Diffusive-viscous Wave Equation Research Articles

Adaptive-coefficient finite difference frequency domain method for time fractional diffusive-viscous wave equation arising in geophysics

The diffusive-viscous wave (DVW) equation is a widely used model to describe frequency dependent attenuation of seismic wave in fluid-saturated porous medium. In this paper taking power law frequency dependent attenuation into account, we first introduce a modified DVW equation (time fractional DVW equation) in which the first order temporal derivative of viscous term is replaced with a fractional order temporal derivative. In consideration of that most of the existing numerical simulations for seismic wave equations are based on time domain methods and truncation with some specific boundary conditions, we incorporate the absorbing boundary condition as complex-frequency-shifted (CFS) perfectly matched layer (PML) into the time fractional DVW equation, and then develop an adaptive-coefficient (AC) finite difference frequency domain (FDFD) method for numerical simulation. The corresponding analytical solution for homogeneous time fractional DVW equation is provided for model validation, and the effectiveness of the developed AC FDFD method is verified by some numerical examples including the homogeneous model and the layered model. Numerical results show that AC FDFD method is more accurate than the traditional 2nd-order FDFD method for numerical modeling of time fractional DVW equation with CFS PML absorbing boundary condition, while requiring similar computational costs.

Complex-valued adaptive-coefficient finite-difference frequency-domain method for wavefield modeling based on the diffusive-viscous wave equation

The diffusive-viscous wave (DVW) equation is an effective model for analyzing seismic low-frequency anomalies and attenuation in porous media. To effectively simulate DVW wavefields, the finite-difference or finite-element method in the time domain is favored, but the time-domain approach proves less efficient with multiple shots or a few frequency components. The finite-difference frequency-domain (FDFD) method featuring optimal or adaptive coefficients is favored in seismic simulations due to its high efficiency. Initially, we develop a real-valued adaptive-coefficient (RVAC) FDFD method for the DVW equation, which ignores the numerical attenuation error and is a generalization of the acoustic adaptive-coefficient FDFD method. To reduce the numerical attenuation error of the RVAC FDFD method, we introduce a complex-valued adaptive-coefficient (CVAC) FDFD method for the DVW equation. The CVAC FDFD method is constructed by incorporating correction terms into the conventional second-order FDFD method. The adaptive coefficients are related to the spatial sampling ratio, number of spatial grid points per wavelength, and diffusive and viscous attenuation coefficients in the DVW equation. Numerical dispersion and attenuation analysis confirm that, with a maximum dispersion error of 1% and a maximum attenuation error of 10%, the CVAC FDFD method only necessitates 2.5 spatial grid points per wavelength. Compared with the RVAC FDFD method, the CVAC FDFD method exhibits enhanced capability in suppressing the numerical attenuation during anelastic wavefield modeling. To validate the accuracy of our method, we develop an analytical solution for the DVW equation in a homogeneous medium. Three numerical examples substantiate the high accuracy of the CVAC FDFD method when using a small number of spatial grid points per wavelength, and this method demands computational time and computer memory similar to those required by the conventional second-order FDFD method. A fluid-saturated model featuring various layer thicknesses is used to characterize the propagation characteristics of DVW.

Local randomized neural networks with discontinuous Galerkin methods for diffusive-viscous wave equation

The diffusive-viscous wave equation is an advancement in wave equation theory, as it accounts for both diffusion and viscosity effects. This has a wide range of applications in geophysics, such as the attenuation of seismic waves in fluid-saturated solids and frequency-dependent phenomena in porous media. Therefore, the development of an efficient numerical method for the equation is of both theoretical and practical importance. Recently, local randomized neural networks with discontinuous Galerkin (LRNN-DG) methods have been introduced in [21] to solve elliptic and parabolic equations. Numerical examples suggest that LRNN-DG can achieve high accuracy, and can handle time-dependent problems naturally and efficiently by using a space-time framework. In this paper, we develop LRNN-DG methods for solving the diffusive-viscous wave equation and present numerical experiments with several cases. The numerical results show that the proposed methods can solve the diffusive-viscous wave equation more accurately with less computing costs than traditional methods.

