-lake system that

has low water-table mounds and aquifers of limited extent in the middle of the system...................................................... 36

24. Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a multiple-lake system that

has high water-table mounds and aquifers of limited extent at the base of the system.......................................................... 37

25. Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a multiple-lake system that

has deep lakes, high water-table mounds, aquifers of limited extent at the base of the system, and Kk /Kv=\,000............... 38

26. Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a multiple-lake system that

has deep lakes, high water-table mounds, aquifers of limited extent at the base of the system, and Kh /Kv=lQO.................. 39

27. Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a multiple-lake system that

has low regional slope and does not contain aquifers............................................................................................................. 40

28. Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a multiple-lake system that

low regional slope and aquifers of limited extent in the middle of the system...................................................................... 41

TABLES

Page 

TABLE 1. Results of digital simulations of the one-lake system comparison of the effects of all combinations of the parameters

that control ground-water flow on lake-ground water interaction......................................................................................... 26

2. Summary of simulations of one-lake system ................................................................................................................................... 28

3. Summary of simulations of multiple-lake system whose water table has moderate regional slope and local relief..................... 32

4. Summary of simulations of multiple-lake system whose water table has low regional slope and local relief............................. 32
NUMERICAL SIMULATION ANALYSIS OF THE 

INTERACTION OF LAKES AND GROUND WATER

By THOMAS C. WINTER

ABSTRACT

Because the interrelationship of lakes and ground water is perhaps 

the least understood aspect of lake hydrology, vertical-section, steadystate, numerical-model simulations were run to evaluate the factors 

that control the interaction of lakes and ground water. The study is 

concerned only with lakes encircled by water-table mounds that are 

at a higher altitude than lake level. Simulations of one-lake and 

multiple-lake systems in vertical sections show that for many hydrogeologic settings, the line (divide) separating local from regional 

ground-water flow systems is continuous beneath individual lakes. If 

the divide is continuous, there exists a point along it at which the 

head is a minimum compared to all other points along the divide. 

This point of minimum head is always greater than the head represented by lake level, therefore in such a setting there can be no movement of lake water through the lake bed to the ground-water system. 

In a setting where the divide is not continuous, the lake loses water 

through part of its bed, but rarely in the littoral zone of the lake.

Factors that strongly influence the position, shape, and continuity 

of the flow-system divide beneath lakes are height of the water table 

on the downslope side of the lake relative to lake level, position and 

hydraulic conductivity of aquifers within the ground-water reservoir, 

ratio of horizontal to vertical hydraulic conductivity of the groundwater system, and lake depth.

INTRODUCTION

BACKGROUND OF THE STUDY, INCLUDING 

PREVIOUS INVESTIGATIONS

Lakes are a valuable natural resource in many areas of 

the United States. In addition to being sources of water 

supply for many communities, they are the focus of 

recreational activity and their aesthetic qualities are 

highly valued. That they are a desirable feature is 

evidenced by the large number of small reservoirs being 

constructed throughout much of the United States for a 

wide variety of purposes. Increasingly, lakes are being 

used as the focal point in planning urban developments.

Unfortunately, the popularity of lakes often leads to 

their deterioration. The increased input of nutrients 

through the activities of man causes the organisms in lakes 

to flourish at rates far in excess of natural conditions. This 

increased productivity in turn causes chemical changes in 

lakes that result in obnoxious odors, fish kills, and

unsightly conditions in the lakes and along their 

shorelines.

The number of problem lakes is sharply increasing. In 

order to evaluate lake problems and the effect of management projects, nutrient budgets and water budgets of lakes 

are needed. Only by thorough understanding of the interrelationships of lake hydrology, chemistry, and biology 

can progress be made in alleviating present lake problems 

and preventing more problems from arising.

In recent years, much money and research effort have 

been expended to understand the principles of lake 

chemistry, biology, and the nutrient balance of lakes. At 

the same time, comparatively little money and effort have 

been devoted to understanding the principles of lake 

hydrology or the water balance of lakes. With few exceptions, the relationship of ground water to lakes has been a 

minor part of the hydrologic studies, and it remains the 

least studied and least understood aspect of lake 

hydrology.

In most studies of lake hydrology, atmospheric water 

(precipitation and evaporation) and surface-water interchange with lakes is measured. Ground water is generally 

calculated as the residual. This practice can lead to serious 

misunderstandings about the interaction of lakes and 

ground water, especially when the error of measurement 

inherent in precipitation and particularly evaporation is 

considered. Further, only through careful field techniques can streamflow measurement error be kept to 

relatively low values.

Some misconception of the interaction of lakes and 

ground water has resulted from lack of understanding of 

ground-water flow, which leads to inadequate instrumentation and data collection. Poorly executed studies 

have led to various erroneous ideas concerning the interaction of lakes and ground water, to the detriment of 

sound lake management. Some scientists and managers 

believe all lakes are discharge points of the ground-water 

system and therefore do not lose water through their beds. 

Others believe that lakes are points of recharge to the 

ground-water system. Some believe ground water flows in
INTERACTION OF LAKES AND GROUND WATER

one side of lakes and out the other, a flow-through condition. Others believe all three situations exist.

Most studies of the interaction of lakes and ground 

water have been in response to a need for water budget 

information for a particular lake or groups of lakes. Some 

studies have not used observation-well instrumentation to 

define the ground-water flow systems, but instead assumed 

certain hydrogeologic conditions. Generally those that 

used wells either used too few wells, that is they assumed a 

simple relationship between lakes and ground water that 

could be determined by one or two wells near the shoreline, or they installed too many wells within the 

immediate drainage basin of the lake. Some studies 

assumed that definition of the water table near a lake was 

sufficient to understand the interactions.

The relation of ground water to prairie potholes in 

North Dakota was studied by Eisenlohr and others (1972). 

Although in the early stages of this study the ground-water 

component was calculated as a residual, a seepage measuring device was subsequently developed. In latter stages 

of the study, wells were placed near some potholes to determine the relationship of ground-water levels to pothole 

water levels (Sloan, 1972).

The ground-water relation to small lakes in Minnesota 

was studied by Manson and others (1968) and Allred and 

others (1971) by placing one or several wells near the lake 

shores. The study approach was similar to that of 

Eisenlohr and others (1972). The general conclusion of 

these studies was that most of the lakes studied had a net 

loss to ground water.

In a study of Lake Sallie, Minnesota (McBride, 1969), the 

ground-water component of the lake-water budget was a 

primary interest. Wells were placed within the entire 

drainage basin of Lake Sallie to define lateral groundwater movement. At some sites several closely spaced wells 

were completed at different depths to define the vertical 

component of ground-water flow. Flow nets were then 

constructed to calculate the quantities of ground water 

moving into and, in one small area during part of the year, 

out of Lake Sallie. McBride used a digital modeling technique to define the vertical distribution of hydraulic head 

in the ground-water system near a lake. His models are of 

the flow within the immediate drainage basin of the lake 

itself from the local ground-water divide to the midpoint of the lake (McBride and Pfannkuch, 1975).

Studies of the interaction between lakes and ground 

water similar to those conducted by McBride (1969), 

although not using simulation modeling, were done in 

Wisconsin by Hackbarth (1968), Hennings (1974), and 

Possin (1973). The lakes in these studies are the flowthrough type ground water enters one side of the lakes, 

and the lakes leak to the ground-water system on the other 

side.

Meyboom (1966, 1967) studied ground-water flow 

systems in the vicinities of lakes and potholes in the prairie

provinces of Canada. Meyboom's work is some of the first 

to examine the problem of determining ground-water 

flow systems and the strength and position of groundwater divides or absence of such divides beneath lakes. 

Studies by Freeze (1969a, 1969b) consider the relationship 

of ground-water flow systems within large drainage basins 

to lakes in the prairie provinces of Canada. The scale of the 

hydrologic sections, however, is such that the detailed flow 

patterns by the lakes can not be defined.

Williams (1968) examined ground-water flow systems 

in vertical sections near small depressions, and the 

relationship of those flow systems to wetlands in northern 

Illinois. The techniques of investigation and results of the 

study were similar to the work of Meyboom. Studies 

similar to and using the techniques of Meyboom were also 

done in East Germany by Schumann (1973).

A general overview of studies of the interaction of lakes 

and ground water, including a literature review, is given 

by Born and others (1974). They also discuss and present 

conceptual diagrams of many variations of ground-water 

flow systems near lakes.

The relationship of ground water to large lakes and 

reservoirs has been studied by several Russian hydrogeologists. Zektzer (1973) discussed the role of groundwater flow in water and salt balances of Lake Baikal and 

the Caspian Sea. Zektzer and Kudelin (1966) discussed the 

methods of determining ground-water flow to lakes with 

special reference to Lake Ladoga, USSR. Payne (1970) 

used radioactive tracer techniques to study the groundwater aspect of the water balance of Lake Challa, Africa. 

Haefeli (1972) has calculated the ground water inflow into 

Lake Ontario from the Canadian side. The study made use 

of the cross-sectional digital ground-water flow model of 

Freeze (1969b). Van Everdingen (1972), as part of an intensive study of the Lake Diefenbaker, Saskatchewan, site, constructed a series of piezometer nests (closely-spaced group 

of wells, each well completed at a different depth) before 

the dam was constructed. The changes in the potentiometric level of different aquifers in the ground-water 

system were observed as the lake filled. The study showed 

reversals of flow within some of the aquifer zones as a 

result of the creation of Lake Diefenbaker. A similar study 

is under way at the Kendrid Lake site, North Dakota, by 

Downey and Paulson (1974).

Careful review of the above literature leads one to the 

conclusion that much work needs to be done to identify 

and evaluate the factors that control lake hydrology, 

especially the factors that control the relationship of 

ground water to lakes. The general principles underlying 

the interaction of lakes and ground water need to be 

defined on theoretical grounds to solve even such basic 

problems as determining the optimum number and placement of observation wells needed to define ground-water 

flow systems near lakes.

A logical first step in defining the general principles
GROUND-WATER FLOW SYSTEMS

controlling the interaction of lakes and ground water is to 

use simulation modeling to examine the detailed patterns 

of ground- water flow around and beneath lakes for a wide 

variety of hydrogeologic settings.

PURPOSE AND SCOPE

The purpose of this report is to examine ground-water 

flow systems near lakes through digital models simulating 

ground- water flow in vertical sections. Use of such models 

makes it possible to examine the general principles of the 

interaction of lakes and ground water. By varying shape 

and altitude of the water table in relation to lake levels, 

lake depth, sediment distribution, size and position of 

aquifers within the ground- water system, ratio of 

horizontal to vertical hydraulic conductivity, ratio of 

hydraulic conductivity of aquifers to that of the surrounding finer-grained materials, and thickness of the 

ground-water reservo-V, it is possible to locate the position and strength (explained later) of the divide that exists 

under a lake or the percentage of the lake bottom through 

which water moves to the ground-water system if there is 

no continuous divide. Although the cross sections are 

hypothetical, the values of the parameters listed above are 

realistic and representative of physiographic and hydrogeologic conditions in glacial terrane, the geologic 

environment in which most natural lakes occur. Lakes in 

this type of terrane are of primary interest in this study, but 

the models used to define the interaction of lakes and 

ground water and the general principles evolved from this 

study should have general application to all types of 

geologic terrane.

ACKNOWLEDGMENT

This study constituted part of a dissertation (Winter, 

1976a), which was submitted to the Faculty of the 

Graduate School at the University of Minnesota in partial 

fulfillment of the requirements for the degree of Doctor of 

Philosophy. Dr. H. O. Pfannkuch provided overall 

guidance during the course of the study. With minor 

modifications the dissertation was released to the open-file 

(Winter, 1976b).

GROUND- WATER FLOW SYSTEMS

THEORETICAL BACKGROUND

This study is concerned only with two-dimensional, 

steady-state, non-homogeneous and (or) anisotropic, 

cross-sectional systems. Thus the basic equation of 

ground-water flow that is used in this report for defining 

the head distribution within the ground-water system is

where K(x,z)=hydraulic conductivity in the two coordinate directions and

ih., ^h= gradient of hydraulic head.

6X 6Z

Equation 1 results from coupling Darcy's equation with 

the equation of continuity. It assumes that fluid density is 

constant and the coordinate axes are aligned collinear 

with the principal directions of anisotropy. Development 

of the equation is discussed in most textbooks of groundwater hydrology (e.g., Domenico, 1972; DeWiest, 1965).

In order to solve equation 1, it is necessary to define 

mathematically the boundary conditions for the region of 

interest. A general model for the type of flow section of 

interest in this report is shown in figure 1. This figure 

represents an x-z coordinate system that has the origin in 

the lower left corner. For any point (P) in the system there 

corresponds a value of hydraulic head. Because there is no 

flow across the vertical boundaries projected down from 

the major divide and the major sink, the head gradient is 

zero f|-~o) along the two sides of the diagram. It is also 

assumed that the base of the system is impermeable, 

thus-|-|-=o. The pressure along the water table is 

atmospheric and the head (h) there is a function of x only. 

