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Predicting Petro-physical Properties using SEM Image Yatin Suri, Indian School of Mines

Copyright 2011, Society of Petroleum Engineers

This paper was prepared for presentation at the SPE Reservoir Characterisation and Simulation Conference and Exhibition held in Abu Dhabi, UAE, 9–11 October 2011.

This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not beenreviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, itsofficers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission toreproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

Abstract

The permeability estimation of reservoir rocks using numerical methods has always been a

challenge for petrophysicist. Several models have been developed for its computation of this parameter.

As the permeability is entirely controlled by the geometry, the possibility arising of estimating the permeability from quantifiable attributes of the space has always has been a approach.

In the present paper, a model is discussed which gives accurate prediction of the permeability

 based on two dimensional SEM image of a core sample of sedimentary rock. The inputs required for themodel are the areas and perimeters measurements from the images of space. The individual

conductances are estimated using hydraulic radius approximation. Before using the data obtained

from images, stereological corrections are used to convert geometries and various hydraulic

corrections are used to account for converging-diverging flow paths. Kirkpatrick's mediumapproximation is finally used to find the value of the hydraulic conductances of the individual pores.

The method has been applied to six data sets of SEM images which include Berea sandstone,

consolidated sandstones and carbonate samples. The laboratory determined air permeabilities of thesesamples ranged from 0.5-400mD. The permeability values predicted by this method are within a factor

of two of the values.

This method requires least data manipulation and computation and is more accurate thanconventional methods such as the Kozeny-Carman equation. The method holds promise of permeability

 predictions on irregular rock samples like drill cuttings which cannot be in standard lab measurements.

Another possible future is to use down-hole borehole imaging technology to provide an image with the

appropriate resolution, thereby allowing in-situ permeability estimation, without the need for core

samples.

Keywords Permeability, Scanning Electron Microscope, Pore Network Modeling, Porosity,Conductance, Sandstones and Carbonates 

Introduction

Being able to relate the transport properties of rocks to their internal pore structure has long been

of great interest to hydrologists, earth scientists and petroleum engineers (Jurgawczynski, 2007).

Permeability is arguably the most important petro-physical property of a reservoir rock and the ability to predict its value without time consuming and expensive laboratory measurement would obviously be of

great practical value (Liang et al, 1997). Scanning Electron Microscope (SEM), images the sample

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surface by scanning it with high energy beam of electrons, which interact with the atoms producingsignals that contain information about the sample’s surface topography, composition and other

 properties such as electrical conductivity.

Petro-physical properties measurements, on samples of porous rocks, are classically performed in the

context of Core Analysis (porosity, permeability, formation factor). On the other hand, qualitative andquantitative informations are obtained from the study of thin sections and SEM analysis of the same

materials (Adler et al, 1991).The purpose of this paper is to present an attempt to establish a link between these two aspects, and totry to solve a long standing problem :to deduce the transport properties of a 3-dimensional sample (plug)

from the corresponding geometrical structures of the pore space, which can be observed and

characterized in 2 dimensions(thin sections).This problem is difficult for two main reasons:

•  It is hard to describe quantitatively the 3-D geometry of the medium in a realistic manner,without introduction of artificial models and parameters (Adler, 1989).

•  The partial differential equations corresponding to these transport phenomena are not easy tosolve, except numerically.

Empirical permeability models such as that of Kozeny-Carman equation predict values of the permeability using knowledge only of porosity and a mean pore diameter or mean grain size. Althoughsimple to implement, the Kozeny-Carman equation is often found to be insufficiently accurate for

reservoir characterization purposes. It is also known that this model, although fairly accurate forunconsolidated sands, tends to become unreliable for consolidated sandstones. The Kozeny-Carman

approach therefore requires some means of estimating the specific surface, which can be problematic

(Berryman and Blair, 1986). The Katz-Thomson equation (1986) can yield accurate estimations of the permeability, using the porosity and the electrical formation factor. However, the requirement of having

a measured value of the electrical formation factor is clearly a disadvantage of this method.

