# River Model Calibration Based on Design of Experiments Theory. A Case Study: Meta River, Colombia

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Modeling Objective and Simplifications

#### 2.2. Parameters Used as Hydrodynamic, Sedimentologic and Morphological Indicators

^{2}), a statistical measure of how close the data fit the regression line, in this case, comparing whether the simulations match real-data. The method admits other statistical measures for goodness-of-fit, like the mean squared error (MSE) and the mean absolute error (MAE).

_{1}·WL + β

_{2}·QL+ β

_{3}·FD + β

_{4}·MP + β

_{5}·SST + β

_{6}·SBT + β

_{7}·ST + β

_{8}·BL + β

_{9}·BE,

- WL = coefficient of determination (R
^{2}) related to water level; - QL = coefficient of determination (R
^{2}) related to flow rate at control cross-sections; - FD = coefficient of determination (R
^{2}) related to flow distribution through island branches; - MP = coefficient of determination (R
^{2}) related to depth through the 2D domain; - SST = coefficient of determination (R
^{2}) related to suspended-load sediment transport; - SBT = coefficient of determination (R
^{2}) related to bed load sediment transport; - ST = coefficient of determination (R
^{2}) related to total load sediment transport; - BL = coefficient of determination (R
^{2}) related to bed level through the 2D sector; - BE = coefficient of determination (R
^{2}) related to bank erosion rate.

_{i}correspond to the relative weight of each indicator on the OWI. These weights are usually chosen by consensus among the modeling team based on experience or based on the uncertainty related to the variables of each indicator. For beginners, the authors of this paper recommend using the same value for all β

_{i}, as explained by Chaves and Alipaz [33] based on Shannon’s principle of maximum entropy, which warrants that the probabilities of underestimating or overestimation the parameters would be the same, considering that the indicators and the parameters are random variables (with also random distributions).

_{HD}); SST, SBT and ST correspond to the sedimentological component (OWI

_{ST}); while BL and BE are calibration parameters accounting for the morphological component (OWI

_{MF}).

_{de}), Critical Shear Stress for erosion (τ

_{er}), Erosion constant (E

_{0}) and Exponent of the erosion. Beck and Basson [36] assessed the hydraulic and sedimentological behavior of the Klein river estuary by means of MIKE-21C, including as calibration parameters: Hydrodynamic time step, Morphological time step, Flooding Depth, Drying depth, Median grain diameter, Mass density of sediment and Porosity, as well as the calibration parameters shown in Table 1. Therefore, if additional calibration parameters are required for a more accurate representation of the process, the authors of the present paper recommend to the reader to conduct a literature review in order to define the range of feasible values for each calibration parameter. For example, the critical shear stress for deposition (τ

_{de}) and the critical shear stress for erosion (τ

_{er}) are very variable parameters which have a great impact on sediment transport mechanisms, but some authors [25,35] have performed sensitivity analysis on these calibration parameters, that could be used as baseline for setting the range of feasible values.

#### 2.3. Experiment Design and Calibration

^{k}factorial type of experiment, which evaluates k calibration parameters, each of them having two alternatives or levels that must be defined according to its corresponding physical or numerical meaning [37,38,40]. In consequence, goodness-of-fit and OWI calculations must be performed for each possible combination defined in the 2

^{k}experiment.

^{k}type of experiment, that means, evaluating these parameters for at least three different representative levels from their previously defined range of possible values [38]. However, following this recommendation depends on available resources. As in the previous DOE, goodness-of-fit and OWI calculations must be performed for each possible combination defined in the 3

^{k}experiment.

_{MF}) and sedimentological (OWI

_{ST}) components of the Overall Weighted Indicator, it is important to maintain the calibration of the hydrodynamic component and its corresponding parameters.

#### 2.4. Case Study: A Reach of the Meta River, Colombia

^{2}in area.

