Finite element analysis of three-dimensional groundwater flow problems

A numerical method is presented for analysing either steady state or transient three-dimensional groundwater flow problems. The governing equation is formulated in terms of the finite element process using the Galerkin approach, and cubic isoparametric elements are used to simulate the flow domain as these permit accurate modelling of curved boundaries. Particular attention is paid to the time dependent movement of the phreatic surface where an iterative technique based on the replacement of the original transient problem by a discrete number of steady state problems is used to effect a solution. Furthermore, in tracing the movement of the surface use is made of the element formulation theory in order to compute the normal to the boundary.The validity of the technique is first established by analysing a radially symmetrical problem for which an alternative analytical solution is available. Finally, a general three-dimensional flow system is studied for which there is no known analytical solution. It is shown that relatively few elements are required to yield practical solutions.     

Computational issues and applications of line-elements to model subsurface flow governed by the modified Helmholtz equation

Two new approaches are presented for the accurate computation of the potential due to line elements that satisfy the modified Helmholtz equation with complex parameters. The first approach is based on fundamental solutions in elliptical coordinates and results in products of Mathieu functions. The second approach is based on the integration of modified Bessel functions. Both approaches allow evaluation of the potential at any distance from the element. The computational approaches are applied to model transient flow with the Laplace transform analytic element method. The Laplace domain solution is computed using a combination of point elements and the presented line elements. The time domain solution is obtained through a numerical inversion. Two applications are presented to transient flow fields, which could not be modeled with the Laplace transform analytic element method prior to this work. The first application concerns transient single-aquifer flow to wells near impermeable walls modeled with line-doublets. The second application concerns transient two-aquifer flow to a well near a stream modeled with line-sinks.     

Time evolution of the first‐order linear acoustic/elastic wave equation using Lie product formula and Taylor expansion

We propose a new numerical solution to the first‐order linear acoustic/elastic wave equation. This numerical solution is based on the analytic solution of the linear acoustic/elastic wave equation and uses the Lie product formula, where the time evolution operator of the analytic solution is written as a product of exponential matrices where each exponential matrix term is then approximated by Taylor series expansion. Initially, we check the proposed approach numerically and then demonstrate that it is more accurate to apply a Taylor expansion for the exponential function identity rather than the exponential function itself. The numerical solution formulated employs a recursive procedure and also incorporates the split perfectly matched layer boundary condition. Thus, our scheme can be used to extrapolate wavefields in a stable manner with even larger time‐steps than traditional finite‐difference schemes. This new numerical solution is examined through the comparison of the solution of full acoustic wave equation using the Chebyshev expansion approach for the matrix exponential term. Moreover, to demonstrate the efficiency and applicability of our proposed solution, seismic modelling results of three geological models are presented and the processing time for each model is compared with the computing time taking by the Chebyshev expansion method. We also present the result of seismic modelling using the scheme based in Lie product formula and Taylor series expansion for the first‐order linear elastic wave equation in vertical transversely isotropic and tilted transversely isotropic media as well. Finally, a post‐stack migration results are also shown using the proposed method.     

Analytic element modelling for strip aquifers

The Analytic Element Method (AEM) provides a convenient tool for groundwater flow analysis in unbounded continuous domains. The AEM is based on the superposition of analytic functions, known as elements, useful at both regional and local scales. In this study, analytic elements for strip aquifers are presented. Such aquifers occur in riverine or coastal deposits and in outcrop zones of confined aquifers. Local flow field is modelled indirectly, using a reference plane related to the aquifer domain through the Schwarz‐Christoffel transform. The regional flow is obtained as a solution of the one‐dimensional flow equation. The proposed methodology was tested by modelling two hypothetical situations, which were compared to exact solutions. It is shown that regional boundaries can be reproduced exactly while local fields are adequately reproduced with analytic elements. The developed elements are applied to simulate a real flow field in northeastern Brazil showing good agreement with measured water levels. Copyright © 2011 John Wiley & Sons, Ltd.     

Laplace-transform analytic element solution of transient flow in porous media

A Laplace-transform analytic element method (LT-AEM) is described for the solution of transient flow problems in porous media. Following Laplace transformation of the original flow problem, the analytic element method (AEM) is used to solve the resultant time-independent modified Helmholtz equation, and the solution is inverted numerically back into the time domain. The solution is entirely general, retaining the mathematical elegance and computational efficiency of the AEM while being amenable to parallel computation. It is especially well suited for problems in which a solution is required at a limited number of points in space–time, and for problems involving materials with sharply contrasting hydraulic properties. We illustrate the LT-AEM on transient flow through a uniform confined aquifer with a circular inclusion of contrasting hydraulic conductivity and specific storage. Our results compare well with published analytical solutions in the special case of radial flow.     

