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Effective-medium theories for fluid-saturated materials with aligned cracks
J.A. Hudson, 1 T. Pointer 2 and E. Liu 3
Volume 49, Issue 5, Date: September 2001, Pages: 509-522

1 Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, 2 BG Geophysics Skill Centre, 100 Thames Valley Park Drive, Reading, Berks RG6 1PT, and 3 British Geological Survey, Murchison House, West Mains Road, Edinburgh EH9 3LA, UK
Copyright European Association of Geoscientists & Engineers
ABSTRACT
There is general agreement between different theories giving expressions for the overall properties of materials with dry, aligned cracks if the number density of cracks is small. There is also very fair agreement for fluid-filled isolated cracks. However, there are considerable differences between two separate theories for fluid-filled cracks with equant porosity. Comparison with recently published experimental data on synthetic sandstones gives a good fit with theory for dry samples. However, although the crack number density in the laboratory sample is such that first-order theory is unlikely to apply, expressions correct to second order (in the number density) provide a worse fit. It also appears that the ratio of wavelength to crack size is not sufficiently great for any detailed comparison with effective-medium theories, which are valid only when this ratio is large. The data show dispersion effects for dry cracks and scattering, neither of which will occur at sufficiently long wavelengths. Data from the water-saturated samples indicate that the effect of equant porosity is significant, although the two theories differ strongly as to just how significant. Once again, and in spite of the reservations mentioned above, a reasonable fit between theory and observation can be shown.

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A LABORATORY INVESTIGATION INTO THE ELASTIC PROPERTIES OF LIMESTONES*
Geophysical Prospecting
Volume 11, Issue 3, Date: September 1963, Pages: 300-312
O. KOEFOED, M. M. OOSTERVELD, I. J. G. ALONS
Measurements have been made by the resonance method of the longitudinal bar velocities and the transverse velocities of 29 chemically pure limestones of varying porosity. The elastic anisotropy factors of the limestones were found to vary over a wide range. For the isotropic limestones, both the longitudinal and the transverse velocity decrease in general with increasing porosity, but this decrease is somewhat irregular. The irregularities in the velocity were found to correlate with the irregularities in the electrical resistivity of the samples when saturated with a standard salt solution, and with those in their tensile strength. This correlation indicates an influence on the velocity of the geometry of the pore space of the rock and of its counterpart: the geometry of the matrix. Indications have been obtained that a condition akin to weathering may be partly responsible for the irregularities in the velocities.

Saturating the samples with water was found to result in a decrease of their velocities. An investigation of the available literature, combined with measurements made in our laboratory with a low frequency pulse method, indicate that water saturation results in a decrease of the velocity at low frequencies and in an increase of the velocity at high frequencies.

Since other investigators have shown that the velocities of dry rock are hardly effected by the frequency, one must conclude that the velocities of water saturated rock do change appreciably with frequency.

The values of Poisson's ratio were found to decrease with increasing porosity, but more so for the dry samples than for the water saturated samples. A possible explanation is that the cross-contraction and cross-dilatation is partly consumed in changing the diameters of the pores, thus reducing the effect on the outer diameter of the sample; this mechanism would be partly resisted by water filling the pores.

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POISSON'S CONSTANT WITH DRY SEDIMENTS AND WITH PACKINGS OF SPHERES*
Geophysical Prospecting
Volume 14, Issue 2, Date: June 1966, Pages: 204-215
H. WACHHOLZ
In order to find a relation between Poisson's constant of dry sediments and the porosity, it is necessary to consider the elastic behavior of the sediment's initial state and the final state during the process of consolidation. Sand and other loose materials of more or less granular substances represent the initial state. As substitute for this zone, elastic constants of simple cubic and hexagonal packing are examined. Supposed is the adhesion for the planes of contact. Besides of Hertz's formula for the contact area of spheres, an elastic function of displacement due to a tangential force for the contact area of the spheres is used. This function has been derived. With the aid of these two. elements, the constants of the mentioned packing are calculated. Poisson's constant will then be calculated from the formula being valid for isotropic solids for three different directions of the sound. The mean value is lying at .

