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**n\PHYSICAL PROPERTIES
Shipboard measurements of physical properties provide quantitative information about the composition and lithology of core material and are used to characterize lithologic units and to correlate core data with downhole logging and seismic reflection data. All physical properties measurements were taken on cores after they equilibrated to room temperature (~25C). Equilibration to room temperature takes 24 hr. Magnetic susceptibility, gamma ray attenuation bulk density, compressional wave (Pwave) velocity, and natural gamma radiation were measured on whole cores using the MST. Thermal conductivity was measured on each core, using the whole core where possible. After core splitting, undrained shear strength, index properties, and additional measurements of Pwave velocity were conducted on the working half.
Multisensor Track Measurements
The MST, which is described in detail by Blum (1997), consists of four sensors: the magnetic susceptibility logger, gamma ray attenuation densiometer (GRA), Pwave logger (PWL), and natural gamma ray detector (NGR). MST data were sampled at discrete intervals along the core. The sample interval and the data acquisition period for each sensor were set to optimize the resolution of data acquired within the sampling time available for each core. MST data are significantly degraded if the core liner is only partially filled or if the core is disturbed. When RCB or XCB drilling was used, the core diameter was less than the nominal 6.6cm core diameter. The reduced core diameter required corrections of the values measured by the MST. The values in the database do not reflect these corrections, but the figures presented in the following chapters show corrected data.
Magnetic Susceptibility Logger
Magnetic susceptibility is the degree to which a material can be magnetized by an external magnetic field. If the ratio of magnetic susceptibility is expressed per unit of volume, volume susceptibility is defined as
INCLUDEPICTURE "http://wwwodp.tamu.edu/publications/chars/kappa.gif" \* MERGEFORMATINET = M/H,
where M = the volume magnetization induced in a material of susceptibility ( INCLUDEPICTURE "http://wwwodp.tamu.edu/publications/chars/kappa.gif" \* MERGEFORMATINET ) by the applied external field (H). Volume susceptibility is a dimensionless quantity. It can be used to help detect changes in magnetic properties caused by variations in lithology or by alteration. Magnetic susceptibility was measured at 5cm intervals along the core using a Bartington meter (model MS2C) with an 88mm coil diameter and a 2s integration period. The Bartington meter operates at a frequency of 0.565 kHz and creates a field intensity of 80 A/m (= 0.1 mT), significantly lower than the field intensity needed to change the field orientation of magnetite grains (~50 mT). The width of the instrument response to a thin layer of material with a high magnetic susceptibility is ~10 cm. For this reason, the first and last measurement of each core section was taken 4 cm from the core section ends.
Gamma Ray Attenuation Densiometer
The GRA densiometer estimates bulk density by measuring the attenuation of gamma rays traveling through the core from a 137Cs source. The gamma rays are attenuated by Compton scattering as they pass through the sample. The transmission of gamma rays through the sample is related to the electron density of the sample by
Yt = Yi x ensd,
where
Yt = the transmitted flux,
Yi = the incident flux on a scatterer of thickness d,
n = the number of scatterers per unit volume or the electron density, and
s = the crosssectional area per electron.
The bulk density ( INCLUDEPICTURE "http://wwwodp.tamu.edu/publications/chars/rho.gif" \* MERGEFORMATINET ) of the material is related to the electron density (n) by
n = INCLUDEPICTURE "http://wwwodp.tamu.edu/publications/chars/rho.gif" \* MERGEFORMATINET x NAV x (Z/A),
where
Z = the atomic number or number of electrons,
A = the atomic mass of the material, and
NAV = Avogadro's number.
Bulk density estimates are therefore accurate as long as the ratio Z/A of the constituent elements is approximately constant and corresponds to the ratio Z/A of the calibration standard. The GRA densiometer was calibrated to a standard consisting of varying amounts of water and aluminum so that the densities of sediments can be accurately determined. GRA density was measured using a 2s integration period at 5cm intervals along the core.
Compressional Wave (PWave) Logger
The compressional wave (Pwave) logger (PWL) measures the ultrasonic traveltime of a 500kHz compressional wave pulse through the core and the core liner. A pair of displacement transducers monitors the separation between the Pwave transducers, and the distance is used to convert ultrasonic traveltime into velocity after correcting for the liner. Good coupling between the liner and the core is crucial to obtaining reliable measurements. The PWL is calibrated by placing a water core between the transducers. The PWL was set to take the mean of 1000 velocity measurements over a 2s period at 5cm intervals along the core.
Natural Gamma Ray Detector
The NGR measures the discrete decay of 40K, 232Th, and 238U, three longperiod isotopes that decay at essentially constant rates within measurable timescales. Minerals that include K, Th, and U are the primary source of natural gamma rays. These minerals are found in clays, arkosic silts and sandstones, potassium salts, bituminous and alunitic schists, phosphates, certain carbonates, some coals, and acid or intermediate igneous rocks (Serra, 1984). The operation of the NGR is outlined by Hoppie et al. (1994). The NGR system contains four scintillation counters arranged at 90 angles from each other in a plane orthogonal to the core track. The counters contain doped sodium iodide crystals and photomultipliers to produce countable pulses. The total response curve of the instrument is estimated to be ~40 cm and so integrates a relatively long length of core in comparison to the other instruments of the MST. Natural gamma ray emissions were measured over a 20s period at 10cm intervals. The NGR was calibrated in port against a thorium source and during Leg 195 by measuring sample standards at the end of operations at every site.
Thermal Conductivity
Thermal conductivity is the measure of the rate at which heat flows through a material. It is dependent on the composition, porosity, density, and structure of the material. Thermal conductivity profiles of sediments and rock sections are used, along with temperature measurements, to estimate heat flow. Thermal conductivity is measured through the transient heating of a core sample with a known geometry using a known heat source and recording the change in temperature with time, using the TK04 system described by Blum (1997). For soft sediment, thermal conductivity measurements are made using a needle probe (Von Herzen and Maxwell, 1959) on wholecore sections; the reported value is the mean of three repeated measurements. For materials too hard for the needle probe to penetrate, thermal conductivity measurements are made after core splitting, using the needle probe in a halfspace configuration (Vacquier, 1985); the reported value is the mean of four repeated measurements. Thermal conductivity measurements were made at an interval of at least one per core unless variations in lithology required more frequent sampling.
Undrained Shear Strength
The undrained and residual shear strength of sediments and serpentinite mud was measured using a WykehamFarrance motorized vane shear apparatus following procedures described by Boyce (1977). In making vane shear measurements, it is assumed that a cylinder of sediment is uniformly sheared around the axis of the vane in an undrained condition. The vane used for all measurements has a 1:1 length to diameter blade ratio with a dimension of 1.28 cm. A high vane rotation rate of 90/min was used to minimize pore fluid expulsion while measurements take place. Torque and strain measurements at the vane shaft were made using a torque transducer and potentiometer. Undrained shear strength measurements were made at least once per core section unless variations in lithology required more frequent sampling.
