Post on 29-Mar-2015
Yhd-12.3105 Subsurface Hydrology
Unsaturated Flow
Teemu Kokkonen Email: firstname.surname@aalto.fiTel. 09-470 23838Room: 272 (Tietotie 1 E)
Water EngineeringDepartment of Civil and Environmental EngineeringAalto University School of Engineering
Yhd-12.3105 Subsurface Hydrology
Transient flow
Soil Moisture Profile – From Groundwater level to soil surface
• Recall some definitions– Groundwater level is defined to be that level
where soil water pressure is atmospheric– Below the groundwater level the soil is
saturated with water and above the groundwater level the soil is unsaturated
– Immediately above the groundwater level there is a capillary fringe that is (amost) fully saturated
Yhd-12.3105 Subsurface Hydrology
Transient flow
Unsaturated Zone
• Water originating from precipitation or irrigation infiltrates through the soil surface and percolates through the unsaturated zone
– This forms recharge to an aquifer– Harmful substances move with water
• In unsaturated zone the water pressure is negative
– Water is retained in soil by capillary forces, which are a combination of cohesive and adhesive forces
Does the percolating water in the figure above enter the subsurface drain? Why / why not?
Yhd-12.3105 Subsurface Hydrology
Transient flow
Tensiometer
Soil sample
hc
Porous plate
Water
1. How can you read the pressure head in the soil sample using the tensiometer shown in the figure?
2. The porous plate needs to be airtight. Why?
3. Why does the water entering the soil sample does not significantly affect the measurement?
• Negative water pressure in soil is measured using a tensiometer
Yhd-12.3105 Subsurface Hydrology
Transient flow
Water Retention Curve
• A graph that shows the relationship between soil water pressure head and moisture content of soil is called the water retention curve
• In the water retention curve the soil water pressure head is typically expressed as a pF value
– pF value is the 10-based logarithm of the absolute value of the pressure head expressed in centimeters of water column height
Pressure head is – 100 cm. What is the corresponding pF value?
Pressure head is – 100 cm => pF value is 2
− As pressure head values range across a large scale taking a logarithm lead s to a garph that is easier to interpret
Yhd-12.3105 Subsurface Hydrology
Transient flow
Water Retention Curve
II
III
I
I: Porosity
II: Air-entry pressure head ha
ha
III: Residual moisture content qres
qres
Yhd-12.3105 Subsurface Hydrology
Transient flow
Water Retention Curve
• It will not be a great surprise that different soils have water retention curves of different shape
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0.0 10.0 20.0 30.0 40.0 50.0 60.0
Volumetric soil water content [%]
pF
Clay
Sand
ClaySand
Which one of the shown water retention curves is for a clay soil and which one for a sand soil? Why?
Yhd-12.3105 Subsurface Hydrology
Transient flow
Water Flow in Unsaturated Zone
• What are the differences to saturated (groundwater) flow?– Hydraulic conductivity is a function of the moisture content of soil
• When moisture content decreases large soil pores are emptied first, which leads both in reduced cross-sectional area of flow and increased tortuosity of the flow paths => hydraulic conductivity drops
– The air-filled pore space is a function of the moisture content of soil• Recall the large difference (several orders of magnitude) in the storativity of confined and
unconfined aquifers
• Recall that moisture content and pressure head are related via the water retention curve
– Hydraulic conductivity and the air-filled pore space can also be expressed as a function of pressure head
Yhd-12.3105 Subsurface Hydrology
Transient flow
Darcy’s Law in Unsaturated Zone
• As presented earlier the hydraulic head H is the sum of pressure head h and gravity head z
• In the unsaturated flow the interest often is to study percolation to groundwater, so let us first write Darcy’s law in one dimension and in vertical direction
1)()()(dz
dhhKhz
dz
dhK
dz
dHhKq zzzz
Here the direction of the z-axis is points downward – hence the negative sign.
Yhd-12.3105 Subsurface Hydrology
Transient flow
Darcy’s Law in Unsaturated Zone: 3D
1)()(
)()(
)()(
z
hhK
z
HhKq
y
hhK
y
HhKq
x
hhK
x
HhKq
zzz
yyy
xxx
Why is the -1 present in the equation of qz missing from the equations for qx and qy?