With the known equation of Poisson's constant of rock having a small porosity, a third-degree polynomial is formed. This polynomial agrees well with the average values of known measurements, and therefore can serve as a guide for the calculation of Poisson's constant of water-saturated sediments.


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SUR LA CORRELATION EXISTANT ENTRE POROSITE ET FAGTEUR DE FORMATION DANS LES SEDIMENTS NON CONSOLIDES*
E. ACCERBONI**
**Osservatorio Geofisico Sperimentale, Trieste, contrib. no 200.

*Presented at the Thirtieth Meeting of the European Association of Exploration Geophysicists at Venice, May 1969.

Copyright 1970 European Association of Geoscientists & Engineers
SUR LA CORRELATION EXISTANT ENTRE POROSITE ET FAGTEUR DE FORMATION DANS LES SEDIMENTS NON CONSOLIDES*
Geophysical Prospecting
Volume 18, Issue 4, Date: December 1970, Pages: 505-515


ABSTRACT


The relationship between porosity and formation factor in unconsolidated homogeneous and anisotropic sediments without granule-liquid interaction is investigated by introducing a parametric model that simulates a variable cellular structure.

In this hypothesis, porosity φ and F-factor are calculated for some fixed values of the parameter, solving numerically two integral expressions.

From these calculations the form of the function φ=f(F), corresponding to the proposed model, has been deduced. This relation is in very good agreement with Archie's empirical law for unconsolidated sands which requires that F=φ--1,3. Therefore it seems that the validity of Archie's law is theoretically confirmed for the unconsolidated sediments considered in this paper.

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M. M. ROKSANDIĆ**
SEISMIC FACIES ANALYSIS CONCEPTS *
Geophysical Prospecting
Volume 26, Issue 2, Date: June 1978, Pages: 383-398

Correspondence to **Present address: SOQUIP, 3340 de la Pérade, Ste-Foy, Québec G1X 2L7, Canada.

*Paper read at the Silver Anniversary Meeting of the European Association of Exploration Geophysicists in The Hague, June 1976.

Copyright 1978 European Association of Geoscientists & Engineers
ABSTRACT


Seismic facies analysis makes use of different seismic parameters in order to get other than structural information. A review is given of possibilities and usefulness of seismic facies analysis in oil exploration.

A seismic facies unit can be defined as a sedimentary unit which is different from adjacent units in its seismic characteristics. Parameters that should be taken into consideration in the seismic facies analysis are as follows: reflection amplitude, dominant reflection frequency, reflection polarity, interval velocity, reflection continuity, reflection configuration, abundance of reflections, geometry of seismic facies unit, and relationship with other units.

Interpretation of seismic facies data may be either direct or indirect. The purpose of the direct interpretation is to find out geological causes responsible for the seismic signature of a seismic facies unit. So, the direct interpretation may be aimed at predicting lithology, fluid content, porosity, relative age, overpressured shales, type of stratification, geometry of the geological body corresponding to the seismic facies unit and its geological setting. The indirect interpretation is intended to reach some conclusions on depositional processes and environments, sediment transport direction, and some aspects of geological evolution (transgression, regression, subsidence, uplift, erosion).

The results of the seismic facies analysis may be shown on seismic facies cross-sections and seismic facies maps. Depending on the available seismic data and geological conditions in the area under consideration, the seismic facies maps may be of different types such as general seismic facies maps showing distribution of different seismic facies units, sand-shale ratio maps, direction of cross-bedding and paleo-transport maps etc.

Several kinds of seismic facies units and their geological interpretation are discussed as examples of seismic facies analysis.

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THE INFLUENCE OF PERMEABILITY ON COMPRESSIONAL WAVE VELOCITY IN MARINE SEDIMENTS*
F. HAMDI 1 D. TAYLOR SMITH**
Volume 30, Issue 5, Date: October 1982, Pages: 622-640

**Marine Science Laboratories, University College of North Wales, Menai Bridge LL59 5EY, UK.
*Received April 1981, revision January 1982.

