# Mathematical and Numerical Modeling of Repeated Floods from the Siret Basin, Romania, a Risk for Population, Environment, and Agriculture

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

^{2}, accounting for 18% of the country’s land. The average flow rate is Q = 225 m

^{3}/s, with a volume of water V

_{w}= 7100 × 10

^{6}m

^{3}, and an annual stock of Q

_{s}= 0.9 km

^{3}/year.

- -
- 79 hydrometric stations; 73 with daily transmissions; 60 automatic stations;
- -
- 110 rain gauges with daily communication;
- -
- 6 evaporimeter stations;
- -
- 256 hydro-geological boreholes, 80 of which are equipped with automatic stations;
- -
- 356 cross-sections, 5 of them are chosen for presentation, as being representative;
- -
- 71 major consumers were tracked systematically.

- -
- Tributary flow rates and flood hydrographs with a specific frequency in time;
- -
- Hydrological data from hydrometric stations for the last decades;
- -
- Precipitation, evaporation, and water consumption for the analyzed basin;
- -
- Environmental parameters such as temperature, wind speed and direction, solar radiation, pressure, and so on.
- -
- Sediments: classification, dimensions, structure, and distribution.

#### 2.1. Local Conditions

- -
- Average annual temperature: T
_{av}= 8.5 °C; - -
- Average minimum temperature, in January T
_{min}= −4.5 °C; average maximum temperature (in July in the south and August in the north) T_{max}= 20 °C; - -
- Relative humidity reaches 80% for approximately 6–8 months per year;
- -
- Average wind speed of 3–6 m/s at 10 m from the ground;
- -
- Annual precipitation averages 500–900 mm, with a wide range from July to November; precipitation is higher here than elsewhere in the country, reaching 450 mm even during the warm seasons.

^{3}/s, a length of L = 283 km, an elevation difference of h = 372 m, and an area of A = 7039 km

^{2}with 193 tributaries. Table 1 shows some measurements realized in 5 Bistrita River monitoring stations during 2019, considered an average year. Values were obtained from the Romanian National Authority Apele Romane, from its report for 2020 [39].

^{2})—the surface from which data is collected; Q

_{av}(m

^{3}/s), Q

_{max}(m

^{3}/s) is the average and maximum flow rate; R (kg/s)—the volume of transported alluvium determined by direct measurements for the mentioned location; and T (°C) is the water temperature.

#### 2.2. Mathematical Model of the Sediment Transport

^{2}]—gravitational acceleration. ν[m

^{2}/s]—kinematic viscosity of water, γ

_{s}, γ—sediment and water specific weight:

_{*}—the shear velocity, U

_{*}=$\sqrt{{\tau}_{0}/\rho}$, τ

_{0}—shear stress in riverbed, V-average velocity, h-depth of water.

_{*}> 60, the coefficients are: C

_{1}= 0.0; C

_{2}= 0.025, C

_{3}= 0.17, C

_{4}= 1.5 and for

_{*}< 60, ${C}_{1}=1-0.56\mathrm{log}{d}_{\ast}$; ${C}_{2}={e}^{2.86\mathrm{log}{d}_{\ast}-3.53-\left(\mathrm{log}{d}_{\ast}\right)}{}^{2}$;

_{gr}and X-the sediments concentration in ppm (parts per million) are:

_{T = X}

^{.}Q, where Q(m

^{3}/s) is the flow rate of the river.

#### 2.3. Mathematical Model of the River Flow

_{0}; width-B; A-cross-section; H-water level; Q—flow rate; and Q = f(T) the flood hydrograph, where T-time. L is thought to be long enough to allow accurate determination of alluvial riverbed changes. As discussed in Section 2.2, the volume rate of transported sediments is Q

_{T}/g

_{s,}where Q

_{T}-is the flow rate of water with sediments. In section 1 (S1-Entrance) it is: ${Q}_{t}-\frac{\partial {Q}_{T}}{\partial x}\frac{dx}{2}$ and in section 2 (S2-Exit): ${Q}_{t}+\frac{\partial {Q}_{T}}{\partial x}\frac{dx}{2}$, where x is the dimension in longitudinal direction. The net amount of sediments that change the river bed between section 1 and section 2 is: $\frac{\partial}{\partial t}\left[AHdx\left(1-\lambda \right)\right]$, where λ—porosity in bed. Combining these relations for unsteady flows, the continuity equation of sediment quantity conservation became:

#### 2.4. Mathematical Model of Fluid Flow and Continuity of the Sediment Transport

_{l}—the flow rate from the main current, t-time, S

_{f}—the friction slope, determined from the Manning formula for rough surfaces, ${S}_{f}={V}^{2}{n}^{2}/{R}^{4/3}$.

