Homotopy perturbation method for solving the space–time fractional advection–dispersion equation

In this paper we present a reliable algorithm, the homotopy perturbation method, to construct numerical solutions of the space–time fractional advection–dispersion equation in the form of a rapidly convergent series with easily computable components. Fractional advection–dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium. The fractional derivatives are described in the Caputo sense. Some examples are given. Numerical results show that the homotopy perturbation method is easy to implement and accurate when applied to space–time fractional advection–dispersion equations.     

The impact of boundary on the fractional advection–dispersion equation for solute transport in soil: Defining the fractional dispersive flux with the Caputo derivatives

The inherent heterogeneity of geological media often results in anomalous dispersion for solute transport through them, and how to model it has been an interest over the past few decades. One promising approach that has been increasingly used to simulate the anomalous transport in surface and subsurface water is the fractional advection–dispersion equation (FADE), derived as a special case of the more general continuous time random walk or the stochastic continuum model. In FADE, the dispersion is not local and the solutes have appreciable probability to move long distances, and thus reach the boundary faster than predicted by the classical advection–dispersion equation (ADE). How to deal with different boundaries associated with FADE and their consequent impact is an issue that has not been thoroughly explored. In this paper we address this by taking one-dimensional solute movement in soil columns as an example. We show that the commonly used FADE with its fractional derivatives defined by the Riemann–Liouville definition is problematic and could result in unphysical results for solute transport in bounded domains; a modified method with the fractional dispersive flux defined by the Caputo derivatives is presented to overcome this problem. A finite volume approach is given to numerically solve the modified FADE and its associated boundaries. With the numerical model, we analyse the inlet-boundary treatment in displacement experiments in soil columns, and find that, as in ADE, treating the inlet as a prescribed concentration boundary gives rise to mass-balance errors and such errors could be more significant in FADE because of its non-local dispersion. We also discuss a less-documented but important issue in hydrology: how to treat the upstream boundary in analysing the lateral movement of tracer in an aquifer when the tracer is injected as a pulse. It is shown that the use of an infinite domain, as commonly assumed in literature, leads to unphysical backward dispersion, which has a significant impact on data interpretation. To avoid this, the upstream boundary should be flux-prescribed and located at the upstream edge of the injecting point. We apply the model to simulate the movement of Cl⁻ in a tracer experiment conducted in a saturated hillslope, and analyse in details the significance of upstream-boundary treatments in parameter estimation.     

On a fractional derivative form of the Green-Ampt infiltration model

The Green-Ampt model for infiltration into homogeneous soils predicts a monotonically decreasing infiltration rate and a wetting front that initially advances as the square root of time. Infiltration in heterogeneous soils, however, can exhibit non-monotonic infiltration rates and wetting front advances that differ from the square root of time (“anomalous diffusion”).Here it is postulated, that if the length scales of the heterogeneities can be assumed to be power law distributed, it may be appropriate to model infiltration in heterogeneous soils in terms of fractional derivatives. Then, by expressing the hydraulic flux as a Caputo fractional derivative (order 0 < α ? 1) of the head, a fractional Green-Ampt infiltration model is obtained. It is shown that solutions of this model predict non-monotonic and anomalous diffusion behaviors consistent with observations in field infiltration trials; a finding that indicates that a non-local moisture flux model, based on fractional derivatives, is a plausible model for describing infiltration into heterogeneous soils.     

Estimating longitudinal dispersion in rivers using Acoustic Doppler Current Profilers

