**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.

280 As discussed in Chapter 7, two statistical approachesâdeterministic frequentist regression and probabilistic (Bayesian) inferenceâwere carried out to identify the best correlation equa- tions between erodibility parameters and 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 which erosion test dataâerosion function apparatus (EFA), jet erosion test (JET), or hole erosion test (HET)âwere used to develop them. Therefore, knowledge of the strength points and limitations of each test is a vital step prior to choosing an equation. Before the most robust correlation equations are presented, the advantages and disadvantages of each test are presented in Section 8.1. Section 8.1 helps readers identify and understand the differences between the erosion tests. The engineer is free to select the best equation according to his or her objective and considering the differences between the equations. Section 8.2 presents the selected correlation equations generated using the deterministic frequentist regression approach in a tabulated format. 8.1 Differences Between the EFA, JET, and HET Table 90 presents a list of advantages and disadvantages associated with the EFA, JET, and HET as well as the applications of each erosion test. The information provided here constitutes recommendations made by the authors on the basis of decades of experience in erosion testing and according to the various challenges confronted in conducting hundreds of erosion tests during this project. The development of the equations proposed in Section 8.2 was based on the data obtained in different erosion tests; therefore, a good knowledge of each test is very useful in choosing the best equation. The content of Table 90 should be studied carefully before the equations proposed in Section 8.2 are used. Chapter 2 discusses all major differences between erosion tests. 8.2 Deterministic Analysis (Frequentist Regression) The best models discussed in Chapter 7 were selected for each erodibility parameter and for each erosion test. The results are shown in Table 91 through Table 101. In each table, the first column gives the accepted range of mean particle size (D50) for the proposed equations. The second column shows the model, a plot of the probability of under- or overpredicting (POU/ POO) for the model, and a plot of predicted versus measured values. Such plots provide great insight into using each equation. For further information on the use of these plots, see Chapter 7, Section 7.3.3.3. The third column gives the value of R2 and the cross-validation (C.V.) score, C H A P T E R 8 Most Robust Correlation Equations

Advantages Drawbacks Applications EFA 1. Minimizes the sample disturbance effect, as it takes the unextruded 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., , Ï , , , and EC). Can give the erosion function directly. 4. Can monitor the erosion rate in real time rather than by 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 (Arneson et al. 2012, Chapter 6). 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 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 cannot be tested with confidence, as the diameter of the sampling tube is 75 mm. 4. The EFA device is fairly expensive (around $50,000 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 with other types of erosion tests. 3. Can be performed on any surfaceâ vertical, horizontal, or inclined. 4. Very good as an index erodibility test. 1. Particles larger than 30 mm cannot be tested with confidence because of the small size of the sample. 2. Coarse-grained soils (i.e., noncohesive sand and gravel) tend to fall back into the open hole during the jet erosion process, thereby making the readings dubious. 3. Very small-scale test application. 4. Typically used for man-made samples. Natural samples are more difficult to test. 5. The flow within the eroded hole and at the soil boundary is complex and difficult to analyze. 6. Gives only three of the five possible erodibility parameters (Ï , , and EC). 7. The elements of erosion are inferred rather than measured directly. 8. There are multiple interpretation techniques for predicting the critical shear stress, and these give significantly different results. 1. Agriculture erosion. 2. Levees. HET 1. Direct similitude with piping erosion in earth dams. 2. Can apply to a wide range of pressure heads and therefore a wide range of hydraulic shear stress at the soilâwater 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 cohesionless soils or soft cohesive soils. 2. Very difficult to run on intact samples in Shelby tubes from the field. Only good for remolded, recompacted samples in the lab. 3. Preparation of the test is difficult and time consuming. 4. No direct monitoring of the erosion process. The erosion rate needs to be inferred and extrapolated. 5. The hydraulic shear stress is inferred rather than directly measured. 6. The data reduction process is quite subjective. 7. Gives only three of the five possible erodibility parameters (Ï , , and EC). 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. Table 90. Comparison of EFA, JET, and HET.

282 Relationship Between Erodibility and Properties of Soils presents remarks on the proposed equation, and gives the correction factor needed to reach 90% confidence in under- or overpredicting the measured erodibility parameter. The units used for each parameter are indicated in Table 47 (Chapter 7) and also in Appendix 4. Table 91 shows the proposed correlation equations for predicting the critical shear stress (tc) based on the EFA test. This table presents the recommended correlation equations for predicting critical shear stress in different D50 ranges. Table 92 and Table 93 show the pro- posed correlation equations for predicting the critical shear stress (tc) based on the JET and HET, respectively. Table 94 shows the proposed correlation equations for predicting the critical velocity (vc) based on the EFA test. Because the HET and the JET do not report critical velocity as an output erodibility parameter, the critical velocity equations are proposed based on the EFA test alone. Table 95 shows the proposed correlation equations for predicting the initial linear slope of the shear stressâerosion rate plot (Et) based on the EFA test. Table 96 and Table 97 show the proposed correlation equations for predicting Et based on the JET and HET, respectively. It must be noted that the proposed equation for D50 < 0.074 mm is based on Group 108 in the HET/Global data set (see Table 68 in Chapter 7). The reason for this selection is that the Refer to Figure 147 Remarks R2 = 0.93 C.V. score = 0.99 1. Refer to Group 77 in Table 50 for further information on the statistical significance of the proposed equation. 2. The POU versus correction factor plot is based on the data used to develop the proposed equation. 3. To reach 90% confidence that the predicted Ïc is less than the actual Ïc , the predicted value should be multiplied by 0.82. (with 0.3 Pa offset). Table 91. Proposed equations for critical shear stress (sc) based on the EFA test data.

