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Generalized conceptual modeling of dimensionless overland flow hydrographs
.••
Journal of
Hydrology
ELSEVIER
Journal of Hydrology 200 (1997) 222-227
Generalized conceptual modeling of dimensionless overland flow hydrographs V.M. Ponce a'*, O.I. Cordero-Brafia b, S.Y.
Hasenin a
~Department of Civil and Environmental Engineering, San Diego State University, San Diego, CA 92182-1324, USA bDepartment of Mathematics and Statistics, American University, Washington, DC 20016-8050, USA
Received 30 October 1996; revised 16 January 1997; accepted 16 January 1997
Abstract
The rising and recession limbs of conceptual dimensionless overland flow hydrographs are calculated for specific values of the rating exponent in the range 1 - m -< 3, including a linear reservoir (m = 1); 100% turbulent Chezy friction (m = 3/2); 100% turbulent Manning friction (m = 5/3); 67% turbulent Chezy (or 75% turbulent Manning) (m = 2); and 100% laminar flow (m = 3). These conceptual overland flow hydrographs show finite amounts of diffusion, increasing with decreasing rating exponent, unlike the kinematic wave hydrograph, which is nondiffusive. © 1997 Elsevier Science B.V. Keywords: Dimensionless; Overland flow; Hydrographs; Conceptual modeling
1. I n t r o d u c t i o n
The calculation of overland flow hydrographs is well established in flood hydrology. Early approaches had a distinct conceptual basis (Horton, 1938, 1945; Izzard, 1944, 1946), while more recent approaches, among them the kinematic wave, have relied on the physics of the phenomena (Wooding, 1965; Woolhiser and Liggett, 1967). The kinematic wave lacks diffusion and, therefore, is not suited for overland flow over mild slopes, where diffusion plays a major role. On the other hand, the conceptual model has an intrinsic diffusion capability, which is underscored by the fact that its reference time-to-equilibrium is twice that of the kinematic wave (Ponce, 1989). A conceptual model o f dimensionless overland flow hydrographs is presented here. The * Corresponding author.
0022-1694/97/$17.00 © 1997- Elsevier Science B.V. All rights reserved PII S0022- 1694(97)00012-7
V.M. Ponce et al./Journal of Hydrology 200 (1997) 222-227
223
procedure generalizes the classical Horton-Izzard approach for a wide range of rating exponents, including m = 3/2 and 5/3, which, to our knowledge, have not been previously calculated. The rising and recession limbs of dimensionless overland flow hydrographs are calculated for the following values of the rating exponent: (1) linear reservoir, m = 1, describing a condition where the overland flow velocity is sensibly constant (Horton, 1938); (2) 100% turbulent Chezy friction, m = 3/2; (3) 100% turbulent Manning friction, m = 5/3; (4) 67% turbulent Chezy friction (equivalent to 75% turbulent Manning), m -- 2: and (5) 100% laminar flow, m = 3.
