# The linear well loss coefficient b collectively

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Unformatted text preview: pecified in the MODFLOW discretization and internal-flow packages [Block-Centered Flow (BCF), Layer-Property Flow (LPF), or Hydrogeologic-Unit Flow (HUF) Packages]. The linear well-loss coefficient (B) collectively defines head loss from flow through formation damaged during well drilling, the gravel pack (representing a possible increased hydraulic conductivity relative to the aquifer), and the well screen. The geometry of the problem and some of the relevant terminology are illustrated in figure 2. The coefficient B can be used directly to define head loss or it can be recast in terms of a dimensionless skin coefficient (Skin), as defined by Earlougher (1977) and Halford and Hanson (2002). The skin coefficient represents a zone of affected hydraulic properties close to the wellbore or well screen. The value of Skin for a model cell depends on the hydraulic conductivity of the skin, the length of the borehole, and the thickness of the skin (which equals rSKIN - rw), among other factors, as , (9) where b is the saturated thickness of the cell (L), bw is the saturated (or active) length of the borehole in the cell (bw = b where Δx is the grid spacing in the x- (column-) direction, and Δy is the grid spacing in the y- (row-) direction. If the grid is square, then this is simplified to (6) However, if the porous medium is anisotropic, then the directional hydraulic conductivities must be considered, and Peaceman (1983) indicates that, in this general case, ro is given by , (7) where Kx and Ky are the values of hydraulic conductivity in the x- and y-directions. The definition of ro given in equation 7 is used in MNW2. For general anisotropic conditions, transmissivity (T) can be written , where b is the saturated thickness of the cell (L). The constant term in equation 4 is the aquifer-loss coefficient (A) for a vertical well, which can be written for anisotropic conditions as Figure 2. Schematic horizontal cross section (plan view) through a vertical well in a finite-difference cell showing some of the factors affecting computed well loss. K is hydraulic conductivity and r is radius. 6 Revised Multi-Node Well (MNW2) Package for MODFLOW Ground-Water Flow Model for a fully penetrating vertical well), and Kh is the effective horizontal hydraulic conductivity of the cell (L/T) when horizontal anisotropy is present, wherein . The skin effect can be pictured as flow occurring across a cylinder having a hydraulic conductance that differs from (and is typically less than) the hydraulic conductance of the media comprising the finite-difference cell in which the well is located. The value of Skin would be negative if KSKIN is larger than Kh. Halford and Hanson (2002) relate the coefficient B to Skin as . terms into the cell-to-well conductance term. The flow to the nth node of the multi-node well is thus defined by the head difference between the cell and the well times a cell-to-well hydraulic conductance as (12) where CWCn is the nth cell-to-well hydraulic conductance (L2/T). After substituting the right side of equation 2 for the term in parentheses, equation 12 can be rewritten to solve for CWCn as (10) The user has the option of directly specifying a value of B (by setting input parameter LOSSTYPE in dataset 2b equal to “GENERAL”) or specifying the characteristics of the skin (if LOSSTYPE is set equal to “SKIN”). In the latter case, MNW2 will automatically calculate a value of Skin and subsequently a value of B using equations 9 and 10. The nonlinear well-loss coefficient (C, with dimensions TP/L(3P-1)) defines head loss from any turbulent flow near the well (Rorabaugh, 1953). The coefficient C and power term (P, dimensionless) typically are estimated at specific wells through the application of step-drawdown tests. There remains some disagreement in the literature about the nonlinear well-loss terms (see, for example, the review by Ramey, 1982). Jacob (1947) states that “the loss of head that accompanies the flow through the screen … is proportional approximately to the square of the discharge.” Rorabaugh (1953) argues that the power (P) is an empirical exponent that “may be unity at very low rates of discharge or it may be in excess of 2” where turbulent flow occurs in or near the well; he provides examples from several field cases for which P varied between 2.4 and 2.8. The higher the value of P, the more likely that the numerical solution will have difficulty converging. If that happens, then the user can reduce the value of P. Because this additional nonlinear term ( ) may cause numerical problems or may not be needed, the user has the option of not including the nonlinear well-loss term in any multi-node well (for example, if LOSSTYPE = GENERAL, then set C = 0.0 in dataset 2c). Equation 2 can be rewritten in terms of the flow rate to each node (Qn). For the simplest case in which only the aquifer-loss term AQn applies, the resulting equation is . (11) The expression in brackets has dimensions of (L2/T) and can be viewed as a hydraulic conductance term, which Halford and Hanson (2002) refer to as the cell-to-well conductance (CWCn). A mor...
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