With the substitution of a new variable defined as

where Cc is an arbitrary constant, Equations 9.3(24) can be simplified since the derivatives of Cc with respect x and y are zero as

The function $ is referred to as the discharge potential for horizontal flow or simply as the potential.

Now the governing equation for horizontal confined flow, Equation 9.2(13), expressed in terms of the head $ is

and can be written in terms of the potential t as —— + —- = 0

ax2 ay2 0

Basic Equations

The fundamental equations of groundwater flow can be derived in terms of the discharge vector Qi rather than the specific discharge qi. For two-dimensional flow, the discharge vector has two components Qx and Qy and is defined as

With the use of Darcy's law

These equations can be rewritten as Q = a(KH-)

Solutions to horizontal confined flow can be obtained when $ is determined from this Laplace's equation with proper boundary conditions satisfied.

The following equations give solutions for horizontal confined flow in terms of (1) One-dimensional flow t = KH- = (2) Radial flow

Two-dimensional flow problems expressed by the differential Equation 9.3(29) are discussed in more detail in Section 9.6.


Gupta, R.S. 1989. Hydrology and hydraulic systems. Prentice-Hall, Inc.

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