On different numerical inverse Laplace methods for solute transport problems

Numerical inversion is required when Laplace transform cannot be inverted analytically by manipulating tabled formulas of special cases. However, the numerical inverse Laplace transform is generally an ill-posed problem, and there is no universal method which works well for all problems. In this study, we selected seven commonly used numerical inverse Laplace transform methods to evaluate their performance for dealing with solute transport in the subsurface under uniform or radial flow condition. Such seven methods included the Stehfest, the de Hoog, the Honig–Hirdes, the Talbot, the Weeks, the Simon and the Zakian methods. We specifically investigated the optimal free parameters of each method, including the number of terms used in the summation and the numerical tolerance. This study revealed that some commonly recommended values of the free parameters in previous studies did not work very well, especially for the advection-dominated problems. Instead, we recommended new values of the free parameters for some methods after testing their robustness. For the radial dispersion, the de Hoog, the Talbot, and the Simon methods worked very well, regardless of the dispersion-dominated or advection-dominated situations. The Weeks method can be used to solve the dispersion-dominated problems, but not the advection-dominated problems. The Stehfest, the Honig–Hirdes, and the Zakian methods were recommended for the dispersion-dominated problems. The Zakian method was efficient, while the de Hoog method was time-consuming under radial flow condition. Under the uniform flow condition, all the methods could present somewhat similar results when the free parameters were given proper values for dispersion-dominated problems; while only the Simon method, the Weeks method, and the de Hoog method worked well for advection-dominated problems.     

A meshless method to simulate solute transport in heterogeneous porous media

We derive a meshless numerical method based on smoothed particle hydrodynamics (SPH) for the simulation of conservative solute transport in heterogeneous geological formations. We demonstrate that the new proposed scheme is stable, accurate, and conserves global mass. We evaluate the performance of the proposed method versus other popular numerical methods for the simulation of one- and two-dimensional dispersion and two-dimensional advective–dispersive solute transport in heterogeneous porous media under different Pèclet numbers. The results of those benchmarks demonstrate that the proposed scheme has important advantages over other standard methods because of its natural ability to control numerical dispersion and other numerical artifacts. More importantly, while the numerical dispersion affecting traditional numerical methods creates artificial mixing and dilution, the new scheme provides numerical solutions that are “physically correct”, greatly reducing these artifacts.     

Parallel finite element algorithm with domain decomposition for three-dimensional magnetotelluric modelling

We present a new finite element (FE) method for magnetotelluric modelling of three-dimensional conductivity structures. Maxwell's equations are treated as a system of first-order partial differential equations for the secondary fields. Absorbing boundary conditions are introduced, minimizing undesired boundary effects and allowing the use of small computational domains. The numerical algorithm presented here is an iterative, domain decomposition procedure employing a nonconforming FE space. It does not use global matrices, therefore allowing the modellization of large and complicated structures. The algorithm is naturally parallellizable, and we show results obtained in the IBM SP2 parallel supercomputer at Purdue University. The accuracy of the numerical method is verified by checking the computed solutions with the results of COMMEMI, the international project on the comparison of modelling methods for electromagnetic induction.     

Investigating the uncertainty of satellite altimetry products for hydrodynamic modelling

Satellite altimetry products are increasingly used in many hydraulic applications, and recent studies demonstrate their suitability for the calibration of hydraulic models. The study investigates the effect of satellite‐data uncertainty on the calibration of a quasi‐two‐dimensional (quasi‐2D) model of the middle‐lower portion of the Po river (~140 km). We refer to extended (~16 years of observations) ERS and ENVISAT altimetry products (i.e. River and Lake Hydrology data, RLH) to investigate the effect of (i) record length (i.e. number of satellite measurements at a given satellite track) and (ii) data uncertainty (i.e. altimetry measurements errors) on the calibration of the quasi‐2D model. We first present an assessment of ERS and ENVISAT altimetry errors and then perform the investigations in a Monte Carlo framework by generating datasets of synthetic altimetry products. The results of our analysis further emphasize the suitability of satellite data for the calibration of hydraulic models, providing also a quantitative assessment of the effect of the uncertainty of altimetry products. The analysis highlights the higher accuracy of ENVISAT data, which ensures a stable calibration with ~1.5 years of data (Mean Absolute Error, MAE, lower than 0.4 m, ~0.2 m of which results directly from the uncertainty of ENVISAT data). ERS‐based calibrations become stable with longer series (~3.5–5 years of data), and the negative effect of uncertainty in ERS data is higher (i.e. MAE of 0.6–0.9 m, of which 0.4–0.6 m results from the uncertainty of ERS measurements). Copyright © 2015 John Wiley & Sons, Ltd.     

