Uncertainty analysis in probabilistic design of detention rockfill dams using Monte-Carlo simulation model and probabilistic frequency analysis of stability factors

The detention rockfill dams are of promising importance in flood control projects, due to their minimal technical requirement, low cost, minimal environmental side effects, and self-automotive operation process. However, due to the complexity of Non-Darcian flow interactions with stability and uncertainties of dam, the reliable design is a challenging topic. This study aimed to examine the effects of uncertainties in probabilistic design of these dams. We proposed a reliable design framework for detention rockfill dams with a focus on the importance of stability analysis. The effects of design uncertainty sources on the stability of dam, safety factors of overturning, sliding and bearing, along with the hydraulic performance of the dam were examined. The results of the model revealed that the uncertainties in input parameters can effectively regenerate uncertainties in the hydraulic performance ranges from − 53.54 to + 110.11%. The safety factor against the sliding (SFS) has maximum dependencies with the uncertainties ranging − 32.63 to + 87.81%. The Monte-Carlo Simulation (MCS) and fitting probability distribution functions to the safety factor histograms, and uncertainty quantifications results in 88.3%in increasing the safety factors as a reliable methodology for stability design of detention rockfill dams. Thus, the study calls for reliable, certain, and safe design of flood protection rockfill ponds. The ecological evaluation and applying more advanced uncertainty assessment methods remains a future research direction of the current study. The developed framework can be used to acquire future detention rockfill dam design/modeling requirements for reliability-based design optimization as a simulation–optimization model coupled whit MCS.


