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
Discharge Potential
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. T=KH.

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 discharge by differentiation.

9. Interpreting Stream Functions
Any Line l
l is an arbitrary line
t is tangent to l
n is normal to l t
Flow Line 2 n
Flow Line 1
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 2 we 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 via the 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 that U is irrotational.
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.

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 for a 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
Circumference=2pr 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