On-chip porous media: Porosity and permeability measurements

Chemical Engineering Science 99 (2013) 274–283

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On-chip porous media: Porosity and permeability measurements Jerry Joseph, Naga Siva Kumar Gunda, Sushanta K. Mitra n Micro and Nano-scale Transport Laboratory, Department of Mechanical Engineering, University of Alberta, Edmonton, Canada T6G 2G8

H I G H L I G H T S

G R A P H I C A L

A B S T R A C T


Conducted experiments to calculate the effective porosity and permeability of on-chip porous media.
Effective porosity is determined by using image analysis technique.
Permeability is estimated by measuring the pressure drop across the pore network and applying Darcy's law.
The flow resistance in the networks is found to decrease with the increase in Darcy number.
Quantification of porosity and permeability would help in future studies of pore-scale transport in reservoir engineering.

art ic l e i nf o

a b s t r a c t

Article history: Received 5 March 2013 Received in revised form 25 May 2013 Accepted 26 May 2013 Available online 7 June 2013

In porous media, accurate determination of petrophysical properties such as effective porosity and permeability are important to understand the porosity–permeability relationships that help in improving the oil extraction mechanisms. Measuring porosity and permeability for on-chip porous media is difficult due to their small length scales. Hence in the present work, we conduct experiments to calculate the effective porosity and permeability of on-chip porous media containing different pore-networks. Four different types of on-chip porous media varying in number of pore bodies and pore throats are fabricated in this work. Porosity for each on-chip porous media is determined by using image analysis technique. Absolute permeability is estimated by measuring the pressure drop across the pore network and applying Darcy's law. The pore sizes of the networks range from 40 μm to 70 μm and the porosity of the on-chip porous media varies from 0.39 to 0.67. The flow resistance in the networks, measured by the quantity ΔP=Q , is found to decrease with the increase in Darcy qffiffiffiffiffiffiffiffiffiffiffi 2 number ( K=h ). The permeability values range between 2.6670.06 and 15.9370.55 Darcy

Keywords: Porosity Permeability On-chip porous media Microfluidics

(2:625  10−12 7 0:059  10−12 and 15:72  10−12 7 0:542  10−12 m2 ). It is found that as the number of pores and throats in on-chip porous media increases, the porosity increases as expected resulting in approximately one order of permeability increase. The results of this study would help in understanding the porosity and the permeability values in on-chip porous media that are miniaturized versions of oil reservoirs, which can be effective in understanding pore scale porosity–permeability relationships. & 2013 Elsevier Ltd. All rights reserved.

1. Introduction It is believed that the key driver to displace oil from the reservoir relies on the fact that the oil–water–gas co-exists in the pore-space of

n

Corresponding author. Tel.: +1 780 492 5017; fax: +1 780 492 2200. E-mail address: [email protected] (S.K. Mitra).

0009-2509/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ces.2013.05.065

the reservoir and one needs to adopt suitable mechanisms to mobilize the oil from the pore space (Kaiser, 2009; Blunt et al., 2002; Jaber et al., 1999; Jamaloei and Kharrat, 2010; Blunt, 2001; Gunde et al., 2010). This brings to the focus of the present work, where a naturally occurring oil reservoir rock, i.e., porous medium, represented on a chip that can be exploited to provide the oil and gas industry with a better tool to understand the pore-scale displacement process relevant to a given geological formation.

J. Joseph et al. / Chemical Engineering Science 99 (2013) 274–283

It is a well known fact that the studies of transport mechanisms (Colombani et al., 2002; Talmon et al., 2010; Ellis and Bazylak, 2012; Ma et al., 2012; Miller and Fogler, 1995) at pore scale have been attempted because of their importance in many engineering applications in engineering. Such studies are highly relevant especially in the energy field since majority of the heavy/light oil is found in carbonate and sandstone formations which consist of solid matrix and pore space. Therefore, the researchers in the past have tried to create micro-models (Karadimitriou and Hassanizadeh, 2012) which consist of simple and regular geometric features, fractal patterns and irregular patterns with characteristic length-scale comparable with the average pore diameter, quite different than the pore geometry of a natural porous media (Jamaloei and Kharrat, 2010; Er et al., 2010; Wu et al., 2012a). However, recent microscopy techniques have made possible to characterize the pore space and the pore connectivity of such reservoir rocks (Rigby et al., 2002; Lindquist et al., 2000; Zhou et al., 2011; Spanne et al., 1994; Bera et al., 2012, 2011; Sok et al., 2002). In parallel, great advancement of micro/ nanofabrication techniques has revolutionized the fabrication of micro-models for energy applications (Berejnov et al., 2008; Fadaei et al., 2011; Bowden et al., 2006). Building on these two advancements, Gunda et al. (2011a) fabricated the Reservoir-on-aChip (ROC), where for the first time the pore network of a naturally occurring oil reservoir rock was replicated on a silicon substrate covered with glass. They conducted oil recovery experiments by water flooding technique and were able to comprehensibly understand the displacement process of oil by water within the pore network. This concept of ROC has now been adopted in a recent paper by Karadimitriou et al. (2012), where they fabricated a similar pore network on a glass substrate. As ROC is becoming a popular tool to characterize a given reservoir, it is imperative that properties like porosity and permeability need to be calculated for such systems. Measurement of effective properties in such microsystems has been difficult in the past due to challenges associated in the measurement techniques at such small length scales. Hence, in this paper, we have elaborated the technique of calculating relevant reservoir properties for on-chip porous medium, often coined as ROC (Gunda et al., 2011a),which can be adopted for other types of micro-models of different geological formations. The fabricated on-chip porous media used in the present work has resemblance with our previous work (Gunda et al., 2011a). Typically, porosity and permeability are measured in a laboratory scale using core-flooding experimental systems. Porosity can be assessed through volumetric measurements of core samples, petrographic image analysis (PIA), or often geological logs. Mercury injection methods (Rigby et al., 2002) and fluid re-saturation method are often used to measure the pore volume of the porous samples. Other advanced and sophisticated techniques like X-ray tomography (Bera et al., 2011; Dong and Blunt, 2009; Okabe and Blunt, 2007; Rigby et al., 2006; Gunde et al., 2010), scanning electron microscopy (Clelland and Fens, 1991; Mattiello et al., 1997; Gunda et al., 2011b), Brunauer–Emmett–Teller (BET) for gas adsorption (Schull, 1948; Gan et al., 1972) and Nuclear Magnetic Resonance (Kenyon, 1992; Timur, 1969) of small sample sizes to estimate the pore distribution and surface area of porous samples. Often these techniques are expensive and are difficult to adopt for pore-scale micro-models. The permeability estimation models in literature can be briefly classified into three classes: (a) Darcy's law (Whitaker, 1986; Vafai and Tien, 1981; Klinkenberg, 1941), (b) Non-linear Darcy's models, e.g. Darcy–Forchheimer equation (Pant et al., 2012; Nield, 1991; Alazmi and Vafai, 2001; Beckermann et al., 1986), and (c) Klinkenberg effect/Knudsen slip models, e.g. modified binary friction model (Pant et al., 2013; Kerkhof, 1996; Carnes and Djilali, 2006). However, the current work deals with liquids in micro-pores

