TY - JOUR
AU - Wang,, Dayong
AB - Abstract The Kozeny–Carman (KC) model is commonly used to predict permeability (k) but sometimes presents obvious predictive deviations. Comparatively, k prediction based on pore networks could more effectively describe the dependence of k on porous structure and accordingly shows high accuracy and reliability. This triggers us to examine the rationality of the KC assumptions about the pore structure and analyze the model applicability according to the pore networks. Two glass bead packs, a sintered glass bead plate and a Berea core are measured using computed tomography imaging and their pore parameters are accordingly calculated. Their pore parameters are obviously distinct, generally reflecting the progressively stronger roles of particle size, compaction-alike sintering and weak cementation in reducing porosity (ϕ), k, pore and pore throat radii (rp and rt). When correlating the pore parameters of the KC model to those of the pore networks, it is found that the specific surface area (av) has no specific relation to rp and rt and that pore structures could be distinct despite the equivalent av. Thus, av is insufficient to distinguish the differences in pore geometry and reflects their influence on k. An analysis of the modified KC equations of our cores and the ϕ−k relationships of some relative homogeneous reservoir rocks (e.g. silty sandstone and Fontainebleau sands) indicates that the variety of the ϕ−k relationships induced by such factors as particle size and pore radius could not be fully predicted by the KC model in some cases, especially when the ϕ is relatively low. Kozeny–Carman model, pore networks, pore characterization, porosity-permeability relationships, reservoir rocks 1. Introduction Estimating permeability (k) of reservoir rocks is one of the most important and fundamental disciplines in engineering geophysics. Pore structure essentially determines the paths of fluid flow inside reservoir rocks and, hence, is closely correlated to their k. Therefore, pore structure characterization is necessary for the establishment of quantitative correlations between pore parameters and k (Bear 1972; Morris et al.2003; Krause et al.2009; Tiab & Donaldson 2015; Srisutthiyakorn & Mavko 2017). As a popular way to predict the k, the Kozeny–Carman (KC) model mainly uses porosity (ϕ) and specific surface area (av) as quantitative descriptions of pore structure (Kozeny 1927; Carman 1937). However, the KC model sometimes presents obvious predictive deviations (Pape et al.1999; Saar & Manga 1999; Xu & Yu 2008; Krause et al.2009). This suggests that the used pore parameters in the model cannot fully reflect the actual variation of k with pore structure in some cases. On the other hand, k prediction based on pore networks, which usually represent the porous medium as an assembly of pores and pore throats (Dong 2007), could more effectively describe the dependence of k on porous structure by introducing some other pore parameters including coordination number (Z), pore and pore throat radii (rp and rt), and accordingly shows high accuracy and reliability (Dmitriy & Tad 2006; Al-Kharusi & Blunt 2007; Moghaddam & Jamiolahmady 2016; Sharqawy 2016; Wang et al.2016; Wang & Sheng 2018). Unfortunately, extraction of pore networks requires high-quality pore structure characterization as a foundation, which is time- and cost-consuming and weakens the popularity of the k prediction based on the pore networks in practical engineering applications. This triggers us to find or construct correlations between the pore parameters of the KC model and those of pore networks, which have the potential to provide a new perspective to further examine the rationality of KC model assumptions about pore structure and analyze model applicability. In this study, we select a Berea core and three artificial analogs with various pore structure characteristics as study cores. Using computed tomography (CT) imaging, a complete set of pore parameters related to both the KC model and pore networks are measured and calculated for a systematic analysis of their interrelations and implications on the reliability of the KC model. Based on such understanding, the applicability of KC model is analyzed by comparing some other relative homogeneous reservoir rocks to the study cores. 