Tortuosity of pore channels in tight rocks as a key parameter in fluid flow ability

Tortuosity of pore channels in tight rocks as a key parameter in fluid flow ability | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Tortuosity of pore channels in tight rocks as a key parameter in fluid flow ability Paulina Krakowska-Madejska This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3500594/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 04 Jan, 2024 Read the published version in Acta Geophysica → Version 1 posted 6 You are reading this latest preprint version Abstract Tortuosity is a significant parameter in porous materials analysis. Not only, when it comes to rocks or soils but also cellular materials, alloys or cells. The multiple definitions exists for tortuosity and several purposes. Geometrical tortuosity describes the pore network paths, on the other hand- thermal, diffusional, electrical and hydraulic tortuosity refers to the transport processes in the pore network. Computed X-ray tomography is the best solution in tortuosity estimation, thanks to the 3D images. In particular, computed X-ray tomography, together with mercury porosimetry, pulse- and pressure-decay permeability methods, as well as electrical parameter measurements, link and expand the information about the tortuosity into the greater meaning. The geological material was composed of tight, low-porosity and low-permeability gas-saturated rocks cored from the present depth of deposition below 3000 m, containing different lithologies, as sandstones, mudstones, limestones and dolomites. The research presents the novel approach in the identification and analysis of the main pore channels based on 3D CT images. Algorithm of the central axis identifies and analyzes the whole main flow path and calculates tortuosity. High correlation was observed between the tortuosity and Swanson parameter from mercury porosimetry data. Moreover, the high correlation was detected between the tortuosity and saturation exponent from electrical parameter measurement in analyzed tight low-porosity and low-permeability deposits. Multilinear regression allows estimating absolute permeability taking CT, MICP and EPM parameters into consideration. Combination of these parameters in one equation with high determination coefficient gives credence in estimating preliminary absolute permeability based on the data which is executed as standard core analysis (MICP, EPM) and data from the non-invasive method (CT). computed X-ray tomography tight rocks pore space tortuosity mercury porosimetry MICP permeability electrical parameters Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Introduction Multidisciplinary laboratory measurements on geological material give the challenging possibility to discover and combine information from different resolutions and physical background. It is extremely important to look closely on the rock by reading petrophysical properties and creating a full image as a one body. Hence, several methods are combined together to check how pore system can behave in low porous and low permeable gas bearing formations. Computed X-ray tomography (CT) is totally safe and high resolution laboratory technique for 3D pore network examination (Ketcham & Carlson 2001 ; Cnudde & Boone 2003; Cubit et al. 2009; Adeleye & Akanji 2022 ). Moreover, it is a matter of scale: do we want to look at the rock in low or high resolution, thus look at the rock in centimeters or nanometers (Cnudde et al. 2011 ; Liu et al. 2018 )? In both case, some of the information is missing. First example do not concentrate on the small pores and in case of tight formation is not acceptable because we lose a huge amount of useful data. The advantage of this resolution is that rock examination is on the core or core plus. Second example, looks deeply into the small pores but we read the information form the small piece of rock, often the rock is the size of a crumb. Tortuosity of the pore ganglia is one of the crucial geometric parameter of pore structure and can be evaluated based on CT (Backeberg et al 2017 ; Mohan et al. 2023 ). There is several methods in tortuosity estimation (Ribeiro et al. 2022 ; Mahmood et al. 2023 ) but only CT gives the chance to conduct the measurement in 3D. Moreover, tortuosity influences electrical parameters (EPM), as formation factor, saturation exponent, and intrusion parameters from mercury porosimetry (MICP). The article presents a new method in tortuosity calculations form the 3D CT image. An attempt was also made to determine the relationship between tortuosity and electrical parameters. Moreover, absolute permeability was evaluated based on the multilinear regression analysis and mentioned parameters. Tortuosity has an enormous impact on rock ability to fluid flow, so the idea of connecting the tortuosity with absolute permeability is significant (Javadpour 2009 ; Berg 2014 ; Kaczmarek et al. 2017 ). Permeability is a challenging parameter, in both measurement and results interpretation (Soulaine et al. 2016 ; Ghanizadeh et al. 2017 ). Many researches were devoted to the permeability determination, often through the pore network modelling, as well as advanced statistical methods (Mostaghimi et al. 2013 ; Krakowska 2019 ; Al Balushi & Taleghani 2022 ). Certainly the tortuosity parameter enriches and adds credibility to the obtained results. Methods and materials The subject of the analysis were tight, low-porosity and low-permeability rocks cored from the present depth of deposition below 3000 m, containing different lithologies, as sandstones, mudstones, limestones and dolomites. The most important link in the research material is connected with the low values of porosity and permeability, meeting the condition of tight, gas-bearing reservoirs. We can expect simplified pore network, revealed in not tortuous pore paths. An example image of the pore space is shown in the Fig. 1 (Feldkamp 1984). Colours refer to the size of the pores. Lots of small objects is visible in the 3D CT image, from red (pores below 99 voxels, while voxel is a pixel in 3D) to green (pores below 99999 voxels). Often in tight formations only few objects are present from the highest volume class (blue colour). It is quite typical in tight formations that single extensive pore networks (dark and light blue) can occur in the whole pore system and simple pore networks (red, orange, yellow, green) predominate. Tortuosity analysis need to connect data from different laboratory methods. First and most precious 3D imaging method is computed X-ray tomography (CT). CT allows to estimate porosity, pore channel size and the most important from the research point – tortuosity. Often tortuosity is determine form the capillary data or petrographic image analysis. CT definitely gives better insight into all pore channels and overview on pore system in 3D (Rabbani et al. 2016 ; Krakowska-Madejska 2022 ; Moosavi et al. 2023 ). Mercury injection capillary pressure data (MICP) answers the question about the potential pore connectivity by mercury injection with the high pressure into the pore space. It appeared that tortuosity can be associated the MICP data, specifically with effective porosity, percentage of pores with diameters above 0.1 µm, percentage of pores with diameters above 1 µm and Swanson parameter (Swanson 1980, Thomeer 1983 , Mao et al. 2013 ). Swanson parameter is directly related to rock permeability and indirectly to tortuosity, because it refers to main point in the injection saturation of mercury, which controls the fluid flow. Pressure and pulse decay methods (PDP) in permeability estimation are crucial in tight rock analysis (Handwerger et al. 2011 ). PDP delivers absolute permeability for low porous and low permeable formations. The last important laboratory analysis is connected with the electrical parameters measurement (EPM). It is carried out on core plugs and determine the electrical resistivity of formation, formation factor, cementation exponent and saturation exponent. This mentioned electrical parameters are directly liked with tortuosity. The more tortuous is the pore network, the worse environment for current flow. Scheme of laboratory measurements on geological materials, as well as equipment description is presented in Fig. 2 . Tortuosity is one of the most important parameters in fluid flow considerations. The more pore space is complex, the greater the difficulty of fluid flow. That is why tortuosity can not be omitted in the tight reservoirs analysis, in which all difficulties count in hydrocarbon exploitation. The geometrical tortuosity is the ratio of the actual flow path or simply saying actual length of pore channel and straight-line distance between the beginning and the end of the pore channel (Thovert et al. 1993 ; Lindquist et al. 1996 ; Ghanbarian et al. 2013 ; Sobieski et al. 2018 ). The novelty in the presented in the paper approach is connected with the identification and analysis of the main pore channels. Skeleton is retrieved from the pore space, hence the pore space is divided into the branches (pore channel) and the junctions (branches connection point). Most algorithms divide the main pore channel info set of branches. Algorithm of the central axis, which is implemented in the poROSE software, does not divide the main pore channel into smaller sections but identify and analyze the whole main flow path. Figure 3 shows main pore channel marked in red and branches marked in green. Often branches are complex, well-built and creates flow paths, but sometimes are the dead ends. The algorithm works for 21-neigbourhood connectivity. The definition of geodesic tortuosity was described by Hormann et al. ( 2016 ), which is defined by the shortest path between two pores that does not intersect the skeleton. Dijkstra's algorithm is most often used for this case (Roque & Costa 2020 ; Pheng et al. 2023), while start and end points are defined by centroid coordinates. The main advantage in the presented approach is identification of main pore channels, without taking into consideration blind pores. The main pore channel is, among other things, identified by searching for the thickest and longest pore channel, by inscribing a 3D sphere into the path. After main pore channel detection, the algorithm calculate tortuosity, as presented in the Fig. 4 . Thousands of main channels can be found in the pore system depending on the number of separate pore networks. It is all connected with the pore space complexity. Multi linear regression (MLR) was implemented in the research to estimate absolute permeability based on variables (parameters) from