Next adventure: The Flow of Water in the Vadose Zone - PowerPoint PPT Presentation

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Next adventure: The Flow of Water in the Vadose Zone

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The Flow of Water in the Vadose Zone The classic solutions for infiltration and evaporation of Green and Ampt, Bruce and Klute, and Gardner The Green & Ampt model for ... – PowerPoint PPT presentation

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Title: Next adventure: The Flow of Water in the Vadose Zone


1
Next adventure The Flow of Water in the Vadose
Zone
  • The classic solutions for infiltration and
    evaporation of Green and Ampt, Bruce and Klute,
    and Gardner

2
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).

3
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.

4
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

5
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

6
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

7
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 ?.

8
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

9
  • 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.

10
So 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

11
Wait, we can solve this!
  • 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.

12
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.

13
Lets look at 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

14
Lets look at this...
  • Infiltrating flux goes with the square root of
    the radius.
  • Compared to conductivity which goes 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.

15
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

16
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

17
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

18
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.

19
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
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