Which of the following is most likely to be associated with a confined aquifer?

Behavior of Aquifers during Pumping

As a confined aquifer is pumped, water is yielded by two primary mechanisms: deformation of the solid matrix and expansion of water. The storativity of a confined aquifer is a dimensionless property defined as:

(6)S=ΔhA/V,

where Δh is the drop in hydraulic head, A is the surface area of the aquifer being pumped, and V is the volume of water withdrawn from storage. In turn, the specific storage of the aquifer is defined as:

(7)SS=S/b=ρwg βP+nβW,

where b is the aquifer thickness, βP is the matrix compressibility, and βW is the compressibility of water. In an unconfined aquifer, water is primarily yielded by desaturation. The specific yield of an unconfined aquifer, which tends to be orders of magnitude greater than storativity, is defined as:

(8)SY=Vd/VT=n–SR,

where Vd is the drainable pore volume and SR is the specific retention, or fraction of porosity in which water is retained by capillary forces.

The matrix compressibility can be defined as the relative change in aquifer thickness divided by the fluid-pressure drop:

(9)βP=Δb/b/ΔP.

If deformation is concurrent with fluid expulsion, normal compaction results, whereas disequilibrium compaction occurs when the rate of loading exceeds the rate of fluid expulsion. In the latter case, fluid pressure (and thus hydraulic head) increases because the fluid temporarily carries part of the load, which is ultimately transferred to the solid matrix. Pumping can withdraw water not only from a confined aquifer but also from an adjoining confining unit, either through vertical leakage from overlying aquifers or from storage within the confining unit, when the hydraulic head in the pumped aquifer becomes less than that in the confining unit. Storativity can be redefined to account for confining-unit compressibility (βP′) in the case of normal compaction as follows:

(10)S=ρwgbβP+nβw+cβP′

where c = H/b and H is the confining-unit thickness (Jacob, 1940). Because fine-grained confining units are more compressible than aquifers, and the lost pore volume is not recoverable, the consolidation of confining units as a result of regionally extensive pumping can result in significant land subsidence. One prominent example is the Central Valley of California (USA), where pumping for irrigated agriculture during the 20th century has caused land surface to subside locally by > 8 m (Galloway, Jones, and Ingebritsen, 1999). Other prominent instances of land subsidence associated with groundwater extraction include Mexico City and metropolitan Jakarta in Indonesia (Galloway and Burbey, 2011).

7.2.2 Solution for Radial Flow to a Well

It is often of interest to know how water is flowing in the vicinity of a pumping well. A very useful solution to Laplace's equation is that for steady radial flow, which applies to flow in the vicinity of a pumping well. This solution assumes radial flow toward a well, so it makes sense to formulate the solution in terms of a radial coordinate r centered on the well, as shown in Figure 7.3. The origin of the coordinate system is taken as the centerline of the well. With this solution, all flow is radially symmetric in the r direction.

The solution for radial flow can be derived directly from the governing Laplace equation (Eq. 6.66), or it can be derived by combining Darcy's law and mass balance. We will take the latter approach, which is straightforward.

Define the discharge of the well as Q [L3/T], which by convention here is positive for a well that removes water from the aquifer and negative for a well that injects water into the aquifer. With mass balance, this same discharge must be flowing through any closed boundary that can be drawn around the well.

Imagine that this boundary is a cylinder of radius r centered on the well. The height of the aquifer is b, so the surface area that flow goes through on this cylinder is 2πrb. The specific discharge in the negative r direction (towards the well) anywhere on this cylindrical surface is −qr=Klpardh/dr). The total discharge through the cylinder is the product of specific discharge and the surface area of the cylinder, and it must equal the discharge of the well:

(7.4)Q=2πrbKd hdr=2πrTdh dr

This equation can be rearranged to separate the variables r and h to give

(7.5)dh=Q2πT drr

Integrating both sides of this equation yields the solution for steady radial flow in an aquifer with constant T:

(7.6)h=Q2πT lnr+C

where C is a constant and r is the radial distance from the center of the well to the point where h is evaluated. This solution satisfies Laplace's equation, which when written in terms of radial coordinates for radially symmetric flow is

(7.7)∇2h=∂2h∂r2+1r∂h∂r

Because of the natural log in Eq. 7.6, the head predicted by this solution has the following behaviors close to and far from the well:

(7.8) asr→0,h→−∞ asr→+∞,h→+∞

Since wells always have some finite radius, the singular behavior as r → 0 is not a concern. On the other hand, the behavior of this solution becomes inappropriate at large distances from the well. In real aquifers, heads do not increase indefinitely with distance from pumping wells because of the existence of features like rivers or lakes that supply water to the aquifer. Since this solution does not incorporate the influence of such far-field boundary conditions, its predictions become inaccurate far from the well. This solution alone is valid only in the region close to the well where the heads and discharges are dominated by the influence of the well.

