CN112099099B - Method for estimating effective magnetization dip angle by magnetic force abnormity in well - Google Patents

Method for estimating effective magnetization dip angle by magnetic force abnormity in well Download PDF

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CN112099099B
CN112099099B CN202010795890.5A CN202010795890A CN112099099B CN 112099099 B CN112099099 B CN 112099099B CN 202010795890 A CN202010795890 A CN 202010795890A CN 112099099 B CN112099099 B CN 112099099B
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史可
刘双
张恒磊
胡祥云
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China University of Geosciences
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Abstract

Compared with the method for estimating the magnetization dip angle by using ground data, the method for estimating the effective magnetization dip angle of the section based on the three-component data in the well has the advantages that the data in the well is related to the relative positions of the well and an abnormal body, so that the data in the well cannot be correctly converted by using the traditional ground frequency domain bit field conversion factor, the frequency domain conversion factor of the magnetic data in the well is provided, and the effective magnetization dip angle is finally estimated by combining the method for cross-correlation estimation of the magnetic total field modulus abnormality and the converted data.

Description

Method for estimating effective magnetization dip angle by magnetic force abnormity in well
Technical Field
The method belongs to the field of exploration geophysics, and particularly relates to a method for estimating an effective magnetization dip angle by magnetic force abnormity in a well.
Background
The magnetization direction and magnitude of underground anomalies are affected not only by the induced magnetization of the earth's magnetic field, but also by remanence. In the presence of residual magnetization, the magnetization direction of the anomaly may deviate from the direction of the geomagnetic field, so that the induced magnetization vectorizes and may distort the morphology of the magnetic anomaly, thereby complicating the generation of the anomalous field and making the data obtained by magnetic prospecting difficult to interpret. Therefore, it is of great significance to estimate the actual magnetization direction of the anomaly. In the problem of estimating the magnetization direction, Gerovska et al propose a method for estimating the magnetization direction based on the cross-correlation between magnetic total field modulus anomaly and magnetic anomaly polarization data, and then propose a method for estimating the effective magnetization direction of a section by using one-dimensional data in a well based on the principle of the method, and obtain the difference from the method for estimating the magnetization direction by using two-dimensional data on the ground.
Disclosure of Invention
A method for estimating effective dip angle of magnetization in a well for magnetic anomalies, comprising the steps of:
s1, based on three components of magnetic anomaly in well including vertical component ZaHorizontal component HaxAnd horizontal component HayCalculating the data to obtain the magnetic abnormal modulus Ta；
S2, for Za、HaxAnd HayThe three components are respectively obtained by Fourier transformA corresponding frequency spectrum;
s3, setting an initial effective magnetization inclination angle degree;
s4, calculating a vertical magnetization direction conversion factor and a horizontal magnetization direction conversion factor based on the effective magnetization inclination angle degree set in the S3 and the relative position relation between the well and the abnormal body;
S5、Zafrequency spectrum multiplied by the perpendicular magnetization direction conversion factor, HaxAnd HayMultiplying by horizontal magnetization direction conversion factors respectively;
s6, respectively carrying out inverse Fourier transform on the frequency spectrums multiplied by the conversion factors to obtain Z converted from the magnetization direction to the vertical directionaData and H with magnetization direction turned to horizontal directionaxAnd HayData;
s7, converting the Z converted in S6a、HaxAnd HayRespectively, the magnetic abnormal modulus T in S1aPerforming correlation calculation, and recording the obtained correlation numerical value;
s8, increasing the effective magnetization tilt angle by n °, 0 ° < n ° <360 °, and repeating S4 to S8, and when the effective magnetization tilt angle increases to a prescribed degree, stopping the circulation;
s9, drawing Z based on the recorded correlation valuea、HaxAnd HayThree correlation curves of (1), find ZaMinimum point of correlation curve and HaxAnd HayAnd the degree corresponding to the maximum point of the correlation curve and the obtained maximum point is the estimated effective magnetization dip angle.
Further, magnetic abnormal modulus T in S1aThe calculation formula of (a) is as follows:
Figure BDA0002625598260000021
wherein Ta represents a magnetic abnormal modulus, ZaRepresents the vertical component, HaxRepresents a horizontal component, HayRepresenting the horizontal component.
Further, the initial effective inclination angle of magnetization in S3 is set to x °, where 0 ° ≦ x ° ≦ 360 °.
Further, the frequency domain conversion factor in S4 is related to the relative positions of the well and the anomaly, and the specific relationship is as follows: in the established coordinate system, when the well is located in the positive direction of the abnormal body, i.e. in the positive direction of the corresponding abscissa axis, in the section under study, the magnetization direction conversion factor is:
Figure BDA0002625598260000031
α2＝cosIs2 γ2＝sinIs2 α1＝cosIs1 γ1＝sinIs1
wherein w represents a magnetization direction conversion factor, Is2To convert the angle of inclination of magnetization, Is1The original magnetization dip angle is obtained;
when the well is located in the negative direction of the abnormal body, namely the negative direction of the abscissa axis, the conversion factor is as follows:
Figure BDA0002625598260000032
α2＝cosIs2 γ2＝sinIs2 α1＝cosIs1 γ1＝sinIs1
wherein, Is2To convert the angle of inclination of magnetization, Is1At an angle of inclination of the original magnetization, α1，γ1Is the cosine of the original magnetization direction, alpha2，γ2Is the new magnetization direction cosine;
when I iss2At 90 °, the conversion factor is the perpendicular magnetization direction conversion factor; when I iss2At 0 °, the conversion factor is the horizontal magnetization direction conversion factor.
