Feasibility Study for Two-Dimensional Frequency Dependent Electromagnetic Sensing Through Casing

Transcription

1 Feasibility Study for Two-Dimensional Frequency Dependent Electromagnetic Sensing Through Casing David Pardo, Carlos Torres-Verdín, and Leszek F. Demkowicz ABSTRACT We perform a numerical sensitivity study of different physical phenomena of interest for borehole resistivity logging applications in steel-cased wells. Specifically, we analyze () the sensitivity of different logging instruments to detecting water invasion behind casing, () the influence of electrical anisotropy behind casing, () the possibility of detecting laminated sands behind casing, and ) the influence of frequency on throughcasing resistivity measurements. The sensitivity analysis is performed using a highly accurate and reliable numerical method based on a D self-adaptive goal-oriented hp-finite element method that can be applied to simulate all types of resistivity logging measurements, including normal/laterolog, induction, and through-casing resistivity measurements. Results quantify the effects of a wide array of physical phenomena that can be sensed through casing and that can be measured with accurate sensors. We find that water invasion and anisotropy effects behind casing can be accurately detected. Measurements can be designed to detect and quantify the properties of shale-laminated sands behind casing. Moreover, numerical simulations indicate that measurements may be extremely sensitivity to variations on frequency and/or casing resistivity. INTRODUCTION Steel casing is the standard means to complete wells drilled into hydrocarbon reservoirs. While steel casing provides mechanical competence that prevents the well from collapsing, especially across soft rock formations, presence of electrically conductive casing poses significant challenges to the probing of electrical properties behind casing. The high electrical conductivity of casing causes the amplitude of electromagnetic (EM) fields to decay exceedingly fast, thereby preventing the accurate sensing and inference of spatial distributions of electrical conductivity behind casing. It was not until the last decade when reliable and accurate instrumentation systems were introduced to probe electrical resistivity behind casing. Interpretation of resistivity logging measurements behind casing was finally possible mainly due to advances in three areas:. Fundamental theoretical work advanced to quantify EM fields measured in the presence of steel casing Kaufman (99) and the design of reliable calibrated instruments Vail et al. (99).. Advances in the manufacturing of EM sensors that minimize the noise-to-signal ratio and enable the reliable measurement of voltages in the nanovolt range. An example of such an advance is the. Advances in the manufacturing of EM sensors that minimize the noise-to-signal ratio and enable the reliable measurement of voltages in the nanovolt range. An example of such an advance is the Cased-Hole Formation Resistivity (CHFR) tool, recently introduced by Schlumberger. Advances in computational electromagnetics necessary for the numerical simulation of resistivity logging measurements acquired in the high-conductivity environment of cased wells. In this area, the contributions by Schenkel and Morrison (99) are remarkable for quantifying the possibility of cross-well resistivity surveys if one well is cased, and by Wu and Habashy (99), who studied the response of a through-casing resistivity tools (TCRT) as a function of frequency. More recent contributions to the field of TCRT include the works of Singer and Strack (99) that develop accurate modeling solutions, and Pardo et al. (6) in the area of computational modeling of TCRT.

