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Abstract

The Bureau of Economic Geology has evaluated hydrate concentrations across deep-water areas of Green Canyon, Gulf of Mexico, using well log data and four-component (4C) seismic data acquired by companies interested in deep oil and gas targets, not in nearseafloor hydrates. Even though these seismic and well log data are not acquired for purposes of studying nearseafloor geology, we have found these off-the-shelf industry data to be invaluable for evaluating hydrate systems positioned immediately below the seafloor. We summarize our data analyses and initial research findings in a two-paper sequence.

In this first paper, we describe how hydrate concentration can be estimated from resistivity logs and then from compressional (VP) and shear (VS) velocities as a joint-inversion approach for quantifying the amount of in-place hydrate. We found no industry well in the Green Canyon area where velocity-log data has been recorded across shallow near-seafloor strata where deep-water hydrates are found. Consequently, we have utilized interval VP and VS velocities obtained by processing deep-water 4C seismic data in our jointinversion hydrate estimations.

The rock physics used to estimate deep-water hydrate concentrations from resistivity logs and from interval velocities is challenging because deep-water, near-seafloor sediments exist in a unique environment characterized by high porosities (greater than 50 percent) and low effective pressures (literally zero at the seafloor). Rock physics analyses are further complicated by the fact that resistivity and velocity responses to the hydrate fraction in seafloor sediments depend on whether the hydrate is layered (either horizontally or vertically) or dispersed, and if dispersed, whether the hydrate is part of the load-bearing matrix or is floating freely in pore spaces. Because the oil and gas industry is not yet focused on hydrate production, there is inadequate core information to define the specific hydrate-sediment morphology that should be used in a rock physics model that is applied to deep-water, near-sea-floor environments in the Green Canyon area. In this first paper we illustrate how hydrate morphology affects the interpretation of resistivity and velocity responses of hydrate-bearing sediment. We have assumed a load-bearing morphology for our inversion work and await specific core information to know if this assumption needs to be modified in future work.

We have found that the Hashin-Shtrikman Lower Bound that can be used to describe the resistivity and elastic moduli of a mixture of arbitrary fractions of quartz, clay, hydrate, and brine is a critical concept for evaluating relationships between resistivity, velocity, and hydrate concentration in deep-water, near-seafloor environments. We discuss our rock physics modeling approach based on the application of Hashin-Shtrikman theory. In the second paper of this series, we describe how we combine the rock physics models that we have developed with velocity attributes determined from 4C seismic data to generate maps of hydrate concentration across our study area.

Introduction

An important aspect of our deep-water hydrate research is our interpretation of the resistivity-log response of high-porosity mixtures of sediment, hydrate, and brine. We develop evidence and logic that the Hashin-Shtrikman Lower Bound (Hashin and Shtrikman, 1962) should dictate the functional behavior for the resistivity of a high-porosity mixture of sediment, hydrate, and conductive brine. In our terminology, “high porosity” means that the porosity of the initial sediment is around critical porosity, which is the porosity condition across the shallower portions of the hydrate stability interval at our study sites.

In this paper, we present evidence that the classical Archie Equation (Archie, 1942) that is used to interpret hydrocarbon concentration from resistivity logs in consolidated media may also be used to predict hydrate concentration in high-porosity, unconsolidated, deep-water sediments if appropriate constants are used to adjust the equation response to the resistivity predicted by the Hashin-Shtrikman Lower Bound for that medium. Neither the Archie Equation nor the Hashin-Shtrikman Lower Bound should be used to interpret hydrate concentration when thin layers of pure hydrate are intercalated with sediment layers (either as horizontal layers or as vertical sheets). The Hashin-Shtrikman Lower Bound should be used to estimate hydrate concentration only when hydrate is uniformly dispersed throughout the sediment and each sediment grain and each clathrate cluster is surrounded by brine. In our methodology, we use the Hashin-Shtrikman Lower Bound to determine the constants in the Archie Equation and then apply this calibrated Archie Equation to resistivity logs acquired across the Green Canyon area to illustrate that significant intervals of hydrate are present at several locations and that some zones within these intervals have hydrate concentrations that exceed 50 percent of the pore space.

Study Sites

Our study was done across two sites in the Green Canyon area of the Gulf of Mexico where 2D profiles of 4C OBC seismic data were acquired by WesternGeco, our industry partner. The locations of these sites are defined on the maps displayed as Figure 1. One study area encompassed Typhoon Field; the second site spanned Genesis Field. There were several reasons for selecting these study sites. First, profiles of low-frequency (10–200 Hz) 4C OBC seismic data and highfrequency (1–10 kHz) autonomous underwater vehicle (AUV) data traversed each site. Second, geotechnical reports were available describing laboratory measurements of seafloor sediment properties made on seafloor borings at Genesis and Typhoon fields. These measurements were critical for calibrating sediment properties to seismic attributes. Third, the seafloor across portions of each study site exhibited bright reflectivity, which is proving to be an excellent proxy that can be used to indicate where a hydrate system exists beneath the seafloor (Roberts et al., 1992, 2006).

Figure 1.

(A) Regional map showing Typhoon study site, Block GC237, and Genesis study site, Block GC204. Locations of Green Canyon area calibration wells, 4C seismic profiles, and AUV profiles used in this study are detailed for (B) Typhoon Field area and for (C) Genesis Field area. AUV data were provided by Dr. Harry Roberts of Louisiana State University.

Figure 1.

(A) Regional map showing Typhoon study site, Block GC237, and Genesis study site, Block GC204. Locations of Green Canyon area calibration wells, 4C seismic profiles, and AUV profiles used in this study are detailed for (B) Typhoon Field area and for (C) Genesis Field area. AUV data were provided by Dr. Harry Roberts of Louisiana State University.

Resistivity Models of Sediment-Hydrate Systems

Log data across our Green Canyon hydrate study areas were acquired by petroleum companies who were interested in deep oil and gas targets, not in near-seafloor gas hydrate systems. Consequently, these companies acquired minimal lithofacies-sensitive log data consisting of only gamma-ray and resistivity measurements across shallow, near-seafloor intervals where hydrates occur. All other logs acquired across hydratebearing zones were measurements such as temperature and rate of penetration, which provide limited lithofacies information. For this reason, any log-based evidence of subseafloor hydrates across Green Canyon lease blocks has to be inferred from resistivity logs.

