Advanced Hydrology Course Note
Lecture 4: Unsaturated-zone Soil Moisture Dynamics
May 6, 2010
Total head: H = h – Z (Z defined as “positive downward”);
if saturated: (pressure); if unsaturated: (suction)
So h is “pressure head” (if +) or “suction head” (if -)
Conservation of Mass
: Volumetric Water content
For unsatureated flow:
is the unsaturated hydraulic conductivity
is capillary pressure head or capillary potential head (a negative sign indicates hydraulic suction)
Define Soil Water Diffusivity :
1. Richards Equation [1931] (Physical)
Buckingham [1907] was probably the first to postulate that Darcy’s Law is also valid for a soil that is only partly saturated with water, and that in this case the hydraulic conductivity is a function of soil water content.
The physics governing the movement of liquid water are described by the Richards equation [1931] derived as follows:
Assume one-dimensional (1-ED) flow, unique relationship (no hysterisis):
or
------Richards Equation
2. Soil Mositure Characteristic Curve
(“Water Retention Curve”)
(The rate of change of soil moisture content with respect to capillary pressure head )
a. Gardner[1958]:
where is the saturated conductivity
is the saturated soil moisture content
is the specific moisture capacity (i.e. the slope of the soil-water retention curve)
represents the relative rate of decrease of hydraulic conductivity with increasing capillary pressure head (becomes drier and drier). It is associated with the width of soil pore size distribution. is the thickness of capillary fringe and it measures the relative importance of capillary force to gravity force for soil moisture movement in a specific soil. Fine-textured soils in which capillary force tends to dominate have greater thickness of capillary fringe than coarse-textured soils, in which gravity effects manifest themselves most readily. The range of covers from 0.01 to 10 meters. However, 0.2-5 meters seems to be the typical values for according to the study of Philip[1969].
b. Brooks an Corey [1964]:
Where is the soil capillary potential () at saturation, also called air-entry capillary potential. B is an empirical constant depending on the soil type. The normal range of B is between 3 ~ 12, and clayey soils have larger B values than sandy soil.
The Richards equation can be re-written in terms of soil saturation degree s
where n is the soil porosity and D is the soil diffusion coefficient. The two terms inside the parenthesis are the the capillary diffusion flux and gravity drainage flux, respectively, both of which has highly nonlinear dependence on the soil saturation s. This form of Rechards Equation is what most of current Land Surface Models (LSMs) have used after Manabe’s Bucket’s Model [1968], which is the first LSM.
c. Mualem–van Genuchten [1980]
where
θis the soil water content [ - ]; ψ is suction pressure head ([L], e.g. cm of water);
θssaturated water content [ - ]; θrresidual water content [ - ];
αis related to the inverse of the air entry suction,α > 0([L−1], or cm−1); and,
nis a measure of the pore-size distribution,n> 1[ - ].
Water retention curve calculated diagram of soil with model formula (van Genuchten, 1980)
ψ
Soil / / / / nSs / 0.0403 / 0.37068 / 0.08742 / 1.57535
Uu / 0 / 0.42125 / 0.00340 / 1.34447
Lu / 0 / 0.42121 / 0.01334 / 1.12614
Tt / 0 / 0.55054 / 0.00681 / 1.08155