^{3}/s, and minimum of around 755 m

^{3}/s. This ratio of 13.67 between maximum and minimum average indicates wide variability in flow rate in any given year. Furthermore, a unimodal trend was found in all hydrometric records; with high flow rates between May and August, and low flow rates between early December and late February.

- Morphological Stability: this section presented a low riverbank variability between 1980 and 2012.
- Hydraulic and sedimentological stability: this section is downstream of the last significant tributary of the Meta River, thus, variations in the hydraulic and sedimentological regime up to its mouth are not significant.
- Morphological typology: presents a mix of braided, straight and sinuous reaches, indicating associated morphological response patterns along much of the river.

#### 2.4.1. Indicators for the Meta River Model

**Adjustment of water level series (WL):**Coefficient of determination (R^{2}) between the hydrograph created from the water level series and the water level simulated by interpolation for the calibration period. The comparison point was set at the abscissa K310 corresponding to the upstream boundary condition.**Adjustment of flow distribution (FD):**Coefficient of determination (R^{2}) between the percentages of flow distribution through river branches around islands, measured on-site and those calculated by the 2D model.**Visual comparison of velocity vectors (Visual Indicator):**A visual comparison was performed between the simulated velocity vectors and those recorded by the ADCP. It was considered that the average velocity value in the measured columns would occur close to 0.6 × H [45], where H is the height of the water column.**Adjustment of suspended sediment transport rate (SST):**The IDEAM station “Aceitico” (K127) holds information regarding suspended sediment transport, estimated through regression models from data between 1996 and 2010. Based on the information available, the following equation was used in this work for estimating suspended sediment transport rate:Q_{S}= 0.0096·Q_{L}^{1.774},_{S}is suspended sediment transport rate (ton/day) and Q_{L}is the river discharge (m^{3}/s). To calculate R^{2}, comparison was made between the Q_{S}values obtained by using discharge measurements and the ones calculated by using the results from the model.**Bed Level Adjustment (BL):**The coefficient of determination (R^{2}) was calculated between the mesh of riverbed levels created from the bathymetric surveys from field campaigns and those simulated by MIKE-21C.**Adequate morphological evolution (Qualitative Indicator):**This visual indicator refers to the morphological evolution of the riverbed and riverbanks. The modeler shall verify if the morphologic changes through time are coherent with the numerical capabilities of the model, for example, no sudden and intense sedimentation or erosion phenomena occurred during the time span considered in the simulation.

_{HD}) and the weighted indicator of sedimentological and morphological adjustment (OWI

_{ST, MF}). The β

_{i}coefficients were defined by the team of modelers participating of this project, based on agreed confidence levels for each variable measured during the field campaigns. Lower β

_{i}coefficients were assigned to variables with higher uncertainty levels.

_{HD}= 0.60 WL + 0.40 FD

_{ST, MF}= 0.30 SST + 0.70 BL

#### 2.4.2. Parameters for Modeling the Meta River

- For the river reach under consideration, the Chézy coefficient is used for quantifying bed resistance. In MIKE-21C, this value is estimated as a function of water depth, following the approach explained by Talmon [46]:Chézy = C·h
^{0.17}^{1/2}/s and h is the water depth in meters. Coefficient C (in m^{1/3}/s) is the parameter to calibrate, and corresponds to the reciprocal of the Manning’s roughness coefficient [47], considered constant by the authors for this study. According to Talmon [46], this approach allows to incorporate small bathymetric irregularities (due to dunes, ripples, etc.) as bed resistance. The adequate calibration of the bed resistance directly impacts on the accuracy of the model to correctly estimate flow distribution and morphologic evolution. For example, if the resistance over a shallow region is too high, too much flow will be deflected and there will be a greater tendency towards developing sandbars [35]. - According to the sedimentological information supplied by IDEAM, the median grain size in the riverbed (D50-bed) is 0.35 mm. Based on this grain size, García [48] suggests using Engelund-Hansen [49], Yang’s [50] and Van Rijn’s [51] equations for estimating sediment transport rates. From modeling studies carried out in 2003 Hidroconsultas LTDA [52] on the same river, it was observed that the sediment transport rates estimated by using Yang’s equation showed a good fit when compared to values from measurements. Based on this and aiming to reduce the number of variables in the study, the modelers decided to use only Yang’s and Van Rijn’s equations.
- From the available imagery, it was found that riverbank variations in time intervals shorter than three years were not significant for the representative section. Islands were defined as covered with vegetation, which favors stability. The erosion rate at the riverbank and the corresponding coefficients were not considered as calibration parameters.
- Due to absence of significant patterns and/or phenomena affecting the morphology, the helical flow coefficient HL was simplified to its default value of 1.00.

