With the methods of construction and simulation of steady-state flow nets in hand, we are now in a position to examine the natural flow of groundwater in hydrogeologic basins.
Let us consider the two-dimensional, vertical cross section of Figure 6.1. The section is taken in a direction perpendicular to the strike of a set of long, parallel ridges and valleys in a humid region. The geologic materials are homogeneous and isotropic, and the system is bounded at the base by an impermeable boundary. The water table is coincident with the ground surface in the valleys, and forms a subdued replica of the topography on the hills. The value of the hydraulic head on any one of the dashed equipotential lines is equal to the elevation of the water table at its point of intersection with the equipotential line. The flowlines and equipotential lines were stretched according to the usual rules for graphical flow-net construction in homogeneous, isotropic media.
It is clear from the flow net that groundwater flow occurs from the highlands toward the valleys. The flow net must fill the entire field of flow, and one consequence of this fact is the occurrence of upward-rising groundwater flow beneath the valleys. The symmetry of the system creates vertical boundaries beneath the valleys and ridges (the dotted lines AB and CD) across which there is no flow. These imaginary impermeable boundaries are known as groundwater divides. In the most symmetric systems, such as that shown on Figure 6.1 they coincide exactly with surface-water divides, and their orientation is precisely vertical. In more complex topographic and hydrogeologic environments, these properties may be lost.
The flowlines in Figure 6.1 deliver groundwater from recharge areas to discharge areas. In a recharge area there is a component to the direction of groundwater flow near the surface that is downward. A recharge area can be defined as that portion of the drainage basin in which the net saturated flow of groundwater is directed away from the water table. In a discharge area there is a component to the direction of groundwater flow near the surface that is upward. A discharge area can be defined as that portion of the drainage basin in which the net saturated flow of groundwater is directed toward the water table. In a recharge area, the water table usually lies at some depth; in a discharge area, it is usually at or very near the surface. For the shaded cell in Figure 6.1, region is the recharge area and region is the discharge area. The line that separates recharge areas from discharge areas is called the hinge line. For the shaded cell, its intersection with the plane of the section is at point E.
The utilization of steady-state flow nets for the interpretation of regional flow deserves some discussion. The approach is technically valid only in the somewhat unrealistic case where the water table maintains the same position throughout the entire year. In most actual cases, fluctuations in the water table introduce transient effects in the flow systems. However, if the fluctuations in the water table are small in comparison with the total vertical thickness of the system, and if the relative configuration of the water table remains the same throughout the cycle of fluctuations (i.e., the high points remain highest and the low points remain lowest), we are within our rights to replace the fluctuating system by a steady system with the water table fixed at its mean position. One should think of the steady system as a case of dynamic equilibrium in which the flux of water delivered to the water table through the unsaturated zone from the surface is just the necessary flux to maintain the water table in its equilibrium position at every point along its length at all times. These conditions are approximately satisfied in many hydrogeologic basins, and in this light, the examination of steady flow nets can be quite instructive. Where they are not satisfied, we must turn to the more complex analyses presented in Section 6.3 for transient regional groundwater flow.
Hubbert (1940) was the first to present a flow net of the type shown in Figure 6.1 in the context of regional flow. He presumably arrived at the flow net by graphical construction. Tóth (1962, 1963) was the first to carry this work forward mathematically. He recognized that the new system in the shaded cell ABCDA of Figure 6.1 could be determined from the solution to a boundary-value problem. The equation of flow is Laplace’s equation [Eq. (2.70)] and the boundary conditions invoke the water-table condition on AD and impermeable conditions on AB, BC, and CD. He used the separation-of-variables technique, similar to that outlined in Appendix III for a simpler case, to arrive at an analytical expression for the hydraulic head in the flow field. The analytical solutions when plotted and contoured provide the equipotential net, and flowlines can easily be added Appendix VII summarizes Tóth’s solutions. The analytical approach has three serious limitations:
- It is limited to homogeneous, isotropic systems, or very simple layered systems.
- It is limited to regions of flow that can be accurately approximated by a rectangle, that is, to water-table slopes, AD, that are very small.
- It is limited to water-table configurations that can be represented by simple algebraic functions. Tóth considered cases with an inclined water table of constant slope, and cases in which a sine curve was superimposed on the incline.
As pointed out by Freeze and Witherspoon (1966, 1967, 1968), all three of these limitations can be removed if numerical simulation, as described in Section 5.3, is used to generate the flow nets. In the following subsections we will look at several flow nets taken from Freeze and Witherspoon’s (1967) numerical results in order to examine the effect of topography and geology on the nature of steady-state regional flow patterns.
Figure 6.2 shows the flow nets for two vertical cross sections that are identical in depth and lateral extent. In both cases there is a major valley running perpendicular to the page at the left-hand side of the system, and an upland plateau to the right. In Figure 6.2(a) the upland water-table configuration, which is assumed to closely follow the topography, has a uniform gentle incline such as one might find on a lacustrine plain. Figure 6.2(b), on the other hand, has a hilly upland water-table configuration such as one might find in glacial terrain.
The uniform water table produces a single flow system. The hinge line lies on the valley wall of the major valley; the entire upland plateau is a recharge area. The hilly topography produces numerous subsystems within the major flow system. Water that enters the flow system in a given recharge area may be discharged in the nearest topographic low or it may be transmitted to the regional discharge area in the bottom of the major valley. Tóth (1963) has shown that as the depth to lateral extent of the entire system becomes smaller and as the amplitude of the hummocks becomes larger, the local systems are more likely to reach the basal boundary, creating a series of small independent cells such as those shown in Figure 6.1. Tóth (1963) suggests that on most flow nets and in most field areas, one can differentiate between local systems of groundwater flow, intermediate systems of groundwater flow, and regional systems of groundwater flow, as schematically illustrated in Figure 6.3. Where local relief is negligible, only regional systems develop. Where there is pronounced local relief, only local systems develop. These terms are not specific, but they provide a useful qualitative framework for discussion.
Figures 6.2 and 6.3 make it clear that even in basins underlain by homogeneous. isotropic geologic materials, topography can create complex systems of groundwater flow. The only immutable law is that highlands are recharge areas and lowlands are discharge areas. For most common topographic configurations, hinge lines lie closer to valley bottoms than to ridge lines. On an areal map, discharge areas commonly constitute only 5–30% of the surface area of a watershed.
Figure 6.4 shows a sampling of numerically simulated flow nets for heterogeneous systems. Comparison of Figures 6.4(a) and 6.2(a) shows the effect of the introduction of a layer at depth with a permeability 10 times that of the overlying beds. The lower formation is an aquifer with essentially horizontal flow that is being recharged from above. Note the effect of the tangent law at the geologic boundary.
If the hydraulic conductivity contrast is increased [Figure 6.4(b)], the vertical gradients in the overlying aquitard are increased and the horizontal gradients in the aquifer are decreased. The quantity of flow, which can be calculated from the flow net using the methods of Section 5.1, is increased. One result of the increased flow is a larger discharge area, made necessary by the need for the large flows in the aquifer to escape to the surface as the influence of the left-hand boundary is felt.
In hummocky terrain [Figure 6.4(c)] the presence of a basal aquifer creates a highway for flow that passes under the overlying local systems. The existence of a high-permeability conduit thus promotes the possibility of regional systems even in areas of pronounced local relief.
There is a particular importance to the position within the basin of buried lenticular bodies of high conductivity. The presence of a partial basal aquifer in the upstream half of the basin [Figure 6.4(d)] results in a discharge area that occurs in the middle of the uniform upland slope above the stratigraphic pinchout. Such a discharge area cannot occur under purely topographic control. If the partial basal aquifer occurs in the downstream half of the system, the central discharge area will not exist; in fact, recharge in that area will be concentrated.
In the complex topographic and geologic system shown in Figure 6.4(e), the two flowlines illustrate how the difference of just a few meters in the point of recharge can make the difference between recharge water entering a minor local system or a major regional system. Such situations have disturbing implications for the siting of waste disposal projects that may introduce contaminants into the subsurface flow regime.
