Transient Radial Flow Toward a Well Aquifer Equation, based on assumptions becomes a 1D PDE for h(r,t) : Transient Radial Flow Toward a Well

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Herbert Holmes
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1 ansien Radial Flw wad a Well Aqife Eqain, based n assmpins becmes a D PDE f h(,) : -ansien flw in a hmgenes, ispic aqife -flly peneaing pmping well & infinie, hiznal, cnfined aqife f nifm hickness, hs essenially hiznal gndwae flw -flw symmey: adially symmeic flw Ss h h K Bnday cndiins - nd de in space PDE f h(,), need w BC s in space, a w s, say and ; in geneal we ll pick w, -One Diichle BC a infiniy and ne Neman a well adis, w, whee we assme we knw he pmping ae, w h h + x y S h h h S h + Diffsin eqains ansien Radial Flw wad a Well Ding a pmping es, we ypically mease dawdwn (as ppsed head), s le s se p he eqain in ems f dawdwn. Assme iniial penimeic sface is hiznal eveywhee. ( ) h h s s h h 0 0 h s, h s s s S s +

2 ansien Flw Well es Analysis slve, we need bnday cndiins and iniial cndiins I.C.: dawdwn is ze, head is nifm iniially. s 0 B.C.s : nd de eqain, we need w BCs Use 0, Use cniniy a he pmping well, eaed as a line sink ( 0) A 0, h s KA KA A b s b K s ansien Flw Well es Analysis s b K s s ( ) s % lim& 0' ) $ (eqies cnsan ) s 0 as (f all )

3 ansien Flw Well es Analysis Slin: e s (, ) d whee $ S e d In mahemaics: he expnenial inegal In hydgelgy: he well fncin e W ( ) d s W ( ) heis Eqain ansien Flw Well es Analysis S : disance fm pmping well bsevain well : ime since pmping saed i $ W ( ) $ % $ ln $ $ Ele s cnsan i i i Vales f W() f vais vales f ae als ablaed in nmes efeences 3

4 ansien Flw Well es Analysis s W ( ) S O gal in aqife esing is find and S; slving he dawdwn eqain shwn abve f is pblemaic appeas wice. One way slve f is cmpae a pl f s vess a nmal gaphic slin (when pling daa fm nmes bsevain wells, we can ypically nmalize s by pling i as a fncin f / insead f ) ansien Flw Well es Analysis ype cve mehd f slving he heis Eqain Pl s vess ( s vess / ) n lg-lg pape On anhe shee f he same pape, pl W() vess / Why /? O pl has in he nmea, b in he eqain defining, appeas in he denmina S Mach daa (keep gaph axes paallel) Pick mach pin easies pick a mach pin wih simple nmbes he mach pin des n have fall n he cve. Slve eqain

5 ansien Flw Well es Analysis (Schwaz and Zhang, 003) ansien Flw Well es Analysis 500 m 3 /d 300 m (Schwaz and Zhang, 003) 5

6 ansien Flw Well es Analysis W ( ) ; 0.; min; s 0.78 m s W ( 3 m 500 m ) d 5 (0.78 m) d S m d 5 & min $ & $ d & % % 0 min $ % ( 300 m) ( 0.) 3.6 x 0-6 B S is had mease ypically shld nly be eped ne significan digi id es 3 x 0-6 ansien Flw Well es Analysis Assmpins: Hmgenes, ispic aqife (we sed a simplified fm f he cniniy eqain insead f inclding he ens f K) Cnsan wae ppeies Infinie aqife (sed develp bnday cndiin) Aqife is cnfined; cnfining nis ae impemeable All flw is adial and hiznal Wae is wihdawn insananesly wih decline in head (n ypically he case f leakage fm cnfining nis lw k lenses) Well diamee is infiniesimal (wae can be sed in he well be, leading delayed dawdwn) Cnsan dischage fm well 6

