Front Cover
Understanding Pore Space through Log Measurements
Understanding Pore Space through Log Measurements
Copyright
Dedication
Contents
Preface
Acknowledgments
1 - Pores and pore space
1.1 The pore space of rocks
1.1.1 The pore space of granular rocks
1.1.1.1 Pore space of some granular rocks not falling within the above categories
1.1.1.1.1 Intercrystalline pore space
1.1.2 The pore space of carbonate rocks—an example of the porosity classification
1.1.3 The pore space of coals
1.1.4 Pore space of shale reservoir rocks
1.1.4.1 Organic pores in shale reservoir rocks
1.1.4.2 Bitumen pores in shale gas reservoirs
1.1.4.3 Kerogen pores in shale gas reservoirs
1.1.4.4 Organic pores in shale oil reservoirs
1.1.5 The pore space of tight sedimentary rocks
1.1.6 The pore space of nonsedimentary rocks
1.2 Classification of pores by size
1.3 Pores and pore throats
1.4 Logs and pore space
1.4.1 Pore attributes used in petrophysical models of pore networks
1.4.2 Pore throat models
1.4.3 A popular and successful model of pore space – the capillary tube bundle for modeling flow of an incompressible fluid throu ...
1.4.4 A generic treatment of transport of an incompressible fluid through a medium such as a porous rock
1.4.5 Modeling the orientation of pores in space
1.5 The fractal model of pore space
1.6 Use of log data
1.6.1 Understanding the degree of connectivity between two pore size classes using NMR log data
1.6.2 Insights from high-resolution resistivity imaging tool data into the pore size heterogeneity
References
Further reading
2 - Inversion of log data to the gross attributes of pore space
2.1 Estimation of the bulk porosity of laminated formations using deterministic approach
2.1.1 Computation of effective porosity using shallow resistivity, neutron capture gamma ray spectrometry, formation bulk density ...
2.1.2 Computation of effective porosity using shallow resistivity, gamma ray, formation bulk density, and neutron porosity data
2.1.2.1 The wet clay content of shale
2.1.3 Total porosity
2.1.3.1 Pore volume not shared with shale, per unit rock volume
2.1.4 The different ways clay/shale is manifest within clastic rocks
2.1.5 The Thomas – Stieber approach
2.1.5.1 Evaluation of ϕmax
2.1.5.2 Simple analysis ignoring structural shale: computation of laminated and dispersed shale volume fractions in the formation
Gamma activity of laminated formation
2.1.5.3 Analysis considering structural shale also
Computation of the structural laminated and dispersed shale volume fractions in the formation
2.1.5.4 Using NMR data
2.1.5.5 Computation of representative resistivity of sand and laminated shale volume using tensor resistivity data
2.2 Stochastic inversion of log data for laminated formation
2.2.1 Basic response equations that lead to the forward model
2.2.1.1 Total porosity of the rock
2.2.1.2 Total porosity of rock within the flushed zone
2.2.1.3 The equation for sand conductivity in the uninvaded zone
2.2.2 The forward model
2.2.3 Essential constraints
2.2.4 Additional outputs computed
2.2.5 Horizontal permeability and vertical permeability
2.2.6 Usage of high-resolution data
2.2.6.1 Solution when constraints are present
2.2.6.2 Solution when constraints are present
2.3 Evaluation of microporosity
2.4 Evaluation of blocky (nonlaminated) reservoirs
Conclusions
Appendix 1
Bulk density and hydrogen index of wet clay and silt and the value of the silt index of shale
Appendix 2
Hydrogen index of dry clay and silt
Appendix 3
Resistivity along and normal to formation bedding
The inversion process
Concluding remarks
Appendix 4
Computation of total porosity from bulk density and magnetic resonance logs
Case of water base mud
Case of oil base mud or SOBM
Appendix 5
The dual water equation and input parameters
Input parameters which are usually given and how they are used
Computation of the value of Cbw
Source of the input parameters
Appendix 6
High-resolution data
High-resolution bulk density and photoelectric factor
High-resolution model
Output of the high-resolution model
Improved-resolution acoustic slowness – multishot processing of sonic data
Signal processing for the different subarrays
S,T plane and S,T plots
Concluding remarks
Improved-resolution density and neutron porosity logs from conventional 2-detector tool data
Deconvolution technique
Alpha processing
Alpha processing for improving the vertical resolution of conventional neutron porosity logs
