Darcy’s Law



Advanced Hydrology (2011) Course Note

Infiltration

May 19, 2011

Definition [Brustaert, 2005, p.307] – Infiltration can e defined as the entry of water into the soil surface and its subsequent vertical motion through the soil profile.

At the local scale, as precipitated water reached the groundwater surface, infiltration into the soil takes place. In between precipitation events, the atmosphere exerts its drying effect, and the water in the soil profile may move to the surface by vapor diffusion and by liquid capillary rise, where it evaporates.

Infiltration capacity [L/T] – maximum (upper limit of) infiltration rate [L/T].

[Brustaert, 2005, p.308] Infiltration can take place in one of two possible ways. When the surface water supply are resulting from precipitation or other sources is intense enough, part of it remains ponded or runs off, and part of it infiltrates at the maximum rate; this maximum rate of infiltration is the infiltration capacity. When the intensity is low, all of the precipitated water seeps into the pores; this is rainfall infiltration.

A. Horton’s Infiltration Capacity (Empirical)

Infiltration Model [Horton, 1937]:

[pic]

[pic]: Infiltration constant

Infiltration Capacity [L/T]:

[pic]

(Cumulative) Infiltration Amount [L]:

[pic]

B. Darcy’s Law [1856] (Physical)

Darcy’s Law for Saturated Flow:

[pic]

[pic]- Discharge [L/T]

H = h – Z (Z defined as “positive downward”);

[pic] if saturated: [pic](pressure); if unsaturated: [pic] (suction)

So h is “pressure head” (if +) or “suction head” (if -)

Conservation of Mass

[pic]

[pic]: Volumetric Water content

For unsatureated flow: [pic]

[pic] is the unsaturated hydraulic conductivity

[pic] is capillary pressure head or capillary potential head (a negative sign indicates hydraulic suction)

Define Soil Water Diffusivity [pic]:

[pic]

C. 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 [pic]relationship (no hysterisis):

[pic]

or

[pic] -----------------Richards Equation

D. Soil Mositure Characteristic Curve (or called “Water Retention Curve”)

(The rate of change of soil moisture content [pic] with respect to capillary pressure head [pic])

1. Gardner[1958]

[pic]

where [pic] is the saturated conductivity

[pic] is the saturated soil moisture content

[pic] is the specific moisture capacity (i.e. the slope of the soil-water retention curve)

[pic] 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. [pic] 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 [pic] covers from 0.01 to 10 meters. However, 0.2-5 meters seems to be the typical values for [pic] according to the study of Philip[1969].

2. Brooks an Corey [1964]:

[pic]

Where [pic] is the soil capillary potential ([pic]) 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

[pic]

[pic]

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.

Mualem–van Genuchten [1980]

[pic]

where

θ is the soil water content [ - ]; ψ  is suction pressure head ([L], e.g. cm of water);

θs saturated water content [ - ]; θr residual water content [ - ];

α is related to the inverse of the air entry suction, α > 0 ([L−1], or cm−1); and,

n is a measure of the pore-size distribution, n > 1 [ - ].

Water retention curve calculated diagram of soil with model formula (van Genuchten, 1980)

ψ [pic][pic]

Soil |[pic] |[pic] |[pic] |n | |Ss |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 | |

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