Significance and Use
Assumptions:
The control well discharges at a constant rate, Q.
The control well is of infinitesimal diameter and fully penetrates the aquifer.
The aquifer is homogeneous, isotropic, and areally extensive.
The aquifer remains saturated (that is, water level does not decline below the top of the aquifer).
The aquifer is overlain, or underlain, everywhere by a confining bed having a uniform hydraulic conductivity and thickness. It is assumed that there is no change of water storage in this confining bed and that the hydraulic gradient across this bed changes instantaneously with a change in head in the aquifer. This confining bed is bounded on the distal side by a uniform head source where the head does not change with time.
The other confining bed is impermeable.
Leakage into the aquifer is vertical and proportional to the drawdown, and flow in the aquifer is strictly horizontal.
Flow in the aquifer is two-dimensional and radial in the horizontal plane.
The geometry of the well and aquifer system is shown in Fig. 1.
Implications of Assumptions:
Paragraph 5.1.1 indicates that the discharge from the control well is at a constant rate. Section 8.1 of Test Method D4050 discusses the variation from a strictly constant rate that is acceptable. A continuous trend in the change of the discharge rate could result in misinterpretation of the water-level change data unless taken into consideration.
The leaky confining bed problem considered by the Hantush-Jacob solution requires that the control well has an infinitesimal diameter and has no storage. Abdul Khader and Ramadurgaiah (5) developed graphs of a solution for the drawdowns in a large-diameter control well discharging at a constant rate from an aquifer confined by a leaky confining bed. Fig. 2 (Fig. 3 of Abdul Khader and Ramadurgaiah (5)) gives a graph showing variation of dimensionless drawdown with dimensionless time in the control well assuming the aquifer storage coefficient, S = 10−3, and the leakage parameter, 
= 10 −3. Note that at early dimensionless times the curve for a large-diameter well in a non-leaky aquifer (BCE) and in a leaky aquifer (BCD) are coincident. At later dimensionless times, the curve for a large diameter well in a leaky aquifer coalesces with the curve for an infinitesimal diameter well (ACD) in a leaky aquifer. They coalesce about one logarithmic cycle of dimensionless time before the drawdown becomes sensibly constant. For a value of rw/B smaller than 10−3, the constant drawdown (D) would occur at a greater value of dimensionless drawdown and there would be a longer period during which well-bore storage effects are negligible (the period where ACD and BCD are coincident) before a steady drawdown is reached.
For values of 
greater than 10−3, the constant drawdown (D) would occur at a smaller value of drawdown and there would be a shorter period of dimensionless time during which well-storage effects are negligible (the period where ACD and BCD are coincident) before a steady drawdown is reached. Abdul Khader and Ramadurgaiah (5) present graphs of dimensionless time versus dimensionless drawdown in a discharging control well for values of S = 10−1, 10−2, 10 −3, 10−4, and 10−5 and rw/B = 10−2, 10−3, 10−4, 10−5, 10−6, and 0. These graphs can be used in an analysis prior to the aquifer test making use of estimates of the hydraulic properties to estimate the time period during which well-bore storage effects in the control well probably will mask other effects and the drawdowns would not fit the Hantush-Jacob solution.
The time required for the effects of control-well bore storage to diminish enough that drawdowns in observation wells should fit the Hantush-Jacob solution is less clear. But the time adopted for when drawdowns in the discharging control well are no longer dominated by well-bore storage affects probably should be the minimum estimate of the time to adopt for observation well data.
The assumption that the aquifer is bounded, above or below, by a leaky layer on one side and a nonleaky layer on the other side is not likely to be entirely satisfied in the field. Neuman and Witherspoon (7, p. 1285) have pointed out that because the Hantush-Jacob formulation uses water-level change data only from the aquifer being pumped (or recharged) it can not be used to distinguish whether the leaking beds are above or below (or from both sides) of the aquifer. Hantush (8) presents a refinement that allows the parameters determined by the aquifer test analysis to be interpreted as composite parameters that reflect the combined effects of overlying and underlying confined beds. Neuman and Witherspoon (7) describe a method to estimate the hydraulic properties of a confining layer by using the head changes in that layer.
