Methods for determining transversely isotropic-elastic constants from borehole sonic velocities in strongly transversely-isotropic formations

1. A method for estimating five transversely-isotropic (TI)-elastic constants using borehole sonic data obtained from at least one subterranean borehole in a transversely isotropic formation, wherein the borehole sonic data is acquired by a borehole sonic tool, the borehole sonic tool comprising a source with a bandwidth in a range 0.2 to 20 kHz, the method comprising:

solving for a quasi-compressional qP-wave velocity V qP (θ) using inversion algorithms based on exact solutions of the Kelvin-Christoffel equations for plane wave velocities in arbitrarily anisotropic formations, wherein the five TI-elastic constants comprise C 11 , C 13 , C 33 , C 55 , and C 66 , wherein the at least one subterranean borehole comprises a horizontal borehole parallel to an X 1 -axis that is perpendicular to a TI-symmetry X 3 -axis and a deviated borehole that makes an angle θ with the TI-symmetry X 3 -axis, and wherein the method further comprises:

recording monopole and cross-dipole waveforms at an array of receivers in the horizontal borehole;

processing monopole refracted headwaves to output compressional modulus C 11 ;

processing fast-dipole waveforms to estimate shear modulus C 66 from a shear wave velocity V SH ;

processing slow-dipole waveforms to estimate shear modulus C 55 from a shear wave velocity V SV ;

solving for the C 33 using the quasi-compressional wave velocity V qP (θ) and quasi-shear wave velocity V qSV (θ) from the deviated borehole;

verifying the estimates of C 55 and C 66 against a measured shear wave velocity V SH (θ); and

solving for C 13; and

outputting the five TI-elastic constants for use in hydraulic fracture design; and

fracturing a formation according to the hydraulic fracture design.

2. The method of claim 1 , wherein the at least one subterranean borehole further comprises a vertical borehole parallel to the TI-symmetry X 3 -axis and the deviated borehole that makes the angle θ with the TI-symmetry X 3 -axis, and wherein the method further comprises:

recording monopole and cross-dipole waveforms at an array of receivers in the vertical borehole;

processing monopole refracted headwaves to output compressional modulus C 33 ;

processing fast-dipole waveforms to estimate shear modulus C 55 from shear wave velocity V SH associated with the deviated borehole;

processing slow-dipole waveforms to estimate shear modulus C 44 from shear wave velocity V SV ;

solving for C 11 using quasi-compressional wave velocity V qP (θ) and quasi-shear wave velocity V qSV (θ) from the deviated borehole;

solving for C 66 using V SH (θ) from the deviated borehole; and

solving for C 13 .

3. The method of claim 1 , wherein the at least one subterranean borehole further comprises a first deviated borehole and a second deviated borehole that make angles θ 1 and θ 2 , respectively, with the TI-symmetry X 3 -axis, and wherein the method further comprises:

recording monopole and cross-dipole waveforms at an array of receivers in the first and second deviated boreholes;

processing the monopole refracted headwaves to output quasi-compressional wave velocities V qP (θ 1 ) and V qP (θ 2 );

processing one or more dipole waveforms associated with the cross-dipole waveforms to estimate shear wave velocities V SH (θ 1 ) and V SH (θ 2 );

processing the one or more dipole waveforms to estimate quasi-shear wave velocities V qSV (θ 1 ) and V qSV (θ 2 );

solving for the C 66 and C 44 using additional shear wave velocities V SH (θ 1 ) and V SH (θ 2 ) from the first and second deviated boreholes, respectively;

solving for the C 33 and the C 11 ; and

solving for the C 13 .

