Polynomial approximate solutions to the Boussinesq equation

We provide closed-form approximate solutions to models of horizontal infiltration described by the Boussinesq equation in a semi-infinite aquifer that is initially dry. The approximations preserve such important qualitative properties as scaling and wetting fronts. They are applicable to four types of boundary conditions, two on head and two on flux, enumerated in the paper. All the considered problems admit self-similar variables that allow reduction to boundary value problems for a nonlinear ordinary differential equation. This work extends recent results by Lockington et al. [Lockington DA, Parlange J-Y, Parlange MB, Selker J. Similarity solution of the Boussinesq equation. Adv Water Resour 2000;23(7):725–9] and Telyakovskiy et al. [Telyakovskiy AS, Braga GA, Furtado F. Approximate similarity solutions to the Boussinesq equation. Adv Water Resour 2002;25(2):191–4], with new approximations developed for two of the four cases and a new extension of a previously existing method for a third case. Numerical results extending the work of Shampine [Shampine LF. Some singular concentration dependent diffusion problems. ZAMM 1973;53:421–2] provide a basis for assessing the accuracy of the new methods.     

A semi-analytical time integration for numerical solution of Boussinesq equation

A semi-analytical time integration method is proposed for the numerical simulation of transient groundwater flow in unconfined aquifers by the nonlinear Boussinesq equation. The method is based on the analytical solution of the system of ordinary differential equations with constant coefficients. While it is unconditionally stable and more accurate than the finite difference methods, the computational cost is much more expensive than (can be more than 10 times) that of the finite difference methods for a single time step. However, by partitioning the nonlinear parameters into linear and nonlinear parts, the costly computation can be performed only once. With larger and less variable time step sizes, the total computational cost can be significantly reduced. Three examples are included to illustrate the advantages and limitations of the proposed method.     

Two-dimensional power series solution for non-axisymmetrical transport in a radially convergent tracer test with scale-dependent dispersion

It has been known for many years that dispersivity increases with solute travel distance in a subsurface environment. The increase of dispersivity with solute travel distance results from the significant variation of hydraulic properties of heterogeneous media and was identified in the literature as scale-dependent dispersion. This study presents an analytical solution for describing two-dimensional non-axisymmetrical solute transport in a radially convergent flow tracer test with scale-dependent dispersion. The power series technique coupling with the Laplace and finite Fourier cosine transform has been applied to yield the analytical solution to the two-dimensional, scale-dependent advection–dispersion equation in cylindrical coordinates with variable-dependent coefficients. Comparison between the breakthrough curves of the power series solution and the numerical solutions shows excellent agreement at different observation points and for various ranges of scale-related transport parameters of interest. The developed power series solution facilitates fast prediction of the breakthrough curves at any observation point.     

Analytical power series solutions to the two-dimensional advection–dispersion equation with distance-dependent dispersivities

As is frequently cited, dispersivity increases with solute travel distance in the subsurface. This behaviour has been attributed to the inherent spatial variation of the pore water velocity in geological porous media. Analytically solving the advection–dispersion equation with distance-dependent dispersivity is extremely difficult because the governing equation coefficients are dependent upon the distance variable. This study presents an analytical technique to solve a two-dimensional (2D) advection–dispersion equation with linear distance-dependent longitudinal and transverse dispersivities for describing solute transport in a uniform flow field. The analytical approach is developed by applying the extended power series method coupled with the Laplace and finite Fourier cosine transforms. The developed solution is then compared to the corresponding numerical solution to assess its accuracy and robustness. The results demonstrate that the breakthrough curves at different spatial locations obtained from the power series solution show good agreement with those obtained from the numerical solution. However, owing to the limited numerical operation for large values of the power series functions, the developed analytical solution can only be numerically evaluated when the values of longitudinal dispersivity/distance ratio e_L exceed 0·075. Moreover, breakthrough curves obtained from the distance-dependent solution are compared with those from the constant dispersivity solution to investigate the relationship between the transport parameters. Our numerical experiments demonstrate that a previously derived relationship is invalid for large e_L values. The analytical power series solution derived in this study is efficient and can be a useful tool for future studies in the field of 2D and distance-dependent dispersive transport. Copyright © 2008 John Wiley & Sons, Ltd.     

Modelling of water table variation in response to time-varying recharge from multiple basins using the linearised Boussinesq equation

A mathematical model is presented to describe the variations of the water table in an unconfined aquifer due to time-varying recharge applied from four rectangular basins. The model is developed by solving the linearised Boussinesq equation using the extended finite Fourier cosine transform. The time-varying recharge rate is approximated by a number of piecewise linear elements of different lengths and slopes depending on the nature of the variation in recharge rate. Application of this model for the prediction of water table fluctuations and in the sensitivity analysis of various controlling parameters on the aquifer response is demonstrated in an example.     

