1. CE 3354 Engineering Hydrology
Lecture 21: Groundwater Hydrology Concepts – Part 1

2. Outline Porous Media Concepts Storage Darcy’s Law
Porosity, Yield, Average Linear Velocity
Heads and gradients
Storage
Confined and Unconfined
Darcy’s Law
Permeability

3. Groundwater Hydrology
Groundwater hydrology is the study of water beneath the surface of the Earth
Groundwater hydrology is the study of water beneath the surface of the earth. Figure
1 is a sketch of a slice into the Earth depicting the different conceptual zones used in
describing the occurance and distribution of groundwater. The uppermost zone is the
unsaturated zone, and in this zone both water and air share pore space. Deeper into
the slice most of the air is excluded and this zone is called the saturated zone. The
interface between the zones is the capillary fringe where water is stored by capillary
forces. The water table is the location of the top of the saturated zone where water
would rise to in a well - and represents the extractable water from a water resource
perspective.

4. Porosity
Groundwater usually is found in porous media (not underground rivers).
A porous medium is comprised of solid space and void or pore space.
Liquids and gasses are found in the pore space, the solid matrix forms the physical structure of aquifers and other geologic formations of interest.
The ratio of total pore volume to bulk medium volume is the porosity
Groundwater usually is found in porous media (not underground rivers). A porous
medium is comprised of solid space and void or pore space. Liquids and gasses are
found in the pore space, the solid matrix forms the physical structure of aquifers and
other geologic formations of interest.
The ratio of total pore volume to bulk medium volume is the porosity, Equation 1
is a typical definition.
Porosity as a concept is important in groundwater hydrology because it is where water
is stored in the subsurface, and establishes a practical upper bound of how much
water can be stored in a porous media. Porosity is also important in mass transport
of materials suspended and dissolved in groundwater because the velocity of these
materials, when computed from discharge of groundwater is inversely proportional
to porosity. This relationship is a consequence of writing flow balance equations
in terms of bulk volume and not liquid volume. The solid structure is not fixed,
and water under pressure deforms the structure slightly thus changing the porosity
slightly, but for most analysis porosity is assumed to the a constant material property
of the porous medium. Table 1 is a list of typical porosity values for different kinds
of geologic materials. The values in the list have considerable variability and the list
is at best just a rough guide.
Table 1: Typical porosity values for selected geologic materials
Soil Description Porosity Range Reference
Volcanic, pumice Walton (1991)
Peat ” ”
Silt ” ”
Clay ” ”
Loess ” ”
Sand, dune ” ”
Sand, fine ” ”
Sand, coarse ” ”
Gravel, coarse ” ”
Sand and gravel ” ”
Till ” ”
Siltstone ” ”
Sandstone ” ”
Volcanic, vesicular ” ”
Volcanic, tuff ” ”
Limestone ” ”
Schist ” ”
Basalt ” ”
Shale ” ”
Volcanic, dense ” ”
Igneous, dense ” ”
Salt bed ” ”

5. Porosity Range in values large for geologic materials
Groundwater usually is found in porous media (not underground rivers). A porous
medium is comprised of solid space and void or pore space. Liquids and gasses are
found in the pore space, the solid matrix forms the physical structure of aquifers and
other geologic formations of interest.
The ratio of total pore volume to bulk medium volume is the porosity, Equation 1
is a typical definition.
Porosity as a concept is important in groundwater hydrology because it is where water
is stored in the subsurface, and establishes a practical upper bound of how much
water can be stored in a porous media. Porosity is also important in mass transport
of materials suspended and dissolved in groundwater because the velocity of these
materials, when computed from discharge of groundwater is inversely proportional
to porosity. This relationship is a consequence of writing flow balance equations
in terms of bulk volume and not liquid volume. The solid structure is not fixed,
and water under pressure deforms the structure slightly thus changing the porosity
slightly, but for most analysis porosity is assumed to the a constant material property
of the porous medium. Table 1 is a list of typical porosity values for different kinds
of geologic materials. The values in the list have considerable variability and the list
is at best just a rough guide.
Table 1: Typical porosity values for selected geologic materials
Soil Description Porosity Range Reference
Volcanic, pumice Walton (1991)
Peat ” ”
Silt ” ”
Clay ” ”
Loess ” ”
Sand, dune ” ”
Sand, fine ” ”
Sand, coarse ” ”
Gravel, coarse ” ”
Sand and gravel ” ”
Till ” ”
Siltstone ” ”
Sandstone ” ”
Volcanic, vesicular ” ”
Volcanic, tuff ” ”
Limestone ” ”
Schist ” ”
Basalt ” ”
Shale ” ”
Volcanic, dense ” ”
Igneous, dense ” ”
Salt bed ” ”