The hydraulic conductivity (K) values are different for 

each geologic unit within the system. For this study, the 

size and position of zones of high hydraulic conductivity 

(termed aquifers for convenience) are varied, and the 

degree of anisotropy is varied.

Recent development of numerical methods as approximate solutions to the equation of ground-water flow 

(Remson and others, 1971; Trescott and Finder, 1975; 

Cooley, 1974; PrickettandLonnquist, 1971; Freeze, 1969b) 

has made it possible to simulate complex ground-water 

flow systems. Although a variety of numerical techniques 

are being developed, the finite-difference method has been 

well documented and has been used successfully in a 

number of studies. The finite-difference models used 

herein have been developed by the U.S. Geological Survey 

over a number of years (Pinder and Bredehoeft, 1968; 

Pinder, 1970; Trescott, 1973; Trescott and Pinder, 1975).

The alternating-direction-implicit method (ADI) was 

used to calculate the distribution of hydraulic potential 

within some of the cross-sectional simulations but, as 

more complex models were considered, it was necessary to 

turn to the strongly implicit procedure (SIP) (Stone, 1968). 

As an example of SIP's greater efficiency, some checkruns 

were made on the same section, using both numerical techniques; ADI did not converge after 99 iterations, whereas 

SIP needed as few as 16. It was found that SIP adequately 

handled all conceivable simulations except one, where 

extremely high hydraulic conductivity contrasts existed in 

close proximity, both in the vertical and horizontal 

directions.

MODELS OF HYDROLOGIC SECTIONS

Only two comprehensive theoretical studies appear to 

have been published on modeling ground-water flow in
INTERACTION OF LAKES AND GROUND WATER

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PRACTICAL CONSIDERATIONS AND ASSUMPTIONS

vertical section (Toth, 1963; Freeze, 1969b). Subsequent 

work based on the report by Freeze has been done by 

Haefeli (1972) and by Freeze (1969a). These studies have 

concentrated on ground-water-flow patterns in vertical 

section in large basins and, although lakes occur along 

some of the sections modeled, the scale of the sections were 

such that detailed flow patterns in the immediate vicinity 

of lakes could not be shown.

Toth's work is based on an analytical solution to the 

ground-water-flow equation, whereas the work of Freeze is 

based on a numerical solution. Freeze (1969b) discusses the 

advantages and disadvantages of each approach. In the 

analytical-solution approach Toth had to make some 

rigid simplifying assumptions: The field has to be approximated by a rectangle; the ratio of horizontal to vertical 

hydraulic conductivity must be 1, that is, the porous media 

must be isotropic; the water table is simulated by superimposing a sine curve on a low regional slope. The 

analytical approach requires that equations be set up for 

each individual case considered according to the boundary 

configurations. There is no general analytical solution to 

the ground-water-flow equation that is valid for all 

boundary conditions at the same time.

The advantages of the numerical solution as outlined by 

Freeze (1969b), on the other hand, are as follows: Many of 

the restrictions of the analytical solution are removed; the 

true shape of the water table can be approximated, that is, 

it is not restricted to the equation for a sine curve; equation 1 can be handled in the numerical solution, that is, 

anisotropic and heterogeneous conditions can be 

simulated. The numerical solution available in the USGS 

program is general; it was designed to handle a wide 

variety of hydrogeologic problems, including the vertical 

section simulations used in this study.

Both basic studies by Toth and Freeze assume no-flow 

boundaries at the base of the system and no-flow 

boundaries beneath the regional topographic highs and 

lows. In addition, the water table is at steady state.

An example of one of Toth's cross-sectional groundwater flow simulations is shown in figure 2. The simulation is of a thick ground-water system that has low 

regional slope and low local relief. Although Toth 

presents many such diagrams showing the effects on 

ground-water flow of many combinations of slope, relief 

and system thickness, this diagram best depicts his 

delineation and definition of local, intermediate, and 

regional flow systems.

An example of the flexibility of numerical models in 

closely simulating field conditions is shown in figure 3 

(Freeze, 1969b). The diagram is of a hydrogeologic setting 

that has a variable water table in different parts of the 

section, anisotropic media, and zones of different 

hydraulic conductivities.

DIGITAL MODELS OF GROUND-WATER FLOW 

NEAR LAKES

PRACTICAL CONSIDERATIONS AND ASSUMPTIONS OF 

THIS STUDY

The boundary conditions assumed for this study relate 

closely to those of Freeze (1969b), that is: (1) The base of the 

system is considered to be impermeable; (2) the vertical noflow boundaries at each end of the section are considered to 

be controlled by the major topographic high and the 

major drain on the ground-water system; and (3) the upper 

boundary of the flow system is the water table and it is at 

steady state. In the model used for this study, the steadystate water table is specified by assigning effectively 

infinite storage to the water-table nodes. Freeze discusses 

thoroughly the justification for these boundary conditions and particularly the assumption that vertical noflow boundaries exist at the two end positions. As he 

points out, many workers have found that vertical no-flow 

boundaries do not occur beneath every topographic high. 

This is undoubtedly true, but the assumption that they do 

exist beneath the major high and low is considered to be 

valid.

Although the sections discussed in this report are hypothetical, they closely approximate field conditions for 

lakes in glacial terrane. The variations in height of the 

water table relative to lake level are reasonable for this type 

of terrane. Although the hydraulic conductivities that 

have been assigned to the geologic matrix in general are 

representative of silty till, in the simulations of this study, 

the relative values of hydraulic conductivity are the controlling factor, not the actual values. The simulated 

sections apply equally to geologic settings that have the 

given relative hydraulic conductivities, whether the 

setting is sand within till, or gravel within sand. The 

hydraulic conductivities assigned to the zones of higher 

relative hydraulic conductivity vary from 100 to 1,000 

times greater than the surrounding geologic materials. 

There are, of course, some zones within glacial drift that 

have hydraulic conductivity values greater than 1,000 

times that of the surrounding materials, but the values 

chosen are representative of a great many geologic 

settings. For convenience in this report, the geologic 

matrix is referred to as "till," and the zones of relatively 

higher hydraulic conductivity are referred to as 

"aquifers."

The hydraulic conductivities assigned to lake sediments are as low as can be assigned with the computer 

program used. In most lakes, the littoral zone is free of fine 

grained sediment; therefore, in the models that considered 

lake sediments, the sediments purposely were not extended 

to the shore line.

Lake water was simulated by assigning very high
INTERACTION OF LAKES AND GROUND WATER

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GROUND-WATER-FLOW DIAGRAMS

hydraulic conductivity values to the nodes within lakes. 

This proved to adequately simulate lakes because the calculated head values were identical to the initial head 

values within each lake simulated in this study.

The least well known parameter needed in the groundwater-flow model is the ratio of horizontal to vertical 

hydraulic conductivity (K^/KV ). This ratio has generally 

not been determined in glacial terrane and it is a point 

around which much of the discussion of the following section of this report centers. It will be shown that it is critical 

to the interaction of lakes and ground water. If the ratio is 

less than 100, lakes rarely lose water to the ground-water 

system under the conditions simulated in the models and if 

the ratio is greater than 1,000, lakes lose water under many 

conditions. If the ratio is between 100 and 1,000, other 

factors that control ground-water flow become more 

important in the relationship of ground water to lakes.

Weeks (1969) measured K^/KV within a single outwash 

sand and gravel aquifer in Wisconsin and found the ratio 

to be not more than 20. Vecchioli and others (1974) calculated Kh/Kv for part of the drift section of Long Island, 

New York as approximately 500. Bennett and Giusti 

(1971), in using electric-analog techniques to study 

ground-water flow in the coastal plain of Puerto Rico, 

show that the ratio had to be 1,000 for the simulations to 

match field data. The importance of this parameter in 

ground-water-flow modeling has been recognized because 

it is the topic of recent papers by Freeze (1972) and Gillham 

and Farvolden (1974). Freeze concludes that Kh /Kv values 

of 100 or larger are not uncommon, and in fact were 

necessary to correlate simulations with field measurements in his study of the Old Wives Lake basin in 

Saskatchewan (Freeze, 1969a). Gillham and Farvolden 

(1974) tested the sensitivity of the ratio and found it to be 

particularly important in areas of recharge and discharge.

Considering the above studies, the models discussed in 

the following section of this report use both 100 and 1,000 

as lower and upper examples.

GROUND-WATER-FLOW DIAGRAMS

Much of the remainder of this report is a discussion of 

ground-water flow systems near lakes, which are best 

illustrated by diagrams showing ground-water flow in 

cross section. Understanding of the important features of 

this type of diagram is basic to the discussion of the 

simulation results. A ground-water-flow diagram shows 

the distribution of hydraulic head within the groundwater system. After the head is calculated for each node, 

lines of equal head, equipotential lines, are drawn. 

Ground-water flow lines are drawn perpendicular to the 

equipotential lines, if the medium is isotropic. Equipotential lines are the projections of water-table contours 

into the subsurface (figure 4).

The finite difference grid used in this study is a network 

of uniform rectangles. About 900 nodes were used for the

one-lake simulations and about 1,800 nodes for the threelake simulations.

To give the most accurate picture of ground-water flow 

systems, and to estimate relative quantities of ground 

water moving through various parts of the ground-water 

reservoir, a flow net should be drawn. Construction of a 

quantitative flow net in a medium that is isotropic (Kh / 

Kv=l), requires that flow lines and equipotential lines be 

orthogonal such that curvilinear squares are formed 

(Harr, 1962).

If the medium is anisotropic (Kh /Kv ^l) the squares are 

deformed according to the degree of anisotropy. In this 

study, Kh /Kv is either 100 or 1,000. To compensate for the 

anisotropy, in homogeneous media equation 1 can be 

transformed to the LaPlace equation by a transformation 

of coordinates. The scale factor used to do this, which was 

not applied to the diagrams in this report (discussed later), 

is the square root of the ratio K^/KV (Harr, 1962, p. 29).

A requirement for constructing precise flow nets is that 

the section have no vertical exaggeration (Van 

Everdingen, 1963). Vertical exaggeration is usually used 

for convenience in illustrating important details of 

geology and ground-water movement that would be lost if 

the true scales were used. The sections discussed in this 

study have a vertical exaggeration of 80:1. Because the 

graphical correction of Van Everdingen (1963) was not 

applied to the sections in this study, they cannot be used as 

exact quantitative diagrams. The diagrams of Van 

Everdingen (1963) provide good examples of the effect of 

the correction on the ground-water-flow fields. The effect 

basically is that the corrected flow nets are much more 

rectangular and the local systems extend deeper into the 

ground-water reservoir than in flow sections not corrected 

for vertical exaggeration. The flow nets (figures 4 and 5) 

were drawn as if the porous medium were isotropic and 

homogeneous merely to show the general concept of flow 

nets and the different-magnitude flow systems that can 

occur near lakes and within the ground-water reservoir. 

Although the two flow nets drawn for this study are not 

quantitatively precise, they do show relative proportions 

of flow in different parts of the ground-water system.

The ground-water system illustrated in figure 4 consists 

of several flow systems of different magnitude. The upper 

part of the ground-water system consists largely of local 

flow systems where water moves from high points on the 

water table (water-table mounds) to adjacent lowlands 

occupied by lakes. Regional ground-water flow occurs 

deep in the system. Recharge to the regional flow system 

occurs at the major drainage divide and discharge from 

this system is to the major drain (stream). A zone of intermediate flow (darkly stippled area) is recharged at the 

water-table mound between lakes 2 and 3 and is discharged into lake 1. It should be noted, but will be discussed in more detail later, that much more ground water 

moves through local flow systems than through the deep
EXPLANATION

Flow line

METERS FEET 

15_r50

250 —————————— 

Line of equal hydraulic potential, in feet above a 

standard datum. 

Interval is variable. 

Dashed lines are supplemental contours

Example of realistic scales that 

could be applied to this diagram

General vicinit

Number is head 

3000 FEET

0 1000 METERS

Zone of intermediate groundwater-flow system

Boundary of flow system

Other patterns are 

explained in fig. 1

Lake number 

Lake altitude

o m o w o m 

• CM CM co co ^r ^r

I CM CM CM CM CMCM 

FIGURE 4.—Hydrologic section showing a quasi-quantitative flow net of ground-water flow near lakes in a multiple-lake system that does no
GROUND-WATER-FLOW DIAGRAMS

regional system as shown by the closer spacing of the flow 

lines in the local systems. A thorough discussion of similar 

type ground-water-flow diagrams is given by Toth (1963), 

Meyboom (1966, 1967), and Freeze (1969b).

