UNITED STATES OEPARTIAENT OF THE I N.TER lOR

t,/ 5 GEOLOGICAL SURVEY . WATER ;~ESOURCES DIVISION

GROUND WATER BRANCH· Washington 25, D.c.

._ ·. ·.

. .,

GROUND-WATER . Tut* --Cd 'l'V · I · HYcRAUL ICS SECTION

· Contrlbutlon .No. Apr II 1952

.··· CYCLIC FLUCTUATIONS OF WATER LEVEL AS A BASIS

FOR DETERMINING AQUIFER TRANSMISSIBILITY

By John G. Ferris

. I\ U G 1 1 19 60

Ll y This contribution was orglnal ly prepared 8S a

paper from the United States for presentation In August &951 of the Brussels Assembly of the lnternation81 Union of _Geo~ desy and Geophysics. Although It will ultimately appear In :1•:.\r ,

· the Assembly proceedings, as published by the International ·Association of Hydrology, the paper Is considered of sufft• clent lnterest to warrant Interim release and distribution by the Geological Survey.

In coostol areas, wells near bodies of tidal water frequently exhibit sinusoidal fluctuations of water level, In response to periodic changes of tidewater stage. Inland, the reg1,1lation of a sur.foce reser­voir often produces correlative changes of ground-water stage In wells adjacent either to the reservoir or to Its attendant streern. As the stage of the surface water rises, the head upon the subaqueous outcrop of the aquifer Increases end thereby either Increases the rate of Inflow to the aquifer or redJces the rete of outflow therefrom. The Increase In recharge or reduction In discharge results In o general recovery of water level in the aquifer. On the subsequent felling stage thls pattern Ia reversed. When the stage of the surface body fluctuates as a simple harmonic motion a train of sinusoidal waves is propagated shoreward through the sub-outcr"op of the aquifer. With Increasing distance from the sub-outcrop, the omplltude of the transmitted wave decreases and the time log of a given maximum or minimum Increases.

( '

j i f : ,) . ~ /

2

If there is no suboutcrop, but the aquifer is confined' by en extensive aquiclude, the rise and fall of the surface-water stage changes the total weight upon the aquifer, Resultant variations In compressive stress ore borne in part by the skeletal aquifer end In part by its con· fined water. The relative compresslbi I !ties of the skeletal mass and the conflned$water determine the ratio of stress assignment and the net response of the piezometric surface to the surface force.

The problem of potential distribution within a semi-infinite solid, with the face at x • o, normal to the infinite dimension and subjected to periodic variations of potential, was long ago analyzed and the solution employed by Angstrom (Carslaw, pp. 41-44) to determine the thermal conductivity ot various sol ids. Similar analyses have been used by other Investigators to determine the condUctivity of the earth, the penetration of diurnal end annual temperature waves, and the flow of heat in the wells of e steam•englno cylinder. The physical nature of these problems is quite analogous to our problem of the aquifer having a sub­outcrop under tidewater or a regulated surface stream. Consequently, those solutions provide a ready pattern for evalueting the hydrologic counter­part.

Assume a homogeneous aquifer of uniform thickness and of great areal extent shoreward • that is, normal to the strike of the sUboutcrop. Assume also that water is released immediately with a decline in pressure and at a rate proportional to that decline. As a further simpl ificatlon, assume thet How is unidimensional ond that the aquifer Is fully penetret• ed by the surface-water body that propagates the cyclic fluctuations. In those situations where the aquifer is not fully penetrated or where It Is under water-table (unconfined) conditions the analysis wl I I still be satisfactory if (1) the observation well is far enough from the subout­crop so that It is unaffected by verticel components of flow and (2) the range in cyclic fluctuation at the observation wei I is only a small frae• tion of the saturated thickness of tho formation. The fund8mental differential equation for the I inear flow of water In an aquifer inter• sected by ~ atreem may be written as follows (Ferris, 1950, P• 286)a

2 ~2 • .§ .22

ax T at

In terminology adapted to this problem

a • net rise or fell of ground-water $tage with refer• ence to the mean stage over an observed period.

x • distance from suboutcrop to observation well. t • time elapsed from convonlent r"eference node within

any cycle. S • cbefficlent of storogo T • coefficl~~t of . tren~misslbll lty

( I )

0

,r-- .

