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3D Multifractal Analysis of Porous Media Using 3D Digital

Images: Considerations for heterogeneity evaluation

Sadegh Karimpouli1 and Pejman Tahmasebi2

1 Corresponding author: Mining Engineering group, Faculty of Engineering,

University of Zanjan, Zanjan, Iran. E-mail: [email protected]

2 Department of Petroleum Engineering, University of Wyoming, Laramie, WY

82071, USA. E-mail: [email protected]

ABSTRACT

Pore structure heterogeneity is a critical parameter controlling mechanical,

electrical and flow transport behavior of rock. Multifractal analysis method

was used for a heterogeneity comparison of 3D rock samples with different

lithology. Six real digital samples, containing three sandstones and three

carbonates were used. Based on the mercury injection capillary pressure test

on these samples we found that the carbonate samples are more

heterogeneous than sandstones, but primary results of multifractal behaviors

for all samples were similar. We show that if multifractal is used to evaluate

and compare heterogeneity of different samples, one needs to follow some

considerations such as: 1. all samples must have the same size in pixel, 2.

samples volume must be bigger than representative volume element, 3.

multifractal dimensions should be firstly normalized to a determined porosity

value and 4. multifractal results should be interpreted based on resolution of

the imaging tool (effects of fine scale sub-resolution pores are missed). Results

revealed that using normalized fractal dimensions, the real samples were

divided to less and high heterogeneous groups. Moreover, study of scale effect

also showed that porous structures of these samples are scale-invariant in a

wide range of scales (from one to eight times bigger).

Keywords: 3D multifractal, 3D Digital Images, heterogeneity evaluation

I. INTRODUCTION

In porous media, heterogeneity is defined as the quality or state of diversity in

characters. More specifically, heterogeneity of porous media represents the

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2

level of dissimilarity of pore-space, pore and throat size distribution,

tortuousness of their connections and their spatial distribution. Heterogeneity

assessment is considered as a critical study in a wide range of applications in

rock physics, especially in petroleum industry and hydrogeology. Some of its

applications are on controlling fluid flow and reservoir production (Edery et al.

2014; Rhodes, Bijeljic and Blunt 2008; Sahimi and Yortsos 1990), CO2 transport

(Oh et al. 2017), fluid-fluid reaction rate (Alhashmi, Blunt and Bijeljic 2016),

elastic wave propagation (Hamzehpour, Asgari and Sahimi 2016) and electrical

properties (Xin et al. 2016). Among all methods, fractal geometry (Mandelbrot

1982) has been widely utilized to characterize the heterogeneity of rock for

more than three decades (Al-Khidir et al. 2013; Chattopadhyay and Vedanti

2016; Hansen and Skjeltorp 1988; Katz and Thompson 1985; Li and Horne

2003; Sahimi 2011; Shen, Li and Jia 1995; Vega and Jouini 2015). Fractal

geometry assumes a self-affine (Simonsen, Hansen and Nes 1998) or self-

similar feature in a system such as porosity in rock medium. The self-similar

feature in this method relates such properties with scaling of measurement by

a power law (Turcotte 1997). The exponent of this power equation is fractal

dimension D which can be used to evaluate the heterogeneity in porous

media. In other words, the greater the fractal dimension, the greater the

heterogeneity of the system (Al-Khidir et al. 2013; Kewen 2004). Since D is an

average characteristic, it cannot fully explain deviations from the average value

and ignore the existent uncertainty. Generally speaking, in box-counting

method (Feder 1988) D is estimated by number of boxes with various sizes

covering all the sample. In this method, all boxes containing even one pixel of

porosity, which we suppose is the main property, is accounted regardless of

the density of porosity (or number of pixels containing porosity) in that box.

Clearly, one cannot assess the variation and amount of pore structure

properly. In multifractal analyses, however, probability of porosity in each box

is considered as a distribution function and, then, represented by a functional

form of fractal dimensions called multifractal (i.e. singularity) spectrum (Jouini,

Vega and Mokhtar 2011; Posadas et al. 2003). Multifractal spectrum and

parameters explain heterogeneity more clearly compared to the traditional

fractal dimension.

Although there is no general model for describing physical processes that

produce self-similarity in soil and rock porous media (Norbisrath et al. 2015),


3

some researchers have accepted and utilized fractal (Curtis et al. 2012; Krohn

1988; Norbisrath et al. 2015; Radlinski et al. 2004) and multifractal law (Dathe

and Thullner 2005; Jouini et al. 2011, 2015; Posadas et al. 2003; Tarquis et al.

2006; Vega and Jouini 2015) in their studies. Some of these researchers applied

soil and sandstone rock samples and showed that self-similar features and

multifractal law is accepted for those samples (Posadas et al. 2003; Tarquis et

al. 2006; Zhang and Weller 2014). Carbonate rock samples have also been used

before (Jouini et al. 2011; Vega and Jouini 2015; Xie et al. 2010) and the results

indicate that all samples do not show self-similarity. Such complex samples

require some pre-studies (e.g., usual cross-plots between multifractal

parameters) to see if they represent the multifractal behavior (see next

section).

