NEW METHODS AND INSTRUMENTS
MATHEMATICAL MODELING OF ELECTROMETRIC DIAGRAMS IN FISSURED MEDIA. PART 2: SYNTHESIZED WELL-LOGGINS CURVES FOR FOUR-ELECTRODE SOUNDING L~C 622.831
V. N. Oparin and G. G. Matasova
The structure of the rock bed affects the stability of the structural elements of a mining system. It is important for the choice of a scheme for the development of a mineral deposit. In this context, it is essential to estimate reliably the degree of cracking of rocks in all stages of mining operations. Geophysical methods (including well logging) have been found to be reliable and effective tools for studying the fissuring of rock beds surrounding underground workings. In the present paper, we develop mathematical tools for modeling electrical curves in fissured rocks and report the results of a computer simulation of synthesized well logging curves. In the first part of this study [i], a general solution was obtained for a field of cylindrical direct current sources on a perimeter of a bore hole with a parallel system of cracks coinciding with the maximum electrical sensitivity surfaces [2]. Taking as an example the simple case of a bore hole of "zero" diameter and cracks of an "infinite" length, we will discuss the following important methodological questions: a) the effec t of asymmetry of an electrical setup on the position of extreme in synthesized diagrams of Pa (ordinate referencing); b) the effect of the size of the receiving dipole MN on the structure of synthesized diagrams; c) the effect of the spacing q on the structure of curves of Pa for a completely symmetric setup; A
d) the effect of the spacing q on the structure of curves of Pa for the normal probe; ^
e) the effect of the spacing q on the structure of Pa diagrams for a lateral probe. The formulas and figures in the present paper are numbered as Allowing the parameter b(the crack length) proach infinity, we write
in expressions
a continuation of [i].
(140)-(144) from [I] to ap-
4. $,(=,)
where $2(a i) is defined according to (139) and Pl according
(145)
to (141).
Numerical calculations on a BESM-6 computer (with a Fortran program) are based on expression (145). For the unit linear dimension we take intercrack spacing A (i.e,. A = i).* The curves are plotted with a discretization of 4/32 (i.e., al = 1/32; K = 0, I, ..., 32). Effect of Asymmetry of the Electrometr_ic Setup on the Extrema of Synthesized Diagrams (Coordinate Referencing) The problem was studied for a spacing q = 1/2 and i. The position of receiving electrodes M--N was determined by the coefficients ~2 and a~ in the following proportions:
Institute of Mining, Siberian Department, Academy of Sciences of the USSR, Novosibirsk. Translated from Fiziko-Tekhnicheskie Problemy Razrabotki Poleznykh Iskopaemykh, No. i, pp. 113-119, January-February, 1990. Original article submitted March ii, 1988.
0038-5581/90/2601-0093512.50
9 1990 Plenum Publishing Corporation
93
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m- Position of cracks ~- Position of electrodes Fig. 3
Fig. 2
Fig. 2. Effect of electrode setup asymmetry on the position of extrema in synthesized well logging curves and coordinate referencing for q = 1/2. Fig. 3. Effect of electrode setup asymmetry on extrema of synthesized diagrams and coordinate referencing for q = i.
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Figures 2 and 3 show fragments of curves synthesized for q = 1/2 and 1 for various q = 1/2 and 1 for various ~2, ~3: the solid rectangles at the base of the abscissa indicate the position of cracks where referencing of Pa was done according to the position of the electrode A(~ z) that was nearest to the mouth of the bore hole, i.e., according to (125). The bars crossing the abscissa indicate the crack positions when ~oordinate referencing was done with respect to the center of the receiving dipole MN, i.e., z = ez + (~2 + ~3)q/2From a comparison of Figs 2 and 3 we see that coordinate referencing of Pa measured in situ should be done relative to the lecation of the center of the receiving dipole MN: in that case, the central part of the main maximum of effective electric resistivity corresponds exactly to the position of the crack 9 Referencing ~ z to any other point of the probe (for instance, to the location of any of the electrodes) makes the relationshi~ unstable and uncertain. Another conclusion that can be drawn from an analysis of curves of Pa according to Figs. 2 and 3 is that preference should be given electrode setups symmetric relative to the spacing: coordinate referencing in that case is stable due to the symmetry of the Pa perturbation from the crack. When the measured electric resistivity Pa in situ is referenced to the center of the receiving dipole MN, there is a one-to-one correspondence between the central part of the main
*All the linear dimensions are valuated by the scale of A.
