The present study was conducted by dimensioning an analysis of urban drainage network projects in two subdivisions in Calçoene and Pracuúba (Figure 1). Calçoene has 10,612 inhabitants and an urbanized area of 3.36 km², while Pracuúba has 3803 inhabitants and an urbanized area of 1.49 km² (IBGE, 2022) .

Both municipalities are partially drained by the Araguari River basin (Figure 1). This is the largest and most important basin in Amapá (Bárbara et al., 2005)

and is more than 300 km long from its source in the Tumucumaque Mountains National Park to its mouth on the right bank of the Amazon River (Marques et al., 2022) . These Amazonian ecosystems and the local population have been impacted by anthropogenic and hydroclimatic processes, of which mining, agriculture, extensive buffalo farming, operation of hydroelectric power plants, and sea level rise stand out. These environmental impacts, combined with climate change, significantly alter some of the hydrological processes and the basin morphology (Cunha, 2013; Santos et al., 2018; Araújo et al., 2020; Silva Júnior et al., 2021; Marques et al., 2022) .

Calçoene and Pracuúba have historically endured a lack of drainage and urban planning infrastructure (IBGE, 2022) . Thus, two subdivisions with subsidized houses were planned to collaborate with the region’s socioeconomic development and implementation of basic infrastructure, enabling assistance to their populations and improving the standard of living (Figure 1) (Brasil, 2019) .

The Calçoene subdivision (Figure 1, upper right image), located at 2˚29'25.2"N and 50˚58'13.4"W, was initially built to accommodate 61 - 62 residential units for families of up to 5 people in an area of 27533.64 m². The Pracuúba subdivision (Figure 1, lower right image) is located at 1˚44'21.9"N and 50˚48'03.8"W, with the same number of residential units as Calçoene, but over a slightly larger area (29377.36 m²). Both are the result of agreements to implement the Calha Norte Program under numbers 895,552/2019 and 905,609/2020 for Calçoene and 895,553/2019 and 915,672/2021 for Pracuúba (Programa Calha Norte, 2021a; Programa Calha Norte, 2021b) .

Through the site plans (Figure S1), it was possible to allow the region to receive rainwater infrastructure intervention in relation to the urban agglomeration and initially visualize the interaction of the future drainage system with the environment (Programa Calha Norte, 2021a; Programa Calha Norte, 2021b) . Then, using the planimetric survey plans (Figure S2), the natural path of rainfall-runoff was identified through visual interpretation. This is a guiding factor in the development of the network layout, since runoff will converge at the same point downstream, the outlet, where it will be discharged (de Souza & Zamuner, 2016; Programa Calha Norte, 2021a; Programa Calha Norte, 2021b) .

The project graphics (see examples in Figure S3), containing the subdivision drainage plans, allowed the verification of distances, areas and identification of the entire project layout, longitudinal profiles, rainwater collection, and conduction devices, as well as construction details (Programa Calha Norte, 2021a; Programa Calha Norte, 2021b) .

Through the Kirpich model (Equation (3)) presented in the form of the California Cuverts Practice Calculus (Miguez et al., 2016) , we established an alternative C_t. The value was then applied in the form of duration in the IDF equations used to characterize new models for estimating diameters (D1, D2, D3, and D4), one of them being the IDF relationship with original parameters.

C t = 57 ⋅ ( L 3 H ) 0.385 (3)

Where:

C_t—concentration time (min)

L—length of the main thalweg of the basin (Km)

H—unevenness between the highest point of the basin and the outlet (m)

The operation of digital remote sensing models, using QGIS version 3.28.2, made it possible to define the values of the length and slope variables applicable to Kirpich’s method. The most relevant watershed upstream of the subdivision and the initial flow collection devices of each subnetwork of Calçoene and Pracuúba were used as a benchmark (Figure 2).

To make the analyses more robust, and due to the lack of diversity of hydrological studies to characterize intense rainfall equations in Amapá (Back & Candorin, 2020) , more scenarios were created using new coefficients applied to the intense rainfall equation, according to Back & Candorin (2020) . The geographical adaptation of using rainfall stations closer to the stations considered in the original project was also performed (Table 3).

