Seismic wave equations in tight oil/gas sandstone media

Tight oil/gas medium is a special porous medium, which plays a significant role in oil and gas exploration. This paper is devoted to the derivation of wave equations in such a media, which take a much simpler form compared to the general equations in the poroelasticity theory and can be employed for parameter inversion from seismic data. We start with the fluid and solid motion equations at a pore scale, and deduce the complete Biot's equations by applying the volume averaging technique.The underlying assumptions are carefully clarified. Moreover, time dependence of the permeability in tight oil/gas media is discussed based on available results from rock physical experiments. Leveraging the Kozeny-Carman equation, time dependence of the porosity is theoretically investigated. We derive the wave equations in tight oil/gas media based on the complete Biot's equations under some reasonable assumptions on the media. The derived wave equations have the similar form as the diffusiveviscous wave equations. A comparison of the two sets of wave equations reveals explicit relations between the coefficients in diffusive-viscous wave equations and the measurable parameters for the tight oil/gas media. The derived equations are validated by numerical results. Based on the derived equations, reflection and transmission properties for a single tight interlayer are investigated. The numerical results demonstrate that the reflection and transmission of the seismic waves are affected by the thickness and attenuation of the interlayer, which is of great significance for the exploration of oil and gas.     

TIME-DOMAIN ANALYSIS OF LINEAR HYSTERETIC DAMPING

Two linear-hysteretic-damping models that provide energy dissipation independent of the deformation frequency, are studied in this paper: a hysteretic Kelvin element and a hysteretic Maxwell element. Both models use the Hilbert transform and yield integro–differential equations for the equations of motion of structures when real-valued signals are utilized in the formulation. It is shown that the use of analytic (complex-valued) signals allows the transformation of these integro–differential equations into differential equations with analytic input signals and complex-valued coefficients. These differential equations show both stable and unstable poles. A technique for the solution of these differential equations is presented; it consists of a conventional modal decomposition of the state-space equations and the integration of the differential equations forward in time for the modal co-ordinates associated with stable poles, and backwards in time for the modal co-ordinates associated with unstable poles. Some numerical examples are presented to illustrate the characteristics of the models and the proposed analysis technique.     

Analytical solution of specific energy and specific force equations: Trapezoidal and triangular channels

The specific energy and specific force equations have many applications in open-channel flow problems. At present, these equations have analytical solution only for rectangular channels. Trial and error procedure also graphical solutions are the existing methods of solving these equations. No analytical solutions are available in the technical literature for these equations in trapezoidal and triangular channels because it is presumed that these equations are quintic equations. The inversion of such equations consists of finding the roots of quintic equations. In the current study for a given channel geometry and discharge, the subcritical (supercritical) depth is analytically found in terms of the other supercritical (subcritical) depth. For this purpose, by considering physically meaningful domains, a quintic equation has been reduced to a quartic equation. In the next step, this quartic equation has been converted to a resolvent cubic equation and two quadratic equations. This research shows these steps clearly to reach an acceptable physical analytic solution for water depth in trapezoidal and triangular channels.     

Existence of the solutions and the attractors for the large-scale atmospheric equations

The atmosphere is a kind of fluid surrounding therotating earth, and its state can be described by thevelocity vector v, the temperature T, the density ρ, andthe pressure p at each point. Its evolution process isessentially governed by the Navier-Stockes equationand the temperature equation. Due to the specialty andthe complexity of the atmospheric problem, althoughsome modified and simplified work have made onthese fluid mechanics and temperature equations, theyare still a very complex forc…     

Low-frequency dilatational wave propagation through fully-saturated poroelastic media

