# Effect of the Non-Stationarity of Rainfall Events on the Design of Hydraulic Structures for Runoff Management and Its Applications to a Case Study at Gordo Creek Watershed in Cartagena de Indias, Colombia

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## Abstract

**:**

_{24h-max}) observations recorded at the synoptic weather station of Rafael Núñez airport (Cartagena de Indias, Colombia) were analyzed, and a linear increasing trend over time was identified. It was also noticed that the occurrence of the rainfall value (over the years of record) for a return period of 10 years under stationary conditions (148.1 mm) increased, which evidences a change in rainfall patterns. In these cases, the typical stationary frequency analysis is unable to capture such a change. So, in order to further evaluate rainfall observations, frequency analyses of P

_{24h-max}for stationary and non-stationary conditions were carried out (by using the generalized extreme value distribution). The goodness-of-fit test of Akaike Information Criterion (AIC), with values of 753.3721 and 747.5103 for stationary and non-stationary conditions respectively, showed that the latter best depicts the increasing rainfall pattern. Values of rainfall were later estimated for different return periods (2, 5, 10, 25, 50, and 100 years) to quantify the increase (non-stationary versus stationary condition), which ranged 6% to 12% for return periods from 5 years to 100 years, and 44% for a 2-year return period. The effect of these findings were tested in the Gordo creek watershed by first calculating the resulting direct surface runoff (DSR) for various return periods, and then modeling the hydraulic behavior of the downstream area (composed of a 178.5-m creek’s reach and an existing box-culvert located at the watershed outlet) that undergoes flooding events every year. The resulting DSR increase oscillated between 8% and 19% for return periods from 5 to 100 years, and 77% for a 2-year return period when the non-stationary and stationary scenarios were compared. The results of this study shed light upon to the precautions that designers should take when selecting a design, based upon rainfall observed, as it may result in an underestimation of both the direct surface runoff and the size of the hydraulic structures for runoff and flood management throughout the city.

## 1. Introduction

^{2}, whose southern shoreline has been populated over the years by illegal low-income settlements that suffer the consequences of both a rising sea level (the swamp is connected to the Caribbean Sea) and an obsolete stromwater system. Furthermore, the cities of Cartagena de Indias (downstream) and Turbaco (upstream) share several watersheds that also drain into the La Virgen swamp. The upstream areas of these watersheds (like the one selected in this study) are mostly rural, which have been gradually converted into impervious areas by local developers that offer more competitive prices than those in Cartagena de Indias. This dynamic will most likely (and rapidly) change the landscape, which brings with it more challenging hydrologic, and hydraulic, conditions.

_{24h-max}) data (from the synoptic weather station located at the Rafael Núñez airport) was used to: (a) assess the trend of the P

_{24h-max}observations over time, specifically for Cartagena de Indias, (b) quantify the rainfall values obtained for several return periods (2, 5, 10, 25, 50, and 100 years) via stationary and non-stationary frequency analyses, (c) carry out a hydrological and hydraulic analysis on the Gordo creek watershed under SC and NSC for different return periods in order to understand the recurring floods reported that affect a commercial and industrial area downstream, and (d) point out the importance of accounting for the effect of climate change in the decision making process in runoff management, especially in flood-prone areas.

## 2. Study Area and Data

#### 2.1. Study Area

#### 2.2. Rainfall Data

_{24h-max}) from 1941 to 2015 (Table 3; gray cell is the maximum value registered) were used. Years 2016 and 2017 have not yet been officially reported by IDEAM.

## 3. Methodology

#### 3.1. Rainfall Stationary Frequency Analysis

_{24h-max}for each of the two methods used. Gray cells show maximum values.

#### 3.2. Rainfall Non-Stationary Frequency Analysis

_{j}) of the CDFs may increase or decrease, and (b) the extreme values of the analyzed variable are independent in time. Parameters of the CDFs are considered time-varying in order to compute the probability of exceeding occurring, which implies that extreme values are not identically distributed over time.

_{NSC}), it is necessary to understand the concept of the waiting time. The waiting time is a random variable (X) defined as the occurrence of a value exceeding the design value for the first time. In stationary conditions, the probability of exceeding remains constant over time, which implies that a geometric distribution can be used to compute the expected value (E(x) = Tr, SC = 1/p). Commonly, this is called the return period [8]. In contrast, under non-stationary conditions, the return period (Tr) is computed with a non-homogeneous geometric distribution (Equation (2)) [18]:

_{j}is the time-varying probability of exceeding, and the subscript j represents the projecting year.

