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through the dam, near the heel, adding to the underground flow. In

other words, the value of — - for a dam of this type is less than for

a dam of the type shown by Fig. 14, even with the same length of
base and the same depth of sheet-piling, or other cut-off wall.



DISCUSSION ON DESIGNING AN EAKTH DAM 00

D. C. Hennv,* M. Am. Sue. C. E. (by letter). — The design of a Mr.
dam in a situation such as the author describes presents an interesting H*^"°J •
and difficult problem. In his conclusions he mentions several causes
of earth dam failures, and states that his paper deals with one cause
only, namely, springs or boils, which might produce piping under the
base of the dam.

The usual percolating velocities are exceedingly low, far below the
power of transporting material. The piping or blow-up phenomenon
implies a combination of circumstances, relating to the material in
place, essentially differing at particular points from the ordinary per-
colating conditions. In the case of clayey foundation, initial cracks
may transmit a large portion of the available pressure to a point close
to free exit, and may set up progressive erosion aided by arching. In
the case of sand and gravel, strata of unusual openness to flow may
be contiguous to layers of very fine sand. In very coarse material, the
open spaces between pebbles and cobbles may be so great as to preclude
true percolation and permit comparatively free flow and high velocity.
The latter case is one which cannot be regarded as applying to founda-
tions for high dams. The other cases are dependent on local devia-
tions from general homogeneity, which for large areas can hardly ever
be known definitely. Moreover, the effect of such deviations cannot be
ascertained by experiments with selected samples of materials placed
in tanks or boxes.

The experiments conducted by the author do not appear to have
had for their object the determination of a maximum gradient which
would be safe against piping, but rather the most economical form of
dam which would produce a maximum reduction of the water gradient,
thereby minimizing the piping danger as well as seepage losses.

It appears to the writer that, whatever may be the danger from
piping, it must be judged by examination of test pits and experience
with existing dams. Usually, with the foundation material which the
author describes, and with ordinary slopes of an earth dam, such
danger is not great, and if necessary can be counteracted economically
by a gravel blanket on the ground below the down-stream toe. Nor
need there be any fear of bank sloughing, if gravelly material similar
to that in the foimdation is used in the down-stream portion of the dam.
The real problem seems to be that of insuring against excessive seepage
losses such as would render the reservoir useless.

^-r. In describing the history of the reservoir, the author states that
when the dam was completed to a height of 30 ft., it was subjected to
a head of 25 ft., at which time water escaped in considerable quantity
from the down-stream toe. Though the quantity of water escaping is
not stated, it is evident that, if this is known or can be ascertained ap-

* Portland, Ore.



56 DISCUSSION ON DESIGNING AN EAETH DAM

Mr. proximately, a full-size experiment is at hand on which to base some
'^°"^' judgment as to the seriousness of the problem.

A portion of this escaping water may have come through the dam
proper. It is certain, however, that by far most of it passed through
the dam foundation, which is described as being of a very porous
nature and of unknown depth, roughly estimated at 240 ft.

If, at that stage of completion, the dam had a full width of base
of approximately 500 ft., the water gradient producing this heavy seep-
age may have been 1 to 20, and a statement of the seepage per linear
foot of dam would permit some judgment as to the practical admissi-
bility, from the storage point of view, of such gradient for a full
reservoir.

The author's determination of a safe gradient of 1 to 9 by vertical
tank test is by no means conclusive. The essential feature, namely,
the quantity of seepage with such gradient, is not stated. However,
even if it were stated and were satisfactorily low, it is necessarily
based on the use of samples of foundation material which, in the
nature of the case, cannot represent any known sort of average of the
deep masses of gravel, sand, cobbles, and boulders as they lie in place
under the dam. Homogeneity cannot exist to any degree in material
of greatly varying sizes deposited by successive floods of varying
intensity.

Assuming, however, that the samples used in the author's tests are
representative of the general foundation, the tests made in a rect-
angular box with a model of a dam with varying depths of tight cut-off
show the quantity of seepage under full head. For the first model
tested, this averaged approximately 0.0040 sec-ft., and, for the second
model, 0.0025 sec-ft. per lin. ft. of model. The writer understands that
these quantities refer to the flow measured in the experiments, on a
scale of y|„ of full size. If this understanding: is correct, then, so
far as the experiment goes, the deduction may be made that, for similar
material in the foundation, the seepage for a dam built on the basis of
the second model would be 120 X 0.0025, or 0.3 sec-ft. per lin. ft. of dam.

