There is a complex interac-tion between soil biology, the crop and the hydrological fac-tors such as soil moisture, percolation, run-off, erosion, and evapo-transpiration.. The discussio
Trang 1THE PURPOSES OF HYDROLOGICAL STUDIES
Hydrology is concerned with all phases of the transport of
water between the atmosphere, the land surface and
sub-surface, and the oceans, and the historical development of
an understanding of the hydrological process is in itself
a fascinating study. 6 As a science, hydrology encompasses
many complex processes, a number of which are only
imperfectly understood It is perhaps helpful in developing
an understanding of hydrological theory to focus attention
not on the individual physical processes, but on the
practi-cal problems which the hydrologist is seeking to solve By
studying hydrology from the problem-solving viewpoint,
we shall see the interrelationship of the physical processes
and the approximations which are made to represent
pro-cesses which are either imperfectly understood or too
com-plex for complete physical representation We shall also
see what data is required to make adequate evaluations of
given problems
A prime hydrological problem is the forecasting of
stream-fl ow run-off Such forecasts may be concerned with
daily fl ows, especially peak fl ows for fl ood warning, or a
seasonal forecast may be required, where a knowledge of
the total volume of run-off is of prime interest More
sophis-ticated forecast procedures are required for the day-to-day
operation of fl ood control reservoirs, hydropower projects,
irrigation and water supply schemes, especially for schemes
which are used to serve several purposes simultaneously
such as hydropower, fl ood control, and irrigation
Hydrologists are also concerned with studying statistical
patterns of run-off A special class of problems is the study
of extreme events, such as fl oods or droughts Such
maxi-mum events provide limiting design data for fl ood spillways,
dyke levels, channel design, etc Minimum events are
impor-tant, for example, in irrigation studies and fi sheries projects
A more complex example of statistical studies is concerned
with sequential patterns of run-off, for either monthly or
annual sequences Such sequences are important when
test-ing the storage capacity of a water resource system, such
as an irrigation or hydropower reservoir, when assessing the
risk of failing to meet the requirements of a given scheme
A specially challenging example of sequential fl ow studies
concerns the pattern of run-off from several tributary areas
of the same river system In such studies it is necessary to try
to maintain not only a sequential pattern but also to model
the cross-correlations between the various tributaries
The question of land use and its infl uences on run-off occupies a central position in the understanding of hydrolog-ical processes Land use has been studied for its infl uence on
fl ood control, erosion control, water yield and agriculture, with particular application to irrigation Perhaps the most marked effect of changed land use and changed run-off char-acteristics is demonstrated by urbanization of agricultural and forested lands The paving of large areas and the infl u-ence of buildings has a marked effect in increasing run-off rates and volumes, so that sewer systems must be designed
to handle the increased fl ows Although not so dramatic, and certainly not so easy to document, the infl uence of trees and crops on soil structure and stability may well prove to be the most far-reaching problem There is a complex interac-tion between soil biology, the crop and the hydrological fac-tors such as soil moisture, percolation, run-off, erosion, and evapo-transpiration Adequate hydrological calculations are
a prerequisite for such studies
A long-term aim of hydrological studies is the clear defi nition of existing patterns of rainfall and run-off Such a defi nition requires the establishment of statistical measures such as the means, variances and probabilities of rate events From these studies come not only the design data for extreme events but also the determination of any changes in climate which may be either cyclical or a longterm trend It is being suggested in many quarters that air pollution may have a gradual effect on the Earth’s radiation balance If this is true
we should expect to see measurable changes in our climatic patterns Good hydrological data and its proper analysis will provide one very important means of evaluating such trends and also for measuring the effectiveness of our attempts to correct the balance
A BRIEF NOTE ON STATISTICAL TECHNIQUES The hydrologist is constantly handling large quantities
of data which may describe precipitation, streamfl ow, climate, groundwater, evaporation, and many other factors
A reasonable grasp of statistical measures and techniques is invaluable to the hydrologist Several good basic textbooks are referenced, 1,2,3,8,9 and Facts from Figures by Moroney, is
particularly recommended for a basic understanding of what statistics is aiming to achieve
The most important aspect of the nature of data is the question of whether data is independent or dependent Very
Trang 2often this basic question of dependence or independence is
not discussed until after many primary statistical measures
have been defi ned It is basic to the analysis, to the
selec-tion of variables and to the choice of technique to have some
idea of whether data is related or independent For example,
it is usually reasonable to assume that annual fl ood peaks
are independent of each other, whereas daily streamfl ows are
usually closely related to preceding and subsequent events:
they exhibit what is termed serial correlation
The selection of data for multiple correlation studies
is an example where dependence of the data is in confl ict
with the underlying assumptions of the method Once the
true nature of the data is appreciated it is far less diffi cult to
decide on the correct statistical technique for the job in hand
For example, maximum daily temperatures and incoming
radiation are highly correlated and yet are sometimes both
used simultaneously to describe snowmelt
