A simple method for designing infiltration low impact development techniques considering effects of urbanization and climate change

Método simplificado de dimensionamento de técnicas de drenagem de baixo impacto baseadas em infiltração considerando efeitos de urbanização e mudanças climáticas

ABSTRACT

Infiltration-based low-impact development (LID) techniques are widely used to reduce peak runoff and enhance groundwater recharge in urban catchments. Their effective design requires selecting an appropriate surface area and maximum ponding depth to ensure sufficient storage volume. However, arbitrary assumptions about the ponding depth can lead to over- or undersizing. This paper introduces MoDOBR, a spreadsheet-based optimization model for designing infiltration basins that accounts for clogging scenarios and variable soil conditions. The model dynamically couples surface water balance with subsurface infiltration using a simplified Green-Ampt formulation. Four numerical examples are presented, including a sensitivity analysis across different soil textures. Results demonstrate that assuming a constant infiltration rate equivalent to the saturated hydraulic conductivity significantly underestimate infiltration dynamics and leads to overdesign basins. In contrast, variability in initial soil moisture conditions has a relatively minor impact on design outcomes. The analysis further reveals that optimal height-area coupling is essential, particularly in low-permeability soils, where the integration of underdrains enables feasible and hydraulically compliant designs. For a 5-year return period event, the required basin area ranges from 2.9% of the catchment for sandy soils to over 50% for clayey soils in the absence of drainage infrastructure.

Keywords:
Design of Retention Ponds; Pre-development; Optimization of Retention Ponds; Low Impact Development

RESUMO

Técnicas de desenvolvimento de baixo impacto (LID) baseadas em infiltração são amplamente utilizadas para reduzir o escoamento de pico e aumentar a recarga de aquíferos em bacias urbanas. Seu dimensionamento eficaz requer a seleção adequada da área superficial e da profundidade máxima de armazenamento temporário para garantir volume de armazenamento suficiente. No entanto, suposições arbitrárias sobre a profundidade de armazenamento podem levar ao superdimensionamento ou subdimensionamento das estruturas. Este artigo apresenta o MoDOBR, um modelo de otimização baseado em planilha eletrônica para o dimensionamento de bacias de infiltração que considera cenários de colmatamento do meio poroso e condições variáveis do solo. O modelo acopla dinamicamente o balanço hídrico superficial com a infiltração no subsolo utilizando uma formulação simplificada de Green-Ampt. São apresentados quatro exemplos numéricos, incluindo uma análise de sensibilidade em diferentes texturas de solo. Os resultados demonstram que assumir uma taxa de infiltração constante equivalente à condutividade hidráulica saturada subestima significativamente a dinâmica da infiltração e leva ao superdimensionamento das bacias. Em contraste, a variabilidade nas condições iniciais de umidade do solo tem impacto relativamente pequeno nos resultados do dimensionamento. A análise revela ainda que o acoplamento ótimo entre altura e área é essencial, especialmente em solos de baixa permeabilidade, onde a integração de drenos inferiores permite projetos viáveis e compatíveis com critérios hidráulicos. Para um evento com período de retorno de 5 anos, a área necessária da bacia varia de 2,9% da bacia de contribuição em solos arenosos a mais de 50% em solos argilosos na ausência de infraestrutura de drenagem.

Palavras-chave:
Dimensionamento de Bacias de Retenção; Pré-urbanização; Otimização de Bacias de Retenção; Desenvolvimento de Baixo Impacto

INTRODUCTION

Retention basins are a type of low-impact development (LID) technique designed to temporarily store surface runoff, thereby reducing peak discharges to drainage systems and enhancing subsurface recharge through infiltration (Baptista et al., 2011; Fletcher et al., 2015). Infiltration-based techniques, such as retention reservoirs, can attenuate both flow rates and peak timings (Winston et al., 2016). When soil infiltration capacity is adequate, these systems can offer technically and economically viable solutions (Mano, 2008), while also aiding in the restoration of the pre-urbanization hydrological regime.

Several hydrological models have been developed to simulate the behavior of retention basins. Among these, mass balance models are commonly used to estimate storage and outflow. One widely used method is the PULS approach (Zoppou, 1999), which solves the mass balance by accounting for precipitation, inflow, and percolation. Due to its simplicity, the PULS method is well suited for spreadsheet-based applications (Gomes Júnior, 2019; Gomes Junior et al., 2023; Ferreira et al., 2019). Based on a prescribed maximum allowable outflow, it determines the minimum volume required for a basin. Other modeling tools include the Stormwater Management Model (SWMM), which provides modules for simulating LID components such as bioretention and rain gardens through hydrological and hydraulic balance equations (Rossman and Huber, 2016), and DRAINMOD, which traditionally operates on a daily time step but has recently been adapted for sub-daily simulations (Skaggs et al., 2012; Braswell et al., 2024).

