Available online at http://ijcpe.uobaghdad.edu.iq and www.iasj.net 

Iraqi Journal of Chemical and Petroleum 
 Engineering  

Vol.21 No.3 (September 2020) 57 – 66 
EISSN: 2618-0707, PISSN: 1997-4884 

 

Corresponding Authors:  Name: Yahya Jirjees Tawfeeq, Email: yahyapetroleum@uokirkuk.edu.iq, Name: Jalal A. Al-Sudani, Email:  

jalsud@uobaghdad.edu.iq  
IJCPE is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. 

 

Digital Rock Samples Porosity Analysis by OTSU Thresholding 

Technique Using MATLAB 

 
Yahya J. Tawfeeq and Jalal A. Al-Sudani 

 
Petroleum Engineering Department, College of Engineering, University of Baghdad, Baghdad, Iraq  

 

Abstract 

 
   Porosity plays an essential role in petroleum engineering. It controls fluid storage in aquifers, connectivity of the pore structure 

control fluid flow through reservoir formations. To quantify the relationships between porosity, storage, transport and rock 

properties, however, the pore structure must be measured and quantitatively described. Porosity estimation of digital image utilizing 

image processing essential for the reservoir rock analysis since the sample 2D porosity briefly described. The regular procedure 

utilizes the binarization process, which uses the pixel value threshold to convert the color and grayscale images to binary images. The 

idea is to accommodate the blue regions entirely with pores and transform it to white in resulting binary image. This paper presents 

the possibilities of using image processing for determining digital 2D rock samples porosity in carbonate reservoir rocks. MATLAB 

code created which automatically segment and determine the digital rock porosity, based on the OTSU's thresholding algorithm. In 

this work, twenty-two samples of 2D thin section petrographic image reservoir rocks of one Iraqi oil field are studied. The examples 

of thin section images are processed and digitized, utilizing MATLAB programming. In the present study, we have focused on 

determining of micro and macroporosity of the digital image. Also, some pore void characteristics, such as area and perimeter, were 

calculated. Digital 2D image analysis results are compared to laboratory core investigation results to determine the strength and 

restrictions of the digital image interpretation techniques. Thin microscopic image porosity determined using OTSU technique 

showed a moderate match with core porosity. 
         
Keywords: Digital rock physics, OTSU thresholding, Thin section image, Porosity, Macro pores, Micro pores 
 

Received on 26/06/2020, Accepted on 22/08/2020, published on 30/09/2020 
 
https://doi.org/10.31699/IJCPE.2020.3.8  

 
1- Introduction 
 

   Image analysis has been used for many years to extract 

relevant information from digital microscopic images. 

Image analysis includes all operations required to obtain 

quantified image information.  

   The typical image analysis sequence involves image 

acquisition, processing, segmentation, measurements, data 

processing and interpretation. Segmentation of images 

considers one of the essential techniques utilized to divide 

the image into its integrated portions for extracting the 

relevant image information [1].  

   Briefly, the segmentation of the image transforms the 

representation of an image into its simplified shape that 

can be examined more critically and naturally [2], [3].   

   Several practical image segmentation applications are 

available such as trace tumors and additional pathologies 

[4], [5], machine vision, object detection [6], face 

detection, medical imaging [7], [8], anatomic building 

studies and diagnoses [9], fingerprint recognition and 

video surveillance.  

   Several techniques of image segmentation, such as 

thresholding [10], edge-based segmentation [11], and 

compression-based methods [12], have been taken during 

recent decades.  

 

   In image processing techniques, many algorithms used, 

such as Artificial Neural Network [13], Convolutional 

Neural Network [14], and K - Nearest Neighbors. In all 

image segmentation methods, the simplest and most 

relevant and useful technique of dividing an image into 

the front class and the background class is thresholding 

technique [15].  

   The thresholding process converts the grayscale image 

into the binary image depending on the threshold values. 

The important of the thresholding process is to select an 

optimal threshold value when there are more threshold 

levels are implemented. Several methods of thresholding 

are currently employed, including OTSU technique, 

clustering [16] and utmost entropy technique [17].  

   OTSU method is fast and ease of coding thresholding 

method among all the purposes mentioned above.   

   Because the OTSU threshold operates on histograms 

(which are integer or float arrays of length 256) it's quite 

fast and approximately 90 lines MATLAB code needed.   

   However, OTSU technique is a histogram-based [18] 

threshold approach for automatic thresholding of the 

image. OTSU's algorithm suggests that the image can be 

divided into two main categories: foreground and 

background.  

 

http://ijcpe.uobaghdad.edu.iq/
http://www.iasj.net/
mailto:yahyapetroleum@uokirkuk.edu.iq
mailto:jalsud@uobaghdad.edu.iq
mailto:jalsud@uobaghdad.edu.iq
http://creativecommons.org/licenses/by-nc/4.0/
https://doi.org/10.31699/IJCPE.2020.3.8


Y. J. Tawfeeq and J.A. Al-Sudani / Iraqi Journal of Chemical and Petroleum Engineering 21,3 (2020) 57 - 66 

 

 

58 
 

   The algorithm designed to find the best threshold value 

that divides the histogram into two classes that maximizes 

the variance between two classes. The development of the 

current OTSU threshold to the multi-level threshold 

referred to as the multi-OTSU thresholds [19]. Porosity is 

the porous space fraction that the rock matrix does not 

occupy [20]. A comprehensive study of the distribution of 

porosity is essential for the reservoir evaluation project 

[21]. Porosity is a crucial property of rock due to 

measuring potential storage volume of hydrocarbons. In 

the carbonate reservoir, porosity ranges about 0.01-0.35 

[22]. The volume fraction of void spaces, i.e. non-rock 

space, divided by the total volume of the sample defined 

as porosity [23]. High porosity values indicate high 
capacities of the reservoir rocks to contain these fluids, 

while low porosity values indicate the opposite [24].  

