1.1.4 Paper [D]
A grey box model describing the hydraulic in a creek.
In Paper[D] the subject is flow routing in a creek in a small watershed, in Northern Zealand in Denmark. The exact size of the watershed is not known.
There are two measuring stations in the creek, see Figure 1.8. The available
L = 2191m
A B
A: Station A
R R
B: Station B R: Rainfall runoff outlet
Kokkedal Nive mølle
Figure 1.8: A sketch of the area in Usserod river.
data are precipitation and depth at two locations in the creek. The goal is to find a relationship between the depth at the upstream station and the depth at the downstream station and to predict the output depth at the downstream station. The Saint Venant equation of mass balance is used as a basis and the lateral inflow between the two measuring stations is modelled as a first order process with precipitation as input. The resulting model is a stochastic linear reservoir model described in continuous time by stochastic differential equations. The model is, however, different from the traditional reservoir model in that the lateral inflow of water between the two measuring stations is a state variable in the model and estimated by use of the Kalman filtering technique.
This can be used in an environmental context so that it might be possible to estimate the concentration of chemical concentrations in the lateral inflow if the corresponding chemical concentrations are estimated both upstream and downstream. This can be very valuable in an environmental analysis. The program CTSM was used to estimate the parameters.
1.2 Comparison of the models
The hydrological subjects in this PhD project are on very different scales and with different aspects. However, all the projects are within the theory of hy-drology. Thus the physical law, conservation of mass is the fundamental law.
In hydrology this can be referred to asthe storage effect, i.e., what comes in is either stored or comes out, see Figure 1.9.
Q
I S
Figure 1.9: The storage effect The storage equation is written
dS
dt =I(t)−Q(t) (1.2)
whereI(t) is the input,Q(t) is the output andS(t) is the storage. The change in storage is the difference of input and output. All the projects/papers have to do with the storage but in different aspects.
The different storage interpretations can be seen graphically in Figure 1.10. In the following the different storage effects are summarized whereas an overview of the included papers are given in section 1.1
The subject of Paper[A] is a rainfall-runoff relationship in a sewage system.
The input is precipitation and the output is excess flow. The storage is twofold Firstly the storage is the time lag between input and output and, secondly, the storage is the long term storage. The model is an input-output model or a black box model, and since the input is precipitation and not effective precipitation the mass balance is not conserved in the model. This can be interpreted so that the storage container either swallows the rain or stores it on a long term basis. However, water comes out of the system eventually. Part of the water evaporates and some is permeated by plants. However, large part infiltrates into the root zone and becomes groundwater and can eventually be observed in creeks and rivers.
The subject in Paper[B] is also a rainfall-runoff relationship, the input is precipi-tation and the output is discharge. The model is not a mass balance model, since evaporation/transpiration are not taken into account, and the base flow is repre-sented by a constant. However in this project there is no swallowing, a balance between input and output exists, only the ”up-scaling” of the precipitation mea-surements is underestimated due to the amount of evaporation/transpiration and groundwater contribution. In this project the storage is time delay. It is a short-term time delay between rain and discharge during the summer and because of the snow storage it is a long-term time delay during winter time.
The topic in Paper[C] is in a different category. The topic is a risk assessment of a water shortage in a hydropower plant, i.e., the risk of emptying the reser-voir. The input data is discharge series and the output is risk assessment, i.e.,
1.2 Comparison of the models 11
Precipitation
Sewage system
Discharge
Time dealy Swallowing
plant hydropower Reservoir in a
shortage water Risk of Discharge
Volume in reservoir
in Iceland Rainfall−runoff
Discharge Precipitation
Time delay Snow contatiner
Storage
[A] [B]
[D] [C]
in a creek Flow routing
upstream waterlevel−
Precipitation Retention time
waterlevel−
downstream
Figure 1.10: Overview of the storage effect.
probabilities versus volume of water shortage. This subject certainly involves storage, and in fact this might be the most obvious form of storage, a storage of water in a reservoir, measured in giga-liters, long-term storage of water from year to year.
Finally the subject in Paper [D] is a flow routing in a creek, the input is both a precipitation and the upstream depth. The water level at the downstream station is modelled as a function of the water level at the upstream station and precipitation. This is a small creek with short distance between the stations and no sub-creek merging in between. Consequently, the upstream water level has the largest impact on the downstream water level. Hence,S the storage is mostly the retention time between the two measuring stations.