The longitudinal section shows the dam to be 400 ft. in length
across the general river bed and 1 600 ft. in length on the adjoining
bench. No data are at hand as to rise of rock under this bench. It
may be interesting, nevertheless, to inquire as to what the total seep-
age under the dam would be if the rock were to rise but slightly away
from the river. In that case the average gradient and seepage per
linear foot for the bench portion of the dam would approximate one-
half that for the full height of the dam. On such assumptions, a

seepage would result of (^400 + w ) X 0..3 = .360 sec-ft. This quantity
of seepage is clearly inadmissible, and the writer deems it likely that



DISCUSSION ON DESIGNING AN EAKTH DAM 57

some of the foregoing assumptions may be known by the author to Mr.

1 Henny.

be erroneous.

Independent of the doubt regarding test samples being representa-
tive of material in place, there must be serious uncertainty as to the
possibility of driving sheet-piling to a depth SO ft. below the bottom
of the cut-off trench, and as to the tightness of such sheet-piling when
driven.

It will be noted that the final design of the dam as presented by
the author shows an approximate gradient of 1 to 10 for full reservoir.
This may be about twice the gradient which prevailed when there was
25 ft. of head against the present dam, at which time heavy seepage
losses occurred.

A detailed study of the pressures as registered in the experimental
box with the model of the dam reveals some marked inconsistencies,
which, if the experimental results are to be made the basis for design,
may require explanation. In considering this subject, the writer has
confined himself to the use of experiments made under a full head of
10 in. representing 100 ft. on the scale of the experiment, and has
selected for this purpose only those numbers, four in the first and three
in the second series, for which pressures are recorded at all points.

Individual differences of flow and pressure are rather large, the
maximvun variations from the average being as follows:

First series: Second series:

Experiments 9, 14, Experiments 4, 8,

19, and 84. and 12.

Flow 7% 20%

Pressure 20% 13%

In order to eliminate individual variations, whatever may be their
cause, pressures and pressure drops are figured on the basis of average
values and are listed for comparison in Table 5.

Table 5 shows the following rat;her surprising results as to pressure
head destroyed by percolation: The pressure drop is greater from
open water to A, from B to Q, and from D to E, with smaller than
with larger flow; the pressure drop is greater from A \o B without
sheet-piling, and with small flow than with sheet-piling and with large
flow; the pressure drop is greater from C to D, in proportion to the
flow, with shallow than with deep sheet-piling.

The drop from open water to the point. A, up stream from the
points of cut-off, is 60% of the total head, in spite of the short dis-
tance of travel; so that the upper portion of the water gradient is
steepest. The great loss of head at entrance appears to be inherent in
experiments of this kind. It may well be doubted, however, whether
such losses occur in the case of actual dams, where the area of entrance
is very extensive, imless it is induced by silt deposits. Should no such



Mr.
Henny,



58 DISCUSSION ON DESIGNING AN EAETH DAM

TABLE 5. — Pressures and Pressure Drops.





Average of


Experiments.




1st Series :
Nos. 9, 14, 19, and 24.


2d Series :

Nos. 4, 8, and 12.




0.0040 sec-ft.


0.0025 sec-ft.








Pressure.


Pressure
drop.


Pressure.


Pressure
drop.




100 ft.

41.6

50 ft. in 240 ft.

37.6

34.8

85 ft. in 240 ft.

12.9

10.1

7.9




100 ft.






58.4 ft.


60.5 ft.


4


39.5
25 ft. in 240 ft.

27.7

■" 22;6

125 ft. in 240 ft.
4.3




Cut-off


4.0


11.8


g




c


2.8


5.7


Cut-otf


21.9


17.7


J)






2.8


3.7


jE .


0.6






2.2


0.6


jr

















92.1




100.0











loss with the actual dam be experienced, and should the pressure at
A be 80 or 90% of the total head instead of 40%, the actual seepage
losses may be double those deduced from the experiments.

In regard to the feature of the design consisting of a tight blanket
up stream intended to lengthen the path of the water, the reasoning
of the author is believed to be sound. The same method was advocated
by the writer and was adopted in the case of the Grand River Diversion
Dam built by the Reclamation Service near Grand Junction, Colo.
The object in this case was the reduction both of uplift and of seepage.
Fig. 18 shows a cross-section of this dam.



