Lecture 14

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					Hydrogeology                                                                         Lecture #14
Neuman solution for non-steady flow in an unconfined aquifer        Reading: Fetter, pp. 164-169

- Last time: Covered non steady-state flow: Theis solution for non-leaky confined aquifers.
- Today: We will continue with another examples of non steady-state flow: pumping an
         unconfined aquifer.
- Remember the issue: For each type of aquifer, how much drawdown would occur at a radial
         distance "r" from the pumped well?

I) Radial flow in an unconfined aquifer:
       - The one case that we haven't dealt with yet
       - Still uses the basic flow equation, but now we have a some new variables:
                - Saturated thickness of the aquifer changes with continued pumping
                - Some water is contributed by specific yield
                        (in confined aquifer: was only contributed from storativity)
                - Introduces a vertical component to flow
                - In confined aquifers: all flow was assumed to be horizontal

       A) Flow description:
              - Initial flow: is Theis-like (horizontal): from release of compression,
                       specific storage component
              - Intermediate flow: has a gravity drainage component (both horizontal
                       and vertical flow)
              - Late-time flow: becomes horizontal again: from specific yield

       B) Flow equation:
              - Solved by Neuman (1972, 1973, 1974, 1987), among others
              - Involves a two-part solution:
                      - Early time solution: for specific storage (storativity) component
                      - Late time solution: for specific yield component
                               ho - h = Q W (uA, uB, gamma)
              - Where:
                      W (ua, ub, gamma) = the well function for a water table aquifer

                      uA = r2S    = early time drawdown data

                      uB = r2Sy = late time drawdown data

                      Γ = r2 Kv
                          b2 Kh
              Variables:             Kv = vertical hydraulic conductivity
                                     Kh = horizontal hydraulic conductivity
                                     b = initial saturated thickness of the
                                     Sy = specific yield (dimensionless)
                                     S = storativity (dimensionless)

C) Steps to a solution:
       1) Solve for uA, uB
       2) Take inverse of uA, uB (see appendices 6A, 6B)
       3) Solve for gamma
       4) Calculate an early-time solution (based on uA)
       5) Calculate a late-time solution (based on uB)

          This is an awkward solution: there isn't a fixed boundary or validation for
            "early time" and "late time"
          There are many (several) different approaches and mathematical solutions
            to this problem.
          In another week: we will construct an empirical solution based on well
            (drawdown) data. Then: we will work the problem backward, generate a
            curve of drawdown vs. time, read parameters from the curve.

D) Assumptions with Neuman's solution:
      1) First water: comes from instantaneous release from elastic storage
      2) Water pumped later in time comes from gravity drainage, storage in
              interconnected pores
      3) Specific yield must be > (10) (elastic storage (Ss))
      4) Hydraulic conductivity MAY be different in horizontal and vertical directions
              (but doesn't have to be different)

- Summary: It very difficult to calculate drawdown in an unconfined aquifer using just
      one well (this requires both an early time and a late time solution).


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