   The porosity evaluated either through formation 

evaluation logs or through laboratory measurements on 

core samples. A general industry practice is to regard core 
measurements as ground truth. However, there can be un-

certainties associated with core measurements especially 

when laboratory conditions ignored under which core 

measurements were made.  
   There are certain factors that control porosity of a 
formation, pore and grain size distribution, mineralogy, 

sorting and diagenesis, etc. [25], [26]. Characterizing 

these controlling factors require advanced logging and 

special analysis on core data (SCAL), yet some of these 

properties require alternate interpretation techniques. 

‘Digital image analysis of thin sections’ is presented as 

this alternate technique.  
   Porosity, mineralogy, pore size distribution and sorting 
analyzed through digital image analysis of thin sections. 

In this study, OTSU's thresholding implemented for 

microscopic image segmentation. The samples of 

microscopic images are processed utilizing MATLAB 

programming. In the present study, microporosity and 

macroporosity of the digital image are determined. Also, 

some pore void characteristics, such as area and 

perimeter, were calculated. 

 

2- Material and Methods 
 

   In this work, twenty-two samples of 2D thin section 

petrographic image used for analyses from the core plugs 

taken from the Buzurgan oil field. Each sample was 

impregnated with blue-dyed epoxy, thin sectioned and 

then was stained for discrimination of carbonate minerals, 

the scanned image has resolution of about 10 𝜇m/pixel.     
   The procedure of scanning and digitizing the image 
called ‘optical microscopy’ and is of lower resolution as 

compared to digital images obtained from ‘scanning 

electron microscopy’. The advantage of the former is that 

it is a fast technique to obtain digital images and a 

disadvantage that pore sizes less than 10 𝜇m cannot be 
quantitatively resolved with optical microscopy. In image 

processing, OTSU's technique used to implement 

automatic image thresholding, and this method named 

after Japanese scientist Nobuyuki OTSU [10].  

   The algorithm in the simplest form returns a single 

threshold of intensity to separate the pixels into two 

groups or classes, foreground and background. The 

algorithm searches deeply for the threshold that 

maximizes the variance of between-class or minimizes the 

variance of within-class.  

   The fundamental concept is that suitable threshold 

classes must separate the intensity values in terms of their 

pixel and, conversely, that the optimum threshold would 

be a threshold providing the best class separation in terms 

of intensity values [27].  

   OTSU's technique has the significant property, in 

relation to its optimality, that it is entirely based on 

computations executed on an image's histogram, an easily 

accessible 1-D array. Let {0, 1, 2 … … … . . , 𝐿 − 1} are the 
separate intensity levels in a digital image of size (M.N) 

(row and column dimensions of the image) pixels, and 

assume (ni) is the number of pixels with intensity (i). The 

total image pixel is; 

𝑀 𝑁 =  𝑛0 + 𝑛1 + 𝑛2+, , , , , , , , , +𝑛𝐿−1. the normalized 
histogram contains parts as [10]; 

 

𝑝𝑖 =
𝑛𝑖

𝑀𝑁
                                                                                                 (1) 

 
∑ 𝑝𝑖 = 1  𝑎𝑛𝑑  𝑝𝑖 ≥ 0

𝐿−1
𝑖=0                                                                       (2) 

 

Where: 

M.N = row and column dimensions of the image 

ni = the number of pixels with intensity (i) 

Pi = probability distribution of intensity (i) 

i = intensity 

   Now, assume that a threshold (T) was selected with 

value (0 < T < L-1), and utilized this threshold to separate 

the digital input image into two classes or groups, (C1) 

and (C2). Where (C1) involves all the image pixels that 

have intensity values ranges (0, T) and (C2) includes all 

the image pixels that have intensity values ranges (T+1, 

L-1). Utilizing this threshold, probability 𝑃1(𝑇) of class 
C1 (i.e. Background class) is given by the cumulative 

sum; 

 

𝑃1(𝑇) = ∑ 𝑝𝑖
𝑇
𝑖=0  =  ∑

𝑛𝑖

𝑀𝑁

𝑇
𝑖=0                                                                  (3) 

  

 

Where: T = threshold. 

In the same way, the probability of the second class 𝑃2(𝑇)  
(class C2, or Foreground class) is given by; 

 

𝑃2(𝑇) = ∑ 𝑝𝑖
𝐿−1
𝑖=𝑇+1 = 1 −  𝑃1(𝑇)                                                   (4) 

 

   The class C1 mean intensity value of the pixels is given 

by; 

 

𝜇1(𝑇) = ∑ 𝑖 𝑝(
𝑖

𝐶1
)𝑇𝑖=0 =  ∑ 𝑖 𝑝(

𝐶1

𝑖
)𝑇𝑖=0 (

𝑃(𝑖)

𝑃(𝐶1)
) =

1

𝑃1(𝑇)
∑ 𝑖 𝑝𝑖

𝑇
𝑖=0              (5) 

 

Where: 𝜇1(𝑇) is mean intensity value for class C1. Where 

𝑃1(𝑇) is given in Eq. (2). The term𝑝 (
𝑖

𝐶1
)is the probability 

of value (i) which derives from class (C1).  



