共查询到20条相似文献，搜索用时 31 毫秒
1.
This work concerns linearization methods for efficiently solving the Richards equation, a degenerate elliptic-parabolic equation which models flow in saturated/unsaturated porous media. The discretization of Richards’ equation is based on backward Euler in time and Galerkin finite elements in space. The most valuable linearization schemes for Richards’ equation, i.e. the Newton method, the Picard method, the Picard/Newton method and the L-scheme are presented and their performance is comparatively studied. The convergence, the computational time and the condition numbers for the underlying linear systems are recorded. The convergence of the L-scheme is theoretically proved and the convergence of the other methods is discussed. A new scheme is proposed, the L-scheme/Newton method which is more robust and quadratically convergent. The linearization methods are tested on illustrative numerical examples. 相似文献
2.
在使用有限元方法求解非饱和土渗流问题时,土-水特征曲线和渗透率函数的强烈非线性经常会造成计算中出现迭代不收敛、计算误差大等问题。基于变量变换的思想,结合时间步长自适应技术提出了一种求解非饱和渗流问题的新方法--欠松弛RFT变换方法（ATUR1）。ATUR1方法通过变量变换,大大降低了Richards方程中未知数在空间和时间上的非线性程度,从而改善这种非线性所带来的计算收敛困难和精度差等问题。欠松弛技术的引入减少了迭代过程中的振荡现象,进一步提高了非线性迭代计算的效率。时间步长自适应技术则有效地控制整个计算过程的误差。数值算例结果说明,ATUR1可以有效地提高计算效率和精度,是一种准确有效的计算方法。 相似文献
3.
An isogeometric analysis (IGA) is introduced to obtain a head-based solution to Richards equation for unsaturated flow in porous media. IGA uses Non-Uniform Rational B-Spline (NURBS) as shape functions, which provide a higher level of inter-element continuity in comparison with Lagrange shape functions. The semi-discrete nonlinear algebraic equations are solved using a combination of implicit backward-Euler time-integration and Newton-Raphson scheme. The time-step size is adaptively controlled based on the rate of changes in the pore pressure. The results from the proposed formulation are compared and verified against an analytical solution for one-dimensional transient unsaturated flow in a homogenous soil column. The proposed method is then applied to four more complex problems including two-dimensional unsaturated flow in a two-layered soil and a semi-circular furrow. The test cases in two-layered soil system involve sharp variations in the pressure gradient at the intersection of the two media, where the pore water pressure abruptly changes. It is shown that the proposed head-based IGA is able to properly simulate changes in pore pressure at the soils interface using fewer degrees of freedom and higher orders of approximation in comparison with the conventional finite element method. 相似文献
4.
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。 相似文献
5.
Iuliu Sorin Pop 《Computational Geosciences》2002,6(2):141-160
We present a numerical analysis of a time discretization method applied to Richards' equation. Written in its saturation-based form, this nonlinear parabolic equation models water flow into unsaturated porous media. Depending on the soil parameters, the diffusion coefficient may vanish or explode, leading to degeneracy in the original parabolic equation. The numerical approach is based on an implicit Euler time discretization scheme and includes a regularization step, combined with the Kirchhoff transform. Convergence is shown by obtaining error estimates in terms of the time step and of the regularization parameter. 相似文献
6.
Domain transformation methods are useful techniques for solving problems on non-stationary domains. In this work, we consider the evolution of the water table in an unconfined aquifer. This nonlinear, time-dependent problem is greatly simplified by using a mapping from the physical domain to a reference domain and is then further reduced to a single, (nonlinear) partial differential equation. We show well-posedness of the approach and propose a stable and convergent discretization scheme. Numerical results are presented supporting the theory. 相似文献
7.
经典的Richards入渗控制方程属于偏微分方程,具有强烈的非线性,难以求得解析解。以入渗时间为最小作用量,基于Richards方程建立关于入渗路径的时间泛函,将考虑重力项的非饱和土垂直入渗问题转化为泛函极值问题,并构造等价的Euler-Lagrange方程进行求解。计算结果表明,扩散系数D（?）与概化湿润锋距离具有函数关系,当扩散系数D（?）形式已知时,可求得最优路径下湿润锋处含水率、较远处湿润锋最小含水率、土壤含水率最大熵分布3个问题,并基于最优路径检验了本研究条件下,Boltzmann变换和线性变换求解Richards方程的精度。求解过程未引进新变量化简Richards方程,不改变原方程结构,因此其解具有普遍性,可作为非饱和土力学计算的一个补充。 相似文献
8.