· Let s0 des I gnete the amp I i tude or he If range of Stage f luctu­etlon of the surface body. The problem resolves then to finding the particular solution of equation (I) that will satisfy t . .he following bcundary condition•

of the Zobel, peeh•d

s • s0 sinwt at x • o (2)

The mathematical development leading to the particular solutlcn differontial e~uation is given in consid~rable detail by Ingersoll, and Ingersoll {1948, PP• 46-47) and only the final form is re­herewith, as

(3)

. If we designate the period of the uniform tide or stage by t 0 In accord with Jacob (1950, p. 365) then w may be expressed in radians per time unit as 2n/t0 end equation (3) becomes

(4)

Equation (4) defines a wove motion whose amplitude rapidly de--xJnS/t0T

creases with distance x as given by the factor s0 e • When the aquifer response is due to loading rather than head change at the sub­outcrop the amplitude hctor should be reduced (Jacob, 1950, p. 356) by the rat lo [a./( a. + e~ )] , where a. Is the vert lea I compress i b iII ty of the skeletal aquifer, p is the compressibi I ity of tlie water, and e is the porosity of the sand. Whtle values of p, the compressibil lty of water con be obtained from published tables of physical data, little Infor­mation Is available to estimate ~. the vertical compressibility of the equifor and this factor may vary considerably. The princ 1pa ! guide to the magnitude of a. would be worrelative data from similar aquifers wh~e pumping test results have e~tablished its value.

From equation (4) the range of ground-water fluctuation at an observation well at a distance x from the suboutcrop is given by

{5)

·~

4

It Is of Interest to note fran the form of equatlon . (5) that .the slower the ftuctuetlon of the surface tide- that Is, the greater the volue of t

0, the greeter Is the ronge of stoga within the oquUer •

. Let t 1 denote the log In time of occurrence of o Given maximum or minimum ground-water stoge following the occurrence of o similar sur­face stoge. Then, from lngersol I, Zobel, and Ingersoll (1948, p. 48), the expression for t1 Is os followsa

(6)

The apparent velocity of tronsmlsslon of the wove through the aquifer Is ·

V • _x -j¥JnT op t l t 0 S (7)

Equation (7) indicates the opparent velocity of a given mexl~ mum or minimum and does not pertain either to the rate of pressure transmission (Noskat, 1939, P• 669) or tho apparent reate of pressure transmission (Jocob, 1940, P• 585) within the oqulfer.

The wove length Is obtained &S

(8)

Physlcoily there v.ould be little opportunity In ground-water hydrology to obtain a snapshot view of the sinusoidal wove troln, as would ~e required to employ eqoatlon (8).

During half the cycle water flows through the ~boutcrop Into the aquifer; In the other half It flows out again. The qucntlty o~ flow .. per half cycle Is llJetermlned with the aid of Dorey's low.

Q • TtL

where L is the length of suboutcrop ocro5S ~ich the flow occurs, and where I, as given by Ingersoll, Zobel, and lngersol I (1948, P• 49), lsa

•x .JwS/2T( )[ ( ~ ( )] &0 e . • ~ aln lt.).t•x'V~)+ cos wt·x.Jifr

.~ ~·

. .'·' ' •

I .

It Is convenient to set up the lntegr~l for the quentlty of flow per unit length of suboutcrop. Bec~use tho gradient as/ax Ia not In phese with s, the limits of lntegretlon ~redetermined by noting from Inspection of equation {9) thd at x • 0 the gr~.dient is zero at t • -n/4w • ·t0 /8, reach~s a minimum at t • n/4ul • to/6, and returns to 0 at t • 3n/4w • lt0 /8.

Q • -T L ( ~=)dt

x•o

-t /8 0

s.. L

~ - s {2t;;ST. L o1j--n

( 10)

(II)

(12)

To II lustrate the appl icabi I ity of these methods to field prob• lems, dat8 are presented for three riverside observation wei Is at the Ashland station of the municipally owned water supply of the City of Lincoln, Nebraska. A map of the wet I station is shown as figure I. Au tomet l c water-stage recorders are operated on observation .we II s I to 3, Inclusive, and at the river gage on the Platte River at the crossing of u. s. Highway 6. A geologic section from west to east through supply wet I 2 is reproduced as figure 2. Typical records from the autographic charts for the river-stage recorder and observation well L are reproduced as figure .3.

Qbservation wells I, 2, and 3 are respectively 42, 106, and 252 feet f'rom the edge of the river at a normal stage. Each welt · is screened end developed in the upper part of' the aquifer. From records lor the river-stage recorder and the three observation wells for the porlod September 23 to 29, 1950, the ratio of ground-water fluctuation to river change was computed for tho ris :ng and fat I ing limb of each cycle. These stage ratios ore summarized in table 1. The period of tho river fluctuation, computed f~r ~~ch I i~b of each cycle, ranged from

~----------------------------------------------------------------------------------~--~m

J-----+-1

LEGEND ~ SUPPLY WELL 8 OBSERVATION WELL

Q TEST WELL

R. 9 E.

I

+ I

24

14 g

25

L 0• STRE:::·:~NO STATIT, L

2poo· 4,00eEET ______ .f---- __ _

R. 10 E.

·~+--N

20

108

0

FIGURE I.-MAP SHOWING A PORTION OF THE CITY OF LINCOLN WELL STATION AT ASHLANDJ NEBR.

