In this paper, the general expression of eccentric electric field distribution of annular single electrode in borehole is derived, and the expressions of electric field distribution and grounding resistance of annular single electrode in three-layer cylindrical medium (double-layer cylindrical interface) and layered medium (one horizontal interface and two horizontal interfaces) are derived.
Resistivity logging method is widely and effectively used in coalfields, oil fields and metal mines. The theory of resistivity logging method (multi-electrode or single-electrode) was formed in 1930s. These theories are all based on the electric field distribution of point source in borehole. Because the electrodes used in electric logging production are not point-shaped, the quantitative interpretation of electric logging is greatly affected. For example, in coal field, single electrode current logging can well divide horizons, but it can't quantitatively explain the resistivity of strata. The electrode actually used in electrical logging is closer to the ring. In order to make the electric logging theory more practical, it is necessary to study the ring electrode logging theory.
Reference [1] calculates the electric field and grounding resistance of annular single electrode in borehole, but the influence of electrode eccentricity, permeable layer and rock interface is not considered in this calculation. In 1957 and 1963, the author derived four topics, such as the general expression of eccentric electric field distribution of annular single electrode in borehole, the expression of electric field distribution and grounding resistance of annular single electrode under the condition of double-layer columnar interface (with permeable layer), and the expression of electric field distribution and grounding resistance of annular single electrode under the condition of one or two horizontal interfaces, among which a digital result was also made for a horizontal interface condition. These derivation and calculation results still have certain theoretical and practical significance, so they are rearranged.
In this paper, the theory of R-ring single electrode logging is systematized, which can also be used as the basis of ring multi-electrode logging theory. If the gauge plate is made by computer, it will be used for quantitative interpretation of electric logging; The calculation results and methods of annular single electrode potential function discussed in this paper will contribute to the development of resistivity logging theory.
1. General expression of electric field distribution of annular single electrode in borehole
The annular single electrode is arranged in a borehole with a radius of a, and the electrode is in a horizontal state. The distance from the electrode center to the borehole axis is L, the conductivity of the medium is σ 1 and σ2, the power supply current is J, the electrode cross-section diameter is 2c, and the distance from the electrode cross-section center to the electrode center is B, as shown in figure 1. Find the expression of electric field distribution.
When the electric field is small, the ring electrode can be replaced by a circular electrode passing through the center of the ring electrode [1]. It is solved by column coordinates (r, θ, z), and the coordinate origin is placed at the intersection of the plane where the electrode is located and the drilling axis.
The potential functions u 1 and u2 will be the integrals of Laplace equation (1).
Zhang Yujun on new methods of geological exploration.
In the medium 1, U 1 can be written as U 1=U0+U 1(2).
Formula (2): uo- potential function of annular single electrode in homogeneous medium;
The function of u 1- interface.
The potential function shall meet the following conditions on the interface:
Zhang Yujun on new methods of geological exploration.
At infinity:
Zhang Yujun on new methods of geological exploration.
Due to symmetry, the following conditions exist:
Zhang Yujun on new methods of geological exploration.
When solving the problem, the integral transformation method used by Bondarev (вибондарев) in solving the disk-shaped single-electrode logging problem will be adopted [2]. Firstly, the expressions of U. and U2 after integral transformation are obtained, and the integral transformation is carried out according to (8).
Zhang Yujun on new methods of geological exploration.
The Laplace equation of the potential function is:
Zhang Yujun on new methods of geological exploration.
The whole process of equation (10) is divided into [[3]].
Zhang Yujun on new methods of geological exploration.
Where In(λг) and kn (λ r) are virtual Bessel functions; It also contains two parts:
Figure 1
Considering that it should be a finite value on the drilling axis and the condition of formula (5) must be satisfied at infinity, it can be written as
Zhang Yujun on new methods of geological exploration.
Now we need to find the expressions of U0 and. The local arc bdφ on the circular electrode is regarded as a point power supply, and its potential function is expressed as [3].
Zhang Yujun on new methods of geological exploration.
formula
Zhang Yujun on new methods of geological exploration.
J0- Bessel function.
Will J. Substitute the value of into the above formula and integrate φ.
Zhang Yujun on new methods of geological exploration.
According to literature [4], J0 (TQ) can be written as follows
Zhang Yujun on new methods of geological exploration.
therefore
Zhang Yujun on new methods of geological exploration.
Then (13) becomes
Transform U0 according to formula (8):
Zhang Yujun on new methods of geological exploration.
Zhang Yujun on new methods of geological exploration.
Then according to the transformation of formula (9)
Zhang Yujun on new methods of geological exploration.
According to literature [4], k0 (λ ρ) can be written as
Zhang Yujun on new methods of geological exploration.
The boundary condition is
Zhang Yujun on new methods of geological exploration.
Substitute the values in (17) and (12) into (18), and then solve.
Zhang Yujun on new methods of geological exploration.
formula
The bit function and U2 should be solved by the following formula
Zhang Yujun on new methods of geological exploration.
