Copyright. Barry Michael Borgman

Copyright by Barry Michael Borgman 2016

The Thesis Committee for Barry Michael Borgman Certifies that this is the approved version of the following thesis: Characterization of the Cana-Woodford Shale using fractal-based, stochastic inversion Canadian County, Oklahoma APPROVED BY SUPERVISING COMMITTEE: Supervisor: Kyle T. Spikes Mrinal K. Sen Clark R. Wilson

Characterization of the Cana-Woodford Shale using fractal-based, stochastic inversion Canadian County, Oklahoma by Barry Michael Borgman, B.S. Thesis Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of Master of Science in Geological Sciences The University of Texas at Austin May 2016

Dedication I dedicate this thesis to my wife, Angela, whose constant support and encouragement through the past few years has made this possible.

Acknowledgements I would like to thank my advisor, Dr. Kyle Spikes, for everything he has done for me over the past few years. His mentorship and guidance has helped me grow as a scientist, and I am grateful for all of the time spent helping me with both classes and research. I would also like to thank Dr. Mrinal Sen and Dr. Clark Wilson for their guidance throughout my time here, both as members on my thesis committee, and as teachers in the classroom. I would like to extend my gratitude to Cimarex Energy for providing me with the data used in this thesis, as well as the rest of the EDGER Forum for their generous support of our research. There are many fellow students who have had a part in making my time at UT quite memorable. I want to thank the members of our group: David Tang, AJ Yanke, Elliot Dahl, Qi Ren, Han Liu, Russell Carter, and Jennifer Beam for their support, advice, and friendship. I would also like to thank Peter Nelson, Jenny Harding, Kelly Regimbal, and the rest of the geophysics students in the fourth-floor lab. I would like to give a special thanks to Tom Hess, for all of his help with data and software issues, as well as his endless nuggets of wisdom, and to Philip Guerrero, for his guidance throughout the process of graduate school. v

Abstract Characterization of the Cana-Woodford Shale using fractal-based, stochastic inversion Canadian County, Oklahoma Barry Michael Borgman, M.S.Geo.Sci. The University of Texas at Austin, 2016 Supervisor: Kyle T. Spikes The past decade has seen a surge in unconventional hydrocarbon exploration and production, driven by advances in horizontal drilling and hydraulic fracturing. Even with such advances, reliable models of the subsurface are crucial in all phases of exploitation. This study focuses on the methods used for estimation of the elastic properties (density, velocity, and impedance), which play a key role in targeting reservoir zones ideal for hydraulic fracturing. Well-log data provides high-resolution vertical measurements of elastic properties, but a relatively shallow depth of investigation imposes spatial limitations. Seismic data provides broader horizontal coverage at lower cost, but sacrifices vertical resolution. Thin beds present in many unconventional reservoirs fall below seismic resolution. In addition, the band-limited nature of seismic data results in the absence of low-frequency content of the Earth model, as well as the high-frequency content present in well logs. Seismic inversion is a process that provides estimates of elastic properties given input seismic and well data. Stochastic inversion is a method that uses well-log data as a priori information, with an added aspect of randomness. The vi

method generates many realizations using the same input model and takes an average of those realizations. We implement two separate stochastic inversion algorithms to estimate P-impedance in the Cana-Woodford Shale in west-central Oklahoma. First, we use a fractal-based, very fast simulated annealing algorithm that exploits the fractal characteristics found in well-log data to build a prior model. The method of very fast simulated annealing optimizes our elastic model by searching for the minimum misfit between observed and synthetic seismic traces. Next, we use a principal component analysis (PCA) based stochastic inversion algorithm to invert for impedance at all traces simultaneously. Comparison of the results with traditional deterministic inversion results shows improved vertical resolution while honoring the low-frequency content of the Earth model. The PCA-based inversion results also show improved lateral continuity of the elastic profile along our 2D line. The impedance profile from the PCA-based approach provides a better representation of the vertical and horizontal variability of the reservoir, allowing for improved targeting of frackable zones. vii

Table of Contents List of Tables...x List of Figures... xi Chapter 1: Introduction...1 1.1 Background of Unconventional Exploration and Production...1 1.2 Geologic Background of the Woodford Shale...3 1.2.1 Depositional Environment...3 1.2.2 Source Rock Potential...4 1.2.3 Lithofacies...4 1.2.4 Rock Properties...5 1.3 Data...5 1.3.1 Well-log Data...5 1.3.2 Seismic Data...14 1.4 Objectives/Organization...16 Chapter 2: Rock-Physics Modeling in the LMR Domain...17 2.1 Introduction...17 2.2 Methodology...18 2.3 Results...20 2.4 Conclusions...23 Chapter 3: Seismic Inversion...25 3.1 Introduction to inverse theory...25 3.2 Inversion of Acoustic Impedance...26 3.3 Deterministic Methods...29 3.3.1 Steepest Descent...29 3.3.2 Conjugate Gradient Method...29 3.4 Stochastic Approach...30 3.4.1 Simulated Annealing...30 3.4.2 Very Fast Simulated Annealing...32 viii

3.4 Summary...34 Chapter 4: Deterministic Inversion of the Cana-Woodford Data Set...36 4.1 Methodology...36 4.2 Results...40 4.3 Conclusions...45 Chapter 5: Fractal-Based, Very Fast Simulated Annealing...46 5.1 Introduction to Fractal Geometry...46 5.1.1 Fractal Behavior of a Time Series...46 5.1.2 Applications in Geophysics...47 5.1.3 Re-scale Range Analysis/Hurst Coefficient...48 5.2 Methodology...49 5.2.1 Fractal-Based Prior Model...49 5.2.2 Very Fast Simulated Annealing...51 5.3 Results...54 5.4 Discussion...62 Chapter 6: Principal Component Analysis Based Stochastic Inversion...64 6.1 Introduction to Principal Component Analysis...64 6.2 Methodology...64 6.3 Results...69 6.4 Discussion...75 Chapter 7: Conclusions and Future Work...77 7.1 Conclusions...77 7.2 Future Work...79 References...80 Vita...84 ix

List of Tables Table 2.1: Values for the bulk and shear moduli, density, and aspect ratios of each mineral represented in the templates built using the SCM....20 x

List of Figures Figure 1.1: A map of the shale plays in the lower 48 states (courtesy of the American Petroleum Institute)....2 Figure 1.2: The full log suite available at the Brooks 1-14H well location in the central part of our study area. The interval of interest at this location lies between ~12,940ft and ~13,250ft, marked by a decrease in density (RHOB) and corresponding increases in gamma ray (GR), p-wave sonic (DT1), and s-wave sonic (DTSM)....6 Figure 1.3: Map view of the survey area in UTM coordinates. Well locations are scattered in blue, and the geometry of the 2D line is plotted in red...7 Figure 1.4: Plot of well-log data from the Harwick 1-1H location. From left to right, gamma ray (GR), density (RHOB), p-wave velocity (Vp), and computed P-impedance (Zp). The red lines mark the Woodford top (upper) and Hunton top (lower)....9 Figure 1.5: Plot of well-log data from the Brooks 1-14H location. From left to right, gamma ray (GR), density (RHOB), p-wave velocity (Vp), and computed P-impedance (Zp). The red lines mark the Woodford top (upper) and Hunton top (lower)....10 Figure 1.6: Plot of well-log data from the Phillips 1-34H location. From left to right, gamma ray (GR), density (RHOB), p-wave velocity (Vp), and computed P-impedance (Zp). The red lines mark the Woodford top (upper) and Hunton top (lower)....11 xi

Figure 1.7: Plot of well-log data from the Justice Trust 1-19H location. From left to right, gamma ray (GR), density (RHOB), p-wave velocity (Vp), and computed P-impedance (Zp). The red lines mark the Woodford top (upper) and Hunton top (lower)....12 Figure 1.8: Plot of well-log data from the Jackson 1-11H location. From left to right, gamma ray (GR), density (RHOB), p-wave velocity (Vp), and computed P-impedance (Zp). The red lines mark the Woodford top (upper) and Hunton top (lower)....13 Figure 1.9: Plot of well-log data from the Hobson 1-26H location. From left to right, gamma ray (GR), density (RHOB), p-wave velocity (Vp), and computed P-impedance (Zp). The red lines mark the Woodford top (upper) and Hunton top (lower)...14 Figure 1.10: 2D CDP stack showing traces 900-950. The Woodford and Hunton horizons are represented by the upper and lower green lines, respectively. The Woodford is interpreted along a continuous trough, whereas the Hunton horizon is a continous peak....15 Figure 1.11: Plot of the 40Hz, 45 degree phase rotated Ricker wavelet (left) and its corresponding amplitude spectrum (right)...15 Figure 2.1: Diagram of the SCM, with grains and pores inserted into a background matrix. The blue and yellow spheres represent quartz and dolomite, respectively, with aspect ratios of 1. The red ellipsoid represents clay, with an aspect ratio of 0.25. The white ellipsoid represents the pore space, with an aspect ratio of 0.1....19 xii

Figure 2.2: (left) Computed P-impedance log is divided into 8 sub-layers. (center) Computed λρ and µρ logs show µρ values to be higher than corresponding λρ values throughout most of the Woodford, with a crossover around 13,200ft in depth. (right) An LMR cross-plot with data points color coded by sub-layer. Layer 2 and layer 8 (circled) separate from the other layers....22 Figure 2.3: (left) An LMR cross-plot of the Brooks well data, with data points color coded by corresponding sub-layers. The dotted lines represent model templates constructed using the self-consistent method. Lines represent varying quartz and clay content, with the line on the upper left corresponding to 100% quartz, and the bottom right corresponding to 100% clay. Porosity for each line starts at 20% at the bottom left, decreasing to 0% at the upper right. (right) The same data points are cross-plotted and color-coded based on GR count. The highest GR values are found along the trend lines corresponding to 50%-80% quartz content....23 Figure 3.1: An example of linear regression using the L2 norm. Blue circles represent denstiy measurements from the Brooks 1-14H well (above the reservoir). The best-fit line using the L2 norm is plotted in red. In this case, the line represents the depth trend of density measurements...26 Figure 3.2: a) Simplified acoustic impedance profile, b) corresponding reflectivity series, (c) Ricker wavelet with a dominant frequency of 40Hz and phase rotation of 45 degrees, d) Synthetic seismic trace resulting from the convolution (Equation 3.6) of the reflectivity series and wavelet....28 xiii

