The Water-Budget Myth
U.S. Geological Survey
STEPHEN 'Sd PAPADOPOLOS
S. S. Papadppulos and Associates, Inc.
H. H. COOPER, JR.
U.S. Geological Survey
Water-resource scientists are concerned that
some basic principles are being overlooked by
water managers. Rather than discuss the scope.
of groundwater hydrology, we have chosen to
focus on a comnpn misconception to illustrate
Perhaps the most common misconception in
groundwater hydrology is that a water budget of
an area determines the magnitude of possible
groundwater development. Several well-known
hydrologists have addressed this misconception
and attempted to dispel it. Somehow, though,
it persists and continues to color decisions by
the water-management community. The laws gov-
erning the development of groundwater in Nevada.
as well as several other states are based on the
idea that pumping within a groundwater basin
shall not exceed the recharge. It is the intent:
of this paper to re-examine the issue.
eis (1940) addressed the subject:
Under natural conditions previous to de-
telopment by wells, aquifers are in a state of
approximate dynamic equilibrium. Discharge by
wells is thus a new discharge superimposed upon
a previously stable system, and it must be bal-
anced by an increase in the recharge of_ the
aquifer, or by a decrease in the old natural
discharge, or by loss of storage in the aquifer,
or by a combination of these.
Brown (1963) attempted to illustrate these
points by demonstrating that (1) under virgin
conditions the height of the water table is a
function of the recharge and transmissivity, and
recharge is balanced by discharge from the aqui-
fer; (2) the effects of groundwater development
are superimposed upon these virgin conditions;
and (3) the rate at which the hydrologic system
reaches a new steady state depends on the rate
at which the natural discharge (in his example
to a stream) can be captured by the cone of de-
pression. Brown's argument, which was highly
technical, was essentially ignored by many hy-
Bredehoeft and young (1970) re-examined the
issue and restated Theis's conclusions:
Under virgin conditions, steady state prevails
in most groundwater systems, and natural re-
I I j
charge is equal to the natural discharge. We
can write the following expression -for the
system as a whole
R0 DO 0,
where R0 is the mean recharge under virgin
conditions and DO is the mean discharge
under virgin conditions.
Some disturbance of the system is necessary
to have a development. At some time after the
start of pumping we can write the following ex-
(0 + AR) (D + d+L ) 0 (4.2)
where AR0 is the change in the mean re-
charge, A DO the change in the mean dis-
charge, Q the rate of withdrawal due to
development, and dV/dt the rate of change in
storage in the system. From Eqs. (4.1) and
(4.2) we can obtain
0 0 d
Assuming water-table conditions we can them
compute an average drawdown for the system as a
whole in the following manner:
Sa = A /(S A),
where sa is average basinwide drawdown,A V the
volume removed from storage at time t, S, the
specific yield of the aquifer, and Ab the area
of the basin. Such an input-output analysis
treats the system just as we would treat a sur-
face water reservoir. The response of the system
is assumed to take place rapidly with effects
equally distributed throughout the basin. In
most groundwater systems the response is not
RESPONSEE OF GROUNDWATER SYSTEMS
In groundwater systems the 'decline of water
levels in a basin because of withdrawal will
occur over a period of years, decades, or even
centuries. Some'water must be taken from stor-
age in the system to create gradients toward a
well. There are two implications to be gathered
from these facts: (1) some water must always be
.ined to create a development, and (2) the time
delays in a groundwater system differ from those
in surface-water systems.
It is apparent from Eq. (4.3) that the vir-
gin rates of recharge R0 or discharge DO are not
of paramount importance in groundwater investi-
gations. For the system to reach some new
equilibrium, which we define as dV/d:= 0, there
.-st be some change in the virgin rate of re-
charge and/or, the rate of discharge DO. It
is these changes, A RO and LD0, that are
The response of groundwater systems depends
on the aquifer parameters (transmissivity and
storage coefficient), the boundary conditions,
and the positioning of the development within
Lohman (1972a), referring to the High Plains
of Texas and New Mexico, made the point again.
The following discussion is a synopsis of Lob-
man's argument taken from Bulletin 16 of the
U.S. Water Resources Council (1973):
WithtrAwAls cannot exceed the rates of re-
S u g "' -- -- -
harge or dis.charge-tfora=-pr oond~ eri of
time ~ wthputresultan t_ nijng" of ground water.
Adjustments in recharge and discharge rates as
a result of pumping can be referred to as cap-
ture, and, inasmuch as sustained yield is limit-
ed by capture and cannot exceed it, estimates
of capture are fundamentally important to quan-
titative groundwater analysis and planning for
long-term water supply.
