承压含水层中非完整井附近“非达西-达西”两区渗流模型近似解析解

抽水井附近由于流速过快往往发生非达西流,而远离抽水井随着流速下降又变为达西流.为了描述这些特征,建立了承压含水层中非完整井附近“非达西-达西”两区渗流模型,即距离抽水井较近的区域由于流速较快假设发生非达西渗流,并利用Izbash公式刻画,而距离抽水井较远由于流速较慢假设仍然满足达西定律,含水层中垂向流速较小也利用达西定律描述.通过线性化近似方法结合Laplace变换和有限Fourier余弦变换对模型进行了求解,分析探讨了该两区模型下水位降深曲线特征.结果表明:抽水初期,非达西渗流区域水位降深与全非达西渗流模型结果吻合,而抽水后期两区模型非达西渗流区域的水位降深与全达西模型水位降深基本一致,但大于全非达西渗流模型的水位降深;抽水初期,两区模型中达西渗流区域的水位降深比全达西渗流模型结果大,但比全非达西渗流模型结果小;对不同时间的水位降深随井距变化曲线分析发现非达西渗流区域水位降深随Izbash公式中的幂指数n增大而减小,而在达西渗流区域水位降深基本不受n值的影响.研究成果对非完整井抽水试验参数反演具有重要理论意义.      

Approximate analytical and numerical solutions for radial non‐Darcian flow to a well in a leaky aquifer with wellbore storage and skin effect

This study investigated non‐Darcian flow to a well in a leaky aquifer considering wellbore storage and a finite‐thickness skin. The non‐Darcian flow is described by the Izbash equation. We have used a linearization procedure associated with the Laplace transform to solve such a non‐Darcian flow model. Besides, the Stehfest method has been used to invert the Laplace domain solutions for the drawdowns. We further analyzed the drawdowns inside the well for different cases. The results indicated that a smaller B_D results in a smaller drawdown at late times and the leakage has little effect on the drawdown inside the well at early times, where B_D is a dimensionless parameter reflecting the leakage. We have also found that the flow for the negative skin case approaches the steady‐state earlier than that for the positive skin. In addition, the drawdown inside the well with a positive skin is larger than that without skin effect at late times, and a larger thickness of the skin results in a greater drawdown inside the well at late times for the positive skin case. A reverse result has been found for the negative skin case. Finally, we have developed a finite‐difference solution for such a non‐Darcian flow model and compared the numerical solution with the approximate analytical solution. It has been shown that the linearization procedure works very well for such a non‐Darcian flow model at late times, and it underestimates the drawdowns at early times. Copyright © 2012 John Wiley & Sons, Ltd.     

Non-Darcian flow to a partially penetrating well in a confined aquifer with a finite-thickness skin

Non-Darcian flow to a partially penetrating well in a confined aquifer with a finite-thickness skin was investigated. The Izbash equation is used to describe the non-Darcian flow in the horizontal direction, and the vertical flow is described as Darcian. The solution for the newly developed non-Darcian flow model can be obtained by applying the linearization procedure in conjunction with the Laplace transform and the finite Fourier cosine transform. The flow model combines the effects of the non-Darcian flow, partial penetration of the well, and the finite thickness of the well skin. The results show that the depression cone spread is larger for the Darcian flow than for the non-Darcian flow. The drawdowns within the skin zone for a fully penetrating well are smaller than those for the partially penetrating well. The skin type and skin thickness have great impact on the drawdown in the skin zone, while they have little influence on drawdown in the formation zone. The sensitivity analysis indicates that the drawdown in the formation zone is sensitive to the power index (n), the length of well screen (w), the apparent radial hydraulic conductivity of the formation zone (K _r2), and the specific storage of the formation zone (S _s2) at early times, and it is very sensitive to the parameters n, w and K _r2 at late times, especially to n, while it is not sensitive to the skin thickness (r _s).     