At the other end of the spectrum are models that attempt to reconstruct the complete 3-D micro-structure of the rock, which can then be used either to compute the properties directly using the Navier-

Stokes equations (Adler et al., 1990), or as a starting point for the development of network models(Blunt, 2001). These network models can then be used to compute various flow properties, such as

absolute or relative permeability (Blunt et al., 2002), as well as to gain insight into the relationship between pore geometry and petro-physical properties (Arns et al., 2004). Spanne et al. (1994), and later

Ferréol and Rothman (1995), used X-ray microtomography to reconstruct the pore structure of a

Fontainebleau sandstone, from which the permeability was calculated numerically, in the latter caseusing the lattice-Boltzmann method. Such approaches are capable of good accuracy, but at the expense

of extensive data collection and computation.

In the present work, we have developed a model for predicting permeability from two-dimensionalSEM images of the pore space, without requiring any computationally intensive procedures of the

aforementioned three-dimensional models. But, unlike simple models such as Kozeny-Carman, we do

make use of rock-specific pore geometry information, obtained from two-dimensional pore images. Thehydraulic conductivities of the individual pores are estimated from their areas and perimeters using thehydraulic radius approximation (Lock, 2001). Stereological correction factors are applied to determine

the true cross-sectional shapes from the images, and to determine the true number density of pores per

unit area. A constriction factor accounts for the variation of the cross-sectional area along the tubelength. The pores are assumed to be arranged in a cubic lattice, after which the effective-medium theory

of Kirkpatrick (1973)  is used to estimate the effective conductance of the pores. Finally, the

 permeability is estimated from the effective pore conductance and the number density of pores.Because of the more complex pore structure found in carbonates, the methodology set up by Lock

(2001) for sandstones had to be entirely reviewed. The areal, thresholding process is adjusted to account

for the wide pore size distributions encountered in carbonates. Finally, in order to improve the

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methodology, a varying co-ordination number is introduced to compensate for an over/under-predictionthat was noticed in the preliminary results. The method is then applied to four samples, with

 permeabilities ranging from 0.5-400 milliDarcies, and the permeabilities are in most cases predicted

within a factor of two.

Analysis of SEM Image

In this approach, the aim is to predict the permeability of sedimentary rocks by collecting area and perimeter measurements from images of the pore-space. Firstly, a two dimensional SEM Image of a rocksample is generated. The images must be converted into digitized grey-scale images in which the pore

space is generally distinguished from the various minerals by having higher grey-scale values. The

images are then threshold to yield binary images in which the pores are black and the mineral grains arewhite. The images were analyzed using the “Image J” Image processing software.

As computers do not process a continuous spectrum of gray values, the input image must be

represented by discrete numerical values, from within some finite range. The image is broken up intosmall areal regions known as pixels, and eight bits per pixel are used to quantify the darkness level. This

raises the question of the appropriate pixel size. Use of a pixel size that is too large, relative to the pore

sizes, will lead to the loss of fine-scale pore features, and consequent inaccuracy in the estimation of the

flow conductances.

Estimating Conductance of Network

For estimating the value of permeability, in this approach, the value of permeability is determined with

the help of conductance of the entire network of pores. And the value of conductance of entire network

is determined with the help of the conductance of individual pores.

Conductance of Individual Pores:

Conductance of a cylindrical poreConsidering an individual pore of length L, with a pressure drop ΔP along its length. The volumetric

flow rate Q through this pore can be written as Q= C ΔP/  μL, where μ is the fluid viscosity and C is the

flow conductance. First, this pore is idealized as a tube having a uniform, although possibly irregular,cross section. If the cross section were circular, with radius r, the flow conductance would be given by

C=πr 4/8, according to Poiseuille’s law. Since the cross-sectional area of the tube is πr 

2 , and the

 perimeter is Π = 2πr. The flow conductance can also be written as

2

3222

28

2

88)(

Π×=

×=⎟

 ⎠

 ⎞⎜⎝ 

⎛ 

Π==

 A R A A A Ar circleC 

h

 .......... (1)Where the hydraulic radius is defined by R h=2*A/Π. According to the hydraulic radius approximation,

we can use equation (1) for the conductance of any cylindrical pore, even if its cross section is not

circular.Koplik et al, (1984) solved the governing laminar flow equation numerically for several pore shapes

found from SEM Microscopy of Massilon sandstone and found that the hydraulic radius approximation

was usually +-30% of the exact conductance. Sisavath et al (2001a)  found similar results for pores inMassilon and Berea sandstone based on comparison with boundary element calculations. Furthermore,

they found that this approximation does not systematically under/over predict the conductance so that

the errors (partially) cancel out. When applied to a network of pores of different cross sections, thesefact in mind and in the spirit of avoiding computationally intensive procedures such as a boundary

elements or numerical conformal mapping (Sisavath et al, 2001a). We use the hydraulic radius

approximation to estimate the flow conductance of the individual pores.