G

X:

1,080

..J 1,0()0

w > w ..J

< w II)

w 1,040

> 0 ::0 <

a:: a: '"' w w ca: > u a: ca: < • w w .., _. Cll .... 0

., ,... u .... 0 0 .., • 0 <

_J x :t:

0..

EXPLANATION

CLAY

SAND

GRAIIE L

HORIZONTAL SCALE

0 2,ooo 4,ooortET .

f"IGURE 2 .- GENERALIZED GEOLOGIC SECTION FROM WEST TO EAST THROUGH SUPPLY WELL 2 OF THE CITY OF LINCOLN AT THE ASHLAND, NEBR. WELL STATION.

., 0

:t:

L---------------------------------------------------------------------------------------------------~~

8

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11\ !\ I 1\ : \ I

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SEPTEMBER 19~0 n

I I I I I

PLATTE! RIVER

1\ I :\

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05 I @ I f7J I \ j I \ I ~ I I I I I

FIGURE 3.-GRAPH SHOWING THE STAGE OF THE PLATTE RIVER AND THE WATER LEVEL IN OBSERVATION WELL I OF THE CITY OF LINCOLN AT THE ASHLAND, NEBR. WELL STATION.

ll

:n

I I

I I I

: @I

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zl 81 Zl

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Table~ Ratio of fhe reco~ded ~ange of g~ound-water stage in obse~­vation wells 1. 2, and 3 as compa~ed to the stage ~ange of the Platte River at Ashland, Nebr.

Reference . We II 1: Well 2 Well 3 numbers

of Rising Fa II i ng Rising Falling Rising Fa IIi ng fluctuation N stage stage stage stage stage stage

l-2 0.73 0.52 0.35

2-3 0.71 0.56 0.4<5

3-4 • 77 .54 .31

4-5 • 76 .56 .29 ·--

. 5-6 .74 .59 .26

6-7 .73 .56 .29

7-8 •69 .47 .. 28

a-g .71 .5<5 .20

g-1.0 ;.72 .52 .33

lO•ll .68 .51 .37

ll-12 .71 .53 .1.4

.73 • 72 .53 .55 .28 .32 Averages

0.72 0.54 0.30

£1 See flgu~• 3.

lO

20.5 hours to 31.0 hours and averaged 24 hoUrs or I day. To apply the stage-ratio data, equation (5) is adapted for use with gallon-per-day­per-foot units, as fol lowsa

( 13)

s,. • e

( 14)

( 15)

The ,logarithmic quantity (sr/2s0 ) is In effect the ratio of the range of ground-water stage to the rang' o{ river or tide stage. The form of equation (I~) suggests the use of a semilogarithmic plot of the logarithm of the range ratio versus the distance x for each observation well. Thus the right-hand member of equation (15) represents the slope of this plot, and if the change in logarithm of the range retto is selected over one log cycle then the numerator of this slope expression reduces to unity. Thus equ&.tton (15) b.ecomes

(tC5)

A more convenl'ent form is gainE-d by removing the radicet

( 17)

As indicated by ~quation (17) it Is necessary that S, th~ coefficient of storage, be known In order to evaluate T, the coefficient of transmlssi,bi I ity. Howev&r, recsonable estimates of S con be mode if It Is known whether the aquifer is locally artesian or nonartesien which generc!llly can be determined from studies of w~ll logs end water-level records.

From the stage-ratio data of table I, a semi logarithmi~ plot, figure 4, was constructed. Using the Ax value indicated for one log cycle, t • I day, end assuming various S values for water-table con­ditions,0as suggested by the section of figure 2, equation (17) yields the followingt

T • 4 • 4- ! 540 )2 s • I ,)00,000 S I

T • 130,000 if s -0.10

T - 190,000 s • 0.15

T • 260,000 s • 0.20

T • 320,000 s • 0.25

At the suboutcrop where x • o, the range of the aquifer respons~, Sr• Js equal to the river or tidal range 2s0, or s,./2s0 approaches unit~ as x approaches zero. Thus, the negative x value noted on figur~ 4 ~t the range ratio of unity represents the ~ffective distance off shore to tfie suboutcrop.

