1. Infiltration into Soils

2. Infiltration
An important class of flow events is related to water entry through the soil surface – in a process known as Infiltration.
The rate of infiltration relative to the rate of water supply on the surface (rain, irrigation) determines how much water enters the soil, how much, if any, will be ponded on surface or create surface runoff.
The leading edge of wetted soil volume, the WETTING FRONT, advances into the drier soil region in response to matric and gravitational potential gradients.

3. Shape of typical infiltration curve
i decreases gradually with time until a constant final rate if is attained.
High initial rate
Ponded infiltration into dry soils follows a predictable temporal pattern – high initial infiltration rates followed by gradual decrease with time to nearly constant final rate.
When water application rate is less than initial infiltration rate i (e.g., rainfall, irrigation) – at a certain time it may exceed surface infiltration capacity and ponding or surface runoff occur.

4. Factors affecting infiltration
Infiltration rate i is dependent on:
1) Initial soil water content
2) Hydraulic conductivity of the surface soil
3) Elapsed time since the onset of water application
4) Presence of impeding layers and other heterogeneities within the soil profile

5. Empirical Infiltration Equations
The predictable and well-behaved shape of infiltration rate vs time led to proposal of several empirical predictive equations.
These were followed by attempts to attach physical significance to the empirical model parameters.
The Lewis-Kostiakov Equation:
An widely used empirical expression was originally proposed by Lewis (1937) but was erroneously attributed to Kostiakov
I is the cumulative depth of infiltration or the volume of water per unit soil surface area, t is the elapsed time, k and a are empirical parameters, and i = dI/dt is the infiltration rate.

6. Empirical Infiltration Equations
Disadvantages of the original Lewis- Kostiakov equation:
It doesn't account for different initial soil water contents
For long infiltration times it erroneously predicts zero rate.
The later problem can be fixed by adding a parameter f0 representing a final infiltration rate to the previous equations.

7. Empirical Infiltration Equations
The Horton Equation:
Horton (1940) proposed another empirical equation based on an exponential form:
where i0 and if are the initial and final infiltration rates, respectively, and d is an empirical parameter

8. Physically Based Infiltration Equations
The Green-Ampt Approximation:
Green and Ampt (1911) adopted a simplified, yet physically-based, approach to describe the infiltration process.
Their solution is useful for cases of infiltration into initially dry soils which exhibit a sharp wetting front.

9. Physically Based Infiltration Equations – Green-Ampt (1911)
Basic Assumptions of the Green-Ampt Approximation:
A distinct wetting front exists such that the water content behind it (q0) remains constant, and abruptly changes to initial water content (qi) ahead of the wetting front
The soil in the wetted region has constant properties (q0, K0, and h0)
The matric potential at the wetting front is constant and equals hf

10. Physically Based Infiltration Equations – Green-Ampt (1911)
Green-Ampt Approximation:
They applied a simplified form of Darcy's law for horizontal flow (no gravitational component) over a wetted region of thickness Lf (i.e., the depth of the wetting front) as:
h0 is matric head at the soil surface (or within the wetted soil volume), hf is the matric head at the wetting front, and K0 is the hydraulic conductivity of the wetted soil (the transmission zone).
Then they invoked conservation of mass by equating cumulative infiltration I and wetting depth (Lf) times change in soil water content (Dq = q0-qi):

11. Physically Based Infiltration Equations – Green-Ampt (1911)
Green-Ampt – Horizontal Infiltration
Equating the two expressions and integrating enable prediction of wetting front distance as a function of time (horizontal flow!):
To simplify the notation we define the effective diffusivity of the wetted soil as: D0 = K0 Dh/Dq. We are now able to predict the wetting front depth, cumulative and instantaneous rates of horizontal infiltration as:

12. Physically Based Infiltration Equations – Green-Ampt (1911)
Green-Ampt Approximation – Vertical Infiltration
For vertical infiltration, we must incorporate the effect of gravity.
Taking soil surface as z=0, Hz=0=h0+0; and at z=-Lf, Hz=Lf=hf-Lf. Introducing these into Darcy's law for vertical infiltration yields:
The solution to the integral equation, found in standard tables of integrals, is given by:

13. Physically Based Infiltration Equations – Green-Ampt (1911)
Green-Ampt Approximation – Vertical Infiltration
The previous equation may be converted to include cumulative infiltration vs. time using I=DqLf:
There is no simple form to express I or i vs. time. However, for short times this solution converges to the solution for the horizontal case, and for long infiltration periods it converges to i=Ko.