In many hydrological studies it has been demonstrated
that the assumption of random processes is not
unreason-able Such an assumption requires an understanding of
sta-tistical distribution and probabilities Real data of different
types has been found to approximate such theoretical
distri-butions as the binomial, the Poisson, the normal distribution
or certain special extreme value distributions Especially, in
probability analysis, it is important that the correct
assump-tion is made concerning the type of distribuassump-tion if
extrapo-lated values are being read from the graphs
Probabilities and return periods are important
con-cepts in design studies and require understanding The term
“return period” can be somewhat misleading unless it is
clearly appreciated that a return period is in fact a
probabil-ity Therefore when we speak of a return period of 100 years
we imply that a magnitude of fl ow, or some other such event,
has a one percent probability of occurring in any given year
It is even more important to realize that the probability of a
certain event occurring in a number of years of record is much
higher than we might be led to believe from considering only its annual probability or return period As an example, the
200 year return period fl ood or drought has an annual ability of 0.5%, but in 50 years of record, the probability that
prob-it will occur at least once is 22% Figure 1 summarizes the probabilities for various return periods to occur at least once
as a function of the number of years of record From such a graph it is somewhat easier to appreciate why design fl oods for such critical structures as dam spillways have return of 1,000 years or even 10,000 years
ANALYSIS OF PRECIPITATION DATA Before analyzing any precipitation data it is advisable to study the method of measurement and the errors inherent
in the type of gauge used Such errors can be considerable (Chow, 1 and Ward 5 )
Precipitation measurements vary in type and precision, and according to whether rain or snow is being measured Precipitation gauges may be read manually at intervals of a day or part of a day Alternatively gauges may be automatic and yield records of short-term intensity Wind and gauge exposure can change the catch effi ciency of precipitation gauges and this is especially true for snow measurements Many snow measurements are made from the depth of new snow and an average specifi c gravity of 0.10 is assumed when converting to water equivalent
Precipitation data is analyzed to give mean annual values and also mean monthly values which are useful in assessing seasonal precipitation patterns Such fi gures are useful for determining total water supply for domestic, agricultural and hydropower use, etc
More detailed analysis of precipitation data is given for individual storms and these fi gures are required for design of drainage systems and fl ood control works Analysis shows the
10 0
.2 4 6 8 1.0
Trang 3relationship between rain intensity (inches per hour) with both
duration and area In general terms, the longer the duration
of storm, the lower will be the average intensity of rainfall
Similarly, the larger the area of land being considered, the
lower will be the average intensity of rainfall For example, a
small catchment area of, say, four square miles may be
sub-jected to a storm lasting one hour with an average intensity of
two inches per hour while a catchment of two hundred square
miles would only experience an average intensity of about one
inch per hour Both these storms would have the same return
period or probability associated with them Such data is
pre-pared by weather agencies like the U.S Weather Bureau and
is available in their publications for all areas of the country
Typical data is shown in Figure 2 The use of these data sheets
will be discussed further in the section on run-off
Winter snowpacks represent a large water storage which
is mainly released at a variable rate during spring and early
summer In general, the pattern of snowfall is less important
than the total accumulation In the deep mountain snowpacks,
snowtube and snowpillow measurements appear to give fairly
reliable estimates of accumulated snow which can be used for
forecasts of run-off volumes as well as for fl ood forecasting
On the fl at prairie lands, where snow is often quite moderate
in amounts, there is considerable redistribution and drifting
of snow by wind and it is a considerable problem to obtain
good estimates of total snow accumulation
When estimates of snow accumulation have been made
it is a further problem to calculate the rate at which the snow
will melt and will contribute to stream run-off Snow
there-fore represents twice the problem of rain, because fi rstly we
must measure its distribution and amount and secondly, it
may remain as snow for a considerable period before it
con-tributes to snowmelt
EVAPORATION AND EVAPO-TRANSPIRATION
Of the total precipitation which falls, only a part fi nally
dis-charges as streamfl ow to the oceans The remainder returns
to the atmosphere by evaporation Linsley 2 points out that ten
reservoirs like Lake Mead could evaporate an amount
equiv-alent to the annual Colorado fl ow Some years ago, studies
of Lake Victoria indicated that the increased area resulting
from raising the lake level would produce such an increase in
evaporation that there would be a net loss of water utilization
in the system
Evaporation varies considerably with climatic zone,
latitude and elevation and its magnitude is often diffi cult to
evaluate Because evaporation is such a signifi cant term in
many hydrological situations, its proper evaluation is often a
key part of hydrological studies
Fundamentally, evaporation will occur when the vapor
pressure of the evaporating surface is greater than the vapor
pressure of the overlying air Considerable energy is required to
sustain evaporation, namely 597 calories per gram of water or
677 calories per gram of snow or ice Energy may be supplied
by incoming radiation or by air temperature, but if this energy
supply is inadequate, the water or land surface and the air will
