River Depletion Due to Pumping of a Well Near a River Glover

River Depletion due to Pumping of a Well Near a River

Glover OFFICERS OF THE UNION, 1953-58 SECTIONS OF THE UNION JOHN A. FLEMING, Honorary President Section of Geodesy JAMES B. MACELWANE, President ALBERT J HosKiNsoN, President MAURICE EWING, Vice President AMERICAN GEOPHYSICAL UNION Section of Seismology Ross JOHN PUTNAM MARBLE, General Secretary R. HEINRICH, President Section of Meteorology WALDO E. SMITH. Executive Secretary EXECUTIVE COMMITTEE PHIL E. CHURCH, President 1530 P Street, N. W. Section of Terrestrial Magnetism Washington 5, D. C. and Electricity VICTOR EX-OFFICIO MEMBERS American National Committee VACQUIER, President of the Section of Oceanography CHAIRMAN, NATIONAL RESEARCH COUNCIL International Union of Geodesy and Geophysics RIciimu) H. FLEMING, President CHAIRMEN, DIVISIONS OF THE and Committee on Geophysics of the Section of Volcanology NATIONAL RESEARCH COUNCIL Wii.i.awm F. Fostiwo, President International Relations NATIONAL RESEARCH COUNCIL Section of Hydrology Physical Sciences HAROLD G. Wrud, President Chemistry and Chemical Technology WASHINGTON, D. C. Geology Section of Tectonophysics and Geography FRANCIS BIRCH, President Biology and Agriculture AMERICAN OFFICERS International Union of Geodesy and 1530 P Street, N.W. Geophysics Washington 5, D. C. October 22, 1954

Messrs. liobert E. Glover & Glenn G. Balmer, Bureau of Reclamation Bldg. 53, Denver Federal Center Denver 2, Colorado

Gentlemen:

Thank you for your letter of October 13 with which you forwarded the closing discussion on your paper relating to river depletion. We are for- warding this, together with Hantush's discussion of your paper, to the Editor and in due course will write to you further. We are calling the Editor's attention to the closing paragraptf your letter.

Sincerely yours,

Waldo E. Smith Executive Secretary

WES/jjc Form OH:- 11 (7 - 52 INFORMATIONAL ROUTING

Bureut: of Recl omnt i on

— — — — _

— —

I. 111114o E Smith asicative Secretary American Geophysical Union AUTHOR 1530 P" art, RN. iftehingto:11 D. C

Deer Igr. Stati,h:

Three copies of our final discussion on our paper relati44 to River Depletion Resulting from Pumpingc Well Naar a River are forwarm"ed herewith.

In the oopy of Dr. Bantushis diseusaioa, forwards& 'with your letter of August 2, 1954, a roman. iu O.ven which he States *bona reduce to our Marsala (12/ if (l/B) radiuses to sem. 144 cannot confirm this ant suspect that the gentiti S has Luadvertently coma omitted from the first radical appearing in this expression.

Sincerely yours gaga'

Lozmrt

Gle= G Balaor

Saolosure (17C:N:))

HUD& to: E. E Glover G G.Balmer REGlover:cah

Interior - Reclamation - Denver, Col, Closing Discussion Relating to the Coolants by Dr. Wadi S. Haritush on the paper on River Depletion ResultIngireek Pimping a Well seer a Elver by Robert n. G1aftrimid.00AmIn0. Balmer vitich appeared in Transactions of the American Geophysical tinien Vol. 35, p. 468, 1954

It is gratifying that Dr. Bantush bee been able to develop a more general solution than that obtained by the Authors, and to

confirm the special case treated by them. In regard to his question

6L.out the position of the image well, it may be stated that thio

wQ11 nay be placed anywhere along the line he describes, and it will

ultimately cause the total yell flow to cross the line representi

the position of the stream. However, the only line alonL which the

drawdown will remain at zero is a straight line drawn zidway between

the pump well and the recharge image well and at right angles to

the line joining them. Since the stream uaintains the condition

of zero dremdown alout, the Dear banh the line of .zero dramdown•

described above, nust coincide with it, and this axes the TvsiticAn

of the iwace well. BUILD YO BUILWJUR FUTURE WISEL WISELY, SAFELY mrc nrwric U.S. SAV TIONT

..iessrs. Robert E. Glover and Glenn G. Balmer Bureau of Reclamation Commissioner's Staff Offices Bldg. 53, Federal Center Denver, Colo. 1530 P Street, N. W. METER01.06Y • Vftshington 5, D. C. 1001.091 Ciust 27, 1954 COMM FPS. Wo.• SEISMOM \IMAM Mina. a MAGNH1Sm Dear Messrs. Glover and Balmer: GEODE5Y Under separate cover, by parcel post, we are mailing dIECTOO you the reprints of"Ri ver Depletion. .River" 7IV5E5 paper in accordance with your order D-210—B Payment for these has been received.