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and therefore the Knudsen slip effects are negligible and hence the binary friction model is not applicable. The Forchheimer effect accounts for the non-liner effects of velocity in pressure drop measurements whereas Darcy's law accounts for linear effects. From physical perspective, Darcy's law accounts for viscous drag whereas the Forchheimer term accounts for the inertial effects of velocity on the pressure drop. Also, there has been an emphasis in extracting pore network information from micro-CT images of sandstone (Al-Kharusi and Blunt, 2007) and carbonate (Okabe and Blunt, 2007; Bera et al., 2012) and then using numerical tools (Prodanovic et al., 2007; Bakke and Oren, 1997; Hazlett, 1995; Singh and Mohanty, 2003; Humby et al., 2002; Jaganathan et al., 2008; Gunda and Mitra, 2012) to calculate porosity and permeability of such extracted networks (Arns et al., 2001; Gervais et al., 2012; Singh and Mohanty, 2000; Wu et al., 2003). However, such technique is limited to the numerical reconstruction method and it is not feasible to visualize the pore-scale displacement process. On the other hand, on-chip porous media gives a tremendous flexibility in observing in situ pore-scale displacement processes and helps in characterizing the porosity–permeability relationship at the pore scale. Thus to further extend the scope of ROC with application to reservoir characterization, effective porosity and permeability are measured for four different pore network structures fabricated on silicon substrates using dry etching. Fabrication procedure for producing such intricate pore network structure has been provided here, which will allow others to replicate the fabrication process relevant to any given extracted pore network. The complete microfluidic chip is fabricated with borofloat glass as covering layer for silicon substrate with proper inlet and outlet for fluidic connections. We characterize the single phase flow properties associated with this on-chip porous media consisting of different pore networks. Porosity is determined by processing the optical images when it is flooded by the dyed fluid. Permeability is calculated by measuring the pressure drop across the chip for different flow rates of deionized (DI) water injected into the chip. The methodology developed for calculating pressure drops in the present work has been adopted from the work reported by Gunda et al. (2013), where they have characterized the structured porous medium (microchannel with integrated micro-pillars). The use of wetting fluid is important to achieve the single phase flow without having any trapped air for porosity and permeability measurements. The present device is made of silicon-glass material and shows water-wet characteristics. The methods for measuring the porosity and the permeability, developed in the present work, can be implemented in other types of micro-models made from different materials such as glass and/or PDMS. PDMS micro-models are easier to make, disposable and less expensive than glass micro-models, and they are widely used in the microfluidic community including porous media micromodels (Bhattacharya et al., 2005; Berejnov et al., 2008; Schneider et al., 2010; Ma et al., 2011; Zhao et al., 2012). The main problem in PDMS micro-models is the wettability, which is not an issue in glass micro-models. PDMS micro-models require some treatment like oxygen–plasma or UV–ozone to convert the surface of PDMS into water-wet (Bhattacharya et al., 2005; Ma et al., 2011). In addition to wettability problem, PDMS has several disadvantages of swelling and sagging when the liquid flows for a longer times or at higher flow rates (Bhattacharya et al., 2005; Berejnov et al., 2008; Schneider et al., 2010; Ma et al., 2011; Zhao et al., 2012). The novelty of the present work lies in conducting experiments to show a detailed analysis of permeability measurement technique for on-chip porous media and to check the accuracy of the porosity measurements using image analysis and saturation methods, which

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is different compared to the material balance approach used in other published works (Gunda et al., 2011a; Jeong and Yavuz Corapcioglu, 2003; Wu et al., 2012b; Zhang et al., 2011). Other highlights of the present work are: (a) Determination of effective porosity using image analysis technique and (b) Estimation of the permeability by measuring the pressure drop across the pore network and applying Darcy's law. We justify the novelty of the present work by putting forward two important facts. First, the increasing popularity of on-chip porous medium or ROC has made it imperative that the flow properties of such pore networks be calculated and documented. Second, calculation of such properties would help in future studies related to the classic problem of porescale to reservoir scale modeling. As it is well known, the properties of any porous media are usually associated with three different length scales: pore scale, macroscopic or lab scale and field scale. An experimental determination of flow properties at the pore scale, in a realistic pore network, would help in documenting pore scale flow properties and can be used for such simulations or for theoretical work. At the same time, fabrication and knowledge of four different porous media geometries and their properties would help other researchers in performing future works related to enhanced oil recovery and multi-phase fluid flow phenomena in similar micro-models. This present paper starts with a brief description of the technique employed for fabricating the four on-chip porous media. This is followed by a description of the experimental procedures for determination of effective porosity and absolute permeability. In the next section, results and discussion for the values of porosity and permeability obtained for different chips are presented. Further characterization of the chip in terms of flow resistance for different Darcy numbers is also presented here.

2. Experimental investigation 2.1. Pore network design Four different pore networks are designed based on the typical sandstone microstructural information. Using Delaunay Triangulation routine (MATLAB, Mathworks Inc., Natick, MA, USA), a pore network of prescribed mean pore size is created for each chip. Mean pore size of these four networks varies from 40 μm to 70 μm. Network 1 and 2 contain mean pore size of 40 μm, Network 3 contain mean pore size of 70 μm and Network 4 containing mean pore size of 50 μm. The aspect ratio (ratio of pore radius to the linked throat radius) for these networks varies from 1.1 to 6. Coordination number (number of connections of a pore) for these networks varies from 3 to 10, based on total number of pores in the network. The pore and throat sizes are based on beta and log-normal distribution to ensure that the size of the throats in the network are different, which is not the case when only log-normal distribution is used. A combination of beta (Keefer and Bodily, 1983) and log-normal distribution helps in creating varied throat sizes that help in obtaining wider range of entrance pressure in the throats during experiments. Fig. 1 shows one of the on-chip porous media considered in this work. As observed here, each chip consists of a pore network, inlet–outlet regions and, entrance and

exit regions. The length and width of the pore network part is 35 mm and 5 mm, respectively. The entrance and exit regions are rectangular in shape and are of 4.6 mm and 3 mm in length, respectively. The inlet/outlet region is circular in shape and 10 mm in diameter. Table 1 provides the number of pores and throats for different chips fabricated in the current work and Fig. 2 shows the nature of the pores and throats, connected with each other through the solid matrices in four separate networks.