CROSS-SECTION OF
GRAND RIVER DIVERSION DAM



Movable Crest




Fig. 18.



It may be stated that in this case measurements of uplift pressures
were made through pipes placed in piers in the dam and ending in



DISCUSSION ON DESIGNING AN EAETH DAM 59

pockets of screened gravel under the fovmdation. The results indicate Mr.
complete absence of entrance losses. They also show drop of pressure ^"°^'
to be closely proportionate to distance along line of creep. It should
be stated, however, that owing to delay in placing movable gates, the
heads at the time of the two measurements were in each case only
between 4 and 5 ft., and that measurements under a full head of
18 ft. may give different results.

Experiments of the character made by the author must always be
of intense interest to hydraulic engineers, and deserve full recognition.

Joseph Jacobs,* M, Am. Soc. C. E. (by letter). — This paper is of Mr.
interest because it deals with an important engineering subject, con- ' ^^*^ ^
cerning which there is all too little reliable information on which
definite theory and judgment may be predicated. The author is entitled
to credit for assembling the results of these tests for the use of the
Profession, and for his frankness in presenting his interpretation and
application of the facts disclosed.

The writer believes that the author failed to reach certain important
conclusions, apparently fully warranted by the tests, and that in some
respects his deductions as to the theory of luiderground water move-
ments are fallacious. Particularly does he believe that the tests do not
justify the conclusion, reached by the author, that the "line of creep"
theory has been disproved.

In describing the initial seepage test in the 30-in. cylindrical tank
(page 7), the author states:

"A valve at the bottom held back the water until the pore spaces
were completely filled; then the valve was opened, and after a few
minutes the relative elevation of the water in the two tubes became
constant. The difference in elevation determined the loss of head, and
from this the hydraulic gradient was computed."

j; *******

"With 6 in. of water on the soil surface, and 8 in. of soil above the
upper tube, the loss of head was 1 ft. in 9."

*******

"Having determined the hydraulic gradient of the underground
material, a trial design was made to find what dimensions would be
necessary in a dam constructed wholly of this gravel."

■ ■These quotations, it seems to the writer, disclose a fundamental
error in the author's application of his test. It will be noted that
there is no mention of the quantity of seepage flow, and, as seepage
flow and hydraulic gradient are inter-dependently related, the mention
of one means nothing unless there is also included a definition of the
other. The hydraulic gradient of 1:9, as found by the author, would
appear to be a purely fortuitous result, for, with a variation in the size

* Seattle, Wash.



J acobfi.



60 DISCUSSION ON DESIGNING AN EARTH DAM

Mr. of valve opening (if it is less than the capacity of the gravel), and with
variation in the depth of water above the soil surface, the ratio of loss
of head to length of travel may be made to vary indefinitely. The
maximum permissible limit of this ratio is that at which the seepage
loss becomes prohibitive as to quantity, or at which the induced veloci-
ties are such that erosion or piping is threatened as a forerunner of
the ultimate failure of the dam. It may be that the author decided
that these requirements had been met in his hydraulic gradient of
1 : 9, but there is nothing in the text indicating this special considera-
tion, nor do the seepage tests on the model dams seera to warrant such
a decision.

It will be noted from the Second Series of Tests, which relate to
the model from which the final dam was designed, that under a 10-in.
head there was a mean seepage loss of approximately 0.0025 cu. ft.
per sec. per lin. ft. of model dam. Bearing in mind that the model
dam was built on a scale of 1 : 120 as compared with the final dam, and
assuming, for the moment, that the hydraulic gradients and therefore
the seepage velocities would be identical in the two, it follows that the
seepage loss for the actual dam, under a condition of full reservoir,
would be 360 cu. ft. per sec, made up as follows :

400 lin. ft. of dam under 100 ft. head, yielding 120 cu. ft. per sec.

1600 " " " " " 100 to ft. head " 240 •' " " '"

Total rate of seepage flow at full reservoir 360 cu. ft. per sec.

This 360 cu. ft. per sec. would be further increased by reason of the
hydraulic gradient in the actual dam being greater than that in the
model. In comparing ratios of head to length of water travel, it is the
friction head and not the total head that must be considered. In the
model dam the 10 in. total head is greatly depleted by entry and exit
heads, thus leaving a greatly reduced head available for friction losses,
whereas, in the actual dam of 100 ft. head, the same quantitative deple-
tion for entry and exit heads has relatively slight effect on the remain-
ing friction head.

