# Box 4 Methods for Estimating Hydraulic Conductivity

Hydraulic conductivity, *K*, is used to describe the capacity of a porous material to transmit water. Estimating representative values of hydraulic conductivity for a wide variety of porous media is required to quantitatively describe groundwater flow rates (*Q*), fluxes (*q*) and velocities (*v*), and determine the spatial and temporal distribution of hydraulic heads. Without hydraulic conductivity data neither simple analytical solutions nor complex computer simulations of groundwater flow are possible. As a result, a number of methods have been developed to characterize the hydraulic conductivity of earth materials. Laboratory and empirical methods are applied to small sample volumes and, although the values provide insight into the general magnitude of hydraulic conductivity at a field site, they may not provide values that are appropriate to fully predict how a groundwater system will behave at the field scale. Generally, field tests involve pumping one well while observing groundwater levels in nearby wells (*aquifer testing*), but some field tests use individual small diameter wells, and/or boreholes, to observe groundwater responses to sudden changes in water levels (*slug testing*). Field tests provide values of *K* that incorporate the broad complexities of the natural system such as the interconnectedness of subsurface zones of high and low hydraulic conductivity (Figure Box 4-1). Field methods for measuring hydraulic conductivity are not presented here. This material focuses on laboratory testing methods and linking these methods with empirically generated relationships between sediment characteristics and hydraulic conductivity.

**Permeameters**

One approach to determining *K* for a sample of earth material in the laboratory is to saturate the sample and drive water through it while observing the flow rate and hydraulic heads. The sample is placed in a cylindrical chamber and water flow is set either at a constant rate (steady state flow) or allowed to decline over time.

The most useful samples are collected using a coring tool or a split-spoon sampler while drilling a borehole because they provide relatively undisturbed samples of the subsurface. Sample collection techniques that drive sample tubes into sediment may compact the sample and change its hydraulic conductivity. Sometimes disturbed granular material is collected and repacked to the approximate consolidation measured in the field. Field consolidation conditions are determined using a resistance-to-penetration test which can be achieved with a variety of tools and procedures, but, in brief, reflects either the pressure required to push a tool into the earth, or a record of the number of blows needed to drive a split spoon coring device a specified distance into the sediments exposed at the bottom of a borehole. Loose repacked samples have lost the layering and complex microstructures that may exist at the field site and influence *K*. Consequently, hydraulic conductivity measurements conducted on such samples provide only a rough estimate of the field value. In some cases, repacked samples are tested a number of times using varying degrees of consolidation and the measured values are averaged.

Standard permeameter operating procedures have been outlined in guidelines provided by the American Society of Testing and Materials (ASTM, 1991). Procedures such as these should be consulted when designing and operating permeameters.

**Constant Head Permeameter**

For a *constant head permeameter* test, an undisturbed core, or repacked sample, is place in the chamber (Figure Box 4-2). Determining *K* from a constant head permeameter test requires application of Darcy’s Law. The apparatus maintains a constant flow of water and hydraulic gradient (*∆h*/*∆L*) as water flows through a fixed length of sample of a given cross-sectional area. Measurements are made under steady-state conditions. *K* is computed by solving Darcy’s law for the hydraulic conductivity as shown in the main text of this book and repeated here for convenience (Figure Box 4-2 and Equation Box 4-1).

(Box 4-1) |

A classroom-scale Darcy apparatus is shown in Figure Box 4-3. Water is pumped from the sump in the lower left to an upper constant-head reservoir in the upper left. That head is maintained by allowing the water to overflow the top of the reservoir where it is collected and returned to the sump. Water flows from the reservoir into the sand-filled column which has an inner diameter of 2 inches (5.08 cm), and an area of 20.27 cm^{2}. There is essentially no resistance to flow in the tube that carries water to the sand and so the head in the reservoir is the head at the face of the sand. Piezometers monitor head at a few locations along the column within the sand. Outflow from the column is controlled by the constant head at the lower reservoir. Once steady flow is established, the outflow is collected in a graduated cylinder. For example, a volume of 148 milliliters (148 cm^{3}) is collected in 90 seconds so the flow rate, *Q*, is 1.64 cm^{3}/s. The gradients between the piezometers vary, being -0.36, -0.34, and -0.44 from left to right, respectively. Rearranging Equation Box 4-1 to solve for *K* = −*Q*/(*Ai*) results in *K* values corresponding to the measured gradients of 0.23, 0.24, 0.18 cm/s respectively. These are a reasonable value for sand. The column can be rotated at the midpoint because the piezometers are flexible tubes. The angle of the column does not affect the head or flow rate because they are controlled only by the reservoir heads, the geometry of the tube (area and length), and the *K* of the sand.

