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Potential and Flow Functions

In the Basic Equations section, the fundamental equation of groundwater flow expressed in terms of discharge potential $ is:

The potential $(x,y) is a single-value function everywhere in the x, y plane. Therefore, lines of constant $2

called equipotential lines, can be drawn in the x, y plane as shown in Figure 9.6.8. When the lines are drawn with a constant interval between the values of the two successive lines (A$ = - $2 = $2 - $3 = . . .), then an equal and constant amount of potential drop is between any two of the equipotential lines.

FIG. 9.6.8 Potential and flow lines.

FIG. 9.6.8 Potential and flow lines.

At any arbitrary point on the equipotential line, flow occurs only in the direction perpendicular to the line (n direction), and no flow occurs in the tangential direction (m direction) as

Accordingly, lines can be drawn perpendicular to the equipotential lines as shown in Figure 9.6.8. These lines are called flow or stream lines.

At this point a second function, called flow or stream function is introduced. Since the specific discharge vector must satisfy the equation of continuity, the function ^ is defined by qx =

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