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The Theis equations are based on the assumption that the water is released from storage in the aquifer in immediate response to the decline in the water table. In unconfined aquifers, gravity drainage may not be immediate, particularly in fine grained formations and with stratifications of slow yielding materials.
When a well is pumped in such formations, a typical time-drawdown curve in an observation well is as shown in Fig. 5.21 (log-log plot, USBR, 1977), which can be divided into three segments:
The first segment, of the observed drawdowns in the first few minutes after pumping began, is similar to the Theis type curve for confined aquifer; the pore water is released instantaneously from storage, due to the compression of the aquifer matrix and expansion of the entrapped air (i.e., not by free gravity drainage of the pore space), and the water level in the observation well drops relatively fast. Transmissibility of the aquifer can be determined by the Theis method, and the storage coefficient is equivalent to that of a confined aquifer.
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(ii) The second segment of the time-drawdown curves indicates a flattening in slope; the water level drops at a slower rate and almost remains constant for a while (after which it begins to drop faster again) because of the replenishment by gravity drainage from the pore space above the cone of depression which is delayed.
(iii) In the third segment, an equilibrium is reached between the gravity drainage and the rate of fall of the water table. This condition is reached after several minutes to several days after pumping began and can be fitted to the Theis type curve for the unconfined condition, which gives the storage coefficient for the unconfined aquifer i.e., specific yield.
Thus, for an unconfined aquifer with delayed yield, a pumping test should be continued sufficiently long to obtain the third segment of the curve and S(= Sy) can be determined by one of the methods of solution of the non-equilibrium equation.
The minimum pumping time required for an accurate estimate of S in an unconfined aquifer depends on the transmissibility (T) of the aquifer. Boulton introduced the term B, defined as-
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B = √T/αSy … (5.32)
Where α is an empirical constant characteristic of the aquifer and its reciprocal 1/α which has the dimension of time, is called the delay index. The delay index can be obtained for the formation material from Fig. 5.22(a). Then knowing r and from an estimate of Sy and T, the minimum pumping time twt can be obtained from the dimensionless plot of ‘αtwt vs r/B’.
Fig. 5.22(b). Values of the minimum pumping time for different aquifer materials are given by USBR (1977) as a guideline as follows:
Delayed yield is not restricted to unconfined aquifers alone. It may occur with leaky aquifers that receive water from the top semi-confining layer with a free water table. The hydraulic characteristics of such an aquifer can be determined by the methods of Boulton (1963) and Kruseman and de Ridder (1970).
The r/B value from the best fitting type curve is used to calculate B and then a from Eq. (5.32) with the computed values of T and Sy; For the same r/B value, αtwt is obtained from Fig. 5.22(b). Knowing α, twt is calculated, which indicates the time that yield is no longer delayed and the time that the observed ‘s vs t curve’ should merge with the Theis type curve for the unconfined aquifer.
Prickett (1965) has given the following values for the delay index:
Example:
For an unconfined aquifer consisting of fine sand, the following hydraulic characteristics are estimated:
What is the minimum pumping time that yield is no longer delayed and Theis equation is applicable.
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Solution:
Bailer Method:
Skibitzke has developed a method for determining the coefficient of transmissibility from the recovery of water level in a well that has been bailed. For large values of t,
S’ = V/4π Tt …(5.33)
Where V = volume of water removed during one bailer cycle, t = length of time since bailer was removed.
If the residual drawdown is observed at some time after completion of n bailer cycles, removing the same quantity of water in each bailer cycle, then-
s’ = V/4π T (1/t1 + 1/t2 + … t/tn) …(5.33a)
The bailer method is thus applied to a single observation of the residual drawdown after the time since bailing was stopped becomes large.
Slug Method:
This is very similar to the bailer method. A quantity of water (V) or slug is injected and the residual heads s at times after injection is completed are noted. An arithmetic plot of residual heads versus reciprocals of times of observation, produces a straight line whose slope, appropriately substituted in the equation-
s’ = V/4π Tt …(5.34)
Permits computation of the coefficient of transmissibility.
This method has serious limitations since the duration of the test is very short and its use is restricted to artesian aquifers of small to moderate transmissibility (< 7.5 × 105 lpd/m).
Application of the Formulae:
The actual field conditions may differ from the basic assumptions on which the formulation has been made. If a field data plot on a logarithmic paper does not coincide with a non-leaky artesian type curve, and is displaced in one direction, it indicates possibly recharge or leakage, while displacement in the opposite direction indicates boundary conditions which are affecting the assumption of constant permeability or constant thickness of the aquifer.
Various approaches have to be tried and compared such as drawing a time-drawdown curve (when the value of ‘u’ is very small). For barrier and recharge boundaries special methods may be adopted such as the ‘method of image wells’. Corrections have to be applied for partial penetrations; however, for distance r > 1.5 b, effects of partial penetration are offset.
Multiple Aquifer Performance Tests:
When a number of aquifers are encountered in a ground water project, it is necessary to determine the hydraulic characteristics of each aquifer, hydraulic interconnections (vertical leakage), and the quality of water from each aquifer. In advanced countries multiple aquifer performance tests are conducted in a single well using a number of piezometers at different depths of each aquifer with a number of packers.
In the Cauvery delta, each aquifer was tested separately with observation wells in the aquifer tested and also in the aquifers lying above and below it. This method of testing gives an idea of the vertical leakage or hydraulic interconnection of the different aquifers by observing the water level fluctuations in the other observation wells during pumping schedule and the results obtained from the multiple aquifer tests conducted in the Cauvery delta are given in Table 5.16 and Fig. 5.23.
