This document presents a graphical method for representing systems of partial differential equations using Petri nets. It uses the 1D Richards equation modeling unsaturated flow as an example. Key aspects covered include representing the state variable, fluxes, boundary conditions, ancillary expressions, and coupling between surface and subsurface domains. The method allows for a complete representation of complex partial differential equation systems through graphical notation.

1. Reservoirology & Graphs
A case of spatially explicit models using 1D Richards equation as an
example
Riccardo Rigon R. & Niccolò Tubini
January 2017
PaulKlee,Fireatfullmoon,1933

2. !2
Rigon & Tubini
A little summary
So, we can say that, the full description of system is a 7-tuple:
where
is the set of place (i.e. the set of state variables with associated
equations)
They are denoted by a circle
The variable name, in this case, is

3. !3
A little summary
So, we can say that, the full description of system is a 7-tuple:
where
is the set of transitions (i.e. the set of fluxes/input-outputs)
They are denoted by a square (and some arcs to connect transitions to
places)
The flux name, in this case, is
Rigon & Tubini

4. !4
A little summary
So, we can say that, the full description of system is a 7-tuple:
where
is the set of connection between a place to a certain transition. If the
transition and the places are indexed by j and i respectively Pre is a
matrix of row index j and column index i
The matrix elements, in principle can be as simple as 1s or/and 0s
indicating the existence or not existence of a connection, but it can
be convenient to have a matrix of expressions and/or values (see
below)
Rigon & Tubini

5. !5
A little summary
So, we can say that, the full description of system is a 7-tuple:
where
is the set of connection between a transition to a certain palce. If the
transition and the places are indexed by j and i respectively Post is a
matrix of row index i and column index j
Adj = Pre -Post is an adjacency matrix -among places- in the
language of graph theory
Rigon & Tubini

6. !6
A little summary
So, we can say that, the full description of system is a 7-tuple:
where
is the set of initial conditions associated to places (in Petri nets
parlor they are “tokens”.
In discrete Petri nets, tokens are represented as small black circles,
and they are consumed at any time step by producing an output. At
least when the time step is explicit.
Rigon & Tubini

7. !7
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. For the
graph below the vocabulary is. The name and the Unit refers to the
semantic (without dive into cognitive sciences) which is supposed to
have a record in our brain.
Rigon & Tubini

8. !8
A little summary
So, we can say that, the full description of system is a 7-tuple:
where
is the set of expressions associate to each non null entry in the Pre
and Post matrixes
The table of expression requires the addition of a few symbol in
the vocabulary
Rigon & Tubini

9. !9
A single reservoir water budget with
Evapotranspiration can be the
represented as
As said, both J and Jg are assigned
Rigon & Tubini
A little summary

10. !10
As a result we are now in the possibility to have a compelete
graphic representation of any
ordinary differential equations sets
For more details see Reservoirology #3 Abouthydrology blogpost
Rigon & Tubini
Introduction

11. !11
partial differential equations sets
Is it possible to use the same characterisation to represent
?
If you want to see the equation first, just continue with the next slide. Otherwise skip to this slide
Rigon & Tubini
A little summary

12. !12
A very simple sketch
Rigon & Tubini

13. !13
Rigon & Tubini
The vocabulary
Let’s start with the vocabulary this time.

14. !14
Now let’s start to consider our graph. Keeping for granted the Tables and
definitions in the previous slides. We start to build a graphical
representation of the equation. Assume for now there is no accumulation
of water on the surface.
So, first, we put a place, and the state variable in.
Dependence on time is assumed (always), therefore we
need to specify just the spatial variable. Actually is
I.e., the water content is function of a pressure variale
called suction, and through it of position (and time).
But putting also this information seems to clutter the
figure too much. One has to assume that spatial
characteristics can depend on some thermodynamical
variable. This graphic symbol corresponds to
Rigon & Tubini
Back to graphs

15. !15
The water budget will require to get a flux expression, here it is as
s obtained by the Darcy-Buckingham law.
In turn, and are parameterised in function of
Rigon & Tubini
Expressions

16. !16
Here it is the additional vocabulary which will be useful later on.
Rigon & Tubini
Additional vocabulary required by the ancillary expressions

17. !17
The traditional divergence of flux can be
represented as a transition, with the caveat that it is
valid for any location (including the boundary) and
it can bring to both positive and negative variation
of the state variable. We used the double arrow to
signify it. Position of the boundary must be
specified elsewhere. We have different types of
transitions. Because in this case we have a flux rate,
it must be intended that the transition symbol
implies a term:
So that
Rigon & Tubini
The internal flux
*see here, if you need e rehearsal on conservations laws

