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**Rock Physics Models for Marine Gas Hydrates**

Darrell A. Terry, Camelia C. Knapp, and James H. Knapp Earth and Ocean Sciences University of South Carolina “Rock Physics Models for Marine Gas Hydrates”, coauthored by myself, Camelia Knapp, and Jim Knapp. Here we describe our recent analysis efforts in rock physics related to Mississippi Canyon Block 118 (MC-118).

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**Long Range Research Goals**

Further develop statistical rock physics to associate seismic properties with lithology in marine gas hydrate reservoirs Investigate AVO and seismic attribute analysis in a marine gas hydrate reservoir Analyze anistropic seismic properties in a marine gas hydrate reservoir to delineate fracture structures and fluid flow pathways Before going into details of recent work, we want to outline our long range direction, which includes 1) Further develop statistical rock physics to associate seismic properties with lithology in marine gas hydrate reservoirs, 2) Investigate AVO and seismic attribute analysis in a marine gas hydrate reservoir, 3) Analyze anisotropic seismic properties in a marine gas hydrate reservoir to delineate fracture structured and fluid flow pathways In today’s discussion we will focus on implementation of the rock physics models as part of the first goal.

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**Outline What is Rock Physics? Models Used by JIP**

Brief Theoretical Background Recent Updates Suggested for Models Candidate Models to Use Role of Well Log Data Future Directions Here is a brief outline of what is to be covered, beginning with a definition of Rock Physics. Then we will show the models used in Dai et al’s earlier work, followed by a brief theoretical background including recent updates suggested for some of the models. Though the actual choice for a model will be driven by the data, we will go through some of the details of our anticipated choice. Then we have a short discussion on the role of well log data to calibrate our model choice. Following this we have short discussions on using velocity profiles from the WesternGeco data, forward seismic modeling, etc.

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What is Rock Physics? Methodology to relate velocity and impedance to porosity and mineralogy Establish bounds on elastic moduli of rocks Effective-medium models Three key seismic parameters Investigate geometric variations of rocks Cementing and sorting trends Fluid substitution analysis Apply information theory Quantitative interpretation for texture, lithology, and compaction through statistical analysis Rock physics is a methodology for applying petroleum systems analysis. In the case of MC-118 the goal is to relate seismic velocity and impedance to porosity and mineralogy in ocean bottom sediments. Given, effective-medium models in conjunction with the three key seismic parameters, compressional velocity, velocity ratio Vp/Vs, and density, we will be able to establish bounds on the elastic moduli of the rocks. We should be able to ascertain some aspects of variations of grain structure, cementation, and possibly sorting trends. In conjunction with probability density functions constructed for appropriate well log datasets and from the seismic datasets, information theory, in the form of Bayesian analysis will be applied to probabilistically estimate lithology, and possibly compaction and texture.

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**Models Used by JIP (from Dai et al, 2004)**

In this and the next slide five effective-medium models are compared to well log data from the Mallik 2L-38 well (Yukon, Canada). This plot is for Compressional velocity vs gas hydrate saturation. The small triangles are values derived from well measurements. The curves marked M1 thru M5 are based on theoretical models. M1 and M2 are cementation models; M3 and M4 are Hertz-Mindlin theory based models for unconsolidated rocks; M5 is an inclusion type model. Dai et al (2004) ascertained that M3 provided the best match in this plot of compression velocity vs gas hydrate saturation level. The inset at the top reveals that in the M3 model gas hydrate contributes to the grain framework stiffness. (from Dai et al, 2004)

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**Models Used by JIP (from Dai et al, 2004)**

Here, similar to the previous slide, we show shear velocity vs gas hydrate saturation. Though the choice for best fit still remains the M3 model, the model overpredicts shear velocity values. Others, notably Sava and Hardage (2006, 2009) and Dutta et al (2009) address this issue. According to Dai et al (2004) the M3 model for shear velocities is sensitive to coordination number, critical porosity, and rock component elastic properties. (from Dai et al, 2004)

