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

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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 U.S. SAV TIONT mrc nrwric ..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 drop- ped 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.
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