*2.2. Theoretical Maximum Flow*

We consider the theoretical maximum flow, as the maximum flow value that can be injected in the network by ensuring that the pressure is not lower than a minimum value established as a constraint. This flow represents the theoretical capacity of the distribution network.

The maximum flow through a simple pipe can be calculated by taking into account the upstream and downstream boundary conditions in terms of gauge pressure or head. In a network the calculation may be similarly performed. The downstream boundary conditions are set by the setting curve, which was built to meet a defined minimum pressure (*Pmin*). The upstream condition, which provides the hydraulic potential of the network, is defined by the hydraulic head available at the source (*Hs*), which can be a reservoir (Figure 3) or a pump. The theoretical maximum flow (*Qmaxt*) is determined by the intersection of the setting curve and the supply curve (see Figure 4).

**Figure 3.** Representation of setting curve and theoretical maximum flow.

**Figure 4.** Theoretical maximum flow for: (**a**) a tank or reservoir; and (**b**) a pump.

The theoretical maximum flow or network capacity can be increased by changes in the supply source: by, for example, building new reservoirs or increasing the pump power or the number of pumps; reducing minimum pressure; and modifying the network (adding reinforcement, greater interconnectivity, making replacements, *etc.*). In our study, we address the latter case.

The theoretical maximum flow is strongly related to the setting curve which varies according to the network configuration, and the changes in the pipes or the service minimum pressure. One way to increase the network capacity without altering *Hs* is by changing the slope of the setting curve. This can be achieved by changing some physical characteristics of the network pipes.

### *2.3. Required Maximum Flow*

For the process of increasing the capacity of the network it is necessary to have a point of reference, a target point defined by the required maximum flow (*Qmaxr*) and its corresponding source pressure head. In CWS, the required maximum flow corresponds to the maximum hourly flow.

The required maximum flow is calculated from the demand, using population growth forecasts, losses, and other future uses and water needs.

#### *2.4. Procedure for Increasing the Capacity of the Network*

The process of increasing the capacity of the network is very useful for the transformation of IWS into CWS. The process is based on changing the characteristics of the network so as to reduce the slope of the setting curve until the value of the required maximum flow is exceeded, see Figure 5.

**Figure 5.** Network capacity increase process.

Because intermittent supply systems have hydraulic, structural [3,19,20], and water quality problems [21–25], we consider that, in the transition process to continuous supply, replacement is more effective than pipe reinforcement.

Starting with the current network, each pipe is replaced with another pipe with a different diameter. This change is evaluated by calculating the theoretical maximum flow. Thus, it is possible to calculate the increase in capacity involved in each of the changes. By incorporating the cost of replacing a pipe *p* an expansion rate is determined:

$$q\_{p,d} = \frac{Q\_{\text{max }t}^{p,d} - Q\_{\text{max }t}^{p}}{\mathbb{C}\left(d\_p\right) \cdot L\_p},\tag{1}$$

where *qp,d* is expansion rate produced by substituting pipe *p* for a new pipe with diameter *d*; *Qo*max *t* is theoretical maximum flow of the original network or the network modified in the previous step; *<sup>Q</sup><sup>p</sup>*,*<sup>d</sup>* max *t* is theoretical maximum flow after substituting pipe *p* for a new pipe with diameter *d*; *C(dp)* is unit length cost of pipe *p* replaced with a new pipe of diameter *d*; and *Lp* is length of modified pipe *p*.

The expansion rate (Equation (1)) is very useful in identifying pipes that constrain the network capacity.

Another approach for selecting the pipes to be replaced, which prioritizes the flow increase over the cost, is by raising the difference in the numerator of the expansion rate (Equation (1)) to an exponent, *n*, as in Equation (2). In this way, the pipes will initially be modified to larger diameters. This will subsequently offset the initial cost as smaller diameters will be required.

$$q\_{p,d}^n = \frac{\left(Q\_{\text{max}\ x}^{p,d} - Q\_{\text{max}\ t}^o\right)^n}{\mathbb{C}\left(d\_p\right) \cdot L\_p},\tag{2}$$

By using the expansion rate in each of the expansion stages, the pipe to be modified or replaced is identified (as the pipe with the largest expansion rate value).