非饱和渗流Richards方程数值求解的欠松弛方法 总被引：1，自引：0，他引：1
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。 相似文献
9.
Richards方程在非饱和渗流模拟及其他相关领域应用广泛。在数值求解过程中,可以采用有限差分方法进行数值离散并迭代求解,为了获得较可靠的数值解,常规的均匀网格空间步长往往是较小的。在一些不利数值条件下,如入渗于干燥土壤,迭代计算费时甚至精度也不能得到很好改善。因此,文章提出Chebyshev空间网格改进方法,结合有限差分方法对Richards方程进行数值离散以获得线性方程组,并通过经典的Picard迭代方法进行迭代求解线性方程组以得到Richards方程的数值解。通过均质土和分层土2个不利情况下的非饱和渗流算例,又结合模型解析解和软件Hydrus-1D,对比研究了改进网格方法与均匀网格方法获得数值解的精度。结果表明,提出的Chebyshev网格方法相较于传统的均匀网格,可以在较少的节点数下获得较高的数值精度,又具有较小的计算开销,有较好的应用前景。 相似文献
10.
A method is presented to estimate actual evapotranspiration (ETA) from potential evapotranspiration (ETP) by numerical modeling of water flow in the unsaturated zone. Water flow is described by the Richards equation with a sink term representing the root water uptake. Evaporation is included in the model as a Neumann boundary condition at the soil surface. The Richards equation is solved in a one-dimensional domain using a mixed finite element method. The values of ETA are obtained by applying a water stress factor to ETP to account for soil moisture changes during the simulation period. The proposed numerical model is used to estimate ETA in an experimental plot located in a flatland area in Buenos Aires (Argentina). Numerical results show that the proposed model is a useful tool for evaluating evapotranspiration under different scenarios. 相似文献
11.
H. Hajinezhad Ali R. Soheili Mohammad R. Rasaei F. Toutounian 《Computational Geosciences》2018,22(6):1433-1444
In this paper, the numerical methods for solving the problem of steam injection in the heavy oil reservoirs are presented. We consider a 3-dimensional model of 3-phase flow, oil, water, and steam, with the effect of 3-phase relative permeability. Interphase mass transfer of water and steam is considered; oil is assumed nonvolatile. We apply the simultaneous solution approach to solve the corresponding nonlinear discretized partial differential equation in the fully implicit form. The convergence of finite difference scheme is proved by the Rosinger theorem. The heuristic Jacobian-Free-Newton-Krylov (HJFNK) method is proposed for solving the system of algebraic equations. The result of this proposed numerical method is well compared with some experimental results. Our numerical results show that the first iteration of the full approximation scheme (FAS) provides a good initial guess for the Newton method. Therefore, we propose a new hybrid-FAS-HJFNK method while there is no steam in the reservoir. The numerical results show that the hybrid-FAS-HJFNK method converges faster than the HJFNK method. 相似文献
12.
Michael D. Tocci C.T. Kelley Cass T. Miller Christopher E. Kees 《Computational Geosciences》1998,2(4):291-309
Richards' equation (RE) is often used to model flow in unsaturated porous media. This model captures physical effects, such as sharp fronts in fluid pressures and saturations, which are present in more complex models of multiphase flow. The numerical solution of RE is difficult not only because of these physical effects but also because of the mathematical problems that arise in dealing with the nonlinearities. The method of lines has been shown to be very effective for solving RE in one space dimension. When solving RE in two space dimensions, direct methods for solving the linearized problem for the Newton step are impractical. In this work, we show how the method of lines and Newton-iterative methods, which solve linear equations with iterative methods, can be applied to RE in two space dimensions. We present theoretical results on convergence and use that theory to design an adaptive method for computation of the linear tolerance. Numerical results show the method to be effective and robust compared with an existing approach. 相似文献
13.