WISE 111114 AMERICAN COOKRATION Very truly yours, GEOPKYSICAl, PESTAItC11 I UNION AMERICAN GEOPHYSICAL UNION

Transactions, American Geophysical Union Volume 35, Number 3 June 1954 41111

RIVER DEPLETION RESULTING FROM PUMPING A WELL NEAR A RIVER

Robert E. Glover and Glenn G. Balmer

Abstract--A well adjacent to a river will take a portion of its supply from the river. A theoretical formula is developed which permits the draft on the river to be computed in terms of the distance of the well from the river, the properties of the aquifer, and time. The formula applies where the river can be considered to flow in a straight course which extends for a considerable distance both upstream and downstream from the well location.

When pumping of a well near a river begins, water is drawn, at first, from the water table in the immediate neighborhood of the well. As the zone of influence widens, however, it begins to draw a part of its flow from the river and, ultimately, the river supplies the entire flow. It is the purpose of this analysis to develop a formula for estimating the amount of flow drawn from the river at any time after pumping begins.

Notation:

D saturated thiclmess of the water-bearing strahim or aquifer, feet •K permeability, ft/sec Q flow of the well ft3/sec xi feet from the well if no river is present, ql flow crossing a straight line at a distance ft3/sec q the flow taken from a river at a distance xi feet from the well, ft3/sec r a radius measured from the center of the well, feet s draw-down at the radius r at the time t, feet t time,from beginning of pumping, seconds ✓ volume of water yielded by a horizontal square foot of the aquifer if the pressure is dropped one foot, dimensionless from the center of the well, feet x and y rectangular coordinates measured • x tistancehe d of a well from a river, measured along a normal to the direction of flow. The 1 river is assumed to extend indefinitely upstream and downstream from the well, feet ot = ICD/V X a time variable running between zero and t, seconds

The differential draw-down ds at the time t due to removal of the quantity of water QcL‘ at the time X is [CARSLAW, 1921]

This expression satisfies the continuity condition

and the condition that the draw-down is zero everywhere when (t - X) = O. Let

Then, by substitution and integration

co 2 s = (Q/2 ir ICD)4/ lircir (Cu /u) du (4)

The relation between the formula for well draw-down presented here with the one used by THEIS [1941] can be established by a simple change of variable. If, in the expression

468 • [Hydrology] RIVER DEPLETION CAUSED BY A WELL 469

co • s = (Q/4 ir ICD) (c "/v)dv r`/4 at

We make the substitution of variable

v = u2

We obtain at once

2 s = (Q/2 ir ICD)/ /_ (c /u)du r/ v4 oct

Eq. (4) is a form of the exponential integral [INGERSOLL and Others, 1948] which can be evaluated from tables [FEDERAL WORKS AGENCY, 1940; JAHNKE and EMDE, 1945] by use of the relation

(e-112/u)du = -0.5 Ei (- r2/4 at) (5) lr/Vc°4-—Frt

The draw-down produced in an aquifer of infinite extent due to a well pumped at the rate Q is given by (4). This expression is valid if s/D is small compared to unity.

We return now to (1) and set r2 = x2 + y2. Then the flow of water across the line x = x1 due to the withdrawal Qc1A. is obtained from the relation

+ CO aqi/ax = - Imsf (a2s/axax) ay -co

D -X) ir e-y2/4oc(t-A) dy =[2Qxe -x2/4 a (t /16 a (t - X)2]f (6) - co • The integral in this last expression is a form of the probability integral [FEDERAL WORKS AGENCY, 1941; IAHNKE and EMDE, 1945; PEIRCE, 1929]. This permits an evaluation of this expression in the form

aq /ax = 2Qaxe-x2/4a(t-X)/ ,-/— vir [4cx(t - X)]3/2 (7) 1 To integrate this expression with respect to A from A=0 to A=t set

v = x/Y4 a(t - A) (8)

Then, by substitution

-v2 q = (Q/ e dv (9) 1 x/ ViTa T

This can be expressed in terms of the probability integral

Z 2 P(Z) = (2/ Irr) e v dv (10) 0

which has been extensively tabulated in terms of the upper limit Z. Then (9) can be put in the form

qi/Q = 0.5 [1 - P(xi/liro—ct)] (11)

This formula implies that there will be a drop in the ground water level along the line x = xl If a river exists at this distance from the well there will be no drop in level but, instead, the river

