**Suggested Citation:**"Chapter 8. Most Robust Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 8. Most Robust Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 8. Most Robust Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 8. Most Robust Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 8. Most Robust Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 8. Most Robust Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 8. Most Robust Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 8. Most Robust Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 8. Most Robust Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 8. Most Robust Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 8. Most Robust Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 8. Most Robust Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 8. Most Robust Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 8. Most Robust Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 8. Most Robust Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 8. Most Robust Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 8. Most Robust Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 8. Most Robust Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 8. Most Robust Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 8. Most Robust Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 8. Most Robust Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 8. Most Robust Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 8. Most Robust Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

301 CHAPTER 8 8. MOST ROBUST CORRELATION EQUATIONS As discussed in the previous chapter, two statistical approaches, deterministic frequentistsâ regression and the probabilistic (Bayesian) inference, were carried out to identify the best correlation equations between the erodibility parameters and the geotechnical properties. In this chapter, the selected models are repeated and tabulated in order to make it easier for practitioners to use them. It should be noted that the erodibility parameters are predicted using different equations depending on what erosion test data (EFA, JET, or HET) are used to develop them. Therefore, the knowledge on the strength points/limitations of each test is a vital step prior to choosing an equation. Before presenting the most robust correlation equations, a summary of the advantages and disadvantages of each test is put together in Section 8.1 by the authors. Section 8.1 helps the readers identify and understand the differences between the erosion tests. The engineer is free to select the best equation according to his/her objective and considering the differences between the equations. Section 8.2 presents the selected correlation equations generated using the deterministic frequentistsâ regression approach in a tabulated format. 8.1. Differences between the EFA, the JET, and the HET In this section, a list of advantages and disadvantages associated with the EFA, the JET, and the HET, as well as the applications of each erosion test are presented. It should be noted that the information provided here are the recommendations made by the authors based on decades of experience in erosion testing, and according to the various challenges confronted in conducting hundreds of erosion tests during this project. The proposed equations in Section 8.2 are developed based on the data obtained in different erosion tests; therefore, a good knowledge of each test is very useful in order to choose the best equation. Chapter 2 discussed all major differences between each erosion test. Table 90 shows a list of advantages, disadvantages, and applications for the EFA, the JET, and the HET. The content of this table should be studied carefully before using the proposed equations in Section 8.2.

302 Table 90. Comparison of the EFA, the JET, and the HET Erosion Test Advantages Drawbacks Applications EFA 1) Minimize the sample disturbance effect, as it takes the un-extruded Shelby Tube sample directly from the field. 2) Can be used on natural samples as well as man-made samples 3) Gives all five erodibility parameters (i.e. , , , , and ). Can give the erosion function directly. 4) Can monitor the erosion rate in real-time rather than interpolating or extrapolating using indirect equations. 5) EFA test results are directly used as input to the TAMU- SCOUR method for bridge scour depth predictions (Chapter 6 of HEC-18). 6) EFA can test the erodibility of the soil at any depth as long as a sample can be recovered. 7) Gives the erosion function which is a fundamental measure of erodibility at the element level. 8) Can be used to test very soft to hard soils. Very broad applications. The velocity range is from 0.2 m/s up to 6 m/s. 1) Shear stress is indirectly measured from velocity using Moody charts which might not be accurate. Also, the average flow velocity is used in the calculation. 2) In some cases, obtaining samples is difficult and costly. The test needs to be done on the sample before the sample is affected by long periods of storage. 3) Particles larger than about 40 mm in size cannot be tested with confidence as the diameter of the sampling tube is 75 mm. 4) The EFA device is fairly expensive (around $50k in 2018). 1) Bridge scour 2) Meander migration 3) Levee overtopping 4) Soil improvement 5) Internal erosion of dams JET 1) Can be run both in the field and in the lab. 2) The latest version of the JET, the mini-JET, is simple, quick, and inexpensive compared to other types of erosion test. 1) Particles larger than 30 mm in size cannot be tested with confidence because of the small size of the sample. 2) Coarse grained soils (i.e. non- cohesive sand and gravel) tend to fall back into the open hole during 1) Agriculture erosion 2) Levees

