The gravity normalized total gradient method proposed by former Soviet scholar Belozkin and others in the late 1960s is a method that uses gravity anomalies measured at a higher precision to determine field sources. , fracture positions and density interface methods can also be used to directly search for oil and gas-bearing structures when conditions are favorable. Judging from the application situation in the former Soviet Union, the effect is still relatively obvious.
The starting point of this method is that the gravitational potential of the remaining mass and its derivatives are analytical functions in the space outside the field source body, but lose their analytical properties at the field source. In an analytic function, the point that loses its analyticity is called the singularity of the function. The problem of determining the field source is to determine the singularity of the function through the analytic continuation of the anomaly. For example, for a horizontal cylinder, the gravity anomaly expression is known to be
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At x=0, extend Δg downward along the z-axis, When D=0, Δg will tend to infinity. Therefore, the axis point of the cylinder is a singular point of equation (2-8-15).
(1) Principle of the method
In actual work, we do not know the expression of the anomaly generated by the research object, and can only use a limited number of discrete data to expand it into levels. represented by numbers. Therefore, due to the finiteness and discreteness of gravity data, the expansion series can only take finite terms. Coupled with errors in observations and the effects of random interference, gravity anomalies can only be approximate in analytical continuation. Especially in downward extension calculations, the extension process is often destroyed before reaching the field source, so singular points related to the field source are often not found. To illustrate this point, we still use the gravity anomaly of a horizontal cylinder to be expanded into a Taylor series for discussion. As shown in Figure 2-8-11, the gravity anomaly extending downward to any point P (x, z) is
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If along the x-axis ( x=0), then the above formula is
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Expand the above formula into a Taylor series near z=0:
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Obviously, when the above equation extends downward and approaches the field source, the (z?D) series diverges, that is, D is a singular point of Δg(z). But when the series does not take infinite terms, the situation is different. At this time
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Even if it passes through the center of the cylinder (z≥D), the factor (2 -8-17) is a finite term series, which only diverges when z?∞, otherwise the series will converge as long as N is a finite term. This shows that it is very difficult to find singularities in actual work.
Figure 2-8-11 Calculation of abnormality of horizontal cylinder at point P (x, z)
Figure 2-8-12 Δg H (ζ) at the axis of the cylinder There is a maximum value
In order to overcome this difficulty, it is necessary to further study the method of using finite series to express gravity. For this reason, the equation (2-8-16) is rewritten as:
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The second term on the right side of the equal sign in the above equation is the remainder of the series, which is That is, discard the error. Let, the above formula can be written as:
Δg(ζ)=ΔgN(ζ)+rN(ζ)
It is proved by series theory that if N takes an appropriate value, then when When ζ<1, there will be rN(ζ)<ΔgN(ζ); conversely, when ζ>1, there will be rN(ζ)>ΔgN(ζ). This is a very important situation, with which singularities related to field sources can be determined. We normalize ΔgN (ζ) with rN (ζ), then we have:
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The change of normalized function ΔgH (ζ) is only related to ΔgN (ζ) is related to the ratio of rN (ζ). As long as the value of N is appropriate, when approaching the singular point from above, rN (ζ) < ΔgN (ζ). And ΔgN (ζ) increases faster than rN (ζ), so ΔgH (ζ) increases accordingly; when passing through the singular point, the situation is reversed, so ΔgH (ζ) gradually decreases, so at ζ? When 1, ΔgN (ζ) will have a maximum value, as shown in Figure 2-8-12.
Therefore, the significance of using the normalized function ΔgH(ζ) is very obvious. It can overcome the difficulty of directly finding singular points using general downward continuation mentioned above.
The current problem is how to determine rN(ζ), because it is still unknown in actual work. Berezkin's research believes that when appropriate mathematical operations (such as Fourier series) are used, rN (ζ) can be replaced by the average value of Δg (ζ) at a certain depth.
(2) Calculation of normalized total gradient
In practical applications, the normalized function ΔgH (ζ) of gravity anomaly is not used. This is because no matter from the perspective of the rounding error rN(ζ) of the series or the random error in the observation value, the downward extension process contains an unsteady sinusoidal function. The amplitude of the sinusoidal quantity becomes larger and larger as z increases, which distorts the continuation result. If the total gradient of the gravity anomaly in the observation plane (xoz plane) (that is, the modulus of the vector sum of gx and gz) is used as the normalization function, the location of the field source or singular point can also be judged based on its extreme value, and Its remainder and random error present a relatively stable quantity in the downward extension, and its influence is much smaller than that reduced by Δg(ζ); as the observation accuracy increases, the calculation results become more reliable.
The expression of the normalized total gradient is as follows:
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Where: GH (x, z) is the xoz vertical plane The total gradient of gravity; G(x, z) is the depth which is the average of the total gradient of M+1 measuring points on z.
It can be seen that GH (x, z) is a ratio of dimension one. In the xoz plane, find the GH (x, z) of each point at a certain depth interval, and finally draw the contour map of GH (x, z).
Since the sine series converges faster than the cosine series under the condition that the gravity anomaly at both end points of the survey line is zero, the sine series is chosen to represent Δg. To this end, the linear term a+bx should be subtracted from each Δg(x,0) of the measured anomaly, where a is the gravity anomaly value Δg(0,0) at the starting point of the measurement line, and b=[Δg(0,0)-Δg (L, 0)]/L, Δg (L, 0) is the gravity value at the end of the survey line, and L is the length of the survey line. In this way, the downward continuation Fourier series expression can be obtained:
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In the formula: N is the number of harmonic terms; Bn is the harmonic, and The expression is
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and is the downward extension factor.
