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A Fortran program for computing the mise-a-la-masse response over a dyke–like body
Computers & Geosciences 28 (2002) 1119–1126
Short Note
A Fortran program for computing the mise-a-la-masse response over a dyke–like body$ Mallika Mullick, R.K. Majumdar* Department of Geological Sciences, Jadavpur University, Kolkata-700 032, India Received 8 July 2001; received in revised form 23 February 2002; accepted 14 March 2002
1. Introduction Mise-a-la-masse is a French phrase meaning excitation at the mass. This method is basically a DC electrical method and has been successfully utilised for locating and delineating shape, size and extent of the ore body (Parasnis, 1967; Jamtild et al., 1984). Here, one of the two current electrodes is placed in a conducting body itself and the other is placed at infinity. Eskola (1979), Eskola and Hongisto (1981), Eloranta (1984, 1985, 1986, 1988), Bowker (1987), Soininen (1987), Dey and Morrison (1979), and Mwenifumbo, (1980) used various techniques to compute surface potential and apparent resistivity profiles over various 2-D bodies for multiple electrode configurations. The present work has followed the numerical method described by Beasley and Ward (1986) (Fig. 1). In the present investigation, responses are obtained as potential and also as resistivity in some instances. The response is obtained in terms of potential, measured with a two-electrode array system. One of the current electrodes is buried in the anomalous body itself and the other is kept at a large distance from the body. Similarly, one of the measuring electrodes is fixed at a large distance from the anomalous body, whereas the other moves on the surface giving almost absolute electrode potential at that point. Mathematical potential responses are calculated for a conducting body at various depths for different positions of the charging point and resistivity contrasts having three different divisions of cells of the body. The strike of the body is along the x-axis and all profiles are
$
Code on server at http://www.iamg.org/CGEditor/index.htm *Corresponding author. E-mail addresses: mallika [email protected] (M. Mullick), [email protected] (R.K. Majumdar).
taken perpendicular to the strike i.e along the y-axis excepting in the situation where the body is divided into 36 cells. The numbering and arrangements of cells are shown (Fig. 2). Here, the profile is taken along the x-axis in order to show the asymmetry in the results due to the asymmetric position of the charging point. To carry out the mathematical calculations, a program is written in Fortran. The program is structurally simple and can be compiled by any Fortran compiler. The main steps in the program are explained and test runs are also provided on the IAMG server. The details about the program have been included in a ‘README’ file also on the server. The run time increases as the body is divided into a greater number of cells. Table 1 shows the times required on two different systems with two different cases with differing numbers of cells.
2. Discussion of results The computation time of mise-a-la-masse response largely depends on the number of cells into which the body is divided. Accuracy as well as computation time increases with increasing number of cells. In order to make a reasonable compromise between the two, most of the cases discussed in this paper are taken with 25 cells. Some cases are shown with 36 and 49 cells. 2.1. Case 1 Here, a body of dimension 2.5 0.1 6.5 units and resistivity r2 ¼ 10 O m is divided into 25 cells and the potential responses for different positions of the charging point at depth 0.15 unit from the surface are obtained (Fig. 3). All responses shown in this paper are obtained for current I ¼ 0:01 A. The response shows a sharp peak when the charging point is at the centre of the topmost part of the body (cell no. 3) and the peak
0098-3004/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 8 - 3 0 0 4 ( 0 2 ) 0 0 0 1 5 - 8
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Fig. 1. Three-dimensional body with conductivity s2 in a half-space with conductivity s1 :
Fig. 2. Order of cell-numbering when the body is divided into (a) 25 cells, (b) 36 cells, (c) 49 cells.
Table 1 Benchmark for comparison of CPU times on different systems Number of cells
49 25
CPU time (s) System 1a
System 2b
146.00 5.54
190.00 6.75
a IBM e-server x200(PIII 866 MHz/256MB SDRAM ECC/ SCSI HDD);compiled and run by Fortran compiler under Linux 7.1 OS. b PC(PIII800 MHz/512MB SDRAM/IDE HDD);compiled and run by gcc compiler under Linux 5.2 OS.