2. Pore structure characterization 2.1. Experimental materials and apparatus A Berea core and three artificial analogs were selected as experimental cores for measurement and calculation of pore parameters and k. Two types of glass bead (GB04 and GB02) with different particle diameters (0.43 and 0.21 mm) were, respectively, stuffed into core tubes with an inner diameter of 12 mm. The formed glass bead packs (GBPs) could be approximately treated as unconsolidated sands (Wang et al.2017). Because there were no diagenetic alterations, the variations in their pore parameters could mainly be attributed to the effects of stochastic packing of glass beads in the core tube (Wang et al.2017). A slate of sintered glass beads (SGB) with the same initial particle size as the GB02 was produced by fusing and melting the glass beads at high temperature (∼700°C) in a muffle furnace (JK-SX2-4-13M, Shanghai Jingke Scientific Instrument Co. Ltd in China). Such a sintering operation was designed to partially mimic particle compaction (Gueven et al.2017), which could also induce porosity reduction (Fortin et al.2009; Eseme et al.2012). Berea cores are frequently used in core flooding experiments and thought to be representative of a standard natural core. Berea sandstones generally consist of smaller particles (e.g. ∼0.15 mm), which are usually sub-angular, moderately spherical and sorted with a mixture of point or concavo-convex contacts (Pini & Benson 2013). Besides, Berea sandstones also comprise certain extents of drusy calcite cementation and some silica cementation in form of quartz overgrowths (Kareem et al.2017). Cementation could decrease the average pore entry radius of sandstones to a low value (Bourbie & Zinszner 1985; Yang & Aplin 2007; Torskaya et al.2014). Although pore structure of Berea cores has been analyzed in some studies previously (Dong 2007; Krause et al.2009; Tanino & Blunt 2012; Sharqawy 2016), these studies only measured a minority of pore parameters. In this study, a Berea core from the Berea Sandstone Petroleum Cores, Vermilion, OH, USA is also selected as an experimental core and a more complete set of pore parameters will be measured and calculated for it. Generally, the pore structures of these four cores can be approximately considered to be subject to the effects of particle size and packing, compaction and cementation. Besides, they are relatively homogenous and free of micro-fractures. Therefore, the influence of micro-fractures on their k (Jin et al.2010; Fan et al.2012) could be reasonably ignored. A micro-focus X-ray CT system (inspeXio SMX-225CTX-SV, Shimadzu Co., Japan) was used to image the pore structures of the four cores (figure 1). To capture the edge of particles as accurately as possible, CT scanning resolution needs to be carefully selected (Beckingham et al.2013). Generally, the high resolution images are superior to the low resolution ones for accurate analysis of transport properties because the former avoids the problem of partial volume effects (Shah et al.2016). The resolution of ∼10.58 μm/voxel was previously used to satisfy the resolution requirement for pore configuration analysis of Berea sandstones (Krause et al.2009). Comparatively, we select a higher resolution of 5.35 μm/voxel to obtain pore parameters of the Berea core, which is also same as the one used by Dong (2007). Although the relatively smooth material surfaces of the three sandstone analogs could lower their resolution requirements to some extent (Berryman & Blair 1987), the scanning resolution for the two GBPs is still controlled to be equal to about 1/10 (i.e. 25 μm/voxel) of the smaller particle diameter (i.e. 0.21 mm) (Wang et al.2016). Moreover, considering the original size or shape of the GB02 would change during the sintering process (Gueven et al.2017), the scanning resolution for the SGB02 is set up to be smaller (i.e. 15 μm/voxel) than that for the GBPs. A series of thin-section CT images could be acquired along the axial direction of the cores located in the rotation stage of the CT