Geofluids

Geofluids / 2020 / Article
Special Issue

Multiscale and Multiphysical Approaches to Fluids Flow in Unconventional Reservoirs

View this Special Issue

Research Article | Open Access

Volume 2020 |Article ID 8866486 | https://doi.org/10.1155/2020/8866486

Taufiq Rahman, Hamed Lamei Ramandi, Hamid Roshan, Stefan Iglauer, "Representative Elementary Volume of Rock Using X-Ray Microcomputed Tomography: A New Statistical Approach", Geofluids, vol. 2020, Article ID 8866486, 13 pages, 2020. https://doi.org/10.1155/2020/8866486

Representative Elementary Volume of Rock Using X-Ray Microcomputed Tomography: A New Statistical Approach

Academic Editor: Jianchao Cai
Received01 Apr 2020
Revised06 Aug 2020
Accepted07 Aug 2020
Published01 Sep 2020

Abstract

Rock heterogeneity is a key parameter influencing a range of rock properties such as fluid flow and geomechanical characteristics. The previously proposed statistical techniques were able to rank heterogeneity on a qualitative level to different extents; however, they need to select a threshold value for determination of representative elementary volumes (REV), which in turn makes the obtained REV subjective. In this study, an X-ray microcomputed tomography (μCT) technique was used to obtain images from different porous media. A new statistical technique was then used to compute REV, as a measure of heterogeneity, without the necessity of defining a threshold. The performance of the method was compared with other methods. It was shown that the calculated sum of the relative errors of the proposed method was lowest compared to the other statistical techniques for all tested porous media. The proposed method can be applied to different types of rocks for more accurate estimation of a REV.

1. Introduction

Accurate determination of rock heterogeneity is critical for a variety of industrial applications; for instance, it plays a key role in determining the reservoir’s ability to recover oil and gas [14], carbon geostorage efficiency [58], contaminant mitigation and natural source zone depletion [9, 10], water discharge and extraction rates [11, 12], or geothermal energy production feasibility [1315]. It is thus essential to understand rock heterogeneity in detail so that reliable predictions can be made and the targeted processes can be further optimised.

Heterogeneity is formed as a result of a verity of geological processes such as deposition, diagenesis, erosion, and structural deformation that ultimately control the geometry of sedimentary deposits [11]. Heterogeneity, thus, is a multiscale phenomenon that reflects the complexity of a geological formation. It is significant at different length scales: from submicron subgrain scale, which may include biogeochemical features such as skeleton structures of biological species forming limestone [16, 17] or grain distribution effects [18, 19], to micrometre heterogeneities induced by different grain shapes and sizes [20], to eventually millimetre to centimetre fractures in the rock generated by geomechanical processes [21, 22] or at even larger scale faults, which have been created by tectonic movements [17, 23].

Knowledge of flow in micrometre pore scale is required to understand macroscopic characteristics of flow in porous media; thus, prior to performing upscaling exercises of petrophysical properties [2427], the accurate characterisation of the porous media at pore scale is crucial for quantifying the heterogeneity of a large system. The pore-scale properties can then be upscaled to core scale or even larger scales. Several works have been conducted to deal with upscaling nonadditive properties of heterogeneous porous media, such as permeability, from the pore scale to the core scale, for example, applying dual-scale pore network models linking pores at different length scales [28] or a direct numerical simulation method for coupling pores at different length scales [4, 29].

One of the methods for quantifying the heterogeneity is using the representative elementary volume (REV), which has been applied to different rock types [3032]. The REV is defined as the minimum volume of a rock that is representative of any larger volume, provided that the next level of heterogeneity at a larger scale has not been reached [33]. This concept is visualised in Figure 1 by plotting porosity against rock volume. The point at which the parameter, i.e., porosity in Figure 1, becomes constant identifies the REV. While a practical REV is required for performing reservoir simulations and providing geological and fluid dynamical predictions, due to experimental limitations originated partly from rock heterogeneity, a constant value cannot be generally obtained. Statistical methods such as direct observation analysis and nonlinear regression modellings, which employ standard deviation of porosity, coefficient of variation of porosity, or relative gradient error of porosity as predictors [20, 3335], are therefore used to provide an estimate of REV. Nevertheless, these methods have some limitations in finding the value of REV. Visual observation, as discussed by Costanza-Robinson et al. [20], has the least accuracy, and the other three methods highly depend on the way of choosing their threshold values [34, 35].