PWave Velocity
Discrete Pwave velocity measurements were made in three directions in the sediments using two pairs of insertion transducers (PWS1 and PWS2) with fixed separations of 7 and 3.5 cm, respectively, and a pair of contact transducers (PWS3) in a modified Hamilton Frame. PWS1, PWS2, and PWS3 use a 500kHz compressional wave pulse to measure ultrasonic traveltimes, which, when combined with transducer separation data, can be used to determine velocity. PWS1 and PWS2 were only used to measure velocity in soft sediments, where they were inserted into the face of the split core. PWS1 is aligned with the core axis (the zdirection), and PWS2 is aligned perpendicular to the core axis (the ydirection). PWS3 is mounted vertically with one transducer fixed and the other mounted onto a screw, allowing the transducer separation to be altered. PWS3 measures velocity in the xdirection in split cores but is also used to measure velocity in discrete samples of hard sediments or crystalline rock. Distilled water is applied to PWS3 to improve the acoustic coupling between the transducers and the sample. Pwave velocity measurements were made at least once per core section.
Index Properties Measurements
Minicore samples of ~10 cm3 were collected using a piston sampler in soft sediment or an electric drill in rocks. Samples were taken at least once per section. Sediment samples were placed in a 20mL beaker and sealed to prevent moisture loss. Rock samples were soaked in seawater for 24 hr before determining the wet mass. Samples were then dried in an oven at 105 5C for 24 hr and allowed to cool in a desiccator before measuring dry weights and volumes (method C in Blum, 1997). Wet and dry sample masses and dry volumes were measured and used to calculate wet bulk density, dry density, grain density, water content, and porosity. Sample mass was determined using two Scientech electronic balances. The balances are equipped with a computerized averaging system that corrects for ship accelerations. The sample mass is counterbalanced by a known mass such that the mass differentials are generally <1 g. Sample volumes were measured at least three times, or until a consistent reading was obtained, using a heliumdisplacement Quantachrome pentapycnometer. A standard reference volume was included with each group of samples during the measurements and rotated among the cells to check for instrument drift and systematic error; each time an error was detected in the measurement of the reference volume, the offending cell was calibrated. The following relationships can be computed from the two mass measurements and dry volume measurements (taken from Blum, 1997, pp. 22 to 23). When a beaker is used, its mass and volume are subtracted from the measured total mass and volume. This results in the following directly measured values:
Mb (bulk mass),
Md (dry mass) = mass of solids (Ms) + mass of residual salt, and
Vd (dry volume) = volume of solids (Vs) + volume of evaporated salt (Vsalt).
Variations in pore water salinity (s) and density ( INCLUDEPICTURE "http://wwwodp.tamu.edu/publications/chars/rho.gif" \* MERGEFORMATINET pw) that typically occur in marine sediments do not affect the calculations significantly, and standard seawater values under laboratory conditions are used:
s = 0.035 wt% and
INCLUDEPICTURE "http://wwwodp.tamu.edu/publications/chars/rho.gif" \* MERGEFORMATINET pw = 1.024 g/cm3.
Pore water mass (Mpw), mass of solids (Ms), and pore water volume (Vpw) can then be calculated:
Mpw = (Mb  Md)/(1  s),Ms = Mb  Mpw = (Md  [s x Mb])/(1  s), andVpw = Mpw / INCLUDEPICTURE "http://wwwodp.tamu.edu/publications/chars/rho.gif" \* MERGEFORMATINET pw = (Mb  Md)/[(1  s) x INCLUDEPICTURE "http://wwwodp.tamu.edu/publications/chars/rho.gif" \* MERGEFORMATINET pw].
Additional parameters required are the mass and volume of salt (Msalt and Vsalt, respectively) to account for the phase change of pore water salt during drying. It should be kept in mind that for practical purposes, the mass of salt is the same in solution and as a precipitate, whereas the volume of salt in solution is negligible. Thus,
Msalt = Mpw  (Mb  Md) = [(Mb  Md) x s]/(1  s), andVsalt = Msalt / INCLUDEPICTURE "http://wwwodp.tamu.edu/publications/chars/rho.gif" \* MERGEFORMATINET salt = {[(Mb  Md) x s]/(1  s)}/ INCLUDEPICTURE "http://wwwodp.tamu.edu/publications/chars/rho.gif" \* MERGEFORMATINET salt,
where the salt density ( INCLUDEPICTURE "http://wwwodp.tamu.edu/publications/chars/rho.gif" \* MERGEFORMATINET salt = 2.20 g/cm3) is a calculated value for average seawater salt.
Moisture content is the pore water mass expressed either as percentage of wet bulk mass or as a percentage of the mass of saltcorrected solids:
Wb = Mpw/Mb = (Mb  Md)/[Mb x (1  s)], andWs = Mpw/Ms = (Mb  Md)/[Md  (s x Mb)].
Calculations of the volume of solids and bulk volume are as follows:
Vs = Vd  Vsalt andVb = Vs + Vpw .
Bulk density ( INCLUDEPICTURE "http://wwwodp.tamu.edu/publications/chars/rho.gif" \* MERGEFORMATINET b), density of solids or grain density ( INCLUDEPICTURE "http://wwwodp.tamu.edu/publications/chars/rho.gif" \* MERGEFORMATINET s), dry density ( INCLUDEPICTURE "http://wwwodp.tamu.edu/publications/chars/rho.gif" \* MERGEFORMATINET d), porosity (P), and void ratio (e) are then calculated according to the following equations:
INCLUDEPICTURE "http://wwwodp.tamu.edu/publications/chars/rho.gif" \* MERGEFORMATINET b = Mb/Vb, INCLUDEPICTURE "http://wwwodp.tamu.edu/publications/chars/rho.gif" \* MERGEFORMATINET s = Ms/Vs, INCLUDEPICTURE "http://wwwodp.tamu.edu/publications/chars/rho.gif" \* MERGEFORMATINET d = Ms/Vb,P = Vpw/Vb, ande = Vpw/Vs.
Electrical Resistivity and Formation Factor
The electrical resistivity of the sediment was measured using a fourelectrode configuration. The instrument used was modified at the University of California, Santa Cruz, from the design of Andrews and Bennett (1981) and was built at the University of Hawaii. The electrodes consisted of four stainless steel pins that are 2 mm in diameter, 15 mm in length, and spaced 13 mm apart. A 20kHz squarewave current was applied on the outer electrodes, and the difference in potential between the two inner electrodes was measured. The size of the current (typically 50 mA) was measured over a resistor in the outer circuit.
