Modelling borehole wave signatures in elastic and poroelastic media with spectral method
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Borehole sonic measurements are an important tool to characterize formation and completion properties of hydrocarbon or water reservoirs. Such measurements can provide direct information about rock physical parameters such as permeability or elastic moduli. These properties are obtained from guided waves propagating along boreholes. The so called tube wave or Stoneley wave is a symmetric mode which compresses the fluid column leading to a piston like motion. If the medium around the borehole wall is permeable, the radial expansion of the fluid column will result in fluid flow across the borehole wall. This results in a sensitivity of the tube wave signature to the permeability of the surrounding formation which manifests itself in a characteristic dispersion and attenuation of the tube wave. Information about the permeability of the surrounding formation provides essential knowledge for reservoir characterization.In addition to the traditional method of using tube wave signatures for formation permeability estimations, the same approach may be used for production monitoring. In sand reservoirs a complicated borehole completion is installed during the production phase for the purpose of controlling sand production. In such a setup highly permeable layers such as a sand screen or a gravel pack are used to prevent sand production.The problem with such completions is that they are very expensive to install and susceptible to plugging or corrosion. No permanent surveillance tool exists to date which allows diagnosis of problems in sandscreened deepwater completions. However, the recently proposed RealTime Completion Monitoring (RTCM) uses the signature of tube waves to identify permeability changes: the increase of the tube wave velocity can indicate a decrease of permeability and vice versa. Therefore, RTCM has potential to identify problems in sandscreened deepwater completions.In order to understand the acoustic response of such deepwater completions, the dispersion and attenuation of tube waves in this complicated setup needs to be studied. To this end I have developed a modelling algorithm based on a spectral method. The developed algorithm computes the dispersion and attenuation of borehole modes propagating in a cylindrically layered structure with an arbitrary number of fluid, elastic and poroelastic layers. The numerical algorithm discretizes the medium along the radial axis using Chebyshev interpolation points derived from Chebyshev polynomials. The differential operators are discretized using spectral differentiation matrices. Thus, for any number of layers, the corresponding equations can be expressed as a generalized algebraic eigenvalue problem. For a given frequency, the eigenvalues correspond to the wavenumbers of different modes. The eigenvectors, computed along with the eigenvalues, correspond to the displacement potentials. They can be used to obtain the variation of displacement and stress components along the radius of the structure.In this thesis the spectral method was first developed for structures with an arbitrary number of fluid and elastic layers. Subsequently, the algorithm was extended for poroelasticity. The results produced by the modelling program are benchmarked against analytical solutions. Such analytical solutions are known for elastic and poroelastic cylinders as well as fluid filled tubes. The tube wave dispersion in a fluidfilled borehole surrounded by an elastic or poroelastic formation obtained with the spectral method was compared to the analytical lowfrequency solution.I obtained the dispersion of the two tube waves propagating in a four layer completion model: fluid – permeable sandscreen – fluid – elastic casing. Varying the permeability of the sandscreen layer allowed me to account for the effect of fluid flow across this layer. Being able to obtain the acoustic response can help to identify broken fluid communication which increases the tube wave velocity. A corroded sandscreen has an extremely attenuated tube wave signature.Furthermore, I have implemented the more complex model of a borehole surrounded by an altered zone in the algorithm. Due to drilling damage the altered zone is an area of reduced permeability. In order to account for the effect of the altered zone on the tube wave signature, up to ten layers were used with stepwise increase of permeability from the borehole towards the formation. Overall, the spectral method proved to be a valuable algorithm to model wave propagation in cylindrical structures.Using borehole modes to evaluate the physical properties of the formation or completions is an important application. However, in borehole seismic modelling, such as crosshole or VSP, it is also important to account for the effect of boreholes and the associated modes. Since the borehole radius is a thousand times smaller then the investigated volume it would require a prohibitively small grid size to explicitly model the borehole. However, it is possible to effectively represent a borehole as a superposition of point sources. This mimics the presence of borehole modes. In order to implement this technique for poroelasticity, it is necessary to model source signatures in poroelastic media. To this end I have analyzed the radiation characteristics and moment tensor solutions for various source types. Together with the spectral method these point source representations can be used to model the effect of boreholes. This will pave the way for more efficient poroelastic seismic modelling in various fluidfilled boreholes and completions.
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