Non-Darcian flow to a well in a leaky aquifer using the Forchheimer equation

Non-Darcian flow to a well in a leaky aquifer was investigated using a finite difference method. Flow in the leaky aquifer is assumed to be non-Darcian and horizontal, while flow in the aquitard is assumed to be Darcian and vertical. The Forchheimer equation was employed to describe the non-Darcian flow in the aquifer. The finite difference solution was compared with the solution of Birpinar and Sen (2004). The latter overestimates the drawdown at early times and underestimates the drawdown at late times; also, the impact of β _D on the drawdown depends on the value of B _D, where β _D is a dimensionless turbulent factor in the Forchheimer equation and B _D is the dimensionless leakage parameter. The impact of leakage on drawdown is similar to that of Darcian flow. A sensitivity analysis indicated that the drawdown is very sensitive to the change in the dimensionless well radius r _cD when B _D is relatively large, while it is sensitive to the change in B _D when B _D is relatively small. The numerical solution has been applied to analyze the pumping test data in Chaj-Doab area of Pakistan. Birpinar ME, Sen Z (2004) Forchheimer groundwater flow law type curves for leaky aquifers. J Hydrol Eng 9(1):51??9     

Analytical solutions for constant-rate test in bounded confined aquifers with non-Darcian effect

This paper proposes a simplified analytical solution considering non-Darcian and wellbore storage effect to investigate the pumping flow in a confined aquifer with barrier and recharge boundaries. The mathematical modelling for the pumping-induced flow in aquifers with different boundaries is developed by employing image-well theory with the superposition principle, of which the non-Darcian effect is characterized by Izbash’s equation. The solutions are derived by Boltzmann and dimensionless transformations. Then, the non-Darcian effect and wellbore storage are especially investigated according to the proposed solution. The results show that the aquifer boundaries have non-negligible effects on pumping, and ignoring the wellbore storage can lead to an over-estimation of the drawdown in the first 10 minutes of pumping. The higher the degree of non-Darcian, the smaller the drawdown.     

侧向有限越流承压含水层中非完整井非稳定流模型及解析解

工程建设中当距离抽水井r=rb处水位基本没有变化或不受抽水影响时,或当此处存在止水帷幕时,含水层系统视为侧向有限延伸,rb为有限半径。为此,构建更加符合工程实际的侧向有限延伸的典型弱透水层-承压水层系统中非完整井非稳定流计算模型,同时考虑井径和井储效应的影响,应用Laplace变换和分离变量法得到了水位降深在拉氏空间下的解析解,并应用拉氏数值逆变换Stehfest法得到真实空间下的水位降深。新建立的解析解可以进一步退化为诸多已有解,并进一步将其与已知解和有限元数值解进行对比,验证了所得解的正确性和可靠性。基于新建解重点分析了侧向边界和井的完整性对承压水层水位降深的影响。结果表明:含水层系统的侧向有限边界仅对抽水后期的水位降深影响明显,含水层系统侧向无限延伸情况下的水位降深要大于情形1(在r=rb处为定水头边界)且明显小于情形2(在r=rb处为不透水边界)下的水位降深,rb越小,两者之间的误差越大;抽水井的完整性对整个抽水期间不同情形下的水位降深均有明显的影响,承压含水层顶板处的水位降深随着抽水井滤管的长度和埋深的增加而减小。     

Numerical modeling of Forchheimer flow to a pumping well in a confined aquifer using the strong-form mesh-free method

A numerical analysis of non-Darcian flow to a pumping well in a confined aquifer using the strong-form mesh-free (MFREE) method is described. This technique is targeted at problems that use advanced numerical approaches for modeling non-Darcian flow and it supports the assumption that the non-Darcian flow follows the Forchheimer equation. Interpolation functions including the multi-quadrics (MQ) basis function (containing shape factors q and α) and the Gaussian (EXP) basis function (with shape factor ω) were found to be important defining parameters which had significant influence on the numerical results. A series of numerical experiments revealed that when q?=?2 and α?=?0.1, the mesh-free method yielded good results and the range of 10^?6?–?10^?3 might be a good choice for the shape factor ω in the EXP basis function. A comparison between the strong-form MFREE method and the finite difference method was done; the results showed that the strong-form MFREE method was very effective for solving non-Darcian flow near a pumping well in a confined aquifer, and was favorable over the finite-difference method, which could undergo oscillation and converging problems at early times.     

