CN1601304A - Ground stress multifrequency reverse demonstration method of dipole trans verse wave well logging - Google Patents

Abstract

The invention is used in the field of oil field logging technology, mainly resolving the defect of existent earth stress detection technology which has not any one non-destructive examination method. Said invention is characterized by that it includes the following steps: making cross dipole array acoustic log data undergo the process of angle rotation treatment, then obtaining two main time sequences, on said basis extracting flexural wave dispersion curves of two principal directions, utilizing data of n frequency points on the flexural wave dispersion curves of two principal polarization directions to create n equalities according to two formulae, in which said formulae can be obtained by inversion of well mode wave acoustic elastic equation, in which n is greater than or equal to 6, and atilizing resolution of sain n equalities so as to can obtain maximum main stress SH and minimum main stress Sh of formation.

Description

Dipole shear wave logging terrestrial stress multifrequency inversion method

Technical field:

The present invention relates to be used in the oil reservoir logging field carrying out carrying out the method that terrestrial stress detects, especially relate to a kind of dipole acoustic log data of utilizing and carry out the method that terrestrial stress detects at oil-water well.

Background technology:

Casing damage in oil-water well is the ubiquitous great petroleum engineering problem in domestic and international each oil field.Sleeve pipe is damaged, loses productive capacity because of the effect that is subjected to unusual external force, not only directly influence the production of this well, economic loss is huge, and casing damaged well also can cause regional injection and extraction system imperfection, production rate decline quickening etc., indirect loss is bigger, and the investment aspect the casing damaged well overhaul is also very big, and therefore prediction and prevention sleeve pipe damage has become petroleum engineering and well logging field question of common concern.Achievement in research is in recent years decreased reason to cover certain understanding, and the factor that causes cover to decrease mainly is divided into three aspects: the mechanical damage of flow-induced corrosion, abnormal formation stress effect, operation.Remove sleeve pipe damage and flow-induced corrosion that uncertain borehole operation causes, the sleeve pipe that other reason causes damages all relevant with the ABNORMAL STRESS variation of well week, and the measurement of therefore carrying out all abnormal formation stress sizes of well has become overlaps the key issue of decreasing prediction.At present both at home and abroad there are two in oil-water well, carrying out the achievement in research that terrestrial stress detects---wellhole burst apart method and hydraulic fracturing, borehole wall avalanche method is with the wellhole stress concentration phenomenon theoretical foundation in the Elasticity.Under plane strain condition, wellhole is subjected to the effect of two-dimensional stress field, infinite distant place, and the shear fracture criterion that stress distribution around the wellhole and rock of borehole are destroyed is determined the stress state equation of breakdown point.Utilize well-calipering apparatus can measure the shape of borehole wall avalanche, width and the degree of depth, thereby determine the size of well week terrestrial stress.The size of utilizing this method to measure terrestrial stress can only be carried out in the few well section that has the wellhole avalanche, and is not only representative poor, and cost dearly, complicated operation.The theoretical foundation of hydraulic fracturing is identical with borehole wall avalanche ratio juris, all is the wellhole stress concentration phenomenon in the Elasticity.If a wellhole section degree of depth packing is got up, and to this section wellhole injection water under high pressure, parting pressure when initial cracking takes place the record borehole wall and the clossing pressure when reducing pressure gradually to crack closure, maximum, the least principal stress in the calculation of Tensile Strength well week by parting pressure, clossing pressure and rock.This method is precondition with the pressure break, also is to carry out well week geostress survey under the situation of destroying wellhole, and because Tensile Strength of Rock T is difficult to measure, is difficult to obtain the accurate stress size.By above narration as can be known, borehole wall avalanche method and hydraulic fracturing all are that the destruction with wellhole is prerequisite, therefore say the terrestrial stress detection that does not also have a kind of lossless detection method to can be used for actual wellhole at present.

Summary of the invention:

In order to overcome the deficiency that does not have a kind of lossless detection method that exists in the existing terrestrial stress detection technique, the invention provides a kind of dipole shear wave logging terrestrial stress multifrequency inversion method, this kind dipole shear wave logging terrestrial stress multifrequency inversion method has the advantages that to utilize the sound logging data to carry out the detection of well week abnormal formation stress under the situation that can be implemented in harmless sleeve pipe.

