Nonlin. Processes Geophys., 15, 893–902, 2008
www.nonlin-processes-geophys.net/15/893/2008/
© Author(s) 2008. This work is distributed under
the Creative Commons Attribution 3.0 License.
Nonlinear Processes
in Geophysics
Multiscaling of porous soils as heterogeneous complex networks
A. Santiago1 , J. P. Cárdenas1 , J. C. Losada2 , R. M. Benito1 , A. M. Tarquis3 , and F. Borondo4
1 Grupo
de Sistemas Complejos, Departamento de Fı́sica, E.T.S.I. Agrónomos, Universidad Politécnica de Madrid,
28040 Madrid, Spain
2 Grupo de Sistemas Complejos, Departamento de Tecnologı́as Aplicadas a la Edificación, E.U. Arquitectura Técnica,
Universidad Politécnica de Madrid, 28040 Madrid, Spain
3 Departamento de Matemática Aplicada, E.T.S.I. Agrónomos, Universidad Politécnica de Madrid, 28040 Madrid, Spain
4 Departamento de Quı́mica, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
Received: 7 August 2008 – Revised: 24 September 2008 – Accepted: 25 September 2008 – Published: 26 November 2008
Abstract. In this paper we present a complex network model
based on a heterogeneous preferential attachment scheme to
quantify the structure of porous soils. Under this perspective
pores are represented by nodes and the space for the flow
of fluids between them is represented by links. Pore properties such as position and size are described by fixed states
in a metric space, while an affinity function is introduced to
bias the attachment probabilities of links according to these
properties. We perform an analytical study of the degree distributions in the soil model and show that under reasonable
conditions all the model variants yield a multiscaling behavior in the connectivity degrees, leaving a empirically testable
signature of heterogeneity in the topology of pore networks.
We also show that the power-law scaling in the degree distribution is a robust trait of the soil model and analyze the influence of the parameters on the scaling exponents. We perform
a numerical analysis of the soil model for a combination of
parameters corresponding to empirical samples with different properties, and show that the simulation results exhibit a
good agreement with the analytical predictions.
1
Introduction
The porous structure of soils has an important influence on
the physical, chemical and biological processes that take
place within them (Young and Crawford, 2004; Blair et al.,
2007). The pores size distribution (PSD) is one important parameter to modeling these processes (Vogel and Roth, 2001;
Correspondence to: R. M. Benito
([email protected])
Lin et al., 1999). However, the difficulty is in understanding
the PSD at the same time that the pore geometry distribution
and their behavior concerning the dynamics fluid transportation. Most of the theoretical approaches to the soil porosity
can not explain them adequately because the available models are idealizations that fail to rescue the complex structure
of the wiring and spatial location of pores (Bird et al., 2006)
and moreover do not consider the mechanisms underlying
the dynamics that leads to the emergence of such structure
(Horgan and Ball, 1994).
During the last years the perspective of complex networks
has successfully been applied to a wide array of scientific
fields (Strogatz, 2001). Network theory describes complex
systems from a purely topological point of view, abstracting away the dynamical processes that take such structure as
substrate (Albert and Barabási, 2002; Newman, 2003). A
complex network can thus be viewed as set of nodes and
links with a non-trivial topology. The study of complex networks has shown the presence of common and hardly intuitive structural properties in both nature and man-made systems. These findings have created a large follow-up debate among physicists because of their ubiquity and peculiar statistical properties, markedly different from the random graphs (Erdös and Rényi, 1959). In particular, complex
networks adjust some statistical properties to power laws, a
characteristic feature of critical-point behavior and a fingerprint of self-organized systems (Bak et al., 1987).
The impact of this new approach has also affected the geosciences. Recent approximations describe the structure of
soils as a network (Vogel and Roth, 2001). Using a network
approach, Valentini et al. (2007) found that rock fractures
exhibit the well known small world phenomenon (Watts and
Strogatz, 1998) when those fractures are considered as nodes
Published by Copernicus Publications on behalf of the European Geosciences Union and the American Geophysical Union.
894
A. Santiago et al.: Multiscaling of porous soils as heterogeneous complex networks
in a network (Valentini et al., 2007). A network approach has
also been applied to the study of climate dynamics by Tsonis
et al. (Tsonis et al., 2006, 2007, 2008; Tsonis and Swanson,
2008; Yamasaki et al., 2008). In this paper we propose a
similar approach to quantify the pore architecture of a soil
through a complex network model.
Network models are prescriptive systems that generate
networks with certain topological traits, such as degree or
clustering distributions. Dynamical network models (Dorogovtsev and Mendes, 2002) are stochastic discrete-time dynamical systems that evolve networks by the iterated addition and subtraction of nodes and links. In our study we interpret porous soils as heterogeneous networks where pores
are represented by nodes with distinct properties (such as
surface and spatial location) and the flow of fluids between
them is represented by links connecting the nodes. The networks are generated by a dynamical model known as heterogeneous preferential attachment (PA) (Santiago and Benito,
2007a,b, 2008a,b), a generalization of the Barabási-Albert
(BA) model (Barabási and Albert, 1999a) to heterogeneous
networks.
The BA model is based on the mechanisms of growth and
preferential attachment (Price, 1965, 1976) and provides a
minimal account of the process leading to the emergence of
scale-free networks (Barabási et al., 1999b). Such networks
are characterized by a degree distribution with an asymptotic
behavior according to a power-law, P (k)∼k −γ , where γ is
known as the scaling exponent. This phenomenon leads to a
non-negligible presence of highly connected nodes or hubs, a
trait commonly referred as fat tails. In the BA model the process starts with a seed of arbitrary size and topology. A new
node is added to the network at each step, bringing a fixed
number m of links attached. These links are preferentially
connected to the already existing nodes following the socalled attachment rule: the linking probability of a network
P
node vi is proportional to its degree ki , 5(vi )=ki / j kj .
This step is iterated until a desired number N of nodes have
been added to the network. Heterogeneous PA models incorporate the influence of node attributes in addition to the
connectivity degree to the attachment rule.
The organization of the paper is as follows. In the next section we describe the formulation of the porous soil model.
The analytical solution for the degree distribution of the
pores generated by the model is presented in the third section.
In the fourth section we present the results of the porous soil
corresponding to two samples of soils of different porosity.
In the last section we present our conclusions derived from
this work.
2
Model formulation
The structure of a porous soil is modeled as a heterogeneous
complex network where nodes vi correspond to pores and
links eij correspond to flow of fluids between them. The links
Nonlin. Processes Geophys., 15, 893–902, 2008
will be considered undirected eij =ej i and thus the connectivity degree ki of a node vi will be a measure of the number of pores that are directly connected with the associated
pore. The properties of a pore are described by the node state
(ri , si ), which account for the position ri of the pore center
in the soil and the pore size si (surface or volume, depending
on whether we are modeling the architecture of a 2-D or 3-D
sample of the system).
The dynamics of the porous structure is modeled as a
stochastic growth process known as dynamic network model
(Dorogovtsev and Mendes, 2002). To motivate the rules that
prescribe the evolution of the model, next we consider the influence of the pore properties on the probability for two pores
being connected in a porous soil. Assume that at a given
time a new pore is created in the medium, the likelihood that
this pore will connect with any of the already existing pores
will be proportional to the size of the older pores and inversely proportional to the distance between them. Likewise,
the higher the number of connections accruing to an existing
pore, the higher the likelihood that a new flow of fluids will
intercept an existing connection and connect the new pore
with the older one. Thus the attachment visibility of an existing network node π(vi ) when a new node va is added will
be proportional to ki , si and d −1 (ri , ra ), where d is the Euclidean metric.