system. Initially, the original 16-bit gray scale images were filtered by Gaussian blur with a radius of 2.00 and denoised using the image processing software Image J (Wang et al.2016) for the best identification of particle edges through the calculated intermediate grayscale value. It needs to be mentioned that the clarity (quality) of our original gray scale images is actually comparable to those obtained by other CT systems with a resolution in the magnitude of microns (e.g. table 3.1 by Dong 2007 and figure 2 by Krause et al.2009). Such images have also been demonstrated to be enough to reveal the details of pore structure, and could be used to extract pore networks (Dong 2007) and determine pore parameters (Krause et al.2009). Subsequently, threshold values between 0 and 65 535 were manually determined by the repeated comparisons of the original gray scale images (figure 2a) with the binary ones that convert pore space into black and particles into white (figure 2b). The specified threshold values guarantee the ratio of the area of all the pore spaces to the total area of the converted binary images to be same as the measured porosity for each core (Leu et al.2014). Then, the binarized images were scaled by the actual size of the core so that the perimeter (Lp,i) and area (Sp,i) of each pore (each closed region) could be calculated by Image J (figure 2c). Figure 1. Open in new tabDownload slide CT system. Figure 1. Open in new tabDownload slide CT system. Figure 2. Open in new tabDownload slide Three stages of image processing (set a slice of SGB02 as an example): (a) 16-bit gray scale CT image, (b) inverted binary image and (c) extracted pore space profile. Figure 2. Open in new tabDownload slide Three stages of image processing (set a slice of SGB02 as an example): (a) 16-bit gray scale CT image, (b) inverted binary image and (c) extracted pore space profile. 2.2. Calculation of pore parameters To eliminate the boundary effect of core-tube wall on the local particle packing and pore structure, the region actually analyzed (i.e. the region of interest (ROI)) is only limited to the central section of the core tube and its margins are at least three times a particle diameter away from the core-tube wall (figure 3). When the ROI sizes lie in a reasonable range, the calculated pore parameters for the ROI could give a dependable illustration for pore characteristics and reflect the permeability magnitude effectively. For each core, the length of the ROI is more than 10 times its average particle size (figure 3), which is approximately twice the suggested minimum size (i.e. 5.15 times of particle size) for a representative element volume (REV) (Clausnitzer & Hopmans 1999; Krause et al.2013; Li & Benson 2015; Saxena et al.2018). Besides, the size of the ROI is also equivalent to the grid sizes (e.g. 1.27 to 5 mm) specified for most core-scale numerical modellings for fluid migration in porous media (Krause et al.2009, 2011; Krevor et al.2011; Chen et al.2017). Figure 3. Open in new tabDownload slide The selected ROI for (a) GBP02, (b) GBP04, (c) SGB02 and (d) Berea core. Figure 3. Open in new tabDownload slide The selected ROI for (a) GBP02, (b) GBP04, (c) SGB02 and (d) Berea core. Possibly a few isolated pores exist in the analyzed cores, which might introduce errors into the calculation of ϕ and av. However, according to Srisutthiyakorn and Mavko (2017), the errors are small and can be reasonably ignored even if the ϕ and av are calculated from the original images containing the isolated pores. Thus, we did not especially exclude these isolated pores from the CT images for the calculation of ϕ and av. For relatively uniform, isotropic and random porous material, the pore surface area per unit volume can be related to the perimeter per unit area (Russ & Dehoff 1999). Accordingly, the three-dimensional specific surface area is usually assumed to be proportional to the two-dimensional specific pore perimeter (Krause et al.2009). Then, the pore perimeter (Lp), pore area (Sp), grain area (Sg), ϕ and av for each ROI can be computed based on the calculated Lp,i and Sp,i for each pore by Image J: $$\begin{equation} {L_p} = \mathop \sum \nolimits {L_{p,i}}, \end{equation}$$ (1) $$\begin{equation} {S_p} = \mathop \sum \nolimits {S_{p,i}}, \end{equation}$$ (2) $$\begin{equation} {S_{\rm{g}}} = {S_{{\rm{ROI}}}} - {S_{\rm{P}}}, \end{equation}$$ (3) $$\begin{equation} \phi = {S_{\rm{p}}}/{S_{{\rm{ROI}}}}, \end{equation}$$ (4) $$\begin{equation} {a_{\rm{v}}} = {L_{\rm{p}}}/{S_{\rm{g}}}. \end{equation}$$ (5) where the SROI denotes the area of the ROI. Besides, the extraction of the pore networks based on a series of CT images is also an alternative approach to analyzing pore characteristics of porous media. The pore network analysis has a wide applicability not only for the relatively simple porous media such as GBPs and Berea sandstones (Sharqawy 2016; Wang et al.2016), but also complicated shale samples (Moghaddam & Jamiolahmady 2016; Wang & Sheng 2018). For instance, the predicted average pore throat radius is of the same magnitude as the corresponding mercury injection capillary pressure (MICP) measurement for Berea cores (Dong 2007; Tanino & Blunt 2012; Sharqawy 2016; Al-Yaseri et al.2017). Both the absolute and relative permeability of Berea cores calculated based on the pore networks can match well with the measured values (Valvatne 2004). Thus, pore network extraction is a suitable method to analyze the pore structure of our four cores with a high accuracy. Here we use the method (i.e. the maximum ball algorithm) and codes of Dong (2007) to extract the pore networks for our cores. The validity and reliability of this algorithm was verified previously by Dmitriy & Tad (2006) and Al-Kharusi & Blunt (2007). 3. Results The calculated pore parameters for the four cores and the statistical results about rp, rt and Z are shown in Table 1 and figure 4. For the randomly packed GBP02 and GBP04, their ϕ values are approximately equal to those of unconsolidated sands (∼45%, Revil & Cathles 1999) and also have a good consistency with the values (i.e. 40.50 and 43.20%) determined by Magnetic Resonance Imaging (MRI) (Wang et al.2017). Besides, their calculated permeability values based on pore networks (i.e. PN permeability) are comparable to the measured values (91.3 D and 50.9 D) (Wang et al.2017) and the deviation lies in 1/3–3 times the measurements; a reasonable range for the predictive errors (Chapuis & Aubertin 2003). For the Berea core, the calculated average rp and rt have the same order of magnitude as the MICP measurements for other Berea samples (Dong 2007; Tanino & Blunt 2012; Sharqawy 2016; Al-Yaseri et al.2017). Moreover, the predictive deviation in its k is also small and only ∼25% of the measured permeability (∼1.0 D). All of these demonstrate the reliability of the pore networks extracted for our cores. Figure 4. Open in new tabDownload slide Statistical distribution of (a) pore radius (solid line) and pore throat radius (dotted line) and (b) coordination number of the four cores. Figure 4. Open in new tabDownload slide Statistical distribution of (a) pore radius (solid line) and pore throat radius (dotted line) and (b) coordination number of the four cores. Table 1. Pore parameters related to the pore networks. Core ϕ (%) rp (μm) Np* rt (μm) Nt* k (D) GBP04 41.19 11.1–219 13 2.5–150 58 213 GBP02 40.24 10.4–169 35 2.5–119 136 59 SGB02 31.87 6.5–93.7 90 1.5–74.9 256 17 Berea 19.76 2.2–66.9 648 0.5–45.1 1296 1.6 Core ϕ (%) rp (μm) Np* rt (μm) Nt* k (D) GBP04 41.19 11.1–219 13 2.5–150 58 213 GBP02 40.24 10.4–169 35 2.5–119 136 59 SGB02 31.87 6.5–93.7 90 1.5–74.9 256 17 Berea 19.76 2.2–66.9 648 0.5–45.1 1296 1.6 * Np and Nt denote to the number of pores and pore throats per cubic millimeter. Open in new tab Table 1. Pore parameters related to the pore networks. Core ϕ (%) rp (μm) Np* rt (μm) Nt* k (D) GBP04 41.19 11.1–219 13 2.5–150 58 213 GBP02 40.24 10.4–169 35 2.5–119 136 59 SGB02 31.87 6.5–93.7 90 1.5–74.9 256 17 Berea 19.76 2.2–66.9 648 0.5–45.1 1296 1.6 Core ϕ (%) rp (μm) Np* rt (μm) Nt* k (D) GBP04 41.19 11.1–219 13 2.5–150 58 213 GBP02 40.24 10.4–169 35 2.5–119 136 59 SGB02 31.87 6.5–93.7 90 1.5–74.9 256 17 Berea 19.76 2.2–66.9 648 0.5–45.1 1296 1.6 * Np and Nt denote to the number of pores and pore throats per cubic millimeter. Open in new tab Some pore parameters