Soil Physics 2010

Outline

• Announcements

• Todd on aggregates

• Saturated flow of water

Soil Physics 2010

Announcements

• No office hours today

• Reminder: Homework due Feb. 10

• Reminder: Exam Feb. 12

• Example exam is now posted. Don’t panic! I covered material in a different order that year, and the class was not dual-listed at the 400-level.

Soil Physics 2010

Todd on Aggregates

Soil Physics 2010

Saturated flow

When is the soil saturated?

Near the surface under heavy rain

Confined aquifer

Under a lake or stream

Recharge basin

Go deep enough…

It turns out that the saturated flow equations are just special (simpler) cases of flow more generally.

Soil Physics 2010

The city of Dijon1800

Ancient capitol fallen on hard times

Soil Physics 2010

Paris

Dijon

Lyon

France, 1800s

Political turmoil:• 1804 - 1815 First Empire - Napoleon

• 1815 - 1830 Restoration - Louis XVIII - Charles X

• 1830 - 1848 Bourgeois Monarchy - Louis-Philippe

• 1848 – 1851 Second Republic

• 1852 - 1870 Second Empire - Napoleon III

Soil Physics 2010

Henry Darcy

Fellow students at L’École des Ponts et Chaussées*:

CauchyChézy

ColiolisDupuitFresnelNavierPitot

St. Venant

Henry graduated 12th in his class. He had been 1st, but he got violently upset

with a chemistry professor over a question about

cooking.* The school of Bridges and Roads

Soil Physics 2010

Most famous publication

Soil Physics 2010

Darcy’s experimentreported in Appendix D ofLes Fontaines Publiques…

Henry Darcy

“One does not get the main square in town named in one’s

honor on the basis of some column experiments.”

-- Allan Freeze, 1994Soil Physics 2010

• Public water supply in Dijon: clean, reliable, gravity flow 12 km to 142 public fountains

• 4 km railroad tunnel allowed the main Paris-Lyon track to pass through Dijon

• Several large bridges near Dijon

• Improvements to the Pitot tube

• Extensive studies on:Flow through open channelsFlow through pipesRoad construction

• Near the end of his life, a few studies on flow through sand

Henry Darcy

Soil Physics 2010

“As much as possible, one should favor the free drawing of water because it is necessary for public health. A city that cares for the interest of the poor class should not limit their water, just as daytime and light are not limited.”

Soil Physics 2010

Darcy’s experiment

1 m3

2.5 m

• Column filled with sand to different depths

• Different water pressures applied across column

• Discharge Q measured (volume of water / time)

• Experiment repeated with different sands

Soil Physics 2010

Darcy’s 3 observations:

1: Q cross-sectional Area2: Q drop in (pressure + elevation)3: Q 1 / flow distance

State of the art

Soil Physics 2010

For slow flow through a pipe,

Q dischargeR radius viscosityp pressure dropL length

L

pRQ

8

4

Mean velocity isL

pRu

8

2

(Poiseuille’s law)

Soil Physics 2010

Darcy’s 1st observation

1: Q cross-sectional Area2: Q drop in (pressure + elevation)3: Q 1 / flow distance

Assume zero resistance in pipes

Darcy’s experiments used a vertical column, but a horizontal column is simpler.

Soil Physics 2010

Darcy’s 2nd observation

1: Q cross-sectional Area

2: Q drop in (pressure + elevation)3: Q 1 / flow distance

Pressure = Elevation?

When you swim underwater, your ears feel pressure

Why doesn’t the water at the bottom of the pool – under lots of pressure – shoot up to the top?

The energy is the same all through the pool. Surface water has elevation; deep water has pressure.

Dep

th

Pressure

Elevat

ion

Soil Physics 2010

Pressure+ Elevation

Energy

Soil Physics 2010

Darcy’s 3rd observation

1: Q cross-sectional Area2: Q drop in (pressure + elevation)

3: Q 1 / flow distance

L

Soil Physics 2010

Darcy’s Law as he wrote it

L

zhzhKAQ 2211

HeightWater

pressure

Water volume / unit

time

Length of flow

Cross-sectional area of flow

Proportionality coefficient:Hydraulic

conductivity

Soil Physics 2010

Units in Darcy’s Law

L

zhzhKAQ 2211

L

LLL

T

L

T

L 2

3

UnitlessVelocity

Soil Physics 2010

Key implications of Darcy’s law: 1

L

hhKAQ 12

For flow through a uniform medium, the hydraulic gradient is constant.

02 h

Taking the derivative of Darcy’s law,

gives the Laplace equation,

hKq

Soil Physics 2010

Key implications of Darcy’s law: 2

The flow is linearly proportional to the gradient.This puts Darcy’s law into the same class as several other equations:

Hooke’s law (elasticity)

Darcy’s law (hydraulic conductivity)

Ohm’s law (electrical conductivity)

Fourier’s law (heat conduction)

Fick’s law (diffusion)

hKq

Vj Tqh CDf

E

1

Soil Physics 2010

Key implications of Darcy’s law: 3

K is a property of the medium*.

The hydraulic conductivity K is not changed by whether the water flows up instead of down, or by having a greater or smaller gradient, or by the pressures or elevations themselves.

In fact, K can often be predicted with reasonable accuracy, given some other information about the medium. For example, the porosity and the grain size or pore size distribution allow a fair estimate.

* Also of the fluid – see later