Hydrology
Lecture 4 Porosity, Permeability and Darcy’s Law
Reading assignment: Chapter 4, pg. 80 - 93, pay close attention to 86-90.
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.
Even rock that was formed as a solid mass will fracture as it is brought towards the
surface. 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.
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:
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.
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.
Primary vs secondary porosity
Primary porosity is the porosity that exists between individual grains in the rock.
Secondary porosity is the porosity that results from fracturing, dissolution, and separation
of the rock after its formation.
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 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, 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.
What effects specific yield?
Grain surface area - as grainsize 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. At larger gravel
sizes specific yield decreases (why???).
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 vessicles, but the vessicles will be isolated, rendering the rock
impermeable.
Grain size affects permeability in a mannor similar to the way that it effects specific
yield.
The thin film of water that clings to the surface of particles is tenaceous. If voidspaces
are large, then additional water can easily move past the water coated particle
surfaces.
However, if voidspaces 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 voidspaces 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:
ha?hb is more generally expressed as dh, and L can be thought of as dL as well.
dh is negative because the change in elevation is from high to low.
We can rearrange this equation to solve for K:
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.
The apparatus used to measure hydraulic conductivity is called a permeameter.
Discussion
Predicting porosity and permeability in subsurface deposits is of prime importance to
Reservoir engineers. These variables are notoriously difficult to predict on account of the
large number of factors that influence the pore-size and shape distributions during and
after deposition.
Effects of Grain size and Grain Size Distribution on Porosity
Porosity is the measure of void volume to total volume, and is independent of grain size.
Porosity depends on:
1) Sorting of the grains or uniformity of grain size. ie. Are the grains all the same size
(well sorted), or are a variety of grain sizes present, with finer grains filling the
spaces between the larger grains (poorly sorted).
2) Shape of the grains.
3) Packing and arrangement of grains
When the variance of the rock size from the mean is large, so that there is very large
grains and very small grains, the porosity decreases. This is due to the small sized grains
filling in the spaces left between the larger ones, thereby cutting down on the available
void space. Well-sorted deposits will have a grain size distribution with the smallest
variation in sizes from the mean and will therefore achieve a higher porosity.
Effects of Grain Size and Grain Size Distribution on Permeability
Permeability is the ease with which fluid flows through a rock or sediment. A rock is
permeable if fluids pass through it and impermeable if fluid-flow through the rock is
negligible. Grain size distribution will affect the permeability of a sample because the
permeability is related to the amount of free space there is for fluids to flow. If the fluid
path is blocked due poorly sorted rock sizes, the effect will in turn cut down on the
amount of fluid that can flow through. This is accounted for mathematically by the
tortuosity constant.
Permeability depends on:
1) Grain size (coarser-grained sediments are more permeable than fine-grained
sediments because the pores between the grains are larger)
2) Sorting
3) Grain shape
4) Packing
Grain size distribution can be used to:
1) Determine filter pack.
2) Estimate hydraulic conductivity.
3) Oil reservoir characteristics.
4) Geo-technical analysis.
Permeability from grain size:
1) As meridian grain size increases, permeability increases due to the larger pore
openings.
2) Permeability decreases for more poorly sorted sediments (well graded).
3) Coarser samples show greater decreases in permeability with increases in standard
deviation of grain sizes than those of fine samples.
Permeability is favored by larger grain sizes because the holes in between the packed
grains are larger than with small ones. Larger spaces will allow for more flow. It can be
seen from Poiseuille’s equation and Darcy’s equation that Q and K are proportional to the
pore throat radius r.
The relationship between porosity and permeability is expressed through the CarmenKozeny relationship that states that permeability is related to porosity as well as the grain
specific area. For certain conditions of cementation and grain size, it is possible that the
porosity be high (as in shales) but this is countered by an increase in grain specific area
and these two effects counter each other. The net result could be a rock with high
porosity but medium to low permeability.
Effects of Fluid Distribution on Resistivity
Some substances, like metals and salt water, conduct electricity well. Others, such as
rubber, plastic, and glass, resist the flow of electricity. Often water has salts dissolved in
it and is a good conductor of electricity. Most rock has a high resistivity. Porous rock
that is soaked in salt water will allow electricity to pass fairly easily because the liquid in
its pores has a low resistivity. Hydrocarbons do not conduct electricity well. This is why
logging is of interest to Reservoir engineers.
Water has a lower resistivity than oil and therefore the water saturation of a rock will
have a large impact on the resistivity. Fluid distribution commonly varies, depending on
the capillarity of the rock and whether it is Water Wetting or Oil Wetting. In the case of
Oil Wetting rocks, water will be trapped in the larger pores and oil in the smaller pores.
