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Aquifer tests (pumping tests, slug tests and constant-head tests) are performed to estimate site-specific values for the hydraulic properties of aquifers and aquitards. Under certain circumstances, however, site-specific hydraulic property data may not be available when needed. For example, reconnaissance studies or scoping calculations may require hydraulic property values before on-site investigations are performed.

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The following sections present representative hydraulic property values reported in the literature for horizontal and vertical hydraulic conductivity, storativity, specific yield and porosity. Refer to these values if site-specific data are unavailable for your study or to check the results of field and laboratory tests conducted at an investigation site.

Try out the interactive calculators for estimating hydraulic conductivity from grain size, specific storage and storativity!

## HydraulicConductivity (K)

Hydraulic conductivity is a measure of a material's capacity to transmit water. It is defined as a constant of proportionality relating the specific discharge of a porous medium under a unit hydraulic gradient in Darcy's law:

$ν=-Ki$

where $\nu$ is specific discharge [L/T], $K$ is hydraulic conductivity [L/T] and $i$ is hydraulic gradient [dimensionless]. Coefficient of permeability is another term for hydraulic conductivity.

Note that hydraulic conductivity, which is a function of water viscosity and density, is in a strict sense a function of water temperature; however, given the small range of temperature variation encountered in most groundwater systems, the temperature dependence of hydraulic conductivity is often neglected.

Transmissivity is the rate of flow under a unit hydraulic gradient through a unit width of aquifer of given saturated thickness. The transmissivity of an aquifer is related to its hydraulic conductivity as follows:

$T=Kb$

where $T$ is transmissivity [L2/T] and $b$ is aquifer thickness [L].

### Representative Values

The following tables show representative values of hydraulic conductivity for various unconsolidated sedimentary materials, sedimentary rocks and crystalline rocks (from Domenico and Schwartz 1990):

 Unconsolidated Sedimentary Materials Material Hydraulic Conductivity(m/sec) Gravel 3×10-4 to 3×10-2 Coarse sand 9×10-7 to 6×10-3 Medium sand 9×10-7 to 5×10-4 Fine sand 2×10-7 to 2×10-4 Silt, loess 1×10-9 to 2×10-5 Till 1×10-12 to 2×10-6 Clay 1×10-11 to 4.7×10-9 Unweathered marine clay 8×10-13 to 2×10-9

 Sedimentary Rocks Rock Type Hydraulic Conductivity(m/sec) Karst and reef limestone 1×10-6 to 2×10-2 Limestone, dolomite 1×10-9 to 6×10-6 Sandstone 3×10-10 to 6×10-6 Siltstone 1×10-11 to 1.4×10-8 Salt 1×10-12 to 1×10-10 Anhydrite 4×10-13 to 2×10-8 Shale 1×10-13 to 2×10-9

 Crystalline Rocks Material Hydraulic Conductivity(m/sec) Permeable basalt 4×10-7 to 2×10-2 Fractured igneous and metamorphic rock 8×10-9 to 3×10-4 Weathered granite 3.3×10-6 to 5.2×10-5 Weathered gabbro 5.5×10-7 to 3.8×10-6 Basalt 2×10-11 to 4.2×10-7 Unfractured igneous and metamorphic rock 3×10-14 to 2×10-10

 To Convert Multiply By To Obtain m/sec 100 cm/sec m/sec 2.12×106 gal/day/ft2 m/sec 3.2808 ft/sec

### Grain Size Relationships

A number of empirical formulas, some dating back over a century, have been proposed which attempt to relate the hydraulic conductivity of an unconsolidated geologic material (granular sediment or soil) to its grain size distribution obtained from sieve analysis. While these formulas can be useful as a first approximation of $K$, one should bear in mind that their generality is limited by a number of factors including the following:

1. the number of sediment samples used to develop the formula
2. the geologic environment(s) comprising the samples used to develop the formula
3. the range of grain size assumed for the formula
4. the uniformity of grain size assumed for the formula

Equations for estimating $K$ from grain size commonly use two metrics from a grain size distribution plot: ${D}_{10}$, the grain diameter for which 10% of the sample is finer (90% is coarser), and ${D}_{60}$, the grain diameter for which 60% of the sample is finer (40% is coarser). ${D}_{10}$ is frequently taken as the effective diameter of the sample while the ratio ${C}_{U}$ = ${D}_{60}$/${D}_{10}$ is known as the coefficient of uniformity.

The following calculator uses formulas by Hazen, Kozeny-Carmen, Beyer and Wang et al. to estimate $K$ from grain size and porosity data.

