Finite difference modeling of Biot's poroelastic equations atseismic frequencies
Abstract
Across the seismic band of frequencies (loosely defined as<10 kHz), a seismic wave propagating through a porous material willcreate flow in the pore space that is laminar; that is, in thislowfrequency "seismic limit," the development of viscous boundary layersin the pores need not be modeled. An explicit time steppingstaggeredgrid finite difference scheme is presented for solving Biot'sequations of poroelasticity in this lowfrequency limit. A key part ofthis work is the establishment of rigorous stability conditions. It isdemonstrated that over a wide range of porous material properties typicalof sedimentary rock and despite the presenceof fluid pressure diffusion(Biot slow waves), the usual Courant condition governs the stability asif the problem involved purely elastic waves. The accuracy of the methodis demonstrated by comparing to exact analytical solutions for both fastcompressional waves and slow waves. Additional numerical modelingexamples are also presented.
 Authors:
 Publication Date:
 Research Org.:
 Ernest Orlando Lawrence Berkeley NationalLaboratory, Berkeley, CA (US)
 Sponsoring Org.:
 USDOE. Assistant Secretary for Fossil Energy.Petroleum
 OSTI Identifier:
 927822
 Report Number(s):
 LBNL61829
Journal ID: ISSN 01480227; JGREA2; R&D Project: G32801; BnR: AC1005000; TRN: US200816%%1074
 DOE Contract Number:
 DEAC0205CH11231
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Geophysical Research; Journal Volume: 111; Journal Issue: B10305; Related Information: Journal Publication Date: 10/14/2006
 Country of Publication:
 United States
 Language:
 English
 Subject:
 54; ACCURACY; ANALYTICAL SOLUTION; BOUNDARY LAYERS; DIFFUSION; POROUS MATERIALS; SEDIMENTARY ROCKS; SEISMIC WAVES; SIMULATION; STABILITY
Citation Formats
Masson, Y.J., Pride, S.R., and Nihei, K.T.. Finite difference modeling of Biot's poroelastic equations atseismic frequencies. United States: N. p., 2006.
Web.
Masson, Y.J., Pride, S.R., & Nihei, K.T.. Finite difference modeling of Biot's poroelastic equations atseismic frequencies. United States.
Masson, Y.J., Pride, S.R., and Nihei, K.T.. Fri .
"Finite difference modeling of Biot's poroelastic equations atseismic frequencies". United States.
doi:.
@article{osti_927822,
title = {Finite difference modeling of Biot's poroelastic equations atseismic frequencies},
author = {Masson, Y.J. and Pride, S.R. and Nihei, K.T.},
abstractNote = {Across the seismic band of frequencies (loosely defined as<10 kHz), a seismic wave propagating through a porous material willcreate flow in the pore space that is laminar; that is, in thislowfrequency "seismic limit," the development of viscous boundary layersin the pores need not be modeled. An explicit time steppingstaggeredgrid finite difference scheme is presented for solving Biot'sequations of poroelasticity in this lowfrequency limit. A key part ofthis work is the establishment of rigorous stability conditions. It isdemonstrated that over a wide range of porous material properties typicalof sedimentary rock and despite the presenceof fluid pressure diffusion(Biot slow waves), the usual Courant condition governs the stability asif the problem involved purely elastic waves. The accuracy of the methodis demonstrated by comparing to exact analytical solutions for both fastcompressional waves and slow waves. Additional numerical modelingexamples are also presented.},
doi = {},
journal = {Journal of Geophysical Research},
number = B10305,
volume = 111,
place = {United States},
year = {Fri Feb 24 00:00:00 EST 2006},
month = {Fri Feb 24 00:00:00 EST 2006}
}

An explicit timestepping finitedifference scheme is presented for solving Biot's equations of poroelasticity across the entire band of frequencies. In the general case for which viscous boundary layers in the pores must be accounted for, the timedomain version of Darcy's law contains a convolution integral. It is shown how to efficiently and directly perform the convolution so that the Darcy velocity can be properly updated at each time step. At frequencies that are low enough compared to the onset of viscous boundary layers, no memory terms are required. At higher frequencies, the number of memory terms required is the samemore »

Poroelastic finitedifference modeling of seismic attenuation anddispersion due to mesoscopic heterogeneity
Seismic attenuation and dispersion are numerically computedfor synthetic porous materials that contain arbitrary amounts ofmesoscopicscale heterogeneity in the porouscontinuum properties. Thelocal equations used to determine the poroelastic response within suchmaterials are those of Biot (1962). Upon applying a step change in stressto samples containing mesoscopicscale heterogeneity, the poroelasticresponse is determined using finitedifference modeling, and the averagestrain throughout the sample is determined, along with the effectivecomplex and frequencydependent elastic moduli of the sample. The ratioof the imaginary and real parts of these moduli determines theattenuation as a function of frequency associated with the modes ofapplied stress (pure compression and pure shear).more » 
Modeling and finite element analysis of the nonstationary action on a multilayer poroelastic seam with nonlinear geomechanical properties
The paper discusses modeling of a multilayer coal seam under hydrodynamic action based on the coupled equations of poroelasticity and filtration with the nonlinear relationship of permeability and porous pressure. The calculations by the finite element method use correspondence between the poroelasticity and thermoelasticity equations. The influence of input data on the size of a degassing hole area is analyzed for the couple problem and pure filtration problem. 
Acoustic and filtration properties of a thermoelastic porous medium: Biot's equations of thermoporoelasticity
A linear system of differential equations describing the joint motion of a thermoelastic porous body and an incompressible thermofluid occupying a porous space is considered. Although the problem is linear, it is very hard to tackle due to the fact that its main differential equations involve nonsmooth rapidly oscillating coefficients, inside the differentiatial operators. A rigorous substantiation based on Nguetseng's twoscale convergence method is carried out for the procedure of the derivation of homogenized equations (not containing rapidly oscillating coefficients), which for different combinations of the physical parameters can represent Biot's system of equations of thermoporoelasticity, the system consisting ofmore »