esm223_07_Other_Reading_Capture_Zone_Curves

esm223_07_Other_Reading_Capture_Zone_Curves - OCT-23-2UUU...

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Unformatted text preview: OCT-23-2UUU MON 10:39 HM BREN SCHOOL UCSB FAX N0. 805 893 7812 I P. 01‘ Capture-Zone Type Curves: A Tool for Aquifer Cleanup by Ira] Javandel and Chin-FL: Tsanga' ABSTRACT Currently a common method of aquifer cleanup is to extract the polluted ground water and, after reducing the concentratiou of contaminants in the water below a certain level. the treated water is either injected baclt into the aquifer, or if it is environmentally and economically feasible. released to a surfacevwater body. The proper design of such an operation is very important. both dramatically and environmentally. in this paper a method is developed Which can assist in the determination of the optimum number of pumping wells. their rates of discharge and locations. such that further degradation of the aquifer is avoided. The complex potential theory has been used to derive the equations for the streamlines separating the capture some of one. two, or more pumping wells from the rest of the aquifer. A. series of capture-zone type curves are presented which can be used as tools for the design of aquifer cleanup projects. The use of these type curves is shown by an hypothetical field case example. INTRODUCTION A recent publication by the Environmental Protection Agency (EPA. 1934) refers to the location of 7'36 hazardous waste sites. out of which 5 38 had met the criteria for inclusion in the National Priorities List (NFL) and another 248 sites had been proposed for addition to the NFL. aEatth Sciences Division. Lawrence Berkeley Laboratory, University of California, 1 Cyclotron, Road, Berkeley. California 94-720. Received July 1935. revised October 1935. accepted ncccmber 1935 , Discussion open until Match 1. 1987. 615 The NFL identifies the targets for long-term attics under the “Superfund” law (CERCLA, 1930). ~ " This list has been continuously growing since ’9 | October 1931 when EPA first published an interim priority lisr of 115 sites. In addition. as of October _ 1984, EPA has inventoried more than 19,000 ‘ uncontrolled hazardous waste sites. The ground | water beneath many of these sites is contaminated with various chemicals. Based on the Sec. 104.(a)(l; of FERCLA. the EPA has the primary responsibility for managing remedial actions at these sites unless it is determined that such actions will be done [ properly by the owner or operator of the facility. l or by any other responsible party. Once a plume of contaminants has been l identified in an aquifer and it has been established that remedial action should be undertaken, the l major task for the person in charge is to determint which remedial alternative is cost-effective. This is required by Sec. 105(7) of CERCLA (1980) and i Sec. 300.580) of the National Contingency Plan § (1 983). One alternative for remedial action is aquifer cleanup. Currently a common method of aquifer cleanup is to cxrract the polluted ground water and- , after reducing the concentration of contaminants ! in the water to a certain level. the treated water 15 either reinjected into the aquifer, or, if it is 136? mitted and feasible, it is released to a surface-watt! body. Given a contaminant plume in the ground _ water and its extent and concentration distribufl‘ln' and. further assuming the scum: of cont'amnation I has been eliminated. one has to choose the least expensive alternative for capturing the plurndi VOL 14, ND. S-GRO U-ND WATER-Sepmmber-Ocrobfl 1986 OCT-23-2UUU MON 10:40 All BREN SCHOOL UCSB “my questions to be answered for the design of “Ch projects include the following: ' 1, What is the optimum number of pumping “115 required? "1. Where should the wells be sited so that no out: "inated water can escape between the pumping Wells? i ' 3. What is the optimum pumping rate for each sell? 4, What is the optimum water treatment :icthod? 