SEP-120 -- TABLE OF CONTENTS

Inversion

3-D subsalt imaging via regularized inversion with model preconditioning (ps.gz 6817K) (pdf 2289K) (src 24567K)
Clapp M. L. and Clapp R. G.
Subsalt imaging is a difficult but increasingly important problem. The poor illumination that occurs when seismic energy is affected by the complex subsurface at and around salt bodies causes significant shadow zones in migration results. These shadow zones may contain real signal, but it is weak compared to the artifacts caused by multipathing and poor illumination. To reduce artifacts and recover this energy, thereby improving the image, we use an imaging (migration) operator in a regularized least-squares inversion. The regularization operator acts on the offset ray parameter (reflection angle) axis of the model space. Performing several iterations of regularized inversion that penalize large changes along the offset ray parameter axis results in an image with recognizable events in the shadow zones and fewer artifacts. We perform regularized inversion with model preconditioning on real 2-D and 3-D data to obtain seismic images that are better in poorly illuminated areas than migration results.
Target-oriented wave-equation inversion (ps.gz 200K) (pdf 330K) (src 835K)
Valenciano A. A., Biondi B., and Guitton A.
A target-oriented strategy can be applied to estimate the wave-equation least-squares inverse image (${<B>m}$) by explicitly computing the Hessian ($<B>H$). The least-squares inverse image is obtained as the solution, using a conjugate gradient algorithm, of a non-stationary least-squares filtering problem ${<B>H} {{<B>m}}={<B>m}_{mig}$ (where ${<B>m}_{mig}$ is the migration image, and the rows of the Hessian are non-stationary filters). This approach allows us to perform the number of iterations necessary to achieve the convergence, by exploiting the sparsity and structure of the Hessian matrix. The results on a constant velocity model and a model with a velocity Gaussian anomaly show the validity of the method.
Inversion and fault tolerant parallelization using Python (ps.gz 102K) (pdf 159K) (src 172K)
Clapp R. G.
Many current areas of research at SEP involve large-scale inversion problems that must be parallelized in order to be tractable. Writing fault-tolerant, parallel code requires significant programming expertise and overhead. In this paper, a library, written in Python, is described that effectively simulates a fault-tolerant parallel code, using simple serial programs. In addition, the library provides the ability to use these parallel objects in out-of-core inversion problems in a fault-tolerant manner.

