Quantifying soil complexity using network models

of soil porous structureMarko Samec (1), Antonio Santiago (2), Juan Pablo Cardenas (2),

Rosa Maria Benito (2), Ana Maria Tarquis (3), Sacha Jon Mooney (4), Dean Korošak (1,5)

(1) Faculty of Civil Engineering, University of Maribor, Maribor, Slovenia (marko.samec@gmail.com) (2) Grupo de Sistemas Complejos, Departamento de Física, Universidad Politécnica de Madrid, 28040 Madrid, Spain

(3) Departamento de Matemática Aplicada, Universidad Politécnica de Madrid, 28040 Madrid, Spain (4) Environmental Sciences Section,University of Nottingham, Nottingham, UK

(5) Institute of Physiology, Faculty of Medicine, University of Maribor, Maribor, Slovenia

SEE INTO

DESCRIB

E

QUANTIF

Y

porous matter

SEE INTO

DESCRIBE

QUANTIFY

3D X-ray CT

porous matter

SEE INTO

DESCRIBE

QUANTIFY

complex network theory

porous matter

SEE INTO

DESCRIB

E

QUANTIF

Y

complexity hsingle parameter

porous matter

how can one see into the porous structure of material

X-RAY COMPUTED TOMOGRAPHY

?

X-ray COMPUTED TOMOGRAPHY

part of x-rays absorbed

higher density of the matter,

more X-rays are absorbed

the ones that manage to pass, create image

pore space

measuring cell

solid space

X-ray COMPUTED TOMOGRAPHY concept

X-ray COMPUTED TOMOGRAPHY image analysis

tomography image

crop and scale

contrast

threshold algorithm

binary image

X-ray COMPUTED TOMOGRAPHY image analysis

sample average pore size

[mm2]

perimeter

[mm]

porosity

[%]

circularity

cement 0 h 0,000946 0,130280 4,70 0,69937

cement 2 h 0,000945 0,123051 2,70 0,72173

cement 24 h 0,000923 0,127010 4,11 0,70707

clay 0 h 0,003331 0,223884 0,83 0,68927

clay 2 h 0,004022 0,247481 1,00 0,68024

clay 24 h 0,003497 0,233920 1,01 0,66495

SC 0 days 0,004285 0,241832 2,56 0,66450

SC 14 days 0,002455 0,191862 2,89 0,68797

SC 28 days 0,003884 0,252968 2,78 0,67895

how can one describethe porous structure of material

COMPLEX NETWORK THEORY

?

BEGININGS

Seven Bridges of Königsberg, Euler (1735)

sociogram, Moreno (1993)

p53 gene network , Volgestein et al. (2000)

medicine, biology, computer science, particle

physics, economics, sociology, ecology,

epidemology, neuroscience,...

node or vertex

link or edge

direction

weight

node degree

degree distribution

BASICS

1 1 0 0 1 0

1 0 1 0 1 0

0 1 0 1 0 0

0 0 1 0 1 1

1 1 0 1 0 0

0 0 0 1 0 0

𝑘𝑖 = 𝑎𝑖𝑗𝑗

BASICS

Erdos-Renyi network (random linking)

scale-free networks (preferential linking)degree distribution

00

number of links k

degr

ee d

istr

ibuti

on P

(k)

many nodes with only a few links

a few hubs with large number of links

𝑃(𝑘)~𝑘−𝛾

X-ray COMPUTED TOMOGRAPHY image analysisCL

AY

φ = 0,9 %

CEM

ENT

φ = 1,8 %

φ = 4,2 %

STAB

ILIZ

ED C

LAY

φ = 2,4 %

φ = 4,2 %

Cumulative size distributions,

determined from X-CT images of different porosity

φ. All samples show scale-free behaviour of size distribution (solid line)

with the exponent α=1,8-2.

NETWORK MODELS for description of porous structure of building materials

threshold network model (SN)

preferential attachement model (EN)

the network heterogeneity can be adjusted by changing the parameters b and m

𝑠𝑖𝑠𝑗𝑑𝑖𝑗𝑚 > 𝜀

𝜋𝑖𝑗~𝑘𝑖 𝜎ሺ𝑖,𝑗ሻ 𝜎ሺ𝑖,𝑗ሻ= 𝑠𝑗𝑏𝑑𝑖𝑗𝑚

can one quantifydifferent porous structure of material

COMPLEXITY

?

COMPLEXITY

correlation matrices

to further explore the effect of the network

parameter m on node correlations we

calculated the complexity h of the pore network

based on node-node link correlations

a) m≪1

b) m=1

c) m>>1

COMPLEXITY

α 1,4 1,5 1,6

h 2,85231 3,06864 3,48277

to further explore the effect of the network

parameter m on node correlations we

calculated the complexity h of the pore network

based on node-node link correlations

by combining X-ray CT and complex network models

we can quantify

complexityof porous structure with single parameter

CONCLUSIONS

Quantifying soil complexity using network models

of soil porous structureMarko Samec (1), Antonio Santiago (2), Juan Pablo Cardenas (2),

Rosa Maria Benito (2), Ana Maria Tarquis (3), Sacha Jon Mooney (4), Dean Korošak (1,5)

(1) Faculty of Civil Engineering, University of Maribor, Maribor, Slovenia (marko.samec@gmail.com) (2) Grupo de Sistemas Complejos, Departamento de Física, Universidad Politécnica de Madrid, 28040 Madrid, Spain

(3) Departamento de Matemática Aplicada, Universidad Politécnica de Madrid, 28040 Madrid, Spain (4) Environmental Sciences Section,University of Nottingham, Nottingham, UK

(5) Institute of Physiology, Faculty of Medicine, University of Maribor, Maribor, Slovenia