The main purpose of measuring sediment resistivity was to determine the formation factor, defined as the ratio of the resistivity of sediment with included pore water divided by the resistivity of the pore water alone. In practice, the formation factor is approximated by measuring the apparent resistivity of the sediment in the split core liner and dividing that value by the apparent resistivity of seawater of similar salinity and the same temperature in a 30cm length of split core liner. Using the same configuration for the measurement of the apparent resistivities removes the effects of geometry from the determination of the formation factor.
Hydraulic Conductivity and Specific Storage
The hydraulic conductivity and specific storage of the serpentinite mud was measured during a consolidation test. In this test, an axial surface load is applied to a laterally constrained sample. The axial load produces an excess pore fluid pressure along the length of the core. The bottom of the sample is drained so that the excess pore fluid pressure at that point is zero. The loads and boundary conditions are applied by a Manheim squeezer, and the amount of fluid displaced is measured as a function of time. Figure HYPERLINK "http://wwwodp.tamu.edu/publications/195_IR/chap_02/c2_f10.htm" \l "738606" \t "_blank" F10 is a cartoon of the apparatus and the boundary conditions. Also shown is the pressure profile along the length of the sample at various times. The assumption of incompressible mineral grains and water, common to soil mechanics (Wang, 2000), allows the volume of water discharged from the sample to be converted to axial displacement using the crosssectional area of the sample. Because the frame, not the mineral grains or the water, is compressed, we can calculate the axial displacement using the following equation:
INCLUDEPICTURE "http://wwwodp.tamu.edu/publications/chars/capdelta.gif" \* MERGEFORMATINET w = volume of water discharged/crosssectional area of sample,
where INCLUDEPICTURE "http://wwwodp.tamu.edu/publications/chars/capdelta.gif" \* MERGEFORMATINET w = the axial displacement. We then use the relationship for displacement in an infinite length cylinder as a function of time (Wang, 2000):
INCLUDEPICTURE "http://wwwodp.tamu.edu/publications/195_IR/chap_02/images/c2_a.gif" \* MERGEFORMATINET ,
to determine the lumped product of constants (on the right hand side of the following equation) by plotting the slope of the displacement over the square root of time
INCLUDEPICTURE "http://wwwodp.tamu.edu/publications/195_IR/chap_02/images/c2_a2.gif" \* MERGEFORMATINET ,
where
cm = the vertical compressibility,
INCLUDEPICTURE "http://wwwodp.tamu.edu/publications/chars/gamma.gif" \* MERGEFORMATINET = the loading efficiency,
INCLUDEPICTURE "http://wwwodp.tamu.edu/publications/chars/sigma.gif" \* MERGEFORMATINET z = the axial load, and
D = the hydraulic diffusivity.
Figure HYPERLINK "http://wwwodp.tamu.edu/publications/195_IR/chap_02/c2_f11.htm" \l "733923" \t "_blank" F11 compares experimentally determined displacements with calculated displacements as a function of time. Only the early time portion of the plot is used to determine the lumped product of constants. Early in the experiment, the decrease in pore pressure has not yet diffused to the end of the sample and so the approximation of an infinite cylinder is still valid. To determine the hydraulic diffusivity (D) from the lumped product, we need to determine the other unknown factors, cm and INCLUDEPICTURE "http://wwwodp.tamu.edu/publications/chars/gamma.gif" \* MERGEFORMATINET .
The vertical compressibility is defined by
INCLUDEPICTURE "http://wwwodp.tamu.edu/publications/195_IR/chap_02/images/c2_a3.gif" \* MERGEFORMATINET .
Because all the components of the equation above, axial displacement ( INCLUDEPICTURE "http://wwwodp.tamu.edu/publications/chars/capdelta.gif" \* MERGEFORMATINET w), axial stress ( INCLUDEPICTURE "http://wwwodp.tamu.edu/publications/chars/sigma.gif" \* MERGEFORMATINET z), and sample length (wo) are measured, it is possible to calculate the vertical compressibility. Also, because the pore pressure throughout the entire length of the sample returns to zero at very long times, the boundary condition of no change in pore pressure ( INCLUDEPICTURE "http://wwwodp.tamu.edu/publications/chars/capdelta.gif" \* MERGEFORMATINET Ppore = 0) is met.
The loading efficiency is defined as
INCLUDEPICTURE "http://wwwodp.tamu.edu/publications/195_IR/chap_02/images/c2_a4.gif" \* MERGEFORMATINET ,
where Ppore = the pore fluid pressure. The assumption of incompressible grains and pore fluid leads to a value = 1 (Wang, 2000).
The specific storage (Ss) is related to the vertical compressibility under the assumptions of incompressible grains and pore fluid by
Ss = cm x INCLUDEPICTURE "http://wwwodp.tamu.edu/publications/chars/rho.gif" \* MERGEFORMATINET f x g,
where
INCLUDEPICTURE "http://wwwodp.tamu.edu/publications/chars/rho.gif" \* MERGEFORMATINET f = the fluid density, and
g = the acceleration of gravity.
Finally, we can determine the hydraulic conductivity from the hydraulic diffusivity and specific storage using
K = D x Ss.
HYPERLINK "http://wwwodp.tamu.edu/publications/195_IR/chap_02/c2_8.htm" INCLUDEPICTURE "http://wwwodp.tamu.edu/publications/195_IR/chap_02/next.gif" \* MERGEFORMATINET
PREDICTING PHYSICAL PROPERTIES OF RESERVOIR ROCKS FROM
THE MICROSTRUCTURAL ANALYSIS OF PETROGRAPHIC THIN
SECTIONS
M.C. Damiani *1 , C.P. Fernandes +2 , A. D. Bueno +3 , L.O. E. Santos +4 , J.A.B. da Cunha Neto +5 ,
P.C. Philippi +6i
(*) Engineering Simulation and Scientific Software (ESSS)
Parque Tecnolgico de Florianpolis  Rodovia SC 401 km 001
88030000 Florianpolis, SC, Brazil
(+) Porous Media and Thermophysical Properties Laboratory (LMPT)
Mechanical Engineering Department. Federal University of Santa Catarina BP476
88040900 Florianpolis, SC, Brazil.
Abstract
Important physical properties of petroleum reservoirs are, usually, obtained by applying standard
experimental tests on rock samples collected along selected depths of the petroleum well. This is,
presently, a laborious and high cost procedure. Present paper describes Imago, which is a new
petrographic analysis software, developed for predicting the physical properties of reservoir rocks,
starting from petrographic thin sections routinely produced in petroleum industry for microscopic
analysis. Imago includes five main modules for i) segmentation of the porous phase from gray level
and/or color digital images, ii) geometrical characterization of the porous microstructure, iii) threedimensional
reconstruction, iv) prediction of phase distribution in wettingnon wetting fluid
equilibrium displacement and v) prediction of intrinsic permeability. Simulation results are
compared with experimental results for Berea and some other petroleum reservoir sandstones.
