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Evan Schankee Um March 2011 Abstract The marine controlled-source electromagnetic (CSEM) method detects and images electrical resistivity associated with hydrocarbon, gas and CO2 reservoirs. Its survey design and data interpretation require modeling of complex and often subtle offshore geology with accuracy and efficiency. In this dissertation, I develop two efficient finite-element time-domain (FETD) algorithms for the simulation of threedimensional (3D) electromagnetic (EM) diffusion phenomena. The two FETD algorithms are used to investigate the time-domain CSEM (TDCSEM) method in realistic shallow offshore environments and the effects of seafloor topography and seabed anisotropy on the TDCSEM method.

The first FETD algorithm directly solves electric fields by applying the Galerkin method to the electric-field diffusion equation. The time derivatives of the magnetic fields are interpolated at receiver positions via Faraday’s law only when the EM fields are output. Therefore, this approach minimizes the total number of unknowns to solve.

To ensure both numerical stability and an efficient time-step, the system of FETD equations is discretized using an implicit backward Euler scheme. A sparse direct solver is employed to solve the system of equations. In the implementation of the FETD algorithm, I effectively mitigate the computational cost of solving the system of equations at every time step by reusing previous factorization results. Since the high frequency contents of the transient electric fields attenuate more rapidly in time, the transient electric fields diffuse increasingly slowly over time. Therefore, the FETD algorithm adaptively doubles a time-step size, speeding up simulations. Comparisons with analytical solutions, 3D finite-difference time-domain (FDTD) solutions and field data demonstrate the accuracy and efficiency of the FETD algorithm.

Although the first FETD algorithm has the minimum number of unknowns, it still requires a large amount of memory because of its use of a direct solver. To mitigate this problem, the second FETD algorithm is derived from a vector-and-scalar potential equation that can be solved with an iterative method. The time derivative of the Lorenz gauge condition is used to split the ungauged vector-and-scalar potential iv equation into a diffusion equation for the vector potential and Poisson’s equation for the scalar potential. The diffusion equation for the time derivative of the magnetic vector potentials is the primary equation that is solved at every time step. Poisson’s equation is considered a secondary equation and is evaluated only at the time steps where the electric fields are output. A major advantage of this formulation is that the system of equations resulting from the diffusion equation not only has the minimum number of unknowns but also can be solved stably with an iterative solver in the static limit. The accuracy and efficiency of the Lorenz-gauge FETD algorithm is verified through comparisons with analytical and 3D FDTD solutions. The detailed comparisons between the two FETD algorithms are presented.

The developed FETD algorithms are used to simulate the TDCSEM method in shallow offshore models that are derived from SEG salt model. In the offshore models, horizontal and vertical electric-dipole-source configurations are investigated and compared with each other. FETD simulation and visualization play important roles in analyzing the EM diffusion of the TDCSEM configurations. The partially-'guided' diffusion of transient electric fields through a thin reservoir is identified on the crosssection of the seabed models. The modeling studies show that the TDCSEM method effectively senses the localized reservoir close to the large-scale salt structure in the shallow offshore environment. Since the reservoir is close to the salt, the non-linear interaction of the electric fields between the reservoir and the salt is observed both on the cross-section and along the seafloor.

Regardless of whether a horizontal or vertical electric-dipole source is used in the shallow offshore models, inline vertical electric fields at intermediate-to-long offsets are approximately an order of magnitude smaller than horizontal counterparts due to the effect of the air-seawater interface. Consequently, the vertical electric-field measurements become vulnerable to the receiver tilt that results from the irregular seafloor topography. The 3D modeling studies also illustrate that the short-offset VED-Ex configuration is very sensitive to a subtle change of the seafloor topography around the VED source. Therefore, the VED-Ex configuration is vulnerable to measurements and modeling errors at short offsets. In contrast, the VED-Ez v configuration is relatively robust to these problems and is considered a practical shortoffset configuration. It is demonstrated that the short-offset configuration can be used to estimate the lateral extent and depth of the reservoir.