The Earth model we used to describe the resistivity properties of deep-water hydrate systems is illustrated as Figure 2. This model assumes that hydrate is uniformly dispersed throughout the sediment and is appropriate for resistivity analyses of hydrate-sediment mixtures whether the hydrate is load-bearing or free-floating. The only requirement is that each clathrate cluster and each mineral grain be surrounded by fluid except at their points of contact with neighboring hydrate clusters and mineral grains.

Figure 2.

Resistivity model for a hydrate system in which hydrate is uniformly distributed throughout its host sediment. A fundamental assumption is that each mineral grain (black) and each clathrate cluster (red) is surrounded by conductive brine (blue), which creates a medium with many pathways for electrical current flow (arrows).

Figure 2.

Resistivity model for a hydrate system in which hydrate is uniformly distributed throughout its host sediment. A fundamental assumption is that each mineral grain (black) and each clathrate cluster (red) is surrounded by conductive brine (blue), which creates a medium with many pathways for electrical current flow (arrows).

Archie Equation Formulated for Clay

The Archie Equation has been used to analyze resistivity responses of fluid-filled porous rocks for more than six decades (Archie, 1942). The clay-free form of this equation with which we begin our analysis can be written as

 

formula

The Archie Equation is an empirical law that was developed to determine water saturation in clean sands from measurements of resistivity and porosity across a sand-fluid mixture. The principal assumption of this empirical law is that electrical current travels only through the brine phase of fluid-saturated sediments because quartz minerals and hydrocarbon fluids are rather good electrical insulators. However, when clay minerals are present in the host sediments, the original form of the Archie Equation (Eq. 1) is no longer accurate. Clay minerals have lower resistivity than clean sands and can have a large impact on the resistivity of a rock formation. If the presence of clay minerals is ignored and the simple form of the Archie Equation is applied to clay-rich sands, water saturation is overestimated at all porosity values. As a result, the saturation of any nonconductive phase that may be in the pores (for example hydrates) will be underestimated.

Schlumberger Wireline Services (1998) proposed a modification to the Archie Equation that takes into account the presence of clay. This modified Archie Equation is valid for several types of clay distribution (clay can be disseminated, structured, or layered). Key parameters required for implementing this modified equation are the volume of clay (Vcl) present in the sediments and the resistivity (Rcl) of the clay minerals. Volume of clay can be estimated from gamma-ray log data, and the resistivity of clay minerals can be measured in the laboratory. If no core samples are available for lab testing, we must use resistivity data measured across pure-clay intervals from nearby geology or rely on published resistivity measurements of clays. Published information sources confirm that Rcl spans a large resistivity range extending from 1 to 1,000 ohm-m (Rider, 1991).

The modified Archie Equation proposed by Schlumberger (1998) is

 

formula

where in our deep-water applications,

  • R is the measured resistivity of sediments containing gas hydrates,

  • Rw is the resistivity of the brine in these hydrate-bearing sediments,

  • ϕ is the porosity of the sediments,

  • a is the geometric factor (a ~1.08),

  • m is the cementation factor (m = 1.2 to 1.7 for unconsolidated sediments),

  • Vcl is the volume of clay estimated from gamma-ray log data,

  • Rcl is the resistivity of clay mineral (Rcl = 1 to 1,000 ohm-m),

  • Sw is the water saturation, and

  • n is the saturation exponent (1.7 ≤ n ≤ 2.2).

Near-Seafloor Sediment and Seismic Velocity

Our rock physics theory that describes how hydrate concentration affects VP and VS velocities in deep-water sediments has been published (Sava and Hardage, 2009; Hardage et al., 2009) and will not be repeated here. In our analysis, we conclude the four rock physics models that can be used to describe deep-water hydrate systems across the Gulf of Mexico:

  • Model A assumes hydrates are uniformly disseminated throughout the sediment and are part of the load-bearing frame of the host sediments.

  • Model B assumes hydrates are also disseminated throughout the sediment, but they float in the porous space and do not change the dry mineral frame of their host sediments.

  • Model C assumes an anisotropic, thin-layered medium in which layers of pure gas hydrate are intercalated with layers of hydrate-free sediments saturated with fluid. These thin layers can be horizontal or vertical. Vertical thin layers approximate thin fractures and veins filled with pure hydrate.

  • Model D is also an anisotropic, thin-layered medium. However, in this model, hydrates are disseminated in thin horizontal layers of sediments in which they occupy 99 percent of the porous space and are part of the load-bearing frame. These thin hydrate-bearing layers are intercalated with thin layers of hydrate-free sediments saturated with fluid.

The key input parameter in all of these models is hydrate concentration. Our goal is to quantitatively relate hydrate concentration to seismic P- and S-wave velocities for each of these four possible hydrate morphologies.

This form of the Archie Equation should be compared with Equation 1, the form of the equation for clean sands. If Vcl = 0, Equation 2 reduces to Equation 1, the classical form of Archie’s Equation.

If we consider the saturation exponent n to be 2, as most published papers suggest, then Equation 2 is quadratic in Sw, and its positive root is

 

formula

This expression for Sw estimates water saturation when the Archie Equation is modified for clay content.

 

formula

If n ≠ 2, the square root term in these equations is replaced with the nth-root equivalent.

Because the Archie Equation is an empirical model, when it is applied to a specific rock matrix and a specific type of pore fluid, parameters a, m, and n must be derived and adjusted to create optimal agreement between resistivity readings and independent knowledge of RW, ϕ, and SW for that rock-fluid system. In typical oil and gas reservoir applications, a ~1.0, n = 2, and m ranges from 2.0 to 2.5.

There is limited experience in applying the Archie Equation to high-porosity mixtures of sediment, brine, and hydrate. In their analysis of Blake Ridge resistivity data, Collett and Ladd (2000) use the following values for their formulation of the Archie Equation: a = 1.05, m = 2.56, n = 2, and RW = 0.23 ohm-m. We cannot find the exact value of RW that was used in their Blake Ridge study. We know only that the pore fluid By definition, the concentration (cgh) of hydrate in the sediments is (1-Sw), or salinity is assumed to be 32,000 ppm. If the hydrate formation temperature is assumed to be 65ºF, then this salinity leads to RW = 0.23 ohm-m. Given our application of the Hashin-Shtrikman bounds that will be discussed in the following sections, we conclude that the parameter values used for the Archie Equation at Blake Ridge are not optimal for high-porosity unconsolidated sediments found in deep water. In our formulation of the Archie Equation, we alter the values to be probability distribution functions in which the expected values of the parameters are: a = 1.0, m = 1.2, n = 2, and RW = 0.17 ohm-m. A value of m = 1.2 for unconsolidated sediments has been suggested by other researchers (Archie, 1942; Sen et al., 1981; Mendelson and Cohen, 1982). Our reasoning for our parameter choices will be further explained as we discuss our second analytical resistivity model, the Hashin-Shtrikman Lower Bound.