7 Head f NM Hydlgy Pgam C.E. Jacb ( ) Hansh s men, well hydalics, hey f leaky aqifes 3 facly, inclding Fank is and William Bsae meged wih Gelgy and Gephysics fm Gesciences Depamen died f hea aack in 970; Hansh ened biefly ansien Radial Flw wad a Well Simplified appach ansien analysis: he Jacb Appximain A simplifying assmpin ha makes slving he heis eqain easie C. E. Jacb (90) Jacb appximain Jacb-Cpe eqain Cpe-Jacb mehd Cpe-Jacb saigh-line mehd i $ W ( ) $ % $ ln $ i i i 3 W ( ) ln

8 8 ansien Flw Well es Analysis ) ( 0.0, F W i i i i << $ % S S we need be lage f be small ( ) ln s 0.0, F ( ) $ % & ' ' ( ) * * +, - - S ln ln.78 s. ansien Flw Well es Analysis ( ) $ % & ' ' ( ) * * +, - - S ln ln.78 s. ( ) ln ln ( ) lg ln.3 ( ) $ % & ' ( ) * +, + S ln ln s - ( ) $ % & S ln s '

9 ansien Flw Well es Analysis s ' & ln$ % ( 0.565) S Jacb-Cpe simplificain ( ) lg ln.3 ansien Flw Well es Analysis Sppse we mease dawdwn a w imes:, & -.5 * -.5 * s ' s $ ln+ ( + ln ' ln+ ( ' ln. %, S ), S ) s s [ ln ] s ln s ln 9

10 ansien Flw Well es Analysis s s ln Pick 0 ; head change pe lg cycle %s lc s - s s s 0 ln.3 ansien Flw Well es Analysis &.5 A, s 0 ln$ ' % S 0 ( ) ' &.5 ln$ % S &.5 0 ln$ % S e 0 &.5 ln$ % S e 0

11 ansien Flw Well es Analysis e 0 &.5 ln$ % S e.5 S ansien Flw Well es Analysis Semi-lg pl (dawdwn vess lg ime) 500 m 3 /d (Schwaz and Zhang, 003)

12 ansien Flw Well es Analysis.3.3 s lc 3 m ( 500 ) d 5 m d (.8m).5 S m.5 5 d S 300 m ( ) 3.min d ( 0min ) 3 x 0 6 ( ) ansien Flw Well es Analysis Remembe ha he Jacb-Cpe mehd is pedicaed n he vale f being less han ab 0.0 S 6 ( 300 m) ( 3 x 0 ) m d ( 5 )( 00 min ) d 0min 0.0 Accding y exbk, 0.0 is clse 0.0, and i is he maximm vale f ve lg cycle f daa, s he Jacb-Cpe mehd is accepable.

ansien Flw Well es Analysis Insead f lking a dawdwn in ne well wih ime, we can als lk a dawdwn meased a he same ime in w me wells: Disance-dawdwn mehd s ( ) s ( ) Meased a he same ime

13 ansien Flw Well es Analysis Insead f lking a dawdwn in ne well wih ime, we can als lk a dawdwn meased a he same ime in w me wells: Disance-dawdwn mehd s ( ) s ( ) Meased a he same ime (Schwaz and Zhang, 003) ansien Flw Well es Analysis s ' &.5 ln$ % S '.5 $ '.5 $ ln % ln ln % i S & & S i &,.5 ),.5 ) s - s $ ln* ' - ln - ln* ' + ln. % + S ( + S ( s s ln + ln [ ] s s ln 3

14 s s ln ansien Flw Well es Analysis s ( s ' $ % ln & ' $ % ln & ( s Disance-dawdwn fmla.5 S ansien Flw Well es Analysis calclae S, we se a fmla simila ha sed in he ime-dawdwn fmla. We sed becase i ld s he ime a which dawdwn was ze; f a disance-dawdwn fmla, we wan knw he disance a which s 0 (we ll call i ) he es f he deivain is he same as f he disance-dawdwn mehd, s we ll jmp igh he fmla:.5 S Disance-dawdwn fmla Remembe ha hee ae w fmlae f applying he Jacb-Cpe mehd ime-dawdwn and disance-dawdwn