1 Marcos Cabral Damiani is a software development engineer from the Engineering Simulation and Scientific Software
(ESSS) staff. Presently, he is working in the software development of Imago at Porous Media and Thermophysical
Properties Laboratory (LMPT). 2 Celso Peres Fernandes is a Visiting Professor from Brazilian Petroleum Association (ANP), working at Porous Media
and Thermophysical Properties Laboratory (LMPT). His fields of interest include image analysis and processing, 3D
geometrical reconstruction and multiphase flow in porous media
3 Andr Duarte Bueno has BS and MSc degrees in Civil Engineering. He is presently making his Doctoral studies in
Porous Media and Thermophysical Properties Laboratory (LMPT) which includes equilibrium phase distribution and
prediction of relative permeability in porous media.
4 Luis Orlando Emerich dos Santos has a BS degree in Electrical Engineering and a MSc degree in Physics. He is
presently making his Doctoral studies in Porous Media and Thermophysical Properties Laboratory (LMPT) which
includes the use of lattice gas automata (LGA) for the analysis of single and twophase flow displacements in 3D
porous representations. 5 Jos A. Bellini da Cunha Neto is Associate Professor at Federal University of Santa Catarina. He specializes in the
macroscopic approach to fluid flow in porous media, image analysis and mercury intrusion processes.
6 Paulo C. Philippi is Professor at Federal University of Santa Catarina where he heads the Porous Media and
Thermophysical Properties Laboratory (LMPT). He specializes in image analysis and processing, single and
multiphase flow in porous media and lattice gas automata (LGA) methods.
i
Corresponding Author (philippi@lmpt.ufsc.br).1. Introduction
Important physical properties of petroleum reservoir rocks are, usually, obtained by
applying standard experimental tests on rock samples collected along selected depths of the
petroleum well. In addition to sampling costs, laboratory routine includes the manufacture of thin
plates for petrographic analysis and the measurement of porosity, intrinsic and relative permeability
and mercury intrusion tests, which are, presently, laborious and high cost procedures.
Recently, the advancement of image analysis methods, applied on thin sections of reservoir
rocks, appears to open a promising fast and low cost technique for predicting these physical
properties, from the solely knowledge of the porous microstructure of the reservoir rock. In this
way, by using statistical homogeneity and spatial isotropy, the threedimensional porous structure is
supposed to be univocally related to twodimensional representation obtained from thin plates.
Furthermore, the physical properties of reservoir rocks, in petroleum extraction, are supposed to be
given from the porous microstructure geometry and the physicochemical interaction between the
involved fluids and the porous surface.
The main aim of porous media study is the description of equilibrium and transfer processes
that are dependent on its porous structure. Common phenomenological methods use adjustable
idealized models and/or experimental data to describe fluid occupation and the rates of fluid
transfer. They are not able to explore local geometric information of the porous structure. Image
analysis techniques, starting from graylevel and/or color pictures taken with an electron scanning
and/or optical microscope of polished porous sections, are suitable to the analysis of the geometrical
structure of porous media and could allow to the knowledge of their influence on fluid equilibrium
and transfer. The main advantage of this approach is that the detailed knowledge of the porous
structure enable to perform this study, starting from very fundamental physical laws which are not
dependent on the porous structure.Virtual core analysis is a relatively recent research field and has
been included in the SPE (Society of Petroleum Engineers) main research lines. At authors
knowledge, the only existing operational software developed to perform core analysis was
constructed by Porous Media Research Institute (PMRI) (http://panda.uwaterloo.ca/pmri/) at
Waterloo University and named VCLab (Virtual Core Laboratory). VCLab starts from binary
segmented images and permeability calculations are based on percolation networks, obtained from
the reconstructed threedimensional representation.
Present paper describes Imago, which is a new petrographic analysis software, developed for
predicting the physical properties of reservoir rocks, starting from petrographic thin sections
routinely produced in petroleum industry for microscopic analysis.
Imago includes six main modules for:
i) filtering and segmentation of the porous phase from gray level and/or color digital
images,
ii) geometrical characterization of the porous microstructure,
iii) threedimensional (3D) reconstruction,
iv) prediction of phase distribution in wetting nonwetting fluid equilibrium
displacement including mercury intrusion,
v) prediction of intrinsic permeability based on skeleton method ,
vi) prediction of intrinsic permeability using lattice gas automata (LGA), by the
integration of local velocity fields.
The segmentation module includes a neural network method for pattern recognition.
Geometrical threedimensional reconstruction uses a new algorithm which starts from the
measured porosity and autocorrelation function and uses a Fourier transform method for reducing
processing time [1].
Phase distribution is predicted by supposing quasistatic displacement by a wetting and/or a
nonwetting invader fluid into the porous structure [2]..Intrinsic permeability is predicted, in a first module, from the 3D representation of the
microstructural skeleton [3]. Another permeability module include, a boolean lattice gas automata
(LGA) for predicting permeability by integrating the velocity field inside the threedimensional
representation of the porous structure.
Constructed in C++ objectoriented language to be a usable tool in petroleum engineering
practice, Imago is multiplatform (Win32 and XWindow) and userfriendly, with tools for
generating and visualizing intermediate results: lists, graphics and image properties, process log and
status. Special tools for threedimensional scientific visualization such as rendering, rotation,
translation, zoom and twodimensional crosssections, are also provided.
Simulation results are compared with experimental results for Berea and some other
petroleum reservoir sandstones.
2. Geometrical analysis and processing: filtering, segmentation and 3D reconstruction
Imago was developed for predicting the physical properties of reservoir rocks, based on
petrographic thin sections routinely produced in petroleum industry for microscopic analysis
(Figure 1). Imago starts from color and/or gray level digital images, obtained after, respectively,
optical and/or electronscanning microscopy performed on reservoir rocks thin plates.
2.1 Filtering
Small features in the image are frequently found as the result of polishing procedures and/or
image acquisition and appear as image noises. These noises are eliminated from the analysis by
using spatial filters. The most commonly used filters are the lowpass filter, which attenuates color
gradients and median filter that eliminates small discontinuities without any geometrical
consequence.
2.2 Segmentation
For fluid flow simulation purpose, it is only necessary to segment the original digital image
in solid and porous phases, resulting in binary black and white images. Thresholding is performed,
by Imago, in accordance with the following methods.
