Development of effective medium models for quantification of elastic properties and modeling of velocity dispersion of saturated rocks
Sayar, Paul Mikhaël
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Elastic effective medium theory (EMT) relates to quantitative rock physics modeling that calculates macroscopic properties of a mixture by incorporating the individual elastic properties, the volume fractions, and the spatial arrangement of the constituents that make up the rock. Despite the valuable merits of effective medium models, these theories exhibit limitations that require further investigation. Common instances are the non-unique configurations of the rock’s elements that give rise to identical wave velocities and the limiting assumption that rocks are purely elastic materials. Consequently, direct applications of classical EMTs can yield inaccurate and non-unique estimates of rock fabric properties that directly affect the assessment of elastic properties. The primary purpose of this dissertation is to improve the reliability of rock physics models based on the use of effective medium theories. In the first part, a rock physics model is developed for reliable estimation of velocities and elastic properties for sandstone-shale laminated rocks that are assumed to be vertical transverse isotropic (VTI). The new model is concerned with the reproduction of typical geological features and petrophysical properties of such formations that exhibit complex rock fabric. Isotropic and anisotropic versions of the self-consistent approximation and the differential effective medium theory, and Backus average are invoked to compute the effective medium’s stiffness tensor. The rock is separated into volumes of sandstone (regarded as isotropic) and shale (regarded as VTI), which are treated separately to reliably reproduce the spatial arrangement of the individual components included in the rock. Shale volumes enclose penny-shaped cracks and clay platelets aligned in the horizontal direction. Total porosity is divided into percolating porosity, isolated pores, and aligned fractures. The new simulation method is implement in three wells in the Haynesville shale and the Barnett shale. Estimates of elastic properties are verified when calculated velocities and sonic logs are in agreement. All relative differences between simulated and measured velocities are below 5.4%. To reduce non-uniqueness, electrical resistivity is calculated with modified effective medium theories and a procedure to compute Stoneley velocity is combined with the rock physics model. A method is advanced to calculate stress distribution and fracture initiation pressure around potential wellbores drilled horizontally in VTI rocks from the stiffness tensor obtained with the improved rock physics model. Effects of degree of anisotropy and elastic properties on fracture initiation pressure are investigated to determine a criterion to locate optimal depths along a vertical well to place a horizontal well. In the second part of the dissertation, an effective medium model is developed for reproduction of four of the main mechanisms of dispersion and attenuation of acoustic waves in saturated rocks. Simple and practical alternatives are introduced for effective medium modeling that account for dispersion mechanisms due to fluid flow inside the pore space. Biot’s flow and squirt flow effects are simulated by the calculation of frequency-dependent equivalent bulk and shear moduli for the solid background of the rock. When equal to the static moduli of minerals that compose the matrix of a rock at low frequencies, dynamic moduli of the solid background become complex at high frequencies and their absolute value increases. Frequency-dependent solid moduli are used as elastic properties of the matrix material in which fluid-filled porous inclusions are then added with dynamic self-consistent approximations for replication of acoustic scattering phenomena due to stiff pores and cracks. Resulting elastic features of the saturated medium calculated with the frequency-dependent effective medium model display viscoelastic behavior. Velocity predictions are conducted on synthetic examples to investigate conditions where dynamic rock physics modeling is necessary to obtain accurate elastic properties.
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