# Relationship between porosity and specific surface area

Surface Area, Pore Size and Density, Kluwer Academic Publisher, Dordrecht, G. Ertl porosity size. Structure of ZSM-5 shape. Aerosil S = m2/g SAXS parameters (mean size / size distribution / specific surface area) are .. Relationship between BET C constant and cross-sectional area. the relationship between the density of the test material and the density of a Porosity. Mass of the solid, g. Total surface area, m2. Specific Pore volume, cm3/ g. There is a reasonable correlation between the porosity and permeability of cores having Several correlations are developed that relate specific surface area of.

In the adsorption based methods, Methylene Blue Dye, as well as gas adsorption, has been used to determine the surface area of clay minerals for several decades [ 5 ]. Fluid flow methods which are basically an application of Kozeny-Carman equation are widely treated in the literature. This equation is the most famous permeability-porosity relation, which is widely used in the field of flow in porous media and is the starting point for many other permeability models [ 78 ].

The equation can be written as follows [ 4 ]: The optical method which is based on statistical image processing has been developed by Chalkley [ 10 ]. The method is based on random tests on two- dimensional images from cross sections of synthetic porous medium [ 10 ].

In this regard a specified needle with length of is randomly thrown on the surface of cross section and the number of pores and grains on which the needle is laying should be counted [ 10 ]. They tried to correlate the porosity of porous structure versus the specific surface of its particles and verified the results by measuring the specific surface using adsorption methods.

Berryman [ 12 ] modified the method used by Debye et al. Berryman and Blair [ 13 ] utilized scanning electron microscopy SEM to estimate porous medium porosity and specific surface by applying two-point correlation functions.

They compared the results obtained from the Kozeny-Carman equation and experimental permeability measurement to adjust the two-point correlation function for finding the specific surface. Berryman and Blair [ 14 ] extended their work to consider tortuosity and electrical formation factor to calculate specific surface using data available for Berea sand stone. Furthermore, they discussed how to choose the optimum magnification of imaging in order to estimate the specific surface from rock cross sections.

They expressed that the specific surface of porous media is a function of the fractal dimensions and microstructural parameters. In addition, they found that the 2D specific surface specific perimeter is in relation to the 3D specific surface, which is dependent on the grain size and fractal dimension.

Okabe and Blunt [ 17 ] utilized multiple-point statistics based on two-dimensional thin sections as training images to generate geologically realistic 3D pore-space representations. As they stated, the thin section images can provide multiple-point statistics, which describe the statistical relation between multiple spatial locations and use the probability of occurrence of particular patterns.

Using the generated 3D structure they predicted the absolute permeability of the sandstone sample using the effective medium theory and lattice Boltzmann simulation method.

Okabe and Blunt [ 18 ] used the approximated value of specific surface from 2D images to reconstruct 3D images of rock.

In this regard, they tried to generate a 3D volume with the same specific surface as the estimated value from 2D images. A method of estimating the specific surface from 2D images has been previously developed by Adler et al. Their method uses binary SEM images of rock to obtain a reconstruction method based on specific surface measurements. One main challenge in the application of multiple-point statistics to the simulation of three-dimensional 3D blocks is the lack of suitable 3D training images.

In order to estimate the specific surface value of rock grains, cross-sectional images of different rocks have been utilized by coupling image analysis methods and statistical calculations.

Initially, the specific surface of different rock structures has been calculated by computational methods with high precision using Micro-CT images provided by Blunt et al. Sahimi [ 24 ] stated that photomicrographs of polished sections of a sample porous medium with sufficient contrast between the pores and the matrix can be used to determine the specific surface of rock grains.

Sahimi [ 24 ] stated that from the relation between 2D surface measurements and the properties of the 3D system, an estimate of specific surface can be obtained. The main objective in this study is to correlate the three-dimensional value of grain specific surface versus 2D characteristics of rock sections. The study here also presents a straightforward relationship between specific surface and the ratio of particle perimeter by its area in 2D images. Methodology One of the main challenges in extracting data from 2D images of porous medium is how to section the medium.

This can be normally done by various angles and positions while the perpendicular orientation is more common. The true values of porosity and specific surface may not be visualized in sectional images because of limited field of view in two dimensions. In this work, the key parameter in estimation of specific surface is the ratio of grain particle perimeter to the area of each grain in 2D section called specific perimeter. The definition of specific perimeter can be expressed as In order to examine the specific surface of grains in more detail, each 3D sample of rock has been gridded to smaller cubic parts.