Anisotropy

Imaging steeply dipping reflectors in TI media by wavefield extrapolation (ps.gz 5564K) (pdf 619K) (src 26978K)
Shan G. and Biondi B.
We develop an anisotropic plane-wave migration method based on wavefield extrapolation. In this new scheme, we decompose both source and receiver wavefields into plane waves by delaying shots. For each plane wave, we design a tilted coordinate system whose tilting angle depends on the propagation direction of the plane wave. The wavefield extrapolation is done by an implicit isotropic operator plus an explicit anisotropic correction operator. We apply this method on a synthetic dataset. The results show that our scheme can accurately handle overturned waves and image steeply dipping reflectors in transversely isotropic media with a vertical axis of symmetry (VTI).
Angle-domain common image gathers for anisotropic migration (ps.gz 804K) (pdf 648K) (src 1841K)
Biondi B.
I present a general methodology for computing and analyzing Angle Domain Common Image Gathers (ADCIGs) in conjunction with anisotropic wavefield-continuation migration. I demonstrate that the aperture angles estimated by transforming prestack images using slant stacks along the subsurface-offset axis are a good approximation of the phase aperture angles, and that they are exactly equal to the phase aperture angles for flat events in VTI media. I introduce a generalization of the concept of migration impulse response for the computation of prestack images function of the subsurface offset that enables a straightforward analytical analysis of the reflector movements caused by perturbations in anisotropic parameters. This analysis shows that the Residual Moveout (RMO) in migrated ADCIGs is function of both the phase aperture angle and the group aperture angle. The dependency of the RMO function on the group angles adds some complexity to the RMO analysis because the computation of group angles from phase angles, which are measured from the ADCIGs, depends on the local background anisotropic velocity at the reflector point. Several numerical examples demonstrate the accuracy of the RMO function predicted by my kinematic analysis, and in contrast, that the approximation of the group angles by the phase angles may lead to substantial errors for events reflected at wide aperture angles.
3D wavefield extrapolation in laterally-varying tilted (ps.gz 2088K) (pdf 843K) (src 7127K)
Shan G. and Biondi B.
We develop a new 3D wavefield-extrapolation method for a transversely isotropic (TI) medium with a symmetry axis. The wavefield extrapolation is done by an implicit isotropic extrapolation operator with an explicit correction operator. The explicit correction is a 2D convolution operator in the space domain, whose coefficients are estimated by a weighted least-squares method in the Fourier domain. The extrapolation operator is stable and suitable for laterally-varying 3D TI media. This new method can be used to extrapolate wavefields in a 3D transversely isotropic medium with a vertical symmetry axis (VTI) in tilted coordinates. We also discuss the effects of the filter length on its accuracy and shorten the filter by changing the least-squares weighting function. We present the impulse response of our algorithm and compare it with the anisotropic phase-shift method.
Common azimuth migration for elliptical and VTI media (ps.gz 66K) (pdf 129K) (src 69K)
Sen S. and Biondi B.
We derive the Commom Azimuth downward continuation operator for elliptically anisotropic and VTI media. For elliptically anisotropic media, the Common Azimuth downward continuation operator derived by a stationary phase approximation of the full 3-D downward continuation operator is exact and it agrees with the constraint imposed by the Common Azimuth approximation on the propagation direction of the source and receiver rays. For VTI media, the dispersion relationship is much more complicated and results in a quartic equation for the stationary path. We introduce a bounded form of Common Azimuth migration for this kind of media, which allows us to develop closed-form analytical solutions without directly solving the quartic equation. Error analysis indicates that the derived analytical solution has similar accuracy as that obtained by solving the full quartic equation. 3-D impulse responses of the anisotropic Common Azimuth downward continuation operator also show significant differences compared to the isotropic operator even at moderate propagation angles.

Automated interpretation

Update on flattening without picking (ps.gz 1762K) (pdf 812K) (src 49160K)
Lomask J., Guitton A., Fomel S., and Claerbout J.
We present a method for efficiently flattening 3D seismic data volumes. First local dips are calculated over the entire seismic volume. The dips are then resolved into time shifts using a Gauss-Newton iterative approach that exploits the Fourier domain to maximize efficiency. To handle faults (discontinuous reflections), we apply a weight inversion scheme. This approach successfully flattens a synthetic faulted model, a field salt peircement dataset, a field dataset with an angular unconformity, and a faulted field dataset.
Flattening without picking faults (ps.gz 232K) (pdf 141K) (src 10015K)
Lomask J., Guitton A., and Valenciano A.
We show that iteratively re-weighted least squares (IRLS) can flatten data cubes with vertically-oriented faults without having to pick the faults. One requirement is that the faults need to have at least part of their tip-lines (fault terminations) encased within the 3D cube. We demonstrate this method's flattening ability on a faulted 3D field data-set.
Non-linear estimation of vertical delays with a quasi-Newtom method (ps.gz 1805K) (pdf 680K) (src 100593K)
Guitton A., Lomask J., and Fomel S.
A local dip (or step out) between two adjacent traces embeds the necessary information to go from one reflection on one trace to the same reflection on the next. In more dimensions, i.e., 3-D, the same result is obtained between distant traces by integrating the local dips in all directions, thus obtaining relative delay maps useful for (1) automatic full-volume picking and (2) automatic flattening of horizons. The estimation of these maps from local dips is a non-linear process. In this paper, this problem is solved with a quasi-Newton technique for 2-D slices and 3-D cubes. Furthermore, the estimation of the relative delays is done globally in a least-squares sense for all reflectors at once. Synthetic and field data examples illustrate the ability of the algorithm to flatten horizon according to their geological time. As a natural extension of our algorithm, any horizon can also be picked automatically at no additional cost.
Image segmentation with bounds (ps.gz 647K) (pdf 301K) (src 5210K)
Lomask J. and Biondi B.
Image segmentation Hale and Emanuel (2002, 2003); Shi and Malik (2000) for tracking salt boundaries Lomask et al. (2004); Lomask and Biondi (2003) is extremely memory intensive. Memory saving measures must be implemented in order to consider applying this technique to 3D seismic cubes. If coarse bounds can be picked, either manually or using another automatic algorithm, this image segmentation algorithm can then be used to partition between the bounds. Unfortunately, the quality of the segmentation result is strongly affected by the shape of the image. For example, elongated images are more likely to be partitioned along their shortest dimension. In this note, we present one such memory saving technique and demonstrate its ability to pick a salt boundary on a 2D seismic section. By imposing bounds, we significantly reduce the size of the problem and, as a result, increase efficiency and robustness. Also, errors created by segmenting thin images can be rectified with novel boundary conditions described here.