格林函数的奇异性处理

格林函数在解决电磁场和直流电场问题的积分方程法和边界单元法等方法中有着广泛的应用.在应用格林函数时会遇到格林函数奇异性的问题,即其在源区域的内部,当场点和源点重合时,会碰到对格林函数为无穷大,积分为奇异积分.对于不同问题中遇到的不同形式的格林函数奇异性问题的各种处理方法进行了分析和评述.对于标量格林函数奇异性问题处理方法有：挖去法、级数展开法、绕开法和解析法等.对于并矢格林函数奇异性问题处理方法有：分量处理法、源并矢法和拟源并矢法等处理方法.通过对实际工作中所遇到的新方法新问题中的不同形式的格林函数的研究,提供了奇异性问题的处理的方法和途径.     

Analysis of steady ground water flow toward wells in a confined-unconfined aquifer

A confined aquifer may become unconfined near the pumping wells when the water level falls below the confining unit in the case where the pumping rate is great and the excess hydraulic head over the top of the aquifer is small. Girinskii's potential function is applied to analyze the steady ground water flow induced by pumping wells with a constant-head boundary in a mixed confined-unconfined aquifer. The solution of the single-well problem is derived, and the critical radial distance at which the flow changes from confined to unconfined condition is obtained. Using image wells and the superposition method, an analytic solution is presented to study steady ground water flow induced by a group of pumping wells in an aquifer bounded by a river with constant head. A dimensionless function is introduced to determine whether a water table condition exists or not near the pumping wells. An example with three pumping wells is used to demonstrate the patterns of potentiometric surface and development of water table around the wells.     

The seepage flow through a free boundary aquifer into a river / L'écoulement d'infiltration d'un aquifère a frontière libre vers une rivière

ABSTRACT

This paper presents a model of the groundwater flow into a river from an aquifer beneath the river. The mathematical problem is to solve Laplace's equation with a free boundary and the solution procedure uses a variational inequality which leads to an approximate solution using finite differences. The method can be used to provide for example, inflow conditions in river modelling calculations.     

A two-grid method for coupled free flow with porous media flow

This paper presents a two-grid method for solving systems of partial differential equations modelling incompressible free flow coupled with porous media flow. This work considers both the coupled Stokes and Darcy as well as the coupled Navier-Stokes and Darcy problems. The numerical schemes proposed are based on combinations of the continuous finite element method and the discontinuous Galerkin method. Numerical errors and convergence rates for solutions obtained from the two-grid method are presented. CPU times for the two-grid algorithm are shown to be significantly less than those obtained by solving the fully coupled problem.     

参考波速线性变化时的声波方程逆散射反演

声波方程的逆散射反演乃是求解双曲型偏微分方程系数项反问题的一种解析方法,一般利用Born近似把这一非线性反问题线性化,并给出了恒参考波速介质中反问题解的解析表达式.由于Born近似假定波速扰动为一级无穷小,因此,在大多数情况下,恒参考波速介质模型的反问题的解无法得以应用.本文研究介质参考波速沿某个方向线性变化时的声散射理论,导出了声波方程逆散射问题解的解析表达式,从而既可使Born近似的假定在大多数情况下能得以满足,又可利用快速Fourier变换快速实现介质波速扰动的反演成象.     

Slopes of an airborne electromagnetic resistivity model interpolated jointly with borehole data for 3D geological modelling

We investigate a novel way to introduce resistivity models deriving from airborne electromagnetic surveys into regional geological modelling. Standard geometrical geological modelling can be strengthened using geophysical data. Here, we propose to extract information contained in a resistivity model in the form of local slopes that constrain the modelling of geological interfaces. The proposed method is illustrated on an airborne electromagnetic survey conducted in the region of Courtenay in France. First, a resistivity contrast corresponding to the clay/chalk interface was interpreted confronting the electromagnetic soundings to boreholes. Slopes were then sampled on this geophysical model and jointly interpolated with the clay/chalk interface documented in boreholes using an implicit 3D potential‐field method. In order to evaluate this new joint geophysical–geological model, its accuracy was compared with that of both pure geological and pure geophysical models for various borehole configurations. The proposed joint modelling yields the most accurate clay/chalk interface whatever the number and location of boreholes taken into account for modelling and validation. Compared with standard geological modelling, the approach introduces in between boreholes geometrical information derived from geophysical results. Compared with conventional resistivity interpretation of the geophysical model, it reduces drift effects and honours the boreholes. The method therefore improves what is commonly obtained with geological or geophysical data separately, making it very attractive for robust 3D geological modelling of the subsurface.     

电导率分段线性变化的水平层的点电源电场的数值解

首先给出柱坐标系中电导率分段线性变化的水平层的点源电场的二维边值问题,然后用变分法将边值问题转变为变分问题。用有限单元法解变分问题,将区域剖分成矩形单元,在单元中进行双线性函数插值,将变分方程化为线性代数方程组。解方程组,得各节点的电位值,由此可计算地表的视电阻率。 算例表明,本方法计算结果与精确解十分符合。本文还举了一个定量分析视电阻率年变化的例子。 本方法占用计算机内存约100K数量级。在MV/6000超小型计算机上计算一条电测深曲线的时间为几十秒钟。     

曲面上的位场转换（二维和三维情况）

在文献[1]中我们利用复变函数论的方法只解决了二维的曲面上的位场转换问题.在本文中,我们利用单层位势理论将它化为一个奇异积分方程求解的问题,并且提出了用奇点分离法来解此奇异积分方程,这对二维和三维情况都适用.通过理论模型的计算,结果的精度是令人满意的.     