CT, MICP, PDP and EPM data. MLR allows finding the relationship between several independent variables and one dependant, in this case absolute permeability (Freund et al. 2006 ): $$\varvec{K}= {\varvec{b}}_{0}+{\varvec{b}}_{1}{\varvec{X}}_{1}+{\varvec{b}}_{2}{\varvec{X}}_{2}+{\varvec{b}}_{3}{\varvec{X}}_{3}+\dots +{\varvec{b}}_{\varvec{n}}{\varvec{X}}_{\varvec{n}}$$ 1 Where: K – dependent variable (absolute permeability); b 0 , b 1 , b 2 , b 3 , …, b n – regression coefficients; X 1 , X 2 , X 3 , …, X n – independent variables; n – number of independent variables. The data was divided into the calibration, validation and testing data sets. Results from MLR can be treated as generalized estimation for tight, low porosity and low permeability rocks. Calculation were carried out in Statistica software (TIBCO 2017) Results and Discussion CT 3D rocks images was transferred into skeleton, which consists of branches (pore channels) and junctions (pore connection point). Basic statistics for parameters from the CT skeleton analysis are depicted in the Table 1 and Fig. 5 . Research material varies in the branches number (pore channels), analysing mean and standard deviation value, what points to diversity in poorly developed pore space. However, this number is still relatively low. On average, five junctions create pore network and three branches merge into the pore junction. Coordination number parameter describes how many branches connect and end in one junction (Wayne 2008 ). Table 1 Basic statistics for parameters from the CT skeleton analysis. Symbols: Ave – average value, CN – coordination number. Parameter N Mean Median Min Max Lower quartile Upper quartile Standard deviation Branches Count 62 2988 1129 23 24428 353 2999 4831 Ave. Junction Sample 62 5 3 1 28 2 6 5 Ave. CN Sample 62 3 3 2 3 3 3 0 Figure 5 shows variety in average number of junctions in the single pore network and consistency in coordination number. Both, average number of junctions and coordination number have similar median value and indicates poorly developed pore system. Table 2 and Figs. 6 – 8 presents basic statistics for parameters from the CT, MICP, PDP and EPM for the analysed rock samples. Tortuosity is around 1.3 (median similar to the average value) for the all analysed samples. It indicates a poorly developed pore space in the all research material. Moreover, average pore diameter from CT is not high, because is around 2 µm. Effective porosity from MICP is quite low, around 3% what is consistent with the information about the percentage of pores with diameters higher than 0.1 µm (important threshold for gas flow regarding gas molecule size) and 1 µm (important threshold for oil flow regarding oil drop size). Swanson parameter was defined for the pore system, not for crack system, and is characteristic for the low porous and low permeable rocks. Absolute permeability is typical as for the tight formations and corresponds with the other parameters from the MICP and CT data. Thus, electrical parameters, as rock formation resistivity, formation factor, cementation exponent and saturation exponent, also assume values reflecting tight rocks. Formation electrical resistivity varies diametrically. Average value and median are not consistent, while standard deviation is quite high. Similarly behaves formation factor, with one difference, the spread in values is even greater than in the rock resistivity. Table 2 Basic statistics for parameters from the CT, MICP, PDP and EPM. Symbols: τ – tortuosity from CT, d CT – pore diameters from CT, Vol – volume of pore space from CT, Φ MICP – effective porosity from MICP, Pores > 0.1 µm – percentage of pores with diameters above 0.1 µm from MICP, Pores > 1 µm – percentage of pores with diameters above 1 µm from MICP, S – Swanson parameter from MICP, K – absolute permeability from pulse or pressure decay measurement, R t – electrical resistivity of formation, F – formation factor, m – cementation exponent, n – saturation exponent, N – number of values, * - geometric mean for absolute permeability Parameter Unit N Mean Median Min Max Standard deviation τ unitless 26 1.365 1.375 1.215 1.487 0.074 d CT µm 62 2.491 2.197 1.092 4.614 0.892 Vol µm 3 62 57 30 8 382 61 Φ MICP frac 51 0.036 0.032 0.002 0.147 0.033 Pores > 0.1 µm % 43 76 81 34 100 21 Pores > 1 µm % 43 58 61 17 100 29 S unitless 25 3.764E-06 3.536E-06 1.350E-07 1.214E-05 3.279E-06 K mD 60 1.463E-02 4.114E-04* 5.180E-05 8.000E-06 2.760E-01 4.987E-02 R t ohmm 49 278 28 7 3279 670 F unitless 49 5244 693 109 81977 16248 m unitless 39 1.665 1.670 1.200 1.980 0.173 n unitless 29 3.228 2.310 1.670 6.200 1.559 Distribution of pore tortuosity is presented in Fig. 6 . Mostly, the tortuosity is in the range of 1.14–1.19. All values of tortuosity can be validated by analysing 3D CT images. In this case, great portion of analysed pore channels have simplified structure, what is connected with the poorly developed pore space and process of deposit sedimentation and consolidation. Lala ( 2020 ) presented micromechanical theory approach to create a novel formula and to estimate the tortuosity in the model from precise experimental measurements. Tortuosity varies from 1.25 to 1.77 for the samples with porosity in the range of 29–44%. Zakirov and Khramchenkov ( 2020 ) found a meaningful effect of pore-level heterogeneity on permeability and tortuosity by investigating fluid flow using lattice Boltzmann simulations. They established the tortuosity in the range of 1.18–1.36 for different types of simulated porous media (nearly round grains). Moreover, Fu et al. ( 2021 ), using 3D CT images of Fontainebleau sandstone, described tortuosity between 1.28 for 24.5% porosity and 1.91 for 8.61% porosity. Figure 7 collectively shows box plots for the logarithm of rock electrical resistivity R t , formation factor F and absolute permeability K , described by center line as median, box edges as quartiles and whiskers as minimum and maximum values. Meanwhile, Fig. 8 presents box plots for the tortuosity τ and cementation factor m in the same box manner. It is worth to mention that cementation factor has low values as expected. Tight formations can be characterized by cementation factor even greater than 2 (around 2.2). Tortuosity corresponds with the amount of pores above 1 µm in diameter from MICP data (Fig. 9 ). There is visible positive relation between the tortuosity and percentages of pores above 1 µm and this relationship indicates that the more pores with diameters greater than 1 µm we observe in the tight rock sample, the more tortuous the pore channels can be expected. Moreover, high correlation is determined between the tortuosity and Swanson parameter (Fig. 10 ). Tortuosity and Swanson parameter is connected with rock ability to fluid flow. Swanson parameter controls the fluid flow because it is main point in the injection saturation of mercury and is estimated in the point at the MICP data curve in which pressure rises rapidly and the mercury fills ever narrower pore channels. Figure 10 interesting relationship for low porous and low permeable rocks. The tortuosity increases with an increase in the Swanson parameter. Swanson parameter is higher when amount of mercury is higher for the lower pressure, what indicates that in low porous and low permeable rocks tortuosity plays a key role. Tortuosity in tight rocks refers to the development of pore network. It is more complicated in conventional deposits. Here, in case of low-porosity and low-permeability rocks it an indicator of the more extensive pore network, if the pore network of tight rocks can be qualified as extensive at all. This relation depicts challenging relationship and can be used as a initial estimation of tortuosity from MICP data in tight rocks. Research was also concentrated on analysis of the connection between the tortuosity from CT and electrical parameters from laboratory measurements on core plugs. Both, tortuosity and electrical parameter are associated with “flow”. The high correlation is observed between the tortuosity and saturation exponent in analyzed tight deposits (Fig. 11 ). Saturation exponent matches dependency on hydrocarbons in the pore system, simply saying it reflects the effect on the resistivity of the sample desaturation process. High values of saturation exponent refers to the oil-wet pore system. Sometimes, the saturation exponent is analyzed qualitatively as a measure of the efficiency for the electrical flow ability within the brine filling a partially saturated rock (Han et al. 2021). Tortuosity and saturation exponent in a sense correspond to each other in the way that high tortuosity leads to complicated flow path and high formation factor (the ratio of the resistivity of a rock filled % with brine to the resistivity of the brine), from the other hand high values of saturation exponent is characteristic for uniform hydrocarbon-wet system described by high values of rock resistivity. Multilinear regression was conducted to estimate absolute permeability taking all parameters into consideration. Absolute permeability form PDP methods were used as a reference. Depending on the data set size one dependent variable – absolute permeability from pulse or pressure decay methods was considered in terms of several independent variables from CT, MICP and EPM. Four different models were obtained including effective porosity from MICP, logarithm of formation factor from EPM, logarithm of rock electrical resistivity from EPM, cementation factor from EPM, saturation exponent from EPM and tortuosity from CT (Table 3 ). Summary results are collected in the Table 3 including parameter used in MLR analysis and determination coefficient for the fit with absolute permeability from PDP. The more weak relation was obtained using effective porosity, logarithm of rock electrical resistivity and tortuosity (R2 = 0.51), while the most strong for the saturation exponent and tortuosity (R2 = 0.77). Moreover, one relationship is worth discussing, namely that consisting of effective porosity, logarithm of formation factor and tortuosity (R2 = 0.72) and presented in Eq. 1 . Usually mercury porosimetry as well as electrical parameters measurement is carried out on every core plug as routine or special core analysis (SCAL). Tortuosity is estimated based on CT measurement and is not a standard laboratory measurement. Combination of these parameters in one equation with quite high determination coefficient gives credence in estimating preliminary absolute permeability based on the data which is executed as standard (MICP, EPM) and data from the non-invasive method (CT). Table 3 Results for MLR