This edition of the Preene Groundwater Consulting blog discusses methods for assessing hydraulic conductivity of soils from particle size data and highlights some of the potential pitfalls if these values are used in dewatering design and other geotechnical problems.
Previous blogs have addressed the question what is hydraulic conductivity? and have clarified the terminology. In geotechnical language hydraulic conductivity is often referred to as coefficient of permeability, most commonly shortened to permeability, but for simplicity we will use the term hydraulic conductivity throughout this blog.
There are several methods for assessing hydraulic conductivity as part of site investigation, including:
It is the last of these – correlations between hydraulic conductivity and particle size distributions in granular soils – that will be discussed here.
SOIL AS A POROUS MEDIUM
Soil is a very complex medium. Conceptually it comprises a skeleton of soil particles in contact with each other, leaving a more or less interconnected system of pore spaces between them. When fluid flows through a soil (and if we assume the soil is saturated then that fluid is water) the flow occurs through the pore space (in the vast majority of soils the soil grains themselves can be considered impermeable). The concept of soils as being a ‘porous medium’ is fundamental to many analysis methods used for groundwater flow problems in soil in the fields of geotechnical engineering and hydrogeology.
Intuitively, it is easy to accept that the ability of a soil to transmit water (i.e. hydraulic conductivity) is controlled, at least in large part, by the nature of the soil pores (the viscosity of water, which will vary with temperature also has an effect, but experience suggests this will be small compared to the effect of the soil type). Features of the soil pores which may have an influence on the flow of water include: the size distribution of the pore space; the tortuosity of the pore space; and the shape and roughness of soil particles forming the edges of the pore space.
Idealised view of soil particles (in black) and surrounding pore space
On a micro-scale the pore spaces are probably a vastly complex hydrodynamic environment, and if it were possible to visualise what they really looked like the scene would probably seem like an alien world out of a science fiction movie. The pragmatic solution for practising engineers and hydrogeologists is to ‘zoom out’ and not to try and discern micro-scale properties but to look for ‘average’ or ‘representative’ parameters or depictions of soil properties. These are the hydraulic conductivity values routinely used in dewatering calculations.
In a porous medium the nature and properties of the pore space will be strongly influenced by the size, shape, roughness and other properties of the soil particles themselves. It is therefore a logical step to think that the hydraulic conductivity must be related in some way to the particle size distribution (and the other properties) of the particles. This has the advantage for practising engineers that we can determine the properties of the particles much more easily than we can determine the properties of the soil pores.
So, beginning in the 19th century, various analysts have developed correlations between the properties of the soil particles and hydraulic conductivity. The most well known is Hazen’s rule, which dates from the 1890s, but there are many others that have been published, and these correlations are still used widely today.
The rest of this blog will discuss some of these hydraulic conductivity correlations. I am not recommending the correlations that are specifically mentioned here, or dismissing any correlations that I do not mention. The examples are simply used to allow discussion of the overall approach of estimating hydraulic conductivity from particle size distributions in granular soils.
WHERE DO THE HYDRAULIC CONDUCTIVITY RELATIONSHIPS COME FROM?
There are some important aspects about this type of hydraulic conductivity correlation that should be remembered when applying these methods for design purposes.
Most of these correlations are not theoretical, but are empirical – in other words they are based on observation. This may involve obtaining a sample of granular material, determining the properties of the particles (for example by sieving to determine particle size) and separately determining the hydraulic conductivity (for example by testing in a permeameter). When this is done for multiple samples it may be possible to identify relationships between hydraulic conductivity and the soil properties across the group of samples.
Furthermore, some of the correlations, including Hazen’s rule, are not for soils at all, in fact they are for granular filter media for water treatment systems. Presumably at some point an enterprising person applied this to a geotechnical problem in sandy soil, liked the results, and the rest is history.
Because these are empirical correlations they are, by definition, applicable only to soils that are similar in nature to those tested in the original study. For example Hazen stated in his work that his rule was applicable over the range of D10 particle size 0.1 mm to 3.0 mm and for soils having a uniformity coefficient (D60/D10) less than five. Unfortunately, this is often forgotten when using Hazen’s rule, and there are many examples of it being applied outside its applicable range, where the results for estimated hydraulic conductivity are likely to be unrealistic. Similar limitations in the range of applicable soils apply to most other correlation methods.
By their nature empirical correlations tend to include some type of correlation factor to relate the particle size factors to hydraulic conductivity. There is a tendency to think of these correlation factors as ‘constants’, while in reality they will rarely be so. Inspection of the original work that developed the correlation often reveals that these factors are not constant but may vary with, for example, temperature and secondary particle characteristics such as angularity and surface roughness.
A final point is that the samples used to develop these correlations were almost certainly not under the same conditions as an in-situ soil. Consider a correlation developed using actual soil samples of a sandy soil (rather than the granular filter media used by, for example, Hazen).
EXAMPLES OF RELATIONSHIPS BETWEEN HYDRAULIC CONDUCTIVITY AND PARTICLE SIZE
Despite these limitations, there are many correlations for granular soils that are widely used, particularly for dewatering design.