When the head is known at some point close to the well, the constant C in Eq. 7.6 can be determined. Say that the head at radius r0 equals h0. The solution at r = r0 is

(7.9) h0=Q2πTlnr0 +C

Solving the above equation for C yields

(7.10)C=h0−Q2 πTlnr0

Substituting this definition of C back into Eq. 7.6 gives a form of the solution for the case where head is known at a point near the well:

(7.11)h=Q 2πTlnrr0+h0

This equation is sometimes referred to as the Thiem equation (Thiem, 1906). The point where r = r0 and h = h0 can be at the radius of the pumping well if you know the head at the pumping well, or it can be at the location of some nearby nonpumping well or piezometer.

Example 7.2

In the confined aquifer illustrated in Figure 7.4, there is a well that pumps water at a steady rate of Q = 500m3/day. Nearby are two observation wells A and B at radial distances of 10m and 25m, respectively. The heads in these wells are hA = 80.0 m, and hB = 82.0 m. Given this information, estimate T and K for the aquifer. Predict the head at the outer wall of the well screen, which has radius rw = 0.5 m.

Since the head at a nearby well is given, use Eq. 7.11 as a starting point. We will use the head at well A in this equation, but we could have used the head at well B and solved things just as easily:

h=Q2πTlnrrA+hA

The only unknown in this equation is T. Use the known head at well B to solve for T:

hB=Q2πTlnrB rA+hA

With a bit of algebraic manipulation, this can be solved for T:

T=Q2πlparhB−hA)lnrBrA=500 m3/day2πlpar 2.0 m)ln25 m10 m =36.5 m2/day

Recalling the definition of T,

K=T/b =lpar36.5 m2/day)/lpar8 m) =4.6 m/day

With T known, the head at the well screen is calculated with the solution

hw=Q2πT lnrwrA+hA =500 m3/day2πlpar36.5 m2/day)ln0.5 m10 m+ 80 m=73.5 m

The pattern of heads for this example is shown in Figure 7.5.

3.1.2.2 Partial Penetration

In practical studies, partial penetration occurs unintentionally most often from the lack of knowledge about the true saturated thickness of aquifer due to economic restrictions, (Figure 3.9).

These wells are common in areas where confined aquifers have large thickness. In a partially penetrating well the flow lines are forced to converge toward the partial penetration well entrance (screens with length, Ls). A water molecule has to travel longer distances than the fully penetrating wells, and therefore, there are bigger drawdowns under the same conditions. In the vicinity of the well the flow lines have a three-dimensional radial but nonplanar rotational shapes provided that the aquifer is isotropic and homogeneous. However, with increase in distance from the well, the flow domain converges to two-dimensional radial flow. For a given discharge, Q, the drawdown around a partially penetrating well is more than that for a fully penetrating well. The analysis of the partially penetration case is more difficult than fully penetrating wells. Practical experiences have shown, as a rule of thumb, that beyond a radial distance almost equal to twice the saturation thickness, the drawdown is approximately the same with a fully penetration case. In practical applications, one tries to minimize the deviations due to a fully penetrating well by considering the following points:

If horizontal bedding is strong in the flow domain and the observation wells are fairly close to the main well the aquifer is assumed to end at the bottom of the main well, and hence, fully penetrating well formulations are used directly on the basis of well penetration length. This approximation leads to underestimations.

Observations may be taken at such radial distances that the effects of partial penetration become negligible and the streamlines are substantially the same as if the well were fully penetrating. This distance, D, (see Figure 3.6) can be calculated in terms of the aquifer thickness, m, horizontal, Kh, and vertical, Kv, hydraulic conductivities as,

(3.6)D=2mK hKv

Jacob (1944) suggested the use of corrected drawdown, sc, by measuring the drawdown at top and bottom of the aquifer separately at a radial distance using a pair of observation wells. The corrected drawdown is calculated as the arithmetic average of top and bottom drawdowns.