Most of the discussed studies also have applied two-dimensional (2D) thin

sections or scanning electronic microscope (SEM) images for the fractal

modeling. Multifractal behavior of a 3D medium, however, can be different

since the complexity of 2D and 3D samples are not always laid similarly. Thus,

in this paper 3D multifractal behaviors of various sandstone and carbonate are

studied. To this end, 3D digital images are obtained using high-resolution 3D

microcomputed tomography (µCT) scanner as they are very common in digital

rock physics (DRP) (Andrä et al. 2013a, b; Karimpouli et al. 2017, 2018;

Karimpouli and Tahmasebi 2015, 2017; Karimpouli, Tahmasebi and Saenger

2018; Tahmasebi et al. 2015, 2016, 2017; Tahmasebi 2017a, b; Tahmasebi

2018a, b). Therefore, six real 3D digital samples, three sandstones

(Bentheimer, Clashach and Doddington) and three carbonates (Estaillades,

Ketton and Portland), are used in this study. These samples are suggested as

standard samples in DRP studies (Alyafei, Mckay and Solling 2016). For

computation of multifractal parameters of these samples, there are some

important factors such as image size, image resolution, sample volume and

sample porosity which their effects have been poorly studied. We aim to study

such factors and their impacts on computation of multifractal dimensions and

heterogeneity evaluation.

This paper is summarized as follows: first, fractal and multifractal basics and

equations are reviewed. Then, six real 3D digital samples are introduced and

their pore structure heterogeneity are evaluated using data from mercury


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injection capillary pressure test. Multifractal behavior of 3D rock porous media

is studied in next section. Results from real samples show that some other

considerations are required for heterogeneity evaluation using multifractal

method. These considerations are considered in discussion section. In this

section, the effects of image size, image resolution, sample volume and

porosity of 3D digital image on multifractal parameters are studied and

discussed.

II. FRACTALS AND MULTIFRACTALS

Let nR be a subset of nR and suppose that ( )N is the number of

spherical pores with the length scale of measurement that can cover . The

scaling law is stated as (Mandelbrot 1982):

( ) ~ DN (1)

where D is the fractal dimension and ~ indicates that ( )N and D are

asymptotically equivalent. D is usually computed using the box-counting

method (Feder 1988). In the box-counting method, boxes with a range of

are considered to cover all the sample. All boxes with even one pixel of

porosity, are counted ( )N and the box-counting dimension 0

D is calculated

as the negative slop of log-log plot of ( )N versus :

00

log ( )lim

log(1/ )

ND

(2)

According to this method, the amount of porosity in each box is not considered

which obviously plays an important role on heterogeneity of pore structure.

However, in multifractal analysis, probability of pore volume in each box is

accounted as a distribution function (Halsey et al. 1986):

( ) ~ i

iP (3)

where i

is the Lipschitz-Holder exponent characterizing scaling in i-th box.

The number of boxes where their probability distributions have an exponent

between and is introduced by scaling law (Halsey et al. 1986):

( )( ) ~ fN (4)


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where ( )f is the fractal dimension of the boxes with exponent . Equation 4

generalizes Eq. 1 where one can use several indices to quantify the scaling of a

system.

Generalized multifractal measures can be obtained over a range of thq moments (positive or negative) of distribution function (Chhabra et al.

1989):

( )( 1)

1

( ) q

Nq Dq

ii

P

(5)

where qD is the multifractal dimension and defined by:

( )

1

0

log ( )1

lim1 log

Nq

ii

q

PD

q

(6)

, and the exponent is defined as the thq order moment (Halsey et al. 1986):

( ) ( 1)q

q q D (7)

( ) ( ) ( )f q q q (8)

In Eq. (6), if 0q , then probability of all boxes is unit and, thus, the equation

would be a special case of mono-fractal dimension (Eq. 2). The 0

D is also called

capacity dimension, which reflects spatial geometry of the system (Voss 1988).

If 1q , the relation represents the special case of 1D which is called entropy

dimension. The new equation is related to Shannon entropy that measures the

uncertainty of the random variable (Vega and Jouini 2015). Equation 6 with a

normalized version of probabilities (( )

1

( . ) ( ) / ( )N

q q

i i ii

q P P

) (Posadas et al.

2003) can be written as:

( )

1

10

( )log ( )

limlog

N

i iiD

(9)

The other special case is 2q . In this case 2

D is called correlation dimension

by which the correlation of measures contained in the boxes of size can be

calculated (Posadas et al. 2003).


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III. REAL 3D DIGITAL SAMPLES

In this paper, six 3D real samples provided by (Alyafei et al. 2016) were used.

This dataset includes three sandstones samples (Bentheimer, Clashach, and

Doddington) and three carbonates samples (Estaillades, Ketton, and Portland).

Some general information of these samples are summarized in Table 1.