Yhd-12.3105 Subsurface Hydrology
Transient flow
Unsaturated Hydraulic Conductivity
Pressure head
Hyd
rau
lic c
on
du
ctiv
ity
Coarse gravel
Fine sand
Clay
PeatRelationship between the pressure head and the hydraulic conductivity for different soil types
Yhd-12.3105 Subsurface Hydrology
Transient flow
Unsaturated Hydraulic Conductivity
• The water retention curve (pF curve) and the unsaturated hydraulic conductivity can be described with the following equations originally proposed by M.Th. van Genuchten and Y. Mualem
aS
aaRSR
hh
hhhh
;
;1)(
a
aa
RS
R
hh
hhhhS;1
;1
2/1½ 11SSKR
RSKKK
Where is the soil moisture (cm3/cm3), R is the residual water content of soil (cm3/cm3), S is the saturated water content of soil (cm3/cm3), S is the saturation of soil (cm3/cm3), h is the pressure head (cm), and ha is the air entry pressure head. Symbols , , and refer to the parameters of the van Genuchten model, and = 1 – 1/. K is the unsaturated hydraulic conductivity, KS is the saturated hydraulic conductivity (cm/h), and KR is the relative conductivity of unsaturated soil (KR = K / KS).
Yhd-12.3105 Subsurface Hydrology
Transient flow
Reminder: Transient Groundwater Flow in 3D
t
HS
zzH
K
y
yH
K
xxH
K
t
HS
zzH
K
y
yH
K
xxH
K zyxzyx
00
Hzyx
VS w
0
Specific storativity S0
volume of water added to storage, per unit volume and per unit rise in hydraulic head
t
HS
z
q
y
q
x
q zyx
0
Yhd-12.3105 Subsurface Hydrology
Transient flow
Flow in Unsaturated Zone: Richards’ Equation
t
hhChK
z
hhK
zy
hhK
yx
hhK
xz
q
y
q
x
qzzyx
zyx
)()()()()(
Specific moisture capacity:
Differential water capacity:
Volume of water released from (or added to) storage per unit decrease (or increase) of pressure head
C [1/m]
t
hhChK
z
hhK
zy
hhK
yx
hhK
x zzyx
)()()()()(
Yhd-12.3105 Subsurface Hydrology
Transient flow
Differential Water Capacity
The definition was:
Differential water capacity: C [1/m]Volume of water released from (or added to) storage per unit decrease (or increase) of pressure head
From the definition above it follows:
dh
dC
,where q is the volumetric moisture content
dt
d
t
h
dh
d
t
hC
So:
Yhd-12.3105 Subsurface Hydrology
Transient flow
Differential Water Capacity
Moisture content q
Pre
ssur
e he
ad
h
dh
dC
Dq
Dh
Yhd-12.3105 Subsurface Hydrology
Transient flow
Numerical Solution – Richards Equation
• Let us discretize the Richards equation in 2D for a longitudial section:
RKz
hK
zx
hK
x
Rz
hK
zx
hK
xt
hC
zzx
zx
1
z (j)
x (i)
Sink / source
Yhd-12.3105 Subsurface Hydrology
Transient flow
Numerical Solution – Richards Equation
tjih 1,
tjih ,1
tjih ,1
tjih 1,
tjih ,
z (j)
x (i)
11,
tjih
1,1
t
jih1,1
t
jih
11,
tjih
1,tjih
Dx
Dz
RKz
hK
zx
hK
xR
z
hK
zx
hK
xt
hC zzxzx
1
Yhd-12.3105 Subsurface Hydrology
Transient flow
Numerical Solution – Richards Equation
jitji
tji
t
ji
t
ji
t
ji
t
ji
tji
tji
ji
RKzKzz
hKz
z
hKz
zx
hK
x
hKx
x
t
hhC
,½,½,½,½,½,½,
,1
,,
11
jitji
tji
tji
tjit
ji
tji
tjit
ji
tji
tjin
ji
tji
tjit
ji
tji
tji
ji
RKzKzz
hhKz
z
hhKz
zx
hhKx
x
hhKx
x
t
hhC
,½,½,1,,
½,,1,
½,,1,
½,,,1
½,
,1
,,
11
RKz
hK
zx
hK
x
Rz
hK
zx
hK
xt
hC
zzx
zx
1
Yhd-12.3105 Subsurface Hydrology
Transient flow
Numerical Solution – Richards Equation
• Approximating the differential water capacity C1. Estimate using the Van Genuchten equation the moisture content that corresponds to
the pressure head at the desired time and location
2. Perturbate the pressure head with a small displacement of Dh
3. Compute the moisture content at h + Dh
Recall that .
How would you approximate C?
dh
dC
4. Now you can estimate C using the difference method as
h
hhhhC
)()(
)(