A nonlinear programming analysis of unconfined steady-state seepage

A finite element, variable mesh analysis of unconfined steady-state seepage problems is presented based on a nonlinear programming algorithm. It is shown that the minimization of an objective function which merely represents a measure of the total flux leaving or entering the mesh at the free surface nodes (except those that belong also to pervious boundaries) does not permit a unique definition of the free surface geometry. This problem, which is apparently related to the numerical instabilities often met when using variable mesh approaches, can be eliminated by adding to the objective function a term representing a sort of overall ‘regularity’ condition for the shape of the free surface. The modified solution procedure turns out to be stable and able to provide meaningful results for practical problems even when rather coarse meshes are adopted.     

A practical and efficient numerical scheme for the analysis of steady state unconfined seepage flows

The scaled boundary finite‐element method (SBFEM), a novel semi‐analytical technique, is applied to the analysis of the confined and unconfined seepage flow. This method combines the advantages of the finite‐element method and the boundary element method. In this method, only the boundary of the domain is discretized; no fundamental solution is required, and singularity problems can be modeled rigorously. Anisotropic and nonhomogeneous materials satisfying similarity are modeled without additional efforts. In this paper, SBFE equations and solution procedures for the analysis of seepage flow are outlined. The accuracy of the proposed method in modeling singularity problems is demonstrated by analyzing seepage flow under a concrete dam with a cutoff at heel. As only the boundary is discretized, the variable mesh technique is advisable for modeling unconfined seepage analyses. The accuracy, effectiveness, and efficiency of the method are demonstrated by modeling several unconfined seepage flow problems. Copyright © 2011 John Wiley & Sons, Ltd.     

无压渗流问题分析的多节点有限元方法

采用基于平均值插值的多节点有限元方法分析有自由面渗流问题。在自由面附近采用多节点单元逼近自由面,利用平均值插值建立多节点单元的形函数,在远离自由面的区域采用四边形/三角形单元剖分。给定一个初始自由面位置,通过对渗流控制方程的多节点有限元求解,根据自由面上节点水头值判断自由面节点的调整方向和大小,最终迭代求出自由面的位置。土坝渗流问题的数值分析表明了所提方法的有效性和足够的计算精度。     

裂隙网络无压渗流分析的初流量法

借鉴初流量法的思想,引入初流速来抵消在裂隙网络干区实际不存在的流速,将Darcy定理扩展到整个区域,定义了在整个区域上的非线性边值问题,并将潜在溢出面边界条件归纳为Signorini型边界条件,建律了等价的变分不等式提法。通过结合连续型的Heaviside函数,给出了基于变分不等式的初流量有限元迭代算法,研制开发了相应的计算程序,通过两个典型算例验证了本文算法在求解复杂裂隙网络渗流自由面的有效性和鲁棒性。     

On the solution of steady unconfined seepage towards semi-infinite slopes

Analytical solutions for the two-dimensional problem of unconfined seepage towards semi-infinite slopes are described. The analysis employs complex variable techniques. Conformal mappings have been used to reduce the problem to solving an ordinary differential equation. The resulting integrals are presented in terms of special functions. The solutions for the location of the phreatic surface and the exit point have been obtained in parametric form. Some additional results and asymptotic expansions of the solutions are also presented. The numerical results have been calculated and plotted for different values of the slope angle in dimensionless co-ordinate space.     

Proof of multiplicity of solutions for groundwater seepage in recharged heterogeneous unconfined aquifers

A new analytical proof is presented for steady‐state seepage in recharged heterogeneous unconfined aquifers. The paper also presents a detailed procedure and important rules for performing correctly numerical studies of unsaturated seepage. Once a numerical solution is calibrated with field data, using a set of spatially distributed values for hydraulic conductivity K and effective infiltration EI, any new numerical analysis with a set of αK and αEI values, where α is a constant, yields an equally good calibration. However, if the effective porosities of each layer are unchanged, the groundwater velocities are multiplied by α, whereas the travel times are divided by α, which may help to select α in order to match known travel time data. This is a clear example of multiple solutions to an inverse problem. The paper underlines the role and the need to finely mesh unsaturated zones and also contacts between layers to reach the asymptotic convergence range, as it was carried out to verify the proof and as it should be completed to study any seepage problem. A few consequences of the new analytical proof and the rigorous procedure are shown with examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Investigation of a hybrid polygonal finite element formulation for confined and unconfined seepage

