An analytical solution to multi-component NAPL dissolution equations

This note presents a novel method for determining the changing composition of a multi-component NAPL body dissolving into moving groundwater, and the consequent changes in the aqueous phase solute concentrations in the surrounding pore water. A canonical system of coupled non-linear governing equations is derived which is suitable for representation of both pooled and residual configurations, and this is solved. Whereas previous authors have handled such problems numerically, it is shown that these governing equations succumb to analytical solution. By a suitable substitution, the equations become decoupled, and the problem collapses to a single first-order equation. The final result is expressed implicitly, with time as a function of the number of moles of the least soluble component, m₁. The number of moles of each other component is expressed explicitly in terms of m₁. It is shown that the time-m₁ relationship has a well behaved inverse. An example is given in which the analytic solution is verified against traditional finite difference analysis, and its computational efficiency is shown.     

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.     

An analytical solution to wave dissipation induced by interacting waves and currents with muddy deposits

Ocean Dynamics - The present study aims to provide a straightforward analytical solution to wave-current-mud interaction by considering the mean shear stress effects in the water layer. A direct...     

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.     

One-dimensional dispersion in unsteady flow in an adsorbing porous medium: An analytical solution

Analytical solutions describing the concentration distribution along one-dimensional unsteady seepage flow through adsorbing saturated finite porous medium have been obtained. An exponential function concentration is enforced at the source of the dispersion, while the change in the concentration is zero at the other boundary. A new time variable has been introduced to solve the unsteady flow problem and the solution is illustrated.     

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.

     

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.     

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 analytical study of general hyper-diffusivity and barotropic eddies

An exact solution to the barotropic potential vorticity equation is used to examine the properties of barotropic vortices under arbitrary nth-order hyper-diffusivity. Analytical expressions are derived for an eddy's lifetime, meridional drift, decay in size, and energy, as functions of the Coriolis parameter, order and magnitude of diffusivity, and the eddy's size, shape and strength. These expressions provide a simple explanation for many observed features of oceanic and atmospheric vortices. For example, the competition between the Coriolis effect and eddy strength in giving permitted eddy geometries; the bias towards a zonal anisotropy for large vortices but not for small ones; energetic preference for axisymmetry; poleward meridional drift of cyclonic vortices; and meridional speed variation depending on eddy geometry and strength.     

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 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 the probabilistic response of sdof non-linear random systems subjected to variable amplitude cyclic loading

Design for a specific ductile failure mode is assuming a rǒle of increasing importance for earthquake-resistant structures. This necessitates an accurate assessment of the distribution of overstrength in the structure, in order that the predefined failure mode can be realized. Consequently, the variability of the response for a given variability in the salient material properties, such as yield strength for steel structures, should be assessed and accounted for. In this paper an analytical method is proposed for the evaluation of the probability density function of the response of a single-degree-of-freedom hysteretic system with random parameters subject to a variable amplitude cyclic load history. A simple algorithm is derived which may be used to obtain the system response as a function of the system parameters. This response function may then be used to evaluate the displacement response probability density function when given the probability density function of the system parameters. Results derived from this procedure are verified against Monte Carlo simulation. It is shown that accurate response statistics are obtained at a fraction of the computing cost of simulation techniques.     

成层粘弹性土中桩土耦合纵向振动时域响应研究

从三维轴对称角度出发,采用粘性阻尼粘弹性连续土介质模型,考虑桩土相互作用效应,对成层土中桩土纵向耦合振动时的桩顶时域响应进行了解析研究。求解时,首先建立定解问题,然后利用拉氏变换先对底部土层进行求解得到其振动位移形式解,然后利用桩土接触界面连续条件来考虑桩土耦合作用,分析底层土中桩段的动力反应,然后利用桩段阻抗函数的传递性,进行逐层递推求解,最终得到桩顶时域和频域响应的半解析解。通过参数影响分析和与工程实测曲线的对比,讨论分析了成层土中桩土耦合振动的响应特性,验证了本文解。基于本文研究可为桩基抗震、防震设计、桩基动力检测提供新的理论支持。     

An analytical solution for predicting transient seepage into ditch drains from a ponded field