Figure 3.3: Flow chart of the fractal-based very fast simulated annealing algorithm (Srivistava and Sen, 2009)....33 Figure 3.4: (left) Comparison of inversion results using a deterministic approach (red) versus a stochastic method (blue) and the observed acoustic impedance (black). (right) 25 realizations of the fractal-based VFSA algorithm are plotted, with their mean represented by the thick black line (from Srivistava and Sen, 2009)....34 Figure 4.1: Log correlation of the Brooks 1-14H well. The synthetic trace computed from the convolution of our wavelet and the reflectivity series is plotted in blue. The observed trace at the well location is plotted in red. 5 traces of each are plotted for visualization purposes. The observed seismic data in the neighborhood of the Brooks well is plotted in black, with the Woodford (pink) and Hunton (green) horizons are superimposed...37 Figure 4.2: Cross correlation window for the Brooks log correlation. Maximum correlation coefficient is 0.928....38 Figure 4.3: Initial Zp model for 301 traces. Red lines represent the Woodford (upper) and Hunton (lower) horizons. Blues correspond to low Zp, and yellows correspond to high Zp....39 Figure 4.4: Inversion analysis window. The Zp comparison shows the measured well-log data (blue curve), initial model (black curve), and deterministic results (red curve). The seismic display compares synthetic (red, left) and observed (black) traces, as well as their difference (red, right). 40 xiv

Figure 4.5: Inverted Zp results using the conjugate gradient method for 301 traces. A time window of 2200-2340ms is shown. Red lines represent the Woodford (upper) and Hunton (lower) horizons. Blues represent low Zp, while yellows represent high Zp. Well locations are superimposed in green....41 Figure 4.6: Comparison of synthetic (blue) and observed (red) traces around the Harwick well location. Green lines represent the Woodford (upper) and Hunton (lower) horizons. Amplitudes match well at the Hunton horizon, as well as the layers above and including the Woodford horizon. The layers within the Woodford reservoir zone are resolved well, but show stronger amplitudes in the synthetic traces....43 Figure 4.7: Comparison of synthetic (blue) and observed (red) traces around the Brooks well location. Green lines represent the Woodford (upper) and Hunton (lower) horizons. Amplitudes match well throughout although some discrepancies are present along the Hunton horizon, as well as in the layers above the Woodford (at CDPs 945-950)....44 Figure 4.8: Comparison of inverted results from the conjugate gradient method (red) with measured well-log data (blue) at the Harwick (left) and Brooks (right) locations....45 Figure 5.1: Cross-plot demonstrating the power-law behavior of the Zp data from the Brooks well location. Log10 of power (y-axis) is plotted against the log10(frequency). The blue line represents the L2 norm of the data points. The slope, β, represents the scaling exponent....47 xv

Figure 5.2: Test of the fractal nature of the Brooks 1-14H well. a) Observed acoustic impedance. b) Histogram and pdf of Zp showing fat-tailed Gaussian distribution. c) Covariance showing power law behavior. d) Semivariogram showing power-law behavior. e) Spectral density showing power-law behavior. f) Plot of log10(r/s) versus log10(bin size). The slope of the best-fit line represents the Hurst coefficient....50 Figure 5.3: The amplitude spectrum of the observed impedance log (left), the fractal-based prior model (center), and the observed seismic data (right). The seismic data is band-limited, with missing low-frequency content below 10Hz, and missing high frequencies above 50 Hz. The impedance log retains the low-frequency content, and the fractal-based prior boosts the high-frequency content....51 Figure 5.4: Pseudo-code for the very fast simulated annealing algorithm (from Sen and Stoffa, 1995)....53 Figure 5.5: 25 realizations of fractal-based VFSA at the Harwick (left) and Brooks (right) locations. Green lines represent the mean of the realizations.55 Figure 5.6: Comparison of synthetic (blue) and observed (red) traces at the Harwick location, as well as comparison of Zp values from the measured log (blue), conjugate gradient method (red), and VFSA (green). Inverted Zp results from the VFSA algorithm show improved resolution relative to the conjugate gradient results. A noticeable mismatch is present at the base of the Woodford, where the Zp curve from VFSA shows a more gradual increase, relative to the abrupt increase seen in the well log data....56 xvi

Figure 5.7: Comparison of synthetic (blue) and observed (red) traces at the Brooks location, as well as comparison of Zp values from the measured log (blue), conjugate gradient method (red), and VFSA (green). Inverted Zp results from the VFSA algorithm show considerable improvement in resolution relative to the conjugate gradient method. This is particularly evident at ~2310ms, where the sharp Zp increase in the layer above the base of the Woodford is better resolved with VFSA. The sharp increase in Zp at the Hunton horizon is better captured in the inversion results at the Brooks well location, although the magnitude of the increase is still off by ~3(km/s)*(g/cc)....57 Figure 5.8: Plots of the objective functions at the Harwick (left) and Brooks (right) locations for 5000 iterations. The objective function represents the misfit between observed and synthetic traces....58 Figure 5.9: The inverted Zp profile using fractal-based VFSA for traces 700-1000. Red lines represent the Woodford (upper) and Hunton (lower) horizons. Blues represent low Zp values, while yellows represent high values. Well locations are superimposed in green. Lateral variability is evident in the coming and going of the distinct low-impedance layers starting at the Brooks location where 3 such layers are present, and moving away in either direction where only 1 or 2 low-impedance layers are visible....59 xvii

Figure 5.10: Comparison of 21 synthetic (blue) and observed (red) traces around the Harwick location. Green lines represent the Woodford (upper) and Hunton (lower) horizons. Similar to the conjugate gradient results, the internal layers show stronger amplitude in the synthetic traces. Amplitudes match well at both the Woodford and Hunton horizons, as well as the layers above the Woodford horizon....60 Figure 5.11: Comparison of 21 synthetic (blue) and observed (red) traces around the Brooks location. Green lines represent the Woodford (upper) and Hunton (lower) horizons. Amplitudes match well for most layers, with the notable exception of the Hunton horizon. The doublet behavior seen in the observed traces is absent in the synthetic traces. Instead, large single amplitudes are present. The peak at ~2290ms shows less lateral continuity in the synthetic traces...61 Figure 5.12: Comparison of 21 synthetic (blue) and observed (red) traces between the Harwick and Brooks locations. Green lines represent the Woodford (upper) and Hunton (lower) horizons. The amplitudes at the Woodford and Hunton horizons match well. Some mismatches are noticeable in the upper most peak....62 Figure 6.1: Six example training images selected at random from the 1000 training images used....66 Figure 6.2: Six examples of the transformed variables, or principal components of the training images. Each plot is color-coded based on the principal component scores. The majority of the variability is captured in the first few PCs....68 xviii

Figure 6.3: Plot of the proportion of the variance explained by each corresponding eigenvector. The first few eigenvectors account for ~80% of the variance....69 Figure 6.4: Comparison of synthetic (blue) and observed (red) traces at the Harwick location, as well as comparison of Zp values from the measured log (blue), conjugate gradient method (red), VFSA (green), and PCA (magenta)....70 Figure 6.5: Comparison of synthetic (blue) and observed (red) traces at the Brooks location, as well as comparison of Zp values from the measured log (blue), conjugate gradient method (red), VFSA (green), and PCA (magenta)....71 Figure 6.6: The inverted Zp profile using PCA-based stochastic inversion for traces 700-1000. Red lines represent the Woodford (upper) and Hunton (lower) horizons. Blues represent low Zp values, while yellows represent high values. Well locations are superimposed in green....72 Figure 6.7: Comparison of 21 synthetic (blue) and observed (red) traces around the Brooks location. Green lines represent the Woodford (upper) and Hunton (lower) horizons. The internal layers at ~2270ms and ~2290ms are resolved well. Discrepancies are present in the amplitudes at the Hunton horizon (doublets in the observed traces, single, large amplitudes in the synthetic traces), as well as the layers above the Woodford horizon....73 xix

Figure 6.8: Comparison of 21 synthetic (blue) and observed (red) traces around the Harwick location. Green lines represent the Woodford (upper) and Hunton (lower) horizons. Amplitudes between the horizons are larger in the synthetic traces than the observed traces, while amplitudes at the horizons match well. The layers above the Woodford horizon show some amplitude mismatch, particularly at higher CDPs....74 Figure 6.9: Comparison of 21 synthetic (blue) and observed (red) traces between the Harwick and Brooks locations. Green lines represent the Woodford (upper) and Hunton (lower) horizons. Amplitudes are larger for the observed traces at all layers, except for the layers at ~2260ms and ~2215ms....75 xx

Chapter 1: Introduction 1.1 BACKGROUND OF UNCONVENTIONAL EXPLORATION AND PRODUCTION The existence of vast unconventional hydrocarbon reserves in the United States is not a new discovery, but with the emergence of new technology in hydraulic fracturing and directional drilling, such reserves have garnered much attention in recent years. Shale plays termed unconventional differ from the conventional oil and gas plays in several respects. A traditional hydrocarbon system has several key components. The source rock contains organic material, which, at the right pressure and temperature, reaches the appropriate maturation window to become oil or gas. Hydrocarbons then migrate upward due to buoyancy, ultimately charging the reservoir rock. A low permeability cap rock above the reservoir restricts further upward flow. Finally, a trap must be present (e.g., anticlinal or fault traps) in order for the hydrocarbons to accumulate in such a way as to be economically exploitable. Unconventional shale plays are a system in which the shale formation acts as source rock, reservoir, and cap rock. Additionally, shale reservoirs often have considerable lateral extent in comparison with conventional reservoirs. The shale boom that began in the early 2000 s saw a dramatic increase in U.S. domestic exploration and production due to wide application of horizontal drilling (Zou et al., 2013). Figure 1.1 maps the shale plays in North America as of 2015. The Marcellus, Bakken, and Eagle Ford shales have gained considerable attention as prolific producers over the past decade, but the available reserves stretch across the southeast, the west, and north to Alaska. 1

