Hydrology
Lecture 4 Porosity, Permeability, and Darcy’s Law
Finally, we begin our study of groundwater. First, we need to consider the variables that
affect the storage and movement of water in aquifers.
Porosity
Groundwater occurs in the void spaces of earth materials: soils, unconsolidated
sediments, and rock.
Unconsolidated and semi-consolidated sediments typically have primary porosity in the
void spaces between the grains. Lithified sedimentary rock is typically low in primary
porosity due to the presence of crystalline cement (quarty or calcite) filling the primary
void spaces. Crystalline igneous and metamorphic rock typically has little primary
porosity. Bedrock with little primary porosity will form secondary porosity as the rock
is fractured due to tectonic stress or pressure. Igneous rock may also have fractures due
to cooling joints and carbonate rocks may have extensive solution cavities and fractures.
These fractures provide space for the storage and movement of groundwater.
Porosity is defined as the percentage of a volume of rock (the all purpose term
hydrologists use for ‘earth materials’) that is empty space.
n=
100V void
Vtotal
Effective porosity: void spaces that are too small to admit water molecules are of little
interest to hydrologists. The amount of voidspace available for fluid flow is the effective
porosity.
Even large voidspaces that are interconnected by small pore throats are unavailable for
fluid flow.
Fortunately, studies have shown that even fine clays have pore throats that are larger than
water molecules, so that, at least in sediments, effective porosity is equal to total porosity.
Total porosity can be computed from measurements of density:
[ (
n = 100 1− ρbulk ρ pd
)]
p bulk is the density of the bulk aquifer material
p pd is the density of the particles that make up the aquifer material
For most rock and soil, the particle density is about 2.65 g/cm3, roughly the density of
quartz and clay minerals.
Factors affecting porosity
Grain size: In and of itself, grain size has no effect on porosity. Well rounded sediments
that are packed into the same arrangement generally have porosities from 26% to 48%
depending on the packing.
A room full of bowling balls and a room full of BBs would have the same porosities if
the spheres were packed the same way.
Sorting: Well sorted sediments generally have higher porosities than poorly sorted
sediments for the simple reason that if a sediment is a range of particle sizes then the
smaller particles may fill in the voids between the larger particles.
Sorting is measured as a ratio of the larger to smaller particle sizes in the sediment. This
measure is called a uniformity coefficient.
Cu =
d60
d10
d60 the grain size below which 60% of the sediment is finer
d10 the grain size below which 10% of the sediment is finer
Grain shape: Irregularly shaped particles tend not to pack as neatly as rounded particles,
resulting in higher proportions of voidspace.
Clay and organic content: Organic particles tend to be irregularly shaped and can
increase voidspace. Clay particles tend to electrostatically repell one-another along the
surface of the particles. This results in a relatively large proportion of voidspace.
Specific yield
Knowing the porosity of a rock will not tell you how much water can be removed from
that rock (sediment, soil, etc..).
Water is sticky. Water molecules are polar - meaning that they carry a net positive and
negative charge across the molecule - so they like to attract each other, and they like to
attract to other charged materials, including most surfaces.
We see the sticky nature of water as surface tension and as the tendency for water to coat
surfaces with a thin film.
If you saturate a volume of sediment and then allow the water to drain out of the
sediment under gravity, all of the sediments will be coated with a thin film of pendular
water (water left ‘hanging’ on the grains).
Specific yield (Sy) is thus a measure of how much water can drain away from the rock
under gravity vs how much water the rock actually holds (the total voidspace).
Specific retention (Sr) , contrariwise, is the amount of water that the rock retains as a
surface film on the sediments.
The sum of specific yield and specific retention is the total porosity.
n = Sy + Sr
What effects specific yield?
Grain surface area - as grain size decreases, total surface area increases, leading to
smaller specific yields at smaller grain sizes.
The highest specific yields come from coarse sands and fine gravels.
The lowest specific yields come from clays, which have small particles and voidspaces
with large surface areas. Clays can have porosities of 50% combined with specific yields
of only 3%.
Permeability
Permeability describes of how easily water is able to move through rock. Permeability is
related to the connectedness of the void spaces and to the grain size of the rock.
Obviously, a rock could be extremely porous, but if each pore was isolated from the
others, the rock would be impermeable and thus make a lousy aquifer. Often volcanic
rocks will have many vesicles, but the vesicles will be isolated, rendering the rock
impermeable.
Grain size affects permeability in a manner similar to the way that it effects specific
yield.
The thin film of water that clings to the surface of particles is tenacious. If void spaces
are large, then additional water can easily move past the water coated particle
surfaces.
However, if void spaces are small, then the surface film of water can actually choke the
movement of additional water through the small spaces.
This explains why clays are so impermeable, even though their porosities can be as high
as 50%. The void spaces in clay are small and clogged with pendular water.
So from the above discussion, we can see why sands make the best aquifers and clays
make the worst (clays are not referred to as aquifers at all, rather they are called
aquicludes).
Hydraulic Conductivity
Hydraulic conductivity is a measure of how easily a particular fluid will pass through a
particular earth material.
It is calculated using a formula widely known as Darcy’s Law (the E=MC2 of
hydrology).