#### Hazen Formula

Hazen (1892; 1911) developed a simple formula for estimating the hydraulic conductivity of a saturated sand from its grain size distribution:

$KH=CHD102$

where ${K}_{H}$ is hydraulic conductivity [cm/s], ${C}_{H}$ is an empirical coefficient equal to 100 cm‑1s‑1 and ${D}_{10}$ is measured in cm.

As reported by Carrier (2003), ${C}_{H}$ is most commonly given as 100 but published values range over two orders of magnitude from 1 to 1000 cm‑1s‑1. The Hazen formula is assumed valid for 0.1 mm ≤ ${D}_{10}$ ≤ 3 mm and ${C}_{U}$Mexican loteria chips. ≤ 5 (Kresic 1997).

#### Kozeny-Carmen Formula

An equation attributed to Kozeny and Carmen (Freeze and Cherry 1979; Rosas et al. 2014) may be used to estimate the hydraulic conductivity of sediments and soils:

$KKC=CKCgνn31-n2D102$

where ${K}_{\mathrm{KC}}$ is hydraulic conductivity [m/s], ${C}_{\mathrm{KC}}$ is an empirical coefficient equal to 1/180 [dimensionless], $g$ is gravitational acceleration [m/s²], $\nu$ is kinematic viscosity of water [m²/s] and $n$ is total porosity [dimensionless]. ${D}_{10}$ is measured in m.

The Kozeny-Carmen formula is assumed valid for sediments and soils composed of silt, sand and gravelly sand (Rosas et al. 2014).

#### Beyer Formula

Beyer (1964) also proposed a simple relationship for estimating hydraulic conductivity from a sediment's grain size distribution:

$KB=CBgνln500D60/D10D102$

where ${K}_{B}$ is hydraulic conductivity [m/s], ${C}_{B}$ is an empirical coefficient equal to 6×10-4 [dimensionless], $g$ is gravitational acceleration [m/s²] and $\nu$ is kinematic viscosity of water [m²/s]. ${D}_{10}$ and ${D}_{60}$ are measured in m.

The Beyer formula is assumed valid for 0.06 mm ≤ ${D}_{10}$ ≤ 0.6 mm and 1 ≤ ${C}_{U}$ ≤ 20 (Kresic 1997).

#### Wang Et Al. Formula

Wang et al. (2017) developed another empirical formula for estimating hydraulic conductivity from the grain size distribution of a sediment or soil:

$KW=CWgνloggD603ν2-1D102$

where ${K}_{W}$ is hydraulic conductivity [m/s], ${C}_{W}$ is an empirical coefficient equal to 2.9×10-3 [dimensionless], $g$ is gravitational acceleration [m/s²] and $\nu$ is kinematic viscosity of water [m²/s]. ${D}_{10}$ and ${D}_{60}$ are measured in m.

The Wang et al. formula is developed from a dataset (Rosas et al. 2014) characterized by 0.05 mm ≤ ${D}_{10}$ ≤ 0.83 mm, 0.09 mm ≤ ${D}_{60}$ ≤ 4.29 mm and 1.3 ≤ ${C}_{U}$ ≤ 18.3.

## Hydraulic Conductivity Anisotropy Ratio (Kz/Kr)

An anisotropy ratio relates hydraulic conductivities in different directions. For example, vertical-to-horizontal hydraulic conductivity anisotropy ratio is given by ${K}_{z}/{K}_{r}$ where ${K}_{z}$ is vertical hydraulic conductivity [L/T] and ${K}_{r}$ is radial (horizontal) hydraulic conductivity [L/T]. Anisotropy in a horizontal plane is given by ${K}_{x}/{K}_{y}$ where ${K}_{x}$ and ${K}_{y}$ are horizontal hydraulic conductivities in the $x$ and $y$ directions, respectively [L/T].

Todd (1980) reports values of ${K}_{z}/{K}_{r}$ ranging between 0.1 and 0.5 for alluvium and possibly as low as 0.01 when clay layers are present.

### Representative Values

The following table shows representative values of horizontal and vertical hydraulic conductivities for selected rock types (from Domenico and Schwartz 1990):

 Material Horizontal Hydraulic Conductivity(m/sec) Vertical Hydraulic Conductivity(m/sec) Anhydrite 10-14 to 10-12 10-15 to 10-13 Chalk 10-10 to 10-8 5×10-11 to 5×10-9 Limestone, dolomite 10-9 to 10-7 5×10-10 to 5×10-8 Sandstone 5×10-13 to 10-10 2.5×10-13 to 5×10-11 Shale 10-14 to 10-12 10-15 to 10-13 Salt 10-14 10-14

## Storativity (S)

### Confined Aquifers

The storativity of a confined aquifer (or aquitard) is defined as the volume of water released from storage per unit surface area of the aquifer or aquitard per unit decline in hydraulic head. Storativity is also known by the terms coefficient of storage and storage coefficient.

