Ex4: Richards’ Equation
Due Date: Nov. 18, 2003
Answer the questions at the end of this document.
Plea follow the format of scientific papers.
(Modified from Hornberger et al., 1998. Elements of Physical Hydrology) Introduction—Vertical Water Movement in the Unsaturated Zone
Section 8.5 prents the Richards' equation which governs vertical flow of water in the unsaturated zone. In partially saturated porous media, Darcy's law is still applicable—water flows from high to low hydraulic head. However, the hydraulic conductivity (as well as the moisture content) are functions of the pressure head. As the pressure head decreas, the moisture content decreas and the hydraulic conductivity decreas. The functions K(ψ) and θ(ψ) are characteristic of a particular material. In this exerci, we will examine the infiltration and re-distribution of water following a precipitation event. The problem domain is shown in Figure 1. The soil column is two meters deep, with the water table (where the pressure is zero) at the bottom. Initially, the hydraulic head everywhere in the column (at and above
the water table) is assumed to be zero. In other words, there is no movement of water. The soil is assumed to be homogeneous.
Figure 1 Depiction of terms appearing in the simulation of vertical flow in the
unsaturated zone.
There are four possible soil types that we can u in our simulations. The are soils that have been well-studied, and for which we have the data necessary to construct the characteristic curves (Figure 2). The characteristic curves shown are functions fitted to the data, using the van Genuchten (1980) approach.
Figure 2 Characteristic curves for four soil types. The relative hydraulic conductivity (K r) is shown in "a"; K r [dimensionless] is multiplied by the saturated hydraulic conductivity (K sat) to obtain the actual hydraulic conductivity (K, shown in "b"). The
volumetric moisture content (θ) is shown in "c". When the pressure is zero (the material is saturated), θ is equal to the porosity of the material; as the pressure gets very small, θ approaches a constant value, the irreducible moisture content. Another variable ud in calculations is the specific moisture capacity, shown in "d", which is
equal to ∂θ/∂ψ.
Using a finite-difference approximation and a predictor-corrector time-stepping scheme, we can solve the one-dimensional Richards' equation and look at the pressure head, hydraulic head, and volumetric moisture content throughout the 2-m soil column. First, we will need to lect the type of material ud for the simulation (1—Hygiene sandstone, 2—Touchet silt loam, 3—Silt loam, 4—Guelph loam). The simulation is for 12 hours, but infiltration occurs only during the first three hours. Once the infiltration rate during the first three hours has been t, all that is left is to execute the MATLAB M-file "RICHARDS.M". Simply lect "Evaluate M-book" from the Notebook menu (or u t
he appropriate keyboard equivalent). Two figures will be shown below the input table; the hydraulic heads on the left and the volumetric moisture content on the right. Results are shown for the following times: 0, 3, 6, 9, and 12 hours. The results for 0 hours reprent the initial conditions. For the volumetric moisture content plots, the x-axis will always range between the irreducible moisture content and the saturated moisture content (or porosity).
Richards' Equation Solution
Parameter Material type, TYPE Infiltration rate (cm hr −1), IN
Range, increment 1–4, 1 0–2.5*, 0.1
MATLAB
TYPE =1;IN = 2.0;>>TYPE=1
>>IN=2.0
>>RICHARDS
* Note that an upper limit for infiltration rate exists for some of the soil types. -120
-100
-120-100
Questions
1. Examine the characteristic curves (Figure 2) for the four soil types. Which soil is the
most permeable (when saturated)? Which is the least permeable? Do you expect a capillary fringe to develop in any of the materials? What can you hypothesize
about the distribution of pore sizes in each soil?
2. Run simulations for each soil type with no infiltration (IN = 0). How thick is the
capillary fringe for each soil type? Is there any vertical movement of water?
3. U a fairly small infiltration rate (IN = 0.1–0.5 cm hr −1) and run the simulation for each soil type. Do any of the soils become completely wet at the surface, that is, does the volumetric moisture content approach the porosity? Does water quickly or slowly reach the water table or capillary fringe?
What happens once infiltration has stopped (after three hours)? What happens if you increa the infiltration rate (IN =
1.0–3.0 cm hr−1; remember that an upper limit for infiltration rate exists for some
soil types)?
References
van Genuchten, M. T. 1980. A clod-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Science Society of America Journal 44: 892-898.