So great a loss as 360 cu. ft. per sec. from any reservoir would, of
course, be prohibitive, even in the absence of any danger of under-
cutting the dam, and it must be assumed that the figures given in the
paper are in error or that the author failed to apply the apparently
clear inference to be drawn from his tests, which latter seems to be
imtenable because the tests appear primarily to have been suggested
by the seepage difficulties encountered in the original dam. It would
be interesting to have now a record of the actual behavior of the dam.
as to seepage losses, if there has been opportunity for test since
construction.



DISCUSSION ON DESIGNING AN EARTH DAM 61

Concerning entry head and exit head there is indication that the Mr.
total head is aflFected by these elements, and it is this fact that renders '^^"■"^*'*-
difficult of interpretation the results obtained from experiments with
small model dams, for in such dams the combined entry and exit heads
may indeed form a relatively large part of the total head, though in
dealing with full-sized dams they would form a relatively small or
negligible part. To determine entry and exit heads from any set of
tests, such as those by the author or by Mr. Colman,* we must first
ascertain the friction head, and this in turn depends on the assumption
as to line and resultant length of water travel along any film or dif-
ferential element of flow. A critical examination of these tests shows
a rate of loss of head due to friction that is not uniform per distance
traveled, as measured between observed tubes. This disparity is due no
doubt to variation in porosity and other mechanical elements of the per-
vious medium, and probably to some extent to observational errors, for
it has been well established that, for an unchanging medium of this char-
acter, the rate of friction head consximption will be uniform. On the
scheme of underground water movement, as hereinafter explained, the
writer finds values of entry and exit heads in the Hays and Colman
tests varying as follows:

In the Hays tests, with a total head of 10 in., entry heads varied
from — 0.90 to -|- 5.98 in., the mean being -|- 5.5 in. ; and
exit heads from — 2.26 to -|- 0.45 in., the mean being — 0.9 in.

In the Colman tests, with an average total head of 54 in., entry
heads varied from — 1.08 to + 21.36 in., the mean being
-|- 7.0 in.; and exit heads from — 7.32 to -|- 13.08 in., the
mean being -\- 3.4 in.

Because velocity is a function of all the elements which characterize
the medium through which the water moves, it was thought that some
relationship might be established between entry head and entry
velocity, and exit head and exit velocity. The graphic method was
used, but there was such lack of consistency in the results deduced
from the tests that no dependable curves could be platted. N'or was
this lack of consistency due to the particular scheme of water move-
ment assumed, for 'other assumptions as to line and length of water
travel showed similar inconsistencies. In the absence of more ex-
tensive data than we now have, we may only say at this time that there
is entry head and exit head of uncertain amounts which must be
reckoned with in experiments on small models, but which are negligibly
small in dealing with full-sized dams.

All the formulas for the flow of water through soils are of the

general form, Q = c - a, in which



* Transactions, Am. Soc. C. E., Vol. LXXX, p. 421.



63 DISCUSSION ox DESIGNING AN EARTH DAM

Mr. Q = discharge ;

'^*''''^^* a = area of cross-section;

— = ratio of loss of head due to friction to length of water

travel ;
and c = a variable coefficient, dependent on water temperature,
porosity, and mechanical analysis of the soil.

Water will follow the line of least resistance, and therefore that
form of section and route of water movement will obtain which, for
a given head, will afford a maximum discharge, or which, for
a given discharge, will consume the minimum head. These conditions,
as will readily be seen from the foregoing equation, in which c for

I
any specific case is a constant, will be satisfied when — is a minimum ;

and, in studying seepage and pressure problems, with respect to
dams on porous foundations, we must first ascertain the form of

I
section and route of water travel which renders — a minimum.

a

It cannot be assumed that the water will be drawn from an in-
definite distance up stream, nor released throughout an indefinite
distance down stream, from the dam, for, if that were the
case, Q would approach its minimum rather than its maximum, and

— would approach its maximum rather than its minimum value.

a

There may be, and doubtless is, some seepage draft from points
far above, and seepage delivery to points far below, the dam, but
the quantities would be negligible in comparison with the main body
of seepage water, which, in the writer's opinion, based on mathematical
analysis, would necessarily be drawn from, and delivered to, points
in the immediate vicinity of the dam. The lengths of river-bed over
which water draft and water release are operative, with respect to
any dam, are important, because these lengths fix the hydraulic
gradients on which pressure and seepage discharge depend.

