HONR 297
Environmental Models
Chapter 2: Ground Water
2.7: Water Table Contour Maps
Images Courtesy: Charles Hadlock, Mathematical Modeling in the Environment
Hydraulic Head Revisited
From our work with
Darcy’s Law and the
interstitial velocity
equation, it should be clear
that a key to ground water
investigation is the
hydraulic head.
 The general method to
determine groundwater
head levels is to drill
several test wells and
measure depths to ground
water to determine the
water table elevation via
known horizontal datum
levels!

Courtesy USGS: http://ga.water.usgs.gov/edu/earthgwwells.html
2
Water Table Contour Map

Once the data from
the test wells is
collected, the
corresponding head
information is usually
put into a water table
contour map, such as
the one shown in
Figure 2.20 of our
text (p. 39).
Courtesy: Charles Hadlock, Mathematical Modeling in the Environment
3
Water Table Contour Map

In a fashion similar to
topographic maps,
which display points
of equal elevation via
contour lines, water
table contour maps
show points of equal
head value via
contour lines!
Courtesy: Charles Hadlock, Mathematical Modeling in the Environment
4
Water Table Contour Map


Remark: Usually, water
table contour maps
are constructed from
test well head levels by
means of interpolation
and extrapolation.
The general
topography of the land
surface above the
water table may also
be taken into account
(especially for a
shallow ground water
aquifer).
Courtesy: Charles Hadlock, Mathematical Modeling in the Environment
5
Key Fact!






When working with water table contour
maps, it is a good idea to keep the following
fact in mind:
Water Table Contour Map Key Fact: At any
point, the ground water always* tends to flow in a
direction that is perpendicular to the head
contour at that point.
Reason: Think of a marble rolling down the
side of a hill – it will always want to move
down the hill in the direction that is always as
steep or directly downward as possible.
Mathematically we say the marble is following
the path of steepest descent, which turns out to
be perpendicular to topographic contour
lines.
Ground water behaves in the same way – it
responds to gravity and wants to follow the
steepest or quickest path down the water
table.
In the rest of Chapter 2 we will assume that
the Key Fact holds – i.e. the flow of
groundwater will be perpendicular to contour
lines.
Courtesy: Charles Hadlock, Mathematical Modeling in the Environment

*The path of the groundwater may be blocked or
deflected in some portion of the aquifer – similar
to a marble rolling downhill and hitting a tree. We
will assume in what follows that this does not
happen.
6
Example 1

Consider the hydraulic head
contour map shown in Fig. 2.2
of our text (p. 41). Assume
the aquifer under
consideration has a hydraulic
conductivity of 50 ft/day and a
porosity of 25%. Assuming
that a source of
contamination exists at the
point marked X, determine
a.
b.
Which of the three wells W1,
W2, or W3 will most likely
be directly affected.
How long will it take for the
contaminated ground water
to travel from the point X to
the affected well?
Courtesy: Charles Hadlock, Mathematical Modeling in the Environment
7
Example 1
Solution:
 (a) Since the head
levels decrease as we
move to the right,
using the key fact, we
can sketch a flow line
starting at X and
ending at W3 (see
Fig. 2.23 in text).

Courtesy: Charles Hadlock, Mathematical Modeling in the Environment
8
Example 1
Courtesy: Charles Hadlock, Mathematical Modeling in the Environment
9
Example 1
Courtesy: Charles Hadlock, Mathematical Modeling in the Environment
10
Example 1
Solution:
 (b) Using the ground water flow
line we drew in part (a), along
with the water table contour
lines, we can estimate travel time
via the interstitial velocity
equation and the relationship

◦ time = distance/velocity.



First, break the flow path into
segments which are determined
by the contour lines, starting
point X, and ending point W3.
In this case, we will have five
segments!
Our goal is to compute travel
time of the groundwater for each
of the five segments.
Courtesy: Charles Hadlock, Mathematical Modeling in the Environment
11
Example 1
S1
S2
S3
S4
Courtesy: Charles Hadlock, Mathematical Modeling in the Environment
S5
12
Example 1


Recall that ν = (K i)/η.
We are given
◦ K = 50 ft/day
◦ η = 0.25.

From the water table contour map we can
find i = Δh/L for each of the segments 2,
3, and 4.

Δh is found from the given contour levels
and L can be measured off of the map using
the given scale!
13
Example 1
For segments 1 and 5, we cannot measure
Δh from the given information, so we will
assume the hydraulic gradient i doesn’t
change drastically from one point to
nearby points.
 Therefore, assume that the hydraulic
gradients in segments 1 and 5 are the
same as the hydraulic gradient in adjacent
segments.

14
Example 1




Let d = length of a given segment.
Then in each calculation that follows, L = d.
Also, let t = travel time in a given segment.
Segment 2:
◦ d = 460 ft
◦ i = Δh/L = (85 ft - 80 ft)/460 ft = 5/460 ≈ 0.011
◦ ν = (K i)/η = ((50 ft/day)(5/460))/0.25 ≈ 2.2 ft/day
◦ t = d/ ν = (460 ft)/(((50 ft/day)(5/460))/0.25) ≈ 209 days.
15
Example 1

Segment 1:
◦ d = 170 ft
◦ i = 5/460 ≈ 0.011 (from segment 2)
◦ ν = (K i)/η = ((50 ft/day)(5/460))/0.25 ≈ 2.2 ft/day
◦ t = d/ ν = (170 ft)/(((50 ft/day)(5/460))/0.25) ≈ 77 days.
Find the time ground water takes to flow
along segments 3, 4, and 5 …
 Our findings can be summarized in a
table:

16
Example 1
Segment
Travel time (days)
1
77
2
209
3
294
4
200
5
123
Total
903 days ≈ 2.5 years
17
Using Excel for Ground Water
Travel Time
We can set up a
table in Excel to
automate the
process used for the
calculations of travel
time in each
segment.
 Hadlock suggests a
set-up on page 43.

18
Excel Table for Example 1
19
Excel Table for Example 1
(Formulas)
20
Example 2

Consider the situation described by Fig.
2.24. The contour lines represent
estimated contours of hydraulic head in
a shallow aquifer composed chiefly of
coarse sand. If a major gasoline spill
onto the ground occurs at point X, as
shown in Fig. 2.24, and some of the
gasoline seeps down to the water table,
indicate on the diagram its likely
migration path with the ground water,
and estimate how long it might take for
the first traces of dissolved gasoline to
reach the boundary of the region
shown in the figure. Pick representative
values of hydrologic parameters you
need from Table 2.1. If you need to
make any additional assumptions, be
sure to explain what you do.
Courtesy: Charles Hadlock, Mathematical Modeling in the Environment
21
Example 2
Courtesy: Charles Hadlock, Mathematical Modeling in the Environment
22
Resources

Charles Hadlock, Mathematical Modeling
in the Environment, Section 2.7
◦ Figures 2.20, 2.21, 2.22, 2.23, and 2.24 used
with permission from the publisher (MAA).

USGS:
◦ http://ga.water.usgs.gov/edu/earthgwwells.html
23