In this work, an X-ray microcomputed tomography (μCT) technique is employed to acquire high-resolution 3D images from different porous media for observing detailed pore morphology at micrometre to millimetre scale. These digital porous media were then subsampled, and their REVs were computed using different methods and compared with a new method proposed in this study. The proposed method uses linear regression to determine the volume at which the standard deviation of porosity is minimised. A weighted regression is then employed in which the estimated REV is imputed, and the corresponding porosity is estimated.

2. Methodology

2.1. Samples

Five porous media with different level of heterogeneities, as presented in Table 1, were selected including a highly homogeneous glass bead pack as a benchmark, a sand pack, two outcrop sandstones (Bentheimer and St. Bees), and one outcrop carbonate (Mount Gambier). In the case of the outcrop rocks, cylindrical core plugs of approximately 5 mm in diameter and 5-10 mm in length were cored from larger blocks. The unconsolidated materials were packed in hollow plastic cylinders.


Porous mediumResolution (μm)Volume (voxels)Volume (mm3)

Glass bead pack3.4016.70
Sand pack4.0928.34
Bentheimer sandstone4.5828.73
St. Bees sandstone3.7820.65
Mount Gambier carbonate4.5831.54

2.2. X-Ray Microcomputed Tomography Imaging

The samples were imaged in a dry state with an Xradia Versa XRM500T μCT instrument at a resolution of ~4 μm3. The subvolumes of the samples (~3 mm3) were imaged to minimise beam hardening, i.e., to improve image quality [36]. The raw images were then filtered using a 3D nonlocal means filter [37] and segmented with a watershed algorithm [38], as shown in Figure 2.

2.3. Digital Measurements

The porosity was calculated by the ratio of the number of voxels in the void phase (pore space) to the total number of voxels (bulk volume) in the segmented image. As shown in Figure 1, the porosity () can vary with sample volume () because of heterogeneity. The images were randomly subsampled into a number of different subvolumes (subsamples), with the condition of no overlap. Figures 3 and 4 exemplify the subsampling method and illustrate the pore space of the digital Bentheimer subsamples with different sizes. Table 2 provides the size of different subsamples. In total, 105 subsamples were acquired. The subsamples ranged from 1003 voxels to voxels, which corresponded to a volume range of ~0.04 mm3 to ~10.38 mm3 for the subsamples with different image resolutions (voxel sizes).


Subsample size (voxels)Number of subsamplesVolume (mm3)
Glass beadsSand packBentheimerSt. BeesMount Gambier

24.257.3910.385.8310.38
32.834.936.923.896.92
42.123.705.192.925.19
41.773.084.332.434.33
61.422.463.461.943.46
61.182.052.881.622.88
81.061.852.601.462.60
80.941.642.311.302.31
80.881.542.161.222.16
80.611.071.500.841.50
100.530.921.300.731.30
100.310.550.770.430.77
120.130.230.320.180.32
200.040.070.100.050.10

3. Results and Discussion

3.1. Visual Interpretation

One of the simplest techniques to determine REV is plotting porosity values against sample volume, as suggested by Bear [33]. Application of this method to the digital porous media is displayed in Figure 5. It is seen that (a) there is a notable variance of porosity associated with the samples from the smallest volume, which is related to heterogeneity [34], and (b), as expected, this variance narrows with the increase of volume. This implies that a REV can be estimated at sufficiently low variance. In terms of point (a), the variance is particularly significant for volumes smaller than 0.1 mm3, and it ranges from ±10% for the highly homogeneous glass bead pack to ±100% for the highly heterogeneous carbonate. The variance of porosity can be thus used as a qualitative indicator for rock heterogeneity.

The results of REV assessment, using Figure 5, by visual inspection, is provided in Table 3. The average porosity (Figure 5) follows the trend described by Bear [33]. However, this is a highly subjective analysis. In particular, the technique of looking at the data points does not provide a clear indication of where exactly the REV need to be placed.


