On reproducing uniform flow exactly on general hexahedral cells using one degree of freedom per surface

The solution of second order elliptic partial differential equations typically arising for flow problems, e.g., in porous media, is frequently expressed as face fluxes of a flow field at hexahedral cells. In this letter, we show by example that the following properties are incompatible: (1) A local velocity reconstruction, (2) reproduction of uniform flow for general hexahedral cells, and (3) a flow field in H(div). This is in particular relevant to mixed finite element methods approximating H(div), and to other methods trying to reconstruct a flow field in a hexahedron using only local face fluxes, e.g., streamline tracing methods for reservoir simulation.     

Streamline-oriented grid generation for transport modelling in two-dimensional domains including wells

Flownets are useful tools for the visualization of groundwater flow fields. Using orthogonal flownets as grids for transport modeling is an effective way to control numerical dispersion, especially transverse to the direction of flow. Therefore tools for automatic generation of flownets may be seen both as postprocessors for groundwater flow simulations and preprocessors for contaminant transport models. Existing methods to generate streamline-oriented grids suffer from drawbacks such as the inability to include sources in the interior of the grid. In this paper, we introduce a new method for the generation of streamline-oriented grids which handles wells in the grid interior, and which produces orthogonal grids for anisotropic systems. Streamlines are generated from an accurate velocity field obtained from the solution of the mixed-hybrid finite element method for flow, while pseudopotentials, which are orthogonal to the streamlines, are obtained by a standard finite element solution of the pseudopotential equation. A comprehensive methodology for the generation of orthogonal grids, including the location of stagnation points and dividing streamlines, is introduced. The effectiveness of the method is illustrated by means of examples. A related paper presents a compatible formulation of the solution for reactive transport, while a second related paper gives a detailed quantitative assessment of the various forms of modelled mixing and their effect on the accuracy of simulations of the biodegradation of groundwater contaminants.     

Modelling the Velocity Field in a Regular Grid in the Area of Poland on the Basis of the Velocities of European Permanent Stations

The paper presents the results of testing the various methods of permanent stations’ velocity residua interpolation in a regular grid, which constitutes a continuous model of the velocity field in the territory of Poland. Three packages of software were used in the research from the point of view of interpolation: GMT (The Generic Mapping Tools), Surfer and ArcGIS. The following methods were tested in the softwares: the Nearest Neighbor, Triangulation (TIN), Spline Interpolation, Surface, Inverse Distance to a Power, Minimum Curvature and Kriging. The presented research used the absolute velocities’ values expressed in the ITRF2005 reference frame and the intraplate velocities related to the NUVEL model of over 300 permanent reference stations of the EPN and ASG-EUPOS networks covering the area of Europe. Interpolation for the area of Poland was done using data from the whole area of Europe to make the results at the borders of the interpolation area reliable. As a result of this research, an optimum method of such data interpolation was developed. All the mentioned methods were tested for being local or global, for the possibility to compute errors of the interpolated values, for explicitness and fidelity of the interpolation functions or the smoothing mode. In the authors’ opinion, the best data interpolation method is Kriging with the linear semivariogram model run in the Surfer programme because it allows for the computation of errors in the interpolated values and it is a global method (it distorts the results in the least way). Alternately, it is acceptable to use the Minimum Curvature method. Empirical analysis of the interpolation results obtained by means of the two methods showed that the results are identical. The tests were conducted using the intraplate velocities of the European sites. Statistics in the form of computing the minimum, maximum and mean values of the interpolated North and East components of the velocity residuum were prepared for all the tested methods, and each of the resulting continuous velocity fields was visualized by means of the GMT programme. The interpolated components of the velocities and their residua are presented in the form of tables and bar diagrams.     

Ray tracing in elliptical anisotropic media using the linear traveltime interpolation (LTI) method applied to traveltime seismic tomography

The linear traveltime interpolation (LTI) method is a suitable ray‐tracing technique for modelling first‐arrival times in isotropic media. LTI is extended to elliptical anisotropic media and applied to a tomographic inversion procedure. A theoretical formulation is first derived and then LTI implementation is discussed in terms of source–receiver arrays and cell size. The method is then combined with the tomographic inversion procedure adopted. The matching of the ray tracing with inversion in elliptical anisotropic media posed a double non‐linear problem. Thus two assumptions were made: the velocity in each cell is uniform and the main directions of anisotropy are known. To take into account the geometrical characteristics of the area under investigation (depth and velocity of the weathering, and thickness of the inner media), cells of varying size were considered. No hypothesis was made on anisotropy weakness.
The algorithm was first tested on synthetic models and then applied to a field survey. On comparing the results of the synthetic models and the field survey with those obtained with a linear raypath approximation, it was found that there were fewer data misfits.     

