6. Groundwater flow equation
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1 6-6. Groundater flo euation Mass conservation: M in M out = M/t Darc s la M- relationsip No e can put tem all togeter. n auifer aving a cross sectional area. Top and bottom are confined b auitards, and one end is connected to a lake. = = = + = M M out in ) ( ) ( )] ( ) ( [ ( ) ( ) = Darc s la: = = In general, and are functions of (,, z, t). In tis case, and depends on (, t) onl. = - Ho is te ater level in ells related to te lake-ater level?
2 Mass cange in a REV aving volume V =. M = V S s ψ from E.[4-] t a fied location, = (z + ψ) = z + ψ = + ψ = ψ M = V S s = S s Putting tem all togeter, Making t ver small, M in M out M = t = = S t S s s [6-] t [6-a] 6- If is constant (i.e. material is uniform), e ave: = S Tis is te one-dimensional groundater flo euation. Note tat [6-] and [6-] represent eactl te same ting. Partial differential euations are noting more tan a language to describe te simple conservation principle. t We can solve [6-] in a number of as. Te net eample sos a simple numerical tecniue to solve [6-b]. s [6-b]
3 Propagation of a draulic pulse 6-3 z = Water level in to lakes as been kept at z = for a long time. Te ater level in te left lake suddenl rises at t =. We divide te auifer into a number of cells and analze te mass balance of eac cell. cell- cell- cell-3 For eample, for cell-: M in M t M out = V = M M 3 = = S s t = S t M M t+ t+ t s + 3 = S t = + + t + t S ( ) s s t+ t 3 ) [6-3]
4 E. [6-3] indicates tat te future state of cell- is dependent on te present state of itself and its neigbors. We can rite [6-3] for all cells and solve tem simultaneousl using a spread seet (lab eercise). 6-4 t = t E. [6-3] is called forard finite difference euation, ic is just anoter a of riting te mass balance euation [6-] and te partial differential euation [6-]. ll of tese give us te increment of storage. To calculate te storage amount itself, e need to specif: Initial condition: = for all > at t = Te to end cells ( and ) onl ave one neigbor, and e cannot appl [6-3]; i.e. tere is no - or. One a to deal it tis problem is to specif: Boundar conditions: = at =, = at = In tis case, e force te boundar cells to take a specified value, ic reflects te psical constraint on te sstem. Tis is called specified ead or st tpe boundar condition.
5 Net, suppose tat te auifer is plugged at =. Te mass balance euation for cell- is: M t = M 9, Terefore, e do not ave to include in our calculation. Tis is anoter a of imposing a psical constraint on te sstem, and called specified flu or nd tpe boundar condition. We can also rite tis as: = at = 6-5 Solution it st tpe boundar conditions (BC s) Tis graps so te solution of [6-b] it te st tpe BC s; = m at = and = at = m.8 (m).6.4 t = min. t = 3 min (m) s t, te grap becomes a straigt line. () Te mass storage does not cange an more. () Specific discarge is constant it respect to bot t and.
6 Solution it st and nd tpe BC s 6-6 Suppose e ave te folloing BC s to go it [6-b]. = m at = and / = at = m.8 (m) (m) t stead state, te solution looks different from te last one. Differential euation Governing processes Boundar conditions Psical constraints Stead-state solution S/t = at stead state, ic means /t = and is a function of onl. d d E.[6-b] is no ritten as: = or = d d It is eas to so tat te stead-state solution is given b: = C + C ere C and C are constants tat are dependent on BC s.
7 Effects of draulic conductivit 6-7 et s go back to te eample it st tpe BC s. Te grap sos te () profile at t = min. In tis case, = -5 m/s. If = -6 m/s, o ould te profile look like at t = min? Ho about = -4 m/s? = = (m) (m) Suppose te middle part of auifer is filled it a material aving a smaller, sa = /. t stead state, o ill te () profile look like? = = (m) (m)
8 Bulk draulic conductivit Water is floing troug a bo at Q (m 3 /s). We are not given te information of te material. We use Darc s la to assign te bulk draulic conductivit ( b ) of te bo. Q = = b b out Q = in = b rea Q = in = out b is dependent on te propert of sediments and o te are arranged in te bo. et s eamine to important cases. Q 6-8 () Parallel laers Q = b = + + Q 3 3 b is given b a eigted aritmetic average of eac material. Te laer tickness serves as a eigting factor. Ticker laers ave a eavier influence on b tan tinner laers.
9 () Serial laers In tis case, e need to ave te same value of in all laers. = b = = = + = Head drop in eac laer must add up to te total ead drop b is given b a eigted armonic average of eac material. 6-9 Eample Suppose = = 3 = /3 = = 3 = /3 = 3, =. Wat is te b for parallel and serial cases?
