Flow
(Unsteady) in a Confined
Aquifer - Theis Method
Fetter 7.3-2 7.4-3
Assumptions for transient drawdown effects (C.V. Theis 1935)
| (1.) Aquifer is confined top and bottom. | |
| (2.) There is no source of recharge to aquifer. | |
| (3.) The Aquifer is compressible and water is | |
| released instantaneously from the aquifer as the head is lowered. | |
| (4.) The well is pumping at a constant rate. |
Theis (Non Equilibrium Equation)
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Replace the integral with a Taylor expansion series.
 |
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| Q = Constant pumping rate (L3/T), | ||
| h = Hydraulic head (L), | ||
| h0 = Hydraulic head before pumping starts (L), | ||
| h0 - h = drawdown (L), | ||
| T = Transmissivity (L2/T), | ||
| t = Time since pumping began (T), | ||
| r = Radial distance from pumping well (L), and | ||
| S = Aquifer storativity. | ||
Sometimes the series is referred to as the Well function W(u).
So  |
Confined aquifer system using Theis analysis
Theis Method for getting aquifer parameters from time - drawdown data.
| Nonequilibrium - cone of depression continues to grow. | ||
| Pumping well - T ® K if know b | ||
| Observation well - T + S | ||
| Rearrange Theis equation | ||
 (ft
or m) |
||
| = s drawdown | ||
 |
||
| u = dimensionless constant | ||
Theis Type Curve - non equilibrium type curve
W(u) vs  |
|||
| 1) Plot s as a function of time on log-log | |||
| 2) Place type curve over field data | |||
| 3) Keep axes parallel | |||
| 4) Overlay the two curves | |||
| 5) Select match point | |||
| usually pick point on field data where W(u) = 1 | |||
 |
|||
| find s and t at this point | |||
| 6) Convert minutes to days | |||
| 7) Substitute values into equations | |||
Values of the function W(u) for various values of u
| u | W(u) | u | W(u) | u | W(u) | u | W(u) |
| 1 X 10-10 | 22.45 | 7 X 10-8 | 15.90 | 4 X 10-5 | 9.55 | 1 X 10-2 | 4.04 |
| 2 | 21.76 | 8 | 15.76 | 5 | 9.33 | 2 | 3.35 |
| 3 | 21.35 | 9 | 15.65 | 6 | 9.14 | 3 | 2.96 |
| 4 | 21.06 | 1 X 10-7 | 15.54 | 7 | 8.99 | 4 | 2.69 |
| 5 | 20.84 | 2 | 14.85 | 8 | 8.86 | 5 | 2.47 |
| 6 | 20.66 | 3 | 14.44 | 9 | 8.74 | 6 | 2.30 |
| 7 | 20.50 | 4 | 14.15 | 1 X 10-4 | 8.63 | 7 | 2.15 |
| 8 | 20.37 | 5 | 13.93 | 2 | 7.94 | 8 | 2.03 |
| 9 | 20.25 | 6 | 13.75 | 3 | 7.53 | 9 | 1.92 |
| 1 x 10-9 | 20.15 | 7 | 13.60 | 4 | 7.25 | 1 X 10-1 | 1.823 |
| 2 | 19.45 | 8 | 13.46 | 5 | 7.02 | 2 | 1.223 |
| 3 | 19.05 | 9 | 13.34 | 6 | 6.84 | 3 | 0.906 |
| 4 | 18.76 | 1 X 10-6 | 13.24 | 7 | 6.69 | 4 | 0.702 |
| 5 | 18.54 | 2 | 12.55 | 8 | 6.55 | 5 | 0.560 |
| 6 | 18.35 | 3 | 12.14 | 9 | 6.44 | 6 | 0.454 |
| 7 | 18.20 | 4 | 11.85 | 1 X 10-3 | 6.33 | 7 | 0.374 |
| 8 | 18.07 | 5 | 11.63 | 2 | 5.64 | 8 | 0.311 |
| 9 | 17.95 | 6 | 11.45 | 3 | 5.23 | 9 | 0.260 |
| 1 X 10-8 | 17.84 | 7 | 11.29 | 4 | 4.95 | 1 X 100 | 0.219 |
| 2 | 17.15 | 8 | 11.16 | 5 | 4.7. | 2 | 0.049 |
| 3 | 16.74 | 9 | 11.04 | 6 | 4.54 | 3 | 0.013 |
| 4 | 16.46 | 1 X 10-5 | 10.94 | 7 | 4.39 | 4 | 0.004 |
| 5 | 16.23 | 2 | 10.24 | 8 | 4.26 | 5 | 0.001 |
| 6 | 16.05 | 3 | 9.84 | 9 | 4.14 |
Source: Adapted from L.K. Wenzel, Methods for Determining Permeability of Water-Bearing
Materials with Special Reference to Discharging Well Methods. U.S. Geological Survey
Water-Supply Paper 887, 1942.
Effects of well-bore storage neglects when
 |
|
| rw = radius of pumping well | |
 |
|
| r = radial distance between pumping well and observation well |
The effect of well-bore storage in the pumped well on the theoretical
time-drawdown plots of observation wells or piezometers. The dashed
curves are those of Parts A and A’ of Figure 2.12
Discharge Measuring Devices
| 1) Commerical water meter | ||
| 2) If a tall ditch ® flume | ||
| 3) Container - bucket, drum | ||
| 4) Orifice Weir | ||
| - perfectly round hole in the center of a circular pipe, fastened to end of level pipe | ||
| - place a piezometer tube 61 cm from orifice plate | ||
| - water level in piezometer represents pressure in discharge pipe | ||
| 5) Open pipe flow | ||
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ENV 302 - Lectures