Dependence of critical porosity on pore geometry and pore structure and its use in estimating porosity and permeability
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Abstract
It is well recognized that the wave velocity is not only influenced by its constituent materials but also by the details of the rock bulk. This situation may bring about data points of Pwave velocity V _{p} measured on a large number of rock samples against either porosity or permeability of the frequently scattered although certain trends may exist. This paper presents the results of a study by employing rock samples on which ϕ, k, and V _{p} are measured in attempt to characterize critical porosity ϕ _{c} and its relation to other rock properties. The approach used in this study is the use of Kozeny equation. The equation is believed to account for all parameters influencing absolute permeability of porous media. A mathematical manipulation done on the equation has resulted in a power law equation that relates pore geometry √(k/ϕ) to pore structure k/ϕ ^{3}. Three different sets of sandstone amounting totally to as many as 716 samples were provided in this study. The properties measured are ϕ, k, and V _{p}, and grain size. For each sandstone data set, at least there are nine groups of the rock samples obtained. When V _{p} is plotted against ϕ, it is found that each group of each sandstone data set has both its own ϕ _{c} and an excellent relation of ϕ, V _{p}, and ϕ _{c}. Furthermore, combining all the basic equation for V _{p}, Kozeny equation, and the empirical relation for porosity results in a model equation to predict permeability. In conclusion, for the sandstones employed, ϕ _{c} is a specific property of a group of rocks having a similar pore geometry.
Keywords
Pore geometry Pore structure Pwave velocity Porosity Permeability Critical porosityIntroduction
Several studies have been conducted to define the relationship between velocity and porosity. The relationship between velocity and porosity for the entire porosity range is not linear as indicated by Wyllie et al. (1956). The velocity value will be maximum at zero porosity that describes mineral velocity and the value of velocity will be minimum when the solid material becomes less and as pore space increases as demonstrated by Raymer et al. (1980). For a porosity greater than 47%, the relationship between velocity and porosity is not linear. Han et al. (1986) showed that the velocity and porosity have a linear relationship and can be separated based on the volume of clay. Furthermore, Nur et al. (1995) shows that sandstone, limestone, dolomite and igneous rocks have different characteristics so that each has its own critical porosity. The value of the critical porosity is determined by the type of mineral, rock texture, and diagenetic processes after deposition (Mavko et al. 2009). Prakoso et al. (2016) shows that the relationship between Pwave velocity with pore geometry and pore structure can be grouped based on the similarity of pore geometry and pore structure. Rock samples with similar pore geometry and pore structure will have their own group referred to as rock type and have different relationships between velocity and porosity.
Several other researchers have arranged the relationship between Pwave velocity and permeability by incorporating the influence of pore geometry. Klimentos (1991) shows that the empirical relationship between Pwave velocity and permeability is directly proportional and influenced by the clay content, the specific surface area and micro porosity. Prasad (2003) used the hydraulic unit to arrange the relationship between Pwave velocity and permeability. Fabricius et al. (2007) shows that the ratio relationship of V _{p} /V _{ s } in dry rocks is strongly influenced by the specific surface area and demonstrated that permeability can be predicted from the relationship between the specific surface area, Vp/Vs and porosity. Weger and Eberli (2009) showed that the relationship between porosity, permeability and V _{p} is influenced by micro porosity, pore network complexity, and pore size.
This paper presents the influence of the rock type on the relationship between Pwave velocity and both porosity and permeability. Rocks grouping was based on the similarity of pore geometry and pore structure. It is evident that each group of rocks has a different critical porosity value. Critical porosity value is not only influenced by lithology of rock as shown by Nur et al. (1995) but also by complexity of pore geometry and pore structure of rocks. By grouping the rocks based on their rock type, the relationship between porosity, permeability and Pwave velocity can be established. Furthermore, these relationships can be used to estimate porosity and permeability based on the Pwave velocity.
Methods and data
Rock grouping
Plotting (k/ϕ)^{0.5} on Y axis against (k/ϕ ^{3}) on X axis on a log–log graph will produce a straight line with maximum slope of line b = 0.5 for capillary tube systems. This means the value of b 0.5 indicates a perfect rounded pore shape. For natural porous media, the value of b is less than 0.5. The more complex the pore shape of the rock, the lower the b value where a is a constant that is interpreted as a correction factor for volumetric fluid flow efficiency for irregular pore systems (Permadi and Wibowo 2013).
Dry bulk modulus and critical porosity
Equation (6) says that plotting B and µ against ϕ will form a straight line with a slope of − 1/ϕ _{c} and will intersect the Y ordinate at B = B _{m} and µ = µ _{m}.
Data used
This study used three data sets of sandstones from three different basins that have variations in porosity, permeability and Pwave velocity. Data sets 1 and 2 were obtained from the publication by Prakoso et al. 2016. Data set 1 are sandstones of the North West Java basin, dominated by finegrained to coarsegrained. The porosity range is 6.76–36.9%, while the permeability data range is 0.12–5713 mD and volume clay is below 20%. The data set 2 sandstones were from the Kutai Basin, dominated by finegrained to coarsegrained. The porosity range is 4.5–36.9%, while the permeability range is 0.05–4504 mD and volume clay is below 10%. The data set 3 sandstones were from the South Sumatra basin, dominated by finegrained to coarsegrained. The porosity range is 6.76–36.9%, while the permeability range is 0.12–5713 mD and volume clay is below 10%. As much as 716 cores of data were used including routine core data analysis, sedimentology analysis and dry Pwave velocity V _{pdry }. The routine core data analysis included permeability, porosity and lithology descriptions. Sedimentology data analysis included petrography (thin section) and XRD. Pwave and Swave velocity data was obtained from measurements using SonicViewerSx equipped with piezoelectric transducer to measure the Pwaves and Swave. Measurements were taken on dry conditions, at atmospheric pressure (1 atm or 0.101 Mpa) and room temperature (25 °C). Prior to measurement, core samples were dried at 150 °F for 12 h.
Results and discussion
Grouping data into rock type
Relation rock type, dry Pwave velocity and critical porosity

