An analytical solution for predicting the transient seepage from a subsurface drainage system

Subsurface drainage systems have been widely used to deal with soil salinization and waterlogging problems around the world. In this paper, a mathematical model was introduced to quantify the transient behavior of the groundwater table and the seepage from a subsurface drainage system. Based on the assumption of a hydrostatic pressure distribution, the model considered the pore-water flow in both the phreatic and vadose soil zones. An approximate analytical solution for the model was derived to quantify the drainage of soils which were initially water-saturated. The analytical solution was validated against laboratory experiments and a 2-D Richards equation-based model, and found to predict well the transient water seepage from the subsurface drainage system. A saturated flow-based model was also tested and found to over-predict the time required for drainage and the total water seepage by nearly one order of magnitude, in comparison with the experimental results and the present analytical solution. During drainage, a vadose zone with a significant water storage capacity developed above the phreatic surface. A considerable amount of water still remained in the vadose zone at the steady state with the water table situated at the drain bottom. Sensitivity analyses demonstrated that effects of the vadose zone were intensified with an increased thickness of capillary fringe, capillary rise and/or burying depth of drains, in terms of the required drainage time and total water seepage. The analytical solution provides guidance for assessing the capillary effects on the effectiveness and efficiency of subsurface drainage systems for combating soil salinization and waterlogging problems.     

An experimental study of water table recharge by seepage losses from a ditch with intermittent flow

Farmed catchments in the Mediterranean area often exhibit dense networks of ditches which are also preferential zones of water table recharge, and thereby of groundwater contamination. This study presents an experimental analysis of seepage losses and related groundwater recharge patterns during a typical Mediterranean runoff event at the scale of a ditch located above a shallow water table. The objectives were (i) to evaluate the patterns of water table recharge by seepage in a ditch, (ii) to study the main flow processes occurring during recharge, and (iii) to estimate solute propagation in case of contaminated flow in the ditch. The field observation indicated three major points. Firstly, they showed that seepage losses during a runoff event in a ditch can rapidly lead to a significant recharge of a shallow water table. Secondly, the recharge induces a groundwater mound much larger than the event plume. The infiltrated water and the accompanying solutes remained in the vicinity of the ditch. The patterns of groundwater recharge and contamination appeared very different. Lastly, both unsaturated and saturated‐piston flow processes were observed which suggests that a variably‐saturated flow modelling approach ought to be used to simulate the ditch‐water shallow table interaction. Finally, the study indicates that the patterns of water table recharge and contamination in Mediterranean catchments with dense ditches network vary largely in space and time, and will require dense monitoring networks to estimate the evolution of the average contamination levels. Copyright © 2008 John Wiley & Sons, Ltd.     

An analytical solution for rapidly predicting post-fire peak streamflow for small watersheds in southern California

Following wildfires, the probability of flooding and debris flows increase, posing risks to human lives, downstream communities, infrastructure, and ecosystems. In southern California (USA), the Rowe, Countryman, and Storey (RCS) 1949 methodology is an empirical method that is used to rapidly estimate post-fire peak streamflow. We re-evaluated the accuracy of RCS for 33 watersheds under current conditions. Pre-fire peak streamflow prediction performance was low, where the average R² was 0.29 and average RMSE was 1.10 cms/km² for the 2- and 10-year recurrence interval events, respectively. Post-fire, RCS performance was also low, with an average R² of 0.26 and RMSE of 15.77 cms/km² for the 2- and 10-year events. We demonstrated that RCS overgeneralizes watershed processes and does not adequately represent the spatial and temporal variability in systems affected by wildfire and extreme weather events and often underpredicted peak streamflow without sediment bulking factors. A novel application of machine learning was used to identify critical watershed characteristics including local physiography, land cover, geology, slope, aspect, rainfall intensity, and soil burn severity, resulting in two random forest models with 45 and five parameters (RF-45 and RF-5, respectively) to predict post-fire peak streamflow. RF-45 and RF-5 performed better than the RCS method; however, they demonstrated the importance and reliance on data availability. The important parameters identified by the machine learning techniques were used to create a three-dimensional polynomial function to calculate post-fire peak streamflow in small catchments in southern California during the first year after fire (R² = 0.82; RMSE = 6.59 cms/km²) which can be used as an interim tool by post-fire risk assessment teams. We conclude that a significant increase in data collection of high temporal and spatial resolution rainfall intensity, streamflow, and sediment loading in channels will help to guide future model development to quantify post-fire flood risk.     

