WO2013148623A1

WO2013148623A1 - Dynamic to static elastic property transforms for tiv rock material

Info

Publication number: WO2013148623A1
Application number: PCT/US2013/033796
Authority: WO
Inventors: Marcelo FRYDMAN; Jorge Aurelio Santa Cruz Pastor
Original assignee: Schlumberger Canada Limited; Services Petroliers Schlumberger; Schlumberger Holdings Limited; Schlumberger Technology B.V.; Prad Research And Development Limited; Schlumberger Technology Corporation
Priority date: 2012-03-26
Filing date: 2013-03-26
Publication date: 2013-10-03

Abstract

A method of performing an oilfield operation associated with an anisotropic rock, including obtaining a plurality of sonic velocities in an inclined borehole through the anisotropic rock, obtaining a plurality of non-linear equations relating the plurality of sonic velocities and a plurality of anisotropic elastic constants of the anisotropic rock, solving the plurality of non-linear equations for the plurality of anisotropic elastic constants using a Newton-Raphson method, obtaining a dynamic to static transform based on a theoretical model for the anisotropic rock, and performing the oilfield operation based on at least one selected from a group consisting of the plurality of anisotropic elastic constants and the dynamic to static transform.

Description

DYNAMIC TO STATIC ELASTIC PROPERTY TRANSFORMS

FOR TIV ROCK MATERIAL

BACKGROUND

[0001] Operations, such as geophysical surveying, drilling, logging, well completion, and production, are performed to locate and gather valuable downhole fluids. Surveys are often performed using acquisition methodologies, such as seismic mapping, resistivity mapping, etc. to generate images of underground formations. These formations are often analyzed to determine the presence of subterranean assets, such as valuable fluids or minerals, or to determine if the formations have characteristics suitable for storing fluids. Although the subterranean assets are not limited to hydrocarbons such as oil, throughout this document, the terms "oilfield" and "oilfield operation" may be used interchangeably with the terms "field" and "field operation" to refer to a site where any types of valuable fluids or minerals can be found and the activities required for extracting them. The terms may also refer to sites where substances are deposited or stored by injecting them into the surface using boreholes and the operations associated with this process. Further, the term "field operation" refers to a field operation associated with a field, including activities related to field planning, wellbore drilling, wellbore completion, and/or production using the wellbore.

SUMMARY

[0002] In general, in one aspect, the present disclosure relates to a method of performing an oilfield operation associated with an anisotropic rock, the method including obtaining a plurality of sonic velocities in an inclined borehole through the anisotropic rock, obtaining a plurality of non-linear equations relating the plurality of sonic velocities and a plurality of anisotropic elastic constants of the anisotropic rock, solving the plurality of non-linear equations for the plurality of anisotropic elastic constants using a Newton-Raphson method, obtaining a dynamic to static transform based on a theoretical model for the anisotropic rock, and performing the oilfield operation based on at least one selected from a group consisting of the plurality of anisotropic elastic constants and the dynamic to static transform.

[0003] In general, in another aspect, the present disclosure relates to a system of performing an oilfield operation associated with an anisotropic rock, the system including a repository storing a plurality of sonic velocities in an inclined borehole through the anisotropic rock, a elastic constant engine configured to solve a plurality of non-linear equations for a plurality of anisotropic elastic constants of the anisotropic rock based on the plurality of sonic velocities, and a transform engine configured to calculate a dynamic to static transform based on a theoretical model for the anisotropic rock, in which the oilfield operation is performed based on at least one selected from a group consisting of the plurality of anisotropic elastic constants and the dynamic to static transform.

