## 4 Results and Discussion
### Basic Geometries of the Wadi Haly Catchment
Numerous workers have examined the idea of the stream order system, most notably in the studies cited in Refs. [9; 11; 16; 40]. Streams can be defined and classified hierarchically using this method, which makes it easier to understand how they relate to their different tributaries. Researchers can successfully show how minor streams feed into larger waterways by providing streams with numerical numbers according to where they are in a drainage system. This technique not only helps in the meaning and visualizing of hydrological information but also improves our understanding of watershed dynamics and the different ecological processes that occur within the proposed systems. The investigation of stream order framework is very helpful in various fields including hydrology and environmental \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Sub-Catchments** & **Total Area** & **Perimeter** & \
\begin{tabular}{c} **Catchment** \\ **Length** \\ \end{tabular} &
\begin{tabular}{c} **Catchment** \\ **Width** \\ \end{tabular} & **H-Average** \\ \hline Sub-C1 & 694.10 & 164.98 & 43.99 & 18.18 & 1017 \\ Sub-C2 & 78.147 & 46.17 & 9.42 & 13.16 & 447 \\ Sub-C3 & 471.69 & 129.03 & 26.34 & 20.88 & 2366 \\ Sub-C4 & 205.23 & 86.66 & 16.26 & 14.72 & 1045 \\ Sub-C5 & 272.85 & 84.61 & 22.42 & 18.5 & 2201 \\ Sub-C6 & 249.97 & 88.42 & 21.2 & 20.34 & 2430 \\ Sub-C7 & 517.87 & 120.12 & 34.2 & 24.91 & 2584 \\ Sub-C8 & 40.063 & 30.82 & 8.1 & 7.36 & 863 \\ Sub-C9 & 76.710 & 53.02 & 20.81 & 5.87 & 787 \\ Sub-C10 & 111.47 & 62.84 & 12.52 & 15.3 & 632 \\ Sub-C11 & 677.99 & 182.41 & 60.28 & 16.82 & 1394 \\ Sub-C12 & 200.39 & 93.75 & 20.25 & 16.79 & 953 \\ Sub-C13 & 326.55 & 84.98 & 29.17 & 17.85 & 2251 \\ Sub-C14 & 691.93 & 211.23 & 58.47 & 24.42 & 2602 \\ Sub-C15 & 456.36 & 122.50 & 36.52 & 14.95 & 2499 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Key geometric characteristics of the Wadi Haly catchment study area. management, because it provides valuable information on water flow systems, sediment carrying, and the overall health of aquatic ecosystems. First-order small rivers are typically the shortest, while second-order streams start running when two first-order branches merge. This hierarchical structure continues, with two second-order streams joining to create a third-order stream, and so forth. Within the Wadi Haly catchment, various sub-catchments exhibit different stream orders, as follows: one sub-catchment has an eighth-order stream (Sub-C1), several sub-catchments contain seventh-order streams (Sub-C2, Sub-C4, Sub-C8, Sub-C10, Sub-C11, Sub-C12, and Sub-C15), and one sub-catchment has a fifth-order stream (Sub-C5) (Table 3; Figure 5b). Stream order is inversely related to the number of streams within a watershed, as higher-order streams are generally formed by the convergence of multiple lower-order streams, resulting in fewer streams as total stream order increases. Table 1 illustrates the relation between total stream order, stream count, and stream length across sub-catchments in the study area, with stream counts ranging from 15,263 for the first order down to 1 at the eighth order, totaling 19,945 streams within the Wadi Haly catchment (Table 3). Sub-catchment 11 has the highest number of streams (2588), while sub-catchment 8 has the fewest (165). This analysis underscores the importance of stream length in evaluating soil erosion potential, as longer stream paths can accelerate erosion. Geographic Information System (GIS) tools, particularly when integrated with digital elevation models (DEM), offer a precise means of measuring stream length and other morphometric characteristics, which are valuable for implementing targeted soil conservation strategies. Many studies indicate that the length of streams generally decreases as total stream order minimizes, with first-order streams typically being the longest and higher-order streams becoming progressively shorter [24; 51; 52; 53; 54; 55].