The process outlined in the flowchart in Figure 6 enables us to define a new network configuration by increasing its capacity after replacing the pipes that represent the lowest costs. The order of priority for the replacement of the pipes is also identified, and this enables us to plan the process of gradually increasing the network capacity. The total cost may be divided into stages based on this prioritization.

**Figure 6.** Flowchart for the process of increasing the network capacity.

### **3. Results and Discussion**

The case of study has two parts:

The first step is to reconstruct the hydraulic conditions of the subsystem in the south of Oruro in the period 1968–2013. Considering that the initial network is the main part of the current network and that it has been maintained since the inauguration of the infrastructure, we will use the current network model. Due to a lack of information, the growth of the urban area is used as a reference to reconstruct the network evolution. As a result, the topology of the network during each of the study periods is estimated. Considering the population and water demand of each of the periods studied, it is possible to calculate the required maximum flow. However, to compare this value with the network capacity it is necessary that both elements have the same dimensions. For this purpose, we propose the use of the theoretical maximum flow indicator.

In the second part, based on the current network of the south Oruro area, we seek to identify and prioritize the order of the pipes that require replacement under economic and hydraulic criteria, in order to gradually increase the network capacity to achieve the necessary infrastructure to transform IWS into CWS (24/7). Besides the theoretical maximum flow, an expansion rate indicator will also be used.

### *3.1. First Part of the Study*

The project "Drinking water for the city of Oruro" [36] was begun in 1968 and included the main infrastructure for water supply in the southern sub-area of the city—a tank and the mains. The subsystem offered continuous supply but is currently an IWS system. There are no records of when the system became intermittent. Based on the theoretical maximum flow, the transition process is analyzed in this part of the study.

Considering the size of the urban area [37], population growth [38], population density, and the number of current users, it is possible to estimate the population growth of the study area, defined by the urban growth (Table 1) during the given periods. Based on the current average supply of the subsystem (84.32 L/capita/d) and a peak factor of 2.5 (representative of other areas of the city that have CWS and similar population growth) the required maximum flow for each year can be calculated.

**Table 1.** Population and required maximum flow for the Oruro southern subsystem.


The urban growth of the city enables us to set growth scenarios for the subsystem supply network, for the changes in the mains and for the inlet to the sectors (see Figure 7). Based on these scenarios, the setting curve and the theoretical maximum flow or capacity of the network (Figure 8) is calculated. A minimum pressure ( *Pmin*) condition of 20 m is adopted; the minimum elevation of the water level in the reservoir ( *Hs*) is 3771 masl (meters above sea level), and the average elevation of the network is 3723.8 masl.

Comparison between the required maximum flow and the theoretical maximum flow or network capacity is shown in Figure 9.

**Figure 7.** Growth of the Oruro southern subsystem network.

**Figure 8.** Setting curve in each of the years of study.

**Figure 9.** Development of the theoretical maximum flow and the required maximum flow in the study area.

The studied system was designed for continuous supply. This is evident because in 1968 the network capacity was greater than the required maximum flow. The network was designed to supply water to a larger population and had a reasonable slack for expansion. This scenario was maintained throughout the first period, 1968–1972; and expansions made during this period were developed in favorable areas. These expansions were developed properly and did not significantly weaken the network capacity.

In the period 1972–1985, the situation changed. The southern region of Oruro grew faster than the rest of the city [37]. The network expansions to cover the new sectors were very large. In addition, the selection of small diameters increased the pressure drop and network capacity was therefore reduced.

In 1985, the required maximum flow exceeded the theoretical maximum flow. This fact imposed a reduction of pressure in the network. The nodes located in unfavorable areas were prone to run out of water during peak consumption. People needed to protect their supply and opted for the use of household tanks. This scenario led to inequity in the supply: nodes located in favorable areas squandered water, due to lack of metering, while users located in unfavorable areas complained about a lack of water. Eventually, a perception of water scarcity prevailed among the population and the operator. A solution was sought.

