EXPERIMENTAL STUDY OF THE INDUCED POLARIZATION EFFECT USING COLE-COLE AND GEMTIP MODELS

Transcription

1 EXPERIMENTAL STUDY OF THE INDUCED POLARIZATION EFFECT USING COLE-COLE AND GEMTIP MODELS by Charles R. Phillips A thesis submitted to the faculty of The University of Utah in partial fulfillment of the requirements for the degree of Master of Science in Geophysics Department of Geology and Geophysics The University of Utah December 2010

2 Copyright c Charles R. Phillips 2010 All Rights Reserved

3 The University of Utah Graduate School STATEMENT OF THESIS APPROVAL The thesis of Charles R. Phillips has been approved by the following supervisory committee members: Michael S. Zhdanov, Chair 07/16/2010 Date Approved Vladimir Burtman, Member 07/16/2010 Date Approved Erich Petersen, Member 07/13/2010 Date Approved and by D. Kip Solomon, Chair of the Department of Geology and Geophysics and by Charles A. Wight, Dean of The Graduate School.

4 ABSTRACT Modeling induced polarization (IP) phenomena is important for developing effective methods for remote sensing of subsurface geology. Two resistivity relaxation models have been developed to describe the IP effect. The Cole-Cole model was formulated empirically more than 50 years ago. A few of the parameters within this model are able to account for IP phenomena and can give useful information for the analysis of bulk rock formations. Forward modeling using synthetic data is done to analyze the three empirical variables of the Cole-Cole model. These empirical variables are the decay coefficient (C), chargeability (m), and the time constant (τ). The generalized effective-medium theory of induced polarization (GEMTIP) model is another resistivity model similar to the Cole-Cole model. However, the GEMTIP model attempts to characterize heterogeneous, multiphase, polarized media using the effective-medium approach and can consider many more useful parameters than the Cole-Cole model. Forward modeling is also done with synthetic data to analyze four GEMTIP parameters. These parameters are the surface polarizability coefficient (α), the decay coefficient (C), the inclusion volume fraction (f), and the ellipticity of the inclusions (ɛ). Complex resistivity data was collected on several rock samples that contain disseminated sulfides, which are conducive to IP phenomena. Inversion of these data were done using both the spherical and elliptical GEMTIP models. In addition, complex resistivity data collected from a carbonate and two shale samples containing pyrite were also inverted using both the Cole-Cole and elliptical GEMTIP models in order to compare their effectiveness in recovering IP parameters. In the case of Cole-Cole inversion, the regularized Newton method was used, and in the case of GEMTIP inversion, the regularized conjugate-gradient method was implemented. Several of the parameters considered in this study were directly measured using X-ray microtomography and the QEMSCAN. Some of these parameters were included in the inversion routines to facilitate accurate inversion results. While the exact cause of the IP effect is quite complicated and still not perfectly understood, it is known that being able to model it can be very useful in improving mineral discrimination techniques. With improved understanding of the IP effect and new advancements in rock physics models, mineral discrimination will become more effective as well as more reliable.

5 CONTENTS ABSTRACT ACKNOWLEDGMENTS iii v 1. INTRODUCTION PRINCIPLES AND APPLICATION OF THE INDUCED POLARIZATION EFFECT History of induced polarization phenomena Cause of induced polarization phenomena Measuring induced polarization phenomena Modeling induced polarization via rock physics models Summary FORWARD MODELING AND INVERSION OF SYNTHETIC DATA Forward modeling: Cole-Cole model Forward modeling: GEMTIP model Regularized conjugate-gradient method Inversion: GEMTIP model MEASURED DATA Samples QEMSCAN Complex resistivity measurements INVERSION RESULTS Spherical GEMTIP model Elliptical GEMTIP model Elliptical GEMTIP and Cole-Cole model comparison Discussion CONCLUSIONS APPENDICES A. COMPLEX RESISTIVITY DATA B. THE LIST OF ELECTRONIC DATA REFERENCES

6 ACKNOWLEDGMENTS I would like to thank the Consortium for Electromagnetic Modeling and Inversion (CEMI) for providing me with financial and technical support. I also give many thanks to my adviser and committee chair, Dr. Michael Zhdanov, for his expert guidance and direction. I am also grateful for the inspiring classes that he taught on electromagnetic and inversion theory. Vladimir Burtman s understanding of rock physics, and in particular, the induced polarization effect was invaluable when it came to learning the theory behind my research. Alexander Gribenko was also instrumental in helping me understand and, when needed, modify the CEMI codes required to conduct my research. In addition I would like to acknowledge the many past and present CEMI researchers for the development of the codes that has enabled this research to move forward. Special thanks must also go to Zonge Engineering and Research Organization, Inc. for their help with complex resistivity measurements of many rock samples, to Dr. Erich Petersen for providing the time on the departments QEMSCAN equipment, and to my wife Annie for her priceless hours of editing what must be very boring text to her.

