Finite-difference algorithm with local time-space grid refinement for simulation of waves

This paper presents a new approach to a local time-space grid refinement for a staggered-grid finite-difference simulation of waves. The approach is based on approximation of a wave equation at the interface where two grids are coupled. As no interpolation or projection techniques are used, the finite-difference scheme preserves second order of convergence. We have proved that this approach is low-reflecting, the artificial reflections are about 10^− 4 of an incident wave. We have also shown that if a successive refinement is applied, i.e. temporal and spatial steps are refined at different interfaces, this approach is stable.     

Numerical modeling of two-phase hysteresis combined with an interface condition for heterogeneous porous media

This paper presents a numerical implementation of two-phase capillary hysteresis and its combination with a capillary interface condition for the treatment of heterogeneities. The hysteresis concepts chosen in this work are first implemented in a node-centered FV discretization scheme and subsequently combined with the interface condition that predicts sharp saturation discontinuities at material interfaces, based on a pressure equilibrium concept. This approach allows for the approximation of history-dependent, and at the same time discontinuous, saturations at material interfaces. The resulting model provides a well-defined evolution of the hysteretic capillary pressure–saturation relationships at material interfaces that is independent of the grid spacing. As demonstrated with a simple 1-D example, this concept therefore offers the advantage that the solution of a two-phase flow problem involving hysteresis does not relate to the grid resolution at the material interfaces.     

Modeling sideswipe in 2D oceanic seismic surveys from sonar data: Application to the Mariana arc

In two-dimensional (2D) marine seismic-reflection surveys, out-of-plane rough seafloor bathymetry can cause multiple ocean-bottom reflections that complicate the interpretation of shallow reflections. Although migration corrects for the in-plane position of reflectors, it cannot resolve the inherent ambiguity in their out-of-plane positions. We show how swath bathymetry, routinely collected in many such surveys, can be used to model out-of-plane seafloor reflections and prevent their misinterpretation as subsurface geology. We use both raw and gridded multi-beam bathymetry data to build images that represent seafloor reflections in migrated seismic data. Comparison of these images to the seismic sections reveals whether suspicious features are out-of-plane water bottom reflections or subsurface reflections. Multi-channel seismic surveys across the Marianas intra-oceanic arc system provide examples where rough seafloor topography produced reflections that were initially misinterpreted. We use our seafloor reflection modeling (SRM) approach to help distinguish a possible landslide from a volcanic cone, to help distinguish real from apparent fault-plane reflections bounding a sediment-filled basin, and to verify that a possible magma chamber reflection results from sub-surface structure, not seafloor sideswipe.     

Impact of cyclonic wind field on interaction of surge–wave computations using finite-element and finite-difference models

Both finite-element and finite-difference numerical models are applied to simulate storm surges and associated currents generated by tropical cyclones that struck the coast of Andhra Pradesh, located on the east coast of India. During a cyclone, the total water level at any location on the coast is made up of the storm surge, surge–wind wave interaction and the tide. The advanced circulation two-dimensional depth-integrated (ADCIRC-2DDI) model based on finite-element formulation and the two-dimensional finite-difference model of storm surges developed at IIT Delhi, hereafter referred as IITD storm surge model, are used. These models are driven by astronomical tides at the open ocean boundary and cyclonic asymmetric winds over the surface of the computational domain. Comparison of model simulated sea-surface elevations with coarse and finer spatial resolutions suggests that the grid resolution near the coast is very crucial for accurate determination of the surges in addition to the local bathymetry. The model underpredicts surges, and the peak surge location shifts more to the right of the landfall as the spatial resolution of the model becomes coarser. The numerical experiments also demonstrate that the ADCIRC model is robust over the IITD storm surge model for surge computations as the coastline is better represented in the former.     