Hashin-Shtrikman Lower Bound

Calculation of Hashin-Shtrikman bounds is a valuable analysis technique for defining the maximum and minimum values that can be observed for magnetic, electrical, and thermal properties of rock systems that are mixtures of several distinct minerals and fluids (Hashin and Shtrikman, 1962).

An analysis by Wempe (2000) has been particularly germane to our study of the resistivity response of hydrate dispersed throughout unconsolidated, high-porosity, near-seafloor sediments. A key graphic of Wempe’s study is reproduced as Figure 3a. Our modification of this graphic is shown as Figure 3b. In these figures, the horizontal axis is porosity and the vertical axis is normalized resistivity R/RW, where R is the resistivity measured across a medium of porosity ϕ and RW is the resistivity of the fluid that fills the pores. The shaded interval labeled ϕC defines the range of critical porosity, which is porosity where the grains of a high-porosity medium convert from a suspended state to a load-bearing condition in which each grain touches at least one other grain. Critical porosity varies from about 0.3 for round, uniform-size grains, to around 0.4 for well sorted rounded grains of variable size, to about 0.6 for highly oblate (flat) grains.

Figure 3.

(Top) Crossplot of normalized resistivity (R/RW) and porosity for a large number of laboratory tests and field-data observations that involve a wide range of conductive media (Wempe, 2000). (Bottom) Our modification of the crossplot to emphasize principles important for deep-water hydrate systems. R is measured resistivity; RW is the resistivity of the pore-filling fluid. The shaded interval ΦC is the range of critical porosity for grains of different geometrical shapes. Note that all data converge to the Hashin-Shtrikman Lower Bound as porosity increases and enters the critical-porosity range.

Figure 3.

(Top) Crossplot of normalized resistivity (R/RW) and porosity for a large number of laboratory tests and field-data observations that involve a wide range of conductive media (Wempe, 2000). (Bottom) Our modification of the crossplot to emphasize principles important for deep-water hydrate systems. R is measured resistivity; RW is the resistivity of the pore-filling fluid. The shaded interval ΦC is the range of critical porosity for grains of different geometrical shapes. Note that all data converge to the Hashin-Shtrikman Lower Bound as porosity increases and enters the critical-porosity range.

The data plotted on Figure 3 are comprehensive and include laboratory measurements and field data gleaned from 11 studies referenced by Wempe (2000). A key concept demonstrated by these data (Fig. 3b) is that the resistivity behavior of all porous media converges to the Hashin-Shtrikman Lower Bound (HS-) when the porosity of the medium equals or exceeds critical porosity. Because the porosity of the deep-water, near-seafloor sediments that span the hydrate stability zone in the Green Canyon area equals or exceeds critical porosity for many tens of meters (sometimes even a few hundred meters) below the seafloor, we are led to an important conclusion: the Hashin-Shtrikman Lower Bound is an ideal function for describing the resistivity of deep-water hydrate systems.

The Hashin-Shtrikman Lower Bound that we calculated is plotted on Figure 4 to illustrate how the resistivity of deep-water sediment varies as a function of hydrate concentration. As hydrate concentration increases from 0 to 60 percent of the pore space, resistivity increases from 0.6 ohm-m to only 2 ohm-m. The implication is that with 60 percent of the pore space occupied by hydrate, a large number of connected paths of conductive brine continue to wend through the sediment-hydrate-brine mixture (Fig. 2).

Figure 4.

The Hashin-Shtrikman Lower Bound (HS-) and two formulations of the clay-free form of the Archie Equation (Eq. 1) displayed as functions of resistivity and hydrate fraction. Hydrate fraction is defined in terms of the pore volume (top axis) or the unit volume (bottom axis). Archie Equation 1 is our formulation for deep-water hydrate systems. Archie Equation 2 was proposed by Collett and Ladd (2000) at Blake Ridge. We stress this fundamental principle: at low hydrate concentrations where porosity is a maximum, deep-water mixtures of sediment and dispersed hydrate must have resistivities that agree with, or approximate, the Hashin-Shtrikman Lower Bound.

Figure 4.

The Hashin-Shtrikman Lower Bound (HS-) and two formulations of the clay-free form of the Archie Equation (Eq. 1) displayed as functions of resistivity and hydrate fraction. Hydrate fraction is defined in terms of the pore volume (top axis) or the unit volume (bottom axis). Archie Equation 1 is our formulation for deep-water hydrate systems. Archie Equation 2 was proposed by Collett and Ladd (2000) at Blake Ridge. We stress this fundamental principle: at low hydrate concentrations where porosity is a maximum, deep-water mixtures of sediment and dispersed hydrate must have resistivities that agree with, or approximate, the Hashin-Shtrikman Lower Bound.

The Hashin-Shtrikman Lower Bound curve displayed on Figure 4 implies that even when hydrate fills 80 percent of the pore space, there are still enough electrical current flow paths so that the resistivity increases to only 4 ohm-m. One factor that may keep the resistivity of this sediment-brine-hydrate mixture at a low value, even though the hydrate concentration is high, is that as hydrate grows, it expels salt into the surrounding brine and makes the brine more conductive. If some of this expelled salt stays close to its point of expulsion (a point of contention among hydrate researchers), the increased salinity of the pore-filling brine will reduce resistivity even at high hydrate concentrations. In terms of electrical conductivity, a smaller number of electrical-current flow paths through higher salinity brine are equivalent to a larger number of flow paths through reduced-salinity brine. In our analysis, however, we do not decrease pore-fluid resistivity as hydrate concentration increases. Using this constraint of a constant porefluid resistivity, the Hashin-Shtrikman Lower Bound implies that a significant increase in resistivity in deepwater hydrate-bearing sediment does not occur until hydrate concentration exceeds 90 percent of the pore space and the number of connected brine-filled paths is severely reduced (Fig. 4).

