Theory Of Constituent Porous Medium Model And Seismic Wave Propagation

Understanding the elastic properties of underground rocks is the requirement of modern oil-gas seismic explorations, which are based on media models and seismic wave propagation theories concerning the elastic properties of subsurface rocks, in which the porous medium model, along with its seismic wave propagation theory, is concerned by many researchers in the field of oil-gas and hydrate explorations. Studies on this subject have great academic significance and applied value for the oil-gas and hydrate explorations.The object of these studies about the porous medium and its seismic wave propagation theory is to obtain the relationships of effective elastic properties between the porous medium and its constituents. With a comprehensive analysis of all the available literature on this subject, it can be found the effective elastic properties of the multi-phase porous medium is a summation of all constituents'elastic properties weighted by their corresponding weighting coefficients. Some researches on the constituent porous medium model and its seismic wave propagation theory are carried out based on this idea and some principles of micro-mechanics, and some key points of these studies are listed below.First, based on those assumptions of the linear elastic and isotropic constituent porous medium model, the constituent equation of effective elastic matrix and constituent elastic matrices, and also the constituent equations of effective elastic moduli and constituent elastic moduli are derived from the average stress and average strain relations between the porous medium and its constituents. Different expressions of the constituent equations of elastic moduli are presented and analyzed, and it can be found that the constituent weighting coefficient plays a very important role in the studies of the constituent model. The theory of seismic wave propagation in the constituent medium is derived on the basis of investigations of the basic points of the linear elastic and isotropic homogenous solid single-phase medium model and its seismic wave propagation theory.Second, combined the diagonalizing conditions of coefficient tensors of the constituent model with the Hashin-Shtrikman bounds, four new formulas of effective elastic modulus are derived to make up for the shortcomings of the Hashin-Shtrikman bounds in the estimation of the effective shear modulus. Based on some comparisons between the constituent model and the Biot-Gassmann equations, the constituent expressions of elastic moduli of a fluid-saturated multi-phase medium and its frame areobtained in order to find the physical meaning of the frame weighting coefficients and the feature of the frame moduli expressions of consolidated and unconsolidated porous rocks. Following the comparison it discusses the weighting coefficients of bulk moduli of rocks saturated with water or gas and the problem of fluid substitution.Third, the conditions fulfilled by orthogonal constituent weighting coefficients are derived after the theory of orthogonal basis functions is introduced into the constituent model. When these coefficients are quadratic functions of the porosity, specific constituent orthogonal weighting coefficients are derived from these orthogonal conditions along with a discussion about how to extend the applicable range of the constituent orthogonal weighting coefficients in the orthogonal coordinates. Based on the combination of the theory of critical porosity and the constituent model, it presents a specific example of the constituent model that includes two transforming points. The reasonability of above-mentioned theories is shown by comparison of theoretical calculations and measured data on effective elastic moduli of clean sandstone or sandstone analogs saturated with pure water.Last, Based on the average stress and strain relations between the anisotropic porous medium and its anisotropic or isotropic constituents, the constituent equation of effective elastic matrix and constituent elastic matrices, and the constituent equations of effective elastic moduli and constituent elastic moduli of the anisotropic constituent model are derived along with a discussion about the weighting pattern of elastic moduli. The correspondence principle between elastic and viscoelastic isotropic constituent porous medium model is derived on the basis of analyses of the Kelvin medium model, and then the constitutive equation and the constituent equations of elastic moduli, and constituent expressions of phase velocity, attenuation coefficient and quality factor are obtained with the aid of this principle.The weighting expressions of the constituent porous medium model in this paper clearly demonstrate the contribution of all constituents to the effective elastic properties of the porous medium, and this model not only provide an idea for the studies on elastic property patterns between the porous medium and its constituents, but also provide a foundation for solving the problem of storage estimation of marine gas hydrate.