WATER BUDGETS
George Ward
Water budgeting is one of the fundamental activities underlying the management of water
resources (as well as much of the science of hydrology). This short issue paper seeks to
formalize the types of water budgets that may be helpful in the present study, particularly in
establishing a quantitative basis for our analyses, as a useful method for incorporating futurescenario conditions such as increased water demands or changed climate, and as a rational
process for cross-basin comparisons.
1. Goals
Consonant with the strategies adopted in the SERIDAS workshop in January 2014, this issue
paper has three main purposes:
● Define the process and assumptions of water budgeting, in the specific context
of long-term water use/water availability, in the particular context of
engineered rivers in arid or semi-arid basins
● Provide guidance on the potential use of water budgets in our second round of
water basin reports
● Lay the groundwork for the potential use of water budgets in cross-basin
analysis
2. Water budgeting as an analytical process
Water budgeting (a.k.a. water accounting, water balancing, stocks and flows) is founded on the
principle of conservation of water mass, or, if we assume the density of water to be constant (a
good assumption for the sort of problems treated in this project), the conservation of water
volume. In fluid mechanics, this is expressed in the continuity equation. But we do not pursue
the formulation of water budgeting from the point of view of the equations of fluid motion,
because it deals with aggregated transports of water, that is, integrations (or averages) over large
areas of space and long intervals of time.* The reasons for aggregating the terms in the
* For the masochistic reader interested in this sort of thing, it is the boundary conditions, not the internal movement
of fluid, that is the primary concern of water budgeting.
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equations of motion include:
(1) In space, the morphological features that confine liquid water may be treated as
geometrical elements in the budget, e.g., river channels, tributaries, lakes,
aquifers;
(2) In time, integration over an interval of time large enough to exceed the travel
times between water-budget elements (most importantly to subsume storm
hydrographs) allows movement of water between the spatial elements to be
represented as simple transfers;
(3) The depiction of time and space behaviors on large scales is appropriate for basinlevel estimation of water volumes and transfers, which is the usual concern of
water planning;
(4) The computational demands for time simulations are greatly reduced, compared
to detailed numerical solutions that frequently require complex special-purpose
codes (such as SWAT, SWMM and MIKE-SHE).
The depiction of individual elements of a river basin as homogeneous entities is frequently
referred to as a water-resources system (e.g., Linsley and Franzini, 1964; Loucks, 1996; Loucks
and van Beek, 2005). When these individual elements are further lumped on a continental or
global scale, the resulting water budget depicts the hydrological cycle (e.g., Chow et al., 1988,
Dingman, 2002). Overviews of water-budget modeling are provided by Zhang et al. (2002) and
Kirby et al. (2008).
A water budget is essentially data-driven: that is to say, it relies upon measurements of key
meteorological, hydrological and hydraulic variables. In setting up a water budget, the level of
spatial aggregation and time integration must be decided. This is largely dictated by the
objectives of the analysis, and by the resolution in available measurement data, which are used to
quantify the various transfers and storages of water (Zhang et al., 2002).
In delineating the spatial elements of the water budget, perhaps the only hard-and-fast rule is that
the boundary of each element outside of the watercourse itself should coincide with the drainage
area. (Otherwise, the flow into the element due to runoff across its boundary must be explicitly
determined, which introduces unnecessary complexity.) Planning and conceptual engineering
generally consider the entirety of a basin or subwatershed, and employ a spatial resolution in
which each structural change, such as a tributary confluence, dam, or fall-line, whether natural or
manmade, defines the boundary of a spatial element. The resulting spatial network is frequently
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formulated as a link-node or a “flow- network” configuration (e.g., Louck, 1996), an analog of
the traditional stem diagram of the hydrologist. For more general hydroclimatological studies, a
greater degree of aggregation can be employed. For example, a water-budget for Texas (Ward,
1993) was based upon subdividing the state into four large regions, each of which was
approximately homogeneous climatologically and contained multiple river basins (or sub
segments thereof).* For water-supply reservoir design and water allocation studies, a time
interval of one month has proven effective, being long enough to satisfy desideratum (2) above
while still depicting the seasonal variations in water budget, and is the predominating
convention. Among the basin-scale models that employ or accommodate a one-month time
series are the State of Oklahoma CRAM (Central Resource Allocation Model), U.S. Forest
Service AQUARIUS (Brown et al., 2002), Colorado State University MODSIM (Labadie, 2006),
the State of Texas Water Availability Model (WAM, Sokulsky, 1998; Wurbs, 2003, 2006), the
Stockholm Environment Institute Water Evaluation And Planning (WEAP) model (Juizo and
Liden, 2010; Seiber and Purkey, 2011), the BRACERO model used in the Schmandt et al. (2000)
study of the Rio Grande, and the CSIRO Murray-Darling Integrated River System Modelling
Framework (IRSMF, Kirby et al., 2012; Yang et al., 2013).