Stereological correction for area and perimeter

The areas and perimeters of the individual pores, as measured from any 2D image, will in general be

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larger   than the actual values for the pore cross-sections. For example, consider a cylindrical pore ofradius r . In general, the plane of the image will intersect this pore at some arbitrary angle θ relative to

the pore axis, and so the pore will appear as an ellipse with a semi-minor axis of r , but a semi-major axis

of r/ cosθ. Hence, the area and perimeter of the image would consequently be larger than the actual area

and perimeter of the pore, and so the estimated hydraulic conductance will be greater than the actualvalue. An approximate stereological correction factor that converts the “measured” values of the

hydraulic conductance into “actual” values can be found by averaging the overestimation in theconductance C over all possible angles, assuming that the pores are randomly oriented with respect tothe plane of the image. The result (Lock et al, 2002) of this calculation is that the pore conductances

estimated from the image must be multiplied by 0.375 to arrive at the “true” conductance of the pore.

Stereological correction for number density

Consideration must also be made of the overestimation in the areal number density of pores that

occurs as a consequence of taking an arbitrary two-dimensional slice that probably does not lie in a plane perpendicular to a lattice direction. If we again consider the idealization of pore microstructure by

a hypothetical cubic lattice, then a slice taken perpendicular to a given lattice direction will only

intersect those pores that lie along that direction. If, however, the slicing plane is not normal to the

lattice direction, it will also intersect some pores that are orthogonal to that first lattice direction.Evaluation of this effect (Lock et al, 2002) leads to the conclusion that the apparent number density of

 pores (i.e., pores per unit area of image) must be divided by 1.47 to yield the actual density of pores thatare oriented in a given lattice direction. An analogous calculation for a rock containing pores having a

random distribution of orientations gives an identical result.

NETWORK OF PORES

 Network models have been used to study the transport properties of porous media. The pore-space

model that we use is that of a network of conductors, connected at volume-less nodes. (At the very lowReynolds numbers encountered in most subsurface flow situations, hydraulic energy losses at

 junctions, which are important in pipe flow at high Reynolds numbers, are negligible). If the

values of all of the conductances were known exactly, and the topology of the network was alsoknown, evaluation of the overall macroscopic conductance of the network would require the solution of

the large system of linear equations. These equations arise by applying the equation Q = C Δ P/L to

each tube, and invoking the fact that, in order to conserve mass, the sum of the fluxes into each node

must be zero.this procedure has been carried out for idealized networks, generally two-dimensional,and can yield much insight into the behavior of network models (Koplik, 1981; David et al., 1990).

An alternative to an exact network calculation is the effective medium approach, in which each

conductor in the network is replaced by a conductor having some effective value C eff . One popularmethod of estimating C eff   is the procedure of Kirkpatrick (1973), who used the criterion that if each

conductor within a certain region is replaced by a conductor with conductance C eff ., the resulting

 perturbations within this region should average out to zero. This consideration leads to the

following equation that implicitly defines C eff  :

0121

=

 

∑

=

N 

i    i eff  

i eff  

C C z 

C C 

)/(………. (2)

Where the co-ordination number  z is the number of pores that meet at each node and the summationis taken over each individual pore in the network. Kirkpatrick (1973)  derived equation (2) under the

assumption that the co-ordination number of each node was the same, but it was later shown (Koplik,

1982) that it can be used for topologically irregular networks by defining  z  to be the mean co-ordination

number of all the nodes in the network. In the present study we idealize the network as being a cubiclattice, and set z  = 6. Other researchers have estimated the mean co-ordination number for sandstones to

 be in the range of 3-8.

