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Q(t) = total rate of groundwater withdrawal
R(t) = total rate of groundwater recharge to the basin
D(t) = total rate of groundwater discharge from the basin
dS/dt = rate of change of storage in the saturated zone of the basin.
Freeze (1971a) examined the response of R(t) and D(t) to an increase in Q(t) in a hypothetical basin in a humid climate where water tables are near the surface. The response was simulated with the aid of a three-dimensional transient analysis of a complete saturated-unsaturated system such as that of Figure 6.10 with a pumping well added. Figure 8.32 is a schematic representation of his findings. The diagrams show the time-dependent changes that might be expected in the various terms of Eq. (8.72) under increased pumpage. Let us first look at the case shown in Figure 8.32(a), in which withdrawals increase with time but do not become excessive. The initial condition at time t₀ is a steady-state flow system in which the recharge, R₀, equals the discharge, R₀. At times t₁, t₂, t₃, and t₄, new wells begin to tap the system and the pumping rate Q undergoes a set of stepped increases. Each increase is initially balanced by a change in storage, which in an unconfined aquifer takes the form of an immediate water-table decline. At the same time, the basin strives to set up a new equilibrium under conditions of increased recharge, R.
Figure 8.32 Schematic diagram of transient relationships between recharge rates, discharge rates, and withdrawal rates (after Freeze, 1971a).
The unsaturated zone will now be induced to deliver greater flow rates to the water table under the influence of higher gradients in the saturated zone. Concurrently, the increased pumpage may lead to decreased discharge rates, D. In Figure 8.32(a), after time t₄, all natural discharge ceases and the discharge curve rises above the horizontal axis, implying the presence of induced recharge from a stream that had previously been accepting its baseflow component from the groundwater system. At time t₅, the withdrawal Q is being fed by the recharge, R, and the induced recharge, D; and there has been a significant decline in the water table. Note that the recharge rate attains a maximum between t₃ and t₄. At this rate, the groundwater body is accepting all the infiltration that is available from the unsaturated zone under the lowered water-table conditions.
In Figure 8.32(a), steady-state equilibrium conditions are reached prior to each new increase in withdrawal rate. Figure 8.32(b) shows the same sequence of events under conditions of continuously increasing groundwater development over several years. This diagram also shows that if pumping rates are allowed to increase indefinitely, an unstable situation may arise where the declining water table reaches a depth below which the maximum rate of groundwater recharge R can no longer be sustained. After this point in time the same annual precipitation rate no longer provides the same percentage of infiltration to the water table. Evapotranspiration during soil-moisture-redistribution periods now takes more of the infiltrated rainfall before it has a chance to percolate down to the groundwater zone. At t₄ in Figure 8.32(b), the water table reaches a depth below which no stable recharge rate can be maintained. At t₅ the maximum available rate of induced recharge is attained. From time t₅ on, it is impossible for the basin to supply increased rates of withdrawal. The only source lies in an increased rate of change of storage that manifests itself in rapidly declining water tables. Pumping rates can no longer be maintained at their original levels. Freeze (1971a) defines the value of Q at which instability occurs as the maximum stable basin yield. To develop a basin to its limit of stability would, of course, be foolhardy. One dry year might cause an irrecoverable water-table drop. Production rates must allow for a factor of safety and must therefore be somewhat less than the maximum stable basin yield.
The discussion above emphasizes once again the important interrelationships between groundwater flow and surface runoff. If a groundwater basin were developed up to its maximum yield, the potential yields of surface-water components of the hydrologic cycle in the basin would be reduced. It is now widely recognized that optimal development of the water resources of a watershed depend on the conjunctive use of surface water and groundwater. The subject has provided a fertile field for the application of optimization techniques (Maddock, 1974; Yu and Haimes, 1974). Young and Bredehoeft (1972) describe the application of digital computer simulations of the type described in Section 8.8 to the solution of management problems involving conjunctive groundwater and surface-water systems.
