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h(z, 0) = h₀
and the boundary conditions are
h(0, t) = h₀ – Δh
h(b’, t) = h₀ – Δh
Terzaghi (1925) provided an analytical solution to this boundary-value problem. He noted that for clays n’β
 α’ in Eq. (8.21). He grouped the remaining aquitard parameters into a single parameter c_v, known as the coefficient of consolidation and defined as
(8.22)
He further defined the dimensionless time factor, T_f, as
(8.23)
Given the aquitard parameter c_v and the geometric parameter b’, one can calculate T_f, for any time t.
Figure 8.17 is a graphical presentation of Terzaghi’s solution h(z, T_f). It allows the prediction of the hydraulic head at any elevation z at any time t in an aquitard sandwiched between two producing aquifers, as long as the drop in hydraulic head Δh can be estimated in the aquifers. It is also possible to interpret this solution for an aquitard that drains to only one aquifer. For example, if the lower boundary of the aquitard on the inset to Figure 8.17 is impermeable, only the upper half of the curves shown in the figure are used for the prediction of h(z, t). The z = 0 line passes through the center of the figure, and the parameters c_y and T_f are defined as above. Wolff (1970) has described a case history that utilizes the concepts of one-dimensional aquitard response.
Predictions of aquitard response, and the inverse application of this theory to determine aquitard parameters, as discussed in Section 8.6, are also important in assessing contaminant migration (Chapter 9) and land subsidence (Section 8.12).
The Real World
Each of the analytical solutions presented in this section describes the response to pumping in a very idealized representation of actual aquifer configurations. In the real world, aquifers are heterogeneous and anisotropic; they usually vary in thickness; and they certainly do not extend to infinity. Where they are bounded, it is not by straight-line boundaries that provide perfect confinement. In the real world, aquifers are created by complex geologic processes that lead to irregular stratigraphy, interfingering of strata, and pinchouts and trendouts of both aquifers and aquitards. The predictions that can be carried out with the analytical expressions presented in this section must be viewed as best estimates. They have greater worth the more closely the actual hydrogeological environment approaches the idealized configuration.
In general, well-hydraulics equations are most applicable when the unit of study is a well or well held. They are less applicable on a larger scale, where the unit of study is an entire aquifer or a complete groundwater basin. Short-term yields around wells are very dependent on aquifer properties and well-field geometry, both of which are emphasized in the well-hydraulics equations. Long-term yields on an aquifer scale are more often controlled by the nature of the boundaries. Aquifer studies on the larger scale are usually carried out with the aid of models based on numerical simulation or electric-analog techniques. These approaches are discussed in Sections 8.8 and 8.9.
The predictive formulas developed in this section and the simulation techniques described in later sections allow one to calculate the drawdowns in hydraulic head that will occur in an aquifer in response to groundwater development through wells. They require as input either the three basic hydrogeological parameters: hydraulic conductivity, K, porosity, n, and compressibility, α; or the two derived aquifer parameters: transmissivity, T, and storativity, S. There is a wide variety of techniques that can be used to measure these parameters. In the next section, we will discuss laboratory tests; in Section 8.5, piezometer tests; and in Section 8.6, pumping tests. In Section 8.7, we will examine some estimation techniques, and in Section 8.8, the determination of aquifer parameters by inverse simulation. The formulas presented in this section are the basis for the pumping-test approach that is described in Section 8.6.
8.4 Measurement of Parameters: Laboratory Tests
The laboratory tests described in this section can be considered as providing point values of the basic hydrogeologic parameters. They are carried out on small samples that are collected during test-drilling programs or during the mapping of surficial deposits. If the samples are undisturbed core samples, the measured values should be representative of the in situ point values. For sands and gravels, even disturbed samples may yield useful values. We will describe testing methods for the determination of hydraulic conductivity, porosity, and compressibility in the saturated state; and we will provide references for the determination of the characteristic curves relating moisture content, pressure head, and hydraulic conductivity in the unsaturated state. We will emphasize principles; for a more complete description of each testing apparatus and more detailed directions on laboratory procedures, the reader is directed to the soil-testing manual by Lambe (1951), the permeability handbook of the American Society of Testing Materials (1967), or the pertinent articles in the compendium of soil analysis methods edited by Black (1965). Our discussions relate more to soils than to rocks, but the principles of measurement are the same. The rock mechanics text by Jaeger (1972) discusses rock-testing procedures.