derived from CT imaging such as grain area (Sg,u) and pore perimeter (Lp,u) per unit area as well as the accordingly calculated ϕ and av for ROI of the four cores are presented in Table 2 and figure 5. Among these parameters, the Sg,u, namely (1−ϕ), expresses the relative pore space portion (figure 5b), while the Lp,u can also reflect the absolute size and geometry of pores (figure 5a). The ROI porosities of the two GBPs vary slightly and are also approximately equal to the measured values by MRI (Wang et al.2017). Figure 5. Open in new tabDownload slide Statistics of (a) pore perimeter per unit area, (b) porosity and (c) specific surface area of the ROI for the four cores. Figure 5. Open in new tabDownload slide Statistics of (a) pore perimeter per unit area, (b) porosity and (c) specific surface area of the ROI for the four cores. Table 2. Pore parameters related to the Kozeny–Carman model. Core Sg,u Lp,u (mm−1) ϕ (%) av (mm−1) GBP04 0.56–0.61 3.86–4.93 38.77–43.77 6.72–8.44 GBP02 0.57–0.64 5.65–6.33 36.61–43.40 9.14–10.86 SGB02 0.63–0.73 4.19–5.16 26.76–36.94 5.80–8.14 Berea 0.76–0.85 11.03–14.43 14.51–24.77 12.99–18.32 Core Sg,u Lp,u (mm−1) ϕ (%) av (mm−1) GBP04 0.56–0.61 3.86–4.93 38.77–43.77 6.72–8.44 GBP02 0.57–0.64 5.65–6.33 36.61–43.40 9.14–10.86 SGB02 0.63–0.73 4.19–5.16 26.76–36.94 5.80–8.14 Berea 0.76–0.85 11.03–14.43 14.51–24.77 12.99–18.32 Open in new tab Table 2. Pore parameters related to the Kozeny–Carman model. Core Sg,u Lp,u (mm−1) ϕ (%) av (mm−1) GBP04 0.56–0.61 3.86–4.93 38.77–43.77 6.72–8.44 GBP02 0.57–0.64 5.65–6.33 36.61–43.40 9.14–10.86 SGB02 0.63–0.73 4.19–5.16 26.76–36.94 5.80–8.14 Berea 0.76–0.85 11.03–14.43 14.51–24.77 12.99–18.32 Core Sg,u Lp,u (mm−1) ϕ (%) av (mm−1) GBP04 0.56–0.61 3.86–4.93 38.77–43.77 6.72–8.44 GBP02 0.57–0.64 5.65–6.33 36.61–43.40 9.14–10.86 SGB02 0.63–0.73 4.19–5.16 26.76–36.94 5.80–8.14 Berea 0.76–0.85 11.03–14.43 14.51–24.77 12.99–18.32 Open in new tab The different kinds of alterations cause the diversity of the pore parameters among these four cores. Ranges of their respective ROI ϕ do not nearly overlap each other and together span a wide interval (i.e. 0.14KC0.45) (figure 6), covering the typical porosity values of most good-quality reservoir sandstones (Revil & Cathles 1999). Generally, the modes of rp, rt and Z move toward the left in the order of GBP04, GBP02, SGB02 to the Berea core (figure 4), presenting a gradually decreasing trend (Table 1). Besides, the portions of isolated pores of these four cores as represented by Z = 0 in figure 4b also increase in such an order, potentially contributing to poor connectivity. The spatial distribution of the KC model pore parameters (figure 5) and their variances (figure 6) could reflect the homogeneity of pore structures caused by the different kinds of alteration. Figure 6. Open in new tabDownload slide (a) Grain area and pore perimeter per unit area (the dotted lines represent the average value of specific surface area of the ROI) and (b) porosity and specific surface area of the ROI for the four cores (D(ϕ) and D(av) denote the variance of ϕ and av, respectively.). Figure 6. Open in new tabDownload slide (a) Grain area and pore perimeter per unit area (the dotted lines represent the average value of specific surface area of the ROI) and (b) porosity and specific surface area of the ROI for the four cores (D(ϕ) and D(av) denote the variance of ϕ and av, respectively.). 4. Discussions 4.1. Interrelation of pore parameters and its implications on the reliability of the KC model The k predictions through the KC model or based on pore networks are generally proposed according to distinct treatments for pore structure. Consequently, they assume different ‘k-dominating’ pore parameters. Because of allowing for the contribution of all pores and pore throats, the PN permeability is closely related to rp and rt (Valvatne 2004; Dong 2007). As shown in Table 1, k presents a strong positive dependence on rp and rt for our cores. Alternatively, the KC model uses ϕ and av to, respectively, describe the effects of pore portion and geometry on permeability. However, there seems no specific relation between the pore network parameters and the