Reservoirology & Graphs A case of spatially explicit models using 1D Richards equation as an

example

Riccardo Rigon R. & Niccolò Tubini

January 2017

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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A very simple sketch

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The vocabulary

Let’s start with the vocabulary this time.

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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

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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

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Expressions

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Here it is the additional vocabulary which will be useful later on.

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Additional vocabulary required by the ancillary expressions

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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

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The internal flux

*see here, if you need e rehearsal on conservations laws

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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)

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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

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According to Mualem (1976) hydraulic conductivity can be derived as a function of the SWRC

van Genuchten

Brooks and Corey

K(Se) = KsSve

⇤1�

�1� S1/m

e

⇥m⌅2

(m = 1� 1/n)

additional symbols are all explained in the next slide

Rigon & Tubini

Ancillary expressions

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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

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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.

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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

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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

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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

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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

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Coupled surface-subsurface

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If

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Coupled surface-subsurface

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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

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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.

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Decorations

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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

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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).

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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 ?

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Find this presentation at

http://abouthydrology.blogspot.com

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Other material at

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Rigon et al.