On the resolving power of the VLF method

In the paper, the mathematical tools, used for the modelling of the electromagnetic field of the harmonic plane wave in the two-dimensional inhomogeneous medium (the case ofE-polarization), are presented. Further, the resolving power of some parameters, that are measured in the VLF and VLF-R methods, is compared in the case of two vertical conductive dykes. To this aim, all the usual parameters of the VLF method and the most important parameter of the VLF-R method () have been chosen. Two groups of models that differ in resistivities have been considered. In each of the groups the distance of the dykes is variable. The parameters |H _y|, which has shown most promising in this respect, is examined also from the point of view of the sensibility to geological noise. The results of the modelling are illustrated by two examples of the results of field measurements.     

An improved interpolation scheme at receiver positions for 2.5D frequency-domain marine controlled-source electromagnetic forward modelling

An improved interpolation scheme is presented for 2.5-dimensional marine controlled-source electromagnetic forward modelling. For the marine controlled-source electromagnetic method, due to the resistivity contrast between the seawater and seafloor sedimentary layers, it is difficult to compute the electromagnetic fields accurately at receivers, which are usually located at the seafloor. In this study, the 2.5-dimensional controlled-source electromagnetic responses are simulated by the staggered finite-difference method. The secondary-field approach is used to avoid the source singularities, and the one-dimensional layered background model is used for calculating the primary fields excited by the source quasi-analytically. The interpolation of electromagnetic fields at the cell nodes for the whole computational domain to the receiver locations is discussed in detail. Numerical tests indicate that the improved interpolation developed is more accurate for simulating the electromagnetic responses at receivers located at the seafloor, compared with the linear or rigorous interpolation.     

Optimal implicit staggered‐grid finite‐difference schemes based on the sampling approximation method for seismic modelling

We propose new implicit staggered‐grid finite‐difference schemes with optimal coefficients based on the sampling approximation method to improve the numerical solution accuracy for seismic modelling. We first derive the optimized implicit staggered‐grid finite‐difference coefficients of arbitrary even‐order accuracy for the first‐order spatial derivatives using the plane‐wave theory and the direct sampling approximation method. Then, the implicit staggered‐grid finite‐difference coefficients based on sampling approximation, which can widen the range of wavenumber with great accuracy, are used to solve the first‐order spatial derivatives. By comparing the numerical dispersion of the implicit staggered‐grid finite‐difference schemes based on sampling approximation, Taylor series expansion, and least squares, we find that the optimal implicit staggered‐grid finite‐difference scheme based on sampling approximation achieves greater precision than that based on Taylor series expansion over a wider range of wavenumbers, although it has similar accuracy to that based on least squares. Finally, we apply the implicit staggered‐grid finite difference based on sampling approximation to numerical modelling. The modelling results demonstrate that the new optimal method can efficiently suppress numerical dispersion and lead to greater accuracy compared with the implicit staggered‐grid finite difference based on Taylor series expansion. In addition, the results also indicate the computational cost of the implicit staggered‐grid finite difference based on sampling approximation is almost the same as the implicit staggered‐grid finite difference based on Taylor series expansion.     

Investigating the use of spatial discretization of hydrological processes in conceptual rainfall runoff modelling: a case study for the meso‐scale

In this study a simple modelling approach was applied to identify the need for spatial complexity in representing hydrological processes and their variability over different scales. A data set of 18 basins was used, ranging between 8 and 4011 km² in area, located in the Nahe basin (Germany), with daily discharge values for over 30 years. Two different parsimoniously structured models were applied in lumped as well as in spatially distributed according to two distribution classifications: (1) a simple classification based on the lithology expressed in three permeability types and (2) a more complex classification based on seven dominating runoff production processes. The objective of the study was to compare the performances of the models on a local and on a regional scale as well as between the models with a view to identifying the accuracy in capturing the spatial variability of the rainfall‐runoff relationships. It was shown that the presence of a specific basin characteristic or process of the distribution classification was not related with higher model performance; only a larger basin size promoted higher model performance. The results of this study also indicated that the permeability generally contained more useful information on the spatial heterogeneity of the hydrological behaviour of the natural system than did a more detailed classification on dominating runoff generation processes. Although model performance was slightly lower for the model that used permeability as a distribution classification, consistency in its parameter values was found, which was lacking with the more complex distribution classification. The latter distribution classification had a higher flexibility to optimize towards the variability of the runoff, which resulted in higher performance, however, process representation was applied inconsistently. Copyright © 2007 John Wiley & Sons, Ltd.     