6. Specific Yield
A concept related to porosity is the specific yield of a material
Sy is the amount of water that will drain from a porous medium under the influence of gravity.
Sr is the amount of water left behind in the material and is called the specific retention
A concept related to porosity is the specific yield of a material; the amount of water
that will drain from a porous medium under the influence of gravity. The amount of
water left behind in the material is called the specific retention - it can be removed
by phase change (heating the material until the water evaporates) but will not drain
otherwise. The forces that control specific retention are capillary forces in the pore
space, largely influenced by the pore diameter. Equation 2 is a mathematical expression
of the definition of the specific yield. Equation 3 is a similar expression for
specific retention.

7. Specific Yield
The specific yield is important in water supply as it represents the amount of water that can drain to wells.
Thus when making groundwater reservoir estimates the water in storage should be based on the specific yield and not porosity.
Two related terms are:
water content
and saturation
The sum of the two terms is the porosity (Sy+Sr = n). The specific yield is important
in water supply as it represents the amount of water that can drain to wells. Thus
when making groundwater reservoir estimates the water in storage should be based
on the specific yield and not porosity. Table 2 is a list of typical values for different
kinds of geologic materials.

8. Measuring Porosity
Porosity is measured in the laboratory from small samples by gravimetric methods and fluid displacement methods.
At the field scale, porosity is measured by geophysical tools calibrated to local geologic media.
Resistivity logging
Acoustic logging
Neutron logging
Gravimetric measurements start with a sample of known volume. The sample is
dried until a constant mass is obtained - this mass is the solids mass. The sample is
then imersed in water that is ionically and mineralogically similar to the formation
water until the sample is saturated. The saturated sample is removed and weighed.
The void volume is determined from the weight of the water absorbed into the sample
and the specific weight of water, as represented by Equation 6
Alternatively, if the desnity of the solids is known, then the porosity can be determined
by two weighings (saturated and dry). The solids density, liquid density and
porosity are realted to each other as in Equation 7.
The determination of porosity can also be directly made using the change in water
volume in the submersion chamber if accurate volume measurements are possible.
Displacement measurements are related to the previous sentence in that volumes are
measured instead of masses. In certain kinds of measurements pressure drop across
a sample is used to measure porosity. Some materials are hydrophibic, and liquids
other than water can be used – but not typically in groundwater (as opposed to
filtration and coalescence applications of porous media). One unusual approach that
is valid is a Karl Fisher titration which uses a water specific chemical reaction (like
an acid-base titration) and could be used to directly measure the amount of water
in a sample. The instrument as well as the reageants are probably too expensive for
use on typical size soil samples, but it is a valid technique.
Geophysical methods are employed to measure porosity in the field. Most evolved
from methods used in oil-exploration and much of the interpretation concepts are
based on finding oil.
Resistivity survey measures the voltage drop between two electrodes with known
spacing. The results are interpreted using the principle that water has low resistivity
while soilds and hydrocarbons have relatively high resistivity. Figure 2 depicts a
typical conceptual model of a resistivity survey.
The formation is treated as an electric circuit, and from this analog the depthporosity
relationship is inferred. Units are ohm-meters, and are equivalent to the
resistance to the flow of electrons provided by a one cubic meter block of aquifer.
Generally the instrument is calibrated to formation samples saturated with demineralized
water and the ratio of this resistivity and the observed resistivity is the
formation factor, F. The relationship of formation factor to various terms is expressed
in Equation 8.
where R100 is the resistivity of the formation saturated with demineralized water
(a laboratory measurement), RA is the resistivity of the formation saturated with
actual formation water (also a laboratory measurement). The formation factor is
further related to the saturation of the formation as expressed in Equation 9.
where Sw is the saturation of the formation and Rm is the measured resistivity. The
formation factor for a fully saturated formation is roughly equivalent to the porosity.
The formation resistivity decreases as porosity and salinity increase thus calibration
is crucial in using resistivity surveys to approximate formation porosity.

