# Combination of Discrete Element Method and Artificial Neural Network for Predicting Porosity of Gravel-Bed River

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Discrete Element Method (DEM)

_{i}= the mass of a particle i; ${\overrightarrow{\mathrm{u}}}_{\mathrm{i}}$ = the velocity of a particle; $\overrightarrow{\mathrm{g}}$ = Gravity acceleration; ${\overrightarrow{\mathrm{f}}}_{\mathrm{i},\mathrm{k}}$ = interaction force between particle i and particle k (contact force); ${\overrightarrow{\mathrm{f}}}_{\mathrm{i},\mathrm{f}}$ = interaction force between the particle i and the fluid; I = moment of inertia; ${\overrightarrow{\mathsf{\omega}}}_{\mathrm{i}}$ = angular velocity; d

_{i}= diameter of the grain i; and ${\overrightarrow{\mathrm{n}}}_{\mathrm{i},\mathrm{k}}$ = directional contact = vector connecting the center of grains i and k.

_{i}= stiffness of grain i; δ

_{i,k}= the characteristic of the contact and displacement (also called the length of the springs in the two directions above); α

_{i}= damping coefficient; and Δu

_{i}= relative velocity of grain at the moment of collision. Following Coulomb, the value of tangential friction is determined by the product of the friction coefficient μ and the orthogonal force component. In the nonlinear contact force, Hertz-Mindlin model, the tangential force component will increase until the ratio (f

^{(}

^{τ)}/f

^{(n)}) reaches a value of μ, and it retains the maximum value until the particles are no longer in contact with each other. A detail of the force models, as well as the method for determining the relevant coefficients, can be found in Reference [28].

#### 2.2. Algorithms for Calculating Grain Size Distribution and Porosity of a Cross Section from DEM Results

_{k}(k = 0, …, K), which intersect the spherical grain matrix. The diameter of generated circle i (i = 1, …, n

_{k}) is dependent on the spherical diameter and the relative position between the k-plane and grain i (Figure 4).

_{s,k}) of all n

_{k}grains in plane k is determined:

_{t}is calculated based on the shape generated by the plane k cut across the grain matrix, whereby porosity of cross section k is calculated by the following equation:

_{k}size fractions with characteristic grain size D

_{j}(j = 1, …, m

_{k}) and D

_{j}< D

_{j+1}, then the area of each fraction is calculated by

#### 2.3. Feed Forward Neural Network (FNN)

#### 2.4. Evaluation of the Model Performance

_{i}is calculated value ith, y

_{i}represents the measured value ith.

## 3. Results and Discussions

#### 3.1. Input Parameters for DEM

_{m}= 0.353 mm and standard deviation σ

^{(d)}= 1.933; Gravel-bed with mean diameter D

_{m}= 7.104 mm and standard deviation σ

^{(D)}=1.375.

_{m}= 0.142 mm and standard deviation σ

^{(d)}= 1.837; Gravel-bed with mean diameter D

_{m}= 7.482 mm and standard deviation σ

^{(D)}= 1.324.

#### 3.2. DEM Verification for Porosity

#### 3.3. DEM Verification for Infiltration

_{m}/d

_{m}) in our simulation (52,69) is in the range of percolation (30–70). In this ratio, fine particles are easy to infill to gravel, a fact consistent with what has been found in previous studies [46]. Figure 8d shows the middle x axis cross-section with most of its void space filled by sediment, however, not all void space was entirely filled. In the bottom of the simulation domain (Figure 8(d1)), fine sediment could not move down because of the bottom walls effect, leading to a sudden increase of fine fraction near the flume bed. This phenomenon usually occurred in flume experiments with gravel and fine sediment [10,42,47]. While there are some limitations in the time and scale of the simulations, it can be said that DEM is suitable for simulating the realistic 3D structure of fine sediment and gravel.

#### 3.4. Input Data for FNN

_{1}(d

_{1}< 0.125 mm), f

_{2}(0.125 mm ≤ d

_{2}< 0.25 mm), f

_{3}(0.25 mm ≤ d

_{3}< 0.5 mm), f

_{4}(0.5 mm ≤ d

_{4}< 1 mm), f

_{5}(1 mm ≤ d

_{5}< 2 mm), f

_{6}(2 mm ≤ d

_{6}< 4 mm), f

_{7}(4 mm ≤ d

_{7}< 8 mm), f

_{8}(8 mm ≤ d

_{8}< 16 mm), and f

_{9}(16 mm ≤ d

_{9}).

