1

Hydraulic head and

Hydraulic Gradient

Hydraulic Head

• Groundwater moves in response to an energy gradient.

• The total energy available for water flow is termed the Hydraulic Head

Hydraulic Head, • and consists of 3 components: • (1) a velocity head, • (2) an elevation head, and • (3) a pressure head.

• Each of these terms is characterized by Bernouli’sequation:

• g=gravity (L/T2)• z=elevation of water mass above a reference elevation (L)• v=velocity (L/T)• ρ for density of water (m/V )• P=pressure (M/LT2; pascals, N/m2)

hvg

zPg

= + +2

2 ρ

• Often, the kinetic energy due to the flowing groundwater is small. Typical groundwater flow rates are about 30 m/y which is about 10-6 m/s.

• Therefore, we can safely drop the kinetic term from the hydraulic head equation to obtain:

h zPg

= +ρ

• For a fluid at rest, the Pressure at a point = weight of the overlying

water per unit cross-sectional area:

• P=ρghp

• hp = height of the water column that provides the pressure head.

• putting this in place of P in our hydraulic head equation:

• h=z+hp

Hydraulic Head

Measured as the height that water will rise in a well relative to a datum (sea level)

Composed of two forces: 1) Elevation head (Z) distance of the bottom of a

well above a datum –often sealevel2) Pressure head (hp) column of water in a well

htotal=z + hp

2

How do we measure hydraulic head?

• With a piezometer, a small-diameter well that is open at its top and its bottom is either open or has a small screen interval.

• Water in the piezometer rises to a height that is directly proportional to the total energy at the bottom of the piezometer.

Hydraulic head measured in a well in a confined aquifer

Hydraulic Headmeasured in wells in an unconfined aquifer

Sand Aquifer

datum

Impermeable Clay

Water Table

Well 1 Well 2

h1to

tal

z = elev. head

hp = pressure head

h2to

tal

Land surface

Groundwater flow

Groundwater flows from high to low total hydraulic head

• Groundwater flows from areas of high total hydraulic head to areas of low total hydraulic head.

• Groundwater does not necessarily flow from areas of high elevation to low elevation or from areas of high pressure head to low pressure head.

• The total hydraulic head must be taken into account to determine groundwater flow direction.

Groundwater flows from high to low total hydraulic

head

z1 z2

hp1

hp2

z1 z2

hp1 hp2

z1 z2

hp1 hp2

z1 z2

hp1

hp2

Groundwater flows in three dimensions.

3

Groundwater flows in three dimensions.

• It is often difficult for us to display groundwater flow in three dimensions, therefore, we often look in two dimensions as in a cross-section or in map view or both.

Map View With only 3 wells, Potentiometric contours Can only be straight lines.

First cut assumption is that Groundwater flow is Perpendicular to Potentiometric contours(Only true for a homogeneous, Isotropic aquifer)

Need a minimum of 3 Wells to obtain a Horizontal groundwater Flow direction.

Vertical Groundwater Flow

A minimum of two wells located in close proximity To one another, but at different depths is needed To determine vertical groundwater flow direction.

Potentiometric Map

• Contours of equal total hydraulic head or equipotential lines.

• Map in a similar fashion as the isohyetal lines for precipitation or elevation contours on a topographic map.

• Determined from water level measurements made in numerous wells across and area.

Potentiometric map of

ENP (m)

• Water Table Map or Potentiometric surface Map. • A plan view of the potentiometric surface of an aquifer.

• For a water table map it is important that the wells are screened across the water table surface. Also that the wells be from the same aquifer, and are not sampling multiple aquifers.

• Only horizontal gradients can be obtained from a water table map.

• For an isotropic aquifer, groundwater flow direction is perpendicular to the equipotentiallines. This is not true for anisotropic mediums.

• At least three wells are needed to determine both groundwater flow direction and gradient.

• This map can also be used to determine areas of groundwater recharge and discharge.

4

Hydraulic gradient

• Hydraulic gradient is the change in head over a distance.

• Hydraulic gradient, dh/dl or (i) can be estimated from 3 or 4 wells.

• When more wells are present, then variations in hydraulic gradient can be visualized.

Hydraulic Gradient

La – Lb = dLLa Lb

ha

hb

ha – hb = dh

Datum h=0

• Groundwater flow direction is vertically downward in areas of groundwater recharge.

• Groundwater flow direction is vertically upward in areas of groundwater discharge.

• A groundwater divide is when groundwater flows in opposite directions away from an area of groundwater recharge.

Localized groundwater flow

Regional groundwater flowStinson Beach, CA

5

Groundwater-surface water interactions Groundwater flow to a stream

Groundwater flow to a well Over Pumping

• Question 1. Three wells are placed side by side at varying depths in the same aquifer as shown on the following figure. You know that the elevation of the ground surface at each well is 225 m.

• What is the total hydraulic head at A, B, and C?• What is the pressure head in each of the wells? • What is the elevation head in each well? • What is the vertical hydraulic gradient between the piezometers?

Surface elevation=225 mMeasured from mean Sea level

A B C

150m

100m

75m80m

77 m60 m

Surface elevation=225 mMeasured from mean Sea level

A B C

150m

100m

75m80m

77 m60 m

• Elevation head is the height of the measuring point above the datum. The elevation at the bottom of the well.

• Subtract the depth of the piezometer from the surface elevation.

• A: h=225 m - 150m = 75 m• B: h=225m - 100m = 125 m• C: h= 225m - 75 m = 150 m

• Note that the hydraulic head is the sum of the elevation head and the pressure head.

• For Darcy’s Law we are interested in the hydraulic gradient, dh/dL.

• It is the difference in the total head divided by the vertical distance between the bottom of two piezometers.

• Between A to B, (148 m - 145 m)/(150-100) = 3m/50m=0.06 downward• Between B to C, (165 m - 148 m)/(100-75) = 17m/25m= 0.68 downward

6

Head of variable Density

• What happens when a well penetrates and aquifer containing brackish or salty water?

• In locations of groundwater of varying salinity, density corrections must be made to the head measured in the well.

• Remember that the pressure at the bottom of the well is related to the density of the water

• P=ρghp

• hp refers to point water pressure head.

• Luscynski (1961) introduced the concepts of point-water head and fresh-water head.

• Point water head is the water level in a well filled with water coming from a point in an aquifer, and which is just enough to balance the pressure in the aquifer at that point.

• Fresh-water head, is a column of water in a well filled with fresh-water that balances the pressure in the aquifer at that point.

point water pressure for the well screened in the salty water:

P1=ρsghp

\Now, let’s assume that the water in the well is now filled with fresh-water, its pressure would be defined as:

P2=ρfghf

Since the pressures P1 and P2represent the same point in the aquifer, basically the bottom of the well, then P1=P2., so

ρsghp=ρfghf

rearranging to solve for hf, the equivalent fresh-water head:

hf = (ρs/ρf)hp

hp hf

• In a fresh water aquifer, all point-water heads are fresh-water heads, and no correction is necessary.

• In an aquifer of variable salinity, correction of point-water heads to an equivalent fresh-water head needs to be made to obtain hydraulic gradients.

• This works best in determining vertical gradients.

• For theoretical reasons, in regions of lateral variations of salinity, the correction can not be used to determine the hydraulic gradient.