Inverse analysis of stochastic moment equations for transient flow in randomly heterogeneous media

We present a nonlinear stochastic inverse algorithm that allows conditioning estimates of transient hydraulic heads, fluxes and their associated uncertainty on information about hydraulic conductivity (K) and hydraulic head (h  ) data collected in a randomly heterogeneous confined aquifer. Our algorithm is based on Laplace-transformed recursive finite-element approximations of exact nonlocal first and second conditional stochastic moment equations of transient flow. It makes it possible to estimate jointly spatial variations in natural log-conductivity (Y=lnK)$(Y=\ln K)$, the parameters of its underlying variogram, and the variance–covariance of these estimates. Log-conductivity is parameterized geostatistically based on measured values at discrete locations and unknown values at discrete “pilot points”. Whereas prior values of Y at pilot point are obtained by generalized kriging, posterior estimates at pilot points are obtained through a maximum likelihood fit of computed and measured transient heads. These posterior estimates are then projected onto the computational grid by kriging. Optionally, the maximum likelihood function may include a regularization term reflecting prior information about Y. The relative weight assigned to this term is evaluated separately from other model parameters to avoid bias and instability. We illustrate and explore our algorithm by means of a synthetic example involving a pumping well. We find that whereas Y and h can be reproduced quite well with parameters estimated on the basis of zero-order mean flow equations, all model quality criteria identify the second-order results as being superior to zero-order results. Identifying the weight of the regularization term and variogram parameters can be done with much lesser ambiguity based on second- than on zero-order results. A second-order model is required to compute predictive error variances of hydraulic head (and flux) a posteriori. Conditioning the inversion jointly on conductivity and hydraulic head data results in lesser predictive uncertainty than conditioning on conductivity or head data alone.     

Multilayer shallow water flow using lattice Boltzmann method with high performance computing

A multilayer lattice Boltzmann (LB) model is introduced to solve three-dimensional wind-driven shallow water flow problems. The multilayer LB model avoids the expensive Navier–Stokes equations and obtains stratified horizontal flow velocities as vertical velocities are relatively small and the flow is still within the shallow water regime. A single relaxation time BGK method is used to solve each layer coupled by the vertical viscosity forcing term. To increase solution stability, an implicit step is suggested to obtain flow velocities. The main advantage of using the LBM is that after selecting appropriate equilibrium distribution functions, the LB algorithm is only slightly modified for each layer and retains all the simplicities of the LBM within the high performance computing (HPC) environment. The performance of the parallel LB model for the multilayer shallow water equations is investigated on CPU-based HPC environments using OpenMP. We found that the explicit loop control with cache optimization in LBM gives better performance on execution time, speedup and efficiency than the implicit loop control as the number of processors increases. Numerical examples are presented to verify the multilayer LB model against analytical solutions. We demonstrate the model’s capability of calculating lateral and vertical distributions of velocities for wind-driven circulation over non-uniform bathymetry.     

An approximate Riemann solver for sensitivity equations with discontinuous solutions

The sensitivity of a model output (called a variable) to a parameter can be defined as the partial derivative of the variable with respect to the parameter. When the governing equations are not differentiable with respect to this parameter, problems arise in the numerical solution of the sensitivity equations, such as locally infinite values or instability. An approximate Riemann solver is thus proposed for direct sensitivity calculation for hyperbolic systems of conservation laws in the presence of discontinuous solutions. The proposed approach uses an extra source term in the form of a Dirac function to restore sensitivity balance across the shocks. It is valid for systems such as the Euler equations for gas dynamics or the shallow water equations for free surface flow. The method is first detailed and its application to the shallow water equations is proposed, with some test cases such as dike- or dam-break problems with or without source terms. An application to a two-dimensional flow problem illustrates the superiority of direct sensitivity calculation over the classical empirical approach.     

利用积分方程的加速迭代算法计算随钻电磁波电阻率测量仪器的响应

采用改进型逐次逼近解法(MSAM)和Aitken加速技术相结合的迭代算法计算二维积分方程.该算法将纵向成层原状地层作为背景地层,将计算区域限制在井眼和侵入带内,具有未知量数目少、收敛速度快、计算精度高的优点.利用该算法对随钻电磁波电阻率测量仪器在轴对称二维地层中的响应进行数值模拟.模拟结果显示,幅度衰减曲线和相位移曲线受井眼、侵入和围岩的影响程度不同,二者径向探测深度和垂向分辨率也有差异,利用补偿后的幅度衰减电阻率曲线和相位移电阻率曲线交叉点的坐标可精确确定地层层界面位置.     

