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Finite-element modeling of the normal resistivity tool in azimuthally inhomogenous formations
PETROLEUM SCIENCE & ENGINEERING
ELSEVIER
Journal of Petroleum Science and Engineering 14 ( 1995) 59-63
Finite-element modeling of the normal resistivity tool in azimuthally inhomogenous formations Michael S. Bittar a,* , David P. Shattuck b, Liang C. Shen b aSperry-Sun Drilling Services, Houston, TX 77032, USA h University of Houston, Houston. Texus. USA
Received 20 July 1994
Abstract A finite-element, three-dimensional, theoretical model of resistivity logging is presented. The method is based on minimizing the total power, or energy, of the system. The derivation of the power equation is outlined. The Rayleigh-Ritz procedure is used for numerical minimization of the total power. For each individual element, the element matrix is obtained first. After that, the overall system of equations is assembled from the individual element matrices. Then, the boundary conditions are enforced and finally, the electric potential is calculated at each node by solving the overall system of equations. The finite-element method is tested for a limiting case with the integral equation method and the agreement is excellent. The finite-element method is used to analyze the response of the 16-inch normal tool in azimuthally inhomogenous formations. The finite-element theoretical model presented can be used to study the responses of all electrode-type resistivity tools, such as the laterolog and the spherically focused log in complex logging situations.
1. Introduction During the past five decades, many sophisticated logging tools have been developed to measure the resistivity of subsurface geological formations. In boreholes drilled with conductive muds, most of the tools used for resistivity logging are the electrode-type. Except for simple geometries, analytic solutions for the responses of these tools are rarely found in the literature. With today’s logging practice, logging tools are being run in complex geometries such as fractured, multilayered and dipping formations. With this practice, the need for techniques that predict the responses of these tools in complex geometries has increased. Simpson et al. (1983), Shattuck et al. (1987), Bittar et al. (1990) studied the induction and the laterolog tools using scale * Corresponding author. 0920.4105/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSD10920-4105(95)00029-1
modeling. Merchant and Thadani ( 1982) developed a technique for the evaluation of the response of finiteelectrode centered resistivity logging tools. Gianzero and Anderson ( 1982) computed the response of a laterolog traversing a thin invaded bed. Zhang and Shen ( 1984) calculated the response of the normal resistivity tool in a borehole crossing a bed boundary. Tsang and Gianzero ( 1984) developed a semi-numerical, hybrid, method to solve the fundamental problem of resistivity logging. Chemali et al. ( 1988) studied the dual laterolog in complex situations. In this paper, a three-dimensional, finite-element theoretical model is constructed to study the response of a normal resistivity tool in complex geometries. The mathematical formulation that governs all electrode-type resistivity tools is derived and the appropriate boundary conditions are given. Several logging examples are also presented (Silvester and Ferrari, 1983).