Included on Figure 4 is a curve labeled Archie Equation 1 that describes the resistivity behavior of the clay-free form of the Archie Equation (Eq. 1) that we think is appropriate for hydrate systems across Green Canyon that are embedded in clean-sand host sediment. Also included is a curve (Archie Equation 2) that describes how the Archie Equation developed by Collett and Ladd (2000) at Blake Ridge would appear. The difference between the responses of these two Archie Equations is created by different choices for m and RW. Specifically, the parameter values used in these two formulations of the Archie Equation are shown in the following table.

Table 1.

Parameter values used in two formulations of the Archie Equation.

We present the following arguments to support our parameter choices for the clay-free form of the Archie Equation:

  • A major contributor to the difference between the two Archie Equations is the different values (2.56 versus 1.2) for the cementation exponent, m. Studies by Sen et al. (1981) and by Mendelson and Cohen (1982) show that m should be reduced to approximately 1.2 for unconsolidated sediments. Cementation exponent values of approximately 2.5 are appropriate for consolidated rocks but appear to be inappropriate for unconsolidated sediments.

  • The study by Wempe (2000) summarized in Figure 3 forces us to conclude that any resistivity equation that is used to analyze deep-water hydrate systems when porosities are equal to or greater than critical porosity must have a functional behavior that approximates the Hashin-Shtrikman Lower Bound. With the above parameters, our version of the Archie Equation is a close approximation of the Hashin-Shtrikman Lower Bound at low hydrate concentrations where porosities exceed the critical porosity value (Fig. 4). We think that an Archie Equation that uses a large value of m deviates too far from the Hashin-Shtrikman Lower Bound and does not represent true resistivity conditions of a deep-water hydrate-sediment-brine mixture.

  • We use a value of 0.17 ohm-m for RW because we assume that the pore fluid has a salinity of 45,000 ppm rather than 32,000 ppm, as was assumed by Collett and Ladd (2000) in their analysis of Blake Ridge resistivity logs. Pore fluid across a hydrate interval should have increased salinity because in converting local brine into clathrate cages, the hydrate-forming process expels NaCl and retains H2O. The magnitude of salinity increase is unknown. We found one resistivity log in our study area to which the logging contractor added the comment that pore fluid salinity was 45,000 ppm. We decided to use that salinity value when we calculated Archie Equation 1 displayed in Figure 4.

  • We have used the clay-free form of the Archie Equation in this discussion only to simplify our arguments that the Hashin-Shtrikman Lower Bound should be the governing physics for describing the resistivity of high-porosity mixtures of sediment, hydrate, and brine. In practice, we used the form (Fig. 2) of the Archie Equation that includes a clay-correction term (Hardage et al., 2009).

Near-Seafloor Sediment and Seismic Velocity

Our rock physics theory that describes how hydrate concentration affects VP and VS velocities in deep-water sediments has been published (Sava and Hardage, 2009; Hardage et al., 2009) and will not be repeated here. In our analysis, we concluded the four rock physics models illustrated as Figure 5 can be used to describe deep-water hydrate systems across the Gulf of Mexico:

Figure 5.

Graphical sketches of the four models of marine hydrate systems assumed in this work: load-bearing hydrates (Model A); free-floating hydrates (Model B); thin layers of horizontal or vertical pure hydrate intercalated with unconsolidated hydrate-free sediments (Model C); thin horizontal layers of disseminated, load-bearing hydrates intercalated with unconsolidated, hydrate-free sediments (Model D). Hydrates are represented in light blue and sediment in black.

Figure 5.

Graphical sketches of the four models of marine hydrate systems assumed in this work: load-bearing hydrates (Model A); free-floating hydrates (Model B); thin layers of horizontal or vertical pure hydrate intercalated with unconsolidated hydrate-free sediments (Model C); thin horizontal layers of disseminated, load-bearing hydrates intercalated with unconsolidated, hydrate-free sediments (Model D). Hydrates are represented in light blue and sediment in black.

  • Model A assumes hydrates are uniformly disseminated throughout the sediment and are part of the load-bearing frame of the host sediments.

  • Model B assumes hydrates are also disseminated throughout the sediment, but they float in the porous space and do not change the dry mineral frame of their host sediments.

  • Model C assumes an anisotropic, thin-layered medium in which layers of pure gas hydrate are intercalated with layers of hydrate-free sediments saturated with fluid. These thin layers can be horizontal or vertical. Vertical thin layers approximate thin fractures and veins filled with pure hydrate.

  • Model D is also an anisotropic, thin-layered medium. However, in this model, hydrates are disseminated in thin horizontal layers of sediments in which they occupy 99 percent of the porous space and are part of the load-bearing frame. These thin hydrate-bearing layers are intercalated with thin layers of hydrate-free sediments saturated with fluid.

  • The key input parameter in all of these models is hydrate concentration. Our goal is to quantitatively relate hydrate concentration to seismic P- and S-wave velocities for each of these four possible hydrate morphologies.

Mixed-Mineralogy Host Sediment

Our theory (Sava and Hardage, 2009; Hardage et al., 2009) allows mixed mineralogy and different saturating fluids to be included in the wave-propagation media that we model. Examples of this mixed-mineralogy modeling capability are shown as Figures 6 and 7. For the curve suites on each plot, the clay content in the sediments varies from 0 to 100 percent at a 25 percent increment. Some parameters used in the modeling, such as critical porosity and coordination number, vary with mineralogy (Murphy, 1982). For clean quartz grains (0 percent clay content), the critical porosity is assumed to be 37 percent, and the coordination number C is considered to be 8 (C is the average number of grains that are physically contacted by any one grain). For pure clay minerals (100 percent clay content), we use a larger critical porosity of 67 percent and a smaller coordination number of 4, as many geotechnical data suggest (Murphy, 1982). For each mixture of quartz and clay minerals, we derive the values for critical porosity and coordination number by doing a linear interpolation between the corresponding values for the two end members of pure quartz and pure clay. Also, at subseafloor depths where hydrates are stable, the porosity of clay-rich sediments is larger than the porosity of pure quartz grains. Therefore, we compute VP and VS as a function of hydrate concentration for sediments having different porosity values: 37 percent for pure quartz and 50 percent for pure clay minerals. For each mixture of quartz and clay, we use again a linear interpolation between the values for the two end members of pure quartz and pure clay. The effective pressure we use in these particular calculations is 0.5 MPa, which corresponds to a depth below seafloor of approximately 60 m.