Salt transport on islands in the Okavango Delta: Numerical investigations

This study uses a numerical model to investigate the groundwater flow and salt transport mechanisms below islands in the Okavango Delta. Continuous evapotranspiration on the islands results in accumulation of solutes and the formation of a saline boundary layer, which may eventually become unstable. A novel Lagrangian method is employed in this study and compared to other numerical methods. The numerical results support the geophysical observations of density fingering on Thata Island. However, the process is slow and it takes some hundreds of years until density fingering is triggered. The results are sensitive to changes of the hydraulic gradient and the evapotranspiration rate. Small changes may lead to different plume developments. Results further demonstrate that density effects may be entirely overridden by lateral flow on islands embedded in a sufficiently high regional hydraulic gradient.     

琼东南盆地块体搬运沉积区多类型水合物赋存特征与数值模拟

南海北部琼东南盆地陵水凹陷的钻探、三维地震与地球化学分析表明, 该区域发育热成因气控制的呈烟囱状反射特征的裂隙充填型和薄砂层的孔隙充填型水合物.不同站位水合物富集程度、厚度与赋存形态不同, 这种复杂水合物富集成藏的控制因素还有待进一步明确.本文利用宽频处理三维地震与随钻测井资料, 精细识别了裂隙充填型水合物发育的空间分布, 分析了断层与块体搬运沉积物对水合物赋存的影响, 通过建立重烃气体影响的Ⅱ型水合物稳定带底界, 并结合数值模拟方法, 定量模拟了多类型水合物形成与富集的控制因素.结果表明: 起源于基底隆起的断层连通了深部气源与浅部水合物稳定带, 控制着深部热成因气向上运移, 发育的多期块体搬运沉积内局部裂隙及下部的有利砂层, 使水合物在垂向上呈多层叠置分布特征.

     

Application of implicit sub-time stepping to simulate flow and transport in fractured porous media

In general, the accuracy of numerical simulations is determined by spatial and temporal discretization levels. In fractured porous media, the time step size is a key factor in controlling the solution accuracy for a given spatial discretization. If the time step size is restricted by the relatively rapid responses in the fracture domain to maintain an acceptable level of accuracy in the entire simulation domain, the matrix tends to be temporally over-discretized. Implicit sub-time stepping applies smaller sub-time steps only to the sub-domain where the accuracy requirements are less tolerant and is most suitable for problems where the response is high in only a small portion of the domain, such as within and near the fractures in fractured porous media. It is demonstrated with illustrative examples that implicit sub-time stepping can significantly improve the simulation efficiency with minimal loss in accuracy when simulating flow and transport in fractured porous media. The methodology is successfully applied to density-dependent flow and transport simulations in a Canadian Shield environment, where the flow and transport is dominated by discrete, highly conductive fracture zones.     

基于保辛算法的声波叠前逆时偏移

叠前逆时偏移是目前成像精度最高的地震偏移方法之一,其实现过程中的一个重要步骤是数值求解全波方程,所以快速有效求解全波方程的数值算法对逆时偏移至关重要. 四阶近似解析辛可分Runge-Kutta (NSPRK) 方法是近年发展的一种具有高效率、高精度的数值求解波动方程的保辛差分方法, 能在粗网格条件下有效压制数值频散, 从而提高计算效率, 节省计算机内存需求量. 本文利用四阶NSPRK方法构造的基本思想,发展了具有六阶空间精度的NSPRK方法,并对新的六阶NSPRK方法进行了详细的稳定性和数值频散分析,以及计算效率比较和波场模拟. 同时将该方法用于声波叠前逆时偏移中, 得到一种时间上保辛、空间具有六阶精度、低数值频散、可应用大步长进行波场延拓并能长时计算的叠前逆时偏移方法,对Sigsbee2B模型进行了偏移成像, 并和四阶NSPRK方法、传统的六阶差分方法、四阶Lax-Wendroff correction (LWC) 方法进行了对比. 数值结果表明, 基于六阶NSPRK方法的叠前逆时偏移能得到更好的成像结果, 是一种优于四阶NSPRK方法、传统的六阶差分方法、四阶LWC叠前逆时偏移的方法, 尤其是在粗网格情况下具有更明显的优越性.     

Three-dimensional finite-element modelling of magnetotelluric data with a divergence correction

A finite-element method for computing the electric field in a 3-D conductivity model of the Earth for plane wave sources, thus enabling magnetotelluric responses to be calculated, is presented. The method incorporates in the iterative solution of the electric-field system of equations the divergence correction technique introduced for finite-difference solutions by Smith (1996). The correction technique accelerates the development of the discontinuity of the normal component of the approximate electric field across conductivity discontinuities. The convergence rate of the iterative solution is improved significantly, especially for low frequencies. The correction technique involves computing the divergence of the current density for the approximate electric field, computing the static potential whose source is this divergence of the current density, and ‘correcting’ the approximate electric field by subtracting from it the gradient of the potential. This is repeated at regular intervals during the iterative solution of the electric-field system of equations. For the method presented here, the Earth model is discretised using a rectilinear mesh comprising uniform cells. Edge-element basis functions are used to approximate the electric field and nodal basis functions are used to approximate the correction potential. The Galerkin method is used to derive the systems of equations for the approximate electric field and correction potential from the respective differential equations. A bi-conjugate gradient solver was found to be adequate for the system of equations for the correction potential; a generalised minimum residual solver was found to be better for the electric-field system of equations. The method is illustrated using the COMMEMI 3D-1A and 3D-2A models.     

