Probabilistic assessment of peak discharge and debris flow concentration in the Surma River using Monte Carlo simulation

A river, driven by gravity and flowing in a ribbon-like way, can vary in size from broad, deep stretches to slender, shallow streams. The headwater, or source, of every river is provided by rainfall, glacier melt, and groundwater flow (Raghunath 2006). Rivers are topographical features whose flow varies on a regular basis due to both natural and human forces (Khublaryan 2002). For Bangladesh, rivers' ever-changing shape is characterized by erosion and sedimentation (Uddin et al. 2017). Studying these changes within particular drainage basins is essential since parameters like debris flow of concentration and discharge fluctuations are a key indicator of climate change's consequences (Lahmer et al. 2001). In places susceptible to floods and heavy precipitation, accurate peak discharge estimation plays an important role in the planning and designing of safe and economical hydraulic systems (Singh et al. 2011; Roy & Mistri 2013). Also, it is required to calculate the debris flow's peak discharge to effectively prevent debris flow disasters. This estimation allows for precise forecasts and evaluations of the impact and risk associated with the flow at different locations (Ikeda et al. 2007). The design event approach and other traditional techniques for estimating peak discharge ignore the randomness of variables; however, Monte Carlo simulation (MCS) provides a more comprehensive solution by treating these elements as probabilities (Charalambous et al. 2013). Therefore, to address uncertainties, peak discharge can be estimated using MCS, thereby enhancing future resilience in flood forecasting. However, few studies have utilized MCS for peak river discharge estimation, and none have specifically examined its application to downstream rivers impacted by debris flow.

A study predicts discharge in the Mahanadi River watershed using four statistical techniques: normal distribution, log Pearson type III, Gumbel distribution, and extreme distribution method (Samantaray & Sahoo 2020). The most successful technique for forecasting peak flood discharge is found to be Gumbel distribution. However, while it is frequently used to analyze severe events in hydrology, its fixed statistical indicators and sensitivity to data fluctuations limit its usefulness (Anghel 2024).

In a research paper of Cimadur River basin's flood peak flow, the method of Thiessen technique, log-normal, log Pearson type III, and Gumbel type 1 were used to analyze daily rainfall data from 2011 to 2020 in order to determine the flood peak discharges (Fitri et al. 2021). The most exact fit was the log Pearson type III distribution. Another study that used data on annual peak discharge and water level found no significant distinction between the log-normal and log Pearson type III techniques in estimating flood discharge (Roy & De 2015). A paper highlights the drawbacks of the log Pearson type 3 (LP3) distribution, especially when dealing with data that is negatively skewed (Bobée 1975). Additionally, its reliance on historical data reduces its reliability for future predictions amid climate change and human impacts on the hydrologic cycle.

There are very few hydrological research projects that use MCS to forecast a flow's peak discharge. According to a study, MCS is an effective method for flood estimation because it incorporates probabilistic input variables, which provide a more accurate picture of flood behavior (Babister et al. 2016). This method makes it possible to comprehend the effects of management actions more clearly. Another study that integrates data from gauged stations with an MCS proposes a way to assess uncertainty (log-normal distribution) in flow-duration curves for ungauged sites, which is helpful for the building of hydropower projects (using average annual flow and an exponent of 1.126) (Murdock & Gulliver 1993). Dastorani et al. (2013) suggest that new machine learning techniques are preferable for calculating the instantaneous peak flow using the average daily flow data. When compared to conventional methods, these techniques are demonstrated to be especially successful in situations when instantaneous peak data is limited. A study seeks to give a probabilistic way of estimating the debris flow peak discharge with the use of the MCS technique (De Paola et al. 2017). In this case, typical MCS is employed to propagate the uncertainty in different parameters related to hydrographic basin modeling and to produce a probability distribution for the peak discharge associated with a specific return time.

MCS in debris flow is used in very few studies. In a research project, a 55-km section of the Bedford Ouse River was given a stochastic water quality model in order to account for uncertainty and forecast the water quality downstream (Whitehead & Young 1979). The model takes into account the effects of upstream effluents and evaluates sensitivity to parameter alterations using Monte Carlo analysis. In a study, the link between peak discharge and debris flow volume was evaluated at Kamikamihorizawa Creek using a debris flow monitoring system with pressure sensors and load cells (henceforth represented as a DFLP system) (Ikeda et al. 2023). The relationship was updated using the data from this creek and earlier observations, demonstrating similar trends between peak discharge and magnitude.