An alternative Boussinesq equation considering the effect of hysteresis on coastal groundwater waves

This paper presents an alternative Boussinesq equation considering hysteresis effect via a third‐order derivative term. By introducing an improved moisture–pressure retention function, this equation describes, with reasonable precision, groundwater propagation in coastal aquifers subject to Dirichlet boundary condition of different oscillation frequencies. Test results confirmed that it is necessary to consider horizontal and vertical flows in unsaturated zone, because of their variable influences on hysteresis. Hysteresis in unsaturated zone can affect the water table wave number of groundwater wave motion, such as wave damping rate and phase lag. Oscillations with different periods exert different hysteresis effect on wave propagation. Truncation/shrinkage of unsaturated zones also affects the strength of hysteresis. These impacts can be reflected in the alternative Boussinesq equation by adjusting the parameter representing the variation rate of moisture associated with pressure change, as opposed to traditional computationally expensive hysteresis algorithms. The present Boussinesq equation is simple to use and can provide feasible basis for future coupling of groundwater and surface water models. Copyright © 2016 John Wiley & Sons, Ltd.     

Exact analytical solutions for two-dimensional advection-dispersion equation in cylindrical coordinates subject to third-type inlet boundary condition

Exact analytical solutions for two-dimensional advection-dispersion equation (ADE) in cylindrical coordinates subject to the third-type inlet boundary condition are presented in this study. The finite Hankel transform technique in combination with the Laplace transform method is adopted to solve the two-dimensional ADE in cylindrical coordinates. Solutions are derived for both continuous input and instantaneous slug input. The developed analytical solutions are compared with the solutions for first-type inlet boundary condition to illustrate the influence of the inlet condition on the two-dimensional solute transport in a porous medium system with a radial geometry. Results show significant discrepancies between the breakthrough curves obtained from analytical solutions for the first-type and third-type inlet boundary conditions for large longitudinal dispersion coefficients. The developed solutions conserve the solute mass and are efficient tools for simultaneous determination of the longitudinal and transverse dispersion coefficients from a laboratory-scale radial column experiment or an in situ infiltration test with a tracer.     

位场曲化平积分方程的迭代解

提出了位场曲化平的新方法. 给定观测曲面S上的位场、S对下方水平面P的相对高程,确定P上的位场. 利用由P向上延拓到S的积分式,建立这两个面上位场及相对高程三者所满足的方程,它是第一类Fredholm积分方程. 用Fourier逆变换式把这一空间域积分式化为波数域积分式,再由指数函数的Taylor展开进一步化为级数式. 积分方程的解采用逐次逼近法迭代计算,即用S上的位场观测值作为P上位场的初始迭代值,用导出的级数式求得S上的位场计算值、由S上的位场观测值与计算值之差校正P上的位场,多次迭代,直到满足迭代终止准则. 我们还给出该积分方程的波数域迭代计算方法. 模型算例表明,重力异常曲化平的均方差和磁异常曲化平的均方差分别为0.0008 mGal和0.0019 nT,在主频为2.26 GHz的笔记本电脑运行,2048×2048数据量,计算时间是975 s. 野外磁场实际资料处理也证实这种方法的有效性.     

Optimal weighting in the finite difference solution of the convection-dispersion equation

Oscillation and numerical dispersion limit the reliability of numerical solutions of the convection-dispersion equation when finite difference methods are used. To eliminate oscillation and reduce the numerical dispersion, an optimal upstream weighting with finite differences is proposed. The optimal values of upstream weighting coefficients numerically obtained are a function of the mesh Peclet number used. The accuracy of the proposed numerical method is tested against two classical problems for which analytical solutions exist. The comparison of the numerical results obtained with different numerical schemes and those obtained by the analytical solutions demonstrates the possibility of a real gain in precision using the proposed optimal weighting method. This gain in precision is verified by interpreting a tracer experiment performed in a laboratory column.     

基于脉冲频响函数的MATLAB动力方程求解方法

提出一种基于脉冲频响函数并利用MATLAB求解的方法，由于脉冲频响函数利用卷积公式进行动力微分方法计算，因此，从理论上讲，它适合于任意外力下的动力方程求解，同时，MATAL又提供了卷积函数conv，所以，在MATLAB下求解动力方程将变得十分容易。     

On the implementation of Prabhu’s exact solution of the stochastic reservoir equation

After 50 years of Prabhu’s paper on the exact solution of the stochastic reservoir equation for the important class of gamma inflow distributions with an integral shape parameter, a detailed implementation of the exact solution is still lacking, despite its potential usefulness from both theoretical and practical points of view. This paper explores some properties of Prabhu’s exact solution and investigates the numerical difficulties associated with its implementation. The solution is also extended to derive the distributions of deficit, spillage, yield, and actual release from the reservoir. Explicit analytical solutions for three relatively simple cases are given in detail as examples and comparisons with approximate numerical solutions are made, which reveal some shortcomings of approximate methods. The implementation of the solution in the general case reveals some numerical problems associated with large values of the shape parameter of the inflow distribution and large ratios of reservoir size to draft, mainly due to accumulation of round-off errors. A Matlab program has been developed to calculate emptying and filling probabilities over a wide range of reservoir parameters using extended precision. Comparison of Prabhu’s solution with the numerical solution of the reservoir integral equation highlights possible problems with the numerical solution, which may produce inaccurate or even invalid results for large reservoirs, large drift, and large skewness of the inflow distribution. A comparison between gamma and lognormal distributions as models of skew revealed that as the reservoir size, drift, and skewness increase, the probability of emptying of the reservoir becomes smaller for the case of gamma inflow than in the case of lognormal flow having the same skewness coefficient.     