In certain cases, the observed drawdown in partially penetrating wells may be adjusted for partial penetration according to the theory and empirical formulations. To this end, various researchers developed methods to correct for partial penetration in order to apply the formulations of full penetration.

Kozeny (1933) and Muskat (1937) developed a dimensionless formula, which relates the partial penetration well discharge, QP, ratio to full penetration well discharge, Q, as,

(3.7)QpQ=p[ 1+7β1/2cos(π2p )]

in which β is referred to as the well slimness, and is defined as a ratio,

β=rw2Ls

where rw is the well radius and Ls is the screen length (see Figure 3.9). Equation (3.7) is valid for penetration ratios p = Ls/m < 0.5 and Ls/m > 30, which implies that β > 0.01667. Graphical representation of Eq. (3.7) for β = 0.01667 is shown in Figure 3.10 for quick calculation of the discharge ratios from the penetration ratio.

Huismann (1972) developed an equation relating the additional drawdown due to the partially penetrating well in terms of the pump discharge, Q, hydraulic conductivity, K, aquifer thickness, m, the penetration ratio, p, screen length, Ls, and the well radius, rw, as,

(3.8)Δsw=2.3Q2πKm1−pp logαLsrw

in which α is the function of p and the amount of eccentricity, e, which is defined as a dimensionless ratio, e = 0.5(1 − Ls/m) = 0.5 (1 − p). Table 3.2 presents α values for sets of p and e values.

TABLE 3.2. α Values

EXAMPLE 3.3 DRAWDOWN CALCULATION

A confined aquifer of 80 m saturation thickness with transmissivity 951 m2/day is tapped through a fully penetrating well at a constant rate, 0.05 m3/sec. The well has 1.5 m diameter and the resulting drawdown after a long abstraction period is 14.7 m. By assuming no well loss, find the drawdown value if the well is screened between 19 and 35 m below the top of the aquifer.

Steady-State Drawdown in an Aquifer or Well

In designing a production or remediation well for an aquifer, it is important to be able to predict total drawdown in the aquifer. Theoretically, in an idealized aquifer having unlimited extent and no recharge, the cone of depression advances outward to infinity. Thus, steady-state analysis can only be approximate, and is useful only after pumping has occurred for some time. Steady-state drawdown at any given radius r1 from the well, relative to drawdown at another radius r2, can be determined by integrating Eq. (3.7b),

(3.8a)s1−s2=Qw2πKb⋅lnr2r1,

where s1 is the drawdown at a radial distance r1 from the well [L], s2 is the drawdown at a radial distance r2 from the well [L], Qw is the rate at which water is pumped from a well [L3/T], K is the hydraulic conductivity [L/T], and b is the aquifer thickness [L].

In practice, absolute drawdown may be estimated if it is possible to define a radius of influence, R, which represents the horizontal distance beyond which pumping of the well has little influence on the aquifer; that is, beyond R, no significant drawdown due to pumping is assumed to exist (see also Section 3.2.4). A variety of formulae for estimating R are given by Lembke (1886 and 1887); one formula is R=bK/2N, where N is annual recharge by precipitation. Sometimes R can be based on hydrogeologic conditions, such as the presence of a known constant-head boundary. Given an appropriate value for R and substituting it in Eq. (3.8a),

(3.8b)s=Qw2πKb⋅lnRr,

where s is the drawdown at a radial distance r from the well [L], Qw is the rate at which water is pumped from a well [L3/T], K is the hydraulic conductivity [L/T], b is the aquifer thickness [L], and R is the radius of influence [L].

