This document describes a graphical language for representing reservoir systems using time-continuous Petri nets (TCPN). Places in the TCPN represent water storages such as volumes of groundwater or energy/momentum contents. Transitions represent fluxes between storages. The TCPN uses colors to distinguish different types of quantities (mass, energy, etc.) and storages. Connections between places and transitions represent differential equations governing the system. An example TCPN represents a system of three differential equations with three storages, inputs, and both linear and nonlinear fluxes. Additional information like parameter values can be provided in tables. Adjacency matrixes describe the connections between places and transitions. TCPNs provide an algebraic framework for conceptual

1. Reservoirology & Graphs
rev 2
Riccardo Rigon R., Marialaura Bancheri, Francesco Serafin
From August 2016
SaradiNambrone,1Agosto2016

2. !2
Rational
In literature we found several representations of the water budget as reservoirs.
However these representations usually are not very explicative.
hillslope to the dynamic saturation area in the riparian zone
and underlying groundwater. The approach connects the two
upper storage units which conceptualize storage in the
riparian peat soils (Ssat) and the freely draining podzols on the
hillslopes (Sup). Direct mapping of the spatial extent of
saturated soils in the valley bottom – that were hydrologically
connected to the stream network during different wetness
conditions (see Ali et al., 2014) – allowed us to develop and
fit a simple antecedent precipitation index-type algorithm
which could explain around >90% of the variability (Birkel
et al., 2010). This algorithm was applied to create a
continuous time series of the expanding and contracting
daily saturation area extent (dSAT) (Figure 4). This dSAT
time series was used as model input to dynamically distribute
daily precipitation inputs between the storage volumes in the
landscape-based (hillslope (Sup) and saturation area (Ssat))
model structure (Soulsby et al., 2015).
Like Birkel et al. (2015), we used reservoirs that could
become unsaturated allowing storage deficits to occur. The
riparian area is normally saturated (i.e. with positive storage),
uppermost storages and captures soil moisture-related
threshold processes of runoff generation (Tetzlaff et al.,
2014). Consequently, Sup was often in deficit, but in wetter
periods would fill and spill into Ssat, which usually has low or
no deficit generating stream flow.
The storages S are state variables in the model, and we
describe the following fluxes and calibrated parameters
shown in Figure 4. The unsaturated hillslope reservoir Sup is
drained (flux Q1 in mm dÀ1
) by a linear rate parameter a (dÀ1
)
and directly contributes to the saturation area store Ssat. The
recharge rate R (mm dÀ1
) to groundwater storage Slow is
linearly calculated using the parameter r (dÀ1
). The Slow store
generates runoff Qlow (mm dÀ1
) contributing to total
streamflow Q (mm dÀ1
) using the linear rate parameter b
(dÀ1
). The runoff component Qsat (in mm dÀ1
), which is
generated nonlinearly from Ssat, conceptualizes saturation
overland flow using the rate parameter k (dÀ1
) and the
nonlinearity parameter α (—) in a power function-type
equation (Figure 2). Q is simply the sum of Qsat and Qlow. The
use of linear or non-linear parameters was based on prior
Figure 4. Conceptual diagram of the model with equations and calibrated parameters (in blue)
2487CONNECTIVITY BETWEEN LANDSCAPES AND RIVERSCAPES
Take an example of the figure above from Birkel et al. 2011. It pretends to
be explicative and from it we should be able to derive easily the set of mass
conservation that rule the system. However, this action requires a little of
analysis. A more complicate set of reservoirs is shown in the next slide.
Rigon et al.
Introduction

3. !3
1
Figure 1. Flow diagram for Prediction in Ungauged Basins2
3
4
Figure 2. Model structure derived from DEM, showing three landscape classes and the5
groundwater system connecting them.6
7
β
D
Ks
Kf
S
i
S
u,max
X
Savenije, H. H. G., & Hrachowitz, M. (2016). Opinion paper: How to make our models more physically-based. Hydrology and Earth System
Sciences Discussions, 1–23. http://doi.org/10.5194/hess-2016-433
Rigon et al.
Introduction