The strong coupling of applied stress and pore fluid pressure, known as poroelasticity, is relevant to a number of applied problems arising in hydrogeology and reservoir engineering. The standard theory of poroelastic behavior in a homogeneous, isotropic, elastic porous medium saturated by a viscous, compressible fluid is due to Biot, who derived a pair of coupled partial differential equations that accurately predict the existence of two independent dilatational (compressional) wave motions, corresponding to in-phase and out-of-phase displacements of the solid and fluid phases, respectively. The Biot equations can be decoupled exactly after Fourier transformation to the frequency domain, but the resulting pair of Helmholtz equations cannot be converted to partial differential equations in the time domain and, therefore, closed-form analytical solutions of these equations in space and time variables cannot be obtained. In this paper we show that the decoupled Helmholtz equations can in fact be transformed to two independent partial differential equations in the time domain if the wave excitation frequency is very small as compared to a critical frequency equal to the kinematic viscosity of the pore fluid divided by the permeability of the porous medium. The partial differential equations found are a propagating wave equation and a dissipative wave equation, for which closed-form solutions are known under a variety of initial and boundary conditions. Numerical calculations indicate that the magnitude of the critical frequency for representative sedimentary materials containing either water or a nonaqueous phase liquid is in the kHz–MHz range, which is generally above the seismic band of frequencies. Therefore, the two partial differential equations obtained should be accurate for modeling elastic wave phenomena in fluid-saturated porous media under typical low-frequency conditions applicable to hydrogeological problems.     

Practical implementation of the fractional flow approach to multi-phase flow simulation

Fractional flow formulations of the multi-phase flow equations exhibit several attractive attributes for numerical simulations. The governing equations are a saturation equation having an advection diffusion form, for which characteristic methods are suited, and a global pressure equation whose form is elliptic. The fractional flow approach to the governing equations is compared with other approaches and the implication of equation form for numerical methods discussed. The fractional flow equations are solved with a modified method of characteristics for the saturation equation and a finite element method for the pressure equation. An iterative algorithm for determination of the general boundary conditions is implemented. Comparisons are made with a numerical method based on the two-pressure formulation of the governing equations. While the fractional flow approach is attractive for model problems, the performance of numerical methods based on these equations is relatively poor when the method is applied to general boundary conditions. We expect similar difficulties with the fractional flow approach for more general problems involving heterogenous material properties and multiple spatial dimensions.     

Development and verification of a 3-D integrated surface water–groundwater model

Coupled modelling of surface and subsurface systems is a valuable tool for quantifying surface water–groundwater interactions. In the present paper, the 3-D non-steady state Navier–Stokes equations, after Reynolds averaging and with the assumption of a hydrostatic pressure distribution, are for the first time coupled to the 3-D saturated groundwater flow equations in an Integrated suRface watEr–grouNdwater modEl (IRENE). A finite-difference method is used for the solution of the governing equations of IRENE. A semi-implicit scheme is used for the discretisation of the surface water flow equations and a fully implicit scheme for the discretisation of the groundwater flow equations. The two sets of equations are coupled at the common interface of the surface water and groundwater bodies, where water exchange takes place, using Darcy’s law. A new approach is proposed for the solution of the coupled surface water and groundwater equations in a simultaneous manner, in such a fashion that gives computational efficiency at low computational cost. IRENE is verified against three analytical solutions of surface water–groundwater interaction, which are chosen so that different components of the model can be tested. The model closely reproduces the results of the analytical solutions and can therefore be used for analysing and predicting surface water–groundwater interactions in real-world cases.     

Automated first and second order moment equations for a set of stochastic differential equations of type AŻ+BZ=C(t)

The generation of the second and higher order moment equations for a set of stochastic differential equations based on Ito's differential lemma is difficult, even for small system of equations. From the knowledge of the statistical properties of the Gaussian white noises associated with the parameters and input coefficients of a set of stochastic differential equations of typeA.+B.Z=C(t), a way to automatically generate the second order moment equations in a computer is presented in this paper. The resulting set of first and second order moment equations is also presented in the same state-space form of the original set of stochastic differential equations through a vectorization of the correlation matrix, which takes advantage of its symmetry. The procedure involved here avoids the inversion of matrixA to apply Ito's differential lemma. Therefore, the presented numerical implementation reduces the computational effort required in the formulation and solution of the moment equations. Moreover, other robust and efficient numerical deterministic integration schemes can be equally applied to the solution of the moment equations.     