Calculation of specific surface from 3D images in each lattice requires counting the pixels of the image. This however depends on the image resolution. Estimation of specific perimeter requires a specified pattern in sectioning. A similar approach of cross sectioning is used in this work to consider various probabilities and complexities of grain structure. Figure 1 illustrates the pattern of cross sections in the plane of - and -axis named as direction.

The black and white parts lying under the cross lines present pores and grains of the rock, respectively. Each cross section indicates a 2D image which was analyzed by computer coding to find the specific perimeter. Next step is to average the data to upscale the results. Averaging specific perimeter in direction can be written as where is normalized specific perimeter and is a function that gives back the single value of specific perimeter for a cross section with an angle.

Here, is the average of specific perimeter for all possible angles of cross sections. Figure 2 presents a 3D view of one grid from a binary image of Berea sandstone as one of the rock types investigated in the current work.

Binary data are provided from Blunt et al. Schematic pattern of degrees of cross sectioning in direction of. Consequently, an averaged value for specific perimeter in each arbitrary section can be obtained as The plane that contains each of, and is clarified in Figure 3.

According to the integral definition, by simplifying the denominator, 6 changes to Figure 3: Clarification of angles and their plates for integral on cross sections.

Although specific perimeter is a function of cross sectioning an angle, its dependency to sectioning angle cannot be formulized easily. Due to the unstructured and random shape of rock grains, the value of specific perimeter fluctuates randomly versus angle of cross section. Figure 4 shows the variations of specific perimeter for 6 different samples of Berea sandstone versus 90 degrees of cross section rotation. Also, there is a similar situation for one of the rock samples Berea sandstone in the different plane of angles labelled as, and.

Figure 5 presents the changes of specific perimeter for different angles of cross section from 0 to degrees for this sample. As clarified in Figure 1in order to have an overall view of rock specific perimeter, all possible angles of cross section should be observed and recorded for each rock piece.

The variations of specific perimeter versus angle of cross section for 6 different samples of Berea sandstone. The fluctuation of specific perimeter for different planes of angles, and in Berea sandstone. Results and discussions where bn is the correlation constant by considering n elements in porosity power series.

This study has been performed on 10 types of sandstones for Fig. Every rock type is divided to increasing the power of porosity in the series n. The average value for correlation constant random cross section images of rock samples used in this study.

### Determination of Specific Surface of Rock Grains by 2D Imaging

In Berea tion of elements in the porosity geometric sequence can be written sandstone and also Sandstones 1, 3 and 4, the granular structure as cannot be easily detected because of compaction or cementation of bulk volume. The length below each sandstone using porosity and average grain size is: Sandstones used for optimizing correlation form and constants. This distribution graphs are obtained by dividing the from 0.

Homogenous samples such as Sandstones 3 and digital rocks to subsections. The porosity in different samples changes These values are considered as real or measured values.

Probability distribution of the correlation constant b for different Fig. Probability distribution of porosity for different rock types. The average of peaks of distributions is about 4. Table 1 presents the summary of results for samples. Grain shape factor of the studied rocks has been estimated in range of about 4. Probability distribution of grain size for different rock types. Grain shape factor of 6 delineates a Again, there is a wide range of variation for different samples porous medium consisting of uniform pack of spheres with no which limits the applicable precision of the presented correla- cementation or compaction.

In real rocks, grains are attached and tion.

If geometry complexity and grain surface A. The correlation presented in this work also is applicable for loose or poorly cemented media. Points which are dropped in the porosity and average grain size of rock.

This relationship, which vicinity of the unit slope line present better predictions.

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Here the value is about and poorly cemented media. Although the variety of fabrics and pore 0. Finally, using the presented correlation, a expands the range of applicability of this relationship. The correlation series of arbitrary datasets have been evaluated and result is has been evaluated for samples of 10 different sandstones and presented in Fig. Though there is ment leads to a new form of Carman—Kozeny equation which is noticeable differences in fabric and pore structure between samples, 32 A.

Principles of heat transfer in porous media.

### The relationship between porosity and specific surface in human cortical bone is subject specific.

This applicability and simplicity can be stated as two main Springer; A method for estimating volume—surface advantages of this correlation. Practical stereological methods for References morphometric cytology. J Cell Biol ; Scattering by an inhomogeneous solid.

The correlation function and its application. J Appl Phys ; J Math Phys ; Kozeny—Carman relations and image processing Harcourt Brace; Unit operations of chemical engineering. Gulf Professional media from two-dimensional cuts. Phys Rev E ; A three-parameter Kozeny—Carman rock physics: Chem Eng Sci ; Pore space reconstruction using multiple-point statistics.