Flow toward Pumping Well, next to river = line source = constant head boundary

Transcription

1 Flow toward Pumping Well, next to river = line source = constant head boundary Plan view River Channel after Domenico & Schwartz (1990) Line Source

2 Leonhard Euler e i" +1 = 0 wikimedia.org

3 Charles V. Theis Analogy between Heat Flow and Groundwater Flow (1935)

4 Darcy s Law q! =!K"h!!!" ' v =!Kh"h Dupuit Eq. Hydrologic Diffusion Equation Ss "h "t = K #2 h S y!h!t =!! #!x hk x "!h!x $ & +!! %!y hk!h $ # y & ='Kh'h "!y % Boussinesq Eq. Above with source or sink =A Ss "h "t = K #2 h + A A =!! #!x hk x "!h!x $ & +!! %!y hk!h $ # y & ='Kh'h "!y % Laplace s Eq: Steady State No sources or sinks " 2 h = 0 = # 2 h + # 2 h + # 2 h #x 2 #y 2 #z 2! 2 h 2 = 0 Poisson s Eq: Above with source or sink =A! 2 h = A / K! 2 h 2 = A / K

5 Potentiometric Surface? Case 1: Confined Aquifer Steady State Radial Flow Constant Pumping Rate Confined Aquifer Confined Aquifer C m

6 Well Drawdown Case 1: Confined aquifer, Constant pumping rate, Steady radial flow q v = "K#h / A = "K #h #r 2" r m 2" mk = #K $h $r r 0 dr h 0 # = $ # dh r r h for radial flow where m=aquifer thickness r m 2! T ln r 0 r = h 0! h = Drawdown Equilibrium eq., or Theim equation e.g., Fetter eq 7-38 D&S eq x

7 Confined Aquifer Steady Pumping Rate /2!T = 5 2" T ln r 0 r = h 0 # h Head, h! Radius, r Theim Eq. confined

8 Potentiometric Surface Case 1: Confined Aquifer Steady State Radial Flow Constant Pumping Rate Confined Aquifer Confined Aquifer m

9 unconfined

10 Darcy s Law q! =!K"h!!!" ' v =!Kh"h Dupuit Eq. Hydrologic Diffusion Equation Ss "h "t = K #2 h S y!h!t =!! #!x hk x "!h!x $ & +!! %!y hk!h $ # y & ='Kh'h "!y % Boussinesq Eq. Above with source or sink =A Ss "h "t = K #2 h + A A =!! #!x hk x "!h!x $ & +!! %!y hk!h $ # y & ='Kh'h "!y % Laplace s Eq: Steady State No sources or sinks " 2 h = 0 = # 2 h + # 2 h + # 2 h #x 2 #y 2 #z 2! 2 h 2 = 0 Poisson s Eq: Above with source or sink =A! 2 h = A / K! 2 h 2 = A / K

11 Well Drawdown Case 2: Unconfined aquifier, Constant pumping rate, Steady radial flow q v = "K#h / A = "K #h #r for radial flow 2" r h 2" K = #K $h $r r 0 dr h 0 # = $ # h dh r r h where m = aquifer thickness /2πr = = flow/unit width! K ln r 0 r = h 0 2! h 2 Unconfined Theim equation e.g., Fetter eq 7-39 D&S eq (Drawdown) 2

12 35 head, h Unconfined 10 5 Theim equations unconfined Radius, r

13 35 head, h Confined Unconfined 10 5 Theim equations Radius, r

14 2! T ln r 0 r = h 0! h confined! K ln r 0 r = h 0 2! h 2 unconfined Swindle: 1) 2) 3) 4)???

15 2! T ln r 0 r = h 0! h confined! K ln r 0 r = h 0 2! h 2 unconfined Swindle: 1) Singularity at r = 0 2) Steady state flux impossible without source - this problem requires annular source term 3) For unconfined case: Purely radial flow impossible Must have z dependence??? 4) Equilibrium => Steady State

16 Potentiometric Surface? Case 2: Confined Aquifer Transient Flow Radial Flow Constant Pumping Rate Confined Aquifer Confined Aquifer C m

17 Darcy s Law q! =!K"h!!!" ' v =!Kh"h Dupuit Eq. Hydrologic Diffusion Equation Ss "h "t = K #2 h S y!h!t =!! #!x hk x "!h!x $ & +!! %!y hk!h $ # y & ='Kh'h "!y % Boussinesq Eq. Above with source or sink =A Ss "h "t = K #2 h + A A =!! #!x hk x "!h!x $ & +!! %!y hk!h $ # y & ='Kh'h "!y % Laplace s Eq: Steady State No sources or sinks " 2 h = 0 = # 2 h + # 2 h + # 2 h #x 2 #y 2 #z 2! 2 h 2 = 0 Poisson s Eq: Above with source or sink =A! 2 h = A / K! 2 h 2 = A / K

18 Diffusion Equation " 2 h = Ss K #h #t Cartesian Coordinates! 2 h!x 2 +!2 h!y 2 +!2 h!z 2 = S s K! h!t Cylindrical Coordinates 1 r!!r r!h!r + 1 r 2! 2 h!" 2 +!2 h!z 2 = S s K! h!t Cylindrical Coordinates, Radial Symmetry h/ φ = 0 K r! 2 h!r 2 + K r r!h!r + K z!2 h!z 2 = S s!h!t Cylindrical Coordinates, Purely Radial Flow h/ φ = 0 h/ z = 0! 2 h!r 2 + 1!h r!r = S s K r!h!t = S T!h!t

19 Radial flow " 2 h "h "h " r r "r = 1 D "t Transient flow, Confined Aquifer, No recharge Constant pumping rate Initial Condition & Boundary conditions: % h(r, 0) = h 0 h(",t ) = h 0 lim r $h ( ' * = r#0 & $r ) 2+T for t > 0 Solution: Problem set 2: Theis equation or Non-equilibrium Eq.

