ERTH403/HYD503, NM Tech Fall 2006

Transcription

1 ERTH43/HYD53, NM Tech Fall 6 Variation from normal rawown hyrograph Unconfine aquifer figure from Krueman an e Rier (99) Variation from normal rawown hyrograph Unconfine aquifer Early time: when pumping tart, rawown ha not reache the water table; water come from elatic torage only (S ) Hyrology Program, Prof. J. Wilon

2 ERTH43/HYD53, NM Tech Fall 6 Variation from normal rawown hyrograph Unconfine aquifer Intermeiate time: now, a moving water table i being raine (S y ); there i a trong 3-D component to flow The vertical flow component mean that more energy i require to get water out of the aquifer flowpath are longer 3 Variation from normal rawown hyrograph Unconfine aquifer Late time: now, mot water i coming from far away from the well; large annular area mean a mall rawown will yiel a lot of water Flow i now cloe to -D Jacob-type flow (or Thei flow) but torage i till from S y 4 Hyrology Program, Prof. J. Wilon

3 ERTH43/HYD53, NM Tech Fall 6 Variation from normal rawown hyrograph Unconfine aquifer Slope are etermine by T; intercept are etermine by S t o early S = -4 t o late S = - bae figure from Krueman an e Rier (99) 5 Variation from normal rawown hyrograph Leakage Flow through aquitar i common An aquifer receiving leakage i a emi-confine aquifer or [ leaky aquifer ] 6 Hyrology Program, Prof. J. Wilon 3

4 ERTH43/HYD53, NM Tech Fall 6 Variation from normal rawown hyrograph Leakage figure from Krueman an e Rier (99) 7 Variation from normal rawown hyrograph Partial penetration (Dawon an Itok, 99) 8 Hyrology Program, Prof. J. Wilon 4

5 ERTH43/HYD53, NM Tech Fall 6 Variation from normal rawown hyrograph Partial penetration (Krueman an e Rier, 99) 9 Variation from normal rawown hyrograph Large iameter well The pecific yiel (even in a confine aquifer, a pumping well mut be open to the atmophere) of a well bore i eentially ; thi high torage term reult in well bore torage effect the water prouce in early time i coming mainly from tore water in the well bore, not inflow to the well bore from the aquifer If T i low, S i very low, or Q i very high, even a mall well bore can be large At low T, well bore above ~ cm can how torage effect; at high T, a well might have to be over.5 m in iameter before torage effect are oberve Hyrology Program, Prof. J. Wilon 5

6 ERTH43/HYD53, NM Tech Fall 6 Variation from normal rawown hyrograph Large iameter well (Krueman an e Rier, 99) Variation from normal rawown hyrograph Bounary effect (Krueman an e Rier, 99) Hyrology Program, Prof. J. Wilon 6

7 ERTH43/HYD53, NM Tech Fall 6 Variation from normal rawown hyrograph In-well pumping tet It bet to avoi uing ata collecte in the pumping well can be ifficult to get meauring intrument in ata can be ba coul amage pump or meauring equipment permeability ajacent to the well bore can be very ifferent than in the bulk of the aquifer bentonite clay kin of low permeability gravel pack well creen 3 Variation from normal rawown hyrograph Skin effect kin or poor creen gravel pack or very well evelope well bae figure from Krueman an e Rier (99) Can get T (lope i fine), but etimate of S will be off you houl never try to etimate S from a pumping well 4 Hyrology Program, Prof. J. Wilon 7

8 ERTH43/HYD53, NM Tech Fall 6 Toay Superpoition Superpoition Image Well Theory Well Tet Analyi Charle Harvey, MIT 5 Motivation: How will the aquifer repon to pumping at a ifferent rate from the ame well? to injection at a contant rate through the ame well? if the pumping rate varie in time? If there are two or more well pumping or injecting at ifferent location? How can we ue the anwer to thee quetion to interpret well tet for moel iagnotic an parameter etimation? For example: Variable pumping rate epecially Recovery Tet Effect of aquifer bounarie Epecially lateral tream an barrier bounarie 6 Hyrology Program, Prof. J. Wilon 8

9 ERTH43/HYD53, NM Tech Fall 6 Linearity The anwer to thee an other quetion relie on the concept of linearity We uperpoe the effect of each event or well eparately, an then um their effect. Many problem of well hyraulic, an inee of grounwater hyrology in general, can be coniere eentially linear an olve via uperpoition. It a very ueful concept But houl be tete to ee if the ytem i actually behaving approximately linear. Teting linearity i not a ubject of thi cla. 7 Reminer: Linear Moel Recall the concept of an operator L( ) Given contant a & b tate (epenent variable) & then an operator L( ) i linear iff L( L( a L( a + + b ) = a L( ) ) = L( ) = a L( '( ) T ' & eg, L( ) = S ( ' ' $ r t r r % ) + L( ) ) + b L( ) '( )# ' r! " 8 Hyrology Program, Prof. J. Wilon 9

