Grain Size Distribution and Hydraulic Properties
by

Prof. H.O. Pfannkuch

Rick Paulson

I. Grain Size Analyses

Since particle diameters typically span many orders of magnitude for natural sediments, we must find a way to conveniently describe wide ranging data sets. The base two logarithmic f (phi) scale is one useful and commonly used way to represent grain size information for a sediment distribution. A tabular classification of grain sizes in terms of f units, millimeters, and other commonly used measurement scales is included for purposes of comparison (see grain size description table appended at the end).

Logarithmic phi values (in base two) are calculated from particle diameter size measures in millimeters as follows:

where:


A grain size separation analysis can be a tedious and time consumming task. The results of grain size distribution analyses on two samples, A and B, taken from standard sieve tests, are given below. On the following pages you are asked to prepare histograms depicting percent frequency of particle size occurance, plots of grain size distribution called cumulative weight percent curves, and other statistical and hydraulic property measures for samples A and B (see requirements for Section I below).

To gain an understanding of how to proceed, look at the results of an example sieve size analysis performed on the MN 104 sample (see Figure 1). Relate those results to the histogram, and the cumulative distribution curves created from the analysis data (Figures 2, and 3) as an example of output to produce. Read Fetter, Sec. 4.2.2, pg. 82, for a discussion of sediment analysis by sieve sifting into particle size fractions. In addition, an in depth description of how grain size separations are performed is given toward the end of this page(see Sec. III, Grain Size Analyses of Sediments).

Study the grain size distribution curves carefully. The curves are cumulative percent frequency distribution curves, that represent the cumulative weight percent by particle size of the sample. In one of the curves (cumulative weight percent passing), the fraction that is finer than each subsequent grain size is shown. In the other curve (cumulative weight percent retained) the fraction that is coarser than each subsequent grain size is shown. Essentially, for each grain size, the curve will tell you how much of the sample was finer or coarser.

MN 104 Sample Analysis Data

Grain Size
(mm)
Grain Size
(f)
Weight of Size
Fraction (g)
Weight
Percent
Cumulative
Weight %Retained
Cumulative
Weight %Passed
0.01
6.64
0.6
0.02
100
0
0.063
4
0.5
0.02
99.98
0.02
0.125
3
0.6
0.02
99.96
0.04
0.25
2
1.2
0.04
99.94
0.06
0.354
1.5
1.8
0.06
99.9
0.1
0.5
1
4.7
0.16
99.84
0.16
0.707
0.5
17.5
0.59
99.68
0.32
1
0
172.2
5.81
99.09
0.91
1.41
-0.5
2570.1
86.74
93.28
6.72
2
-1
152.7
5.15
6.54
93.46
2.83
-1.5
41.2
1.39
1.39
98.61
4
-2
0
0
0
100

Figure 1


Figure 2


Figure 3


Requirements for Section I:
(As always, please include one representative sample calculation for each procedure).

  1. Complete the missing information in the tables of the grain size distribution analyses for Sample A and Sample B.

  2. Using the grain size analysis data for Samples A and B, construct an histogram of the distribution. You may use grain size in f units or in mm for the size classes.

  3. Using the same A and B sample data, construct cumulative percent frequency distribution curves for the grain size distribution of both samples. Please use the f scale on the grain size axis. Include both weight percent retained, and weight percent passing curves on the same plot.

  4. Calculate the following four descriptive parameters using the cumulative percent curve for each sample respectively. Refer to the end of this lab handout for further discussion of the statistical parameters. In the first three formulas below, the subscript in the phi terms (fx) refers to the grain size at which x% of the sample is coarser than that size, or also, the size at which x% of the sample is retained on that particular sieve size and any coarser screened sieves above that particular sieve. Please show your determination of the various fx graphically on the cumulative curves.


  5. Calculate the hydraulic conductivity (K) for samples A and B using the Hazen approximation (see Fetter, Sec. 4.4.3, pg. 98, for a discussion of the Hazen approximation). In your calculations use the d10 particle size in millimeters that you obtained from the grain size curves in Part 4d above. The Hazen approximation of hydraulic conductivity is applicable when the d10 effective particle size is between 0.1 and 3.0 mm, and is calculated as follows:

    where:


  6. The Krumbein and Monk equation is used to estimate the permeability (in darcies) of a sediment from a grain size analysis. This equation was developed empirically using very well sorted (see Figure 4 for a depiction of a poorly sorted sediment) sediment samples ranging from -0.75 to 1.25f in mean grain size, and with standard deviations ranging from 0.04 to 0.80f. Calculate the Krumbein and Monk intrinsic permeability (k) of samples A and B.

