RUNOFF
Excerpt from Lars Bengtsson ”Hydrologi – Teori och
processer”, Dept of Water Resources Engineering Lund
University. Translated by Rolf Larsson Sept 2004.
This chapter treats runoff from catchments. The focus is on modelling with box
models and numerical methods. A catchment is topographically separated from its
surroundings. It may contain flat land, slopes, forests, wetlands, lakes and
rivers as shown in Figure 1.precipitationwhich does not evaporate will eventually
– after short or long time storage – reach the outlet from the catchment. Runoff
as a function of time can be estimated with more or less sophisticated models.
Over long time periods the runoff can be estimated through a water balance. When
calculating runoff it is generally necessary to take into account the relationship
between storage and runoff.
Precipitation is transformed into runoff “after” water has been evaporated and
transpirated to the atmosphere. Precipitation or meltwater infiltrates into the
soil. Groundwater moves towards ditches and streams. If much water is added to
the groundwater runoff becomes more intensive. If groundwater rises to the soil
surface rain water has to runoff on the surface. Surface runoff also occurs if
the rain intensity exceeds the infiltration capacity. From ditches and small
streams water moves via larger streams and rivers to the outlet from the catchment.
Figure 1 Catchment with two precipitation gauges.
Figure 2
Conceptual description of the runoff process.
A conceptual description of the runoff process is given in Figure 2. In order
to describe the runoff process in a physically proper way one, has to describe
the water movement at all points in the catchment. One should – see Figure 3
– describe how some precipitation infiltrates into the soil while some forms
surface runoff, how water goes to the atmosphere as evaporation and transpiration,
how water percolates to the groundwater and flows via small streams and rivers
to the catchment outlet. One should describe how the groundwater level rises
in a hillslope and some water forms surface runoff. One should take into account
that some land is impermeable and that some water remains in puddles on the ground.
Percolation through the unsaturated zone shall be described everywhere with the
Richards equation while taking into account the variation of the groundwater
level. Groundwater flow shall be described everywhere with the Darcy equation.
Surface runoff should be described with the Manning eq everywhere. Flow in streams
and rivers should be calculated using the full momentum equations. However all
of this is not possible because the catchment cannot be described in necessary
detail. Instead one normally chooses to describe the runoff process in a very
simplified way.
In this chapter some different ways of calculating runoff are presented. The
focus is on conceptual box models. Traditional methods like the water balance,
unit hydrograph and the – in US much used – API/SCS curve methods are presented.
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Figure 3
Schematic picture of the runoff process in a catchment.
WATER BALANCE
The water balance for a catchment with area A is
dS
=p-e-q
dt
(1a)
with specific runoff
q=
Q
A
(1b)
where Q = runoff at the outlet, p = precipitation intensity (e.g. mm/d),
e = rate of evaporation, t = time and S = water storage in the area expressed
as volume/catchment area.
For long time periods, years, storage change is small. Runoff can then be estimated
as p - e. However, it is usually the areal evaporation that is calculated from
the water balance since precipitation and discharge are more easily measured.
Water storage in a catchment consists of snow, surface water (rivers and lakes)
soil water in the unsaturated zone and groundwater. Therefore with h denoting
storage (volume per area),
S = h = hsnow + hsurface + hUZ + hgw
(2)
If the storages are measured, which is quite difficult since they should represent
averages over large areas, while also measuring river discharge and precipitation,
then evaporation can be estimated also over shorter time scales; or vice versa
if the evaporation has been measured then the runoff can be estimated from water
balance.
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RESERVOIR MODEL
The more water there is in a lake the higher the outflow from that lake. In analogy:
the more water stored in a catchment the higher the runoff. This does not apply
when water is stored as snow. A catchment can be symbolically represented by
a box with a hole and partly filled with water, Figure 4. The higher the water
level the higher the outflow. The water level in the box represents the amount
of water stored in the catchment. However, it is an imaginary value essentially
corresponding to near surface groundwater. More discussion will be presented
further down. There is a relationship between runoff and water storage in
catchment:
q = f (h)
(3)
This relationship may be different depending on whether the water level is rising
or sinking. It may also vary depending on different vegetation during the year
and depending on where in the catchment water is stored. For the box with a hole
there are no such variations. Accordingly- when using this box analogy – the
relationship is static. If the runoff relationship (3) is combined with the
continuity equation (1) and storage is denoted h (not S) you get an equation
with h as the only unknown
Figure 4
Outflow from a reservoir is a function of the water level.
dh
= p - e - f(h)
dt
(4)
Since precipitation usually infiltrates through the soil to the groundwater
before contributing to runoff, it is usually not enough to use only one box model
to describe runoff in a catchment where water flows via groundwater to river
discharge. If the soil in the catchment is very wet – at field capacity – so
that precipitation forms runoff very quickly then the reservoir theory can be
applied. Reservoir theory is also directly applicable to lakes.
