Lecture Packet 7.Transient Systems and Groundwater Storage

  1.72, Groundwat er Hydrology Prof. Charles Harvey

  Le ct u r e Pa ck e t # 7 : Tr a n sie n t Syst e m s a nd Gr oun dw a t e r St or a ge H ow doe s t h e m a ss of w a t e r st or e d in a n a qu ife r ch a n ge ?

  Confined Aquifer – Com pression of bot h t he m at erial and t he wat er Unconfined Aquifer – Wat er drains out pores at t he wat er t able when t he wat er t able drops and fills pores w hen t he w at er t able rises. The change in st ore from com pression is negligible. Ret urn t o t he t im e deriv at ive or change in st orage t erm , w hich m ay be w rit t en as

  ∂ n ∂ ∂n ρ ρ

• = n ρ ∂t ∂t ∂t

  Densit y changes Porosit y changes ( com pressibilit y)

  Overall definit ion: S s , spe cific st or a ge , ( 1/ L) of a sat urat ed aquifer is t he volum e t hat a unit volum e of aquifer released from st orage for a unit decline in head. ( Volum e per Volum e per head change) .

  Confin e d Aqu ife r s

  Pot ent iom et ric Unit Cross

  Surface Sect ion

  Unit decline in pot ent iom et ric surface b

  For a confined aquifer we m ust consider com pr e ssibilit y . How can a reduct ion in aquifer volum e occur?

• Com pression of the individual grains or rock skeleton (assum ed negligible – individual grains are incom pressible)
• Rearrangem ent of the grains – m ore com pact
• Com pression of the water in the pores

• We will get S ρgα + ρgβn _{s =}

  St ress at any dept h is due t o:

  Tot a l st r e ss act ing downward on a plane

  σ _{T = w eight of rock and w eight of w at er} Som e of t he st ress is borne by rock skelet on; som e by wat er. σ = σ + P Æ effect ive st ress ( borne by rock) + fluid pressure _{T T} When pum ping an aquifer t he change in st ress is:

  σ σ d T = d T + dP Tot al St ress

  Fluid Pressure Effect ive St ress But t he w eight of overlying w at er and rock is essent ially const ant over t im e. So t he change in t ot al st ress is zero. d σ T = d σ T + dP = 0

  σ d T = - dP Fluid pressure decreases, t he st ress on t he grains becom es great er ( im agine we t ook away t he fluid) .

  Fluid pressure cont rols t he volum et ric deform at ion.

  ρgh - ρgz = ρg(h – z) --- at a point, z is constant At a point , P = dP = d ρgh and substituting (dρgz = 0 = derivative of a constant)

  σ ρgh d e = - dP = - d I f pum ping increases, head goes down, and t he effect ive st ress goes up. Consider w hat happens w hen σ e goes up.

  W a t e r Pr odu ce d fr om Aqu ife r Com pa ct ion ₂

• Aquifer com pressibility, α, [ L / M] is defined as follow s ( corresponds t o shift ing of grains and reduct ion in porosit y.

  − (dV ) /V _{t t} α

  = σ d _e

  V t is t he original volum e ( t hickness) dV t is t he change in volum e ( t hickness) σ d e is t he change in effect ive st ress

  Aquifer get s sm aller w it h increase in effect ive st ress. Consider a unit volum e V t = 1. αdσ _{e = - dV t} Volum e of wat er pum ped t hat com es from aquifer com pact ion: We also know t hat t he change in volum e of wat er produced by aquifer com pact ion m ust equal t he volum e of wat er and sand account ing for t he reduct ion. dV w = - dV t subst it ut ion for t he t erm on t he right dV w = αdσ e = - α(dρgh) = -αρg(dh) For a unit decrease in head due t o pum ping dh = - 1

  αρg dV w = because t his is for a unit decrease in head, t his is a com ponent of our S s , specific st orage.

  W a t e r Pr odu ce d by t h e Ex pa n sion of W a t e r

  We can also define fluid com pressibilit y

  − (dV ) /V _{W W} β = dP

  β is the fluid com pressibility - com pressibility of water n is t he porosit y ( volum e of w at er is t ot al volum e t im es n) so, for a unit t ot al volum e, nV t = n( 1) = n dV = - βndP = -βn(dρgh – dρgz) = -βnρgdh _W

  For a unit decline in head, dh = - 1, dV w = + βnρg which is anot her com ponent of S s

  Specific St orage is sum : ρ g(α + β n) _{- 1} S s = I t has unit s [ L ] . I t is volum e produced per aquifer volum e per head decline.

• First term is from aquifer com pressibility.
• Second term is from water expansion. Let ’s ret urn t o flow equat ion ( t im e derivat ive t erm ) :

  n ∂ ρ ∂ ρ ∂ n

• = n ρ ∂ t ∂ t ∂ t

  The t ot al change in m ass wit h t im e is due t o • A change in density with tim e term associated with water com pressibility.

• A change in porosity with tim e term associated with the com pressibility of the aquifer.

