考虑关断时间的回线源激发TEM三维时域有限差分正演

从麦克斯韦旋度方程出发可以直接导出瞬变电磁场扩散方程,然而扩散方程不含电场对时间的一阶导数,不能构成显式的时域有限差分方程,借鉴du Fort-Frankel有限差分离散方法引入虚拟位移电流项构建显式时域有限差分方程.对Wang和Hohmann的经典时域算法进行了两点改进:第一,通过将矩形回线源电流密度加入麦克斯韦方程组的安培环路定理方程,实现回线源瞬变电磁激发源加入;第二,在计算中考虑关断时间.第一点改进使时域有限差分方程考虑了一次场的计算,并且源的计算不再依赖均匀半空间模型响应作为初始条件,使算法能够适应表层电阻率不均匀时的三维复杂模型.由于实际观测中不可能出现阶跃电流的关断形式,第二点改进可以方便设置发射电流下降沿.采用改进的三维时域有限差分正演算法对均匀半空间模型、四类三层模型、均匀半空间中含有低阻块体模型进行了计算并分别与解析解、线性数字滤波解、积分方程解和Wang的三维时域有限差分解进行了对比验证.以H模型为例,采用建立的三维时域有限差分正演算法计算了不同关断时间的斜阶跃脉冲回线源瞬变电磁中心点感应电动势衰减曲线.以实际地质资料为基础,构建包含两层采空区的三维复杂模型,以1 μs的极短关断时间进行了复杂模型定回线源瞬变电磁响应计算,并计算了该复杂模型的视电阻率曲线.     

A comparison of a semi-implicit with an explicit scheme in a three-dimensional hydrodynamic model

As the capacity of computers increases, the size and resolution of numerical models can be increased. In tidal models, however, using an explicit scheme together with high spatial resolution results in an unreasonably small time-step demanded by the CFL condition for stability. This condition is usually the most restrictive and applies to the propagation of the gravity wave.A fully nonlinear three-dimensional model has been developed, using the Galerkin method in the vertical dimension, in which the gravity wave terms are treated by an alternating-direction implicit scheme, while the friction, viscous and advective terms are treated explicitly. This permits a stable solution with a longer time-step than that required in an equivalent explicit scheme, while not needing as much computational effort as a fully implicit scheme. This semi-implicit model is compared with an explicit model in terms of efficiency, accuracy, and stability. Tidal and wind-driven flows and free oscillations in a rectangular sea model are examined, using various boundary conditions. The semi-implicit scheme takes approximately 1.1 times as long to run (per time-step) on the CRAY-1 computer as the explicit scheme.     

Behaviour of a hydrodynamic finite element model

A finite element model which solves the vertically integrated momentum and continuity equations is described. Linear triangular elements are used to describe the geometry and parameter variations. The Galerkin method of weighted residuals is employed to cast the equations in a form amenable to numerical solution. The model is based on a fully implicit formulation using finite differences for the temporal derivatives.Means of evaluating the non-linear terms of the governing equations are described, and model results are presented for a frictionless tidal channel. The example is chosen such that the non-linearities have a large influence on the solution, and as a result the linearization scheme significantly affects the model's behaviour.Suppression of the non-linear instabilities generated by the convective terms in the momentum equations is examined for the case of flow around a 180° bend. Both the imposition of artificially high roughness coefficients and the use of an effective eddy viscosity are examined in terms of their ability to damp the oscillations which arise for this example.Finally, model results are presented for a case study involving determination of remedial measures to improve flow conditions at a river outfall in Southern Ontario.     

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.     

PUBLICATIONS OF THE INTERNATIONAL ASSOCIATION OF HYDROLOGICAL SCIENCES

Abstract

Explicit algebraic expressions of influence coefficients of an aquifer system are addressed. The methodology is straightforwardly exemplified using a one-dimensional (1D) semi-confined homogeneous aquifer model. It consists of the integration of the governing equation from two different viewpoints. On the one hand, the analytical solution to the governing equation is presented under an integral form involving the appropriate Green function. On the other hand, equations resulting from a finite difference discretization are processed to deduce a fully explicit algebraic expression of the nodal hydraulic head. This latter, arranged after the previously developed continuum solution, helps with deriving an explicit algebraic expression of influence coefficients corresponding to the studied aquifer model. Finally, the study is extended to an inhomogeneous aquifer model.     