The longitudinal dispersion coefficient (D) is an important parameter needed to describe the transport of solutes in rivers and streams. The dispersion coefficient is generally estimated from tracer studies but the method can be expensive and time consuming, especially for large rivers. A number of empirical relations are available to estimate the dispersion coefficient; however, these relations are known to produce estimates within an order of magnitude of the tracer value. The focus of this paper is on using the shear-flow dispersion theory to directly estimate the dispersion coefficient from velocity measurements obtained using an Acoustic Doppler Current Profiler (ADCP). Using tracer and hydrodynamic data collected within the same river reaches, we examined conditions under which the ADCP and tracer methods produced similar results. Since dead zones / transient storage (TS) are known to influence the dispersion coefficient, we assessed the relative importance of dead zones in different stream reaches using two tracer-based approaches: (1) TS modeling which explicitly accounts for dead zones and (2) the advection–dispersion equation (ADE) which does not have separate terms for dead zones. Dispersion coefficients based on the ADE tend to be relatively high as they describe some of the effects of dead zones as well. Results based on the ADCP method were found to be in good agreement with the ADE estimates indicating that storage zones play an important role in the estimated dispersion coefficients, especially at high flows. For the river sites examined in this paper, the tracer estimates of dispersion were close to the median values of the ADCP estimates obtained from multiple datasets within a reach. The ADCP method appears to be an excellent alternative to the traditional tracer-based method if care is taken to avoid spurious data and multiple datasets are used to compute a distance-weighted average or other appropriate measure that represents reach-averaged conditions.     

A two-sided fractional conservation of mass equation

A two-sided fractional conservation of mass equation is derived by using left and right fractional Mean Value Theorems. This equation extends the one-sided fractional conservation of mass equation of Wheatcraft and Meerschaert. Also, a two-sided fractional advection-dispersion equation is derived. The derivations are based on Caputo fractional derivatives.     

一种全局优化的隐式交错网格有限差分算法及其在弹性波数值模拟中的应用

在数值模拟中,隐式有限差分具有较高的精度和稳定性.然而,传统隐式有限差分算法大多由于需要求解大型矩阵方程而存在计算效率偏低的局限性.本文针对一阶速度-应力弹性波方程,构建了一种优化隐式交错网格有限差分格式,然后将改进格式由时间-空间域转换为时间-波数域,利用二范数原理建立目标函数,再利用模拟退火法求取优化系数.通过对均匀模型以及复杂介质模型进行一阶速度-应力弹性波方程数值模拟所得单炮记录、波场快照分析表明:这种优化隐式交错网格差分算法与传统的几种显式和隐式交错网格有限差分算法相比不但降低了计算量,而且能有效的压制网格频散,使弹性波数值模拟的精度得到有效的提高.     

Time–space fractional governing equations of transient groundwater flow in confined aquifers: Numerical investigation

In order to model non‐Fickian transport behaviour in groundwater aquifers, various forms of the time–space fractional advection–dispersion equation have been developed and used by several researchers in the last decade. The solute transport in groundwater aquifers in fractional time–space takes place by means of an underlying groundwater flow field. However, the governing equations for such groundwater flow in fractional time–space are yet to be developed in a comprehensive framework. In this study, a finite difference numerical scheme based on Caputo fractional derivative is proposed to investigate the properties of a newly developed time–space fractional governing equations of transient groundwater flow in confined aquifers in terms of the time–space fractional mass conservation equation and the time–space fractional water flux equation. Here, we apply these time–space fractional governing equations numerically to transient groundwater flow in a confined aquifer for different boundary conditions to explore their behaviour in modelling groundwater flow in fractional time–space. The numerical results demonstrate that the proposed time–space fractional governing equation for groundwater flow in confined aquifers may provide a new perspective on modelling groundwater flow and on interpreting the dynamics of groundwater level fluctuations. Additionally, the numerical results may imply that the newly derived fractional groundwater governing equation may help explain the observed heavy‐tailed solute transport behaviour in groundwater flow by incorporating nonlocal or long‐range dependence of the underlying groundwater flow field.     

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.     

Variable spatial and temporal weighting schemes for use in multi-phase compositional problems

Variable spatial and temporal weighting of the advective contaminant mole fraction term is explored as a means of reducing numerical dispersion of contaminant plumes in a multi-phase compositional simulator. The spatial schemes considered are upstream, central and a non-linear flux limiter, while fully-implicit and Crank-Nicolson time weighting are examined. The performance of each weighting scheme, in terms of stability of the Newton iteration and computational cost, is assessed for simplified problems designed to be representative of various aspects of more complex subsurface remediation problems. Results indicate that for problems with an homogeneous permeability field, the non-linear flux limited along with fully-implicit weighting gives superior performance to any other combination of spatial and temporal weighting schemes. For heterogeneous permeability fields, the macrodispersion imparted by heterogeneity dominates numerical dispersion so that smearing of contaminant mole fraction fronts does not appear to be a serious problem.     