Subsurface stratigraphy and the resulting subsurface variations in hydraulic conductivity can exist in an infinite variety. It should be clear from these few examples that geological heterogeneity can have a profound effect on regional groundwater flow. It can affect the interrelationship between local and regional systems, it can affect the surficial pattern of recharge and discharge areas, and it can affect the quantities of flow that are discharged through the systems. The dramatic effects shown on Figure 6.4 are the result of conductivity contrasts of 2 orders of magnitude or less. In aquifer-aquitard systems with greater contrasts, flow patterns become almost rectilinear, with horizontal flow in the aquifers and vertical flow across the aquitards.
Flowing wells (along with springs and geysers) symbolize the presence and mystery of subsurface water, and as such they have always evoked considerable public interest.
The classic explanation of flowing wells, first presented by Chamberlain (1885) and popularized by Meinzer (1923) in connection with the Dakota sandstone, proposed an outcrop-related geologic control. If, as shown in Figure 6.5(a), an aquifer outcrops in an upland and is recharged there, an equipotential net can develop whereby the hydraulic head in the aquifer downdip from the recharge area is higher than the surface elevation. A well tapping the aquifer at such a location, and open at the surface, will flow.
However, it is not necessary to have this geologic environment to get flowing wells, nor is it a particularly common control. The primary control on flowing wells is topography. As shown in Figure 6.5(b), a well in a discharge area that has an intake at some depth below the water table will tap a hydraulic head contour with a head value that lies above the land surface, even in homogeneous, isotropic terrain. If there were a horizontal aquifer at depth beneath the valley in Figure 6.5(b), it need not outcrop to give rise to flowing wells. A well tapping the aquifer in Figure 6.4(b) beneath the valley at the left of the diagram would flow.
Any hydrogeologic system that leads to hydraulic-head values in an aquifer that exceed the surface elevation will breed flowing wells. The importance of topographic control is reflected in the large numbers of flowing wells that occur in valleys of rather marked relief. The specific location of areas of flowing wells within topographically low basins and valleys is controlled by the subsurface stratigraphy.
The Dakota sandstone configuration of Figure 6.5(a) has also been overused as a model of the regional groundwater-recharge process. Aquifers that outcrop in uplands are not particularly ubiquitous. Recharge regimes such as those shown in Figures 6.4(c), 6.4(d), and 6.7(b) are much more common.
Meyboom (1966a) and Tóth (1966) have shown by means of their work in the Canadian prairies that it is possible to map recharge areas and discharge areas on the basis of field observation. There are five basic types of indicators: (1) topography, (2) piezometric patterns, (3) hydrochemical trends, (4) environmental isotopes, and (5) soil and land surface features.
The simplest indicator is the topography. Discharge areas are topographically low and recharge areas are topographically high. The most direct indicator is piezometric measurement. If it were possible to install piezometer nests at every point in question, mapping would be automatic. The nests would show an upward-flow component in discharge areas and a downward-flow component in recharge areas. Such a course is clearly uneconomical, and in any case comparable information can often be gleaned from the available water-level data on existing wells. A well is not a true piezometer because it is usually open all along its length rather than at one point, but in many geologic environments, especially those where a single aquifer is being tapped, static water-level data from wells can be used as an indicator of potentiometric conditions. If there are many wells of various depths in a single topographic region, a plot of well depth versus depth to static water level can be instructive. Figure 6.6 defines the fields on such a plot where the scatter of points would be expected to fall in recharge areas and discharge areas.
Geochemical interpretation requires a large number of chemical analyses carried out on water samples taken from a representative set of wells and piezometers in an area. Groundwater as it moves through a flow system undergoes a mechanical evolution that will be discussed in Chapter 7. It is sufficient here to note the general observation that salinity (as measured by total dissolved solids) generally increases along the flow path. Water from recharge areas is usually relatively fresh; water from discharge areas is often relatively saline.
Information on groundwater flow systems is also obtained from analyses of well or piezometer samples for the environmental isotopes 2H, 3H, 18O, and 14C. The nature of these isotopes is described in Section 3.8. Tritium (3H) is used to identify water that has entered the groundwater zone more recently than 1953, when weapons testing in the atmosphere was initiated (Figure 3.11). The distribution of 3H in the groundwater flow system can be used to outline the subsurface zone that is occupied by post-l953 water. In that the tritiated zone extends into the flow system from the recharge area, this approach provides a basis for estimating regional values of the average linear velocity of groundwater flow near the recharge area. Peak concentrations of 3H in the groundwater can sometimes be related to peaks in the long-term record of 3H concentration in rain and snow.
The distribution of 14C can be used to distinguish zones in which old water occurs (Section 3.8). This approach is commonly used in studies of regional flow in large aquifers. 14C is used in favorable circumstances to identify zones of water in the age range several thousand years to a few tens of thousands of years. Case histories of 14C studies of regional flow in aquifers are described by Pearson and White (1967) and Fritz et al. (1974). Hydrochemical methods of interpretation of 14C data are described In Section 7.6.
Particularly in arid and semiarid climates, it is often possible to map discharge areas by direct field observation of springs and seeps and other discharge phenomena, collectively labeled groundwater outcrops by Meyboom (1966a). If the groundwater is highly saline, “outcrops” may take the form of saline soils, playas, salinas, or salt precipitates. In many cases, vegetation can provide a significant clue. In discharge areas, the vegetative suite often includes salt-tolerant, water-Ioving plants such as willow, cottonwood, mesquite, saltgrass, and greasewood. Most of those plants are phreatophytes. They can live with their roots below the water table and they extract their moisture requirements directly from the saturated zone. Phreatophytes have been studied in the southwestern United States by Meinzer (1927) and Robinson (1958, 1964) and in the Canadian prairies by Meyboom (1964, 1967). In humid climates, saline and vegetative groundwater outcrops are less evident, and field mapping must rely on springs and piezometric evidence.
As an example of an actual system, consider the flow system near Assiniboia, Saskatchewan (Freeze, 1969a). Figure 6.7(a) shows the topography of the region and the field evidence of groundwater discharge, together with a contoured plot of the hydraulic head values in the Eastern sand member, based on available well records. The stratigraphic position of the Eastern sand is shown along section in Figure 6.7(b). Meyboom (1966a) refers to this hydrogeologic environment, which is rather common in the Great Plains region of North America, as the prairie profile.
The steady-state flow-net approach to the analysis of regional groundwater flow has now been applied in many parts of the world in a wide variety of hydrogeological environments. The approach has generally been applied in drainage basins of small to moderate size, but it has also been utilized on a much larger scale by Hitchon (1969a, b). His analysis of fluid flow in the western Canadian sedimentary basin considered systems that extend from the Rockies to the Canadian Shield. The analysis was carried out to shed new light on the nature of petroleum migration and accumulation. It is discussed more fully in Chapter 11.
6.2 Steady-State Hydrologic Budgets
Steady-state flow nets of regional groundwater flow, whether they are developed on the basis of piezometric measurements and field observations or by mathematical or analog simulation, can be interpreted quantitatively to provide information that is of value in the determination of a hydrologic budget for a watershed.
Figure 6.8 shows a quantitative flow net for a two-dimensional, vertical cross section through a heterogeneous groundwater basin. This particular water-table configuration and set of geologic conditions gives rise to two separate flow systems: a local how system that is shallow but of large lateral extent (subsystem B), and a larger regional system (subsystem A). The local system is superimposed on the regional system in a way that could hardly have been anticipated by means other than a carefully constructed flow net. With the methods of Section 5.1, we can easily calculate the discharge through each flow system. For s = 6000 m, the total relief is 100 m, and since there are 50 increments of potential, Δh = 2 m. Assuming hydraulic conductivities of 10–4 and 10–5 m/s, the discharge through each stream-tube is 2.0 × 10–4 m3/s (per meter of thickness of the flow system perpendicular to the diagram). Counting the flow channels in the two subsystems leads to the values: QA = 2.8 × 10–3 m3/s, QB = 2.0 × 10–4 m3/s. Quantities calculated in this way represent the regional discharge through an undeveloped basin under natural conditions. As we shall see in Section 8.10, the development of groundwater resources by means of wells leads to new regional systems that may allow total basin yields much greater than the virgin flow rates.