Three-dimensional modelling of estuarine turbidity maxima in a tidal estuary

A new numerical model for simulating estuarine dynamics is introduced here. This model, called General Estuarine Transport Model (GETM), has been specifically designed for reproducing baroclinic, bathymetry-guided flows where the tidal range may exceed the mean water depth in large parts of the domain such that drying and flooding processes are relevant. Several physical and numerical features of the model support exact and stable results for such domains. For the physics, high-order turbulence closure schemes guarantee proper reproduction of vertical exchange processes. Among the specific numerical features, generalised vertical coordinates, orthogonal curvilinear horizontal coordinates, high-order TVD advection schemes and stable drying and flooding algorithms have been implemented into GETM. The model is applied here to simulate the dynamics of estuarine turbidity maxima (ETMs), a complex feature present in most tidal estuaries. First, idealised simulations for a two-dimensional domain in the xz space will be shown to reproduce the basic generation mechanisms for ETMs. Then, a realistic three-dimensional simulation of the Elbe estuary in Northern Germany will be carried out. It is demonstrated that for a given forcing situation the model reproduces a stable ETM at the correct location.Responsible Editor: Phil Dyke     

Capturing the spatiotemporal variability of bedload transport: A time‐lapse imagery technique

Monitoring sediment transport in morphologically complex and labile channels remains a difficult task, even at the laboratory scale. To address this challenge, a fully automated imagery technique for continuously mapping the spatial and temporal variability of bedload transport is proposed. This method uses differentiated time‐lapse imagery taken from a fixed camera to detect bed variations induced by grain displacement. The technique is not based on tracking the individual particles; rather, it evaluates macroscopic colour changes within a region that contains several grains, which depend on the occurrence and intensity of the bedload transport. Image‐derived data were compared with the sediment flux measured during four flume experiments, and produced good correspondence. The method provides continuous tracking of the location of the transporting channels, and enables estimation of local variations in the magnitude of the bedload flux. Moreover, the spatial extent of the monitoring area offers an unprecedented opportunity to aggregate spatially dense and continuous data at the reach scale, as needed to properly capture the full range of variability of morphologically complex and rapidly evolving gravel‐bed rivers. Despite being limited to laboratory‐scale physical experiments, the method provides useful data to investigate fundamental morphodynamic processes such as bar migration, bank erosion, anabranches opening/closure, and the associate spatial and temporal scales. Further, the data obtained have the potential to enhance numerical model calibration and improve our understanding of the complex dynamics of real‐world settings. Copyright © 2017 John Wiley & Sons, Ltd.     

基于高铁震源简化桥墩模型激发地震波的数值模拟

高铁列车运行过程中绝大部分时间都是行驶在高架桥上的,高铁列车经过桥梁时,通过与大地耦合的桥墩激发地震波发震过程和平地不完全一样.本文探索高铁列车行驶经过高架桥桥墩,通过桥墩对地下介质激发地震波的机理及过程.为了便于理论分析,文中将高铁列车简化为在高架桥上沿一个方向运动的移动线源,通过每节车厢前后组轮对,对每一个桥墩施加力的作用,而桥墩插入地面几十米深至围岩,与表层土壤和深层围岩双重耦合,由此给出高铁列车通过桥墩激发地震波的震源时间函数.同时,基于广义连续介质力学框架下的修正偶应力理论,推导包含介质特征尺度的弹性波动方程,并应用此弹性波动方程以及构建的高铁震源时间函数,采用优化的交错网格有限差分算法,实现数值模拟,将合成的地震记录与实际地震记录对比分析,其结论将为进一步的基于高铁震源的成像和反演研究提供理论依据.

     

A mixed-dimensional finite volume method for two-phase flow in fractured porous media

We present a vertex-centered finite volume method for the fully coupled, fully implicit discretization of two-phase flow in fractured porous media. Fractures are discretely modeled as lower dimensional elements. The method works on unstructured, locally refined grids and on parallel computers with distributed memory. An implicit time discretization is employed and the nonlinear systems of equations are solved with a parallel Newton-multigrid method. Results from two-dimensional and three-dimensional simulations are presented.