2.2.1 Color images
Color digital images are derived from optical microscopy. Thin plates are prepared by
introducing a colored resin into the porous structure and the porous phase is identified as the
geometrical region related to the particular resins color. Digital images are frequently acquired and
stored as 24 bits files in the RGB color model. In this model, each color is the result of a
combination of components red, green and blue [4]. Imago presents two packages for color images
segmentation. The first one works directly with RGB color model and the second one starts from
HSI model, where the information of color is stored in the hue (H) component, while components S
and I give information on saturation and intensity, respectively. Thresholding is performed using
interclass variance maximization method [5], applied to each component histogram in both color
models. Taking petroleum industry needs, segmentation method was automated, although manual
segmentation is also enabled in this module, when, due to polishing and/or acquisition problems,
original digital image has not a good quality. In this way, zoom, hand and preview tools were
included in this module to simplify manual segmentation work (Figure 2)..2.2.2 Graylevel images
Imago includes three graylevel segmentation modules. The first module is manual and
based on the histogram threshold which appears to best separate the graylevel classes associated to
solid and porous phase, respectively. The second module is automatic and based on the interclass
variance maximization method. The third module uses the global entropy method introduced by [6]
and was, also, automated.
Imago also includes a neural network module for both gray level and color images that is,
presently, in its developing phase. Preliminary results are very promising and the intention is to
validate this method, which is very suitable for automating.
2.3 Measure of geometrical parameters
2.3.1 Autocorrelation function
In porous materials, one can theoretically distinguish between the solid and pore phase. The
pore space of porous media can be characterized by the phase function Z(x) as follows:
= otherwise 0
porespace the to belongs when 1 x Z(x) (1)
where x denotes the position with respect to an arbitrary origin.
The porosity e, the autocorrelation function C(u), and the normalized autocovariance
function RZ(u) can be, respectively, defined by the following statistical averages (denoted by an
overbar):
e= Z(x) (2)
u) + = (x)Z(x Z C(u) (3)
) R 2 Z
e (e
e]

 + e]  = u) [Z(x [Z(x) (u) (4)
Porosity e is obviously a positive quantity which is limited to the (0, 1)interval. It can be
shown that a function Rz(u) is a normalized covariance function if all its Fourier components are
nonnegative. This work is restricted to homogeneous media, where statistical parameters are
assumed to be independent of position x in space. Thus, the porosity is constant and Z R (u)depends
only upon the vector u being independent of position x. Moreover, when the porous media is
isotropic, Z R is a function of only u = u , and does not depend on the direction of u.
A simple method to calculate the correlation function and normalized covariance function
was used by [7], [8]. Let S be a section of a porous medium, given by a 2D binary representation,.porous phase is represented in black and the solid matrix in white. The binary image S is divided
into two halves S1 and S2. Hence,
S=S1S2, S1S2= (5)
In order to calculate Rz(u), S1 is first translated by a distance u along the xaxis; yielding
S1(+u). The spatial average indicated in Eq. (3) is calculated as an intersection of images,
Z(x y Z(x u y , ) , ) + = S1(+u) S (6)
giving the correlation function.
Since C(u) is related to the probability of finding two points separated by u and belonging to
the same phase, it is, however, advantageous to calculate the autocorrelation function C(u) as a
function of the twodimensional vector u =(x,y) and, then, to take its mean value around a circle
with radius u= u . This last procedure produces more reliable C(u) values because it increases the
number of realizations needed to calculate this probability. For an image f(x,y), the Fourier
transform of the autocorrelation function is also the power spectrum of f(x,y) (WienerKhinchin
theorem). The 2D autocorrelation function is calculated in Imago by using the Fourier transform.
Fluctuations are drastically reduced, when C(u) is calculated using Fourier transform.
In this way, with the Fourier transform of phase function, the correlation function can be
obtained, rapidly, using Fourier transform methods. Figure 3 show, a surface display representation
of the power spectrum of a porous section binary image. The correlation function is the inverse
Fourier transform of the power spectrum.
Given the two dimensional estimate ) y , x ( C , we can obtain the desired one dimensional
(isotropic) correlation function C(u) by averaging over the ) y , x ( C values at a fixed radius u.
Except for the cases (0,u) and (u, 0), ) y , x ( C will not generally be known at the points of interest
(see Figure 4).
Figure 5 shows the correlation function obtained with Fourier transform method, compared
with the previous displacement method.
2.3.2 Pore Size Distribution
In Imago, pore size distribution of binary 2D sections is obtained by successive opening,
derived from mathematical morphology and using balls with increasing radius [5]. The resulting
image can be viewed as the union of balls completely enclosed in the porous phase. Thus, after
opening, porous phase lost all the features smaller than the opening ball. The cumulative porous
distribution is, then, given by:
( ) ( )
e
e e r r F  = (7)
where e is the total porosity of the original image and ( ) r e is the volume fraction of the porous
phase after opening with a radius r ball..To reduce time processing, opening operation is not applied directly to the binary image, but
to a transformed image called background distance image. In this image, each pixel is labeled with
the smaller distance from it to the neighbor background. This labeling technique uses a sequential
algorithm [9], where the Euclidean distance is approximated by a discrete integer distance. The
most commonly used discrete distance is the chamfer distance known as d34, where each neighbor
from a given point, taken following the horizontal and vertical coordinated axis, are considered to
be 3 measuring units distant from the starting point. The diagonal neighbors are considered to be at
4 measuring units from that point. Thus, this discrete distance gives the numerical approximation of
4/3 = 1.333...to the square root of 2 (1.4142...). The main advantage of using this discrete distance is
related to the lower computer storage as only integers are stored in resident memory. Balls
generated with d34 distance that are used to perform opening operation are octagonal in shape.