Further, the specific derivation procedure of the vertical magnetization direction conversion factor and the horizontal magnetization direction conversion factor in S4 is as follows:
firstly, according to the extension of the geologic body in three directions, the magnetic body is divided into a two-degree body and a three-degree body, for the ground, a data acquisition surface and a line are always positioned in the upper half space of the abnormal body, namely, under the established space rectangular coordinate system, the data acquisition is carried out on a plane with Z being 0, the center of the abnormal body is positioned on a plane with Z being 300, the Z axis is vertically and downwards positive, for the well, the abnormal body is assumed to be the two-degree body, the established coordinate system X axis is perpendicular to the trend of the abnormal body, the Y axis is parallel to the trend of the abnormal body, the data acquisition is carried out along the well axis, namely, the Z axis direction, because the well axis is in the positive direction of the abnormal body, namely, the positive X direction, and the negative direction of the abnormal body, namely, the negative X direction, therefore, the data in the well is related to the relative position of the well and the abnormal body, therefore, when the frequency domain conversion factor of the two-degree body is deduced, namely, under, the well is in both the positive and negative directions of the anomaly;
for the second degree body, the horizontal component HayZero, its potential-field conversion factor is derived from the dirichlet problem of the two-dimensional laplace equation:
Figure BDA0002625598260000041
wherein X and Z represent X-axis and Z-axis coordinates, uxxIs a quadratic derivative of the function u over x, uzzIs the second derivative of the function u over z,
Figure BDA0002625598260000042
as boundary conditions, i.e. measurement data;
case 1 well is in the negative direction of the anomaly:
for such cases, the solution of the dirichlet problem of the two-dimensional laplace equation is:
Figure BDA0002625598260000043
wherein u is a solution of the dirichlet problem, δ represents a Z-axis coordinate of any point on the boundary, T (X, Z) satisfies the above formula, T is a magnetic anomaly, and X and Z represent X-axis and Z-axis coordinates, that is:
Figure BDA0002625598260000044
for T (0, z) and
Figure BDA0002625598260000045
making a one-dimensional Fourier transform on Z:
the obtained magnetic anomaly frequency spectrum expression is as follows:
Figure BDA0002625598260000046
wherein S isTIs a magnetic anomaly spectrum; f is frequency, i is imaginary unit;
the equation (1) is differentiated by the differential theorem:
Figure BDA0002625598260000047
Figure BDA0002625598260000048
it can be found that for case 1: the X-direction derivative factor is 2 pi f, and the Z-direction derivative factor is 2 pi if;
giving a two-degree poisson formula:
Figure BDA0002625598260000051
Figure BDA0002625598260000052
Zaperpendicular component of magnetic anomaly, HaIs a magnetic abnormal horizontal component, M is a mode of magnetization, alpha and gamma are direction cosines of magnetization, delta is density of a geologic body, G is a universal gravitation constant, and v is a gravitational potential; v. ofxzAnd vzxIs the second partial derivative of v in the X and Z directions, vzzIn the Z direction of vSecond derivative, vxxIs the second derivative in the X direction of v;
let the gravitational potential frequency spectrum be SvAnd carrying out derivation in X and Z directions on the gravitational potential spectrum:
Figure BDA0002625598260000053
Figure BDA0002625598260000054
to pair
Figure BDA0002625598260000055
Carrying out second-order derivation:
Figure BDA0002625598260000056
Figure BDA0002625598260000057
Figure BDA0002625598260000058
substituting the above formula into (4) and (5) can obtain:
Figure BDA0002625598260000059
Figure BDA00026255982600000510
wherein S iszFor the magnetically anomalous vertical component spectrum, SHIs a magnetic anomaly horizontal component spectrum;
can be obtained from the formulae (6) and (7) Sz，SHThe interconverting component conversion factors are:
Sz→SH：-i (8)
SH→Sz：i (9)
new magnetization direction magnetic potential of two-degree body, i.e. alpha2，γ2For the new magnetization direction cosine is:
Figure BDA0002625598260000061
U2is the magnetic potential of the new magnetization direction, t1Indicating the original direction of magnetization, i.e. the new magnetic potential is along the original direction of magnetization t1Integral of curve in the reverse direction
Order SHIs HaSpectrum of (1), HaThe spectrum expression is as follows:
Figure BDA0002625598260000062
order SZIs ZaSpectrum of (2), ZaThe spectrum expression is as follows:
Figure BDA0002625598260000063
the component conversion factor obtained by using the equations (8) and (9) can be obtained:
Figure BDA0002625598260000064
the expression (10), the expression (11), the expression (13) can be used for obtaining:
Figure BDA0002625598260000065
wherein alpha is1，γ1Is the original magnetization direction cosine;
the new component (H) can be deduced from the relationship between the magnetic potential and the componenta2，Za2) Spectrum meterThe expression is as follows:
Figure BDA0002625598260000068
Figure BDA0002625598260000066
thus for wells located in the negative direction of the anomaly, the arbitrary magnetization direction conversion factor is:
Figure BDA0002625598260000067
wherein alpha is1，γ1Is the cosine of the original magnetization direction, alpha2，γ2Is the new magnetization direction cosine;
case 2 well in the positive direction of the anomaly:
for such cases, the solution of the dirichlet problem of the two-dimensional laplace equation is:
Figure BDA0002625598260000071
t (x, z) satisfies the above equation, T being a magnetic anomaly, i.e.:
Figure BDA0002625598260000072
for T (0, z) and
Figure BDA0002625598260000073
making a one-dimensional Fourier transform on Z:
the obtained magnetic field spectrum expression is as follows:
Figure BDA0002625598260000074
STin the case of a spectrum of magnetic anomalies,f is frequency, i is imaginary unit;
equation (17) is differentiated by the differential theorem:
Figure BDA0002625598260000075
Figure BDA0002625598260000076
it can be found that for case 2: the X-direction derivative factor is-2 pi f and the Z-direction derivative factor is 2 pi if
Giving a two-degree poisson formula:
Figure BDA0002625598260000077
Figure BDA0002625598260000078
Zaperpendicular component of magnetic anomaly, HaIs the horizontal component of magnetic anomaly, M is the modulus of magnetization, α, γ are the direction cosines of magnetization, δ is the density of the geologic body, G is the universal gravitation constant, and v is the gravitational potential.