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The effective-medium approximation becomes exact in the two limiting cases z=2 and z∞ whichcorresponds to series and parallel network. For z=2 equation (2) reduces to the harmonic mean. Ceff =N /

∑  (1/Ci), whereas z∞, it yields the arithmetic mean, Ceff =∑Ci / N. For 2

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Figure 1: SEM Image of Sandstone Sample A

Sample B: (Figure 2) Sample B is sandstone  with sample depth of 1489.32m and reservoir petrography as follows: 

The rock is dominantly fine grained, sub angular to sub rounded with good sorting. The rock matrix isclay. Preservation of fair amount of intergranular porosity has been observed with moderate to good

interconnectivity of pores. However, moderate to poor intergranular porosity has also been observed at

some places due to bounding of the grains with clay matrix. The clay is appearing to be detrital innature. Formation of pyrite nodules within the sandstone has also been observed. The average porosity

and permeability of the rock is good and displays good reservoir characteristics.

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Figure 2: SEM Image of Sandstone Sample B

Sample C: (Figure 3)

Sample C is limestone  with sample depth of 1476.14m and reservoir petrography as follows:

The rock appears compact to well compact, semi-crystalline nature with poor porosity under SEM. Therock is tending towards crystalline with formation of calcite crystal surface which reduces porosity

significantly. Formation of tiny pyrite framboidal balls have been observed as floating on the limestone

surface and also the presence of framboidal ball indicates the deep sea sedimentary origin. At places, presence of foraminifera and subordinate amount of authigenic chlorite clay with chloritic structure has

also been observed .The average porosity of the rock is poor. However, moderate development of micro

 pores has also been observed at few places. The presence of authigenic chlorite clay has beenconfirmed by EDX analysis.

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Figure 3: SEM Image of Limestone Sample C

Sample D: (Figure 4)

Sample D is limestone  with sample depth of 1478.95m and reservoir petrography as follows:

The limestone is tight and compact, semi crystalline under SEM at different magnifications. The Figure4 shows development of crystalline calcite surface which reduces the porosity significantly. The

limestone has also undergone for some digenetic changes with formation a rhombic dolomite crystal.

The benthic foraminifera (miliolid) have also been observed at places which indicate the deep seaenvironment origin. The foraminifer’s chambers are filled by calcite crystal which reduces the organic

 porosity considerably. The average porosity of the rock is poor.

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Figure 4: SEM Image of Limestone Sample D

Discussion of Results and Conclusions

The results for the samples are summarized in Table 1. The permeability is, in these cases, predicted

within about a percentage error of two.

Table 1: Results for rock samples of Indian Fields

Sample LABporosity

MODELporosity

LABpermeability

MODELPermeability

A 23.84 26.10 118.17 119

B 22.58 21.90 17.88 11.54

C 7.21 7.80 2.16 3.27

D 9.74 9.70 4.67 4.96

D1-I5-25 28.9 26.5 18.33 20

D1-I5-26 31.9 29 17.12 18.07

L3-30 23.58 24 12.56 13.1

L3-33 19 20 38.55 37.9L6-57 17.86 19 6.56 7.26

L6-58 16.61 16 6.65 7.08

L7-60 16.28 16.5 10.71 10.98

L7-62 16.64 16.2 11.73 12.01

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Figure 5: Crossplot of predicted and measured permeabilities for both the sandstone and carbonatedata set.

Figure 6: Crossplot of predicted and measured porosity values for both the sandstone and carbonate

data set.

After the generation of the SEM Image, Image analysis is done with the help of Image processing

software, ImageJ, which analysis the thresholded image and total number of pores in it and gives the areaand perimeter of each individual pores. Area and perimeter values are then used to determine the hydraulicconductance of each individual pore and then, after applying stereological corrections and constrictionfactor, kirkpatrik effective medium theory is used to determine the hydraulic conductance of the entirenetwork. With the help of overall conductance value, permeability of the sample is determined.

With the above described methodology the results of permeability and porosity predicted anddetermined by laboratory is very close within about a percentage factor of two (Figure 5 and Figure 6).Sample C, the carbonate, appeared not to have been carefully handled during the laboratory procedures,resulting in the formation of fractures within the rock, which very likely caused an increase in thelaboratory permeability. Carbonate rocks are much more heterogeneous than sandstones, and so it isquite possible that the small area of our image did not capture the full pore size distribution that existedin the core. In addition, there is also the ongoing challenge of addressing the impact of vugs and their

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connectivity on permeability calculations based on image analysis. Detailed geological input inunderstanding the nature and origin of the observed vugs, combined with statistical means to definethe percolation threshold of vugs from core/log images, offers a pragmatic solution. Based on ouranalysis of a number of samples, including a few not described in this paper, it seems that the methodworks quite well and helps us to determine quantitative values of petrophysical properties.