A comparison of two numerical models for the prediction of vibrations from underground railway traffic

Two prediction models for calculating vibration from underground railways are developed: the pipe-in-pipe model and the coupled periodic finite element–boundary element (FE–BE) model.The pipe-in-pipe model is a semi-analytical three-dimensional model that accounts for the dynamic interaction between the track, the tunnel and the soil. The continuum theory of elasticity in cylindrical coordinates is used to model two concentric pipes: an inner pipe to represent the tunnel wall and an outer pipe to represent the surrounding soil. The tunnel and soil are coupled accounting for equilibrium of stresses and compatibility of displacements at the tunnel–soil interface. This method assumes that the tunnel is invariant in the longitudinal direction and the problem is formulated in the frequency–wavenumber domain using a Fourier transformation. A track, formulated as an Euler–Bernoulli beam, is then coupled to this model. Results are transformed to the space domain using the inverse Fourier transform.The coupled periodic FE–BE model is based on a subdomain formulation, where a boundary element method is used for the soil and a finite element method for the tunnel. The Craig–Bampton substructuring technique is used to efficiently incorporate the track in the tunnel. The periodicity of the tunnel is exploited using the Floquet transformation to formulate the track–tunnel–soil interaction problem in the frequency–wavenumber domain and to compute the wave field radiated into the soil.An invariant concrete tunnel, embedded in a homogeneous full space is analyzed using both approaches. The pipe-in-pipe model offers an exact solution to this problem, which is used to validate the coupled periodic FE–BE model. The free field response due to a harmonic load in the tunnel is predicted and results obtained with both models are compared. The advantages and limitations of both models are highlighted. The coupled periodic FE–BE model has a greater potential as it can account for the complex periodic geometry of the tunnel and the layering in a soil medium. The effect of coupling a floating slab to the tunnel–soil system is also studied with both models by calculating the insertion gain.     

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.     

Stability conditions of explicit integration algorithms when using 3D viscoelastic artificial boundaries

Viscoelastic artificial boundaries are widely adopted in numerical simulations of wave propagation problems. When explicit time-domain integration algorithms are used, the stability condition of the boundary domain is stricter than that of the internal region due to the influence of the damping and stiffness of an viscoelastic artificial boundary. The lack of a clear and practical stability criterion for this problem, however, affects the reasonable selection of an integral time step when using viscoelastic artificial boundaries. In this study, we investigate the stability conditions of explicit integration algorithms when using three-dimensional (3D) viscoelastic artificial boundaries through an analysis method based on a local subsystem. Several boundary subsystems that can represent localized characteristics of a complete numerical model are established, and their analytical stability conditions are derived from and further compared to one another. The stability of the complete model is controlled by the corner regions, and thus, the global stability criterion for the numerical model with viscoelastic artificial boundaries is obtained. Next, by analyzing the impact of different factors on stability conditions, we recommend a stability coefficient for practically estimating the maximum stable integral time step in the dynamic analysis when using 3D viscoelastic artificial boundaries.

     

Recursive evaluation of interaction forces of unbounded soil in the time domain

The interaction forces representing the contribution of the linear unbounded soil to the equations of motion of a nonlinear soil-structure-interaction analysis are specified in the form of convolution integrals. They can be evaluated recursively in the time domain. In this procedure, the forces at a specific time are computed from the displacements at the same time and from the most recent forces and most recent past displacements. It is, in principle, only approximate. When the dynamic-stiffness coefficients can be expressed as the ratios of two polynomials in frequency, the appropriately chosen recursive equations are exact. Two possibilities of choosing a recursive equation are discussed.

• (i) The impulse-invariant method, where the unknown recursive coefficients are calculated by solving a system of equations which are established by equating the rigorous and recursive formulations for a discretized unit impulse displacement.
• (ii) In the segment approach, the dynamic-stiffness coefficients in the time domain are interpolated piecewise. Applying the z-transformation analytically then results in an explicit recursive equation without solving a system of equations.

The recursive evaluation of the convolution integrals in the time domain leads to a dramatic reduction in the computational effort up to two and three orders of magnitude and in the storage requirement. This makes the time-domain analysis using the substructure method computationally competitive with the corresponding direct (non-recursive) frequency-domain procedure of determining the complex response which is, however, applicable only to a linear (total) system.     