9. Measuring Porosity
At the field scale, porosity is measured by geophysical tools calibrated to local geologic media.
Resistivity logging
Acoustic logging
Neutron logging
Resistivity survey measures the voltage drop between two electrodes with known
spacing. The results are interpreted using the principle that water has low resistivity
while soilds and hydrocarbons have relatively high resistivity. Figure 2 depicts a
typical conceptual model of a resistivity survey.
The formation is treated as an electric circuit, and from this analog the depthporosity
relationship is inferred. Units are ohm-meters, and are equivalent to the
resistance to the flow of electrons provided by a one cubic meter block of aquifer.
Generally the instrument is calibrated to formation samples saturated with demineralized
water and the ratio of this resistivity and the observed resistivity is the
formation factor, F. The relationship of formation factor to various terms is expressed
in Equation 8.
where R100 is the resistivity of the formation saturated with demineralized water
(a laboratory measurement), RA is the resistivity of the formation saturated with
actual formation water (also a laboratory measurement). The formation factor is
further related to the saturation of the formation as expressed in Equation 9.
where Sw is the saturation of the formation and Rm is the measured resistivity. The
formation factor for a fully saturated formation is roughly equivalent to the porosity.
The formation resistivity decreases as porosity and salinity increase thus calibration
is crucial in using resistivity surveys to approximate formation porosity.

10. Measuring Porosity
At the field scale, porosity is measured by geophysical tools calibrated to local geologic media.
Resistivity logging
Acoustic logging
Neutron logging
Resistivity survey measures the voltage drop between two electrodes with known
spacing. The results are interpreted using the principle that water has low resistivity
while soilds and hydrocarbons have relatively high resistivity. Figure 2 depicts a
typical conceptual model of a resistivity survey.
The formation is treated as an electric circuit, and from this analog the depthporosity
relationship is inferred. Units are ohm-meters, and are equivalent to the
resistance to the flow of electrons provided by a one cubic meter block of aquifer.
Generally the instrument is calibrated to formation samples saturated with demineralized
water and the ratio of this resistivity and the observed resistivity is the
formation factor, F. The relationship of formation factor to various terms is expressed
in Equation 8.
where R100 is the resistivity of the formation saturated with demineralized water
(a laboratory measurement), RA is the resistivity of the formation saturated with
actual formation water (also a laboratory measurement). The formation factor is
further related to the saturation of the formation as expressed in Equation 9.
where Sw is the saturation of the formation and Rm is the measured resistivity. The
formation factor for a fully saturated formation is roughly equivalent to the porosity.
The formation resistivity decreases as porosity and salinity increase thus calibration
is crucial in using resistivity surveys to approximate formation porosity.

11. Measuring Porosity
At the field scale, porosity is measured by geophysical tools calibrated to local geologic media.
Resistivity logging
Acoustic logging
Neutron logging
Acoustic surveys are based on the principle that sound travels through solids and fluids
at speeds characteristic of the materials. For a porous material a simple additive
model to predict sound travel time is expressed in Equation 10.
where t is the time of flight of the acoustic pulse (transmitter to receiver) and v
is the speed of sound in the various materials. Rearrangement and substitution of
time of flight values for individual materials provides an estimation of porosity as
expressed in Equation 11.

12. Measuring Porosity
At the field scale, porosity is measured by geophysical tools calibrated to local geologic media.
Resistivity logging
Acoustic logging
Neutron logging
Acoustic surveys are based on the principle that sound travels through solids and fluids
at speeds characteristic of the materials. For a porous material a simple additive
model to predict sound travel time is expressed in Equation 10.
where t is the time of flight of the acoustic pulse (transmitter to receiver) and v
is the speed of sound in the various materials. Rearrangement and substitution of
time of flight values for individual materials provides an estimation of porosity as
expressed in Equation 11.

13. Measuring Porosity
At the field scale, porosity is measured by geophysical tools calibrated to local geologic media.
Resistivity logging
Acoustic logging
Neutron logging
Neutron surveys are based on the principle that hydrogen is a good neutron reflector.
Water is a dense source of hydrogen (11% by mass) and it will scatter high-energy
neutrons more than a material that is a poor neutron reflector. A neutron log measures
the energy of neutrons scattered by a material. High reflection (backscatter)
implies mostly water, therefore a high porosity. As with resistance tools, neutron
tools need to be calibrated to local material conditions. Neutron logs also can be used
above the water table to determine water content of unsaturated materials.

