Brine transport in porous media: On the use of Von Mises and similarity transformations

In this paper we use a Von Mises transformation to study brine transport in porous media. The model involves mass balance equations for fluid and salt, Darcy's law and an equation of state, relating the salt mass fraction to the fluid density. Application of the Von Mises transformation recasts the model equations into a single nonlinear diffusion equation. A further reduction is possible if the problem admits similarity. This yields a formulation in terms of a boundary value problem for an ordinary differential equation which can be treated by semi‐analytical means. Three specific similarity problems are considered in detail: (i) one‐dimensional, stable displacement of fresh water and brine in a porous column, (ii) flow of fresh water along the surface of a salt rock, (iii) mixing of parallel layers of brine and fresh water. This revised version was published online in July 2006 with corrections to the Cover Date.     

Analysis of DNAPL infiltration in a medium with a low-permeable lens

In this paper we study the infiltration of DNAPL in a porous medium containing a single low-permeable lens. Our aim is to determine whether or not DNAPL infiltrates into the lens. A key role is played by the capillary pressure: DNAPL cannot infiltrate into the lens unless the capillary pressure exceeds the entry pressure of the lens. In the model this is reflected by an interface condition, the extended capillary pressure condition. To derive analytical approximations we first consider a steady-state DNAPL plume in a homogeneous medium. This results in an estimate of the DNAPL plume width as a function of depth, and an asymptotic solution for small saturations. Assuming that the extent of the lens is much larger than the width of the unperturbed DNAPL plume in the homogeneous medium, we derive an explicit criterion for DNAPL infiltration into the lens in terms of a critical inflow rate. A numerical algorithm is presented in which the extended capillary pressure condition is incorporated. The numerical and analytical results show good qualitative agreement. This revised version was published online in August 2006 with corrections to the Cover Date.     

Solving stiff mass transfers in compositional multiphase flow models: Numerical stability and spurious solutions

This paper points out two numerical problems linked to the resolution of compositional multiphase flow models for porous media with the finite‐volume technique. In particular, we consider fluid mixtures featuring fast mass transfers between the phases, hence stiff. In this context, we show how the computation of mass exchange kinetics can be expensive and that erroneous saturation front locations arise. A numerical splitting method is developed which is proven to be stable with advection‐type time steps, whatever the stiffness of the mass transfer. The erroneous front location problem is illustrated and shown to be intrinsically linked to the numerical diffusion. If we assume that the fluids are in thermodynamical equilibrium, we find that spurious solutions can be avoided by deriving and solving a new uncoupled hyperbolic equation for the saturation. This revised version was published online in July 2006 with corrections to the Cover Date.     

A mixed finite element method for Hele-Shaw cell equations

A new numerical method to solve the system of equations describing two phase flow in a Hele-Shaw cell is presented. It combines a mixed finite element method, the method of subtraction of the singularity and a front tracking grid in a single computational strategy. This choice of discretization techniques is well motivated by the difficulties present in the system of equations and the physics of the problem. The new method was tested against analytical solutions and also by solving the Saffman–Taylor viscous fingering problem for finite and infinite mobility ratios. In both cases convergence under mesh refinement is achieved for the fingers developed from an initial sinusoidal interface. Finger splitting is observed for low values of the surface tension and high mobility ratio. Different explanations, based in our results, are provided for this phenomenon. This revised version was published online in July 2006 with corrections to the Cover Date.     

Identification of the diffusion coefficient for a semi-linear hyperbolic-parabolic problem with discontinuous nonlinear terms

Numerical computations concerning geochemical processes in porous media require a knowledge of the values of the macroscopic parameters. In the context of the interpretation of leaching tests one such parameter is the apparent macroscopic diffusivity. In this paper, a mathematical model for dissolution in the presence of diffusion is considered. It is shown that the constant diffusion coefficient can be identifiable when the time integral of the flux is known. Moreover, a practical algorithm for computing the diffusion coefficient is proposed. This revised version was published online in August 2006 with corrections to the Cover Date.     

均质弹性地基中单桩的扭转振动特性研究

利用积分方程的方法研究了均质弹性地基中单桩的扭转振动问题。在分析过程中,首先利用积分变换的方法得出均质弹性地基内作用一埋置扭矩时的基本解。基于基本解,根据变形协调条件建立了控制单桩扭转振动特性的第2类Fredholm积分方程。对所得积分方程进行数值求解得到单桩扭转振动时的扭矩和扭转角及动力柔度系数,并对其进行了参数分析。所得结论对桩基础设计与计算以及桩基低应变扭转波动测技术有一定的指导意义。     

