A mass conservative and water storage consistent variable parameter Muskingum-Cunge approach

Abstract

Abstract. The variable parameter Muskingum-Cunge (MC) flood routing approach, together with several variants proposed in the literature, does not fully preserve the mass balance, particularly when dealing with very mild slopes (<10−3). This paper revisits the derivation of the MC and demonstrates (i) that the loss of mass balance in MC is caused by the use of time variant parameters which violate the implicit assumption embedded in the original derivation of the Muskingum scheme, which implies constant parameters and at the same time (ii) that the parameters estimated by means of the Cunge approach violate the two basic equations of the Muskingum formulation. The paper also derives the modifications needed to allow the MC to fully preserve the mass balance and, at the same time, to comply with the original Muskingum formulation in terms of water storage. The properties of the proposed algorithm have been assessed by varying the cross section, the slope, the roughness, the space and the time integration steps. The results of all the tests also show that the new algorithm is always mass conservative. Finally, it is also shown that the proposed approach closely approaches the full de Saint Venant equation solution, both in terms of water levels and discharge, when the parabolic approximation holds.

Reply to the Discussion of “Assessment and review of the hydraulics of storage flood routing 70 years after the presentation of the Muskingum method” by M. Perumal

“…This is also demonstrated in comparisons of storage routing results with solutions of the complete equations (e.g. Koussis, 1975;Miller & Cunge, 1975;Weinmann, 1977;Weinmann & Laurenson, 1979;Todini, 2007). …”

Discussion of “Assessment and review of the hydraulics of storage flood routing 70 years after the presentation of the Muskingum method”

“…The only way to overcome this paradoxical illusion is to develop the linear form of the Muskingum storage equation using hydrodynamic principles based on the extension of the Kalinin-Miljukov (K-M) theory (Apollov et al, 1964), as illustrated by the author in his paper, and by this writer (Perumal, 1992b). Alternatively, the justification for the linear storage equation in the form of weighted discharge can also be arrived at directly from the St Venant equations based on the theory advocated by this writer (Perumal, 1994a), or by the method advocated by Todini (2007) and Price (2009). Detailed description of these developments is beyond the scope of this Discussion.…”

“…It is this writer's hunch that Cunge (1969) had openly discredited McCarthy's Muskingum storage theory, by equating the reach storage to sum of prism and wedge storages, based on Chow's (1959) representation of the storage equation. But a careful study of interpreting McCarthy's (1938) Muskingum method based on either the Kalinin-Miljukov extension theory (Apollov et al, 1964), or the approaches advocated by Perumal (1994a), Todini (2007) and Price (2009), shows that the discharge q m at the centre of the Muskingum reach length L can be expressed as:…”

“…But, unfortunately, the author has mostly oriented the review based on the studies carried out by himself until the year 1986. The only recent relevant literature quoted in the paper pertains to the work of Todini (2007). However the author chose to ignore a significant body of literature presented by Price (1985), and this writer and his co-workers (Perumal 1992a(Perumal ,b, 1994a(Perumal ,b, 2003Perumal & RangaRaju, 1999, 2007Perumal et al, 2001 on improving the understanding of the Muskingum method directly, and indirectly (Perumal & Ranga-Raju, 1998a, including the study of unification of matched diffusivity theories (Perumal, 1989) advocated by Cunge (1969) and the author (Koussis, 1978).…”

“…The only recent relevant literature quoted in the paper pertains to the work of Todini (2007). However the author chose to ignore a significant body of literature presented by Price (1985), and this writer and his co-workers (Perumal 1992a(Perumal ,b, 1994a(Perumal ,b, 2003Perumal & RangaRaju, 1999, 2007Perumal et al, 2001 on improving the understanding of the Muskingum method directly, and indirectly (Perumal & Ranga-Raju, 1998a, including the study of unification of matched diffusivity theories (Perumal, 1989) advocated by Cunge (1969) and the author (Koussis, 1978). In Perumal (1989), it was shown that both the approaches of matched diffusivity theories lead to the same coefficients C 1 , C 2 and C 3 of the Muskingum routing equation in terms of channel and flow characteristics, and the differences in the form of the weighting parameters of both schemes are only apparent.…”