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Interpolation and amplitudes regularization

Data regularization: inversion with azimuth move-out (ps.gz 322K) (pdf 334K) (src 1337K)
Clapp R. G.
Data regularization is cast as a least-squares inversion problem. The model space is a five-dimensional (t,cmpx, cmpy, hx, hy) hypercube. The regularization minimizes the difference between various (t, cmpx, cmpy) cubes by applying a filter that acts in (hx,hy) plane. Azimuth Move-out is used transform the cubes to the same ( hx,hy) before applying the filter. The methodology is made efficient by a Fourier-domain implementation, and preconditioning the problem. The methodology, along with two approximations is demonstrated on 3-D dataset from the North Sea. The inversion result proves superior at a reasonable cost.
Interpolation and signal extraction of teleseismic wavefields with the linear radon transform (ps.gz 2365K) (pdf 917K) (src 46149K)
Wilson C. K. and Guitton A.
We present a new method for data interpolation and signal/noise separation of teleseismic wavefields recorded by regional seismic arrays. The method exploits the plane wave nature of direct arrivals and receiver-side arrivals from regional scale structure by decomposing the recorded wavefield into a plane wave basis using the linear radon transform. Casting the radon transform as an inversion problem allows the incorporation of time dependent weighting schemes and model variance tuning which are helpful in minimizing artifacts related to the transform process while enhancing lower amplitude arrivals. Following radon transformation, we mute portions of the radon panel that represent plane waves with significantly different moveout ($$.1 s/km) relative to the direct arrival. Transformation back to the data domain from the muted radon domain gives the original signal without (1) plane waves following undesired moveouts, (2) white ambient noise, and/or (3) arrivals not represented well by plane waves (diffractions). Interpolation follows from the inverse data spray operation computed upon return to the data domain and the implicit assumption that a plane wave basis provides the most compact representation of the teleseismic wavefield.
Focusing-effect AVO/AVA: overview of results and assessment of problems (ps.gz 1525K) (pdf 676K) (src 3174K)
Vlad I.
Small-scale heterogeneities in the Earth produce visible focusing of seismic wavefield amplitudes with offset, but minimal variations in traveltimes. These effects are called Focusing-Effect AVO (FEAVO) or AVA (FEAVA) for avoiding confusion with lithology-caused AVO/AVA. FEAVO/FEAVA is not an unpredictable phenomenon that occurs at random. It appears in a number of well-defined geological settings, it can be modeled with appropriate precautions, it can be identified by its spatially predictable patterns, and can be removed in a manner that takes into account the specific physics involved. This paper summarizes work published over the course of several years in seven different SEP reports, providing an overview of the results obtained up to date and an assessment of the most critical problems to be solved.
Interpolating with data-space prediction-error filters (ps.gz 1078K) (pdf 455K) (src 6362K)
Curry W.
The Madagascar sea elevation dataset presents a problem where data are collected along crossing tracks. These tracks are not straight, and are therefore irregular in the model space. Previous methods assumed that the data were regularly sampled in the model space coordinate system, or did not take into account the regularities in the acquisition of the data. Instead of attempting to find a prediction-error filter in the model space, I estimate two prediction-error filters in a coordinate system based on the data's spatial distribution, and show how to regularize the data with these filters with promising results. I then show how this strategy can be applied to 2D and 3D land surveys when data predicted by reciprocity is included.
Non-stationary PEFs and large gaps (ps.gz 421K) (pdf 390K) (src 1139K)
Curry W.
Prediction-error filters (PEFs) may be used to interpolate missing data, either to increase the sampling of data that are regularly-sampled Spitz (1991), as well as to interpolate larger gaps in data Claerbout (1992, 1999). In addition to using multi-dimensional PEFs, non-stationary PEFs Crawley et al. (1998) have been used to interpolate regularly-sampled data Crawley (2000). Non-stationary ...