起伏地表条件下频率-空间域声波介质正演模拟

频率-空间域正演模拟是频率域及Laplace-Fourier域全波形反演的基础,起伏地表条件下波形反演算法的关键是正演算法中考虑起伏地表的影响。基于带PML吸收边界的声波波动方程,在已有最优9点有限差分正演算法的基础上构建了起伏地表条件下频率-空间域正演算法。通过应用变网格技术,进一步提高算法的计算效率、降低内存开销,使得大规模起伏地表模型的频率域正反演问题成为可能。理论分析及数值测试表明：通过对近地表区域进行局部网格加密,可有效地压制由于矩形网格离散引起的角点散射;结合变网格技术可较易获得5倍以上计算效率的提高及内存占用的降低,且随着模型尺度的增加及地表起伏高程差的减小,倍数将显著增加;在细网格与粗网格交界处产生的虚假反射振幅幅值控制在原始波场的2%以内,满足地震波场正反演的需求。     

Numerical simulation of seismic waves in models with anisotropic formations: coupling Virieux and Lebedev finite-difference schemes

Anisotropy is widespread in the Earth’s interior. However, there is a number of models where anisotropic formations comprise as few as 10–20?% of the volume, and this includes fractured reservoirs, thin-layered packs, etc. while the major part of the medium is isotropic. In this situation, the use of computationally intense anisotropy-oriented approaches throughout the computational domain is prodigal. So this paper presents an original advanced finite-difference algorithm based on the domain decomposition technique with individual scheme used inside subdomains. It means that the standard staggered grid scheme or the Virieux scheme is used in the main part of the model which is isotropic, while the anisotropy-oriented Lebedev scheme is utilized inside domains with anisotropic formations. Finite-difference consistency conditions at the artificial interface where the schemes are coupled are designed to make the artificial reflections as low as possible, namely, for the second-order scheme, the third order of convergence of the reflection coefficients is proved.     

Finite element methods for variable density flow and solute transport

Saltwater intrusion into coastal freshwater aquifers is an ongoing problem that will continue to impact coastal freshwater resources as coastal populations increase. To effectively model saltwater intrusion, the impacts of increased salt content on fluid density must be accounted for to properly model saltwater/freshwater transition zones and sharp interfaces. We present a model for variable density fluid flow and solute transport where a conforming finite element method discretization with a locally conservative velocity post-processing method is used for the flow model and the transport equation is discretized using a variational multiscale stabilized conforming finite element method. This formulation provides a consistent velocity and performs well even in advection-dominated problems that can occur in saltwater intrusion modeling. The physical model is presented as well as the formulation of the numerical model and solution methods. The model is tested against several 2-D and 3-D numerical and experimental benchmark problems, and the results are presented to verify the code.     

非均匀网格层析静校正技术在地震资料处理中的应用

由于近地表模型速度结构的非均匀性,导致基于均匀网格的近地表速度模型很难达到最优,并同时存在信息冗余的弊端,由此提出了非均匀网格层析静校正方法。非均匀网格化建模方法基于背景速度场、地质构造带速度结构,以及目标地质图速度模型进行三级网格划分,运用一定的追踪方法获取方位角信息与地表初至波走时,将初至波观测走时与计算走时之间的时差分至追踪到各网格中,经近地表速度模型的多次迭代,构建最终速度模型。采用该方法在近地表速度纵、横向变化剧烈地区得到的静校正时间剖面,与其他常规校正方法比较,静校正效果较好。     

Nearly perfectly matched layer boundary conditions for operator upscaling of the acoustic wave equation