Alpha processing for improving the vertical resolution of conventional density logs
Enhancement of vertical resolution of gamma ray logs
High-resolution NMR data
References
Further reading
3 - Pore attributes of conventional reservoirs
3.1 The pore space of intergranular rocks
3.2 Attributes of pore space
3.2.1 Pore shape, pore size, and pore throat size
3.2.2 Pores as the building blocks of pore space
3.2.3 Geometry of the pore body
3.2.4 Size of a pore
3.2.5 The concept of pore class
3.2.6 Surface area to volume ratio
3.2.7 The characteristic length scale of pore space
3.2.8 Hydraulic radius measure of the pore space
3.2.9 The pore shape factor
3.2.10 T2 log mean
3.3 Distribution of incremental porosity over pore radius
3.3.1 Computation of CP(r) using NMR data
3.3.2 Distribution of incremental porosity over pore throat radius
3.3.3 Ratio of pore size to pore throat size
3.3.4 Hard data on the distribution of pore throat size over incremental porosity (CPT(R))
3.3.5 Obtaining CPT(R) from mercury injection data
3.3.6 Obtaining CPT(R) from log data
3.4 Computation of CPT(R) from the NMR data
3.4.1 The linear conversion work flow
3.4.2 Nonlinear conversion work flow
3.5 Pore shape factor through integrating NMR and MICP (mercury intrusion data)
3.5.1 Frequency distribution of pore radius
3.6 A simple visualization of constriction and its effect on the gross permeability of pore space
3.6.1 Model prediction of permeability
3.6.2 Timur–Coates permeability predictor from the perspective of constriction
3.7 Fractal attributes of pore space
3.7.1 The fractal model of the pore space, based on a pore–pore throat assemblage visualization of the physical pore space
3.7.2 A fractal model of the pore space
3.7.3 Permeability from the perspective of the fractal model of the pore space
3.7.4 Cumulative pore volume
3.7.5 Representative hydraulic tortuosity and cumulated surface area to cumulated volume of the capillaries
3.7.5.1 Brookes–Corey permeability and the fractal model of pore space
3.8 Electrical formation factor from the perspective of the fractal model of the pore space
Appendix 1
Relation between tortuosity (τ), porosity (ϕ), and formation factor (F)
References
Further reading
4 - Pore space attributes of nonconventional reservoirs
4.1 CBM reservoirs
4.1.1 The components of the space occupied by fluids in coals
4.1.1.1 Hodot's pore classification scheme
4.1.2 Characterization of the pore space of coals – cleats and fractures that are not cleats
4.1.2.1 Extraction of fractures and cleats from borehole images
4.1.2.1.1 The process for low-angle events
4.1.2.1.2 The process for high-angle events
4.1.2.2 Computation of fracture/cleat aperture
4.1.2.3 Computation of cleat density from images
4.1.2.4 Computation of cleat density from acoustic logs
4.1.2.4.1 Stoneley reflection coefficient, transmission coefficient, and energy loss
4.1.2.5 Cleat volume per unit rock volume
4.1.2.6 Noncleat fracture volume per unit rock volume
4.1.2.7 Cleat orientation and the direction of maximum principal horizontal stress
4.1.3 Characterization of the pore space of coals using NMR data
4.1.3.1 Partitioning of the pore space of coals using NMR data by pore size
4.1.3.2 Interpore-class connectivity
4.1.3.2.1 Degree of connectivity between mesopores and macropore assemblages
4.1.3.2.2 Degree of connectivity between micropore and mesopore assemblages
4.1.3.3 Partitioning of the pore space of coals as bound fluid and free fluid volumes using NMR data
4.1.4 Permeability of coal—measurement
4.1.4.1 Measurement of the permeability of coal using log data
4.1.4.1.1 Permeability from stoneley full waveform inversion
4.1.4.2 Modeling the permeability of coal using log data
4.2 Shale reservoirs
4.2.1 Pore size encountered within shale reservoirs
4.2.2 Differentiation of pore classes for shale reservoirs
4.2.2.1 Bound fluid and free fluid porosity
4.2.3 Multidimensional inversion of NMR echo data using maximum entropy principle
4.2.3.1 Application of MEP
4.2.4 Presentation of the results of inversion
4.2.5 Porosity partition
4.2.5.1 Shale reservoirs bearing oil
4.2.6 The method of diffusion editing
4.2.7 The method of Laplace Inversion with regularization
4.2.8 D-T2 plots (more familiarly known as D-T2 maps) – forward models
4.2.8.1 Gas diffusion and the role played by adsorption of gas in the modeling
T1, T2 are relaxivity for gas shales.