Figure 6.

P-wave velocity as a function of the volumetric fraction of hydrate (cgh) in a sediment mixture having variable clay content and an effective pressure of 0.5 MPa, which simulates a subseafloor depth of approximately 60 m. Model A: load-bearing hydrates disseminated in the sediments; Model B: free-floating hydrates disseminated in the sediments; Model C: horizontal or vertical layers of pure hydrates producing slow P-waves (dotted lines) and fast P-waves (solid lines); Model D: horizontal or vertical layers of disseminated, load-bearing hydrates producing slow P-waves (dotted lines) and fast P-waves (solid lines). Each curve corresponds to a different clay content (ranging from 0 to 100 percent at a 25 percent increment), different critical porosity values, and different coordination numbers. The curves are computed at increasing porosity values as the clay content increases. Arrows indicate increasing clay content.

Figure 6.

P-wave velocity as a function of the volumetric fraction of hydrate (cgh) in a sediment mixture having variable clay content and an effective pressure of 0.5 MPa, which simulates a subseafloor depth of approximately 60 m. Model A: load-bearing hydrates disseminated in the sediments; Model B: free-floating hydrates disseminated in the sediments; Model C: horizontal or vertical layers of pure hydrates producing slow P-waves (dotted lines) and fast P-waves (solid lines); Model D: horizontal or vertical layers of disseminated, load-bearing hydrates producing slow P-waves (dotted lines) and fast P-waves (solid lines). Each curve corresponds to a different clay content (ranging from 0 to 100 percent at a 25 percent increment), different critical porosity values, and different coordination numbers. The curves are computed at increasing porosity values as the clay content increases. Arrows indicate increasing clay content.

Figure 7.

S-wave velocity as a function of the volumetric fraction of hydrate (cgh) in a sediment mixture having variable clay content and an effective pressure of 0.5 MPa, which simulates a subseafloor depth of approximately 60 m. Model A: load-bearing hydrates disseminated in the sediments; Model B: free-floating hydrates disseminated in the sediments; Model C: horizontal or vertical layers of pure hydrates producing slow S-waves (dotted lines) and fast S-waves (solid lines); Model D: horizontal or vertical layers of disseminated, load-bearing hydrates producing slow S-waves (dotted lines) and fast S-waves (solid lines). Different curves correspond to different clay content (from 0 to 100 percent at a 25 percent increment), different critical porosity values, and different coordination numbers. The curves are computed at increasing porosity values as clay content increases. Arrows indicate increasing clay content.

Figure 7.

S-wave velocity as a function of the volumetric fraction of hydrate (cgh) in a sediment mixture having variable clay content and an effective pressure of 0.5 MPa, which simulates a subseafloor depth of approximately 60 m. Model A: load-bearing hydrates disseminated in the sediments; Model B: free-floating hydrates disseminated in the sediments; Model C: horizontal or vertical layers of pure hydrates producing slow S-waves (dotted lines) and fast S-waves (solid lines); Model D: horizontal or vertical layers of disseminated, load-bearing hydrates producing slow S-waves (dotted lines) and fast S-waves (solid lines). Different curves correspond to different clay content (from 0 to 100 percent at a 25 percent increment), different critical porosity values, and different coordination numbers. The curves are computed at increasing porosity values as clay content increases. Arrows indicate increasing clay content.

As expected, the P- and S-wave velocities decrease with increasing clay content (and implicitly with increasing porosity), as we observe from each panel of Figures 6 and 7. For Model A (load-bearing hydrates), as porosity and clay content of the sediments increase, it becomes more challenging to estimate small hydrate concentrations, especially using S-wave velocity data (Fig. 7). For layered model D, we observe that both P- and S-wave anisotropy decreases with increasing clay content because the elastic properties of clay minerals are closer to those of hydrates than are the elastic properties of quartz minerals (Table 1). This modeling shows that we should expect larger anisotropy in P- and S-wave velocities if hydrate layers are intercalated with clean sands than with clay-rich sediments. Models C and D take into account only the anisotropy due to thin layers and consider the clay-rich sediments to be isotropic. This assumption may hold for sediments immediately below seafloor. However, as depth increases, the stress-induced anisotropy of clays will increase. At large depths, Models C and D associated with clay-rich sediments should be adjusted to account for the additional anisotropy caused by the anisotropy of clay minerals.

Uncertainty in Estimating Hydrate Concentration

Our approach to estimating the uncertainty in hydrate concentration calculated from resistivity logs is based on stochastic simulations. We represent input parameters used in the deterministic Archie’s Law and in its modified version for clay content by various probability distribution functions (PDFs) that express mathematically the variation and uncertainty of these parameter values. These probability distribution functions are either: (1) uniform distributions over the possible range of variability for each input parameters, or (2) Gaussian distributions. A uniform distribution assumes that any value for an input parameter is equally likely over the range of variability that is allowed. A Gaussian distribution suggests that the most likely value for the parameter is the mean of its associated Gaussian distribution and that the variance of its distribution function is a measure of the uncertainty of the parameter value about the mean.

Therefore, we represent each input parameter in the Archie Equation not by a single number, but by a probability distribution function that allows us to incorporate the inherent uncertainty about that input parameter into the calculation of hydrate concentration. These distribution functions permit us to use constraints on each parameter that are based on measurements or on knowledge acquired over similar environments.

After we assign a probability distribution function to each input parameter, we then run Monte Carlo simulations over these distributions. We randomly draw a set of values of R, Rw, ϕ, a, m, Vcl, and Rcl from their respective PDFs and compute the hydrate concentration using the modified Archie Equation (Eq. 2). Then we draw again, randomly and independently, another set of values for these input parameters and obtain another possible realization of the hydrate concentration using the same Archie Law Equation. We repeat this procedure many times (5,000 or more), and we end up with many possible realizations for hydrate concentration at a certain subseafloor coordinate. From these many realizations of the possible hydrate concentration at a certain location, we derive a probability distribution function of the estimated hydrate concentration, which mathematically represents the uncertainty of our prediction of hydrate concentration at that target point. From this distribution of hydrate concentration we derive our best estimate of the hydrate concentration, which we express as the expected concentration value (defined as the mean value of the PDF) and the uncertainty of the estimate (the standard deviation of the PDF).