8.11 Artificial Recharge and Induced Infiltration
In recent years, particularly in the more populated areas of North America where water resource development has approached or exceeded available yield, there has been considerable effort placed on the management of water resource systems. Optimal development usually involves the conjunctive use of groundwater and surface water and the reclamation and reuse of some portion of the available water resources. In many cases, it involves the importation of surface water from areas of plenty to areas of scarcity, or the conservation of surface water in times of plenty for use in times of scarcity. These two approaches require storage facilities, and there is often advantage to storing water underground where evaporation losses are minimized. Underground storage may also serve to replenish groundwater resources in areas of overdraft.
Any process by which man fosters the transfer of surface water into the groundwater system can be classified as artificial recharge. The most common method involves infiltration from spreading basins into high-permeability, unconfined, alluvial aquifers. In many cases, the spreading basins are formed by the construction of dikes in natural channels. The recharge process involves the growth of a groundwater mound beneath the spreading basin. The areal extent of the mound and its rate of growth depend on the size and shape of the recharging basin, the duration and rate of recharge, the stratigraphic configuration of subsurface formations, and the saturated and unsaturated hydraulic properties of the geologic materials. Figure 8.33 shows two simple hydrogeological environments and the type of groundwater mound that would be produced in each case beneath a circular spreading basin. In Figure 8.33(a), recharge takes place into a horizontal unconfined aquifer bounded at the base by an impermeable formation. In Figure 8.33(b), recharge takes place through a less-permeable formation toward a high-permeability layer at depth.
Figure 8.33 Growth of a groundwater mound beneath a circular recharge basin.
Both cases have been the subject of a large number of predictive analyses, not only for circular spreading basins but also for rectangular basins and for recharge from an infinitely long strip. The latter case, with boundary conditions like those shown in Figure 8.33(b), also has application to canal and river seepage. It has been studied in this context by Bouwer (1965), Jeppson (1968), and Jeppson and Nelson (1970). The case shown in Figure 8.33(a), which also has application to the development of mounds beneath waste disposal ponds and sanitary landfills, has been studied in even greater detail. Hantush (1967) provides an analytical solution for the prediction of h(r, t), given the initial water-table height, h₀, the diameter of the spreading basin, a, the recharge rate, R, and the hydraulic conductivity and specific yield, K and S_y, of the unconfined aquifer. His solution is limited to homogeneous, isotropic aquifers and a recharge rate that is constant in time and space. In addition, the solution is limited to a water-table rise that is less than or equal to 50% of the initial depth of saturation, h₀. This requirement implies that R   K. Bouwer (1962) utilized an electric-analog model to analyze the same problem, and Marine (1975a, 1975b) produced a numerical simulation. All three of these analyses have two additional limitations. First, they neglect unsaturated flow by assuming that the recharge pulse traverses the unsaturated zone vertically and reaches the water table unaffected by soil moisture-conditions above the water table. Second, they utilize the Dupuit-Forchheimer theory of unconfined flow (Section 5.5) which neglects any vertical flow gradients that develop in the saturated zone in the vicinity of the mound. Numerical simulations carried out on the complete saturated-unsaturated system using the approaches of Rubin (1968), Jeppson and Nelson (1970), and Freeze (1971a) would provide a more accurate approach to the problem, but at the expense of added complexity in the calculations.
Practical research on spreading basins has shown that the niceties of predictive analysis are seldom reflected in the real world. Even if water levels in spreading ponds are kept relatively constant, the recharge rate almost invariably declines with time as a result of the buildup of silt and clay on the basin floor and the growth of microbial organisms that clog the soil pores. In addition, air entrapment between the wetting front and the water table retards recharge rates. Todd (1959) notes that alternating wet and dry periods generally furnish a greater total recharge than does continuous spreading. Drying kills the microbial growths, and tilling and scraping of the basin floor during dry periods reopens the soil pores.