analysis based on rock electrical parameters and tortuosity from CT. Symbols: R2 MLR – determination coefficient of MLR, b* – standardized partial regression coefficients, F – formation factor, τ – tortuosity, n – saturation factor, m – cementation factor, R t – electrical resistivity of formation Petrophysical parameter, b* R2 MLR Φ MICP, 0.388; logF, 0.089; τ, 0.789 0.72 Φ MICP, 0.321; m, 0.630; τ, 0.251 0.56 Φ MICP, 0.319; logR t , 0.020; τ, 0,746 0.51 n, 0.730; τ, 0.430 0.77 $$\varvec{l}\varvec{o}\varvec{g}\varvec{K}= -22.1295+\left(19.5795\varvec{*}{\varvec{\phi }}_{\varvec{e}\varvec{f}\varvec{f}}\right)+\left(0.1541\varvec{*}\varvec{l}\varvec{o}\varvec{g}\varvec{F}\right)+\left(14.6972\varvec{*}\varvec{\tau }\right)$$ 1 The best solution was obtained for the Eq. 2 , based on tortuosity and saturation exponent. Absolute permeability can be easily estimated using these two parameters. Detail results of MLR analysis for the best solution is presented in the Table 4 including standardized partial regression coefficient and partial regression coefficient. $$\varvec{l}\varvec{o}\varvec{g}\varvec{K}= -4.18069+\left(-0.00712\varvec{*}\varvec{\tau }\right)+(-0.03334\varvec{*}\varvec{n})$$ 2 Table 4 Results of MLR analysis for the best solution Parameter Standardized partial regression coefficient b * Partial regression coefficient b Intercept -4.181 τ -0.429 -0.007 n -0.727 -0.033 Figure 12 illustrates the comparison between the logarithm of absolute permeability form PDP (logK) and estimated absolute permeability based on MLR (logK MLR). Relationship is good taking into consideration heterogeneous material as tight low-porosity and low-permeability rocks and different lithologies. Al-Anazi and Gates ( 2010 ) presented the result of permeability estimation from well logs using core clustering and BPNN (a nonparametric, nonlinear statistical models for both regression and classification purposes), GRNN (a single-pass nonlinear learning algorithm in a neural network architecture for continuous variables estimation) and SVM (Super Vector Machines) methods. They obtained correlation coefficient for permeability prediction as follows: BPNN – 0.71, for GRNN – 0.73 and for SVM – 0.74. It is worth mentioning, that it is quite high correlation taking into consideration comparison between the core and well logs results. Performed calculations on core samples allowed receiving correlation coefficient equal to 0.88 using only MLR analysis and only core data. Gholami et al ( 2014 ) showed applications of artificial intelligence methods in prediction of permeability in hydrocarbon reservoirs. Two methods were used to predict permeability in this case: RVR (Relevance Vector Regression) and SVR (Support Vector Regression). They compared the result of permeability estimation with the core permeability and obtained determination coefficient about 0.9 for both methods. Tight, low-porosity and low-permeability gas-saturated rocks, cored from the present depth of deposition below 3000 m, are unique and quite difficult geological material, hence the prediction in permeability in the presented MLR case is lower than in mentioned paper. Presented formulas have a limitation and can be tested on tight, gas-bearing formations. Moreover, the equations were calibrated, validated and tested on data from four different lithologies, hence it can have a positive or negative influence. Conclusions The research presents the novel approach in the identification and analysis of the main pore channels based on 3D CT images. Algorithm of the central axis, which was implemented in the research, identifies and analyzes the whole main flow path and calculates tortuosity. The main flow path plays a key role in hydrocarbons and water exploitation. Tight, low-porosity and low-permeability gas-saturated rocks are extremely hard geological material because the variability of petrophysical parameters in the well profile is significant. 3D CT images, together with mercury porosimetry data, pulse- and pressure-decay permeability data, as well as electrical parameters can give a detailed information about the specific parameter distribution. Calculated geometrical tortuosity is around 1.3 for the all analyzed core samples. It indicates a poorly developed pore space in the all research material. Mostly, the detected tortuosity is in the range of 1.14–1.19. High correlation was observed between the tortuosity and Swanson parameter from mercury porosimetry data. Tortuosity and Swanson parameter is connected with rock ability to fluid flow. Moreover, the high correlation was detected between the tortuosity and saturation exponent from electrical parameter measurement in analyzed tight low-porosity and low-permeability deposits. Multilinear regression allows estimating absolute permeability taking CT, MICP and EPM parameters into consideration. Four different models were obtained including effective porosity from MICP, logarithm of formation factor from EPM, logarithm of rock electrical resistivity from EPM, cementation factor from EPM, saturation exponent from EPM and tortuosity from CT. Combination of these parameters in one equation with high determination coefficient gives credence in estimating preliminary absolute permeability based on the data which is executed as standard core analysis (MICP, EPM) and data from the non-invasive method (CT). Declarations Acknowledgements Autor thank Magdalena Habrat for implementing the algorithm to the poROSE software. The author are is grateful to the Polish Oil and Gas Company PKN Orlen Group for providing the data. Funding Research was financed by the National Centre for Research and Development in Poland, program LIDER VI, project no. LIDER/319/L–6/14/NCBR/2015: Innovative method of unconventional oil and gas reservoirs interpretation using computed X-ray tomography. Moreover, paper was financially supported by the Ministry of Education and Science in Poland subsidy 16.16.140.315 for Faculty of Geology Geophysics and Environmental Protection AGH University of Krakow in 2023 year. The research project was partly supported by program “Excellence initiative – research university” IDUB for the AGH University of Krakow (project number 1591). 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Sobieski W, Matyka M, Gołembiewski J, Lipiński S (2018) The path tracking method as an alternative for tortuosity determination in granular beds. Granul. Matter 20(4):72. Soulaine C, Gjetvaj F, Garing C, Roman S, Russian A, Gouze P, Tchelepi HA (2016) The Impact of Sub-Resolution Porosity of X-ray Microtomography Images on the Permeability. Trans Por Media 113:227–243. Swanson BF (1981) A simple correlation between permeabilities and mercury capillary pressures. J Petrol Technol 33(12):2498–2504. Thomeer JH (1983) Air permeability as a function of three pore-network parameters. J Petrol Technol 35(4):809–814. Thovert JF, Salles J, Adler PM (1993) Computerized characterization of the geometry of real porous media: their discretization, analysis and interpretation. J. Microsc. 170(1):65–79. TIBCO Software (2017). Statistica help. On-line version Wayne MA (2008) Geology of Carbonate Reservoirs: The Identification, Description, and Characterization of Hydrocarbon Reservoirs in Carbonate Rocks. Willey & Sons Inc., Hoboken. Zakirov TR, Khramchenkov MG (2020) Prediction of permeability and tortuosity in heterogeneous porous media using a disorder parameter. Chem Eng Sci 227:115893. Cite Share Download PDF Status: Published Journal Publication published 04 Jan, 2024 Read the published version in Acta Geophysica → Version 1 posted Editorial decision: Minor revisions 19 Nov, 2023 Reviewers agreed at journal 07 Nov, 2023 Reviewers invited by journal 06 Nov, 2023 Editor invited by journal 06 Nov, 2023 Editor assigned by journal 01 Nov, 2023 First submitted to journal 27 Oct, 2023 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3500594","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":246643213,"identity":"443044f0-8f46-4305-b48a-fba0b928b87f","order_by":0,"name":"Paulina Krakowska-Madejska","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAy0lEQVRIiWNgGAWjYBACCQYGZiBlA2Yx8AAxH5Fa0hBa2IjUcpgELZKzDz82+Nh2Xl4+uoHxwds2hjyCWqT50owTZ7bdNtx45wCz4dw2hmKCWuR4GIwP8267zbhxRgKbNG8bQ2IbYS3sn4FaztkDtbD/JkqLNA+PcTLvtgOJ8yUS2JiJ0iLZw1NsOPNfcvIGicRmyTnnJAj7ReIM+2aJD2fsbOfPSD744U2ZTR4/IS1wYHCAsQFkRALROhjkGyA0CVpGwSgYBaNgpAAAEY85LHrJaugAAAAASUVORK5CYII=","orcid":"https://orcid.org/0000-0002-8261-4350","institution":"AGH University of Krakow: Akademia Gorniczo-Hutnicza im Stanislawa Staszica w Krakowie","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"Paulina","middleName":"","lastName":"Krakowska-Madejska","suffix":""}],"badges":[],"createdAt":"2023-10-27 20:26:38","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3500594/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3500594/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11600-023-01262-7","type":"published","date":"2024-01-04T15:01:15+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":46102063,"identity":"134a95ef-eec8-474d-a40f-bf7aee419d48","added_by":"auto","created_at":"2023-11-08 16:04:04","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":324602,"visible":true,"origin":"","legend":"\u003cp\u003eExemplary 3D image of pore space of tight, low-porosity and low-permeability carbonates. Colours refer to the pore sizes in voxels (pixel in 3D with the size of 0.5x0.5x0.5 µm) detected in the pore space: red – 832 pores, orange – 706, yellow – 366, green – 175, light blue – 18, dark blue – 1.c\u003c/p\u003e","description":"","filename":"floatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-3500594/v1/9749d9065b3839734dc47b2f.png"},{"id":46102059,"identity":"66228516-f7fd-4bdb-9c28-81d402d04e25","added_by":"auto","created_at":"2023-11-08 16:04:04","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":208494,"visible":true,"origin":"","legend":"\u003cp\u003eScheme of laboratory measurements on geological materials. Symbols: CT – computed X-ray tomography, MICP – mercury injection capillary pressure data (mercury porosimetry), PDP – pulse- / pressure-decay methods for absolute permeability, EPM – electrical parameters laboratory measurement\u003c/p\u003e","description":"","filename":"floatimage2.png","url":"https://assets-eu.researchsquare.com/files/rs-3500594/v1/f022490cadbdc9dd26a531d8.png"},{"id":46103019,"identity":"e0e466cf-fc6e-4ce3-b423-d7e9d5c8d5c0","added_by":"auto","created_at":"2023-11-08 16:12:04","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":253885,"visible":true,"origin":"","legend":"\u003cp\u003eProducts of skeleton analysis on 3D