A Decision Analytic Approach for Inspecting Leaking Hydraulic Distribution Systems that Relies on Bayesian Updating of Limited Information

1.1. Literature Review

Published research on algorithms for leak location in hydraulic networks can be divided into two categories: those based on pressure and flow gauges, and those that measure other variables. Among the former are Anfinsen and Aamo [2], who developed a model based on hyperbolic equations to detect the location and size of leaks; Blesa et al. [3] used a linear model with variable parameters and EPANET software to estimate the location of leaks by comparing estimated and actual pressure heads; Adachi et al. [4] estimated leak locations by minimizing the difference between measured and modeled pressures and flows, and Ruzza et al. [5] applied Kalman filtering to estimate leak locations. The inclusion of uncertainty to the leak location problem is presented by Liao et al. [6] who applied frequency response analysis and deep learning, by Morosini et al. [7] and Costanzo et al. [8] who applied model calibration, by Tylman et al. [9] who developed an automated system for detecting dielectric fluid leaks from buried cables using Bayesian networks and artificial intelligence, and by Malekpour et al. [10] and Lazhar et al. [11] who used an inverse transient analysis model and genetic algorithms to detect leaks in oil lines, taking into account the viscoelasticity of the pipe walls. Recently, Moosavian et al. [12] applied a neuro-fuzzy inferential model for leak location, aiming to minimize the number of head measurements needed, and Vorathin et al. [13] compared sensor types for leak location purposes in low-pressure gas distribution networks.

Regarding methods for locating fluid leaks, measuring variables additional to pressures and flows, Tariq et al. [14] suggest using accelerometers to detect leaks in water networks through neural networks and decision trees, while Wang et al. [15], within the framework of the homogeneous transformation theory, combined magnetic markers and internal detectors. On the use of sound wave propagation measurements, Zeng et al. [16] proposed using a “coherence-gram” to locate leaks in buried pipes, Scussel et al. [17] used acoustic correlators and Monte Carlo simulation to incorporate the effect of the uncertainties in pipe geometry and material properties on the propagation rate of the noise provoked by the leak while Li et al. [18] resorted to Fourier transformations for analyzing the acoustical waves produced by the lost fluid.

Recently, several research groups have published algorithms that use acoustic gauges (hydrophones) to detect leaks in water distribution networks. Satterlee et al. [19] coupled acoustic measurements with vibration meters to feed a neural network, Sitaropoulos et al. [20] and Uchendu et al. [21] applied, respectively, multifractal analysis and a multipath model to the task, while the simultaneous location of several leaks is discussed by Zeng et al. [22] and Liu et al. [23]. Additionally, Xu et al. [24] dealt with locating leaks in hot water networks, Islam et al. [25] employed thermal images and geo-acoustic probes for their leak searching algorithm, Sohn et al. [26] showed the operation of hydrophones tagged to fire hydrants to realistically test their frequency analysis-based leak location algorithm and Jacobsz [27] suggested the installation of fiber optic cables close to buried water pipes, to detect the temperature and stress changes provoked by a leak. Regarding leaks in gas distribution systems, Xiao and Li [28] compared acoustic techniques for detecting leaks in complex low-pressure networks, while Ali et al. [29] applied empirical model decomposition and K-clustering to improve the signal-to-noise ratio of the acoustic measurements.

There are also studies focused on optimally locating sensors in fluid distribution networks to better detect potential leaks. Li et al. [30] developed a clustering algorithm to optimally place pressure sensors in water distribution networks for leak detection, Hu et al. [31] applied multi-objective optimization to locate pressure sensors considering the severity of leaks in different places, while Goulet et al. [32] used model falsification to optimize the position of a minimum number of flow velocity meters.

Finally, regarding the planning of inspections of the network, Mancuso et al. [33] applied the Robust Portfolio Modeling approach and multi-attribute risk-based value theory to schedule inspections and maintenance operations in underground pipelines. Khaleel and Simonen [34] analyzed the selection of ultrasonic inspection strategies and inspection intervals to maximize tube reliability, and Ossai et al. [35] addressed inspection and repair decisions for lines prone to internal corrosion, using Markov chains and Monte Carlo simulation.

