Continuum Equation and

Basic Equation of Water Flow in Soils

Continuum Equation and

Basic Equation of Water Flow in Soils

January 28, 2002

Elementary Volume - 1Elementary Volume - 1

Create a volume with imaginary boundaries within a pool of water (our fluid system)

Call it “elementary volume”

Elementary Volume - 2Elementary Volume - 2

What is the scale of elementary volume ?

H2O

Elementary Volume - 3Elementary Volume - 3 On molecular level, there are molecules and voids. Pick

a point in the molecular volume, and your sample is H, O or void

If we take a larger volume, chance is better that we get a sample of water “as a fluid”

Each point in our Representative Elementary Volume (REV) should give us the same properties

Representative Elementary Volume Representative Elementary Volume

Volume large enough to be representative of the fluid (same properties everywhere)

Small compared to the fluid system as a whole

Can have any shape

REV REV

Assume simple shape: The Cube

The CubeThe Cube

Imagine X-Y-Z axis

Z

X

Yx

x

yy

z

z

Describe volume of water flowing INTO cube

x

x

yy

z

z

Q = q * A

Qx = qx * y * z

Same for Qy and Qz inflow

Qx = qx * y * z

Qy = qy * x * z

Qz = qz * x * y

x

x

yy

z

z

Describe volume of water flowing OUT of the cube

x

x

yy

z

z

Q = q * A + Change in flow

Qx = qx * y * z + ( * x )* y * z x

qx

Qx = qx * y * z + ( * x ) * y * z x

qx

Qy = qy * x * z + ( * y ) * x * z y

qy

Qz = qz * x * y + ( * z ) * x * y z

qz

Outflow in 3 directions gives:

Mass BalanceMass Balance

All that flows in must flow out, except for the storage within the volume

Or:

SOutIn

Mass Balance AssumptionsMass Balance Assumptions Water is incompressible

No compression of water and storage in our “elemental volume”

No sources or sinks in our “elemental volume”

Steady State (no changes over time)Water flowing in equals water flowing out

Thus:Thus:

0 OutIn

All Inflow:

Qx = qx * y * z

Qy = qy * x * z

Qz = qz * x * y

In (qx * y * z) + (qy * x * z) + (qz * x * y)

qz * x * y + ( * z ) * x * y z

qz

qx * y * z + ( * x ) * y * zx

qx

+

qy * x * z + ( * y ) * x * z y

qy

+

Out

0 OutIn(qx * y * z) + (qy * x * z) + (qz * x * y)

-

qx * y * z + ( * x ) * y * zx

qx

qy * x * z + ( * y ) * x * z

qz * x * y + ( * z ) * x * y

y

qy

z

qz

-

-

- ( * x ) * y * z x

qx

- ( * y ) * x * z

- ( * z ) * x * y = 0

y

qy

z

qz

0

z

q

y

q

x

q zyx

0

z

q

y

q

x

q zyx

OROR

Now consider when S 0Now consider when S 0 For example, our REV is a cube of soil where

the change in volumetric water content ( during time (t) is

t

Rate of gain (or loss) of water by our REV of soil is the rate of change in volumetric water content multiplied by the volume of our REV:

zyxt

S

*

Thus:Thus:

SOutIn

Becomes:Becomes:

zyxt

OutIn

*

z

q

y

q

x

q

tzyx

“Continuity Equation of water”

x

qx

- ( * x ) * y * z

- ( * y ) * x * z y

qy

z

qz

- ( * z ) * x * y =

zyxt

*

Proceeding as before we obtain:

z

q

y

q

x

q

tzyx

3-D form of Continuity Equation of water is :

Where: is the change in volumetric water content with time; t

qx, qy and qz are fluxes in the x, y and z directions, respectively.

qt

.In shorthand mathematical notation:

Where the symbol (del) is theVector differential operator, representing the 3-D gradient in space.

OR

qdivt

Where div is the scalar product of the del

operator and a vector function called the divergence.

Now apply Darcy’s law and substitute : Now apply Darcy’s law and substitute :

z

HKq

y

HKq

x

HKq

zz

yy

xx

Into the Continuity Equation, we get : Into the Continuity Equation, we get :

Basic Equation for Water Flow in Basic Equation for Water Flow in SoilsSoilsBasic Equation for Water Flow in Basic Equation for Water Flow in SoilsSoils

z

HK

zy

HK

yx

HK

xt zyx

Food for Thought:Food for Thought: Now that we have an expression for water flow

involving hydraulic conductivity (K) and hydraulic head gradient (H), ….

What about case with constant hydraulic conductivity, K?Flow in Saturated Zone!

What about when K and H is a function of and matric suction head?

Flow in Unsaturated Zone!

Food for Thought:Food for Thought:

An expression exists to define q in steady state…

0

z

q

y

q

x

q zyx