Patent Application
20080178668
The present invention relates generally to measurement and analysis of formation. More particularly, the present invention relates to a method of evaluating elastic properties of a transversely isotropic formation.
It is well known that the laminated formation rock presents anisotropic mechanical properties. This anisotropic behavior is due to formation's sedimentary structures, such as the fine layers, oriented fissures/fractures, or anisotropy fibers/grains. The effects of this anisotropy on seismic shear anisotropy have been well documented since the 1970's, e.g. “Weak Elastic Anisotropy” by Leon Thomsen (Geophysics, Vol. 51, 1986). One common form of the anisotropy model, the Transversely Isotropic (TI) model, has been widely used in geophysical and geomechanical applications, e.g. “A model for bedding related formation failure” by Atkinson, C. and Bradford, 2001 (I.D.R.: OFSR/RN/2001/005/RDV/C). However, one of the major difficulties constraining the applications of the anisotropy model is how to determine the elastic constants from seismic or borehole sonic measurements. This constraint effects even the simplest anisotropy models, such as the TI model, with five independent elastic constants.
Recently, with the development of measuring tools, the borehole's four velocities (a compressional velocity VP, a tube wave velocity VT, a shear horizontal wave velocity VSH, and a shear vertical wave velocity VSV) can be measured with more accuracy, for example, via Schlumberger's new sonic tool SONIC SCANNER. However, it is still impossible to determine the TI properties directly from the measured four velocities.
The current invention provides methods and apparatus for the determination of the transversely isotropic (TI) formation elastic properties directly from the borehole measurements. In accord with the objects of the invention which will be discussed in more detail below, a method of TI formation evaluation comprises receiving a plurality of borehole measurements; deriving a correlation between a first TI stiffness parameter and other TI stiffness parameters where the first and other TI stiffness parameters representing mechanical behavior of the TI formation; and computing the first and other TI stiffness parameters based on the borehole measurements and the derived correlation. The method further comprises evaluating TI formation elastic properties based on the computed first and other TI stiffness parameters. The method further comprises assuming that the shear modulus parallel to TI symmetric axis can be approximated from other moduli. The method further comprises assuming the Shear Modulus G′ (parallel to TI symmetric axis) is proportional to the shear module in the plane that inclined to TI symmetric axis with about 45 degree.
Additional objects and advantages of the invention will become apparent to those skilled in the art upon reference to the detailed description taken in conjunction with the provided figures.
The present invention is illustrated by way of example and not intended to be limited by the figures of the accompanying drawings in which like references indicate similar elements and in which:
a is a diagram used to illustrate an embodiment of a correlation between two shear moduli G′ and G′45 of sand formation;
b is a diagram used to illustrate another embodiment of a correlation between two shear moduli G′ and G′45 of shale formation;
a is a diagram of a sample of four velocities measured from sonic tool of wireline logging device;
b is a diagram of TI formation elastic moduli calculated from the measurements form the borehole after applying the present invention; and
In the early part of the 20th century anisotropy was more a topic of scientific research than a property used in engineering design. Nye gave an excellent introduction to anisotropy in crystals from a material scientist perspective and it was in Lekhnitski's paper “Theory of Elasticity of an Anisotropic Body” [Lekhnitski, 1963], that the mechanical properties of anisotropy material was first addressed in engineering design.
We start by reviewing the classical TI theory and the relations of stiffness tensor (c) and the compliance tensor (α). This is followed by giving the variation of elastic moduli (Young' Modulus and Shear Modulus) along a specific line versus the inclination of this line to TI symmetric axis. From elastic theory, the deformation constitution of elastic medium can be described with the generalized Hook' law as:
σij=cijklεkl (i, j, k=1,2,3) (1)
where σij and εkl are stress and elastic strain tensor respectively, and cijkl is the fourth order (3×3×3×3) elastic stiffness tensor.
With the consideration that stress and strain tensors are symmetric tensors (σij=σji, εij=εji), the above relation can be represented with compacted indices, following Voigt's recipe:
{σ}=[c]{ε} (2)
where
{σ}={σ11,σ22,σ33,τ23,τ31,τ12}T (3)
{ε}={ε11,ε22,ε33,γ23,γ31,γ12} (4)
here c is the compacted 2nd order stiffness tensor (6×6).
For the TI medium, without loss of generality, we assume the symmetric axis is parallel to X3 axis, as shown in
The five independent constants are c11, c33, c44, c66 and c13, and c12 is a dependent constant where c12=c11−2c66. All these parameters have been well documented in the geophysical area.