Let the gravitational potential frequency spectrum be SvAnd carrying out derivation in X and Z directions on the gravitational potential spectrum:
Figure BDA0002625598260000079
Figure BDA0002625598260000081
to pair
Figure BDA0002625598260000082
Carrying out second-order derivation:
Figure BDA0002625598260000083
Figure BDA0002625598260000084
Figure BDA0002625598260000085
substituting the above formula into (20) and (21) can obtain:
Figure BDA0002625598260000086
Figure BDA0002625598260000087
from the formulae (22) and (23), Sz，SHThe interconverting component conversion factors are:
Sz→SH：i (24)
SH→Sz：-i (25)
the new magnetization direction magnetic potential of the second-degree body is as follows:
Figure BDA0002625598260000088
α2，γ2is the new magnetization direction cosine, t1Indicating the original direction of magnetization, i.e. the new magnetic potential is along the original direction of magnetization t1Integral of curve in the reverse direction
HaThe spectrum expression is as follows:
Figure BDA0002625598260000089
Zaexpression of frequency spectrumComprises the following steps:
Figure BDA00026255982600000810
the component conversion factor obtained by using equations (24) and (25) can be obtained:
Figure BDA00026255982600000811
from formula (26), formula (28), formula (29):
Figure BDA0002625598260000091
wherein alpha is1，γ1Is the original magnetization direction cosine;
the spectral expression of the new component can be deduced from the relationship between the magnetic bit and the component:
Figure BDA0002625598260000092
Figure BDA0002625598260000093
thus in the positive direction of the anomaly for the well: the arbitrary magnetization direction conversion factor is:
Figure BDA0002625598260000094
wherein alpha is1，γ1Is the cosine of the original magnetization direction, alpha2，γ2Is the new magnetization direction cosine;
for the three-degree volume, since the three-degree volume has is not zero, the potential-field conversion factor is derived from the dirichlet problem of the three-dimensional laplace equation:
Figure BDA0002625598260000095
one-dimensional data in the well cannot be used as a boundary condition to solve the above equation,
Figure BDA0002625598260000096
the method for reducing errors comprises the steps of selecting an average value of a plurality of well estimation results as a final estimation result and selecting a group of three components with Hay close to zero for estimation.
Further, the correlation calculation in S7 is specifically as follows:
Figure BDA0002625598260000101
COV denotes covariance, D denotes variance, Z denotesaRepresenting the vertical component, TaRepresenting magnetic abnormal modulus, N representing ZaAnd TaCorrelation value of (1), Hax、HayAnd TaAnd calculating the correlation similarly.
Further, the effective magnetization inclination angle specified degree in S9 is set to be between 0 ° and 360 °.
The technical scheme provided by the invention has the beneficial effects that:
(1) the effective magnetization dip angle of the section of the abnormal body is quickly and efficiently estimated by utilizing the logging data, and particularly, the estimation result of the second-degree body is accurate;
(2) errors exist in the three-dimensional estimation result, the errors are reduced by means of averaging the multi-well data estimation result, and the practicability is high.
Drawings
FIG. 1 is a flow chart of a method of estimating effective dip angle of magnetization in a borehole for magnetic anomalies according to the present invention;
FIG. 2 is a cross-sectional view of a cylinder model well site;
FIG. 3 is a top view of a sphere model well site;
FIG. 4 is a diagram showing the results of effective magnetization tilt estimation of a cylinder model;
fig. 5 is a diagram showing the results of estimating the effective magnetization tilt angle of the sphere model.
Detailed Description
A method for estimating effective dip angle of magnetization in a well by magnetic anomaly, as shown in fig. 1, comprises the following steps:
s1, based on three components of magnetic anomaly in well including vertical component ZaHorizontal component HaxAnd horizontal component HayCalculating the data to obtain the magnetic abnormal modulus Ta(ii) a The formula is as follows:
Figure BDA0002625598260000111
wherein Ta represents a magnetic abnormal modulus, ZaRepresents the vertical component, HaxRepresenting the horizontal component in the x-direction, HayRepresenting the horizontal component in the y-direction.