- Chézy Roughness coefficient as a function of depth; where C is the calibration parameter.
- Sediment Transport Equation;
- Riverbed Load Factor (Kb);
- Suspended Load Factor (Ks);
- Transverse Slope coefficient (TSC);
- Transverse Slope power (TSP).

## 3. Results

#### 3.1. Hydrodynamic Calibration

• Sediment Transport Equation: | Van Rijn |

• Riverbed Load Factor (Kb): | 0.100 |

• Suspended Load Factor (Ks): | 0.300 |

• Transverse Slope coefficient (TSC): | 0.625 |

• Transverse Slope power (TSP): | 0.500 |

_{HD}= 0.9062, as shown in Table 3.

#### 3.2. Screening Design—2^{k} Experiment

^{k}) has the function of detecting the statistically significant factors. The parameters and levels at which the first design was evaluated are shown in Table 4.

^{2}statistic indicates that the adjusted model explains 96.91% of the variability in OWI

_{ST, MF}.

^{k}experiment shall be completed with the graphic verification of assumptions of Homoscedasticity, Normality and Independence. These charts are presented in Figure 5.

#### 3.3. Fit Design—3^{k} Experiment

^{k}type, allows specifying the calibration value of the parameters. The parameters and levels at which the second design was evaluated are shown in Table 4. However, since some 2nd and 3rd order significant interactions were found at Table 5, they cannot be excluded from the experiment. Nevertheless, if all these interactions are considered, the necessary degrees of freedom in the residues will not be enough to perform the analysis of variance. Therefore, the modelers decided to vary the TSP as a replica generator, because it is one of the original main parameters whose effect was recognized as not significant at Figure 4. As previously defined at Table 4, the TSP was varied within the range from 0.5 to 1.

^{2}statistic indicates that this fit model explains 97.69% of the variability in OWI

_{ST, MF}. As in the 2

^{k}experiment, verification tests were performed for Homoscedasticity, Normality and Independence assumptions. Homoscedasticity and Independence charts are presented in Figure 6.

_{ST, MF}values were obtained when using the Van Rijn Transport equation.

_{ST, MF}within the boundaries of the experimental region. From this, the best OWI

_{ST, MF}result found was 0.864, associated with the following conditions: Van Rijn transport equation; Ks = 0.5; Kb = 0.1.

_{ST, MF}, the hydrodynamic component of OWI has to be checked in order to determine if its calibration remains valid.

_{ST, MF}. An OWI

_{HD}of 0.90 was obtained from using those results. The qualitative calibration parameters related to velocity vectors and morphological evolution were also evaluated for this optimal case. For two sections of the river reach used as case study, Figure 7 displays the comparison between observed and modeled velocity vectors, where a good fit was observed both for magnitude and direction.

## 4. Discussion

#### 4.1. Hydrodynamic Calibration

#### 4.2. Screening Design—2^{k} Experiment

_{ST, MF}variability and to evaluate the statistical significance of each effect by comparing its mean square against an estimate of the experimental error.

_{ST, MF}. Based on this, only those three parameters were used for the second 3

^{k}type experiment design, or fit design.

#### 4.3. Fit Design—3^{k} Experiment

_{ST, MF}, when considering a confidence level of 95%.

