Государственный комитет Республики Узбекистан
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- Figure 8.8
Table 8.1 Values of W(u) for Various Values of u
SOURCE: Wenzel, 1942. Figure 8.5(b) shows a calculated plot of h0 – h versus t for the specific set of parameters noted on the figure. A set of field measurements of drawdown versus time measured in a piezometer that is set in an ideal confined aquifer with these properties would show this type of record. The shape of the function h0 – h versus t, when plotted on log-log paper as in Figure 8.5(b), has the same form as the plot of W(u) versus 1/u shown in Figure 8.5(a). This is a direct consequence of the relations embodied in Eqs. (8.6) and (8.7), where it can be seen that h0 – h and W(u), and t and 1/u, are related to one another through a constant term. It is also possible to calculate values of h0 – h at various values of r at a given time t. Such a calculation leads to a plot of the cone of depression (or drawdown cone) in the potentiometric surface around a pumping well. Figure 8.4 provides a schematic example. The steepening of the slope of the cone near the well is reflected in the solution, Eq. (8.7). The physical explanation is clear if one carries out the simple flow-net construction shown in the plan view of Figure 8.4 and then carries the hydraulic head values down onto the section. For a given aquifer the cone of depression increases in depth and extent with increasing time. Drawdown at any point at a given time is directly proportional to the pumping rate and inversely proportional to aquifer transmissivity and aquifer storativity. As shown in Figure 8.6, aquifers of low transmissivity develop tight, deep drawdown cones, whereas aquifers of high transmissivity develop shallow cones of wide extent. Transmissivity exerts a greater influence on drawdown than does storativity. In that geologic configurations are seldom as ideal as that outlined above, the time-drawdown response of aquifers under pumpage often deviates from the Theis solution shown in Figure 8.5. Figure 8.5 (a) Theoretical curve of W(u) versus 1/u. (b) Calculated curve of h0 – h versus t. Theis solution shown in Figure 8.5. We will now turn to some of the theoretical response curves that arise in less ideal situations. Specifically, we will look at (1) leaky aquifers, (2) unconfined aquifers, (3) multiple-well systems, (4) stepped pumping rates, (5) bounded aquifers, and (6) partially penetrating wells. Figure 8.6 Comparison of drawdown cones at a given time for aquifers of (a) low transmissivity; (b) high transmissivity; (c) low storativity; (d) high storativity. Leaky Aquifers The assumption inherent in the Theis solution that geologic formations overlying and underlying a confined aquifer are completely impermeable is seldom satisfied. Even when production wells are screened only in a single aquifer, it is quite usual for the aquifer to receive a significant inflow from adjacent beds. Such an aquifer is called a leaky aquifer, although in reality it is the aquitard that is leaky. The aquifer is often just one part of a multiple-aquifer system in which a succession of aquifers are separated by intervening low-permeability aquitards. For the purposes of this section, however, it is sufficient for us to consider the three-layer case shown in Figure 8.7. Two aquifers of thickness b1 and b2 and horizontal hydraulic conductivities K1 and K2 are separated by an aquitard of thickness b’ and vertical hydraulic conductivity K’. The specific storage values in the aquifers are SS1 and SS2, while that in the aquitard is S’S. Since a rigorous approach to flow in multiple-aquifer systems involves boundary conditions that make the problem intractable analytically, it has been customary to simplify the mathematics by assuming that flow is essentially horizontal in the aquifers and vertical in the aquitards. Neuman and Witherspoon (1969a) report that the errors introduced by this assumption are less than 5% when the conductivities of the aquifers are more than 2 orders of magnitude greater than that of the aquitard. Figure 8.7 Schematic diagram of a two-aquifer “leaky” system. Recall that T = Kb and S = S1b. The development of leaky-aquifer theory has taken place in two distinct sets of papers. The first, by Hantush and Jacob (1955) and Hantush (1956, 1960), provided the original differentiation between the Theis response and that for leaky aquifers. The second, by Neuman and Witherspoon (1969a, 1969b, 1972) evaluated the significance of the assumptions inherent in the earlier work and provided more generalized solutions. The analytical solution of Hantush and Jacob (1955) can be couched in the same form as the Theis solution [Eq. (8.7)] but with a more complicated well function. In fact, Hantush and Jacob developed two analytical solutions, one valid only for small t and one valid only for large t, and then interpolated between the two solutions to obtain the complete response curve. Their solution is presented in terms of dimensionless parameter, r/B, defined by the relation (8.8) In analogy with Eq. (8.7), we can write their solution as (8.9) where