None of the reported procedures applies Decision Analysis, in the sense introduced by Howard [1], to the leak location problem. Additionally, previous published research does not consider subjective knowledge relative to the pipes' conditions nor use it to obtain “a posteriori” probability distributions, which are used to solve a decision problem under uncertainty to produce a plan to locate the leak. Finally, the application of the procedure shown here is much simpler than the methods previously reported, which, due to their mathematical and computational complexity, are not feasible for most engineers supervising hydraulic networks, given the time and tools available.

3.1. Case Study Description

A simple hydraulic system, depicted in Fig. (1), is considered, consisting of three sections: Section 1 (S₁) is a horizontal section of length l₁, Section 2 (S₂) corresponds to a vertical section of length l_2, and Section 3 (S₃) is a final horizontal section of length l₃. The diameters of the respective sections are D₁, D_2, and D₃. An inlet stream of volumetric flow rate Q₁ and pressure P₁ enters the pipe from its left end, while the volumetric flow Q₂ comes out of the right end at pressure P₂.

The leak is located at a linear distance X from the inlet and occurs through a circular orifice of diameter d_X. The flow rate of the leak q_X (volume/time) is calculated by a discharge equation:

Where ρ is the density of the liquid, A_X the cross-sectional area of the fracture, assuming a circular hole of diameter d_X, P_X and P_ATM are, respectively, the pressure inside the pipe at position X, and the atmospheric pressure. The valve constant K is set so that when d_X and P_X correspond, respectively, to a leak hole diameter equal to the largest pipe diameter in the network and to the highest pressure in the system, all inlet flow would be lost (q_X equals Q₁). The pressures at the point of the leak, P_X, and the outlet P₂ are calculated from the Bernoulli equations, where g is the gravitational acceleration constant.

Where V₁ and V₂ are the linear velocities of the flow at the inlet and outlet of the pipe (V₁=Q₁/A₁ and V₂=Q₂/A₃, where A₁ and A₃ are respectively the cross-sectional areas of pipes of diameter D₁ and D₃ and Q₂=Q₁−q_X), V_X,1 and V_X,2 are the linear velocities of the liquid, in the direction parallel to the pipe, just before and after the leak. Variables ΔP_F1,X, and ΔP_FX,2 are, respectively, the frictional pressure losses from the inlet to the leak position and from there to the outlet, and Z₁, Z_X, and Z₂ are the vertical heights, with respect to a horizontal reference line, of the inlet stream, leak position, and outlet stream. The other terms of the equations depend on the leak position as shown below, where the mass flows W₁ and W₂ are calculated respectively as ρ×Q₁ and ρ×Q₂, υ is the specific volume of the fluid (1/ρ) and f is the friction factor.

If l₁+l₂≤X

The lengths l₁, l_2, and l₃, diameters D₁, D_2, and D_3, and the average density ρ and viscosity of the liquid are known, as well as an estimate of the friction factor f. With the pressure P₁ and the flow Q₁, the Bernoulli equation allows the highest pressure of the system to be calculated, based on which the discharge constant K of Eq. (1) can be calculated. The unchanged system parameters were set to D₁=D₂=D₃=5 cm, l₁=l₃=20 m, l₂=10 m, Q₁=10.65 m³/h, P₁=2 atm, ρ=997 kg/m³ and f=0.0045.

3.2. A Priori Information

It is believed that the leak is equally likely to appear in any position and that its diameter is similarly expected to be anywhere between 1 and 5 cm. So, X is a uniformly distributed variable between 0 and 50 m, while d_X is a uniform variable between 1 and 5 cm. While the literature reports specific probability distributions for pipe crack sizes across different materials, the uniform distribution is a “minimum information” distribution and may be used to reflect operators’ immediate knowledge once a leak is suspected. Simulating X and d_X from their distributions and, using equations 1 to 18, the cumulative probability distribution of the outlet flow Q2 is calculated and shown in Fig. (2). Figs. 2 and 3 were produced by simulating 20’000 instances of X and d_X from their probability distributions. The number of replications was found to be large enough that further increases did not noticeably change the cumulative probability distribution shape.

The cumulative probability distributions of Q₂ conditional on whether the leak lies in the first (0<X<20), second (20<X<30), or third (30<X<50) pipe section are shown in Fig. (3). The section containing the leak alters the shape of the cumulative probability distribution of the outlet flow. Thus, observing the outlet flow should change the probability distribution of which section contains the leak. In the following, it is shown how an observation of Q₂ can be used to update the initial beliefs about the possible leak location, and how this updated knowledge is used to decide how to search for the leak.