#### 3.5. Porosity Prediction Based on FNN Model

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Frings, R.M.; Kleinhans, M.G.; Vollmer, S. Discriminating between pore-filling load and bed-structure load: A new porosity-based method, exemplified for the river Rhine. Sedimentology
**2008**, 55, 1571–1593. [Google Scholar] [CrossRef] - Nunez-Gonzalez, F.; Martin-Vide, J.P.; Kleinhans, M.G. Porosity and size gradation of saturated gravel with percolated fines. Sedimentology
**2016**, 63, 1209–1232. [Google Scholar] [CrossRef] - Gayraud, S.; Philippe, M. Influence of Bed-Sediment Features on the Interstitial Habitat Available for Macroinvertebrates in 15 French Streams. Int. Rev. Hydrobiol.
**2003**, 88, 77–93. [Google Scholar] [CrossRef] - Verstraeten, G.; Poesen, J. Variability of dry sediment bulk density between and within retention ponds and its impact on the calculation of sediment yields. Earth Surf. Process. Landf.
**2001**, 26, 375–394. [Google Scholar] [CrossRef] - Wilcock, P.R. Two-fraction model of initial sediment motion in gravel-bed rivers. Science
**1998**, 280, 410–412. [Google Scholar] [CrossRef] [PubMed] - Driscoll, F.G. Groundwater and Wells, 2nd ed.; Johnson Filtration Systems Inc.: Saint Paul, MN, USA, 1986. [Google Scholar]
- Bui, V.H.; Bui, M.D.; Rutschmann, P. Advanced Numerical Modeling of Sediment Transport in Gravel-Bed Rivers. Water
**2019**, 11, 550. [Google Scholar] [CrossRef] - Doherty, J. Calibration and Uncertainty Analysis for Complex Environmental Models; Watermark Numerical Computing: Brisbane, Queensland, Australia, 2015. [Google Scholar]
- Wu, W.; Wang, S.S. Formulas for sediment porosity and settling velocity. J. Hydraul. Eng.
**2006**, 132, 858–862. [Google Scholar] [CrossRef] - Wooster, J.K.; Dusterhoff, S.R.; Cui, Y.T.; Sklar, L.S.; Dietrich, W.E.; Malko, M. Sediment supply and relative size distribution effects on fine sediment infiltration into immobile gravels. Water Resour. Res.
**2008**, 44. [Google Scholar] [CrossRef] [Green Version] - Peronius, N.; Sweeting, T.J. On the correlation of minimum porosity with particle size distribution. Powder Technol.
**1985**, 42, 113–121. [Google Scholar] [CrossRef] - Carling, R.C.; Templeton, D. The effect of carbachol and isoprenaline on cell division in the exocrine pancreas of the rat. Q. J. Exp. Physiol.
**1982**, 67, 577–585. [Google Scholar] [CrossRef] - Komura, S. Discussion of “Sediment transportation mechanics: Introduction and properties of sediment”. J. Hydraul. Div.
**1963**, 89, 263–266. [Google Scholar] - Frings, R.M.; Schuttrumpf, H.; Vollmer, S. Verification of porosity predictors for fluvial sand-gravel deposits. Water Resour. Res.
**2011**, 47. [Google Scholar] [CrossRef] - Ouchiyama, N.; Tanaka, T. Porosity estimation for random packings of spherical particles. Ind. Eng. Chem. Fundam.
**1984**, 23, 490–493. [Google Scholar] [CrossRef] - Yu, A.B.; Standish, N. Estimation of the Porosity of Particle Mixtures by a Linear-Mixture Packing Model. Ind. Eng. Chem. Res.