It is also possible to calculate the rate of recharge or discharge at the water table at any point along its length. If the hydraulic conductivities at each point are known and the hydraulic gradient is read off the flow net, Darcy’s law can be invoked directly. If the recharge and discharge rates are plotted above the flow net as in figure 6.8, the smooth line that joins the points is known as a recharge-discharge profile. It identities concentrations of recharge and discharge that would be difficult to predict without the use of a quantitative flow net. The crosshatched area above the horizontal zero line in the recharge-discharge profile represents the total groundwater recharge; the crosshatched area below the line represents the total groundwater discharge. For steady flow the two must equal each other.
The three-dimensional equivalent of a recharge-discharge profile is a contoured map of a drainage basin showing the areal distribution of the rates of recharge and discharge. The preparation of such a map in the field would require measurements of the saturated hydraulic conductivity of the near-surface geologic formations, and measurements or estimates of the hydraulic gradient at the water table.
There is one aspect of the arguments presented in this section that leads into a vicious circle. We have noted that the existing water-table configurations, which control the nature of the groundwater flow patterns, will influence rates of groundwater recharge. But it is also true that the patterns and amounts of recharge will control to a certain degree the configuration of the water table. Thus far we have assumed a fixed position of the water table and developed the recharge and discharge patterns. In reality, both the water-table configurations and the recharge patterns are largely controlled by the spatial and temporal patterns of precipitation and evapotranspiration at the ground surface. In the analyses of Sections 6.3 through 6.5, we will look at the saturated-unsaturated interactions that control the response of the water table under various climatic conditions.
The recharge-discharge regime has important interrelationships with the other components of the hydrologic cycle. For example, in Figure 6.8 the entire regional flow from subsystem A discharges into the major valley at the left of the diagram. For any given set of topographic and hydrogeologic parameters we can calculate the average rate of discharge over the discharge area in, say, cm/day. In humid areas, this rate of upward-rising groundwater would be sufficient to keep water tables high while satisfying the needs of evapotranspiration, and still provide a base-flow component to a stream flowing perpendicular to the cross section. If such a stream had a tributary flowing across basin A from right to left, parallel to the cross section of Figure 6.8, one would expect the stream to be influent (losing water to the subsurface system) as it traverses the recharge area, and effluent (gaining water from the subsurface system) as it crosses the discharge area.
Quantification of these concepts requires the introduction of a hydrologic budget equation, or water balance, that describes the hydrologic regime in a watershed. If we limit ourselves to watersheds in which the surface-water divides and groundwater divides coincide, and for which there are no external inflows or outflows of groundwater, the water-balance equation for an annual period would take the form
P = Q + E + ΔSS + ΔSG (6.1)
where P is the precipitation, Q the runoff, E the evapotranspiration, ΔSS the change in storage of the surface-water reservoir, and ΔSG the change in storage of the groundwater reservoir (both saturated and unsaturated) during the annual period.
If we average over many years of record, it can be assumed that ΔSS = ΔSG = 0, and Eq. (6.1) becomes
P = Q + E (6.2)
where P is the average annual precipitation, Q the average annual runoff, and E the average annual evapotranspiration. The values of Q and E are usually reported in centimeters over the drainage basin so that their units in Eq. (6.2) are consistent with those for P. For example, in Figure 6.9(a), if the average annual precipitation, P, over the drainage basin is 70 cm/yr and the average annual evapotranspiration, E, is 45 cm/yr, the average annual runoff, Q, as measured in the stream at the outlet of the watershed but expressed as the equivalent number of centimeters of water over the drainage basin, would be 25 cm/yr.
Let us consider an idealization of the watershed shown in Figure 6.9(a), wherein most of the watershed comprises a recharge area, and the discharge area is limited to a very small area adjacent to the main stream. The groundwater flow net shown in Figure 6.8 might well be along the section X-X’. We can now write two hydrologic-budget equations, one for the recharge area and one for the discharge area.
In the recharge area [Figure 6.9(b)],
where QS is the surface-water component of average annual runoff, R the average annual groundwater recharge, and ER the average annual evapotranspiration from the recharge area.
In the discharge area [Figure 6.9(b)],
where D is the average annual groundwater discharge (and equal to R) and ED the average annual evapotranspiration from the discharge area. For a discharge area that constitutes a very small percentage of the basin area, P need not appear in Eq. (6.4).
If we set
Eq. (6.4) becomes
where QG is the groundwater component of average annual runoff (or average annual baseflow). Equation (6.5) reflects the earlier statement that groundwater discharge in a valley goes to satisfy both evapotranspiration demands and the subsurface component of streamflow. Equation (6.6) suggests that it might be possible to separate streamflow hydrographs into their surface-water and groundwater components; further consideration of this point is deferred until Section 6.6.
The application of the steady-state hydrologic-budget equations provides only a crude approximation of the hydrologic regime in a watershed. In the first place it is a lumped-parameter approach (rather than a distributed-parameter approach), which does not take into account the areal variations in P, E, R, and D. On an average annual basis, in a small watershed, areal variations in P and E may not be large, but we are aware, on the basis of Figure 6.8, that real variations in R and D can be significant. Second, the average annual approach hides the importance of time-dependent effects. In many cases, the groundwater regime is approximated quite closely by a steady-state regime, but P, E, and Q are strongly time-dependent.
The foregoing discussion of steady-state hydrologic budgets is instructive in that it clarifies many of the interactions between groundwater flow and the other components of the hydrologic cycle. The application of Eqs. (6.2), (6.3), and (6.4) in practice, however, is fraught with problems. One needs several years of records of precipitation, P, and stream runoff, Q, at several sites. In principle, the groundwater components, R and D, can be determined by flow-net analysis, but in practice, the uncertainty surrounding hydraulic conductivity values in heterogeneous groundwater basins leads to a wide range of feasible R and D values. The evapotranspiration parameters, ER and ED, must be estimated on the basis of methods of questionable accuracy.
Of all these questions, it is the evapotranspiration estimates that pose the greatest problem. The most widely used methods of calculation utilize the concept of potential evapotranspiration (PE), which is defined as the amount of water that would be removed from the land surface by evaporation and transpiration processes if sufficient water were available in the soil to meet the demand. In a discharge area where upward-rising groundwater provides a continuous moisture supply, actual evapotranspiration (AE) may closely approach potential evapotranspiration. In a recharge area, actual evapotranspiration is always considerably less than potential. Potential evapotranspiration is dependent on the evaporative capacity of the atmosphere. It is a theoretical calculation based on meterological data. AE is the proportion of PE that is actually evapotranspired under the existing soil moisture supply. It is dependent on the unsaturated moisture storage properties of the soil. It is also affected by vegetative factors such as plant type and stage of growth. The most common methods of calculating potential evapotranspiration those of Blaney and Criddle (1950), Thornthwaite (1948), Penman (1948), and Van Bavel (1966). The first two of these are based on empirical correlations between evapotranspiration and climatic factors. The last two are energy-budget approaches that have better physical foundations but require more meterological data. Pelton et al. (1960) and Gray et al. (1970) discuss the relative merits of the various techniques. The conversion of PE rates to AE rates in a recharge area is usually carried out with a soil-moisture budget approach. The Holmes and Robertson (1959) technique has been widely applied in the prairie environment.
For the specific case of phreatophytic evapotranspiration from a discharge area with a shallow water table, direct measurements of water-level fluctuations, as outlined in Section 6.8, can be used to calculate the actual evapotranspiration.
For examples of hydrologic-budget studies on small watersheds, in which special attention is paid to the groundwater component, the reader is directed to the reports of Schicht and Walton (1961), Rasmussen and Andreasen (1959) and Freeze (1967).
6.3 Transient Regional Groundwater Flow
Transient effects in groundwater flow systems are the result of time-dependent changes in the inflows and outflows at the ground surface. Precipitation rates evapotranspiration rates, and snowmelt events are strongly time-dependent. Their transient influence is felt most strongly near the surface in the unsaturated zone, so any analysis of the transient behavior of natural groundwater flow must include both saturated and unsaturated zones.