W(r/B) is known as the leaky well function. Hantush (1956) tabulated the values of W(r/B). Figure 8.8 is a plot of this function against 1/u. If the aquitard is impermeable, then K’ = 0, and from Eq. (8.8), r/B = 0. In this case, as shown graphically in Figure 8.8, the Hantush-Jacob solution reduces to the Theis solution. If T1 (= K1b1) and S1 (= S1b1) are known for the aquifer and K’ and b’ are known for the aquitard, then the drawdown in the hydraulic head in the pumped aquifer for any pumpage Q at any radial distance r at any time t can be calculated from Eq. (8.9), after first calculating u for the pumped aquifer from Eq. (8.6), r/B from Eq. (8.8), and W(u, r/B) from Figure 8.9. Figure 8.8 Theoretical curves of W(u,r/B) versus 1/u for a leaky aquifer (after Walton, 1960). Figure 8.9 Theoretical curves of W(u,r/B11, r/B21, β11, β21) versus 1/u for a leaky two-aquifer system (after Neuman and Witherspoon, 1969a). The original Hantush and Jacob (1955) solution was developed on the basis of two very restrictive assumptions. They assumed that the hydraulic head in the unpumped aquifer remains constant during the removal of water from the pumped aquifer and that the rate of leakage into the pumped aquifer is proportional to the hydraulic gradient a across the leaky aquitard. The first assumption implies that the unpumped aquifer has an unlimited capacity to provide water for delivery through the aquitard to the pumped aquifer. The second assumption completely ignores the effects of the storage capacity of the aquitard on the transient solution (i.e., it is assumed that S’S = 0). In a later paper, Hantush (1960) presented a modified solution in which consideration was given to the effects of storage in the aquitard. More recently, Neuman and Witherspoon (1969a, 1969b) presented a complete solution that includes consideration of both release of water from storage in the aquitard and head drawdowns in the unpumped aquifer. Their solutions require the calculation of four dimensionless parameters, which, with reference to Figure 8.7, are defined as follows: (8.10) Neuman and Witherspoon’s solutions provide the drawdown in both aquifers as a function of radial distance from the well, and in the aquitard as a function of both radial distance and elevation above the base of the aquitard. Their solutions can be described in a schematic sense by the relation (8.11) Tabulation of this well function would require many pages of tables, but an indication of the nature of the solutions can be seen from Figure 8.9, which presents the theoretical response curves for the pumped aquifer, and at three elevations in the aquitard, for a specific set of r/B and β values. The Theis solution is shown on the diagram for comparative purposes. Because of its simplicity, and despite the inherent dangers of using a simple model for a complex system, the r/B solution embodied in Figure 8.8 is widely used for the prediction of drawdowns in leaky-aquifer systems. Figure 8.10 shows an h0 – h versus t plot for a specific case as calculated from Eq. (8.9) with the aid of Figure 8.8. Figure 8.10 Calculated curve of h0 – h versus t for a leaky aquifer, based on Hantush-Jacob theory. The drawdown reaches a constant level after about 5 × 103 seconds. From this point on, the r/B solution indicates that steady-state conditions hold throughout the system, with the infinite storage capacity assumed to exist in the upper aquifer feeding water through the aquitard toward the well. If the overlaying aquitard were impermeable rather than leaky, the response would follow the dotted line. As one would expect, drawdowns in leaky aquifers are less than those in non-leaky aquifiers, as there is now an additional source of water over and above that which can be supplied by the aquifer itself. Predictions based on the Theis equation therefore provide a conservative estimate for leaky systems; that is, they over-predict the drawdown, or, put another way, are unlikely to reach the values predicted by the Theis equation for a given pumping scheme in a multiaquifer system. Unconfined Aquifers When water is pumped from a confined aquifer, the pumpage induces hydraulic gradients toward the well that create drawdowns in the potentiometric surface. The water produced by the well arises from two mechanisms: expansion of the water in the aquifer under reduced fluid pressures, and compaction of the aquifer under increased effective stresses (Section 2.10). There is no dewatering of the geologic system. The flow system in the aquifer during pumping involves only horizontal gradients toward the well; there are no vertical components of flow. When water is pumped from an unconfined aquifer, on the other hand, the hydraulic gradients that are induced by the pumpage create a drawdown cone in the water table itself and there are vertical components of flow (Figure 8.11). The water produced by the well arises from the two mechanisms responsible for confined delivery plus the actual dewatering of the unconfined aquifer. There are essentially three approaches that can be used to predict the growth of unconfined drawdown cones in time and space. The