On the contrary, in a study on the Surma River in Bangladesh, peak flow was analyzed and predicted using Radial Basis Function (RBF) and Multi-Layer Perceptron (MLP) artificial neural networks. The RBF model outperformed the MLP, attaining 99.55% efficiency (Ahmed & Shah 2017). However, the study was hampered by the possibility of overfitting from a small dataset. Another study focused on the Surma River, crucial for the Sylhet region of Bangladesh, examining flood vulnerability in Sylhet city using the Soil Conservation Service Curve Number (SCS-CN) method for runoff estimation and HEC-RAS software for floodplain mapping, revealing inundation areas of 6.34–6.88% for different return periods (Munna et al. 2021). Yet, accuracy may have been impacted by limitations such as the use of unsupervised image classification, assumptions regarding direct runoff to the Surma River, and steady flow analysis.

According to a paper, the Surma River, which is categorized as a downstream river, gathers different types of pollutants and sediments, indicating particular debris flow conditions that necessitate a careful evaluation of peak discharge (Kadir et al. 2022). Therefore, the main goal of this paper is to use a probabilistic technique called the MCS to determine the peak discharge of the Surma River in Sylhet, Bangladesh, within a range of debris flow concentration (C_v). When utilizing various methods to forecast peak discharges considering debris flow, there are significant uncertainties and fluctuations. However, the MCS ultimately reduces these uncertainties. Also, the research determines the C_v value for the rainy season in the Surma River basin. This will shed light on the possibility of flooding and fluctuations in a debris flow during the months with heavy precipitation. Satisfying all of these objectives will enable the research to offer important insights into the debris flow behavior of the river all year long, encompassing both the high-flowing rainy season and the dry season. A thorough understanding of peak discharges during the non-rainy season can assist in detecting possible risks and direct the management of water resources outside of the typical flood season. This study considers return periods of 5, 10, 25, 50, and 100 years when determining peak discharge (Dhital et al. 2011; Munna et al. 2021; Islam & Tingsanchali 2024). Furthermore, this research paper presents a versatile and comprehensive model created for ungauged catchments in data-scarce regions to overcome the difficulties of sparse data networks and challenging catchment access. This method is designed for regions with restricted access and limited data, in contrast to earlier research that frequently relies on large hydrological datasets. The methodology's flexible and comprehensive design is especially well-suited for comparable downstream rivers with limited data coverage, and it uses uncertainty analysis to improve comprehension and accuracy of the peak discharge of debris flows.

This research has employed several significant parameters, such as (1) peak discharge (Q_p), (2) debris flow concentration (C_v), (3) time of concentration (t_c), (4) meandering length (L), (5) total length of the segment (l), (6) runoff coefficient (C), (7) speed of flow (V), (8) return period (T), and (9) catchment area. The meandering length and segment length were calculated in this study using a geographic information system (GIS).

Debris flow concentration (C_v): The debris flow concentration, or coefficient C_v, is a measure to represent the homogeneity of water distributed over a river channel. This is indicated by the volumetric concentration of sediment within the flow. C_v can be utilized by calculating the ratio of sediment flow (Q_s) to sediment–water mixture total flow (Q_t) (De Paola et al. 2014). The volumetric concentration of suspended materials has an impact on the delayed settling effect (Richardson & Zaki 1954; Camp 1968; Byun et al. 2014). Thus, it is widely known that the debris flow concentration (C_v) of a flow plays a key role in determining its settling velocity.

Monte Carlo simulations: MCSs treat model parameters as random variables, reflecting real-world uncertainty, in contrast to classical simulations that rely on fixed values (Allen 1995). This effective method models and predicts complicated systems using random sampling. Based on statistical sampling and probability theory, the simulation provides a range of possible outcomes by iteratively running with different random values for uncertain variables (Blondeel et al. 2019; Moghadam et al. 2019). It enables a thorough evaluation of the uncertainties related to different parameters in hydrological studies, which helps provide more reliable and accurate modeling results (Bonate 2001).

Debris flow concentration (C_v) is a significant factor in this study that affects the MCS method's probabilistic peak discharge estimation. Table 1 shows the range of C_v values for the mud flood and mudflow classifications, which range from 0.2 to 0.76 (O'Brien 2009). For evaluating peak discharge scenarios, this range offers a realistic parameter scope and is used as a foundation for developing the variability in debris flow characteristics.

Through the integration of these classifications, a thorough evaluation of possible peak discharge values under different debris flow intensities is guaranteed.

In Equations (4)–(6), K₁ and K₂ are constant within the entire regional territory, respectively, equal to 0.456 and 0.11; d_c, C₁, and D are constant within individual rainfall homogeneous areas; z is the average height of basin expressed in meters; d is the critical duration of the precipitation; and is usually 0.45.

A closed-form relationship for estimating a mudflow's peak flow given a return period T can be found in Equation (9). The parameters used to calculate the mudflow discharge in the equation above are typically impacted by a number of sources of uncertainty, including the topographical nature of the parameters (A_m, l, L, and H), flow velocity (v), runoff coefficient (C), and sediment concentration (C_v) of debris–mud flow.

