{"id":176,"date":"2020-10-12T16:45:16","date_gmt":"2020-10-12T16:45:16","guid":{"rendered":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/?post_type=chapter&p=176"},"modified":"2020-12-29T17:46:05","modified_gmt":"2020-12-29T17:46:05","slug":"analyzing-grain-size-distribution","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/chapter\/analyzing-grain-size-distribution\/","title":{"raw":"Box 2 Analyzing Grain-size Distribution","rendered":"Box 2 Analyzing Grain-size Distribution"},"content":{"raw":"The standard approach to describing the grain-size distribution<\/em> of a granular, unconsolidated sample is to start with a dry volume of granular material, weigh it, and place it on top of a nest of sieves that have progressively smaller mesh openings (Figure\u00a0Box\u00a02\u20111). The sieves are shaken for a period of time so each sieve holds only grains with diameters larger than its mesh size and smaller than the mesh size of the sieve immediately above it. Then the material on each sieve is collected and weighed. Sieve analyses methods are outlined in the American Society of Testing and Materials ASTM C136\/C136M-19 Standard Test Method for Sieve Analysis of Fine and Coarse Aggregates (ASTM, 2019).\r\n\r\n \r\n\r\n[caption id=\"attachment_179\" align=\"alignnone\" width=\"868\"]\"photographs Figure Box2-1 -<\/strong> To determine grain size distribution, the sample is dried, weighed and placed in a nest of sieves with the largest openings on top and the smallest on the bottom. A pan with a solid bottom is placed to catch any portion of the sample not retained on the sieves. The sample is shaken and the weight of the sample on each sieve is recorded (images from https:\/\/pavementinteractive.org\/reference-desk\/testing\/aggregate-tests\/gradation-test\/, accessed July 10, 2020).[\/caption]\r\n\r\nA table is created (Figure Box 2-2) that shows the cumulative percent retained<\/em> (weight of the sample on the coarsest sieve plus the weight on consecutive sieves of decreasing size). A synonymous term is cumulative percent coarser than<\/em>. Values can also be reported as cumulative percent finer than<\/em> (100% minus cumulative percent coarser than) or cumulative percent passing<\/em> (100% minus cumulative percent retained). These data provide the grain-size distribution as shown in (Figure Box 2-3).\r\n\r\n \r\n\r\n[caption id=\"attachment_181\" align=\"alignnone\" width=\"954\"]\"Table Figure Box2-2<\/strong> - Example of grain-size distribution data from a sieve analysis. The cumulative percent retained and\/or finer-than is computed from the weights on each sieve relative to the total sample weight. Note that in this example 98 grams of sample were recovered from the sieves suggesting either 2 grams entered the pan or part of the sample stuck to the sieves (Woessner and Poeter, 2020, gw-project.org).[\/caption]\r\n\r\nOnce the grain size distribution table is generated, the cumulative percent data are plotted on a standard grain size distribution curve (Figure Box 2-3).\r\n\r\n \r\n\r\n[caption id=\"attachment_182\" align=\"alignnone\" width=\"1024\"]\"Graph Figure Box2-3 -<\/strong> A cumulative percent grain size distribution curve for the data presented in Figure Box 2-2. The grain size (mm) is on a log scale. The green line represents the effective grain size: d<\/em>90<\/sub> retained (d<\/em> = diameter) or d<\/em>10<\/sub> finer-than. The value for this distribution is taken from where the green, dashed line crosses the red line. The black line at 50% provides the median grain size where it crosses the red line, d<\/em>50<\/sub>. The sample uniformity is computed using the value where the distribution crosses the yellow dotted line, d<\/em>40<\/sub> retained and\/or d<\/em>60<\/sub> finer-than, and the effective grain size (e.g., cumulative percent retained, uniformity coefficient = d<\/em>40<\/sub>\/d<\/em>90<\/sub>).[\/caption]\r\n\r\nGrain-size distributions (Figure Box 2-3) are most commonly described by their effective grain size<\/em> (size of particle for which 90% of the sample is coarser than that value, which is d<\/em>90<\/sub> cumulative percent retained or d<\/em>10<\/sub> cumulative percent finer-than) as defined in Equations Box 2-1 and Box 2-2, and the median grain size<\/em> (d<\/em>50<\/sub> for both cumulative percent retained and cumulative percent finer-than) as defined in Equation Box 2-3. A third parameter, the uniformity coefficient<\/em>, provides information about how uniform (with uniform meaning similar size) the grain sizes are within the sample: uniformity coefficient = d<\/em>40<\/sub>\/d<\/em>90<\/sub> for cumulative percent retained data and d<\/em>60<\/sub>\/d<\/em>10<\/sub> for cumulative percent finer-than data (Equations Box 2-4 and Box 2-5). If the ratio is less than two, the sample is considered fairly uniform. A perfectly uniform sample size distribution curve is a vertical line on a grain-size distribution graph (all the grains are the same size) and has a uniformity coefficient of one.\r\n<\/a>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Effective grain size<\/em> = d<\/em>90<\/sub> cumulative percent retained<\/strong><\/em><\/td>\r\n(Box 2-1)<\/td>\r\n<\/tr>\r\n
Effecitve grain size<\/em> = d<\/em>10<\/sub> cumulative percent finer than<\/strong><\/em><\/td>\r\n(Box 2-2)<\/td>\r\n<\/tr>\r\n
Median grain size<\/em> = d<\/em>50<\/sub><\/td>\r\n(Box 2-3)<\/td>\r\n<\/tr>\r\n
Uniformity Coefficient<\/em> = [latex]\\frac{d_{40}}{d_{90}}[\/latex] cumulative percent retained<\/strong><\/em><\/td>\r\n(Box 2-4)<\/td>\r\n<\/tr>\r\n
Uniformity Coefficient<\/em> = [latex]\\frac{d_{60}}{d_{10}}[\/latex] cumulative percent finer than<\/strong><\/em><\/td>\r\n(Box 2-5)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIn some empirical equations the mean particle grain size<\/em> and\/or the inclusive standard deviation<\/em> are used to estimate sample hydrogeological properties. These values cannot be derived directly from the standard grain-size distribution curve. Folk and Ward (1957) developed a method to compute these values. Their method requires converting cumulative percent grain size distribution data to what they call phi units, [latex]\\phi[\/latex]. Phi units can be calculated from grain size as shown in Equation Box 2-6. Consequently, values of phi can be positive and negative.\r\n\r\n\r\n\r\n
[latex]\\displaystyle \\phi =-\\log _{2}\\left ( \\frac{size\\ mm}{1\\ mm} \\right )[\/latex]<\/td>\r\n(Box 2-6)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nwhere\r\n\r\n\r\n\r\n\r\n
log2<\/sub>(x<\/em>)<\/td>\r\n=<\/td>\r\n[latex]\\frac{\\ln (x)}{\\ln (2)}[\/latex]<\/td>\r\n<\/tr>\r\n
x<\/em><\/td>\r\n=<\/td>\r\ndiameter in millimeters divided by 1 millimeter in order to render the value dimensionless<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe mean phi size is computed as shown in Equation Box 2-7.\r\n\r\n\r\n\r\n
Mean phi size<\/em> = [latex]\\displaystyle \\left(\\frac{\\phi_{16}+\\phi_{50}+\\phi_{84}}{3}\\right)[\/latex]<\/td>\r\n(Box 2-7)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe mean equation can use either cumulative percent retained or finer-than data. The inclusive standard deviation of a size distribution, \u03c3<\/em>I<\/em><\/sub> is defined for cumulative percent retained as shown in Equation Box 2-8.\r\n\r\n\r\n\r\n
Inclusive standard deviation<\/em> = [latex]\\displaystyle \\sigma_I\\ =\\ \\left(\\frac{\\phi_{84}-\\phi_{16}}{4}\\right)+\\ \\left(\\frac{\\phi_{95}-\\phi_5}{6.6}\\right)[\/latex]<\/td>\r\n(Box 2-8)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWith a grain size distribution in hand, a number of empirical equations can be applied to estimate the value of hydrogeologic properties.\r\n