14. Average Linear Velocity
The discharge Q divided by cross sectional flow area A in a pipe or open channel is the velocity V
In groundwater, some of the area is solid, so the porosity enters the equation
Consider Figure 3 and assume the flow of water into the garbage can is 1-liter/minute.
Assume the garbage can can hold up to 1000-liters. Then we expect the can to take
1000 minutes to fill before overflowing. Furthermore, because the area of the bottom
of the can is 1 m2, then the water surface rises at a rate of 1 mm per minute. Now
assume we fill the can with marbles whose total volume is 500-liters, but that they
pack in such a way that they ”fill” the can to the top. The void space in the can
is now 500-liters, but the overall can dimensions are unchanged. The porosity of
the porous medium is now If we now fill the can at a rate of 1-liter/minute,
the water will overflow after only 500 minutes (half the time), and the rate that the
water surface rises is 2 mm per minute.
Expressing the velocity of the rate of water rise in terms of the volumetric fill (flow)
rate, as in Equation 12 relates between porosity, discharge, and velocity.
where, u is the average linear velocity, Q is the volumetric flow rate (discharge), A is
the cross sectional area perpendicular to flow, n is the porosity. In the garbage can,
the can without marbles has porosity of 1.0, while the marble filled can has porosity
0.5.
Groundwater flow equations almost always are constructed in terms of the discharge
term, Q, thus when these results are used to compute speeds, the porosity relationship
needs to be explicitly stated.

15. Energy and Head Hydraulic gradient is change in head per unit distance
Consider the well in Figure 1. The water level represents the energy available to
drive groundwater through a porous medium and the height of this column of water
above some reference elevation is called the head. In the context of classic hydraulics
the head in a groundwater system is the sum of elevation and pressure head. Because
flow speeds are usually quite small the kinetic head term is usually neglected.
As water moves through a groundwater system it loses energy, hence its head will
decrease in the direction of flow. The actual change in head per unit distance of travel
is called the hydraulic gradient. In terms of calculus, the hydraulic gradient and the
gradient of head are the same vector, but opposite sense. For two wells, any gradient
will be a directional derivative unless it is known that both wells are on the same
flowline. For three wells, a plane can be passed throuhg the head values, and the
negative of the gradient of the equation of the plane is the hydraulic gradient.
A simple method that can be used with three wells is triangulation. Suppose three
wells are completed in an aquifer as depicted in Figure 4.
From head measurements in each well, the groundwater hydraulic gradient can be
computed and flow directions predicted. This triangulation approach has some special
assumptions
1. Piezometric surface in the vicinity of the three wells can be approximated by
a plane.
2. All three wells sample the same aquifer unit.
3. The wells all measure vertically averaged head, or head at identical elevations
in the aquifer.
4. Head measurements are taken at essentially the same time
The objective is to determine the direction and magnitude of the hydraulic gradient.
Heath (1989) illustrates a graphical method to solve the equation of the plane
passing through the three wells - here the mathemitical procedure is presented. Of
importance is that the wells are reasonably cloees and that a plane is an appropriate
conceptial model of the water table.

16. Energy and Head
Piezometric surface in the vicinity of the three wells can be approximated by a plane.
All three wells sample the same aquifer unit.
Wells measure vertically averaged head
Head measurements at same time
Consider the well in Figure 1. The water level represents the energy available to
drive groundwater through a porous medium and the height of this column of water
above some reference elevation is called the head. In the context of classic hydraulics
the head in a groundwater system is the sum of elevation and pressure head. Because
flow speeds are usually quite small the kinetic head term is usually neglected.
As water moves through a groundwater system it loses energy, hence its head will
decrease in the direction of flow. The actual change in head per unit distance of travel
is called the hydraulic gradient. In terms of calculus, the hydraulic gradient and the
gradient of head are the same vector, but opposite sense. For two wells, any gradient
will be a directional derivative unless it is known that both wells are on the same
flowline. For three wells, a plane can be passed throuhg the head values, and the
negative of the gradient of the equation of the plane is the hydraulic gradient.
A simple method that can be used with three wells is triangulation. Suppose three
wells are completed in an aquifer as depicted in Figure 4.
From head measurements in each well, the groundwater hydraulic gradient can be
computed and flow directions predicted. This triangulation approach has some special
assumptions
1. Piezometric surface in the vicinity of the three wells can be approximated by
a plane.
2. All three wells sample the same aquifer unit.
3. The wells all measure vertically averaged head, or head at identical elevations
in the aquifer.
4. Head measurements are taken at essentially the same time
The objective is to determine the direction and magnitude of the hydraulic gradient.
Heath (1989) illustrates a graphical method to solve the equation of the plane
passing through the three wells - here the mathemitical procedure is presented. Of
importance is that the wells are reasonably cloees and that a plane is an appropriate
conceptial model of the water table.