10 ERTH43/HYD53, NM Tech Fall 6 Example application of Linear Moel Rule Let & be the rawown ue to unit (e.g. m 3 /) pumping at each of two well, Let contant a & b, repectively, repreent the multiplier to fin the actual pumping rate at each well. Then the total rawown r ue to pumping rate a at welli = a ue to unit pumping at the two well i = ue to arbitrary pumping at the two well i = a + + b r Pumping well Pumping well Point of interet 9 Define i a the rawown caue by a ingle well, well inex i, ay ue to a unit pumping rate (eg, Q=Q u =m 3 /) i ometime calle the influence or repone function. Let illutrate how we ue linearity to anwer our quetionfor well hyraulic where the repone of an aquifer to pumping by a well at a contant rate i ecribe by the Thei moel, or it Cooper-Jacob logarithmic approximation. Moel Due to linearity: Drawown ue to arbitrary pumping at two well i = a + b Q Q = W ( u) + W ( u ) 4! T 4! T aqu bqu = W ( u) + W ( u ) 4! T 4! T Mot of what follow can be generalize to other aquifer/well moel; you can fin thi in your text an variou reference. where : r S r S u = ; u = 4Tt 4Tt r, r = itance to well&, repectively r r Pumping well Pumping well Point of interet Hyrology Program, Prof. J. Wilon

11 ERTH43/HYD53, NM Tech Fall 6 Anwer to quetion: How will the aquifer repon to pumping at a ifferent rate from the ame well? to injection at a contant rate through the ame well? aqu = W ( u) 4! T $ Qu = W ( u) 4! T We ll take a look at a imple verion of thi problem, tep change in pumping, not involving a convolution integral. if the pumping rate varie in time? If there are two or more well pumping or injecting at ifferent location? = 4! T r S # W ( ) t 4T ( t $ " ) Q( " ) " # t % aqu bqu = W ( u) + W ( u ) 4! T 4! T Varying pumping rate Can be hanle by uperpoing pumping an injection in our well o a to recreate the eire pumping hitory. In practice aume pumping hitory i piecewie contant, i.e. tep change in pumping rate. Suppoe, e.g.: Q =, t<, Q = Q, <t <t, pumping perio Q = Q, t <t, pumping perio where Q an Q are, repectively, contant pumping rate uring the two pumping perio. Q Q Q t t Then uperpoe rawown. Thi i mot clearly een in the imple cae of a recovery tet (next lie). t t Hyrology Program, Prof. J. Wilon

12 ERTH43/HYD53, NM Tech Fall 6 Well Tet Analyi: How can we ue the anwer to thee quetion to interpret well tet for moel iagnotic an parameter etimation? For example: Variable pumping rate epecially Recovery Tet Effect of aquifer bounarie epecially lateral tream an barrier bounarie Q What happen when we hut off a well that ha been pumping? Doe rawown recover intantly? t t? 3 Recovery Tet For a realitic recovery moel, we ue uperpoition. Preten the well keep pumping at the ame contant rate +Q, but, at the time it hut off a an imaginary well injecting water into the aquifer at contant rate Q at the ame location; ue uperpoition to a the reult. Becaue Q + -Q = for t >, we are moeling the ituation where then Q=. (Freeze an Cherry, 979) 4 Hyrology Program, Prof. J. Wilon

13 ERTH43/HYD53, NM Tech Fall 6 Recovery Tet t - Harvey, MIT 5 Recovery Tet Recovery tet can be important, a there i no problem maintaining a contant pumping rate. Uing emilog moel, the forwar moel i: Time < t Q &.5T t # = ln$! 4' T % r S " Time < t Superpoing olution: = + Q &,.5T t ),.5T t' )# = $ ln* ' - ln * ' 4. T! % + r S ( + r S (" total pumping " injection" total Q & t # = ln$! 4 ' T % t' " t: time ince pumping tarte t : time ince pumping toppe 6 Hyrology Program, Prof. J. Wilon 3

14 ERTH43/HYD53, NM Tech Fall 6 Inverion: total Q & t # = ln$! 4 ' T % t' ".3Q & t # T = log$! 4' % t' " Recovery Tet To be ue with emilog plot: T =.3 Q $ log t ' & ) =.3 Q 4 " # lc % t' ( 4 " # lc Q & t # T = ln$! 4' % t' " late time log (t/t ) When the pump i hut off, t/t =! While at very large time, t/t! early time The curve line at early time i ue to the fact that the injection well ha u >.. 7 Multiple Well If there are two or more well, ay with well number inex ( ) i an each pumping at (time varying) rate Q i then (t) = (t)+ (t) + 3 (t)+ =! i which work if each of the Q i are contant or if they are time varying Q i (t) with ifferent time hitorie. If the pumping rate are contant wrt time then the rawown i given by Q Q Q3 = W ( u) + W ( u) + W ( u3) " T 4" T 4" T ri S ui = ; toi = tart time for well i 4T ( t! t ) oi r r Pumping well Pumping well Point of interet 8 Hyrology Program, Prof. J. Wilon 4

