Water flow in soil, hydraulic conductivity

Soil Science and Soil Physics

Lecture 8

Saturated flow

Darcy, H., 1856. Les Fountaines de la Ville de Dijon

Henry Darcy (1856) filtration of water for fountains in

Dijon

After many experiments he found that water flow

through the soil column depends on:

• directly proportional to pressure drop

• inversely proportional to the length

• directly proportional to the crossectional area

• dependent on coefficient which is specific for

each media

Henry Darcy

Darcy law

Q = flow [L3.T-1]

A = crossectional area [L2]

Ks = saturated hydraulic conductivity [L.T-1]

�H = H1 – H2 (hydraulic head drop) [L]

L= sample lenght [L]

valid in fully

saturated porous

media

For example: under

the ground water

level

L

Α∆ΗΚQ S=

LH2

H1

datum

z1

z2

h1

h2

Hi = hi+zi

for:

Transforms to the:

More gereneral form:

Pro 1D vertical flow

kde:

q ... Volume flux [L.T-1]

Q ... Flow rate [L3.T-1]

A ... Crossectional area [L2]

L

HKq s

∆=

A

Qq =

dl

dHKq s=

HKdl

dHKq Ss ∇−=−=

poznámka: negative

sign due to the fact

grad H aims against

flow direction

Coefficient or the saturated hydraulic conductivity KsAlso called (sometimes) filtration coefficient, darcy’s coefficient or

permeability (incorrect)

commonly used units Ks (m.s-1), (cm.d-1), (cm.s-1)

Ks is property of water-solid interaction. Parameter which is related only to the porous media (independently on flowing liquid)is:

k = Ksν/g [ L2]

kde ν kinematic viscosity

Permeability k

Ks for different media

k (c

m2)

Ks

(cm

.s-1

)

Ks

(m.s

-1)

Zdroj: Císlerová a Vogel, 1998

clay

silt sa

nd

y lo

amsa

nd

gra

vel

Example 1 :Vertical column: q=?

b=

10cm

q

const. head

seepage face

1) datum definition

H = 0

H2

= H1 = z1

2

1

2) points 1 and 2 with known hydraulic

heads

= z2

+z

3) Darcy’s law

( )

1

12

.110100

)0110(100 −−=

−−=

=−

−=∆

−=

dcm

L

HHK

L

HKq ss

WATER

SOIL

Ks=100cm/dL=

100cm

dH=10 cm

Water

headinflow

Free

outflow

100 cm

Ks

H1

H2

Example 2 horizontal column: q = ?

1) Step 1, definition of datum and coordination system

2) Definition of points 1 and 2). Then x1 = 0 and h1 = 10 cm,

x2=100cm, h2 = 0, z1 = z2 = 0, L = x2 - x1 = 100 cm

3) Hydraulics heads are then H1 = h1 + z1 = 10 cm, H2 = h2 + z2 =

0 cm

5) Darcy’s law

( ) 112 .10100

)100(100 −=

−−=

−−=

∆−= dcm

L

HHK

L

HKq ss

1) Measurements of Ks using constant head permeameter

VODA

PŮDA

Ks= ?

b

L

q

Const. head

Seepage face

Soil column

H1 = 0 + 0 (lower end)

H2 = b + L (upper end)

∆H = (b+L) - 0

then:

Ks measurements

( )Lb

qL

H

qLKs +

−=∆

−=

( )LbAt

VL

HAt

VLKs +

=∆

=

In practice we measure Q, resp. V/t, then:

Q

A

sample

po

rou

s d

iscs

Q

measurement

overflow

inflowconstant head

b

L

Fig. : http://www.utdallas.edu/~brikowi/Teaching/Geohydrology/LectureNotes/Darcy_Law/Permeameters.html

Constant head permeameter

soil sample must be carefully saturated with water to

measure “real” Ks.