3.3. Selection of the Section Review Sequence

Once it is noticed that the outlet flow decreases, indicating a leak, the pipe sections need to be checked to identify the section containing the leak. The system is divided into three sections:

(1) Section 1 (S₁): the first horizontal section (with a length of 20m)

(2) Section 2 (S₂): the central vertical section of 10m in length

(3) Section 3 (S₃): the final 20 m pipe section.

The time needed to inspect sections 1, 2, and 3 is, respectively, Δt₁, Δt₂, and Δt₃. When a leak is detected, a decision must be made regarding the order of inspection: which section to check first, which section to check next if no leak is found, and which section to inspect last if the leak remains undetected in the previously reviewed sections. The sequence of revisions is chosen through decision trees. Two cases are treated: an “a priori” case, in which the revision sequence is selected based on the initial beliefs on the leak location, and a “a posteriori” case, for which the a priori beliefs are updated using an observation of the outlet flow size, and the new knowledge state is used to choose the review sequence.

3.3.1. Recommended Review Sequence Based on A Priori Information

If no observations of the leak effects are made, the section review sequence is chosen based on a priori information, assuming that all points are equally prone to leaks, and the hole size can be anywhere between 1 and 5 cm. The checking of sections during the search for the leak is represented by the decision tree of (Fig. 4). Decision trees are read from left to right; boxes represent decisions, and circles represent uncertainties. The lines emanating from decisions stand for alternatives, while those emanating from uncertainties represent possible outcomes with their corresponding probabilities. The first decision is which section to inspect first, and the first uncertainty is whether the leak is found there. If the leak is not in the visited section, a decision is made on which section to visit next, which in turn leads to uncertainty about whether the leak is found in the newly visited section. The lines at the right end of the tree, which are not followed by any circle or square, are called tree tips, and the decision consequences are written next to them. Decision trees are solved from right to left by calculating the value of the uncertainties and decisions (called collectively nodes). The value of an uncertainty is the expected value of the set of nodes that are connected to its possibilities, while the value of a decision is the minimum value (as in this case, the problem aims to minimize the amount of lost liquid) among the set of nodes that are connected to its alternatives. The value of a decision situation is the value of the initial tree node. While there are several software packages in which decision trees can be implemented and solved, for this work, the relevant calculations were set up in MS Excel, using Macros explicitly programmed for the problem. The consequences are the result accrued by the decision maker for the alternatives and the outcomes leading to a specific tree tip. For example, if it is decided to inspect section 1 first (S₁) and the leak is located here (X∈S₁) the total volume of liquid lost is ∆t₁E[q_X|X∈S₁], where E[q_X|X∈S₁] is the expected value of the leak flow rate if the leak is in Section 1. In this case, the consequences would be measured in cubic meters.

Since it was assumed that the leak has an equal probability of occurring at any point along the system, the probability that the break is in a section depends on the section length. The probabilities that the break is in one section, conditional on not being in another, are calculated by conditioning, e.g. P(X∈S₂|X∉S₁)=P(X∈S₂∧ X∉S₁)/P(X∉S₁ )=P(X∈S₂)/(1−P(X∈S₁)). The probabilities and expected values of the leak flow rate conditional on the section in which the leak lies are presented in Table 1.

Table 1.

P(X∈S1)	0.4	E[qX|X∈S1]	3.21
P(X∈S2)	0.2	E[qX|X∈S2]	3.71
P(X∈S3)	0.4	E[qX|X∈S3]	4.30

By noting the wide range of outlet flow values resulting from a leak (Fig. 2), it can be concluded that the leak flow rate also shows significant variability. The decision, however, is based on the expected value of the leak flow rate, as it is assumed that the pipe owner has a linear preference for the volume of fluid lost (i.e., it is “risk neutral” regarding this quantity), in which case, the minimization of the expected value is an adequate decision objective. When the time to review each section is set as ∆t₁ = ∆t₂ = ∆t₃ = 1 hour, solving the decision tree in Fig. (4) provides an optimal review sequence starting with Section 3, continuing with Section 1, and finishing with Section 2, corresponding to the expected amount of lost liquid of 6.4 m³.

