An analytical solution for non-Darcian flow in a confined aquifer using the power law function

We have developed a new method to analyze the power law based non-Darcian flow toward a well in a confined aquifer with and without wellbore storage. This method is based on a combination of the linearization approximation of the non-Darcian flow equation and the Laplace transform. Analytical solutions of steady-state and late time drawdowns are obtained. Semi-analytical solutions of the drawdowns at any distance and time are computed by using the Stehfest numerical inverse Laplace transform. The results of this study agree perfectly with previous Theis solution for an infinitesimal well and with the Papadopulos and Cooper’s solution for a finite-diameter well under the special case of Darcian flow. The Boltzmann transform, which is commonly employed for solving non-Darcian flow problems before, is problematic for studying radial non-Darcian flow. Comparison of drawdowns obtained by our proposed method and the Boltzmann transform method suggests that the Boltzmann transform method differs from the linearization method at early and moderate times, and it yields similar results as the linearization method at late times. If the power index n and the quasi hydraulic conductivity k get larger, drawdowns at late times will become less, regardless of the wellbore storage. When n is larger, flow approaches steady state earlier. The drawdown at steady state is approximately proportional to r¹⁻ⁿ, where r is the radial distance from the pumping well. The late time drawdown is a superposition of the steady-state solution and a negative time-dependent term that is proportional to t^{(1−n)/(3−n)}, where t is the time.     

Hybrid Laplace transform finite element method for solving the convection–dispersion problem

It can be very time consuming to use the conventional numerical methods, such as the finite element method, to solve convection–dispersion equations, especially for solutions of large-scale, long-term solute transport in porous media. In addition, the conventional methods are subject to artificial diffusion and oscillation when used to solve convection-dominant solute transport problems. In this paper, a hybrid method of Laplace transform and finite element method is developed to solve one- and two-dimensional convection–dispersion equations. The method is semi-analytical in time through Laplace transform. Then the transformed partial differential equations are solved numerically in the Laplace domain using the finite element method. Finally the nodal concentration values are obtained through a numerical inversion of the finite element solution, using a highly accurate inversion algorithm. The proposed method eliminates time steps in the computation and allows using relatively large grid sizes, which increases computation efficiency dramatically. Numerical results of several examples show that the hybrid method is of high efficiency and accuracy, and capable of eliminating numerical diffusion and oscillation effectively.     

Numerical analysis of multicomponent responses of surface-hole transient electromagnetic method

We calculate the multicomponent responses of surface-hole transient electromagnetic method. The methods and models are unsuitable as geoelectric models of conductive surrounding rocks because they are based on regular local targets. We also propose a calculation and analysis scheme based on numerical simulations of the subsurface transient electromagnetic fields. In the modeling of the electromagnetic fields, the forward modeling simulations are performed by using the finite-difference time-domain method and the discrete image method, which combines the Gaver–Stehfest inverse Laplace transform with the Prony method to solve the initial electromagnetic fields. The precision in the iterative computations is ensured by using the transmission boundary conditions. For the response analysis, we customize geoelectric models consisting of near-borehole targets and conductive wall rocks and implement forward modeling simulations. The observed electric fields are converted into induced electromotive force responses using multicomponent observation devices. By comparing the transient electric fields and multicomponent responses under different conditions, we suggest that the multicomponent-induced electromotive force responses are related to the horizontal and vertical gradient variations of the transient electric field at different times. The characteristics of the response are determined by the varying the subsurface transient electromagnetic fields, i.e., diffusion, attenuation and distortion, under different conditions as well as the electromagnetic fields at the observation positions. The calculation and analysis scheme of the response consider the surrounding rocks and the anomalous field of the local targets. It therefore can account for the geological data better than conventional transient field response analysis of local targets.     

Discontinuous Galerkin methods for simulating bioreactive transport of viruses in porous media

Primal discontinuous Galerkin (DG) methods are formulated to solve the transport equations for modeling migration and survival of viruses with kinetic and equilibrium adsorption in porous media. An entropy analysis is conducted to show that DG schemes are numerically stable and that the free energy of a DG approximation decreases with time in a manner similar to the exact solution. Combining results for free and attached virus concentrations, we establish optimal a priori error estimates for the coupled partial and ordinary differential equations of virus transport. Numerical results suggest that DG can treat bioreactive transport of viruses over a wide range of modeling parameters, including both advection- and dispersion-dominated problems. In addition, it is shown that DG can sharply capture local phenomena of virus transport with dynamic mesh adaptation.     