Retention basin design typically requires the specification of surface area, maximum ponding depth, and, when applicable, overflow structures. In some cases, these systems are designed for extreme events with return periods of up to 200 years (Fletcher et al., 2015). Among the established design strategies, the zero-impact criterion (County, 2007) is one of the most widely adopted. This approach compares pre and post-urbanization conditions, where the pre-development state reflects natural land use and higher infiltration potential, and the post-development state includes increased imperviousness and runoff generation. The design aims to match post-development outflows to pre-development peak flows by sizing the storage volume based on the excess runoff and sizing overflow structures accordingly (Rosa, 2016).

Another common method is the envelope curve approach (Santos et al., 2021; Silveira & Goldenfum, 2007), which assumes a constant outflow rate and an inflow hydrograph based on rainfall intensity from the Intensity-Duration-Frequency (IDF) curve. Runoff is calculated using the rational method, and the maximum difference between accumulated inflow and outflow determines the required storage volume, similar to the traditional Rippl method (Tomaz, 2003).

Both the zero-impact and envelope curve methods are suitable for preliminary design and provide volumetric control for urban runoff. However, they do not capture the temporally dynamic and non-linear behavior of infiltration systems. Infiltration-based LIDs often exhibit strong temporal dependencies that influence their effectiveness and require more detailed modeling. For example, Duke et al. (2024) showed that a lack of dynamic information on ponding levels within LIDs can compromise their mitigation performance, particularly under conditions with variable rainfall and antecedent moisture states.

While dynamic models better capture these behaviors, many are too complex for routine use during predesign stages. The Richards equation (Richards, 1931), the standard for simulating vadose zone processes, is computationally intensive and generally unsuitable for manual or spreadsheet-based implementation. Furthermore, numerically stable explicit time-step solutions of the Richards equation are rarely practical for design settings. Alternatively, the Green-Ampt model (Green & Ampt, 1911), derived as a simplification of the Richards equation, provides a physically-based yet tractable framework for modeling infiltration. It captures non-linear behavior and can be implemented efficiently in spreadsheets (Gomes Junior et al., 2023).

The objective of this work is to present a fast and efficient method for designing retention basins and other infiltration techniques, considering the temporal variation of infiltration at the technique surface. By obtaining the minimum depth necessary to guarantee the storage of the minimum volume to combat the effects of excess urbanization, the designs provided by the developed model also allow a scenario of full clogging of the media and its hydraulic devices. It is important to emphasize that the minimum storage volume of surface runoff does not necessarily coincide with the flooding volume modeled by the non-linear hydrological simulation, so the solution to the problem must be found by trial and error or by optimization algorithms.

In response to the growing challenges posed by climate change and the non-stationarity of IDF relationships (Paiva et al., 2024), the method also includes a simplified procedure for designing the basin’s freeboard to account for intensified rainfall events. Four examples are presented to demonstrate the application of the proposed method. The first example addresses a typical design scenario for a 5,000 m² urban catchment on sandy soil. The second example explores the influence of soil texture on optimal design height and area. The third example incorporates bottom orifices to regulate drainage and ensure compatibility with an imposed height of 80 cm. The final example presents a local sensitivity analysis of the model parameters to assess their influence on peak infiltration, ponding depth, and drainage time, in addition to provide an evaluation of initial soil moisture conditions sensitivity in the simulation dynamics.

METHODS

Model for Optimized Design of Retention Basins (MoDOBR)

This section presents the mathematical approach for determining the inflow hydrograph and the dissipation of the inflow through infiltration at the base of the reservoir. The conceptual model represents a small upstream catchment, modeled using the rational method, which discharges into a retention basin that gradually recharges inflow, normalized by the retention basin surface area, to the underlying water table.

The peak discharge from the rational method is given by (Mulvaney, 1851; Kuichling, 1889):

$Q^p=C\cdot i\cdot A_c$,(1)

where Q^p is the peak discharge of the triangular hydrograph [L · T⁻³], C is the average runoff coefficient [-], i is the precipitation intensity [L · T⁻¹], and A_c is the upstream drainage area [L²]. It is assumed that the surface area of the retention basin is negligible relative to the contributing catchment and does not alter A_c. Accordingly, the following formulations apply under the assumption that the retention basin is located downstream and hydraulically connected to the upstream catchment outlet.

