A simpler solution which greatly reduces the time necessary to compute the specific yield by the pumping-test method of Remson and Lang (1955) is presented. The method consists of computing the volume of dewatered material in the cone of depression and comparing it with the total volume of discharged water. The original method entails the use of a slowly converging series to compute the volume of dewatered material. The solution given herein is derived directly from Darcy's law. INTRODUCTION A pumping-test method to determine the specific yield of a watertable aquifer was presented by Remson and Lang (1955). The method involves the determination of the volume of dewatered material in the cone of depression during the course of a pumping test. The specific yield is then determined by comparing the volume of dewatered material with the total volume of discharged water. The calculation of the volume of dewatered material requires the solution of an exponential series that converges very slowly and is, therefore, time consuming. The example presented by Remson and Lang needed 60 terms of the series and required more than ? hours of computation. This paper presents a more easily evaluated equation for rapidly computing the volume of dewatered material in the cone of depression. The work was carried out under the supervision of Aller Sinnott, district geologist, as part of the investigation of the ground-water resources of New Jersey in cooperation with the New Jersey Department of Conservation and Economic Development. The senior author, a hydrologist-geophysicist with the British Guiana Geological Survey, participated in this study as a foreign trainee unde^ the program sponsored by the International Cooperation Administration. THEORY As pointed out by Remson and Lang, it may not be possible to apply the standard formulas to data from a pumping test in a shallow watertable aquifer because of the slow drainage of the aquifer material 41 42 GROUND-WATER HYDRAULICS during the test and (or) because of a varying rate of discharge. However, the general equilibrium formula can be applied if a pumping rate Q is constant for a long enough period so that the cone of depression reaches approximate equilibrium form and is declining only very slowly. The condition of approximate equilibrium is described by Wenzel (1942, p. 98-99), ... as pumping continues, a hydraulic gradient that is essentially an equilibrium gradient will be established close to the pumped well, and water will be transmitted to the well through the water-bearing material in approximately the amount that is being pumped. The decline of the water table and the resulting unwatering of material in this area will then be much slower. The assumptions used in the development of the general equilibrium formula and those used by Remson and Lang also apply here. The following is quoted from Remson and Lang (1955, p. 322). "Although the water table continues to decline slowly, the assumption that steady-state conditions have been reached involves only £, slight error no greater than that resulting from such a cause as fluctuation in pump discharge." The following paragraph, adapted from Wenzel (1942, p. 77), describes the requisite conditions of the test: An isotropic and homogeneous water-bearing bed of infinite a.real extent is assumed to rest on a relatively impervious formation. The discharging well, equipped with a pump, is fully screened to the bottom of the water-bearing material. It is assumed that water movement from the outer radiu? of the screen Pumped well Ground surface FIGUBE 11. Diagram showing drawdown of the water table In the vicinity of a prmped well. METHOD FOR DETERMINING SPECIFIC YIELD FROM PUMPING TESTS 43 to the pump intake occurs without loss of head or with a head loss that is regligible compared with the drawdown in the well. The water table before pumping, and the underlying impervious bed, are assumed to be horizontal. It is assumed also that there is no recharge to the aquifer during the test and that all the water pumped is removed from storage. Figure 11 shows the shape of the cone of depression in the vicinity of a well pumping from a water-table aquifer. The following symbols or nomenclature are used in the mathematical derivations in this report : Q=the discharge rate of the pumped well in gallons per day P=the field coefficient of permeability of the aquifer in gallons per day per square foot under a unit hydraulic gradient and at the prevailing water temperature r = the horizontal distance from the axis of the pumped well to a point on the cone of depression, in feet s = the drawdown at distance r, in feet stt = the drawdown just outside the screen of the pumped well, in fe°t m=the thickness of the zone of saturation before pumping or the height of the static water table above the aquifer bottom, in feet T Pm= the coefficient of transmissibility of the aquifer in gallons pe^ day per foot. It is the flow through a vertical strip of the aquifer 1 foot wide and extending the saturated height of the aquifer, at unit hydraulic gradient. From Darcy's law: Q=27rrP(-^)(m~S) (1) Therefore Q where dr , . T , x 7 /c^. (m s)ds= a(m s)ds (2) Integrating, as2 Inr ams+-=-f InB (3) z where B is the constant of integration. Then r^Be'***2'* (4) Equation 4 describes the cone of depression when it has reached virtually an equilibrium shape or position. The volume of dewatered material in cubic feet, V, within the cone of depression is (5) 44 GROUND-WATER HYDRAULICS the limits of integration being chosen at zero drawdown (for example, the extent of the cone at equilibrium) and the drawdown outside the screen of the pumped well. The value for r in equation 4 may be substituted in equation 5 to give