Decline of water levels in response to sus-
tained withdrawal may continue over a long
period of time. At first, some water must be
taken from storage in the system to create gra-
dients toward pumping wells. Two important
implications of these statements concerning a
long-term water supply are that (1) some water
must be removed from storage in the system to
develop a groundwater supply, and (2) time de-
lays in areal distribution of pumping effects
in many groundwater systems demonstrate that
balanced (equilibrium or steady-state). condi-
tions-of flow do not ordinarily exist. In the
clearest examples, water levels decline drasti-
cally, and some wells go dry long before the
system as a whole reaches a new equilibrium
balance between replenishment and natural and
imposed discharge rates.
The most well-known example of such a condi-
tion.of nonequilibrium is the major groundwater
development of the southern High Plains of Texas
and New Mexico. Water is contained in extensive
deposits (the Ogallala formation) underlying the
plains (Figure 4.1). Average thickness of these
deposits is about 300 feet. They consist of
silt, sand, and gravel and form a groundwater
reservoir of moderate permeability. The reser-
voir rests on relatively impermeable rock and
constitutes the only large source of g:>Dndwater
available to the area.
The southern High Plains slcpe gently from
west to east, cut off from external sources of
water upstream and downstream by escarpments,
Triassic and O;der Rocks
I little or no usab;e grounawaler)
FICGE 4.1 Development of groundwater in the southern
Figh Plains of Texas and New Mexico. Withdrawal has
resulted i. a pronounced decline of water levels in the
riddle of the so-thern Hich Plains, but it has had
little effect on the gradient to the east (n.tc:al dis-
charge) or on .at.ral rec:ar.e.
JOHN D. BREDEHOEFT, S. S. PAPADO and H. H COOPER, JR.
JOHN D. BREDEHOEFT, S. S. PAPADOl JC, anid H. H. COOPER, JR.
GCoundwater: The Water-Budget I )
as illustrated in Figure 4.1. Replenishment is
dependent on the scanty precipitation, and total
0Oecharge in the southern High Plains is extreme-
y small in comparison with the enormous imposed
discharge (pumping for irrigation). Total re-
charge is equivalent to only a fraction of an
inch of water per year over the whole of the
High Plains. The natural discharge, of the same
order as the recharge, continues from seeps and
springs along the eastern escarpment.
Withdrawal by pumping has increased rapidly
in the past 50 years and at present amounts to
about 1.5 trillion gallons per year (4.6 million
acre-feet per year). The withdrawal has result-
ed in a pronounced decline of water levels in
the middle of the Plains, where pumping is heav-
iest (and where the increase in cost of pumping
has been greatest). Little additional natural
recharge can be induced into the system because
the water table lies 50 feet or more beneath the
land surface in most of the area, the unsaturat-
ed volume of aquifer available for possible re-
charge is more than ample.
Nor has natural discharge been salvaged by
the lowered water levels. As may be noted in
Figure 4.1, the hydraulic gradient, or water-
table slope, toward the eastern escarpment has
been virtually unchanged. Even if all discharge
could be salvaged by pumping, however, the sal-
vaged water would be only a small percentage of
!HSE CIRCULAR ISLAND
Perhaps the easiest way to illustrate our point
further is to consider pumping groundwater on
an island situated in a freshwater lake. The
situation is shown schematically in Figure 4.2.
An alluvial aquifer overlies bedrock of low per-
meability on the island. Rainfall directly on
the island recharges the aquifer. Under virgin
conditions, this recharge water is discharged
by outflow from the aquifer into the lake. The
height of the water table beneath the island is
determined by the rate of recharge, the area of
the island, and the transmissivity.
.Under virgin conditions, we can determine a
water balance for the island. From our previous
..tation, recharge to the island is
/ r"a dA = R
where ra is the average rate of recharge and A
is the total area of the island. Discharge from
the island is and
f h k Lcr D,
is Is '
where k is the hydraulic conductivity of the
aquifer, h the height of the water table de-
O""ed to be equal to the hydraulic head), and
the gradient in hydraulic head taken at the
shoreline of the island (defined to be normal
to the shoreline), L the total length of the
shoreline, and RO DO.
We drill a well and begin to pump water from
the aquifer on the island. A cone of depression
develops and expands outward from the well.
Figure 4.3 shows this cone of depression a short
time after pumping has begun.
if we look at the periphery of the island,
we see that until the pumping causes a signifi-
cant change in the gradient in head at the
shoreline the discharge continues unchanged.
Gradients in hydraulic head, or saturated thick-
ness, must be changed at the shoreline in order
to change the discharge.
If we write the system balance for the en-
tire island, at some time before the cone
expands to the shoreline, we see that
Ra DO- 0 # 0,
where Q is the rate of pumping. As neither the
recharge, R0, nor the discharge, DO, has changed
from its initial value, the water pumped, Q, is
balanced by the water removed from storage.