The previous considerations prompt us to model the dynamics of the porous structure of soils as a particular case of
heterogeneous preferential attachment (PA) defined by three
elements:
1. R is an arbitrary space. The elements x∈R are referred
as node states.
2. ρ is a nonnegative real function with unit measure over
R referred as node state distribution.
3. σ is a nonnegative real function over R 2 referred as
affinity of the interactions.
This formalism prescribes the evolution of a network according to the following rules:
(i) The nodes vi are characterized by their state xi ∈R. The
node states describe intrinsic properties deemed constant in the timescale of evolution of the network.
(ii) The growth process starts with a seed composed by N0
nodes (with arbitrary states xi ∈R) and L0 links.
(iii) A new node va (with m links attached) is added to the
network at each iteration. The number m is common
for all the added nodes and remains constant during
the evolution of the network. The newly added node is
randomly assigned a state xa following the distribution
ρ(x).
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A. Santiago et al.: Multiscaling of porous soils as heterogeneous complex networks
(iv) The m links attached to va are randomly connected
to the network nodes following a distribution {5(vi )}
given by an extended attachment rule,
π(vi )
5(vi ) = P
,
j π(vj )
π(vi ) = ki · σ (xi , xa ).
(1)
The attachment kernel or visibility π of a node vi in the
rule is given by the product of its degree ki and its affinity
σ with the newly added node va , which is itself a function
of the states xi and xa . It thus can be seen that for each interaction σ biases the degree ki of the candidate node. Steps
(iii) and (iv) are iterated until a desired number of nodes has
been added to the network. To sum up, the choice of the
triple (R, ρ, σ ) determines the form of heterogeneity in the
attachment mechanism.
Consistently with the previous formalism, the porous soil
model will be defined by a state space R=M×S, where M
is a Euclidean box with dimension 2 or 3 that represents the
medium geometry and S is an interval of the real line that
represents the spectrum of possible pore sizes; a state distribution ρ(x)=ρ(r, s) that represents the probability for a new
pore having a certain position r and size s; and an affinity
function
σ (x, y) =
sxα
,
(δ + |rx − ry |)β
(2)
where δ is a small nonnegative offset that ensures the continuity of σ on R 2 , while α and β are free parameters that
measure the relative importance of the size and distance of
the pores in the affinity of the attachment mechanism.
R DefinP
ing the porosity of the medium φ=Sv /ST = i si / M dr as
the fraction of void space in the material, then the desired
number of nodes N that ends the growth process is the least
number that yields φ>φ0 for a certain threshold φ0 .
3
Analysis
In this section we derive an analytical solution for the stationary degree distribution P (k) of the proposed soil model. The
solution is obtained by rate equations (Dorogovtsev et al.,
2000; Krapivsky and Redner, 2001) which establish a balance in the flows of degree densities over a partition of the
network. The exposition will be brief therefore we submit the
reader to (Santiago and Benito, 2008a) for a more detailed
discussion of the solution for the general class of PA models.
Let us first define a sequence of functions {f (k, x, N )}N >0
which measure the probability density of a randomly chosen
node having degree k and state x in a network at the iteration
t=N . The degree densities are local metrics, thus they uniformly converge when N→∞ to a stationary density function f (k, x). Finally, the stationary degree distribution P (k)
measures the probability of a randomly chosen node having
degree k in the thermodynamic limit.
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895
We will denote by V (x)={vi , xi =x} the subset of nodes in
the network with state x=(rx , sx ). Assuming that the assignation of states xi is uncorrelated with the topology of the
growing network, and that there are no linking events between existing nodes, the sequence {f } can be modeled on
each V (x). For each x, the form of the equation will be
L1 −L2 =R1 −R2 , where:
L1 =density of nodes with degree k at t=N+1;
L2 =density of nodes with degree k at t=N;
R1 =increase in density due to nodes with degree k−1
that have gained a link at t=N;
R2 =decrease in density due to nodes with degree k that
have gained a link at t=N .
The resulting density rate equation for k>m is
(N + 1)f (k, x, N + 1) − Nf (k, x, N ) =
σ (x, y)
[(k − 1)f (k − 1, x, N ) − kf (k, x, N )]. (3)
=m
ψ(y, N ) y
where ψ(y, N ) is defined as the partition factor
X Z
σ (x, y)f (k 0 , x, N ) dx,
ψ(y, N ) =
k0
(4)
R
k0
and the brackets mean averaging over the random variable y,
Z
hgiy =
g ρ(y)dy.
(5)
R
For k=m the resulting density rate equation is
(N + 1)f (m, x, N + 1) − Nf (m, x, N ) =
σ (x, y)
= ρ(x) − m
m f (m, x, N ).
ψ(y, N ) y
(6)
There are no nodes with degree k<m, since when N→∞ all
the links attached to newly added nodes find receptive nodes
in the network, therefore the previous equations define all the
possible cases in each iteration.
In the thermodynamic limit N→∞, f (k, x, N +
1)=f (k, x, N )=f (k, x) and the rate equations become
(7)
f (k, x) =
D
E
σ (x,y)
m ψ(y) [(k − 1)f (k − 1, x)−
y
−kf (k, x)]
D
ρ(x) − m
for k > m
E
σ (x,y)
ψ(y) y mf (k, x)
for k = m
where ψ(y) is the stationary partition factor, defined as
ψ(y) = lim ψ(y, N ) =
N→∞
X Z
0
σ (x, y)f (k 0 , x)dx.
=
k
k0
(8)
R
Nonlin. Processes Geophys., 15, 893–902, 2008
896
A. Santiago et al.: Multiscaling of porous soils as heterogeneous complex networks
Notice that the stationary rate equations are coupled via
ψ(y). In order to decouple Eq. 7, let us first assume that
the variation of ψ(y) over the subset Rρ ={y∈R:ρ(y)>0} is
small enough. This can be expressed as the following approximation criterium
h(ψ(y) − ψ)2 iy < ε1 ,
(9)
where ψ=hψ(y)iy and ε1 is an accuracy bound. In that case
we may approximate the coupling term as a quotient of two
mean-field measures,
hσ (x, y)iy
w(x)
σ (x, y)
=
'
,
(10)
ψ(y) y
hψ(y)iy
ψ
where w(x) can be interpreted as the mean-field fitness of
network nodes with state x,
Z
ρ(y)
α
w(x) = hσ (x, y)iy = sx
dy,
(11)
β
R (δ + |rx − ry |)
and ψ can be interpreted as the mean-field partition factor of
the network,
ψ = hψ(y)iy =
Z X Z
sxα
k0
f (k 0 , x) dx ρ(y) dy,
=
β
R (δ + |rx − ry |)
R k0
(12)
both defined for a given distribution ρ(y) of the incoming
node states. Furthermore, let us assume that the variation of
w(x) over R is small enough, which can be expressed as a
second approximation criterium
h(w(x) − w̄)2 ix < ε2 ,
(13)
where w̄=hw(x)ix and ε2 is a second accuracy bound. In that
case the mean-fieldP
stationary partition factor ψ can be approximated by ψ' k kP (k)w̄'2 mw̄. Under the assumption that both approximation criteria (Eqs. 9 and 13) hold, the
coupling factor may then be estimated as
σ (x, y)
w(x)
w(x)
ŵ(x)
'
'
=
,
(14)
ψ(y) y
ψ
2mw̄
2m
where ŵ(x)=w(x)/w̄ is a mean-field normalized fitness,
which measures the fitness of network nodes with state x
relative to the average fitness of all the network nodes, for
a given distribution ρ(x). The resulting decoupled system
becomes then
f (k, x) =
ŵ(x)[(k − 1)f (k − 1, x) − kf (k, x)]/2 for k > m,
ρ(x) − ŵ(x)mf (k, x)/2
for k = m.