A new 3-D ray tracing method based on LTI using successive partitioning of cell interfaces and traveltime gradients

We present a new method of three-dimensional (3-D) seismic ray tracing, based on an improvement to the linear traveltime interpolation (LTI) ray tracing algorithm. This new technique involves two separate steps. The first involves a forward calculation based on the LTI method and the dynamic successive partitioning scheme, which is applied to calculate traveltimes on cell boundaries and assumes a wavefront that expands from the source to all grid nodes in the computational domain. We locate several dynamic successive partition points on a cell's surface, the traveltimes of which can be calculated by linear interpolation between the vertices of the cell's boundary. The second is a backward step that uses Fermat's principle and the fact that the ray path is always perpendicular to the wavefront and follows the negative traveltime gradient. In this process, the first-arriving ray path can be traced from the receiver to the source along the negative traveltime gradient, which can be calculated by reconstructing the continuous traveltime field with cubic B-spline interpolation. This new 3-D ray tracing method is compared with the LTI method and the shortest path method (SPM) through a number of numerical experiments. These comparisons show obvious improvements to computed traveltimes and ray paths, both in precision and computational efficiency.     

Mixed multiscale finite elements and streamline methods for reservoir simulation of large geomodels

We present a new approach to reservoir simulation that gives accurate resolution of both large-scale and fine-scale flow patterns. The method uses a mixed multiscale finite-element method (MMsFEM) to solve the pressure equation on a coarse grid and a streamline-based technique to solve the fluid transport on a fine-scale subgrid. The MMsFEM is based on the construction of special approximation velocity spaces that are adaptive to the local properties of the differential operator. As such, MMsFEM produces a detailed subgrid velocity field that reflects the impact of the fine-scale heterogeneous structures. By combining MMsFEM with rapid streamline simulation of the fluid transport, we aim towards a numerical scheme that facilitates routine reservoir simulation of large heterogeneous geomodels without upscaling. The new method is applied to two different test cases. The first test case consists of two (strongly) heterogeneous quarter five-spot problems in 2D. The second test case is a 3D upscaling benchmark taken from the 10th SPE Comparative Solution Project, a project whose purpose is to compare and validate upscaling techniques. The test cases demonstrate that the combination of multiscale methods and streamlines is a robust and viable alternative to traditional upscaling-based reservoir simulation.     

Improved spatial delineation of streambed properties and water fluxes using distributed temperature sensing

A new method was developed for analysing and delineating streambed water fluxes, flow conditions and hydraulic properties using coiled fibre‐optic distributed temperature sensing or closely spaced discrete temperature sensors. This method allows for a thorough treatment of the spatial information embedded in temperature data by creating a matrix visualization of all possible sensor pairs. Application of the method to a 5‐day field dataset reveals the complexity of shallow streambed thermal regimes. To understand how velocity estimates are affected by violations of assumptions of one‐dimensional, saturated, homogeneous flow and to aid in the interpretation of field observations, the method was also applied to temperature data generated by numerical models of common field conditions: horizontal layering, presence of lateral flow and variable streambed saturation. The results show that each condition creates a distinct signature visible in the triangular matrices. The matrices are used to perform a comparison of the behaviour of one‐dimensional analytical heat‐tracing models. The results show that the amplitude ratio‐based method of velocity calculation leads to the most reliable estimates. The minimum sensor spacing required to obtain reliable velocity estimates with discrete sensors is also investigated using field data. The developed method will aid future heat‐tracing studies by providing a technique for visualizing and comparing results from fibre‐optic distributed temperature sensing installations and testing the robustness of analytical heat‐tracing models. Copyright © 2016 John Wiley & Sons, Ltd.     

复杂地表条件下基于线性插值和窄带技术的地震波走时计算

为了在复杂地表条件下实现地震波走时计算,提出了一种基于线性插值和窄带技术的走时计算新方法.其中,线性插值用于局部走时计算,窄带技术用于局部波前捕获和追踪.为了逼近起伏地表,采用三角网和矩形网相结合的方法对速度模型进行剖分.为了得到局部走时计算公式,利用费马（Fermat）原理和关于入射点位置的限定条件.有关编程实践和数值试验表明：新方法不仅可以有效、灵活地处理地表高程的剧烈变化,而且还具有很好的适应性和稳定性,得到的计算结果满足波前传播规律.     