Technical scheme of the present invention is: this kind dipole shear wave logging terrestrial stress multifrequency inversion method, it is characterized in that: cross dipole array sonic log data are carried out obtaining two main time serieses after the angle rotation, extract the flexural wave dispersion curve of two principal directions on this basis, utilize the data based formula (5) and the formula (6) of two n Frequency points on the flexural wave dispersion curve of main polarization direction to set up n equation, wherein n is more than or equal to 6;

$v_{+}\left({\mathrm{\ω}}_n\right)-v_P\left({\mathrm{\ω}}_n\right)-2=(S_H+S_h)\left[C_{1i}\right]\left(\right[{\hat{R}}_{i,j}^S\left({\mathrm{\ω}}_n\right)\left]\right[2\mathrm{\π},\mathrm{0,2}\mathrm{\π},\mathrm{0,0}{]}^T)---\left(5\right)$

$v_{-}\left({\mathrm{\ω}}_n\right)=(S_H-S_h)\left[C_{\mathrm{li}}\right]\left(\right[{\hat{R}}_{i,j}^S\left({\mathrm{\ω}}_n\right)\left]\right[0,\mathrm{\π},0,-\mathrm{\π},\mathrm{\π}{]}^T)---\left(6\right)$

In the formula: $v_P\left({\mathrm{\ω}}_n\right)=P_0\left[C_{\mathrm{li}}\right]\left[{\hat{R}}_{i,j}^P\right][2\mathrm{\π},\mathrm{0,2}\mathrm{\π},\mathrm{0,0}{]}^T$ Represent the influence of wellbore pressure, v ₊(ω _n)=[v _x(ω _n)+v _y(ω _n)]/v _R(ω _n) be two main polarization directions flexural wave speed relatively and, v _-(ω _n)=[v _x(ω _n)-v _y(ω _n)]/v _R(ω _n) be the relative mistake of the flexural wave speed of two main polarization directions, wherein v _x(ω _n) and v _y(ω _n) the parallel flexural wave frequency dispersion phase velocity that reaches when vertical with biggest principal stress direction with biggest principal stress direction of expression respectively, corresponding with a point on the corresponding dispersion curve, v _R(ω _n) be the flexural wave phase velocity of reference state, [C _1i]=[1, c ₁₁₁, c ₁₁₂, c ₁₂₃] be stratum three-order elastic modulus matrix,

With

Be respectively P ₀And S _HGet unit pressure (S _h=0) the integral coefficient R the time _Ij, S _HAnd S _hRepresent maximum, least principal stress of well week respectively; Just can obtain the major principal stress S on stratum by finding the solution this n equation _HWith least principal stress S _hThereby, just can realize the abnormal formation stress inverting of well week.

The present invention has following beneficial effect: because when taking said method to measure, can utilize the lithology and density logging data on cross dipole acoustic logging data and stratum to be finally inversed by well week abnormal formation stress, realize under the situation of not damaging sleeve pipe, detecting well week abnormal formation stress, preferably resolved the key issue in the cover damage prediction of oil field.

Description of drawings:

Accompanying drawing 1 is a process flow diagram of the present invention;

Accompanying drawing 2 is to utilize the present invention that the XX of Daqing oil field Portugal well is carried out well week abnormal formation stress inversion result displayed map.

Embodiment:

The present invention carries out under following enlightenment: Rai, C.S., and Hanson, K.E. wait people's anisotropic measurement of shear wave velocity in 1988 pass through sedimentogeneous rock to find, the acoustic elasticity phenomenon of rock is very obvious, acoustic velocity is subjected to the influence of applied stress very big in the rock, and p-and s-wave velocity increases along with the increase of applied stress, and dipole shear wave can split into the speed dipole shear wave parallel with vertical with applied stress under the effect of applied stress.Based on this, the inventor considers based on acoustic theory, selects for use cross dipole SWAL data to be finally inversed by the size of terrestrial stress, how to carry out correct inverting certainly, needs the problem of considering exactly.