This equation is valid for both pre- and post-urbanization conditions, with adjustments made through different values of C and t_c. By the rational method assumptions, the precipitation intensity depends on the time of concentration, which can be estimated by various empirical expressions. One widely used formulation is the SCS-Lag equation (Natural Resources Conservation Service, 2010):

$t_c=23.19\left[\frac{l^{0.8}{\left(S_{\text{SCS}}+1\right)}^{0.7}}{1140Y^{0.5}}\right]$,(2)

where t_c is the time of concentration [min], l is the flow path length [m], Y is the average slope of the basin [%], and S_SCS is the potential maximum retention [mm], computed as S_SCS = 25400/CN − 254 using the Curve Number (CN) method.

Among the various formulations for estimating design rainfall intensity, this study adopts a Sherman-type intensity-duration-frequency (IDF) equation (Sherman, 1931):

$i=\frac{K\cdot {\text{TR}}^{\text{a}}}{{\left(b+t_c\right)}^c}$,(3)

where K, a, b, and c are empirical IDF parameters, and TR is the return period [years].

Given Q^p and t_c, the triangular hydrograph can be represented by the ascending and descending parts:

$Q\left(t\right)=Q_p\cdot \left(\frac{t}{t_c}\right),\text{if}t\le t_c$ (4)

$Q\left(t\right)=Q_p\cdot \left[\max \left(1-\frac{t-t_c}{t_c}\mathrm{,0}\right)\right],\text{if}t>t_c$

where Q(t) is the instantaneous discharge [L³ · T⁻¹].

The difference in volume between the pre- and post-urbanization hydrographs defines the minimum storage volume required:

$V_{\text{min}}=\arg \underset{t\in \text{τ}}{\max }\left[\underset{0}{\overset{t}{{\displaystyle \int }}}\left(Q_{\text{pos}}\left(t\right)-Q_{\text{pre}}\left(t\right)\right)dt\right]$,(5)

where τ is the simulation period. This volume V_min ensures that, even in the event of complete clogging and non-functioning drainage, the system can store the excess runoff generated by urbanization.

Drainage may occur via orifices, weirs, or pumps. In infiltration basins, the primary outlet is bottom infiltration. Since such basins typically have a much larger horizontal area than depth, lateral infiltration is neglected. For smaller systems (e.g., rain gardens or permeable pavements), lateral infiltration may be important (Lee et al., 2015). For simplicity, MoDOBR assumes only vertical infiltration.

Infiltration capacity is estimated using the Green-Ampt model (Green & Ampt, 1911):

$C\left(t\right)=k_{sat}\cdot \left[\frac{\Delta \theta \left(\psi +h_p\left(t\right)\right)}{F\left(t\right)}\right]$,(6)

where k_sat is the saturated hydraulic conductivity, ∆θ is the effective porosity, ψ is the suction head [L], h_p(t) is the ponding depth [L], and F(t) is the cumulative infiltration [L].

To approximate Green-Ampt dynamics with explicit time steps, infiltration rate is defined as:

$f\left(t\right)=\min \left(C\left(t\right),\frac{h_p\left(t\right)}{\Delta t}\right)$,(7)

where the factor h_p/∆t represents the available infiltration rate within a computational time-step. Cumulative infiltration is then updated as:

$F\left(t+\Delta t\right)=F\left(t\right)+\Delta t\cdot f\left(t\right)$ (8)

A mass balance at the basin surface yields:

$h_p\left(t+\Delta t\right)=h_p\left(t\right)+\Delta t\cdot \left(\frac{Q\left(t\right)}{A}-f\left(t\right)-S\left(t\right)\right)$ , (9)

where A is the surface area [L²] and S(t) includes any other inflow/outflow terms (e.g., from orifices, evapotranspiration, irrigation) [L·T^-1].

The maximum ponding depth is:

$h_p^{\text{max}}=\underset{t\in \text{τ}}{\max }\left(h_p\left(t\right)\right)$,(10)

where τ is the simulation duration [T].

Detention time restriction

An important criterion in LID design is limiting the duration of surface ponding to reduce health risks such as mosquito proliferation. This study adopts a maximum detention time t_d = 24 hours. Numerically, we calculate the time t_v at which h_p(t) ≤ δ, where δ is a small depth (e.g., 1 mm).

Coupling design heights

For a given V_min and design height h, assuming a prismatic reservoir:

$A=\frac{V_{\text{min}}}{h}$,(11)

where A is the average retention basin surface area [L²]. Herein throughout the paper a constant average area is assumed, but other assumptions can be readily implemented in the dynamical model.