_{ST, MF}residues were distributed into 16 equally probable classes, and each class was then compared with the expected number of observations. Using Statgraphics software for this analysis, a P-value was obtained equal to 0.9273 for this normality test, which is greater than or equal to 0.05. This prevents rejecting the hypothesis that OWI

_{ST, MF}residues do not come from a normal distribution, with statistical confidence level of 95%.

^{k}experiment, the results shown in this figure indicate that the residuals did not follow a trend in terms of their execution order. The plot of residues in Figure 6 suggests that there is no autocorrelation among the residuals of the ANOVA.

_{ST, MF}= 0.8293 and a global OWI = 0.8571.

#### 4.4. Additional Remarks

- The domain of quantitative variables must be continuous;
- It is possible to discard a significant calibration parameter value, because of an inadequate choice of levels or alternatives during the screening design;
- Some combinations of calibration parameter values might generate numeric instability during the simulation;
- The lack of replicates (with different results) for the same configuration limits the degrees of freedom of the experiment.

## 5. Conclusions

- Compare the efficiency of the method and calibration performance by using other DOE types (Fractional Factorials, Taguchi, Latin Square, etc.) and statistical measures of goodness of fit (Nash-Sutcliffe’s efficiency coefficient (NSE), P-Bias, etc.).
- Compare the performance of the method with other calibration approaches for different conditions: numerical models, river characteristics, quantity and type of calibration parameters, etc.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Church, M.; Ferguson, R.I. Morphodynamics: Rivers beyond steady state. Water Resour. Res.
**2015**, 51, 1883–1897. [Google Scholar] [CrossRef] - Yadav, B.; Eliza, K. A hybrid wavelet-support vector machine model for prediction of lake water level fluctuations using hydro-meteorological data. Measurement
**2017**, 103, 294–301. [Google Scholar] [CrossRef] - Zhu, Q.; Wang, Y.P.; Gao, S.; Zhang, J.; Li, M.; Yang, Y.; Gao, J. Modeling morphological change in anthropogenically controlled estuaries. Anthropocene
**2017**, 17, 70–83. [Google Scholar] [CrossRef] - Pascolo, S.; Petti, M.; Bosa, S. On the wave bottom shear stress in shallow depths: The role ofwave period and bed roughness. Water
**2018**, 10, 1348. [Google Scholar] [CrossRef] - Logan, B.L.; McDonald, R.R.; Nelson, J.M.; Kinzel, P.J.; Barton, G.J. Use of Multidimensional Modeling to Evaluate a Channel Restoration Design for the Kootenai River, Idaho; Scientific Investigations Report 2010–5213; U.S. Geological Survey: Reston, VA, USA, 2011.
- Stewart, G.; Anderson, R.; Wohl, E. Two-dimensional modelling of habitat suitability as a function of discharge on two Colorado rivers. River Res. Appl.
**2005**, 21, 1061–1074. [Google Scholar] [CrossRef] - Ouédraogo, W.; Raude, J.; Gathenya, J. Continuous modeling of the Mkurumudzi River catchment in Kenya using the HEC-HMS conceptual model: Calibration, validation, model performance evaluation and sensitivity analysis. Hydrology
**2018**, 5, 44. [Google Scholar] [CrossRef] - Refsgaard, J.C.; Henriksen, H.J. Modelling guidelines—Terminology and guiding principles. Adv. Water Resour.
**2004**, 27, 71–82. [Google Scholar] [CrossRef] - Kannan, N.; Santhi, C.; White, M.J.; Mehan, S.; Arnold, J.G.; Gassman, P.W. Some Challenges in hydrologic model calibration for large-scale studies: A case study of SWAT model application to Mississippi–Atchafalaya River basin. Hydrology
**2019**, 6, 17. [Google Scholar] [CrossRef] - Arsenault, R.; Brissette, F.; Martel, J.L. The hazards of split-sample validation in hydrological model calibration. J. Hydrol.
**2018**, 566, 346–362. [Google Scholar] [CrossRef] - Hernandez-Suarez, J.S.; Nejadhashemi, A.P.; Kropp, I.M.; Abouali, M.; Zhang, Z.; Deb, K. Evaluation of the impacts of hydrologic model calibration methods on predictability of ecologically-relevant hydrologic indices. J. Hydrol.
