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