Traveltime computation by wavefront-orientated ray tracing

For multivalued traveltime computation on dense grids, we propose a wavefront‐orientated ray‐tracing (WRT) technique. At the source, we start with a few rays which are propagated stepwise through a smooth two‐dimensional (2D) velocity model. The ray field is examined at wavefronts and a new ray might be inserted between two adjacent rays if one of the following criteria is satisfied: (1) the distance between the two rays is larger than a predefined threshold; (2) the difference in wavefront curvature between the rays is larger than a predefined threshold; (3) the adjacent rays intersect. The last two criteria may lead to oversampling by rays in caustic regions. To avoid this oversampling, we do not insert a ray if the distance between adjacent rays is smaller than a predefined threshold. We insert the new ray by tracing it from the source. This approach leads to an improved accuracy compared with the insertion of a new ray by interpolation, which is the method usually applied in wavefront construction. The traveltimes computed along the rays are used for the estimation of traveltimes on a rectangular grid. This estimation is carried out within a region bounded by adjacent wavefronts and rays. As for the insertion criterion, we consider the wavefront curvature and extrapolate the traveltimes, up to the second order, from the intersection points between rays and wavefronts to a gridpoint. The extrapolated values are weighted with respect to the distances to wavefronts and rays. Because dynamic ray tracing is not applied, we approximate the wavefront curvature at a given point using the slowness vector at this point and an adjacent point on the same wavefront. The efficiency of the WRT technique is strongly dependent on the input parameters which control the wavefront and ray densities. On the basis of traveltimes computed in a smoothed Marmousi model, we analyse these dependences and suggest some rules for a correct choice of input parameters. With suitable input parameters, the WRT technique allows an accurate traveltime computation using a small number of rays and wavefronts.     

Comparison of sparse‐grid geometric and random sampling methods in nonlinear inverse solution uncertainty estimation

A new uncertainty estimation method, which we recently introduced in the literature, allows for the comprehensive search of model posterior space while maintaining a high degree of computational efficiency. The method starts with an optimal solution to an inverse problem, performs a parameter reduction step and then searches the resulting feasible model space using prior parameter bounds and sparse‐grid polynomial interpolation methods. After misfit rejection, the resulting model ensemble represents the equivalent model space and can be used to estimate inverse solution uncertainty. While parameter reduction introduces a posterior bias, it also allows for scaling this method to higher dimensional problems. The use of Smolyak sparse‐grid interpolation also dramatically increases sampling efficiency for large stochastic dimensions. Unlike Bayesian inference, which treats the posterior sampling problem as a random process, this geometric sampling method exploits the structure and smoothness in posterior distributions by solving a polynomial interpolation problem and then resampling from the resulting interpolant. The two questions we address in this paper are 1) whether our results are generally compatible with established Bayesian inference methods and 2) how does our method compare in terms of posterior sampling efficiency. We accomplish this by comparing our method for two electromagnetic problems from the literature with two commonly used Bayesian sampling schemes: Gibbs’ and Metropolis‐Hastings. While both the sparse‐grid and Bayesian samplers produce compatible results, in both examples, the sparse‐grid approach has a much higher sampling efficiency, requiring an order of magnitude fewer samples, suggesting that sparse‐grid methods can significantly improve the tractability of inference solutions for problems in high dimensions or with more costly forward physics.     

Development and evaluation of a local grid refinement method for block-centered finite-difference groundwater models using shared nodes

A new method of local grid refinement for two-dimensional block-centered finite-difference meshes is presented in the context of steady-state groundwater-flow modeling. The method uses an iteration-based feedback with shared nodes to couple two separate grids. The new method is evaluated by comparison with results using a uniform fine mesh, a variably spaced mesh, and a traditional method of local grid refinement without a feedback.Results indicate: (1) The new method exhibits quadratic convergence for homogenous systems and convergence equivalent to uniform-grid refinement for heterogeneous systems. (2) Coupling the coarse grid with the refined grid in a numerically rigorous way allowed for improvement in the coarse-grid results. (3) For heterogeneous systems, commonly used linear interpolation of heads from the large model onto the boundary of the refined model produced heads that are inconsistent with the physics of the flow field. (4) The traditional method works well in situations where the better resolution of the locally refined grid has little influence on the overall flow-system dynamics, but if this is not true, lack of a feedback mechanism produced errors in head up to 3.6% and errors in cell-to-cell flows up to 25%.     