The observed variation in gradients along the flow column could be due to slightly different packing of the sand, or errors in the piezometer readings which might result from something like an air bubble in one of the tubes. For practical groundwater work, this sand would be assigned a *K* of 0.2 cm/s. Often knowing *K* within an order of magnitude is beneficial in groundwater analysis. There is much uncertainty in quantifying groundwater flow, and the range of the *K* values from this experiment is small compared with the many other sources of error in groundwater analysis.

When water is following though a constant head permeameter the flow needs to be low enough that unconsolidated sediment grains are not being separated and frictional losses in the apparatus are too large. Cormican et al. (2020) suggest that a reasonable gradient can be determined by making *K* measurements with progressively lower gradients until two consecutive runs yield the same value. Klute (1965) noted that for most constant head permeameters the lower limit of measurement was about 0.00016 cm/s (0.14 m/d).

**Falling Head Permeameter**

The constant head permeameter is useful for measuring *K* of samples with higher values of hydraulic conductivity. When K is low, it is difficult to establish a constant flow rate over a relatively short time interval (minutes-hours), so instead, a *falling head permeameter* is used. In this method the water levels and flow rates change over time. Data requirements include the dimensions of the sample and connected tube, and the change in water level over time (Figure Box 4-4).

Over a specified period of time, *t*_{n}* − t*_{0}, the flow rate from the tube, *Q*_{tube}, and the flow of water through the sample, *Q*_{sample}, are equal.

= | (Box 4-2) |

where:

Q_{tube} |
= | average volumetric discharge from tube during test (L^{3}/T) |

A_{tube} |
= | area of the tube is 0.7854d_{tube}^{2} (L^{2}) |

h_{0} |
= | height of water in the tube relative to the outlet at start time (L) |

h_{n} |
= | height of water in the tube relative to the outlet at end time (L) |

t_{n} |
= | end time (T) |

t_{0} |
= | start time (T) |

= | change in head over the duration of the test (L/T) |

(Box 4-3) |

where:

Q_{sample} |
= | average volumetric discharge through sample during test (L^{3}/T, volume over time) |

A_{sample} |
= | area of the sample is 0.7854d_{sample}^{2} (L^{2}) |

L |
= | length of sample (L) |

= | average gradient across the sample during the test because outlet elevation is the datum (L/L, unitless) |

Equating Equation Box 4-3 to Equation Box 4-2 results in Equation Box 4-4.

(Box 4-4) |

where:

K |
= | hydraulic conductivity of the sample (L/T) |

Rearranging Equation Box 4-4 leads to Equation Box 4-5.

(Box 4-5) |

where:

h |
= | hydraulic head measured from the outlet datum (L) |

Integrating with the conditions that *h* = *h*_{0} at *t*_{0} and *h* = *h*_{n} at *t*_{n}, provides a solution for *K* as Equation Box 4-6.

(Box 4-6) |

Substituting the areas of the sample and tube with *A*_{sample} = 0.7854*d*_{sample}^{2} and *A*_{tube} = 0.7854*d*_{tube}^{2} because the area is *πr*^{2}, or *π*(*d*/2)^{2}, which is (*π*/4)*d*^{2} as indicated in (Figure Box 4-4), the hydraulic conductivity is computed as shown in Equation Box 4-7.

(Box 4-7) |

**Empirical Relationships Used to Estimate Hydraulic Conductivity**

Over time researchers have attempted to generate relationships between physical parameters of unconsolidated samples (especially grain size) and hydraulic conductivity. Researchers often perform grain size distribution analysis of granular samples to determine engineering and hydrogeologic properties. The procedure for determining grain-size distribution is described in Box 2 of this book.

The most commonly developed empirical relationships relate *K* to grain-size distribution data (median grain size, effective grain size, uniformity of grain size and mean grain size). A correlation supported by observing functional relationships between measurements of different related parameters on many individual samples are referred to as *empirical relationships* or *empirical equations*. Such relationships were developed by conducting both grain size analysis and a permeameter tests on the same sample for 10’s to 100’s of samples. Most all of the equations were developed for sand-rich samples. Four of these relationships are included here: Slichter (1899), Hazen (1911), Terzaghi (1925), and the USBR (United States Bureau of Reclamation) method as described in Vukovic and Soro (1992). Devlin (2015) developed an Excel based tool to estimate hydraulic conductivity from grain size analysis data sets that applies up to 15 empirical equations. The tool for estimating hydraulic conductivity from grain size can be accessed by clicking here. To properly apply this public domain software tool, the documentation should be carefully reviewed in order to understand the equations and their associated assumptions and limitations.It is important to know that empirical expressions do not usually rely on use of consistent units. Consequently, it is imperative that the user of the expression take note of the input units for each variable and the associated output units for *K*.