KC ones. Apparently, ϕ could reduce with the decreased rp and rt. After all, ϕ is calculated as the sum of all pores and pore throats. Nevertheless, ϕ does not positively relate to rp and rt for our cores. For instance, the GBP02 has almost the same ϕ as the GBP04 despite its much smaller rp and rt (Table 1). Besides, av does not show specific relations to rp or rt for all the four cores either (Tables 1 and 2). There could be a distinct difference in rp and rt despite the approximate av; for example, between the GBP04 and SGB02 (figures 3 and 5). We randomly select four ROI from the GBP04, GBP02, SGB02 and Berea core, respectively (figure 7). The SGB02 has smaller Lp,u than the GBP02 due to its smaller pore size, implying that in this case the pore size dominates the difference of Lp,u. However, for the GBP02 and GBP04, the greater pore distribution density of the former offsets the influence of its smaller pore size, resulting in bigger Sg,u and Lp,u than the latter (same for the Berea core and other three cores). Under this condition, their difference in Sg,u and Lp,u (accordingly av) can be mainly attributed to the effects of the pore distribution density. Actually, Sg,u, Lp,u, ϕ and av are calculated based on the summation of all the individual pores and pore throats in a unit area or volume. Thus, they not only depend on the size of individual pore, but also the pore distribution density (Srisutthiyakorn & Mavko 2017). Differently, the permeability is mainly determined by the relatively big pores and pore throats and some small pores and pore throats contribute little to it. This, however, cannot be effectively reflected by any of the KC model pore parameters (e.g. Sg,u, Lp,u, ϕ and av), especially in the case of the inhomogeneous pore structure. Figure 7. Open in new tabDownload slide Pore distribution (the black color) in randomly selected ROI for (a) GBP04, (b) GBP02, (c) SGB02 and (d) Berea core. Figure 7. Open in new tabDownload slide Pore distribution (the black color) in randomly selected ROI for (a) GBP04, (b) GBP02, (c) SGB02 and (d) Berea core. 4.2. Applicability of the KC model in porosity–permeability relationship prediction One of the most important roles of the KC model in engineering geophysics is to analyze and predict ϕ−k relationships for reservoir rocks (Tiab & Donaldson 2015). In this case, the original KC model is usually simplified into an only-ϕ dependent (high-order) form by calculating other pore parameters (e.g. av, tortuosity and shape factor) using ϕ (Tiab & Donaldson 2015). An exponential relationship could be used to relate the av to ϕ (Krause et al.2009) and the modified KC equations for the GBP02, SGB02 and Berea core are accordingly derived by substituting the term av (Table 3). Noticeably, the data pairs of av and ϕ for the GBP04 present a relatively concentrated distribution, indicating that no obvious relationship could exist between av and ϕ (figure 6b). Thus, the GBP04 will be excluded in the subsequent analysis. Those equations are used to analyze the ϕ–k relationships of some other relatively homogeneous cores, including the silty sandstones, Fontainebleau sand (Bourbie & Zinszner 1985; Revil & Cathles 1999) and GBPs (Wang et al.2017) (figure 8) to understand the applicability of the KC model. The GBPs could be roughly treated as idealized porous media (Wang et al.2017). The high consistency of their ϕ–k relation with the KC model for GBP02 reveals the good predictive capacity of the KC model for such granular porous media (Peyman et al.2013; Torskaya et al.2014; Taheri et al.2017; Zheng & Tannant 2017). The Fontainebleau sands consist of almost 100% quartz and are generally well sorted (Bourbie & Zinszner 1985). When the porosity ranges between 0.15 and 0.25, their permeability is slightly higher than the prediction by KC model of the Berea core as shown in figure 8. The mercury porosimetry diagram reveals that their average pore entry radius varies from 10 to 20 μm (Bourbie & Zinszner 1985), which is greater than the average pore throat size of the Berea core (7.2 μm) (Table 1). Beside, thin sections of the Fontainebleau sands show no (or very little) inter-granular cements and almost