5. Where should one reinject the treated water mic into the aquifer? 'F'he purpose of this paper is to introduce a w 5:11an. method for answering four of the above :uestlons which are of hydraulic nature. I First, we shall develop the theory and give a series Of sample type curves which can he used as I zools for aquifer restoration. Then, the procedure {or application of the curves will be given in an action ‘80). answering the above questions. 71cc . n interim THEORY October 1 Lonsidet-a homogeneous and isotropic aquifer 300 with a uniform thickness 13. A uniform and steady :ound regional flow with a Darcy velocity U is parallel to minmd and in the direction of the negative x-axis. Let us 104(3)“; propose that a series of n. pumping wells penetrating must};an :hc full thickness of the aquifer and located on the :3 unless j-‘attis are used for extracting the contaminated _Dne :i l water, For n greater than one we want to find the facility.” I maximum distance between any two wells such ‘ -- > i'nat flow is permitted from the interval between the wells. Once such distances are determined we :en ablished i are interesred in separating the capture zone of _, thg i those wells from the rest of the aquifer. We shall gtgrminc 1 Start with n = 1 and expand the theory for larger .' This is I values of n. The following development is bated on 3) and linlllieation of the complex potential theory 'y Plan l Milne-Thomson, 1968). 1 is l Case 1, n = 1 for l in this case for the sake of simplicity and vater and. Without losing the generality, we shall assume that tinants the pumping well is located at the origin of the iflordinate system. The equation of the dividing water is 5 Per. ‘I. streamlines which separate the capture zone of this cwwam at” from the rear of the aquifer is "ind ‘ r=i Q — Q tan" 3’. <1) :rihution. ‘ ZBU erBU x math?" I Where a = aquifer thickness (or). Q = well diseharge 1‘35: ' lite (ma/sec), and U = regional flow velocity 1‘3- ‘l‘n/See). One may note that the only parameter in me. oher 019333 FAX N0. 805 893 7812 P. 02 SINGLE-WELL CAPTURE-ZUNE TYPE CURVES '1000. -500. 0. 500.1000.1500.EODO.EEDO. Meters Fig. 1. A set of type curves showing the capture zones of a single pumping well located at the origin for various values (If .l equation (1) is the ratio (QIBU) which has the dimension of length (m). Figure 1 illustrates a set a p of type curves for five Values of parameter (Q/BU). For each value of (Q/BU), all the water particles 3' within the corresponding type curve will eventually go .to the pumping well. Figure 2 illustrates the paths of some of the water particles within the capture zone with (Q/BU) = 2000. leading to the pumping well located at the origin. The intersection of each of the curves shown in Figure 1 and the ( ') x-axis is the position of the stagnation point whose distance from the well is equal to Q12 aBU. In fact, equation (1) may be written in nondimensional formas 1 1 - Yn =:_—_,,,_rt 1—— 2 YD 2 211 an xD () where yD = Elly/Q, dimensionless, and Me tors Cl- "500. “1000. '500. 0. 500. 1000. t500. 2000. 2500. Meters Fig. 2. The paths of some water particles within the capture I zone with (DIBUl I 2000, leading to the pumping well ~ located at the origin. 617 r OCT-23-2UUU MON 10:40 All BREN SCHOOL UCSEl l‘ 0.50 D . ES _ Pumping Well )9 0.00 x Regional Flew 3.0 4.0 Fig. 3. Nondimonsional-form of the capture-zone type curve for a single pumping well. XI) = BUx/Q, dimensionless. Figure 3 shows the nondimensional form of the captutevzone type 'curve for a single pumping well. Case 2, n = 2 Here, we shall consider two pumping wells located on the y-axis, each at a distance d from the origin. Each well is being pumped at a constant ratel‘Q. The complex potential representing the combination of flow toward these tWo wells and the ‘bniform regional flow is given by i. l W=Uz+ a 2,3 [his - id) +1n<z +id>1 + c (3) where z is a complex variable which is defined as x + iy and i = x/Tl'. ‘ The velocity potentian and stream function w for such