[1] Across the seismic band of frequencies (loosely defined as <10 kHz), a seismic wave propagating through a porous material will create flow in the pore space that is laminar; that is, in this low-frequency ''seismic limit,'' the development of viscous boundary layers in the pores need not be modeled. An explicit time stepping staggered-grid finite difference scheme is presented for solving Biot's equations of poroelasticity in this low-frequency limit. A key part of this work is the establishment of rigorous stability conditions. It is demonstrated that over a wide range of porous material properties typical of sedimentary rock and despite the presence of fluid pressure diffusion (Biot slow waves), the usual Courant condition governs the stability as if the problem involved purely elastic waves. The accuracy of the method is demonstrated by comparing to exact analytical solutions for both fast compressional waves and slow waves. Additional numerical modeling examples are also presented.

This paper investigates the details of reflection, transmission, and conversion of plane waves incident upon a fracture at arbitrary angles. The elastic compliance of fractures that is produced by the presence of a planar collection of void spaces and asperities of contact is modeled as a displacement‐discontinuity boundary condition between two elastic half‐spaces. Closed‐form expressions for the transmission and reflection coefficients on a fracture are derived by replacing the boundary conditions for a welded interface by those for a fracture into the standard procedure for plane wave analysis. The closed‐form expressions reveal that a single fracture can produce a variety of potentially diagnostic waves such as transmitted waves, reflected waves, converted waves, head waves, and P interface waves and introduce a finite group time delay to all these waves with respect to the incident wave. The amplitude and group time delay of the fracture‐induced waves are controlled by the fracture stiffness, wave frequency, and the Poisson's ratio of the medium. The head wave and inhomogeneous P interface waves are generated when an SV wave is incident upon a fracture, at and beyond a critical angle, respectively, which is determined by Snell's law. For some combinations of the fracture stiffness and the Poisson's ratio of the half‐spaces, no reflection or transmission of a P wave or an SV wave occurs.

S U M M A R YThis paper describes an efficient approach for computing the frequency response of seismic waves propagating in 2-and 3-D earth models within which the magnitude and phase are required at many locations. The approach consists of running an explicit finite difference time domain (TD) code with a time harmonic source out to steady-state. The magnitudes and phases at locations in the model are computed using phase sensitive detection (PSD). PSD does not require storage of time-series (unlike a fast Fourier transform), reducing its memory requirements. Additionally, the response from multiple sources can be obtained from a single finite difference run by encoding each source with a different frequency. For 2-D models with many sources, this time domain phase sensitive detection (TD-PSD) approach has a higher arithmetic complexity than direct solution of the finite difference frequency domain (FD) equations using nested dissection re-ordering (FD-ND). The storage requirements for 2-D finite difference TD-PSD are lower than FD-ND. For 3-D finite difference models, TD-PSD has significantly lower arithmetic complexity and storage requirements than FD-ND, and therefore, may prove useful for computing the frequency response of large 3-D earth models.

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There are increasing numbers of published examples from around the world in which significant 4D time shifts have been observed in the overburden above producing reservoirs.Indeed, this topic prompted the TLE special section "Rocks under strain" in December 2005. The significance of these 4D observations is that, if we wish to fully understand the 4D signature of compacting reservoirs, we can no longer think of the reservoir in isolation. The seismic response outside the reservoir changes because the nonreservoir rocks deform in response to reservoir activity. While these nonreservoir 4D seismic changes can obscure or contribute to the reservoir-level signal, making the 4D interpretation uncertain, if utilized appropriately they may also be used to provide information on the actual reservoir pressure changes. This new pressure information can thus be used to complement well measurements or other 4D seismicbased methods (such as the multi-attribute pressure and saturation inversion outlined by Floricich et al., 2006), providing valuable data for reservoir monitoring and management. In this paper, we build on the work of several authors who have presented methods to invert surface deformation measurements for reservoir volume or pressure change. We show how 4D seismic can extend this approach by focusing on the inversion of 3D strain deformation estimates for the overburden derived directly from the repeated seismic data. The method is then applied to Genesis Field in the Gulf of Mexico, in which there are series of compacting unconsolidated stacked turbidite reservoirs. Time-lapse time strain.Rock velocities are sensitive to changes in stress and strain so that if a volume of rock strains, the change in traveltime through it will be made up of a contribution due to the change in distance traveled by the seismic wave and a contribution due to the change in velocity. A perturbation formula relating changes in vertical traveltime t, velocity v, and vertical layer thickness z, assuming small changes in thickness and velocity, is given by Landrø and Stammeijer (2004) (1)Hatchell and Bourne (2006) went on to make the assumption that changes in thickness and velocity can be linearly related by a constant of proportionality, R, which relates the fractional change in velocity and vertical strain so that ∆v/v = -Rε zz , resulting in the following relationship (2) where we have replaced ∆z/z by ε zz , signifying the vertical component of the strain tensor. The left side of Equation 2 is the derivative of the time-shift field, which we term time strain. A method to obtain time-lapse time strains from 4D seismic is described in the companion paper in this special section ("4D time strain and the seismic signature of geomechanical compaction at Genesis"). If we have knowledge of the magnitude of R, we can obtain estimates of vertical strain directly from 4D seismic observations. With estimates of vertical strain for the overburden, a linearized inversion can be employed to obtain reservoir pressure change. Segall (1992) shows ...