# An Analysis of the Potential Impact of Climate Change on the Structural Reliability of Drinking Water Pipes in Cold Climate Regions

^{*}

## Abstract

**:**

## 1. Introduction

- ○
- Hypothesis: Pipe failures and failure rates are correlated to frost heave of the ground, and thereby also correlated to air temperature.

#### Reliability of Pipes

## 2. Method

- •
- Failure rates (number of failures per temperature per day) will be plotted against same day temperature and against the average temperature of the preceding week. Plotting failure rates instead of number of failures will adjust the data correctly since there are more days registered with warm temperatures than with cold temperatures. The potential correlation will be tested with a linear regression model, and the model will be statistically tested for a certain confidence level.

- Failures were analyzed directly against the temperature on the day the failure occurred.
- Failures were analyzed against the average temperature one week preceding (including the day of the failure) the relevant failure.

#### 2.1. Preparing and Plotting Failure Data

- (a)
- Gathering of failure data. Identification of reliable failure data for each city, which is the relevant observation window. This is based on an assessment of when the utility started to collect failure data in a structured way.
- (b)
- Historical temperature data for the relevant cities was collected from a national database (at www.senorge.no).
- (c)
- Failures were correlated to temperature through dates given for the failure events.
- (d)
- Number of failures per temperature were counted = F/temp.
- (e)
- Number of days per temperature were counted = D/temp.
- (f)
- Historical failure rates (failures per day) per temperature were calculated by Equation (1).
- (g)
- Failure rates (step f) were plotted against temperature.

_{r}= Historical failure rate for a given temperature.

#### 2.2. Analyzing Correlation between Failure Rates and Temperature

- (h)
- Calculation of average number of days registered per temperature according to Equation (2).
- (i)
- Calculation of standard deviation for the number of days registered per temperature.
- (j)
- Calculation of the minimum number of days needed for being part of the correlation calculation, based on the numbers found in points h and i.
- ○
- More specifically: minimum number of days registered for a single temperature = average number of days registered per temperature (1834 days)—standard deviation (1762 days) = 72 days.
- ○
- In order to include a temperature in the correlation analysis, we needed 72 days or more registered of the temperature.

- (k)
- Plotting failure rates (failures/temperature/day) versus temperature.
- (l)
- Calculating linear regression with R and R
^{2}values. - (m)
- Statistically quantify the uncertainty of the regression line by performing a test of hypothesis of the linear correlation.
- (n)
- If statistically viable, establish the regression line as a model on the impact of temperature on failure rates, with a given uncertainty.
- (o)
- We assume that failures occurring during the summer, i.e., during warm temperatures, are not dependent on temperature. The failures occurring during these temperatures are therefore exclusively dependent on other parameters, like deterioration, hydraulic pressure, corrosion etc. The failure rate during these temperatures therefore constitutes the ‘baseline’ failure rate of the pipes. This approach is also considered in Le Gauffre et al. [20]. The failure rate for the warmest temperature was at 0.21 failures/temperature/day (based on the linear regression), constituting the baseline failure rate.

_{average}= Average number of days registered per temperature.

#### 2.3. Preparing Data for Expected Future Temperature Increase

- (p)
- From the Norwegian Environment Agency report, we gathered reliable data on the potential future temperature increase. The estimated future temperature increase was based on the following reasoning:
- ○
- Values from three scenarios for future greenhouse gas emissions were used. They represent different expected temperature increases.
- ○
- Only median values of the climate scenario predictions were used, as they refer to the value in which 50% of the projection values are larger and 50% of the projection values are smaller. The median therefore represents a most probable outcome, as it represents a middle way between more extremes.
- ○
- T downscaling processes were used in the National Agency report. One downscaling process modelled three of the RCP scenarios, while the other downscaling process modelled two of the scenarios. Where the two downscaling processes modelled the same RCP scenario, an average of the expected temperature value was used.
- ○
- The estimated temperature increase varies by the season, so an expected temperature increase was calculated per season for the three RCP scenarios.
- ○
- The three RCP scenarios represent the following [2]:
- ◾
- RCP 2.6: Emissions are reduced drastically after 2020, and will be close to 0 within 2080.
- ◾
- RCP 4.5: Emissions are reduced after 2040, and in 2080 emissions are on a level that correspond to 40% of emissions in 2012.
- ◾
- RCP 8.5: ‘Business as usual’ scenario, where the increase in emissions follow the same pattern as today.
- ◾
- The values given after RCP relates to the estimated extra heat supply, given in W/m
^{2}, for the emission scenario.

- ○
- A linear increase in temperature from the reference period (1971–2000) to the projection period (2071–2100) was assumed. This was done since no time series of the future temperature increase is available for the Norwegian-based temperature data (only the estimate for the projection period). Even though historic Norwegian temperature data fluctuates, it is possible to create a linear approximation to the data series for the past 50 years or so [2], supporting our claim that it can also be done for future temperature increase.
- ○
- The modelling in Hanssen-Bauer et al. [2] is based on a temperature increase from the reference period of climate in Norway, which is from 1971 to 2000. The increased temperature is projected towards the period of 2071 to 2100. From the middle of the reference period to the middle of the projection period there is 100 years (1985 to 2085). The objective of this paper is to predict the impact of climate 50 years into the future. Our year of focus is therefore roughly 2070. In 2070 we assumed that about 85% of the expected temperature increase has occurred. We based this assumption on the following principles:
- ◾
- We assumed that the temperature increase from today to the projection period is linear, as argued in the previous point.
- ◾
- 2070 is 85% towards 2085 (when the linear increase is assumed).