For this reason, the inventor at first sets up and is subjected to even hydrostatic force active state, i.e. reference state wellhole mode wave acoustic elasticity equation.The wellhole state that selection is subjected to even hydrostatic force effect is a reference state, and with respect to reference state, the outer homogeneous solid medium of well is subjected to the even static stress S of single shaft that unlimited distance applies along x direction and y direction simultaneously _HAnd S _hEffect.Under the acting in conjunction of these quiet bias stress, set out by the equation of motion, consider the boundary condition at borehole wall place, deriving obtains expression matrix form along the DIPOLE FLEXURAL WAVES acoustic elasticity equation of maximum (φ=0 °), minimum (φ=90 °) principal direction of stress polarization:

$\frac{v_x\left(\mathrm{\ω}\right)-v_R\left(\mathrm{\ω}\right)}{v_R\left(\mathrm{\ω}\right)}=\mathrm{CR}{\mathrm{\Θ}}_x---\left(1\right)$

$\frac{v_y\left(\mathrm{\ω}\right)-v_R\left(\mathrm{\ω}\right)}{v_R\left(\mathrm{\ω}\right)}=\mathrm{CR}{\mathrm{\Θ}}_y---\left(2\right)$

C＝[C _1i]＝[1，C ₁₁₁，c ₁₁₂，c ₁₂₃]，i＝1，2，3，4

$R={\∫}_{r_0}^{\∞}\mathrm{rdr}\left[I_{\mathrm{ij}}(r,n,P_0,S_H,S_h)\right],i=\mathrm{1,2},\mathrm{3,4},j=\mathrm{1,2,3,4},6$

Θ _x＝[π，π/2，π，-π/2，π/2] ^T

Θ _y＝[π，-π/2，π，π/2，-π/2] ^T

The v of equation left end _x(ω) and v _yFlexural wave frequency dispersion phase velocity when (ω) representing φ=0 ° (with well week maximum horizontal principal stress parallel direction) and φ=90 ° (with all maximum horizontal principal stress parallel directions of well) respectively, corresponding with a point on the corresponding dispersion curve, be subjected to the flexural wave speed v of two main polarization directions behind the stress _x(ω) and v _y(ω) can utilize cross dipole array sound acoustic wave train logging data to extract by phase velocity obtains.v _RBe the flexural wave phase velocity of reference state (ω), equal the flexural wave phase velocity before wellhole is subjected to the abnormal formation stress effect.The C of equation right-hand member is the three-order elastic modulus matrix of formation rock, and three-order elastic modulus wherein is a unknown quantity.Θ _x, Θ _yIt is the column vector that characterizes doublet source side.R is (4*5) matrix, and representative comprises well week major principal stress S _H, least principal stress S _hAnd in-hole fluid pressure P ₀The integrand integration radially of influence has wherein comprised physical quantity major principal stress S to be asked _HWith least principal stress S _hCan calculate integral coefficient R=[R by reference state wellhole and formation parameter _Ij].Equation (1) and (2) have been set up by measurable cross dipole array sound all-wave can extract the speed shear wave velocity and maximum to be asked, the relation of least principal stress that obtains.

Owing in acoustic elasticity equation (1), (2) 6 inverted parameters: c are arranged ₁₁₁, c ₁₁₂, c ₁₂₃, S _H, S _hAnd P ₀This 6 parameters and frequency-independent, and the speed shear wave velocity that is extracted by cross dipole array sonic log full-wave train is the function of frequency, be that equation (1), (2) are the functions of frequency, so consider to utilize two a plurality of Frequency points on the flexural wave dispersion curve of main polarization direction to set up inversion equation.

Θ _xAnd Θ _yIn the symbols of element of the 2nd, 4 and 5 row opposite, and other symbol of element is identical.Integral coefficient R=[R _Ij], i=1,2,3,4, j=1,2,3,4,6.R _IjSubscript j when getting odd number, only comprise stress P ₀(S _H+ S _h); When subscript j gets even number, only comprise (S _H-S _h); So have do signed magnitude arithmetic(al) respectively by (1) and (2) two formulas

$\frac{v_x\left(\mathrm{\ω}\right)+v_y\left(\mathrm{\ω}\right)}{v_R\left(\mathrm{\ω}\right)}=2\left[C_{1i}\right]\left\{P_0\right[{\hat{R}}_{i,j}^P]+[{\hat{R}}_{i,j}^S\left](S_H+S_h)\right\}[2\mathrm{\π},\mathrm{0,2}\mathrm{\π},\mathrm{0,0}{]}^T--\left(3\right)$

$\frac{v_x\left(\mathrm{\ω}\right)-v_y\left(\mathrm{\ω}\right)}{v_R\left(\mathrm{\ω}\right)}=\left[C_{1i}\right]\left[{\hat{R}}_{i,j}^S\right][0,\mathrm{\π},0,-\mathrm{\π},\mathrm{\π}{]}^T(S_H-S_h)--\left(4\right)$

The inversion equation of following form will just can be obtained after equation (3) and (4) expansion

