Schön, J: Physical Properties of Rocks

Conversions

Content

Units

Units: Conversion for temperature, length, volume, mass, density, pressure, velocity and slowness, thermal conductivity, specific heat capacity between SI and other used units.

Elastic parameters

Moduli: Conversion of any combination of two elastic parameters in a different combination for isotropic materials (Table 6-1 in text).
In a second worksheet the moduli are calculated from velocities and density.

File

Content

Figure in text

Nuclear

Vsh-GR

Vsh-GR: Relationship between Gammaray Index and shale content for different empirical equations.

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Electrical

Fractured conductivity

Calculation of electrical conductivity and formation factor as function of fracture porosity. The rock consists of two pore systems:

• Matrix porosity
• Fracture porosity (oriented).

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Laminated sediment

Worksheet gives a forward calculation of the vertical and horizontal resistivity as function of volumetric composition for

• laminated shaly sand with shale resistivity and sand parameters (water resistivity, porosity, water saturation, Archie-exponents) as input;
• laminated bimodal sand with parameters for a coarse and a fine sand (water resistivity, porosity, water saturation, Archie-exponents) as input.

Shaly sand equations

There are two worksheets:
Poupon eq: Forward calculation of formation resistivity as function of water saturation for different shale content Vsh. Calculation based on Poupon's equation for laminated shaly sand.
Shaly sand eq: For a given formation resistivity and water resistivity as input you can calculate the resulting water saturation for the following equations: Poupon, Simandoux, and Indonesia. Inputs are also Archie parameters, porosity and shale resistivity.

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Permittivity models

Calculation of relative permittivity as function of porosity (2-component composite) with the input parameters permittivity of solid and of fluid. Worksheet for the following models:

• Layered model,
• Generalized Lichtenecker-Rother,
• CRIM equation,
• Inclusion model (Clausius-Mossotti) for sphere,
• Inclusion model (Hanai-Bruggeman) for ellipsoids with depolarization exponent as parameter.

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Elastic_Mechanical

Bound models

There are two worksheets:
Voigt_Reuss_Hashin: For a two component layered model (solid, fluid) the elastic moduli (compressional modulus, shear modulus) are calculated as function of fluid volume fraction (porosity) based on the model of Voigt (upper bound) and Reuss (lower bound), and the Voigt-Reuss-Hill mean value. With density the velocities are calculated.
For the same 2-component model the Hashin-Shtrikman upper and lower bounds are calculated.
Generalized eq: For a two component layered model (solid, fluid) the elastic compressional modulus is calculated as function of fluid volume fraction (porosity) based on the generalized Lichtenecker-Rother equation.

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Inclusion isotropic

There are two worksheets:
Kuster_Toksöz: Calculation of normalized compressional and shear wave velocity for a two component material (solid, fluid) as function of porosity. Two models are used: Inclusions are spheres and inclusions are penny shaped cracks. Calculations for gas and water filled inclusion.
Budiansky: Calculation of compressional and shear wave velocity as function of fracture parameter epsilon and porosity. For calculation penny shaped random distributed inclusions are assumed. Additional input parameter is the aspect ratio.

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Inclusion anisotropic

Hudson: Calculation of the components of the tensor of elasticity based on the assumption of a VTI medium (horizontal cracks). Additional input parameter is the aspect ratio.

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Fluid replacement

There are two worksheets:
Gassmann: The worksheet allows a fluid replacement based on Gassmann's equation. Input: compressional and shear wave velocity measured for rock saturated with fluid 1, porosity, compressional modulus and density of fluid 1 and fluid 2. Output: compressional and shear wave velocity for the rock saturated with fluid 2.
Example: The worksheet gives a log example for a fluid replacement gas → water.

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Structured model

There are five worksheets for different velocity influences:
Porosity: Calculation of the porosity effect upon velocity with the quotient of pore aspect ratio to grain aspect ratio as parameter.
Pressure: Calculation of pressure effect upon velocity controlled by parameters of the contact elasticity.
Tensor: Calculation of the components of the "structure tensor" as function of structure angle and contact properties.
Derivation of velocity ratios (V_p/V_s) and Thomson's anisotropy parameters for the dry rock.
Velocity grids: Grids are calculated with the equations from table "Tensor". As input two sets of parameters are used (parameter ƒ, angle α).
Vp vs. strength: Calculation of the relationship between velocity (in this case compressional wave velocity) and compressional strength.

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Shuey AVO

Using Shuey's equation R_pp is calculated as function of the angle Θ for different cases of wet sand, gas sand and shale. Input: material parameters of the layers.

6-46

Thermal

Layered models

There two worksheets:
2-components: Calculation of thermal conductivity of a 2-component material (solid, fluid) as function of porosity. The following equations are used:
Voigt model (parallel, upper bound), Reuss model (series, lower bound), arithmetic mean, geometric mean, Krischer and Esdorn model with parameter a, generalized Lichtenecker-Rother model with parameter α.
10-components: Worksheet for calculation of thermal conductivity of a material consisting of (maximum) 10 components. Input: volume fraction and conductivity of components.
Calculation for following models: Voigt model (parallel, upper bound), Reuss model (series, lower bound), Krischer and Esdorn model with parameter a, generalized Lichtenecker-Rother model with parameter α.

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Table 9-14

Inclusion models

There are three worksheets:
Spheres: Calculation of thermal conductivity as function of volume fraction for a 2-component material under assumption of

• spherical pores as inclusion in a solid host material,
• spherical grains as inclusion in a fluid host material.

Disk-random: Calculation of thermal conductivity as function of volume fraction for a 2-component material under assumption of

• spherical pores,
• disk shaped random oriented pores

as inclusion in a solid host material.
Ellipsoids-oriented: Calculation of thermal conductivity as function of volume fraction for a 2-component material under assumption of ellipsoidic inclusions with orientation. Calculations deliver thermal conductivity for x-, y-axis and z-axis. Additional input parameter is aspect ratio.

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Relationships

VR_Hill_mean

Voigt-Reuss-Hill mean value for elastic properties, velocities, and thermal conductivity is calculated for a 10-component mineral composite. Input: elastic parameters, density, thermal conductivity. Variable: volume fractions.

Defectmodel

Based on the Defectmodel the relationship between thermal conductivity and compressional wave velocity is calculated. The controlling input parameter A_solid describes the influence of mineral composition whereas the defect parameter D controls the effects of fractures etc.. The calculated curves are compared with some experimental data.

11-10

Inclusion model

A correlation between thermal conductivity and compressional wave velocity is calculated based on the application of

• the Budiansky and O'Connell model for elastic properties,
• the Clausius-Mossotti model for thermal conductivity.

Both models have as input the properties of the components (solid, inclusion), porosity and aspect ratio.
The calculated curves for two cases are compared with on some experimental data.

11-13

The example from the textbook (Darling, 2005) is used to demonstrate core and log data analysis.
Core analysis:

• Porosity-permeability regression
• Capillary pressure analysis
• Application of Leverett's and Thomeer's equation
• Derivation of Archie-parameters

Log analysis:

• Calculation of V-shale
• Calculation of Porosity
• Calculation of Water saturation
• Permeability estimate.

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Example-Carbonate

The example demonstrates the calculation of mineral fraction (calcite, dolomite) and porosity from Neutronlog and Densitylog using matrix inversion.
Table crossplot presents the density and neutron data; you can move the plot upward to fit with the chartbook-plot.
Table analysis gives the calculation for mineral composition and porosity.

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