大步长波场深度延拓的理论

波场延拓是地震偏移成像的基础. 快速进行目标区波场延拓对石油勘探中急需发展的深部地震勘探和无组合海量地震数据的成像有重要意义. 在目标区成像中,目前已有的波场延拓方法,包括基于走时计算的Dix方法和射线追踪方法,以及基于小步长波场递推的方法,在适应复杂介质、计算精度和计算效率的某一方面还不能完全满足实际需要. 本文提出一种基于“算子相位”李代数积分的快速计算延拓算子的方法,称为大步长波场延拓方法. 在该方法中,指向目标区的波场延拓算子象征的复相位被表示成波数的线性组合. 线性组合的系数是层速度函数及其导数的深度积分,计算和存储较为方便. 波场延拓算子通过相移算子加校正的方法,利用快速Fourier变换在空间域和波数域予以实现. 利用动力学等价关系导出了便于计算的表达式. 本文比较了算子主象征函数用一步法展开和用两步法展开的精度,从而说明大步长方法的精度要高于递推方法. 在横向和纵向线性变化介质中,将大步长方法的脉冲响应与递推法做了比较,说明大步长延拓算子的走时精度主要取决于相移因子中的横向变速校正项;且在各种近似下,大步长算子发生的频散都非常小.     

Partial derivatives of dispersion curves for higher modes of love waves in a single-layered medium

Summary The partial derivatives of the dispersion curves have been calculated by means of a method founded on the theorem of implicit functions, which was described in[1]. The partial derivatives of the dispersion curves of Love waves are studied in detail for a model of a single-layered medium. Analytical formulae for the partial derivatives are used for deriving the limits of these partial derivatives when the period approaches zero or the critical period.     

Hybrid upwind discretization of nonlinear two-phase flow with gravity

Multiphase flow in porous media is described by coupled nonlinear mass conservation laws. For immiscible Darcy flow of multiple fluid phases, whereby capillary effects are negligible, the transport equations in the presence of viscous and buoyancy forces are highly nonlinear and hyperbolic. Numerical simulation of multiphase flow processes in heterogeneous formations requires the development of discretization and solution schemes that are able to handle the complex nonlinear dynamics, especially of the saturation evolution, in a reliable and computationally efficient manner. In reservoir simulation practice, single-point upwinding of the flux across an interface between two control volumes (cells) is performed for each fluid phase, whereby the upstream direction is based on the gradient of the phase-potential (pressure plus gravity head). This upwinding scheme, which we refer to as Phase-Potential Upwinding (PPU), is combined with implicit (backward-Euler) time discretization to obtain a Fully Implicit Method (FIM). Even though FIM suffers from numerical dispersion effects, it is widely used in practice. This is because of its unconditional stability and because it yields conservative, monotone numerical solutions. However, FIM is not unconditionally convergent. The convergence difficulties are particularly pronounced when the different immiscible fluid phases switch between co-current and counter-current states as a function of time, or (Newton) iteration. Whether the multiphase flow across an interface (between two control-volumes) is co-current, or counter-current, depends on the local balance between the viscous and buoyancy forces, and how the balance evolves in time. The sensitivity of PPU to small changes in the (local) pressure distribution exacerbates the problem. The common strategy to deal with these difficulties is to cut the timestep and try again. Here, we propose a Hybrid-Upwinding (HU) scheme for the phase fluxes, then HU is combined with implicit time discretization to yield a fully implicit method. In the HU scheme, the phase flux is divided into two parts based on the driving force. The viscous-driven and buoyancy-driven phase fluxes are upwinded differently. Specifically, the viscous flux, which is always co-current, is upwinded based on the direction of the total-velocity. The buoyancy-driven flux across an interface is always counter-current and is upwinded such that the heavier fluid goes downward and the lighter fluid goes upward. We analyze the properties of the Implicit Hybrid Upwinding (IHU) scheme. It is shown that IHU is locally conservative and produces monotone, physically-consistent numerical solutions. The IHU solutions show numerical diffusion levels that are slightly higher than those for standard FIM (i.e., implicit PPU). The primary advantage of the IHU scheme is that the numerical overall-flux of a fluid phase remains continuous and differentiable as the flow regime changes between co-current and counter-current conditions. This is in contrast to the standard phase-potential upwinding scheme, in which the overall fractional-flow (flux) function is non-differentiable across the boundary between co-current and counter-current flows.     