Y. J. Tawfeeq and J.A. Al-Sudani / Iraqi Journal of Chemical and Petroleum Engineering 21,3 (2020) 57 - 66 

 

 

59 
 

   The second part of the Equation comes from Bayes' 

formula. The third part comes from the fact 𝑝 (
𝐶1

𝑖
) =

1, probability of (C1) for given (i) equal to (1), since only 
class (C1) values is considered. While 𝑃(𝑖)is the 
probability of the (ith) value, which is simply the (ith) 

component of histogram(𝑝𝑖 ). Finally, 𝑃(𝐶1) is the 
probability of class (C1) that is equal to Eq. (3). In the 

same way, the class (C2) means intensity value of pixels 

given by; 

 

𝜇2(𝑇) = ∑ 𝑖 𝑝(
𝑖

𝐶2
)𝐿−1𝑖=𝑇+1 =

1

𝑃2(𝑇)
∑ 𝑖 𝑝𝑖

𝐿−1
𝑖=𝑇+1                                   (6) 

 

Where: 𝜇2(𝑇) is mean intensity value for class C2. The 

term𝑝 (
𝑖

𝐶2
)is the probability of value (i) which derives 

from class (C2). 

The average intensity of the entire image (i.e., the global 

mean) given by; 

 

𝜇𝐺 = ∑ 𝑖 𝑝(𝑖)
𝐿−1
𝑖=0                                                                                    (7) 

 

Where: 𝜇𝐺  = is a global mean or average intensity of the 
entire image 

By substitute of the previous results, the validity of the 

following two equations can be confirmed: 

 

𝑃1𝜇1 + 𝑃2𝜇2 = 𝜇𝐺                                                                                 (8) 

 

And 

 

𝑃1 + 𝑃2 = 1                                                                                           (9) 
 

Where:  

𝑃1 = Background class (class C1) probability. 
𝑃2= Foreground class (class C2) probability. 
 𝜇1 = mean intensity value for class C1 
𝜇2 = mean intensity value for class C2 
 𝜇𝐺  = is a global mean or average intensity of the entire 
image 

We can use the normalized, dimensionless metric to 

assess the "goodness" of the threshold at level (T)[28]; 

 

𝜂 =
𝜎𝐵

2

𝜎𝐺
2                                                                                                 (10) 

 

Where;  

(𝝈𝑮
𝟐 )= global variance (the variance in intensity of all 

pixels in image) 

 𝜂 = dimensionless metric 
(𝜎𝐵

2) = between-class variance 
 

The global variance and it is given in Equation below 

[10]; 

 

𝜎𝐺
2 = ∑ (𝑖 − 𝜇𝐺 )

2𝑝𝑖
𝐿−1
𝑖=0                                                                         (11) 

 

And between-class variance  (𝜎𝐵
2) is given in Equation 

below; 

 

𝜎𝐵
2 =  𝑃1(𝜇1 − 𝜇𝐺 )

2 + 𝑃2(𝜇2 − 𝜇𝐺 )
2                                                 (12) 

 

Equation (12) can also be written as; 

 

𝜎𝐵
2(𝑇) = 𝑃1  𝑃2[𝜇1 − 𝜇2]

2                                            (13) 

 

   Eq. (13) showed that the 𝜎𝐵
2 will be larger whenever the 

two means (𝜇1) and (𝜇2) are farther from each other, 
demonstrating that (between-class variance) separability 

measure between classes. Since (𝜎𝐺
2)  is constant for a 

given image; therefore the (𝜂)is a measure of separability 
also, and maximizing this dimensionless metric is 

equivalent to maximizing (𝜎𝐵
2). Note that Eq. (10) 

assumes implicitly that (𝜎𝐺
2 > 0). This variance can only 

be zero if all intensity levels in the image are the same, 

which implies that only one class of pixels exists. This, in 

turn, means that (𝜂 = 0) for a constant image since 
separability of single class from itself is zero. the final 

results yield when (T)reintroduced again: 

 

𝜂(𝑇) =
𝜎𝐵

2 (𝑇)

𝜎𝐺
2                                                                                        (14) 

 

And 

 

𝜎𝐵
2(𝑇) = 𝑃1(𝑇)[(𝜇1(𝑇) − 𝜇𝐺 )]

2 + 𝑃2(𝑇) [(𝜇2(𝑇) − 𝜇𝐺 )]
2               (15) 

 

Then, the best threshold value is the, (T*) that maximizes 

𝜎𝐵
2(𝑇) : 

 

𝜎𝐵
2(𝑇∗) = 𝑚𝑎𝑥0≤𝑇≤𝐿−1𝜎𝐵

2(𝑇)                                                             (16) 

 

OTSU's algorithm also is defined as a weighted sum of 

the two classes' variances [10]: 