$v_{+}\left({\mathrm{\ω}}_n\right)-v_P\left({\mathrm{\ω}}_n\right)-2=(S_H+S_h)\left[C_{\mathrm{li}}\right]\left(\right[{\hat{R}}_{i,j}^S\left({\mathrm{\ω}}_n\right)\left]\right[2\mathrm{\π},\mathrm{0,2}\mathrm{\π},\mathrm{0,0}{]}^T)---\left(5\right)$

$v_{-}\left({\mathrm{\ω}}_n\right)=(S_H-S_h)\left[C_{\mathrm{li}}\right]\left(\right[{\hat{R}}_{i,j}^S\left({\mathrm{\ω}}_n\right)\left]\right[0,\mathrm{\π},0,-\mathrm{\π},\mathrm{\π}{]}^T)---\left(6\right)$

In the formula: $v_P\left({\mathrm{\ω}}_n\right)=P_0\left[C_{\mathrm{li}}\right]\left[{\hat{R}}_{i,j}^P\right][2\mathrm{\π},\mathrm{0,2}\mathrm{\π},\mathrm{0,0}{]}^T$ Represent the influence of wellbore pressure.v ₊(ω _n)=[v _x(ω _n)+v _y(ω _n)]/v _R(ω _n) be two main polarization directions flexural wave speed relatively and, v _-(ω _n)=[v _x(ω _n)-v _y(ω _n)]/v _R(ω _n) be the relative mistake of the flexural wave speed of two main polarization directions, wherein v _x(ω _n) and v _y(ω _n) the parallel flexural wave frequency dispersion phase velocity that reaches when vertical with biggest principal stress direction with biggest principal stress direction of expression respectively, corresponding with a point on the corresponding dispersion curve, v _R(ω _n) be the flexural wave phase velocity of reference state.[C _1i]=[1, c ₁₁₁, c ₁₁₂, c ₁₂₃] be stratum three-order elastic modulus matrix, With Be respectively P ₀And S _HGet unit pressure (S _h=0) the integral coefficient R the time _Ij, S _HAnd S _hIt is respectively maximum, least principal stress of well week.According to equation (5), (6), utilize the data of two individual Frequency points of n (n=6) on the flexural wave dispersion curve of main polarization direction to set up n equation, just can obtain the major principal stress S on stratum by finding the solution these equatioies _HWith least principal stress S _hThereby, just can realize the abnormal formation stress inverting of well week.

The invention will be further described below in conjunction with accompanying drawing:

As shown in Figure 1, this kind dipole shear wave logging terrestrial stress multifrequency inversion method, its step obtains two main time serieses at first cross dipole array sonic log data being carried out after the angle rotation, extract the flexural wave dispersion curve of two principal directions on this basis, its specific implementation process is: adopt spectral domain weighting analogue method to carry out the flexural wave dispersion curve to fast, the slow flexural wave that utilizes the cross dipole acoustic logging method to obtain and extract, corresponding data point promptly is (ω on the flexural wave dispersion curve _n, v _x(ω _n)).The array waveform of the phase vectors that this method utilization makes up after to Fourier transform carries out phase matching, and with Gaussian window weighting structure fitness function, thereby determine that by the maximal value of fitness function under each frequency the slowness of flexural wave is the inverse of speed, the specific implementation method is:

Time domain data by the n road of N receiver record is dn (t), at first it is obtained Dn (ω) as Fourier transform.

The structure objective function:

$f(\mathrm{\ω},p_z)=\frac{\left|D^{+}\left(\mathrm{\ω}\right)S(\mathrm{\ω},p_z)\right|}{\sqrt{D^{+}\left(\mathrm{\ω}\right)D\left(\mathrm{\ω}\right)}\sqrt{S^{+}(\mathrm{\ω},p_z)S(\mathrm{\ω},p_z)}}---\left(7\right)$

Wherein

$D\left(\mathrm{\ω}\right)=\left[\begin{array}{c}{D}_1\left(\mathrm{\ω}\right)\\ D_2\left(\mathrm{\ω}\right)\\ M\\ D_N\left(\mathrm{\ω}\right)\end{array}\right]$ $S(\mathrm{\ω},p_z)=\left[\begin{array}{c}1\\ \exp \left(\mathrm{i\ω}{p}_z\mathrm{\Δz}\right)\\ \exp \left(2\mathrm{i\ω}{p}_z\mathrm{\Δz}\right)\\ M\\ \exp \left((N-1)\mathrm{i\ω}{p}_z\mathrm{\Δz}\right)\end{array}\right]$

