Hydraulic conductivity

Allen Hazen derived an empirical formula for approximating hydraulic conductivity from grain size analyses:

${\displaystyle K=C(D_{10})^{2}}$

where

${\displaystyle C}$ Hazen's empirical coefficient, which takes a value between 0.0 and 1.5 (depending on literatures), with an average value of 1.0. A.F. Salarashayeri & M. Siosemarde give C as usually taken between 1.0 and 1.5, with D in mm and K in cm/s.

${\displaystyle D_{10}}$ is the diameter of the 10 percentile grain size of the material

A pedotransfer function (PTF) is a specialized empirical estimation method, used primarily in the soil sciences, however has increasing use in hydrogeology.[1] There are many different PTF methods, however, they all attempt to determine soil properties, such as hydraulic conductivity, given several measured soil properties, such as soil particle size, and bulk density.

The constant-head method is typically used on granular soil. This procedure allows water to move through the soil under a steady state head condition while the volume of water flowing through the soil specimen is measured over a period of time. By knowing the volume ${\displaystyle \Delta V}$ of water measured in a time ${\displaystyle \Delta t}$, over a specimen of length ${\displaystyle L}$ and cross-sectional area ${\displaystyle A}$, as well as the head ${\displaystyle h}$, the hydraulic conductivity, ${\displaystyle K}$, can be derived by simply rearranging Darcy's law:

${\displaystyle K={\frac {\Delta V}{\Delta t}}{\frac {L}{Ah}}}$

Proof: Darcy's law states that the volumetric flow depends on the pressure differential, ${\displaystyle \Delta P}$, between the two sides of the sample, the permeability, ${\displaystyle k}$, and the viscosity, ${\displaystyle \mu }$, as: [2]

${\displaystyle {\frac {\Delta V}{\Delta t}}=-{\frac {kA}{\mu L}}\Delta P}$

In a constant head experiment, the head (difference between two heights) defines an excess water mass, ${\displaystyle \rho Ah}$, where ${\displaystyle \rho }$ is the density of water. This mass weighs down on the side it is on, creating a pressure differential of ${\displaystyle \Delta P=\rho gh}$, where ${\displaystyle g}$ is the gravitational acceleration. Plugging this directly into the above gives

${\displaystyle {\frac {\Delta V}{\Delta t}}=-{\frac {k\rho gA}{\mu L}}h}$

If the hydraulic conductivity is defined to be related to the hydraulic permeability as

${\displaystyle K={\frac {k\rho g}{\mu }}}$,

this gives the result.'

In the falling-head method, the soil sample is first saturated under a specific head condition. The water is then allowed to flow through the soil without adding any water, so the pressure head declines as water passes through the specimen. The advantage to the falling-head method is that it can be used for both fine-grained and coarse-grained soils. .[3] If the head drops from ${\displaystyle h_{i}}$ to ${\displaystyle h_{f}}$ in a time ${\displaystyle \Delta t}$, then the hydraulic conductivity is equal to

${\displaystyle K={\frac {L}{\Delta t}}\ln {\frac {h_{f}}{h_{i}}}}$

Proof: As above, Darcy's law reads

${\displaystyle {\frac {\Delta V}{\Delta t}}=-K{\frac {A}{L}}h}$

The decrease in volume is related to the falling head by ${\displaystyle \Delta V=\Delta hA}$. Plugging this relationship into the above, and taking the limit as ${\displaystyle \Delta t\rightarrow 0}$, the differential equation

${\displaystyle {\frac {dh}{dt}}=-{\frac {K}{L}}h}$

has the solution

${\displaystyle h(t)=h_{i}e^{-{\frac {K}{L}}(t-t_{i})}}$.

Plugging in ${\displaystyle h(t_{f})=h_{f}}$ and rearranging gives the result.

In compare to laboratory method, field methods gives the most reliable information about the permeability of soil with minimum disturbances. In laboratory methods, the degree of disturbances affect the reliability of value of permeability of the soil.

Pumping test is the most reliable method to calculate the coefficient of permeability of a soil. This test is further classified into Pumping in test and pumping out test.

There are also in-situ methods for measuring the hydraulic conductivity in the field.
When the water table is shallow, the augerhole method, a slug test, can be used for determining the hydraulic conductivity below the water table.
The method was developed by Hooghoudt (1934)[4] in The Netherlands and introduced in the US by Van Bavel en Kirkham (1948).[5]
The method uses the following steps:

1. an augerhole is perforated into the soil to below the water table
2. water is bailed out from the augerhole
3. the rate of rise of the water level in the hole is recorded
4. the ${\textstyle K}$-value is calculated from the data as:[6]