(15)
Solving Eq. 15 we obtain f (m, x)=2ρ/(ŵm+2) and the solution for the stationary density for k>m is
!
k
Y
ŵ(j − 1)
2ρ
f (k, x) =
(16)
ŵj + 2 ŵm + 2
j =m+1
Nonlin. Processes Geophys., 15, 893–902, 2008
and integrating the density in Eq. 16 over the state space R
the stationary degree distribution is
!
Z
k
Y
ŵ(j − 1)
2ρ
P (k) =
dx.
(17)
ŵj
+
2
ŵm
+2
R j =m+1
The solution given by Eq. 17 is valid for any variant of the
soil model, irrespective of the geometry of the medium M or
the pore sizes S, the distribution of pore properties ρ or the
dependence of the affinity σ on the pore properties via α and
β. Nevertheless, the validity of the solution is restricted by
the extent to which the approximation criteria in Eqs. 9 and
13 hold. The mean-field approximation can be considered
accurate in the measure that the averages of σ (x, y) along its
two arguments do not exhibit large variations over R. This
does not exclude the possibility of large fluctuations in the
affinity σ (x, y) over R 2 , provided that the variability of the
averages is kept within the desired bounds.
Furthermore, we may assume that any pore in the soil has
a non-zero size, thus from Eq. 11 it follows that the meanfield fitness w(x) is strictly positive over R and the stationary
degree density in Eq. 16 may be expressed for k≥m as
f (k, x) =
2ρ/ŵ B(k, 1 + 2/ŵ)
,
m + 2/ŵ B(m, 1 + 2/ŵ)
(18)
R1
where Legendre’s Beta function B(y, z) = 0 t y−1 (1 −
t)z−1 dt for y, z>0 satisfies the functional relation
0(a)/ 0(a+b)=B(a, b)/ 0(b) for Euler’s Gamma function
0(x). Likewise, integrating the density in Eq. 18 over R we
obtain an expression of the stationary degree distribution
P (k) for k≥m in terms of the Legendre’s Beta function
which is equivalent to Eq. 17.
Notice that the stationary densities for a given pore state in
the soil model, as given by Eq. 18, follow the form of a Beta
function with arguments (k, 1+2/ŵ). Given that the Beta
function behaves as B(y, z)∼y −z when y→∞, this implies
that the degree densities exhibit a multiscaling according to
power laws k −γ (ŵ) along the continuous spectrum of normalized fitness ŵ, with scaling exponents γ (ŵ)=1+2/ŵ spanning themselves a continuum. The multiscaling phenomenon
is a general property of heterogeneous PA networks (Santiago and Benito, 2007a, 2008a) that is exhibited by any variant of the proposed soil model, irrespective of the particular
details. As ŵ increases (resp. decreases), the exponent γ of
f decreases (resp. increases). Nodes with states more fit than
the average (ŵ>1) adopt densities with γ <3, which exhibit
a slower asymptotical decay, and tend to produce more hubs.
Nodes with states less fit than the average (ŵ<1) adopt densities with γ >3, which exhibit a faster asymptotical decay.
Notice also that the mean-field fitness in Eq. 11 may be
factorized into w(x)=w1 (rx )·w2 (sx ), where the first term accounts for the fitness dependence on the pore position, and
the second one accounts for the dependence on the pore size.
We may use this fact to simplify the analysis of the influence
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A. Santiago et al.: Multiscaling of porous soils as heterogeneous complex networks
of the pore properties (rx , sx ) on the scaling exponents in the
density f (k, x). The influence of the pore position r will
depend largely on
R the probability distribution of the pore positions ρM (r)= S ρ(r, s)ds. When the pore position is uniformly distributed over the medium M, for a given pore size
s0 the mean-field fitness w will be highest on the medium
center and will decrease as the pore position approaches the
boundary of M, since w1 (r) is performing an averaging of
the reciprocal of distance over M. However, as the distribution of the pore positions ρM becomes increasingly inhomogeneous, the situation of the pores with respect to the regions
with higher densities will become progressively more important in the determination of the fitness factor w1 . For highly
variable distributions ρM this trait will prevail over the centrality of the pores in the medium geometry. Irrespective of
the inhomogeneity of ρM , notice that as β is increased the
variability of the factor w1 over M will grow, as the contribution to w1 by distant pores becomes progressively lower.
With regards to the influence of the pore size s, for a given
pore position r0 the fitness factor will increase with the size
as w2 ∼s α irrespective of the distribution ρ of pore properties. Notice that again as α is increased the variability of the
factor w2 over S will grow. It should be also pointed out
that with the general form of σ adopted for the soil model
the scaling exponents of f (k, x) will exhibit themselves a
power-law scaling in their dependence on s according to
γ (r0 , s)∼s −α . To sum up, the increase of either α or β results in an increase in the variability of w over R, and in the
case of β such increase is more acute as the inhomogeneity
of ρM increases. Given the dependence of the scaling exponent γ on the normalized fitness w̄, the higher variability
of w will translate into a larger spread of the distribution of
exponents γ of the density components.
The described behavior evidences a signature of heterogeneous PA in the structure of porous soils, by which the
density components f (k, x) of pores in the underlying network will exhibit different scaling exponents according to the
pore properties such as size or position. This signature may
be empirically detected by selecting subsets of pores within
the soil sample so that one of their properties takes a similar value, common to all the subsets, while the other property takes a similar value within each subset, but differing
across the other subsets: {(ri , s0 )}i , or conversely, {(r0 , si )}i .
If the partial degree distributions for these subsets {P (k|xi )}
exhibit power laws with different exponents, it is then feasible to assume that this pore property (either position or size)
is biasing the attachment mechanism. Under equal circumstances, a higher variation in the empirically obtained exponents would point to a higher exponent α or β in the affinity
function σ . Furthermore, to check the existence of heterogeneity it would be sufficient to find one such subset of nodes
whose partial distribution {P (k|xi )} exhibits a power-law behavior with an exponent different to the one in the global distribution P (k).
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897
With regards to the stationary degree distribution P (k) of
the soil model, Eq. 17 shows that it is obtained by the integration of the density components, which form a set of
power-laws with varying exponents γ (x). As we have seen,
little fluctuations of w over R will yield density components
with similar scaling exponents and thus degree distributions
similar to the homogeneous PA case, the so-called BarabásiAlbert model. On the other hand, large fluctuations of w
over R (as in the case of large α or large β parameters, particularly with inhomogeneous ρ distributions) will yield a
wider spectrum of exponents γ and thus a larger deviation of
the degree distribution from the Barabási-Albert model. The
asymptotic behavior of P (k) will be dominated by the slowest decaying components, associated to pore properties with
highest mean-field fitness w. Furthermore, given that ŵ will
be distributed by definition around 1, the highest value of the
spectrum will necessarily verify ŵmax ≥1 and therefore the
degree distribution P (k) of all the variants of the soil model
will exhibit scaling exponents satisfying 1<γ ≤3.
Finally a caveat should be raised about the accuracy of
the preceding discussion regarding small pore networks. The
analytical results have been derived for the thermodynamic
limit, where the degree densities (and thus the degree distribution) reach stationarity. Strictly this does not apply in the
context of pore networks, where there exist intrinsic limitations to the growth process. Given that each pore has a nonnegligible size and the medium size is constant, the porosity
of generated soils grows monotonically with time, eventually
reaching the desired porosity of the modeled system. This
circumstance yields a requisite size of the generated network
for a medium
sample of a given size. Being the average pore
R
size s̄= R sρ(r, s)dx, then the requisite network size will be
given by N0 'φ0 ST /s̄, where ST is the total medium size and
φ0 is the desired porosity. As the requisite size N0 becomes
lower the finite-size effects will become more important, in
particular when N0 103 . This may be the case when the
medium porosity is very low or when the pore size is large
relative to the medium size. The finite-size effects will yield
a growing discrepancy between some network metrics and
the analytical results, such as a decrease in the scaling exponent γ of the degree distribution and the presence of an
exponential cutoff regime. This problem could nevertheless
be circumvented by modeling a larger soil sample.