Flux-Corrected Transport with MT3DMS for Positive Solution of Transport with Full-Tensor Dispersion

Solute transport is usually modeled by the advection-dispersion-reaction equation. In the standard approach, mechanical dispersion is a tensor with principal directions parallel and perpendicular to the flow vector. Since realistic scenarios include nonuniform and unsteady flow fields, the governing equation has full tensor mechanical dispersion. When conventional grid-based numerical methods are used, approximation of the cross terms arising from the off-diagonal terms cause nonphysical solution with oscillations. As an example, for the common scenario of contaminant input into a domain with zero initial concentration, the cross-dispersion terms can result in negative concentrations that can wreak havoc in reactive transport applications. To address this issue, we use the well-known flux-corrected-transport (FCT) technique for a standard finite volume method. Although FCT has most often been used to eliminate oscillations resulting from discretization of the advection term for explicit time stepping, we show that it can be adapted for full-tensor dispersion and implicit time stepping. Unlike other approaches based on new discretization techniques (e.g., mimetic finite difference, nonlinear finite volume), FCT has the advantage of being flexible and widely applicable. Implementation of FCT requires solving an additional system of equations at each time step, using a modified “low order” matrix and a modified right-hand-side vector. To demonstrate the flexibility of FCT, we have modified the well-known and widely used groundwater solute transport simulator, MT3DMS. We apply the new simulator, MT3DMS-FCT, to several benchmark problems that suffer from negative concentrations when using MT3DMS. The new results are mass conservative and strictly nonnegative.     

饱和多孔介质三维时域黏弹性人工边界与动力反应分析的显式有限元法

本文基于Biot的饱和多孔介质本构方程,考察具有辐射阻尼的外行球面波,推导了饱和多孔介质三维黏弹性人工边界的法向和切向边界方程;在已有的饱和多孔介质二维显式有限元数值计算方法基础上,提出该理论的三维方法,并开发了实现该三维方法的有限元程序.算例表明饱和多孔介质三维时域黏弹性人工边界与动力反应分析的显式有限元法具有较好的精度和稳定性.     

On a monotonicity preserving Eulerian–Lagrangian localized adjoint method for advection–diffusion equations

Eulerian–Lagrangian localized adjoint methods (ELLAMs) provide a general approach to the solution of advection-dominated advection–diffusion equations allowing large time steps while maintaining good accuracy. Moreover, the methods can treat systematically any type of boundary condition and are mass conservative. However, all ELLAMs developed so far suffer from non-physical oscillations and are usually implemented on structured grids. In this paper, we propose a finite volume ELLAM which incorporates a novel correction step rendering the method monotone while maintaining conservation of mass. The method has been implemented on fully unstructured meshes in two space dimensions. Numerical results demonstrate the applicability of the method for problems with highly non-uniform flow fields arising from heterogeneous porous media.     

Iterated extended Kalman filter method for time-lapse seismic full-waveform inversion

Time-lapse seismic data is useful for identifying fluid movement and pressure and saturation changes in a petroleum reservoir and for monitoring of CO₂ injection. The focus of this paper is estimation of time-lapse changes with uncertainty quantification using full-waveform inversion. The purpose of also estimating the uncertainty in the inverted parameters is to be able to use the inverted seismic data quantitatively for updating reservoir models with ensemble-based methods. We perform Bayesian inversion of seismic waveform data in the frequency domain by combining an iterated extended Kalman filter with an explicit representation of the sensitivity matrix in terms of Green functions (acoustic approximation). Using this method, we test different strategies for inversion of the time-lapse seismic data with uncertainty. We compare the results from a sequential strategy (making a prior from the monitor survey using the inverted baseline survey) with a double difference strategy (inverting the difference between the monitor and baseline data). We apply the methods to a subset of the Marmousi2 P-velocity model. Both strategies performed well and relatively good estimates of the monitor velocities and the time-lapse differences were obtained. For the estimated time-lapse differences, the double difference strategy gave the lowest errors.     

基于二次场的海洋可控源电磁法三维有限元正演

根据库伦规范势的定义,推导出关于磁矢量势和电标量势的偏微分方程,为了克服由电流源引起的奇异性和数值模拟计算困难,将电磁总场分解为一次场和二次场,一次场由基于Schelkunoff势函数的一维正演算法得到,二次场由有限元法计算得到,实现了海洋可控源电磁法三维有限元正演算法。通过一维数值模拟实例,验证该算法的计算精度。然后,利用该算法对带海底地形的三层储层模型进行正演,分析了海底地形对海洋控源电磁场各分量产生的影响。      

Transport in chemically and mechanically heterogeneous porous media. I: Theoretical development of region-averaged equations for slightly compressible single-phase flow