18. !18
Assuming that we want a Dirichlet boundary
condition, we have to specify it. We use a place sign,
with the variable estimated at the proper position,
in this case. Dashed line means the boundary
condition. The square edge means that is given. In
this case the information given is:
With an assigned f(t) time series
Rigon & Tubini
Boundary condition (if you really wants to make it explicit)

19. !19
To fully specify the fluxe, we can use the van Genuchten or Brooks and
Corey parameterisation of the so called water retention curves (SWRC, i.e.
the functional relation between water content and suction)
van Genuchten
Brooks and Corey
additional symbols are explained below
Rigon & Tubini
Ancillary expressions

20. !20
According to Mualem (1976) hydraulic conductivity can be derived as a
function of the SWRC
van Genuchten
Brooks and Corey
K(Se) = KsSv
e
⇤
1 1 S1/m
e
⇥m⌅2
(m = 1 1/n)
additional symbols are all explained in the next slide
Rigon & Tubini
Ancillary expressions

21. !21
It can be observed that the Expressions Table should be enriched by ancillary
expressions, containing the parameterisations:
Arrows mean assignment, “:=“ indicates a definition. Overscripts indicate a
specification of a particular parameterisation. Different parameterisation
cannot be mixed
Rigon & Tubini
Ancillary expressions in a Table

22. !22
We can then add the other inputs and outputs,
which are in this case boundary conditions.
Direction of the arrow and the position of the
square serve to indicate that precipitation is an
input and evapotranspiration an output. Because
these input boundary conditions (bc) are given to a
specific location they are marked by a 0 or zb
In this case the equation becomes:
Rigon & Tubini
Richards equation (well, without water on the surface!)
Differently that previously we put a Neumann bc.

23. !23
The graph in the previous slide assumes that all the water inputed by
rainfall infiltrates in soil. If rainfall water does not infiltrate all it
accumulates on top of the soil and form a pond. The situation is
therefore that we will have a free water surface at
For
water is is present. We can
assume that for that interval
is not varying in time in this region. What
is time varying is, in case,
the water level
Rigon & Tubini
Richards equation with water at the surface

24. !24
For this region, if existing, the graph is
The resulting equation is an ordinary equation one:
Rigon & Tubini
We have a new domain and a new equation
Please notices (and you will realise after that the constrain is not cosmetics

25. !25
Equation:
It comes with a constraint:
If null, this means that the water table is in the soil. This suggests
a generalisation of the equation with switching boundary conditions, which we
discuss later on.
Rigon & Tubini
We have a new domain and a new equation

26. !26
If we consider the full picture, in fact,
the situation becomes more complicate.
The flux at the interface must now account for
the fact there is an inferior domain now.
If
Rigon & Tubini
Coupled surface-subsurface

27. !27
If
Rigon & Tubini
Coupled surface-subsurface

28. !28
Please observe that if
Also observe that the fluxes enters with a derivative for the subsurface and
without for the surface flow. This formal change is left implicit in the
diagrams.
Rigon & Tubini
Coupled surface-subsurface

29. !29
Because ET depends upon radiation
also the graph on the left could be
enhanced with this information
Therefore the symbol
Identifies a parameter that enters in
the equation.
Rigon & Tubini
Decorations

30. !30
There is a further aspect of the equation that needs to be investigated.
This is connected with saturation. If the soil column (or part of it)
is saturated is Richards equation still valid ?
In this case
for some z. Then,
and the flux is stationary, unless the column desaturate. For these instants
i.e., it switches to positive pressure (negative suctions)
For completeness
Rigon & Tubini

31. !31
Then:
However, groundwater studies show that even is saturated
conditions the flux cab be non stationary. In fact, at saturation, the
medium, assumed rigid, shows property of elasticity. This can be
described by letting, when
Rigon & Tubini
and the equation becomes
For completeness
In most of the 1D applications, however, steady state approximation,
would be fine (if an appropriate integration method is chosen).

32. !32
Conclusion
We show that the Petri net graphical notation is actually useful to draw partial
differential equations (pdes). It is obviously necessary to keep in mind the
peculiarity of pdes, meaning that they require, for instance boundary
conditions. Another peculiarity is that internal variables can appear (in our case
thermodynamical ones, like suction) that mediate the spatio-temporal
dependence of the main variable (e.g. the water contentent).
Rigon et al.
Questions ?

33. !33
Find this presentation at
http://abouthydrology.blogspot.com
Ulrici,2000?
Other material at
Domande
Rigon et al.