One special case of Eq. (3.8b) is the drawdown in the well casing itself; Eq. (3.8b) gives the drawdown, sw, in the well if r is set equal to rw, the radius of the well:

(3.8c)sw=Qw2πKb⋅lnRrw

Eqs. (3.8a), (3.8b), and (3.8c) are all variations of the Thiem equation, which estimates drawdown in an aquifer or well under steady-state conditions. One assumption invoked in the Thiem equation is that changes in saturated aquifer thickness due to drawdown are small compared with the total saturated depth, and therefore can be neglected. Note that a confined aquifer has a constant saturated thickness, but an unconfined aquifer does not. For an unconfined aquifer in which drawdown is a significant fraction of the saturated thickness, Eq. (3.8a) must be expressed in terms of head instead of drawdown:

(3.9)h22−h22=QwKπ⋅lnr2r1,

where h2 is the hydraulic head at a radial distance r2 from the well [L], and h1 is the hydraulic head at a radial distance r1 from the well [L]. See Bear (1979) for further discussion.

In the various applications of the Thiem equation shown in Eqs. (3.8a)–(3.8c), the product (Kb) appears. This quantity measures the ability of an aquifer to deliver water to a well and is called transmissivity, T, with units [L2/T].

Groundwater flow into a well also can be analyzed by using a flow net. If there is no regional flow in the aquifer, lines of constant drawdown (isopotentials) are circles centered on the well, and streamlines are arranged radially around the well (Fig. 3.11).

For a confined aquifer, even if there is flow in the aquifer in the absence of the well, the resulting flow field due to both the well pumping and the pre-existing flow can be determined by applying a simple but powerful concept called superposition. Given a flow net showing head values for preexisting flow in a confined aquifer, one can subtract the drawdown at each point in the aquifer caused by well pumping (as shown in Fig. 3.11) to create the hydraulic head distribution that would actually result from placement and operation of a well in the aquifer. When the resultant isopotentials are drawn, and streamlines are extended through these new isopotentials following the rules for constructing flow nets, an accurate flow net that includes the effect of the pumping well results. The technique of superposition often allows quantitative flow solutions to be obtained in even complex situations.

Superposition in phreatic aquifers is more problematic than superposition in confined aquifers, because changes in saturated thickness affect aquifer transmissivity and thereby create a nonlinear relationship between head gradient and total flow. However, for unconfined aquifers in which drawdown is a sufficiently small fraction of aquifer thickness, the technique of superposition can be a useful approximation. The reader is referred to Strack (1989) for further details on superposition in unconfined aquifers.

Example 3.3

The hydraulic conductivity in a confined aquifer 3 m thick is estimated to be 10− 2 cm/sec. The hydraulic gradient is approximately 0.0012, and the porosity is 0.25. A well 20 cm in diameter is installed in the aquifer and is pumped at an average rate of 10 liter/min.

What is the drawdown in this well if the radius of influence is 75 ft?

Estimate the hydraulic gradient in the aquifer at distances of 5 and 20 m, both directly upstream and downstream of the well.

Drawdown in the well at steady state can be estimated using the Thiem equation in the form of Eq. (3.8c):

sw=10liter /min2π10−2cm/sec⋅3m⋅1min60sec⋅1000 cm31liter⋅1m100cm ⋅ln75ft⋅0.3048m/ft0.1m=48cm.

Gradients due to the well being pumped can be superimposed on the aquifer’s hydraulic gradient in the absence of pumping. Use Eq. (3.7b) to estimate the gradient, ds/dr, created by the well.

At 5 m:

dsdr=10liter/ min2π3cm2/sec500cm⋅1min60sec⋅ 1000cm31liter=0.018.

At 20 m, using Eq. (3.7b), ds/dr is equal to 0.0044.

Directly upstream of the well, the hydraulic gradient will be increased due to pumping.

At5m:gradient =0.0012+0.018=0.019.At20m:gradient=0.0012+0.0044=0.0056.

Within the radius of influence downstream of the well, the hydraulic gradient of the aquifer is sloping away from the well, but maintains the same sign; in contrast, the pumping of the well tends to cause water to flow in the opposite direction, toward the well, and thus the gradient sign is reversed.

At5m:gradient=0.0012−0.018=−0.017.At20m:gradient=0.0012−0.0044 =−0.0032.

At all four points, the hydraulic gradient is toward the well.

4.9.1 Confined Aquifer Approximation

The basic idea of these models stems from a practical question as how can one apply confined aquifer models to the unconfined aquifers? It is necessary that three-dimensional flows should be approximately planar-radial flow. For the application of available models unconfined aquifer must have approximately similar physical conditions as confined aquifers. The following points are worthy to notice in the applications.

