# The Impact Assessment of Water Supply DMA Formation on the Monitoring System Sensitivity
## 2 Materials and Methods
### Model Network  
The model grid network selected for study consisted of 25 nodes with the same elevations, located symmetrically in five columns and five rows. The distances between the rows and columns were 100 m each. A water demand of 4 dm\({}^{3}\)/s was provided for each node which corresponds to the simultaneous withdrawal of water from the tap above the tub (0.2 dm\({}^{3}\)/s) by 20 customers. The total water consumption in the entire network was 100 dm\({}^{3}\)/s. A total of 12 variants (\(A\)-\(L\), Figure 1) of connections between nodes with different geometric structures were used in the research. The total length of the pipes ranged from 2400 to 4000 m. In all variants, the same method of supplying the network through a water source located 40 m above the level of nodes was adopted. The internal diameters of the pipes varied, depending on the variant, from 80 (assumed minimum diameter) to 300 mm. Diameters were selected to maintain a water flow velocity as close as possible to 1.0 m/s. The roughness of the pipes was assumed as 0.01 mm. Steady state simulations of the hydraulic conditions of the model grid networks were carried out using the Bentley WaterGEMS software. A reservoir, the water source for the model network, was located 40 m higher than the network nodes. Initially, simulations of operation in normal hydraulic conditions for each of the network variants were carried out. The obtained results of the flow rate in the mains and the pressure in the nodes constituted the comparative material for further analyses. A significant water leak, corresponding to the opening of a fire hydrant (10 dm\({}^{3}\)/s, 36 m\({}^{3}\)/h), was assumed in simulation studies as an additional water demand in selected nodes. Each time, only one fire hydrant was considered opened at the time. In all variants, the same locations of the opening hydrants H1, H2, and H3 were assumed (Figure 2). In the first phase of the research, it was assumed that the pressure gauges used are characterised by an uncertainty of readings of \(\pm\)2.0%. Assuming the maximum pressure in the analysed network (40 m H\({}_{2}\)O), this uncertainty can be defined as \(\pm\)0.8 m H\({}_{2}\)O. The problem of uncertainty of the flowmeter sensors was omitted due to the fact that the simulated leakage increases the flow by 10%, which is much higher than the accuracy of most devices for measuring flow rate. At this stage, the number of nodes was sought where the difference in the pressure head in normal conditions and after opening the hydrant was equal to or greater than this uncertainty. At this stage of the research, the problem of the number and location of measuring sensors was not considered, assuming that all  
Figure 1: Geometric structures of the model grid network (\(A\)–\(L\)). Total pipe lengths (m): (\(A\)): 2400; (\(B\)): 2400; (\(C\)): 2400; (\(D\)): 2600; (\(E\)): 2600; (\(F\)): 2600; (\(G\)): 2700; (\(H\)): 2800; (\(I\)): 2900; (\(J\)): 3200; (\(K\)): 3200; and (\(L\)): 4000 m.  
nodes are subject to the assessment of pressure differences. The calculated number of nodes was compared with the total length of all pipes and the number of loops in all considered network variants. Due to the fact that in several variants the length of the network pipelines and the number of loops were repeated, the calculated number of nodes was also compared with the parameter describing the degree of complexity (\"density\") of the geometric structure of the considered network variants. The fractal dimension (Equation (1)) was selected for this purpose, calculated by the box-counting method [36], often used in image analysis. The calculations of the fractal dimension were carried out with the Fractalvese 3.0 software [37]. \[D=\lim_{\delta\to 0}\frac{logN_{\delta}(F)}{-log\delta} \tag{1}\]  
where \(D\)--fractal dimension, \(F\)--analysed fractal set, \(\delta\)--image resolution, and \(N_{\delta}(F)\)--minimum number of sets of maximum \(\delta\) diameter which can cover the \(F\) set.