60
MS. Bittar et al. /Journal of Petroleum Science and Engineering 14 (1995) 59-43
2. Theoretical formulation
P=
The finite-element method incorporates the use of the principle of minimization of a power or energy function. This principle, which is well known in mechanics, also applies to electromagnetic problems since electromagnetic fields behave in such a way that the total energy is minimized (Brauer and Vander Heiden, 1988). Since most electrode-type logging tools utilize very low frequency ( < 1000 Hz) current sources, it is assumed in this theoretical formulation that these current sources are DC. The power, P’, per unit volume is given by (Harrington, 1961) : dB dD P’=H.z+E.x+J.E with the constitutive
(7)
The electric field, E, is equal to the negative gradient of the electric potential V: E=-vV Substituting
(8) Eq. 8 into Eq. 7 yields:
In cylindrical coordinates, ten finally as:
the total power can be writ-
relations expressed as: (9) @a)
D=&
(2b)
J=aE
(2c)
where p is the permittivity, o electric current assumed to be to time is equal
magnetic permeability, E is the electric is the electric conductivity and J is the density. Since the current sources are DC, the partial derivative with respect to zero and Eq. 1 becomes:
P’= J.E
P’ = uE2
- I,V,
(1)
B=pH
Substituting
aE2d0 R
In order to uniquely define the problem, it is necessary to specify the following boundary conditions: ( 1) The electric potential, V, is continuous at any point in the formation. (2) The electric potential, V, vanishes at infinity. (3) The electric potential, V, on the surface of the metal electrodes is constant. (4) At the boundary adjacent to the current electrode (electrode delivers current 1,) :
(3)
(10)
Eq. 2c into Eq. 3 yields: (4)
The dissipated power, P,, is obtained by integrating Eq. 4 over the volume to obtain: (5)
where n is a unit outward normal vector on the electrode surface S. (5) At the boundary adjacent to the measured electrode (electrode delivers no current) :
n where the domain 0 is the volume of the region of interest. The power, P,, supplied by the source is given by: P,=
-I,V,
(6) At the boundary mandrel:
adjacent to the tool insulating
(6)
where Z, is the current source and V, is the electric potential at the surface of the current electrode. The minus sign indicates that power is delivered by the source. The total power, P, in the region of interest is equal to the sum of the power delivered by the source and the dissipated power:
8V -= dn
0
(12)
Eq. 9 is solved numerically, by dividing the domain R into a number of finite-elements (each with electric conductivity F). Using the Rayleigh-Ritz procedure, the derivatives of P with respect to V are set to zero.
M.S. Bittar et al. /Journal
of Petroleum Science and Engineering 14 (1995) 59-63
ces. Then, the above boundary conditions are enforced and, finally, the electric potential is calculated at each node by solving the over-all system of equationsThe apparent resistivity of the formation is then determined by:
-160 -120
h%l=+ s
’
1
“‘1
10
Resistivity
“‘I
100
” ‘4
1000
(ohm-m)
Fig. 1. Response of the normal resistivity tool for normal tool diameter. The log drawn as a solid line was generated by the integral equation method. The log drawn in small circles was computed by the finite-element method. The borehole is 8 inches and the mud resistivity is I R-m. In the finite-element calculation, the tool diameter is 3.5 inches.
For each individual element, the element matrix is obtained first. After that, the overall system of equations is assembled from the individual element matri-200 -160 -120 -80
120 160 1
(13)
where K is the tool’s geometric constant that will give the true formation resistivity when the tool is in a homogenous medium. Z, is the current flowing from the current electrode and V is the electric potential calculated at a measured electrode.
160 200
61
10 Resistivity
100
1000
(ohm-m)
Fig. 2. Response of the normal resistivity tool for very small tool diameter. The log drawn as a solid line was generated by the integral equation method. The log drawn in small circles was computed by the finite-element method. The borehole is 8 inches and the mud resistivity is 1 .0-m. In the finite-element calculation, the tool diameter is 0.2 inches.
3. Finite-element method verification The finite-element method was tested for a limiting case with the integral equation method. For this limiting case, the integral equation method provides an accurate solution (Shen et al., 1990). The response of the 16inch normal resistivity tool in a borehole crossing two thick beds is computed. The distance between the current and the measure electrode is 16 inches and the tool diameter is 3.5 inches. The borehole is 8 inches in diameter and is filled with conductive mud of 1 0-m resistivity. The resistivity of the upper bed is 10 L&m and the resistivity of the lower bed is 100 0-m. Fig. 1 shows two logs superimposed on each other. The log drawn as a solid line was generated by the integral equation method and the log drawn in small circles was computed by the finite-element method. The agreement between the two logs is satisfactory in the 10 0-m bed. However, the resistivity reading calculated with the finite-element method in the 100 L&m bed is slightly greater than that calculated by the integral equation method. This discrepancy is due to the fact that in the integral equation method a point electrode is assumed for the normal tool and in the finite-element method a 3.5 inch diameter tool is assumed. A second log was calculated by the finiteelement method. In this case, the diameter of the tool was reduced to 0.2 inch. Fig. 2 shows the comparison between this log and the log generated by the integral equation method. Here, the agreement between the two methods is excellent in both the 10 0-m and the 100 L&n-mbeds.