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**Theoretical Background**

Effective-medium models for unconsolidated sediments Mindlin, 1949 (Hertz-Mindlin Theory) Digby, 1981; Walton, 1987 Dvorkin and Nur, 1996 Jenkins et al, 2005 Sava and Hardage, 2006, 2009 Dutta et al, 2009 The most widely used effective-medium models for unconsolidated sediments are based on what is known as Hertz-Mindlin theory (Mindlin, 1949). The first practical models were by Digby (1981) and Watson (1987), each using different grain-, or particle-, packing arrangements. Watson (1987) more closely followed Mindlin (1949) by developing two models, one for smooth grains allowing for slip among the particles (perfectly smooth spheres) and one for infinitely rough grains not allowing slip among the particles (infinitely rough spheres). Separately, and using somewhat different assumptions from Watson (1987) developed a model for infinitely rough spheres that is identical to Watson (1987). Most models we have encountered in the literature were based on the Dvorkin and Nur (1996) paper. Sava and Hardage (2006, 2009) departed from that and based their analysis on Watson’s (1987) perfectly smooth sphere model in order to better fit shear velocities. Later we found the paper by Dutta et al (2009) that also preferred Watson’s smooth sphere model as updated by Jenkins et al (2005). Jenkins et al (2005) consolidated Watson’s two models into a single model using a parameter alpha to specify the percentages of perfectly smooth vs infinitely rough spheres.

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**Theoretical Background**

The mathematics of Hertz-Mindlin theory is based on the mechanics of applying the stress-strain relationship to compression of two spheres, and then averaging over a large volume of spheres for a choice of grain packing. In addition an assumption has to be made as to the smoothness of the sphere surface, either infinitely rough which prevents relative rotation between the spheres, or perfectly smooth which allows slippage of the spheres. (from Walton, 1987) (from Mindlin, 1949)

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**Theoretical Background**

Modifications for saturation conditions and presence of gas hydrates Dvorkin and Nur, 1996 Helgerud et al, 1999; Helgerud, 2001 Dvorkin and Nur (1996), in addition to laying out an effective-medium mode based on Hertz-Mindlin theory, also described a framework predicting effective moduli for saturated conditions, which has also become the framework for predicting effective moduli for unconsolidated sediments emplaced with gas hydrates. This framework is better described in Helgerud et al (1999), and best described in Helgerud (2001).

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**Why Use Jenkins’ Update?**

Hertz-Mindlin theory often under predicts Vp/Vs ratios in comparison with laboratory rocks and well log measurements (Dutta et al, 2009) for unconsolidated sediments. A similar problem is noted in Sava and Hardage (2006, 2009). Additional Degree-of-Freedom Noted in Dai et el (2004), Hertz-Mindlin theory based models may over predict elastic wave properties. Sava and Hardage (2008, 2009), to address this issue, used the perfectly smooth sphere model from Watson (1987). Jenkins et al (2005) by developing Watson’s (1987) models into a single model, provides an additional degree-of-freedom. Jenkins et al (2005) suggests-s that in most cases, the parameter partitioning infinitely rough to perfectly smooth spheres should be 0.5 to 0.6.

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**Comparisons with Jenkins’ Update**

Here in these four plots we show a parametric test comparing Jenkins et al (2005) updated model with the more commonly referenced models (Watson, 1987; Dvorkin and Nur, 1996). The upper plots are for compressional velocity; the lower plots are for shear velocity. In each of the four plots blue is used for infinitely rough sphere model, green for perfectly smooth sphere model, and red for Jenkins et al (2005) updated model with alpha being the partitioning parameter. In the left plots alpha is set to 0.2; in the right plots alpha is set to In all plots, if alpha is set to 0.0, the red curve will perfectly match the green curve; if alpha is set to 1.0, the red curve will perfectly match the blue curve.

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**Baseline Model Hertz-Mindlin theory (Jenkins et al, 2005)**

Effective dry-rock moduli (Helgerud, 2001) For our baseline model, we calculate effective-medium bulk and shear moduli using the formulation of Hertz-Mindlin theory with the enhancements from Jenkins et al (2005). Of significance is that the model is a function of pressure (i.e P). To predict properties away from the critical porosity, typically called dry-rock moduli, we use the heuristic approach outlined in Dvorkin and Nur (1996) that is more fully described in Helgerud (2001).

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**Baseline Model Gassmann’s equations Velocity equations Poisson’s ratio**

Bulk density To complete the baseline model we need equations to represent saturation (Gassmann’s equation, Poisson’s ration, and bulk density. Also, we must specify the relationships between the elastic moduli and the velocities. The baseline model represents unconsolidated sediments fully saturated.

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**Model Configurations Gas Hydrate Models (for solid gas hydrate)**

Rock Matrix (Supporting Matrix / Grain) Pore-Fluid (Pore Filling) Rock Matrix Pore-Fluid We mentioned previously that JIP, based on the Mallik 2L-38 well log data, selected the rock matrix grain arrangement as providing the best match. In the rock matrix configuration, the gas hydrate contributes directly to the stiffness of the grain framework. In the pore-fluid configuration small gas hydrate particulates that do not contribute to the stiffness of the grain frame. We have also developed Partial Gas Saturation models (for free gas), one for homogeneous gas saturation and another for patchy gas saturation.