4.2.8.2 Relaxation of bulk gas
4.2.8.3 Relaxation of gas within organic pores
4.2.8.4 Relaxation of gas within inorganic nanopores
4.2.8.5 Adsorption of methane gas into kerogen and the role played by this adsorption in relaxing spins
4.2.9 D-T2 maps and other plots related to the results of echo data inversion—field examples
4.2.9.1 Burst mode activation
4.2.10 Partitioning of total gas into free and adsorbed gas components using only NMR data
4.2.10.1 Partitioning of methane into free and adsorbed methane using relaxation data from echo amplitude inversion
4.3 Characterization of fractured reservoirs
Appendix 1
Kherroubi's work flow for trace extraction for low-angle events
Extraction of pixels which form part of a fracture
Obtaining fracture traces from the extracted pixel sets
Finding the set of vectors (segments) that best fits a trace when placed end to end
Computing the main fracture/cleat orientation mentioned above
Obtaining the final cleat/fracture traces of high confidence
Appendix 2
A derivation of Eq. (4.21)
Appendix 3
Computation of kerogen volume, gas volume, and total porosity in shale gas reservoir
The density of the adsorbed methane (ρgad)
Computation of adsorbed methane volume per unit rock volume
Total gas volume within the kerogen pores and its partition
Total porosity
References
Further reading
5 - Log measurements commonly used for finding the Bulk porosity of conventional reservoirs
5.1 Pore space attributes of conventional reservoirs
5.2 Measurement of bulk porosity
5.2.1 Measurement of formation density for bulk porosity
5.2.1.1 Single scattering of gamma photons
5.2.1.2 Multiple-scattered photons play a big role in the formation density measurement
5.2.1.2.1 Dual detector data processing using single window (per detector) count rates
5.2.1.2.1.1 Graphical method of solving the detector response equations
5.2.1.2.1.2 Computing density and density correction without using graph
5.2.1.2.1.3 Final correction to computed bulk density to account for electron density of water not being half of the bulk density of water
5.2.1.2.2 Computation of ρb using multiwindow count rate inversion
5.2.1.2.2.1 Photon energy spectrum versus number of scatterings
5.2.1.2.2.2 Photoelectric index
5.2.1.2.2.3 Equations of “spine”
5.2.1.2.2.4 Ratio of count rate in lithology window to that in compton window
5.2.1.2.2.5 The quantity (SCHC) of Eq. (5.49)
5.2.1.2.2.6 The quantity (HCH) of Eq. (5.49):
5.2.1.2.2.7 The quantity (SSC) of Eq. (5.49)
5.2.1.2.2.8 Effect of mud cake
5.2.1.2.2.9 The borehole effect
5.2.1.2.3 General method of simultaneous inversion of multienergy window count rates
Appendix 1
Simple approach to the ratio (SSC)
Measurement of slowing down length and diffusion length of neutrons for bulk porosity
Kinematics of neutron transport
Apparent hydrogen index
Slowing down length, diffusion length, and migration length
The two-group model of neutron transport – group 1 transport and slowing down length
Energy partitioning defining the groups – rationale
The thermal neutron energy spectrum
Neutron flux distribution for point source in a homogeneous isotropic medium for group 1
In the two-group model of neutron transport, inelastic collisions of neutrons are ignored
Removal cross section for group 1
L1 is known as slowing down length of neutrons
Physical meaning of the term “neutron slowing down length”
Neutron flux distribution for point source in a homogeneous isotropic medium for group 2
Diffusion length
Physical meaning of diffusion length
Neutron migration length
Neutron detectors
Neutron detector count rates
Group 1 neutron flux and epithermal neutron detector counts
