# 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
component

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)
4T
- Where:
W (ua, ub, gamma) = the well function for a water table aquifer

uA = r2S    = early time drawdown data
4Tt

uB = r2Sy = late time drawdown data
4Tt

Γ = r2 Kv
b2 Kh
2
Variables:             Kv = vertical hydraulic conductivity
Kh = horizontal hydraulic conductivity
b = initial saturated thickness of the
aquifer
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)

Comments:
 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).

2

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