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Unformatted text preview: column, and layer directions,
respectively; these also correspond with terms DELR, DELC,
and THCK, as used by Harbaugh and others (2000) to represent
the widths of the cell in the row and column directions, and the
vertical thickness of the cell, respectively. In a finitedifference
grid used for a regional groundwater simulation, the grid spacing in the vertical direction is usually much smaller than in the
horizontal directions. Thus, the length within a cell of a vertical
borehole is usually equal to the smallest possible dimension
of the cell. The length within a cell of a horizontal borehole
would typically be much larger than that of a vertical borehole.
The longest possible length of a borehole within a cell would
occur if it connects the opposite corners of the cell and passes
diagonally through the node (fig. 34). Therefore, the borehole
length in any direction through a finitedifference cell cannot
be characterized by an ellipsoid that has principal directions
aligned with the finitedifference grid coordinate axes. Also, if
all else is the same, then the celltowell conductance increases
proportionately with the length of a borehole within a given cell
(analogous to the streambed conductance).
If it is assumed that the celltowell conductance per unit
length in each principal direction is an “intrinsic property” for
which the effective value for well alignments other than in the
x, y, and zdirections can be estimated using ellipsoidal interpolation (analogous to the hydraulic conductivity tensor), then
the effective value of CWC in a cell for a nonvertical linear
well or well segment can be estimated from the effective value
of the celltowell conductance per unit length multiplied by
the length of the borehole (as described below).
The linear aquiferloss coefficient (A), as described for a
vertical well by equation 8, is described first. For a horizontal Figure 33. Plot showing results
of applying the pumpcapacity
relations to the modified Reilly
problem in which the desired
discharge equals 7,800 cubic
feet per day (ft3/d) for two 365day transient stress periods with
three nearby wells pumping
at 4,000 ft3/d during the first
transient stress period. When
the head in the well equals
or exceeds 3.65 feet, the lift
is equivalent to that for the
maximum discharge of the pump.
When the head in the well drops
below 23.75 ft, the lift is greater
than the maximum capacity of
the pump and the net discharge
becomes zero. Model Features and Processes Figure 34. Schematic threedimensional perspective drawing of a
representative finitedifference cell connected to a nonvertical multinode well passing through the blockcentered node, the lowerleft
corner on the front face, and the upperright corner of the back face
of the cell. For visual clarity, vertical spacing (Dz) is exaggerated
relative to horizontal dimensions compared to a typical grid used in a
regional groundwater simulation. well, the value of ro is recomputed by replacing values of b,
Kx, and Ky with the appropriate corresponding terms for the
new orientation. For example, if a horizontal well extends
the full length of the cell and is aligned with (parallel to) the
columns of the grid (ydirection) (fig. 35), then (26)
where Ay is the value of A for a well oriented parallel to the
ydirection, and roy is the effective radius of the cell when
a horizontal well is oriented parallel to the ydirection, also
expressed as . (27) Similarly, if a horizontal well is aligned with the rows of
the grid (xdirection), then . 33 Figure 35. Schematic crosssectional view of a finitedifference
cell containing a horizontal well aligned with the ydirection of the
grid, showing approximate relation between cell size and equivalent
gridcell radius (ro), although Dz is usually much smaller than Dx in
regional groundwater models. assuming that the basic relations represented in equation 10
for a vertical well would also be applicable for a horizontal
well. If the properties of the well (including rw, bw, rskin, and
KSKIN) are known (or estimated), then extending equation 10
to a horizontal well requires the determination of the proper
components of hydraulic conductivity to substitute in the
equation on the basis of the orientation of the wellbore in relation to the grid.
For well orientations aligned with the x, y, and zdirections, it is assumed that flow into or out of the well is locally
in the plane perpendicular to the well and has radial symmetry.
In equation 9, the average effective hydraulic conductivity
(Keff = Kh) within the horizontal plane (which is perpendicular
to the axis of a vertical well) is taken as the geometric mean
of the principal values of the hydraulic conductivity tensor (Kx
and Ky); that is,
. Then by analogy, for a horizontal well oriented in the ydirection, the effective hydraulic
conductivity would be a function of Kx and Kz. Assuming that
flow radially into or out of the well is parallel to this plane, the
average effective hydraulic conductivity within the plane of
radial flow can be computed as the geometric mean of Kx and
Kz, or
. Substituting...
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 Winter '14

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