MS. Bittar et al. /Journal of Petroleum Science and Engineering I4 (1995) 59-63
62
4. Response of the normal tool in azimuthally inhomogenous formations
Fig. 3. A 16 inch normal tool in a borehole with three bed boundaries. The borehole is 8 inches in diameter and tilled with conductive mud of 1 R-m resistivity. The resistivity of the top and bottom beds is 75 R-m. The middle bed is 8 inches thick with a resistivity of 5 L&m.
The integral equation method can only solve for simple cases such as a normal resistivity tool in a borehole with a thick bed or a normal resistivity tool in a borehole crossing a boundary separating two beds. More complex cases such as a normal tool in a borehole with three bed boundaries or a tool in an azimuthally inhomogenous formation is solved with the finite-element method. The case that involves a normal tool in a borehole with three bed boundaries is shown in Fig. 3 and the case that involve the tool in an azimuthally inhomogenous bed is shown in Fig. 4. In all these cases, the borehole is 8 inches in diameter and filled with conductive mud of 1 0-m resistivity. The middle bed is 8 inches thick. The responses of the 16-inch normal tool for these five cases are shown in Fig. 5. The resistivity of the thick top and bottom beds is 75 am. The resistivity of the middle bed is 5 0-m for case A, and varies azimuthally for cases B, C, D, and E as shown in Fig. 4. Log A in Fig. 5 is the response of the normal tool in a borehole with three bed boundaries with azimuthally homogenous beds. This log does not reach full response in the middle bed because of borehole and thin bed effects. As an azimuthally resistive (75 0-m) inhomogeneity is introduced for the B, C, D and E cases, the 80
48 32
Case
B
Case
C
90
-48 -64 -8C
Case
D
Case
E
Fig. 4. Top views of the middle bed for cases A, B, C, D, and E. The borehole is 8 inches in diameter and filled with conductive mud of 1 0-m resistivity. The resistivity of the bed varies in the azimuthal direction as shown above. The resistivity contrast is 75 to 5 0-m.
35
45
55
65
75
Resistivity (n-m) Fig. 5. Response of the 16-inch normal tool in complex geometry. This figure shows five logs: log A, B, C, D and E. These logs represent the response of the tool in cases A, B, C, D and E. The resistivity contrast is 75 n-m to 5 am.
MS. Bittar et al. /Journal of Petroleum Science and Engineering 14 (1995) 5963 80
63
6. Nomenclature
64
D
48
E
32
B
H
16
J P'
0
1s
-1 6 -32
VS PS Pd
-48 -64
P Y”
0
60
120
180
240
300
Resistivity (&II)
I-L E
u Fig. 6. Response of the 16-inch normal tool in complex geometry. This figure shows five logs: log A, B, C, D and E. These logs represent the response of the tool in cases A, B, C, D and E. The resistivity contrast is 1000 0-m to IO in 0-m.
tool starts to read higher resistivity. In the middle bed, Log A is reading 23 C&m and Log E is reading 50 0m. Fig. 6 shows the response of the 16-inch normal tool in five complex cases with the same geometry as in the previous study. Here, the contrast between the beds is 1000 to 10. In the middle bed, Log A is reading 185 0-m and Log E is reading 980 0-m.
5. Conclusions
In this paper, a three-dimensional theoretical model of resistivity logging by the finite-element method is presented. The finite-element model is verified by comparing it to an integral equation model for a limiting case and good agreement is shown. The finite-element method is used in complex geometry to analyze the response of a 16-inch normal resistivity tool in azimuthally inhomogenous formations. The outlined finite-element theoretical model can be made more general to study the responses of all electrode-type tool in many complex logging situations.
