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NUMERICAL MODELING OF POROSITY WAVES AS A

MECHANISM FOR RAPID FLUID TRANSPORT

IN ELASTIC POROUS MEDIA

______

A Dissertation

Presented to

The Faculty of the Graduate School

At the University of Missouri-Columbia

______

In Partial Fulfillment

Of the Requirements for the Degree

Doctor of Philosophy

______

AJIT JOSHI

Dr. Martin Appold, Dissertation Supervisor

DECEMBER 2015

The undersigned, appointed by the dean of the Graduate School, have examined the dissertation entitled

NUMERICAL MODELING OF POROSITY WAVES AS A MECHANISM FOR RAPID FLUID TRANSPORT IN ELASTIC POROUS MEDIA presented by Ajit Joshi, a candidate for the degree of

Doctor of Philosophy, and hereby certify that, in their opinion, it is worthy of acceptance.

Dr. Martin Appold

Dr. Mian Liu

Dr. Peter Nabelek

Dr. Sergei Kopeikin

Dedicated to My Beloved parents, Sapana and Aasha

ACKNOWLEDGEMENTS

I would like to acknowledge and express my sincere gratitude to all those people who helped me, encouraged me and motivated me during my doctoral program at the University of

Missouri-Columbia. First of all, I am very thankful to my advisor, Dr. Martin Appold, for his continuous motivation, encouragement and guidance from the very beginning until the successful completion of this dissertation. I sincerely thank my committee members, Dr. Mian Liu, Dr.

Peter Nabelek and Dr. Sergei Kopeikin for their valuable feedbacks, suggestions and comments that improved this dissertation.

I am also very grateful to Nancy Hunter from Platte River Associates for her help in running the BasinMod2D™ software and providing prompt responses to all of my questions regarding the software. It is my pleasure to thank the Department of Geological Sciences at the

University of Missouri-Columbia for funding my doctoral study. My sincere thanks also go to

Marsha Huckabey and Tammy Bedford at the Department of Geological Sciences, for the administrative assistance. I would also like to thank all faculties, graduate students and staffs of the Department of Geological Sciences at the University of Missouri-Columbia who provided me with any kind of help and support to complete my doctoral research. I extend my special note of gratitude to all my friends for their support during the study.

Finally, I am also indebted to my parents and all the family members, who persistently encouraged me for successful completion of this degree without expressing their difficulties to me.

TABLE OF CONTENTS

ACKNOWLEDGEMENTS ...... ii LIST OF FIGURES ...... v CHAPTER 1: INTRODUCTION ...... 1 1.1. Purpose of the study ...... 1 References ...... 4 CHAPTER 2: FLUID PRESSURE EVOLUTION DURING SEDIMENTARY BASIN DIAGENESIS: IMPLICATIONS FOR HYDROCARBON TRANSPORT BY POROSITY WAVES ...... 7 Abstract ...... 7 2.1. Introduction ...... 8 2.2. Theoretical background ...... 15 2.3. Model construction ...... 16 2.4. Results ...... 18 2.4.1. Base case scenario ...... 18 2.4.2. High pressure generation rate scenario ...... 23 2.4.3. Low pressure generation rate scenario ...... 26 2.5. Discussion ...... 28 2.6. Conclusions ...... 33 References ...... 34 CHAPTER 3: POTENTIAL OF POROSITY WAVES FOR METHANE TRANSPORT IN THE EUGENE ISLAND FIELD OF THE GULF OF MEXICO BASIN ...... 43 Abstract ...... 43 3.1. Introduction ...... 44 3.2. Geological setting ...... 47 3.3. Model set-up ...... 49 3.3.1. Theoretical background ...... 49 3.3.2. Model description ...... 50 3.4. Model results ...... 54 3.4.1. Gradual pressure generation scenario ...... 54 3.4.2. Instantaneous pressure generation scenario ...... 56

iii

3.5. Discussion ...... 65 3.6. Conclusions ...... 69 References ...... 70 Chapter 4: Numerical modeling of porosity waves in the Nankai accretionary wedge décollement: implications for slow slip events ...... 74 Abstract ...... 74 4.1. Introduction ...... 75 4.2. The Nankai accretionary wedge, offshore Japan ...... 78 4.3. Model theory and set-up ...... 80 4.4. Results ...... 84 4.4.1. Base case scenario ...... 84 4.4.2. High fluid pressure source scenario...... 88 4.4.3. Optimal velocity scenario ...... 90 4.4.4 High compaction factor scenario ...... 93 4.4.5. Variation of porosity wave velocity with décollement distance ...... 95 4.5. Discussion ...... 96 4.6. Conclusions ...... 100 References ...... 101 APPENDIX ...... 109 FORTRAN program code for methane porosity wave calculations ...... 109 VITA ...... 120

LIST OF FIGURES

Figure Page

Figure 2.1. Conceptual diagram of a porosity wave ascending through a sedimentary column viewed at times, t1, t2, and t3. The circle size in the wave is proportional to porosity, which increases with increasing pressure...... 9 Figure 2.2. Model cross-section of the hypothetical sedimentary basin with hydrocarbon source rock at the bottom and composed entirely of shale...... 17 Figure 2.3. Model stratigraphic evolution of the hypothetical sedimentary basin at (a) 3.1 Ma, (b) 1.5 Ma, (c) 1.0 Ma, (d) 0 Ma. Violet blue and light gray colors represents shale and hydrocarbon source rock, respectively...... 19 Figure 2.4. Temperature evolution at the center of the source rock for three different model scenarios. The source rock temperature change is greatest for the high pressure generation rate scenario and lowest for the low pressure generation rate scenario...... 20 Figure 2.5. Saturation evolution of (a) oil and natural gas in the middle of the source rock. Oil generation started at 1.68 Ma, reached a peak saturation of 0.15 at 1.02 Ma, and declined to zero saturation at the present time. Gas production started at 1.56 Ma and reached a peak saturation of 0.2 at the present time. Saturation plots for (b) oil and (c) natural gas as a function of age and depth. Neither oil nor gas migrated significantly from the source rock...... 21 Figure 2.6. Evolution of overpressure in the model basin at (a) 3.1 Ma, (b) 1.5 Ma, (c) 1.0 Ma, (d) 0 Ma for the base case scenario...... 22 Figure 2.7. Evolution of (a) pore pressure and (b) pressure generation rates in the hydrocarbon source rock resulting from compaction disequilibrium and hydrocarbon formation for the base case scenario...... 23 Figure 2.8. Evolution of oil and natural gas saturation for the (a) high and (b) low pressure generation rate scenarios...... 25 Figure 2.9. Evolution of (a) pore pressure and (b) pressure generation rates in the hydrocarbon source rock resulting from compaction disequilibrium and hydrocarbon formation for the high pressure generation rate scenario...... 26 Figure 2.10. Evolution of (a) pore pressure and (b) pressure generation rates in the hydrocarbon source rock resulting from compaction disequilibrium and hydrocarbon formation for the low pressure generation rate scenario...... 27 Figure 2.11. (a) Ratio of pressure generation rate to hydraulic diffusivity in the hydrocarbon source rock as a function of time for the three model scenarios. (b) Ratio of pressure generation rate to hydraulic diffusivity as a function of depth at the present day for the high pressure generation rate scenario. The solid purple line represents the minimum ratio needed for the formation of porosity waves saturated with oil...... 30

Figure 3.1. Location map of the Eugene Island minibasin (Alexander and Handschy 1998) ...... 48 Figure 3.2. Sensitivity of porosity wave velocity to time step size and nodal spacing. .. 51 Figure 3.3. Plots of the three permeability-depth profiles used in the modeling...... 54 Figure 3.4. Plots of pore fluid pressure as a function of depth after (a) 2400 years, (b) 4800 years, (c) 7200 years, (d) 12,000 years, (e) 18,000 years, and (f) 24,000 years for the gradual pressure generation scenario...... 56 Figure 3.5. Plots of pore fluid pressure as a function of depth after (a) 0 years, (b) 25 years, (c) 55.6 years, (d) 68.1 years, (e) 69.5 years, and (f) 71 years for the permeability-depth profile 1 in the instantaneous pressure generation scenario. . 58 Figure 3.6. Temporal variation of the (a) wave pore volume, (b) wave velocity, and (c) wave volumetric flow rate for permeability–depth profiles 1 and 2 in the instantaneous pressure generation scenario...... 59 Figure 3.7. Plots of pore fluid pressure as a function of depth after (a) 0 years, (b) 1 year, (c) 5 years, (d) 7 years, (e) 7.8 years, and (f) 7.9 years for permeability- depth profile 2 for the instantaneous pressure generation scenario...... 61 Figure 3.8. Plots of pore fluid pressure as a function of depth after (a) 0 hours, (b) 0.88 hours, (c) 4.38 hours, and (d) 9.64 hours for permeability-depth profile 3 for the instantaneous pressure generation scenario...... 63 Figure 3.9. Plots of pore fluid pressure as a function of depth after (a) 0 hours, (b) 0.044 hours, (c) 0.39 hours, (d) 4.82 hours, (e) 9.2 hours, and (f) 10.51 hours for permeability-depth profile 3 and instantaneous pressure generation, where fluid pressure at the source was raised to 104% of lithostatic at 4.5 km depth, keeping all other model parameters the same as for Figure 3.8...... 64 Figure 4.1. (a) Location map of the Nankai accretionary wedge near Shikoku Island in Japan with a NW-SE trending transect (white) corresponding to the cross section in (b) the modified two-dimensional cross-section employed in the pressure wave model by Bourlange and Henry (2007). One-dimensional model in the current study was constructed based on the geometry of the décollement zone in figure b...... 79 Figure 4.2. Plots showing permeability as a function of the effective stress for four model scenarios...... 84 Figure 4.3. Plots of pore fluid pressure as a function of décollement distance after (a) 0 year and (b) 333,000 years for instance A of the base case scenario...... 85 Figure 4.4. Sensitivity analysis of nodal and time spacing to the wave velocity for (a) the base case scenario, (b) the high fluid pressure source scenario, and (c) the optimal velocity scenario...... 86 Figure 4.5. Plots of pore fluid pressure as a function of décollement distance after (a) 0 year, (b) 45 years, (c) 180 years, (d) 270 years, (e) 315 years, and (f) 342 years for the base case scenario...... 87

Figure 4.6. Plots of pore fluid pressure as a function of décollement distance after (a) 0 year, (b) 5 years, (c) 10 years, (d) 15 years, (e) 24 years and (f) 25 years for the high fluid pressure source scenario...... 91 Figure 4.7. Plots of pore fluid pressure as a function of décollement distance after (a) 0 year, (b) 0.55 years, (c) 0.57 years, (d) 0.596 years, (e) 0.6 years and (f) 0.62 years for the optimal velocity scenario...... 92 Figure 4.8. Plots of pore fluid pressure as a function of décollement distance after (a) 0 year, (b) 30 years, (c) 90 years, (d) 120 years, (e) 135 years and (f) 150 years for the high compaction factor (∗) scenario...... 94 Figure 4.9. Plots of pressure wave velocity as a function of décollement distance for all four scenarios...... 95

vii

LIST OF TABLES Table Page

Table 2.1. Case studies from various sedimentary basins predicted high pore fluid pressures from conversion of oil to natural gas...... 11 Table 2.2. Case studies of pore fluid pressure evolution from hydrocarbon generation in various sedimentary basins around the world...... 12 Table 2.3. Parameter values used in the three model scenarios...... 18 Table 4.1. Model parameter values for all the scenarios...... 82

viii

CHAPTER 1: INTRODUCTION

1.1. Purpose of the study

The anomalously rapid migration of overpressured fluids through low permeability geologic porous media has been well documented in a variety of geological settings including hydrocarbon migration in sedimentary basins (Young et al. 1977;

Anderson et al. 1994; Cartwright 1994; Whelan et al. 2001; Xie et al. 2001; Boles et al.

2004) and the ascent of water in subduction zones (Henry 2000; Husen and Kissling

2001; Davis et al. 2006; Bourlange and Henry 2007; John et al. 2012). The Eugene

Island hydrocarbon field in the Gulf of Mexico basin is a well characterized site where hydrocarbon source rocks are separated from shallow reservoirs by several kilometers of low permeability rocks and sediments (Alexander and Handschy 1998). Despite low expected Darcy fluxes, hydrocarbons sourced at depths of 4.5 km appear to have migrated to shallow reservoirs at rates of millimeters to kilometers per year (Holland et al. 1990; Losh et al. 1999; Revil and Cathles 2002; Haney et al. 2005; Joshi et al. 2012).

Seismic and core drilling analysis of an accretionary wedge in the Nankai subduction zone by the Ocean Drilling Program (ODP) suggested fluids produced by mineral dehydration from the subducting oceanic plate were overpressured close to the lithostatic pressure (Screaton et al. 2002; Kodaira et al. 2004; Gamage and Screaton

2006; Tobin and Saffer 2009). Hydrologic observations from the Nankai accretionary wedge by the ODP have suggested fluid overpressures thus generated in the décollement could have propagated, as pressure pulses, up the dip of the décollement at rates of ~1 km per day (Davis et al. 2006). Rapid propagation of these pressure pulses up the dip of the

Nankai décollement are hypothesized to have caused aseismic slip that has also been

observed to propagate at rates of kilometers per day (Kodaira et al. 2004).

Overpressured fluids that are close to the lithostatic pressure can reduce the

effective normal stress and dilate the porosity, permeability, and pressure region termed

porosity waves. Thus formed porosity waves can travel significantly faster than fluids in

the background Darcian flow regime, so that they can serve as an efficient means of fluid

transport in a low permeability geologic media. Porosity waves have been studied

previously in order to explain transport of mantle magma (Richter and McKenzie 1984;

Scott and Stevenson 1984; Spiegelman 1993a; Spiegelman 1993b; Wiggins and

Spiegelman 1995; Richardson et al. 1996), flow of CO2 and water (Connolly 1997;

Fontaine et al. 2003; Miller et al. 2004; Bourlange and Henry 2007; Connolly 2010; Tian

and Ague 2014) and expulsion of hydrocarbons in low permeability rocks in a

sedimentary basin (Connolly and Podladchikov 2000; Appold and Nunn 2002; Revil and

Cathles 2002; Joshi et al. 2012).

Joshi et al. (2012) carried out one-dimensional numerical modeling of behavior of

porosity waves in oil-saturated elastic media. Porosity waves were spawned in

hydrocarbon source rock because of sediment compaction and hydrocarbon formation

that increase fluid pressures at gradual rates of 30 Pa year−1, which is higher than the rate

of pressure diffusion. They found that porosity waves are unlikely to have been an

important oil transport mechanism at Eugene Island because the porosity waves could

only exist within a narrow range of geological conditions and had low frequencies of formation. However, they suggested that porosity waves could be an efficient means for

methane transport at Eugene Island because of methane’s lower viscosity and density

compared to oil. The results of their study suggest that high rates of fluid pressure

generation on the order of 2000 Pa year−1 would required to form porosity waves in methane-saturated sediments because of the much higher hydraulic diffusivity of methane-saturated sediments compared to oil-saturated sediments.

Thus, in order to explore the ability of porosity waves to transport overpressured fluids through low permeability geologic media, three hypotheses were proposed and tested in the present study. The first hypothesis is that a high fluid pressure generation rate of ~2000 Pa year−1, considered necessary to form porosity waves in

methane-saturated sediments, is achievable under geologically reasonable conditions in a

compacting kerogen-rich sedimentary basin. In order to test this hypothesis, a two- dimensional basin evolution model was constructed using the commercial software

package BasinMod 2D™, that simulated sediment deposition and compaction, heat

transport, hydrocarbon formation from kerogen maturation, and associated flow of oil,

gas and water. The first hypothesis is explored in Chapter 2, which discusses possible

ranges of pressure generation rates resulting from sediment diagenesis and hydrocarbon

formation in a generic organic rich sedimentary basin and implications for porosity wave

formation.

The second hypothesis of the present research was that porosity waves can form

from fluid pressure generation rates of ~2000 Pa year−1 in methane-saturated low

permeability sediments at the Eugene Island in the Gulf of Mexico basin. In order to test

second hypothesis, a one-dimensional numerical model was constructed that solved a

pore fluid mass conservation equation for elastic porous media and Darcy’s law. Chapter

3 summarizes and discusses results from this one-dimensional porosity wave model. The

model was used to evaluate the genesis of porosity waves and their ability to transport

methane through low permeability sediments in the Eugene Island hydrocarbon field.

The model addressed the research questions of how large porosity waves can get, how frequently they can form, how fast and far they can travel and how much methane they

can transport.

The third hypothesis of the present research was that fluid pressure generation

from sediment diagenesis and mineral dehydration in the subducting oceanic plate in the

Nankai accretionary wedge can form porosity waves that can travel fast enough to

account for aseismic slip at the Nankai accretionary wedge décollement. The hypothesis

was tested using a separate one-dimensional numerical solution to the pore fluid mass

conservation equation and Darcy’s law to solve for fluid pressure evolution along the

décollement. Chapter 4 summarizes the model results regarding porosity waves genesis

from overpressures in the décollement and the geologic conditions under which they are

able to transport fluids at rates fast enough to account for aseismic slip.

References

Alexander LL & Handschy JW (1998) Fluid flow in a faulted reservoir system: fault trap analysis for the Block 330 field in Eugene Island, south addition, offshore Louisiana. AAPG Bulletin, 82,387-411. Anderson RN, Flemings P, Losh S, Austin J & Woodhams R (1994) Gulf of Mexico growth fault drilled, seen as oil, gas migration pathway. Oil and Gas Journal, 92,97-104. Appold M & Nunn J (2002) Numerical models of petroleum migration via buoyancy‐driven porosity waves in viscously deformable sediments. Geofluids, 2,233-247. Boles JR, Eichhubl P, Garven G & Chen J (2004) Evolution of a hydrocarbon migration pathway along basin-bounding faults: Evidence from fault cement. AAPG Bulletin, 88,947-970. Bourlange S & Henry P (2007) Numerical model of fluid pressure solitary wave propagation along the décollement of an accretionary wedge: application to the Nankai wedge. Geofluids, 7,323-334. Cartwright JA (1994) Episodic basin-wide fluid expulsion from geopressured shale sequences in the North Sea basin. Geology, 22,447-450.