**2018**, 564, 758–772. [Google Scholar] [CrossRef] - Kavetski, D.; Kuczera, G.; Franks, S.W. Calibration of conceptual hydrological models revisited: 1. Overcoming numerical artefacts. J. Hydrol.
**2006**, 320, 173–186. [Google Scholar] [CrossRef] - Guerrero, M.; Di Federico, V.; Lamberti, A. Calibration of a 2-D morphodynamic model using water–sediment flux maps derived from an ADCP recording. J. Hydroinforma.
**2013**, 15, 813–828. [Google Scholar] [CrossRef] - Troy, T.J.; Wood, E.F.; Sheffield, J. An efficient calibration method for continental-scale land surface modeling. Water Resour. Res.
**2008**, 44, 1–13. [Google Scholar] [CrossRef] - Getirana, A.C.V. Integrating spatial altimetry data into the automatic calibration of hydrological models. J. Hydrol.
**2010**, 387, 244–255. [Google Scholar] [CrossRef] - Francés, F.; Vélez, J.I.; Vélez, J.J. Split-parameter structure for the automatic calibration of distributed hydrological models. J. Hydrol.
**2007**, 332, 226–240. [Google Scholar] [CrossRef] - Singh, S.; Bárdossy, A. Hydrological model calibration by sequential replacement of weak parameter sets using depth function. Hydrology
**2015**, 2, 69–92. [Google Scholar] [CrossRef] - Beven, K. Prophecy, reality and uncertainty in distributed hydrological modelling. Adv. Water Resour.
**1993**, 16, 41–51. [Google Scholar] [CrossRef] - Wright, K.A.; Goodman, D.H.; Som, N.A.; Alvarez, J.; Martin, A.; Hardy, T.B. Improving hydrodynamic modelling: An analytical framework for assessment of two-dimensional hydrodynamic models. River Res. Appl.
**2017**, 33, 170–181. [Google Scholar] [CrossRef] - Paarlberg, A.J.; Guerrero, M.; Huthoff, F.; Re, M. Optimizing dredge-and-dump activities for river navigability using a hydro-morphodynamic model. Water
**2015**, 7, 3943–3962. [Google Scholar] [CrossRef] - Wu, K.; Yeh, K.C.; Lai, Y.G. A combined field and numerical modeling study to assess the longitudinal channel slope evolution in a mixed alluvial and soft bedrock stream. Water
**2019**, 11, 735. [Google Scholar] [CrossRef] - Guan, M.; Liang, Q. A two-dimensional hydro-morphological model for river hydraulics and morphology with vegetation. Environ. Model. Softw.
**2017**, 88, 10–21. [Google Scholar] [CrossRef] [Green Version] - Kang, T.; Kimura, I.; Shimizu, Y. Responses of bed morphology to vegetation growth and flood discharge at a sharp river bend. Water
**2018**, 10, 223. [Google Scholar] [CrossRef] - Castro-Bolinaga, C.F.; Fox, G.A. Streambank erosion: Advances in monitoring, modeling and management. Water
**2018**, 10, 1346. [Google Scholar] [CrossRef] - Bosa, S.; Petti, M.; Pascolo, S. Numerical modelling of cohesive bank migration. Water
**2018**, 10, 961. [Google Scholar] [CrossRef] - Klein, A. Verification of Morphodynamic Models on Channels, Trenches, and Pits. Master’s Thesis, TU Delft, Delft, The Netherlands, March 2004. [Google Scholar]
- Van Waveren, R.H.; Groot, S.; Scholten, H.; van Geer, F.; Wösten, H.; Koeze, R.; Noort, J. Good Modelling Practice Handbook; RWS-RIZA: Lelystad, The Netherlands, 1999. [Google Scholar]
- DHI. MIKE 21C Curvilinear model for river morphology—Scientific Documentation. Available online: http://manuals.mikepoweredbydhi.help/2017/Water_Resources/MIKE21C_Scientific_documentation.pdf (accessed on 11 November 2018).
- Papanicolaou, A.N.T.; Krallis, G.; Edinger, J. Sediment transport modeling review—Current and future developments. J. Hydraul. Eng.
**2008**, 134, 1–14. [Google Scholar] [CrossRef] - Mueller, E.R.; Pitlick, J. Sediment supply and channel morphology in mountain river systems: 1. Relative importance of lithology, topography, and climate. J. Geophys. Res. Earth Surf.
**2013**, 118, 2325–2342. [Google Scholar] [CrossRef] - Sear, D.A.; Newson, M.D.; Thorne, C.R. Guidebook of Applied Fluvial Geomorphology; Thomas Telford Ltd: London, UK, 2010. [Google Scholar]
- Matte, P.; Secretan, Y.; Morin, J. Hydrodynamic modeling of the St. Lawrence fluvial estuary. I: Model setup, calibration, and validation. J. Waterw. Port Coast. Ocean Eng.