The embodiment is carried out on the basis of the established cylinder model and the established sphere model, the actual magnetization direction of the model, the effective magnetization inclination angle of the section and the relative position relationship between the well and the model are respectively shown in fig. 2 and 3, and the N direction in the figure is the positive direction of the abscissa axis of the section;
s2, for Za、HaxAnd HayFourier transform is carried out on the three components to respectively obtain corresponding frequency spectrums;
s3, setting an initial effective magnetization inclination angle degree, where x ° is 1 ° in this embodiment;
s4, calculating a vertical magnetization direction conversion factor and a horizontal magnetization direction conversion factor based on the effective magnetization inclination angle degree set in the S3 and the relative position relation between the well and the abnormal body;
the frequency domain conversion factor is related to the relative positions of the well and the abnormal body, and the specific relationship is as follows: in the established coordinate system, when the well is located in the positive direction of the abnormal body, i.e. in the positive direction of the corresponding abscissa axis, in the section under study, the magnetization direction conversion factor is:
Figure BDA0002625598260000112
α2＝cosIs2 γ2＝sinIs2 α1＝cosIs1 γ1＝sinIs1
wherein w represents a magnetization direction conversion factor, Is2To convert the angle of inclination of magnetization, Is1The original magnetization dip angle is obtained;
when the well is located in the negative direction of the abnormal body, namely the negative direction of the abscissa axis, the conversion factor is as follows:
Figure BDA0002625598260000113
α2＝cosIs2 γ2＝sinIs2 α1＝cosIs1 γ1＝sinIs1
wherein, Is2To convert the angle of inclination of magnetization, Is1At an angle of inclination of the original magnetization, α1，γ1Is the cosine of the original magnetization direction, alpha2，γ2Is the new magnetization direction cosine;
when I iss2At 90 °, the conversion factor is the perpendicular magnetization direction conversion factor; when I iss2At 0 deg., the conversion factor is the horizontal magnetization direction conversion factor; the table below shows the well two-degree volume frequency domain potential-field conversion factor,
TABLE 1 well-in-well two-degree-volume frequency domain potential-field conversion factor
Figure BDA0002625598260000121
The specific derivation process of the vertical magnetization direction conversion factor and the horizontal magnetization direction conversion factor is as follows:
firstly, according to the extension of the geologic body in three directions, the magnetic body is divided into a two-degree body and a three-degree body, for the ground, a data acquisition surface and a line are always positioned in the upper half space of the abnormal body, namely, under an established space rectangular coordinate system, data acquisition is carried out on a plane with Z being 0, the center of the abnormal body is on a plane with Z being 300, the Z axis is vertically and downwards positive, for a well, the abnormal body is assumed to be the two-degree body, the X axis of the established coordinate system is perpendicular to the trend of the abnormal body, the Y axis is parallel to the trend of the abnormal body, data acquisition is carried out along the well axis, namely, the Z axis direction, because the well axis can be in the positive direction of the abnormal body, namely, the positive X direction, and can also be in the negative direction of the abnormal body, namely, the relative position of the well and the abnormal body, therefore, the data in the well is related to the relative position of the well and the abnormal body, when the frequency domain conversion factor of the two-degree body is, the well is in both the positive and negative directions of the anomaly;
for the second degree body, the horizontal component HayZero, its potential-field conversion factor is derived from the dirichlet problem of the two-dimensional laplace equation:
Figure BDA0002625598260000131
wherein X and Z represent X-axis and Z-axis coordinates, uxxIs a quadratic derivative of the function u over x, uzzIs the second derivative of the function u over z,
Figure BDA0002625598260000132
as boundary conditions, i.e. measurement data;
case 1 well is in the negative direction of the anomaly:
for such cases, the solution of the dirichlet problem of the two-dimensional laplace equation is:
Figure BDA0002625598260000133
wherein u is a solution of the dirichlet problem, δ represents a Z-axis coordinate of any point on the boundary, T (X, Z) satisfies the above formula, T is a magnetic anomaly, and X and Z represent X-axis and Z-axis coordinates, that is:
Figure BDA0002625598260000134
for T (0, z) and
Figure BDA0002625598260000135
making a one-dimensional Fourier transform on Z:
the obtained magnetic anomaly frequency spectrum expression is as follows:
Figure BDA0002625598260000136
wherein S isTIs a magnetic anomaly spectrum; f is frequency, i is imaginary unit;
the equation (1) is differentiated by the differential theorem:
Figure BDA0002625598260000137
Figure BDA0002625598260000138
it can be found that for case 1: the X-direction derivative factor is 2 pi f, and the Z-direction derivative factor is 2 pi if;
giving a two-degree poisson formula:
Figure BDA0002625598260000141
Figure BDA0002625598260000142
Zaperpendicular component of magnetic anomaly, HaIs a magnetic abnormal horizontal component, M is a mode of magnetization, alpha and gamma are direction cosines of magnetization, delta is density of a geologic body, G is a universal gravitation constant, and v is a gravitational potential; v. ofxzAnd vzxIs the second partial derivative of v in the X and Z directions, vzzIs the second derivative of v in the Z direction, vxxIs the second derivative in the X direction of v;