Constant head permeameter

2) Measurement of Ks falling-head permeameter

Ks= ?

b(t)

L

q

Falling head

Seepage face

Experiment is done on soil sample in the

laboratory. Initial water level is equal to b0

H1 = 0, H2(t) = L+b(t), ∆H(t) = [b(t) + L] - 0

becomes:

b0

( )L

LbK

dt

dbq s

+−==

dtL

K

Lb

db s−=+

( )Lb

LbLb

Lb

db b

b

b

b++

=+=+∫

0

1lnln 1

0

1

0

Integration of the left side

b1WATER

SOIL

∫∫ −=−=−11

0

1

0

t

ss

t

s

L

tKdt

L

Kdt

L

K

Right side integration

dtL

K

Lb

db s−=+

L

tK

Lb

Lb s 1

0

1ln −=++

then:

Lb

Lb

t

LKs +

+=

1

0

1

ln

Zdroj: http://www.utdallas.edu/~brikowi/Teaching/Geohydrology/LectureNotes/Darcy_Law/Permeameters.html

Falling-head permeameter

sample

Po

rou

s d

iscs

Q

wa

ter

leve

l dro

p

b0

L

b1

Lb

Lb

t

L

A

AK bs +

+=

1

0

1

ln

Consider different

crossectional areas of

burette and column:

db

d

where:t1:. timeAb : cross. area of buretteA ... cross. area of the sample

VODA

PŮDA

b

L

h0

h2 = b, z2=L

h1 = 0, z1 = 0

Ks is constant, h(z) = ?

Darcy’s law:

z

zhK

L

LbK

L

HKq sss

+−=

+−=

∆−=

zh

zL

bz

L

hhh =

−= 12

In this case, pressure head

profile is linear .

Example 3 :Pressure head along the soil column h(z)=?

q

q12 q1z

Flux across the surface is called infiltration rate i.

Total amount of water infiltrated is I [L]

Infiltration rate can be steady or unsteady

Measurements of Ks in field – infiltration experiments

Double-ring infiltration:• Two rings• Spike inserted inside of the

inner ring• water batches are being

added to the inner ring, every time the water level reaches the spike

• The time between spike appearance is measured for each batch and flux is then calculated for each time interval

Ideal case Common practice

Measurements of Ks in field

0

5

10

15

20

25

30

35

40

45

0:00:00 0:30:00 1:00:00 1:30:00 2:00:00

kum

ula

tivn

ivn

í mn

ožs

tví [

l]

0.00E+0

3.00E-5

6.00E-5

9.00E-5

1.20E-4

1.50E-4

1.80E-4

infil

tra

ční r

ych

lost

[m

/s]

Kumulativní infiltrace

Infiltrační rychlost

( )dttqIt

∫=0

( ) sKtq =lim

dt

dIq =

Evaluation of double ring infiltration

cumulative infiltrated amountinfiltration velocity

Zdroj: E. Sulzman

• θ changes in time and space

• relationship θ(h) e.g. retention curve

exist

• hydraulic conductivity depends on θθθθresp. on h for h < 0

• K(θ), resp. K(h)is called hydraulic conductivity function

Unsaturated hydraulic conductivity

H)(Kq ∇−= θ

where:q fluxH hydraulic head

Darcy-Buckingham lawEdgar Buckingham (1907)

Zdroj: Kutílek et al. 1994

flow direction at dhoriginal level

Based on assumption that from known of pore size distribution (from retention curve) and equation and Laplace equation, and Equation describing the flow in individual capillary the unsaturated Hydraulic conductivity can be predicted

Capillary models predict relative hydraulic conductivity

where Ks is measured saturatedhydraulic conductivity

Capillary models (Childs and Collis-George, 1950)

Capillary models

( ) ( ) sr KKK θθ =

Then for relative hydraulic conductivity Kr

where: Is saturated hydraulic conductivity

constants C12 a C2 are canceled

Relative hydraulic conductivity

( ) θθθ

dh

CCK ∫=0

2221

1

( ) ( )( )

∫

∫==

S

h

d

h

d

K

KK

S

r θ

θ

θ

θ

θθ

θ

02

02

( ) SS KK =θ

We will obtain the hydrulic conductivity of the capillary model as follows:

Burdine (1953):

∫

∫θ

θ

θ

θ

θθ

=θs

s

r

h

d

h

d

K

0

2

0

22

)(

Mualem (1976): 2

0

0

2/1

)(

θ

θ

θ

θ=θ

∫

∫θ

θ

s

s

r

h

d

h

d

K

(θ/θs)b ..... Relative tortuosity is included

Retention curves functions are substituted:::.