The rocks are composed by the dominant minerals quartz.

Assumed bulk modulus mineral modulus (B _{m}) 37 Gpa
The relationship between porosity with bulk modulus and critical porosity value for each rock
Rock  Data Set 1  Data Set 2  Data Set 3  

Type  Equation  ϕ _{c}, fraction  Equation  ϕ _{c}, fraction  Equation  ϕ _{c}, fraction 
4  B = − 96.425ϕ + 37  0.3837  –  –  B = − 110.45ϕ + 37  0.3350 
5  B = − 110.45ϕ + 37  0.3350  B = − 125.42ϕ + 37  0.2950  B = − 132.14ϕ + 37  0.2800 
6  B = − 123.33ϕ + 37  0.3000  B = − 48.00ϕ + 37  0.2500  B = − 148ϕ + 37  0.2500 
7  B = − 137.04ϕ + 37  0.2700  B = − 176.19ϕ + 37  0.2100  B = − 185ϕ + 37  0.2000 
8  B = − 154.17ϕ + 37  0.2400  B = − 205.56ϕ + 37  0.1800  B = − 205.56ϕ + 37  0.1800 
9  B = − 173.06ϕ + 37  0.2138  B = − 246.94ϕ + 37  0.1498  –  – 
10  B = − 189.74ϕ + 37  0.1950  B = − 283.7ϕ + 37  0.1304  B = − 246.67ϕ + 37  0.1500 
11  B = − 212.25ϕ + 37  0.1743  B = − 356.09ϕ + 37  0.1039  –  – 
12  B = − 259.22ϕ + 37  0.1427  B = − 414.06ϕ + 37  0.0894  B = − 296ϕ + 37  0.1250 
13  B = − 333.4ϕ + 37  0.1110  B = − 486.94ϕ + 37  0.0760  –  – 
14  B = − 523.98ϕ + 37  0.0706  B = − 616.67ϕ + 37  0.0600  –  – 
Different ranges of porosity, permeability and V _{ pdry } of rock type 5 for the data sets 1, 2 and 3
RT  Data set 1  Data set 2  Data set 3  

ϕ fraction  k mD  V _{ pdry } m/s  ϕ fraction  k mD  V _{ pdry } m/s  ϕ fraction  k mD  V _{ pdry } m/s  
5  0.346  273  1219  0.260  1348  1880  0.225  1846  2249 
0.294  298  1707  0.241  856  2116  0.235  1505  2360  
0.291  525  2063  0.248  1704  1943  0.247  1147  2111  
0.308  236  1418  0.277  4162  1713  0.259  1192  1998 
Some data points of neighboring rock types overlap and some data points of a certain rock type fall on the trend line of a next rock type. These indicate that some members of two closest neighbors may have the same porosity but different pore geometry and structure. Such behavior implies that the dry bulk modulus and Pwave velocity of the sandstones are not only controlled by porosity but also by pore geometry and structure. In other words, dry bulk modulus and Pwave velocity of the sandstones in relation to porosity are specifically characterized by the similarity in pore geometry and structure. This results is a certain critical porosity value for a given rock type.
Relationship of specific surface area, pore geometry and Pwave velocity
Value of constants a and b in Eq. 10 for the data sets 1, 2 and 3
Rock  Data set 1  Data set 2  Data set 3  