**1991**, 30, 1372–1385. [Google Scholar] [CrossRef] - Yu, A.B.; Standish, N. Limitation of Proposed Mathematical-Models for the Porosity Estimation of Nonspherical Particle Mixtures. Ind. Eng. Chem. Res.
**1993**, 32, 2179–2182. [Google Scholar] [CrossRef] - Suzuki, M.; Oshima, T. Estimation of the co-ordination number in a multi-component mixture of spheres. Powder Technol.
**1983**, 35, 159–166. [Google Scholar] [CrossRef] - Nolan, G.; Kavanagh, P.E. Computer simulation of random packings of spheres with log-normaldistributions. Powder Technol.
**1993**, 76, 309–316. [Google Scholar] [CrossRef] - Desmond, K.W.; Weeks, E.R. Influence of particle size distribution on random close packing of spheres. Phys. Rev. E
**2014**, 90, 022204. [Google Scholar] [CrossRef] [Green Version] - Saljooghi, B.S.; Hezarkhani, A. Comparison of WAVENET and ANN for predicting the porosity obtained from well log data. J. Pet. Sci. Eng.
**2014**, 123, 172–182. [Google Scholar] [CrossRef] - Bui, M.D.; Kaveh, K.; Rutschmann, P. Integrating Artificial Neural Networks into Hydromorphological Model for Fluvial Channels. In Proceedings of the 36th IAHR World Congress, Hague, The Netherlands, 28 June–3 July 2015; pp. 1673–1680. [Google Scholar]
- Link, C.A.; Himmer, P.A. Oil reservoir porosity prediction using a neural network ensemble approach. In Geophysical Applications of Artificial Neural Networks and Fuzzy Logic; Springer: Berlin/Heidelberg, Germany, 2003; pp. 197–213. [Google Scholar]
- Bagheripour, P.; Asoodeh, M. Fuzzy ruling between core porosity and petrophysical logs: Subtractive clustering vs. genetic algorithm-pattern search. J. Appl. Geophys.
**2013**, 99, 35–41. [Google Scholar] [CrossRef] - Kraipeerapun, P.; Fung, C.C.; Nakkrasae, S. Porosity prediction using bagging of complementary neural networks. In Proceedings of the International Symposium on Neural Networks; Springer: Berlin/Heidelberg, Germany, 2009; pp. 175–184. [Google Scholar]
- Toro-Escobar, C.M.; Parker, G.; Paola, C. Transfer function for the deposition of poorly sorted gravel in response to streambed aggradation. J. Hydraul. Res.
**1996**, 34, 35–53. [Google Scholar] [CrossRef] - Cundall, P.A.; Strack, O.D. A discrete numerical model for granular assemblies. Geotechnique
**1979**, 29, 47–65. [Google Scholar] [CrossRef] - Johnson, K.L. Contact Mechanics; Cambridge University Press: Cambridge, UK, 1985. [Google Scholar]
- Fleischmann, J.; Serban, R.; Negrut, D.; Jayakumar, P. On the Importance of Displacement History in Soft-Body Contact Models. J. Comput. Nonlinear Dyn.
**2016**, 11, 044502. [Google Scholar] [CrossRef] - Landau, L.; Lifshitz, E. Theory of Elasticity, 3rd ed.; Pergamon Press: Oxford, UK, 1986. [Google Scholar]
- Mindlin, R. Compliance of elastic bodies in contact. J. Appl. Mech. Trans. ASME
**1949**, 16, 259–268. [Google Scholar] - Zell, A. Simulation Neuronaler Netze; Addison-Wesley: Bonn, Germany, 1994; Volume 1. [Google Scholar]
- Haykin, S. Neural Networks: A Comprehensive Foundation; Prentice Hall: Upper Saddle River, NJ, USA, 1999. [Google Scholar]
