Elastic-wave evaluation of downhole hydraulic fracturing: Modeling and field applications

We numerically simulate elastic-wave propagation along a fluid-filled borehole with a hydraulically fractured formation. The numerical model is based on the results of hydraulic fracturing on laboratory specimens. Two typical models are simulated: a main fracture crossing the borehole and a fracture network extending from the borehole. In addition, both models contain small, secondary fractures surrounding the borehole. Our result indicates that wave propagation in the main-fracture model is characterized by significant S-wave anisotropy for polarization along and normal to the fracture orientation, with the magnitude of anisotropy depending on the fracture aperture and filling material. In contrast, no significant anisotropy is observed for the fracture network model. In both models, wave propagation is significantly affected by small-fracture-induced near-borehole velocity variation. Our modeling results provide a theoretical foundation for evaluating hydraulic fracturing using the borehole acoustic logging. The hydraulic fracture-induced S-wave anisotropy can be evaluated with the cross-dipole S-wave logging, and the fracturing-induced velocity change can be detected by acoustic traveltime tomography. We used field data examples to demonstrate the effectiveness and practicality of using the borehole acoustic techniques for hydraulic fracturing evaluation.

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Temporal high-order staggered-grid finite-difference schemes for elastic wave propagation

Geophysics ◽

10.1190/geo2017-0005.1 ◽

2017 ◽

Vol 82 (5) ◽

pp. T207-T224 ◽

Cited By ~ 15

Author(s):

Zhiming Ren ◽

Zhen Chun Li

Keyword(s):

Wave Propagation ◽

Wave Equation ◽

Finite Difference ◽

Elastic Wave ◽

High Order ◽

Elastic Wave Propagation ◽

Staggered Grid ◽

S Wave ◽

Order Accuracy ◽

Spatial Derivatives

The traditional high-order finite-difference (FD) methods approximate the spatial derivatives to arbitrary even-order accuracy, whereas the time discretization is still of second-order accuracy. Temporal high-order FD methods can improve the accuracy in time greatly. However, the present methods are designed mainly based on the acoustic wave equation instead of elastic approximation. We have developed two temporal high-order staggered-grid FD (SFD) schemes for modeling elastic wave propagation. A new stencil containing the points on the axis and a few off-axial points is introduced to approximate the spatial derivatives. We derive the dispersion relations of the elastic wave equation based on the new stencil, and we estimate FD coefficients by the Taylor series expansion (TE). The TE-based scheme can achieve ([Formula: see text])th-order spatial and ([Formula: see text])th-order temporal accuracy ([Formula: see text]). We further optimize the coefficients of FD operators using a combination of TE and least squares (LS). The FD coefficients at the off-axial and axial points are computed by TE and LS, respectively. To obtain accurate P-, S-, and converted waves, we extend the wavefield decomposition into the temporal high-order SFD schemes. In our modeling, P- and S-wave separation is implemented and P- and S-wavefields are propagated by P- and S-wave dispersion-relation-based FD operators, respectively. We compare our schemes with the conventional SFD method. Numerical examples demonstrate that our TE-based and TE + LS-based schemes have greater accuracy in time and better stability than the conventional method. Moreover, the TE + LS-based scheme is superior to the TE-based scheme in suppressing the spatial dispersion. Owing to the high accuracy in the time and space domains, our new SFD schemes allow for larger time steps and shorter operator lengths, which can improve the computational efficiency.

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Anisotropic finite-difference algorithm for modeling elastic wave propagation in fractured coalbeds

Geophysics ◽

10.1190/geo2010-0240.1 ◽

2012 ◽

Vol 77 (1) ◽

pp. C13-C26 ◽

Cited By ~ 12

Author(s):

Zhenglin Pei ◽

Li-Yun Fu ◽

Weijia Sun ◽

Tao Jiang ◽

Binzhong Zhou

Keyword(s):

Wave Propagation ◽

Finite Difference ◽

Elastic Wave ◽

Taylor Series Expansion ◽

Real Data ◽

Elastic Wave Propagation ◽

Numerical Dispersion ◽

Staggered Grid ◽

S Wave ◽

Data Set

The simulation of wave propagations in coalbeds is challenged by two major issues: (1) strong anisotropy resulting from high-density cracks/fractures in coalbeds and (2) numerical dispersion resulting from high-frequency content (the dominant frequency can be higher than 100 Hz). We present a staggered-grid high-order finite-difference (FD) method with arbitrary even-order ([Formula: see text]) accuracy to overcome the two difficulties stated above. First, we derive the formulae based on the standard Taylor series expansion but given in a neat and explicit form. We also provide an alternative way to calculate the FD coefficients. The detailed implementations are shown and the stability condition for anisotropic FD modeling is examined by the eigenvalue analysis method. Then, we apply the staggered-grid FD method to 2D and 3D coalbed models with dry and water-saturated fractures to study the characteristics of the 2D/3C elastic wave propagation in anisotropic media. Several factors, like density and direction of vertical cracks, are investigated. Several phenomena, like S-wave splitting and waveguides, are observed and are consistent with those observed in a real data set. Numerical results show that our formulae can correlate the amplitude and traveltime anisotropies with the coal seam fractures.

