Resource Potential of DeepWater Hydrates Across the Gulf of Mexico: Part 1, Estimating Hydrate Concentration from Resistivity Logs and Seismic Velocities

Published:December 01, 2009
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Sava Diana, Hardage Bob, 2009. "Resource Potential of DeepWater Hydrates Across the Gulf of Mexico: Part 1, Estimating Hydrate Concentration from Resistivity Logs and Seismic Velocities", Unconventional Energy Resources: Making the Unconventional Conventional, Tim Carr, Tony D’Agostino, William Ambrose, Jack Pashin, Norman C. Rosen
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Abstract
The Bureau of Economic Geology has evaluated hydrate concentrations across deepwater areas of Green Canyon, Gulf of Mexico, using well log data and fourcomponent (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 offtheshelf 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 twopaper sequence.
In this first paper, we describe how hydrate concentration can be estimated from resistivity logs and then from compressional (V_{P}) and shear (V_{S}) velocities as a jointinversion approach for quantifying the amount of inplace hydrate. We found no industry well in the Green Canyon area where velocitylog data has been recorded across shallow nearseafloor strata where deepwater hydrates are found. Consequently, we have utilized interval V_{P} and V_{S} velocities obtained by processing deepwater 4C seismic data in our jointinversion hydrate estimations.
The rock physics used to estimate deepwater hydrate concentrations from resistivity logs and from interval velocities is challenging because deepwater, nearseafloor 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 loadbearing 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 hydratesediment morphology that should be used in a rock physics model that is applied to deepwater, nearseafloor environments in the Green Canyon area. In this first paper we illustrate how hydrate morphology affects the interpretation of resistivity and velocity responses of hydratebearing sediment. We have assumed a loadbearing 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 HashinShtrikman 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 deepwater, nearseafloor environments. We discuss our rock physics modeling approach based on the application of HashinShtrikman 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 deepwater hydrate research is our interpretation of the resistivitylog response of highporosity mixtures of sediment, hydrate, and brine. We develop evidence and logic that the HashinShtrikman Lower Bound (Hashin and Shtrikman, 1962) should dictate the functional behavior for the resistivity of a highporosity 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 highporosity, unconsolidated, deepwater sediments if appropriate constants are used to adjust the equation response to the resistivity predicted by the HashinShtrikman Lower Bound for that medium. Neither the Archie Equation nor the HashinShtrikman 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 HashinShtrikman 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 HashinShtrikman 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 lowfrequency (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).
Resistivity Models of SedimentHydrate 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 nearseafloor gas hydrate systems. Consequently, these companies acquired minimal lithofaciessensitive log data consisting of only gammaray and resistivity measurements across shallow, nearseafloor 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 logbased 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 deepwater 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 hydratesediment mixtures whether the hydrate is loadbearing or freefloating. 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.
Archie Equation Formulated for Clay
The Archie Equation has been used to analyze resistivity responses of fluidfilled porous rocks for more than six decades (Archie, 1942). The clayfree form of this equation with which we begin our analysis can be written as
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 sandfluid mixture. The principal assumption of this empirical law is that electrical current travels only through the brine phase of fluidsaturated 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 clayrich 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 (V_{cl}) present in the sediments and the resistivity (R_{cl}) of the clay minerals. Volume of clay can be estimated from gammaray 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 pureclay intervals from nearby geology or rely on published resistivity measurements of clays. Published information sources confirm that R_{cl} spans a large resistivity range extending from 1 to 1,000 ohmm (Rider, 1991).
The modified Archie Equation proposed by Schlumberger (1998) is
where in our deepwater applications,
R is the measured resistivity of sediments containing gas hydrates,
R_{w} is the resistivity of the brine in these hydratebearing 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),
V_{cl} is the volume of clay estimated from gammaray log data,
R_{cl} is the resistivity of clay mineral (R_{cl} = 1 to 1,000 ohmm),
S_{w} is the water saturation, and
n is the saturation exponent (1.7 ≤ n ≤ 2.2).
NearSeafloor Sediment and Seismic Velocity
Our rock physics theory that describes how hydrate concentration affects V_{P} and V_{S} velocities in deepwater 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 deepwater hydrate systems across the Gulf of Mexico:
Model A assumes hydrates are uniformly disseminated throughout the sediment and are part of the loadbearing 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, thinlayered medium in which layers of pure gas hydrate are intercalated with layers of hydratefree 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, thinlayered 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 loadbearing frame. These thin hydratebearing layers are intercalated with thin layers of hydratefree 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 Swave 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 V_{cl} = 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 S_{w}, and its positive root is
This expression for S_{w} estimates water saturation when the Archie Equation is modified for clay content.