Porous mediumPorosity sample (%)VariationREV (mm3)

Glass bead pack35.35±10%2.8
Sand pack35.18±67%4.9
Bentheimer sandstone18.07±30%>10
St. Bees sandstone16.27±100%>7.5
Mt. Gambier carbonate40.54±100%>10

Of the largest volume investigated. Porosity variation for 0.1 mm3 subvolumes. By visual inspection.
3.2. Standard Deviation Analysis

Another method for identifying the cut-off (i.e., the REV value) is applying a mathematical criterion. The unbiased standard deviation , Equation (1), [39] was used to perform the analysis on the samples. Recalling that the sample volume was subsampled into the th size, where is the standard deviation (unbiased estimate), is the porosity measured for the th subvolume (), is the arithmetic average of the porosity of the th subvolume, is the total number of subsamples taken, is the smallest subsample, and is the largest subsample; the porosity variance as a function of volume can then be assessed by plotting versus subsample volume (Figure 6).

As seen in Figure 6, the standard deviation rapidly drops with increasing subvolume, but the slope of the decay also rapidly decreases, approximately following a power-law relation , where and are the least square nonlinear regression fitting parameters and is the subsample volume (Table 4). While the power-law exponents are all in the range (), coefficient causes the largest difference in terms of REV. Note that the most homogenous sample, i.e., the glass bead pack, has the smallest coefficient and exponent, and the most heterogeneous sample, i.e., the carbonate, has the largest coefficient . The exponent is likely related to the grain size as the sand pack has the largest and the largest grains.


Porous mediumFitting equation

Glass bead pack V-0.2980.3785-0.298
Sand pack V-0.7442.1761-0.697
Bentheimer sandstone V-0.3621.2637-0.362
St. Bees sandstone V-0.4211.2302-0.421
Mt. Gambier carbonate V-0.3824.3488-0.382

The standard deviation curves in general give a good indication of heterogeneity, and different samples can be readily compared. However, a threshold value is required to determine the REV. Table 5 shows that a smaller threshold value resulted in a larger REV. At the same time, a smaller threshold had higher accuracy. REV thus strongly depends on the required accuracy. The experimental accuracy of a standard helium pycnometer, ±0.5% [40], was used as a threshold value, Figure 6 (dashed black line). It was found that the glass bead pack consistently had the smallest REV (0.4 mm3 at 0.5% threshold) indicating that the sample was very homogenous, in contrast to Mt. Gambier, which always had the highest REV and thus was a very heterogeneous rock.


Porous mediumThreshold (% porosity)REV (mm3) from data curveREV (mm3) from fitting equation

Glass bead pack0.50.30.40
Sand pack0.54.857.40
Bentheimer0.54.310.40
St. Bees sandstone0.55.1>6
Mt. Gambier carbonate0.5>10.5>10.5
Glass bead pack10.070.05
Sand pack12.853.10
Bentheimer12.32.00
St. Bees sandstone12.11.60
Mt. Gambier carbonate1>10.5>10.5
Glass bead pack20.050.05
Sand pack20.81.13
Bentheimer20.450.29
St. Bees sandstone20.150.32
Mt. Gambier carbonate24.17.50

The results were consistent with data reported by Zhang et al. [35], who measured a REV of 5 mm3 for Brent Triassic sandstone and a REV of 2 mm3 for a crushed glass bead pack (0.5% threshold), and Stroeven et al. [39] conclude that the REV for a densely packed system is significantly lower than for a less dense pack, as presented in Figure 2 and Table 5. It was found that the grain size had likely a major impact on REV, which was also reflected in the relatively high REV values of the sand pack (Table 5); even though the sand pack was comparatively significantly more symmetrically formed, it had large grains.

3.3. Coefficient of Variation

The coefficient of variation, CV, Equation (2) [41], is unitless and therefore useful when comparing variation between datasets with substantial differences in their means. CV can be used for a REV analysis in a similar way as .

The plot of CV as a function of subvolume (Figure 7) shows that CV rapidly decreases with increasing sample volume. The CV values again can be approximated using power-law relations (Table 6), similar to , as CV and are closely related (Equations (1) and (2)). The exponents and the ratios between the coefficients of CV (Table 6) and (Table 4) fitting equations are similar, although the nominal values are different. Similar to the standard deviation analysis, a cut-off value needs to be selected to obtain REV. Three thresholds (Table 7) were tested, and it was again observed that REV strongly depended on the threshold value.


