共查询到20条相似文献，搜索用时 15 毫秒
1.
This methods note examines the use of adaptive underrelaxation of Picard iterations to accelerate the solution convergence for nonlinear ground water flow problems. Ground water problems are nonlinear when drains, phreatophytes, stream aquifer, and similar features are simulated. Typically, simple Picard iterations are used to address such nonlinear problems. Nevertheless, the convergence rate can be slow, or convergence cannot be obtained. However, convergence often can be accelerated using Picard iterations with adaptive underrelaxation, and convergence often can be obtained where it otherwise would not occur. 相似文献
2.
Xiaoxian Zhang A. Glyn Bengough John W. Crawford Iain M. Young 《Journal of Hydrology》2002,260(1-4):75-87
The non-linear solvers in numerical solutions of water flow in variably saturated soils are prone to convergence difficulties. Many aspects can give rise to such difficulties and in this paper we address the gravity term and the prescribed-flux boundary in the Picard iteration. The problem of the gravity term in the Picard iteration is iteration-to-iteration oscillation as the gravity term is treated, by analogy with the time-step advance technique, ‘explicitly’ in the iteration. The proposed method for the gravity term is an improvement of the ‘implicit’ approach of Zhang and Ewen [Water Resour. Res. 36 (2000) 2777] by extending it to heterogeneous soil and approximating the inter-nodal hydraulic conductivity in the diffusive term and the gravity term with the same scheme. The prescribed-flux boundary in traditional methods also gives rise to iteration-to-iteration oscillation because there is no feedback to the flux in the solution at the new iteration. To reduce such oscillation, a new method is proposed to provide such a feedback to the flux. Comparison with traditional Picard and Newton iteration methods for a wide range of problems show that a combination of these two proposed methods greatly improves the stability and consequently the computational efficiency, making the use of small time step and/or under-relaxation solely for convergence unnecessary. 相似文献
3.
《Advances in water resources》1998,21(3):237-249
The transport and fate of reactive chemicals in groundwater is governed by equations which are often difficult to solve due to the nonlinear relationship between the solute concentrations for the liquid and solid phases. The nonlinearity may cause mass balance errors during the numerical simulation in addition to numerical errors for linear transport system. We have generalized the modified Picard iteration algorithm of Celia et al.5 for unsaturated flow to solve the nonlinear transport equation. Written in a ‘mixed-form’ formulation, the total solute concentration is expanded in a Taylor series with respect to the solution concentration to linearize the transport equation, which is then solved with a conventional finite element method. Numerical results of this mixed-form algorithm are compared with those obtained with the concentration-based scheme using conventional Picard iteration. In general, the new solver resulted in negligible mass balance errors (< ∥10−8∥%) and required less computational time than the conventional iteration scheme for the test examples, including transport involving highly nonlinear adsorption under steady-state as well as transient flow conditions. In contrast, mass balance errors resulting from the conventional Picard iteration method were higher than 10% for some highly nonlinear problems. Application of the modified Picard iteration scheme to solve the nonlinear transport equation may greatly reduce the mass balance errors and increase computational efficiency. 相似文献
4.
We develop a new approach for solving the nonlinear Richards’ equation arising in variably saturated flow modeling. The growing complexity of geometric models for simulation of subsurface flows leads to the necessity of using unstructured meshes and advanced discretization methods. Typically, a numerical solution is obtained by first discretizing PDEs and then solving the resulting system of nonlinear discrete equations with a Newton-Raphson-type method. Efficiency and robustness of the existing solvers rely on many factors, including an empiric quality control of intermediate iterates, complexity of the employed discretization method and a customized preconditioner. We propose and analyze a new preconditioning strategy that is based on a stable discretization of the continuum Jacobian. We will show with numerical experiments for challenging problems in subsurface hydrology that this new preconditioner improves convergence of the existing Jacobian-free solvers 3-20 times. We also show that the Picard method with this preconditioner becomes a more efficient nonlinear solver than a few widely used Jacobian-free solvers. 相似文献
5.
An exact, closed-form analytical solution is developed for calculating ground water transit times within Dupuit-type flow systems. The solution applies to steady-state, saturated flow through an unconfined, horizontal aquifer recharged by surface infiltration and discharging to a downgradient fixed-head boundary. The upgradient boundary can represent, using the same equation, a no-flow boundary or a fixed head. The approach is unique for calculating travel times because it makes no a priori assumptions regarding the limit of the water table rise with respect to the minimum saturated aquifer thickness. The computed travel times are verified against a numerical model, and examples are provided, which show that the predicted travel times can be on the order of nine times longer relative to existing analytical solutions. 相似文献
6.