 

𝜎𝑊
2 = 𝑃1(𝑇) 𝜎1

2(𝑇) + 𝑃2(𝑇)𝜎2
2(𝑇)                                                     (17) 

 

Where: weights  (𝑷𝟏) and (𝑷𝟐) are the probabilities of 
the background (class C1) and foreground(class C2) 

classes respectively, separated by a threshold (T) are 

stated previously. While𝝈𝟏
𝟐(𝑇)is the variance of the pixels 

in the background (below threshold), 𝝈𝟐
𝟐(𝑇) is the 

variance of the pixels in the foreground (above threshold), 

given in Equations below; 

 

𝜎1
2(𝑇) =

1

𝑃1(𝑇)
∑ [𝑖 − 𝜇1(𝑇)]

2 𝑝(𝑖) 𝑇𝑖=0                                            (18)                                                     

 

𝜎2
2(𝑇) =

1

𝑃2(𝑇)
∑ [𝑖 − 𝜇2(𝑇)]

2 𝑝(𝑖) 𝐿−1𝑖=𝑇+1                                 (19) 

 

 

And 𝝈𝑾
𝟐 (𝑻) is within-class variance.  

Then, the best threshold value is the, T* that 

minimizes𝜎𝐵
2(𝑇) ; 

 

𝜎𝑊
2 (𝑇∗) = 𝑚𝑖𝑛0≤𝑇≤𝐿−1𝜎𝑤

2 (𝑇)                                                             (20) 



Y. J. Tawfeeq and J.A. Al-Sudani / Iraqi Journal of Chemical and Petroleum Engineering 21,3 (2020) 57 - 66 

 

 

60 
 

   For two classes, OTSU showed that minimizing the 

within-class variance is the same as maximizing within-

class variance, in another meaning; subtracting within-

class variance from the total variance get something 

called the between-class varianceσB
2 (T) [29].The best 

threshold using the OTSU method is the one maximizing 

overall between classes variance or minimizing overall 

within-class variance. In Fig. 1-a there is a simple 

bimodal distribution with two homogeneous classes 

where the threshold value (T) can be easily determined. If 

there is no valley one as shown in Fig. 1-b the method of 

determining threshold value (T) is minimizing the total 

variance within both classes or maximizes the overall 

variance between both classes. The best threshold is 

maximizing the between classes variance or, contrariwise, 

minimizing the within-class variance[30-31]. 

 

 
Fig. 1. Typical Image Histogram Show (1-a) Simple 

Bimodal Distribution (1-b) No-Bimodal Distribution 

 

   In this study, OTSU algorithm implementation started 

with converting the color image to grayscale image and 

plotting the input image's normalized histogram. 

According to the threshold value, the histogram pixels are 

separated into two clusters or class. The cumulative sums 

and the cumulative means for each class are calculated 

using Eq. (3) through (6). The total (global) intensity 

means it is calculated using Eq. (7). The between-class 

variance 𝜎𝐵
2(𝑇) is calculated using Eq. (13), where the 

farther apart the means, the larger will be 𝜎𝐵
2(𝑇). The 

maximum"between-class variance 𝜎𝐵
2(𝑇) is set as an 

optimum OTSU threshold value (T*). When the 

maximum is not unique, the values corresponding to the 

detected maximum can be (T *) averaged. Finally, 

optimum separability measure, (𝜂 ∗)at (T = T*) is 
obtained using Eq. (14). The separability is a measure of 

how easily separable the classes are. A uniform 

distribution is (0), and a clear, bimodal is (1). 

 

3- Calculations and Analysis 

 

3.1. OTSU Thresholding Implementation 

 

   The objective of this section is to introduce and apply 

the binary segmentation algorithm of OTSU on the 

samples under study. MATLAB code created which 

automatically segment and determine the digital rock 

porosity, based on the OTSU's thresholding algorithm 

using the following main MATLAB code; 

 

 

 

 
 

   As a first step, a pixel-value histogram using MATLAB 

image analysis toolbox plotted for each digital image, as 

shown in Fig. (2), because the used image type is an 8-bit 

image, there were (256) possible pixel values. The 

histogram height with 256 bins is calculated, where each 

bin's height is equal to the number of pixels with that 

pixel value from (0) to (255). For each given image, the 

probability of the pixel value (i) calculated using Eq. (3) 

by separating the height of bin by the complete number of 

pixels in the histogram. Since the goal is to maximize 

between-class variance, each class mean, global intensity 

means and variances of both classes calculated using Eq. 

(5), (7) and (15), respectively. 