The 3D datasets were obtained using micro-CT scanner in digital core

laboratory at Maersk Oil Research and Technology Centre (MORTC), Qatar. A

4.8 mm plug of each sample was scanned with an image resolution of about 3

µm (Alyafei et al. 2016). The size of these samples is 1024×1024×1024 voxels.

However, for the sake of low computation, smaller samples with the size of

512×512×512 voxels were extracted from the center of each sample. The

extracted samples are then segmented and used in this study. These samples

are illustrated in Figure 1. More details about the pre-processing and

segmentation methods can be found in Alyafei et al. (2016).

Figure 1 obviously shows that the porous structure of these samples is

different with various levels of heterogeneity. Heterogeneity of porous

structure, here, refers to dissimilarity of the pore-throat shape, size and sorting

distributions and tortuousness of their interconnectivity. A well-known data

revealing heterogeneity is mercury injection capillary pressure (MICP) test (Al-

Khidir et al. 2013; Kewen 2004; Leal et al. 2001; Pini, Krevor and Benson 2012;

Swanson 1981; Thomeer 1960). MICP experiments of these samples were

conducted by Alyafei et al. (2015) and are presented in Figure 2. MICP curve

initiates from a displacement injection pressure at zero volume porosity and

ends with a high value pressure when all connected porosities are filled by

mercury. These two points defines the location of the curve on the MICP plot,

however, curvature amount of this curve is related to the pore-throat size and

sorting and interconnectivity. This parameter is known as pore geometrical

factor (PGF) (Thomeer, 1960) and is used to evaluate heterogeneity of the

porous media (Al-Khidir et al. 2013; Kewen 2004; Pini et al. 2012). The higher

curvature of MICP curve, the lower PGF and vice versa. It means a sample with

a low value of PGF contains well-sorted and interconnected pores that possess

a narrow pore size distribution. On the contrary, a steady increase curve with

high PGF suggests poor sorting pores or let say more heterogeneous sample


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(Kewen 2004; Pini et al. 2012). Bimodality of MICP curve also implies a higher

level of heterogeneity (Leal et al. 2001; Thomeer 1960).

MICP curves of real samples (Figure 2) show that porous structure of

sandstone samples (Bentheimer, Clashach and Doddington) are somehow the

same since they have similar curves. The wide plateau of these curves depicts

well-sorted narrow pore-throat size distribution. However, MICP curve of two

carbonate samples i.e., Estailades and Ketton are bimodal and the other

sample (i.e, Portland) shows a steady increasing curve with increasing injected

mercury, which indicates a poor-sorted with wider pore-throat size

distribution. These evidences imply that porous media of carbonate samples

are more complex and more heterogeneous than sandstone samples.

IV. 3D MULTIFRACTAL ANALYSIS

A critical step for multifractal analysis is selecting a proper range for moment

orders and box sizes (Saucier and Muller 1999). The basic criteria are linear

behavior of multifractal parameters. Clearly, a suitable range of q is defined

when the ( )q linearly changes with moment order q . Similarly, a proper

range of box sizes is where the partition function ( . )q linearly changes with

box size (Dathe, Tarquis and Perrier 2006) in logarithmic scale. With a

reasonable threshold value of R2 = 90% (goodness of fitness) for both criteria

(Vega and Jouini 2015), we found that a range of 3 92 2 for box size (Figure

3a) and 3 3q (Figure 3b), with empirical increment of 0.5 for moment

orders are appropriate for all of the real samples.

Subsequently, we implemented a 3D version of the box-counting method

(Posadas et al. 2003) and computed all multifractal parameters

( , , ( ), , , ( . )q qD q f q ) for each sample. The cross-plot of some of these

variables are shown in Figure 4. The accuracy of these parameters can be

verified using the qD value and ( )f curve. According to the basic

concepts of fractal law (Mandelbrot 1982), 0

D changes in a range of 2 to 3 for

3D porous media. As it is illustrated in Figure 4(a), 0

D for all samples lied on a

valid range. Moreover, ( )f is a hyperbolic curve and touches the internal

bisector ( )f , which are observed in Figure 4(b). These behaviors confirm


8

validity of our computations. An interpretation of these plots will be discussed

in next sections.

V. DISCUSSION

A. Heterogeneity Evaluation

Based on the previous studies on 2D multifractals (Jouini et al. 2011; Vega and

Jouini 2015; Xie et al. 2010), it has been shown that qD q curve can be used

for heterogeneity evaluation. Generally, qD decreases when q increases.

However, such a reduction in curve’s slope highly depends on sample

heterogeneity. Since homogeneous samples are mono-fractal, they show a

relatively flat curve, particularly for 2q (Jouini et al. 2011). Yet,

heterogeneous samples represent a steeper curve. Another heterogeneity

indicator is the ( )f curve, which represent a hyperbolic curve for

multifractal samples. Heterogeneity of porous structure can be evaluated by

aperture and symmetry of this curve. The wider aperture or range of and

the more asymmetrical curve, the more heterogeneous porous medium

(Posadas et al. 2003; Vega and Jouini 2015).