4. !4
The idea is to build an algebra of objects to represent (water) budgets
giving a clear idea of the type of interactions that the budget is subject
to.
Any symbol should correspond to a mathematical term or a group of
mathematical terms. The number and the collocation of parameters of
the models should be clear.
There is a better way to represent
reservoir interactions ?
Rigon et al.
Introduction

5. !5
Introduction
Rigon et al.
At the beginning,
I was trying to develop my own algebra
However I realised soon that Petri Nets cover the same area. I
had to adjust somewhat my perception because in Petri Nets,
storages (called places) are represented as circles, and fluxes
(called transitions) are represented as small rectangle. I
made, in our case, the rectangles small square. But the
concept and the rules remains the same. All of it resulted in a
graphical algebra a little more verbose than my original one,
but, at the end, more explicative.

6. !6
The other relevant difference
Introduction
Rigon et al.
The normal Petri Nets, are discrete Petri net, and do not have
time. Instead, we are dealing with dynamical systems and, having
time inside, is a necessity for us. So, our, are time continuous
Petri nets (TCPN).
Blätke, M. A., Heiner, M., & Marwan, W. (2011). Petri Nets in System Biology (pp. 1–108). Msgdeburg Universität.
2.2. Standard Petri Net 5
the integration of qualitative, continuous and stochastic information. This allows the representation
of di↵erent kinetic processes and di↵erent data types. Petri nets link structural and dynamic analysis
techniques to investigate and validate a model such as graph theory, application of linear algebra to
check a model and simulation methods. This facilitates the performance of simulation studies to explore
the time-dependent dynamic behaviour, the in-depth analysis of structural criteria and the state space
of a model.
2.2 Standard Petri Net
A Petri net is represented by a directed, finite, bipartite graph, typically without isolated nodes. The
four main components of a general Petri net are: places, transitions, arcs and tokens; see Figure 2.3,
A.
Figure 2.3: Petri Net Formalism. Petri nets consist of places, transitions, arcs and tokens (A). Just
places are allowed to carry tokens (B). Two nodes of the same type can not be connected with each other
(C). The Petri net represents the chemical reaction of the water formation (D). A transition is enabled and
may fire if its pre-places are su ciently marked by tokens.
Places are passive nodes. They are indicated by circles and refer to conditions or states. In a
biological context, places may represent: populations, species, organisms, multicellular complexes,
single cells, proteins (enzymes, receptors, transporters, etc.), molecules or ions. But places could also
embody temperature, pH-value or membrane potential; see also Section 3.3.1. Only places are allowed
to carry tokens; see Figure 2.3, B.

7. !7
For information about
normal Petri Nets, a good reference is
Murata, T. (1989). Petri Nets: Properties, Analysis and Applications. Proceedings of the IEEE,
77(4), 1–40.
Petri Nets
Rigon et al.

8. !8
So …. our graphical language
Places correspond to storages, for instance the volume of
water in groundwater, or the energy content of the same
groundwater, or its momentum content. To distinguish the
various “storages” the circle has a variable specification. It
is intended that a graph is composed of places that
contains the same type of quantities, mass, momentum or
energy. Places can be coloured, and the same color used to
represent the same physical place but in different types of
budgets.
Places
Rigon et al.

9. !9
So …. our graphical language
Each one of the places depicted represents the time
variation of the named quantity. For instance the place at
the center represents
Places
Rigon et al.

10. !10
Symbols for reservoirs systems
an arc (positive in the direction of the arrow) connect
a place with a transition and viceversa.
In our case a transition is a flux. By convention, the arc is
of the same color of the place from where it exits.
Rigon et al.
Transitions or Fluxes
A flux (transition) is represented by a square the
symbols inside represents the type of flux

11. !11
So, a linear reservoir
is represented as follows:
Rigon et al.
A simple example to begin with
with a one-to one correspondence with the equation below:

12. !12
or, BTW
Rigon et al.
A simple example to begin with
if we want to use colors

13. !13
Symbols
a linear flux (positive in the direction of the arrow).
an external forcing, for instance precipitation but can be
evapotranspiration, if this is measured is represented by
a
It is a term of the r.h.s of the budget equation
a non linear flux (positive in the direction of the arrow).
The symbol indicating non-linearity alone
Rigon et al.
Arcs