[0004] In general, in yet another aspect, the present disclosure relates to A non- transitory computer readable medium (CRM) storing a plurality of instructions for performing an oilfield operation associated with an anisotropic rock, the instructions including functionality for obtaining a plurality of sonic velocities in an inclined borehole through the anisotropic rock, obtaining a plurality of nonlinear equations relating the plurality of sonic velocities and a plurality of anisotropic elastic constants of the anisotropic rock, solving the plurality of nonlinear equations for the plurality of anisotropic elastic constants using a Newton- Raphson method, obtaining a dynamic to static transform based on a theoretical model for the anisotropic rock, and performing the oilfield operation based on at least one selected from a group consisting of the plurality of anisotropic elastic constants and the dynamic to static transform.

[0005] Other aspects of the disclosure will be apparent from the following description and the appended claims.

BRIEF DESCRIPTION OF DRAWINGS

[0006] FIG. 1 shows a system in accordance with one or more embodiments of the disclosure.

[0007] FIG. 2 shows a well in accordance with one or more embodiments of the disclosure.

[0008] FIG. 3 shows a flowchart in accordance with one or more embodiments of the disclosure.

[0009] FIG. 4 shows a graph in accordance with one or more embodiments of the disclosure.

[0010] FIG. 5 shows a computer system in accordance with one or embodiments of the disclosure.

DETAILED DESCRIPTION

[0011] Specific embodiments of the disclosure will now be described in detail with reference to the accompanying figures. Like elements in the various figures are denoted by like reference numerals for consistency.

[0012] In the following detailed description of embodiments of the disclosure, numerous specific details are set forth in order to provide a more thorough understanding of the disclosure. However, it will be apparent to one of ordinary skill in the art that the disclosure may be practiced without these specific details. In other instances, well-known features have not been described in detail to avoid unnecessarily complicating the description. [0013] FIG. 1 shows a system (100) in accordance with one or more embodiments of the disclosure. As shown in FIG. 1, there exists an oilfield (102), a repository (1 12), an elastic constant engine (1 14), and a transform engine (1 16). The repository (1 12), the elastic constant engine (1 14), and the transform engine (1 16) may be located on the same or different hardware devices (e.g., personal computer (PC), server, laptop, mainframe, cable box, kiosk, smart phone, tablet PC, etc.) connected by a network having wired and/or wireless segments.

[0014] In one or more embodiments of the disclosure, the oilfield (102) includes one or more boreholes through anisotropic rock (e.g., shale). In addition, in one or more embodiments of the disclosure, the oilfield (102) includes tools to measure/acquire the sonic velocities of the waves in the inclined borehole(s) to generate sonic well logs.

[0015] In one or more embodiments of the disclosure, the repository (1 12) stores the sonic velocities of the waves in the inclined borehole(s). The repository may be a database or a flat file. The repository may include storage devices and/or arrays of any number or size.

[0016] Rock is described as elastic and isotropic if its elastic properties are constant in all directions. Although most sedimentary rocks are anisotropic, modeling techniques often assume the rock is isotropic. As a result, the calculated elastic properties and the horizontal far field stresses are oversimplified due to the isotropic assumption, and inconsistent with the anisotropic characteristics of the formation. These anisotropic rock properties and horizontal stresses play a non- trivial role in one or more aspects of the exploitation of hydrocarbon reservoirs. The reasons for the petroleum industry's use of isotropic simplification are related to a limitation in measuring the dynamic elastic constants. For the isotropic material, just two elastic constants are needed, and we can fully characterize those with vertical P-wave (V_p) and S-wave (V_s) data from a sonic log and with density data p), as shown, for example, in the stiffness matrix below:

[0018] Here, σ is the stress tensor and ε is the strain tensor. Additionally, isotropic theories and modeling techniques require less data and therefore, are much easier to implement. However, in laboratory measurements of shale, anisotropy as high as 100% has been widely reported for both static and dynamic conditions. Accordingly, shale is better described as a transversely isotropic rock that has symmetric axes perpendicular to the layering of the shale.

[0019] Shale is characterized by thin laminate or parallel layering or bedding and is a component of sedimentary basins. Clastic sediments, e.g., shale, are composed of rock and/or mineral fragments. Further, clastic sediments exhibit properties that are symmetric about an axis normal to the plane of isotropy. These properties are known as transversely isotropic properties and their symmetric axes are perpendicular to the bedding, known as TIV.