    The Krumbein and Monk equation is:


    where:


  7. Convert the Krumbein and Monk permeabilities (k) (in darcies) calculated in Part 6 above into hydraulic conductivities (K) (in cm/sec) using the relation:

    where:


    Present all your results above in the form of a single table. Remember to please include one representative sample calculation for each procedure.



Discussion Questions for Section I:

  1. From a simple visual inspection of the histograms and cumulative frequency curves, which of the samples is most poorly sorted i.e., the greatest distribution of size classes, (see Figure 4)? Justify your answer i.e., refer to what you look for in both types of diagrams.


    Figure 4 Depiction of a poorly sorted sediment sample.

  2. What is the verbal classification of sorting for these samples (refer to the discussion of Inclusive Graphic Standard Deviation in the Statistical Parameters Section at the end of the handout)?

  3. According to Krumbein and Monk, permeability decreases with an increase in standard deviation. Explain in physical terms why this should be true.

  4. Standard deviation is a statistical concept which assumes that the sample for which you are performing the calculations has a normal distribution (bell-shaped curve) about a mean value. How well do sediment samples A and B fit this model (refer to your histograms)? Evaluate the validity of using the calculated standard deviation to estimate the Krumbein and Monk permeability of the sediments you are dealing with in samples A and B.

  5. Compare the permeabilities and hydraulic conductivities calculated using the Krumbein and Monk equation and the Hazen method respectively on samples A and B, to the permeabilities and hydraulic conductivities measured using the permeameter test data for samples A and B from Lab 2 (Lab 2 results will be posted on the class web site "q and a" page). Are there significant differences? Which methods yield the highest and lowest results? Offer explanations as to why the results may differ. With which results do you feel most comfortable?




SAMPLE A

Grain Size
(mm)
Percent of sample
retained (by weight)
Cumulative
% finer
Cumulative
% coarser
Grain Size
(f)
10 0      
9 2      
8 2      
7 6      
5 5      
1 15      
0.75 10      
0.5 10      
0.3 15      
0.09 25      
0.05 5      
0.01 3      
0.009 2      



SAMPLE B

Grain Size
(mm)
Percent of sample
retained (by weight)
Cumulative
% finer
Cumulative
% coarser
Grain Size
(f)
15 0      
10 2      
8 1.5      
6 1.5      
3 7      
1 10      
0.8 3      
0.5 10      
0.2 10      
0.1 20      
0.06 10      
0.03 15      
0.01 10      



II. More Grain Size Analyses

The purpose of this next portion of the exercise is to aquaint you with the variations in grain size distributions for different sedimentary deposits. You are given the data obtained by grain size analysis of three different sediment samples: a very well sorted friable sandstone, a well sorted sand from a flowing spring, and a poorly sorted glacial till.

The St. Peter Sandstone has undergone an extensive multicyclic depositional history. It's last depositional episode was as a beach sand along a transgressing sea. During at least one of its previous depositional episodes the sand grains were eolian (wind transported) deposits. Consequently, the range of particle sizes is somewhat restricted owing to the narrow range of particle sizes that can be transported by wind. Due to its extensive depositional history the St. Peter Sandstone is texturally, and mineralogically very mature.

The Boiling Springs sample comes from a fluvial environment of deposition. As such, the sediment tends to be fairly well sorted. The energy of deposition in a fluvial environment can fluctuate widely. Consequently, the range of particle sizes can also be somewhat wide spread. In this particular case, the energy of deposition was fairly low as evidenced by the fairly fine size of the particles in the sample.

The Grantsburg Sublobe Till is a glacially deposited sediment. The till was laid down by the Grantsburg Sublobe of the Des Moines Lobe during the Late Wisconsinin glaciation approximately 14,000 years ago. As is typically the case with ice transported sediments, the till is poorly sorted with a concurrent wide range of particle sizes.