The withdrawal of ground water from the numerous municipal wei Is nearby is relatively steady, except for minor changes in rate and distribution of pumping. These changes are more ept to disturb the time of maxima or minima observed than the ratio of wei I-to-river change. Further, the compressed time scale of the woter-l~vel recorders limits interpolation for this purpose to a greater degree than does th~

gage•height scale. Any variations in the effective screen resist~nce of each observation wei I would also tend to distort observations of maxima or minima timing. The wide rTange In the observed lag of moxlma and mInima shown by tab I e 2 may resu It from any one or a combination of several of the aforementioned causes. To apply the time-leg dcta, equation (6) is modified as followsa

t I .. x§ '!-. n.T

2 x 2's t t . • 0

I' 4nT

01 600

500 ~ ...

""' ""' .... ~ .. ""' c.) 0 w 400

~~~~ a::

""' > a: 0

300 .... ..J ..J w 3:: z 2 1-

200 ~ rr:

""' Ill £0 0

:1 0

100 ~ a.J \) z ~ on 0

0

02 .

~ ~ ~

~ ~

--

. 06 . 0.7 0.8 0.9 1.0

~'\WE:LL 3

~ '-....

~ ~ ~L2

- ~ ~ WELL I

~~ ,.., OffSHORE D!STANC~-1"- """'

"' TO SUB-OUTCROP ~

- I ___ ___! ____ -----1-

-100

fiGURE 4.-SEMILOGARITHMIC PLOT OF THE RATIO OF GROUND-WATER STAGE TO STREAM STAG~ VS. THE ~I STANCE oF JHE ossE~VATION

1wELL FROM jfHE cvcL

11c STRE

1AM. 1

Tabl~ 2. Time lag, in hours, of mtntmum or maximum ground-water stage relative to Platte River stage, as recorded by observation wei Is 1. 2, and 3 at the Ashland wei I station of the City of Lincoln, Nebr.

Reference Well l Well 2 Wei I 3 number of

fluctuation s1 · Minimum Maximum Minimum N1ax I mum N1 in imum lt'.ax imum -

l 1.25 3.75 --2 2.50 3.50 6.00 .

3 2.00 4.00 7.50

4 2.25 2.75 6.75

5 L.15 3.75 6.75

0: 2.25 3.25 5,.75 -

7 L.50 4.00 6.50

6 2.00 2.50 5.50

9 2.50 4.00 7.00

10 2.75 2.25 6.75

II 2.25 3.75 6.75

12 2.50 2.50 5.SO

1.90 2.40 3.90 2.80 6.70 6.00 Averages

2.1 3.3 5.3

AI See figure 3•

,

-

14'

ForT In g~l Ions per doy per foot end t0

end t 1 in doys, eq.J~tion (18) becomes

( 18)

(19)

The overoge values of t 1• the time log, ore plotted for . eoch well 1n figure 5. The slope of this groph is xftl which oppears In equation (19) ot the sq.Jore exponent. Substituting In eq.Jotlon (19) the slope coordlnotes noted on figure 5 there results

2 T - 0160 X 250 X

(5/24)2 l s

T - eco.ooo s

T • 86.000 if s • o.to

T • 130,000 s a 0.15

T • 170.000 s • 0.20

T • 220,000 s • 0.25

At the suboutcrop. where x • o. the time log, tl • o. The neg• otlve volue of x indicoted by figure 5 ot the t 1 • 0 oxls Is the effec­tive dlstonce off shore to the suboutcrop.

The Iorge difference between the coefficients of tronsmlssi­bil ity indicated by the stege-rotlo method os compared to the time-log method moy reflect the influence of the nearby pumping in distorting the goge height ond time of eoch maximum or minimum. The T estimotes obtained by both methods ond resultant everoges are summarized In toble 3.

Consideroble refinement of the T estimates would be possible by expending the time scale on the woter-stoge recorders through the use of daily time gears In lieu of the normal weekly gears. It would also be desirable to make the studies during periods when the rote of with­drawal by the nearby supply wells Is constant. A more adequate test of these methods might be possible withe profile I ine of observetion wei Is

15

t, TIME LAC IN HOURS

0 2 3 4 5 6

30~~---T----r---,---~----.----.--~----~---T----r---,---~--~

I ~ i w--~---+----~--~--~----+---~---;----~---+----~--4---~

~ i '

250 i w~v1 ----!----+---~---+----~~

~ /v 200~~--4----+----~--~---4----+,----~--~---4----+~~~~--~--~

~ /1

8 v ~I !

IOO~S·--+---+---~--~---+-W_E_L4-L2~p/~~--+---4---~~ll~--+---~~---l / ~~ I

~ / <li

~ / I ~Q~Q--~--~--+---~~/~--~--~---~--+---+-1--~--~~

WEL/0 i /v l I

0~-+--~v~+-~--~--~--~~--~--+~--~--+,~,

/ 1:\ 1 I I · / ~ [>orrsHOAE: DISTANCE I

/ II' TO SUB·OUTC ROP I

-50 I I I L ____ ---------.--J

At1 : 5 HRS.

