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Marine and Petroleum Geology

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Research paper

A comprehensive study on geometric, topological and fractalcharacterizations of pore systems in low-permeability reservoirs based onSEM, MICP, NMR, and X-ray CT experiments

Yuqi Wua,b, Pejman Tahmasebib,∗, Chengyan Lina, Muhammad Aleem Zahidc, Chunmei Donga,Alexandra N. Golabd, Lihua Rena

a School of Geosciences, China University of Petroleum (East China), Qingdao, 266580, ChinabDepartment of Petroleum Engineering, University of Wyoming, Laramie, 82071, USAc Faculty of Marine Sciences, Lasbela University, Balochistan, 90250, Pakistand Thermo Fisher Scientific, Canberra, ACT, 2600, Australia

A R T I C L E I N F O

Keywords:Pore systemTopology characteristicsFractal dimensionNMRX-ray CT

A B S T R A C T

Characterization of pore systems in subsurface systems is of great importance for predicting the properties ofrocks and classifying the subsurface systems. Geometric features have been used widely for this aim, but to-pological characteristics of the pore structures are not studied much. In fact, accurate characteristics of porespace should comprise both its geometric and topological properties. In this paper, the above vital characteristicsare comprehensively studied based on direct experimental results. Besides, previous studies aiming at linkingfractal dimension analysis to pore space are often based on limited sources of information, which are themercury injection capillary pressure (MICP) and nuclear magnetic resonance (NMR). In this paper, the scanningelectron microscope (SEM), MICP, NMR, and X-ray computed tomography (X-ray CT) experiments are all used tocharacterize the geometric and topological properties of pore space of several low-permeability porous media.Based on our observations, the advantages and disadvantages of the above techniques in characterizing the porestructure are also summarized. Moreover, the differences of these three experiments are quantified using fractaldimension. The results indicate that the NMR technique is a promising tool for characterizing geometric featuresof pore systems as it can cover more details than other techniques. Most of geometric, topological, fractal andtransport properties of pore space can be obtained from X-ray CT method, which is unique among all themethods. In addition, the 3D fractal dimensions of pore systems obtained from NMR is smaller than that fromMICP, which is due more ultra-micropores captured by NMR that smooth the surface of the pore systems. Finally,a novel method constrained by a new pore shape factor for calculating the pore size from 2D images is proposedby which the pore-size distributions are compared more effectively.

1. Introduction

In subsurface characterization, finding the high-quality andpermeable regions (e.g. sweet spots of reservoirs) is very crucial(Clarkson et al., 2013; Nelson, 2011, 2009; Tahmasebi et al., 2017; Xiaoet al., 2017). The quality of the reservoirs was affected by many mac-roscopic factors, such as depth, porosity, and permeability of the re-servoirs, hydrocarbon content, and hydrocarbon quality in the re-servoirs. In fact, most of these macroscopic properties of reservoirs (e.g.porosity and permeability) are mainly controlled by the microscopicpetrophysical properties of the rocks, such as the features of pore sys-tems (Blunt et al., 2013; Daigle et al., 2017; Daigle and Dugan, 2011;

Fagbemi et al., 2018b, 2018a; Hu et al., 2017, 2012; Hu and Brusseau,1994; Jiang, 2008; Tahmasebi and Kamrava, 2018; Xiong et al., 2016).As such, the characterization of pore systems is of great importance forporous media modeling. The pore structure properties of porous mediainclude the geometrical and topological characteristics (Jiang, 2008).For the reservoirs with few or no fractures, the geometrical propertiesof pore space consist of volume, surface area, radius and shape of thepores and throats. The topological parameters comprise the con-nectivity, tortuosity, Euler number, coordination number, and two-point correlation function. As to the fractured reservoirs, such as shalesand coal seams, the characterization of pore systems should include thedescription and measurement of the fractures in the rock because the

https://doi.org/10.1016/j.marpetgeo.2019.02.003Received 28 November 2018; Received in revised form 19 January 2019; Accepted 2 February 2019

∗ Corresponding author.E-mail addresses: ptahmase@uwyo.edu (P. Tahmasebi), lincy@upc.edu.cn (C. Lin).

Marine and Petroleum Geology 103 (2019) 12–28

Available online 06 February 20190264-8172/ © 2019 Elsevier Ltd. All rights reserved.

T

fractures also make a considerable effect on the macroscopic propertiesof the rocks (Clarkson and Bustin, 1996; Ramandi et al., 2018, 2016a;2016b; Tan et al., 2018; Zhang et al., 2019).

To date, numerous techniques have been presented to describe andcharacterize the pore structures of porous media (Gao et al., 2019; Huet al., 2017; Kibria et al., 2018; Li et al., 2015; Nelson, 2009; Shao et al.,2017). These methods can be divided into two groups, namely directimaging measurements and indirect techniques. The former representsa set of methods by which the pore structure features are directlyanalyzed and characterized using imaging techniques, such as variousmicroscopes, X-ray computed tomography (X-ray CT) (Huang et al.,2015; Peng et al., 2012; Wu et al., 2018; Zhang et al., 2016) and laserscanning confocal instruments (Minsky, 1988). The images obtainedfrom these techniques can be two- and three-dimensional. These ima-ging tools are the optical microscope, common scanning electron mi-croscope (SEM) (Henares et al., 2014), back-scattered scanning electronmicroscope (BSEM) (Letham and Bustin, 2018; Li et al., 2018; Van Geetet al., 2001; White et al., 1984), transmission electron microscopy(TEM) (Chalmers et al., 2012), environmental scanning electron mi-croscope (ESEM) (Romero and Simms, 2008), field-emission scanningelectron microscopy (FE-SEM) (Chalmers et al., 2012), focused ionbeam scanning electron microscope (FIB-SEM) (Tahmasebi et al., 2015;Zhang et al., 2017) and broad ion beam scanning electron microscope(BIB-SEM) (Hemes et al., 2015; Norbisrath et al., 2015). Each micro-scope has its specific pros and cons, which is determined by imagingprinciple and work performance of the microscope. Owing to theavailability of 2D SEM images, some researchers measured the pore sizedistribution based on 2D images. However, several assumptions need tobe taken into account when the imaging is limited to 2D sections. Forexample, often the shape of pores is assumed as a circle for evaluatingthe pore size pore distribution (Pengfei et al., 2018; Su et al., 2018),which clearly is unreasonable (Dong and Blunt, 2009). In our research,thus, a novel method constraint by the pore shape factor for calculatingthe pore radius will be proposed.

On the other hand, the indirect measurement techniques, for ex-ample, mercury injection capillary pressure (MICP) (Favvas et al., 2009;Lai et al., 2018a; Okolo et al., 2015), nuclear magnetic resonance(NMR) (Lai et al., 2018b; Li et al., 2015; Yao et al., 2010), gas (i.e. N2

and CO2) adsorption (Favvas et al., 2009) and small-angle neutronscattering (SANS) (Favvas et al., 2009; Radlinski and Mastalerz, 2017;Sun et al., 2018), have been widely used to acquire pore-space prop-erties. 2D SEM imaging method can obtain high-resolution details ofpore system, but it only covers a small field of view in the sample andcannot fully describe the realistic geometric and topological char-acteristics of 3D pore space. In contrast, the indirect measurementtechniques can capture such information regarding the 3D pore struc-ture of a large sample.

The previous studies about pore system characterization are mostlyfocused on the geometric characteristics of pores using one type oftechnique or a combination of some techniques of SEM, MICP, NMR, N2

adsorption, SANS, and X-ray CT (Fink et al., 2018; Lai et al., 2018b;Sarkar et al., 2018; Zhang et al., 2016), but the analyses of the topo-logical features of the pore systems are often neglected. Whereas, thetopological parameters such as coordination number and tortuosity alsohave a very significant influence on the transport properties (Cai andYu, 2011). Hence, in this work, the topological features of the poresystem will be given more attention to. In addition, the advantages anddisadvantages of calculating the pore radius from SEM, MICP, NMR andX-ray CT will also be summarized.

The fractal theory has been widely used in many fields includinggeology, materials science, architecture, economic management,biology, medicine and soil science (Cai et al., 2015; Daigle et al., 2014;Ge et al., 2016; Li et al., 2017; Mandelbrot and Wheeler, 1983). Thedifferences between the fractal theory and the traditional classical Eu-clidean geometry are that the dimension of the classical geometry is aninteger, such as 1 and 2, but the dimension of fractal geometry can be a

decimal (Ge et al., 2016). Fractal theory can describe complex and ir-regular geometry whereas traditional geometry cannot characterize. Onthe topic of our paper for pore structure, several studies have been donebased on the MICP, NMR and CT experiments (Cai et al., 2015; Ge et al.,2016; Lai et al., 2018b; Serrano et al., 2013). For example, Li (2010)proposed to calculate the fractal dimension of the pore structure withthe help of MICP test (Li, 2010). This method is criticized elsewhere (Geet al., 2016; Lai et al., 2018b; Liu et al., 2017; Sun et al., 2018). Ge et al.(2015) used the NMR data to analyze the fractal characteristics andevaluate reservoir quality (Ge et al., 2015). Cai et al., 2015 made areview about fractal modeling of permeability for porous media (Caiet al., 2015). However, their studies are mostly based on one techniqueof MCIP and NMR (Lai and Wang, 2015; Li et al., 2017). The benefitsand drawbacks of calculating the fractal dimension of pore structurebased on MICP, NMR and CT have not yet been compared and studied.Thus, in this paper, an accurate formula derivation about computingthe fractal dimension of pore space from MICP and NMR will be pre-sented. Moreover, we analyze the fractal dimension of the pore struc-ture from MICP, NMR, and X-ray CT tests and the results from thesethree methods will be discussed accordingly.