**K estimated from grain size data using the Slichter method**

Slichter (1899) developed an empirical equation for *K* that related the square of the mean grain diameter, *d*, to a constant related to porosity. Fraser (1935) provides additional details on the development of the equation as shown in Equation Box 4-8.

(Box 4-8) |

where:

K |
= | estimated hydraulic conductivity in centimeters per second |

d |
= | mean grain diameter in centimeters |

µ |
= | dynamic viscosity at a given temperature in gram/(cm sec) |

C_{S} |
= | a constant for a given porosity (n) based on the following non-linear relationship: (n=26% C_{S}=84.3; n=36% C_{S}=28.8; n=47% C_{S}=11.8) |

The relationship of *C*_{S} to porosity is shown in Figure Box 4-5.

Slichter’s relationship was formulated assuming grains are uniform spheres regularly distributed so the value *d* represents the mean diameter of the grains. He stated that the value of *d* for a for a sample of grains of various sizes should be equal to the diameter of a sample with grains of a single size that would yield an appropriate coefficient of permeability (hydraulic conductivity). The mean grain diameter is computed from the grain size distribution curve and is presented in Box 2 of this book.

Vukovic and Soro (1992) formulated *C*_{S} as 1/*n*^{3.287} (error +/- 5%) and interpreted *d* as the effective grain size (*d*_{10} percent finer than grain size distribution, see Box 2 of this book) when they presented a formulation of the Slichter equation. The use of the effective grain size, which is directly derived from the grain size distribution curve, is smaller than the mean grain diameter and its use reduces the computed hydraulic conductivity value. They suggest that the use of the Slichter method is appropriate when a sample uniformity coefficient is less than 5.

**K estimated from grain size data using the Hazen method**

Hazen (1911) developed a relatively simple equation for *K* based on the effective grain size (*d*_{90} cumulative percent retained or *d*_{10} finer-than) and a constant that varied with the dominate material type and sorting as shown in Equation Box 4-9.

(Box 4-9) |

where:

K |
= | estimated hydraulic conductivity in centimeters per second |

d_{10} |
= | effective grain size in centimeters (d_{10} finer-than, d_{90} retained) |

C |
= | dimensionless coefficient defined as follows: |

Material |
C |

Very fine sand, poorly sorted | 40-80 |

Fine sand with appreciable fines | 40-80 |

Medium sand, well sorted | 80-120 |

Coarse sand, poorly sorted | 80-120 |

Coarse sand, well sorted, clean | 120-150 |

Often, as a first estimate, the Hazen approximation constant, *C*, is set to 100. Professional judgement is used in applying this approach. Sorting used here is assumed to refer to the general uniformity of grain sizes. A uniformity coefficient analysis of the size distribution data is useful when selecting a value of *C*. Uniformity coefficient is discussed in Box 2 of this book. Vukovic and Soro (1992) suggest the Hazen method is applicable for samples with a uniformity coefficient of less than 5.

**K estimated from grain size data using the Terzaghi method**

Terzaghi (1925) worked in soil mechanics and developed an equation for *K* of sand samples that accounted for sample porosity and the effective grain diameter, while adjusting for the difference in fluid viscosity between the testing temperature and typical field temperature, as shown in Equation Box 4-10.

(Box 4-10) |

where:

K |
= | estimated hydraulic conductivity in centimeters per second (L/T) |

C |
= | Terzaghi coefficients ranging from 600 to 800 to reflect variation from irregular to smooth particles respectively (dimensionless) |

V_{0} |
= | viscosity at 10°C in the same units used for V_{T} |

V_{T} |
= | viscosity at the test temperature in the same units used for V_{0} |

n |
= | porosity (dimensionless) |

d_{10} |
= | effective grain diameter (d_{10} finer-than, d_{90} retained) in centimeters (L) |

The constants (Terzaghi coefficients) represent a range of particle conditions, generally, 800 is used for samples with rounded and smooth grains and 600 for coarse grains with irregular shapes. Vukovic and Soro (1992) summarized the literature and found the following values for Terzaghi coefficients: sea sand, 750-663; dune sand 800; pure river sand, 696-460; muddy river sand 203.