Comparing both cross-plots of qD q and ( )f , in Figure 4, suggests all

samples have a similar heterogeneity level except for the Ketton, which is less

heterogeneous. Obviously, these results are not supported by the findings

from MICP curves shown in Figure 2. So, the questions are: Although

heterogeneity of samples are totally different, why do they show similar

multifractal behavior? And, how samples with different heterogeneities would

be distinguishable from each other by multifractal analysis? To answer these

crucial questions, we need to consider some other effects such as sample

volume, scale and porosity.

B. Effect of Sample Volume and Porosity

To explore the effect of representative volume element and porosity on

multifractal parameters, we produced new sub-samples from each original

sample, but with smaller sizes. The original sample size was 512×512×512, and

three other sub-samples with size of 256×256×256, 128×128×128 and

64×64×64 were extracted by decreasing sample size toward the center of the


9

original sample. Before going thorough multifractal computations, there is an

important point that should be considered. Multifractal parameters computed

from the box-counting method are sensitive to the box sizes. Therefore,

consideration #1 is: all samples must have the same size. As mentioned earlier,

we found a range of 3 92 2 as suitable lengths for minimum and maximum

box sizes. According to consideration #1, various sub-samples were resampled

to the size of 512×512×512. For preserving the binary image, resampling must

be carried out without interpolation with different sizes. This allows us to

preserve both porous structures and computation elements simultaneously.

Then, multifractal parameters were computed accordingly. The results are

shown in Figure 5.

It is believed that porous structures and complexity of samples are lost due to

decreasing in sample size or volume. Thus, a more homogeneous sample is

expected to be generated. The presented results in Figure 5 shows this and

indicates that heterogeneity of all samples decreases by decreasing the sample

size. This is a correct behavior until a sample, which is not representative, is

generated. If the sample size is too small, it would not show multifractal

behavior. For example, in cross-plot of qD q , shown in Figure 5(a), the

qD

curve does not represent a declining trend for sample size of 64×64×64 in

almost all cases. Furthermore, the ( )f curve is distorted, especially for small

and narrow range of values shown in Figure 5(b) for this sample size.

Therefore, in all cases, samples with smaller size of 128×128×128 are not

representative anymore. However, this behaviour in Estailades can be seen for

sample size of 32×32×32, which means representative sample size in this case

is at least 64×64×64. Subsequently, consideration #2 can be stated as: samples

with sizes smaller than representative volume have not to be considered for

heterogeneity evaluation.

All representative sub-samples from an original sample were expected to show

similar multifractal behavior, but this is not implied from Figure 5. We thought

that the only parameter, which causes multifractal properties to be variant,

may be porosity. Therefore, porosity values for all sub-samples were computed

and shown in Figure 5(a). It is obvious that multifractal dimensions change by

porosity. Now, let us to go back to one of the questions that why samples with

different heterogeneities do not show different multifractal behavior? Based on


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our findings in Figure 5, it can be concluded that this is due to different

porosity values of samples. Therefore, one needs to consider the porosity

effect on multifractal dimensions before going through comparison step.

Posadas et al. (2003) showed that 0

D and 1D are highly correlated with

porosity values. Vega and Jouini (2015) suggested a logarithmic trend of 0

D

with porosity, which was also implemented in this study. We found the trends

of 0

D and 1D with porosity by fitting a logarithmic equation to results

obtained in Figure 5. All samples with different sizes were resampled to the

same size of 512×512×512 according to consideration #1, and then were used

for this analysis. Table 2 summaries these relations and their R2 values for all

samples. It should be noted that in presence of additional and larger images

one could calculate a more accurate trend. These trends are used for

computing normalized values of 0

D and 1D in each porosity value for every

sample, even if that sample has a different porosity. The obtained trends were

used to compute the normalized values of 0

D and 1D in each porosity in range

of 10 to 14%, for all samples (Figure 6).

Based on the normalized values for 0

D and 1D in Figure 6, samples are divided

into two groups of less and more heterogeneous samples. Three sandstone

samples and one carbonate (Ketton) are in the less heterogeneous group,

while two other carbonate samples are in the more heterogeneous group.

According to the results in section II, this grouping is now more rational than

the results in Figure 4. To explain about the Ketton, we refer back to MICP

curves in Figure 2. Although all carbonate samples are more heterogeneous in

reality, most of fine pores of these samples are sub-resolution regarding to the

image resolution. This means that they are not detected in digital images using

an imaging tool with such a resolution limit (2.9 µm). To show detectable range

in the digital samples, image resolution line was computed and shown in Figure

2 (solid line). Detectable part of MICP curves falls below the image resolution

line. This indicates that fine scale pores of Ketton and Portland are missed and,

therefore, multifractal calculations cannot cover them. After removing MICP

curves higher than image resolution line, it is observed that MICP curve of

Ketton behaves similar to sandstone samples. This is why Ketton is not

classified as a regular carbonate (more heterogeneous group). However,

steady increasing behaviour of Portland MICP curves cause it to be more


11

heterogeneous. Also bimodality of Estailades curve remains detectable, which

made it also to be more heterogeneous. These results show that our grouping

strategy, based on normalized porosity samples, was completely valid. Thus,

consideration # 3 is: multifractal dimensions should be firstly normalized to a

determined porosity value and, then, compared with each other. In addition,

consideration #4 is: multifractal results are based on detectable pore according

to the resolution of the imaging tool. Fine scale sub-resolution pores are

missed and, thus, multifractal study is not able to cover them.

