The TBIE method and its applications To borehole acoustics in rocks with parallel fractures or tilted anisotropy Pei-cheng Xu Datatrends Research Corp. April 14, 2009

TBIE Transformed Boundary Integral Equations

Model I - Borehole in rocks with parallel fractures fluid Receiver Source fractures rock

Model II - Borehole in rocks with tilted anisotropy Axis of borehole Symmetry axis of anisotropy of surrounding medium Receiver fluid Source rock

Objectives Develop an analytical formulation to predict the full acoustic waves in a fluid-filled borehole surrounded by rocks with parallel fractures or tilted anisotropy. Implement robust numerical solution for this formulation. Study the effect of the fractures or tilted anisotropy on the borehole acoustic waves.

Borehole in exploration geophysics Technical background Borehole in exploration geophysics Borehole acoustics is used in exploration geophysics to estimate petrophysics parameters of rocks in the scale of a foot. Anisotropy of rock properties can be a result of vertical fractures (HTI) or laminated thin bedding (VTI) or both. Deviated boreholes are often drilled from offshore platforms and some in-land sites.

Horizontal, deviated and curved boreholes in oil and gas exploration fractures

acoustic tool Technical background monopole dipole Source-receiver offsets（m) （1）（2） （3）（4） （5） （6） 2.70 2.85 3.00 3.15 3.30 3.45 （7）（8）（9）（10）（11）（12) 3.60 3.75 3.90 4.05 4.20 4.35 Receivers borehole Dipole source frequency = 3000 Hz Source time function：Ricker wavelet fluid monopole dipole Source

Types of borehole waves Technical background Types of borehole waves GENERATION OF BOREHOLE SONIC WAVES z r q W– water wave P– head P wave S – head S wave receiver G – guided waves rock borehole W zr-zs G S P water source 2a

Types of borehole waves Technical background Types of borehole waves TYPICAL SEQUENCE OF BOREHOLE SONIC WAVES G P S W Pressure Time (ms) P – Head P wave S – Head S wave W– Water wave G – Guided waves (pseudo-Rayleigh, Stoneley, flexural)

Example of borehole full waveform due to a monopole VP=3.305 m/ms VS=1.969 m/ms Pseudo-Rayleigh Stoneley phase group P S Water

Special case: vertical borehole in VTI rock When the axis of borehole and axis of anisotropy symmetry coincide, and there no fractures, classic analytical solution is available in the form of wavenumber integrals. The wavenumber integrals have irregularly oscillatory integrands and infinite integration domains. They must be evaluated numerically. We have developed the Modified Clenshaw-Curtis (MCC) integration method to evaluate wavenumber integrals accurately and efficiently.

Wavenumber integrals Irregularly oscillatory Regularly oscillatory

The MCC integration method F(kr, z) is fitted by Chebyshev polynomials in each interval. An infinite interval is transformed to finite through change of variable. Then the integration is carried out exactly or asymptotically with desired accuracy in each interval. When subdividing the interval or doubling the order of polynomials, no previous sampling is wasted. The fitting is independent of x (2D case) or r (3D case). This method is most efficient when involving a large number of different x or r.

Existing approaches to the boundary value problems of Models I and II The Finite Difference method (Leslie and Randall, 1991; Sinha et al. , 2006). The Variational method (Ellefsen et al., 1991) The Perturbation method (Sinha et al. ,1994). The conventional Boundary Integral Equations (BIE) method (Bouchon, 1993)

The conventional BIE method The original 3D problem becomes a 2D problem on the cylindrical surface. The coefficients in the boundary integrals involve fundamental solutions in the full spaces of the solid and fluid. The fundamental solutions (Green’s functions and associated stresses) in the solid are wavenumber integrals (I3D) when the rock is layered or anisotropic. Has difficulty handling the infinity in z.