At the end of the 19th Century, Allen Hazen, a waterworks and sanitary engineer from New England in the United States was probably the first to propose an empirical correlation for the hydraulic conductivity of sand from its particle size distribution (PSD) curve. Probably due to its simplicity, Hazen’s rule is widely used by today’s geotechnical practitioners, often without due regard to the limitations that Hazen himself stated in his study, which was intended to determine guidelines for suitable sand gradings for water supply filtration. He determined that the D10 particle size (called the ‘effective grain size’) and D60/D10 (the ‘uniformity coefficient’) were both important factors. Hazen’s rule to estimate hydraulic conductivity k is commonly expressed as:
Where C is a correlation factor and D10 is the 10 per cent particle size taken from the particle size distribution curves (see image below).
Example of particle size distribution curve
Hazen also stated that (when k is in m/s and D10 is in millimetres) the correlation factor C could vary between about 0.007 and 0.014. In geotechnical practice, presumably for reasons of simplicity, C is commonly taken to be 0.01. It cannot be stressed too strongly that, even within its range of application, Hazen’s rule gives approximate hydraulic conductivity estimates only.
In the century following Hazen’ s original work several others have developed expressions which relate particle size distributions of sands to hydraulic conductivity. This includes Slichter, Terzaghi, Kozeny and Rose (all reported in Loudon, 1952), Kozeny-Carman (reported in Carrier, 2003), Masch and Denny (reported in Trenter, 1999) and Prugh (originally reported in the first editions of Powers et al, 2007 and included in textbooks such as Cashman and Preene, 2012). Unlike Hazen, who did not seek to address in-situ soils, some correlations include for effects of porosity, angularity of the grains and specific surface of the grains. None claim to be relevant to soils other than ‘a wide range of sands’.
One interesting source is Loudon (1952), which reviewed various published formulae and supplemented the review with laboratory investigations. This concluded that the error prediction using Hazen’s rule could be of the order of plus or minus 200 per cent but that Kozeny’s formula – which is similar to that of Terzaghi, though more complicated – was to be preferred to the various others. Loudon stated that an accuracy of about plus or minus 20 per cent can be expected from Kozeny’s formula.
Loudon also proposed that his own formula, based on Kozeny, should be used for reasons of simplicity, where k is the hydraulic conductivity (in cm/s), n is the porosity of the soil (expressed as a fraction not a percentage), S is the specific surface of the particles (surface area per unit volume of particles, in units of cm2 per cm3) and a and b are correlation factors with values of 1.365 and 5.15 respectively.
The porosity of a sample can be very difficult to determine either in the laboratory or in-situ. This is a limitation on the usefulness of Loudon and other similar works and may be an explanation for the somewhat erratic results that they sometimes give.
POTENTIAL PITFALLS OF THE APPROACH
Even where hydraulic conductivity correlations are applied carefully and to high standards, there are several potential pitfalls to be aware of:
Applying the method to an inappropriate soil type: Any method for correlating hydraulic conductivity with particle size will have a corresponding range of granular soil types to which it is applicable. This will normally be stated in the original source references, and may be defined in terms of ranges of soil parameters such as D10, D50, D60, etc. If a correlation method is applied outside of its range of validity, then significant mis-estimates of hydraulic conductivity may result.
Samples tested for particle size may be unrepresentative of in-situ soil: The samples used for particle size testing may be unrepresentative. When bulk or disturbed samples are recovered from below the water level in a borehole there is a risk that finer particles will be washed from the sample. This is known as ‘loss of fines’. Samples affected in this way will tend to give over-estimates of hydraulic conductivity. Loss of fines is particularly prevalent in disturbed samples taken from the drilling tools. Loss of fines is usually less severe for tube samples; these methods may give more representative samples in fine sands. Conversely, invasion of the samples by drilling mud during sampling may increase the fines content and result in under-estimation of hydraulic conductivity.
Effect of soil structure or fabric: Any soil structure or fabric (e.g. thin silt layers or laminations within a sand bed) present in the in-situ soil may be disturbed during sampling. Even if the fabric is well preserved in the sample itself, it will be destroyed by the process of test specimen preparation for particle size testing, when the sample is effectively homogenised. Hydraulic conductivity estimates based on the PSD curve of the resulting homogenised sample are likely to be unrepresentative of the in-situ hydraulic conductivity. For example, if a clean sand deposit does contain laminations of silt or clay, these will become mixed into the mass of the sample during preparation and the PSD curve will indicate clayey or silty sand; hydraulic conductivity may be under-estimated.
Effect of cementing of soil pores: In many parts of the world, such as the Middle East or locations with a tropical climate, some granular soils may have some weak cementing present between the soil particles, due to mineral deposits such as calcite. These mineral deposits forming the cement will take up some of the space within the soil pores, potentially reducing hydraulic conductivity. This cementing effect will be lost when the sample is broken up during test specimen preparation for particle size testing, and hydraulic conductivity correlations may give erroneous results.
Most projects that involve excavations in granular soils will have some particle size distribution (PSD) data available as part of the site investigation. Correlations with hydraulic conductivity are easy to apply, and are likely to remain part of dewatering design practice. The objective of this blog was to describe the background to these methods and discuss potential pitfalls. As stated earlier, I am not recommending any correlations that are specifically mentioned in this blog, or dismissing any correlations that are not mentioned, the examples are simply used to allow a discussion of the basis and validity of the approach.