Simple reservoir
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Catchment runoff can usually be modeled as a linear reservoir.
q = f (h) = h/T
(4)
where T = reservoir time constant. Sometimes, maybe more usual, but less good
from an understanding point of view, the reservoir constant is defined as 1/T.
The smaller T, the faster the catchment responds to precipitation, i.e. the faster
the runoff increases. For a small steep area the constant is smaller than for
a large flat area. The time constant is constant only within intervals of h or
runoff. His will be explained further on.
The equation for a linear reservoir is produced when f(h) in eq. (4) is substituted
in eq. (1).
dh
h
=p-edt
T
(5)
p-e is substituted by an effective p. The analytical solution for constant p
is
h = ho e
- t/T
+ T p (1 - e
- t/T
)
(6)
where ho is h(t = 0). If you choose to eliminate h instead of q you get
q = qo e
- t/T
+ p (1 - e
- t/T
)
(7)
where qo = q(t=0).
If the value of p varies then q(t) is calculated stepwise in time from q (t dt) and the average value of p over the time step dt
q (t) = a q (t - dt) + b p
(8)
with
a = e
- dt/T
; b = 1 - e
- dt/T
(9a)
or if outflow during the period dt is found as the average value of q at the
beginning and the end of the time step, i.e. as 0.5/T (h(t) + h(t-dt)),
T
- 0.5
1
a=
;b = d t
T
T
+ 0.5
+ 0.5
dt
dt
(9b)
With coefficients according to eq. (9b) you get essentially an implicit solution,
see chapter on surface runoff, even if , due to the linear relationship between
q and h one gets an explicit expression, eq. (9), for q.
Example
Runoff from a moist area is 10 mm/d when it starts to rain. The rain continues
for three days with totally 60 mm. The time constant for the catchment is four
days. Find a) expected maximum runoff if there is no evaporation and b) the time
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when runoff is back at 10 mm/d if evaporation after the rain is 3 mm/d.
Solution
Precipitation is p = 20 mm/d. The solution for a linear reservoir is
q = qo e
- t/T
+ p (1 - e
- t/T
)
Maximum runoff occurs when the rain stops
q = 10 e
- 3/4
+ 20 (1 - e
- 3/4
) = 15.3 mm/d
The recession is
q = qo e
- t/T
- t/T
- evap (1 - e
)
with t = 0 and qo = q (t = 0) when the rain stops. The time when q = 10 mm/d
can be solved from eq.
10 = 15.3 e
e
- t/4
- t/4
- 3 (1 - e
= 13/18.3 = 0.71;
- t/4
)
t = 1.4 days
i.e at time 4.4 days after the rain started.
Reservoir theory can be given a physical explanation. The flow along mildly
sloping ground towards a stream usually occurs as groundwater flow. The runoff
per unit width, see Figure 5, is according to Darcy eq. Q/B = v H = KIH with
H = depth of groundwater layer, v = velocity of flow, K = hydraulic conductivity
and I = slope of the groundwater table, which is approximately equal to the ground
slope. If the soil is homogeneous then K = constant. Then also the flow velocity
is constant. Downstream of the slope whose length is L, the flow equals specific
runoff for the surface with area A = BL and is q = K I H/L. If this expression
is substituted into the water balance
dh
1 = p - e - K I H/L
dt
where h = average storage in the catchment and H = groundwater level at the outlet.
Level multiplied by porosity, n, equals water volume.
If groundwater level at the downstream point can represent groundwater level
in the whole area
dh n
KI
= (p - e) h
dt S
LS
(14)
which is the same expression as for linear reservoir, eq. (5) with time constant
T=
LS
KI
(15)
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Figure 5
Water flow per unit width, vH, and per unit area vH/L towards
draining stream; v = Darcy velocity
It is therefore physically relevant to model catchment runoff according to linear
reservoir theory. Now you cannot find the time constant T by using eq. (15) but
you can get an idea of its order of magnitude. L and I are length and slope of
the area respectively, n and K are porosity and hydraulic conductivity for the
soil layer where the groundwater exists.