**K estimated from grain size data using the USBR method**

The United States Bureau of Reclamation (1978, as reported in Vukovic and Soro, 1992) developed a relationship shown in Equation Box 4-11 for medium sands with a uniformity coefficient of less than 5.

K = 0.36 (d_{20}^{2.3}) |
(Box 4-11) |

where:

K |
= | estimated hydraulic conductivity in centimeters per second (L/T) |

d_{20} |
= | grain size in centimeters (d_{20} finer-than) (L) |

**Other empirical equations for estimating K**

Other authors have formulated empirical equations, though most are complex or require professional judgement in their use. Shepherd (1989) used median gain size raised to an exponent between 1.5 and 2, and a shape factor to estimate K. Masch and Denny (1966, pp. 665-677) developed a graphical empirical relationship based on the median grain diameter, *d*_{50} in phi units, and the inclusive standard deviation, *σ*_{I}. Using their graph, the median grain diameter is projected from the *x* axis to the appropriate plotted line of measured intrinsic standard deviation, then the corresponding *K* is read from the *y*-axis. Slichter (1899), Terzaghi (1925), the Kozeny-Carmen equation (Bear, 1972), Fair and Hatch (1933) and others have attempted to use porosity as a factor in grain size relationships for computing hydraulic conductivity. However, porosity is poorly correlated to *K* and relationships that are proposed are non-linear. Also, most empirical equations based on grain size require a coefficient to bring grain size data into line with permeameter results. Most empirical relationships have been developed for unconsolidated sand rich materials. Cedergren (1989, page 42) describes the use of laboratory consolidation tests to estimate the hydraulic conductivity of clays and silts. The hydraulic conductivity is directly proportional to a time factor for a given percent of consolidation.

Clearly, estimates of *K* can be made based on grain size analysis, however, if multiple methods are used, they may not agree. Cormican et al. (2020) evaluated the use permeameter and grain size analyses to determine the hydraulic conductivity of engineered porous media used in groundwater remediation. They compared the results of 16 empirical methods with permeameter results. For the material they tested, the methods of Slichter (1898) and Shepard (1989) were most representative. If empirical equations are used to characterize *K* for a groundwater project, it is recommended that some site samples be analyzed with both a laboratory-based permeameter test and a grain size distribution analysis so that the most representative of the empirical equations can be used and their coefficient can be adjusted to better represent site-specific conditions.

As stated previously, a challenge in using both permeameter and grain size analyses to estimate *K* arises from the fact that they represent only a small portion of the field setting, essentially a point sample. Also, disturbed samples don’t include the micro structures and packing of the field sediments. Thus, the values may not appropriately represent conditions at the field site. However, they are commonly used to establish the range *K* values for field sites and to estimate groundwater flow rates.

**Field Tests to Establish Hydraulic Conductivities**

Characterizing the hydraulic conductivity of unconsolidated and consolidated earth materials in a field setting is often accomplished by using boreholes and wells to observe the response of a groundwater system to the withdrawal of water. The general principle of deriving a representative field-scale value of hydraulic conductivity is to use boreholes or wells installed in such a way that the earth material properties are not altered. Then the groundwater system is manipulated, usually by pumping one of the wells, and monitoring water level declines in nearby wells. The water level declines are called drawdowns and the approximate cone-like shape of the head distribution around the pumping well is called a drawdown cone. When the pumping is stopped, water levels rise (recover). If the *K* value of the aquifer is high, water levels will not change as much in response to pumping and recovery as when the same process is performed in material with a low *K*. Thus, observing groundwater responses to changes in the groundwater system yields information on hydraulic conductivities. Analytical equations, numerical methods, and computer simulations of groundwater flow, are used in conjunction with the measurements of drawdown versus time to characterize hydraulic conductivities at both local and regional scales. A schematic of a field test in a simple setting is shown in Figure Box 4-6.

Field methods used to determine hydraulic conductivity are explained in a number of modern and older references. Foundational material on groundwater hydraulics including the presentation of a wide range of aquifer test methods can be found in the United States Geological Survey Professional Paper 708 by Lohman (1972). A good description of how field-scale *K* values are derived is also found in the publicly available Ground Water Manual: A Water Resources Technical Publication, (United States Bureau of Reclamation, 1995).