Poroelasticity is a field in materials science and mechanics that studies the interaction between fluid flow and solids deformation within a linear porous medium and it is an extension of elasticity and porous medium flow diffusion equation. Propagation of tidal fluctuations through groundwater aquifers or wave induced pressure fluctuations at ocean bottoms are. We shall discuss the basic properties of solutions to the wave equation 1. Examplesincludewaterwaves,soundwaves,electromagneticwavesradiowaves. The fabric dependence of tensors appearing in the poroelastic model of wave propagation is summarized in section 3. Without the pore pressure, equation 4a degenerates to the classical elastic relation. Poroelastic waves with thermal and viscous losses biot. Throughout this paper we aim to present a straightforward derivation of the main equations describing wave propagation in poroelastic media, with a particular.
In the mathematical sense, a wave is any function that moves. Abstract acoustic waves can be a viable tool for the detection and identi cation of. A nodal discontinuous galerkin finite element method for the poroelastic wave equation article pdf available in computational geosciences february 2019 with 574 reads how we measure reads. Seismic reflection dispersion due to waveinduced fluid flow in heterogeneous reservoir rocks luanxiao zhao1, dehua han 2, qiuliang yao3, rui zhou, and fuyong yan2 abstract we have investigated the impact of waveinduced fluid flow.
Poroelastic longitudinal wave equation for soft living tissues. Springdamper equivalents of the fractional, poroelastic. Role of structural anisotropy of biological tissues in. Pdf poroelastic longitudinal wave equation for soft. This model includes the aforementioned viscous and thermal. The string has length its left and right hand ends are held. Numerical simulation in coupled elastic and poroelastic media is important in oil and gas exploration. Journal of computational physics vol 403, 15 february. Seepage force on a buried submarine pipeline induced by a. The new model gives estimates of the vertical and the horizontal permeabilities, as well as. The wave equation is a secondorder linear hyperbolic pde that describesthe propagation of a variety of waves, such as sound or water waves. In this way, our effective elastic tensor will depend on both. Equation chapter 1 section 1 body waves propagation in a.
Velocitystress finitedifference modeling of poroelastic wave. The frequency equation each for a pervious and an impervious. We develop a numerical solver for threedimensional wave propagation in coupled poroelasticelastic media, based on a highorder discontinuous galerkin dg method, with the biot poroelastic wave equation formulated as a first order conservative velocitystrain hyperbolic system. Timedomain numerical modeling of poroelastic waves. Pdf 2d poroelastic wave modelling with a topographic free. The mathematics of pdes and the wave equation michael p. The wellposedness of the poroelastic system is proved by adopting an. Simple derivation of electromagnetic waves from maxwells.
Wavessuch as these water wavesspread outward from a source. Simple derivation of electromagnetic waves from maxwell. The theory was proposed by maurice anthony biot 1935, 1941. Porous media, biots theory, wave propagation, direct numerical methods. Vibration analysis of an infinite poroelastic circular. Vibration analysis of an infinite poroelastic circular cylindrical shell immersed in fluid. However, the interface between elastic and poroelastic media is a challenge to handle. Obrien1 abstract afourthorderinspaceandsecondorderintime3dstaggered sg and rotatedstaggeredgrid rsg method for the solution of biots equation are presented. University of calgary seismic imaging summer school august 711, 2006, calgary abstract abstract. To derive an upwind numerical flux, we find an exact solution to the riemann problem, including the poroelastic. This equation is solved into a polynomial equation of degree eight, whose roots represent the vertical.
Body waves in poroelastic media saturated by two immiscible. The system of equation is solved for displacement field variables, expressed by a set of second order partial differential equations. Partial differential equations generally have many different solutions a x u 2 2 2. Pdf a weightadjusted discontinuous galerkin method for. Derivation of a microstructural poroelastic model mark chapman. Chapter 2 wave propagation in viscous fluid this chapter summarizes with the derivation of the mathematical form of the acoustic wave propagation in the fluid. Stability condition and dispersion analysis gareth s. The global fluid flow results into the dissipation of energy due to the relative motion between the solid and fluid particles.