Hydraulic Conductivity
The hydraulic conductivity, K, was defined in Section 2.1, and its relationship to the permeability, k, was explored in Section 2.3. The saturated hydraulic conductivity of a soil sample can be measured with two types of laboratory apparatus. The first type, known as a constant-head permeameter, is shown in Figure 8.18(a); the second type, a falling-head permeameter, is shown in Figure 8.18(b).
Figure 8.18 (a) Constant-head permeater; (b) falling-head permeater (after Todd, 1959).
In a constant-head test, a soil sample of length L and cross-sectional area A is enclosed between two porous plates in a cylindrical tube, and a constant-head differential H is set up across the sample. A simple application of Darcy’s law leads to the expression
(8.24)
where Q is the steady volumetric discharge through the system. It is important that no air become entrapped in the system, and for this reason it is wise to use deaired water. If disturbed samples are being tested in the permeameter, they should be carefully saturated from below as they are emplaced.
In a falling-head test [Figure 8.18(b)], the head, as measured in a tube of cross-sectional area a, is allowed to fall from H₀ to H₁ during time t. The hydraulic conductivity is calculated from
(8.25)
This equation can be derived (Todd, 1959) from the simple boundary-value problem that describes one-dimensional transient flow across the soil sample. In order that the head decline be easily measurable in a finite time period, it is necessary to choose the standpipe diameter with regard to the soil being tested. Lambe (1951) suggests that for a coarse sand a standpipe whose diameter is approximately equal to that of the permeameter is usually satisfactory, whereas a fine silt may necessitate a standpipe whose diameter is one-tenth the permeameter diameter. Lambe also suggests that the point   be marked on the standpipe. If the time required for the head decline from H₀ to   is not equal to that for the decline from   to H₁, the test has not functioned correctly and a check should be made for leaks or entrapped air.
Klute (1965a) notes that the constant-head system is best suited to samples with conductivities greater than 0.01 cm/min while the falling-head system is best suited to samples with lower conductivity. He also notes that elaborate, painstaking measurements are not generally required for conductivity determinations on field samples. The variability among samples is usually large enough that precise determination of the conductivity of a given sample is not warranted.
For clayey materials the hydraulic conductivity is commonly determined from a consolidation test, which is described in the subsection on compressibility below.
Porosity
In principle, the porosity, n, as defined in Section 2.5, would be most easily measured by saturating a sample, measuring its volume, V_T, weighing it and then oven drying it to constant weight at 105°C. The weight of water removed could be converted to a volume, knowing the density of water. This volume is equivalent to the volume of the void space, V_v; and the porosity could be calculated from n = V_v/V_T.
In practice, it is quite difficult to exactly and completely saturate a sample of given volume. It is more usual (Vomocil, 1965) to make use of the relationship
(8.26)
which can be developed by simple arithmetic manipulation of the basic definition of porosity. In Eq. (8.26), ρ_b is the bulk mass density of the sample and ρ_s is the particle mass density. The bulk density is the oven-dried mass of the sample divided by its held volume. The particle density is the oven-dried mass divided by the volume of the solid particles, as determined by a water-displacement test. In cases where great accuracy is not required, ρ_s = 2.65 g/cm³ can be assumed for most mineral soils.
Compressibility
The compressibility of a porous medium was defined in Section 2.9 with the aid of Figure 2.19. It is a measure of the relative volumetric reduction that will take place in a soil under an increased effective stress. Compressibility is measured in a consolidation apparatus of the kind commonly used by soils engineers. In this test, a soil sample is placed in a loading cell of the type shown schematically in Figure 2.19(a). A load L is applied to the cell, creating a stress σ, where σ = L/A, A being the cross-sectional area of the sample. If the soil sample is saturated and the fluid pressure on the boundaries of the sample is atmospheric (i.e., the sample is free-draining), the effective stress, σ_e, which leads to consolidation of the sample, is equal to the applied stress, σ.
The reduction in sample thickness, b, is measured after equilibrium is achieved at each of several loading increments, and the results are converted into a graph of void ratio, e, versus effective stress, σ_e, as shown in Figure 2.19(b). The compressibility, α, is determined from the slope of such a plot by
(8.27)
where e₀ is the initial void ratio prior to loading. As noted in Section 2.9, α is a function of the applied stress and it is dependent on the previous loading history.