Connolly JAD (1997) Devolatilization-generated fluid pressure and deformation-propagated fluid flow during prograde regional metamorphism. Journal of Geophysical Research: Solid Earth, 102,18149-18173. Connolly JAD (2010) The Mechanics of Metamorphic Fluid Expulsion. Elements, 6,165-172. Connolly JAD & Podladchikov YY (2000) Temperature-dependent viscoelastic compaction and compartmentalization in sedimentary basins. Tectonophysics, 324,137-168. Davis EE, Becker K, Wang K, Obara K, Ito Y & Kinoshita M (2006) A discrete episode of seismic and aseismic deformation of the Nankai trough subduction zone accretionary prism and incoming Philippine Sea plate. Earth and Planetary Science Letters, 242,73- 84. Fontaine FJ, Rabinowicz M & Boulègue J (2003) Hydrothermal processes at Milos Island (Greek Cyclades) and the mechanisms of compaction-induced phreatic eruptions. Earth and Planetary Science Letters, 210,17-33. Gamage K & Screaton E (2006) Characterization of excess pore pressures at the toe of the Nankai accretionary complex, Ocean Drilling Program sites 1173, 1174, and 808: Results of one- dimensional modeling. Journal of Geophysical Research: Solid Earth, 111,B04103. Haney MM, Snieder R, Sheiman J & Losh S (2005) A moving fluid pulse in a fault zone. Nature, 437,46-46. Henry P (2000) Fluid flow at the toe of the Barbados accretionary wedge constrained by thermal, chemical, and hydrogeologic observations and models. Journal of Geophysical Research: Solid Earth, 105,25855-25872. Holland DS, Leedy JB & Lammlein DR (1990) Eugene Island Block 330 Field--USA Offshore Louisiana. In: Structural traps III: Tectonic fold and fault traps: AAPG Treatise of Petroleum Geology (eds Beaumont EA & Foster NH), Atlas of Oil and Gas Fields, 103- 143. Husen S & Kissling E (2001) Postseismic fluid flow after the large subduction earthquake of Antofagasta, Chile. Geology, 29,847-850. John T, Gussone N, Podladchikov YY, Bebout GE, Dohmen R, Halama R, Klemd R, Magna T & Seitz H-M (2012) Volcanic arcs fed by rapid pulsed fluid flow through subducting slabs. Nature Geoscience, 5,489-492. Joshi A, Appold MS & Nunn JA (2012) Evaluation of solitary waves as a mechanism for oil transport in poroelastic media: A case study of the South Eugene Island field, Gulf of Mexico basin. Marine and Petroleum Geology, 37,53-69. Kodaira S, Iidaka T, Kato A, Park J-O, Iwasaki T & Kaneda Y (2004) High Pore Fluid Pressure May Cause Silent Slip in the Nankai Trough. Science, 304,1295-1298. Losh S, Eglinton L, Schoell M & Wood J (1999) Vertical and lateral fluid flow related to a large growth fault, South Eugene Island Block 330 Field, offshore Louisiana. AAPG Bulletin, 83,244-276. Miller SA, Collettini C, Chiaraluce L, Cocco M, Barchi M & Kaus BJ (2004) Aftershocks driven by a high-pressure CO2 source at depth. Nature, 427,724-727. Revil A & Cathles L (2002) Fluid transport by solitary waves along growing faults: a field example from the South Eugene Island Basin, Gulf of Mexico. Earth and Planetary Science Letters, 202,321-335.

Richardson CN, Lister JR & McKenzie D (1996) Melt conduits in a viscous porous matrix. Journal of Geophysical Research: Solid Earth, 101,20423-20432. Richter FM & McKenzie D (1984) Dynamical Models for Melt Segregation from a Deformable Matrix. The Journal of Geology, 92,729-740. Scott DR & Stevenson DJ (1984) Magma solitons. Geophysical Research Letters, 11,1161-1164. Screaton E, Saffer D, Henry P & Hunze S (2002) Porosity loss within the underthrust sediments of the Nankai accretionary complex: Implications for overpressures. Geology, 30,19-22. Spiegelman M (1993a) Flow in deformable porous media. Part 1 Simple analysis. Journal of Fluid Mechanics, 247,17-38. Spiegelman M (1993b) Flow in deformable porous media. Part 2 numerical analysis–the relationship between shock waves and solitary waves. Journal of Fluid Mechanics, 247,39-63. Tian M & Ague JJ (2014) The impact of porosity waves on crustal reaction progress and CO2 mass transfer. Earth and Planetary Science Letters, 390,80-92. Tobin HJ & Saffer DM (2009) Elevated fluid pressure and extreme mechanical weakness of a plate boundary thrust, Nankai Trough subduction zone. Geology, 37,679-682. Whelan JK, Eglinton L, Kennicutt MC & Qian Y (2001) Short-time-scale (year) variations of petroleum fluids from the US Gulf Coast. Geochimica et Cosmochimica Acta, 65,3529- 3555. Wiggins C & Spiegelman M (1995) Magma migration and magmatic solitary waves in 3-D. Geophysical Research Letters, 22,1289-1292. Xie X, Li S, Dong W & Hu Z (2001) Evidence for episodic expulsion of hot fluids along faults near diapiric structures of the Yinggehai Basin, South China Sea. Marine and Petroleum Geology, 18,715-728. Young A, Monaghan PH & Schweisberger R (1977) Calculation of ages of hydrocarbons in oils-- physical chemistry applied to petroleum geochemistry I. AAPG Bulletin, 61,573-600.

CHAPTER 2: FLUID PRESSURE EVOLUTION DURING SEDIMENTARY

BASIN DIAGENESIS: IMPLICATIONS FOR HYDROCARBON TRANSPORT

BY POROSITY WAVES

Abstract

Numerous sedimentary basins around the world show evidence of hydrocarbons having ascended, commonly episodically, through kilometers of low permeability sediments at rates much higher than expected from Darcy’s Law mechanics. Solitary waves, which are zones of elevated fluid pressure and porosity discharged from compressible sediments undergoing rapid increases in pore fluid pressure, have been hypothesized to be responsible for this anomalously rapid fluid transport. Previous research has shown that solitary waves capable of transporting oil can only form if the ratio of fluid pressure generation rate (Pg) to hydraulic diffusivity (D) exceeds a value of

about 1.1×108 Pa m−2. The purpose of the present study was to explore the rate and magnitude of fluid pressure generation during sedimentary basin diagenesis to determine under what geologic conditions Pg/D ratios can be high enough to form solitary waves.

For the study, a cross-sectional model was constructed to estimate the pressure generation

rate during the evolution of a generic sedimentary basin as a result of sediment

deposition, burial, and compaction, heat transport, kerogen maturation to form

hydrocarbons, quartz cementation, and the flow of oil, gas, and water.

Pressure generation rates were found to vary from 1’s to a maximum of ~510 Pa

year-1, and were caused by compaction disequilibrium and hydrocarbon formation. The

highest pressure generation rates of 100’s of Pa year−1 were achieved under optimal

conditions of high sedimentation rate, sediment compressibility, and basin temperatures,

and low porosity and permeability. These highest pressure generation rates were

sufficient to form oil-transporting solitary waves at depths greater than ~4 km. To form

solitary waves capable of transporting methane, pressure generation rates ~2000 Pa year-1 times greater than those needed for oil are expected to be needed, which are unlikely to be achievable from compaction disequilibrium and hydrocarbon formation, but may be achievable from earthquakes.

2.1. Introduction

The rapid, kilometer-scale ascent of hydrocarbons and other fluids through low permeability sediments has been observed in many sedimentary basins around the world including the Gulf of Mexico basin (Young et al. 1977, Anderson 1993, Anderson et al.

1994, Losh et al. 1999, Roberts 2001, Whelan et al. 2001, Revil & Cathles 2002, Haney et al. 2005), the Yinggehai basin, China (Xie et al. 2001, Xie et al. 2003), the Niger

Delta, Nigeria (Caillet and Batiot 2003), the Santa Barbara Basin, California (Boles et al.

2004), the Uinta basin, Utah (Cathles and Adams 2005), and the North Sea basin

(Cartwright 1994). Previous research suggests that solitary waves could be responsible for this rapid fluid migration, as solitary waves are capable of traveling at velocities orders of magnitude higher than the fluid flow velocities predicted from Darcy’s law

(Rice 1992, Appold and Nunn 2002, Revil 2002, Revil and Cathles 2002, Bourlange and

Henry 2007, Joshi et al. 2012). Solitary waves in elastic porous media are manifest as regions of higher fluid pressure and porosity relative to their surroundings (Figure 2.1).

However, solitary waves are only able to form and propagate when specific conditions are met: (1) the rate of fluid pressure generation is high relative to the rate of fluid pressure diffusion, (2) fluids are highly overpressured, typically greater than 90% of

lithostatic pressure, and (3) the permeability of the

geologic medium is a sensitive function of

effective stress. For solitary waves transporting

oil, the first condition has been found to be

satisfied when the pressure generation rate (Pa

s−1) is at least a factor of 1.1×108 greater than the

hydraulic diffusivity (m2 s−1) (Joshi et al. 2012).

Critical thresholds in the ratio of pressure

generation rate to diffusion rate for the formation Figure 2.1. Conceptual diagram of a porosity wave ascending through a of solitary waves transporting fluids other than oil sedimentary column viewed at times, t1, t2, and t3. The circle size in the wave is probably also exist, though they have not yet been proportional to porosity, which increases with increasing pressure. determined. Therefore, in order to evaluate the feasibility of the solitary wave mechanism, the rate at which fluid pressure can increase to significantly overpressured levels must be known. The purpose of the present study was to explore the rate and magnitude of fluid pressure generation that could develop in a sedimentary basin during deposition, burial, and diagenesis under a geologically reasonable range of conditions.

The subject of fluid pressure evolution in sedimentary basins has long been of interest to the geological community, particularly overpressure development.

Overpressure is important for many reasons, including that: (1) it is commonly associated with the presence of oil and gas (Timko and Fertl 1971, Spencer 1987, Luo and Vasseur 1996, Xie et al. 2001), (2) it retards kerogen maturation, oil-gas cracking,

and clay mineral transformation reactions (Carr 1999, Zou and Peng 2001, Carr et al.

2009, Ugana et al. 2012, Feyzullayev 2013), (3) it can lead to the formation of micro- cracks and veins (Bons 2001, Ferket et al. 2004, Hilgers et al. 2006), (4) it may activate and reactivate faults (Hubbert and Rubey 1959, Raleigh et al. 1976, Rice 1992, Miller et al. 2004, Miller 2007, Calderoni et al. 2009, Garagash et al. 2012), and (5) it can cause landslides (Chapron et al. 2004, Mourgues and Cobbold 2006, Mourgues et al. 2009,

Lacoste et al. 2012, Mourgues et al. 2014). In addition, as noted above, overpressure may lead to the formation of solitary waves, which may lead to enhanced fluid transport in low permeability media and contribute to the formation of ore deposits (Cathles 2007,

Chi et al. 2013), mud volcanoes (Revil 2002, Davies et al. 2011, Feyzullayev 2013), and hydrocarbon fields (Losh et al. 1999, Revil and Cathles 2002, Haney et al. 2005, Jung et al. 2014).

Various factors play a role in forming overpressure in sedimentary basins.

Perhaps the most important is compaction disequilibrium (Dickinson 1953, Chapman

1980, Chapman 1982, Sharp 1982, Hunt 1990, Powley 1990, Hart et al. 1995, Gordon and Flemings 1998, Swarbrick et al. 2000, Vejbæk 2008, Marίn-Moreno et al. 2013,

Zhang 2013). According to this mechanism, as compressible, low permeability sediment is buried, pore fluid is unable to escape the pore space rapidly enough to keep pace with the pore space collapse, leading to pore pressure increase. In the Gulf of Mexico basin, for example, compaction disequilibrium may have generated pore fluid pressures close to lithostatic (Bredehoeft and Hanshaw 1968, Sharp and Domenico 1976, Bethke 1986,

Harrison and Summa 1991, Hart et al. 1995). Overpressure generation can be augmented by the existence of fluid sources. The maturation of kerogen to form hydrocarbons and thermal cracking of oil to gas can be important contributors to overpressure (Barker 1990,

Luo and Vasseur 1996, Swarbrick and Osborne 1998, Swarbrick et al. 2002) in many sedimentary basins (Table 2.1). Overpressure development may also be augmented by processes that reduce porosity and permeability, for example, through the cementation of pore space by clay minerals, quartz, or calcite and the conversion of smectite to illite during diagenesis (Towe 1962, Boles and Frank 1979, Walderhaug 1996, Bjorlykke and

Hoeg 1997, Bjorkum et al. 2001, Broichhausen et al. 2005, Nadeau et al. 2005, Thyberg et al. 2010, Thyberg and Jharen 2011, Lahann and Swarbrick 2011, Opara 2011, Goulty et al. 2012, Taylor and Macquaker 2014).

Table 2.1. Case studies from various sedimentary basins predicted high pore fluid pressures from conversion of oil to natural gas.

Location References

Anadarko basin, Oklahoma, USA Lee and Deming (2002), Deming et al. (2002)

Athabasca basin, Canada Chi et al. (2013)

Danish North Sea Vejbæk (2008)

Delaware basin, Delaware, USA Lee and Williams (2000)

Laramide basin, Wyoming, USA Surdam et al. (1997)

Malay basin, Malaysia Tingay et al. (2013)

Precaspian basin, Kazakhstan Anissimov (2001)

Paleozoic Anticosti basin, eastern Canada Chi et al. (2010)

Uinta basin, Utah, USA McPherson and Bredehoeft (2001)

Numerous case studies have attempted to model fluid pressure evolution in sedimentary basins (Table 2.2). These studies suggest pore fluid pressure generation rates of 1’s to 10’s of Pa year−1 to be common in sedimentary basins, though significantly

11 higher rates may be possible. For example, a study of the Neuquen basin in Argentina by

Monreal et al. (2009) predicted overpressures of 5.5 MPa to have been reached nearly instantaneously with the onset of hydrocarbon generation.

Table 2.2. Case studies of pore fluid pressure evolution from hydrocarbon generation in various sedimentary basins around the world.

Location References

Anadarko basin, Oklahoma, USA Deming et al. (2002), Lee and Deming (2002)

Athabasca basin, Canada Chi et al. (2013)

Beaufort-Mackenzie basin, Canada Chen et al. (2010)

Big Horn basin, Wyoming, USA Beaudoin et al. (2013)

Bohai bay basin, northern China Xie et al. (2001), Guo et al. (2010)

Bonaparte basin, northern Australia Fadul et al. (2012)

Danish central graben, Danish North Vejbæk (2008) Sea, Denmark

Delaware basin, Delaware, USA Lee and Williams (2000), Hansom and Lee (2005)

East Java basin, Indonesia Tanikawa et al. (2010)

Eastern Black Sea basin, SE Europe Scott et al. (2009), Marin-Moreno et al. (2013)

Gulf of Mexico basin, USA Bethke (1986), Hart et al. (1995), Gordon and Flemings (1998), Stump and Flemings (2000), Finkbeiner et al. (2001)

Norwegian Sea basin, Norway Borge (2002)

Junggar basin, northwest China Luo et al. (2007)

Laramide basin, Wyoming, USA Surdam et al. (1997)

Lower Kutai basin, Indonesia Ramdhan and Goulty (2010), Goulty et al. (2012)

Malay basin, Malaysia Oconnor et al. (2011), Tingay et al. (2013)

Navarin basin, Alaska, USA Bour & Lerche (1994)

Neuquen basin, Argentina Monreal et al. (2009)

North Louisiana salt basin, Louisiana, Nunn (2012) USA

Ordos basin, northwest China Xue et al. (2011)

Paleozoic Anticosti basin, eastern Chi et al. (2010) Canada

Qiongdongnan basin, South China Shi et al. (2013)

San Joaquin basin, California, USA Wilson et al. (1999)

Sichuan basin, southwestern China Liu et al. (2014)

South Caspian basin Feyzullayev (2013)

South Oman salt basin, Oman Kukla et al. (2011)

Taranaki basin, New Zealand Webster et al. (2011)

Uinta basin, Utah, USA McPherson and Bredehoeft (2001)

Viking graben, southeast Norway Swarbrick et al. (2000), Bolas et al. (2004), Broichhausen et al. (2005)

Yinggehai basin, South China Zhang et al. (2013)

In order to define more clearly the geologic conditions under which solitary waves could form in sedimentary basins, a detailed sensitivity analysis was undertaken to investigate the effect of variations in basin hydromechanical and geochemical parameters on pressure generation rate. A previous study of this type has been performed by

Hansom and Lee (2005). Part of their study consisted of a generic one-dimensional model of the deposition and compaction of a 10 km thick shale sequence. Pressure generation rates were highest during periods of oil and gas generation, and reached 50 Pa year−1. Hansom and Lee (2005) also carried out a two-dimensional model of the evolution of the Delaware Basin. Here too, pressure generation rates were highest during

periods of oil and gas generation, and reached 2 Pa year−1. Sensitivity analysis took into account the effects of variations in sedimentation rate, heat flow, and the kinetics of the

conversion of kerogen to hydrocarbons, but did not take into account the effects of

variations in sediment permeability, compressibility, or the conversion of smectite to

illite and quartz cementation, all of which can significantly affect hydraulic diffusivity,

i.e. the diffusion of fluid pressure in porous media. In addition, it is unclear whether or

not Hansom and Lee (2005) included a petroleum expulsion mechanism (e.g. capillary

pressure or fracture induced) in their model, which would further affect pressure

evolution in the basin.

The present manuscript presents the results of a two-dimensional numerical

modeling study of a hypothetical 20 km long × 5 km deep sedimentary basin in which

pore fluid pressure evolution, multi-phase fluid flow, and heat transport were calculated

as a function of sediment deposition, burial, and diagenetic processes including

compaction, kerogen maturation, hydrocarbon formation and quartz cementation. The

results include an analysis of the sensitivity of pore pressure evolution to permeability,

sediment compressibility, total organic carbon content, heat flow, and sedimentation rate.

Though the study is generic in design rather than based on a specific field site, the results

are broadly representative of sedimentary basinal processes and thus yield insights into the rates of pressure generation that are possible in sedimentary basins within geologically reasonable limits, which have important implications for solitary wave genesis.

2.2. Theoretical background

Pore fluid pressure evolution was modeled using the BasinMod 2DTM software.

The model reconstructed the chronologic deposition of petroleum-rich source rock in a shale dominated sedimentary basin and calculated the dynamic evolution of pore fluid pressures in the basin, taking into account the effects of sediment compaction, diagenesis, kerogen maturation, hydrocarbon formation, heat flow, and associated multi-phase fluid flow of water, oil, and gas.