This study presents a formulation for field problems using hybrid polygonal finite elements, taking steady state seepage through a porous material as the focus. We make comparisons with a conventional finite element formulation based on a single primary variable, focussing on the advantages of the hybrid formulation in terms of flux field accuracy and extension to convex polygonal shaped elements. For the unconfined case, we adopt a head dependent hydraulic conductivity that does not require remeshing. The performance of the hybrid polygonal element formulation is demonstrated through a series of numerical examples. The results show a sensitivity of the location of the free surface in unconfined seepage to mesh configuration for hybrid quadrilateral meshes with various aspect ratios, but not for hybrid polygonal meshes with various orientations and irregularity. Examination of the free surface location results for several conforming shape function options shows an insensitivity to choice of interpolation function, provided that it conforms with the assumptions in the formulation. Copyright © 2016 John Wiley & Sons, Ltd.     

A new method for mapping variability in vertical seepage flux in streambeds

A two-step approach was used to measure the flux across the water-sediment interface in river channels. A hollow tube was pressed into the streambed and an in situ sediment column of the streambed was created inside the tube. The hydraulic gradient between the two ends of the sediment column was measured. The vertical hydraulic conductivity of the sediment column was determined using a falling-head permeameter test in the river. Given the availability of the hydraulic gradient and vertical hydraulic conductivity of the streambed, Darcy’s law was used to calculate the specific discharge. This approach was applied to the Elkhorn River and one tributary in northeastern Nebraska, USA. The results suggest that the magnitude of the vertical flux varied greatly within a short distance. Furthermore, the flux can change direction from downward to upward between two locations only several meters apart. This spatial pattern of variation probably represents the inflow and outflow within the hyporheic zone, not the regional ambient flow systems. In this study, a thermal infrared camera was also used to detect the discharge locations of groundwater in the streambed. After the hydraulic gradient and the vertical hydraulic conductivity were estimated from the groundwater spring, the discharge rate was calculated.     

A novel finite element method for seepage analysis

A novel finite element method has been proposed in this paper for the solution of seepage problems economically and accurately. In this method the governing equation and the prescribed boundary conditions are transformed so that they refer to a suitable logarithmically condensed ‘image’ space; the physical problem domain is also mapped into the image space. The transformed equation is then solved in the image space using standard finite elements, subject to the transformed boundary conditions. Because physical space is logarithmically condensed in the image space, the proposed method is capable of dealing with large or very large aspect ratio seepage problems economically and accurately. The validity of the method has been demonstrated by means of a number of examples including anisotropy and non-linearity. In all cases an excellent degree of accuracy was achieved, efficiently and economically.     

非承压含水层底部单孔疏放水渗流特征

由含水层底部向上施工疏放水孔（井）进行疏水降压是保证煤矿安全开采的重要措施之一,该类疏放水孔与地面抽水井的渗流特征具有显著差别。为了研究井下疏放水孔的渗流特征,以非承压含水层底部单井疏放水孔为例,采用数值模拟的方法对其渗流特征进行研究。研究结果表明,当疏放水井的井长l_w小于临界长度l_c时,即l_w<l_c,疏放水井上部含水层孔隙水压力大于0,且由下而上呈先增加后减小的分布规律,渗流量与井长呈指数函数关系增加;当l_w ≥ l_c时,疏放水非完整井的渗流特征及渗流量与完整井相同;渗流量随井半径增加呈指数关系增加,且指数小于1;将s+l_w代替水位降深s对裘布依（Dupuit）潜水完整井计算公式进行修正,当l_w<l_c时,采用修正的Dupuit潜水完整井计算公式计算疏放水非完整井渗流量更精确。该研究结果对认识井下疏放水井上部孔隙水压力分布特征及合理布置疏放水井具有重要的借鉴意义。     

Singularity elements for seepage analysis

A technique for locating the nodal points of standard isoparametric quadrilateral elements properly around a discharge velocity singularity point in a soil seepage flow problem is presented. It is based on the analytical solution of the problem in an infinite medium and gives simple formulae for calculating the nodal coordinates. The effectiveness of the method is justified with examples.     