Lambe (1951) describes the details of the testing procedure. The most common loading method is a lever system on which weights of known magnitude are hung. There are two types of loading cell in common use. In the fixed-ring container [Figure 8.19(a)], all the sample movement relative to the container is downward. In the floating-ring container [Figure 8.19(b)], compression occurs toward the middle from both top and bottom. In the floating-ring container, the effect of friction between the container wall and the soil specimen is smaller than in the fixed-ring container. In practice, it is difficult to determine the magnitude of the friction in any case, and because its effect is thought to be minor, it is normally neglected. Cohesionless sands are usually tested as disturbed samples. Cohesive clays must be carefully trimmed to fit the consolidometer ring.
Figure 8.19 (a) Fixed-ring consolidometer; (b) floating-ring consolidometer (after Lambe, 1951).
In soil mechanics terminology, the slope of the e – σ_e curve is called the coefficient of compressibility, a_v. The relationship between a_v and α is easily seen to be
(8.28)
More commonly, soils engineers plot the void ratio, _e, against the logarithm of σ_e. When plotted in this manner, there is usually a significant portion of the curve that is a straight line. The slope of this line is called the compression index, C_c, where
(8.29)
In most civil engineering applications the rate of consolidation is just as important as the amount of consolidation. This rate is dependent both on the compressibility, α, and the hydraulic conductivity, K. As noted in connection with Eq. (8.22), soils engineers utilize a grouped parameter known as the coefficient of consolidation, C_v, with is defined as
(8.30)
At each loading level in a consolidation test, the sample undergoes a transient drainage process (fast for sands, slow for clays) that controls the rate of consolidation of the sample. If the rate of decline in sample thickness is recorded for each loading increment, such measurements can be used in the manner described by Lambe (1951) to determine the coefficient of consolidation, C_v, and the hydraulic conductivity, K, of the soil.
In Section 8.12, we will further examine the mechanism of one-dimensional consolidation in connection with the analysis of land subsidence.
Unsaturated Characteristic Curves
The characteristic curves, K(ψ) and θ(ψ), that relate the moisture content, θ, and the hydraulic conductivity, K, to the pressure head, ψ, in unsaturated soils were described in Section 2.6. Figure 2.13 provided a visual example of the hysteretic relationships that are commonly observed. The methods used for the laboratory determination of these curves have been developed exclusively by soil scientists. It is not within the scope of this text to outline the wide variety of sophisticated laboratory instrumentation that is available. Rather, the reader is directed to the soil science literature, in particular to the review articles by L. A. Richards (1965), Klute (1965b), Klute (1965c), and Bouwer and Jackson (1974).
8.5 Measurement of Parameters: Piezometer Tests
It is possible to determine in situ hydraulic conductivity values by means of tests carried out in a single piezometer. We will look at two such tests, one suitable for point piezometers that are open only over a short interval at their base, and one suitable for screened or slotted piezometers that are open over the entire thickness of a confined aquifer. Both tests are initiated by causing an instantaneous change in the water level in a piezometer through a sudden introduction or removal of a known volume of water. The recovery of the water level with time is then observed. When water is removed, the tests are often called bail tests; when it is added, they are known as slug tests. It is also possible to create the same effect by suddenly introducing or removing a solid cylinder of known volume.
The method of interpreting the water level versus time date that arise from bail tests or slug tests depends on which of the two test configurations is felt to be most representative. The method of Hvorslev (l951) is for a point piezometer, while that of Cooper et al. (l967) is for a confined aquifer. We will now describe each in turn.
The simplest interpretation of piezometer-recovery data is that of Hvorslev (1951). His initial analysis assumed a homogeneous, isotropic, infinite medium in which both soil and water are incompressible. With reference to the bail test of Figure 8.20(a), Hvorslev reasoned that the rate of inflow, q, at the piezometer tip at any time t is proportional to the hydraulic conductivity, K, of the soil and to the unrecovered head difference, H – h, so that
(8.31)
where F is a factor that depends on the shape and dimensions of the piezometer intake. If q = q₀ at t = 0, it is clear that q(t) will decrease asymptotically toward zero as time goes on.
Figure 8.20 Hvorslev piezometer test. (a) Geometry; (b) method of analysis.