4
Model application
In this section we present results concerning the numerical simulation of the porous soil model using as input data
from intact soil samples with different physical properties,
obtained from the same profile but at different depths. We
simulate pore networks with the same porosity, size distribution and spatial location of the empirical 2-D soil samples.
Nonlin. Processes Geophys., 15, 893–902, 2008
6 898
A. Santiago
al.: Multiscaling
of porous
soils as
as heterogeneous
heterogeneous complex
A. Santiago
et al.: et
Scaling
and multiscaling
of soils
complexnetworks
networks
D1
Z=100
Table 1. Physical properties of two soil samples (D1 and D3) obtained in the same profile at different depths. Note: C. Sand and
F. Sand correspond to coarse and fine particles of sand respectively.
The names D1 and D3 indicate the sequence of cuts in depth, thus,
in this soil, there is an intermediate cut D2 between D1 and D3.
Soil
Horizon
Depth
(cm)
D1
D3
A2
Bt2
D1
10–35
98–152
D1
Z=150
= - 1.77
= - 1.69
NºPORES=580
Nº PORES=668
MEAN SIZE=8.00
MEAN SIZE=11.80
MAX SIZE=371
MAX SIZE=2497
Particle (%)
C.sand
62
21
F.sand
24
40
Silt
3
4
10000
Clay
D1
Z=200
D3
11
35
1000
D1
Z=250
F(size)
= - 1.82
6
A. Santiago et al.: Scaling and multiscaling of soils as heterogeneous
complex networks
= - 1.71
Fig. 1. 3D representation of soils D1 (left) and D3 (right) obtained
100
Nº PORES=652
Nº PORES=755
using Computed Tomography (CT).
D1
D1
MEAN SIZE=10.48
Z=100
MAX SIZE=645
10
= - 1.77
threshold value was identified as the equi-probability value
for air and solid.
As our model considers the pore size (surface=S) we analyze the distribution of pore sizes in both soil samples making
1
1
10
= - 1.69
NºPORES=580
Nº PORES=668
100 1000 10000
size
MAX
SIZE=371
MEAN SIZE=11.80
MEAN SIZE=8.00
MAX SIZE=2497
Fig.
2.2.Pore
for soil
soil sample
sampleD1
D1obtained
obtained
Fig.10000
Poresize
sizedistribution,
distribution,FF(size),
(size), for
D1 cuts
D1 also
atatdifferent
longitudinal
cutsZ).
Z).The
Thegraphics
figures
alsodisdisdifferentdepths
depths (i.e.
(i.e. longitudinal
22),),maximum
Z=200
Z=250
1000
playthe
thenumber
numberofofpores,
pores,
meansize
size of
of pores
pores (pixels
(pixels
play
mean
maximum
= - 1.82
= - 1.71
sizeofofpores
pores(pixels
(pixels2 ) and
scaling
each
size
scaling exponent
exponentφφ for
for
eachdistribution.
distribution.
100
PORES=652
Thestraight
straight
linescorrespond
correspond
to the power law
Nº PORES=755
The
lines
to
law Nºfittings.
fittings.
F(size)
tained. Imaging parameters were 155keV and 25 A. Proprietary software (GE Medical) was used to reconstruct the
16-bit 3D imagery from the sequence of axial views. The
resulting voxel size was 45.1 µm. File sizes ranged from
70 to 200 Mb, which made subsequent
processing of theD3
enD1
tire volume practically infeasible. Accordingly, three subvolumes were extracted from each of the two original volFig. 1. 3-D
3D representation
and
D3D3
(right)
obtained
Fig.
representationofofsoils
soilsD1
D1(left)
(left)
and
(right)
obtained
umes
(using
GE Medical
Microview) and care was taken to
using
Computed
Tomography
using
Computed
Tomography(CT).
(CT).
ensure no overlay of the subvolumes. The subvolumes measured 256x256x256 units, corresponding to about 16.8 million4.1
voxels.
A 3D Gaussian
filter
in
MicroView
HealthEmpirical
analysis
of
soil
samples
tained.
Imaging
parameters
were
155keV and(GE
25 A.
Procareprietary
, 2006)software
was also(GE
runMedical)
on each subvolume
to
reduce
was used to reconstructnoise
the
andThe
beam-hardening
artifacts,
typical
CTaxial
imagery.
16-bit
3Dsamples
imageryused
frominthe
sequence
of
views.
The
soil
this
study of
were
collected
from
two
resulting
voxel
size
FileSOLOS,
sizes ranged
from
horizons
of
an
(EMBRAPA
2006).
PhysCT
imagery
of Argissolo
soil, was
like45.1
otherµm.
digital
imagery,
typically
70 tocharacteristics
200
Mb, proportion
whichofmade
of the
en- be
ical
relevance
to theprocessing
current
study,
can
contains
a large
ofsubsequent
mixed-voxels
(voxels
whose
tire
volume
practically
infeasible.
Accordingly,
three
subobserved
in
Table
1.
This
soil
is
characteristic
of
the
coastal
digital number is the weighted average of more than one convolumes
were
extracted
from
each
of the
voltablelands
ofasnortheast
Brazil,
formed
ontwo
theoriginal
Tertiary
Barstituent
- such
a solid/air
interface).
To
facilitate
identiumes (using
GE
Medical Microview)
and carestate
was (Itapirema
taken to
reiras
group
of
formations
in
Pernambuco
fication of constituent peaks in the gray-scale histogram, a
ensure no overlay
of the subvolumes.
The subvolumes
meaa hardsetting
behavior.
3D Experimental
filter
executedStation)
in NIHpresenting
ImageJ (Rasband
, 2006)
was run
sured 256x256x256 units, corresponding to about 16.8 milThesubvolume
images of to
themask
soil samples
D1 anddiffered
D3 were obtained
on each
voxels
more
lion voxels.
A 3D Gaussian
filter inwhich
MicroView (GEby
Healthusing
an
EVS
(now
GE
Medical)
MS-8
MicroCT
scanner
thancare
0.1%
from
the
surrounding
neighborhood
of
124
vox, 2006) was also run on each subvolume to reduce noise
1).unit
Though
someFull
samples
required
paring
to fitcan
the
els (Fig.
(5x5x5
volume).
details
of
this
technique
and beam-hardening artifacts, typical of CT imagery.
64
mm
diameter
imaging
tubes,
field
orientation
was
mainbe found
in (Elliot
and
, 2007). Histograms
of the
CT imagery
ofparameters
soil, Heck
like other
imagery,
typically
tained.
Imaging
were digital
155
keV
and
25OriginPro
A. Propriunmasked
voxels
were
subsequently
ported
into
contains
a large(GE
proportion
ofwas
mixed-voxels
(voxels whose
etary
software
Medical)
used smoothing
to reconstruct
16(Origin
Lab
Corporation
, 2006);
after
thethe
hisdigital
number
is the weighted
average
of more than one
conbit 3-D imagery from the sequence of axial views. The retograms
(adjacent
averaging
25 levels),Topeaks
wereidentiidenstituent
- such as
a solid/airofinterface).
facilitate
sulting voxel size was 45.1 µm. File sizes ranged from 70 to
fication
constituent
in the
gray-scale
histogram,
a
tified
in theofPeak
Fitting peaks
Module.