2.2. Fabrication This section provides a brief description of the fabrication procedure used for on-chip porous media. Detailed information on the fabrication process can be found elsewhere (Gunda et al., 2011a). A 4″ silicon substrate (Silicon Valley Microelectronics Inc., Santa Clara, CA, USA) is used for etching the pore networks. The silicon substrate is initially cleaned in a piranha solution (H2 SO4 and H2 O2 in a volumetric ratio of 3:1) for 30 min and then dried. The cleaned substrates were then placed in HMDS Oven (Yield Engineering System/YES, Livermore, US) for 17 min to coat a thin layer of hexamethyldisilizane (HMDS) for proper adhesion of positive photo-resist (PPR) on silicon. A 2:5 μm thick layer of PPR HPR 506 (Fuji-film Electronic Materials Inc., Mesa, Arizona, USA) is spin coated on the substrate with the following settings; Spread cycle: 10 s at 500 rpm and Spin cycle: 40 s at 4000 rpm using Solitec headway spinner (WT Services, Inc., Fitchburg, MA, USA). Next the substrate was soft baked for 90 s at 115 1C using the Solitec hotplate (WT Services, Inc., Fitchburg, MA, USA). Then the substrates are kept in open atmosphere for approximately 15 min, in order to ensure re-hydration of PPR for proper exposure and developing of PPR during lithography. This is followed by exposing PPR to UV light (350 mJ=cm2 energy) for 4 s and the development of structures in Developer 354 for 35 s. The silicon substrate is then etched for about 40 μm using inductively coupled plasma reactive ion etching (ICPRIE). The PPR is then stripped off using acetone and then properly cleaned using oxygen plasma treatment (Branson Barrel etcher). A complete chip is fabricated by closing the open networks in silicon using a glass covering layer. Each chip has one inlet and outlet port on its covering layer, located centrally with respect to the circular inlet and outlet regions. The volume of the inlet–outlet regions along with that of the inlet/outlet ports is larger than the volume of the network, and hence, with a suitably chosen volume flow rate, one would achieve laminar flow in the entire pore network. Borofloat glass (Borofloat 33, Schott North America, Inc., Louisville, KY, USA) is selected as the covering material, and holes are drilled using abrasive water-jet machining process. Both the bottom layer with fabricated pore network and the covering top layer are piranha-cleaned again in similar composition as before (H2 SO4 and H2 O2 in a volumetric ratio of 3:1). Anodic bonding of the top layer of glass (with inlet and outlet holes) is performed in SUSS Bonder (CB6L, SUSS Microtec, Garching, Germany) with the bottom layer of silicon containing the pore network. Fig. 3 contains an exploded view of the chip placed in a microfluidic casing with connecting tubes and O-rings. Table 1 Quantification of on-chip porous media in terms of number of pores and throats.

Fig. 1. The on-chip porous medium with pore network, entrance and exit regions and circular inlet/outlet regions.

Network type

Number of pores

Number of throats

Network Network Network Network

2000 3000 3000 6000

6000 9000 9000 20 000

1 2 3 4

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Fig. 2. Different configuration of on-chip porous media: (a) Network 1. (b) Network 2. (c) Network 3, and (d) Network 4.

Fig. 3. Schematic of the experimental set-up us for porosity and permeability measurements.

Scanning Electron Microscope (ZEISS, Germany) has been used to characterize the on-chip porous media, before the bonding step is performed. The pore and throat size at various location within the chip are measured to determine the precision of the network. An SEM image of the fabricated network (Network 3) is shown in Fig. 4(a). The pores and throats in the image (three dimensional) are represented by circular cylinder and rectangular box shapes, respectively. The fabricated network contains complex connectivity of pores and throats (as shown in Fig. 4(a)) which closely resembles the reservoir characteristics, as observed by Bera et al. (2011, 2012) and others (Lindquist et al., 2000). Fig. 4(b) shows another SEM image which is the magnified image of Network 4.

It is evident from the figure that precise vertical wall profiles have been achieved in the fabricated pore networks. Both SEM images were taken with secondary electron detector at 20 kV and 50 μA beam current. Fig. 4(a) was taken at magnification of 100  with a view field of ∼1720 μm  1262 μm whereas Fig. 4(b) was taken at magnification of 200  with a view field of ∼842 μm  618 μm. The entire network has been observed at lower magnification and approximately 10–12 locations were magnified to see the precision of the network at higher magnification. To further characterize the etched networks, parameters like the mean depth, width and surface roughness of these networks are measured at several cross-sections (entrance and exit regions,

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Fig. 4. (a) SEM image of Network 3. (b) Enlarged view of pores and throats for Network 4.

inlet–outlet regions and within the network) using surface profilometer (Ambios XP 300, Ambios Technology Inc, Santa Cruz, CA, USA). Table 2 compares the average values of these parameters for the fabricated chips. Fig. 5 shows an example of the surface-profile reading for Network 2. As observed from the figures, the depth of the network remains constant throughout the chip i.e., ∼42 μm. It is to be noted that the stylus of the profilometer is not able to capture the depth information at the throats whose width is less than 2:5–4 μm. Hence, the average depth of the etched network measured at other locations was verified from direct measurements of the SEM images.

