Horizontal Infiltration using Richards Equation.

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Horizontal Infiltration using Richards Equation.

• The Bruce and Klute approach for horizontal
  infiltration

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Richards Eq lets derive it again for kicks

• Richards Equation is easy to derive, so let's do
  it here for one-dimensional horizontal flow.
• For horizontal flow Darcys law says
• The conservation of mass tells us
• the time rate of change in stored water is equal
  to the negative of the change in flux with
  distance (i.e., an increase or decrease in flux
  with distance results in respective depletion or
  accumulation of stored water)

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recall and

• Taking the first derivative of Darcys law with
  respect to position,
• and substituting the result from the conservation
  of mass for the left side
• Richards Equation for horizontal flow.

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Richards Eq

• Using the definition of soil-water diffusivity
• this may be written in the form of the diffusion
  equation
• Our goal to solve this rascal for horizontal
  infiltration into dry soil

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So what are the rules here?

• If we find a solution to Richards Equation
  which satisfies the boundary and initial
  conditions, then we have the unique solution.
• The Green and Ampt solution had a one-to-one
  relationship between the square root of time and
  position.
• With that motivation, lets introduce a
  similarity variable" referred to as either the
  Boltzman or Buckingham transform

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Now putting the equation in terms of ?

• We are going to write Richards Eq. with ? in
  place of t and z. We need to calculate the
  substitutions for the derivatives. For z
• For t

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We have and
and

• Using the expressions for dz and dt, we see that
  the right side of Richards Eq is
• NOTE The partial derivatives are now simple
  derivatives since there is only one variable in
  the similarity version of the equation.
• The left side can be put in terms of ?

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Finishing the substitution

• Putting this all together we find
• multiply
• by t

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We have in terms of ??

• Multiplying each side by d? and integrating from
  ? ? i to ? we obtain
• where ? is the dummy variable of integration.
  This may be rearranged
• The Bruce and
• Klute Eq.!

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Wait! That integral is constant!

• If ?i is zero (initially dry soil), the integral
  is identified as a soil-water parameter
• which is referred to as the soil sorptivity.
• For infiltration into initially dry soil the BK
  Eq. is
• Pretty Simple! Clearly solutions to will depend
  on the form of D(?) and S(?).

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Now what? Need D and S!

• What forms of the function D(?) allow for
  analytical solutions?
• Philip (1960 a, b) developed a broad set of forms
  of D(?) which produce exact solutions.
• Brutsaert (1968) then provided an expression for
  diffusivity which fit well to natural soils and
  allow solution

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Using the Brutsaert DB

• n and Do determined experimentally for soil.
• 1lt n lt 10, depending on pore size distribution
• Do is the diffusivity at saturation
• Using this for D(?), BK is generally solved by

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THE SOLUTION!!!

• This may be easily checked by putting the
  solution into BK and turning the crank.
• This equation gives the exact solution for the
  shape of the wetting front as a function of time.
  
• Exactly the information that we could not get out
  of the Green and Ampt approach.

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• Horizontal infiltration as a function of n for a
  Brutsaert soil with Do 1, and n 2, 5, and 10.
  The wetting front becomes increasingly sharp as
  n increases, making the pore size distribution
  narrower.

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HYDRUS-2D Simulations of Horz. Infil.

• Plotting moisture content vs position above and
  moisture content vs x/t1/2 on the lower plot
• They fit the Boltzman transform!
• Also recall that in Miller similarity time scales
  with the square root of macroscopic length
  scalenot bad!

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Now about that sorptivity ...

• Can also solve for the S. Using Brutsaerts
  equation and the definition of S
• Lets pull out the constant

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Computing the Sorptivity ...

• So we have the result
• Which is easy to integrate to obtain
• Most often sorptivity is reported for saturated
  soil, ? 1

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Why bother (with S)?

• Suppose we want to calculate the infiltration
• Integrating the moisture content over all
  positions at a given time
• We can evaluate the same integral by switching
  the bounds of integration so that we integrate
  all positions over the moisture content
• or in
• of ??

?
x
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Computing cumulative infiltration

• Which is just
• which is exactly the form obtained by Green and
  Ampt! (i.e. square root of time)
• Can calculate the rate of infiltration
• Again, identical Green and Ampt!

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OK, but what is sorptivity?

• A parameter which expresses the macroscopic
  balance between the capillary forces and the
  hydraulic conductivity.
• Recall From the discussion of the Green and Ampt
  results that
• Ksat goes up with ?2 and ?f goes with 1/? (? is
  the characteristic microscopic length scale, for
  instance d50), then we can guess that sorptivity
  will get larger for coarser soils, but only with
  ?1/2

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Miller Scaling Big Time

• From the definition of S(?) we know that
• where S is the sorptivity, D is the diffusivity,
  K is the conductivity, ? is the moisture content
  and ? is the Boltzman transform variable, ?
  xt-1/2

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Miller Scaling S

• To derive the scaled value of sorptivity we must
  replace the variables with the appropriate scaled
  quantities
• (L is macro-
• scopic scale)
• We find

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Wrapping this up ...

• S scales with ?1/2, (Just as we saw in the Green
  and Ampt Sorptivity).
• As a little bonus, we see what the effect of
  changing fluid properties would produce in the
  value of sorptivity