Copyright 1982 European Association of Geoscientists & Engineers
ABSTRACT


To investigate the effect of permeability on the propagation of seismo-acoustic waves through marine sediments, a theoretical model based on Biot's equations is established which relates the compressional wave velocity measured at a fixed frequency to computed velocities at zero and infinite frequencies in terms of sediment porosity and permeability. The model is examined experimentally in a standard soil mechanics consolidation test (itself dependent, among other things, on sediment porosity and permeability) which has been modified to include measurements of compressional wave velocity at 1 MHz and shear-wave velocity at 5 kHz. This test allows the elastic modulus of the sediment frame to be assessed under different load conditions simultaneous with the velocity determinations.

From a number of tests on different samples, five samples are chosen to typify the range of sediment sizes. The results show that the difference between the measured velocity at 1 MHz and the model-derived velocity at zero frequency increases with increasing particle size (from clays to fine sand), with decreasing porosity, and with increasing permeability. For sediments coarser than fine sand the simple model breaks down, possibly because of the dominance of scattering/diffraction effects at the high frequency of the experiment. Within this limitation the model seems satisfactory to offer a capability of predicting the permeability of a sea floor sediment to an order of magnitude by the in situ measurement of seismic velocities over a wide range of frequencies; the prediction process requires a good in situ determination of sediment porosity such as that offered by electrical formation factor measurements.
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APPLICATION OF PETROPHYSICAL MEASUREMENTS TO THE PREDICTION OF SEISMIC RESPONSES OF DIFFERING LITHOLOGYOR PORE FLUIDS1
B. J. RAFISON 2
Volume 36, Issue 8, Date: November 1988, Pages: 847-856

2 Unocal Science and Technology Division, P.O Box 76, Brea, CA 92621, U.S.A.
1 Received March 1987, revision accepted May 1988.

Copyright 1988 European Association of Geoscientists & Engineers
ABSTRACT


Determination of petrography and pore fluid content is an ultimate goal of an integrated seismic-petrophysical study. For lack of a general inversion technique, forward modelling is useful in studying the relations between lithology, stratigraphy, pore fluid content and the seismic response. This report describes a study of two clastic sequences in Utah, from which 32 rock samples were analysed. A detailed petrographic study was done. Laboratory measurements were made of ultrasonic compressional- and shear-wave velocity as a function of pressure. We computed the velocities at seismic frequencies for the samples when dry, over-pressured, brine saturated, and oil saturated. The velocities were sensitive to the porosity, carbonate cementation and the depositional facies. We generated velocity profiles for hypothetical reservoirs for a range of saturation states. The velocity profiles were used to generate synthetic seismic shot gathers to study the seismic response of these clastic reservoirs. The fluid-saturation strongly affects the seismic respone, as does the presence of a coal seam. An amplitude change with offset is often observed. However, stratigraphy appears to have a stronger effect on the seismic response


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SEISMIC AND ELECTRICAL PROPERTIES OF UNCONSOLIDATED PERMAFROST1
M. S. KING 2 R. W. ZIMMERMAN 3 R. F. CORWIN 4
Volume 36, Issue 4, Date: May 1988, Pages: 349-364


2 Department of Mineral Resources Engineering, Imperial College, London SW7 2BP, UK. 3 Department of Mechanical Engineering, University of California, Berkeley, CA 94720, USA. 4 Harding Lawson Associates, Novato, CA 94948, USA.

1 Paper read at the 48th EAEG meeting, Ostend, June 1986; last material received December 1987.

Copyright 1988 European Association of Geoscientists & Engineers
ABSTRACT


A model has been developed to relate the velocities of acoustic waves Vp and Vs in unconsolidated permafrost to the porosity and extent of freezing of the interstitial water. The permafrost is idealized as an assemblage of spherical quartz grains embedded in a matrix composed of spherical inclusions of water in ice. The wave-scattering theory of Kuster and Toksoz is used to determine the effective elastic moduli, and hence the acoustic velocities. The model predicts Vp and Vs to be decreasing functions of both the porosity and the water-to-ice ratio. The theory has been applied to laboratory measurements of Vp and Vs in 31 permafrost samples from the North American Arctic. Although no direct measurements were made of the extent of freezing in these samples, the data are consistent with the predictions of the model. Electrical resistivity measurements on the permafrost samples have demonstrated their essentially resistive behaviour. The ratio of resistivity of permafrost in its frozen state to that in its unfrozen state has been related to the extent of freezing in the samples.