2.3. Experimental setup for porosity and permeability The experimental setup, as shown in Fig. 3, consists of a custom made microscope for taking optical images of the chip when it is flooded with dyed fluid for the purpose of porosity calculation. Custom made microscope set-up contains modular zoom lens system with 1  adapter tube, Zoom 6000 body tube, C-mount couplers, accessory optics, motorized configurations, collimated light source, coaxial illumination fiber optics, infinity corrected objective lenses, 2=3″ CMOS camera (5 M pixel CMOS image sensor), computer and data acquisition system. This system is configured to provide high magnification, high resolution and large view field images. It provides high contrast image through the dynamic magnification range of 0.09–393  , covers a field of view which ranges from 0.01 to 125.68 mm, the working distance can be varied from 34 to 390 mm, and the edge flatness and clarity are obtained using infinity corrected objective lenses (Navitar, 2013). It is to be noted that the entire network needs to be filled with dyed fluid before any kind of measurements are taken and the microscope helps in visualization of the filling of the pore network. We used commercially available white dye Keyfluor White OB (915-426-51, Keystone Aniline Corporation, Chicago, IL, USA). The chemical composition of dye is Benzoxazole. It is water soluble liquid dye. It shows water wettability characteristic with silicon and glass. Since our chip is fabricated with silicon and glass, the fluid used in the present work shows the water-wet characteristics. Because of this water-wet property of the present device, wetting fluid such as water can achieve single phase flow without the presence of trapped air for porosity and permeability measurements. The chip is placed inside a solid casing which contains inlet and outlet ports. The inlet of the chip is connected to a syringe pump (Harvard Apparatus, MA) which controls the flow of liquid. For measuring the inlet pressure, pressure transducer (0–2.5 PSI, 0–5 PSI and 0–15 PSI gauge, Omega Engineering, Inc., Laval, Quebec, Canada) is used at the inlet port. The pressure is recorded by using a data acquisition device (DAQ) (720 mA/710 V, 24-Bit Analog Input, National Instruments, Austin, TX, USA).

Table 2 Surface profile properties (width, average roughness and average depth) of fabricated chips. Network

Width (mm)

Average roughness (μm)

Average depth (μm)

1 2 3 4

4.89 4.95 4.96 4.96

4.54 6.17 5.24 6.67

40.44 40.76 41.44 41.10

The chip is firmly placed inside the solid casing and care is taken that no leakage occurs during the liquid flow by placing O-rings at the interface between inlet and outlet ports of the casing and the chip. Teflon tubing is selected to connect the syringe pump, the pressure transducer and the chip. The electric terminals of the pressure transducer are connected to the DAQ device, which is then connected to a computer. DI Water is injected by the syringe pump at different flow rates to conduct the desired experiments. Porosity is a ratio of the pore volume over the total volume. In designed images, the porosity is calculated by dividing the number of pixels which belongs to the pore space with the total number of pixels of a given image. The porosity of the fabricated chip was determined using the image analysis technique. Authors in the past have mentioned about the difficulty in measuring porosity by using experimental material balance procedures in micro-models (Buchgraber et al., 2012). Image analysis is a simple and less time consuming method for calculating porosity compared to the material balance calculations. The images of each chip were obtained along the length of its network using the microscope after flooding it with a white colored dye. For clarity and demonstration purpose, a part of the whole image of Network 3 is shown in Fig. 6. The image analysis technique mainly consists of three steps: normalization, segmentation or thresholding, and filtering. The obtained images of the present chip are first converted to 8 bit images from 16 bit, which render them as grey scale images, by importing them into the image processing software FIJI (Open Source image processing package based on ImageJ). This is followed by the normalization process for improving the resolution and contrast of the 8 bit images, which includes changing the range of pixel intensity values. A normalized image is shown in Fig. 6(b). These normalized images are then segmented for clear distinction between different phases. The different phases are distinguished and identified based on the intensity of its pixels by adjusting the threshold of the intensity of the pixels. For the images obtained in this experiment, the threshold value changes based on the individual image. As an example, an image after thresholding of its pixel intensity has been provided in Fig. 6(c) for Network 3, where the threshold value is 49. The pixels that have a value below 49 are kept as black and ones that were above 49 are kept as white. The black color in Fig. 6(c) represents the solid

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It is important to consider different pressure drops that occur during the flow in the experimental setup described here. The total pressure drop is given by Akbari et al. (2009) ΔP total ¼ ΔP C;I þ ΔP C;O þ ΔP D þ ΔP FD þ ΔP minor þ ΔP ev

ð2Þ

where ΔP C;I is the pressure loss in the inlet tube between the pressure transducer and the inlet port of the chip, ΔP C;O is the pressure drop in the outlet tube connected to the outlet port of the chip, ΔP D comprises of the pressure drop in the entrance region where the flow is still not fully developed and in the exit region right after the pore network, ΔP FD is the pressure drop in the fully developed region of the pore network, ΔP minor is the pressure drop due to the 901 bends at the inlet and the outlet ports and due to the sudden expansion and contraction at the inlet and outlet ports respectively, and ΔP ev is the pressure drop due to electroviscous effect. The pressure loss in the inlet and outlet tubes, ΔP C;I and ΔP C;O respectively, have been calculated using the Hagen–Poiseuille equation for pressure drops in a tube and have been included in the net pressure drop while calculating the permeability values. ΔP D has been calculated for the entrance and exit regions which are rectangular in shape with the same width and height but different length as that of the chip. ΔP minor can be ignored as they are calculated to be very less (of the order of 10−3 Pa) and ΔP ev can be ignored for rectangular chip used in the present work. Details for calculating such pressure drops can be found elsewhere (Akbari et al., 2009; Gunda et al., 2013). Due to the variation in flow rates and the time taken to attain steady state flow in each case, errors and uncertainties can occur during the pressure drop measurement and hence in the permeability calculation. Pressure drop measurements for each chip are repeated three times and the flow rates are varied between 30 μl=min and 200 μl=min. The readings were taken once the temporal fluctuations in the pressure transducer voltage readings were stabilized. The final value of the pressure drops for each chip is the mean of these three measurements.

3. Results and discussion

Fig. 5. Surface profile measurements for Network 2: (a) At the entrance. (b) Inside the network.

matrix and the white color refers to the pore space. The images are then smoothened by applying median filter to reduce any noise, as shown in Fig. 6(d). A detailed description of the image processing method can be found elsewhere (Gunda et al., 2011b). Once the noise in the image is reduced, it is then imported to MATLAB and is converted to a binary image, which contains the information of black and white pixels as 0 s and 1 s, respectively. Porosity is calculated by counting the number of pixels that represent the pore space and dividing it by the total number of pixels. The absolute permeability is determined by injecting deionized water through the chip at different flow rates and measuring the corresponding pressure drop across the pore network length. Once the pressure drop is recorded, the permeability is calculated by using Darcy's law which is given by (Dullien, 1992; Gostick et al., 2006) K¼

Q μL A  ΔP FD

ð1Þ

where K is the permeability, Q is the flow rate of water, μ is the viscosity of water, L is the length of the porous network, A is the flow cross-sectional area which is the product of the width (W) and height (h) of the ROC and ΔP FD is the pressure difference across the pore network.