A new solution for a partially penetrating constant‐rate pumping well with a finite‐thickness skin

A mathematical model describing the constant pumping is developed for a partially penetrating well in a heterogeneous aquifer system. The Laplace‐domain solution for the model is derived by applying the Laplace transforms with respect to time and the finite Fourier cosine transforms with respect to vertical co‐ordinates. This solution is used to produce the curves of dimensionless drawdown versus dimensionless time to investigate the influences of the patch zone and well partial penetration on the drawdown distributions. The results show that the dimensionless drawdown depends on the hydraulic properties of the patch and formation zones. The effect of a partially penetrating well on the drawdown with a negative patch zone is larger than that with a positive patch zone. For a single‐zone aquifer case, neglecting the effect of a well radius will give significant error in estimating dimensionless drawdown, especially when dimensionless distance is small. The dimensionless drawdown curves for cases with and without considering the well radius approach the Hantush equation (Advances in Hydroscience. Academic Press: New York, 1964) at large time and/or large distance away from a test well. Copyright © 2007 John Wiley & Sons, Ltd.     

How appropriate is the Thiem equation for describing groundwater flow to actual wells?

The Thiem equation of radial groundwater flow to a well is more than 100 years old and is still commonly used. Here, deviations caused by some of its simplifications are quantified by comparing the analytical to a numerical model that allows the implementation of more complex geometries. The assumption of horizontal flow in the Thiem equation, which necessitates uniform inflow over the entire screen length of the fully penetrating well, was found to cause deviations from actual pumping wells where the pump is placed above the screen, resulting in non-uniform inflow and additional drawdown. The same applies to partially penetrating wells, where inflow peaks and additional drawdown occur, especially when the well is screened in the lower part of the aquifer. The use of the Thiem equation in the near-field of a well should thus be restricted to situations where the screen inflow is relatively uniformly distributed, e.g. when it covers large portions of the aquifer thickness. The presence of a gravel pack and a background gradient, on the other hand, are of limited importance.     

水资源和环境工程中水平井研究简介

简单介绍了水平井在水资源和环境工程中的近期研究进展, 重点介绍了在不同含水层(如承压含水层、潜水含水层、越流含水层及河下含水层) 中, 水平井抽水条件下降深的半解析解, 同时分析了降深随抽水时间变化的标准曲线和微分标准曲线.这些半解析解可用于分析小流量水平井在中长时段的降深特性.分析了用于排水和供水的大流量水平井的水力学特征, 并介绍了求解渗流-管流耦合井流系统的一种新方法.同时介绍了在非饱和含水层中水平井抽取气体的动力学特征, 并分析了地面覆盖和未覆盖2种情况下的气体清除效率, 报告了在不同饱和含水层条件下水平井的捕获区和捕获时间的计算.      

An approach to interpretation of step drawdown tests

To understand the behaviour of drawdown with discharge under site-specific conditions, a step drawdown test (SDT) was conducted on a tubewell in alluvium near Delhi, India, and the data were analyzed to find the value of well efficiency. The entrance velocity of groundwater into the well screen was computed to see if it exceeded the safety limit of 0.03 m/s. Reynold's number (Re) was also calculated to study the nature of flow at the well screen. The decrease in efficiency of the well at higher discharge was attributed to an increase in turbulence and curvature of the flow path of water. An alternative, more efficient approach has been put forward, using a spreadsheet programme to fit a polynomial trend line on a plot of drawdown versus discharge and deriving a polynomial trend line equation. This equation predicts the behaviour of drawdown with discharge under site-specific conditions. The calculated efficiency of the tubewell can, at best, be regarded as a reflection of head loss on account of the laminar flow from the aquifer.     

A “fractal” modification of Torricelli’s formula

A modification is proposed of Torricelli’s (1608–1647) formula for the velocity of water discharging from a small hole at the bottom of a large tank filled with fractal solid material. The new formula takes proper account of the mechanical energy losses due to flow in the solid matrix, thus expanding the area of validity of the classical Torricelli’s formula. Moreover, it offers a convenient alternative to Darcy’s law for estimating the discharge rate from an aquifer. The new formula was derived from laboratory experiments, with a low-Reynolds number discharge flow (Darcian flow). It was tested in a natural karst aquifer where the flow is non-Darcian, at Almiros spring on the island of Crete (Greece). In both cases, the predictive capability of the modified formula is established.     