This procedure allows us to incorporate the inherent uncertainty of all of the input parameters into our final calculation result and to estimate the impact of all these uncertainties on our final estimate of hydrate concentration. Another advantage of our approach is that it allows us to understand the sensitivity of hydrate concentration to each of the individual input parameters. In this way we can decide which parameters are the most critical for reducing the inherent uncertainty associated with our predictions of hydrate concentration.

Our definitions of the probability distribution function (PDF) associated with each parameter used in the modified Archie Equation (Eq. 2) at one analysis site are illustrated in Figure 8. These distribution functions form the basis of the Monte Carlo-based random and independent “draws” of parameter values that we used to calculate hydrate concentration at that location. In addition to the uncertainties associated with the parameters used in the hydrate estimations, we also assign an uncertainty to the resistivity log readings that we use in the Archie Equation calculations. For example, the PDF used for a log reading of 2 ohm-m is shown in Figure 9.

Figure 8.

Distribution functions used to define the uncertainty of each parameter involved in the modified Archie Equation across one depth interval at one calibration well. The scale on the vertical axis is normalized by the number of Monte Carlo realizations (10,000 in this case). If the number of realizations is reduced to 100, the decimal points would move two places to the right, changing 0.0004 to 0.04.

Figure 8.

Distribution functions used to define the uncertainty of each parameter involved in the modified Archie Equation across one depth interval at one calibration well. The scale on the vertical axis is normalized by the number of Monte Carlo realizations (10,000 in this case). If the number of realizations is reduced to 100, the decimal points would move two places to the right, changing 0.0004 to 0.04.

Figure 9.

Example of a Gaussian distribution function used to describe the uncertainty of a resistivity log measurement. In this example, the log reading is 2 ohm-m.

Figure 9.

Example of a Gaussian distribution function used to describe the uncertainty of a resistivity log measurement. In this example, the log reading is 2 ohm-m.

Joint Inversion of Resistivity and Velocity

Our approach to predicting hydrate concentration is based on the concept that all of the parameters used in our rock physics elastic modeling (velocity estimation) and in our applications of the Archie Equation (resistivity estimation) are uncertain. Probability theory enables us to quantify this uncertainty and to combine various types of information, particularly velocity data and resistivity data, into a joint inversion for hydrate concentration. The attraction of a joint inversion approach to estimating hydrate concentration is that joint inversion reduces the uncertainty of the estimation that is made.

To implement a joint-inversion technique, each parameter in our rock-physics elastic modeling and in our formulation of the Archie Equation is expressed as a probability density function (PDF). Gaussian distributions are used when the expected value for the model parameter is known, with the mean of the Gaussian function being the expected value of the parameter, and the standard deviation of the function defining the uncertainty associated with this expected parameter value.

In contrast to a Gaussian distribution, a uniform distribution is used when the value of a parameter is not known, but the range of variability for the parameter can be defined. A uniform distribution assumes that within the range of variability being considered, any value of the described parameter is equally probable.

The parameters we used in our joint inversion were assigned the following PDFs:

  • Gaussian distribution: Porosity (ϕ); clay fraction (Vcl); bulk and shear moduli for quartz, clay, and brine (Kq, Kcl, Kw, Gq, Gcl); density of brine, quartz, and clay (ρwq, ρcl); effective pressure (Peff); coordination number (C); cementation exponent (m); geometrical factor (a); and pore-fluid resistivity (Rw).

  • Uniform distribution: Porosity (ϕ); clay fraction (Vcl); bulk and shear moduli for quartz, clay, and brine (Kq, Kcl, Kw, Gq, Gcl); density of brine, quartz, and clay (ρwq, ρcl); effective pressure (Peff); coordination number (C); cementation exponent (m); geometrical factor (a); and pore-fluid resistivity (Rw).

It is important to note that the VP velocity profile at this well exhibits an increasing trend The parameters listed here encompass all of the variables involved in predicting hydrate concentration for: (1) resistivity-log measurements, and (2) interval-velocity behavior. The manner in which these parameters were segregated between these two data-inversion domains (resistivity and velocity) is shown graphically as Figures 10 and 11.

Figure 10.

Types of PDFs used to describe the parameters needed to invert resistivity data to hydrate concentration. (top) Xm = mean of Gaussian distribution; σx = standard deviation. (bottom) X1 to X2 = range of uniform distribution.

Figure 10.

Types of PDFs used to describe the parameters needed to invert resistivity data to hydrate concentration. (top) Xm = mean of Gaussian distribution; σx = standard deviation. (bottom) X1 to X2 = range of uniform distribution.

Figure 11.

Types of PDFs used to describe the parameters needed to invert velocity data to hydrate concentration. (top) Xm = mean of Gaussian distribution; σx = standard deviation. (bottom) X1 to X2 = range of uniform distribution.

Figure 11.

Types of PDFs used to describe the parameters needed to invert velocity data to hydrate concentration. (top) Xm = mean of Gaussian distribution; σx = standard deviation. (bottom) X1 to X2 = range of uniform distribution.

Our probabilistic approach to estimating hydrate concentration is based on the concept that all parameters used in a joint inversion can be described by PDFs that account for the natural variability in the elastic properties of the mineral, hydrate, and fluid constituents of seafloor sediments, as well as for the variability in brine resistivity, cementation exponent, clay mineral resistivity, and other petrophysical parameters involved in a joint inversion of resistivity and seismic velocity to hydrate concentration.

It is important to note that probability density functions describing porosity, effective pressure, mineralogy, coordination number, cementation exponent,geometric factor, resistivity of brine, and most other sediment variables needed in an inversion for hydrate concentration vary with depth. In our method, we update the PDFs for these parameters at each depth coordinate, with these updates based on depth variations of parameters observed from geotechnical borings at Typhoon and Genesis fields and on parameter behavior determined a priori (reasoning based on theoretical deduction, not on observation).

At each depth coordinate, we model the joint theoretical relations between hydrate concentration cgh (the model parameter we need to calculate) and the resistivity R and seismic propagation velocity (both VP and VS) of subseafloor strata (which represent the observed parameters). We refer to the parameters involved in both our rock physics elastic modeling and in our Archie Equation (which must be corrected for clay content) as common parameters.

As shown on Figure 12, there are three of these common parameters in our two inversion algorithms: porosity (ϕ), hydrate concentration (cgh), and volume fraction of clay (Vcl). Clay fraction is estimated from local gamma-ray logs. We use a Monte Carlo procedure to draw values for common parameters ϕ and Vcl from their associated PDFs and then compute the corresponding velocity and resistivity values using Monte Carlo draws from the PDFs for each of the model parameters that are required for calculating hydrate concentration (Fig. 12).