FIGURE 5.- PLOT OF TIME LAG VS. DISTANCE OF OBSERVATION WELL FROM SUB-OUTCROP.

16

Tflb 1 S'~ Summery of de term l Mt i ens of the coefficient of transmissibi I ity by stage-ratio and time-log m~thods.

Method T, coefficient of trensmlsslbi I lty, in gal lor.s per day per foot

- -Coefficient of storage S•O. 10 S•O .15 5•0.20 5•0.25

Stage-rot io method 1.30,000 190,000 260,000 320,000

' Time-log method 86,000 1.30,000 170,000 220,000

-Average 110,000 160,000 220,000 270,000

at right ongles to the edge of the fluctuating surface-water body and in on area remote from heavy pumping.

It Is reported that the SDturated thickness of al luviel de­p0slts in the well field area overages about 70 feet. Using this thickness and the averege T values of toble 3, the average permeability in gallons per day per square foot is 1,600 for S•O.IO; 2,300 for 5•0.15; 3,100 for 5•0.20; and 3,900 for 5•0.25. In comparison, gradient · studies by tho city, bosod on water-table contour maps, indicate on average permeabi I lty of 2,200 gal Ions per day per square foot.

The obove-lndlcoted values of the coefficients of transmissi­bility and storage should be considered as tentative, pending the com­pl~tlon of other hydrologic studi~u ln the Ashland area. However, these data serve to demonotrate the oppl icebl I ity and usefulness of the methods d~scrlbed for analyzing cyclic fluctuations of ground-water level. Although their greatest us~ wi I I be in ar~~s of tldol streems and seas or near regulated streams and l~kes, it was shown by Rambaut (H~OI, p. 2.35) that these methods con also be q:>plied with fair results to vDriotlons that resonble periodic motion but oro I imlted ln duration to a single mDximum or minimum. Thus, the response of on aquifer to the passage of o flood crest In a hydr~ul icol ly connected stream moy lend itself to this analysis. A logical extension from this generalized problem of a simple sinusoidal motion v.ould be to study the oppl icabll ity of the unit functions or delta functions of ~loctrical-network analysis to the response of aquifers to comp I e)( pot terns of recharge from precipitation or to the response of a stream to vorious relnfal 1-runoff patterns.

AcknowiE:>dgmcnts

The wthor is greatly lndobtcd to H. A. Welte, district · geologist of the u. 5. Geologic~! 5urvC!y ot Lincoln, Nabr., for his

ossi&tonce in assembling the basic deto for this report and to D. L. Erickson, Director of Parks, Public Property and lmprovaments and City Enginc!er r•f the City of Lincoln, Nebr._ end members of his steff for

· ~ ... ---·

17

· tnelr hearty cooperation end considerable effort In making these data . evallable.

Authors nots;

Before preparation of this report w~s completed, an excel lent report on cyclic fluctuations of ground-water level was published by Werner and Noren (1951, PP• 238-244). The work of the author, though of subsequent date, represents an independent analysis made without prior knowledge. oi- advance notice of their coincidental research.

References.

CAASLAW, H. ~ s., 1945, Introduction to the mathematical theory of the conduction of heat in solids, pp. 41-44, New York, Dover Publications, Inc.

FERRIS, J. G., 1950, A quantitative method for determining ground-water characteristics for drainage designa Agr. Eng., vot. 3t, no. 6, PP• 285-269, 29r, June.

INGERSOLL, L. R., ZOBEL, O. J;., AND INGERSOLL, A. C., 1948, Heat conduc­tion, with engineering and geological appl !cations, PP• 45-57, New Y~rk, McGraw - HI I I Co., Inc.

JACOB, C. E., 1950, Flow of ground water, Chapter 5 in Engineering Hydraul lcs, p. 365, New York, John Wiley and Sons, Inc.

--------~-------• 1940, On the flow of water in an elastic artesian aqulfera Am. Geophys. Union Trans., vel. 21, pp. 585-566.

' NIJSKAT, NORRIS, l937, The flow cf homogeneous· fluids through porous media, p. 669, New York, McGraw- Hi I I Co., Inc.

WERNER, P. W., and NOREN, D., 1951, Progressive waves In non-artesian · aquifers a Am • Geophys. Union Trans., . vo I. 32, pp. 230•244.

t267

t : < ; . '