In rather thick saturation thickness, m, the maximum drawdown, sM, is relatively small. In practice, (sM/m) × 100 < 10 is a good criterion for the application of confined aquifer models to unconfined aquifers. Under these conditions, the three-dimensional flow is approximated as radial flow and gravity drainage contribution is not significant.

Based on the theoretical studies of Boulton (1963), the TD record must be taken at observation wells.

Boulton (1963) gave the minimum time interval after the start of pumping for unconfined aquifer models to be applicable as,

(4.29)t≥5SymK

For thin saturation thickness aquifers, Jacob (1940) suggested correction of drawdowns prior to application of confined aquifer models (Chapter 3, Section 3.9). Hence, the equivalent confined aquifer drawdown, sc, is expressed as,

(4.30)sc=su−su22 m

None of the above mentioned conditions guarantee the full applicability of confined aquifer models. Most often, in practice, there are difficulties as far as the physical plausibility of the final parameter estimates is concerned.

Conclusion

The hydrogeological typology of groundwater systems rests on two pillars, the distinction between unconfined and confined aquifers as well as the differentiation of porosity types. The first is defined by the position of the water table relative to the hydrogeological units; the second is determined by the characteristics of the geological material. The framework provided by this typology supports the understanding of aquifer properties governing flow and storage processes in groundwater systems. Groundwater systems can be further classified based on other aspects such as the chemical or thermal properties of the water. Possible interrelations between groundwater flow, chemistry and temperature need to be considered particularly in the hydrogeological assessment of deep groundwater systems, coastal aquifers, or settings affected by human impacts.

5.1.3 Water Table and Potentiometric Surface Maps

A map of the water table surface (unconfined aquifer) or the potentiometric surface (confined aquifer) is a good tool for understanding the patterns of horizontal flow in aquifers. These surfaces represent the horizontal distribution of hydraulic head in the aquifer. Since water always flows towards lower hydraulic head, horizontal flow directions can be inferred from these maps. If the aquifer materials have isotropic hydraulic conductivity in the horizontal plane, then flow will be perpendicular to the contours of hydraulic head, as discussed in Section 3.5.

To construct one of these maps, you need water level measurements from a number of observation wells screened in the same aquifer. If there are significant vertical head gradients in the aquifer, the data should come from wells that are all screened at approximately the same level in the aquifer. For a water table map, the data should be from wells that are screened across or just below the water table.

Surface water elevations are another source of useful data for water table maps. If the aquifer and surface water are in direct contact, the water table will intersect the surface water at an elevation close to the elevation of the surface water. Along steep stream banks, there is often seepage because the water table is slightly higher than the stream surface. The elevation of the water table can differ from the elevation of a surface water body if there is a layer of low-conductivity sediment on the bottom of the surface water body.

Figure 5.7 shows an example water table map based on surface water and observation well levels. Contours are drawn by interpolating between the data points, a process that is a bit subjective and guided by experience. The pond level is higher than the surrounding water table, indicating that the pond may be perched. The inferred flow directions near the pond also indicate that it is a source of water to the aquifer.

On the other hand, the inferred groundwater flow directions near the larger stream indicate that groundwater is discharging into it from both sides. This is a gaining stream, one that picks up groundwater discharge. Where water table contours cross a gaining stream, they form a “V” that points in the upstream direction. Losing streams are just the opposite: contours form a “V” that points in the downstream direction.

12.4.3.6 Porewater pressure

Groundwater flow within subglacial substratum is sometimes responsible for the development of overpressure and underpressure zones. Within confined aquifers, the pressure exerted by overlying ice associated with the infiltration of meltwater favours the development of high porewater pressure in the substratum, often referred to as fluid overpressure. Overpressurized groundwater is common beneath and at the ice sheet margin (Boulton and Caban, 1995). Overpressurized zones formed during the Pleistocene glaciations have been determined using zones of anomalously high-amplitude seismic reflections in the continental shelf of Massachusetts (Siegel et al., 2014). In this case, overpressure subsists in modern times as the result of ice loading from the late Pleistocene glaciation and rapid sedimentation. The development of marginal permafrost also enhances overpressure development as permafrost prevents groundwater from welling up directly beyond the ice sheet margin (Piotrowski, 1994, 1997). This effect generates high heads and water overpressures in the proglacial zone in which pressures would be atmospheric in the absence of permafrost.