Porous mediumFitting equation

Glass bead pack V-0.2950.0106-0.295
Sand pack V-0.6860.0611-0.686
Bentheimer sandstone V-0.3740.0705-0.374
St. Bees sandstone V-0.3660.0716-0.366
Mt. Gambier carbonate V-0.3890.1058-0.389


Porous mediumThreshold (-)REV (mm3) from data curveREV (mm3) from fitting equation

Glass bead pack0.0250.090.05
Sand pack0.0252.953.55
Bentheimer0.025>10.50>10.50
St. Bees sandstone0.0255.40>6.00
Mt. Gambier carbonate0.025>10.50>10.50
Glass bead pack0.0500.010.01
Sand pack0.0501.951.33
Bentheimer0.0502.702.50
St. Bees sandstone0.0502.232.70
Mt. Gambier carbonate0.0504.006.90
Glass bead pack0.0750.010.01
Sand pack0.0750.500.74
Bentheimer0.0750.850.85
St. Bees sandstone0.0750.180.85
Mt. Gambier carbonate0.0751.952.45

Table 7 indicates that the carbonate rock had the largest heterogeneity as expected, followed by St. Bees sandstone, while the glass bead pack was the most homogeneous media. By looking at the μCT images, it seemed that the Bentheimer was more homogeneous than St. Bees because of its more symmetrical structure. However, the grain size in St. Bees was substantially smaller than in Bentheimer, as seen in Figure 2; thus, the grain size was a key factor, which strongly influenced REV.

Zhang et al. [35] measured a REV of 0.02 mm3 for a Brent Triassic sandstone () at a . Their result is consistent with the observations in this study; nevertheless, using delivers results that are more compatible with pycnometric measurements (Table 7). The consequence of using such a CV cut-off is, however, noticeable; REV increases ~100-fold, assuming the sandstones are somewhat similar, which seems likely. This threshold value can thus considerably change the REV and has to be selected carefully, and it needs to be compatible with any subsequent analysis, e.g., NMR measurements, which can observe individual atoms [42], and may need a different accuracy level than for instance capillary pressure-water saturation measurements where volume balances in core plugs are considered [43].

3.4. Relative Gradient Error Criterion

An alternative method for determining REV is the relative gradient error () analysis, Equation (3) [20]: where ϕ is the porosity, is the subsample number, and is the volume of the subsample.

Figure 8 presents the plot of computed for all samples versus sample volume; again, it is seen that rapidly drops with increasing subvolume, and the data are much closer to each other especially after 1 mm3 subvolume. This is also reflected in the statistical fits through the data points (Table 8), where generally high power-law exponents (~-0.5 to -2.2 with a median value of -1.6) were calculated.


Porous mediumFitting equation

Glass bead pack V-1.6750.0014-1.675
Sand pack V-1.6290.0124-1.629
Bentheimer sandstone V-0.5910.0102-0.591
St. Bees sandstone V-2.1950.0083-2.195
Mt. Gambier carbonate V-1.5730.0100-1.573

Similar to the other statistical methods, to determine REV, a threshold value had to be selected, which similarly showed a significant influence of the threshold value (Table 9). This influence was, however, smaller than where or CV analysis was used, mainly because of the lower sensitivity of analysis, i.e., more similar numbers and curves generally need to be compared in the analysis. For instance, the glass bead pack had values smaller than 0.1 mm3 for all thresholds tested while the REVs of the other four samples could hardly be accurately distinguished with the approach. Note that Costanza-Robinson et al. [20] prescribed as a threshold value, which, however, led to low accuracy in this case (Table 8). It, therefore, seems that the and CV analysis are both superior to the method while the relative gradient error method is still superior to the visual method, consistent with Costanza-Robinson et al. [20].


Porous mediumThreshold (-)REV (mm3) from data curveREV (mm3) from fitting equation

Glass bead pack0.0250.100.10
Sand pack0.0251.501.25
Bentheimer0.0251.352.25
St. Bees sandstone0.0250.851.00
Mt. Gambier carbonate0.0251.221.28
Glass bead pack0.0500.100.10
Sand pack0.0500.850.82
Bentheimer0.0501.100.74
St. Bees sandstone0.0500.800.72
Mt. Gambier carbonate0.0501.100.71
Glass bead pack0.0750.100.10
Sand pack0.0750.720.65
Bentheimer0.0750.920.61
St. Bees sandstone0.0750.760.60
Mt. Gambier carbonate0.0750.990.56

3.5. Regression Modelling

To avoid the threshold choice in the above methods and therefore have a unique solution to REV value, a new method to estimate the volume of each rock is proposed in this study, at which point the variance of the porosity is minimised. To achieve this, a linear regression model is used, where volume is modelled as a function of the standard deviation of porosity. The standard deviations of the smallest subvolumes are identified as outliers and excluded from the model as they added unnecessary noise. To ensure the normality of the residual errors, the volume is modelled as : where is the volume, is the standard deviation of the corresponding volume, is the constant, is the slope parameters, and is the error term of the regression equation.