Electromagnetic and seismic reflection surveys can be used together in areas of permafrost: firstly an EM survey to determine the extent of freezing and then the acoustic velocity model to predict the velocities in the permafrost. The necessary transit time corrections can thus be made on seismic reflection records to compensate for the presence of permafrost.

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POROSITY AND PORE STRUCTURE FROM ACOUSTIC WELL LOGGING DATA1
Geophysical Prospecting
Volume 41, Issue 4, Date: May 1993, Pages: 435-451

G. TAO 2 M.S. KING 2
2 Department of Mineral Resources Engineering, Imperial College of Science, Technology and Medicine, Prince Consort Road, London SW7 2BP, U.K.

1 Paper read at the 53rd EAEG meeting, Florence, May 1991. Received February 1992, revision accepted October 1992.

Copyright 1993 European Association of Geoscientists & Engineers
ABSTRACT


Wyllie's time-average equation and subsequent refinements have been used for over 20 years to estimate the porosity of reservoir rocks from compressional (P)-wave velocity (or its reciprocal, transit time) recorded on a sonic log. This model, while simple, needs to be more convincingly explained in theory and improved in practice, particularly by making use of shear (S)-wave velocity. One of the most important, although often ignored, factors affecting elastic velocities in a rock is pore structure, which is also a controlling factor for transport properties of a rock. Now that S-wave information can be obtained from the sonic log, it may be used with P-waves to provide a better understanding of pore structure. A new acoustic velocities-to-porosity transform based on an elastic velocity model developed by Kuster and Toksöz is proposed. Employing an approximation to an equivalent pore aspect ratio spectrum, pore structure for reservoir rocks is taken into account, in addition to total pore volume. Equidimensional pores are approximated by spheres and rounded spheroids, while grain boundary pores and flat pores are approximated by low aspect ratio cracks. An equivalent pore aspect ratio spectrum is characterized by a power function which is determined by compressional-and shear-wave velocities, as well as by matrix and inclusion properties. As a result of this more sophisticated elastic model of porous rocks and a stricter theory of elastic wave propagation, the new method leads to a more satisfactory interpretation and fuller use of seismic and sonic log data. Calculations using the new transform on data for sedimentary rocks, obtained from published literature and laboratory measurements, are presented and compared at atmospheric pressure with those estimated from the time-average equation. Results demonstrate that, to compensate for additional complexity, the new method provides more detailed information on pore volume and pore structure of reservoir rocks. Examples are presented using a realistic self-consistent averaging scheme to consider interactions between pores, and the possibility of extending the method to complex lithologies and shaly rocks is discussed.
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Seismic velocities in fractured rocks: an experimental verification of Hudson's theory1
S. Peacock 2, 3 ,C. McCann 2 , J. Sothcott 2 T.R. Astin 2
Geophysical Prospecting
Volume 42, Issue 1, Date: January 1994, Pages: 27-80

2 Postgraduate Research Institute for Sedimentology, University of Reading, Whiteknights, P.O. Box 227, Reading RG6 2AB, U.K.
1 Paper presented at the 53rd EAEG meeting, Florence, Italy, May 1991. Received September 1992, revision accepted August 1993.
3 School of Earth Sciences, University of Birmingham, Edgbaston, Birmingham B15 2TT, U.K.