Boundary conditions

The conventional BIE for borehole acoustics fundamental solutions unknowns

Integral transform of BIE: from z to kz original unknowns original known coefficients

Transformed BIE in Cartesian coordinates

Transformed BIE in cylindrical coordinates

Angular phase transform Transformed BIE in matrix form

Summary of the TBIE approach Set up conventional BIE: reducing the domain of unknowns from 3D full space to the cylindrical surface. From BIE to TBIE: replacing z by kz; reducing cylindrical surface to a line circle. Replace Cartesian (x,y,z) by cylindrical (r,q,z). From TBIE to linear system of equations. Solve TBIE for unknown nodal displacements and pressure on the line circle. Obtain displacements and pressure at any field location from the displacements and pressure on the line circle through direct evaluation of boundary integrals. Take inverse integral transform of the above result: from kz back to z.

The triple-fold infinite integration

Model I geometry in the kz domain water q water x d fracture reduced borehole rock

Symmetry axis of the fractured rock Borehole in HTI formation Symmetry axis of the borehole z x Symmetry axis of the fractured rock

Effect of a fracture on borehole waves Borehole and fracture form a composite waveguide. Fracture causes wave anisotropy. Distinguish a fracture from anisotropy: dual flexural waves and leaky fracture mode. Dual flexural waves - channel flexural wave followed by borehole flexural wave in the waveform. Leaky fracture mode - sharp dip in the spectrum Effects of fracture aperture, orientation and distance are as expected.

Effects of a fracture 0.5 cm 0.5 cm Uniform Fractured Fractured

Effects of a fracture 0.5 cm 0.3 m 0.5 cm 0.3 m Uniform Fractured

Effects of a fracture 1 cm Uniform Fractured Fractured Uniform 1 cm

Effects of a fracture Flexural wave azimuthally anisotropic z=4.35 m d = 0 m h=0.5 cm d d = 0.2 m d = 0.3 m d = 0.5 m d = 2 m 5 % 10 % azimuthally anisotropic 15 % 20 %

Effects of a fracture Flexural wave azimuthally anisotropic ISO z=4.35 m d = 0 h=0.5 cm d = 0.2 m d d = 0.3 m d = 0.5 m d = 2 m 5 % 10 % azimuthally anisotropic 15 % 20 %

Model II geometry in the kz domain Top View (against z-axis) x’ q x water rock reduced borehole x’ z’ z Side View (along y-axis) rock formation x reduced borehole

Transformation between coordinate systems: the borehole and the rock x3 x2 x’3 x’2 Rock x1 x’1

Borehole in rocks with tilted anisotropy Symmetry axis of the borehole z z Symmetry axis of the rock x x

Oblique body waves in TI media Technical background Oblique body waves in TI media

Study of the effect of tilted anisotropy On borehole waves Amplitude spectrum: magnitude and shape change gradually with increased tilted angle. Waveforms: arrivals of events shift gradually with increased tilted angle. Azimuthal anisotropy reaches maximum at f=90o and reduces to none at f=0o.

Dipole spectra at different tilted angles (q=0o-0o) Effects of tilted anisotropy Dipole spectra at different tilted angles (q=0o-0o) f=0o f=10o f=80o f=90o

Dipole spectra at different tilted angles (q=90o-90o) Effects of tilted anisotropy Dipole spectra at different tilted angles (q=90o-90o) f=0o f=10o f=80o f=90o

Dipole spectra at different tilted angles Effects of tilted anisotropy Dipole spectra at different tilted angles (q=90o-90o vs 0o-0o) f=0o f=10o f=80o f=90o

Dipole amplitude spectra at different tilted angle Effects of tilted anisotropy Dipole amplitude spectra at different tilted angle q=0o-0o

at the fast and slow principal azimuths Effects of tilted anisotropy Dipole waveforms at the fast and slow principal azimuths f=90o

Dipole waveforms at different tilted angle Effects of tilted anisotropy Dipole waveforms at different tilted angle q=0o-0o

Conclusions The Integral transform successfully overcomes the numerical difficulty of other methods in dealing with the infinitely long borehole. The MCC method is ideal for handling the three-fold infinite, irregularly oscillatory integrals involved in the TBIE approach. The TBIE method enables us to study the effects of a vertical fracture on the borehole waves, which no other researchers have been able to do. The TBIE method enables us to produce synthetic borehole waves in tilted anisotropic rocks more accurately and efficiently than other methods.

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