This defines the area required to store V_min in case of zero infiltration. In practical applications, infiltration contributes to volume reduction and affects the required height.

The design objective is to find a pair (A,h) such that: - The volume V_min is satisfied, - The maximum ponding depth $h_p^{\text{max}}\le h$, - The drainage time t_v ≤ t_d. This can be posed as an optimization problem:

$$\begin{array}{c}minimiz{e}_h\left|h-h_{\max }\right|\\ subjectto:h_{\min }\le h\le h_{\max }\\ \text{Eqs}\text{. (5) - (11)}\text{,}\\ t_v\le t_d\end{array}$$ (12)

If no feasible solution exists within [h_min,h_max], a different LID technique may be needed. The maximum tolerable depth $h_p^{\text{max}}$ must account not only for construction constraints, but also for safety ones.

This optimization ensures that the smallest possible surface area is used while satisfying design constraints. MoDOBR solves this problem using the Generalized Reduced Gradient (GRG) nonlinear solver in Excel (Smith and Lasdon, 1992). An example spreadsheet is shown in Figure 1.

Figure 1
Example spreadsheet for MoDOBR implementation. Formulae correspond to the equations described in the manuscript.

Freeboard and protective structure to convey larger floods due to climate change

Climate change is expected to increase rainfall intensities by 20–40% in South America (Paiva et al., 2024). To account for this, a scaling factor γ is applied to the design intensity:

$i_d\left(TR,t_c\right)=\gamma \cdot i\left(TR,t_c\right)$ (13)

The freeboard height h_b required to offset this additional volume is: (13)

$h_b=\frac{C_{pos}\cdot \left(\gamma -1\right)\cdot i\left(TR\right)\cdot A_c\cdot t_c}{A}$ (14)

To prevent overtopping, a surface spillway must be included. The design discharge is:

$Q^{p*}=C_d\cdot L_{\text{ef}}\cdot {\left(h+h_b-h_s\right)}^{3/2}$,(15)

where C_d ≈ 1.8, L_ef is the effective spillway length [L], and h_s is the spillway crest height [L]. Rearranging gives: (16)

$L_{\text{ef}}=\frac{C_{\text{pos}}\cdot i_d\cdot A}{C_d\cdot {\left(h+h_b-h_s\right)}^{3/2}}$ (16)

Although the weir flow S(t) can be incorporated into Equation (9), this paper assumes a conservative scenario where the weir is not considered to estimate the maximum depth. The designed freeboard thus fully retains the excess volume.

The model requires the following input parameters:

• Saturated hydraulic conductivity, k_sat

• Soil suction head, ψ

• Initial infiltration depth, F(0)

• Initial ponding depth, h_p(0)

• Time step, ∆t

• Total simulation time, t_f

• IDF curve parameters: K, a, b, c

• Pre- and post-urbanization runoff coefficients

• Pre- and post-urbanization times of concentration

EXAMPLE 1 – TYPICAL DESIGN CONDITION

A commercial area in São Carlos, São Paulo, with a total area of 5,000 m², consists of 40% impervious surfaces (C = 1) and 60% permeable surfaces (C = 0.6). An infiltration basin is to be designed to handle runoff from a 5-year return period storm event. The time of concentration for the post-urbanization scenario is 10 minutes. Under pre-urbanization conditions (pastureland), the runoff coefficient was C = 0.35 and the time of concentration was 25 minutes. The rainfall intensity-duration-frequency (IDF) curve parameters are K = 819.67, a = 0.138, b = 10.77, and c = 0.75 (Gomes Junior et al., 2021). The underlying soil is predominantly sandy, with hydraulic parameters k = 120.4 mm · h⁻¹, ∆θ = 0.42, and ψ = 49.5 mm. The initial infiltration and ponding depths are F(0) = 5 mm and h(0) = 0 mm, respectively.

The goal is to determine the area and height of the infiltration basin, assuming that the maximum ponding depth equals the design height, as per Equation (12). The results are compared with a case where the height is arbitrarily fixed at 0.80 m.