• 470 GLOVER and BALMER [Trans. AGU, V. 35 - 3]

will supply an additional flow to the well. These factors can be accounted for if a recharge well is placed at the point where the pumped well is imaged by the riverbank. A similar form to (11) would apply to the recharge well also, with the result that the flow across the line x = x1 is doubled. Then the total flow across the line if the river maintains the level is

q/Q = 1 - P(x1/1/4-6tt)— (12)

These results may be summarized as follows: The draw-down in an aquifer of infinite extent due to a well discharging at the rate Q is given by (4). This formula is valid, as a first approxi- mation, if the draw-down s is small compared to the depth D. A similar well located at the dis- river. The part of the total tance x1 from a river will draw the flow q from the flow of the well which comes from the river is q/Q. The value of this ratio is given by (12).

Suppose we have three wells at distances of 1000, 5000, and 10,000 ft from a river, respectively, which have been pumped for a period of five years. It is desired to estimate the part of their flow which comes from the river. If the aquifer data are: D = 100 ft, K = 0.001 ft/sec, V = 0.2, a = ICD/V = 0.5, t = 157,770,000 sec, and -V4--67a.= 17, 763; then the values for the three wells are as shown in Table 1.

Table 1--Values for the three wells

Well x1/ VTIFet— P(xi/V7-1-0—ct)a q/Q b

1 1,000 0.0563 0.0635 0.9365 2 5,000 0.2815 0.3094 0.6906 3 10,000 0.5630 0.5741 0.4259 - aFrom tables of the probability integral. bFrom Eq.(12) or from Figure 1 directly.

+0. Then at the end of five years the three wells are drawing, respectively, 93, 69, and 42 pct of OA their flow from the river.

0.3 While English units have been used here, the formulas given are valid when used in any con- 0.2 sistent unit system. Such a system permits the• use of only one unit of a kind. Since, in our case, 0.1 only units of length and time are involved, change to another system of units can be accomplished 00 0.1 02 03 0• 0 S 1.0 1 .11 by replacement of the foot and second units, used

(-At) herein, with the new units. Fig. 1--The part of the well flow taken from the References river as function of the parameter x1/1/4 at CARSLAW,H. S., The conduction of heat, MacMillan and Co., p. 152, 1921. FEDERAL WORKS AGENCY, Work Projects Administration for the City of New York, Tables of sine, cosine, and exponential integrals, v. 1 and 2, Tables MT5 and MT6, Superintendent of Documents, Washington, D. C., 1940. FEDERAL WORKS AGENCY, Work Projects Administration for the City of New York, Tables of probability functions, v. 1, Table MT8, Superintendent of Documents, Washington, D. C. 1941. INGERSOLL, L. R., 0. J. ZOBEL, and A. C. INGERSOLL, Heat conduction, McGraw-Hill, App. F, p. 253, 1948. JAHNKE, EUGENE, and FRITZ EMDE, Tables of functions with formulas and curves, Dover Pub., Table 1, pp. 6-8, 1945. PEIRCE, B. 0., A short table of integrals, Ginn and Co., pp. 116-120, 1929. THEIS, C. V., The effect of a well on the flow of a nearby stream Trans. Amer. Geophys. Union, pt. 3, pp. 734-738, 1941.

U. S. Bureau of Reclamation, Denver Federal Center, Denver 2, Colorado

(Communicated manuscript received May 15, 1953; open for formal discussion until November 1, 1954.) • OFFICERS OF THE UNION, 1950-58 SECTIONS Of THE 4UNION Jolust A. FLEMING, Honorary President Section of Geodesy HARRY H. HESS, President WALTER H. BUCHER, President Section of Seismology F. W. REICHELDERFER, Vice President AMERICAN GEOPHYSICAL UNION PERRY BYERLY, President BEIJ, General Secretary Section of Meteorology EXECUTIVE COMMITTEE BENJAMIN HOLZMAN, President 1111WALD() E. SMITH, Executive Secretary 1630 P Street, N. W. Section of Terrestrial Magnetism Washington 5, D. C. and Electricity American National Committee of the J. WALLACE JOYCE, PreSident EX-OFFICIO MEMBERS Union of Geodesy and Geophysics Section of Oceanography CHAIRMAN, NATIONAL RESEARCH COUNCIL International ALFRED C. REDFIELD, PreSident and Committee on Geophysics of the Section of Volcanology CHAIRMEN, DIVISIONS OF THII ESPER S. LARSEN, JR., President NATIONAL RESEARCII COUNCIL NATIONAL RESEARCH COUNCIL International Relations Section of Hydrology FRANK J. VEIHMEYER, President Physical Sciences WASHINGTON, D. C. Chemistry and Chemical Technology Section of Tectonophysics Geology and Geography 1530 P Street, N. W. ROBERT BALK, President Biology and Agriculture Washington 5, D. C. AMERICAN OFFICERS International Union of Geodesy and July 14, 1954 Geophysics

Dear Sir:

In compliance with your request we are returning the illustrative

material that was used in connection with the publication of your paper

that appeared recently in the Transactions.