If n ≠ 2, the square root term in these equations is replaced with the nthroot 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 R_{W}, ϕ, and S_{W} for that rockfluid 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 highporosity 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 R_{W} = 0.23 ohmm. We cannot find the exact value of R_{W} that was used in their Blake Ridge study. We know only that the pore fluid By definition, the concentration (c_{gh}) of hydrate in the sediments is (1S_{w}), or salinity is assumed to be 32,000 ppm. If the hydrate formation temperature is assumed to be 65ºF, then this salinity leads to R_{W} = 0.23 ohmm. Given our application of the HashinShtrikman 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 highporosity 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 R_{W} = 0.17 ohmm. 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 HashinShtrikman Lower Bound.
HashinShtrikman Lower Bound
Calculation of HashinShtrikman 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, highporosity, nearseafloor 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/R_{W}, where R is the resistivity measured across a medium of porosity ϕ and R_{W} is the resistivity of the fluid that fills the pores. The shaded interval labeled ϕ_{C} defines the range of critical porosity, which is porosity where the grains of a highporosity medium convert from a suspended state to a loadbearing condition in which each grain touches at least one other grain. Critical porosity varies from about 0.3 for round, uniformsize grains, to around 0.4 for well sorted rounded grains of variable size, to about 0.6 for highly oblate (flat) grains.
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 HashinShtrikman Lower Bound (HS) when the porosity of the medium equals or exceeds critical porosity. Because the porosity of the deepwater, nearseafloor 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 HashinShtrikman Lower Bound is an ideal function for describing the resistivity of deepwater hydrate systems.
The HashinShtrikman Lower Bound that we calculated is plotted on Figure 4 to illustrate how the resistivity of deepwater 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 ohmm to only 2 ohmm. 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 sedimenthydratebrine mixture (Fig. 2).
The HashinShtrikman 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 ohmm. One factor that may keep the resistivity of this sedimentbrinehydrate 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 porefilling brine will reduce resistivity even at high hydrate concentrations. In terms of electrical conductivity, a smaller number of electricalcurrent flow paths through higher salinity brine are equivalent to a larger number of flow paths through reducedsalinity brine. In our analysis, however, we do not decrease porefluid resistivity as hydrate concentration increases. Using this constraint of a constant porefluid resistivity, the HashinShtrikman Lower Bound implies that a significant increase in resistivity in deepwater hydratebearing sediment does not occur until hydrate concentration exceeds 90 percent of the pore space and the number of connected brinefilled paths is severely reduced (Fig. 4).
Included on Figure 4 is a curve labeled Archie Equation 1 that describes the resistivity behavior of the clayfree form of the Archie Equation (Eq. 1) that we think is appropriate for hydrate systems across Green Canyon that are embedded in cleansand 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.
We present the following arguments to support our parameter choices for the clayfree 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 deepwater hydrate systems when porosities are equal to or greater than critical porosity must have a functional behavior that approximates the HashinShtrikman Lower Bound. With the above parameters, our version of the Archie Equation is a close approximation of the HashinShtrikman 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 HashinShtrikman Lower Bound and does not represent true resistivity conditions of a deepwater hydratesedimentbrine mixture.
We use a value of 0.17 ohmm for R_{W} 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 hydrateforming process expels NaCl and retains H_{2}O. 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 clayfree form of the Archie Equation in this discussion only to simplify our arguments that the HashinShtrikman Lower Bound should be the governing physics for describing the resistivity of highporosity mixtures of sediment, hydrate, and brine. In practice, we used the form (Fig. 2) of the Archie Equation that includes a claycorrection term (Hardage et al., 2009).
NearSeafloor Sediment and Seismic Velocity
Our rock physics theory that describes how hydrate concentration affects V_{P} and V_{S} velocities in deepwater 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 deepwater hydrate systems across the Gulf of Mexico:
Model A assumes hydrates are uniformly disseminated throughout the sediment and are part of the loadbearing 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, thinlayered medium in which layers of pure gas hydrate are intercalated with layers of hydratefree 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, thinlayered 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 loadbearing frame. These thin hydratebearing layers are intercalated with thin layers of hydratefree 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 Swave velocities for each of these four possible hydrate morphologies.
MixedMineralogy Host Sediment
Our theory (Sava and Hardage, 2009; Hardage et al., 2009) allows mixed mineralogy and different saturating fluids to be included in the wavepropagation media that we model. Examples of this mixedmineralogy 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 clayrich sediments is larger than the porosity of pure quartz grains. Therefore, we compute V_{P} and V_{S} 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.