Numerical approximation based on different forms of the governing partial differential equation can lead to significantly
different results for two-phase flow in porous media. Selecting the proper primary variables is a critical step in efficiently
modeling the highly nonlinear problem of multiphase subsurface flow. A comparison of various forms of numerical approximations
for two-phase flow equations is performed in this work. Three forms of equations including the pressure-based, mixed pressure–saturation
and modified pressure–saturation are examined. Each of these three highly nonlinear formulations is approximated using finite
difference method and is linearized using both Picard and Newton–Raphson linearization approaches. Model simulations for several
test cases demonstrate that pressure based form provides better results compared to the pressure–saturation approach in terms
of CPU_time and the number of iterations. The modification of pressure–saturation approach improves accuracy of the results.
Also it is shown that the Newton–Raphson linearization approach performed better in comparison to the Picard iteration linearization
approach with the exception for in the pressure–saturation form. 相似文献
14.
The evolution of a gravity-driven free-surface flow of varying horizontal extent which couples with a field evolving within the flow is solved using a finite difference discretization of a mapping of the problem onto the unit square. Since the size of the solution domain may show several orders of magnitude of variation, while the normalized geometry of the domain and the internal field may not vary significantly, this procedure avoids excessively fine or coarse discretizations, as well as interpolations at the boundary. The parabolic and hyperbolic evolution equations for the internal field are considered. The evolution of the coupled system is solved by an implicit marching scheme. The discretizations in space and in time are accurate to second order. Multipoint upwinding is used to avoid an instability arising advective terms are large. The evolution equations are nonlinear, and are solved using a nested Newton–Raphson procedure. The nesting is achieved by using successively better approximations to the ture evolution equations. The matrix equation that arises is solved by a conjugate-gradient-like (ORTHOMIN) iteration procedure with an incomplete Cholesky factorization preconditioning. The method has a wide variety of potential applications in the earth sciences, with the ability to describe glacier flow, lava flow, avalanching and landslides. Some calculations of the thermomechanical evolution of ice-sheets are given as illustrations, and the possible existence of thermally induced instabilities is considered. 相似文献
15.
Multi-phase flow in porous media in the presence of viscous, gravitational, and capillary forces is described by advection diffusion equations with nonlinear parameters of relative permeability and capillary pressures. The conventional numerical method employs a fully implicit finite volume formulation. The phase-potential-based upwind direction is commonly used in computing the transport terms between two adjacent cells. The numerical method, however, often experiences non-convergence in a nonlinear iterative solution due to the discontinuity of transmissibilities, especially in transition between co-current and counter-current flows. Recently, Lee et al. (Adv. Wat. Res. 82, 27–38, 2015) proposed a hybrid upwinding method for the two-phase transport equation that comprises viscous and gravitational fluxes. The viscous part is a co-current flow with a one-point upwinding based on the total velocity and the buoyancy part is modeled by a counter-current flow with zero total velocity. The hybrid scheme yields C1-continuous discretization for the transport equation and improves numerical convergence in the Newton nonlinear solver. Lee and Efendiev (Adv. Wat. Res. 96, 209–224, 2016) extended the hybrid upwind method to three-phase flow in the presence of gravity. In this paper, we present the hybrid-upwind formula in a generalized form that describes two- and three-phase flows with viscous, gravity, and capillary forces. In the derivation of the hybrid scheme for capillarity, we note that there is a strong similarity in mathematical formulation between gravity and capillarity. We thus greatly utilize the previous derivation of the hybrid upwind scheme for gravitational force in deriving that for capillary force. Furthermore, we also discuss some mathematical issues related to heterogeneous capillary domains and propose a simple discretization model by adapting multi-valued capillary pressures at the end points of capillary pressure curves. We demonstrate this new model always admits a consistent solution that is within the discretization error. This new generalized hybrid scheme yields a discretization method that improves numerical stability in reservoir simulation. 相似文献
16.