Solution

The post-urbanization runoff coefficient is the weighted average of the surface types:

$C_{\text{pos}}=\left(0.4\times 1\right)+\left(0.6\times 0.6\right)=0.76$ (17)

Substituting into the IDF equation, the rainfall intensity for a 5-year event is:

$i_{\text{pos}}=\frac{819.67\cdot 5^{0.138}}{{20.77}^{0.75}}=69.68\text{mm}\cdot {\text{h}}^{\text{-1}}$ (18)

The post-urbanization peak flow is then:

$Q_{\text{pos}}^p=0.76\cdot \left(\frac{69.68}{1000\times 3600}\right)\times 5000=0.111{\text{m}}^{\text{3}}\cdot {\text{s}}^{\text{-1}}$ (19)

The pre-urbanization peak flow, similarly calculated, is:

$Q_{\text{pre}}^p=0.034{\text{m}}^{\text{3}}\cdot {\text{s}}^{\text{-1}}$ (20)

Using Equation (4), the pre- and post-urbanization hydrographs are generated. The resulting storage volume required to match the pre-urbanization condition is V_min = 52.14 m³. If the design height is arbitrarily set to 0.80 m, the surface area required would be:

$A=\frac{52.14}{0.8}=65.18{\text{m}}^{\text{2}}$ (21)

This area corresponds to 1.3% of the total contributing basin area. However, hydrological simulation results show that this configuration leads to undersizing, as the maximum ponding depth exceeds the design height. This is illustrated in Figure 2.

Figure 2
Simulation results comparing an arbitrary height of 0.80 m with the height calculated using Equation (12). (a) Temporal evolution of infiltration alongside pre- and post-urbanization inflows; (b) Ponding depth compared to design height; (c) Accumulated infiltration; (d) Temporal variation of stored volume. The inset chart in (a) shows a detail of only the infiltration rate.

In the fixed-height case, the simulation shows a maximum ponding depth of 88 cm—exceeding the design height and indicating a failure to meet performance criteria.

Solving the optimization problem via Equation (12), the ideal design height is 0.36 m, and the corresponding required surface area is 143 m², or 2.9% of the catchment area. This lower height increases the required surface area but ensures compliance with storage and drainage constraints. The increased area also alters the infiltration dynamics, allowing the excess volume generated by urbanization to be accommodated without overflow. The full state evolution for this case is also shown in Figure 2.

EXAMPLE 2 – EFFECT OF DIFFERENT SOIL TYPES FOR THE SAME PROJECT

Using the same catchment and rainfall conditions from Example 1, this example evaluates how the performance of the system varies for different soil types listed in Table 1, assuming a fixed design height of 0.8 m for practical feasibility. Next, the compatible heights for each soil type are computed using Equation (12) to assess which soils can meet the design constraints without overflow.

Table 1
Soil parameter values adopted for the simulation, adapted from (Rossman & Huber, 2016).

Soil Type	ksat [mm · h-1]	∆θ	ψ[mm]
Sandy Loam	120.4	0.42	49.5
Loamy Sand	30.0	0.40	61.2
Sandy Clay	10.9	0.41	110.0
Clay	3.3	0.43	88.9
Silty Clay	6.6	0.49	166.9

Solution

The simulation results for the five soil types are shown in Figure 3, based on a total simulation time of t_f = 1440 minutes (24 hours). It is observed that only the sandy loam and sandy clay soil types are able to drain completely within the 24-hour detention limit when the height is fixed at 0.8 m. However, even for these soils, the maximum ponding depths exceed the design height—reaching 0.88 m and 0.95 m, respectively—indicating non-compliance with the geometric constraint.

Figure 3
Simulation results for a fixed height of 0.8 m across different soil types listed in Table 1, assuming a required detention volume of 52.14 m³. (a) Accumulated infiltration, (b) ponding depth, and (c) infiltration rate, computed using Equation (7).

As previously discussed, only sandy soils are able to meet the emptying time requirement when the height is arbitrarily fixed. However, if the design height is optimized using Equation (12), all soil types yield viable solutions with detention times below 24 hours. These optimized solutions are shown in Figure 4, and the corresponding compatible design heights and required surface areas (expressed as a percentage of the contributing basin area) are summarized in Figure 5.

Figure 4
Simulation results using optimized heights from Equation (12) for different soil types to meet a detention volume of 52.14 m³. (a) Accumulated infiltration, (b) ponding depth, and (c) infiltration rate.

Figure 5
(a) Compatible heights that avoid overflow and guarantee the required detention volume; (b) Percentage of the contributing basin area (A_c) that must be allocated for the retention basin area (A).

EXAMPLE 3 – USE OF DRAINS IN CASES WHERE AREA AND HEIGHT ARE CONSTRAINED

Returning to Example 1, where the design height was arbitrarily set to 0.80 m, it was observed that the simulated ponding depth exceeded this value, resulting in an incompatible design. Solving Equation (12) for this scenario yields an optimal height of 36 cm and a corresponding area of 143 m². However, assuming that the design height must be fixed at 0.80 m, the required area becomes 65.18 m², as shown in Example 1.