Most Robust Correlation Equations 283 Remarks R2 = 0.94 C.V. score = 0.66 1. Refer to Group 124 in Table 48 for further information on the statistical significance of the proposed equation. 2. The POU versus correction factor plot is based on the data used to develop the proposed equation. 3. There is almost 90% chance that the predicted value is less than the actual (with 0.5 Pa offset). Ïc Refer to Figure 147 Table 91. (Continued). data used in Group 108 were all related to soils with mean particle size less than 0.074 mm. That is, Group 108 in both HET/Global and HET/Fine would lead to the same selected equation. Table 98 shows the proposed correlation equations for predicting the initial linear slope of the velocityâerosion rate plot (Ev) based on the EFA test. Because the HET and the JET do not report Ev as an output erodibility parameter, the proposed equations are based on the EFA test alone. Table 99 shows the proposed correlation equations for predicting the erosion category (EC) based on the EFA test. Table 100 and Table 101 show the proposed correlation equations for predicting EC based on the JET and HET, respectively.

284 Relationship Between Erodibility and Properties of Soils Remarks R2 = 0.50 C.V. score = 0.10 1. Refer to Group 113 in Table 51 for further information on the statistical significance of the proposed equation. 2. The POU versus correction factor plot is based on the data used to develop the proposed equation. 3. To reach 90% confidence that the predicted Ïc is less than the actual Ïc, the predicted value should be multiplied by 0.6 (with 1 Pa offset). Table 92. Proposed equation for critical shear stress (sc) based on the JET data.

Most Robust Correlation Equations 285 Remarks R2 = 0.64 C.V. score = 0.43 1. Refer to Group 19 in Table 54 for further information on the statistical significance of the proposed equation. 2. The POU versus correction factor plot is based on the data used to develop the proposed equation. 3. To reach 90% confidence that the predicted Ïc is less than the actual Ïc, the predicted value should be multiplied by 0.6 (with 1 Pa offset). Table 93. Proposed equation for critical shear stress (sc) based on the HET data.

286 Relationship Between Erodibility and Properties of Soils Refer to Figure 146 Remarks R2 = 0.88 C.V. score = 0.72 1. Refer to Group 27 in Table 58 for further information on the statistical significance of the proposed equation. 2. The POU versus correction factor plot is based on the data used to develop the proposed equation. 3. To reach 90% confidence that the predicted is less than the actual , the predicted value should be multiplied by 0.7 (with 0.1 m/s offset). Table 94. Proposed equations for critical velocity (vc) based on the EFA test data.

Most Robust Correlation Equations 287 Refer to Figure 146 Remarks R2 = 0.80 C.V. score = 0.80 1. Refer to Group 117 in Table 56 for further information on the statistical significance of the proposed equation. 2. The POU versus correction factor plot is based on the data used to develop the proposed equation. 3. To reach 90% confidence that the predicted is less than the actual , the predicted value should be multiplied by 0.8 (with 0.2 m/s offset). Table 94. (Continued).

288 Relationship Between Erodibility and Properties of Soils Remarks R2 = 0.91 C.V. score = 0.64 1. Refer to Group 77 in Table 61 for further information on the statistical significance of the proposed equation. 2. The POO versus correction factor plot is based on the data used to develop the proposed equation. 3. To reach 80% confidence that the predicted is greater than the actual , the predicted value should be multiplied by 2.5 (with 15 mm/h-Pa offset). Table 95. Proposed equations for shear stress slope (Es) based on the EFA test data.

Most Robust Correlation Equations 289 Remarks R2 = 0.90 C.V. score = 0.53 1. Refer to Group 134 in Table 59 for further information on the statistical significance of the proposed equation. 2. The POO versus correction factor plot is based on the data used to develop the proposed equation. 3. To reach 87% confidence that the predicted is greater than the actual , the predicted value should be multiplied by 2 (with 4 mm/h-Pa offset). Table 95. (Continued).