7 CHAPTER 1 INTRODUCTION The induced polarization (IP) effect, in general, is related to the complex resistivity of rocks. The Cole-Cole relaxation model was determined empirically and is used to relate resistivity measurements to the petrophysical properties of rock. The generalized effective-medium theory of induced polarization (GEMTIP) has been derived based on the effective-medium approach to the characterization of heterogeneous, multiphase, polarized mediums typical in rock formations. It describes the relationships between petrophysical and structural properties of rock and the parameters of the corresponding resistivity relaxation model (Zhdanov, 2008). The Cole-Cole model, while limited to only a few parameters, has been tested for many years and is still very commonly used; therefore, we will apply it in this study and compare the results with the GEMTIP model. The parameters of the GEMTIP model are determined by the intrinsic petrophysical and geometrical characteristics of the medium: the mineralization and/or fluid content of the rocks, the matrix composition, porosity, anisotropy, and polarizability of the formations. Therefore, in principle, these parameters may serve as a basis for determining the intrinsic characteristics of a polarizable rock formation from observed electrical data. Zhdanov (2008) proved that the GEMTIP model simplifies to the Cole-Cole model in the case of a two phase medium of spherical inclusions. In the case of a multiphase medium or a medium containing nonspherical inclusions, the Cole-Cole model may not adequately model the IP effect. Consequently, both the Cole-Cole and GEMTIP models are tested and applied. Forward modeling and inversion schemes were developed based on the regularized Newton method and the regularized conjugate-gradient method (Zhdanov, 2002). Synthetic resistivity data as well as measured resistivity data from many shale, carbonate, and rock samples were studied. Using the resistivity data, we inverted for three IP parameters: the time constant (τ), decay coefficient (C), and the chargeability (m) in the case of the Cole-Cole inversion, and the volume fraction (f), surface polarizability (α) and decay (C) coefficients in the case of GEMTIP inversion. This thesis is organized into six chapters. It begins logically with this introduction, giving interests and motivations behind the problems considered in this research project. Chapter 2

8 2 describes the principles and application of induced polarization (IP) phenomena, including a brief history of the IP effect. Chapter 3 is broken into four sections. The first two sections utilize synthetic data to illustrate forward modeling using both the Cole-Cole and GEMTIP rock physics models. The third section discusses the theory behind the regularized conjugategradient (RCG) method used to invert the complex resistivity data sets. The final section of Chapter 3 applies the RCG method to invert synthetic data using the GEMTIP rock physics model. The fourth chapter introduces the many rock samples studied, what kind of measurements were taken, and the sources of the measured data. Chapter 5 combines everything from the previous chapters into inversion of the measured data using the various rock physics models. The results are analyzed and the differences and similarities between the rock physics models are discussed. The final chapter ends with conclusions.

9 CHAPTER 2 PRINCIPLES AND APPLICATION OF THE INDUCED POLARIZATION EFFECT 2.1 Introduction to induced polarization phenomena The discovery of induced polarization (IP) dates back as early as 1913 to the French geophysicist, Conrad Schlumberger. He observed that if he passed a dc current through rocks containing metallic sulphides and interrupted the current abruptly, the resultant voltages in the Earth decayed slowly rather than instantly (Seigel et al., 2007). This was the first recorded measurement of the IP effect. Over the following decades, the development and testing of the IP effect expanded into Russia, Europe and the USA. In recent years, the IP method has been used primarily for bulk mineral exploration. However, there have been cases of using the IP method for hydrocarbon exploration (e.g., Davydycheva 2004, 2006). With all of these studies of IP, it is still considered a phenomenon that is not entirely understood, and while there has been a recent resurgence in its study, there is much yet to learn about it. 2.2 Cause of induced polarization phenomena James Wait describes his first exposure to the IP effect when Professor Yanzhong Luo and Dr. Guiqing Zhang, after carrying out careful sample measurements on the effective resistivity of a wetted rock sample, showed him that there was a strange frequency dependence. Wait then coined the phrase complex resistivity because it had both amplitude and phase (Luo et al., 1998). The exact cause of the IP effect is complicated. It can be basically defined as current flow accompanied by complex electrochemical reactions (Frasier, 1964). Figure 2.1 (a) shows current being applied across a host rock with mineral (metallic) inclusions. The porous region around the inclusion is generally saturated with some fluid, allowing ions to travel across. In the case of a medium with metallic inclusions these become polarized, and attract positive ions to form an electrical double layer at the boundary of the inclusion. This causes charge build up to occur, thus impeding current flow. Finally, when the current is abruptly terminated, instead of the voltage immediately dropping to zero, there is a gradual decay of the charge that has been built up. This is analogous to the capacitor effect. This effect occurs primarily anywhere the inclusions are good conductors.

10 Figure 2.1. Conceptual illustrations of surface polarization. a) Surface polarization of disseminated minerals in a uniformly conductive host rock. b) Surface polarization of a mineralized vein. Modified from Frazier (1964). 4

11 2.3 Measuring induced polarization phenomena Induced polarization is manifested as frequency dependent complex resistivity of rocks. Thus, in order to measure the IP effect, one has to measure the resistivity (both real and imaginary) of rocks at multiple frequencies. This thesis is concerned with measuring this effect in small rock samples (roughly an inch square in size) that can be analyzed in a controlled environment. Zonge Engineering and Research Organization has a laboratory setup called a GDP16 which allows small rock samples to be subject to multifrequency measurements while recording complex resistivity values. In this way, IP phenomena can be identified anywhere a resistivity response is seen as a function of frequency. Photographs of the measurement system are shown in Figures 2.2 and 2.3. Figure 2.2 shows how the rock sample is placed between two current carrying electrodes. The electrodes are placed in a electrolyte solution to help facilitate current flow. This solution will percolate throughout the rock, carrying current with it. Figure 2.3 shows the entire measurement setup. 5 Figure 2.2. Sample holder, rock sample, and receiving and transmitting electrodes.

