To further compound the complexity of a groundwater system, we recognize that it suffers from a decided limitation: The size of the control volume is not fixed. Since all groundwaters are connected, defining the limits of a control volume is perforce an arbitrary exercise. Unlike surface water, which is constrained to the applicable watershed/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).

Following 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 (Ponce, 2006) increasing progressively in time.

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 (Sophocleous, 1997), entering along the upstream boundary in the presence of pumping;

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 (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, joining 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 graphical portrayal of the various relevant fluxes in groundwater flow.

Fig. 3  Inputs and outputs in an aquifer's control volume. [In preparation 180203]

3.  SUSTAINABLE USE OF GROUNDWATER

The central issue of sustainability is the question of how much water to pump from an aquifer while still remaining 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 can 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.

If an aquifer can be readily 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 judged against groundwaters. This approach, while apparently sensical, 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 cannot and should not be altogether eliminated; rather, it should be regulated with the aim to manage, mitigate, and/or minimize depletion in order to 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.

Irrigation is the classical example of consumptive use, particularly when the drainage waters (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 consumptive or partially nonconsumptive, 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 older concept of safe yield has been around for more than 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.

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 (springs, wetlands, and baseflow), it follows that very little or none of the nearly horizontal recharge could be captured without encrochment on the rights of others. Under nonequilibrium conditions, pumping one-hundred percent of the natural recharge could theoretically result in the capturing of up to 50% of the natural discharge (Fig. 2). At the asymptotic limit, i.e., under near equilibrium conditions, any amount of capture would come from discharge and encroach upon established downstream rights.

This bleak picture is somewhat ameliorated when it is recognized that the vertical discharge (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 to capture any and all of the vertical recharge. This shift from capturing nearly horizontal to capturing vertical recharge is predicated upon the fact that nearly horizontal recharge is regional, while vertical recharge is local. Under this spatial optic, all vertical recharge could be subject to capture. Therefore, the quantification of vertical recharge takes on a renewed importance. Moreover, basing the determination of sustainable yield solely on vertical recharge should (all but) put to rest the argument that capture by pumping is negatively affecting 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 the cybernetic hydrologic balance, an approach better suited for yield hydrology than the conventional one (L'vovich, 1979; Ponce, 2018).

In the cybernetic approach to the hydrologic balance, 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).

---

5.  EXAMPLE APPLICATION

The methodology described in this article is applied to data from the Sarada river basin, in Andhra Pradesh, India. Eight (8) years of rainfall-runoff data are available. The precipitation data consists of a daily rainfall hyetograph; 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). Hydrograph separation is accomplished using established principles to obtain surface runoff S (mm) (Ponce, 2014). The matrix P-R-S is used to run online water balance 2.

Figure 7 shows the tabular results of the online program. Figure 8 plots groundwater recharge coefficient K_g vs mean annual precipitation P. Despite the noise in the data, we note the general trend to an increase in K_g with an increase in P.

Fig. 7  Results of cybernetic hydrologic balance for Sarada river basin data.

Fig. 7  Graph of groundwater recharge coefficient vs mean annual precipitation.

---

6.  RECHARGE COEFFICIENTS AND SUSTAINABLE YIELD

The vertical groundwater recharge, characterized by K_g, is the fraction of precipitation that reaches the groundwater table. For any mean annual precipitation, and, for that matter, for any annual precipitation, the recharge coefficient may be interpreted as the amount of water that could be captured by pumping while assuring a sustainable yield. Ostensibly, this strategy does not compromise any part of the (nearly horizontal) recharge or discharge, a claim which has been a point of contention in groundwater resource evaluations since the 1950s (Kazmann, 1956).

For any one basin, we recommend that the recharge coefficients be evaluated following the cybernetic hydrologic balance approach described herein. Given the intrinsic natural hydrologic variability, in any one year, the sustainable groundwater yield, i.e., the permissible amount of capture, is practically based on the amount of precipitation occurring in the previous year. Amounts of artificial replenishment, rejected discharge, and/or return flows, if any, could be added to the analysis at this time (Ponce, 2007).

---

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 recharges and discharges 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 baseflow. It is surmised that vertical recharge, i.e., the recharge originating in local precipitation, is the only recharge that may be tapped for capture by groundwater to avoid a significant encroachment on established rights.

A methodology is developed and tested to evaluate vertical recharge. The methodology is based on L'vovich's cybernetic hydrologic balance, 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. An amount of groundwater capture over and above the vertical recharge would have to demonstrate, by means of relevant ecohydrological and baseflow studies, that it does not significantly affect established rights.

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

---

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., R. P. Pandey, and S. Ercan. 2000. Characterization of drought across climatic spectrum. Journal of Hydrologic Engineering, ASCE, Vol. 5, No. 2, April, 222-224.

Ponce, V. M. 2006. Groundwater utilization and sustainability. Online article.

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

Ponce, V. M. 2014. Effect of groundwater pumping on the healht 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.

---