Introduction
Floods in Iran are highly hazardous with a high occurrence frequency due to the destructive effects of climate change, heavy rainfall patterns and weak watershed management practices (Nikoo et al. 2015;Parsaie et al. 2020;Riahi-Madvar et al. 2019). Tens of thousands of floods have occurred in recent years, causing more than 868 casualties and economic losses worth US$2.2 billion in 2019 based on the Iran energy ministry. The establishment of flood control strategies is classified into structural and non-structural methods, that are crucial in reliable reduction of flood disasters (Kanani-Sadat et al. 2019). The main action in Non-structural flood control includes the development of flood warning systems, decision support systems, integrated watershed management plans, and public training (Chuntian and Chau 2002;Li et al. 2006;Mosavi et al. 2018;Wang et al. 2017;Wu and Chau 2006;Yaseen et al. 2019). On the other hand, the structural Responsible Editor: Philippe Garrigues actions use physical based methods for flood mitigation and flood peak reduction, include detention ponds, rockfill dams, and reservoir management. The detention rock fill dams due to the use of natural materials, easy construction and environment friendly concept are interested in structural flood control methods (Michioku et al. 2005). The control mechanism in detention rock fill dams composed of storing the inlet flood volume to reduce the flood peak output with a delay in flood depletion that increase the reaction time for downstream areas. In these ponds, the permissible and safe flood outlet discharge is determined based on the downstream condition, and safe river conveyance capacity that is used as the design discharge of flood control structure (Riahi Madvar et al. 2009). In the rockfill dams, because of the porous materials and coarse voids of the rockfills body, the depletion process is done automatically in a safe operation condition without any need for operation management (Riahi-Madvar et al. 2019).
The main steps in detention rock fill dam designs include the acquisition of the safe discharge for the downstream river, the determination of dam storage volume based on the permissible downstream peak discharge, the determination of best location for the dam that satisfy this required storage, preliminary designing of the detention dam, and finalizing the design dimension using an optimization procedure (Riahi-Madvar et al. 2019).
With reference to the rockfill detention ponds for flood control, in the rockfill dams, two types of flow occur, one is the non-Darcian through flow inside the coarse porous media and the other is open channel flow over the crest of the rockfill. The through flow inside the coarse porous media of rockfill is turbulent, non-linear and non-Darcian flow that differs from the other flood control structures. Disregarding the special features of through flow in rockfill dams, the conventional design studies of detention ponds ignore the non-Darcian flow in the body of rockfills (Akan 1990;McEnroe 1992;Abt and Grigg 1978;Wycoff and Singh 1976;Li and Gowing 2005). Moreover, the errors due to the ignoring non-Darcian flow in coarse porous media in the designing of detention rockfill dams are very crucial, because this term of flow and flood depletion affect the routed output flood peak and downstream inundations, and further affect the reliability in dam designs. It is very important to develop design models for detention rockfill dams based on the hydraulic behavior of non-Darcian flow in rockfills.
In the designing of detention rockfill dams, the uncertainties in the flood prediction and dam dimensions are important sources of the reliability of dam operation. With respect to the importance of uncertainty in reliable design of flood control strategies, previous studies have made effort to address this issue recently. First, from the perspective of the source and definition of uncertainty, researchers have proposed various sources and methods for uncertainty quantification. They concluded that the uncertainties were caused by a lack of model parameter (data), scenario, model conceptualization, empirical inaccuracy, assumptions in programming, spatial, temporal and technological features, that could be assessed by probabilistic, stochastic, scenario analysis, quantitative or qualitative methods (Zhang et al. 2019;Hariri-Ardebili 2018). The uncertainty in design model can be initiated from the model parameters, rockfill characteristics, hydrologic uncertainties and further affect the stability and safety factors of dam. The uncertainty categorized into two classes of aleatory uncertainty that originates from the natural behavior and intrinsic complexity of the phenomenon, and the epistemic uncertainty that originates from the unknowns in the theoretical basis and design parameters of the model (Doyle et al. 2019;Huang et al. 2021;Castaldo et al. 2022;Rahimi et al. 2021;Zhang et al. 2021). The uncertainty analysis can be done using fuzzy logic, Monte-Carlo simulation (MCS), entropy method, Bayesian information, generalized likelihood uncertainty estimator methods (Noori et al. 2010;Dehghani et al. 2014;Hariri-Ardebili 2018;Zhang et al. 2019;Guo et al. 2018Guo et al. , 2020Vema et al. 2020;Najafzadeh et al. 2021;Ji et al. 2021;Sevieri et al. 2021;Seifi et al. 2021;Rahimi et al. 2021;Ghiasi et al. 2021;Sharafati et al. 2021;Zhao et al. 2021;Tang et al. 2022;Riahi and Bahrami Chegeni 2022;Moradi Kia et al. 2022). The MCS method is frequently used as the simulator engine in uncertainty quantification that identifies the uncertainties by analyzing probability distribution of model parameters and random number generations. The output uncertainty of model results is determined and calculated using statistical indices by successive generation and sampling of input parameters. The main advantages of MCS uncertainty analyzing are high calculation speed, ease of use, not need to use partial differential equations, insensitivity to the probability distribution of response parameter, possibility of generating outputs with specified probability level (Couto et al. 2013;Pinheiro et al. 2019;Zheng and Han 2016;Riahi-Madvar et al. 2011;Ebtehaj et al. 2020).
The modeling of uncertainty in flood control strategies can be explored by considering various types and sources of uncertainty in models. Generally, datadriven flood prediction models have received increasing attention for quantitative uncertainty analysis (Høybye and Rosbjerg 1999;Maskey et al. 2003;Safavi 2009;Abrishamchi et al. 2005;Micovic et al. 2015, Riahi-Madvar et al. 2021. Furthermore, the investigation of dam stability regarding the uncertainties is interesting. In the uncertainty quantification of dams regarding the structural stability of dam, Altarejos-García et al. (2012a, b) investigated the probabilistic response of gravity dams under specified loads and quantified the reliability results in combination with numerical modelling. Peyras et al. (2012) combined the risk analysis with reliability of dam and developed a risk model for gravity dam based on the resistance probability and input loads. Su and Wen (2013) used probabilistic theory with fuzzy mathematics to investigate the uncertainty in stability of gravity dams by considering the fuzziness and randomness of the effective stability parameters. Alembagheri and Seyedkazemi (2015) used the static and dynamic conditions of parameters and seismic loads to investigate the seismic behavior of Pine Flat gravity dam based on the MCS model and approximate moment estimation technique. Morales-Torres et al. (2016) investigated the uncertainty in the fragility of gravity dam using the compatible fragility analysis method and the fragility curves used for the risk determination. Jia et al. (2018) analyzed the shear stability of cemented sand, gravel and rock (CGSR) dam based on the Mohre-Columb and limit region equations with expansion resistance combined with First-order second-moment method (FOSM). Shu et al. (2020) used the evidence theory and fuzzy evidence theory in determination of the safety of dam considering nonhomogeneous information uncertainty.
However, an in-depth evaluation of the uncertainty in designing detention rockfill dams is missing and rarely considered and studies relevant to detention rockfill dams were inadequate. In light of the above knowledge gaps in the uncertainty analysis of detention rockfill dams design, it can be concluded that the uncertainty quantification in detention rockfill dams needs to be investigated and conceptualized furthermore. In this study, the most commonly used uncertainty analysis method, MCS is adopted with hydraulic models of flood routing in coarse porous media of rockfills for reliable dam designing.
Moreover, relevant studies worldwide have not yet developed a unified standard framework for uncertainty quantification in the rockfill dams' stability against flood events. Also, the improvements in safety factors of rockfill porous dams using the probabilistic design procedure have not yet been studied. Hence the authors in the current study, were encouraged to combine MCS uncertainty quantification with designing equation of rockfill detention dams to develop a probabilistic framework for detention rockfill dam designing. In summary, this study aims to develop a reliability-based stability design framework for detention rockfill dams regarding the dominant uncertain components and parameters, including inlet hydrograph, elevation-storage curve, rockfill stage-discharge relations, nonlinear flood routing in rockfills. The uncertainties in the stability factors, safety factor against overturning (SFO), safety factor against the sliding (SFS), and the safety factor against friction (SFF) is quantified and by fitting the probability distribution to the MCS results the precent of improvements in reliability designs is calculated.