Figure 1.1: A map of the shale plays in the lower 48 states (courtesy of the American Petroleum Institute). Several factors dictate the viability of unconventional resources, such as market prices of oil and gas, and the availability of drilling and completions technologies and methods that facilitate efficient extraction of hydrocarbons. Reservoir conditions often complicate the exploitation of unconventional plays. Low permeability often seen in shale reservoirs restricts the flow of hydrocarbons due to pore-throat sizes on the scale of nanometers. The advances in hydraulic fracturing and horizontal (directional) drilling have helped to overcome permeability issues in reservoir rock (Ross and Bustin, 2008). Horizontal drilling allows for greater access along the lateral extent of the reservoir from a single well, decreasing the distance of flow necessary to reach the borehole. Hydraulic fracturing increases permeability by opening pre-existing fractures, as well as creating new ones. Operators open fractures by injecting fluid and sand into the well at pressures 2

as high as 20,000psi (Montgomery and Smith, 2010). The ability to re-fracture the same reservoir many times plays an important role in prolonging the life of the well and increasing viability (Zou et al., 2013). Zou et al. (2013) cite three essential characteristics of a shale gas system: total organic carbon (TOC), maturity, and brittle mineral content. Source rock maturity determines whether the desired conditions are present. Brittle mineral content is shown to dictate the feasibility of hydraulic fracturing in a reservoir, though significant ambiguity exists regarding the definition of brittleness. Typical gas shales in the U.S. have quartz and carbonate content ranging from 28-52% and 4-16%, respectively (Zou et al., 2013). The exploitation of unconventional reserves requires a joint effort between engineers and geoscientists. Despite the lateral continuity often seen in shale reservoirs, they exhibit significant heterogeneity. A single shale unit can contain many interbedded layers with varying porosity, permeability, and elastic properties. Understanding and accurately predicting these properties improves exploration and production, and helps to identify areas ideal for hydraulic fracturing (Jiang and Spikes, 2013). An accurate model will implement information about the geological, petrophysical, and geophysical nature of the reservoir (Varga et al., 2012). Zou et al. (2013) use several examples of unconventional reservoirs to conclude that no one model can be used in rock mineral composition of shales. 1.2 GEOLOGIC BACKGROUND OF THE WOODFORD SHALE 1.2.1 Depositional Environment The late Devonian saw widespread deposition of organic rich shales throughout present-day North America (Lambert, 1993). During this time, the North American craton was located in the tropics below the paleoequator (Heckel and Witzke, 1979; 3

Ettensohn and Elam, 1985; Roberts and Mitterer, 1992; Callner, 2012). This period marked a transition from worldwide accumulation of carbonates to widespread clastic wedge encroachment due to orogenic events, and subsequent deposition of black shales on the carbonate banks (Callner, 2012). The Woodford Shale was deposited in an epicontinental sea along the southern margin of the North American Craton (Roberts and Mitterer, 1992; Callner 2012). Callner (2012) cites a range of previous estimates of water depth during the time of deposition, from 150ft to over 1,000ft, though current estimates show 1,000ft to be too deep. 1.2.2 Source Rock Potential The Chattanooga (Woodford) Shale refers to the organic rich late Devonian shales that span the midcontinent of North America (Lambert, 1993). In Nebraska, Kansas, and eastern Oklahoma it is referred to as the Chattanooga Shale (Carlson, 1963; Goebel, 1968; Amsden, 1980). The name Woodford Shale is used in the Anadarko basin in western Oklahoma, and the Permian basin in western Texas and New Mexico (Ellison, 1950; Amsden, 1975). The Bakken in North and South Dakota s Williston basin (Meissner, 1978) and shales in the Illinois basin (Conant and Swanson, 1961) are also equivalent forms. Known to be a major source rock in the Permian basin (Wright, 1963), as well as the Anadarko, Williston, and Illinois basins (Fertl and Chilingarian, 1990; Williams, 1974; Dow, 1974; Bethke et al., 1991), the Chattanooga (Woodford) shale and its equivalent forms are estimated to account for the generation of nearly 8% of the world s original petroleum reserves (Fritz, 1991). 1.2.3 Lithofacies Caldwell (2013) divided the Woodford Shale in the Anadarko Basin into seven lithofacies that make up the basal, lower, middle, and upper Woodford. The basis for 4

defining these lithofacies is the varying proportions of silica, clay, dolomite, and total organic carbon (TOC). The basal unit is composed primarily of organic-poor, clayey mudrock (OPCM) with 41-52% clay content. Three main lithologies dominate the lower and middle Woodford: clayey mudrock (CM) (38-41% clay), clayey siliceous mudrock (CSM) (27% clay and 55% quartz), and dolomitic clayey mudrock (DCM) (33% clay, 32% quartz, and 15% dolomite). The upper Woodford consists primarily of CSM and siliceous mudrock (14% clay and 75% quartz) (Caldwell, 2013). 1.2.4 Rock Properties Understanding the rock properties of a reservoir is crucial to the design of all phases of hydrocarbon exploitation. Completions engineers look to geophysicists to provide specific zones of the reservoir that are ideal for hydraulic fracturing. Many look for the brittleness index (a term not without controversy, as many definitions of brittleness exist) to predict the ease with which both preexisting and induced fracture networks can be opened, allowing for the flow of oil and gas. Caldwell (2013) notes the presence of silica as a key determinant in the fracability of reservoir rock higher silica content corresponds to lower frac pressures and more proppant placed during completions. 1.3 DATA 1.3.1 Well-log Data The data set used in this thesis consists of well-log measurements from seven wells in the Cana-Woodford field, as well as a 2D seismic line that intersects six of those wells. Each well contains eight curves: simple gamma ray (2), density (2), p-wave sonic, fast and slow s-wave sonic, and depth (Figure 1.2). Figure 1.3 shows the location of each well, as well as the geometry of the 2D seismic survey line (the Brown 1-19H well 5

location lies away from the 2D line; therefore, we exclude it from this study). Noticeable variations exist among different well locations. We will discuss the data from each well in further detail. Figure 1.2: The full log suite available at the Brooks 1-14H well location in the central part of our study area. The interval of interest at this location lies between ~12,940ft and ~13,250ft, marked by a decrease in density (RHOB) and corresponding increases in gamma ray (GR), p-wave sonic (DT1), and s- wave sonic (DTSM). 6

Figure 1.3: Map view of the survey area in UTM coordinates. Well locations are scattered in blue, and the geometry of the 2D line is plotted in red. The Harwick 1-1H well (Figure 1.4) is the first well encountered along our 2D line. The top of the Woodford Shale is picked at ~13,000ft (marked by the upper red line), corresponding to a sharp increase in gamma ray values and sharp decreases in density (RHOB), P-wave velocity (Vp), and P-impedance (Zp). The Woodford-Hunton interface is marked by a return to high Zp, Vp, and RHOB values and low gamma ray values at ~13,270ft. These characteristics of the top and base of the Woodford persist throughout the data set. Data from the Harwick well also show the Woodford to possess significant internal variability. The high gamma ray values at the top give way to a thin layer (~5ft) of relatively low values near 13,020ft. This corresponds to a small spike in Vp and Zp. This is followed by an interval approximately 115ft thick characterized by a 7

series of thin layers with high gamma ray values, but relatively uniform Vp and Zp. The next interval of interest spans approximately 60ft, from ~13,140ft to ~13,200ft. The gamma ray values are considerably lower than the previous interval, though at 200-250API, they are still considered high. It is important to note that the simple gamma ray log does not differentiate among potassium, uraniam, and thorium. Attributing high gamma ray values to clay content would almost certainly prove to be misguided in this data set. A layer measuring approximately 40ft thick with gamma ray values above 300API gives way to the basal unit, which is characterized by decreasing gamma ray and increasing Vp, Zp, and density values. 8

Figure 1.4: Plot of well-log data from the Harwick 1-1H location. From left to right, gamma ray (GR), density (RHOB), p-wave velocity (Vp), and computed P- impedance (Zp). The red lines mark the Woodford top (upper) and Hunton top (lower). The internal layers present in the Woodford at the Brooks 1-14H well location (Figure 1.5) differ from those seen at the Harwick 1-1H well in several important respects. The high-velocity layer (relative to the reservoir) near the top of the reservoir (just below the initial drop in Vp) at ~12,965ft is more pronounced in the Brooks well than the corresponding layer in the Harwick well. The high-density layer from 13,200-13,220ft present in the Brooks well is missing in the Harwick well. Overall, the higherfrequency internal variability in the reservoir is more apparent at the Brooks location. The Phillips 1-34H well location (Figure 1.6) shows similar internal layers as the Brooks 9

location, with the same pronounced high-velocity layer and high-density layer near the top and base of the reservoir, respectively. Figure 1.5: Plot of well-log data from the Brooks 1-14H location. From left to right, gamma ray (GR), density (RHOB), p-wave velocity (Vp), and computed P- impedance (Zp). The red lines mark the Woodford top (upper) and Hunton top (lower). 10

Figure 1.6: Plot of well-log data from the Phillips 1-34H location. From left to right, gamma ray (GR), density (RHOB), p-wave velocity (Vp), and computed P- impedance (Zp). The red lines mark the Woodford top (upper) and Hunton top (lower). The Justice-Trust 1-19H, Jackson 1-11H, and Hobson 1-26H well locations (Figure 1.7, 1.8, and 1.9, respectively) possess the same high-velocity layer near the top of the reservoir. However, the high-density layer near the base of the reservoir is missing at these locations, and the variability in the middle of the reservoir is less pronounced. 11