CT images\u003c/p\u003e","description":"","filename":"floatimage3.png","url":"https://assets-eu.researchsquare.com/files/rs-3500594/v1/22088321a43c5bc1902ce2fe.png"},{"id":46103016,"identity":"01ccf11d-9a4b-4e30-819b-2254341464f9","added_by":"auto","created_at":"2023-11-08 16:12:04","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":85385,"visible":true,"origin":"","legend":"\u003cp\u003eTortuosity definition based on 3D CT images. Symbols: \u003cem\u003eτ – \u003c/em\u003etortuosity of the main pore channel, L\u003csub\u003ea\u003c/sub\u003e – actual flow path (actual length of pore channel), L – straight-line distance between the beginning and the end of the pore channel\u003c/p\u003e","description":"","filename":"floatimage4.png","url":"https://assets-eu.researchsquare.com/files/rs-3500594/v1/2b1fb569c4bbc80e26c0a62c.png"},{"id":46102070,"identity":"8691543f-063d-45fd-94fc-84b2e7d7056e","added_by":"auto","created_at":"2023-11-08 16:04:05","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":80462,"visible":true,"origin":"","legend":"\u003cp\u003eBox plots for the average number of junctions and the average coordination number in the samples from CT. Description: center line – median, box edges – quartiles, whiskers – percentiles\u003c/p\u003e","description":"","filename":"floatimage5.png","url":"https://assets-eu.researchsquare.com/files/rs-3500594/v1/ad3589d94109b15358f378e6.png"},{"id":46104138,"identity":"4675981e-40f7-4506-932f-9333dbebc17e","added_by":"auto","created_at":"2023-11-08 16:20:04","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":58427,"visible":true,"origin":"","legend":"\u003cp\u003eTortuosity distribution for all samples\u003c/p\u003e","description":"","filename":"floatimage6.png","url":"https://assets-eu.researchsquare.com/files/rs-3500594/v1/5c5dfa1a6c446ffe544cc09a.png"},{"id":46102068,"identity":"1246515e-3fa2-4826-8bbe-ecd6b53d7664","added_by":"auto","created_at":"2023-11-08 16:04:04","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":69245,"visible":true,"origin":"","legend":"\u003cp\u003eBox plots for the logarithm of rock electrical resistivity \u003cem\u003eR\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e, formation factor \u003cem\u003eF\u003c/em\u003e, absolute permeability \u003cem\u003eK\u003c/em\u003e. Description: center line – median, box edges – quartiles, whiskers – minimum and maximum value\u003c/p\u003e","description":"","filename":"floatimage7.png","url":"https://assets-eu.researchsquare.com/files/rs-3500594/v1/bfd4b2542853b9281cc5d9d0.png"},{"id":46103020,"identity":"787e4e9a-c4a4-40fd-863f-7cd275d32fe6","added_by":"auto","created_at":"2023-11-08 16:12:04","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":73713,"visible":true,"origin":"","legend":"\u003cp\u003eBox plots for the tortuosity \u003cem\u003eτ\u003c/em\u003e and cementation factor \u003cem\u003em\u003c/em\u003e. Description: center line – median, box edges – quartiles, whiskers – minimum and maximum value\u003c/p\u003e","description":"","filename":"floatimage8.png","url":"https://assets-eu.researchsquare.com/files/rs-3500594/v1/c1e913d61d6c44d2216216b4.png"},{"id":46103018,"identity":"59b41912-59b7-4000-9c4c-aaa41004728e","added_by":"auto","created_at":"2023-11-08 16:12:04","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":85541,"visible":true,"origin":"","legend":"\u003cp\u003eRelation between the tortuosity from CT and percentage of pores with dimeter above 1 µm in diameter from MICP\u003c/p\u003e","description":"","filename":"floatimage9.png","url":"https://assets-eu.researchsquare.com/files/rs-3500594/v1/32556ff102ba1588dd69af6f.png"},{"id":46102064,"identity":"56c6375e-3c8f-42d6-a9b1-2f4ce6ae3b8e","added_by":"auto","created_at":"2023-11-08 16:04:04","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":91972,"visible":true,"origin":"","legend":"\u003cp\u003eRelation between the tortuosity from CT and Swanson parameter from MICP\u003c/p\u003e","description":"","filename":"floatimage10.png","url":"https://assets-eu.researchsquare.com/files/rs-3500594/v1/35819ea58666da094655aa54.png"},{"id":46102066,"identity":"c0b197a2-3315-4615-ab9f-ea27cc9a2129","added_by":"auto","created_at":"2023-11-08 16:04:04","extension":"png","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":112361,"visible":true,"origin":"","legend":"\u003cp\u003eRelation between the tortuosity from CT and saturation exponent from electrical measurements on core samples\u003c/p\u003e","description":"","filename":"floatimage11.png","url":"https://assets-eu.researchsquare.com/files/rs-3500594/v1/3563967ac45b7de0dae6842c.png"},{"id":46102065,"identity":"5698b2b2-d7ec-46ef-ad1e-3a253e85af50","added_by":"auto","created_at":"2023-11-08 16:04:04","extension":"png","order_by":12,"title":"Figure 12","display":"","copyAsset":false,"role":"figure","size":109544,"visible":true,"origin":"","legend":"\u003cp\u003eLogarithm of absolute permeability from PDP (logK) versus logarithm of absolute permeability from MLR\u003c/p\u003e","description":"","filename":"floatimage12.png","url":"https://assets-eu.researchsquare.com/files/rs-3500594/v1/207afe5aab294488e3b24b32.png"},{"id":49315491,"identity":"eda61086-9a27-4e4d-9036-fe292799302a","added_by":"auto","created_at":"2024-01-08 15:08:12","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1143299,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3500594/v1/8fca4846-7392-4556-96ce-8e2bbff5c8f4.pdf"}],"financialInterests":"","formattedTitle":"Tortuosity of pore channels in tight rocks as a key parameter in fluid flow ability","fulltext":[{"header":"Introduction","content":"\u003cp\u003eMultidisciplinary laboratory measurements on geological material give the challenging possibility to discover and combine information from different resolutions and physical background. It is extremely important to look closely on the rock by reading petrophysical properties and creating a full image as a one body. Hence, several methods are combined together to check how pore system can behave in low porous and low permeable gas bearing formations. Computed X-ray tomography (CT) is totally safe and high resolution laboratory technique for 3D pore network examination (Ketcham \u0026amp; Carlson \u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e2001\u003c/span\u003e; Cnudde \u0026amp; Boone 2003; Cubit et al. 2009; Adeleye \u0026amp; Akanji \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). Moreover, it is a matter of scale: do we want to look at the rock in low or high resolution, thus look at the rock in centimeters or nanometers (Cnudde et al. \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2011\u003c/span\u003e; Liu et al. \u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e2018\u003c/span\u003e)? In both case, some of the information is missing. First example do not concentrate on the small pores and in case of tight formation is not acceptable because we lose a huge amount of useful data. The advantage of this resolution is that rock examination is on the core or core plus. Second example, looks deeply into the small pores but we read the information form the small piece of rock, often the rock is the size of a crumb.\u003c/p\u003e \u003cp\u003eTortuosity of the pore ganglia is one of the crucial geometric parameter of pore structure and can be evaluated based on CT (Backeberg et al \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2017\u003c/span\u003e; Mohan et al. \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e2023\u003c/span\u003e). There is several methods in tortuosity estimation (Ribeiro et al. \u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e2022\u003c/span\u003e; Mahmood et al. \u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e2023\u003c/span\u003e) but only CT gives the chance to conduct the measurement in 3D. Moreover, tortuosity influences electrical parameters (EPM), as formation factor, saturation exponent, and intrusion parameters from mercury porosimetry (MICP).\u003c/p\u003e \u003cp\u003eThe article presents a new method in tortuosity calculations form the 3D CT image. An attempt was also made to determine the relationship between tortuosity and electrical parameters. Moreover, absolute permeability was evaluated based on the multilinear regression analysis and mentioned parameters. Tortuosity has an enormous impact on rock ability to fluid flow, so the idea of connecting the tortuosity with absolute permeability is significant (Javadpour \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2009\u003c/span\u003e; Berg \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2014\u003c/span\u003e; Kaczmarek et al. \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2017\u003c/span\u003e). Permeability is a challenging parameter, in both measurement and results interpretation (Soulaine et al. \u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e2016\u003c/span\u003e; Ghanizadeh et al. \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2017\u003c/span\u003e). Many researches were devoted to the permeability determination, often through the pore network modelling, as well as advanced statistical methods (Mostaghimi et al. \u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e2013\u003c/span\u003e; Krakowska \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e2019\u003c/span\u003e; Al Balushi \u0026amp; Taleghani \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). Certainly the tortuosity parameter enriches and adds credibility to the obtained results.\u003c/p\u003e"},{"header":"Methods and materials","content":"\u003cp\u003eThe subject of the analysis were tight, low-porosity and low-permeability rocks cored from the present depth of deposition below 3000 m, containing different lithologies, as sandstones, mudstones, limestones and dolomites. The most important link in the research material is connected with the low values of porosity and permeability, meeting the condition of tight, gas-bearing reservoirs. We can expect simplified pore network, revealed in not tortuous pore paths. An example image of the pore space is shown in the Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e (Feldkamp 1984). Colours refer to the size of the pores. Lots of small objects is visible in the 3D CT image, from red (pores below 99 voxels, while voxel is a pixel in 3D) to green (pores below 99999 voxels). Often in tight formations only few objects are present from the highest volume class (blue colour). It is quite typical in tight formations that single extensive pore networks (dark and light blue) can occur in the whole pore system and simple pore networks (red, orange, yellow, green) predominate.