Redefinition of coefficient of storage.


                           UNITED STATES
                    DEPARTMENT OF THE INTERIOR
                         GEOLOGICAL SURVEY
                     Water Resources Division
                        Washington 25, D.C.

                                               February 1, 1955
                                               Code No. 45030

GROUND WATER BRANCH MEMORANDUM NO. 55.28

To:       Field offices of the Ground Water Branch

From:     Chief, Ground Water Branch

Subject:  Redefinition of coefficient of storage.

It has been evident for a long time that some of the hydrologic
concepts used widely in ground-water investigations have not been
clearly understood or consistently applied.  The resultant
confusion that has existed in the minds of many of us too
frequently has been reflected in the way our field studies have
been carried on and analyzed and in the way our investigative
reports have been written and reviewed.  There has been a growing
desire, therefore, to put our house in better order, to refine our
appreciation and understanding of certain important hydrologic
concepts, and so to improve the effectiveness with which we
examine, analyze, and report on problems of ground-water
occurrence and movement.  Paramount among the concepts needing
reappraisal is that of the coefficient of storage, and the results
of work directed toward this objective are the basis of this
memorandum.  Additional memoranda will follow as promptly as other
concepts can be scrutinized and reappraised.

It should be made clear that the redefinition given here, and the
others that may follow, are not to be regarded as sacrosanct and
incapable of further change.  Any and all of our definitions and
concepts are subject to change in the light of new information.
Each redefinition, therefore, is to be regarded as a carefully
thought cut interim term which is to be used until and unless
restudy warrants a further change.

In the use of the term coefficient of storage there has been
uncertainty as to whether the term applied to an aquifer only, or
to the aquifer and the adjacent confining beds; there has been
uncertainty also as to whether it applied to artesian aquifers
only or to water-table aquifers as well.  As you will see, the
proposed new definition applies to the aquifer only, and it
applies to both artesian and water-table aquifers.  Detailed
explanations, complete with diagrams, of its application to
artesian and to water-table situations have been prepared for
inclusion in the summary of John G. Ferris' hydraulics lectures,
to be duplicated for internal distribution in the near future.
These explanations will indicate, among other things, why the
specific yield is essentially equal, but not exactly equivalent,
to the coefficient of storage for water-table conditions.

The proposed definition of the coefficient of storage follows.
The first paragraph beneath it constitutes explanatory material
which, in this or some modified form, may appropriately be
included in most reports in which the term coefficient of storage
is defined.  The paragraphs beyond are additional explanatory
material for your information.

     "The coefficient of storage of an aquifer is the volume of
     water it releases from or takes into storage per unit surface
     area of the aquifer per unit change in the component of head
     normal to that surface."

A simple way of visualizing this concept is to imagine an artesian
aquifer which is elastic and is uniform in thickness, and which is
assumed, for convenience, to be horizontal.  If the head of water
in that aquifer is decreased there will be released from storage
some finite volume of water that is proportional to the change in
head.  Because the aquifer is horizontal, the full observed head
change is evidently effective perpendicular to the aquifer
surface.  Imagine further a representative prism extending
vertically from the top to the bottom of the aquifer, and
extending laterally so that its cross-sectional area is
coextensive with the aquifer-surface area over which the head
change occurs.  The volume of water released from storage in that
prism, divided by the product of the prism's cross-sectional area
and the change in head, results in a dimensionless number which is
the coefficient of storage.  If this example were revised
slightly, it could be used to demonstrate the same concept of
coefficient of storage for a horizontal water-table aquifer or for
a situation in which the head of water in the aquifer is
increased.

As with almost any concise definition of a basic concept, it is
necessary to develop its full significance, its limitations, and
its practical use and application through elaborative discussion.
The coefficient of storage is not exception in this respect, and
the following discussion will serve to bring out a few ideas that
are important in applying the concept to artesian and water-table
aquifers in horizontal or inclined attitudes.

Observe that the statement of the storage-coefficient concept
first focuses attention on the volume of water that the aquifer
releases from or takes into storage.  Identification and
measurement of this volume poses no particular problem but it
should be recognized that it is measured outside the aquifer under
the natural local conditions of temperature and atmospheric
pressure; it is not the volume that the same amount of water would
occupy if viewed in place of the aquifer.

Although the example used to depict the concept of the storage
coefficient was arbitrarily developed around a horizontally
disposed artesian aquifer, the concept applies equally well to
water-table aquifers and is not compromised by the attitude of the
aquifer.  This flexibility of application relies importantly,
however, an relating the storage-coefficient concept to the
surface area of the aquifer and to the component of head change
that is normal to that surface.  In turn, this relationship
presupposes that the particular aquifer prism involved in the
movement of water into or out of storage is that prism whose
length equals the saturated into or out of storage is that prism
whose length equals the saturated thickness of the aquifer,
measured normal to the aquifer surface, and those cross-sectional
area equals the area of the aquifer surface over which the head
change occurs.  Furthermore, water moves into or out of storage in
this prism in direct proportion only to that part of the head
change that acts to compress or distend the length of the prism.
In other words, the component of the head change to be considered
in the release or storage of water is that which acts normal to
the aquifer surface.  The mathematical models devised for
analyzing ground-water flow usually require uniform thickness of
aquifer.  However, the storage coefficient concept, as defined
here, applies equally well to aquifers that thicken or thin
substantially, if the "surface area" is measured in the plane that
divides the aquifer into upper and lower halves that are
symmetrical with respect to flow.  The imaginary prism would then
be taken perpendicular to this mean plane of flow.