cool, thus slowing down the evaporation process In the long term the total energy supply is a function of the net radiation balance which, in turn, is a function of latitude There is there-fore a tendency for annual evaporation to be only moderately variable and to be a function of latitude, whereas short term evaporation may vary considerably with wind, air temperature, air vapor pressure, net radiation, and surface temperature The discussion so far applies mainly to evaporation from
a free water surface such as a lake, or to evaporation from a saturated soil surface Moisture loss from a vegetated land surface is complicated by transpiration Transpiration is the term used to describe the loss of water to the atmosphere from plant surfaces This process is very important because the plant’s root system can collect water from various depths
of the underlying soil layers and transmit it to the atmosphere
In practice it is not usually possible to differentiate between evaporation from the soil surface and transpiration from the plant surface, so it is customary to consider the joint effect and call it evapo-transpiration This lumping of the two processes has led to thinking of them as being identical, however, we do know that the evaporation rate from a soil surface decreases
as the moisture content of the soil gets less, whereas there
is evidence to indicate that transpiration may continue at a nearly constant rate until a plant reaches the wilting point
To understand the usual approach now being taken to the calculation of evapo-transpiration, it is necessary to appreciate
what is meant by potential evapo-transpiration as opposed to
actual evapo-transpiration Potential evapo-transpiration is the
moisture loss to the atmosphere which would occur if the soil layers remained saturated Actual evapo-transpiration cannot exceed the potential rate and gradually reduces to a fraction
of the potential rate as the soil moisture decreases Various formulae exist for estimating potential evapo-transpiration in terms of climatic parameters, such as Thornthwaites method,
or Penman or Turk’s formulae Such investigations have shown that a good fi eld measure of potential evapo-transpiration is pan evaporation from a standard evaporation-pan, such as the Class A type, and such measurements are now widely used To turn these potential estimates into actual evapo-transpiration
it is commonly assumed that actual equals potential after the soil has been saturated until some specifi c amount of mois-ture has evaporated, say two inches or so depending on the soil and crop It is then assumed that the actual rate decreases exponentially until it effectively ceases at very low moisture contents In hydrological modeling an accounting procedure can be used to keep track of incoming precipitation and evapo-ration so that estimates of evapo-transpiration can be made The potential evapo-transpiration rate must be estimated from one of the accepted formulae or from pan-evaporation mea-surements, if available Details of such procedures are well illustrated in papers by Nash 17 and by Linsley and Crawford 44
in the Stanford IV watershed model
RUN-OFF: RAIN
It is useful to imagine that we start with a dry catchment, where the groundwater table is low, and the soil moisture
Trang 4has been greatly reduced, perhaps almost to the point where
hygroscopic moisture alone remains When rain fi rst starts
much is intercepted by the trees and vegetation and this
inter-ception storage is lost by evaporation after the storm Rain
reaching the soil infi ltrates into pervious surfaces and begins
to satisfy soil moisture defi cits As soil moisture levels rise, water percolates downward toward the fully saturated water table level If the rain is heavy enough, the water supply may exceed the vertical percolation rate and water then starts to
fl ow laterally in the superfi cial soil layers toward the stream
Trang 5channels: this process is termed interfl ow and is much debated
because it is so diffi cult to measure At very high rainfall rates,
the surface infi ltration rate may be exceeded and then direct
surface run-off will occur Direct run-off is rare from soil
sur-faces but does occur from certain impervious soil types, and
from paved areas Much work has been done to evaluate the
relative signifi cance of these various processes and is well
documented in references (1,2,3)
Such qualitative descriptions of the run-off process are
helpful, but are limited because of the extreme complexity
and interrelationship of the various processes Various
meth-ods have been developed to by-pass this complexity and to
give us usable relationships for hydrologic calculations
The simplest method is a plot of historical events, showing
run-off as a function of the depth of precipitation in a given
storm This method does not allow for any antecedent soil
moisture conditions or for the duration of a particular storm
More complex relationships use some measure of soil
moisture defi ciency such as cumulative pan-evaporation or
the antecedent precipitation index Storm duration and
pre-cipitation amount is also allowed for and is well illustrated
by the U.S Weather Bureau’s charts developed for various
areas (Figure 2) It is a well to emphasize that the
anteced-ent precipitation index, although based on precipitation, is
intended to model the exponential decay of soil moisture
between storms, and is expressed by
I N ( I 0 k M I M ) k (N − M) where I 0 is the rain on the fi rst day and no more rain occurs
until day M, when I M falls If k is the recession factor, usually
about 0.9, then I N will be the API for day N The
expres-sion can of course have many more terms according to the
number of rain events
Before computers were readily available such
calcu-lations were considered tedious Now it is possible to use
more complex accounting procedures in which soil moisture
storage, evapo-transpiration, accumulated basin run-off,
percolation, etc can all be allowed for These procedures
are used in more complex hydrological modeling and are
proving very successful
RUN-OFF: SNOWMELT
As a fi rst step in the calculation of run-off from snow,
meth-ods must be found for calculating the rate of snowmelt This