Figure 1.7: Plot of well-log data from the Justice Trust 1-19H location. From left to right, gamma ray (GR), density (RHOB), p-wave velocity (Vp), and computed P-impedance (Zp). The red lines mark the Woodford top (upper) and Hunton top (lower). 12

Figure 1.8: Plot of well-log data from the Jackson 1-11H location. From left to right, gamma ray (GR), density (RHOB), p-wave velocity (Vp), and computed P- impedance (Zp). The red lines mark the Woodford top (upper) and Hunton top (lower). 13

Figure 1.9: Plot of well-log data from the Hobson 1-26H location. From left to right, gamma ray (GR), density (RHOB), p-wave velocity (Vp), and computed P- impedance (Zp). The red lines mark the Woodford top (upper) and Hunton top (lower). 1.3.2 Seismic Data The 2D seismic data used in our algorithm is a CDP stack (Figure 6) consisting of 2,669 traces. Recording time was 0 6000 ms, at a sample rate of 2 ms. We made several attempts to extract both a statistical wavelet and a wavelet using the Brooks well, but log correlation results were less than satisfactory. After trial and error in the log-correlation process, a 200-ms Ricker wavelet with a dominant frequency of 40 Hz and phase rotation of 45 o (Figure 7) is input into our algorithm. 14

Figure 1.10: 2D CDP stack showing traces 900-950. The Woodford and Hunton horizons are represented by the upper and lower green lines, respectively. The Woodford is interpreted along a continuous trough, whereas the Hunton horizon is a continous peak. Figure 1.11: Plot of the 40Hz, 45 degree phase rotated Ricker wavelet (left) and its corresponding amplitude spectrum (right). 15

1.4 OBJECTIVES/ORGANIZATION The objective of this thesis is to characterize the Cana-Woodford Shale by generating estimates of the acoustic impedance profile along a 2D seismic survey line. This is achieved through several stages. First, we construct rock-physics model templates using a self-consistent method (SCM) to examine the expected behavior of different lithologies and pore fluids in the lambda-rho mu-rho (LMR) domain. Next, we examine P-impedance estimates generated using a deterministic approach. We then compare results from the deterministic approach to results from two separate stochastic inversion algorithms. In the first method, we implement a fractal-based, very fast simulated annealing (VFSA) algorithm in an effort to recover low- and high-frequency content absent from the observed seismic data. In the second method, we combine the fractalbased VFSA approach with principal component analysis (PCA) in an effort to improve the lateral continuity of our model estimates between well locations. We then discuss the merits of each approach, as well as possible future work. 16

Chapter 2: Rock-Physics Modeling in the LMR Domain 2.1 INTRODUCTION The application of rock physics models can often serve as a means to overcome the spatial limitations imposed by well log data. Avseth et al. (2010) discuss a variety of rock physics models, including theoretical bounds, inclusion models, and contact theory models, among others. They note the ability of these models to aid in extrapolation of rock properties away from well locations and the positive impact this has on exploration and appraisal. A method of analyzing the relationship between the Lamé parameters, λ and µ, is discussed in great detail by Goodway et al. (2010), Perez et al. (2011), and Close et al. (2012). Goodway et al. (2010) discuss the importance of λ (or incompressibility) and µ (rigidity) in determining optimal zones for hydraulic fracturing. They provide examples of data from the Barnett Shale in Texas, as well as shales and carbonates from Western Canada. Empirical observation of the data shows λ to be the most important factor of minimum closure stress a measure of the minimum amount of pressure needed to open existing or induced fractures (Goodway et al., 2010). Perez et al. (2011) present a method for the integration of rock physics models with cross-plots of λρ and µρ (LMR, where ρ is bulk density). They utilize both the Hertz-Mindlin contact theory and Hashin- Shtrikman bounds to build templates for expected LMR trends of various rock compositions at different porosity values. Close et al. (2012) discuss the applications of such templates in determining sweet spots (zones that are highly susceptible to hydraulic fracturing). Their analysis of data from the Horn River Basin in British Columbia, Canada, shows low λρ values paired with high µρ values as a good indicator of reservoir quality. 17

2.2 METHODOLOGY The model chosen for this study is the self-consistent method described by Berryman (1980), in which he outlines a means of estimating elastic constants in heterogeneous media using elastic-wave scattering theory. This method seeks to simplify the equations used to compute the elastic moduli by addressing the displacement field, rather than the stress and strain tensors. We insert grains and pores into a background matrix whose parameters we are able to vary. Ignoring multiple scattering, we attempt to match impedances between the inclusions and the background matrix by adjusting the parameters of the matrix, at which point the plane wave is no longer affected by the inclusions (Berryman, 1980). It will likely serve as a good starting point for future work on the Woodford Shale, using a variety of models. Estimating the effective elastic constants requires the input of both the measured bulk and shear moduli of each inclusion, as well as their aspect ratios and a range of porosities. Figure 2.1 gives a schematic of the medium containing inclusions of quartz, clay, dolomite, and porosity. For each well location, the self-consistent method is used to model elastic moduli for six different mineral compositions, each with porosities ranging from 0 to 20%. The compositions contain quartz percentages of 100,75,50,25 and 0%, with clay volume ranging from 0-100%. A sixth model is constructed using 100% dolomite to serve as a reference for the expected trend of carbonates. We fix the pore aspect ratios at 0.1, and set brine saturation and gas saturation at 20% and 80%, respectively. Once the bulk and shear moduli are computed for each case, λ is calculated using the relationship: l = K - 2 3 m (2.1) Next, we model densities for each composition, and calculate values for both λρ and µρ along the range of porosities: 18

lr = I 2 2 P - 2I S (2.2) 2 mr = I S (2.3) This allows us to build a template in which lines representing the trend of λρ and µρ values are super-imposed on the values derived from measured well log data. Table 2.1 shows the values used for the elastic moduli and densities of each mineral in the templates. Figure 2.1: Diagram of the SCM, with grains and pores inserted into a background matrix. The blue and yellow spheres represent quartz and dolomite, respectively, with aspect ratios of 1. The red ellipsoid represents clay, with an aspect ratio of 0.25. The white ellipsoid represents the pore space, with an aspect ratio of 0.1. 19

Constituent Bulk Modulus (GPa) Shear Modulus (GPa) Density (g/cc) Aspect Ratio Quartz 36.6 45 2.65 1 Clay 21 7 2.9 0.25 Dolomite 95 45 2.87 1 Brine 2.4 0 1.05 0.1 Gas 0.0702 0 0.1543 0.1 Table 2.1: Values for the bulk and shear moduli, density, and aspect ratios of each mineral represented in the templates built using the SCM. 2.3 RESULTS Figure 2.2a shows the computed P-impedance curve for the reservoir interval at the Brooks 1-14H well location. Figure 2.2b shows the corresponding λρ and µρ curves for the same interval. We divide the reservoir into 8 sublayers in order to visualize the behavior of different zones in the LMR domain (Figure 2.2c). Two layers stand out in both Figures 2.2b and 2.2c. Layer 2 (light blue) corresponds to the largest separation of λρ and µρ, with µρ values as high as 47GPa*g/cc, and λρ values between 10 and 20GPa*g/cc. Points from this layer lie along the y-axis in the LMR domain. In contrast, layer 8 (red) shows the only zone of λρ and µρ crossover in the reservoir. These points lie closer to the x-axis in the LMR domain. Remaining layers show λρ values lower than µρ values, and fall along a linear trend up the y-axis in the LMR domain. 20

We build model templates for varying proportions of quartz and clay and varying porosity using the SCM. Figure 2.3a shows the data points superimposed on the model templates in the LMR domain. The dotted line on the far left of the template represents 100% quartz, while the dotted line on the bottom right represents 100% clay; lines in between represent smoothly varying compositions of quarts and clay. Porosity decreases from the origin toward the upper right of each line. Porosity ranges from 20% (bottom left) to 0% (top right), though not all porosities are visible for each line. Figure 2.3b plots the templates and data points, but color-codes data points based on their corresponding GR values. We scale the color bar from 75-200 for visualization purposes, but GR values exceed 800 API in some places. The lowest GR values correspond to layer 2, while layer 8 shows GR values between ~130API and ~200API. The highest GR values correspond to the remaining layers. 21

Figure 2.2: (a) Computed P-impedance log is divided into 8 sub-layers. (b) Computed λρ and µρ logs show µρ values to be higher than corresponding λρ values throughout most of the Woodford, with a crossover around 13,200ft in depth. (c) An LMR cross-plot with data points color coded by sub-layer. Layer 2 and layer 8 (circled) separate from the other layers. 22

Figure 2.3: (a) An LMR cross-plot of the Brooks well data, with data points color coded by corresponding sub-layers. The dotted lines represent model templates constructed using the self-consistent method. Lines represent varying quartz and clay content, with the line on the upper left corresponding to 100% quartz, and the bottom right corresponding to 100% clay. Porosity for each line starts at 20% at the bottom left, decreasing to 0% at the upper right. (b) The same data points are cross-plotted and color-coded based on GR count. The highest GR values are found along the trend lines corresponding to 50%-80% quartz content. 2.4 CONCLUSIONS The Cana-Woodford Shale contains significant heterogeneity throughout its 300ft interval at the Brooks well location. Layer 2 shows high µρ values and low λρ values compared to the rest of the reservoir. This suggests that this is the most brittle zone (favorable to hydraulic fracturing). Layer 8 not only shows λρ and µρ values that indicate a more ductile material, but the model template suggests low porosity (<10%) as well. The remaining points follow a trend of higher µρ and lower λρ values. The model template suggests a relatively high quartz content (~75%) and porosity values greater than 10%. Figure 2.3b illustrates the potential danger of interpreting simple GR measurements as a sure-fire indicator of clay content. The highest GR values fall along model lines corresponding to relatively high quartz content, which indicates there may be 23

a different source of radiation within these zones. The exceptionally high values in layers 1 and 3-7 could be due, in part, to kerogen-hosted uranium. Additional logs, such as spectral gamma ray or elemental capture spectroscopy (ECS) logs are needed for accurate assessment of the mineralogy. Comparison of the results with empirical observations by Goodway et al. (2010) indicates that layer 2 lies in the most brittle region of the LMR domain, while layer 8 is in the most ductile region. 24