Acknowledgements

The author thanks Institute of Reservoir Studies, ONGC Ahmedabad, India for allowing to carryout this research and analysis of different samples of Indian fields.

References: 1.  Jurgawczynski, M.: “Predicting absolute and relative permeabilities of carbonate rocks using Image

analysis and effective medium theory”, Ph.D. Thesis, Imperial College, London (2007).2.   Prediction of permeability from the skeleton of 3-D Pore Structure  by Z.R. Liang, P.C.Philippi,

C.P.Fernandes and F.S.Magnani; SPE Paper No. 39017, Fifth Latin American and Carribbean

Petroleum Engineering Conference and Exhibition, Brazil 1997.

3.  Three-Dimensional Reconstructed Porous Media - Application To The Study Of Transport

 Mechanisms In Sandstones by Adler P.M. ,Jacquin C.G. ,Rahon D Society of Core Analysis 19914.  ADLER P.M. (1989) Flow in Porous Media-In:The "Fractal Approach to Heterogeneous

Chemistry", edited by D. Avnir,Wiley,New York.

5.  Berryman, J. G., and Blair, S. C., Use of digital image analysis to estimate fluid permeability of

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 B, (1986) 34, 8179-81.7.  Adler, P. M., C. G. Jacquin, and J. A. Quiblier, Flow in simulated porous media,  International

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8.  Blunt, M.J.: “Flow in porous media: pore network models and multiphase flow”, current opinion incolloid and Interface science (2001) 6:197-207

9.  Blunt, M. J., Jackson, M. D., Piri, M., and Valvatne, P. H., Detailed physics, predictive capabilitiesand macroscopic consequences for pore-network models of multiphaseflow,  Adv. Water Resour.,

vol. 25, pp. 1069-1089, 2002.

10. Arns, C. H., Bauget, F., Limaye, A., Sakellariou, A., Senden, T. J., Sheppard, A. P., Sok, R. M.,Pinczewski, W. V., Bakke, S., Berge, L. I., Øren, P. E., Knackstedt, M. A., Pore scale

characterisation of carbonates using x-ray microtomography, SPE Paper 90368, presented at theSPE Ann. Tech. Conf., Houston, 2004.

11. Spanne, P., J.-F. Thovert, C. J. Jacquin, W. B. Lindquist, K. W. Jones, and P. M. Adler, Synchrotron

computed tomography of porous media – topology and transports,  Phys. Rev. Letts., (1994) 73,2001-04.

12. Ferréol, B. and D. H. Rothman, Lattice-Boltzmann simulations of flow through Fontainebleau

sandstone, Transp. Porous Media, (1995) 20, 3-20.13. Lock, P. A,  Estimating the Permeability of Reservoir Sandstones using Image Analysis of Pore

Structure, Ph.D. thesis, Imperial College, London, 2001.14. Kirkpatrick, S., Percolation and conduction , Rev. Mod. Phys., vol. 45, pp. 574-588, 1973.

15. Sisavath, S., Jing, X.D. and Zimmerman, R.W.,  Laminar flow through irregularly shaped pores in sedimentary rocks, Transport in Porous Media (2001) 45:41-62

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sandstone from image analysis of pore structure, J. Appl. Phys., vol.92, pp. 6311–6319, 2002.

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18. Koplik, J., On the effective medium theory of random linear networks, J. Phys. C , vol.14, pp. 4821-1981, 1981.

19. David, C., Gueguen, Y. and Pampoukis, G.: “Effective medium theory and network theory appliedto the transport properties of rock”, Journal of Geophysical research (1990) 95: 6993-7005.

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Nomenclatures:

Q Volumetric flow rate

C Flow conductance

ΔP Pressure dropμ  Fluid viscosity

L length of the individual pore

r RadiusΠ  Perimeter

A Area

R h Hydraulic radiusCeff Effective Conductance

z Coordination Number

Φ  Porosityk Permeability