2. The conceptual model Runoff in the overland flow plane is described by the differential equation of storage: dS I - 0 =
(1)
--
dt
in which I = inflow, O = outflow, and d S / d t = rate of change of storage in the plane. A mass balance leads to the equilibrium outflow: q~ = i L
(2)
in which i = rainfall excess, and L = length of the plane. For the rising limb, I = iL, and O = q, from which dS dt
iL-q=
(3)
--
For the receding limb, I = 0, and O = q, from which -q=
dS d~-
(4)
The Horton-Izzard model is based on the discharge-storage rating (5)
q = aS"
in which a and rn are coefficient and exponent, respectively. Eq. (5) is assumed to be applicable for any discharge, including that at equilibrium. Additionally, the equilibrium storage is equal to: (6)
Se = qet~
2
in which te = reference time-to-equilibrium. Because of runoff diffusion, the equilibrium outflow is approached asymptotically, i.e. q - . qe as t ~ oo. Thus, the reference time-toequilibrium is less than the actual time-to-equilibrium, which approaches infinity. For the rising limb, the generalized conceptual model is: dS
aS m - aS m = -dt
(7)
224
V.M. Ponce et al./Journal of Hydrology 200 (1997) 222-227
The solution of Eq. (7) is (Dooge, 1973; Ponce, 1989):
1 1
t,=~
l
1 d(qm ) l-q,
(8)
in which t, = rite, with t = the accumulated time from the start of rising, and t e = the reference time-to-equilibrium, and q, =q/qe, with q = the outflow at time t, and qe = the equilibrium outflow. For the receding limb, the generalized conceptual model is: (9)
-aS m=dS
dt The solution for the receding limb depends on whether the rising limb has reached equilibrium or not. First, assuming that the rising limb is at equilibrium, the solution of Eq. (9) is (Dooge, 1973):
1 -~-d(q m) q,
t,=-~
(10)
in which t, = fit e, with t = the accumulated time from the start of recession, and te = the reference time-to-equilibrium (of the rising limb); and q, = q/qe, with q = the outflow at time t, and qe = the outflow at the start of recession, i.e. the equilibrium outflow. Assuming that the peak outflow at the end of the rising limb is qp, less than the equilibrium outflow (qe), the solution of Eq. (9) is: l-m
t,=-~
()j-
1 qp ~ ~
1
1 (q,m)
(ll)
q*
in which t, is the same as in Eq. (10), but now q, = q/qp. Eq. (8) is evaluated herein for values of m equal to 1, 3/2, 5/3, 2, and 3. 1. m = l
2. m = 3 / 2
(12)
t,=-~ln(1-q,)
t, --
(')
arctan ~
1
1
1 - arctan ~(1 +2q 3~, / - ( ~ ) l n ( 1 - q ; ) 2
+ ( ~ ) l n ( 1 +q,3+q, ~ )
(13)
V.M. Ponce et al./Journal of Hydrology 200 (1997) 222-227 1
2
225
1
3. m=5/3 t, = - 0.3 ln(l - q5) +0.3 cos(O.67r)ln[1+q5 +2q5cos(O.27r)] 2 1 +0.3cos(1.87r)ln[1+q5 +2q5cos(O.67r)] +0.6sin(O.67r){arctan[q~+c°s(O' ~ 27r)l[j -0.3}
+0"6sin(l'87r){ arctan[ q51+c°s(0"610]
}
4. m=2 t,= hn
(14)
(15)
\1-q71 1° I 0.9"
t I t
0.8-
t
KW
I'
0.7. ~ / I
~
m = 5/3
m= 3/2 0.6"
°sI 0.4-
/' Z
m=l
£
""/
0.3" 0.2-
0.0, 0.0
ols
11o
lls
21o
21s
3.0
Fig. 1. Rising limb of conceptual dimensionless overland flow hydrograph for selected values of the rating exponent m.
V.M. Ponce et alJJournal of Hydrology 200 (1997) 222-227
226
5. m = 3
t, =
In
-
g+arctan
- ~-(3 1 + 2 q
L (q?-1)2 j F o l l o w i n g D o o g e (1973), the solution of Eq. (10) for m = 1 and m > 1 is: 1. m = l
1
t,=-~lnq,
(17) 1-m
2. m > l
1
t , - ~ q ,
m
-1)
(18)
where q, = 1 at t, = 0, i.e. the outflow is at equilibrium at the start of the recession. The solution of Eq. (11) parallels that o f Eq. (10), but in this case q, = q/qp, and t, is affected by an appropriate dimensionless outflow factor (compare Eqs. ( 1 0 ) a n d (1 1)). Figs. 1 and 2 show the rising and recession d i m e n s i o n l e s s overland flow hydrographs
1.0.
0.9- I 0.8. 0.7. 0.6. 3/2
~- 0.50.4" 0.3.
--1 0.20.1.
/ m= 5/3 m=2
0.0.40.0
r 0.6
/ m=3
-r 1.0
i 1.5
v 2.0
2.5
3.0
t.
Fig. 2. Receding limb of conceptual dimensionless overland flow hydrograph for selected values of the rating exponent m.
