RUNOFF This chapter trmodels and numesurroundings. Irivers as shown – after short oras a function oOver long time pcalculating runbetween storage Precipitation itranspirated tosoil. Groundwatthe groundwatersurface rain wathe rain intensstreams water mo

Figure 1 Catchm

Excerpt from Lars Bengtsson ”Hydrologi – Teori och processer”, Dept of Water Resources Engineering LundUniversity. Translated by Rolf Larsson Sept 2004.

eats runoff from catchments. The focus is on modelling with box rical methods. A catchment is topographically separated from its t may contain flat land, slopes, forests, wetlands, lakes and in Figure 1.precipitationwhich does not evaporate will eventually long time storage – reach the outlet from the catchment. Runoff f time can be estimated with more or less sophisticated models. eriods the runoff can be estimated through a water balance. When off it is generally necessary to take into account the relationship and runoff.

s transformed into runoff “after” water has been evaporated and the atmosphere. Precipitation or meltwater infiltrates into the er moves towards ditches and streams. If much water is added to runoff becomes more intensive. If groundwater rises to the soil ter has to runoff on the surface. Surface runoff also occurs if ity exceeds the infiltration capacity. From ditches and small ves via larger streams and rivers to the outlet from the catchment.

ent 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

q - e - p = td

dS (1a)

with specific runoff

AQ = q (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.

f(h) - e - p = td

dh (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).

Th - e - p =

tddh

(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)),

0.5 + td

T

0.5 - td

T

= b; 0.5 +

tdT

1 = a (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 and h one gets an explicit expression, eq. (9), for q. 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 - evap (1 - e - t/T) 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 - t/4 - 3 (1 - e - t/4) e - t/4 = 13/18.3 = 0.71; 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

tddh

1 = p - e - K I H/L

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

h S L IK - e) - (p

Sn =

tddh

(14)

which is the same expression as for linear reservoir, eq. (5) with time constant

IK S L = T (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..

Figure 6 Reservoir with three outlets and two thresholds 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

Th -h = q threshold (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

Th -h +

Th = q

0

0

1

(19)

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

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T1

tddh =

T1 +

T1

tddh =

tdq d

eff01⎟⎟⎠

⎞⎜⎜⎝

⎛ (21)

You can introduce an effective time constant which is valid when h > h0,

T1 +

T1

1 = T = T

01

eff (22)

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) = 1 e -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 7

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

qqln

t -t = T0

0 (23)

Figure 8 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.

q - f - e - p = td

h dyt

mark (23)N.B. mark = soil

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

⎟⎠⎞

⎜⎝⎛

C Fh p = f

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 h ; e p

h <h ; hh e p

= )hf( e p = e

p

ppmark (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

h h ;C F

h p - e p - p = td

h d

h <h ;C F

h p - hh e p - p =

tdh d

p

bmark

p

b

p

mark

≥⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

(31a)

(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

0 = give - f 0.5 + e 0.5 + td

hdtt+dtt+

dtt+ n (32a)

where ’given’ = p + 0.5 et + 0.5 ft (19b) which becomes (with h = ht+dt )

h <h ; 0 = give - C F

h p 0.5 + hh e p 0.5 +

tdh

p

b

p

n⎟⎠⎞

⎜⎝⎛

(32c)

h <h ; 0 = give - C F

h p 0.5 + e p 0.5 + tdh

p

b

n⎟⎠⎞

⎜⎝⎛

(32d)

The solution procedure is shown in Fig 12.

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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):

tdh d mark 2 = p - e - f; e, f = funk(hmark) (20a)

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tddh

3 = f - q; q = funk(h) (20b)

Figure 13 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:

tddh

4 = f - q - perk (21) djup = deep

tdh d djup

5 = perk - qdjup (22)

qdjup = Th

djup

djup6 (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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