As with steady-state regional flow, the main features of transient regional flow are most easily illustrated with the aid of numerical simulations carried out in hypothetical groundwater basins. Freeze (1971a), building on the earlier work of Rubin (1968), Hornberger et al. (1969), and Verma and Brutsaert (1970), described a mathematical model for three-dimensional, transient, saturated-unsaturated flow in a groundwater basin. His equation of flow couples the unsaturated flow equation [Eq. (2.80)] and the saturated flow equation [Eq. (2.74)] into an integrated form that allows treatment of the complete subsurface regime. The numerical solutions were obtained with a finite-difference technique known as successive over relaxation. The model allows any generalized region shape and any configuration of time-variant boundary conditions. Here, we will look at the transient response in a two-dimensional cross section to a snowmelt-type infiltration event.
The region of flow is shown in Figure 6.10(a) (at a 2:1 vertical exaggeration). The boundaries comprise a stream AB at constant hydraulic head, an impermeable basement AFED, and the ground surface BCD. The region contains a homogeneous and isotropic soil whose unsaturated characteristic curves are those of Figure 2.13.
As we have seen in Sections 2.6 and 5.4, saturated-unsaturated flow conditions can be presented in three ways: as a pressure-head field, as a moisture-content field, and as a total hydraulic-head field. From the first we can locate the position of the water table, and from the last we can make quantitative flow calculations. Figure 6.10(a), (b), and (c) shows these three fields at time t = 0 for initial conditions of steady-state flow resulting from the imposition of a constant hydraulic head along CD. The initial conditions feature a deep, nearly flat water table and very dry surface moisture conditions. At all times t > 0, a surface flux equivalent to 0.09 K0 (where K0 is the saturated hydraulic conductivity of the soil) is allowed to enter the flow system on the upper boundary. As shown on Figure 6.10(d), this rate of inflow creates a water-table rise that begins after 100 h and approaches the surface after 400 h. Figure 6.10(e) and (f) shows the moisture-content and total hydraulic-head fields at t = 410 h.
Figure 6.11 shows the effect on the flow system of the introduction of a heterogeneous geological configuration. The unstippled zone has the same soil properties as those for the homogeneous case of Figure 6.10, but a low-permeability clay layer has been inserted near the surface and a high-permeability basal aquifer at depth. The permeability and porosity relationships are noted in Figure 6.11(a). Figure 6.11(b) illustrates the transient response of the water table to the same surface inflow conditions as those of Figure 6.10. Figure 6.11(c) shows the total hydraulic head pattern at t = 460 h. This set of diagrams serves to clarify the saturated-unsaturated mechanisms that are operative in the formation of a perched water table.
If the hydraulic-head field in a watershed can be determined at various times by field measurement or mathematical simulation, it becomes possible to make a direct calculation of the amount of water discharging from the system as a function of time. If the discharge area is limited to a stream valley, the transient rate of groundwater discharge provides a measure of the baseflow hydrograph for the stream. Increased baseflow is the result of increased hydraulic gradients in the saturated zone near the stream, and, as the theoretical models show, this is itself a consequence of increased up-basin gradients created by a water-table rise. The time lag between a surface-infiltration event and an increase in stream baseflow is therefore directly related to the time required for an infiltration event to induce a widespread water-table rise. Figure 6.12 is a schematic illustration of the type of baseflow hydrograph that results from a hydrologic event of sufficient magnitude to exert a basin-wide influence on the water table. Baseflow rates must lie between Dmaximum, the maximum possible baseflow, which would occur under conditions of a fully saturated basin, and Dmimimum, the minimum likely baseflow, which would occur under conditions of the lowest recorded water-table configuration.
Quantitative calculations can also be carried out at the inflow end of the system to examine the interrelationship between infiltration and groundwater recharge. The concepts are clearest however, when one works with the one-dimensional system outlined in the following section.
6.4 Infiltration and Groundwater Recharge
In Section 6.1, we defined the terms recharge area and discharge area; in Section 6.2, we first calculated recharge and discharge rates. Let us formalize these concepts with the following definitions for the processes of recharge and discharge.
Groundwater recharge can be defined as the entry into the saturated zone of water made available at the water-table surface, together with the associated flow away from the water table within the saturated zone.
Groundwater discharge can be defined as the removal of water from the saturated zone across the water-table surface, together with the associated flow toward the water table within the saturated zone.
It should be clear from the previous section that these two saturated processes are intimately interrelated to a pair of parallel processes in the unsaturated zone. Let us define the process of infiltration as the entry into the soil of water made available at the ground surface, together with the associated flow away from the ground surface within the unsaturated zone.
In a similar fashion. we will define exfiltration as the removal of water from the soil across the ground surface, together with the associated flow toward the ground surface within the unsaturated zone. This term was coined by Philip (1957f), but it is not yet widely used. The process is often referred to as evaporation, but this leads to confusion as to whether the meteorological processes in the atmosphere are included.
The Theory of Infiltration
The process of infiltration has been widely studied by both hydrologists and soil physicists. In hydrology, Horton (1933) showed that rainfall, when it reaches the ground surface. infiltrates the surface soils at a rate that decreases with time. He pointed out that for any given soil there is a limiting curve that defines the maximum possible rates of infiltration versus time. For heavy rains, the actual infiltration will follow this limiting curve, which he called the curve of infiltration capacity of the soil. The capacity decreases with time after the onset of rainfall and ultimately reaches an approximately constant rate. The decline is caused mainly by the filling of the soil pores by water. Controlled tests carried out on various soil types by many hydrologists over the years have shown that the decline is more rapid and the final constant rate is lower for clay soils with fine pores than for open-textured sandy soils. If at any time during a rainfall event the rate of rainfall exceeds the infiltration capacity excess water will pond on the soil surface. It is this ponded water that is available for overland flow to surface streams.
The hydrologic concept of infiltration capacity is an empirical concept based on observations at the ground surface. A more physically based approach can be found in the soil physics literature, where infiltration is studied as an unsaturated subsurface flow process. Most analyses have considered a one-dimensional vertical flow system with an inflow boundary at the top. Bodman and Colman (1943) provided the early experimental analyses, and Philip (1957a, 1957b, 1957c, 1957d, 1957e, 1958a, 1958b), in his classic seven-part paper, utilized analytical solutions to the one-dimensional boundary-value problem to expose the basic physical principles on which later analyses rest. Almost all of the more recent theoretical treatments have employed a numerical approach to solve the one-dimensional system. This approach is the only one capable of adequately representing the complexities of real systems. Freeze (1969b) provides a review of the numerical infiltration literature in tabular form.
From a hydrologic point of view, the most important contributions are those of Rubin et al. (1963, 1964). Their work showed that Horton’s observed curves of infiltration versus time can be theoretically predicted, given the rainfall intensity, the initial soil-moisture conditions, and the set of unsaturated characteristic curves for the soil. If rainfall rates, infiltration rates, and hydraulic conductivities are all expressed in units of [L/T], Rubin and his coworkers showed that the final constant infiltration rate in the Horton curves is numerically equivalent to the saturated hydraulic conductivity of the soil. They also identified the necessary conditions for ponding as twofold: (1) the rainfall intensity must be greater than the saturated hydraulic conductivity, and (2) the rainfall duration must be greater than the time required for the soil to become saturated at the surface.
These concepts become clearer if we look at an actual example. Consider a one-dimensional vertical system (say, beneath point A in Figure 6.10) with its upper boundary at the ground surface and its lower boundary just below the water table. The equation of flow in this saturated-unsaturated system will be the one-dimensional form of Eq. (2.80):
where ψ (= h – z) is the pressure head, and K(ψ) and C(ψ) are the unsaturated functional relationships for hydraulic conductivity K and specific moisture capacity C. In the saturated zone below the water table (or more accurately, below the point where ψ = ψa, ψa being the air-entry pressure head), K(ψ) = K0 and C(ψ) = 0, where K0 is the saturated hydraulic conductivity of the soil.