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eTortuosity analysis need to connect data from different laboratory methods. First and most precious 3D imaging method is computed X-ray tomography (CT). CT allows to estimate porosity, pore channel size and the most important from the research point \u0026ndash; tortuosity. Often tortuosity is determine form the capillary data or petrographic image analysis. CT definitely gives better insight into all pore channels and overview on pore system in 3D (Rabbani et al. \u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e2016\u003c/span\u003e; Krakowska-Madejska \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e2022\u003c/span\u003e; Moosavi et al. \u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e2023\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eMercury injection capillary pressure data (MICP) answers the question about the potential pore connectivity by mercury injection with the high pressure into the pore space. It appeared that tortuosity can be associated the MICP data, specifically with effective porosity, percentage of pores with diameters above 0.1 \u0026micro;m, percentage of pores with diameters above 1 \u0026micro;m and Swanson parameter (Swanson 1980, Thomeer \u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e1983\u003c/span\u003e, Mao et al. \u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e2013\u003c/span\u003e). Swanson parameter is directly related to rock permeability and indirectly to tortuosity, because it refers to main point in the injection saturation of mercury, which controls the fluid flow.\u003c/p\u003e \u003cp\u003ePressure and pulse decay methods (PDP) in permeability estimation are crucial in tight rock analysis (Handwerger et al. \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2011\u003c/span\u003e). PDP delivers absolute permeability for low porous and low permeable formations. The last important laboratory analysis is connected with the electrical parameters measurement (EPM). It is carried out on core plugs and determine the electrical resistivity of formation, formation factor, cementation exponent and saturation exponent. This mentioned electrical parameters are directly liked with tortuosity. The more tortuous is the pore network, the worse environment for current flow. Scheme of laboratory measurements on geological materials, as well as equipment description is presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eTortuosity is one of the most important parameters in fluid flow considerations. The more pore space is complex, the greater the difficulty of fluid flow. That is why tortuosity can not be omitted in the tight reservoirs analysis, in which all difficulties count in hydrocarbon exploitation. The geometrical tortuosity is the ratio of the actual flow path or simply saying actual length of pore channel and straight-line distance between the beginning and the end of the pore channel (Thovert et al. \u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e1993\u003c/span\u003e; Lindquist et al. \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e1996\u003c/span\u003e; Ghanbarian et al. \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2013\u003c/span\u003e; Sobieski et al. \u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e2018\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eThe novelty in the presented in the paper approach is connected with the identification and analysis of the main pore channels. Skeleton is retrieved from the pore space, hence the pore space is divided into the branches (pore channel) and the junctions (branches connection point). Most algorithms divide the main pore channel info set of branches. Algorithm of the central axis, which is implemented in the poROSE software, does not divide the main pore channel into smaller sections but identify and analyze the whole main flow path. Figure\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e shows main pore channel marked in red and branches marked in green. Often branches are complex, well-built and creates flow paths, but sometimes are the dead ends. The algorithm works for 21-neigbourhood connectivity.\u003c/p\u003e \u003cp\u003eThe definition of geodesic tortuosity was described by Hormann et al. (\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2016\u003c/span\u003e), which is defined by the shortest path between two pores that does not intersect the skeleton. Dijkstra's algorithm is most often used for this case (Roque \u0026amp; Costa \u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e2020\u003c/span\u003e; Pheng et al. 2023), while start and end points are defined by centroid coordinates.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe main advantage in the presented approach is identification of main pore channels, without taking into consideration blind pores. The main pore channel is, among other things, identified by searching for the thickest and longest pore channel, by inscribing a 3D sphere into the path. After main pore channel detection, the algorithm calculate tortuosity, as presented in the Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e. Thousands of main channels can be found in the pore system depending on the number of separate pore networks. It is all connected with the pore space complexity.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eMulti linear regression (MLR) was implemented in the research to estimate absolute permeability based on variables (parameters) from CT, MICP, PDP and EPM data. MLR allows finding the relationship between several independent variables and one dependant, in this case absolute permeability (Freund et al. \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2006\u003c/span\u003e):\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\varvec{K}= {\\varvec{b}}_{0}+{\\varvec{b}}_{1}{\\varvec{X}}_{1}+{\\varvec{b}}_{2}{\\varvec{X}}_{2}+{\\varvec{b}}_{3}{\\varvec{X}}_{3}+\\dots +{\\varvec{b}}_{\\varvec{n}}{\\varvec{X}}_{\\varvec{n}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere: \u003cem\u003eK\u003c/em\u003e \u0026ndash; dependent variable (absolute permeability); \u003cem\u003eb\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e, \u003cem\u003eb\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e, \u003cem\u003eb\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e, \u003cem\u003eb\u003c/em\u003e\u003csub\u003e\u003cem\u003e3\u003c/em\u003e\u003c/sub\u003e, \u003cem\u003e\u0026hellip;, b\u003c/em\u003e\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e \u0026ndash; regression coefficients; \u003cem\u003eX\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e, \u003cem\u003eX\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e, \u003cem\u003eX\u003c/em\u003e\u003csub\u003e\u003cem\u003e3\u003c/em\u003e\u003c/sub\u003e, \u003cem\u003e\u0026hellip;, X\u003c/em\u003e\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e \u0026ndash; independent variables; \u003cem\u003en\u003c/em\u003e \u0026ndash; number of independent variables.\u003c/p\u003e \u003cp\u003eThe data was divided into the calibration, validation and testing data sets. Results from MLR can be treated as generalized estimation for tight, low porosity and low permeability rocks. Calculation were carried out in Statistica software (TIBCO 2017)\u003c/p\u003e"},{"header":"Results and Discussion","content":"\u003cp\u003eCT 3D rocks images was transferred into skeleton, which consists of branches (pore channels) and junctions (pore connection point). Basic statistics for parameters from the CT skeleton analysis are depicted in the Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e and Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e. Research material varies in the branches number (pore channels), analysing mean and standard deviation value, what points to diversity in poorly developed pore space. However, this number is still relatively low. On average, five junctions create pore network and three branches merge into the pore junction.\u003c/p\u003e \u003cp\u003eCoordination number parameter describes how many branches connect and end in one junction (Wayne \u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e2008\u003c/span\u003e).\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eBasic statistics for parameters from the CT skeleton analysis. Symbols: Ave \u0026ndash; average value, CN \u0026ndash; coordination number.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"9\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c9\" colnum=\"9\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eParameter\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eN\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eMean\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eMedian\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eMin\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eMax\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003eLower quartile\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c8\"\u003e \u003cp\u003eUpper quartile\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c9\"\u003e \u003cp\u003eStandard deviation\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBranches Count\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e62\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e2988\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1129\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e23\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e24428\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e353\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e2999\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e4831\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAve. Junction Sample\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e62\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e28\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAve. CN Sample\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e62\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e shows variety in average number of junctions in the single pore network and consistency in coordination number. Both, average number of junctions and coordination number have similar median value and indicates poorly developed pore system.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e and Figs.