Copyright 1994 European Association of Geoscientists & Engineers
ABSTRACT


Flow of fluids in many hydrocarbon reservoirs and aquifers is enhanced by the presence of cracks and fractures. These cracks could be detected by their effects on propagation of compressional and shear waves through the reservoir: several theories, including Hudson's, claim to predict the seismic effects of cracks. Although Hudson's theory has already been used to calculate crack densities from seismic surveys, the predictions of the theory have not yet been tested experimentally on rocks containing a known crack distribution. This paper describes an experimental verification of the theory. The rock used, Carrara marble, was chosen for its uniformity and low porosity, so that the effect of cracks would not be obscured by other influences. Cracks were induced by loading of laboratory specimens. Velocities of compressional and shear waves were measured by ultrasound at 0.85 MHz in dry and water-saturated specimens at high and low effective pressures. The cracks were then counted in polished sections of the specimens. In 'dry' specimens with both dry and saturated cracks, Hudson's theory overpredicted observed crack densities by a constant amount that is attributed to the observed value being systematically underestimated. The theory made poor predictions for fully saturated specimens. Shear-wave splitting, caused by anisotropy due to both crystal and crack alignment, was observed. Cracks were seen to follow grain boundaries rather than the direction of maximum compression due to loading. The results demonstrate that Hudson's theory may be used in some cases to determine crack and fracture densities from compressional- and shear-wave velocity data.

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A new velocity model for clay-sand mixtu res1
Geophysical Prospecting
Volume 43, Issue 1, Date: January 1995, Pages: 91-118
Shiyu Xu, Roy E. White
ABSTRACT


None of the standard porosity-velocity models (e.g. the time-average equation, Raymer's equations) is satisfactory for interpreting well-logging data over a broad depth range. Clays in the section are the usual source of the difficulty through the bias and scatter that they introduce into the relationship between porosity and P-wave transit time. Because clays are composed of fine sheet-like particles, they normally form pores with much smaller aspect ratios than those associated with sand grains. This difference in pore geometry provides the key to obtaining more consistent resistivity and sonic log interpretations.

A velocity model for Clay–sand mixtures has been developed in terms of the Kuster and Toksöz, effective medium and Gassmann theories. In this model, the total pore space is assumed to consist of two parts: (1) pores associated with sand grains and (2) pores associated with clays (including bound water). The essential feature of the model is the assumption that the geometry of pores associated with sand grains is significantly different from that associated with clays. Because of this, porosity in shales affects elastic compliance differently from porosity in sand-Stones. The predictive power of the model is demonstrated by the agreement between its predictions and laboratory measurements and by its ability to predict sonic logs from other logs over large depth intervals where formations vary from unconsolidated to consolidated sandstones and shales.
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Application of acoustic full wavetrains for the determination of lithology, reservoir and mechanical parameters of formation1
Geophysical Prospecting
Volume 44, Issue 5, Date: September 1996, Pages: 761-787
Maria Bała, Jadwig A. Jarzyna
ABSTRACT


The significant development in acoustic full waveform logging during the last ten years has made it increasingly possible for log analysts to determine the physical properties of a rock formation in situ.

Parallel to the methods applied to a single wavetrain during seismic processing, the new techniques, used for sets of wavetrains, have been successfully tested with acoustic full waveforms. Instantaneous characteristics analysis is included in this group of methods. This approach, leading to qualitative and quantitative interpretation, reveals the influence of small changes in physical properties on acoustic full wavetrains.

Applications of complex acoustic waveform analysis for the determination of inhomogeneous zones are presented. Colour diagrams of instantaneous characteristics are used for the detection of fractured regions and slow formations with increased attenuation of acoustic waves.

Results of the interpretation of individual acoustic full waveforms, based on cross-correlation and spectral analysis, using the IDNP and IDNS computer programs, e.g. velocities of compressional waves, shear and Stoneley waves, are presented. Since the bulk density of the rocks was known, it was possible, using the velocities of P- and S-waves obtained, to calculate the dynamic elastic moduli. We used the interpretation of acoustic full wavetrains to calculate porosity. The sonic porosity is compared to the porosity obtained from other logs and to that obtained from core sample analysis.

The examples of acoustic full wavetrains were recorded in the Miocene sulphur-bearing limestones in central Poland. Field measurements were made using the domestic prototype equipment for well log recordings in shallow boreholes.