_{f}of sediments is:

_{g}

_{—}weight of sediments absorbed in the mainstream during a time interval ∆t and transported downstream in the cross-section S

_{2}, G

_{f}—the weight of sediments which further, in the next sector may be transported or deposited, G

_{i}

_{—}the total weight of the entered sediments in the cross-section S

_{1}, and G’

_{f}—only the transported sediments. The constant C’ = 0.043 for the first two sectors, C’= 0.06 for the third, and C’ = 0.084 for the last one, selected based on the sediment characteristics for each zone.

_{i}fractions in the solid discharge, the sediments removed from the bed is $C\u2019\left({t}_{1}^{\left(i\right)}-{t}_{2}^{\left(i\right)}\right)/2$, while ${b}_{1}^{\left(i\right)}{G}_{f}={b}_{2}^{\left(i\right)}\left({G}_{i}-{C}_{g}\right)$ represents the weight of fractions remained in bed; b

^{(i)}is the weight of fraction “i” in riverbed [47].

_{1}, Q

_{2}—the inflow and outflow discharge:

_{l}(x) that passes through the cross-section in a time interval equal to dx/V [27]. The weigh balance of the liquid became:

#### 2.5. Numerical Modeling of Floods

- -
- -
- Selection of structures with direct impact on the water flow for these sectors, such as spillways, dams, weirs, bridges, and decks, required as initial conditions;
- -
- Granulometry, type, and nature of sediments from the minor and major riverbeds in the selected sections, to assign appropriate values for sediment transport and floods.

^{3}/s and reaching up to 4600 m

^{3}/s for floods on Siret. Some rules must be respected when schematizing the numerical model:

- -
- Schematization methodology is chosen so the level of accuracy remains the same, with good precision both at low and high flow rates, when the cross-section is partially or full of water and overflows in the major riverbed, localities, or agricultural lands;
- -
- The cross-sections in the minor riverbed are chosen so the dams that are above the elevations in the major riverbed are represented as correctly as possible;
- -
- Different average roughness values are used in the minor and major riverbed starting from the upstream to downstream sector, considering the composition of the alluvial bed, potential vegetation islands, and sinuosity of the water course. For better precision, the roughness was determined using the values obtained by measurements with non-homogeneous Manning values in the minor bed of 0.018–0.025 s/m
^{1/3}, the major bed of 0.032–0.05 s/m^{1/3}, and flooded localities of up to 0.08–1 s/m^{1/3}; - -
- For the input upstream limit condition, the second recorded flood hydrograph from 2020 was used, as being the worst case. The level of the reservoir near Bacau city was considered the output, the downstream border.

^{3}/s before being run for the flood mode. These tests were carried out to validate the model, the selected sections and roughness in the cross-sections. It was assured the flow continuity for the selected river sector, considering the main tributaries and sediments characteristics. The differences obtained compared to direct measurements in the field were approximately 40–60 cm, indicating that the model was calibrated.

^{.}∆t for grid point x

_{j}. The superscript (m) represents the time moment $(m+1)/2\cdot \mathit{\Delta}t$, where ${A}_{j+1}^{(m)a}$ represents the cross-section area for the grid point x

_{j}

_{+ 1}, which is considered, at this time, an average water elevation between two adjacent h-grid points values. To avoid computational instability, the terms are expressed as a combination of known values Q

^{m}and unknown terms Q

^{m + 1}. Relations (11) and (12) become:

_{j}, and water level h

_{j}for a gradual flow rate, the computational procedure begins from the downstream boundary conditions, where the water level h

_{L}is maintained constant.