Of special interest to this study are the lines, hereafter 

referred to as divides, separating the several types of flow 

systems. In following the line dividing the zone of local 

flow from zones of larger magnitude flow (by lake 3, fig. 4, 

for example), it should be noticed that there is a point on 

the divide at which the head is a minimum compared to 

every other point along the divide. This point of 

minimum head occurs beneath the shoreline on the downslope side of the lake. In the case illustrated, the hydraulic 

head at this point is 2.2 feet (0.7 m) higher than the waterlevel altitude of lake 3. The hydraulic head everywhere else 

along the divide is greater than 2.2 feet (0.7 m) above the 

lake-level altitude. Thus, under the given conditions, it is 

impossible for water to move from the lake to the groundwater system because the hydraulic gradient is upward 

toward the lake bottom.

This point of minimum head is the stagnation point 

commonly referred to in the ground-water-flow literature 

(for example, Harr, 1962). It is a point in the flow field at 

which vectors of flow are equal in opposite directions and 

therefore cancel. A value of head exists at the stagnation 

point and, as will be shown below, it is this value of head 

relative to the head represented by lake level that is of 

prime interest in understanding the interaction of lakes 

and ground water. If a stagnation point exists that has a 

head greater than that of lake level, a continuous groundwater divide exists beneath the lake making it impossible 

for water to move against the hydraulic gradient from the 

lake to the ground-water system. The position of the 

stagnation point is determined by the distribution of head 

within the ground-water system.

The stagnation point is a point of diversion of groundwater-flow paths (fig. 4). Water moving downward from 

the water-table mound on the downslope side of the lake is 

diverted upward toward the lake. A small amount of water 

that moves beneath the lake from the upslope side is 

diverted upward toward the lake on the downslope side. 

Water moving in the local or intermediate flow systems of 

the lakes at lower altitudes is diverted downslope and 

water moving in the regional flow system is diverted 

deeper into the ground-water reservoir.

Much of the remainder of the report is concerned with 

the effect of varying the height of the water table in 

relation to lake level, K^/KV , Kaq/Kt (ratio of the 

hydraulic conductivity of aquifers to that of-the surrounding till), position and size of aquifers, and lake 

depth, on the position and head value of the stagnation 

point relative to lake level.

To illustrate the effect of varying several parameters on 

the ground-water flow systems in general and the stagnation point in particular, figures 4 and 5 can be compared.

The presence of aquifers of limited extent, which have 

hydraulic conductivity 1,000 times greater than the till and 

are at an intermediate depth within the ground-water 

system, change the configuration of the flow systems. The 

aquifers also decrease the head at the stagnation points 

relative to the level of each lake and move the points closer 

to the bottom of lakes 1 and 2. In addition to having a 

limited aquifer beneath the water-table mound between 

lakes 2 and 3, the height of the water table was decreased on 

both sides of lake 3. These two changes near lake 3 result in 

reduced local ground-water flow into one-half of lake 3 on 

the upslope side, a small amount of ground-water flow 

into the littoral zone of the lake on the downslope side, and 

loss of lake water to the ground-water system through 

most of the downslope half of the lake bottom.

To appreciate fully the discussion in the following part 

of the report, the diversion of ground-water-flow lines 

caused by the presence of aquifers of limited extent within 

the till should be noted (fig. 5). It should be noted also that 

an intermediate flow system was sketched in figure 4, but 

not in figure 5. Although an intermediate flow system 

could have been included in figure 5 and deleted from 

figure 4, the intermediate system was included in figure 4 

to illustrate where such systems can occur, where they are 

recharged, and where they discharge. The intermediate 

system could be recharged at any of the upper three water 

table mounds or discharged into any of the lower lakes or 

the major drain. Such systems have importance especially 

in explaining differences in water chemistry in lakes 

occurring at different altitudes in a region.

Ground-water chemistry is controlled partly by the 

length of time a particle of water is in contact with 

minerals in the geologic framework through which the 

water flows. Thus, the longer the flow path, or the lower 

the hydraulic conductivity, the longer the residence time, 

and the greater will be the opportunity for chemical interchange. For lakes situated on geologic materials of similar 

hydraulic conductivity, those that receive ground water 

from intermediate or large local ground-water flow 

systems are more likely to contain more, and a wider 

variety of, dissolved minerals than lakes that receive 

ground water from small local flow systems. It will be seen 

in the remainder of the report that the size of local flow 

systems alone can vary widely. Therefore, the large variation in lake-water chemistry that is so frequently noted in 

lakes in close proximity in many lake regions is not at all 

surprising. On the contrary, after becoming aware of the 

factors that control the interaction of lakes and ground 

water discussed in this report, and realizing that virtually 

every lake has its unique ground water contribution, a 

wide variety in the chemistry of lakes should be expected.

Because the stagnation point is the key to understanding the relation of ground-water to lakes and it is 

determined by the distribution of head, the remainder of 

the illustrations in the report are contoured computer
EXPLANATION

-————<———— 

Flow line

250—————————— 

Line of equal hydraulic po 

tential, in feet above a 

standard datum 

Interval is variable. 

Dashed lines are supplemental contours

METERS FEET 

15-T50

-UP

Example of realistic scales that 

could be applied to this diagram

General vicinit

Number is head 

3000 FEET

1000 METERS

Boundary of flow system

Other patterns are 

explained in fig. 1

Kh /Kv = 1,000 

= 1,000

in

FIGURE 5.—Hydrologic section showing a quasi-quantitative flow net of ground-water flow near lakes in a multiple-lake system that contains 
ONE-LAKE SYSTEM 11

printouts that show only isopotential lines. The isopleth 

interval is variable; only enough lines were drawn to 

depict the general distribution of hydraulic head, the 

specific location and head of the stagnation point, or the 

percent of lake bottom through which the lake loses water 

to the ground-water system. Figure 4, a specially constructed flow net, can be compared with figure 6, a contoured computer printout, to aid in understanding the 

remainder of the illustrations in this report. Both 

diagrams are of identical settings.

The following sections of the report discuss two general 

settings. In the first, one lake is situated within the side of a 

drainage basin. The interaction of lakes and ground water 

is thoroughly studied for this type of setting by examining 

all possible combinations of the parameters mentioned 

previously. In the second general setting, three lakes are 

situated within the side of a drainage basin to examine the 

relationship of the lakes to each other and to compare the 

results to the one-lake setting.

The diagrams of vertical ground-water flow are discussed in dimensionless terms. But because lakes in glacial 

terrane are of primary interest in this study, the 

dimensions expected to be of most general interest are also 

given.

It must be kept in mind, however, that the specific 

results of this study apply to ground-water systems that 

approximate the given conditions. If the boundaries were 

greatly changed relative to each other, the problem would 

then be changed. However, if the new problem were analyzed as in this study, the general conclusions would be 

similar to this study.

ONE-LAKE SYSTEM

The approximate dimensions, relative to total length 

and thickness of the ground-water system, of features in 

the one-lake system are given in the following paragraphs 

and in figure 7. Realistic examples of the dimensions are 

shown in parentheses.

The thickness of the system ranges from 0.35T (70 ft; 21 

m) at the lower end of the basin to T (as much as 200 ft; 61 

m) at the upper end. The length of the system (L) is 

approximately 6 miles (9 km). The height of the water 

table above lake level is varied between 0.20T and 0.35 T 

(40 and 70 ft; 12 and 21 m) on the upslope side of the lake 

and between 0.05 T and 0.10T (10 and 20 ft; 3 and 6 m) on 

the downslope side. Lake depths simulated are 0.05T and 

0.25 T (10 and 50 ft; 3 and 15 m). Values of Kh /Kv 

(horizontal to vertical hydraulic-conductivity ratio) used 

in the simulations are 100 and 1,000. Simulated aquifers 

vary from full aquifers (0.05TxL) that extend the entire 

length of the basin to aquifers of limited extent 

(0.05Tx0.25L) that occur upslope from the lake (center 

point at 0.85L), beneath the lake (center point at 0.60L),

and downslope from the lake (center point at 0.35L). The 

aquifers are placed both at the base and at an intermediate 

level (center point at 0.25 T above the base of the system) of 

the ground-water reservoir. The lake surface is at 0.65T 

above the base and the midpoint is about 0.60L from the 

downslope end of the section for all simulations of the 

one-lake system. The system simulated can be thought of 

as a ground-water reservoir that has a datum of 100 feet (30 

m) (bottom of the ground-water reservoir). The values of 

head are in feet relative to that datum, thus the lake surface 

has an altitude of 230 feet (70 m), the higher water table at 

the upslope end of the section 300 feet (100 m), and the 

water table at the downslope end 170 feet (52 m).

A large number of simulations could be run varying 

each of these parameters. Generally, the parameters are 

varied between two values so the general direction of 

change of the stagnation point could be determined. If a 

change in a parameter lowers the head at the stagnation 

point relative to lake level, an even greater change of that 

parameter in the same direction would lower the head at 

the point even more.

The pattern of ground-water flow near a lake that has a 

water table 0.35T higher than lake level on the upslope 

side and 0.1T higher on the downslope side, has no sediments, and Kh/Kv=l,QQQ, is shown in figure 8. A groundwater divide occurs deep beneath the lake that has a head at 

the stagnation point 0.009T (1.8 ft; 0.5 m) higher than lake 

level. The position of the stagnation point is approximately 0.2T above the base of the system. In contrast, if 

lake sediments are simulated, and all other parameters are 

held constant, the ground-water divide shifts slightly 

upward, and the head at the stagnation point increases to 

0.02 T (4.2 ft; 1.3 m) higher than lake level (fig. 9). Thus, it 

is evident that the presence of lake sediments has a significant effect on the position and head of the stagnation 

point. Because lakes without sediments are virtually 

unknown, much of the remainder of this report is concerned with lakes that have a sediment layer.

The effect on the interaction of lakes and ground water 

of lowering the water table 0.05 T (10 ft; 3 m) on the downslope side and holding all other parameters as in figure 9, 

is shown in figure 10. The head at the stagnation point 

drops to 0.007T (1.4 ft; 0.4 m) above lake level and the 

point moves closer to the bottom of the lake. Again, 

holding all parameters as in figure 10, but lowering the 

water table on the upslope side of the lake to 0.20T (40 ft; 

12m) above lake level, decreases the head at the stagnation 

point to 0.004T (0.9 ft; 0.3 m) above lake level (fig. 11). The 

stagnation point remains at about the same position as in 

figure 10. This suggests that the height of the water table 

relative to lake level on the upslope side of a lake has less 

influence on the stagnation point of the ground-water 

divide beneath a lake than that on the downslope side.
EXPLANATION

Water table

METERS FEET 

15H-50

230 ——————

Line of equal hydraulic potential, in feet above a 

standard datum. 

Interval is variable.

Other patterns are 

explained in fig. 1

Example of realistic scales that 

could be applied to this diagram

General vicinit

Number is head 

Lake number 

Lake altitude

FIGURE 6.—Hydrologic section showing distribution of hydraulic head in the ground- water reservoir of a multiple-lake system that does no
ONE-LAKE SYSTEM 13

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EXPLANATION

Water table

230

Line of equal hydraulic potential, in feet above a 

standard datum. 

Interval is variable.

Other patterns are 

explained in fig. 1

METERS FEET 

15-T50 Example of realistic scales that 

could be applied to this diagram General vicin

Number is head 

FIGURE 8.—Hydrologic section showing distribution of hydraulic head in the ground- water reservoir of a one-lake system that does not contain 
ONE-LAKE SYSTEM 15

The presence of extensive aquifers within the groundwater reservoir has a significant effect on the interaction of 

lakes and ground water as seen in figure 12. Given the 

same setting as in figure 9, but simulating an aquifer at the 

base of the ground-water system that extends the full 

length of the basin and has Kaq /Kt =\QQ, decreases the 

head at the stagnation point from 0.02T (4.2 ft; 1.3 m) 

(figure 9) to 0.001 T (0.2 ft; 0.1 m) higher than lake level. 

Again, holding all parameters constant, but increasing 

Kaq /Kt to 1,000 causes the lake to lose water to the groundwater system through its entire bed (fig. 13).

The effect on the interaction of lakes and ground water 

of raising the extensive aquifer vertically to the 0.25T 

(intermediate) level in the ground-water system, where the 

setting is otherwise similar to figure 12, is shown in figure 

14. The weak ground-water divide (head at the stagnation 

point is 0.001 T (0.2 ft; 0.1 m) higher than lake level) in the 

former is obliterated and the lake loses water through 

three-eighths of its bed.

The position of smaller aquifers of limited extent 

within the ground-water reservoir also has varying 

influences on the interaction of lakes and ground water. 