Figure 1 depicts a basic “module” of a water-budget, a stream channel terminating in a reservoir,
showing both a sketch of the watershed surface and a stem-diagram depiction. The same
schematic module is shown in Figure 2 with the principal components of the water budget
indicated. All terms in volume of water are accumulated over the accounting time interval. The
water budget is expressed through several equations. Change in storage in the reservoir is
∆V = Qi + R + (P – E) - Qo
(1)
Terms may be added to the right side of this equation for diversions (negative) and return flows
(positive). The term (P – E) is the net volume added at the water surface by direct precipitation
and evaporation. Typically these are measured as water depths, so must be multiplied by the
*
The hierarchy of regional boundaries was: (1) the state boundary; (2) physiographic discontinuity, viz. the
caprock escarpment; (3) drainage divides (as approximated by county boundaries).
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Figure 1 - Sketch of watershed “module” (left) and corresponding stem diagram (right)
surface area of the watercourse. If the stream or reservoir serves as cooling water for steamelectric power generation, then E must include both natural and “forced” evaporation (Ward,
1980).
Usually, the single most important term in the equation is runoff R from the watershed. This is,
in turn, given by
R = (P–i - I)
(2)
where i denotes infiltration into the soil (exchanges with the subsurface are indicated by
lowercase letters), including the initial abstraction from a rainfall event, and I is the sum of
interception on surfaces, mainly the leaves of vegetation, and ponding. (I is not shown in Fig. 2.)
Typically i is measured in depth of water, and must be multiplied by the surface area of the
watershed to determine the total volume. Much of this infiltrated water is temporarily stored in
the near-surface layer, some of which percolates more slowly to deeper layers and deep aquifers.
The change in water stored in the soil over the accounting time interval is:
∆s = i - ET - p
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(3)
Figure 2 - Same as Figure except showing principal components of water balance
where p is the deep percolation rate (not shown in Fig. 2), and ET denotes evapotranspiration, i.e.
the flux from the watershed surface due to direct evaporation from the soil and transpiration by
plants. (Some workers include I in ET.) Further exchanges with the stream channel (e.g.,
interflow) or the reservoir bed (e.g., leakage) would be accommodated by additional terms in
both (1) and (3).
The upstream inflow term Qi is required when that boundary of the element intersects the stream
channel. If the element area encompasses the entirety of the watershed area upstream from the
dam, then Qi is clearly not needed because the upstream terminus of the stream is contained
within the element area. Otherwise, Qi is the outflow from the next element upstream, and is the
means by which two successive elements are coupled. The downstream outflow term Qo
represents any release from the reservoir, including flood spills, gate leakage, and deliberate
releases through the dam. There are two broad categories of deliberate release: (1) service
releases, e.g., hydropower generation, water for downstream users for whom the stream channel
is part of the delivery system, or for makeup of downstream storage facilities, including other
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reservoirs; (2) environmental flows, i.e., flows required to maintain or enhance aquatic and
riparian ecosystems located downstream. A river with a sequence of mainstem dams may be
represented by a concatenation of the modules depicted in Figs. 1 and 2.