Using the coefficients and according to Queiroz Júnior et al. (2013) and Back & Candorin (2020) , three additional dimensioning process models were created with a focus on the diameter found for each stretch of the networks, considering different input parameters in the scenarios (Table 4 and Table 5). The objective was to evaluate the changes in diameter, flow, C_t, and rainfall intensity to carry

K, a, b, c—constant coefficients.

out a comparative study and verify the possible significance of the variations in results in each subnetwork of Pracuúba and Calçoene.

For the scenarios proposed using Kirpich’s method to define the C_t, the flow coefficient was not fully preserved in relation to the original scenario. This was due to the forest areas upstream of the subdivisions having a greater impact on soil permeability, so a coefficient of 0.2 was adopted. In the other areas, the coefficient of 0.8 was maintained.

The catchment areas affecting each subnetwork’s initial collection devices were also modified. In general, they were expanded to include the area between the watershed and the start of the subdivision’s rainwater collection.

The pipe utilization rate, that is, the maximum filling of the pipes, was also adapted to the set of parameters indicated by Tucci (2004) in models where the use of the circular culvert stretch is full (100%). It is worth noting that the original model used a 95% rate. However, no references were found in the literature for this specific value, which was, standardized at 100% or 90% by Azevedo Netto & Araújo (1998) , 100% by Wilken (1978) , 85% by Methods and Durrans (2003) and 82% by DAEE/CETESB (1980) . The minimum and maximum velocities were maintained at 0.6 m/s and 5.0 m/s, respectively, in all scenarios (Tucci, 2004) . Return times were maintained at 10 years in all scenarios.

After processing the dimensioning, complete hydraulic calculation tables were generated in Microsoft Excel 365 for each model, named M1, M2, M3, and M4. From these results, a summary was then extracted for each stretch of the project, with a description of the devices linked to the stretch, their length and diameter as ballast for evaluating the capacity of the real system and comparing outputs.

As a complement to the estimated diameters resulting from different inputs and hydrological models, a series of statistical tests were developed to test the impact of these independent variables on the estimates of the project diameters (D1, D2, D3, and D4).

The following were performed: descriptive statistical analyses, tests of assumptions of normality (Shapiro-Wilk), homoscedasticity of variances and residual variation, paired Wilcoxon test for nonparametric comparisons in pairs or double entries, and tests of multiple regression analyses. To this end, R 4.0.3 (Crawley, 2007; R Development Core Team 2020) was used.

The dependent variable was the diameter (D), and the independent variables were the catchment area (A), flow coefficient (C), rainfall intensity (I), and concentration time (C_t). It is important to note that the “municipality” factor was used to test whether there is any effect of “locality” on the variations of inputs in the calculated diameters (D).

In this regard, the research questions are summarized as follows: 1) Did the estimated diameters (D) show significant differences in the drainage network when the models (original and new proposed ones) were tested? 2) Are the responses of the different models sensitive to oversimplification errors of drainage and rainwater project parameters, estimated at M1, when considering different approaches in relatively geographically close Amazonian municipalities (Calçoene and Pracuúba)?

In summary, the statistical significance of the network diameter dimensioning outputs (D) (α < 0.05) was tested, considering the following assumptions: 1) study of the effective benchmark project approved by the Ministry of Defense (MD) and 2) analysis of alternative hydrological-hydraulic scenarios modeled. As a practical effect, the multi-comparative analysis of the results served to test whether there would actually be a need for optimization (or not) in the dimensioning of the original project (M1 or MD), with potential impacts on the technical and economic sizing, when proposing new models.

Based on the complete processing of the dimensioning of each model for each subnetwork analyzed (Table S1), the following diameters were found, broken down according to the project stretch, which represents the route between two system devices. “CU” represents culvert, “CB” means connection box, “MH” indicates maintenance hole, and “ES” symbolizes the energy sink positioned at the end of the network in the outlet.

The initial stretches of the catchment network, starting with the acronym “CU,” were hidden (Table 6 and Table 7), as there was no change in diameters, and all (by project premise and convention) are at least 400 mm, a value that persisted in all models and all networks analyzed. The project assumptions also predicted a minimum diameter of 600 mm for the main culverts, represented by the connections between MH-CB, MH-MH, and MH-ES.