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e\u0026ndash;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e presents basic statistics for parameters from the CT, MICP, PDP and EPM for the analysed rock samples. Tortuosity is around 1.3 (median similar to the average value) for the all analysed samples. It indicates a poorly developed pore space in the all research material. Moreover, average pore diameter from CT is not high, because is around 2 \u0026micro;m. Effective porosity from MICP is quite low, around 3% what is consistent with the information about the percentage of pores with diameters higher than 0.1 \u0026micro;m (important threshold for gas flow regarding gas molecule size) and 1 \u0026micro;m (important threshold for oil flow regarding oil drop size). Swanson parameter was defined for the pore system, not for crack system, and is characteristic for the low porous and low permeable rocks. Absolute permeability is typical as for the tight formations and corresponds with the other parameters from the MICP and CT data. Thus, electrical parameters, as rock formation resistivity, formation factor, cementation exponent and saturation exponent, also assume values reflecting tight rocks. Formation electrical resistivity varies diametrically. Average value and median are not consistent, while standard deviation is quite high. Similarly behaves formation factor, with one difference, the spread in values is even greater than in the rock resistivity.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eBasic statistics for parameters from the CT, MICP, PDP and EPM. Symbols: \u003cem\u003eτ\u003c/em\u003e \u0026ndash; tortuosity from CT, \u003cem\u003ed CT\u003c/em\u003e \u0026ndash; pore diameters from CT, \u003cem\u003eVol\u003c/em\u003e \u0026ndash; volume of pore space from CT, \u003cem\u003eΦ MICP\u003c/em\u003e \u0026ndash; effective porosity from MICP, \u003cem\u003ePores\u0026thinsp;\u0026gt;\u0026thinsp;0.1 \u0026micro;m\u003c/em\u003e \u0026ndash; percentage of pores with diameters above 0.1 \u0026micro;m from MICP, \u003cem\u003ePores\u0026thinsp;\u0026gt;\u0026thinsp;1 \u0026micro;m\u003c/em\u003e \u0026ndash; percentage of pores with diameters above 1 \u0026micro;m from MICP, \u003cem\u003eS\u003c/em\u003e \u0026ndash; Swanson parameter from MICP, K \u0026ndash; absolute permeability from pulse or pressure decay measurement, \u003cem\u003eR\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e \u0026ndash; electrical resistivity of formation, \u003cem\u003eF\u003c/em\u003e \u0026ndash; formation factor, \u003cem\u003em\u003c/em\u003e \u0026ndash; cementation exponent, \u003cem\u003en\u003c/em\u003e \u0026ndash; saturation exponent, \u003cem\u003eN\u003c/em\u003e \u0026ndash; number of values, * - geometric mean for absolute permeability\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"8\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eParameter\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eUnit\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eN\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eMean\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eMedian\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eMin\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003eMax\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c8\"\u003e \u003cp\u003eStandard deviation\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eτ\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eunitless\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e26\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1.365\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e1.375\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e1.215\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e1.487\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e0.074\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003ed CT\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u0026micro;m\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e62\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e2.491\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e2.197\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e1.092\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e4.614\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e0.892\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eVol\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u0026micro;m\u003csup\u003e3\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e62\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e57\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e30\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e382\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e61\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eΦ MICP\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003efrac\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e51\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.036\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.032\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.002\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.147\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e0.033\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003ePores\u0026thinsp;\u0026gt;\u0026thinsp;0.1 \u0026micro;m\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e%\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e43\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e76\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e81\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e34\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e100\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e21\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003ePores\u0026thinsp;\u0026gt;\u0026thinsp;1 \u0026micro;m\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e%\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e43\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e58\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e61\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e17\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e100\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e29\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eS\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eunitless\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e25\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e3.764E-06\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e3.536E-06\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e1.350E-07\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e1.214E-05\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e3.279E-06\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eK\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003emD\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e60\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1.463E-02\u003c/p\u003e \u003cp\u003e\u003cem\u003e4.114E-04*\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e5.180E-05\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e8.000E-06\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e2.760E-01\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e4.987E-02\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eR\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eohmm\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e49\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e278\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e28\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e3279\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e670\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eF\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eunitless\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e49\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e5244\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e693\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e109\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e81977\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e16248\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003em\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eunitless\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e39\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1.665\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e1.670\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e1.200\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e1.980\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e0.173\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003en\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eunitless\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e29\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e3.228\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e2.310\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e1.670\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e6.200\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e1.559\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eDistribution of pore tortuosity is presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e. Mostly, the tortuosity is in the range of 1.14\u0026ndash;1.19. All values of tortuosity can be validated by analysing 3D CT images. In this case, great portion of analysed pore channels have simplified structure, what is connected with the poorly developed pore space and process of deposit sedimentation and consolidation.\u003c/p\u003e \u003cp\u003eLala (\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e2020\u003c/span\u003e) presented micromechanical theory approach to create a novel formula and to estimate the tortuosity in the model from precise experimental measurements. Tortuosity varies from 1.25 to 1.77 for the samples with porosity in the range of 29\u0026ndash;44%. Zakirov and Khramchenkov (\u003cspan citationid=\"CR44\" class=\"CitationRef\"\u003e2020\u003c/span\u003e) found a meaningful effect of pore-level heterogeneity on permeability and tortuosity by investigating fluid flow using lattice Boltzmann simulations. They established the tortuosity in the range of 1.18\u0026ndash;1.36 for different types of simulated porous media (nearly round grains). Moreover, Fu et al. (\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2021\u003c/span\u003e), using 3D CT images of Fontainebleau sandstone, described tortuosity between 1.28 for 24.5% porosity and 1.91 for 8.61% porosity.