Lecture 9

Darcy’s Law

Fetter 4.4

The flow of ground water, or any other fluid moving through rock (oil, brine) is

governed by an empirical law, one derived from experimental observation, not

from theory. This expression for ground water flow is known as Darcy’s Law.

Henry Darcy formulated this law in the 1850’s and published his observations in

1856. Darcy was born in Dijon, France in 1803. To help Dijon receive a dependable

supply of water, Darcy designed a collection system for water from a large spring

more than 10km from Dijon. The water was piped to the city and was conveyed to

homes through a distribution system designed by Darcy.

Darcy became very knowledgeable about the flow of fluids in pipes. He also worked

with the flow of fluids through pipes filled with sand. The sand acted as a purification

system.

Darcy designed the following experiment to describe flow of fluids through sand

filled pipes.

This is an example of a pipe which can be used to demonstrate Darcy’s

experiment:

Figure 4.12 Horizontal pipe filled with sand to demonstrate Darcy’s experiment.

(Darcy’s original equipment was actually vertically oriented.)

 - cylinder filled with sand. - circular area = r2 = A. - Water flows through the cylinder at a constant rate Q, maintained by keeping ha constant (constant head). -Head loss is produced across the sand tube () (ha - hb) = friction - Darcy noted that if he doubled Q, that the head loss also doubled, this implies that their is a direct relationship between Q and the gradient.

Q is proportional to the head loss divided by the flow length à the hydraulic

Q as well is directly proportional to the cross section area. If a tube of larger

diameter is used, then a larger Q must be maintained to keep the same head.

Darcy called this proportionality constant K, the permeability of the medium.

K depends on the size, shape, packing and orientation of the material in the

sand tube.

When these portions are combined, we are left with Darcy’s Law to describe

fluid flow through porous media.

The negative sign is used by convention to note that water is flowing from

Darcy’s Law states that fluid will flow through a porous media at a rate which

is proportional to the product of the cross sectional area through which flow

can occur, or the hydraulic gradient and the hydraulic conductivity.

Hydraulic Conductivity is a term which has replaced what Darcy called

the permeability.

If Darcy would have investigated fluids other than water, he would have

realized that the viscosity of the fluid influences Q.

Viscosity is the resistance of a fluid to flow.

For instance, molasses has a higher viscosity than water. Molasses is more

resistant to fluid flow.

Viscosity is indirectly proportional to Q.

If a more viscous fluid is used Darcy’s experiment, then Q decreases.

 µ = Dynamic viscosity

Fetter calls µ the Dynamic Viscosity.

Not all water is pure. Much water contains dissolved solids or other material

dissolved in solution. This can be expressed as the specific weight g . The

Specific weight is the force exerted by gravity on a unit volume of the fluid.

Thus, the g is a function of r and g.

 g = r g where g = acceleration due to gravity.

Q is directly proportional to g

 Hence:

Q then is a function of not only the medium but of the fluid used.

Intrinsic Permeability - representative of the properties of the porous media alone

 Ki = Cd2 Where: Ki = intrinsic permeability C = shape factor properties of Media d = mean pore diameter

K- Hydraulic Conductivity

 -property of media and fluid

Ki = Darcy’s (unit)

à Term typically used in petroleum.

Aquifer - a geologic unit that can store and transmit water at rates fast enough

to supply reasonable amount to wells.

 Equivalent to an ore an economic deposit of a mineral.

Most rock units near the earth’s surface have hydraulic conductivity sufficient

to make them aquifers.

 Aquifers have porosity and permeability.

An average house using 150 - 300 gallons of water a day requires a well

yielding > 2-3 gallons per minute.

Confining layer - a geologic unit having little to no intrinsic permeability < approximately

10-2 Darcy’s

 dependent on the permeability contrast between rocks units - 2 orders of magnitude.
 Water usually moves through confining layers, although very slowly.

Leaky Confining layer - a confining layer that can store and transmit water.

ENV 302 - Lectures