Imaging

ADCIGs for forward-scattered wavefields (ps.gz 275K) (pdf 222K) (src 3479K)
Shragge J., Artman B., and Biondi B.
We extend the 2-D theory of angle-domain common-image gathers (ADCIGs) to forward-scattered wavefields, and present a method for extracting reflectivity as a function of either the reflected or converted-wave receiver-side scattering angle. We use the shot-profile configuration of wave-equation migration along with planar source and receiver wavefields to generate an analytic hyper-plane surface in the intermediate offset-domain common-image gather space. Geometrical relations and partial derivatives of the hyper-plane function generate six constraint equations for the the unknown six parameters, allowing us to solve for the source- and receiver-side reflection angles and geologic dip angle. Results of numerical experiments indicate that information on wavefield focusing is present in forward-scattered ADCIGs, which suggests that this algorithm may be useful tool for improving wave-equation based tomography of transmission wavefields.
Time windowing passive seismic data in the frequency domain (ps.gz 115K) (pdf 137K) (src 245K)
Artman B.
One of the principle obstacles to the utility of passive seismic data is its bulk. With several hundred to thousands of geophones, we are able to generate mountains of data in a very short time. The simplest method of trimming down this volume is to keep only the recorded wavefield around times when usable source energy is known to be present. In the case of teleseismic imaging, or when utilizing unconventional, but known sources, this is easily done. However, if one hopes to image with the truly ambient noise field, time windowing amounts to removing needed signal. ...
Converted-wave common-azimuth migration (ps.gz 384K) (pdf 491K) (src 682K)
Rosales D. A. and Biondi B.
Multicomponent seismic data may hold a wealth of information for oil exploration and reservoir characterization. Multicomponent seismic contains energy from converted waves that is not seen in conventional seismic; therefore, the development of new techniques to process converted-wave data is important. Much progress has been made in many areas of converted-wave seismic processing, such as stacking, DMO, migration and velocity analysis Alfaraj (1992); Harrison and Stewart (1993); Huub Den Rooijen (1991); Iverson et al. (1989); Tessmer and Behle (1988). ...
Converted-mode angle-domain common-image gathers for migration velocity analysis (ps.gz 969K) (pdf 1120K) (src 2071K)
Rosales D. A. and Biondi B.
Common-image gathers are very useful for velocity and petrophysical analysis. Wavefield-extrapolation methods produce Angle-Domain Common-Image Gathers (ADCIGs). For the conventional PP case, ADCIGs are a function of the opening angle. The representation of ADCIGs for PS data (PS-ADCIGs) is more elaborate than for conventional ADCIGs. In PS-ADCIGs, the P-to-S velocity ratio is a major variable in transforming the subsurface offset to the opening angle, and in transforming this opening angle to either the P-incidence or the S-reflection angle. Numerical studies show that when the P-to-S velocity ratio and image midpoint information are not incorporated the error in computing PS-ADCIGs is enough to introduce artifacts in the velocity model.
Orthogonal mesh generation for Riemannian wavefield extrapolation (ps.gz 481K) (pdf 666K) (src 5043K)
Shragge J.
This paper presents a general method for generating 2D or 3D orthogonal coordinate systems. Developed coordinate systems are triplication free and appropriate for use in Riemannian wavefield extrapolation. This method exploits properties of potential function solutions of Laplace's equation. I show that certain specifications of a potential function's boundary conditions lead to a physical representation of equipotential surfaces where there are equivalent to extrapolation steps. Potential function solutions, obtained through conjugate gradient solvers, are used subsequently in a phase-ray-tracing procedure that generates geometric rays orthogonal to the equipotential surfaces. These rays collectively define an orthogonal coordinate system linked to the underlying Cartesian mesh through definable one-to-one mappings. The utility of this approach in generating coordinate systems is tested on a 2D model of rugged topography from the Canadian Foothills, and on 3D topography of the San Francisco Bay area.
Wavefield extrapolation in frequency-wavenumber domain for spatially-varying velocity models (ps.gz 65K) (pdf 60K) (src 241K)
Alvarez G. and Artman B.
Mixed domain wavefield extrapolation methods can handle, to a large extent, spatial velocity variations by performing part of their computation in the $$-${k}$ domain and part of the computation in the $$-${x}$ domain. The best-known mixed domain methods are ...
Migration and modeling of seismic data affected by focusing-effect AVO/AVA (ps.gz 146K) (pdf 98K) (src 133K)
Vlad I.
