# Flow to wells from a circular source

(a) Equations for flow and drawdown produced by a single well supplied by a circular source are given in figures 4-10 through 4-12. It is apparent from fig­ure 4-11 that considerable computation is required to determine the height of the phreatic surface and re­sulting drawdown in the immediate vicinity of a grav­ity well (r/h less than 0.3). The drawdown in this zone usually is not of special interest in dewatering systems and seldom needs to be computed. However, it is al­ways necessary to compute the water level in the well for the selection and design of the pumping equip­ment.

(b) The general equations for flow and draw­down produced by pumping a group of wells supplied by a circular source are given in figure 4-13. These equations are based on the fact that the drawdown at any point is the summation of drawdowns produced at that point by each well in the system. The drawdown factors F to be substituted into the general equations in figure 4-13 appear in the equations for both arte­sian and gravity flow conditions. Consequently, the factors given in figure 4-14 for commonly used well arrays are applicable for either condition.

(c) Flow and drawdown for circular well arrays can also be computed, in a relatively simple manner, by first considering the well system to be a slot, as shown in figure 4-15 or 4-16. However, the piezo­metric head in the vicinity of the wells (or wellpoints)

will not correspond exactly to that determined for the slot due to convergence of flow to the wells. The piezo­metric head in the vicinity of the well is a function of well flow Qw; well spacing a; well penetration W; effec­tive well radius rw; aquifer thickness D, or gravity head H; and aquifer permeability k. The equations given in figures 4-15 and 4-16 consider these variables.

(1) Flow to wells from a line source,

(a) Equations given in figures 4-17 through 4-19 for flow and drawdown produced by pumping a single well or group of fully penetrating wells supplied from an infinite line source were developed using the method of image wells. The image well (a recharge well) is located as the mirror image of the real well with respect to the line source and supplies the per­vious stratum with the same quantity of water as that being pumped from the real well.

(b) The equations given in figures 4-18 and 4-19 for multiple-well systems supplied by a line source are based on the fact that the drawdown at any point is the summation of drawdowns produced at that point by each well in the system. Consequently, the drawdown at a point is the sum of the drawdowns pro­duced by the real wells and the negative drawdowns produced by the image or recharge wells.

(c) Equations are given in figures 4-20 through 4-22 for flow and drawdown produced by pumping an infinite line at wells supplied by a (single) line source, The equations are based on the equivalent slot assump­tion. Where twice the distance to a single line source or 2L is greater than the radius of influence R, the value of R as determined from a pumping test or from figure 4-23 should be used in lieu of L unless the exca­vation is quite large or the tunnel is long, in which case equations for a line source or a flow-net analysis should be used.

(d) Equations for computing the head midway between wells above that in the wells Ahm are not given in this manual for two line sources adjacent to a single line of wells. However, such can be readily de­termined from (plan) flow-net analyses.

(2) Limitations on flow to a partially penetrating well. Theoretical boundaries for a partially penetrat­ing well (for artesian flow) are approximate relations intended to present in a simple form the results of more rigorous but tedious computations. The rigorous computations were made for ratios of R/D = 4.0 and 6.7 and a ratio R/rw = 1000. As a consequence, any agreement between experimental and computed values cannot be expected except for the cases with these particular boundary conditions. In model studies at the U. S. Army Engineer Waterways Experiment Station (WES), Vicksburg, Mississippi, the flow from a partially penetrating well was based on the formula:

q.,=

or

Qwp = kD(H – h.)# (4-2a)

with

2ttG

* ln(R/rw)

where

G = quantity shown in equation (6), figure 4-10 / = geometric shape factor Figure 4-26 shows some of the results obtained at the WES for £ for wells of various penetrations centered inside a circular source. Also presented in figure 4-26 are boundary curves computed for well-screen penetra­tions of 2 and 50 percent. Comparison of j computed from WES model data with § computed from the boundary formulas indicates fairly good agreement for well penetrations > 25 percent and values of R/D be­tween about 5 and 15 where R/rw 1 200 to 1000. Other empirical formulas for flow from a partially penetrating well suffer from the same limitations.

(3) Partially penetmting wells. The equations for gravity flow to partially penetrating wells are only considered valid for relatively high-percent penetra­tions.