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Steady Evaporation from a Water Table

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lower bound of this integral is taken. at the water table where h(0) = 0. 7. What next? ... Our integral becomes. which may be integrated to obtain. 10. All Right! ... – PowerPoint PPT presentation

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Title: Steady Evaporation from a Water Table


1
Steady Evaporation from a Water Table
  • Following Gardner
  • Soil Sci., 85228-232, 1958

2
Why pick on this solution?
  • Of interest for several reasons
  • it is instructive in how to solve simple
    unsaturated flow problems
  • it provides very handy, informative results
  • introduced widely used conductivity function.

3
The set-up
  • The problem we will consider is that of
    evaporation from a broad land surface with a
    water table near by.
  • Assume
  • The soil is uniform,
  • The process is one-
  • dimensional (vertical).
  • The system is at steady state
  • Notice that since the system is at steady state,
  • the flux must be constant with elevation, i.e.
    q(z) q.

4
Getting down to business ..
  • Richards equation is the governing equation
  • At steady state the moisture content is constant
    in time, thus d?/ dt 0, and Richards equation
    becomes a differential equation in z alone

5
Simplifying further ...
  • Since both sides are first derivatives in z, this
    may be integrated to recover the Buckingham-Darcy
    Law for unsaturated flow
  • or
  • where the constant of integration q is the
    vertical flux through the system. Notice that q
    can be either positive or negative corresponding
    to evaporation or infiltration.

6
Solving for pressure vs. elevation
  • We would like to solve for the pressure as a
    function of elevation. Solving for dz we find
  • which may be integrated to obtain
  • h' is the dummy variable of integration
  • h(z), or h is the pressure at the elevation
    z
  • lower bound of this integral is taken at the
    water table where h(0) 0.

7
What next? Functional forms!
  • To solve need a relationship between
    conductivity and pressure.
  • Gardner introduced several conductivity
    functions which can be used to solve this
    equation, including the exponential relationship
  • Simple, non-hysteretic, doesnt deal with hae,
  • is only accurate over small pressure ranges

8
Now just plug and chug
  • which may be re-arranged as
  • To solve this we change variables and let
  • or

9
Moving right along ...
  • Our integral becomes
  • which may be integrated to obtain

10
All Right!
  • Solution for pressure vs elevation for steady
    evaporation (or infiltration)from the water table
    for a soil with exponential conductivity.
  • Gardner (1958) notes that the problem may also
    be solved in closed form for conductivitys of
    the form K a/(hn b) for n 1, 3/2, 2, 3, and
    4

11
Rearranging makes it more intuitive
  • We can put this into a more easily understood
    form through some simple manipulations. Note
    that we may write h (1/?)Lnexp(? h), so
    adding and subtracting h
  • gives us a useful form

12
We now see ...
  • Contributions of pressure and flux separately.
  • As the flux increases, the argument of Ln
    gets larger, indicating that at a given
    elevation, the pressure potential becomes more
    negative (i.e., the soil gets drier), as expected
    for increasing evaporative flux.
  • If q0, the second term on the right hand side
    goes to zero, and the pressure is simply the
    elevation above the water table (i.e.,
    hydrostatic, as expected).

13
Another useful form
  • May also solve for pressure profile
  • Although primarily interested in upward flux,
    note that if the flux is -Ks that the pressure is
    zero everywhere, which is as we would expect for
    steady infiltration at Ks.

14
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15
The Maximum Evaporative Flux
  • At the maximum flux, the pressure at the soil
    surface is -infinity, so the argument of the
    logarithm must go to zero. This implies
  • solving for qmax, for a water table at depth z

16
So what does this tell us?
  • Considering successive depths of
  • z 1/?, 2/? , 3/?
  • we find that
  • qmax(z)/Ks 0.58, 0.16, and 0.05,
  • very rapid decrease in evaporative flux as the
    depth to the water table increases.

17
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