《Advances in water resources》2005,28(10):1091-1101
Certain nonlinear diffusion equations of degenerate parabolic type display a finite speed of propagation of disturbances. This mathematical behavior can be used to describe a wide range of nonlinear phenomena such as the penetration distance of a thermal layer, the boundary of a reaction zone, or a wetting front in unsaturated soil moisture flow. However, there are two main difficulties in obtaining solutions to problems of this class. One is that the location of the interface is not known a priori and must be discovered during the analysis. The other is the fact that the differential equation is singular in the neighborhood of the interface. The solution technique developed and presented in this work overcomes these difficulties by extracting a local solution of the differential equation in the neighborhood of the diffusing front. One profound result is the discovery that the velocity of the front is entirely controlled by the first term of the spectral series expansion. Also, by capturing the critical behavior of the solution in the region of the singularity and incorporating the behavior as a dominant factor, the series expansion is provided a means for very rapid convergence. The versatility of the solution technique is demonstrated by solving various boundary value problems covering a broad range of interest and the solutions are tested against previously published results. 相似文献
7.
B.H. Gilding 《Advances in water resources》1983,6(1):36-43
The present paper describes an approach to modelling the unsaturated soil-moisture zone in the framework of an integrated physically-based hydrologic response model. It is supposed that the subsurface flow regime may be viewed as two separate entities — a saturated flow system which may be modelled by standard two-dimensional regional techniques, and a single overlying unsaturated zone in which the flow is essentially vertical. Coupling takes place via the definition of saturation at the lower boundary of the unsaturated zone, and via a conservative water balance. Attention is focused on the computational procedure for the unsaturated zone as a self-contained module. The major difficulties are the definition of the interface between the saturated and unsaturated zones, the nonlinear character of the equation used to describe unsaturated flow, the inclusion of realistic atmospheric boundary conditions, and, the interaction between water uptake by plants and available soil-moisture. Each of these points is discussed, in turn, with the emphasis on mathematically formulating the problem in such a way that the most important physical features are reproduced with a minimal amount of computational effort. The text concludes with a few illustrative examples. 相似文献
8.
Laboratory and numerical investigation of flow and transport near a seepage-face boundary 总被引：3，自引：0，他引：3
Laboratory and numerical modeling investigations were completed to study the unconfined ground water flow and transport processes near a seepage-face boundary. The laboratory observations were made in a radial sand tank and included measurements of the height of the seepage face, flow velocity near the seepage face, travel time distribution of multiple tracer slugs, and streamlines. All the observations were reliably reproduced with a three-dimensional, axi-symmetric, variably saturated ground water flow model. Physical data presented in this work demonstrate and quantify the importance of three-dimensional transport patterns within a seepage-face zone. The results imply that vertically averaged flow models that employ Dupuit approximations might introduce error in the analysis of localized solute transport near a seepage-face boundary. The experimental dataset reported in this work will also be of interest for those who are attempting to validate a numerical algorithm for solving ground water and contaminant discharge patterns near a surface-water boundary. 相似文献
9.
Rutledge AT 《Ground water》2006,44(3):483-487
Basic concepts are illustrated for the display of ground water level recession as a linear plot on a semilog graph, as first described by Rorabaugh. This exponential decay function can be achieved if there is a definable outflow boundary such as a lake or river and if water levels are expressed relative to the altitude of the boundary. The model can be used to estimate aquifer hydraulic diffusivity. Concepts are illustrated using three finite-difference simulations. One represents the ideal case as described by Rorabaugh, in which the altitude of the outflow boundary is uniform along its length. Another simulation includes a sloping boundary with simple geometry and demonstrates that the model can be used accurately. Based on this simulation, it appears that the ground water level must be expressed relative to the closest point on the outflow boundary. The third simulation includes a sloping boundary and complex boundary shape, and demonstrates departures from the linear model of recession and errors in the estimate of hydraulic diffusivity. Another cause of nonlinearity is the instability of the ground water head profile soon after a recharge event. The nature of these early-time departures will vary depending on the location of the water level observation site relative to the outflow boundary and the hydrologic divide of the ground water flow system. 相似文献
10.
《Advances in water resources》1999,23(3):271-301
Primary variable switching appears as a promising numerical technique for variably saturated flows. While the standard pressure-based form of the Richards equation can suffer from poor mass balance accuracy, the mixed form with its improved conservative properties can possess convergence difficulties for dry initial conditions. On the other hand, variable switching can overcome most of the stated numerical problems. The paper deals with variable switching for finite elements in two and three dimensions. The technique is incorporated in both an adaptive error-controlled predictor–corrector one-step Newton (PCOSN) iteration strategy and a target-based full Newton (TBFN) iteration scheme. Both schemes provide different behaviors with respect to accuracy and solution effort. Additionally, a simplified upstream weighting technique is used. Compared with conventional approaches the primary variable switching technique represents a fast and robust strategy for unsaturated problems with dry initial conditions. The impact of the primary variable switching technique is studied over a wide range of mostly 2D and partly difficult-to-solve problems (infiltration, drainage, perched water table, capillary barrier), where comparable results are available. It is shown that the TBFN iteration is an effective but error-prone procedure. TBFN sacrifices temporal accuracy in favor of accelerated convergence if aggressive time step sizes are chosen. 相似文献
11.