Figure 12.

Joint inversion of VP and R to estimate hydrate concentration, cgh. The parameter PDFs [a total of 21 inputs and one output (cgh)] involved in the inversion are listed in the boxes. (top) Joint inversion at a shallow subseafloor depth. (bottom) Joint inversion at a deeper depth.

Figure 12.

Joint inversion of VP and R to estimate hydrate concentration, cgh. The parameter PDFs [a total of 21 inputs and one output (cgh)] involved in the inversion are listed in the boxes. (top) Joint inversion at a shallow subseafloor depth. (bottom) Joint inversion at a deeper depth.

In this fashion we obtain many possible realizations of the functions relating hydrate concentration, resistivity, and seismic propagation velocity. This joint relation is nonunique, uncertain, and can be expressed mathematically as a probability density function in three-dimensional (cgh, VP, R) model space [or in (cgh, VS, R) model space if preferred]. This 3D joint-theoretical PDF, which we will denote as ξ(cgh, VP, R), changes with depth and defines the correlation (and the inherent uncertainty) between hydrate concentration and the velocity and resistivity properties of hydrate-bearing sediments (Fig. 12). We emphasize VP velocities rather than VS velocities in our inversion because we found that across most of the OBC seismic grid we analyzed, hydrate fills less than 25 percent of the available pore space of the host sediment. For this range of hydrate fraction, there is a quasi-linear relationship between VP and cgh (Models A and B of Fig. 6); whereas, VS exhibits little sensitivity to changes in cgh when cgh is less than 25 percent (Models A and B of Fig. 7).

To estimate hydrate concentration using seismic and resistivity data, we implement a Bayesian approach formulated in the context of an inverse problem, as proposed by Tarantola (1987). First, we express our prior information about hydrate concentration (information obtained before analyzing any seismic data or resistivity data) as a PDF. We denote this prior PDF as ΛM(cgh), where subscript M stands for “model” parameter. In our study, this prior PDF is assumed to be a uniform distribution over all physically possible values for the hydrate pore-space fraction, meaning we allow this uniform distribution to range from 0 to 100 percent.

Second, we combine this prior PDF of hydrate concentration, Λ M(cgh), with information provided by seismic and resistivity measurements at calibration wells to create a three-dimensional PDF spanning cgh, VP, and R parameter space. Our prior information and any information obtained from seismic and resistivity data are assumed to be statistically independent. This assumption allows the prior joint PDF that combines hydrate concentration and data, Λ(cgh,VP,R), to be written as:

 

formula

In this equation, subscript D stands for data, and ΛD(VP) and ΛD(R) are Gaussian PDFs that account, respectively, for measurement uncertainties in the seismic P-wave velocity data and resistivity log data we use in our hydrate inversion. Our assumption of statistical independence between seismic and resistivity measurements is logical because velocity and resistivity data are obtained at different calendar times and using different field procedures and equipment.

Third, we use Tarantola’s (1987) strategy that states that the posterior PDF combining hydrate concentration and data, Ψ(cgh,VP, R), is proportional to the prior joint PDF for hydrate concentration and data, Λ(cgh, VP, R), multiplied by the joint theoretical PDF, ξ(cgh, VP, R), which we derive using stochastic rock physics modeling. Therefore, we can write:

 

formula

From this posterior joint PDF, Ψ (cgh,VP, R), we derive the marginal distribution of hydrate concentration, ΨM(cgh), by integrating the posterior joint PDF over velocity and resistivity data space. This marginal distribution, ΨM(cgh), represents the posterior PDF for hydrate concentration in the pore space of the host sediment, and the mean of this distribution is the parameter that we display along our OBC profiles to represent the amount of in-place hydrate.

At each calibration well, we apply this Bayesian inversion procedure to estimate the posterior PDF of hydrate concentration, using both local seismic velocity values and local resistivity-log data in the inversion. When we leave a calibration well and calculate hydrate concentration along an OBC profile, our hydrate estimate is expressed at each depth location along the OBC line as a posterior PDF that involves only VP velocities. We define the mean value of this posterior PDF as the expected value for hydrate concentration at each OBC line coordinate. In addition we produce a measure of the uncertainty associated with this estimate of hydrate concentration, which is the standard deviation of the posterior PDF.

Joint Inversion Examples

The hydrate prediction concepts described in the preceding section have been applied to create joint-inversion estimates of hydrate concentration at calibration wells inside our study area. The input data for these inversions are the resistivity log acquired in the calibration well and seismic-based VP and VS interval velocities determined from raytrace modeling local to each well.

The estimation of hydrate concentration at Well B, Genesis Field, is illustrated on Figure 13. The function labeled NC on the data panels of this figure defines the effect of normal compaction on the rock property that is illustrated in each panel. The method used to calculate these normal compaction curves is described in Hardage et al., 2009.

Figure 13.

(Top) Seismic-based VP and VS interval velocities, resistivity log, and their respective estimates of hydrate concentration at Well B, Genesis Field. The BHSZ boundary is defined as the top of the layer where VP velocity exhibits a reversal in magnitude. The increase in resistivity below the BHSZ boundary is caused by free gas. (Bottom) Joint inversion of resistivity and VP velocity indicates that hydrate occupies 14.4 percent of the pore space (mean value of the PDF). The estimation error is ±2.6 percent (standard deviation of the PDF).

Figure 13.

(Top) Seismic-based VP and VS interval velocities, resistivity log, and their respective estimates of hydrate concentration at Well B, Genesis Field. The BHSZ boundary is defined as the top of the layer where VP velocity exhibits a reversal in magnitude. The increase in resistivity below the BHSZ boundary is caused by free gas. (Bottom) Joint inversion of resistivity and VP velocity indicates that hydrate occupies 14.4 percent of the pore space (mean value of the PDF). The estimation error is ±2.6 percent (standard deviation of the PDF).

Intervals above the base of the hydrate stability zone boundary, where both velocity and resistivity have values greater than those associated with normal compaction, are assumed to be zones of hydrate concentration. Using this normal-compaction behavior as one constraint for our joint inversion, the mean value of the probability distribution function (PDF) in Figure 13B indicates that hydrate occupies approximately 14 percent of the pore space in the local vicinity of Well B. Similar joint inversions of resistivity and velocity data have been done to estimate hydrate concentrations at all of the calibration wells across the Genesis Field and Typhoon Field areas shown on Figure 1.