Double reservoir
The soil usually has different conductivity at different levels. Close to the
soil surface the soil usually has higher conductivity and higher porosity. Close
to the ground surface there are holes created by roots and animals. Also frost
causes the soil to become more porous. The hydraulic conductivity can for example
be 10 - 100 times higher 10 cm from the ground surface than 100 cm from the ground
surface in moraines. The time constant is therefore smaller the more water there
is in the soil and the higher the groundwater level is. This relationship, which
mainly depends on the fact that hydraulic conductivity is higher near the soil
surface, can be modelled with a non-linear relationship between discharge and
storage..
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Reservoir with three outlets and two thresholds
Figure 6
However it is more common to describe the runoff as as the sum of several reservoirs.
Symbolically and conceptually the catchment is a box with a number of holes in
it. The higher the water rises the more (and bigger) holes the water flows from,
c.f. Figure 6. For each hole there is a relationship
q=
h - h threshold
T
(11)
where hthreshold is the level, expressed as water level, at which the water can leave
or the storage level at which the part of the reservoir in question contributes
to the runoff. More physically expressed ( even if there is no direct physical
explanation) this means that when the groundwater level rises the water reaches
material with higher conductivity and the outflow will increase.
For a linear double reservoir, with two holes and one threshold, the runoff is
described by the continuity equation and the runoff equation
q=
h
T1
+
h - h0
(19)
T0
where, see Figure 6 for principle but with only two outlets: T1 is reservoir-time
constant for the lower and slower storage, T0 time constant for the upper faster
part and h0 storage volume when the upper part starts being active.
Double linear reservoir can be solved numerically using explicit or implicit
method. There are some possibilities for analytical solutions if you can keep
track of whether h is larger or smaller than the threshold h0.
Analytical solution
As long as h < ho then equations (1) and (11) correspond to a linear reservoir
for which the analytical solution as given by (6). If h > ho you can see that
after taking the derivative of eq. (11), that
8
d q dh ⎛ 1
1 ⎞ dh 1
=
⎜⎜ + ⎟⎟ =
d t d t ⎝ T1 T0 ⎠ d t Teff
(21)
You can introduce an effective time constant which is valid when h > h0,
T = Teff =
1
1
T1
+
(22)
1
T0
Then q = h/T which in combination with the continuity equation is a linear
reservoir with solution according to eq. (6).
When h passes ho during a period dt one should divide dt in two parts so that
h is equal to ho after the first of these parts. Below you can find an example
solved analytically and a numerical scheme.
Example
Effective precipitation is 10 mm/d during 1 day followed by 3 days at 3 mm/d.
What is the runoff after 1 and 4 days respectively if runoff at the start is
1 mm/d and and the time constants in a double reservoir are 20 and 4 days
respectively and the threshold value when runoff starts from the upper reservoir
is 40 mm.
Solution
The threshold level 40 mm is reached when
q = h0/T1 = 40/20 = 2 mm/d
The effective time constant for full storage is
1/T = 1/T1 + 1/T0 = 1/20 + 1/4 = 6/20;
T = 3.33 days
Day 1
q = q0 e
-t/T
1
+ p (1 - e
-t/T
) = 1 e
1
-1/4
+ 10 (1 - e
-1/4
) = 2.99
This is higher than the threshold value 2 mm/d. The time is calculated for which
h = ho. Therefore
2 = 1 e
-t/4
+ 10 (1 - e
-t/4
);
which gives
t = 0.47 days
during the later part (1 - 0.47 = 0.53) of the day the effective time constant
is T = 3.33. Runoff is then
q = 2 e
- 0.53/3.33
+ 10 (1 - e
- 0.53/3.33
) = 3.18 mm/d
Day 2 - 3 - 4
After that q is larger than the threshold value 2 mm/d and after 4 days, i.e.
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after another three days
q (after rain stopped) = 3.18 e
- 3/3.33
+ 3 (1 - e
- 3/3.33
) = 3.07 mm/d
Explicit numerical solution
Explicit numerical solution is straightforward, but the computational time step
must be short. dh/dt is calculated as p-q for old p and q values. h(t+dt) then
becomes h(t) + dteff ⋅ dh/dt. Thereafter the new h is used to calculate from the
outflow expression q (t+dteff). One has to separate the time step, dt, for input
data and output and the much shorter computational time step, dteff. The
computational scheme becomes (with nt = number of dteff per dt) as shown in Figure
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Figure 7.