  Our analysis of “ where wat er from st orage com es from ” gives a t erm involving ρ g(α + β n) to replace the change in m ass with tim e term above. S s = Going back t o t he cont inuit y equat ion:

  ∂ ρ q ρ q [ ] ∂ ρ q ⎤ ∂ ρ n ⎡∂ [ ]

  y [ ] x z − = ⎢ ⎥

  ∂ t dy dz ⎣ dx ⎦

  Rat e of Change of Mass st ored:

  q ∂ ρ ∂ n ∂ ∂ n ⎡∂ ρ q [ y ] ∂ q ⎤

  ρ ρ [ ] [ ρ ] x z

  = n ρ − + + + = ⎢ ⎥ dy dz

  ∂ t ∂ t ∂ t ⎣ dx ⎦

  Subst it ut ing in for Darcy’s law , pulling out K ( hom ogeneous, opt ional) , and subst it ut ing for t he st orage t erm s, we obt ain: _{2 2 2}

  ∂ h ∂ h ⎡∂ h ∂ h ∂ h ⎤ = + + n g = ρ S ρ β ρ s _{2 2 2}

  ρ K ⎢ ⎥ ∂ t ∂ t dy dz

  ⎣ dx ⎦

Not e st orage t erm m ust convert a volum e produced t o a m ass produced, so m ult iply ρ

  by Cancel first ρ , giving our final confined flow equation as: _{2 2 2}

  ∂ h ∂ h ⎡∂ h ∂ h ∂ h ⎤ S = ρ (β α ) + + + g n s ^{2 2 2}

  ⎥ = K ⎢ ∂ t ∂ t dy dz ⎣ dx ⎦

  Sum m a r y of St or a ge M e ch a n ism s – Con fin e d Aqu ife r

  Wat er is released from st orage during a decrease in h by t wo m echanism s:

• Com pact ion of aquifer caused by increasing effect ive st ress and rearrangem ent of grains.
• Expansion of w at er caused by decreasing fluid pressure.

  α and This gives rise t o a specific st orage coefficient w it h t w o t erm s and a funct ion of β. ₂ Mat erial Com pressibilit y [ m / N or 1/ Pa] _{- 6 - 8}

  Clay 10 – 10 _{- 7 - 9} Sand 10 – 10 _{- 8 - 10} Gravel 10 – 10 _{- 9 - 11} Sound Rock 10 – 10 _{- 10} Wat er 4.4 x 10

  Where does st ored wat er com e from in confined aquifers? ρg(α + βn)

  Assum e: S s =

• 10 ft of draw down ( reduct ion in head)
• Porosit y of 30%
₂ Com pressibilit y ( m / N) Wat er from St orage, S s x ∆H _{- 6}

  Clay 10 0.0303 _{- 8} Sand 10 0.000694 _{- 10} Granit e 10 0.000398 _{- 10} Wat er 4.4 x 10

  e 100 g ra to

  80 From expansion of

  s m

  wat er

  o fr r

  60

  te

  From m edia

  a

  com pression

  w f

  40

  o t n e

  20

  rc e P

  clay gravel granit e Wit h m ore realist ic porosit ies, result s are sim ilar

  Porosit y Com pression Expansion Tot al Wat er Clay 45% 98.06% 1.94% 0.0305 Gravel 35% 39.37% 60.63% 0.00759 Granit e 25% 0.90% 99.10% 0.00332

• 3D flow equation, Hom ogeneous I sotropic
2<
• 3D Heterogeneous, Anisotropic
• dz dz
• 3D Heterogeneous, I sotropic
• dy ⎜
• dz

  = 0 for any equat ion. ∂ t

  For St e a dy St a t e , S _s

  ⎝ dz ⎠⎦ ⎝ ∂ h

  ⎟ ⎠

  ⎜ K dy ⎟

  ⎟ ⎠

  ⎝ K dx

  ∂ t = ⎢ ⎣ dx ⎜

  ⎦ ⎥

  ∂ h ⎡ ∂ ⎛ ∂ h ⎞ ∂ ⎛ ∂ h ⎞ ∂ ⎛ ∂ h ⎞⎤ _{s ⎜ K ⎟⎥} S

  ∂ t ⎣⎢ dx K dx dy dy

  ∂ h ⎡ ∂ ∂ h ∂ ∂ h ∂ ∂ h⎤ S = + K K _{s x y z}

  ⎥ ⎦ ∂ t

  ∂ t = K ⎢ ⎣ dx ² dy ² dz ²

  or S = h K ∇ ² _{s s}

  ∂ h ⎡∂ h ∂ ² h ∂ ² h ⎤ ∂ h S

  Ke y Equa t ion s for a con fin e d a qu ife r

  For st eady st at e:

• No water from storage
• S values doesn’t m atter

  _{2 2}

  ⎡∂ h ∂ ∂ h h ⎤

• S
_{2 2} = T ⎢ ⎥ ∂ t dy

  ⎣ dx ⎦ _{3 3 - 2 - 6}

  S = S s b = St orat ivit y [ L / L ] Å Values like 10 t o 10 ₂ T = Kb = Transm issivit y [ L / T] Com pare values t o S y ! B = aquifer t hickness [ L]

• Confined aquifer, Heterogeneous, Anisotropic

  ∂ h ⎡ ∂ ∂ h ∂ ∂ h⎤ S = T + T _{x y}

  ⎥ dx dx dy dy

  ∂ t ⎣⎢ ⎦

• Confined aquifer, Heterogeneous, I sotropic

  ∂ h ⎡ ∂ ⎛ ∂ h ⎞ ∂ ⎛ ∂ h ⎞⎤ S = ⎜T

  ⎜ T ⎟ + ⎢ ⎥ ⎜ ⎟⎟ ∂ t dx dx dy dy

  ⎝ ⎠ ⎝ ⎠ ⎣ ⎦

• Confined aquifer, Hom ogeneous, Anisotropic
_{2 2}

  ∂ h ⎡ ∂ h ∂ h ⎤ S T _{x y 2} + T ₂ = ⎢ ⎥

  ∂ t dx dy ⎣ ⎦