In this study, in order to fully characterize the geometrical and to-pological properties and analyze the fractal dimension of the porestructure of the porous media, a variety of experiments, including SEM,MICP, low field NMR and X-ray CT, are conducted on five samples fromthe Shahejie formation in the Wangjiagang field, Dongying depression,Bohai Bay Basin, China. The rest of this paper is organized as follows:Section 2 gives a brief introduction of the geological background whileSection 3 mainly describes the utilized samples and the processes of ourexperiments. Section 4 consists of two parts, where the first part willdemonstrate the methodologies, results and pros and cons of calculatinggeometrical and topological properties based on SEM, MICP, NMR, andX-ray CT techniques. As to the second part, the formula derivationabout calculating the fractal dimension of the pore structure will begiven, and discussions about the results of the fractal dimension fromdifferent techniques are presented. A summary of our findings will begiven in Section 5.

2. Geological background

In this study, five low-permeability sandstone samples are collectedfrom Es4s of the Shahejie formation in the Wangjiagang field, Dongyingdepression, Bohai Bay Basin, China, which is located in the south of theBohai Bay Basin of China, shown in Fig. 1. The Eocene Shahejie For-mation (Es) contains the main source and reservoir rocks. The organicmatter of the source is dominated by type I kerogens (oil prone) and Eshas four major divisions; Es4, Es3, Es2, and Es1 (from bottom to top).The organic-rich lowermost interval (Es4) is further divided into upper(Es4s) and lower parts (Es4x). Es4x consists of interbedded turbiditicfan sandstones deposited in a semi-deep and deep lacustrine environ-ment. Es4s is composed of gray to brown-gray and black mudstones,shales, dolomites, and turbiditic fan sandstone interbeds which aredeposited in a semi-deep and deep lacustrine environment. Es4s de-posited in variable depth ranges from 1500m to 2800m in differentsub-depressions of the Dongying Depression having overall depthranges. More detail about the sequence stratigraphy and sedimentaryfacies of Es4s, Wangjiagang field, Dongying Depression, Bohai BayBasin can be found in Zahid (Zahid et al., 2016). In order to char-acterize and analyze the pore system of Es4s beach-bar reservoir, fivesamples, representing characteristic beach-bar reservoir, were chosento perform the SEM, MICP, low field NMR experiment, and one samplewas studied using X-ray CT scanning.

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3. Experiments

3.1. Sample collection and preprocessing

The basic information of the five experimental samples is shown inTable 1. These core samples represent a specific but to the hetero-geneous nature of the samples. SEM, MICP and NMR tests were im-plemented on all these samples. Especially, X-ray CT experiment wascarried out for S5. Regarding sample preprocessing, the remnants of thereservoir fluids and the drilling mud caused by the mud invasion shouldbe cleaned before the experiments. After cutting and polishing, samplesare reshaped into cylinders with the diameter of 2.5 cm and the lengthof 5 cm for NMR. After we conducted NMR experiments, these cylinderswere cut into two small cylinders with the length of 3 cm and 1.8 cm,respectively. These cylinders with the length 3 cm are used for MICPtests. The others were prepared for the SEM experiments. For S5, a tinycylinder with the diameter 3mm and the length of 1 cm that was ex-tracted from the small cylinder with the length of 1.8 cm was used forX-ray CT.

3.2. Experiments

3.2.1. SEM experimentSEM is a commonly used direct method to separate pore types,

observe pore morphology and calculate pore radius (Nelson, 2009).Although one 2D SEM image only reveals one cross-section pore mor-phology of a three-dimensional (3D) pore geometry, it is easy to obtain,and it can reflect plenty of information about the pore structures.Therefore, it is still a popular method to characterize the features ofpore space. The SEM experiment was carried out at State Key Labora-tory of Heavy Oil, China University of Petroleum (East China). Thesteps of SEM experiment consist of polishing, milling by high-energyargon ion beaming, coating a conductive carbon film on the samplesurface and observing using the electron microscopy. SEM images werephotographed using the Hitachi S-4800 SEM.

3.2.2. MICP experimentMICP is a popular indirect method for measuring pore and throat

size distributions (Okolo et al., 2015; Yao and Liu, 2012). The MICPexperiment in our research was conducted at China National PetroleumCorporation (CNPC) Key Well Logging Laboratory, China University ofPetroleum (East China) using AutoPore™ IV9510 (Micromeritics In-strument Corporation). The sample with a diameter of 2.5 cm was firstwashed for removing the oil and then dried at 95 °C for 48 h and sub-jected to a vacuum. Finally, liquid mercury was injected into the samplewith the pressure from 0 to 207MPa while the amount of mercury intothe rock under different pressures is recorded.

3.2.3. NMR experimentThe NMR method has been widely used in pore structure analysis

(Lai et al., 2018b; Li et al., 2015; Yao et al., 2010; Yao and Liu, 2012).The detailed principle is shown in Lai et al. (2018b). The NMR

Fig. 1. (A) Location map representing the subtectonic units of the Bohai Bay Basin; (I) Jiyang Depression, (II) Huanghua Depression, (III) Bozhong Depression, (IV)Zhezhong Depression, (V) Liaohe Depression (VI) Dongpu Depression (B) Structural map of the Dongying Depression, study area with well locations are in the dashedgreen box. (C) Cross section (Q-Q/) showing the various tectonostructural zones and key stratigraphic intervals within the Dongying Depression (Zahid et al., 2016).(D) Generalized Cenozoic-Quaternary sequence framework and stratigraphy showing tectonic and sedimentary evolution stages and the major petroleum systemelements of the Dongying Depression, Bohai, Bay Basin, East China. The yellow dashed box represents the target layer (Zahid et al., 2016). (For interpretation of thereferences to colour in this figure legend, the reader is referred to the Web version of this article.)

Table 1Information of samples and the experimental methods.

Sample ID Well Depth (m) Analyses

S1 W122 2812.2 SEM, MICP, NMRS2 W125 2798.5 SEM, MICP, NMRS3 W149 1686.1 SEM, MICP, NMRS4 W149 1689.7 SEM, MICP, NMRS5 F142 3116.2 SEM, MICP, NMR, X-ray CT

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experiment in our research was conducted at China National PetroleumCorporation (CNPC) Key Well Logging Laboratory, China University ofPetroleum (East China). First, five samples were dried for 24 h at 95 °Cto remove the fluid in the pore space and subjected to a vacuum. Then,these samples were fully saturated with brine and centrifuged to reachthe bound water status. Meanwhile, the NMR T2 relaxation time wasmeasured on brine saturated and irreducible saturated core plugs usingthe MARAN-II ultra-rock spectrometer (Oxford Instrument Incorpora-tion).

3.2.4. X-ray CT experimentWith the rapid development of X-ray CT equipment, CT has been

applied to more and more industries (Blunt et al., 2013; Bultreys et al.,2016; Mathews et al., 2017; Ren et al., 2015; Thompson et al., 2016;Wang et al., 2008). The major advantage of the X-ray CT method is thatit can perform three-dimensional, non-destructive imaging (Chengyanet al., 2018). Many petroleum geologists have used CT equipment toanalyze the pore structure inside the cores. The technique is not only

beneficial to preserve the precious core but also can be used to describethe realistic three-dimensional pore system of porous media (Chengyanet al., 2018). Due to it, the digital core analysis got rapid development.In our study, S5 was selected for X-ray CT experiment. 2020 CT to-mograms were obtained using the FEI HeliScanTM helical μ-CT imagingfacility in Canberra. During the scanning process, the X-ray source andthe detector remain stationary and the images of different angles wereobtained by rotating the turn table. The turn table were moved ac-cording to a pre-defined helical trajectory, along which a series ofradiographs were recorded uniformly around the sample at differentviewing angles (Ramandi et al., 2016a; Sheppard et al., 2014). Theresolution of the images is 2.4 μm per voxel. The digital core analysiswill be explained in the following sections.

Fig. 2. Workflow of calculating the pore size from S1 SEM image using PRCSF.