Dynamic sediment compaction in the model was based on the static exponential porosity-depth relationship of Sclater and Christie (1980), modified to incorporate the effect of temporal changes in fluid pressure. Chemical compaction was expressed as

porosity reduction caused by quartz cementation following the theory of Walderhaug

(1996). Coupled compaction-driven fluid pressure evolution, fluid flow, and heat

transport were calculated based on the theory presented by Bethke (1985), with the addition of saturation-dependent relative permeability and capillary pressure relationships to account for the presence of multiple fluid phases. Sediment permeability was calculated as a function of sediment grain size using a modified Kozeny-Carman equation

(Ungerer et al. 1990). Bulk thermal conductivity was calculated from the temperature dependent water and rock matrix thermal conductivities (Deming and Chapman 1989).

The formation of hydrocarbons from kerogen maturation was based on the kinetic model of Burnham (1989) for Type II kerogen. Expulsion of hydrocarbons from the source rock occurred in the model when one of the following two conditions was met: (1) the pore fluid pressure exceeded the capillary pressure (Schowalter 1979) or, (2) the pore fluid pressure exceeded a critical pressure for fracture formation,

(2.1)

which led to an instantaneous increase in the permeability of the source rock, followed by

a gradual decrease in permeability according to a prescribed fracture closure rate of 0.05 per million years (Myr) (BasinMod 1-D® 2009, BasinMod 2-D® 2009, Düppenbecker et

al. 1991). In equation (1), is the pore fluid pressure, is the effective stress, and is

a fracture gradient factor that varies from 0 to 1 and expresses the ratio of pore fluid

pressure to lithostatic pressure.

2.3. Model construction

The hypothetical model sedimentary basin in the present study consisted of a

rectangular space with 5 km depth × 20 km length filled entirely with shale (Figure 2.2).

The cross-sectional model domain was discretized to form a numerical grid with a constant horizontal nodal spacing of 50 m. The vertical nodal spacing was a function of

sedimentation rate, which was varied in the model, and time step size, which was held

constant in the model at 20,000 years.

In order to evaluate the possible ranges of pressure generation rates in the model

sedimentary basin, three different scenarios were investigated (Table 2.3), beginning with

a “base case scenario” based on conditions in the Eugene Island 330 minibasin in the

Gulf of Mexico, which has been the focus of much previous solitary wave research

(Appold and Nunn 2002, Revil and Cathles 2002a, b, Joshi et al. 2012). A “high pressure

generation rate” and a “low pressure generation rate” scenario were also simulated in

order to estimate maximum and minimum possible pressure generation rates,

respectively, within geologically reasonable ranges of parameter values.

Figure 2.2. Model cross-section of the hypothetical sedimentary basin with hydrocarbon source rock at the bottom and composed entirely of shale.

A constant matrix thermal conductivity value was applied throughout the model basin in each scenario. For the base case scenario, the matrix thermal conductivity value

was calculated based on a geothermal gradient of 33°C km−1 for the north-central Gulf of

Mexico basin (Nunn et al. 1984), a basement heat flow of 60 mW m−2, and an initial

porosity of 40%. For the base case scenario, permeability was set to 10−18 m2 and

compressibility to 10−8 Pa−1, which are values near the middle of the logarithmic ranges

for shale (Freeze and Cherry 1979, Neuzil 1994).

Similar boundary conditions were applied in all of the model scenarios. The base of the model domain was assigned a no-flow boundary condition for all fluids based on the compacted nature of the sediments, and was also assigned a specified heat flux, which

for the base case scenario was 60 mW m−2. The lateral boundaries were assigned to be

insulating with respect to heat flow but open with respect to fluid flow, with pressure assigned to be hydrostatic. The upper boundary of the model was assigned hydrostatic

pressure and a temperature of 5° C.

Table 2.3. Parameter values used in the three model scenarios.

Model Parameters Base Case Low Pressure High Pressure Generation Rate Generation Rate

Sedimentation rate (mm yr−1) 1.39 0.139 13.9

Initial Porosity (fraction) 0.4 0.5 0.2

Shale Compressibility (Pa-1) 10−8 (0.16 km−1) 10-9 (0.016 km−1) 10-7 (1.6 km−1)

Basement heat flow (mW m−2) 60 50 120

Kerogen Type Type II Type II Type II (Espitalie ’88 Viking Graben) (Burnham ’89) (Burnham ‘89)

% TOC 5 5 0.5

Shale Permeability (m2) 10−18 10−17 10−19

Surface Temperature (°C) 5 5 10

Matrix Thermal Conductivity 1.82 3.64 0.91 (W m−1 C−1)

In all of the model scenarios, hydrocarbons originated from a 300 m thick interval

of source rock that was buried to a final depth of 4.5 km, analogous to the Eugene Island

330 minibasin (Holland et al. 1990). In the base case scenario, the total organic carbon

content (TOC) of the source rock was 5% and consisted of Type II kerogen.

2.4. Results

2.4.1. Base case scenario

Simulation results for the base case scenario are shown in Figures 2.3‒2.7. Figure 2.3

shows sediment accumulation in the basin at 3.1 Ma, 1.5 Ma, 1.0 Ma and 0 Ma.

Deposition of the shale in the simulation began at 3.6 Ma at a rate of 1.39 mm yr−1 and continued until the present day, producing a five kilometer-thick basin. Deposition of

hydrocarbon-rich source rock in the simulation began at 3.46 Ma and continued until 3.24

Ma, producing a 300 m thick interval at a final depth of 4.5 to 4.8 km.

Figure 2.3. Model stratigraphic evolution of the hypothetical sedimentary basin at (a) 3.1 Ma, (b) 1.5 Ma, (c) 1.0 Ma, (d) 0 Ma. Violet blue and light gray colors represents shale and hydrocarbon source rock, respectively.

Figure 2.4 shows the evolution of subsurface temperature at the thickness

midpoint of the source rock. The temperature of the source rock increased by

approximately 70°C Myr−1 in the simulation, bringing the source rock into the oil

window by about 1.7 Ma, after which hydrocarbon formation could augment fluid

pressure generation previously caused only by compaction disequilibrium.

Figure 2.4. Temperature evolution at the center of the source rock for three different model scenarios. The source rock temperature change is greatest for the high pressure generation rate scenario and lowest for the low pressure generation rate scenario.

Figure 2.5a shows a plot of oil and gas saturation as a function of time. Although the source rock entered the oil window at about 2 Ma, chemical reaction kinetics delayed meaningful oil formation until about 1.68 Ma. A peak oil saturation of 0.15 was reached at 1.02 Ma. At later simulation times, oil was gradually destroyed by higher temperatures such that oil saturation became zero by about 0.25 Ma. Formation of natural gas began at

1.56 Ma, with saturation increasing steadily to 0.2 by 0.22 Ma, remaining constant thereafter until the present day. Figures 2.5b and 2.5c, which show oil and gas saturation in the context of basin thickness over time, indicate that oil and natural gas remained confined to the source rock interval during the simulation and did not migrate to shallower depths.

Figure 2.6 shows the evolution of overpressure at 3.1 Ma, 1.5 Ma, 1.0 Ma and 0

Ma. Pore fluid pressures were close to hydrostatic (zero overpressure) throughout the 1 km depth of the basin at 3.1 Ma, and remained close to hydrostatic in the upper 500 m of

20 the basin thereafter in the simulation. At greater depths by 1.5 Ma and later in the simulation, significant overpressures developed as a result of compaction disequilibrium.

Figure 2.5. Saturation evolution of (a) oil and natural gas in the middle of the source rock. Oil generation started at 1.68 Ma, reached a peak saturation of 0.15 at 1.02 Ma, and declined to zero saturation at the present time. Gas production started at 1.56 Ma and reached a peak saturation of 0.2 at the present time. Saturation plots for (b) oil and (c) natural gas as a function of age and depth. Neither oil nor gas migrated significantly from the source rock.

The formation of hydrocarbons after 1.68 Ma produced a sharp further increase in overpressure in the hydrocarbon source interval in the lower 500 m of the basin. By 0

Ma in the simulation, overpressures reached ~42 MPa in the lower 500 m of the basin,

where compaction disequilibrium accounted for about 92% of the total overpressure with

the remaining overpressure coming from hydrocarbon generation. Overpressure from

hydrocarbon generation reached a maximum value of 5.6 MPa at 0.24 Ma.

Figure 2.6. Evolution of overpressure in the model basin at (a) 3.1 Ma, (b) 1.5 Ma, (c) 1.0 Ma, (d) 0 Ma for the base case scenario.

Figures 2.7a and 2.7b show the evolution of pore fluid pressure and pressure

generation rate resulting from compaction disequilibrium and hydrocarbon formation.

Prior to about 1.68 Ma, fluid pressure was generated primarily by compaction

disequilibrium at a rate of ~20 to 25 Pa year−1.

Figure 2.7. Evolution of (a) pore pressure and (b) pressure generation rates in the hydrocarbon source rock resulting from compaction disequilibrium and hydrocarbon formation for the base case scenario.

After about 1.68 Ma, the kerogen-rich source rock was buried deeply enough to allow

hydrocarbons to start forming, causing episodic increases in fluid pressure and fluid

pressure generation rate up to ~50 Pa year−1, at least twice as high as produced by

compaction disequilibrium alone. The episodic increases in fluid pressure generation rate

were followed by strong decreases caused by hydrofracturing when pore pressure reached the critical pressure of equation (2.1). Hydrofracturing increased the permeability of the

source rock, causing both fluid pressure and fluid pressure generation rate to decline.

Once permeability had decreased sufficiently due to closure of the fractures, fluid

pressures gradually increased again. A total of three such episodes of fluid pressure

build-up and dissipation instigated by hydrocarbon formation were predicted in the base

case simulation.

2.4.2. High pressure generation rate scenario

In order to calculate the likely maximum pressure generation rate possible from compaction disequilibrium and hydrocarbon formation, a “high pressure generation rate”

scenario was simulated in which model parameters were altered to values at or near the

limits of documented geologic ranges so as to maximize overpressure and the rate of fluid pressure generation. Compared to the base case scenario, the sedimentation rate and

shale compressibility were increased by an order of magnitude, the % TOC and shale

permeability were decreased by an order of magnitude, basement heat flow and surface

temperature were doubled, and the initial porosity and matrix thermal conductivity were

halved.

Higher sedimentation rates lead to higher rates of build-up of total stress, which provided that hydraulic diffusivity is low, will translate to higher rates of fluid pressure

generation. High basin sedimentation rates are on the order of 10’s of mm year−1 (Soria et al. 1999, Vallius 1999, White et al. 2002). As compressibility increases, hydraulic diffusivity decreases, which prevents the diffusion of pressure during compaction and thus increases fluid pressure generation rate. A compressibility value of 10−7 Pa−1 is near

the maximum for geologic media and corresponds to unconsolidated clay (Ingebritsen

and Appold 2012). Fluid pressure generation is increased by rapid reaction kinetics for

kerogen maturation, leading to the choice of Espitalie et al.’s (1988) kinetic model.

Sensitivity analysis showed that the lower % TOC used in the present scenario compared

to the base case scenario maximized pressure generation from hydrocarbon formation.

Permeability in the model decreases with decreasing porosity according to a modified

Kozeny-Carman equation. As permeability decreases, hydraulic diffusivity decreases,

increasing fluid pressure generation rate until permeability is so low that fluids are

effectively immobile, after which no further increases in fluid pressure generation rate

occur. The shale permeability value of 10−19 m2 in the present scenario is near the lower

limit of 10−20 m2 reported by Freeze and Cherry (1979). High basement heat flow and

low sediment thermal conductivity increase the geothermal gradient so that kerogen-rich

source rock would be heated more quickly during burial, increasing the rate of

hydrocarbon generation and thus, the rate of fluid pressure generation.

Figure 2.4 shows that the rate of source rock temperature change in the present

scenario was 410°C Myr−1, which is about six times greater than the rate in the base case

scenario. Figure 2.8a shows plots of oil and natural gas saturation as a function of time.

Formation of oil began at about 0.09 Ma, reaching a peak saturation of 0.14 at 0.01 Ma,

and declining to a saturation of 0.09 at the present day.

Figure 2.8. Evolution of oil and natural gas saturation for the (a) high and (b) low pressure generation rate scenarios.

Natural gas formation began at about 0.04 Ma and continued to increase to the present

day to a saturation of 0.11. In contrast to the base case scenario, oil generation

dominated over natural gas generation throughout the period of hydrocarbon generation.

This is because the source rock in the high pressure generation rate scenario did not

become hot enough to generate much gas, in contrast to the base case scenario, in which

the present day source rock temperature was approximately 90° C higher. Although the

basement heat flow was higher, rapid deposition of cooler sediments in the basin could

have decreased source rock temperature in the high pressure generation rate scenario

compared to the base case scenario.

Figure 2.9 shows the evolution of pore pressure and pressure generation rate in

the hydrocarbon source rock resulting from compaction disequilibrium and hydrocarbon

formation. Pore pressure from hydrocarbon generation alone reached a maximum value

of 6.1 MPa at the present day. Pore pressures from compaction disequilibrium and

hydrocarbon generation combined reached a maximum value of ~105 MPa at the present

day. Pressure generation rate from compaction disequilibrium and hydrocarbon

formation combined reached a maximum value of 510 Pa year−1 approximately twice that

attained from compaction disequilibrium alone.

Figure 2.9. Evolution of (a) pore pressure and (b) pressure generation rates in the hydrocarbon source rock resulting from compaction disequilibrium and hydrocarbon formation for the high pressure generation rate scenario.

2.4.3. Low pressure generation rate scenario

Simulation results for the low pressure generation rate scenario are shown in

Figures 2.8b and 2.10. Model parameter value changes included an order of magnitude

26 decrease in sedimentation rate and shale compressibility, an order of magnitude increase in shale permeability, a two-fold increase in matrix thermal conductivity, an increase in initial porosity from 0.4 to 0.5, and a decrease in basement heat flow from 50 to 40 mW m−2. Parameter values were chosen so as to depress fluid pressure generation from compaction disequilibrium and hydrocarbon formation. The TOC, kerogen type, and surface temperature were the same as in the base case scenario.

Figure 2.10. Evolution of (a) pore pressure and (b) pressure generation rates in the hydrocarbon source rock resulting from compaction disequilibrium and hydrocarbon formation for the low pressure generation rate scenario.

Consistent with the lower rate of sedimentation and burial, the 4°C Myr−1 rate of temperature change in the hydrocarbon source rock for the low pressure generation rate scenario was much lower than the 70°C Myr−1 rate in the base case scenario (Figure 2.4).

The lower rate of temperature increase in the low pressure generation rate scenario delayed hydrocarbon generation (Figure 2.8b). Oil formation began at about 14.25 Ma, reaching a maximum saturation of 0.15 at 2.4 Ma, and then declining slightly thereafter to a saturation of 0.14 at the present day. Generation of natural gas began at about 12.3

Ma, increasing steadily to a maximum saturation of 0.06 at the present day, which is

about half of the maximum saturation reached in the base case scenario.

Figure 2.10 shows the evolution of pore pressure and pressure generation rate

resulting from compaction disequilibrium and hydrocarbon formation. The results show

that most of the pore pressure is generated from compaction disequilibrium, with

hydrocarbon formation contributing a maximum of 0.67 MPa at about 4.65 Ma,

decreasing thereafter to 0.43 MPa at the present day. Overall pore pressure reaches a

maximum value of approximately 80 MPa at the present day. Pore pressure generation

rate declines slightly but steadily throughout the simulation from about 2.6 Pa year−1 at

30 Ma to about 1.8 Pa year−1 at the present day. This contrasts with the rapid episodes of

pore pressure increase caused by hydrocarbon formation in the base case scenario, which

were suppressed in the present scenario by the low heating rate.

The results of this model scenario show that when sedimentation rate and

compressibility are low, the rate of sediment compaction is low which slows the rate of

pore pressure generation. Low sediment compressibility combined with high

permeability leads to high hydraulic diffusivity, which reduces pore pressure build-up in

the hydrocarbon source rock. Lower temperatures caused by higher thermal conductivity and lower basement heat flux suppress hydrocarbon formation, another source of pore pressure generation. Thus, sedimentary basins with these characteristics, despite having

high organic carbon content, would be unfavorable for solitary wave formation.

2.5. Discussion

Previous research by Joshi et al. (2012) showed that porosity waves whose pore

space was saturated with oil could form and propagate in sedimentary basins undergoing

28 compaction disequilibrium and hydrocarbon formation where the ratio of pressure

8 −2 generation rate (Pg) to hydraulic diffusivity (D) exceeded 1.1×10 Pa m . Figure 2.11a shows the Pg/D ratio as a function of time for the three model scenarios. The Pg/D ratio for the base case scenario remained relatively constant at about 102 Pa m−2 throughout the simulation and never reached the threshold value for porosity wave formation. In contrast to the base case scenario, the Pg/D ratio for the low pressure generation rate scenario decreased over time from ~ 10 Pa m−2 at the start of the simulation to a value of

−2 ~ 0.1 Pa m because of the decrease in Pg in the source rock. The trend of decreasing

Pg/D ratio over time is probably mainly a function of the trend of continuously decreasing

Pg over time in the absence of significant hydrocarbon production. Because in general the rate of compaction decreases over time, pressure generation rate in all three model scenarios also decreases over time while compaction disequilibrium dominates as the source of fluid pressure generation. In the base case and high pressure generation rate scenarios, the trend is reversed when hydrocarbon generation occurs. In the high

5 −2 pressure generation rate scenario, Pg increased enough from 3×10 Pa m at the start of

10 −2 the simulation to a maximum of 3.5×10 Pa m at 0.05 Ma to raise the Pg/D ratio above the threshold value of 1.1×108 Pa m−2 for porosity wave formation by 0.07 Ma.

Figure 2.11b shows the Pg/D ratio as a function of depth at the present day for the high pressure generation rate scenario. The threshold value for porosity wave formation is only exceeded at depths below 4.2 km. Because porosity and permeability increase with decreasing depth, hydraulic diffusivity becomes too high for porosity waves to exist at shallower depths given the Pg values possible from compaction disequilibrium and hydrocarbon generation.

Figure 2.11. (a) Ratio of pressure generation rate to hydraulic diffusivity in the hydrocarbon source rock as a function of time for the three model scenarios. (b) Ratio of pressure generation rate to hydraulic diffusivity as a function of depth at the present day for the high pressure generation rate scenario. The solid purple line represents the minimum ratio needed for the formation of porosity waves saturated with oil.

