Iii

ALIRAN 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 waktu 2. 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

P h= z+ ρg P = ρgh p h = z + hp

h

h = hydraulic head z = elevation head hp= pressure head

Definisi hydraulic head pada sebuah titik

P( A) h( A) = + z ( A) γw

(1)

A z(A)

Datum

z diukur vertikal ke atas Terhadap bidang datum

Example: Static water table 1. Calculation of head at A Choose datum at the top of the impermeable layer

2m

=

P (A)

z (A) =

1m X A

1m Impermeable stratum

5 m thus

h( A)

=

4γ w 1 4γ w + 1 = 5m γw

Example: Static water table 2. Calculation of head at X Choose datum at the top of the impermeable layer

2m 1m X A

1m Impermeable stratum

P ( X) = z ( X) =

5 m thus

h ( X)

=

γw 4 γw γw

+ 4 = 5m

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

Example: Static water table 3. Calculation of head at A Choose datum at the water table

P (A) =

2m

z( A)

1m X A

1m Impermeable stratum

5m

=

4γ w -4

thus h (A) =

4γ w γw

- 4 = 0m

Example: Static water table 4. Calculation of head at X Choose datum at the water table

P (X) z ( X)

2m 1m X A

1m Impermeable stratum

5m

= 1γ w = −1

thus h ( X)

=

γw − 1 = 0m γw

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

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

Differences in head are required for flow (not pressure)

2m 1m X A

1m Impermeable stratum

5m

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 in water of variable density Point-water head

Fresh-water head

P2 = ρ f gh f

hf hp

P1

z

P1 = ρ p gh p

P2

 ρp hf =  ρ  f

 hp  

Water flow through soil ∆h

Soil Sample

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

Darcy’s Experiment 

The first systematic study of the movement water through a porous medium is made Henry Darcy. ha

hb L

Q



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

dh  hA − hB  Q = − KA  = − KA dl  L 

Hydraulic conductivity 

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

−Q K= A( dh / dL )

− ( L3 / T ) K= 2 L ( L / L)

( )

γ   ρg   K = k   = k  µ  µ  

γ = specific weight, µ = dynamic viscosity.

Measurement of permeability constant head device

inlet load H

outlet

device for flow measurement

sample

Manometers

L porous disk

Fig. 4 Constant Head Permeameter

Constant head permeameter 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 V H = kA T L It can be seen that in a constant head permeameter:: k =

VL AHT

(3)

Measurement of permeability Standpipe of cross-sectional area a

porous disk H

H1

Sample H2 L of area A

Fig. 5 Falling Head Permeameter

Falling head permeameter Solution −a

dH H = kA dt L

(4a)

Standpipe of area a

Equation (4a) has the solution:

H1

kA − an( H ) = t + cons tan t (4b) L

H L

Initially H=H1 at time t=t1 Finally H=H2 at time t=t2. aL n( H1 / H 2 ) k= A t 2 − t1

(4c)

Sample of area A

H2

Typical permeability values 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 Clays

Fissured & Weathered Clays

Typical Permeability Ranges (metres/second) 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.

Definition of Hydraulic Gradients For horizontal flow v=vx and k=kH and thus

z

v x = − k H ix

A ∆z

B O

∆x

C x

dimana h(C ) − h( B ) ix ≈ ∆x jadi ∂h vx = −k H ∂x

Definition of Hydraulic Gradients For vertical flow v=vz and k=kV and thus

z

v z = −k z iz

A

dimana

∆z

B O

∆x

C x

h( A) − h( B ) iz ≈ ∆z jadi ∂h v z = −kv ∂z

Gradient of the potentiometric surface 92 90

81

70 63

80

Aquifer characteristics 

Transmissivity(T) is a measure of the amount of water that can be transmitted horizontally 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 ;

S s = ρ w g ( α + nβ )

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

z

x

aquifer

Flow

Impermeable bedrock

Flow into a soil element

vz C

vx

D

Soil Element

B

∆z

A ∆ x

Netflow = (v x ( B) 　 − v x ( D))∆y∆z + (v z (C ) − v z ( A))∆x∆y Untuk aliran tunak, netflow menjadi nol;

∂v x ∂v z + =0 ∂x ∂z

Continuity Equation Continuity Equation

+

Flow equation

= 0

+

Darcy's Law

vx Darcy’s Law

∂v x ∂v z + ∂x ∂z

vz

∂h = −k H ∂x ∂h = −k V ∂z

∂ ∂h ∂ ∂h (k H ) + (k V ) = 0 ∂x ∂x ∂z ∂z

Flow equation

∂ ∂h ∂ ∂h (k H ) + (k V ) = 0 ∂x ∂x ∂z ∂z

2

For a homogeneous soil

∂ h ∂ h kH 2 + kV 2 = 0 ∂x ∂z 2

For an isotropic soil

2

2

∂ h ∂ h + 2 =0 2 ∂x ∂z

Equations of Groundwater flow 

Confined aquifer ;



Unconfined aquifer;

∂ 2 h ∂ 2 h S ∂h + 2 = 2 ∂x ∂y T ∂t

∂ 2 h ∂ 2 h S y ∂h + 2 = 2 ∂x ∂y Kb ∂t

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 equipotential 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;

Kph q' = f

q’ = the total volume discharge per unit width of aquifer. K = the hydraulic conductivity p = the number of flow paths bounded by adjacent pairs of streamlines h = the total head loss over the length of the streamlines f = 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;

dh q ' = Kb dl   

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

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 and the equipotential lines are vertical.

Dupuit equation;

2 2  1 h1 − h2   q ' = K  2  L 

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.  



ρ = 0.999 x 103 kg/cm3 µ = 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)



dynamic viscosity (µ) Ns/m2.



bulk modulus.

ρ = m /V

γ = ρg