Let us specify a rainfall rate R at the upper boundary. From Darcy’s law,
If the rate of groundwater recharge to the regional flow system is Q, then, by analogy to Eq. (6.9), the condition at the saturated base of the system is
The boundary-value problem defined by Eqs. (6.7), (6.9), and (6.10) was solved by Freeze (1969b) with a numerical finite-difference method that is briefly outlined in Appendix VIII. Figure 6.13 shows the results of a representative simulation of a hypothetical infiltration event. The three profiles show the time-dependent response of the moisture content, pressure head, and hydraulic head in the upper 100 cm of a soil with unsaturated hydrologic properties identical to those shown in Figure 2.13. The transient behavior occurs in response to a contant-intensity rainfall that feeds the soil surface at the rate
R = 0.13 cm/min. This rate is 5 times the saturated hydraulic conductivity of the soil, K0 = 0.026 cm/min. The initial conditions are shown by the t = 0 curves, and subsequent curves are labeled with the time in minutes.
The left-hand diagram shows how the moisture content increases down the profile with time. The surface becomes saturated after 12 min, and the soil pores in the entire profile are almost filled with water after 48 min.
The central diagram shows the pressure-head changes. The pressure-head curve for t = 12 min does not reach the ψ = 0 point, so the upper few centimeters of surface saturation, indicated by the moisture-content profile, must be “tension-saturated.” By the 24-min mark the pressure head at the ground surface has reached +10 cm, the indication being that a 10-cm-deep layer of water is ponded on the surface at this point in time. (In this simulation, the maximum allowable ponding depth had been set to 10 cm.) There is also and inverted water table 5 cm below the ground surface which propagates down the profile with time. The true water table, which is initially set at 95 cm depth, remains stationary through the first 36 min but then begins to rise in response to the infiltrating moisture from above.
The hydraulic head profiles near the surface on the right-hand diagram provide the hydraulic-gradient values that can be inserted in Darcy’s law to calculate the rate of infiltration at various times. The datum for the values appearing on the horizontal scale at the top was arbitrarily chosen as 125 cm below the ground surface.
Figure 6.14 shows the time-dependent infiltration rate at the ground surface for the constant-rainfall case shown in Figure 6.13. As predicted by Rubin and Steinhardt (1963), the infiltration rate is equal to the rainfall rate until the soil becomes saturated at the surface (and the 10-cm deep pond has been filled); then it decreases asymptotically toward a value equal to K0. During the early period, as the soil pores are filling up with water, the moisture contents, pressure heads, and hydraulic heads are increasing with time and the downward hydraulic gradient is decreasing. This decrease is balanced by an increase in the hydraulic conductivity values under the influence of the rising pressure heads. The decrease in infiltration rate occurs at the point when the combination of gradients and conductivities in the soil can no longer accept all the water supplied by the rainfall. The rainfall not absorbed by the ground as infiltration nor stored in the 10-cm-deep pond is available for land overflow.
A similar approach can be used to simulate cases with evaporation at the surface (R negative) or discharge at depth (Q negative), or to analyze redistribution patterns that occur between rainfall events.
The question of whether a given input and a given set of initial conditions and soil type will give rise to groundwater recharge is actually a question of whether this set of conditions will result in a water-table rise. The rise provides the source of replenishment that allows the prevailing rate of recharge to continue. The possibility of a water-table rise is greater for (1) low-intensity rainfalls of long duration rather than high-intensity rainfalls of short duration, (2) shallow water tables rather than deep, (3) low groundwater recharge rates rather than high, (4) net antecedent moisture conditions rather than dry, and (5) soils whose characteristic curves show high conductivity, low specific moisture capacity, or high moisture content over a considerable range of pressure-head values.
In some hydrogeological environments, cases of recharge-sustaining infiltration to the water table are isolated in time and space. In such cases, the types of hydrologic events that lead to recharge are best identified on the basis of field measurement. In the past this was often done on the basis of observation-well hydrographs of water-table fluctuations. However, as indicated in Section 6.8, there are a variety of phenomena that can lead to water-table fluctuations, and not all represent true groundwater recharge. The safest course is to supplement the observation-well records with measurements of hydraulic head both above and below the water table. Figure 6.15 shows a set of field instrumentation designed to this end. Figure 6.16 displays the soil-moisture and water-table response recorded at an instrumented site in east-central Saskatchewan during a dry period punctuated by a single heavy rainfall. The water-table rise is the result of direct infiltration from above.
At another site nearby, the same rainfall did not result in infiltration to the water table, despite the fact that the saturated hydraulic conductivity there was much higher than at the site shown in Figure 6.16. The characteristic curves of the sandy soil at the second site gave rise to a deep water table and very dry near surface soil-moisture conditions. As noted by Freeze and Banner (1970), estimates of the infiltration and recharge properties of a soil based only on knowledge of the saturated hydraulic conductivity of the soil and its textural classification can often be misleading. One should not map a sand or gravel plain as an effective recharge area without first investigating the water-table depth and the nature of the unsaturated functional relationships for the soil. Small differences in the hydrologic properties of similar field soils can account for large differences in their reaction to the same hydrologic event.
The mechanisms of infiltration and groundwater recharge are not always one-dimensional. In hilly areas certain portions of a groundwater recharge area may never receive direct infiltration to the water table. Rather, recharge may be concentrated in depressions where temporary ponds develop during storms or snowmelt periods. Lissey (1968) has referred to this type of recharge as depression-focused. Under such conditions, the water table still undergoes a basinwide rise. The rise is due to vertical infiltration beneath the points of recharge and subsequent horizontal flow toward the water-table depressions created between these points. Further discussion on the interactions between groundwater and ponds is withheld until Section 6.7.
6.5 Hillslope Hydrology and Streamflow Generation
The relationship between rainfall and runoff is at the very core of hydrology. In a scientific sense, there is a need to understand the mechanisms of watershed response. In an engineering sense, there is a need for better techniques for the prediction of runoff from rainfall. We know, of course, that the larger rivers are fed by smaller tributaries, and it is this network of small tributary streams that drains by far the largest percentage of the land surface. We will therefore focus on the ways in which water moves into small stream channels in upstream tributary drainage basins during and between rainfall events.
The path by which water reaches a stream depends upon such controls as climate, geology, topography, soils, vegetation, and land use. In various parts of the world, and even in various parts of the same watershed, different processes may generate streamflow, or the relative importance of the various processes may differ. Nevertheless, it has been recognized that there are essentially three processes that feed streams. As illustrated in Figure 6.17, these are overland flow, subsurface stormflow (or interflow), and groundwater flow. An insight into the nature of the subsurface flow regime is necessary for understanding the production of runoff by any of these three mechanisms.
The role of the regional groundwater flow system in delivering baseflow to a stream was covered in Sections 6.2 and 6.3. Although it may sometimes contribute to runoff during storms, its primary role is in sustaining streams during low-flow periods between rainfall and snowmelt events. We will focus our interest in this section on overland flow and subsurface stormflow.
The classic concept of streamflow generation by overland flow is due to Horton (1933). The dependence of overland flow on the infiltration regime in the unsaturated surface soils of a watershed has been discussed in the previous section. The concepts are summarized in Figure 6.14.
As originally presented, Horton’s theory implied that most rainfall events exceed infiltration capacities and that overland flow is common and areally widespread. Later workers recognized that the great heterogeneity in soil types at the ground surface over a watershed and the very irregular patterns of precipitation in both time and space create a very complex hydrologic response on the land surface. This led to the development of the partial-area-contribution concept (Betson, 1964; Ragan, 1968), wherein it is recognized that certain portions of the watershed regularly contribute overland flow to streams, whereas others seldom or never do. The conclusion of most recent field studies is that overland flow is a relatively rare occurrence in time and space, especially in humid, vegetated basins. Most overland flow hydrographs originate from small portions of the watershed that constitute no more than 10%, and often as little as 1–3%, of the basin area, and even on these restricted areas only 10–30% of the rainfalls cause overland flow.
Freeze (1972b) provided a heuristic argument based on the theory of infiltration and the ponding criteria of Rubin and Steinhardt (1963) to explain the paucity of overland-flow occurrences.