${\displaystyle K=F\left(H_{o}-H_{t}\right)/t}$

Cumulative frequency distribution (lognormal) of hydraulic conductivity (X-data)

where: ${\textstyle K=}$ horizontal saturated hydraulic conductivity (m/day), ${\textstyle H=}$ depth of the waterlevel in the hole relative to the water table in the soil (cm), ${\textstyle H_{t}=H}$ at time ${\textstyle t}$, ${\textstyle H_{o}=H}$ at time ${\textstyle t=0}$, ${\textstyle t=}$ time (in seconds) since the first measurement of ${\textstyle H}$ as ${\textstyle H_{o}}$, and ${\textstyle F}$ is a factor depending on the geometry of the hole:

${\displaystyle F=4000r/h'(20+D/r)(2-h'/D)}$

where: ${\displaystyle r=}$ radius of the cylindrical hole (cm), ${\displaystyle h'}$ is the average depth of the water level in the hole relative to the water table in the soil (cm), found as ${\textstyle h'=(H_{o}+H_{t})/2}$, and ${\textstyle D}$ is the depth of the bottom of the hole relative to the water table in the soil (cm).

The picture shows a large variation of ${\textstyle K}$-values measured with the augerhole method in an area of 100 ha.[7] The ratio between the highest and lowest values is 25. The cumulative frequency distribution is lognormal and was made with the CumFreq program.

The transmissivity is a measure of how much water can be transmitted horizontally, such as to a pumping well.

Transmissivity should not be confused with the similar word transmittance used in optics, meaning the fraction of incident light that passes through a sample.

An aquifer may consist of ${\displaystyle n}$ soil layers. The transmissivity ${\displaystyle T_{i}}$ of a horizontal flow for the ${\textstyle i{\mbox{-th}}}$ soil layer with a saturated thickness ${\displaystyle d_{i}}$ and horizontal hydraulic conductivity ${\displaystyle K_{i}}$ is:

${\displaystyle T_{i}=K_{i}d_{i}}$

Transmissivity is directly proportional to horizontal hydraulic conductivity ${\displaystyle K_{i}}$ and thickness ${\displaystyle d_{i}}$. Expressing ${\displaystyle K_{i}}$ in m/day and ${\displaystyle d_{i}}$ in m, the transmissivity ${\displaystyle T_{i}}$ is found in units m²/day.
The total transmissivity ${\displaystyle T_{t}}$ of the aquifer is:[6]

${\displaystyle T_{t}=\sum T_{i}}$ where ${\displaystyle \sum }$ signifies the summation over all layers ${\displaystyle i=1,2,3,\dots ,n}$.

The apparent horizontal hydraulic conductivity ${\displaystyle K_{A}}$ of the aquifer is:

${\displaystyle K_{A}=T_{t}/D_{t}}$

where ${\displaystyle D_{t}}$, the total thickness of the aquifer, is ${\displaystyle D_{t}=\sum d_{i}}$, with ${\displaystyle i=1,2,3,\dots ,n}$.

The transmissivity of an aquifer can be determined from pumping tests.[8]

Influence of the water table
When a soil layer is above the water table, it is not saturated and does not contribute to the transmissivity. When the soil layer is entirely below the water table, its saturated thickness corresponds to the thickness of the soil layer itself. When the water table is inside a soil layer, the saturated thickness corresponds to the distance of the water table to the bottom of the layer. As the water table may behave dynamically, this thickness may change from place to place or from time to time, so that the transmissivity may vary accordingly.
In a semi-confined aquifer, the water table is found within a soil layer with a negligibly small transmissivity, so that changes of the total transmissivity (${\textstyle D_{t}}$) resulting from changes in the level of the water table are negligibly small.
When pumping water from an unconfined aquifer, where the water table is inside a soil layer with a significant transmissivity, the water table may be drawn down whereby the transmissivity reduces and the flow of water to the well diminishes.

The resistance to vertical flow (${\textstyle R_{i}}$) of the ${\textstyle i{\mbox{-th}}}$ soil layer with a saturated thickness ${\displaystyle d_{i}}$ and vertical hydraulic conductivity ${\textstyle K_{v_{i}}}$ is:

${\displaystyle R_{i}=d_{i}/K_{v_{i}}}$

Expressing ${\textstyle K_{v_{i}}}$ in m/day and ${\displaystyle d_{i}}$ in m, the resistance (${\textstyle R_{i}}$) is expressed in days.
The total resistance (${\textstyle R_{t}}$) of the aquifer is:[6]

${\displaystyle R_{t}=\sum R_{i}=\sum d_{i}/K_{v_{i}}}$

where ${\textstyle \sum }$ signifies the summation over all layers: ${\textstyle i=1,2,3,...,n.}$
The apparent vertical hydraulic conductivity (${\textstyle K_{v_{A}}}$) of the aquifer is:

${\displaystyle K_{v_{A}}=D_{t}/R_{t}}$

where ${\textstyle D_{t}}$ is the total thickness of the aquifer: ${\textstyle D_{t}=\sum d_{i}}$, with ${\textstyle i=1,2,3,...,n.}$

The resistance plays a role in aquifers where a sequence of layers occurs with varying horizontal permeability so that horizontal flow is found mainly in the layers with high horizontal permeability while the layers with low horizontal permeability transmit the water mainly in a vertical sense.