There are several excellent case histories that provide an account of specific projects involving artificial recharge from spreading basins. Seaburn (1970) describes hydrologic studies carried out at two of the more than 2000 recharge basins that are used on Long Island, east of New York City, to provide artificial recharge of storm runoff from residential and industrial areas. Bianchi and Haskell (1966, 1968) describe the piezometric monitoring of a complete recharge cycle of mound growth and dissipation. They report relatively good agreement between the field data and analytical predictions based on Dupuit-Forchheimer theory. They note, however, that the anomalous water-level rises that accompany air entrapment (Section 6.8) often make it difficult to accurately monitor the growth of the groundwater mound.
While water spreading is the most ubiquitous form of artificial recharge, it is limited to locations with favorable geologic conditions at the surface. There have also been some attempts made to recharge deeper formations by means of injection wells. Todd (1959) provides several case histories involving such diverse applications as the disposal of storm-runoff water, the recirculation of air-conditioning water, and the buildup of a freshwater barrier to prevent further intrusion of seawater into a confined aquifer. Most of the more recent research on deep-well injection has centered on utilization of the method for the disposal of industrial wastewater and tertiary-treated municipal wastewater (Chapter 9) rather than for the replenishment of groundwater resources.
The oldest and most widely used method of conjunctive use of surface water and groundwater is based on the concept of induced infiltration. If a well produces water from alluvial sands and gravels that are in hydraulic connection with a stream, the stream will act as a constant-head line source in the manner noted in Figures 8.15(d) and 8.23(d). When a new well starts to pump in such a situation the pumped water is initially derived from the groundwater zone, but once the cone of depression reaches the stream, the source of some of the pumped water will be streamflow that is induced into the groundwater body under the influence of the gradients set up by the well. In due course, steady-state conditions will be reached, after which time the cone of depression and the drawdowns within it remain constant. Under the steady flow system that develops at such times, the source of all the pumped groundwater is streamflow. One of the primary advantages of induced infiltration schemes over direct surface-water utilization lies in the chemical and biological purification afforded by the passage of stream water through the alluvial deposits.
8.12 Land Subsidence
In recent years it has become apparent that the extensive exploitation of groundwater resources in this century has brought with it an undesired environmental side effect. At many localities in the world, groundwater pumpage from unconsolidated aquifer-aquitard systems has been accompanied by significant land subsidence. Poland and Davis (1969) and Poland (1972) provide descriptive summaries of all the well-documented cases of major land subsidence caused by the withdrawal of fluids. They present several case histories where subsidence has been associated with oil and gas production, together with a large number of cases that involve groundwater pumpage. There are three cases—the Wilmington oil field in Long Beach, California, and the groundwater overdrafts in Mexico City, Mexico, and in the San Joaquin valley, California—that have led to rates of subsidence of the land surface of almost 1 m every 3 years over the 35-year period 1935–1970. In the San Joaquin valley, where groundwater pumpage for irrigation purposes is to blame, there are three separate areas with significant subsidence problems. Taken together, there is a total area of 11,000 km² that has subsided more than 0.3 m. At Long Beach, where the subsiding region is adjacent to the ocean, subsidence has resulted in repeated flooding of the harbor area. Failure of surface structures, buckling of pipe lines, and rupturing of oil-well casing have been reported. Remedial costs up to 1962 exceeded $100 million.
Mechanism of Land Subsidence
The depositional environments at the various subsidence sites are varied, but there is one feature that is common to all the groundwater-induced sites. In each case there is a thick sequence of unconsolidated or poorly consolidated sediments forming an interbedded aquifer-aquitard system. Pumpage is from sand and gravel aquifers, but a large percentage of the section consists of high-compressibility clays. In earlier chapters we learned that groundwater pumpage is accompanied by vertical leakage from the adjacent aquitards. It should come as no surprise to find that the process of aquitard drainage leads to compaction^* of the aquitards just as the process of aquifer drainage leads to compaction of the aquifers. There are two fundamental differences, however: (1) since the compressibility of clay is 1–2 orders of magnitude greater than the compressibility of sand, the total potential compaction of an aquitard is much greater than that for an aquifer; and (2) since the hydraulic conductivity of clay may be several orders of magnitude less than the hydraulic conductivity of sand, the drainage process, and hence the compaction process, is much slower in aquitards than in aquifers.