## 3. Results

- -
- Maximum flow rate, using the Bistrita River as an example, and minimum volume of transported sediments in the uncertainty interval;
- -
- Maximum volume of transported sediments based on the hypothesis of flood hydrographs with lower flow rates but longer duration, using the Siret tributaries. Table 5 shows the results for the uncertainty intervals for maximum flood discharge and entrained/deposited sediment volume up to a probability, P = 10%.

_{t}—is typically recorded during the increasing period of the flood hydrograph, while the sediment deposits—V

_{d}—are typically recorded when the flow rate decreases. When the hydrographs show multiple flood peaks or when successive floods are recorded at short intervals, as in 2020, new problems arise.

^{3}/s for the Siret River, with historical minimum Q

_{min}= 45 m

^{3}/s and maximum Q

_{max}= 4650 m

^{3}/s. The transported solid sediments have an average velocity of 95 kg/s and a volume of 5.98 million tons per year. It contains about 10% dragged alluvium. Coarse alluvial deposits have the greatest thickness in the Marasesti-Doaga area, reaching over 100 m, and decreasing to about 40 m in the Jorasti-Vulturu area and 15–20 m in the Milcov-Bordeasca area to the south.

^{3}/s. At the Frumosu station, a historical flow rate of Q

_{max}= 772 m

^{3}/s was recorded in 2020. The second scenario assumes a longer but less intense flood. Figure 7 illustrates the area chosen for modeling.

_{0}= 1–8 m, cross-sections dimensions, hydraulic conductivity K = 10–300 m/day along the watershed, with average values K = 30–100 m/day in the north part and average transmissivity θ = 100–500 m

^{2}/day with higher values θ = 1000–3000 m

^{2}/day in the Focsani area.

- -
- Initial cross-sections: S
_{1}: length l_{1}= 70 m, S_{2}: l_{2}= 100 m, S_{3}: l_{3}= 140 m, S_{4}: l_{4}= 200 m; - -
- Distances between these sections: d
_{1}= 1200 m, d_{2}= 2350 m, d_{3}= 3500 m, d_{4}= 1750 m; - -
- River slope: 0.5
^{o}/_{oo}; - -
- The average annual rainfall is 500–600 mm, with 450 mm in the summer.

#### 3.1. The Risk Zones

_{i}= 37.4 kg/s, with C

_{g}= 19.8 kg/s of suspended sediments; the rest was transported. After 52 h, the flow rate in the Panagarati area was Q = 2280 m

^{3}/s, Q = 1720 m

^{3}/s in Vaduri, and Q = 2325 m

^{3}/s in Reconstructia.

^{3}/s without forecast implies the discharge of a flow of 1480 m

^{3}/s on Siret. If high flow rates from downstream Pangarati tributaries are superposed, the discharge capacity at all other dams is exceeded and the flood wave cannot be transited. Lake Pangarati no longer has an attenuation volume; however, it has a small, useful volume so it can no longer mitigate the high flows from the captured basin. The flood flows are discharged by opening the spillways at a lifting speed of 0.4 m/s. Its initial utile volume of approximately 6.4 million m

^{3}is today only 2.1 million m

^{3}. Under these conditions, the evacuation capacity of the Pangarati Lake is exhausted. The Vaduri Lake has a relatively small volume and cannot mitigate the high flows from the basin captured by this lake either. The util volume of the initial lake of 5.60 million m

^{3}is now only 2.39 million m

^{3}due to massive silting. Lake Reconstructia was supposed to have a regularizing effect on the upstream accumulation. Since the upstream lakes only allow the passage of flood flows, it can no longer take over from the volume of the flood either. Its initial useful volume of 5.6 million m

^{3}is also only 2.4 million m

^{3}today. They currently operate under difficult conditions. It is impossible to capture large amounts of water from floods due to their massive siltation (some over 70%, e.g., Pangarati and Vaduri). The floods cannot be stopped. Additionally, human errors in the exploitation of these lakes in 2005 produced the amplification of flood waves by superimposing the discharged flows above the naturally maximum-formed flood. According to recent analysis, most hydropower lakes in Romania have only 20–40% utile water storage capacity for water supply and electricity production. This means that the additional flows can no longer be stored and floods move downstream, inundating large areas where the transverse riverbeds are not high enough to transport the excess water.

