To further compound the problem's complexity, we reckon that a groundwater analysis suffers from a decided limitation:  The size of the control volume is not fixed. Since all groundwaters are ultimately connected, defining the limits of a control volume is perforce an arbitrary exercise. Unlike surface water, which is constrained to the applicable watershed or basin boundary, there is no similar boundary for groundwater flow. Pumping without limit will generally result in an ever increasing size of the cone of depression (Prudic and Herman, 1996; Ponce and Vuppalapati, 2015).

Echoing Theis' statement, we reiterate that groundwater is not a volume, but a flow. An arbitrarily defined control volume will have: (1) inflow (recharge), (2) outflow (discharge), and (3) stored volume of groundwater (water filling the soil or rock voids). Pumping constitutes an external demand, with its discharge originating in any or all of the three components mentioned above. Under this optic, three scenarios are possible (Fig. 2):

1. A pristine system, in its natural condition, in dynamic equilibrium, in the absence of pumping;

2. A developed system, where the amount of pumping is equal to the increase in recharge (captured recharge) plus the decrease in discharge (captured discharge).

3. A depleted system, where an additional fraction of the pumped discharge is actually being extracted from the stored volume. In this case, aquifer depletion takes place, with the size of the cone of depression increasing progressively in time (Ponce, 2006).

Fig. 2  Pristine, developed, and depleted groundwater systems.

In a typical groundwater system, recharge consists of all flow entering the control volume. This amounts to:

1. The natural (equilibrium or pristine) nearly horizontal recharge, entering along the upstream boundary in the absence of pumping;

2. The captured (nonequilibrium or induced) nearly horizontal recharge, entering along the upstream boundary in the presence of pumping (Sophocleous, 1997);

3. The captured nearly horizontal discharge, converted into recharge as a direct result of pumping; and

4. The vertical recharge, defined as the fraction of precipitation that reaches the top of the control volume (i.e., the groundwater table) within the timeframe of analysis.

Note that in a highly developed system, the vertical recharge may include amounts of artificial groundwater replenishment and/or soil leaching amounts (Ponce, 2007).

Discharge from the control volume consists of:

1. The natural, nearly horizontal discharge, in the absence of pumping; or the residual nearly horizontal discharge, in the presence of pumping, exiting along the downstream boundary (Fig. 2); and

2. Deep percolation, i.e., the fraction of precipitation that leaves the control volume as a net vertical discharge (positive flux) at its bottom, only to join deeper groundwaters.

Amounts of deep percolation are largely intractable and generally considered to be a relatively small fraction of precipitation (a global average of about 2%) and, therefore, negligible on practical grounds (L'vovich, 1979). In other words, deep percolation is the (small) fraction of precipitation that is effectively lost from the surface waters. Figure 3 shows a schematic of the various fluxes relevant to groundwater flow.

Fig. 3  Inputs and outputs in an aquifer's control volume.

3.  SUSTAINABLE USE OF GROUNDWATER

The central issue of sustainability is the question of how much water to pump from an aquifer and still remain sustainable (Alley et al., 1999). In the typical case, aquifer replenishment is slow, taking decades, if not hundreds or thousands of years. Thus, it follows that excessive pumping of groundwater may lead to depletion and the consequent lack of sustainability. A depleted aquifer is one that cannot recover fast enough to continue to be of use; therefore, it is unsustainable.

Since an aquifer may be depleted through excessive pumping, it may be argued that the solution is to leave the aquifers alone and use only surface waters. Unlike groundwaters, the recycling time of surface water is a global average of 11 days (L'vovich, 1979; Ponce et al., 2000); therefore, all surface waters are sustainable when compared to groundwaters. This approach, while apparently logical, does away with the practice of groundwater pumping established for the past century in developed societies. We argue here that this extreme solution is politically incorrect. Groundwater use should not be altogether suspended; rather, it should be regulated with the aim to manage, mitigate, and/or minimize depletion and other negative effects in order to best meet the challenge of sustainability.

The discussion then shifts to the components of the groundwater balance. Where should the pumped water come from? We note that there is a caveat in the hydrologic balance. The water extracted by pumping could be either:

• Consumptive, when none of the pumped water returns to the aquifer; or

• Partially nonconsumptive, when a certain fraction of the pumped water returns to the aquifer at some point in time or space.

Irrigated agriculture is the classical example of consumptive use, particularly when the drainage waters (drainage is ostensibly a necessity in arid/semiarid regions) are collected and removed from the premises via surface flow (Fig. 4). Other uses (domestic or industrial) may be totally or partially consumptive, depending on the local situation.

Fig. 4  Irrigation canal (left) and drainage canal (right), Wellton-Mohawk irrigation, Arizona.