94
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Fig. 4. Effect of receiving dipole size on the structure of synthesized charts (a - for q = 1/2; b - for q = i).
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L Fig. 5. Influence of sounding base on the structure of synthesized well logging curves for "normal" setup (MN = 7/9q). maxima of well logging curves and the location of cracks in the bed (homogeneous model). The distance between the principal peaks of Pa is equal to the distance between the cracks. Effect of the Size of the ReceivinK Dipole on the Structure of the Curves This question was studied for a symmetric setup with a reduction of the receiving dipole size from 7/9 to i/gq. Figure 4a, b presents calculated curves of Pa for q = 1/2 and I, respectively (fragments of computer diagrams between two cracks). As seen from these curves, a reduction in the receiving dipole width leads to an increased absolute maximum of Pa perturbation from the crack accompanied by a reduction in its width. The width of the principal maximum is determined by (and virtually equal to) the width of the measurement dipole. Similar conclusions also hold for symmetric probes. Of interest is another feature that follows from a comparison of these diagrams: with a transition to lateral setups (small MN) the zone of lower Pa values near the principal maxima at the location of the cracks is larger. This
95
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Fig. 7
Fig. 6. Effect of sounding base on the structure of synthesized well logging curves for a symmetric setup (MN = i/3q). Fig. 7. Influence of sounding base on the structure of synthesized well logging curves for the "lateral" setup (MN = i/9q). means that it is more appropriate to correlate the center of the principal maximum of in situ well logging curves to the center of anomalies of a negative sign rather than the center of the principal maximum for natural well logging curves with lateral setups (!). For cracks with spacings comparable to the spacing or larger than it, the width of these negative anomalies is equal to the spacing. The significance of this conclusion is due to the fact that, as the width of the receiving dipole is reduced due to the discreteness of the measurement of geophysical information as determined by the well logging step, the probability that an individual crack will be situated between receiving electrodes is reduced. Effect of the Spacing q on the Structure of Computer-Generated
Curves p= for a Symmetric Setup
In this section, which presumes answers to questions c, d, and e from the Introduction, we will discuss the more general problem of the effect of the spacing q on the structure of the electrical diagram for a symmetric electrometric sertup. Figures 5-7 present synthesized diagrams of well logging for q = 1/9, 1/4, 1/3, i, 2 ..... 9 for a receiving dipole similar in size to the base of the probe-potential-probe (Fig. 5) which is equal to 1/3 of the spacing (Fig. 6) and much smaller than the lateral array spacing (Fig. 7). The graphs reveal a peculiarity in the structure of the synthesized curves: with increasing spacings, zones of zero effective resistivity appear. The smaller the receiving dipole, all other conditions being equal, the wider this zone (see Figs. 5-7). The synthesized curves become identically zero if the minimum distance between the emitter and receiver electrodes becomes greater compared to the intercrack spacing. The intermediate case (existence of zones with zero 0a) takes place when, as the probe moves in the well, both receiving electrodes become separated simultaneously from the emitter electrodes by cracks. Physically, this fact can easily be explained: since the cracks here are "infinite," the drop in potential in the zone separated from the emitter electrode by, for example, two cracks is equal to zero, as there is no electric field in this region. ReceiVing electrodes will always be in such a "dead" zone, regardless of the position of the probe in the well, if the minimum distance between the emitter and receiving electrodes is larger than the intercrack distance. It should be clear that an increase of the spacing will result in a degeneration of the synthesized curve to zero if the ratios of interelectrode spacings are fixed in the fol-
96