[0020] Transverse isotropy in sedimentary sequences can be related to the following: a) intrinsic anisotropy that is due to the constituent plate-shaped clay particles oriented parallel to each other, b) horizontally or titled layered sedimentary rocks (each layer can be isotropic on a small scale), and c) a system of parallel fractures or microcracks within the rock. [0021] FIG. 2 shows a well (201) in accordance with one or more embodiments of the disclosure. As shown in FIG. 2, a deviated well (201) extends through formation (203). The formation may include anisotropic rock. Dipole sources (205) are arranged along the deviated well (201). The dipole sources (205) may produce any signal known in the art, for example, acoustic waves. Additionally, the dipole sources (205) may be configured to produce a signal for a specific duration or at a specific intervals. The dipole sources (205) may produce a signal to correspond with one another or the dipole sources may produce a signal at the same time as one another or at a different time with respect to each other. Moreover, the dipole sources (205) may be arranged at any position along the deviated well (201) or at either or both ends of the deviated well (201). One of ordinary skill in the art would appreciate that the dipole sources 205 are not limited to acoustic waves or the arrangement as shown in FIG. 2.

[0022] Further, dipole receivers (207) may also be arranged along the deviated well (201). The dipole receivers (207) may be an array of receivers and the dipole receivers may be arranged any position along the deviated well (201) or at either or both ends of the deviated well (201). The dipole receivers (207) may be configured to receive any signal known in the art, such as acoustic wave signals. One of ordinary skill in the art would appreciate that the dipole receivers (207) are not limited to acoustic waves or the arrangement as shown in FIG. 2.

[0023] In deviated wells (wells that deviate from a vertical trajectory through a formation) (e.g., Well (201)), the axes of the isotropy planes (bedding planes) are not normal to the borehole axis, and thus wave propagation within the borehole is not axis-symmetric. The deviated well (201) may also be referred to as an inclined borehole. Therefore, sonic well logs acquired in deviated boreholes will reflect different velocity data than those acquired in vertical boreholes drilled through the same formation. As a result, calculated elastic constants are inconsistent across the formation.

[0024] In one or more embodiments of the disclosure, five independent elastic constants describe the stiffness matrix for a transverse isotropic material Mathematically, just 3 of the 5 independent elasticity constants may be solved for in the transverse isotropic scenario. In particular, the stiffness matrix for the transverse isotropic condition may be expressed as:

Γ

[0026] However, determining the five dynamic elastic constants may be based on the borehole dipole sonic anisotropy and by solving an inverse problem. Based on the stiffness matrix in material coordinates, it is possible to calculate sonic velocities in any direction. The Biot's theory of poroelasticity describes the mechanical and pore fluid diffusion behavior of an elastic porous medium. The porous medium is an elastic solid (often called matrix) permeated by an interconnected network of pores (voids) filled with a fluid (liquid or gas) under pressure. Many natural substances such as rocks, soils, biological tissues, and manmade materials such as foams and ceramics can be considered as porous media.

[0027] The concept of a porous medium originally emerged in soil mechanics. The general concept of a poroelastic medium and the theory of poroelasticity (now known as Biot theory) gives a complete and general description of the mechanical behavior of a poroelastic medium.

[0028] In one or more embodiments of the disclosure, the presence of a free- moving fluid in rock modifies its mechanical behavior. This is a time-dependent problem as a result of fluid flow in response to pressure gradients. Analyses of geomechanics problems in the oil industry often ignore pore fluid effects or assume that the pore pressures can be found independently of the mechanical deformation. Although such assumptions are sometimes acceptable, there are many conditions in which the coupling between mechanical deformation and pore pressure diffusions should be considered.

