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.     

Two-region non-Darcian flow toward a well in a confined aquifer

We have derived an analytical solution for two-region flow toward a well in a confined aquifer based on a linearization method. The two-region flow includes Izbash non-Darcian flow near the well and Darcian flow in the rest of the aquifer. The wellbore storage is also considered. The type curves in the non-Darcian and Darcian flow domains are obtained by a numerical Laplace inversion method incorporated in MATLAB programs. We have compared our results with the one-region Darcian flow model (Theis). Our solutions agree with those of Sen [Sen Z. Type curves for two-region well flow. J Hydr Eng 1988;114(12):1461–84] which were obtained using the Boltzmann transform at late times for fully turbulent flow, while some difference has been found at early and moderate times. We have defined a dimensionless non-Darcian hydraulic conductivity term which is shown to be a key parameter for analyzing the two-region flow. A smaller dimensionless non-Darcian hydraulic conductivity results in a larger drawdown in the non-Darcian flow region at late times. However, the dimensionless non-Darcian hydraulic conductivity does not affect the slope of the dimensionless drawdown versus the logarithmic dimensionless time in the non-Darcian flow region at late times. The dimensionless non-Darcian hydraulic conductivity does not affect the late time drawdown in the Darcian flow region.     

Spatio-temporal modeling of PM2.5 concentrations with missing data problem: a case study in Beijing,China

ABSTRACT

One of the major challenges in conducting epidemiological studies of air pollution and health is the difficulty of estimating the degree of exposure accurately. Fine particulate matter (PM_2.5) concentrations vary in space and time, which are difficult to estimate in rural, suburban and smaller urban areas due to the sparsity of the ground monitoring network. Satellite retrieved aerosol optical depth (AOD) has been increasingly used as a proxy of ground PM_2.5 observations, although it suffers from non-trivial missing data problems. To address these issues, we developed a multi-stage statistical model in which daily PM_2.5 concentrations can be obtained with complete spatial coverage. The model consists of three stages – an inverse probability weighting scheme to correct non-random missing patterns of AOD values, a spatio-temporal linear mixed effect model to account for the spatially and temporally varying PM_2.5-AOD relationships, and a gap-filling model based on the integrated nested Laplace approximation-stochastic partial differential equations (INLA-SPDE). Good model performance was achieved from out-of-sample validation as shown in R² of 0.93 and root mean square error of 9.64 μg/m³. The results indicated that the multi-stage PM_2.5 prediction model proposed in the present study yielded highly accurate predictions, while gaining computational efficiency from the INLA-SPDE.     

Sloshing waves in electrified fluid layers in an excited rectangular container

This work aims to demonstrate the dynamical properties of the free interface separating two bounded layers of electrified Newtonian fluids inside a parametrically excited boxed basin. The mathematical model of the investigated problem is contacted with the linearized Navier–Stokes equation of viscous fluids and Maxwell equations together with the boundary conditions that are solved utilizing Laplace transform. Solutions of the governing equations in time domain are then numerically calculated by using Durbin's numerical inverse Laplace transform scheme. Graphical illustrations of the accomplished numerical results are provided to examine the influence of some selected parameters on the stability picture of the interfacial waves as well as the electric surface charge distribution. The numerical results proposed that the electric Euler number Eu, promotes the instability of the wave motion whatever the values of the frequency of oscillation and the destabilizing effect of Eu becomes weaker at larger values of the frequency; while a dual effect (in) stabilizing of the frequency of oscillations is reported whether the electric effects exist or not. This irregular influence of the frequency oscillation has also been monitored regarding to the forces exerted on the side wall of the container.     

三维完全非线性波浪水槽的数值模拟

用有限元求解拉普拉斯方程,建立了三维完全非线性数值波浪水槽.跟踪流体自由表面的方法为满足完全非线性自由表面条件的半拉格朗日法,对离散单元采用20节点的六面体二次等参数单元.并把数值计算结果与水面初始升高产生箱体内流体运动解析解和二阶斯托克斯波理论解进行了对比,结果表明该模型是稳定的、守恒的,能精确模拟非线性波浪的产生和传播.     

Three-Dimensional Boundary Element Method Applied to Nonlinear Wave Transformation

For higher accuracy in simulating the transformation of three dimensional waves,in consid-eration of the advantages of constant panels and linear elements,a combined boundary elements is appliedin this research.The method can be used to remove the transverse vibration due to the accumulation ofcomputational errors.A combined boundary condition of sponge layer and Sommerfeld radiation condi-tion is used to remove the reflected waves from the computing domain.By following the water particle onthe water surface,the third order Stokes wave transform is simulated by the numerical wave flume tech-nique.The computed results are in good agreement with theoretical ones.     

GEO卫星Hill方程的分析型解法

提出并实现了一种求解静地轨道GEO(geostationary orbit)卫星Hill方程摄动解的分析型方法。根据地固坐标系下GEO卫星运动的特点,在其定点处把摄动力进行泰勒展开,通过选择特定的参考轨道获得了GEO卫星的地球扁率摄动解。在此基础上,成功地将拉普拉斯变换运用于GEO卫星的Hill方程,得到了一组可用于求解摄动分析解的递推积分公式。通过逐次趋近的方法,利用这组积分公式可以有效地实现由低阶解推求高阶解。     

重力扰动对惯性导航系统的位置误差影响分析

从重力学和牛顿力学的基本概念出发,给出了包含重力扰动影响的惯导误差力学编排方程,以单通道惯导系统为例,讨论了三种变化情况下,由垂线偏差引起的惯导位置误差及其误差传播特性,并以分辨率为1′×1′的某区域垂线偏差数据为背景场进行仿真。由仿真结果可以看出,在设定航线上,垂线偏差引起的惯导系统水平误差最大可达3 km。     

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 .     

用于势场反演的特殊解法

本文以无旋场的积分路径为射线,以边界上已知的势值为输入、输出量,利用反拉东变换求一个梯度场的势函数,用反演的方法求解可以化为拉普拉斯方程的一类二阶偏微分方程。其思路新颖,原理独特,运算正确,边界条件处理方便,数值实现容易。在重力场和地电场的研究中可能得到应用。