Chris Estevez Hydrogeology

Measuring Hydraulic Conductivity Using Petro Physical Measurements: An Analysis of Methods to

Improve Accuracy and Precision on the Field

Abstract:

Hydraulic conductivity is always a challenge to estimate. The environmental clean-up industry

benefits from accurate and precise estimations of hydraulic conductivity in order to develop methods to

clean up contaminants such as heavy metals. In order to speed up the removal of contaminants from land,

different methods based on the principles of hydraulic conductivity can be utilized. Gravitational based

methods can be used for saturated soils, Proportional Cylinder, and Arya-Dierolf Models can be used for

unsaturated soils, saturated soils, and the principles of DC resistivity can be used. All of these methods

can provide effective purposes when it comes to large scale or small scale clean-up operations at

contaminant sites.

Introduction:

Hydraulic conductivity or “K” can be extremely difficult to measure due to the random

probability of factors which can make measurements difficult, inaccurate, and not precise. Vegetation,

soil texture, subterranean bedrock, and fractures just to name a few can all make precision, and accuracy

of measuring hydraulic conductivity problematic on the field. The aim of this paper serves to explore the

physical properties which govern the estimation of hydraulic conductivity, and explore applications which

serve to increase the efficiency of measuring Hydraulic conductivity in a natural environment. An

environment in which the elements can all influence accuracy. Through the exploration of current

hydrogeology models, equations, and physical methods, an analysis of the most effective field techniques,

models, and equations can be made which will aid in the precision, and accuracy needed with respect to

groundwater contamination cleanup. This paper will present concepts, and an analysis of the

effectiveness, and when the concepts presented are applicable with respect to and when to Hydraulic

conductivity can be estimated using a variety of models which have recently been developed with a fair

degree of accuracy. These models correspond to data based on unsaturated and saturated soil.

Darcy’s Law and the Principles of Gravity with Respect to Steady State Flow

In terms of measuring hydraulic conductivity, steady state methods can be used for accuracy of K.

Steady state methods are preferable for measuring K because the method tends to be simpler compared to

transient flow. It is easier to calculate K because less variables and considerations are required.

Considerations such as flux density, and the angle of the radius make transient flow methods difficult to

use (Nimmo 1987, p. 124). With steady state methods, simpler, yet more time consuming methods may be

used. Methods based on Darcy’s Law can be considered. Darcy’s law which is:

Q=-k(dH/dt)(A) (1)

Where: Q discharge, K is hydraulic conductivity, dH/dT is the hydraulic gradient, A is the cross

sectional area

The principles of steady state flow can be used to estimate K with a fair degree of accuracy in

homogenous and saturated samples. Steady state flow is based on the idea of conservation of mass. The

amount of water which flows through a sample that is saturated will at some distance be the same amount

which will be discharged (EPA, 2004). Based on steady state flow, there have been experiments which

have developed methods to estimate K in mediums which are saturated. Since contaminants in the soil at

time may be saturated with water if precipitation occurred or if the ground is near a flood plain, the

consideration for saturated medium must occur. Another consideration is if the ground is relatively

homogenous or uniform with the same type of soil such as sand, gravel or silt for example. Darcy’s law

meets these considerations.

Darcy’s law is effective for finding out K in soil which is completely saturated as well as partially

saturated. The formula takes into consideration that the total driving force is proportional to the flux. This

consideration has led to several tests which provide methods for measuring K, and some are more

effective than others. These methods depend on the current environment which is being tested. Whether

the environment is uniformed, saturated, unsaturated, or mixed all modify Darcy’s Law and the inherent

accuracy of K. A steady state method that can be used is based on centrifugal forces (Nimmo 1987,

p.125). A centrifugal field which is based on the principles of gravity can be shown in figure 1. In Figure

1 illustrates a very important concept in a method which pertains to accurate measurement of fluid in

saturated medium. In the real world, water movement is based on gravity. Gravity which is a force that

governs almost every single movement in terms of Hydrogeology that occurs. Since water is driven by a

centrifugal force or the density of the water, angular speed of rotation, and the distance from the axis of

rotation, overtime the flow will become steady (Nimmo 1987, p.126) This will allow Darcy’s Law to be

used to estimate K. Darcy’s law with consideration to centrifugal force is now:

q = - K (dH/dR) – pw^2*r (2)

Where: Q is discharge, K is the hydraulic conductivity, dH/dR is the hydraulic gradient, and pw^2*r is

centrifugal force where p is the density of water, w is the angular speed of rotation, and R is the distance

from a centrifuge (Nimmo 1987, p.126).