An analytical solution for flow in a manifold

An analytical solution is developed for flow in a manifold. The interest is primarily for trickle irrigation laterals, but the solution has broader applications including those for which pressure increases in the direction of flow and for intake manifolds. Both velocity head losses and variable discharge along the manifold are considered in the fundamental analysis. The appropriate second order, nonlinear equation is solved for two flow regimes, laminar and fully turbulent. Results indicate that for most trickle irrigation laterals the velocity head loss is negligible, but for an example from a chemical processing system the effect is important.     

瞬变电磁场时域格林函数解

近源时间域电磁场具有信号强、探测深度大、精度高等优点,但传统勘探电磁场理论中偶极子近似在近源会引起较大误差,导致这一优势的发挥受到了制约.开展直接时间域电磁场解析式研究,是解决这一问题的途径之一.本文提出在点电荷微元假设下,引入时域格林函数,求取瞬变电磁场时间域解析解.采用积分运算法,把电磁场阻尼波动方程的求解问题转化为求其格林函数积分形式解的问题;建立辅助路径解决奇点问题,利用复分析中的约当引理、留数定理和广义函数等理论和方法,推导计算出时间域格林函数的时空四重广义积分.得到达朗贝尔方程的直接时域格林函数精确解析式,与传统方法“比拟”出的公式具有相同的形式,验证了本文推导的时域格林函数解析公式的正确性;推导出扩散方程的直接时间域解析解.通过与时变点电荷源时间域的电磁响应近似表达式进行对比,得出本文所推导的公式计算精度较高的结论;建立了全空间回线源瞬变电磁场问题的直接时间域求解公式.为解决全场区瞬变电磁场精细探测直接时域解析问题提供了基础理论.     

An analytical solution for describing the transient temperature distribution in an aquifer thermal energy storage system

A mathematical model is developed for predicting the temperature distribution in an aquifer thermal energy storage (ATES) system, which consists of a confined aquifer bounded from above and below by the rocks of different geological properties. The main transfer processes of heat include the conduction and advection in the aquifer and the conduction in the rocks. The semi‐analytical solution in dimensionless form for the model is developed by Laplace transforms and its corresponding time‐domain solution is evaluated by the modified Crump method. Field geothermal property data are used to simulate the temperature distribution in an ATES system. The results show that the heat transfer in the aquifer is fast and has a vast effect on the vicinity of the wellbore. However, the aquifer temperature decreases with increasing radial and vertical distances. The temperature in the aquifer may be overestimated when ignoring the effect of thermal conductivity. The temperature distribution in an ATES system depends on the vertical thermal conduction in the rocks and the horizontal advection and thermal conduction in the aquifer. The present solution is useful in designing and simulating the heat injection facility in the ATES systems. Copyright © 2010 John Wiley & Sons, Ltd.     

瞬变电磁扩散场到虚拟波场的精细积分变换算法

现有的瞬变电磁解释与反演方法主要对电阻率参数进行解释,较难给出准确的地质构造信息.为了能够较多获得地质构造信息,可将瞬变电磁扩散场信号转换为虚拟波场信号,提高瞬变电磁法的分辨率.本文采用精细积分法,实现了瞬变电磁扩散场到虚拟波场的转换.将一个高度病态的线性方程组转换成求积分的过程,大大降低了解决病态问题的难度.积分步长...     

从瞬变电磁场到波场的优化算法

在波场的正变换中,保证瞬变电磁场的计算精度的同时,应用两步最优化算法,成功地控制和减少了积分系数个数和离散数字积分的采样点个数,解决了在波场反变换式中,由于积分系数过多而产生的欠定方程组的问题,同时改善了第一类算子方程的不适定性.在波场反变换中,将正则化算法用于逆变换过程的计算中,通过采用偏差原理和Newton迭代格式选出最优的正则化参数,使得反变换所得到的波场稳定、可靠.通过对数值计算结果与已知波场函数对比,证明了该方法的有效性和实用性.     