Figure 2.11 has important implications for the role of porosity waves as agents of

oil transport in sedimentary basins. The results indicate that porosity waves are unlikely

to be able to form in oil-rich porous media over a wide range of common sedimentary

basin conditions. Although compaction disequilibrium is able to produce the high

magnitude of overpressure needed for porosity wave formation, it is generally not able to

produce the high rates of pressure generation needed for porosity wave formation. These

high rates are only achieved when hydrocarbon formation begins contributing to pressure

generation. Moreover, porosity wave formation is expected to be limited to deep parts of

sedimentary basins, because with decreasing depth, porosity, permeability, and therefore

hydraulic diffusivity will increase to a point where the Pg/D ratio is too small for porosity

waves to form. Increasing hydraulic diffusivity with decreasing depth would be expected

to limit not only porosity wave formation but also porosity wave migration to greater depths within sedimentary basins, because at shallower depths the waves’ pressure

diffuses away faster than the waves ascend, causing the waves to fade into the

background. Thus, porosity waves may be effective at transporting oil from source rocks

to relatively deep reservoirs, but not to shallow reservoirs in sedimentary basins.

The base case and high pressure generation rate scenarios show mild episodicity in fluid pressure generation rate, with three and two spikes in fluid pressure generation rate predicted in the simulations, respectively (Figures 2.7 and 2.9). This episodicity in pressure generation rate may have implications for porosity wave formation frequency, as porosity waves are more likely to form when pressure increases sharply rather than gradually. The frequency of these sharp pressure generation rate episodes is a function of

the pressure threshold for fracture opening and the rate of fracture closure. The low

frequency of sharp pressure generation rate increases suggests that the frequency of

porosity wave formation is also likely to be low in sedimentary basins, which would limit

porosity waves’ effectiveness at charging reservoirs with oil.

The results of the present study also have implications for the question of whether or not porosity waves could be effective at transporting methane. If the Pg/D ratio of

1.1×10−8 Pa m−2 is the critical threshold for the formation of porosity waves carrying

methane as it was for porosity waves carrying oil, then much higher pressure generation

rates would be needed to form porosity waves in methane-saturated sediments than oil-

saturated sediments because of the much higher hydraulic diffusivity of methane-

saturated sediments. The higher hydraulic diffusivity is a function of the lower viscosity

and higher compressibility of methane as can be seen from equations (2.2)-(2.4), where

hydraulic diffusivity is defined as

(2.2)

where K is the hydraulic conductivity defined as,

(2.3)

and Ss is the specific storage defined as,

. (2.4)

where g is gravitational acceleration, k is intrinsic permeability, ρf is fluid density, μf is fluid viscosity, βb is the compressibility of the bulk porous medium, βf is fluid

compressibility, and φ is porosity. Thus, because the viscosity of methane is about two

orders of magnitude lower than that of oil, and the compressibility of methane is about 20

times greater than that of oil at a pressure of 80 MPa and a temperature of 150° C, the

formation of porosity waves in methane-saturated sediments would require a pressure

generation rate ~2000 Pa year−1 times greater than the rate needed to form porosity waves

in oil-saturated sediments. Because the maximum pressure generation rates predicted in

the current modeling are only on the order of ~510 Pa year−1, compaction disequilibrium

and hydrocarbon formation would seem unlikely to be able to generate pressure fast

enough to form porosity waves in methane-rich sediments.

However, a hypothesis proposed here for further testing is that earthquakes could cause pressure generation rates high enough in sedimentary basins to produce porosity waves in methane- as well as oil-saturated sediments. Observations of pore fluid pressure fluctuation in monitoring wells from around the world show that earthquake could raise the pore fluid pressure virtually instantaneously from a geologic perspective (Kobayashi

1971, Manga et al. 2003, Wang and Chia 2008, Cox et al. 2012, Shi et al. 2014).

Episodic expulsion of methane from compacted lacustrine sediments in the Lake Baikal

(Rensbergen et al. 2002) and the glacial sediments in the southeast Bering Sea (Cook et

al. 2011) could have triggered by seismic shaking or earthquake induced instantaneous

rise in pore fluid pressures.

2.6. Conclusions

Two-dimensional numerical modeling of the evolution of a generic sedimentary

basin shows that the combination of compaction disequilibrium and hydrocarbon

formation can increase pore pressures at rates fast enough relative to the rate of pore

pressure diffusion to form porosity waves capable of transporting oil under only a

relatively narrow range of geologic conditions. These conditions include high

sedimentation rate, sediment compressibility, and basin temperatures, and low porosity

and permeability. Conditions for forming porosity waves capable of transporting oil

were predicted to occur mainly at depths below about 4 km and late in the evolution of

the basin. Compaction disequilibrium and hydrocarbon formation were found to be

−1 capable of generating fluid pressure at a rate of up to 510 Pa year . If the same Pg/D ratio of 1.1×108 Pa/m2 is needed to form porosity waves transporting methane as is

needed to form porosity waves transporting oil, then because of the much higher

hydraulic diffusivity of methane-saturated sediments compared to oil-saturated

sediments, pressure generation rates needed to form porosity waves to transport methane would be about 2000 times greater than the pressure generation rates needed to form

porosity waves to transport oil. Thus, based on the results of the present study, a more

rapid pressure generating mechanism than compaction disequilibrium and hydrocarbon

formation is needed to form porosity waves capable of transporting methane in

sedimentary basins, as well as oil at shallow depths in sedimentary basins.

References

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CHAPTER 3: POTENTIAL OF POROSITY WAVES FOR METHANE

TRANSPORT IN THE EUGENE ISLAND FIELD OF THE GULF OF MEXICO

BASIN

Abstract

The Eugene Island minibasin and hydrocarbon field in the northern Gulf of

Mexico basin is a well-documented occurrence of anomalously rapid fluid migration. A variety of evidence suggests that hydrocarbons ascended through kilometers of very low permeability sediments separating source rocks and reservoirs at velocities up to kilometers per year. Previous research has shown that porosity waves could transport petroleum at millimeter per year velocities under a narrow range of low permeabilities.

The purpose of the current research was to test the hypothesis that porosity waves could transport methane much more rapidly than oil.

To test this hypothesis, a one-dimensional numerical model of porosity wave behavior was constructed for Eugene Island sediments saturated with methane. The results show that the gradual rates of pore fluid pressure generation typically caused by diagenesis are too slow for porosity waves to form. Instead, essentially geologically instantaneous pore fluid pressure increases are needed, which then could allow porosity waves to ascend 1-2 km at velocities >10’s of m/year. Thus, porosity waves could only reach the lower reservoirs at Eugene Island, but could transport methane orders of magnitude faster than the background Darcian flow regime. Based on their predicted size, 10’s of porosity waves would have been needed to charge these lower reservoirs with their original amount of methane. Whether a mechanism for instantaneous pore fluid pressure generation exists at Eugene Island is unclear. Earthquakes are capable of

generating essentially instantaneous pore fluid pressure increases of order 1’s of MPa or

greater, although Eugene Island’s seismic history is thought to have been relatively quiet.

Thus, despite their high velocities, porosity waves are unlikely to have played a major

role in transporting methane at Eugene Island, but could have in other more seismically

active locations where required methane transport distances are smaller.

3.1. Introduction

The seeming ability of hydrocarbons to ascend quickly through thick sequences

of low permeability sediments that separate hydrocarbon source rocks from reservoirs has

been widely reported for the northern Gulf of Mexico basin (Young et al. 1977; Anderson

et al. 1994; Losh et al. 1999; Whelan et al. 2001; Haney et al. 2005). One of the best

documented examples is the Eugene Island minibasin and hydrocarbon field, where the

principal hydrocarbon reservoirs are Plio-Pleistocene sands lying at depths of about 0.5 to

3 km, but the principal hydrocarbon sources are no younger than Early Tertiary sediments

located at depths of at least 4.5 km. The hydrocarbon sources and reservoirs are

separated by muds and shales whose permeabilities decrease with depth to perhaps as low

as 10−24 m2 at 4.5 km depth (Revil and Cathles 2002). Despite such low permeabilities, a variety of geochemical and geophysical evidence suggests that hydrocarbons may have

ascended through these sediments at rates of at least millimeters per year to as high as

kilometers per year (Roberts and Nunn 1995; Losh et al. 1999; Guerin 2000; Roberts

2001; Whelan et al. 2001; Revil and Cathles 2002; Nunn 2003; Haney et al. 2005)

compared to rates of millimeters per million years predicted from Darcy’s law (Joshi et

al. 2012). This research and that of Lin and Nunn (1997) suggest that most of the

hydrocarbon migration is likely to have been focused along the Red fault, a major growth

fault along the eastern margin of the Eugene Island minibasin.

Previous researchers have proposed that porosity waves could be responsible for the anomalously rapid hydrocarbon migration observed at Eugene Island. Appold and

Nunn (2002) investigated the capability of porosity waves to transport oil in viscous sediments by building a one–dimensional numerical model that simulated the burial of an organic rich sedimentary layer and the conversion of kerogen to oil. Their results showed porosity waves to form as zones of elevated liquid fraction (i.e. fluid–filled porosity)

where rates of hydrocarbon generation were high. The waves ascended due to buoyancy at rates up to about one millimeter per year. Revil and Cathles (2002) derived an analytical solution for porosity wave velocity from differential equations for fluid mass conservation in elastic porous media. Porosity waves in elastic porous media are manifest as discrete pulses of elevated fluid pressure and porosity that rise along the pressure gradient in the sedimentary succession. Revil and Cathles (2002) predicted that porosity waves traveling along the Red fault under high overpressures and whose pore

space was saturated with a dense (e.g. 800 kg m−3) fluid could reach velocities of 1’s to

10’s of km year−1. Joshi et al. (2012) carried out a numerical modeling investigation of

porosity wave behavior in highly overpressured, oil-saturated, elastic porous media.

Porosity waves formed where the hydraulic diffusivity was low relative to the rate of pore

pressure generation, which was caused by compaction disequilibrium and hydrocarbon

generation. Joshi et al. (2012) found that porosity waves could only form under a narrow

range of low permeabilities between 10−25 and 10−24 m2, could only ascend distances of

between one and two kilometers before diffusing into the background, traveled at

maximum velocities of millimeters per year, and had a low frequency of formation.

Thus, they concluded that porosity waves are unlikely to have been an important

mechanism for transporting oil at Eugene Island. However, Joshi et al. (2012)

hypothesized that porosity waves may be much more effective at transporting methane

because of its lower density and viscosity compared to oil. The purpose of the present

study was to investigate that hypothesis.

The generation of porosity waves in elastic porous media is principally governed

by the ratio of the pressure generation rate (Pg) to the hydraulic diffusivity (D). Joshi et al. (2012) found that in oil–saturated porous media, porosity waves could form when the

8 −2 Pg/D ratio exceeds about 10 Pa m . When the Pg/D ratio is lower, then pore pressures

diffuse away into the surroundings before a coherent porosity wave can coalesce and

depart from its pressure generation source zone. Joshi et al. (2012) predicted an average

pressure generation rate of about 30 Pa year−1 in the hydrocarbon source rocks at Eugene

Island, caused primarily by compaction disequilibrium with minor contributions from

hydrocarbon generation late in the history of the minibasin. This low pressure generation

rate is why Joshi et al. (2012) found porosity wave formation to be limited to sediments

with the very low permeabilities on the order of 10−24 to 10−25 m2, as noted above, in

order to raise sufficiently the Pg/D ratio, where

(3.1)

where, k is the intrinsic permeability, ρf and μf are density and viscosity of fluid

respectively, g is the acceleration due to gravity, and Ss is the specific storage of the

porous medium. If a similar Pg/D ratio were needed to form porosity waves in methane–

saturated sediments, then given the high hydraulic diffusivity of methane–saturated porous media, a pressure generation rate of ~2000 Pa year−1 would be required.

The purpose of the current research was to test the hypothesis that porosity waves could be effective at transporting methane from deep source rocks to much shallower reservoirs in the Eugene Island minibasin. To test that hypothesis, the research sought to

answer the questions of how large porosity waves can get, how frequently can they form,

how fast and far can they travel, and how much methane can they transport. The research

results potentially have implications not only for Eugene Island but also for other

locations where methane has apparently moved rapidly through low permeability,

overpressured sediments.

3.2. Geological setting

The Eugene Island minibasin hosts one of the largest Plio–Pleistocene oil and gas

fields in the outer continental shelf of the Gulf of Mexico basin and has been extensively

studied by previous researchers (Holland et al. 1980; Holland et al. 1990; Anderson et al.

1994; Alexander and Flemings 1995; Hart et al. 1995; Alexander and Handschy 1998;

Gordon and Flemings 1998; Losh et al. 1999; Roberts 2001; Losh et al. 2002; Haney et

al. 2005; Joshi et al. 2012). The minibasin is about 19 × 15 km in dimensions and is

located about 270 km southwest of New Orleans (Figure 3.1). The minibasin formed

during the time of maximum sedimentation in the Pliocene–Pleistocene that shifted the

center of the deposition over 320 km southwestward from just west of the present mouth

of the Mississippi River to 160 km south of the present shoreline of the Louisiana–Texas

border (Woodbury et al. 1973). The rapid deposition of sediments caused the withdrawal

of the underlying thick Miocene salt layer, leading to the development of listric normal

47 growth faults and salt diapirs that bound the minibasin. Evidence from pore fluid chemistry, kerogen maturity indicators, hydrocarbon composition, borehole logging, fault-plane-section-analysis, and three dimensional seismic surveys indicates that the growth faults could have served as preferential conduits for the episodic ascent of fluids

(Anderson et al. 1994; Lin and Nunn 1997; Losh 1998; Losh et al. 1999; Whelan et al.

2001; Haney et al. 2005). Some of the hydrocarbon fluids were trapped in rollover anticlines formed in the downthrown blocks of the growth faults (Holland et al. 1990).

Figure 3.1. Location map of the Eugene Island minibasin (Alexander and Handschy 1998)

Alexander and Flemings (1995) subdivided the structural and stratigraphic evolution of the Eugene Island minibasin into three phases: prodelta, proximal deltaic, and fluvial. During the early prodelta phase, the deposition of bathyal and prodelta shales, turbidites, and distal deltaic sands on top of the salt layer caused the salt layer to migrate outward laterally, creating more accommodation space above the layer for sedimentation. The principal sand reservoir deposited during this phase was the Lentic sand, which is overpressured. The proximal deltaic phase is characterized by the

deposition of low stand, shelf–margin deltas that consist of alternating sequences of sand

and mud in distributary channel, distributary channel–mouth bar, and delta–front environments. The sand reservoirs, such as OI, MG, LF, KE, JD and HB, deposited in this phase were moderately overpressured. During the late fluvial phase, the underlying salt layer was completely evacuated from the minibasin such that little further accommodation space for sedimentation was created. The deltaic system then prograded southward, forming an erosional unconformity on the exposed shelf. A major reservoir, the GA sand, was deposited early during the fluvial phase when faults were active and created structural closure for the reservoir. Later deposited sands lacked this structural closure and were poorer in hydrocarbons (Alexander and Flemings 1995). Late fluvial phase sediments are hydrostatically pressured (Lin and Nunn 1997).

3.3. Model set-up

3.3.1. Theoretical background

Previous researchers have predicted that porosity waves can form and propagate in elastic porous media if permeability is a sensitive function of effective stress or fluid pressure, the effective stress is low (meaning that fluid pressure is high), and that the rate of fluid pressure generation is high relative to the hydraulic diffusivity (Rice 1992; Revil and Cathles 2002; Bourlange and Henry 2007; Joshi et al. 2012). Also, the higher the rate of pressure generation relative to the hydraulic diffusivity, then the higher the amplitude of the porosity waves. Porosity wave behavior in the present study was modeled using the theory employed by Joshi et al. (2012). The fundamental equation is a one–dimensional form of the conventional differential equation of fluid mass balance

used in hydrology to describe the diffusion of single-phase fluid pressure in a poroelastic medium (Freeze and Cherry 1979; Schwartz and Zhang 2003; Ingebritsen et al. 2006),

(3.2)

In this equation, P is the pore fluid pressure, t is time, and z is the vertical spatial

coordinate. The specific storage () of the porous medium is defined as,

(3.3)

Here, and represent sediment and fluid compressibility, respectively, and represents porosity. Sediment porosity () is a function of effective stress according to

(Revil and Cathles 2002),

exp (3.4)

where is the surface porosity at zero depth and represents the effective stress. The

sediment permeability was calculated based on an exponential relationship of

permeability with effective stress (Rice 1992; Revil and Cathles 2002)

∗ exp / (3.5)

∗ where is the sediment permeability at zero depth and is a compaction factor. The

velocity of methane in the background flow regime was calculated using Darcy’s law,

(3.6)

where q is the specific discharge.

3.3.2. Model description

To evaluate the ability of porosity waves to transport methane in the Eugene

Island minibasin, equations (3.2) – (3.6) were solved over a five kilometer thick sequence of methane–saturated sediment using an implicit finite difference method. Results were

found to be sensitive to nodal spacing (Δz) and time step size (Δt). This is illustrated by

Figure 3.2, in which porosity wave velocity is plotted as a function of Δz and Δt over a 50

m depth interval in the hydrocarbon source region, corresponding to a permeability variation of 1.0×10−24 to 2.0×10−24 m2.

Figure 3.2. Sensitivity of porosity wave velocity to time step size and nodal spacing.

Figure 3.2 shows that porosity wave velocity increases with decreasing Δz and Δt. The

change in velocity decreases with decreasing Δz and Δt, indicating that the numerical

solution is convergent in both the spatial (z) and temporal (t) dimensions. However, in

the present study full numerical convergence could not be achieved within practical time

limits. The best compromise between accuracy and simulation time that could be

achieved was for Δt ≤ 10−5 year and Δz ≤ 10−3 m, the values used in all of the results

presented below, so that simulations could be completed on time scales no longer than

months. Thus, the porosity wave velocities presented must be regarded as minimum

velocities where the actual velocities are likely to be at least half an order of magnitude

higher.

The initial pore fluid pressure in the model was assigned to be 93% of the

lithostatic pressure (Hart et al. 1995) and was calculated using the following equation

0.93 (3.7)

−3 where is the bulk density (2200 kg m ) and d is the depth. The fluid pressures at the

top and the bottom of the model were held constant over time. The initial permeability

−13 2 profile was calculated with equation (3.5) using a constant value of 6.6×10 m and a

∗ value of 0.25 MPa, both derived from well core analysis of the sediments from the

Red fault (Elliott 1999; Revil and Cathles 2002; Nunn 2003), and effective stress () values calculated from

(3.8)

where σT is the total stress.