**2017**, 143, 04017010. [Google Scholar] [CrossRef] - Chaves, H.M.L.; Alipaz, S. An integrated indicator based on basin hydrology, environment, life, and policy: The watershed sustainability index. Water Resour. Manag.
**2007**, 21, 883–895. [Google Scholar] [CrossRef] - DHI Water & Environment. MIKE 21 Flow Model FM—User Guide: Sand Transport Module, incl. Shoreline Morphology; DHI Water & Environment: Hørsholm, Denmark, 2017. [Google Scholar]
- De Villiers, J. 2D Modelling of Turbulent Transport of Cohesive Sediments in Shallow Reservoirs. Master’s Thesis, University of Stellenbosch, Stellenbosch, South Africa, 2006. [Google Scholar]
- Beck, J.S.; Basson, G.R. Klein River estuary (South Africa): 2D numerical modelling of estuary breaching. Water SA
**2008**, 34, 33–38. [Google Scholar] - Jain, R. The Art of Computer Systems Performance Analysis: Techniques for Experimental Design, Measurement, Simulation, and Modeling, 1st ed.; John Wiley & Sons: Hoboken, NJ, USA, 1991. [Google Scholar]
- Montgomery, D.C. Design and Analysis of Experiments, 9th ed.; John Wiley & Sons: Hoboken, NJ, USA, 2017. [Google Scholar]
- Kleijnen, J.P.C. Experimental design for sensitivity analysis, optimization, and validation of simulation models. In Handbook of Simulation; Banks, J., Ed.; John Wiley & Sons: Hoboken, NJ, USA, 1998; pp. 173–223. [Google Scholar]
- Lee, R. Statistical design of experiments for screening and optimization. Chemie-Ingenieur-Technik
**2019**, 91, 191–200. [Google Scholar] [CrossRef] - Dorfmann, C.; Knoblauch, H. ADCP measurements in a reservoir of a run-of-river Hydro Power Plant. In Proceedings of the 6th International Symposium on Ultrasonic Doppler Methods for Fluid Mechanics and Fluid Engineering, Prague, Czech Republic, 9–11 September 2008; pp. 45–48. [Google Scholar]
- Universidad del Norte. Actualización de los Estudios y Diseños para la Navegabilidad del río Meta entre Cabuyaro (K804) y Puerto Carreño (K0); Universidad del Norte: Barranquilla, Colombia, 2013. [Google Scholar]
- Plan Maestro Fluvial de Colombia 2015; Ministerio de Transporte: Bogotá, Colombia, 2015.
- Caracterización del Transporte en Colombia—Diagnóstico y Proyectos de Transporte e Infraestructura; Ministerio de Transporte: Bogotá, Colombia, 2005.
- Pasternack, G.B.; Gilbert, A.T.; Wheaton, J.M.; Buckland, E.M. Error propagation for velocity and shear stress prediction using 2D models for environmental management. J. Hydrol.
**2006**, 328, 227–241. [Google Scholar] [CrossRef] - Talmon, A.M. Bed Topography of River Bends with Suspended Sediment Transport. Ph.D. Thesis, Delft University of Technology, Delft, The Netherlands, 1992. [Google Scholar]
- DHI Water & Environment. MIKE 21 Flow Model—User Guide: Hydrodynamic Module; DHI Water & Environment: Hørsholm, Denmark, 2017. [Google Scholar]
- García, M.H. Sediment transport and morphodynamics. In Sedimentation Engineering; American Society of Civil Engineers: Reston, VA, USA, 2008; pp. 21–163. [Google Scholar]
- Engelund, F.; Hansen, E. A monograph on sediment transport in alluvial streams; Technical University of Denmark: Copenhagen, Denmark, 1967. [Google Scholar]
- Yang, C.T. Incipient motion and sediment transport. J. Hydraul. Div.
**1973**, 99, 1679–1704. [Google Scholar] - Van Rijn, L.C. Sediment transport, part II: Suspended load transport. J. Hydraul. Eng.
**1984**, 110, 1613–1641. [Google Scholar] [CrossRef] - Hidroconsultas LTDA. Estudios básicos en el río Meta para la línea base de ingeniería tendiente a definir el sistema más adecuado para el mantenimiento de un canal navegable, obras de encauzamiento y demás obras fluviales entre la desembocadura del río Casanare y Puerto Texas; Hidroconsultas LTDA: Bogota, Colombia, 2003. [Google Scholar]