From 1975 to 2022, the average water level was 25.77 m, with the highest and lowest recorded at 46.28 and 9.82 m, respectively. The maximum depth of the Surma River was 170 m, with an average depth of 86 m. During the monsoon season, the river's peak discharge reached 5,199.12 m³/s. The minimum discharge was 583.50 m³/s. Analysis of annual discharge data from this period indicates a mean discharge of 2,194.08 m³/s. Table 2 presents the annual maximum water data and the annual maximum discharge data for 45 years, from 1975 to 2022.

Table 3 shows a median water level of 25.8 m, with a standard deviation of 12.16 and a coefficient of variation of 47.18%, demonstrating high variability based on an analysis of 45 years of data. The distribution is flatter and platykurtic, as indicated by a kurtosis of −1.611, but a minor positive skewness (0.0823) indicates some higher values that influence the distribution (Hatem et al. 2022). A reliable mean estimate is provided by the 95% confidence interval (22.24–29.30), highlighting the significance of water level monitoring for flood risk management.

Similarly, discharge statistics, which exhibit significant fluctuation, have a median of 2,052.25 m³/s, a standard deviation of 1,069.26, and a coefficient of variation of 48.73%. Some higher values with a platykurtic distribution are also indicated by positive skewness (0.45) and kurtosis of −0.62 (Hatem et al. 2022). The 95% confidence interval (1,883.59–2,504.5) highlights the need for close monitoring of discharge for water resource and flood management.

The values of l and L were found from GIS analysis.

In this study, to calculate L and l for Equation (7), a clipped rectangular region covering the study area was used for collecting streams and waterways in Sylhet city from the OpenStreetMap.org website. ArcMap 10.8.2 shapefile import was the first step in the procedure. GIS techniques were then used to extract the study region from the shapefile. The segment lengths and meandering lengths of the river within the chosen study area were calculated after this extraction.

Here, the total length of the segment (L) and the meandering length (l) were collected using the GIS ArcMap tool. And velocity (v) 3 m s⁻¹ or 0.18 km/min was taken (BWDB).

Here, t_c was found in minute.

The MCS approach was employed to find the time of concentration (t_c) using Equation (10) and the water level dataset. Python was used as the programming language for MCSs, and Visual Studio was used as the development environment. Numerous t_c values were obtained from various water level datasets after the relationship between time of concentration (t_c) and water level (H) was established. The peak time of concentration was then determined by MCS using these datasets for H. This yielded the following results: water level (H) = 0.0064802 km and time of concentration (t_c) = 77.17 min (1.286 h).

The depth of rainfall during a certain return period is determined by the duration of rainfall. If the return period remains constant, the rainfall depth increases over the duration of rainfall (Munna et al. 2018). The rainfall depth of Sylhet for t-minutes is given in Table 4.

From Table 4, by interpolation, rainfall depths (d_c) were found for return periods (T) of 5, 10, 25, 50, and 100 years and for time of concentration (t_c) of 77.17 min (1.286 h). The values of rainfall depths were 87.83, 98.298, 111.52135, 121.337, and 131.0767 mm, respectively.

The rainfall intensities were 68.297, 76.437, 94.35, and 101.9259 mm/h for return periods of 5, 10, 50, and 100 years, respectively.

This is the relationship between debris flow concentration (C_v) and discharge (Q).

By using datasets of Q and a range of C_v (0.2–0.55), MCS was used to derive the peak discharge (Q) and C_v from Equation (12). For a 25-year return period, the simulation predicted a peak discharge of 0.0091571 km³/h or 2,543.63 m³/s and a corresponding C_v of 0.5499967. Peak discharges were determined to be 1,904.66, 2,174.77, 2,846.72, and 3,124.25 m³/s for return periods of 5, 10, 50, and 100 years, respectively, with corresponding C_v values of 0.5499989, 0.5499976, 0.5499940, and 0.5499147. This optimized value offers an in-depth understanding of the possible variations in peak discharge under conditions of variable flow concentration.

The maximum peak discharge (Q) and accompanying C_v (without range) for the monsoon season were calculated by MCS using the datasets for Q and Equation (11) for return periods of 5, 10, 25, 50, and 100 years. The simulation revealed a maximum peak discharge of 0.0183441 km³/h or 5095.58 m³/s with a corresponding C_v of 0.7886368.

According to Christopher (2004), uncertainty can be reduced by using discharge-probability curves, differentiating between knowledge uncertainty and natural variability, and improving estimate methods in order to comprehend the application's optimal performance. Furthermore, an uncertainty bandwidth for flood hydrographs was created by establishing a threshold for estimating flows (Jokar et al. 2021). By applying similar methodologies, this paper demonstrates in Figure 10 that uncertainty in the model has been effectively removed. The analysis shows that a C_v value of 0.54 yields a lower uncertainty bound than the other tested values (0.2, 0.35, 0.45, 0.65, and 0.9), which exhibit higher uncertainty. Figure 10 provides proof that a C_v value of 0.54 maintains an effective connection with observed data and greatly improves the reliability of predicted discharge values. These results point to a significant improvement in enhancing the reliability of hydrological predictions.