Parametric infinite element for seepage analysis

A set of mapping functions in the form of convergent series for an infinite element, which is capable to include the infinitely distanced constant head boundary condition from the area of disturbance (e.g. pumping), is proposed based on the asymptotic far-field behaviour of typical seepage flow problems. The derived mapping functions have been successfully used in three-dimensional point symmetric, two-dimensional axi-symmetric and one-dimensional unidirectional flow for the fixed head boundary at infinite distance. The result shows excellent agreement with analytical solution. For the first time, the mapping function of an infinite element is presented in the form of a convergent series. The infinite elements are really capable of reducing the cost and efficiency of conventional finite element analysis. Finally, a figure is also proposed to indicate the required size of the near field to obtain accurate drawdown at specified locations based on some calculations for two-dimensional radial flow case.     

Steady flow rate to a partially penetrating well with seepage face in an unconfined aquifer

The flow rate to fully screened, partially penetrating wells in an unconfined aquifer is numerically simulated using MODFLOW 2000, taking into account the flow from the seepage face and decrease in saturated thickness of the aquifer towards the well. A simple three-step method is developed to find the top of the seepage face and hence the seepage-face length. The method is verified by comparing it with the results of previous predictive methods. The results show that the component of flow through the seepage face can supply a major portion of the total pumping rate. Variations in flow rate as a function of the penetration degree, elevation of the water level in the well and the distance to the far constant head boundary are investigated and expressed in terms of dimensionless curves and equations. These curves and equations can be used to design the degree of penetration for which the allowable steady pumping rate is attained for a given elevation of water level in the well. The designed degree of penetration or flow rate will assure the sustainability of the aquifer storage, and can be used as a management criterion for issuing drilling well permits by groundwater protection authorities.     

二维各向异性岩土介质中渗流分析的等参有限元方法

为了建筑环境的保护,软土地区基坑围护中的坑内降水与坑外水位保持常是一对矛盾的课题.而坑内外的水位差,极易形成止水结构两侧的渗流运动.对饱和各向异性、水平延伸、厚度大致相等、且存在一、二类边界时的承压含水层中的承压水二维非稳定流渗流问题,本文推导出了二维、各向异性渗流等参有限元方程,用以进行基坑围护止水结构附近渗流场分析,并给出了算例.     

潜水回水初始阶段的定性半定量分析及对水跃成因的解释

文章提出并采用水力坡度分析方法,对潜水回水初始阶段进行了定性-半定量分析,为了解潜水回水过程提供了新的视角。通过分析潜水回水初始时刻的水力坡度,河渠侧的界面被分为2段:河间潜水位之上的铅直段和之下的水平段。水平段上点的水力坡度最大且相等,方向为水平向右;铅直段上点的水力坡度方向趋于铅直向下,越向上水力坡度越小。经过微小时间段Δt后,水力坡度均相应变小,铅直段相邻点间地下水运动发生冲突,水力坡度方向由趋于铅直向下向右偏转。将水力坡度分析方法用于分析潜水井流的过程和特征,水跃的成因可解释为:在相同水力坡度和过水断面面积下,井内大空间水流速度远大于孔隙中的潜水流速度,井内水位急速下降,从而形成水跃;水跃越大,进入井内的流量就越大。     