Vertical anisotropy (i.e. transverse isotropy) in background also significantly affects the pattern in electric field diffusion by elongating and strengthening the electric field in the horizontal direction. As the degree of vertical anisotropy increases, the vertical resistivity contrast across the reservoir interface decreases. As a result, the week reservoir response is increasingly masked by the elongated and strengthened background response. Consequently, the TDCSEM method loses its sensitivity to the reservoir. The modeling studies show that correct interpretations of TDCSEM measurements require accurately modeling of irregular seafloor topography and seabed anisotropy.

vi Acknowledgements First, I would like to express my sincere gratitude to my advisor, Jerry Harris for his guidance, support, and encouragements during my graduate studies. Through numerous research meetings with him, I learned not only geophysical insights and wisdom but also how to communicate scientific ideas.

My sincere gratitude goes out to my co-advisor, David Alumbaugh at SchlumbergerEMI and UC-Berkeley. He has provided crucial guidance and support at every moment during my graduate studies since 2003. Without his guidance, I could not have turned myself into an electromagnetic geophysicist.

I would like to express my sincere gratitude to my academic committee members, Margot Gerritsen and Rosemary Knight. They were willing to serve as my committee members although my research topics were rare in school of earth sciences. It was always wonderful to have their insightful suggestions and criticisms even during such short annual reviews and qualifying exam. Tapan Mukerji kindly served as a university chair in my defense.

Several other people also deserve my thanks for their help. Nestor Cuevas at EMI provided profound discussions about electromagnetic theories. Jiuping Chen at EMI taught me about how to use and modify his inversion code. Discussions with Seunghee Lee, a former EMI scientist helped me to understand electromagnetic geophysics in many different ways. My thanks also extend to other EMI people for their kindness whenever I visited EMI.

Professor Andreas Hördt at Institute for Geophysics and Extraterrestrial Physics kindly provided field data for me. Mike Commer and Gregory Newman at Lawrence Berkeley National Laboratory allowed me to use their finite-difference codes since

2003. Ryang-Hyun Cho, a computer scientist at Daum answered my numerous questions. Claude Reichard, the director at Technical Writing Program kindly helped me to improve my journal articles. Gboyega Ayeni (Stanford Exploration Project) kindly provided the SEG Salt Model.

vii I express my thanks to Stanford Wave Physics (SWP) people: Yemi Arogunmati, Claudia Baroni, Jolene Robin-McCaskill, Armando Sena, Bouko Vogelaar, Youli Quan, and Tieyuan Zhu. I express my special thanks to Yemi, Jolene, Armando and Bouko for proofreading my dissertation and articles. The collaboration with Professor Haohuan Fu at Tsinghua University was an excellent opportunity to learn his advanced programming practices. I also thank to Tara Ilich, Student Service Manager in Geophysics for her kind help through my PhD studies.

I wish to express my thanks for financial support during my graduate studies. The support includes a Littlefield Fellowship from the School of Earth Sciences (2006), a ConocoPhilips Fellowship (2007), the Mrs. C. J. Belani Fellowship for Computational Geosciences from the Stanford Center for Computational Earth and Environmental Science (2008), and a Chevron Stanford Graduate Fellowship (2008 to 2011).

Last but not the least, I would like to express my love and gratitude to my parents, Chung-In and Kwang-Sook. Their emphasis on education drove me to the completion of my graduate studies. I always thank my brother Eugene for his emotional support and for taking care of my parents during my long absence. I also express my deep gratitude to my parents-in-law for their support and prayer for my wife Bo Young and me. Finally, my gratitude goes to Bo Young. I have always felt happy to share my life with you **.

