Thus the length within a cell of a vertical borehole

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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 finite-difference grid used for a regional ground-water 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 finite-difference cell cannot be characterized by an ellipsoid that has principal directions aligned with the finite-difference grid coordinate axes. Also, if all else is the same, then the cell-to-well conductance increases proportionately with the length of a borehole within a given cell (analogous to the streambed conductance). If it is assumed that the cell-to-well 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 z-directions 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 cell-to-well conductance per unit length multiplied by the length of the borehole (as described below). The linear aquifer-loss 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 pump-capacity 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 three-dimensional perspective drawing of a representative finite-difference cell connected to a nonvertical multinode well passing through the block-centered node, the lower-left corner on the front face, and the upper-right 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 ground-water 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 (y-direction) (fig. 35), then (26) where Ay is the value of A for a well oriented parallel to the y-direction, and roy is the effective radius of the cell when a horizontal well is oriented parallel to the y-direction, also expressed as . (27) Similarly, if a horizontal well is aligned with the rows of the grid (x-direction), then . 33 Figure 35. Schematic cross-sectional view of a finite-difference cell containing a horizontal well aligned with the y-direction of the grid, showing approximate relation between cell size and equivalent grid-cell radius (ro), although Dz is usually much smaller than Dx in regional ground-water 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 z-directions, 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 y-direction, 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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