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4. Methodologies, results and discussion

4.1. Geometric and topological characteristics of pore system

4.1.1. Pore space features from SEM4.1.1.1. Methodology. For 2D images, the conventional methodology ofcalculating the pore radius is to assume the pores as the equivalentcircles and get pore size r using =r A

π(1) where A is the area of the

pore (Pengfei et al., 2018; Su et al., 2018). In fact, many pores in 2Dimages are not circular. To get a more realistic result, a novel method isproposed to compute the pore size from 2D slices which is called poreradius calculation based on shape factor (PRCSF). The detailed steps areas follows:

A) Binarize SEM images to obtain the pore space using the thresholdingmethod, perform the median filter to reduce the noises and applythe watershed algorithm to separate the connected pores and cal-culate the area and perimeter of each pore.

B) Obtain the shape factor G of the pore using =G AP2 (2), where A and

P is the area and perimeter of the pore, respectively. The shapefactor is a common parameter for characterizing the pore mor-phology (Dong and Blunt, 2009). The larger the shape factor, thesmoother the pore tends to be and the closer the pore is to circle.Furthermore, the pore shape factor is also a vital parameter forcomputing the permeability of the rock using the high-resolutionimages.

C) For the equilateral triangle, square, and circle, the shape factors are0.0481, 0.071, and 0.0796, respectively. To accurately get the hy-draulic radius of the pores, the pores are equated with circular,square and triangular pores according to G, respectively. Then thecorresponding radius calculation formulas are used to extract theradii of the inscribed circle of the equivalent pores as their poreradii. Taking the S1 sample as an example, the workflow of PRCSF isshown in Fig. 2 schematically.

4.1.1.2. Pore size distribution from SEM. The pore size distributions ofS1-S5 were computed based on the PRCSF algorithm and the results aredisplayed in Fig. 3. As can be seen, the pore size distributions of S1 andS2 are bimodal. For S1-S4, there are small proportions of small pores(less than 1 μm) in the pore system, the minimum pore radii of them are0.2 μm, which is the resolution of SEM images, and their maximum poreradii are distributed from 8 to 10 μm. Besides, all of the curves showthat the main pore radii of S5 are larger than the primary pore radii ofother samples. Moreover, the corresponding pore radii of their peaks

corresponding to S1, S2, S3, S4, and S5 are 4.05 μm, 3.29 μm, 3.02 μm,2.35 μm and 4.99 μm, respectively.

4.1.2. Pore space features from MICPMICP curves of the five samples are shown in Fig. 4. Inspecting the

results shows that more mercury gets into S5 when the pressure reaches1MPa, which indicates that there are many large pores in this sample.

4.1.2.1. Methodology. According to the classical capillary pressuretheory, pore-throat radius (called pore radius hereafter) can beobtained from the varied pressure as:

=R σcosθP S2

( ),

c Hg (3)

where σ is the interfacial tension (0.48 N/m for a mercury/air system).Pc is capillary pressure. θ is contact angle, 42.87° from the experimentmeasure and SHgis mercury saturation. Then, the probabilitydistribution function pdf of pore radius can be calculated using(Alyafei et al., 2016):

= = − = −PDF RdSdR

PdSdP

dSdlnP

.HgC

Hg

c

Hg

c (4)

Using Eq. (4), pore radius distributions of all the samples are cal-culated and the results are shown in Fig. 5. It is manifested from Fig. 5that the pore distributions of S1, S2 and S4 have similar characteristicsand pore volume fraction of their peaks are all more than 0.15 while thescores of S4 and S5 are around 0.1. The major pore radii of S1, S2 andS4 are more concentrated at 1.8–3.7 μm, 1.2–2.4 μm and 2.7–3.8 μm,respectively. Besides, the average pore radii of S1-S5 derived fromMICP are 1.68 μm, 1.36 μm, 0.28 μm, 2.18 μm, and 2.67 μm, respec-tively.

4.1.2.2. Pore size distribution from MICP. In this research, according tothe pore size distribution characteristics and fractal dimension featuresof pore space, the pores are divides into macropores, mesopores,micropores, and ultra-micropores. The fractal dimension of differentpores will be described in detail. The radii of macropores, mesopores,micropores and ultra-micropores are more than 5 μm, 1–5 μm, 0.1–1 μmand less than 0.1 μm, respectively. Porosity in each of the above groupwas computed and listed in Table 2. It can be observed that themacropores in S5 have a higher proportion than other samples. Thepercentage of mesopores for S1, S2, S4, and S5 are more than 40%,which implies many pores are mesopores.

Fig. 3. Pore size distributions of S1-S5 obtained from the SEM images using PRCSF.

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4.1.3. Pore space features from NMRThe NMR experimental results are displayed in Fig. 6. It can be

observed that there are two peaks for all the samples. The change ofincremental porosity with the relaxation time (T2) from S1 and S3 issimilar. The pore size distributions will be approximately obtained fromthese figures, which will be described below.

4.1.3.1. Methodology. Regarding low field NMR, transverse relaxationtime T2 consists of bulk relaxation time (T2B), diffusion relaxation time(T2D) and surface relaxation time (T2S):

= + +T T T T1 1 1 1

B D S2 2 2 2 (5)

T2B is usually much larger than T2, so it can be ignored. When usinga sufficiently small echo interval, one can simplify Eq. (5) as the fol-lowing (Zhao et al., 2017):

= =T T

ρ SV

1 1S2 2 (6)

where S is the pore surface area in μm2, V is the pore volume in μm3 andρ is the transversal surface relaxivity in μm/ms.

The relationship between S and V can be expressed by (Zhao et al.,

2017):

=SV

mr (7)

where r is the pore radius, m is a constant and it is a function of theshape of the pore. Thus, the relationship between the pore radius r andthe relaxation time of T2 is as follow (Lai et al., 2018b; Li et al., 2015):

=r nT2 (8)

where n is related to ρ and m, with the unit of μm/ms.One of the issues regarding the NMR technique to characterize the

pore structure of the rock is the determination of the coefficient n. Thecoefficient n is generally obtained by comparing the cumulative prob-ability curve of the T2 spectrum of the NMR experiment with the cu-mulative probability curve of the pore radius from MICP test when twocurves are tangent (Analysis et al., 2017).

The coefficient n values of the five samples are determined as shownin Fig. 7. The result is listed in Table 3. As can be clearly seen, the poreradius cumulative probability curves of S5 from NMR and MICP have abig gap representing the heterogeneity of S5.

4.1.3.2. Pore size distribution from NMR. The pore size distributions of

Fig. 4. The MICP curves of all the samples.

Fig. 5. Pore size distributions of S1-S5 from MICP tests.

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all the samples can be acquired by solving Eq. (8), which is shown inFig. 8. According to Fig. 8, it is demonstrated that the pore sizedistribution curve of S5 has only one single peak and the curves of othersamples are bimodal. For S1-S4, their main pore radii are distributedfrom 1 to 9 μm and the secondary peaks are distributed from 0.03 to0.2 μm. The curve of S5 shows that some of the pores are larger than100 μm, a possible reason for small fractures in the sample. Moreover,the corresponding pore radii of the peaks of S1-S5 are 2.81 μm,2.31 μm, 2.14 μm, 3.31 μm and 2.56 μm, respectively. In addition, theproportion of macropores, mesopores, micropores, and ultra-micropores are shown in Table 4. There are similar proportions ofdifferent kind of pores between the MICP and NMR tests.

4.1.4. Pore space features from X-ray CTIn the CT experiment, the resolution is set to 2.4 μm, so the mi-

cropores and ultra-micropores of S5 will not be covered. When it comesto the pore space analysis using X-ray CT technique, the representativeelementary volume (REV) is one of the parameters that should be de-fined ahead. In this paper, based on the porosity change of subvolumewith its volume (voxels) that is shown in Fig. 9, a sample with the sizeof 600× 600×600 voxels is chosen as the REV size. In terms of theprocessing steps of the CT image, the REV was firstly extracted from theoriginal grayscale CT images, subsequently filtered by median filtering,and segmented using an interactive threshold with the threshold value(11000, under the 16-bit unsigned greyscale) for separating solid(Fig. 10(a)) and pore space (Fig. 10(b)). Next, the closing algorithm wasperformed on the segmented images to get the smoothed pore space.Finally, the porosity of the REV is 7.67%. According to whether thepore space is connected or not, the pore space can be divided intoconnected and isolated pore space (Wu et al., 2018). Connected porespace means pore bodies are connected with other pore bodies bythroats. Subsequently, the connected pore space was acquired by op-erating the connected algorithm. Isolated pore space was also obtainedby subtracting the connected pore space from the total pore space,Fig. 10(c). Isolated pore space occupies 3.16% of the total volume and41.20% of the volume of pore space, which indicates there is the badconnectivity for the pore space of S5. The reason is that macropores andmesopores may be connected by the micropores and ultra-micropores,but the tiny pores are not contained in the pore space from CT test.Besides, the centroid path tortuosity is introduced to characterize thecomplexity of the pore space. It is defined as the ratio between actualpath length of pore space along a direction and the distance of twocentroids of the curved pore space. The greater the tortuosity value, themore complicated the pore structure is (Wu et al., 2018). The centroidpath tortuosity of S5 is 3.28, which reveals that the existing pore systemin S5 is complicated. After the pore space was separated, the pores andthroats were distinguished using the maximal ball algorithm (Dong andBlunt, 2009). Moreover, the pore network model of the S5 sample wasextracted, Fig. 10(d).