To ensure volume compatibility with the adopted height of 0.80 m, a drainage system composed of perforated underdrains can be introduced. These drains regulate the outflow to prevent overtopping, ensuring that the ponding depth does not exceed the design height while meeting the volume criteria defined by the zero-impact condition.

The flow through the orifices can be modeled using the classical orifice equation (Porto, 2004):

$S\left(t\right)=n_o\cdot C_d\cdot A_{\text{ef}}\cdot \sqrt{2g\cdot \max \left(h_p\left(t\right)-h_o\mathrm{,0}\right)},$ (22)

where A_ef = πD²/4 is the effective cross-sectional area of each orifice [L²]. D is the orifice diameter [L], g is the gravitational acceleration [L·T^-2], h_o is the orifice elevation from the basin floor [L], and n_o is the number of orifices [-].

The goal is to determine the number of underdrains n_o with an adopted diameter of 25.4 mm that can reduce a maximum ponding depth of 88 cm to the target height of 80 cm. Additionally, the effective width of the surface spillway is calculated assuming a climate change scaling factor (γ) of 1.2.

Solution

The orifice flow S(t) is incorporated as a source term in Equation (9). Using a drain diameter of 25.4 mm, the solution involves trial-and-error simulations with varying numbers of orifices. Figure 6 presents the ponding depth evolution for 0, 2, 4, 6, and 7 orifices. In all cases, the basin surface area is held constant.

Figure 6
Influence of the number of orifices on the maximum ponding depth. The discharge coefficient was set to 0.6, and n indicates the number of orifices. The dashed black line represents the design height of 0.80 m. The inset chart shows the dynamic behavior within 15 to 20 minutes, showing the differences near the peak.

The results show that 7 perforated underdrains with a 25.4 mm diameter are sufficient to reduce the maximum ponding depth to the target of 0.80 m. Increasing the number of drains promotes faster emptying, which can reduce the residence time but may also shift peak flows and accelerate the hydrograph response.

Therefore, a balance must be achieved between drainage efficiency and peak attenuation. For perforated drains embedded in gravel or filter material, discharge coefficients between 0.3 and 0.6 are recommended. The freeboard height, calculated from Equation (14), is:

$h_b=\frac{0.76\cdot \left[\left(1.2-1\right)\cdot 105.2\right]\cdot 5000\cdot 10\cdot 60}{65.18}=0.20\text{m}$ (23)

The surface spillway is designed for a 10-year return period—higher than the 5-year return period used for the retention basin sizing. Applying Eq. (15), the required effective width of the spillway is:

$L_{\text{ef}}=\frac{0.76\cdot 115.76\cdot 1.20\cdot 5000}{1.6\cdot {\left(0.8+0.2-0.8\right)}^{3/2}}=100\text{cm}$ (24)

LOCAL SENSITIVITY ANALYSIS OF PARAMETERS

Considering the peak infiltration rate, the maximum ponding depth, and the emptying time, a sensitivity analysis was conducted on the soil parameters that characterize the hydrological behavior of the filter medium. This analysis was performed using the soil types listed in Table 1. Six levels of percentage variation from a baseline value were evaluated: −75%, −50%, −25%, 0%, +25%, +50%, and +75%. For each case, the relative percentage deviations in the model outputs were calculated.

The baseline scenario for this analysis corresponds to Example 1, in which a 5-year return period rainfall event is simulated over a basin with an area of 5,000 m² and an average runoff coefficient of 0.76. The objective of this analysis is to evaluate the individual impact of each soil parameter on model performance and to identify which parameters require more precise specification during the design process. The results of the local sensitivity analysis are presented in Figure 7. Among the infiltration parameters, saturated hydraulic conductivity and moisture deficit showed the highest sensitivity to change. However, a different sensitivity profile may be observed if the base parameters correspond to a clayey soil.

Figure 7
Results of the local sensitivity analysis for each of the soil types tested. Each row represents a soil type. The columns from left to right represent the sensitivity of the peak infiltration flow rate, ponding depth, and emptying time, respectively, in relation to variations in saturated hydraulic conductivity, effective porosity, and matric suction potential.

In addition to the local sensitivity analysis of the parameters, we evaluate the effect of different initial soil moisture conditions, and we compare the developed approach with a dynamic model assuming a constant infiltration capacity equal to the saturated hydraulic conductivity. This analysis aims to evaluate the role of uncertain initial conditions and the effectiveness of simplified approaches such as assuming a constant infiltration rate over time. These results are shown in Figure 8.