The initial porosity in the model was calculated using equation (3.4) with a

−8 constant surface porosity () of 30% and a bulk sediment compressibility of 1.0×10

Pa−1 (Revil and Cathles 2002). The density, viscosity and compressibility of the methane

were held constant in all of the simulations at 250 kg m−3, 3×10−5 Pa s, and 10−8 Pa−1,

respectively.

Using the theory and parameter values described above, numerical models

simulating porosity wave behavior were carried out for two different scenarios: (1) a

scenario of gradual pressure generation caused by sedimentary diagenesis, and (2) a

scenario of instantaneous pressure generation. For the gradual pressure generation

scenario, the pressure generation rate was assumed to vary in a Gaussian fashion with

depth and to be constant over time. The maximum pressure generation rate was set at

513 Pa year−1 at a depth of 4.5 km and decreased both upward and downward to a value of zero from the central maximum value over a total depth interval of 300 m. For the instantaneous pressure generation scenario, lithostatic fluid pressure was assigned as an initial condition to the same 300 m depth interval as in the gradual pressure generation scenario.

Three different permeability–depth profiles were tested in the instantaneous pressure generation scenario (Figure 3.3). In the first profile, permeability decreased with increasing depth according to equation (3.5) until reaching a value of ~10−24 m2 at

the 4.5 km depth of the hydrocarbon source rocks. This profile was fit to data acquired

from drill stem tests performed in the Pathfinder well of the Red fault to a depth of 2 km

(Elliott 1999; Revil and Cathles 2002; Nunn 2003), and extrapolated to a depth of 5 km.

Thus, permeability in the lower 3 km of the Eugene Island minibasin is not known with

great certainty. Because the first profile predicts permeabilities that are so low in the

hydrocarbon source region—near the minimum reported for shales (Neuzil 1994)—a

second permeability–depth profile was considered in which permeability decreased with

depth as in the first profile until reaching a value of 10−21 m2 at a depth of 3.3 km, below

which permeability remained constant with depth. In the third profile, the initial background pore fluid pressure was raised to 96% of lithostatic and the permeability was allowed to vary continuously with increasing depth according to equation (3.5), reaching

a value of 10−18 m2 at 4.5 km depth, the approximate average permeability for shale

(Freeze and Cherry 1979).

Figure 3.3. Plots of the three permeability-depth profiles used in the modeling.

3.4. Model results

3.4.1. Gradual pressure generation scenario

Figure 3.4 presents simulation results for the gradual pressure generation scenario

at six times ranging from 2400 to 24,000 years. The results show a build–up of pore fluid

pressure at the hydrocarbon source rock depth of 4.5 km based on the prescribed pressure

generation rate of 513 Pa year−1, the maximum value reached from a combination of compaction disequilibrium and hydrocarbon formation in the basin evolution modeling of described in the second chapter. However, the region of increased fluid pressure never

54 coalesces into a discrete wave that ascends with a relatively constant wavelength. Rather, fluid pressure diffuses both upward and downward from the pressure generation source, but much more rapidly upward because of the increasing hydraulic diffusivity with decreasing depth, caused mainly by increasing permeability with decreasing depth. Thus, compaction disequilibrium and hydrocarbon formation are unlikely to generate fluid pressure quickly enough to overcome fluid pressure dissipation by diffusion to allow porosity wave formation. This is consistent with the findings mentioned in chapter two where it was hypothesized that a pressure generation rate (Pg) to hydraulic diffusivity (D) ratio of 1.1×108 Pa m−2 must be exceeded for solitary waves to form, which would require fluid pressure generation rates of at least ~2000 Pa year−1 for porosity waves saturated with methane because of methane’s low viscosity. In fact, separate calculations carried out in the present study using fluid pressure generation rates up to 150× greater than the 2000 Pa year−1 threshold value were still too low to overcome the rate of pressure diffusion and form porosity waves in methane–saturated porous media. These results suggest that in order to form solitary waves in methane–saturated porous media, essentially instantaneous rates of fluid pressure generation are needed.

Figure 3.4. Plots of pore fluid pressure as a function of depth after (a) 2400 years, (b) 4800 years, (c) 7200 years, (d) 12,000 years, (e) 18,000 years, and (f) 24,000 years for the gradual pressure generation scenario.

3.4.2. Instantaneous pressure generation scenario

Solitary wave behavior resulting from instantaneous fluid pressure generation was investigated for three different permeability–depth profiles. Figure 3.5 shows results for

permeability–depth profile 1 in which permeability decreased with increasing depth

according to equation (3.5) fit to the data of Revil and Cathles (2002) and illustrated in

Figure 3.3. The results show pore fluid pressure as a function of depth for six times

ranging from 0 to 71 years. Fluid pressure was initially set to lithostatic over a 300 m pressure source interval from a depth of 4.5 to 4.8 km. Elsewhere in the model, the initial fluid pressure was 93% of lithostatic. The higher initial pressures in the pressure source interval correspondingly raised permeabilities there relative to the surroundings according to equation (3.5). Fluid pressure diffused toward the upper boundary of the pressure source interval, beyond which pressure diffusion slowed because of the lower permeability of the overlying sediments. As a result, fluid pressure built up at the top boundary of the pressure source interval, creating a discrete waveform with a peak amplitude and wave length, i.e. a porosity wave. Once the amplitude of the porosity wave had grown large enough, the wave ascended rapidly from the pressure source interval, leaving behind a wake of slightly elevated pressures. Figure 3.5 shows that as the wave ascended, its amplitude decreased as a result of increasing hydraulic diffusivity with decreasing depth. This means that wave amplitude decreased with time, before finally completely disappearing after about 71 years. Because wave amplitude decreased with time, wave pore volume also decreased with time (Figure 3.6a). Wave velocity increased with time from a few meters per year to about 320 m year−1 at the time of its

disappearance into the background. Because the rate of velocity increase is much faster

than the rate of wave pore volume decrease, the wave volumetric flow rate increased with

time from 5×104 m3 year−1 up to a maximum of 2.5×106 m3 year−1 until the wave

disappeared.

Figure 3.5. Plots of pore fluid pressure as a function of depth after (a) 0 years, (b) 25 years, (c) 55.6 years, (d) 68.1 years, (e) 69.5 years, and (f) 71 years for the permeability-depth profile 1 in the instantaneous pressure generation scenario.

Figure 3.6. Temporal variation of the (a) wave pore volume, (b) wave velocity, and (c) wave volumetric flow rate for permeability–depth profiles 1 and 2 in the instantaneous pressure generation scenario.

As for solitary waves in oil-saturated porous media (Joshi et al. 2012), the

porosity wave traveling through methane-saturated porous media represented in Figure

3.5 also leaves behind a wake of elevated fluid pressure. For oil-saturated porous media,

the corresponding increase in permeability and thus, hydraulic diffusivity in the wake

prevented the formation of further porosity waves despite ongoing pressure generation.

Elevated fluid pressures in methane-saturated porous media left behind after the passage

of porosity waves would inhibit, though not necessarily prevent, the formation of further

porosity waves until fluid pressures in the wave diffused to background values after about

250,000 years. Thus, a subsequent episode of pressure generation while the wake of

elevated fluid pressure of the original wave was still in place would have to be stronger

than the original episode of pressure generation in order to form another wave.

Figure 3.7 shows simulation results for permeability-depth profile 2, in which

fluid pressure was generated instantaneously but permeability was constant at 10−21 m2 below 3.3 km depth instead of continuing to decrease with increasing depth according to equation (3.5) as for profile 1 (Figure 3.3). This profile choice was based on the observation that porosity and therefore permeability commonly become constant in sedimentary basins below depths of 2–4 km (Selley 1985). Porosity wave behavior in

Figure 3.7 is qualitatively similar to that of profile 1 for instantaneous pressure generation. However, for profile 2, the porosity wave travels about 10× faster than for profile 1 and about 500 m further before dissipating into the background compared to profile 1.

The results also show that sediment permeability must not necessarily be a sensitive function of effective stress in order for porosity waves to propagate, because at depths below 3.3 km, permeability was constant and dependent on effective stress only within the high fluid pressure porosity wave region where permeability was greater than

10−21 m2. Porosity waves in profile 2 reached a wave pore volume of 2.5×106 m3 after 1 year, shrinking in size to a pore volume of 1.8×106 m3 after 7.8 years and to zero

immediately thereafter (Figure 3.6a). Compared to the variable permeability–depth

profile for the instantaneous pressure generation scenario, the wave pore volume was

60 lower due to the greater hydraulic diffusivity of the constant permeability–depth profile at greater depths.

Figure 3.7. Plots of pore fluid pressure as a function of depth after (a) 0 years, (b) 1 year, (c) 5 years, (d) 7 years, (e) 7.8 years, and (f) 7.9 years for permeability-depth profile 2 for the instantaneous pressure generation scenario.

Wave velocity increased sharply from about 130 m year−1 in the deep, constant

permeability zone to a value of about 2700 m year−1 after 7.8 years after the wave had moved into shallower, more permeable sediments (Figure 3.6b). The volumetric flow rate increased correspondingly from about 1.5×106 m3 year−1 in the deep, constant

permeability zone to a maximum of 2.5×107 m3 year−1 at shallower depths after 7.8 years.

Porosity waves were predicted to travel a vertical distances of ~2 km from their source

before disappearing as a result of pressure diffusion from the wave. As the wave

ascended, it left behind a wake of elevated fluid pressure that persisted for about 200,000

years before pressures returned to their initial values as a result of diffusion.

Figure 3.8 shows simulation results for profile 3 (Figure 3.3), in which fluid

pressure in the hydrocarbon source rock was increased instantaneously, and permeability

decreased more gradually with increasing depth than in profile 1 according to an assumed

initial fluid pressure of 96% of lithostatic such that permeability in the source rock

reached the average value for shale of 10−18 m2. At the higher permeabilities in this

simulation, the rate of fluid pressure diffusion was too high for porosity waves to form.

The swelling of fluid pressures visible in Figures 3.5 and 3.7 at the upper boundary of the

high pressure zone was prevented in this simulation by the higher permeability.

To test if higher initial fluid pressures in the source rock could overcome the

higher permeabilities of profile 3 to form porosity waves, a simulation was run in which

fluid pressure was raised to 104% of lithostatic in the source rock. Figure 3.9 shows

results from this simulation. A porosity wave was able to form and depart from the high

pressure zone, traveling ~500 m at an average velocity of 242 km year−1 before diffusing

completely into the background porous medium. The higher permeability in this

62 simulation was responsible for the rapid wave velocity compared to the previous simulations in which porosity waves traveled at velocities of 10’s to 100’s of meters per year. However, this higher permeability also reduced the vertical travel distance relative to the 1–2 km travel distances predicted in the previous simulations because of the correspondingly higher hydraulic diffusivity.

Figure 3.8. Plots of pore fluid pressure as a function of depth after (a) 0 hours, (b) 0.88 hours, (c) 4.38 hours, and (d) 9.64 hours for permeability-depth profile 3 for the instantaneous pressure generation scenario.

Figure 3.9. Plots of pore fluid pressure as a function of depth after (a) 0 hours, (b) 0.044 hours, (c) 0.39 hours, (d) 4.82 hours, (e) 9.2 hours, and (f) 10.51 hours for permeability-depth profile 3 and instantaneous pressure generation, where fluid pressure at the source was raised to 104% of lithostatic at 4.5 km depth, keeping all other model parameters the same as for Figure 3.8.

3.5. Discussion

The results of the present study have implications for the ability of porosity waves to transport methane at Eugene Island and other hydrocarbon basins. The results predict

that porosity waves can transport methane at rates orders of magnitude higher than the

millimeter per year rates predicted by Joshi et al. (2012) for petroleum transport via

porosity waves. The porosity waves also travel at rates 14 orders of magnitude greater

than the 5×10−4 millimeter per million year’s rate of methane flow predicted from

Darcy’s law mechanics. Thus, porosity waves could transport methane at the very rapid

rates predicted from field observations at Eugene Island (Anderson et al. 1991; Anderson

et al. 1994; Whelan et al. 1994; Losh et al. 1999; Whelan et al. 2001; Haney et al. 2005).

As for the petroleum-bearing porosity waves studied by Joshi et al. (2012),

methane-bearing porosity waves are also able to ascend about 1–2 km before diffusing

into the background. This means that porosity waves could only transport methane to the

deeper Pleistocene reservoirs at Eugene Island such as OI, MG and JD that lie at depths

of 2–3 km below the sea floor, but could not transport methane up to the shallowest

reservoirs that lie at depths of 1–2 km below the sea floor. This is consistent with the

observation that the deeper reservoirs are more methane-rich (Holland et al. 1980;

Holland et al. 1990) than the shallower reservoirs. The distance traveled by the porosity

waves is directly proportional to the size of the initial pressure perturbation and inversely

proportional to the hydraulic diffusivity.

Porosity waves traveling through methane-saturated sediments were predicted in

the present study to have volumes of about 3×106 m3, assuming a spherical geometry for

the waves. As for travel distance, wave volume is directly proportional to the size of the

65 initial pressure perturbation and inversely proportional to hydraulic diffusivity. Holland et al. (1990) estimated the deeper sand reservoirs (OI, MG, and JD) at Eugene Island to contain about 3.5×1010 m3 of natural gas (reported for standard temperature of 16° C and pressure of 0.1 MPa). At a depth of about 3 km where pressure and temperature would be about 60 MPa and 104° C, this volume of natural gas in OI, MG, and JD reservoirs would reduce to 108 m3. Thus, on the order of 10’s of porosity waves would have been needed to charge the OI, MG, and JD reservoirs at Eugene Island. In order for porosity waves to have been effective at charging the deeper reservoirs at Eugene Island with methane, a repeated source of essentially instantaneous pressure generation would had to have been present. The number of pressure generation events needed would decrease the more porosity waves were simultaneously formed from each event. That is, it is possible that multiple porosity waves could form simultaneously along the strike of a fault, a hypothesis that is not testable in the present one–dimensional model.

Based on the results of Figure 3.4, typical basin diagenetic processes like compaction disequilibrium and hydrocarbon generation from kerogen are likely to increase pore pressure too gradually to allow porosity waves to form in methane- saturated porous media. Likewise, the cracking of petroleum to natural gas and ductile shear, which although are able to generate overpressures around lithostatic or greater, also probably occur too gradually to allow porosity waves to form in methane-saturated porous media (Barker 1990; Nunn 1997; Berg and Gangi 1999; Carcione and Gangi

2000).

An alternate hypothesis is that earthquakes could increase pressure rapidly enough, to a high enough magnitude, and with a high enough frequency of occurrence for

porosity wave to have been important agents of methane transport to charge reservoirs at

Eugene Island. That earthquakes can cause essentially instantaneous increases in pore

fluid pressure has been well documented from observations of shallow groundwater wells

that show that hydraulic head can increase by meters to even tens of meters following

earthquakes (Wood et al. 1985; Rojstaczer and Wolf 1992; Masterlark and Wang 2002;

Jonsson et al. 2003; Wang and Chia 2008; Wang and Manga 2010; Wang 2011; Cox et

al. 2012). Ge and Stover (2000) calculated hydraulic head changes resulting from 1 m

slip along a steeply dipping normal fault extending to a depth of 10 km. They predicted

maximum hydraulic head increases of about 130 m near the fault, which equates to pore

fluid pressure increases of about 1.3 MPa. Zhou and Burbey (2014) modeled pore fluid

pressure responses to sudden slip on normal, strike-slip, and reverse faults as a result of

sudden fault plane slip at depths of about 5 km. They predicted maximum pore pressure

increases near the fault on the order of 1’s of MPa for decimeter-scale fault plane slip,

and 10’s of MPa for meter-scale fault plane slip.

Whether earthquakes have been large and frequent enough at Eugene Island for porosity waves to have been important agents of methane transport is questionable.

According to Losh and Haney (2006), slip along the Red fault, and along growth faults throughout the northern Gulf of Mexico basin generally, is probably aseismic, based on the absence of felt earthquakes over decades of offshore hydrocarbon production, the low shear strength and high porosity of the sediments, and the high degree of overpressuring

of fluids in the sediments. However, earthquakes could be an important mechanism for

generating porosity waves in other hydrocarbon basins where the sediments are stronger.

The present study has some important limitations that affect the interpretation of

the results. First, the calculations were performed for a porous medium saturated with

methane, whereas oil and water are also known to be present at Eugene Island. The

coexistence of other pore fluids with methane could enhance the ability of porosity waves

to transport methane by lowering the relative permeability for methane and thus the

hydraulic diffusivity of the porous media. Lower hydraulic diffusivity would lower the

pressure generation rate needed to form porosity waves and allow them to ascend further before diffusing into the background.

However, porosity wave ascent could be reduced if the waves encountered

permeable horizons that could bleed off fluid pressure laterally. This illustrates a second

limitation of the current model, which is its one-dimensional nature and its inability to treat lateral pore pressure diffusion resulting from permeability heterogeneity.

A third important limitation of the current model is that it did not take into account the change in methane density and viscosity caused by changes in temperature and pressure associated with changes in depth. The condition of constant methane density and viscosity was stipulated to facilitate numerical stability, as simulations that incorporated an equation of state for methane density and viscosity rapidly became unstable before producing useful results. Methane density and viscosity would be expected to decrease by 8 and 10 % respectively over ascents of 1 km, which could increase porosity wave velocities by 15%. However, the decrease in density and viscosity with depth would also increase hydraulic diffusivity, which would decrease the distance that the porosity wave ascended.

Although the results of the present study suggest that porosity waves are unlikely

to have played an important role in transporting methane to reservoirs at Eugene Island,

they also leave open the possibility that porosity waves could be important agents of the

transport of methane or other gases in other locations where fluid pressures approach lithostatic pressure, where earthquake frequency is high, or where the required transport distances are less than 1–2 km. For example, rapid and episodic release of methane at the seafloor in the southeast Bering Sea originating from overpressured sediments at depths possibly greater than 1 km is likely to have been triggered by earthquakes (Cook et al.

2011) could possibly result from porosity waves.

The present study leaves some important questions unanswered that could be targeted by future work. These include the ability of high permeability facies to produce lateral pressure diffusion, the lateral spacing of porosity waves that could be expected along a fault plane, the effects of variable methane density and viscosity, and the effects of the presence of other pore fluids such as oil and water.

3.6. Conclusions

Despite important limitations, the present study allows some conclusions to be drawn about the ability of porosity waves to transport methane at Eugene Island.