The porosity measurement data for the different chips are provided in Table 3. The four different networks that we created represent porous media over a wide range of porosity. Apart from calculating porosity from image analysis presented in the earlier section, the porosity of the networks were also calculated using the design images (AUTOCAD drawing file), one of which is shown in Fig. 1. This was done to distinguish between the true porosity of the network and the effective porosity after etching. The fabricated networks may contain isolated or dead end pores (due to some inaccuracies of the fabrication process) and may result into a reduced value of porosity as compared to the true porosity. The design images were imported to MATLAB and the porosity was calculated by counting the number of pixels that belonged to the pore space and dividing it by the total number of pixels. A comparison of the porosity obtained from image analysis with the porosity of the design images is provided in Table 3. There is a slight variation between the two porosity values. In general, porosity varies between the least dense network (Network 1: design porosity – 0.42, chip porosity – 0.39) to the most dense network (Network 4: design porosity – 0.69, chip porosity – 0.67). The deviations could occur due to the difference in thresholding limits that have been applied to the optical images of the networks. To determine the effect of thresholding on the values of porosity, the images were processed for porosity calculation within 75% of the base thresholding limit. Table 3 provides the variation in the effective porosity values, obtained by image analysis, due to the change in thresholding limits applied.

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Fig. 6. A part of the image of Network 3 is shown here for the purpose of explaining the porosity calculation procedure; (a) 8 bit optical image. (b) Normalized image with better resolution and contrast. (c) Image after thresholding; and (d) Thresholded image after filtering.

Table 3 Porosity and permeability values of different networks. Network type

Design Ćporosity

Porosity from Ćoptical images

Permeability Ć(Darcy)

Permeability

Network 1 Network 2 Network 3 Network 4

0.42

0.39 70.04

2.667 0.06

2.625 70.059

0.44

0.42 70.04

5.50 70.40

5.428 70.395

0.66

0.65 70.05

10.88 7 1.03

10.7387 1.025

0.69

0.677 0.03

15.93 7 0.55

15.7237 0.543

Ćð10−12 m2 Þ

For example shown in Fig. 6(c) for Network 3, if one selects the threshold value as 39 instead of 49, the estimated porosity increases by approximately 5.7%. If one selects the threshold value as 59 instead of 49, the estimated porosity decreases by approximately 4.3%. This change in porosity happens due to the improper selection of threshold value where most part of the image is assigned to the wrong phase. For Network 3, an optimal threshold value is 49. More details on the effect of threshold value on measuring porosity of other types of porous media can be found elsewhere (Gunda et al., 2011b). Fig. 7 presents the variation of the pressure drop with flow rate for four different chips. It is observed that the pressure drop increases with the flow rate. It is also found that the, Network 1 has a higher pressure drop value for different flow rates. As the porosity increases, the slope of the pressure gradient decreases thus signifying less resistance to the flow. Network 4, with maximum porosity, has the least pressure drop values.

Permeability values for the four different chips are shown in Table 3. The permeability varies between 2:66 7 0:06 and 15:93 7 0:55 Darcy (2:625  10−12 7 0:059  10−12 to 15:72  10−12 7 0:542  10−12 m2 ) and the error estimates have been provided in Table 3. As expected, the Network 1 has the least permeability due to lowest porosity. Network 4 has the highest permeability as it contains the maximum number of pores and throats. Though Network 1 and Network 2 contain the same average pore size, permeability in Network 2 is two times that of Network 1 due to the increase in number of pores and throats. The same trend is seen in permeability change from Network 2 to Network 3 where the number of pores and throats remain the same but the average pore size increases from 40 μm to 70 μm. The increase in permeability from Network 3 to Network 4 is not as much as seen in the previous trends even though the number of throats and pores has increased with a reduction in the average pore size. Overall, with increase in porosity values from 0.39 to 0.67, the permeability increase is approximately of one order of magnitude. As per authors knowledge, there is no theoretical model available to calculate the permeability value for such unstructured porous media, which is presented here. The existing correlations, like Kozeny–Carman, etc. have little relevance in terms of the complex on-chip porous media studied here (Cornell and Katz, 1953; Carman and Carman, 1956; Alzaydi, 1975; Geertsma, 1974; Carrier, 2003; Xu and Yu, 2008; ValdesParada et al., 2009). Hence we have not compared the permeability results presented here with theoretical models. But the methodology to calculate the permeability is well studied for structured porous medium such as the recent work by Gunda et al. (2013), where they have shown that even for structured microfluidic porous system (microchannels with integrated pillars

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Fig. 8. Variation of the flow resistance with different flow rates for each fabricated network.

Fig. 7. Variation of the pressure drop with flow rates for different networks.

forming a structured micro-scale porous media), the experimental permeability values deviate quite significantly with respect to some of the most sophisticated permeability–porosity correlations developed till date. Fig. 8 represents the variation of flow resistance, ΔP=LμU, for different flow rates. Here μ is the viscosity of DI water, which is 1  10−3 Pa s, U is the velocity which is obtained by dividing the flow rate Q with the cross-sectional area A of the chip and L is the length of the network, which is 35 mm. It is evident from the figure that Network 1 offers the maximum resistance to flow due to the least porosity and Network 4 offers the least resistance. Fig. 9 shows another indication of the resistance to flow, given by the ratio of pressure drop to the flow rate value (ΔP=Q ), with qffiffiffiffiffiffiffiffiffiffiffi 2 varying Darcy number ( K=h ). Here K is the permeability of the chip, which has been obtained experimentally and h is the height of the etched network. As seen in Fig. 9, the flow resistance decreases as the Darcy number increases. Here the flow resistance is maximum for the network with the least porosity (Network 1) and is minimum for the network with the maximum porosity (Network 4). The flow resistance gradually decreases as the porosity increases in the network. In case of inertial flow, one may need to consider the modified version of Eq. (1), which is often referred as, Darcy–Forchheimer equation, which can be written as (Gurau et al., 2007) ΔP FD μ ρ ¼ v þ v2 K Ki L

ð3Þ

Fig. 9. Variation of pressure drop per unit flow with changes in Darcy number for different networks.