All parameters other than the location of the limited 

aquifer at the base of the ground-water system are held 

constant in figures 15, 16, and 17. In the simulation that 

has a limited basal aquifer upslope from the lake (0.85L) 

(figure 15), the head at the stagnation point is 0.006T (1.3 

ft; 0.4 m) above lake level. With the limited basal aquifer 

beneath the lake (0.60L) (figure 16), the head at the stagnation point is 0.009T (1.8 ft; 0.5 m) above lake level. The 

limited basal aquifer underlying the water-table mound 

on the downslope side of the lake (0.35L) causes the lake to 

lose water through three-eighths of its bed (figure 17).

Similar patterns of ground-water flow near lakes hold 

for settings that have aquifers at the same position 

laterally but moved vertically to 0.25 T above the base of 

the ground-water system. One difference is that, if a 

limited aquifer directly beneath a lake is raised vertically, 

it tends to increase the head at the stagnation point. In a 

section simulating this situation, the head at the stagnation point increases from 0.009T (1.8 ft; 0.5 m) (figure 16) 

to 0.01 2T (2.5 ft; 0.8 m) higher than lake level with the 

aquifer at the higher position. If the limited aquifer is 

raised to directly under, and in contact with, the lake sediments, the head at the stagnation point increases to 0.016T 

(3.3 ft; 1.0 m) higher than the lake level. If the aquifer is at 

an intermediate level but downslope from the lake (0.25T, 

0.35L), a situation parallel to that shown in figure 17, the 

percentage of lake bottom through which water moves to 

the ground-water system increases from three-eighths to 

four-eighths.

These simulations suggest that the position of limited 

aquifers beneath or upslope from a lake has little influence 

on the interaction of lakes and ground water. But if the 

limited aquifer underlies the water-table mound on the

downslope side of the lake, there is a strong tendency for 

the head at the stagnation point to be only slightly above 

lake level, if at all; in the latter case the lake loses water 

through part of its bed.

The effect of lake depth on the interaction of lakes and 

ground water can be seen by comparing figure 18 with 

figure 12. All parameters in these two simulations, except 

depth, are identical. The small difference between lake 

level and head at the stagnation point associated with the 

shallow lake (fig. 12) is eradicated in the deep-lake setting, 

and the lake loses water through about six tenths of its bed. 

This example is one of the more dramatic in comparing 

ground-water flow near shallow and deep lakes. In some 

simulations the difference is not as great, but in all cases 

the difference between lake level and the head at the 

stagnation point for deep lakes is less, or they have more 

tendency to lose water, than shallow lakes in equivalent 

settings.

It is evident that a large number of simulations would be 

needed to examine all possible combinations of the 

variables considered in this study. To keep the analysis as 

simple as possible, yet show the underlying relationships 

of the interaction of lakes and ground water, a parallel 

series of lake-ground water settings are compared and 

summarized in table 1. At first glance the table might seem 

complex, but with a little study it is not so formidable. It 

should be realized that many variables are summarized in 

this c»ne table.

The table consists of four quadrants, each of which is 

subdivided into four subquadrants. Each subquadrant has 

an upper and a lower part. For ease of discussion, the 

different parts of the table are referred to as shown in figure 

19. The major quadrants are the combinations of 

hydraulic conductivity (K^/KV and Kaq /Kt ) used in the 

study. In quadrant I, both ratios are 100 and in quadrant 

IV, both are 1,000. Within the subquadrants are combinations of the height of the water table above lake level for 

both sides of the lake. In subquadrant 1-1 the water table is 

0.35T (70 ft; 21 m) higher on the upslope side and 0.1OT (20 

ft; 6 m) higher on the downslope side. In subquadrant 1-4, 

the water table on the upslope side is 0.20T (40 ft; 12 m) 

higher than lake level and 0.05T (10 ft; 3 m) higher on the 

downslope side. Within each of the subquadrants, the 

upper part (a) refers to shallow lakes and the lower part (b), 

refers to deep lakes. Within each part (a and b), the 

position and size of the aquifers are listed in a parallel 

fashion as follows:

FB-Full (extensive) aquifer at the base of the groundwater system. 

FM-Full (extensive) aquifer at an intermediate level

(0.25T) of the ground-water system. 

PBU-Partial (limited extent) aquifer at the base of the

ground-water system upslope from the lake (0.85L). 

PBB-Partial (limited extent) aquifer at the base of the 

ground-water system beneath the lake (0.60L).
EXPLANATION

Water table

230————

Line of equal hydraulic potential, in feet above a 

standard datum. 

Interval is variable.

Other patterns are 

explained in fig 1

METERS FEET 

15-r50 Example of realistic scales that 

could be applied to this diagram

General vicin

Number is head 

FIGURE 9.—Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a one-lake system that contains lake s

contain aquifers.
EXPLANATION

Water table

230————

Line of equal hydraulic potential, in feet above a 

standard datum. 

Interval is variable.

Other patterns are 

explained in fig. 1

Kh /Kv = 1 '000

METERS FEET 

15-r50 Example of realistic scales that 

could be applied to this diagram

General vicinit

Number is head 

0-L-O 0 

0

3000 FEET

1000 METERS

FIGURE 10.—Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a one-lake system that has a low water-table 

side of the lake and does not contain aquifers.
EXPLANATION

Water table

230————

Line of equal hydraulic potential, in feet above a 

standard datum. 

Interval is variable.

Other patterns are 

explained in fig. 1

K/K= 1,000

METERS FEET 

15-r50

0-L-O 0 

0

CD 

O

ro 

o 

o

ro 

o

Example of realistic scales that 

could be applied to this diagram

General vicinity 

Number is head a

3000 FEET

1000 .METERS

10 

Ol

o

FIGURE 11.—Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a one-lake system that has low water-table m

the lake and does not contain aquifers.
EXPLANATION METERS FEET 

15-r50

Water table

230-

Line of equal hydraulic potential, in feet above a 

standard datum. 

Interval is variable.

Other patterns are 

explained in fig. 1

Kh /Kv = 1 '000

Example of realistic scales that 

could be applied to this diagram

,''

General vicinity 

Number is head a

FIGURE 12.—Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a one-lake system that has an extensive aquifer o

conductivity at the base of the system.
EXPLANATION 

Water table

METERS FEET 

15-r50

230-

Line of equal hydraulic potential, in feet above a 

standard datum. 

Interval is variable.

Other patterns are 

explained in fig. 1

Example of realistic scales that 

could be applied to this diagram

(H-0 Q 3000 FEET

FIGURE 13.—Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a one-lake system that has an extensive aquifer o

conductivity at the base of the system.
ONE-LAKE SYSTEM 21
EXPLANATION

Water table

230 ————

Line of equal hydraulic potential, in feet above a 

standard datum. 

Interval is variable.

Other patterns are 

explained in fig. 1

K

METERS FEET

15-1-50

Example of realistic scales that 

could be applied to this diagram

C

General vicinit

Number is head 

0^0 0 3000 FEET

FIGURE 15.—Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a one-lake system that has an aquifer of lim

the lake at the base of the system.
EXPLANATION 

Water table

230————

Line of equal hydraulic po 

tential, in feet above a 

standard datum. 

Interval is variable.

Other patterns are 

explained in fig. 1

Kh /Kv = 1 '000 

Kaq /Kt = 1,000

METERS FEET 

15-r50 Example of realistic scales that 

could be applied to this diagram

3

General vicinit

Number is head 

FIGURE 16.—Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a one-lake system that has an aquifer of li

lake at the base of the system.
EXPLANATION

Water table

230

Line of equal hydraulic potential, in feet above a 

standard datum. 

Interval is variable.

Other patterns are 

explained in fig. 1

Kh /Kv =

Kgq /Kt = 1,000

METERS FEET 

r50 Example of realistic scales that 

could be applied to this diagram

0-1-0 0 3000 FEET

FIGURE 17.—Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a one-lake system that has an aquifer of limit

the lake at the base of the system.
EXPLANATION 

Water table

METERS FEET

15-1r50

230-

Line of equal hydraulic potential, in feet above a 

standard datum 

Interval is variable.

Other patterns are 

explained in fig. 1

Kh / Kv = 1,000

Example of realistic scales that 

could be applied to this diagram

FIGURE 18.—Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a one-lake system that has a deep lake a

relatively low hydraulic conductivity at the base of the system.
26 INTERACTION OF LAKES AND GROUND WATER

TABLE I.—Results of digital simulations of the one-lake system—comparison of the effects 

of all combinations of the parameters tested that control ground-water flow on lakeground water interaction.

[0.35T 0.1 OT: height of water table above lake level on upslope and downslope sides, respectively]

10' /c Itf

2

10

// Koq

L/ Kt

_

10

r 0.35T, 0.10T ,

13)' 10* -FB3 (4)" (0.1)* m

34) FM (5.0) D

PBU D

PBB D

PBD D

PMU D

PMB D

PMD D 

16) FM (2) (0.7) D 1 

15) FM (4) (C3)7 D/LJ

50-FB D

FM D

PBU D

PBB D

PBD D

PMU D

PMB D

79) PMD (RF*0.5)D

0.2QT, 0.10T

28) 10-FB (2.8) D

FM D

PBU D

PBB D

44) PBD (RF 0.5)D

PMU D

PMB D

PMD D

68) 50-FB (1.3) D

71) FM (2.1) D

PBU D

PBB D

77) PBD (0.3) D

PMU D 

PMB D

80) PMD (RF 0.6)D

0.35T, 0.10T

22) 10-FB (2/8) L

20) FM (7/8) L

PBU D

PBB D

41) PBD (0.1) D

PMU D

PMB D

PMD L

(17) FM (4) (8/8) L) 

|18) FM (2) (7/8) Lj

50-FB L

FM L

PBU D

PBB D

75) PBD (5/10) L

PMU D

PMB D

PMD L

0.2QT, 0.10T

30) 10-FB (6/8) L

FM L

PBU D

PBB D

46) PBD (cr>) D/L

PMU D

PMB D

PMD L

50-FB L

FM L

PBU D

PBB D

PBD L

PMU D

PMB D

PMD L

0.35T, 0.05T

24) 10-FB (1.3) D

36) FM (1.7) D

PBU D

PBB D

PBD D

PMU D

PMB D

56) PMD (0.1) D

66) 50-FB (0.7) D

70) FM (0.8) D

PBU D

PBB D

76) PBD (3/10) L

PMU D

PMB D

PMD L

0.20T, 0.05T

31) 10-FB (0.5 D

37) FM (0.6) D

PBU D

PBB D

48) PBD (CD) D/L

PMU D

PMB D

58) PMD (CD) D/L

69) 50-FB (0.1) D

72) FM (0.4) D

PBU D

PBB D

PBD L

PMU D 

PMB D

81) PMD (2/10) L

0.35T, 0.05T

26) 10-FB (3/8) L

FM L

PBU D

PBB D

PBD L

PMU D

PMB D

PMD L

50-FB L

FM L

PBU D

PBB D

PBD L

PMU D

PMB D

PMD L

0.20T, 0.05T

33) 10-FB (7/8) L

FM L

PBU D

PBB D

49) PBD (2/8) L

PMU D

PMB D

PMD L

50-FB L

FM L

PBU D

PBB D

PBD L

PMU D

PMB D

PMD L

0.35T, O.JOT

21) 10-FB (0.2) D

35) FM (3/8) L

PBU D

PBB D

40) PBD (0.8) D

PMU D

PMB D

55) PMD (1/8) L 

|14) FB (6.8) Lj

65) 50-FB (6/10) L

FM L

PBU D

PBB D

74) PBD (4/10) L

PMU D

PMB D

PMD L

0.20T, 0.1DT

29) 10-FB (5/8) L

FM L

PBU D

PBB D

45) PBD (0.2) D

PMU D

PMB D

57) PMD (2/8) L

50-FB L

FM L

PBU D

PBB D

PBD L

PMU D 

PMB D

PMD L

0.35T, 0.1DT

23) 10-FB (8/8) L

19) FM(2)(8/8) L

PBU D

PBB D

42) PBD (0.4) D

PMU D

PMB D

r PMD L

[l9a) FM(4)(8/8) L

50-FB L

FM L

PBU D

PBB D

PBD L

PMU D

PMB D

PMD L

0.20T, O.JOT

10-FB L

FM L

PBU D

PBB D

47) PBD (2/8) L

PMTJ D

PMB D

PMD L

50-FB L

FM T.