the constant grain size (∼250 μm) greater than that of the Berea core (∼150 μm) (Bourbie & Zinszner 1985; Pini & Benson 2013). Both of these factors could account for the difference between the Fontainebleau sands and Berea core. For the silty sandstones, when their porosities are equivalent to that of the SGB02 (0.275–0.375), the measured permeability is somewhat under the KC model of the SGB02. Because of both the smaller grain size (∼100 vs. ∼210 μm) and more diverse diagenetic alterations than those of the SGB02, there is a tendency to lower the permeability of silty sandstones (Chilingar 1964). In a smaller porosity range (i.e. 0.175–0.25), which overlaps that of the Berea core, the measured k of the silty sandstones is just a bit higher than prediction of the KC model of the Berea core. These two types of sandstone have close grain sizes (∼100 vs. ∼150 μm) and the slight-medium level diagenetic alteration (Pini & Benson 2013). If the porosity of the silty sandstones is below 0.175, their ϕ–k relation obviously deviates from that of the Berea core. In this case, the decreasing pore entry radius is considered to significantly reduce the permeability of the silty sandstones (Bourbie & Zinszner 1985), which cannot be effectively accounted for by the KC model (Peyman et al.2013; Wang et al.2017). Figure 8. Open in new tabDownload slide Analysis of the modified KC equations for the study cores and the measured porosity–permeability relationships of some other reservoir rocks. Figure 8. Open in new tabDownload slide Analysis of the modified KC equations for the study cores and the measured porosity–permeability relationships of some other reservoir rocks. Table 3. The modified KC equations for the study cores. Core KC models Only ϕ dependent KC models GBP02 |$k = {\rm{3}}{\rm{.17}} \,\times\, {10^{ - {\rm{8}}}} \,\times\, {{{{\phi ^{\rm{3}}}} /{{a_{\rm{v}}}^{\rm{2}} \cdot ( {1 - \phi } )}}^2}$| |$k = 6.{\rm{08}} \,\times\, {10^{ - 11}} \,\times\, {{{{\phi ^{1.1924}}} / {( {1 - \phi } )}}^2}$| SGB02 |$k = {\rm{1}}{\rm{.13}} \,\times\, {10^{ - {\rm{8}}}} \,\times\, {{{{\phi ^{\rm{3}}}} /{{a_{\rm{v}}}^{\rm{2}} \cdot ( {1 - \phi } )}}^2}$| |$k = 3.{\rm{1}}6 \,\times\, {10^{ - 11}} \,\times\, {{{{\phi ^{1.2446}}} /{( {1 - \phi } )}}^2}$| Berea |$k = {\rm{2}}{\rm{.98}} \,\times\, {10^{ - {\rm{8}}}} \,\times\, {{{{\phi ^{\rm{3}}}} /{{a_{\rm{v}}}^{\rm{2}} \cdot ( {1 - \phi } )}}^2}$| |$k = 2.{\rm{05}} \,\times\, {10^{ - 11}} \,\times\, {{{{\phi ^{1.9164}}} / {( {1 - \phi } )}}^2}$| Core KC models Only ϕ dependent KC models GBP02 |$k = {\rm{3}}{\rm{.17}} \,\times\, {10^{ - {\rm{8}}}} \,\times\, {{{{\phi ^{\rm{3}}}} /{{a_{\rm{v}}}^{\rm{2}} \cdot ( {1 - \phi } )}}^2}$| |$k = 6.{\rm{08}} \,\times\, {10^{ - 11}} \,\times\, {{{{\phi ^{1.1924}}} / {( {1 - \phi } )}}^2}$| SGB02 |$k = {\rm{1}}{\rm{.13}} \,\times\, {10^{ - {\rm{8}}}} \,\times\, {{{{\phi ^{\rm{3}}}} /{{a_{\rm{v}}}^{\rm{2}} \cdot ( {1 - \phi } )}}^2}$| |$k = 3.{\rm{1}}6 \,\times\, {10^{ - 11}} \,\times\, {{{{\phi ^{1.2446}}} /{( {1 - \phi } )}}^2}$| Berea |$k = {\rm{2}}{\rm{.98}} \,\times\, {10^{ - {\rm{8}}}} \,\times\, {{{{\phi ^{\rm{3}}}} /{{a_{\rm{v}}}^{\rm{2}} \cdot ( {1 - \phi } )}}^2}$| |$k = 2.{\rm{05}} \,\times\, {10^{ - 11}} \,\times\, {{{{\phi ^{1.9164}}} / {( {1 - \phi } )}}^2}$| Open in new tab Table 3. The modified KC equations for the study cores. Core KC models Only ϕ dependent KC models GBP02 |$k = {\rm{3}}{\rm{.17}} \,\times\, {10^{ - {\rm{8}}}} \,\times\, {{{{\phi ^{\rm{3}}}} /{{a_{\rm{v}}}^{\rm{2}} \cdot ( {1 - \phi } )}}^2}$| |$k = 6.{\rm{08}} \,\times\, {10^{ - 11}} \,\times\, {{{{\phi ^{1.1924}}} / {( {1 - \phi } )}}^2}$| SGB02 |$k = {\rm{1}}{\rm{.13}} \,\times\, {10^{ - {\rm{8}}}} \,\times\, {{{{\phi ^{\rm{3}}}} /{{a_{\rm{v}}}^{\rm{2}} \cdot ( {1 - \phi } )}}^2}$| |$k = 3.{\rm{1}}6 \,\times\, {10^{ - 11}} \,\times\, {{{{\phi ^{1.2446}}} /{( {1 - \phi } )}}^2}$| Berea |$k = {\rm{2}}{\rm{.98}} \,\times\, {10^{ - {\rm{8}}}} \,\times\, {{{{\phi ^{\rm{3}}}} /{{a_{\rm{v}}}^{\rm{2}} \cdot ( {1 - \phi } )}}^2}$| |$k = 2.{\rm{05}} \,\times\, {10^{ - 11}} \,\times\, {{{{\phi ^{1.9164}}} / {( {1 - \phi } )}}^2}$| Core KC models Only ϕ dependent KC models GBP02 |$k = {\rm{3}}{\rm{.17}} \,\times\, {10^{ - {\rm{8}}}} \,\times\, {{{{\phi ^{\rm{3}}}} /{{a_{\rm{v}}}^{\rm{2}} \cdot ( {1 - \phi } )}}^2}$| |$k = 6.