# Box 2 Analyzing Grain-size Distribution

The standard approach to describing the *grain-size distribution* 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 Box 2‑1). 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).

A table is created (Figure Box 2-2) that shows the *cumulative percent retained* (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*. Values can also be reported as *cumulative percent finer than* (100% minus cumulative percent coarser than) or *cumulative percent passing* (100% minus cumulative percent retained). These data provide the grain-size distribution as shown in (Figure Box 2-3).

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).

Grain-size distributions (Figure Box 2-3) are most commonly described by their *effective grain size* (size of particle for which 90% of the sample is coarser than that value, which is *d*_{90} cumulative percent retained or *d*_{10} cumulative percent finer-than) as defined in Equations Box 2-1 and Box 2-2, and the *median grain size* (*d*_{50} for both cumulative percent retained and cumulative percent finer-than) as defined in Equation Box 2-3. A third parameter, the *uniformity coefficient*, provides information about how uniform (with uniform meaning similar size) the grain sizes are within the sample: uniformity coefficient = *d*_{40}/*d*_{90} for cumulative percent retained data and *d*_{60}/*d*_{10} 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.

Effective grain size = d_{90} cumulative percent retained |
(Box 2-1) |

Effecitve grain size = d_{10} cumulative percent finer than |
(Box 2-2) |

Median grain size = d_{50} |
(Box 2-3) |

Uniformity Coefficient = cumulative percent retained |
(Box 2-4) |

Uniformity Coefficient = cumulative percent finer than |
(Box 2-5) |

In some empirical equations the *mean particle grain size* and/or the *inclusive standard deviation* 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 units can be calculated from grain size as shown in Equation Box 2-6. Consequently, values of phi can be positive and negative.

(Box 2-6) |

where

log_{2}(x) |
= | |

x |
= | diameter in millimeters divided by 1 millimeter in order to render the value dimensionless |

The mean phi size is computed as shown in Equation Box 2-7.

Mean phi size = |
(Box 2-7) |

The mean equation can use either cumulative percent retained or finer-than data. The inclusive standard deviation of a size distribution, *σ*_{I} is defined for cumulative percent retained as shown in Equation Box 2-8.

Inclusive standard deviation = |
(Box 2-8) |

With a grain size distribution in hand, a number of empirical equations can be applied to estimate the value of hydrogeologic properties.