14. !14
so, the previous linear budget could be
Rigon et al.
A simple example to begin with
when rainfall is an assigned time series, i.e. a flux boundary condition

15. !15
Who is connected with whom ?
Rigon et al.
Allowed connections (with colors)
So we can have
but not other type of connections
transition —> place
place —> transition
place —> transition —> place
transition —> place —> transitions

16. !16
Who is connected with whom ?
Rigon et al.
One transition can be connected with more than one place,
implying the existence of a partition coefficient
One place can have more that one transitions, also implying
some partition coefficient

17. !17
One flux that interacts back:
simply means that
where B is some metastable place
Who is connected with whom ?
Rigon et al.

18. !18
Soulsby, C., Birkel, C., & Tetzlaff, D. (2016). Modelling storage-driven connectivity between landscapes and riverscapes: towards a simple
framework for long-term ecohydrological assessment. Hydrological Processes, 30(14), 2482–2497. http://doi.org/10.1002/hyp.10862
hillslope to the dynamic saturation area in the riparian zone
and underlying groundwater. The approach connects the two
upper storage units which conceptualize storage in the
riparian peat soils (Ssat) and the freely draining podzols on the
hillslopes (Sup). Direct mapping of the spatial extent of
saturated soils in the valley bottom – that were hydrologically
connected to the stream network during different wetness
conditions (see Ali et al., 2014) – allowed us to develop and
fit a simple antecedent precipitation index-type algorithm
which could explain around >90% of the variability (Birkel
et al., 2010). This algorithm was applied to create a
continuous time series of the expanding and contracting
daily saturation area extent (dSAT) (Figure 4). This dSAT
time series was used as model input to dynamically distribute
daily precipitation inputs between the storage volumes in the
landscape-based (hillslope (Sup) and saturation area (Ssat))
model structure (Soulsby et al., 2015).
Like Birkel et al. (2015), we used reservoirs that could
become unsaturated allowing storage deficits to occur. The
riparian area is normally saturated (i.e. with positive storage),
but can have small deficits following prolonged dry periods
in summer. In the upper stores, water levels below a certain
threshold can only be further depleted by transpiration and no
lateral flow to the riparian area will be generated. Incoming
precipitation fluxes are first intercepted and reduced by PET –
if available. The remaining effective precipitation fills the
uppermost storages and captures soil moisture-related
threshold processes of runoff generation (Tetzlaff et al.,
2014). Consequently, Sup was often in deficit, but in wetter
periods would fill and spill into Ssat, which usually has low or
no deficit generating stream flow.
The storages S are state variables in the model, and we
describe the following fluxes and calibrated parameters
shown in Figure 4. The unsaturated hillslope reservoir Sup is
drained (flux Q1 in mm dÀ1
) by a linear rate parameter a (dÀ1
)
and directly contributes to the saturation area store Ssat. The
recharge rate R (mm dÀ1
) to groundwater storage Slow is
linearly calculated using the parameter r (dÀ1
). The Slow store
generates runoff Qlow (mm dÀ1
) contributing to total
streamflow Q (mm dÀ1
) using the linear rate parameter b
(dÀ1
). The runoff component Qsat (in mm dÀ1
), which is
generated nonlinearly from Ssat, conceptualizes saturation
overland flow using the rate parameter k (dÀ1
) and the
nonlinearity parameter α (—) in a power function-type
equation (Figure 2). Q is simply the sum of Qsat and Qlow. The
use of linear or non-linear parameters was based on prior
systematic tracer-aided multi-model testing for similar
catchments in the Scottish Highlands (Birkel et al., 2010;
Capell et al., 2012). In particular, the non-linear conceptu-
alization of Qsat has a physical basis in the dynamic
expansion of the saturation area and fluxes that generate
storm runoff. Likewise, the linear nature of Q1 and R reflect
Figure 4. Conceptual diagram of the model with equations and calibrated parameters (in blue)
2487CONNECTIVITY BETWEEN LANDSCAPES AND RIVERSCAPES
Copyright © 2016 The Authors Hydrological Processes Published by John Wiley & Sons Ltd. Hydrol. Process. 30, 2482–2497 (2016)
Birkel, C., Soulsby, C., & Tetzlaff, D. (2011). Modelling catchment-scale water storage dynamics: reconciling dynamic storage with tracer-inferred passive
storage. Hydrological Processes, 25(25), 3924–3936. http://doi.org/10.1002/hyp.8201
Rigon et al.
A more complicate example
We can represent the initial system as follows:

