WO2014000815A1

WO2014000815A1 - Anisotropy estimation

Info

Publication number: WO2014000815A1
Application number: PCT/EP2012/062726
Authority: WO
Inventors: Kenneth Duffaut; Ole Petter DYBVIK; Lasse RENLI; Anders DRÆGE
Original assignee: Statoil Petroleum ASA
Current assignee: Equinor Energy AS
Priority date: 2012-06-29
Filing date: 2012-06-29
Publication date: 2014-01-03
Anticipated expiration: 2014-12-29

Description

Anisotropy Estimation

TECHNICAL FI ELD

The invention relates to the field of estimating elastic anisotropy, and in particular to method of estimating an anisotropy parameter in a geological subsurface.

BACKGROUND

When modelling the properties of a geological subsurface, the anisotropy of the subsurface must be taken into account. Some subsurfaces are relatively isotropic, but other subsurfaces, such as those composed of shale, display anisotropic properties. Failure to take into account the anisotropy of the subsurface can lead to errors and misinterpretation of modelled properties of the geological subsurface.

There are different ways to characterize anisotropy. In the field of geophysics, anisotropy is commonly characterized using Thomsen parameters, as described in Thomsen, L., 1986, Weak elastic anisotropy: Geophysics, 51 , 1954-1966. Thomsen parameters are dimensionless ratios of components of the elastic stiffness tensor. The Thomsen parameters are:

C_u - C 33

ε = (Eq. 1 )

Cij is an elastic stiffness tensor that characterizes the elasticity of the medium. The Thomsen parameters typically have an absolute value less than 1 for subsurface rocks.

For some lithologies, such as shale, the degree of anisotropy is directly affected by the porosity. This is because, as a shale formation is compacted, porosity decreases and platelets align preferentially, leading to more pronounced anisotropy. Older shale formations therefore tend to exhibit a higher degree of anisotropy than young shale formations. The following equations (Wang, Z., 2002, Seismic anisotropy in sedimentary rocks, part 2: Laboratory data: Geophysics 67, 1423-1440) illustrate how Thomsen parameters can be derived using porosity:

z = Ae^~Bi> (Eq. 4) y = Ae^~B (Eq. 5)

Φ represents porosity, and A and B are different for ε and γ. A and B are derived empirically using regression (for example, linear regression) on core plug measurements. Using this model it is possible to predict ε and γ only using porosity. While this model requires only porosity as input to find ε and γ, it is intended to be used for pure shale formations, and the modelled anisotropy has a high degree of uncertainty. Furthermore, there may be circumstances where porosity data for the subsurface is not readily obtainable or available, in which case Equations 4 and 5 can not be used.

Bandyopadhyay, K., 2009, "Seismic Anisotropy: Geological Causes and its Implications to Reservoir Geophysics", PhD Theses; Stanford University describes various approximations and empirical relations for considering anisotropy.

SUMMARY

It is an object of the invention to estimate elastic anisotropy parameters even where porosity data is not available. According to a first aspect, there is provided a method of estimating an elastic anisotropic parameter of a rock physics model for a geological subsurface. P-wave and S-wave velocity values are obtained for the subsurface. A ratio of P-wave velocity and S-wave velocity is used to derive a first elastic stiffness tensor ratio. The first elastic stiffness tensor ratio and at least one empirically derived constant is used to determine a second elastic stiffness tensor ratio. The elastic anisotropic parameter is then derived using the first and second elastic stiffness tensor ratios.

In an optional embodiment, the anisotropic parameter is a Thomsen parameter. If an S-wave velocity value is not otherwise available, it may optionally be derived using the P-wave velocity and at least an empirically derived constant.

As an option, the first elastic stiffness tensor ratio is C33/C44, and the first elastic stiffness tensor ratio is derived using the square of the ratio of the P-wave velocity to the S-wave velocity. As a further option, the second elastic stiffness tensor ratio is any of C11/C44, C66 C33 and C13/C44.

The second elastic modulus tensor ratio is optionally determined from the first elastic stiffness tensor by performing regression on previously obtained data for a known lithology to determine the empirically derived constant. As a further option, the second elastic stiffness tensor is determined from the first elastic stiffness tensor by further performing regression of previously obtained data for a second known lithology to determine a second empirically derived constant, and using weighted values of the first and second empirically derived constant to determine the second elastic stiffness tensor.