In addition, the single-phase fluid flow in pore space was simulatedbased on the Navier-Stokes and Darcy's Law. Navier-Stokes equation isexpressed as (Tahmasebi, 2018a; Zhang and Tahmasebi, 2018):

⎧⎨⎩

∇ − ∇ =∇ =

μ V PV

0. 0

2

(9)

where μ is the fluid viscosity, v represents the velocity of the fluid and prepresents the pressure. The inlet pressure is 100,000 Pa and the outletpressure is 80,000 Pa. The fluid viscosity μ is set as 0.001 Pa.S. Finally,the effective permeability was calculated to be 16.89×10−3 μm2.

The coordination number and specific Euler number (SEN) are im-portant parameters for characterizing the topological characteristics.These parameters can describe the connectivity of a specific componentin the porous medium (Jiang, 2008). The coordination number dis-tribution was computed using the maximal ball algorithm (Dong andBlunt, 2009) and the results are shown in Fig. 11(a). From the co-ordination number distribution, it can be obtained that 18.02% poresare isolated while 79.17% pores are connected by 1–5 throats. Themaximum of the coordination number is 21, which confirms that thereare many small throats around the pore. With regard to SEN, it was firstintroduced by Vogel and Roth (2001). SEN can overcome the weak-nesses of the conventional Euler number (Jiang, 2008). SEN is definedby:

=χ χV

,V (10)

where χ is the conventional Euler number of a porous medium withvolume V. SEN makes it possible for comparing the connectivity amongrock images of different volume (Jiang, 2008). More information aboutSEN can be found in Vogel and Roth (2001) and Jiang (2008). The SENcurve of S5 is shown in Fig. 11(b), which illustrates that the con-nectivity of pore space increases as the minimum (>5.0 μm) diameterof pores decreases. In other words, the pore connectivity increaseswhen the smaller pores are added. Furthermore, two-point correlationfunction (TPCF) was used to describe the probability of finding twovoxels in the 3D models at different positions both in the pore space(Huang et al., 2015; Tahmasebi, 2018b, 2017). The definition of TPCFis:

+ = +hS x x I x I x h( , ) ( ) ( ), (11)

where x and x + h are two voxels in the 3D porous media. h is themagnitude of the vector. I x( )is an indicator function. + =I x h( ) 1whenin the void space and + =I x h( ) 0otherwise. +I x I x h( ) ( )representsthe average of the multiplication of two indicator functions (Wu et al.,2018). From Fig. 11(c), it can be clearly seen that the correlation of twovoxels decreases with the increase of their distance. The TPCF curvelevels off at 0.0076 when the distance of two voxels is more than150 μm.

In terms of pore geometry system of S5, distributions of the pore andthroat size, pore-throat aspect ratio, pore, and throat shape factor werecomputed using the maximal ball algorithm (Dong and Blunt, 2009).These distributions are demonstrated in Fig. 11(d-h). The pore andthroat shape factor have been defined in Section 4.1.1. The aspect ratioindicates the ratio of the pore radius to the linked average value of allthe throat radii (Wu et al., 2018). As for the pore size distribution, mostpore radii (64.18%) are distributed from 8 to 18 μm. Relating to thethroat radius distribution, about 73.82% throat radii are constitutedbetween 4 and 12 μm. The pore and throat shape factor distributions

Table 2Porosity of different pores from MICP tests.

Sample ID Porosity Macropores Mesopores Micropores Ultra-micropores

Proportion Porosity Proportion Porosity Proportion Porosity Proportion Porosity

S1 12.00 0.07 0.84 0.46 5.52 0.26 3.12 0.21 2.52S2 14.6 0.02 0.29 0.41 5.99 0.35 5.11 0.22 3.21S3 11.4 0.04 0.46 0.10 1.14 0.60 6.84 0.22 2.51S4 15.5 0.05 0.78 0.47 7.29 0.28 4.34 0.20 3.10S5 13.6 0.20 2.72 0.47 6.39 0.24 3.26 0.09 1.22

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imply that the throats are smoother than the pores because the shapefactors of the throats are overall larger than them of the pores. Oncemore, the most pore and throat shape factors are less than 0.071, whichreveals that the pores and throats in the cross sections are not circular,which also proves our statement in Section 4.1.1. When it comes topore-throat aspect ratio distribution, 92.86% aspect ratio can be found

between 2 and 6, and the largest aspect ratio is 44.09.

4.1.5. Comparison of pore size distributions: SEM, MICP, NMR, and X-rayCT

In order to better summarize the advantages and disadvantages ofthe SEM, MICP, NMR, and X-ray CT experiments in characterizing pore

Fig. 6. NMR relaxation time (T2) distributions of saturated and irreducible saturated five samples from the study area with their total porosity (phi in %) from NMR.Figures (a), (b), (c), (d) and (e) are from S1, S2, S3, S4 and S5, respectively.

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systems, pore size distributions obtained from these experiments arecompared in Fig. 12. It is evident that NMR and MICP can cover morepore sizes than SEM because the experimental sample size (2.5 cm) of

NMR and MICP is much larger than the utilized sample in SEM (about2mm). More accurate aperture distributions can be obtained if a largerSEM is available. Apart from this, more features of pores and throatscan be attained through SEM images. For example, one can find themicropores and ultra-micropores in clay minerals, measure the poresize and the morphological characteristics. In addition, NMR can obtainmore information about smaller pores (less than 0.004 μm) than MICP,because it is easier for water to enter tiny pores of water-wet rocks thanmercury. In fact, mercury in MICP test was the non-wetting phase so itcannot go into these very tiny pores even under the maximal pressure.

Fig. 7. The cumulative probability curves of the T2 relaxation time and NMR. When the cumulative probability curve of the T2 relaxation time is tangent with one ofMICP, the coefficient n will be determined. Figures (a), (b), (c), (d) and (e) are from S1, S2, S3, S4 and S5, respectively.

Table 3n values of the five samples.

Sample ID S1 S2 S3 S4 S5

n (μm/ms) 0.027 0.0225 0.0318 0.050 1.895

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Therefore, the porosity of the same sample from NMR is more than thatfrom MICP. Besides, the pore radii measured by MICP tend to be smallerthan NMR and SEM. In fact, the reason is that Pc and SHgmeasured byMICP refer to the throat radius and the volume of the pores connectedby the throat, respectively. In other words, the radius obtained from Eq.(3) is the throat radius, not the pore size. The volume of the injectedmercury in the MICP test is the total volume of the throat and the pores.This is pore shielding in the MICP test (Li et al., 2015). The effect ofpore shielding leads to uncertainty in pore size distribution from MICP,which generally results in underestimating the aperture distributions.

But overall, the pore radius distributions measured by the threemethods are similar, especially SEM and NMR. As to NMR, it is a goodtechnique to measure aperture distributions. However, the coefficientbetween transverse relaxation time (T2) and pore/throat radius (R)varies for different samples. It is cumbersome to get the coefficient ofevery sample. As such, finding a good solution to calculate is necessary.In terms of X-ray CT technique for S5, compared the former threemethods, it can measure more pore structure parameters, including thegeometry and topology properties such as connectivity, coordinationnumber, and pore and throat shape factor, but also can be used for

Fig. 8. Pore size distributions of S1-S5 from NMR experiment.

Table 4The porosity of four kinds of pores from NMR experiment.

Sample ID Porosity Macropores Mesopores Micropores Ultra-micropores.

Proportion Porosity Proportion Porosity Proportion Porosity Proportion Porosity

S1 12.25 0.06 0.74 0.51 6.25 0.21 2.57 0.22 2.70S2 14.82 0.02 0.30 0.42 6.23 0.36 5.34 0.20 2.96S3 12.09 0.01 0.12 0.52 6.29 0.23 2.78 0.24 2.90S4 15.38 0.07 1.08 0.54 8.30 0.20 3.08 0.19 2.92S5 14.03 0.32 4.49 0.49 6.87 0.19 2.66 0.00 0.00

Fig. 9. Porosity change of subvolume with the volume (voxels).

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simulating single-phase and multi-phase flow in the pore system. Theseadvantages of X-ray CT cannot be replaced by other methods. However,the imaging size of the X-ray CT technique is controlled by its resolu-tion, which results in ignoring micro-pores features smaller than theresolution and missing the ones larger than the size of the image. Thus,for rocks with high heterogeneity or fractures, it is difficult to capturethe full pore structure details because the sample size of CT scanning isgenerally small. Therefore, it is necessary to model a multi-scale digitalrock which should contain micropores and macropores. One can in-tegrate several digital rocks of different resolutions into one large-scaledigital rock (Tahmasebi, 2018c).