Group 2 neutron flux and thermal neutron detector counts
Effect of the borehole and factoring-in of the borehole effect
Standard conditions
How the count rate data are inverted to porosity
Apparent water-filled limestone porosity for standard conditions
Appendix 2
Mean squared displacement of fast neutrons
Appendix 3
Solution of Eqs. (5.79) and (5.103)
Solution of Eq. (5.79)
Solution of Eq. (5.103)
Measurement of porosity using acoustic wave slowness data
Slowness porosity relations
Elastic moduli and wave propagation speeds
Porosity dependence of shear wave propagation speed for rocks where the pores are largely interconnected
Empirical and semiempirical relations
Porosity dependence of compressional wave propagation speed for rocks whose pores are largely connected to one another
Model-based inversion of compressional slowness to porosity for sandstones
Modeling of p and q: some insights
The case of spherical pores
Model-based inversion of acoustic slowness to porosity using Xu – White scheme
Evaluation of the exponents p and q
Fluid properties and grain properties input to the model-based inversion described above
Empirical relations connecting porosity and compressional slowness in rocks
The Raymer–Hunt–Gardner relation
Extending Raymer–Hunt–Gardner equation to rocks containing clay
A relation like Raymer–Hunt–Gardner relation for shear wave speed
Wyllie's relation
A generalization of Wyllie's relation
The acoustic body wave slowness measurement
Monopole excitation
Dipole excitation
Recorded waveforms time evolution of wave field illustrated for the slow formation case
Recorded dipole waveforms for the case of a fast formation
The importance of frequency of bandwidth of excitation
Appendix 4
A brief discussion on STC and DSTC
Nondispersive STC or simply STC
Dispersive STC
The effect of noise
Quadrupole excitation
Appendix 5
An illustration of array receiver signal processing using semblance
Tool modes interference or otherwise with the borehole modes
The dispersion model used in quadrupole waveform data inversion
The inversion method
References
Further reading
6 - Log measurements essential for characterizing the pore space of unconventional reservoirs
6.1 Measurement of total porosity using nuclear magnetic resonance
6.1.1 NMR theory
6.1.1.1 Perturbation
6.1.1.1.1 The purpose of applying the perturbation
6.1.1.2 Perturbation in the classical picture of perturbation
6.1.1.2.1 Dephasing of spins
6.1.1.2.2 Rephasing of spins
6.1.1.2.3 Acquisition of spin echoes
6.1.1.2.4 Temporal decay of echo amplitude
6.1.1.2.5 Effect of spatial variation of the B0 field
6.1.1.2.6 Bloch's equations with diffusion for the case of the static magnetic field varying spatially
6.1.1.2.7 Effect of the spin flips (180 degrees tipping of spins) on phase
6.1.1.2.7.1 Evaluation of A((2n−1)τ)
6.1.1.2.8 The amplitude of the nth echo
6.1.2 Pore space attributes and the relaxation of transverse magnetization
6.1.2.1 Surface relaxation (spin–lattice relaxation) and bulk relaxation (spin–spin relaxation)
6.1.2.2 Relaxation of the transverse magnetization of a pore saturated with a grain-wetting fluid
6.1.2.3 Evaluation of porosity
6.1.2.3.1 Input data
6.1.2.4 Obtaining CPBj from mj(T2j)
6.1.2.5 Quality control of the inversion
6.1.3 Total porosity and the bin porosities
6.1.4 Estimation of total porosity directly from the echo data
6.1.5 Obtaining total porosity using the formation density and NMR data
6.2 NMR and the porosity of CBM reservoirs
6.2.1 Coal pores
6.2.2 The T2 relaxation spectra of coals
6.2.3 Total porosity and gas volume
6.2.4 Porosity available within the rock for holding free gas
6.2.5 Cleat volume per unit rock volume
