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Water scarcity is the major problem confronting both urban and rural dwellers in Enugu State. This scarcity emanated from indiscriminate pipe failure, lack of adequate maintenance, uncertainty on the time of repair or replacement of pipes etc. There is no systematic approach to determining replacement or repair time of the pipes. Hence, the rule of thumb is used in making such a vital decision. The population is increasing, houses are built but the network is not expanded and the existing ones that were installed for no less than two to three decades ago are not maintained. These compounded the problem of scarcity of water in the state. Replacement or repair of water pipes when they are seen spilling water cannot solve this lingering problem. The solution can be achieved by developing an adequate predictive model for water pipe replacement. Hence, this research is aimed at providing a solution to this problem of water scarcity by suggesting a policy that will be used for better planning. The interests in this paper were to obtain a water pipe failure model, the intensity function
*λ*(t) [failure rate], the reliability R(t) and the optimal time of replacement and they were achieved. It was observed that the failure rate of the pipes increases with time while their reliability deteriorates with time. Hence, the Optimal replacement policy is that each pipe should be replaced after 4
^{th} break when the reliability = 0.0011.

Water distribution network is a network consisting of pipes, valves and pumps that distribute water to the end users [

1) Asbestos-concrete (AC) pipes,

2) Unplastisized polyvinyl chloride (UPVC) or plastic pipes and

3) Galvanized Iron pipes (GVI).

These pipes are further divided into mains, branches and terminals. Mains are of different lengths and diameters ranging from 13 ft to 20 ft and 200 mm to 600 mm. They carry water directly from the pumping station and the distribution storage. Branches have a diameter ranging from 75 mm to 150 mm, but of the same length with the mains. They carry water from the mains to various terminals. Terminals also have the same length as others but have a diameter ranging from 25 mm to 50 mm. They are called service lines in the network. The terminals convey water to the end-users and they are the smallest in size but the largest in number.

Each of the types of pipes in

The major problems of the GVI pipes are bulkiness, rusting and expensiveness compared to the other two types. Majority of their failure result from rusting. UPVC is fast replacing GVI pipes because of their portability and durability.

Many factors are responsible for water pipe breaks. For the purpose of this paper, these factors are divided into internal and external factors, which are collectively called covariates. Internal factors: Internal factors include the pressure exerted by the flowing water in the pipe; the action of chemicals used in the water treatment; the diameter of the pipes and the intrinsic material of the pipes etc. Pressure exerted by water over a long period of time on pipes leads to their break and this pressure is inversely related to the diameter of pipes. The pressure

Pipes | |||
---|---|---|---|

Types | AC | UPVC | GVI |

Mains | yes | Yes | yes |

Branches | yes | Yes | yes |

Service (Terminals) | Non | Yes | yes |

Pipes | ||
---|---|---|

Types | Advantages | Disadvantages |

AC | Durable, cheap, | Brittle, bulky, Cancerous, Contaminate water running through them |

PVC | Durable, cheap, portable, safer, does not contaminate water running through them | Non |

GVI | Durable, withstand internal and external pressure | Bulky, rusting, expensive, contaminate water running through them |

is also intensified, as the pipes are located nearer to the source and reduces as the length of the pipes increases. External factors: External factors include the soil type, topography, lay-dept, shock and rust. These factors are called external because they occur outside the pipes. The nature of soil in which the pipe is laid determines the rate of pipe’s break etc. The rates at which these factors occur influence the maintenance decisions.

Enugu State Water Corporation EnuguWater generation and distribution are the sole concern of the water Generation unit, Enugu State Water Corporation Enugu. The replacement and repair functions are the sole responsibilities of the engineering and plumbing units. The pressure under which the pumps distribute water ranges from 6 bars to 16 bars. Frequency of pipe breaks depend on the following factors namely; lay depth, topography, intrinsic material of the pipes, pressure, diameter of the pipes, lay location etc. Period of replacement: pipes are replaced as they break and the rule of thumb is extensively used. Maintenance actions include periodic greasing of nuts and bolts, replacement or repair of failed pipes. In most cases, the maintenances are done on demand, as there is no regularity on the time limit for the maintenance services. But since pipes constitute the bulk of the money cost of the water network distribution system, there is a need to develop a failure model to detect the probability of failure at any given time, the failure rate, the reliability of the pipes and the optimal replacement time of the pipes to avoid waste of water resources.