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A generalized Biot–Gassmann model for the acoustic properties of shaley sandstones1
Geophysical Prospecting
Volume 48, Issue 3, Date: May 2000, Pages: 539-557
José M. Carcione, Boris Gurevich, Fabio Cavallini
ABSTRACT
We obtain the wave velocities of clay-bearing sandstones as a function of clay content, porosity and frequency. Unlike previous theories, based simply on slowness and/or moduli averaging or two-phase models, we use a Biot-type three-phase theory that considers the existence of two solids (sand grains and clay particles) and a fluid. The theory, which is consistent with the critical porosity concept, uses three free parameters that determine the dependence of the dry-rock moduli of the sand and clay matrices as a function of porosity and clay content.

Testing of the model with laboratory data shows good agreement between predictions and measurements. In addition to a rock physics model that can be useful for petrophysical interpretation of wave velocities obtained from well logs and surface seismic data, the model provides the differential equation for computing synthetic seismograms in inhomogeneous media, from the seismic to the ultrasonic frequency bands.

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A semi-empirical velocity-porosity-clay model for petrophysical interpretation of P- and S-velocities
Geophysical Prospecting
Volume 46, Issue 3, Date: May 1998, Pages: 271-285
Igor Goldberg, Boris Gurevich
ABSTRACT
We design a velocity–porosity model for sand-shale environments with the emphasis on its application to petrophysical interpretation of compressional and shear velocities. In order to achieve this objective, we extend the velocity–porosity model proposed by Krief et al., to account for the effect of clay content in sandstones, using the published laboratory experiments on rocks and well log data in a wide range of porosities and clay contents.

The model of Krief et al. works well for clean compacted rocks. It assumes that compressional and shear velocities in a porous fluid-saturated rock obey Gassmann formulae with the Biot compliance coefficient. In order to use this model for clay-rich rocks, we assume that the bulk and shear moduli of the grain material, and the dependence of the compliance on porosity, are functions of the clay content.

Statistical analysis of published laboratory data shows that the moduli of the matrix grain material are best defined by low Hashin–Shtrikman bounds. The parameters of the model include the bulk and shear moduli of the sand and clay mineral components as well as coefficients which define the dependence of the bulk and shear compliance on porosity and clay content. The constants of the model are determined by a multivariate non-linear regression fit for P- and S-velocities as functions of porosity and clay content using the data acquired in the area of interest.

In order to demonstrate the potential application of the proposed model to petrophysical interpretation, we design an inversion procedure, which allows us to estimate porosity, saturation and/or clay content from compressional and shear velocities.

Testing of the model on laboratory data and a set of well logs from Carnarvon Basin, Australia, shows good agreement between predictions and measurements. This simple velocity-porosity-clay semi-empirical model could be used for more reliable petrophysical interpretation of compressional and shear velocities obtained from well logs or surface seismic data.

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Fracture-frequency prediction from borehole wireline logs using artificial neural networks
Geophysical Prospecting
Volume 47, Issue 6, Date: November 1999, Pages: 1031-1044
Elaine M. FitzGerald, Christopher J. Bean, Ronan Reilly
ABSTRACT
Borehole-wall imaging is currently the most reliable means of mapping discontinuities within boreholes. As these imaging techniques are expensive and thus not always included in a logging run, a method of predicting fracture frequency directly from traditional logging tool responses would be very useful and cost effective. Artificial neural networks (ANNs) show great potential in this area. ANNs are computational systems that attempt to mimic natural biological neural networks. They have the ability to recognize patterns and develop their own generalizations about a given data set. Neural networks are trained on data sets for which the solution is known and tested on data not previously seen in order to validate the network result. We show that artificial neural networks, due to their pattern recognition capabilities, are able to assess the signal strength of fracture-related heterogeneity in a borehole log and thus fracture frequency within a borehole. A combination of wireline logs (neutron porosity, bulk density, P-sonic, S-sonic, deep resistivity and shallow resistivit ...