Pumping a well in a confined aquifer releases water from aquifer storage by two mechanisms: compression of the aquifer and expansion of water.

In a confined aquifer (or aquitard), storativity is defined as

$S=Ssb$

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where $S$ is storativity [dimensionless], ${S}_{s}$ is specific storage[L-1] and $b$ is aquifer (or aquitard) thickness [L].

The typical storativity of a confined aquifer, which varies with specific storage and aquifer thickness, ranges from 5×10-5 to 5×10-3 (Todd 1980).

Specific storage is the volume of water that a unit volume of aquifer (or aquitard) releases from storage under a unit decline in head. Specific storage is related to the compressibilities of water and the aquifer (or aquitard) as follows:

$Ss=ρgα+nβ$

where $\rho$ is mass density of water (= 1000 kg/m³) [M/L³], $g$ is gravitational acceleration (= 9.8 m/sec²) [L/T²], $\alpha$ is aquifer (or aquitard) compressibility [T²L/M], $n$ is total porosity [dimensionless], and $\beta$ is compressibility of water (= 4.4×10-10 m sec²/kg or Pa-1) [T²L/M].

### Unconfined Aquifers

The storativity of an unconfined aquifer includes its specific yield or drainable porosity:

$S=Sy+Ssb$

where ${S}_{y}$ is specific yield [dimensionless].

Lowering of the water table in an unconfined aquifer leads to the release of water stored in interstitial openings by gravity drainage.

Compared to gravity drainage, aquifer compression and water expansion in a water-table aquifer yield relatively little water from storage; hence, ${S}_{y}\gg {S}_{s}b$ and $S\cong {S}_{y}$ in unconfined aquifers.

Storativity in unconfined aquifers typically ranges from 0.1 to 0.3 (Lohman 1972).

### Representative Values

The following table provides representative values of specific storage for various geologic materials (Domenico and Mifflin [1965] as reported in Batu [1998]):

 Material Ss (ft-1) Plastic clay 7.8×10-4 to 6.2×10-3 Stiff clay 3.9×10-4 to 7.8×10-4 Medium hard clay 2.8×10-4 to 3.9×10-4 Loose sand 1.5×10-4 to 3.1×10-4 Dense sand 3.9×10-5 to 6.2×10-5 Dense sandy gravel 1.5×10-5 to 3.1×10-5 Rock, fissured 1×10-6 to 2.1×10-5 Rock, sound < 1×10-6

 To Convert Divide By To Obtain ft-1 0.3048 m-1

Freeze and Cherry (1979) provided the following compressibility values for various aquifer materials:

 Material Compressibility, α (m2/N or Pa-1) Clay 10-8 to 10-6 Sand 10-9 to 10-7 Gravel 10-10 to 10-8 Jointed rock 10-10 to 10-8 Sound rock 10-11 to 10-9

Pa-1 = m2/N = m sec2/kg

Example Calculations
1. Use compressibility data to estimate the storativity of a 35-ft thick confined sand aquifer (assume $\rho$ = 1000 kg/m3 and $n$ = 0.3).

$S={S}_{s}b=\rho g\left(\alpha +n\beta \right)b$ = (1000 kg/m3)(9.8 m/sec2) [10-8 m2/N + (0.3) (4.4×10-10 m2/N)](35 ft)(0.3048 m/ft) = 1.1×10-3

How much does the expansion of water contribute to the total storativity in this example?

${S}_{w}=\rho gn\beta b$ = (1000 kg/m3)(9.8 m/sec2)(0.3)(4.4×10-10 m2/N)(35 ft)(0.3048 m/ft) = 1.4×10-5

2. Use specific storage data to estimate storativity for the same confined sand aquifer given in the preceding example.

$S={S}_{s}b$ = (5×10-5ft-1)(35 ft) = 1.8×10-3

#### Specific Storage Calculator

$Ss=ρgα+nβ$

Enter values for aquifer compressibility and porosity to compute specific storage.

#### Storativity Calculator

$S=Ssb=ρgα+nβb$

Enter values for aquifer compressibility, porosity and thickness to compute storativity.

## Specific Yield (Sy)

Specific yield is defined as the volume of water released from storage by an unconfined aquifer per unit surface area of aquifer per unit decline of the water table.

