1. Chapter 3: Confined Aquifers
Matt Thomas
WASR 8730
1/11/2018

2. Overview Review of confined aquifers
Thiem’s method – (transient) Steady-state
Theis’s method – Unsteady-state
Jacob’s method – Unsteady-state

3. Confined aquifers
Aquifer unit bounded above and below by low permeability confining units
Water is generally pressurized, causing well water levels to often rise above the upper confining unit
Investigate piezometric surface as opposed to the water table (for unconfined aquifers)

4. Confined aquifers Important quantities
Distance from well to piezometer i (ri)
Drawdown in piezometer i (si)
Depth of aquifer (D)
Discharge from pumping well (Q)

5. Assumptions Aquifer is confined Infinite areal extent
Homogenous, isotropic, constant thickness
Piezometric surface is horizontal prior to pumping
Aquifer pumped at constant rate
Well is fully penetrating and screened over entire thickness of aquifer
Water removed from storage is discharged instantaneously with head decline (unsteady-state)
Well diameter is small (unsteady-state)

6. Note In a confined aquifer, water must come from storage
In theory, this means must always have unsteady-state flow
In practice, if Δs is small can treat conditions as steady-state

7. Study area
We’ll use data from a pump test in the Oude Korendijk polder in The Netherlands
Aquifer unit is from 18 m to 25 m depth

8. Thiem’s method (1906) Well discharge may be expressed as
More commonly expressed in terms of drawdown (sm1 – sm2) instead of head
Assumptions: 1-6, steady-state flow

9. Thiem’s method (1906) – one piezometer
If drawdown data is only available from one piezometer and the well itself, can use
Where smw is drawdown in the well
Not an ideal method; always drill multiple piezometers if possible

10. Thiem’s method (1906) – procedures
Plot time-drawdown curves on semi-log plot (time on log scale)
Check that late-times are approx. parallel for all curves
This means steady-state assumption is met
Plug drawdown data for a given late-time into Thiem equation in terms of drawdown and solve for KD (recall T = KD)
Repeat for all combinations of piezometers and calculate the mean KD

11. Thiem’s method (1906) - procedures
Plot distance-drawdown curve on semi-log plot (dist. on log scale)
Calculate slope of line as drawdown per log-cycle r (Δsm)
Plug into Thiem equation as , solve for KD

12. Thiem’s method (1906) - notes
Can use Thiem’s method with either time-drawdown or distance-drawdown data
Often times the cone of depression will continue to deepen over whole pumping period
Thiem’s method still valid as long as rate is constant in all piezometers
This is called transient steady-state flow
Usually quicker, easier, and cheaper to achieve in a pump test

13. Theis’s method (1935) For unsteady-state flow
Includes time and storativity
Recall: Storativity is a function of specific storage and the area of the aquifer
Specfic storage is in turn a function of the elasticities of the mineral grains and water, as well as the porosity of the material
General idea: discharge can be calculated as the product of rate of head decline and storativity summed over the area of effect
Assumptions: 1-8, unsteady-state flow

14. Theis’s method (1935)
Mathematically,

15. Theis’s method (1935) – well function
W(u) is called the “well function of u”
W(u) is called “dimensionless drawdown”, u is called “dimensionless time”
Can generally calculate u as in the previous slide, then look up the value of W(u) for a given u

16. Theis’s method (1935) – curve fitting
If have s, r, and t values then can determine S and KD
Problem: have two unknowns and an exponential integral
This prevents a traditional explicit solution
Must use the curve-fitting method to solve

17. Theis’s method (1935) – curve fitting
Rewrite as
Rewrite as
Note that the first term in each equation is constant
Plot s vs. t/r2 and W(u) vs. 1/u on same log-log plot
The two curves will be of the same shape
Offset horizontally by Q/4πKD and vertically by 4KD/S
See pg. 63 for more info

18. Theis’s method (1935) - notes
Don’t consider early-time data
Assumptions involved in this and many other models requiring curve-fitting are not valid in early-times
If plot of data produces a flat curve, the solution is indeterminant

19. Jacob’s method (1946) For unsteady-state flow
Assumptions: 1-8, unsteady-state, u < 0.1
Theis:
As t increases and r decreases, u decreases
When t sufficiently large enough and r sufficiently small enough, all terms after ln(u) become negligible
For u < 0.1, can approximate Theis equation as

20. Jacob’s method (1946) From here, can go three different directions
r = constant
t = constant
Use t/r2
All follow same general pattern:
Plot X-drawdown data, where X is the non-constant variable and on log scale
Fit a straight line to the data
Find the x-intercept (X0) and slope (Δs per log-cycle X)
Plug into relevant equations and solve for KD and S
Be sure to verify that u < 0.1 with the calculated values!

21. Jacob’s method (1946) Data Procedure in text Equations Notes
r = constant
3.4
Use piezometers at moderate distances
Should be close agreement among KD and S values
t = constant
3.5
Need data from at least three piezometers
Repeat for several t
Should have agreement among calculated values
s vs. t/r2
3.6

22. Summary
Three methods shown generally agree with each other

23. AQTESOLV