Porosity waves are unlikely to be able to form as a result of gradual pressure generation typical of basin diagenetic processes such as sediment compaction and hydrocarbon formation because the high hydraulic diffusivity of methane-saturated sediments causes fluid pressure to diffuse from its generation source more quickly than it can build up in order for a porosity wave to coalesce. In order for porosity waves to form in methane- saturated sediments, geologically instantaneous pressure generation is needed that raises

fluid pressure on the order of 1’s of MPa above near-lithostatic background pressures.

Earthquakes are a potential mechanism for rapid, high magnitude pore pressure generation, though presently available evidence suggests that Eugene Island’s geological history and that of the surrounding northern Gulf of Mexico basin has been relatively quiet seismically.

Once formed, the porosity waves could be expected to ascend distances of 1-2 km at very rapid rates on the order of at least 10’s of m year−1. Uncertainties in velocity are

based on uncertainties in the value of permeability at depths below three kilometers and

on uncertainties in the magnitude of effects of further reductions in time step size and

nodal spacing, which could not be investigated because of practical limitations in

numerical simulation time. Thus, porosity waves would only be able to reach the lower

reservoirs at Eugene Island and 10’s of porosity waves would have been needed to charge

the reservoirs with the amount of methane that they were observed initially to contain.

Though porosity waves are unlikely to have been important agents of methane transport

at Eugene Island, they could have been in other settings that are more seismically active

and where required methane transport distances are less than 1-2 km.

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CHAPTER 4: NUMERICAL MODELING OF POROSITY WAVES IN THE

NANKAI ACCRETIONARY WEDGE DÉCOLLEMENT: IMPLICATIONS FOR

SLOW SLIP EVENTS

Abstract

Previous interpretations of seismic and hydrologic observations in the Nankai subduction zone show fluid overpressures in the Nankai décollement at depths below ~2 km from the seafloor could have increased to near lithostatic values due to sediment diagenesis and dehydration of the subducting oceanic plate. Gradual increases in décollement overpressures might have generated pressure pulses that could have ascended up the dip of the décollement at rates of ~1 km day−1. Upward ascent of these pressure pulses might reduce effective normal stress decoupling the accretionary wedge from underthrusted sequences and initiate aseismic slip that also propagate at rates of ~1 km day−1. High fluid pressures could dilate the porosity and form porosity waves that have capability of transporting fluids at rates significantly greater than the background

Darcy’s flow. The aim of the present study was to test a theoretical verification of the field observations that porosity waves can travel at rates fast enough to account for aseismic slip in the Nankai accretionary wedge décollement. The hypothesis was tested using a transient one-dimensional numerical solution to the fluid mass conservation equation and the Darcy’s law with an assumption of the elastic porous medium.

Results from the present study show porosity waves at depths of ~2 km in the décollement can be generated from overpressures in excess of lithostatic pressure by 3%.

Thus formed porosity waves can propagate at rates sufficient enough to account for aseismic slip along the décollement over a wide range of geological conditions.

Simulation results from sensitivity analysis show porosity wave velocity to be strongly

dependent on specific storage, fluid viscosity and permeability-depth gradient along the

décollement. Thus, overpressures less than the lithostatic pressure (99%) could also propagate porosity waves at speeds adequate to cause aseismic slip, if the décollement

conditions were near the limits of the geologically reasonable ranges.

4.1. Introduction

Mechanically weak décollement zones in many accretionary wedges have been recognized as pathways for rapid and episodic migration of deep overpressured fluids that have been generated by dehydration of the subducting oceanic plate (Fisher and

Hounslow 1990; Moore and Vrolijk 1992; Brown et al. 1994; Bekins et al. 1995; Sample

1996; Carson and Screaton 1998; Henry 2000; Brown et al. 2005; Bourlange and Henry

2007). These overpressured fluids may travel at velocities of up to 1’s of km per day.

For example, Sano et al. (2014) estimated fluid velocities of this magnitude after the 2011

magnitude 9.0 Tohoku-Oki earthquake based on temporal changes in helium isotope

composition of ocean bottom water in the Japan trench. In the Nankai accretionary

wedge, hydrologic observations from the Ocean Drilling Program (ODP) show that

current plate convergence could increase pore fluid pressures at rates of 1’s to 100’s of kPa per year at a depth of ~ 1 km beneath the seafloor, from where transient pulses of elevated pressure could discharge and propagate up the dip of the decollement at rates of

1’s of km per day (Davis et al. 2006). Other researchers have suggested that these

pressure pulses could originate from even greater depths of 10’s of km beneath the

seafloor in the decollement, and as the pressure pulses propagate, they could trigger

aseismic slip events through a local reduction of the effective stress (Dragert et al. 2001;

Kodaira et al. 2004; Obara et al. 2004; Davis et al. 2006; Ide et al. 2007; Wech and

Creager 2011; Katayama et al. 2012).

A consequence of the high rates of fluid pressure generation noted above is that

fluid pressures reach high values near lithostatic in the décollement by depths of only ~1

km below the seafloor (Screaton et al. 2002; Gamage and Screaton 2006; Tobin and

Saffer 2009). High overpressures likely persist to much greater depths in the décollement. For example, seismic reflection imaging has revealed a high reflectivity zone at depths of ~35 km that could correspond to a high Poisson’s ratio, possibly caused by high pore fluid pressure (Kodaira et al. 2004). The decollement’s separation of low permeability folded thrusted wedge sediments from underlying less deformed underthrusted sediments suggests that the decollement is mechanically weak because of elevated pore fluid pressures, possibly as a result of sediment compaction and diagenesis

(Ujiie et al. 2003; Tsuji et al. 2008; Tobin and Saffer 2009), fluid generation from smectite dehydration (Saffer and Bekins 1998), or permeability reduction by shearing along the décollement (Ikari and Saffer 2012). These elevated fluid pressures could create fractures within the decollement zone, allowing high fluid pressures to propagate up-dip, decoupling the accretionary wedge from the underthrusted sequence, and increasing effective stress while decreasing pore fluid pressure downdip (Morgan and

Karig 1995). As fluid pressure increases and effective stress decreases, porosity typically increases, giving rise to the term “porosity waves” for propagating high fluid pressure pulses (Richter and McKenzie 1984; Connolly and Podladchikov 1998; Connolly and

Podladchikov 2000; Connolly and Podladchikov 2015).

Because porosity waves can travel at rates significantly faster than fluids in the

background Darcian flow regime, porosity waves have been invoked to explain

anomalously rapid fluid transfer in a variety of geological settings, including magma

transport in the mantle (Richter and McKenzie 1984; Scott and Stevenson 1984;

Spiegelman 1993a; Spiegelman 1993b; Wiggins and Spiegelman 1995; Richardson et al.

1996), the flow of water and carbon dioxide during metamorphism (Connolly 1997;

Fontaine et al. 2003; Miller et al. 2004; Connolly 2010; Tian and Ague 2014), and the

expulsion of hydrocarbons from low permeability rocks in sedimentary basins (Connolly

and Podladchikov 2000; Appold and Nunn 2002; Revil and Cathles 2002; Joshi et al.

2012). Porosity waves have also been invoked to explain the existence of pressure

compartments in sedimentary basins (Connolly and Podladchikov 1998).

Porosity waves have previously been studied in the Nankai wedge décollement by

(Bourlange and Henry 2007). They reported that porosity waves could discharge from a

source of high fluid pressure of 97.5% of lithostatic pressure, equivalent to 8 MPa of

overpressure, at a decollement permeability of 10−18 m2, and could propagate at velocities

of 100’s of m year−1. These velocities are significantly lower than the km day−1 rates they reported for aseismic slip on the Nankai décollement, so they concluded that porosity waves are not responsible for aseismic slip there. However, based on the work of Seno (2009) and Tsuji et al. (2014), overpressures in the Nankai décollement may actually reach lithostatic pressure. Because porosity wave velocity is a sensitive function of source fluid pressure (Connolly and Podladchikov 2000; Joshi et al. 2012; Yarushina et al. 2015), it is possible that porosity waves may be able to travel faster than Bourlange and Henry (2007) calculated. In addition, Bourlange and Henry did not consider the

sensitivity of their model results to other parameters such as specific storage and fluid

viscosity. Thus, the purpose of the present research was to test the hypothesis that porosity waves could travel at velocities fast enough to account for aseismic slip in the

Nankai accretionary wedge décollement under different conditions than considered by

Bourlange and Henry (2007) but still within the geologically reasonable range for the

setting. In order to test this hypothesis, the present study used the same transport

equations as Bourlange and Henry (2007) but solved them using an implicit finite

difference method in one spatial dimension along the décollement instead of the finite

element method in two spatial dimensions.

4.2. The Nankai accretionary wedge, offshore Japan

The Nankai accretionary wedge was formed as a result of stacking of imbricate

thrusts within accreted sediments from the subducting Philippine Sea plate underneath

the Eurasian plate at a rate of ~ 4 cm year−1 (Seno et al. 1993) (Figure 4.1). The low taper angle of the wedge of ~4-5° (Saffer 2010) may be a function of the mechanical weakness of the thrusted sediments as a result of elevated pore fluid pressures (Davis et al. 1983).

The lithostratigraphic study across the Nankai trough by ODP Leg 190 shows sediments from the subducting Philippine plate to consist of four lithostratigraphic units:

(1) Early-Middle Miocene volcaniclastic facies overlying basaltic oceanic basement, (2)

Middle Miocene to Early Pliocene Lower Shikoku Basin facies (LSB), (3) Early Pliocene to Quaternary Upper Shikoku Basin facies (USB), and (4) the Quaternary trench-wedge facies (Moore et al. 2001). The volcaniclastic facies is composed of large amounts of siliciclastic silt and sand along with volcanic sand and ash. The Lower Shikoku basin

facies is composed entirely of hemipelagic mud. The Upper Shikoku Basin facies

consists of smectite-rich hemipelagic mudstone interbedded with thin layers of volcanic

ashes and tuffs. The trench-wedge facies consists of coarsening upward sand and silt turbidites interbedded with hemipelagic mud. The decollement zone, which serves as a mechanical boundary between the upper wedge sediments and the lower underthrust sediments, lies entirely within the Lower Shikoku Basin facies (Moore et al. 2001).

Figure 4.1. (a) Location map of the Nankai accretionary wedge near Shikoku Island in Japan with a NW-SE trending transect (white) corresponding to the cross section in (b) the modified two-dimensional cross-section employed in the pressure wave model by Bourlange and Henry (2007). One-dimensional model in the current study was constructed based on the geometry of the décollement zone in figure b.

Heat flow decreases landward across the Nankai trough from a maximum of ~

200 mW m−2 near the deformation front to ~ 50 mW m−2 about 50 km from the

deformation front. This high heat flow near the deformation front could have resulted

from the presence of a now-extinct Shikoku basin back-arc spreading center (i.e. Kinan

seamount) (Yamano et al. 2003). Alternatively, the observed anomalous heat flow might

be due to heat conduction from low permeability sediments beneath the seafloor that

impeded hotter fluids ascending upward through faults (Yamano et al. 1992; Kawada et

al. 2014; Yamano et al. 2014). High heat flow in the Nankai trough may drive smectite

to illite transformation reactions that generate enough fresh water to cause the low

chlorine anomaly observed in pore waters of the accretionary wedge sediment

(Underwood et al. 1993; Brown et al. 2001; Henry and Bourlange 2004). In contrast to

the typical pattern of exponentially decreasing porosity with depth, the underthrust

sediments have higher porosity than immediately overlying sediments, which may be a

result of high pore fluid pressures within the underthrust sediments (Morgan and Ask

2004). The high fluid pressures within the underthrust sediments may have migrated into

the décollement zone by hydrofracturing, which resulted in weak coupling of the two

plates along the décollement (Hubbert and Rubey 1959; Davis et al. 1983; Morgan and

Karig 1995; Ujiie et al. 2003; Moore et al. 2005; Tsuji et al. 2008).

4.3. Model theory and set-up

In order to model porosity wave behavior in the Nankai accretionary wedge

décollement, a theoretical approach similar to that of Bourlange and Henry (2007) was

followed. Pore fluid pressure (P) as a function of time () and distance () along the

décollement was calculated according to the conventional differential equation of single-

phase pore fluid mass conservation

(4.1)

In this equation, is the specific storage of the porous medium, and are the

density and viscosity of the water, respectively, is the intrinsic permeability, and is

the acceleration due to gravity. Porosity wave formation has been found to require

permeability to be a sensitive function of effective stress () (Rice 1992). In the

modeling, the following permeability-effective stress relationship was used

∗ exp / (4.2)

where, is the permeability at the sea floor, i.e. the permeability at zero effective stress,

and ∗ is a compaction factor. Decollement porosity () decreased exponentially with depth () below the seafloor and was based on an averaged porosity-depth profile for

accretionary wedges by Le Pichon et al. (1990),

exp /1500 (4.3)

where, is the porosity at the seafloor held constant at 0.7 in all of the simulations.

Porosity change in the sediments as a result of fluid pressure change can be computed

from the conventional definition of compressibility of the porous media (),

∆ ∆ (4.4)

Equation (4.1) was solved over one spatial dimension over a distance of 32 km

along the décollement using an implicit finite difference method. Four model scenarios

for porosity wave behavior were investigated: (1) a base case scenario simulated for two

instances (A and B), where instance A was simulated using the same model parameter

values as those of simulation I of Bourlange and Henry (2007), whereas instance B was simulated at a smaller time step and nodal spacing, (2) a high fluid pressure source scenario set to 103.5% of lithostatic pressure but whose other hydrologic parameter

values were otherwise identical to the base case scenario, (3) an optimal velocity scenario

that employed model parameter values within the limits of geological uncertainty needed

to maximize porosity wave velocity, and (4) a high compaction factor scenario that

81 employed the maximum possible value of the compaction factor keeping other parameter values the same as in the base case scenario. Model parameter values and their sources used in the four scenarios are shown in Table 4.1.

Table 4.1. Model parameter values for all the scenarios. Model parameters Base case scenario High Optimal High

fluid velocity compaction

pressure scenario factor

Instance Instance source scenario

A B scenario

∆s, m 320 0.01 0.01 0.01 0.01

∆t, year 0.3 3×10−5 10−5 10−7 3×10−4

∗, MPa a0.26 a0.26 a0.26 *0.36 b0.334

−1 c −4c−4 c −4 d −6 c −4 , specific storage, m 10 10 10 3×10 10

a −3a−3 a −3 e −4 a −3 , fluid viscosity, Pa s 10 10 10 10 10

Minimum décollement a10−18 a10−18 a10−18 f10−21 a10−18

permeability, m2

Pore fluid pressure source, a97.5 a97.5 **103.5 g99 a97.5

% of lithostatic pressure

Note: Alphabetical superscripts show sources of parameter values as indicated below. Symbol ‘*’ represents parameter value at which rock permeability in the lower boundary reached a lower limit in the geological range. Symbol ‘**’ represents parameter value at which peaked porosity wave can be formed with other parameter values the same to that of base case scenario. aBourlange and Henry (2007), bFisher and Zwart (1997), cBourlange et al. (2004), dGe and Screaton (2005), dKitajima et al. (2012), eKorosi and Fabuss 1968, eKorson et al. 1969, eKestin et al. 1978, fGamage and Screaton (2006), gSeno (2009), gTsuji et al. (2014)

In all of the model scenarios, lithostatic pressure () at any distance (s) along the

décollement was computed from

(4.5)

where ρb is the bulk density, defined by

1 (4.6)

where and represent the water and grain densities, held constant at 1024 and 2700

kg m−3, respectively in the simulations. Effective stress is related to pore fluid pressure by

(4.7)

Figure 4.2 shows décollement permeability as a function of effective stress as

−12 2 calculated from equation (2), where equals 5×10 m . Following Bourlange and

Henry (2007), permeability was allowed to decrease with increasing effective stress to a

minimum value of 10−18 m2 for the base case, high fluid pressure source, and high compaction factor scenarios, and to 10−21 m2 for the optimal velocity scenario, after

which permeability was held constant with further increases in effective stress (depth).

The northwest and southeast boundaries of the model were assigned a constant

fluid pressure throughout the simulation time. An initial background pore fluid pressure

of 80% of lithostatic was assigned over the 32 km distance of the décollement (Gamage

and Screaton 2006; Bourlange and Henry 2007; Skarbek and Saffer 2009). A constant

elevated fluid pressure source of 97.5% of lithostatic was assigned over the first 260 m of

the grid extending from its northwest boundary. This constant elevated fluid pressure

source is considered a product of the continual deposition and compaction of accretionary

83 wedge sediments and the release of water from mineral dehydration in the underthrusted rocks (Bekins et al. 1995; Saffer and Bekins 1998; Kufner et al. 2014).

Figure 4.2. Plots showing permeability as a function of the effective stress for four model scenarios.

The same model grid with a constant nodal spacing (∆s) of 0.01 m was used in all model scenarios except for instance A of the base case scenario. The time step size (∆t) used for the four different scenarios varied, as shown in Table 4.1.

4.4. Results

4.4.1. Base case scenario

Figure 4.3 presents simulation results for pore fluid pressure as a function of décollement distance at time = 0 and after 333,000 years for instance A of the base case scenario. The model employed a boundary condition at the NW boundary of the model domain in which fluid pressure was set to 39 MPa. Fluid pressure decreased to 32 MPa

at the first grid node beyond the NW boundary, and decreased linearly thereafter with

increasing decollement distance according to the corresponding burial depth shown in

Figure 4.1b and equation (4.5). Figure 4.3 shows fluid pressure to have propagated from

the source at the NW domain boundary for a distance of 4.6 km after 330,000 years,

corresponding to an average velocity of ~1.5×10−2 m year−1. This value is much lower

than the velocity of ~220 m year−1 predicted by Bourlange and Henry (2007) over the

same travel distance for the same model parameter values as used to generate the results

shown in Figure 4.3. This discrepancy may result from the high sensitivity of the finite

difference solution used in the present study to nodal spacing and time step size.

Figure 4.3. Plots of pore fluid pressure as a function of décollement distance after (a) 0 year and (b) 333,000 years for instance A of the base case scenario.

This sensitivity is demonstrated in Figure 4.4a, in which average porosity wave velocity over the first 4.6 km of travel distance in the base case scenario is plotted as a

function of ∆s and ∆t. The results show porosity wave velocity to increase by orders of

magnitude with decreasing ∆s and ∆t, eventually converging to about 50 m year−1 at ∆s ≤

0.01 m and ∆t ≤ 3×10−4 year. Thus, these smaller ∆s and ∆t values reduced the

85 discrepancy between the present model and Bourlange and Henry (2007) finite element model from a factor of about 15,000 to about 4.4.