- Bhattacharya, B.; Price, R.; Solomatine, D.P. Machine learning approach to modeling sediment transport. J. Hydraul. Eng.
**2007**, 133, 440–450. [Google Scholar] [CrossRef] - Kingma, D.P.; Ba, J. Adam: A method for stochastic optimization. In Proceedings of the 3rd International Conference for Learning Representations, San Diego, CA, USA, 7–9 May 2015. [Google Scholar]
- Jangid, M.; Srivastava, S. Handwritten Devanagari Character Recognition Using Layer-Wise Training of Deep Convolutional Neural Networks and Adaptive Gradient Methods. J. Imaging
**2018**, 4, 41. [Google Scholar] [CrossRef] - He, Z.; Zhang, X.; Cao, Y.; Liu, Z.; Zhang, B.; Wang, X. LiteNet: Lightweight neural network for detecting arrhythmias at resource-constrained mobile devices. Sensors
**2018**, 18, 1229. [Google Scholar] [CrossRef] - Bui, M.D.; Kaveh, K.; Penz, P.; Rutschmann, P. Contraction scour estimation using data-driven methods. J. Appl. Water Eng. Res.
**2015**, 3, 143–156. [Google Scholar] [CrossRef] - Heaton, J. Introduction to Neural Networks with Java; Heaton Research, Inc.: St. Louis, MO, USA, 2008. [Google Scholar]
- McGeary, R.K. Mechanical packing of spherical particles. J. Am. Ceram. Soc.
**1961**, 44, 513–522. [Google Scholar] [CrossRef] - Navaratnam, C.U.; Aberle, J.; Daxnerová, J. An Experimental Investigation on Porosity in Gravel Beds. In Free Surface Flows and Transport Processes; Springer: Cham, Switzerland, 2018; pp. 323–334. [Google Scholar]
- Gibson, S.; Abraham, D.; Heath, R.; Schoellhamer, D. Vertical gradational variability of fines deposited in a gravel framework. Sedimentology
**2009**, 56, 661–676. [Google Scholar] [CrossRef] - Gibson, S.; Abraham, D.; Heath, R.; Schoellhamer, D. Bridging Process Threshold for Sediment Infiltrating into a Coarse Substrate. J. Geotech. Geoenviron. Eng.
**2010**, 136, 402–406. [Google Scholar] [CrossRef] - Holdich, R.G. Fundamentals of Particle Technology; Midland Information Technology and Publishing: Nottingham, UK, 2002. [Google Scholar]
- Valdes, J.R.; Santamarina, J.C. Clogging: Bridge formation and vibration-based destabilization. Can. Geotech. J.
**2008**, 45, 177–184. [Google Scholar] [CrossRef] - Leonardson, R. Exchange of Fine Sediments with Gravel Riverbeds. Ph.D. Thesis, University of California, Berkeley, CA, USA, 2010. [Google Scholar]
- Seal, R.; Parker, G.; Paola, C.; Mullenbach, B. Laboratory experiments on downstream fining of gravel, narrow channel runs 1 through 3: Supplemental methods and data. In External Memo M-239; St. Anthony Falls Hydraulic Lab, University of Minnesota: Minneapolis, MN, USA, 1995. [Google Scholar]
- Ridgway, K.; Tarbuck, K. Voidage fluctuations in randomly-packed beds of spheres adjacent to a containing wall. Chem. Eng. Sci.
**1968**, 23, 1147–1155. [Google Scholar] [CrossRef] - Sulaiman, M.; Tsutsumi, D.; Fujita, M. Porosity of sediment mixtures with different type of grain size distribution. Annu. J. Hydraul. Eng.
**2007**, 51, 133–138. [Google Scholar] [CrossRef]