4.2. Fractal dimension analysis of pore structure

4.2.1. MethodologyFor pores with fractal features, fractal dimensions are used to de-

scribe the regularity and fragmentation of the pore space. The re-lationship between fractal dimension Df and pore radius r of the porespace is as follows (Ge et al., 2016; Li et al., 2017):

∝ −N r r( ) ,Df (12)

where N r( )is the number of pores of which pore radius is more than r.Differentiating Eq. (12), one can obtain the pore radius distributionfunction f r( ),

= ∝ − −f r dN Rdr

r( ) ( ) .D 1f(13)

Assuming that the pore volume of a pore v and its pore radius can beexpressed as

∝v r .3 (14)

Thus, the pore accumulative volume of the pore space where thepore radius is more than r can be obtained by:

∫ ∫= = = −− − −( )V r f r vdr ar dr b r r(¿ ) ( ) .r

r

r

rD max

D D2 3 3max max

f f f

(15)

where rmaxis the maximum pore radius, a and b are constants and theyare related to the shape of the pores. When =r rmin, the whole porevolume can be gotten by:

> = −− −( )V r b r r( ) .min maxD

minD3 3f f

(16)

Combining Eqs. (3) and (4), one can obtain the pore volume cu-mulative frequency >F r( )

> = >>

= −−

− −

− −F r V rV r

r rr r

( ) ( )( )

( )( )

.min

maxD D

maxD

minD

3 3

3 3

f f

f f (17)

Owing to ≪r rmin max , Eq. (6) can be simplified as

⎜ ⎟> = − = − ⎛⎝

⎞⎠

− −

−

−

F r r rr

rr

( ) ( )( )

1 .maxD D

maxD

max

D3 3

3

3f f

f

f

(18)

Thus, pore volume accumulation frequency F(< r) with pore radiusless than r

⎜ ⎟< = − > = ⎛⎝

⎞⎠

−

F r F r rr

( ) 1 ( ) .max

D3 f

(19)

4.2.2. Fractal dimension from MICPIn the MICP experiment, the pore volume cumulative frequency

with pore radius less than r can be written as:

Fig. 10. Digital rock analysis of S5 (a) Solid space, (b) All pore space, (c) Isolated pore space, (d) Pore network model.

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< = −F r S( ) 1 .Hg (20)

Therefore, the following formula is reasonable,

− = − − −log S D logP D logP(1 ) ( 3) ( 3) .Hg f C f min (21)

One can determine Df based on the slope of the double logarithmicplot with −log(1 S )Hg and logPC. When we computed the Df of thesesamples, we noticed that the pores with different sizes have fractalcharacteristics. Thus, we divided the pores into four groups based on

Fig. 11. Topology and geometry features of pore space of S5 from X-ray CT. (a) Coordination number distribution, (b) Specific Euler number, (c) Two-pointcorrelation function (d) Pore radius distribution, (e) Throat radius distribution, (f) Aspect ratio distribution, (g) Pore shape factor distribution, and (h) Throat shapefactor distribution.

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the fractal dimension and the pore size distributions, as mentionedearlier in Section 4.1.2. The fractal dimensions of macropores (Df 1),mesopores (Df 2), micropores (Df 3) and ultra-micropores (Df 4) are becomputed based on Eq. (21). The fractal dimension Df of the entirepores will be obtained using the weighted average of Df1, Df2 Df3, an-d Df4 as following:

=× + × + × + ×

DD φ D φ D φ D φ

φ.f

f f f f1 1 2 2 3 3 4 4

(22)

In Eq. (22), φ1, φ2, φ3 φ4, and φare the porosity of the macropores,mesopores, micropores, ultra-micropores and all the pores,

respectively. Taking the S1 an example, one can get Df1, Df2 Df3, and Df4

from Fig. 13. The detailed parameters about the fractal dimension of allthe samples are listed in Table 5. It can be clearly seen that the Df1, Df2Df3, and Df4 of the same sample gradually decrease, which indicatesthat the large pores are more irregular, fragmentized and hetero-geneous because Df1is the largest for the same sample. Overall, thesurface of the pore space of all of the samples is rough as the fractaldimensions Df are more than 2.4.

4.2.3. Fractal dimension from NMRRegarding the NMR test, the following equation can be derived from

Fig. 12. Comparison on pore size distributions of S1-S5 from SEM, MICP and NMR.

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Eqs. (8) and (19):

< = − − −log F r D logT D logT( ( )) (3 ) (3 ) .f f max2 2 (23)

Eq. (23) is similar to Eq. (21), so one can calculate Df for NMR usinga similar method described in Section 4.2.1. The schematic diagram ofthe fractal dimensions of four kinds of pores in S1 are shown in Fig. 14.The fractal dimensions of all the samples from NMR tests are presentedin Table 6. As can be seen, the fractal dimension of macropores is

relatively large. But, for S1-S4 samples, the fractal dimensions of mi-cropores are greater than the fractal dimensions of mesopores. Fur-thermore, the fractal dimension of the entire pores for the same samplefrom NMR experiments is approximate to, but smaller than that fromthe MICP tests, which may be for the reason that more ultra-microporescan be captured by NMR and these pores smooth the surface of pothe resystems.

Fig. 13. Schematic of determining the fractal dimension of S1 from the MICP test.

Table 5Fractal dimension parameters of all the samples from the MICP test.

Sample ID Df Macropores Mesopores Micropores Ultra-micropores.

Df1 φ1 Df2 φ2 Df3 φ3 Df4 φ4

S1 2.60 2.99 0.84 2.56 5.52 2.61 3.12 2.54 2.60S2 2.61 2.99 0.29 2.89 5.99 2.40 5.11 2.40 2.61S3 2.68 2.99 0.46 2.66 1.14 2.84 6.84 2.72 2.68S4 2.68 2.97 0.78 2.61 7.29 2.81 4.34 2.58 3.10S5 2.62 2.97 2.72 2.51 6.39 2.56 3.26 2.62 2.62

Fig. 14. Schematic of determining the fractal dimension of S1 from NMR experiment.

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4.2.4. Fractal dimension from X-ray CTAs mentioned, the pores from X-ray CT cannot cover all types of

pores. Therefore, only the fractal dimensions of the 3D image and 2Dimages of S5 are computed in the part. The fractal dimension of the 3Dimage of S5 is analyzed to describe the roughness of the pore surfacewhile Df of 2D images is calculated to understand the irregularity ofpore boundaries in each layer. The less smooth the surface is, the biggerthe fractal dimension (Cai et al., 2015; Ge et al., 2016). The fractaldimension of 3D pore systems is 2.45, which reveals the surface of porespace is not smooth. The fractal dimensions of 2D images demonstratedin Fig. 15 are more than 1.3. Moreover, the porosity of each layerFig. 15 was also calculated to show the heterogeneity of the pore space.The pore space of S5 is heterogeneous because the porosity of each slicefluctuates significantly. Interestingly, the fractal dimension and por-osity of each layer have similar variation trend, which can be due to thefact that the high porous layer with large pores tend to has high fractaldimension.

In summary for Section 4.2, the fractal dimensions of all types ofpores can be calculated using MICP and NMR techniques. Comparingthe fractal dimensions obtained by MICP and NMR, one can discoverthat the fractal dimension from MICP is larger. As to the X-ray CTmethod, the fractal dimensions of the 2D and 3D pore systems both areobtained. Moreover, the fractal dimension of the pores and the porosityof each layer in the 2D images show similar variation trend.

5. Conclusion

The geometrical, topological and fractal characteristics of the porestructure of Es4s reservoir from the Shahejie formation in theWangjiagang field, Dongying depression, Bohai Bay Basin, China wereanalyzed based on SEM, MICP, NMR, and X-ray CT experiments.

Besides, a novel method of calculating pore size based on pore shapefactor from 2D images was put forward. According to pore size dis-tributions measured through these techniques, the advantages anddisadvantages of SEM, MICP, NMR, and X-ray CT were summarized.NMR and MICP can cover more pores (ultra-micropores, micropores,and macropores) in the rock than SEM and X-ray CT. Nevertheless, thepore size obtained from MICP tends to be smaller than NMR and X-rayCT because the pore shielding exists in the MICP test. In addition, NMRis an effective method to characterize the pore structure. Unfortunately,the coefficient between pore radius and T2 relaxation time is difficult tocompute, so it is necessary to find a better method to calculate it.Moreover, X-ray CT is a promising technique that can acquire a largenumber of geometry and topology properties of the pore system and thedigital rock model can also be used to perform the single-phase ormulti-phase flow simulation. Whereas, for low-permeability and tightrocks, micro-CT is not enough to fully characterize the pore systemsbecause sub-resolution micropores in the cores cannot be captured. Infact, those micropores have a very crucial influence on the transportproperties of the porous media. Therefore, it is indispensable to modelthe multiscale digital core that should contain large and small pores forcomplex rocks, which will be significant to analyze pore systems andsimulate multi-phase flow in porous media. Furthermore, character-ization of the pore systems should also be linked to mineralogicalanalysis for better analyzing the effects of mineral wettability on thetransport of hydrocarbons in such media.