Figure 4.4. Sensitivity analysis of nodal and time spacing to the wave velocity for (a) the base case scenario, (b) the high fluid pressure source scenario, and (c) the optimal velocity scenario.

The behavior of porosity waves using the convergent ∆s and ∆t values noted above was investigated further in instance B of the base case scenario and is shown in Figure 4.5 in plots of fluid pressure versus distance at times ranging from 0 to 342 years.

Figure 4.5. Plots of pore fluid pressure as a function of décollement distance after (a) 0 year, (b) 45 years, (c) 180 years, (d) 270 years, (e) 315 years, and (f) 342 years for the base case scenario.

The high fluid pressure assigned at the NW boundary of the grid raised permeability there by a factor of ~105 over that of the background. This greatly increased

permeability, implicitly accompanied by increased porosity, allowed a porosity wave to

propagate rapidly along the décollement, accelerating from a velocity of about 50 m

year−1 to a final velocity of about 90 m year−1 after about 30 km, by which point the

porosity wave had dissipated to background fluid pressure values.

The shape of the porosity wave in Figure 4.5 differs from that of the porosity waves in Joshi et al. (2012), which had distinct peaks in their pressure. These peaks

develop if the hydraulic diffusivity is low relative to the maximum pressure in the wave.

However, the existence of a clearly defined peak in pressure is not a requirement for

porosity waves (Connolly and Podladchikov 1998). The porosity waves in Figure 5

resemble ordinary pressure diffusion fronts. However, in comparison to pressure diffusion fronts that migrate through constant-permeability porous media, the deformability of the modeled porous media, both in terms of porosity and permeability, allows the pressure front to move much more quickly, transporting fluid as it moves because of its corresponding increased porosity relative to the background.

4.4.2. High fluid pressure source scenario

The porosity wave velocities of 10’s of m year−1 predicted for the base case

scenario are much lower than the km day−1 velocities reported by Bourlange and Henry

(2007) to be needed for aseismic slip. A further scenario was therefore modeled to test the effect of increasing fluid pressure on porosity wave velocity. The results of a model

in which fluid pressure was set to 41 MPa (about 103.5% of lithostatic pressure) over the first 260 m thick decollement, from the NW boundary of the model domain are shown in

Figure 4.4b. This fluid pressure is the lowest at which a peaked porosity wave was found to form. All other hydrologic model parameters were the same as in the base case scenario. Model results were even more sensitive to ∆s and ∆t than in the base case

scenario. In fact, values of ∆s and ∆t small enough to obtain a convergent solution could

not be found within practical limits of model execution time. Because porosity wave

velocity was found to increase with decreasing ∆s and ∆t, the highest velocity reached by the model (~4×105 m year−1, equivalent to ~1 km day−1) should be regarded as a

minimum velocity for the hydrologic conditions modeled. This result is important

because it suggests that for high overpressures of at least 103.5% of lithostatic pressure, porosity wave migration at a velocity high enough to account for aseismic slip on the

Nankai decollement is possible. The reason for this rapid porosity wave velocity compared to that in the base case scenario is that the high fluid pressure lowered effective stress enough to raise the permeability of the décollement by a factor of ~109 relative to

the background décollement permeability.

Figure 4.6 shows the pressure of the porosity wave as a function of travel distance, calculated at ∆t = 10−5 year and a large ∆s of 0.01 m. At this coarse spatial resolution, the velocity of the wave is much slower than would be expected for a convergent solution at finer spatial resolution (Fig. 4.4b). However, at this coarse spatial

resolution, a simulation can be executed quickly enough within practical time limits in

which the porosity wave traverses the entire model grid length of 32 km, providing

further qualitative insights into porosity wave behavior. In contrast to the porosity wave

generated in the base case scenario, the porosity wave in the high fluid pressure source

scenario developed a discrete peak due to the high ratio of maximum fluid pressure in the

wave to the background hydraulic diffusivity. The porosity wave peak amplitude

diminished with travel distance, disappearing completely after about 18 km. Thereafter,

the porosity wave continues to travel without a distinct peak until the end of the model

grid. As for the base case scenario, the porosity wave in the high fluid pressure source

scenario accelerates as it travel as a result of increasing permeability with increasing

distance, but leaves behind a wake of much higher fluid pressures than in the base case

scenario, which is maintained as long as the high fluid pressure at the source is

maintained.

4.4.3. Optimal velocity scenario

In order to test whether porosity waves could travel at velocities high enough to

account for aseismic slip if the waves were sourced from geologically more realistic

pressures below lithostatic, an optimal velocity scenario was simulated in which fluid

pressure at the source was set to 99% of lithostatic pressure over the first 260 m from the

northwest grid boundary. In addition, the threshold at which permeability remained

constant and no longer decreased with depth was lowered from 10−18 to 10−21 m2, σ* in

equation (2) was increased from 0.26 to 0.36 MPa, and specific storage and fluid

viscosity were reduced by factors of 33 and 10, respectively compared to the base case

scenario.

As for the previous scenarios, porosity wave velocity was found to increase

strongly with decreasing ∆s and ∆t (Fig. 4.4c). Furthermore, as for the high fluid

pressure source scenario, values of ∆s and ∆t small enough to obtain a convergent

solution could not be found within practical limits of model execution time. Thus, as in

the high fluid pressure source scenario, the highest velocity reached by the model

(~8×105 m year−1, equivalent to ~2 km day−1) should be regarded as a minimum velocity

for the hydrologic conditions modeled, which is a velocity high enough to account for aseismic slip according to Bourlange and Henry (2007).

Figure 4.7 shows porosity wave pressure as a function of distance, calculated at a coarse time step and nodal spacing of ∆t = 10−7 year and ∆s = 0.01 m to allow execution

of a simulation within practical time limits in which the porosity wave traverses the entire length of the model grid. The porosity wave generated in the optimal fluid velocity

scenario resembled that of the high fluid pressure source scenario in that the wave

developed a discrete peak due to the high fluid pressure in the wave relative to the

hydraulic diffusivity of the background porous medium.

Figure 4.7. Plots of pore fluid pressure as a function of décollement distance after (a) 0 year, (b) 0.55 years, (c) 0.57 years, (d) 0.596 years, (e) 0.6 years and (f) 0.62 years for the optimal velocity scenario.

The amplitude of the porosity wave peak diminished as the wave traveled,

disappearing after the wave had traveled about 18 km. As for the previous scenarios, the

porosity wave accelerated as it traveled but left behind a wake of less elevated pressure.

4.4.4 High compaction factor scenario

The high compaction factor scenario was simulated in order to test the effects of

uncertainty in the value of the compaction factor (σ*), keeping all other hydrologic model

parameters the same as in the base case scenario. Bourlange and Henry (2007) calculated

a σ* value of 0.26 MPa from Fisher and Zwart (1997) fault sediment data from the

Barbados accretionary wedge but calculations from the present study indicate a value of

0.334 MPa to be more likely, which is the value that was used in the present high

compaction factor scenario.

Raising σ* from 0.26 to 0.334 MPa increases the average porosity wave velocity

by a factor of ~2.5 over that predicted in instance B of the base case scenario over a

travel distance of ~4.6 km (Fig. 4.8). The porosity wave travels about 30 km before

disappearing into the background due to the higher hydraulic diffusivity at shallower depths. Thus porosity wave velocity is highly sensitive to the value of the compaction factor, and small uncertainties in the value of the compaction factor can lead to large changes in porosity wave velocity.

Figure 4.8. Plots of pore fluid pressure as a function of décollement distance after (a) 0 year, (b) 30 years, (c) 90 years, (d) 120 years, (e) 135 years and (f) 150 years for the high compaction factor (∗) scenario.

4.4.5. Variation of porosity wave velocity with décollement distance

All of the simulated model scenarios predicted porosity wave velocity to increase as a function of travel distance up-dip along the décollement due to increasing permeability and decreasing effective stress (Fig. 4.9). A slight exception to this trend is the optimal velocity scenario, which showed a drop in porosity wave velocity at a distance of about 15 km before increasing again. This is most likely caused by the disappearance of the peak in the porosity wave at that distance (Fig. 4.7), which lowered the wave’s permeability. These results suggests that the potential for porosity waves to cause aseismic slip increases as they ascend to shallower depths along the décollement.

Figure 4.9. Plots of pressure wave velocity as a function of décollement distance for all four scenarios.

4.5. Discussion

Despite limitations from lack of convergence, the model results from the present

study have implications for fluid pressure and mass propagation and aseismic slip along

the Nankai accretionary wedge décollement. Results from the present study demonstrate

that overpressures generated at depths of ~2 km could have migrated along the

décollement as pressure pulses (i.e. porosity waves) at rates much faster than expected

Darcy fluxes. However, model results also suggest that porosity waves only reach the km day−1 velocities needed to cause aseismic slip under special conditions.

One such condition is overpressure exceeding lithostatic pressure. In the present

study, when fluid pressure exceeded lithostatic pressure by 3.5%, no other changes to the

base case scenario were needed, indicating that at high overpressures, porosity waves

capable of traveling at velocities great enough to cause aseismic slip could form over an

otherwise broad range of geological conditions. Porosity waves may be able to reach km

day−1 velocities at lower overpressures above lithostatic pressure than 103.5%, as a

convergent solution at 103.5% of lithostatic pressure could not be found due to run-time

limitations. Thus, the true porosity wave velocity for the 103.5% of lithostatic pressure

case must be higher than modeled, making it difficult to project what the porosity wave

velocities would be at lower overpressures above lithostatic, and leaving open the

possibility that they may be fast enough to account for aseismic slip. The limiting factor

for this high overpressure scenario may be the length of time that fluid pressures greater

than lithostatic pressure can be maintained at the porosity wave source, as rocks should fracture at such high fluid pressures, causing fluid pressure to decrease, most likely to

sub-lithostatic values.

However, the simulation results show that porosity waves can reach km day−1 velocities at fluid pressures below lithostatic pressure (~99% of lithostatic pressure) in the Nankai décollement provided that hydrologic parameter values are near the limits of their geologically reasonable ranges. These include low specific storage of 3×10−6 m−1, a particularly high sensitivity of permeability with respect to effective stress, manifest by a high σ* value of 0.36, low minimum décollement permeability of 10−21 m2, and a low pore fluid viscosity of 10−4 Pa s.

Support for these values comes from several previous studies. Kitajima et al.

(2012) determined specific storage values as low as 3×10−6 m−1 from drill core samples from depths of ~500 m in the Nankai accretionary wedge. A similar specific storage value of 10−6 m−1 for Nankai accretionary wedge sediments was determined by (Ge and

Screaton 2005) from numerical modeling of pore pressure changes resulting from seismic dislocation. Décollement permeability values as low as 10−21 m2 were determined by

(Gamage and Screaton 2006) from analyses of drill core and numerical modeling. For this permeability to be reached at a depth of 2 km in the model requires a σ* value of 0.36 according to equation (4.2). A fluid viscosity of 10−4 Pa s is actually more appropriate for the expected 100-200° C temperatures at 2 km depth in the Nankai wedge (Korosi and

Fabuss 1968; Korson et al. 1969; Kestin et al. 1978) than 10−3 Pa s, which was used in the base case, high fluid pressure source, and high compaction factor scenarios to remain consistent with Bourlange and Henry’s (2007) choice.

A check on the accuracy of the numerical models in the present study can be made by comparing the simulated porosity wave velocities to those predicted using

(Connolly and Podladchikov 1998) analytical solution

/ (4.8)

where represents porosity wave velocity, , and represent Darcy flux and porosity

within the maximum pressure in the porosity wave respectively, whereas and

represent Darcy flux and porosity outside the porosity wave in the background porous

medium respectively. Equation (4.8) assumes that the porosity wave is stationary, i.e.

does not change in form over time. For unpeaked porosity waves, velocities computed

from the numerical and analytical solutions agreed within an order of magnitude with one

another. In contrast, for peaked porosity waves, velocities computed from the numerical

and analytical solutions differed by more than an order of magnitude with one another.

The larger discrepancy in wave velocities for the peaked porosity waves most likely

stems from the fact that the assumption of stationarity is violated, i.e. amplitude of

peaked porosity waves diminishes with increasing travel distance.

The numerical models in the present study have some important limitations that

should be considered in interpretations of the results. One limitation is the one-

dimensional geometry of the models. Limiting the models to one spatial dimension

parallel to the décollement effectively means that the décollement is impermeable to fluid

flow and pressure diffusion in the second and third spatial dimensions. If the

décollement and surrounding geologic media were significantly permeable in the second

or third spatial dimensions, then fluid and pore pressure could escape the décollement,

attenuating the porosity waves’ amplitude, velocity, and fluid transport capability.

However, previous studies of the Nankai accretionary wedge suggest that fluid flow is

mainly focused up-dip along the décollement with minimal lateral diffusion because the

décollement is overlain by low-permeability sediments (Davis et al. 2006; Tsuji et al.

2008; Tanikawa et al. 2014).

Another possible limitation of the models is the assumption of an elastic rheology.

The validity of this assumption can be tested by calculating the Deborah number (De)

(Poliakov et al. 1996),

∆/ (4.9)

where ∆ represents the density contrast between rock matrix and fluid and L is the

length of the problem domain under consideration. Sediment core samples obtained by

the Integrated Ocean Drilling Program (IODP) from depths of ~1 km in the Nankai

accretionary wedge have been found to have shear modulus values ranging between 0.79

and 1.88 GPa (Hashimoto et al. 2011). Using these shear modulus values for L = 2.3 km

leads to Deborah numbers in the range of 2×10−2 to 4×10−2. These numbers are close to the viscous limit of 10−2, suggesting that the Nankai accretionary wedge sediments

should have significant viscous character. Another test of rheology is the Maxwell body

relaxation time (t0)

2/ (4.10)

where μ is the material viscosity and E is the Young’s modulus. Using viscosities

in the range of 1017 – 1020 Pa s for sedimentary basins (Biot and Odé 1965; Selig and

Wermund 1966; Mazariegos et al. 1996; Poliakov et al. 1996) and a Young’s modulus of

108 Pa determined (Bourlange et al. 2004) leads to Maxwell body relaxation times

ranging from 60 to 60,000 years. Most of this time range is much longer than the length

of time of the porosity wave behaviors, indicating a likely significant elastic character of

the Nankai accretionary wedge sediments. Thus, the overall rheology of the Nankai

accretionary wedge sediments is likely viscoelastic. Including a viscous component in the décollement rheology would likely dampen the porosity wave behavior simulated in

the models. However, this effect would have to be orders of magnitude in size to lower

the convergent velocities of the porosity waves in the high fluid pressure source and

optimal velocity scenarios to values too low to cause aseismic slip. Nonetheless, testing

the effects of a viscous component to the Nankai décollement rheology on porosity wave

behavior would be a useful topic for future research.

4.6. Conclusions

Results from the present study support the hypothesis that porosity waves originating from regions of high overpressure in the Nankai accretionary wedge could travel much more rapidly than Darcy fluxes of groundwater. If fluid pressures exceed lithostatic pressure by 3.5% and possibly less at depths below ~2 km in the décollement, then the porosity waves originating from there could propagate at the ~1 km day−1 or faster velocities needed to cause aseismic slip. Sustaining fluid pressures in excess of lithostatic pressure, however, may not be geologically realistic over the long term. Fluid pressures less than lithostatic pressure can also generate porosity waves that can travel at rates fast enough to cause aseismic slip provided that hydrogeological parameters in the décollement are near the limits of their ranges of uncertainty. These include a low specific storage of about 3×10−6 m−1, a particularly high sensitivity of permeability to

effective stress manifest by a compaction factor value of about 0.36, a corresponding low permeability of 10−21 m2 below depths of 2 km, and low fluid viscosity of order 10−4 Pa s appropriate for the originating depth of the porosity waves. Thus, the present model

100 results suggest that porosity waves could cause aseismic slip events in the Nankai décollement and could also be important in other accretionary wedges where overpressures in the décollement have reached near lithostatic pressures.