According to a second aspect, there is provided a computer device that has a device for receiving P-wave velocity data for a geological subsurface and a device for obtaining S-wave velocity data for the subsurface. A processor is arranged to use a ratio of P-wave velocity and S-wave velocity to derive a first elastic stiffness tensor ratio. The processor is further arranged to use the first elastic stiffness tensor ratio and at least one empirically derived constant to determine a second elastic stiffness tensor ratio. The processor is further arranged to derive an elastic anisotropic parameter relating to the geological subsurface using the first and second elastic stiffness tensor ratios.

The computer device may also be provided with a database for storing any of the P- wave velocity data and S-wave velocity data.

As an option, the processor is arranged to derive the S-wave velocity using the P-wave velocity and at least an empirically derived constant.

The first elastic stiffness tensor ratio is optionally C33/C44, in which case the processor is arranged to derive the first elastic stiffness tensor ratio using the square of the ratio of the P-wave velocity to the S-wave velocity. As a further option, the second elastic stiffness tensor ratio is selected from any of C11/C44, C₆6 C₃3 and Ci₃/C₄₄-

The processor is optionally arranged to determine the second elastic stiffness tensor ratio from the first elastic stiffness tensor ratio by performing a regression on previously obtained data for a known lithology to determine the empirically derived constant. The processor is optionally arranged to determine the second elastic stiffness tensor ratio from the first elastic stiffness tensor ratio by further performing regression of previously obtained data for a second known lithology to determine a second empirically derived constant, and using weighted values of the first and second empirically derived constant to determine the second elastic stiffness tensor.

According to a third aspect, there is provided a computer program, comprising computer readable code which, when run on a computer apparatus, causes the computer apparatus to perform the method as described above in he first aspect.

According to a fourth aspect, there is provided a computer program product comprising a computer readable medium and a computer program as described above in the third aspect, wherein the computer program is stored on the computer readable medium.

BRIEF DESCRIPTION OF THE DRAWINGS

Figure 1 is a graph showing a comparison of Ci₃/C₄₄ and C33/C44 ratios for a shale lithology according to an embodiment of the invention;

Figure 2 is a graph showing a comparison of Ci₃/C₄₄ and C33/C44 ratios for a sandstone lithology according to an embodiment of the invention;

Figure 3 is a graph showing a comparison of Cn/C₄₄ and C33/C44 ratios for a shale lithology according to an embodiment of the invention;

Figure 4 is a graph showing a comparison of Cn/C₄₄ and C33/C44 ratios for a sandstone lithology according to an embodiment of the invention; Figure 5 is a graph showing a comparison of C₆6 C₃3 and C33/C44 ratios for a shale lithology according to an embodiment of the invention;

Figure 6 is a graph showing a comparison of C₆6 C₃3 and C33/C44 ratios for a sandstone lithology according to an embodiment of the invention;

Figure 7 is a flow diagram illustrating steps of an embodiment of the invention; and

Figure 8 illustrates schematically in a block diagram an apparatus according to an embodiment of the invention.

DETAILED DESCRIPTION

A model is described that can be used to predict anisotropy parameters. In the examples given below, the term anisotropy is used to refer to elastic dynamic vertical transverse (VTI) anisotropy. Note that similar techniques can be used on other types of anisotropy of lower orders. Shales and sandstones are provided as examples of geological subsurfaces to which the model applies, but it will be appreciated that the model may be applied to any type of geological subsurface that displays anisotropic properties. Thomsen parameters, described above, are used as examples of anisotropic parameters that can be predicted.

It has been observed that some elastic stiffness tensor ratios can be empirically related to a ratio of P-wave velocity (V_p) and S-wave velocity (V_s). V_p is readily available from well logs or can be relatively easily obtained where well logs are not available. V_s data may be available. If V_s data is not available, then a reasonably accurate approximation may be made of V_s using the following Equation:

^ = 0.86^ - 1.17 (Eq. 6) Note that Equation 6 is only one of many different ways that V_s can be estimated. For example, a very simple model to obtain V_s is to divide V_p by 3. Equation 6 is one example way of finding V_s, taken from Castagna, J. P., Batzle, M.L., and Eastwood, R.L., Relationships between compressional-wave and shear-wave velocities in clastic silicate rocks, Geophysics, Vol 50, pg. 571 -581 ,1985. The elements of the elastic stiffness tensor C₃₃ and C₄₄ can be related to the ratio of V_p and V_s using the following Equation:

The Thomsen parameters can be re-written as a function of the elastic stiffness tensor ratio shown in Equation 7 as follows:

c, 33

Γ ^66 Γ ^ 44

C 66 - C ^■"44 c 33 c "33

r* (Eq. 9)

