z = -50 cm, ? = -100 cm,

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define z 0 at soil surface h z ? 0
-200cm -200cm
25 cm
z -25 cm, ? -100 cm h z ? -25 cm
-100cm -125 cm
25 cm
z -50 cm, ? -100 cm, h z ? -50cm
-100cm -150 cm Which direction will water
flow?
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Water Balance of a soil element Input
- Output
? Storage ?w? qz ? ?x? ?y??t - ?w? (qz
?qz )? ?x? ?y??t ?w? ????x??y ??t ??z ?z
? t ?qz ?? ?z ? t
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Darcys Law describing the rate of water movement
in the vertical (z) direction through an
unsaturated porous medium, upward flow is
negative, downward flow is positive qz Kh(?)
d(z ?(?)) Kh(?) dz d ?(?) Kh (?) Kh
(?) d ?(?) dz dz
dz dz Where qz rate
of water movement per unit cross sectional area
(length/time) Kh(?) unsaturated hydraulic
conductivity (length/time), a function of soil
moisture content (? ) z elevation above an
arbitrary benchmark (length) ?(?) matric
potential (length), a function of soil moisture
content(?) ?qz ? Kh (?) ? Kh (?) ?
?(?) ?z ?z ?z ?qz ?? ?z
? t
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• Richards Equation of soil moisture change and
  movement over time
• ?? ? Kh(?) ? ?(?) - ? Kh(?)
• ?t ?z ?z ?z
• ? volumetric moisture content
• t time
• Kh(?) hydraulic conductivity at the current
  moisture content
• ?(?) soil matric potential at the current
  moisture content
• z positive distance in the downward flow
  direction
• z -z, where z elevation, so that
• ?z -1
• ?z

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• Richards Equation, difference approximation for
  small changes
• ?? ? ? Kh(?) ??(?) - ? Kh(?)
• ?t ?z ?z
• ?z
• ? volumetric moisture content
• t time
• Kh(?) hydraulic conductivity at the current
  moisture content
• ?(?) soil matric potential at the current
  moisture content
• z positive distance in the downward flow
  direction
• z -z, where z elevation, so that
• ?z -1
• ?z

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One Dimensional Soil Profile Continuous variation
in soil properties can be represented by nodes or
points that represent the center of a soil layer
For each node, the elevation and soil properties
(? , K(? ), ?(?)) are defined to reflect the
actual conditions, and the Richards Equation used
to describe how soil moisture at each level will
change as function of the points above and below
for a small increment of time. The resulting
equations (one for each node) are solved
simultaneously for each time step. The results
for each time step are used to calculate how soil
moisture will move in the next time increment.
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Green and Ampt Approach to simulating
infiltration of water into the soil
surface Initial assumption is soil water content
is uniform in the profile at ? ?o Stage 1
Infiltration rate water input rate Stage 2
Infiltration rate lt water input rate, soil
surface becomes saturated (? ?), and the
wetting front moves into the soil profile.
Above the wetting front, the soil is saturated,
below the wetting front the soil is at ?o. The
depth of water infiltrated at a given time will
be F(t) (?- ?o)zf(t) where zf(t) the
depth that the wetting front has penetrated into
the soil
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To apply Darcys law, the saturated soil at the
surface has zero tension, but it will have a
positive pressure if there is water ponded on
the surface. At the leading edge of the wetting
front, the water is being drawn into the soil by
the soil water tension at the wetting front ?f
which is considered to be a function of the soil
properties.
Soil surface
h depth of ponding H
? 0
K saturated hyd. Cond.
Wetting front
z zf(t), ? ?f h zf(t) ?
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Applying Darcys law in difference
formulation qz Kh(?) ?(z ?(?)) ?
z Kh(?) can be replaced with saturated
hydraulic conductivity Kh And also
recognizing that ? z zf(t) where zf(t) the
depth that the wetting front has penetrated into
the soil And recalling that F(t) (?-
?o)zf(t) And doing a number of substitutions
and reorganizations leads too..
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• Green and Ampt Infiltration Equation after
  ponding at the soil surface
• f(t) f(t) Kh 1 ?f (? - ?0)
• F(t)
• Where
• f(t) rate of water infiltration into the soil
  (cm/sec)
• f(t) rate of water infiltration into the soil
  after ponding (cm/sec)
• Kh saturated hydraulic conductivity (cm/sec)
• ?f matric potential at the wetting front
• porosity of the soil
• ? 0 initial soil moisture content
• F(t) cumulative water infiltrated into the
  soil.

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Influence of water input rate (e.g., rainfall or
irrigation)
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Influence of initial water content on
infiltration rates
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• In practice many hydraulic parameters such as
• Kh(?) hydraulic conductivity as a function of
  moisture content
• ?(?) soil matric potential as a function of
  moisture content
• Kh saturated hydraulic conductivity (cm/sec)
• ?f matric potential at the wetting front
• porosity of the soil
• are estimated from pedotransfer functions which
  relate the
• above quantities, which are unmeasured in most
  soils, to
• some characteristics that are commonly measured.
  So
• known characteristics, such as clay content,
  silt content,
• organic matter content, or soil depth may be used
  to estimate
• these difficult to measure and often unmeasured
  characteristics
• such as Kh(?), the hydraulic conductivity as a
  function of
• moisture content.

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Limitations of applying Darcys Law in
unsaturated soils with vegetation (e.g., Richards
Eq. And Green and Ampt Eq.) These are data
intensive, and there are uncertainties about
variations in soil properties with depth and
time. Even when soil properties are reasonably
well known, there is uncertainty about how
Kh(?) and ?(?) vary with soil properties.
Existence of large macro-pores caused by
living, dead and decaying plant roots insects,
worm, and other animal burrows cracks due to
drying or freezing, which can transmit large
volumes of water rapidly may be non-Darcian flow
(rapid and independent of soil
matrix) Nonetheless, Richards equation and the
Green and Ampt equations can simulate average
water flow rates by adjusting Kh(?) and ?(?) to
fit observed water flow, but difficulties are
often encountered when these approaches are used
to simulate water quality if macro-pores are not
taken into account. Flow in macropores carries
the contaminants much, much faster than the
simulated average flow through the soil matrix.
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Examples of spatial heterogeneity in infiltration
Fingered Flow in sandy laboratory medium
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Richards Equation Small time steps are used
(generally less than 1 hour), so the approach
can be computationally intensive, and
requires hourly precipitation data. This can
be valuable for simulating surface runoff
generated by intense storms. While such events
are important, they tend to be rare in well
vegetated landscapes.
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Simulated and measured stream flow using a single
soil profile water balance to represent the
Vermilion River watershed, without using
Richards or Green-Ampt equations.
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Where does surface runoff come from?
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Distribution of Saturated Soils near Danville,
VT March 21, 1973
Source Dunne and Leopold
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Distribution of Saturated Soils Near Danville,
VT August 25, 1973
Variable Source Area Concept
Source Dunne and Leopold, 1978