非饱和渗流Richards方程数值求解的欠松弛方法

非饱和土渗流理论是岩土工程问题的基础理论,在土石坝渗流、污染物传输、冻土渗流相变和边坡稳定分析等领域有着广泛的应用。非饱和土渗流Richards方程的数值求解过程中,某些参数如水力传导系数计算不当可能引起非线性方法,如Picard方法或Newton方法的迭代收敛震荡,从而导致非线性迭代方法收敛缓慢和精度降低。为了消除或降低迭代收敛震荡对求解精度和计算性能的影响,目前主要采用欠松弛方法。通过一维入渗算例和二维非均质土坝渗流算例演示已有欠松弛方法的局限性,进而提出新的短项混合欠松弛法,并对其实用性和可靠性进行验证。     

Theoretical study and simulation of moisture transport in unsaturated porous media by periodic homogenization

In this work, the macroscopic Richards equation for moisture transport is established in unsaturated porous media using periodic homogenization. By performing dimensional analysis on microscopic equations of moisture transfer, dimensional numbers characterizing moisture transport appear. The application of the asymptotic homogenization leads to the classical Richards equation, which is justified rigorously this way. Moreover, we obtain an accurate definition of the homogenized diffusion tensor of moisture involving the geometric properties of the microstructure and known transport properties of the material. A different behavior for the transport of water vapor between hygroscopic and super‐hygroscopic region is revealed. Finally, a simple 2D example where an analytical solution exists is addressed. Copyright © 2014 John Wiley & Sons, Ltd.     

加速型改进迭代法在非饱和土渗流中的应用研究

Richards方程常用于非饱和土渗流问题,并且应用广泛。在数值求解中,对Richards方程线性化,进而采用有限差分法进行数值离散以及迭代计算。其中传统的迭代法比如Jacobi迭代、Gauss-Seidel迭代法（GS）和连续超松驰迭代法（successive over-relaxation method,简称SOR）迭代收敛率较慢,尤其在离散空间步长较小以及离散时间步长较大时。因此,采用整体校正法以及多步预处理法对传统迭代法进行改进,提出一种基于整体校正法的多步预处理Gauss-Seidel迭代法（improved Gauss-Seidel iterative method with multistep preconditioner based on the integral correction method,简称ICMP(m)-GS）求解Richards方程导出的线性方程组。通过非饱和渗流算例,并与传统迭代法和解析解对比,对改进算法的收敛率和加速效果进行了验证。结果表明,提出的ICMP(m)-GS可以很大程度地改善线性方程组的病态性,相较于常规方法GS,SOR以及单一改进方法,ICMP(m)-GS具有更快的收敛率,更高的计算效率和计算精度。该方法可以为非饱和土渗流的数值模拟提供一定参考。     

Coupling water flow and solute transport into a physically-based surface-subsurface hydrological model

A distributed-parameter physically-based solute transport model using a novel approach to describe surface-subsurface interactions is coupled to an existing flow model. In the integrated model the same surface routing and mass transport equations are used for both hillslope and channel processes, but with different parametrizations for these two cases. For the subsurface an advanced time-splitting procedure is used to solve the advection-dispersion equation for transport and a standard finite element scheme is used to solve Richards equation for flow. The surface-subsurface interactions are resolved using a mass balance-based surface boundary condition switching algorithm that partitions water and solute into actual fluxes across the land surface and changes in water and mass storage. The time stepping strategy allows the different time scales that characterize surface and subsurface water and solute dynamics to be efficiently and accurately captured. The model features and performance are demonstrated in a series of numerical experiments of hillslope drainage and runoff generation.     

Numerical simulation of Richards equation in partially saturated porous media: under-relaxation and mass balance

Numerical simulation of Richards equation in unsaturated soil is known to be difficult because of the highly non-linear material properties involved. An extensive study was conducted to clarify the role of mass balance and under-relaxation on the rate in which the correct solution is approached. Both finite element and finite difference techniques were considered. For the former, the h-based formulation was compared with a mass-conservative mixed form. The conductivity function (K) was under-relaxed in two ways while the capacity function (C) is computed following the standard mass or non-mass conservative schemes recommended in literature. For fairly coarse discretisation, it was found that large errors were produced when K was under-relaxed by evaluating it at the average of heads from current and previous time step (UR1), regardless of the numerical scheme used. Maintaining global mass balance is found to have little impact on the accuracy. All numerical schemes that under-relaxed K by computing it at the average of two most recent iterations in the current time step (UR2) converged quicker to the correct solution with increasing discretisation, although more iterations per time step than UR1 is needed to achieve a stable solution. An important practical ramification is that it appears to be possible to achieve reasonably accurate and oscillation-free results using fairly coarse discretisation by making only minor modifications (namely, using UR2 for conductivity function) to the h-based finite element formulation and applying some minimum time step criteria.     