In Calçoene Subnetworks I and II (Table S2 and Table S3), the diameter outputs were identical in all four scenarios. However, in Subnetwork III, there were changes in the results for concrete pipes, mainly in the final stretches of the network, compared to the original model (M1) (Table 6). Although quantitatively, there are many more unchanged stretches when considering the importance of the stretch given by its length in meters; in model M2, there were changes in 11.41% of the total length of the network, especially the reduction in the diameter of the MH-13-ES stretch from 1000 mm to 800 mm.

In the M3 model, we found a 9.02% change in the length of the network, highlighted by an increase in the diameter of the MH-15-MH-5 stretch from 600 mm to 800 mm. The results of the M4 model indicate a 20.44% change in the network’s total length, with an increase in the MH-15-MH-5 stretch from 600 mm to 800 mm and a reduction in diameter in the MH-13-ES stretch from 1000 mm to 800 mm.

Despite flow differentiations in all models, Calçoene Subnetworks I and II data were insufficient to determine differences in diameter dimensioning results. These are the smallest networks analyzed, with lengths ranging from 180 m to 294 m, which suggests that the size of the systems was not enough to increase variations in the culverts.

In these cases, the minimum diameters of the project premise were able to hide any variations in a large part of the route, as observed in the complete subnetwork dimensioning spreadsheets (Table S1).

The M2 models used Kirpich’s method to obtain the C_t and, consequently, rainfall duration (Table 4). The reduction in diameter in the last stretch of M2 of Calçoene Subnetwork III is justified by the changes in the catchment area, flow coefficient, and C_t, which are the main variables for obtaining peak flow (Wang & Wang, 2018; Alamri et al., 2023) .

In Calçoene, the rate of original rainfall duration and that obtained through Kirpich’s method were very similar: 10 minutes and 9.6 minutes, respectively, which did not occur in Pracuúba. However, using the IDF ratios of Back & Candorin (2020) impacted and considerably increased the rainfall intensity in the M3 and M4 models.

For Calçoene Subnetwork III, Kirpich’s method and the Back & Candorin (2020) IDF ratios behaved inversely in terms of their influence on diameters. Compared to the original model (M1), changing the way the C_t was obtained generated the smallest set of diameters (M2) and changing the equation for defining standard rainfall (M3) generated the largest set of diameters. The simultaneous replacement of the C_t calculation method and the intense rainfall equation (M4) generated an intermediate set of diameter values with a reduction in one stretch and an increase in another.

For Pracuúba (Table 7), all the alternative models, M2, M3, and M4, showed increased diameters from 1000 m to 1200 mm in the MH-2-MH-7, MH-7-MH-15, and MH-15-ES stretches. Added to these changes, in the M2 model, there was an increase from 600 mm to 800 mm in the MH-9-CB-13 stretch, totaling an increase of 28.38% in the total length of the subdivision drainage network.

In the M3 model, there was a change from 600 mm to 800 mm in the MH-11-MH-2 stretch, which, together with changes in the final stretches, represented 28.84% of the change in diameters concerning the total length of the system. In the M4 model, in addition to the changes in the last three stretches of the system, there were also increases from 800 mm to 1000 mm in the CB-12-MH-2 stretch and 600 mm to 800 mm in the MH-9-CB-13 and CB-13-MH-1 stretches, totaling 29.81% of the total length of the rainwater culvert network.

In Pracuúba, all diameters increased. This occurred due to the change in how the C_t was obtained and the variation in the IDF ratios used. That is, both behaved similarly, enhancing rainfall intensity. Table 5 evidences this information, given that M2 generated 182.21 mm/h of rainfall and M3 generated 180.85 mm/h of rainfall intensity.

The slope found upstream of the subdivision, resulting from the characteristics of the relief at the site, was a preponderant factor since at a length (L) similar to Calçoene, the unevenness (H) generated a much shorter C_t. This corroborates with Back (2014) , Leal & Tonello (2016) , and Souza et al. (2018) , who stated that the physiographic characteristics of a basin are of great importance in determining C_t.