2 sensitivity of through-casing resistivity measurements to different physical effects often encountered while probing rock formations behind casing. This is achieved with the use of highly accurate numerical simulation software specifically designed for that purpose. Our numerical methodology is based on a self-adaptive, goal-oriented hp-finite Element Method (FEM) that converges exponentially in terms of the quantity of interest vs. the problem size (as well as CPU time). This highly accurate, and very reliable adaptive algorithm has been successfully applied to a variety of resistivity logging devices, including simple TCRT Pardo et al. (6), Induction Pardo et al. (b), and Normal/Laterolog instruments. A detailed description of the numerical methodology can be found in Pardo et al. (6) and Pardo et al. (a), and, for the sake of brevity, is not presented here. For the problems considered in this paper, our numerical methodology guarantees relative errors below %, thereby enabling the analysis of small variations (perhaps of just %) of the results. This paper is a continuation of the results presented in Pardo et al. (6). Here, we consider more challenging problems, and we analyze physically relevant effects through casing, such as water invasion, detection of electrical anisotropy, detection of thin layers, and influence of frequency on the measurements. The organization of this paper is as follows. First, we introduce the logging applications of interest that are driving our research. These resistivity logging applications are governed by Maxwell s equations, that we describe in the next Section, along with the corresponding variational formulation for axisymmetric problems. Then, we illustrate our findings with numerical results. Conclusions are presented thereafter. Finally, we include an Appendix containing validation results for our Finite Element software. More precisely, we compare results obtained with our self-adaptive goaloriented hp-refinement strategy against those obtained with a semi-analytic D radial software for a number of model problems. All problems and applications considered in this article are assumed to be axisymmetric, that is, the geometry, material properties, and distribution of antennas is axial symmetric with respect to the axis of the borehole. cm.. The first receiver antenna is located feet=.9 m. above the transmitter antenna, and foot=.8 m. below the second receiver antenna. All antennas are moving simultaneously along the z-axis, and they are located on the borehole wall. A borehole with resistivity R= Ω m. A steel casing of width. =.7 cm. surrounding the borehole, with resistivity R=. Ω m = Ω m. A cylindrical layer of cement surrounding the casing, with resistivity R= Ω m. Three formation materials, as described on Table. Closure of the Domain. We consider a bounded computational domain Ω. A variety of boundary conditions (BC s) can be imposed on the boundary Ω such that the difference between the solution of such a problem and the solution of the original problem defined over the whole space is small. For example, it is possible to use an infinite element technique as described in Cecot et al. () or a perfect matched layer (PML). Also, since the electromagnetic fields and their derivatives decay exponentially in the presence of lossy media (non-zero conductivity), we may simply impose a homogeneous Dirichlet BC on the boundary of a sufficiently large computational domain. In this paper, we will follow this Dirichlet BC approach, selecting a domain of size feet and 8 inches = 7. m. in the horizontal direction, and 6 feet = 8.9 m. in the vertical direction (Fig. ). TCRT APPLICATIONS In this section, we describe three TCRT axisymmetric model problems in a borehole environment that are of great interest in a variety of logging applications. Problem I: Using cylindrical coordinates (ρ, φ, z), we consider the following geometry, sources, receivers and materials (illustrated in Fig. ): One transmitter toroidal antenna of dimensions cm. Figure : Geometry of Problem. Measurements based on one transmitter (toroidal) antenna and two receiver antennas. Formation composed of a borehole, a casing, a cement, a target (oil) layer, and a background layer.

3 simulations is to compute the first difference of the vertical component of the electric field at the receiving toroidal coils (l and l ) of radius a =.97 cm. divided by the (vertical) distance z between them and the perimeter of the toroids, i.e. [ ] E z (l) dl E z (l) dl /[( z)(πa)] = l l = z [E z(l ) E z (l )]. (.) Then, we will study the sensitivity of this quantity with respect to variations of frequency (from Hz up to Khz) and water invasion (of resistivity.8ω m) in the target (oil) zone, as indicated in Fig.. Problem II: In this problem, we consider the following geometry, sources, receivers and materials (illustrated in Fig. ): The borehole, casing, cement, and antennas considered in problem I. Several formation materials, as described on Table. We shall consider a computational domain of the same dimensions as the one used for problem I. For details, see Fig.. the possibility of sensing through casing the different layers of laminated shales and sands existing in the formation. We also want to study the sensitivity of quantity. with respect to variations of frequency (from Hz up to Khz) and the effect of water invasion (of resistivity.8 Ω m) into the sand layers, as indicated in Fig.. Problem III: In this problem, we consider a typical Gulf of Mexico sedimentary sequence. We select the following geometry, sources, receivers and materials (illustrated in Fig. ): The transmitter and receiver antennas considered in problem I. A borehole with resistivity R= Ω m. A steel casing with resistivity R=. Ω m = Ω m. A cylindrical layer of cement, with resistivity R= Ω m. Several formation materials, as described on Table. We shall consider a computational domain of similar dimensions to the one used for problem I. See Fig. for additional details. Figure : Geometry of Problem. Measurements based on one transmitter (toroidal) antenna and two receiver antennas. Formation composed of a borehole, a casing, a cement, a background layer, and a target zone formed by thin layers of shales and sands. Figure : Geometry of Problem. Measurements based on one transmitter (toroidal) antenna and two receiver antennas. Formation composed of a borehole, a casing, a cement, a background layer (shale), and a target zone formed of an oil layer, an oil-water layer, and a water layer, all of them separated by shale layers.