snowmelt can then be treated similarly to a rainfall input
Snowmelt will also be subject to soil moisture storage effects
and evapo-transpiration
The earliest physically-based model to snowmelt was
the degree-day method which recognized that, despite the
complexity of the process, there appeared to be a good
cor-relation between melt rates and air temperature Such a
relationship is well illustrated by the plots of cumulative
degree-days against cumulative downstream fl ow, a rather
frustrating graph because it cannot be used as a
forecast-ing tool This cumulative degree-day versus fl ow plot is an
excellent example of how a complex day-to-day behavior yields a long-term behavior which appears deceptively simple Exponential models and unit hydrograph methods have been used to turn the degree-day approach into a work-able method and a number of papers are available describing such work (Wilson, 38 Linsley 32 ) Arguments are put forward that air temperature is a good index of energy fl ux, being
an integrated result of the complex energy exchanges at the snow surface (Quick 33 )
Light’s equation 31 for snowmelt is based on physical reasoning which models the energy input entirely as a tur-bulent heat transfer process The equation ignores radiation and considers only wind speed as the stirring mechanism, air temperature at a standard height as the driving gradient for heat fl ow and, fi nally, vapour pressure to account forcondensation–evaporation heat fl ux It is set up for 6 hourly computation and requires correction for the nature of the forest cover and topography It is interesting to compare Light’s equation with the U.S Crops equation 36 for clear weather to see the magnitude of melt attributed to each term
By far the most comprehensive studies of snowmelt have been the combined studies by the U.S Corps of Engineers and the Weather Bureau (U.S Corps of Engineers 36,37 ).They set up three fi eld snow laboratory areas varying in size from 4 to 21 square miles and took measurements for periods ranging from 5 to 8 years Their laboratory areas were chosen to be representative of certain climatic zones Their investigation was extensive and comprehensive, rang-ing from experimental evaluation of snowmelt coeffi cient
in terms of meteorological parameters, to studies of mal budgets, snow-course and precipitation data reliability, water balances, heat and water transmission in snowpacks, streamfl ow synthesis, atmospheric circulations, and instru-mentation design and development
A particularly valuable feature of their study appears to have been the lysimeters used, one being 1300 sq.ft in area and the other being 600 sq.ft (Hilderbrand and Pagenhart 30 ).The results of these lysimeter studies have not received the attention they deserve, considering that they give excellent indication of storage and travel time for water in the pack It may be useful to focus attention on this aspect of the Corps work because it is not easy to unearth the details from the somewhat ponderous Snow Hydrology report Before leav-ing this topic it is worth mentioning that the data from the U.S studies is all available on microfi lm and could be valu-able for future analysis It is perhaps useful at this stage to write down the Light equation and the clear weather equa-tion from the Corps work to compare the resulting terms Light’s equation 31 (simple form in °F, inches of melt and standard data heights)
DU 0 001 84T a 10 0 00001560 00578 e 6 11
where
U = average wind speed (m.p.h.) for 6 hr period
T = air temperature above 32°F for 6 hr period
Trang 6e vapor pressure for 6 hr period
h station elevation (feet)
D melt in inches per 6 hr period
The U.S Corps Equation is 36
M = Incident Radiation incoming clear air longwave
cloud longwave [Conduction Condensation]
k ′ and k are approximately unity
N fraction of cloud cover
I i incident short wave radiation (langleys/day)
a albedo of snow surface
T a daily mean temperature °F above 32°F at 10′ level
T c cloud base temperature
T d dew point temperature °F above 32°F
U average wind speed—miles/hour at 50′ level
Putting in some representative data for a day when the
mini-mum temperature was 32°F and the maximini-mum 70°F,
incom-ing radiation was 700 langleys per day and relative humidity
varied from 100% at night to 60% at maximum temperature,
the results were:
Light Equation
D Air temp melt and Condensation melt
1.035 0.961 inches/day
1.996 inches/day
U.S Corps Equation
M incoming shortwave incoming longwave
air temp Condensation
1.424 0.44 0.351 0.59
1.925 inches/day
Note the large amount attributed to radiation which the Light
equation splits between air temperature and radiation It is
a worthwhile operation to attempt to manufacture data for
these equations and to compare them with real data The
high correlations between air temperature and radiation is
immediately apparent, as is the close relationship between
diurnal air temperature variation and dewpoint temperature
during the snowmelt season Further comparison of the
for-mulae at lower temperature ranges leave doubts about the
infl uence of low overnight temperatures
There is enough evidence of discrepancies between real
and calculated snowmelt to suggest that further study may
not be wasted effort Perhaps this is best illustrated from
some recent statements made at a workshop on Snow and Ice
Hydrology. 39 Meier indicates that, using snow survey data,
the Columbia forecast error is 8 to 14% and occasionally
40 to 50% Also these errors occurred in a situation where
the average deviation from the long-term mean was only
12 to 20% For a better comparison of errors it would be interesting to know the standard error of forecast compared with standard “error” of record from the long-term mean Also, later in the same paper it is indicated that a correct heat exchange calculation for the estimation of snowmelt cannot
be made because of our inadequate knowledge of the eddy convection process At the same workshop the study group
on Snow Metamorphism and Melt reported: “we still cannot measure the free water content in any snow cover, much less the fl ux of the water as no theoretical framework for fl ow through snow exists.”