The
major peak
with the
200 Mb, which made subsequent processing of the entire vol3D mean
filter executed
in NIH was
ImageJ
(Rasband
, 2006)
was run
lowest
digital number
taken
to be that
correspondumeeach
practically
infeasible.
Accordingly,
three subvolumes
subvolume
mask
by more
ing on
to the void
space; tothe
nextvoxels
majorwhich
peakdiffered
was considered
were
extracted
from
each
of
the
two
original
volumes
(usthan
0.1%
from
the
surrounding
neighborhood
of
124
voxto be
solid
material.
Based on the
central
tendency
and
dising
GE
Medical
Microview)
and
care
was
taken
to
ensure
els (5x5x5
unit volume).
Full
details of this
technique
can
persions
(assuming
distributions)
of the
two peaks,
no
theGaussian
subvolumes.
The subvolumes
be overlay
found inof(Elliot
and Heck , 2007).
Histogramsmeasured
of the
one256×256×256
for air (µCT solid ) andcorresponding
the other fortosolid
(µCT solid
), a
about
million
unmasked voxelsunits,
were subsequently ported
into 16.8
OriginPro
threshold
value
was
identified
as
the
equi-probability
value
voxels.
3-DCorporation
Gaussian filter
in MicroView
(GE Healthcare,
(Origin A
Lab
, 2006);
after smoothing
the hisfor air
andwas
solid.
2006)
also run
on each of
subvolume
reduce
noise
tograms
(adjacent
averaging
25 levels),topeaks
were
iden-and
beam-hardening
artifacts,
typical
of
CT
imagery.
As
our in
model
considers
pore size
(surface=S)
we anatified
the Peak
Fittingthe
Module.
The
major
peak with
the
mean digital
number
was
tosoil
be imagery,
that correspondlyzelowest
the
porelike
sizes
intaken
both
samples
making
CTdistribution
imagery
ofofsoil,
other
digital
typically
ing to theacuts
void
the next
peakinwas
considered
contains
large
proportion
of major
mixed-voxels
longitudinal
ofspace;
256x256
units
(pixels)
the(voxels
3D
CTwhose
imto
be
solid
material.
Based
on
the
central
tendency
and
disdigital
number
is
the
weighted
average
of
more
than
one
conagery following the algorithm proposed by Vogel and Roth
persions
(assuming
Gaussian
distributions)
of
the
two
peaks,
(2001). This algorithm basically consists in marking the
one for
air (µ
and the
other for solid
(µ
), a
CT solid
solid
adjacent
black
pixels
in) two
consecutive
lines
onCTthe
entire
Nonlin.
Processes
Geophys.,
15, 893–902,
2008
MEAN SIZE=30.74
Z=150
MAX SIZE=9958
10
MEAN SIZE=10.48
MEAN SIZE=30.74
MAX SIZE=645
MAX SIZE=9958
stituent1 - such as a solid/air interface). To facilitate iden10
100 1000
10000
tification1 of constituent
peaks
in the gray-scale histogram,
plane and then performing
an overall study to recognize the
size
a 3-D filter executed in NIH ImageJ (Rasband, 2006) was
pore entities and in that way to obtain sizes and locations.
run
on eachdistribution,
subvolume
to mask
voxels
which
differed by
Fig.
2. PSD,
Pore size
F (size),
sample
D1 obtained
The
given by F (size),
forfor
D1soil
can
be observed
in Fig. 2
more
than
0.1%
from
the
surrounding
neighborhood
of 124
atfor
different
depths
(i.e.
longitudinal
cuts
Z).
The
figures
also
disfour profile
cuts
at volume).
different depths
(Z=100,150,200,250).
(5×5×5
unit
this technique
playvoxels
the number
of pores,
mean size of Full
poresdetails
(pixels2of
), maximum
Asofcan
seen,in2all
the cuts
aφdistribution
that follows
bebe
found
anddisplay
Heck, 2007).
Histograms
sizecan
pores
(pixels
)(Elliot
and scaling
exponent
for each
distribution.of the
a unmasked
power
denoting
a
scale
free
character
in
the
The
straightlaw
lines
correspond
to
the
power
law
fittings.
voxels were−φsubsequently ported intodistribution
OriginPro
of(Origin
pore sizes,
(s) ∼ s . 2006); after smoothing the hisLab fCorporation,
tograms (adjacent averaging of 25 levels), peaks were idenThe in
same
be observed
in the
deepest
soil
tified
the behaviour
Peak Fittingcan
Module.
The major
peak
with the
sample
D3
exhibited
by
Fig.
3,
nevertheless,
in
this
scenario
lowest
digital number
was taken
that correspondplane
and mean
then performing
an overall
studytotobe
recognize
the
the
number
of pores
andthe
its next
size are
opposite
to D1.
The difingentities
to the and
void
major
peak
considered
pore
inspace;
that way
to obtain
sizes
andwas
locations.
ferences
in
the
3D
CT
imagery
of
both
soil
samples
can be
be solid
Based
on can
the central
tendency
ThetoPSD,
givenmaterial.
by F (size),
for D1
be observed
in Fig.and
2 disobserved
in
Fig.
1.
Due
to
the
fact
that
our
model
consid(assuming
Gaussian
distributions)
of the two peaks,
forpersions
four profile
cuts at different
depths
(Z=100,150,200,250).
ers
also
the
pores,
analyzed
spatial
one
forseen,
air spatial
(µ
anddisplay
oneoffor
solidwe
(µCTsolid
a the
threshold
As
can
be
allCTair
the )location
cuts
a distribution
that),follows
distribution
of
the
centers
of
mass
of
pores
(CMP)
(Vogel
valuelaw
wasdenoting
identified
as the
equi-probability
for air and
a power
a scale
free
character in the value
distribution
Roth,
the. same cut previously mentioned for
solid.
ofand
pore
sizes, 2001)
f (s) ∼ins−φ
bothAssoil
Fig. 4A
the(surface=S)
distribution
the
oursamples.
model considers
the shows
pore size
weofanaThe same
behaviour
be observed in the
deepest
soil soil
pore
in four can
superimposed
sample
lyze centers
the distribution
of pore sizes incuts
bothfor
soilthe
samples
maksample
D3results
exhibited
byD3,
Fig.not
3, nevertheless,
in this scenario
D1
forcuts
shown, units
are similar).
Thethespatial
ing(the
longitudinal
of 256×256
(pixels) in
3-D
the
number
of
pores
and
its
size
are
opposite
to
D1.
Theuniformity
difdistributions
of
CMP were
analyzed
toproposed
study
their
Vogel
CT
imagery
following
the
algorithm
by
ferences in the 3D CT imagery of both soil samples can be and
atRoth
each(2001).
depth considered.
We basically
counted the
number
of CMP
This
algorithm
in marking
observed
in Fig. 1.
Due
to the fact
that ourconsists
model considfound
within
each
box,
varying
the
box
length
size
from
the
adjacent
black
pixels
in
two
consecutive
lines
on
the
en-2,
ers also the spatial location of pores, we analyzed the spatial
4,
8,
up
to
32
pixels.
In
the
longest
size
length
used
64
data
tire
plane
and
then
performing
an
overall
study
to
recognize
distribution of the centers of mass of pores (CMP) (Vogel
points
were
obtained
hence
nopreviously
bigger
sizes
were
the
pore
entities
thatcut
way
to obtain
sizes
andused
locations.
and
Roth,
2001)
in and
the in
same
mentioned
for to assure
enough
data
available
for
statistics.
