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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). Variable-discharge tests Confined Aquifers, Birsoy-Summer’s 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 Jacob’s approximation of the Theis equation (3.7) (shown below) Variable-discharge tests Confined Aquifers, Birsoy-Summer’s Method The drawdown in the aquifer at time t during the nth pumping period of intermittent pumping is shown by the following expression: where where where Intermittent Pumping Variable-discharge tests Confined Aquifers, Birsoy-Summer’s Method For step-wise (uninterrupted pumping): t’(i-1) = ti, and the ‘adjusted time’ where becomes Variable-discharge tests Confined Aquifers, Birsoy-Summer’s Method If the intermittent pumping rate is constant (Q=Q1 =Q2=…….Qn) then the adjusted time becomes: Dividing both sides of equation 12.1 by Qn gives an expression for drawdown: Step-wise Pumping Variable-discharge tests Confined Aquifers, Birsoy-Summer’s 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 Variable-discharge tests Confined Aquifers, Aron-Scott’s Method •The sharpest decrease in discharge occurs soon after the start of pumping. Aron and Scott take this into account. They show that when: where sn=drawdown at a certain moment tn,; se=excess drawdown caused by the earlier higher discharge Variable-discharge tests Confined Aquifers, Birsoy-Summer’s Method •Determine the slope of the straight line Variable-discharge tests Confined Aquifers, Aron-Scott’s Method If the fully developed drawdown is considered to extend to the distance ri at which then the se (excess drawdown) can be approximated by: Variable-discharge tests Confined Aquifers, Aron-Scott’s Method 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: 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. Free-flowing wells Confined aquifer, unsteady-state flow, Hantush’s Method where Free-flowing wells Confined aquifer, unsteady-state flow, Hantush’s Method 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 Leaky aquifer, steady-state flow, Hantush-DeGlee’s method where 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 it’s discharge is variable. Well-fields