Figure 8
Influence of assuming a dynamic infiltration calculated with the Green-Ampt model compared to assuming a constant infiltration rate given by the minimum between the saturated hydraulic conductivity (k_sat) and the ponding depth stored depth over time (a) - (c). Part (a) represents the evolution of the infiltration rate, (b) the cumulative infiltration, and (c) the evolution of the ponding depth, with the black dashed line representing the optimized height calculated from MoDOBR. Parts (d) - (f) show the effect of different initial conditions of infiltrated depth at the beginning of the simulation. Values tested represent the boundaries of the antecedent moisture condition (AMC) classification for dormant phase for AMC I, AMC II, and AMC III (Calero Mosquera et al., 2021).

DISCUSSION

The results from Example 1 demonstrate that arbitrarily specifying the height of a retention basin, even for a known soil type, can lead to either under- or over-dimensioned designs. While in purely detention basins (i.e., systems with no infiltration), excess runoff can often be compensated by adjusting surface area alone, this is not the case for retention basins. The design height directly influences infiltration capacity, and its incorrect specification may produce solutions that overflow or fail to meet volumetric targets, especially due to the non-linearity of the infiltration process. The examples presented in this study adopt a conservative criterion: the minimum volume necessary to store post-urbanization runoff, even under full clogging and non-functioning drainage conditions.

Example 2 highlights the sensitivity of different soil types to design performance when subjected to the same hydrograph and design volume. Again, arbitrary height choices proved inadequate across all tested soils. In three of the five soil types, the basins failed to meet the 24-hour maximum detention time, and even in the two cases where drainage time was acceptable, the ponding depth exceeded the assumed height—indicating an infeasible or unsafe design. These findings reinforce the need for height-area coupling through an iterative or optimization-based process.

The use of underdrains, particularly in low-permeability soils such as clay or silty clay, was shown to be an effective solution for reducing the required surface area and achieving feasible heights. The optimized solutions generated by the MoDOBR model yielded viable basin designs for all tested soils, with emptying times under 2.5 hours for a 10-minute storm. However, for soils with low hydraulic conductivity (e.g., clayey or sandy clay textures), design requirements became significantly more demanding, with 20%–50% of the total catchment area needed and design heights dropping below 10 cm. In contrast, sandy soils required only 2.9%–9.5% of the catchment area, making them more favorable for underdrain-free retention designs.

Example 3 explored how underdrains and freeboard can be used to adapt a fixed design height. The results confirmed that by selecting the appropriate number and diameter of perforated drains, it is possible to align ponding depth with the adopted design height while still ensuring drainage performance. However, the system is highly sensitive to these drain parameters. As the orifices are typically embedded in filter materials such as gravel, a discharge coefficient between 0.3 and 0.6 is recommended to account for head losses and partial clogging. In addition, a spillway design considering climate-change effects to convey a larger flood (i.e., the 25-yr event) was performed, indicating a simple and rapid manner to consider effects of climate change in the design of LIDs.

The sensitivity analysis further emphasizes the influence of soil parameters on model outcomes. Saturated hydraulic conductivity (k_sat) was consistently the most sensitive parameter across all performance metrics—peak infiltration rate, ponding depth, and emptying time. Effective porosity (∆θ) had a similar but slightly lesser impact, while the influence of matric suction head (ψ) became more pronounced in finer-textured soils. For these soils, all three parameters exhibited comparable sensitivity, underscoring the importance of accurate soil characterization. These findings also provide insights into the impact of aging and performance degradation, as clogging tends to reduce ∆θ and k_sat, potentially leading to significant deviations from the original design performance.

One of the key contributions of the methods presented in this paper is the incorporation of the time-varying nature of infiltration, modeled using the Green-Ampt infiltration capacity formulation. In contrast, a simplified assumption often used for ease of calculation is to treat infiltration as a constant rate equal to the soil’s saturated hydraulic conductivity (k_sat). However, the results shown in Figure 8(a)–(c) highlight that this simplification leads to a consistent overestimation of infiltration capacity.

The dynamic ponding depth captured in the Green-Ampt formulation introduces a time-dependent hydraulic gradient that enhances infiltration modeling, thereby lowering the maximum ponding depth, increasing infiltration discharge, and raising cumulative infiltration. Neglecting this dynamic behavior introduces greater uncertainty in the results than does uncertainty in the initial soil moisture condition. Initial conditions to any simulation are a problem that hydrologists typically solve by performing a warming-up phase in hydrological models to ensure more realistic initial values for states. However, when using design events, warming-up conditions are not available for design purposes. We show, however, that under relatively large events (i.e., ≥ 5 years), the effect of the initial soil moisture condition in the retention basin is less relevant than the assumption of considering a constant infiltration rate.