Then,
the
variance
The
PSD,
given
by
F
(size),
for
D1
can
be
observed
in
Fig. 2
both soil samples. Fig. 4A shows the distribution of the
for
four
profile
cuts
at
different
depths
(Z=100,
150,
for
each
side
length
was
calculated
and
is
shown
in
Fig.
4B.
pore centers in four superimposed cuts for the sample soil 200,
250).
As
can
be
seen,
all
the
cuts
display
a
distribution
that
The
thatshown,
the spatial
distribution
of pores is
D1
(theresults
resultsevidence
for D3, not
are similar).
The spatial
distributions
CMP were
analyzed
to study
reasonablyofuniform
over
the surfaces
fortheir
all uniformity
the cuts considatered.
each depth considered. We counted the number of CMP
www.nonlin-processes-geophys.net/15/893/2008/
found within each box, varying the box length size from 2,
4, 8, up to 32 pixels. In the longest size length used 64 data
points were obtained hence no bigger sizes were used to assure enough data available for statistics. Then, the variance
A. Santiago et al.: Scaling and multiscaling of soils as heterogeneous complex networks
A. Santiago et al.: Multiscaling of porous soils as heterogeneous complex networks
4.2 Results of the model
D3
Z=100
D3
Z=150
7
899
Fig. 5 shows two pore networks generated by the soil model
assuming parameters corresponding to soil samples D1 and
= - 1.83
= - 1.87
D3. Each new node added to the growing network is assigned
200
Nº PORES=1292
NºPORES=2191
a characteristic surface and spatial location. Following the
MEAN SIZE=4.90
MEAN SIZE=5.19
empirical results obtained regarding the PSD in soil samples
MAX SIZE=614
MAX SIZE=296
D1 and D3 (see Fig. 6), the pore surfaces in both networks
are distributed according to power-laws with scaling expo10000
nents φD1 = −1.7 (D1) and φD3 = −2.0 (D3). Likewise
D3
D3
100 pore size in the distributions is bounded by
the maximum
Z=200
Z=250
1000
9958 pixels2 (D1) and 1837 pixels2 (D3), and in both cases
= - 2.03
= - 2.00
we consider the minimum pore size equal to 1 pixel2 . It
100
should be pointed out that the nodes in the networks correNº PORES=2302
Nº PORES=2147
spond to the centers of mass of pores and the size of pores
MEAN SIZE=3.81
MEAN SIZE=4.98
10
MAX SIZE=291
0
is not represented.
The spatial location of the pore centers
MAX SIZE=1837
is uniform for
both
networks,
0
100again in accordance
200 with the
1
empirical
results
obtained
in
soil
samples
D1 and D3 (see
1
10
100 1000 10000
x
size
Fig.4). The surfaces of the soil samples simulated are chosen
120in order to yield the porosity coefficients observed
differently
Fig.3.3.Same
SameasasFig.
Fig.22for
for soil
soil sample D3.
Fig.
D3.
considering Z = 11z=100
cuts (10.3% for D1 and 15.8% for D3)
irrespective of the network size. For instance, given a netz=150pores we simulate soil surfaces of
work size of N = 10000
follows a power law denoting a scale free character in the
2
3326976
pixels
(D1)
and 473344 pixels2 (D3).
80
distribution of pore sizes, f (s)∼s −φ .
The parameters αz=200
and β were also chosen differently in
The same behaviour can be observed in the deepest soil
the simulated networks. Considering that the soil sample D3
sample D3Aexhibited by Fig. 3, nevertheless,D1
in this scenario
z=250
has a higher percentage
of clay (see Table 1) and a lower size
the number of 200
pores and its size are opposite to D1. The difof pores (see Fig. 1) we can assume that it is more comferences in the 3-D CT imagery of both soil samples can be
40
pact. For
this reason we assign to the distance between pores
observed in Fig. 1. Due to the fact that our model consida
higher
importance
in our model. Thus, for the soil D3 we
ers also the spatial location of pores, we analyzed the spatial
used
α
=
1.0
whereas
in D3 α = 0.5. On the other hand,
distribution of100
the centers of mass of pores (CMP) (Vogel
due
to
the
fact
that
the
soil
D1 has a higher percentage of sand
and Roth, 2001) in the same cut previously mentioned for
(see
Table
1)
and
a
higher
size
of pores (see Fig. 1) we can as0
both soil samples. Figure 4A shows the distribution of the
sume
that
the
bigger
particles
in
the sandy soil prevent higher
pore centers in four superimposed cuts for the sample soil
0
10
20
30
connectivities
in
pores
formed
by
sand, contrary to the pores
0 D3, not shown, are similar). The spatial
D1 (the results for
Length
0 were analyzed
100 to study
200 their uniformity
formed by clay. In a clay
pore its(pixels)
surface (perimeter in our
distributions of CMP
x
2D simulation) can be occupied by a higher number of partiat each depth considered. We counted the number of CMP
120
Fig.
4.2,A: Spatial
location
of
the
center
ofthus
pores
in four long
cles,
to alocation
sand
pore
of mass
similar
size,
the
found within each box, varying the box length size
from
Fig.
4. contrary
(A):
Spatial
ofmass
the
center
of
pores
in number
four
z=100
B
cutsconnections
Z colors)
(different colors)
for
soilsample
sample
D1.
(B):B: Varian
4, 8, up to 32 pixels. In the longest size length used
64 datacutslongitudinal
of potential
(space
thatthe
separates
two
particles)
tudinal
Z
(different
for the
soil
D1.
Variance
of the
number
of CMP
in boxes
of different
length Bez=150
points were obtained hence
no bigger sizes were used to asof a clay
pore
is higher
in founded
comparison
with
a sand pore.
ofvariance
the number
ofthe
CMP
founded
in boxes of different length in d
incause
different
longitudinal
80available for statistics. Then, the
sure enough data
surface ofcuts
the Z.
pore is more important in a clay soil,
z=200
for each side length was calculated and is shown in
Fig. 4B.
ferent
longitudinal
cutsweZ.use β = 1.0 for D1 and β = 2 for D3.
in our model
The results evidence thatz=250
the spatial distribution of pores is
In the model both networks start with a seed of N0 = 10
ing exponents φD1 =−1.7 (D1) and φD3 =−2.0 (D3). Likereasonably uniform
40 over the surfaces for all the cuts considpores connected by L0 = 9 links. In both cases the the numwise the maximum pore size in the distributions is bounded
ered.
ber of potential links for each new node added is m = 3
by 9958 pixels2 (D1) and 1837 pixels2 (D3), and in both
and it remains unaltered during the evolution of the network.
cases we consider the minimum pore size equal to 1 pixel2 .
4.2 Results of the model
0
The aggregation process is iterated until 200 nodes have been
It should be pointed out that the nodes in the networks cor0
10
20
30
added to
networkof(the
low
is the
chosen
forpores
the sake
Figure 5 shows two poreLength
networks
generated by the soil
respond
to the
the centers
mass
of number
pores and
size of
(pixels)
of
visibility).
The
simulation
results
in
Fig.
5
show
that in
model assuming parameters corresponding to soil samples
is not represented. The spatial location of the pore centers
the
top
network
the
model
parameters
tend
to
prevent
the
Fig.
A: Spatial
location
the mass
of growing
pores in four
longiD14.and
D3. Each
newofnode
addedcenter
to the
network
is uniform for both networks, again in accordance with the formation ofresults
hubs obtained
more drastically
than theD1
bottom
one.(see
Fig. 7
tudinal
cuts Z a(different
colors)surface
for the and
soil sample
B: Variance
is assigned
characteristic
spatial D1.
location.
Folempirical
in soil samples
and D3
oflowing
the number
of CMP founded
boxes ofregarding
different length
in difdepicts
the surfaces
cartesianofpore
networks
generated
model
the empirical
results in
obtained
the PSD
in
Fig.
