Simulation of the effect of stress-induced anisotropy on borehole compressional wave propagation

Formation elastic properties near a borehole may be altered from their original state due to the stress concentration around the borehole. This can lead to an incorrect estimation of formation elastic properties measured from sonic logs. Previous work has focused on estimating the elastic properties of the formation surrounding a borehole under anisotropic stress loading. We studied the effect of borehole stress concentration on sonic logging in a moderately consolidated Berea sandstone using a two-step approach. First, we used an iterative approach, which combines a rock-physics model and a finite-element method, to calculate the stress-dependent elastic properties of the rock around a borehole subjected to an anisotropic stress loading. Second, we used the anisotropic elastic model obtained from the first step and a finite-difference method to simulate the acoustic response of the borehole. Although we neglected the effects of rock failure and stress-induced crack opening, our modeling results provided important insights into the characteristics of borehole P-wave propagation when anisotropic in situ stresses are present. Our simulation results were consistent with the published laboratory measurements, which indicate that azimuthal variation of the P-wave velocity around a borehole subjected to uniaxial loading is not a simple cosine function. However, on field scale, the azimuthal variation in P-wave velocity might not be apparent at conventional logging frequencies. We found that the low-velocity region along the wellbore acts as an acoustic focusing zone that substantially enhances the P-wave amplitude, whereas the high-velocity region caused by the stress concentration near the borehole results in a significantly reduced P-wave amplitude. This results in strong azimuthal variation of P-wave amplitude, which may be used to infer the in situ stress state

Investigation of borehole cross-dipole flexural dispersion crossover through numerical modeling

Crossover of the dispersion of flexural waves recorded in borehole cross-dipole measurements is interpreted as an indicator of stress-induced anisotropy around a circular borehole in formations that are isotropic in the absence of stresses. We have investigated different factors that influence flexural wave dispersion. Through numerical modeling, we determined that for a circular borehole surrounded by an isotropic formation that is subjected to an anisotropic stress field, the dipole flexural dispersion crossover is detectable only when the formation is very compliant. This might happen only in the shallow subsurface or in zones having high pore pressure. However, we found that dipole dispersion crossover can also result from the combined effect of formation intrinsic anisotropy and borehole elongation. We found that a small elongation on the wellbore and very weak intrinsic anisotropy can result in a resolvable crossover in flexural dispersion that might be erroneously interpreted as borehole stress-induced anisotropy. A thorough and correct interpretation of flexural dispersion crossover thus has to take into account the effects of stress-induced and intrinsic anisotropy and borehole cross-sectional geometry

Investigation of the high-frequency wavefield of an off-center monopole acoustic logging-while-drilling tool

During logging-while-drilling (LWD) operations, complex drill string movements and the weight of the drill pipe often lead to a measurement tool that is not centralized. Therefore, studies of the response of an off-center acoustic LWD tool are essential to facilitate better interpretation of measurements made in an actual drilling environment. Such studies will be helpful for tool design and data processing. We used a finite-difference method to simulate the response of a noncentralized monopole acoustic LWD tool at high frequency (10 kHz). We analyzed the effects on the waveforms for receivers at different azimuths caused by an off-center tool with differing amounts of offset. We used velocity-time semblance and dispersion analysis methods to help us to understand the modes in the waveforms at different azimuth receivers for different tool offsets. We have found that the waveforms in the direction of the tool offset, that is, where the fluid column is smallest, were affected the most. Waveforms in the orthogonal direction were less affected by tool offset. Collar flexural and collar quadrupole modes appear when the tool is off center. In addition, the formation flexural and quadrupole modes contaminate the Stoneley wave. Waveforms in a fast formation are more strongly affected by the offset of the tool than those in a slow formation. In a fast formation, the new collar modes make it difficult to determine the P-wave velocity in the direction of tool offset whereas it is easier in the orthogonal direction. However, P-waves are less contaminated by new modes in a slow formation. Due to the significant changes in waveforms with azimuth when the tool is off center, the simple addition of all waveforms from an azimuthal distribution of receivers will not result in a clean waveform that is sensitive to only the surrounding formation.National Natural Science Foundation (China) (41174118)National Natural Science Foundation (China) (41404100)S&T Special Project (2011ZX05020-009)China Postdoctoral Science Foundation (2013M530106)International Postdoctoral Exchange Fellowship Progra