An analytical solution is provided for predicting time dependent seepage into an array of equally spaced parallel ditch drains in a homogeneous and anisotropic soil medium underlain by an impervious layer and receiving water from a ponded horizontal field of infinite extent. The solution can account for both unequal levels of water in the adjacent drains and variable depths of ponding at the soil surface. The validity of the developed model is tested by first reducing it to a steady state solution and then comparing predictions obtained from it for a few flow situations with corresponding predictions obtained from the analytical works of others. A numerical comparison of the developed model for a flow situation is also carried out using MODFLOW. The surface discharge distribution is found to show relatively greater uniformity at the early stages of simulation but with the progress of time, the extent of uniformity is found to reduce particularly for cases where the soil is subjected to a uniform depth of ponding. However, even when a soil surface is subjected to a constant depth of ponding, a high anisotropy ratio (ratio of horizontal to vertical hydraulic conductivity of soil) of the soil alone may lead to a marked improvement on the uniformity of the surface discharge distribution at all times in comparison to a soil having a low anisotropy ratio. A better uniformity of surface discharge may also be achieved by suitably adjusting the depths of ponding over the surface of the soil – regions close to the ditches be provided with zero or negligible depths of ponding and the ponding depths may be made to progressively increase with the increase in distance from the ditch faces. As the developed analytical model is of a general nature, it is hoped that the solution provided herein will lead to a better and realistic design of ditch drainage networks for controlling waterlogged areas and in reclaiming salt affected soils.     

Scattering of plane P waves by circular-arc layered alluvial valleys: An analytical solution

IntroductionThe site effect of local inhomogeneity and irregularity on seismic wave propagation is one of the most attractive topics in seismology. The problem may be resolved by either analytical method or numerical method. Here, the analytical method is the wave function expansion method; numerical methods include the finite difference, finite element, boundary integration equation, and discrete wave number method, etc. Although these numerical methods can be applied for sites of arbitrary s…     

Application of a new analytical method to a model of non-Darcian consolidation in clay soils

A model of consolidation in clay soil is studied by analytical means and the results compared with a recent numerical solution. Good agreement is obtained when the soil is only weakly non-Darcian. As the non-Darcian behavior of the soil becomes more pronounced the numerical solution is shown to be in error.     

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.     

Soil–pile interaction during the passage of rayleigh waves: An analytical solution

A simple analytical solution is presented to calculate the single-pile response when excited by the passage of Rayleigh seismic waves. Closed-form expressions for the horizontal and vertical displacement distributions are presented for piles with finite or infinite length. The analytical results for both free-head and fixed-head piles are obtained through a dynamic Winkler model, with realistic frequency-dependent ‘springs’ and ‘dashpots’. The results of the presented method are in excellent agreement with results of a rigorous solution. It is shown that in vertical motion, the differences between pile and soil displacements are far more significant than in horizontal motion, and therefore, further work is needed to investigate the importance of pile-soil-pile interaction (group effects), because of the vertical component of Rayleigh seismic waves.     

An analytical study on turbulence with mixed-flow model and two-phase model

Suspended sediment is carried by turbulent water flows in rivers.Traditional sediment-laden flow analysis treats the suspension as a mixed liquid,and recent two-phase flow model enables separate velocity measurement of the two coupling phases.A simplified theoretical analysis was presented to discuss the differences between the two models in reporting turbulence intensity in experimental research.The turbulence intensity of the mixture is lower than the weighted average of those of the two phases in mixed-flow experiments.The mixture’s turbulence intensity becomes higher than the average of the two phases in two-phase experiments due to the presence of velocity lag.The same set of data may lead to either an underestimation or an overestimation of actual turbulence levels when different models are used.     

BIOSCREEN-AT: BIOSCREEN with an exact analytical solution

BIOSCREEN is used extensively for screening-level evaluation of the transport of dissolved contaminants in ground water. The code has an effective graphical user interface that makes it ideal for use in both professional practice and as a teaching aid. BIOSCREEN implements the approximate transport solution of Domenico (1987). This note describes an enhanced version of the program, BIOSCREEN-AT, which supplements the Domenico solution with an exact analytical solution. The exact analytical solution has been integrated seamlessly within the BIOSCREEN interface and provides a simple and direct way to calculate an exact solution to the transport equation and, if desired, to assess the significance of the errors introduced by the Domenico (1987) solution for site-specific applications. The modified version of BIOSCREEN is designated BIOSCREEN-AT and can be downloaded free of charge from http://www.sspa.com/./software/BIOSCREEN.htm.