Although limitations of data often preclude the use of the complex melt equations, various investigators have used the simple degree-day method with good success (Linsley 32 and Quick and Pipes 40,46,47 ) There may be reasonable justifi ca-tion for using the degree-day approach for large river basins with extensive snowfi elds where the air mass tends to reach a dynamic equilibrium with the snowpck so that energy supply and the resulting melt rate may be reasonably well described
by air temperature In fact there seems to be no satisfactory compromise for meteorological forecasting; either we must use the simple degree-day approach or on the other hand we must use the complex radiation balance, vapour exchange and convective heat transfer methods involving sophisticated and exacting data networks
COMPUTATION OF RUN-OFF—
SMALL CATCHMENTS Total catchment behavior is seen to be made up of a number
of complex and interrelated processes The main processes can be reduced to evapo-transpiration losses, soil moisture and groundwater storage, and fl ow of water through porous media both as saturated fl ow and unsaturated fl ow To describe this complex system the hydrologist has resorted to a mix-ture of semi-theoretical and empirical calculation techniques Whether such techniques are valid is justifi ed by their abil-ity to predict the measured behavior of a catchment from the measured inputs
The budgeting techniques for calculating transpiration losses have already been described From an estimation of evapo-transpiration and soil moisture and mea-sured precipitation we can calculate the residual precipita-tion which can go to storage in the catchment and run-off
evapo-in the streams A method is now required to determevapo-ine at what rate this effective precipitation, as it is usually called, will appear at some point in the stream drainage system The most widely used method is the unit hydrograph approach
fi rst developed by Sherman in 1932. 16
To reduce the unit hydrograph idea to its simplest form, consider that four inches of precipitation falls on a catch-ment in two hours After allowing for soil moisture defi cit and evaporation losses, let us assume that three inches of this precipitation will eventually appear downstream as run-off Effecitvely this precipitation can be assumed to have fallen
on the catchment at the rate of one and a half inches per hour
Trang 7for two hours This effective precipitation will appear some
time later in the stream system, but will now be spread out
over a much longer time period and will vary from zero fl ow,
rising gradually to a maximum fl ow and then slowly
decreas-ing back to zero Figure 3 shows the block of uniform
precip-itation and the corresponding outfl ow in the stream system
The outfl ow diagram can be reduced to the unit hydrograph
for the two hour storm by dividing the ordinates by three
The outfl ow diagram will then contain the volume of run-off
equivalent to one inch of precipitation over the given
catch-ment area For instance, one inch of precipitation over one
hundred square miles will give an area under the unit
hydro-graph of 2690 c.f.s days
When a rainstorm has occurred the hydrologist must fi rst
calculate how much will become effective rainfall and will
contribute to run-off This can best be done in the framework
of a total hydrological run-off model as will be discussed later
The effective rainfall hydrograph must then be broken down
into blocks of rainfall corresponding to the time interval for
the unit hydrograph Each block of rain may contain P inches
of water and the corresponding outfl ow hydrograph will have
ordinates P times as large as the unit hydrograph ordinates
Also, several of these scaled outfl ow hydrographs will have to
be added together This process is known as convolution and
is illustrated in Figure 4 and 5
The underlying assumption of unit hydrograph theory is
that the run-off process is linear, not in the trivial straight line
sense, but in the deeper mathematical sense that each
incre-mental run-off event is independent of any other run-off In
the early development, Sherman 16 proposed a unit
hydro-graph arising from a certain storm duration Later workers
such as Nash 17,23 showed that Laplace transform theory, as
already highly developed for electric circuit theory, could be used This led to the instantaneous unit hydrograph and gave rise to a number of fascinating studies by such workers as Dooge, 18 Singh, 19 and many others They introduced expo-nential models which are interpretable in terms of instanta-neous unit hydrograph theory Basically, however, there is
no difference in concept and the convolution integral, Eq (1) can be arrived at by either the unit hydrograph or the instan-taneous unit hydrograph approach The convolution integral can be written as:
0
0
Figure 4 shows the defi nition diagram for the formulation is
only useful if both P, the precipitation rate, and u, the
instan-taneous unit hydrograph ordinate are expressible as ous functions of time In real hydrograph applications it is more useful to proceed to a fi nite difference from of Eq (1)
continu-in which the continu-integral is replaced by a summation, Eq (2), and Figure 5
3 ins of Rain
1000 2000
Actual Run-off Q
P
Unit Hydrograph Area equals 1 inch Rain
FIGURE 3 Hydrograph and unit hydrograph of run-off from effective rain.
