2012 SEG SEG Las Vegas 2012 Annual Meeting Page 1

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1 Frequency-domain EM modeling of 3D anisotropic magnetic permeability and analytical analysis Jiuping Chen, Michael Tompkins, Ping Zhang, Michael Wilt, Schlumberger-EMI, and Randall Mackie, Formerly Schlumberger-EMI, presently Fugro-EM SUMMARY 3D NUMERICAL MODELING OF µ In this paper we present the modification and verification of a 3D frequency-domain electromagnetic modeling algorithm for dealing with heterogeneous, anisotropic magnetic permeability. To test the modifications, the code is verified against a 1D layered earth response, and 3D axially-symmetric doughnut model. In addition, to better understand the magnetic field responses to both conductivity and permeability, analytical analysis is performed and demonstrates that in a low induction number ( k r 1) regime the magnetic current or magnetization may be more important thant the induced electric current in determining the EM response to a medium. INTRODUCTION In applied EM geophysics, estimation of electrical resistivity has been widely used for a variety of applications such as mineral prospecting, well-logging, deep earth structure exploration and environmental monitoring. On the other hand, magnetic permeability or susceptibility, one of the most fundamental physical properties of rock, has usually been limited to studies in static magnetic exploration for massive ore deposits and UneXplored Ordnance (UXO) classification and identification (i.e., Huang and Won, 2003). In addition to limited application, few 3D numerical algorithms are available for use in simulating effects due to magnetic permeability. Most well-known and widely used solutions are based on the analytic expressions of a conductive and permeable sphere in air, excited by a magnetic dipole source (e.g., Grant and West, 1965). In a recently published paper by Mukherjee and Everett (2011), they tried to verify a 3D finite-element solution dealing with both conductivity and permeability with the Grant and West (1965) solution. Other algorithms, as far as we know, such as Wang and Hohmann (1993), Alumbaugh et al. (1996), Haber et al. (2000), and Lovell (1990) are capable of simulating both conductivity and permeability. However no detailed studies exist from these authors on the effect of permeability. In this study, we aim to extend an existing 3D finite-difference EM forward modeling algorithm to cover heterogeneous, anisotropic magnetic permeability. In addition, we carry out an analytical analysis by using the integral representation of the EM response for a spherical model to help understand the physics of the secondary field. Our goal is to get a sense what role the magnetic current or magnetization plays relative to the induced electric current in generating secondary magnetic fields for low induction numbers, which has implications for how interpretations can be made. Modification of the 3D algorithm Here, we modify the 3D numerical EM code as presented in a series of journal papers (Mackie et al.,1994; Madden and Mackie, 2002). In this algorithm, 3D earth models are divided into regular prism cells. The forward modeling algorithm uses a finite-difference method to solve for the magnetic fields along the edges of the cells; the electric fields can be obtained by interpolation. The resistivity at each cell can be diagonally anisotropic, i.e., ρ xx,ρ yy,and ρ zz. The magnetic permeability is assumed to be constant everywhere, with a freespace value of µ 0 = 4π 10 7 H/m. The code is parallelized using the open source PETSc (Portable, Extendable Toolkit for Scientific Computation) platform, a suite of data structures and routines for the scalable (parallel) solution of scientific applications modeled by partial differential equations (Balay et al., 2011). A Krylov-based linear iterative solver with a pre-conditioner matrix is used to solve the assembled linear equations. To deal with magnetic permeability, in the region where there is no source term, we have to go back to the integral forms of Maxwell s equations, which (following Mackie et al., 1994) are H dl = σe ds, (1) E dl = iωµh ds, (2) where σ and µ are the electric conductivity and magnetic permeability, and both can be tensor quantities. Since here we are concerned with the magnetic permeability, let us only focus on equation 2. With this formulation and the discretization scheme shown in Figure 1, for example, the H y -component of equation 2 can be written as {[E x (i, j,k) E x (i, j,k 1)] [E z (i, j,k) E z (i 1, j,k)]}l = iω < µ yy > H y (i, j,k)l 2, (3) where L is the grid spacing in the transformed system, and the average magnetic permeability < µ yy > is defined as: < µ yy >= 1 4 [µ yy(i 1, j,k 1) +µ yy (i, j,k 1) + µ yy (i 1, j,k) + µ yy (i, j,k)]. (4) Similarly we can work on H x and H z -component of equation 2, and then combine the discretized forms of electric field in equation 1, grouping these second-order equations together in the form: M xx N xy N xz H x Ax = N yx M yy N yz = b (5) N zx N zy M zz H y H z SEG Las Vegas 2012 Annual Meeting Page 1