Bear (1979) relates specific yield to total porosity as follows:

$n=Sy+Sr$

where $n$ is total porosity [dimensionless], ${S}_{y}$ is specific yield [dimensionless] and ${S}_{r}$ is specific retention [dimensionless], the amount of water retained by capillary forces during gravity drainage of an unconfined aquifer. Thus, specific yield, which is sometimes called effective porosity, is less than the total porosity of an unconfined aquifer (Bear 1979).

### Representative Values

Heath (1983) reports the following values (in percent by volume) for porosity, specific yield and specific retention:

 Material Porosity (%) Specific Yield (%) Specific Retention (%) Soil 55 40 15 Clay 50 2 48 Sand 25 22 3 Gravel 20 19 1 Limestone 20 18 2 Sandstone (unconsolidated) 11 6 5 Granite 0.1 0.09 0.01 Basalt (young) 11 8 3

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The following table shows representative values of specific yield for various geologic materials (from Morris and Johnson 1967):

 Material Specific Yield (%) Gravel, coarse 21 Gravel, medium 24 Gravel, fine 28 Sand, coarse 30 Sand, medium 32 Sand, fine 33 Silt 20 Clay 6 Sandstone, fine grained 21 Sandstone, medium grained 27 Limestone 14 Dune sand 38 Loess 18 Peat 44 Schist 26 Siltstone 12 Till, predominantly silt 6 Till, predominantly sand 16 Till, predominantly gravel 16 Tuff 21

## Porosity (n)

Porosity is defined as the void space of a rock or unconsolidated material:

$n=Vv/Vt$

where $n$ is porosity [dimensionless], ${V}_{v}$ is void volume [L3] and ${V}_{t}$ is total volume [L3].

### Representative Values

The following tables show representative porosity values for various unconsolidated sedimentary materials, sedimentary rocks and crystalline rocks (from Morris and Johnson 1967):

 Unconsolidated Sedimentary Materials Material Porosity (%) Gravel, coarse 24 - 37 Gravel, medium 24 - 44 Gravel, fine 25 - 39 Sand, coarse 31 - 46 Sand, medium 29 - 49 Sand, fine 26 - 53 Silt 34 - 61 Clay 34 - 57

 Sedimentary Rocks Rock Type Porosity (%) Sandstone 14 - 49 Siltstone 21 - 41 Claystone 41 - 45 Shale 1 - 10 Limestone 7 - 56 Dolomite 19 - 33

 Crystalline Rocks Rock Type Porosity (%) Basalt 3 - 35 Weathered granite 34 - 57 Weathered gabbro 42 - 45

'Bingo'
Nursery rhyme
Songwriter(s)Unknown

'Bingo', also known as 'Bingo Was His Name-O', 'There Was a Farmer Had a Dog', or informally 'B-I-N-G-O', is an English language children's song of obscure origin. Additional verses are sung by omitting the first letter sung in the previous verse and clapping instead of actually saying the word. It has a Roud Folk Song Index number of 589.

## Lyrics

The contemporary version generally goes as follows:[1]

There was a farmer had a dog,
and Bingo was his name-o.
B-I-N-G-O
B-I-N-G-O
B-I-N-G-O
And Bingo was his name-o.
There was a farmer had a dog,
and Bingo was his name-o.
(clap)-I-N-G-O
(clap)-I-N-G-O
(clap)-I-N-G-O
And Bingo was his name-o.
There was a farmer had a dog,
and Bingo was his name-o.
(clap)-(clap)-N-G-O
(clap)-(clap)-N-G-O
(clap)-(clap)-N-G-O
And Bingo was his name-o.
There was a farmer had a dog,
and Bingo was his name-o.
(clap)-(clap)-(clap)-G-O
(clap)-(clap)-(clap)-G-O
(clap)-(clap)-(clap)-G-O
And Bingo was his name-o.
There was a farmer had a dog,
and Bingo was his name-o.
(clap)-(clap)-(clap)-(clap)-O
(clap)-(clap)-(clap)-(clap)-O
(clap)-(clap)-(clap)-(clap)-O
And Bingo was his name-o.
There was a farmer had a dog,
and Bingo was his name-o.
(clap)-(clap)-(clap)-(clap)-(clap)
(clap)-(clap)-(clap)-(clap)-(clap)
(clap)-(clap)-(clap)-(clap)-(clap)
And Bingo was his name-o.

## Earlier forms

The earliest reference to any form of the song is from the title of a piece of sheet music published in 1780, which attributed the song to William Swords, an actor at the Haymarket Theatre of London.[2][3] Early versions of the song were variously titled 'The Farmer's Dog Leapt o'er the Stile', 'A Franklyn's Dogge', or 'Little Bingo'.