Numerical simulation of NAPL flow in the subsurface

A three-dimensional, three-phase numerical model is presented for simulating the movement of immiscible fluids, including nonaqueous-phase liquids (NAPLs), through porous media. The model is designed to simulate soil flume experiments and for practical application to a wide variety of contamination scenarios involving light or dense NAPLs in heterogeneous subsurface systems. The model is derived for the three-phase flow of water, NAPL, and air in porous media. The basic governing equations are based upon the mass conservation of the constitutents within the phases. The descretization chosen to transform the governing equations into the approximating equations, although logically regular, is very general. The approximating equations are a set of simultaneous coupled nonlinear equations which are solved by the Newton-Raphson method. The linear system solutions needed for the Newton-Raphson method are obtained using a matrix of preconditioner/accelerator iterative methods. Because of the special way the governing equations are implemented, the model is capable of simulating many of the phenomena considered necessary for the sucessful simulation of field problems including entry pressure phenomena, entrapment, and preferential flow paths. The model is verified by comparing it with several exact analytic test solutions and three soil flume experiments involving the introduction and movement of light nonaqueous-phase liquid (LNAPL) or dense nonaqueous-phase liquid (DNAPL) in heterogeneous sand containing a watertable. This revised version was published online in August 2006 with corrections to the Cover Date.     

Node-centered finite volume discretizations for the numerical simulation of multiphase flow in heterogeneous porous media

Three node-centered finite volume discretizations for multiphase porous media flow are presented and compared. By combination of these methods two additional discretization methods are generated. The ability of these schemes to describe flows at textural interfaces of different geologic formations is investigated. It was found that models with nonzero-entry pressures for the capillary pressure-saturation relationship in conjunction with the Box discretization may give rise to spurious oscillations for flows around low permeable lenses. Furthermore, the applicability and sensitivity of the discretization methods with regard to the used computational grids is discussed. The schemes are used for the numerical study of two-phase flow in porous media with zones of different material properties. This revised version was published online in July 2006 with corrections to the Cover Date.     

A locally conservative Eulerian–Lagrangian numerical method and its application to nonlinear transport in porous media

Eulerian-Lagrangian and Modified Method of Characteristics (MMOC) procedures provide computationally efficient techniques for approximating the solutions of transport-dominated diffusive systems. The original MMOC fails to preserve certain integral identities satisfied by the solution of the differential system; the recently introduced variant, called the MMOCAA, preserves the global form of the identity associated with conservation of mass in petroleum reservoir simulations, but it does not preserve a localized form of this identity. Here, we introduce an Eulerian-Lagrangian method related to these MMOC procedures that guarantees conservation of mass locally for the problem of two-phase, immiscible, incompressible flow in porous media. The computational efficiencies of the older procedures are maintained. Both the original MMOC and the MMOCAA procedures for this problem are derived from a nondivergence form of the saturation equation; the new method is based on the divergence form of the equation. A reasonably extensive set of computational experiments are presented to validate the new method and to show that it produces a more detailed picture of the local behavior in waterflooding a fractally heterogeneous medium. A brief discussion of the application of the new method to miscible flow in porous media is included. This revised version was published online in July 2006 with corrections to the Cover Date.     

孔边线状裂纹断裂扩展的研究

本文通过叠加原理和梅林(Mellin)变换,将无限弹性体内孔边线状裂纹的控制方程和边界条件化成对偶积分方程,又通过希尔伯特(Hilbert)有限变换,将对偶积分方程化为关于S(t~2)的第二类弗雷德霍姆(Fredholm)积分方程。通过弗雷德霍姆积分方程的解表示了孔边裂纹扩展时的动态应力强度因子,并得出了数值结果。在特殊情况下,数值结果与文献〔2〕的结果相吻合。应力强度因子随着裂纹的扩展而下降。同时,通过实验,测定了孔边线状裂纹的断裂扩展速度。     

The Study of Propagation of Linear Cracks Intersection Hole

In this paper, the superimposing principle and the theory of Mellin transforms are used to reduce the controlling equations and boundary conditions of linear cracks intersecting hole into dual integral equations in an infinite elastic body. With finite Hilbert transforms, the dual integral equations can be reduced into a Fredhole integral equation of the second kind which is related to S(t2 ). The dynamic stress intensity factor can be expressed with the solution of integral equation and its numerical results in obtained. In the particular case, the numerical results are the same as those in reference[2] . the intensity factor decreases with the propagation of cracks. With experiments, the propagation velocity of these linear cracks are measured.     

新型边界积分方程的无单元解及其应用

目前隧道及大型地下工程往往在裂隙岩体中开挖,而裂隙与地下空间的距离及裂隙的扩展条件,制约着隧道及地下工程的稳定性。应用能考虑孔洞和裂纹问题的新型边界积分方程与无网格加辽金法结合,建立一种新型的边界无单元法。在该方法中基本的未知量是由边界上的面力和边界上位移密度函数构成的复变量边界函数 。文中应用的边界积分公式和Muskhelishvili的积分公式直接相关。将无网格构造方法引入新型的边界积分方程,建立了新型的边界无单元法。应用该方法详细分析了含隧道和裂纹间相互关系等问题,其数值结果与解析结果吻合很好,说明该方法的正确性和可行性。     