PBU D

pun D

PBD L

PMIT D

PMB D

PMD L

0.35T, 0.05T

25) 10-FB (4/8) L

FM L

PBU D

PBB D

43) PBD (1/8) L

PMU D

PMB D

PMD L

67) 50-FB (6/10) L

FM L

PBU D

PBB D

PBD L

PMU D

PMB D

PMD L

0.20T, 0.05T

32) 10-FB (7/8) L

FM L

PBU D

PBB D

PBD L

PMU D

PMB D

PMD L

50-FB L

FM L

PBU D

PBB D

PBD L

PMU D 

PMB D

PMD L

0.35T, 0.05T

27) 10-FB (8/8) L

FM L

PBU D

PBB D

PBD L

PMU D

PMB D

PMD L

50-FB L

FM L

PBU D

PBB D

PBD L

PMU D

PMB D

PMD L

0.2QT, 0.05T

10-FB L

FM L

38) PBU (1.3) D

39) PBB (1.8) D

50) PBD (3/8) L

52) PMU (1.4) D

53) PMB (2.5) D

59 PMD (4/8) L

T54) PMB (3.3) D"]

PBB

50-FB L

FM L

PBU D

 »? » PBB (1.6) D

PBD L

78) PMD (1.2) D

PMB D

PMD L

Simulation number, keyed to table 2.

2Lake depth. In each cell, entries above dashed line are shallow lakes (0.05T) and those below it 

are deep lakes (0.35T).

Aquifer position. (See text for explanation.) 

Thickness of aquifer if greater than 0.05T; 2-0.10T, 4-0.20T.

5 If decimal, height of head, in feet, of stagnation point above lake level; if fraction, proportion 

of lake bed that leaks.

6D, divide exists beneath lake; L, lake loses water through its bed.

7 era indicates the stagnation point is at the threshold of being obliterated the head value is the 

same as lake level.

RF, Influence of the water-table mound on the downslope side of a lake extends to the base of the 

ground-water system, such that at the base the gradient of head is to the right (upslope direction) and 

there is no regional movement beyond that point.
ONE-LAKE SYSTEM 27

a

— 1 •—

b

——— I

a

. _ q . . o

b

a

— 1 —

b

——— II

a

— 3 — -

b

a

— 2 — •

b

————

a

— 4 —

b

a

-- 2 —

b

I ———

a

— 4 — • b

a

— 1 —

b

——— I

a

n

&

b

a

__. H .__

b

—— r

a

-r 3 ——

b

a

— 2--

b

I ———

a

4 . -.

b

a

__ 2 —

b

V ———

a

-- -4 —

b

FIGURE 19.—Sketch showing key to table 1.

PBD-Partial (limited extent) aquifer at the base of the

ground-water system downslope from the lake

(0.35L). 

PMU-Partial (limited extent) aquifer at an intermediate

level of the ground-water system upslope from 

the lake (0.25T, 0.85L). 

PMB-Partial (limited extent) aquifer at an intermediate

level of the ground-water system beneath the

lake (0.25T, 0.60L). 

PMD-Partial (limited extent) aquifer at an intermediate

level of the ground-water system downslope from

the lake (0.25 T, 0.35L).

The simulations shown in brackets in table 1 are 

included for comparison purposes. They are generally of 

settings similar to one in the subquadrant but, for 

example, the aquifer might be more than one node thick.

Table 2 is a general summary of the one-lake simulations run for this study, and it includes those simulations 

in which aquifers were not included.

The parallel construction of table 1 makes it relatively 

easy to compare the effect of changes in the various parameters on the interaction of lakes and ground water. If 

interest is in the effect of different hydraulic conductivity 

ratios for given water-table combinations, aquifer size and 

position, and lake depth, that setting is found in 

equivalent positions in the major quadrants. For example, 

if one wishes to compare settings where Kaq /Kt is 100 to 

another where it is 1,000, and the setting has a water table 

0.35 T (70 ft; 21 m) higher than lake level on the upslope 

side and 0.05T (10 ft; 3 m) higher on the downslope side, 

Kh /Kv is 1,000, and the lake is 0.05 T (10 ft; 3 m) deep, subquadrant II-2a can be compared with subquadrant IV-2a 

for whatever aquifer situation is of interest. If interest is in 

comparison of different water table configurations for 

given hydraulic conductivity ratios, this can be done 

within a major quadrant.

The arrangement of table 1 is such that there is a general 

trend for the difference between lake level and head at the 

stagnation point to become less and for the potential for 

water loss from the lake to increase as one moves from 

quadrant I to quadrant IV. An example of this can be seen 

by comparing the PBD setting for shallow lakes in each 

subquadrant 3. In subquadrant 1-3, a ground-water divide 

exists that extends to the base of the ground-water reservoir 

(RF 0.5) (RF indicates reverse flow along the base of the 

system). In subquadrant II-3 the ground-water divide 

weakens to where the head at the stagnation point is 

0.001 T (0.2 ft; 0.1 m) higher than lake level. In subquadrant III-3 the ground-water divide is at the threshold 

of being obliterated (CD) and the lake losing water. And in 

subquadrant IV-3, where the hydraulic conductivity ratios 

are highest, the lake loses water through two-eighths of its 

bed.

For the type of hydrogeologic settings considered in this 

report, knowing the effect of a certain direction of change 

of a given parameter on the head at the stagnation point 

makes simulation of every possible combination of parameters unnecessary. If it can be shown that a ground-water 

divide exists beneath a lake in a setting that has greatest 

potential for water loss (for example, PBU in subquadrant IV-4) then an even stronger divide will exist in 

settings where this potential is less. Conversely, if a lake 

loses water in a setting that has greatest potential for a 

divide to exist (for example, PBD in sub-quadrant I-2b 

then it should lose even more water in settings where the 

potential for water loss is greater. In these cases, one 

simulation is sufficient to determine the relationship for 

eight others. In some situations, the simulation may have 

even greater implications because if a shallow lake loses 

water for a given setting, a deep lake in a similar setting 

will lose even more water.
28 INTERACTION OF LAKES AND GROUND WATER 

TABLE 2.—Summary of simulations of one-lake system

hi » 2 £ i

i

2 

3 

4 

5

6 

7 

8 

9 

10

11 

12 

13 

14 

15

16 

17 

18 

19 

20

21 

22 

23 

24 

25

26 

27 

28 

29 

30

31 

32 

33

34 

35

36 

37 

38 

39 

40

41 

42 

43 

44 

45

47 

48 

49 

50

51 

52 

53 

54 

55

56 

57 

58 

59 

.60

61 

62 

63 

64 

65

66 

67 

68 

69 

70

71 

72 

73 

74 

75

76 

77 

78 

79 

80 

81

Simulation 

identification 

number

3-11:20.71 

3-11:17.24 

3-12: 3.80 

3-13:14.95 

3-14: 9.31

3-14:20.74 

3-24: 4.11 

3-21: 8.58 

3-21: 7.43 

3-27: 8.99

3-28: 4.71 

4-1 : 9.16 

3-17:23.54 

3-18:24.67 

3-17:22.66

3-18: 5.60 

3-17: 8.17 

3-18:23.72 

3-14:10.57 

3-19:23.22

3-28:10.08 

3-19:25.64 

3-31: 9.19 

4-11: 7.31 

3-28: 9.55

4-11: 3.37 

3-28: 9.51 

4-10: 3.79 

3-31: 5.28 

4-10: 6.05

8-4 :16.83 

4-1 : 9.73 

4-14: 3.41

8-4 :16.88 

8-4 : 4.11

8-5 :17.21 

8-5 :16.60 

4-4 : 9.43 

4-1 :12.80 

4-4 : 8.02

8-6 :16.17 

8-5 :20.62 

8-5 :18.13 

4-3 :11.40 

4-4 : 4.46

4-3 : 9.54 

4-4 :10.74 

4-3 :10.65 

4-3 : 5.08 

4-3 :10.15

4-7 :10.38 

8-4 :16.90 

4-8 : 4.98 

4-9 : 4.78 

4-4 : 8.96

8-5 :17.07 

4-4 : 8.00 

8-4 :16.87 

4-2 :11.88 

3-24: 7.79

3-24: 7.95 

3-22: 8.99 

3-27: 4.18 

8-7 :16.73 

8-7 : 4.70

8-6 :15.40 

8-6 :19.32 

8-7 :15.83 

8-7 :15.36 

8-6 :15.61

8-7 :15.30 

8-7 :14.90 

8-7 :21.19 

8-6 :16.11 

8-7 :15.98

8-6 :16.16 

8-7 :16.50 

8-7 :21.18 

8-6 :17.23 

8-7 :17.45 

8-7 :16.30

Lake 

Depth

) OST'O 251

X 

X 

X 

X 

X

X 

X 

X

X 

X

X 

X 

X 

X 

X

X 

X

X 

X

x

X 

X 

X 

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X

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X

X 

X 

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X 

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X

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X 

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X 

X 

X

X 

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X 

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X 

X 

X 

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X 

X 

X 

X 

X

X 

X 

X 

X 

X

X 

X 

X 

X 

X

X 

X 

X 

X 

X 

X

Lake 

Sediments

Yes ] No

X 

X 

X 

X 

X

X 

X 

X 

X 

X

X 

X 

X 

X 

X

X 

X 

X 

X 

X

X 

X 

X 

X 

X

X 

X 

X 

X 

X

X 

X 

X

X 

X

X 

X 

X 

X

X

X

X 

X 

X 

X

X 

X 

X 

X 

X

X

X 

X 

X 

X

X 

X 

X 

X 

X

X 

X 

X 

X 

X

X 

X 

X 

X 

X

X 

X 

X 

X 

X

X 

X 

X 

X 

X 

X

VKv

i!io2 !io3 !io4

1 *!1 X

1 x 

1 x

1 X 

1 X 

1 X 

1 X 

1 X

1

X

1

1 x

1 X

1 x

X

1 X

X 

1 X

1 X

1 x

1 X 

1 X

1 x

X

X 

X 

X 

X

X

1 X 

X 

X

X 

X

X 

X 

X 

X

X 

X 

X

1 x 

x

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1 x 

1 x

X 

X

1 

X

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X I

1 X 

1 X 

1 X 

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Ix

X* .

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,X |

'x ' 1 I* 1

10 21 103

X 

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X 

X 

X

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X 

X 

X 

X

X 

X

X 

X

X

X 

X 

X 

X 

X

X 

X

X 

X 

X

X 

X 

X 

X 

X 

X

Aqui

Fuir

I

X

X 

X 

X 

X

X 

X 

X 

X 

X

X 

X

X

X

X 

X 

X

Base Middle

X

fer 

Partial

 1 i a "

1 | 0. a) i s: , o

O 1 10 i »

r^ ' v IB 

8. ' § 'J

Base J __

'

x£

X*

X 

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X 

X

X 

X

X 

X 

X

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X 

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X 

X 

X 

X

X X

X 

X

X

X 

X

01 J= 0

o ' to »

"to 1 6 3 

o. . 5 5 3 1 to D

0.25T

X 

X 

X5 

X

X 

X 

X 

X

X 

X 

X 

X

Height of 

ablet abo relative 

ystem tn

V

1 3

0 «

\re l|ke j

icknessj

1 V

B a »

3 O T3

0.35|0.20J0.1Cj0.05i'

X 

X 

X 

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X 

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X 

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X

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Results

c > ""* to 

1 to .-flu w ]jj

1

RF(20)| 

RF(12), 

1.8 

1 5/8 

RF(4) I

0.2 ,

R*'4 ' 

*o!l} ' 

4.2 '

1.4 1 

0.9 | 

0.1 | 

1 6/8

0.7 { 

1 8/8 

1 7/8 

1 8/8 

1 7/8

0.2 i 

2/8 

1 8/8 

1.3 1 

1 4/8

| 3/8 

8/8 

2.8 1 

1 5/8 

1 6/8

0.5 1 

1 7/8 

| 7/8

5.0 j 

1 3/8

1.7 I 

0.6 1 

1.3 I 

1.8 | 

0.8 | o.i 1

0.4 | 

| 1/8 

RF(0.5). 

0.2 j

I 2/8

1 2/8 

1 3/8

1 2/8 

1.4 1 

2.5 1 

3.3 I 

1 1/8

0.1 | 

I 2/8

] 4/8

0.9 , 

RF(4) | 

CTD 

0.5 ' 

1 6/10

0.7 | 

 6/10 

1.3 ' 

0.1 | 

0.8 |

0.4 1 

1.6 I 

1 4/10 

1 5/10

1 3/10 

0.3 | 

1.2 |

RF(0.6) | 

_______ |_ 2/10

2Aquifer does not extend full length of the 

3Aquifer thickness is 0.20T.

^Aquifer thickness is 0.10T.

5Aquifer is in contact with base of sediments.
MULTIPLE-LAKE SYSTEM 29

Some general relationships concerning the one-lake 

system can be deduced from table 1. For most hydrogeologic settings as simulated in this study, the following 

changes in parameters tend to weaken the difference in 

head between the lake and the stagnation point beneath a 

lake or tend to cause a lake to lose water: (1) Lowering the 

water table on either side of a lake (lowering it on the 

downslope side has a greater effect than lowering it on the 

upslope side), (2) increasing K^/KV , (3) increasing the 

hydraulic conductivity of aquifers relative to the till, 

which has a slightly greater effect than increasing Kh /Kv , 

(4) increasing the depth of a lake, and (5) raising aquifers 

from the base to an intermediate level in the ground-water 

system. An exception to statement 5 is the situation where 

an extensive aquifer that has low Kh /Kv and low Kaq /Kt 

(quadrant I, table 1) is raised from the base to an intermediate level. In this case the head at the stagnation point 

increases slightly.

