# The Dynamics of Water Wells Efficiency Reduction and Ageing Process Compensation

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}[5]. In some cases, changes of the drawdown can be explained by the Rorabaugh formula, where the power exponent p in the expression CQ is different from 2. Mackie, as cited by Atkinson et al. [6], reviewed the results of more than 20 carefully conducted step-drawdown tests of wells completed in fractured rock aquifers and concluded that most of the responses fell into one of the three categories where specific drawdown versus discharge rate is: linear, polynomial with power exponent equal to 2, or different from 2. The results of a study by Motyka and Wilk regarding the determination of the non-linear flow zone around several dozen wells drilled in fractured rocks indicate that the radius of turbulent flow zones are usually from 0.5 to 5.0 m, although in most cases they do not exceed 1 m [7].

## 2. Materials and Methods

- η hydraulic efficiency [-];
- s
_{1}aquifer loss [L]; - s drawdown observed in the pumping well [L];
- s
_{2}well loss in the well-screen adjacent zone [L]; - Q volumetric flow rate [L
^{3}/T]; - B aquifer resistance coefficient for laminar water flow [T/L
^{2}]; - C well resistance coefficient for turbulent flow [T
^{2}/L^{5}]; - α parameter describing the prevalence of turbulent flow [T/L
^{3}].

**Assumption**

**1.**

**Assumption**

**2.**

_{2}) should not significantly exceed the value of aquifer loss (s

_{1}). This assumption is usually relevant to new wells where the well-loss is insignificant in comparison to total drawdown. However, such criteria is also applicable to other wells in good hydraulic condition.

_{2}) should not significantly exceed the value of aquifer loss (s

_{1}). This means that:

## 3. Results

#### 3.1. First-Order Approximation

_{0}; η(t

_{0})):

_{0}equals:

_{0}. However, this fact should be taken into account when performing numerical calculations.

_{age}< 0 over time will always be satisfied.

_{cmp}> 0 over time will always be satisfied.

_{0}after the expiry of the period of ageing Δt

_{age}reaches a state with the hydraulic efficiency of η

_{1}. This state represents the initial state for the later compensation of the well loss value. After the compensation time Δt

_{cmp}the well reaches an efficiency of η. The number of iterations depends on the manner of transition between the extreme states of the well desired by the user.

#### 3.2. Second-Order Approximation

_{0}are equal to:

_{age}< 0) if the passage of time is expressed in a limited range, given as the strict inequality $\Delta {t}_{age}<{(\dot{\alpha}Q\xb7{\eta}_{0})}^{-1}$.

_{cmp}> 0 with time and decreasing flow rate in the well over time. In other words, it is possible to recover at least a part of the hydraulic efficiency lost. As before, after time Δt

_{age}, well efficiency decreased from η

_{0}to η

_{1}

_{,}and it constitutes the initial value for the compensation of the well loss through a reduction in the flow rate of the borehole at time Δt

_{cmp}. The number of iterations depends on the manner of transition between the extreme states of the well desired by the user.

## 4. Discussion

#### 4.1. First Order Approximation

#### 4.1.1. Compensation of Well Ageing

_{1}is:

_{1}results in a change in efficiency as a result of compensation as:

#### 4.1.2. Full Compensation of Well Ageing

_{0}, i.e., t → t

_{0}, so the time range becomes infinitesimal (Δt

_{i}→ 0) then the decrease in well efficiency as a result of ageing also becomes infinitesimal (Δη

_{age}→ 0 or η

_{1}→ η

_{0}). Furthermore, an increase in well efficiency as a result of compensating the well loss will also be infinitesimal (Δη

_{cmp}→ 0 or η → η

_{1}). As for the rate of each of the above changes, it will be given using the “0/0” indeterminate form. However, it follows from the principle of transitivity of implication that η → η

_{0}also occurs, therefore in general the transition of the function to the limit Δη → 0 where Δt → 0 must occur. This means that the equality of function limits will be satisfied.

_{0}= const and assuming for further simplicity constancy of coefficient β will result:

_{1}< η

_{0}. Therefore, their quotient is $\frac{{\eta}_{1}}{{\eta}_{0}}<1$ and the square of quotient is ${(\frac{{\eta}_{1}}{{\eta}_{0}})}^{2}\ll 1$. Furthermore, it can be assumed that the time available for compensating well loss is shorter than the well ageing time. It is therefore realistic to assume that the relationship Δt

_{cmp}< Δt

_{age}will be true each time. This means that the exponent in Equation (31) will be equal to β << 1. In the context of mathematical analysis, it proceeds in its limit to the value of β → 0. Hence, expression (30) must assume the form:

_{0}) parameter does depend on the property of the area directly adjacent to the functional part of the well screen and the permeability of the rock-soil medium conducting the fluid, as well as the type of the flowing reservoir medium.

_{age}≈ Δt

_{cmp}) and the well efficiencies compared to the limits of the structure’s ageing period range are almost identical (η

_{0}≈ η

_{1}), then the exponent from Equation (31) equals β ≈ 1. Obviously, this case will occur for initial moments—in situations where Δt

_{age}→ 0 and Δt

_{cmp}→ 0, i.e., where η

_{1}→ η

_{0}(or Δη

_{age}→ 0). Therefore, the limit:

_{1}for the initial moment under the conditions described above defines the mutual relationships between well loss and aquifer depression (or generally for a pool of any reservoir medium).