_{t}) can be expressed as (Equation (3)) [2,18]:

_{24h-max}data at the Rafael Núñez airport weather station (Table 4) exhibits an increasing linear trend over time (dotted black line in Figure 5), which confirms the non-stationary occurrence of the analyzed variable.

_{24h-max}values, the non-stationary frequency analysis followed these steps [4,18]: (a) selecting the CDF under non-stationary condition (GEV or Gumbel); (b) defining a model (constant, linear, or establishing other models) of each parameter (location, shape, and scale) with its trend (increasing or decreasing); (c) estimating parameters by likelihood method; (d) calculating the goodness-of-fit test of the Akaike Information Criterion (AIC) [19] and then selecting the CFD with the minimum value of it; and (e) estimating the P

_{24h-max}for various return periods.

_{mean}– 1978)], α = 31.1 mm, and k = −0.106. The t

_{mean}term in the previous expression represents an analyzed year centered over its own mean between 1941 and 2015. A sensitivity analysis showed that variations of both the scale and the shape parameters do not bring an adequate trend for the P

_{24h-max}when the results were compared with the findings of IDEAM. The AIC test was used in order to evaluate the GEV distribution goodness-of-fit by comparing both scenarios (stationary and non-stationary). The obtained values were 747.5103 (non-stationary) and 753.3721 (stationary). The results of the P

_{24h-max}for several return periods are presented in Table 5.

#### 3.3. Curve Number (CN) Estimation

_{5}) is estimated to establish whether the soil is in dry, average/normal or wet (Table 6) [21]; a set of equations are then proposed to adjust the CN accordingly (Equations (4) and (5)) [8,22].

_{III}to mimic a saturated soil as the most critical scenario in the Gordo creek watershed in that season, during which more than 53 mm of rain fell in 5 consecutive days. Table 7 shows the values of CN composite obtained for average and wet conditions (gray cells).

#### 3.4. Time of Concentration Estimation

#### 3.5. Direct Surface Runoff (DSR) Estimation

_{24h-max}, and its distribution was estimated via multiannual probabilistic analysis of 30 pluviographs [8] (Figure 6) (P90% was the one used; it indicates the probability that the rainfall pattern observed falls to the left of the P90% curve); (c) no rainfall area reduction was needed due to the size of the watershed; (d) hydrograph generation via SCS-unit hydrograph method; and (e) a lag time of 0.6 Tc (Tlag = 0.6Tc). Table 8 summarizes the peak flow values for stationary and non-stationary conditions.

#### 3.6. Channel Hydraulic Modeling

## 4. Results and Discussion

_{24h-max}observations recorded at the Rafael Núñez airport weather station not only evidenced an increasing trend (Figure 5), but also, in Table 3, it could be noticed that, before 1985 rainfall events above 148.1 mm (estimated rainfall value for a 10-year return period under stationary conditions in Table 4) just occurred once (in 1970). After 1985, rainfall events of that magnitude (or higher) have occurred nine times (2011 rainfall was included as it is close enough). Colombian legislation recommends a 50-year return period for the design of open channels for watersheds of an area less than 1000 ha [5]. In Table 4 and Table 5, the rainfall for this return period, even for stationary conditions, was estimated to be nearly 200 mm. However, in Table 3, a rainfall of 201.8 mm has been already recorded in 1989. From this, it may be inferred that any hydraulic structure to be designed for such return period should be also evaluated for a higher value to test its hydraulic performance.

_{24h-max}(Section 3.1 and Section 3.2) and the peak flow (Section 3.5) for several return periods under stationary and non-stationary conditions depicted in Figure 10 also showed an increase, from 6% to 44% for P

_{24h-max}(Table 10) and from 8% to 77% for peak flow (Table 11), depending upon the return period. The increase seen in P

_{24h-max}for return periods of 5 to 100 years are quite similar to each other, with an average value of 1.09. This value is in line with those observed in some areas of the Caribbean region of Colombia [6,7]. Despite the fact that a 44% increase in P

_{24h-max}for the 2-year return period looks, at a glance, to be too high when compared to the remaining values, it may be construed as an indication that a more conservative approach might be conducted when designing with this low return period as the probability of exceeding is high.

_{24h-max}, with the maximum value occurring at a 2-year return period. A 44-percent increase in rainfall (Table 10) caused a 77-percent peak flow rise (Table 11). Once again, this indicates the precautions that the designers must take when using this return period. For the other return periods, it may be observed that the increase in P

_{24h-max}resulted in a peak flow increase of a similar range.

_{24h-max}and peak flow for different return periods (DPmax, DQp, Pratio and Qratio). As expected, both show how the behavior of P

_{24h-max}, and the watershed’s response to it (the peak flow), have the same trend.