4 problem is to study the sensitivity of quantity. with respect to variations of frequency (from Hz up to Khz), anisotropy effects, and water invasion (of different resistivities) in the oil-, oil-water-, and water-saturated layers, as indicated in Fig.. Remark. In all problems considered here, we assume a casing of uniform thickness and resistivity. Although this is almost never the case in practical applications (due to joints and corrosion effects in the casing) it is possible to use calibrated tools in order to account for these effects. A design of a calibrated tool has been shown, for instance, in Pardo et al. (6) and, for simplicity of presentation, we will not consider calibrated tools in this paper. MAXWELL S EQUATIONS All problems described above are governed by Maxwell s equations. Assuming a time-harmonic dependence of the form e jωt, where j = is the imaginary unit, t denotes time, and ω is angular frequency, Maxwell s equations can be written as H = ( σ + jω ǫ)e + J imp E = jω µ H M imp (.) ( ǫe) = ρ ( µh) = Here, H and E denote the magnetic and electric field, respectively, J imp is a prescribed, impressed electric current density, M imp is a prescribed, impressed magnetic current density, tensors ǫ, µ, and σ stand for dielectric permittivity, magnetic permeability, and electrical conductivity of the medium, respectively, and ρ denotes the electric charge distribution. We assume that the determinants of µ and σ + jω ǫ are nonzero. The equations described in (.) are to be understood in the distributional sense, i.e. they are satisfied in the classical sense in subdomains of regular material data, and they also imply appropriate interface conditions across material interfaces. Source Antennas Toroid antennas are modeled by prescribing an impressed volume magnetic current M imp. Using the equivalence principle (see, for example, Harrington (96)), we can substitute the original impressed magnetic volume current M imp by an equivalent magnetic surface current M imp Γ = [ˆn E] Γ, (.) defined on an arbitrary surface Γ enclosing the spatial support of M imp. Variational Formulation. For axisymmetric fields, the corresponding variational formulation in terms of the azimuthal component of the magnetic field H φ = (,H φ,) Find H φ H φ,d + H D (Ω) such that: [ ( σ ρ,z + jω ǫ ρ,z ) ] H φ ( Fφ ) dv Ω +jω ( µ φ H φ ) F φ dv = Ω M imp φ F φ dv + M imp F φ,γ φ ds Ω Γ F φ H D(Ω), (.) where H φ,d represents the Dirichlet data -typically H φ,d = -, F φ = (,F φ,) H D (Ω) is a test function, H D (Ω) = {H φ : (,H φ,) H D (curl;ω)} = {H φ L (Ω) : ρ H φ + H φ L H φ (Ω), L (Ω), H φ ΓD = }, ρ z σ ρ,z, ǫ ρ,z are the meridian components of the conductivity and permittivity tensors, respectively, and µ φ is the azimuthal component of the magnetic permeability tensor. For a more detailed derivation of the variational formulation, see Pardo et al. (a). NUMERICAL RESULTS On this Section, we obtain numerical results for the three problems of interest introduced above using the self-adaptive goal-oriented hp-fem described in Pardo et al. (a). More precisely, we shall analyze a number of computergenerated logs displaying the absolute value of the normalized first vertical difference of E z. in logarithmic scale as a function of the vertical position (in m) of the first receiver antenna. Problem I In Fig. left panel we display three logs. Different logs (curves) correspond to different frequencies: Hz (solid line), Hz (dashed line) and Khz (triangles). Results are very sensitive to the target (oil) layer, which can be clearly identified from the logging results. For different frequencies, results are similar to each other in the target layer less than % variations. However, shoulder effects are more frequency sensitive up to % variations as we modify the frequency from Hz up to khz. Figure right panel displays four logs, at Hz and Khz with and without a water invasion radius into the target (oil) zone. From these results, we conclude that the water invasion through casing has an effect of decreasing the measured signal by as much as one order of magnitude. Measurements of Resistivity Logging Tools with Mandrel Through Casing. Now, we consider problem I with the mandrel described in Fig.. Fig. 6 left panel displays three curves that correspond to different frequencies: Hz (solid line), Hz (dashed line) and Khz (triangles). From the physical point of view, results

