1. Unsaturated Flow
Water beneath the land surface occurs in two principal zones, the unsaturated (vadose) zone and the saturated zone
In the unsaturated zone, the spaces between particle grains and the cracks in rocks contain both air and water. Although a considerable amount of water can be present in the unsaturated zone, this water cannot be pumped by wells because capillary forces hold it too tightly.
In contrast to the unsaturated zone, the voids in the saturated zone are completely filled with water.

2. Unsaturated Flow
The approximate upper surface of the saturated zone is referred to as the water table.
Water in the saturated zone below the water table is referred to as ground water.
Between the unsaturated zone and the water table is a transition zone, the capillary fringe. In this zone, the voids are saturated or almost saturated with water that is held in place by capillary forces.
Height of capillary fringe may vary from a few millimeters for a coarse sand to several meters for clay soil.

3. Unsaturated Flow
water
content
qsat qf z
capillary fringe
Intermediate zone
soil water zone
unsaturated
zone
saturated
Water table is the surface where the water in a saturated porous medium is at atmospheric pressure. Below the water table P > Patm.
In capillary fringe P < Patm. Water is held by capillary forces acting against gravity. It is a saturated zone
Soil water zone is the uppermost belt in which water is extracted by action of plants or by soil evaporation

4. Soil Horizons
The soil profile typically consists of a succession of more or less distinct strata.
O Horizon: Contains large amount of organic material. Also known as humus.
A Horizon: Zone of major biological activity
E Horizon: Present only in older, well- developed soils, and generally occur between the A and B Horizons
B Horizon: Generally ticker than A. Clay accumulation and pressure from above layers reduce its porosity and increase density.
C Horizon: consists of weathered and fragmented rock material. It could also contain alluvial aeolian or glacial deposits
Bedrock

5. Soil Texture (USDA)
Soil texture is a soil property used to describe the relative proportion of different grain sizes of mineral particles in a soil.
12 soil texture classifications are defined by the USDA
They are classified by the fractions of each soil separate (sand, silt, and clay) present in a soil
Clay : < mm
Silt : mm
Sand : mm
Gravel : > 2 mm

6. Soil Properties
Porosity (f ): Ratio of voids (water + air) to total volume. It depends on soil texture with an approximate range of < f < 0.75

7. Soil Properties
Moisture content (q ): Parts of the voids are filled with water. Moisture content is the ratio of water volume to total volume.
Thus, 0 ≤ q ≤ f , When soil is saturated q = qs = f
Field Capacity (FC): In practice it means the moisture content of the field soil 2 to 3 days after a soaking rain or heavy irrigation. Theoretically it is the moisture a particular soil can hold against gravity. Usually measured at bar (-33 kpa).
It really is an effort to define an arbitrary point on a time versus drainage curve. It is the moisture content where rapid drainage has ceased. That point is fairly clear in sand but unclear in clay loam.
sand: ~0.15 loam : ~0.30 clay: ~0.40
Gravitational Water: Water drained from soil before FC is reached.

8. Unsaturated Flow
Permanent Wilting Point (PWP): Moisture content of soil when plants can no longer maintain turgid leaves. It is soil specific and assumed to occur at the same matric potential (-15 bar = 1500 kpa)
Available Moisture: Water held between FC and PWP and it is the water available to plants. Easier to determine in agricultural land but more difficult in forests. Silt and clay loam have the greatest capacity to hold water, whereas coarse sand has the least and pure clay is intermediate.
Hygroscopic Moisture: Thin moisture films around soil particles below PWP. This water is unavailable to plants
Retention Storage: Sum of available and hygroscopic moisture. It is a quantity that presumably must be satisfied by rainfall before a soil can yield water to groundwater or streamflow.

9. Unsaturated Flow
Relationships between soil types and total available soil moisture holding capacity, field capacity and wilting point (from Walker and Skogerboe, 1987)

10. Capillary
If water is withdrawn from a soil matrix that does not shrink upon drying, air enters the pore space, and air-water interfaces (menisci) are present in the pore space. Such curved interfaces are maintained by capillary forces.
Glass attracts water molecules (adhesion) more strongly than do other molecules themselves (cohesion).
Thus, water is pulled up inside the tube
PB = PC = Patm = 0 (relative)
hB = hC = 0
hB = hA + h2  hA= -h2 (capillary head)

11. Capillary s s = Surface tension (F/L)
Capillary rise is inversely proportional to pore diameter
In soil physics: Smaller diameter pores retain water against higher suctions than do larger pores. Thus, when water drains from soil or rock, large pores empty first.
A suction (negative pressure relative to Patm) must be applied to withdraw water from unsaturated zone above the water table.