Scheme for double reservoir
By studying runoff recession during periods with no input of effective rain
(p-e=0), i.e. recession analysis, one can find the time constants for the
recession. Long before the use of linear reservoir theory for runoff recession
analysis was made. The recession was often found to be exponential so that as
in Figure 8, one could plot the logarithm of the discharge as a linear function
of time. From eq. (7) (q = q0 e-t/T) one can see that
10
T=
t - t0
q
ln 0
q
Figure 8
(23)
Recession curves for a river plotted in a semilogarithmc diagram,
after Harlin (1992)
Knowing how the runoff decreases from q0 at time t0 to q at time t in a not too
large interval (q0-q) one can describe T for discharge in this interval. Examples
of observed recessions are shown in Figure 8, after Harlin (1992). One can see
that the recession process is linear in a semilogarithmic diagram with one slope
for high flows and another for low flows. This points to the fact that runoff
can be described using a double linear reservoir model. That is exactly the
procedure used in the Swedish HBV model, Bergström (1976), which is decribed
further below.
CALCULATION OF EFFECTIVE PRECIPITATION-SOIL WATER ROUTINE
In the reservoir models that have been described above the effective precipitation
has been given, defined as the water that reaches the runoff reservoir and
contributes to runoff. In order for a model to be useful in real life it has
to calculate runoff from real precipitation or snow melt. A reservoir model
therefore has to be combined with a model that converts precipitation to effective
precipitation. This is done with a soil water model/routine. Such a model is
shown in Figure 9. All the soil water located between the soil surface and the
groundwater level is treated as if it were in a box (or reservoir). Continuity
demands
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Figure 9
Soilwater model with precipitation, evaporation, surface runoff and
percolation to groundwater.
d h mark
= p - e - f - q yt
dt
N.B. mark = soil
(23)
where hmark = amount of water in soilwater storage, p = intensity of real
precipitation or snowmelt intensity, e = rate of evaporation, f = groundwater
generation which is modelled as rate of filling of simple or double reservoir
for which q = f(h), and qyt = surface runoff.
The rate of evaporation depends on potential evaporation and soil water content
e = pe f(hmark) and rate of percolation depends on rain intensity and soilwater
content so that f = f (p, hmark).
The more moisture in the soil the faster the water reaches the groundwater for
runoff. At every point in the catchment the soil can be represented by a box.
The level in the box rises due to precipitation and sinks due to evaporation.
Not until the box is full, which corresponds to water contents reaching field
capacity, can the water from the box contribute to runoff. Water surplus
percolates to groundwater. However the box must represent all points in the
catchment, each with different soil depth (field capacity). One could see this
as a row of boxes with different FC, field capacity. When the smallest of these
boxes is full there is a little contribution to runoff; when all boxes are full
the contribution is great. Therefore percolation from one box representing many
boxes, as in the Swedish HBV-model, Bergström (1976), can be expressed with the
equation
⎛ h ⎞
f =p⎜
⎟
⎝ FC⎠
b
(28)
where both h and FC are related to the wilting point (set at zero) and where
b is a coefficient. Now h is an average soil water content for the whole area
and FC a representative value of field capacity in those parts of the area where
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field capacity is high. h cannot surpass FC. Even if FC conceptually should equal
the field capacity of the soil, the value used as FC in modeling may deviate
from the real value. Eq. (28) describes many points which have different field
capacity and at which the groundwater generation is different
Figure 10
Approximate relationship between evaporation and soil water content.
The ratio between potential and real evaporation, Figure 9, depends on relative
soil moisture, i.e. the ration between soil water content and field capacity,
h/FC. In sandy soils the relationship is rather linear e/pe = hmark/FC but in clay
soils the evaporation rate stays close to the potential rate even when hmark goes
below FC. For most soils one can find a relationship which stays linear for h
< hp, with hp denoting the soil water content over which e = pe.
h
⎧
⎪p e ; h < h p
e = p e f(h mark) = ⎨ h p
⎪ p e ;h ≥ h
p
⎩
(30)
In the Swedish HBV-model there is no surface runoff included. If it is assumed
that there is no surface runoff (qyt = 0) eq. (23) for the soilwater box – after
e and f have been substituted, becomes
b
d h mark
h
⎛ h ⎞
=p-pe -p⎜
⎟ ; h < hp
dt
hp ⎝ F C ⎠
(31a)
b
d h mark
⎛ h ⎞
=p-pe-p⎜
⎟ ; h ≥ hp
dt
⎝ FC⎠
(31b)
It is not evident whether there is evaporation or not during periods of rain.
If calculations are made for a daily time step and rain falls as storm showers,
then evaporation occurs. If the rain is a longer lasting frontal rain then
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evaporation is very limited.
The soilwater equation (23) can be solved numerically using an explicit scheme.