Algorithms for staggered-grid computations for poroelastic,elastic, acoustic,and scalar wave equations

Heterogeneous wave equations are more complicated numerically than homogeneous wave equations, but are necessary for physical validity. A wide variety of numerical solutions of seismic wave equations is available, but most produce strong numerical artefacts and local instabilities where model parameters change rapidly. Accuracy and stability of heterogeneous equations is achieved through staggered-grid formulations. A new pseudospectral staggered-grid algorithm is developed for the poroelastic (Biot) equations. The algorithm may be reduced to handle the elastic and acoustic limits of the Biot equations. Comparisons of results from poroelastic, elastic, acoustic and scalar computations for a 2D model show that porous medium parameters may affect amplitudes significantly. The use of homogeneous wave equations for modelling of a heterogeneous medium, or of a centred rather than a staggered grid, or of simplified (e.g. acoustic) wave equations when elastic or poroelastic media are synthesized, may produce erroneous or ambiguous interpretations.     

Thermodynamics and constitutive theory for multiphase porous-media flow considering internal geometric constraints

This paper provides the thermodynamic approach and constitutive theory for closure of the conservation equations for multiphase flow in porous media. The starting point for the analysis is the balance equations of mass, momentum, and energy for two fluid phases, a solid phase, the interfaces between the phases and the common lines where interfaces meet. These equations have been derived at the macroscale, a scale on the order of tens of pore diameters. Additionally, the entropy inequality for the multiphase system at this scale is utilized. The internal energy at the macroscale is postulated to depend thermodynamically on the extensive properties of the system. This energy is then decomposed to provide energy forms for each of the system components. To obtain constitutive information from the entropy inequality, information about the mechanical behavior of the internal geometric structure of the phase distributions must be known. This information is obtained from averaging theorems, thermodynamic analysis, and from linearization of the entropy inequality at near equilibrium conditions. The final forms of the equations developed show that capillary pressure is a function of interphase area per unit volume as well as saturation. The standard equations used to model multiphase flow are found to be very restricted forms of the general equations, and the assumptions that are needed for these equations to hold are identified.     

APPROXIMATIONS TO SHEAR-WAVE VELOCITY AND MOVEOUT EQUATIONS IN ANISOTROPIC MEDIA1

Backus and Crampin derived analytical equations for estimating approximate phase-velocity variations in symmetry planes in weakly anisotropic media, where the coefficients of the equations are linear combinations of the elastic constants. We examine the application of similar equations to group-velocity variations in off-symmetry planes, where the coefficients of the equations are derived numerically. We estimate the accuracy of these equations over a range of anisotropic materials with transverse isotropy with both vertical and horizontal symmetry axes, and with combinations of transverse isotropy yielding orthorhombic symmetry. These modified equations are good approximations for up to 17% shear-wave anisotropy for propagations in symmetry planes for all waves in all symmetry systems examined, but are valid only for lower shear-wave anisotropy (up to 11%) in off-symmetry planes. We also obtain analytical moveout equations for the reflection of qP-, qSH-, and qSV- waves from a single interface for off-symmetry planes in anisotropic symmetry. The moveout equation consists of two terms: a hyperbolic moveout and a residual moveout, where the residual moveout is proportional to the degree of anisotropy and the spread length of the acquisition geometry. Numerical moveout curves are computed for a range of anisotropic materials to verify the analytical moveout equations.     

对波达波夫和Pride震电波方程组的对比分析

用Biot介质参数说明了波达波夫震电波方程组中弹性动力学 参数的含义，解释了第一类和第二类震电效应的意义，在忽略第一类震电效应条件下将该方 程组与Pride方程组进行比较，说明了二者在描述第二类震电效应方面的异同点. 同时指出 ：波达波夫方程组忽略了流体与固体的耦合质量；方程中的黏性耗散项丢掉了一个孔隙度因 子，依据该方程组计算出的弹性波和转换电场的幅度将偏大；边界条件之一存在错误，会影 响对波在界面上的反射透射规律的描述.     