The cone of depression will eventually
change the gradients in hydraulic head at the
shoreline significantly. At this time, dis-
charge from the system begins to change. This
is shown schematically in Figure 4.4. The dis-
charge can be changed by pumping so that -the
system is brought into balance. At some time
kh2 I dL = Q.
Since the virgin rate of recharge, t0'
equals the virgin rate of discharge, DO, we
where the quantity
:..: hi i = I .. c- = .3,
cS !s C s s
which we define as the "capture." The system
is in balance when the capture is equal to the
water pumped, i.e., 0 = D.
The term capture is defined and discussed
in Definitions of Selected Ground-Water Terms--
Revisions and Conceptual Refinements (Lohman,
Water withdrawn artificially from an aquifer is
derived from a decrease in storage in the aqui-
fer, a reduction in the previous discharge from
the aquifer, an increase in the recharge, or a
combination of these changes. The decrease in
discharge plus the increase in recharge is term-
ed capture. Capture may occur in the form of
decreases in the groundwater discharge into
L -J h i I dL ,
streams, lakes, and the ocean, or fr. decreases
in that component of evapotranspirati derived
from the saturated zone. After a new artificial
withdrawall from the aquifer has begun, the head
in the aquifer will continue to decline until
the new withdrawal is balanced by capture.
For the island system chosen, we can induce
flow from the lake into the aquifer. In fact,
the capture can be greater than the virgin re-
charge of discharge
j dh Id>
kh dL D
's Is 0
kh dL > RO.
In fact, the magnitude of pumpage that can be
sustained is determined by (1) the hydraulic
conductivity of the aquifer and (2) the avail-
able drawdown, which are independent of other
factors (Figure 4.5).
At first glance, 'this island aquifer system
seems much too simple for general conclusions;
however,: the principles that apply to this sys-
tem apply to most other aquifer systems. The
ultimate production of groundwater depends on
S11 I tl T I
<'-'^ ae aI "'"'
FIGURE 4.2 Cross section of an alluvial
aquifer, underlain by bedrock of low
permeability, on an island in a fresh-
FIGURE 4.3 Cross section of the island
depicting the cone of depression soon
after .p=i.-.; has beca-..
f:JR' 4.4 Cross seci:on of the island
aquifer sys:es wnen the influence cf
ur-,;-.i has :a:r.ted the shoreline.
r":R; 4.5 S:neartic cross section of
the island aquifer system, which illus-
t:ates that the .ar.nitude of pu.page
from tnis system is dependent on the
available dravdo.n, aquifer thickness,
and the hydraulic conductivity of the
,roi,;dwacer: The Water-Budget K J
Schematic map of an intermon-
shoving areas of recharge,
and two hypothetical water-
schemes, Case I and Case II.
how much the rate of recharge and (or) discharge
can be changed--how much water can be captured.
Although knowledge of the virgin rates of re-
charge and discharge is interesting, such knowl-
edge is almost irrelevant in determining the
sustained yield of a particular groundwater
reservoir. We recognize that such a statement
is contrary to much common doctrine. Somehow,
we have lost or misplaced the ideas Theis stated
in 1940 and before.
)S:roundwater systems generally respond much slow-
-er than other elements of the hydrologic cycle.
It can take long periods of time to establish a
new steady state. For this reason, groundwater
hydrologists are concerned with the time-depend-
ent dynamics of the system.
To illustrate the influence of the dynamics
of a groundwater system, we have chosen a rather
simple system for analysis. Consider a closed
intermontane basin of the sort one might find
in the western states. Under virgin conditions
the system is in equilibrium: phreatophyte
evapotranspiration in the lower part of the
basin is equal to recharge from the two streams
a: the upper end (Figure 4.6).
Pumping begins in the basin, and, for sin-
plicity, we assume tie pu.T.page equals the re-
charge. The fcllcwin- two assu.pticns regarding
the hydrology are made:
1. Recharge is independent of the pumping
in the basin, a typical condition, especially
in the arid west.
2. Phreatophyte use decreases in a linear
manner (Figure 4.7) as the water levels in the
vicinity decline by 1-5 ft. Phreatophyte use
of water is assumed" to cease when the water
level is lowered 5 ft below the land surface.
The geometry and pertinent hydrologic parameters
assumed for the system are shown in Table 4.1.
0 The system was simulated mathematically by
a finite-difference approximation to the equa-
tions of flow. The equations are nonlinear and
of the following form.:
V (khh) = S + W (x y t)
where k is the hydraulic conductivity, h the hy-
draulic head (which is equal in our case to the
saturated aquifer thickness), S the storage
coefficient, and W the source function. (time
dependent). In essence, this is a two-dimen-
sional water-table formulation of the problem in
which the change in saturated thickness within
the aquifer is accounted for.