12 Figure 2.3. Recording system used at Zonge Engineering and Research Organization Inc. to obtain EM measurements. 6

13 2.4 Modeling induced polarization via rock physics models An important problem of electromagnetic geophysics is to study the frequency-dependent complex resistivity of rocks, in which the IP phenomenon is often manifested. As mentioned before, over the last 50 years several conductivity relaxation models have been developed. Such models include the empirical Cole-Cole model (Cole and Cole, 1941; Pelton et al., 1978), electrochemical models developed by Ostrander and Zonge (1978), and the generalized effective-medium theory of induced polarization (GEMTIP). GEMTIP is a new, rigorous, mathematically formulated conductivity model introduced by Zhdanov (2008). The widely accepted Cole-Cole model uses bulk rock variables and does not address rock composition, while the GEMTIP model uses effective-medium theory to describe the complex resistivity of heterogeneous rocks. The GEMTIP resistivity model incorporates the physical and electrical characteristics of rocks at the grain scale into an analytic expression. These characteristics include grain size, grain shape, mineral conductivity, porosity, anisotropy, polarizability, mineral volume fraction, pore fluids, and more (Zhdanov, 2008). Pelton in 1977 first demonstrated that the relaxation model derived by Cole and Cole (1941) is an excellent representation of the complex conductivity (both real and imaginary) of polarized rock formations. This frequency dependent complex resistivity is shown in Figure 2.4, which is described in the following formula: ρ(ω) = ρ DC (1 m ( 1 )) ( iωτ) C 7 (2.1) where ρ DC is the dc resistivity (Ohm-m); ω is the angular frequency (rad/sec); the time constant, τ, determines at what frequency the peak response in the imaginary resistivity will occur, whereas the dimensionless intrinsic chargeability, m, characterizes the intensity of the IP effect and the relaxation parameter, C, depicts its magnitude. These effects can been seen graphically in Figure 2.4 (Emond, 2007). Characterizing observed IP responses in terms of their Cole-Cole parameters has proven useful in resolving different rocks, but primarily through differences in their average particle size. Grounded metallic structures have been easily recognized by the long time constant of their IP responses. However, despite much effort, attempts to predict the mineralogical composition of rocks by analysis of IP response characteristics have not been very fruitful (Seigel et al., 2007). The generalized effective-medium approach is a new innovative method for determining the same parameters as the Cole-Cole model, and much more. GEMTIP computes the

14 Real Imaginary.. _-- decreasing..... time constant /..., /...,.' '..' ' ; '11' / "..'.' ' ; ;. >T,. / /,.,; " / /.oj'~.. 0 0,, :.,.,., " " I... ; oj' _II''''',.,.; " , d. ".0""'".. :::-:.~... ecreaslng... _.--."" decay coefficient decreasing chargeability Frequency ---+unequal scales Figure 2.4. Complex resistivity behavior of the Cole-Cole model. The effect of changing chargeability (m), time constant (τ), and decay coefficient (C) from Equation 2.1 is illustrated. 8

15 9 effective resistivity based on many rock microparameters. In the case of a multiphase medium with spherical inclusions, the GEMTIP resistivity relaxation model can be described by the following formula: where ( ))) 1 N 1 ρ ef = ρ 0 (1 + (f l m l 1 (2.2) l=1 1 + ( iωτ l ) C l τ l = m l = 3 ρ 0 ρ l 2ρ l + ρ 0 (2.3) ( ) 1 al C l (2ρ l + ρ 0 ) (2.4) 2α l with the parameters being defined in Table 2.1. More sophisticated GEMTIP resistivity models exist for more complex situations. In this project we used a GEMTIP model that allowed us to describe mineral shapes by ellipsoids with any degree of ellipticity, from oblate to prolate. While the application of the elliptical model is very similar to the one shown above, its derivation is more complicated and may be found in Zhdanov (2008). The simplest analytic solution of this general theory is for spherical grains in a homogeneous matrix given by formula 2.2. One can see when comparing formula 2.2 and formula 2.1 that they are similar. In fact, the Cole-Cole model appears as a special case of the GEMTIP model for a two-phase medium with spherical inclusions (Zhdanov, 2008). As a result, Cole-Cole parameters can be explained via the relationships seen within GEMTIP. 2.5 Summary Understanding induced polarization phenomena is important for developing the methods of subsurface geophysical exploration. With growing interest in this area of exploration geophysics, considerable research has been invested in the understanding of IP phenomena. Only recently has a breakthrough in the application of IP methods developed. This breakthrough was introduced initially by Zhdanov (2008) who developed the generalized effective-medium theory of induced polarization, a relaxation model that computes resistivity as a function of frequency. We now have the theoretical tools to not only model the IP effect for bulk mineral properties (provided by the Cole-Cole model), but to model IP for more accurate mineral discrimination.

16 10 Table 2.1. Description of GEMTIP parameters. Variable Units Name Description ρ ef Ohm-m effective resistivity resulting effective resistivity ρ 0 Ohm-m matrix resistivity matrix resistivity of rock being modeled f l - grain volume fraction volume fraction of each grain type m l - grain chargeability grain chargeability of each grain type ω Hertz angular frequency angular frequency of EM signal τ l second time constant time constant for each grain C l - decay coefficient decay coefficient determined from empirical data ρ l Ohm-m grain resistivity resistivity of each grain type a l meter grain radius radius of each grain type α l Ohm m 2 sec c l surface polarizability coefficient behavior of charges on grain surface determined from empirical data