4. The method of claim 1 , wherein the at least one subterranean borehole further comprises a vertical borehole parallel to the TI-symmetry X 3 -axis, the horizontal borehole parallel to an X 1 -axis and perpendicular to the TI-symmetry X 3 -axis, and the deviated borehole that makes the angle θ with the TI-symmetry X 3 -axis, and wherein the method further comprises:

recording monopole and cross-dipole waveforms at an array of receivers in the vertical, horizontal, and deviated boreholes;

calculating the C 33 and C 44 using first additional quasi-compressional VP, first additional quasi-shear V qSV , and first additional shear V SH wave velocities from the vertical borehole;

estimating the C 11 , the C 55 , and the C 66 using second additional quasi-compressional V q , second additional quasi-shear V qSV , and second additional shear V SH wave velocities, respectively, from the horizontal borehole; and

estimating the C 13 from at least one of a group consisting of: compressional V qP and quasi-shear V qSV wave velocities along the deviated borehole.

5. The method of claim 1 , further comprising:

recording monopole and cross-dipole waveforms at an array of receivers in the horizontal borehole;

processing the monopole refracted headwaves to output monopole compressional modulus C 11 ;

processing fast-dipole waveforms to estimate shear modulus C 66 ;

processing slow-dipole waveforms to estimate shear modulus C 55 ;

solving three travel-time equations for C 33 , C 13 , and reflector distance D, using as inputs C 11 , C 55 , C 66 from the borehole sonic data from the horizontal borehole, wherein reflection data analysis provides estimates of arrival times t 1 , t 2 , t 3 ; sub-array offsets z 1 , z 2 , and z 3 ; and ray-path directions θ 1 , θ 2 , and θ 3 ; and wherein the method further comprises:

outputting the five TI-elastic constants and the reflector distance D based on an exact solution for the quasi-compressional wave velocity V qP (θ).

6. The method of claim 1 , wherein the at least one subterranean borehole comprises a horizontal borehole parallel to the X 1 -axis that is perpendicular to the TI-symmetry X 3 -axis, and wherein the method further comprises:

recording monopole and cross-dipole waveforms at an array of receivers in the horizontal borehole;

processing monopole refracted headwaves to output monopole compressional modulus C 11 ;

processing fast-dipole waveforms to estimate shear modulus C 66 ;

processing the slow-dipole waveforms to estimate the shear modulus C 55 ;

recording the first quasi-compressional refracted waveforms from a proximate stiff layer at an array of receivers in the horizontal borehole;

processing a proximate stiff layer quasi-compressional arrival using a Slowness-Time Coherence (STC) algorithm to output an arrival time t 2 ;

determining an offset time t 1 of a proximate stiff layer arrival from a T-R line from the STC algorithm;

estimating at least one reflector distance D of a proximate stiff layer reflector from a surface of the borehole;

for each reflector distance D, estimating a sequence of V qP (θ) as a function of the angle θ based on T-R spacings;

plotting a velocity corresponding to each T-R pair as a function of the angle θ;

solving for Thomsen parameters ε and δ and the modulus C 13 using quasi-compressional wave velocity V qP (θ=45 degrees); and

outputting the five TI-elastic constants and reflector distance D based on a weak anisotropy assumption for the quasi-compressional wave velocity V qP (θ), wherein the proximate stiff layer is stiffer than a formation between the proximate stiff layer and the horizontal borehole.

7. The method of claim 6 , wherein the proximate stiff layer comprises limestone.

8. The method of claim 1 , wherein the five TI-elastic constants are determined with a resolution of a borehole sonic tool in an axial direction of one of the at least one subterranean boreholes.

9. The method of claim 1 , further comprising:

measuring a true dip of the transversely-isotropic formation; and

determining a relative dip of the transversely-isotropic formation with respect to an axial direction of the horizontal borehole.

10. A method for estimating five transversely-isotropic (TI)-elastic constants using borehole sonic data obtained from a horizontal borehole, wherein the borehole sonic data is acquired by a borehole sonic tool, the borehole sonic tool comprising a source with a bandwidth in a range 0.2 to 20 kHz, in a transversely isotropic formation, the method comprising:

recording monopole and cross-dipole waveforms at an array of receivers in the horizontal borehole parallel to an X 1 -axis that is perpendicular to a TI-symmetry X 3 -axis;

processing monopole refracted headwaves to output monopole compressional modulus C 11 ;

processing fast-dipole waveforms to estimate shear modulus C 66 ;