Based on drill stem test data from wells located in the Cretaceous aquifers of the Alberta Basin, zones of anomalously low fluid pressures have also been measured (Bachu and Underschultz, 1995). The numerical modelling of these underpressure zones shows that basin hydrodynamics cannot produce these zones without mechanical unloading processes (Bekele et al., 2003). Although a significant component of the low-pressure zones is related to unloading by high erosion rates (from Early Eocene), an additional component such as glacial unloading is needed to fit the low-pressure values recorded in the Cretaceous aquifer of the Alberta Basin. Since ice sheets retreat much faster than they advance, the resulting high unloading rates could have led to the formation of underpressure zones within confined aquifers (Vinard et al., 2001; Bekele et al., 2003) (Box 12.1).

Box 12.1

Two Models Based on Groundwater Flow Simulations in North America During The Last Glaciation Illustrating Changes of Hydraulic Heads Values Over a Whole Glaciation/Deglaciation Cycle (A), and Changes in Groundwater Flow Patterns, Directions, and Velocities Between Glacial and Interglacial Periods (B) (Fig. 12.8A and B)

(A) Changes in hydraulic heads in intracratonic sedimentary basins (based on the study of Bense, V.F., Person, M.A., 2008. Transient hydrodynamics within intercratonic sedimentary basins during glacial cycles. J. Geophys. Res.-Earth Surf., 113 (F4)).

The simulations apply to intracratonic sag basin basins with maximum depths of 5000 m and a width (length) of 700 km at the southern edge of the LIS during the last glacial maximum (20 ka BP), in the Williston, Michigan, and Illinois basins. The sedimentary basin consists of four aquifers separated by three semiconfining units (shale beds). The results reveal characteristic spatial patterns of underpressure and overpressure that occur in aquitards and aquifers as a result of recent glaciation and deglaciation. Four main stages can be defined:

Stage I: Ice Sheet Advance. During the advance of the ice sheet toward the south, the flexural response to ice sheet loading decreases the effective ice surface elevation, thus causing a reduction of hydraulic head at the surface. This process occurs in combination with the generation of elevated fluid pressures in the low-permeability confining units due to mechanical loading. This loading triggers hydraulic gradients that force fluid flow toward the surface.

Stage II: Last Glacial Maximum (LGM). During the LGM, hydraulic head disturbances induced by mechanical loading propagate rapidly through the aquifers. High fluid pressures in the aquifers will diffuse into the confining units in between the aquifers. When compared to the initial steady-state conditions, horizontal fluid fluxes in the deeper aquifers steadily rise by up to two orders of magnitude by the end of the LGM.

Stage III: Ice Sheet Retreat. Once the ice sheet starts to retreat, hydraulic heads start to decline. This is a result of the unloading of the basin by the rapid reduction in ice sheet thickness. Below, where the ice sheet was covering the basin, hydraulic heads in the confining units become subhydrostatic because of mechanical unloading.

Stage IV: Present-Day Conditions. In the present-day, low and high hydraulic heads are found in the basin. Near the centre of the basin, overpressures occur at all depths. The location of these anomalous heads is near the position of maximum extent of the former ice sheet. The presence of these anomalously high hydraulic heads can be understood by realizing that lateral migration of high hydraulic heads occurred beyond the toe of the ice sheet during the LGM.

(B) Variations in groundwater flow patterns, directions, and velocities between glacial and nonglacial times (based on the study of Breemer, C.W., Clark, P.U., Haggerty, R., 2002. Modeling the subglacial hydrology of the late Pleistocene Lake Michigan Lobe, Laurentide Ice Sheet. Geol. Soc. Am. Bull. 114 (6), 665–674).

Groundwater flow simulations beneath the Lake Michigan Ice Lobe are based on a two-dimensional flow line parallel to ice flow. The model includes a hydrostratigraphic description of six regional aquifer units and four regional confining units that were present below the Lake Michigan Ice Lobe. Under nonglacial conditions (i.e., modern conditions), groundwater is typically recharged in topographically high areas and discharged from topographic lows. However, simulations under glacial conditions indicate that the Lake Michigan Ice Lobe altered or reversed topographically driven pressure gradients, resulting in groundwater flow patterns and velocities that differed substantially from those of current modern conditions. Under glacial conditions, simulated groundwater flow velocities are high. Beneath the ice, groundwater has a relatively strong downward component in the aquifer although an upward component is observed close to the margin. The upward component is more expressed in the case of a 7-mm meltwater film at the ice–bed interface (water film thickness is based on results from Ice Stream B in Alley, 1989). Under glacial conditions, when a subglacial water film is absent, groundwater velocities are higher, and flow is consistently directed toward the south.