Darcy was a French engineer who made some experiments in the mid-nineteeth century
on how fluids move through sediments. What Darcy did was to measure the rate of flow
of water draining by gravity through pipes filled with different types of materials. The
relationships that he discovered now bear his name.
What did Darcy discover?
First, flow is directly proportional to the area of the pipe (no surprise there - the bigger
the pipe, the more water will flow).
Flow is also proportional to the difference in height of the water from when it begins
flowing through the pipe to when it comes out of the pipe. We call the height of water in
an aquifer (measured from some defined datum) the hydraulic head.
Obviously, it is the difference in hydraulic head from the start of the pipe to the end of
the pipe that is generating the pressure pushing the water through the pipe. We call this
difference in height = pressure the hydraulic gradient.
Finally, flow is inversely proportional to the length of the pipe. The more sediment that
the water has to flow through, the more it will be impeded.
So, Darcy made his measurements and wrote the following formula:
h −h
Q = −KA a b 
 L 
ha–hb is more generally expressed as dh, and L can be thought of as dL as well.
dh 

Q = −KA 
dL
dh is negative because the change in elevation is from high to low.
We can rearrange this equation to solve for K:
−Q
K=
A( dh dl )
The coefficient K is called the Hydraulic Conductivity and is particular to specific
combinations of fluid and earth material. The greater the value of K, the higher will be
the rate of flow of a fluid through a material.
Hydraulic conductivity is similar to permeability. The difference is that permeability is a
property that is specific to the sediment, while hydraulic conductivity describes the
behavior of a specific fluid and a specific sediment.
The units of hydraulic conductivity can be obtained from the above equation - L/T.
However, commonly hydraulic conductivity is reported as cubic units per time per area.
One can see that this conversion is accomplished easily by multiplying L/T X L2/L2.
In lab today, we will experimentally determine the hydraulic conductivities of three
different sediments using a simple version of Darcy’s original experiment.
Intrinsic Permeability
As I mentioned previously, hydraulic conductivity, even though it describes the
movement of a fluid through a geological material, is not strictly a measure of the
permeability of the material.
This is because the hydraulic conductivity value derived is dependent on the properties of
the fluid as well as the sediment. If you ran the same Darcian experiment twice using an
identical set-up and appartatus, but replacing water with motor oil, you would get very
different measures of hydraulic conductivity.
For hydrologists working with water under different temperature, and therefore, different
viscosity conditions, and for folks such as petroleum engineers who often have different
substances (water, brine, petroleum, gas) moving through the same rock, a measure of
permeability is needed that is dependent only on the properties of the geological material.
Experiments by Hubbert (1956) and others have allowed hydrologists to separate the
effects on flow of the sediments from those of the fluid. Experiments carried out using
various fluids flowing through glass beads demonstrates:
a. Flow is proportional to the weight of the fluid. That is, a heavier (denser) liquid will
generate more flow force under the pull of gravity.
flow ∝ γ
the specific weight of the fluid. This is a force per cubic volume.
Specific wt can be more usefully expressed as density X little g, the acceleration due to
gravity.
γ = ρg
b. Flow is inversely proportional to the dynamic viscosity of the fluid
flow ∝
1
µ
the dynamic viscosity of the fluid.
These two properties are fluid properties.
c. Flow is proportional to the square of the diameter of the particles.
flow ∝ d 2
d. Flow is also related to the shape of the particle openings. This can be expressed as a
dimensionless constant, C.
flow ∝ C
So, we can now write a new version of Darcy’s law.
Cd 2ρ g dh
Q=−
A
µ dl
Q is the flow, A is the area, dh / dl is familiar, this is the hydraulic gradient.
If we combine the sediment characteristics, we can create a new measure called intrinsic
permeability that is only dependent on these characteristics.
k = Cd 2
little k is the intrinsic permeability. Its dimensions are L2 (area).
Our equation for Darcy’s law now becomes:
v=−
kρ g dh
µ dl
The original equation in this form is:
v = −K
dh
dl
Therefore, the relation between K and k, hydraulic conductivity and intrinsic
permeability is:
K=k
ρg
µ
Where rho and mu are properties of the fluid in question, and g is the acceleration due to
gravity.
In the petroleum industry, the Darcy is used as a standard unit of intrinsic permeability
(often just called permeability).
1 darcy = 9.87 x 10-9 cm2
One darcy is the permeability that will lead to a specific discharge of 1 cm/s for a fluid of
1 cP (centipoise) of dynamic viscosity under a hydraulic gradient that makes pg dh/dl = 1
atmosphere.
If the characteristics of a fluid in an aquifer change, the hydraulic conductivity will
change. For example, as the temperature and/or salinity of groundwater changes so does
its density and viscosity. To be able to predict the effect these changes have on hydraulic
conductivity, we need to be able to estimate the relationship between hydraulic
conductivity and intrinsic permeability.
What are the general rules of permeability?
a.As the median grain size increases, so does permeability.
b.Poorly sorted sediments are less permeable than well sorted sediments.
c.The decrease in permeability associated with poor sorting is more severe in coarse
sediments than it is in fine sediments