C. Effect of Sample Scale

For investigating the effect of scale or resolution (i.e. voxel length in

µm/voxel), we resampled all the real samples with scale factors (S.F) of 1/1,

1/2, 1/4 and 1/8 and new samples with different resolutions of about 3, 6, 12

and 24 µm/voxel were generated. Figure 7 shows an example from Benheimer

sample with different SFs. To remove effects of different image sizes, all

samples were again resized to 512×512×512. Then the multifractal parameters

of each new sample were computed accordingly.

According to the results in Figure 8, it can be seen that the fractal dimension

0D remains almost constant for all samples even with eight times different

resolution. This verifies the concept of scale-invariant or self-similar behavior

of rock porous structure in a 3D medium. The variation of other multifractal

dimensions (qD ) with positive moments order ( 0q ) is also less than 1%

(Figure 7a). It should be noted that porosity values are very close in samples

with different scales.

VI. CONCLUSIONS

In this paper, multifractal properties of rock porous media were studied using

3D digital images of different rock types. Multifractals are believed to be an

efficient way for heterogeneity evaluation, but our primary results in real

samples showed a similar multifractal behavior for almost all samples even

with different lithology and their obvious different heterogeneities. It was

found that at least four considerations should be taken into account before

using multifractal parameters to evaluate heterogeneity:

1) All samples must have the same size in pixel.


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2) Samples volume must be bigger than representative volume.

3) Multifractal dimensions should be firstly normalized to a determined

porosity value.

4) Multifractal results are based on detectable pore according to resolution of

the imaging tool. Fine scale sub-resolution pores are missed.

According to these considerations, one requires normalizing the multifractal

dimensions to a porosity value. To do that, in the samples with the same size

and resolution, the variation of multifractal dimensions against porosity was

investigated and some useful logarithmic trends were extracted accordingly.

Comparison of normalize multifractal dimensions revealed that, unlike the

primary results, different multifractal behaviors are obtained for distinct

samples as expected. Results showed a promising grouping of the real samples.

Sandstone samples were all categorized in less heterogeneous group, while the

carbonates samples, except the Ketton, were considered in the more

heterogeneous group. MICP test and resolution limit of imaging tool showed

that most of fine-pores in this sample were sub-resolution and were not

detectable in the utilized images. Therefore, its MICP curve reduced to a

sample with lower heterogeneity such as sandstone samples.

The effect of representative volume element was also studied. The results

indicated that the samples with the size of 64×64×64, except for the case of

Estaillades being with the size of 32×32×32, do not behave as a multifractal

medium. In other words, those samples cannot be considered as

representative samples for evaluating the heterogeneity. Moreover, the effect

of image scale was also investigated in this study. The results demonstrated

scale-invariant behavior of porous media, since fractal dimension remains

constant for a large variation of scales (from one to eight times bigger).

This study represents a thorough investigation among the important factors

affecting the calculation of multifractal parameters, particularly when one

utilizes these parameters for the heterogeneity evaluation and the comparison

of various samples.

Conflicts of Interest


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The authors certify that they have NO affiliations with or involvement in any

organization or entity with any financial interest in the subject matter or

materials discussed in this manuscript.

References

Al-Khidir K. E., Benzagouta M. S., Al-Qurishi A. A., and Al-Laboun A. A. 2013.

Characterization of heterogeneity of the Shajara reservoirs of the

Shajara formation of the Permo-Carboniferous Unayzah group. Arabian

Journal of Geosciences 6, no. 10, 3989–3995.

Alhashmi Z., Blunt M. J. and Bijeljic B. 2016. The Impact of Pore Structure

Heterogeneity, Transport, and Reaction Conditions on Fluid–Fluid

Reaction Rate Studied on Images of Pore Space. Transport in Porous

Media 115, no. 2, 215–237.

Alyafei N., Raeini A. Q., Paluszny A. and Blunt M. J. 2015. A Sensitivity Study of

the Effect of Image Resolution on Predicted Petrophysical Properties.

Transport in Porous Media 110, no. 1, 157–169.

Alyafei N., Mckay T. J., and Solling T. I. 2016. Characterization of petrophysical

properties using pore-network and lattice-Boltzmann modelling: Choice

of method and image sub-volume size. Journal of Petroleum Science

and Engineering 145, 256–265.

Andrä H., Combaret N., Dvorkin J., Glatt E., Han J., Kabel M., Keehm Y., Krzikalla

F., Lee M., Madonna C., Marsh M., Mukerji T., Saenger E. H., Sain R.,

Saxena N., Ricker S., Wiegmann A. and Zhan X. 2013a. Digital rock

physics benchmarks-Part I: Imaging and segmentation. Computers and

Geosciences 50, 25–32.