where, K is the classical Darcy or viscous permeability, KI is the inertial permeability resulting from the non-linear effect, ρ is the density of fluid and v ¼Q/A is the velocity of fluid flow. The flow rates used in the present study are relatively low (∼30 μl=min to 200 μl=min, resulting in very low velocities), which results in laminar flow within the network and hence non-linear effects of velocity, accounted through Forchheimer effect, may not play any significant role. However, for the sake of completeness, the experimental data has been fitted to Eq. (3). The pressure gradient vs velocity profile for Network 1 is shown in Fig. 10 using water as the working fluid. It is observed that the Darcy–Forchheimer equation is in good agreement with the experimental data. For Network 1, the viscous or Darcy permeability is 2.92 70.5 Darcy (2:88  10−12 70:494  10−12 m2 ) and the inertial permeability is 0:143 7 0:29ð10−6 Þ m. There is a slight difference in permeability calculated using classical Darcy's law 2.66 7 0.06 Darcy (2:625  10−12 7 0:059  10−12 m2 ) as compared to the

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References

Pressure gradient, ΔP/L (x106 Pa.s.m-1)

3

2

1

0 0.000

0.002

0.004 Fluid velocity, v (m/s)

0.006

0.008

Fig. 10. Variation of pressure gradient (ΔP=L) with respect to change in the fluid velocity (v) flowing through Network 1. Line indicates the data fit for Darcy– Forchheimer equation and symbols indicate the experimental data of the present work.

Darcy–Forchheimer equation (2:92 70:5 Darcy (2:88  10−12 7 0:494  10−12 m2 )). Neglecting such small difference, for simplicity, we have only considered the Darcy (or viscous) permeability for our networks.

4. Conclusion In this work, a novel on-chip porous medium was fabricated and characterized using SEM and surface profilometer. Four different types of chips varying in number of pore bodies and pore throats were considered in this work. Properties like porosity and permeability were calculated to characterize such on-chip porous media. A very simple image analysis technique, as opposed to a complex material balance approach, was used to determine the porosity and it is found that the porosity values ranged between 0.39 and 0.67. Pressure drops across the pore networks were measured and were used for calculating permeability values. It was observed that as the flow rate increased, the pressure drop increased linearly in the networks. The flow resistance was maximum for Network 1 and was minimum for Network 4. It was also seen that the resistance to flow offered by the networks decreases with increase in Darcy number. Permeability for each of the chips was calculated and was found to be ranging between 2.66 and 15.93 Darcy (2:625  10−12 and 15:72  10−12 m2 ). It is concluded that the quantification of these properties onchip porous media opens up new opportunities in pore-scale modeling and simulation of different oil recovery processes.

Acknowledgments The authors thank Nikolaos K. Karadimitriou and Dr. S.M. Hassanizadeh, Department of Earth Sciences, Universiteit Utrecht, for providing the network design. The authors also thank Bijoyendra Bera for his help and support in developing the chip. The authors gratefully acknowledge Dr. Siddhartha Das and Lalit Pant, Department of Mechanical Engineering, University of Alberta, for their helpful comments and suggestions. Financial support from Natural Sciences and Engineering Council (NSERC) is greatly acknowledged by the authors.

Akbari, M., Sinton, D., Bahrami, M., 2009. Pressure drop in rectangular microchannels as compared with theory based on arbitrary cross section. J. Fluids Eng. 131, 0412021–0412028. Al-Kharusi, A.S., Blunt, M.J., 2007. Network extraction from sandstone and carbonate pore space images. J. Pet. Sci. Eng. 56, 219–231. Alazmi, B., Vafai, K., 2001. Analysis of fluid flow and heat transfer interfacial conditions between a porous medium and a fluid layer. Int. J. Heat Mass Transfer 44, 1735–1749. Alzaydi, A.A., 1975. Flow of Gases through Porous Media. Ohio State University. Arns, C., Knackstedt, M., Val Pinczewski, W., Lindquist, W., 2001. Accurate estimation of transport properties from microtomographic images. Geophys. Res. Lett. 28, 3361–3364. Bakke, S., Oren, P.E., 1997. 3-d pore-scale modelling of sandstones and flow simulations in the pore networks. SPE J. 2, 136–149. Beckermann, C., Viskanta, R., Ramadhyani, S., 1986. A numerical study of nondarcian natural convection in a vertical enclosure filled with a porous medium. Numer. Heat Transfer 10, 557–570. Bera, B., Gunda, N.S.K., Mitra, S.K., Vick, D., 2012. Characterization of nanometer-scale porosity in reservoir carbonate rock by focused ion beam – scanning electron microscopy. Microsc. Microanal. 18, 171–178. Bera, B., Mitra, S.K., Vick, D., 2011. Understanding the micro structure of Berea Sandstone by the simultaneous use of micro-computed tomography (micro-CT) and focused ion beam-scanning electron microscopy (FIB-SEM). Micron 42, 412–418. Berejnov, V., Djilali, N., Sinton, D., 2008. Lab-on-chip methodologies for the study of transport in porous media: energy applications. Lab Chip 8, 689–693. Bhattacharya, S., Datta, A., Berg, J.M., Gangopadhyay, S., 2005. Studies on surface wettability of poly (dimethyl) siloxane (PDMS) and glass under oxygen–plasma treatment and correlation with bond strength. J. Microelectromech. Syst. 14, 590–597. Blunt, M., 2001. Flow in porous media-pore network models and multiphase flow. Curr. Opinion Colloid Interface Sci. 6, 197–207. Blunt, M., Jackson, M., Piri, M., Valvatne, P., 2002. Detailed physics, predictive capabilities and macroscopic consequences for pore-network models of multiphase flow. Adv. Water Resour. 25, 1069–1089. Bowden, S., Monaghan, P., Wilson, R., Parnell, J., Cooper, J., 2006. The liquid–liquid diffusive extraction of hydrocarbons from a north sea oil using a microfluidic format. Lab Chip 6, 740–743. Buchgraber, M., Al-Dossary, M., Ross, C., Kovscek, A., 2012. Creation of a dualporosity micromodel for pore-level visualization of multiphase flow. J. Pet. Sci. Eng. 86–87, 27–38. Carman, P.C., Carman, P.C., 1956. Flow of Gases through Porous Media. Butterworths Scientific Publications London. Carnes, B., Djilali, N., 2006. Analysis of coupled proton and water transport in a PEM fuel cell using the binary friction membrane model. Electrochim. Acta 52, 1038–1052. Carrier Jr., W.D., 2003. Goodbye, Hazen; Hello, Kozeny-Carman. J. Geotech. Geoenviron. Eng. 129, 1054–1056. Clelland, W., Fens, T., 1991. Automated rock characterization with SEM/imageanalysis techniques. SPE Formation Eval. 6, 437–443. Colombani, J., Galliéro, G., Duguay, B., Caltagirone, J., Montel, F., Bopp, P., et al., 2002. A molecular dynamics study of thermal diffusion in a porous medium. Phys. Chem. Chem. Phys. 4, 313–321. Cornell, D., Katz, D.L., 1953. Flow of gases through consolidated porous media. Ind. Eng. Chem. 45, 2145–2152. Dong, H., Blunt, M.J., 2009. Pore-network extraction from micro-computerizedtomography images. Phys. Rev. E 80, 036307. Dullien, F., 1992. Porous Media: Fluid Transport and Pore Structure, vol. 26. Academic Press. Ellis, J., Bazylak, A., 2012. Dynamic pore network model of surface heterogeneity in brine-filled porous media for carbon sequestration. Phys. Chem. Chem. Phys. 14, 8382–8390. Er, V., Babadagli, T., Xu, Z., 2010. Pore-scale investigation of the matrix–fracture interaction during CO2 injection in naturally fractured oil reservoirs. Energy Fuels 24, 1421–1430. Fadaei, H., Scarff, B., Sinton, D., 2011. Rapid microfluidics-based measurement of CO2 diffusivity in bitumen. Energy Fuels 25, 4829–4835. Gan, H., Nandi, S., Walker Jr., P., 1972. Nature of the porosity in American coals. Fuel 51, 272–277. Geertsma, J., 1974. Estimating the coefficient of inertial resistance in fluid flow through porous media. Old SPE J. 14, 445–450. Gervais, P.C., Bardin-Monnier, N., Thomas, D., 2012. Permeability modeling of fibrous media with bimodal fiber size distribution. Chem. Eng. Sci. 73, 239–248. Gostick, J., Fowler, M., Pritzker, M., Ioannidis, M., Behra, L., 2006. In-plane and through-plane gas permeability of carbon fiber electrode backing layers. J. Power Sources 162, 228–238. Gunda, N.S.K., Bera, B., Karadimitriou, N., Mitra, S.K., Hassanizadeh, S., 2011a. Reservoir-on-a-chip (ROC): a new paradigm in reservoir engineering. Lab Chip 11, 3785–3792. Gunda, N.S.K., Choi, H.W., Berson, A., Kenney, B., Karan, K., Pharoah, J., Mitra, S.K., 2011b. Focused ion beam-scanning electron microscopy on solid-oxide fuel-cell electrode: image analysis and computing effective transport properties. J. Power Sources 196, 3592–3603.