[^{b}^{−1}) can be used to describe the intensity of a NHPP. They define intensity as the mean number of failure, N(t), per unit time and that once the intensity has been estimated the number of failures in a given time period is Poisson.

[

Water utilities are concerned with large number of main breaks and the resulting direct and indirect costs due to deterioration. These contribute to frequent breaks and resulting repair/replacement and rehabilitation costs [

Water distribution pipes are repairable system because some maintenance action of repair can be done on them to restore the operation of the pipes after failures. [^{B}^{−1} and V(t; M, β) = Mt^{β}; M, β > 0. β determines how reliability decays or grows. If β < 1, the intensity function decreases over time and the reliability improves. If β = 1, the reliability remains the same over time, but if β > 1, the reliability declines and intensity increases. Again, if β > 1, we have NHPP and for β = 1, we have HPP. λ(t) is an increasing concave (straight, convex) curve if 1 < β < 2 [

The Poisson Process provides a realistic model for many random phenomena. Since values of a Poisson random variable are the non-negative integers, any random phenomenon for which count of some sort is of interest is a candidate for modelling by assuming a Poisson distribution [_{s}(.) takes place after an exponential amount of time with parameter λ; the rest of the inter event times are iid exponential with parameter λ [_{1}, X_{2}, …, which, with probability 1, are not all zeros [

In this section, we present the data that is used for this work in

t = 2005 - 2015 = 10 years (1)

λ = ∑ x i N = 18 20 = 0.9 ; i = 1 , 2 , ⋯ , 18 ; N = 1 , 2 , ⋯ , 20 (2)

λ c = 0.9 10 = 0.09 ; λ c = failureperunittime (3)

We present how the parameters of the model were determined in

The graphic method of determining the parameters of the water pipe failure model is presented in

ln a = − 2.4 ⇒ a = exp ( − 2.4 ) = 0.09 (4)

b = 1.60 1.575 = 1.02 (5)

S/No | Items | Values |
---|---|---|

1 | Type of pipe | 50 mm GVI |

2 | Installation period | 2005 |

3 | Period of replacement | 2015 |

4 | Length of pipe | 18 ft |

5 | Number of pipes | 20 |

6 | Record of breaks within the period | 18 |

7 | Location | Uduma Street N/Haven |

l^{c} | t | lt | ln(λt) = lnΛ(t) | ln(t) |
---|---|---|---|---|

0.09 | 1 | 0.09 | −2.41 | 0 |

0.09 | 2 | 0.18 | −1.7 | 0.7 |

0.09 | 3 | 0.27 | −1.3 | 1.1 |

0.09 | 4 | 0.36 | −1 | 1.4 |

0.09 | 5 | 0.45 | −0.8 | 1.6 |

0.09 | 6 | 0.54 | −0.6 | 1.8 |

0.09 | 7 | 0.63 | −0.46 | 2 |

0.09 | 8 | 0.72 | −0.3 | 2.1 |

0.09 | 9 | 0.81 | −0.2 | 2.2 |

0.09 | 10 | 0.9 | −0.11 | 2.3 |

Hence

p n ( t ) = exp ( − 0.09 t 1.02 ) ( 0.09 t 1.02 n ! ) n ; ( n , t ) = 0 , 1 , 2 , ⋯ , 10 (6)

p 0 ( 1 ) = exp ( − 0.09 ) = 0.9139 (7)

p 1 ( 1 ) = exp ( − 0.09 ) ( 0.09 ) = 0.0823 (8)

p 2 ( 2 ) = exp ( − 0.1825 ) ( 0.1825 ) 2 2 ! = 0.0139 (9)

⋮

p 10 ( 10 ) = exp ( − 0.9424 ) ( 0.9424 ) 10 10 ! = 0.000 (10)

Since b > 1, then, our model assumes NHPP (λ(t)).

Equation (6) is the water pipe failure model, where p = probability; n = the number of failures; t = the time in years; a = 0.09 and b = 1.02 are the parameters of the model and a, b, t > 0.