response falls off rapidly and the curve becomes more flat when the charging point is shifted to the centre of the body (cell no. 13). There is a further rapid reduction of the response when the charging point is further
lowered down to the centre of the lowermost portion of the body (cell no. 23). Here, the amplitude of the response is reduced, but the anomaly still persists showing the presence of the body. The potential responses for the same body with the same charging point at cell no. 3 are obtained for different depths of burial from the surface (Fig. 4). The peak response shows a sharp peak for depth 0.15 units and falls off rapidly as the depth of burial increases and it is undetected when depth of burial goes beyond 0.85 units. The potential responses for the same body for different resistivity contrasts are obtained with same charging point (cell no. 3) at depth 2.35 units (Fig. 5). Here, the response is maximum for the highest resistivity contrast (100:1) and anomaly reduces as the contrast decreases but even at depth 2.35 units, the body can be detected with this method for resistivity contrast 300:25. Asymmetric responses over a 301 inclined dyke with dimension 4 0.8 8.0 units are obtained for various
M. Mullick, R.K. Majumdar / Computers & Geosciences 28 (2002) 1119–1126
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Fig. 3. Potential response over a vertical body of dimension 2.5 0.1 6.5 units for various positions of charging point. Depth to top of body from surface=0.15 units.
Fig. 4. Potential response over a vertical body of dimension 2.5 0.1 6.5 units for various depths to top of body from surface. Charging point at cell no. 3.
depths of burial with the fixed charging point (cell no. 3) at the centre of the top of the body (Fig. 6). The technique of cell division for an inclined body is shown
(Figs. 6 and 7). The order of the cell numbering is the same as shown for a vertical body with 25 cells. Here, the volume of the body is increased and hence the
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Fig. 5. Potential response over a vertical body of dimension 2.5 0.1 6.5 for various resistivity contrasts with host. Charging point at cell no. 3 and depth to top of body from surface=2.35 units.
Fig. 6. Potential response over an inclined conducting body having dip=301, size of body=4 0.8 8.0, for various depths to top of body from surface. Charging point at cell no. 3.
response is decreased. It shows that the peak response decreases with an increase in depth of burial of the inclined body and is shifted towards the downdip side (Fig. 7). Here also, the peak response decreases sharply with depth and response with inclination 601 is much less than that of inclination 301 for other conditions remaining the same.
2.2. Case 2 Here, a vertical tabular body of dimension 6 0.2 12 units is considered and divided into 36 cells. The potential responses over this vertical body are obtained along the strike with different positions of the charging point with depth to its top=1.0 unit (Fig. 8).
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Fig. 7. Potential response over an inclined conducting body having dip=601, size of body=4 0.8 8.0 units for various depths to top of body from surface. Charging point at cell no. 3.
The response decreases sharply when the charging point is shifted from cell nos. 3–21. The magnitude of the peak anomaly here is less when compared with Case 1 (Figs. 3 and 4) due to larger size and depth of the anomalous body. The responses of the same body with different depths of burial are obtained for the charging point at cell no. 3 (Fig. 9). The response decreases sharply as depth of burial increases from 0.5 to 1.7 units and responses become flat at depth of 3.0 units.
2.3. Case 3
Fig. 8. Potential response over a vertical tabular body of dimension 6 0.2 12 units for various positions of charging point.
Here, the response is little asymmetric as the charging point is not just at the centre of the body.
Here, a body of dimension 7 0.4 7 units with resistivity 10 O m is divided into 49 cells and resistivity response for different positions of charging point are discussed. The apparent resistivity values over this body have been obtained with depth to its top 1.1 units and charging point at cell no. 4 (Fig. 10). Here, a symmetric resistive peak is obtained over the conducting vertical dyke and response falls sharply away from the body. The apparent resistivity anomaly with a depth of 2 units is obtained when the charging point is at cell no. 25 (Fig. 11). Here, the resistivity anomaly over the body is found to have been increased as the charging point is placed at a lower level.
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Fig. 9. Potential response over a vertical tabular body of dimension 6 0.2 12 units for various depths to top of body from surface. Charging point at cell no. 3.
Fig. 10. Apparent resistivity (ra ) values over vertical body of size 7 0.4 7 units, having depth to its top from surface=1.1 units and charging point at cell no. 4.
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Fig. 11. Apparent resistivity (ra ) values over vertical body of size 7 0.4 7 units, having depth to its top from the surface=2 units and charging point at cell no. 25.
3. Conclusions
References
The following general conclusions can be drawn from this theoretical study.