17 CHAPTER 3 FORWARD MODELING AND INVERSION OF SYNTHETIC DATA Forward modeling using the Cole-Cole and GEMTIP resistivity models is an important step in understanding how IP parameters relate to complex resistivity data. Buist (2009) demonstrates forward modeling using three IP parameters of the Cole-Cole model. These results are compared with forward modeling results using the GEMTIP model. By comparing the two models we can determine how both can describe the IP effect. At the same time, it is shown how varying IP parameters themselves, within each model, change the IP response in the data. As mentioned before, the Cole-Cole model is primarily used for bulk mineral discrimination and contains no information about rock composition. This model is given by equation 2.1. The GEMTIP model on the other hand incorporates both physical and electrical characteristics of rocks at the grain scale into one analytic expression. These characteristics include grain size, grain shape, mineral conductivity, porosity, anisotropy, polarizability, mineral volume fraction, pore fluids, and more (Zhdanov, 2008). The first analytic solution of this general theory is for spherical grains in a homogeneous matrix given by Equation 2.2. Each variable of the model is described in Table 2.1. It should be mentioned again that when comparing formula 2.2 and formula 2.1 that they are similar. In fact, the Cole-Cole model appears as a special case of the GEMTIP model for a two-phase medium with spherical inclusions (Zhdanov, 2008). 3.1 Forward modeling: Cole-Cole model To understand in more detail what effect the three different IP parameters within the Cole-Cole model has on synthetic data, a series of scenarios shown in Tables 3.1, 3.2 and 3.3 have been modeled. Within these tables, two of the three parameters are kept constant while the third parameter is varied. All modeling parameters were based on averaging the previous Cole-Cole parameters obtained from literature results for a variety of common minerals (Buist, 2009).

18 12 Table 3.1. Cole-Cole parameters used for synthetic modeling when varying m. Model ρ 0 (DC) m τ C No IP 500 NA NA NA Table 3.2. Cole-Cole parameters used for synthetic modeling when varying τ. Model ρ 0 (DC) m τ C No IP 500 NA NA NA Table 3.3. Cole-Cole parameters used for synthetic modeling when varying C. Model ρ 0 (DC) m τ C No IP 500 NA NA NA

19 13 Figure 3.1 shows that when the chargeability factor (m) is varied (from 0.1 or model 1, to 0.6 or model 6) it increases the slope of decent within the real part (a) of the resistivity value, while in the imaginary part (b) it increases or intensifies the resistivity value. Changing m does not change the location (at what frequency) of the IP response, but only its amplitude. In both the real (a) and imaginary (b) parts of the resistivity value, adjusting the time constant τ (from seconds in model 1 to 100 seconds in model 6) shifts the peak response to a lower frequency (Figure 3.2). Emond (2007) showed that decreasing the grain size was analogous to decreasing the time parameter, which causes the IP peak to be at a lower frequency. The final figure (3.3) depicts the effect that changing the relaxation parameter C, (from 0.1 in model 1 to 0.6 in model 6) has on the magnitude of the induced polarization response. Laboratory studies (Pelton, 1978) of induced polarization have suggested that the decay coefficient is not equal to 1.0, but is typically in the range of 0.1 to 0.6 like those modeled. 500 a Real ρ ef (Ωm) b Imag. ρ ef (Ωm) Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 No IP Frequency (Hz) Figure 3.1. Complex resistivity behavior of the Cole-Cole model when the chargeability (m) factor varies from 0.1 to 0.6. Both the time constant (τ), and relaxation parameter (C) are held constant (1 second and 0.4, respectively). a) Real resistivity, b) Imaginary resistivity.

20 14 a 500 Real ρ ef (Ωm) b Imag. ρ ef (Ωm) Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 No IP Frequency (Hz) Figure 3.2. Complex resistivity behavior of the Cole-Cole model when the time constant (τ) varies from to 100 seconds. Both the chargeability (m), and relaxation parameter (C) are held constant (0.4 and 0.4, respectively). a) Real resistivity, b) Imaginary resistivity. As a result, the induced polarization decay is slower than exponential decay, but one can also clearly see that as the value of C increases, so does its magnitude in the imaginary part (b). Meanwhile, the real part of the resistivity value (a) has a sign reversal about the frequency of 0.1 Hertz. From these results it is shown that all three parameters affect the complex resistivity signature. However, the most important, and often the only one looked at in real life induced polarization surveys, is the chargeability factor, which has a greater effect on the data over a wider range of frequencies. The chargeability factor is also the most useful in identifying the amount of a given geophysical substance that is being mined. This is very useful for mining companies that are only interested in mining large quantities of minerals. The chargeability can also be used to help identify mineral types, along with the relaxation coefficient. In general, however, all three parameters (chargeability, time constant, and decay coefficient) will be measured and used for acquiring as much information as possible.

21 15 a 500 Real ρ ef (Ωm) b Imag. ρ ef (Ωm) Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 No IP Frequency (Hz) Figure 3.3. Complex resistivity behavior of the Cole-Cole model when the relaxation parameter (C) factor varies from 0.1 to 0.6. Both the time constant (τ), and chargeability (m) are held constant (1 second and 0.4, respectively). a) Real resistivity, b) Imaginary resistivity. 3.2 Forward modeling: GEMTIP model Right now, there exists two analytical GEMTIP models that are used for modeling rock resistivity. The first is called the spherical GEMTIP model, and the second is called the elliptical GEMTIP model. Their use and purpose is identical apart from the elliptical model having the ability to consider inclusion shapes of arbitrary ellipticity, using three different radii values. In contrast, the spherical GEMTIP model only allows for isometric inclusions to be appropriately modeled, using a single radius value. Several synthetic models have been created to show how many of the parameters contained in these GEMTIP models affect real and imaginary resistivity data. In particular, I focused on modeling the effects of volume fraction (f), surface polarizability coefficient (α), decay coefficient (C), and ellipticity (ɛ). Table 3.4 shows the values used for the nonvarying parameters associated with Figures 3.4 through 3.6. Table 3.5 shows the values used for the parameters associated with Figure 3.7. Table 3.6 shows the values used for the parameters associated with Figures 3.8 through 3.10.