A comparative modeling study of a dual tracer experiment in a large lysimeter under atmospheric conditions

In this paper, five model approaches with different physical and mathematical concepts varying in their model complexity and requirements were applied to identify the transport processes in the unsaturated zone. The applicability of these model approaches were compared and evaluated investigating two tracer breakthrough curves (bromide, deuterium) in a cropped, free-draining lysimeter experiment under natural atmospheric boundary conditions. The data set consisted of time series of water balance, depth resolved water contents, pressure heads and resident concentrations measured during 800 days. The tracer transport parameters were determined using a simple stochastic (stream tube model), three lumped parameter (constant water content model, multi-flow dispersion model, variable flow dispersion model) and a transient model approach. All of them were able to fit the tracer breakthrough curves. The identified transport parameters of each model approach were compared. Despite the differing physical and mathematical concepts the resulting parameters (mean water contents, mean water flux, dispersivities) of the five model approaches were all in the same range. The results indicate that the flow processes are also describable assuming steady state conditions. Homogeneous matrix flow is dominant and a small pore volume with enhanced flow velocities near saturation was identified with variable saturation flow and transport approach. The multi-flow dispersion model also identified preferential flow and additionally suggested a third less mobile flow component. Due to high fitting accuracy and parameter similarity all model approaches indicated reliable results.     

Response of a non-linear system with restoring forces governed by fractional derivatives—Time domain simulation and statistical linearization solution

In this paper, the random response of a non-linear system comprising frequency dependent restoring force terms is examined. These terms are accurately modeled in seismic isolation and in many other applications using fractional derivatives. In this context, an efficient numerical approach for determining the time domain response of the system to an arbitrary excitation is first proposed. This approach is based on the Grunwald–Letnikov representation of a fractional derivative and on the well-known Newmark numerical integration scheme for structural dynamic problems. Next, it is shown that for the case of a stochastic excitation, in addition to the time domain solutions, a frequency domain solution can be readily determined by the method of statistical linearization. The reliability of this solution is established in a Monte Carlo simulation context using the herein adopted time domain solution scheme. Furthermore, several related parameter studies are reported.     

Dispersion-relationship-preserving seismic modelling using the cross-rhombus stencil with the finite-difference coefficients solved by an over-determined linear system

Finite-difference modeling with a cross-rhombus stencil with high-order accuracy in both spatial and temporal derivatives is a potential method for efficient seismic simulation. The finite-difference coefficients determined by Taylor-series expansion usually preserve the dispersion property in a limited wavenumber range and fixed angles of propagation. To construct the dispersion-relationship-preserving scheme for satisfying high-wavenumber components and multiple angles, we expand the dispersion relation of the cross-rhombus stencil to an over-determined system and apply a regularization method to obtain the stable least-squares solution of the finite-difference coefficients. The new dispersion-relationship-preserving based scheme not only satisfies several designated wavenumbers but also has high-order accuracy in temporal discretization. The numerical analysis demonstrates that the new scheme possesses a better dispersion characteristic and more relaxed stability conditions compared with the Taylor-series expansion based methods. Seismic wave simulations for the homogeneous model and the Sigsbee model demonstrate that the new scheme yields small dispersion error and improves the accuracy of the forward modelling.     