The apparent simplicity of the diagrams of Figs. 1 and 2 and the corresponding expressions of
the water budget in the above equations is belied by the complexity of the processes they
represent. First, it should be noted that each term of the water budget is in fact a function of
time. Evapotranspiration and infiltration, for example, involve the same water, except for what
is retained in storage in the soil or percolated to a deeper aquifer, so that approximately ET = i,
but they operate on very different time scales. Infiltration due to storm rainfall occurs only
during and immediately after the event (when the rainfall remains ponded or temporarily in the
distributaries of the watershed) and is otherwise zero, while evapotranspiration operates long
after the storm event and depends upon the growth and metabolism of vegetation. Even if a time
period as long as a month is used in the water budget, it may be necessary to account for the
“carry-over” effect of temporary storage in the watershed and subsequent uptake by plants.
Second, equations (1) – (3) all involve rate terms. This might suggest that the water budget is, in
effect, a balancing of rates. This has a certain cogency, analogous to a balancing of expenses and
income: if negative, the money will eventually be depleted, so the account is unsustainable, and
likewise if ∆V in (1) is negative, the reservoir will eventually fail. For the water budget,
however, some of the terms depend upon the availability of stored water, in the stream channels
and lakes, in the river’s bed and banks, and in the soil. Inflow into a reservoir, for instance,
cannot be retained unless the reservoir is first drawn down below capacity. One of the
advantages of using large increments of time, and for that matter of space, is that the effects of
these latent dependencies tend to “average out,” and a water budget consisting only of rates
becomes more meaningful.
The employment of equations (1) – (3) is dependent upon which terms may be best estimated
from available data (or models) for a given basin. Precipitation data is generally widely
available but may be reported in aggregated amounts, such as monthly or annually.
Evapotranspiration, on the other hand, is difficult to measure, and data are generally not
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available at a watershed scale, and not for extended time periods. It was noted above that runoff
R, which includes any tributaries that are not explicitly separated (and therefore treated as
subwatersheds), is typically the most important term in the water budget. This is not because it
is the largest term, but because it is the principal source for surface water, and therefore is central
to a riverine water budget. If a rigorous stream-gauging network has been in place for a long
period, then this data can be used as a direct estimate of runoff. For large-scale hydroclimate
studies, a runoff-to-rainfall coefficient may be determined from the averages of streamflow and
precipitation (e.g., Sellers, 1965; Ward, 1993). A better approach, if the data are available, is to
regress streamflow against precipitation (e.g., Sellers, 1965; Lanning-Rush, 2000).
Evapotranspiration may then be estimated as ET = P – R, which is equations (2) and (3)
combined, I included in ET and recharge to deep groundwater p neglected. Evaporation from the
water surface is best estimated from pan data, provided the pan data has been well-calibrated
against rigorous field measurements. Lacking pan data, evaporation may be estimated by the
Dalton equation with bulk-aerodynamic coefficient (see Penman, 1948; Ward, 1980; Singh and
Xu, 1997; Sartori, 1999; Brutsaert, 2005; McJannet et al., 2008, among others).
3. Potential use of water budgets in SERIDAS project
From the standpoint of analysis and presentation, the use of a “standard” template for a basinwide annual water budget is recommended, following the general format shown in Table 1. The
details of this template will evolve during the course of the project, and some categories will be
specific, even unique, to particular basins. For purposes of this water budget, the basin is
subdivided into the reach above the downstreammost reservoir (the “impounded” reach), and the
reach below (the “unimpounded” reach). The latter is important only if there is an environmental
flow issue, e.g., the need for freshwater delivery to the estuary of the river. Otherwise, the
template simplifies to a budget for the impounded reach only.
