1. Chapter 12:Variable-discharge tests and tests in well fields Aquifers may be pumped at variable discharge rates either deliberately or due to characteristics of the pump. Aquifers can be pumped step-wise (always pumped but pumping rates vary) or may be pumped intermittently (not always pumped and pumping rates can vary).

2. Confined Aquifers, Birsoy-Summers Method Birsoy and Summers present an analytical solution for the drawdown response in a confined aquifer that is pumped step-wise or intermittently. They apply the principle of superposition (ch.6) to Jacobs approximation of the Theis equation (3.7) (shown below) Variable-discharge tests

3. The drawdown in the aquifer at time t during the nth pumping period of intermittent pumping is shown by the following expression: where Variable-discharge tests Confined Aquifers, Birsoy-Summers Method where

4. Intermittent Pumping

5. For step-wise (uninterrupted pumping): Variable-discharge tests t (i-1) = t i, and the adjusted time becomes where Confined Aquifers, Birsoy-Summers Method

6. If the intermittent pumping rate is constant (Q=Q 1 =Q 2 =…….Q n ) then the adjusted time becomes: Dividing both sides of equation 12.1 by Q n gives an expression for drawdown: Variable-discharge tests Confined Aquifers, Birsoy-Summers Method

7. Step-wise Pumping

8. Variable-discharge tests Confined Aquifers, Birsoy-Summers Method Assumptions (from Ch.3): 1.)The aquifer is confined 2.)The aquifer has a seemingly infinite areal extent 3.)The aquifer is homogeneous, isotropic, and of uniform thickness over the area influenced by the test 4.)Prior to pumping, the piezometric surface is horizontal (or nearly so) over the area influenced by the test 5.)The aquifer is pumped step-wise or intermittently at a variable discharge rate or is intermittently pumped at a constant discharge rate 6.)The well penetrates the entire thickness of the aquifer and thus receives water by horizontal flow The following conditions are added: the flow to the well is in an unsteady state R is small and t is sufficiently large

9. The sharpest decrease in discharge occurs soon after the start of pumping. Aron and Scott take this into account. They show that when: Variable-discharge tests Confined Aquifers, Aron-Scotts Method where s n= drawdown at a certain moment t n, ; s e= excess drawdown caused by the earlier higher discharge

10. Variable-discharge tests Confined Aquifers, Birsoy-Summers Method Determine the slope of the straight line

11. Confined Aquifers, Aron-Scotts Method Variable-discharge tests If the fully developed drawdown is considered to extend to the distance r i at which then the s e (excess drawdown) can be approximated by:

12. Assumptions: All same as in chapter 3, except: 5.)The discharge rate decreases with time, the sharpest decrease occurring soon after pumping the following condition is added: Variable-discharge tests Confined Aquifers, Aron-Scotts Method

13. Free-Flowing Wells Based on the conditions that the drawdown in the well is constant and discharge decreases with time. To satisfy these conditions, the well is shut until pressure becomes static, then at t=0 the well is opened and the water level in the well drops instantaneously to a constant drawdown level which is equal to the outflow. The well discharges at a decreasing rate.

14. Confined aquifer, unsteady-state flow, Hantushs Method where Free-flowing wells

15. Assumptions All same as in chapter 3, except: 5.)At the start of the test (t=0), the water level in the free-flowing well drops instantaneously. At t>0, the drawdown in the well is constant, and its discharge is variable. The following condition is added: the flow to the well is in an unsteady state Free-flowing wells Confined aquifer, unsteady-state flow, Hantushs Method

16. Leaky aquifer, steady-state flow, Hantush-DeGlees method where Free-flowing wells

17. Assumptions All assumptions that underlie the standard methods for leaky aquifers, except: 5.)At the beginning of the test (t=0), the water level in the well drops instantaneously. At t>0, the drawdown in the well is constant, and its discharge is variable.

18. Well-fields