Andrä H., Combaret N., Dvorkin J., Glatt E., Han J., Kabel M., Keehm Y., Krzikalla

F., Lee M., Madonna C., Marsh M., Mukerji T., Saenger E. H., Sai, R.,

Saxena N., Ricker S., Wiegmann A. and Zhan X. 2013b. Digital rock

physics benchmarks-part II: Computing effective properties. Computers

and Geosciences 50, 33–43.


14

Andrew M., Bijeljic B. and Blunt M. J. 2014. Pore-scale imaging of trapped

supercritical carbon dioxide in sandstones and carbonates.

International Journal of Greenhouse Gas Control 22, 1-14.

Brenchley P. J. and Rawson P. F. 2006. The geology of England and Wales.

Geological Society of London.

Chattopadhyay P. B. and Vedanti N. 2016, Fractal Characters of Porous Media

and Flow Analysis, in Dimri, V.P. ed., Fractal Solutions for

Understanding Complex Systems in Earth Sciences. Springer

International Publishing, Cham, p. 67–77.

Chhabra A. B., Meneveau C., Jensen R. V. and Sreenivasan K. R. 1989. Direct

determination of the f (α) singularity spectrum and its application to

fully developed turbulence. Physical Review A 40, no. 9, 5284.

Curtis M. E., Sondergeld C. H., Ambrose R. J. and Rai C. S. 2012. Microstructural

investigation of gas shales in two and three dimensions using

nanometer-scale resolution imaging. AAPG Bulletin 96, no. 4, 665–677.

Dathe A., Tarquis A. M. and Perrier E. 2006. Multifractal analysis of the pore-

and solid-phases in binary two-dimensional images of natural porous

structures. Geoderma 134, no. 3, 318–326.

Dathe, A. and Thullner M. 2005. The relationship between fractal properties of

solid matrix and pore space in porous media. Geoderma 129, no. 3,

279–290.

Dubelaar C. W. and Nijland T. G. 2015. The bentheim sandstone: geology,

petrophysics, varieties and its use as dimension stone. Engineering

Geology for Society and Territory 8, 557-563.

Edery Y., Guadagnini A., Scher H. and Berkowitz B. 2014. Origins of anomalous

transport in heterogeneous media: Structural and dynamic controls.

Water Resources Research 50, no. 2, 1490–1505.

Feder J. 1988, Fractals. Springer Science & Business Media.


15

Halsey T. C., Jensen M. H., Kadanoff L. P., Procaccia I. and Shraiman B. I. 1986.

Fractal measures and their singularities: the characterization of strange

sets. Physical Review A 33, no. 2, 1141.

Hamzehpour H., Asgari M. and Sahimi M. 2016 Acoustic wave propagation in

heterogeneous two-dimensional fractured porous media. Physical

Review E 93, no. 6, 63305.

Hansen J. P. and Skjeltorp A. T. 1988. Fractal pore space and rock permeability

implications. Physical Review B 38, no. 4, 2635.

Jouini M. S., Vega S. and Mokhtar E. A., 2011, Multiscale characterization of

pore spaces using multifractals analysis of scanning electronic

microscopy images of carbonates. Nonlinear Processes in Geophysics

18, no. 6, 941–953.

Jouini M. S., Vega S., Al‐Ratrout A. and Al-Ratrout A. 2015. Numerical

estimation of carbonate rock properties using multiscale images.

Geophysical Prospecting 63, no. 2, 405–421.

Karimpouli S., Khoshlesan S., Saenger E. H., and Koochi H. H. 2018. Application

of alternative digital rock physics methods in a real case study: a

challenge between clean and cemented samples. Geophysical

Prospecting. DOI: 10.1111/1365-2478.12611

Karimpouli, S., and Tahmasebi, P. 2015. Conditional reconstruction: An

alternative strategy in digital rock physics. Geophysics, 81, no. 4.

Karimpouli S., Tahmasebi P., Ramandi H. L., Mostaghimi P. and Saadatfar M.

2017. Stochastic modeling of coal fracture network by direct use of

micro-computed tomography images. International Journal of Coal

Geology, 179, 153–163.

Karimpouli S. and Tahmasebi P. 2017. A Hierarchical Sampling for Capturing

Permeability Trend in Rock Physics. Transport in Porous Media 116, no.

3, 1057–1072.

Karimpouli, S., Tahmasebi, P. and Saenger, E.H., 2018. Estimating 3D elastic

moduli of rock from 2D thin-section images using differential effective

medium theory. Geophysics, 83, no. 4, pp. MR211-MR219.


16

Katz A. J. and Thompson A. H. 1985. Fractal sandstone pores: implications for

conductivity and pore formation. Physical review letters 54, no. 12,

1325.

Kewen L. 2004. Characterization of rock heterogeneity using fractal geometry,

in SPE International Thermal Operations and Heavy Oil Symposium and

Western Regional Meeting: Society of Petroleum Engineers.