 

 
Fig. 2. Pixel-Value Image Histogram for Sample No.(7) 

 

 

function level = otsu(histogramCounts) 

total = sum(histogramCounts); % total number of pixels in 

the image  

%% OTSU automatic thresholding 

top = 256; 

sumB = 0; 

wB = 0; 

maximum = 0.0; 

sum1 = dot(0:top-1, histogramCounts); 

for ii = 1:top 

    wF = total - wB; 

    if wB > 0 && wF > 0 

        mF = (sum1 - sumB) / wF; 

        val = wB * wF * ((sumB / wB) - mF) * ((sumB / wB) - 

mF); 

        if ( val >= maximum ) 

            level = ii; 

            maximum = val; 

        end 

    end 

    wB = wB + histogramCounts(ii); 

    sumB = sumB + (ii-1) * histogramCounts(ii); 

end 

end 

 



Y. J. Tawfeeq and J.A. Al-Sudani / Iraqi Journal of Chemical and Petroleum Engineering 21,3 (2020) 57 - 66 

 

 

61 
 

   Finally, the OTSU threshold value (TB) obtained as a 

value of for which 𝜎𝐵
2(𝑇) is maximum. For not unique 

maximum threshold value, the OTSU threshold value (TB) 

obtained by averaging the values of corresponding to the 

various maxima detected. Additionally, the separability 

measure, 𝜼 ∗  was calculated using Eq. (14) at (T = TB).  It 
is sufficient to increase between class variance Eq. (13) 
this will decrease within-class variance too. Therefore, 

only the "between-class variance” calculated for each 

threshold and picked the threshold that maximizes the 

variance. Depending on each microscopic image quality, 

OTSU's algorithm was run several times for better results. 

The MATLAB function (im2bw) is used to convert an 

intensity image to a binary image. The binary image level 

which is a normalized intensity value that lies in the range 

(0, 1) was calculated depending on optimal threshold 

value (TB). The results of threshold, class variance and 

separability criterion are for twenty-two used samples are 

listed in the Table 1. The results of OTSU thresholding 

are exposed in Fig. 3 for some analyzed samples, for 

example. It can be observed that the threshold of the 

OTSU has divided the digital image into two levels: white 

(porous) and Black (matrix) background. 

 

3.2. Porosity and Pore Space Characteristics 

Determination 

 

   A digital image comprises pixels, which are building 

blocks of an image. Core samples image used in this 

study were cropped at (637x478) pixels; hence the total 

number of pixels in a sample is 304486 pixels. Core thin 

section samples consist of empty pore space filled with 

the blue liquid epoxy and solid grains comprising of 

different minerals colors, as shown in Fig. 3. The 

definition of porosity from image analysis can be written 

in pixels term as [32-33]; 

 

𝑃𝑜𝑟𝑜𝑠𝑖𝑡𝑦 (𝐼𝑚𝑎𝑔𝑒) =
∑ 𝑝𝑖𝑥𝑒𝑙𝑠 𝑖𝑛 𝑝𝑜𝑟𝑒 𝑠𝑝𝑎𝑐𝑒

𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑖𝑥𝑒𝑙𝑠
                                        (21) 

 

Table 1. Results of OTSU Thresholding Method 

Parameters 

CoreID Threshold 
Separability 

Criterion 

Between 

Class  

Variance 

Level 
Global 

Mean 

7 116 0.738 2014.095 0.453 77.549 

19 70 0.655 857.701 0.273 51.699 

23 82 0.673 977.849 0.32 60.731 

24 98 0.651 352.521 0.383 99.982 

25 130 0.6607 1290.457 0.508 108.541 

27 60 0.692 901.321 0.234 31.857 

30 63 0.642 843.274 0.246 47.497 

31 79 0.665 1249.349 0.309 51.538 

35 94 0.797 1968.251 0.367 52.731 

36 83 0.751 1673.708 0.324 55.031 

37 108 0.794 2587.521 0.422 58.826 

38 67 0.621 615.282 0.262 49.961 

39 117 0.814 2494.852 0.457 60.336 

45 65 0.614 628.847 0.254 47.842 

46 79 0.639 652.216 0.309 48.881 

47 82 0.640 1128.102 0.32 48.569 

48 55 0.604 280.360 0.215 35.338 

49 113 0.838 2917.877 0.441 56.905 

50 109 0.473 321.355 0.426 108.324 

51 71 0.617 564.410 0.277 52.768 

52 67 0.620 554.612 0.262 42.295 

53 64 0.5693 443.973 0.25 40.040 

 

   Porosity also defined in terms of pore sizes as micro and 

macropores. Core thin section samples used in the current 

study were scanned with optical microscopy having a 

pixel resolution of (10 𝜇m).  
   Substantial porosity may be residing in pore sizes less 

than (10 𝜇m), i.e. sub-resolution pores. Such sub-
resolution pores were visually observe-able on thin-

section images but with a mixed response of clay-silt 

matrix and porosity.   

   In the current study, sub-resolution pores are defined as 

(Micro Pores), and pore sizes larger than (10 𝜇m) is 
defined as (Macro Pores). A subjective adjustment factor 

was used to extract matrix effect from sub-resolution 

pores as [25]; 

 

𝜑𝑖𝑚𝑎𝑔𝑒 = 𝐴 ∗ 𝜑𝑚𝑖𝑐𝑟𝑜 + 𝜑𝑚𝑎𝑐𝑟𝑜 = 𝜑𝑡𝑜𝑡𝑎𝑙                                            (22) 

 

Where: 

𝜑𝑖𝑚𝑎𝑔𝑒  = porosity derived from image analysis 

𝜑𝑚𝑖𝑐𝑟𝑜  = microporosity derived from image analysis 
𝜑𝑚𝑎𝑐𝑟𝑜  = macroporosity derived from image analysis 
𝜑𝑡𝑜𝑡𝑎𝑙  = total porosity derived from image analysis 
Ain = adjustment factor (between 0 and 1) to remove 

matrix effect from sub-resolution pores. 