St. Peter Sandstone
Grain Size
(f)
Percent
by weight
Cumulative
% Coarser
Cumulative
% Finer
Grain Size
(mm)
1.25
0.38
0.38
99.62
0.420
1.50
2.23
2.61
97.39
0.354
1.75
9.11
11.72
88.28
0.297
2.00
14.48
26.21
73.79
0.250
2.25
15.02
41.22
58.78
0.210
2.50
16.61
57.83
42.17
0.177
2.75
19.75
77.58
22.42
0.149
3.00
13.50
91.08
8.92
0.125
3.25
6.08
97.16
2.84
0.105
3.50
1.10
98.27
1.73
0.088
3.75
0.74
99.00
1.00
0.074
4.00
0.37
99.37
0.63
0.062
4.25
0.63
100.00
0.00
0.053



Boiling Springs
Grain Size
(f)
Percent
by weight
Cumulative
% Coarser
Cumulative
% Finer
Grain Size
(mm)
-1.00
0.18
0.18
99.82
2.000
-0.50
0.13
0.32
99.68
1.410
0.00
0.34
0.65
99.35
1.000
0.50
0.78
1.43
98.57
0.707
1.00
3.12
4.55
95.45
0.500
1.50
17.88
22.43
77.57
0.354
2.00
41.68
64.11
35.89
0.250
3.00
35.63
99.74
0.26
0.125
3.51
0.22
99.96
0.04
0.088
3.99
0.02
99.98
0.02
0.063
5.64
0.02
100.00
0.00
0.020



Grantsburg Sublobe Till
Grain Size
(f)
Percent
by Weight
Cumulative
% Coarser
Cumulative
% Finer
Grain Size
(mm)
-1.00
9.60
9.60
90.40
2.000
-0.77
1.40
11.00
89.00
1.705
-0.27
2.20
13.20
86.80
1.205
0.23
3.30
16.50
83.50
0.854
1.00
14.30
30.80
69.20
0.500
1.73
6.80
37.60
62.40
0.302
2.22
9.40
47.00
53.00
0.214
2.73
9.50
56.50
43.50
0.151
3.24
4.00
60.50
39.50
0.106
3.75
7.40
67.90
32.10
0.075
4.20
2.80
70.70
29.30
0.055
4.76
6.00
76.70
23.30
0.037
5.23
5.50
82.20
17.80
0.027
5.72
11.50
93.70
6.30
0.019
6.27
2.30
96.00
4.00
0.013
6.73
0.80
96.80
3.20
0.009
7.24
0.70
97.50
2.50
0.007
7.73
0.60
98.10
1.90
0.005
8.97
1.90
100.00
0.00
0.002



Requirements for Section II:

  1. Prepare histograms and cumulative percent coarser and finer plots for each of the three sediment samples.

  2. Calculate the four statistical parameters for each of the three samples as in Section I.

  3. Calculate the hydraulic conductivities and convert to permeabilities as in Section I.

Present your results in the form of a single table.

Discussion Questions for Section II:

  1. Discuss the differences you see in the grain size curves for the three sediments. Relate this to their geologic history. Use the statistical parameters as a reference.

  2. Provide the verbal classification of the sorting of these sediments (refer to the discussion of Inclusive Graphic Standard Deviation in the Statistical Parameters Section at the end of the handout).

  3. What kind of relationship do you see between grain size distribution and the hydraulic properties? How much do you think grain size distribution controls conductivity?


III. Grain Size Analysis Of Sediments

The objective of this section is to give you a sense of the physical procedures and statistical parameters which are involved in performing a sediment analysis. This section may come in handy if you ever have to do one of these procedures.

The purpose of this procedure is to determine the distribution of grain sizes in an unconsolidated sediment sample. This analysis is broken down into several steps dealing first with the coarse fraction (sand and gravel) and then with the fine fraction (silt and clay).

Procedure:

A. Removal of organic material
The purpose of removing organic carbon particles is that they do not represent clastic grains and are therefore not to be considered in a grain size analysis. Depending on the percentage of organics in the sediment this step may be quite lengthy. In brief, the procedure involves adding water and hydrogen peroxide to help remove the organics and disaggregate the sample.