Dewatered or "dry" grid cells in the USGS ground water modeling software MODFLOW may cause nonphysical artifacts, trigger convergence failures, or interfere with parameter estimation. These difficulties can be avoided in two dimensions by modifying the spatial differencing scheme and the iterative procedure used to resolve nonlinearities. Specifically, the spatial differencing scheme is modified to use the water level on the upstream side of a pair of adjacent cells to calculate the saturated thickness and hence intercell conductance for the pair. This makes it possible to explicitly constrain the water level in a cell to be at or above the cell bottom elevation without introducing nonphysical artifacts. Thus constrained, all initially active cells will remain active throughout the simulation. It was necessary to replace MODFLOW's Picard iteration method with the Newton-Raphson method to achieve convergence in demanding applications involving many dry cells. Tests using a MODFLOW variant based on the new method produced results nearly identical to conventional MODFLOW in situations where conventional MODFLOW converges. The new method is extremely robust and converged in scenarios where conventional MODFLOW failed to converge, such as when almost all cells dewatered. An example application to the Edwards Aquifer in south-central Texas further demonstrates the utility of the new method. 相似文献
12.
Horizontal gravity filtration of groundwater in soil is considered. Under Boussinesq approximation, the problem is reduced to a one-dimensional nonlinear parabolic equation in phreatic water level. The problem of linearizing the original equation is discussed. The comparison of gravity-filtration problem solutions in the nonlinear and linearized formulations shows considerable discrepancies to exist between the solutions, especially, for boundary problems with mixed boundary conditions, when the value of the function is not fixed on the right boundary. An analytical solution is obtained for steady-state flow from a water body into the soil with subsequent leakage into underlying beds. Two regimes are shown to exist: one with an infinite exponential tail, and another in the form of a finite groundwater mound. A new approach is proposed to the linearization problem—quasilinearization with the use of the Burgers equation. 相似文献
13.
14.
One approach for simulating ground water–lake interactions is to incorporate the lake into the ground water solution domain as a high-conductivity region. Previous studies have developed this approach using fully saturated models. This study extends this approach to variably saturated models, so that ground water–lake interactions may be more easily simulated with commonly used or public domain variably saturated codes that do not explicitly support coupled lake–water balance modeling. General guidelines are developed for the choices of saturated hydraulic conductivity and moisture retention and relative permeability curves for the lake region. When applied to an example ground water–lake system, model results are very similar to those from a model in which the lake is represented as a specified head boundary continuously updated by a lake mass balance. The high-conductivity region approach is most suitable for relatively simple geometries and lakes with slower and smaller fluctuations when the overall flow pattern and system fluxes, rather than the detailed flow pattern around the intersection of the lake and land surfaces, are of interest. 相似文献
15.
《Advances in water resources》2005,28(10):1133-1141
We study the motion of wetting fronts for vertical infiltration problems as modeled by Richards’ equation. Parlange and others have shown that wetting fronts in infiltration flows can be described by traveling wave solutions. If the soil layer is not initially dry, but has an initial distribution of water content then the motion of the wetting front will change due to the interaction of the infiltrating flow with the pre-existing soil conditions. Using traveling wave profiles, we construct simple approximate solutions of initial-boundary value problems for Richards’ equation that accurately describe the position and moisture distribution of the wetting front. We show that the influences of surface boundary conditions and initial conditions produce shifts to the position of the wetting front. The shifts can be calculated by examining the cumulative infiltration, and are validated numerically for several problems for Richards’ equation and the linear advection–diffusion equation. 相似文献
16.
将土体视为固-液两相介质,基于饱和土体有效应力原理,建立饱和土体-地下综合管廊结构体系相互作用动力模型：在地应力平衡的静力状态下,采用Duncan-Chang非线性弹性本构模型,在地震波作用的动力状态下,采用Davidenkov非线性黏弹性本构模型;考虑饱和土体黏弹性动力人工边界条件,并将地震动作用转化为作用在人工边界节点上的动力荷载。模型考察不同土体材料、结构特性以及土-结构接触摩擦对结构地震响应的影响,得出如下结论：（1）地震波的卓越周期与场地卓越周期相近时,引起结构上的变形最大;（2）综合管廊结构管廊壁厚越薄,埋深越深,结构尺寸越大,结构刚度越小,结构变形越大;（3）不考虑土-结构接触面的状态非线性将会增大结构变形。 相似文献
17.