Krohn C. E. 1988. Fractal measurements of sandstones, shales, and carbonates.

Journal of Geophysical Research: Solid Earth 93, no. B4, 3297–3305.

Leal L., Barbato R., Quaglia A., Porras J. C. and Lazarde H. 2001. Bimodal

Behavior of Mercury-Injection Capillary Pressure Curve and Its

Relationship to Pore Geometry, Rock-Quality and Production

Performance in a Laminated and Heterogeneous Reservoir, in SPE Latin

American and Caribbean Petroleum Engineering Conference: Society of

Petroleum Engineers.

Li K., and Horne R. N. R. 2003. Fractal characterization of the geysers rock, in

Proceedings of the GRC 2003 annual meeting.

Mandelbrot B. B. 1982. Multifractal measures, especially for the geophysicist.

Pure and applied geophysics 131, no. 1, 5–42.

Ngwenya B. T., Elphick, S. C. and Shimmield G. B. 1995. Reservoir sensitivity to

water flooding: An experimental study of seawater injection in a North

Sea reservoir analog. AAPG Bulletin 79, 285-303.

Norbisrath J. H., Eberli G. P., Laurich B., Desbois G., Weger R. J. and Urai J. L.

2015. Electrical and fluid flow properties of carbonate microporosity

types from multiscale digital image analysis and mercury injection.

AAPG Bulletin 99, no. 11, 2077–2098.

Oh J., Kim K.-Y., Han W. S., and Park E. 2017. Transport of CO2 in

heterogeneous porous media: Spatio-temporal variation of trapping

mechanisms. International Journal of Greenhouse Gas Control 57, 52–

62.


17

Pini R., Krevor S. C. M., and Benson S. M. 2012. Capillary pressure and

heterogeneity for the CO2/water system in sandstone rocks at reservoir

conditions. Advances in Water Resources 38, 48–59.

Posadas A. N. D., Giménez D., Quiroz R. and Protz R. 2003. Multifractal

characterization of soil pore systems. Soil Science Society of America

Journal 67, no. 5, 1361–1369.

Radlinski A. P. P., Ioannidis M. A. A., Hinde A. L. L., Hainbuchner M., Baron,M.,

Rauch H. and Kline S. R. R. 2004. Angstrom-to-millimeter

characterization of sedimentary rock microstructure. Journal of colloid

and interface science 274, no. 2, 607–612.

Rhodes M. E., Bijeljic B. and Blunt M. J. 2008. Pore-to-field simulation of single-

phase transport using continuous time random walks. Advances in

Water Resources 31, no. 12, 1527–1539.

Sahimi M. 2011. Flow and transport in porous media and fractured rock: from

classical methods to modern approaches. John Wiley & Sons.

Sahimi M. and Yortsos Y. C. 1990. Applications of fractal geometry to porous

media: a review, in Annual Fall Meeting of the Society of Petroleum

Engineers, New Orleans, LA: Society of Petroleum Engineers.

Santarelli F. and Brown E. 1989. Failure of three sedimentary rocks in triaxial

and hollow cylinder compression tests. International Journal of Rock

Mechanics and Mining Sciences & Geomechanics Abstracts 26, 401-413.

Saucier A. and Muller J. 1999. Textural analysis of disordered materials with

multifractals. Physica A: Statistical Mechanics and its Applications

267(1), 221–238.

Simonsen, I., Hansen, A., and Nes, O. M. 1998. Determination of the Hurst

exponent by use of wavelet transforms. Physical Review E, 58(3), 2779.

Shen P., Li K. and Jia F. 1995. Quantitative description for the heterogeneity of

pore structure by using mercury capillary pressure curves, in

International Meeting on Petroleum Engineering: Society of Petroleum

Engineers.


18

Swanson B. F. 1981. A Simple Correlation Between Permeabilities and Mercury

Capillary Pressures. Journal of Petroleum Technology 33, no. 12, 2498–

2504.

Tahmasebi, P. 2017a. Structural adjustment for accurate conditioning in large-

scale subsurface systems. Advances in Water Resources, 101.

Tahmasebi, P. 2017b. HYPPS: A hybrid geostatistical modeling algorithm for

subsurface modeling. Water Resources Research, 53, 7, 5980–5997.

Tahmasebi, P. 2018a. Accurate modeling and evaluation of microstructures in

complex materials. Physical Review E, 97, 2, 023307.

Tahmasebi, P. 2018b. Packing of discrete and irregular particles. Computers

and Geotechnics, 100, 52–61.

Tahmasebi P., Sahimi M., Kohanpur A. H. and Valocchi A. 2016b. Pore-scale

simulation of flow of CO 2 and brine in reconstructed and actual 3D

rock cores. Journal of Petroleum Science and Engineering 155, 21–33.

Tahmasebi P., Sahimi, M. and Andrade, J.E. 2017. Image‐based modeling of

granular porous media. Geophysical Research Letters, 44(10), 4738-

4746.