- ○
- According to Hanssen-Bauer et al. [2], the expected temperature increase during spring looks like the expected increase during fall, with small deviations. For this paper, we therefore looked at spring and fall as a common group representing a single season. We looked at temperature increases during three seasons; summer, winter and spring + fall.

- (q)
- In order to relate temperature to season, we assumed a temperature range for each season. We divided the temperature data into three intervals to represent the three seasons:
- ○
- Summer: warmer than 12 degree Celsius.
- ○
- Winter: colder than 0 degree Celsius.
- ○
- Spring + fall: between 0 and 12 degree Celsius.

#### 2.4. Analyzing the Future Impact of Climate on Failures

- (r)
- The temperature range was divided into the three seasons as stated in segment q. The temperature range was from −23 to 25 degrees Celsius, and is illustrated in Figure 2. We now also looked at temperatures outside of the standard deviation calculated in segment j. An expected average future temperature increase was established based on the process described in segment p. These temperatures are given in Table 1.
- (s)
- Expected temperatures for 2070 were calculated by increasing the historical summer, winter and spring + fall seasons with the temperatures given in Table 1.
- (t)
- Calculation of the historical failure rate for each temperature by Equation (1), same as step f.
- (u)
- Calculation of the share of the failure rates that are caused or driven by temperature, as given by Equation (3). The baseline failure rate was set to 0.21 failures/temperature/day (from segment o).
- ○
- The total failure rate at each temperature was given by the linear regression equation (Equation (9)).
- ○
- The share of the failure rate which is driven by temperature increases with negative temperatures in a linear fashion, in the exact same rate as the linear regression line. This is illustrated in Figure 2.

- (v)
- For the historic data, an expected number of failures (for each temperature) driven by temperature was calculated according to Equation (4). This gave us a historic number for failures, which were probably temperature driven.
- (w)
- For 2070 we calculated an expected number of failures caused by temperature (for the range of temperatures from −23 to25 degrees Celsius) with Equation (5).
- ○
- For this we took into consideration the ‘movement’ of the temperature curve to the right, as illustrated in Figure 2. The curve is ‘moved’ by the temperature increase calculated in segment s.

- (x)
- An increase or reduction in expected number of failures in 2070 was calculated for each temperature (from −23 to 25 degrees Celsius) with Equation (6). This gave us a comparison between historical failures driven by temperature and expected number of failures in 2070 driven by temperature.
- (y)
- By accumulating the numbers found for each temperature in segment x, we could calculate the total number of increased or reduced failures across the entire temperature range, in total in 2070, compared to the historical data. This was done with Equation (7).
- (z)
- The accumulated number in segment y was then used to calculate the % of increase or decrease of expected failures within 2070. Equation (8) was used for this. This calculation was done for each of the three climate scenarios. The results can be used to discuss the impact of climate on the reliability of the drinking water network through the impact on failures and failure rates.

_{r%}= share of failure rate (given in %) caused by temperature for a given temperature.

_{change per temp}= Change in expected number of failures per temperature in 2070.

_{change total}= Change in total expected number of failures in 2070. The number of failures are accumulated across the temperature range.

_{change total}= the percent of total increase or decrease of expected failures (across the temperature range) within 2070 compared to historical number of failures.

## 3. Results

#### 3.1. Pipe Data

#### 3.2. The Baseline Failure Rate

#### 3.3. Correlation of Pipe Temperature

^{2}value. The linear regression shows a trend of an inverse correlation between increasing failure rates and temperatures. The R

^{2}value shows that 78.2% of the total variation in the data can be explained by the linear regression model.

#### 3.3.1. Linear Regression Model

^{2}for the regression line is −0.884 and 0.782, respectively. An R

^{2}value of 0.782 means that 78.2% of the variations in y can be explained with x, while 21.8% of the variations are caused by other factors. Since we only have estimates for the standard deviations, hypothesis testing for the a and b values must in this case be based on the t-distribution [25].

_{0}= 0 means that we are testing for any linear correlation, and setting ρ < 0 means that we are testing for a left tailed test. Since we are first and foremost interested in looking for a linear correlation in the data, we have to set r = 0. The null hypothesis was therefore that the population correlation coefficient equals 0, while the working hypothesis was that the coefficient was negative. The test parameter t for the hypothesis test is calculated with the following equation:

_{0}= 0, and our r and r

^{2}calculated values gave us a t = −11.52, allowing us to reject the null hypothesis and conclude that there is statistical evidence of a negative linear correlation. This correlation can be calculated down to a certainty greater than a = 0.0005, meaning a certainty higher than 99.9995%. The conclusion is therefore that this is a very good model of linear correlation.