Field measurement of the velocity of overland flow using dye tracing

A new method of computing the mean velocity of overland flow using dye tracing is proposed in which a specified cross-section is divided into zones of relatively uniform flow characteristics, termed partial sections. The mean surface velocity for each partial section is determined by timing the passage of injected dye, and this figure is multiplied by 0.67, 0.70, or 0.80, depending on whether the flow is laminar, transitional, or turbulent, to give mean velocity. The mean velocity for the entire cross-section is calculated by multiplying the mean velocity of each partial section by its cross-sectional area, summing the products for all the partial sections, and dividing by the total area of the cross-section. A field test shows that mean velocity derived in this manner differs from mean velocity derived by the discharge method (i.e. by dividing discharge measured volumetrically by cross-sectional area) by an average of only 7.07 per cent. Thus the partial-section technique appears to provide a reliable method of estimating mean velocity of overland flow.     

Comparison of local grid refinement methods for MODFLOW

Many ground water modeling efforts use a finite-difference method to solve the ground water flow equation, and many of these models require a relatively fine-grid discretization to accurately represent the selected process in limited areas of interest. Use of a fine grid over the entire domain can be computationally prohibitive; using a variably spaced grid can lead to cells with a large aspect ratio and refinement in areas where detail is not needed. One solution is to use local-grid refinement (LGR) whereby the grid is only refined in the area of interest. This work reviews some LGR methods and identifies advantages and drawbacks in test cases using MODFLOW-2000. The first test case is two dimensional and heterogeneous; the second is three dimensional and includes interaction with a meandering river. Results include simulations using a uniform fine grid, a variably spaced grid, a traditional method of LGR without feedback, and a new shared node method with feedback. Discrepancies from the solution obtained with the uniform fine grid are investigated. For the models tested, the traditional one-way coupled approaches produced discrepancies in head up to 6.8% and discrepancies in cell-to-cell fluxes up to 7.1%, while the new method has head and cell-to-cell flux discrepancies of 0.089% and 0.14%, respectively. Additional results highlight the accuracy, flexibility, and CPU time trade-off of these methods and demonstrate how the new method can be successfully implemented to model surface water-ground water interactions.     

地震立体层析成像的实现方法及效果分析

立体层析成像是一种新的地震反射波层析成像方法,能为叠前深度偏移提供较为精确的宏观速度模型。本文研究了立体层析成像的实现方法,包括斜率与走时数据的拾取、离散速度模型构建和初始化、射线参数的确定、斜率和走时及射线计算以及反演问题解法等,建立立体层析成像的算法流程。并通过对Marmousi模型试验,对立体层析成像运行所需的主要参数,如初始速度模型、拾取数据量、离散网格尺寸、速度平滑权重等进行测试和分析,总结这些不同参数对立体层析反演结果的影响规律,用以指导生产实践。     

A Particle Tracking Algorithm for Arbitrary Unstructured Grids

A new semi-analytical algorithm for generating the local groundwater velocity field within irregular three-dimensional grid cells (such as a Voronoi cell) is presented. The method supports particle tracking in the MODFLOW-USG software package for a variety of supported unstructured grid formats. It is shown to replicate the results from the fully analytical Pollock's method when used on a rectilinear grid, and is demonstrated for difficult test cases where there is high spatial variability in the flow field within individual cells. The method is now available as an option in the mod-PATH3DU software package and may be used with most MODFLOW-USG packages without additional intervention.     

青藏高原东部及其边缘地区的地壳上地幔三维速度结构

利用在青藏高原东部及其邻近地区记录到的１万余条近震到时资料,反演该地区的地壳上地幔三维速度结构。采用网格点模型描述三维速度结构,模型维数为２２２２６,网格点间距水平向为１００ｋｍ,垂直向为２０ｋｍ,网格点之间的速度值通过线性插值给出。采用改进了的快速三维射线追踪方法,确定三维非均匀介质中的地震射线路径和理论走时。反演结果显示,青藏高原南部的上地壳中（３０ｋｍ左右的深度）存在一低速区,这和面波反演的结果一致,羌塘块体下地壳有明显的低速异常带,青藏公路沿线的垂直速度剖面显示出岩石层受挤压增厚的构造特征。     