22 16 Table 3.4. GEMTIP parameters used for synthetic modeling when varying α, C, and f. Radius 10 4 ρ 0 (DC) 450 ρ 1 1 α 0.10 C 0.7 f 0.02 Table 3.5. Elliptical GEMTIP parameters used for synthetic modeling when varying ɛ. ρ 0 (DC) 450 ρ 1 1 α 0.5 C 0.8 f 0.02 Table 3.6. Elliptical GEMTIP parameters used for synthetic modeling when varying ɛ for different values of α. ρ 0 (DC) 300 ρ ρ α 1 2 α 2.1, 1, 10 C C f f Figure 3.4 shows the GEMTIP resistivity model plotted, keeping all parameters constant but the surface polarizability coefficient. In this example, five different values of α are chosen to represent the range of reasonable values. It is very clearly shown that by varying α, the response of the computed resistivity shifts in frequency. Low values of α correspond to low frequency response, while high values of α correspond to high frequency response. Also, varying α does not appear to affect the amplitudes of the computed real or imaginary resistivities. Typical values for α may range anywhere from 0 to 10 Ω m2 sec c l. Figure 3.5 shows how varying the decay coefficient (C) affects the computed resistivity values (while holding all other parameters constant). Interestingly, C appears to affect both the location of the resistivity response (in frequency) as well as its amplitude (both real and imaginary). Typical values for C range from 0 to 1. Because this is the same IP parameter as C in the Cole-Cole model, the forward modeling results can be directly compared.

23 17 Real 445 ρ ef (Ωm) Imaginary ρ ef (Ωm) α = 10e 3 α = 10e 2 α = 10e 1 α = 1 α = Frequency, Hz Figure 3.4. GEMTIP resistivity response obtained by keeping all parameters constant but the surface polarizability coefficient (α). Similarly, Figure 3.6 shows the resistivity response of GEMTIP when all parameters are kept constant but the volume fraction (f). By increasing f over several magnitudes, it becomes clear that the amplitude of the resistivity response significantly decreases in the real and increases in the imaginary. Understanding the effects of the volume fraction is particularly significant because it is one of the parameters that can be measured directly. Figure 3.7 gives important information that can come only from the elliptical GEMTIP rock physics model. Previously, GEMTIP modeling was limited to approximating each inclusion as spheres. This new elliptical model provides three different radii to be specified for each inclusion a, b, and c, where a and b lie in the xy plane and c extends in the z. Ellipticity in this case is defined as ɛ = a = b. Therefore, in Figure 3.7, a value of ɛ = 0.5 means c c that the ratio of the radii of the inclusion is such that a = b and that a c range of ellipticity values is anywhere from ɛ = 1 8 = 0.5. A typical to ɛ = 8. The figure shows that slightly prolate (ɛ = 0.5) and slightly oblate (ɛ = 2) spheroids produce similar resistivity curves as do highly prolate (ɛ = 0.125) and highly oblate (ɛ = 8) spheroids. One can see how adding this

24 18 Real 445 ρ ef (Ωm) Imaginary ρ ef (Ωm) Frequency, Hz C = 0.1 C = 0.3 C = 0.5 C = 0.7 C = 0.9 Figure 3.5. GEMTIP resistivity response obtained by keeping all parameters constant but the decay coefficient (C). new capability to account for more arbitrarily shaped inclusions can dramatically affect ones ability to model rock resistivity. Work has also been done to examine the effects of including two inclusion phases in addition to the matrix phase. A similar study may be found in Zhdanov (2008); however, only the spherical GEMTIP model was used in that study, and it was not able to examine the contribution of elliptical inclusions. Figure 3.8 now shows the combined effects of varying the surface polarizability coefficient at a very low ellipticity. It is important to note that all parameters were kept constant other than the surface polarizability of the third phase. Also, the effects of the two inclusion types are manifested in the double peaks that are prominent in the imaginary resistivity. One indication of multiple inclusions in actual EM surveys of a formation is these multiple resistivity peaks that arise in the imaginary resistivity data. Figures 3.9 and 3.10 show similar plots for α but use different values for ɛ. In fact, Figure 3.9 has an ellipticity of ɛ = 1, which coincides with the spherical GEMTIP three-phase model.

25 19 Real 440 ρ ef (Ωm) ρ ef (Ωm) Imaginary f = 10e 4 f = 10e 3 f = 10e 2 f = 10e Frequency, Hz Figure 3.6. GEMTIP resistivity response obtained by keeping all parameters constant but the volume fraction (f).

26 Real 440 ρ ef (Ωm) Imaginary ρ ef (Ωm) ε = ε = 0.5 ε = 2 ε = Frequency, Hz Figure 3.7. Elliptical GEMTIP resistivity response obtained by keeping all parameters constant but the ratio of the radii of the mineral inclusion.

27 21 Ellipticity (ε) = Real ρ ef (Ωm) Imag. ρ ef (Ωm) α = 0.1 α = 1 α = Frequency (Hz) Figure 3.8. Three-phase elliptical GEMTIP model with varying values for α of only the third phase. The ellipticity of the second phase was set to ɛ = 1, and ɛ = for the third phase.