Field experiments and numerical simulations of whirlpool foundation pit dewatering

A whirlpool foundation pit is a small-diameter, deep circular pit. Because of its depth and small diameter, a large drawdown is required, and a limited number of wells can be installed inside the pit. During excavation, partially penetrating wells inside and outside the foundation pit have to be installed to lower the water level when the aquifer is too thick. However, partially penetrating wells near partially penetrating curtains cannot be treated by analytical methods. Therefore, it is necessary to use numerical methods to predict dewatering during excavation. Field experiments were performed on whirlpool foundation pit 1880 of Baosteel Group, Shanghai, China, to obtain pumping rates and drawdown, pumping with a single well and two wells in the confined aquifer. The results indicate that the drawdown inside the pit induced by pumping wells outside the foundation pit was small, whereas it was large for pumping wells inside the pit. The pumping wells inside and outside the pit had to be combined to lower the water level. A three-dimensional numerical model was developed to simulate the dewatering process. The hydraulic conductivities of the confined aquifers were inversed by using the pumping tests. Operation schedules were simulated with the corrected model for different combinations of wells inside and outside the pit. The results suggest that different schedules and operation conditions affect drawdown. The monitored results during dewatering indicate that the simulation and field measurements were in agreement. The results can be applied to similar situations.     

非达西裂隙流对渗透性基岩中流场及溶质羽的影响

以基岩渗流方向与裂隙轴向呈45度角为例,探讨了当裂隙轴向与基岩水流斜交时,裂隙流的非达西程度对流场及溶质运移的影响.使用Comsol Multiphysics多物理场仿真软件构建了一个在中部包含水平单裂隙的正方形多孔介质模型,裂隙中的非达西流用Izbash方程去刻画,裂隙的上游基岩中存在持续的溶质源.随着裂隙水流的非达西程度逐渐增强,流场及污染物分布表现出如下特征:（1）裂隙中水流流速逐渐增大;流线在裂隙与基岩界面处的折射逐渐偏离折射定律;（2）裂隙水流的流向逐渐偏向基岩水流的渗流方向;（3）溶质羽宽度变宽但对称性逐渐降低;（4）溶质在水平方向上的浓度峰值逐渐降低,右侧浓度逐渐升高;（5）裂隙产生的回弥散对溶质运移作用逐渐增强,使裂隙中更多的溶质运移到了上层基岩中.总体而言,裂隙流的非达西程度对流场及污染物分布有着显著的影响.      

承压含水层井流-盖层弯曲效应的解析理论

指出承压含水层盖层的弯曲变形与开采井周围的径向地下水运动存在相互作用, 而这一效应在传统的井流理论中没有被认识到.通过引入弹性薄板理论, 建立了无越流的承压含水层井流-顶板弯曲效应的解析模型, 同时考虑了含水层和水的压缩性, 结果表明Theis井流方程给出的抽水降深偏小.在此基础上推导了有越流承压含水层井流-盖层弯曲效应的偏微分方程, 求出了解析解, 并与传统理论的结果进行了对比, 表明Hantush-Jacob公式计算的降深也是偏小的.在抽水井附近和抽水初期, 传统理论可能导致显著的相对误差.      

Analysis of Steady-State Flow To Multiscreened Wells Under Natural Vertical Flow Conditions

Water levels measured at multiscreened wells in unconfined aquifers may not coincide, in general, with the elevation of the water table. The presence of vertical gradients (as often is the case in recharge areas) or the existence of confining layers may cause the water levels to differ from local hydraulic heads in the aquifer. In these cases, a misinterpretation of water levels may lead to the erroneous conclusion that observed drawdowns are provoked by overpumping. In this paper, we analyze the effect that a natural vertical gradient has on water levels in wells screened over their entire saturated thickness. As one would expect, it is observed that, even without pumping, the water level in the wells lies below the water table. Type curves relating the steady-state drawdown to the vertical gradient and to the hydraulic conductivity anisotropy are presented. These curves were obtained using a groundwater flow numerical model (FREESURF: Neuman and Witherspoon, 1970). The theoretical results are checked with field data from deep wells in the detrital Madrid aquifer. In this particular aquifer, it is observed that the effect of vertical gradients is important both in terms of drawdowns and flow rates.