地下水位变化对透—阻型岩溶塌陷影响的分析

文章从一维地下水运动和渗透力学的角度,分析比较潜水位上升与承压水位下降对岩溶地区透—阻型盖层中阻水层渗透稳定性的影响,重点讨论了承压水位下降速度(降速)与下降深度（降深）对阻水层中渗透坡降的影响,结果表明:地下水位变化（潜水位上升或承压水位下降）产生的非稳定渗流不利于岩溶洞穴开口上方阻水层的稳定,承压水位的下降对岩溶开口附近处阻水层中渗透力的影响远大于潜水位的变化;在承压层水位最大降深确定的条件下,承压水位下降速度愈快,岩溶开口附近处阻水层中向下渗透力愈大。因此,在覆盖型岩溶地区抽取地下水时,为了减缓或避免覆盖型塌陷的发生,应同时控制好最大降深和最大开采速度。      

复杂地形尾矿坝的三维渗流分析

对于复杂地形的尾矿坝,由于二维模型不能充分反映复杂多变的渗流域,也不能考虑到三个方向渗流因素同时作用的影响,从而无法体现真正的渗流场。通过建立三维渗流数学模型,并考虑复杂断面概化和地形的简化,对实际工程进行了数值计算,其结果与试验结果吻合较好。结果表明,对复杂地形的尾矿坝渗流分析时必须采用三维模型,三维数值计算中对复杂地形进行适当的简化和概化对结果的影响较小,可以满足精度要求,因此大大减小了计算的复杂程度和难度。     

An improved buffer analysis technique for model-based 3D mineral potential mapping and its application

Buffer zones in bo\th two-dimensional (2D) and three-dimensional (3D) spaces are commonly used in prospectivity mapping. The method completes a modelling that starts with a real example and progresses to the development of a virtual model. This includes the consideration of lithological or structural contacts at depth, which is a theoretical concept based on extrapolation of data collected in the field, rather than an empirical observation of the feature based on physical samples. This contribution documents an improved buffer analysis method for the study of 3D-space that is implicit (rapid), precise (smooth) and based on triangulated characteristics, which can be used to construct influence domains of geological models. As traditional 2D GIS-based mineral potential mapping is gradually becoming limited with time, mineral potential mapping in three dimensions (3D) is increasingly becoming an important tool in finding concealed economic mineralization. This contribution documents an improved methodology of buffer analysis for prospectivity mapping processing mineralized favourable models rather than describing an advance in the geometry of surface rendering of “geological complexity”. Measures used in this buffer analysis include the: (1) voxelization of geological objects (i.e. assigning numerical values of features on a regular cube in 3D-space); (2) revision of the 3D Euclidean distance transform and the calculation of signed distance field; (3) extracting surfaces from the field; and (4) construction of a buffer-surface based on a “discrete smooth interpolation” (DSI) algorithm. Furthermore, this contribution constructs 3D models using a buffer analysis algorithm and prospectivity mapping introduced here, which is based on real data from the Jiama Cu-polymetallic deposit in Tibet and Daye Fe deposit in the Hubei Province, China. This contribution also presents a comparison between voxel and irregular triangle models, which illustrate that irregular triangle mesh buffer analysis (ITB) can improve modelling techniques for GIS-based 3D mineral potential mapping. The outcome is an increase in the accuracy of prospectivity mapping.     

浑水渗流理论及平原水库防渗技术研究

浑水渗流不同于清水渗流,在其渗流过程中,能够形成一层弱透水性的防渗层,减少渗漏水量。根据浑水渗流理论,提出采用引黄河浑水入库在库底形成淤积层来减少水库的渗漏。在分析了黄河浑水泥沙特点的基础上,研究黄河浑水引入平原水库形成的淤积层对库底渗漏的影响,并经室内试验研究得出渗漏量与淤积层的相关关系、不同水头对渗流量的影响关系,得出一定厚度的淤积层能够起到较好的防渗效果。将研究成果应用于平原水库库底防渗,经理论计算和工程应用评价,淤积层防渗效果能够满足设计要求,可为同类工程提供参考     

有自由面渗流分析的高斯点法

有自由面的渗流问题是比较复杂的非线性问题，这类问题的有限元求解一般采用固定网格法。本文根据饱和区和非饱和区的渗流特点，提出了高斯点法，该算法原理简单，易于实现。