To minimise the standard deviation in this equation, it is assumed to have a theoretical value of zero. When is zero, the equation becomes , and therefore, the volume at which the standard deviation of the porosity is minimised is , i.e., the exponential of the -intercept. While this provides an estimated REV, the percent porosity of which the REV corresponds to is also calculated by employing a weighted linear regression of present porosity as a function of : where is the percent porosity, represent the constant, is the slope term of this model, and is the error term.

The regression is weighted by the reciprocal of the standard deviation at each volume. Once the constant and slope terms are calculated, the REV for that particular rock is substituted into the equation, thus returning its corresponding percent porosity. The results for each rock are presented in Table 10 in their untransformed original units. The back-transformed regression equations are then shown visually in Figure 9.


RockEstimated REV (mm3)Porosity (%) estimate at estimated REV

Glass beads6.235.3
Sand pack6.135.0
Bentheimer14.918.6
St. Bees6.915.3
Mt. Gambier19.142.3

3.6. Comparison of the Techniques

Sum of the relative errors (unitless ratio) of regression [44], obtained from each technique, is used to compare the above techniques appropriately: where abs is the absolute value, Re is the sum of the relative error of regression, is the model estimate, and is the exact value.

Table 11 lists the sum of relative errors obtained from each model fit. As seen in Table 11, the proposed regression technique presented the lowest errors of the regression. Note that the average porosity was used in the model to have only one measurement per subvolume to be consistent with other techniques. In addition, in the majority of cases, the proposed method predicted a REV larger than the other methods. However, the comparison of the estimates with the plotted data indicated that the proposed regression method was very sensible both in terms of prediction and error propagation. Many of the subsamples from the rocks still showed a considerable variation of the porosity at the estimates of REV obtained by other techniques. A notable advantage of the proposed method is that it removes the subjectivity of either visually estimating REV or choosing a threshold to estimate REV.


Porous mediumStandard deviation ()Coefficient of variation (CV)Relative gradient error ()Regression modelling ()

Glass bead pack3.31163.284222.72270.0552
Sand pack3.94853.85697.62580.1872
Bentheimer sandstone2.91532.94829.52890.2919
St. Bees sandstone4.7374.961240.51220.2660
Mt. Gambier carbonate2.62842.682119.7690.3842

4. Conclusion

Quantification of rock heterogeneity can be accomplished using a REV, i.e., the larger the REV, the higher the heterogeneity. A range of porous media was imaged using the μCT technique and then subsampled (105 subsamples for each porous medium, a total of 525 subsamples). Different statistical methods with which REV can be estimated, including visual interpretation, standard deviation analysis, coefficient of variation analysis, and relative gradient error criterion analysis, were tested on the subsamples and compared with the new regression method proposed in this study.

The results indicate that the visual inspection approach is the least accurate approach. The relative gradient error, the standard deviation, and the coefficient of variation analyses have a higher degree of accuracy than the visual inspection approach. However, the results highly depend on the selected threshold values. Furthermore, it is not possible to determine a true REV in a theoretical sense or base on selecting threshold values. The proposed regression modelling method, however, does not rely on a visual inspection and threshold value selection. The sum of relative errors of regression is also the lowest using the proposed technique. The method gives larger REV, which is satisfactory as many of the subsamples from different rocks show considerable variation in porosity at the estimated REV obtained by other techniques. In addition, it is shown that the grain size has a profound impact on REV, i.e., the larger the grains, the larger the REV, and the samples with a very ordered and symmetric structure can have a large REV if they contain large grains.

Data Availability

The data supporting the conclusions of the study is already presented in the manuscript, and readers can download from the paper or can contact the authors.

Conflicts of Interest

There are no conflicts of interest.