Since the smaller pores determine the passage of current (path of least resistance),
resistivity will be high until the oil in the small pores is substituted by water. At that
condition, even though the water saturation shows only a small change, the corresponding
change in resistivity could be significant. The profile of resistivity versus water saturation
therefore shows a slight curvature.
In the case of Water Wetting rocks, however, the water resides in the smallest pores. In
this case as saturation increases, water starts to force the oil out of the big pores. The
saturation of water continually increases by significant amounts and this accompanied by
decreasing resistivity of the rock. In this case the profile of rock resistivity versus
saturation is more linear.
Darcy’s Experiment and Hydraulic Conductivity
Prof. Patrick M. Colgan This page was last updated on 10-17-99
Flow through a porous medium
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The volume of water that can be drained from a porous rock is less than the pore
volume of the rock, and it may be considerably less in rocks with small pores.
In studying permeability (or the ability of a material to transmit a fluid) we find
that the character of the pore openings rather than their total volume exerts the
chief control on the rate of flow of water through a sediment or rock.
The driving force for all groundwater flow is gravity. The resisting force is
friction between water and the pore surfaces as well as friction within the fluid
itself (viscosity).
Quantitative studies of the flow of water through natural porous materials are
based on an empirical relationship known as Darcy’s Law.
Darcy's Law only holds for low velocities generally much less than 1 cm/s where
flow is laminar (see p. 143-144 for details).
Darcy’s Law
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In 1856 Henri Darcy described an experiment of flow through a porous media in a
paper called, “Water Supply of the City of Dijon, France”.
The experiment measured the rate of water flow (Q = discharge = m3/s) through a
porous sand filter. Darcy related Q to cross-sectional area and hydraulic gradient.
The constant relating these variables is a material property that we now call
hydraulic conductivity. This material property varies enormously (over 14 orders
of magnitude) in different types of natural porous media.
The results of Darcy's experiment were:
1. Q is proportional to the cross sectional area of the filter(A).
2. Q is proportional to ha - hb.
3. Q is inversely proportional to L the length of the filter.
4. The constant of proportionality (K) for a given porous material can be
found by such an experiment.
5. K depends on grain size and pore shape. It is assumed that water is used in
this experiment.
This can be summarized into what is now known as Darcy’s Law:
Q=-KAI
where I = hydraulic gradient = (ha - hb) / L
or more generally as
Q = -K A dh/dl
the negative sign indicates that flow is in the direction of decreasing hydraulic
head
Intrinsic Permeability and Hydraulic Conductivity
hydrualic conductivity = K = -Q / (A dh/dl)
The units of K are L/T
Q is also inversely proportional to the viscosity of the fluid and proportional to the
specific weight of the fluid. Q is also a function of grain shape and particle diameter.
Because of this a better description of an intrinsic property of the porous media is called
the intrinsic permeability.
intrinsic permeability - takes into account, specific weight (Ws) of fluid and viscosity (m)
of the fluid.
Q = C d2 Ws/ m (dh/dl)
where C is shape factor and d is particle diameter
intrinsic permeability = Ki = C d2
K = Ki ( Ws/m )
or K = Ki ( D x g/ m)
where D is density and g is acceleration of gravity
The dimensions for Ki are L2.
1 darcy is a unit of intrinsic permeability used by petroleum engineers.
lab standard for K is pure water at T = 15.5 degrees C
See table from Cherry for how units are related to each other.
Measurement of Hydraulic Head
The piezometer is a tube that measures the total hydraulic head at a point in an aquifer.
The gradient is equal to the change in head divided by the distance between the screens in
the piezometers.
Geometric mean vs arithmetic mean for conductivity values
The variation of K in natural materials is over 14 orders of magnitude. Because of this we
can't use an arithmetic average of K values like we do with other numbers. We must use a
geometric mean. an arithmetic average of numbers that vary by several orders of
magnitude will be influenced by the largest numbers. This is why it is better to use a
geometric mean.
Study the example problem on page 102 for calculating a geometric mean.
Permeameters
Permeameters are used for calculating the hydraulic conductivity of natural materials in
the lab. There are two major problems with using these numbers. One is that sands must
be recompacted to approximate their original packing, this is hard to do and will
significantly change the lab value from the field value. The other more basic problem is
that lab values of K are commonly one to three orders of magnitude lower than values
derived in the field by tests such as the slug test. This is because the size of the sample in
field tests is much larger and may incorporate secondary porosity such as fractures or
bedding planes and other high K layers.
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constant head - this method is used for relativity for permeable sediments such as
sand and fine gravel. See pages 103-107 in the text for an explanation of this test.
falling head - this method is used for low permeability sediments such as silts and
clays. See pages 103-107 in the text for an explanation of this test.
Study the handout from the lab manual for an example of how the tests are run and how
K and Ki can be calculated.