V.M. Ponce et aL/Journal of Hydrology 200 (1997) 222-227
227
calculated with eqns (12)-(16) and eqns (17) and (18), respectively. Also included in Fig. 1 is the rising limb of the kinematic wave overland flow model, for m = 3/2, expressed in terms of t, for comparison (Ponce, 1989): 1 1 --
t. = ~q,m
As are for for
(19)
Fig. 1 shows, the conceptual models of overland flow have asymptotic solutions, and therefore, able to simulate runoff diffusion. Fig. 1 also shows that diffusion is greatest m = 1, the case of a linear reservoir. Within the range 1 <- m <-- 3, diffusion is smallest m - - 3.
3. S u m m a r y
We calculate conceptual dimensionless overland flow hydrographs for five rating exponents in the range m = 1 (linear reservoir) to m = 3 (laminar flow). The resulting hydrographs show that it is possible to simulate varying amounts of diffusion with the conceptual overland flow models, unlike the kinematic wave model, which is nondiffusive.
References Dooge, J.C.I., 1973. Linear theory of hydrologic systems. Technical Bulletin 1468, US Department of Agriculture, Washington, DC, pp. 327. Horton, R.E., 1938. The interpretation and application of runoff plot experiments with reference to soil erosion problems. Proc. Soil Sci. Soc. Am. 3, 340-349. Horton, R.E,, 1945. Erosional development of streams and their drainage basins: hydrophysical approach to quantitative geomorphology. Bull. Geol. Soc. Am. 56, 275-370. lzzard, C.F., 1944. The surface profile of overland flow. Trans. Am. Geophys. Union 25 (6), 959-968. lzzard, C.F., 1946. Hydraulics of runoff from developed surfaces. Proc. Highway Res, Board, Washington, DC 26, 129-146. Ponce, V.M., 1989. Engineering Hydrology, Principles and Practices. Prentice Hall, Engtewood Cliffs, NJ. Wooding, R.A., 1965. A hydraulic model for the catchment-stream problem. J. Hydrol. 3, 254-267. Woolhiser, D.A., Liggett, J.A., 1967. Unsteady one-dimensional flow over a plane--the rising hydrograph. Water Resour. Res. 3 (3), 753-771.
Journal of
Hydrology
ELSEVIER
Journal of Hydrology 200 (1997) 222-227
Generalized conceptual modeling of dimensionless overland flow hydrographs V.M. Ponce a'*, O.I. Cordero-Brafia b, S.Y.
Hasenin a
~Department of Civil and Environmental Engineering, San Diego State University, San Diego, CA 92182-1324, USA bDepartment of Mathematics and Statistics, American University, Washington, DC 20016-8050, USA
Received 30 October 1996; revised 16 January 1997; accepted 16 January 1997
Abstract
The rising and recession limbs of conceptual dimensionless overland flow hydrographs are calculated for specific values of the rating exponent in the range 1 - m -< 3, including a linear reservoir (m = 1); 100% turbulent Chezy friction (m = 3/2); 100% turbulent Manning friction (m = 5/3); 67% turbulent Chezy (or 75% turbulent Manning) (m = 2); and 100% laminar flow (m = 3). These conceptual overland flow hydrographs show finite amounts of diffusion, increasing with decreasing rating exponent, unlike the kinematic wave hydrograph, which is nondiffusive. © 1997 Elsevier Science B.V. Keywords: Dimensionless; Overland flow; Hydrographs; Conceptual modeling
1. I n t r o d u c t i o n
The calculation of overland flow hydrographs is well established in flood hydrology. Early approaches had a distinct conceptual basis (Horton, 1938, 1945; Izzard, 1944, 1946), while more recent approaches, among them the kinematic wave, have relied on the physics of the phenomena (Wooding, 1965; Woolhiser and Liggett, 1967). The kinematic wave lacks diffusion and, therefore, is not suited for overland flow over mild slopes, where diffusion plays a major role. On the other hand, the conceptual model has an intrinsic diffusion capability, which is underscored by the fact that its reference time-to-equilibrium is twice that of the kinematic wave (Ponce, 1989). A conceptual model o f dimensionless overland flow hydrographs is presented here. The * Corresponding author.