processing slow-dipole waveforms to estimate shear modulus C 55 ;

solving three travel-time equations for C 33 , C 13 , and reflector distance D, using as inputs C 11 , C 55 , C 66 from the borehole sonic data from the horizontal borehole, wherein reflection data analysis provides estimates of arrival times t 1 , t 2 , t 3 ; sub-array offsets z 1 , z 2 , and z 3 ; and ray-path directions θ 1 , θ 2 , and θ 3 , wherein the horizontal borehole is parallel to an X 1 -axis that is perpendicular to a TI-symmetry X 3 -axis, and wherein the method further comprises:

outputting the five TI-elastic constants and the reflector distance D based on an anisotropy assumption for a quasi-compressional wave velocity V qP (θ) for use in hydraulic fracture design; and fracturing a formation according to the hydraulic fracture design.

11. A system for locating hydrocarbons in a transversely-isotropic (TI) formation, comprising:

a borehole sonic tool that measures borehole sonic data along at least one subterranean borehole in the formation wherein the borehole sonic tool comprises a source with a bandwidth in a range 0.2 to 20 kHz;

a processor that processes the borehole sonic data to determine five TI-elastic constants as a function of position along the at least one subterranean borehole;

an output device that outputs the five TI-elastic constants as the function of position along the at least one subterranean borehole, wherein determining the five TI-elastic constants comprises:

solving for a quasi-compressional qP-wave velocity V qP using inversion algorithms based on exact solutions of Kelvin-Christoffel equations for plane wave velocities in arbitrarily anisotropic formations, and wherein the five TI-elastic constants comprise C 11 , C 13 , C 33 , C 55 , and C 66 ; and

outputting the five TI-elastic constants for use in hydraulic fracture design; and fracturing a formation according to the hydraulic fracture design.

12. The system of claim 11 , wherein the at least one subterranean borehole comprises a vertical borehole parallel to a TI-symmetry X3-axis and a deviated borehole that makes an angle θ with the TI-symmetry X 3 -axis, and wherein determining the five TI-elastic constants further comprises:

recording monopole and cross-dipole waveforms at an array of receivers in the vertical borehole;

processing monopole refracted headwaves to output compressional modulus C 33 ;

processing fast-dipole waveforms to estimate shear modulus C 55 from a shear wave velocity V SH ;

processing slow-dipole waveforms to estimate shear modulus C 44 from a shear wave velocity V SV ;

solving for C 11 using the quasi-compressional qP-wave velocity V qP (θ) and a quasi-shear wave velocity V qSV (θ) from the deviated borehole;

solving for C 66 using a measured shear wave velocity V SH (θ) from the deviated borehole; and

solving for C 13 .

13. The system of claim 11 , wherein the at least one subterranean borehole comprises a horizontal borehole parallel to an X 1 -axis that is perpendicular to the TI-symmetry X 3 -axis and a deviated borehole that makes an angle θ with the TI-symmetry X 3 -axis, and wherein determining the five TI-elastic constants further comprises:

recording monopole and cross-dipole waveforms at an array of receivers in the horizontal borehole;

processing monopole refracted headwaves to output compressional modulus C 11 ;

processing fast-dipole waveforms to estimate shear modulus C 66 from a shear wave velocity V SH ;

processing slow-dipole waveforms to estimate shear modulus C 55 from a shear wave velocity V SV ;

solving for C 33 using the quasi-compressional qP-wave velocity V qP (0) and a quasi-shear wave velocity V qSV (θ) from the deviated borehole;

checking to verify estimates of C 55 and C 66 against a measured shear wave velocity V SH (θ); and

solving for C 13 .