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**Model Configurations Pore-Fluid Rock Matrix**

For the Pore-fluid configuration we need to update calculation of the fluid bulk moduli and the bulk density. Adaptation for the Rock-Matrix configuration requires calculation of a reduced porosity, new material moduli, and bulk density.

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**Well Log Data Mallik 2L-38 JIP Wells Keathley Canyon Atwater Valley**

(Data Digitized from Collett et al, 1999) Compressional and shear velocities for Mallik2L-38 were digitized from Collett et al (1999) and are shown here. We have also obtained image files of the paper logs from MMS for JIP wells in Keathley Canyon and Atwater Valley. Our intention is to digitize the JIP well records if we are unable to obtain the records otherwise.

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**Well Log Data: Crossplot**

Mallik 2L-38 Other logs for crossplots Porosity Resistivity Gas Hydrate Saturation Crossplots with third attribute Generate probability distribution functions (PDFs) A crossplot of the compressional and shear velocities from Mallik 2L-38 is shown here. With this and crossplots with the other downhole measurements from the Mallik well, we hope to be able to separate out different formation types and map them to seismic characteristics.

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**MC-118 Stacking Velocities**

WesternGeco: locations of stacking velocity profiles for 3D stack 253 profiles Spaced 40 CMPs apart, inline and crossline Convert to interval velocities Along with the WesternGeco 3D stack data we have obtained a file containing stacking velocity profiles (stacking velocity vs time) at 253 locations spaced 40 CMP apart, inline and crossline. The red crosses mark the locations of the profiles with the dashed blue rectangular area marking the boundaries of the MC-118 block. Blue circles identify locations of profiles that include a velocity reversal identified from a quick review of all 253 profiles. Magenta asterisks identify the three vents known to exist on the mound. The right plot, with the green line, identifies a slice taken through the MC-118 block on which the profiles on the next slide are from.

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**MC-118 Stacking Velocities**

The axis of the three profiles shown here are time versus stacking velocity.

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**Future Directions: Synthetic Seismic Models**

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**Future Directions Create Rock Physics Templates**

Amplitude Variation with Offset (AVO) Seismic Inversion (WesternGeco data, Pre-Stack Gathers) Acoustic impedance Elastic Impedance Attribute analysis Assign Lithology and Estimate Gas Hydrate Probabilities Based on Information Theory

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References Dai, J.; Xu, H.; Snyder, F.; Dutta, N.; Detection and estimation of gas hydrates using rock physics seismic inversion: Examples from the northern deepwater Gulf of Mexico. The Leading Edge, January 2004, p Digby, P. J.; The effective elastic moduli of porous granular rocks. J. Appl. Mech., v. 48, p Dutta, T.; Mavko, G.; Mukerji, T.; Improved granular medium model for unconsolidated sands using coordination number, porosity and pressure relations. Proc. SEG 2009 International Exposition and Annual Meeting, Houston, p Dvorkin, J.; Nur, A.; Elasticity of high-porosity sandstones: Theory for two North Sea data sets. Geophysics, v. 61, p Helgerud, M. B.; Dvorkin, J.; Nur, A.; Sakai, A.; Collett, T.; Elastic-wave velocity in marine sediments with gas hydrates: Effective medium modeling. Geophys. Res. Lett., v. 26, n. 13, p Helgerud, M. B.; Wave Speeds in Gas Hydrate and Sediments Containing Gas Hydrate: A Laboratory and Modeling Study. Ph.D. Dissertation, Stanford University, April Jenkins, J.; Johnson, D.; La Ragione, L.; Maske, H.; Fluctuations and the effective moduli of an isotropic, random aggregate of identical, frictionless spheres. J. Mech. Phys. Solids, v. 53, pp Mindlin, R. D.; Compliance of elastic bodies in contact. J. Appl. Mech., v. 16, p Sava, D.; Hardage, B.; Rock physics models of gas hydrates from deepwater, unconsolidated sediments. Proc. SEG 2006 Annual Meeting, New Orleans, p Sava, D.; Hardage, B.; Rock-physics models for gas-hydrate systems associated with unconsolidated marine sediments. In: Collett, T.; Johnson, A.; Knapp, C.; Boswell, R.; eds. Natural gas Hydrates – Energy Resource Potential and Associated Geologic Hazards. AAPG Memoir 89, p Walton, K.; The effective elastic moduli of a random packing of spheres. J. Mech. Phys. Solids, v. 35, n. 2, pp

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**Model Configurations Partial Gas Saturation Models (for free gas)**

Homogeneous Gas Saturation Patchy Gas Saturation The baseline model can also be used to develop partial gas saturation models for investigating the presence of free gas.

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