lowing fashion for lateral probes: according to Figs. 5-7 for a gradient probe (~3 - a2 = 1/9) the synthesized curve degenerates already for a spacing equal to 3A; for the normal probe (a3 - ~2 = 7/9) the degeneration takes place with a spacing of 9&. The case considered in the present section explains certain odd results of experiments conducted in Norilsk mines. Well logging is sometimes accompanied by a peculiar "superconductivity" effect near single workings during well logging. At the location of disrupted zones, given a sufficiently large intensity of current fed to the bed, a potential difference at certain points in the well is either nonexistent or infinitesimal. The preceding findings provide a natural explanation for this effect. According to the computer simulation, the case of an "infinite" crack takes place virtually for parameter b on the order of 1.5-2. At b = 0, the synthesized curve, by virtue of the lemma from [I], becomes a constant (unity), i.e., the cases studied above of b ~ ~ (specifically, 0 < b < 2) are intermediate cases for the extreme situations of b = 0 and b = ~. If we reduce b from 2 to O, only the deviations of Pa from unity with the above curves in Figs. 2-7 will be affected: the smaller the value of b, the smaller the difference between Pa and unity. Importantly, the structure of the curves in Figs. 2-7 remains practically unchanged, i.e., the preceding conclusions still hold. The statement of the problem and the analysis of formulas for its solution in [i] indicate that, by including the degree of opening of cracks 6 i, we obtain mainly an increased width of the principal maxima of Pa above cracks, incremented by 6 i and retaining the structural features depicted in Figs. 2-7. The analytic curves presented here are perfect in a certain sense. Since in field experiments geophysical information is presented discretely according to the well logging step, discretization inevitably "smooths out" these graphs. Figure 8 shows, as an illustration, several such curves plotted with elementary linear interpolation for a logging step that is in a certain proportion to the intercrack spacing (see notations in Fig. 8). The figure confirms the hypothesis used in [3] concerning the probability (in first approximation) of approximating perturbations of geophysical (electric) parameters as a cosinusoid or a sinusoid with a period equal to the intercrack spacing. In order to create a method of reliable and unequivocal determination of structural hierarchy of beds and solve related problems, it is of great interest to analyze a different hypothesis advanced in [2]: the validity and scope of the superposition principle of perturba-
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97
tions of measured geophysical parameters from cracks of different levels. An analytic study of this issue will be the subject of a future publication. Importance is also attached to a phenomological approach, which establishes correlations between the spectral characteristics of geophysical curves and prior data on the structural hierarchy of geological formations [4]. CONCLUSIONS i. It has been proved that the distance between cracks can be determined by the distance between the principal maxima of electrical well logging diagrams. 2. Coordinates of the measured electric resistivity should be tied to the center of the receiving dipole of the electrometric sounding probe. There is a direct correlation between the location of cracks and the extrema on geophysical well logging curves. 3. The correlation between the position of the principal bendpoints on geophysical well logging diagrams and the location of cracks (according to conclusion 2 above) is especially stable for a symmetric four-electrode setup. ^
4. With decreasing length of the receiving dipole the absolute maximum of Pa from the crack grows while its width decreases. The width of the principal maximum is practically equal to the length of the measurement dipole. 5. With a lateral setup (small MN) the position of the cracks should be determined by the position of the principal minima on pa" 6) When the minimal distance between emitter and receiver electrodes is larger than the distances between long cracks (in areas of fractured beds), an effect may appear where the intensely fissured areas exhibit abnormally low effective electric resistivity. 7. A theoretical substantiation is given to the hypothesis at the basis of the phenomenological theory of geophysical bore hole defectoscopy described in [3] as it applies to the electrometric method: a) there is a correlation between the positions of extrema on geophysical well logging curves and the cracks in the bed; and b) perturbations of a parameter being measured from the cracks of a given level can be approximated by the sinusoid or cosinusoid curve of appropriate amplitude and period. LITERATURE CITED i.
2.
3.
4.
98
V. N. Oparin, "Mathematical modeling of electromechanical well logging curves for fissured media, Part I: The field of direct current cylindrical sources," Fiz.-Tekh. Probl. Razrab. Polezn. Iskop., No. 1 (1989). V. N. Oparin et al., "Interpretation of ultrasonic and electromechanical measurements in studies of ores in the Talnakh deposit," in: Methodology of Stress Measurements in a Rock Bed [in Russian], Institute of Mining, Siberian Department, USSR Academy of Sciences, Novosibirsk (1978). V. N. Oparin, "Principles of bore hole geophysical defectoscopy, Part i: Spectral analysis and defect measures," Fiz.-Tekh. Probl. Razrab. Polezn. Iskop., No. 6 (1982). N. A. Sadovskii, "Natural granularity of rocks," Dokl. Akad. Nauk SSSR, 247, No. 4 (1979).