  

   As a pixel representing pore size of less than 10 𝜇m 
may consist of both a grain and a pore. Ф𝑚𝑖𝑐𝑟𝑜 is micro 

pores porosity, and Ф𝑚𝑎𝑐𝑟𝑜 is macro pores porosity. 

Microporosity was adjusted by the factor (𝐴 = 0.75) to 
exclude matrix effect from micropores, and image 

porosity is calculated using Equation (22).  

   The results of image porosity obtained by OTSU's 

thresholding techniques are listed in the Table 2 with the 

comparison with core porosity. Fig. 4 shows a 

comparison of porosity obtained OTSU's thresholding 

techniques vs core porosity. The task of digital rock 

analysis needs to be quantitatively measured in an area of 

interest, whether it is pores or grains, extracted from a 

digital rock image.  

   The extracted objects are binary objects, where interest 

object with object label map is presented. Binary objects 

characteristically interpreted to get a value of (1), and the 

residual pixels to reach a value of (0). A binary object can 

be described by size, shape or distance from other objects. 

An object's size can be defined by area and perimeter. 

   The area is a suitable measure of the total size. 

Perimeter is mainly ideal for discriminating between 

objects with simple shapes and those with complex 

shapes. Consider the function In (i, j) described for the 

object of an (MxN) image: 

 

𝐼𝑛 (𝑖, 𝑗) = {
1     𝑖𝑓 𝐼(𝑖, 𝑗) = 𝑛𝑡ℎ  𝑜𝑏𝑗𝑒𝑐𝑡 𝑛𝑢𝑚𝑏𝑒𝑟
0                                            𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

}                              (23) 

 

The area in pixels is then specified by; 
 

𝐴𝑛 = ∑ ∑ 𝐼𝑛 (𝑖, 𝑗)
𝑁−1
𝑗=0

𝑀−1
𝑖=0                                                                      (25) 

 

 

 



Y. J. Tawfeeq and J.A. Al-Sudani / Iraqi Journal of Chemical and Petroleum Engineering 21,3 (2020) 57 - 66 

 

 

62 
 

   The simple calculation of the perimeter takes from the 

number of boundary pixels belonging to an object. This 

achieved thru calculating pixels number with a value of 

(1) and at least one adjacent pixel of (0).  

   Another problem in the measurement of the perimeter is 

to separate an object's internal and external perimeter 

(segmented pores).  

   The boundary pixel exact vertex points usually 

understood to be at the center of that pixel. The boundary 

pixels location for perimeter measurement yields internal 

perimeter; external perimeter yields with the boundary of 

pixels in the background around the object; as shown in 

Fig. 5.  

 

 

 
Fig. 3. Original and Segmented Binary Rock Images 

Thresholds Using OTSU Method (Samples 25,35,39, and 

49 respectively from top to bottom) 

 

 

 

 

Table 2. Image porosity analysis results using the OTSU 

thresholding method 
Core 
ID 

total pores macropores micropores Phi_Macro Phi_Micro PHI_Im PHI_Core 

7 304486 41837 9137 0.137 0.030 0.159 0.182 

19 304486 29577 30561 0.097 0.100 0.172 0.193 

23 304486 33739 8747 0.110 0.028 0.132 0.145 

24 304486 22551 8474 0.074 0.027 0.094 0.095 

25 304486 36559 9924 0.120 0.032 0.144 0.158 

27 304486 14852 13283 0.048 0.043 0.081 0.085 

30 304486 48714 14699 0.159 0.048 0.196 0.207 

31 304486 40496 16749 0.132 0.055 0.174 0.199 

35 304486 33036 6687 0.108 0.021 0.124 0.142 

36 304486 35322 9609 0.116 0.031 0.139 0.144 

37 304486 42083 3843 0.138 0.012 0.147 0.184 

38 304486 47138 15888 0.154 0.052 0.193 0.212 

39 304486 50218 12294 0.164 0.040 0.195 0.217 

45 304486 48565 13666 0.159 0.044 0.193 0.213 

46 304486 21743 13485 0.071 0.044 0.104 0.147 

47 304486 17129 27177 0.056 0.089 0.123 0.147 

48 304486 18437 15278 0.060 0.050 0.098 0.17 

49 304486 38680 3195 0.127 0.010 0.134 0.167 

50 304486 13801 9660 0.045 0.031 0.069 0.12 

51 304486 25996 13299 0.085 0.043 0.118 0.128 

52 304486 28605 12458 0.093 0.040 0.124 0.139 

53 304486 21650 14724 0.071 0.048 0.107 0.156 

 

 
Fig. 4. Comparison of image-based total porosity using 

OTSU thresholding method with core porosity 

 

   The circular equivalent diameter defined as the diameter 

of a circle with the same area as the region. Thus, once 

the area of the pore measured, the equivalent diameter 

(Deq) calculated as [26]; 

 

𝐷𝑒𝑞 = √
4𝐴

𝜋
                                                                                          (25) 

 

   Specific surface area or surface to volume ratio is 

approximated by the ratio of pore perimeter to pore area. 