Transport processes in heterogeneous porous media are often treated in terms of one-equation models. Such treatment assumes that the velocity, pressure, temperature, and concentration can be represented in terms of a single large-scale averaged quantity in regions having significantly different mechanical, thermal, and chemical properties. In this paper we explore the process of single-phase flow in a two-region model of heterogeneous porous media. The region-averaged equations are developed for the case of a slightly compressible flow which is an accurate representation for a certain class of liquid-phase flows. The analysis leads to a pair of transport equations for the region averaged pressures that are coupled through a classic exchange term, in addition to being coupled by a diffusive cross effect. The domain of validity of the theory has been identified in terms of a series of length and timescale constraints.In Part II the theory is tested, in the absence of adjustable parameters, by comparison with numerical experiments for transient, slightly compressible flow in both stratified and nodular models of heterogeneous porous media. Good agreement between theory and experiment is obtained for nodular and stratified systems, and effective transport coefficients for a wide range of conditions are presented on the basis of solutions of the three closure problems that appear in the theory. Part III of this paper deals with the principle of large-scale mechanical equilibrium and the region-averaged form of Darcy's law. This form is necessary for the development and solution of the region-averaged solute transport equations that are presented in Part IV. Finally, in Part V we present results for the dispersion tensors and the exchange coefficient associated with the two-region model of solute transport with adsorption.     

A lattice Boltzmann-finite element model for two-dimensional fluid–structure interaction problems involving shallow waters

In this paper, a numerical method for the modeling of shallow waters interacting with slender elastic structures is presented. The fluid domain is modeled through the lattice Boltzmann method, while the solid domain is idealized by corotational beam finite elements undergoing large displacements. Structure dynamics is predicted by using the time discontinuous Galerkin method and the fluid–structure interface conditions are handled by the Immersed Boundary method. An explicit coupling strategy to combine the adopted numerical methods is proposed and its effectiveness is tested by computing the error in terms of the energy that is artificially introduced at the fluid–solid interface.     

湖泊三维风生流隐式差分模型的研究

建立了湖泊三维风生流的隐格式差分模型,通过对该模型计算稳定性的分析发现,它的时间步长的取值具有较大的灵活性．该模型波用于武汉市墨水湖的民生流模拟,实例计算表明该模型计算结果合理,能较好地反映湖泊风生流的流态特征．     

Conjunctive surface-subsurface modeling of overland flow

In this paper, details of a conjunctive surface-subsurface numerical model for the simulation of overland flow are presented. In this model, the complete one-dimensional Saint-Venant equations for the surface flow are solved by a simple, explicit, essentially non-oscillating (ENO) scheme. The two-dimensional Richards equation in the mixed form for the subsurface flow is solved using an efficient strongly implicit finite-difference scheme. The explicit scheme for the surface flow component results in a simple method for connecting the surface and subsurface components. The model is verified using the experimental data and previous numerical results available in the literature. The proposed model is used to study the two-dimensionality effects due to non-homogeneous subsurface characteristics. Applicability of the model to handle complex subsurface conditions is demonstrated.     

Transient foundation solution of saturated soil to impulsive concentrated loading

The transient dynamic response of saturated soil under suddenly applied normal and horizontal concentrated loading is studied in this paper. The behavior of saturated soil is governed by Biot's consolidation theory. The general solutions for Biot equations of equilibrium are derived in terms of displacements and variations of fluid volume, using Laplace–Hankel integral transforms. The solutions in the time domain can be evaluated by numerical inverse Laplace–Hankel transforms. Selected numerical results for displacements, stresses, and pore pressures are presented. Comparisons with existing closed-form solutions for the elastic half-space are made to confirm the accuracy of the present solutions. The solutions can be used to study a variety of transient wave propagation problems and dynamical interactions between saturated soil and structures.     

Explicit infiltration equations and the Lambert W-function

The Green and Ampt infiltration formula, as well as the Talsma and Parlange formula, are two-parameter equations that are both expressible in terms of Lambert W-functions. These representations are used to derive explicit, simple and accurate approximations for each case. The two infiltration formulas are limiting cases that can be deduced from an existing three-parameter infiltration equation, the third parameter allowing for interpolation between the limiting cases. Besides the limiting cases, there is another case for which the three-parameter infiltration equation yields an exact solution. The three-parameter equation can be solved by fixed-point iteration, a scheme which can be exploited to obtain a sequence of increasingly complex explicit infiltration equations. For routine use, a simple, explicit approximation to the three-parameter infiltration equation is derived. This approximation eliminates the need to iterate for most practical circumstances.