Material and methods
Uncertainty in the detention rockfill dam's output and safety factor of stability was calculated by Monte Carlo simulation method, applying uncertainty in input hydrograph equations, reservoir depth-volume relationship, stage-discharge relationship of rockfill dams, and nonlinear flood routing in rockfills. Moreover, safety factors were evaluated against overturning, sliding, and friction to obtain the stability of the detention rockfill dam under the uncertainty conditions. This section summarizes the material/methods of the developed model framework.

Inflow flood hydrograph
The inflow hydrograph is simulated using a classical probabilistic method. The gamma distribution function is one of the frequently used methods for the inflow flood hydrograph regeneration. This equation has been utilized in many studies to simulate flood hydrographs (Riahi-Madvar et al. 2021;Machajski and Kostecki 2018).
where t p is the time when the inflow flood hydrograph reaches the inflow flood peak discharge, m is the dimensionless coefficient of the hydrograph shape, I represent inlet hydrograph discharge, and I p indicates inlet hydrograph peak. In this equation, m is the shape parameter; as the value of m increases, the flood volume decreases, the descending branch of the hydrograph is shortened, and the hydrograph shape is sharpened. The flood volume can be calculated by integrating Eq. (1) over time: where V f is the flood volume and Γ is the gamma function.

Depth-volume relationship in the reservoir
The depth-volume relationship in the reservoir is achieved by combining the depth-area relationship of the reservoir (A = k H + z 0 n ) with the continuity, A = dS/dh as follows: (1) where S is the reservoir volume, H is the depth of water in the reservoir, A is the water surface in the reservoir, and Z 0 , k, and n are constant coefficients of the reservoir.

Stage-discharge in detention rockfill dam
The analytical stage-discharge relation in a detention rockfill dam is obtained by combining the non-Darcian equation of flow velocity in coarse porous media with the continuity equation and nonlinear flow resistance equations (Samani et al. 2003).
In which, α is: where α and β are constant coefficients, d shows the dam material size (grain size), σ is the standard deviation of the dam material size, g is the gravity acceleration, and n p indicates material porosity, v represents fluid velocity, L demonstrates the dam length in the flow direction, W is the dam width perpendicular to the flow direction, H 1 and H 2 are water depths upstream and downstream of the dam, and θ is the slope angle of the upstream and downstream. The optimal calculated values of α and β are 54 and − 0.077, respectively (Samani et al. 2003).

Flow routing in detention rockfill dam
A nonlinear ordinary differential equation of unsteady flow in detention rockfill dams is obtained by combining Eqs.
According to this equation, the storage volume of water in the reservoir is a nonlinear function of the storage volume of water in the reservoir and water depth upstream and downstream. A total of 36,000 simulations were conducted in the developed mathematical model of Riahi-Madvar et al. (2019) by parametric analysis on a wide range of m, n, Z 0 , k, I p , t p , d, L, θ, S parameters. They finally proposed a simple, practical equation to design detention rockfill dams.
where S f is the required reservoir volume, and Q p is the outlet discharge peak.