4). The
the soil
samples
simulatedbyarethechoferent
longitudinal
cuts D3
Z. (see Fig. 6), the pore surfaces in both
representing
soil D1
(top)the
andporosity
the soilcoefficients
D3 (bottom).
sen
differently the
in order
to yield
ob- The
soil samples
D1 and
cartesian
layouts also
evidence
that the
of the
poresfor
tends
networks are distributed according to power-laws with scalserved
considering
Z=11
cuts (10.3%
forsize
D1 and
15.8%
to be a more significant trait in the final connectivity of the
D1
F(size)
y
A
Variance
y
Variance
B
www.nonlin-processes-geophys.net/15/893/2008/
Nonlin. Processes Geophys., 15, 893–902, 2008
8
900
A. Santiago
et al.:etScaling
andand
multiscaling
soilsasasheterogeneous
heterogeneous
complex
networks
A. Santiago
al.: Scaling
multiscaling of
of soils
complex
networks
A. Santiago et al.: Multiscaling of porous soils as heterogeneous complex networks
10000
10000
D1
D1
MAX PORE SIZE: 9958
MAX PORE SIZE: 9958
POROSITY: 10.3 %
1000
POROSITY: 10.3 %
F(size)
F(size)
1000
: - 1.7
: - 1.7
D3
100
MAX PORE SIZE:
D3 1837
100
POROSITY: 15.8 %
MAX PORE SIZE: 1837
10
: - 2.0
POROSITY: 15.8 %
10
: - 2.0
1
1
10
100
size
1000
10000
1
10F (size),100
1000 D1 (black
10000
Fig. 6. Pore size1distribution,
for soil samples
size at different depths (i.e.
triangles) and D3 (blue triangles) obtained
longitudinal cuts Z = 11). Inside the figures can be observed the
Fig. 6. Pore
size
distribution,
(size),
soil samples D1
maximum
of pores
(pixels2F
and
thefor
scaling
φ (black
for
Fig.
6. Poresize
size
distribution,
F) (size),
for soilexponent
samples
D1
(black
triangles)
and
D3
(blue
triangles)
obtained toat the
different
depths
each distribution. The straight
lines correspond
power law
triangles)
and
D3
(blue
triangles)
obtained
at
different
depths
(i.e. longitudinal cuts Z=11). Inside the figures can be observed (i.e.
fittings.
longitudinal
cuts size
Z =
11). (pixels
Inside2 )the
canexponent
be observed
the maximum
of pores
andfigures
the scaling
φ for the
2
maximum
size of pores
) and
the scaling
each distribution.
The (pixels
straight lines
correspond
to theexponent
power lawφ for
eachfittings.
distribution. The straight lines correspond to the power law
fittings.
Fig. 5. Layouts of two pore networks generated by the soil model
corresponding
D1 (top)generated
and D3 (bottom).
Node
color
Fig.
5. Layoutstoofsoil
twosample
pore networks
by the soil
model
corresponds totothe
degree
colors
mean
high
corresponding
soilconnectivity
sample D1 (top)
andk:D3warm
(bottom).
Node
color
degrees, coldtocolors
mean low degrees.
The
spatial
position
pores
corresponds
the connectivity
degree k:
warm
colors
meanofhigh
degrees,
colors
degrees.
The spatial
position
of pores
in thesecold
layouts
hasmean
beenlow
chosen
according
to the
Kamada-Kawai
in
these layouts has been chosen according to the Kamada-Kawai
algorithm.
algorithm.
g. 5. Layouts of two pore networks generated by the soil model
rrespondingD3)
to soil
sample D1
and D3 (bottom).
Node color
irrespective
of(top)
the network
size. For instance,
given a
the pore
position,
aspores
evidenced
by amean
large
share
of
network
size
of N=10
000k:
wecolors
simulate
soilhigh
surfaces
rresponds topore
thethan
connectivity
degree
warm
2
2
highly
connected
nodes
being
located
in
peripheral
positions
of 332mean
6976 low
pixels
(D1) and
344 position
pixels (D3).
grees, cold colors
degrees.
The473
spatial
of pores
of the
soil
surface.
This
insight
may
be
explained
by the factin
parameters
and β were
chosen differently
these layouts The
has been
chosenα according
to also
the Kamada-Kawai
that pore sizes follow highly inhomogeneous distributions,
the simulated networks. Considering that the soil sample D3
gorithm. while
pore positions follow completely homogeneous distrihas a higher percentage of clay (see Table 1) and a lower size
butions. The choice of the affinity parameters α and β also
of pores (see Fig. 1) we can assume that it is more compact.
play a role in this trait, as the introduction of a superlinear β
For this reason we assign to the distance between pores a
re than thetends
poreto position,
as evidenced
by abetween
large share
of and
further decrease
the correlation
position
higher importance in our model. Thus, for the soil D3 we
connectivity
inbeing
the cartesian
layouts,
as shownpositions
by the bottom
ghly connected
nodes
located
in
peripheral
used α=1.0 whereas in D3 α=0.5. On the other hand, due to
networkThis
in Fig.insight
7.
the soil surface.
may
explained
by the
the fact that the
soil D1
hasbe
a higher
percentage
of fact
sand (see
Fig. 8 depicts the degree distributions P (k) obtained
at pore sizes
follow
inhomogeneous
1) andhighly
a higher
size of pores (seedistributions,
Fig. 1) we can asTable
through numerical simulation of the D1 and D3 networks
sume that the bigger
particles inhomogeneous
the sandy soil prevent
higher
hile pore positions
distriwith a sizefollow
of N =completely
10000 pores.
The results show
that the
connectivities
in
pores
formed
by
sand,
contrary
to
the
pores
tions. Thedistributions
choice of do
thenot
affinity
and β
fit wellparameters
a power-lawαalong
thealso
studied
formed
by
clay.
In
a
clay
pore
its
surface
(perimeter
in
however
they
evidence a progressively
better agreeay a role ininterval,
this trait,
as the
introduction
of a superlinear
β our
2-D
simulation)
can
be
occupied
by
a
higher
number
of
mentdecrease
for higher
degrees.
This can
be explained
by and
the partifact
nds to further
the
correlation
between
position
cles,
contrary
to
a
sand
pore
of
similar
size,
thus
the
number
that
the
power-law
components
in
the
density
spectrum
with
nnectivity of
in potential
the cartesian
layouts,
as shown
by the two
bottom
connections
(space
that separates
particles)
twork in Fig.
of a7.
clay pore is higher in comparison with a sand pore. Be-
Fig. 8 depicts the degree distributions P (k) obtained
rough numerical
of the 15,
D1893–902,
and D32008
networks
Nonlin. simulation
Processes Geophys.,
th a size of N = 10000 pores. The results show that the
stributions do not fit well a power-law along the studied
terval, however they evidence a progressively better agree-
cause the surface of the pore is more important in a clay soil,
in our model we use β=1.0 for D1 and β=2.0 for D3.
In the model both networks start with a seed of N0 =10
pores connected by L0 =9 links. In both cases the the number of potential links for each new node added is m=3 and it
remains unaltered during the evolution of the network. The
aggregation process is iterated until 200 nodes have been
added to the network (the low number is chosen for the sake
of visibility). The simulation results in Fig. 5 show that in the
top network the model parameters tend to prevent the formation of hubs more drastically than the bottom one. Figure 7
depicts the cartesian pore networks generated by the model
representing the soil D1 (top) and the soil D3 (bottom). The
cartesian layouts also evidence that the size of the pores tends
to be a more significant trait in the final connectivity of the
pore than the pore position, as evidenced by a large share of
highly connected nodes being located in peripheral positions
of the soil surface. This insight may be explained by the fact
that pore sizes follow highly inhomogeneous distributions,
while pore positions follow completely homogeneous distributions. The choice of the affinity parameters α and β also
play a role in this trait, as the introduction of a superlinear β
tends to further decrease the correlation between position and
connectivity
the cartesian
the bottom
Fig.