0-m
electric flux density (coulomb/mete?) electric field strength (volt/meter) magnetic flux density, weber/mete? magnetic field strength (ampere/meter) electric current density (ampere/mete?) power per unit volume (watt/mete?) current delivered by current electrode (ampere) voltage at current electrode (volt) source power (watt) disspative power (watt) total power (watt) permeability (henry/meter) permittivity (farad/meter) conductivity (siemens/meter) resistivity (ohm-meter)
References Bittar, M., Shattuck, D. and Shen, L.C., 1990. Laboratory study of the shallow laterolog in high and low resistivity contrast. The Log Analyst, 3 1. Braurer, J.R. and Vander Heiden, R.H., 1988. Finite element modeling of electromagnetic resonators and absorbers. J. Appl. Phys.. 63(8): 3197-3199. Chemali, R., Gianzero, S.C. and Su, SM., 1988. The dual laterolog in common complex situations. Proc. 29th Annu. Meet. Sot. Prof. Well Log Analysts, Vol. 1, Paper N. Gianzero, S.C. and Anderson, B., 1982. An integral transform solution to the fundamental problem in resistivity logging. Geophysics, 47: 946956. Hanington, R.F., 1961. Time-Harmonic Electromagnetic Fields. McGraw-Hill Book Company, New York, N.Y. Merchant, G.A. and Thadani S.G., 1982. Finite electrode resistivity tool modelling. Proc. 23rd Annu. Meet. Sot. Prof. Well Log Analysts, Vol. 1, Paper Q. Shattuck, D., Bittar, M. and Shen, L.C., 1987. Scale modeling of the laterolog using synthetic focusing methods, The Log Analyst, 28. Simpson, RX, Shang, H.C. and Shen, L.C., 1983. A laboratory study of induction logs using physical modeling. Proc. 24th Annu. Meet. Sot. Prof. Well Log Analysts, Vol. 1, Paper M. Silvester, P.P. and Femui, R.L., 1983. FiniteElements for Electrical Engineers. Cambridge University Press, Cambridge. Tsang, L. and Gianzero, S.C., 1984. Solution of the fundamental problem in resistivity logging with a hybrid method. Geophysics, 49: 159&1604. Zhang, G.J. and Shen, L.C., 1984. Response of a normal resistivity tool in a borehole crossing a bed boundary. Geophysics, 49: 142149.
ELSEVIER
Journal of Petroleum Science and Engineering 14 ( 1995) 59-63
Finite-element modeling of the normal resistivity tool in azimuthally inhomogenous formations Michael S. Bittar a,* , David P. Shattuck b, Liang C. Shen b aSperry-Sun Drilling Services, Houston, TX 77032, USA h University of Houston, Houston. Texus. USA
Received 20 July 1994
Abstract A finite-element, three-dimensional, theoretical model of resistivity logging is presented. The method is based on minimizing the total power, or energy, of the system. The derivation of the power equation is outlined. The Rayleigh-Ritz procedure is used for numerical minimization of the total power. For each individual element, the element matrix is obtained first. After that, the overall system of equations is assembled from the individual element matrices. Then, the boundary conditions are enforced and finally, the electric potential is calculated at each node by solving the overall system of equations. The finite-element method is tested for a limiting case with the integral equation method and the agreement is excellent. The finite-element method is used to analyze the response of the 16-inch normal tool in azimuthally inhomogenous formations. The finite-element theoretical model presented can be used to study the responses of all electrode-type resistivity tools, such as the laterolog and the spherically focused log in complex logging situations.
1. Introduction During the past five decades, many sophisticated logging tools have been developed to measure the resistivity of subsurface geological formations. In boreholes drilled with conductive muds, most of the tools used for resistivity logging are the electrode-type. Except for simple geometries, analytic solutions for the responses of these tools are rarely found in the literature. With today’s logging practice, logging tools are being run in complex geometries such as fractured, multilayered and dipping formations. With this practice, the need for techniques that predict the responses of these tools in complex geometries has increased. Simpson et al. (1983), Shattuck et al. (1987), Bittar et al. (1990) studied the induction and the laterolog tools using scale * Corresponding author. 0920.4105/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSD10920-4105(95)00029-1
modeling. Merchant and Thadani ( 1982) developed a technique for the evaluation of the response of finiteelectrode centered resistivity logging tools. Gianzero and Anderson ( 1982) computed the response of a laterolog traversing a thin invaded bed. Zhang and Shen ( 1984) calculated the response of the normal resistivity tool in a borehole crossing a bed boundary. Tsang and Gianzero ( 1984) developed a semi-numerical, hybrid, method to solve the fundamental problem of resistivity logging. Chemali et al. ( 1988) studied the dual laterolog in complex situations. In this paper, a three-dimensional, finite-element theoretical model is constructed to study the response of a normal resistivity tool in complex geometries. The mathematical formulation that governs all electrode-type resistivity tools is derived and the appropriate boundary conditions are given. Several logging examples are also presented (Silvester and Ferrari, 1983).