6.3 Porosity of shale gas/shale oil reservoirs
6.3.1 Estimation of effective porosity
6.3.1.1 Estimation of vcbw
6.4 Estimation of elemental concentration in rocks
6.4.1 Neutron capture gamma spectrometry
6.4.1.1 The capture gamma yield
6.4.1.2 Relative yield of capture gamma rays of an element
6.4.2 Inelastic gamma spectrometry
6.4.2.1 Inelastic gamma yield
6.4.2.2 Relative yield of inelastic gamma rays of an element
6.4.3 Computation of the average density of solid part of the formation
6.4.4 The acquisition of inelastic and capture gamma ray spectra
6.4.4.1 Transforming a pulse height spectrum to an energy spectrum fit for spectral decomposition
6.5 Characterizing the pore space of CBM reservoirs using image data
6.5.1 Cleat/fracture aperture and volume, and matrix porosity of CBM reservoirs
6.5.1.1 Estimation of cleat/fracture aperture and volume
6.5.1.1.1 Extraction of fracture segments from image data
6.5.1.1.2 Estimation of cleat/fracture aperture
6.5.1.1.3 Cleat volume per unit rock volume
6.5.1.1.4 Fracture volume per unit rock volume
6.5.1.2 Evaluation of matrix pore volume
6.5.1.2.1 Removal of the isolated conductive anomalies
6.5.1.2.2 Removal of the isolated resistive anomalies
6.5.1.2.3 Matrix conductivity and matrix pore volume
6.6 Characterizing the pore space of shale reservoirs using image data
6.6.1 Shale pores
6.6.2 Delineation of fractures within shale reservoirs using image data
6.6.3 Borehole electric images and distribution of organic matter
6.7 Generation of high-resolution electrical images of the borehole wall
6.7.1 Sensors
6.7.2 Position of each sensor in space
6.7.3 Data acquisition
6.7.4 Process flow for creating borehole images from the button current maps
6.7.4.1 EMEX correction
6.7.4.2 Data equalization
6.7.4.3 Speed and depth corrections
6.7.4.4 Magnetic declination correction
6.7.4.5 The button conductivity matrix
6.7.4.6 Normalization
6.7.4.7 Scaled button conductivity data
Appendix 1
Porosity calibration for NMR
Appendix 2
Certain aspects of relaxation of magnetization within fluid-filled porous media (after Brownstein and Tarr (1979))
Cylindrical geometry
Spherical geometry
Appendix 3
Computation of porosity using density and NMR log data (after Freedman et al., 1997)
Appendix 4
Kerogen property model and hydrocarbon property model used (Mosse et al., 2016)
Kerogen property model
Hydrocarbon property model
Appendix 5
Decomposition of acquired gamma ray spectra using the standard spectra
Spectral decomposition
Appendix 6
Permeability prediction for CBM reservoirs using image data
Analysis
Appendix 7
Histogram equalization
Proof of Eq. (A7.5):
References
Further reading
7 - Characterizing pores and grains using logs
7.1 Pore facies
7.1.1 Pore size distribution
7.1.1.1 Pore size distribution from NMR
7.1.2 Pore shapes from logs
7.1.2.1 Pore aspect ratio and acoustic logs
7.1.2.1.1 Inversion of acoustic slowness data to pore aspect ratio in sandstones
7.1.2.1.1.1 Differential effective medium model
7.1.2.1.1.2 Spherical pores
7.1.2.1.1.3 Evaluation of representative grain-shear modulus
7.1.2.2 Inversion of shear wave slowness log data to pore aspect ratio
7.1.2.3 Inversion of compressional wave slowness log data to pore aspect ratio
Appendix 1
Case of dry frame
The case of dry frame with no compliant pores and stiff pores spherical in shape
Appendix 2
Forward model of the bulk modulus of water saturated rock
Grain modulus
Inversion of acoustic slowness data to pore aspect ratio in carbonates
Differential effective medium model
Evaluating αP from shear wave velocity
Evaluating αP from shear wave velocity and compressional wave velocity
Aspect ratio of intergranular pores