Beasley, C.W., Ward, S.H., 1986. Three dimensional mise-a-lamasse modelling applied to mapping fracture zones. Geophysics 51 (1), 98–113. Bowker, A., 1987. Size determination of slab like ore bodies— an interpretation scheme for single hole mise-a-la-masse anomalies. Geoexploration 24, 07–218. Dey, A., Morrison, H.F., 1979. Resistivity modelling for arbitrarily shaped three-dimensional structure. Geophysics 44, 753–780. Eloranta, E.H., 1984. A method for calculating mise-a-la-masse anomalies in the case of high conductivity contrast by integral equation technique. Geoexploration 22, 77–88. Eloranta, E.H., 1985. A comparison between mise-a-la-masse anomalies near vertical contact. Geoexploration 23, 471– 481. Eloranta, E.H., 1986. The behaviour of mise-a-la-masse anomalies near vertical contact. Geoexploration 24, 1–4. Eloranta, E.H., 1988. The modelling of mise-a-la-masse anomalies in an anisotropic half space by integral equation method. Geoexploration 25, 93–101. Eskola, L., 1979. Calculation of galvanic effects by means of the method of subareas. Geophysical Prospecting 27, 616–627. Eskola, L., Hongisto, H., 1981. The solution of stationary electric field strength and potential of a current source in a 2.5 dimensional environment. Geophysical Prospecting 29, 260–273.
1. The larger the body, the smaller will be the mise-a-lamasse anomaly. 2. For a fixed depth of burial, the anomaly is reduced when the charging point is shifted deeper inside the body. 3. The response is asymmetric on the inclined body and the peak response is shifted towards the downdip side with increase in depth of burial. 4. High resistivity values are obtained over a conducting body. However, the resistivity values are dependent on the depth of the charging point.
Acknowledgements The authors are grateful to Dr. Kalyan Kumar Das, Department of Physical Chemistry, Jadavpur University for helping with software development and providing computer facilities.
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M. Mullick, R.K. Majumdar / Computers & Geosciences 28 (2002) 1119–1126
Jamtild, A., Magmisson, K.-A., Olsson, O., Stenberg, L., 1984. Electrical borehole measurements for the mapping of fracture zones in crystalline rock. Geoexploration 22, 203–216. Mwenifumbo, C.J., 1980. Interpretation of mise-a-la-masse data for vein type bodies. Ph.D. Dissertation, University of Western Ontario, Canada, 245pp.
Parasnis, D.S., 1967. Three dimensional electric mise-a-la-masse survey of an irregular lead–zinc– copper deposit in central Sweden. Geophysical Prospecting 15, 407–473. Soininen, H., 1987. Mise-a-la-masse modelling by integral equation with the continuity equation as a constraint. Geoexploration 24 (6), 455–460.
Short Note
A Fortran program for computing the mise-a-la-masse response over a dyke–like body$ Mallika Mullick, R.K. Majumdar* Department of Geological Sciences, Jadavpur University, Kolkata-700 032, India Received 8 July 2001; received in revised form 23 February 2002; accepted 14 March 2002
1. Introduction Mise-a-la-masse is a French phrase meaning excitation at the mass. This method is basically a DC electrical method and has been successfully utilised for locating and delineating shape, size and extent of the ore body (Parasnis, 1967; Jamtild et al., 1984). Here, one of the two current electrodes is placed in a conducting body itself and the other is placed at infinity. Eskola (1979), Eskola and Hongisto (1981), Eloranta (1984, 1985, 1986, 1988), Bowker (1987), Soininen (1987), Dey and Morrison (1979), and Mwenifumbo, (1980) used various techniques to compute surface potential and apparent resistivity profiles over various 2-D bodies for multiple electrode configurations. The present work has followed the numerical method described by Beasley and Ward (1986) (Fig. 1). In the present investigation, responses are obtained as potential and also as resistivity in some instances. The response is obtained in terms of potential, measured with a two-electrode array system. One of the current electrodes is buried in the anomalous body itself and the other is kept at a large distance from the body. Similarly, one of the measuring electrodes is fixed at a large distance from the anomalous body, whereas the other moves on the surface giving almost absolute electrode potential at that point. Mathematical potential responses are calculated for a conducting body at various depths for different positions of the charging point and resistivity contrasts having three different divisions of cells of the body. The strike of the body is along the x-axis and all profiles are
$
Code on server at http://www.iamg.org/CGEditor/index.htm *Corresponding author. E-mail addresses: mallika [email protected] (M. Mullick), [email protected] (R.K. Majumdar).
taken perpendicular to the strike i.e along the y-axis excepting in the situation where the body is divided into 36 cells. The numbering and arrangements of cells are shown (Fig. 2). Here, the profile is taken along the x-axis in order to show the asymmetry in the results due to the asymmetric position of the charging point. To carry out the mathematical calculations, a program is written in Fortran. The program is structurally simple and can be compiled by any Fortran compiler. The main steps in the program are explained and test runs are also provided on the IAMG server. The details about the program have been included in a ‘README’ file also on the server. The run time increases as the body is divided into a greater number of cells. Table 1 shows the times required on two different systems with two different cases with differing numbers of cells.