A new benchmark semi-analytical solution for density-driven flow in porous media

A new benchmark semi-analytical solution is proposed for the verification of density-driven flow codes. The problem deals with a synthetic square porous cavity subject to different salt concentrations at its vertical walls. A steady state semi-analytical solution is investigated using the Fourier–Galerkin method. Contrarily to the standard Henry problem, the cavity benchmark allows high truncation orders in the Fourier series and provides semi-analytical solutions for very small diffusion cases. The problem is also investigated numerically to validate the semi-analytical solution. The obtained results represent a set of new test case high quality data that can be effectively used for benchmarking density-driven flow codes.     

Galerkin finite element method and the groundwater flow equation: 1. Convergence of the method

This paper is concerned with the convergence of the Galerkin finite element method applied to a groundwater flow problem containing a borehole, with special reference to quadrature effects and the accuracy of the solution. It is shown that there exists an optimal quadrature rule for every choice of piecewise polynomial basis functions. Another interesting result proved here is that, in a direct application of the method the accuracy is very nearly independent of the degree of the polynomial basis functions, but strongly dependent on the distance of the borehole from the boundary if this is small.     

半无限空间中圆形孔洞周围SH波的散射

建立了求解在含有圆形孔洞的弹性半空间中SH波散射与圆形孔洞附近动应力集中问题的解析方法。利用SH波散射的对称性和多极坐标的方法，构造了一个可以预先满足半空间自由表面上应力自由边界条件的圆形孔洞对SH波散射的波函数。利用这一波函数，则可将该问题转化成对一个圆形孔洞散射的求解问题。该问题的解答最终又可归结为对一组无究代数方程组的求解问题，并可利用截断有限项的方法对其进行计算，最后给出了有关圆形也洞附近动应力集中问题的算例和数值结果，并讨论了波数与圆孔至自由边界距离变化对动应力集中的影响。     

Non-linear three-dimensional inversion of cross-well electrical measurements

Cross-well electrical measurement as known in the oil industry is a method for determining the electrical conductivity distribution between boreholes from the electrostatic field measurements in the boreholes. We discuss the reconstruction of the conductivity distribution of a three-dimensional domain. The measured secondary electric potential field is represented in terms of an integral equation for the vector electric field. This integral equation is taken as the starting point to develop a non-linear inversion method, the so-called contrast source inversion (CSI) method. The CSI method considers the inverse scattering problem as an inverse source problem in which the unknown contrast source (the product of the total electric field and the conductivity contrast) in the object domain is reconstructed by minimizing the object and data error using a conjugate-gradient step, after which the conductivity contrast is updated by minimizing only the error in the object. This method has been tested on a number of numerical examples using the synthetic 'measured' data with and without noise. Numerical tests indicate that the inversion method yields a reasonably good reconstruction result, and is fairly insensitive to added random noise.     

On a method of computing the gravitational fields of inhomogeneous bodies

Summary A solution of the direct gravity problem for a finite body with variable density is given. The method is based on Green's formula and is applicable when a particular solution of Poisson's equation is known. The attraction due to the body is expressed by integrals over its surface The exact solution of the direct gravity problem, as known from the theory of two-dimensional fields [1–3], is closely connected with the problem of the analytic continuation of the exterior field of the attracting mass system into its interior. In the first place, this is a problem of determining the singularities of the exterior field, their distribution within the system and their nature. This approach to the solution of the direct problem is also meaningful from the point of view of determining the characteristics of the attracting system and, therefore, also of solving the inverse problem. In the case of two-dimensional fields the methods of analytical continuation were widely developed in a series of well-known papers by V. N. Strakhov, and they are mainly based on the methods of the theory of the functions of the complex variable. These methods were also successfully applied by Tsirulskii and Golizdra [1, 2] in treating the homogeneous and inhomogeneous, two-dimensional direct problem by means of Cauchy's integrals. However, as regards three-dimensional fields a number of fundamental problems has not been solved in this respect.Dedicated to 90th Birthday of Professor Frantiek Fiala     

基于复阻尼模型水平剪切型结构的时域解析算法

近些年来提出的数值方法虽克服了应用复阻尼模型求解动力响应时可能出现的发散现象,但其计算过程繁杂且不能表达出结构动力特性及响应随结构参数的变化规律.论文参照三对角To-eplitz矩阵特征值问题的数学解法,推导出水平剪切型结构各阶自振频率、振型函数的解析形式.通过对运动方程进行Fourier变换,得到复阻尼理论下结构的传...     

A fast direct solution method for nonlinear equations in an analytic element model

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.