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212 GROUNDWATER AND WELLS ot..en came, I ~ o=zao 9om p.3ID mbaarl w,=~W nl~u m)-a.+ ~Cane3 H,=5]On (t)9 m) ~ \ CCOne2 H •COOM \ cope t nzz ml !~_ '-'' -_.~_~ eml \. I ~% ~ \` ' oz No os m~ ~ _ U3 h1009 m1 ~- i has only extended outward an additional 170 ft (51.8 m) and deepened by an ad- ditional 0.3 ft (0.09 m). An additional ra- dial expansion of only 130 ft (39.6 m) and an increase in depth of only 0.2 ft (0.06 m) occurs in the next ]0 hours. Calculations of the volume of each of the cones would show that cone 2 has twice the volume of cone 1, and cone 3 has three times the vol- ume of cone 1. This occurs because, at a constant pumping rate, the same volume of water is discharged from the well dur- ing each 10-hour interval Thus, the in- crease in volume of the cone of depres- Figure 9.7. Changes in radius and depth of cone of Sion is constant over time if the well is depression after equal intervals of time, at rnnstant being pumped at a constant rate and the pamPing rate. aquifer is homogeneous. It is evident from this example that after some hours deepening or expansion of the cone during short intervals of pumping is barely discernible. This often leads observers to conclude that the cone has stabilized and will not expand or deepen as pumping continues. The cone of depression will continue to enlarge, however, until one or more of the following conditions is met: 1. It intercepts enough of the flow in the aquifer to equal the pumping rate. 2. It intercepts a body of surface water from which enough additional water will enter the aquifer to equal the pumping rate when combined with all the flow toward the well. 3. Enough vertical recharge from precipitation occurs within the radius of influence to equal the pumping rate. 4. Sufl"icient leakage occurs through overlying or underlying formations to equal the pumping rate. When the cone has stopped expanding because of one or more of the above con- ditions, equilibrium exists. There is no further drawdown with continued pumping. In some wells, equilibrium occurs within a few hours after pumping begins; in others, it never occurs even though the pumping period may be extended for years. EQUILIBRIUM WELL EQUATIONS More than a hundred years ago, engineers began work on adapting Darcy's basic flow equation to groundwater flow toward a pumping well. The objective was to derive simple mathematical expressions for describing the flow regime of water in the ground. Because direct observation of groundwater movement is impossible, mathematical analysis ofTers a convenient and reliable way to predict what happens to water in the i ground. Well discharge equations for equilibrium conditions were derived by various in- vestigators (Slichter, 1899; Turneaure and Russell, 1901; Thiem, 1906). These equa- l lions relating well discharge to drawdown assumed two-dimensional radial flow to- ward swell (the vertical component of flow is ignored). There are two basic equations; one for unconfined conditions and the other for confined conditions. For both equa-