Focusing-effect AVO or AVA is the phenomenon of velocity and/or absorption lenses creating substantial amplitude variations, but only small traveltime anomalies Kjartansson (1979). The patterns thus created can interfere significantly with AVO/AVA caused by lithological contrasts at the reflector. To render amplitude analysis feasible, these patterns need to be removed from the image. I will use the acronym ``FEAVO'' to refer to focusing-effect AVO or AVA in general, reserving ``FEAVA'' only for specific references to the angle domain. These terms refers only to amplitudes focusing through heterogeneities smaller than the Fresnel zone, as formalized by Spetzler et al. (2004), and which do not cause energy to be lost by sending it outside the finite spatial extent of the seismic survey (i.e., ``illumination problems''). Focusing can be positive (usual meaning of term) or negative (i.e., in the case of absorption). Vlad and Biondi (2002), Vlad (2002), ...
Fourier-domain imaging condition for shot-profile migration (ps.gz 582K) (pdf 316K) (src 1398K)
Artman B. and Fomel S.
Cross-correlating up-coming and down-going wavefields inherently applies a spatial multiplication. This multiplication could be performed in the wave-number domain as a convolution. However, the full imaging condition, including subsurface offset, transforms to a Fourier domain equivalent that is also a lagged multiplication. This fact allows for the simple analysis of anti-aliasing criteria. Migrations with synthetic data with flat and dipping reflectors in a homogeneous medium are produced to evaluate the Fourier domain algorithm and shots from the Marmousi data set are shown as examples of its efficacy. Periodic replications in the image space are introduced when solving the imaging condition in the Fourier domain which make results unsatisfactory. The cost of computing the imaging condition in the Fourier domain is much higher than its space domain equivalent since very few subsurface offsets need to be imaged if the velocity model is reasonably accurate. Analysis of the Fourier domain imaging condition leads to the conclusion that anti-aliasing efforts can be implemented post-migration.
A self-adaptive algorithm for choosing reference velocities in the presence of lateral velocity variations (ps.gz 1615K) (pdf 372K) (src 10661K)
Wang H. and Shan G.
Seismic wave propagation depicted with the perturbation theory has important and wide-spread uses in reflection seismology. As we know, in perturbation theory, wave propagation needs a reference velocity. The closer the reference velocity is to the true velocity, the more accurate the wave propagation is. However, it is not easy to choose reasonable reference velocities in the presence of severe lateral velocity variations. Assigning a reference velocity value at each spatial point is not computationally feasible, because there is a trade-off between the calculation cost and the number of reference velocities. We show that the accuracy of seismic wave propagation can be more easily improved by choosing a set of reasonable reference velocities rather than by optimizing a one-way wave propagator. Therefore, we introduce a self-adaptive approach to choose a set of reference velocities for an extrapolation layer, in the presence of lateral velocity variations. Through sorting the velocity data an array with increasing values, and by setting a threshold average-velocity ratio or velocity- variance ratio, we can choose a set of reasonable reference velocities for wavefield extrapolation. This method can also be used for image edge detecting. It is flexible and computationally cost-effective.
Unified seismic-wave imaging - from data space to model space (ps.gz 53K) (pdf 109K) (src 40K)
Wang H. and Shan G.
Under operator, matrix and inverse theory, seismic-wave imaging can be considered a unified process-mapping from data space to model space. The main topics in seismic-wave imaging include (1) seismic-data interpolation, regularization and redatuming, which mainly decrease the imaging noise; (2) seismic-wave illumination analysis, which predicts whether a target reflector can be imaged and evaluates the suitability of an acquisition configuration in the case of rugged topography and severe lateral velocity variations; and (3) seismic-wave migration/inversion imaging algorithms, which give an imaging result with the help of a wave propagator, known a macro-velocity model. The last and most important thing is to build an accurate macro-velocity model. All of the processes can be considered with the conjugate operator/matrix under least-squares theory. In this article, we review the following topics: (1) expression of data space and model space; (2) affiliation between data space and model space; (3) seismic-data preprocessing; (4) seismic-data illumination; (5) migration imaging and inversion imaging as least-squares inverse problems; (6) amplitude-preserving migration imaging with wavefield extrapolation; (7) migration velocity analysis and inversion and (8) some related topics. We express the imaging process with the operator or matrix theory and give some directions for further research.