14. The system of claim 11 , wherein the at least one subterranean borehole comprises a first and a second deviated borehole that make angles θ 1 and θ 2 , respectively, with a TI-symmetry X 3 -axis, and wherein determining the five TI-elastic constants further comprises:

recording monopole and cross-dipole waveforms at an array of receivers in the first deviated borehole and second deviated borehole;

processing monopole refracted headwaves to output quasi-compressional wave velocities V qP (θ 1 ) and V qP (θ 2 );

processing dipole waveforms to estimate shear wave velocities V SH (θ 1 ) and V SH (θ 2 );

processing the dipole waveforms to estimate quasi-shear wave velocities V qSV (θ 1 ) and V qSV (θ 2 );

solving for C 66 and C 44 using the shear wave velocities V SH (θ 1 ) and V SH (θ 2 ) from the first and second deviated boreholes, respectively;

solving for C 33 and C 11 ; and

solving for C 13 .

15. The system of claim 11 , wherein the at least one subterranean borehole comprises a vertical borehole parallel to a TI-symmetry X 3 -axis, a horizontal borehole parallel to an X 1 -axis and perpendicular to the TI-symmetry X 3 -axis, and a deviated borehole that makes an angle θ with the TI-symmetry X 3 -axis, and wherein determining the five TI-elastic constants further comprises:

recording monopole and cross-dipole waveforms at an array of receivers in the vertical, horizontal, and deviated boreholes;

calculating C 33 and C 44 using compressional V P , slow shear V SV , and fast shear V SH wave velocities from the vertical borehole;

estimating C 11 , C 55 , and C 66 using compressional V P , slow-shear V SV , and fast-shear V SH wave velocities, respectively, from the horizontal borehole; andYes

estimating C13 from at least one of a group consisting of: quasi-compressional V qP (θ) and quasi-shear V qSV (θ) wave velocities along the deviated borehole.

16. The system of claim 11 , wherein the at least one subterranean borehole comprises a horizontal borehole parallel to an X 1 -axis that is perpendicular to a TI-symmetry X 3 -axis, and wherein determining the five TI-elastic constants further comprises:

recording monopole and cross-dipole waveforms at an array of receivers in the horizontal borehole;

processing monopole refracted headwaves to output monopole quasi-compressional modulus C 11 ;

processing fast-dipole waveforms to estimate shear modulus C 66 ;

processing slow-dipole waveforms to estimate shear modulus C 55 ;

solving three travel-time equations for C 33 , C 13 , and reflector distance D, using as inputs C 11 , C 55 , C 66 from the borehole sonic data from the horizontal borehole, wherein reflection data analysis provides estimates of arrival times t 1 , t 2 , t 3 ; sub-array offsets z 1 , z 2 , and z 3 , and ray-path directions θ 1 , θ 2 , and θ 3 ; and

outputting the five TI-elastic constants and the reflector distance D based on an exact solution for the quasi-compressional wave velocity V qP (0).

17. The system of claim 11 , wherein the at least one subterranean borehole comprises a horizontal borehole parallel to an X 1 -axis that is perpendicular to a TI-symmetry X 3 -axis, and wherein determining the five TI-elastic constants further comprises:

recording monopole and cross-dipole waveforms at an array of receivers in the horizontal borehole;

processing monopole refracted headwaves to output monopole compressional modulus C 11 ;

processing fast-dipole waveforms to estimate shear modulus C 66 ;

processing slow-dipole waveforms to estimate shear modulus C 55 ;

recording first quasi-compressional refracted waveforms from a proximate stiff layer at an array of receivers in the horizontal borehole;

processing a proximate stiff layer quasi-compressional arrival using a Slowness-Time Coherence (STC) algorithm to output an arrival time t 2 ;

determining an offset time t 1 of a proximate stiff layer arrival from a T-R line from the STC algorithm;

estimating at least one reflector distance D of a proximate stiff layer reflector from a surface of the horizontal borehole;

for each reflector distance D, estimating a sequence of V qP (θ) as a function of the angle θ based on T-R spacings;

plotting a velocity corresponding to each T-R pair as a function of an angle θ;

solving for Thomsen parameters ε and δ and modulus C 13 using quasi-compressional wave velocity V qP (θ=45 degrees); and

outputting the five TI-elastic constants and the at least one reflector distance D based on an anisotropy assumption for the quasi-compressional wave velocity V qP (θ), wherein the proximate stiff layer is stiffer than a formation between the proximate stiff layer and the horizontal borehole.