It is easy to derive the expressions of gx and gz by deriving the derivative of equation (2-8-21):
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Calculation of Bn: Assume that the survey line L is composed of M+1 survey points with a point distance of Δx, then x=jΔx (j=0, 1, 2,...,M), L= MΔx, so Bn can be expressed as
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Use equation (2-8-22) to equation (2-8-23′) to calculate GH (x, The results of z) show that troubles caused by random errors and other disturbances will be encountered during the downward continuation process. At the same time, since the discontinuity of the field at the end of the survey line will cause false anomalies in the GH (x, z) curve, Bn must also be multiplied by a smoothing factor qm to enhance the stability of the continuation process:
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In the formula: N is the total number of harmonic terms; m=1, 2, 3,...
qm has a smoothing effect, and its value changes from 1 to zero, the larger n (high-frequency components) are, the stronger the smoothing effect will be. In oil exploration, m=2 is more appropriate. Therefore, in the end there is:
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In the subsequent research and experiments of many people, following the example of B. Cianciara et al. The phase concept proposed by others adds a phase diagram corresponding to the total gradient diagram, and its calculation formula is:
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(3) Theoretical Model and Examples
p>Figure 2-8-13 shows the calculation results of an anticline model with a top depth of 1km, a bottom depth of 1.8km, and a width of 3km.
It can be seen that at the center of gravity of the anticline (field source center), GH (x, z) has a maximum value of 8.86, which confirms the theoretical effect of this method.
Figure 2-8-13 Δg (x, 0), gx, gz curves and GH (x, z) equivalent profile of the anticlinal body
N=50; 1 —Δg curve; 2—GH (x, 0) curve; 3—gz curve; 4—gx curve
Figure 2-8-14 shows the curve of the same horizontal cylinder (radius 0.5km , the center burial depth is 2km, σ = 1.0g/cm3), the profile length L = 20km, the results of the total number of terms (i.e. harmonic number) N of different series. The choice of N is a critical parameter. When N is from 60 to 20, the position of the extreme value of GH (x, z) moves from top to bottom, and the maximum value is from small to large and then to small. When N=40, the center of the anomaly coincides with the cylinder, and the extreme value of GH (x, z) is also the largest. Therefore, in actual work, different N values ??should be used for calculation to find the largest one among the extreme values. The value position is the depth position closest to the field source. At this time, the contours on both sides of the anomaly center are also closest to the center.
Figure 2-8-15 shows the total gradient and phase diagram corresponding to the vertical step model with an upper top burial depth of 3km, a lower bottom burial depth of 6km, and σ = 0.5g/cm3. When N is appropriate, the extreme position is basically consistent with the position of the corner point of the step, and the phase diagram very accurately indicates the position of the vertical plane of the step using the zero value between the minimum value and the maximum value.
At the top of the anticline structure containing oil and gas, due to the loss of mass, the maximum value of the gravity height formed by the anticline structure will be reduced (about 10g.u.), but this reduction It's impossible to tell. On the total gradient map, two maximum values ??appear at the edge of the structure, and a minimum value appears at the top of the structure, forming the characteristic of "two highs and one low", which has become a typical sign for searching for oil and gas structures ( See Figure 2-8-16).
Before 1970, the former Soviet Union used this method on 45 known geological structures. There are 30 oil and gas fields and 15 metal deposits in the known areas; the results show that in 42 known areas, the GH (x, z) field has been reliably displayed, and only 3 are unreliable. Figure 2-8-17 The total gradient diagram of the Hotibai oil and gas field. When N=30, there is an obvious image of two highs and one low. On the left side of the figure, similar features appear at about the same depth, so it is inferred that there is a smaller oil and gas reservoir with similar burial depth, which has been confirmed by subsequent seismic work and drilling.
Figure 2-8-14 GH (x, z) contour profile at different N values
(a) N=60; (b) N=50; (c) N=40; (d) N=30; (e) N=20
Figure 2-8-15 Model diagram of plumb steps
(a) Return Normalized total gradient isoline profile, the dotted line position is the vertical step model; (b) Normalized total gradient phase diagram
Figure 2-8-16 Normalization of the hydrocarbon-bearing anticline structural model Total chemical gradient diagram
Figure 2-8-17 Δg (x, 0), GH (x, 0) curve and GH (x, z) field of the Hotibai oil and gas field
N=30; 1-seismic marker layer; 2-oil and gas reservoir
Based on theory and practical examples, Belozkin made the following conclusions about the practical capabilities of this method:
(1) The GH (x, z) method can divide the gently structured gravity field with an amplitude of 20m to 30m and a depth of more than 3km to 5km from the observed gravity field, and determine its depth. Theoretically, the accuracy can reach 5～50.
(2) The GH (x, z) method can also divide the gravity field of the oil layer with a thickness of more than 50m and the gas layer with a thickness of more than 30m from the observed gravity field, which is used for gravity method and other geophysical prospecting methods. Comprehensive and direct search for oil and gas reservoirs with relatively small reserves provides an effective way.
(3) In metal ore exploration, this method can find the depth of the top surface or center of gravity of the ore body and make some analysis of its tendency.
(4) The GH (x, z) method can be used to study the deep structure of the earth's crust and determine the location of deep faults and their distribution directions.