Preliminary Estimate of Detention Basin Efficiency at Watershed Scale

Abstract

Urban encroachment in natural floodplain areas and infrastructures interfering with watercourses have caused higher flood risks in lowland areas. In this context, detention basins have become a fundamental instrument for stormwater and environmental management at watershed scale. Numerical methods of flood routing are generally coupled with optimization algorithms to investigate the factors that affect the overall efficiency of detention basins in controlling the peak flows throughout a watershed. To overcome the procedure effort due to numerical integration, a simple innovative approach, based on the linear system theory applied to the solution of hydrologic flood routing, is proposed for a preliminary estimate of overall efficiency. First a numerical analysis is performed to ensure that the schematization of the detention basin as a linear system leads to technically acceptable approximation. Then, a simple analytical equation is provided that allows a preliminary estimate of detention basin efficiency in downstream river reaches. Sensitivity analysis of the above equation provides information about the factors that most contribute to the downstream flow reduction variability. Finally, the proposed methodology, adequately extended to a parallel system of stormwater detention basins within a watershed, can be easily integrated in optimization algorithms.

Appendix

Solution of Eq. 17 is a piecewise function on intervals $[0, \overline {t}_{li} + \overline {t}_{ci}]$, $[\overline {t}_{li} + \overline {t}_{ci}, \infty ]$ which correspond, in order, the following solutions

$$u_{\Delta Q}(t-t_{li})=\int_{0}^{t} u_{\Delta Q}(\epsilon)\ \delta(t-t_{li})d\epsilon=\frac{1}{t_{ci}}-\frac{1}{t_{ci}} \left(1-e^{-\frac{t-t_{li}}{k_{i}}}\right)$$

(21.a)

$$u_{\Delta Q}(t-t_{li})=\int_{0}^{t} u_{\Delta Q}(\epsilon)\ \delta(t-t_{li})d\epsilon=\frac{1}{t_{ci}}-\frac{1}{t_{li}} e^{-\frac{t-t_{li}}{k_{i}}} \left(e^{\frac{t_{ci}}{k_{i}}}-1\right)$$

(21.b)

Solution of convolution integral of Eq. 18 is a piecewise function on intervals $[0, \overline {t}_{li}], [\overline {t}_{li}, \overline {t}_{ci}+\overline {t}_{li}], [\overline {t}_{ci}+\overline {t}_{li}, \overline {t}_{r}+\overline {t}_{li}], [\overline {t}_{r}+\overline {t}_{li}, \overline {t}_{r}+\overline {t}_{ci}+\overline {t}_{li}], [\overline {t}_{r}+\overline {t}_{ci}+\overline {t}_{li}, \infty ]$ because the response function (17), the inflow hydrograph Q _i (t) to detention basin and the outflow hydrograph Q _o (t) are discontinuous (Fig. 5a). Thus, the outlet dimensionless hydrograph difference at WS _f outlet point f are listed in the following, according to the specified interval order

$$\Delta q(\overline{t}-\overline{t}_{li})= 0$$

(22.a)

$$\Delta q(\overline{t}-\overline{t}_{li})=S_{i} \overline{t}_{p}^{n-1} \left[\frac{K_{i}}{\overline{t}_{ci}} \left(1-e^{-\frac{\overline{t}-\overline{t}_{li}}{K_{i}}} \right)\right]$$

(22.b)

$$\Delta q(\overline{t}-\overline{t}_{li})=S_{i} \overline{t}_{p}^{n-1} \left[\frac{K_{i}}{\overline{t}_{ci}} e^{-\frac{\overline{t}-\overline{t}_{li}}{K_{i}}} \left(e^{\frac{\overline{t}_{ci}}{K_{i}}}-1\right)\right]$$

(22.c)

$$\Delta q(\overline{t}-\overline{t}_{li})=S_{i} \overline{t}_{p}^{n-1} \frac{K_{i}}{\overline{t}_{ci}} \left[1- e^{-\frac{\overline{t}-\overline{t}_{li}}{K_{i}}}\right]$$

(22.d)

$$\Delta q(\overline{t}-\overline{t}_{li})=S_{i} \overline{t}_{r}^{n-1} \frac{K_{i}}{\overline{t}_{ci}} \left(e^{\frac{\overline{t}_{r}}{K_{i}}}-1\right) \left(e^{\frac{\overline{t}_{ci}}{K_{i}}}-1\right)e^{-\frac{\overline{t}-\overline{t}_{li}}{K_{i}}}$$