We appreciate very much your aid in this regard. • Sincerely yours,

Waldo E. Smith Executive Secretary

WES/sp

• • D-210-B

APR 27 1914

Mr. Waldo E. Smith Executive Secretary American GeJphysical Union 1530 P Street, N. W. Washington, D. C.

Dear Mr. Smith:

The carbon copy for our paper on "River Depletion Re- sulting from Pumping a Well Near a River" is returned herewith as requested in your letter of April 23, 1954. Correction of four minor errors are indicated in green on the carbon copy.

A check for $8.20 is enclosed to cover the cost of 200 reprints. • Sincerely yours Maim Robert E. Glover ie727 Glenn G. Balmer

Enclosures (2968075)Ni

Blind to: 209-1 209-2

RbGlover:h1m-s

• D-210-11

Mr. Waldo 2. Smith Lxecutive Secretary American Geophysical Union 1530 P Street, N. W. Washington, D. C.

Dear Mr. Smith:

The carbon copy for our paper on "River Depletion Re- --. ' sulting from Pumping a .4ell Near a River" is returned herewith as requested in your letter of April 23, 1954. Correction of four minor errors arc indicated in green on the carbon copy.

A check for $8.20 is enclosed to cover the cost of 2C0 reprints.

Sincerely yours

(SiGNEM Robert E. Glover (SIGNEN Glenn G. Balmer

Enclosures (2968075) III, 000010, 114 iplied to: 209-1 209-2 00100010001161.,...b1.-. OFFICERS OF THE UNION, 1953-56 SECTIONS OF THE UNION JOHN A. FLEMING, Honorary President Section of Geodesy JAMES 13. 11,1AcELwANE, President ALBERT J HOSKINSON, President MAURICE EWING, Vice President AMERICAN GEOPHYSICAL UNION Section of Seismology ROSS R. HEINRICH, President N PUTNAM MARBLE, General Secretary Section of Meteorology ALDO E. SMITH. Executive Secretary EXECUTIVE COMMITTEE PHIL E. CHURCH, President 1530 P Street, N. W. Section of Terrestrial Magnetism Washington 5, D. C. and Electricity VICTOR VACQUIER, President EX-OFFICIO MEMBERS National Committee American of the Section of Oceanography CHAIRMAN, NATIONAL RESEARCH COUNCIL International Union of Geodesy and Geophysics RICHARD H. FLEMING, President CHAIRMEN, DIVISIONS OF THE and Committee on Geophysics of the Section of Volcanology NATIONAL RESEARCH COUNCIL WILLIAM F. FOSHAG, President International Relations NATIONAL RESEARCH COUNCIL Section of Hydrology Physical Sciences HAROLD G. WILM, President Chemistry and Chemical Technology WASHINGTON, D. C. Section of Tectonophysics Geology and Geography FRANCIS BIRCH, President Biology and Agriculture AMERICAN OFFICERS International Union of Geodesy and Geophysics 13C P tr7et, Lashingto 5, Ti. C. April 23, 1954 Dear Messrs. Glover and Balmer: Master Copy for_Tralsactions--Attached please find caron copy of your paper as prepared for Transactions of the American Geophysical Union. The original of the carbon copy enclosed will be reduced one-third in the published volume (size 5 3/5 by 8 3/8 inches). Symbols or Jreek characters which are only roughly made on the carbon will be properly entered on the original. Webster's Dictionary, the Chicago Manual Style, of and the Suggestions to Authors of the American Geophysical Union (Transactions, February, 1949, PP. 159-160), have been used as the basis for editing your paper.

Correction and Return of Carbon Copy--Kindly read carefully and indicate in or-(na-i=ed) any-typographical errors including a marginal mark, and return in lOsed iii envelope immediately. Please note it will be possible only to make correction for typographical errors. Changes that are absolutely necessary and in- volve the same length of line or paragraph can be made, but typist's time required will have to be charged at cost (average cost for full page is 43.00). The cor- rections indicated on the enclosed carbon copy have been made on the oviginal master copy.