Hvorslev defined the basic time lag, T₀, as
(8.32)
When this parameter is substituted in Eq. (8.31), the solution to the resulting ordinary differential equation, with the initial condition, h = H₀ at t = 0, is
(8.33)
A plot of field recovery data, H – h versus t, should therefore show an exponential decline in recovery rate with time. If, as shown on Figure 8.20(b), the recovery is normalized to H – H₀ and plotted on a logarithmic scale, a straight-line plot results. Note that for H – h/H – H₀ = 0.37, ln(H – h/H – H₀) = –1, and from Eq. (8.33), T₀ = t. The basic time lag, T₀, can be defined by this relation; or if a more physical definition is desired, it am be seen, by multiplying both top and bottom of Eq. (8.32) by H – H₀, that T₀ is the time that would be required for the complete equalization of the head difference if the original rate of inflow were maintained. That is, T₀ = V/q₀ where V is the volume of water removed or added.
To interpret a set of field recovery data, the data are plotted in the form of Figure 8.20(b). The value of T₀ is measured graphically, and K is determined from Eq. (8.32). For a piezometer intake of length L and radius R [Figure 8.20(a)], with L/R > 8, Hvorslev (1951) has evaluated the shape factor, F. The resulting expression for K is
(8.34)
Hvorslev also presents formulas for anisotropic conditions and for a wide variety of shape factors that treat such cases as a piezometer open only at its basal cross section and a piezometer that just encounters a permeable formation underlying an impermeable one. Cedergren (1967) also lists these formulas.
In the field or agricultural hydrology, several in situ techniques, similar in principle to the Hvorslev method but differing in detail, have been developed for the measurement of saturated hydraulic conductivity. Boersma (1965) and Bouwer and Jackson (1974) review those methods that involve auger holes and piezometers.
For bail tests of slug tests run in piezometers that are open over the entire thickness of a confined aquifer, Cooper et al. (1967) and Papadopoulos et al. (1973) have evolved a test-interpretation procedure. Their analysis is subject to the same assumptions as the Theis solution for pumpage from a confined aquifer. Contrary to the Hvorslev method of analysis, it includes consideration of both formation and water compressibilities. It utilizes a curve-matching procedure to determine the aquifer coefficients T and S. The hydraulic conductivity K can then be determined on the basis of the relation, K = T/b. Like the Theis solution, the method is based on the solution to a boundary-value problem that involves the transient equation of groundwater flow, Eq. (2.77). The mathematics will not be described here.
For the bail-test geometry shown in Figure 8.21(a), the method involves the preparation of a plot of recovery data in the form H – h/H – H₀ versus t. The plot is prepared on semilogarithmic paper with the reverse format to that of the Hvorslev test; the H – h/H – H₀ scale is linear, while the t scale is logarithmic. The field curve is then superimposed on the type curves shown in Figure 8.21(b).
Figure 8.21 Piezometer test in a confined aquifer. (a) Geometry; (b) type curves (after Papadopoulos et al., 1973).
With the axes coincident, the data plot is translated horizontally into a position where the data best fit one of the type curves. A matchpoint is chosen (or rather, a vertical axis is matched) and values of t and W are read of the horizontal scales at the matched axis of the field plot and the type plot, respectively. For ease of calculation it is common to choose a matched axis at W = 1.0. The transmissivity T is then given by
(8.35)
where the parameters are expressed in any consistent set of units.
In principle, the storativity, S, can be determined from the a value of the matched curve and the expression shown on Figure 8.21(b). In practice, since the slopes of the various a lines are very similar, the determination of S by this method is unreliable.
The main limitation on slug tests and hail tests is that they are heavily dependent on a high-quality piezometer intake. If the wellpoint or screen is corroded or clogged, measured values may be highly inaccurate. On the other hand, if a piezometer is developed by surging or backwashing prior to testing, the measured values may reflect the increased conductivities in the artificially induced gravel pack around the intake.
It is also possible to determine hydraulic conductivity in a piezometer or single well by the introduction of a tracer into the well bore. The tracer concentration decreases with time under the influence of the natural hydraulic gradient that exists in the vicinity of the well. This approach is known as the borehole dilution method, and it is described more fully in Section 9.4.
8.6 Measurement of Parameters: Pumping Tests
In this section, a method of parameter measurement that is specifically suited to the determination of transmissivity and storativity in confined and unconfined aquifers will be described. Whereas laboratory tests provide values of the hydrogeological parameters, and piezometer tests provide in situ values representative of a small volume of porous media in the immediate vicinity of a piezometer tip, pumping tests provide in situ measurements that are averaged over a large aquifer volume.