Multiples and noise attenuation

Water-bottom and diffracted 2D multiple reflections in data space and image space (ps.gz 840K) (pdf 467K) (src 5304K)
Alvarez G.
Water-bottom multiples from a dipping interface have the same kinematics in a Common Midpoint (CMP) gather as a primary from a reflector with twice the dip at twice the perpendicular depth at the CMP location. When migrated with the velocity of the primaries, these multiples are overmigrated just as primaries migrated with higher velocity, and their moveout is thus predictable in image space. Diffracted multiples, on the other hand, have an apex-shifted moveout in CMP gathers and a more complicated, also apex-shifted, residual moveout in image space when migrated with the velocity of the primaries. I illustrate the moveout of water-bottom and diffracted multiples in image space with a simple 2D synthetic dataset.
Multiple attenuation: data space vs. image space - A real data example (ps.gz 6241K) (pdf 1413K) (src 14598K)
Rosales D. A. and Guitton A.
Multiples can be attenuated either before or after the imaging process. Adaptive subtraction can be applied after migration to eliminate the multiples in the image space. We build a multiple model based on the sea floor reflection, that is kinematically correct. We then perform multiple attenuation to remove the water-bottom multiple in both the image space and the data space.
Sparse radon transforms with a bound-constrained approach (ps.gz 534K) (pdf 268K) (src 1192K)
Guitton A.
Radon transforms are popular operators for velocity analysis Guitton and Symes (2003); Taner and Koehler (1969), noise attenuation Foster and Mosher (1992), and data interpolation Hindriks and Duijndam (1998); Trad et al. (2002). One property that is often sought in radon domains is sparseness, where the energy in the model space is well focused for each corresponding event in the data space. ...
Removal of linear events with combined radon transforms (ps.gz 2293K) (pdf 1333K) (src 6155K)
Artman B. and Guitton A.
We explore the classic signal and noise separation problem of removing linear events from shot-gathers through several inversion schemes using a combined modeling operator composed of both hyperbolic and linear radon transforms. Data are inverted simultaneously for both linear and hyperbolic moveout which provides two model-space outputs. These are forward modeled seperately to provide an output data-space devoid of one of the model-space components. We employ this approach to analyze the removal of direct arrivals and ground-roll from shot-gathers. Inversion schemes used imnclude: 1) bound-constrained, 2) Cauchy norm regularization, 3) Huber norm approximating the l₁ norm, and 4) the l₂ norm using linear least-squares. Synthetic tests and four field shot-gathers are used to demonstrate the approach. Methods 1, 2, and 3 are designed to provide sparse model-space inversions. In the real data examples, the least-squares solution is able to better achieve the signal to noise separation goal despite its model-space being often less appealing.