B. Shaking and Centrifuging
The purpose of this step is to wash the sample with distilled water to remove salts and help break down agglomeration in the mud.

C. Dispersion
The purpose of this step is to disperse the aggregates of clay size particles so that they will be sized as individuals, not as aggregates. This procedure consists of mixing the sediment with a peptizer solution, or dispersing agent.

D. Wet Sieving
The purpose of this step is to separate the sediment sample into coarse and fine fractions. To do this, select a 4 phi size sieve and place the sediment sample in it. Under the sieve place a tightly fitting funnel with a 1000 ml beaker attached to it. Wash the sample until the stream leaving the funnel contains no fines and is clear. The material passing into the beaker is the fine fraction which should be saved for pipette analysis later. The material on the sieve is the coarse fraction which will be sieved.

E. Sieving the coarse fraction
Select the screens to be used. For accurate work use the 1/4 phi set. Nest the sieves in order, the coarsest on the top, the finest on the bottom. Pour the weighed dry sample into the top sieve, cover, and set the timer for ten minutes. Upon completion of the sieving, the contents of each sieve are weighed. Be sure to tap the screens firmly on a sheet of paper to remove lodged grains from the screen and add them to the appropriate sample. Be sure to strike the table evenly with the rim of the screen. Make sure that all loose grains get accounted for by repeating this procedure. Weigh to the 0.0001 gram. Each phi size is weighed in this manner and the weights are recorded on a data sheet.

F. Pipetting the fine fraction
The pipetting technique of size analysis is based on the differential settling velocities of the particles. Stoke's Law of Settling Velocities is the basis for this method. Calculations have been made as to the time a certain grain size will be at a certain depth. After the initial shaking to homogenize the sediment, aliquots are drawn from a depth and at a time so that the last particles of a given size are just settling past the sampling depth. The resulting aliquot gives a weight of the material finer than the given size. Successive aliquots give the weights in each phi size.

STATISTICAL PARAMETERS

Mode: Mode is the most frequently-occurring particle diameter. It is the diameter corresponding to the steepest point (point of inflection) on the cumulative curve (only if the curve has an arithmetic frequency scale). It also corresponds to the highest point on the distribution curve. Advantages: the mode is quite valuable in sediment genesis transport studies, especially when two or more sources are contributing. The mode diameter often stays fairly constant in an area while the other, more "synthetic" measures tend to vary more erratically. It deserves more common use. The disadvantages are its lack of common usage, and in fact that it is difficult to determine. Also it is independent of the grain size of the rest of the sediment, therefore it is not a good measure of overall average size.

Median: Half of the particles by weight are coarser than the median, and half are finer. It is the diameter corresponding to the 50% mark on the cumulative frequency curve and may be expressed either in phi or in mm. The advantage is that it is by far the most commonly used measure and the easiest to determine. The disadvantage is that it is not affected by the extremes of the curve, therefore does not reflect the overall size of sediments (especially skewed ones) well. For bimodal sediments it is almost worthless. Its use is not recommended.

Geometric (Graphic) Mean: The best graphic measure for determining overall size is the graphic mean. It corresponds very closely to the mean as computed by the method of moments, yet is much easier to find. It is much superior to the median because it is based on three points and gives a better overall picture.

Inclusive Graphic Standard Deviation: This formula includes 90% of the distribution and is the best overall measure of sorting. Measurement of sorting values for a large number of sediments has suggested the following verbal classification for sorting for each value of inclusive graphic standard deviation:


phi (f)Size Range
Verbal Description of Sorting
under .35 phi very well sorted
0.35 - 0.50 phi well sorted
0.50 - 0.71 phi moderately well sorted
0.71 - 1.0 phi moderately sorted
1.0 - 2.0 phi poorly sorted
2.0 - 4.0 phi very poorly sorted
over 4.0 phi extremely poorly sorted


The best sorting attained by natural sediments is about .20-25 phi, and Texas dune and beach sands run about .25-.35 phi. Texas river sediments so far measured range between .40-2.5 phi, and pipetted flood plain or neritic silts and clays average about 2.0-3.5 phi. The poorest sorted sediments, such as glacial tills, mudflows, etc., have values in the neighborhood of 5 phi to 8 phi or even 10 phi.