15 ansien Flw Well es Analysis 0 gpm 9. f 3 /min,350 f 3 /d Dawdwns meased a 0 min (Schwaz and Zhang, 003) ansien Flw Well es Analysis.3 s lc.3 3 f ( 350 ) d 987 f d ( 5.7 f).5 S S S ( ) 0min f d 987 f ( ) 9 x 0 5 ( 900 f) ( ) d [( )( 0min) ] 0.5 ( 900 f) 9 x 0 5 ( d ) 0min 5

16 ansien Flw Well es Analysis ' $ % ln & s ' 0 $ ln % & ( h ( lc Disance-dawdwn fmla hiem eqain ansien Flw Well es Analysis ' $ % ln & s ' 0 $ ln % & ( h ( lc Disance-dawdwn fmla hiem eqain Fmlain f he DDF is idenical ha f he hiem eqain his means ha when Jacb s appximain applies, he diffeence in dawdwn beween any w pins sabilizes he penimeic sface is being dawn dwn nifmly wih ime 6

17 ansien Flw Well es Analysis Why desn his apply in ealy ime? Sabiliy is achieved when he slpe f he penimeic sface has he cec gadien spply he well We ae cnsanly lweing he penimeic sface we ms d his elease wae fm sage. In ealy ime, wae is nly eleased fm sage nea he well, b as ime ges lage, wae is spplied fm a lage aea. As he vlme f aqife spplying inceases, less change in head is needed yield he same amn f wae, s he cne f depessin inceases a a deceasing ae. 7

18 Hw d cnes f depessin vay fm nmal wih changes in aqife ppeies, and hw des his affec he dawdwn hydgaph f a mniing well? Hw migh dawdwn hydgaphs vay fm nmal in siains whee pmping es assmpins ae n me? Vaiains in dawdwn Cne f depessin as a fncin f ime min 0 min 0 min 000 gpd/f; S 0 - ; 300 gpm (Hall, 996) 8

19 Vaiains in dawdwn Vaiains wih pmping ae 00 gpm 300 gpm 600 gpm 000 gpd/f; S 0-; 00 f (Hall, 996) Vaiains in dawdwn Vaiains wih saiviy S 0. S S S S 0-; 300 gpm; 00 f (Hall, 996) 9

Vaiains in dawdwn Vaiains wih ansmissiviy 500 gpd/f, 000 gpd/f,500 gpd/f S 000 gpd/f; 300 gpm; 00 f (Hall, 996) Sensiiviy and S smmaized Lw S: Cne shape emains simila, he cne is js bigge High S:

20 Vaiains in dawdwn Vaiains wih ansmissiviy 500 gpd/f, 000 gpd/f,500 gpd/f S 000 gpd/f; 300 gpm; 00 f (Hall, 996) Sensiiviy and S smmaized Lw S: Cne shape emains simila, he cne is js bigge High S: Feeze and Chey (979) Lw : igh, deep cne; High : wide, shallw cne Lw S, ceaes geae dawdwn pdce a given vlme f wae; cne f depessin shape says he same Lw : had ansmi wae, ms efficien pdcin is fm clse in, ceaes seepe gadiens f s pdce wae fm clse in. 0

11. HAFAT İş-Enerji Power of a force: Power in the ability of a force to do work

11. HAFAT İş-Enerji Power of a force: Power in the ability of a force to do work MÜHENDİSLİK MEKNİĞİ. HFT İş-Eneji Pwe f a fce: Pwe in he abiliy f a fce d wk F: The fce applied n paicle Q P = F v = Fv cs( θ ) F Q v θ Pah f Q v: The velciy f Q ÖRNEK: İŞ-ENERJİ ω µ k v Calculae he pwe