Conclusions

The hydrate inversion results at calibration wells across our study area showed close agreements between hydrate concentrations predicted from resistivity log data acquired in calibration wells and from seismic interval velocities calculated local to these wells. As a result, we concluded that the extension of our velocity-based inversion methodology to OBC receiver stations positioned considerable distances from a calibration well produced reliable hydrate estimates along each OBC seismic profile.

We must stress that our hydrate estimates involve an inescapable bias that comes into play when we impose a specific hydrate-sediment morphology in order to formulate the inversion algorithms that we used. For example, our resistivity inversion is based on the assumption that hydrate exists in subseafloor sediment as a disseminated morphology rather than as a thin-layered morphology. Thus a disseminated-hydrate bias is in-grained in the selection of parameter values that we use when inverting resistivity log data. Similarly, our velocity inversion assumes that this disseminated hydrate exists as a load-bearing morphology, not as a free-floating morphology or as a thin-layered morphology. Thus a load-bearing, disseminated-hydrate bias is embedded in our inversion algorithm that relates velocity to hydrate concentration.

If seafloor cores were collected and analyzed to determine the true nature of the hydrate morphology across our study area, a morphology-driven bias would still have to be incorporated into our inversion results. However, that bias would be based on hard evidence, not on assumptions.

In the second paper that follows of this two-paper series, we show how our joint inversion of resistivity and velocity (Fig. 13) is applied to create maps of hydrate concentration across several Green Canyon lease blocks.

References

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G.E.
,
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Collett
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T.S.
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DeAngelo
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Graebner
, and
D.E.
Wagner
,
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,
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, and in
situ data to estimate gas-hydrate concentrations in deep-water, near-seafloor strata of the Gulf of Mexico: Final report for DOE Contract DE-FC26-05NT42667
 ,
255
p. (available as a pdf file at http://www.netl.doe.gov/technologies/oil-gas/FutureSupply/MethaneHydrates/projects/DOEProjects/MH_42667GOMSeismic.html
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Roberts
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Cook
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362
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Roberts
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,
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Hardage
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Shedd
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Hunt
Jr.
,
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,
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, Rock-Physics Models for Gas-Hydrate Systems Associated with Unconsolidated Marine Sediments, in
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Collett
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Scala
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Figures & Tables

Table 1.

Parameter values used in two formulations of the Archie Equation.

Contents

GeoRef

References

References

Archie
,
G.E.
,
1942
,
The electric resistivity log as an aid in determining some reservoir characteristics
:
AIME Trans.
 , v.
46
, p.
54
.
Collett
,
T.S.
, and
J.
Ladd
,
2000
,
Detection of gas hydrate with downhole logs and assessment of gas hydrate concentrations (saturations) gas volumes on the Blake Ridge with electrical resistivity log data
, in
C.K.
Paull
,
R.
Matsumoto
,
P.J.
Wallace
, and
W.P.
Dillon
, eds.:
Proceedings of the Ocean Drilling Program, scientific results
, v.
164
, p.
179
191
.
Hardage
,
B.A.
,
P.E.
Murray
,
D.
Sava
,
M.M.
Backus
,
M.V.
DeAngelo
,
R.J.
Graebner
, and
D.E.
Wagner
,
2009
,
Combining multicomponent seismic attributes, new rock physics models
, and in
situ data to estimate gas-hydrate concentrations in deep-water, near-seafloor strata of the Gulf of Mexico: Final report for DOE Contract DE-FC26-05NT42667
 ,
255
p. (available as a pdf file at http://www.netl.doe.gov/technologies/oil-gas/FutureSupply/MethaneHydrates/projects/DOEProjects/MH_42667GOMSeismic.html
Hashin
,
Z.
, and
S.
Shtrikman
,
1962
,
A variational approach to the theory of the effective magnetic permeability of multiphase materials
:
Jour. Appl. Phys.
 , v.
33
, no.
10
, p.
3125
3131
.
Mendelson
,
K.S.
, and
M.H.
Cohen
,
1982
,
The effect of grain anisotropy on the electrical properties of sedimentary rocks
:
Geophysics
 , v,
47
, p.
257
263
.
Murphy
,
W.F.
,
1982
,
Effects of microstructure and pore fluids on the acoustic properties of granular sedimentary materials
: Stanford University PhD dissertation.
Rider
,
M.H.
,
1991
, The geologic interpretation of well logs, revised edition:
Cathness
 ,
Scotland
,
Whittles Publishing
.
Roberts
,
H.H.
,
D.J.
Cook
, and
M.K.
Sheedlo
,
1992
,
Hydrocarbon seeps of the Louisiana continental slope: Seismic amplitude signature and sea floor response
:
GCAGS Transactions
 , v.
42
, p.
349
362
.
Roberts
,
H.H.
,
B.A.
Hardage
,
W.W.
Shedd
, and
J.
Hunt
Jr.
,
2006
,
Seafloor reflectivity—an important seismic property for interpreting fluid/gas expulsion geology and the presence of gas hydrate
:
The Leading Edge, Society of Exploration Geophysicists
 , v.
25
, p.
620
628
.
Sava
,
D.C.
, and
B.A.
Hardage
,
2009
, Rock-Physics Models for Gas-Hydrate Systems Associated with Unconsolidated Marine Sediments, in
T.
Collett
,
A.
Johnson
,
C.
Knapp
, and
R.
Boswell
eds.,
Natural Gas Hydrates: Energy Resource Potential and Associated Geologic Hazards
 :
AAPG Special Publication
.
Schlumberger Wireline Services
,
1998
, Log interpretation principles/applications:
Schlumberger Educational Services
,
Houston, TX
.
Sen
,
P.N.
,
C.
Scala
, and
M.H.
Cohen
,
1981
,
A self similar model of sedimentary rocks with application to the dielectric constant of fused glass beads
:
Geophysics
 , v.
46
, p.
781
796
.
Tarantola
,
A.
,
1987
, Inverse problem theory: Methods for data fitting and model parameter estimation:
Elsevier Science
,
613
p.
Wempe
,
W.
,
2000
,
Predicting flow properties using geophysical data—improving aquifer characterization
: Stanford University PhD dissertation,
157
p.

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