R ( t ) = p n ( t ) λ ( t ) (11)

A reliability approach is used here to determine the optimal time of replacement of pipes. Reliability approach has never been used to determine the optimal water pipe replacement policy. The pipe should be replaced a little time before the reliability becomes zero. Continuous used of the pipe when the reliability become zero add more to the cost because the failure rate λ(t) increases rapidly as the reliability R(t) becomes zero. Hence, t (time in years) is optimal at the least reliability R(t).

Form the data collected from Enugu State Water Corporation Enugu we obtain the parameters of our model “a” and “b” and use them to generate a table for the probability of number of failure at any given time t. Also failure rate and reliability of the pipes were determined and presented in Tables 4-7 respectively.

Result A:

p n ( t ) = exp ( − 0.09 t 1.02 ) ( 0.09 t 1.02 n ! ) n ; ( t , n ) = 1 , ⋯ , 10 (12)

The probability of the number of failure of pipes at a given time is presented in

Result B:

λ ( t ) = ( 0.0918 ) t 0.02 (13)

The computation of the intensity function λ(t) is presented in

Result C:

R ( t ) = p n ( t ) λ ( t ) (14)

In

N | t (yrs) | a | b | P_{n}(t) |
---|---|---|---|---|

0 | 1 | 0.09 | 1.02 | 0.9139 |

1 | 1 | 0.09 | 1.02 | 0.0823 |

2 | 2 | 0.09 | 1.02 | 0.0139 |

3 | 3 | 0.09 | 1.02 | 0.0027 |

4 | 4 | 0.09 | 1.02 | 0.0005 |

5 | 5 | 0.09 | 1.02 | 0.0001 |

6 | 6 | 0.09 | 1.02 | 0 |

7 | 7 | 0.09 | 1.02 | 0 |

8 | 8 | 0.09 | 1.02 | 0 |

9 | 9 | 0.09 | 1.02 | 0 |

10 | 10 | 0.09 | 1.02 | 0 |

t (years) | ab | t^{0.02} | λ(t) |
---|---|---|---|

1 | 0.0918 | 1 | 0.0918 |

2 | 0.0918 | 1.014 | 0.0931 |

3 | 0.0918 | 1.0222 | 0.0938 |

4 | 0.0918 | 1.0281 | 0.0944 |

5 | 0.0918 | 1.0327 | 0.0948 |

6 | 0.0918 | 1.0365 | 0.0952 |

7 | 0.0918 | 1.0397 | 0.0954 |

8 | 0.0918 | 1.0425 | 0.0957 |

9 | 0.0918 | 1.0449 | 0.0959 |

10 | 0.0918 | 1.0471 | 0.0961 |

n, t | P_{n}(t) | λ(t) | R(t) |
---|---|---|---|

1 | 0.0823 | 0.0918 | 0.8965 |

2 | 0.0139 | 0.0931 | 0.1493 |

3 | 0.0027 | 0.0938 | 0.0053 |

4 | 0.0005 | 0.0944 | 0.0011 |

5 | 0.0001 | 0.0948 | 0 |

6 | 0 | 0.0952 | 0 |

7 | 0 | 0.0954 | 0 |

8 | 0 | 0.0957 | 0 |

9 | 0 | 0.0959 | 0 |

10 | 0 | 0.0961 | 0 |

^{th} break when the reliability of the pipes is 0.0011. Hence, it is most reasonable to replace the pipe when the reliability is very close to zero.

In this paper, we developed water pipe failure model and the optimal water pipe replacement time (policy) of the pipes. The water pipe failure model was a Non-homogenous Poisson process and reliability has never been used to determine optimal replacement policy on a repairable system. We also obtained the intensity function λ(t) [failure rate] of the pipes. From this work, it was observed that the failure rate of the pipes increases with time while their reliability deteriorates with time. Hence, the Optimal replacement policy is that the pipe should be replaced after the 4^{th} break when the reliability = 0.0011.

Amuji, H.O., Ogbonna, C.J., Ugwuanyim, G.U., Iwu, H.C. and Nwanyibuife, O.B. (2018) Optimal Water Pipe Replacement Policy. Open Journal of Optimization, 7, 41-49. https://doi.org/10.4236/ojop.2018.72002