HYDRAULIC HEAD

In last month's newsletter (Vol. 5 No. 3) we discussed Darcy's Law, which includes hydraulic head.  This month we will discuss the following about hydraulic head and it's properties:

What does hydraulic head represent?

What is the equation for hydraulic head?

Does the equation for hydraulic head equal the total energy of a groundwater system?

How do I determine hydraulic head in the field?

Why is hydraulic head important?

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What does hydraulic head represent?

Hydraulic head is a measurement of the total mechanical energy per weight of the groundwater flow system.  In other words, it is the fluid potential for flow through porous media.   It is predominantly comprised of pressure and elevation heads.

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What is the equation for hydraulic head?

The equation for hydraulic (h) head has two components; elevation head and pressure head. 

h = z + h_p

where:

z = elevation of the fluid above a reference elevation (L)

h_p = pressure head (L), calculated by:

h_p = P/ρg

where:

P = pressure (M/LT²)

ρ = density of fluid (M/L³)

g = acceleration of gravity (L/T²)

 

For example, in the given scenario hydraulic head would be calculated as follows:

 

 

 

 

Does the equation for hydraulic head equal the total energy of a groundwater system?

This is a tricky question.  The value calculated for hydraulic head is representative of the total energy, however it is not the exact value.  This is because the derivation of the total energy of a groundwater flow system also includes a term for kinetic energy.  Kinetic energy represents the energy obtained from the movement of the water (Newtonian physics:  a body in motion tends to remain in motion).  The equation for total mechanical energy of a groundwater flow system is:

 E_tm = (v²/(2g)) + z + h_p

where v = velocity (L/T)

             g =  acceleration of gravity(L/T²)

             z = elevation of the fluid (L)

             h_p = pressure head (L)

The equation for hydraulic head does not include the kinetic energy term because the velocity of groundwater flowing in porous media under natural hydraulic gradients is very low, and hence the value for kinetic energy is significantly smaller than the elevation and pressure head, and can be safely ignored. 

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How do I determine hydraulic head in the field?

Hydraulic head can be determined by measuring the depth of the water table in a groundwater well, usually a piezometer.  A piezometer is a non-pumping well, typically with a small diameter and a short well screen through which water can enter.  A piezometer nest is a group of two or more piezometers which are set very close to each other but are screened at different depths.  The use of a piezometer nest is an excellent way to determine the vertical hydraulic gradient (vertical change in hydraulic head) at a location.

When a piezometer is installed there is some information about it that needs to be recorded for determining hydraulic head.  The depth of the piezometer (i.e. depth to the screened section) and the elevation at the surface are two examples of data that may be needed for calculating hydraulic head.

The following example illustrates how hydraulic head (and other related parameters) would be calculated from the depth of a water table in a piezometer:

The data below were collected from a piezometer nest:

	Piezometer A	Piezometer B	Piezometer C
Elevation at Surface (m.a.s.l.)*	250	250	250
Depth of piezometer (m)	150	100	80
Depth to Water Table (m.b.s.)**	84	79	65

*  meters above sea level    ** meters below surface

Part i)  Calculate the hydraulic head

The hydraulic head is the elevation of the water in the piezometer.  It is calculated by subtracting the depth to water from the surface elevation:

A:  250 - 84 = 166 m    B:  250 - 79 = 171 m    C:  250 - 65 = 185 m

Part ii) Calculate the pressure head at each well

The pressure head is the height of the water above the depth of the piezometer.  Hence it is calculated by:

A:  150 - 84 = 66 m     B:  100 - 79 = 21 m      C:  80 - 65 = 15 m

Part iii)  Calculate the elevation head at each well

The elevation head is the height of the measuring point (i.e. the piezometer depth) above the datum.  In our case we have taken our datum to be mean sea level, therefore the elevation head for each well is calculated as:

A:  250 - 150 = 100m    B:  250 - 100 = 150 m    C:  250 - 80 = 170 m

Notice how the hydraulic head is the sum of the pressure head and the elevation head!

Part iv)  Calculate the vertical hydraulic gradient between the piezometers

The hydraulic gradient is the difference in hydraulic head divided by the vertical distance between the two piezometers.

From piezometer A to B we have:

171 - 166  = 5 m difference in head

150 - 100 = 50 m vertical distance

So, the vertical gradient is 5/50 = 0.1

And from B to C we have:

185 - 171 = 14 m difference in head

100 - 80 = 20 m vertical distance

So, the vertical gradient is 14/20 = 0.7

In both cases we see that the gradient is downwards, meaning the water is flowing down through the porous media.

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Why is hydraulic head important?

The analysis of virtually every physical process involves a potential gradient.  For example, an electrical current flows through circuits from higher voltages to lower.  The same can be said for water flow, there is a potential gradient that determines the direction of flow, and this potential gradient is determined by hydraulic head.  Hydraulic head is the cornerstone of groundwater flow knowledge, a key component in Darcy's law which describes fluid flow through porous media.  It is incorporated into every groundwater flow model including the groundwater flow model, Visual MODFLOW, and the pumping/slug test analysis programs, Aquifer Test, AquiferWin32 and Single Well Solutions.

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There are several excellent references available for more information on hydraulic head and it's related properties.  References used for this newsletter were:

Bear, J. (1972).  Dynamics of Fluids in Porous Media.  Published by Dover Publications Inc., New York.

Fetter, C.W. (1988).  Applied Hydrogeology 3rd edition.  Published by Prentice Hall, Inc., New Jersey.

Freeze, R.A. and J.A. Cherry (1979).  Groundwater.  Published by Prentice Hall, Inc., New Jersey.

 

AquiferTest, AquiferWin32 & Single Well Solutions

Which one is for me?

 

With such excellent selection available for programs which perform pumping/slug test analysis, how can you ensure that you are selecting the right program for your needs?

If you do not have any data from observation wells, and will be conducting an analysis based solely on data from a single well, Single Well Solutions is an excellent option.

If you are not a frequent analyzer of pumping/slug tests, or if you know exactly which solution method you like to use, AquiferWin32 may be the program best suited for you.

If you often analyze pumping/slug test data, are uncertain about, or like to experiment with, different solution methods, AquiferTest would be a great program for use.

 

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