Click to go down to [Part A]  [Part B]  [Part C]

Correction:  In Part C, Di and D50 should be in meters, not millimeters.

This exercise is due in class on Thursday, Feb. 18. 

Exercise 3:  Sediment Transport

In Part A of this exercise you will analyze a sample of bed sediment data. Part B focuses on calculating shear stress to determine whether entrainment of bed material will occur. Part C focuses on variations in suspended sediment discharge. Be sure to include all worked spreadsheets when you turn in your exercise.

Part A. Bed Sediment Size Analysis

A natural gravel-bedded stream usually is not dominated by a single size of gravels, but has on its bed a heterogeneous population of gravels of different sizes. To characterize the bed material of a stream cross-section, the bed material is sampled by wading across the stream, picking gravels from the bed using some random or regular sampling scheme, and measuring the size (diameter) of each gravel. To obtain a representative sample, typically 100 gravels are measured. This is called doing a gravel count.

Gravel diameter is measured (usually in mm) in the field using a ruler, hand tape or calipers. Since natural gravels are not spheres, the diameter will be different depending on where on the gravel you measure it. Diameter measured along the longest axis of the gravel is called the A-diameter (measured on the A-axis). The B-axis is the longest axis that occurs at right angles to the A-axis. The C-axis is the axis that occurs at right angles to both the A-axis and B-axis. In doing a gravel count, the diameter of the B-axis is usually measured, since this is considered the single most representative measure of diameter. The gravel count data can be recorded in two different ways: 1) write down the measured diameter of each gravel, or 2) start with a set of pre-determined size classes and record the number of gravels that fall within each size class. In this section you will work with a data set recorded in the second way. Size classes were defined based on the phi size scale used in sedimentology (a log base 2 transformation of diameter in millimeters). The phi (F ) scale was designed to allow simpler analysis of data in the pre-computer days. On the phi scale, bigger particles have more negative numbers; clay particles are ³ 8F , the boundary between sand and gravel is –1F (2 mm), and gravels have phi values <-1.

Columns A and B define the size classes used. Column C, to be used in plotting the data, contains the upper limit of each size class. Sediment size data are best analyzed according to the total weight of gravel, not the number of gravels, within each size class. Column D contains the average weight of a gravel (a typical rock density is assumed) for each size class. These data must be corrected for a type of sampling bias. When sampling gravels on river beds, larger gravels are favored. Because they are larger, they occupy a greater surface area on the bed than smaller gravels, and they are therefore more likely to be picked up in sampling than smaller, but possibly more numerous, gravels. This sampling bias must be corrected. Column E contains a correction factor that is based on the inverse of the square of the mean diameter of each size class (Leopold, 1970). Columns B through E are standard data that would be used in any gravel analysis. Column F is specific to this particular site -- it contains the number of gravels in each size class.

You will do a cumulative frequency analysis and determine the particle size of the 84th percentile (D84). Do your analysis on a spreadsheet, in the following way:

1. In Column G, calculate the total weight (uncorrected) of gravels in each size class, by multiplying the count times the average weight.

2. In Column H, calculate the corrected total weight for each size class by multiplying the uncorrected total weight by the size correction factor. Be sure to put titles for columns G and H in the rows above the data.

3. Sum the numbers in Column H to get the corrected total weight of the entire sample. Put the sum in a cell at the bottom of Column H.

4. In Column I, calculate the percentage of total sample weight that occurs in each size class. This should be expressed as a percent, not a decimal.

5. In Column J, calculate cumulative percentage of total weight, starting from the smallest size class. For the smallest diameter class (4 – 5.6 mm), the cumulative percent is equal to the percent in that class. For the next diameter class (5.6 – 8 mm), the cumulative percent is equal to the percent in 5.6 - 8 mm plus the percent in 4 – 5.6 mm. For each successive diameter class, the cumulative percent is equal to the percent in that class plus the percents in all smaller diameter classes. Thus for each size class, the cumulative percent represents percent finer -- the percentage of total sample weight in grains that are equal to or finer (smaller) than that size class. The values should be 100% when you get down to the larger sizes in Column J.

6. Make a cumulative frequency plot of size versus % finer. In Excel, use the XY (scatter) chart with markers and a line connecting the markers. Plot cumulative percent (column J) on the Y (vertical) axis against upper size limit (column C) on the X (horizontal) axis. Label the axes.