Acknowledgments

The authors wish to acknowledge the financial assistance provided through the Australian National Low Emissions Coal Research and Development (ANLEC R&D), project 3-0911-0155. ANLEC R&D is supported by Australian Coal Association Low Emissions Technology Limited and the Australian Government through the Clean Energy Initiative. The μCT measurements were performed using the μCT system courtesy of the National Geosequestration Laboratory (NGL) of Australia. The NGL is a collaboration between Curtin University, CSIRO, and the University of Western Australia established to conduct and deploy critical research and development to enable commercial-scale carbon storage options. Funding for this facility was provided by the Australian Federal Government.

References

  1. N. R. Morrow, Interfacial Phenomena in Petroleum Recovery, CRC Press, 1991.
  2. M. Blunt, F. J. Fayers, and F. M. Orr Jr., “Carbon dioxide in enhanced oil recovery,” Energy Conversion and Management, vol. 34, no. 9-11, pp. 1197–1204, 1993. View at: Publisher Site | Google Scholar
  3. S. Du, S. Pang, and Y. Shi, “Quantitative characterization on the microscopic pore heterogeneity of tight oil sandstone reservoir by considering both the resolution and representativeness,” Journal of Petroleum Science and Engineering, vol. 169, pp. 388–392, 2018. View at: Publisher Site | Google Scholar
  4. S. Norouzi Apourvari and C. H. Arns, “Image-based relative permeability upscaling from the pore scale,” Advances in Water Resources, vol. 95, pp. 161–175, 2016. View at: Publisher Site | Google Scholar
  5. J.-C. Perrin and S. Benson, “An experimental study on the influence of sub-core scale heterogeneities on co2 distribution in reservoir rocks,” Transport in Porous Media, vol. 82, no. 1, pp. 93–109, 2009. View at: Google Scholar
  6. S. C. M. Krevor, R. Pini, L. Zuo, and S. M. Benson, “Relative permeability and trapping of CO2and water in sandstone rocks at reservoir conditions,” Water Resources Research, vol. 48, no. 2, 2012. View at: Publisher Site | Google Scholar
  7. T. Rahman, M. Lebedev, A. Barifcani, and S. Iglauer, “Residual trapping of supercritical co2 in oil-wet sandstone,” Journal of Colloid and Interface Science, vol. 469, pp. 63–68, 2016. View at: Publisher Site | Google Scholar
  8. N. Wei, M. Gill, D. Crandall et al., “CO2flooding properties of liujiagou sandstone: influence of sub-core scale structure heterogeneity,” Greenhouse Gases: Science and Technology, vol. 4, no. 3, pp. 400–418, 2014. View at: Publisher Site | Google Scholar
  9. K. Sookhak Lari, G. B. Davis, J. L. Rayner, T. P. Bastow, and G. J. Puzon, “Natural source zone depletion of lnapl: a critical review supporting modelling approaches,” Water Research, vol. 157, pp. 630–646, 2019. View at: Publisher Site | Google Scholar
  10. K. Sookhak Lari, J. L. Rayner, and G. B. Davis, “Towards characterizing lnapl remediation endpoints,” Journal of Environmental Management, vol. 224, pp. 97–105, 2018. View at: Publisher Site | Google Scholar
  11. C. E. Koltermann and S. M. Gorelick, “Heterogeneity in sedimentary deposits: a review of structure-imitating, process-imitating, and descriptive approaches,” Water Resources Research, vol. 32, no. 9, pp. 2617–2658, 1996. View at: Publisher Site | Google Scholar
  12. J. C. Maréchal, S. Lanini, B. Aunay, and P. Perrochet, “Analytical solution for modeling discharge into a tunnel drilled in a heterogeneous unconfined aquifer,” Groundwater, vol. 52, no. 4, pp. 597–605, 2014. View at: Publisher Site | Google Scholar
  13. Q. Zhou, J. T. Birkholzer, C.-F. Tsang, and J. Rutqvist, “A method for quick assessment of co2 storage capacity in closed and semi-closed saline formations,” International Journal of Greenhouse Gas Control, vol. 2, no. 4, pp. 626–639, 2008. View at: Publisher Site | Google Scholar
  14. T. R. Elliot, T. A. Buscheck, and M. Celia, “Active co2reservoir management for sustainable geothermal energy extraction and reduced leakage,” Greenhouse Gases: Science and Technology, vol. 3, no. 1, pp. 50–65, 2013. View at: Publisher Site | Google Scholar