The second widely held concept of surface-runoff generation promotes subsurface stormflow as a primary source of runoff. Hewlett and Hibbert (1963) showed the feasibility of such flow experimentally, and Whipkey (1965) and Hewlett and Hibbert (1967) measured lateral inflows to streams from subsurface sources in the field. The prime requirement is a shallow soil horizon of high permeability at the surface. There is reason to suppose that such surface layers are quite common in the form of the A soil horizon, or as agriculturally tilled soils or forest litter.
On the basis of simulations with a mathematical model of transient, saturated-unsaturated, subsurface flow in a two-dimensional, hillslope cross section, Freeze (1972b) concluded that subsurface stormflow can become a quantitatively significant runoff component only on convex hillslopes that feed deeply incised channels, and then only when the permeabilities of the soils on the hillslope are in the very highest bracket of the feasible range. Figure 6.18 shows three simulated hydrographs for the hillslope cross section shown in the inset.
The three cases each differ by an order of magnitude in the saturated hydraulic conductivity, K0, of the hillslope soil. The line below the stippled regions represents the subsurface stormflow contribution. In each case, one result of the saturated-unsaturated process in the hillslope is a rising water table near the valley (as indicated for t = 5 h on the inset). Overland flow from direct precipitation on the saturated wetland created on the streambank by the rising water tables is shown by the stippled portions of the hydrographs. Only curves A and B show a dominance of the storm hydrograph by subsurface stormflow, and the K0 values for these curves are in the uppermost range of reported held measurement. On concave slopes the saturated valley wetlands become larger more quickly and overland flow from direct precipitation on these areas usually exceeds subsurface stormflow, even where hillslope soils are highly permeable.
In the Sleepers River experimental watershed in Vermont (Figure 6.19), Dunne and Black (1970a, b), working with an integrated set of surface and subsurface instrumentation, including an interceptor trench [Figure 6.19(b)], were able to measure simultaneous hydrographs of each of the three component outflows from the hillslope to the stream. The example shown in Figure 6.19(c) displays the preponderance of overland flow that was a recurring feature of measurements in the Sleepers River watershed. Auxiliary instrumentation showed that the contributing areas, as in case C in Figure 6.18, were limited to topographically low wetlands created by rising water tables adjacent to the stream channel.
One feature of the streamflow-generating mechanism uncovered at the Sleepers River watershed has been widely reported (Hewlett and Nutter, 1970) in many other watersheds in humid climates. We refer to the expansion and contraction of wetlands during and following storms under the influence of the subsurface flow system. The resulting variation with time in the size of contributing areas is often referred to as the variable-source-area concept. It differs from the partial-area concept in two ways. First, partial areas are thought of as being more-or-less fixed in location, whereas variable areas expand and contract. Second, partial areas feed water to streams by means of Hortonian overland flow, that is, by water that ponds on the surface due to saturation of the soils at the surface from above, whereas variable areas are created when surface saturation occurs from below. In the Sleepers River watershed, the majority of the overland flow that arrived at the stream from the variable source areas was created by direct precipitation on the wetlands. In many forested watersheds (Hewlett and Nutter, 1970), a significant proportion of the water arising from variable source areas arrives there by means of subsurface stormflow. Table 6.1 provides a summary of the various storm runoff processes in relation to their major controls.
Source: Dunne, 1978.
In recent years there has been rapid growth in the development of physically based hydrologic prediction models that couple surface and subsurface flow. Smith and Woolhiser (1971) have produced a model for the simulation of overland flow on an infiltrating hillslope, and Freeze (1972a) has produced a model that couples saturated-unsaturated flow and streamflow. Stephenson and Freeze (1974) report on the use of the latter model to complement a field study of snowmelt runoff in a small upstream source area in the Reynolds Creek experimental watershed in Idaho.
Chemical and Isotopic Indicators
There are three main approaches that can be used in studies of the processes of streamflow generation during storm runoff: (1) hydrometric monitoring using instruments such as current meters, rain gages, observation wells, and tensiometers; (2) mathematical simulations; and (3) monitoring of dissolved constituents and environmental isotopes, such as 2H, 3H, and 18O. Information obtained from the first two methods served as the basis for the discussion presented above. We will now focus on the hydrochemical and isotope approach.
The chemical mass-balance equation of dissolved constituents in streamflow at a particular sampling location at the stream at a specified time can be expressed
where C is the concentration in the stream water of the constituent under consideration, such as Cl–, SO42–, or , and Q is the stream discharge [L3/T]. Qp, Qo, Qs, and Qg, represent the contributions to the streamflow from direct rainfall on the stream, overland flow, subsurface stormflow, and groundwater flow, respectively Cp, Co, Cs, and Cg, are the concentrations of the chemical constituent in these streamflow components. The mass-balance equation for streamflow at the same location is
Values for Q are obtained by measuring the streamflow. C is obtained by chemical analysis of samples from the stream at the location where Q is measured. For narrow streams in headwater basins, Qp is often negligible relative to Q. This leaves two equations with six unknown quantities, Co, Cs, and Cg, Qo, Qs, and Qg. A pragmatic approach at this point is to lump Qo and Qs together as a component referred to as direct runoff (Qd), which represents the component of rainfall that moves rapidly across or through the ground into the stream. Cd is defined as the representative concentration in this runoff water. Substitution of these terms in Eqs. (6.11) and (6.12) and combining these equations yields
Values of Cg are normally obtained by sampling shallow wells or piezometers near the stream or by sampling the stream baseflow prior to, or after, the storm. The second method is appropriate if the stream is fed only by shallow groundwater during baseflow periods. Values of Cd are obtained by sampling surface drainage or soil-zone seepage near the stream during the storm-runoff period. If analyses of these samples yield no excessive variation in space and time, the choice of a representative or average concentration is not unduly subjective. In sedimentary terrain, Cd is generally small relative to Cg because the groundwater has traveled deeper and has a much longer residence time. Substitution of the values of Cd and Cg along with the stream-water parameters, C and Q into Eq. (6.13) yields a value of Qg, the groundwater component of the streamflow. If C and Q are measured at various times during the storm-runoff period, the variation of Qg can be computed, as shown schematically in Figure 6.20.
Pinder and Jones (1969) used variations in Na2+, Ca2+, Mg2+, Cl–, SO42–, and in their study of storm-runoff components in small headwater basins in sedimentary terrain in Nova Scotia. In a similar investigation in Manitoba, Newbury et al. (1969) found SO42– and electrical conductance to be the best indicators for identification of the groundwater component in that area. In these and many other investigations using the hydrochemical method, it is commonly concluded that the groundwater-derived component of streamflow during peak runoff is appreciable. Pinder and Jones, for example, reported values in the range 32–42%.
One of the main limitations in the hydrochemical method is that the chemical concentrations used for the shallow groundwater and to represent the direct runoff are lumped parameters that may not adequately represent the water that actually contributes to the stream during the storm. The chemistry of shallow groundwater obtained from wells near streams is commonly quite variable spatially. Direct runoff is a very ephemeral entity that may vary considerably in concentration in time and space.
To avoid some of the main uncertainties inherent in the hydrochemical method, the naturally occurring isotopes 18O, 2H, and 3H can be used as indicators of the groundwater component of streamflow during periods of storm runoff. Fritz et al. (1976) utilized 18O, noting that its concentration is generally very uniform in shallow groundwater and baseflow. Although the mean annual values of 18O in rain at any given location have little variation, the 18O content of rain varies considerably from storm to storm and even during individual rainfall events. The 18O method is suited for the type of rainfall event in which the 18O content of the rain is relatively constant and is much different from the shallow groundwater or baseflow. In this situation the 18O of the rain is a diagnostic tracer of the rainwater that falls on the basin during the storm. From the mass-balance considerations used for Eq. (6.13), the following relation is obtained:
where ∂18O denotes the 18O content in per mille relative to the SMOW standard (Section 3.8) and the subscripts w, g, and R, indicate stream water, shallow groundwater, and runoff water derived from the rainfall (Qw = Qg + QR). This relation provides for separation of the rain-derived component from the component of the streamflow represented by water that was in storage in the groundwater zone prior to the rainfall event. Fritz et al. (1976), Sklash et al. (1976), and Sklash (1978) applied this method in studies of streamflow generation in small headwater basins in several types of hydrogeologic settings. They found that even during peak runoff periods, the groundwater component of streamflow is considerable, often as much at half to two-thirds of the total streamflow. The resolution of the apparent contradictions between the mechanisms of streamflow generation suggested by the hydrochemical and isotopic approaches and those suggested by hydrometric measurements remains a subject of active research.