28 22 Ellipticity (ε) = 1 Real ρ ef (Ωm) Imag. ρ ef (Ωm) α = 0.1 α = 1 α = Frequency (Hz) Figure 3.9. Three-phase elliptical GEMTIP model with varying values for α of only the third phase. The ellipticity of both the second and third phase was set to ɛ = 1.

29 23 Ellipticity (ε) = 8 Real ρ ef (Ωm) Imag. ρ ef (Ωm) α = 0.1 α = 1 α = Frequency (Hz) Figure Three-phase elliptical GEMTIP model with varying values for α of only the third phase. The ellipticity of the second phase was set to ɛ = 1, and ɛ = 8.0 for the third phase.

30 3.3 Regularized conjugate-gradient method The purpose of inversion is to recover model parameters from measured data. The inverse problems are all ill-posed and require regularized inversion through minimization of the Tikhonov parametric functional. The regularization ensures that a unique and stable solution is obtained from the measured data. Both the regularized Newton method and the regularized conjugate-gradient (RCG) method are utilized in this research; however, only the RCG method will be derived here. Derivations and numerical schemes of both methods can be found in Zhdanov (2002). The RCG method is an iterative solver, which updates the model parameters on each iteration using conjugate gradient directions l α, according to the following formula: 24 m n+1 = m n + δm n k α n l α (m n ). (3.1) First, the direction of regularized steepest ascent is used: lα (m 0 ) = l α (m 0 ). (3.2) The next direction is a linear combination of the regularized steepest ascent in this step, and the conjugate gradient direction in the previous step: lα (m 1 ) = l α (m 1 ) + β α 1 l α (m 0 ). (3.3) The steps k α n are selected based on the minimization of the parametric functional: P α (m n+1 ) = P α (m n ) k nl α α (m n ) = Φ( k n). α (3.4) Minimization of this functional gives the following best estimation for the length of the step using a linear line search: kn α = ( l α n,l α n)/( l α n,fmnf mn + α(w W) l α n) = ( l α n,l α n)/[(f mn lα n,f mn lα n ) + α(w l α n,w l α n)] (3.5) = ( l α n,l α n)/[ F mn lα n 2 + α W l α n 2 ]. The β n coefficients are determined by the following formula: β α n = l α (m n ) 2 / l α (m n 1 ) 2. (3.6) The final numerical scheme for the RCG method can be summarized as follows:

31 25 r n = A(m n ) d, l α n = l α (m n ) = F mnr n + αw W(m n m apr ), β α n = l α n 2 / l α n 1 2, lα n = l α n + β α n l α n 1, (3.7) lα 0 = l 0, k α n = ( l α n,l α n)/[ F mn lα n 2 + α W l α n 2 ], m n+1 = m n k α n l α n. The initial regularization parameter α is calculated using the following formula: α = A(m 1) d 2 m 1 m apr 2. (3.8) We also implement the adaptive regularization; therefore, α n becomes where: α n = αq n, (3.9) 0 < q < 1. (3.10) This numerical scheme has been implemented using a code written in MATLAB. 3.4 Inversion: GEMTIP model The synthetic data set used is obtained considering one disseminated phase that occurs at one grain size. The resistivity of the phase is chosen to represent pyrite inclusions. These inclusions are approximated in shape by a spherical ball and all have the same radius. This is a very simple scenario and has been purposefully chosen to minimize inversion complexity. It comprises a rock matrix with the resistivity 400 Ωm. The volume fraction of pyrite in the sample is 8.0%. The known values of the decay and surface polarizability coefficients are 0.5 and 0.4 Ω sec m2 respectively. The radius of the approximated pyrite balls is 12.5 mm. The c l resistivity of the pyrite inclusions is 0.3 Ωm. With these variables defined, the effective resistivity can be determined by GEMTIP. Figure 3.11 shows both the real effective resistivity and the imaginary effective resistivity plotted against frequency. The two data sets shown are observed and predicted data. The observed data came from using the true model parameter values, while the predicted data were

32 26 Real ρ ef (Ωm) ρ ef (Ωm) Imaginary Original Predicted Frequency (Hz) % Misfit Iteration Figure Synthetic sample. a) Real effective resistivity plotted over a range of frequencies. b) Imaginary effective resistivity plotted over a range of frequencies. c) Percent misfit vs. number of iterations from the inversion. determined by minimizing the Tikhonov parametric functional using the RCG method with initial conditions. It is obvious that the original observed data fit the predicted data very well. After inversion, the correct surface polarizability and decay coefficients were recovered. Figure 3.12 shows the converging model steps plotted over the misfit functional. In the regularized inversion, a relaxation coefficient of q = 0.7 was used. The misfit threshold was set to 0.01 and it took 44 iterations to reach this threshold. The initial regularization parameter calculated was α = 8719 which decreased to a final value of α = 14. The parameters are summarized in Table 3.7. It shows that the two parameters that were inverted for (the decay coefficient and the surface polarizability coefficient) were given initial values of 1.0. The final recovered value for the decay coefficient is 0.5. The final recovered value for the surface polarizability coefficient is 0.4. Table 3.7 also shows that the final inverted values are exactly equal to the true model values, showing the the inversion worked perfectly. This is primarily due to the simplicity of this particular problem.

