Uniform asymptotic Green's functions for efficient modeling of cracks in elastic layers with relative shear deformation controlled by linear springs

We present a uniform asymptotic solution (UAS) for a displacement discontinuity (DD) that lies within the middle layer of a three‐layer elastic medium in which relative shear deformation between parallel interfaces is controlled by linear springs. The DD is assumed to be normal to the two interfaces between the elastic media. Using the Fourier transform method we construct a leading term in the asymptotic expansion for the spectral coefficient functions for a DD in a three‐layer‐spring medium. Although a closed‐form solution will require a solution in terms of an infinite series, we demonstrate how this UAS can be used to construct highly efficient and accurate solutions even in the case in which the DD actually touches the interface. We compare the results using the Green's function UAS solution for a crack crossing a soft interface with results obtained using a multi‐layer boundary element method. We also present results from an implementation of the UAS Green's function approach in a pseudo‐3D hydraulic fracturing simulator to analyze the effect of interface shear deformation on the fracture propagation process. These results are compared with field measurements. Copyright © 2008 John Wiley & Sons, Ltd.     

Note on the Generalized Hansen and Laplace Coefficients

Recently, Breiter et al. [Celest. Mech. Dyn. Astron., 2004, 88, 153–161] reported the computation of Hansen coefficients X _k ^{γ ,m} for non-integer values of γ. In fact, the Hansen coefficients are closely related to the Laplace b _s ^(m), and generalized Laplace coefficients b _s,r ^(m) [Laskar and Robutel, 1995, Celest. Mech. Dyn. Astron., 62, 193–217] that do not require s,r to be integers. In particular, the coefficients X ₀ ^{γ ,m} have very simple expressions in terms of the usual Laplace coefficients b _{γ +2} ^(m), and all their properties derive easily from the known properties of the Laplace coefficients.     

Modelling poroelastic hollow cylinder experiments with realistic boundary conditions

A general poroelastic solution for axisymmetrical plane strain problems with time dependent boundary conditions is developed in Laplace domain. Time‐domain results are obtained using numerical inversion of the Laplace transform. Previously published solutions can be considered as special cases of the proposed solution. In particular, we could reproduce numerical results for solid and hollow poroelastic cylinders with suddenly applied load/pressure (Rice and Cleary, Rev. Geophys. Space Phys. 1976; 14 :227; Schmitt, Tait and Spann, Int. J. Rock Mech. Min. Sci. 1993; 30 :1057; Cui and Abousleiman, ASCE J. Eng. Mech. 2001; 127 :391). The new solution is used to model laboratory tests on thick‐walled hollow cylinders of Berea sandstone subjected to intensive pressure drawdown. In the experiments, pressure at the inner boundary of the hollow cylinder is observed to decline exponentially with a decay constant of 3–5 1/s. It is found that solutions with idealized step‐function type inner boundary conditions overestimate the induced tensile radial stresses considerably. Although basic poroelastic phenomena can be modelled properly at long time following a stepwise change in pressure, realistic time varying boundary conditions predict actual rock behaviour better at early time. Experimentally observed axial stresses can be matched but appear to require different values for α and ν than are measured at long time. The proposed solution can be used to calculate the stress and pore pressure distributions around boreholes under infinite/finite boundary conditions. Prospective applications include investigating the effect of gradually changing pore pressure, modelling open‐hole cavity completions, and describing the phenomenon of wellbore collapse (bridging) during oil or gas blowouts. Copyright © 2004 John Wiley & Sons, Ltd.     

Radial groundwater flow to a finite diameter well in a leaky confined aquifer with a finite‐thickness skin

A mathematical model that describes the drawdown due to constant pumpage from a finite radius well in a two‐zone leaky confined aquifer system is presented. The aquifer system is overlain by an aquitard and underlain by an impermeable formation. A skin zone of constant thickness exists around the wellbore. A general solution to a two‐zone leaky confined aquifer system in Laplace domain is developed and inverted numerically to the time‐domain solution using the modified Crump (1976) algorithm. The results show that the drawdown distribution is significantly influenced by the properties and thickness of the skin zone and aquitard. The sensitivity analyses of parameters of the aquifer and aquitard are performed to illustrate their effects on drawdowns in a two‐zone leaky confined aquifer system. For the negative‐skin case, the drawdown is very sensitive to the relative change in the formation transmissivity. For the positive‐skin case, the drawdown is also sensitive to the relative changes in the skin thickness, and both the skin and formation transmissivities over the entire pumping period and the well radius and formation storage coefficient at early pumping time. Copyright © 2009 John Wiley & Sons, Ltd.     