P- and S-wavefield simulations using both the first- and second-order separated wave equations through a high-order staggered grid finite-difference method

In seismic exploration, it is common practice to separate the P-wavefield from the S-wavefield by the elastic wavefield decomposition technique, for imaging purposes. However, it is sometimes difficult to achieve this, especially when the velocity field is complex. A useful approach in multi-component analysis and modeling is to directly solve the elastic wave equations for the pure P- or S-wavefields, referred as the separate elastic wave equations. In this study, we compare two kinds of such wave equations: the first-order (velocity–stress) and the second-order (displacement–stress) separate elastic wave equations, with the first-order (velocity–stress) and the second-order (displacement–stress) full (or mixed) elastic wave equations using a high-order staggered grid finite-difference method. Comparisons are given of wavefield snapshots, common-source gather seismic sections, and individual synthetic seismogram. The simulation tests show that equivalent results can be obtained, regardless of whether the first-order or second-order separate elastic wave equations are used for obtaining the pure P- or S-wavefield. The stacked pure P- and S-wavefields are equal to the mixed wave fields calculated using the corresponding first-order or second-order full elastic wave equations. These mixed equations are computationally slightly less expensive than solving the separate equations. The attraction of the separate equations is that they achieve separated P- and S-wavefields which can be used to test the efficacy of wave decomposition procedures in multi-component processing. The second-order separate elastic wave equations are a good choice because they offer information on the pure P-wave or S-wave displacements.     

Validation of new equations for dynamic analyses of tall frame-type structures

The accuracy of the new equations for long frame-type structures, derived by Kerr and Zarembski, and extended recently by Kerr and Accorsi also to dynamic analyses, is investigated. At first, the natural frequencies of a tall frame-type structure, with twenty floors, are determined using the new equations, and then they are compared with the corresponding results calculated using the finite element method and the common shear-building analysis. The found agreement with the finite element analysis is close. To check further the accuracy of the new equations, small scale models are tested on a shake table. The natural frequencies recorded in the tests are then compared with those determined from the new equations, for a wide range of geometrical parameters. The agreement with the test data is good. The presented study indicates that the new equations are well suited for analyzing the dynamic response of tall frame-type structures.     

An approximate Riemann solver for sensitivity equations with discontinuous solutions

The sensitivity of a model output (called a variable) to a parameter can be defined as the partial derivative of the variable with respect to the parameter. When the governing equations are not differentiable with respect to this parameter, problems arise in the numerical solution of the sensitivity equations, such as locally infinite values or instability. An approximate Riemann solver is thus proposed for direct sensitivity calculation for hyperbolic systems of conservation laws in the presence of discontinuous solutions. The proposed approach uses an extra source term in the form of a Dirac function to restore sensitivity balance across the shocks. It is valid for systems such as the Euler equations for gas dynamics or the shallow water equations for free surface flow. The method is first detailed and its application to the shallow water equations is proposed, with some test cases such as dike- or dam-break problems with or without source terms. An application to a two-dimensional flow problem illustrates the superiority of direct sensitivity calculation over the classical empirical approach.     

一种TI介质纯qP波正演方法及其在逆时偏移中的应用

基于Tsvankin提出的精确频散关系,利用近似展开的方法,推导出解耦合的TTI介质纯qP波近似方程,并将方程中的偏微分算子分解成一个laplace算子和一个标量算子,用于代表qP波的精确传播方向,构建时间域二阶纯qP波方程.此推导过程无需设置横波速度为零,能够更加精确地描述qP波的运动学特征.这个方程相比于求解波数域二阶解耦qP波方程,计算效率高,存储需求小;相比于基于Alkhalifah频散关系推导的时间域二阶纯qP波方程,假象干扰压制好,数值误差小,更具一般性.但此方法求解波矢量时采用波场梯度一阶渐近近似,会造成垂直于对称轴方向的波场振幅不准确.为了较正振幅,将椭圆分解方法应用于此方程中,构建纯qP波椭圆分解方程,使得振幅更加均衡,并与Xu等提出的方程比较分析,应用本文构建的纯qP波椭圆分解方程得到的波场振幅值更加准确.本文首先选取了均匀TI介质模型进行了qP波正演模拟,并抽取波场单道波形进行振幅分析,验证了本文构建的纯qP波方程和纯qP波椭圆分解方程的正确性及有效性;然后选取BP TTI模型进行了qP波正演模拟,将其qP波正演结果和均匀TI介质模型振幅分析结果相结合,突出了本文构建的纯qP波椭圆分解方程的优势及适应性;最后选取逆冲模型和BPTTI模型,应用本文构建的纯qP波椭圆分解方程对其进行逆时偏移成像,验证了本文构建的纯qP波椭圆分解方程在逆时偏移中的可行性和适用性.     