  15. T. Dewers, P. Eichhubl, B. Ganis et al., “Heterogeneity, pore pressure, and injectate chemistry: control measures for geologic carbon storage,” International Journal of Greenhouse Gas Control, vol. 68, pp. 203–215, 2018. View at: Publisher Site | Google Scholar
  16. J. R. Jones, G. Poologasundarampillai, R. C. Atwood, D. Bernard, and P. D. Lee, “Non-destructive quantitative 3d analysis for the optimisation of tissue scaffolds,” Biomaterials, vol. 28, no. 7, pp. 1404–1413, 2007. View at: Publisher Site | Google Scholar
  17. K. Bjorlykke, Petroleum Geoscience: From Sedimentary Environments to Rock Physics, Springer Science & Business Media, 2010.
  18. R. Al-Raoush and A. Papadopoulos, “Representative elementary volume analysis of porous media using x-ray computed tomography,” Powder Technology, vol. 200, no. 1-2, pp. 69–77, 2010. View at: Publisher Site | Google Scholar
  19. Y. Bazaikin, B. Gurevich, S. Iglauer et al., “Effect of CT-image size and resolution on the accuracy of rock property estimates,” Journal of Geophysical Research: Solid Earth, vol. 122, no. 5, pp. 3635–3647, 2017. View at: Publisher Site | Google Scholar
  20. M. S. Costanza-Robinson, B. D. Estabrook, and D. F. Fouhey, “Representative elementary volume estimation for porosity, moisture saturation, and air-water interfacial areas in unsaturated porous media: data quality implications,” Water Resources Research, vol. 47, no. 7, 2011. View at: Publisher Site | Google Scholar
  21. S. Iglauer, S. Wang, and V. Rasouli, “X-ray micro-tomography measurements of fractured tight sandstone,” in SPE Asia Pacific Oil and Gas Conference and Exhibition, Society of Petroleum Engineers, 2011. View at: Google Scholar
  22. J. P. Morris, R. L. Detwiler, S. J. Friedmann, O. Y. Vorobiev, and Y. Hao, “The large-scale geomechanical and hydrogeological effects of multiple co2 injection sites on formation stability,” International Journal of Greenhouse Gas Control, vol. 5, no. 1, pp. 69–74, 2011. View at: Publisher Site | Google Scholar
  23. E. Dombrádi, D. Sokoutis, G. Bada, S. Cloetingh, and F. Horváth, “Modelling recent deformation of the pannonian lithosphere: lithospheric folding and tectonic topography,” Tectonophysics, vol. 484, no. 1-4, pp. 103–118, 2010. View at: Publisher Site | Google Scholar
  24. S. Rolland du Roscoat, M. Decain, X. Thibault, C. Geindreau, and J. F. Bloch, “Estimation of microstructural properties from synchrotron x-ray microtomography and determination of the rev in paper materials,” Acta Materialia, vol. 55, no. 8, pp. 2841–2850, 2007. View at: Publisher Site | Google Scholar
  25. T. Kanit, F. N’Guyen, S. Forest, D. Jeulin, M. Reed, and S. Singleton, “Apparent and effective physical properties of heterogeneous materials: representativity of samples of two materials from food industry,” Computer Methods in Applied Mechanics and Engineering, vol. 195, no. 33-36, pp. 3960–3982, 2006. View at: Publisher Site | Google Scholar
  26. M. J. King, P. R. King, C. A. McGill, and J. K. Williams, “Effective properties for flow calculations,” Transport in Porous Media, vol. 20, no. 1-2, pp. 169–196, 1995. View at: Publisher Site | Google Scholar
  27. O. Rozenbaum and S. R. du Roscoat, “Representative elementary volume assessment of three-dimensional x-ray microtomography images of heterogeneous materials: application to limestones,” Physical Review E, vol. 89, no. 5, 2014. View at: Publisher Site | Google Scholar
  28. A. Moctezuma, S. Bekri, C. Laroche, and O. Vizika, “A dual network model for relative permeability of bimodal rocks: application in a vuggy carbonate,” in Proceedings of the International Symposium of the Society of Core Analysts, pp. 1–13, Pau, France, 2003. View at: Google Scholar