Zhang Yujun on new methods of geological exploration.
Put one in (19). And b, substituting the value of (12), and then finding the potential function and U2 by (20); Then substitute the values in formula (15) and U0 into formula (2), and finally get the general expression of electric field distribution of annular single electrode in borehole:
Zhang Yujun on new methods of geological exploration.
When the electrode is located in the center of the borehole, that is, l=0, a conclusion can be drawn.
Zhang Yujun on new methods of geological exploration.
(23) This formula is consistent with the result obtained by Ovqinnikov (иковчинников) in reference [1]. Literature [[ 1] has derived the expression of grounding resistance in this case, and made a digital calculation.
The influence of electrode eccentricity can be evaluated by (2 1) and (22). These two formulas are very complicated, and digital calculation needs the help of electronic computers.
2. Expressions of electric field distribution and grounding resistance of annular single electrode under the condition of double-layer columnar interface.
If in the above problem, there is a permeable layer (medium 2) between mud (medium 1) and rock (medium 3), with an outer diameter of D and a conductivity of σ2 (see Figure 2), it can be concluded that,
Figure 2
Zhang Yujun on new methods of geological exploration.
Using the four boundary conditions of the potential function on two interfaces, we can find the four constant coefficients in the above formula. Because we mainly want to find the expression of U 1, we only write the following:
Zhang Yujun on new methods of geological exploration.
formula
Zhang Yujun on new methods of geological exploration.
The potential function U 1 is obtained by formula (20).
Zhang Yujun on new methods of geological exploration.
Zhang Yujun on new methods of geological exploration.
Formula (26) is the electric field distribution of annular single electrode in borehole under the condition of two-layer columnar interface; Obviously, when P23=0 or d→∞, equation (26) becomes equation (23).
When l=0, that is, when the electrode center coincides with the drilling axis, Equation (26) becomes the following form:
Zhang Yujun on new methods of geological exploration.
formula
Zhang Yujun on new methods of geological exploration.
If the equipotential surface passing through the circle (z=0, r=b+c) is approximately coincident with the surface of the annular electrode, that is, the annular electrode is regarded as the equipotential surface in the electric field of the circular electrode passing through the center of its cross section, then in the case of two-layer interface, when the electrode is located in the center of the borehole, the grounding resistance of the annular single electrode can be obtained by Formula (27).
Zhang Yujun on new methods of geological exploration.
In formula (28), if P23=0 or d→∞, the result in [1] can be obtained. Equation (28) can be used to evaluate the influence of permeable layer on annular single-electrode logging. After the computer obtains the digital results, the gauge plate for quantitative interpretation of annular single-electrode logging under the condition of two-layer columnar interface can be made.
3. Expressions of electric field distribution and grounding resistance of annular single electrode under horizontal interface condition.
In order to understand the influence of formation on annular single electrode logging, we first discuss the situation under a horizontal interface condition (see Figure 3). The plane SS is the horizontal interface of two kinds of media (σ 1 and σ2), and the electric field distribution and grounding resistance of the ring electrode are obtained.
Figure 3
When solving the problem, we will still use the method in the reference [1], that is, the annular electrode is replaced by a circular virtual electrode with radius b, and the expression of electric field distribution is obtained; When calculating the grounding resistance, the surface of the annular electrode is approximately regarded as the equipotential surface passing through the outer circle of the electrode (z=0, r=b+c). The column coordinate system (r, θ, z) is still used when solving problems. Part of the arc bdθ is regarded as a point source, and its potential function is [3]
Zhang Yujun on new methods of geological exploration.
Zhang Yujun on new methods of geological exploration.
formula
Zhang Yujun on new methods of geological exploration.
Substitute the value of j' into equations (29) and (30) and integrate θ to obtain the expression of potential function:
Zhang Yujun on new methods of geological exploration.
Where F(δ, k)- elliptic integral of the first kind;
Zhang Yujun on new methods of geological exploration.
Let z = 0 and r = b+c in formula (3 1), and the expression of grounding resistance of annular electrode under the condition of one layer of horizontal interface is obtained.
Zhang Yujun on new methods of geological exploration.
formula
Zhang Yujun on new methods of geological exploration.
If a=b+c is used as the unit of length, b is 0.9a, and c is 0. 1a, then in this case, the grounding resistance r can be written as
Zhang Yujun on new methods of geological exploration.
formula
Zhang Yujun on new methods of geological exploration.
If d→∞ in equation (34), the grounding resistance of annular electrode in uniform medium can be obtained:
Zhang Yujun on new methods of geological exploration.
This result is consistent with икккккккккккккккккккккк.
According to Equation (34), the change of grounding resistance when the annular electrode passes through the interface is calculated numerically. In calculation, d is equal to 0, 0.2, 0.5, 0.7, 1.0, 1.5, 2.0, 2.5, 3, 4, 5, 6, 7, 10, 15, ∞ K 12 takes 0.4, 0.6, 0.8, 0.9, +0.95, 0.98,+1 etc. 14 respectively. Mathematical and astronomical tables [6] are used in the calculation; In the calculation, r is expressed in units, that is, 2π2aσ 1.