#### 3.4. Correlation of Pipe Breaks vs. Average Temperature the Preceding Week

^{2}value. The linear regression shows a trend of an inverse correlation between increasing failure rates and weekly average temperatures, with a higher R

^{2}value than the correlation of failure rates and temperature (calculated to be 0.782 in Section 3.1). This means that about 3% more of the variation in the data can be explained by the average weekly temperature than the temperature on the day of the failure.

#### Linear Regression Model

^{2}for the regression line is −0.901 and 0.813, respectively. In order to determine if there is statistically proof to state that there is a negative linear correlation between failure rates and average weekly temperatures, we performed a test of hypothesis according to the process described in Section 3.3.1. For the weekly temperature analysis, we put z = 30 since number of representative samples for weekly average temperature was n = 32. Calculating the test parameter t with n = 32, ρ

_{0}= 0, and our R and R

^{2}calculated values gave us a t = −11.54, which meant that we could reject the null hypothesis with a certainty of 99.9995% (a = 0.0005) and conclude that there is statistical evidence of a negative linear correlation.

#### 3.5. Expected Impact of Climate Change on Reliability

## 4. Discussion

^{2}values of 78.2% and 81.3%. The quantitative correlations that are defined by Equations (9) and (12) can thus be used to estimate failure rates and number of failures in a network based on air temperature levels. According to the linear correlation of temperature vs. failure rates, defined by Equation (9), the failure frequency (observed failures per day) is 86% higher at −15 degrees Celsius than at 23 degrees Celsius, and it is 52% higher at 0 degrees Celsius than at 23 degrees Celsius. This means that we are observing almost twice as many failures in the coldest part of the year as in the warmest part. According to the linear regression model of temperature vs. failure rates (Equation (9)) the failure frequency increases with 0.0235 for each 5 degree Celsius temperature increase, which equates to 0.047 for each 10 degree increase.

## 5. Conclusions

- -
- Failure rates increase during cold winter months.
- -
- Frost loading and frost heave is an important factor for increased failure rates during winter months.
- -
- Changed soil temperatures cause thermal stresses in the ground, which impact pipes.
- -
- A large portion of failures in cold climates are transverse, which are normally caused by pipe-soil interactions and pipe bending.
- -
- Transverse fractures can be doubled during winter months.
- -
- Pipes in trenches which are more vulnerable to frost heave experience more failures during winter than the average of the network, thus verifying the frost heave effect on pipes during winter.
- -
- Grey cast iron pipes are more vulnerable to failures during winter months than other materials, being the only material that has a substantially increased failure rate during winter months. These pipes are also often laid in trenches exposed to frost heave (certain construction periods), showing that there might be a connection between high failure rates of grey cast iron pipes and frost vulnerable construction periods.

^{2}value than the model for correlation of failure rates vs. temperature, showing a slightly stronger explanation of the variation in the data. The difference in R

^{2}value is 0.03. We therefore conclude that 3% more of the variation in data can be explained by weekly average temperatures than by daily temperatures. This also shows us that failure rates are more dependent on type of season (cold vs. warm season) than the length of the cold period. A general conclusion is that the frequency of a failure event increases drastically in cold weather/cold seasons, and that the length of a cold period within the season further increases the frequency of an event, although in a small measure.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Overview of the method used to assess potential impact of future temperature change on failure rates.

**Figure 2.**An illustration of data used to predict future impact of temperature increase on failures.

**Figure 6.**Linear regression showing the correlation between failure rates and average temperature the preceding week.

**Table 1.**Expected temperature increase used in the three scenarios, distributed across the three seasons. The numbers are based on calculations in Hanssen-Bauer et al. [2].

Temperature Increase [Degree C] | RCP Scenarios | ||
---|---|---|---|

Season | RCP 2.6 | RCP 4.5 | RCP 8.5 |

Winter | 1.9 | 3.1 | 5.35 |

Fall + spring | 1.9 | 2.95 | 4.6 |

Summer | 0.7 | 2.0 | 3.4 |

**Table 2.**Expected change in number of failures/failure rates within 2070 due to temperature increase. The change is given in % compared to historical failure data.

Scenario | RCP 2.6 | RCP 4.5 | RCP 8.5 |
---|---|---|---|

% change | −2.7 | −4.4 | −7.2 |

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**MDPI and ACS Style**

Bruaset, S.; Sægrov, S. An Analysis of the Potential Impact of Climate Change on the Structural Reliability of Drinking Water Pipes in Cold Climate Regions. *Water* **2018**, *10*, 411.
https://doi.org/10.3390/w10040411

**AMA Style**

Bruaset S, Sægrov S. An Analysis of the Potential Impact of Climate Change on the Structural Reliability of Drinking Water Pipes in Cold Climate Regions. *Water*. 2018; 10(4):411.
https://doi.org/10.3390/w10040411

**Chicago/Turabian Style**

Bruaset, Stian, and Sveinung Sægrov. 2018. "An Analysis of the Potential Impact of Climate Change on the Structural Reliability of Drinking Water Pipes in Cold Climate Regions" *Water* 10, no. 4: 411.
https://doi.org/10.3390/w10040411