The Dynamics of Water Wells Efficiency Reduction and Ageing Process Compensation

Water wells play an increasingly important role in providing water for the civilian population all over the world. Like other engineering structures, wells are subject to ageing processes resulting in degradation, which is observed as a reduction in hydraulic efficiency throughout their lifespan. To date, it has been found that the ageing process of a well is determined by a number of factors. The mathematical description of this process can be simplified. Drawing on Jacob’s equation, this paper presents the course of the degradation process as a variable depending on operation time, well loss and flow rate. To apply the determined relationships in practice, simplifying assumptions were adopted, which make it possible to determine the moment of ageing compensations of the degradation processes. It was also demonstrated that the degradation process may be slowed down by the appropriate selection of initial operating parameters. The presented discussion highlights the significance of parameters α, δ and exponent β. The relation between hydraulic resistances in an aquifer and in the engineering structure is closely connected with these values. The presented arguments indicate that step drawdown tests provide the necessary information which allows tracking changes in the ageing processes occurring in the engineering structure. The analysis of the drawdown test results makes it possible to determine the moment when the necessary adjustments in the operating parameters of a water well should be performed. Eventually, it allows maintaining the high hydraulic efficiency of the intake and extending the lifespan of the well in accordance with the principle of sustainability.


32
The groundwater extraction is still growing all over the world. It is estimated that within for half 33 a century, i.e. between 1960 and 2010, groundwater use doubled [1]. The drilling of new groundwater in a schematic figure (Fig. 1).  pumping rate was defined as the discharge rate that will not cause the water level in the well to drop 179 below a prescribed limit. In conclusions, authors emphasizes that in some cases the constant-head 180 pumping can be an alternative method of well management of the well. Analysis of the recorded data 181 changes shown that the discharge rate of the well trend with time was similar to that of the springs' 182 hydrograms. Moreover, the water volume extracted doesid not exceed the recharge [40].

194
Literature on hydrogeological issues states that the efficiency of a well can be described using an

219
For this purpose, the initial assumptions relating to the ageing process of the well must could be

236
The well efficiency function can be expanded into a Taylor series around moment = 0 as: or by writing the above equation as follows: The time derivative of well efficiency is: The second-order time derivative can be described by the expression: The above can be expressed in a simplified manner as: In assumption 2 the value of well loss (s2) should not significantly exceed the value of aquifer 245 loss (s1). This means that: Therefore, the rate of change is small in comparison with other elements in the expression for the 247 second-order time derivative of well efficiency: Including in the expression the first-order derivative in the form - The above considerations can be simplified by applying approximations in the solution, in which 251 higher-order terms are negligible.
In this case the flow rate decreases with time and the well resistances do not increase 288 significantly. This leads to the compensation of ageing in the operated pumping well by reducing the 289 flow rate. For ̇< 0 ̇≈ 0 the following can be specified: Obviously, in the above case the inequality Δηcmp > 0 over time will always be satisfied.
It should be pointed out here that a well with an initial efficiency of η0 after the expiry of the

297
In this case, it was assumed that higher-order terms, starting from the third order, are negligible.

298
Therefore, the power series expansion of the well efficiency function is as follows: while ensuring that the time derivatives at the moment of time t = t0 are equal to: Therefore, the relationship for the well efficiency change will be equal to: The first element of the difference provided in square brackets could be significantly smaller 302 than one. This is possible because the time derivative of the product ̇+̇= ( ) for a short 303 time (where Δt → 0) is low or slow-variable. Therefore, for ( ) ≈ 0 or ≈ and Δt → 0 304 an accurate expression for well efficiency change can be obtained at first-order approximation.

309
For ̇≈ 0 ̇> 0 the simplified form of relationship (37) applies: The above equation will be satisfied for the loss of hydraulic efficiency of a well (Δηage < 0) if the 311 passage of time is expressed in a limited range, given as the strict inequality ∆ < (̇• 0 ) −1 .
Sformatowano: Nie Wyróżnienie Sformatowano: Nie Wyróżnienie flow rate in the well over time. In other words, it is possible to recover at least a part of the hydraulic 317 efficiency lost.

318
As before, after time Δtage, well efficiency decreased from η0 to η1, and it constitutes the initial  It follows from case 1 that efficiency η1 is: while obviously always ∆ = 1 − 0 < 0. Case 2 indicates that the final efficiency η value is: while this time always ∆ = − 1 > 0.

336
The insertion of the expression for efficiency η1 results in a change in efficiency as a result of 337 compensation as: Full compensation of well ageing

339
It is readily visible that the total change of well efficiency equals: the sign of the total change (difference) in efficiency is determined by the size of the individual 341 elements.

342
Assuming that time moves towards moment t0, i.e. t → t0, so the time range becomes infinitesimal 343 (Δti → 0) then the decrease in well efficiency as a result of ageing also becomes infinitesimal 344 (Δηage → 0 or η1 → η0). Furthermore, an increase in well efficiency as a result of compensating the 345 well loss will also be infinitesimal (Δηcmp → 0 or η → η1). As for the rate of each of the above changes, 346 it will be given using the "0/0" indeterminate form. However, it follows from the principle of 347 transitivity of implication that η → η0 also occurs, therefore in general the transition of the function 348 to the limit Δη → 0 where Δt → 0 must occur. This means that the equality of function limits will be 349 satisfied.

353
The insertion of expressions (13) and (14) The separation of variables makes it possible to express the formula in the following form:

380
Taking into account equations (1) and (33), the hydraulic efficiency of the well can be equal to:

390
The hydraulic efficiency of a water supply well in accordance with the initial Jacob's formula -

408
The above discussion clearly demonstrates that periodical step drawdown tests play a   To reduce operating costs, the company conducting groundwater extraction constructed a 457 replacement well. The methods presented in this article were used during its operation (Fig 3.).

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Initially, the maintenance of an efficiency of 80% was established, which corresponded to the flow 459 rate of 9.4x10 -3 m 3 /s, however parameter Q0 =37.110 -3 m 3 /s. After a further 8 years the reduced flow  led to gravel-pack clogging. Finally, the value δ1 decreased to 2.410 -3 m 3 /s, which corresponds to an

524
In other words, the time available for taking corrective actions for the well becomes extended and is 525 similar in its order of magnitude to the time of operation of the wellbore.

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The solution presented in this paper involves ageing compensation and maintaining as high 529 hydraulic efficiency as possible. It concerns water supply wells in which well-loss has a minor share 530 in the drawdown generated by water pumping. So, the presented methodology is valid to both o all 531 new wells and to other wells in good hydraulic condition, where hydrogeological conditions do not has beenwas minimized. The upcoming rehabilitation of the Well 2 probably will be much more 585 effective in this case. It can be also be noticed that theoretical considerations presented in this paper 586 were verified by step-drawdown tests at the beginning and also at the end of research period. So, the 587 method can be applied in practical solution in well management to prevention of inefficiencies.

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Acknowledgements. The authors are grateful for the constructive comments made by 590 anonymous reviewers which helped in improvingto improve the manuscript.