J. Joseph et al. / Chemical Engineering Science 99 (2013) 274–283

Gunda, N.S.K., Joseph, J., Tamayol, T., Akbari, M., Mitra, S.K., 2013. Measurement of pressure drop and flow resistance in microchannels with integrated micropillars. Microfluidics Nanofluidics 14, 711–721. Gunda, N.S.K., Mitra, S.K., 2012. Direct simulation of transport properties from three-dimensional (3D) reconstructed solid-oxide fuel-cell (SOFC) electrode microstructures. J. Phys. Conf. Ser. 362, 012001. Gunde, A.C., Bera, B., Mitra, S.K., 2010. Investigation of water and CO2 (carbon dioxide) flooding using micro-CT (micro-computed tomography) images of berea sandstone core using finite element simulations. Energy 35, 5209–5216. Gurau, V., Bluemle, M., De Castro, E., Tsou, Y.M., Zawodzinski Jr., T., Mann Jr., J., 2007. Characterization of transport properties in gas diffusion layers for proton exchange membrane fuel cells. 2. Absolute permeability. J. Power Sources 165, 793–802. Hazlett, R., 1995. Simulation of capillary-dominant displacements in microtomographic images of reservoir rocks. Transp. Porous Media 20, 21–35. Humby, S., Biggs, M., Tuzun, U., 2002. Explicit numerical simulation of fluids in reconstructed porous media. Chem. Eng. Sci. 57, 1955–1968. Jaber, J., Probert, S., Williams, P., 1999. Evaluation of oil yield from jordanian oil shales. Energy 24, 761–781. Jaganathan, S., Tafreshi, H.V., Pourdeyhimi, B., 2008. A realistic approach for modeling permeability of fibrous media: 3-D imaging coupled with CFD simulation. Chem. Eng. Sci. 63, 244–252. Jamaloei, B., Kharrat, R., 2010. Analysis of pore-level phenomena of dilute surfactant flooding in the presence and absence of connate water saturation. J. Porous Media 13, 671–690. Jeong, S.W., Yavuz Corapcioglu, M., 2003. A micromodel analysis of factors influencing NAPL removal by surfactant foam flooding. J. Contaminant Hydrol. 60, 77–96. Kaiser, M., 2009. Hydrocarbon production forecast for committed assets in the shallow water outer continental shelf of the Gulf of Mexico. Energy 34, 1813–1825. Karadimitriou, N.K., Hassanizadeh, S., 2012. A review of micromodels and their use in two-phase flow studies. Vadose Zone J. 11, 1–10. Karadimitriou, N.K., Niasar, V., Hassanizadeh, S.M., Kleingeld, P.J., Pyrak-Nolte, L.J., 2012. A novel deep reactive ion etched (DRIE) glass micro-model for two-phase flow experiments. Lab Chip 12, 3413–3418. Keefer, D.L., Bodily, S.E., 1983. Three-point approximations for continuous random variables. Manage. Sci. 29, 595–609. Kenyon, W.E., 1992. Nuclear magnetic resonance as a petrophysical measurement. Nucl. Geophys. 6, 153–171. Kerkhof, P.J., 1996. A modified Maxwell–Stefan model for transport through inert membranes: the binary friction model. Chem. Eng. J. Biochem. Eng. J. 64, 319–343. Klinkenberg, L., 1941. The permeability of porous media to liquids and gases. In: Drilling and Production Practice. Lindquist, W., Venkatarangan, A., Dunsmuir, J., Wong, T.F., 2000. Pore and throat size distributions measured from synchrotron x-ray tomographic images of Fontainebleau sandstones. J. Geophys. Res. B Solid Earth 105, 21509–21527. Ma, K., Liontas, R., Conn, C., Hirasaki, G., Biswal, S., 2012. Visualization of improved sweep with foam in heterogeneous porous media using microfluidics. Soft Matter 8, 10669–10675. Ma, K., Rivera, J., Hirasaki, G.J., Biswal, S.L., 2011. Wettability control and patterning of PDMS using UV–ozone and water immersion. J. Colloid Interface Sci. 363, 371–378. Mattiello, D., Balzarini, M., Ferraccioli, L., Brancolini, A., SpA, A., 1997. Calculation of constituent porosity in a dual-porosity matrix: MRI and image analysis integration. SCA-9706. Miller, M.J., Fogler, H., 1995. Prediction of fluid distribution in porous media treated with foamed gel. Chem. Eng. Sci. 50, 3261–3274. Navitar Inc., 2013. High magnification imaging. URL: 〈http://www.navitar.com/ categories/zoom-6000.aspx〉. Nield, D., 1991. The limitations of the Brinkman–Forchheimer equation in modeling flow in a saturated porous medium and at an interface. Int. J. Heat Fluid Flow 12, 269–272. Okabe, H., Blunt, M.J., 2007. Pore space reconstruction of vuggy carbonates using microtomography and multiple-point statistics. Water Resour. Res. 43, 2–12.