3.3.2. Recommended Review Sequence Based on A Posteriori Information

A precise definition of a “rough observation of the size of the outlet flow” is provided by observing whether flow Q₂ is bigger or smaller than a threshold value Q_U. This observation is used to update the a priori probabilities of which section is leaking, and the resulting a posteriori probabilities are used to select the section review sequence that minimizes the expected volume of liquid lost. The decision tree for this modification, shown in Figs. (5 and 6), starts with the uncertain event of observing a “large” (Q₂≥Q_U) or “small” (Q₂<Q_U) outlet flow. The tree substructures following the outcomes of this uncertainty are analogous to the tree based on a priori information, except that the probabilities and expected flow rates of lost liquid are conditioned by whether Q₂ is small or large. The update of the probabilities is done through Bayes’ theorem.

(19) (20)

P(Q₂<Q_U) can be calculated from the marginal distribution of Q₂. As an example, let Q_U equals 3.2 m³/h and the event Q₂<Q_U, which has a marginal probability of P(Q₂<Q_U)=0.1 (Fig. 2) and a probability conditional on the leak being in Section 1 of P(Q₂<Q_U|X∈S₁)=0.005 (Fig. 3). Based on its length, the a priori probability of Section 1 leaking is P(X∈S₁)=0.4, so the corresponding a posteriori probability given Q₂<Q_U is P(X∈S₁|Q₂<Q_U)=P(Q₂<Q_U|X∈S₁)×P(X∈S₁)/P(Q₂<Q_U)=0.005×0.4/0.1=0.02. This means that if the outlet flow is smaller than 3.2 m³/h, it is extremely unlikely that the leak lies in Section 1.

The threshold value Q_U should be selected so that the expected amount of lost liquid (i.e., the value calculated by solving the tree) is minimized. If ∆t₁, ∆t₂, and ∆t₃ are set to one hour, (Fig. 7) shows that Q_U should be set to 3.2 m³/h to reduce the expected amount of fluid lost.

For Q_U=3.2 m³/h, the conditional probabilities of the leak being in each section and expected flow rates of lost fluid are shown in Tables 2 and 3. The calculated probabilities of the outlet flow being bigger or smaller than Q_U are, respectively, P(Q₂≥3.2)=0.90 and P(Q₂<3.2)= 0.10.

Table 2.

P(X∈S1| Q2≥3.2)	0.44	E[qX|X∈S1,Q2≥3.2]	3.08
P(X∈S2| Q2≥3.2)	0.20	E[qX|X∈S2,Q2≥3.2]	3.11
P(X∈S3| Q2≥3.2)	0.36	E[qX|X∈S3,Q2≥3.2]	3.17

Table 3.

P(X∈S1|Q2<3.2)	0.02	E[qX|X∈S1,Q2<3.2]	6.58
P(X∈S2|Q2<3.2)	0.22	E[qX|X∈S2,Q2<3.2]	7.37
P(X∈S3|Q2<3.2)	0.76	E[qX|X∈S3,Q2<3.2]	7.94

Table 4.

Decision based on information	Recommended section review sequence		Expected volume of liquid lost (VT) in m3
A priori	S3→S1→S2		6.43
A posteriori	Large outlet flow (Q2≥3.2)	S1→S3→S2	5.49	6.03
	Small outlet flow (Q2<3.2)	S3→S2→S1	10.90

If the time needed to check the sections is taken as ∆t₁ = ∆t₂ = ∆t₃ = 1 hour, Table 4 presents the recommended review sequences with a priori information, and the recommended ones based on a posteriori information for the cases when (Q₂≥3.2) and (Q₂<3.2). The notation S_i→S_j→S_k is used to represent a review sequence beginning in Section i, continuing with Section j, and finishing with Section k.

Let V_{T, APR} be the expected amount of lost liquid when the recommended sequence is based on a priori information, and V_{T, POS} be the corresponding value when the sequence is chosen based on updated information. In the table V_{T, APR} = 6.43 m³ and V_{T, POS} = 6.03 m³_. The advantage of observing Q₂ in choosing the review sequence, instead of relying solely on prior knowledge, is denoted as ∆V_T and calculated by V_{T, PR} −V_{T, POS}. From the table, ∆V_T is 0.40 m³. ∆V_T is called “observation value”.

3.3.4. Effect of Section Inspection Times

To analyze the effect of the section inspection times on the recommended sections review sequence, ∆t₁ was set to one, while the ratios ∆t₃/∆t₁ and ∆t₂/∆t₁ were varied. For each case, the decision trees of (Figs. 4-6) were solved to determine the recommended inspection order.