In the M4 model, when Kirpich’s method and the IDF ratios of Back & Candorin (2020) were combined, the greatest impact of changing diameters was generated. Compared to the alternative models, the original project (M1) was undersized by at least 28.38% of its length. Adugna et al. (2019) in a water capacity assessment in Addis Ababa, Ethiopia, attested to a flash flood scenario in a system where more than 72% of drainage channels were oversized due to obstructions from pollution. This underscores the attention needed for recurring maintenance and cleaning of the system, especially in systems with partial undersizing scenarios (de Souza & Zamuner, 2016; Daltoé et al., 2016; da Silva et al., 2020) .

An undersized system does not have the same technical efficiency and support capacity and can cause socio-environmental and even economic impacts (Alamri et al., 2023) , such as increased risks of flooding, property damage, and material and human losses (Sabóia et al., 2017; Christofidis et al., 2019; Sousa et al., 2021; Hou et al., 2021) . Therefore, it is important to ensure the appropriate dimensioning, considering the specificities of each region. This premise is fundamental in the Amazon, as this region still has low population contingents and potential areas to be rapidly densified and urbanized.

Based on analysis of the hydrological and hydraulic dimensioning variables of the rainwater conduits for Calçoene and Pracuúba (Amapá/Brazil), Wilcoxon tests (bi-caudal) and multiple linear regressions were performed.

The Wilcoxon nonparametric test showed in the paired comparisons of the “D” medians that the comparative responses between the D1 networks do not differ significantly from the D: D1 - D2 network à (V = 3.2, p-value = 0.1294, not significant).

The remaining paired comparisons resulted in the following parameters D: D1 - D3 à (V = 0.0, p-value = 0.0369, Significant); D2 - D3 à (V = 3.2, p-value = 0.1294, not significant); D1 - D4 à (V = 5.0, p-value = 0.0234, Significant); D2 - D4 à (V = 0.0, p-value = 0.1489, not significant); D3 - D4 à (V = 7.0, p-value = 0. 484, not significant).

Figure 3 shows the discrete variations of diameters “D” calculated for each network in the respective municipality (CAL1, CAL2, CAL3, and PRA), considering four models (D1, D2, D3, and D4) in the dimensioning of the rainwater network.

The Wilcoxon tests compared the Calçoene × Pracuúba outputs with significant results for D1 - D3 (p < 0.05) and D1 - D4 (p < 0.05). The differences between the M1 model and the M3 and M4 models are due to the IDF relationships, with the original model based on the coefficients of Queiroz Júnior et al. (2013) and the M3 and M4 models based on the studies by Back & Candorin (2020) .

These results indicate a significant influence from adopting coefficients with different values for the same location, showing that the lack of studies of intense rainfall for Amapá can generate significant variations in rainfall (Back & Candorin, 2020) . However, the dimensioning results for Calçoene when related to Pracuúba distorted the impact of the intrinsic characteristics of each subdivision, which generated a loss of expressiveness in some analyzes due to relief characteristics, slope, and area of the basin, which reduced flow peaks in Calçoene and

increased in Pracuúba. Put on the same plane, these relationships lessen the impact of scenarios that involve the analysis of replacing the method for estimating concentration times, which is highly dependent on the physiographic characteristics of the basin (Leal & Tonello, 2016; Souza et al., 2018) .

In each of the subnetworks, the multiple regressions showed the significant explainability of the diameters (D1, D2, D3, D4) as a function of the independent parameters considered in this research: catchment area (A), flow coefficient (C), rainfall intensity (I), and concentration time (C_t) (p < 0.05). Therefore, all diameters (D1, D2, D3, D4) showed the expected significant dependence (p < 0.0001). For example, on the one hand, D1 was explained by three basic dimensioning variables (A, I, C_t), and its explainability resulting from the analysis was 71.97% R a j 2 . On the other hand, D2 was explained by only two variables (A, C_t), with R a j 2 explainability equal to 74.91%. Similar to D1, D3 was explained by three variables (A, I, C_t), showing the best explainability, with R a j 2 equal to 78.82%. Like D2, D4 was explained by only two variables (A, C_t), and its explainability was R a j 2 equal to 75.63%.

The variables A and C_t had a significant influence on all the dimensioning models. This behavior highlights the relevance and sensitivity of these variables in the design and dimensioning of rainwater networks in the Amazon. In addition, the results suggest the need to pay attention to them more accurately; only in this way would it be possible to adequately estimate peak flows and safely dimension urban drainage networks in the region (Rocha et al., 2022; Alamri et al., 2023) .