Stability analysis of detention rockfill dams
The forces acting on detention rockfill dams, their torque responded to dam toe (D), and the coordinates of the torque points are set according to Fig. 1, where D represents dam toe (Table 1).
The forces acting on dams are horizontal and vertical, tipper or resistant factors. The forces, equations, explanations, and types of effects of each force on the structure stability are given in Table S1 (Supplementary).
Dam safety coefficients can be achieved by forces and moments in Table S1. The dam safety factor equations and their allowed limits are given as follows:

Safety factor against overturning
The total resistant moments of dam toe toward the total overturning moments of dam toe are equal to the safety factor against overturning for the rockfill dam as follows:

Safety factor against sliding
The total vertical forces toward the total horizontal are equal to the safety factor against the slide of the rockfill dam as follows: F V and F H are listed below.
In addition, μ is the static friction coefficient as much as 0.77.

Safety factor against friction
The total ratio of vertical force and allowed shear stress of rockfill dam foundation toward total horizontal force is equal to the safety factor against friction.
In which q is the allowed shear stress of materials at the shear surface.

Integrated design framework of MCS with uncertainty and structural stability
Uncertainty is inevitable in engineering, whether due to a lack of engineering knowledge in that field or inherent uncertainty. In the current model, the uncertainties propagate through the model, resulting in uncertainty in the model's output are evaluated. Uncertainty propagation (UP) analysis used to evaluate the quantification of uncertainty propagation of input parameters in the model outputs. The Monte Carlo simulation method in the developed integrated model generates parameter samples using the density function of the input parameters and then runs the simulation model for the generated parameters. In the generation samples, the rejection rule of predefined certainty in input parameters is used to improve the design's stability. Finally, the output uncertainty is determined and quantified at the end of resampling without replacement. The Monte Carlo simulation method used as a numerical method, sampling-based, and non-intrusive procedure. This process in the current paper, aims to use the Monte Carlo simulation approach to assess the stability of a detention rockfill dam. The developed framework uses three primary steps as shown in Fig. 2. The first step in this research is to identify the input parameters of input hydrograph equations, reservoir depth-height relationship, stage-discharge relationship from rockfill dam, and flow routing equation in detention rockfill dam based on the prior probability distribution of parameters. Then, each of these parameters' prior probability distribution function is selected. The water depth behind the reservoir (H 1 ) and the cross-sectional thickness (L) of the dam are designed in the second phase, utilizing the applicable uncertainties on the parameters of the given equations. As a result of the uncertainty in the input parameters, these values, the uncertainties in H 1 and L, also quantified. In the final step (Fig. 2), the safety factor of the dam stability is determined based on the uncertainty in the design parameters of hydrologic model, water depth, and dam thickness. The dam safety factors derived using this method include a probability distribution function that can be used to evaluate the uncertainties created in the outputs. An example for design of a detention rockfill dam provided by Riahi-Madvar et al. (2019) is utilized to evaluate the proposed method. The values used in this example can be found in the crisp value column of Table 2 and the corresponding prior probability distribution of parameters. According to this example, the V f , S f , H 1 , and L values are 80,569.1, 53,862, 3.76, and 3.44, respectively. The safety factors for SFS, SFO, and SFF are also 1.0274, 1.1276, and 3.5567, respectively. The proposed method is classified into three steps (Fig. 2), and the output, for example, is divided into three sections below. In the current study, the uncertainty quantification of the design model is developed in MATLAB environment. The model firstly develops the hydrologic and hydraulic behavior of the detention rock fill dam under the condition of flood occurrence. The hydraulic behavior of the dam is simulated using Eqs. (1-7) and the structural behavior the dam is simulated by Eqs. (8-10). For all of the model parameters, an appropriate probability distribution function is used to evaluate the parametric uncertainties. In the MCS phase, the hydrologic component determines the flood volume resulted from the detention dam and based on the safe values of flood depletion to the downstream the reservoir volume is optimized. The height and thickness of dam are determined using Eqs. (3) and (4) and the calculated reservoir volume. The effects of parameter's uncertainties on the design values are quantified using 10,000 random samples. The model is executed on a computer with Intel(R) Core i5, 2.6 GHz CPU, 6 GB of RAM.