Consider the vertical cross section shown in Figure 8.34. A well pumping at a rate Q is fed by two aquifers separated by an aquitard of thickness b.
Figure 8.34 One-dimensional consolidation of an aquitard.
Let us assume that the geometry is radially symmetric and that the transmissivities in the two aquifers are identical. The time-dependent reductions in hydraulic head in the aquifers (which could be predicted from leaky-aquifer theory) will be identical at points A and B. We wish to look at the hydraulic-head reductions in the aquitard along the line AB under the influence of the head reductions in the aquifers at A and B. If h_A(t) and h_B(t) are approximated by step functions with a step Δh (Figure 8.34), the aquitard drainage process can be viewed as the one-dimensional, transient boundary-value problem described in Section 8.3 and presented as Eq. (8.21). The initial condition is h = h₀ all along AB, and the boundary conditions are h = h₀ – Δh at A and at B for all t > 0. A solution to this boundary-value problem was obtained by Terzaghi (1925) in the form of an analytical expression for h(z, t). An accurate graphical presentation of his solution appears as Figure 8.17. The central diagram on the right-hand side of Figure 8.34 is a schematic plot of his solution; it shows the time-dependent decline in hydraulic head at times t₀, t₁ . . . , t ∞ along the line AB. To obtain quantitative results for a particular case, one must know the thickness b’, the vertical hydraulic conductivity K’, the vertical compressibility α’, and the porosity n’ of the aquitard, together with the head reduction Δh on the boundaries.
In soil mechanics the compaction process associated with the drainage of a clay layer is known as consolidation. Geotechnical engineers have long recognized that for most clays α   nβ, so the latter term is usually omitted from Eq. (8.21). The remaining parameters are often grouped into a single parameter c_v, defined by
(8.73)
The hydraulic head h(z, t) can be calculated from Figure 8.17 with the aid of Eq. (8.23) given c_v, Δh, and b.
In order to calculate the compaction of the aquitard given the hydraulic head declines at each point on AB as a function of time, it is necessary to recall the effective stress law: σ_T = σ_e + p. For σ_T = constant, dσ_e = -dp. In the aquitard, the head reduction at any point z between the times t₁ and t₂ (Figure 8.34) is dh = h₁(z, t₁) – h₂(z, t₂). This head drop creates a fluid pressure reduction: dp = ρg dψ = ρg d(h – z) = pg dh, and the fluid pressure reduction is reflected by an increase in the effective stress dσ_e = -dp. It is the change in effective stress, acting through the aquitard compressibility α’, that causes the aquitard compaction Δb’. To calculate Δb’ along AB between the times t₁ and t₂, it is necessary to divide the aquitard into m slices. Then, from Eq. (2.54),
(8.74)
where dh_i is the average head decline in the ith slice.
For a multiaquifer system with several pumping wells, the land subsidence as a function of time is the summation of all the aquitard and aquifer compactions. A complete treatment of consolidation theory appears in most soil mechanics texts (Terzaghi and Peck, 1967; Scott, 1963). Domenico and Mifflin (1965) were the first to apply these solutions to cases of land subsidence.
It is reasonable to ask whether land subsidence can be arrested by injecting groundwater back into the system. In principle this should increase the hydraulic heads in the aquifers, drive water back into the aquitards, and cause an expansion of both aquifer and aquitard. In practice, this approach is not particularly effective because aquitard compressibilities in expansion have only about one-tenth the value they have in compression. The most successful documented injection scheme is the one undertaken at the Wilmington oil field in Long Beach, California (Poland and Davis, 1969). Repressuring of the oil reservoir was initiated in 1958 and by 1963 there had been a modest rebound in a portion of the subsiding region and the rates of subsidence were reduced elsewhere.