Acoustic imaging and sensor modeling are processes that require repeated solution of the acoustic wave equation. Solution of the wave equation can be computationally expensive and memory intensive for large simulation domains. One scheme for speeding up solution of the wave equation is the operator-based upscaling method. The algorithm proceeds in two steps. First, the wave equation is solved for fine grid unknowns internal to coarse blocks assuming the coarse blocks do not need to communicate with neighboring blocks in parallel. Second, these fine grid solutions are used to form a new problem which is solved on the coarse grid. Accurate and efficient wave propagation schemes also must avoid artificial reflections off of the computational domain edges. One popular method for preventing artificial reflections is the nearly perfectly matched layer (NPML) method. In this paper, we discuss applying NPML to operator upscaling for the wave equation. We show that although we only apply NPML to the first step of this two step algorithm (directly affecting the fine grid unknowns only), we still see a significant reduction of reflections back into the domain. We describe three numerical experiments (one homogeneous medium experiment and two heterogeneous media examples) in which we validate that the solution of the wave equation exponentially decays in the NPML regions. Numerical experiments of acoustic wave propagation in two dimensions with a reasonable absorbing layer thickness resulted in a maximum pressure reflection of 3–8%. While the coarse grid acceleration is not explicitly damped in our algorithm, the tight coupling between the two steps of the algorithm results in only 0.1–1% of acceleration reflecting back into the computational domain.     

基于TIN的三维地层建模及可视化技术研究

基于多层格网模型的表面建模方法,以及以此为基础形成的类三棱柱体元建模方法,要产生大量冗余的三角形面片或者细长的三棱柱单元,既不利于三维模型的快速浏览,也不能满足数值计算单元剖分的原则。采用TIN（不规则三角网）来表达地层面模型可以消除数据冗余。根据数据源的特点,实现了直接通过逐点插入法构造TIN模型,或者先通过Kriging插值法构造格网模型,再转化为TIN模型,以及利用等高线数据构造TIN模型。对TIN的裁剪算法进行了探讨,实现了地层面裁剪和断层的三维建模技术。借助可视化开发类库VTK,实现了三维地层模型的多种可视化表达方法。根据龙滩水电工程地质勘查情况,建立了坝址区域的三维地层模型。     

基于频率域高阶有限差分法的正演模拟及并行算法

在弹性波频率空间域有限差分数值模拟方面,差分网格及边界条件是影响弹性波模拟成功与否的关键,为了压制数值模拟中的网格频散,采用25点有限差分算子,建立了有限差分矩阵方程,且借鉴匹配层衰减边界条件思想,设计了弹性波频率空间域有限差分数值模拟算法。由于采用高阶有限差分法来提高差分格式的精度,将会导致计算量显著增加,为此,对频率空间域有限差分弹性波数值模拟方法,采用流水线技术与分治策略进行了并行算法研究,提高了计算效率,使得在合理的计算时间内更精确地模拟弹性波在弹性介质中的传播过程。     

Method of calculating tsunami travel times in the Andaman Sea region

A new model to calculate tsunami travel times in the Andaman Sea region has been developed. The model specifically provides more accurate travel time estimates for tsunamis propagating to Patong Beach on the west coast of Phuket, Thailand. More generally, the model provides better understanding of the influence of the accuracy and resolution of bathymetry data on the accuracy of travel time calculations. The dynamic model is based on solitary wave theory, and a lookup function is used to perform bilinear interpolation of bathymetry along the ray trajectory. The model was calibrated and verified using data from an echosounder record, tsunami photographs, satellite altimetry records, and eyewitness accounts of the tsunami on 26 December 2004. Time differences for 12 representative targets in the Andaman Sea and the Indian Ocean regions were calculated. The model demonstrated satisfactory time differences (<2 min/h), despite the use of low resolution bathymetry (ETOPO2v2). To improve accuracy, the dynamics of wave elevation and a velocity correction term must be considered, particularly for calculations in the nearshore region.     

FDwave3D: a MATLAB solver for the 3D anisotropic wave equation using the finite-difference method

Seismic modeling plays an important role in geophysics and seismology for estimating the response of seismic sources in a given medium. In this work, we present a MATLAB-based package, FDwave3D, for synthetic wavefield and seismogram modeling in 3D anisotropic media. The seismic simulation is carried out using the finite-difference method over the staggered grid, and it is applicable to both active and passive surveys. The code package allows the incorporation of arbitrary source mechanisms and offers spatial derivative operators of accuracy up to tenth-order along with different types of boundary conditions. First, the methodological aspects of finite-difference method are briefly introduced. Then, the code has been tested and verified against the analytical solutions obtained for the homogeneous model. Further, the numerical examples of layered and overthrust models are presented to demonstrate its reliability.