An Amplitude And Traveltime Calculation Using A Higher-Order Parabolic Equation

A higher-order parabolic equation is used to compute the traveltime (phase) and the amplitude in constant density acoustic media. This approach is in the frequency domain, thereby avoiding the high frequency approximation inherent in the Eikonal equation. Intrinsic attenuation can be naturally incorporated into the calculation. The error at large angles of propagation caused by the expansion of the square root operator can be virtually eliminated by adding more terms to the expansion. An efficient algorithm is obtained by applying the alternate direction method. Our results are in excellent agreement with the finite element approach for the range-dependent wedge-shaped benchmark problem. The amplitude and the phase are calculated for a syncline and the Marmousi models

Fracture clustering effect on amplitude variation with offset and azimuth analyses

Traditional amplitude variation with offset and azimuth (AVOAz) analysis for fracture characterization extracts fracture properties through analysis of reflection AVOAz to determine anisotropic parameters (e.g., Thomsen’s parameters) that are then related to fracture properties. The validity of this method relies on the basic assumption that a fractured unit can be viewed as an equivalent anisotropic medium. As a rule of thumb, this assumption is taken to be valid when the fracture spacing is less than λ/10. Under the effective medium assumption, diffractions from individual fractures destructively interfere and only specular reflections from boundaries of a fractured layer can be observed in seismic data. The effective medium theory has been widely used in fracture characterization, and its applicability has been validated through many field applications. However, through numerical simulations, we find that diffractions from fracture clusters can significantly distort the AVOAz signatures when a fracture system has irregular spacing even though the average fracture spacing is much smaller than a wavelength (e.g., ≪λ/10). Contamination by diffractions from irregularly spaced fractures on reflections can substantially bias the fracture properties estimated from AVOAz analysis and may possibly lead to incorrect estimates of fracture properties. Additionally, through Monte Carlo simulations, we find that fracture spacing uncertainty inverted from amplitude variation with offset (AVO) analysis can be up to 10%–20% when fractures are not uniformly distributed, which should be the realistic state of fractures present in the earth. Also, AVOAz and AVO analysis gives more reliable estimates of fracture properties when reflections at the top of the fractured layer are used compared with those from the bottom of the layer

Seismological investigation of the mechanical properties of a hot dry rock geothermal system

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Earth and Planetary Sciences, 1979.Microfiche copy available in Archives and Science.Vita.Bibliography: leaves 316-327.by Michael Fehler.Ph.D

3D Weak-Dispersion Reverse-Time Migration with a StereoModeling Method

The finite difference method has been widely used in seismic modeling and reverse time migration. However, it generally has two issues: large computational cost and numerical dispersion. Recently, a nearly-analytic discrete operator was developed to approximate the partial differential operators. Based on this spatial discretization, many weak-dispersion and efficient StereoModeling methods have been developed, which are found to be superior to conventional algorithms in suppressing numerical dispersion. In this paper, we generalize one StereoModeling method, the nearly-analytic central difference method (NACD), from 2D to 3D and apply it to 3D reverse-time migration. Numerical results show that the NACD can be used effectively as a new tool for seismic modeling and migration. The reverse time migration (RTM) results for the 3D SEG/EAGE Phase A classic dataset 1 show that the NACD can get a much better image than the Lax-Wendroff correction (LWC) method particularly when using a coarse grid size.Massachusetts Institute of Technology. Earth Resources Laboratory (Founding Members Consortium