GROUNDWATER SUPPLIES

Groundwater is an important direct source of supply that is tapped by wells, as well as a significant indirect source since surface streams are often supplied by subterranean water.

Near the surface of the earth, in the zone of aeration, soil pore spaces contain both air and water. This zone, which may have zero thickness in swamplands and be several hundred feet thick in mountainous regions, contains three types of moisture. After a storm, gravity water is in transit through the larger soil pore spaces. Capillary water is drawn through small pore spaces by capillary action and is available for plant uptake. Hygroscopic moisture is held in place by molecular forces during all except the driest climatic conditions. Moisture from the zone of aeration cannot be tapped as a water supply source.

In the zone of saturation, located below the zone of aeration, the soil pores are filled with water, and this is what we call groundwater. A stratum that contains a substantial amount of groundwater is called an aquifer. At the surface between the two zones, called the water table or phreatic surface, the hydrostatic pressure in the groundwater is equal to the atmospheric pressure. An aquifer may extend to great depths, but because the weight of overburden material generally closes pore spaces, little water is found at depths greater than 600 m (2000 ft). The amount of water that will drain freely from an aquifer is known as specific yield.

The flow of water out of a soil can be illustrated using Figure 5-2. The flow rate must be proportional to the area through which flow occurs times the velocity, or

where

Q = flow rate, in m3/sec

A = area of porous material through which flow occurs, in m2

v = superficial velocity, in m/sec

The superficial velocity is of course not the actual velocity of the water in the soil, since the volume occupied by the soil solid particles greatly reduces the available area for flow. If a is the area available for flow, then

where

v′ = actual velocity of water flowing through the soil

a = area available for flow

Solving for v′,

If a sample of soil is of some length L, then

(5.4)v'= Ava=AvaL=Lporosity

since the total volume of the soil sample is AL and the volume occupied by the water is aL.

Water flowing through the soil at a velocity v′ loses energy, just as water flowing through a pipeline or an open channel does. This energy loss per distance traveled is defined as

where

h = energy, measured as elevation of the water table in an unconfined aquifer or as pressure in a confined aquifer, in m

L = horizontal distance in direction of flow, in m

The symbol (delta) simply means “a change in,” as in “a change in length, L.” Thus this equation means that there is a change (loss) of energy, h, as water flows through the soil some distance, L.

In an unconfined aquifer, the drop in the elevation of the water table with distance is the slope of the water table in the direction of flow. The elevation of the water surface is the potential energy of the water, and water flows from a higher elevation to a lower elevation, losing energy along the way. Flow through a porous medium such as soil is related to the energy loss using the Darcy equation,

where

K = coefficient of permeability, in m/day

A = cross-sectional area, in m2

The Darcy equation makes intuitive sense, in that the flow rate (Q) increases with increasing area (A) through which the flow occurs and with the drop in pressure, Δh/ΔL. The greater the driving force (the difference in upstream and downstream pressures), the greater the flow. The fudge factor, K, is the coefficient of permeability, an indirect measure of the ability of a soil sample to transmit water; it varies dramatically for different soils, ranging from about 0.0005 m/day for clay to over 5000 m/day for gravel. The coefficient of permeability is measured commonly in the laboratory using permeameters, which consist of a soil sample through which a fluid such as water is forced. The flow rate is measured for a given driving force (difference in pressures) through a known area of soil sample, and the permeability calculated.

Example 5.1

Problem. A soil sample is installed in a permeameter as shown in Figure 5-3. The length of the sample is 0.1 m, and it has a cross-sectional area of 0.05 m2. The water pressure on the upflow side is 2.5 m and on the downstream side it is 0.5 m. A flow rate of 2.0 m3/day is observed. What is the coefficient of permeability?