The advantages of a basin-wide summary include the following: (1) the general distribution of
water supply can be exhibited in a compact form; (2) several terms in the water budget involving
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Table 1 - Suggested template for annual water-budget
(Water volumes in million cubic metres unless indicated otherwise)
Watershed area
Reservoir capacity
Impounded reach:
Precipitation
Runoff
Total diversions
Lake evaporation
Discharge downstream
00.0
00.0
km2
Mm3
Mean air temperature
Water-supply capacity
000.0
00.0
0.0
0.0
00.0
Water uses, impounded reach:
M&I
Surface water use
0.0
Forced evaporation
Groundwater use
0.0
Return flows
0.0
Evapotranspiration
Recharge to groundwater
Total returns
Total diversions
agriculture
0.0
0.0
0.0
Downstream of impounded reach:
Precipitation
000.0
Runoff
00.0
Total diversions
0.0
Flow to estuary or mouth
00.0
Water uses, unimpounded reach:
M&I
Surface water use
0.0
Groundwater use
0.0
Return flows
0.0
00.0
00.0
°C
Mm3
000.0
00.0
0.0
0.0
hydroelectric steam-electric
0.0
0.0
0.0
0.0
0.0
0.0
0.0
Evapotranspiration
Recharge to groundwater
Total return flows
000.0
00.0
0.0
agriculture
0.0
0.0
0.0
intrabasin transfers are eliminated; (3) dominant or critical processes may be readily identified,
motivating specific trends and policy analyses; (4) specific terms in the water budget for which
data are lacking or their estimation is dubious may be readily identified and the probable
influence on the overall water budget appraised. It may be desirable in certain basins to
supplement this presentation with subcatchment analyses. Similarly, use of an annual water
budget obviates the complexity of depicting seasonal behavior, and further eliminates “holdover” terms in the water budget. For specific terms in specific basins, supplementary
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information on seasonal variation, such as wet-season precipitation, and dry-season shortfalls can
be presented separately as necessary.
The issue of the time period of aggregation needs to be given more consideration by the team as
the work on individual basins proceeds. Generally a period of some years must be aggregated so
that bias from variance in individual years will be minimized. Two aggregation periods are
suggested: a climatological “normal” and a critical drought.
The climatological “normal”, the averaged values, both annual and monthly, over a 30-year
period beginning with year 1 of a decade (e.g., 1901, 1911, etc.), is a reference average for
climate studies that is widely observed internationally. The use of 30 years as a baseline period
can be traced back to the meeting of the International Meteorological Committee meeting in
1872, though it was not until 1935, at the Commission for Climatology of the International
Meteorological Organization (the predecessor to the World Meteorological Organization, WMO)
in Sopot-Danzig, that the decision was reached to establish the 1901-1930 period as the baseline
for climatological studies (World Climate Programme, 1989; Commission for Climatology,
2011). Later, the WMO specified that the normals would be re-calculated every 30 years, so that
1901-30, 1931-60, etc. are the “standard normals”. (Several countries recalculate normals at a
higher frequency than this.)
As our basins are arid to semi-arid, the occurrence of drought is an immediate demonstration of
the limits of water supply, and the results of this project’s analyses and policy recommendations
will therefore be immediately meaningful to water managers. There is a tendency to focus on the
worst one or two years in a drought period. However, the budget should be constructed for an
entire drought period starting from the last point in time when all reservoirs were full to the point
of minimum available water in storage.
The water-budget format immediately facilitates the construction of alternative scenarios, and
their ceteris paribus evaluation, because the individual terms may be modified to reflect a
specific scenario while the other terms are held constant. Future population growth, new
reservoir construction, increases or decreases in irrigated agriculture, and climate change may be
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individually addressed in this manner. Moreover, the use of a uniform reporting format like
Table 1 will also facilitate cross-basin comparisons.
Sources of data for the terms of Table 1 will vary by basin depending upon the extent and detail
of available information. Some basins have a rich and accessible resource of hydrometeorological data and the benefit of past modeling studies from which the individual terms of Table 1
may be drawn. For others, limited data and the lack of past analytical studies will entail at best
rough estimates of some of the terms.
Uncertainty is ubiquitous in the analysis of natural systems, especially those like hydrometeorology in which the measurements of the key terms are sparse in space and time, and may be
relatively crude. The metric and interpretation of this uncertainty is a separate topic, but in the
context of a water-budget tabulation such as the template of Table 1, it is simple to include
standard error bands (i.e., ± 0.0) after each term, from which the overall uncertainty magnitude
as well as the principal contributors to this uncertainty may be readily identified.
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