60
MS. Bittar et al. /Journal of Petroleum Science and Engineering 14 (1995) 59-43
2. Theoretical formulation
P=
The finite-element method incorporates the use of the principle of minimization of a power or energy function. This principle, which is well known in mechanics, also applies to electromagnetic problems since electromagnetic fields behave in such a way that the total energy is minimized (Brauer and Vander Heiden, 1988). Since most electrode-type logging tools utilize very low frequency ( < 1000 Hz) current sources, it is assumed in this theoretical formulation that these current sources are DC. The power, P’, per unit volume is given by (Harrington, 1961) : dB dD P’=H.z+E.x+J.E with the constitutive
(7)
The electric field, E, is equal to the negative gradient of the electric potential V: E=-vV Substituting
(8) Eq. 8 into Eq. 7 yields:
In cylindrical coordinates, ten finally as:
the total power can be writ-
relations expressed as: (9) @a)
D=&
(2b)
J=aE
(2c)
where p is the permittivity, o electric current assumed to be to time is equal
magnetic permeability, E is the electric is the electric conductivity and J is the density. Since the current sources are DC, the partial derivative with respect to zero and Eq. 1 becomes:
P’= J.E
P’ = uE2
- I,V,
(1)
B=pH
Substituting
aE2d0 R
In order to uniquely define the problem, it is necessary to specify the following boundary conditions: ( 1) The electric potential, V, is continuous at any point in the formation. (2) The electric potential, V, vanishes at infinity. (3) The electric potential, V, on the surface of the metal electrodes is constant. (4) At the boundary adjacent to the current electrode (electrode delivers current 1,) :
(3)
(10)
Eq. 2c into Eq. 3 yields: (4)
The dissipated power, P,, is obtained by integrating Eq. 4 over the volume to obtain: (5)
where n is a unit outward normal vector on the electrode surface S. (5) At the boundary adjacent to the measured electrode (electrode delivers no current) :
n where the domain 0 is the volume of the region of interest. The power, P,, supplied by the source is given by: P,=
-I,V,
(6) At the boundary mandrel:
adjacent to the tool insulating
(6)
where Z, is the current source and V, is the electric potential at the surface of the current electrode. The minus sign indicates that power is delivered by the source. The total power, P, in the region of interest is equal to the sum of the power delivered by the source and the dissipated power:
8V -= dn
0
(12)
Eq. 9 is solved numerically, by dividing the domain R into a number of finite-elements (each with electric conductivity F). Using the Rayleigh-Ritz procedure, the derivatives of P with respect to V are set to zero.
M.S. Bittar et al. /Journal
of Petroleum Science and Engineering 14 (1995) 59-63
ces. Then, the above boundary conditions are enforced and, finally, the electric potential is calculated at each node by solving the over-all system of equationsThe apparent resistivity of the formation is then determined by:
-160 -120
h%l=+ s
’
1
“‘1
10
Resistivity
“‘I
100
” ‘4
1000
(ohm-m)
Fig. 1. Response of the normal resistivity tool for normal tool diameter. The log drawn as a solid line was generated by the integral equation method. The log drawn in small circles was computed by the finite-element method. The borehole is 8 inches and the mud resistivity is I R-m. In the finite-element calculation, the tool diameter is 3.5 inches.