Tahmasebi P. 2018. Nanoscale and multiresolution models for shale samples.

Fuel, 217, 218-225.

Tarquis A. M. M., McInnes K. J. J., Key J. R. R., Saa A., García M. R. R., Díaz M. C.

C., Garcia M. R. and Diaz M. C. 2006. Multiscaling analysis in a

structured clay soil using 2D images. Journal of hydrology 322, no. 1,

236–246.

Thomeer J. H. M. 1960. Introduction of a Pore Geometrical Factor Defined by

the Capillary Pressure Curve. Journal of Petroleum Technology 12, no. 3,

73–77.

Turcotte D. D. L. 1997. Fractals and chaos in geology and geophysics.

Cambridge university press.


19

Watson J. 1911. British and Foreign Building Stones. A Descriptive Catalogue of

the Specimens in the Sedgwick Museum. Cambridge Univ. Press,

Cambridge, UK

Vega S. and Jouini M. S. S. 2015. 2D multifractal analysis and porosity scaling

estimation in Lower Cretaceous carbonates. Geophysics 80, no. 6,

D575–D586.

Voss R. F. 1988. Fractals in nature: from characterization to simulation, in The

science of fractal images: Springer, New York, NY, p. 21–70.

Xie S., Cheng Q., Ling Q., Li B., Bao Z. and Fan P. 2010. Fractal and multifractal

analysis of carbonate pore-scale digital images of petroleum reservoirs.

Marine and Petroleum Geology 27, no. 2, 476–485.

Xin N., Changchun Z., Zhenhua L., Xiaohong M. and Xinghua Q. 2016. Numerical

simulation of the electrical properties of shale gas reservoir rock based

on digital core. Journal of Geophysics and Engineering 13, no. 4, 481.

Zhang Z. and Weller A. 2014. Fractal dimension of pore-space geometry of an

Eocene sandstone formation. Geophysics 79, no. 6, D377–D387.

Tables

Table 1. Summary of real samples used in this study.

Sample Location Porosity (%) Composition Reference

Bentheimer Germany 18.1 97.5% quartz

2% feldspar

0.5% kaolinite

Dubelaar and

Nijland (2015)

Clashach England 15.5 90% quartz and

10% feldspars

Ngwenya,

Elphick and

Shimmield

(1995)

Doddington England 18.4 95.2% quartz

4.8% white

micas and

feldspars

Santarelli and

Brown (1989)

Estaillades France 10.8 99% calcite Watson (1911)


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1% dolomite

and silica

Ketton UK 11.1 99.1% calcite

0.9% quartz

Andrew, Bijeljic

and Blunt

(2014)

Portland UK 7.1 96.6% calcite

3.4% quartz

Brenchley and

Rawson (2006)

Table 2. Trends of 0

D and 1D to porosity for each sample with value.

Sample 0

D trend R2 1D trend R2

Benheimer 0.91ln 4.45 0.66 0.62ln 3.37 0.64

Clashach 0.35ln 3.39 0.58 0.32ln 3.21 0.78

Doddington 0.58ln 3.79 0.83 0.40ln 3.35 0.91

Estaillades 0.32ln 3.47 0.74 0.23ln 3.20 0.79

Ketton 0.31ln 3.31 0.51 0.61ln 2.82 0.92

Portland 0.90ln 4.76 0.62 0.71ln 4.29 0.70

Figures captions:

Figure 1. Real 3D segmented samples used in this study: (a) Bentheimer, (b)

Clashach, (c) Doddington, (d) Estaillades, (e) Ketton, and (f) Portland.


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Figure 2. Injection pressure of mercury vs fractional pore volume occupied by

mercury for six real samples. Solid line shows detection limit of imaging tool.

The upper part of image resolution line is not detectable in digital samples

(modified from Alyafei et al. (2015)).

Figure 3. Cross plot of log .q to log( ) and ( )q to q for Bentheimer

sample. Linear relationship in these plots reveals the proper range for box size

and exponent q moment. Labels in part ‘a’ are also valid for par ‘b’.


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Figure 4. Multifractal parameters of all real samples. Cross plots of: (a) qD q

and (b) ( )f . Labels in part ‘a’ are also valid for par ‘b’.

Figure 5. Multifractal parameters of real samples with different sizes. Cross-

plots of: (a) qD q and (b) ( )f . indicates the length of sample

(sample size is 3 ) and Phi is the porosity in percent. Labels in part ‘a’ are

also valid for par ‘b’.


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Figure 6. Normalized values of 0

D and 1D for all samples in a range of porosity

from 10 to 14% using porosity trends obtained for each sample in Table 2. Two

different groups from heterogeneity point of view are divided based on these

results. Labels in part ‘a’ are also valid for par ‘b’.

Figure 7. A 2D section of Bentheimer sandstone with different scale factors

(S.F).


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Figure 8. Multifractal parameters of rescaled real samples. Cross plots of: (a)

qD q and (b) ( )f . Labels in part ‘a’ are also valid for par ‘b’.

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