The determination of T and S from a pumping test involves a direct application of the formulas developed in Section 8.3. There, it was shown that for a given pumping rate, if T and S are known, it is possible to calculate the time rate of drawdown, h₀ – h versus t at any point in an aquifer. Since this response depends solely an the values of T and S, it should be possible to take measurements of h₀ – h versus t at some observational point in an aquifer and work backward through the equations to determine the values ofTand S.
The usual course of events during the initial exploitation of an aquifer involves (1) the drilling of a test well with one or more observational piezometers, (2) a short-term pumping test to determine the values of T and S, and (3) application of the predictive formulas of Section 8.3, using the T and S values determined in the pumping test, to design a production well or wells that will fulfill the pumpage requirements of the project without lending to excessive long-term drawdowns. The question of what constitutes an “excessive” drawdown and how drawdowns and well yields are related to groundwater recharge rates and the natural hydrologic cycle are discussed in Section 8.10.
Let us now examine the methodology of pumping-test interpretation in more detail. There are two methods that are in common usage for calculating aquifer coefficients from time-drawdown data. Both approaches are graphical. The first involves curve matching on a log-log plot (the Theis method), and the second involves interpretations with a semilog plot (the Jacob method).
Log-Log Type-Curve Matching
Let us first consider data taken from an aquifer in which the geometry approaches that of the idealized Theis configuration. As was explained in connection with Figure 8.5, the time-drawdown response in an observational piezometer in such an aquifer will always have the shape of the Theis curve, regardless of the values of T and S in the aquifer. However, for high T a measurable drawdown will reach the observation point faster than for low T, and the drawdown data will begin to march up the Theis curve sooner. Theis (1935) suggested the following graphical procedure to exploit this curve-matching property:

1. 
   Plot the function W(u) versus 1/u on log-log paper. (Such a plot of dimensionless theoretical response is known as a type curve.)
   
2. 
   Plot the measured time-drawdown values, h₀ – h versus t, on log-log paper of the same size and scale as the W(u) versus 1/u curve.
   
3. 
   Superimpose the field curve on the type curve keeping the coordinate axes parallel. Adjust the curves until most of the observed data points fall on the type curve.
   
4. 
   Select and arbitrary match point and read off the paired values of W(u), 1/u, h₀ – h, and t at the match point. Calculate u from 1/u.
   
5. 
   Using these values, together with the pumping rate Q and the radial distance r from well to piezometer, calculate T from the relationship
   

(8.36)

6. 
   Calculate S from the relationship
   

(8.37)
Equations (8.36) and (8.37) follow directly from Eqs. (8.7) and (8.6). They are valid for any consistent system of units. Some authors prefer to present the equations in the form
(8.38)
(8.39)
where the coefficients A and B are dependent on the units used for the various parameters. For SI units, with h₀ – h and r measured in meters, t in seconds, Q in m³/s, and T in m²/s, A = 0.08 and B = 0.25. For the inconsistent set of practical units widely used in North America, with h₀ – h and r measured in feet, t in days, Q in US. gal/min, and T in U.S. gal/day/ft, A = 114.6 and B = 1.87. For Q and T in terms of Imperial gallons, A remains unchanged and B = 1.56.
Figure 8.22 illustrates the curve-matching procedure and calculations for a set of field data. The alert reader will recognize these data as being identical to the calculated data originally presented in Figure 8.5(b). It would probably be intuitively clearer if the match point were taken at some point on the coincident portions of the superimposed curves. However, a few quick calculations should convince doubters that it is equally valid to take the matchpoint anywhere on the overlapping fields once they have been fixed in their correct relative positions. For ease of calculation, the matchpoint is often taken at W(u) = 1.0, u = 1.0.
Figure 8.22 Determination of T and S from h₀ – h versus t data using the log-log curve-matching procedure and the W(u) versus 1/u-type curve.
The log-log curve-matching technique can also be used for leaky aquifers (Walton, 1962) and unconfined aquifers (Prickett, 1965; Neuman, 1975a). Figure 8.23 provides a comparative review of the geometry of these systems and the types of h₀ – h versus t data that should be expected in an observational piezometer in each case. Sometimes time-drawdown data unexpectedly display one of these forms, thus indicating a geological configuration that has gone unrecognized during the exploration stage of aquifer evaluation.
Figure 8.23 Comparison of log-log h₀ – h versus t data for ideal, leaky, unconfined, and bounded systems.