2.4 Threedimensional reconstruction
Threedimensional reconstruction is based on a modification of Adler et al. Gaussian
truncated method [7] proposed by [1]. In Adler s method, threedimensional stochastic simulation
of porous structure is based on the generation of a random noncorrelated field X(x), which gives
the correct binary phase function Z(x), after the successive application of a linear and a nonlinear
filters. Linear filter gives Y(x), which is a convolution
Y(x)=a*X= +
s
s) a(s)X(x (8)
inside a spherical domain with radius lequal to the correlation length, measured at the original 2D
binary image. Nonlinear filter transforms Y(x), which is a real variable field to a binary field
Z(x){0,1}. By assuming homogeneity and isotropy, 3D pore structure can thus be constructed
from 2D porous sections, preserving porosity and autocorrelation function. Another way to carry
out linear filter is to generate Y(x) from X(x) using Fourier transform [10]. From a computational
point of view, the use of the fast Fourier transform algorithm, instead of laborious solution of
nonlinear equation, makes the Fourier transform superior to linear filter method. Further, although
Fourier transform (p) a of a real field a(r) is complex, it can be easily show that in discrete
representations, due to periodicity and conjugate symmetry properties of Fourier transform,
computer resident memory requirements are the same for both fields. A truncated Gaussian method
by using Fourier transform was proposed by [1]. The difference between this method and the
previous ones is that the field Y(x) is directly generated from its autocovariance function RY(u). In
fact, by the WienerKhinchin theorem, the Fourier transform of the autocovariance of a function is
the power spectrum of this function, i.e.,
2 ^ 2
Y
^
Y Y  R = (Y) = (R = (u)) (p) (9)
Therefore, if the autocorrelation function is known for an arbitrary field Y(x), Fourier
transform can be used to generate this field, with the same autocorrelation function. In fact, the
above equation means that the Fourier transform R
^
Y (p) of RY (u) is only related to the magnitude
^
Y of
^
Y =(Y) . This means that the phase angle of
^
Y does not depend on the autocovariance.R
^
Y (p) , or, in other words, that any two functions,
^
Y , with different phase angle may give the
same autocorrelation. This method does not need the linear filter and so avoids solving the
nonlinear equations, reducing computer running time. Using the fast Fourier transform makes this
algorithm more efficient. This idea was also used by [11] in reconstructing twodimensional serial
sections in their hybrid approach. However, their method needs linear filter to generate 3D porous
structure representations from these noncorrelated serial sections. Present approach avoids solving
a set of nonlinear equations associated with linear filter transform. In addition, when compared with
Adlers method [10], it reduces the resident memory requirements, because the independent
gaussian field data X(x) are not needed. Therefore, both operating time and computer memory
requirements are improved. These are the advantages of the truncated Gaussian method by using
Fourier transform used by Imago.
Figure 6 shows a schematic description of Liang et al.s method used in the reconstruction
module of Imago and Figure 7 presents an example of output of this package, showing a
comparison for the cumulative distribution of pore volume fraction between the values measured on
the original binary image and the mean value obtained by measuring on several 2D cross sections
of the reconstructed microstructure.
Opening operation, section 2.3.2, was used in this computation step. Factor N is related to
the linear size of the 3D spatial representation and factor n is a sampling factor taken over RZ(u)
data: sampling factor n means that the only points considered for RZ(u) data were the ones separated
by n1 pixels.
Results show that, when sampling factor n is reduced, threedimensional reconstruction
gives a systematic deviation with a larger amount of small size pores when compared to the original
binary image. The better agreement was obtained, in present sample case, for n=5 and n=6. In fact,
it must be remembered that present reconstruction method is based: i) on the hypothesis that the
target original twodimensional representation, is a realization of a stochastic process which is
considered to be ergodic and stationary and ii) on the hypothesis that this process is Gaussian and
can be, inherently, described by its only two first moments.
Imago has a three dimensional window for visualizing the threedimensional reconstructed
microstructure presented by isosurfaces and/or isolines, with important operational tools including
orthogonal cuts, zoom, translation and rotation. Figure 8 shows an example of this softwares
feature.
3. Prediction of phase distribution in immiscible displacement
Determination of phase distribution inside the porous structure is an important problem in
oil recovery, giving reliable information for the prediction of recovery ratio and rate on secondary
extraction.
Imagos package for the prediction of phase distribution is based on [2], [12]. Phase
distribution is simulated by supposing that equilibrium states are attained through quasistatic
processes. Method is based on the relationship between capillary pressure and the curvature radius
of the interface given by YoungLaplace s equation, when this interface is supposed to be spherical:
i i
o B A
AB i
P P
(d r 
1)s  = (10).where d is the Euclidean dimension of the space, sAB is the interfacial tension between fluids A
and B, i
o A P is the pressure at the domain of fluid A which is connected to a free region L, and i B P is
the pressure of fluid B, at step i.
In this way, for a given capillary pressure i
o A P  i B P , nonwetting phase will be found in the
geometrical region where it is possible to be place a ball with radius r i , given by the above equation,
and problem is reduced to a geometrical problem that can be solved by image analysis methods.
Method is fully described in [2], [12] and readers are referred to these papers for further details.
Imagos 3D visualization window enable the software operator to follow fluid displacement
in the course of invasion process. An example of such a visualization is given in Figure 9 showing,
wetting fluid location in an imbibition process, at an intermediate pressure step.
3.1 Mercury intrusion simulation
Mercury intrusion on samples of reservoir rocks is a routinely performed experimental test
in petroleum industry, giving information on porosity, size distribution of pore constrictions and
thresholding constriction diameter.
Imagos package for the simulation of mercury intrusion is the same used for the prediction
of phase distribution explained above considering a quasistatic displacement of nonwetting fluid
into a reconstructed porous structure, initially filled with an ideally compressible fluid. Contact
angle with respect to nonwetting fluid is further corrected from the 180 o value, used in the
simulation, to the 140 o value that is commonly accepted between experimenters.
Figure 10 gives a comparison of simulation results with experimental results for Berea500
sandstone. Best sampling factor was n=5. For testing statistical homogeneity, simulation was
performed for three different linear sizes: N=60, 80 and 100.
4. Flow simulation
4.1 Intrinsic permeability
The simplest approaches for the calculation of intrinsic permeability, based on the idea of
conduit flow, ignore the fact that different pores are interconnected with each other. These are
called capillary permeability models, among which the socalled CarmanKozeny model enjoys
much greatest popularity. Another permeability model is the cutandrandomrejoin model. The
sample is sectioned into two by a plane perpendicular to the flow direction, and the two parts are
rejoined together in a random fashion. The flow rate in the capillaries is assumed to be described by
a HagenPoiseuille relationship. In empirical permeability models permeability is usually correlated
with some characteristic parameters. Network models have been used in the last four decades. The
simplest network is a regular lattice, which consists of a regular arrangement of sites or pores,
connected by bonds or throats. Pores and throats are characterized by their radii, with an
independent probability distribution function for each, and by the centertocenter distance.
Furthermore, the lattice is characterized by its coordination number, the number of throats meeting
at a pore. Regular networks such as square, hexagonal, kagom, trigonal, cube, and cross square
were treated. Network models are all based on the some information of pore structure, such as pore
size distribution and coordination numbers. But, these data are almost assumed in the previous
works.
Imago has two packages for permeability calculation. The first package is based on the
Skeleton Method and the second one is based on a Lattice Boolean Model..4.2 Skeleton Method
Imago uses skeleton method [3], where a detailed analysis and validation results can be
found. The skeleton of a porous structure is a connected line composed by all points p that are
equally distanced from porous surface. In this way, it can be considered as the line giving a first
approximation to the main flow line through the porous structure. Skeleton is, thus, an intermediate
level model, between permeability models based on percolation networks and models based on the
integration of local velocity fields. In this way, considering, in addition, the meaningless of
secondary flows for the calculation of fluid transfer rate, in low Reynolds number flows, Imago
calculation method is based on the following assumptions:
i) Skeleton nodes are considered as flow bifurcation points;
ii) Flow between two successive nodes is considered to be onedimensional, through
a variable area cylindrical duct.