17. 3-Well Gradient Example

18. 3-Well Gradient Exampley
2000ft C B
3000ft A
(0,0) x

19. 3-Well Gradient ExampleC B A
3000ft
2000ft
(0,0) x y

20. 3-Well Gradient ExampleC B A
3000ft
2000ft
(0,0) x y
470
460
2000ft
440

21. 3-Well Gradient ExampleC B A
3000ft
2000ft
(0,0) x y
470
460
~1789ft
2000ft
440

22. Groundwater Storage
Storage refers to the ability of a porous medium to store water within its bulk.
The mechanisms of storage are:
draining and filling of the pore space
compression of the water, and
compression of the solids.
In an unconfined aquifer the draining and filling of the pore space is the most significant mechanism.
In a confined aquifer, the compression and decompression of the solids structure is the primary mechanism of storage.
Storage refers to the ability of a porous medium to store water within its bulk. The
mechanisms of storage are draining and filling of the pore space, compression of the
water, and compression of the solids. In an unconfined aquifer the draining and
filling of the pore space is the most significant mechanism. In a confined aquifer,
the compression and decompression of the solids structure and to a lesser extent the
water compressibility is the primary mechanism of storage.
Figure 1 is a sketch of the unconfined storage mechanism in terms of a unit area of
aquifer and unit change of head. The storage coefficient (S) is the volume of water
added to or removed from storage per unit change in head, per unit area of porous
material. Equation 1 is the classical definition of storage coefficient.
In an unconfined aquifer the value of storage coefficient and the specific yield are for
all practical purposes the same and the terms are sometimes used interchangeably
(although Sy only occurs at the water table and not throughout the entire aquifer
thickness).

23. Unconfined Aquifer Storage in Unconfined is by drain/fill pore space.
In an unconfined aquifer the value of storage coefficient and the specific yield are for
all practical purposes the same and the terms are sometimes used interchangeably
(although Sy only occurs at the water table and not throughout the entire aquifer
thickness).

24. Confined Aquifer
Storage in confined is by compression/decompression of the aquifer and water
In a confined aquifer, the pore space is entirely filled (by definition), thus the compression
and decompression of the water and solids matrix are the only remaining
mechanisms of storage. The storage coefficient is identical (in mathematical construction)
but is much smaller in magnitude (by several powers of ten) than unconfined
storage coefficients. The corresponding sketch is displayed in Figure 2.

25. Specific Storage
Storage per unit thickness of aquifer is called the specific storage

26. Estimating Storage
Estimate by making head measurements at two different times and apply the storage equation

27. Estimating Storage Same idea for multiple blocks
Estimate each block and then average
Observe that the individual changes in head and volumes in/out are assumed to
be known. Maps can be also be used to determine the denominator in the above
relationship for determining storage coefficients.
Consider the two maps in Figures 5 and 6. The maps are supposed to be the same
size; and assume the scale is correct. To compute storage changes one superimposes
a grid system onto both maps (software can be used if the maps are digital
elevation maps and properly geo-referenced) to perform the analog of a cut-and-fill
calculation.
Figures 5 and 6 already have a grid system superimposed, but usually this step is
the analysts job - next the analyst would estimate the average water level in each
cell, then take the difference between the two maps to determine the head changes
in each cell of the grid, and finally use these head changes to determine the change
in aquifer volume represented in the maps. Then the analyst would compute the
storage coefficient from the many-blocks equation as in Equation ??.

