Seismic Upscaling using Pair- and Multi correlation Function Approach
TIWARY, D.K., Institute for Theoretical Geophysics, University of Oklahoma, Norman, OK 73019, USA, firstname.lastname@example.org, BAYUK, I.O., Institute of Physics of the Earth, Russian Academy of Sciences, Moscow, Russia, email@example.comAmmerman, DEVON, M., Devon Energy Inc., Oklahoma City, OK 73102, USA, firstname.lastname@example.orgChesnokov, Evgeni M., Institute for Theoretical Geophysics, University of Oklahoma, Norman, OK 73019, USA, email@example.com
Random presence of various minerals, cracks and voids in different proportion make a rock inhomogeneous. These heterogeneities may be characterized by their different elastic constants. The size of heterogeneities can vary from micro-scale to macro-scale, and therefore the elastic properties become scale dependent. There has been great deal of effort to estimate the effective properties of the heterogeneous materials, such as upper and lower bounds, self-consistent theory, differential effective medium theory; some are based on the assumption of particular shaped inclusions. The mathematical formulations used to estimate effective properties are based upon the solution of singularly perturbed linear equations approximation, using partial differential equations, stochastic differential equations, ordinary differential equations and Markov chains. But these approximation methods, generally, do not take into account the interaction between heterogeneities. The reasons for dispersion of the elastic properties are: elastic scattering and intrinsic attenuation. When the size of the heterogeneity is comparable to the wavelength of the propagating wave, scattering attenuation becomes dominant. Our methodology accounts for the interaction between different inclusions, based on pair correlation and multicorrelation function approach, to upscale heterogeneous media while estimating dynamic physical properties. The static physical properties of the heterogeneous materials can be estimated provided we have detailed information on (i) the volume fraction of the various phases, (ii) the elastic properties of the various phases, and (iii) the geometric details of how the different phases are arranged relative to each other. The static physical properties estimated thus, will belong to the zero frequency. A link between static and dynamic measurements can be established if the Green's function and the correlation function corresponding to zero frequency can be estimated.