Parameter inversion of the diffusive–viscous wave equation based on Gaussian process regression

Abstract The diffusive–viscous wave (DVW) equation is used to characterize the relationship between frequency-dependent seismic responses and saturated fluids by incorporating the frictional dissipation and viscous damping to the scalar wave equation. Simultaneous inversion of three model parameters in DVW equation is essential for seismic interpretations. Traditional inversion methods require continuous forward-modeling updates, resulting in low computational efficiency. Moreover, the traditional methods have limitations in simultaneously inverting multi-parameters of wave equations such as the DVW equation, usually fixing one parameter to invert the other two parameters. Gaussian process regression (GPR) is a kernel-based non-parametric probabilistic model that introduces prior variables through Gaussian processes (GP). We present a method for the inversion of the three parameters (velocity, diffusive and viscous attenuation coefficients) of the DVW equation based on GPR. The procedure consists of initially implementing the central finite difference approximation to discretize the DVW equation in the time domain. Subsequently, a Gaussian prior is provided on two snapshots of the DVW equation to obtain the corresponding kernel functions. Furthermore, the hyperparameters in kernel functions and the three model parameters are simultaneously trained by minimizing the negative logarithmic marginal likelihood with few training samples while incorporating the underlying physics in terms of encoding the DVW equation into the kernel functions. It is worth noting that it is the first time of implementing three-parameter simultaneous inversion based on the DVW equation. The numerical examples in homogeneous, layered and heterogeneous media demonstrate the effectiveness of this method.

Analysis and Hermite Spectral Approximation of Diffusive-Viscous Wave Equations in Unbounded Domains Arising in Geophysics

The diffusive-viscous wave equation (DVWE) is widely used in seismic exploration since it can explain frequency-dependent seismic reflections in a reservoir with hydrocarbons. Most of the existing numerical approximations for the DVWE are based on domain truncation with ad hoc boundary conditions. However, this would generate artificial reflections as well as truncation errors. To this end, we directly consider the DVWE in unbounded domains. We first show the existence, uniqueness, and regularity of the solution of the DVWE. We then develop a Hermite spectral Galerkin scheme and derive the corresponding error estimate showing that the Hermite spectral Galerkin approximation delivers a spectral rate of convergence provided sufficiently smooth solutions. Several numerical experiments with constant and discontinuous coefficients are provided to verify the theoretical result and to demonstrate the effectiveness of the proposed method. In particular, We verify the error estimate for both smooth and non-smooth source terms and initial conditions. In view of the error estimate and the regularity result, we show the sharpness of the convergence rate in terms of the regularity of the source term. We also show that the artificial reflection does not occur by using the present method.

Discontinuous Galerkin method for the diffusive-viscous wave equation

This paper develops and analyzes the discontinuous Galerkin method for the diffusive-viscous wave equation. The proposed numerical scheme is established by using the discontinuous Galerkin discretization for the spatial variable and a tailored finite difference scheme to approximate the first and second order temporal derivative terms. A second order temporal convergence rate and the optimal order spatial error estimate in the DG norm are derived. We also provide the optimal spatial error estimate in the L2(Ω)-norm under the extra parameter assumption. Numerical tests are presented to illustrate the predicted convergence behaviours and the versatility of the proposed method.

A new approximation to the reflection coefficient of the diffusive-viscous wave equation and its evaluation for frequency-dependent AVA inversion

The frequency-dependent seismic anomalies related to the hydrocarbon reservoirs have lately attracted wide attentions, and the diffusive-viscous wave equation was built to explain these anomalies. Frequency-dependent amplitude versus angles of incidence (FAVA) inversion is a commonly used and effective reservoir prediction technique. The existing frequency-dependent reflection coefficient formula based on the diffusive-viscous wave equation is complex and highly nonlinear, containing eight model parameters, then it is difficult to apply for FAVA inversion. In this paper, we firstly apply the exact frequency-dependent reflection coefficient formula to FAVA inversion for two typical models. The inversion results indicate that the reflection coefficient formula can be reasonably simplified. Through a series of mathematical derivations, we obtain a new approximate frequency-dependent reflection coefficient formula. The new formula contains half of the model parameters of the exact formula, which greatly shrinks the search space of the global optimization algorithm and improves computational efficiency in FAVA inversion. Finally, the accuracy of the approximate formula is analyzed by the forward modeling and FAVA inversion. The inversion results show that our formula is applicable within incident angle of 35 degrees and the frequency band range of seismic exploration, which supplies a theoretical foundation for the practical application of the FAVA inversion based on the diffusive-viscous wave equation.

Numerical analysis of the diffusive-viscous wave equation

The diffusive-viscous wave equation arises in a variety of applications in geophysics, and it plays an important role in seismic exploration. In this paper, semi-discrete and fully discrete numerical methods are introduced to solve a general initial-boundary value problem of the diffusive-viscous wave equation. The spatial discretization is carried out through the finite element method, whereas the time derivatives are approximated by finite differences. Optimal order error estimates are derived for the numerical methods. Numerical results on a test problem are reported to illustrate the numerical convergence orders.

