Chapter 3: Flow of Water in the Vadose Zone

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Chapter 3 Flow of Water in the Vadose Zone

• The classic solutions for infiltration and
  evaporation by Green and Ampt, Bruce and Klute,
  and Gardner

Williams, 2002
http//www.its.uidaho.edu/AgE558 Modified after
Selker, 2000
http//bioe.orst.edu/vzp/
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The Green Ampt model for Infiltration

• Conceived of in the framework of a capillary tube
  bundle concept
• Media is modeled as a bundle of capillary tubes
  oriented in the direction of infiltration.
• Tubes fill in parallel to the same depth (i.e.,
  the wetting front is at a well-defined position,
  the same in all tubes).

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Goals

• May be employed in calculating the infiltration
  into soils which proceed at any angle relative to
  vertical by simply adjusting the magnitude of
  gravity.
• We will carry out some 1-dimensional calculations
• Calculate the horizontal rate of infiltration
  into a dry soil.
• Solve for ponded vertical infiltration.
• Solve for falling head vertical infiltration.
• Note This conceptual approach can be
  applied to multidimensional problems.

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The basic idea

• Infiltration is driven by two forces
• Force of gravity acting on the water.
• Pressure drop produced by a sharp wetting front

• Assumptions
• Wetting front is completely sharp
• Soil is saturated behind wetting front

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The wetting front potential term

• Where does this term come from and what is its
  magnitude?
• Lets take the case of soil made up capillary
  tubes of N sizes,
• Each of the sizes makes up a fraction ai of the
  total cross-sectional pore area. The porosity n
  is

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Continuing on the wetting front potential

• Part of the potential gradient is generated by
  the capillary contacts.
• Pressure due to capillary contacts will result in
  a force per unit area.
• The force is proportional to the length of
  solid/liquid contact per unit area.
• For a unit of cross-sectional area, total length,
  la, of water/solid contact is

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Wetting Front Potential, cont.

• Assume a contact angle of ?f
• The pressure per unit area, Pf, at the
  water/solid interface is
• Resistance to flow? Poiseuille capillary flow!
  For the ith tube, we have the flow per unit area,
  qi, of
• Where ?P is the pressure drop across a capillary
  tube of length L with a fluid flowing with
  viscosity ?.

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Computing flux

• For our capillary bundle the total flux would be
• If the flow is horizontal, then the driving force
  is simply the surface tension, so the pressure
  drop in the tubes is Pf and the flux into the
  system is given by
• For convenience, lets define the system
  constant

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• We recognize K as the saturated hydraulic
  conductivity. In the customary units of pressure
  head ?ghf Pf 3.8 is
• Now hang on there! That looks like Darcys Law!
  If we have a soil with wetting front potential hf
  and conductivity K, then the flux would be
• So we have just found Darcy and seen how the
  terms relate to capillary properties.

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.. Solving for infiltration

• The soil is going from dry to saturation, we can
  relate the flux to the rate of movement of the
  sharp wetting front
• Combining 3.10 and 3.11 we obtain a
  differential equation for the position of the
  wetting front in time

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Solve for the position of the wetting front in
time

• integrate to find that
• or, solving for the length of infiltration
• the wetting front advances with the square root
  of time.
• the total infiltration is also proportional to
  the square root of time.

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We may now compute flux

• Recall
• So we may put in the result for L and take the
  derivative
• where SGA (referred to as the sorptivity) is a
  constant dependent on the media, and equal to
  (2nKhf)1/2.
• Key observation
• Horizontal flux decreases proportionally to one
  over the square root of time.

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What are effects of soil texture

• Plugging in our previous results
• To help see how SGA relates to soil
  characteristics, make the vastly simplifying
  assumption that the soil has only one pore
  radius. SGA becomes
• where C is a constant which depends only on the
  fluid. This then tells us that for our
  simplified soil that the infiltrating flux will
  be given by

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Infiltrating flux varies as ?r
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Lets look at this...

• Infiltrating flux goes down with the square root
  of the radius.
• Compared to conductivity which decreases with r2.
  
• Why the difference?
• The larger surface area of the particles in the
  finer soil provides a large capillary pull into
  the soil, which is almost enough to overcome the
  lower hydraulic conductivity.

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Next stop Vertical infiltration

• Essentially as easy as solving for horizontal
  infiltration.
• Ponded infiltration Darcy's law for the system
  is simplified by the fact that the wetting front
  is assumed to be sharp, with saturation behind
  the front
• The head loss across the system is the sum of the
  potential at the wetting front, hf, the depth of
  ponding, d, plus the depth of infiltration, L,
  while the length of travel is L

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Vertical, constant head infiltration

• Separating variables
• which may be integrated to obtain
• Square root of time behavior for horizontal
  infiltration replaced with logarithmic
  relationship.
• Note taking Taylor expansion at short time, the
  horizontal and vertical results become identical

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Vertical falling head...

• Same procedure can solve for infiltration under
  falling head.
• In this case, we replace the depth of ponding, d,
  with this depth minus the depth of water which
  has infiltrated, nL (where nLltd, after which the
  pond is gone). Using (3.20) as before, we obtain

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Typical Results

• Green and Ampt model predictions for infiltration
  in a soil with 30 pore space, Ks 0.03 cm/sec,
  hf 25 cm and d 20 cm.

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Field Measurement of GA Parameters

• Bouwer Infiltrometer
• Get Ks by falling head
• Get hf by shutting valve and measuring the vacuum
  created by the wetting front pressure
• Measured Ks about 1/2 fully saturated Ks
• Measured hf really the air entry pressure which
  is 2 times water entry pressure