Given flow movement is based on gravity, using a modified Darcy’s Law for purposes of analysis

allows for the measurement of K to be more accurate compared to equation 1 which does not consider

centrifugal force. The more parameters or variables, the degree of K estimation or accuracy increases.

This modification of Darcy’s law is best used in saturated environments as parameters such as the density

of water at the location of a measured site is needed. This density may be affected by other factors such as

contaminants which may be present due to pollutants such as mercury or lead which may alter the density

of the fluid. This version of Darcy’s law is accurate because on the field, fluid is not the same density

everywhere. Since gravity and the density of a fluid play a role in underground movement, by including

centrifugal force as a substitution for the area variable, the formula considers gravitational movement in a

saturated, steady flow system. This can apply to real world scenarios such as a contaminant site which is

constantly saturated or most of the time at least such as a flood plain, or a swamp. Further increasing the

accuracy of K estimation.

The Proportional Cylinder (PC) model and the Arya-Dierolf (AD) Model

With contaminants, measuring hydraulic conductivity can be important by knowing the pore size of

the soil. The Proportional Cylinder (PC) model (figure 2) and the Arya-Dierolf (AD) Model (figure 3) can

be used to determine unsaturated hydraulic conductivity. The PC model is based on certain assumptions.

These assumptions are that a fixed cylinder shape is applied to each pore in which the radius and length

are proportional. The Arya-Dierolf Model comes with the assumption that the pores corresponding to the

grain size have equal length and a radius which allows for a certain value to correspond to particle size,

and void ratio (Nimmo, Herkelrath 2007; p. 768).

The Proportional Cylinder (PC) model is based on the equation:

h =CpcR (3)

Where: h is a geometric constant( figure 2), Cpc is a proportionality constant, R is the effective

radius. (Nimmo, Herkelrath 2007, p. 767)

Equation 3 shows the relationship for the PC model, and equation 4 and 5 listed below determines

the volumetric water content of a given pore.

θ ψ =ᶲ Mcum ((Ce/ ψ)^.66)*(3hPC/4e)^.33) (4)

Where: θ ψ is the volumetric water content of a given pore based on water head (Sharp 2007, p.

25), ᶲ Mcum is the cumulative mass of particle distribution, Ce/ ψ)^.66) is the capillary behavior,

e is the void ratio, and hPC is the geometric constant based on the The Proportional Cylinder

Model. (Nimmo, Herkelrath 2007, p. 768)

The Arya-Doerolf (AD) model is based on the equation:

θ ψ =ᶲ Mcum ((Ce/ ψ)^.66)*(3hAD/4e)^.33) (5)

Where: θ ψ is the volumetric water content of a given pore based on water head (Sharp 2007, p.

25), ᶲ Mcum is the cumulative mass of particle distribution, Ce/ ψ)^.66) is the capillary behavior,

e is the void ratio, and hAD is the geometric constant based on the Arya-Doerolf Model (Nimmo,

Herkelrath 2007, p. 767-68)

These models can provide clues to estimate hydraulic conductivity on a scale needed to remove

contaminants because they focus on determining the size of each individual pore in a unsaturated

medium. One of the problems that face waste removal is what if they ground is not saturated. With

both of these models, both the Proportional Cylinder and the, unsaturated pore size can be

determined. This size is based on the volumetric water content of a given pore based on water head,

which is an estimate of the pore size if it is filled with water. This could also apply to saturated soils

as well. As it is an estimate, the model can be used to determine the geometric constant. The model as

shown in figure 3 shows on the y-axis a numerical number for a fraction and an effective radius for

three types of soil: small, medium, and coarse, on the x-axis. Based on the typical effective radius

typical for a specific type of soil, a constant can be generated. This constant along with typical

capillary behavior, and void ratio common for these types of soil can provide a means for estimating

K once the volume for a pore is known (Nimmo, Herkelrath 2007, p. 771)

The Proportional Cylinder model works in a similar fashion, but the geometric constant, or hPC is

based on the Arya-Dierolf model. So in essence, the models work side by side but the benefit of the

PC model is that it provides information on the matric potential of the soil (Nimmo, Herkelrath 2007,

p.770).

The matric potential accounts for the moisture content. This moisture content is essentially the

effect of water molecules building up along the surface of a pore. (Taiz, Lincoln 2010). What the PC

model does is based on the water content or the pore or θ ψ calculated using the AD model, the matric

potential of a sample of soil in this case can be found, based on the size of particles: fine, medium,

and coarse grained. The PC model itself along with the AD model is based on empirical data based on

countless trials of different grain sizes tested using the relationship based on equation 3, 4, and 5.

(Nimmo, Herkelrath 2007, p.772).