let the gravitational potential frequency spectrum be SvAnd carrying out derivation in X and Z directions on the gravitational potential spectrum:
Svz＝2πifSv
Figure BDA0002625598260000143
to pair
Figure BDA0002625598260000144
Carrying out second-order derivation:
Figure BDA0002625598260000145
Figure BDA0002625598260000146
Figure BDA0002625598260000147
substituting the above formula into (4) and (5) can obtain:
Figure BDA0002625598260000148
Figure BDA0002625598260000149
wherein S iszFor the magnetically anomalous vertical component spectrum, SHIs a magnetic anomaly horizontal component spectrum;
can be obtained from the formulae (6) and (7) Sz，SHThe interconverting component conversion factors are:
Sz→SH：-i (8)
SH→SZ：i (9)
new magnetization direction magnetic potential of two-degree body, i.e. alpha2，γ2For the new magnetization direction cosine is:
Figure BDA0002625598260000151
U2is the magnetic potential of the new magnetization direction, t1Indicating the original direction of magnetization, i.e. the new magnetic potential is along the original direction of magnetization t1Integral of curve in the reverse direction
Order SHIs HaSpectrum of (1), HaThe spectrum expression is as follows:
Figure BDA0002625598260000152
order SZIs ZaSpectrum of (2), ZaThe spectrum expression is as follows:
Figure BDA0002625598260000153
the component conversion factor obtained by using the equations (8) and (9) can be obtained:
Figure BDA0002625598260000154
the expression (10), the expression (11), the expression (13) can be used for obtaining:
Figure BDA0002625598260000155
wherein alpha is1，γ1Is the original magnetization direction cosine;
the new component (H) can be deduced from the relationship between the magnetic potential and the componenta2，Za2) The spectrum expression of (1):
Figure BDA0002625598260000156
Figure BDA0002625598260000157
thus for a well located in an anomalous negative direction, any magnetization direction conversion factor is:
Figure BDA0002625598260000158
wherein alpha is1，γ1Is the cosine of the original magnetization direction, alpha2，γ2Is the new magnetization direction cosine;
case 2 well in the positive direction of the anomaly:
for such cases, the solution of the dirichlet problem of the two-dimensional laplace equation is:
Figure BDA0002625598260000161
t (x, z) satisfies the above equation, T being a magnetic anomaly, i.e.:
Figure BDA0002625598260000162
for T (0, z) and
Figure BDA0002625598260000163
making a one-dimensional Fourier transform on Z:
the obtained magnetic field spectrum expression is as follows:
Figure BDA0002625598260000164
STis a magnetic anomaly frequency spectrum, f is frequency, and i is an imaginary number unit;
equation (17) is differentiated by the differential theorem:
Figure BDA0002625598260000165
Figure BDA0002625598260000166
it can be found that for case 2: the X-direction derivative factor is-2 pi f and the Z-direction derivative factor is 2 pi if
Giving a two-degree poisson formula:
Figure BDA0002625598260000167
Figure BDA0002625598260000168
Zaperpendicular component of magnetic anomaly, HaIs the horizontal component of magnetic anomaly, M is the modulus of magnetization, α, γ are the direction cosines of magnetization, δ is the density of the geologic body, G is the universal gravitation constant, and v is the gravitational potential.
Let the gravitational potential frequency spectrum be SvAnd carrying out derivation in X and Z directions on the gravitational potential spectrum:
Figure BDA0002625598260000171
Figure BDA0002625598260000172
to pair
Figure BDA0002625598260000173
Carrying out second-order derivation:
Figure BDA0002625598260000174
Figure BDA0002625598260000175
Figure BDA0002625598260000176
substituting the above formula into (20) and (21) can obtain:
Figure BDA0002625598260000177
Figure BDA0002625598260000178
from the formulae (22) and (23), Sz，SHThe interconverting component conversion factors are:
Sz→SH：i (24)
SH→SZ：-i (25)
the new magnetization direction magnetic potential of the second-degree body is as follows:
Figure BDA0002625598260000179
α2，γ2is the new magnetization direction cosine, t1Indicating the original direction of magnetization, i.e. the new magnetic potential is along the original direction of magnetization t1Integral of curve in the reverse direction
HaThe spectrum expression is as follows:
Figure BDA00026255982600001710
Zathe spectrum expression is as follows:
Figure BDA00026255982600001711
the component conversion factor obtained by using equations (24) and (25) can be obtained:
Figure BDA00026255982600001712
from formula (26), formula (28), formula (29):
Figure BDA0002625598260000181
wherein alpha is1，γ1Is the original magnetization direction cosine;
the spectral expression of the new component can be deduced from the relationship between the magnetic bit and the component:
Figure BDA0002625598260000182
Figure BDA0002625598260000183
thus in the positive direction of the anomaly for the well: the arbitrary magnetization direction conversion factor is:
Figure BDA0002625598260000184
wherein alpha is1，γ1Is the cosine of the original magnetization direction, alpha2，γ2Is the new magnetization direction cosine;
for the three-degree volume, since the three-degree volume has is not zero, the potential-field conversion factor is derived from the dirichlet problem of the three-dimensional laplace equation:
Figure BDA0002625598260000185
one-dimensional data in the well cannot be used as a boundary condition to solve the above equation,
Figure BDA0002625598260000186
the method for reducing errors comprises the steps of selecting an average value of a plurality of well estimation results as a final estimation result and selecting a group of three components with Hay close to zero for estimation.