FIGURE 5-3. Permeameter used for measuring coefficient of permeability using the Darcy equation

Solution. The pressure drop is the difference between the upstream and downstream pressures, or Δh = 2.5 − 0.5 = 2.0 m. Using the Darcy equation, and solving for K,

(5.7)K=QAΔhΔL=2.00.05×20.1=2m/day

If a well is sunk into an unconfined aquifer, shown in Figure 5-4, and water is pumped out, the water in the aquifer will begin to flow toward the well. As the water approaches the well, the area through which it flows gets progressively smaller, and therefore a higher superficial (and actual) velocity is required. The higher velocity of course results in an increasing loss of energy, and the energy gradient must increase, forming a cone of depression. The reduction in the water table is known in groundwater terms as a drawdown. If the rate of water flowing toward the well is equal to the rate of water being pumped out of the well, the condition is at equilibrium, and the drawdown remains constant.

FIGURE 5-4. Drawdown in water table due to pumping from a well

If, however, the rate of water pumping is increased, the radial flow toward the well has to compensate, and this results in a deeper cone or drawdown.

Consider a cylinder, shown in Figure 5-5, through which water flows toward the center. Using Darcy's equation,

FIGURE 5-5. Cylinder with flow through the surface

(5.8)Q =KAΔhΔL=K(2πrh)ΔhΔr

where r is the radius of the cylinder, and 2πrh is the cross-sectional surface area of the cylinder. If water is pumped out of the center of the cylinder at the same rate as water is moving in through the cylinder surface area, the above equation can be integrated to yield

where h1 and h2 are the height of the water table at radial distances r1 and r2 from the well.

This equation can be used to estimate the pumping rate for a given drawdown any distance away from a well, using the water level measurements in two observation wells in an unconfined aquifer, as shown in Figure 5-6. Also, knowing the diameter of a well, it is possible to estimate the drawdown at the well, the critical point in the cone of depression. If the drawdown is depressed all the way to the bottom of the aquifer, the well “goes dry”—it cannot pump water at the desired rate. Although the derivations of the above equations are for an unconfined aquifer, the same situation would occur for a confined aquifer, where the pressure would be measured by observation wells.

FIGURE 5-6. Two monitoring wells define the extent of drawdown during extraction.

Example 5.2

Problem. A well is 0.2 m in diameter and pumps from an unconfined aquifer 30 m deep at an equilibrium (steady state) rate of 1000 m3 per day. Two observation wells are located at distances 50 m and 100 m, and they have been drawn down by 0.2 m and 0.3 m, respectively. What is the coefficient of permeability and estimated drawdown at the well?

Solution

(5.10)K=QInr1r2π(h22−h22)=1000In(100/50)3.14[(24.8)2−(29.72)] =37.1m/day

If the radius of the well is 0.2/2 = 0.1 m, this can be plugged into the same equation, as

(5.11)Q=πK(h22−h22)lnr1r2=3.14×37.1[(29.72)−h22] ln500.1=1000

and solving for h2,

Since the aquifer is 30 m deep, the drawdown at the well is 30 − 28.8 = 1.2 m.

Multiple wells in an aquifer can interfere with each other and cause excessive drawdown. Consider the situation in Figure 5-7, where a single well creates a cone of depression. If a second production well is installed, the cones will overlap, causing greater drawdown at each well. If many wells are sunk into an aquifer, the combined effect of the wells can deplete the groundwater resources and all wells will “go dry.”

FIGURE 5-7. Multiple wells and the effect of extraction on the groundwater table

The reverse is also true, of course. Suppose one of the wells is used as an injection well, then the injected water flows from this well into the others, building up the groundwater table and reducing the drawdown. The judicious use of extraction and injection wells is one way that the flow of contaminants from hazardous waste or refuse dumps can be controlled, as discussed further in Chapter 15.

Finally, many assumptions are made in the above discussion. First, we assume that the aquifer is homogeneous and infinite—that is, it sits on a level aquaclude and the permeability of the soil is the same at all places for an infinite distance in all directions. The well is assumed to penetrate the entire aquifer and is open for the entire depth of the aquifer. Finally, the pumping rate is assumed to be constant. Clearly, any of these conditions may cause the analysis to be faulty, and this model of aquifer behavior is only the beginning of the story. Modeling the behavior of groundwater is a complex and sophisticated science.