Reprints--Eeprints can be supplied, on the basis of two pages, four pages, eight pages, twelve pages, etc., provided your request is received before ay l4 at the following rates:

-- Number of reprints 2 pp. 4 pp. 8 pp. 12 pp. 16 pp. 20 pp. 24 pp. First 100 43.50 47.00 10.4.00 421.00 428.00 05.00 442.00 Additional 100's 0.60 1.20 2.4o 3.60 4.80 6.00 7.20

In addition there will be cost of postage or of expressage (the "latter charge collect). If puenis maa_with order, reprints will be sent postage or express - .paid. If postage can not be charged directly as in the case of some government orders, the rates above must be increased slightly tb cover this cost.

Your article will make a j,..-page reprint. How many do you desire? Covers may be supplied at additional cost of *9.50 ler the first 100; 403.50.X.or each added 100.

Thanking you in anticipation of your preolpt reply, • Sincerely yours

Waldo E. Smith Executive Secretary April 23, /954

Mr. Waldo E. Smith Executive Secretary American Geophysical Union 1530 P Street N. W. Washington, D. C.

Dear Mr. Smith:

The tracing for the figure to accompany the paper on "River Depletion Resulting from Pumping and Well Hear a River," by Robert E. Glover and Glenn G. Balmer, is forwarded herewith as requested in your letter of April 21, 1954.

Sincerely yours

Robert E. Glover inclosure (15/4-trey OFFICERS OF THE UNION, 1953-56 SECTIONS OF THE UNION JOHN A. FLEmnia, Honorary President Section of Geodesy JAMES B. MACELWANE, President ALBERT J HOSKINSON, President MAURICE EWING, Vice President AMERICAN GEOPHYSICAL UNION Section of Seismology Ross R. HEINRICH, President IN FUTNAbI MARBLE, General Secretary Section of Meteorology ALDO E. SMITH. Executive Secretary EXECUTIVE COMMITTEE PHIL E. Cuuacii, President 1530 P Street, N. W. Section of Terrestrial Magnetism Washington 5, D. C. and Electricity VICTOR VACQUIER, President EX-OFFICIO MEMBERS American National Committee of the Section of Oceanography CHAIRMAN, NATIONAL RESEARCH COUNCIL International Union of Geodesy and Geophysics RICHARD H. FLEMING, President CHAIRMEN, DIVISIONS OF THE and Committee on Geophysics of the Section of Volcanology NATIONAL RESEARCH COUNCIL WILLIAM F. FOSHAG, President International Relations NATIONAL RESEARCH COUNCIL Section of Hydrology Physical Sciences HAROLD G. WILim, President Chemistry and Chemical Technology WASHINGTON, D. C. Section of Tectonophysics Geology and Geography FRANCIS Wiwi!, President Biology and Agriculture AMERICAN OFFICERS International Union of Geodesy and Geophysics 1530 P Street, N.W. Washingtcn 5, D. C. April 21, 1954

Messrs. Robert E. Glover and Glenn G. Balmer, Bureau of Reclamation, Commissioner's Staff Offices, Building 53, Denver Federal Center, Denver 2, Colorado

Gentlemen:

In preparing for publication your paper entitled "aver depletion resulting from pumping a well near a river" we find that you have only sent us prints of the figure accompanying it. Please let • us have the tracing by return mail or, even better, a glossy photo- graphic print with overall width of ink (a little wider than the border) of 4 inches and a vertical dimension of about 4.1 inches. If you send the print, we do not need it by return mail, but we will appreciate word from you that it is on its way and that we may ex- pect it for sure within ten days or two weeks.

Sincerely yours,

Waldo E. Smith Executive Secretary

WES/jjc

• Form DFC- 11 (7-52) INFORMATIONAL ROUTING Bureau of Reclamation

Mr. litildo L. Built ICIOCUtivre thweretary FEB 26 Igit American 044150g$1 5a1 Union 1530 P Meet, Mr. W. Washington, D. C. AUTHOR Dear Mr. Smiths

The ebonies isquested in your letter of February 80 1954, are made below. \ letters in The revised biblioc;raphy is as falows. au f- I.;saientheses are thee* uged in the oriainal draft.

(d) Caralav, H. S., The Conduction of Heat. MacMillan and Co., p. 152, 1921 e cif• v (a) Federal Works Agena7, %Irk Projects Adisisilltretioo for the \ City of New York, Tables of Sine, Cosine, IM14 asstial. Integrals, Volumes I eat II, fables M.T. sad M.T.6, Superintends* of Documents, 1,Lahington, D. C. 1940 ) (b) Federal Works Asesey• Week Projects Administration for the City of jay York, Tables of Probability Functions Volume I, T4ble M.T.0 Superintendent of Documents, lissbisitea• D. C. 1941 :11 N (c) L. B. Label, 0. J., Ingereell, A. C., Neat Cesillagtion, $4i11, Appendix I. 253* 1948 4:7 (e) &hake, ilUgene, and Rade, Frits, Tables of Ihinctions with Foingulae and Curves, Dever Publications, Table 1, pp. 643, 194,•