The reconstructed 3D porous structure is usually a cube with side length Lx Ly Lz. The
xaxis of the medium is chosen to be parallel to the direction of macroscopic flow. Impervious
boundary conditions are applied to the external surfaces of the cube that are parallel to axes y and z.
The skeleton of the pore structure is obtained by using Mas thinning algorithms [3]. The 3D
skeleton of pore structure provides a real network. The hydraulic conductance of the fluid in the
network can be calculated from the local hydraulic radius. Therefore, permeability can be estimated
from the above data.
To compute intrinsic permeability, we need to define the hydraulic conductance of the fluid
in the network. Flow is assumed to be sufficiently slow. For each point p in a given link, there is
associated a pore section in a plane which contains p and is orthogonal to the link. This section is
obtained by performing an oblique slice of unity thickness in the reconstructed 3D structure,
orthogonal to the link. Furthermore, one makes the simplification that the resistance to fluid flow in
the link point p may be characterized in terms of the hydraulic radius rH of this pore section:
section cross of perimeter of length
section cross of area rH =
(11)
Thus, the conductance of a fluid in a point of pore link, gL, is given by Poiseuille s equation and
may be written as:
(p) g
4
H
L
hl
2pr =
(12)
where h is the fluid viscosity and l is the length of the pore (here l =1).
The overall resistance to flow between the two neighboring nodes i and j, is
=
j
p=i L ij (p) g
1
g
1 (13)
The flow rate of fluid between the two nodes,.) P (P g Q j i ij ij  = (14)
where Pi and Pj are the nodal pressures. Since the fluid is incompressible, mass conservation
requires that
0 Q
j
ij = (15)
where j runs over all the links connected to node i. Equation (15) together with the appropriate
boundary conditions form a complete solution to the steady flow of an incompressible fluid in the
pore skeleton network. The equations are solved using successive overrelaxation:
i
j ij
j j ij
i
g
P g P b)P (1 + b =
(16)
where b is a relaxation parameter.
By imposing a pressure drop DP across the skeleton network and computing the resulting
single phase flow rate Q, the intrinsic permeability k of the pore network is calculated from Darcy s
law:
L L
Q k
z y
x
DP
hL = ADP
hLQ = (17)
where Q is the volumetric flow rate, A is the normal crosssectional area of the sample, L is the
length of the sample in the macroscopic flow direction, DP P1P2 is hydrostatic pressure drop, and
h is the viscosity of the fluid.
To illustrate the process to calculate permeability, a 2D skeleton of the pore space is shown
in Figure 11, calculated at each porous point, by solving Stokes equations (Present paper used a
latticeBoltzmann method for the calculation of full velocity field).The links and nodes of the graph
are, respectively, composed of points with exactly two neighbors and three or more neighbors. Dead
ends are associated with stagnated flows and are eliminated from the skeleton. Figure illustrates a
detailed porous region showing the skeleton superposed to the local velocity field, and depicts a
geometrical region where flow is stagnated, presenting a low speed vortex. It is seen that, although,
with a strictly geometrical background, skeleton method can capture stagnation regions by
associating deadends in the geometrical description of these regions.
Table 1 gives a sample of validation result for Berea sandstone with a nominal permeability
of 500 mD. Best results where found for sampling factors n=5 and n=6 which corresponds to the
sampling factors giving the best agreement for pore size distribution (see Figure 7).
As secondary flows associated to flow deviations inside complex porous structures do not
significantly contribute to the main flow resistance when the Reynolds number is very low, the
present method appears to be very suitable and easier to use when compared with methods based on
numerical solutions of Stokes equation, giving the full velocity field. In addition, it does not suffer.from the wellknown limitations of methods based on percolation networks. In fact, the skeleton is
constructed trying to preserve the fine details of the pore structure along the flow path and can, thus,
better describe its influence on main flow.
4.3 Lattice Gas Cellular Automata Model
Imago includes a package for predicting intrinsic permeability of porous media based on
Lattice Gas Cellular Automata (LGA) methods. Details about the model and validation results are
given in [13], a companion paper submitted to present meeting. LGA is a relatively recent method
developed to perform hydrodynamic calculations being object of considerably interest in the last
years. The method is due originally to Frish, Hasslacher and Pomeau, [14]. In its simplest form, it
consists of a regular lattice populated with particles that hop from site to site in discrete time steps
in a process, often, called propagation. After propagation, the particles in each site interact with
each other in a process called collision, in which the number of particles and momentum are
preserved. An exclusion principle is imposed in order to achieve better computational efficiency.
In despite of its simplicity, this model evolves in agreement with Stokes equation for low
Mach and low Reynolds numbers. Some aspects make LGA methods very attractive for simulating
flows through porous media. In fact, the method is explicit in time and boundary conditions in
flows through complex geometry structures are very easy to describe in LGA simulation.
Simulations are performed with integers, needing less resident memory capability and boolean
arithmetic reduces running time. LGA is, furthermore, very suitable for parallel processing.
Taking into account the above features, LGA model was introduced as a package of Imago
software:
i) to increase the precision in permeability calculations, using a model based on the full
velocity field,
ii) for validation purposes in the construction of simpler (and faster) permeability
models.
Figure 12 shows a typical output from Imagos LGA package for the prediction of a
Brazilian sandstone intrinsic permeability. Figure also shows pore size distribution measured as the
mean from 10 original 2D binary image, compared with the mean value obtained by measuring on
several 2D cross sections taken from the reconstructed microstructures. Sampling factor that gave
the best agreement for pore size distribution was n=5, which was taken as a pattern value for the
simulation with LGA model on reconstructed microstructures. Experimental permeability value was
k=441 mD. Predicted permeability value was k=368 mD for a reconstructed porous structure with
linear size N=100 and sampling factor n=5 and k=462 mD for N=200 and n=5. In fact, in addition
to sampling factor, the correct choice of linear size N is important to ensure statistical homogeneity
in permeability calculations. In general, statistical homogeneity in flow calculations is, only,
ensured, at a linear size N that is, often, greater than the value that assures geometrical homogeneity
in the calculation of pore size distribution. Further validation results can be found in [13].
5. Conclusions
Prediction of physical properties of reservoir rocks needs the correct knowledge of their
threedimensional porous structure and the correct understanding of fluids behavior inside porous
space and of its physicochemical interaction with solid surfaces. These are, yet, very difficult and
actual problems, which call for a very intensive research work.