28. Estimating Storage Use of groundwater elevation maps
Observe that the individual changes in head and volumes in/out are assumed to
be known. Maps can be also be used to determine the denominator in the above
relationship for determining storage coefficients.
Consider the two maps in Figures 5 and 6. The maps are supposed to be the same
size; and assume the scale is correct. To compute storage changes one superimposes
a grid system onto both maps (software can be used if the maps are digital
elevation maps and properly geo-referenced) to perform the analog of a cut-and-fill
calculation.
Figures 5 and 6 already have a grid system superimposed, but usually this step is
the analysts job - next the analyst would estimate the average water level in each
cell, then take the difference between the two maps to determine the head changes
in each cell of the grid, and finally use these head changes to determine the change
in aquifer volume represented in the maps. Then the analyst would compute the
storage coefficient from the many-blocks equation as in Equation ??.

29. Darcy’s Law
Permeability refers to the ease which water can flow through a porous material under a specified gradient.
Permeable materials offer little resistance, while impermeable materials offer a lot of resistance.
Permeability refers to the ease which water can flow through a porous material under
a specified gradient. Permeable materials offer little resistance, while impermeable
materials offer a lot of resistance. All materials have some permeability, although it
can be quite small (a Styrofoam cup is a good example they will leak water, but it
takes the better part of a week).
Darcys law was established empirically by Henri Darcy in Figure 7 is a sketch
of a filter column, similar to Darcys experimental apparatus. The distance from the
outlet to the height of water in the outlet piezometer is h2. The distance from the
outlet to the height of water in the inlet piezometer is h1. The porous medium has
length, L. The cross sectional area of the filter perpendicular to the direction of flow
is A.

30. Darcy’s Law Established experimentally 1856
Total discharge through a filter, Q, was proportional to:
cross sectional area of flow, A,
head loss h1 − h2.
Q, was inversely proportional to:
the length of the filter column, L.
Permeability refers to the ease which water can flow through a porous material under
a specified gradient. Permeable materials offer little resistance, while impermeable
materials offer a lot of resistance. All materials have some permeability, although it
can be quite small (a Styrofoam cup is a good example they will leak water, but it
takes the better part of a week).
Darcys law was established empirically by Henri Darcy in Figure 7 is a sketch
of a filter column, similar to Darcys experimental apparatus. The distance from the
outlet to the height of water in the outlet piezometer is h2. The distance from the
outlet to the height of water in the inlet piezometer is h1. The porous medium has
length, L. The cross sectional area of the filter perpendicular to the direction of flow
is A.

31. Darcy’s Law
The constant of proportionality is called the hydraulic conductivity
Permeability is sometimes used interchangeably
In reservoir engineering the permeability is related to K, but not numerically identical.
Permeability refers to the ease which water can flow through a porous material under
a specified gradient. Permeable materials offer little resistance, while impermeable
materials offer a lot of resistance. All materials have some permeability, although it
can be quite small (a Styrofoam cup is a good example they will leak water, but it
takes the better part of a week).
Darcys law was established empirically by Henri Darcy in Figure 7 is a sketch
of a filter column, similar to Darcys experimental apparatus. The distance from the
outlet to the height of water in the outlet piezometer is h2. The distance from the
outlet to the height of water in the inlet piezometer is h1. The porous medium has
length, L. The cross sectional area of the filter perpendicular to the direction of flow
is A.

32. Measuring Permeability
Permeameters
Constant head
Falling head
On Sept. 10, 1977, convicted murderer Hamida Djandoubi became the last person executed by guillotine in France.

33. Constant Head Permeameter
A sample is placed in the permeameter
Constant head gradient is maintained across the sample.
Flow rate is measured
Darcy’s law applied:
The constant head permeameter is simply a device that exactly duplicates Darcys
original experiment. A sample is placed in the permeameter and constant head
gradient is maintained across the sample. The hydraulic conductivity is inferred by
direct application of Darcys Law.
Figure 9 is a schematic diagram of a constant head permeameter. A sample of porous
medium is placed the device. The length of the sample is L, the cross sectional area
of the sample is A. The head is measured at the inlet and outlet of the sample. h1
is the head at the inlet of the sample as measured from the outlet of the sample. h2
is the head at the outlet of the sample. Q is the rate of flow through the sample
measured by recording the time required for a known volume of water to pass through
the sample.
Darcys law for this experiment is exactly Equation 8. Rearranging this equation
gives the following formula for hydraulic conductivity;

