ANALYTICAL SOLUTIONS FOR WELL DRAWDOWN WITH WELL LOSSES 2. REAL WELL NEAR BOUNDARY - SOLUTION BY IMAGE WELL

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1 JOURNAL OF ENVIRONMENTAL HYROLOGY The Electroic Joural of the Iteratioal Aociatio for Evirometal Hydrology O the World Wide Web at VOLUME 3 5 ANALYTICAL SOLUTIONS FOR WELL RAWOWN WITH WELL LOSSES. REAL WELL NEAR BOUNARY - SOLUTION BY IMAGE WELL Radimir Novoty Pavel Pech epartmet of Cotructio epartmet of Water Reource Czech Uiverity of Agriculture Prague, Czech Republic The olutio i the precedig Paper were baed o the aumptio of a ifiite aquifer. Thi aumptio, however, i ot valid i the cae a well i located ear a ifiite boudary, which may be impermeable or have differet permeability, ad the boudary i withi the drawdow coe of the pupig well. A olutio i preeted for drawdow i a pumpig well that iclude well ki loe ad a ifiite boudary. The olutio i baed o the theory of mirror reflectio of image well. Joural of Evirometal Hydrology Volume 3 Paper 8 November 5

2 INTROUCTION The ifluece of a ifiite impermeable boudary i olved by the applicatio of the mirror reflectio theory. We aume that the well drawdow durig the dicharge from the well i idetical with the ifluece reultig from the uperpoitio of the effect of dicharge,, from the give real well ad the effect of imultaeou dicharge of the ame from a imagiary well. Thi image well i a mirror image of the real well beyod the impermeable boudary at the ame ditace from the impermeable boudary a the real well. THEORY Real Well i the Proximity of Lateral Impermeable Boudary With the dicharge of a well i the proximity of a lateral impermeable boudary, which i withi reach of the well, the reultig groud water drawdow,, i determied by the um of the theoretical drawdow, re, due to the dicharge from the real well ad the drawdow, f, due to the dicharge from the image well placed i a mirror-reflected poitio. for the real well = re + f () re = te + SKIN () te = drawdow at a ideal well (W = ) SKIN = additioal drawdow at a well caued by additioal reitace or well lo. If we coider the o-tatioary flow regime, the water level drawdow i the real well i give by olvig the equatio i dimeiole form (Agarwal et al., 97): r + = r r t (3) The complete olutio of Equatio (3) may be obtaied by the applicatio of the Laplace traform. The Laplace traform i iverted umerically uig the Stehfet algorithm 368 (Wei Chu Chu, 98; Pech, 986): V = π T m i co( j, k) ( ). i i= i= / / / K ( c ) W c K ( c ) ( ) + ( ( ) + ( )) 3 c c K c C c K c W c K c [ ] / / / / / / / / ( r c ) / ( ) K + c K c (4) T = tramiivity; S = torativity; = well dicharge; C = a dimeiole well bore torage cotat (Ramey, 97); c = the variable i Laplace pace ; r = x /r W ; x (ee Figure ), Joural of Evirometal Hydrology Volume 3 Paper 8 November 5

3 Figure. Schematic drawig of the well i the proximity of impermeable lateral boudary ( - drawdow due to imagiary well, - drawdow due to the real well, 3 - overall drawdow. r w = well radiu; W = ki factor; K (x) i a Beel fuctio of the ecod kid of order zero; K (x) i a Beel fuctio of the ecod kid of order oe; = ; ad co j (, k) j ( ) k ( m) k l! jm k j t r m ( m ) =!! (5) t = dimeiole time (/r w ) or, for t < 5, i.e. i the coditio whe the Jacob emilogarithmic approximatio of the Thei well fuctio i applicable, it i,46,46 V = l + W + l π T r S r S W (6) Joural of Evirometal Hydrology 3 Volume 3 Paper 8 November 5