For leaky aquifers the time-drawdown data can be matched against the leaky type curves of Figure 8.8. The r/B value of the matched curve, together with the matchpoint values of W(u, r/B), u, h₀ – h and t can be substituted into Eqs. (8.6), (8.8), and (8.9) to yield the aquifer coefficients T and S. Because the development of the r/B solutions does not include consideration of aquitard storativity, an r/B curve matching approach is not suitable for the determination of the aquitard conductivity K’. As noted in the earlier subsection on aquitard response, there are many aquifer-aquitard configurations where the leakage properties of the aquitards are more important in determining long-term aquifer yields than the aquifer parameters themselves. In such cases it is necessary to design a pumping-test configuration with observational piezometers that bottom in the aquitards as well as in the aquifers. One can then use the pumping-test procedure outlined by Neuman and Witherspoon (1972), which utilizes their more general leaky-aquifer solution embodied in Eqs. (8.6), (8.10), and (8.11). They present a ratio method that obviates the necessity of matching field data to type curves as complex as those of Figure 8.9. The method only requires matching against the Theis curve, and calculations are relatively easy to carry out.
As an alternative approach (Wolff, 1970), one can simply read off a T_f value from Figure 8.17 given a hydraulic head value h measured in an aquitard piezometer at elevation z at time t. Knowing the aquitard thickness, b’, one can solve Eq. (8.23) for c_v. If an α value can be estimated, Eq. (8.22) can be solved for K’.
For unconfined aquifers the time-drawdown data should be matched against the unconfined type curves of Figure 8.12. The η value of the matched curve, together with the match-point values of W(u_A, u_B, η), u_A, u_B, h₀ – h and t can be substituted into Eqs. (8.13) through (8.15) to yield the aquifer coefficients T, S, and S_y. Moench and Prickett (1972) discuss the interpretation of data at sites where lowered water levels cause a conversion from confined to unconfined conditions.
Figure 8.23(d) shows the type of log-log response that would be expected in the vicinity of an impermeable or constant-head boundary. However, bounded systems are more easily analyzed with the semilog approach that will now be described.
Semiology Plots
The semilog method of pump-test interpretation rests on the fact that the exponential integral, W(u), in Eqs. (8.5) and (8.7) can be represented in infinite series. The Theis solution then becomes
(8.40)
Cooper and Jacob (1946) noted that for small u the sum of the series beyond ln u becomes negligible, so that
(8.41)
Substituting Eq. (8.6) for u, and noting that ln u = 2.3 log u, that –ln u = ln u, and that ln 1.78 = 0.5772, Eq. (8.41) becomes
(8.42)
Since Q, r, T, and S are constants, it is clear that h₀ – h versus log t should plot as a straight line.
Figure 8.24(a) shows the time-drawdown data of Figure 8.22 plotted on a semilog graph. If Δh is the drawdown for one log cycle of time and t₀ is the time intercept where the drawdown line intercepts the zero drawdown axis, it follows from further manipulation with Eq. (8.42) that the values of T and S, in consistent units, are given by
(8.43)
(8.44)
As with the log-log methods, these equations can be reshaped as
(8.45)
(8.46)
where C and D are coefficients that depend on the units used. For Δh and r in meters, t in seconds, Q in m³/s, and T in m²/s, C = 0.18 and D = 2.25. For Δh and r in feet, t in days, Q in U.S. gal/min, and T in U.S. gal/day/ft, C = 264 and D = 0.3. For and T in terms of Imperial gallons, C = 264 and D = 0.36.
Figure 8.24 (a) Determination of T and S from h₀ – h versus t data using the semilog method; (b) semilog plot in the vicinity of an impermeable boundary.
Todd (1959) states that the semilog method is valid for u < 0.01. Examination of the definition of u [Eq. (8.6)] shows that this condition is most likely to be satisfied for piezometers at small r and large t.
The semilog method is very well suited to the analysis of bounded confined aquifers. As we have seen, the influence of a boundary is equivalent to that of a recharging or discharging image well. For the case of an impermeable boundary, for example, the effect of the additional imaginary pumping well is to double the slope of the h₀ – h versus log t plot [Figure 8.24(b)]. The aquifer coefficients S and T should be calculated from Eqs. (8.43) and (8.44) on the earliest limb of the plot (before the influence of the boundary is felt). The time, t₁, at which the break in slope takes place can be used together with Eqs. (8.19) to calculate r_i, the distance from piezometer to image well [Figure 8.15(c)]. It takes records from three piezometers to unequivocally locate the position of the boundary if it is not known from geological evidence.