解扩散方程初边值问题的数值边界元积分方程法

在应用中(如数值天气预报等),经常需要对时空偏微分方程进行数值求解,通常大多采用有限差分计算或有限元法,虽然它们是应用得最广的数值解法(如差分方法),但也有某些局限和不足,本文提出了一种边界元积分方程法。作为示例,我们对扩散方程的初边值问题进行了基本原理和方法的讨论,其基本思想是通过积分变换,消除对时间的依赖性,再在变换空间中,用边界元法对积分后的方程进行数值处理,最后用数值逆变换以完成该问题的数值求解,本方法可对更为复杂的依赖于时间的方程进行类似处理,它具有不同于传统有限差分法和有限单元法特点的优越性,可供有关工作者解初边值问题试用和参考。     

改进的 Clough-Penzien地震地面运动模型

简要介绍了Kanai-Tajimi模型、Clough-Penzien模型和改进的Kanai-Tajimi模型。基于Clough-Penzien和改进的Kanai-Tajimi模型，提出了改进的Clough-Penzien模型。利用拉普拉斯变换。导出了改进的Clough-Penzien模型。比较了Kanal-Tajimi模型、Clough-Penzien模型和改进的Kanai-Tajimi模型和改进的Clough-Penzien模型的统计特性。     

Laplace method to investigate subsurface geologic structures and its application

The ages of layered strata show a particular spatial distribution, and the bedding plane is isopotential surface of a scalar T(x, y, z) correlative to the age of a given stratigraphic horizon. The scalar T is obtained as the solution of boundary value problem of Laplace equation in power series, and the solution describes the three-dimensional geologic structures within an analytical space bounded by faults or unconformities on which the value of T jumps. The solution is termed the horizon function and consists of the datum succession and the structural part. The datum succession defines the relationship between the relative age (time-scale; T) and the thickness of strata (spatial scale; z) for undeformed horizontal strata. Geologic structures are described by the structural part of the solution composed by Fourier series, and the coefficients included there are determined by the least-square method using the dip and strike defined by particular combinations of the derivatives of T or the horizon data obtained in the area. The undersampled nature of geologic data is overcome by the selection of solution type, Eigenvalues, and boundary condition. Geological map and cross-sections are composed quantitatively and automatically by combining the spatial distribution of investigated T with the digital map defining the landform of the area. Test results were examined from this point of view. Improvement of the structural part to have the result fit completely the measured data is possible by introduction of the multiplying polynomials, although this concerns mathematical nature of the potential T.     

基于常函数的抗几何变换的矢量地理数据水印算法

基于常函数对于某些变换具有的不可变性,本文对抗几何变换的矢量地理数据水印算法进行研究。首先,对常函数的概念及其特征进行阐述,并推证了一个抗旋转、平移和放缩攻击的常函数;然后,基于建立的常函数,通过把水印信息嵌入到几何不变量中,提出了一种抗几何变换的矢量地理数据水印算法;最后,对提出的水印算法的鲁棒性进行了实验分析。实验结果表明,提出的算法能抵抗压缩、增点、删点、裁剪、平移、旋转、放缩以及由以上攻击组成的复合攻击,具有较好的鲁棒性。     

总强度磁异常各阶垂向导数换算新方法

磁异常垂直导数换算在磁性目标解释推断过程中具有重要意义。分析了总强度磁异常各阶垂向导数频率转换因子的滤波特性,指出常规傅立叶变换法在求解总强度磁异常各阶垂向导数时,观测数据中高频噪声会显著放大,甚至会淹没掉真实信息。从理论上证明了总强度磁异常沿垂直方向的积分和各阶导数均为调和函数,在此基础上,提出联合采用空间域和频率域运算求解其沿垂直方向的各阶导数。该方法的基本思路是：利用频率域转换关系,计算平面上总强度磁异常沿垂直方向的积分值;选择三点二阶中心差分或双三次样条曲线函数法计算总强度磁异常垂向积分值沿水平方向的二阶导数;最后将二阶水平导数代入拉普拉斯方程求解出总强度磁异常的垂向一阶导数;以总强度磁异常及其垂向一阶导数沿水平方向的二阶导数为基础,结合拉普拉斯方程,可进一步求解出总强度磁异常任意阶垂向导数。同时,为了研究提出方法的有效性,采用球体磁场模型进行验证,并首次推导了地磁场方向和磁化强度方向不一致时,球体总强度磁异常沿垂直方向的一阶和二阶导数表达式。经研究表明：提出的方法换算得到的垂向导数结果精度明显优于常规傅立叶变换法换算结果,且具有较强的抗噪能力,尤其是在计算高阶导数时效果更加明显。     

MODIS数据并行计算方法研究与实现

详细分析了MODIS数据辐射定标、投影变换实际操作过程中的并发性,提出了三种不同的任务分解策略,实现了两种预处理操作,并对试验数据进行分析比较,发现投影变换更适合于并行化处理,而按行和按列任务分解的效率要高于按波段的任务分解。