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e collectively shows box plots for the logarithm of rock electrical resistivity \u003cem\u003eR\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e, formation factor \u003cem\u003eF\u003c/em\u003e and absolute permeability \u003cem\u003eK\u003c/em\u003e, described by center line as median, box edges as quartiles and whiskers as minimum and maximum values. Meanwhile, Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e presents box plots for the tortuosity \u003cb\u003eτ\u003c/b\u003e and cementation factor \u003cem\u003em\u003c/em\u003e in the same box manner. It is worth to mention that cementation factor has low values as expected. Tight formations can be characterized by cementation factor even greater than 2 (around 2.2).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eTortuosity corresponds with the amount of pores above 1 \u0026micro;m in diameter from MICP data (Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e). There is visible positive relation between the tortuosity and percentages of pores above 1 \u0026micro;m and this relationship indicates that the more pores with diameters greater than 1 \u0026micro;m we observe in the tight rock sample, the more tortuous the pore channels can be expected.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eMoreover, high correlation is determined between the tortuosity and Swanson parameter (Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e). Tortuosity and Swanson parameter is connected with rock ability to fluid flow. Swanson parameter controls the fluid flow because it is main point in the injection saturation of mercury and is estimated in the point at the MICP data curve in which pressure rises rapidly and the mercury fills ever narrower pore channels. Figure\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e interesting relationship for low porous and low permeable rocks. The tortuosity increases with an increase in the Swanson parameter. Swanson parameter is higher when amount of mercury is higher for the lower pressure, what indicates that in low porous and low permeable rocks tortuosity plays a key role. Tortuosity in tight rocks refers to the development of pore network. It is more complicated in conventional deposits. Here, in case of low-porosity and low-permeability rocks it an indicator of the more extensive pore network, if the pore network of tight rocks can be qualified as extensive at all. This relation depicts challenging relationship and can be used as a initial estimation of tortuosity from MICP data in tight rocks.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eResearch was also concentrated on analysis of the connection between the tortuosity from CT and electrical parameters from laboratory measurements on core plugs. Both, tortuosity and electrical parameter are associated with \u0026ldquo;flow\u0026rdquo;. The high correlation is observed between the tortuosity and saturation exponent in analyzed tight deposits (Fig.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e11\u003c/span\u003e). Saturation exponent matches dependency on hydrocarbons in the pore system, simply saying it reflects the effect on the resistivity of the sample desaturation process. High values of saturation exponent refers to the oil-wet pore system. Sometimes, the saturation exponent is analyzed qualitatively as a measure of the efficiency for the electrical flow ability within the brine filling a partially saturated rock (Han et al. 2021). Tortuosity and saturation exponent in a sense correspond to each other in the way that high tortuosity leads to complicated flow path and high formation factor (the ratio of the resistivity of a rock filled % with brine to the resistivity of the brine), from the other hand high values of saturation exponent is characteristic for uniform hydrocarbon-wet system described by high values of rock resistivity.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eMultilinear regression was conducted to estimate absolute permeability taking all parameters into consideration. Absolute permeability form PDP methods were used as a reference. Depending on the data set size one dependent variable \u0026ndash; absolute permeability from pulse or pressure decay methods was considered in terms of several independent variables from CT, MICP and EPM. Four different models were obtained including effective porosity from MICP, logarithm of formation factor from EPM, logarithm of rock electrical resistivity from EPM, cementation factor from EPM, saturation exponent from EPM and tortuosity from CT (Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e). Summary results are collected in the Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e including parameter used in MLR analysis and determination coefficient for the fit with absolute permeability from PDP. The more weak relation was obtained using effective porosity, logarithm of rock electrical resistivity and tortuosity (R2\u0026thinsp;=\u0026thinsp;0.51), while the most strong for the saturation exponent and tortuosity (R2\u0026thinsp;=\u0026thinsp;0.77). Moreover, one relationship is worth discussing, namely that consisting of effective porosity, logarithm of formation factor and tortuosity (R2\u0026thinsp;=\u0026thinsp;0.72) and presented in Eq.\u0026nbsp;\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e1\u003c/span\u003e. Usually mercury porosimetry as well as electrical parameters measurement is carried out on every core plug as routine or special core analysis (SCAL). Tortuosity is estimated based on CT measurement and is not a standard laboratory measurement. Combination of these parameters in one equation with quite high determination coefficient gives credence in estimating preliminary absolute permeability based on the data which is executed as standard (MICP, EPM) and data from the non-invasive method (CT).\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eResults for MLR analysis based on rock electrical parameters and tortuosity from CT. Symbols: R2 MLR \u0026ndash; determination coefficient of MLR, b* \u0026ndash; standardized partial regression coefficients, F \u0026ndash; formation factor, τ \u0026ndash; tortuosity, n \u0026ndash; saturation factor, m \u0026ndash; cementation factor, R\u003csub\u003et\u003c/sub\u003e \u0026ndash; electrical resistivity of formation\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"2\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePetrophysical parameter, b*\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eR2 MLR\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eΦ\u0026nbsp;MICP, 0.388; logF, 0.089; τ, 0.789\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.72\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eΦ\u0026nbsp;MICP, 0.321; m, 0.630; τ, 0.251\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.56\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eΦ\u0026nbsp;MICP, 0.319; logR\u003csub\u003et\u003c/sub\u003e, 0.020; τ, 0,746\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.51\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003en, 0.730; τ, 0.430\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.77\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003cdiv id=\"Equ2\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\varvec{l}\\varvec{o}\\varvec{g}\\varvec{K}= -22.1295+\\left(19.5795\\varvec{*}{\\varvec{\\phi }}_{\\varvec{e}\\varvec{f}\\varvec{f}}\\right)+\\left(0.1541\\varvec{*}\\varvec{l}\\varvec{o}\\varvec{g}\\varvec{F}\\right)+\\left(14.6972\\varvec{*}\\varvec{\\tau }\\right)$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe best solution was obtained for the Eq.\u0026nbsp;\u003cspan refid=\"Equ3\" class=\"InternalRef\"\u003e2\u003c/span\u003e, based on tortuosity and saturation exponent. Absolute permeability can be easily estimated using these two parameters. Detail results of MLR analysis for the best solution is presented in the Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e including standardized partial regression coefficient and partial regression coefficient.\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\varvec{l}\\varvec{o}\\varvec{g}\\varvec{K}= -4.18069+\\left(-0.00712\\varvec{*}\\varvec{\\tau }\\right)+(-0.03334\\varvec{*}\\varvec{n})$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eResults of MLR analysis for the best solution\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eParameter\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eStandardized partial regression coefficient\u003c/p\u003e \u003cp\u003e\u003cem\u003eb\u003c/em\u003e\u003csup\u003e\u003cem\u003e*\u003c/em\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003ePartial regression coefficient\u003c/p\u003e \u003cp\u003e\u003cem\u003eb\u003c/em\u003e\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eIntercept\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e-4.181\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eτ\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-0.429\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e-0.007\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003en\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-0.727\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e-0.033\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig12\" class=\"InternalRef\"\u003e12\u003c/span\u003e illustrates the comparison between the logarithm of absolute permeability form PDP (logK) and estimated absolute permeability based on MLR (logK MLR). Relationship is good taking into consideration heterogeneous material as tight low-porosity and low-permeability rocks and different lithologies.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eAl-Anazi and Gates (\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2010\u003c/span\u003e) presented the result of permeability estimation from well logs using core clustering and BPNN (a nonparametric, nonlinear statistical models for both regression and classification purposes), GRNN (a single-pass nonlinear learning algorithm in a neural network architecture for continuous variables estimation) and SVM (Super Vector Machines) methods. They obtained correlation coefficient for permeability prediction as follows: BPNN \u0026ndash; 0.71, for GRNN \u0026ndash; 0.73 and for SVM \u0026ndash; 0.74. It is worth mentioning, that it is quite high correlation taking into consideration comparison between the core and well logs results. Performed calculations on core samples allowed receiving correlation coefficient equal to 0.88 using only MLR analysis and only core data.\u003c/p\u003e \u003cp\u003eGholami et al (\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2014\u003c/span\u003e) showed applications of artificial intelligence methods in prediction of permeability in hydrocarbon reservoirs. Two methods were used to predict permeability in this case: RVR (Relevance Vector Regression) and SVR (Support Vector Regression). They compared the result of permeability estimation with the core permeability and obtained determination coefficient about 0.9 for both methods. Tight, low-porosity and low-permeability gas-saturated rocks, cored from the present depth of deposition below 3000 m, are unique and quite difficult geological material, hence the prediction in permeability in the presented MLR case is lower than in mentioned paper.