_{24h-max}, peak flow, and water elevation) and Figure 13, where the bankfull level (9.94 m) is reached at a lower return period (floods occur more frequently) under non-stationary conditions (Tr < 2-year). Under stationary conditions, a 2-year return period flow, defined as the mean annual flow (or annual maximum daily flow) [28,29], did not result in a bankfull section, which is not what is currently occurring in the study area (floods every year). The change in the rainfall pattern evidenced in previous sections may be the reason for a shift towards more recurrent and higher-than-usual events that cause floods more frequently than in the past. Likewise, statistically speaking, the ACI results obtained in Section 3.2 (747.5103 for non-stationary and 753.3721 for stationary) demonstrated that the non-stationary condition is more adequate. The AIC test serves to evaluate how good a model is for predicting future values [30]. The lower the value of the AIC the better the model.

_{24h-max}pattern and the Gordo creek watershed’s hydrological response, the fact that almost 85% of the watershed is still rural indicates that any increase in the impervious area will both raise the peak flow and reduce the time of concentration. These two variables will worsen the situation at the outlet of Gordo creek watershed unless a series of measures are implemented. For instance, the combination of sustainable urban drainage systems (SUDS) [31,32], stormwater storage vaults/tanks (both online and offline), aquifer storage and recovery (ASR) wells, infiltration ponds, among others, have proved to be effective at managing stormwater by: (a) keeping the peak flow at the same magnitude (or lower) when pre-development and post-development scenarios are compared, (b) avoiding the design of larger hydraulic structures, (c) minimizing the risk of floods, and (d) recharging aquifers, as a plus.

## 5. Conclusions

_{24h-max}values associated to a given return period are a key variable when estimating design DSR for hydraulic structures for stormwater management, especially in ungauged watersheds. The values obtained in the AIC test carried out in this study, indicated that a frequency analysis under non-stationary conditions represents best the behavior of the rainfall patterns of the P

_{24h-max}observations at the Rafael Núñez station. A stationary scenario is, in this case, further from the natural reality, when compared to non-stationary conditions. This was also confirmed by both the increase in the occurrence of P

_{24h-max}events greater than or equal to 148.1 mm (value for a 10-year return period under stationary conditions), and the increasing linear trend over time of the overall P

_{24h-max}observations. These findings may be an indication that the typical frequency analysis of rainfall under stationary conditions may no longer be applicable when calculating the design rainfall associated with a given return period for sizing hydraulic structures throughout Cartagena de Indias. Furthermore, designing under stationary conditions may have direct implications in the decisions local authorities will be taking in the coming years, given that millions of dollars will be invested in upgrading the city’s stormwater system. The 1-D hydraulic simulation performed herein revealed that the Gordo creek watershed outlet area can be flooded even with an event of 2-year return period (every year)—a situation that had never been observed in the past according to what local people have affirmed. The increase in the P

_{24h-max}, though, is not the sole factor to be taken into account when evaluating the reasons behind the more frequent floods reported. Uncontrolled and unplanned urbanization of vegetated areas may and will exacerbate the recurrent floods registered within the study area.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 3.**(

**a**) Access road to industrial area (2007); (

**b**) Bus stranded in flooded area (2007); (

**c**) Box-culvert (watershed outlet at cross-section K0+00) (2015); (

**d**) Area downstream of the watershed outlet (2010). The locations where the pictures were taken are shown in Figure 2.

**Figure 5.**P

_{24h-max}and GEV parameters (location and scale) behavior over time (Rafael Núñez airport station).

**Figure 8.**Natural cross-sections of Gordo creek. (

**a**) Cross-section at K0+018.5; (

**b**) cross-section at K0+178.5. WS indicates the water surface level.

**Figure 9.**Hydraulic modeling with HEC–RAS: (

**a**) Plan view; (

**b**) longitudinal profile. WS indicates the water surface level.

**Figure 11.**P

_{24h-max}and peak flow variation for various return periods. DPmax and DQp are the P

_{24h-max}and peak flow difference between non-stationary and stationary conditions.

**Figure 12.**P

_{24h-max}and peak flow ratio variation for various return periods. Qratio and Pratio are the ratio of non-stationary to stationary values of peak flow and P

_{24h-max}respectively.

**Figure 13.**Water level at cross-section K0+018.5 for different return periods under stationary and non-stationary conditions.