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Identification of a hydrodynamic threshold in karst rocks from the Biscayne Aquifer, south Florida, USA

A hydrodynamic threshold between Darcian and non-Darcian flow conditions was found to occur in cubes of Key Largo Limestone from Florida, USA (one cube measuring 0.2 m on each side, the other 0.3 m) at an effective porosity of 33% and a hydraulic conductivity of 10 m/day. Below these values, flow was laminar and could be described as Darcian. Above these values, hydraulic conductivity increased greatly and flow was non-laminar. Reynolds numbers (Re) for these experiments ranged from <0.1 to 7. Non-laminar flow conditions observed in the hydraulic conductivity tests were observed at Re close to 1. Hydraulic conductivity was measured on all three axes in a permeameter designed specifically for samples of these sizes. Positive identification of vertical and horizontal axes as well as 100% recovery for each sample was achieved. Total porosity was determined by a drying and weighing method, while effective porosity was determined by a submersion method. Bulk density, total porosity and effective porosity of the Key Largo Limestone cubes averaged 1.5 g/cm³, 40 and 30%, respectively. Two regions of anisotropy were observed, one close to the ground surface, where vertical flow dominated, and the other associated with a dense-laminar layer, below which horizontal flow dominated.     

基坑潜水疏干的简化计算在昊华水泥厂中的应用

以流线、流面、汇点的概念为基础,对稳定流双井干扰和直线隔水边界附近涌水量理论公式进行对比分析,提出了二个虚拟界面,其中虚拟界面Ⅰ,运用流线、流面的性质,流线方程等给出证明;虚拟界面Ⅱ则通过半无限条形降落漏斗的分析,应用元流和总流的能量方程得到流量为零,流线为零的平面。在同样条件下,条形无限涌水量是半无限潜含水层涌水量的二倍。应用总流能量方程对三种情况水头损失的分析,解释了这种关系存在的合理性,得出虚拟界面Ⅱ,并以此得出该界面内的最大残余水头计算公式。将基坑降水运用虚拟界面简化为扇形,条形半无限含水层,从而实现单井预测,该方法应用到昊华水泥厂基坑降水中,预测效果理想。     

Anisotropy and bedding effects on the hydro geological regime in a confined aquifer to design an appropriate dewatering system

Prediction of groundwater inflow into mining excavations is very important in order to design an effective dewatering system to keep the mine workings dry and create prolonged cone of depression. The effects of anisotropy ratio and bedding on the hydraulic head and drawdown curves of a dewatering test carried out in a fully penetrating well in a confined aquifer have been investigated. An existing numerical finite element model has been used to perform the simulations. The results of the numerical model are compared to those from analytical Jacob and Lohman solution for estimating hydraulic heads and drawdown curves. It was found that the anisotropy ratio and bedding should not have a significant effect on drawdown and the quantity of inflow into a confined aquifer. It was further found that taking the simultaneous effects of anisotropy and bedding into account reduces the differences in the results of analytical and numerical methods. Comparison of the field data and model predictions showed that, the modelling results for a three layer anisotropic aquifer fit well to the field data than those results obtained for a single layer aquifer and the relative error decreased from 4.81 % to 2.98 %.     

Analytical solutions for the capture zone of a pumping well near a stream

The capture zone for a fully penetrating well in an aquifer with regional flow to a stream boundary under steady-state conditions was delineated using complex algebra and image well theory. Regional flow in the aquifer was allowed to take different directions relative to the stream axis. Two critical pumping rates, Q _C1 and Q _C2, produce three capture-zone pattern scenarios: (1) at low pumping rates (Q?<?Q _C1) water is solely withdrawn from the aquifer and no water from the stream enters the aquifer, (2) at medium pumping rates (Q _C1?<?Q?<?Q _C2) a portion of stream water enters the aquifer but it is not captured by the well, and (3) at high pumping rates (Q?>?Q _C2) pumped water is supplied from both the aquifer and the stream with different proportions. For the second and third scenarios, the stream length interval through which stream water enters the aquifer was determined and found to be more sensitive to pumping rate as the regional flow direction approaches the stream axis. The portion of pumped water supplied by the stream was determined in the third scenario. Finally, the capture-zone asymmetry with respect to its axial line was delineated.