Trang 8Expanding Eq (2) for a particular value of R,
The whole family of similar equations for Q may be expressed
in matrix form (Snyder 20 )
P
n n n
u
columns
1 2 3 (5)
Or more briefl y
Equation (6) specifi es the river fl ow in terms of the
precipita-tion and the unit hydrograph In practice Q and P are sured and u must be determined Some workers have guessed
mea-a suitmea-able functionmea-al form for u with one or two unknown
parameters and have then sought a best fi t with the available data For instance, Nash’s series of reservoirs yields 17,23
u t n
t
n n
t k
1
in which there are two parameters, K and n Another approach
is to solve the matrix Eq (6) as follows (Synder 21 )
{ }⎡⎣ ⎤⎦1 { } (8)
It has already been demonstrated that R m + n − 1 so that
there are more equations available than there are unknowns The solution expressed by Eq (8) therefore automatically
yields the least squares values for u This result will be
referred to after the next section
t- τ
u(t- τ) u
O
Q(t)= τ<4 0 u(t- τ)P(τ)dτ
FIGURE 4 Determination of streamflow from precipitation input
using an instantaneous unit hydrograph.
Trang 9MULTIPLE REGRESSION AND STREAMFLOW
The similarity of the fi nite difference unit hydrograph approach
to multiple regression analysis is immediately apparent The
fl ow in terms of precipitation can be written as
The similarity with Eq (6) is obvious and may be complete
if we have selected the correct precipitation data to correlate
precipitation at 6 a.m with downstream fl ow at 9 a.m when
we know that there is a 3-hour lag in the system Therefore,
using multiple regression as most hydrologists do, the method
can become identical with the unit hydrograph approach
LAKE, RESERVOIR AND RIVER ROUTING
The run-off calculations of the previous sections enable
esti-mates to be made of the fl ow in the headwaters of the river
system tributaries The river system consists of reaches of
channels, lakes, and perhaps reservoirs The water travels
downstream in the various reaches and through the lakes
and reservoirs Tributaries combine their fl ows into the main
stream fl ow and also distributed lateral infl ows contribute to
the total fl ow This total channel system infl uences the fl ow
in two principal ways, fi rst the fl ow takes time to progress
through the system and secondly, some of the fl ow goes into
temporary storage in the system Channel storage is usually
only moderate compared with the total river fl ow quantities,
but lake and reservoir storage can have a considerable infl
u-ence on the pattern of fl ow
Calculation procedures are needed which will allow for
this delay of the water as it fl ows through lakes and
chan-nels and for the modifying infl uence of storage The problem
is correctly and fully described by two physical equations,
namely a continuity equation and an equation of motion
Continuity is simply a conversion of mass relationship while
the equation of motion relates the mass accelerations to the
forces controlling the movement of water in the system Open
channel fl uid mechanics deals with the solution of such
equa-tions, but at present the solutions have had little application
to hydrological work because the solutions demand detailed
data which is not usually available and the computations are
usually very complex, even with a large computer
Hydrologists resort to an alternative approach which is
empirical; it uses the continuity equation but replaces the
equa-tion of moequa-tion with a relaequa-tionship between the storage and the
fl ow in the system This assumption is not unreasonable and is
consistent with the assumption of a stage–discharge ship which is widely utilized in stream gauging
RESERVOIR ROUTING The simplest routing procedure is so-called reservoir rout-ing, which also applies to natural lakes The continuity equa-tion is usually written as;
The second equation relates storage purely to the outfl ow, which
is true for lakes and reservoirs, where the outfl ow depends only
on the lake level The outfl ow relationship may be of the form:
if the outfl ow is controlled by a rectangular weir, or:
where K ′ and n depend on the nature of the outfl ow channel
Such relationships can be turned into outfl ow—storage
relationships because storage is a function of H, the lake level
The Eqs (12) and (13) can then be rewritten in the form
Alternatively, there may be no simple functional relationship,
but a graphical relationship between O and S can be plotted
or stored in the computer The continuity equation and the outfl ow storage relationship can then be solved either graph-ically or numerically, so that, given certain infl ows, the out-
fl ows can be calculated Notice the assumption that a lake or reservoir responds very rapidly to an infl ow, and the whole lake surface rises uniformly
During the development of the kinematic routing model described later, a reservoir routing technique was developed which has proved to be very useful Because reservoir rout-ing is such an important and basic requirement in hydrology, the method will be presented in full