On the other hand, the fractal dimension of pore space was calcu-lated based on MICP, NMR, and X-ray CT tests. MICP and NMR tech-niques can be used to analyze the fractal dimension all types of pores,from ultra-micropores to macropores. The macropores tend to be moreirregular and heterogeneous than ultra-micropores and micropores. Thefractal dimension from MICP is larger than one from NMR for the

Table 6Parameters about the fractal dimension of all the samples from NMR experiment.

Sample ID Df Macropores Mesopores Micropores Ultra-micropores

Df1 φ1 Df2 φ2 Df3 φ3 Df4 φ4

S1 2.47 2.90 0.74 2.49 6.25 2.75 2.57 2.03 2.70S2 2.50 2.96 0.30 2.62 6.23 2.61 5.34 2.01 2.96S3 2.44 2.98 0.12 2.47 6.29 2.77 2.78 2.02 2.90S4 2.43 2.81 1.08 2.42 8.30 2.73 3.08 2.01 2.92S5 2.50 2.92 4.49 2.41 6.87 2.05 2.66 0.00 0.00

Fig. 15. Change of fractal dimension and porosity of each layer from X-ray CT.

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sample. Besides, the fractal dimensions of the 2D and 3D pore space canbe obtained from the X-ray CT data. However, the fractal dimension ofmicroporosity and ultra-micropores cannot be obtained by this methodbecause its resolution is low. It can be proved that the fractal dimensionof pore pace from 2D CT images changes with the area of pore space, sothe fractal theory is an efficient tool for classifying pore types and as-sessing reservoir quality.

Acknowledgments

This work was funded by the Graduate School Innovation Programof China University of Petroleum (18 C×06024 A) and TechnologyMajor Project, P.R. China (2016ZX05054012, 2017ZX05009001). Thefirst author would like to acknowledge Muhammad Jawad Munawarfrom University of the Punjab for his help in preparation of themanuscript and the China Scholarship Council (CSC) for its financialsupport for his living expenses at the University of Wyoming as a vis-iting Ph.D. student.

Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.marpetgeo.2019.02.003.

References

Alyafei, N., Mckay, T.J., Solling, T.I., 2016. Characterization of petrophysical propertiesusing pore-network and lattice-Boltzmann modelling: Choice of method and imagesub-volume size. J. Petrol. Sci. Eng. 145, 256–265. https://doi.org/10.1016/j.petrol.2016.05.021.

Analysis, A.L., Xiao, S.U.N., Xian, Z., Cheng, L.I.N., Method, Q.E., Pore, H., Based, D.,Analysis, M., 2017. 2 2665803 1–7.

Blunt, M.J., Bijeljic, B., Dong, H., Gharbi, O., Iglauer, S., Mostaghimi, P., Paluszny, A.,Pentland, C., 2013. Pore-scale imaging and modelling. Adv. Water Resour. 51,197–216. https://doi.org/10.1016/j.advwatres.2012.03.003.

Bultreys, T., De Boever, W., Cnudde, V., 2016. Imaging and image-based fluid transportmodeling at the pore scale in geological materials: A practical introduction to thecurrent state-of-the-art. Earth Sci. Rev. 155, 93–128. https://doi.org/10.1016/j.earscirev.2016.02.001.

Cai, J., Luo, L., Ye, R., Zeng, X., Hu, X., 2015. Recent Advances on Fractal Modeling ofPermeability for Fibrous Porous Media. Fractals 23, 1540006. https://doi.org/10.1142/S0218348X1540006X.

Cai, J., Yu, B., 2011. A Discussion of the Effect of Tortuosity on the Capillary Imbibition inPorous Media. Porous Media, Transp. https://doi.org/10.1007/s11242-011-9767-0.

Chalmers, G.R., Bustin, R.M., Power, I.M., 2012. Characterization of gas shale pore sys-tems by porosimetry, pycnometry,surfacearea,andfield emission scanning electronmicroscopy/transmission electron microscopy image analyses: Examples from theBarnett, Woodford, Haynesville, Marcellus,andDoig units. Am. Assoc. Petrol. Geol.Bull. 96, 1099–1119. https://doi.org/10.1306/10171111052.

Chengyan, L., Yuqi, W., Lihua, R., Yang, W., Weichao, Y., Xiaolong, S., Xianguo, Z., Yimin,Z., 2018. Review of digital core modeling methods. Prog. Geophys. 33, 0679–0689.

Clarkson, C.R., Marc Bustin, R., 1996. Variation in micropore capacity and size dis-tribution with composition in bituminous coal of the Western Canadian SedimentaryBasin: Implications for coalbed methane potential. Fuel 75, 1483–1498. https://doi.org/10.1016/0016-2361(96)00142-1.

Clarkson, C.R., Solano, N., Bustin, R.M., Bustin, A.M.M., Chalmers, G.R.L., He, L.,Melnichenko, Y.B., Radliński, A.P., Blach, T.P., 2013. Pore structure characterizationof North American shale gas reservoirs using USANS/SANS, gas adsorption, andmercury intrusion. Fuel 103, 606–616. https://doi.org/10.1016/j.fuel.2012.06.119.

Daigle, H., Dugan, B., 2011. Capillary controls on methane hydrate distribution andfracturing in advective systems. Geochem. Geophys. Geosyst. 12. https://doi.org/10.1029/2010GC003392.

Daigle, H., Hayman, N.W., Jiang, H., Tian, X., Jiang, C., 2017. Multiscale pore networksand their effect on deformation and transport property alteration associated withhydraulic fracturing. Energy Procedia 125, 71–79. https://doi.org/10.1016/J.EGYPRO.2017.08.067.

Daigle, H., Johnson, A., Thomas, B., 2014. Determining fractal dimension from nuclearmagnetic resonance data in rocks with internal magnetic field gradients. Geophysics.https://doi.org/10.1190/Geo2014-0325.1.

Dong, H., Blunt, M.J., 2009. Pore-network extraction from micro-computerized-tomo-graphy images. Phys. Rev. E 80, 036307. https://doi.org/10.1103/PhysRevE.80.036307.

Fagbemi, S., Tahmasebi, P., Piri, M., 2018b. Interaction Between Fluid and Porous Mediawith Complex Geometries: A Direct Pore-Scale Study. Water Resour. Res. https://doi.org/10.1029/2017WR022242.

Fagbemi, S., Tahmasebi, P., Piri, M., 2018a. Pore-scale modeling of multiphase flowthrough porous media under triaxial stress. Adv. Water Resour. 122, 206–216.https://doi.org/10.1016/J.ADVWATRES.2018.10.018.

Favvas, E.P., Sapalidis, A.A., Stefanopoulos, K.L., Romanos, G.E., Kanellopoulos, N.K.,Kargiotis, E.K., Mitropoulos, A.C., 2009. Characterization of carbonate rocks bycombination of scattering, porosimetry and permeability techniques. MicroporousMesoporous Mater. 120, 109–114. https://doi.org/10.1016/j.micromeso.2008.09.015.

Fink, R., Amann-Hildenbrand, A., Bertier, P., Littke, R., 2018. Pore structure, gas storageand matrix transport characteristics of lacustrine Newark shale. Mar. Petrol. Geol. 97,525–539. https://doi.org/10.1016/J.MARPETGEO.2018.06.035.

Gao, Z., Yang, X., Hu, C., Wei, L., Jiang, Z., Yang, S., Fan, Y., Xue, Z., Yu, H., 2019.Characterizing the pore structure of low permeability Eocene Liushagang Formationreservoir rocks from Beibuwan Basin in northern South China Sea. Mar. Petrol. Geol.99, 107–121. https://doi.org/10.1016/J.MARPETGEO.2018.10.005.

Ge, X., Fan, Y., Deng, S., Han, Y., Liu, J., 2016. An improvement of the fractal theory andits application in pore structure evaluation and permeability estimation. J. Geophys.Res. Solid Earth 121, 6333–6345. https://doi.org/10.1002/2016JB013074.

Ge, X., Fan, Y., Li, J., Aleem Zahid, M., 2015. Pore structure characterization and clas-sification using multifractal theory-an application in santanghu basin of westernchina. J. Petrol. Sci. Eng. 127, 297–304. https://doi.org/10.1016/j.petrol.2015.01.004.

