The calculation of overland flow hydrographs is well established in flood hydrology. Early approaches had a distinct conceptual basis (Horton, 1938,1945; lzzard, 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 of dimensionless overland flow hydrographs is presented here. The procedure generalizes the classical Horton-Izzard approach for a wide range of rating exponents, including
Runoff in the overland flow plane is described by the differential equation of storage:
in which A mass balance leads to the equilibrium outflow:
in which
For the receding limb,
The Horton-Izzard model is based on the discharge-storage rating:
in which
in which q → q as _{e}t → ∞. Thus, the reference time-to-equilibrium is less than the actual time-to-equilibrium, which approaches infinity. For the rising limb, the generalized conceptual model is:
The solution of Eq. 7 is (Dooge, 1973; Ponce, 1989):
in which t = the accumulated time from the start of rising, and t = the reference time-to-equilibrium, and _{e}q_{ *} = q / q, with _{e}q = the outflow at time t, and q = the equilibrium outflow.
_{e}For the receding limb, the generalized conceptual model is:
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):
in which t = the accumulated time from the start of recession, and t = the reference time-to-equilibrium (of the rising limb); and _{ *}q_{ *} = q /q, with _{e}q = the outflow at time t, and q = the outflow at the start of recession, i.e. the equilibrium outflow.
_{e}
Assuming that the peak outflow at the end of the rising limb is q), the solution of Eq. 9 is:
_{e}
in which
Equation 8 is evaluated herein for values of
Following Dooge (1973), the solution of Eq. 10 for
where
The solution of Eq. 11 parallels that of Eq. 10, but in this case t _{ *} is affected by an appropriate dimensionless outflow factor (compare Eq. 10 and 11).
Figures 1 and 2 show the rising and recession dimensionless overland flow hydrographs calculated with Eqs. 12-16 and Eqs. 17 and 18, respectively. Also included in Fig. 1 is the rising limb of the kinematic wave overland flow model, for
As Fig. 1 shows, the conceptual models of overland flow have asymptotic solutions, and am therefore, able to simulate runoff diffusion. Figure 1 also shows that diffusion is greatest for
We calculate conceptual dimensionless overland flow hydrographs for five rating exponents in the range
Dooge, J.C.I., 1973. Linear theory of hydrologic systems.
Horton, R. E., 1938. The interpretation and application of runoff plot experiments with reference to soil erosion problems.
Horton, R. E., 1945. Erosional development of streams and their drainage basins: Hydrophysical approach to quantitative geomorphology. Izzard, C. F., 1944. The surface profile of overland flow. Trans. Am. Geophys. Union 25 (6), 959-968.
Izzard, C. F., 1946. Hydraulics of runoff from developed surfaces.
Ponce, V. M., 1989.
Wooding, 1965. A hydraulic model for the catchment-stream problem.
Woolhiser, D. A., Liggett, J. A., 1967. Unsteady one-dimensional flow over a plane - the rising hydrograph. |

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