Stream Function Definitions

1
Stream Function Definitions
For Confined Aquifer
U is aquifer flux. It is a vector. U qb,
where q is flux per unit depth and b is aquifer
thickness
DischargePotential
Then
2
Governing Equations
Continuity Equation becomes
Zero Divergence implies no sources or sinks
Get LaPlace Equation by substitution into above
Solutions to LaPlace equation are potential
functions. When Divergence of a vector field is
zero, flux is the gradient of the potential.
3
Unconfined Aquifer (redefine T)
Base of Aquifer is datum, H is saturated
thickness. TKH.
4
Potentials are Constant Heads
Because , 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.
b is the aquifer thickness for a confined aquifer.
Substituting for U and H.
For an Unconfined Aquifer
5
Parallel Vectors
In both cases (confined, unconfined) the
following vector fields are parallel
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 unconfined aquifers.
6
Flow along a line l
Consider a fluid particle moving along a line l.
For each small displacement, dl,
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
dx
dy
dl
And that
and
Then
And
7
Stream Function
Since must be satisfied along a line l,
such a line is called a streamline or flow line.
A mathematical construct called a stream
function can describe flow associated with these
lines.
The Stream Function is defined as the function
which is constant along a streamline, much as a
potential function is constant along an
equipotential line.
Since is constant along a flow line, then for
any dl,
Along the streamline
8
Stream Functions
Rearranging
and
we get
and
from which we can see that
So that if one can find the stream function, one
can get the dischargeby differentiation.
9
Interpreting Stream Functions
l
Any Line
l is an arbitrary line t is tangent to l n is
normal to l
t
Flow Line 2
Flow Line 1
n
What is the flow that crosses l between Flow
Lines 1 and 2? For each increment dl
10
Discharge Between Lines 1, 2
If we integrate along line l, between Flow Lines
1 and 2we will get the total flow across the
line.
This is true even if K or T is heterogeneous
11
Conjugate Functions
We have already shown that
and since
The rotation of the discharge potential field may
be calculated viathe Curl as follows
Since the Curl of U is zero, the flow field is
irrotational.
12
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
13
Conjugate Functions
From this, we see that ,
because we know thatU is irrotational.
Thus, the Potential Function and Stream Function
both satisfyLaPlaces Equation. By definition,
flow lines are parallel to streamlines (lines of
constant stream function value), and
perpendicular to lines ofconstant
potential. Thus, the streamlines and potential
lines are also perpendicular.
14
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.
If and Then
This implies that if one can find a solution for
uniform flow and fora point source or sink, then
they can be added together to get a solution for
(uniform flow) (source) (sink)
15
Main Equations - Aquifer Flux
In Cartesian Coordinates
In Polar Coordinates
These are the Cauchy-Riemann Equations.
16
Uniform Flow
For a uniform flow rate U (L2/T) at an angle a
with respect to the x-axis
a
17
Radial Flow Source
For an injection well at the origin, flow Q
across any circle with radius r is equal to due
to continuity.
r
Circumference2pr
Q
U Flow per unit length
There is no rotational flow
18
Develop F and Y Lines
Separate
Equipotential
Integrating
Streamline
19
and Y Lines for Capture Zone Theory