Item | Developed Areas | Rural Areas | Total | |||
---|---|---|---|---|---|---|

Impervious | Urban | Residential | Bush | Forest | ||

Area (ha) | 2.9 | 12.7 | 23.4 | 109.5 | 109.5 | 258 |

Station | Type | IDEAM ID N° | Height (masl) | Geographic Coord. | Operating Period | ||
---|---|---|---|---|---|---|---|

North | West | Start | Finish | ||||

Rafael Núñez Airport | Synoptic (*) | 14015020 | 2 | 10°26′31′′ | 75°31′13′′ | 15 March 1941 | Still active |

Year | 1941 | 1942 | 1943 | 1944 | 1945 | 1946 | 1947 | 1948 | 1949 | 1950 | 1951 |

P_{24h-max} (mm) | 60.0 | 35.0 | 30.0 | 89.0 | 71.0 | 60.0 | 60.0 | 107.0 | 54.0 | 85.0 | 93.0 |

Year | 1952 | 1953 | 1954 | 1955 | 1956 | 1957 | 1958 | 1959 | 1960 | 1961 | 1962 |

P_{24h-max} (mm) | 41.0 | 51.0 | 90.0 | 110.0 | 95.0 | 40.0 | 109.0 | 68.0 | 109.0 | 65.0 | 75.0 |

Year | 1963 | 1964 | 1965 | 1966 | 1967 | 1968 | 1969 | 1970 | 1971 | 1972 | 1973 |

P_{24h-max} (mm) | 59.0 | 69.0 | 89.0 | 76.0 | 67.0 | 89.0 | 129.0 | 157.0 | 104.7 | 120.0 | 74.1 |

Year | 1974 | 1975 | 1976 | 1977 | 1978 | 1979 | 1980 | 1981 | 1982 | 1983 | 1984 |

P_{24h-max} (mm) | 126.4 | 101.6 | 54.4 | 60.5 | 68.6 | 120.7 | 135.9 | 124.4 | 98.0 | 63.4 | 102.7 |

Year | 1985 | 1986 | 1987 | 1988 | 1989 | 1990 | 1991 | 1992 | 1993 | 1994 | 1995 |

P_{24h-max} (mm) | 164.5 | 64.9 | 171.3 | 115.0 | 201.8 | 77.8 | 32.5 | 161.5 | 133.4 | 54.8 | 76.3 |

Year | 1996 | 1997 | 1998 | 1999 | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 |

P_{24h-max} (mm) | 99.4 | 99.6 | 85.6 | 108.5 | 116.2 | 76.2 | 73.5 | 161.8 | 149.0 | 76.4 | 122.3 |

Year | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | - | - |

P_{24h-max} (mm) | 183.1 | 95.3 | 61.3 | 150.7 | 146.1 | 94.2 | 88.0 | 116.0 | 51.8 | - | - |

Tr (year) | P_{24h-max} | |
---|---|---|

Gumbel | Weibull | |

2 | 88.6 | 89.2 |

5 | 124.4 | 125.0 |

10 | 148.1 | 148.1 |

25 | 178.1 | 176.4 |

50 | 200.3 | 196.9 |

100 | 222.4 | 216.8 |

Tr (year) | P_{24h-max} (mm) |
---|---|

2 | 128.1 |

5 | 140.6 |

10 | 161.5 |

25 | 189.1 |

50 | 213.1 |

100 | 244.8 |

ARC Type | Dormant Season | Growing Season |
---|---|---|

I (dry) | <12.7 mm | <36 mm |

II (normal) | 12.7–27.9 mm | 36–53 mm |

III (wet) | >27.9 mm | >53 mm |

Item | CN Developed Areas | CN Rural Areas | |||
---|---|---|---|---|---|

Impervious | Urban | Residential | Bush | Forest | |

CN_{II} | 98 | 88 | 85 | 86 | 60 |

CN_{III} | 99.1 | 94.4 | 92.9 | 93.4 | 77.5 |

Area (ha) | 2.9 | 12.7 | 23.4 | 109.5 | 109.5 |

CN_{II} sub-areas | 86.9 | 73 | |||

CN_{II} composite | 75.1 | ||||

CN_{III} sub-areas | 93.8 | 85.5 | |||

CN_{III} composite | 86.7 |

Tr (year) | Peak Flow (m^{3}/s) | |
---|---|---|

SC | NSC | |

2 | 14.3 | 25.3 |

5 | 24.3 | 29.0 |

10 | 31.2 | 35.2 |

25 | 40.2 | 43.6 |

50 | 47.1 | 51.0 |

100 | 53.9 | 60.9 |

Channel | Section | Slope (%) | |||
---|---|---|---|---|---|

From | To | Type | Number of Cells | Dimensions | |

K0+000 | K0+008.44 | Box culvert | 2 | 2.5 m × 2.0 m | 1.25 |

K0+012.44 | K0+062.44 | Pipe culvert | 2 | Diameter of 1.6 m | 0.37 |

K0+067.11 | K0+097.11 | Box culvert | 2 | 2.0 m × 2.0 m | 0.033 |

Tr (year) | P_{24h-max} (mm) | Ratio (NSC/SC) | Avg. Ratio | ||
---|---|---|---|---|---|

