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ALIRAN AIR TANAHALIRAN AIR TANAH

MACAM ALIRAN AIRTANAH.

1. Aliran laminer; aliran yang partikel-partikel airnya bergerak sejajar dengan kecepatan relatif lambat.2. Aliran turbulent, aliran yang yang partikel-partikel airnya bergerak secara berputar (bergolak), biasanya mempunyai kecepatan yang besar.

Aliran laminer:

1. Aliran tetap ( Steady Flow ), aliran tidak berubah karena waktu2. Aliran tidak tetap ( unsteady flow), aliran yg berubah karena waktu.

Kecepatan aliran airtanah tergantung pada Gravitasi (landaian hidrolik) dan friksi (gesekan).

Grafitasi akan memdorong airtanah bergerak dari tempat yang tinggi ke Tempat yang rendah. Besarnya dinyatakan sebagai Landaian Hidrolik.

Landaian Hidrolik;

i = dh / dl

Friksi (gesekan ) sebagai penghambat lajunya aliran airtanah. - Gesekan Dalam tergantung pada kekentalan air, suhu air, semakin kental semakin lambat alirannya.- Gesekan luar tergantung pada partikel-partikelnya. Pada batuan yang berbutir halus akan mempunyai permukaan luas sehingga banyak air yang menempel atau melekat pada bitran(adhesi), maka gesekan luar semakin besar akibatnya aliran menjadi lambat.

Dengan demikian aliran airtanah tergantung pada landaian hidrolik dan Kesarangan efektif atau kelulusan air.

Liran airtanah dalam akuifer (media berpori) akan mengikuti hukum Darcy

Hydraulic head

hp

z

hh = hydraulic head z = elevation headhp= pressure head

p

p

hzh

ghP

g

Pzh

A

z(A)

Datum

Definisi hydraulic head pada sebuah titik)(

)()( Az

APAh

w

(1)

z diukur vertikal ke atas Terhadap bidang datum

1. Calculation of head at A

Choose datum at the top of the impermeable layer

2 m

5 mX

A

Impermeable stratum1 m

1m

Example: Static water table

z (A)

thus

h A) mw

w

(

1

41 5

P w(A) 4

The heads at A and X are identical does this imply that the head is constant throughout the region below a static water table?

2. Calculation of head at X

Choose datum at the top of the impermeable layer


2 m

5 mX

A


1m

P X

z X

thus

h X m

w

w

w

( )

( )

( )

4

4 5

2 m

5 mX

A


1m

3. Calculation of head at A

Choose datum at the water table


P (A)

z A

thus

h (A) m

w

w

w

( )

=

= -

= - =

4

4

44 0

Again, the head at P and X is identical, but the value is different

2 m

5 mX

A


1m

4. Calculation of head at X

Choose datum at the water table


P (X)z X

thus

h X m

w

w

w

( )

( )

1

1

1 0

• The value of the head depends on the choice of datum

• Differences in head are required for flow (not pressure)

2 m

5 mX

A


1m

It can be helpful to consider imaginary standpipes placed in the soil at the points where the head is required

The head is the elevation of the water level in the standpipe above the datum

Head

Head in water of variable density

Point-water head Fresh-water head

hp

hf

zP1 P2

pf

pf

ff

pp

hh

ghP

ghP

2

1

Darcy found that the flow (volume per unit time) was

proportional to the head difference h

proportional to the cross-sectional area A

inversely proportional to the length of sample L

Water flow through soil

h

Soil Sample

Darcy’s Experiment The first systematic study of the movement water through a porous

medium is made Henry Darcy.

The discharge (Q) is proportional to the difference in the height of the water (hydraulic head , h) between the end and inversely proportional to the flow length (L).

The flow is obviously proportional to the cross sectional area of the pipe. When combined with the proportional constant, K, the result is the expression known as Darcy’s law

ha hb

L Q

dl

dhKA

L

hhKAQ BA

Hydraulic conductivity

Hydraulic conductivity may be referred to as the coefficient of permeability;

= specific weight, = dynamic viscosity.

g

kkK

LLL

TLK

dLdhA

QK

/

)/(

/ 2

3

Manometers

L

inlet

outlet

H

constant headdevice

device for flow measurement

load

porous disk

Fig. 4 Constant Head Permeameter

sample

Measurement of permeability

The volume discharge X during a suitable time interval T is collected.

The difference in head H over a length L is measured by means of manometers.