     

降水空间不均匀性对径流过程模拟的影响

应用数字高程流域水系模型，通过子流域和网格2种空间离散方式，采用传统的子流域法、雨量插值子流域法、逐个网格法3种雨量数据输入处理方法，分别作日流量模拟与洪水过程模拟，模拟过程与实测水文过程相比较表明：日模中3种雨量处理方法计算精度相当，次模中逐个网格法优于传统子流域法和雨量插值子流域法；降雨空间分布不均匀性考虑得越充分，水文过程模拟精度越高。     

Biogeochemical Zones Within a Macrotidal,Dry-Tropical Fluvial-Marine Transition Area: A Dry-Season Perspective

The Fitzroy River delivers large amounts of nutrients and fine sediments to Keppel Bay (contiguous with the Great Barrier Reef Lagoon) during intermittent flow events. This study explores sources, forms and transformations of nutrients in Keppel Bay, and develops a functional process zonation that integrates seabed geochemistry and water column nutrient characteristics which are controlled by suspended sediment. The water column and seabed properties were investigated over two dry seasons, with supplementary core incubations taken to measure carbon decomposition rates and nutrient fluxes. Keppel Bay can be divided into three zones, the: zone of maximum resuspension (ZMR); coastal transitional zone (CTZ); and blue water zone (BWZ). Mineralisation of predominantly terrestrial organic matter occurs in the ZMR where nutrient uptake by phytoplankton is light limited. The CTZ and BWZ had higher light penetration and phytoplankton growth was likely limited by N and P, respectively. The identified zones conform to the bathymetry and hydrodynamic characteristics of the bay, allowing for the development of an integrated conceptual model accounting for the benthic and pelagic biogeochemical processes. Recognition of these different zones shows that considerable variation in benthic and water column properties is possible within a small system with the bathymetric and hydrodynamic characteristics of the fluidized bed reactor.     

曲线坐标系下的完全匹配层吸收边界条件

在地震波数值模拟中,需要采用吸收边界条件以吸收人为边界反射。本文针对曲线坐标系下的二阶弹性波方程提出了一种完全匹配层(PML)吸收边界条件。与直角坐标系下的PML吸收边界条件类似,曲线坐标系下的PML吸收边界条件是一种在频率域中给出的人工边界条件,由相应的复坐标变换得到。在变换到时间域后,完全匹配层中将出现复杂的卷积运算。为了避免这些卷积运算,引入了4个中间变量。为了简化自由边界条件,采用正交贴体网格对起伏地表模型进行网格剖分。数值算例表明,该方法可以有效消除人为边界反射。     

Model reduction and discretization using hybrid finite volumes for flow in porous media containing faults

In this paper, we study two different model reduction strategies for solving problems involving single phase flow in a porous medium containing faults or fractures whose location and properties are known. These faults are represented as interfaces of dimension N ? 1 immersed in an N dimensional domain. Both approaches can handle various configurations of position and permeability of the faults, and one can handle different fracture permeabilities on the two inner sides of the fracture. For the numerical discretization, we use the hybrid finite volume scheme as it is known to be well suited to simulating subsurface flow. Some results, which may be of use in the implementation of the proposed methods in industrial codes, are demonstrated.     