7. Now format your chart to look like a standard cumulative frequency plot. Make the X-axis a log scale. In Excel, do this by selecting the axis, then use the Format-Selected Axis-Scale command. To help you in reading values off the chart, you may want to add minor tick marks on both axes, at appropriate intervals. Do this with the Format-Selected Axis-Scale command. To help you read values off the chart, add gridlines. Do this using the Chart-Chart Options-Gridlines command; in gridlines, select major and minor units for both the X and Y axes.

8. Print out your chart. Read the D₅₀ and D₈₄ values off the chart. The D₅₀ is the diameter that corresponds to the 50th percentile; fifty percent of the sample is equal to or finer than this diameter. The D₈₄ is the diameter that corresponds to the 84th percentile. State the D₅₀ and D₈₄ diameters in your answers, and include a printout of your worked spreadsheet and your chart when you turn in this exercise.

Part B. Bed Sediment Entrainment

In this section, you will evaluate two flow events to determine whether entrainment of bed material will occur or not. There are three major steps to the calculations: 1) calculate boundary shear stress, t ₀, for a specific flow event at the cross section; 2) following the Shields criterion approach, calculate critical shear stress, t _cr, using sediment characteristics at the cross-section; 3) following the Komar selective entrainment approach, calculate critical shear stress, t _cr, using sediment characteristics at the cross-section. The final step is compare t ₀ to each of the t _cr estimates to see if entrainment will occur.

1. Set up a spreadsheet with columns as shown above. Add columns for the values of r , r _s, and g (r = density of water = 1000 kg/m³, r _s = density of sediment = 2650 kg/m³, and g = acceleration of gravity = 980.7 cm/sec²). Make sure there is a label at the top of each column (use the first row for labels), and make sure the label includes the units for each term.

2. In a new column, calculate slope of the water surface (S). (No need to do this step if you have it in your spreadsheet already.)

3. Convert R from feet to meters, putting the values in a new column (labelled). Convert g from cm/sec² to m/sec², putting the values in a new column (labelled). Convert D₅₀ and D₈₄ to meters, putting the values in new columns (labelled).

4. Calculate boundary shear stress using the DuBoys equation, t₀, = r gRS. In a new column (labelled), enter the Duboys equation as an Excel formula, and calculate t ₀ .

5. Calculate critical shear stress for D₅₀ and for D₈₄, using the Shield criterion approach, t_cr = q _cgD(r _s-r ). What value should be used for q_c? The Shields diagram (Fig. 4.5A) shows that for hydraulically rough beds, q _c is a constant. Both of these cross-sections have hydraulically rough beds. Shields originally proposed q_c= 0.06, but recent work has shown that q_c= 0.045 is more appropriate for gravel bed channels. (See the discussion of this point on p. 109-111.) In a new column (labelled), enter the Shields equation as an Excel formula, and calculate t_CR for D₅₀. Then do this in another column for D₈₄.

6. Another approach to estimating critical shear stress is Komar’s (1988) selective entrainment function:
t_cr = 0.045(r _s-r )gD₅₀^0.6D_i^0.4.   
In this equation, D₅₀ = diameter in mm of the 50th percentile, and D_i = diameter in mm of a specific particle size (i.e., a particle of the ith percentile). The purpose of this equation is to estimate critical shear stress for any specific particle size (ith percentile) within a population of gravels of different sizes. Calculate critical shear stress for D₅₀ and for D₈₄, using:
t_cr = 0.045(r _s-r )gD₅₀^0.6D_i^0.4.    In a new column (labelled), enter the Komar equation as an Excel formula, and calculate t_CR for D₅₀. Then do this in another column for D₈₄.

7. Print and turn in your spreadsheet showing your calculations for part B.

In this section you will analyze a set of suspended sediment data for one year. The data for Badwater Creek at Lysite, Wyoming, Water Year 1969, are in a file called X3partC.xls or X3partC.txt. A new version of the file, in Excel 5.0, has just been added (X3partC2.xls).   The file contains values for suspended sediment concentration and water discharge (Q) for each day of WY 1969. The file consists of three columns: date in column A, water discharge (Q) in cfs in column B, and suspended sediment concentration (SSC) in mg/L in column C. Each suspended sediment concentration value is the average suspended sediment concentration for that 24-hour period, and each Q value is the average discharge rate for that 24-hour period.

1. First make an annual hydrograph showing both Q and SSC. Make a chart with date on the X axis (horizontal axis) and Q on the Y axis (vertical axis). This should be an XY (scatter) chart, with the series represented by a line without markers.

2. Then add SSC to this chart as a second data series. To do this, on the chart sheet use the command Chart-Add Data, and enter the data range in the box. To enter the data range, you can type in a range that you previously wrote down, or after getting in the Add Data box go to the sheet with the data and select the cells that you want to enter. (You have entered the Y data range, but Excel knows that you want to use the same X value range as for the first data series.)