[0029] In the inclined borehole problem, there is information on borehole coordinates (Vp, Vsl, Vs2) and the material angles (dip and dip-direction). Accordingly, the Newton-Raphson method may be used to solve the set of non linear equations and that results in an iterative process with the following recurrence formula:

[0030]

[0031] Further, the Jacobian matrix of first-order partial derivations of F(X) with respect to X evaluated at X¹ may be expressed as follows:

Κ,(θ) C₄₄-2C₆₆)sm²(20) sin² Θ + cos θ ² -^ _ι+κ₃(θ) 2(C_n+

2Κ(Θ) Κ(θ) 2Κ(Θ) Κ(θ)

Κ_γ(θ) _ι Κ₃(θ) 2(C_{11 +}C₄₄-2C₆₆)sin²f2#)

J = sm²0- cos θ+ ²

2Κ(Θ) Κ(θ) 2Κ(Θ) Κ(θ)

0 0 cos²0 ύη²θ sin⁴6> sin⁴6> sin⁴6> sin⁴6>

l-cos²# cos Θ +

8 8

Κ(θ) = - C₄₄ )sin² Θ - (C₃₃ - C₄₄ )cos² θ + (C_n + C₄₄ - Κ,(θ) = 2 sin ² ^[(C_n-C₄₄)sin² ^-(C₃₃-C₄₄)cos² ^]+2( _n + C₄₄ - 2C₆₆)sin²(2^)

A: ( ) = [(C_n-C₄₄ )sin² #-(C₃₃-C₄₄)cos² ^]cos² Θ

Κ₃(θ) = 2[(C_n-C₄₄ )sin²0-(C₃₃-C₄₄ )cos² θ\- sin² Θ + cos² θ)+ 2(C, 2C₆₆)sin²(2^)

[0032] In one or more embodiments of the disclosure, the elastic constant engine (114) is configured to calculate the anisotropic elastic constants (e.g., C_u, C₃₃, C₄₄, C₆₆, etc.) for the anisotropic rock based on the sonic velocities. Specifically, the anisotropic elastic constants are calculated by solving multiple non-linear equations using the Newton-Raphson method as shown above.

[0033] In one or more embodiments of the disclosure, the undrained response characterizes the condition where the time scale characteristic of the loading is too short to allow fluid movement to take place, while drained response characterizes conditions where the induced pore pressure dissipate and return to its original value. The undrained and drained responses also characterize the instantaneous and long-term behaviors of the poroelastic material under the particular conditions of a suddenly applied constant loading.

[0034] Dynamic or wave propagation phenomena is frequency dependent, but can be approximated by undrained conditions (fast loading), while static properties are the drained properties (slow loading). The induced pore pressure stiffens the elastic response, (i.e., both the undrained bulk modulus and undrained Poisson's ratio are greater than their drained values). In general, the drained properties are required for the analyses of geomechanics problems.

[0035] In one or more embodiments of the disclosure, the transform engine (116) is configured to generate a dynamic to static transform based on a theoretic model of the anisotropic rock. For TIV assumption, the Reuss average of the drained bulk modulus (KdR) can be obtained from the Reuss average of the undrained modulus (KuR) by:

[0037]

1 -

K ....... =

[0038]

[0039] !=^<s

[0040] where K^§ _R is the Reuss average of the the grain bulk modulus, vm is the volume fraction (∑ vm = 1) of the m-th isotropic grain having bulk modulus Km; Kf is the pore fluid modulus (oil, water and gas mixture); Φ is the total porosity; is s^Ui_j the undrained The drained (v ) and undrained (v_u ) Poisson ratio can be obtained as:

[0041] δ-C *

[0042] where K and K_u are the drained and undrained bulk modulus respectively, and G is the shear modulus. For a linear isotropic solid, shearing cannot induce a pore pressure change, and thus the undrained shear modulus is identical to the drained shear modulus.

[0043] The dynamic/undrained elastic constants can be derived from measurements (log derived) with appropriate equations, using sonic log compressional and shear slowness along with density log data. Dynamic elastic constants can also be determined in the laboratory using high frequency acoustic pulses on core samples and through seismic.