An early transcription of the song (without a title) dates from the 1785 songbook 'The Humming Bird',[4] and reads: This is how most people know the traditional children's song:

The farmer's dog leapt over the stile,
his name was little Bingo,
the farmer's dog leapt over the stile,
his name was little Bingo.
B with an I — I with an N,
N with a G — G with an O;
his name was little Bingo:
B—I—N—G—O!
His name was little Bingo.
The farmer loved a cup of good ale,
he called it rare good stingo,
the farmer loved a cup of good ale,
he called it rare good stingo.
S—T with an I — I with an N,
N with a G — G with an O;
He called it rare good stingo:
S—T—I—N—G—O!
He called it rare good stingo
And is this not a sweet little song?
I think it is —— by jingo.
And is this not a sweet little song?
I think it is —— by jingo.
J with an I — I with an N,
N with a G — G with an O;
I think it is —— by jingo:
J—I—N—G—O!
I think it is —— by jingo.

A similar transcription exists from 1840, as part of The Ingoldsby Legends, the transcribing of which is credited in part to a 'Mr. Simpkinson from Bath'. This version drops several of the repeated lines found in the 1785 version and the transcription uses more archaic spelling and the first lines read 'A franklyn's dogge' rather than 'The farmer's dog'.[5] A version similar to the Ingoldsby one (with some spelling variations) was also noted from 1888.[6]

The presence of the song in the United States was noted by Robert M. Charlton in 1842.[7] English folklorist Alice Bertha Gomme recorded eight forms in 1894. Highly-differing versions were recorded in Monton, Shropshire, Liphook and Wakefield, Staffordshire, Nottinghamshire, Cambridgeshire, Derbyshire and Enborne. All of these versions were associated with children's games, the rules differing by locality.[8] Early versions of 'Bingo' were also noted as adult drinking songs.[9]

Variations on the lyrics refer to the dog variously as belonging to a miller or a shepherd, and/or named 'Bango' or 'Pinto'. In some variants, variations on the following third stanza are added:

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The farmer loved a pretty young lass,
and gave her a wedding-ring-o.
R with an I — I with an N,
N with a G — G with an O;
(etc.)

This stanza is placed before or substituted for the stanza starting with 'And is this not a sweet little song?'

Versions that are variations on the early version of 'Bingo' have been recorded in classical arrangements by Frederick Ranalow (1925), John Langstaff (1952), and Richard Lewis (1960). Under the title 'Little Bingo', a variation on the early version was recorded twice by folk singer Alan Mills, on Animals, Vol. 1 (1956) and on 14 Numbers, Letters, and Animal Songs (1972).

### 123 Bingo Flash

The song should not be confused with the 1961 UK hit pop song 'Bingo, Bingo (I'm In Love)' by Dave Carey, which originated as a jingle for radio station Radio Luxembourg.

## References

1. ^Fox, Dan (2008). World's Greatest Children's Songs. ISBN978-0-7390-5206-8., p. 17.
2. ^Gilchrist A. G., Lucy E. Broadwood, Frank Kidson. (1915.) 'Songs Connected with Customs'. Journal of the Folk-Song Society 5(19):204–220, p. 216–220.
3. ^Highfill, Philip H., Kalman A. Burnim, Edward A. Langhans. (1991.) 'Swords, William'. In: A Biographical Dictionary of Actors, Vol 14, p 355.
4. ^n.a. (1785). The Humming Bird : Or, a Compleat Collection of the Most Esteemed Songs. Containing Above Fourteen Hundred of the Most Celebrated English, Scotch, and Irish Songs. London and Canterbury: Simmons and Kirkby, and J. Johnson. p. 399.
5. ^Barham, Richard. (1840). 'A Lay Of St. Gengulphus'. The Ingoldsby Legends. (Full PDF, p. 162)
6. ^Marchant, W. T. (1888). In praise of ale: or, Songs, ballads, epigrams, & anecdotes relating to beer, malt, and hops; with some curious particulars concerning ale-wives and brewers, drinking-clubs and customs. p. 412.
7. ^Charlton, Robert M. (1842). 'Stray Leaves From the Port-Folio of a Georgia Lawyer, part 2', The Knickerbocker 19(3):121–125. p. 123–125.
8. ^Gomme, Alice Bertha (1894). The Traditional Games of England, Scotland, and Ireland: With Tunes, Singing-rhymes, and Methods of Playing According to the Variants Extant and Recorded in Different Parts of the Kingdom. vol 1.