7. Cartesianin
networks
of poreslayouts,
generatedasbyshown
the soilby
model
con7.
network
in
Fig.
sidering real data from soil samples. The top network corresponds
Figure
depicts
degree
distributions
P (k)code
obtained
to soil
sample8D1
and thethe
bottom
one to
sample D3. Color
as
Fig.
5.
through
numerical simulation of the D1 and D3 networks
with a size of N=10 000 pores. The results show that the
distributions do not fit well a power-law along the studied
www.nonlin-processes-geophys.net/15/893/2008/
Fig. 7. Cartesian networks of pores generated by the soil model considering real data from soil samples. The top network corresponds
longitudinal cuts Z = 11). Inside the figures can be observed the
maximum size of pores (pixels2 ) and the scaling exponent φ for
each distribution. The straight lines correspond to the power law
fittings.
A. Santiago et al.: Multiscaling of porous soils as heterogeneous complex
networks
901
A. Santiago
et al.: Scaling and multiscaling of soils as heterogeneous
comple
D1
D3
model
color
n high
pores
Kawai
are of
itions
e fact
tions,
distriβ also
ear β
n and
ottom
ained
works
at the
udied
greee fact
with
ity in the to
tested in re
We have
ity in the sc
affinity fun
butions of
the asympt
ular, it is w
all the varia
with expon
networks.
soil model
empirical s
that the sim
analytical p
degree met
exponents
Fig.
Distributions of the
the number
number of
of nodes
nodes with
with degree
k connections,
Fig. 8. Distributions
k, N ·P (k),
N
· P (k),with
obtained
the for
soilthe
model
for the soil
samplesand
D1 D3 Acknowledg
obtained
the soilwith
model
soil samples
D1 (black)
MEC under
(black)
and
D3
(blue).
The
network
size
is
N
=
10000
in
both
(blue). The network size is N=10 000 in both cases.
No. MTM20
cases.
Project TAG
5
Fig. 7. Cartesian networks of pores generated by the soil model considering real data from soil samples. The top network corresponds
to soil sample D1 and the bottom one to sample D3. Color code is
Fig.
7.Fig.
Cartesian
networks of pores generated by the soil model conas in
5.
sidering real data from soil samples. The top network corresponds
to soil sample D1 and the bottom one to sample D3. Color code as
Fig.
5.
interval,
however they evidence a progressively better agree-
ment for higher degrees. This can be explained by the fact
that the power-law components in the density spectrum with
lower scaling exponents dominate the distribution decay, so
that the asymptotic behavior of P (k) over arbitrarily high
degrees would fit the power-law as analytically predicted.
The behavior of the distribution for lower degrees can be
explained by the inhomogeneity of the distribution of pore
sizes, which yields a non-negligible presence of very large
pores in the network. The distributions also evidence that
the sample D3 tends to yield a higher probability of hubs
than the sample D1, as was expected from the network layouts represented by Figs. 5 and 7. We have to remark that
the connectivity pattern between pores was not observed empirically, the scale-free distributions obtained (both numerically and analytically) are the Q
result of the implementation of
a reasonable attachment rule (vi ) and the adoption of real
parameters of soil samples. The distributions P (k) generated
by the soil model can be tested in future research.
www.nonlin-processes-geophys.net/15/893/2008/
Conclusions
lower scaling exponents dominate the distribution decay, so
that
the asymptotic
behavior
of P (k)
over arbitrarily
In summary,
we have
presented
a complex
networkhigh
model
degrees
would
fit
the
power-law
as
analytically
based on a heterogeneous PA scheme in order topredicted.
quantify the
The behavior of the distribution for lower degrees can be
structure of porous soils. The proposed soil model allows
explained by the inhomogeneity of the distribution of pore
the specification of different medium geometries and pore
sizes, which yields a non-negligible presence of very large
dynamics through the choice of different underlying spaces,
pores in the network. The distributions also evidence that the
distributions of pore properties and affinity functions. We
sample D3 tends to yield a higher probability of hubs than
havesample
obtained
solutions
degreelayouts
densities
the
D1, analytical
as was expected
fromforthethenetwork
and
degree
distribution
of
the
pore
networks
generated
represented by Figs. 5 and 7. We have to remark that by
thethe
model
in
the
thermodynamic
limit.
We
have
shown
that
these
connectivity pattern between pores was not observed empirinetworks
exhibit
a
multiscaling
of
their
degree
densities
cally, the scale-free distributions obtained (both numericallyaccording
to power are
lawsthe
with
exponents spanning a continuum,
and
analytically)
result
Q of the implementation of a
and
that
such
phenomenon
a signature
of heterogenereasonable attachment rule leaves
(vi ) and
the adoption
of real
ity
in
the
topology
of
pore
networks
that
can
empirically
parameters of soil samples. The distributions Pbe(k)
genertested
real
samples.
ated
byinthe
soilsoil
model
can be tested in future research.
We have also shown the relationship between the variability in the scaling exponents and the parameters regulating the
function, as well as the inhomogeneity of the distri5affinity
Conclusions
butions of pore properties, and the consequences of this on
In
webehavior
have presented
a complex
network In
model
thesummary,
asymptotic
of the degree
distribution.
particbased
heterogeneous
PA scheme
in degree
order todistributions
quantify the of
ular, itonisaworth
emphasizing
that the
structure
of porous
soils.
The proposed
soil model behaviors
allows
all the variants
of the
soil model
exhibit power-law
the
of different
medium
geometries
and pore
withspecification
exponents within
the limits
empirically
observed
in real
dynamics
the performed
choice of different
underlying
spaces,
networks.through
We have
a numerical
analysis
of the
distributions
ofapore
propertiesofand
affinity functions.
We to
soil model for
combination
parameters
corresponding
have
obtained
analytical
solutionsproperties.
for the degree
densities
empirical
samples
with different
We have
shown
and degree distribution of the pore networks generated by the
that the simulation results exhibit a good agreement with the
model in the thermodynamic limit. We have shown that these
analytical predictions, concerning the scaling behavior of the
networks exhibit a multiscaling of their degree densities acdegree metrics as well as the ranges of values of the scaling
cording to power laws with exponents spanning a continuum,
exponents
obtained.
and
that such
phenomenon leaves a signature of heterogene-
Nonlin. Processes Geophys., 15, 893–902, 2008
Reference
Albert, R. a
networks
Bak, P., Tan
explanati
Barabási, A
networks
Barabási, A
scale-free
Bird, N., Cr
multifrac
211—219
Blair, J. M.,
ford, J. C
erogenou
Cislerova, N
ings of th
cesses in
gium, 24–
Dorogovtsev
Adv. Phys
Dorogovtsev
of growin
85, 4633–
Elliot, T.R. a
ing of CT
EMBRAPA
Empresa
Janeiro, 2
Erdős, P. an
6, 290–29
GE Healthc
and Anal
902
A. Santiago et al.: Multiscaling of porous soils as heterogeneous complex networks
Acknowledgements. This work has been supported by the Spanish
MEC under Project “i-MATH” No. CSD2006-00032 and Project
No. MTM2006-15533, GESAN and Comunidad de Madrid under
Project TAGRALIA-CM-P-AGR-000187-0505.
Edited by: A. Tsonis
Reviewed by: J. Garcia Miranda and another anonymous referee
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