A general expression is derived for the dispersion relations and the impulse response of a radially layered borehole. The model geometry consists of a central fluid cylinder surrounded by an arbitrary number of solid annuli. A Thomson-Haskell type propagator matrix is used to relate stresses and displacements across the layers. Although the model is completely general, the geometries considered here are restricted to those of a cased bole. Layers of steel, cement, and formation surround the innermost fluid layer. Synthetic microseismograms containing all body and interface waves are calculated for a variety of model parameters.Formation body wave arrivals are relatively unaffected by the presence of a casing. They may, however, be hard to identify if cement velocities are close to or larger than those of the formation. The Stoneley and pseudoRayleigh wave arrivals are strongly influenced by the casing parameters. They respond to the combined effects of the steel, the cement, and the formation.

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A generalization of the technique of Tubman et al. (1984) allows the inclusion of intermediate fluid layers in the theoretical study of elastic wave propagation in a layered borehole. The number and location of fluid layers are arbitrary. The only restrictions are that the central cylinder is fluid and the outermost formation is solid. Synthetic full‐waveform microseismograms in poorly bonded cased holes can be generated, allowing investigation of free pipe and cement sheathed pipe with no bond to the formation. If there is a fluid layer between the steel and the cement, the steel is free to ring. The first arrival in this situation is from the casing, even with an extremely thin fluid layer or microannulus. The amplitude and duration of the pipe signal depend upon the thickness of the fluid layer. While the first arrival is from the casing, the formation body‐wave energy is present. The character of the waveform will vary as the formation parameters vary. If the duration of the steel arrival is small, it is possible to distinguish the formation P-wave arrival. If the fluid layer is between the cement and the formation, then the steel is well bonded to the cement but the cement is not bonded to the formation. In this case the thicknesses of the fluid and cement layers are important in determining the nature of the first arrival. If there is a large amount of cement bonded to the steel, the cement can damp out the ringing of the pipe and make it possible to distinguish formation arrivals. If there is less cement bonded to the steel, the cement does not damp out the steel ringing but the cement rings along with the steel and the first arrival is from the combination of the steel and the cement. The velocity of this wave depends upon the velocities and thicknesses of the steel and cement layers.

An important factor in the simulation of reservoir performance is the input reservoir parameters. Layers are often modeled with Constant parameters (porosity, permeability, etc.) or with only minor variations. The failure to incorporate appropriate heterogeneity into reservoir performance predictions can be a significant problem. This paper examines the variability in reservoir parameters as measured with wireline logs. Both vertical and horizontal well data are utilized for this purpose. The nature of the log variability in both a sandstone and a carbonate reservoir are evaluated using rescaled range (R/S) analysis in conjunction with power spectrum analysis, similar to the methodology of Hewett et al. The R/S analysis yields a fractal dimension that can be used to model the reservoir parameters. Both vertical and horizontal well log data can be characterized as fractional Gaussian noise of similar fractal dimension. Fractal reservoir realizations are generated using Fourier transform techniques. This technique is straightforward and successfully maintains the desired spatial dependencies of the data. The fractal dimension is derived from the results of the statistical analysis. Some potential problems with this determination will be highlighted and discussed. Examples of simulation results demonstrate improved reservoir performance predictions using the fractal realizations. Reservoir sweep efficiencies are altered and a different recovery performance is obtained relative to linearly interpolated realizations. Introduction Improvements in reservoir characterization can yield major improvements in reservoir performance predictions. However it is usually not possible to completely describe fine scale reservoir variability. Statistical modeling is a way to incorporate heterogeneities consistent with observations into reservoir models. A number of approaches have been proposed to accomplish this goal. Stochastic modeling incorporates a variety of techniques to generate reservoir realizations (see Halderson for a good summary). For example, average bed thicknesses, widths, observed distributions, etc. can be used to produce Markov fields. Gathering and interpreting the data for input is normally quite labor intensive and requires a high skill level. This approach, if used properly, can provide good representations of the reservoir. Geostatistics provides a number of useful tools for modeling. Semi-variograms represent the variance as a function of lag spacing (separation between points). The assumption for semi-variograms, as usually applied in petroleum reservoir characterization, is of fields with no spatial correlation at long lag spacings. This can be a problem in the presence of long range cyclic structures. Improvements such as indicator kriging can more accurately represent the reservoir structure and variation. These geostatistical approaches also require a significant amount of data and effort. Fractal modeling can be used to include heterogeneities in a limited data environment. P. 803^