For each individual element, the element matrix is obtained first. After that, the overall system of equations is assembled from the individual element matri-200 -160 -120 -80
120 160 1
(13)
where K is the tool’s geometric constant that will give the true formation resistivity when the tool is in a homogenous medium. Z, is the current flowing from the current electrode and V is the electric potential calculated at a measured electrode.
160 200
61
10 Resistivity
100
1000
(ohm-m)
Fig. 2. Response of the normal resistivity tool for very small tool diameter. The log drawn as a solid line was generated by the integral equation method. The log drawn in small circles was computed by the finite-element method. The borehole is 8 inches and the mud resistivity is 1 .0-m. In the finite-element calculation, the tool diameter is 0.2 inches.
3. Finite-element method verification The finite-element method was tested for a limiting case with the integral equation method. For this limiting case, the integral equation method provides an accurate solution (Shen et al., 1990). The response of the 16inch normal resistivity tool in a borehole crossing two thick beds is computed. The distance between the current and the measure electrode is 16 inches and the tool diameter is 3.5 inches. The borehole is 8 inches in diameter and is filled with conductive mud of 1 0-m resistivity. The resistivity of the upper bed is 10 L&m and the resistivity of the lower bed is 100 0-m. Fig. 1 shows two logs superimposed on each other. The log drawn as a solid line was generated by the integral equation method and the log drawn in small circles was computed by the finite-element method. The agreement between the two logs is satisfactory in the 10 0-m bed. However, the resistivity reading calculated with the finite-element method in the 100 L&m bed is slightly greater than that calculated by the integral equation method. This discrepancy is due to the fact that in the integral equation method a point electrode is assumed for the normal tool and in the finite-element method a 3.5 inch diameter tool is assumed. A second log was calculated by the finiteelement method. In this case, the diameter of the tool was reduced to 0.2 inch. Fig. 2 shows the comparison between this log and the log generated by the integral equation method. Here, the agreement between the two methods is excellent in both the 10 0-m and the 100 L&n-mbeds.
MS. Bittar et al. /Journal of Petroleum Science and Engineering I4 (1995) 59-63
62
4. Response of the normal tool in azimuthally inhomogenous formations
Fig. 3. A 16 inch normal tool in a borehole with three bed boundaries. The borehole is 8 inches in diameter and tilled with conductive mud of 1 R-m resistivity. The resistivity of the top and bottom beds is 75 R-m. The middle bed is 8 inches thick with a resistivity of 5 L&m.
The integral equation method can only solve for simple cases such as a normal resistivity tool in a borehole with a thick bed or a normal resistivity tool in a borehole crossing a boundary separating two beds. More complex cases such as a normal tool in a borehole with three bed boundaries or a tool in an azimuthally inhomogenous formation is solved with the finite-element method. The case that involves a normal tool in a borehole with three bed boundaries is shown in Fig. 3 and the case that involve the tool in an azimuthally inhomogenous bed is shown in Fig. 4. In all these cases, the borehole is 8 inches in diameter and filled with conductive mud of 1 0-m resistivity. The middle bed is 8 inches thick. The responses of the 16-inch normal tool for these five cases are shown in Fig. 5. The resistivity of the thick top and bottom beds is 75 am. The resistivity of the middle bed is 5 0-m for case A, and varies azimuthally for cases B, C, D, and E as shown in Fig. 4. Log A in Fig. 5 is the response of the normal tool in a borehole with three bed boundaries with azimuthally homogenous beds. This log does not reach full response in the middle bed because of borehole and thin bed effects. As an azimuthally resistive (75 0-m) inhomogeneity is introduced for the B, C, D and E cases, the 80
48 32
Case
B
Case
C
90
-48 -64 -8C
Case
D
Case
E
Fig. 4. Top views of the middle bed for cases A, B, C, D, and E. The borehole is 8 inches in diameter and filled with conductive mud of 1 0-m resistivity. The resistivity of the bed varies in the azimuthal direction as shown above. The resistivity contrast is 75 to 5 0-m.