283

Pant, L.M., Mitra, S.K., Secanell, M., 2012. Absolute permeability and Knudsen diffusivity measurements in PEMFC gas diffusion layers and micro porous layers. J. Power Sources 206, 153–160. Pant, L.M., Mitra, S.K., Secanell, M., 2013. A generalized mathematical model to study gas transport in PEMFC porous media. Int. J. Heat Mass Transfer 58, 70–79. Prodanovic, M., Lindquist, W., Seright, R., 2007. 3d image-based characterization of fluid displacement in a berea core. Adv. Water Resour. 30, 214–226. Rigby, S., Fletcher, R., Raistrick, J., Riley, S., 2002. Characterisation of porous solids using a synergistic combination of nitrogen sorption, mercury porosimetry, electron microscopy and micro-focus x-ray imaging techniques. Phys. Chem. Chem. Phys. 4, 3467–3481. Rigby, S.P., Watt-Smith, M.J., Chigada, P., Chudek, J.A., Fletcher, R.S., Wood, J., Bakalis, S., Miri, T., 2006. Studies of the entrapment of non-wetting fluid within nanoporous media using a synergistic combination of MRI and microcomputed x-ray tomography. Chem. Eng. Sci. 61, 7579–7592. Schneider, M.H., Willaime, H., Tran, Y., Rezgui, F., Tabeling, P., 2010. Wettability patterning by uv-initiated graft polymerization of poly (acrylic acid) in closed microfluidic systems of complex geometry. Anal. Chem. 82, 8848–8855. Schull, C., 1948. The determination of pore size distribution from gas adsorption data. J. Am. Chem. Soc. 70, 1405–1410. Singh, M., Mohanty, K., 2000. Permeability of spatially correlated porous media. Chem. Eng. Sci. 55, 5393–5403. Singh, M., Mohanty, K.K., 2003. Dynamic modeling of drainage through threedimensional porous materials. Chem. Eng. Sci. 58, 1–18. Sok, R., Knackstedt, M., Sheppard, A., Pinczewski, W., Lindquist, W., Venkatarangan, A., Paterson, L., 2002. Direct and stochastic generation of network models from tomographic images; effect of topology on residual saturations. Transp. Porous Media 46, 345–372. Spanne, P., Thovert, J., Jacquin, C., Lindquist, W., Jones, K., Adler, P., 1994. Synchrotron computed microtomography of porous media: topology and transports. Phys. Rev. Lett. 73, 2001–2004. Talmon, Y., Shtirberg, L., Harneit, W., Rogozhnikova, O., Tormyshev, V., Blank, A., 2010. Molecular diffusion in porous media by PGSE ESR. Phys. Chem. Chem. Phys. 12, 5998–6007. Timur, A., 1969. Pulsed nuclear magnetic resonance studies of porosity, movable fluid, and permeability of sandstones. J. Pet. Technol. 21, 775–786. Vafai, K., Tien, C., 1981. Boundary and inertia effects on flow and heat transfer in porous media. Int. J. Heat Mass Transfer 24, 195–203. Valdes-Parada, F.J., Ochoa-Tapia, J.A., Alvarez-Ramirez, J., 2009. Validity of the permeability Carman–Kozeny equation: a volume averaging approach. Phys. A Stat. Mech. Appl. 388, 789–798. Whitaker, S., 1986. Flow in porous media I: a theoretical derivation of Darcy's law. Transp. Porous Media 1, 3–25. Wu, M., Xiao, F., Johnson-Paben, R., Retterer, S., Yin, X., Neeves, K., 2012a. Singleand two-phase flow in microfluidic porous media analogs based on Voronoi tessellation. Lab Chip 12, 253–261, Miniaturisation for Chemistry and Biology. Wu, M., Xiao, F., Johnson-Paben, R.M., Retterer, S.T., Yin, X., Neeves, K.B., 2012b. Single-and two-phase flow in microfluidic porous media analogs based on voronoi tessellation. Lab Chip 12, 253–261. Wu, W., Lei, S.Y., Du, J.H., Wang, B.X., 2003. Relationship of threshold diameter and Darcean permeability in unconsolidated porous structures. Chem. Eng. Sci. 58, 3565–3570. Xu, P., Yu, B., 2008. Developing a new form of permeability and Kozeny–Carman constant for homogeneous porous media by means of fractal geometry. Adv. Water Resour. 31, 74–81. Zhang, C., Oostrom, M., Grate, J.W., Wietsma, T.W., Warner, M.G., 2011. Liquid CO2 displacement of water in a dual-permeability pore network micromodel. Environ. Sci. Technol. 45, 7581–7588. Zhao, L.H., Lee, J., Sen, P.N., 2012. Long-term retention of hydrophilic behavior of plasma treated polydimethylsiloxane (PDMS) surfaces stored under water and Luria–Bertani broth. Sensors Actuators A Phys. 181, 33–42. Zhou, N., Hosokawa, T., Suekane, T., Wang, Q., 2011. Experimental study of capillary trapping on the pore scale for various sandstone cores. Energy Procedia 4, 5017–5023.