Lecture Packet 7.Transient Systems and Groundwater Storage

(1)

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

∂ρ

n

₌

∂ρ

∂

n

+

ρ

∂

t

∂

t

∂

t

Densit y changes ( com pressibilit y)

Porosit y changes

Overall definit ion: Ss, 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}

surface

b

Unit decline in pot ent iom et ric

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 t he individual grains or rock skelet on ( assum ed negligible – individual grains are incom pressible)

(2)

• We w ill get Ss = ρgα + ρgβn

St ress at any dept h is due t o:

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

σT = 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.

σT = σT + P Æ effect ive st ress ( borne by rock) + fluid pressure

When pum ping an aquifer t he change in st ress is: d σT = dσT + 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 σT = dσT + dP = 0

dσT = - 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.

At a point , P = ρgh - ρgz = ρg( h – z) - - - at a point , z is const ant dP = dρgh and subst it ut ing ( dρgz = 0 = derivat ive of a const ant ) dσe = - dP = - dρgh

I f pum ping increases, head goes down, and t he effect ive st ress goes up. Consider w hat happens w hen σe goes up.

1.72, Groundw at er Hydrology Lect ure Packet 7

(3)

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

• Aquifer com pressibilit y, α, [ L2_{/ M] is defined as follow s ( corresponds t o shift ing}

of grains and reduct ion in porosit y.

−

(

dV

_t

) /

V

_t

Aquifer get s sm aller w it h increase in effect ive st ress. Consider a unit volum e Vt = 1. We can also define fluid com pressibilit y

(4)

(5)

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

This gives rise t o a specific st orage coefficient w it h t w o t erm s and a funct ion of Mat erial Com pressibilit y [ m / N or 1/ Pa]

Clay

Sound Rock

Where does st ored wat er com e from in confined aquifers?

10 ft of draw down ( reduct ion in head)

Wit h m ore realist ic porosit ies, result s are sim ilar Porosit y Com pression

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

(6)

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

• 3D flow equat ion, Hom ogeneous I sot ropic

2

∂

h

⎡∂

h

∂

2

h

∂

2

h

⎤

∂

h

S

s

+

or

S

s

=

K

∇

2

h

∂

t

=

K

⎢

_⎣

dx

2

dy

2

dz

2

⎥

_⎦

∂

t

• 3D Het erogeneous, Anisot ropic

∂

h

⎡ ∂

∂

h

∂

h

∂

h

⎤

S

s

=

x

+

K

y

K

z

∂

t

_⎣

⎢

dx

K

dx

dy

+

dz

_⎦

⎥

• 3D Het erogeneous, I sot ropic

∂

h

⎡ ∂ ⎛

∂

h

⎞

∂ ⎛ ∂

h

⎞

∂ ⎛ ∂

h

⎞⎤

s

⎜

K

⎟⎥

S

∂

t

=

⎢

_⎣

dx

⎜

⎝

K

dx

⎟

⎠

+

dy

⎜

_⎝

K

dy

⎟⎟

_⎠

+

dz

⎝

dz

⎠⎦

∂

h

For St e a dy St a t e ,

S

s

=

0

for any equat ion.

∂

t

For st eady st at e:

• No wat er from st orage

• S values doesn’t m at t er

1.72, Groundw at er Hydrology Lect ure Packet 7

(7)

2 D Flow Equ a t ion

• Confined aquifer, hom ogeneous, isot ropic.

2

∂

h

⎡∂

h

∂

2

h

⎤

S

+

∂

t

=

T

⎢

_⎣

dx

2

dy

2

⎥

_⎦

S = Ssb = St orat ivit y [ L3/ L3] Å Values like 10- 2 t o 10- 6

T = Kb = Transm issivit y [ L2/ T] Com pare values t o Sy !

B = aquifer t hickness [ L]

• Confined aquifer, Het erogeneous, Anisot ropic

∂

h

⎡ ∂

∂

h

∂

h

⎤

S

=

x

+

T

y

∂

t

_⎣

⎢

dx

T

dx

dy

_⎦

⎥

• Confined aquifer, Het erogeneous, I sot ropic

∂

h

⎡ ∂ ⎛

∂

h

⎞

∂ ⎛ ∂

h

⎞⎤

S

∂

t

=

_⎣

⎢

dx

⎝

⎜

T

dx

⎟

⎠

+

dy

_⎝

⎜

T

dy

_⎠

⎟⎟

_⎦

⎥

• Confined aquifer, Hom ogeneous, Anisot ropic

∂

h

⎡ ∂

2

h

∂

2

h

⎤

S