35
45
55
65
75
Resistivity (n-m) Fig. 5. Response of the 16-inch normal tool in complex geometry. This figure shows five logs: log A, B, C, D and E. These logs represent the response of the tool in cases A, B, C, D and E. The resistivity contrast is 75 n-m to 5 am.
MS. Bittar et al. /Journal of Petroleum Science and Engineering 14 (1995) 5963 80
63
6. Nomenclature
64
D
48
E
32
B
H
16
J P'
0
1s
-1 6 -32
VS PS Pd
-48 -64
P Y”
0
60
120
180
240
300
Resistivity (&II)
I-L E
u Fig. 6. Response of the 16-inch normal tool in complex geometry. This figure shows five logs: log A, B, C, D and E. These logs represent the response of the tool in cases A, B, C, D and E. The resistivity contrast is 1000 0-m to IO in 0-m.
tool starts to read higher resistivity. In the middle bed, Log A is reading 23 C&m and Log E is reading 50 0m. Fig. 6 shows the response of the 16-inch normal tool in five complex cases with the same geometry as in the previous study. Here, the contrast between the beds is 1000 to 10. In the middle bed, Log A is reading 185 0-m and Log E is reading 980 0-m.
5. Conclusions
In this paper, a three-dimensional theoretical model of resistivity logging by the finite-element method is presented. The finite-element model is verified by comparing it to an integral equation model for a limiting case and good agreement is shown. The finite-element method is used in complex geometry to analyze the response of a 16-inch normal resistivity tool in azimuthally inhomogenous formations. The outlined finite-element theoretical model can be made more general to study the responses of all electrode-type tool in many complex logging situations.
0-m
electric flux density (coulomb/mete?) electric field strength (volt/meter) magnetic flux density, weber/mete? magnetic field strength (ampere/meter) electric current density (ampere/mete?) power per unit volume (watt/mete?) current delivered by current electrode (ampere) voltage at current electrode (volt) source power (watt) disspative power (watt) total power (watt) permeability (henry/meter) permittivity (farad/meter) conductivity (siemens/meter) resistivity (ohm-meter)
References Bittar, M., Shattuck, D. and Shen, L.C., 1990. Laboratory study of the shallow laterolog in high and low resistivity contrast. The Log Analyst, 3 1. Braurer, J.R. and Vander Heiden, R.H., 1988. Finite element modeling of electromagnetic resonators and absorbers. J. Appl. Phys.. 63(8): 3197-3199. Chemali, R., Gianzero, S.C. and Su, SM., 1988. The dual laterolog in common complex situations. Proc. 29th Annu. Meet. Sot. Prof. Well Log Analysts, Vol. 1, Paper N. Gianzero, S.C. and Anderson, B., 1982. An integral transform solution to the fundamental problem in resistivity logging. Geophysics, 47: 946956. Hanington, R.F., 1961. Time-Harmonic Electromagnetic Fields. McGraw-Hill Book Company, New York, N.Y. Merchant, G.A. and Thadani S.G., 1982. Finite electrode resistivity tool modelling. Proc. 23rd Annu. Meet. Sot. Prof. Well Log Analysts, Vol. 1, Paper Q. Shattuck, D., Bittar, M. and Shen, L.C., 1987. Scale modeling of the laterolog using synthetic focusing methods, The Log Analyst, 28. Simpson, RX, Shang, H.C. and Shen, L.C., 1983. A laboratory study of induction logs using physical modeling. Proc. 24th Annu. Meet. Sot. Prof. Well Log Analysts, Vol. 1, Paper M. Silvester, P.P. and Femui, R.L., 1983. FiniteElements for Electrical Engineers. Cambridge University Press, Cambridge. Tsang, L. and Gianzero, S.C., 1984. Solution of the fundamental problem in resistivity logging with a hybrid method. Geophysics, 49: 159&1604. Zhang, G.J. and Shen, L.C., 1984. Response of a normal resistivity tool in a borehole crossing a bed boundary. Geophysics, 49: 142149.