6.6 Baseflow Recession and Bank Storage
It should now be clear that streamflow hydrographs reflect two very different types of contributions from the watershed. The peaks, which are delivered to the stream by overland flow and subsurface stormflow, and sometimes by groundwater flow, are the result of a fast response to short-term changes in the subsurface flow systems in hillslopes adjacent to channels. The baseflow, which is delivered to the stream by deeper groundwater flow, is the result of a slow response to long-term changes in the regional groundwater flow systems.
It is natural to inquire whether these two components can be separated on the basis of a direct examination of the hydrograph alone without recourse to chemical data. Surface-water hydrologists have put considerable effort into the development of such techniques of hydrograph separation as a means of improving streamflow-prediction models. Groundwater hydrologists are interested in the indirect evidence that a separation might provide about the nature of the groundwater regime in a watershed. The approach has not led to unqualified success, but the success that has been achieved has been based on the concept of the base-flow recession curve.
Consider the stream hydrograph shown in Figure 6.21.
Flow varies through the year from 1 m3/s to over 100 m3/s. The smooth line is the baseflow curve. It reflects the seasonally transient groundwater contributions. The flashy flows above the line represent the fast-response, storm-runoff contributions. If the stream discharge is plotted on a logarithmic scale, as it is in Figure 6.21, the recession portion of the baseflow curve very often takes the form of a straight line or a series of straight lines, such as AB and CD. The equation that describes a straight-line recession on a semilogarithmic plot is
where Q0 is the baseflow at time t = 0 and Q is the baseflow at a later time, t.
The general validity of this equation can be confirmed on theoretical grounds. As first shown by Boussinesq (1904), if one solves the boundary-value problem that represents free-surface flow to a stream in an unconfined aquifer under Dupuit-Forchheimer assumptions (Section 5.5), the analytical expression for the outflow from the system takes the form of Eq. (6.15). Singh (1969) has produced sets of theoretical baseflow curves based on analytical solutions to this type of boundary-value problem. Hall (1968) provides a complete historical review of baseflow recession.
In Figure 6.21 the rising portions of the baseflow hydrograph must fit within the conceptual framework outlined in connection with Figure 6.12. Many authors, among them Farvolden (1963), Meyboom (1961), and Ineson and Downing (1964), have utilized baseflow-recession curves to reach interpretive conclusions regarding the hydrogeology of watersheds.
In the upper reaches of a watershed, subsurface contributions to streamflow aid in the buildup of the flood wave in a stream. In the lower reaches, a different type of groundwater-streamflow interaction, known as bank storage, often moderates the flood wave. As shown in Figure 6.22(a), if a large permanent stream undergoes an increase in river stage under the influence of an arriving flood wave, flow may be induced into the stream banks. As the stage declines, the flow is reversed. Figure 6.22(b), (c) and (d) shows the effect of such bank storage on the stream hydrograph, on the bank storage volume, and on the associated rates of inflow and outflow.
Bank-storage effects can cause interpretive difficulties in connection with hydrograph separation. In Figure 6.22(e) the solid line might represent the actual subsurface transfer at a stream bank, including the bank-storage effects. The groundwater inflow from the regional system, which might well be the quantity desired, would be as shown by the dashed line.
The concept of bank storage was clearly outlined by Todd (1955). Cooper and Rorabaugh (1963) provide a quantitative analysis based on an analytical solution to the boundary-value problem representing groundwater flow in an unconfined aquifer adjacent to a fluctuating stream. The numerical solutions of Pinder and Sauer (1971) carry the quantitative analysis a step further by considering the pair of boundary-value problems representing both groundwater flow in the bank and open-channel flow in the stream. The two systems are coupled through the inflow and outflow terms that represent the passage of water into and out of bank storage.
6.7 Groundwater-Lake Interactions
Stephenson (1971) has shown that the hydrologic regime of a lake is strongly influenced by the regional groundwater flow system in which it sits. Large, permanent lakes are almost always discharge areas for regional groundwater systems. The rates of groundwater inflow are controlled by watershed topography and the hydrogeologic environment as outlined in Section 6.1. Small, permanent lakes in the upland portions of watersheds are usually discharge areas for local flow systems, but there are geologic configurations that can cause such lakes to become sites of depression-focused recharge.
Winter (1976), on the basis of numerical simulations of steady-state lake and groundwater flow systems, showed that where water-table elevations are higher than lake levels on all sides, a necessary condition for the creation of a recharge lake is the presence of a high-permeability formation at depth. His simulations also show that if a water-table mound exists between two lakes, there are very few geologic settings that lead to groundwater movement from one lake to the other.
A recharging lake can leak through part or all of its bed. McBride end Pfannkuch (1975) show, on the basis of theoretical simulation, that for cases where the width of a lake is greater than the thickness of associated high-permeability surficial deposits on which it sits, groundwater seepage into or out of a lake tends to be concentrated near the shore. Lee (1977) has documented this situation by a field study using seepage meters installed in a lake bottom. The design and use of simple, easy-to-use devices for monitoring seepage through lake beds in nearshore zones is described by Lee and Cherry (1978).
In many cases. a steady-state analysis of groundwater-lake interaction is not sufficient. In the hummocky, glaciated terrain of central-western North America, for example, temporary ponds created by runoff from spring snowmelt lead to transient interactions. Meyboom (1966b) made field measurements of transient groundwater flow in the vicinity of a prairie pothole. Figure 6.23 shows the generalized sequence of flow conditions he uncovered in such an environment. The upper diagram shows the normal fall and winter conditions of uniform recharge to a regional system. The middle diagram illustrated the buildup of groundwater mounds beneath the temporary ponds. The third diagram shows the water-table relief during the summer under the influence of phreatophytic groundwater consumption by willows that ring the pond. Meyboom‘s careful water balance on the willow ring showed that the overall effect of the transient seasonal behavior was a net recharge to the regional groundwater system.
The measurement of water-level fluctuations in piezometers and observation wells is an important facet of many groundwater studies. We have seen in Section 6.4, for example, how a water-table hydrograph measured during an infiltration event can be used to analyze the occurrence of groundwater recharge. We will discover in Chapter 8 the importance of detecting long-term regional declines in water levels due to aquifer exploitation. Water-level monitoring is an essential component of field studies associated with the analysis of artificial recharge (Section 8.11), bank storage (Section 6.6), and geotechnical drainage (Chapter 10).
Water-level fluctuations can result from a wide variety of hydrologic phenomena, some natural and some induced by man. In many cases, there may be more than one mechanism operating simultaneously and if measurements are to be correctly interpreted, it is important that we understand the various phenomena. Table 6.2 provides a summary of these mechanisms, classified according to whether they are natural or man-induced, whether they produce fluctuations in confined or unconfined aquifers, and whether they are short-lived, diurnal, seasonal, or long-term in their time frame. It is also noted that some of the mechanisms operate under climatic influence, while others do not. Those checked in the “confine” column produce fluctuations in hydraulic head at depth, and it should be recognized that such fluctuations must be measured with a true piezometer, open only at its intake. Those checked in the “unconfined” column produce fluctuations in water-table elevation near the surface. This type of fluctuation can be measured either with a true piezometer or with a shallow observation well open along its length.
|Groundwater recharge (infiltration to the water table)||✔||✔||✔||✔|
|Air entrapment during groundwater recharge||✔||✔||✔||✔|
|Evapotranspiration and phreatophytic consumption||✔||✔||✔||✔|
|Bank-storage effects near streams||✔||✔||✔||✔|
|Tidal effects near oceans||✔||✔||✔||✔|
|Atmospheric pressure effects||✔||✔||✔||✔||✔||✔|
|External loading of confined aquifers||✔||✔||✔|
|Artificial recharge; leakage from ponds, lagoons, and landfills||✔||✔||✔|
|Agricultural irrigation and drainage||✔||✔||✔||✔|
|Geotechnical drainage of open pit mines, slopes, tunnels, etc.||✔||✔||✔|
Several of the natural phenomena listed in Table 6.2 have been discussed in some detail in earlier sections. Many of the man-induced phenomena will come into focus in later chapters. In the following paragraphs we will zero in on four types of fluctuations: those caused by phreatophytic consumption in a discharge area, those caused by air entrapment during groundwater recharge, those caused by changes in atmospheric pressure, and those caused by external loading of elastic confined aquifers.