0022-1694/97/$17.00 © 1997- Elsevier Science B.V. All rights reserved PII S0022- 1694(97)00012-7
V.M. Ponce et al./Journal of Hydrology 200 (1997) 222-227
223
procedure generalizes the classical Horton-Izzard approach for a wide range of rating exponents, including m = 3/2 and 5/3, which, to our knowledge, have not been previously calculated. The rising and recession limbs of dimensionless overland flow hydrographs are calculated for the following values of the rating exponent: (1) linear reservoir, m = 1, describing a condition where the overland flow velocity is sensibly constant (Horton, 1938); (2) 100% turbulent Chezy friction, m = 3/2; (3) 100% turbulent Manning friction, m = 5/3; (4) 67% turbulent Chezy friction (equivalent to 75% turbulent Manning), m -- 2: and (5) 100% laminar flow, m = 3.
2. The conceptual model Runoff in the overland flow plane is described by the differential equation of storage: dS I - 0 =
(1)
--
dt
in which I = inflow, O = outflow, and d S / d t = rate of change of storage in the plane. A mass balance leads to the equilibrium outflow: q~ = i L
(2)
in which i = rainfall excess, and L = length of the plane. For the rising limb, I = iL, and O = q, from which dS dt
iL-q=
(3)
--
For the receding limb, I = 0, and O = q, from which -q=
dS d~-
(4)
The Horton-Izzard model is based on the discharge-storage rating (5)
q = aS"
in which a and rn are coefficient and exponent, respectively. Eq. (5) is assumed to be applicable for any discharge, including that at equilibrium. Additionally, the equilibrium storage is equal to: (6)
Se = qet~
2
in which te = reference time-to-equilibrium. Because of runoff diffusion, the equilibrium outflow is approached asymptotically, i.e. q - . qe as t ~ oo. Thus, the reference time-toequilibrium is less than the actual time-to-equilibrium, which approaches infinity. For the rising limb, the generalized conceptual model is: dS
aS m - aS m = -dt
(7)
224
V.M. Ponce et al./Journal of Hydrology 200 (1997) 222-227
The solution of Eq. (7) is (Dooge, 1973; Ponce, 1989):
1 1
t,=~
l
1 d(qm ) l-q,
(8)
in which t, = rite, with t = the accumulated time from the start of rising, and t e = the reference time-to-equilibrium, and q, =q/qe, with q = the outflow at time t, and qe = the equilibrium outflow. For the receding limb, the generalized conceptual model is: (9)
-aS m=dS
dt The solution for the receding limb depends on whether the rising limb has reached equilibrium or not. First, assuming that the rising limb is at equilibrium, the solution of Eq. (9) is (Dooge, 1973):
1 -~-d(q m) q,
t,=-~
(10)
in which t, = fit e, with t = the accumulated time from the start of recession, and te = the reference time-to-equilibrium (of the rising limb); and q, = q/qe, with q = the outflow at time t, and qe = the outflow at the start of recession, i.e. the equilibrium outflow. Assuming that the peak outflow at the end of the rising limb is qp, less than the equilibrium outflow (qe), the solution of Eq. (9) is: l-m
t,=-~
()j-
1 qp ~ ~
1
1 (q,m)
(ll)
q*
in which t, is the same as in Eq. (10), but now q, = q/qp. Eq. (8) is evaluated herein for values of m equal to 1, 3/2, 5/3, 2, and 3. 1. m = l
2. m = 3 / 2
(12)
t,=-~ln(1-q,)
t, --
(')
arctan ~
1
1
1 - arctan ~(1 +2q 3~, / - ( ~ ) l n ( 1 - q ; ) 2
+ ( ~ ) l n ( 1 +q,3+q, ~ )
(13)
V.M. Ponce et al./Journal of Hydrology 200 (1997) 222-227 1
2
225
1
3. m=5/3 t, = - 0.3 ln(l - q5) +0.3 cos(O.67r)ln[1+q5 +2q5cos(O.27r)] 2 1 +0.3cos(1.87r)ln[1+q5 +2q5cos(O.67r)] +0.6sin(O.67r){arctan[q~+c°s(O' ~ 27r)l[j -0.3}
+0"6sin(l'87r){ arctan[ q51+c°s(0"610]
}
4. m=2 t,= hn
(14)
(15)
\1-q71 1° I 0.9"
t I t
0.8-
t
KW
I'
0.7. ~ / I
~
m = 5/3
m= 3/2 0.6"
°sI 0.4-
/' Z
m=l
£
""/
0.3" 0.2-
0.0, 0.0
ols
11o
lls
21o
21s
3.0
Fig. 1. Rising limb of conceptual dimensionless overland flow hydrograph for selected values of the rating exponent m.