2 Figure 1: The finite-difference geometry for the integral forms of Maxwell s equations. where b contains the terms associated with the known boundary values and source field,and M i j and N i j are the sub-stiff coefficient matrices. Accordingly, the original coefficient matrix A, the preconditioning matrix P, the right-hand side vector b in equation 5, as well as the new static correction term ρ µh need to be changed by incorporating the diagnostically anisotropic magnetic permeability. EM reciprocity helps to reduce computational time in the case that the number of transmitters is much greater than the number of receivers, such as in many marine CSEM applications. In addition, generally the reciprocal relationship holds for magnetic H field if µ is the same as µ 0. However, when the magnetic permeability varies, the reciprocity is valid only for magnetic induction and B field. For an i-oriented magnetic dipole source, and measuring j-oriented magnetic induction component, where i and j can be pointing x,y,or z, according to Chen et al.(2005), we have the following reciprocal expression B i j (r tx,r rx ) = B ji (r rx,r tx ), (6) where B ji (r rx,r tx ) is the i-component of B field observed at the artificial location r tx, generated by a j-oriented artificial magnetic dipole source at location r rx. This means µ j j (r rx )H i j (r tx,r rx ) = µ ii H ji (r rx,r tx ). (7) By using the reciprocity, the solution obtained by solving equation 5 is H ji, therefore, we perform the following step Verification of the algorithm H i j (r tx,r rx ) = µ ii µ j j H ji (r rx,r tx ). (8) We present two test examples to verify the modifications made to include magnetic permeability. These two examples are (1) a layered 1D µ medium; and (2) a 3D doughnut µ model (axial symmetry) in a whole space. Figure 2: Comparison of CSEM3Dmu vs semi-analytical solutions in a 1D layered µ medium. As shown in the inset in Figure 2, the resistivity and magnetic permeability of the uniform whole space are 10 Ω.m and µ 0. The 1D layered model contains a µ layer with a thickness of 75 m, and a relative permeability of 1.0 and 1.2. We use a cross-well configuration for the test. The separation between the vertical Tx well (vertical magnetic dipole, VMD) and the Rx well is 200m, and the applied frequency is 500 Hz. We show the computed magnetic field H z plot along the Rx profile, generated by a VMD at a constant depth 1100 m for both the 1D, and 3D codes. The semi-analytical solutions were obtained by running a numerical 1D code (Tompkins, 2003), which are displayed in black dashed lines (free space µ 0 ), and green solid line (µ r = 1.2), respectively. The numerical responses obtained from running the modified code CSEM3Dmu are plotted in blue dashed lines (free space) and red solid line (µ r = 1.2). It can be seen that for the free space, the black and blue lines match well; for the µ r model, the green and red lines are in a fairly good agreement. The relative difference between them is less than 2%, which can be attributed to the grid size used in the 3d modeling. The sharp drop of the vertical H z component at depth from 1075 to 1100m is primarily caused by the µ r change, because we know that the normal magnetic induction B z is continuous cross the boundary layer, while H z is not (i.e., it is scaled by µ r ). The second test example is a 3D doughnut µ model imbedded in a uniform wholespace of resistivity (10 Ω.m) and magnetic permeability (µ 0 ). The 3D doughnut is displayed in Figure 3, with an inner and outer radius of 25 and 100 m, respectively. The thickness of the doughnut is 75 m, and µ r = 1.2. Again, the Tx (VMD) is located at depth 1100 m (inside the doughnut), and the source frequency is 500 Hz. The comparison was run against the code CWNLAT (Lovell, 1990). CWNLAT is a finite element program, and allows electromagnetic simulation in media with variable resistivity, magnetic permeability, and dielectric properties. Assuming an axiallysymmetric TM excitation in an azimuthally symmetric (2 dimensions) cylindrical geometry. The doughnut model is a perfect example for this code. SEG Las Vegas 2012 Annual Meeting Page 2