**Figure 4.**Normal Probability of Effects for the weighted indicator of sedimentological and morphological adjustment (OWI

_{ST, MF}) in experimental design 2

^{k}.

**Table 1.**Typical calibration parameters. (Source: Adapted from [28]).

Component | Parameter | Description |
---|---|---|

Hydrodynamics | - Bed resistance to flow
| This is related to the riverbed’s resistance to flow. For MIKE-21C, the value at each cell is determined as a function of water depth and Manning’s or Chézy’s coefficient. The model uses as input a constant value or maps corresponding to their spatial distribution. |

- Eddy viscosity—K
| Parameter related to the calibration of the velocity vector and depends on the turbulence state. For MIKE-21C, usually between 0.2 m^{2}/s and 5.0 m^{2}/s. | |

Sedimentology/ Morphology | - Sediment transport equation
| The selection of sediment transport equation defines the parameters to be used in the model, and it mainly depends on granulometry (i.e., particle size and shape). A complete description of the transport formulae can be found in the MIKE-21C user’s guide [28]. |

- Bed load factor—Kb
- Suspended load factor—Ks
| The total sediment load is estimated through a sediment transport equation (MIKE-21C includes a set of options for this purpose). | |

- Longitudinal slope coefficient—LSC
- Transverse slope coefficient—TSC
- Transverse slope power—TSP
| These parameters modify sediment transport rates in the stream current, considering morphological changes affecting transverse and longitudinal slopes. | |

- Helical flow calibration constant—HL
| In MIKE-21C this parameter defines the intensity of the helical flow on the riverbed due to the secondary currents. It is usually assumed to be equal to 1; however and ranges from 0.4 to 1.2, according to [34]. | |

- Erosion rate on riverbanks
| According to MIKE-21C, the erosion rate on the riverbanks can be calibrated using three different parameters, $\alpha $: Transversal slope of the riverbank. $\psi $: Fraction of sediment transport rate near the bank. $\gamma $: Erosion constant which does not depend on the hydraulic condition. |

Abscissa (km) | Arm | Observed | C = 50 | C = 55 | C = 60 |
---|---|---|---|---|---|