Uncertainty in model input parameters
The first step in determining the uncertainty of model outputs is to look at the input parameters with uncertainty. The input parameters, their classification, unit of measurement, standard deviation, the prior probability distribution function, and mean value are listed in Table 2. As seen in the last column of Table 2, the uncertainty of input parameters in model design consideration is investigated including the uncertainties of inlet hydrograph, volume-elevation relation of reservoir, non-Darcian stage-discharge equation in coarse rockfill material, non-linear flood routing in detention rockfill dams. The uniform distribution function, Gaussian distribution function, and generalized extreme value (GEV) distribution function are the three probability distribution functions used in this research. Generalized extreme value distribution function are used for extreme values such as maximum inlet hydrograph  peak, maximum time of the hydrograph reaches the peak and the maximum outlet hydrograph peak. The Gaussian distribution is mainly used for coefficients, and the uniform distribution is applied for materials. A significant number of outputs from the model are required to examine the uncertainty using the Monte Carlo simulation method. For this study, 100,000 values are produced for the input parameters based on the prior probability distribution functions and then they are given to the simulated model and finally, the desired outputs are calculated. In the generation samples, based on the Latin hypercube sampling (LHS), the rejection rule of certainty in input parameters is used.

Hydraulic performance of detention rockfill dam under uncertainty
The hydraulic performance of the dam in flood control, under uncertain condition, is evaluated using MCS combined with simulation model. The model determines the water depth behind the dam according to the allowable downstream flood discharge and the dam length after applying the uncertainty to the input parameters which are mentioned in the previous section. The volume of inlet flood to the reservoir is the first value obtained using Eq. (2) after applying the uncertainty in the input parameters. Figure 3 shows the V f histogram, which indicates the volume of the inlet flood and should follow an extreme probability distribution function. As shown in Fig. 3, the GEV distribution fits well with the V f values among the various type of distributions. The values of the parameters of shape (k), scale (σ 1 ), and location parameter (μ 1 ) are − 0.05518809, 6558.68, and 81,937, respectively, for this distribution. The equation of the GEV probability distribution function is presented in the following.
Following the calculation of V f , the values of S f ، H 1 , and L are calculated, and the Sf, H 1 , and L histograms are shown in Fig. 4. The elements that cause uncertainty in the stability of detention rockfill dam are the depth of the water behind the dam and the thickness of the dam, which are obtained in this step. Now it is time to investigate the effects of input uncertainties on the stability of the detention rockfill dam. In this way, the design stability and its collapse is evaluated based on the uncertainty of parameters. Figure 5 shows the violin plot for the V f , S f , H 1 , and L, in which the effect of uncertainty on the hydraulic performance of the rockfill dam can be examined. The upper and lower bounds are 6.31 and 2.23 for H 1 , 143,650 and 63,048 for V f , 113,170  These values present high level of uncertainty in the flood control variables (V f , S f and H 1 ) and indicate that using the rejection rule of only 15% for input parameter's uncertainty results in considerable reduction in reliability of hydraulic performance of the detention rockfill dam. Therefore, simple designs based on presumed parameters in the filed designs have the deficiency that they cannot accurately guarantee the acceptable performance for flood reduction; as a result, the performance of preliminary designing procedure will be low, and the associated uncertainty will increase. Therefore, we have proposed designing based on improved stability analysis for probabilistic designs of detention rockfill dams, as follows.

Uncertainty results in safety factors and stability of the structure
The safety factors and the influence of uncertainty on the structure's stability are calculated after obtaining the optimal values of V f , S f , H 1 , and L with applying the effect of uncertainty of input parameters on these values. Figure 6 displays the resulted output histogram of the safety factors. The effect of uncertainties on SFO has caused uncertainty propagation of + 47.61% and − 21.78% compared to 1.1276 which is a mean value in SFO. In addition, the effect of uncertainties on SFS and SFF values have caused uncertainty propagation as much as + 87.81%, − 32.63%, and + 65.07%, − 28.05% compared to 1.0274 and 3.5567. Therefore, the maximum uncertainty in the safety factors is + 87.81% and − 32.63%, which is related to the SFS. Table 4 presents the mean, median, mode, standard deviation, and upper and lower boundaries of each safety factor. From Table 3 and Fig. 6, it can be noticed that the SFF histogram has a left-skewed distribution and for this reason, the mean value is greater than the median and mode values. But the SFO and SFS histograms have a left-skewed distribution so the mode value is greater than the     Figure 7 also illustrates the violin plot of safety factors observed in each shape method's upper and lower bounds, mean, distribution shape, and standard deviation. Figure 8 shows the changes in safety factors against the water depth of the dam reservoir. As shown, the trend of SFS against H 1 is constant. This indicates that the horizontal force due to the upstream water depth is equal to its vertical force. The variation of SFF relative to H 1 is also descending and according to Eq. (10), this equation can be divided into two parts (μFV)/FH and qL/FH. The fraction (μFV)/ FH is SFS which has a constant trend, and the numerator of qL/FH is constant, which decreases with increasing H 1 and increases with decreasing H 1 . Hence, the SFF value against H 1 is descending. The SFO value against H 1 is ascending. According to Eq. (8-a), the vertical force torque of the reservoir water is less than the torque resulting from its horizontal force relative to the dam toe. These results indicate the in the preliminary design of detention rockfill dams, due to the uncertainty, the structure may collapse due the poor stability and stability analysis regarding these uncertainties need to be recalculated for reliable operation of the dam. Next section, the authors have used rejection rule of 15% uncertainty bound in input parameters, to improve the destinies and precising preliminary designs.