Confined flow into a tunnel during progressive drilling: an analytical solution

A convolution integral is developed to evaluate transient, drilling speed-dependent discharge rates into a tunnel gradually excavated in a homogeneous infinite aquifer. Comparison with the classical instantaneous-drilling evaluation commonly used in practice reveals drastically reduced early-time discharge rates, higher maximum rates, and similar long-term rates. Dimensionless-type curves are provided to help assess total discharge sensitivity to drilling time and predict safer maximum flow rates.     

An analytical solution for ground water transit time through unconfined aquifers

An exact, closed-form analytical solution is developed for calculating ground water transit times within Dupuit-type flow systems. The solution applies to steady-state, saturated flow through an unconfined, horizontal aquifer recharged by surface infiltration and discharging to a downgradient fixed-head boundary. The upgradient boundary can represent, using the same equation, a no-flow boundary or a fixed head. The approach is unique for calculating travel times because it makes no a priori assumptions regarding the limit of the water table rise with respect to the minimum saturated aquifer thickness. The computed travel times are verified against a numerical model, and examples are provided, which show that the predicted travel times can be on the order of nine times longer relative to existing analytical solutions.     

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.     

An analytical solution for heterogeneous and anisotropic anticline reservoirs under well injection

This study develops a mathematical model for describing the steady-state head response to fluid injection into a fully penetrating well in a heterogeneous and anisotropic anticline reservoir. In the model, the upper boundary of the anticline reservoir is approximated by a form of step change in reservoir thickness and the domain of the reservoir is divided into two regions with different hydraulic conductivities. By virtue of the properties of Fourier series, the method of separation of variables is employed to develop the analytical solution of the model.     

瞬变电磁波场转换算法的改进

地球物理电磁场数据与虚拟地震波场数据之间存在数学上的等效转换关系,通过这种等效转换,可有效提高地球物理电磁法对地下目标体分界面的辨识度.但是这种转换在数学上属于不适定问题,可采用奇异值分解法处理.由于大奇异值控制计算矩阵的主要信息,小的奇异值控制计算矩阵的次要信息,传统的截断奇异值分解法只保留大奇异值,而忽略小的奇异值,导致数值解不够精确.本文提出一种新的修正方案——改进截断奇异值法,采用岭估计方法计算由小的奇异值引起的虚拟波场.模型计算结果表明：改进截断奇异值法比传统的奇异值分解法得到的波场转换结果更好,对某煤矿采空区探测数据进行了处理,成功分辨出采空区分界面.

     

平面SV波在饱和土半空间中圆柱形孔洞周边的散射

在Biot饱和多孔介质动力学理论的基础上,利用Fourier—Bessel级数展开法,得到SV波在饱和土半空间中圆柱形孔洞周边的散射问题的解析解答。与已有相关问题的解析解答进行对比,验证了此解的正确性,并给出算例,分析了入射频率对柱面上的应力集中因子的影响。     

A new method for the analytical solution of a degenerate diffusion equation

Certain nonlinear diffusion equations of degenerate parabolic type display a finite speed of propagation of disturbances. This mathematical behavior can be used to describe a wide range of nonlinear phenomena such as the penetration distance of a thermal layer, the boundary of a reaction zone, or a wetting front in unsaturated soil moisture flow. However, there are two main difficulties in obtaining solutions to problems of this class. One is that the location of the interface is not known a priori and must be discovered during the analysis. The other is the fact that the differential equation is singular in the neighborhood of the interface. The solution technique developed and presented in this work overcomes these difficulties by extracting a local solution of the differential equation in the neighborhood of the diffusing front. One profound result is the discovery that the velocity of the front is entirely controlled by the first term of the spectral series expansion. Also, by capturing the critical behavior of the solution in the region of the singularity and incorporating the behavior as a dominant factor, the series expansion is provided a means for very rapid convergence. The versatility of the solution technique is demonstrated by solving various boundary value problems covering a broad range of interest and the solutions are tested against previously published results.     