Peirce, B. 0.„ A Short Table of Integrals, Ginn and Co., pp• 116420, 1929

Mein, C. V., The Ittect of a Well on the noir of a Nearby t3tress, Transactions Anstiemn Georhisical Union, Part III, Pp. 734-738, 1941

Interior - Reclamation - Denver, Colo. As a title to the figure we would suggest:

"rhe pert of the well floe taken from the river as a function of the parameter ;V 044- -"

Since papers which bare been set in type in previous issues Indicate that the American Geogbysical Union is careful to moilth ambiguities which can occur *bon the solidus is used, we believe that it would be preferable to redoes the equations .1.1 the single-line form.

It is noted that an Shildiract is included as an item in the Suggestions to Authca's. If one Should be desired for a piper as brief as ours, the following say be used: A vela located in an aquifer throuot which a river flows will take a Art of its WYPIY from the river. The present a_:41yais develops a DareiWia for the part of the veil flag taken &oaths stress in terns Otte probability integral.

Sincerely yours AWED)

X. E Glover

Glenn G. &Auer

2 OFFICERS OF THE UNION, 1953-56 SECTIONS OF THE UNION JOHN A. FLEMING, Honorary President Section of Geodesy JAMES B. MACELWANE, President ALBERT J HOSKINSON, President liamicE3 EwiNa, Vice President AMERICAN GEOPHYSICAL UNION Section of Seismology Ross R. HEINRICH, President uN PUTNAM MARBLE, General SecretarY Section of Meteorology WALDO E. SMITH. Executive Secretary EXECUTIVE COMMITTEE PHIL E. CHURCH, President I 1530 P Street, N. W. Section of Terrestrial Magnetism Washington 5, D. C. and Electricity VICTOR VACQUIER, President EX-OFFICIO MEMBERS American National Committee of the Section of Oceanography CHAIRMAN, NATIONAL RESEARCH COUNCIL International Union of Geodesy and Geophysics RICHARD H. FLEmING, President CHAIRMEN, DIVISIONS OF TliE and Committee 4an Geophysics of the Section of Volcanology NATIONAL RESEARCH COUNCIL WILLIAM F. FOSHAG, President International Relations NATIONAL RESEARCH COUNCIL Section of Hydrology Physical Sciences HAROLD G. Witm, President Chemistry and Chemical Technology WASHINGTON, D. C. Section of Tectonophysics Geology and Geography FRANCIS BIRCH, President Biology and Agriculture AMERICAN OFFICERS International 1530 1-) Street, Union of Geodesy and • Geophysics ':ashington 5, D. • October 23, 1953

hessrs. Robert T. Glovc,r and GTenn G. 13almer U. 3. Department of the Interior Commissioner's Staff Uffices Buildini; 53, Denver Federal Center Denver 2, Colo.

Gentlemen:

This will aknowledLrc, receiot of your letter of October 8 relative to the questions raised in our letter of September 17. e are forwari- ing your letter to the editor and in due course we will trite to you • further.

Sincerely yours,

.4aluo jmith 2.1xecutive Secretary

, 1530 P Street, N. W. )1 usemi..001 Washington 5, D. C. - 'Feb. 2,1954 - • Dear lEssrs. Gloydr awl Balmer:

This is just a periodic note to let you know that work

on your paper entitled River depletion resulting from pumping a well is progressing. As we have questions or as it approaches publication, we will write to you further. • Sincerely yours, AMERICAN GEOPHYSICAL UNION Form DFC- 11 (:'-52) INFORMATIONAL ROUTING Bureau of Reclamation

OCT -8 1953

kkr. WA* S. itiottki holeative Beerotery Alaarteapw Goophyalgal Union P intraet L. lateholarnes. D. C. I AUTHOR L _____ paw lir. alit&

Mg, gisetione raised is your latter of lhopteriber WIN be lkasivered se folilsems

Ito relation beteeea the teeaula far vo3.1 4inargolosa 9oyeet),4 here 140 the ow wed by Ilhele e sea be estabilehed by a elatple cheap mrsitiegi. Is, is the eormateas oh _ tr • ie r "v ( ) 1- av 41TXP 2 foe Vs min the eilletitutiort or variable

We olvtain ttA once: -i• te '116' 2 •0 la 4 dui (a) —E.- 2rK V 1)-- I r This type of espreaetea will. he Steal to be tukiloyal is ileammee C of our Apace

tee meet east* to ewe ilktrficeilAy ismahlai proper relermee t.43 the slevidossent at the Met of than too thwas. Is Theis' inti Aiwa be gigolos credit to kroteeeer lade of the Delverelty of Claelaimal develogeint of this ram of the lisixriMMOSAlik. ilhos asking the cloaverelea to the care of a Avoyst vela. be &twee it in two texas in his equatleas (4)ma ( :3).4 lila apaties(0 is the ease se eipaties (1) share.