3 Figure 3: Comparison of CSEM3Dmu vs CWNLAT solutions for a 3D doughnut µ model in a uniform whole space. As shown in Figure 3, the computed H z field from CSEM3Dmu (red solid line) is in good agreement with the CWNLAT responses which are in green. In order to improve the numerical accuracy of CSEM3Dmu, we used denser grids around both Tx and Rx locations. This was required, because the Rx is treated as an artificial source in the code, and each artificial source is distributed at the 8 nodes surrounding the source point by volume weighting. Similarly, the field computed at the artificial Rx (actually the real Tx location) is also interpolated by volume weighting at the 8 nodes around that point. ANALYTICAL ANALYSIS OF MAGNETIC RESPONSES Integral representation of magnetic responses The general EM induction phenomenon for a conductive target buried in an extremely resistive background is well known, and has been clearly understood and conceptually illustrated in Figure 4 (adapted from Grant and West, 1965). When the resistivity contrast between the target and surrounding host is limited, besides the vortex current induced in the target, there is also channeled current or galvanic current generated around the body (McNeill et al., 1984). If the target is also magnetically permeable, more physical phenomena are present. We illuminate these phenomena through analysis of analytical solutions in the low frequency case. For a general medium (with conductivity σ 0 and magnetic permeability µ 0 ) containing both a conductive and permeable object (σ and µ), and excited by an external source, the magnetic field at an observation point consists of incident field H inc and scattered or secondary field H s and the secondary field H s can be represented as (de Hoop, 1995) H s (r) = g(r r )[σ(r ) σ 0 (r )]E(r )dv(r ) v +[k0 2 + ] g(r r )[ µ 1]H(r )dv(r ) v µ 0 = H se + H sm, (9) Figure 4: A carton showing concept EM induction for both a conductive and permeable body. Adapted from Grant and West(1965) where k 0 = iωµ 0 σ 0, is the wave number, and the homogenerous scalar Green s function is given by g(r) = e ik 0 r 4π r. (10) In equation 9, the first term is the magnetic field generated by the excessive electrical current density [σ σ 0 ]E(r ), and thus we denote it by H se. The second term is the magnetic field generated by the magnetic current (Wait, 1953) or magnetization [ µ µ 0 1]H(r ), and therefore can be distinguished by H sm. We would like to analyze how these two terms behave in a typical low frequency regime. Analytical solutions To simplify our analysis, let us take the Tx-Rx configuration as shown in Figure 5 as an example. The Tx is a VMD with a magnetic moment m, and is put on the same vertical plane xoz as Rxs, which only measure the vertical field H z. The sphere is located at the origin of the system, with a radius a, conductivity σ, and permeability µ, Assume a is much smaller than the distance r 0. Then the electric field inside the sphere can be approximated by the y component of the incident electric field, i.e., and for a VMD source, we know that E(r i ) Eyŷ, (11) Ey i = iωµ 0m 4πr0 2 (ik 0 r 0 + 1)e ik 0r 0. (12) Under the assumptions that both induction numbers k 0 r 0, k 0 r 1, and the vertical distance z is much less than r, without dwelling on detailed derivations, the vertical component Hz se of the secondary magnetic field due to the electric current can be simplified as H se z iωµ 0m 4πr 2 0 }{{} [σ σ 0 ] πa3 }{{}} 4πr {{ 2 } (13) Apparently, the first bracket denotes the incident E field at the sphere. Then when multiplied by the second bracket, it gives SEG Las Vegas 2012 Annual Meeting Page 3

4 Figure 5: A schematic spherical model for understanding the magnetic response from both conductive and magnetically permeable target. rise to a current segment with a unit of A.m. This current segment produces the magnetic field at the Rx with the geometric term in the third bracket. Following similar algebraic operations, the secondary vertical magnetic field Hz sm from the magnetic current can be approximated as Hz sm ( m 4πr0 3 ) [µ r 1] 4 3 πa3 ( 1 }{{}}{{} 4πr 3 ) }{{} (14) Again the physical meaning can be inferred from the three brackets shown in equation 14. Here we see that µ r 1 = χ is the suscepibility of the sphere, and then when excited by the external magnetic field, it produces the magnetization, which in turn generates another secondary magnetic field as well. To quantify the above analysis, a spherical model was examined, and the computed results are shown in Figure 6. In this model, the resistivities for the background and the sphere are 100 and 10 Ω. m, respectively. The relative permeability of the sphere