Reservoir routing can be greatly simplifi ed by ing that complex stage–discharge relationships can be lin-earised for a limited range of fl ows It is even more simple
recogniz-to relate stage levels recogniz-to srecogniz-torage and then recogniz-to linearise the storage–discharge relationship The approach described below can then be applied to any lake or reservoir situation, ranging from natural outfl ow control to the operation of gated spill-ways and turbine discharge characteristics
Trang 10From a logical point of view, it is probably easier to
develop the routing relationship by considering storage, or
volume changes In a fi xed time interval ∆ T, the reservoir
infl ow volume is VI ( J ), where J indicates the current time
interval The corresponding outfl ow volume is VO ( J ) and the
reservoir storage volume is S ( J ) If the current infl ow volume
VI ( J ) were to equal the previous outfl ow value VO ( J −1), then
the reservoir would be in a steady state and no change in
res-ervoir storage would occur Using the hypothetical steady
state as a datum for the current time interval, we can defi ne
changes in the various fl ow and storage volumes, where ∆
where QO ( J ) is the outfl ow which is equal to VO ( J )/ ∆ T The
corresponding equation for the previous time interval is
This equation can be rewritten for fl ows by substituting
∆ VO ( J ) equals ∆ QO ( J )* ∆ T and ∆ VI ( J ) equals ∆ QI ( J )* ∆ T,
Equation (25) to (27) represent an extremely simple
reser-voir or lake routing procedure To achieve this simplicity,
the change in infl ow, ∆ QI ( J ), and the change in outfl ow,
∆ QO ( J ), must each be changes from the outfl ow, QO ( J − 1),
in the previous time interval, as defi ned in a similar manner
to Eqs (15) and (16) The value of K is determined from the storage-discharge relationship, where K is the gradient,
d S /d Q This storage factor, K, which has dimensions of time,
can be considered constant for a range of outfl ows
When the storage–discharge relationship is non-linear, which is usual, it is necessary to sub-divide into linear seg-
ments The pivotal values of storage, S ( P,N ), and discharge,
QO ( P,N ), where N refers to the N th pivot point, are
tabu-lated Calculations proceed as described until a pivotal value
is approached, or is slightly passed The next value of K is
calculated, not from the two new pivotal values, but from the
latest outfl ows QO ( J ) and from the corresponding storage
S ( J ) The current value of storage is calculated from,
is always a direct and unique relationship between storage,
S ( J ), and outfl ow, QO ( J ).
In summary, the factor 1/(1 + K / ∆ t ) in Eq (25),
repre-sents the proportion of the infl ow change, ∆ QI ( J ) which
becomes outfl ow The remaining infl ow change becomes storage The process is identical for increasing or decreas-ing fl ows: when fl ows decrease, the changes in outfl ow and storage are both negative Eqs (25) and (27), the heart of the matter, are repeated for emphasis,
to an input Also, storage is a function of conditions at each end of the length of channel being considered, rather than just the conditions at the outfl ow end
The simplest channel routing procedure is the so-called Muskingum method developed on the Muskingum River (G.T McCarthy, 24 Linsley 2 )
Trang 11Channel routing is also based on continuity, Eq (11)
Before it can be utilized this equation must be rewritten in
fi nite difference form
Also, some assumption must be made from which storage
can be computed The Muskingum method linearizes the
problem and assumes that storage in a whole channel reach
is completely expressible in terms of infl ow and outfl ow
from the reach, namely
S K [ xI + (1 − x ) O ] (30)
Substituting from (16) in (15), following Linsley, 2 the result
obtained is,
O2c I0 2c I1 1c O2 1 (31)
where c 0 , c 1 and c 2 are functions of K, x and t and c 0 + c 1 +
c 2 = 1 The infl uence of x is illustrated in Figure 6 Extending
this equation to N days:
c I c N O
1
(32)
Usually, because c 0 , c 1 and c 2 are less than one, terms in c 3
and higher are ignored, but whatever the simplifi cation, the
result is of the form
con-null matrix.) It must be admitted that, from the theory, only
a 1 , a 2 and a 3 are independent, but in practice the precision
with which a solution can be obtained for c 0 , c 1 and c 2 does
not justify calculating a 4 , a 5 etc from the fi rst three The best approach at this stage is once more to resort to least squares
fi tting, recasting Eq (34) in the form of Eq (8.) Many hydrologists calculate the routing coeffi cients by
evaluating K and x in the traditional graphical method
Many assumptions are made in the Muskingum method, such as the linear relationship between storage and discharge, and the implied linear variation of water surface along the reach In spite of these assumptions the method has proved its value
