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Integral Electric Current Method in 3-D Electromagnetic Modeling for Large Conductivity Contrast Michael S

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Integral Electric Current Method in 3-D Electromagnetic Modeling for Large Conductivity Contrast Michael S Integral electric current method in 3-D electromagnetic modeling for large conductivity contrast Michael S. Zhdanov, University of Utah, Vladimir I. Dmitriev, Moscow State University, and Alexander Gribenko¤, University of Utah SUMMARY FORMULATION OF THE IEC METHOD We introduce a new approach to 3-D electromagnetic (EM) modeling for models with large conductivity contrast. It is based on the equa- The conventional IE method is based on the following equation for the tions for integral current within the cells of the discretization grid, in- total electric E fields: stead of the electric field or electric current themselves, which are used in the conventional integral equation (IE) method. We obtain these in- ZZZ ¡ 0¢ ¡ 0 ¢ b ¡ 0¢ tegral currents by integrating the current density over each cell. The E r = Gb E r j r ¢ [Dse (r)E(r)]dv + E r : (1) integral currents can be found accurately for the bodies with any con- D ductivity. As a result, the method can be applied, in principle, for the ¡ ¢ models with high conductivity contrast. At the same time, knowing the where Gb E r j j r is the electric Green’s tensor defined for an un- integral currents inside the anomalous domain allows us to compute bounded conductive medium with the complex background conduc- b the EM field components in the receivers using the standard integral tivity seb = s ¡ iwe; E is the background electric field; domain D representations of the Maxwell’s equations. We call this technique an corresponds to the volume with the anomalous conductivity distribu- integral electric current (IEC) method. The method is carefully tested tion se (r) = seb + Dse (r); r 2 D: by comparison with an analytical solution for a model of a sphere with large conductivity embedded in the homogenous whole space. The conventional approach to discretization of the integral equation (1) is based on dividing domain D into N elementary cells, Dn; formed N [ by some rectangular grid in the domain D = n=1 Dn; and assuming that Dse (r) has the constant value Dse within the cell. INTRODUCTION n We also assume that each cell Dn is so small that the electric field is One of the difficult problems in electromagnetic (EM) modeling is ac- approximately constant within the cell, E(r) ¼ E(rn); where rn is a curate numerical solution for models with large conductivity contrast. center point of rectangular cell Dn: Under this condition the discrete This problem appears, for example, in modeling EM data for mineral analog of equation (1) can be written as (Zhdanov, 2002): exploration when we have a conductive target embedded in relatively resistive host rocks. The study of the topography effect on EM data b b requires the solution of a similar problem, because the contrast in con- eD = GDsbeD+eD; (2) ductivity between the conductive earth and nonconductive air can be 8 10 as large as 10 ¡ 10 times. Well-logging is another area where one where sb is a (3N £ 3N) diagonal matrix of anomalous conductivities, should take into account a strong contrast between the cased borehole, b eD and eD are the vectors of the total and background electric fields for example, and surrounding rock formations. formed by the x, y and z components of these fields at the centers of b In this paper, we introduce a new approach to the solution of this prob- the cells Dn of the anomalous domain D. The 3N £ 3N matrix GD lem based on the integral equation (IE) method. The conventional IE is formed by the volume integrals over the elementary cells Dn of the b algorithms are usually written for the electric field or electric current components of the corresponding electric Green’s tensor GE , acting components within the domain with anomalous conductivity. This do- inside domain D. main is divided in the number of cells, which are selected to be so Note that equation (1), or equivalent matrix equation (2) provides an small that the field components vary slowly within the cell. If the con- adequate approximation of the original integral equation, if the follow- ductivity of the body and/or frequency are high, it is difficult to satisfy ing conditions hold: 1) the linear size h of elementary cell D is much this condition. The EM field varies extremely fast within a good con- n smaller than the wave length l of the EM field in the background ductor, which may result in errors of numerical modeling. In order to b medium, overcome this difficulty, Newman and Hohmann (1988) used a special grouping of the boxcar basis functions to form current loops within the conductor. Farquharson and Oldenburg (2002) implemented the more h << lb; (3) sophisticated edge element basis functions