The Hantush-Jacob theoretical development requires that the leakage into the aquifer is proportional to the drawdown, and that the drawdown does not vary in the vertical in the aquifer. These requirements are sometimes described by stating that the flow in the confining beds is essentially vertical and in the aquifer is essentially horizontal. Hantush's (9) analysis of an aquifer bounded only by one leaky confining bed suggested that this approximation is acceptably accurate wherever 
The Hantush-Jacob method requires that there is no change in water storage in the leaky confining bed. Weeks (10) states that if the “leaky” confining bed is thin and relatively permeable and incompressible, the solution of Hantush and Jacob (2) will apply, whereas the solution of Hantush (8), which is described in Test Method D6028, that considers storage in confining beds will apply if at least one confining bed is thick, of low permeability, and highly compressible. For the case where one layer confining the aquifer is sensibly impermeable, and the other confining bed is leaky and bounded on the distal side by a layer in which the head is constant it follows from Hantush (8) that when time, t, satisfies 
the drawdowns in the aquifer will be described by the equation
Note that in Hantush's (8) solution, the term 
appears instead of the expression given for u in Eq 3, namely 
The implication being from Hantush (8) that after the time criterion given by Eq 9 is satisfied, the apparent storage coefficient of the aquifer will include the aquifer storage coefficient and one third of the storage coefficient for the confining bed. If the storage coefficient of the confining bed is very much less than that of the aquifer, then the effect of storage in the confining bed will be very small or sensibly nil. To illustrate the use of Hantush's time criterion, suppose a confining bed is characterized by b′ = 3 m, K′ = 0.001 m/day, and S′s = 3.6 × 10 −6 m−1, then the Hantush-Jacob solution Eq 10 would apply everywhere when 
If the vertical hydraulic conductivity of the confining bed was an order of magnitude larger, K′ = 0.01 m/day, then the Hantush-Jacob (2) solution would apply when t > 23 min.
It should be noted that the Hantush (8) analysis assumes that well bore storage is negligible.
Moench (11) presents numerical results that give insight into the effects of control well storage and changes in storage in the confining bed on drawdowns in the aquifer for various parameter values. However, Moench does not offer an explicit formula for when those effects diminish enough for subsequent drawdown data to fit the Hantush-Jacob solution.
The assumption stated in 5.1.5, that the leaky confining bed is bounded on the other side by a uniform head source, the level of which does not change with time, was considered by Neuman and Witherspoon (12, p. 810). They considered a confined system of two aquifers separated by a confining bed as shown schematically in Fig. 3. Their analysis concluded that the drawdowns in an aquifer in response to discharging from a well in that aquifer would not be affected by the properties of the other, unpumped, aquifer for times that satisfy 
FIG. 2 Time
FIG. 3 Schematic Diagram of Two-Aquifer System
Scope
1.1 This test method covers an analytical procedure for determining the transmissivity and storage coefficient of a confined aquifer and the leakance value of an overlying or underlying confining bed for the case where there is negligible change of water in storage in a confining bed. This test method is used to analyze water-level or head data collected from one or more observation wells or piezometers during the pumping of water from a control well at a constant rate. With appropriate changes in sign, this test method also can be used to analyze the effects of injecting water into a control well at a constant rate.
1.2 This analytical procedure is used in conjunction with Test Method D4050.
1.3 Limitations—The valid use of the Hantush-Jacob method is limited to the determination of hydraulic properties for aquifers in hydrogeologic settings with reasonable correspondence to the assumptions of the Theis nonequilibrium method (Test Method D4106) with the exception that in this case the aquifer is overlain, or underlain, everywhere by a confining bed having a uniform hydraulic conductivity and thickness, and in which the gain or loss of water in storage is assumed to be negligible, and that bed, in turn, is bounded on the distal side by a zone in which the head remains constant. The hydraulic conductivity of the other bed confining the aquifer is so small that it is assumed to be impermeable (see Fig. 1).
1.4 The values stated in SI units are to be regarded as standard. The values given in parentheses are mathematical conversions to inch-pound units, which are provided for information only and are not considered standard.
1.4.1 The converted inch-pound units use the gravitational system of units. In this system, the pound (lbf) represents a unit of force (weight), while the unit for mass is slugs. The converted slug unit is not given, unless dynamic (F = ma) calculations are involved.
1.5 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility of the user of this standard to establish appropriate safety and health practices and determine the applicability of regulatory limitations prior to use.
FIG. 1 Cross Section Through a Discharging Well in a Leaky Aquifer (from Reed (1)).4 The Confining and Impermeable Bed Locations Can Be Interchanged