2. Discussion of results The computation time of mise-a-la-masse response largely depends on the number of cells into which the body is divided. Accuracy as well as computation time increases with increasing number of cells. In order to make a reasonable compromise between the two, most of the cases discussed in this paper are taken with 25 cells. Some cases are shown with 36 and 49 cells. 2.1. Case 1 Here, a body of dimension 2.5 0.1 6.5 units and resistivity r2 ¼ 10 O m is divided into 25 cells and the potential responses for different positions of the charging point at depth 0.15 unit from the surface are obtained (Fig. 3). All responses shown in this paper are obtained for current I ¼ 0:01 A. The response shows a sharp peak when the charging point is at the centre of the topmost part of the body (cell no. 3) and the peak
0098-3004/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 8 - 3 0 0 4 ( 0 2 ) 0 0 0 1 5 - 8
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M. Mullick, R.K. Majumdar / Computers & Geosciences 28 (2002) 1119–1126
Fig. 1. Three-dimensional body with conductivity s2 in a half-space with conductivity s1 :
Fig. 2. Order of cell-numbering when the body is divided into (a) 25 cells, (b) 36 cells, (c) 49 cells.
Table 1 Benchmark for comparison of CPU times on different systems Number of cells
49 25
CPU time (s) System 1a
System 2b
146.00 5.54
190.00 6.75
a IBM e-server x200(PIII 866 MHz/256MB SDRAM ECC/ SCSI HDD);compiled and run by Fortran compiler under Linux 7.1 OS. b PC(PIII800 MHz/512MB SDRAM/IDE HDD);compiled and run by gcc compiler under Linux 5.2 OS.
response falls off rapidly and the curve becomes more flat when the charging point is shifted to the centre of the body (cell no. 13). There is a further rapid reduction of the response when the charging point is further
lowered down to the centre of the lowermost portion of the body (cell no. 23). Here, the amplitude of the response is reduced, but the anomaly still persists showing the presence of the body. The potential responses for the same body with the same charging point at cell no. 3 are obtained for different depths of burial from the surface (Fig. 4). The peak response shows a sharp peak for depth 0.15 units and falls off rapidly as the depth of burial increases and it is undetected when depth of burial goes beyond 0.85 units. The potential responses for the same body for different resistivity contrasts are obtained with same charging point (cell no. 3) at depth 2.35 units (Fig. 5). Here, the response is maximum for the highest resistivity contrast (100:1) and anomaly reduces as the contrast decreases but even at depth 2.35 units, the body can be detected with this method for resistivity contrast 300:25. Asymmetric responses over a 301 inclined dyke with dimension 4 0.8 8.0 units are obtained for various
M. Mullick, R.K. Majumdar / Computers & Geosciences 28 (2002) 1119–1126
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Fig. 3. Potential response over a vertical body of dimension 2.5 0.1 6.5 units for various positions of charging point. Depth to top of body from surface=0.15 units.
Fig. 4. Potential response over a vertical body of dimension 2.5 0.1 6.5 units for various depths to top of body from surface. Charging point at cell no. 3.
depths of burial with the fixed charging point (cell no. 3) at the centre of the top of the body (Fig. 6). The technique of cell division for an inclined body is shown
(Figs. 6 and 7). The order of the cell numbering is the same as shown for a vertical body with 25 cells. Here, the volume of the body is increased and hence the
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Fig. 5. Potential response over a vertical body of dimension 2.5 0.1 6.5 for various resistivity contrasts with host. Charging point at cell no. 3 and depth to top of body from surface=2.35 units.
Fig. 6. Potential response over an inclined conducting body having dip=301, size of body=4 0.8 8.0, for various depths to top of body from surface. Charging point at cell no. 3.
response is decreased. It shows that the peak response decreases with an increase in depth of burial of the inclined body and is shifted towards the downdip side (Fig. 7). Here also, the peak response decreases sharply with depth and response with inclination 601 is much less than that of inclination 301 for other conditions remaining the same.
2.2. Case 2 Here, a vertical tabular body of dimension 6 0.2 12 units is considered and divided into 36 cells. The potential responses over this vertical body are obtained along the strike with different positions of the charging point with depth to its top=1.0 unit (Fig. 8).
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Fig. 7. Potential response over an inclined conducting body having dip=601, size of body=4 0.8 8.0 units for various depths to top of body from surface. Charging point at cell no. 3.