Vector S (ω, p _z) in element be the phase shift of receiver with respect to first receiver, Δ z receives spacing.Have only in the uniform dielectric under a kind of mode wave situation of propagating along the z direction, the slowness of this mode wave is Pz when frequency is ω.Do normalization make fitness function f (ω, maximal value Pz) is 1, (ω, maximal value Pz) will provide true slowness value to different frequency minor function f.In actual well-log information is handled, at first wave train data are encrypted sampling again, do the frequency domain data that FFT obtains encrypting sampling again, then with following function replacement f (ω Pz) obtains fitness function:

$F({\mathrm{\ω}}_m,p_z)=\underset{j=l-m}{\overset{l+m}{\mathrm{\Σ}}}W({\mathrm{\ω}}_j,{\mathrm{\ω}}_m)f({\mathrm{\ω}}_m,p_z)--\left(8\right)$

Wherein, W (ω _j, ω _m)=exp[-(ω _m-ω _j) ²/ (2 σ ²)] be gaussian weighing function.Under each frequency, above-mentioned function is scanned, determine its maximal value, thereby obtain the slowness dispersion curve with certain slowness scope.

Then, utilize the data based formula (5) of two n Frequency points on the flexural wave dispersion curve of main polarization direction and formula (6) to set up n equation, wherein n is more than or equal to 6;

$v_{+}\left({\mathrm{\ω}}_n\right)-v_P\left({\mathrm{\ω}}_n\right)-2=(S_H+S_h)\left[C_{\mathrm{li}}\right]\left(\right[{\hat{R}}_{i,j}^S\left({\mathrm{\ω}}_n\right)\left]\right[2\mathrm{\π},\mathrm{0,2}\mathrm{\π},\mathrm{0,0}{]}^T)---\left(5\right)$

$v_{-}\left({\mathrm{\ω}}_n\right)=(S_H-S_h)\left[C_{\mathrm{li}}\right]\left(\right[{\hat{R}}_{i,j}^S\left({\mathrm{\ω}}_n\right)\left]\right[0,\mathrm{\π},0,-\mathrm{\π},\mathrm{\π}{]}^T)---\left(6\right)$

In the formula: $v_P\left({\mathrm{\ω}}_n\right)=P_0\left[C_{\mathrm{li}}\right]\left[{\hat{R}}_{i,j}^P\right][2\mathrm{\π},\mathrm{0,2}\mathrm{\π},\mathrm{0,0}{]}^T$ Represent the influence of wellbore pressure, v+ (ω _n)=[v _x(ω _n)+v _y(ω _n)]/v _R(ω _n) be two main polarization directions flexural wave speed relatively and, v-(ω _n)=[v _x(ω _n)-v _y(ω _n)]/v _R(ω _n) be the relative mistake of the flexural wave speed of two main polarization directions, wherein v _x(ω _n) and v _y(ω _n) the parallel flexural wave frequency dispersion phase velocity that reaches when vertical with biggest principal stress direction with biggest principal stress direction of expression respectively, corresponding with a point on the corresponding dispersion curve, v _R(ω _n) be the flexural wave phase velocity of reference state, [C _1i]=[1, c ₁₁₁, c ₁₁₂, c ₁₂₃] be stratum three-order elastic modulus matrix,

With

Be respectively P ₀And S _HGet unit pressure (S _h=0) the integral coefficient R the time _Ij, S _HAnd S _hRepresent maximum, least principal stress of well week respectively;

Just can obtain the major principal stress S on stratum by finding the solution this n equation _HWith least principal stress S _hThereby, just can realize the abnormal formation stress inverting of well week.

Fig. 2 utilizes SWAL multifrequency terrestrial stress inversion method the XX of Daqing oil field Portugal well to be carried out the test result of well week abnormal formation stress inverting, at first utilize spectral domain weighting analogue method that the dipole array all-wave of cross dipole array sonic log instrument record is carried out dispersion analysis, carried out the inverting of multifrequency terrestrial stress on the basis of dispersion analysis, specific practice is:

1,, eight array dipole sound all-waves of cross dipole array sonic log instrument record is obtained Dn (ω) as Fourier transform in each depth point.To each Frequency point, in certain velocity range, ask objective function F (ω _m, p _z) maximal value, calculated the speed shear wave velocity of each depth point, thereby the speed dispersion curve dispersion curve that obtains each depth point is shown in Fig. 2 three roads with frequency change.