Well-posedness of the diffusive-viscous wave equation arising in geophysics

The diffusive-viscous wave equation arises in a variety of applications in geophysics such as the attenuation of seismic waves propagating in fluid-saturated solids, and frequency-dependent phenomena in porous media. This paper provides a well-posedness analysis on the initial-boundary value problem of the diffusive-viscous wave equation.

Seismic numerical analysis for partially gas-saturated porous rocks by an extended local Rytov Fourier approximation based on the diffusion-viscous theory

The frequency-dependent attenuation and velocity dispersion of seismic responses are closely related to hydrocarbon reservoirs. To further investigate the characteristics of seismic responses caused by pore fluid-bearing reservoirs, the role of gas saturation is analyzed in seismic responses of sand reservoirs characterized by the patchy saturation model. To this end, a novel wave extrapolation method is developed based on the diffusive-viscous wave equation (DVWE) as well as a scheme for an extended local Rytov Fourier (ELRF) approximation within the extrapolation depth interval. Our proposed method considers the presence of fluid mixtures in the porous media, resulting in seismic attenuation and dispersion by the mechanism generally known as wave-induced fluid flow (WIFF). This method enables an accommodation for the lateral variations in slowness, diffusion coefficient and viscosity. Subsequently, the extrapolation is adopted to model the synthetic seismic data of a distributary channel model. During this modeling, a gas-water saturated sand reservoir embedded into one of the channels was used to comparatively analyze the distinct features on its seismic synthetic data. We exhibited the numerical simulation results using the proposed wave extrapolation method here and the traditional acoustic wave equation (AWE) method. A comparison of the simulation results, demonstrates that our proposed numerical method can depict the seismic dispersion and frequency-dependent attenuation as well as the phase delay effects associated with gas-water-saturated sand reservoirs. Furthermore, we compare the seismic responses by changing the gas saturations of the sand reservoir. The gas saturation of the reservoir has significant effects on the seismic characteristics of the numerical modeling data. The numerical modeling method improves our understanding of the mechanisms of seismic frequency-dependent characteristics associated with gas saturations and potentially contributes to better insights into gas reservoir indicators derived from seismic field data.

Properties of seismic absorption induced reflections

Seismic reflections at an interface are often regarded as the variation of the acoustic impedance (product of seismic velocity and density) in a medium. In fact, they can also be generated due to the difference in absorption of the seismic energy. In this paper, we investigate the properties of such reflections. Based on the diffusive-viscous wave equation and elastic diffusive-viscous wave equation, we investigate the dependency of the reflection coefficients on frequency, and their variations with incident angles. Numerical results at a boundary due to absorption contrasts are compared with those resulted from acoustic impedance variation. It is found that, the reflection coefficients resulted from absorption depend significantly on the frequency especially at lower frequencies, but vary very slowly at small incident angles. At the higher frequencies, the reflection coefficients of diffusive-viscous wave and elastic diffusive-viscous wave are close to those of acoustic and elastic cases, respectively. On the other hand, the reflections caused by acoustic impedance variation are independent of frequency but vary distinctly with incident angles before the critical angle. We also investigate the difference between the seismograms generated in the two different media. The numerical results show that the amplitudes of these reflected waves are attenuated and their phases are shifted. However, the reflections obtained by acoustic impedance contrast, show no significant amplitude attenuation and phase shift.

Extended reflectivity method for modelling the propagation of diffusive–viscous wave in dip‐layered media

ABSTRACTThe reflectivity method plays an important role in seismic modelling. It has been used to model different types of waves propagating in elastic and anelastic media. The diffusive–viscous wave equation was proposed to investigate the relationship between frequency dependence of reflections and fluid saturation. It is also used to describe the attenuation property of seismic wave in a fluid‐saturated medium. The attenuation of diffusive–viscous wave is mainly characterised by the effective attenuation parameters in the equation. Thus, it is essential to obtain those parameters and further characterise the features of the diffusive–viscous wave. In this work, we use inversion method to obtain the effective attenuation parameters through quality factor to investigate the characteristics of diffusive–viscous wave by comparing with those of the viscoacoustic wave. Then, the reflection/transmission coefficients in a dip plane‐layered medium are studied through coordinate transform and plane‐wave theory. Consequently, the reflectivity method is extended to compute seismograms of diffusive–viscous wave in a dip plane multi‐layered medium. Finally, we present two models to simulate the propagation of diffusive–viscous wave in a dip plane multi‐layered medium by comparing the results with those in a viscoacoustic medium. The numerical results demonstrate the validity of our extension of reflectivity method to the diffusive–viscous medium. The numerical examples in both time domain and time–frequency domain show that the reflections from a dip plane interface have significant phase shift and amplitude change compared with the results of horizontal plane interface due to the differences in reflection/transmission coefficients. Moreover, the modelling results show strong attenuation and phase shift in the diffusive–viscous wave compared to those of the viscoacoustic wave.