12. Soil-Water Potential
In an unsaturated porous medium, the part of the total energy possessed by the fluid due to the soil suction forces is referred to as the suction head, y
Also called capillary-pressure head
Initially it is easier to withdraw water. As q gets smaller much greater suction is needed
It is function of moisture content
Total head is sum of suction and gravity heads h = y + z

13. Tensiometer
The water in U-tube comes to equilibrium potential with the water in the soil because water moves in or out of the porous ceramic cup.
P: matric potential
Z-P: total head
If water level raises above the cup then tensiometer operates as a manometer
In commercial tensiometers Hg replaces water in the U-tube; cm of Hg must be multiplied by 13.

14. Example

15. Hysteresis Effect
Ink bottle effect
As suction increases smaller and smaller pores are drained and the films of water around particles become thinner.
If water is fed to the soil, thus reducing suction, smaller pores refill but larger pores resist adsorption.

16. Darcy’s Law
A porous system is an interconnected structure of tiny conduits of various shapes and sizes. Thus, we can conceptualize it as a system of pipes.
Hagen-Poisulle Equation for laminar flow in a circular conduit:
where m is dynamic viscosity,
For porous medium part of the cross-sectional area A is occupied by soil rock strata, so the ratio Q/A is not equal to actual fluid velocity
Q/A defines a volumetric flux called Darcy flux.

17. Darcy’s Law Actual average velocity is
where K is hydraulic conductivity and f is porosity.
Darcy’s law is valid as long as flow is laminar
Example: Water is percolating through a fine sand aquifer with hydraulic conductivity 10-2 cm/s and porosity 0.4 toward a stream 100 m away. If the slope of the water table is 1% calculate travel time to the stream.

18. Darcy’s Law For unsaturated conditions
where is the hydraulic gradient, and h = y + z is the total head (suction head + elevation head).
(-) sign indicates that total head decreases in the direction of flow due to friction.
Both K and y are function of moisture content.

19. Soil Moisture Flux
The flow of moisture through the soil can be calculated using Darcy’s equation given measurements of soil suction head y at different depths z in the soil and knowledge of the relationship between hydraulic conductivity K an y.
Total head is found by adding suction head y to the depth z
Note that both are negative
z is (-) because it is taken (+) upward with 0 at the soil surface
y is (-) because it is a suction force which resists flow of moisture away from location

20. Example
Calculate the soil moisture flux q between depths 0.8 m and 1.8 m. For this soil K = 250(-y )-2.11, where K is in cm/day and y is in cm.

21. Example (cont’d) Flux is always downward
Because h1 > h2 for the whole period
Flux initially increases in general and reaches its maximum at week 6
It diminishes after week 6 because both head difference and conductivity diminish as soil dries out
Note that as the soil becomes wetter its hydraulic conductivity increases because there are more continuous fluid-filled pathways through which fluid can move

22. Example (complete picture)
Rainfall during April and May flows down the soil, reducing soil suction head
Later soil dries out due to ET causing the soil suction head to increase again
h shows greatest variability at z = 0.4 m.
It falls below the profile at z=0.8 m from the beginning of June onward 
Moisture flux is upward between these two depths

23. 1-D Richard’s Equation Darcy’s equation:
Combining: where D is soil diffusivity given by

24. Infiltration The process of water entering the soil
Factors influencing infiltration:
Condition of soil surface & vegetation cover
Soil properties such as porosity and hydraulic conductivity
Current moisture content (antecedent or initial moisture content)
Infiltration rate, f(t) is the rate at which water enters the soil [L/T]
It is a flux: Amount of water entering soil per unit time and per unit area Q/A  (cm3/hr )/(cm2) = cm/hr
Potential infiltration rate (infiltration capacity), fc: Infiltration rate when water supply is not limiting.

