Notes on Overland Flow

By

Ric Soulis, et. al

1.1Flat CLASS 3.3

Overland flow is generated if the depth of the ponded water goes above a hard-coded limit of 10 cm (in APREP). Excess water is put directly into runoff (in ICEBAL and TMCALC).

1.2Sloped CLASS 3.3

1.2.1Definitions

B = Area

= length of stream segment ‘i' in the grid

= flow entering a channel of length from both banks.

= Drainage Density

= slope

n = Manning’s n

S = storage

h = height above max ponded water (S = Bh)

q* = Manning approximation of the kinematic wave velocity

1.2.2Derivation

Starting with Manning’s equation for the kinematic wave velocity

(1)units: LT-1

In the code:
Equation (1) is calculated directly to simplify the writing of equations (15) and (29), which are calculated later in the code.

is the flow entering a channel of length from both banks. Assuming a stream is not on the grid-square boundary, which doesn’t happen because the grid square is really just an approximation of a mini-sub-watershed, we can assume that two hill-slopes are contributing to the flow entering the channel.

(2)

(area of one bank = area of flux boundary at one bank – a diagram would be useful)

units: L3T-1

Substituting (1) into (2)

(3)

Summing over the channels in the grid,

(4)

(5)

Calculating the flow per unit horizontal area

(6)

(equation 7 in Ric’s article in the 2000 AO paper)

Now considering the variation in storage:

(7)

An explanation of why
Let be the storage at the end of time step, , and be the storage at the beginning of time step, . Additionally, storage refers to the volume of water on the land surface beyond the ponding limit, so Q refers to flow
Assumption: because of the model dynamics, we assume that all water (rain, snowmelt) is dumped onto the land surface prior to runoff. Therefore, the storage at the beginning of the time step will always be greater than the storage at the end of the time step.
In reality, the flow could be doing a lot of things within a time step. Here are a few options:

Figure 2: Types of flow
The areas under the curves represent how much water constitutes overland flow. Unconstrained overland flow assumes constant flow throughout the time step. Constraining the flow by available water simple limits the overland flow by what is available.
Using manning’s approximation of the kinematic wave velocity, we may assume that the flow decreases predictably from the beginning of the time step to the end

Figure 3: Manning approximation of a kinematic wave – Predictable curve.
The flow cumulated over the time step (which is the area under the curve) represents a decrease in storage.
Therefore:, where the negative sign is because the flow is being removed from storage.

Table 1: an explanation of the evaluation of the storage change in overland flow

Getting back to equation (7)

(7)

Hence

(8) and (9)

(unconstrained change in storage)

(10)

(11) and (12)

(unconstrained Manning’s overland flow)

(13) and (14)

(unconstrained Manning’s overland flow)

Note that

In the code:
Equation (14) can be simplified using equation (1).
(15)

1.2.3Unconstrained Flow vs. Constrained Flow

Within the model, it is important to constrain the amount of flow that goes to the stream. With unconstrained flow, we assume a constant flow rate throughout the timestep, making it possible to calculate more overland flow than is physically available or possible.

Constrained by Available Water

There are two ways in which the overland flow can be constrained in this version of sloped CLASS 3.3. The first is that flow is constrained by the available water. Within the code, constraining the flow by the available water is done by using either the amount of water available (h), or the amount of flow calculated (-∆hu).

Overland flow = min(h, -∆hu) (16) (constrained by available water)

Constrained by Physically Possible Flow

Constraining overland flow by physically possible flow requires further manipulation of equation (13). To simplify the manipulation of equation (13), we say that

and b = 5/3

(17)

(18)

Integrating equation (18) provides

(19)

At h = h0 and t = t0, (20)

Substituting (20) back into (19) gives

(21)

(22)

(23)

(24)

(25)

Substituting (23) into (24) gives

(26)

Substituting into equation (25) gives

(27)

Substituting and b = 5/3 into equation (27) gives

(28)

Equation (28) represents overland flow constrained by physically possible flow.

In the code:
Equation (28) can be simplified.
Normalizing equation (14), the unconstrained Manning’s overland flow, to avoid division by numbers close to zero in the code results in
(29)
Substituting equation (29) into equation (28) gives:
[MSOffice1] (30)

1.3Implementation – Sloped CLASS 3.3

The code for overland flow is implemented directly into WATROF.

1.3.1Overland flow FORTRAN code

C * PART 1 - OVERLAND FLOW
C * (MODELLED USING MANNINGS EQUATION).
C * CALCULATED USING THREE PARAMETERS "XSLOPE, MANNING_N AND DD"
C * XSLOPE = AVERAGE SLOPE OF GRU
C * MANNING_N = MANNING'S 'N'
C * DD = DRAINAGE DENSITY
C * TWO OPTIONS ARE AVAILABLE TO CONSTRAIN THE FLOW
DO 100 I=IL1,IL2
IF(FI(I).GT.0.0) THEN
IF(ZPOND(I).GT.ZPLIM(I))THEN
C Calculate the depth of water available for overland flow. Units: L
DOVER(I)=ZPOND(I)-ZPLIM(I)
C Calculate the flow velocity at the beginning of the timestep
C (based on kinematic wave velocity) Units: LT-1
VEL_T0(I)=DOVER(I)**(2./3.)*SQRT(XSLOPE(I))/(MANNING_N(I))
C Eqn (1) in spec doc
C Calculate a normalized unconstrained overland flow to avoid numerical
C problems with a division of small DOVER(I) values.
NUC_DOVER(I) = -2*DD(I)*VEL_T0(I)*DELT
C Eqn (29) in spec doc
C Constrained Overland Flow - Limited by physically possible flow
DODRN(I)=DOVER(I)*(1.0-1./((1.0-(2./3.)*NUC_DOVER(I))
+ **(3./2.)))
C Eqn (30) in spec doc
IF(RUNOFF(I).GT.1.0E-08) THEN
TRUNOF(I)=(TRUNOF(I)*RUNOFF(I)+(TPOND(I)+TFREZ)*
1 DODRN(I))/(RUNOFF(I)+DODRN(I))
ENDIF
RUNOFF(I)=RUNOFF(I)+DODRN(I)
IF(DODRN(I).GT.0.0)
1 TOVRFL(I)=(TOVRFL(I)*OVRFLW(I)+(TPOND(I)+TFREZ)*
2 FI(I)*DODRN(I))/(OVRFLW(I)+FI(I)*DODRN(I))
OVRFLW(I)=OVRFLW(I)+FI(I)*DODRN(I)
ZPOND(I)=ZPOND(I)-DODRN(I)
ENDIF
ENDIF
100 CONTINUE

Table 2: Overland code for sloped CLASS 3.3

1.3.2Variable definition

ZPOND: depth of ponded water

ZPLIM: ponding limit

DOVER: sub-area (C,G,CS,GS) water available for overland flow

MANNING_OVR: unconstrained depth of sub-area (C,G,CS,GS) water contributing to overland flow

XSLOPE: effective slope of the hill

MANNING_N: manning’s n

E_HILL_LEN: effective hillslope length (Ls)

DELT: delta time

DODRN: constrained depth of sub-area (C,G,CS,GS) water contributing to overland flow

TOVRFL: temperature of overland flow

OVRFLW: total constrained GRU/Tile overland flow (C+G+CS+GS)

1.3.3Options for Constraining the Flow

The above code contains both versions of the sloped CLASS 3.3. If the flag iwfoflw (in WATROF) is set to 1, then the new sloped CLASS 3.3 is used. Otherwise the 2000 AO paper implementation is used.

[MSOffice1]Nhu will never equal 3/2 as it is the fractional change in h given a certain ∆t.