SOIL 6010
Project Proposal
Geano Dong
Matlab Functions for Richards Equation - Philip Infiltration Solution (RE-PIS)
Introduction
As an important part of the theory of soil water movement, the Richards equation and the Philip
solution for infiltration should be developed as the basic functions in the soil physics toolbox. The main
task that the tool will perform is to use the same numerical methods that Philip used in his research
(Philip, 1957) to solve the Richards equation.
There are some existing Matlab infiltration models based on the solution equation.
(Na’amani)While parameters like 𝐾(𝜃) have to be assigned before running. They are usually calculated
or referred from other resource. The tool will use the initial and boundary conditions to estimate the
cumulative infiltration I and the infiltration rate i. The sorptivity S will be calculated in the running
process, which will be easily viewed. The Philip infiltration functions will also draw figures showing soil
infiltration profiles.
Theory
The primary equation that will be employed is the Richards equation for transient water flow,
𝜕𝜃
𝜕
𝜕𝜃
𝜕𝐾(𝜃)
=
(𝐷𝑤 (𝜃) ) +
𝜕𝑡
𝜕𝑧
𝜕𝑧
𝜕𝑧
(1)
where 𝜃 is the soil water content, 𝑡 is the time, 𝑧 is the gravitational potential head, 𝐷𝑤 (𝜃) is the soil
water diffusivity, and 𝐾(𝜃)is the unsaturated water conductivity.
The soil hydraulic properties 𝐾(𝜃) and 𝐷𝑤 (𝜃) can be estimated from many different models.
The van Genuchten model (van Genuchten, 1980)will be coded and used in this study.
Set 𝑥 = −𝑧 for convenience, then Equation (1) becomes
𝜕𝜃
𝜕
𝜕𝜃
𝜕𝐾(𝜃)
=
(𝐷𝑤 (𝜃) ) −
𝜕𝑡
𝜕𝑥
𝜕𝑥
𝜕𝑥
(2)
By applying the transformations 𝑦 = 𝑥 − 𝑥 ′ , 𝜒 = 𝑦 ′ 𝑡 −1, the solution is found as a power series
in 𝑡1/2 :
𝑚
𝑥 = 𝜑𝑡1/2 + 𝜒𝑡 + 𝜓𝑡 3/2 + 𝜔𝑡 2 + ⋯ + 𝑓𝑚 (𝜃)𝑡 2 + ⋯
(3)
in which the coefficients 𝜑, 𝜒, 𝜓, 𝜔 ⋯ 𝑓𝑚 are functions of 𝜃, which can be solved by numerical
methods.(Philip, 1955) Then the cumulative infiltration I and the infiltration rate i can also be expressed
as infinite series:
𝐼 = 𝑆𝑡1/2 + 𝐴1 𝑡 + 𝐴2 𝑡 3/2 + ⋯
(4)
𝜃
in which the sorptivity 𝑆 = ∫𝜃 𝑛 ∫ 𝜑 𝑑𝜃 (𝜃0 𝑎𝑛𝑑 𝜃𝑛 are boundary conditions). 𝐴1 and 𝐴2 can be solved in
0
the same way.
Applications
As some of the basic functions in the soil physics toolbox, RE-PIS will contain to main functions –
the “vanGenuchtent” and the “RePis”. The first one will be a complementary function for an existing
function “vanGenuchten”. (Flores A.N., 2010) The input parameters will be soil water content instead of
the potential head.
The expected end-users for this tool will be researchers, teachers and students who need to use
the Philip infiltration model. The functions could also be called by other soil physics functions.
References
Philip, J. R. (1955). Numerical Solution of Equations of the Diffusion Type with Diffusivity ConcentrationDependent. Transactions of the Faraday Society, 51(7), 885-892.
Philip, J. R. (1957). The Theory of Infiltration: 1. the Infiltration Equation and Its Solution. Soil Science,
83(5), 345-358.
van Genuchten, M. T. (1980). A Closed-Form Equation for Predicting the Hydraulic Conductivity of
Unsaturated Soils. Soil Science Society of America Journal, 44(5), 892-898.
Na’amani, L. MATLAB - Horton & Philips Models. Retrieved from http://www.loainaamani.com/Academics/MATLAB_Hoton_and_Philips.htm
Flores, A.N. van Genuchten code. (2010). Retrieved from
http://earth.boisestate.edu/lejoflores/2010/11/07/van-genuchten-code/
Please also provide a short name or acronym for the tool.
Matlab Functions for Richards Equation - Philip Infiltration Solution (RE-PIS)
Please provide one or two specific examples and explain how your tool will be an
improvement.
The MATLAB – Horton & Philip Models (http://www.loainaamani.com/Academics/MATLAB_Hoton_and_Philips.htm) provides the Matlab codes for determining
ponding time, infiltration rate etc. The hydraulic conductivity K in the program is approximated by users
by using the saturated hydraulic conductivity. These script files don’t have a very good compatibility for
other programs.
The input parameters of RE-PIS functions are easier for users to determine. The parameters of van
Genuchten equations can be referred from other resource easily. The van Genuchten functions can also
be used in other soil physics problems.
There is another existing Richards equation solver
(http://earth.boisestate.edu/lejoflores/2010/11/07/richards-solver-code/), which can solve the mixed
form of the Richards equation. The RE-PIS solves the water content form of the Richards equation that
can be an addition to Richards equation solvers.
How will K of theta and theta of h (or D of theta) be determined?
They will be determined by the van Genuchten functions, so some additional parameters need to be
provided by users, such as Ks, θs, θr.
Will the boundary conditions be provided by the user?
Yes. The boundary conditions can be soil water content (preferred) or potential head.
Will the tool be for vertical flow only?
Yes. The tool will be developed for vertical flow specifically. If time permits, the function could be
revised to be compatible for the horizontal flow.