[0044] The above equations provide insight into the relationship between drained and undrained moduli for granular systems. These concepts are related to dynamic to static empirical correlations used to estimate elastic properties from measurements (log or seismic derived). Empirical models are widely used for that due to their simplicity in fitting a relationship between parameters. Current practice suggests a reduction of the Young's modulus and a strong assumption that the dynamic and static Poisson's ratio are identical. However, it is more rigorous to use a dynamic to static transform based on a theoretical model validated through laboratory testing. With a theoretical model, once the parameters describing the physical controlling factors are determined, the model can be applied anywhere that the controlling parameters can be estimated. This achieves better predictions and deeper understanding of the rocks than empirical models. The equations can be extended to other types of isotropy considered in rock mechanics, especially for transverse isotropy .

[0045] In one or more embodiments of the disclosure, one or more oilfield operations are performed based on the anisotropic elastic constants and/or the dynamic to static transform. For example, one oilfield operation may include analyzing plane of weakness failure in anisotropic rock.

[0046] FIG. 3 shows a flowchart for performing an oilfield operation associated with an anisotropic rock. One or more boxes in FIG. 3 may be executed by the system (100) (e.g., transform engine (1 16), elastic constant engine (1 14)). The sequence of boxes shown in FIG. 3 may differ among embodiments of the disclosure, and one or more of the boxes may be performed in parallel and/or may be optional. Moreover, one or more boxes in FIG. 3 may be repeated. Accordingly, the scope of the disclosure should not be considered limited to the specific arrangement of boxes shown in FIG. 3.

[0047] Initially, sonic velocities in an inclined borehole through the anisotropic rock are obtained (BOX 305). The sonic velocities may be stored in a repository.

[0048] In BOX 310, non-linear equations relating sonic velocities and anisotropic elastic constants of the anisotropic rock are obtained (discussed above).

[0049] In BOX 315, the non-linear equations are solved for the anisotropic elastic constants using the Newton-Raphson method (discussed above).

[0050] In BOX 320, a dynamic to static transform based on a theoretical model for the anisotropic rock is obtained (discussed above).

[0051] In BOX 325, an oilfield operation based on the anisotropic elastic constants and/or the dynamic to static transform is performed. One example of an oilfield operation includes analyzing plane of weakness failure in anisotropic rock.

[0052] FIG. 4 shows a graph of the effect of sonic velocity anisotropy on the estimation of pore pressure in accordance with one or more embodiments of the present disclosure. Specifically, the graph shows the relationship between the estimated pore pressure, calibrated pore pressure, and compressional velocity with respect to borehole angle. Initially, the dynamic elastic constants are calculated for a transverse isotropic rock in an inclined borehole following the methodology described above. Further, using a isotropic simplification, the calibrated pore pressure is constant. As shown in FIG. 4, taking sonic anisotropy into account, the compressional velocity changes with respect to borehole angle and the estimated pore pressure also changes.

[0053] Embodiments of the disclosure may be implemented on virtually any type of computer regardless of the platform being used. For example, as shown in FIG. 5, a computer system (500) includes one or more processor(s) (502) (such as a central processing unit (CPU), integrated circuit, etc.), associated memory (504) (e.g., random access memory (RAM), cache memory, flash memory, etc.), a storage device (506) (e.g., a hard disk, an optical drive such as a compact disk drive or digital video disk (DVD) drive, a flash memory stick, etc.), and numerous other elements and functionalities of today's computers (not shown). The computer system (500) may also include input means, such as a keyboard (508), a mouse (510), or a microphone (not shown). Further, the computer system (500) may include output means, such as a monitor (512) (e.g., a liquid crystal display (LCD), a plasma display, or cathode ray tube (CRT) monitor). The computer system (500) may be connected to a network (514) (e.g., a local area network (LAN), a wide area network (WAN) such as the Internet, or any other type of network) via a network interface connection (not shown). Those skilled in the art will appreciate that many different types of computer systems exist, and the aforementioned input and output means may take other forms. Generally speaking, the computer system (500) includes at least the minimal processing, input, and/or output means to practice embodiments of the disclosure.

