Darcy’s Law and Hydraulic Head

Darcy's Law and Hydraulic Head

1. Hydraulic Head

Q

=

K

h1 - h2 L

A

h1 and h2 are hydraulic heads associated with

points 1 and 2.

The hydraulic head, or total head, is a measure of the potential of the water fluid at the measurement point.

hp1 h1

Q z1

datum

h2 hp2

z2

"Potential of a fluid at a specific point is the work required to transform a unit of mass of fluid from an arbitrarily chosen state to the state under consideration."

Three Types of Potentials

A. Pressure potential

work required to raise the water pressure

W1

=

1 m

PV dP

0

=

1 m

P m dP = P

0 w

w

w : density of water assumed to be independent of pressure

V: volume

z=0 P=0 v=0 Reference state

z=z P=P v=v

Current state

B. Elevation potential work required to raise the elevation

W 2

=

1 m

Zmgdz = gz

0

C. Kinetic potential work required to raise the velocity (dz = vdt)

W3

=

1 m

Zmadz

0

=

1 m

Z 0

m

dv dt

dz

=

vvdv = v 2

0

2

Total potential:

=

P

w

+

gz

+

v2 2

Unit [L2T-1]

Total [hydraulic] head:

h

=

g

=

P

w g

+

z

+

v2 2g

Unit [L]

Total head or hydraulic head:

h

=

P

w g

+

z

+

v2 2g

Piezometer

pressure head [L]

elevation [L]

Kinetic term

P1

g

h1

P2

g

h2

z1

z2

datum

A fluid moves from where the total head is higher to where it is lower. For an ideal fluid (frictionless and

incompressible), the total head would stay constant.

(Fetter, p141) Low elevation to high elevation Low pressure head to high pressure head Flow between points of same elevation Flow between points of same pressure head

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