一种时空域优化的高精度交错网格差分算子与正演模拟

如何有效压制数值频散是有限差分正演模拟研究中的关键问题之一.近年来,许多学者对二阶声波方程的差分算子开展了大量的优化工作,在压制频散方面取得不错的效果.一阶压强-速度方程广泛用于研究地震波在地下变密度模型中传播规律,目前针对一阶方程的优化工作大多只是在空间差分算子上展开.本文在前人研究的基础上,推导出一阶声波方程中压强场与偏振速度场之间的解析关系,据此在传统交错网格基础上给出一种高精度的显式时间递推格式,该递推格式将时间差分与空间差分算子结合在一起,并采用共轭梯度法得到精确时间递推匹配系数,实现时空差分算子的同时优化.在编程实现算法的基础上,通过频散分析与三个典型模型测试表明:本文方法能够较为有效地压制时间频散与空间频散,提高数值计算精度;同时对复杂模型也有很好适用性.     

Solute transport routing in a small stream

Abstract

The impact of pollution incidents on rivers and streams may be predicted using mathematical models of solute transport. Practical applications require an analytical or numerical solution to a governing solute mass balance equation together with appropriate values of relevant transport coefficients under the flow conditions of interest. This paper considers two such models, namely those proposed by Fischer and by Singh and Beck, and compares their performances using tracer data from a small stream in Edinburgh, UK. In calibrating the models, information on the magnitudes and the flow rate dependencies of the velocity and the dispersion coefficients was generated. The dispersion coefficient in the stream ranged between 0.1 and 0.9 m²/s for a flow rate range of 13–437 L/s. During calibration it was found that the Singh and Beck model fitted the tracer data a little better than the Fischer model in the majority of cases. In a validation exercise, however, both models gave similarly good predictions of solute transport at three different flow rates.     

Using high resolution tracer data to constrain water storage,flux and age estimates in a spatially distributed rainfall‐runoff model

Models simulating stream flow and conservative tracers can provide a representation of flow paths, storage distributions and mixing processes that is advantageous for many predictive purposes. Compared with models that only simulate stream flow, tracer data can be used to investigate the internal consistency of model behaviour and to gain insight into model performance. Here, we examine the strengths and weaknesses of a data‐driven, spatially distributed tracer‐aided rainfall‐runoff model. The model structure allowed us to assess the influence of landscape characteristics on the routing and mixing of water and tracers. The model was applied to a site in the Scottish Highlands with a unique tracer data set; ~4 years of daily isotope ratios in stream water and precipitation were available, as well as 2 years of weekly soil and ground water isotopes. The model structure was based on an empirically based, lumped tracer‐aided model previously developed for the catchment. The best model runs were selected from Monte Carlo simulations based on dual calibration criteria using objective functions for both stream isotopes and discharge at the outlet. Model performance for these criteria was reasonable (Nash–Sutcliffe efficiencies for discharge and isotope ratios were ~0.4–0.6). The model could generally reproduce the variable isotope signals in the soils of the steeper hill slopes where storage was low, and damped isotope responses in valley bottom cells with high storage. The model also allowed us to estimate the age distributions of internal stores, water fluxes and stream flow. Average stream water age was ~1.6 years, integrating older groundwater in the valley bottom and dynamic younger soil waters. By tracking water ages and simulating isotopes, the model captured the changes in connectivity driven by distributed storage dynamics. This has substantially improved the representation of spatio‐temporal process dynamics and gives a more robust framework for projecting environmental change impacts. Copyright © 2016 The Authors Hydrological Processes Published by John Wiley & Sons Ltd.     

Numerical modelling of suspended sediments

The turbulent advection-diffusion mathematical model in three-dimensional space is solved by a mixed finite element finite difference method. Linear finite elements in the vertical direction and central finite differences in the horizontal directions are used coupled with the Galerkin error minimization procedure. The integration in time is performed in fractional steps (one explicit one implicit) by splitting the differential operator. The method is illustrated by application to the three-dimensional movement of suspended sediment. Its accuracy is checked by comparison to analytical solutions and its efficiency is gauged relative to finite elements and implicit finite difference solutions for two-dimensional suspended sediment transport over a dredged channel.