                        The Artesian Case

Consider an artesian aquifer, in any given attitude, in which the
head of water is changed, but which remains saturated before,
during, and after the change.  It is assumed that the beds of
impermeable material confining the aquifer are fluid in the sense
that they have no inherent ability to absorb or dissipate changes
in forces external to or within the aquifer.  Inasmuch as no
dewatering or filling of the aquifer is involved, the water
released from or taken into storage can be attributed only to the
compressibility of the aquifer material and of the water.  By
definition the term "head of water" and any changes therein
connotes measurements in a vertical direction with reference to
some datum.  In a practical field problem the change in head very
likely would be observed as a change in water-level elevation in a
well.  The change in head is an indication of the change in
pressure in the aquifer prism, and the total change in force
tending to compress the prism is equal to the product of the
change in pressure multiplied by the end area of the prism.
Obviously this change in force is not affected by the inclination
of the aquifer, inasmuch as a confined pressure system is involved
and the component of force due to pressure always acts normal to
the confining surface.  Thus any conventional method of observing
head change will correctly identify the change in pressure normal
to the aquifer surface and may be considered as a component of
head acting normal to that surface.

Summary statement -- For an artesian aquifer, regardless of its
attitude, the water released from or taken into storage, in
response to a change in head, is attributed solely to
compressibility of the aquifer material and of the water.  The
volume of water (measured outside the aquifer) thus released or
stored, divided by the product of the head change and the area of
the aquifer surface over which it is effective, correctly
determines the storage coefficient of the aquifer.


                       The Water-Table Case

Application of the storage-coefficient concept to water-table
aquifers is more complex, although reasoning similar to that
developed in the preceding paragraphs can be applied to the
saturated zone of an inclined water-table aquifer.  Consider a
water-table aquifer, in any given attitude, in which the head of
water is changed.  Obviously there will now be dewatering or
refilling of the aquifer, inasmuch as it is an open gravity system
with no confinement of its upper surface.  Thus the volume of
water released from or taken into storage must not be attributed
not only to the compressibility of the aquifer material and the
water, in the saturated zone of the aquifer, but also to gravity
drainage or refilling in the zone through which the water table
moves.  The volume of water involved in the gravity drainage or
refilling, divided by the volume of the zone through which the
water table moves, is the specific yields.  Except in aquifer of
low porosity the volume of water involved in gravity drainage or
refilling will ordinarily be so many hundreds or thousands of
times greater than the volume attributable to compressibility that
for practical purposes it can be said that the coefficient of
storage equals the specific yield.  The conventional method of
measuring change in head by observing change in water level
elevation in a well evidently identifies the vertical change in
position of the water table.  In other words, head change equals
vertical movement of the water table.  It can be seen that the
volume of the zone through which the water table moves is equal to
the area of the aquifer surface over which the head change occurs,
multiplied by the head change, multiplied by the cosine of the
angle of inclination of the water-table.  The product of the last
two factors is the component of head change acting normal to the
aquifer surface.  The importance of interpreting correctly the
phrase "component of head change" which appears in the definition
of the storage coefficient cannot be overemphasized.

Summary statement -- For a water-table aquifer, regardless of its
attitude, the water released from or taken into storage, in
response to a change in head, is attributed partly to gravity
drainage or refilling of the zone through which the water table
moves, and partly to compressibility of the water and aquifer
material in the saturated zone.  The volume of water thus released
or stored, divided by the product of the area of aquifer surface
over which the head change occurs, and the component of head
change normal to that surface, correctly determines the storage
coefficient of the aquifer.  Usually the volume of water
attributable to compressibility is a negligible proportion of the
total volume of water released or stored and can be ignored.  The
storage coefficient then is sensibly equal to the specific yield.


                            Conclusion

It should be understood that the numerical results obtained by
substituting aquifer-test data in an appropriate mathematical
model indicate the transmissibility and storage coefficients for
an ideal aquifer.  The hydrologist must judge how closely the real
aquifer resembles this particular ideal.  It is usually recognized
that in short pumping tests under water-table conditions the water
does not drain from the smaller openings in the unwatered portion
of the aquifer in any manner even approximating the instantaneous
release assumed in devising the mathematical model (the  Theis
nonequilibrium formula).  Similarly, in testing artesian aquifers
it is recognized that the aquifer skeleton does not adjust
instantaneously to the change in head, that considerable water is
often contributed by intercalated clay beds, and furthermore that
water leaks through the confining beds which, in the mathematical
model, have been assumed to be impermeable.  These recognized
departures from the ideal, however, do not constitute grounds for
modifying the definition given for the coefficient of storage.
The definition does not, and cannot assure anyone that any pumping
test in any aquifer will result in establishing the correct
coefficient of storage for that aquifer.

Pertinent comments on the redefinition of the coefficient of
storage are solicited.  The form given above may be considered
official, however, until further notice.



                                  (s) A. N. Sayre