2 44 ₂ ^44

r ^*

When re-writing the Thomsen parameters in form shown in Equations 8 to 10, it can be seen that every Thomsen parameter is a function of a few ratios of elastic stiffness tensors: ε is a function of C11 C33 and C₃3/C₄ ; δ is a function of C13/C33 and C /C₃3; and γ is a function of C₆6 C₃3 and C₃3/C . Common to all of the Thomsen parameters is the C₃₃/C ratio which, as described above in Equation 7, can be found directly from the vertical P- and S-wave velocities V_p and V_s

Even with knowledge of C₃₃/C obtained from V_p and V_s, in order to find the Thomsen parameters it is necessary to obtain values for Cn (to obtain ε), C₆6 (to obtain γ) and Ci₃ (to obtain δ). Ratios of Cn , C₆6 and C1₃ to other elastic stiffness tensors can be found by obtaining known anisotropic velocities and finding a correlation between C₃3 C and other elastic stiffness tensor ratios. By way of example, a database of measurements of anisotropic velocities from core plugs was developed using published data. The data sources in this example are as follows: Dewhurst, D.N. et al., 201 1 , Geomechanical and ultrasonic characterization of a Norwegian Sea shale: Geophysics, 76, WA101 -WA1 1 1 (referred to as Dewhurst);

Domnesteanu, P. et al., 2002, Velocity anisotropy and attenuation of shale in under- and overpressured conditions: Geophysical Prospecting, 50, 487-503 (referred to as Dommesteanu);

Grande, L, 2008, Acoustic Properties of a Shallow Mudstone Core, NGI Report No. 20071 121 -1 (referred to as Grande); Hornby, B.E., 1998, Experimental laboratory determination of the dynamic elastic properties of wet, drained shales: Journal of Geophysical Research, 103, 29945-29964 (referred to as Hornby);

Jakobsen, M. and Johansen, T.A., 2000, Anisotropy approximations for mudrocks: A seismic laboratory study: Geophysics 65, 171 1 -1725 (referred to as Jakobsen);

Sarout, J. and Gueguen Y., 2008, Anisotropy of elastic wave velocities in deformed shales: Part 1 - Experimental results: Geophysics, 73, D75-D89 (referred to as Sarout);

Bhuiyan, M. H., 2009, Stress Dependent P- and S-Wave Velocities in Brine Saturated Sand-Clay Mixtures, Diploma Thesis, Department of Petroleum Engineering and Applied Geophysics, NTNU, Trondheim, Norway Vernik, L. and Liu X.,1997, Velocity anisotropy in shales: A petrophysical study: Geophysics, 62, 521 -532 (referred to as Vernik); and

Wang, Z., 2002, Seismic anisotropy in sedimentary rocks, part 2: Laboratory data: Geophysics 67, 1423-1440 (referred to as Wang). It has been found that the ratio of C33/C44 to other elastic modulus stiffness ratios is not independent, and a good correlation can be made between C33/C44 and other elastic modulus stiffness ratios. Figure 1 shows a comparison between the model and available core data for the prediction of the Ci₃/C₄₄ ratio for shales. It can be seen that there is a good correlation between C13/C44 and C₃3 C₄4- The range of values for C33/C44 arises from data obtained at different depths in the subsurface. Figure 2 shows a comparison between the model and available core data for the prediction of the Ci₃/C₄₄ ratio for sandstones. Again, it can be seen that there is a good correlation between C13/C44 and C₃3 C₄4-

Figure 3 shows a comparison between the model and available core data for the prediction of the Cn/C₄₄ ratio for shales. It can be seen that there is a good correlation between C11/C44 and C₃3 C₄4-

Figure 4 shows a comparison between the model and available core data for the prediction of the Cn/C₄₄ ratio for sandstones. Again, it can be seen that there is a good correlation between C11/C44 and C₃i/C₄4-

Figure 5 shows a comparison between the model and available core data for the prediction of the C₆6 C₃₃ ratio for shales. It can be seen that there is a good correlation between C₆6 C₃3 and C₃3 C₄4-

Figure 6 shows a comparison between the model and available core data for the prediction of the C₆6 C₃3 ratio for sandstones. Again, it can be seen that there is a good correlation between C₆6 C₃3 and C₃i/C₄4-

Linear regression was used on the data shown in Figures 1 to 6 to correlate C13/C44, C11/C44 and C₆6 C₄4 respectively to C₃3 C₄4- Linear equations can be used to express these relationships as follows:

11 = A_£ +B_£ c 33

(Eq. 1 1 )

44 44 C, 66 c 44

c (Eq. 12)

33 c 33

C 33

c (Eq. 13)

44 c 44

In Equations 1 1 , 12 and 13, Α_ε, A_Y, A₅ ,and Β_ε, B_Y, B₅ are constants that were found when assuming a linear relation between them by using linear regression on the data obtained from the core plug database.