Seismology and the earths deep interior the elastic wave equation the elastic wave equationthe elastic wave equation elastic waves in infinite homogeneous isotropic media numerical simulations for simple sources plane wave propagation in infinite media frequency, wavenumber, wavelength conditions at material discontinuities snell. In this paper, the pml technique is extended to wave problems with an ap interface or an ep interface. Solution of the wave equation by separation of variables the problem let ux,t denote the vertical displacement of a string from the x axis at position x and time t. Liua adepartment of electrical and computer engineering, duke university, durham, north carolina 27708, u. This will result in a linearly polarized plane wave travelling in the x direction at the speed of light c. A porous medium or a porous material is a solid often called matrix permeated by an interconnected network of pores voids filled with a fluid liquid or gas. Two sample double porosity materials are denoted as material a and. The wave phase is constant along a planar surface the wavefront. For full access to this pdf, sign in to an existing account, or purchase an annual subscription. In particular, we examine questions about existence and. We use the nodal discontinuous galerkin method with a laxfriedrich flux to model the wave propagation in transversely isotropic and poroelastic media.
Yet, the poroacoustics model does not capture all of the effects and sometimes it is necessary to also include elastic waves in the porous matrix, solving a full poroelastic waves problem. The additional constitutive constants h0, h00 and r0 characterize the coupling betweenthesolidand. Wave equations, examples and qualitative properties. Seismic reflection dispersion due to waveinduced fluid. Solution of the wave equation by separation of variables. A staggeredgrid finitedifference method with perfectly. The effect of dissipation due to global fluid flow causes a stiff relaxation term, which is incorporated in the numerical scheme through an operator splitting approach. First, a set of secondorder memoryefficient unsplit pmls for the acoustic, elastic and poroelastic wave equations is independently proposed.
Poroelastic wave equations of biots theory are expressed in curvilinear coordinates and solved on a collocated grid by utilizing the fourthorder. The application of the nearly optimal sponge boundary conditions for seismic wave propagation in poroelastic media jingyi chen department of earth sciences, memorial university of newfoundland, st. Pdf a nodal discontinuous galerkin finite element method. The transversely isotropic poroelastic wave equation. Poroelastic wave equation including the biotsquirt. The deformation of the medium influences the flow of the fluid and vice versa. Laplaces equation recall the function we used in our reminder. We examined two examples, where the first was a single.
The 3d wave equation plane wave spherical wave mit 2. This is covered by the poroelastic waves interface, where the biotallard model can be selected. In equation 5, is the frequency of wave, k is wavenumber, c1, c2. Chapter maxwells equations and electromagnetic waves. A weightadjusted discontinuous galerkin method for the poroelastic wave equation. The new poroelastic wave equation is an improved attempt to quantitatively and consistently relate wave propagation in anisotropic poroelastic rocks with. Incidence of acoustic wave through the liquid at the interface results in its re. Furthermore, coupling bcs in the absorbing layer are consistently derived.
Chapter 4 the wave equation another classical example of a hyperbolic pde is a wave equation. Chapter 4 derivation and analysis of some wave equations wavephenomenaareubiquitousinnature. Evidently, the sum of these two is zero, and so the function ux,y is a solution of the partial differential equation. The transversely isotropic poroelastic wave equation can be formulated to include the biot and the squirtflow mechanisms to yield a new analytical solution in terms of the elements of the squirtflow tensor. The wave equation outline of mechanical waves longitudinal and transverse waves waves in a string, sound waves the wave equation description of waves using functions of two variables travelling waves the wave equation 0 y v y 1 2 2 2 2 2 x t waves in a string. Therefore, our present bisq model is a generalized bisq model, which the solidfluid coupling anisotropy and both mechanisms are simultaneously included in a unified poroelastic wave equation. Poromechanics is a branch of physics and specifically continuum mechanics and acoustics that studies the behaviour of fluidsaturated porous media. Before we derive the final form of the wave propagation equation in viscous fluid, we first look at two conservation mass and momentum of equations and state equation in the fluid. Spherical wave propagation in a poroelastic medium with. The generalized poroelastic wave equation is derived from the kinetic energy of the twophase system. A nodal discontinuous galerkin finite element method for. Unlike the poroelastic model used in here, dupuy et al. Spectralelement simulations of wave propagation in porous media. The slow pwave is clearly seen, as opposed to finite permeability case.
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