Continuum Equation
and
Basic Equation of Water Flow in Soils
January 28, 2002
Elementary Volume - 1
 Create a volume with imaginary boundaries
within a pool of water (our fluid system)
 Call it “elementary volume”
Elementary Volume - 2
 What is the scale of elementary volume ?
 H2O
Elementary 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
 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
 Assume simple shape: The Cube
The Cube
 Imagine X-Y-Z axis
Z
z
z
X
Y
x
y
x
y
 Describe volume of water flowing INTO cube
z
z
x
y
x
Q=q*A
Qx = qx * y * z
y
 Same for Qy and Qz inflow
Qx = qx * y * z
Qy = qy * x * z
Qz = qz * x * y
z
z
x
y
x
y
 Describe volume of water flowing OUT of the
cube
z
z
x
y
y
x
Q = q * A + Change in flow
Qx = qx * y * z + (
q x
x
* x )* y * z
Outflow in 3 directions gives:
Qx = qx * y * z + (
Qy = qy * x * z + (
Qz = qz * x * y + (
q x
x
q y
y
q z
z
* x ) * y * z
* y ) * x * z
* z ) * x * y
Mass Balance
 All that flows in must flow out, except for the
storage within the volume
 Or:
In  Out  S
Mass 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:
In  Out  0
 All Inflow:
Qx = qx * y * z
Qy = qy * x * z
Qz = qz * x * y
In 
(qx * y * z) + (qy * x * z) + (qz * x * y)
Out 
qx * y * z + (
q x
x
* x ) * y * z
+
qy * x * z + (
q y
y
* y ) * x * z
+
qz * x * y + (
q z
z
* z ) * x * y
In  Out  0
(qx * y * z) + (qy * x * z) + (qz * x * y)
qx * y * z + (
q x
x
* x ) * y * z
qy * x * z + (
q y
y
* y ) * x * z
qz * x * y + (
q z
z
* z ) * x * y
-(
q x
x
* x ) * y * z
-(
q y
y
* y ) * x * z
-(
q z
z
* z ) * x * y = 0
 q x q y qz 
0
 




x

y

z


OR
q x
x

q y
y

q z
z
0
Now consider when S  0
 For
example, our REV is a cube of soil where
the change in volumetric water content (q)
during time (t) is q
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:
 q 
S  
* xyz )
 t 
Thus:
In  Out  S
Becomes:
In  Out 
q
t
* xyz )
Proceeding as before we obtain:
-(
q x
x
* x ) * y * z
-(
q y
y
* y ) * x * z
-(
q z
z
* z ) * x * y = q * xyz 
t
“Continuity Equation of water”
 q x q y q z 

 



t

x

y

z


q


3-D form of Continuity Equation of water is :
q
t
Where:
q
t
q y
 q x
q z
 


 x
y
z





is the change in volumetric water content with time;
qx, qy and qz are fluxes in the x, y and z directions, respectively.
In shorthand mathematical notation:
q
 .q
t
Where the symbol  (del) is the
Vector differential operator, representing the 3-D gradient in space.
OR
q
t
  div q
Where div is the scalar product of the del
operator and a vector function called the
divergence.
Now apply Darcy’s law and substitute :
qx

 Kx
qy

 K
qz

 Kz
H
x
H
y
y
H
z
Into the Continuity Equation, we get :
q
 
H   
H   
H 
 K y
 

 Kx

 Kz

t
x 
x  y 
y  z 
z 
Basic Equation for Water Flow in Soils
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 q and matric
suction head?
Flow in Unsaturated Zone!
Food for Thought:
 An expression exists to define q in steady
state…
q x
x

q y
y

q z
z
0