(22.e)

Fig. 5

Outflow hydrographs a of SW _i translated at WS _f outlet point f without detention basin (q _i ) and with detention basin (q _o ); b at WS _f outlet point f without detention basin at sub-watershed outlet (q _f ) and with (q _fd ) detention basin Δq = q _i −q _o

Solution of convolution integral Eq. 20 is a piecewise function on intervals $[0, \overline {t}_{li}], [\overline {t}_{li}, \overline {t}_{ci}+\overline {t}_{li}], [\overline {t}_{ci}+\overline {t}_{li}, 1], [1, \overline {t}_{r}], [\overline {t}_{p}, \overline {t}_{r}+\overline {t}_{li}], [\overline {t}_{r}+\overline {t}_{li}, \overline {t}_{r}+\overline {t}_{ci}+\overline {t}_{li}], [\overline {t}_{r}+\overline {t}_{ci}+\overline {t}_{li}, 1+\overline {t}_{r}], [1+\overline {t}_{r}, \infty ]$ (Fig. 5b).

The different solutions are listed in the following, according to the specified interval order

$$q_{fd}(\overline{t})=\overline{t}_{r}^{n-1} \overline{t}$$

(23.a)

$$q_{fd}(\overline{t})=\overline{t}_{r}^{n-1} \overline{t} - S_{i} \overline{t}_{r}^{n-1} \left[\frac{K_{i}}{\overline{t}_{ci}} \left(1-e^{-\frac{\overline{t}-\overline{t}_{li}}{K_{i}}} \right)\right]$$

(23.b)

$$q_{fd}(\overline{t})=\overline{t}_{r}^{n-1} \overline{t} - S_{i} \overline{t}_{r}^{n-1} \left[\frac{K_{i}}{\overline{t}_{ci}} e^{-\frac{\overline{t}-\overline{t}_{li}}{K_{i}}} \left(e^{\frac{\overline{t}_{ci}}{K_{i}}}-1\right)\right]$$

(23.c)

$$q_{fd}(\overline{t})=\overline{t}_{r}^{n-1} - S_{i} \overline{t}_{r}^{n-1} \left[\frac{K_{i}}{\overline{t}_{ci}} e^{-\frac{\overline{t}-\overline{t}_{li}}{K_{i}}} \left(e^{\frac{\overline{t}_{ci}}{K_{i}}}-1\right)\right]$$

(23.d)

$$q_{fd}(\overline{t})=\overline{t}_{r}^{n-1} (1-\overline{t}+\overline{t}_{r}) - S_{i} \overline{t}_{r}^{n-1} \left[\frac{K_{i}}{\overline{t}_{ci}} e^{-\frac{\overline{t}-\overline{t}_{li}}{K_{i}}} \left(e^{\frac{\overline{t}_{li}}{K_{i}}}-1\right)\right]$$

(23.e)

$$q_{fd}(\overline{t})=\overline{t}_{r}^{n-1} (1-\overline{t}+\overline{t}_{r}) + S_{i} \overline{t}_{r}^{n-1} \frac{K_{i}}{\overline{t}_{ci}} \left[1-e^{-\frac{\overline{t}-\overline{t}_{li}}{K_{i}}} \left(e^{\frac{\overline{t}_{li}}{K_{i}}}+e^{\frac{\overline{t}_{li}}{K_{i}}}-1\right)\right]$$

(23.f)

$$q_{fd}(\overline{t})=\overline{t}_{r}^{n-1} (1-\overline{t}+\overline{t}_{r}) + S_{i} \overline{t}_{r}^{n-1} \frac{K_{i}}{\overline{t}_{ci}} \left(e^{-\frac{\overline{t}_{r}}{K_{i}}}-1\right) \left(e^{-\frac{\overline{t}_{ci}}{K_{i}}}-1\right)e^{-\frac{\overline{t}-\overline{t}_{li}}{K_{i}}}$$

(23.g)

$$q_{fd}(\overline{t})=S_{i} \overline{t}_{r}^{n-1} \frac{K_{i}}{\overline{t}_{ci}} \left(e^{\frac{\overline{t}_{r}}{K_{i}}}-1\right) \left(e^{\frac{\overline{t}_{ci}}{K_{i}}}-1\right)e^{-\frac{\overline{t}-\overline{t}_{li}}{K_{i}}}$$

(23.h)