is 2. The radius is 50 m. The distance between the VMD to the center of the sphere is 200 m, and the horizontal distance from the origin to the Rx well is also 200m. The depth of the Rxs range from -100 to 100 m. At 10 Hz, the assumptions made in the previous section are roughly satisfied. Figure 6a shows the primary, secondary, and the total H z field. In this case, the secondary field is about 3% of the primary. Therefore, both primary and the total field are almost completely overlapping at the top. Even though the secondary field is relatively small, Figure 6b shows that the magnetization field Hz sm is about 3 times as large as the magnetic field Hz se generated by the induced current. The point we would like to make here is that at low frequency and in a relatively resistive host, magnetization, rather than induced current, appears to dominate the observed secondary field as opposed to what is observed in conventional cross-well EM surveys. Figure 6: An example of the approximate analytical analysis on the sphere. (a) the primary, secondary, and total H z. (b) the ratio of Hsm z H in this model. z se CONCLUSION We have made modifications to an existing 3D EM algorithm to accommodate the need for simulating the effect of magnetic permeability heterogeneities including anisotropy. Two test examples have been presented, and demonstrate that the modified code compares well with semi-analytical and alternative numerical solutions, providing a useful tool for the simulation and monitoring of magnetically permeable formations. We also have carried out analysis of an analytical solution for magnetic field responses using an integral representation for a conductive and permeable sphere in a dipole field. The approximate analysis shows that the secondary magnetic field can be attributed to two parts: one is generated by the induced electric current, and the other by the magnetic current or magnetization. At a low induction number regime ( k r 1), the magnetization part may dominate in the observed anomaly over the induced electric current. This physical understanding will help us to better apply this technique to different applications, for example, hydraulic fracturing by injection of magnetically permeable particles into formations. SEG Las Vegas 2012 Annual Meeting Page 4

5 EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2012 SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web. REFERENCES Alumbaugh, D. L., G. A. Newman, L. Prevost, and J. N. Shadid, 1996, Three-dimensional, wideband electromagnetic modeling on massively parallel computers: Radio Science, 31, Balay, S., J. Brown, K. Buschelman, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. Mclnnes, B. F. Smith, and H. Zhang, 2011, PETSc Users Mannual, ANL-95/11-Revision 3.2, Argonne National Laboratory. Chen, J., D. W. Oldenburg, and E. Haber, 2005, Reciprocity in electromagnetics: Applications to marine MMR: Physics of the Earth & Planetary Interiors, 150, Grant, F. S., and G. F. West, 1965, Interpretation theory in applied geophysics: McGraw -Hill. Haber, E., U. M. Ascher, D. A. Aruliah, and D. W. Oldenburg, 2000, Fast simulation of 3D electromagnetic problems using potentials: Journal of Computational Phys ics, 163, Huang, H., and I. J. Won, 2003, Characterization of UXO -like targets using broadband electromagnetic induction sensors: IEEE Transactions on Geosciences and Remote Sensing, 41, Lovell, J., 1990, Modeling frequency effects on laterologs: Reserch note, Schlumberger, 10 July Mackie, R. L., J. T. Smith, and T. R. Madden, 1994, Three-dimensional electromagnetic modeling using finite difference equations: The magnetotelluric example: Radio Science, 29, no. 4, Madden, T. M., and R. L. Mackie, 2002, Three-dimensional magnetotelluric modeling and inversion: Proceeding of the IEEE, 77, no. 2, McNeill, J. D., R. N. Edwards, and G. M. Levy, 1984, Approximate calculations of the transient EM response from buried conduc tors in a conductive half -space: Geophysics, 49, Mukherjee, S., and M. E. Everett, 2011, 3D controlled-source EM edge-based finite element modeling of conductive and permeable heterogeneities: Geophysics, 76, no. 4, F215 F226. Tompkins, M., 2003, Quantitative analysis of mulicomponent borehole electromagnetic induction responses using anisotropic forward modeling and inve rsion: P.h.D. thesis, University of Wisconsin. Wait, J., 1953, A conducting permeable sphere in the presence of a coil carry ing an oscillating current: Canadian Journal of Physics, 31, Wang, T., and G. W. Hohmann, 1993, A finite -difference time domain solution for 3D EM modeling: Geophysics, 58, SEG Las Vegas 2012 Annual Meeting Page 5