Rock Physics

Using knowledge of microstructure to improve estimates and bounds on elastic constants and transport coefficients in heterogeneous media (ps.gz 99K) (pdf 226K) (src 86K)
Berryman J. G.
The most commonly discussed measures of microstructure in composite materials are the spatial correlation functions, which in a porous medium measure either the grain-to-grain correlations, or the pore-to-pore correlations in space. Improved bounds based on this information such as the Beran-Molyneux bounds for bulk modulus and the Beran bounds for conductivity are well-known. It is first shown how to make direct use of bounds and spatial correlation information to provide estimates that always lie between these upper and lower bounds for any microstructure whenever the microgeometry parameters are known. Then comparisons are made between these estimates, the bounds, and two new types of estimates. One new estimate for elastic constants makes use of the Peselnick-Meister bounds (based on Hashin-Shtrikman methods) for random polycrystals of laminates to generate self-consistent values that always lie between the bounds. A second new type of estimate for conductivity assumes that measurements of formation factors (of which there are at least two distinct types in porous media, associated respectively with pores and grains for either electrical and thermal conductivity) are available, and computes new bounds based on this information. The paper compares and contrasts these various methods in order to clarify just what microstructural information - and how accurately that information - needs to be known in order to be useful for estimating material constants in random and heterogeneous media.
Geomechanical constants of heterogeneous reservoirs: pore fluid effects on shear modulus (ps.gz 82K) (pdf 194K) (src 77K)
Berryman J. G.
To provide quantitative measures of the importance of fluid effects on shear waves in the heterogeneous reservoirs, a model material called a ``random polycrystal of porous laminates'' is introduced. This model poroelastic material has constituent grains that are layered (or laminated), and each layer is an isotropic, microhomogeneous porous medium. All grains are composed of exactly the same porous constituents, and have the same relative volume fractions. But the order of lamination is not important because the up-scaling method used to determine the transversely isotropic (hexagonal) properties of the grains is Backus averaging, which - for quasi-static or long-wavelength behavior - depends only on the volume fractions and layer properties. Grains are then jumbled together totally at random, filling the reservoir, and producing an overall isotropic poroelastic medium. The poroelastic behavior of this medium is then analyzed using the Peselnick-Meister-Watt bounds (of Hashin-Shtrikman type). We study the dependence of the shear modulus on pore fluid properties and determine the expected range of behavior. In particular we compare and contrast these results with those anticipated from Gassmann's fluid substitution formulas, and to the predictions of Mavko and Jizba for very low porosity rocks with flat cracks. This approach also permits the study of arbitrary numbers of constituents, but for simplicity the numerical examples are restricted here to just two constituents. This restriction also permits the use of some special exact results available for computing the overall effective stress coefficient in any two-component porous medium. The bounds making use of polycrystalline microstructure are very tight. Results for shear modulus demonstrate that the ratio of compliance differences R (i.e., shear compliance changes over bulk compliance changes) is usually nonzero and can take a wide range of values, both above and below the value R = 4/15 for low porosity, very low aspect ratio flat cracks. Results show the overall shear modulus in this model can depend relatively strongly on mechanical properties of the pore fluids, sometimes (but rarely) more strongly than the dependence of the overall bulk modulus on the fluids.