5 NO WATER INVASION Hz Hz Hz NO WATER INVASION NO WATER INVASION Hz NO WATER INVASION Hz " WATER INVASION Hz " WATER INVASION Hz cm cm. Ohm m 6.67 cm cm Mandrel. Ohm m cm Magnetic Buffer Ohm m Relative Permeability Borehole. Ohm m Radius =.79 cm 8 6 (V/m ) (V/m ) Figure : Simulated TCRT measurements for Problem I. Absolute value of the first vertical difference of E z (normalized) against the vertical position (in m) of the first receiver antenna. Different curves correspond to different frequencies and different radii of water invasion into the target (oil) layer. Left panel: no invasion at Hz (solid line), no invasion at Hz (dashed line) and no invasion at Khz (triangles). Right panel: no invasion at Hz (solid line), no invasion at Khz (dashed line), = 6.96 cm. invasion at Hz (triangles), and = 6.96 cm. invasion at Khz (circles). are not proportional to the resistivity of the formation, and a physical interpretation of the results could not be performed by these authors. Nevertheless, the possibility of modeling mandrel and casing in the same simulation could be used as a tool for designing new logging instruments. Problem II Fig. 6 right panel displays three curves corresponding to different frequencies: Hz (solid line), Hz (dashed line) and Khz (triangles). Results are sensitive to the layers of laminated shales and sands existing in the formation. More precisely, we can clearly identify the vertical position of the ten layers of sands and shales. For different frequencies, results are only slightly different to each other. More precisely, we observe a slightly weaker signal (up to % variation) for the Khz case. Figure 7 left panel shows the effect of water invasion of resistivity.8ω m into the sand layers at a frequency of Hz. We observe a considerable increase of the first vertical difference of E z measured at the receiver antennas as we invade the sand layers with water. We also observe that, as we invade the sands with water, the Radius 7.6 cm Figure : Left panel: Geometry of a 7.6 cm thick mandrel, equipped with one transmitter antenna and two receiver antennas. Right panel: Amplification of the geometry toward the transmitter antenna. All antennas include magnetic buffers. NO WATER INVASION, TOOL DESIGN Hz Hz Hz Amplitude of first difference of E (V/m ) z NO WATER INVASION Hz Hz Hz 7 6 (V/m ) Figure 6: Absolute value of the first vertical difference of E z (normalized) against the vertical position (in m) of the first receiver antenna. Different curves correspond to different frequencies: Hz (solid line), Hz (dashed line) and Khz (triangles). Left panel: Problem I with mandrel. Right panel: Problem II. diffentiation between different sand-shale layers on the formation becomes unclear. Radius of invasion ranges from (no invasion) up to 6 = 9. cm. Figure 7 right panel displays four logs, at Hz and Khz with and

6 without a water invasion radius into the sand layers. From these results, we conclude that as we increase the frequency, the received signal slightly decreases. On the other hand, the water invasion through casing has the effect of increasing the measured signal as well as modifying the overall shape of the solution. Notice that the peaks observed on the curve formed by triangles - water invasion at Hz- are due to physical effects, and not to numerical errors, since our software delivers results with guaranteed numerical errors below %. Hz NO WATER INVASION 6" WATER INVASION " WATER INVASION " WATER INVASION 6" WATER INVASION 7 6 (V/m ) Hz NO WATER INVASION Hz NO WATER INVASION Hz " WATER INVASION Hz " WATER INVASION Hz 7 6 Amplitude of first difference of E (V/m ) z Figure 7: Problem II. Absolute value of the first vertical difference of E z (normalized) against the vertical position (in m) of the first receiver antenna. Different curves correspond to different frequencies and different radi of water invasion into the target (oil) layers. Left panel: no invasion at Hz (solid line), 6 =. cm. invasion at Hz (dashed line), =.8 cm. invasion at Hz (triangles), = 6.96 cm. invasion at Hz (circles), and 6 = 9. cm. invasion at Hz (pluses + ). Right panel: no invasion at Hz (solid line), no invasion at Khz (dashed line), = 6.96 cm. invasion at Hz (triangles), = 6.96 cm. invasion at Khz (circles). In Fig. 8, we analyze the sensitivity of the measurements with respect to the conductivity of casing. More precisely, Fig. 8 left panel display results for casing resistivity equal to Ω m (solid and dashed lines), and 6 Ω m (triangles and circles), respectively. At first glance, it seems that as we increase the casing conductivity, we cannot distinguish between different materials in the formation. However, Fig. 8 right panel shows that by plotting the curves composed of circles and triangles in a more detailed scale (see Fig. 8 left panel ), we observe substantial variations on the measurements as we move the antennas in the vertical direction. These variations correspond to the different materials in the forimum values in the x-axis of Fig. 