33 27 C Misfit m True Model α Figure Synthetic sample. The misfit functional is plotted with shaded isolines signifying the direction of decreasing misfit. Model steps are also plotted and are shown to converge after several iterations. Table 3.7. GEMTIP inversion values. Variable Units True model Initial Recovered ρ matrix Ωm f C Seconds ρ phase1 Ωm a mm α Ω m 2 sec c l

34 CHAPTER 4 MEASURED DATA 4.1 Samples This section analyzes the experimental complex resistivity data obtained for a representative set of rock samples. The viability of the Cole-Cole and GEMTIP conductivity models was tested with multifrequency EM measurements acquired for three shale, one carbonate, and four igneous rock samples at Zonge Engineering. The carbonate and two of the shale samples were previously measured and studied by Buist (2009). His description and analysis of these three samples will be reiterated here. The other shale and four igneous rock samples will also be described below and analyzed in the following chapter Shale and Carbonate samples Comprised of an organic-rich shale source rock from the Haynesville formation (Utah), sample M3-4 was chosen for analysis because of its pyrite grains (Figure 4.1). Davydycheva et al. (2006) suggested that shales with pyrite veins or lenses are the most likely candidate to produce a recordable IP measurement at low frequencies (Buist, 2009). Sample M1-3 (Figure 4.2) originated from the same well. It is of the same organic content and as the previous sample but with very little in the way of pyrite grains. This sample was chosen to see the effect of pyrite inclusions on the measured IP response (Buist, 2009). Sample D4-3 (Figure 4.3) is from the Akah Formation (Utah). It is dominated by evaporites that are interbedded with open marine carbonate rocks and shoaling-up carbonate buildups to the west, and terrigenous clastic rocks to the north-northeast. The pyrite inclusions are thought to be deposited as a replacement mineral for some of the organic material and for this reason are disseminated. Complex resistivity values were collected in orthogonal directions on sample D4-3 to view the effect of anisotropy on the recorded IP measurement. The core was taken from a depth of 6095 feet, well within the Akah shale formation (Buist, 2009). Sample D2-1 is shale similar to sample D4-3, taken from the same core but from a slightly shallower depth. Resistivity measurements were collected on this sample along the bedding. This sample is shown in Figure 4.4.

35 29 Figure 4.1. Sample M3-4, location, organic rich shale source rock with visible pyrite inclusions and veins. Sampled from a well depth of feet. Figure 4.2. Sample M1-3, location, organic rich shale source rock with no visible pyrite inclusions and veins. Sampled from a well depth of feet.

36 30 Figure 4.3. Sample D4-3, Akah Formation, Suan Juan County, Utah. It is an organic rich carbonate source rock with visible pyrite lenses under a hand lens. Sampled from a well depth of 6095 feet. Figure 4.4. Shale sample D2-1. Resistivity measurements were collected parallel to bedding.

37 Igneous rocks Four rock samples were chosen to study: K01, K02, B1b, and B5-1. The first two samples (K01 and K02) are monzonites from the Kori Kollo mine in Bolivia with disseminated pyrite (FeS 2 ) and a predominantly sericite and quartz matrix. Pictures of K01 and K02 are shown in Figures 4.5 and 4.6, respectively. The other two rock samples (B1b and B5-1) were taken from the Bingham Canyon Mine in Utah. The first of these two, sample B1b, is a Monzonite rock containing disseminated pyrite and chalcopyrite (CuFeS 2 ), and is shown in Figure 4.7. The second of these, sample B5-1, is a Quartz Monzonite Porphyry rock and contains disseminated chalcopyrite and bornite (Cu 5 FeS 4 ), and is shown in Figure 4.8. These four samples were selected because of the mineral inclusions that can be potentially ideal for analysis by the GEMTIP rock physics model. Sample K01 was analyzed using three-dimensional (3D) X-ray microtomography (Emond, 2007). The X-ray microtomography created a 3D volume of attenuation coefficients. The attenuation coefficients were used to distinguish the pyrite from the rock matrix. In addition to the 3D image, the volume fraction, inclusion size, and surface area can be determined by X-ray microtomography. A summary of the X-ray microtomography can be seen in Table 4.1. For example, a 3D microtomographic image of sample K01 is shown in Figure 4.9, highlighting the sulfide phases. Figure 4.5. Sample K01. Monzonite from the Kori Kollo mine, Bolivia. Contains disseminated pyrite.

38 32 Figure 4.6. Sample K02. Monzonite from the Kori Kollo mine, Bolivia. Contains disseminated pyrite. Figure 4.7. Sample B1b. Monzonite from the Bingham Canyon Mine, Utah. Contains disseminated pyrite and chalcopyrite.

39 33 Figure 4.8. Quartz monzonite porphyry from the Bingham Canyon Mine, Utah. Contains disseminated chalcopyrite and bornite. Table 4.1. K01 mineralogical summary (after Emond, 2007). Method Optical Matrix composition Sericite, Quartz Pyrite vol. fraction Pyrite radius 10% 1 to 2.5 mm. X-ray, quantitative - 7% 478 inclusions. 90% volume from 15 inclusions of 0.5 to 1.3 mm.

40 34 Figure 4.9. X-ray tomography image of Sample K01, Korri Kollo, Bolivia, pyrite in a sericite and quartz matrix. The image has been optimized to show the pyrite. 4.2 QEMSCAN The carbonate and two of the shale samples have been analyzed using the QEMSCAN housed at the University of Utah department of Geology and Geophysics. Using the QEM- SCAN allows one to obtain a detailed mineralogical analysis of a sample. The QEMSCAN is an automated mineral analysis system based on a Zeiss EVO 50 scanning electron microscope. Four Energy Dispersive Spectrometry (EDS) detectors are used simultaneously to decrease the time required to analyze a sample. Polished samples coated in epoxy are scanned for mineral composition, size, and orientation. A color coded map of mineral composition is created as well as a quantitative analysis of each mineral type and inclusion size. Buist (2009) acquired QEMSCAN measurements on the carbonate and two shale samples, with a quantitative mineral analysis of each shown in Table 4.2. Figure 4.10 is a representative section of sample M3-4. Figures 4.11 and 4.12 are representative sections of samples M1-3 and D4-3 respectively.

