Transcript Stream Functions - Rice University

Stream Function Definitions
For Confined Aquifer
U  Th
  Th
Then
Discharge
Potential
U is aquifer flux.
It is a vector.
U = qb,
where
q is flux per unit
depth and b is
aquifer thickness
U  Th  
Governing Equations
Continuity Equation becomes
divergence U  U  0
Zero Divergence implies no sources or sinks
Get LaPlace Equation by substitution into above
U       0
2
Solutions to LaPlace equation are potential functions. When
Divergence of a vector field is zero, flux is the gradient of the
potential.
Unconfined Aquifer (redefine T)
Base of Aquifer is datum, H is saturated thickness. T=KH.
U  Th  KHH
U    KHH
KH

2
2

Potentials are Constant Heads
Because U   , gradient of potential field gives aquifer
flux. Lines along which potential is constant are called
Equipotential lines, and are like elevation contours.
Darcy velocity may be calculated from the aquifer flux U.

U
q
b
b is the aquifer thickness for a confined aquifer.
KH 2
2

implies H 
. Also U  .
2
K
Substituting for U and H.
U

1/ 2
q 
  2K   2K
H
2
For an Unconfined Aquifer
K
Parallel Vectors
In both cases (confined, unconfined) the following vector fields are
parallel
v || q || U
Note: v is the seepage velocity v = q/n, where n is porosity
Since the divergence is zero, the streamlines cannot cross or join.
U is convenient, since its formulation is the same for confined and
 aquifers.
unconfined
Flow along a line l
Consider a fluid particle moving along a line l. For each small
displacement, dl,
dl  idx  jdy
Where i and j are unit vectors in the x and y directions, respectively.
Since dl is parallel to U, then the cross product must be zero.
Remembering that:

And that:
Then


ii 0
U  iU x  jU y
and
i jk
U  dl  iUx  jUy  idx  jdy

And
 Ux dy  Uy dx k  0

dx dy

Ux Uy
dx
dy
dl
Stream Function
Since
dx dy
must be satisfied along a line l, such a line is

Ux Uy
called a streamline or flow line. A mathematical construct called
a stream function can describe flow associated with these lines.

The Stream Function x, y  is defined as the function which
is constant along a streamline, much as a potential function is
constant along an equipotential line.
y  is constant along a flow line, then for any dl,
Since x,



d 
dx 
dy  0
x
y
Along the streamline
Stream Functions
Rearranging:
we get



d 
dx 
dy  0
x
y


dx  
dy
x
y
from which we can see that

and

dx dy

and
Ux Uy
U y dx  U x dy


Ux  
and U y 
x
 y
So that if one can find the stream function, one can get the discharge
by differentiation.

Interpreting Stream Functions
l
Any Line
t
n
2
1
l is an arbitrary line
t is tangent to l
n is normal to l
dx
dy
t  i
j
dl
dl
dy
dx
n i
j
dl
dl
Flow Line 2
Flow Line 1
t n  0
What is the flow that crosses l

between Flow Lines 1 and 2?
dy
dx 


For each 
increment dl:
dQ  U  ndl  U x i  U y j  i 
jdl
dl
dl 
 U x dy  U y dx



dy 
dx
y
x
 d

Discharge Between Lines 1, 2
If we integrate along line l, between Flow Lines 1 and 2
we will get the total flow across the line.
Q12 
2
2
2
1
1
1
 dQ   d  
 2  1  1  2
This is true even if K or T is heterogeneous
Conjugate Functions
We have already shown that:  2  0 and since U  
Ux  


; Uy  
x
y


The rotation of the discharge potential field may be calculated via
the Curl as follows:
 U U          
  U   y  x k      k  0
y  x  y  y  x 
 x
Since the Curl of U is zero, the flow field is irrotational.


Conjugate Functions
Since the Curl of U is zero, the flow field is irrotational.
Doing the same calculation, but using the relationship between the
Stream Function and U, we get:
U y U x         
2
  U  

k      k   k  0
y  x x  y  y 
 x
Conjugate Functions
From this, we see that
U is irrotational.
 2   0 , because we know that
Thus, the Potential Function and Stream Function both satisfy
LaPlace’s Equation.

By definition, flow lines are parallel to streamlines (lines of
constant stream function value), and perpendicular to lines of
constant potential.
Thus, the streamlines and potential lines are also perpendicular.
Superposition
One special property of solutions to the LaPlace Equation is that it is
linear. Thus, solutions to the equation may be added together, and
the sum of solutions will also be a solution.
2
2
2 1
If    0 and    0
Then
 2 1  2  0
This implies that if one can find a solution for uniform flow and for
a point source
 or sink, then they can be added together to
get a solution for (uniform flow)+ (source) + (sink)
Main Equations - Aquifer Flux
In Cartesian Coordinates:

In Polar Coordinates:


Ux  

x
y
 
Uy  

y x

1 
Ur  

r
r 
1  
U  

r  r
These are the Cauchy-Riemann Equations.
Uniform Flow
For a uniform flow rate U (L2/T) at an angle a with respect to the x-axis:
Ux  
a


= U cosa
x
Uy  

 U sin a
y


d 
dx 
dy
x 
y
 U x dx  U y dy
 U cos a dx  U sin a dy
  0  U x cosa  y sin a 
  0  U y cosa  x sin a 


Radial Flow Source
For an injection well at the origin, flow Q across any circle with
radius r is equal to due to continuity.
Ur 
Q

1 


2r
r
r 
Circumference=2r
U = Flow per unit length
There is no rotational flow:
1  
U  

0
r  r
r
Q
Develop and  Lines
Q

1 
Ur 


2r
r
r 
Separate

Q

r
2r
Equipotential
Q
Integrating
  0 
ln r
2
1 
Q


r  2r

Q


2
Streamline
Q
  0 

2
and  Lines
for Capture Zone Theory
Combine a uniform field and a well
Q
  0  Ux 
ln r
2
Q

  Uy
2
Q
1 y
or 
tan
 Uy
2
x