Flow through a reservoir is governed by the one-dimensional equation of storage (Fig. 2):

in which I = inflow, O = outflow, and dS/dt = rate of change of storage, expressed in L³ T ^-1 units.

Figure 2  Inflow, outflow, and rate of change of storage in a control volume.

A commonly used relationship between outflow and storage is the following:

in which K = storage coefficient and m = exponent. For m = 1, Eq. 2 reduces to the linear form:

in which K is a proportionality constant or linear storage coefficient, which has the units of time (T). When combined with Eq. 1, either Eq. 2 or Eq. 3 will provide runoff diffusion (Ponce, 1989).

3.  DIFFUSION IN OPEN-CHANNEL FLOW

Channels are surface-water hydraulic features which may or may not provide runoff diffusion, depending on the relative scale of the flood wave. The amount of wave diffusion is characterized by the dimensionless wavenumber σ_*, as shown in Fig. 3. The dimensionless wavenumber is defined as:

in which L = wavelength of the disturbance, and L_o = the length of channel in which the equilibrium flow drops a head equal to its depth (Lighthill and Whitham, 1955):

Four types of waves are identified:

1. Kinematic waves,

2. Diffusion waves,

3. Mixed kinematic-dynamic waves, and

4. Dynamic waves.

Kinematic waves lie on the left side of the wavenumber spectrum, featuring constant dimensionless relative wave celerity and zero attenuation. Dynamic waves lie on the right side, featuring constant dimensionless relative wave celerity and zero attenuation. Mixed kinematic-dynamic waves lie in the middle of the spectrum, featuring variable dimensionless relative wave celerity and medium to high attenuation. Diffusion waves are intermediate between kinematic and mixed kinematic-dynamic waves, featuring mild attenuation. [Note that in hydraulic engineering practice, dynamic waves are commonly referred to as Lagrange or "short" waves, while the mixed kinematic-dynamic waves are commonly referred to as "dynamic waves," fueling a semantic confusion].

Fig. 3  Celerity of wave propagation in open-channel flow (Ponce and Simons, 1977).

For flood routing computations, the governing equations of continuity and motion, commonly referred to as the Saint Venant equations, may be linearized and combined into a convection-diffusion equation with discharge Q as the dependent variable (Hayami, 1951; Dooge, 1973; Dooge et al., 1982; Ponce, 1991a ; Ponce, 1991b):

in which V = Vedernikov number, defined as the ratio of relative kinematic wave celerity to relative dynamic wave celerity (Ponce, 1991b):

in which β = exponent of the discharge-flow area rating Q = A^β, V_o = mean flow velocity, D_o = mean flow hydraulic depth, and g = gravitational acceleration.

In Eq. 6, for V = 0, the coefficient of the second-order term reduces to the kinematic hydraulic diffusivity, originally due to Hayami (1951). On the other hand, for V = 1, the coefficient of the second-order term reduces to zero, and the diffusion term vanishes. Under this flow condition, all waves, regardless of scale, travel with the same speed, fostering the development of roll waves (Fig. 4).

Fig. 4  Roll waves in a steep irrigation canal.

4.  DIFFUSION IN CATCHMENT DYNAMICS

Surface runoff in catchments may be one of three types (Ponce, 1989a; 2014a):

1. Concentrated flow, when the effective rainfall duration is equal to the time of concentration.

   Fig. 5 (a)  Concentrated flow.

2. Superconcentrated flow, when the effective rainfall duration is longer than the time of concentration.

   Fig. 5 (b)  Superconcentrated flow.

3. Subconcentrated flow, when the effective rainfall duration is shorter than the time of concentration.

   Fig. 5 (c)  Subconcentrated flow.

Figure 6 shows a typical open-book schematization for overland flow modeling. Input is effective rainfall on two planes adjacent to a channel. Output is the outflow hydrograph at the catchment outlet.

Fig. 6  Open-book catchment schematization.

Figure 7 shows dimensionless catchment outflow hydrographs for the three cases described above (Ponce and Klabunde, 1999). The maximum possible peak outflow is: Q_p = I_e A, in which I_e = effective rainfall intensity, and A = catchment area. By definition, the maximum possible peak outflow is reached for concentrated and superconcentrated flows. However, in the case of subconcentrated flow, the peak outflow fails to reach the maximum possible value. Effectively, this amounts to runoff diffusion, because the flow has actually been spread in time (and space).

Thus, runoff diffusion is produced for all waves when the time of concentration exceeds the effective rainfall duration. This is typically the case of midsize and large basins, for which the catchment slope (along the hydraulic length) is usually sufficiently mild (small). The time of concentration is directly related to catchment hydraulic length and bottom friction, and inversely related to bottom slope and effective rainfall intensity (Ponce, 1989b; 2014b).

Fig. 7  Dimensionless catchment runoff hydrographs (Ponce and Klabunde, 1999).