Aquifers that extend the full length of the ground-water 

basin have a greater effect on the interaction of lakes and 

ground water than limited aquifers. If no aquifer is 

present, a lake generally will not lose water even if Kh /Kv 

is 1,000 (table 2). Limited aquifers upslope and beneath a 

lake have little effect on the ground-water divide beneath a 

lake. Under the conditions simulated in this study, 

aquifers in these positions will not cause the divide to 

weaken regardless of the hydraulic conductivity of the 

medium. Limited aquifers downslope from a lake, 

whether at the base or at an intermediate level of the 

ground-water system, have a significant effect on the interaction of lakes and ground water because under many 

conditions of high Kh /Kv and Kaq /Kt and low water-table 

configurations, a lake will have a weak ground-water 

divide or will lose water. In every simulation run for this 

study, the stagnation point associated with the groundwater divide is always under the downslope littoral or 

shoreline zone of a lake. This has particular significance 

when designing an observation-well program for studying 

the interaction of lakes and ground water because determining the position and head, or absence, of the stagnation point is the key to the relationship.

MULTIPLE-LAKE SYSTEM

In many geologic environments, lakes occur at different 

altitudes within major drainage basins. Although it is a 

common belief that water moves successively from higher 

lakes to lower lakes through the ground-water system 

within a major drainage basin—a situation where all the 

lakes would be the flow-through type, it is shown in the 

following discussion that this may not be the case. Digital 

simulations were run of a number of variations in the 

hydrogeologic setting of three lakes along a regional slope 

from a major divide to a major drain. The vertical 

exaggeration of the sections are the same as those in the 

one-lake system (80:1).

MODERATE REGIONAL SLOPE AND LOCAL RELIEF

The length and thickness dimensions of the multiplelake system are different from those of the one-lake system, 

thus the proportions of the features are different. As before, 

the dimensions in parentheses are included as examples of 

realistic field situations.

A three-lake simulation set will be discussed later in 

which the models are of a system that has a water table 

with very low regional slope, a physiographic condition 

common in many geologic terranes.

One type of lake-ground water setting not simulated in 

this study is that of a straight-line, sloping water table 

between adjacent lakes. As will be apparent in the 

following sections of this report, the height of the water 

table mound on the downslope side of a lake relative to 

lake level is an important control on the interaction of 

lakes and ground water. Lakes in a setting of a straightline water table simply would gain ground-water inflow 

on one side and lose water out the other.

The thickness of the ground-water reservoir of the threelake system ranges from T (as much as 250 ft; 76 m) on the 

upslope end of the section to 0.40T (100 ft; 33 m) on the 

downslope end. The height of the water table above lake 

level on the regional upslope sides of the two lower lakes is 

0.20T (50 ft; 15m) and on the downslope side of all three 

lakes is 0.08T (20 ft; 6 m). The highest lake in the section 

has the water table 0.12T (30 ft; 9 m) above lake level on the 

upslope side. The difference between the level of the 

highest lake and the elevation of the water table on the 

upslope side of that lake was purposely made less than the 

two lower lakes because it is believed that lakes that occur 

high topographically are more likely to have lower water 

tables adjacent to them than the lower lakes. The watertable altitudes were varied in some of the simulations. The 

dimensions of the shallow lakes are about O.IL wide and 

0.20T deep. The aquifers are about 0.04T thick and those 

of limited areal extent are 0. IbL long. As before, the datum 

for the head values is 100.

The positions of the three lakes are as follows: The 

surface of lake 1 is about 0.65 T above the base and 0.32L 

from the left side of the section, lake 2 is at 0.75T and 

0.60L, and lake 3 is at 0.88T and 0.85L. The aquifers are 

positioned at the base and, if at an intermediate level, at 

0.30T above the base. The lateral position of the limited 

aquifers varies as noted in the text.

A simulation of a three-lake system in which each lake is 

shallow, is underlain by sediment, has a 0.08T (20 ft; 6 m) 

water-table mound on the downslope side of each, has a 

ground-water system whose Kh /Kv is 1,000, and contains 

no aquifers, is shown in figure 6. The position and head of 

the stagnation point on the ground-water divide beneath 

each lake is shown as in the discussion of the one-lake 

simulations. Under the given conditions, each lake gains 

ground-water discharge from its own local ground-water 

flow system, and each is completely isolated from the 

others.
30 INTERACTION OF LAKES AND GROUND WATER

For the three-lake system, comparison of all possible 

combinations of different water-table altitudes, lake 

depths, hydraulic conductivity ratios, and size and position of aquifers, as was done for the one-lake system, 

would lead to a far more complex summary table. The 

basic goal of simulating the multiple-lake system is to 

determine if the lakes have interrelationships with each 

other that would not be evident in simulations of the onelake system. The results of the multiple-lake simulations 

in which the water table has moderate regional slope and 

local relief are summarized in table 3.

The effect on the interaction of lakes and ground water 

of an extensive aquifer at the base of the system can be seen 

by comparing row 1 with rows 2 and 3 of table 3. If an 

extensive aquifer at the base of the system has a hydraulic 

conductivity 1,000 times greater than till, and the Kh /Kv of 

the ground-water system is 100, the difference in head 

between the lake and the stagnation point beneath lake 2 is 

very small. In this simulation, there is very little change in 

the head at the stagnation point for lake 1 compared to the 

simulation listed in row 1, but there is water loss through 

the bottom of lake 3. If Kh /Kv is increased to 1,000 (row 3, 

table 3), holding all other parameters as in row 2 of table 3, 

a very strong divide is established beneath lake 1, and lakes 

2 and 3 lose water through most, or all, of their bed (fig. 

20).

Evidence that the lakes are independent of each other 

can be seen by comparing rows 6 and 7 of table 3. In these 

simulations, Kh /Kv and Kaq /Kl are both 1,000 and two 

limited aquifers are at the base of the ground-water system 

at the 0.45L and 0.75L positions. The only difference 

between the two is that the water-table mound between 

lakes 2 and 3 was decreased to only 0.047 (10 ft; 3 m) above 

the level of lake 3, and the upslope side of lake 3 was 

decreased to 0.087 (20 ft; 6 m) higher than lake level for the 

latter simulation (row 7). The effect of this change in the 

water-table near lake 3 on lakes 1 and 2 is very little, 

whereas the head at the stagnation point beneath lake 3 

decreases 0.0037 (0.8 ft; 0.2 m). If a limited aquifer is added 

at the 0.15L position, and all other parameters are as in the 

simulation listed in row 7 (table 3), the head at the stagnation point on the ground-water divide beneath lake 1 

decreases considerably, and there is little effect on lakes 2 

and 3 (compare rows 8 and 7).

The effect on the interaction of lakes and ground water 

of moving the position of limited aquifers laterally can be 

seen by comparing rows 14 and 15 of table 3. In the simulation listed in row 14, the aquifer at the 0.15L position is 

downslope from the edge of lake 1 (fig. 5). In the simulation listed in row 15 (fig. 21), this aquifer is shifted slightly 

in the upslope direction (to 0.22L), so that the upslope part 

of the aquifer is partially beneath the downslope side of 

lake 1. All other parameters for these two simulations are 

held constant. This change lowers the head at the stagnation point near lake 1 by 0.0027 (0.5 ft; 0.2 m). Thus, it is 

evident that the interaction of lakes and ground water is

sensitive to the lateral position of limited aquifers downslope from a lake. The maximum effect is felt if the 

upslope part of an aquifer is beneath the downslope side 

of a lake, and the aquifer underlies the water-table mound 

downslope from a lake.

The effect on the interaction of lakes and ground water 

of moving the limited aquifers vertically from the base of 

the system to a middle position is similar to the one-lake 

situation; that is, the head at the stagnation point is 

lowered. (Compare rows 6 and 12, and rows 11 and 19 of 

table 3, and see also figs. 22 and 23.) Note that the middle 

aquifer in these two simulations is shifted to the right, 

from the 0.45L to about the 0.53L position, compared to 

the other simulations discussed to this point. This was 

done for most of the multiple-lake simulations because of 

the findings mentioned in the previous paragraph.

The effect of deep lakes compared to shallow lakes is 

also similar to the simulations of the one-lake system. 

Comparing rows 9 and 22 of table 3, in which all parameters are similar except lake depth, shows the relatively 

greater influence on the ground-water flow systems near 

deep lakes compared to shallow lakes. The ground-water 

divide beneath lake 1 in the simulation of shallow lakes 

(fig. 24) has a head at the stagnation point 0.0047 (1 ft; 0.3 

m) greater than lake level, whereas in the simulation of 

deep lakes (fig. 25), the lake loses water through twoeighths of its bed. Beneath lake 2, the head at the stagnation point decreases from 0.0147 (3.4 ft; 1.0 m) greater than 

lake level in the shallow-lake simulation to 0.0087 (2.1 ft; 

0.6 m) in the deep-lake simulation. Beneath lake 3, the 

difference in head between the lake and the stagnation 

point of 0.0047 (0.9 ft; 0.3 m) for the shallow lake, can be 

compared to the deep lake which loses water through 

three-eighths of its bed.

In most of the simulations of three-lake systems 

discussed thus far, Kaq /Kt and Kh /Kv are 1,000. These 

values were chosen because, as shown in the one-lakesystem simulations, they tend to create conditions for 

maximum water loss from the lake to the ground-water 

system. It is of interest, therefore, to compare rows 21 and 

22 (table 3) and figures 25 and 26. The simulation 

summarized in row 21 has Kh /Kv =\00 (fig. 26), whereas in 

the latter this ratio is 1,000 (fig. 25). The stagnation points 

occur very deep in the system in the former and, in fact, the 

local flow cells around each of the lakes extend nearly the 

full thickness of the ground-water system and little ground 

water flows past the middle lake.

It is evident from the above discussion that, in most 

cases, lakes in a multiple-lake system act essentially 

independently. Changing the height of a water-table 

mound or the position of aquifers by one lake and not by 

the others has a considerable effect on that one lake that is 

similar to the equivalent one-lake simulation. The change 

may have a minor effect on an adjacent lake, changing the 

head at the stagnation point, for example, by perhaps 

0.00047 to 0.00087 (0.1 to 0.2 ft; 0.03 to 0.06 m).
MULTIPLE-LAKE SYSTEM 31

The only situation where a considerable effect is felt on 

the lakes is in the case of extensive aquifers within the 

ground-water system, and whereKh /Kv andKaq /Kt are 

large (row 3 of table 3) (fig. 20). In this case, and there are 

similar cases in the one-lake settings, there is massive 

downward movement of water in the upper region of the 

ground-water basin and massive upward movement in the 

lower region. The great difference in head between the 

lake and the stagnation point associated with the lowest 

lake seems to be established because of the large influence 

of the water-table mound on the downslope side of that 

lake. If the water-table mound were not there, there would 

be massive upward movement on the lower side of the 

diagram and the hinge line (the point at the water table 

separating regional recharge from regional discharge) 

would be near the middle of the diagram. Apparently the 

mound is sufficiently strong to cause downward movement that will not be overcome, but instead, will be 

responsible for a strong ground-water divide that has a 

head at the stagnation point of very large magnitude. It is 

interesting to note in this case that the stagnation point is 

not beneath the lake. Rather, it is downslope from the lake 

and is nearly directly beneath the highest part of the watertable mound.

LOW REGIONAL SLOPE AND LOCAL RELIEF

A series of simulations were run of a multiple-lake 

system in which regional slope and local relief were kept 

relatively low (summarized in table 4). The length and 

thickness dimensions of the multiple-lake system that has 

low regional slope and local relief are unlike those for the 

system of moderate slope and relief, thus the proportions 

of the features again are different. The thickness of the 

system ranges from T ( 160 ft; 49 m) on the upslope end to 

0.60T (100 ft; 30 m) on the downslope end of the section.

The water table on the upslope side of all the lakes is 

0.13T (20 ft; 6 m) higher than the adjacent lake level, and 

on the downslope side is 0.06T (10 ft; 3 m) higher. The 

difference in altitude between the lakes is 0.06T (10 ft; 3 m). 

Only shallow lakes 0.06 T (10 ft; 3 m) deep are simulated. 

The aquifers are 0.06T (10 ft; 3 m) thick, and the limited 

aquifers are 0.13L long.