含峭壁V形峡谷对地震SH波散射的解析解

地表地形常引起地震动的局部放大,这是由于地震波传播至局部地形时产生了散射现象.本文利用波函数展开方法和区域匹配技术,提出了含峭壁V形峡谷对平面SH波散射问题的解析解,并进行了退化验证.通过频域内的参数分析,揭示了峭壁深度、入射波频率和角度等因素对峡谷场地地面运动的影响规律,发现上部峭壁会增强峡谷对地震动的地形放大效应.研究结果不仅为数值方法提供了验证基准,还可为含峭壁峡谷周边建筑物的抗震设计提供顺河向地震动输入.

     

Time-domain solution for transient dynamic response of a large-diameter thin-walled pipe pile

The propagation of stress waves in a large-diameter pipe pile for low strain dynamic testing cannot be explained properly by traditional 1D wave theories. A new computational model is established to obtain a wave equation that can describe the dynamic response of a large-diameter thin-walled pipe pile to a transient point load during a low strain integrity test. An analytical solution in the time domain is deduced using the separation of variables and variation of constant methods. The validity of this new solution is verifi ed by an existing analytical solution under free boundary conditions. The results of this time domain solution are also compared with the results of a frequency domain solution and fi eld test data. The comparisons indicate that the new solution agrees well with the results of previous solutions. Parametric studies using the new solution with reference to a case study are also carried out. The results show that the mode number affects the accuracy of the dynamic response. A mode number greater than 10 is required to enable the calculated dynamic responses to be independent of the mode number. The dynamic response is also greatly affected by soil properties. The larger the side resistance, the smaller the displacement response and the smaller the refl ected velocity wave crest. The displacement increases as the stress waves propagate along the pile when the pile shaft is free. The incident waves of displacement and velocity responses of the pile are not the same among different points in the circumferential direction on the pile top. However, the arrival time and peak value of the pile tip refl ected waves are almost the same among different points on the pile top.     

An analytical solution for the head distribution in a tidal leaky confined aquifer extending an infinite distance under the sea

A mathematical model is developed to investigate the effects of tidal fluctuations and leakage on the groundwater head of leaky confined aquifer extending an infinite distance under the sea. The leakages of the offshore and inland aquitards are two dominant factors controlling the groundwater fluctuation. The tidal influence distance from the coast decreases significantly with the dimensionless leakage of the inland aquitard (u_i). The fluctuation of groundwater level in the inland part of the leaky confined aquifer increases significantly with the dimensionless leakage of the offshore aquitard (u_o). The influence of the tidal propagation parameter of an unconfined aquifer on the head fluctuation of the leaky confined aquifer is comparatively conspicuous when u_i is large and u_o is small. In other words, ignoring water table fluctuation of the unconfined aquifer will give large errors in predicting the fluctuation, time lag, and tidal influence distance of the leaky confined aquifer for large u_i and small u_o. On the contrary, the influence of the tidal propagation parameter of a leaky confined aquifer on the head fluctuation of the leaky confined aquifer is large for large u_o and small u_i.     

A novel analytical solution for a slug test conducted in a well with a finite-thickness skin

An aquifer containing a skin zone is considered as a two-zone system. A mathematical model describing the head distribution is presented for a slug test performed in a two-zone confined aquifer system. A closed-form solution for the model is derived by Laplace transforms and Bromwich integral. This new solution is used to investigate the effects of skin type, skin thickness, and the contrast of skin transmissivity to formation transmissivity on the distributions of dimensionless hydraulic head. The results indicate that the effect of skin type is marked if the slug-test data is obtained from a radial two-zone aquifer system. The dimensionless well water level increases with the dimensionless positive skin thickness and decreases as the dimensionless negative skin thickness increases. In addition, the distribution of dimensionless well water level due to the slug test depends on the hydraulic properties of both the wellbore skin and formation zones.