\u003c/p\u003e \u003cp\u003ePresented formulas have a limitation and can be tested on tight, gas-bearing formations. Moreover, the equations were calibrated, validated and tested on data from four different lithologies, hence it can have a positive or negative influence.\u003c/p\u003e"},{"header":"Conclusions","content":"\u003cp\u003eThe research presents the novel approach in the identification and analysis of the main pore channels based on 3D CT images. Algorithm of the central axis, which was implemented in the research, identifies and analyzes the whole main flow path and calculates tortuosity. The main flow path plays a key role in hydrocarbons and water exploitation.\u003c/p\u003e \u003cp\u003eTight, low-porosity and low-permeability gas-saturated rocks are extremely hard geological material because the variability of petrophysical parameters in the well profile is significant. 3D CT images, together with mercury porosimetry data, pulse- and pressure-decay permeability data, as well as electrical parameters can give a detailed information about the specific parameter distribution.\u003c/p\u003e \u003cp\u003eCalculated geometrical tortuosity is around 1.3 for the all analyzed core samples. It indicates a poorly developed pore space in the all research material. Mostly, the detected tortuosity is in the range of 1.14\u0026ndash;1.19.\u003c/p\u003e \u003cp\u003eHigh correlation was observed between the tortuosity and Swanson parameter from mercury porosimetry data. Tortuosity and Swanson parameter is connected with rock ability to fluid flow. Moreover, the high correlation was detected between the tortuosity and saturation exponent from electrical parameter measurement in analyzed tight low-porosity and low-permeability deposits.\u003c/p\u003e \u003cp\u003eMultilinear regression allows estimating absolute permeability taking CT, MICP and EPM parameters into consideration. Four different models were obtained including effective porosity from MICP, logarithm of formation factor from EPM, logarithm of rock electrical resistivity from EPM, cementation factor from EPM, saturation exponent from EPM and tortuosity from CT. Combination of these parameters in one equation with high determination coefficient gives credence in estimating preliminary absolute permeability based on the data which is executed as standard core analysis (MICP, EPM) and data from the non-invasive method (CT).\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003eAcknowledgements\u003c/p\u003e\n\u003cp\u003eAutor thank Magdalena Habrat for implementing the algorithm to the poROSE software. The author are is grateful to the Polish Oil and Gas Company PKN Orlen Group for providing the data.\u003c/p\u003e\n\u003cp\u003e\u0026nbsp;\u003cstrong\u003eFunding\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eResearch was financed by the National Centre for Research and Development in Poland, program LIDER VI, project no. LIDER/319/L\u0026ndash;6/14/NCBR/2015: Innovative method of unconventional oil and gas reservoirs interpretation using computed X-ray tomography.\u003c/p\u003e\n\u003cp\u003eMoreover, paper was financially supported by the Ministry of Education and Science in Poland subsidy 16.16.140.315 for Faculty of Geology Geophysics and Environmental Protection AGH University of Krakow in 2023 year. The research project was partly supported by program \u0026ldquo;Excellence initiative \u0026ndash; research university\u0026rdquo; IDUB for the AGH University of Krakow (project number 1591).\u003c/p\u003e\n\u003cp\u003e\u0026nbsp;\u003cstrong\u003eEthics declarations\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConflict of Interest\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe author declare that have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n \u003cli\u003eAl-Anazi A, Gates ID (2010) A support vector machine algorithm to classify lithofacies and model permeability in heterogeneous reservoirs. Eng Geol 114(3\u0026ndash;4):267\u0026ndash;277.\u003c/li\u003e\n \u003cli\u003eArchie GE (1942) The electrical resistivity log as an aid in determining some reservoir characteristics. 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Appl Comp and Geosci 5:100019.\u003c/li\u003e\n \u003cli\u003eSobieski W, Matyka M, Gołembiewski J, Lipiński S (2018) The path tracking method as an alternative for tortuosity determination in granular beds. Granul. Matter 20(4):72.\u003c/li\u003e\n \u003cli\u003eSoulaine C, Gjetvaj F, Garing C, Roman S, Russian A, Gouze P, Tchelepi HA (2016) The Impact of Sub-Resolution Porosity of X-ray Microtomography Images on the Permeability. Trans Por Media 113:227\u0026ndash;243.\u003c/li\u003e\n \u003cli\u003eSwanson BF (1981) A simple correlation between permeabilities and mercury capillary pressures. J Petrol Technol 33(12):2498\u0026ndash;2504.\u003c/li\u003e\n \u003cli\u003eThomeer JH (1983) Air permeability as a function of three pore-network parameters. J Petrol Technol 35(4):809\u0026ndash;814.\u003c/li\u003e\n \u003cli\u003eThovert JF, Salles J, Adler PM (1993) Computerized characterization of the geometry of real porous media: their discretization, analysis and interpretation. J. Microsc. 170(1):65\u0026ndash;79.\u003c/li\u003e\n \u003cli\u003eTIBCO Software (2017). Statistica help. On-line version\u003c/li\u003e\n \u003cli\u003eWayne MA (2008) Geology of Carbonate Reservoirs: The Identification, Description, and Characterization of Hydrocarbon Reservoirs in Carbonate Rocks. Willey \u0026amp; Sons Inc., Hoboken.\u003c/li\u003e\n \u003cli\u003eZakirov TR, Khramchenkov MG (2020) Prediction of permeability and tortuosity in heterogeneous porous media using a disorder parameter. Chem Eng Sci 227:115893.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"acta-geophysica","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"agph","sideBox":"Learn more about [Acta Geophysica](http://link.springer.com/journal/11600)","snPcode":"11600","submissionUrl":"https://www.editorialmanager.com/agph/default2.aspx","title":"Acta Geophysica","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"computed X-ray tomography, tight rocks, pore space, tortuosity, mercury porosimetry MICP, permeability, electrical parameters","lastPublishedDoi":"10.21203/rs.3.rs-3500594/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3500594/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Tortuosity is a significant parameter in porous materials analysis. Not only, when it comes to rocks or soils but also cellular materials, alloys or cells. The multiple definitions exists for tortuosity and several purposes. Geometrical tortuosity describes the pore network paths, on the other hand- thermal, diffusional, electrical and hydraulic tortuosity refers to the transport processes in the pore network. Computed X-ray tomography is the best solution in tortuosity estimation, thanks to the 3D images. In particular, computed X-ray tomography, together with mercury porosimetry, pulse- and pressure-decay permeability methods, as well as electrical parameter measurements, link and expand the information about the tortuosity into the greater meaning. The geological material was composed of tight, low-porosity and low-permeability gas-saturated rocks cored from the present depth of deposition below 3000 m, containing different lithologies, as sandstones, mudstones, limestones and dolomites. The research presents the novel approach in the identification and analysis of the main pore channels based on 3D CT images. Algorithm of the central axis identifies and analyzes the whole main flow path and calculates tortuosity. High correlation was observed between the tortuosity and Swanson parameter from mercury porosimetry data. Moreover, the high correlation was detected between the tortuosity and saturation exponent from electrical parameter measurement in analyzed tight low-porosity and low-permeability deposits.\nMultilinear regression allows estimating absolute permeability taking CT, MICP and EPM parameters into consideration. Combination of these parameters in one equation with high determination coefficient gives credence in estimating preliminary absolute permeability based on the data which is executed as standard core analysis (MICP, EPM) and data from the non-invasive method (CT).","manuscriptTitle":"Tortuosity of pore channels in tight rocks as a key parameter in fluid flow ability","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2023-11-08 16:03:59","doi":"10.21203/rs.3.rs-3500594/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Minor revisions","date":"2023-11-20T02:06:35+00:00","index":"","fulltext":""},{"type":"reviewerAgreed","content":"","date":"2023-11-07T15:14:40+00:00","index":0,"fulltext":""},{"type":"reviewersInvited","content":"","date":"2023-11-06T10:27:53+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"Acta Geophysica","date":"2023-11-06T06:25:59+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2023-11-01T07:33:48+00:00","index":"","fulltext":""},{"type":"submitted","content":"Acta Geophysica","date":"2023-10-27T04:14:16+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"acta-geophysica","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"agph","sideBox":"Learn more about [Acta Geophysica](http://link.springer.com/journal/11600)","snPcode":"11600","submissionUrl":"https://www.editorialmanager.com/agph/default2.aspx","title":"Acta Geophysica","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"c22a2072-4a96-4594-9ec5-a086f02458cc","owner":[],"postedDate":"November 8th, 2023","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2024-01-08T15:03:55+00:00","versionOfRecord":{"articleIdentity":"rs-3500594","link":"https://doi.org/10.1007/s11600-023-01262-7","journal":{"identity":"acta-geophysica","isVorOnly":false,"title":"Acta Geophysica"},"publishedOn":"2024-01-04 15:01:15","publishedOnDateReadable":"January 4th, 2024"},"versionCreatedAt":"2023-11-08 16:03:59","video":"","vorDoi":"10.1007/s11600-023-01262-7","vorDoiUrl":"https://doi.org/10.1007/s11600-023-01262-7","workflowStages":[]},"version":"v1","identity":"rs-3500594","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3500594","identity":"rs-3500594","version":["v1"]},"buildId":"7rjqhiLT3MXkJMwkYKINL","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}