Chapter 3: Seismic Inversion 3.1 INTRODUCTION TO INVERSE THEORY The estimation of physical parameters from a set of observations is a problem that arises in many fields of science and engineering (Aster et al., 2012). The set of observations, d, results from some operator, G, acting on the set of model parameters, m: d = Gm (3.1) Equation 3.1 represents the forward problem. Assuming the underlying physics are well understood, we are able to compute the expected observations, d syn, from a given model. The inverse problem seeks to estimate the model parameters from the set observations by starting with a model estimate, m 0, and perturbing the model until a minimum misfit is found between the observed and predicted data: E = d obs d syn (3.2) E = d obs Gm (3.3) There are many methods in practice to minimize the misfit between observed and predicted data (Equations 3.2,3.3). We provide the example of the linear regression as an introduction to parameter estimation. The linear regression computes a best-fit line to a set of observations by inverting for m in Equation 3.1. We multiply both sides of Equation 3.1 by G T in order to make the forward operator invertible, then solve for m: G T Gm = G T d (3.4) m = (G T G) 1 G T d (3.5) This method also called the L2 norm (Figure 3.1) largely ignores outliers in the observed data. Higher-order norms give more weight to outliers. For further reading on the method of linear regressions, we refer the reader to Späth (1991). 25

Figure 3.1: An example of linear regression using the L2 norm. Blue circles represent denstiy measurements from the Brooks 1-14H well (above the reservoir). The best-fit line using the L2 norm is plotted in red. In this case, the line represents the depth trend of density measurements. 3.2 INVERSION OF ACOUSTIC IMPEDANCE Data acquired from borehole tools both during and after drilling operations provide measurements of the Earth s subsurface. Sonic logs measure p- and s-wave slowness at a typical sampling interval of 0.5ft measured depth (MD). Density measurements sampled at the same interval allow for the computation of impedance curves, which we use to infer many important rock properties, such as porosity, fluid saturation, and fluid type. The depth of investigation away from the borehole, however, limits horizontal resolution. Measurements are reliable only in the immediate vicinity of well locations. Lateral heterogeneities in reservoir rock necessitate broader coverage, but high drilling costs reduce the feasibility of achieving such coverage through well logs alone. 26

Seismic data offers the widespread coverage not found in well-log data, but sacrifices the high vertical resolution of sonic logs. The typical layer thicknesses that are resolved by surface-seismic data are in the tens of meters. The relatively thin vertical heterogeneities often found in unconventional reservoirs lie below seismic resolution, but play an important role in net pay calculation, fluid flow, and completions procedures. In addition, seismic data provides a measure of interface properties of the subsurface a contrast between overlying and underlying layers, rather than properties of the layers themselves. The goal of the exploration geophysicist is to provide intelligent estimates of interval properties from interface measurements, often with the aid of some sparse well data. Seismic inversion is a process that seeks to model the interior of the Earth using surface seismic and well data as input (Russell, 1988). Seismic reflection data, s, is the result of a convolution of a source wavelet, w (i.e. from a vibroseis or explosive source), with the Earth s reflectivity series, r: s(t) = w(t) * r(t) (3.6) In the case of well-log data, we have the reflectivity series and are able to compute the expected seismic response using Equation 3.6 (Figure 3.2). 27

Figure 3.2: a) Simplified acoustic impedance profile, b) corresponding reflectivity series, (c) Ricker wavelet with a dominant frequency of 40Hz and phase rotation of 45 degrees, d) Synthetic seismic trace resulting from the convolution (Equation 3.6) of the reflectivity series and wavelet. Inversion allows us to estimate the elastic properties of the subsurface using measured seismic data coupled with some a priori information (often well-log data). An objective function represented by the misfit between observed and synthetic traces is minimized through an iterative process of model perturbation, forward modeling of the synthetic seismic response, and either acceptance or rejection of the model update. Many commercial software packages currently offer some form of inversion algorithm (e.g., conjugate gradient method), but often a mismatch exists in the frequency content between the inverted volume and the earth model, due to the bandwidth limitations of the seismic data. The low-frequency Earth model is lost, as seismic amplitudes oscillate around the value zero and fail to record the overall trend of increasing density and impedance with 28

depth. The high-frequency content seen in well-log data, as previously discussed, is also lost. Many approaches have been developed to broaden the frequency content of inverted models, while attempting to minimize computational cost (Russell, 1988; Ingber, 1989; Srivistava and Sen, 2009; Sen and Stoffa, 2013; Xue, 2013). Local optimization methods search for the minimum of the objective function by searching only in the downhill direction (model perturbations that result in a lower objective function), while global optimization methods allow for both uphill (increasing objective function) and downhill moves. We will discuss several of these approaches in the following section. 3.3 DETERMINISTIC METHODS 3.3.1 Steepest Descent Shewchuk (1994) provides a thorough overview of gradient methods such as Steepest Descent and Conjugate gradient. These methods seek to solve systems of linear equations of the form Ax = b, where A is a known nxn matrix, b is a known nx1 vector, and x is an unknown nx1 vector. The Steepest Descent method starts at some initial model, x o, and searches for the solution, x, by moving along the opposite direction of maximum slope, f. This is called the residual, r, given by: r i = b Ax i (3.7) Each move toward the minimum value is given by the equation: where α represents the step size: 3.3.2 Conjugate Gradient Method x i+1 = x i + α i r i (3.8) α i = r i T r i r i T Ar i (3.9) Unlike the steepest descent method, which uses search directions in the same direction as the residuals, the conjugate gradient method computes search directions that 29

are A-orthogonal (or conjugate) to the previous residuals and search directions. A- orthogonality of two vectors is defined by: d i T Ad j = 0 (3.10) The method begins by computing the residual and search direction, which are the same for the first iteration: d 0 = r 0 = b Ax 0 (3.11) The step length, α, is now computed using both the residual and search direction: The new model is computed by: Each subsequent iteration uses the following steps: α i = r i T r i d i T Ad i (3.12) x i+1 = x i + α i d i (3.13) r i+1 = r i α i Ad i (3.14) T ri+1 β i+1 = r i+1 (3.15) r T i r i d i+1 = r i+1 + β i+1 d i (3.16) (Algorithm from Shewchuk, 1994) 3.4 STOCHASTIC APPROACH 3.4.1 Simulated Annealing Commercial software packages often implement a local optimization method, (e.g., the conjugate gradient method) to invert for the elastic properties of reservoir rock. Methods such as steepest descent and conjugate gradient, however, have their limitations. They seek to find the minimum of the objective function by moving downhill. This type of method can be useful when the starting model is very close to the global minimum. In certain situations, the limitation of only accepting downhill moves can lead to the 30

optimization algorithm becoming trapped in a local minimum. Global optimization methods seek to avoid this pitfall by allowing for the possibility of an uphill move. The global optimization method of simulated annealing comes from the physical process of annealing, in which a solid is heated to the point that it transitions to the liquid state (Ingber, 1993, Sen and Stoffa, 1995). At this point it is allowed to cool toward the minimum energy state (crystallization), achieving thermal equilibrium at each temperature along the way. The Boltzmann PDF (Equation 3.17) gives the probability of being in state i with energy Ei: P(Ei) = exp( E i KT ) Σexp( E j KT ), (3.17) where K represents the Boltzmann Constant, and T is the temperature. The gradual reduction of T increases the probability of reaching the minimum energy state as T 0. Sen and Stoffa (2009) stress the importance of requiring equilibrium. Cooling too quickly (quenching) can freeze the material (our objective function) at a local minimum. By implementing a slow cooling schedule (annealing), the material will freeze at or near the global minimum. Several algorithms utilize the concept of simulated annealing. The Metropolis algorithm (Metropolis et al., 1953, Kirkpatrick et al., 1983) generates a model update through small perturbations of the model parameters. Acceptance of the updated model depends on meeting one of two criteria. If the energy, or objective function E(mnew) of the updated model is less than that of the previous model, then the new model is accepted. Unlike greedy algorithms, however, an uphill move, or increase in the objective function, does not automatically disqualify the new model. A probability (that decreases with temperature) exists that a model in which E(mnew) is greater than E(mold) 31

can still be accepted. This probability, known as the Metropolis criterion, overcomes the trapping in local minima. 3.4.2 Very Fast Simulated Annealing The method of very fast simulated annealing (VFSA) used by Srivistava and Sen (2009) implements changes to the simulated annealing algorithm proposed by Ingber (1989, 1993) to overcome some of the more cumbersome calculations and improve computational efficiency (Sen and Stoffa, 1995). Imposing constraints on each model parameter through a minimum and maximum value improves the method of model perturbation. A new probability distribution (Equation 3.18; cumulative probability given by Equation 3.19) is used to draw the parameter yi for the generation of random moves: gt(y) = 1 2( y i +T)ln(1+ 1 T i ) (3.18) GTi(yi) = 1 + sign(y ln(1+ y i ) i) T i 2 2 ln(1+ 1 ) T i (3.19) where Ti is the temperature at iteration i. The use of the above distribution, as well as an improved cooling schedule (which will be discussed later) are two of the main features of the VFSA algorithm. In addition, the acceptance criteria and model perturbation need separate temperatures. Srivistava and Sen (2009) obtain improved results from the use of the fractalbased VFSA algorithm (Figure 3.3) compared to results from a traditional deterministic inversion. The high-resolution inversion results show similar frequencies to the measured log impedances, with most of the realistic peaks from the well data shown in the estimates. Multiple realizations of the estimated impedance log show a standard deviation 32

below 20%. Figure 3.4a shows a comparison between the observed data, results from deterministic inversion, and results from stochastic inversion. Compared with the deterministic approach, results from the stochastic inversion capture the major peaks and troughs seen in the well data. The frequency content of the inverted impedance log is also similar to that of the measured data. Figure 3.4b shows 25 realizations of the estimated impedance log using VFSA. Figure 3.3: Flow chart of the fractal-based very fast simulated annealing algorithm (Srivistava and Sen, 2009). 33