{\rm{08}} \,\times\, {10^{ - 11}} \,\times\, {{{{\phi ^{1.1924}}} / {( {1 - \phi } )}}^2}$| SGB02 |$k = {\rm{1}}{\rm{.13}} \,\times\, {10^{ - {\rm{8}}}} \,\times\, {{{{\phi ^{\rm{3}}}} /{{a_{\rm{v}}}^{\rm{2}} \cdot ( {1 - \phi } )}}^2}$| |$k = 3.{\rm{1}}6 \,\times\, {10^{ - 11}} \,\times\, {{{{\phi ^{1.2446}}} /{( {1 - \phi } )}}^2}$| Berea |$k = {\rm{2}}{\rm{.98}} \,\times\, {10^{ - {\rm{8}}}} \,\times\, {{{{\phi ^{\rm{3}}}} /{{a_{\rm{v}}}^{\rm{2}} \cdot ( {1 - \phi } )}}^2}$| |$k = 2.{\rm{05}} \,\times\, {10^{ - 11}} \,\times\, {{{{\phi ^{1.9164}}} / {( {1 - \phi } )}}^2}$| Open in new tab 5. Conclusion Based on CT imaging, pore parameters related to both the KC model and pore networks of a Berea core, a slate of SGB and two glass bead packs were measured and calculated for a systematic analysis of their interrelations and the implications for the reliability and applicability of the KC model. These study cores were subject to the effects of different kinds of alteration. Their respective ranges for local porosities did not quite overlap each other and some other pore parameters were also obviously distinct among them, generally reflecting the progressively stronger roles of particle size, compaction-like sintering and weak cementation in reducing porosity, permeability, pore and pore throat radii. As a main pore parameter used to describe the dependence of permeability on pore geometry in the KC model, the specific surface area was found to have no specific relation to pore and pore throat radii and pore structure could be distinct despite the equivalent specific surface area. Thus, it was insufficient to distinguish the differences in pore geometry and this reflected their influence on permeability. Our different samples exhibited the various dependences of the permeability on pore configuration. Correspondingly, the different KC expressions were constructed for them, respectively. When selecting the porosity as the primary dependent variable, their KC equations constituted the piecewise functions covering a wide porosity range (i.e. 0.14–0.45) and also matched well with the porosity–permeability relationship of some reservoir rocks (e.g. silty sandstone and Fontainebleau sands), indicating wide applicability and reliability for reservoir rocks with high porosity, no or little cement and relatively homogenous particle sizes. However, with further decreases in porosity, the KC model would potentially become inapplicable, which has also been suggested by other studies. In this case, the KC parameters could not effectively reflect the difference in the pore parameters such as particle size and pore radius among the different reservoir rocks, especially when the heterogeneity of their pore structure was obvious. Acknowledgements This work was financially supported by the National Science Foundation of China (No. 51776030, 51436003 and 51376033), the PetroChina Innovation Foundation (No. 2017D-5007-0210), the Foundation Research Funds for the Central Universities (No. DUT18ZD207) and National Key Research and Development Program (No. 2016YFB0600804). Conflict of interest statement. None declared. References Al-Kharusi A.S. , Blunt M.J. , 2007 . Network extraction from sandstone and carbonate pore space images , Journal of Petroleum Science and Engineering , 56 , 219 – 231 . Google Scholar Crossref Search ADS WorldCat Al-Yaseri A.Z. , Zhang Y. , Ghasemiziarani M. , Sarmadivaleh M. , Lebedev M. , Roshan H. , Iglauer S. , 2017 . Permeability evolution in sandstone due to CO2 injection , Energy & Fuels , 31 , 12390 – 12398 . Google Scholar Crossref Search ADS WorldCat Bear J. , 1972 . Dynamics of Fluids in Porous Media , Elsevier , New York . 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TI - Analysis of the Kozeny–Carman model based on pore networks
JF - Journal of Geophysics and Engineering
DO - 10.1093/jge/gxz089
DA - 2019-12-01
UR - https://www.deepdyve.com/lp/oxford-university-press/analysis-of-the-kozeny-carman-model-based-on-pore-networks-0zwefXNUXL
SP - 1191
VL - 16
IS - 6
DP - DeepDyve
ER -