Evaporation and Phreatophytic Consumption
In a discharge area it is often possible to make direct measurements of evapotranspiration on the basis of water-table fluctuations in shallow observation wells. Figure 6.24 (after Meyboom, 1967) displays the diurnal fluctuations observed in the water-table record in a river valley in western Canada.
The drawdowns take place during the day as a result of phreatophytic consumption (in this case by Manitoba maple); the recoveries take place during the night when the plant stomata are closed. White (1932) suggested an equation for calculating evapotranspiration on the basis of such records. The quantity of groundwater withdrawn by evapotranspiration during a 24-h period is
where E is the actual daily evapotranspiration ([L]/day), Sy the specific yield of the soil (% by volume), r the hourly rate of groundwater inflow ([L]/h), and s the net rise or fall of the water table during the 24-h period [L]. The r and s values are graphically illustrated in Figure 6.24. The value of r, which must represent the average rate of groundwater inflow for the 24-h period, should be based on the water-table rise between midnight and 4 A.M. Meyboom (1967) suggests that the Sy value in Eq. (6.16) should reflect the readily available specific yield. He estimates that this figure is 50% of the true specific yield as defined in Section 2.10. If laboratory drainage experiments are utilized to measure specific yield, the value used in Eq (6.16) should be based on the drainage that occurs in the first 24 h. With regard to Figure 6.24, the total evapotranspiration for the period July 2–8 according to the White method is 1.73 ft (0.52 m).
Many field workers have observed an anomalously large rise in water levels in observation wells in shallow unconfined aquifers during heavy rainstorms. It is now recognized that this type of water-level fluctuation is the result of air entrapment in the unsaturated zone (Bianchi and Haskell, 1966; McWhorter, 1971). If the rainfall is sufficiently intense, an inverted zone of saturation is created at the ground surface, and the advancing wet front traps air between itself and the water table. Air pressures in this zone build up to values much greater than atmospheric.
As a schematic explanation of the phenomenon, consider Figure 6.25(a) and (b).
In the first figure, the air pressure, pA, in the soil must be in equilibrium both with the atmosphere and with the fluid pressure, pw. This will hold true at every point X on the water table within the porous medium and at the point Y in the well bore. If, as is shown in Figure 6.25(b), the advancing wet front creates an increase, dpA, in the pressure by an equivalent amount, dpw. The pressure equilibrium in the well at point Y is given by
Since pA = pW and dpA = dpw, we have
For dpA > 0, ψ > 0, proving that an increase in entrapped air pressure leads to a rise in water level in an observation well open to the atmosphere.
This type of water-level rise bears no relation to groundwater recharge, but because it is associated with rainfall events, it can easily be mistaken for it. The most diagnostic feature is the magnitude of the ratio of water-level rise to rainfall depth. Meyboom (1967) reports values as high as 20:1. The anomalous rise usually dissipates within a few hours, or at most a few days, owing to the lateral escape of entrapped air to the atmosphere outside the area of surface saturation.
Changes in atmospheric pressure can produce large fluctuations in wells or piezometers penetrating confined aquifers. The relationship is an inverse one; increases in atmospheric pressure create declines in observed water levels.
Jacob (1940) invoked the principle of effective stress to explain the phenomenon. Consider the conditions shown in Figure 6.25(c), where the stress equilibrium at the point X is given by
In this equation, pA is atmospheric pressure, σT the stress created by the weight of overlying material, σe the effective stress acting on the aquifer skeleton, and pw the fluid pressure in the aquifer. The fluid pressure, pw, gives rise to a pressure head, ψ, that can be measured in a piezometer tapping the aquifer. At the point Y in the well bore,
If, as is shown in Figure 6.25(d), the atmospheric pressure is increased by an amount dpA, the change is stress equilibrium at X is given by
from which it is clear that dpA > dpw. In the well bore, we now have
Substitution of Eq. (6.20) in Eq. (6.22) leads to
since dpA – dpw > 0, so too is ψ – ψ’ > 0, proving that an increase in atmospheric pressure leads to a decline in water levels.
In a horizontal, confined aquifer the change in pressure head, dψ = ψ – ψ’ in Eq. (6.23), is numerically equivalent to the change in hydraulic head, dh. The ratio
is known as the barometric efficiency of the aquifer. It usually falls in the range 0.20–0.75. Todd (1959) provides a derivation that relates the barometric efficiency, B, to the storage coefficient, S, of a confined aquifer.
It has also been observed that changes in atmospheric pressure can cause small fluctuations in the water table in unconfined aquifers. As the air pressure increases, water tables fall. Peck (1960) ascribes the fluctuations to the effects or the changed pressures on air bubbles entrapped in the soil-moisture zone. As the pressure increases, the entrapped air occupies less space, and it is replaced by soil water, thus inducing an upward movement of moisture from the water table. Turk (1975) measured diurnal fluctuations of up to 6 cm in a fine-grained aquifer with a shallow water table.
It has long been observed (Jacob, 1939; Parker and Stringfield, 1950) that external loading in the form of passing railroad trains, construction blasting, and earthquakes can lead to measureable but short-lived oscillations in water levels recorded in piezometers tapping confined aquifers. These phenomena are allied in principle with the effects of atmospheric pressure. Following the notation introduced in Figure 6.25(c) and (d), note that a passing train creates transient changes in the total stress, σT. These changes induce changes in pw, which are in turn reflected by changes in the piezometric levels. In a similar fashion, seismic waves set up by earthquakes create a transient interaction between σe and pw, in the aquifer. The Alaskan earthquake of 1964 produced water-level fluctuations all over North America (Scott and Render, 1964).
Time Lag in Piezometers
One source of error in water-level measurements that is often overlooked is that of time lag. If the volume of water that is required to register a head fluctuation in a piezometer standpipe is large relative to the rate of entry at the intake, there will be a time lag introduced into piezometer readings. This factor is especially pertinent to head measurements in low-permeability formations. To circumvent this problem, many hydrogeologists now use piezometers equipped with down-hole pressure transducers that measure head changes directly at the point of measurement without a large transfer of water. Reducer tubes that decrease the diameter of the standpipe above the intake have also been suggested (Lissey, 1967). In cases where these approaches are not feasible, the time-lag corrections suggested by Hvorslev (1951) are in order.
FREEZE, R. A. 1969. The mechanism of natural groundwater recharge and discharge: 1. One-dimensional, vertical, unsteady, unsaturated flow above a recharging or discharging groundwater flow system. Water Resources Res., 5, pp. 153–171.
FREEZE, R. A. 1974. Streamflow generation. Rev. Geophys. Space Phys., 12. pp. 627–647.
FREEZE, R. A., and P. A. WITHERSPOON. 1967. Theoretical analysis of regional groundwater flow: 2. Effect of water-table configuration and subsurface permeability variation. Water Resources Res., 3, pp. 623–634.
HALL. F. R. 1968. Baseflow recessions-a review. Water Resources Res., 4, pp. 973–983.
MEYBOOM, P. 1966. Unsteady groundwater flow near a willow ring in hummocky morraine. J. Hydrol., 4, pp. 38–62.
RUBIN, J., and R. STEINHARDT. 1963. Soil water relations during rain infiltration: I. Theory. Soil Sci. Soc. Amer. Proc., 27, pp. 246–251.
TÓTH, J. 1963. A theoretical analysis of groundwater flow in small drainage basins. J. Geophys. Res., 68, pp. 4795–4812.