As expected, the P and Swave 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 (loadbearing hydrates), as porosity and clay content of the sediments increase, it becomes more challenging to estimate small hydrate concentrations, especially using Swave velocity data (Fig. 7). For layered model D, we observe that both P and Swave 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 Swave velocities if hydrate layers are intercalated with clean sands than with clayrich sediments. Models C and D take into account only the anisotropy due to thin layers and consider the clayrich sediments to be isotropic. This assumption may hold for sediments immediately below seafloor. However, as depth increases, the stressinduced anisotropy of clays will increase. At large depths, Models C and D associated with clayrich 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, R_{w}, ϕ, a, m, V_{cl}, and R_{cl} 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 Carlobased 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 ohmm is shown in Figure 9.
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 jointinversion technique, each parameter in our rockphysics 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 (V_{cl}); bulk and shear moduli for quartz, clay, and brine (K_{q}, K_{cl}, K_{w}, G_{q}, G_{cl}); density of brine, quartz, and clay (ρ_{w},ρ_{q}, ρ_{cl}); effective pressure (P_{eff}); coordination number (C); cementation exponent (m); geometrical factor (a); and porefluid resistivity (R_{w}).
Uniform distribution: Porosity (ϕ); clay fraction (V_{cl}); bulk and shear moduli for quartz, clay, and brine (K_{q}, K_{cl}, K_{w}, G_{q}, G_{cl}); density of brine, quartz, and clay (ρ_{w},ρ_{q}, ρ_{cl}); effective pressure (P_{eff}); coordination number (C); cementation exponent (m); geometrical factor (a); and porefluid resistivity (R_{w}).
It is important to note that the V_{P} 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) resistivitylog measurements, and (2) intervalvelocity behavior. The manner in which these parameters were segregated between these two datainversion domains (resistivity and velocity) is shown graphically as Figures 10 and 11.
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 c_{gh} (the model parameter we need to calculate) and the resistivity R and seismic propagation velocity (both V_{P} and V_{S}) 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 (c_{gh}), and volume fraction of clay (V_{cl}). Clay fraction is estimated from local gammaray logs. We use a Monte Carlo procedure to draw values for common parameters ϕ and V_{cl} 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).
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 threedimensional (c_{gh}, V_{P}, R) model space [or in (c_{gh}, V_{S}, R) model space if preferred]. This 3D jointtheoretical PDF, which we will denote as ξ(c_{gh}, V_{P}, R), changes with depth and defines the correlation (and the inherent uncertainty) between hydrate concentration and the velocity and resistivity properties of hydratebearing sediments (Fig. 12). We emphasize V_{P} velocities rather than V_{S} 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 quasilinear relationship between V_{P} and c_{gh} (Models A and B of Fig. 6); whereas, V_{S} exhibits little sensitivity to changes in c_{gh} when c_{gh} 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 Λ_{M}(c_{gh}), 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 porespace 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(c_{gh}), with information provided by seismic and resistivity measurements at calibration wells to create a threedimensional PDF spanning c_{gh}, V_{P}, 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(c_{gh},V_{P},R), to be written as:
In this equation, subscript D stands for data, and Λ_{D}(V_{P}) and Λ_{D}(R) are Gaussian PDFs that account, respectively, for measurement uncertainties in the seismic Pwave 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(c_{gh},V_{P}, R), is proportional to the prior joint PDF for hydrate concentration and data, Λ(c_{gh}, V_{P}, R), multiplied by the joint theoretical PDF, ξ(c_{gh}, V_{P}, R), which we derive using stochastic rock physics modeling. Therefore, we can write:
From this posterior joint PDF, Ψ (c_{gh},V_{P}, R), we derive the marginal distribution of hydrate concentration, Ψ_{M}(c_{gh}), by integrating the posterior joint PDF over velocity and resistivity data space. This marginal distribution, Ψ_{M}(c_{gh}), 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 inplace 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 resistivitylog 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 jointinversion 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 seismicbased V_{P} and V_{S} 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.
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 normalcompaction 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 velocitybased 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 hydratesediment 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 thinlayered morphology. Thus a disseminatedhydrate bias is ingrained 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 loadbearing morphology, not as a freefloating morphology or as a thinlayered morphology. Thus a loadbearing, disseminatedhydrate 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 morphologydriven 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 twopaper 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
Figures & Tables
Contents
Unconventional Energy Resources: Making the Unconventional Conventional
 Abstract
 Introduction
 Study Sites
 Resistivity Models of SedimentHydrate Systems
 Archie Equation Formulated for Clay
 NearSeafloor Sediment and Seismic Velocity
 HashinShtrikman Lower Bound
 NearSeafloor Sediment and Seismic Velocity
 MixedMineralogy Host Sediment
 Uncertainty in Estimating Hydrate Concentration
 Joint Inversion of Resistivity and Velocity
 Joint Inversion Examples
 Conclusions
 References