喜马拉雅西构造结及邻区岩石圈演化三维有限元数值模拟

用数值模拟的方法探讨了自10 Ma以来喜马拉雅弧形造山带的西构造结及邻区岩石圈的形变和位移变化特征,并采用不同的流变参数和流变结构模拟了前2 Ma内的形变和应力场的演化过程,通过对不同黏性系数、本构关系和初始地壳厚度的改变在演化过程中所起的作用研究表明：（1）采用黏弹性模型时,西构造结区黏性参数的选取对西构造结及邻区隆升高度和隆升范围有重要影响,计算结果给出西构造结区地壳黏性系数应小于10²³Pa·s;（2）相对刚性的塔里木盆地采用弹性结构时,对整个西构造结区域的应力和应变场均能引致明显变化;（3）力学性质稳定块体的存在并没有使应力场的传播出现明显的解耦现象.     

Pressure and shear stress caused by raindrop impact at the soil surface: Scaling laws depending on the water depth

Raindrop impact is an important process in soil erosion. Through its pressure and shear stress, raindrop impact causes a significant detachment of the soil material, making this material available for transport by sheet flow. Thanks to the accurate Navier–Stokes equations solver Gerris, we simulate the impact of a single raindrop of diameter D, at terminal velocity, on water layers of different thickness h: , , D, 2D, in order to study pressures and shear stresses involved in raindrop erosion. These complex numerical simulations help in understanding precisely the dynamics of the raindrop impact, quantifying in particular the pressure and the shear stress fields. A detailed analysis of these fields is performed and self‐similar structures are identified for the pressure and the shear stress on the soil surface. The evolution of these self‐similar structures are investigated as the aspect ratio h/D varies. We find that the pressure and the shear stress have a specific dependence on the ratio between the drop diameter and the water layer thickness, and that the scaling laws recently proposed in fluid mechanics are also applicable to raindrops, paving the road to obtain effective models of soil erosion by raindrops. In particular, we obtain a scaling law formula for the dependence of the maximum shear stress on the soil on the water depth, a quantity that is crucial for quantifying erosion materials. Copyright © 2016 John Wiley & Sons, Ltd.     

考虑微观结构的非饱和渗透系数计算公式

分维理论是预测非饱和渗透系数的一种常用方法。在对有侧限条件下高庙子膨润土的非饱和渗透系数的试验结果分析后,发现分维理论并不适用,其缺陷在于不能够反映膨润土这种特殊粘土在水化过程中的微观结构变化。因为膨润土是一种纳米材料,其小孔隙和大孔隙分布在水化过程中都会发生变化,而一般性粘土和砂土没有这种特殊的物理化学特性。结合Kozeny—Carman关于多孔介质的半经验公式,提出了半经验一半理论的考虑微结构的膨润土的非饱和渗透系数计算公式。在对高庙子膨润土的扫描电镜试验和压汞试验资料分析的基础上,定性验证了所提出公式的正确性。     

Analysis of water seepage in a pavement system using the particulate approach

A particulate model has been developed to analyze the effects of transient and steady state seepage of water through a randomly-packed coarse-grained soil as an alternative to conventional seepage analysis based on continuum models. In this model, the soil skeleton and the pore water are volumetrically coupled in the transient and steady-state conditions. The concept of relative density has been used to define different compaction levels of the soil layers forming a pavement filter system and observe the seepage response to compaction. First, Monte–Carlo simulation is used to randomly pack discrete spherical particles from a specified particle size distribution (PSD) to achieve a desired relative density based on the theoretical minimum and maximum void ratios. Then, a water pressure gradient is applied across one two-layer unit to trigger water seepage. The interstitial pore water motion is idealized using Navier–Stokes (NS) equations with provision to incorporate the drag forces acting between the pore fluid and soil particles. The NS equations are discretized using finite differences and applied to discrete elements in a staggered, structured grid. The model predicted hydraulic conductivities are validated using widely used equations.     

子午线弧长的计算方法及精度分析

计算子午线弧长除了采用经典的级数展开算法之外,还可通过数值积分与常微分方程数值解法进行求解。为评价各种算法的精度,本文选取大地纬度自0°-90°、间隔距离为1°、1'、1″的3组样本数据,分别基于传统算法、数值积分算法和常微分方程数值算法3大类11种算法计算得到各组样本所对应的子午线弧长结果,并从算法精度和运算速度两个方面对各种数值算法进行了分析与评价。实例表明三阶、四阶Runge-Kutta算法不仅精度高,而且运算效率是其他算法的2倍多,研究结果为计算子午线弧长的提供了有效的算法模型。