Figure 3.4: (left) Comparison of inversion results using a deterministic approach (red) versus a stochastic method (blue) and the observed acoustic impedance (black). (right) 25 realizations of the fractal-based VFSA algorithm are plotted, with their mean represented by the thick black line (from Srivistava and Sen, 2009). 3.4 SUMMARY Deterministic approaches such as the conjugate gradient method have proven very effective in minimizing objective (misfit) functions in a wide variety of problems. Commercial software packages have implemented such algorithms with tremendous success. Solutions to geophysical inverse problems do carry certain caveats, however. Due to the non-unique nature of inverted impedance models an infinite combination of impedances can result in the same seismic response it is desirable to produce a series of realizations and take an average. Deterministic methods rely entirely on the input model, returning the same answer each time the same model is used. Stochastic methods 34

introduce randomness into the model in an effort to account for that non-uniqueness. Additionally, the conjugate gradient method runs the risk of becoming trapped in a local minimum, should the starting model not be close enough to the solution. Global optimization methods seek to mitigate this problem by allowing for some probability of accepting a model that increases the objective function. 35

Chapter 4: Deterministic Inversion of the Cana-Woodford Data Set 4.1 METHODOLOGY We us the workflow available in Hampson-Russell Software (HRS) to carry out a post-stack deterministic inversion on the Cana-Woodford data set. The first steps in the process involve selecting the various input data. We use our CDP stack as input seismic data. P-wave sonic logs, density logs, and P-impedance logs are selected from each of the seven available wells to build the initial model. We select horizons for the Woodford Shale and underlying Hunton group. The HRS workflow calls for the extraction of a statistical wavelet for log correlation. We executed several trials of log correlations using different extracted statistical wavelets, as well as the Extract Wavelet Using Wells function, but were unable to reasonably match amplitudes between synthetic and observed traces. We then generated a series of Ricker wavelets with varying dominant frequencies and phase rotations. Figure 4.1 shows the log correlation using a 40Hz, 45- degree phase rotated Ricker wavelet. Amplitudes match reasonably well, with a crosscorrelation of 0.912 (Figure 4.2). 36

Figure 4.1: Log correlation of the Brooks 1-14H well. The synthetic trace computed from the convolution of our wavelet and the reflectivity series is plotted in blue. The observed trace at the well location is plotted in red. 5 traces of each are plotted for visualization purposes. The observed seismic data in the neighborhood of the Brooks well is plotted in black, with the Woodford (pink) and Hunton (green) horizons are superimposed. 37

Figure 4.2: Cross correlation window for the Brooks log correlation. Maximum correlation coefficient is 0.912. The initial models (Vp, density, and Zp) are constructed using an extrapolation algorithm that projects well data along the given horizons. A high-cut frequency of 10/15Hz is selected for the model. Figure 4.3 shows the Zp model with Woodford and Hunton horizons super-imposed in red. The frequency filter smooths the well-log data to frequencies similar to that of seismic data. We carry out inversion analysis (Figure 4.4) to analyze the match between measured and inverted impedances, as well as synthetic and observed seismic traces. The inverted Zp curve (red) matches the low-frequency behavior of the measured Zp (blue) fairly well in the Woodford interval, but it fails to match the thin variations seen at the 38

well-log scale. The synthetic (red) and observed (black) seismic traces have a correlation of 0.96. Figure 4.3: Initial Zp model for 301 traces. Red lines represent the Woodford (upper) and Hunton (lower) horizons. Blues correspond to low Zp, and yellows correspond to high Zp. 39

Figure 4.4: Inversion analysis window. The Zp comparison shows the measured welllog data (blue curve), initial model (black curve), and deterministic results (red curve). The seismic display compares synthetic (red, left) and observed (black) traces, as well as their difference (red, right). 4.2 RESULTS We apply the inversion results to the volume for 301 traces (CDPs 700-1000) and a time interval of 2100-2500ms. A window of 2200-2340ms is shown (Figure 4.5) to better visualize the reservoir interval. We super-impose the Woodford and Hunton horizons in red. The color bar shows high-impedance values represented by bright yellow and low-impedance values shown in dark blue. The final volume shows improved resolution from the initial model, resolving distinct low-impedance layers just above the Hunton horizon between CDPs 700 and 750, as well as CDPs 850 and 900. The frequency content still falls well below the frequencies seen at the well-log scale. Inverted impedance values in the Woodford generally fall between 8 and 11 km/s * g/cc, 40

with a few low- and high-impedance zones at the base and top of the Woodford, respectively. Figure 4.5: Inverted Zp results using the conjugate gradient method for 301 traces. A time window of 2200-2340ms is shown. Red lines represent the Woodford (upper) and Hunton (lower) horizons. Blues represent low Zp, while yellows represent high Zp. Well locations are superimposed in green. Figures 4.6 and 4.7 show comparisons of the observed and synthetic traces around the Harwick 1-1H and Brooks 1-14H well locations, respectively. The synthetic traces around the Harwick location are able to resolve the same internal layers in the reservoir. The amplitudes at these events are larger in the synthetic traces. The synthetic traces around the Brooks location do a better job of matching amplitudes, although some discrepancies exist, particularly at the Hunton horizon. Figure 4.8 shows comparisons between the well-log data and inverted Zp curves at the Harwick and Brooks locations. 41

Both show inverted Zp curves that are much smoother than the well-log curves. The inverted results are resampled to a 0.05ms interval for a quantitative comparison with well-log values. The maximum deviation from the log values at the Harwick location from 2200-2330ms is 47%. The mean deviation for the same interval is 11%. The maximum deviation at the Brooks location is 38%, with a mean deviation of 12%. We take a closer look at the reservoir interval from 2268-2318ms at the Harwick location, and 2260-2318ms at the Brooks location. The maximum deviation and mean deviation for this interval at the Harwick location are 45% and 9%, respectively. The maximum and mean deviations for the reservoir at the Brooks location are 38% and 12%, respectively. 42

Figure 4.6: Comparison of synthetic (blue) and observed (red) traces around the Harwick well location. Green lines represent the Woodford (upper) and Hunton (lower) horizons. Amplitudes match well at the Hunton horizon, as well as the layers above and including the Woodford horizon. The layers within the Woodford reservoir zone are resolved well, but show stronger amplitudes in the synthetic traces. 43

Figure 4.7: Comparison of synthetic (blue) and observed (red) traces around the Brooks well location. Green lines represent the Woodford (upper) and Hunton (lower) horizons. Amplitudes match well throughout although some discrepancies are present along the Hunton horizon, as well as in the layers above the Woodford (at CDPs 945-950). 44

Figure 4.8: Comparison of inverted results from the conjugate gradient method (red) with measured well-log data (blue) at the Harwick (left) and Brooks (right) locations. 4.3 CONCLUSIONS The deterministic approach succeeds in matching observed and synthetic seismic traces fairly well, but the difference in amplitudes might be a result of wavelet issues. The final volume in Figure 4.5 fails to capture the variability within the reservoir interval, which is evident when comparing curves at the trace locations. Moreover, the contrast in the impedance values of the reservoir and neighboring carbonates, as well as the reservoir thickness, results in inverted values that are almost exclusively overestimated. Improved well planning requires higher-resolution results. 45

Chapter 5: Fractal-Based, Very Fast Simulated Annealing 5.1 INTRODUCTION TO FRACTAL GEOMETRY The concept of fractal geometry arises from the failure of classical geometry to describe natural patterns (Mandelbrot, 1983). Previously, patterns were described using integer dimensions, such as a one-dimensional line, a two-dimensional plane, or a threedimensional solid (Dimri, 2005). Fractal geometry seeks to describe natural patterns using non-integer or fractional dimensions. Dimri (2005) provides the example of two different coastlines to illustrate fractional dimensions. The eastern coast of Florida shows much less complexity (fractional dimension D 1) than the Norwegian coastline, in which fjords increase the fractional dimension to D=1.52. The fractal behavior or characteristic is often associated with self-similarity seen at different scales (Peterson, 1984). In other words, they are non-differentiable due to variations at all scales (Hewett, 1986). Furthermore, the number of distinct scales of length found in natural patterns is essentially infinite (Mandelbrot, 1983). Hewett (1986) points at that although the original fractals described by mathematicians were truly self-similar, natural distributions are better described by stochastic functions with statistical self-similarity. 5.1.1 Fractal Behavior of a Time Series The three components that characterize a geophysical time series are the stochastic, trend, and periodic components (Dimri et al., 2012). Quantifying the strength of persistence of the data is necessary to quantify its stochastic component. The most common technique is plotting power-spectrum against frequency. The slope, β, represents the scaling exponent, which characterizes the strength of persistence. Dimri (2005) gives the definition of the self-affine time series as one that follows power-law behavior. In 46

other words, power as function of frequency is proportional to frequency raised to some exponent: P(f) f β (5.1) Figure 5.1: Cross-plot demonstrating the power-law behavior of the Zp data from the Brooks well location. Log10 of power (y-axis) is plotted against the log10(frequency). The blue line represents the L2 norm of the data points. The slope, β, represents the scaling exponent. 5.1.2 Applications in Geophysics One of the first applications of fractal statistics in the geosciences was in describing the annual flows of rivers (Hewett, 1986). Subsequently it was realized that the resulting sediment distribution of such fractal processes might exhibit fractal characteristics themselves. Data from the German Continental Deep Drilling Programme (KTB) demonstrates the power-law behavior of several physical properties of the subsurface, such as density, magnetic susceptibility, resistivity, and conductivity (Dimri, 47

