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W <br />214 GROUNDWATER AND WELLS <br />R = radius of the cone of depression, R = radius of the cone of depression, in m <br />in ft <br />r = radius of the well, in ft r = radius of the well, in m <br />Equation 9.1 is often called the equilibrium, or Thiem, equation. <br />Figure 9.9 is a vertical section of a well pumping from a confined aquifer. The <br />equation for a well operating under confined conditions is: <br />Kb(H - h) 2.73Kb(H -h) <br />Q= 528 log R/r Q - log RIr (v'`� <br />where where <br />b = thickness of aquifer, in ft h = thickness of aquifer, in m <br />All other terms are as defined for All other terms are as defined for <br />Equation 9.1 Equation 9.1 <br />Derivations of the foregoing equations are based on the following simplifying <br />assumptions: <br />1. The water - bearing materials have a uniform hydraulic conductivity within the <br />radius of influence of the well. <br />2. The aquifer is not stratified. <br />3. For an unconfined aquifer, the saturated thickness is constant before pumping <br />starts; for a confined aquifer, the aquifer thickness is constant. <br />4. The pumping well is 100- percent efficient, that is, the drawdown levels inside <br />and just outside the well bore are at the same elevation (see Chapter 16). (Head losses <br />in the vicinity of the well are minimal.) <br />5. The intake portion of the well penetrates the entire aquifer. <br />6. The water table or potentiometric surface has no slope. <br />7. Laminar flow exists throughout the aquifer and within the radius of influence of <br />the well. <br />8. The cone of depression has reached equilibrium so that both drawdown and ra- <br />dius of influence of the well do not change with continued pumping at a given rate. <br />These assumptions appear to limit severely the use of the two equations. In reality, <br />however, they do not. For example, uniform hydraulic conductivity is rarely found in <br />a real aquifer, but the average hydraulic conductivity as determined from pumping <br />tests has proved to be reliable for predicting well performance. In confined aquifers <br />where the well is fully penetrating and open to the formation, the assumption of no <br />stratification is not an important limitation. <br />Assumption of constant thickness is not a serious limitation because variation in <br />aquifer thickness within the cone of depression in most situations is relatively small, <br />especially in sedimentary rocks. Where changes in thickness do occur, as in glacial <br />sediments, for example, they can be taken into account. The assumption that a well <br />is 100- percent efficient can cause the calculated well yield to be seriously in error if <br />the real well is inefficient because of improper design or construction. Factors con- <br />tributing to inefficiency are discussed in Chapter 16. <br />The assumption that the water table or potentiometric surface is horizontal before <br />pumping begins is not correct. The slope or hydraulic gradient, however, is usually <br />almost flat and the effect on calculation of well yield is negligible in most cases. Slope <br />