Multi‐scale assimilation of root zone soil water predictions

When hydrology model parameters are determined, a traditional data assimilation method (such as Kalman filter) and a hydrology model can estimate the root zone soil water with uncertain state variables (such as initial soil water content). The simulated result can be quite good. However, when a key soil hydraulic property, such as the saturated hydraulic conductivity, is overestimated or underestimated, the traditional soil water assimilation process will produce a persistent bias in its predictions. In this paper, we present and demonstrate a new multi‐scale assimilation method by combining the direct insertion assimilation method, particle swarm optimisation (PSO) algorithm and Richards equation. We study the possibility of estimating root zone soil water with a multi‐scale assimilation method by using observed in situ data from the Wudaogou experiment station, Huaihe River Basin, China. The results indicate there is a persistent bias between simulated and observed values when the direct insertion assimilation surface soil water content is used to estimate root zone soil water contents. Using a multi‐scale assimilation method (PSO algorithm and direct insertion assimilation) and an assumed bottom boundary condition, the results show some obvious improvement, but the root mean square error is still relatively large. When the bottom boundary condition is similar to the actual situation, the multi‐scale assimilation method can well represent the root zone soil water content. The results indicate that the method is useful in estimating root zone soil water when available soil water data are limited to the surface layer and the initial soil water content even when the soil hydraulic conductivities are uncertain. Copyright © 2011 John Wiley & Sons, Ltd.     

采动区地表动态沉降预测的Richards模型

针对Knothe时间函数在采动区地表动态沉降预测中存在着较大偏差的问题,提出了地表动态沉降预测的Richards模型。基于开采过程中的地表沉降变形规律,给出了能够反映地表沉降全过程的理想时间函数模型所具有的基本特征,引入Richards生长曲线模型,分析了模型参数与地质采矿条件之间的定性、定量关系,并探讨了模型的适用范围,指出地表发生连续的、渐变的沉降变形时,模型预测效果较好。实例分析表明,Richards模型符合理想时间函数模型的要求,可较好地模拟采动区地表动态沉降过程,能够计算出地表移动持续时间和地表点在某一时刻的下沉值、下沉速度、加速度等动态参数,且模型的可塑性较强,具有较广泛的适用性     

Green-Ampt模型渗透系数取值方法研究

格林-安姆普特（Green-Ampt）模型原理简单、使用方便,在浅层滑坡的降雨入渗分析中有很大的应用潜力。渗透系数是Green-Ampt模型中的一个主要参数,该渗透系数并不一定等于土体的饱和渗透系数。通过与理查茲（Richards）方程求解进行比较,研究了Green-Ampt模型中渗透系数的取值方法。研究发现,为获得与Richards方程相同的入渗量计算结果,需对饱和渗透系数进行修正,该修正系数的大小与入渗深度和土体类型有关。对于文中研究的土体,当Green-Ampt模型中渗透系数取为0.7倍饱和渗透系数时,由Green-Ampt模型计算的孔隙水压力分布与Richards方程计算结果最为接近,建议Green-Ampt模型中渗透系数修正系数取0.7。     

Solution of the unsaturated soil moisture equation using repeated transforms

An alternative method of solution for the linearized ‘theta‐based’ form of the Richards equation of unsaturated flow is developed in two spatial dimensions. The Laplace and Fourier transformations are employed to reduce the Richards equation to an ordinary differential equation in terms of a transformed moisture content and the transform variables, s and ξ. Separate analytic solutions to the transformed equation are developed for initial states which are either in equilibrium or dis‐equilibrium. The solutions are assembled into a finite layer formulation satisfying continuity of soil suction, thereby facilitating the analysis of horizontally stratified soil profiles. Solution techniques are outlined for various boundary conditions including prescribed constant moisture content, prescribed constant flux and flux as a function of moisture change. Example solutions are compared with linearized finite element solutions. The agreement is found to be good. An adaptation of the method for treating the quasilinearized Richards equation with variable diffusivity is also described. Comparisons of quasilinear solutions with some earlier semi‐analytical, finite element and finite difference results are also favourable. Copyright © 2001 John Wiley & Sons, Ltd.     

Isogeometric analysis for unsaturated flow problems

Unsaturated flow problems in porous media often described by Richards’ equation are of great importance in many engineering applications. In this contribution, we propose a new numerical flow approach based on isogeometric analysis (IGA) for modeling the unsaturated flow problems. The non-uniform rational B-spline (NURBS) basis is utilized for spatial discretization whereas the stable implicit backward Euler method for time discretization. The nonlinear Richards’ equation is iteratively solved with the aid of the Newton–Raphson scheme. Owing to some desirable features of an efficient numerical flow approach, major advantages of the present formulation involve: (a) numerical oscillation at the wetting front can be avoided or facilitated, simply by using either an h-refinement or a lumped mass matrix technique; (b) higher-order exactness can be obtained due to the nature of the IGA features; (c) the approach is straightforward to implement and it does not need any transformation, e.g., Kirchhoff transformation or filter algorithm; and (d) in contrast to the Picard iteration scheme, which forms linear convergences, the proposed approach can however yield quadratic convergences by using the Newton–Raphson method for solving resultant nonlinear equations. Numerical model validation is analyzed by solving a three-dimensional unsaturated flow problem in soil, and its derived results are verified against analytical solutions. Numerical applications are then studied by considering three extensive examples with simple and complex configurations to further show the accuracy and applicability of the present IGA.