Knowing the cross-sectional area A, Darcy’s law gives

It can be seen that in a constant head permeameter::

(3)

Constant head permeameter

V

Tk A

H

L

kV L

A H T

H2

H1H

L

Fig. 5 Falling Head Permeameter

Standpipe ofcross-sectionalarea a

Sample of area A

porous disk

Measurement of permeability

H2

H1

H

L

Standpipeof area

a

Sample of area

A

Solution

adH

dtkA

H

L(4a)

Equation (4a) has the solution:

a n HkA

Lt cons t ( ) tan (4b)

Initially H=H1 at time t=t1Finally H=H2 at time t=t2.

kaL

A

n H H

t t

( / )1 2

2 1(4c)

Falling head permeameter

10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9 10-10 10-11 10-12

Gravels Sands Silts Homogeneous ClaysFissured & Weathered Clays

Typical Permeability Ranges (metres/second)

Typical permeability values

Soils exhibit a wide range of permeabilities and while particle size may vary by about 3-4 orders of magnitude permeability may vary by about 10 orders of magnitude.

z

xx

z

A

B C

O

For horizontal flow v=vx

and k=kH and thus

Definition of Hydraulic Gradients

x

hkv

x

BhChi

ikv

Hx

x

xHx

jadi

)()(

dimana

z

xx

z

A

B C

O

Definition of Hydraulic Gradients

z

hkv

z

BhAhi

ikv

vz

z

zzz

jadi

)()(

dimana

For vertical flow v=vz

and k=kV and thus

Gradient of the potentiometric surface

70

80

90

63

92

81

Aquifer characteristics Transmissivity(T) is a measure of the amount of water that can be transmitted horizo

ntally through a unit width of the full saturated thickness of the aquifer under a hydraulic gradient of 1.

T = Kb. K= hydraulic conductivity, b = saturated thickness of the aquifer

Storativity (S) or coefficient of storage; is the volume of water that a permeable unit will absorb or expel from storage per unit surface are per unit change in head.

Specific storage (Ss) is the amount of water per unit volume of a saturated formation that is stored or expelled from storage owing the compressibility of the mineral skeleton and the pore water per unit change in head. Jacob expression ;

ngS ws w = the density of the water (ML-3),g = the acceleration of gravity (LT-2), = the compressibility of the aquifer skeleton (1/(M/LT2)), = the compressibility of the water (1/(M/LT2)),n = the porosity (L3/L3)

Impermeable bedrock

aquifer Flow

z

x

A

B

C

D

x

z

vz

vxSoilElement

Flow into a soil element

yxAvCvzyDvBv zzxx ))()(())()((Netflow 　

Untuk aliran tunak, netflow menjadi nol;

0

z

v

x

v zx

Continuity Equation

v

x

v

zx z 0

x

kh

x zk

h

zH V( ) ( ) 0

Darcy's Law

v kh

x

v kh

z

x H

z V

+

Continuity Equation

Darcy’s Law

Flow equation

+

x

kh

x zk

h

zH V( ) ( ) 0Flow equation

kh

xk

h

zH V

2

2

2

20 For a homogeneous soil

For an isotropic soil

2

2

2

20

h

x

h

z

Equations of Groundwater flow

Confined aquifer ;

Unconfined aquifer;

t

h

Kb

S

y

h

x

h

t

h

T

S

y

h

x

h

y

2

2

2

2

2

2

2

2

Flow net

The method of flow-net construction presented here is based on the following assumptions; The aquifer is homogeneous The aquifer is fully saturated The aquifer is isotropic There is no change in the potential field with time The soil and water are incompressible Flow is laminar, and Darcy’s law is valid All boundary conditions are known

Flow net (continued)

Steps in making a flow net Sketch the flow system and identify prefixed flow lines and equipot

ential lines. Identify prefixed end positions of flow lines and equipotential lines. Draw trial set of flow lines Draw trial set of equipotential lines orthogonal to flow lines.

Water flowing by using the completed flow net can be quantified by using formula;

f

Kphq '

q’ = the total volume discharge per unit width of aquifer.K = the hydraulic conductivityp = the number of flow paths bounded by adjacent pairs of streamlinesh = the total head loss over the length of the streamlinesf = the number of squares bounded by any two adjacent streamlines and covering the entire length of flow

Steady flow in a confined aquifer

The quantity of flow per unit width, q’, may be determined from Darcy’s law;

K is hydraulic conductivity b is aquifer thickness dh/dl is slope of potentiometric surface

dl

dhKbq '

Steady flow in an unconfined aquifer

Employ Dupuit assumptions; The hydraulic gradient is equal to the slope of the water table For small water table gradient, the streamlines are horizontal an

d the equipotential lines are vertical.

Dupuit equation;

L

hhKq

22

21

2

1'

Example problems

A sand aquifer has a median grain diameter of 0.050 cm. For pure water at 150C, what is the greatest velocity for which Darcy’s law is valid. = 0.999 x 103 kg/cm3

= 1.14 x 10-2 g/s.cm

If hydraulic conductivity is 23 ft/day, what is the discharge per unit width of the flow system in figure below.

Sifat-sifat fluida The density of a fluid ; (Nm-3)

The specific weight(Nm-3)

Vm /

g dynamic viscosity () Ns/m2.

bulk modulus.

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