**Figure 2.**Contact forces revised from Fleischmann [29].

**Figure 5.**Three-layer feed forward neural network (

**a**) where input layer has p input nodes, hidden layer has h activation functions, and output layer has q nodes. (

**b**) a node of the network.

**Figure 6.**Porosity of packing of binary mixtures of spheres with size ratio (d/D) 0.14 in comparison with McGeary’s measurement [40] (

**a**) pure coarse grain, (

**b**) fine sediment fraction 0.4, and (

**c**) porosity of mixture.

**Figure 7.**Porosity obtained from DEM simulations in comparison with the porosity measurement of Navaratnam et at. [41] (

**a**) gravel flume, (

**b**) vertical porosity variation.

**Figure 8.**Bed structure of filled gravel at final computational time step. (

**a**) bridging and (

**b**) percolation and of the middle x-axis cross-section, (

**c**) bridging and (

**d**) percolation.

**Figure 9.**Simulated fine fraction variation by the depth in comparison with Gibson measurement [42]. (

**a**) Bridging and (

**b**) percolation.

**Figure 10.**The cumulative grain-size distributions of bed materials at ten different represented cross-sections in z-direction (

**a**) bridging, (

**d**) percolation and bed structures at cross section 480th in z-direction, (

**b**) Dataset-1, (

**c**) Dataset-2, (

**e**) Dataset-3, and (

**f**) Dataset-4.

**Figure 11.**FNN predicted porosity along the depth (z-direction) (

**a**) Datasets 1 and 2, (

**d**) Datasets 3 and 4, and scatter plot (

**b**) Dataset-1, (

**c**) Dataset-2, (

**e**) Dataset-3, and (

**f**) Dataset-4.

**Figure 12.**FNN predicted porosity along x-directio (

**a**) Datasets 5 and 6, (

**b**) Datasets 7 and 8, and Scatter plots (

**c**) Dataset-5, (

**d**) Dataset-6, (

**e**) Dataset-7, and (

**f**) Dataset-8.

Density of Sphere (kg/m^{3}) | Density of Water (kg/m^{3}) | Young’s Modulus (Pa) | Poisson Ratio | Friction between Grains | Coefficient of Restitution |
---|---|---|---|---|---|

2350 | 1000 | 5.0 × 10^{6} | 0.45 | 0.35 | 0.40 |

Statistical Indicators | Case-1 | Case-2 | |
---|---|---|---|

Run 1 | Run 2 | ||

R | 0.9857 | 0.957526 | 0.908266 |

RMSE | 0.0165 | 0.048585 | 0.059763 |

MAE | 0.0125 | 0.036198 | 0.05138 |

Process | Pair Time | Neigh Time | Comm Time | Outpt Time | Other Time |
---|---|---|---|---|---|

Case-3 (Bridging) | |||||

Insert | 14,607.7 | 16,269.9 | 19.4313 | 8.31424 | 2593.69 |

Settle | 8319.46 | 2372.5 | 5.84765 | 4.09249 | 1035.83 |

Case-4 (Percolation) | |||||

Insert | 24,954.6 | 43,270.4 | 42.0592 | 8.5656 | 5287.41 |

Settle | 14,112.2 | 12,281.2 | 11.608 | 3.28125 | 2083.9 |

**Table 4.**Verification of Fine Sediment Distribution with Gibson’s Measurement [42].

Statistical Indicators | Case-3 (Bridging) | Case-4 (Percolation) |
---|---|---|

R | 0.969191 | 0.940474 |

RMSE | 0.128067 | 0.261443 |

MAE | 0.066765 | 0.121255 |

Statistical Indicators | Bridging | Percolation | ||
---|---|---|---|---|

Dataset-1 | Dataset-2 | Dataset-3 | Dataset-4 | |

R | 0.965968 | 0.989206 | 0.990841 | 0.994024 |

RMSE | 0.015736 | 0.008786 | 0.007807 | 0.005753 |

MAE | 0.009580 | 0.006548 | 0.004898 | 0.003155 |

Statistical Indicators | Bridging | Percolation | ||
---|---|---|---|---|

Dataset-5 | Dataset-6 | Dataset-7 | Dataset-8 | |

R | 0.9298 | 0.9786 | 0.9236 | 0.9748 |

RMSE | 0.0113 | 0.0063 | 0.0097 | 0.0060 |

MAE | 0.0080 | 0.0050 | 0.0056 | 0.0041 |

© 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**

Bui, V.H.; Bui, M.D.; Rutschmann, P.
Combination of Discrete Element Method and Artificial Neural Network for Predicting Porosity of Gravel-Bed River. *Water* **2019**, *11*, 1461.
https://doi.org/10.3390/w11071461

**AMA Style**

Bui VH, Bui MD, Rutschmann P.
Combination of Discrete Element Method and Artificial Neural Network for Predicting Porosity of Gravel-Bed River. *Water*. 2019; 11(7):1461.
https://doi.org/10.3390/w11071461

**Chicago/Turabian Style**

Bui, Van Hieu, Minh Duc Bui, and Peter Rutschmann.
2019. "Combination of Discrete Element Method and Artificial Neural Network for Predicting Porosity of Gravel-Bed River" *Water* 11, no. 7: 1461.
https://doi.org/10.3390/w11071461