Perimeter and area of each pore (i) are outputs of binary 

image analysis as discussed previously. The specific 

surface area of digital binary rock sample image is written 

as [26]; 

 

 



Y. J. Tawfeeq and J.A. Al-Sudani / Iraqi Journal of Chemical and Petroleum Engineering 21,3 (2020) 57 - 66 

 

 

63 
 

𝑆𝑖 =
𝑃𝑜𝑟𝑒 𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟

𝑃𝑜𝑟𝑒 𝐴𝑟𝑒𝑎
                                                                              (26) 

 

   The specific area of the analyzed sample is 

approximated as the average specific area of all pores. 

 

𝑆 =
1

𝑁
∑ 𝑆𝑖                                                                                           (27) 

 

   The results of image pore space characteristics obtained 

by OTSU's thresholding techniques are listed in the table 

(3). 

 

 
Fig. 5. Perimeter measurement by counting the number of 

object boundary pixels 

 

Table 3. Image pore space characteristics results using 

OTSU thresholding method 

Core 

ID 

Avg. Area 

(µm
2
) 

Avg. Equiv. 
Diameter 

(µm) 

Avg. 
Perimeter 

(µm)  

Avg. Specific 
surface area 

(1/ µm) 

7 49.45 3.774 18.355 1.088 

19 10.911 2.659 10.927 1.2691 

23 31.01 3.412 18.228 1.178 

24 28.788 3.754 17.583 1.136 

25 42.531 4.403 22.288 1.092 

27 25.809 3.131 12.076 1.064 

30 32.995 3.413 18.412 1.111 

31 24.72 3.364 14.549 1.1028 

35 51.983 3.489 19.649 1.201 

36 44.861 3.172 15.724 1.185 

37 92.241 4.973 26.83 1.077 

38 29.816 3.443 17.315 1.140 

39 102.234 5.658 27.138 1.009 

45 35.332 3.436 18.804 1.169 

46 21.252 3.390 13.455 1.137 

47 7.144 2.165 6.754 1.249 

48 15.916 3.061 11.361 1.138 

49 116.055 5.648 26.785 1.004 

50 23.206 3.619 14.817 1.101 

51 18.550 3.139 13.178 1.181 

52 26.980 3.474 15.193 1.134 

53 19.365 3.190 12.951 1.1583 

 

3.3. Results and Discussion  

 

In this study, three statistical parameters are considered 

for the analysis of image porosity resulted from a digital 

image.  

   These statistical parameters utilized to assess the 

accuracy of porosity predicted from the digital rock 

analysis. Absolute average percent relative error (AARE) 

used to quantify the average value of the absolute relative 

deviation of measured porosity value from experimental 

core porosity data. 

   The standard deviation of the estimated image porosity 

relative to the experimental values is essential to measure 

the accuracy of the correlation and used algorithm. The 

value of standard deviation is usually expressed in 

percent, and the small value indicates higher accuracy.   

   The purpose of performing the correlation coefficient is 

to describe the strength of the association between two 

variables, namely experimental and calculated values.  

   The correlation coefficient (R) expresses the presence or 

non-presence of a linear interrelationship between the two 

observed variables. If the linear interrelationship is 

positive, the correlation coefficient will be a positive 

number between 0 and 1.0. If, on the other hand, it is 

negative, the number will be between (0) and (1).  

   The coefficient of determination (R
2
) is the square of 

the coefficient of correlation (R) shows percentage 

variation in the y-axis that described by all x-axis 

variables collected. It is varied between (0) and (1) with 

higher values is better. The results of AARE, standard 

deviation, correlation coefficient, and coefficient of 

determination are 14.66, 0.029, 0.892 and 0.796, 

respectively. 

 

4- Discussions and Conclusions 
 

   Porosity from image analysis was compared against 

core porosity to validate the goodness of porosity 

prediction from image analysis. However, uncertainties 

associated with both measurements shall be considered as 

well. Porosity from image analysis is limited to pixels 

resolution of optical microscopy and represents a very 

small section of the rock sample.  

   Core porosity is determined on the 1-inch cylindrical 

plug while the dimensions of thin section sample are only 

35 𝜇m thick with a diameter of 1-inch.  
   The volume investigated is different. Studied scanned 

samples with optical microscopy had a pixel resolution of 

10 𝜇m. Pore sizes larger than (10 𝜇m) (i.e. macropores) 
were correctly resolved, but there was a significant 

quantity of sub-resolution pores (micropores) with mixed 

response of pore and matrix.  

   A subjective adjustment factor used to take out this 

matrix effect from micro-pores. This single value of 

adjustment factor was determined while comparing image 

porosity against core porosity for all samples. This factor 

has selected as (0.75) in this study to correct the matrix 

effect during OTSU segmentation method.  

   The suggested value may have worked for the analyzed 

samples of the current research and can differ in other 

environments. Uncertainty analysis can also be analyzed 

for porosity from thin section image analysis.  

 

  



Y. J. Tawfeeq and J.A. Al-Sudani / Iraqi Journal of Chemical and Petroleum Engineering 21,3 (2020) 57 - 66 

 

 

64 
 

   As follows from error analysis results, due to an 

appropriate choice and capture of blue color using OTSU 

algorithm segmentation, the porosity obtained by image 

analysis is quite close to the core porosity. The errors are 

about 14.66% with a standard deviation of about 0.029 

and high correlation coefficient of (R = 0.892). 

Furthermore, binary images capture the pore distribution 

excellently without counting matrix material as pore 

space.  