^{3}/s can only be discharged at the limit if the downstream lakes have been emptied beforehand; moreover, in the area of the confluence between Bistrita and Siret the flow rate is below 600 m

^{3}/s. Evacuation of these flows must be less than 6 h, otherwise the flood wave will become uncontrollable. A flood greater than 2835 m

^{3}/s cannot be discharged under any circumstances without endangering downstream structures and communities unless more than 12 h of flood mitigation is possible and special consideration is given to discharging the flows from other dams.

#### 3.2. Proposed Solutions

## 4. Discussion

^{3}/s in 1991, and Bistrita of 4680 m

^{3}/s in 2020. The Trotuș River, together with its tributary, Tazlau, has the highest recorded maximum specific discharge in Romania, of 16 m

^{3}/s per km

^{2}. In total, 5 flow rates higher than 1500 m

^{3}/s have been recorded in the last 20 years on this river, corresponding to a probability of 5%. The Belci dam was destroyed by the flood of 1991, and even today it is not restored. As a consequence, the Siret riverbed was massively affected.

## 5. Conclusions

- -
- Excessive deforestation in large areas of watercourses reception basins;
- -
- Inadequate works, such as bridges, footbridges and dams, location of houses, or economic objectives—such as sawmills, gravel and sand mining stations, wood and reed processing—that occupy the minor riverbed;
- -
- Household annexes and fences in major riverbeds, as well as the storage of household waste, sawdust, and wood. During floods, household garbage or wood left on illegally deforested slopes is transported and accumulated in meandering sectors or at narrow bridge openings;
- -
- Under-sizing of bridges and decks.

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The hydrographic basin of the Siret River: (

**a**) nationally occupied surface; (

**b**) data recorded in the last decades.

**Figure 2.**Floods on Bistrita River: (

**a**) 2005, near Batca Doamnei; (

**b**) 2005, near Prelunca; (

**c**) 2020, near Piatra Neamt.

**Figure 3.**Floods on Siret River: (

**a**) 2005, near Piatra Neamt; (

**b**) 2020, after the third flood, near Ramnicu Sarat; (

**c**) 2015, near Bacau; (

**d**) 2020, near Focsani.

**Figure 5.**Area of interest: (

**a**) Longitudinal profile of tributaries; (

**b**) Longitudinal profile Siret; (

**c**), (

**d**) Cross-sections.

**Figure 6.**Numerical modeling of the sediment transport: (

**a**) Discretization of the interest area; (

**b**) Sediment transport-initial phase; (

**c**) Sediment transport during the flood.

**Figure 8.**Images during modeling the sediment transport phenomena: (

**a**) Small flow rates, only few sediment transported; (

**b**) High flow rates entrained and deposited sediments; (

**c**) Very high flow rates, rapid changes in the riverbed.

**Figure 12.**Numerical results for the flooded surfaces: (

**a**) Covered surfaces after the first flood; (

**b**) After the second flood; (

**c**) after the third flood; (

**d**) Covered surfaces after the third flood; (

**e**) Comparison between the first and the third flood.

**Figure 13.**Management plans to transient the floods in Siret basin: (

**a**) Temporary dams; (

**b**) Lateral bank slopes; (

**c**) different proposed solutions; (

**d**) Buffer zones, reservoirs.

Sections/Q[m^{3}/s] | SH Carlibaba | SH Carnu | SH Brosteni | SH Vaduri | SH Bacau |
---|---|---|---|---|---|