20 Radial flow " 2 h "h "h " r r "r = 1 D "t Initial Condition & Boundary conditions: % h(r, 0) = h 0 h(", t ) = h 0 lim r $h ( ' * = r#0 & $r ) 2+T for t > 0 Solution: Theis equation or Non-equilibrium Eq. Drawdown = h 0 " h = 4#T W (u) where W (u) = "Ei("u) = $ % e"# r 2 S d# where u = u # 4tT = r 2 4Dt and where W (") = 0 W( 0) = "

21 Theis Eq.: Drawdown = h 0 " h = Approximation for t >> 0 Drawdown = h 0 " h # 4$T 4#T W (u) % 2.25 D t ( ln' * & ) W (u) = "Ei("u) = " " ln u + u " u2 4 + u 3 3# 3! " u4 4 # 4! + u5 5 # 5! "... W (u ) " # # lnu for small u < 0.1 ; i.e., long times or small r where u = r 2 /4Dt r 2 D&S p. 151 = -ln (1.7811) -ln u = -ln ( u) $ = -ln r2 S ' $ & ) = ln& % 4 T t ( % 2.25 Dt r 2 ' ) (

22 -2 0 approx apx t=1 Drawdown =0.1 m 3 /s t=100 t=10 T =0.015 m 2 /s 8 S =0.006 D=25 m 2 /s Confined radius, r

23 Pumping of Confined Aquifer Not GW level Potentiometric sfc! USGS Circ 1186

24 Pumping of Unconfined Aquifer USGS Circ 1186

25 8 6 W(u) 4 2 Well Function W(u) = - Ei (-u) 0-2 W(u) ~ ln(u) OK for u < u

26 5 4 "Well Function" W(u) W(u) ln(u) + u -u 2 / ln(u) + u 2 -/4 u + u 3 /18 - u 4 /96 + u 5 /600 + u 3 /18 -u 4 /96 +u 5 / ln(u) u W(u)

27 Potentiometric Surface Pumping Well Observation Well Confined Aquifer m Time-Drawdown Method

28 Theis Eq. Drawdown = h 0 " h = 4#T W (u) for u < 0.1 Drawdown = h 0 " h # 4$T % 2.25 D t ( ln' * & ) r 2 Time Drawdown Method: Use the approximation to calculate T from the head drop in a single observation well at two times, t 1 and t 2 h 0 " h 1 # h 0 " h 2 # 4$T ln % 2.25 D t ( 1 ' * & r 2 ) 4$T ln % 2.25 D t 2 ' & r 2 ( * ) Subtract: h 2 " h 1 # 4$T ln % t 1 ' & t 2 ( * for u < 0.1 ) t >> 0 and/or small r

29 20 15 Well Function W(u) y = x R= 1 Drawdown, m Assumed values: = 0.1 m 3 /s T = m 2 /s S = D = 25 m 2 /s 2.303" slope = 4#T so T = 1295 m 2 /d = m 2 /s t, Log T, days

30 Potentiometric Surface Pumping Well Observation Well #1 Observation Well #2 Confined Aquifer m Distance-Drawdown Method

31 Theis Eq. Drawdown = h 0 " h = 4#T W (u) for t >> 0 Drawdown = h 0 " h # 4$T % 2.25 D t ( ln' * & ) r 2 Distance Drawdown Method: Use above to calculate T from head the difference between two different observation wells located at r 1 and r 2 Subtract: h 0 " h 1 # h 0 " h 2 # h 2 " h 1 # 4$T 4$T 2$T ln % r ( 2 ' * & ) % 2.25 D t ( ln' 2 * & r 1 ) % 2.25 D t ( ln' 2 * & r 2 ) r 1 for t >> 0 Steady state approximated => Theim eq (confined)

32 Swindle? 1) Singularity at r = 0? 2) Steady state flux impossible without source - this problem requires annular source term 3) Purely radial flow impossible for unconfined case Steady state?!!

33 Swindle? 1) Singularity at r = 0? OK, required condition " lim r!h % r!0 $ ' = #!r & 2"T for t>0 2) Steady state flux impossible without source - this problem requires annular source term 3) Purely radial flow impossible for unconfined case Steady state?!! Steady Shape

34 -2 0 approx t=1 2 t=10 Drawdown 4 6 =0.1 m 3 /s t=100 T =0.015 m 2 /s 8 S =0.006 D=25 m 2 /s Confined radius, r