Hemes, S., Desbois, G., Urai, J.L., Schröppel, B., Schwarz, J.O., 2015. Multi-scale char-acterization of porosity in Boom Clay (HADES-level, Mol, Belgium) using a combi-nation of X-ray μ-CT, 2D BIB-SEM and FIB-SEM tomography. MicroporousMesoporous Mater. 208, 1–20. https://doi.org/10.1016/j.micromeso.2015.01.022.

Henares, S., Caracciolo, L., Cultrone, G., Fernández, J., Viseras, C., 2014. The role ofdiagenesis and depositional facies on pore system evolution in a Triassic outcropanalogue (SE Spain). Mar. Petrol. Geol. 51, 136–151. https://doi.org/10.1016/J.MARPETGEO.2013.12.004.

Hu, Q., Brusseau, M.L., 1994. The effect of solute size on diffusive-dispersive transport inporous media. J. Hydrol 158, 305–317. https://doi.org/10.1016/0022-1694(94)90059-0.

Hu, Q., Ewing, R.P., Dultz, S., 2012. Low pore connectivity in natural rock. J. Contam.Hydrol. 133, 76–83. https://doi.org/10.1016/J.JCONHYD.2012.03.006.

Hu, Q., Zhang, Y., Meng, X., Li, Z., Xie, Z., Li, M., 2017. Characterization of micro-nanopore networks in shale oil reservoirs of Paleogene Shahejie Formation in DongyingSag of Bohai Bay Basin, East China. Petrol. Explor. Dev. 44, 720–730. https://doi.org/10.1016/S1876-3804(17)30083-6.

Huang, Z.-Q., Winterfeld, P.H., Xiong, Y., Wu, Y.-S., Yao, J., 2015. Parallel simulation offully-coupled thermal-hydro-mechanical processes in CO2 leakage through fluid-driven fracture zones. Int. J. Greenh. Gas Control 34, 39–51. https://doi.org/10.1016/J.IJGGC.2014.12.012.

Jiang, Z., 2008. Quantitative Characterisation of the Geometry and Topology of PoreSpace in 3d Rock Images.

Kibria, M.G., Hu, Q., Liu, H., Zhang, Y., Kang, J., 2018. Pore structure, wettability, andspontaneous imbibition of Woodford Shale, Permian Basin, West Texas. Mar. Petrol.Geol. 91, 735–748. https://doi.org/10.1016/J.MARPETGEO.2018.02.001.

Lai, J., Wang, G., 2015. Fractal analysis of tight gas sandstones using high-pressuremercury intrusion techniques. J. Nat. Gas Sci. Eng. 24, 185–196. https://doi.org/10.1016/j.jngse.2015.03.027.

Lai, J., Wang, G., Cao, J., Xiao, C., Wang, S., Pang, X., Dai, Q., He, Z., Fan, X., Yang, L.,Qin, Z., 2018a. Investigation of pore structure and petrophysical property in tightsandstones. Mar. Petrol. Geol. 91, 179–189. https://doi.org/10.1016/j.marpetgeo.2017.12.024.

Lai, J., Wang, G., Wang, Z., Chen, J., Pang, X., Wang, S., Zhou, Z., He, Z., Qin, Z., Fan, X.,2018b. A review on pore structure characterization in tight sandstones. Earth Sci.Rev. 177, 436–457. https://doi.org/10.1016/j.earscirev.2017.12.003.

Letham, E.A., Bustin, R.M., 2018. Quantitative validation of pore structure character-ization using gas slippage measurements by comparison with predictions from bundleof capillaries models. Mar. Petrol. Geol. 91, 363–372. https://doi.org/10.1016/J.MARPETGEO.2018.01.014.

Li, J., Yin, J., Zhang, Y., Lu, S., Wang, W., Li, J., Chen, F., Meng, Y., 2015. A comparison ofexperimental methods for describing shale pore features - A case study in the BohaiBay Basin of eastern China. Int. J. Coal Geol. 152, 39–49. https://doi.org/10.1016/j.coal.2015.10.009.

Li, K., 2010. Analytical derivation of Brooks-Corey type capillary pressure models usingfractal geometry and evaluation of rock heterogeneity. J. Petrol. Sci. Eng. 73, 20–26.https://doi.org/10.1016/j.petrol.2010.05.002.

Li, P., Zheng, M., Bi, H., Wu, S., Wang, X., 2017. Pore throat structure and fractal char-acteristics of tight oil sandstone: A case study in the Ordos Basin, China. J. Petrol. Sci.Eng. 149, 665–674. https://doi.org/10.1016/j.petrol.2016.11.015.

Li, Z., Wu, S., Xia, D., He, S., Zhang, X., 2018. An investigation into pore structure andpetrophysical property in tight sandstones: A case of the Yanchang Formation in thesouthern Ordos Basin, China. Mar. Petrol. Geol. 97, 390–406. https://doi.org/10.1016/J.MARPETGEO.2018.07.014.

Liu, H., Zheng, J., Wang, K., You, Z., 2017. Comments on “Comments on ‘Reply tocomments on “Analytical derivation of Brooks–Corey type capillary pressure modelsusing fractal geometry and evaluation of rock heterogeneity. J. Petrol. Sci. Eng. 159,614–616. https://doi.org/10.1016/j.petrol.2017.09.050.

Mandelbrot, B.B., Wheeler, J.A., 1983. The Fractal Geometry of Nature. Am. J. Phys. 51,286–287. https://doi.org/10.1119/1.13295.

Mathews, J.P., Campbell, Q.P., Xu, H., Halleck, P., 2017. A review of the application of X-ray computed tomography to the study of coal. Fuel 209, 10–24. https://doi.org/10.1016/J.FUEL.2017.07.079.

Minsky, M., 1988. Memoir on inventing the confocal scanning microscope. Scanning.https://doi.org/10.1002/sca.4950100403.

Nelson, P.H., 2011. Pore-throat sizes in sandstones, siltstones, and shales: Reply. Am.Assoc. Petrol. Geol. Bull. https://doi.org/10.1306/12141010159.

Y. Wu et al. Marine and Petroleum Geology 103 (2019) 12–28

27

Nelson, P.H., 2009. Pore-throat sizes in sandstones, tight sandstones, and shales. Am.Assoc. Petrol. Geol. Bull. 93, 329–340. https://doi.org/10.1306/10240808059.

Norbisrath, J.H., Eberli, G.P., Laurich, B., Desbois, G., Weger, R.J., Urai, J.L., 2015.Electrical and fluid flow properties of carbonate microporosity types from multiscaledigital image analysis and mercury injection. Am. Assoc. Petrol. Geol. Bull. 99,2077–2098. https://doi.org/10.1306/07061514205.

Okolo, G.N., Everson, R.C., Neomagus, H.W.J.P., Roberts, M.J., Sakurovs, R., 2015.Comparing the porosity and surface areas of coal as measured by gas adsorption,mercury intrusion and SAXS techniques. Fuel 141, 293–304. https://doi.org/10.1016/j.fuel.2014.10.046.

Peng, S., Hu, Q., Dultz, S., Zhang, M., 2012. Using X-ray computed tomography in porestructure characterization for a Berea sandstone: Resolution effect. J. Hydrol472–473, 254–261. https://doi.org/10.1016/j.jhydrol.2012.09.034.

Pengfei, Z., Shuangfang, L.U., Junqian, L.I., Haitao, X.U.E., Wenbiao, L.I., Siyuan, W.,2018. Quantitative characterization of microscopic pore structure for shales usingscanning electron microscopy.

Radlinski, A.P., Mastalerz, M., 2017. Neutron scattering study of vitrinite: Insights intosub-micrometer inclusions in North American Carboniferous coals of bituminousrank. Int. J. Coal Geol. 186, 145–154. https://doi.org/10.1016/j.coal.2017.10.012.

Ramandi, H.L., Armstrong, R.T., Mostaghimi, P., 2016a. Micro-CT image calibration toimprove fracture aperture measurement. Case Stud. Nondestruct. Test. Eval. 6, 4–13.https://doi.org/10.1016/j.csndt.2016.03.001.

Ramandi, H.L., Liu, M., Tadbiri, S., Mostaghimi, P., 2018. Impact of dissolution of syn-genetic and epigenetic minerals on coal permeability. Chem. Geol. 486, 31–39.https://doi.org/10.1016/J.CHEMGEO.2018.03.015.

Ramandi, H.L., Mostaghimi, P., Armstrong, R.T., Saadatfar, M., Pinczewski, W.V., 2016b.Porosity and permeability characterization of coal: a micro-computed tomographystudy. Int. J. Coal Geol. 154–155, 57–68. https://doi.org/10.1016/j.coal.2015.10.001.

Ren, W., Yang, Z., Sharma, R., Zhang, C., Withers, P.J., 2015. Two-dimensional X-ray CTimage based meso-scale fracture modelling of concrete. Eng. Fract. Mech. 133,24–39. https://doi.org/10.1016/J.ENGFRACMECH.2014.10.016.