To solve the water shortage in less favorable points, there are two potential solutions: the first option, though not very evident in scenarios of economic scarcity and poor management, is to expand the capacity of the network by replacing and reinforcing the mains; and the second option is to opt for intermittent supply (a widespread misconception derived from a lack of water and funding).

The adoption of intermittent supply limits the hours of supply in different areas, setting schedules that seek to reduce the required maximum flow. This action may be useful initially as the main sections transport lower flow rates during peak consumption and therefore sufficient flow reaches the points located in unfavorable areas. However, population growth will require expansion and this condemns the system to intermittent supply.

In this second period, the theoretical maximum flow and the required maximum flow curves intersect. A point was reached at which the capacity of the network no longer met the demand of the population with continuous supply (this situation has been maintained since then). Accordingly, we can say that a policy based on an IWS approach was implemented in the south of Oruro starting between 1970 and 1985 (see Figure 9).

In the third study period, 1985–2007, intermittent supply was consolidated. New expansion further reduced the capacity of the network. The expansion of the S01-16 sector used diameters that were too small and, in addition, supply was extended to an even higher point. These features made it a critical point, which conditioned the setting curve severely and, consequently, the value of the theoretical maximum flow.

Between 2007 and 2013, the subsystem was expanded with the S01-18 sector, which did not affect the capacity of the network because the sector is in a lower area and the installed diameters were suitable for the needs of the sector. It proved to be a good expansion, since it did not greatly affect the setting curve and the theoretical maximum flow.

The reduction of network capacity is a problem not perceived when the network works with intermittent supply, because water reaches all sectors, although at very low pressures; however, this situation will be primarily responsible for inequitable supply.

### *3.2. Second Part of the Study*

The capacity of the subsystem network in the south of Oruro is currently insufficient and, as a result, supply is intermittent. It is necessary to increase this capacity if the aim is to achieve continuous supply. Due to a shortage of funds, the expansion is bound to be gradual and staged. Accordingly, it is necessary to know the order of pipe replacement. The budget available annually for the process of increasing the network capacity is Bs. 700,000 (seven hundred thousand Bolivian boliviano, equivalent to €89,172). The transition from IWS to CWS requires that the network has a capacity for continuous supply of 91.98 L/s.

Unit costs, which include all the elements necessary for the replacement of pipes, and which depend on the diameter, are presented in Table 2.


**Table 2.** Unit costs for replacing pipes as a function of diameter.

When large diameter pipes are changed, the theoretical network maximum flow is increased (see Figure 10). This increase is more pronounced in pipes P-1, P-2, P-3, P-4, P-11, and P-13, and even more so in pipes P-12 and P-17. These last mentioned pipes are the most critical for the network and represent bottlenecks; so, any action to expand the network must take them into account. Augmenting the diameter of the remaining pipes causes minimal increases in the network capacity, so their importance in increasing network capacity is minimal. Generally, the first upgrade of diameter produces a significant increase in network capacity, while subsequent diameter upgrades do not produce such a large effect. The curve tends to reflect an asymptotic behavior (Figure 10).

**Figure 10.** Network capacity increase based on diameter change.

The expansion process was developed using the expansion ratio *q* given in Equation (2) for *n* = 1, 2 and 3. The exponent was chosen depending on the theoretical maximum flow (Figure 11). The use of the indicator with an exponent greater than 1 enables lower costs to be reached when the expansion flow rate is larger; however, *n* = 1 can be used when the maximum flow requirement is lower. In any case, it is adequate for making a comparison with indicators.

**Figure 11.** Comparison of cumulative costs depending on the expansion rate used.

To reach the required maximum flow of 91.98 L/s, the lowest cost is produced by the indicator *q* with *n* = 2. The setting curve is gradually changed in five steps that define the order of actions for the gradual improvement (Figure 12). With this prioritization, three stages of investment (last column of Table 3 and Figure 13) are defined.

**Figure 12.** Setting curve evolution in the capacity increase process.


**Table 3.** Results of the network capacity increase process.

Due to increasing water-losses produced after transition from IWS to CWS [3], and which consequently increase the required maximum flow, it is necessary to implement an active leakage control [39] simultaneously with the network capacity expansion to avoid oversizing the network and wasting water.

**Figure 13.** Stages of investment.