It is more convenient to rewrite the TI medium's stress and strain relation (equation (2)) as:
{ε}=[α]{σ} (6)
where α is defined as elastic compliance tensor, it relates to the stiffness tensor c as:
[α]=[c]−1 (7)
Therefore, the elements in compliance tensor α can be represented by the elastic moduli of the medium as:
where:
The physical meanings of these elastic constants are shown in
From equations (7) and (8), we can derive the relations of Cij and elastic moduli as:
Once the stiffness parameters Cij have been derived from sonic or seismic measurements, the elastic moduli E, E′, υ, υ′ and G′ can be computed from the above equations (10) to (14). It should be noted that the above equations are defined at the material Cartesian coordinate system (X3 axis parallel to the material symmetric axis). The current invention is related to a method to derive the stiffness parameters Cij from sonic or seismic measurements thus further to evaluate the elastic moduli E, E′, υ, υ′ and G′.
Some research has been done to build equations between borehole measurements and the TI stiffness parameters Cij. For example, Sinha and Norris [1993] gave the equations of these four velocities with the TI elastic constants as:
The five unknown independent constants are c11, c33, c44, c66 and C13. What can be obtained from the borehole measurements are VP, VT, VSH, VSV, θ, ρ, Vf and ρƒ. Thus the four equations (15) to (18) are not enough to solve the five stiffness parameters c11, c33, c44, c66 and c13 and further the elastic moduli E, E′, υ, υ′ and G′ based on borehole measurements.
In the Derive a Correlation Step 32, we derive a correlation between the five stiffness parameters c11, c33, c44, c66 and c13 based on core data. In one embodiment, which will be detailed later, we derive an equation between the five stiffness parameters c11, c33, c44, c66 and c13 as:
where fgain is a gain factor which will be detailed and verified later.
The details of the Derive a Correlation Step 32 will be explained later. At this stage, the five equations (15) to (19) are enough to solve the five stiffness parameters c11, c33, c44, c66 and c13. Thus, in the following Compute Stiffness Parameters Step 34, the five stiffness parameters c11, c33, c44, c66 and c13 are computed based on the borehole measurements by applying the five equations (15) to (19). Further, in the Calculate Elastic Moduli Step 36, the elastic moduli E, E′, υ, υ′ and G′ are computed by applying the equations (10) to (14).
Now we turn to details of the Derive a Correlation Step 32. We start with examining the variation of these elastic moduli along any specific direction inclined to the TI symmetric axis, which is corresponding to the case of wellbore penetration in TI formation, as shown in
Since both stress and strain are second order tensors and they transform as second order tensors, the compliance tensor α (and also the stiffness tensor c) must transform as the fourth order tensor. The transformation equation is:
α′ijkl=bipbjqbkrblsαpqrs (20)
where α
By rotating the coordinate space along X1 axis with an angle of θ to the borehole coordinate system, we get the new coordinate system X′1X′2X′3 where X′3 is the borehole axis, as shown in
Combing equations (20) and (21) and compacting the indices as described previously (equations (2) and (6)), we can get the transformation equation, as given by Lekhniskii [1963]:
α′ij=qmiqnjαmn (i,j,m,n=1,2,3,4,5,6) (22)
where qij takes the form:
This gives the TI formation elastic compliance tensor in X′1X′2X′3 coordinate system (borehole coordinate system) as:
with:
α′11=α11 (25)
α′12=cos2 θ·α12+sin2 θ·α13 (26)
α′13=sin2 θ·α12+cos2 θ·α13 (27)
α′14=0.5 sin 2θ(−α12+α13) (28)
α′22=cos4 θ·α11+0.5 sin2 2θ·α13+sin4 θ·α33+sin2 2θ·α44 (29)
α′23=0.25 sin2 2θ(α11+α33)+(sin4 θ+cos4 θ)·α13−sin2 2θ·α44 (30)
α′24=−sin θ cos3 θ·α11+0.25 sin 4θ·α13+sin3 θ cosθ·α33+0.5 sin 4θ·α44 (31)
α′33=sin4 θ·α11+0.5 sin2 2θ·α13+cos4 θ·α33 +sin2 2θ·α44 (32)
α′34=−sin3 θ cos θ·α11−0.25 sin 4θ·α13+sin θ cos3 θ·α33−0.5 sin 4θ·αa44 (33)
α′44=cos2 2θ·α44+sin2 2θ·(α11+α33−2α13) (34)
α′55=cos2 θ·α44+sin2 θ·α66 (35)
α′56=0.5 cos 2θ·(α44−α66) (36)
α′66=sin2 θ·α44+cos2 θ·α66 (37)
The non-zero a′14, α′24 and a′34 illustrate that normal stress can induce not only normal strains, but also shear strains, and versa visa. The non-zero a′56 represents that applying a shear stress in one direction can also induce shear strain in another direction. This kind of complexity can explain clearly the complexity of seismic/sonic waves transmitting in layered rock formation. Parameter a′44 represents the shear compliance in the plane of X′2X′3 along the X′3 axis (borehole axis). By comparing with other components of a′ij, we find that variation of a′44 versus θ is relatively small. Specifically, in one embodiment, it takes the value between a44 (where θ=0 or θ=90 in equation 34) and (a11+a33−2a13) (where θ=45 in equation 34).
