Государственный комитет Республики Узбекистан

Figure 8.12 Theoretical curves of W(u_A, u_B, η) versus 1/u_A and 1/u_B for an unconfined aquifer (after Neuman, 1975a).
For an anisotropic aquifer with horizontal hydraulic conductivity K_r and vertical hydraulic conductivity K_z, the parameter η is given by
(8.15)
If the aquifer is isotropic, K_z = K_r, and η = r²/b². The transmissivity T is defined as T = K_rb. Equations (8.12) through (8.15) are only valid if S_y   S and h₀ – h   b.
The prediction of the average drawdown at any radial distance r from a pumping well at any time t can be obtained from Eqs. (8.13) through (8.15) given Q, S, S_y, K_r, K_z, and b.
Multiple Well Systems, Stepped Pumping Rates, Well Recovery, and Partial Penetration
The drawdown in hydraulic head at any point in a confined aquifer in which more than one well is pumping is equal to the sum of the drawdowns that would arise from each of the wells independently. Figure 8.13 schematically displays the drawdown h₀ – h at a point B situated between two pumping wells with pumping rates, Q₁ = Q₂. If Q₁ ≠ Q₂, the symmetry of the diagram about the plane A – A’ would be lost but the principles remain the same.
Figure 8.13 Drawdown in the potentiometric surface of a confined aquifer being pumped by two wells with Q₁ = Q₂.
For a system of n wells pumping at rates Q₁, Q₂, . . . Q_n, the arithmetic summation of the Theis solutions leads to the following predictive equation for the drawdown at a point whose radial distance from each well is given by r₁, r₂, . . . r_n,
(8.16)
where

and t_i, is the time since pumping started at the well whose discharge is Q_i.
The summation of component drawdowns outlined above is an application of the principle of superposition of solutions. This approach is valid because the equation of flow [Eq. (8.1)] for transient flow in a confined aquifer is linear (i.e., there are no cross terms of the form ∂h/∂r · ∂h/∂t). Another application of the principle of superposition is in the case of a single well that is pumped at an initial rate Q₀ and then increased to the rates Q₁, Q₂, . . . Q_m in a stepwise fashion by the additions ΔQ₁, ΔQ₂, . . . ΔQ_m. Drawdown at a radial distance r from the pumping well is given by
(8.17)
where

and t_j, is the time since the start of the pumping rate Q_j.
A third application of the superposition principle is in the recovery of a well after pumping has stopped. If t is the time since the start of pumping and t’
is the time since shutdown, then the drawdown at a radial distance r from the well is given by
(8.18)
where

Figure 8.14 schematically displays the drawdowns that occur during the pumping period and the residual drawdowns that remain during the recovery period.
Figure 8.14 Schematic diagram of the recovery in hydraulic head in an aquifer after pumping is stopped.
It is not always possible, or necessarily desirable, to design a well that fully penetrates the aquifer under development. This is particularly true for unconfined aquifers, but may also be the case for thick confined aquifers. Even for wells that are fully penetrating, screens may be set over only a portion of the aquifer thickness.
Partial penetration creates vertical flow gradients in the vicinity of the well that render the predictive solutions developed for full penetration inaccurate. Hantush (1962) presented adaptations to the Theis solution for partially penetrating wells, and Hantush (1964) reviewed these solutions for both confined and leaky-confined aquifers. Dagan (1967), Kipp (1973), and Neuman (1974) considered the effects of partial penetration in unconfined aquifers.
Bounded Aquifers
When a confined aquifer is bounded on one side by a straight-line impermeable boundary, drawdowns due to pumping will be greater near the boundary [Figure 8.15(a)] than those that would be predicted on the basis of the Theis equation for an aquifer of infinite areal extent. In order to predict head drawdowns in such systems, the method of images, which is widely used in heat-flow theory, has been adapted for application in the groundwater milieu (Ferris et al., 1962).
Figure 8.15 (a) Drawdown in the potentiometric surface of a confined aquifer bounded by an impermeable boundary; (b) equivalent system of infinite extent; (c) plan view.
With this approach, the real bounded system is replaced for the purposes of analysis by an imaginary system of infinite areal extent [Figure 8.15(b)]. In this system there are two wells pumping: the real well on the left and an image well on the right. The image well pumps at a rate, Q, equal to the real well and is located at an equal distance, x₁, from the boundary. If we sum the two component drawdowns in the infinite system (in identical fashion to the two-well case shown in Figure 8.13), it becomes clear that this pumping geometry creates an imaginary impermeable boundary (i.e., a boundary across which there is no flow) in the infinite system at the exact position of the real impermeable boundary in the bounded system. With reference to Figure 8.15(c), the drawdown in an aquifer bounded by an impermeable boundary is given by
(8.19)
where

