## Flow Equation For An Unconfined Aquifer

In contrast to confined aquifers, an unconfined aquifer has a free surface (water table) boundary, a boundary at atmospheric pressure. Water released from storage occurs due to gravity drainage as the water table in the aquifer responds to pumping, drainage, or natural or artificial recharge. The unconfined flow problem is commonly analyzed using the Dupuit assumptions: (1) uniform and horizontal flow within any vertical cross section, and (2) the velocity at the free surface may be expressed as qx = —K(dh/dx). The second assumption implies small slopes of the free surface.

Using the concept of vertical averaging, the governing equation characterizing two dimensional horizontal flow in an unconfined aquifer can be expressed as:

where Kxx, Kvv = components of hydraulic conductivity along the x and y coordinate axes (L/T)

SY = specific yield of the aquifer (dimensionless) Ss = specific storage of the aquifer (1 /L) R — net recharge (L/T)

The specific storage effect is generally negligible when compared to the specific yield and can be dropped to give

which is the nonlinear Boussinesq equation. Pumping and injection wells may also be incorporated via Eq. (10) in the recharge term of the equation as point sources or sinks. There are several ways to linearize Eq. (13). The first method is based on the assumption that the depth of the flow varies slightly in the flow domain, e.g., mildly sloping aquifers. The head may then be expressed by h = h + h

where h is the average depth of flow and h is the derivation of the head from h. if we assume h the Boussinesq equation becomes

where Txx = Kxxh and Tyy = Kyyh. It can be seen that Eq. (16) is identical to the governing equation of the confined flow.

The second method of linearization is based on the temporal variation of the temporal derivative. Rewriting the temporal derivative as dh_SLdh? y dt~ 2h dt

and assuming S = Sy/2h is approximately constant and equal to Sy/2h, the Bous-sinesq equation is intrinsically linear in h2,

If the initial and boundary conditions are also linear in h2, Eq. (18) has been shown to be more accurate in predicting the water level.

The third method of linearization, as is used in MODFLOW (McDonald and Harbaugh, 1988), is to use the calculated head value from the last iteration h' to replace h, i.e., Txx = Kxxh' and Tyy = Kyyh', and iteration continues until a convergence criterion is met.

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