TRANSMISSIVITY
DISTRIBUTION IN THE TUCSON BASIN AQUIFER
Donald J. Supkow, Ph.D.
Ed. By: William M. Turner, Ph.D.
INTRODUCTION
This study is an outgrowth of an effort by the University of Arizona, Department of
Hydrology and Water Resources to construct a digital model of the Tucson Basin
aquifer. Of particular importance in the construction of digital and electric analog
models of nonhomogeneous aquifers is the spatial distribution of aquifer
transmissivity.
A casual inspection of the areal distribution of aquifer transmissivity within the
Tucson Basin aquifer as determined by aquiferperformance tests (University of Arizona
Agricultural Engineering files) and as reported by Anderson (1970) indicates that the
values of transmissivity are not randomly distributed in space.
Low values of aquifer transmissivity (<5,000 gallons per day per foot (<62 m^{2}/d)
occur around the periphery of the basin whereas larger values of aquifer transmissivity
(greater than 100,000 gpd/ft (1,240 m^{2}/d) occur within the central and outlet
parts of the basin. This is manifested by a general trend of increasing aquifer
transmissivity in the direction of groundwater flow.
Gates (1972) found the frequency distribution of aquifer transmissivity in the Tucson
Basin, as determined by aquiferperformance tests, to be normal or lognormal. Such
a sample is biased in that people tend to drill wells in known areas of high
transmissivity and avoid drilling wells in areas of known low transmissivity. The
frequency distribution of transmissivity as derived from a calibrated and validated
digital computer model of the Tucson Basin aquifer is thought to be more nearly
representative of the real world because the aquifer sample is without bias. That
is, the sample comes from nearly the entire aquifer and not just a small part of it.
GEOHYDROLOGIC SETTING
The Tucson groundwater basin is located in the Basin and Range Physiographic Province
of the arid American southwest. This area is characterized by infrequent but
sometimes violent local thunderstorms in the summer and widespread frontal storms in the
winter. Recharge to the groundwaterflow system is primarily by infiltration of
runoff in the various stream channels throughout the area. The stream channels
generally flow only after rainfall events. Average yearly precipitation range from
10 to 12 inches (254 to 305 mm).
The aquifer is comprised of Tertiary to Quaternary alluvium that occupies downfaulted
blocks between the surrounding mountains.
Ground water is usually under unconfined conditions. The direction of
groundwater flow is generally from south to north. In some areas the alluvium may
be as much as 7,000 feet (2,134 m) thick (Davis, 1965). The alluvium ranges from
finegrained silt and clay through poorly to wellsorted sand to poorly to wellsorted
gravel.
ELECTRIC ANALOG MODEL
An electric analog model study of the Tucson Basin aquifer was conducted by Anderson
(1970). His model included an area of about 750 square miles and was
constructed with a nodal spacing of onehalf mile (0.8 km).
DIGITAL MODEL
A digital model of the Tucson Basin was developed by Supkow using the same nodal
spacing as the electric analog model and covering approximately the same area. The
digital model has a total of 2,497 nodes each covering an area of 1/4 square mile (0.65
km2).
Supkow used calibration process and adjusted aquifertransmissivity values at
individual nodes by trial and error until calculated waterlevel changes matched
historical waterlevel changes. He stopped the calibration process when the
average nodal error (the difference between calculated and historical waterlevel changes
over the 19 year period from 1947 to 1966) for the entire model was reduced to 5.65 feet
(1.72 m).
Supkow prepared an aquifertransmissivity isopleth map of the Tucson Basin as
determined by the digital modeling process. In some parts of the aquifer the
transmissivities are somewhat less than they were prior to the present period of heavy
pumping that caused a decrease in saturated thickness.
THEORY OF TRANSMISSIVITY DISTRIBUTION IN AN ALLUVIAL AQUIFER
The alluvial sediments within the Tucson aquifer were deposited on some sort of
preexisting erosional surface. We can expect that this erosional surface, prior to
being covered by alluvium, had some sort of drainage network. If we assume that
Horton's Laws of stream orders were as valid in the past as they are at present, we can
expect that they applied to the drainage network of the paleoerosional surface.
Horton's Law states that there is an exponential relationship between stream order and
drainage network parameters such as area of drainage basin and length of stream.
Stream order is a measure of the hierarchy of stream tributaries in which (for a given map
scale) the smallest stream is assigned the lowest order of "1" and the larger
streams are assigned succeedingly higher orders.
For example, a firstorder stream has no tributaries. A secondorder stream has
only firstorder streams as tributaries. A thirdorder stream has only first and
secondorder streams as tributaries and so on. If we prepare a semilogarithmic plot
of stream order versus length of stream or area drained for any catchment area in the
world, we obtain a straight line where the highest order stream has the longest length and
drains the largest area. The lowestorder streams have the shortest length and drain
the smallest areas. Likewise, a semilogarithmic plot of stream order versus number
of streams is also a straight line. The firstorder streams for any particular
drainage net are most numerous.