Interior - Reclamation - Denver, Colo. •

sommuns his oqpi‘ifo() he states that the sphols need with it have the Mae assiellass given with his equation (5). This **peers to be inerror eadi. is taut, the two following stalimmests are eoutradlotory. If it is missested that the idisstitir of the tvo Zmneslea be **Seined Is a toothlike oontataleg the first t* SIMS at our soma fanifra". retwommed to solistak (4) or our Apar Iblobnoto aboatil Sall with Saniala (2) More), Rd that the telltales reterimoo be laalAwled.

The Itteet of &Well an the Flow of a 1111101, Strom, C. V. Theis, :Latione. 411111rIcan Geolttweical Union, 1.01, lagt In* iffry;,„

Livxx; to two u.t tripe of expression riamessmied b:/*

eillatles( 2)Oen belasas the oiestity ,D wears in other mall

turmulas fad ito use ei.latestes a poseibla ~as of vieirrioal errors vibes miss tie egpostion. name note that equation 03 rewires

the faster dip shlAih differs by a teether of 2. It is also beliotreit that the serseeter wili bold to as usable regnitudas Um the v Owe; seilleetior'&* The adramtemes of the writer's selvaten ower the grist prervious4 gereseatail lig Mate SIM that it is mare esiselses that it is aoressed terms of a vellAssua tabulated timetlea. aid that it is in sensistent tare sma therefore sot subs)** to the limitations %Lich itr. is hoe Wend as his 1941 greph. The writer's solattiso will ter aramplit, work in the metric system without ahem. uherels Ur• graphviii sot,

Your *Mr to go to the ovum of including a gragh is ilea* oelproolated. Bemuse the prcbabilitg isitogral. is as well lam end tablas It are se tradel$ dietrkuutat we *coda that a gosh wag at* be assottliory• If pm delfts o lookade ones however, the enalopedaj ray be we& Ms hes been constructed to tit late your forest, mak is made to read direst], is teas of I". It it is set used, the rellimiass to it Is the falliaring memple shoat& be deleted. amius The spiatoetiaaIs ficlA coeditions My be art by the itollasisa ampfeeliw eibree wale st diseassee of 1.000, 5040, az4 20,000 test tree * river, titepeetiveirs shish hoe bees meet tor period et 5 pure. It is desired to estimate the gait at their flow videti sem tram the river. If Os aquifer asts ores

D feet 0/14.‘ * 0401 feet per meaced ✓ IN 0.2

1.57,170,000 secomie vi-447; 4,763.

That tor the tkrati walla c/14 *

WW1 sa ‘[167ti)-

• 1,000 0.043 0. 0036 - 2 5,0:0 .3094 .6906 v 3 1/40Co .Yria .4259

Uk4111 at lib* end or five years tbe three *WU core driarlege reopeettvely. 93, 6,end uSperaeat of their ritim troos the Timor.

* tibias Otthe prebehility integral. ** Fria lhanmalit (lit) or fres Paws 3 direetlf•

3 It is suggested that this example be inserted just ahead of the last paragraph in our paper. Sincerely yours

Robert R. Glover

Glens G.

Enclosure

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/1r. VOA* Salitetima alltissal, sat Publisittion Cosallitss berissam esspbmIssl, Mates 1330 P Ihrrtbrast Ilsoldagtos 10 D•C• AUMOR Sew lir. fhwitht _ Um ono trassolittatt berevitta Uwe Iota of a poor as issaltisig Pao Pimplas Vail Naar a. Um° by Mort II. Glow sad Glom G. Ilelsor. Ms piper la Wag Awarded for aels&WelliWas 0, par etamdttss as to its saitatillit, tor vilb1186111.0. Ilinears4 yours

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interior-Re,hmation-Denver,Coi, • RPITT. DEPIaPION RZULTING FROM PUMPING A 1-L;L1 TEAR A RIVER

Robert L,;. Glover*and Glenn G. Balmer**

When pumping of a well near a river begins, water is drawn, at first, from the water table in the immediate neighborhood of the well. As the zone of influence widens, however, it begins to draw a rart of its flow from the river and,ultimately,the river supplies the entire flow. It is the purpose of this analysis to develop a formula for estimating the amount of flow drawn from the river at any time after pumping begins.'