A ground-water system that has no aquifers is shown in 

figure 27. Ground-water divides of equal size and strength 

(the difference in head between the lake and the stagnation point is 0.02T (3.0 ft; 0.9 m) by each lake) occur 

beneath each lake. It should be noted that simulations of 

this system of lower slope and relief differ somewhat from 

the other three-lake system discussed previously in that the 

water table relative to lake level by the highest lake is the 

same as by the two lower lakes.

Simulations that have extensive aquifers at the base of 

the system, Kaq /Kt =100, and Kh /Kv =l,WO (row 27, table 

4), show no water loss from any of the lakes for any of the 

simulations. This is true also if the hydraulic conductivity

ratios are reversed; that is, Kaq /Kt =1,000 and Kh /Kv =WO 

(row 28, table 4). In simulations that have both ratios at 

1,000, a ground-water-flow pattern similar to the other 

three-lake system occurs—there is water loss through the 

entire bed of the higher two lakes and a divide established 

beneath the lowest lake (row 29, table 4).

In simulations that have limited aquifers beneath the 

water-table mounds between the lakes, and the up-slope 

edge of the aquifers are partly beneath the lake, a groundwater divide occurs beneath the lake, although the 

difference in head between the lake and the stagnation 

point of each is rather small (fig. 28). This result holds 

even when the limited aquifer is moved to about 0.13 T (20 

ft; 6 m) below the bed of the lower lake.

In the three-lake system that has low regional slope and 

local relief, it appears that the ground-water divides tend 

to be slightly stronger in most cases than where a higher 

regional slope is simulated. The slope of the water table 

from the lowermost water-table mound to the left edge of 

the hydrologic sections in the simulation of moderate 

regional slope, shows a drop of several tens of feet. The 

water-table slope in this same part of the section, in the 

simulations of low regional slope, is essentially flat, 

resulting in very slow ground-water movement. This very 

flat water table near the valley of the major drain on the 

system acts as a "check valve" on the entire system and 

tends to increase the tendency of the lakes to have stronger 

local ground-water-flow systems than in conditions of a 

higher-sloping water table.

In none of the simulations does water move from one 

lake to the next downslope if there is a water-table mound 

between the two lakes. Even if an upper lake loses water to 

the ground-water system, the movement generally is deep 

to the system and the water moves beneath the local flow 

systems associated with lower lakes. Thus, even in settings 

where lakes occur at various altitudes down a sloping 

valley side, each of the lakes, for most settings, acts as an 

independent entity and has little relationship to lakes 

either upslope or downslope from it. This conclusion 

gives increased importance to the summary table (table 1) 

of the one-lake system.

The effect of reservoir thickness on ground-water-flow 

systems is extensively discussed by Toth (1963), and to a 

lesser extent by Freeze (1969b). The general effect of a thin 

ground-water reservoir is that regional flow does not occur 

for many settings, especially if regional slope is low. To 

check the effect of a thin ground-water system on the interaction of lakes and ground water, a setting was simulated 

that was 100 feet (30 m) thick on the upper end of the 

section and 50 feet (15 m) thick on the lower end. The 

difference between the section and figure 29 is that the 

local flow systems associated with each lake extend to the 

base of the ground-water reservoir, and regional flow does 

not occur (row 33, table 4).
TABLE 3.—Summary of simulations of multiple-lake system whose water table has moderate regional slope and local relief

Row Number

1 

2 

3 

4 

5

6 

7 

8 

9 

10

11 

12 

13 

14 

15

16 

17 

18 

19 

?0

21 

22 

23 

24

Simulation 

identificatioi 

number

4-22:28.67 

4-23: 5.70 

4-25:14.96 

4-24: 5.74 

4-25:10.08

4-24:10.95 

5-1:21.20 

5-2:10.85 

5-22:20-95 

5-16:6.77

5-14:10.51 

4-25:20.58 

5-7:19.57 

5-5:4.99 

5-8:11.22

5-12:5.60 

5-22:18.29 

5-15:6.70 

5-13:10.17 

5-19:7.43

5-22:11.13 

5-21:13.49 

5-28:8.05 

5-23:26.79

Lake 

depth

3 &

& Z

x

X

X

X 

X

X

Kh/Kv 

ID2 10 3

X 

X 

X 

X 

X

X 

X X

X

X 

X 

X 

X 

X

X

X 

X

x

102 10 3

X 

X 

X 

X

X 

X 

X 

X 

X

X

x

X 

X 

X 

X

Aquifer position

§ 

o

z

x

X

Full1

H

<u o

i d

X 

X

Partia 

Base

————

_) j _> 

in iA m rH •* fddo"

X X

X X 

X X

xxx

XS 4 XR5 X 

XS XR X

XS XR X

————

XS XR X 

XS XR X

.*• 

0.30T

r77

o o d

————

X X 

X X

xxx

XS X X

XS X X 

XS XR X 

XS XR X 

XS XR X

XS XR X 

XS XR X

Height of water table above lake level 

(relative to total svstem thickness}

Lake 1 

Upslope

O VO CM 00 

CM i-l .-1 0

O O O O

X 

X 

X 

X 

X

X 

X 

X 

X 

X

X 

X 

X 

X 

X

X 

X 

X 

X 

X

X 

X 

X 

X

down00 -3- 

0 0

0 0

X

X 

X

X 

X

X 

X

x 

x 

x

x 

x 

x 

x

Lake 2

CM f"l i-l O

O O O O

X 

X 

X 

X 

X

X 

X 

X 

X 

X

x 1

X 

X 

X 

X

X 

X 

X 

X 

X

X 

X 

X 

X

down00 -3- 

0 0

0 0

X

X 

X

X 

X

X 

X 

X 

X 

X

1 X

Lake 3 

Upslope

O VO CM 00 

CM i-l i-H 0

0000

X 

X 

X 

X 

X

X 

X 

X 

X 

X

X 

X 

X 

X 

X

X 

X 

X 

X 

X

X 

X

X

downslope

0 0

0 0

X

X 

X 

X 

X

x

X 

XXX

X 

X 

X 

X 

X

1 x

Res

Lake 1 

Height of Percent 

head of of bed 

stagnation that 

point loses 

above lake water 

level 

(in feet)

3.1 

3.4 

8.3 

RF(8.9) 

RF(6.7)

4.2 

4.0 

1.7 

1.0 

1.0

0.6 

4.4 

4.5 

0.7 

0.1

2/6 

1.3 

2/6 

2.2

1.6 

2/8 

2.0 

4/8

Lak

Height of 

head of 

stagnation 

point 

above lake 

level 

(in feet)

3.8 

0.2 

RF(6.4) 

1.4

2.8 

2.7 

2.5 

3.4 

3.2

2.7 

1.8 

1.7 

1.5 

1.5

1.1 

2.4 

3.4 

1.8 

2.7

4.2(RF) 

2.1 

RF(O.l) 

1.0

Aquifer extends full length of the basin. 

2Aquifer does not extend full length of the basin. 

•^Partial aquifer at base upslope from lake 3.

4XS, aquifer shifted from 0.15L to 0.22L. 

5XR, aquifer shifted from 0.45L to 0.53L.

TABLE 4.—Summary of simulations of multiple-lake system whose water table has low regional slope and local relief

H

9 °

st

|

3 3 & IB -H £

"aS §

Sue

S S 

3

25 6-4: 5. 312

26 6-17:9.36*

27 6-5:7.56

28 6-9:2.38

29 6-11:5.14

30 6-11:6.92

31 6-16:25.29 

32 6-16:26.10 

33 6-18:6. 49 5

Lake

depth

0

i-l O.

a o>

JS 0)

en o

X

X

X

X

X

X

> XXX

Kh/Kv

102 10 3

X

X

X

X

X

X 

X

X

Kaq/Kt

102 10 3

X

X

X

X

X 

X

A ir —

0)

c

z

x

x

X

nHu-*-->- Ci IMja JL I J.U11

Full' Partial2

S

X

X

X

Base

>j >-> J

o o o

XXX

0.45T

S3 tf rf 

en in oo

o o o

XXX

xxx

Height of water table above lake level

(relative to total system thickness)

Lake 1

downUpslope slope

en vo

•-I 0

0 0

X X

X X

X X

X X

X X

X X

X X

1 x x

X X

Lake 2

downUpslope slope

en *

i-H O

0 0

X X

X X

X X

X X

X X

X X

X X 

X X 

X X

Lake 3

downUpslope slope

en vo

i-l O

0 0

X X

X X

X X

X X

X X

X X

X X 

X X 

X X

Results

Lake 1

Height of

head of 

stagnation

point above 

lake level

(in feet)

Percent

of bed 

that

loses 

water

3.0

3.0

2.7

2.8

3.9

1.6

2.1 

0.4 

RF(4)

Lake 2

Height of Pe

head of of 

stagnation th

point above los

lake level wa

(in feet)

3.0

3.0

1.9

1.8

7

1.9

2.2 

0.8 

RF(3)

basin. 5Shallow ground-water system.
EXPLANATION

Water table

230———————— 

Line of equal hydraulic po 

tential, in feet above a 

standard datum. 

Interval is variable.

Other patterns are 

explained in fig. 1

METERS FEET 

15-r50 Example of realistic scales that 

could be applied to this diagram

General vicini

Number is head 

3000 FEET

1000 METERS

FIGURE 20.—Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a multiple-lake system that has an extensiv

hydraulic conductivity at the base of the system.
EXPLANATION

Water table

230———————— 

Line of equal hydraulic po 

tential, in feet above a 

standard datum. 

Interval is variable.

Other patterns are 

explained in fig. 1

Kh /Kv = 1,000

METERS FEET 

15-rBO Example of realistic scales that 

could be applied to this diagram General vicini

Number is head 

3000 FEET

O O O O

CM CO "fr iO

CM CM CM CM

FIGURE 21.—Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a multiple-lake system that has water-table 

and aquifers of limited extent in the middle of the system.
MULTIPLE-LAKE SYSTEM 35
EXPLANATION

Water table

230———————— 

Line of equal hydraulic potential, in feet above a 

standard datum. 

Interval is variable.

Other patterns are 

explained in fig. 1

Kh / Kv = 1,000

METERS FEET 

15-T50 Example of realistic scales that 

could be applied to this diagram General vicin

Number is head 

3000 FEET

1000 METERS

Lake number 

J Lake altitude

260

o o o o 

CM co -<fr in

CM CM CM <N

FIGURE 23.—Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a multiple-lake system that has low water

of limited extent in the middle of the system.
EXPLANATION

Water table

230———————— 

Line of equal hydraulic potential, in feet above a 

standard datum. 

Interval is variable.

Other patterns are 

explained in fig. 1

METERS FEET 

15-r50 Example of realistic scales that 

could be applied to this diagram

General vicin

Number is head 

3000 FEET

1000 METERS

FIGURE 24.—Hydrologic section showing distribution of hydraulic head in the ground- water reservoir of a multiple-lake system that has high water

of limited extent at the base of the system.
EXPLANATION

Water table

230———————— 

Line of equal hydraulic potential, in feet above a 

standard datum. 

Interval is variable.

Other patterns are 

explained in fig. 1

METERS FEET 

15-T50

0-M)

Example of realistic scales that 

could be applied to this diagram General vicin

Number is head 

3000 FEET

Lake number 

ke altitude

O O O O

CM oo

CM CM CM CM

O CMCO O

O> O>O> O

CM CMCM CO 00

I

T

0

FIGURE 25.—Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a multiple-lake system that has deep lake

aquifers of limited extent at the base of the system, and Kh /Kv=\,QQQ.
EXPLANATION

Water table

230———————— 

Line of equal hydraulic potential, in feet above a 

standard datum. 

Interval is variable.

Other patterns are 

explained in fig. 1

METERS FEET 

15-r50

o-i-o

Example of realistic scales that 

could be applied to this diagram

3000 FEET

1000 METERS

General vicin

Number is hea

Lake number 

1 Lake altitude x 

260 '

OY—

CM

O OOO

CM COrJ-lO

CM CMCMCM

CM 

CD 

CM

O

N. OOOX3)

CM CMCMCM

O O 

O *- 

CO CO

FIGURE 26.—Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a multiple-lake system that has deep lake

aquifers of limited extent at the base of the system, and Kh /Kv=lOO.
EXPLANATION

Water table

230———————— 

Line of equal hydraulic potential, in feet above a 

standard datum. 

Interval is variable.

Other patterns are 

explained in fig. 1

Kh /Kv = 1,000

METERS FEET 

15-T50 Example of realistic scales that 

could be applied to this diagram

General vicin

Number is head 

0_10 0 3000 FEET

0 1000 METERS

Lake number 

Lake altitude

220

FIGURE 27.—Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a multiple-lake system that has low regiona

aquifers.
EXPLANATION

Water table

230——————— 

Line of equal hydraulic potential, in feet above a 

standard datum. 