34. Constant Head Permeameter
Spreadsheet model to make computations
The constant head permeameter is simply a device that exactly duplicates Darcys
original experiment. A sample is placed in the permeameter and constant head
gradient is maintained across the sample. The hydraulic conductivity is inferred by
direct application of Darcys Law.
Figure 9 is a schematic diagram of a constant head permeameter. A sample of porous
medium is placed the device. The length of the sample is L, the cross sectional area
of the sample is A. The head is measured at the inlet and outlet of the sample. h1
is the head at the inlet of the sample as measured from the outlet of the sample. h2
is the head at the outlet of the sample. Q is the rate of flow through the sample
measured by recording the time required for a known volume of water to pass through
the sample.
Darcys law for this experiment is exactly Equation 8. Rearranging this equation
gives the following formula for hydraulic conductivity;

35. Falling Head Permeameter
The head is measured at the inlet of the sample as the height of water in the tube above the sample
Change in this height with time is the flow rate
Head and the flow rate vary with time.
The falling head permeameter was presented in the infiltration lecture as a way to
intorduce the Green-Ampt infiltration model. The discussion is repeated in this
section, but without reference to runoff.
Figure 10 is a schematic diagram of a falling head permeameter. A sample of porous
medium is placed the device. The length of the sample is L, the cross sectional area
of the sample is A. A smaller area tube rises above the sample. This tube provides
the driving force required to move water through the porous sample with reasonably
small water volume. The area of this tube is a. The head is measured at the inlet
of the sample as the height of water in the tube above the sample, h(t). Q(t) is the
varying rate of flow through the sample. Observe that in a falling head permeameter,
both the head and the flow rate vary with time.
Darcys law for this situation is 9

36. Falling Head Permeameter
Volume balance
Separate and integrate
Simplify
The falling head permeameter was presented in the infiltration lecture as a way to
intorduce the Green-Ampt infiltration model. The discussion is repeated in this
section, but without reference to runoff.
Figure 10 is a schematic diagram of a falling head permeameter. A sample of porous
medium is placed the device. The length of the sample is L, the cross sectional area
of the sample is A. A smaller area tube rises above the sample. This tube provides
the driving force required to move water through the porous sample with reasonably
small water volume. The area of this tube is a. The head is measured at the inlet
of the sample as the height of water in the tube above the sample, h(t). Q(t) is the
varying rate of flow through the sample. Observe that in a falling head permeameter,
both the head and the flow rate vary with time.
Darcys law for this situation is 9

37. Falling Head Permeameter
Analysis by plotting and find slope of line in log-linear space
The falling head permeameter was presented in the infiltration lecture as a way to
intorduce the Green-Ampt infiltration model. The discussion is repeated in this
section, but without reference to runoff.
Figure 10 is a schematic diagram of a falling head permeameter. A sample of porous
medium is placed the device. The length of the sample is L, the cross sectional area
of the sample is A. A smaller area tube rises above the sample. This tube provides
the driving force required to move water through the porous sample with reasonably
small water volume. The area of this tube is a. The head is measured at the inlet
of the sample as the height of water in the tube above the sample, h(t). Q(t) is the
varying rate of flow through the sample. Observe that in a falling head permeameter,
both the head and the flow rate vary with time.
Darcys law for this situation is 9

38. Falling Head Permeameter
Spreadsheet tool to help with the analysis. Use trial- error with K to get computed to fit observed
The falling head permeameter was presented in the infiltration lecture as a way to
intorduce the Green-Ampt infiltration model. The discussion is repeated in this
section, but without reference to runoff.
Figure 10 is a schematic diagram of a falling head permeameter. A sample of porous
medium is placed the device. The length of the sample is L, the cross sectional area
of the sample is A. A smaller area tube rises above the sample. This tube provides
the driving force required to move water through the porous sample with reasonably
small water volume. The area of this tube is a. The head is measured at the inlet
of the sample as the height of water in the tube above the sample, h(t). Q(t) is the
varying rate of flow through the sample. Observe that in a falling head permeameter,
both the head and the flow rate vary with time.
Darcys law for this situation is 9

39. Field Methods

40. Next Time Direct Application of Darcy’s Law Steady flow solutions
Rectilinear flow
Flow to wells