Application of the discontinuous spectral Galerkin method to groundwater flow

The discontinuous spectral Galerkin method uses a finite-element discretization of the groundwater flow domain with basis functions of arbitrary order in each element. The independent choice of the basis functions in each element permits discontinuities in transmissivity in the flow domain. This formulation is shown to be of high order accuracy and particularly suitable for accurately calculating the flow field in porous media. Simulations are presented in terms of streamlines in a bidimensional aquifer, and compared with the solution calculated with a standard finite-element method and a mixed finite-element method. Numerical simulations show that the discontinuous spectral Galerkin approximation is more efficient than the standard finite-element method (in computing fluxes and streamlines/pathlines) for a given accuracy, and it is more accurate on a given grid. On the other hand the mixed finite-element method ensures the continuity of the fluxes at the cell boundaries and it is particular efficient in representing complicated flow fields with few mesh points. Simulations show that the mixed finite-element method is superior to the discontinuous spectral Galerkin method producing accurate streamlines even if few computational nodes are used. The application of the discontinuous Galerkin method is thus of interest in groundwater problems only when high order and extremely accurate solutions are needed.     

Three-dimensional local grid refinement for block-centered finite-difference groundwater models using iteratively coupled shared nodes: a new method of interpolation and analysis of errors

This paper describes work that extends to three dimensions the two-dimensional local-grid refinement method for block-centered finite-difference groundwater models of Mehl and Hill [Development and evaluation of a local grid refinement method for block-centered finite-difference groundwater models using shared nodes. Adv Water Resour 2002;25(5):497–511]. In this approach, the (parent) finite-difference grid is discretized more finely within a (child) sub-region. The grid refinement method sequentially solves each grid and uses specified flux (parent) and specified head (child) boundary conditions to couple the grids. Iteration achieves convergence between heads and fluxes of both grids. Of most concern is how to interpolate heads onto the boundary of the child grid such that the physics of the parent-grid flow is retained in three dimensions. We develop a new two-step, “cage-shell” interpolation method based on the solution of the flow equation on the boundary of the child between nodes shared with the parent grid. Error analysis using a test case indicates that the shared-node local grid refinement method with cage-shell boundary head interpolation is accurate and robust, and the resulting code is used to investigate three-dimensional local grid refinement of stream-aquifer interactions. Results reveal that (1) the parent and child grids interact to shift the true head and flux solution to a different solution where the heads and fluxes of both grids are in equilibrium, (2) the locally refined model provided a solution for both heads and fluxes in the region of the refinement that was more accurate than a model without refinement only if iterations are performed so that both heads and fluxes are in equilibrium, and (3) the accuracy of the coupling is limited by the parent-grid size—a coarse parent grid limits correct representation of the hydraulics in the feedback from the child grid.     

ACOUSTIC MODELLING ON A GRID OF VERTICALLY VARYING SPACING1

The Fourier spectral method and high-order differencing have both been shown to be very accurate in computing spatial derivatives of the acoustic wave equation, requiring only two and three gridpoints per shortest wavelength respectively. In some cases, however, there is a lack of flexibility as both methods use a uniform grid. If these methods are applied to structures with high vertical velocity contrasts, very often most of the model is oversampled. If a complicated interface has to be covered by a fine grid for exact representation, both methods become less attractive as the homogeneous regions are sampled more finely than necessary. In order avoid this limitation we present a differencing scheme in which the grid spacings can be extended or reduced by any integer factor at a given depth. This scheme adds more flexibility and efficiency to the acoustic modelling as the grid spacings can be changed according to the material properties and the model geometry. The time integration is carried out by the rapid expansion method. The spatial derivatives are computed using either the Fourier method or a high-order finite-difference operator in the x-direction and a modified high-order finite-difference operator in the z-direction. This combination leads to a very accurate and efficient modelling scheme. The only additional computation required is the interpolation of the pressure in a strip of the computational mesh where the grid spacing changes.     

基于LTI和网格界面剖分的三维地震射线追踪算法

将二维线性走时插值射线追踪算法(LTI)推广应用至三维模型,并结合网格界面剖分方式,提出了一种三维射线追踪算法．该算法既可获得高精度的全局最小走时和射线路径,又具有快速稳定的特点．三维模型计算结果表明,在模型参数包括网格密度完全相同情况下,本文算法较传统的三维最短路径算法在计算效率、走时和射线的计算精度上均有明显改进．