Return to where text links to Box 2<\/a><\/p>","rendered":"

The standard approach to describing the grain-size distribution<\/em> of a granular, unconsolidated sample is to start with a dry volume of granular material, weigh it, and place it on top of a nest of sieves that have progressively smaller mesh openings (Figure\u00a0Box\u00a02\u20111). The sieves are shaken for a period of time so each sieve holds only grains with diameters larger than its mesh size and smaller than the mesh size of the sieve immediately above it. Then the material on each sieve is collected and weighed. Sieve analyses methods are outlined in the American Society of Testing and Materials ASTM C136\/C136M-19 Standard Test Method for Sieve Analysis of Fine and Coarse Aggregates (ASTM, 2019).<\/p>\n

 <\/p>\n

\"photographs
Figure Box2-1 –<\/strong> To determine grain size distribution, the sample is dried, weighed and placed in a nest of sieves with the largest openings on top and the smallest on the bottom. A pan with a solid bottom is placed to catch any portion of the sample not retained on the sieves. The sample is shaken and the weight of the sample on each sieve is recorded (images from https:\/\/pavementinteractive.org\/reference-desk\/testing\/aggregate-tests\/gradation-test\/, accessed July 10, 2020).<\/figcaption><\/figure>\n

A table is created (Figure Box 2-2) that shows the cumulative percent retained<\/em> (weight of the sample on the coarsest sieve plus the weight on consecutive sieves of decreasing size). A synonymous term is cumulative percent coarser than<\/em>. Values can also be reported as cumulative percent finer than<\/em> (100% minus cumulative percent coarser than) or cumulative percent passing<\/em> (100% minus cumulative percent retained). These data provide the grain-size distribution as shown in (Figure Box 2-3).<\/p>\n

 <\/p>\n

\"Table
Figure Box2-2<\/strong> – Example of grain-size distribution data from a sieve analysis. The cumulative percent retained and\/or finer-than is computed from the weights on each sieve relative to the total sample weight. Note that in this example 98 grams of sample were recovered from the sieves suggesting either 2 grams entered the pan or part of the sample stuck to the sieves (Woessner and Poeter, 2020, gw-project.org).<\/figcaption><\/figure>\n

