SEP-89 (1995)

Reservoir Monitoring and Rockphysics

Sediments with gas hydrates: Internal structure from seismic AVO

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Ecker C., Dvorkin J., and Nur A.
We use rock-physics-based synthetic seismic modeling to interpret AVO data of methane hydrate structures. Two micromechanical models of gas hydrate deposition are examined: (1) the hydrate cements grain contacts and strongly reinforces the sediment; and (2) the hydrate is located away from the grain contacts and only weakly affects the stiffness of the sediment's frame. We calculate the effect of pore fluid on seismic velocities from Gassmann's equation. Synthetic seismograms are generated from 1-D elastic modeling (Thompson-Haskell reflectivity method) and the AVO responses at the BSR are obtained from Zoeppritz equations. Comparison with real seismic AVO data from the Blake Outer Ridge, offshore Florida and Georgia, shows that the second model of hydrate deposition can reproduce the real AVO effect qualitatively. The inferred topology of hydrate deposition leads to two important conclusions: (1) the sediment containing hydrate is uncemented and thus mechanically weak, and (2) the permeability of this sediment is very low because hydrate clogs large pore-space conduits. This second conclusion explains the fact that free gas is trapped underneath the BSR at the Blake Outer Ridge. The data furthermore indicate the absence of strong reflections at the top of the hydrate. We conclude that the high concentration of hydrate in the sediment immediately above the BSR gradually decreases with decreasing depth.

4-D seismic steamflood monitoring

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E. Lumley D.
Six repeated 3-D seismic data sets are analyzed at a single steamflood site in Indonesia. Migrated images and velocity analysis illuminate fluid-flow features of the steamflood in time-lapse mode. The steam front is visible seismically as large reflection and diffraction responses, and velocity decreases of up to 40%, in a disk centered at the steam injection well. The steam front intensifies in place at a radius of less than 50 m for the first nine months of injection, and then breaks out and rapidly propagates to the northwest of the injection pattern at and beyond the 13-month survey. The steam movement mapped by time-lapse seismic correlates with two temperature observation wells and oil production data from the surrounding producing wells. A thin annulus of hot water (steam condensate) is visible in time slices and a 10% increase in velocity. A transient pressure front is observed at 2 months of steam injection to propagate towards the northwest, but not toward the southeast. Mapping the early pressure front movement predicts where the thermal and steam fronts will follow. The steam front is shown to follow the leading pressure front movement to the northwest one full year after the pressure front was initially mapped.

Equations of motion for isotropic porous media with multiple solid constituents

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Berryman J. G. and Pride S. R.
The volume averaging technique for obtaining macroscopic equations of motion for materials that are microscopically inhomogeneous is extended to the situation in which multiple solid constituents form a porous matrix while a uniform fluid fills the pores. Previous volume averaging efforts of Pride et al. (1992) and others have concentrated on single solid constituent porous media. The analysis for multiple solid constituents is complicated by the presence of internal interfaces between the solid constituents within the averaging volume. These interfaces are characterized by constants that measure the fraction of the interface on which solid touches solid, fluid touches one solid, fluid touches the other solid, or fluid lies on both sides of the interface. These fractions are easily computed if the interface fractions are assumed to be uncorrelated, but real materials may be expected to exhibit some correlation. On the other hand, these interface fractions do not appear in the volume average equations at the macroscopic level. To complete the analysis, it is found that the jacketed and unjacketed tests of Biot and Willis (1957) together with the thought experiments of Berryman and Milton (1991) for solid matrix composed of two constituents are required in order to obtain definite results. Results are found to be in complete agreement with earlier work of Brown and Korringa (1975) concerning the most general possible form of the quasistatic equations for volume deformation and therefore of the equations of motion for wave propagation through such media.

AMO

Azimuth moveout: the operator parameterization and antialiasing

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Fomel S. and Biondi B. L.
A practical implementation of azimuth moveout (AMO) must be both computationally efficient and accurate. We achieve computational efficiency by parameterizing the AMO operator with the help of a transformed midpoint coordinate system. To achieve accuracy, the AMO operator needs to be carefully designed for antialiasing. We propose a modified version of Hale's antialiasing algorithm, which switches between interpolation in time and interpolation in space depending on the operator dips. The method is applicable to a vide variety of integral operators and compares favorably with the triangle filter technique. A simple synthetic example tests the applicability of the method to the AMO case.

Amplitude preserving offset continuation in theory Part 2: Solving the equation

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Fomel S.
I consider an initial value problem for the offset continuation (OC) equation introduced in Part One of this paper (SEP-84). The solutions of this problem create integral-type OC operators in the time-space domain. Moving to the frequency-wavenumber and log-stretch domain, I compare the obtained operators with the well-known Fourier DMO operators. This comparison links the theory of DMO with the advanced theory of offset continuation.

Model Building

Velocity model building in AVS

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Mora C. B., Clapp R. G., and Biondi B.
For the last three years SEP has been developing tools within Advanced Visual System (AVS), to better handle 3-D seismic data. Our newest module attempts to include the entire velocity model building process within our AVS environment. It combines some of the best elements of GOCAD with the higher level of flexibility offered by AVS. An interactive model building process was tested by combining the new module, along with other AVS modules developed at SEP, to build a 3-D velocity model from a land survey dataset provided by INTEVEP S.A.