303 3) Can be performed on any surface vertical, horizontal, and inclined. 4) Very good as an index erodibility test. the jet erosion process thereby making the readings dubious. 3) Very small-scale test application. 4) Typically used for man-made samples. Natural are more difficult to test 5) The flow within the eroded hole and at the soil boundary is complex and difficult to analyze. 6) Only gives three of the erodibility parameters ( , , and ) out of the five possible parameters. 7) The elements of erosion are inferred rather than measured directly. 8) There are multiple interpretation techniques to predict the critical shear stress which give significantly different results. HET 1) Direct similitude with piping erosion in earth dams. 2) Can apply to a wide range of pressure heads and therefore wide range of hydraulic shear stress at the soil-water interface. 1) The sample needs to be cohesive and strong enough to stand under its own weight. Therefore, the test cannot be run on loose cohesion-less soils or soft cohesive soils. 2) Very difficult to run on intact samples in Shelby tubes from the field. Only good for remolded re- compacted samples in the lab. 3) Difficult and time-consuming preparation of the test. 4) No direct monitoring of the erosion process. The erosion rate needs to be inferred and extrapolated. 5) The hydraulic shear stress is inferred, and not directly measured. 6) The data reduction process is quite subjective. 7) Only gives three of the erodibility parameters ( , , and ) out of the five possible parameters. 8) The flow within the eroded hole and at the soil boundary is complex 1) Internal erosion of earth dams 2) Suffusion 3) Levee breach 4) Soil improvement

304 8.2. Deterministic (Frequentistsâ Regression) Analysis The best models as discussed in the previous chapter are selected for each erodibility parameter and for each erosion test. The results are shown in the following tables. The first column from left shows the erosion test data used to develop the equations. The second column from left indicates the accepted mean particle size (D50) range for the proposed equations. The third column from left shows the model, and the âprobability of under/over-predicting (POU/POO)â plots for the model. For further information on the use of these plots, please refer to the Section 7.3.3.3 of this report. Finally, the last column from left presents some remarks on the proposed equation. It also presents the correction factor to reach a 90% confidence in under/over-predicting of the measured erodibility parameter. It should be noted that the units used for each parameter are indicated in Appendix 4 as well as in Table 47. Table 91 shows the proposed correlation equations to predict the critical shear stress ( ) based on the EFA test. This table presents the recommended correlation equations to predict critical shear stress in different D50 ranges. It should be noted that alongside with each proposed equation, these tables often give two plots showing the POU (or POO, where applicable) vs. correction factor, as well as the âpredictedâ vs. âmeasuredâ. Such plots provide great insight on using each equation. Also, a column containing some remarks is provided on the right side of each equation. This column includes the values of R2 and the cross-validation score (named as C.V in the tables). Table 92 and Table 93 also show the proposed correlation equations to predict the critical shear stress ( ) based on the JET and HET, respectively.

305 Table 91. Proposed equations for critical shear stress ( ) based on the EFA Test data EFA 50 0.3 Refer to Figure 147 0.074 50 0.3 . . . . Remarks R2 = 0.93 C.V Score = 0.99 1- Refer to the Group 77 in Table 50 for further information on the statistical significance of the proposed equation. 2- The âPOU vs. Correction Factorâ plot is based on the data used to develop the proposed equation. 3- In order to reach a 90% confidence that the predicted is smaller than the actual , the predicted value should be multiplied by 0.82. (with 0.3 Pa offset). 50 0.074 . . . . . . Remarks R2 = 0.94 C.V Score = 0.66 1- Refer to the Group 124 in Table 48 for further information on the statistical significance of the proposed equation. 2- The âPOU vs. Correction Factorâ plot is based on the data used to develop the proposed equation.

306 3- There is almost 90% chance that the predicted value is smaller than the actual (with 0.5 Pa offset).

307 Table 92. Proposed equations for critical shear stress ( ) based on the JET data JET 50 0.3 . . . . . . Remarks R2 = 0.50 C.V Score = 0.10 1- Refer to the Group 113 in Table 51 for further information on the statistical significance of the proposed equation. 2- The âPOU vs. Correction Factorâ plot is based on the data used to develop the proposed equation. 3- In order to reach a 90% confidence that the predicted Ïc is smaller than the actual Ïc, the predicted value should be multiplied by 0.6 (with 1 Pa offset).