3.3.4.1. Inspection Sequences Based on a Priori Information

The recommended sequences of visits are shown chromatically in Fig. (8), for the case in which only a priori knowledge is used. If the time it takes to check Section 1 is small relative to that of the other sections (large values of ∆t₃/∆t₁ and ∆t₂/∆t₁, near the top right corner of (Fig. 8), Section 1 should be checked first. Similarly, if Section 2 can be checked quickly than the other sections (near the bottom right corner), this section is reviewed first. Generally, if the time to review a given section is considerably shorter than the other sections, the model recommends checking that section first. This is because if the leak is there, it can be stopped promptly, and if it is not, little time has been invested. The area of the graph for which the model recommends visiting Section 2 first is smaller than the corresponding areas for which the first-visited section is either Section 1 or Section 3, as Section 2 is the shortest and thus the least likely to harbor the leak. Moreover, among the sequences starting with either Section 1 or 3, those reviewing Section 2 next are restricted to a smaller figure area than those checking Section 2 last.

On the other hand, the area of (Fig. 8) where it is recommended to start reviewing Section 3 is larger than the corresponding area recommending starting in Section 1, even though the two sections have the same size and, therefore, the same a priori probability of leaking. This is because leaks in Section 3 tend to have higher flow rates than those in Section 1, so they must be addressed earlier.

3.3.4.2. Inspection Sequences Based on A Posteriori Information

The case in which the observation of the outlet flow being bigger or smaller than a threshold value Q_U is used to update the a priori information is treated next. Q_U is selected to minimize the expected value of the total liquid lost, and may differ depending on the values of ∆t₂/∆t₁ and ∆t₃/∆t₁. However, (Fig. 9) shows that the minimizing Q_U value remains approximately equal to 3.2 m³/h for the range of ∆t₂/∆t₁ and ∆t₃/∆t₁ values of interest.

The recommended inspection sequences for large outflows (Q2(QU), indicating a small leak) are shown in Fig. (10). Since a small leak is more likely to be in Section 1, this is the section to be visited first for most of the (∆t₂/∆t₁, ∆t₃/∆t₁) space, save for the case in which the review time of the other sections is sufficiently low relative to that of Section 1, specifically if ∆t₂/∆t₁<0.45 or ∆t₃/∆t₁<0.85. This condition is more restrictive for sequences that begin with Section 2, as the leak is less likely to occur in this section.

If the outflow is less than Q_U, indicating a significant leak, the recommended inspection sequences, as ∆t₂/∆t₁ and ∆t₃/∆t_1, are varied and shown in Fig. (11). In this case, the probability that the leak is in Section 3 is higher, so Section 3 is reviewed first in most cases, except when ∆t₂ is small and ∆t₃ is large, in which case Section 2 is reviewed first. Although the a priori probability of the leak being in Section 1 is twice that of Section 2, if the leak is large, it is very unlikely to be in Section 1, so Section 2 is the last to be checked.

The difference between the expected value of lost liquid when the inspection sequence is decided with a posteriori information and the corresponding quantity using a priori information is called “observation value” and provides a measurement of the worth of the measurement used for updating the a priori information. This quantity is shown in Fig. (12). When the observation value is null, it means that the recommended inspection sequence is the same for the a priori decision and for both instances of the a posteriori one. If Section 3 can be reviewed swiftly (low value of ∆t₃), the observation value is small, as the a priori and a posteriori inspection sequence selections recommend reviewing this section first, given that the most significant leaks are expected to be in here. The important values of the observation value are associated with cases in which reviewing Section 3 would take a lot of time (big value of ∆t₃), as in this case, the solution of the decision problem with a posteriori information changes the recommended sequence depending on the observed outlet flow.

Figure 13 shows a plot of the observation value when ∆t₂/∆t₁ is fixed to one and ∆t₃/∆t₁ is varied. Below this plot, color bars indicate the inspection sequence recommended for the ∆t₃/∆t₁ value directly above on the horizontal axis, the top bar stand for decisions based on a priori information, while the middle and bottom horizontal bars show, respectively, the sequence based on a posteriori information, for the cases of a big (Q₂>Q_U) or small (Q₂≤Q_U) outlet flow. For example, for ∆t₃/∆t₁ below approximately 0.82, all recommended sequences start reviewing Section 3, with the recommended sequence based on a posteriori information if the outlet flow is small, selecting Section 2 to be visited next.