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Approximation of pure acoustic seismic wave propagation in TTI media

Geophysics ◽

10.1190/geo2011-0092.1 ◽

2011 ◽

Vol 76 (5) ◽

pp. WB97-WB107 ◽

Cited By ~ 28

Author(s):

Chunlei Chu ◽

Brian K. Macy ◽

Phil D. Anno

Keyword(s):

Wave Propagation ◽

Wave Equation ◽

Elastic Wave ◽

Wave Equations ◽

Computational Cost ◽

P Wave ◽

Anisotropy Axis ◽

S Wave ◽

Time Migration ◽

Tti Media

Pseudoacoustic anisotropic wave equations are simplified elastic wave equations obtained by setting the S-wave velocity to zero along the anisotropy axis of symmetry. These pseudoacoustic wave equations greatly reduce the computational cost of modeling and imaging compared to the full elastic wave equation while preserving P-wave kinematics very well. For this reason, they are widely used in reverse time migration (RTM) to account for anisotropic effects. One fundamental shortcoming of this pseudoacoustic approximation is that it only prevents S-wave propagation along the symmetry axis and not in other directions. This problem leads to the presence of unwanted S-waves in P-wave simulation results and brings artifacts into P-wave RTM images. More significantly, the pseudoacoustic wave equations become unstable for anisotropy parameters [Formula: see text] and for heterogeneous models with highly varying dip and azimuth angles in tilted transversely isotropic (TTI) media. Pure acoustic anisotropic wave equations completely decouple the P-wave response from the elastic wavefield and naturally solve all the above-mentioned problems of the pseudoacoustic wave equations without significantly increasing the computational cost. In this work, we propose new pure acoustic TTI wave equations and compare them with the conventional coupled pseudoacoustic wave equations. Our equations can be directly solved using either the finite-difference method or the pseudospectral method. We give two approaches to derive these equations. One employs Taylor series expansion to approximate the pseudodifferential operator in the decoupled P-wave equation, and the other uses isotropic and elliptically anisotropic dispersion relations to reduce the temporal frequency order of the P-SV dispersion equation. We use several numerical examples to demonstrate that the newly derived pure acoustic wave equations produce highly accurate P-wave results, very close to results produced by coupled pseudoacoustic wave equations, but completely free from S-wave artifacts and instabilities.

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Experimental study on the effects of fractures on elastic wave propagation in synthetic layered rocks

Geophysics ◽

10.1190/geo2015-0661.1 ◽

2016 ◽

Vol 81 (4) ◽

pp. D441-D451 ◽

Cited By ~ 9

Author(s):

Tianyang Li ◽

Ruihe Wang ◽

Zizhen Wang ◽

Yuzhong Wang

Keyword(s):

Wave Propagation ◽

Fractal Dimension ◽

Power Spectrum ◽

Elastic Wave ◽

Power Law ◽

Layered Media ◽

Physical Models ◽

P Wave ◽

Fracture Density ◽

S Wave

Fractures greatly increase the difficulty of oil and gas exploration and development in reservoirs consisting of interlayered carbonates and shales and increase the uncertainty of highly efficient development. The presence of fractures or layered media is also widely known to affect the elastic properties of rocks. The combined effects of fractures and layered media are still unknown. We have investigated the effects of fracture structure on wave propagation in interlayered carbonate and shale rocks using physical models based on wave theory and the similarity principle. We have designed and built two sets of layered physical models with randomly embedded predesigned vertically aligned fractures according to the control variate principle. We have measured the P- and S-wave velocities and attenuation and analyzed the effects of fracture porosity and aspect ratio (AR) on velocity, attenuation, and power spectral dimension of the P- and S-waves. The experimental results indicated that under conditions of low porosity ([Formula: see text]), Han’s empirical velocity-porosity relations and Wang’s attenuation-porosity relation combined with Wyllie’s time-average model are a good prediction for layered physical models with randomly embedded fractures. When the porosity is constant, the effect of different ARs on elastic wave properties can be described by a power law function. We have calculated the power spectrum fractal dimension [Formula: see text] of the transmitted signal in the frequency domain, which can supplement the S-wave splitting method for estimating the degree of anisotropy. The simple power law relation between the power spectrum fractal dimension of the P-waveform and fracture density suggests the possible use of P-waves for discriminating fracture density. The high precision and low error of this processing method give new ideas for rock anisotropy evaluation and fracture density prediction when only P-wave data are available.

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Compliance estimation and multiscale seismic simulation of hydraulic fractures

Interpretation ◽

10.1190/int-2017-0218.1 ◽

2018 ◽

Vol 6 (4) ◽

pp. T951-T965 ◽

Cited By ~ 3

Author(s):

Edith Sotelo ◽

Yongchae Cho ◽

Richard L. Gibson Jr.

Keyword(s):

Wave Propagation ◽

Hydraulic Fracturing ◽

Fracture Network ◽

Production Performance ◽

Sensitivity Analyses ◽

Hydraulic Fractures ◽

Hydrocarbon Production ◽

Fine Mesh ◽

Multiscale Finite Element ◽

Seismic Wavefield

Hydraulic fracturing is a common stimulation technique in unconventional reservoirs to create fractures systems and allow hydrocarbon production. Proppant (granular material) is normally injected during hydraulic fracturing to keep open the fracture network and enhance hydrocarbon production performance. Proppant has a strong influence on fracture compliance and therefore will affect the characteristics of the generated seismic wavefield. To account for the effect of proppant in fracture compliance, we have developed new analytical formulations to obtain normal and tangential compliance for the case of dry and fluid-saturated fractures. We derive these expressions based on Hertz-Mindlin contact theory. Results from the compliance sensitivity analyses provide insights into the effects of proppant distribution and mechanical properties on fracture compliance. We also applied the innovative generalized multiscale finite-element method (GMsFEM) to simulate wave propagation through discrete hydraulic fractures filled with proppant. The GMsFEM approach represents individual fractures on a finely discretized mesh; this fine mesh is used to capture fracture properties by generating quantities (basis functions) that are used for modeling wave propagation on a much coarser grid. This methodology reduces the size of the computational problem, allowing faster results. Simulation results indicate the changes of the scattered wavefield as the proppant placement varies in different parts of the fractures and as the number of fracture stages increases.

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