Now, the results obtained only when K 12 = 0.8 are shown in fig. 4.
Figure 4
These results can be used to evaluate the boundary impact. It can be seen from the calculation results that, if the rock thickness is ≥7a, the boundary influence is less than 10% under any resistivity difference condition, that is, when it is 3.5a away from the boundary, the boundary influence is less than 10%. Then, when measuring true resistivity with a single ring electrode, if the measurement accuracy is 10% and the thickness is more than 3.5 times of the maximum diameter of the ring electrode, the calculation results also show that the rock interface shows the inflection point of the curve on the single electrode logging curve.
4. Expressions of electric field distribution and grounding resistance of annular single electrode under the condition of two horizontal interfaces.
Using the potential function formula [7] of point power supply under two horizontal interface conditions, the potential function of annular single electrode is obtained by the same calculation method as in the previous section, which is expressed by formula (35-37), and only the potential function in the medium where the electrode is located is written here:
1. When the electrode is located in the first medium (Figure 5):
The value we are mainly concerned about has practical significance only in multi-electrode logging, so in order to discuss the problem of single-electrode logging, we only write the expression:
Zhang Yujun on new methods of geological exploration.
formula
Zhang Yujun on new methods of geological exploration.
Figure 5
2. When the electrode is located in the second medium (Figure 6):
Zhang Yujun on new methods of geological exploration.
Zhang Yujun on new methods of geological exploration.
For the same reason, we only write expressions:
Zhang Yujun on new methods of geological exploration.
formula
Zhang Yujun on new methods of geological exploration.
3. When the electrode is in the third medium (Figure 7):
For the same reason, we only write expressions:
Zhang Yujun on new methods of geological exploration.
Figure 7
Zhang Yujun on new methods of geological exploration.
formula
Zhang Yujun on new methods of geological exploration.
If R 1, R2 and R3 respectively represent the grounding resistance of the annular single electrode in 1, 2 and 3, their expressions can be obtained by the same method as in the previous section, where a=b+c is the unit of length.
Zhang Yujun on new methods of geological exploration.
formula
Zhang Yujun on new methods of geological exploration.
Zhang Yujun on new methods of geological exploration.
formula
Zhang Yujun on new methods of geological exploration.
Zhang Yujun on new methods of geological exploration.
formula
Zhang Yujun on new methods of geological exploration.
If σ 1 = σ 3, k2 1=k23=-k 12, the expression of potential function and grounding resistance of annular single electrode under the condition of plate-like rock stratum can be obtained.
The series in equation (35-40) can converge quickly when k2 1 and k23 are greater than-1 less than 1. When one of them is equal to 1, the series in the above expression may lose convergence, and the potential function and grounding resistance will depend on the value of dielectric conductivity σ.
refer to
[ 1]Овчинников И.К.,К теории однозлектродного каротажа,Изв.АН СССР.сер.геоф.，No3， 1958。
[2]Бондарев В.И.,Полелискового злехтрода,расположенного в скважине,Изв.АН СССР,сер.геоφ.，No3， 1963。
[3] Sabolovsky, special function for geophysical exploration, 1957.
[4] Watson G.N., paper on Bessel function theory, 1945.
[5]Градлгейн И.С.,Таблицы интегралов сумм,рддов и произведений, 1962.
[6]ГлазенапС.П.,Математические и астрономические таблиды, 1932.
[7]Дахнов В.Н.,Интерпретадиярезультатов геофизических исследрваний разрезов скважин, 1955.
Originally published in Geophysical and Geochemical Exploration, 1979, No.2..
Supplement: The author paid attention to the quantitative interpretation of single-electrode logging during the graduation practice of 1956 (comprehensive logging in coal fields with γ-γ logging method as the main method): the theory of resistivity logging method in the former Soviet Union was formed in 1930s, and its main contributions are: ваок (1933). The theory they deduced is the electric field distribution of point power supply in borehole. Because the electrode used in electrical logging is not a point electrode, but a ring electrode wrapped by cable, the single electrode current logging in coal field can well divide horizons, but it can not quantitatively explain the resistivity of rock strata.
After returning to school, I asked the field theory teacher иковчиников, and the document [1] was produced. The author participated in the numerical calculation of this paper. Inspired by this work, the author deduces the expression of electric field distribution and grounding resistance of annular single electrode under one or two horizontal interface conditions by using Bessel function, and makes a digital result for one horizontal interface condition, which is written in the author's graduation thesis 1957, an academician of the former Soviet Academy of Sciences. In 1963, the author also derived the general expression of eccentric electric field distribution of annular single electrode in borehole, as well as the expression of electric field distribution and grounding resistance when there is a double-layer columnar interface formed by permeable layer, and formed a complete annular single electrode logging theory, which was published in the second phase of Geophysical Exploration 1979.