S5、ZaFrequency spectrum multiplied by the perpendicular magnetization direction conversion factor, HaxAnd HayMultiplying by horizontal magnetization direction conversion factors respectively;
s6, respectively carrying out inverse Fourier transform on the frequency spectrums multiplied by the conversion factors to obtain Z converted from the magnetization direction to the vertical directionaData and H with magnetization direction turned to horizontal directionaxAnd HayData;
s7, converting the Z converted in S6a、HaxAnd HayRespectively, the magnetic abnormal modulus T in S1aPerforming correlation calculation, and recording the obtained correlation numerical value; the correlation calculation is specifically as follows:
Figure BDA0002625598260000191
COV denotes covariance, D denotes variance, Z denotesaRepresenting the vertical component, TaRepresenting magnetic abnormal modulus, N representing ZaAnd TaCorrelation value of (1), Hax、HayAnd TaAnd calculating the correlation similarly.
S8, increase the effective magnetization tilt angle by n °, in this embodiment, n ° -1 °, and loop S4 to S8.
S9, when the effective magnetization tilt angle increases to a specified degree m, the cycle is stopped, in this embodiment,360 degrees when m degrees; plotting Z based on recorded relevance valuesa、HaxAnd HayThree correlation curves of (1), find ZaMinimum point of correlation curve and HaxAnd HayAnd the degree corresponding to the maximum point of the correlation curve and the obtained maximum point is the estimated effective magnetization dip angle. In this embodiment, the result of estimating the effective magnetization tilt angle of the cylinder model is shown in fig. 4, and table 2 is an error table for estimating the effective magnetization tilt angle of the cylinder model. The result of estimating the effective magnetization tilt angle of the sphere model is shown in fig. 5, and table 3 is an error table for estimating the effective magnetization tilt angle of the sphere model.
TABLE 2 error table for estimating effective magnetization tilt angle of cylinder model (actual value: 54.73 degree)
Number of well Za estimation Error of the measurement Ha estimation Error of the measurement
N1
55° 0.27° 54° 0.73°
N2 56° 1.27° 53° 1.73°
N3 58° 3.27° 52° 2.73°
S1 55° 0.27° 54° 0.73°
S2 56° 1.27° 53° 1.73°
S3 58° 3.27° 51° 2.73°
Average 1.6° 1.7°
TABLE 3 table of effective magnetization dip angle estimation error of sphere model (actual value: 63 degree)
Number of well Za estimation Error of the measurement Hax (Hay) estimate Error of the measurement
1 28° 35° 33° 30°
2 24° 39° 33° 30°
3 47° 16° 162° 99°(discarding)
4 65° 2° 84° 21°
5 27° 36° 33° 30°
6 24° 39° 32° 31°
7 45° 18° 162 99°(discarding)
8 66° 3° 85° 22°
Average 40.8° 22.2° 50° 13°

Claims (6)

1. A method for estimating effective dip angle of magnetization in a well by magnetic anomaly, comprising the steps of:
s1, based on three components of magnetic anomaly in well including vertical component ZaHorizontal component HaxAnd horizontal component HayCalculating the data to obtain the magnetic abnormal modulus Ta；
S2, for Za、HaxAnd HayFourier transform is carried out on the three components to respectively obtain corresponding frequency spectrums;
s3, setting an initial effective magnetization inclination angle degree;
s4, calculating a vertical magnetization direction conversion factor and a horizontal magnetization direction conversion factor based on the effective magnetization inclination angle degree set in the S3 and the relative position relation between the well and the abnormal body;
S5、Zafrequency spectrum multiplied by the perpendicular magnetization direction conversion factor, HaxAnd HayRespectively multiplying the frequency spectra by horizontal magnetization direction conversion factors;
s6, respectively carrying out inverse Fourier transform on the frequency spectrums multiplied by the conversion factors to obtain Z converted from the magnetization direction to the vertical directionaData and H with magnetization direction turned to horizontal directionaxAnd HayData;
s7, converting the Z converted in S6a、HaxAnd HayRespectively, the magnetic abnormal modulus T in S1aPerforming correlation calculation, and recording the obtained correlation numerical value;
s8, increasing the effective magnetization inclination angle by n degrees, wherein the effective magnetization inclination angle is more than or equal to 0 degree and less than or equal to 360 degrees, and circulating from S4 to S8;
s9, when the effective magnetization inclination angle is increased to a specified degree, stopping the circulation, and drawing Z based on the recorded correlation valuea、HaxAnd HayThree correlation curves of (1), find ZaMinimum point of correlation curve and HaxAnd HayThe degree corresponding to the obtained maximum point of the correlation curve is the estimated effective magnetization dip angle;
the frequency domain conversion factor in S4 is related to the relative positions of the well and the abnormal body, and the specific relation is as follows: in the established coordinate system, when the well is located in the positive direction of the abnormal body, i.e. in the positive direction of the corresponding abscissa axis, in the section under study, the magnetization direction conversion factor is:
Figure FDA0003002954410000021
α2＝cosIs2γ2＝sinIs2α1＝cosIs1γ1＝sinIs1
wherein w represents a magnetization direction conversion factor, Is2To convert the angle of inclination of magnetization, Is1The original magnetization dip angle is obtained;
when the well is located in the negative direction of the abnormal body, namely the negative direction of the abscissa axis, the conversion factor is as follows:
Figure FDA0003002954410000022
α2＝cosIs2γ2＝sinIs2α1＝cosIs1γ1＝sinIs1
wherein, Is2To convert the angle of inclination of magnetization, Is1At an angle of inclination of the original magnetization, α1，γ1Is the cosine of the original magnetization direction, alpha2，γ2Is the new magnetization direction cosine;
when I iss2At 90 °, the conversion factor is the perpendicular magnetization direction conversion factor; when I iss2At 0 °, the conversion factor is the horizontal magnetization direction conversion factor.