The matric potential is useful because it can account for the saturation of the pore, and this can be

useful for hydraulic conductivity estimates because for one to understand how easy it is for water or

any fluid for this matter to flow through a medium, the matric potential help give ranges based on the

moisture content in percentage, and with a certain matric potential in units of kPa, the critical pre

radius can be determined. Radius can also be determined from θ ψ using the AD model, but the

matric potential can provide a farther degree of accuracy. One, it can provide a estimate for moisture

content, and the critical pore radius which can all aid in estimating the hydraulic conductivity

(Richard, 2000) Especially out in the environment where ground moisture may be important to

understanding how contaminants move about in the ground. The whole purpose behind the AD and

PC models is to provide more information using common hydrogeological parameters or in this case

the equation variables presented in the equations for both models. Table 1 illustrates an example of

matric potential, moisture content, and critical pore radius measured in mm (Richard, 2000).

Electrical properties of hydraulic conductivity are important to understand as they can help

estimate K. Particularly using the geophysical method of resistivity which focus on the physical

property of electrical conductivity. These methods may provide accurate estimates of K then

conventional means such as bore-holes or slug tests. Resistivity methods also provide a cost-effective,

efficient, and the ability to cover a large area (Slater 2007, p. 177)

Geophysics can provide a way to accurately measure or account for K in the soil in a way that it

accounts for the spatial intricacy movement of fluid in a ground (Slater 2007, p.171). One of the

problems with K estimation with respect to contaminants is that a level which is precise and accurate

is needed to isolate contaminants that can be extremely toxic to the environment. Say a contaminant

near a major drinking water such as sulphates. Tests such as slug test, and borehole methods give a

biased view of hydraulic conductive, and this bias can cause errors in terms of accuracy. These errors

can occur from how the bore-hole method is dug, or even how the slug test is set up (Slater 2007,

p.170). With the AD and PC model mentioned previously, the models can be used with a sample, but

using a sample is only good when trying to find K at a level which is somewhat limited in the area

that it covers. K estimates need to be covered at a macroscopic level meaning they cover larger areas.

An oil spill for example from a pipeline somewhere Midwest would benefit from geophysical

methods more so then smaller isolated cases in which K estimates from Darcy’s law using

gravitational methods, or models such as the PC and AD model presented would be more appropriate.

Electrical properties are important because they relate the Hydrogeological concepts to the

Physics connect of current, and resistivity. There is a similarity between K and how it can be

estimating using resistivity. Resistivity has been shown to have a similarity to giving an estimate to

the interconnected pore volume (Slater 2007, p.177). Using DC restively over a large area can give a

lot of data on the sub terrain. It can provide information on the bedrock, deposits of ore, or even

aquifer locations. When applied over a large area, an estimate of the interconnected pore volume can

be determined. Using the relationship in equation 4:

ρ’ = (ρ’’ int) (6)

Where: ρ’ is the resistivity, and (ρ’’ int) is the resistivity which can be connected to the

interconnected pore space (Slater 2007, p.177).

This relationship using resistivity techniques can provide an estimation of the pore space, which

can provide a method for estimating K over a large area. There is one drawback to using resistivity

though. Resistivity can depend on a couple of factors that might be present in the soil tested. These

factors include the geochemical composition of the soil, the mineralogy of the soil, and the

distribution of grains at a sample site (Slater 2007, p.177). These estimates of resistivity can provide

an idea of what type of rock, and even chemical is present in the soil. The conductivity derived from

resistivity can be seen in figure 4 (Wightman, Sirles, K. 2003).

Conclusion:

The problem with estimating hydraulic conductivity can be a challenge, especially for the

environmental industry. Estimating the proper value for K can be important in order to devise methods

to properly clean sites. There have been terrible man-made disasters which have caused contamination

over a large region. Some examples include the Love Canal neighborhood which is in Niagara Falls, New

York which had thousands of tons of highly toxic waste underneath it or the Pitcher lead contamination

site in Pitcher, Oklahoma in which contaminated water from the nearby coal mines spread onto nearby

rivers, and creeks, polluting the environment and devastating the town (Mother Nature Network, 2013).