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Chapter 1. Introduction

1.1 Brief introduction to the marine CSEM method

1.2 FDTD and FETD modeling

1.3 Chapter description

Chapter 2. Marine Controlled-Source Electromagnetic Method

2.1 Electromagnetic diffusion physics of the marine CSEM method

2.2.1 Maxwell’s equations

2.2.2 EM diffusion equations

2.2.3 Electromagnetic detection processes of the marine CSEM method............ 12

2.3 Diffusion of Transient Electric field in Earth

2.4 CSEM acquisition system and survey configuration

Chapter 3. 3D FETD Formulation of the Electric-Field Diffusion Equation.

.............. 19

3.1 Development of FETD formulation

3.2 Discretization in time and space

3.3 Boundary conditions

3.4 Initial condition and FEDC formulation

3.5 Conclusions

Chapter 4. Solutions of the Electric-Field Diffusion-Equation-based FETD Formulation

4.1 Numerical implementation approaches

4.2 Validation and performance analysis

4.2.1 Homogeneous seafloor model

4.2.2 Seafloor model with resistive layer

ix 4.2.3 Land model with 3D gas reservoir

4.2.4 Dipping seafloor model

4.2.5 Interpretation of TDCSEM field data

4.4 Conclusions

Chapter 5. 3D FETD Formulation of the Lorenz-Gauge Vector-Potential Equation.

.. 68

5.1 Theoretical development of the Lorenz-gauge vector potential equation and its characteristics

5.1.1. Lorenz-gauge vector potential equation

5.1.2. Interface conditions under the Lorenz gauge condition

5.1.3. Physical meaning of gauge conditions

5.2 FETD formulation of the vector-potential equation

5.3 FE formulation of the scalar potential equation

5.4 Conclusions

Chapter 6. Solutions of the Lorenz-Gauge FETD Formulation

6.1 Numerical implementation

6.2 Validation and performance analysis

6.2.1 Homogeneous seafloor model

6.2.2 Seafloor model with 3D hydrocarbon reservoir

6.3 Comparison of the two FETD formulations

6.4 Impact of time-step doubling on conditioning of matrix

6.5 Conclusions

Chapter 7. 3D FETD Model Construction for TDCSEM Simulations from a Realistic Seismic Model

7.1 FETD model construction

7.2 FETD mesh calibration

7.3 Conclusions

x Chapter 8. 3D FETD Modeling of the TDCSEM Method in Realistic Shallow Offshore Environments

8.1 Horizontal electric-dipole source configuration

8.1.1 Analysis of the electric fields on cross-sections

8.1.2 Analysis of the electric fields at seafloor receivers

8.2 Vertical electric-dipole source configuration

8.2.1 Analysis of the electric fields on cross-sections

8.2.2 Analysis of the electric fields at seafloor receivers

8.3 Effects of a vertically anisotropic seabed

8.4 Conclusions

Chapter 9. References

9.1. Appendix A: edge- and node-based functions

9.2. Appendix B: assembly of the global matrix equation

9.3. Appendix C: 3D FEFD formulation of the electric-field diffusion equation.. 168

9.4. Appendix D: backward Euler method

9.5. Appendix E: double-curl electric-field diffusion equation

9.6. Appendix F: spectral analysis for determining initial time step size............... 174

9.7. Appendix G: aspect ratio

9.8. Bibliography

xiList of Tables

Table 6.1.

A summary of the direct FETD simulations with various air resistivities.

When a direct FETD simulation fails, the unavailable information is denoted as •. In the seventh column, × and ○ denote accurate and inaccurate solutions, respectively.

Table 6.2.

A summary of the Lorenz-gauge FETD simulations with various air resistivities. Symbols are the same as in Table 6.1.

Table 8.1.

The average slope angle and the direction at each VED source position along the survey line (Figure 7.2a). The left and right directions indicate the negative and positive x-direction along the survey line.

Table 9.1.

Connectivity information based on Figure 9.2. The connectivity information is used to map local descriptions to a global description.

Table 9.2.

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