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APPENDIX

FORTRAN program code for methane porosity wave calculations

C TITLE: Porosity Wave Propagation in Poroelastic Media

C This program calculates pore fluid pressure in the elastic porous media using the mass balance C equation and the Darcy's law with an exponential permeability-effective stress relationship C from Revil and Cathles (2002)

PROGRAM SOLWAVE

INTEGER*8 NNODES, P1, PRTSTP(100), SIMPRO, TIMSTP, ITOP, IPK, IBOT, + NMTOUT, TPRIVL, N

CHARACTER INIDAT

PARAMETER (P1=5000000)

DOUBLE PRECISION DELZ, DELT, PERMEA(P1), GRAVAC, VISLIQ, + SIMLNS, DEPTH(P1), Z(P1), ZMAX, DENLIQ, INTIME, PRRATE, + LODIAG(P1), MNDIAG(P1), UPDIAG(P1), LOAD(P1), + PRSNEW(P1), FLUPRE(P1), TIME, TEMP(P1), MDEPTH, + PERME0, SIGSTR, DARVEL(P1), EFFSTR(P1), PREMAX, + TOTSTR(P1), PRETOP, PREBOT, SIMLNY, BKGPRE(P1), + CMPBLK, CMPFLD, POROS(P1), SPECST(P1), LITHFR, + PORBOT, PORTOP, WAVTOP, WAVBOT, WAVEPK(2), + WAVLEN, WAVVOL, PI, PORSUM, WAVPOR, WVPVOL, + WAVVEL, WAVTIM, WAVVFL, MAXPRE, STEMP, + PREGEN(P1), DENGRN, DELPOR(P1), EFFPRE(P1)

OPEN (UNIT=12, FILE='solwave-node-data.dat',STATUS='UNKNOWN') OPEN (UNIT=13, FILE='solwave-time-data.dat',STATUS='UNKNOWN') OPEN (UNIT=14, FILE='last-timestep-data.dat',STATUS='UNKNOWN') OPEN (UNIT=15, FILE='solwave-initial-data.dat',STATUS='UNKNOWN')

10 FORMAT(/, A45, 2X, A10, /) 20 FORMAT(A24, 1X, A22) 25 FORMAT(A24,1X,A23,1X,A20,1X,A25,1X,A15,1X,A25,1X,A17,1X,A23) 30 FORMAT(A9, I10, 1X, A22, 1X, I3, A7) 40 FORMAT(/, A21, E15.7, /, /) 50 FORMAT(A24, 1X, A18, 1X, A15, 1X, A25, 1X, A23, 1X, A18, 1X, A28, + 1X, A24, 1X, A35) 60 FORMAT(9(E15.7, 2X)) 70 FORMAT(A8, I5)

C Define parameter values (SI units)

NNODES = 5000000 PERME0 = 6.6D-13 SIGSTR = 0.25D+06 GRAVAC = 9.81D+00 DENLIQ = 0.25D+03 VISLIQ = 3.0D-05 DENGRN = 2.2D+03

109

ZMAX = 5.0D+03 DELZ = ZMAX/(NNODES-1) CMPBLK = 1.0D-08 CMPFLD = 1.0D-08 STEMP = 0.05D+02 PI = 3.141592654D+00 PREMAX = 1.0D+00 PRRATE = 9.59D-07 LITHFR = 0.93D+00 MDEPTH = 1.0D+03

C Define simulation length in years

SIMLNY = 7.1D+01

SIMLNS = SIMLNY*3.1536D+07

TIMSTP = 7100000 DELT = SIMLNS/TIMSTP

WRITE(*,40) ' Delta t (years) = ', DELT/3.1536D+07 WRITE(*,40) ' Delta Z (meters) = ', DELZ

C Define time steps for which output is to be printed to nodal output file

DO I = 1, 100

PRTSTP(I) = TIMSTP*I/100

END DO

C Define number of time steps for which output is to be printed to time output file

NMTOUT = 20

C Define time step interval for which output is to be printed to time output file

TPRIVL = TIMSTP/NMTOUT

C Assign elevation, depth, total stress, initial fluid pressure, permeability, C effective stress, porosity

DO I = 1, NNODES

C ...define elevation and depth

Z(I) = (I-1)*DELZ IF (I .EQ. NNODES) THEN DEPTH(I) = 0.0D+00 + MDEPTH ELSE DEPTH(I) = (MDEPTH + ZMAX) - Z(I) ENDIF

C ...compute total stress

TOTSTR(I) = DENGRN*GRAVAC*DEPTH(I)

110

C ...compute specific storage

SPECST(I) = DENLIQ*GRAVAC*(CMPBLK + POROS(I)*CMPFLD)

C ...assign initial pressure generation rate and fluid pressure

IF (DEPTH(I) .GE. 4.5D+03 .AND. DEPTH(I) .LE. 4.8D+03) THEN

FLUPRE(I) = PREMAX*TOTSTR(I) PREGEN(I) = PRRATE

ELSE

FLUPRE(I) = LITHFR*TOTSTR(I) PREGEN(I) = 0.0D+00

END IF

C ...assign a reference unperturbed background fluid pressure

BKGPRE(I) = LITHFR*TOTSTR(I)

C ...compute effective stress

EFFSTR(I) = TOTSTR(I) - FLUPRE(I)

C ...assign initial porosity (Revil and Cathles, 2002)

POROS(I) = 0.3D+00*(1 - CMPBLK*EFFSTR(I))

C ...compute permeability (Revil and Cathles, 2002)

PERMEA(I) = PERME0*(EXP(-EFFSTR(I)/SIGSTR))

C ...compute subsurface temperature(Temperature gradient of 33 degree C Celsius/km)(Holland et al., 1990; Gordon and Flemings, 1998))

TEMP(I) = STEMP + 0.33D-01*DEPTH(I)

END DO

C Calculate initial Darcy velocity

DO I = 1, NNODES

IF (I .LE. NNODES-1) THEN

DARVEL(I+1) = -(PERMEA(I+1)/VISLIQ))*((FLUPRE(I+1) - + FLUPRE(I))/DELZ - DENLIQ*(-1.0D+00*GRAVAC))

ENDIF

END DO

DARVEL(1) = DARVEL(2)

111

C Write header for TECPLOT-formatted nodal output file

PRINT*, "Writing Initial Conditions....."

WRITE(12,10) 'TITLE = "1D implicit fluid pressure solution:', + 'node data"'

WRITE(12,20) 'VARIABLES = "Depth (m)",','"Fluid Pressure (MPa)"'

WRITE(15,10) 'TITLE = "1D implicit fluid pressure solution:', + 'node data"'

WRITE(15,25) 'VARIABLES = "Depth (m)",', '"Fluid Pressure (MPa)",' + ,'"Permeability (m2)",', '"Effective stress (MPa)",', + '"Porosity (%)",', '"Specific storage (1/m)",', + '"Temperature(C)",','"Darcy velocity (m/yr)"'

C Write header for TECPLOT-formatted time output file

WRITE(13,10) 'TITLE = "1D implicit fluid pressure solution:', + 'time data"'

WRITE(13,50) 'VARIABLES = "Time (yr)",', + '"Wave bottom (m)",', '"Wave top (m)",', + '"Wave peak position (m)",', '"Wave velocity (m/yr)",', + '"Wave length (m)",', '"Average wave porosity (%)",', + '"Wave pore volume (m3)",', + '"Wave volumetric flow rate (m3/yr)"'

WRITE(13,70) 'ZONE, I=', NMTOUT+1

C Write initial conditions to TECPLOT-formatted nodal output file

INTIME = 0.0D+00

C Assign "Y" or "y" if provided TECPLOT-formatted input file

INIDAT = 'N'

IF (INIDAT .EQ. 'Y' .OR. INIDAT .EQ. 'y') THEN

CALL INPTDT(P1, Z, DEPTH, FLUPRE, NNODES, INTIME)

END IF

TIME = INTIME

CALL TECWRT(P1, DEPTH, FLUPRE, TIME, NNODES)

C Write initial conditions for other variables in a separate file

CALL INIWRT(P1, DEPTH, FLUPRE, PERMEA, EFFSTR, POROS, + SPECST, TEMP, DARVEL, TIME, NNODES)

112

CLOSE(12) CLOSE(13) CLOSE(14) CLOSE(15)

C Solve for pore fluid pressures from the combined mass balance equation and the Darcy’s law using C implicit finite difference method

C ...Define upper and lower boundary conditions

PREBOT = FLUPRE(1) PRETOP = FLUPRE(NNODES)

PORBOT = POROS(1) PORTOP = POROS(NNODES)

WAVVEL = 0.0D+00

DO 500 N = 1, TIMSTP

TIME = N*(DELT/3.1536D+07) + INTIME

DO I = 1, NNODES

IF (I .EQ. 1) THEN LOAD(I) = PREBOT MNDIAG(I) = 1.0D+00 UPDIAG(I) = 0.0D+00 END IF

IF (I .EQ. NNODES) THEN LOAD(I) = PRETOP MNDIAG(I) = 1.0D+00 LODIAG(I) = 0.0D+00 END IF

IF ((I .GT. 1) .AND. (I .LT. NNODES)) THEN

LODIAG(I) = PERMEA(I)/(DELZ*DELZ)

MNDIAG(I) = -2.0D+00*PERMEA(I)/(DELZ*DELZ) - + (PERMEA(I+1) - PERMEA(I))/(DELZ*DELZ) - + SPECST(I)*VISLIQ/(DENLIQ*GRAVAC*DELT)

UPDIAG(I) = PERMEA(I)/(DELZ*DELZ) + + (PERMEA(I+1) - PERMEA(I))/(DELZ*DELZ)

LOAD(I) = DENLIQ*GRAVAC*((PERMEA(I+1) - PERMEA(I))/DELZ) + - SPECST(I)*VISLIQ*FLUPRE(I)/(DENLIQ*GRAVAC*DELT) + - SPECST(I)*VISLIQ*PREGEN(I)/(DENLIQ*GRAVAC)

ENDIF

END DO

113

CALL TRIDIA(P1, LODIAG, MNDIAG, UPDIAG, LOAD, NNODES, PRSNEW)

C ...Update fluid pressure, effective stress, permeability and porosity

DO I=1, NNODES

IF (I .GE. 2 .AND. I .LE. NNODES-1) THEN

FLUPRE(I) = PRSNEW(I) EFFPRE(I) = EFFSTR(I) IF (FLUPRE(I) .LT. BKGPRE(I)) THEN FLUPRE(I) = BKGPRE(I) END IF EFFSTR(I) = TOTSTR(I) - FLUPRE(I) PERMEA(I) = PERME0*(EXP(-EFFSTR(I)/SIGSTR))

C ...Compute porosity change (Revil and Cathles, 2002)

DELPOR(I) = -0.3D+00*CMPBLK*(EFFSTR(I) - EFFPRE(I))

C ...Update porosity

POROS(I) = POROS(I) + DELPOR(I)

ELSE POROS(I) = POROS(I)

ENDIF END DO

DO I=1, NNODES

C ...Update specific storage

SPECST(I) = DENLIQ*GRAVAC*(CMPBLK + POROS(I)*CMPFLD)

C ...Update Darcy velocity

IF (I .LE. NNODES-1) THEN

DARVEL(I+1) = -(PERMEA(I+1)/VISLIQ)*((FLUPRE(I+1) - + FLUPRE(I))/DELZ - DENLIQ*(-1.0D+00*GRAVAC))

ENDIF

END DO

DARVEL(1) = DARVEL(2)

C ...Check to see if wave flux-related data should be sent to time output file

IF ((TIMSTP .LE. NMTOUT) .OR. (MOD(N,TPRIVL) .EQ. 0) .OR. + (N .EQ. TIMSTP) .OR. (N .EQ. 1)) THEN

C ...Find bottom of wave by comparing current fluid pressure to

114

C background fluid pressure

I=1

120 IF ((FLUPRE(I) - BKGPRE(I)) .GT. 0) THEN WAVBOT = Z(I) IBOT = I ELSE I=I+1 GO TO 120 END IF

C ...Find peak of wave by locating maximum fluid pressure

MAXPRE = FLUPRE(I) - BKGPRE(I)

130 IF ((FLUPRE(I) - BKGPRE(I)) .LT. MAXPRE) THEN WAVEPK(2) = Z(I-1) IPK = I-1 ELSE MAXPRE = FLUPRE(I) - BKGPRE(I) I=I+1 GO TO 130 END IF

C ...Find top of wave by comparing current fluid pressure to background C fluid pressure: top defined by location where fluid pressure is C less than 0.1% greater than the background fluid pressure

140 IF (FLUPRE(I) .LT. 1.001D+00*BKGPRE(I)) THEN WAVTOP = Z(I) ITOP = I ELSE I=I+1 GO TO 140 END IF

C ...Compute wave length

WAVLEN = ABS(WAVTOP - WAVBOT)

C ...Compute wave volume, assuming spherical geometry

WAVVOL = (4.0D+00/3.0D+00)*PI*(WAVLEN/2.0D+00)**3

C ...Compute the weighted average porosity of the wave

PORSUM = 0.0D+00

DO I = IBOT, ITOP PORSUM = PORSUM + POROS(I)*(Z(I) - Z(I-1)) END DO

WAVPOR = PORSUM/WAVLEN

C ...Compute the wave pore volume

115

WVPVOL = WAVVOL*WAVPOR

C ...Compute wave velocity

IF (N .GE. 2) THEN WAVVEL = (WAVEPK(2) - WAVEPK(1))/(TPRIVL*DELT) ELSE WAVVEL = 0.0D+00 END IF

C ...Set current wave peak location to old wave peak location

WAVEPK(1) = WAVEPK(2)

C ...Compute the length of time needed for the wave to traverse its C wavelength

WAVTIM = WAVLEN/WAVVEL

C ...Compute volumetric flow rate of wave

WAVVFL = WVPVOL/WAVTIM

C ...Write flux-related data to time output file

OPEN (UNIT=13,FILE='solwave-time-data.dat',ACCESS='APPEND', + STATUS='OLD')

PRINT*, "Writing Time Output File for Timestep:",N

WRITE(13,60) TIME, WAVBOT, WAVTOP, WAVEPK(2), + WAVVEL*3.1536D+07, WAVLEN, WAVPOR, WVPVOL, + WAVVFL*3.1536D+07

CLOSE(13) END IF

C ...Check to see if results should be printed to nodal output file

OPEN (UNIT=12,FILE='solwave-node-data.dat',ACCESS='APPEND', + STATUS='OLD')

DO I = 1, 100

IF (N .EQ. PRTSTP(I)) THEN

PRINT*, "Writing Nodal Output File for Timestep:",N CALL TECWRT(P1, DEPTH, FLUPRE, TIME, NNODES)

ENDIF END DO CLOSE(12)

C ...Write the last time step results to a separate nodal output file

116

DO I = 1, 100

IF (N .EQ. PRTSTP(I)) THEN OPEN (UNIT=14,FILE='last-timestep-data.dat', + STATUS='REPLACE') PRINT*, "Writing Nodal Output File for Last Timestep:",N CALL FNLWRT(P1, DEPTH, FLUPRE, TIME, NNODES)

ENDIF END DO CLOSE(14)

C ...Report simulation progress at 1% increments

SIMPRO = TIMSTP/100

IF (MOD(N,SIMPRO) .EQ. 0) THEN

WRITE(*,30) ' Timestep', N, 'completed. Simulation', + INT(100*FLOAT(N)/FLOAT(TIMSTP)), '% done.'

ENDIF

500 CONTINUE

STOP

END PROGRAM SOLWAVE

SUBROUTINE TRIDIA(P1, A, B, C, F, N, X)

C This is the Thomas algorithm solution for tri-diagonal matrices given by Wang & Anderson (1982)

INTEGER*8 N, NU, P1

DOUBLE PRECISION A(P1), B(P1), C(P1), X(P1), F(P1), ALPHA(P1), + BETA(P1), Y(P1)

ALPHA(1) = B(1) BETA(1) = C(1)/ALPHA(1) Y(1) = F(1)/ALPHA(1)

DO I=2, N ALPHA(I) = B(I) - A(I)*BETA(I-1) BETA(I) = C(I)/ALPHA(I) Y(I) = (F(I) - A(I)*Y(I-1))/ALPHA(I) END DO

C Begin backward substitution from last row

X(N) = Y(N) NU = N-1 DO I=1, NU J = N-I X(J) = Y(J) - BETA(J)*X(J+1) END DO

117

RETURN END

SUBROUTINE TECWRT(P1, DEPTH, FLUPRE, TIME, NNODES)

C Subroutine to write results to TECPLOT-formatted output file

INTEGER*8 NNODES, P1

DOUBLE PRECISION DEPTH(P1), FLUPRE(P1), TIME

10 FORMAT(A13, E15.7, 1X, A7, 1X, A4, I8) 20 FORMAT(2(E15.7,2X))

WRITE(12,10) 'ZONE T = "', TIME, 'years",', 'I = ', NNODES

DO I = 1, NNODES

WRITE(12,20) DEPTH(I), FLUPRE(I)/1.0D+06

END DO

RETURN END

SUBROUTINE INIWRT(P1, DEPTH, FLUPRE, PERMEA, EFFSTR, POROS, + SPECST, TEMP, DARVEL, TIME, NNODES)

C Subroutine to write initial data to TECPLOT-formatted output file

INTEGER*8 NNODES, P1

DOUBLE PRECISION DEPTH(P1), FLUPRE(P1), TIME, PERMEA(P1), + EFFSTR(P1), SPECST(P1), TEMP(P1), DARVEL(P1), + POROS(P1)

10 FORMAT(A13, E15.7, 1X, A7, 1X, A4, I8) 20 FORMAT(8(E15.7,2X))

WRITE(15,10) 'ZONE T = "', TIME, 'years",', 'I = ', NNODES

DO I = 1, NNODES

WRITE(15,20) DEPTH(I), FLUPRE(I)/1.0D+06, PERMEA(I), EFFSTR(I), + POROS(I), SPECST(I), TEMP(I), DARVEL(I)*3.1536D+07

END DO

RETURN END

SUBROUTINE FNLWRT(P1, DEPTH, FLUPRE, TIME, NNODES)

C Subroutine to write last time-step results to TECPLOT-formatted output file

118

INTEGER*8 NNODES, P1

DOUBLE PRECISION DEPTH(P1), FLUPRE(P1), TIME

10 FORMAT(/, A45, 2X, A10, /) 20 FORMAT(A24, 1X, A22) 30 FORMAT(A8, E15.7, 1X, A7, 1X, A3, I8) 40 FORMAT(2(E15.7,2X))

WRITE(14,10) 'TITLE = "1D implicit fluid pressure solution:', + 'node data"' WRITE(14,20) 'VARIABLES = "Depth (m)",','"Fluid Pressure (MPa)"' WRITE(14,30) 'ZONE T="', TIME, 'years",', 'I= ', NNODES

DO I = 1, NNODES

WRITE(14,40) DEPTH(I), FLUPRE(I)/1.0D+06

END DO

RETURN END

SUBROUTINE INPTDT(P1, Z, DEPTH, FLUPRE, NNODES, INTIME)

C Subroutine to read initial data from TECPLOT-formatted input file

INTEGER NNODES, P1, N CHARACTER TXT DOUBLE PRECISION Z(P1), DEPTH(P1), FLUPRE(P1), INTIME

10 FORMAT(2(E15.7,2X)) 20 FORMAT(A4,1X, A3, E15.7, 1X, A7,1X, A2, 3X, I8)

READ(14,*) READ(14,*) READ(14,*) READ(14,*) READ(14,*) TXT, TXT, INTIME, TXT, TXT, N

IF (N .NE. NNODES) THEN

PRINT*,"Error: Please use input file with same nodal size" STOP

ELSE

DO I = 1, NNODES

READ(14,10) DEPTH(I), FLUPRE(I) FLUPRE(I) = FLUPRE(I)*1.0D+06

END DO END IF RETURN END SUBROUTINE INPTDT

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VITA

Ajit Joshi was a born in Kavre, Nepal, ‘The Kingdom of the Himalayas’. He received his Bachelors and Master’s degrees from the Tribhuvan University in Nepal. After that he moved to the USA where he completed his second Master’s degree in geological sciences with a specialization in hydrogeology from the University of Missouri-Columbia in 2012. He then started a doctoral program at the University of Missouri-Columbia, continuing his research on the origin and hydrocarbon transport capability of porosity waves in sedimentary basins. He published a manuscript on petroleum transport by porosity waves in Marine and Petroleum Geology in 2012. In addition to his research modeling porosity waves, he also has served as a Teaching Assistant for two introductory courses in physical and environmental geology since 2011. He also likes to teach different aspects of environmental geology. His research interests are in numerical modeling of deep subsurface fluid flow, groundwater contamination, interactions of groundwater and surface water. Besides, he is also interested in modeling climate change impact on groundwater systems and its response to various other systems such as natural ecosystems, agriculture, society, etc.

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