25. Infiltration When i < fc then f = i
i.e., when rate of supply is less than infiltration capacity, infiltration rate is equal to supply (rainfall intensity, rate of irrigation)
Cumulative infiltration rate, F(t): Accumulated depth of water infiltrated to the soil by a given time

26. Percolation and Redistribution
Rapid infiltration and rainfall continues to add to water content
Surface is approaching saturation, water percolates deeper into soil
Rain ceases and water redistributes in a few hours
In a day or so, gravity and matric potentials draw the water deeper

27. Horton Model f0 : infiltration rate at t = 0
fc : infiltration rate at t = ∞
k : decay constant [1/T]

28. Philip’s Model S : Sorptivity K : Conductivity
First term: Suction effect
Second term: Gravity effect
For horizontal infiltration suction is the only effect
 last term disappears

29. Green-Ampt Model
Horton and Philip’s equations are obtained from approximate solutions of Richard’s equations.
Green-Ampt is a more physically based approach with an analytical solution

30. Green-Ampt Model
As the wetting front passes a location, q changes from qi to f

31. Green-Ampt Model
Continuity eq’n: Total amount of water that has entered the soil at time t is
Darcy’s Law:
( f is (+) downward, q is (+) upward)
h1 (at surface) = h0 ; h2 (in dry soil below wetting front) = -y – L
(usually h0 is negligible)

32. Green-Ampt Model If h0 is not eligible, use (y - h0) for y
Note that this is a nonlinear equation in F; it requires iterations to solve for F.
Given K, t, y, and Dq, start with a trial F (usually second term on right hand side, i.e. Kt) and iterate.
Infiltration rate is given by

33. Green-Ampt Parameters
Application of Green-Ampt model requires estimates of hydraulic conductivity K, porosity f, and wetting front suction head y.
Brooks and Corey
Effective porosity: fe = f – qr, where qr is residual water content
Effective saturation:
It can be shown that
, where l and yb are constants
Bouwer (1966) found that K  Ks / 2

35. Example
Compute infiltration rate f and cumulative infiltration rate F after 1 hr of infiltration into a silt loam soil that initially had an effective saturation of 30%. Assume water is ponded to a small but negligible depth
Solution: For silt loam soil qe= 0.486, y =16.7 cm, and K = 0.65 cm/hr
Take F(t) = Kt as first guess  F(1 hr) = 0.65 cm
Substitute F =1.27 cm  F=  ……  F = 3.17 cm
 f = 1.81 cm/h

36. Ponding
So far we assumed that water is ponded to a small depth on the soil surface, so all the water soil can infiltrate is available at the surface
In reality ponding only occurs if i > fc
The ponding time is the elapsed time between the time rainfall begins and the time water begins to pond on the soil surface

37. Ponding Time
Until t = tp, cumulative infiltration F will be equal to rainfall depth
Three principles to determine tp
Prior to ponding all rainfall is infiltrated
fc is a function of F
Ponding occurs when i > fc

38. Example
Compute ponding time and depth of water infiltrated at ponding for silt loam at 30% initial effective saturation with (a) i= 1 cm/h, (b) i=5 cm/hr
Solution: From previous example y ∙Dq = 5.68 cm and K = 0.65 cm/hr
(a)
(b)
In each case f = i

39. Double Ring Infiltrometer
Because the inner and outer rings are filled with water, water flows virtually vertically through the inner ring into the soil
Drop in water level is measured with time
Best results are obtained at field capacity
Draw-backs: Very time consuming, requiring frequent attention, either by recording measurements or by maintaining equilibrium in the height between the rings. The practicality of the instrument is reduced by the fact the rings are extremely heavy to move. It also requires a flat undisturbed surface which sometimes is not available.

40. Infiltration in Forests
Forest floors absorb the energy of falling rain and permits clean water to penetrate to mineral soil layers
Infiltration capacity of forest soils is almost always greater than rainfall intensity due to highly permeable top soil and macropores
Therefore, infiltration excess overland flow is extremely rare in forested landscapes
However, forest floor disturbance by cultivation, grazing, repeated burning, logging and road building may impair infiltration and cause overland flow
Subsurface stormflow or lateral flow is quite common in forested watersheds. Water infiltrating quickly reaches low permeability soil or bedrock and start moving parallel to surface in the soil