Interval is variable.

Other patterns are 

explained in fig. 1

Kh /Kv = 1,000

METERS FEET 

15-rBO Example of realistic scales that 

could be applied to this diagram General vicin

Number is hea

3000 FEET 

1000 METERS

FIGURE 28.—Hydrologic section showing distribution of hydraulic head in the ground-water reservoir of a multiple-lake system that has low reg

limited extent in the middle of the system.
42 INTERACTION OF LAKES AND GROUND WATER

QUANTITATIVE ASPECTS OF GROUND-WATER 

INTERCHANGE WITH LAKES

Most of the discussion to this point is concerned with 

the qualitative aspects of the relationship of ground water 

to lakes. Attention is focused on factors that control the 

presence and strength, or the absence of a ground-water 

divide beneath a lake. The importance of the position and 

head relative to lake level of the stagnation point of a 

ground-water divide, if present, is stressed. Not only is the 

basic presence or absence of a divide of interest to those 

concerned with lake hydrology, but the quantities of water 

involved are also of great interest.

The basic concept of a flow net is presented in the discussion of ground-water-flow diagrams. It is pointed out 

in that section that figures 4 and 5 are not quantitative 

flow nets in the precise meaning of the term, because the 

ground-water reservoirs they illustrate are considerably 

anisotropic and vertical exaggeration is great.

A quantitative flow net can be used to calculate groundwater discharge through an isotropic medium using the 

following form of Darcy's Law (Freeze, 1969b):

Q=K&$zv&m, (2)

where Q =discharge through a segment of the flow net,

K =hydraulic conductivity,

A$ =drop in hydraulic potential between equipotential surfaces,

As length of flow path in the segment of the flow 

net,

Am =width of the segment of the flow net perpendicular to the flow direction, and 

Au> =thickness of the flow system perpendicular to 

the plane of the diagram.

Taking a unit thickness of the system (if=l), and 

considering that for the squares of the netAs=Am, equation 

2 reduces to

Useful properties of a flow net are that discharge in each 

flow channel (zone located between two adjacent flow 

lines) remains constant throughout its length, and equal 

quantities of water move through each flow channel. In 

spite of the fact that figures 4 and 5 cannot be used to 

accurately calculate discharge, they are useful in 

estimating relative quantities because the proportions of 

flow between the various flow systems (local, intermediate, and regional) remain constant.

Referring to figure 4, it is shown by relative spacing of 

stream lines that much more water moves through the 

local flow systems than the intermediate or regional 

systems. The difference in quantities of flow between the 

various flow systems is not as great in the situation represented by figure 5, partly because of the relatively weak 

local flow near, and the leakage from, lake 3.

The relationship of ground water to lake 3 of figure 5 is 

of interest when considering quantities of flow into and 

out of the lake. Despite the fact that the lake loses water 

through at least half of its bed, the flow net indicates more 

ground-water gain into the lake than loss out of it. Not 

only is the loss relatively low when considering only the 

hydraulic potential distribution, but most of the outflow 

must be through the bottom sediments rather than 

through the littoral zone, which is true of nearly all situations where lakes lose water to the ground-water system. 

From equation 3, it is clear that Q must be rather low if 

both K andA0 are low. This pattern of inflow in the littoral 

zone of a lake, even on the downslope side, and loss, if 

present, through the lake sediments is a common 

phenomenon in nearly all the simulations in which loss 

occurs.

McBride (1969) found that the inflow to Lake Sallie, 

Minnesota, is greatest in the shallows near the shoreline 

and decreases logarithmically toward the deeper part of 

the lake. He later showed, through a modeling approach 

(McBride and Pfannkuch, 1975), that this is a general 

phenomenon. It can also be corroborated by a detailed 

flow net analysis.

The results of this study and those of McBride are 

important to the calculation of water budgets for lakes. 

Because even if a lake loses water through much of its bed, 

the quantity of ground-water gain in the littoral zone may 

be considerably greater than loss through the sediments of 

the lake.

The other important factor in calculating the quantities 

of ground-water interchange with a lake is the hydraulic 

conductivity (K) of the ground-water system. Specific 

values of hydraulic conductivity have been mentioned 

little in the discussion up to this point, because in the 

models all K values are relative. If hydraulic conductivity 

is changed from 0.2 ft/day (0.06 m/day) to 200.0 ft/day (61 

m/day), but all the other K values in each simulation are 

changed accordingly, the resulting head configurations 

are identical. (Compare rows 25 and 26 of table 4.)

If interest is in quantities of ground-water discharge, for 

use in determining water budgets for example, specific 

values of K are extremely important. (See equation 3.) 

Obviously, far more ground water will interact with a lake 

situated in highly conductive rock material such as sand 

and gravel than a lake situated in silty till, provided the 

hydraulic gradients (ftp) are similar.

AREAL AND TEMPORAL VARIATIONS

The hydrologic sections discussed in this report 

represent a two-dimensional vertical view of the groundwater system. By carefully selecting the line of section on a 

map, the general qualitative interrelationship of lakes and 

ground water can be adequately determined. To assure 

areal coverage of the lake's entire ground-water basin, 

however, a more accurate analysis of the entire drainage
IMPLICATIONS FOR FIELD STUDIES

basin of a lake would have to consider radial flow. Perhaps 

the best approach would be to use three-dimensional 

modeling techniques which have recently received much 

research attention and are becoming available for general 

use. Until three-dimensional models of ground water flow 

near lakes are operational, an alternate approach to 

providing areal coverage is to run two-dimensional 

models of selected lines of section. (The lines of section 

must be chosen so they radiate from a lake and are representative of a pie-shaped part of the ground-water system 

near a lake.)

Because the ground-water system is dynamic, a steadystate hydrologic section can be representative of a system 

in which changes occur very slowly over a period of time. 

If lake and water-table altitudes change rapidly, the most 

accurate analysis would be to solve the equations of 

transient ground-water flow. Examples are lakes in a 

setting of widely fluctuating water-table and lake levels, 

especially if the changes are out of phase with one another, 

or are much greater in one than the other. Not only would 

flow nets be needed to calculate the changed rates of 

ground-water discharge but the basic relationship of a 

losing versus a gaining lake might be changed. An 

example of this is given by Meyboom (1967), who found 

that some lakes in the prairie provinces of Canada are 

underlain by ground-water divides for part of the year 

(spring and summer), but lose water to the ground-water 

system for the remainder of the year (fall and winter). 

Depending on the accuracy needed for calculation of 

ground-water discharge, sections would have to be drawn 

to reflect the changing conditions.

There are probably many lakes in hydrogeologic 

settings where the trends in lake levels and water-table 

levels are essentially parallel for most of the year (Winter 

and Pfannkuch, 1976; McBride, 1969). In such situations, 

hydrologic sections would be needed for one time or only a 

few times during the year. Freeze (1969b) argues, in 

considering large basins, that the effect of small-scale 

cycles of wetting and drying during the year can be 

approximated by a steady-state average.

The most likely situations where hydrologic sections 

would be needed for more than one or two times during a 

year would be if: (1) A large rise in lake level would not be 

accompanied by a similar rise in the water table—for 

example, a flood wave passing through a lake that has 

streamflow in and out or a large drop in lake level caused 

by release of water through a dam, or (2) a large drop in the 

water table were caused by discharge of ground water by 

phreatophytes, or by dissipation of water-table mounds 

because of lack of rainfall. Both latter conditions had a 

bearing on the results of Meyboom's study.

IMPLICATIONS FOR FIELD STUDIES OF THE 

INTERACTION OF LAKES AND GROUND WATER

It was pointed out in the introductory paragraphs of this 

report that a number of different approaches to studying

the relationship of ground water to lakes have been tried. 

Some workers have thought that a single piezometer 

(observation well) beside a lake would adequately define 

the relationship, while others have placed large numbers 

of piezometers near a lake. Knowledge of the possible 

variations of ground-water flow systems near lakes, as 

demonstrated in the simulations of this study, suggest 

placement of piezometers that would optimize information on the interaction of lakes and ground water using a 

minimum number of piezometers.

Rather than place piezometers within the drainage 

basin of a lake, most effort should be on finding the 

maximum altitude of the water-table along the areal 

ground-water divide around a lake where such a divide 

exists. For flow-net analysis it is necessary to find this 

water-table-mound maximum and to place a piezometer 

on it for each segment of the lake's drainage basin where 

different ground-water-flow patterns are suspected. As 

seen from the simulations, one of the most important 

locations is the height of the water table above lake level on 

the downslope side of a lake.

Finding the maximum height of a water-table mound 

can be difficult. It has been demonstrated a number of 

times in the field that water-table mounds do not underlie 

directly topographic highs. Topography is not shown on 

the hydrologic sections—one can visualize topographic 

highs anywhere along the sections. In addition, the 

accessibility of roads and the need to obtain the land 

owner's permission is an important constraint in locating 

drilling sites and observation wells. It is easily demonstrated in any of the simulations of a three-lake system (fig. 

4, for example) that a piezometer placed closer to lake 2 

than lake 3 along the water-table profile between lakes 2 

and 3 would record water-table altitudes lower than the 

level of lake 3. One would conclude in this case that, since 

the water table at that point is at an intermediate altitude 

between the two lakes, water flows from lake 3 to lake 2, an 

erroneous conclusion reached in many studies of lakeground water interaction in which few wells are placed 

and the principles of ground-water flow near lakes are not 

understood.

In addition to locating key points along the water-table 

divide around a lake, it is important to know the location 

and head relative to lake level of the stagnation point on 

the ground-water divide beneath the lake, if such a point 

exists. This is best accomplished by placing a nest of 

closely spaced piezometers, each completed at a different 

depth, in the most appropriate location. As suggested by 

the simulations of this study, the most appropriate 

location to look for the stagnation point is beneath the 

shoreline on the downslope side of the lake.

Data on the other factors that control ground-water flow 

(hydraulic conductivity of the various geologic units, 

hydrogeologic boundaries, location of aquifers) are not 

easily obtained, but there are accepted methods for 

acquiring them.
44 INTERACTION OF LAKES AND GROUND WATER

Field determination of the critical Kh/Kv ratio, on the 

other hand, is a serious problem. Although it has not been 

field tested, the method of determining Kh/K-v from 

measured head values proposed by Gillham and 

Farvolden (1974) may prove useful.

SUMMARY

The relationship of ground water to lakes is the least 

well known aspect of lake hydrology. This report discusses 

steady-state digital-model simulations of ground-water 

flow near lakes as a first step toward a general theoretical 

understanding of the interaction of lakes and ground 

water. Numerical-model simulations were run to define 

the principles of the interaction of lakes and ground water, 

and to evaluate parameters that control ground-water flow 

near lakes.

The numerical models, using the alternating-directionimplicit (ADI) and strongly implicit procedure (SIP) 

methods to solve the steady-state ground-water-flow 

equation for vertical sections, were used to simulate onelake and multiple-lake systems.

The report demonstrates by analysis of ground-water 

flow in vertical sections that the existence, position, and 

head value at the stagnation point of the ground-waterflow field, relative to the head represented by lake level, is 

the key to understanding the interaction of lakes and 

ground water. The stagnation point is the point of least 

head along the divide between ground-water-flow cells 

beneath a lake. Therefore, if a stagnation point exists, the 

divide is continuous, the lake cannot leak, and it is the discharge point of the ground-water flow system. If there is 

no stagnation point, the ground-water divide related to the 

lake is not continuous, and the lake can leak through part 

or all of its bed.

Factors that strongly influence the interaction of lakes 

and ground water are height of the water table on the 

downslope side of the lake relative to lake level, position 

and relative hydraulic conductivity of aquifers within the 

ground-water reservoir, ratio of horizontal to vertical hydraulic conductivity of the ground-water system, regional 

slope, and lake depth. Of lesser significance to lakeground water interaction are height of the water table on 

the upslope side of a lake relative to lake level, groundwater reservoir thickness, and presence of lake sediments.

The models are believed to be the first to show the 

detailed patterns of ground-water flow beside and beneath 

lakes in a wide variety of hydrogeologic settings. They are 

unique because they show ground-water-flow patterns 

beneath and between a series of lakes, each at a different 

altitude, along a valley side. A significant finding is that, 

for most settings, water does not move from higher lakes 

through the ground to lower lakes if a water-table mound 

exists between the lakes. This can happen only if one 

presumes a straight-line water-table profile connects the 

lakes.

In field studies of the interaction of lakes and ground 

water it is especially important to define the geologic 

framework including the hydraulic conductivities of the 

geologic units—the boundaries of the system, the ratio of 

horizontal to vertical hydraulic conductivity, the height of 

the ground-water watershed relative to lake level, and the 

strength (head at the stagnation point relative to that 

represented by lake level) and position of the stagnation 

point.

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