It is known that sandstones and shales generally show different anisotropic behaviour. Sandstones are usually reasonably isotropic, while shales typically exhibit a greater degree of anisotropy. The constants are therefore valid only for a particular lithology. In Figures 1 to Figure 6, the linear fit after fitting the above model to the core plug data is illustrated.

Regression parameters Α_ε, Α_γ, Α_δ, Β_ε, B_Y, and B₅ may be determined when fitting the model (Equations 1 1 to 13) to the core data shown in Figures 1 to 6.

As C13/C44, C11 C44 and C₆6 C₃3 can all be expressed in terms of C33/C44, it will be apparent that the Thomsen parameters can be expressed solely in terms of the ration

ε is determined by substituting Equation 1 1 into Equation 8 as follows:

Y is determined by substituting Equation 12 into Equation 9 as follows:

δ is determined by substituting Equation 13 into Equation 10 as follows:

As described above with reference to Equation 7, C33/C44 can be expressed solely in terms of V_p and V_s. This allows the Thomsen parameters to be expressed in terms of Vp and V_s, using the constants A and B that have been derived for ε, γ and δ for a particular lithology, as follows:

The Thomsen parameters can then be used to express the anisotropy of the geological subsurface in a forward rock physics model.

It will be appreciated that in some circumstances, a subsurface may have a mixed lithology. For example, it may comprise both shale and sandstone. In this case, the predictions can be weighted using values for Α_ε, A_Y, A₅ ,and Β_ε, B_Y, B₅ for both sandstone and shale. The weighting can be done using the volume fraction of shale (V_sh) or the v

Turning now to Figure 7, there is shown a flow diagram illustrating steps of the invention. The following numbering corresponds to that of Figure 7:

P-wave velocity data is obtained.

S2. S-wave velocity data is obtained. This may be from measurements, or estimated from V_p, for example by using Equation 6. S3. C33/C44 (a first elastic stiffness tensor ratio) is derived using V_p and V,

54. Regression is performed on known C33/C44, and Cn/C₄₄, Ci₃/C₄₄ and C₆6 C₃3 data in order to find constants for a particular lithology that allow Cn/C₄₄, Ci₃/C₄₄ and C66 C33 to be expressed in terms of C₃3/C₄₄.

55. In the event of mixed lithologies, the volume fraction of one of the lithologies (for example V_sh, the volume fraction of shale) is used to obtain constants that allow C11/C44, C13 C44 and C₆6 C₃3 to be expressed in terms of C33/C44 for a single, mixed lithology subsurface.

56. ε is obtained using C33/C44 and Cn/C₄₄ expressed in terms of C33/C44 and empirically derived constants.

S7. δ is obtained using C33/C44 and Ci₃/C₄₄ expressed in terms of C33/C44 and empirically derived constants. S8. Y is obtained using C33/C44 and derived C₆6 C₃3 expressed in terms of C33/C44 and empirically derived constants.

Turning now to Figure 8, there is illustrated a computer device 1 according to an embodiment of the invention. The computer device is provided with an input device 2 for receiving data. Examples of such a device include a keyboard or other user input device, and a receiver for receiving data from a remote device. The data may include P-wave data, S-wave data or any other geological data relating to the geological subsurface. A database 3 is stored on a computer readable medium in the form of a memory 4, which may be disposed locally at the computer device 1 as shown, or may be disposed remotely but accessible from the computer device 1 (not shown). A processor 5 is provided that is arranged to perform the steps described above. The computer device 1 may also be provided with an output device 6 to output the results of the determination of the Thomsen parameters. Examples of an output device 6 include a transmitter, a display screen and a link to a printer.

A computer program 7 may also be provided which, when executed by the processor 5, causes the computer device to perform the steps illustrated in Figure 7. The computer program 7 may be stored on a computer readable medium such as the memory 4.

To summarize, correlations between ratios of elastic stiffness tensors can be found which allows the Thomsen parameters of a geological subsurface to be determined using P-wave velocity, S-Wave velocity, and constants used to derive ratios of elastic stiffness tensors. This is possible because Thomsen parameters can be written as ratios of the elastic stiffness tensors and, for typical rocks, these ratios are not independent but can be correlated to other, known rations such as C33/C44 ratio.