Improving preliminary design regarding the stability under uncertainty
As concluded in previous sections, the preliminary designs are not certain and reliable for flood reduction and need to be refined based on the rejection rule in uncertainty of input parameters. In this section, the effect of input parameter uncertainties is investigated to determine how much can they improve the safety factors. Input parameters are imported to Monte Carlo simulation method with the rejection rule about 15% uncertainty to assess the probability of solution improvement and weakening. Therefore, to  calculate the probability of improvement and failing, a probability distribution should first be fitted to each safety factor histogram, including Normal, Kernel, GEV, and Gama. Moreover, goodness-of-fit tests-Chi-square and Kolmogorov-Smirnov-were used to obtain the best fit among the four probability distributions. In all tests, if the test accepts the null hypothesis, which means the distribution is fitted well with the histogram, at 5% confidence level, h is 0, and if the test rejects the null hypothesis at 5% confidence level, h is 0. The p-value was used to examine the probability distribution functions for which the null hypothesis was accepted. Table 4 shows the goodness-of-fit in each probability distribution for each safety factor. In Table 4, the kernel probability distribution function could be accepted at the 5% confidence level on all safety factors. It is worth mentioning that both hypothesis test, chi-square and Kolmogorov-Smirnov, accepted that the kernel probability distribution function is fitted well with SFO, SFS, and SFF histograms. Because only one probability distribution function passed the hypothesis tests and the other ones could not pass them, the P-value is not used in this case. Figure 9 indicates the fit of the kernel probability distribution function on the safety factors histograms and the cumulative safety factors distribution function against the kernel cumulative distribution function, which shows the excellent fit of this function on safety factors.
Given that the area under the graph of the probability distribution function is equal to 1, it is possible to calculate whether safety factors are more or less than the mean. The safety factor SFF is used to determine this issue (for example, the SFF value is 3.5567), which can be separated into two sections using the probabilistic distribution function diagram (Fig. 10). The area below the probability distribution graph, which is more than this value of 3.5567 (Area 2), indicates that the uncertainty with this probability has improved the solutions. The other area, which is less than 3.5567 (Area 1), indicates that uncertainty with this probability has made the solutions worse. Table 5 shows all safety factors, the percentage of probability that uncertainty causes to improve and worsen the solutions, and the total mean As shown, SFO safety factor (93.16%) had the most significant improvement, and the SFF safety factor (16.64%) had the highest decrease in safety factor. According to this table, uncertainty in a detention rockfill dam design can have positive consequences.

Conclusion
The detention rockfill dam is one of the most effective structural options for delaying the flood peak and reducing the occurrence of floods. Detention rockfill dams are simple, execute and manage. During the design of these dams, suitable locations are selected to conduct the initial design of the dam. Then, the dam height and length are obtained, and in the last step, the final design and optimization of the dam are proposed. Uncertainties in the dam design process led to unreliable outputs. The effect of uncertainty of design parameters was investigated on the output of the initial design of the detention rockfill dam (height and length of the dam) and safety factors. Then, the input parameters of input hydrograph equation, reservoir depth-volume relationship, stage-discharge relationship of rockfill dam, and flow routing equation in rockfill dam detention were replaced with appropriate probability distribution function to investigate the uncertainty in these parameters. The effect of uncertainties on the output of the equations, which was the water depth behind the reservoir and the thickness of the dam as the main design features, was investigated by combining these uncertainties in the rockfill dam mathematical design model. The results indicated that about 15% uncertainty in design input parameters might cause + 87.81 and − 32.63% uncertainty in safety factors. In addition, the negative or positive effect of uncertainty on increasing or decreasing safety factors was examined. The uncertainty of input parameters increased the safety factor with an average probability of 88.3% and decreased it with 11.7%.