Table 5 contrasts model predicted discharge levels for 1970–2020 with actual discharge values from a monitoring station. The close alignment between predicted and observed discharges for the majority of years shows how accurately the model captures changes in river discharge and faithfully replicates overall discharge patterns observed in the field.

Figure 11 displays the patterns of peak discharge values predicted by the model and observed between 1970 and 2020. The model effectively captures the observed discharge trend, demonstrating its ability to simulate hydrological processes and reliably represent the system's behavior. Based on the best-fitted line in Figure 12, a scatter plot of observed and predicted discharge values demonstrates a high correlation (R² = 0.7097). The model's ability to correctly forecast discharge and reflect hydrological processes is demonstrated by the line equation, y = 1.0005x + 59.209, which shows the model's accurate discharge predictions with a little positive bias.

The purpose of this study was to guarantee predictive modeling accuracy in the context of peak discharge analysis by choosing performance metrics like MAE, MSE, and RMSE. Similar approaches were observed in project management applications, as demonstrated by the performance evaluation of the Kotay Bridge Project using earned value analysis (EVA) (Ugural & Burgan 2021). These project assessments highlight the importance of selecting appropriate metrics for thorough performance monitoring. In a paper, heavy metal hazards, such as those in the Surma River, were evaluated using hydrological research and MCS (Acharjee et al. 2022). This shows that MCS is dependable for capturing variability and uncertainty in discharge estimates in river systems, in addition to being efficient at simulating environmental risk.

After calculating the R-squared value, the model performance metrics were determined. The effectiveness of the suggested model in predicting peak discharge is seen in Table 6, where its mean squared error (MSE) of 0.430 shows a notable decrease in prediction errors and agreement with observed values (Frost 2023). Furthermore, the model's dependability is reinforced by the mean absolute error (MAE) of 0.580, which shows that predictions are only 0.580 m³/s away from observed values in terms of average absolute error. The model's effectiveness in accurately projecting peak discharge is further supported by the root mean squared error (RMSE) of 0.655, which further indicates decreased error magnitude and variability. When combined, these measures validate the model's resilience for real-world hydrological and water resource management uses.

The F test was performed to determine any variations between the predicted value and the observed value. The results of the F test comparing the observed discharge with the predicted discharge are displayed in Table 7. The p-value is found to be 0.197852 which is higher than the level of significance 0.05 (Siegel 2012). Therefore, it is impossible to reject the null hypothesis. As a result, the study's observed discharge and predicted discharge are similar.

The peak discharge assessment of the Surma River in Sylhet, Bangladesh, using an MCS method is presented in this study. The approach effectively determined the time of concentration and optimized it against water level height. For the 25-year return period, the MCS yielded a peak discharge of 2,543.63 m³/s and a corresponding C_v of 0.5499967. Peak discharges for 5-, 10-, 50-, and 100-year return periods were 1,904.66, 2,174.77, 2,846.72, and 3,124.25 m³/s, respectively, with corresponding C_v values of 0.5499989, 0.5499976, 0.5499940, and 0.5499147. During the monsoon season, the maximum peak discharge was 5,095.58 m³/s, with a C_v of 0.7886368. The C_v value graph (0.54) significantly reduced uncertainties compared with other C_v values. The model's validation against observed discharge values from the BWDB indicated a coefficient of determination (R²) value of 0.7097, highlighting the model's accuracy and reliability in forecasting peak discharge in the urbanized river segment. The MSE, MAE, and RMSE were nearly zero, indicating acceptable accuracy and minimal errors. The observed and predicted discharges were similar, as indicated by the p-value from the F test (0.197852), which was greater than the significance level of 0.05. Considering hydrological process uncertainties, the application of MCS in both time of concentration and peak discharge calculation with debris flow concentration is essential. The probabilistic aspect of the model provides valuable insights for managing water resources, urban planning, and infrastructure design, especially in areas prone to heavy precipitation events. This work advances peak discharge estimation techniques by offering a practical method for urban river systems. The results provide a foundation for better water resource management decision-making, ensuring the adaptability of urban areas to fluctuating hydrological conditions.

The authors thank USGS (United States of Geological Survey) for the SRTM DEM Image, HDx (Humanitarian Data Exchange) for the administrative map of Sylhet city, OpenStreetMap.org for stream data, and Bangladesh Water Development Board (BWDB) for providing rainfall and discharge data.

This study is done by self-funding.

This study did not involve human or animal subjects, and no ethical approval was required.

Data cannot be made publicly available; readers should contact the corresponding author for details.

The authors declare there is no conflict.