An approximation to the reflection coefficient of plane longitudinal waves based on the diffusive-viscous wave equation

The frequency-dependent seismic anomalies related to hydrocarbon reservoirs have lately attracted wide interest. The diffusive-viscous model was proposed to explain these anomalies. When an incident diffusive-viscous wave strikes a boundary between two different media, it is reflected and transmitted. The equation for the reflection coefficient is quite complex and laborious, so it does not provide an intuitive understanding of how different amplitude relates to the parameters of the media and how variation of a particular parameter affects the reflection coefficient. In this paper, we firstly derive a two-term (intercept-gradient) and three-term (intercept-gradient-curvature) approximation to the reflection coefficient of the plane diffusive-viscous wave without any assumptions. Then, we study the limitations of the obtained approximations by comparing the approximate value of the reflection coefficient with its exact value. Our results show that the two approximations match well with the exact solutions within the incident angle of 35°. Finally, we analyze the effects of diffusive and viscous attenuation parameters, velocity and density in the diffusive-viscous wave equation on the intercept, gradient and curvature terms in the approximations. The results show that the diffusive attenuation parameter has a big impact on them, while the viscous attenuation parameter is insensitive to them; the velocity and density have a significant influence on the normal reflections and they distinctly affect the intercept, gradient and curvature term at lower acoustic impedance.

Frequency-dependent reflection coefficients in diffusive-viscous media

Amplitude variation with offset/angle of incidence inversion has been playing a key role in hydrocarbon detection and reservoir characterization. Traditionally, it has been limited to elastic even viscoelastic cases. We investigated the reflectivity in diffusive-viscous media. Our study demonstrated that the magnitude of the reflection coefficients in such a media is not only related to the incident angle and the parameters of the medium but also strongly depends on the frequency. We first demonstrated the validity of diffusive-viscous wave equation, proposed empirically, based on the continuum equations and basic laws of physics. Then, we investigated the frequency-dependent phase velocity and the quality factor [Formula: see text], whose impacts typically increase toward high frequencies, and we studied the magnitude and phase of the reflection coefficient at an interface between two diffusive-viscous media. A general equation of the reflection coefficients was also obtained for a stack of arbitrary number of plane layers in such media. Finally, we computed synthetic vertical seismic profile seismograms using the reflectivity method for a model consisting of five fluid-saturated layers to demonstrate that the diffusive and viscous terms in diffusive-viscous theory have a big impact on the synthetic seismograms, and stronger attenuation is observed in the layer having larger attenuation parameters.

Modeling the Propagation of Diffusive-Viscous Waves Using Flux-Corrected Transport–Finite-Difference Method

Seismic numerical modeling is a technique for simulating the wave propagation in the earth. The aim is to predict the seismogram, given an assumed structure of the subsurface. Real subsurface structure is complex and often multi-phase media because of fluid saturation, so the commonly used models such as acoustic, elastic media, etc., cannot characterize the information of real subsurface structure. The anelastic attenuation occurs when the waves propagate in fluid-saturated media. The diffusive-viscous model can be used to describe the attenuation of seismic waves propagating in fluid-saturated rocks, and it is also used to investigate the relationship between the frequency dependence of reflections and fluid saturation in a porous medium. In this paper, we derive the finite-difference scheme for the diffusive-viscous wave equation and simulate the propagation of seismic waves in fluid-saturated media based on the diffusive-viscous model, using the flux-corrected transport-finite-difference method (FCT-FDM). The numerical results show that the propagating waves in fluid-saturated media greatly attenuate by comparing with those of acoustic case.

Numerical simulation of seismic low-frequency shadows and its application

Strong low-frequency energy beneath a hydrocarbon reservoir is called a seismic low-frequency shadow and can be used as a hydrocarbon indicator (Taner et al., 1979) but the physical mechanism of the observed low-frequency shadow is still unclear. To study the mechanism, we performed seismic numerical simulation of geological models with a hydrocarbon-bearing zone using the 2-D diffusive-viscous wave equation which can effectively model the characteristics of velocity dispersion and transform the seismic data centered in a target layer slice within a time window to the time-frequency domain by using time-frequency signal analysis and sort the frequency gathers to common frequency cubes. Then, we observe the characteristics of the seismic low-frequency shadow in the common frequency cubes. The numerical simulations reveal that the main mechanism of seismic low-frequency shadows is attributed to high attenuation of the medium to high seismic frequency components caused by absorption in the hydrocarbon-filled reservoir. Results from a practical example of seismic low-frequency shadows show that it is possible to identify the reservoir by the low-frequency shadow with high S/N seismic data.