SC | NSC | P_{24h-max} Diff. (NSC-SC) | |||

2 | 89.2 | 128.1 | 38.9 | 1.44 | |

5 | 125.0 | 140.6 | 15.6 | 1.12 | 1.09 |

10 | 148.1 | 161.5 | 13.4 | 1.09 | |

25 | 178.1 | 189.1 | 11.0 | 1.06 | |

50 | 200.3 | 213.1 | 12.8 | 1.06 | |

100 | 222.4 | 244.8 | 22.4 | 1.10 |

Tr (year) | Peak Flow (m^{3}/s) | Ratio (NSC/SC) | Avg. Ratio | ||
---|---|---|---|---|---|

SC | NSC | Flow Diff. (NSC-SC) | |||

2 | 14.3 | 25.3 | 11.0 | 1.77 | |

5 | 24.3 | 29.0 | 4.7 | 1.19 | 1.12 |

10 | 31.2 | 35.2 | 4.0 | 1.13 | |

25 | 40.2 | 43.6 | 3.4 | 1.08 | |

50 | 47.1 | 51.0 | 3.9 | 1.08 | |

100 | 53.9 | 60.9 | 7.0 | 1.13 |

Bankfull Elev. Is at 9.94 m | Max. Water Level Elevation Reached at Peak Flow | ||||||
---|---|---|---|---|---|---|---|

Tr | |||||||

2 | 5 | 10 | 25 | 50 | 100 | ||

SC | P_{24h-max} (mm) | 89.2 | 125.0 | 148.1 | 178.1 | 200.3 | 222.4 |

Q (m^{3}/s) | 14.3 | 24.3 | 31.2 | 40.2 | 47.1 | 53.9 | |

Water elev. (m) | 9.56 | 10.18 | 10.70 | 10.70 | 11.02 | 11.22 | |

NSC | P_{24h-max} (mm) | 128.1 | 140.6 | 161.5 | 189.1 | 213.1 | 244.8 |

Q (m^{3}/s) | 25.3 | 29.0 | 35.2 | 43.6 | 51.0 | 60.9 | |

Water elev. (m) | 10.24 | 10.56 | 10.75 | 10.92 | 11.13 | 11.29 | |

Water elev. difference (m) | 0.68 | 0.38 | 0.05 | 0.22 | 0.11 | 0.07 |

Flood Depth (m) | |||
---|---|---|---|

Tr (year) | SC | NSC | NSC-SC |

2 | 0.00 | 0.30 | 0.30 |

5 | 0.24 | 0.62 | 0.38 |

10 | 0.76 | 0.81 | 0.05 |

25 | 0.76 | 0.98 | 0.22 |

50 | 1.08 | 1.19 | 0.11 |

100 | 1.28 | 1.35 | 0.07 |

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**MDPI and ACS Style**

Gonzalez-Alvarez, A.; Coronado-Hernández, O.E.; Fuertes-Miquel, V.S.; Ramos, H.M.
Effect of the Non-Stationarity of Rainfall Events on the Design of Hydraulic Structures for Runoff Management and Its Applications to a Case Study at Gordo Creek Watershed in Cartagena de Indias, Colombia. *Fluids* **2018**, *3*, 27.
https://doi.org/10.3390/fluids3020027

**AMA Style**

Gonzalez-Alvarez A, Coronado-Hernández OE, Fuertes-Miquel VS, Ramos HM.
Effect of the Non-Stationarity of Rainfall Events on the Design of Hydraulic Structures for Runoff Management and Its Applications to a Case Study at Gordo Creek Watershed in Cartagena de Indias, Colombia. *Fluids*. 2018; 3(2):27.
https://doi.org/10.3390/fluids3020027

**Chicago/Turabian Style**

Gonzalez-Alvarez, Alvaro, Oscar E. Coronado-Hernández, Vicente S. Fuertes-Miquel, and Helena M. Ramos.
2018. "Effect of the Non-Stationarity of Rainfall Events on the Design of Hydraulic Structures for Runoff Management and Its Applications to a Case Study at Gordo Creek Watershed in Cartagena de Indias, Colombia" *Fluids* 3, no. 2: 27.
https://doi.org/10.3390/fluids3020027