*2.3. Pipe Failure Modeling*

Once the status of each pipe is determined, the information is entered into a network solver, EPANET, for hydraulic simulation (*i.e.*, solving for WDS system equations consisting of conservations of mass and energy). To simulate the pipe leakage, pressure-dependent flows (*Ql*) are assigned to the closest node to the damaged pipe using the following equation:

$$\mathbf{Q}\_l = \mathbf{C}\_\mathbf{D} \mathbf{p}^\alpha \tag{6}$$

where CD = the discharge coefficient (= ˆ 2gγw ˙α A) (the emitter coefficient in EPANET), where g = gravitational acceleration and α = the exponent of the power function; γw = the specific weight of water; A = the opening area of the damaged pipe; and p = pressure at the closest node.

Lambert [45] conducted an experimental study to investigate the accuracy of the power function model in Equation (6) and provided a guideline on using exponent α based on the pipe material and level of leakage. It was suggested that the exponent of 0.5 would be used to simulate large detectable leakages in metal pipes; the exponent of 1.0 (linear relationship between discharge and pressure) would be used if no information on the pipe material is available. Puchovsky [46] theoretically derived a discharge coefficient and validated it using sprinkler data.

The shape and area of the opening on ruptured pipe varies depending on pipe material, origin of the pipe damage, and direction of external force. Large opening area is more likely to occur in the failure of large pipes than smaller pipes. To that end, it was assumed that the total opening area is equivalent to 10% of the entire cross-sectional pipe area during leaks. In case of breakage, the entire cross-sectional area was used as the opening area.

If a pipe was tagged as leaking, it was modeled in a hydraulic simulation, as illustrated in Figure 2a, and the discharge coefficient was assigned to the downstream node. The pipe breakage was modeled as shown in Figure 2b. The discharge coefficient was assigned to the upper node in flow direction. Then, the broken pipe was set to "closed" to disconnect the water flow. The demand of the node connected to a broken pipe was modified to consider the degraded delivery capacity due to disconnection. In the model, the nodal demands of both end junctions of the broken pipe were reduced by degrees of node (DoN). Here, DoN is defined as the number of connections/edges that a node has to other nodes in a network. As shown in Figure 2b, the base demand of the upstream node is reduced by 25% (= 1/DoN = 1/4), while the downstream node is reduced by 33.3% (= 1/DoN = 1/3).

**Figure 2.** A schematic diagram to describe pipe damage modeling: (**a**) leakage condition and (**b**) breakage condition.