Romero, E., Simms, P.H., 2008. Microstructure investigation in unsaturated soils: A re-view with special attention to contribution of mercury intrusion porosimetry andenvironmental scanning electron microscopy. Geotech. Geol. Eng. 26, 705–727.https://doi.org/10.1007/s10706-008-9204-5.

Sarkar, P., Kumar, A., Singh, K.H., Ghosh, R., Singh, T.N., 2018. Pore system, micro-structure and porosity characterization of Gondwana shale of Eastern India usinglaboratory experiment and watershed image segmentation algorithm. Mar. Petrol.Geol. 94, 246–260. https://doi.org/10.1016/J.MARPETGEO.2018.04.006.

Serrano, S., Perán, F., Jiménez-Hornero, F.J., Gutiérrez de Ravé, E., 2013. Multifractalanalysis application to the characterization of fatty infiltration in Iberian and Whitepork sirloins. Meat Sci. 93, 723–732. https://doi.org/10.1016/j.meatsci.2012.11.015.

Shao, X., Pang, X., Li, Q., Wang, P., Chen, D., Shen, W., Zhao, Z., 2017. Pore structure andfractal characteristics of organic-rich shales: A case study of the lower SilurianLongmaxi shales in the Sichuan Basin, SW China. Mar. Petrol. Geol. 80, 192–202.https://doi.org/10.1016/J.MARPETGEO.2016.11.025.

Sheppard, A., Latham, S., Middleton, J., Kingston, A., Myers, G., Varslot, T., Fogden, A.,Sawkins, T., Cruikshank, R., Saadatfar, M., Francois, N., Arns, C., Senden, T., 2014.Techniques in helical scanning, dynamic imaging and image segmentation for im-proved quantitative analysis with X-ray micro-CT. Nucl. Instrum. Methods Phys. Res.Sect. B Beam Interact. Mater. Atoms 324, 49–56. https://doi.org/10.1016/j.nimb.2013.08.072.

Su, Y., Zha, M., Ding, X., Qu, J., Wang, X., Yang, C., Iglauer, S., 2018. Pore type and poresize distribution of tight reservoirs in the Permian Lucaogou Formation of the JimsarSag, Junggar Basin, NW China. Mar. Petrol. Geol. 89, 761–774. https://doi.org/10.1016/j.marpetgeo.2017.11.014.

Sun, M., Yu, B., Hu, Q., Yang, R., Zhang, Y., Li, B., Melnichenko, Y.B., Cheng, G., 2018.Pore structure characterization of organic-rich Niutitang shale from China: Smallangle neutron scattering (SANS) study. Int. J. Coal Geol. 186, 115–125. https://doi.org/10.1016/j.coal.2017.12.006.

Tahmasebi, P., 2018a. Packing of discrete and irregular particles. Comput. Geotech. 100,52–61. https://doi.org/10.1016/J.COMPGEO.2018.03.011.

Tahmasebi, P., 2018b. Accurate modeling and evaluation of microstructures in complexmaterials. Phys. Rev. E 97, 023307. https://doi.org/10.1103/PhysRevE.97.023307.

Tahmasebi, P., 2018c. Nanoscale and multiresolution models for shale samples. Fuel 217,

218–225. https://doi.org/10.1016/j.fuel.2017.12.107.Tahmasebi, P., 2017. HYPPS: A hybrid geostatistical modeling algorithm for subsurface

modeling. Water Resour. Res. 53, 5980–5997. https://doi.org/10.1002/2017WR021078.

Tahmasebi, P., Javadpour, F., Sahimi, M., 2017. Data mining and machine learning foridentifying sweet spots in shale reservoirs. Expert Syst. Appl. 88, 435–447.

Tahmasebi, P., Javadpour, F., Sahimi, M., 2015. Multiscale and multiresolution modelingof shales and their flow and morphological properties. Sci. Rep. 5, 16373. https://doi.org/10.1038/srep16373.

Tahmasebi, P., Kamrava, S., 2018. Rapid multiscale modeling of flow in porous media.Phys. Rev. E 98, 052901. https://doi.org/10.1103/PhysRevE.98.052901.

Tan, Y., Pan, Z., Liu, J., Feng, X.-T., Connell, L.D., 2018. Laboratory study of proppant onshale fracture permeability and compressibility. Fuel 222, 83–97. https://doi.org/10.1016/J.FUEL.2018.02.141.

Thompson, A., Maskery, I., Leach, R.K., 2016. X-ray computed tomography for additivemanufacturing: A review. Meas. Sci. Technol. https://doi.org/10.1088/0957-0233/27/7/072001.

Van Geet, M., Swennen, R., David, P., 2001. Quantitative coal characterisation by meansof microfocus X-ray computer tomography, colour image analysis and back-scatteredscanning electron microscopy. Int. J. Coal Geol. 46, 11–25. https://doi.org/10.1016/S0166-5162(01)00006-4.

Vogel, H.J., Roth, K., 2001. Quantitative morphology and network representation of soilpure structure, vol. 24. pp. 233–242.

Wang, G., Yu, H., De Man, B., 2008. An outlook on x-ray CT research and development.Med. Phys. https://doi.org/10.1118/1.2836950.

White, S.H., Shaw, H.F., Huggett, J.M., 1984. The use of back-scattered electron imagingfor the petrographic study of sandstones and shales. J. Sediment. Res. https://doi.org/10.1306/212F844F-2B24-11D7-8648000102C1865D.

Wu, Y., Lin, C., Ren, L., Yan, W., An, S., Chen, B., Wang, Y., Zhang, X., You, C., Zhang, Y.,2018. Reconstruction of 3D porous media using multiple-point statistics based on a3D training image. J. Nat. Gas Sci. Eng. 51, 129–140. https://doi.org/10.1016/j.jngse.2017.12.032.

Xiao, D., Lu, S., Yang, J., Zhang, L., Li, B., 2017. Classifying Multiscale Pores andInvestigating Their Relationship with Porosity and Permeability in Tight SandstoneGas Reservoirs. Energy Fuel. 31, 9188–9200. https://doi.org/10.1021/acs.energyfuels.7b01487.

Xiong, Q., Baychev, T.G., Jivkov, A.P., 2016. Review of pore network modelling of porousmedia: Experimental characterisations, network constructions and applications toreactive transport. J. Contam. Hydrol. 192, 101–117. https://doi.org/10.1016/J.JCONHYD.2016.07.002.

Yao, Y., Liu, D., 2012. Comparison of low-field NMR and mercury intrusion porosimetryin characterizing pore size distributions of coals. Fuel 95, 152–158. https://doi.org/10.1016/j.fuel.2011.12.039.

Yao, Y., Liu, D., Che, Y., Tang, D., Tang, S., Huang, W., 2010. Petrophysical character-ization of coals by low-field nuclear magnetic resonance (NMR). Fuel 89, 1371–1380.https://doi.org/10.1016/j.fuel.2009.11.005.

Zahid, M.A., Chunmei, D., Lin, C., Gluyas, J., Jones, S., Zhang, X., Munawar, M.J., Ma, C.,2016. Sequence stratigraphy, sedimentary facies and reservoir quality of Es4s,southern slope of Dongying Depression, Bohai Bay Basin, East China. Mar. Petrol.Geol. 77, 448–470. https://doi.org/10.1016/j.marpetgeo.2016.06.026.

Zhang, S., Canter, L., Sonnenfeld, M., 2017. Capillary fluid dynamics within unconven-tional rocks investigated by scanning electron microscopy. Am. Assoc. Petrol. Geol.Bull. 101, 1759–1765. https://doi.org/10.1306/01111716095.

Zhang, X., Tahmasebi, P., 2018. Micromechanical evaluation of rock and fluid interac-tions. Int. J. Greenh. Gas Control 76, 266–277.

Zhang, Y., He, Z., Jiang, S., Lu, S., Xiao, D., Chen, G., Li, Y., 2019. Fracture types in thelower Cambrian shale and their effect on shale gas accumulation, Upper Yangtze.Mar. Petrol. Geol. 99, 282–291. https://doi.org/10.1016/J.MARPETGEO.2018.10.030.

Zhang, Y., Xu, X., Lebedev, M., Sarmadivaleh, M., Barifcani, A., Iglauer, S., 2016. Multi-scale x-ray computed tomography analysis of coal microstructure and permeabilitychanges as a function of effective stress. Int. J. Coal Geol. 165, 149–156. https://doi.org/10.1016/j.coal.2016.08.016.

Zhao, P., Wang, Z., Sun, Z., Cai, J., Wang, L., 2017. Investigation on the pore structureand multifractal characteristics of tight oil reservoirs using NMR measurements:Permian Lucaogou Formation in Jimusaer Sag, Junggar Basin. Mar. Petrol. Geol. 86,1067–1081. https://doi.org/10.1016/j.marpetgeo.2017.07.011.

Y. Wu et al. Marine and Petroleum Geology 103 (2019) 12–28

28