to avoid the inaccuracy of the conventional boxcar basis function approach. and 2) h is much smaller than the wave length la of the EM field in a In this paper we consider a novel approach for solving this problem. medium with anomalous conductivity: We develop a new form of the IE method, which is based on the equa- tions for integral current within the cells, instead of the electric field or electric current themselves. We obtain these integral currents by in- h << la: (4) tegrating the current density over each cell. The integral currents can be found accurately for a body with any conductivity. We do not use The first condition (3) usually holds for typical geophysical EM mod- anymore the requirements that the field varies slowly inside the cell, eling problems. The second condition (4) may fail in the case of high because we deal with the integral of this field. As a result, the method anomalous conductivity, which is the subject of this paper. can be applied, in principle, for models with arbitrary conductivity contrast. At the same time, knowing the integral currents inside the Our goal is to construct a discrete analogue of integral equation (1), anomalous domain allows us to compute the EM field components in which would provide an accurate approximation only under condition the receivers using the standard integral representations of Maxwell’s (3). In order to obtain a system of linear equations with respect to 0 equations. We call this technique an integral electric current (IEC) integral currents, let us multiply both sides of equation (1) by Dse (r ). method. As a result we have: ZZZ ¡ 0¢ ¡ 0¢ ¡ 0 ¢ b ¡ 0¢ j r = Dse r Gb E r j r ¢ j(r)dv + j r ; (5) D Integral electric current method in 3-D electromagnetic modeling for large conductivity contrast where: ¡ ¢ ¡ ¢ ¡ ¢ E(rn); was calculated in the center of each elementary cell from this jb r0 =Dse r0 Eb r0 ; r0 2 D; column according to the following formula: b is the induced current due to background field E . E(rn) = In =(DsenDn); (8) Integrating both sides of equation (5) over elementary cell Dp and using the integral electric current, In; computed for this cell with the assuming that anomalous conductivity is constant within the cell Dp, IEC method (where Dn and Dsen are the volume and the anomalous Dse = Dsep, we find conductivity of the corresponding elementary cell, respectively). Fig- ure 1, panel b, shows that the electric field computed for the finest dis- · ¸ N ZZZ ZZZ ¡ ¢ cretization (25 cells in the vertical direction) describes well the skin b 0 0 b Ip = Dsep ∑ GE r j r dv ¢ j(r)dv + Ip; (6) effect within the conductive body, while the field on the coursed dis- D D n=1 n p cretization of 5 vertical cells is practically insensitive to the skin ef- fect. At the same time, the difference between the observed EM field b where Ip is the integral current in the cell Dp due to background field components at the surface is within just 1.5% (Figures 2 and 3). This b E : ZZZ remarkable property of the IEC solution is related to the main principle b b ¡ 0¢ 0 Ip = j r dv : of the IEC method, which is based on computing the integral electric D p current, In; within every cell. In this case the electric field computed according to formula (8) should also describe the averaged electric The last equation can be written using matrix notation: field within the cell, which corresponds well to the plots shown in Fig- ure 1, panel b. One can see that the plots of the horizontal electric field components for the coarser discretization describe the average values be b ID = sbGDID+ID; (7) of the same plot for the finer discretization. The plots of the vertical component of the electric field behaves a little bit differently, because where sb is a (3N £ 3N) diagonal matrix of anomalous conductivities, the vertical field is 103 times smaller than the horizontal fields. Nev- b ID and ID are the vectors of the total and background electric field ertheless, the plots for 15-cell and 25-cell discretizations practically intensities formed by the x, y and z components of these fields at the coincide, which is a clear manifestation that we reached the optimal e centers of the cells Dn of the anomalous domain D. Matrix Gb D is a level of discretization at 25 cells in the vertical direction. The solution transposed matrix of the original linear system (2), Gb D; for the vector will not change if we will use the finer discretization. Thus, another e of electric current e (which justifies the notation we use for Gb ). important property of the new IEC method is that it does not require a D D very fine discretization to produce an accurate result, because it does Thus, forward electromagnetic modeling based on the IE method is re- not operate with the discretized electric field but with the integral cur- duced to the solution of the matrix equation (7) for the unknown vector rents, instead.
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