The computational routine is as shown in Fig 11.
Figure 11.
Soilwater routine.
The soilwater box is a non-linear reservoir. Implicit numerical solution is
possible, in a similar way as shown in the chapter [not included] on surface runoff.
Newton-Raphsons method is given there. Average values of e and f over the period
dt are taken to be the arithmetic average between the values at the start and
at the end of the time step. It follows that
h t+ dt + 0.5
et+dt + 0.5 f t+dt - give n = 0
dt
(32a)
where ’given’ = p + 0.5 et + 0.5 ft
(19b)
which becomes (with h = ht+dt )
b
h
h
⎛ h ⎞
+ 0.5 p e + 0.5 p ⎜
⎟ - give n = 0 ; h < h p
dt
⎝FC⎠
hp
(32c)
b
h
⎛ h ⎞
+ 0.5 p e + 0.5 p ⎜
⎟ - give n = 0 ;
dt
⎝FC⎠
h < hp
The solution procedure is shown in Fig 12.
14
(32d)
Figure 12.
Soilwater calculation-implicit method
Since some of the inflows/outflows to/from the box, sometimes are negligible
there are simplified solutions. If it does not rain and h ≥ hp, see eq. (31b),
then h decreases with the constant rate pe. If it does not rain and h < hp then
eq. (31a) is essentially the equation for a linear reservoir.
Surface runoff
Surface runoff may occur as Hortonian runoff or as saturated soil runoff.
Contribution to saturated soil surface runoff is rain falling on the soil surface
where the groundwater level is at the soil surface. The areal extension of
saturated soil at the surface can be related to the soil water content. The part
of the rain which has a higher intensity than the maximum infiltration capacity
of the soil, i.e. after some time the difference between the rain intensity and
the hydraulic conductivity at saturation , contributes to Hortonian surface
runoff. This runoff contribution must first fill up depression storage before
real surface runoff occurs. Surface runoff has been treated in a separate chapter
[not included]. Calculation of surface runoff can be made using a non-linear
reservoir or a reservoir with a small time constant.
RUNOFF MODEL
Outflow at the bottom of the soilwater box, f, forms inflow to the runoff box,
i.e it corresponds to peff, as it has been described in the section on reservoir
models. A conceptual runoff model can therefore be illustrated as in Figure 13.
One has to solve the continuity equations for soilwater eq. (31) and for the
runoff box eq. (1) and the accompanying runoff (the equations are given new
numbering here):
d h mark
2 = p - e - f; e, f = funk(hmark)
dt
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(20a)
dh
3 = f - q; q = funk(h)
dt
Figure 13
(20b)
Runoff model with soilwater reservoir and double runoff reservoir
The solution is most easily done with an explicit method according to the scheme
given in Figure 13. hmark is related to hwilt, which is taken to be zero.
Figure 14.
Explicit solution in runoff model
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The runoff model can be made more complex than shown above. In the model shown
in Figure 13 there are 5 parameters, hp; FC; To; T1; ho. The more parameters
introduced the more difficult it is to adjust the model to real general situations
even if special events can be modeled almost perfectly. One can include surface
runoff from the soilwater box, one can use several holes, i.e. more time constants,
in the runoff box. One can allow evaporation directly from parts of the runoff
box and one can let some precipitation go straight to the runoff box. One can
let runoff go into another box. There are several modifications all of which
can be motivated in special situations.
Figure 15
Runoff model with soilwater box , runoff reservoir and a deep water
storage
An addition to the runoff model shown in Figure 13 which is very often used is
a deep groundwater box , which is used to model slow groundwater flow forming
base flow. Water is supposed to percolate from the runoff box into the deep
groundwater box, as shown in Figure 15. In the Swedish HBV-model smaller lakes
are taken as part of the groundwater storage in the sense that precipitation
on the lake surface goes directly to deep storage and evaporation from the lake
surface constitutes a loss from the deep storage. The rate of percolation, perk,
from groundwater box to deep groundwater-box can be constant or depend on amount
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of stored water. The continuity equations for the runoff box and the deep
groundwater box becomes respectively:
dh
4 = f - q - perk
dt
djup = deep
d h djup
5 = perk - qdjup
dt
qdjup =
(21)
(22)
h djup
6
T djup
(23)
The runoff is the summation of q and qdjup.
Before one has a complete runoff model the three boxes described here for soilwater,
groundwater and deep groundwater must be supplemented with interception- and
snow models for calculation of inflow to the soilwater box, and with a routing
model which describes how the runoff is delayed in the system of streams and
rivers in the catchment.
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