V.M. Ponce et alJJournal of Hydrology 200 (1997) 222-227
226
5. m = 3
t, =
In
-
g+arctan
- ~-(3 1 + 2 q
L (q?-1)2 j F o l l o w i n g D o o g e (1973), the solution of Eq. (10) for m = 1 and m > 1 is: 1. m = l
1
t,=-~lnq,
(17) 1-m
2. m > l
1
t , - ~ q ,
m
-1)
(18)
where q, = 1 at t, = 0, i.e. the outflow is at equilibrium at the start of the recession. The solution of Eq. (11) parallels that o f Eq. (10), but in this case q, = q/qp, and t, is affected by an appropriate dimensionless outflow factor (compare Eqs. ( 1 0 ) a n d (1 1)). Figs. 1 and 2 show the rising and recession d i m e n s i o n l e s s overland flow hydrographs
1.0.
0.9- I 0.8. 0.7. 0.6. 3/2
~- 0.50.4" 0.3.
--1 0.20.1.
/ m= 5/3 m=2
0.0.40.0
r 0.6
/ m=3
-r 1.0
i 1.5
v 2.0
2.5
3.0
t.
Fig. 2. Receding limb of conceptual dimensionless overland flow hydrograph for selected values of the rating exponent m.
V.M. Ponce et aL/Journal of Hydrology 200 (1997) 222-227
227
calculated with eqns (12)-(16) and eqns (17) and (18), respectively. Also included in Fig. 1 is the rising limb of the kinematic wave overland flow model, for m = 3/2, expressed in terms of t, for comparison (Ponce, 1989): 1 1 --
t. = ~q,m
As are for for
(19)
Fig. 1 shows, the conceptual models of overland flow have asymptotic solutions, and therefore, able to simulate runoff diffusion. Fig. 1 also shows that diffusion is greatest m = 1, the case of a linear reservoir. Within the range 1 <- m <-- 3, diffusion is smallest m - - 3.
3. S u m m a r y
We calculate conceptual dimensionless overland flow hydrographs for five rating exponents in the range m = 1 (linear reservoir) to m = 3 (laminar flow). The resulting hydrographs show that it is possible to simulate varying amounts of diffusion with the conceptual overland flow models, unlike the kinematic wave model, which is nondiffusive.
References Dooge, J.C.I., 1973. Linear theory of hydrologic systems. Technical Bulletin 1468, US Department of Agriculture, Washington, DC, pp. 327. Horton, R.E., 1938. The interpretation and application of runoff plot experiments with reference to soil erosion problems. Proc. Soil Sci. Soc. Am. 3, 340-349. Horton, R.E,, 1945. Erosional development of streams and their drainage basins: hydrophysical approach to quantitative geomorphology. Bull. Geol. Soc. Am. 56, 275-370. lzzard, C.F., 1944. The surface profile of overland flow. Trans. Am. Geophys. Union 25 (6), 959-968. lzzard, C.F., 1946. Hydraulics of runoff from developed surfaces. Proc. Highway Res, Board, Washington, DC 26, 129-146. Ponce, V.M., 1989. Engineering Hydrology, Principles and Practices. Prentice Hall, Engtewood Cliffs, NJ. Wooding, R.A., 1965. A hydraulic model for the catchment-stream problem. J. Hydrol. 3, 254-267. Woolhiser, D.A., Liggett, J.A., 1967. Unsteady one-dimensional flow over a plane--the rising hydrograph. Water Resour. Res. 3 (3), 753-771.