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The ultimate drawdown in a well measured at the designed rate Q of continuous pumping for a given period is different from the theoretical drawdown due to well losses.
The total drawdown sw is made up of:
(i) Head loss resulting from laminar flow in the formation (formation loss),
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(ii) Head loss resulting from the turbulent flow in the zone close to the well face,
(iii) Head loss through the well screen, and
(iv) Head loss in the well casing-
and sw = BQ + CQ2 …(5.89)
sw/Q t= B + CQ …(5.90)
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Where B = coefficient of formation loss due to- (i); C = coefficient of well losses due to (ii), (iii) and (iv) and has the dimensions of T2/L5; sw/Q = specific drawdown (reciprocal of specific capacity Q/sw); BQ = formation loss and CQ2 = well loss.
A plot of sw/Q versus Q gives a straight line, Fig. 5.48, whose slope gives the well loss coefficient C and the intercept on the sw/Q axis gives the formation loss coefficient B.
Rorabaugh (1953) had modified this graphical method when the drawdown is given by sw = BQ + CQn, ‘n’ deviating significantly from 2.
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Demonstration of Well Losses:
1. From the Recovery Immediately after Pumping is Stopped:
If well losses occur in disproportionately high gradients in the immediate vicinity of the well, the resulting picture will be that of a small, very steep cone of depression, superimposed on the main cone during pumping, Fig. 5.49. When the pumping stops, this small cone will fill up rapidly as the volume of storage is very small and subsequently the main cone will fill up. It can be observed that in the first 1-5 minutes after pumping is stopped, there is rapid recovery after which the recovery pattern simulates the inverse of drawdown.
2. From the Distance-Drawdown Curve if an Observation Well is Available:
If the aquifer transmissibility is known from a pump test on the well, the drawdown per log cycle of distance ∆s can be calculated from Eq. (4.82).
T = 2.3Q/2π∆s
The distance-drawdown graph may be plotted and extrapolated towards the well at a distance re the effective radius of the well, i.e., the radius of the actual well + gravel pack. It will normally be noticed that the observed drawdown is more than read from the graph, the difference giving the actual well losses.
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3. From a Step-Drawdown Test:
A step-drawdown test is carried out at three to five different rates of pumping or steps, each step being of the same duration, say 60 or 90 min. The maximum pumping rate preferably exceeds the designed pumping rate, though not too much, as too high pumping rates may adversely affect the aquifer framework permanently by excessive turbulence in the immediate vicinity of the well; for instance the soil grains around the well may resettle, but tighter than before. The well loss coefficient is given by-
C = ∆s2/∆Q2 – ∆s1/∆Q1/∆Q1 + ∆Q2 …(5.91)
Where ∆Q = increase in pumping rate at each step; ∆Q1 = Q2 – Q1, ∆Q2 = Q3 – Q2, ∆s = difference in drawdown after completion of each step and ∆s1 = s – s1, ∆s2 = s3 – s2.
A step-drawdown test in normally carried out on a well designed as a production well to determine the ‘specific capacity versus yield’ relationship. Evaluation of C gives the well efficiency. It is included in production well drilling contracts where it is generally prescribed to be less than 2000 sec2/m5.
Example:
A variable rate well production test was conducted at Sagar, Madhya Pradesh with the results given in Table 5.18. Determine the coefficient of well loss and formation loss and the corresponding percentages of total drawdown in the last case- (a) by drawing the specific drawdown curve, (b) by using Eqs. (5.90) and (5.91).
Solution:
(a) Data processed to draw the specific drawdown curve is given in Table 5.19.
A plot of sw/Q versus Q gives a straight line graph, Fig. 5.50. The vertical intercept on the sw/Q axis gives B = 1.3 min/m2 and the slope of the straight line graph gives C = 0.65 min2/m5. For the last case, step 5, for a pumping rate of 3270 1pm, formation loss = BQ = 1.3 (3.27) = 4.25 m, well loss = CQ2 = 0.65 (3.27)2 = 6.95 m sum of formation and well losses = 11.20 m against a measured drawdown of 11.29 m; percentage of formation loss × 4.25/11.20 × 100 = 37.9% of total drawdown percentage of well loss = 6.95/11.20 × 100 = 62.1% of total drawdown.
The average value of C = 0.65 min2/m5 = 2340 sec2/m5 which is higher than the prescribed limit of 2000 sec2/m5.
(b) Data processed to determine the well loss coefficient C by Eq. (5.91) is given in Table 5.20.
The average values of well loss and formation loss are about 58 and 42% respectively. The well should be properly designed and developed to keep the well loss to a minimum. The step-drawdown test enables the determination of the well loss coefficient and also the well efficiency.
Eq. (5.93a): Well efficiency = BQ/Sw × 100 = 5.49/11.29 × 100 = 48.7%.