Another diffi culty which occurs with the Muskingum method usually occurs when the travel time in the reach and the time increments for the data are approximately equal, such as when the travel time is about one day and the data available is mean daily fl ows From the strict mathematical viewpoint, the time ∆ t should be a fraction of the travel time
K, otherwise fl ow gradients such as d O /d t are not well
rep-resented on a fi nite difference basis However, it is still sible to use time increments ∆ t greater than K if the fl ow is
pos-only changing slowly, but caution is necessary The reason
for caution is that the C values in Eq (31) are a function of ∆ t / K and are not constant
Solving for the C ’s in terms of K, x and ∆ t:
(35c) Hence,
C0 C1 C2 1 (35d)
To illustrate the infl uence of ∆ t and K, some synthetic data as
used to construct Figure 7 Values for K and x were chosen and when values of C 0 , C 1 , and C 2 were calculated for differ-ent values of ∆ t In addition, an assumed infl ow was routed using the K and x values and using a time interval ∆ t which was small compared with K These resulting infl ows and outfl ows were then reanalyzed for C 0 , C 1 , and C 2 using time increments fi ve times greater than the original ∆ t Such ∆ t values exceeded K The new estimates of C 0 , C 1 , and C 2 are Time
FIGURE 6 Muskingum routing: to illustrate the influence of x on
Trang 12also plotted in Figure 7 Note the general agreement in shape
of the C 0 and curves, etc Exact agreement is lost because
of the diminishing accuracy of the d O /d t terms, etc as ∆ t
increases Note also how rapidly the C values change when
∆ t is approximately equal to K
These remarks are not necessarily meant to dissuade
hydrologists from using the Muskingum method The
inten-tion is to illustrate some of the pitfalls so that it may be
pos-sible to evaluate the probable validity or constancy of the
coeffi cients (Laurenson 22 ) The worst situation appears to be
when ∆ t ; K, because in real rivers K decreases with rising
stage and the C values are very sensitive to whether ∆ t is just
less than or greater than K
A very real problem in the application of the Muskingum
method, and in fact of any channel routing procedure, is the
problem of lateral infl ow to the channel reach Given the
infl ows and outfl ows for the reach as functions of time, it
is necessary to separate out the lateral fl ows before best fi t
values of K and x can be determined The lateral fl ows will,
in general, bear no relationship to the pattern of main
stream-fl ows Sometimes it is possible to use stream-fl ow measurements
on a local tributary stream as an index of total lateral fl ow
The cumulative volume of lateral fl ow can be determined
by subtracting summed infl ows from summed outfl ows The
measured tributary fl ow can then be scaled up to equal the
total lateral fl ow and these fl ows can then be subtracted from the reach outfl ows This residual outfl ow can be used in the determination of the Muskingum coeffi cients
Sometimes it is possible to fi nd periods of record where lateral fl ows are small or perhaps have a more predictable pattern, such as during recession periods Also, the routing coeffi cients can be refi ned by an iterative procedure and by using various sets of data, although not infrequently it is found necessary to defi ne Muskingum coeffi cients for dif-ferent ranges of fl ow, because real stream fl ow is not linear
as is assumed in the model
KINEMATIC WAVE THEORY
It has been known for many years that fl ood wave movement
is much slower that would be expected from the shallow
water wave theory result of gy Seddon 25 showed as early
as 1900 that fl ood waves on the Mississippi moved at about 1.5 of river velocity The theoretical reasons for this slowness
of fl ood wave travel have only been realized during the last twenty years The theory describing these waves is known either as monoclinic wave theory or kinematic wave theory Kinematic wave theory has suffered from neglect because of its usual presentation in the literature The common approach is to start with the continuity equation and a simplifi ed Manning equation These equations are combined to yield Seddon’s Law that fl ood waves travel at about 1.5 times the mean fl ow velocity Such an approach restricts the method to an unalterable fl ood wave translating through the channel system Although such a simple model has the benefi t of approximating the real situation, much more useful and general results are available by setting up the two equations in fi nite difference form and carrying out simultaneous solution on a computer In particular the method handles lateral infl ow with great ease and also cal-culates discharge against time, eliminating the guess work from the time calculation
Kinematic wave theory can be seen to be a simplifi cation
of the more general unsteady fl ow theory in open channels, for which the equation of motion and the continuity equation can be written (Henderson and Wooding 4,26 )
x
v g
v
v t
v
c R
0
2 21
C0C'0
C'2
C1C'1
C2
0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6
FIGURE 7 Muskinghum coefficients as a function of storage
and routing period.