8 right panel are close to each other, and in order to obtain physically consistent results, we need to utilize very accurate electronic sensors as well as numerical methods, i.e. the noise-tosignal ratio has to remain below %. Otherwise, measurement errors will dominate, and physical conclusions will not be obtained. Notice the similarities between these results and the ones observed for problem I. NO WATER INVASION Ohm m, Hz Ohm m, Hz 6 Ohm m, Hz 6 Ohm m, Hz 7 (V/m ) NO WATER INVASION 6 Ohm m, Hz 6 Ohm m, Hz Amplitude of first difference of E (V/m ) z Figure 8: Problem II. Absolute value of the first vertical difference of E z (normalized) against the vertical position (in m) of the first receiver antenna. Different curves correspond to different frequencies and different resistivities of the casing. Left panel: casing resistivity equal to Ω m at Hz (solid line) and Khz (dashed line), and casing resistivity equal to 6 Ω m at Hz (triangles) and Khz (circles). Right panel: casing resistivity equal to 6 Ω m. Different curves correspond to different frequencies: Hz (solid line) and Khz (dashed line). Problem III In Fig. 9 left panel different curves correspond to different frequencies with and without presence of anisotropy effects: Hz (solid line) without anisotropy, Khz (dashed line) without anisotropy, Hz (triangles) with anisotropy, and Khz (circles) with anisotropy. Results clearly suggest the presence of the various layers in the formation. The received signal becomes more sensitive to frequency variations depending upon the vertical positions of the antennas. In particular, results are sensitive to frequency variations in the water-bearing layer. This results indicates the possibility of estimating the electrical properties of the formation by generating several measurements at different frequencies. Anisotropy effects translate into

7 small variations on the measurements, ranging from % up to %. In Fig. 9 right panel we display another comparison of results with and without presence of anisotropy. In this case, results without anisotropy correspond to a resistivity of the shale equal to Ω m vertical resistivity, as opposed to Ω m horizontal resistivity. Measurements for the case of no anisotropy are quite different from those with anisotropy. More precisely, differences due to anisotropy effects for this case may be as large as 8%. NO INVASION. ANISOTROPY STUDY Hz Hz Hz (ANISOTROPY) Hz (ANISOTROPY) (V/m ) NO INVASION. ANISOTROPY STUDY Hz (ANISOTROPY) Hz (ANISOTROPY) Hz (Rv) Hz (Rv) (V/m ) Figure 9: Problem III. Absolute value of the first vertical difference of E z (normalized) against the vertical position (in m) of the first receiver antenna. Different curves correspond to different frequencies and to the effect (or not) of anisotropy. Left panel: no anisotropy - horizontal resistivity only- at Hz (solid line), no anisotropy -horizontal resistivity only- at Khz (dashed line), anisotropy at Hz (triangles), and anisotropy at Khz (circles). Right panel: no anisotropy -vertical resistivity only- at Hz (solid line), no anisotropy -vertical resistivity only- at Khz (dashed line), anisotropy at Hz (triangles), and anisotropy at Khz (circles). In Fig. we study the effect of water invasion in presence of anisotropy. More precisely, we study the effect of a 6 =. cm. water invasion into the oil-, oil-water- and water- saturated layers, with resistivity equal to Ω m, Ω m, and.ω m, respectively. Fig. shows a stronger signal by about % % in the oil-, oilwater- and water- saturated layers when we consider water invasion at Hz as well as at Khz. Measurements across the shale layers are similar to each other (with and without water invasion), since we are not considering any type of water invasion into the shale. ANISOTROPIC Hz (NO INVASION) Hz (6" INVASION) Hz (NO INVASION) Hz (6" INVASION) (V/m ) Figure : Problem III. Absolute value of the first vertical difference of E z (normalized) against the vertical position (in m) of the first receiver antenna. Different curves correspond to different frequencies and to the effect (or not) of water invasion: no water invasion at Hz (solid line), no water invasion at Khz (dashed line), a 6 layer of water invasion at Hz (triangles), and a 6 layer of water invasion at Khz (circles). CONCLUSIONS In this paper, we have numerically studied a number of relevant physical effects occurring in resistivity logging through casing, by utilizing a self-adaptive goal-oriented hp-finite Element Method. The performed simulations indicate the following physical conclusions about through casing resistivity logging: As we increase the frequency (from Hz up to Khz) the received signal may decrease by as much as %-% in some areas (for example, where shoulder effects occur). Nevertheless, the overall shape of the solution does not experience drastic variations as we modify the frequency. Water invasion effects can be clearly sensed through casing. More precisely, variations due to water invasion on the model problems we have considered in this paper are large (up to one order of magnitude). The use of mandrel instruments with casing led to measurements that are not proportional to the formation resistivity, and therefore, a physical interpretation of the results could not be performed. Nevertheless, the possibility of combining a mandrel and a casing into the same simulation may serve as a tool for designing new logging instruments. As we increase the conductivity of the casing, mea-

8 ations. Also, as we increase the casing conductivity, sensitivity with respect to formation conductivity decreases, and more accurate sensors and numerical methods are needed to sense these variations. Otherwise, readings will be dominated by noise, and physical interpretation of results will not be possible. We can identify thin layers of shales and sands in the formation. Nevertheless, when the casing conductivity is high, we need to use accurate sensors. More precisely, for a casing resistivity equal to 6 Ω m, variations in measurements corresponding to shaleand sand-layers are below %. Anisotropy effects should be taken into account in order to accurately reproduce the actual measurements of a logging instrument through casing. ACKNOWLEDGMENTS We are grateful to Dr. Tarek Habashy, of Schlumberger- Doll Research, for providing us with his FORTRAN code to simulate the electromagnetic response of a vertical magnetic dipole in the presence of D radial media. This work was financially supported by Baker Atlas and The University of Texas at Austin s Joint Industry Research Consortium on Formation Evaluation sponsored by Aramco, Baker Atlas, BP, British Gas, Chevron, ConocoPhillips, ENI E&P, ExxonMobil, Halliburton, Marathon, Mexican Institute for Petroleum, Norsk-Hydro, Occidental Petroleum, Petrobras, Schlumberger, Shell E&P, Statoil, TOTAL, and Weatherford International Ltd. VALIDATION OF RESULTS In this appendix, we validate our software by comparing the self-adaptive goal-oriented hp-finite Element Method with a radial code. A radial code is a software that utilizes a semi-analytical method based on the Fourier transform. Solutions provided by the radial code shall be identical to the analytical solutions without integration errors, which may be large in problems with highly conductive materials (for example, problems with casing). On the other hand, numerical errors arising on finite element methods are of different nature: Modeling Errors. Errors due to the fact that we are solving similar but different problems to the original ones. These include:. The use of finite computational domains for solving problems on unbounded domains. In our case, we shall consider a computational domain of m in the horizontal direction and 8 m in the vertical direction.. The use of different types of source antennas. For modeling the effect of a Vertical Magnetic Dipole (VMD), we will consider a solenoidal antenna with finite cross-section of dimensions.. m and a radius equal to a =. m with an impressed current I = /(πa ) A. Approximation Errors. Errors due to the fact that we are not solving the modeled problems exactly. We are only approximating them using grids with a finite size h and a finite order of approximation p. Model Problems Considered for the Comparison Using cylindrical coordinates (ρ, φ, z), we consider the following domains: Ω I = {u = (u ρ,u φ,u z ) :. =.m u ρ. =.97m}. Ω II = {u = (u ρ,u φ,u z ) :. =.97m u ρ 6. =.m}. Ω III = {u = (u ρ,u φ,u z ) : 6. =.m u ρ 8. =.m}. Ω IV = {u = (u ρ,u φ,u z ) : 8. =.m u ρ =.8m}. Ω V = {u = (u ρ,u φ,u z ) : =.8m u ρ }. The source consists of a Vertical Magnetic Dipole (VMD) antenna located at the origin (,,), as shown in Fig. A-. We will simulate the electric field response at point (.m,, m). In Table, we describe two different problems that consider the geometry described above and different resistivities for each subdomain: Comparison Results Results have been compared to each other by computing the following relative errors in percentage: Error = u u u, Error = phase(u ) phase(u ) phase(u ) Error = real(u ) real(u ) real(u ) Error = imag(u ) imag(u ) imag(u ),, and, (A-) (A-) (A-) (A-) where u and u are the solutions provided by the radial code and the finite element (FE) software, respectively. Numerical results displaying quantities A-, A-, A-, and A- are shown in Fig. A-. The left panel corresponds to problem I, and the right panel to problem II.