X | 27.96 | 33.25 | 33.52 | 33.85 | 37.99 |

XI | 26.42 | 30.99 | 31.23 | 31.50 | 40.40 |

XII | 21.48 | 25.36 | 25.57 | 25.81 | 33.83 |

I | 17.96 | 21.25 | 21.38 | 21.59 | 28.49 |

II | 18.13 | 21.46 | 21.63 | 21.85 | 27.05 |

III | 32.11 | 38.41 | 38.77 | 39.19 | 44.61 |

IV | 80.30 | 97.35 | 98.11 | 99.28 | 107.17 |

V | 92.21 | 109.53 | 110.47 | 111.55 | 118.18 |

VI | 73.15 | 87.65 | 88.41 | 89.34 | 96.22 |

VII | 59.37 | 70.91 | 71.49 | 72.22 | 76.38 |

VIII | 43.96 | 52.56 | 53.02 | 53.56 | 57.15 |

IX | 33.10 | 39.51 | 39.85 | 40.29 | 41.87 |

Annual Average | 43.85 | 52.35 | 52.79 | 53.34 | 59.11 |

Nr. | River | Station | F (km^{2}) | Hydrologic Parameters | Water Parameters | |||||
---|---|---|---|---|---|---|---|---|---|---|

Q_{av} (m^{3}/s) | Q_{max} (m^{3}/s) | R (kg/s) | Date | T (°C) | pH | Organic Mg KmnO_{4}/L | ||||

1 | Siret | Lespezi | 58,744 | 36.6 | 1825 | 67.8 | 04.03 | 6.1 | 7.5 | 10.97 |

2 | Siret | Dragesti | 11,811 | 76.8 | 2650 | 126 | 06.05 | 6.4 | 7.5 | 12.09 |

3 | Siret | Lungoci | 36,030 | 208 | 3950 | 349 | 06.05 | 6.8 | 7.6 | 14.16 |

4 | Suceava | Itcani | 2330 | 16.4 | 1725 | 15.1 | 11.08 | 14.0 | 7.2 | 9.1 |

5 | Bistrita | Roman | 4285 | 32 | 1925 | 40.1 | 11.08 | 15.0 | 7.3 | 9.4 |

6 | Bistrita | Frumosu | 2816 | 37.5 | 1320 | 7.42 | 11.08 | 15.5 | 7.3 | 9.7 |

7 | Trotus | Vranceni | 4077 | 35.4 | 2500 | 37.8 | 12.09 | 14.8 | 7.4 | 6.73 |

8 | Putna | Botariu | 2518 | 16.5 | 1790 | 87.2 | 12.09 | 13.4 | 7.3 | 8.1 |

Nr. | Parameter | SH Lespezi | SH Dragesti | SH Lungoci |
---|---|---|---|---|

1 | Humidity (105 °C) % | 55.46 | 53.38 | 55.66 |

2 | Organic substances (%) | 6.95 | 5.93 | 8.36 |

3 | Mineral substances (%) | 93.05 | 94.07 | 91.64 |

4 | NH^{+}_{4} (mg/100 g) | 5.85 | 7.27 | 8.01 |

5 | NO^{−} _{3} (mg/100 g) | 0.33 | 0.41 | 0.19 |

6 | PO^{3−}_{4} (mg/100 g) | 0.068 | 0.072 | 0.086 |

7 | N- NH^{+}_{4} +N- NO^{−} _{3} (mg/100 g) | 4.61 | 5.74 | 6.27 |

8 | P- PO^{3−}_{4} (mg/100 g) | 0.022 | 0.023 | 0.028 |

10 | N _{included}/P_{included} | 209.54 | 249.56 | 223.93 |

Year | 1989 | 1990 | 1991 | 1992 | 1993 | 1994 | 1995 | 1996 | 1997 | 1998 | 1999 | 2000 | 2001 | 2002 | 2003 | 2004 |

Q_{max} | 1535 | 1260 | 1860 | 889 | 1320 | 2460 | 1400 | 334 | 277 | 2620 | 1370 | 275 | 3270 | 2045 | 1020 | 603 |

2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 |

1614 | 1120 | 1040 | 1380 | 830 | 447 | 435 | 2200 | 796 | 727 | 4650 | 1375 | 785 | 2068 | 3024 | 2022 | 4653 |

P (%) | Gamma | GEV | Frechet | LogPearson | ${\mathit{Q}}_{\mathbf{inf}}^{\mathbf{max}}$ | ${\mathit{Q}}_{\mathbf{sup}}^{\mathbf{max}}$ | Gamma | Weibull | LogPearson | InvGaussian | V_{min} | V_{max} |
---|---|---|---|---|---|---|---|---|---|---|---|---|

0.1 | 5937 | 7157 | 7449 | 6293 | 5937 | 7449 | 2468 | 2245 | 2468 | 2608 | 2245 | 2608 |