#### 4.1.3. Compensation Time

#### 4.2. Second Order Approximation

_{0}is numerically related to the flow of the reservoir medium for which the flow rate of a water supply well ${Q}_{0}={(C/B)}^{-1}$ results in the hydraulic efficiency of the well equal to 0.5 (50%), i.e., where well loss (s

_{2}) is equal to aquifer loss (s

_{1}). An increase in the value of parameter δ

_{0}means that the well reaches the same level of hydraulic efficiency with a lower volumetric flow rate. This is combined with a progressive increase in the resistance coefficient value of the well in turbulent flow at the well-screen near zone in relation to the resistance coefficient of laminar flow in the aquifer. In other words, parameter δ

_{0}may constitute a reliable indicator of well condition.

_{cmp}= 0, i.e., in the conditions of well operation, is Δt

_{cmp}<< Δt

_{age}. This means that for the long and intensive operation of water supply wells, flow rate reductions and compensation of hydraulic efficiency loss must not be delayed. For exponent β = 1, i.e., at the initial stage of the lifespan of a water supply well, factor $\delta ={\delta}_{1}={s}_{2}/{s}_{1}=\alpha Q$ ensures compensation time at a level equal to the time of ageing, i.e., operation of the structure—Δt

_{cmp}≈ Δt

_{age}, provided that the efficiencies assume similar values (η

_{0}≈ η

_{1}). In other words, the time available for taking corrective actions for the well becomes extended and is similar in its order of magnitude to the time of operation of the wellbore.

#### 4.3. Case Study

^{−3}m

^{3}/s, however parameter α

^{−1}= B/C = Q

_{0}= 37.4 × 10

^{−3}m

^{3}/s. After 10 years of uninterrupted operation there was an unforeseen failure connected with the exposure of the pumping unit. A pumping test demonstrated a well loss of up to 80% of total drawdown. To protect the new pump unit, the pumping rate was limited to 5.6 × 10

^{−3}m

^{3}/s, which corresponded to the hydraulic efficiency of the well of 30%. The conducted rehabilitation procedures failed to bring the expected results. The tests demonstrated that the final low efficiency of the well was caused by the relatively low initial well efficiency and the lack of efficiency compensation during operation. It has led to gravel-pack clogging and particulate damage. Finally, the value Q

_{0}decreased to 2.4 × 10

^{−3}m

^{3}/s, which corresponds to an efficiency of 50%.

^{−3}m

^{3}/s, however parameter Q

_{0}= 37.1 × 10

^{−3}m

^{3}/s. After a further eight years the reduced flow rate was 5.8 × 10

^{−3}m

^{3}/s with a constant efficiency. In this time, the value Q

_{0}decreased to 23.9 × 10

^{−3}m

^{3}/s, which corresponds to an efficiency of 50%.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Head losses in pumped well [4]; where: R

_{t}—distance from center of well to effective point formation where transition from laminar to turbulent flow takes place.

**Figure 2.**The hydraulic efficiency of the well without flow rate compensation: t

_{0}—at the beginning of operation time; t

_{1}—at the end of operation time.

**Figure 3.**The hydraulic efficiency of the well with flow rate compensation: t

_{0}—the working point at the beginning of operation; t

_{1}—the working point before first flow rate compensation; t

_{1′}—the working point after first flow rate compensation; t

_{2}—the working point before second flow rate compensation; t

_{2′}—the working point after second flow rate compensation; η

_{0}—the origin efficiency curve; η

_{2}—the current efficiency curve.

**Table 1.**A listing of the calculation results for the full compensation of ageing-related well efficiency decrease, relationship of the type $\delta =\alpha \xb7{Q}^{\beta}$ for $\Delta \eta =\Delta {\eta}_{age}+\Delta {\eta}_{cmp}=0$.

β | δ_{β} | η | Note |
---|---|---|---|

0 | ${Q}_{0}^{-1}=\alpha =C/B$ | $1/(1+{\delta}_{0}\xb7Q)$ | Full compensation—long period of well operation |

1 | ${s}_{2}/{s}_{1}=\alpha \xb7Q$ | $1/(1+{\delta}_{1})$ | Full compensation—initial moments of well operation |

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**MDPI and ACS Style**

Polak, K.; Górecki, K.; Kaznowska-Opala, K.
The Dynamics of Water Wells Efficiency Reduction and Ageing Process Compensation. *Water* **2019**, *11*, 117.
https://doi.org/10.3390/w11010117

**AMA Style**

Polak K, Górecki K, Kaznowska-Opala K.
The Dynamics of Water Wells Efficiency Reduction and Ageing Process Compensation. *Water*. 2019; 11(1):117.
https://doi.org/10.3390/w11010117

**Chicago/Turabian Style**

Polak, Krzysztof, Kamil Górecki, and Karolina Kaznowska-Opala.
2019. "The Dynamics of Water Wells Efficiency Reduction and Ageing Process Compensation" *Water* 11, no. 1: 117.
https://doi.org/10.3390/w11010117