2005; Dimri et al., 2012). Browaeys and Fomel (2009) extend the concept of fractals to studying the effects of heterogeneities on wave scattering and attenuation, while Srivistava and Sen (2009) exploit fractal behavior to generate intelligent starting models for seismic inversion algorithms. 5.1.3 Re-scale Range Analysis/Hurst Coefficient The generation of our fractal-based prior model begins with computation of several statistics of our well-log data, such as mean, covariance, and the Hurst coefficient (Caccia et al., 1997; Srivistava and Sen, 2009; Xue, 2013). Srivistava and Sen (2009) provide the process of estimation of the Hurst coefficient, H, which results from the power-law relation found in many datasets in nature (Hurst et al., 1965): R S = (N 2 )H (5.2) The process of re-scaled range (RS) analysis allows for the determination of the range, R, and standard deviation, S. The number of points in a given bin size is represented by N. The dataset is partitioned into bins of varying size while the cumulative summation is computed for each bin relative to its mean (Caccia et al., 1997; Srivistava and Sen, 2009; Xue, 2013): n y n = i=1 (y i y n), n = 2, N 1 (5.3) R N = (y n ) max (y n ) min (5.4) S N = σ N (5.5) Taking the linear regression of the cross-plot of (R/S) versus (N/2) provides a line with slope equal to H. For the method of generating fractional Gaussian noise, the reader is referred to Caccia (1997) and Srivistava and Sen (2009). 48

5.2 METHODOLOGY 5.2.1 Fractal-Based Prior Model We started our inversion algorithm by constructing a fractal-based prior model using the power-law behavior. We loop over every CDP along our 2D line, and assign each CDP the impedance curve corresponding to the nearest well. Figure 5.2 demonstrates the fractal nature of the acoustic impedance log from the Brooks 1-14H well in the central part of our study area. Figure 5.2a shows the measured acoustic impedance, and Figure 5.2b displays the distribution of the impedance values in the Brooks well. In addition, Figures 5.2c and 5.2d exhibit the covariance and the semivariogram of the impedance data, respectively. The covariance starts at a value of 1 at zero lag and heads toward zero as the lag moves toward its maximum. The semivariogram increases with lag but never reaches a well-defined sill. The power spectrum is shown in Figure 5.2e. Figure 5.2f shows the cross-plot of log10(r/s) vs. log10(n/2). The slope of the best-fit line represents the Hurst coefficient, H. Using H, we generate fractional Gaussian noise (fgn) to build our prior model that includes the low- and highfrequency information missing from the seismic data, while maintaining the statistics of our measured log data. Figure 5.3 shows the amplitude spectrum of the log data, the prior model using fgn, and the measured seismic data. The amplitude spectrum of the observed impedance data shows low-frequency content (<10Hz), as well as content at frequencies above 100Hz. The log itself has frequency content beyond ~5KHz, but log data is resampled to 2ms for easier implementation into the algorithm. The fgn amplitude spectrum also contains low-frequency content below 10Hz, but also boosts highfrequency content. The amplitude spectrum of the observed seismic data shows the absence of low-frequency (<10Hz) content and high-frequency (>50Hz) content. 49

Figure 5.2: Test of the fractal nature of the Brooks 1-14H well. a) Observed acoustic impedance. b) Histogram and pdf of Zp showing fat-tailed Gaussian distribution. c) Covariance showing power law behavior. d) Semi-variogram showing power-law behavior. e) Spectral density showing power-law behavior. f) Plot of log10(r/s) versus log10(bin size). The slope of the bestfit line represents the Hurst coefficient. 50

Figure 5.3: The amplitude spectrum of the observed impedance log (left), the fractalbased prior model (center), and the observed seismic data (right). The seismic data is band-limited, with missing low-frequency content below 10Hz, and missing high frequencies above 50 Hz. The impedance log retains the low-frequency content, and the fractal-based prior boosts the highfrequency content. 5.2.2 Very Fast Simulated Annealing We insert the generated fractional Gaussian prior into the VFSA algorithm as our initial model. The first step in this process involves computing the initial objective function, E0. The initial model is used to compute a reflectivity series, which is then convolved with our phase-rotated Ricker wavelet. Equation 5.6 is the objective function that we seek to minimize: E = Σ(obs syn) 2 Σ(obs+syn ) 2 +Σ(obs syn) 2 (5.6) Next we perturb our model by drawing a random number, u, from a uniform distribution, U[0,1], and inserting it into the following relationship: y = sign(u - 1 )T[(1 + 1 2 T ) 2u 1 1] (5.7) where T represents the sampling temperature, given by: T(k) = T0exp(-ck 1/NM ) (5.8) 51

The cooling schedule (Equation 5.8) uses the initial temperature, T0, multiplied by an exponential function where c represents a decay parameter, k is the iteration number, and NM is the number of model parameters. In order to achieve faster convergence, we set NM equal to 2. The updated model comes from Equation 5.9: mnew = mold + y(mmax mmin) (5.9) A moving Backus average on the initial model defines the range of m. We then allow for a maximum perturbation of 20% above and below the Backus average value for each cell in the input vector. This constraint ensures that our model does not exceed our lowfrequency model. After model perturbation the new objective function, Enew, is computed. The acceptance of mnew depends on the value of ΔE, or Enew E0. In the event that ΔE is negative, the objective function has decreased, and the new model is accepted. If ΔE is positive, the objective function has increased, indicating an uphill move. In this situation, a random number, r, is drawn from a uniform distribution, U[0,1], and compared to a value, P, given by the relationship P = exp( ΔE T ) (5.10) T(k) = exp(-ck 1/NM ) (5.11) If P>r, the new model is accepted. Otherwise, the new model is rejected, and a new perturbation is computed. Figure 5.4 shows the workflow of very fast simulated annealing. The number of iterations is set to 5000, with an initial temperature of 350. We set the maximum number of moves per temperature to 3. 52

*Loop over temperature (maximum number of iterations) *Loop over number of random moves per temperature *Loop over number of model parameters Draw u ϵ U[0,1] y=sign(u 1 2 )T[(1 1 T ) 2u 1 1] mnew = mold + y(mmax mmin) mmin mnew mmax end ΔE = E(mnew) E(mold) if ΔE 0 mold = mnew E(mold) = E(mnew) end if ΔE > 0 P = exp( ΔE T ) Draw random number r ϵ U[0,1] if P > r mold = mnew E(mold) = E(mnew) end end end end Figure 5.4: Pseudo-code for the very fast simulated annealing algorithm (from Sen and Stoffa, 1995). 53

5.3 RESULTS The VFSA algorithm was applied to a 301-trace volume around the Harwick 1-1H and Brooks 1-14H well locations (CDPs 700-1000). Figure 5.5 shows the results of 25 realizations of VFSA at the Harwick and Brooks wells. The green line represents the mean of the realizations. Figure 5.6 shows the comparison between observed and synthetic traces at the Harwick well location, as well as the comparison between observed impedances and results from both deterministic and VFSA approaches. The VFSA resolves the same events in the seismic traces, but some mismatch exists between amplitudes. The inverted Zp values from VFSA (green line) show improved resolution relative to the deterministic results, and are able to resolve the major variations within the reservoir. However, some thinner layers in the reservoir still lie below resolution, and some layers that are resolved show values that are overestimated. The maximum deviation at the Harwick location from 2200-2330ms is 46%. The mean deviation for this interval is 9%. The maximum deviation and mean deviation for the reservoir interval at this location are 46% and 7%, respectively. Figure 5.7 shows the same plot for the Brooks well location. Again, the synthetic traces resolve the same events as the observed traces. The inverted impedance values from VFSA do a much better job resolving internal layers at the Brooks location. The maximum deviation for the interval 2200-2340ms is 61%, while the mean deviation is 9%. The maximum and minimum deviation for the reservoir interval are 25% and 7%, respectively. 54

Figure 5.5: 25 realizations of fractal-based VFSA at the Harwick (left) and Brooks (right) locations. Green lines represent the mean of the realizations. 55

Figure 5.6: Comparison of synthetic (blue) and observed (red) traces at the Harwick location, as well as comparison of Zp values from the measured log (blue), conjugate gradient method (red), and VFSA (green). Inverted Zp results from the VFSA algorithm show improved resolution relative to the conjugate gradient results. A noticeable mismatch is present at the base of the Woodford, where the Zp curve from VFSA shows a more gradual increase, relative to the abrupt increase seen in the well log data. 56

Figure 5.7: Comparison of synthetic (blue) and observed (red) traces at the Brooks location, as well as comparison of Zp values from the measured log (blue), conjugate gradient method (red), and VFSA (green). Inverted Zp results from the VFSA algorithm show considerable improvement in resolution relative to the conjugate gradient method. This is particularly evident at ~2310ms, where the sharp Zp increase in the layer above the base of the Woodford is better resolved with VFSA. The sharp increase in Zp at the Hunton horizon is better captured in the inversion results at the Brooks well location, although the magnitude of the increase is still off by ~3(km/s)*(g/cc). The minimization of the objective function (Equation 5.6) is shown at the Harwick and Brooks locations in Figure 5.8. The ability of the algorithm to accept uphill moves is evident in the oscillatory behavior seen at early iterations. As the number of iterations increases, the probability of accepting an increase in the objective function tapers toward zero. Figure 5.9 shows the final inverted impedance profile in the neighborhood of the Harwick and Brooks wells. Bright yellows represent high impedance 57

values, and dark blues correspond to low values. The overlying and underlying carbonate groups are shown by the highest values of Zp. The Woodford and Hunton horizons are superimposed in red. Three distinct low-impedance layers are visible around the Brooks location (CDP 942). As we move away from the Brooks well toward lower CDPs, the layers become less apparent. The area between the Brooks and Harwick wells (CDPs ~800-900) shows a thick, low-impedance layer near the base of the Woodford, and another thick, relatively high-impedance layer near the top of the Woodford. The Harwick well location (CDP 761) shows less layering, with Zp values ranging from ~7.5-10(km/s)*(g/cc). Figure 5.8: Plots of the objective functions at the Harwick (left) and Brooks (right) locations for 5000 iterations. The objective function represents the misfit between observed and synthetic traces. 58