3. You will find that SSC has taken over the Y axis scale and Q is no longer visible. SSC goes up to much higher values than Q, so Q plots as a small line at the bottom. In Excel, you can add a secondary axis, so that each data series is plotted on an axis appropriate for its range. To add a secondary axis, select the data series that you wish to plot on a secondary axis (i.e., the sediment concentration series). Then use the command Format – Data Series – Axis, and select Secondary Axis.

4. Label the axes, using the command Chart – Chart Options - Titles. Make any changes in axis scale, font size, line weight or pattern, etc. that are needed to make the chart clearly readable and attractive. To edit axes, first select the axis, then use the command Format – Selected Axis – Scale (or other options). Print our your annual hydrograph chart.

5. Now make a suspended sediment rating curve chart. This is an XY chart with SSC on the Y axis and Q on the X axis. The data should be represented by markers only, no line. With standard axes, you will see a broad scatter of data points without much shape. SSC vs. Q is usually plotted on a log-log chart. Select each axis and change it to a logarithmic scale. This should make the relationship between SSC and Q more evident.

6. You may find that the axes cross at 1 or 0.1, in the middle of your plot. Select the axes and change them so the axes cross at 0.01. Label the axes. Make any changes in axis scale, font size, line weight or pattern, etc. that are needed to make the chart clearly readable and attractive. Print out your suspended sediment rating curve chart.

7. Now calculate suspended sediment discharge for each day in the record. The basic approach is: 1) determine the total volume of discharge in a day (discharge rate per unit time x time per day), and then 2) multiply sediment concentration per unit volume of water by total volume of water, to get total amount of sediment in the day. "Amount"of sediment means mass or weight. Your answer should be sediment discharge in tons, for each day. In an empty column, set up a formula to calculate suspended sediment discharge for each day. (Hint: It may be easier to find any errors if you do this in steps, in several formulas each in a different column.) Use the conversion factors given below.  (Note: If a cell is formatted as General, a number <<1 is displayed in base E (for example, 0.001 = 1E-03). To view it as a decimal number, select the cell and use the Format – Cell command to format it as Number. To see the number as a decimal, you may need to increase the decimal places in the Format - Cell command, or increase the width of the column.)  For your information, for Oct. 1, 1968, Q = 7.7 cfs, SSC = 47 mg/l, and SS discharge = about 0.98 tons.

Conversion factors for suspended sediment calculations:
1 liter = 0.035315 cubic feet
1 mg = 0.000001 kg
1 kg = 0.0011023 tons

8. Determine the total suspended discharge at Badwater Creek at Lysite, Wyoming for WY 1969, by summing all the daily values.

9. Determine the WY 1969 suspended sediment yield for Badwater Creek. Suspended sediment yield is measured in tons per square mile. The drainage area of Badwater Creek at Lysite is 415 square miles.

10. Turn in copies of all of your spreadsheets and charts with your answers.

Part C Questions:
1. What is the WY 1969 total suspended discharge for Badwater Creek? What is the WY 1969 suspended sediment yield for Badwater Creek watershed?
2. Using your annual hydrograph of discharge and suspended sediment concentration, describe variation in suspended sediment concentration throughout the year. In which season is SSC the highest? The lowest?
3. From your suspended sediment rating curve chart, briefly state, qualitatively (in words), the relationship between discharge and SSC at this station.

Part C extra credit:
Do a regression analysis to estimate the suspended sediment rating curve equation. First, make a new column containing log values of Q and a new column containing log values of SSC. Use a formula = log(Q or SSC cell) to calculate the log values. In Excel, you can do regression using the command Tools – Data Analysis – Regression. You will input the Y data range (all the cells containing values of log SSC), then the X data range (all the cells containing values of log Q). Under output options, select New Worksheet Ply. When the regression model is done, look at the R² to see how much of the variance in the data is explained by the regression model. You will also have a table that gives the coefficients of the regression model. Write out the regression model, in both forms:
log SSC = intercept coefficient + (X variable coefficient)logQ

SSC = intercept coefficient · Q^{(X var. coefficient)}

Question for extra credit section: Examine your regression model and your suspended sediment rating curve chart. Do think the model is a good fit or not? Do you notice any unusual points (i.e., SSC values that outliers, either quite high or quite low compared to other SSC values in the same range of Q)?

References:

Komar, P.D., 1987. Selective gravel entrainment and the empirical evaluation of flow competence. Sedimentology 34: 1165-1176.