Kurtosis: In the normal probability curve, defined by the gaussian formula; the phi diameter interval between the 5 phi and 95 phi points should be exactly 2.44 times the phi diameter interval between the 25 phi and 75 phi points. If the sample curve plots as a straight line on probability paper (i.e., if it follows the normal curve), this ratio will be obeyed and we say it has normal kurtosis (1.00). Departure from a straight line will alter this ratio, and kurtosis is the quantitative measure used to describe this departure from normality. It measures the ratio between the sorting in the "tails" of the curve and the sorting in the central portion. If the central portion is better sorted than the tails, the curve is said to be excessively peaked or leptokurtic; if the tails are better sorted than the central portion, the curve is deficiently or flat-peaked and platykurtic. Strongly platykurtic curves are often bimodal with subequal amounts of the two modes; these plot out as a two-peaked frequency curve, with the sag in the middle of the two peaks accounting for its platykurtic character. For normal curves, kurtosis equals 1.00. Leptokurtic curves have a kurtosis over 1.00 (for example a curve with kurtosis=2.00 has exactly twice as large a spread in the tails as it should have, hence it is less well sorted in the tails than in the central portion); and platykurtic have kurtosis under 1.00. Kurtosis involves a ratio of spreads; hence it is a pure number and should not be written with a phi attached. The following verbal limits are suggested for values of kurtosis:


Kurtosis Value
Verbal Description
of Kurtosis
under 0.67 very platykurtic
0.67 - 0.90 platykurtic
0.90 - 1.11 mesokurtic
1.11 - 1.50 leptokurtic
1.50 - 3.00 very leptokurtic
over 3.00 extremely leptokurtic


The distribution of kurtosis values in natural sediments is itself strongly skewed, since most sediments are around .85 to 1.4, yet some values as high as 3 or 4 are not uncommon.

Skewness: This formula simply averages the skewness obtained using the 16 phi and 84 phi points with the skewness obtained by using the 5 phi and 95 phi points, both determined by exactly the same principle. This is the best skewness measure to use because it determines the skewness of the "tails" of the curve, not just the central portion, and the "tails" are just where the most critical differences between samples lie. Furthermore, it is geometrically independent of the sorting of the sample.

Symmetrical curves have skewness=0.00; those with excess fine material (a tail to the right) have positive skewness and those with excess coarse material (a tail to the left) have negative skewness. The more the skewness value departs from 0.00, the greater the degree of asymmetry. The following verbal limits on skewness are suggested: for values of skewness:


Skewness
Verbal Description
of Skewness
from +1.00 to +0.30 strongly fine skewed
from +0.30 to +0.10 fine skewed
from +0.10 to -0.10 near symmetrical
from -0.10 to -0.30 coarse skewed
from -0.30 to -1.00 strongly coarse skewed


The absolute mathematical limits of the measure are +1.00 to -1.00, and few curves have skewness values beyond +0.80 to -0.80.

Coefficient of Uniformity: This is a nonstatistical measure of the spread of the curve. It is similar to the standard deviation, but is used for samples that don't follow a normal curve. This parameter is defined in different ways by different people.


IV. Grain Size Scales and Naming Classifications For Sediments

The objective of this section is to show you a few of the many grain size and textural naming classification schemes in widespread use today.

Udden-Wentworth Classification

The grade scale that has traditionally been used for sediments is the Udden-Wentworth (1922) size class scale (see classification table appended to the end of this lab). This scale is a geometric series in which each grade limit is twice as large as the next smaller grade limit. The scale starting at 1 mm and changing by a fixed ratio of 2 was first introduced by J. A. Udden (1898), who also named the sand grades we use today. However, Udden drew the gravel/sand boundary at 1 mm and used different terms in the gravel and mud divisions. For more detailed work, sieves were used at intervals of 20.5 and 20.25.

The base two logarithmic f (phi) scale, devised by Krumbein (1934) and based on the Udden-Wentworth geometric series scale, is a much more convenient way of presenting data than if the values are expressed in millimeters, and is used almost exclusively in recent work in sedimentology. By transforming the millimeter scale into phi units, size class divisions of equal width are created.

Unified Soil Classification System

A commonly used soil classification system found in engineering disciplines is the Engineering Unified Soil Classification System or simply the Unified System. This classification system is mainly intended for soil classification for foundations and hydraulic structures. It is based on grain size and soil saturation to liquid limit.

