Soil Physics 2010

Outline

• Announcements

• Richards’ equation

• Unsaturated flow

Soil Physics 2010

Announcements

• Homework 4 due March 3

• Excel Solver demo on course website

• Quiz!

h

Soil Physics 2010

Question 1

Drying

Wetting

h

0

0

h

0

0

Soil Physics 2010

Question 2 Different lines show different possibilities

Soil Physics 2010

Why different flow equations?

Steady-state Transient

Saturated

Unsaturated

Darcy’s law

Darcy’s law (with K())

N/A

Richards’ equation

Darcy’s law:L

AKq

changes with time

No K()

No No ()

Soil Physics 2010

Equation of Continuity(Conservation of Mass)

Steady-state Transient

Saturated

Unsaturated

Darcy’s law

Darcy’s law (with K())

Richards’ equation

Input – Output = Change in Storage

x

q

=t

tx

q

Soil Physics 2010

Richards’ equation

LKq

Given Darcy’s law:

xK

xx

q

Let things change from place to place (say, in the x-direction)

tx

q

We also want

conservation of mass

xK

xt

So we substitute it in

to the left-hand side

Soil Physics 2010

Richards’ equation

xK

xt

But this doesn’t allow

K to change with

xK

xt

So we permit that, and…

voilà: Richards’ equation

zK

zyK

yxK

xt zyx

We can generalize it to 2 or 3 dimensions…

… and add in anisotropy

Soil Physics 2010

Richards’ equation

xK

xt

Remember that the

potential gradient, ,

combines elevation, osmotic, pressure, and matric components (among others).

x

Sometimes it’s convenient to separate out the elevation part:

1 x

Kxt

Vertical

0 x

Kxt

Horizontal

Just remember that this doesn’t include elevation!

Soil Physics 2010

Topp & Dane, Methods of soil analysis

K(), averages by texture

Coarse soils:Lower Higher Ks

More abrupt drop

At low :Small → big KHuge range of KHuge uncertainty in K

Soil Physics 2010

K() and K() for 3 textures(Mualem-van Genuchten functions)

1.E-16

1.E-14

1.E-12

1.E-10

1.E-08

1.E-06

1.E-04

1.E-02

1.E+00

1.E+02

0 0.1 0.2 0.3 0.4 0.5 0.6

Sand

clay

loam

K()

1.E-16

1.E-14

1.E-12

1.E-10

1.E-08

1.E-06

1.E-04

1.E-02

1.E+00

1.E+02

1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04

Sand

clay

loam

K()

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

0 0.1 0.2 0.3 0.4 0.5 0.6

Sand

clay

loam

()

K() has more hysteresis

Soil Physics 2010

How do we measure K() in the lab?Ks is pretty easy.K() is slow, and hard to control.

• Apply water at steady q < Ks

• Wait till outflow = inflow

• Measure and/or across a “test interval”• Prevent evaporation

• Water evenly, no disturbance

• Tall column, or tension at bottom

• Tensiometer can change flow

• Measure with gamma-rays

Soil Physics 2010

How do we measure K() in the lab?

K() is slow, and hard to control.

Other methods:

• Centrifuge

• Evaporation

• One-step

• Multi-step

As decreases:SlowerHarder to controlMore uncertainty

Soil Physics 2010

How do we measure K() in the field?

• Instantaneous profile• Various others• Best solved with Inverse methods

The “forward problem”:Given the parameters and boundary conditions, simulate what happened (or will happen).

The “inverse problem:Given the data and the boundary conditions, estimate the parameter values. (A spreadsheet’s Solver solves an inverse problem.)