Figure 8 (d)–(f) highlight the influence of initial soil moisture on hydrological responses by evaluating four distinct conditions: (i) 5 mm (used as the default in Examples 1 to 3), (ii) AMC I (12.7 mm), (iii) AMC II (25 mm), and (iv) AMC III (53.4 mm). While the early stages of the simulation show moderate variation in infiltration and discharge rates, the results quickly converge after approximately 30 minutes. Additionally, cumulative infiltration values remain relatively stable across all scenarios, suggesting that the system is only moderately sensitive to initial moisture conditions.

More importantly, these results reveal that simplifying assumptions—such as treating infiltration as constant or neglecting its dynamic behavior—introduce substantially greater errors into the design process than variations in initial conditions. These oversimplifications consistently underestimate the true infiltration potential of the system, leading to conservative and often oversized basin designs. This reinforces the value of incorporating physically based, time-varying infiltration models such as Green-Ampt into even simplified design tools.

Ultimately, the findings presented in this paper emphasize that retention basin design is inherently a coupled, nonlinear problem—where infiltration dynamics, geometric constraints, and drainage performance are interdependent and cannot be treated in isolation. Oversimplified assumptions may offer initial convenience but come at the cost of accuracy, efficiency, and long-term resilience.

CONCLUSIONS

The Model for Optimized Sizing of Retention Basins (MoDOBR) was developed and applied across three numerical examples and two sensitivity analyses. The model is relatively easy to use and can be implemented in electronic spreadsheets. An open-source tool was developed and made available to replicate all results and assist in the design of infiltration-based techniques. The model requires only basic catchment data and the physical properties of the soil at the base of the retention basin. The methods presented in this study can be extended to other infiltration techniques such as rain gardens, retention basins, detention basins (assuming no infiltration), and permeable pavements.

The results of calculation examples 1, 2, 3, and the sensitivity analyses support the following conclusions:

• The proposed method ensures that, in the event of complete soil clogging, the available surface volume is sufficient to store the excess runoff caused by urbanization.

• The method for designing the freeboard and surface spillway, while accounting for the effects of climate change, is simple and practical. It requires only a coefficient that represents the expected increase in design rainfall intensity, typically ranging from 1.2 to 1.4, based on recent studies.

• Saturated hydraulic conductivity is the most sensitive parameter, followed by effective porosity and matric suction potential. In predominantly clayey soils, these three parameters have roughly equal effects. In predominantly sandy soils, the influence of matric potential is less significant.

• Assuming constant infiltration (e.g., f = k_sat) substantially underestimate infiltration dynamics resulting in oversized designs, while variation in initial soil moisture has a relatively minor impact in the model’s performance. This underscores the importance of using time-dependent infiltration formulations over simplified static assumptions.

• Incompatibility between the design height of the retention basin and the maximum ponding depth can lead to either undersizing or oversizing. If it is necessary to adjust the height or surface area due to site-specific constraints, the use of perforated underdrains can help make the heights compatible.

• There is only one compatible design height, and the proposed problem has a unique solution, provided the minimum and maximum design height bounds are broad enough to contain the solution space. This formulation enables the use of gradient-based solvers. In the case of Excel, the nonlinear GRG solver is efficient and well-suited for solving the problem.

The MoDOBR model also supports climate-resilient infrastructure design through integrated freeboard and overflow structure sizing, allowing for rainfall intensification scenarios based on user-defined coefficients. Although the hydrograph tested in this study was the triangular hydrograph of the rational method, the model is compatible with any input hydrograph. A rainfall duration equal to the basin’s time of concentration was assumed in this analysis, but future studies may explore the impact of longer storm durations. Additional research could also incorporate cost factors—such as land use opportunity cost, excavation, and labor—into a comprehensive cost function to be minimized, while treating the compatibility of basin height as a design constraint. The spreadsheet used to perform all calculations in this study is freely available in an online repository at (Gomes Júnior, 2024).

Data Avaiability

Research data is available in an open repository published in: https://github.com/marcusnobrega-eng/MoDOBR

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Edited by

• Editor in-Chief:
  Adilson Pinheiro

• Associated Editor:
  Priscilla Macedo Moura

Publication Dates

• Publication in this collection
  10 Nov 2025
• Date of issue
  2025

History

• Received
  09 Oct 2024
• Reviewed
  20 Apr 2025
• Accepted
  26 June 2025