0.5 | 4847 | 5372 | 5481 | 5073 | 4846 | 5800 | 2004 | 1879 | 2013 | 2070 | 1879 | 2070 |

1 | 4363 | 4676 | 4734 | 4535 | 4050 | 4734 | 1799 | 1712 | 1809 | 1840 | 1580 | 1750 |

3 | 3578 | 3651 | 3656 | 3667 | 3400 | 3850 | 1468 | 1430 | 1477 | 1477 | 1400 | 1477 |

5 | 3200 | 3203 | 3194 | 3255 | 3030 | 3255 | 1310 | 1290 | 1317 | 1308 | 1230 | 1317 |

10 | 2668 | 2617 | 2597 | 2685 | 2597 | 2960 | 1089 | 1088 | 1094 | 1078 | 1088 | 1094 |

20 | 2104 | 2045 | 2024 | 2094 | 2024 | 2104 | 856 | 866 | 860 | 844 | 860 | 870 |

25 | 1912 | 1859 | 1840 | 1897 | 1840 | 1912 | 778 | 789 | 781 | 767 | 760 | 790 |

30 | 1750 | 1706 | 1689 | 1732 | 1689 | 1750 | 712 | 723 | 715 | 703 | 701 | 723 |

40 | 1480 | 1455 | 1443 | 1463 | 1443 | 1480 | 603 | 612 | 607 | 598 | 582 | 612 |

50 | 1254 | 1249 | 1243 | 1240 | 1240 | 1254 | 512 | 519 | 517 | 512 | 511 | 519 |

60 | 1053 | 1065 | 1065 | 1046 | 1046 | 1066 | 433 | 435 | 437 | 436 | 433 | 440 |

70 | 864 | 889 | 896 | 866 | 864 | 896 | 359 | 357 | 363 | 366 | 357 | 366 |

75 | 770 | 800 | 811 | 778 | 770 | 811 | 323 | 319 | 327 | 331 | 319 | 332 |

80 | 674 | 707 | 723 | 687 | 674 | 723 | 286 | 281 | 290 | 295 | 281 | 295 |

90 | 462 | 488 | 516 | 491 | 462 | 516 | 207 | 200 | 208 | 215 | 200 | 215 |

95 | 328 | 332 | 369 | 367 | 328 | 369 | 158 | 153 | 156 | 163 | 153 | 163 |

97 | 259 | 240 | 283 | 302 | 240 | 302 | 134 | 131 | 129 | 133 | 129 | 134 |

River | Siret | Putna | Bistrita | Bistrita | Cuejd | Cracau | Racaciuni | Trotus | Oituz |

Place | Movileni | Putna | Lunca | Costisa | Garceni | Magazia | Racaciuni | Faget | Oituz |

Distance (km) | 21 | 7 | 32 | 22 | 10 | 14 | 11 | 5 | 11 |

River | Dragomirna | Bistrita | Slanic | Siret | Tazlau | Siret | Siret | Siret | Siret |

Place | Mitoc | Piatra Neamt | Slanic Moldova | Oituz | Tazlau | Solont | Caiut | Milcov | Ramnicu Sarat |

Distance (km) | 42 | 82 | 20 | 25 | 72 | 13 | 5 | 63 | 120 |

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## Share and Cite

**MDPI and ACS Style**

Radulescu, V.
Mathematical and Numerical Modeling of Repeated Floods from the Siret Basin, Romania, a Risk for Population, Environment, and Agriculture. *Water* **2023**, *15*, 1103.
https://doi.org/10.3390/w15061103

**AMA Style**

Radulescu V.
Mathematical and Numerical Modeling of Repeated Floods from the Siret Basin, Romania, a Risk for Population, Environment, and Agriculture. *Water*. 2023; 15(6):1103.
https://doi.org/10.3390/w15061103

**Chicago/Turabian Style**

Radulescu, Victorita.
2023. "Mathematical and Numerical Modeling of Repeated Floods from the Siret Basin, Romania, a Risk for Population, Environment, and Agriculture" *Water* 15, no. 6: 1103.
https://doi.org/10.3390/w15061103