9 . m Absolute Value Phase Real Part Imag Part Absolute Value Phase Real Part Imag Part m Trasmitter Receiver Relative Error (in %) Relative Error (in %) II IV 6 6 I III." " " 8" Figure A-: D cross section of the geometry of our problems of interest, composed of five different subdomains, a transmitter antenna (VMD), and a receiver antenna. V Frequency (in Hz) Frequency (in Hz) Figure A-: Relative Error in percent as a function of frequency. We display the relative errors of the absolute values (solid line), the phases (dashed line), the real (triangles), and the imaginary parts (circles) of the solution. Left panel: problem I. Right panel: problem II. In all cases, we observe an excellent agreement between solutions obtained with the radial code and solutions obtained with the hp-fe software. More precisely, the relative error remains below.% in most cases. We also observe a slight increase of the error as we increase the frequency. This error increase is due to integration errors on the radial code. Table corresponding to problem I illustrates that the radial code has not fully converged on the imaginary part at Mhz. In Fig. A- we display the solution of problem I as a function of the frequency. Solution of problem I as a function of the resistivity of layer is shown in Fig. A- for frequencies at Hz left panel and Mhz right panel. In summary, we showed in this appendix that numerical errors associated with our FE software are negligible (below.%). At the same time, the FE code enables accurate simulations of realistic logging problems with possibly complex geometries. REFERENCES Cecot, W., W. Rachowicz, and L. Demkowicz,, An hp-adaptive finite element method for electromagnetics. III: A three-dimensional infinite element for Maxwell s A limit of 6 integration intervals with a Gaussian integration rule of degree 8 has been imposed as the maximum accuracy for integration on the radial code. Numerical results indicate that this limit is not enough at high frequencies to obtain an error below.%. Amplitude of E φ (V/m) Real Part (FEM) Imag Part (FEM) Real Part (Radial) Imag Part (Radial) 6 Frequency (in Hz) Figure A-: Problem I. Azimuthal component of the electric field E φ (in V/m) as a function of frequency. We display the real (solid line and triangles) and imaginary (dashed line and circles) parts of the solutions provided by the radial (triangles and circles) and Finite Element (solid and dashed lines) codes. equations.: International Journal of Numerical Methods in Engineering, 7, Harrington, R. F., 96, Time-harmonic electromagnetic

10 7 8 Hz Real Part (FEM) Imag Part (FEM) Real Part (Radial) Imag Part (Radial) Mhz Real Part (FEM) Imag Part (FEM) Real Part (Radial) Imag Part (Radial) Wu, X. and T. M. Habashy, 99, Influence of steel casings on electromagnetic signals.: Geophysics, 9, Amplitude of E φ (V/m) 9 Amplitude of E φ (V/m) 6 8 Resistivity of Layer (in Ω m) Resistivity of Layer (in Ω m) Figure A-: Problem I. Azimuthal component of the electric field E φ (in V/m) as a function of the resistivity of layer. We display the real (solid line and triangles) and imaginary (dashed line and circles) parts of the solutions provided by the radial (triangles and circles) and Finite Element (solid and dashed lines) codes. Left panel: Hz. Right panel: Mhz. fields.: McGraw-Hill. Kaufman, A. A., 99, The electrical field in a borehole with casing.: Geophysics,, 9 8. Pardo, D., L. Demkowicz, C. Torres-Verdin, and M. Paszynski, a, A goal oriented hp-adaptive finite element strategy with electromagnetic applications. Part II: electrodynamics.: Submitted to Computational Methods on Applied Mechanics and Engineering (CMAME). b, Simulation of resistivity logging-whiledrilling (LWD) measurements using a self-adaptive goal-oriented hp-finite element method.: SIAM Journal on Applied Mathematics (in press). Preprint available at: Pardo, D., C. Torres-Verdin, and L. Demkowicz, 6, Simulation of multi-frequency borehole resistivity measurements through metal casing using a goal-oriented hp-finite element method.: IEEE Transactions on Geosciences and Remote Sensing,, no. 8,. Schenkel, C. J. and H. F. Morrison, 99, Electrical resistivity measurement through metal casing.: Geophysics, 9, 7 8. Singer, B. S. and K. M. Strack, 99, New aspects of through-casing resistivity theory.: Geophysics, 9, 7 8. Vail, W., S. Momii, R. Woodhouse, M. Alberty, R. Peveraro, and J. Klein, 99, Formation resistivity measurements through metal casing.: SPWLA th Annual

11 Formation materials for problem I. Formation materials for problem II. Formation materials for problem III. Resistivities (in Ω m). Imaginary part of E φ for Problem I (in V/m). We compare finite element results (right column) against those obtained with the radial code with 6 integration intervals (center column) and integration intervals (left column), respectively. Table : Formation materials for problem I. Material, Vertical Location -Layers- Resistivity, Horizontal Location Oil =.8 m. z =.8 m. R= Ω m. ρ >. cm. Background z < =.8 m. R= Ω m. =.8 m. < z ρ >. cm. Table : Formation materials for problem II. Material, Vertical Location -Layers- Resistivity, Horizontal Location Sands =.8 m. z 8 =.89 m. R= Ω m. 8 =.8 m. z 6 =.8796 m. ρ >. cm. 6 =.888 m. z =.7 m. =.9 m. z =.66 m. =.696 m. z =.8 m. =. m. z 8 =.8 m. =.696 m. z 8 =.76 m. =.9 m. z 8 =.77 m. 6 =.888 m. z 7 8 =.68 m. 8 =.8 m. z 9 8 =.96 m. Shales 8 =.89 m. z 8 =.8 m. R= Ω m. 6 =.8796 m. z 6 =.888 m. ρ >. cm. =.7 m. z =.9 m. =.66 m. z =.696 m. =.8 m. z =. m. 8 =.8 m. z =.696 m. 8 =.76 m. z =.9 m. 8 =.77 m. z 6 =.888 m. 7 8 =.68 m. z 8 =.8 m. 9 8 =.96 m. z =.8 m. Background z < =.8 m. R= Ω m. =.8 m. < z ρ >. cm. Table : Formation materials for problem III. Material, Vertical Location -Layers- Resistivity, Horizontal Location Oil 8 =.996 m. z =.896 m. R=6 Ω m. ρ > 7.78 cm. Oil-Water 7 =.776 m. z 67 =.6 m. R= Ω m. ρ > 7.78 cm. Water =. m. z = 7.6 m. R=. Ω m. ρ > 7.78 cm. Shale z < =. m. R=x Ω m. = 7.6 m. z 7 =.776 m. ρ > 7.78 cm. 67 =. m. z 8 =.996 m. =.9 m. < z

12 Table : Resistivities (in Ω m). Resistivities Ω Ω Ω Ω Ω Problem I 6 Problem II.. Table : Imaginary part of E φ for Problem I (in V/m). We compare finite element results (right column) against those obtained with the radial code with 6 integration intervals (center column) and integration intervals (left column), respectively. Frequency Radial Code Radial Code FE Code (in Hz) intervals 6 intervals E E E E E E-9-7.8E-8-7.8E-8-7.8E E E-8-9.8E-8 +.6E-7 +.E-7 +.9E-7 -.7E-9 -.6E E E E E- -.76E- -.79E- -.76E E E- -.76E E- -.86E- -.8E-