Transient vibrations of an isotropic elastic sphere

Summary An attempt is made to study the dynamic response of a thick, homogeneous isotropic elastic sphere under the action of uniformly distributed internal and external pressure distributions which vary sinusoidally with time. An exact solution of the problem is obtained with the aid of the Laplace transform method and the theory of residues. Some main properties of the radial vibration field are examined. The frequency equation for the free vibration is derived explicitly. Several limiting cases of interest are recovered.     

Errors in response calculations for beams

When the finite element method is used to idealize a structure, its dynamic response can be determined from the governing matrix equation by the normal mode method or by one of the many approximate direct integration methods. In either method the approximate data of the finite element idealization are used, but further assumptions are introduced by the direct integration scheme. It is the purpose of this paper to study these errors for a simple structure. The transient flexural vibrations of a uniform cantilever beam, which is subjected to a transverse force at the free end are determined by the Laplace transform method. Comparable responses are obtained for a finite element idealization of the beam, using the normal mode and Newmark average acceleration methods; the errors associated with the approximate methods are studied. If accuracy has priority and the quantity of data is small, the normal mode method is recommended; however, if the quantity of data is large, the Newmark method is useful.     

Analytical model for fully three‐dimensional radial dispersion in a finite‐thickness aquifer

Analytical solutions for contaminant transport in a non‐uniform flow filed are very difficult and relatively rare in subsurface hydrology. The difficulty is because of the fact that velocity vector in the non‐uniform flow field is space‐dependent rather than constant. In this study, an analytical model is presented for describing the three‐dimensional contaminant transport from an area source in a radial flow field which is a simplest case of the non‐uniform flow. The development of the analytical model is achieved by coupling the power series technique, the Laplace transform and the two finite Fourier cosine transform. The developed analytical model is examined by comparing with the Laplace transform finite difference (LTFD) solution. Excellent agreements between the developed analytical model and the numerical model certificate the accuracy of the developed model. The developed model can evaluate solution for Peclet number up to 100. Moreover, the mathematical behaviours of the developed solution are also studied. More specifically, a hypothetical convergent flow tracer test is considered as an illustrative example to demonstrate the three‐dimensional concentration distribution in a radial flow field. The developed model can serve as benchmark to check the more comprehensive three‐dimensional numerical solutions describing non‐uniform flow contaminant transport. Copyright © 2009 John Wiley & Sons, Ltd.     

Uniform half-plane elastodynamic problems by an approximate boundary element method

In this paper the applicability of an approximate Boundary Element Method to uniform half-plane elastodynamic problems is investigated. This method employs the concept of images to construct approximate fundamental solutions for the half-plane and does not require any half-plane surface discretization. The method is formulated in the frequency domain for the case of harmonic disturbances or the Laplace transform domain for the case of transient disturbances. In the latter case a numerical inversion of the transformed solution is necessary to obtain the time domain response. The proposed method can be used as an alternative to boundary element methods that either utilize the infinite plane fundamental solution and thus require a half-plane surface discretization, or employ the exact half-plane fundamental solution, which even though leads to no surface discretization, is of a very lengthy and complicated form. Two characteristics numerical examples are used to illustrate the proposed method and study its advantages and disadvantages.     

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.     

G-S变换的快速算法

在电磁场瞬变响应的数值计算中 ,常采用G S变换法作逆拉氏变换 .它是纯实数运算 ,而且只需对较少的拉氏变换变量s值作计算 （通常对每一采样时间选用 1 2个s值 ） ,因而是一种计算速度较快的算法 .但是 ,要对大量采样时间作计算 ,其计算量仍太大 .本文基于拉氏变换的延迟定理 ,建立了一种新的G S变换算法 .数值检验结果表明 ,新算法可成级次地减少对大量采样时间作G S变换的计算量 ,显著提高电磁场瞬变响应的计算速度 .     

Dynamic poroelastic soil column and borehole problem analysis

The one-dimensional dynamic column and borehole problems of soil mechanics formulated on the basis of the poroelastic theory of Vardoulakis and Beskos are solved analytically-numerically. The quasi-static counterparts of these problems are analysed as special cases of the dynamic ones. Use of Laplace transform with respect to time reduces the column and borehole problems to ordinary differential equations with constant and variable coefficients, respectively. The transformed solution of these problems is obtained analytically for the column and by finite differences for the borehole problem, and after, a numerical Laplace transform inversion produces the time domain response. Both a suddenly applied and a harmonically varying with time load are considered. It is concluded that the significance of inertial effects depends on the kind of loading and that the degree of saturation for the nearly saturated case greatly affects the response.     

Note on the propagation of waves in infinite spherically-aeolotropic medium produced by a blast in a spherical cavity

Summary By using the Laplace transform method, solution has been obtained for the elastodynamic problem due to blast pulse on the inner surface of a spherical cavity in an infinite spherically aeolotropic elastic solid. Both types of blast producing radial and rotational waves have been considered.     