激光测速粒子对复杂流动的响应研究——Ⅰ颗粒非恒定运动数学模型及其数值方法

流体激光测速的精度与示踪粒子的跟随特性即流体中异质粒子的非恒定运动特性密切相关。首先对粒子非恒定运动方程进行了探讨,着重考虑了在高颗粒雷诺数时该方程的修正问题,简要分析了该方程的数学属性,并构造了这类方程的数值计算方法。分析表明,高颗粒雷诺数下的粒子非恒定运动方程为非线性奇异积分方程,而当颗粒雷诺数小于1时,则线性化为第二类渥尔特拉(Volterra)积分方程。以几种均匀流中球形小颗粒的非恒定运动为算例,计算结果与其解析解及有关实验数据的比较表明,数值方法具有良好的计算精度。     

混合法计算三维非均匀介质中的电磁场

介绍了联合运用积分方程法与有限元法(简称混合法)来计算三维非均匀介质中电磁场分布的理论方法,并进行了数值模拟实验。混合法的原理是引入一个包围非均匀目标体的虚构边界,在边界内部的场用有限元法模拟,在边界外部(包括边界)的场用积分方程表达,二者在边界上通过场的连续性耦合起来。数值实验结果表明,混合法既能显著地减小网格规模,又能灵活地模拟复杂的介质情况,且计算精度较高。     

交错网格下的浅水方程高分辨Gauss型格式

在交错网格上,基于高精度的Gauss积分公式,针对浅水波方程设计了对模拟涌波具有高分辨率的完全二阶精度的数值计算格式。由于采用了交错网格,差分格式不需要解Riemann问题,因此本文格式具有计算简单、工作量少、编程简便等特点。另外,在一维单个方程时,本文格式在CFL(Courant Friedrich Lewy)条件限制下为TVD(Total Variation Diminishing)格式,在二维和三维情况下格式具有MmB(Maximum and Minimum Bounds Preserving)性质。利用国家高性能计算中心(合肥)的曙光1000型分布存储大规模并行机,对在交错网格下所构造的求解浅水方程的高分辨差分格式进行了并行实现,几个算例的计算结果令人满意。     

Influences of axial load on the lateral response of single pile with integral equation method

This article revisits the influences of axial load on the lateral response of single pile with integral equation method. The problem is formulated by decomposing the pile soil system into an extended elastic soil and a fictitious pile, the former of which is analyzed by making use of the fundamental Mindlin's solution for a concentrated horizontal load whereas the latter is modeled by the conventional beam bending theory. According to the rotation compatibility condition between the fictitious pile and the extended soil, a Fredholm integral equation of the second kind is established with the shear strain and rotation angle of the fictitious pile being the basic unknowns. The bending moment and displacement distribution along the pile are subsequently obtained. Comparison with existing solutions validates the accuracy and applicability of the present formulation. The results of parametric analysis indicate that the influences of axial load on the lateral response of single piles could be significant, and in general, the bending moment and horizontal displacement distributions along the pile increase considerably with the increase of axial load. Copyright © 2011 John Wiley & Sons, Ltd.     

An indirect boundary integral equation method for poroelasticity

This paper presents an indirect boundary integral equation method for analysis of quasi-static, time-harmonic and transient boundary value problems related to infinite and semi-infinite poroelastic domains. The present analysis is based on Biot's theory for poroelastodynamics with fluid viscous dissipation. The solution to a given boundary value problem is reduced to the determination of intensities of forces and fluid sources applied on an auxiliary surface defined interior to the surface on which the boundary conditions are specified. A coupled set of integral equations is established to determine the intensities of forces and fluid sources applied on the auxiliary surface. The integral equations are solved numerically in the Laplace domain for quasi-static and transient problems, and in the frequency domain for time-harmonic excitations. The kernel functions of the integral equation correspond to appropriate Green's functions for a poroelastic full space or half-space. The convergence and numerical stability of the present scheme are established by considering a number of bench mark problems. The versatility of the present method is demonstrated by studying the quasi-static response of a rigid spheroidal anchor, and time-harmonic and transient response of a rigid semi-circular tunnel.     

变降深抽水附近地下水非稳定运动的积分方程解

根据完整井变流量抽水附近地下水运动的解析解,利用积分方程的方法,推导出了水井水位降深以任意一种函数关系随时间变化时含水层中地下水非稳定运动积分方程形式的解,并提出了该积分解相应的近似计算方法.最后通过一个实例验证了该方法的实用性.     

A quadratic variation indirect boundary element method for traction boundary-value problems of two-dimensional elastostatics

Boundary integral equations for traction boundary-value problems of two-dimensional elastostatics are derived by the indirect boundary element method. Quadratic variation functions for the representation of geometry, fictitious forces and displacements over each boundary element are described. A system of equations approximating to the boundary integral equations is obtained by a Galerkin formulation in which the integral equation is written at Gauss integration points of elements. The method of computation of the Cauchy principal value is described. Examples of application to the analysis of stress and displacement around underground excavations demonstrate the accuracy and efficiency of the formulation.