NOTES ON OVERLAND FLOW
By
Ric Soulis, et. al
1.1 Flat 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).
1.2 Sloped CLASS 3.3
1.2.1 Definitions
B = Area
L(iv ) = length of stream segment ‘i' in the grid
Q (i ) = flow entering a channel of length L(iv ) from both banks.
Dd  
L(vi )
= Drainage Density
B
 = slope
n = Manning’s n
S = storage
h = height above max ponded water (S = Bh)
q* = Manning approximation of the kinematic wave velocity
1.2.2 Derivation
Starting with Manning’s equation for the kinematic wave velocity
q* 
 2/3
h
n
(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.
Q (i ) is the flow entering a channel of length L(iv ) 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.


Q (i )  q *  h  L(vi )  2
velocity area of
one
2 banks
(2)
(area of one bank = area of flux boundary at one bank – a diagram would be useful)
units: L3T-1
Substituting (1) into (2)
Q (i ) 
2  5 / 3 (i )
h Lv
n
(3)
Summing over the channels in the grid,

Q  Q ( i )  2 L(vi )
 Q  2 Dd B

 5/3
h
n
(4)
 5/3
h
n
(5)
Calculating the flow per unit horizontal area
qover 
Q 2 Dd  5 3

h
B
n
(6)
(equation 7 in Ric’s article in the 2000 AO paper)
Now considering the variation in storage:
t1
t1
t0
t0
S t1  St 0    Qdt    qover Bdt
(7)
t1
An explanation of why
St1  St0   Qdt
to
St1 be the storage at the end of time step, t1 , and Sto be the storage at the
beginning of time step, t o . Additionally, storage refers to the volume of water on the
Let
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:
Qo
Q1
Q1
Qo
Qo, Q1
to
t1
time
to
Constant flow
t1
time
to
Linear decline
t1
time
Linear incline
Qo
Q1
Qo, Q1
to
t1
Non-Linear Decline
time
to
t1
time
Non-Linear
Figure 1: 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
Qo
Predictable curve
(Manning approx of Kinematic wave)
Q1
to
t1
time
Figure 2: 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.
t1
Therefore: St1  St0    Qdt , where the negative sign is because the flow is being
to
removed from storage.
Table 1: an explanation of the evaluation of the storage change in overland flow
Getting back to equation (7)
t1
t1
t0
t0
S t1  St 0    Qdt    qover Bdt
(7)
Hence
dS
 q over B
dt
(8)
and
S  qover t 0 Bt  S u
(unconstrained change in storage)

d hB 
 q over B
dt
(10)
(9)

dh
 q over
dt

S u
 qu (12)
B
(unconstrained Manning’s overland flow)
(11)
2 Dd  5 3
dh

h
dt
n
and
(13)
h  qover t 0 t 
2 Dd  5 / 3
 h0 t  hu (14)
n
(unconstrained Manning’s overland flow)
and
h  
Note that hu  qu
In the code:
Equation (14) can be simplified using equation (1).
hu  2 Dd  q *  h0 t
(15)
1.2.3 Unconstrained 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
A
2 Dd 
and b = 5/3
n
b
dh

 A h
dt
(17)

dh
  A  dt
hb
(18)
Integrating equation (18) provides
 1  1b

h   At  C
1 b 
(19)
 1  1b
At h = h0 and t = t0, C  
h0  At 0
1 b 
(20)
Substituting (20) back into (19) gives
 1  1b
 1  1b

h   At  
h0  At 0
1 b 
1 b 
 1  1b
 1  1b

h   At  t 0   
h0
1 b 
1 b 

 1  1b 
 h1b  1  b  At  t 0   
h0 
1 b 


 h01b  At  t 0 1  b 


(21)
(22)
(23)
1
 h  h01b  At  t 0 1  b 1b
1

 At  t 0 1  b  

1b
 h01b 1 


h01b





1
 At  t 0 1  b  1b
 h0 1 

h01b


(24)
h  h0  h
(25)
Substituting (23) into (24) gives
 At  t0 1  b  
h  h0  h0 1 

h01 b


1
1 b
1


  At  t0 1  b  1 b 
h  h0 1  1 
 
h01 b
 
 


1


  At  t0 h0b 1  b  1b 
h  h0 1  1 
  (26)
h0
 
 


Substituting t  t  t 0 into equation (25) gives
1


b
1b




Ah

t
1

b


0
h  h0 1  1 
 
h0
 
 


Substituting A 
(27)
2 Dd 
and b = 5/3 into equation (27) gives
n
1


5/3
15 / 3


2 Dd   h0  t  1  5 / 3


h  h0 1  1 


n  h0
 




3 / 2
  4 D   h 2 / 3  t  
d
0
h  h0 1  1 
 
n
  3
 
3/ 2
 
 
 
 
1

h  h0 1  
 
2/3
4
D


h


t

 
d
0

1

 
  3
n

(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
Nhu 
hu
2 Dd  2 / 3

 h0 t
h0
n
(29)
Substituting equation (29) into equation (28) gives:
3/ 2
 
 
 
 
1

 h0 1  
 
 1  2  Nhu  
  3
 
(30)
1.3 Implementation – Sloped CLASS 3.3
The code for overland flow is implemented directly into WATROF.
1.3.1 Overland flow FORTRAN code
C
C
C
C
C
C
C
C
*
*
*
*
*
*
*
DO 100 I=IL1,IL2
IF(FI(I).GT.0.0) THEN
IF(ZPOND(I).GT.ZPLIM(I))THEN
Calculate the depth of water available for overland flow. Units: L
DOVER(I)=ZPOND(I)-ZPLIM(I)
C
C
Calculate the flow velocity at the beginning of the timestep
(based on kinematic wave velocity) Units: LT-1
VEL_T0(I)=DOVER(I)**(2./3.)*SQRT(XSLOPE(I))/(MANNING_N(I))
Eqn (1) in spec doc
C
C
C
C
C
+
C
PART 1 - OVERLAND FLOW
(MODELLED USING MANNINGS EQUATION).
CALCULATED USING THREE PARAMETERS "XSLOPE, MANNING_N AND DD"
XSLOPE = AVERAGE SLOPE OF GRU
MANNING_N = MANNING'S 'N'
DD = DRAINAGE DENSITY
TWO OPTIONS ARE AVAILABLE TO CONSTRAIN THE FLOW
Calculate a normalized unconstrained overland flow to avoid numerical
problems with a division of small DOVER(I) values.
NUC_DOVER(I) = -2*DD(I)*VEL_T0(I)*DELT
Eqn (29) in spec doc
Constrained Overland Flow - Limited by physically possible flow
DODRN(I)=DOVER(I)*(1.0-1./((1.0-(2./3.)*NUC_DOVER(I))
**(3./2.)))
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
1.3.2 Variable 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
MANNING_N
E_HILL_LEN
DELT
DODRN
TOVRFL
OVRFLW
: effective slope of the hill
: manning’s n
: effective hillslope length (Ls)
: delta time
: constrained depth of sub-area (C,G,CS,GS) water contributing to
overland flow
: temperature of overland flow
: total constrained GRU/Tile overland flow (C+G+CS+GS)
1.3.3 Options 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.