10 nisotrop of draulic conductivit 6- In laered sediments, te value of b varies it direction. Tis is called te anisotrop of draulic conductivit. Note tat eterogeneit at a small scale appears as anisotrop at a larger scale. Darc s la in anisotropic material is given b: = = = Majorit of sallo unconsolidated sediments ave orizontal strata, and teir draulic conductivit is iger in orizontal direction ( and ) tan in vertical direction ( z ). z z z
11 Similarit beteen Darc s and Om s a 6- We sa tat te b is te aritmetic average for te parallel case, and te armonic average for te serial case. Does tis remind us of someting? V V R R R I = V/R R= /( E ) E : electrical conductivit R Electricit and groundater obe te same form of euation. Darc s la Om s la Fourier s la = d d i = E dv d i : current densit H = H dt d H : eat flu densit as like Darc s la are called constituitive euations or penomenological las. In psical sciences, e combine tem it te conservation euation: Q in Q out S = t Tis concept is te foundation of man scientific disciplines.
12 6- To-dimensional flo euation Suppose a confined auifer aving a constant tickness (b). We can analze te mass balance of a bo in tis auifer in a a similar to te analsis of one-dimensional flo. We sa tat M in - M out in -direction is given b (see p.6-): Similarl, M in - M out in -direction is given b: [ ] = = b M M out in [ ] = = b M M out in = - = - b Storage cange is (see p.6-): M = V bo S s = b S s Putting tem all togeter into M in - M out = M/t, t S b b b s = + [M in M out ] [M in M out ] M/t
13 6-3 Dividing bot sides b b + = S s t [6-4] Tis is te -D groundater flo euation. t stead state, E. [6-4] becomes: + = [6-5] Grapical solution of -D flo euation Suppose a to-dimensional, confined auifer of omogeneous, isotropic sediments. Nort and sout boundaries of te auifer are impermeable, and east and est boundaries are connected to rivers. Water levels in rivers are constant. Ho does draulic ead cange from te est to east? 5 m Cross section Plan vie = 5 m = m m
14 Observe: () Flo lines are normal to contour lines. () Contours meets te impermeable boundaries at 9º. Tese are common features of te stead-state flo in isotropic auifers. Water level in groundater ells indicate draulic ead in te confined auifer. Joining te ell ater levels, e can define an imaginar surface of draulic ead. Te contours in te flo diagram above so te sape of te surface.
15 Euipotential and flo line Te potentiometric surface is an imaginar surface defined b te ater level in ells (draulic ead, ) in a single auifer. ike an oter surface, e can dra contours of constant, called euipotentials. potentiometric surface sand impermeable cla From Darc s la: =, = In vector form, e can rite, (, ) = (- /, - /) 8 78 = 8 m If te material is isotropic ( = ), e can just use and (, ) = (-/, -/) i.e. te flo direction is parallel to te gradient vector. Wat does tis vector represent? = 75 m
16 Groundater flo direction is is normal to euipotentials en =. 6-6 Hdraulic conductivit of anisotropic materials varies it direction, and flo lines ma not be normal to euipotentials.
17 Flonet construction Cross section 6-7 et s go back to te analsis of te confined auifer. Tis time it as someat irregular sape. Hdraulic ead is m in te est river and 7 m in te east river. Plan vie Can e dra euipotentials? Remember te flo is at stead state. Is constant trougout te auifer? Q in Q out 5 m impermeable = m = 7 m impermeable m
18 Euipotentials and are son belo. Note tat te meet te impermeable boundar at 9º. W? 6-8 Can e dra a fe flo lines? () Flo lines are normal to euipotentials. () Spacing beteen flo lines? l b b l Te strip beteen to flo lines is called a flo tube. It is customar to dra flo lines so tat eac flo tube as te same Q. Suppose a orizontal laer in te auifer aving a tickness (normal to te page) of m. Te flo rate troug a -m tick tube is given b: Q i = ) d b i ( b i = ere =.5 m dl li If e cose te spacing so tat b i = l i, ten Q in eac tube is eual to. Flo nets constructed tis a provide a useful tool for te analsis of stead-state flo in omogeneous and isotropic materials.
19 bove is an eample of a properl constructed flo net. Suppose tat = -4 m/s. We can sa tat: () Tere are four and alf flo tubes. () Eac tube carries Q =.5-4 m 3 /s. (3) -m tick laer as a total flo rate of. -4 m 3 /s. (4) If te auifer tickness is 5 m, te total flo rate ill be m 3 /s. Effects of eterogeneit Can e construct a flo net for tis case? 5 m = -3 m/s = 5 m =.3-3 m/s = m = -3 m/s m
20 4 Flo lines and euipotentials are still normal to eac oter. Hoever, eterogeneit creates some features tat ere not seen in te omogeneous case. () Flo lines refract at te zone boundaries. () Euipotentials are muc denser in te lo- zone Groundater takes te pats of least resistance b taking te sortest pat troug te lo- material. Te sortest pat troug te lo- zone is acieved b floing straigt across te zone. Ho ill te flo net look like in tis case? 5 m = -3 m/s = 5 m = 5-3 m/s = m m
21 In tis case, te pats of least resistance are acieved b canneling te flo troug te ig- zone < Tangent la (optional) Note tat, at stead state, te flo volume (Q = ) crossing te interface is eual on bot sides. θ + et s sa and are te specific discarges along flo directions. From mass balance principle, = θ = ere is te ead difference beteen te to contours. Note tat contours are continuous across te interface. = / / = tanθ tanθ is greater tan, ic means θ must be greater tan θ. Te flo lines refract.
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