[0054] Further, in one or more embodiments of the disclosure, one or more elements of the aforementioned computer system (500) may be located at a remote location and connected to the other elements over a network. Further, embodiments of the disclosure may be implemented on a distributed system having a plurality of nodes, where each portion of the disclosure may be located on a different node within the distributed system. In one embodiment of the disclosure, the node corresponds to a computer system. The node may correspond to a processor with associated physical memory. The node may correspond to a processor or micro-core of a processor with shared memory and/or resources. Further, software instructions in the form of computer readable program code to perform embodiments of the disclosure may be stored, temporarily or permanently, on a tangible computer readable storage medium, such as a compact disc (CD), a diskette, a solid state memory device, a tape, memory, or any other non-transitory tangible computer readable storage device.

[0055] Although most of the sedimentary rocks are anisotropic, most of the

Geomechanics modeling currently being used by the industry is based on simple isotropic assumptions. The methodology of the present disclosure moves toward a more realistic anisotropic (TIV) model. The elastic stiffness for these anisotropic rocks plays a non-trivial role in one or more aspects of hydrocarbon exploitation. During geophysical investigations, elastic properties are used to obtain reliable time-to-depth conversion, pore pressure from seismic data, and seismic imaging in sedimentary basins. Drilling problems associated with wellbore instability are normally aggravated when crossing shale layers. During a hydraulic fracture operation, both orientation and propagation of the fractures will be strongly influenced by the stress distribution (magnitude and direction) in the reservoir and surrounding rocks (mainly shales). During production, the pressure depletion/injection will affect the state of stress inside the reservoir and its overburden (mainly shales). These stress changes may cause reservoir compaction, changes to reservoir performance, movement of the overburden, re-activation of faults, and compromise to the well integrity.

[0056] The obtained results bring a contribution to the understanding of the material properties and stress development in a transverse isotropic rock. The methodology developed has allowed the full determination of the dynamic elastic constants of a transverse isotropic rock based on borehole dipole sonic anisotropy in inclined boreholes. While the disclosure has been described with respect to a limited number of embodiments, those skilled in the art, having benefit of this disclosure, will appreciate that other embodiments can be devised which do not depart from the scope of the disclosure as disclosed herein. Accordingly, the scope of the disclosure should be limited only by the attached claims.

Claims

CLAIMS What is claimed is:

1. A method of performing an oilfield operation associated with an anisotropic rock (203), comprising:

obtaining a plurality of sonic velocities in an inclined borehole (201) through the anisotropic rock (203);

obtaining a plurality of non-linear equations relating the plurality of sonic velocities and a plurality of anisotropic elastic constants of the anisotropic rock (203);

solving the plurality of non-linear equations for the plurality of anisotropic elastic constants using a Newton- Raphson method;

obtaining a dynamic to static transform based on a theoretical model for the anisotropic rock; and

performing the oilfield operation based on at least one selected from a group consisting of the plurality of anisotropic elastic constants and the dynamic to static transform.

2. The method of claim 1, wherein obtaining the dynamic to static transform comprises: obtaining a Reuss average of a drained bulk modulus from a Reuss average of an undrained modulus.

3. The method of claim 1, wherein obtaining the dynamic to static transform comprises: approximating wave propagation phenomena based on undrained conditions.

4. The method of claim 3, wherein static properties are drained properties.

5. The method of claim 1, wherein the anisotropic rock (203) is shale.

6. The method of claim 1, wherein the plurality of anisotropic elastic constants comprises compressional, slow-shear, fast-shear, and Stoneley-shear.