Haitjema HM 《Ground water》2006,44(1):102-105
The analytic element method, like the boundary integral equation method, gives rise to a system of equations with a fully populated coefficient matrix. For simple problems, these systems of equations are linear, and a direct solution method, such as Gauss elimination, offers the most efficient solution strategy. However, more realistic models of regional ground water flow involve nonlinear equations, particularly when including surface water and ground water interactions. The problem may still be solved by use of Gauss elimination, but it requires an iterative procedure with a reconstruction and decomposition of the coefficient matrix at every iteration step. The nonlinearities manifest themselves as changes in individual matrix coefficients and the elimination (or reintroduction) of several equations between one iteration and the other. The repeated matrix reconstruction and decomposition is computationally intense and may be avoided by use of the Sherman-Morrison formula, which can be used to modify the original solution in accordance with (small) changes in the coefficient matrix. The computational efficiency of the Sherman-Morrison formula decreases with increasing numbers of equations to be modified. In view of this, the Sherman-Morrison formula is only used to remove equations from the original set of equations, while treating all other nonlinearities by use of an iterative refinement procedure. 相似文献
18.
The transformation of a weakly nonlinear interfacial solitary wave in an ideal two-layer flow over a step is studied. In the vicinity of the step the wave transformation is described in the framework of the linear theory of long interfacial waves, and the coefficients of wave reflection and transmission are calculated. A strong transformation arises for propagation into shallower water, but a weak transformation for propagation into deeper water. Far from the step, the wave dynamics is described by the Korteweg-de Vries equation which is fully integrable. In the vicinity of the step, the reflected and transmitted waves have soliton-like shapes, but their parameters do not satisfy the steady-state soliton solutions. Using the inverse scattering technique it is shown that the reflected wave evolves into a single soliton and dispersing radiation if the wave propagates from deep to shallow water, and only dispersing radiation if the wave propagates from shallow to deep water. The dynamics of the transmitted wave is more complicated. In particular, if the coefficient of the nonlinear quadratic term in the Korteweg-de Vries equation is not changed in sign in the region after the step, the transmitted wave evolves into a group of solitons and radiation, a process similar to soliton fission for surface gravity waves at a step. But if the coefficient of the nonlinear term changes sign, the soliton is destroyed completely and transforms into radiation. The effects of cubic nonlinearity are studied in the framework of the extended Korteweg-de Vries (Gardner) equation which is also integrable. The higher-order nonlinear effects influence the amplitudes of the generated solitons if the amplitude of the transformed wave is comparable with the thickness of lower layer, but otherwise the process of soliton fission is qualitatively the same as in the framework of the Korteweg-de Vries equation. 相似文献
19.
Discharges of polluted water from abandoned mines are a major cause of degradation of water resources worldwide. Pollution arises after abandoned workings flood up to surface level, by the process termed ground water rebound. As flow in large, open mine voids is often turbulent, standard techniques for modeling ground water flow (which assume laminar flow) are inappropriate for predicting ground water rebound. More physically realistic models are therefore desirable, yet these are often expensive to apply to all but the smallest of systems. An overall strategy for ground water rebound modeling is proposed, with models of decreasing complexity applied as the temporal and spatial scales of the systems under analysis increase. For relatively modest systems (area < 200 km2), a physically based modeling approach has been developed, in which 3-D pipe networks (representing major mine roadways, etc.) are routed through a variably saturated, 3-D porous medium (representing the country rock). For systems extending more than 100 to 3000 km2, a semidistributed model (GRAM) has been developed, which conceptualizes extensively interconnected volumes of workings as ponds, which are connected to other ponds only at discrete overflow points, such as major inter-mine roadways, through which flow can be efficiently modeled using the Prandtl-Nikuradse pipe-flow formulation. At the very largest scales, simple water-balance calculations are probably as useful as any other approach, and a variety of proprietary codes may be used for the purpose. 相似文献
20.
An effective stress method is presented for the analysis of liquefaction of ground including soil-structure interaction, based on an explicit-implicit finite element method. A simple constitutive model is developed to be incorporated in the effective stress method. The constitutive model consists of the Ramberg-Osgood model extended to two-dimensional problems and a new dilatancy model. The effectiveness of the constitutive model is examined with results of a simple shear test. Besides, the effective stress method is verified by comparing its numerical results with results of a shaking table test. It is found that the present method can simulate well the response of a saturated dense sand-structure system. The difference of the response computed by the effective stress method and the total stress method is discussed. It is found that the total stress method can simulate the response of the saturated sand within an accumulating excess pore water pressure of less than 70 per cent of the initial overburden stress. 相似文献