Pore aspect ratio, Grain aspect ratio, and dielectric logs
Pore aspect ratio and grain aspect ratio
Appendix 3
Klein and Swift model for the dispersion of the dielectric permittivity of brines
Appendix 4
Wu's tensor
References
Further reading
8 - Archie's cementation exponent
8.1 Introduction
8.2 Prediction of the value of Archie cementation exponent “m” using effective medium theories
8.3 Approaches used in modelling Archie's m parameter—General Remarks
8.4 The approach for computing “m” using single-frequency dielectric data and using Archie's equation
8.5 The approach for computing “m” using multifrequency dielectric data and using Archie's equation
8.6 The approach for computing “m” from grain attributes obtained through multifrequency dielectric data inversion
8.6.1 Inversion of Archie “m” from single-frequency dielectric data
8.6.2 Inversion of Archie “m” from multifrequency dielectric data
8.6.3 Work flow for the generation of the dispersion model of dielectric permittivity ε
8.6.4 The dispersion model for rock conductivity
8.6.5 Applicability of the model to clayey rocks
8.6.6 The role of contribution to complex permittivity, coming from unconnected pores
8.6.7 Approaches to estimation of Archie “m” based on differential effective medium theory
8.6.7.1 A symmetric formulation of the effective medium of a composite
8.6.7.2 An illustration of modeling “m” using the differential effective medium theory
8.6.7.3 Salient features of Sheng's model for sedimentary rocks
8.6.7.4 Sheng's model and Archie's “a”.
8.6.7.5 Sheng's model and Archie's “m”.
Appendix 1
Derivation of Eq. (8.22) which is in fact modified Maxwell Garnett mixing law
Case of aligned inclusions (identical spatial orientation of principle axes of inclusions)
Appendix 2—differential effective medium theory for aligned inclusions case
Appendix 3
Depolarization factors
Introduction
Depolarization factors of an ellipsoid in general and a spheroid in particular
Approaches to model “m”: in case of Shaly rocks using the Bergman spectral density representation of the effective permitti ...
Sum rules to be obeyed by (s')
Constraint on g(s')
Further necessary condition that the spectral density function is expected to satisfy
Appendix 4
Application of Maxwell's Equations applied to binary mixtures: results in the Quasi-Static limit—Bergman's theorem
The poles of f (s) are real numbers
The fact of positive residues of the poles of f(s) arises naturally in this analysis
The fact of every pole of f(s) being a simple pole arises naturally in this analysis
The essence of the analysis made, and its importance
An approach to Archie “m” through NMR data analysis
Percolation theories and Archie “m” factor
Appendix 5: logarithmic mixing law for effective permittivity of a mixture
Charge distribution within the medium
Behavior of the mixing law in the zero-frequency limit
Approaches to estimate “m” through fractal model of pore space
Concluding remarks
References
Further reading
9. Permeability of unimodal pore system
9.1 Introduction
9.2 Response of local pressure field, local fluid velocity field, and average fluid velocity field to changes in driving pressure
9.3 A simple model of pore space presented
9.4 Flow through a capillary
9.4.1 How pore space attributes influence permeability
9.4.1.1 Preamble
9.5 Surface area or representative pore dimension or characteristic length scale driven approaches to permeability
9.5.1 The “bundle of capillary tubes” model of pore space
9.5.2 Forward model of permeability in terms of pore space attributes
9.5.3 The stream tube model of flow through a porous medium whose grain- and pore-arrangement of a macro level volume segment, is ...
