University of calgary seismic imaging summer school august 711, 2006, calgary abstract abstract. Wave equations, examples and qualitative properties. The generalized poroelastic wave equation is derived from the kinetic energy of the twophase system. 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. In order to deal with the coupled model, the first.
Partial differential equations generally have many different solutions a x u 2 2 2. Schanz and cheng 11 have studied the transient wave propagation in a onedimensional poroelastic column. We use the nodal discontinuous galerkin method with a laxfriedrich flux to model the wave propagation in transversely isotropic and poroelastic media. The wave equation is a secondorder linear hyperbolic pde that describesthe propagation of a variety of waves, such as sound or water waves. 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 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. Throughout this paper we aim to present a straightforward derivation of the main equations describing wave propagation in poroelastic media, with a particular. Seismic wave propagation and modelling in poroelastic media.
The theory was proposed by maurice anthony biot 1935, 1941. 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. This equation is solved into a polynomial equation of degree eight, whose roots represent the vertical. 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. Chapter 4 derivation and analysis of some wave equations wavephenomenaareubiquitousinnature. Two sample double porosity materials are denoted as material a and. Obrien1 abstract afourthorderinspaceandsecondorderintime3dstaggered sg and rotatedstaggeredgrid rsg method for the solution of biots equation are presented. Pdf a nodal discontinuous galerkin finite element method. Numerical simulation in coupled elastic and poroelastic media is important in oil and gas exploration. The fabric dependence of tensors appearing in the poroelastic model of wave propagation is summarized in section 3. The wellposedness of the poroelastic system is proved by adopting an. For full access to this pdf, sign in to an existing account, or purchase an annual subscription. The deformation of the medium influences the flow of the fluid and vice versa. Simple derivation of electromagnetic waves from maxwell.
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. 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. A nodal discontinuous galerkin finite element method for. Springdamper equivalents of the fractional, poroelastic.
Examplesincludewaterwaves,soundwaves,electromagneticwavesradiowaves. A weightadjusted discontinuous galerkin method for the poroelastic wave equation. Nowadays, the us propagation in natural hydrogels, mostly composed of water, is usually modeled by. The frequency equation each for a pervious and an impervious. We examined two examples, where the first was a single. The additional constitutive constants h0, h00 and r0 characterize the coupling betweenthesolidand. Simple derivation of electromagnetic waves from maxwells. Pdf poroelastic longitudinal wave equation for soft. Poroelastic waves with thermal and viscous losses biot. 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. Laplaces equation recall the function we used in our reminder. Wavessuch as these water wavesspread outward from a source. In this way, our effective elastic tensor will depend on both. In this paper, the pml technique is extended to wave problems with an ap interface or an ep interface.
Seepage force on a buried submarine pipeline induced by a. The slow pwave is clearly seen, as opposed to finite permeability case. In particular, we examine questions about existence and. Abstract acoustic waves can be a viable tool for the detection and identi cation of. Chapter maxwells equations and electromagnetic waves. Porous media, biots theory, wave propagation, direct numerical methods.
The mathematics of pdes and the wave equation michael p. Poroelastic wave equation including the biotsquirt. To derive an upwind numerical flux, we find an exact solution to the riemann problem, including the poroelastic. In equation 5, is the frequency of wave, k is wavenumber, c1, c2. First, a set of secondorder memoryefficient unsplit pmls for the acoustic, elastic and poroelastic wave equations is independently proposed. Stability condition and dispersion analysis gareth s.
Pdf 2d poroelastic wave modelling with a topographic free. This will result in a linearly polarized plane wave travelling in the x direction at the speed of light c. As seen from the figure, the primary wave in poroelastic medium arrives earlier than the elastic wave, and the slow poroelastic wave follows the latter with smaller amplitude. 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. Incidence of acoustic wave through the liquid at the interface results in its re. However, the interface between elastic and poroelastic media is a challenge to handle. Liua adepartment of electrical and computer engineering, duke university, durham, north carolina 27708, u. Timedomain numerical modeling of poroelastic waves. The global fluid flow results into the dissipation of energy due to the relative motion between the solid and fluid particles. Propagation of tidal fluctuations through groundwater aquifers or wave induced pressure fluctuations at ocean bottoms are. Journal of computational physics vol 403, 15 february. Body waves in poroelastic media saturated by two immiscible. 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. 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.
The string has length its left and right hand ends are held. Vibration analysis of an infinite poroelastic circular. 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. Poroelastic wave equations of biots theory are expressed in curvilinear coordinates and solved on a collocated grid by utilizing the fourthorder. Furthermore, coupling bcs in the absorbing layer are consistently derived. The system of equation is solved for displacement field variables, expressed by a set of second order partial differential equations. Seismic reflection dispersion due to waveinduced fluid. Derivation of a microstructural poroelastic model mark chapman. The transversely isotropic poroelastic wave equation. A staggeredgrid finitedifference method with perfectly. This is covered by the poroelastic waves interface, where the biotallard model can be selected. This model includes the aforementioned viscous and thermal. Evidently, the sum of these two is zero, and so the function ux,y is a solution of the partial differential equation. In the mathematical sense, a wave is any function that moves.
Without the pore pressure, equation 4a degenerates to the classical elastic relation. Solution of the wave equation by separation of variables. Spherical wave propagation in a poroelastic medium with. We shall discuss the basic properties of solutions to the wave equation 1. The wave phase is constant along a planar surface the wavefront. Poroelastic longitudinal wave equation for soft living tissues. Pdf a weightadjusted discontinuous galerkin method for. Vibration analysis of an infinite poroelastic circular cylindrical shell immersed in fluid. Equation chapter 1 section 1 body waves propagation in a. The 3d wave equation plane wave spherical wave mit 2. Velocitystress finitedifference modeling of poroelastic wave. The new model gives estimates of the vertical and the horizontal permeabilities, as well as. The new poroelastic wave equation is an improved attempt to quantitatively and consistently relate wave propagation in anisotropic poroelastic rocks with.
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