7. The method of claim 1, wherein solving the plurality of non-linear equations comprises generating a Jacobian matrix (J) given by: sin²# + K 2(C_{11 +}C₄₄-2C₆₆)sin²f2#) cos θ

2Κ(Θ) Κ(θ) 2Κ(Θ) Κ(θ)

Κ,(θ) _ι Κ₃(θ) 2(C_{11 +}C₄₄-2C₆₆)sin²^)

J = sin²#- cos θ+ ²

2Κ(Θ) Κ(θ) 2Κ(Θ) Κ(θ)

0 0 cos²0 ύη²θ sin⁴6> sin⁴6> _{2 Λ} sin⁴ θ sin

l-cos²#- cos θ +

8 8

Κ(θ) = [(Qi -C₄₄)sin²^-(C₃₃ -C₄₄)cos²^]² +(C_n +C₄₄ -2C₆₆)² sin²(2#)

Κ,(θ) = 2 sin ² #[(C_n-C₄₄)sin² #-(C₃₃-C₄₄)cos² #]+2(^Cn + C₄₄ - 2C₆₆)sin²(2#)

Κ₂(θ) = [(C_n-C₄₄ )sin² #-(C₃₃-C₄₄)cos² #]cos² #

^ #) = 2[(C_n-C₄₄)sin² #-(C₃₃-C₄₄)cos² θ\- sin² Θ + cos² #)+2(C_n +C₄₄ - 2C₆₆)sin²(2#)

8. A system of performing an oilfield operation associated with an anisotropic rock, comprising:

a repository (112) storing a plurality of sonic velocities in an inclined borehole through the anisotropic rock;

a elastic constant engine (114) configured to solve a plurality of non-linear equations for a plurality of anisotropic elastic constants of the anisotropic rock based on the plurality of sonic velocities; and

a transform engine (116)configured to calculate a dynamic to static transform based on a theoretical model for the anisotropic rock,

wherein the oilfield operation is performed based on at least one selected from a group consisting of the plurality of anisotropic elastic constants and the dynamic to static transform.

9. The system of claim 8, wherein the transform engine (116) is further configured to: obtain a Reuss average of a drained bulk modulus from a Reuss average of an undrained modulus.

10. The system of claim 8, wherein the transform engine (116) is further configured to: approximate wave propagation phenomena based on undrained conditions.

11. The system of claim 10, wherein static properties are drained properties.

12. The system of claim 8, wherein the anisotropic rock is shale.

13. The system of claim 8, wherein the plurality of anisotropic elastic constants comprises compressional, slow-shear, fast-shear, and Stoneley-shear.

14. The system of claim 8, wherein the elastic constant engine (114) is further configured to solve the plurality of non-linear equations by generating a Jacobian matrix (J) given by:

ι₊κ₃(θ) 2(C_{11 +}C₄₄-2C₆₆)sin²f20) sin 0 +— ^l—^L cos 0

2Κ(Θ) Κ(θ) 2Κ(Θ) Κ(θ)

l Κ₃(θ) 2(C_{11 +}C₄₄-2C₆₆)sin²f20)

J = COS 0 +

2Κ(Θ) Κ(θ) 2Κ(Θ) Κ(θ)

0 0 cos²0 sin²0 sin⁴0 sin⁴0 2 sin⁴0 2 _n sin⁴0

1-cos 0 cos 0 +

8 8 2 2

Κ(θ) = [(Q_i -C₄₄)sin² -(C₃₃ -C₄₄)cos²0]² +(C_n +C₄₄ -2C₆₆)² sin²(20)

Κ,(θ) = 2 sin ²0[(C_n-C₄₄)sin²0-(C₃₃-C₄₄)cos²0]+2( _n + C₄₄ - 2C₆₆)sin²(20)

Κ₂(θ) = [(C_n-C₄₄ )sin²0-(C₃₃-C₄₄)cos²0]cos²0

0) = 2[(C_n-C₄₄)sin²0-(C₃₃-C₄₄)cos² θ\- sin²0 + cos² e)+2(C_n +C₄₄ - 2C₆₆)sin²(20)

15. A computer program product comprising computer readable program code embodied therein for performing a method according to any of claims 1-7.