9.6 Depiction of pore space in the model
9.6.1 Stream tubes
9.7 Elemental stream tube
9.8 Integral representation of macroscopic permeability (or simply “permeability”), and the concept of microscopic permeability ...
9.9 Integral representation of permeability
9.10 Average permeability field ks, also referred to as the “effective permeability factor”
9.11 Integral representation of average permeability field (effective permeability factor) ks and relation between permeability ...
9.12 The elemental stream tube permeability factor field
9.12.1 Explicit representation of permeability factor of a streamline (elemental stream tube) in terms of some of its attributes
9.12.2 Decomposition of permeability into macroscopic pore space attributes, namely, hydraulic tortuosity, hydraulic constriction ...
9.13 Insights from the simplest possible pore space model and the role played by “hydraulic radius” in permeability modeling—der ...
9.14 Concept of hydraulic radius
9.14.1 How microscopic streamline attributes and macroscopic pore space attributes are related in a porous medium
9.14.1.1 Precise definition of the term “representative tortuosity (τ) of pore space”
9.14.2 Hydraulic Constriction factor (Cs) of connected pore space, which is the macroscopic counterpart of streamline attribute C(S)
9.15 Generalized Kozeny-Carman equation
9.15.1 Conventional form of Kozeny-Carman equation predicting permeability
9.15.2 Well known equations for permeability prediction from log measurements. The equations discussed below are for water wet roc ...
9.15.2.1 The SDR equation for permeability
9.15.2.2 Timur equation for permeability
9.15.2.3 Coates equation for permeability
9.15.2.4 Berg's equation
9.15.2.5 Computation of average grain size and grain size distribution from NMR data
9.15.2.6 Relation between representative or effective pore dimension and the representative or effective grain dimension—Van Baaren' ...
9.15.2.6.1 Van Baaren's equation
9.16 From Kozeny-Carman equation, to Van Baaren's equation
9.17 The RGPZ equation
Appendix 1
Hydraulic constriction factor
Appendix 2
Estimation of bound fluid volume used in Coates equation from NMR data
Appendix 3
A brief derivation of RGPZ equation
Rock attributes which control permeability, and their inversion from log measurements –some challenges
Pore space attributes, and log measurements (in the context of nonfractal modeling of pore space)
Calculation of Cs
Prediction of effective permeability factor ks, and permeability k using log measurements (in the context of nonfractal mod ...
Eq. (9.21) and the Kozeny-Carman equation—interpreting the fitting-constant C in Kozeny-Carman equation
Why the “bundle of capillary tubes” models of pore space succeed in predicting the fluid transport properties of porous media
The pores and pore throats model of pore space
Fractal dimension of connected pore space and its influence on permeability
Some important relations
Pore volume Vp and porosity ϕ
Representative hydraulic tortuosity τ
Ratio of cumulative surface area of connected pore space, to cumulative volume of connected pore space Spv
Relationship between Df, Dt and ϕ, and between Df, F, and ϕ
Relation of the results from fractal model of pore space, as arrived at above, with Brooks-Corey equation, but with pore sp ...
Relation of the results from fractal model of a pore space comprised of pores and pore throats, with the Brooks-Corey equation
Permeability from Brooks-Corey equation—the impact of the fractal model
Estimation of rmax, rmin, and γ1 (pore size to pore throat size ratio) from NMR log data
Estimation of rmax from NMR log data
Estimation of rmin from NMR log data
Estimation of γ1 (pore size to pore throat size ratio)
Generation of a valid permeability predictor as a continuous log
Estimation of pore size heterogeneity index λ from logs
Computation of electric formation factor log
Estimation of the (fractal) dimension of pore radius fractal (Df) from logs
Estimation of Df from λ
Estimation of Df from rmax, rmin
Estimation of Df from rmax, and NMR (1T2)mean
Estimation of Df from rmax, and k
Estimation of Df from F
Significance of the parameter Df
The fractal dimension of the pore radius fractal and Archie's a,m parameters
Estimation of the fractal dimension for lengths of capillaries (Dt) using log measurements
Estimation of τ the hydraulic tortuosity associated with connected pore space, assuming pore space is not a fractal object
Estimation of τ the tortuosity associated with connected pore space, assuming pore space is fractal
Appendix 4
Fluid transport through a perfectly rigid framework of grains, with no isolated pore space
Formal averaging procedure applied on local fluid velocity field and local fluid pressure field, that are related through l ...