While the concept of sustainable yield has only been recently recognized (Alley et al., 1999), the admittedly older concept of safe yield has been around for nearly a century. Lee (1915) defined "safe yield" as the limit to the quantity of water which can be withdrawn from an aquifer, regularly and permanently, without dangerous depletion of the storage reserve. He noted that water permanently extracted from an underground reservoir reduces the volume of water passing through the basin by way of natural channels, i.e., the natural discharge. To illustrate the existence of this natural discharge, Lee observed that heavy pumping would commonly result in the drying up of springs and wetlands (Ponce, 2014). Thus, he distinguished between a theoretical safe yield, equal to the natural recharge, and a practical safe yield, a lower value which takes into account the need to maintain a residual discharge (Fig. 2). According to Lee, the residual discharge must be ascertained and deducted from the theoretical safe yield in order to obtain the practical safe yield. Over the past two decades, the latter has morphed into sustainable yield, which reaches beyond hydrogeology.

How much should the residual discharge be when assessing sustainable yield? In cases when the captured discharge must be minimized due to the existence or claims of downstream water rights (i.e., springs, wetlands, and baseflow), it follows that very little or none of the nearly horizontal recharge could be captured without encroachment on the rights of others. Under nonequilibrium conditions, pumping one-hundred percent of the natural recharge could approximately result in the capturing of up to 50% of the natural discharge (Fig. 2). Under equilibrium conditions, any amount of capture would come from discharge and possibly encroach upon established downstream rights.

This bleak picture is somewhat ameliorated when it is recognized that the vertical discharge, i.e., the fraction of local precipitation that manages to reach the water table, is not specifically included in the determination of (nearly horizontal) recharge. Thus, a resolution of the conflict of rights between surface water and groundwater would be the capture of any of the vertical recharge. This shift from capturing nearly horizontal to capturing vertical recharge is predicated upon the fact that nearly horizontal recharge is of regional scale, while vertical recharge is essentially of local scale. Under this focused spatial optic, all vertical recharge could be subject to capture. Moreover, basing the determination of sustainable yield solely on vertical recharge is likely to counteract the argument that capture by pumping could negatively affect neighboring ecoystems and surface waters.

4.  THE CYBERNETIC HYDROLOGIC BALANCE

Ir order to properly quantify vertical recharge and, therefore, gain a handle on sustainable groundwater yield, we resort to L'vovich's cybernetic hydrologic balance, an approach better suited for yield hydrology than the conventional one (L'vovich, 1979; Ponce, 2018).

In L'vovich's approach, annual precipitation is separated into two components (Fig. 5):

P  =  S  +  W

(1)

in which S = surface runoff, i.e., the fraction of runoff originating on the land surface, and W = catchment wetting, or simply, wetting, the fraction of precipitation not contributing to surface runoff.

Fig. 5  The cybernetic hydrologic balance (L'vovich, 1979).

In turn, wetting is separated into two components:

W  =  U  +  V

(2)

in which U = baseflow, i.e., the fraction of wetting which exfiltrates as the dry-weather flow of streams and rivers, and V = vaporization, i.e., the fraction of wetting returned to the atmosphere as water vapor.

Runoff (i.e., total runoff) is the sum of surface runoff and baseflow:

R  =  S  +  U

(3)

Combining Eqs. 1 to 3:

P  =  R  +  V

(4)

Equations 1 to 4 constitute a set of water balance equations. Four water balance coefficients may be defined: (1) runoff coefficient, (2) baseflow coefficient, (3) wetting coefficient, and (4) recharge coefficient.

The runoff coefficient is:

R Kr  =  _____             P

(5)

The baseflow coefficient is:

U Ku  =  _____             W

(6)

The wetting coefficient is:

W Kw  =  _____              R

(7)

The groundwater recharge coefficient is:

U Kg  =  _____              P

(8)

Figure 6 shows runoff and baseflow coefficients calculated by Ponce and Shetty (1995), based on data reported by L'vovich (1979). It is seen that in five all cases, runoff and baseflow coefficients increase with annual precipitation.

[Click on either image to display]

(a)      (b)

Fig. 6 (a) Runoff coefficients and (b) baseflow coefficients, for the following data: 1. Africa: evergreen sclerophyll forests and scrub. 2. Africa: mountain conifer forests. 3. North America (Canada); subarctic forests (taiga). 4. South America: wet evergreen forests in the mountains. 5. Asia (india): semideciduous forests in the mountains (Western Ghats).

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5.  EXAMPLE APPLICATION

The methodology described in the previous section is applied herein to data from the Sarada river basin, upstream of Anakapalli, in Andhra Pradesh, India (Fig. 7). The basin is positioned between the Eastern Ghats and the eastern coastline of India, and features a subhumid climate (Ponce et al., 2000), with a drainage area of 1,980 km².

Wikimedia Commons Fig. 7  Geographical location of Anakapalli, in Andhra Pradesh, India.

Eleven (11) years of rainfall-runoff data are analyzed. The precipitation data consists of a daily rainfall hyetograph and the runoff data consists of the hydrograph measured at the basin outlet. The annual hyetographs are used to calculate annual rainfall P (mm). Each annual hydrograph is integrated to obtain runoff R (mm). Surface runoff S (mm) is obtained by hydrograph separation using established principles (Ponce, 2014). The matrix of P-R-S is used to run online water balance 2.