Sensitivity Analysis of a Combined Groundwater Flow and Solute Transport Model Using Local-Grid Refinement: A Case Study

Combining groundwater flow models with solute transport models represents a common challenge in groundwater resources assessments and contaminant transport modeling. Groundwater flow models are usually constructed at somewhat larger scales (involving a coarser discretization) to include natural boundary conditions. They are commonly calibrated using observed groundwater levels and flows (if available). The groundwater solute transport models may be constructed at a smaller scale with finer discretization than the flow models in order to accurately delineate the solute source and the modeled target, to capture any heterogeneity that may affect contaminant migration, and to minimize numerical dispersion while still maintaining a reasonable computing time. The solution that is explored here is based on defining a finer grid subdomain within a larger coarser domain. The local-grid refinement (LGR) implemented in the Modular 3D finite-difference ground-water flow model (MODFLOW) code has such a provision to simulate groundwater flow in two nested grids: a higher-resolution sub-grid within a coarse grid. Under the premise that the interface between both models was well defined, a comprehensive sensitivity and uncertainty analysis was performed whereby the effect of a parameter perturbation in a coarser-grid model on transport predictions using a higher-resolution grid was quantified. This approach was tested for a groundwater flow and solute transport analysis in support of a safety evaluation of the future Belgian near-surface radioactive waste disposal facility. Our reference coarse-grid groundwater flow model was coupled with a smaller fine sub-grid model in two different ways. While the reference flow model was calibrated using observed groundwater levels at a scale commensurate with that of the coarse-grid model, the fine sub-grid model was used to run a solute transport simulation quantifying concentrations in a hypothetical well nearby the disposal facility. When LGR coupling was compared to a one-way coupling, LGR was found to provide a smoother flow solution resulting in a more CPU-efficient transport solution. Parameter sensitivities performed with the groundwater flow model resulted in sensitivities at the head observation locations. These sensitivities identified the recharge as the most sensitive parameter, with the hydraulic conductivity of the upper aquifer as the second most sensitive parameter in regard to calculated groundwater heads. Based on one-percent sensitivity maps, the spatial distribution of the observations with the highest sensitivities is slightly different for the upper aquifer hydraulic conductivity than for recharge. Sensitivity analyses were further performed to assess the prediction scaled sensitivities for hypothetical contaminant concentrations using the combined groundwater flow and solute transport models. Including all pertinent parameters into the sensitivity analysis identified the hydraulic conductivity of the upper aquifer as the most sensitive parameter with regard to the prediction of contaminant concentrations.     

Development of a discontinuous approach for modeling fluid flow in heterogeneous media using the numerical manifold method

In the numerical modeling of fluid flow in heterogeneous geological media, large material contrasts associated with complexly intersected material interfaces are challenging, not only related to mesh discretization but also for the accurate realization of the corresponding boundary constraints. To address these challenges, we developed a discontinuous approach for modeling fluid flow in heterogeneous media using the numerical manifold method (NMM) and the Lagrange multiplier method (LMM) for modeling boundary constraints. The advantages of NMM include meshing efficiency with fixed mathematical grids (covers), the convenience of increasing the approximation precision, and the high integration precision provided by simplex integration. In this discontinuous approach, the elements intersected by material interfaces are divided into different elements and linked together using the LMM. We derive and compare different forms of LMMs and arrive at a new LMM that is efficient in terms of not requiring additional Lagrange multiplier topology, yet stringently derived by physical principles, and accurate in numerical performance. To demonstrate the accuracy and efficiency of the NMM with the developed LMM for boundary constraints, we simulate a number of verification and demonstration examples, involving a Dirichlet boundary condition and dense and intersected material interfaces. Last, we applied the developed model for modeling fluid flow in heterogeneous media with several material zones containing a fault and an opening. We show that the developed discontinuous approach is very suitable for modeling fluid flow in strongly heterogeneous media with good accuracy for large material contrasts, complex Dirichlet boundary conditions, or complexly intersected material interfaces. Copyright © 2015 John Wiley & Sons, Ltd.     

Symmetry and M-Matrix Issues for the O-Method on an Unstructured Grid

More sophisticated discretization methods than the traditional control-volume finite-difference methods, have been proposed by Aavatsmark et al. in recent papers for solving the mass balance equations for porous media flow. These methods are based on a local representation of fluxes across cell-edges of control volumes (CVs). This paper will focus on mathematical properties of the discrete operator that arises when an elliptic term of the form ???(K?p) is discretized based on these discretization principles.