However estimating hf is problematic for use in a

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Unformatted text preview: pumped well under fully penetrating conditions, Δhp is the additional drawdown in the well due to the effects of partial penetration, α is the fraction of partial penetration (ratio of open interval to thickness), rw is the radius of the well, and b is the saturated thickness of the aquifer. However, estimating Δhf is problematic for use in a simple numerical approximation. Prickett (1967) states that a good approximation for the drawdown due to laminar flow under fully penetrating conditions can be made using the Thiem (1906) equation for steady-state flow in an areally extensive, homogeneous, isotropic aquifer, which can be expressed as , (21) where re is the radial distance to a point where drawdown is negligible. Prickett (1967) indicates that a value of re = 2,000 ft would be adequate for most regional simulation models of confined aquifers. Walton (1962, p. 8) assumes that the partial penetration correction can be based on values of re = 10,000 ft for a confined aquifer and re = 1,000 ft for an unconfined aquifer. The approach of assuming a universal value of re and not adjusting it for variations in aquifer dimensions or properties, however, is overly simplistic and arbitrary. A difficulty arises from the assumptions about the aquifer that are consistent with the applicability of the Thiem (1906) equation. Kruseman and De Ridder (1970, p. 51) note that a true steady state (where drawdown is zero) is impossible in a nonleaky confined aquifer, so that an exact value for re is indeterminate. Nevertheless, the method of Prickett (1967) was implemented and evaluated for a range of assumptions about the value of re in a series of numerical experiments. The results indicated that it worked acceptably well in some tested cases, but yielded unacceptably erroneous values in too many other cases that included plausible combinations of parameter values and boundary conditions. To yield greater accuracy in estimating the drawdown due to partial penetration effects for a broader range of conditions, an analytical solution within the MNW2 Package was implemented to calculate the additional drawdown in a well that does not fully penetrate the saturated thickness of an aquifer. Following Moench (1993), the solution for drawdown (Δh) in a well pumping from a nonleaky confined aquifer can be expressed as the sum of two components (22) where ΔhT represents the drawdown computed with the Theis (1935) solution for flow to a well in a nonleaky confined aquifer. The WTAQ source code for the confined aquifer solutions (for both fully and partially penetrating conditions) has been extracted and incorporated as a subroutine internally within the MNW2 Package to calculate Δhp; this coding implementation is invisible to the user of MODFLOW and MNW2. Note that figures 8 and 9 indicate that, if the partial penetration fraction (α) is large, say greater than 0.90, then there is only a very small partial penetration effect on drawdown, and if α ≥ 0.99, then the effect is negligible. Therefore, to avoid potential numerical problems when α ≥ 0.99, it is automatically assumed that the well is effectively fully penetrating whenever α ≥ 0.99 for a node in a multi-node well and that computation of drawdown due to partial penetration is not necessary under these conditions. When the partial penetration fraction is very low, the drawdown correction may be very large (fig. 9) and very sensitive to small changes in the penetration fraction. Under these conditions, it is possible that the analytically based WTAQ solution for partial penetration effects will fail to converge. If that happens for a node of a multi-node well with a relatively low partial penetration fraction (α < 0.20), then the cell-towell conductance will be set equal to zero, shutting off flow between the aquifer and that node of the well, and an appropriate informational message will be written to the output file; if it fails to converge for a relatively large partial penetration fraction (α ≥ 0.20), then no partial penetration correction will be made (equivalent to assuming α = 1.0), and an appropriate informational message will be written to the output file. It is assumed that the same relations used to estimate partial penetration effects in an aquifer are applicable to a well that is only open to a fraction of the saturated thickness of a model layer (or cell). The general well-loss equation introduced in equation 2 can be expanded to account for additional drawdown due to partial penetration effects in a vertical well as , (23) where Δhp is the additional drawdown in the well due to partial penetration. Following the development of Halford and Hanson (2002, p. 8), the flow between the cell and the well can be expressed in terms of the head difference (equation 23) and a cell-to-well conductance, CWC (L2/T), as (24) Model Features and Processes Assuming that the additional drawdown due to partial penetration is additive with the other head loss terms, substituting the right side of equation 23 into equation 24, and solving for CWC results in (25) The acc...
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This document was uploaded on 01/20/2014.

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