14. Physically Based Infiltration Equations – Philip (1957)
Philip's Solution For Vertical and Horizontal Infiltration
J.R. Philip (1957, 1969) presented the first analytical solution to the Richards Equation for vertical and horizontal infiltration.
Horizontal Infiltration
For horizontal infiltration Philip showed that the cumulative and instantaneous infiltration rates are given by:
S stands for SORPTIVITY which is a function of initial and boundary water contents, S=S(q0,qi), and t is the time elapsed since water application.

15. Physically Based Infiltration Equations – Philip (1957)
Philip's Solution - Horizontal Infiltration
When a sharp wetting front exists, the sorptivity may be approximated by:
Lf is the distance from the boundary to the wetting front.
Vertical Infiltration
Philip's solution of the Richards equation for vertical infiltration describes the time dependence of cumulative infiltration as an infinite series in powers of t1/2
A1, A2,... are parameters dependent upon soil properties and on initial and boundary water contents.

16. Philip’s Infiltration Equation
Philip's Solution – Vertical Infiltration
For practical purpose, the series in the previous equation is commonly truncated and only the first two terms are retained:
Due to sorptive forces
of relatively dry soil
Contribution of gravity
The influence of sorptivity diminishes with time reflecting the reduction in hydraulic gradient as the soil becomes wetter.

17. Philip’s Infiltration Equation
Philip's Solution – Vertical Infiltration
For long infiltration times when water is ponded on the soil surface (q0=qs at the surface), the final infiltration rate approaches K(qs)=Ks.
The ratio A1/Ks is bounded by 1/3≤A1/Ks≤2/3.
For flux-limited infiltration rate such as low intensity rainfall P we may approximate the equivalent time to ponding te from the time at which i = P (note that P is a flux)
This is particularly useful for predicting whether and when surface runoff will occur.

18. Time to ponding observed predicted
For flux-limited infiltration rate such as low intensity rainfall P we may approximate the time to ponding te from the time at which i = P:
To obtain estimates whether and when surface runoff will occur.
observed
predicted

19. Time compression correction..

20. Physically Based Infiltration Equations
Philip's Solution – Example - Continued
Finally, we apply Philip's Solutions for horizontal and vertical cumulative infiltration, respectively.

21. Infiltration Rates for Different Soils
Cumulative infiltration and infiltration rates for different textured soils (Hillel 1998):
sand
loam

22. Infiltration from surface pond or point source
In some instances we may be interested in infiltration in 2 or 3 dimensions to predict:
Water distribution from drip irrigation systems.
Flow and wetting patterns from an irrigated furrow.
Flow rates and patterns from leaking underground tanks.

23. Infiltration From Surface Disk Source (3-D Flow)
Detailed solutions for 2- or 3-D flow using the Richards equation are feasible only by means of numerical methods.
However, under steady state flow conditions and certain assumptions regarding K(h) it is possible to arrive at approximate analytical solutions to flow from various source geometries.
The Gardner hydraulic conductivity model:
Philip (1969) discusses a variety of multidimensional solutions from surface and subsurface point and line sources.

24. Multidimensional Infiltration
In contrast with 1-D vertical flow where a wetting front advances indefinitely (capillarity and gravity operate in the same direction), multidimensional steady flows attain a finite and constant distribution of matric head and water content about the water source.
The 2- or 3-D steady distributions of h or q reflect a balance between sorptive forces (matric potential) & gravitational forces.
capillarity
gravity

25. Wooding’s solution – flow from a shallow pond
An approximate solution to steady state infiltration rate from a shallow and circular pond of water on the soil surface was derived by Wooding (1968):
Sorptive Forces
Term
Gravity
rs is the radius of the pond.
The two terms in above equation represent the contribution of gravity to the flow (Ks), and contributions due to sorptive forces.
Unlike one-dimensional flow, 3-D steady state infiltration rate exceeds Ks (unless rs infinity i.e., 1-D flow).
This approximate solution provides a simple means by which Ks and b may be measured in situ.

26. Infiltration Measurement - Double ring infiltrometer
Simple and easy to install, well-defined infiltration geometry (known area of entry) and measurable water amounts and rates.
Buffer zone formed by outer ring ensures 1-D flow.
Requires large amounts of water, simple analyses.

27. Disk infiltrometer - ponded
Step 1:
Step 2:
Step 3:
Using:
; and b [shape factor] = 0.55

28. Tension infiltrometer

29. Tension infiltrometer
Definitions
Step 1 – measure Q1 and h1 and h2
Step 2 –
Step 3 –
Step 4 –
Alternate analysis:

30. Other methods for measuring K(h)

31. Other methods for K(h) estimation