290 Relationship Between Erodibility and Properties of Soils Remarks R2 = 0.90 C.V. score = 0.67 1. Refer to Group 5 in Table 65 for further information on the statistical significance of the proposed equation. 2. The POO versus correction factor plot is based on the data used to develop the proposed equation. 3. To reach 90% confidence that the predicted is greater than the actual , the predicted value should be multiplied by 1.4 (with 5 mm/h-Pa offset). Table 96. Proposed equations for shear stress slope (Es) based on the JET data.

Most Robust Correlation Equations 291 Remarks R2 = 0.93 C.V. score = 0.23 1. Refer to Group 15 in Table 63 for further information on the statistical significance of the proposed equation. 2. The POO versus correction factor plot is based on the data used to develop the proposed equation. 3. To reach 88% confidence that the predicted EÏ is greater than the actual EÏ, the predicted value should be multiplied by 2 (with 6 mm/h-Pa offset). Table 96. (Continued).

292 Relationship Between Erodibility and Properties of Soils Use this equation when PF > 30%. Remarks R2 = 0.86 C.V. score = 0.55 1. Refer to Group 40 in Table 66 for further information on the statistical significance of the proposed equation. 2. The POO versus correction factor plot is based on the data used to develop the proposed equation. 3. To reach 80% confidence that the predicted is greater than the actual , the predicted value should be multiplied by 2 (with 10 mm/h-Pa offset). Table 97. Proposed equations for shear stress slope (Es) based on the HET data.

Most Robust Correlation Equations 293 Remarks R2 = 0.81 C.V. score = 0.531 1. Refer to Group 108 in Table 68 for further information on the statistical significance of the proposed equation. 2. The POO versus correction factor plot is based on the data used to develop the proposed equation. 3. To reach 90% confidence that the predicted is greater than the actual , the predicted value should be multiplied by 1.45 (with 0 mm/h-Pa offset). Table 97. (Continued).

294 Relationship Between Erodibility and Properties of Soils Remarks R2 = 0.86 C.V. score = 0.64 1. Refer to Group 86 in Table 71 for further information on the statistical significance of the proposed equation. 2. The POO versus correction factor plot is based on the data used to develop the proposed equation. 3. To reach 80% confidence that the predicted is greater than the actual , the predicted value should be multiplied by 5 (with 10 mm-s/m-h offset). Table 98. Proposed equations for velocity slope (Ev) based on the EFA test data.

Most Robust Correlation Equations 295 Remarks R2 = 0.79 C.V. score = 0.52 1. Refer to Group 126 in Table 69 for further information on the statistical significance of the proposed equation. 2. The POO versus correction factor plot is based on the data used to develop the proposed equation. 3. To reach 80% confidence that the predicted is greater than the actual , the predicted value should be multiplied by 2 (with 10 mm-s/m-h offset). Table 98. (Continued).

296 Relationship Between Erodibility and Properties of Soils Remarks R2 = 0.92 C.V. score = 0.80 1. Refer to Group 91 in Table 75 for further information on the statistical significance of the proposed equation. 2. The POU versus correction factor plot is based on the data used to develop the proposed equation. 3. To reach 90% confidence that the predicted EC is less than the actual EC, the predicted value should be multiplied by 0.84. Table 99. Proposed equations for erosion category (EC) based on the EFA test data.

Most Robust Correlation Equations 297 Remarks R2 = 0.55 C.V. score = 0.53 1. Refer to Group 132 in Table 73 for further information on the statistical significance of the proposed equation. 2. The POU versus correction factor plot is based on the data used to develop the proposed equation. 3. To reach 90% confidence that the predicted EC is less than the actual EC, the predicted value should be multiplied by 0.75. Table 99. (Continued).

298 Relationship Between Erodibility and Properties of Soils < . = â . Ã + . Ã â . Ã + . Remarks R2 = 0.70 C.V. score = 0.58 1. Refer to Group 88 in Table 76 for further information on the statistical significance of the proposed equation. 2. The POU versus correction factor plot is based on the data used to develop the proposed equation. 3. To reach 90% confidence that the predicted EC is less than the actual EC, the predicted value should be multiplied by 0.85. Table 100. Proposed equation for erosion category (EC) based on the JET data.

Most Robust Correlation Equations 299 Remarks R2 = 0.77 C.V. score = 0.78 1. Refer to Group 48 in Table 81 for further information on the statistical significance of the proposed equation. 2. The POU versus correction factor plot is based on the data used to develop the proposed equation. 3. To reach 90% confidence that the predicted EC is less than the actual EC, the predicted value should be multiplied by 0.85. Table 101. Proposed equations for erosion category (EC) based on the HET data. (continued on next page)

300 Relationship Between Erodibility and Properties of Soils Remarks R2 = 0.70 C.V. score = 0.54 1. Refer to Group 12 in Table 79 for further information on the statistical significance of the proposed equation. 2. The POU versus correction factor plot is based on the data used to develop the proposed equation. 3. To reach 100% confidence that the predicted EC is less than the actual EC, the predicted value should be multiplied by 0.95. Table 101. (Continued).