In the examples given above, a model is presented that uses the observed correlations to predict the ratios: Cn /C44, C13 C₄4 and C₆6 /C33, from the C33/C44 ratio. As C33/C44 is relatively easy to measure (from V_p and V_s logs), and often logged, the method can be used for many wells an exploration areas. Unlike prior art models, the model presented above is independent of porosity. The Thomsen parameters obtained can be used directly in rock physics models, seismic processing and imaging, and seismic amplitude interpretations to obtain accurate models of a geological subsurface. It will be appreciated by the person of skill in the art that various modifications may be made to the above described embodiments without departing from the scope of the present invention as defined in the appended claims. For example, the Thomsen parameters are used as exemplary parameters describing anisotropic parameters of a geological subsurface. However, it will be appreciated that the invention may be applied to finding other types of parameters that can be used to characterise the anisotropy of a geological subsurface. Furthermore, the Thomsen parameters are expressed above in terms of the ratio of C33/C44, but it will be appreciated that they may be expressed as any elastic stiffness tensor ratio used in determining the Thomsen parameters. In addition, the above description refers to elastic dynamic vertical transverse anisotropy, but it would be a simple matter to adapt the techniques described above to estimate parameters for other types of anisotropy of lower orders.

Claims

1. A method of estimating an elastic anisotropic parameter of a rock physics model for a geological subsurface, the method comprising:

obtaining a P-wave velocity value for the subsurface;

obtaining an S-wave velocity value for the subsurface;

using a ratio of P-wave velocity and S-wave velocity to derive a first elastic stiffness tensor ratio;

using the first elastic stiffness tensor ratio and at least one empirically derived constant to determine a second elastic stiffness tensor ratio;

deriving the anisotropic parameter using the first and second elastic stiffness tensor ratios.

2. The method according to claim 1 , wherein the anisotropic parameter is a Thomsen parameter.

3. The method according to claim 1 or 2, further comprising deriving the S-wave velocity using the P-wave velocity and at least an empirically derived constant.

4. The method according to any of claims 1 to 3, wherein the first elastic stiffness tensor ratio is C33/C44, and the first elastic stiffness tensor ratio is derived using the square of the ratio of the P-wave velocity to the S-wave velocity.

5. The method according to claim 4, wherein the second elastic stiffness tensor ratio is any of C11/C44, C₆6 C₃3 and Ci₃/C₄4-

6. The method according to any of claims 1 to 5, wherein the second elastic stiffness tensor ratio is determined from the first elastic stiffness tensor by performing regression on previously obtained data for a known lithology to determine the empirically derived constant.

7. The method according to claim 6, wherein the second elastic stiffness tensor is determined from the first elastic stiffness tensor by further performing regression of previously obtained data for a second known lithology to determine a second empirically derived constant, and using weighted values of the first and second empirically derived constant to determine the second elastic stiffness tensor.

8. A computer device comprising:

a device for receiving P-wave velocity data for a geological subsurface;

a device for obtaining S-wave velocity data for the subsurface;

a processor arranged to use a ratio of P-wave velocity and S-wave velocity to derive a first elastic stiffness tensor ratio;

the processor being further arranged to use the first elastic stiffness tensor ratio and at least one empirically derived constant to determine a second elastic stiffness tensor ratio;

the processor being further arranged to derive an elastic anisotropic parameter relating to the geological subsurface using the first and second elastic stiffness tensor ratios.

9. The computer device according to claim 8, further comprising a database for storing any of the P-wave velocity data and S-wave velocity data.

10. The computer device according to claim 8 or 9, wherein the processor is further arranged to derive the S-wave velocity using the P-wave velocity and at least an empirically derived constant.

1 1 . The computer device according to any of claims 8 to 10, wherein the first elastic stiffness tensor ratio is C33/C44, the processor being arranged to derive the first elastic stiffness tensor ratio using the square of the ratio of the P-wave velocity to the S-wave velocity.

12. The computer device according to claim 1 1 , wherein the second elastic stiffness tensor ratio is any of C11/C44, C₆6 C₃3 and Ci₃/C₄4-

13. The computer device according to any of claims 8 to 12, wherein the processor is arranged to determine the second elastic stiffness tensor ratio from the first elastic stiffness tensor ratio by performing regression on previously obtained data for a known lithology to determine the empirically derived constant.

14. The computer device according to claim 13, wherein processor is arranged to determine the second elastic stiffness tensor ratio from the first elastic stiffness tensor ratio by further performing linear regression of previously obtained data for a second known lithology to determine a second empirically derived constant, and using weighted values of the first and second empirically derived constant to determine the second elastic stiffness tensor.

15. A computer program, comprising computer readable code which, when run on a computer apparatus, causes the computer apparatus to perform the method of any of claims 1 to 7.

16. A computer program product comprising a computer readable medium and a computer program according to claim 15, wherein the computer program is stored on the computer readable medium.