Engineering Unified Soil Classification System

(Unified System)

Category
Group Symbol
Coarse Gravels Clean Gravel Gravel, well graded
GW
Grained     Gravel, poorly graded
GP
Soils   Gravel with Fines Gravels, mixed, non plastic, fines
GM
      Gravels, clayey-plastic, fines
GC
  Sands Clean Sands Sands, well graded
SW
      Sands, poorly graded
SP
    Sand with Fines Sands, mixed-plastic, fines
SM
      Sands, clayey-plastic, fines
SC
Fine Silts Liquid Limit < 50 Mineral silts, low plasticity
ML
Grained and   Clays (mineral), low plasticity
CL
Soils Clays   Organic silts, low plasticity
OL
    Liquid Limit > 50 Mineral silts (high plasticity)
MH
      Clays (mineral), low plasticity
CH
      Organic clays, high plasticity
OH
Highly Organic Soils
Organic soils as Peat
Pt



Soil Triangle Of Basic Soil Textural Classes

Another widely used classification system is the soil triangle of basic soil textural classes. This naming system is commonly used by the U.S. Department of Agriculture and the U.S. Soil Conservation Service. In this system the relative percentages of three particle size catagories are considered. The three catagories of particles are sand, silt, and clay. The triangular diagram is subdivided into several soil textural classification types, with each soil type comprising a range of percentages of the three particle types. The soil classification type is determined by plotting the percentages of each of the three soil particle classes found within the soil sample on the triangular diagram. The point of intersection of each of these three particle class percentages will fall within one of the soil classification types (see Soil Classification Triangle).

As can be seen by inspection of the triangle, each of the three particle types can vary from zero to 100% of the content of a sediment sample. For example, a 100% clay textural composition would plot at the apex at the top of the triangle. Lesser percentages of clay content would plot somewhere between the top apex and the base of the triangle. The same technique holds true for sand or silt content except that the 100% sand or silt content points are located at the left and right bottom apexes of the triangle respectively. Lesser contents of sand or silt would plot somewhere between the 100% apex and the side of the triangle opposite that apex.




Grain Size Classification Table

U. S. Standard
Sieve Mesh
Millimeters
(fractional)
Millimeters Microns Phi (f) Wentworth Size Class
    (1 Kilometer)   -20  
Use   4096   -12  
    1024   -10 Boulder (-8 to -12f)
wire   256   -8 Cobble (-5 to -8f)
    64   -6  
squares   16   -4 Pebble (-2 to -5f)
5   4   -2  
6   3.36   -1.75  
7   2.83   -1.5 Granule (-1 to -2f)
8   2.38   -1.25  
10   2   -1  
12   1.68   -0.75  
14   1.41   -0.5 Very coarse sand
(0 to -1f)
16   1.19   -0.25  
18   1   0  
20   0.84   0.25  
25   0.71   0. 5 Coarse sand (1 to 0f)
30   0.59   0.75  
35 1/2 0.5 500 1  
40   0.42 420 1.25  
45   0.35 350 1.5 Medium sand (2 to 1f)
50   0.3 300 1.75  
60 1/4 0.25 250 2  
70   0.21 210 2.25  
80   0.177 177 2.5 Fine sand (3 to 2f)
100   0.149 149 2.75  
120 1/8 0.125 125 3  
140   0.105 105 3.25  
170   0.088 88 3.5 Very fine sand (4 to 3f)
200   0.074 74 3.75  
230 1/16 0.0625 62.5 4  
270   0.053 53 4.25  
325   0.044 44 4.5 Coarse silt (5 to 4f)
Analyzed   0.037 37 4.75  
  1/32 0.031 31 5  
by 1/64 0.0156 15.6 6 Medium silt (6 to 5f)
  1/128 0.0078 7.8 7 Fine silt (7 to 6f)
Pipette 1/256 0.0039 3.9 8 Very fine silt (8 to 7f)
    0.002 2 9  
or   0.00098 0.98 10 Clay
    0.00049 0.49 11 (Some use 2m or 9f
Hydrometer   0.00024 0.24 12 as the clay boundary)
    0.00012 0.12 13