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Entrainment/Competence Calculation
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Entrainment
Entrainment/Competence Calculation 
The great importance of critical shear stress calculations for sediment transport models was discussed earlier. The conditions of incipient motion for various flow depths are not only important for sediment transport formulae, but also for sediment competence calculations. Modifications to previous approaches were tested to provide observers with field methods for computation. Bar core samples were obtained on point bars on the lower 1/3 planview position through a bend and 0.5 elevation of the bankfull stage (Rosgen, 1996, chap.7, page 8, and 2001b). The core sample is sieved in the field, with the largest size particle (D_{i}) recorded and the D_{50} of the sample. The size distribution of the bar sample is similar to the gradation of particles associated with bedload transport at the bankfull stage (Rosgen 1996, page 79).
The work by Andrews on Sagehen Creek, CA, established an equation for critical dimensionless shear stress (Equation II8). The largest particle in the bar sample, (D_{i}) was substituted for the (d_{i}) and the median particle diameter of the bed at the riffle (D_{50}) is substituted for the subpavement ( ). Since subpavement samples are difficult to obtain, the bar and bed material sampling on the riffle was used instead. The relations using bar samples instead of subpavement and the use of the largest particle on the bar fir the (D_{i}) agree quite well with the original relation presented by Andrews and Erman (1986) (Figure 64).
Figure 64. Relative protrusion of bed surface.
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Once the bankfull critical dimensionless shear stress is determined, the competence calculation is used to assess the depth and/or slope necessary to move the largest particle made available on point bar D_{i} using Equation II9. An example of the entrainment and competence calculations is shown in Table 5.
Table 5. Sediment competence calculations for Upper Wolf (C4 stream type) and Lower Wolf Creek (D4 stream type). 



Upper Wolf Creek
(C4 Stream Type)
(eq. II8)
Where:  =  critical dimensionless shear stress 
D_{i} =  the largest particle on bar  
D_{50} =  the medium diameter of bed material on riffle 
To calculate depth required to move largest size of bed load using Shields relation:
Where:
Lower Wolf Creek
(D4 Stream Type)
(eq. II8)
(eq. II9)
The interpretations of this analysis indicate that upper Wolf Creek has sufficient shear stress to move the 77 mm particle (.097 or 1.0 foot of depth required; actual bankfull depth is 1.85 ft.) The available bankfull shear stress could move the D90 of the bed material. If the shear stress is greater than that required to move the D_{100}, then excess bed scour would be anticipated. Immediately downstream on the lower Wolf Creek (D4 stream type), the required depth (for a flatter slope) to move the 38 mm particle in the bar is 2.9 feet, whereas the actual mean bankfull depth of the D4 stream type is 0.81 feet. This indicates that the stream would not have the competence to move the larger sizes. As a result, the prediction would be aggradation or excess deposition, and field observations confirm the prediction. The increase in sediment supply in lower Wolf is also associated with an increase in sand sizes from streambank erosion and a reduction in bankfull shear stress and stream power. If an increase in sediment transport occurs due to greater sediment supply, it is usually related to the sand sizes rather than larger particles because it requires less stream power to transport these smaller sizes. These calculations are adapted not only for stability examinations, but for natural channel design to ensure the new channel has the competence (depth and slope combination) to move the sediment size made available.
West Fork San Juan River
Entrainment calculations using Equation II7, were made on the West Fork San Juan River, a much larger river (1150 cfs) than Wolf Creek (Table 6).
Table 6. Entrainment computation for Lower West Fork of the San Juan River  


Step 1: Calculate bankfull critical dimensionless shear stress
(eq.II7)
Critical Dimensionless Shear Stress  
d_{i}  Bed material D_{50} (from riffle pebble count)  = 70mm 
d_{50}  Subsurface D_{50} or Bar Sample D_{50}  = 20mm 
= 0.028
Step 2: Calculate the mean bankfull depth required to move the largest particle from bar sample
transformed to
d:  Mean bankfull depth at riffle (feet)  
:  Critical Dimensionless Shear Stress  = 0.028 
S:  Bankfull water surface slope  = 0.007 ft/ft 
Submerged specific weight of sediment  = 1.65  
D_{i}:  Largest particle in Bar Sample 130mm*  = 0.43 feet 
Comparison of the particle protrusion ratio of the using Equation II7 with measured bedload and scour chain data indicated that the equation predicts within the ratio values of 37. The range of appropriate ratio values using Equation II8, is 1.33.0. The bar sample D_{50} is substituted for subpavement , in Equation II7. The prediction of the largest size on the bar (D_{i}) of 130 mm was very close to the largest particle of 140 mm measured by the 6 inch HellySmith bedload sampler. The predicted depth of 2.8 feet, necessary to move a 140 mm particle, using the relations shown in Table 6 approximates the measured bankfull depth of 2.6 feet.
Bedload Transport
Sediment load needs to be considered in addition to sediment competence or size of sediment moved. Bedload formulas (e.g., Bagnold 1980), are designed to predict sediment load, generally by size fraction for the energy available for each flow stage on an annual basis. The Bagnold equation was selected to test against the measured data since it uses unit stream power () (Equation II11), or shear stress times velocity. Since the discharge between upper and lower Wolf Creek is similar, the unit stream power would differ at the two reaches due to reductions of depth, slope and velocity in the braided reach. A plot showing the relation for discharge and unit stream power for the two river types is presented in Figure 65 (PDF, 18 kb, 1 p.).
For the same discharge between the two rivers, the unit stream power is an order of magnitude lower for the braided (D4) reach at the bankfull stage. The two variables in the Bagnold relation that are generally not changed are the value of 0.04, and the use of D_{50} of bed material for transport size. In this example, the D_{i} of the bar sample was used in place of D_{50} for the calculation of total bedload transport, and the dimensionless shear stress value (), computed as in Table 6, was used to substitute for the Shields threshold criterion () value in the Equation II13. The results of the application of the Bagnold equation to Upper Wolf Creek is shown in Figure 66 (PDF, 27 kb, 1 p.), where predicted values and actual values are in close agreement.
These relations would indicate that the reach is supplylimited. The total bedload transport for Lower Wolf Creek shows an approximate 50 percent reduction in unit transport rate as measured, compared to the upstream reach. The Bagnold relation slightly overpredicted the total bedload transport of the lower reach compared to measured values, but it showed a reduction in transport rate as compared to the upper reach (Figure 67 (PDF, 22 kb, 1 p.)).
The measured values indicate the amount of material in deposition compared to Upper Wolf Creek (C4) stream type, Figure 66. Measured bed material size at this crosssection indicates a shift to finer bed material and excess sediment deposition. The stream type is associated with a braided form (D4) and is contributing to excess sediment storage in the reach.
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