15 ERTH43/HYD53, NM Tech Fall 6 Image Well Theory With multiple well, not all of the well nee be real. We can ue the concept of virtual or image well to uperpoe rawown in orer to mimic Dirichlet an Neumann bounary conition. -Q(t) image well Q(t) real well contant hea bounary 9 How woul repreent a traight line contant hea Dirichlet bounary locate a itance from the pumping well? If the well i pumping at rate Q(t)=Q (t), cauing rawown (t)= (t), then place a mirrior image injection well, injecting at the ame rate Q (t) = - Q (t), at a itance on the other ie of the bounary, cauing rawown (t) = - (t) [ie, with rawup (t)]. Then (t)= (t)+ (t) Often calle a recharge bounary -Q(t) image well Along the bounary, where r =r, the rawown from well i balance by the rawup from well, uch that (t)=, preerving the contant hea BC. contant hea bounary Q, &.5T t # &.5T t #) Q & r # = real + image. * ln $! - ln $!' = ln $! 4 / T + % r S " % r S "( / T % r " Q(t) real well 3 Hyrology Program, Prof. J. Wilon 5

16 ERTH43/HYD53, NM Tech Fall 6 How woul repreent a traight line contant hea Dirichlet bounary locate a itance from the pumping well? -Q(t) Q(t) image well real well contant hea bounary (Freeze an Cherry, 979) 3 How woul repreent a traight line no-flux Dirichlet bounary locate a itance from the pumping well? If the real well i pumping at rate Q(t)=Q r (t), where r = real, cauing rawown (t)= r (t), Often calle a barrier bounary then place a mirrior image pumping well, pumping at the ame rate Q i (t)= Q(t) = +Q r (t), where i = image, at a itance on the other ie of the bounary, cauing rawown i (t) Then (t)= r (t)+ i (t) Along the bounary, where r r =r i, the graient from real well i balance by the graient from image well, uch that!/!x=, preerving the no flux BC. Q(t) image well Q(t) real well Uing the emilog moel: = + Q &,.5T t ),.5T t )# - $ ln * ' + ln * '! 4. T % + r S ( + ri S (" real image no flow bounary y x 3 Hyrology Program, Prof. J. Wilon 6

17 ERTH43/HYD53, NM Tech Fall 6 How woul repreent a traight line no-flux Dirichlet bounary locate a itance from the pumping well? Q(t) Q(t) image well real well (Freeze an Cherry, 979) (Schwartz an Zhang, 3) no flow bounary y x 33 How woul repreent a traight line no-flux Dirichlet bounary locate a itance from the pumping well? image well Q(t) no flow bounary Q(t) real well (De Wiet, 965) 34 Hyrology Program, Prof. J. Wilon 7

18 ERTH43/HYD53, NM Tech Fall 6 What about other geometrie? You can ue image well in other geometric pattern to mimic more complex geometric omain (ee e.g., Bear, 97) Well near a corner : Image pumping well Real pumping well Image pumping well Image pumping well 35 Well Tet Analyi with Bounarie How can we ue imagewell theory to interpret well tet? For example: Variable pumping rate epecially Recovery Tet Effect of aquifer bounarie epecially lateral tream an barrier bounarie Previou clae: the well-tet equation mae the aumption that the aquifer i of infinite horizontal extent. We ue image well theory to hanle nearby bounarie. The image well reult can be repreente in log-log, emilog, an erivative plot an ue to iagnoe the preence of ifferent type of BC, an to etimate their parameter, uch a itance above, a well a T an S. 36 Hyrology Program, Prof. J. Wilon 8

19 ERTH43/HYD53, NM Tech Fall 6 Contant hea or recharge bounary Well Tet Analyi with Bounarie log t Influence of recharge bounary = real + image Q & r # ' ln $! ( T % r " Barrier or no-flow bounary = + Slope = m log t Influence of barrier bounary Slope = m Q &,.5T t ),.5T t )# - $ ln * ' + ln * '! 4. T % + r S ( + ri S (" real image 37 Well Tet Analyi with Bounarie How far to the bounary? We pick o that R = I R = I = Q 4"T ln.5tt R r R S Q 4"T ln.5tt I r I S log t " ln.5tt R r R S =.5Tt I r I S " r I = r R $ # t I t R % ' & " t R r R = t I r I log t 38 Hyrology Program, Prof. J. Wilon 9

20 ERTH43/HYD53, NM Tech Fall 6 Well Tet Analyi with Bounarie Where i the bounary locate? obervation well Q real pumping well obervation well 39 Hyrology Program, Prof. J. Wilon