flow system are the real and imaginary parts of W in equation (3) which can be written as p =Ux+4 Bfln[x2+(y-d)3]+111[X1+(y‘l'd):‘ll‘a"C Tn" ..... (4) — d o =Uy+ {tan“ y d-l-tan'l Y+ } (5) 2173 x x In general, when the distante between one wells is too large for a given diacharge rate Q, a stagnation point will be formed behind each pumping well. In this case some fluid particles are able to escape from the interval between the two Wells. When the distance between these two wells is reduced while keeping Q constant, eventually a position will be reached where only one stagnation point will ; appear and that would be on the negative xeaxis. In this case no fluid particles can escape from the space between the two wells. If we keep reducing 6133 FAX N0. 805 893 7812 P. 03 the distance between the two wells, again two . . . . $55 meiosis stagnation points Will appear on the negative 1;. - L of I: one moving toward the origin and the other away? in: r or from it, and still no fluid particles could escape :‘hE/fl from the space between the wells. The followingm’: I l'l by derivation gives the reaton for such behavior. ‘ Ta’stlLJ' To find the position of the stagnation points" ‘st 'JL one must set the derivative of W to zero: ‘3: e11 dw Q 1 1 i ' u— = U + w—vc , l = 0 dz zit-B ZHICl z+1d The roots of equation (6) are given by ' Th Q ,1. a. imit “'1' g _ m WT? I p 3 5“ wBU i [Q liwfiU) 1 4d l by “En. _ squat When 2d :5 Q/aBU, that is, the distance between»; ,Iugnat. the two wells is larger than Q/aBU, equation (7) p ‘ would give two complex roots. Each of these mi v +_ corresponds to the position of a stagnation point: ' " behind each pumping well. The coordinates of the“ . " 0' 'rgr‘ two stagnation points are . Lqumfl all ~ a Q i /"—"—‘—"—*2 a 2 {g of a P - l, identa.‘ Q “iii” lllulilil'flt d _ -1 W . an ( ZnBU ’ “é 4d {Q “T’Bm H lens at , tine ms Note that only when 2d at Q/aBU the Coordinatl‘fl“ mitten of these two stagnation points become approxi- _g;,j mately [~(Q/2nBU), d] and [*(QIZnBU), -d]. I 1 When 2d 2;- Q/nBU, contaminated water can escape l ’3' E from the space between the We pumping wells; at the larger the distance, the more fluid will escape. It is apparent from equation (7) that if the distant: between the We wells 2d is equal to Q/nBU, than both roots of equation (6) are equal and real such that ‘ ‘- — Q is) ' ZWBU where 3 \‘D = B 21222: in this case we shall have one stagnation point on the negative xvaxis whose distance from the origin is Q/erU. Under this condition no flow can pass betWeen the two pumping wells. Finally, if 2d s‘: Q/aBU, equation (6) would yield two real roots. The coordinates of the WW Ht! Lair" 5 -_;_..._._ —.—_..t— stagnation points corresponding to these two roots “50* are Q 2 I we: _ + 3.5 f 3 u. '2‘ O H: { zfiBU V [Q (nBU) ] ‘l-Cl l' I and Q r Fig. 4. C {— -- - vs V {Q3/(trBUlzl - 4612.0} 45 1° “'3‘” Walls. EtrBU OCT-23-2UUU MON 10:41 All BREN SCHOOL UCSB agam W0 mviously, when 2d becomes smaller and smaller, “game . km: Of these points tends to the origin and the e Other aw? ‘ ‘ I" he; one tends to the point with coordinates of mm fiscal?“ .flQh-rBU), 0] .When 2d re Q/rr BU, no flow can 1e foliflwmgi ‘ .5 between the two pumping wells. Therefore, it )ehavror. “ “3 3,. fimbiishEd that the condition for preventing the we of contaminated fluid between twn pump- .3! wells separated by a distance Ed is )= 0 Q 9 201 fi t—HBU ( 3 y The optimum condition is achieved at the 1.3.?) mi: when 2d = Q/cBU and the distance of the gagnarion point from the origin is (Q/ZNBU). The :quai'; . a of the streamlines passing through this use between mgriatiofl point is quation (7) g d of Fhese moo c 4- (tan‘1 y d + tan"1 y + ) = t Bu (10) iation Pomp; I - tnBU x' x no 1 line may note that again the only parameter in equation (10) is (Q/BU). Figure 4 shows the plot of a pair of these streamlines for (Q/BU) = 800; linates of D2] ) _ ‘ some H-‘eful distances on this figure are also ‘. identimd. Figure 5 gives a set of type curves m) i I din-strafing the capture zones for two pumping wells and for several values of parameter (CZ/BU). . ‘- i-I- (Jnc may note that equation (10) also can be : coordinates . on written in nondimensional form as e approxi- . 