Figure 5.9: The inverted Zp profile using fractal-based VFSA for traces 700-1000. Red lines represent the Woodford (upper) and Hunton (lower) horizons. Blues represent low Zp values, while yellows represent high values. Well locations are superimposed in green. Lateral variability is evident in the coming and going of the distinct low-impedance layers starting at the Brooks location where 3 such layers are present, and moving away in either direction where only 1 or 2 low-impedance layers are visible. Figure 5.10 shows a comparison of 21 synthetic and observed traces in the neighborhood of the Harwick well location (CDPs 750-770). The Woodford and Hunton horizons are superimposed in green. Amplitudes match well along the Hunton horizon, but the peaks of the internal layers are considerably larger in the synthetic traces. We show a 21-trace comparison around the Brooks well (CDPs 930-950) in Figure 5.11. At this location the internal layers show a better match, while the peaks along the Hunton horizon show different amplitude characteristics. The observed traces show doublets along the interface, while the synthetic traces show larger single amplitudes. A plot of 21 59

traces midway between well locations is shown in Figure 5.12. Synthetic traces in this section show good amplitude matches with the observed traces. Fewer internal layers are visible, both in the observed and synthetic traces. Figure 5.10: Comparison of 21 synthetic (blue) and observed (red) traces around the Harwick location. Green lines represent the Woodford (upper) and Hunton (lower) horizons. Similar to the conjugate gradient results, the internal layers show stronger amplitude in the synthetic traces. Amplitudes match well at both the Woodford and Hunton horizons, as well as the layers above the Woodford horizon. 60

Figure 5.11: Comparison of 21 synthetic (blue) and observed (red) traces around the Brooks location. Green lines represent the Woodford (upper) and Hunton (lower) horizons. Amplitudes match well for most layers, with the notable exception of the Hunton horizon. The doublet behavior seen in the observed traces is absent in the synthetic traces. Instead, large single amplitudes are present. The peak at ~2290ms shows less lateral continuity in the synthetic traces. 61

Figure 5.12: Comparison of 21 synthetic (blue) and observed (red) traces between the Harwick and Brooks locations. Green lines represent the Woodford (upper) and Hunton (lower) horizons. The amplitudes at the Woodford and Hunton horizons match well. Some mismatches are noticeable in the upper most peak. 5.4 DISCUSSION The very fast simulated annealing algorithm provides higher-resolution results than traditional deterministic inversion. Analysis at the well locations shows that we capture a significant portion of the variability seen in the well-log data. Comparison of the synthetic and observed traces shows a good match with respect to layer resolution. Amplitude mismatches are still present, most notably in the Harwick comparison (Figure 5.10) between 2280ms and 2300ms. There are some mismatches in the log data and inverted Zp curves at the top and base of the Woodford. Inversion results fail to line up the sharp decrease in Zp at the top of the reservoir (Brooks 1-14H, Figure 5.7), as well as the sharp increase at the base (Harwick 1-1H, Figure 5.6). This could be due to phase 62

issues arising from the extrapolation method. Overall, the results using the fractal-based VFSA algorithm show considerable improvement. 63

Chapter 6: Principal Component Analysis Based Stochastic Inversion 6.1 INTRODUCTION TO PRINCIPAL COMPONENT ANALYSIS Joliffe (2002) describes principal component analysis (PCA) as a way of reducing the dimensionality of a large set of interrelated variables by transforming to the principal components (PCs) a new set of uncorrelated variables that are reordered so that the first few PCs retain the majority of the variation seen in the original variables. For the observed model space with d-dimensional vectors, the principal axes refer to the set of orthonormal axes in which the variance under projection is maximal (Tipping and Bishop, 1999). These principal axes are shown to be the dominant eigenvectors of the sample covariance matrix. Many applications utilize this technique, such as data compression, image processing, visualization, exploratory data analysis, pattern recognition and time series prediction (Tipping and Bishop, 1999). Kim et al. (2012) use a form of PCA to improve face-recognition algorithms. Xue (2013) addresses the issue of preserving lateral continuity in seismic inversion results by implementing PCA on a set of training images generated from fractal-based VFSA. Transforming the training images into a set of principal components reduces the dimensions of the model space. This allows for the inversion of the elastic profile along a 2D line at every trace simultaneously. The original model space is then reconstructed using a linear combination of the PCs. Xue shows that using only the PCs associated with >80% of the variability of the original data, the error in the reconstructed model can be considered negligible. 6.2 METHODOLOGY Following the work of Xue (2013) we implement a PCA-based stochastic inversion algorithm to invert simultaneously for acoustic impedance at all traces along 64

our 2D line. The first step in this process is to generate training images from many independent threads of fractal-based VFSA. This method uses the fractal-based prior distribution to construct an intermediate distribution (training images) as input for the PCA algorithm. Using 10 independent threads, each with a different starting temperature, we generate many realizations by indexing the model at each iteration along the way. After a suitable number of realizations are stored (e.g., 5000), 1000 training images are selected at random (100 realizations from each starting temperature). Each training image is reshaped into a column vector and stored in an n p matrix, in which n represents the total number of data points from each image, and p represents the total number of images. 65

Figure 6.1: Six example training images selected at random from the 1000 training images used. Once we have our training images, we use principal component analysis to reduce the dimensions of our model space. The following steps outline the process of computing the principal components. We start by centering our data matrix: Next, we compute the covariance matrix of our centered data: X c = X X (6.1) Cov = 1 n 1 X c T X c (6.2) After computing the eigenvectors, a, and eigenvalues, λ, of the covariance matrix, we obtain the principal components (Figure 6.2) using the following equation: z j = a j T (x x ) (6.3) 66

The eigenvectors, eigenvalues, and principal components are then sorted based on descending eigenvalues. A plot of the eigenvalues against the eigenvector index (Figure 6.3) shows that the first few PCs explain a majority of the variability. For the purposes of this study, we use the first 2 PCs. The reconstruction of our model space is then carried out using the following equation: X = X + z 2 (a 2 c) T (6.4) in which z 2 and a 2 represent the first 2 PCs and corresponding eigenvectors, respectively, and c represents a coefficient vector we attach to the eigenvectors. Attaching c to the eigenvectors allows us to perturb our model in the reduced model space, rather than perturbing our impedance profile directly (Xue, 2013). Our method uses a coefficient vector that has the same dimensions as the selected eigenvectors. We iteratively update c one column at a time, starting with the column corresponding to the first PC. After a certain number of iterations, we store the value of c for that column and move to the next. After we reconstruct the model space, we weight each column vector from 1 to p based on its corresponding eigenvalue. The sum of all weighted vectors returns the final model. 67

Figure 6.2: Six examples of the transformed variables, or principal components of the training images. Each plot is color-coded based on the principal component scores. The majority of the variability is captured in the first few PCs. 68

Figure 6.3: Plot of the proportion of the variance explained by each corresponding eigenvector. The first few eigenvectors account for ~80% of the variance. 6.3 RESULTS Comparison of the synthetic and observed seismic traces, as well as comparisons of the well-log data with deterministic, VFSA, and PCA results are shown for the Harwick and Brooks well locations in Figure 6.4 and Figure 6.5, respectively. Similar amplitude issues are present in the synthetic traces from both PCA and VFSA results. The PCA results at Harwick show some improvement in matching Zp values, particularly at 2265-2275ms. The results at the Brooks location show significant improvement in Zp estimation near the base of the Woodford. The layer at ~2310ms is resolved quite well, and the algorithm better predicts the true value of Zp at the Woodford-Hunton interface as well. The maximum deviation for the 2200-2330ms interval at the Harwick location is 43%. The mean deviation for this interval is 7%. The maximum deviation and mean deviation for the reservoir interval at the Harwick location is 42% and 6%, respectively. 69

At the Brooks location, the maximum deviation for the interval 2200-2340ms is 37%. The mean deviation is 8%. The maximum and minimum deviations for the reservoir interval are 28% and 7%, respectively. Figure 6.4: Comparison of synthetic (blue) and observed (red) traces at the Harwick location, as well as comparison of Zp values from the measured log (blue), conjugate gradient method (red), VFSA (green), and PCA (magenta). 70

Figure 6.5: Comparison of synthetic (blue) and observed (red) traces at the Brooks location, as well as comparison of Zp values from the measured log (blue), conjugate gradient method (red), VFSA (green), and PCA (magenta). The final impedance profile for CDPs 700-1000 (Figure 6.6) shows inversion results in the neighborhood of both the Harwick (CDP 761) and Brooks (CDP 942) well locations. The overlying carbonates show Zp values from ~13(km/s)*(g/cc) to over 14(km/s)*(g/cc). The area around the Brooks well location shows three distinct lowerimpedance layers within the Woodford. This trend continues to the right and left of the image for several CDPs. At CDPs lower than ~920, however, the bottom low-impedance layer tapers off. This is also seen at CDPs above ~970. The reservoir thickness decreases toward the Harwick well location. The three distinct lower-impedance layers give way to two distinct layers, with the uppermost layer showing greater thickness than in the 71

Brooks location. The distinction between individual layers becomes less clear when looking directly between the Brooks and Harwick wells (CDP ~850). There still exists a visible pattern of at least two lower-impedance layers, though their edges are less defined. Figure 6.6: The inverted Zp profile using PCA-based stochastic inversion for traces 700-1000. Red lines represent the Woodford (upper) and Hunton (lower) horizons. Blues represent low Zp values, while yellows represent high values. Well locations are superimposed in green. We plot a 21-trace window around the Brooks and Harwick locations (Figures 6.7 and 6.8) to examine closer the match between synthetic and observed traces. The synthetic traces around the Brooks location match fairly well with the observed traces. Some noticeable issues arise at the base of the Woodford, where the observed traces show doublets around the Brooks location, while the synthetic traces show single peaks 72