One-thousand years of operation were simu-
lated. Stream recharge, phreatophyte-water use,
pumping rate, and change in storage for the en-
tire basin were graphed as functions of time.
Two development schemes were examined: Case I,
in wnich the pumping was more or less centered
within the valley, and Case II, in which the
pu;.ping was adjacent to the phreatophyte area.
The system does not reach equilibrium until
the phreatophyte-water use (the natural dis-
0 20 40 60 80 *100
PERCENT OF INITIAL PHREATOPHYTE USE
FIGU?. 4.7 Assi-re lniear function relating9 -reatc-
pnyve-water use to dr:awown.
I I J.
56 JOHN D.
charge) is entirely salvaged (capt ) by pump-
ing, i.e., phreatophyte water use equals zero
(we define equilibrium as 6 V/ 6 t 0). In
Case I, phreatophyte-water use (Figure 4.8) is
still approximately 10 percent of its initial
value at year 1000. In Case II it takes 500 yr
for the phreatophyte-water use to be completely
We can illustrate the same point by looking
at the total volumes pumped from the system,
along with the volume taken from storage "mined*
(Figure 4.9) ...
In both cases, for the first 100 yr, nearly
all of the water comes from storage. Obviously,
as the system approaches equilibrium, the rate
of change of the volume of water removed from
storage also approaches zero. If the aquifer
was thin, it is apparent that wells could go dry
long before the system could approach equilib-
This example illustrates three important
1. The rate at which the hydrologic system
can be brought into equilibrium depends on the
rate at which the discharge can be captured.
2. The placement of pumping wells in the
system significantly changes the dynamic res-
ponse and the rate at which natural discharge
can be captured.
3. Some groundwater must be mined before the
system can be brought into equilibrium.
We have attempted to make several important
1. Magnitude of development depends on hy-
drologic effects that you want to tolerate,
TABLE 4.1 Aquifer Parameters
Storage coefficient (s)
Average use annuall)
Average recharge rate
Average pumping rate
50 x 25 miles
0.5 x 10- ft2/sec
aEoEDHOEFT, S. S. .PAP'OPULUS, and H. H. COOPER, JR.
FIGURE 4.8 Plot of the rate of recharge, pumping, and
phreatophyte use versus time.
ultimately or at any given time (which could be
dictated by economics or other factors). To
calculate hydrologic effects you need to know
the hydraulic properties and boundaries of the
aquifer. Natural recharge and discharge at no
time enter these calculations. Hence, a water
budget is of little use in determining magnitude
2. The magnitude of sustained groundwater
pumpage generally depends on how much of the
natural discharge can be captured.
3. Steady state is reached only when pumping
is balanced by capture (.6R0 + ADO), in
most cases the change in recharge, LR0, is
small or zero, and balance must be achieved by
a change in discharge, tDO. Before any natural
discharge can be captured, some water must be
removed from storage by pumping. In many cir-
cumstances the dynamics of the groundwater
system are such that long periods of time are
necessary before any kind of an equilibrium
condition can develop. In some circumstances
FIC'G? 4.9 Total vclu-e
ace versJs time.
pumped and the change i- stc:-
Groundw'aer: The W'aer-Budget Myth
the system response is so slow that mining will
continue well beyond any reasonable planning
These concepts must be kept in mind to man-
age groundwater resources adequately. Unfortu-
nately, many of our present legal institutions
do not adequately account for them.
Bredehoeft, J. D., and R. A. Young (1970). A temporal
allocation of ground-vater--A simulation approach,
water Resour. Aes. 6, 3-21.
Brown, R. R. (1963). The cone of depression and the
area of diversion around a discharging well in an
infinite strip aquifer subject to uniform recharge,
U.S. Jecl. Surv; 'Wter-Supply -ap. 1S45C, pp. C69-
Lohman-. S. W. (1972a). Ground-water hydraulics, U.S.
Geol. Su:v. Prof. Pap. 708, 70 pp.
Lohma., S. k. (1972b). Definitions of selected ground-
va:er terms--Revisions and conceptual refinements,
U.S. Geol. Suzv. a':e-Supply Pap. 1988, 21 pp.
Theis, C. V. (1940). The source of water derived from
wells: Essential factors controlling the response
of an aquifer to development, Civil Eng. 10, 277-
U.S. waterr Resources Council (1973). Essentials of
ground-water hydrology pertinent to water-resources
plan-ing, U.S. Hater Resour. Council Bull. No. 16,