308 Table 93. Proposed equations for critical shear stress ( ) based on the HET data HET 50 0.3 . . . . Remarks R2 = 0.64 C.V Score = 0.43 1- Refer to the Group 19 in Table 54 for further information on the statistical significance of the proposed equation. 2- The âPOU vs. Correction Factorâ plot is based on the data used to develop the proposed equation. 3- In order to reach a 90% confidence that the predicted Ïc is smaller than the actual Ïc, the predicted value should be multiplied by 0.6 (with 1 Pa offset).

309 Table 94 shows the proposed correlation equations to predict the critical velocity ( ) based on the EFA test. It is noteworthy that since the HET and the JET do not report the critical velocity as an output erodibility parameter, the critical velocity equations are proposed only based on the EFA Test. Table 94. Proposed equations for critical velocity ( ) based on the EFA Test data EFA 50 0.3 . . Refer to Figure 146 0.074 50 . . . . Remarks R2 = 0.88 C.V Score = 0.72 1- Refer to the Group 27 in Table 58 for further information on the statistical significance of the proposed equation. 2- The âPOU vs. Correction Factorâ plot is based on the data used to develop the proposed equation. 3- In order to reach a 90% confidence that the predicted is smaller than the actual , the predicted value should be multiplied by 0.7 (with 0.1 m/s offset). 50 0.074 . . . . . Remarks R2 = 0.80 C.V Score = 0.80 1- Refer to the Group 117 in Table 56 for further information on the

310 statistical significance of the proposed equation. 2- The âPOU vs. Correction Factorâ plot is based on the data used to develop the proposed equation. 3- In order to reach a 90% confidence that the predicted is smaller than the actual , the predicted value should be multiplied by 0.8 (with 0.2 m/s offset).

311 Table 95 shows the proposed correlation equations to predict the initial linear slope of shear stress-erosion rate plot ( ) based on the EFA test. Table 96 and Table 97 also show the proposed correlation equations to predict based on the JET and HET, respectively. It must be noted that the proposed equation for the D50 < 0.074 mm is based on the Group 134 in HET/Global dataset (See Table 59). The reason for this selection is that the data used in Group 134 were all related to the soils with mean particle size smaller than 0.074 mm. In other words, the Group 134 in both HET/Global and HET/Fine would lead to the same selected equation. Table 95. Proposed equations for shear stress slope ( ) based on the EFA Test data EFA 50 0.074 mm . . . . Remarks R2 = 0.91 C.V Score = 0.64 1- Refer to the Group 77 in Table 61 for further information on the statistical significance of the proposed equation. 2- The âPOO vs. Correction Factorâ plot is based on the data used to develop the proposed equation. 3- In order to reach a 80% confidence that the predicted is greater than the actual , the predicted value should be multiplied by 2.5 (with 15 mm/hr-Pa offset). 50 0.074 . . . . . Remarks R2 = 0.90 C.V Score = 0.53 1- Refer to the Group 134 in Table 59 for further information on the

312 statistical significance of the proposed equation. 2- The âPOO vs. Correction Factorâ plot is based on the data used to develop the proposed equation. 3- In order to reach a 87% confidence that the predicted is greater than the actual , the predicted value should be multiplied by 2 (with 4 mm/hr-Pa offset).

313 Table 96. Proposed equations for shear stress slope ( ) based on the JET data JET 50 0.074 mm . . . Remarks R2 = 0.90 C.V Score = 0.67 1- Refer to the Group 5 in Table 65 for further information on the statistical significance of the proposed equation. 2- The âPOO vs. Correction Factorâ plot is based on the data used to develop the proposed equation. 3- In order to reach a 90% confidence that the predicted is greater than the actual , the predicted value should be multiplied by 1.4 (with 5 mm/hr-Pa offset). 50 0.074 . . . . Remarks R2 = 0.93 C.V Score = 0.23 1- Refer to the Group 15 in Table 63 for further information on the statistical significance of the proposed equation. 2- The âPOO vs. Correction Factorâ plot is based on the data used to develop the proposed equation. 3- In order to reach a 88% confidence that the predicted EÏ is greater than the actual EÏ, the predicted

314 value should be multiplied by 2 (with 6 mm/hr-Pa offset).