The response decreases sharply when the charging point is shifted from cell nos. 3–21. The magnitude of the peak anomaly here is less when compared with Case 1 (Figs. 3 and 4) due to larger size and depth of the anomalous body. The responses of the same body with different depths of burial are obtained for the charging point at cell no. 3 (Fig. 9). The response decreases sharply as depth of burial increases from 0.5 to 1.7 units and responses become flat at depth of 3.0 units.
2.3. Case 3
Fig. 8. Potential response over a vertical tabular body of dimension 6 0.2 12 units for various positions of charging point.
Here, the response is little asymmetric as the charging point is not just at the centre of the body.
Here, a body of dimension 7 0.4 7 units with resistivity 10 O m is divided into 49 cells and resistivity response for different positions of charging point are discussed. The apparent resistivity values over this body have been obtained with depth to its top 1.1 units and charging point at cell no. 4 (Fig. 10). Here, a symmetric resistive peak is obtained over the conducting vertical dyke and response falls sharply away from the body. The apparent resistivity anomaly with a depth of 2 units is obtained when the charging point is at cell no. 25 (Fig. 11). Here, the resistivity anomaly over the body is found to have been increased as the charging point is placed at a lower level.
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Fig. 9. Potential response over a vertical tabular body of dimension 6 0.2 12 units for various depths to top of body from surface. Charging point at cell no. 3.
Fig. 10. Apparent resistivity (ra ) values over vertical body of size 7 0.4 7 units, having depth to its top from surface=1.1 units and charging point at cell no. 4.
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Fig. 11. Apparent resistivity (ra ) values over vertical body of size 7 0.4 7 units, having depth to its top from the surface=2 units and charging point at cell no. 25.
3. Conclusions
References
The following general conclusions can be drawn from this theoretical study.
Beasley, C.W., Ward, S.H., 1986. Three dimensional mise-a-lamasse modelling applied to mapping fracture zones. Geophysics 51 (1), 98–113. Bowker, A., 1987. Size determination of slab like ore bodies— an interpretation scheme for single hole mise-a-la-masse anomalies. Geoexploration 24, 07–218. Dey, A., Morrison, H.F., 1979. Resistivity modelling for arbitrarily shaped three-dimensional structure. Geophysics 44, 753–780. Eloranta, E.H., 1984. A method for calculating mise-a-la-masse anomalies in the case of high conductivity contrast by integral equation technique. Geoexploration 22, 77–88. Eloranta, E.H., 1985. A comparison between mise-a-la-masse anomalies near vertical contact. Geoexploration 23, 471– 481. Eloranta, E.H., 1986. The behaviour of mise-a-la-masse anomalies near vertical contact. Geoexploration 24, 1–4. Eloranta, E.H., 1988. The modelling of mise-a-la-masse anomalies in an anisotropic half space by integral equation method. Geoexploration 25, 93–101. Eskola, L., 1979. Calculation of galvanic effects by means of the method of subareas. Geophysical Prospecting 27, 616–627. Eskola, L., Hongisto, H., 1981. The solution of stationary electric field strength and potential of a current source in a 2.5 dimensional environment. Geophysical Prospecting 29, 260–273.
1. The larger the body, the smaller will be the mise-a-lamasse anomaly. 2. For a fixed depth of burial, the anomaly is reduced when the charging point is shifted deeper inside the body. 3. The response is asymmetric on the inclined body and the peak response is shifted towards the downdip side with increase in depth of burial. 4. High resistivity values are obtained over a conducting body. However, the resistivity values are dependent on the depth of the charging point.
Acknowledgements The authors are grateful to Dr. Kalyan Kumar Das, Department of Physical Chemistry, Jadavpur University for helping with software development and providing computer facilities.
1126
M. Mullick, R.K. Majumdar / Computers & Geosciences 28 (2002) 1119–1126
Jamtild, A., Magmisson, K.-A., Olsson, O., Stenberg, L., 1984. Electrical borehole measurements for the mapping of fracture zones in crystalline rock. Geoexploration 22, 203–216. Mwenifumbo, C.J., 1980. Interpretation of mise-a-la-masse data for vein type bodies. Ph.D. Dissertation, University of Western Ontario, Canada, 245pp.
Parasnis, D.S., 1967. Three dimensional electric mise-a-la-masse survey of an irregular lead–zinc– copper deposit in central Sweden. Geophysical Prospecting 15, 407–473. Soininen, H., 1987. Mise-a-la-masse modelling by integral equation with the continuity equation as a constraint. Geoexploration 24 (6), 455–460.