Figure 8 shows the tabular results of the online program, where the groundwater recharge coefficient K_g is given in Col. 11. Figure 9 plots K_g vs mean annual precipitation P, in ascending order of precipitation. Despite the apparent noise in the data, the general trend is for an increase in K_g with an increase in P.

Fig. 8  Cybernetic hydrologic balance for Sarada river basin data.

Fig. 9  Groundwater recharge coefficient vs mean annual precipitation, Sarada river basin data.

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6.  RECHARGE COEFFICIENTS AND SUSTAINABLE YIELD

The vertical groundwater recharge coefficient is the fraction of precipitation that manages to reach the groundwater table. Given mean annual precipitation, and, for that matter, given any amount of annual precipitation, the recharge coefficient may be interpreted as the amount of water that could be captured by pumping while assuring a sustainable yield. This strategy does not compromise any part of the (nearly horizontal) recharge or discharge. Note that the conversion of discharge to recharge (in a control volume) has been a point of contention in groundwater resource evaluations for many decades (Kazmann, 1956).

For any one watershed/basin, the recharge coefficient may be evaluated following the cybernetic hydrologic balance described herein (Ponce, 2018). Given the natural hydrologic variability, in any one year, the sustainable groundwater yield, i.e., the permissible amount of capture, could be practically based on the amount of precipitation occurring in the previous year. Amounts of artificial replenishment and/or return flows, if any, may be added to the (natural) vertical recharge at this time (Ponce, 2007).

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7.  SUMMARY AND CONCLUSIONS

The concepts of safe yield and sustainable yield of groundwater are analyzed and compared in the context of a hydrologic balance. All recharge and discharge fluxes are duly accounted for. Since groundwater is a flow, not a volume, tapping the nearly horizontal natural recharge may arguably compromise the rights of other users in the vicinity, including natural ecosystems, wetlands, and neighboring baseflow. It is proposed here that vertical recharge, i.e., the recharge originating in local precipitation, is the only recharge that could be freely tapped for capture by groundwater in order to avoid encroachment on established rights.

A methodology to evaluate vertical recharge is developed and tested. The methodology is based on the cybernetic hydrologic balance of L'vovich (1979), including the definition of a groundwater recharge coefficient. This coefficient represents the fraction of precipitation that reaches the water table; therefore, it may be used to evaluate and assess sustainable groundwater yield. Any amount of groundwater capture over and above the vertical recharge would have to demonstrate, by means of suitable hydrological (baseflow) and ecohydrological (ecosystems) studies, that it does not significantly affect established rights.

The online calculator online water balance 2 rounds up the analysis.

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ACKNOWLEDGMENTS

The authors wish to recognize Donna Tisdale and the people of the community of Boulevard, in southeast San Diego County, California, for their support of theoretical and applied research in groundwater sustainability throughout the past 12 years. The Sarada river basin data used in this study was provided by Dr. Y. R. S. Rao, National Institute of Hydrology, Roorke, India.

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REFERENCES

Alley, W. M., T. E. Reilly, and O. E. Franke. 1999. Sustainability of groundwater resources. U.S. Geological Survey Circular 1186, Denver, Colorado, 79 p.

Kazmann, R. G. (1956). "Safe yield" in ground water development: Reality or illusion? Journal of the Irrigation and Drainage Division, American Society of Civil Engineers, Vol. 82, No. IR3, November, Paper 1103.

Lee, C. H. (1915). The determination of safe yield of underground reservoirs of the closed-basin type. Transactions, American Society of Civil Engineers, Vol. LXXVIII, Paper No. 1315, 148-218.

L'vovich, M. I. 1979. World water resources and their future. Translation from Russian by Raymond L. Nace, American Geophysical Union.

Ponce, V. M., and A. V. Shetty. 1995. A conceptual model of catchment balance: 2. Application to runoff and baseflow modeling. Journal of Hydrology, 173, 41-50.

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Ponce, V. M. 2007. Sustainable yield of groundwater. Online article.

Ponce, V. M. 2014. Effect of groundwater pumping on the health of arid vegetative ecosystems. Online article.

Ponce, V. M. 2014. Engineering hydrology: Principles and practices. Online textbook.

Ponce, V. M. and B. Vuppalapati. 2015. The myth of groundwater resource evaluation. Online article.

Ponce, V. M. 2018. Why is the cybernetic hydrologic balance better suited for yield hydrology than the conventional approach? Online article.

Prudic, D. E., and M. E. Herman. 1996. Ground-water flow and simulated effects of development in Paradise Valley, a basin tributary to the Humboldt River, in Humboldt County, Nevada. U.S. Geological Survey Professional Paper 1409-F.

Sophocleous, M. 1997. Managing water resources systems: Why "safe yield" is not sustainable. Editorial, Ground Water, Vol. 35, No. 4, July-August, 561.

Theis, C. V. 1940. The source of water derived from wells: Essential factors controlling the response of an aquifer to development. Civil Engineering, Vol. 10, No. 5, May, 277-280.

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