'BU3’“d]- iii 1 - to cum) _ yD +(1/21r) :er can escape ‘Fn +gltan 1-«——--:;---7--~+tan L T] :21 sing wells; I (11) will escape." f the distance where yD = BUy/Q, dimensionless; and l/nBU, then in = BUx/Q, dimensionless. nd real such I (53 * I 1000. In on Q/EU = T1 the origin i ow can pass i n . E D (6) would I - g If the two se two roots ] “500. l r H ‘1000. -9} sec. 0. see. 1000. 150d. 2000. secs. 2 Meters 3 -' HQ. 4. Capture tone of mo pumping wells properly located l i O} ‘9 Prevent any leakage from the space between the two WelIsv FAX N0. 805 893 7812 P. 04 DOUBLEHWELL CAPTURE-ZONE TYPE CURVES 1500. 1000. } EDD. {fl 5;, Re ionel Flow H 0 --—9—-———~—— g '500. “1000 '1500. Ih—- 'SOU. Cl. 500‘ 1000. 1500. 2000. 2500. 3000. Hetero Fig. 5. A set of type curves showing the capture zones of two pumping wells located on the y-exis for various values of (Cl/BU}. Case 3, n = 3 In this case we shall consider three pumping wells, one at the origin and two on the y-axis at (0, d) and (D, —d). The regional flow, as before, has a velocity of U and is parallel to and in the direction of the negative x-axis. The complex potential repre- senting flow toward these three Wells and the i uniform regional flow is given by WeUa+~2—9—Bv [1n a+ln(z-id)+ln(a+id)] +C (12) TI“ The velocity potential it and the stream function ill for this flow system are given by o=Ux+flg{ln(xi‘ +y2)+ 4n ln[x2 +(y—d)31+ln[x1 ‘+(y+d)‘i}+C (13) ‘ —d d W=Uy+B-(tan"z+tan"’Z—+tan“Z:-) (14) 21rB x x X I—Iere also, when d is large, fluid will escape between the wells and three stagnation points will be formed, one behind each well. Keeping the rate of discharge of each well constant and reducing the distance between each pair of wells, eventually a position will appear where no flow will pass in between the Wells. Again, to find the position of the stagnation points one must set the derivative of W in equation (12) equal to zero: } dW Q 1 1 1 —= +e—[—+ .+ . dz 21713 2 zflid z+id i=0 <15) 619 r— OCT-23-2UUU MON 10:41 All BREN SCHOOL UCSB Equation (15) may be written as 3 - m d2 w m- = 16) a + z A ( where A = *(ZnBUNQ. The disotiminant of equation (16) may be Written as d4 d1 1 D=d2('fi‘w+'A-T) It can be shown easily that D is positive, except for the limiting case when d = 0. In that case D vanishes, too. As a result, when d 9% 0 equation (16) has one real root and two other roots which ' are complex conjugates of each other. When d or Q/EW‘BU we obtain three stagnation points located at Q o O. _ O)Iz2‘( zflBuld)lz3'( ‘F—h ZnBU ’ .. ZnBU’ When d becomes smaller and smaller, that is, the distance between the wells decreases, the stagnation point on the x-axis moves away from the origin and the ether two tend to come closer to the ydaxis while appraoehing the x-axis. Such that for d = (2- 3x/El) Q/ZWBU the position of Stagnation points are 2;:(— ' Q Q Q z“(_l'54aaau’0)'z”‘l O'Hawau’l'gaweul' Q Q zi'l 0'73 aneu’ 9ai-riaul' The value of d = (2 3x72") Q/ZnBU is the maximum distance betWeen two pumping wells where no fluid could escape between the wells. One may note that this distance is approximately 1.2 times the optimum distance between two Wells for the ease of n = 2. Eventually, when d becomes zero, that is, when the outer two wells coineide with the middle one, three roots of equation (16) correspond to one stagnation point on the negative x-axis with a distance of SQ/EWBU from the origin and the other two collapse at the origin. At the optimum condi- tion, the equation for the streamlines passing through the stagnation point on the negative x-axis becomes -d +d 3 y+ zfilEUuan'hi-s- tan" yx +tan 1 a—w—yx )== TIPS-I (18) where d -~ 3 2 Q/(rBU). Since d is only a function of (QIBU), it is apparent that once again equation 620 on NO. 805 893 7512 g H P. 05 N i THREE-NEH. cantons-zone TVPE waves”, an 1500. . QIBU=|DODm Y 1000. 