   Digital thin section image analysis can be considered as 

an alternate technique to evaluate porosity and pore space 

rock properties rather than experimental core analysis. 

Histogram thresholding established method had the 

element of subjectivity in it where the threshold on pixel 

intensity histogram had to be manually adjusted till the 

analyst is visually satisfied that pore space adequately 

captured.  

   This visual analysis was challenging, as optically 

scanned images used in the current study had a pixel 

resolution of 10𝜇m/pixel, and there were a significant 
number of pores with size less than the pixel resolution. 

In another study, regression equations used to achieve a 

good correlation of porosity between image analysis and 

routine core analysis data.  

   These adjustments and regression lost the predictive 

power of image analysis. OTSU clustering introduced as 

an automatic technique to separate the intensity histogram 

into two parts and segment the pores from the matrix. 

Thin sections image porosity using the OTSU technique 

showed a good match with core porosity; with the 

additional benefit, that workflow now automated. 

Moreover, the OTSU method can predict threshold values 

if it desired to make image interpretation with a 

thresholding technique. In the current study, porosity is 

the main petro physical property determined from thin 

section images. For future work, the permeability as a 

function of porosity and pore space characteristics can be 

estimated.  

   The predictive power of the OTSU method is 

encouraging, as it can be applied on vastly available drill 

cuttings as a secondary means of porosity data. However, 

for the wells where conventional core data is not available 

or possible, porosity can be determined from thin section 

images for its integration with well logs interpretation to 

reduce uncertainties. Some limitations to thin section 

image analysis were also observed. For optically scanned 

images, pore sizes less than ten𝜇m had a mixed response 
of matrix and porosity.  

   A subjective but a single adjustment factor was required 

to remove the matrix effect from such pores for all 

analyzed samples.  

   This is equally applicable for both automatic and 

manual thresholding techniques. Clustering analyzes 

porosity from pore filling blue epoxy, i.e., a blue cluster; 

it was observed that clustering over-estimates porosity if 

blue color is also present as a matrix color. Such a 

situation will be equally challenging for manual 

thresholding and hence, can be concluded as a general 

limitation of thin section image analysis. 

 

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Y. J. Tawfeeq and J.A. Al-Sudani / Iraqi Journal of Chemical and Petroleum Engineering 21,3 (2020) 57 - 66 

 

 

66 
 

 

باستخدام  OTSUتحليل مسامية النماذج الصخرية الرقمية باالستناد على تقنيةعتبة
 الماتالب

 

 يحيى جرجيس توفيق وجالل عبد الواحد السوداني
 

 سة، جامعة بغداد، بغداد، العراققسم هندسة النفط، كلية الهند
 

 الخالصة
 

تلعب المسامية دوًرا أساسًيا في الجيولوجيا وهندسة البترول. وهي تتحكم في تخزين السوائل في طبقات المياه    
الجوفية وحقول النفط والغاز وتوصيل هيكل المسام للتحكم في تدفق السوائل وانتقالها من خالل تكوينات الخزان. 

   الصخور، يجب قياس بنية المسام ووصفها كميًا. لتحديد العالقات بين المسامية والتخزين والنقل وخصائص 
حيث يتم فيها  حساب مسامية الصور الرقمية باستخدام معالجة الصور يعد امرا مهما لتحليل صخور المكامن   

الوتينية يستخدم عملية تحويل الصورة الى ابيض واسود حيث  وصف المسامية ثنائية األبعاد للعينة. العمليات
تستخدم حد قيمة البيكسل لتحويل الصور الملونة والرمادية إلى صور ثنائية. والفكرة هي استيعاب المناطق 
الزرقاء بالكامل مع المسام وتحويلها إلى اللون األبيض في صورة ثنائية ناتجة. تقدم هذه الدراسة إمكانات 

م معالجة الصور لتحديد مسامية عينات الصخور الرقمية ثنائية األبعاد في صخور مكامن الكربونية. تم استخدا
إنشاء كود ماتالب، والذي يقوم تلقائًيا بتقطيع مسامية الصخور الرقمية وتحديدها، استناًدا إلى خوارزمية عتبة 

OTSU ات مقطع رقيق ثنائي األبعاد لحقل عينة من صخور مكمن الصورة ذ 22. في هذا العمل، تمت دراسة
نفط عراقي. تتم معالجة أمثلة صور القسم الرقيق ورقمنتها باستخدام برمجة ماتالب. في الدراسة الحالية، ركزنا 

للصورة الرقمية. أيضا، تم حساب بعض خصائص الفراغ المسامي،  مسامات المايكرو والماكرو على حساب
المختبري  ائج تحليل الصور الرقمية ثنائية األبعاد بنتائج المسامية من الفحصمثل الحجم والمحيط. تم مقارنة نت

لتحديد قوة تقنيات تفسير الصور الرقمية المستخدمة. أظهرت مسامية الصورة الدقيقة المجهرية التي تم حسابها 
 تطابًقا جيًدا مع المسامية المحسوبة من الفحوصات المختبرية. OTSUباستخدام تقنية 

 
 : فيزياء الصخور الرقمية، عتبة أوتسو، صورة المقطع الرقيق، المسامية، مسامات المايكرو، مسامات الماكرو.الدالةلمات الك