One can use the same approach to predict the decreased drawdowns that occur in a confined aquifer in the vicinity of a constant-head boundary, such as would be produced by the slightly unrealistic case of a fully penetrating stream [Figure 8.15(d)]. For this case, the imaginary infinite system [Figure 8.15(e)] includes the discharging real well and a recharging image well. The summation of the cone of depression from the pumping well and the cone of impression from the recharge well leads to an expression for the drawdown in an aquifer bounded by a constant head boundary:
(8.20)
where u_r and u_i are as defined in connection with Eq. (8.19).
It is possible to use the image well approach to provide predictions of drawdown in systems with more than one boundary. Ferris et al. (1962) discuss several geometric configurations. One of the more realistic (Figure 8.16) applies to a pumping well in a confined alluvial aquifer in a more-or-less straight river valley. For this case, the imaginary infinite system must include the real pumping well R, an image well I₁ equidistant from the left-hand impermeable boundary, and an image well I₂ equidistant from the right-hand impermeable boundary. These image wells themselves give birth to the need for further image wells. For example, I₃ reflects the effect of I₂ across the left-hand boundary, and I₄ reflects the effect of I₁ across the right-hand boundary. The result is a sequence of imaginary pumping wells stretching to infinity in each direction. The drawdown at point P in Figure 8.16 is the sum of the effects of this infinite array of wells. In practice, image wells need only be added until the most remote pair produces a negligible effect on water-level response (Bostock, 1971).
Figure 8.16 Image-well system for pumpage from a confined aquifer in a river valley bounded by impermeable boundaries.
The Response of Ideal Aquitards
The most common geological occurrence of exploitable confined aquifers is in sedimentary systems of interbedded aquifers and aquitards. In many cases the aquitards are much thicker than the aquifers and although their permeabilities are low, their storage capacities can be very high. In the very early pumping history of a production well, most of the water comes from the depressurization of the aquifer in which the well is completed. As time proceeds the leakage properties of the aquitards are brought into play and at later times the majority of the water being produced by the well is aquitard leakage. In many aquifer-aquitard systems, the aquitards provide the water and the aquifers transmit it to the wells. It is thus of considerable interest to be able to predict the response of aquitards as well as aquifers.
In the earlier discussion of leaky aquifers, two theories were introduced: the Hantush-Jacob theory, which utilizes the W(u, r/B) curves of Figure 8.8, and the Neuman-Witherspoon theory, which utilizes the W(u, r/B₁₁, r/B₂₁, β₁₁, β₂₁) curves of Figure 8.9. In that the Hantush-Jacob theory does not include the storage properties of the aquitard, it is not suitable for the prediction of aquitard response The Neuman-Witherpoon solution, in the form of Eq. (8.11) can be used to predict the hydraulic head h(r, z, t) at any elevation z in the aquitard (Figure 8.7) at any time t, at any radial distance r, from the well. In many cases, however, it may be quite satisfactory to use a simpler approach. If the hydraulic conductivity of the aquitards is at least 2 orders of magnitude less than the hydraulic conductivity in the aquifers, it can be assumed that flow in the aquifers is horizontal and leakage in the aquitards is vertical. If one can predict, or has measurements of, h(r, t) at some point in an aquifer, one an often predict the hydraulic head h(z, t) at an overlying point in the aquitard by the application of a one-dimensional flow theory, developed by Karl Terzaghi, the founder of modern soil mechanics.
Consider an aquitard of thickness b’ (Figure 8.17) sandwiched between two producing aquifers. If the initial condition is a constant hydraulic head h = h₀ in the aquitard, and if the drawdowns in hydraulic head in the adjacent aquifers can be represented as an instantaneous step function Δh the system can be represented by the following one-dimensional boundary-value problem.
Figure 8.17 Response of an ideal aquitard to a step drawdown in head in the two adjacent aquifers.
From Eq. (2.76), the one-dimensional form of the flow equation is
(8.21)
where the primed parameters are the aquitard properties. The initial condition is