300 500 ‘ ‘ I 400 I fl, - 505 . A I I ' D V g o x Regional Flow l g 44 41—...»- ._._ ‘ I L. a w 2 '500. "1000. -1EDD. ‘ g -500. a. son. soon. 1500. anon. anon, 3gb. . Meters Q “#6, Fig. 5. A set of type curves showing the capture zones of \ + m l three wells all located on the y-axis for various values of”? i I (QIBU). ' I Hi where y: + } (13) is dependent on one parameter (Q/BU). Figigt we”, a _ a 6 shows a set of type Curves illustrating the capth ‘ : Md; zones for three pumping wells located on the deaccm p- y-axis for several values of parameter (Q/BU). N 1:; {our beam. that one of the pumping wells is located at the " indicates t origin and the other two are on the positive and J :he optimi negative y-axis with a distance of 3V3 Q/nBU front Dumping V the origin. 43%.; g is about tl' Here, one can also write equation (18) in a a}. Mug p131, nondimensional form as me - " ten 1 _ yo . Yo - (WE/t) sue- tan‘“ +tan1—---+ YD 2a [ xD X13 I _,_ a 2/ 3 l 1500. tan“1 M ] = i Er i x . D iooo. where x13, and yD are dimensionless coordinates as F defined before. =gg_ l General Case is , J | l | -soo. J i We shall now attempt to extend the solution. for a larger number of pumping wells. Table 1 55* shows some characteristic distances for the cases that we have already discussed. There are two generalizations that one can infer from Table 1. (1) The distance betWeen dividing Streamlines fa: upstream from the wells is equal to (nQ/BU) and” it is twice the distance between these streamlines at the line of wells. (2) The equation of the dividillfi streamlines for the case of n pumping wells can by; [ Fig.1 A . written down by comparing the corresponding ‘ equations for one, two, and three pumping wellsfig‘i IIo- "1000. OCT-23-2UUU MON 10:42 All BREN SCHOOL UCSB 1“ some Characteristic. Distances in Flow Regime FYF'E cu ‘ HVESJW ‘ Tabla 1- _ p in, One, Two, and Three Pumping Wells Under a ‘”“' _—‘--, l j”;- Uniform Regional Ground-Water Flow Optimum distance Distance between Distance between hetWetn each pair dividing streamlines streamlines of pumping wells at the line of fer upstream from #___._._ Q _ '— _ — : ture zonesqf I -,- + -—-—- {tan 1 y 1 + tan 1 Y . tut values of"" I Zflgu x x a _ “Q Han-1 3—15 } st (20) x ZBU r when: . ya, . . . yn are y-coordinates of pumping El/fhm' F1315“ I! wells 1, 2, . . . , and n. g ‘3 CaPtlii‘i Finding the optimum distance between two l as - ' on the dill: ‘ adjacent pumping wells when it gets larger than Egg/gag”: four becomes quite cumbersome. Our investigation mitive an .ndicates that for the case of four pumping Wells, I :he optimum distance between two adjacent QMBU {WP jumping wells is approximately 1.2 Q/(WBU) which ‘ :sabout the same as for the case of three pumping wells ' igure 7 shows a set of type curves for the i an o: four pumping wells for several values of FOUR-WELL CAPTURE-ZONE TYPE CURVES 1500. (Hi _ loco. lordinatcs as l D Rational Flow .he solution l g ' Table 1 r the cases '50“- are two - l Table 1. i '1000- amlines fat,” f ill/BU) andw "-"" . gtreamlines “Eco. u. see. loco. secs. acne. aeoo. seen. f the dwelth Meters wells can hi. i F; . ondfi J 9- 7. A set of type curves showrng capture zones of four P, 1 g Tl" “meme Wells, all leaned on the y-exis for several values of ping wells: lu/BuL 0‘ FAX N0. 805 893 7812 P. 08 parameter (Q/BU). Note that two of the wells are on the positive and the other two are on the negative y-axis. The distance between each pair of wells depends on the type curve (i.e., Q/BU value) chosen. Once the type curve is selected, the optimum disrance between each pair is d = 1.2 Q/(aBU). APPLICATION As was discussed earlier, presently a common method of aquifer cleanup is extracting the polluted ground water, removing from it the contaminants, and disposing or reinjecting the treated water. Naturally, the cost of such operation is a function of the exrent of cleanup. However,...
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