Inversion

Flexible seismic traveltime tomography applied to diving waves

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Harlan W. S.
To prepare for 3D anisotropic applications of traveltime tomography I found it necessary to reformulate the estimation of raypaths and the parameterization of velocities. Raypaths should be efficient to calculate and store in memory without recalculation. Velocities should be able to change with angle when necessary. Ray relaxation can optimize ray paths through an anisotropic medium described only by group velocities. Three parameters adequately describe transversely isotropic group velocities with a vertical axis of symmetry. One parameter changes most arbitrarily, one changes only along the vertical axis of symmetry, and one remains a constant. 3D raypaths are described as sums of smooth curves with a small number of coefficients. A generic Gauss-Newton algorithm perturbs these coefficients to minimize traveltimes between endpoints. A small number of coefficients are saved in memory to describe paths efficiently. Diving wave tomography can use conventional traveltime tomographic algorithms to model and invert the traveltimes of direct arrivals. Early synthetic tests minimized the complexity of velocity anomalies necessary to explain the data. Raypaths robustly converge to their appropriate distribution.

Iterative least-square inversion for amplitude balancing

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Berlioux A. and Harlan W. S.
Variations in source strength and receiver amplitude can introduce a bias in the final AVO analysis of prestack seismic reflection data. In this paper we tackle the problem of the amplitude balancing of the seismic traces from a marine survey. We start with a 2-D energy map from which the global trend has been removed. In order to balance this amplitude map, we first invert for the correction coefficients using an iterative least-square algorithm. The coefficients are calculated for each shot position along the survey line, each receiver position in the recording cable, and each offset. Using these coefficients, we then correct the original amplitude map for amplitude variations in the shot, receiver, and offset directions.

Linearized elastic inversion in the domain

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Bernasconi G., Drufuca G., and Rocca F.
We present an algorithm for elastic linearized inversion in the wavenumber-frequency domain. The medium is considered to be composed of a known constant background plus an unknown small variation of P-impedance, S-impedance and density. The linearized approach enables us to use the tools of linear algebra to analyze the ill-conditioning of the inversion. Operating in the wavenumber-frequency domain, we gain the advantage of the simplicity of the theory, as the scattering process becomes the interaction of monochromatic plane waves with sinusoidal variations of the elastic parameters. Inversion and analysis of the information give a smoothed but reliable image of the original model. We test the algorithm on synthetic data. Because of the ill-conditioning of the inverse problem, we find that even small numerical errors cause deterioration of the quality of inversion: P-P reflections allow an accurate estimation of P-impedance perturbation only. Multicomponent data are necessary to recover S-impedance perturbation with this approach.

Adjoints

The adjoint of the viscous wave equation

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Claerbout J.
The scalar wave equation is a linear operator whose output is any subset of the wavefield solution and whose input is the source function. Here we show how to compute the adjoint operator, i.e., that whose input is the wavefield subset and whose output is the sources.

Ellipsoids versus hyperboloids

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Claerbout J.
Many operators can be coded in either ``push'' or ``pull'' forms, namely, you can loop over the space of outputs and find the inputs (pull), or loop over the space of inputs and find the outputs (push). The adjoint of a push operator is a pull operator and vice versa. Migration can be done with either hyperbolas or diffractors. Inversions impose extra accuracy requirements that may lead to forward modeling being a pull operator. A DMO operator is a product of two operators, either of which is a push or pull, leading to four possible representations.

Approximate vs. exact adjoints in inversion

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Crawley S.
Gradient-based inversion methods employ adjoint operators to find the gradient direction in the objective function. Generally we assume that the exact adjoint is the optimal choice. In this paper I discuss one type of approximate adjoint, called a pull adjoint because of the way it discretizes the output space. For operators which sum and spray data along curved trajectories, such as Kirchoff and moveout operators, pull adjoints more closely approximate an operator in a continuous space. This characteristic is important because we use these operators to emulate the wave equation, which is a continuous and unitary operator. This paper presents a theoretical argument for the use of pull adjoints in some inversions, and presents a simple inversion example where the use of pull adjoints is of benefit.

Computer

Reproducible research documents using GNU make

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Schwab M. and Schroeder J.
The Stanford Exploration Project (SEP) publishes scores of reproducible electronic research documents each year. Traditionally, researchers at SEP have used the makefile dialect cake to maintain their documents. Because of cake's limited distribution, we decided to replace it by GNU make. GNU make is an alternative make dialect that is widely used in software maintenance and is well documented. To allow an author to cope with document maintenance as opposed to software maintenance, Richard Stallman amended the original GNU make source code: a new GNU make special built-in target, .SECONDARY, enables the author to specify how to deal with missing files when checking if a target is up-to-date. Furthermore, the maintenance rules common to all reproducible documents were rethought when we translated them from cake to GNU make. The most significant change is the introduction of three result lists that distinguish between easily-reproducible, conditionally-reproducible, and non-reproducible results. A declaration of the results' reproducibility in the makefile simplified most of the crucial reproducibility rules (such as make burn, which removes all easily-reproducible results). We tested the new GNU make rules by translating Claerbout's three most recent textbooks. SEP plans to publish all its future reproducible documents using GNU make.

Interactive web pages with Java

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Lenga K., Clapp R. G., and Claerbout J. F.
Recently, Sun introduced Java, a new programming language with machine independent object files. Java, and the Java Browser, provides the next step in web browser technology by bringing over machine independent byte codes and interpreting them on the local machine rather than the host machine. This capability opens up a new realm of possibilities in information transfer and interactive displays. This summer SEP tested Java's capabilities by developing an interactive web-based demo of wave propagation, a frame viewer (similar to SEP's old Animator), and a cube viewer.