This paper presented Imago, a new petrographic analysis software that includes very recent
analysis methods, derived from geometrical stochastic analysis and fluid mechanics, intended to be
a usable tool in petroleum industry..Imago includes as it main new features:
i) a fast algorithm for threedimensional geometrical reconstruction based on Fourier
transform, with optimized computer resident memory storage requirements;
ii) a fast skeleton method for predicting intrinsic permeability, avoiding the wellknown
limitations of percolation networks;
iii) a very precise method based on lattice gas cellular automata for the prediction of
intrinsic permeability, calculated from the integration of full velocity field inside porous space.
All these methods were integrated in a single software written in C++ object oriented
language and based on a multiplatform library (Win32 and XWindow) which enables Imago to be
run in a very large class of computers, ranging from the very diffused personal computers to highlevel
Unixbased workstations. A very elaborated graphical interface with online output windows,
hand tools, lists, export/import modules and 3D visualization make Imago userfriendly and
flexible.
Validation results were also presented for Berea and some brazilian sandstones, including
pore size distribution, mercury intrusion and intrinsic permeability.
Acknowledgements
The authors would like to thanks CENPES/PETROBRAS (Centro de Pesquisas e
Desenvolvimento Leopoldo A. Miguez de Mello) for providing the images and experimental data of
reservoir rocks. C.P. Fernandes gratefully acknowledges financial support of ANP (Agncia
Nacional do Petrleo).
References
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qpZPP 7$8$H$gd&Q[1] Liang, Z. R., Fernandes, C.P., Magnani, F.S. & Philippi, P.C., 1998, A Reconstruction
Technique of 3D Porous Media by using Image Analysis and Using Fourier Transform, Journal of
Petroleum Science and Engineering, 21, 34, 273283.
[2] F.S. Magnani P.C. Philippi, Liang Z.R. & C.P. Fernandes, 2000, Modelling twophase
equilibrium in threedimensional porous microstructures, International Journal of Multiphase
Flow, 26 (1), 99123.
[3] Z.R. Liang, P.C. Philippi, C.P. Fernandes, and F.S. Magnani, 1999, SPE Paper 56006:
Prediction of Permeability From the Skeleton of 3D Pore Structure, SPE Reservoir Evaluation &
Engineering, 2 (2), 161168 .
[4] Gonzalez, R. C., Wood, R.E. (1992) Digital Image Processing, AddisonWesley Publishing
Company.
[5] Coster, M. and Chermant, J.L., 1989, Precis Danalyse Dimages. Presses du CNRS, Paris.
[6] Kapur, J.N., Sahoo D.K., Wong A.K.C., 1981, A New Method for Gray Level Picture
Thresholding using the Entropy of the Histogram Computer Graphics, Vision and Image
Processing 29, 273285.
[7] Adler, P.M., Jacquin C.G. and Quiblier J.A., 1990. Flow in Simulated Porous Media. Int. J.
Multiphase Flow, 16: 691712..[8] Philippi, P.C., Yunes, P.R., Fernandes, C.P., Magnani, F.S., 1994, The Microstructure of Porous
Building Materials: Study of a Cement and Lime Mortar, Transport in Porous Media, 14, 219245
[9]. J. M. Chassery, A. Montanvert, Gometrie Discrte en Analyse dImages, Editions Hermes,
Paris, 1991.
[10] Adler, P.M., 1992. Porous Media: Geometry and Transports. ButterworthHeinemann, New
York.
[11] Ioannidis, M.A., Kwiecien M. and Chatzis I., 1995. Computer Generation and Application of
3D Model Porous Media: From PoreLevel Geostatistics to the Estimation of Formation Factor.
Paper SPE 30201 presented at the Petroleum Computer Conference, Houston, TX.
[12] P.C. Philippi, F. S. Magnani and A.D. Bueno Two Phase Equilibrium Distribution in ThreeDimensional
Porous Microstructures. Submitted to Produccion 2000 / Aplicaciones de la
ciencia en la ingeniera de petrleo, May 0812 /2000, Foz de Iguau.
[13] L.O. E. Santos, P.C. Philippi, M.C. Damiani. A Boolean Lattice Gas Method for Predicting
Intrinsic Permeability of Porous Media. Submitted to Produccion 2000 / Aplicaciones de la
ciencia en la ingeniera de petrleo, May 0812 /2000, Foz de Iguau.
[14] Frisch, U., Hasslacher B., Pomeau Y., 1986, LatticeGas Automata for the NavierStokes
Equation, Physical Review Letters, 56, 15051508..Figures and Tables
Figure 1. Imago methodology.
Figure 2. HSI segmentation module for color images..Figure 3. Surface display of power spectrum for a porous section binary image.
Figure 4. Obtaining the isotropic correlation function from discrete values for a particular image..0
0.05
0.1
0.15
0.2
0.25
0 50 100 150 200 250
displacement u (pixels)
C( u)
x direction
y direction
average
Fourier transform
Figure 5. Comparison of correlation function.
RZ(u) RY(u) R Y
^
(p)
^
Y(p)
+
Random phase
angle
^
Y(p)
Y(x) Z(x)
1
Non linear
filter
Figure 6. Schematic description of Liang et al. s method for 3D reconstruction [1]..0
0.2
0.4
0.6
0.8
1
1.2
0.00 20.00 40.00 60.00 80.00 100.00
Pore radius mm
Cumulative volume fraction
n=6, N=200
n=5, N=200
n=4, N=200
n=3, N=200
n=2, N=200
original
Figure 7. Comparison for the cumulative distribution of pore volume fraction of Berea sandstone
500mD, between values measured on the original binary image and the mean value obtained by
measuring on several 2D cross sections of the reconstructed microstructure.
Figure 8. Imago threedimensional visualization window..Figure 9. Simulation of wettingfluid invasion (in light) into a reconstructed 3D structure.
0
0.2
0.4
0.6
0.8
1
1.2
0 10 20 30 40 50 60 70 80
Ball radius ( mm )
Hg saturation
Experimental
N=100, n=5
N=80, n=5
N=60, n=5
Figure 10. Comparison of simulation results with experimental data for Berea 500 mD sandstone..Figure 11. 2D skeleton of the pore space superposed to the local velocity field, illustrating
a geometrical region where flow is stagnated, presenting a low speed vortex.
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25 30 35
radius [pixel]
Cumulative volume fraction
Original, k=441
n=5, N=100, k=368
n=5, N=200, k=462
Figure 12. Pore size distribution and predicted LGA permeability values for a
typical Brazilian sandstone. Experimental value measured at
CENPES/Petrobras was k=441 mD..Table 1. Comparison of the result for Berea
sandstone 500 mD for different sampling factors n
and linear size N. Table also shows the number of
skeleton nodes where pressure is to be calculated in
the simulation.
n N Node
numbers
K (mD)
60 2,098 169.1
80 5,187 181.4
100 10,957 226.4 4
120 19,323 335.7
40 697 209.4
100 13,125 465.4
110 18,390 478.3
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