first, which might be termed the complete analysis, recognizes that the unconfined well-hydraulics problem (Figure 8.11) involves a saturated-unsaturated flow system in which water-table drawdowns are accompanied by changes in the unsaturated moisture contents above the water table (such as those shown in Figure 2.23). Figure 8.11 Radial flow to a well in an unconfined aquifer. The complete analysis requires the solution of a boundary-value problem that includes both the saturated and unsaturated zones. An analytical solution for this complete case was presented by Kroszynski and Dagan (1975) and several numerical mathematical models have been prepared (Taylor and Luthin, 1969; Cooley, 1971; Brutsaert et al., 1971). The general conclusion of these studies is that the position of the water table during pumpage is not substantially affected by the nature of the unsaturated flow above the water table. In other words, while it is conceptually more appealing to carry out a complete saturated-unsaturated analysis, there is little practical advantage to be gained, and since unsaturated soil properties are extremely difficult to measure in situ, the complete analysis is seldom used. The second approach, which is by far the simplest, is to use the same equation as for a confined aquifer [Eq. (8.7)] but with the argument of the well function [Eq. (8.6)] defined in terms of the specific yield Sy, rather than the storativity S. The transmissivity T must be defined as T = Kb, where b is the initial saturated thickness. Jacob (1950) has shown that this approach leads to predicted drawdowns that are very nearly correct as long as the drawdown is small in comparison with the saturated thickness. The method in effect relies on the Dupuit assumptions (Section 5.5) and fails when vertical gradients become significant. The third approach, and the one most widely used in practice, is based on the concept of delayed water-table response. This approach was pioneered by Boulton (1954, 1955, 1963) and has been significantly advanced by Neuman (1972, 1973b, 1975a). It can be observed that water-level drawdowns in piezometers adjacent to pumping wells in unconfined aquifers tend to decline at a slower rate than that predicted by the Theis solution. In fact, there are three distinct segments that can be recognized in time-drawdown curves under water-table conditions. During the first segment, which covers only a short period after the start of pumping, an unconfined aquifer reacts in the same way as does a confined aquifer. Water is released instantaneously from storage by the compaction of the aquifer and by the expansion of the water. During the second segment, the effects of gravity drainage are felt. There is a decrease in the slope of the time-drawdown curve relative to the Theis curve because the water delivered to the well by the dewatering that accompanies the falling water table is greater than that which would be delivered by an equal decline in a confined potentiometric surface. In the third segment, which occurs at later times, time-drawdown data once again tend to conform to a Theis-type curve. Boulton (1963) produced a semiempirical mathematical solution that reproduces all three segments of the time-drawdown curve in an unconfined aquifer. His solution, although useful in practice, required the definition of an empirical delay index that was not related clearly to any physical phenomenon. In recent years there has been a considerable amount of research (Neuman, 1972; Streltsova, 1972; Gambolati, 1976) directed at uncovering the physical processes responsible for delayed response in unconfined aquifers. It is now clear that the delay index is not an aquifer constant, as Boulton had originally assumed. It is related to the vertical components of flow that are induced in the flow system and it is apparently a function of the radius r and perhaps the time t. The solution of Neuman (1972, 1973b, 1975a) also reproduces all three segments of the time-drawdown curve and it does not require the definition of any empirical constants. Neuman’s method recognizes the existence of vertical new components, and the general solution for the drawdown, h0 – h, is a function of both r and z, as defined in Figure 8.11. His general solution can be reduced to one that is a function of r alone if an average drawdown is considered. His complex analytical solution can be represented in simplified form as (8.12) where W(uA, uB, η) is known as the unconfined well function and η = r2/b2, Figure 8.12 is a plot of this function for various values of η. The type A curves that grow out of the left-hand Theis curve of Figure 8.12, and that are followed at early time, are given by (8.13) where and S is the elastic storativity responsible for the instantaneous release of water to the well. The type B curves that are asymptotic to the right-hand Theis curve of Figure 8.12, and that are followed at later time, are given by (8.14) where and Sy is the specific yield that is responsible or the delayed release of water to the well. Yüklə 0,7 Mb. Dostları ilə paylaş:
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