4 If r =, the r = x. (ee Figure ). After modificatio, the drawdow i the well i expreed by the relatio,46 V = l W πt rw + ( ) x S The drawdow i ay poit B determied by the coordiate x, y (Figure ) for a o-tatioary flow regime, i B = π T (, ) ( ) co j k m i= i= / ( ) / ( ) m K r c K r c i + i / (8) c K ( c ) (7) r = r/r W ad r t = r t /r W The ditace r ad r are determied from the relatio a f r = x + x + y (9) a f () r = x x + y If we coider the applicatio of the Jacob emilogarithmic approximatio, the water level drawdow i poit B i B =, 46 T r S + 46 l l, () π r S or, for the ake of implificatio, B = T l, 46 π rr S () Well i the Proximity of Lateral Cotat Head Recharge Boudary By way of example let u ue a well i the proximity of a cotat head recharge boudary (urface flow) fully peetratig the aquifer. The olutio i aalogou with that applied to the impermeable boudary. The differece i that i the cae of a imagiary well we coider cotat ijectio (it i a ijectio well). Overall drawdow produced by the dicharge from the real well i determied a the um of the drawdow i the real well ad the egative drawdow due to ijectio ito the imagiary well. Aalogou to the impermeable lateral boudary (Equatio ) the overall drawdow i = re - f (3) The drawdow i the dicharge well i give by Equatio 3. I cae of applicatio of the Jacob emilogarithmic approximatio, the drawdow i the dicharge Joural of Evirometal Hydrology 4 Volume 3 Paper 8 November 5

5 Figure. Schematic drawig of the well i the proximity of lateral feedig boudary ( drawdow due to imagiary well, drawdow due to real well, 3 overall drawdow). well i,46 = l + W π T rw S,46 l r S V (4) or, after modificatio, V r = l + W π T rw (5) Joural of Evirometal Hydrology 5 Volume 3 Paper 8 November 5

6 The drawdow, V, doe ot deped o time ad the curve, V = f(log t), proceed horizotally, which mea that the ifluece of the lateral boudary will maifet itelf after certai time by the tabilizatio of the drawdow. The tramiio capacity ad torage capacity computatio ca be baed oly o the iitial part of the iflow tet before the part which ca be evaluated by the Jacob emilogarithmic approximatio i reached. The drawdow i ay poit B (aalogou to the cae of impermeable boudary) i B = π T (, ) ( ) co j k m i= i= m i / ( ) / ( ) K r c K r c / c K ( c ) (6) i ad, whe coiderig the applicability of the Jacob emilogarithmic approximatio, B =, 46 T r S 46 l l, (7) π r S or, after modificatio, B = r l π (8) T r CONCLUSION I thi cotributio the relatio are derived for the determiatio of the drawdow i a well with well loe ituated i the proximity of the permeable or impermeable boudary. By mea of the derived relatiohip we ca determie drawdow at a arbitrary place whe the real well, with additioal reitace ad wellbore torage take ito accout, i ituated ear boudary. ACKNOWLEGMENTS The author would like to thak Prof. Ig. Vladimir Havlik, Ph.., Czech Techical Uiverity, Prague, ad oc. Ig. Joef Buchtele, Ititute of Hydrodyamic, CAV for their review of thi paper. REFERENCES Agarwal, R. G., R. Al-Huaiy, ad H. J. Ramey Jr. 97. A ivetigatio of wellbore torage ad ki effect i uteady liquid flow.. Aalytical treatmet. AIME Tra Bear, J Hydraulic of groudwater. New York; McGraw-Hill. Pech, P Evaluatio of the hydraulic parameter from the aquifer tet. Czech Uiv. Of Agriculture, Prague. Ramey, Jr., H.J. 97. Short-time well tet data iterpretatio i the preece of ki effect ad wellbore torage. J. Pet. Tech.; :97 4. Stehfet, H. 97. Algorithm 368: Numerical iverio of Laplace traform. Comm. ACM. 3: Todd,.K. 98. Groudwater Hydrology. Secod Ed. New York; Wiley. Walto, W.C., 97. Groudwater Reource Evaluatio. McGraw-Hill, New York. Chu, W.C., J. Garcia-Rivera, ad R. Raghava. 98. Aalyi of iterferece tet data iflueced by wellbore torage ad ki at the flowig well. JPT. Tra. AIME 49 p. Joural of Evirometal Hydrology 6 Volume 3 Paper 8 November 5

7 ARESS FOR CORRESPONENCE r. Pavel Pech epartmet of Water Reource Uiverity of Agriculture Prague Kamycka ulice 65 Prague 6 Czech Republic pech@lf.czu.cz Joural of Evirometal Hydrology 7 Volume 3 Paper 8 November 5