  29. S. N. Apourvari and C. H. Arns, “An assessment of the influence of micro-porosity for effective permeability using local flux analysis on tomographic images,” in IPTC 2014: International Petroleum Technology Conference: European Association of Geoscientists & Engineers, Pau, France, 2014. View at: Google Scholar
  30. S. M. Shah, J. P. Crawshaw, F. Gray, J. Yang, and E. S. Boek, “Convex hull approach for determining rock representative elementary volume for multiple petrophysical parameters using pore-scale imaging and Lattice–Boltzmann modelling,” Advances in Water Resources, vol. 104, pp. 65–75, 2017. View at: Publisher Site | Google Scholar
  31. J. O. Adeleye and L. T. Akanji, “Pore-scale analyses of heterogeneity and representative elementary volume for unconventional shale rocks using statistical tools,” Journal of Petroleum Exploration and Production Technology, vol. 8, no. 3, pp. 753–765, 2018. View at: Publisher Site | Google Scholar
  32. S. Jackson, Q. Lin, and S. Krevor, “Representative elementary volumes, hysteresis, and heterogeneity in multiphase flow from the pore to continuum scale,” Water Resources Research, vol. 56, no. 6, article e2019WR026396, 2020. View at: Google Scholar
  33. J. Bear, Dynamics of Fluids in Porous Media, Courier Corporation, 2013.
  34. M. Halisch, “The rev challenge–estimating representative elementary volumes and porous rock inhomogeneity from high resolution micro-ct data sets,” in Proceedings of International Symposium of the Society of Core Analysts, Napa Valley, California, USA, 2013. View at: Google Scholar
  35. D. Zhang, R. Zhang, S. Chen, and W. E. Soll, “Pore scale study of flow in porous media: scale dependency, rev, and statistical rev,” Geophysical Research Letters, vol. 27, no. 8, pp. 1195–1198, 2000. View at: Publisher Site | Google Scholar
  36. S. R. Stock, Microcomputed Tomography, CRC Press, 2018.
  37. A. Buades, B. Coll, and J. M. Morel, “A non-local algorithm for image denoising,” in 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05), IEEE, vol. 2, pp. 60–65, San Diego, CA, June 2005. View at: Publisher Site | Google Scholar
  38. S. Schlüter, A. Sheppard, K. Brown, and D. Wildenschild, “Image processing of multiphase images obtained via x-ray microtomography: a review,” Water Resources Research, vol. 50, no. 4, pp. 3615–3639, 2014. View at: Publisher Site | Google Scholar
  39. M. Stroeven, H. Askes, and L. J. Sluys, “Numerical determination of representative volumes for granular materials,” Computer Methods in Applied Mechanics and Engineering, vol. 193, no. 30-32, pp. 3221–3238, 2004. View at: Publisher Site | Google Scholar
  40. S. Iglauer, M. A. Fernø, P. Shearing, and M. J. Blunt, “Comparison of residual oil cluster size distribution, morphology and saturation in oil-wet and water-wet sandstone,” Journal of Colloid and Interface Science, vol. 375, no. 1, pp. 187–192, 2012. View at: Publisher Site | Google Scholar
  41. M. G. Vangel, “Confidence intervals for a normal coefficient of variation,” The American Statistician, vol. 50, no. 1, pp. 21–26, 1996. View at: Google Scholar
  42. O. Talabi, S. AlSayari, S. Iglauer, and M. J. Blunt, “Pore-scale simulation of nmr response,” Journal of Petroleum Science and Engineering, vol. 67, no. 3-4, pp. 168–178, 2009. View at: Publisher Site | Google Scholar
  43. C. H. Pentland, R. El-Maghraby, S. Iglauer, and M. J. Blunt, “Measurements of the capillary trapping of super-critical carbon dioxide in Berea sandstone,” Geophysical Research Letters, vol. 38, no. 6, 2011. View at: Publisher Site | Google Scholar
  44. M. ÇIMEN, “Estimation of daily suspended sediments using support vector machines,” Hydrological Sciences Journal, vol. 53, no. 3, pp. 656–666, 2009. View at: Publisher Site | Google Scholar

Copyright © 2020 Taufiq Rahman et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


More related articles

 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views551
Downloads342
Citations

Related articles

Article of the Year Award: Outstanding research contributions of 2020, as selected by our Chief Editors. Read the winning articles.