Figure 8 shows dimensionless rising overland flow hydrographs for a kinematic wave model (labeled KW) and for several storage-concept models, for the rating exponent m (Eq. 2) ranging from m = 1, corresponding to a linear reservoir, to m = 3, corresponding to laminar flow (Ponce et al., 1997). The kinematic wave time-to-equilibrium, akin to the time of concentration, is theoretically equal to one-half of the time of concentration of the storage-based models (Ponce, 1989; 2014). It is seen that the storage models spread the hydrograph and, consequently, produce diffusion, while the kinematic wave model lacks runoff diffusion altogether. The kinematic time-to-equilibrium is the shortest possible value of time of concentration, resulting, in the aggregate, in the largest peak flows. Thus, under pure kinematic flow, runoff diffusion vanishes.

Fig. 8  Dimensionless rising hydrographs of overland flow (Ponce et al., 1997).

In practice, a numerical kinematic wave model may not be entirely devoid of diffusion, due to the appearance of numerical diffusion (Ponce, 1991a). In fact, first-order schemes of the kinematic wave equation produce numerical diffusion. This diffusion, however, is uncontrolled, not based on physical parameters and, therefore, unrelated to the true diffusion of the physical problem.

5.  DIFFUSION IN ONLINE COMPUTATIONS

Examples of runoff diffusion can be calculated with the following online scripts:

• Program ONLINE ROUTING01 for linear reservoir routing.

• Program ONLINE ROUTING05 for channel routing using the Muskingum-Cunge method.

• Program ONLINE OVERLAND for overland flow using the diffusion wave method.

6.  SUMMARY

The concept of runoff diffusion is reexamined in order to clarify its role in connection with unsteady free-surface flow routing in reservoirs, channels, and catchments. All flood routing calculations are effectively computing the amount of wave diffusion. Diffusion is the spreading of the hydrograph in time and, therefore, in space. Diffusion is inherent to reservoirs and it is always produced in flow through reservoirs.

In channel flow, diffusion is produced in the absence of kinematic or dynamic wave conditions, i.e., under diffusion or mixed kinematic-dynamic wave conditions, the latter provided the Vedernikov number is less than 1. At the threshold V = 1, all diffusion disappears and this flow condition promotes the development of roll waves.

In catchment runoff, diffusion is produced: (1) for all wave types, when the time of concentration exceeds the effective rainfall duration, i.e., for subconcentrated catchment flow, a condition which is usually associated with midsize and large basins, or (2) for all effective rainfall durations, when the wave is a diffusion wave (or mixed kinematic-dynamic wave type), which is usually associated with a sufficiently mild catchment slope. In conclusion, in catchment runoff, diffusion is produced in: (1) midsize and large basins, with long time of concentration, and/or (2) mild catchment slopes, which typically feature diffusion waves.

REFERENCES

Dooge, J. C. I. (1973). Linear theory of hydrologic systems, Technical Bulletin No. 1468, Agricultural Research Service, U.S. Department of Agriculture, Washington, D.C.

Dooge, J. C. I., W. B. Strupczewski, and J. J. Napiorkowski. (1982). Hydrodynamic derivation of storage parameters of the Muskingum model, Journal of Hydrology, Vol. 54, 371-387.

Hayami, S. (1951). On the propagation of flood waves. Bulletin No. 1, Disaster Prevention Research Institute, Kyoto University, Kyoto, Japan, December.

Lighthill, M. J., and G. B. Whitham. 1955. On kinematic waves: I. Flood movement in long rivers. Proceedings, Royal Society of London, Series A, 229, 281-316.

Ponce, V. M., and D. B. Simons. (1977). Shallow wave propagation in open channel flow. Journal of the Hydraulics Division, ASCE, Vol. 103, No. HY12, pages 1461-1476, December.

Ponce, V. M. (1986). Diffusion wave modeling of catchment dynamics. Journal of Hydraulic Engineering, ASCE, Vol. 112, No. 8, pages 716-727, August.

Ponce, V. M. (1989). Engineering hydrology: Principles and practices, Prentice Hall, Englewood Cliffs, New Jersey.

Ponce, V. M. (1991a). The kinematic wave controversy. Journal of Hydraulic Engineering, ASCE, Vol. 117, No. 4, pages 511-525, April.

Ponce, V. M. (1991b). New perspective on the Vedernikov number. Water Resources Research, Vol. 27, No. 7, pages 1777-1779, July.

Ponce, V. M., O. I. Cordero-Braña, and S. Y. Hasenin. (1997). Generalized conceptual modeling of dimensionless overland flow hydrographs. Journal of Hydrology, 200, pp. 222-227.

Ponce, V. M., and A. C. Klabunde. (1999). Parking lot storage modeling using diffusion waves. Journal of Hydrologic Engineering, Vol. 4, No. 4, October, pp. 371-376.

Ponce, V. M. (2014). Engineering hydrology: Principles and practices, Second edition, online.