Horizontal vibrations of a disk on a poroelastic half-space

This paper analytically examines the horizontal vibration of a rigid disk on a saturated poroelastic half-space. The pressure-solid displacement form of the harmonic equations of motion for asymmetric dynamic problem are developed from the form of the equations originally presented by Biot. Making use of a new method the solution of the above equations is obtained. According to the mixed boundary -value conditions, the dual integral equations of the horizontal vibration of a rigid disk on a saturated poroelastic half-space are established. By appropriate transforms, it is shown that the dual integral equations can be reduced to a pair of Fredholm integral equations of the second kind, whose solutions are then computed. Numerical results for the horizontal dynamic compliance coefficient are given at the end of this paper.     

Style-of-Faulting in Ground-Motion Prediction Equations

Equations for the prediction of response spectral ordinates invariably include magnitude, distance and site classification as independent variables. A few equations also include style-of-faulting as a fourth variable, although this has an almost negligible effect on the standard deviation of the equation. Nonetheless, style-of-faulting is a useful parameter to include in ground-motion prediction equations since the rupture mechanism of future earthquakes in a particular seismic source zone can usually be defined with some confidence. Current equations including style-of-faulting use different schemes to classify fault ruptures into various categories, which leads to uncertainty and ambiguity regarding the nature and extent of the effect of focal mechanism on ground motions. European equations for spectral ordinates do not currently include style-of-faulting factors, and seismic hazard assessments in Europe often combine, in logic-tree formulations, these equations with those from western North America that do include style-of-faulting coefficients. In this article, a simple scheme is provided to allow style-of-faulting adjustments to be made for those equations that do not include coefficients for rupture mechanism, so that style-of-faulting can be fully incorporated into the hazard calculations. This also considers the case of normal fault ruptures, not modelled in any of the current Californian equations, but which are the dominant mechanism in many parts of Europe. The scheme is validated by performing new regressions on a widely used European attenuation relationship with additional terms for style-of-faulting. This revised version was published online in July 2006 with corrections to the Cover Date.     

Vertical dynamic response of a disk on a saturated poroelastic half-space

This paper considers the vertical dynamic response of a disk on a saturated poroelastic half-space. Firstly the pressure-solid displacement form of the harmonic equations of motion for a poroelastic solid are developed from the form of the equations originally presented by Biot. These equations are solved by a new method. Then the mixed boundary value problem for the vertical harmonic vibration of a disk on a poroelastic half-space is studied. The two types of drainage conditions at the surface of the poroelastic half-space are considered: (a) the surface of the poroelastic half-space is assumed to be completely pervious both within and exterior to the plate; (b) The interface between the plate and the poroelastic half-space is assumed to be impervious and the exterior region is assumed to be pervious. By using the Hankel transform techniques, the paper develops the governing dual integral equations. These governing integral equations are further reduced to systems of standard Fredholm integral equations of the second kind by Abel transform.     

A discontinuous Galerkin method for two-dimensional flow and transport in shallow water

A discontinuous Galerkin (DG) finite element method is described for the two-dimensional, depth-integrated shallow water equations (SWEs). This method is based on formulating the SWEs as a system of conservation laws, or advection–diffusion equations. A weak formulation is obtained by integrating the equations over a single element, and approximating the unknowns by piecewise, possibly discontinuous, polynomials. Because of its local nature, the DG method easily allows for varying the polynomial order of approximation. It is also “locally conservative”, and incorporates upwinded numerical fluxes for modeling problems with high flow gradients. Numerical results are presented for several test cases, including supercritical flow, river inflow and standard tidal flow in complex domains, and a contaminant transport scenario where we have coupled the shallow water flow equations with a transport equation for a chemical species.