As an approximation, we propose the assumption:
α44=ƒgain(α11+α33−2α13) (38)
where ƒgain is the gain factor which can be derived with published core and field test data. From equation (34), equation (38) assumes the Shear Modulus G′ (parallel to TI symmetric axis) is proportional to the shear module in the direction that inclined to TI symmetric axis with an angle of about 45 degree in the plane perpendicular to formation isotropy plane.
From Equations (7), (10), (11) and (12), Equation (38) can be rewritten as:
or equation (19) as we stated before:
Thus, with the derived equation (19) together with equations (15)-(18), we can solve the five stiffness parameters c11, c33, c44, c66 and C13, based on the borehole measurements VP, VT, VSH, VSV, θ, ρ, Vf and ρ71 , with the assumption that shear module parallel to symmetric axis has a correlation with other moduli. Therefore, the invention proposes a method to evaluate the TI formation elastic properties directly from sonic measurement. The model is based on mechanical deformation analysis and it assumes that the shear modulus parallel to TI symmetric axis can be approximated from other moduli.
The following part checks the accuracy of the assumption and further derives the value of the gain factor fgain from the statistical analysis of a variety of published core data and field measurements. Here we define G′45 as the shear modulus in the plane that inclined to TI symmetric axis with the angle of about 45 degree. From equation (34), we will have:
Comparing with equation (39) with equation (40), we understand that the gain factor fgain can be calculated from:
G′45=fgain G′ (41)
Three groups of data have been collected and used for check: core data published by Zhijing Wang, in the article “Seismic anisotropy in sedimentary rocks” (Geophysics, Vol. 67, NO. 5, 2002); field and core data published by Leon Thomsen, in the article “Weak Elastic Anisotropy” (Geophysics, Vol. 51, 1986); and core data published by Lev Vernik and Xingzhou Liu, in the article “Velocity anisotropy in shales: A petrophysical study” (Geophysics. Vol. 62, No. 2, 1997).
Table 1a shows measured formation rock TI stiffness tensor C, elastic moduli, and shear modulus G′45, on sand and shale formation published by Zhijing Wang, where Lith=1 and 2 represents the sand formation and shale formation respectively.
Table 1b shows measured formation rock TI stiffness tensor C, elastic moduli and shear modulus G′45, on sand and shale at various condition based on test data from Leon Thomsen, where Lith=1 and 2 represents the sand formation and shale formation respectively.
Table 1c shows measured formation rock TI stiffness tensor C, elastic moduli, and shear modulus G′45, on sand and shale at various condition based on test data from Lev Vernik and Xingzhou Liu, where Lith=1 and 2 represents the sand formation and shale formation respectively.
Results of G′ and G′45 are plotted in
The statistical study has been carried out to build the correlation gain factor fgain versus formation type and Thomsen's parameter γ. The results are shown in Table 2.
A case study has been attached to demonstrate the process of determining the TI formation elastic properties using the above invention. Specifically, we used the field measurement of sonic data from SONIC SCANNER of Schlumberger (as shown in
The foregoing description of the preferred and alternate embodiments of the present invention has been presented for purposes of illustration and description. It is not intended to be exhaustive or limit the invention to the precise examples described. Many modifications and variations will be apparent to those skilled in the art. The embodiments were chosen and described in order to best explain the principles of the invention and its practical application, thereby enabling others skilled in the art to understand the invention for various embodiments and with various modifications as are suited to the particular use contemplated. It is intended that the scope of the invention be defined by the accompanying claims and their equivalents.