The actual form of approach and exit channel for seepage, from
free water above to free water below the dam, will depend on the base
width of the dam, the depth of the pervious medium, the position of
the cut-off wall, and the ratio of the depth of the cut-off wall to the
depth of the pervious medium. If the approach is from free water
above the dam to a full vertical section beneath the heel of the dam,

I
the writer finds that, whatever its form, the minimum value of - j

occurs when the distance up stream from the dam equals the depth
of the pervious medium beneath the dam; that, of the various curves



DISCUSSION ON DESIGNING AN EAETH DAM



63



CASE A




Fig. 19



CASE B.




FiQ. 20



CASE C.




Fig. 21




Fig. 22



64 DISCUSSION ON DESIGNING AN EARTH DAM

Mr. which might define the form of approach, the ellipse affords the mini-
Jacobs. 7

mum : and, tinallv, that the circle, as a special form of ellipse,
a

affords the absolute minimum. This, then, means a circle having its

center at the heel of the dam and having a radius equal to the depth

of the pervious medium beneath the dam.

In Figs. 19, 20, and 21 are shown three forms of seepage channel,
any one of which, assuming that the dam itself is impervious, might
be the actual for a particular case, depending on the position of the
cut-off wall and other conditions, as already stated. These will now
be examined: Eef erring to Figs. 19 to 21, consider a section of unit
thickness normal to the paper. Dividing the seepage channel into
longitudinal sections, as obviously demanded by their change in cross-
sectional form, the mean lengths of these sections would be l^, l^, etc.,
as shown, and, for a unit thickness, their mean areas would be
some function of D, the depth of the pervious medium beneath the
dam. The total seepage flow would be the aggregate of the approxi-
mately parallel films of flow, each film having its individual length,
individual discharge, and individual hydraulic gradient. The mean
film would control as to the total seepage loss, and the hydraulic grade
line of the top film would control as to the pressure on the base of
the dam, except that, at and near the lower toe of the dam, in some
cases, the hydraulic grade line of the bottom film might control, as
will be explained later.

Case A. — In this case the seepage flow occupies the full prism sec-
tion between the base of the dam and the base of the pervious stratum.
The mean lengths of its several sections, as is obvious from Fig. 19,
would be:

Zj = /g = 0.7sr)4 Z), ?2 = m D, ?3 = 2, = '-^, 1^ = (w — m) D.

The mean areas of these sections would be:

D-{-a—n)D D

«1 = «2 = «5 = «6 = ^ ; «3 = «4 = -^ 2 = y (^ — ")•

h ^ .

From the original equation for seepage How, (J = c — o, we obtain,
h = - X -; or, for any section, n, h„ = — X -^. The total liead,



therefore, would be, h = h^ -\- h^ + • • • • ^'e

r \ Oj a., "g/ '• ^ 2 — n/

Case B. — In this case (Fig. 20) the seepage flow occupies the wedge-
shaped sections beneath the dam, and, in the space between the base



DISCUSSION ON DESIGNING AN EAETH DAM 65

of the dam and the top of the seepage channel, there would be dead Mr.
water, or approximately dead water. For this case we would have: "^° ^'



?, = /^ = 0.7854 D,l^^ —- V4 m^ + n\ l^ = -~ V4 (w — mf + «^

D

«i = 04 = D, ttj = flg = — (2 — n).

c \ Oj «., a^ /

Q / sj A:rf^ + «2 + V 4 («' — mf + n\



A. 5708 +



2—7?



Cose C. — In this case (Fig. 21) the form of approach and exit is
defined by ellipses, as shown, and, as in the previous case, the space
between the base of the dam and the top of the seepage channel would
be approximately dead water. The distance up stream from which
seepage water would be drawn is determined as follows :

The mean film, l^ = fc, is the quarter circumference of an ellipse

D

the major and minor semi-axes of wliich are a c = -— (2 m + x) and

a f = (1 + «), respectively, and its length, according to the usually
accepted approximate formula, would be

Tt D 1(2 m + xf + (1 + n)-



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