A Bayesian framework for fracture characterization from surface seismic data

We describe a methodology for quantitatively characterizing the fractured nature of a hydrocarbon or geothermal reservoir from surface seismic data under a Bayesian inference framework. Fractures provide pathways for fluid flow in a reservoir, and hence, knowledge about a reservoir’s fractured nature can be used to enhance production of the reservoir. The fracture properties of interest in this study (to be inferred) are fracture orientation and excess compliance, where each of these properties are assumed to vary spatially over a 2D lateral grid which is assumed to represent the top of a reservoir. The Bayesian framework in which the inference problem is cast has the key benefits of (1) utilization of a prior model that allows geological information to be incorporated, (2) providing a straightforward means of incorporating all measurements (across the 2D spatial grid) into the estimates at each grid point, (3) allowing different types of measurements to be combined under a single inference procedure, and (4) providing a measure of uncertainty in the estimates. The observed data are taken from a 2D array of surface seismic receivers responding to an array of surface sources. Well understood features from the seismic traces are extracted and treated as the observed data, namely the P-wave reflection amplitude variation with acquisition azimuth (amplitude versus azimuth, or AvAz, data) and fracture transfer function (FTF) data. AvAz data are known to be more sensitive to fracture properties when the fracture spacing is significantly smaller than the seismic wavelength, whereas fracture transfer function data are more sensitive to fracture properties when the fracture spacing is on the order of the seismic wavelength. Combining these two measurements has the benefit of allowing inferences to be made about fracture properties over a larger range of fracture spacing than otherwise attainable. Geophysical forward models for the measurements are used to arrive at likelihood models for the data. The prior distribution for the hidden fracture variables is obtained by defining a Markov random field (MRF) over the lateral 2D grid where we wish to obtain fracture properties. The fracture variables are then inferred by application of loopy belief propagation (LBP) to yield approximations for the posterior marginal distributions of the fracture properties, as well as the maximum a posteriori (MAP) and Bayes least squares (BLS) estimates of these properties. Verification of the inference procedure is performed on a synthetic dataset, where the estimates obtained are shown to be at or near ground truth for a large range of fracture spacings

3D weak-dispersion reverse time migration using a stereo-modeling operator

Reliable 3D imaging is a required tool for developing models of complex geologic structures. Reverse time migration (RTM), as the most powerful depth imaging method, has become the preferred imaging tool because of its ability to handle complex velocity models including steeply dipping interfaces and large velocity contrasts. Finite-difference methods are among the most popular numerical approaches used for RTM. However, these methods often encounter a serious issue of numerical dispersion, which is typically suppressed by reducing the grid interval of the propagation model, resulting in large computation and memory requirements. In addition, even with small grid spacing, numerical anisotropy may degrade images or, worse, provide images that appear to be focused but position events incorrectly. Recently, stereo-operators have been developed to approximate the partial differential operator in space. These operators have been used to develop several weak-dispersion and efficient stereo-modeling methods that have been found to be superior to conventional algorithms in suppressing numerical dispersion and numerical anisotropy. We generalized one stereo-modeling method, fourth-order nearly analytic central difference (NACD), from 2D to 3D and applied it to 3D RTM. The RTM results for the 3D SEG/EAGE phase A classic data set 1 and the SEG Advanced Modeling project model demonstrated that, even when using a large grid size, the NACD method can handle very complex velocity models and produced better images than can be obtained using the fourth-order and eighth-order Lax-Wendroff correction (LWC) schemes. We also applied 3D NACD and fourth-order LWC to a field data set and illustrated significant improvements in terms of structure imaging, horizon/layer continuity and positioning. We also investigated numerical dispersion and found that not only does the NACD method have superior dispersion characteristics but also that the angular variation of dispersion is significantly less than for LWC. Read More: http://library.seg.org/doi/abs/10.1190/geo2013-0472.1National Natural Science Foundation (China) (Grant 41230210)Massachusetts Institute of Technology. Earth Resources Laboratory (Founding Members Consortium

Estimation of fracture compliance from tubewaves generated at a fracture intersecting a borehole

Understanding fracture compliance is important for characterizing fracture networks and for inferring fluid flow in the subsurface. In an attempt to estimate fracture compliance in the field, we developed a new model to understand tubewave generation at a fracture intersecting a borehole. Solving the dispersion relation in the fracture, amplitude ratios of generated tubewave to incident P-wave were studied over all frequency ranges. Based on the observations from the model, we propose that measuring amplitude ratios near a transition frequency can help constrain fracture compliance and aperture. The transition frequency corresponds to the regime where the viscous skin depth in the fracture is comparable to its aperture. However, measurements in the high frequency limit can place a lower bound on fracture compliance. Comparing the model to a previously published VSP dataset, we argue that compliance values of the order 10[superscript −10] −10[superscript −9] m/Pa may be possible in the field