2, obtain dispersion curve after, get the data point (ω on fast, the slow flexural wave dispersion curve _n, v ₊(ω _n)) and (ω _n, v _-(ω _n)), four Frequency points (both n=4) are got in each depth point, bring into and set up inversion equation in the inverse model.

Utilization is carried out the inverting of multifrequency terrestrial stress according to the program of inverse model establishment, inverting obtains maximum, the least principal stress that well week exists, shown in Fig. 2 four roads, the 4th road provided successively from left to right stress that superstratum pressure, multifrequency inverting obtain and with stress difference (second and third curve) and normalized speed flexural wave rate curve.The abnormal formation stress that institute's corresponding well section inverting obtains among the figure is 2.89Mpa, and this well section is corresponding to the PI6 oil reservoir, and its offset well occurred cover in 2003 in this layer position to be decreased, and testing result and geological analysis coincide.

Use the present invention and realized utilizing under the lossless case sound logging data to carry out the detection of well week abnormal formation stress, need not destroy wellhole and can realize the terrestrial stress size detection.Can utilize continuous SWAL data to carry out the well week terrestrial stress inverting of full well, cost is low, and is simple to operate.The well week that has overcome present existence can not continuous coverage in wellhole avalanche method and the hydraulic fracturing in the abnormal formation stress detection method and the shortcoming that must detect under wellhole destruction situation.In addition, set up the multi-frequency inverse model that utilizes cross dipole sound logging inverting abnormal formation stress by the present invention, model form is succinct, the inverting meaning is clear and definite, utilize inverse model inverting well week terrestrial stress of can directly programming, because directly utilize and the closely-related SWAL data of terrestrial stress size, inversion result is more accurate.

Claims (1)

1, a kind of dipole shear wave logging terrestrial stress multifrequency inversion method, it is characterized in that: cross dipole array sonic log data are carried out obtaining two main time serieses after the angle rotation, extract the flexural wave dispersion curve of two principal directions on this basis, utilize the data based formula (5) and the formula (6) of two n Frequency points on the flexural wave dispersion curve of main polarization direction to set up n equation, wherein n is more than or equal to 6;

$v_{+}\left({\mathrm{\ω}}_n\right)-v_P\left({\mathrm{\ω}}_n\right)-2=(S_H+S_h)\left[C_{1i}\right]\left({[\hat{R}}_{i,j}^S\left({\mathrm{\ω}}_n\right)\right]{[2\mathrm{\π},\mathrm{0,2}\mathrm{\π},\mathrm{0,0}]}^T)----\left(5\right)$

$v_{-}\left({\mathrm{\ω}}_n\right)=(S_H-S_h)\left[C_{1i}\right]\left(\right[{\hat{R}}_{i,j}^S\left({\mathrm{\ω}}_n\right)\left]{[0,\mathrm{\π},0,-\mathrm{\π},\mathrm{\π}]}^T\right)----\left(6\right)$

In the formula: $v_P\left({\mathrm{\ω}}_n\right)=P_0\left[C_{\mathrm{li}}\right]\left[{\hat{R}}_{i,j}^P\right]{[2\mathrm{\π},\mathrm{0,2}\mathrm{\π},\mathrm{0,0}]}^T$ Represent the influence of wellbore pressure, v+ (ω _n)=[v _x(ω _n)+v _y(ω _n)]/v _R(ω _n) be two main polarization directions flexural wave speed relatively and, v-(ω _n)=[v _x(ω _n)-v _y(ω _n)]/v _R(ω _n) be the relative mistake of the flexural wave speed of two main polarization directions, wherein v _x(ω _n) and v _y(ω _n) the parallel flexural wave frequency dispersion phase velocity that reaches when vertical with biggest principal stress direction with biggest principal stress direction of expression respectively, corresponding with a point on the corresponding dispersion curve, v _R(ω _n) be the flexural wave phase velocity of reference state, [C _1i]=[1, c ₁₁₁, c ₁₁₂, c ₁₂₃] be stratum three-order elastic modulus matrix,

With

Be respectively P ₀And S _HGet unit pressure (S _h=0) the integral coefficient R the time _Ij, S _HAnd S _hRepresent maximum, least principal stress of well week respectively; Just can obtain the major principal stress S on stratum by finding the solution this n equation _HWith least principal stress S _hThereby, just can realize the abnormal formation stress inverting of well week.

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