315 Table 97. Proposed equations for shear stress slope ( ) based on the HET data HET 0.3 50 0.074 mm Use this equation when PF > 30% . . . . . Remarks R2 = 0.86 C.V Score = 0.55 1- Refer to the Group 40 in Table 66 for further information on the statistical significance of the proposed equation. 2- The âPOO vs. Correction Factorâ plot is based on the data used to develop the proposed equation. 3- In order to reach a 80% confidence that the predicted is greater than the actual , the predicted value should be multiplied by 2 (with 10 mm/hr-Pa offset). 50 0.074 . . . . . Remarks R2 = 0.81 C.V Score = 0.531 1- Refer to the Group 108 in Table 68 for further information on the statistical significance of the proposed equation. 2- The âPOO vs. Correction Factorâ plot is based on the data used to develop the proposed equation. 3- In order to reach a 90% confidence that the predicted is greater than the actual , the predicted

316 value should be multiplied by 1.45 (with 0 mm/hr-Pa offset).

317 Table 98 shows the proposed correlation equations to predict the initial linear slope of velocity- erosion rate plot ( ) based on the EFA test. It is noteworthy that since the HET and the JET do not report as an output erodibility parameter, the equations are proposed only based on the EFA Test. Table 98. Proposed equations for velocity slope ( ) based on the EFA Test data EFA 50 0.074 mm . . . . . Remarks R2 = 0.86 C.V Score = 0.64 1- Refer to the Group 86 in Table 71 for further information on the statistical significance of the proposed equation. 2- The âPOO vs. Correction Factorâ plot is based on the data used to develop the proposed equation. 3- In order to reach a 80% confidence that the predicted is greater than the actual , the predicted value should be multiplied by 5 (with 10 mm-s/m-hr offset). 50 0.074 . . . . . . Remarks R2 = 0.79 C.V Score = 0.52 1- Refer to the Group 126 in Table 69 for further information on the statistical significance of the proposed equation. 2- The âPOO vs. Correction Factorâ plot is based on the

318 data used to develop the proposed equation. 3- In order to reach a 80% confidence that the predicted is greater than the actual , the predicted value should be multiplied by 2 (with 10 mm-s/m-hr offset).

319 Table 99 shows the proposed correlation equations to predict the erosion category (EC) based on the EFA test. Table 100 and Table 101 also show the proposed correlation equations to predict EC based on the JET and HET, respectively. Table 99. Proposed equations for erosion category (EC) based on the EFA Test data EFA 0.3 50 0.074 mm . . . . . Remarks R2 = 0.92 C.V Score = 0.80 1- Refer to the Group 91 in Table 75 for further information on the statistical significance of the proposed equation. 2- The âPOU vs. Correction Factorâ plot is based on the data used to develop the proposed equation. 3- In order to reach a 90% confidence that the predicted EC is smaller than the actual EC, the predicted value should be multiplied by 0.84. 50 0.074 . . . . . Remarks R2 = 0.55 C.V Score = 0.53 1- Refer to the Group 132 in Table 73 for further information on the statistical significance of the proposed equation. 2- The âPOU vs. Correction Factorâ plot is based on the

320 data used to develop the proposed equation. 3- In order to reach a 90% confidence that the predicted EC is smaller than the actual EC, the predicted value should be multiplied by 0.75.

321 Table 100. Proposed equations for erosion category (EC) based on the JET data JET 50 0.3 . . . . Remarks R2 = 0.70 C.V Score = 0.58 1- Refer to the Group 88 in Table 76 for further information on the statistical significance of the proposed equation. 2- The âPOU vs. Correction Factorâ plot is based on the data used to develop the proposed equation. 3- In order to reach a 90% confidence that the predicted EC is smaller than the actual EC, the predicted value should be multiplied by 0.85.

322 Table 101. Proposed equations for erosion category (EC) based on the HET data HET 0.3 50 0.074 mm . . . . Remarks R2 = 0.77 C.V Score = 0.78 1- Refer to the Group 48 in Table 81 for further information on the statistical significance of the proposed equation. 2- The âPOU vs. Correction Factorâ plot is based on the data used to develop the proposed equation. 3- In order to reach a 90% confidence that the predicted EC is smaller than the actual EC, the predicted value should be multiplied by 0.85. 50 0.074 . . . . Remarks R2 = 0.70 C.V Score = 0.54 1- Refer to the Group 12 in Table 79 for further information on the statistical significance of the proposed equation. 2- The âPOU vs. Correction Factorâ plot is based on the data used to develop the proposed equation. 3- In order to reach a 100% confidence that the predicted EC is smaller than the actual EC, the

323 predicted value should be multiplied by 0.95.