含峭壁V形峡谷对地震SH波散射的解析解

地表地形常引起地震动的局部放大,这是由于地震波传播至局部地形时产生了散射现象.本文利用波函数展开方法和区域匹配技术,提出了含峭壁V形峡谷对平面SH波散射问题的解析解,并进行了退化验证.通过频域内的参数分析,揭示了峭壁深度、入射波频率和角度等因素对峡谷场地地面运动的影响规律,发现上部峭壁会增强峡谷对地震动的地形放大效应.研究结果不仅为数值方法提供了验证基准,还可为含峭壁峡谷周边建筑物的抗震设计提供顺河向地震动输入.

     

A series solution for surface motion ampli-fication due to underground group cavities: Incident P waves

Introduction In recent years some big cities in China commonly build subways in central urban areas to improve crowded surface transportation, and the subways usually are large-scale group cavities composed by two parallel cavities. Construction of large-scale group cavities will certainly affect the design ground motion near the cavities, and further affect seismic safety of the existing build-ings nearby. So it is of significance to estimate this effect on seismic design of the buildings. H…     

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.     

A new solution and data analysis for gas flow to a barometric pumping well

When an open well is installed in an unsaturated zone, gas will flow to/from the well depending on the pressure difference between the well and the surrounding media. This process is called barometric pumping and the well is called a barometric pumping well (BPW). Understanding subsurface gas pressure distribution and gas flow rate to/from a BPW is indispensable to optimize passive soil vapor extraction. This study presents a 2-D semi-analytical solution to calculate the subsurface gas pressure and gas flow rate to/from a BPW with and without a check valve. The problem is conceptualized as a mixed-type boundary value problem. The solution for pumping without a check valve is used to analyze the behavior of the radius of influence (ROI). Results show that ROI is time-dependent. It increases with radial gas permeability and decreases with vertical gas permeability. Field application of the solution without a check valve demonstrates the high accuracy of the developed solution.     

Application of the iterative solution method with Schur complement reduction to mixed finite elements based in a tetrahedral discretization

The iterative solution method for mixed finite element methods is applied to a 3-D domain partitioned with tetrahedral elements. For the particular discretization technique of first partitioning the domain with hexahedral cells, and then subsequently partitioning cells with five tetrahedral elements, a Schur complement decomposition is devised wherein the actual number of equations solved is reduced by 80%. Although this Schur complement reduction requires a fair amount of computational overhead, its application within the iterative solution method can reduce overall solution time by about 44%, depending on closure criterion and other factors.     

水平界面上P-SV转换波转换点的精确解

转换波共转换点的叠加和道集选取都需要准确地计算转换点的位置.Tessmer和Behle、Taylor分别给出了水平单层介质中P-SV转换波转换点坐标的解析解,由于其表达式的复杂性,在应用中几乎不被采用.在本文中,运用Snell定律重新建立了在水平反射界面上反射的P-SV转换波的转换点坐标的四次方程,并严格地推导出与纵波速度、横波速度、炮检距和反射深度有关的转换点坐标的解析解,确定了惟一的解析表达式.将这一结果应用于P-SV转换波的速度分析和叠加处理中.简化的公式有较好的应用价值.     

大型积分方程降阶解法与重力资料曲面延拓

分析了等效源曲面延拓方法积分方程核函数的自相似性、冗余性,根据Fredholm积分方程核函数特征,提出了一种小波余弦非线性阈值压缩算法,实现了大型Fredholm积分方程的降阶,使得Bhattacharyya等（1977年）提出的等效源曲面延拓方法能够处理大面积、大数据量的资料. 理论模型结果表明,当压缩比为40.5%时硬阈值压缩方法的曲面延拓可以达到很高的精度,当压缩比为81.1%时, 硬阈值压缩方法精度降低,而采用我们提出的余弦非线性阈值可以明显提高曲面延拓的精度. 将硬阈值、软阈值和非线性阈值压缩三种不同方法用于川东北气田MT－1线资料的处理,当压缩比达79%时,三种不同方法曲化平结果都可以达到很好的效果,但非线性阈值压缩的曲化平结果失真最小,它能够客观地反映杨家河局部隆起,为在该区寻找与油气有关的局部构造提供重要依据.