1. Yhd-12.3105 Subsurface Hydrology
Unsaturated Flow
Teemu Kokkonen
Tel
Room: 272 (Tietotie 1 E)
Water Engineering
Department of Civil and Environmental Engineering
Aalto University School of Engineering

2. 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

3. Unsaturated Zone
Does the percolating water in the figure above enter the subsurface drain? Why / why not?
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

4. Tensiometer
Negative water pressure in soil is measured using a tensiometer
Soil sample
Porous plate
Water hc
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?

5. 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
As pressure head values range across a large scale taking a logarithm lead s to a garph that is easier to interpret
Pressure head is – 100 cm. What is the corresponding pF value?
Pressure head is – 100 cm => pF value is 2

6. Water Retention Curve III III: Residual moisture content qres qres
II: Air-entry pressure head ha ha II I
I: Porosity

7. Water Retention Curve
It will not be a great surprise that different soils have water retention curves of different shape
Clay
Sand
Which one of the shown water retention curves is for a clay soil and which one for a sand soil? Why?

8. 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

9. 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
Here the direction of the z-axis is points downward – hence the negative sign.

10. Darcy’s Law in Unsaturated Zone: 3D
Why is the -1 present in the equation of qz missing from the equations for qx and qy?

11. Unsaturated Hydraulic Conductivity
Coarse gravel
Relationship between the pressure head and the hydraulic conductivity for different soil types
Fine sand
Peat
Clay
Hydraulic conductivity
Pressure head

12. 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
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).

13. Reminder: Transient Groundwater Flow in 3D
Specific storativity S0
volume of water added to storage, per unit volume and per unit rise in hydraulic head

14. Flow in Unsaturated Zone: Richards’ Equation
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]

15. 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:
,where q is the volumetric moisture content
So:

16. Differential Water Capacity
Moisture content q
Pressure head h Dq Dh

17. Numerical Solution – Richards Equation
Let us discretize the Richards equation in 2D for a longitudial section:
x (i)
Sink / source
z (j)

18. Numerical Solution – Richards Equation
z (j)
x (i) Dx Dz

19. Numerical Solution – Richards Equation

20. Numerical Solution – Richards Equation
Approximating the differential water capacity C
Estimate using the Van Genuchten equation the moisture content that corresponds to the pressure head at the desired time and location
Perturbate the pressure head with a small displacement of Dh
Compute the moisture content at h + Dh
Now you can estimate C using the difference method as
Recall that
How would you approximate C?