The tensor field α҃ (r) is entirely defined by the medium properties
Appendix 5
Stoneley wave slowness and Stoneley mobility
Appendix 6
Estimation of γ1 (pore size to pore throat size ratio)
References
Further reading
10. Permeability and electrical conductivity of rocks hosting multimodal pore systems and fractures
Preamble
10.1 Pore size nomenclature
10.1.1 Micropores, mesopores, and macropores
10.2 Pore classification using NMR T2 distribution
10.2.1 Relation between pore sizes and NMR T2: the fast diffusion limit
10.2.2 Limits of validity of a T2 threshold-based porosity partition
10.2.3 Model-based porosity partition for the case of rocks having intragranular porosity
10.2.4 Case where vugs are not sparsely distributed in the rock
10.3 Pore classification using well bore images
10.3.1 Preconditioning of data
10.3.2 Dip picking
10.3.3 Closing the data gap
10.3.3.1 Training pattern classification
10.3.3.2 Score class prototype
10.3.3.3 Simulation to populate the pixels which have no data
10.3.3.4 Image rescaling
10.3.4 Extraction of fracture segments
10.3.4.1 Extraction of low apparent dip fracture segments
10.3.4.2 Spatial orientation of traces
10.3.4.3 Differentiating bedding planes and facture planes
10.3.4.4 Main orientation of fractures
10.3.4.5 Obtaining high confidence fracture traces
10.3.4.5.1 Extraction of high apparent dip fracture segments
10.3.5 Matrix extraction
10.3.6 The problem of computing the pore volume contribution of heterogeneities
10.3.6.1 The challenge of extraction of heterogeneities
10.3.7 An efficient methodology for extracting heterogeneities
10.3.7.1 Classification of the mosaic pieces
10.3.7.2 Connectedness attribute of a conductive heterogeneity
10.3.7.3 Final porosity partition
10.3.7.4 The mosaic image and the heterogeneity image
10.3.7.5 Classification of heterogeneities and quantification of the porosity associated with them
10.3.7.6 Porosity association of the different types of spots (heterogeneities)
Appendix 1
10.4 Porosity partition using acoustic logs
10.4.1 Challenges of porosity partition using acoustic logs
10.4.2 A work flow of porosity partition using acoustic logs
What is a “vug” in the model?
Sensitivity of model results to the shape of the vug
Appendix 2
The self-consistent theory of Berryman in the long wavelength limit
Introduction to Kuster–Toksoz model
The Kuster–Toksoz estimates
Invoking the assumption of long wavelength limit
Self-consistent estimates in the long wavelength limit
Invoking the long wavelength limit
10.5 Electrical conductivity of an unfractured composite hosting dual porosity
Modeling σ2 for fully water saturated component 2 case
Evaluation of σ the electrical conductivity of a composite hosting dual porosity
Case of partial saturation
10.6 Permeability of an unfractured composite hosting dual porosity
Base rock permeability
Permeability of the composite rock
10.7 Electrical conductivity of fractured rocks
Partial saturation case
10.8 The variable cementation exponent method of computing water saturation
Case of connected pore space having tortuosity unity
Pore volumes
Rationale behind Equation (10.89)
Level by level evaluation of the Archie cementation exponent of the formation
Discussion
Computation of water saturation
Appendix 3
Rationale behind Equation (10.89)
10.9 Permeability of rocks hosting connected vugs/fractures
Alternate simpler way of estimating Qcon
Permeability kb of the base rock
Permeability of gross rock
10.10 Bray – Smith method of computing permeability
10.11 Another approach to permeability modeling which also relies on NMR log data
References
Further reading
Index
A
B
C
D
E
F
G
H
I
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Z
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