With both these polluted sites, one question arises. Which method is the best with respect to remove

contaminants within the soil with respect to hydraulic conductivity? Since hydraulic conductivity

describes the ability or ease of fluid to move in the soil (Taiz, Lincoln 2010), the various methods

presented in this paper based on the foundations of K can be grouped into two groups. One group being

estimating K in a smaller area, and the second being estimating K over a large area. For estimating K in a

smaller area which is saturated, Darcy’s law using gravitational methods is the flexible choice along with

the developed AD and PC models by John R. Nimmo, J. Rubin, and D. P. Hammermeister because they

provide the crucial saturated pore size and radius information which is crucial to estimating K. This can

be useful because the ease of which a fluid moves through the ground is dependent on these

parameters. The EPA for example would find this information useful or any Hydrogeological engineer in

order to devise a way to clean up a contaminated site. This information, coupled with the type of

contaminant can make the development of such clean-up methods possible or easier. For large scale

areas, DC resistivity provides a promising approach. Although the interconnected pore size is subjected

to the chemistry, and distribution of particles in the ground, it is a means to get a general idea of the K

over a vast area. But when it comes to estimating K with respect to contaminants, smaller scale methods

are better as precision and accuracy is needed because with some contaminants, a high enough particles

per million(ppm) can mean life or death for animals or humans.

Figures/Tables:

Method using Darcy’s Law which is based on centrifugal forces.

Figure 1: (Nimmo 1987, p.125)

In figure 1, gravity plays a crucial role in proving that Darcy’s Law can be used for steady flow situations. Overtime the porous medium will come to an equilibrium in which the flow is steady (Nimmo 1987, p.126). In this experiment, the material in between the ceramic plates is rotated in a centrifuge, similar to pendulum. Overtime, gravity will lose energy, and the flow will become steady. This also can show how the conservation of mass applies to Darcy’s Law in the case of Darcy’s Law including centrifugal force (Nimmo 1987, p.127)

Figure 2: Arya-Dierolf (AD) model (Nimmo, Herkelrath 2007, p.768)

Figure 2 shows that the effective radius based on a sample (fine, medium, or coarse) is directly related to a geometric function, which can be used in the Arya-Dierolf (AD) model formula to find the water content that a pore takes up.

Figure 3: The Proportional Cylinder (PC) model (Nimmo, Herkelrath 2007, p.770)

Table 1: (Richard, 2000)

Moisture Content(%)

Matric Potential(kPa)

Critical poreradius (mm)

30 -239.8 0.0005640 -169.4 0.0008050 -98.9 0.0013655 -63.7 0.0021260 -28.5 0.0047362 -14.4 0.0093564 -0.3 0.391

64.048 -0.00007 19.2

This table shows the relationship between matric potential, moisture content, and critical pore radius at a certain kPA value. Note that this relationship is a not proportional as the PC and AD model graphs are not linear.

Figure 4: (Wightman, Sirles, K. 2003)

References:

1)  EPA. "General Approach." Further Description:- Hydrogeology. EUGRIS, 2004. Web. 16 Apr. 2013.

2) Mother Nature Network. "America's 10 Worst Man-made Environmental Disasters."MNN - Mother Nature Network. MNN HOLDINGS, LLC., 2013. Web. 20 Apr. 2013.

3)  Nimmo, J. R., J. Rubin, and D. P. Hammermeister. "Unsaturated Flow in a Centrifugal Field' Measurement of Hydraulic Conductivity and Testing of Darcy's Law."WATER RESOURCES RESEARCH 23.1 (1987): 124-34. USGS. Web. 15 Apr. 2013. <http://wwwrcamnl.wr.usgs.gov/uzf/abs_pubs/papers/WRR.23.1.pdf>.

4) Nimmo, John R., William N. Herkelrath, and Ana M. Laguna Luna. "Physically Based Estimation of Soil Water Retention from Textural Data: General Framework, New Models, and Streamlined Existing Models." National Research Program, Western Region. USGS, 22 Jan. 2007. Web. 15 Apr. 2013.

5) Richard, Tom. "Capillary Theory and Matric Potential."Http://compost.css.cornell.edu/oxygen/capillary.html#2. Cornell Composting Science and Engineering, 8 Nov. 2000. Web. 20 Apr. 2013.

6) Sharp Jr., John M. "A GLOSSARY OF HYDROGEOLOGICAL TERMS."Http://www.geo.utexas.edu/faculty/jmsharp/sharp-glossary.pdf. Department of Geological Sciences Jackson School of Geosciences The University of Texas Austin, Texas, USA, 2007. Web. 16 Apr. 2013.

7) Slater, Lee. "Near Surface Electrical Characterization of Hydraulic Conductivity: From Petrophysical Properties to Aquifer Geometries—A Review." Mtnet.dias. Springer Science+Business Media B.V., 7 July 2007. Web. 19 Apr. 2013.

8) Taiz, Lincoln, and Eduardo Zeiger. Plant Physiology. 5th ed. Sunderland, Massachusetts: Sinauer Associates, 2010. Print.

9) Wightman, W. E., Jalinoos F. Sirles, and Hanna K. "Induced Polarization (IP) and Complex Resistivity." U.S. Environmental Protection Agency. Federal Highway Administration, Sept. 2003. Web. 10 Apr. 2013.