2. The method for estimating the effective magnetization dip angle in the well by the magnetic anomaly in the well according to claim 1, wherein the magnetic anomaly modulus T in S1aThe calculation formula of (a) is as follows:
Figure FDA0003002954410000023
wherein Ta represents a magnetic abnormal modulus, ZaRepresents the vertical component, HaxRepresents a horizontal component, HayRepresenting the horizontal component.
3. The method of claim 1, wherein the initial effective inclination angle of magnetization at S3 is set to x °, wherein 0 ° ≦ x ° ≦ 360 °.
4. The method for estimating the effective magnetization dip angle according to the magnetic force anomaly in the well as the claim 1, wherein the vertical magnetization direction conversion factor and the horizontal magnetization direction conversion factor are derived in the following steps in S4:
firstly, according to the extension of the geologic body in three directions, the magnetic body is divided into a two-degree body and a three-degree body, for the ground, a data acquisition surface and a line are always positioned in the upper half space of the abnormal body, namely, under the established space rectangular coordinate system, the data acquisition is carried out on a plane with Z being 0, the center of the abnormal body is positioned on a plane with Z being 300, the Z axis is vertically and downwards positive, for the well, the abnormal body is assumed to be the two-degree body, the established coordinate system X axis is perpendicular to the trend of the abnormal body, the Y axis is parallel to the trend of the abnormal body, the data acquisition is carried out along the well axis, namely, the Z axis direction, because the well axis is in the positive direction of the abnormal body, namely, the positive X direction, and the negative direction of the abnormal body, namely, the negative X direction, therefore, the data in the well is related to the relative position of the well and the abnormal body, therefore, when the frequency domain conversion factor of the two-degree body is deduced, namely, under, the well is in both the positive and negative directions of the anomaly;
for the second degree body, the horizontal component HayZero, its potential-field conversion factor is derived from the dirichlet problem of the two-dimensional laplace equation:
Figure FDA0003002954410000031
wherein X and Z represent X-axis and Z-axis coordinates, uxxIs a quadratic derivative of the function u over x, uzzIs the second derivative of the function u over z,
Figure FDA0003002954410000032
as boundary conditions, i.e. measurement data;
case 1 well is in the negative direction of the anomaly:
for such cases, the solution of the dirichlet problem of the two-dimensional laplace equation is:
Figure FDA0003002954410000033
wherein u is a solution of the dirichlet problem, δ represents a Z-axis coordinate of any point on the boundary, T (X, Z) satisfies the above formula, T is a magnetic anomaly, and X and Z represent X-axis and Z-axis coordinates, that is:
Figure FDA0003002954410000034
for T (0, z) and
Figure FDA0003002954410000035
making a one-dimensional Fourier transform on Z:
the obtained magnetic anomaly frequency spectrum expression is as follows:
Figure FDA0003002954410000041
wherein S isTIs a magnetic anomaly spectrum; f is frequency, i is imaginary unit;
the equation (1) is differentiated by the differential theorem:
Figure FDA0003002954410000042
Figure FDA0003002954410000043
it can be found that for case 1: the X-direction derivative factor is 2 pi f, and the Z-direction derivative factor is 2 fif;
giving a two-degree poisson formula:
Figure FDA0003002954410000044
Figure FDA0003002954410000045
Zaperpendicular component of magnetic anomaly, HaIs a magnetic abnormal horizontal component, M is a mode of magnetization, alpha and gamma are direction cosines of magnetization, delta is density of a geologic body, G is a universal gravitation constant, and v is a gravitational potential; v. ofxzAnd vzxIs the second partial derivative of v in the X and Z directions, vzzIs the second derivative of v in the Z direction, vxxIs the second derivative in the X direction of v;
let the gravitational potential frequency spectrum be SvAnd carrying out derivation in X and Z directions on the gravitational potential spectrum:
Figure FDA0003002954410000046
Figure FDA0003002954410000047
to pair
Figure FDA0003002954410000048
Carrying out second-order derivation:
Figure FDA0003002954410000049
Figure FDA00030029544100000410
Figure FDA00030029544100000411
substituting the above formula into (4) and (5) can obtain:
Figure FDA0003002954410000051
Figure FDA0003002954410000052
wherein S iszFor the magnetically anomalous vertical component spectrum, SHIs a magnetic anomaly horizontal component spectrum;
can be obtained from the formulae (6) and (7) Sz，SHThe interconverting component conversion factors are:
Sz→SH：-i (8)
SH→SZ：i (9)
new magnetization direction magnetic potential of two-degree body, i.e. alpha2，γ2For the new magnetization direction cosine is:
Figure FDA0003002954410000053
U2is the magnetic potential of the new magnetization direction, t1Indicating the original direction of magnetization, i.e. the new magnetic potential is along the original direction of magnetization t1Curve integration in the reverse direction;
order SHIs HaSpectrum of (1), HaThe spectrum expression is as follows:
Figure FDA0003002954410000054
order SZIs ZaSpectrum of (2), ZaThe spectrum expression is as follows:
Figure FDA0003002954410000055
the component conversion factor obtained by using the equations (8) and (9) can be obtained:
Figure FDA0003002954410000056
the expression (10), the expression (11), the expression (13) can be used for obtaining:
Figure FDA0003002954410000057
wherein alpha is1，γ1Is the original magnetization direction cosine;
the new component (H) can be deduced from the relationship between the magnetic potential and the componenta2，Za2) The spectrum expression of (1):
Figure FDA0003002954410000061
Figure FDA0003002954410000062
thus for wells located in the anomalous negative direction: the arbitrary magnetization direction conversion factor is:
Figure FDA0003002954410000063
wherein alpha is1，γ1Is the cosine of the original magnetization direction, alpha2，γ2Is the new magnetization direction cosine;
case 2: well is located in the positive direction of the anomaly:
for such cases, the solution of the dirichlet problem of the two-dimensional laplace equation is:
Figure FDA0003002954410000064
t (x, z) satisfies the above equation, T being a magnetic anomaly, i.e.:
Figure FDA0003002954410000065
for T (0, z) and
Figure FDA0003002954410000066
making a one-dimensional Fourier transform on Z:
the obtained magnetic field spectrum expression is as follows:
Figure FDA0003002954410000067
STis a magnetic anomaly frequency spectrum, f is frequency, and i is an imaginary number unit;
equation (17) is differentiated by the differential theorem:
Figure FDA0003002954410000068
Figure FDA0003002954410000069
it can be found that for case 2: the X-direction derivative factor is-2 pi f and the Z-direction derivative factor is 2 pi if
Giving a two-degree poisson formula:
Figure FDA0003002954410000071
Figure FDA0003002954410000072
Zaperpendicular component of magnetic anomaly, HaIs a magnetic abnormal horizontal component, M is a mode of magnetization, alpha and gamma are direction cosines of magnetization, delta is density of a geologic body, G is a universal gravitation constant, and v is a gravitational potential;
let the gravitational potential frequency spectrum be SvAnd carrying out derivation in X and Z directions on the gravitational potential spectrum:
Figure FDA0003002954410000073
Figure FDA0003002954410000074
to pair
Figure FDA0003002954410000075
Carrying out second-order derivation:
Figure FDA0003002954410000076
Figure FDA0003002954410000077
Figure FDA0003002954410000078
substituting the above formula into (20) and (21) can obtain:
Figure FDA0003002954410000079
Figure FDA00030029544100000710
from the formulae (22) and (23), Sz，SHThe interconverting component conversion factors are:
Sz→SH：i (24)
SH→SZ：-i (25)
the new magnetization direction magnetic potential of the second-degree body is as follows:
Figure FDA00030029544100000711
wherein alpha is2，γ2Is the new magnetization direction cosine;
t1indicating the original direction of magnetization, i.e. the new magnetic potential is along the original direction of magnetization t1Integral of curve in the reverse direction
HaThe spectrum expression is as follows:
Figure FDA0003002954410000081
Zathe spectrum expression is as follows:
Figure FDA0003002954410000082
the component conversion factor obtained by using equations (24) and (25) can be obtained:
Figure FDA0003002954410000083
from formula (26), formula (28), formula (29):
Figure FDA0003002954410000084
wherein alpha is1，γ1Is the original magnetization direction cosine;
the spectral expression of the new component can be deduced from the relationship between the magnetic bit and the component:
Figure FDA0003002954410000085
Figure FDA0003002954410000086
thus in the positive direction of the anomaly for the well: the arbitrary magnetization direction conversion factor is:
Figure FDA0003002954410000087
wherein alpha is1，γ1Is the cosine of the original magnetization direction, alpha2，γ2Is the new magnetization direction cosine;
for the three-degree volume, since the three-degree volume has is not zero, the potential-field conversion factor is derived from the dirichlet problem of the three-dimensional laplace equation:
Figure FDA0003002954410000091
one-dimensional data in the well cannot be used as a boundary condition to solve the above equation,
Figure FDA0003002954410000092
is a boundary condition of the dirichlet problem of the three-dimensional Laplace equation, therefore, the conversion factor of the second-degree volume is directly utilized to process the data of the third-degree volume, which means that the default of the third-degree volume section is the second-degree volume section, the result has errors, and the method for reducing the errors comprises the step of selecting the average value of the estimation results of a plurality of wells as the final estimation resultThe result is evaluated by selecting a set of three components with Hay close to zero.
5. The method for estimating the effective magnetization dip angle in the well for the magnetic anomaly according to the claim 1, wherein the correlation calculation in the step S7 is as follows:
Figure FDA0003002954410000093
COV denotes covariance, D denotes variance, Z denotesaRepresenting the vertical component, TaRepresenting magnetic abnormal modulus, N representing ZaAnd TaCorrelation value of (1), Hax、HayAnd TaAnd calculating the correlation similarly.
6. The method of claim 1, wherein the effective inclination angle of magnetization is set to m ° in S9 in a prescribed degree of 0 ° < m ° <360 °.
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