( )

≥

<

=θ

λ

b

b

b

e

Hh

Hhh

H

h

1

( ) ( )( )

≥

<α−+=θ

01

01

1

h

hh

h

mn

e

Bro

oks

a C

ore

yva

n G

en

uch

ten

Most commonly used models of Kr prediction

From Brooks and Coreyaλb

eer θ)(θK +=

...... and the final expressions for Kr predictions are then

=(h)Kr

b

λba

b Hhh

H<

+ /

bHh ≥1

Where parameters a and b are for Burdine model Burdine a=2 a b=3Mualem a=2 a b=2.5

rs

reθ θθ

θθ−−

= Efective water content

21

5.0 11 ])θ([θ)(θK mmeeer −−=from van Genuchten

vztahu

Mualem

rs

re

θθ

θθθ

−−

=

......

=(h)Kr

( ) ( )[ ]{ }( )[ ] 0

αh1

αh1αh1m/2n

2mnmn

<−+

−+−−−

h

01 ≥h

-2

-1

0

1

2

3

4

0.00 0.10 0.20 0.30 0.40 0.50

θ (-)

log

hc

písekhlinitopísčitá půdajílovitohlinitá půdajílovitohlinitá půda

-2

-1

0

1

2

3

4

0.00 0.20 0.40 0.60 0.80 1.00

Kr (-)

log

hc

Retention curve Hydraulic conductivity

function

VG

BC

VGM

BCM

Retention curve and hydraulic conductivity function for different soils

Unsaturated hydraulic conductivity

písek jíl

písčitá hlína

jílovitá hlína

0 20 40 60 80 100

kapilární tlaková výška [cm]

hydr

aulic

ká v

odiv

ost

[cm

/den

]

1000

100

10

1

0.1

0.01

Tension infitlrometer

Measurements of I under

infitrometer with fixed

tension head

Measurements of hydraulic conductivity in field

Tension infiltrometer

Jury a Horton 2004

hlo

ub

ka p

ůd

níh

o p

rofi

lu (

cm)pressure head (cm)

clayLoamy soil

dep

th(c

m)

Infiltration experiment – the wetting front

0

5

10

15

20

25

30

35

40

45

0:00:00 0:30:00 1:00:00 1:30:00 2:00:00

kum

ula

tivn

ivn

í mn

ožs

tví [

l]

0.00E+0

3.00E-5

6.00E-5

9.00E-5

1.20E-4

1.50E-4

1.80E-4

infil

tra

ční r

ych

lost

[m

/s]

Kumulativní infiltrace

Infiltrační rychlost

( )dttqIt

∫=0

( ) )h(Ktvlim i=

dt

dIq =

Result of one infiltration experiment

- analogous to Ks

- One K(hi) value for each infiltration done with the pressure head hi

cumulative infiltrated amountinfiltration velocity

Measurement of K(h) in field

- Clay loam soil

1.00E-07

1.00E-06

1.00E-05

1.00E-04

1.00E-03

-0.20-0.18-0.16-0.14-0.12-0.10-0.08-0.06-0.04-0.020.00soil-water suction at the tension disc [m]

K(h

) [m

/s]

soil surface

35 cm below surface

60 cm below surface

Measurements of K(h) in laboratory

- evaluation using inverse modeling ::

Kutílek, M., Kuráž, V., Císlerová, M. Hydropedologie, skriptum ČVUT 1994

Císlerová, M. Inženýrská hydropedologie, skriptum ČVUT 2001

Císlerová M., Vogel T. Transportní procesy, skriptum ČVUT

Jury, W.A. and R. Horton, Soil Physics. Sixth Edition, 2004.

M. E Sumner, Handbook of Soil Science 1998

http://edis.ifas.ufl.edu/AE266

References