41 Table 4.2. Summary mineralogy of shale and carbonate samples (modified from Buist, 2009). QEMSCAN Sample M3-4 Sample Composition Plagioclase 29% Feldspar 22% Mica 15% Quartz 12% Calcite 8% Dolomite 6% Pyrite 3.3% Pyrite Diameter Porosity Clay Content 3.5% total pyrite. Largest grain 35µm. 20% above 5µm and 80% less than 5µm. Disseminated thoughout sample. Large vein 2mm wide 10mm long Low porosity at 2% with no connecting veins causing low permeability Total clay content less than 1%. Sample D4-3 Dolomite 66% Clays + Quartz 30% Pyrite 2% Other 2% Total pyrite 2%, Largest grain is 90µm. 50% above 10µm, 30% 5-10µm, 20% less than 5µm. Disseminated with no visible veins. Almost no porosity (0.6%) and not interconnected Very fine-grained and very difficult to separate the quartz from the clay. Estimated clay: 3%. Sample M1-3 Plagioclase 35% Feldspar 22% Micas 19% Quartz 10.6% Pyrite 3.5% Disseminated throughout, 5% above 10µm, 50% 5-10µm, 45% less than 5µm. Largest grain 40µm. Interconnected veins creating high permeability with a porosity of 1.5%. Low volumn of clay (less than 1% total) 35

42 Figure Sample M3-4 mineralogy images from QEMSCAN. a) Mineralogy map. b) Pyrite highlighted in black. c) Largest pyrite particles listed (not to scale). 36

43 Figure Sample M1-3 mineralogy images from QEMSCAN. a) Mineralogy map. b) Largest pyrite particles listed (not to scale). 37

44 . ~, Carbonates Quartz Pyrite....,, i...,,~,.....' "" ;, ~. \..:',.' '. ;'".'..." '" '. '. ~" ~. '.. ~. ~.. 4.'.,... ~..,.,..., '.. " ~,'",." '.,... " " '.'" : : ~,.... '. I '...'. ; '.,,.,... ". '.'.. '....!'...,.~ " '...,'. '..... ' ",,...~.....'. ','.,. ",. '...., '.,..",..... ;,..... ~ :, ".. '. ' \. " "..... ~. ' ~ i.,, ". :'..,'C " -",,,,...,-,,,,,,.. ; ~.''''.,.. '''.. """ 'lip,.""',,,. _...,... ~.... ".. :'.....:.', ~. '. ~, '...,. " -.. ',._- _.... "'.,- ' '.. " ",.-.:.'..(',,. :-...., Figure Sample D4-3 mineralogy images from QEMSCAN. a) Mineralogy map. b) Pyrite highlighted in black. c) Largest pyrite particles listed (not to scale). 38

45 4.3 Complex resistivity measurements Complex resistivity measurements were obtained from each sample at Zonge Engineering and Research Organization, Inc. Frequency domain data were collected over a range from about.01 Hz to about 300 Hz. Multiple measurements were acquired for a few of the samples to test the repeatability of the measurement procedure. In addition, data were collected in two orthogonal directions on sample D4-3. This was done to test whether the anisotropy of the bedding in that sample would have an actual effect on the measured resistivity. The equipment used to perform these measurements are shown in Figures 2.3 and 2.2. Analyzing these samples with the GDP16 required special preparation procedures. First, each sample has to be trimmed to approximately one inch square to fit appropriately between the current electrodes. Second, if the sample was measured dry it would yield resistivity values in the Megaohms. Thus, in order for the electrolytes to flow through the porous region of the sample, it had to be soaked in water for approximately three days prior to analysis. This procedure was followed for each sample that was measured. Plots of the raw data recorded at Zonge are shown in Figures 4.13 through 4.18 for samples D2-1x, B1b, and B5-1. Multiple sets of data were recorded for each sample and are numbered accordingly on each figure. Unfortunately, one difficulty arises when using the GDP16 to record complex resistivity, which is that the resistivity measurements for a given sample tend to shift to lower and lower values as more and more measurements are recorded. This is possibly due to the increased penetration of the electrolytes via the copper sulfate solution flowing into the porous regions of the sample as time passes. However, it appears that this shift in resistivity only occurs inbetween measurements, and that the frequency dependent resistivity for each measurement has a similar trend and is only offset by some dc resistivity value. Two methods have been implemented in order to account for this data shift. In the case of shale sample D2-1x, the data of each set is averaged to create one data set. Figure 4.13 shows the raw data for three different measurements of sample D2-1x, while Figure 4.14 shows the same raw data, but averaged. In addition, the error bars on Figure 4.14 show the standard deviation of the data at each frequency. For rock samples B1b and B5-1 the differences between each measurement s dc resistivity and the dc resistivity of a reference measurement were subtracted. This effectively shifted each measurement to the same dc resistivity of the reference measurement. The shifted resistivity data (and the data that was used for analysis in later chapters) can be seen in Figure 4.16 for sample B1b, and Figure 4.18 for sample B

46 Figure Sample D2-1x raw data plotted: a) Real resistivity, b) Imaginary resistivity. 40

47 Figure Sample D2-1x raw data averaged. Error bars show standard deviation of the data at each frequency: a) Real resistivity, b) Imaginary resistivity. 41