Clint N. Dawson Héctor Klíe Mary F. Wheeler Carol S. Woodward 《Computational Geosciences》1997,1(3-4):215-249
A new parallel solution technique is developed for the fully implicit three‐dimensional two‐phase flow model. An expandedcell‐centered finite difference scheme which allows for a full permeability tensor is employed for the spatial discretization, and backwardEuler is used for the time discretization. The discrete systems are solved using a novel inexact Newton method that reuses the Krylov information generated by the GMRES linear iterative solver. Fast nonlinear convergence can be achieved by composing inexact Newton steps with quasi‐Newton steps restricted to the underlying Krylov subspace. Furthermore, robustness and efficiency are achieved with a line‐search backtracking globalization strategy for the nonlinear systems and a preconditioner for each coupled linear system to be solved. This inexact Newton method also makes use of forcing terms suggested by Eisenstat and Walker which prevent oversolving of the Jacobian systems. The preconditioner is a new two‐stage method which involves a decoupling strategy plus the separate solutions of both nonwetting‐phase pressure and saturation equations. Numerical results show that these nonlinear and linear solvers are very effective. 相似文献
17.
Florian Frank Chen Liu Faruk O. Alpak Beatrice Riviere 《Computational Geosciences》2018,22(2):543-563
A numerical method is formulated for the solution of the advective Cahn–Hilliard (CH) equation with constant and degenerate mobility in three-dimensional porous media with non-vanishing velocity on the exterior boundary. The CH equation describes phase separation of an immiscible binary mixture at constant temperature in the presence of a conservation constraint and dissipation of free energy. Porous media / pore-scale problems specifically entail images of rocks in which the solid matrix and pore spaces are fully resolved. The interior penalty discontinuous Galerkin method is used for the spatial discretization of the CH equation in mixed form, while a semi-implicit convex–concave splitting is utilized for temporal discretization. The spatial approximation order is arbitrary, while it reduces to a finite volume scheme for the choice of element-wise constants. The resulting nonlinear systems of equations are reduced using the Schur complement and solved via inexact Newton’s method. The numerical scheme is first validated using numerical convergence tests and then applied to a number of fundamental problems for validation and numerical experimentation purposes including the case of degenerate mobility. First-order physical applicability and robustness of the numerical method are shown in a breakthrough scenario on a voxel set obtained from a micro-CT scan of a real sandstone rock sample. 相似文献
18.
We consider two-component (typically, water and hydrogen) compressible liquid–gas porous media flows including mass exchange between phases possibly leading to gas-phase (dis)appearance, as motivated by hydrogen production in underground repositories of radioactive waste. Following recent work by Bourgeat, Jurak, and Smaï, we formulate the governing equations in terms of liquid pressure and dissolved hydrogen density as main unknowns, leading mathematically to a nonlinear elliptic–parabolic system of partial differential equations, in which the equations degenerate when the gas phase disappears. We develop a discontinuous Galerkin method for space discretization, combined with a backward Euler scheme for time discretization and an incomplete Newton method for linearization. Numerical examples deal with gas-phase (dis)appearance, ill-prepared initial conditions, and heterogeneous problem with different rock types. 相似文献
19.
Zero effective stress boundary condition along with constant fluid flux is commonly encountered in geotechnical applications such as uncased borehole stability, fluid injection and production at an uncased borehole, hydraulic fracturing and sand production. This complex boundary condition introduces high nonlinearity in the numerical simulation. Conventional iterative methods such as Newton–Raphson method are required to solve this nonlinear problem iteratively, which involve huge computing time and also pose numerical difficulties on the convergence. To overcome this numerical difficulty and hence reduce the computing time, a novel numerical technique is proposed in this paper. Its performance is evaluated using a numerical example simulating fluid injection around an uncased borehole. Copyright © 2009 John Wiley & Sons, Ltd. 相似文献
20.
This work is intended for the development of a numerical method to simulate flows and solute transport in multiphasical porous medium taking into consideration the interaction of solid/solute. More precisely, the studied problem is modeled by a coupled system composed of an elliptical equation (for the flow) and an equation convection–diffusion–reaction (for the transfer). Numerical simulations were realistic for two-dimensional problems confirming the stability and efficiency of the combined scheme in the characterization of a pollutant transport through an unsaturated zone of an industrial site. 相似文献