Type  a  b  a  b  a  b 
4  0.2321  − 0.0008  0.2679  − 0.0010  
5  0.3746  − 0.0006  0.5143  − 0.0010  0.2889  − 0.0009 
6  0.972  − 0.0007  0.8216  − 0.0010  0.8006  − 0.0010 
7  4.0132  − 0.0009  1.5589  − 0.0010  0.8329  − 0.0006 
8  7.0156  − 0.0009  1.8662  − 0.0009  1.5428  − 0.0007 
9  8.4532  − 0.0008  4.0296  − 0.0009  –  – 
10  10.326  − 0.0007  4.2154  − 0.0008  –  – 
11  11.276  − 0.0006  5.8732  − 0.0009  –  – 
12  12.363  − 0.0007  6.4773  − 0.0008  11.2240  − 0.0007 
13  16.669  − 0.0008  7.3995  − 0.0009  
14  7.7684  − 0.0008 
Porosity and permeability estimation using V _{pdry}
In the previously section, it has been discussed that each rock type has similar pore geometry and pore structure which is characterized by the critical porosity value. The relationship between porosity and Pwave velocity for each rock type is limited by Pwave velocity value of the mineral at porosity zero and zero Pwave velocity at critical porosity value that reflects a condition of suspension. Thus, the critical porosity value is the maximum porosity value in natural porous media.

The rocks are composed by the dominant minerals quartz.

Assumed that the value of bulk mineral modulus (Bm) 37 Gpa and the shear modulus mineral (μm) value 44 Gpa
Equation 15 can be used to estimate porosity using a different value of critical porosity for each rocks group. Critical porosity for each rock type can be estimated using Nur Equation as shown in Fig. 2 or Table 1. Equation 15 shows that if the value of dry Pwave velocity 0, then the porosity is equal to the critical porosity which reflects a condition of the suspension.
Equation 15 is derived based on rock type and critical porosity. It means that porosity estimation is performed for specific rock groups that have similar pore geometry and pore structure. Thus, using Eq. 15 will obtain accurate porosity estimation results. Porosity estimation using Eq. 15 requires measurement of V _{p} and V _{s} data. The value of bulk modulus and shear modulus of mineral are assumed 37 and 44 Gpa, respectively, for quartz mineral. The critical porosity value can be simply estimated by the linear equation of Nur et al. 1995 for each rock type as shown in Fig. 2 and Table 1. Equation 15 contains information about the acoustic impedance shown in term of V _{p} × ρ. Thus, acoustic impedance data can be integrated directly with Eq. 15 for porosity estimation.
In reservoir modeling, the accuracy of permeability estimation is very important because it will greatly affect the fluid flow. Equation 16 is intended to estimate permeability from V _{p} based on rock type. Equation 16 is arranged base on Kozeny equation (Eq. 9). The value of porosity is estimated using Eq. 15. Porosity estimation using Eq. 15 requires measurement data of V _{p} and V _{s}, while the specific surface area of each rock type is estimated empirically using Eq. 12. As discussed earlier, Eq. 12 is obtained based on the relationship of V _{p} with S _{b} as shown in Fig. 2. The values of a and b of Eq. 12 for each rock type are shown in Table 3. The term of V _{p} x ρ is usually known as acoustic impedance. Thus, if 3D cube acoustic impedance and V _{p} are available, then the permeability can easily be estimated from Eq. 16 with a good degree of accuracy.
Conclusions
 1.
Specific relations of dry bulk modulus (B _{dry}) and dry Pwave velocity (V _{pdry }) to porosity are obvious when the rock samples are grouped on the basis of similarity in pore attributes, pore geometry and pore structure. This led to the results exhibiting that a critical porosity is a specific property.
 2.
For any given rock group, specific internal surface area increases as pore geometry in terms of mean hydraulic radii decreases, resulting in a decrease in permeability. The results show that Pwave velocity decreases as both porosity and specific internal surface area increase. These all explain why Pwave velocity decreases with permeability but reversely with porosity.
 3.
Porosity and permeability of rocks can be predicted on the basis of Pwave velocity once the rock type and the corresponding critical porosity are established. For all the rock samples employed, very good results of the predicted porosity and permeability have been demonstrated.
Notes
Acknowledgments
One of the authors, Suryo Prakoso, would like to thank the universitas Trisakti, Geophysical Laboratory FTKE Trisakti University for provided SonicViewer tool, and PPPTMGB “Lemigas” for the core samples and the corresponding petrography analysis data provided for his research work.
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