19. !19
Now, start to read it. There are three ordinary differential equations
(ODEs), represented by the three places, and here also colored in different
ways. Moreover, there is a given input (J). One of the equations, contains a
non linear term. The others are linear. Because J is split into two directions,
a partition coefficient is necessary.
Rigon et al.
One circle is one ODE

20. !20
The equations, in one-to-one correspondence to the net are:
Where the fluxes form is note yet specified.
Rigon et al.
One circle is one ODE

21. !21
The missing knowledge can be given by a table of this type:
where 7 parameters are given, and the type on non linearity present in equation
(2) is specified. J(t) is given. If also ET is measured (as actually happened in the
original paper, parameters reduce to 5).
Rigon et al.
Topology must be completed

22. !22
As in any other graph, we also have an adjacency matrix (AM) to exploit
the connections. Because in TCPN the connections are between transition
and places and places to transitions, we can split the AM into two
matrixes. We can see them from the point of view of transition (fluxes).
This is called Pre These are called Post
is the set of places
is the set of transitions
Formalities
Rigon et al.

23. !23
is the set of places
is the set of transitions
Pre is a matrix n x l (rows are for the places, column are for the
transitions), and to each couple is associated a flux, or, in a more abstract
way to say it, it is:
Where represents a space of allowed fluxes expressions
Post is a matrix l x n
Both Pre and Post are adjacency matrixes with weighted entries
Rigon et al.
Formalities
If

24. !24
So, we can try a first definition by saying that a TCPN is a 5-tuple:
where S is the set of tokens present in places (at any specific time, including
the initial one).
The set
is said the preset of . The set:
the postset of
Navarro-Gutierrez, M., Ramirez-Trevino, A., & Gomez-Gutierrez, D. (2016). Modelling the behaviour of a class of dynamical systems with
Continuous Petri Nets (pp. 1–6). Presented at the 2013 IEEE 18th Conference on Emerging Technologies & Factory Automation (ETFA),
IEEE. http://doi.org/10.1109/ETFA.2013.6647992
Rigon et al.
Formalities

25. !25
Actually, we introduced a few new elements here (besides places, transitions, arcs,
tokens):
a Table of association between fluxes and their expression:
Expressions or, possible algebraic form of the fluxes
Rigon et al.
Table of association

26. !26
Rigon et al.
Table of association
So, we can say that, the full description of system is a 7-tuple:
where
is a vocabulary of fluxes and symbols, with their semantic
is the set of expressions where the symbols are combined to produce
the fluxes form and, ultimately the fluxes values
To build the vocabulary, we used simple table, but a more rigorous
approach could be envisioned by looking at the Basic Modelling
Interface (BMI)

27. !27
Introducing some of the next networks we will also show explicitly
what the previous sets and matrixes are.
A TCPN identifies a a coupled set of ordinary differential equations, in number
equal to the rank of the places. They are:
Rigon et al.
Formalities
In practice, Post and Pre are sparse matrixes, and it could be convenient to
store them in triplets
where expression, eventually are mapped to real values when expressions are
evaluated.

28. !28
Every modelling solution is actually a compound. So the blue graph
below on the right can be seen as a “coarse graining” of the one on the left.
To signify compounds the normal Petri net graphics use a sign, which we
will use, instead to indicate sink/source effects.
Rigon et al.
Compounds

29. !29
thus, if the linear water budget below contains also a sink/source one
Rigon et al.
Sources/Sinks
it can be represented as follows:

30. !30
A sink - source term could always be represented with a coupled equation.
Therefore, the previous graphs could be
However, not always one wants to explicit all the chain of interrelationships.
The symbol
was used to mean an unspecified number of interrelation and fluxes.
Rigon et al.
Sources/Sinks