In consideration of the above relationships, Supkow hypothesized that there should be
an exponential relationship between transmissivity of an alluvial aquifer and the area of
the aquifer having the specified transmissivity. We would expect the alluvial
aquifer to have more lowvalued transmissivities than highvalued transmissivities.
In terms of a probability density function (PDF) this would be
f(T) = 1/l e ^{T/l} for T>O
f(T) = 0 otherwise
where
T = transmissivity
l = arithmetic mean value of transmissivity.
This PDF refers only to the frequency distribution of transmissivity and not to the
locations of any particular transmissivity in space.
TEST OF HYPOTHESIS
To test this hypothesis, a statistical study was made of the transmissivity values used
in the digital model of the Tucson Basin aquifer. Values of aquifer transmissivity
ranged from <1,000 gallons per day per foot (<124 m2/d) to as high as 450,000 gpd/ft
(5,580 m^{2}/d). The arithmetic mean of all 2,497 transmissivity values in
the model was 54,848 gpd/ft (680 m2/d).
The probability P that T should lie between specified limits T_{1} and T_{2},
is given by:
A plot of probability versus transmissivity with l
= 54,848 gpd/ft (705 m2d) is given by the solid curved line in the Figure
1. For example, the probability of transmissivity values less than or equal to
10,000 gpd/ft (<=124 m^{2}/d) is 0.1667, the probability of transmissivity
values less than or equal to 50,000 gpd/ft (<=620 m^{2}/d) is 0.5981, and the
probability of transmissivity values less than or equal to 450,600 gpd/ft (<=5,587 m^{2}/d)
is 0.9997. This means that chance of successfully drilling randomly to find areas of
the aquifer with transmissivity greater than 450,000 (5,580 m2/d) is practically nil.
A plot of probability versus transmissivity with l = 54,848
gpd/ft (705 m2d) is given by the solid curved line in the Figure 1.
For example, the probability of transmissivity values less than or equal to 10,000 gpd/ft
(<=124 m^{2}/d) is 0.1667, the probability of transmissivity values less than
or equal to 50,000 gpd/ft (<=620 m^{2}/d) is 0.5981, and the probability of
transmissivity values less than or equal to 450,600 gpd/ft (<=5,587 m^{2}/d) is
0.9997. This means that chance of successfully drilling randomly to find areas of
the aquifer with transmissivity greater than 450,000 (5,580 m2/d) is practically nil.
The plotted points represent probabilities of transmissivity as
determined by actual count of the number of nodes in the digital model having
transmissivity values less than or equal to the plotted transmissivity value. We can
see that there is a reasonably close fit between the calculated curve and the observed
data points. The data points used to plot Figure 1 are in Table
1.
DISCUSSION OF RESULTS
It is common knowledge that alluvial basins consisting of coalesced alluvial fans have
the coarsest, mostpoorly sorted detritus around the periphery of the basins.
Because of the poorlysorted detritus and the thinner saturated thickness around a basin
periphery, the transmissivity there is generally lower. Although transmissivity,
(the product of permeability and saturated thickness), is a derived mathematical
parameter, it is a parameter that has real meaning in the real world of alluvial
aquifers. It is a quantity that results from the various inputs that form a dynamic
alluvial aquifer system.
Given the present aquifer, an increase in average annual rainfall will increase the
average recharge rate. This will result in more ground water flowing through the
system. Under similar hydraulic gradients and greater saturated thickness, the
aquifer will have higher transmissivity. Conversely, a decrease in the quantity of
ground water flowing through the aquifer will result in lower aquifer
transmissivities.
It thus appears that the average value of transmissivity for an alluvial basin aquifer
given by "l" is a function of the climate and
the drainage area of the basin. If is true, it then might be possible to predict the
mean value of transmissivity within an alluvial basin if we have the climatic data and the
area of the rainfall catchment for the basin, even though no information is available on
the thickness or permeability of the sediments. In following this reasoning, if we
find another groundwater basin similar in size and climatic conditions to the Tucson
Basin, we would expect that basin to have an average transmissivity similar to that of the
Tucson Basin.
Now that more and more aquifer systems are being modeled by means of digital computers,
analyses such as the one described here could be carried out routinely. It would be
a simple matter to check on the validity of the relationships postulated herein.
Certainly, the a priori knowledge that the transmissivity must be exponentially
distributed would be a powerful constraint on construction of the model.
REFERENCES
Anderson, T. W., Electricalanalog analysis of the hydrologic system in Tucson
basin,
Arizona: U.S. Geol. Survey WaterSupply Paper 1939C, p. 1524,
1970.
Davis, R. W., A geophysical investigation of hydrologic boundaries in the Tucson
basin, Pima County, Arizona: unpublished Ph.D.
dissertation, University of
Arizona, 64 p., 1965.
Gates, J. S, Worth of data used in digitalcomputer models of groundwater basins:
unpublished Ph.D. dissertation, University of Arizona, 214 p.,
1972.
Supkow, D.J., 1973, Transmissivity Distribution in the Tucson Basin Aquifer,
Proceedings of the Arizona Section of the American Water Resources
Association
and the Hydrology Section of the Arizona Academy of Science, May 56,
1973,
Prescott, Arizona, PP. 113123.