To provide a notation let:

D represent the saturated thickness of thz- water bearing stratum or aquifer. (feet)

K the permeability. (ft/see)

Q the flow of the well. (1t3/sec)

• ql the flow crossing a straight line at a distance xl feet from the well if no river is present. (ft3/sec)

the flow taken from a river at a distance xl feet from the well. (ft3/see)

✓ a radius measured from the center of the well. (feet)

s the draw-down at the radius r at the time t. (feet)

t time, countcd from the instant when pwnping begins. (seconds)

✓ the volume of water yielded by a horizontal square foot of the aquifer if the pressure is dropped 1 foot. (dimensionless)

x and y rectani;ular coordinates measured from the center of the well. (feet)

111 *Engineer, Bureau of Reclamation **Physicist, Bureau of Reclamation • the. distanei. of a 1-.7-c.11 from a xi:lifer, measured alonc, !lormal to the direction of flow. i'he river is assulned to extend Indefinitely upstream and downstream from the (feet)

a a time variable running between 0 and t. (seconds) d ffere:-:t: The/d.raw-don ds at the time t due to removal of the quantity of water Oa at the tirtic a is:di

0. 40L(t-A) d ds a Tr n A) C A

This expression satisfies the continuity condition:

77d)s6._. cc(a23 1 as • • (2) • r d r and the _condition that th draT-don iS zero everywhere whcn (t-

Let

• (3) (t - ) ▪

Then, by substitution and integranon

s oo

• at ( . (4) '7. TT 17

A-OCt C' 'his is a'form of the expoaential Intcra1/7hLch can be evaluated from tables a/c/ by use of tik. ••••••• cc, n -u-- 2 C 1 . r ) F ...... --du. ± • (5) u ,. i— OT C—t'

V/. at 1/Refer to T3ibliocraphy. •

• The draw-down produced in an aquifer of infinite extent due to a well pumped at the rate Q is given by Equation (1r). This expression is valid if s/D is small compared to unity.

We return now to Equation (1) and set r2 • x2 + y2. Then the flow of water across the line x 11$ xl due to the withdrawal is obtained from the relation: +00 +00 x2 2 40C(t-"A) focCt- Nxe dy . 16troc(t- A)2

-00

The integral in this last expression is a form of the probability integral.b/2/1/ This permits an evaluation of this expression in the form: 2 x 4CC(t- (7) • VT? To integrate this expression with respect to A from C to

A ai t WA.

V • • • (

\./ 1 Oqt-

Thcn, by substitution

Q _ 1/2- a S e dv • • (9) ql CR-C. .

AOCt

This can bc expn!ssed in terns of the probability integral

2 p(Z)*$ •

3 S • which has been extensively tabulated 12/2/f/ in terms of the upper limit Z . Then Equation (9) can be put in the form:

r( . (II) CsC t

This formula implies that there will be a drop in thc ground water level along the line x = xl. If a river exists at this distance from the well there will be no drop in level but, instead, the river will supply an additional flow to the well. These factors can be accounted for if a recharge well is p1 need at the point where the pumped well is imaged by the riverbank. A similar form to Equation (11) would apply to the recharge well also, with the result that the flow across the line x = xl is doubled. Then the total flow across the line if the river maintains the level is:

xi . (12) 140C t

These results may be summarized as follows: The draw-down in an aquifer of infinite extent due to a well discharging at the rate Q • is given by Expression (4). This formula Is valid, as a first approxi- mation, if the draw-down s is small compered to the depth D. A similar well located at the distance xi from a river will dray the flow q from the river. The part of the total flow of the well which comes from the river is q/Q. The value of this ratio is given by Equation (12).

While English units have been used herein, the formulas given are valid when used in any consistent unit system. Such a system permits the use of only one unit of a kind. Since, in our case, only units of length and time are involved, change to another system of units can be accomplished by replacement of the foot and second units, used herein, with the new units.

• I • BILLIOGRAPFY

a. Wbls of Sine, CosiJc and '.i:gioncIrC_al Int ixals, Volums I II, .1'1:!dral, Works AL- cy, Surirte;:dt. cf Documr::As, T;'ashington, D. C •

b. TabL:s of Probabfl.:Ity ?unctions, Volum:: 1, Fedcral Works Agcncy, Sup,;rintendt of Docunts, WashIngton, 7.T.. C.

c. iLat Co,Iductlovi, 7.obc1 and Inz;crsoll, Company,

d. Thc: Conductlon of Tat, Carslaw,Mccmillan and Conrany,

e. Tabl,isof Functions, Jahnlw and ilfac, Uovr, 1(:*5.

f. A Short Tablc of I::trals, P..)irce,Cinn and Company, thii'd revised edition,

•