31. !31
We can have also systems of equations, formally of the same type, but
parametrised by some variable, for instance, one case is that of the age-ranked
water budget:
e.g. Rigon and Bancheri, 2016,
where the water volumes are separated according to water age
Rigon et al.
Parameterised equations

32. !32
In this case, instead of representing the equations with (unspecified) many
graphs, we use only one, but with shadows or borders.
Rigon et al.
Parameterised equations

33. !33
When dealing with cases in which some physics-chemical reaction can be reverted,
the case can be represented as a loop:
And simplified, as:
Rigon et al.
More complicate topologies

34. !34
Rigon et al.
Coupled equations: the single reservoir water budget
The single reservoir water budget
has been a little complicated
with respect to the simpler one
in previous slides, to introduce
evapotranspiration ad
percolation loss. Both
precipitation and percolation are
assigned as flux boundary
conditions.

35. !35
Rigon et al.
Coupled equations: the single reservoir energy budget
Graphical figures are of the same
color of the ones in the mass
budget when referring to the same
location, and just the “token” type
change. Some quantities have the
same name than those in the mass
budget, but are underlined. In fact,
any advection brings energy from
one place to another. To obtain
from mass advection the energy
advection, we need to multiply the
internal energy per unit mass
transported for the mass, which is,
for instance for
evapotranspiration:
Rigon et al.

36. !36
Rigon et al.
Coupled equations
The energy budget contains other
terms, beyond the advective fluxes.
• Radiation,
• thermal convective fluxes,
• conduction
Radiation is given here as an
external flux. This is not
completely true, because it is
actually a net budget which
involve also the temperature of
surfaces.
Rigon et al.

37. !37
Rigon et al.
Coupled equations
A single reservoir water budget with
Evapotranspiration. Its equation is
As said, both J and Jg are assigned
Rigon et al.

38. !38
Rigon et al.
Coupled equations
The fact that Evapotranspiration expression contains the net radiation, suggests
that the budget can be modified as follows
where the symbol
tells that Rn does not enter in the budget but it is one of its parameters
the dotted line means that it is an assigned boundary condition.
Rigon et al.

39. !39
Rigon et al.
Coupled equations
Rigon et al.

40. !40
Rigon et al.
Coupled equations
The two coupled budget can be represented as above. Arcs of the energy
budget are weighted, in order to transform the quantity advected in energy. J in
this case is not partitioned because the weight also implies it. The global graph
look simpler than the single two.
Rigon et al.

41. !41
Rigon et al.
Coupled equations
Shadows in the water budget mean that the water budget is parametrised, and
represents actually a group of equations.

42. !42
Now assume to have a river network
Consider the path starting in A1, for example.
It can be decomposed into steps (states)
and we can write the water budget for each
of them.
Rigon et al.
River Networks

43. !43
The Petri scheme can be
Rigon et al.
River Networks

44. !44
The full network interactions can be represented as follows
Rigon et al.
River Networks

45. !45
It can be simplified as shown above
Rigon et al.
Simplifications

46. !46
And further simplified as:
meaning that all the subnet of i goes into 3, and those of k into 5
Rigon et al.
Simplifications

47. !47
Here we introduce a space explicit place/
The difference is that the storage in channels has a spatial curvilinear
coordinate. Tributary’s water enters the channel at some “x”, and exit
at the last downstream x.
Rigon et al.
Space explicit places

48. !48
The full network interactions or as below.
A typical case is when channel is described by a width function, or
when the dynamics of water is modeled by a 1d - de Saint-Venant
equation.
Rigon et al.
Space explicit places

49. !49
The network above can be simplified as
where it is assumed that each i outflow has a coordinate x associated.
Rigon et al.
Space explicit places simplified

50. !50
The network with a space explicit channel is, at the end,
particularly simple. But it is easy to produce more complicate
configurations, especially if a single hydrologic response unit
(HRU) is subdivided in many interconnected reservoirs.
Rigon et al.
Comments

51. !51
Conclusion
We hope to have defined a group of signs able to simplify the understanding of
reservoir interaction and the way we build our model.
Rigon et al.
Questions ?

52. !52
Find this presentation at
http://abouthydrology.blogspot.com
Ulrici,2000?
Other material at
Domande
Rigon et al.