A closed-form solution for a double infinite Euler-Bernoulli beam on a viscoelastic foundation subjected to harmonic line load

The dynamic response of a double infinite beam system connected by a viscoelastic foundation under the harmonic line load is studied. The double infinite beam system consists of two identical and parallel beams, and the two beams are infinite elastic homogeneous and isotropic. A viscoelastic layer connects the two beams continuously. To decouple the two coupled equations governing the response of the double infinite beam system, a variable substitution method is introduced. The frequency domain solutions of the decoupled equations are obtained by using Fourier transforms as well as Laplace transforms successively. The time domain solution in the generalized integral form are then obtained by employing the corresponding inverse transforms, i.e. Fourier transform and inverse Laplace transform. The solution is verified by numerical examples, and the effects of parameters on the response are also investigated.     

Errors in response calculations for extensional vibrations of bars

The transient extensional vibrations of a slender and uniform bar, which is clamped at one end and is subjected to an axial force at the other free end, are investigated by applying three methods: the Laplace transform method, the normal mode and Newmark β methods in conjunction with the finite element method (FEM); the errors caused by the spatial discretization of the FEM and the direct integration of the Newmark β method are studied and compared with those of the previous paper where the flexural vibrations of a cantilever beam were considered. The reason why the extensional vibration problem is investigated here is that the condition seems to be severe due to the closeness of adjacent natural frequencies, the larger values of natural frequencies and the smaller number of nodal variables of the FEM in comparison with those of the flexural vibration problem. The numerical results show that the errors in response of the extensional vibration problem are large. However, if one follows the criterion proposed in this paper, accurate response is obtainable by the Newmark β method, which requires less computer time than either of the other methods mentioned above.     

A fundamental solution due to a periodic point force in the interior of an elastic half-space

The fundamental solution for a periodic point force in the interior of a three-dimensional, homogeneous, isotropic, elastic half-space is derived. The method of synthesis and superposition is employed to obtain the solution in the Laplace transform as well as the frequency domain. These correspond to the dynamic equivalent of Mindlin's static half-space point force solutions. It is reduced, for certain limiting conditions, to the dynamic equivalent of Boussinesq's and Cerruti's problems of a normal and tangential periodic point force respectively, on the boundary of a half-space. Also, static solutions of Mindlin, Boussinesq and Cerruti are recovered for small frequency parameters. Finally, results are presented and compared with other available solutions.     

Displacement produced in an elastic half-space by the impulsive torsional motion of a circular ring source

Summary In this paper the problem of disturbance in an elastic semi-infinite medium due to the torsional motion of a circular ring source on the free surface of a medium are studied. Two cases, when the medium is either homogeneous or inhomogeneous, are treated. In order to solve the problem, the Laplace transform and the Hankel transform and the Laplace inversion by Cagniard's method as modified byDe Hoop (1959) are applied. Finally, the integrals for displacement are evaluated numerically. The displacement on the free surface as a function of time is shown by means of graphs, in the case of both a homogeneous and an inhomogeneous medium, indicating clearly the variation in displacement due to the presence of an inhomogeneity.     

A Neumann expansion approach to flow through heterogeneous formations

A stochastic approach is used for the study of flow through highly heterogeneous aquifers. The mathematical model is represented by a random partial differential equation in which the permeability and the porosity are considered to be random functions of position, defined by the average value, constant standard deviation and autocorrelation function characterized by the integral scale. The Laplace transform of the solution of the random partial differential equation is first written as a solution of a stochastic integral equation. This integral equation is solved using a Neumann series expansion. Conditions of convergence of this series are investigated and compared with the convergence of the perturbation series. For mean square convergence, the Neumann expansion method may converge for a larger range of variability in permeability and porosity than the classic perturbation method. Formal expressions for the average and for the correlation moments of the pressure are obtained. The influence of the variability of the permeability and porosity on pressure is analyzed for radial flow. The solutions presented for the pressure at the well, as function of the permeability coefficient of variation, may be of practical interest for evaluating the efficiency of well stimulation operations, such as hydraulic fracturing or acidizing methods, aimed at increasing the permeability around the well.     

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.     

电磁波对轴对称二维层状介质的散射

本文讨论了任意N层平面分层,每层又可具有任意柱面分层的轴对称二维非均匀有耗介质结构,分析电磁波在其中的传播、散射特性.由于采用电磁场的数值模式匹配理论,径向用离散数值本征模式替代连续模,纵向则用模式匹配方法导出场递推关系的解析表达式,大大减少了数值计算量,而计算精度不变.场的数值结果可很好的再现地下电参数剖面.相应的数值分析程序可用于从感应测井到电磁波测井的计算机辅助解释.