Elastic wave attenuation, dispersion and anisotropy in fractured porous media
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Development of a hydrocarbon reservoir requires information about the type of fluid that saturates the pore space, and the permeability distribution that determines how the fluid can be extracted. The presence of fractures in a reservoir can be useful for obtaining this information. The main objectives of this thesis are to investigate how fracturing can be detected remotely using exploration seismology. Fracturing will effect seismic data in a number of ways. Firstly, if the fractures are aligned preferentially in some direction, the medium will exhibit long wavelength anisotropy. In turn, if wave propagation is not aligned with one of the symmetry axes of the effective medium then shear wave splitting will depend upon the properties of the fracture filling fluid. Secondly, elastic waves will experience attenuation and dispersion due to scattering and wave-induced fluid flow between the fractures and matrix porosity. This occurs because the fractures are more compliant than the background medium and therefore there will be a pressure gradient formed during passage of the wave, causing fluid to flow between fractures and background. If the direction of shear-wave propagation is not perpendicular or parallel to the plane of fracturing, the wave polarized in the plane perpendicular to the fractures is a quasi-shear mode, and therefore the shear-wave splitting will be sensitive to the fluid bulk modulus.The magnitude of this sensitivity depends upon the extent to which fluid pressure can equilibrate between pores and fractures during the period of the deformation. In this thesis I use the anisotropic Gassmann equations and existing formulations for the excess compliance due to fracturing to estimate the splitting of vertically propagating shear-waves as a function of the fluid modulus for a porous medium with a single set of dipping fractures and with two conjugate fracture sets dipping with opposite dips to the vertical. This is achieved using two alternative approaches. In the first approach it is assumed that the deformation taking place is quasi-static. That is, the frequency of the elastic disturbance is low enough to allow enough time for fluid to flow between both the fractures and the pore space throughout the medium. In the second approach I assume that the frequency is low enough to allow fluid flow between a fracture set and the surrounding pore space, but high enough so that there is not enough time during the period of the elastic disturbance for fluid flow between different fracture sets to occur. It is found that the second approach yields a much stronger dependency of shear-wave splitting on the fluid modulus than the first one. This is a consequence of the fact that at higher wave frequencies there is not enough time for fluid pressure to equilibrate and therefore the elastic properties of the fluid have a greater effect on the magnitude of the shear-wave splitting. I conclude that the dependency of the shear-wave splitting on the fluid bulk modulus will be at its minimum for quasi-static deformations, and will increase with increasing wave frequency.In order to treat the problem of dispersion and attenuation due to wave-induced fluid flow I consider interaction of a normally incident time-harmonic longitudinal plane wave with a circular crack imbedded in a porous medium governed by Biot’s equations of dynamic poroelasticity. The problem is formulated in cylindrical coordinates as a system of dual integral equations for the Hankel transform of the wave field, which is then reduced to a single Fredholm integral equation of the second kind. It is found that the scattering that takes place is predominantly due to wave induced fluid flow between the pores and the crack. The scattering magnitude depends on the size of the crack relative to the slow wave wavelength and has its maximum value when they are of the same order. I conclude that this poroelastic effect should not be neglected, at least at seismic frequencies. Using the solution of the scattering problem for a single crack and multiple-scattering theory I estimate the attenuation and dispersion of elastic waves taking place in a porous medium containing a sparse distribution of such cracks. I obtain from this analysis an effective velocity which at low frequencies reduces to the known static Gassmann result and a characteristic attenuation peak at the frequency such that the crack size and the slow wave wavelength are of the same order.When comparing with a similar model in which multiple scattering effects are neglected I and that there is agreement at high frequencies and discrepancies at low frequencies. I conclude that the interaction between cracks should not be neglected at low frequencies, even in the limit of weak crack density. Since the models only agree with each other at high frequencies, when the time available for fluid diffusion is small, I conclude that the interaction between cracks that takes place as a result of fluid diffusion is negligible at high frequencies. I also compare my results with a model for spherical inclusions and find that the attenuation for spherical inclusions has exactly the same dependence upon frequency, but a difference in magnitude that depends upon frequency. Since the attenuation curves are very close at low frequencies I conclude that the effective medium properties are not sensitive to the shape of an inclusion at wavelengths that are large compared to the inclusion size. However at frequencies such that the wavelength is comparable to or smaller than the inclusion size the effective properties are sensitive to the greater compliance of the flat cracks, and more attenuation occurs at a given frequency as a result.
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