Once the grain size distribution table is generated, the cumulative percent data are plotted on a standard grain size distribution curve (Figure Box 2-3).<\/p>\n

 <\/p>\n

\"Graph
Figure Box2-3 –<\/strong> A cumulative percent grain size distribution curve for the data presented in Figure Box 2-2. The grain size (mm) is on a log scale. The green line represents the effective grain size: d<\/em>90<\/sub> retained (d<\/em> = diameter) or d<\/em>10<\/sub> finer-than. The value for this distribution is taken from where the green, dashed line crosses the red line. The black line at 50% provides the median grain size where it crosses the red line, d<\/em>50<\/sub>. The sample uniformity is computed using the value where the distribution crosses the yellow dotted line, d<\/em>40<\/sub> retained and\/or d<\/em>60<\/sub> finer-than, and the effective grain size (e.g., cumulative percent retained, uniformity coefficient = d<\/em>40<\/sub>\/d<\/em>90<\/sub>).<\/figcaption><\/figure>\n

Grain-size distributions (Figure Box 2-3) are most commonly described by their effective grain size<\/em> (size of particle for which 90% of the sample is coarser than that value, which is d<\/em>90<\/sub> cumulative percent retained or d<\/em>10<\/sub> cumulative percent finer-than) as defined in Equations Box 2-1 and Box 2-2, and the median grain size<\/em> (d<\/em>50<\/sub> for both cumulative percent retained and cumulative percent finer-than) as defined in Equation Box 2-3. A third parameter, the uniformity coefficient<\/em>, provides information about how uniform (with uniform meaning similar size) the grain sizes are within the sample: uniformity coefficient = d<\/em>40<\/sub>\/d<\/em>90<\/sub> for cumulative percent retained data and d<\/em>60<\/sub>\/d<\/em>10<\/sub> for cumulative percent finer-than data (Equations Box 2-4 and Box 2-5). If the ratio is less than two, the sample is considered fairly uniform. A perfectly uniform sample size distribution curve is a vertical line on a grain-size distribution graph (all the grains are the same size) and has a uniformity coefficient of one.
\n<\/a><\/p>\n\n\n\n\n\n\n\n
Effective grain size<\/em> = d<\/em>90<\/sub> cumulative percent retained<\/strong><\/em><\/td>\n(Box 2-1)<\/td>\n<\/tr>\n
Effecitve grain size<\/em> = d<\/em>10<\/sub> cumulative percent finer than<\/strong><\/em><\/td>\n(Box 2-2)<\/td>\n<\/tr>\n
Median grain size<\/em> = d<\/em>50<\/sub><\/td>\n(Box 2-3)<\/td>\n<\/tr>\n
Uniformity Coefficient<\/em> = \"\frac{d_{40}}{d_{90}}\" cumulative percent retained<\/strong><\/em><\/td>\n(Box 2-4)<\/td>\n<\/tr>\n
Uniformity Coefficient<\/em> = \"\frac{d_{60}}{d_{10}}\" cumulative percent finer than<\/strong><\/em><\/td>\n(Box 2-5)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

In some empirical equations the mean particle grain size<\/em> and\/or the inclusive standard deviation<\/em> are used to estimate sample hydrogeological properties. These values cannot be derived directly from the standard grain-size distribution curve. Folk and Ward (1957) developed a method to compute these values. Their method requires converting cumulative percent grain size distribution data to what they call phi units, \"\phi\". Phi units can be calculated from grain size as shown in Equation Box 2-6. Consequently, values of phi can be positive and negative.<\/p>\n\n\n\n
\"\displaystyle \phi =-\log _{2}\left ( \frac{size\ mm}{1\ mm} \right )\"<\/td>\n(Box 2-6)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

where<\/p>\n\n\n\n\n
log2<\/sub>(x<\/em>)<\/td>\n=<\/td>\n\"\frac{\ln (x)}{\ln (2)}\"<\/td>\n<\/tr>\n
x<\/em><\/td>\n=<\/td>\ndiameter in millimeters divided by 1 millimeter in order to render the value dimensionless<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The mean phi size is computed as shown in Equation Box 2-7.<\/p>\n\n\n\n
Mean phi size<\/em> = \"\displaystyle \left(\frac{\phi_{16}+\phi_{50}+\phi_{84}}{3}\right)\"<\/td>\n(Box 2-7)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The mean equation can use either cumulative percent retained or finer-than data. The inclusive standard deviation of a size distribution, \u03c3<\/em>I<\/em><\/sub> is defined for cumulative percent retained as shown in Equation Box 2-8.<\/p>\n\n\n\n
Inclusive standard deviation<\/em> = \"\displaystyle \sigma_I\ =\ \left(\frac{\phi_{84}-\phi_{16}}{4}\right)+\ \left(\frac{\phi_{95}-\phi_5}{6.6}\right)\"<\/td>\n(Box 2-8)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

With a grain size distribution in hand, a number of empirical equations can be applied to estimate the value of hydrogeologic properties.<\/p>\n

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