204 | Right | 0.7076 | 0.6875 | 0.6915 | 0.6962 |

Left | 0.2924 | 0.3125 | 0.3085 | 0.3038 | |

289 | Right | 0.1598 | 0.1757 | 0.1694 | 0.1514 |

Left | 0.8402 | 0.8243 | 0.8306 | 0.8486 | |

267 | Right | 0.2580 | 0.2718 | 0.2453 | 0.2516 |

Left | 0.7420 | 0.7282 | 0.7547 | 0.7484 | |

248 | Right | 0.0560 | 0.0527 | 0.0436 | 0.0327 |

Left | 0.9440 | 0.9473 | 0.9564 | 0.9673 |

C | R^{2} | OWI_{HD} | |
---|---|---|---|

Water Level (WL) | Flow Distribution (FD) | ||

50 | 0.7893 | 0.9979 | 0.8727 |

55 | 0.8450 | 0.9981 | 0.9062 |

60 | 0.8272 | 0.9984 | 0.8957 |

Level | Sediment Transport Equation (A) * | Suspended Load Factor—Ks (B) * | Riverbed Load Factor Kb (C) * | Transverse Slope Coeff.—TSC (D) * | Transverse Slope Power—TSP (E) * |
---|---|---|---|---|---|

High | Yang | 0.1 | 0.1 | 0.625 | 0.5 |

Low | Van Rijn | 0.9 | 0.5 | 1.250 | 1.0 |

Source | Sum of Squares | Df | Mean Square | F-Ratio | P-Value |
---|---|---|---|---|---|

A: Transport Eq | 0.084400 | 1 | 0.084400 | 245.66 | 0.0000 |

B: Ks | 0.106300 | 1 | 0.106300 | 309.35 | 0.0000 |

C: Kb | 0.007400 | 1 | 0.007400 | 21.76 | 0.0001 |

AB | 0.040200 | 1 | 0.040200 | 117.23 | 0.0000 |

BC | 0.010200 | 1 | 0.010200 | 29.90 | 0.0000 |

ABC | 0.010200 | 1 | 0.010200 | 29.67 | 0.0000 |

Total error | 0.008200 | 24 | 0.000340 | ||

Total (corrected) | 0.267200 | 31 |

Level | Sediment Transport Equation (A)* | Suspended Load Factor—Ks (B)* | Riverbed Load Factor Kb (C)* |
---|---|---|---|

High | Yang | 0.1 | 0.1 |

Mid | - | 0.5 | 0.3 |

Low | Van Rijn | 0.9 | 0.5 |

Source | Sum of Squares | Df | Mean Square | F-Ratio | P-Value |
---|---|---|---|---|---|

A: Transport Eq | 0.1033940 | 1 | 0.1033940 | 91.53 | 0.0000 |

B: Ks | 0.0666866 | 2 | 0.0333433 | 29.52 | 0.0000 |

C: Kb | 0.0196977 | 2 | 0.0098489 | 8.72 | 0.0016 |

AB | 0.0281542 | 2 | 0.0140771 | 12.46 | 0.0002 |

BC | 0.0134977 | 2 | 0.0067489 | 5.97 | 0.0085 |

ABC | 0.0231324 | 4 | 0.0057831 | 5.12 | 0.0045 |

Total error | 0.0248521 | 22 | 0.0011296 | ||

Total (corrected) | 0.2794150 | 35 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Acuña, G.J.; Ávila, H.; Canales, F.A.
River Model Calibration Based on Design of Experiments Theory. A Case Study: Meta River, Colombia. *Water* **2019**, *11*, 1382.
https://doi.org/10.3390/w11071382

**AMA Style**

Acuña GJ, Ávila H, Canales FA.
River Model Calibration Based on Design of Experiments Theory. A Case Study: Meta River, Colombia. *Water*. 2019; 11(7):1382.
https://doi.org/10.3390/w11071382

**Chicago/Turabian Style**

Acuña, Guillermo J., Humberto Ávila, and Fausto A. Canales.
2019. "River Model Calibration Based on Design of Experiments Theory. A Case Study: Meta River, Colombia" *Water* 11, no. 7: 1382.
https://doi.org/10.3390/w11071382