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.     

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.     

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.     

A new analytical solution for assessing climate change impacts on subsurface temperature

Groundwater temperature is an important water quality parameter that affects species distributions in subsurface and surface environments. To investigate the response of subsurface temperature to atmospheric climate change, an analytical solution is derived for a one‐dimensional, transient conduction–advection equation and verified with numerical methods using the finite element code SUTRA. The solution can be directly applied to forward model the impact of future climate change on subsurface temperature profiles or inversely applied to produce a surface temperature history from measured borehole profiles. The initial conditions are represented using superimposed linear and exponential functions, and the boundary condition is expressed as an exponential function. This solution expands on a classic solution in which the initial and boundary conditions were restricted to linear functions. The exponential functions allow more flexibility in matching climate model projections (boundary conditions) and measured temperature–depth profiles (initial conditions). For example, measured borehole temperature data from the Sendai Plain and Tokyo, Japan, were used to demonstrate the improved accuracy of the exponential function for replicating temperature–depth profiles. Also, the improved accuracy of the exponential boundary condition was demonstrated using air temperature anomaly data from the Intergovernmental Panel on Climate Change. These air temperature anomalies were then used to forward model the effect of surficial thermal perturbations in subsurface environments with significant groundwater flow. The simulation results indicate that recharge can accelerate shallow subsurface warming, whereas upward groundwater discharge can enhance deeper subsurface warming. Additionally, the simulation results demonstrate that future groundwater temperatures obtained from the proposed analytical solution can deviate significantly from those produced with the classic solution. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

Sediment delivery from agricultural land to rivers via subsurface drainage

Diffuse sources of sediment and sediment‐associated nutrients are of increasing environmental concern because of their impacts on receiving water courses. The aim of the research reported here was to monitor the outflow from four field (land) drains at two farms in the English Midlands in order to estimate the quantity of sediment delivered to the local rivers and the most likely sources and processes involved. A multiparameter sediment unmixing model was employed, using environmental magnetic, geochemical and radionuclide tracers in order to determine the most likely origin of sediments transported through the drains. Results demonstrated that there was a generally linear relationship between drainflow sediment loss and drainflow volume and that the majority (>70%) of the sediment exported from the drains was derived from topsoil. Macropore flow through heavily cracked soils is supported by the data to be the most likely means of sediment delivery to the drains. In one catchment, drains contributed over 50% of the annual sediment budget. Spatial and temporal variations in the sources of sediment reaching one drain outlet were investigated in detail. A link between soil moisture deficit (SMD) and the frequency of high‐intensity rainfall events was used to explain the appearance and persistence of a new sediment source in this drain after October 1998. It is concluded that field drains have the potential to be significant conduits of sediment and agrochemicals in a wide variety of environments in the UK. It is also suggested that this potential may increase if projected climate change leads to more intense rainfall events and increases in SMD across a greater area of the UK. Copyright © 2005 John Wiley & Sons, Ltd.     

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 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.     

基于等值反磁通原理的浅层瞬变电磁法

基于等值反磁通原理的瞬变电磁法是一种新的探测地下纯二次场的方法.该方法采用上下平行共轴的两个相同线圈通以反向电流作为发射源,且在该双线圈源合成的一次场零磁通平面上,测量对地中心耦合的纯二次场.理论计算和物理实验论证了该方法能够有效消除接收线圈本身的感应电动势,从而获得地下纯二次场的响应.理论推导和数值计算证明了该方法采用的双线圈源比传统瞬变电磁法采用的单线圈源对地中心耦合场能量更集中,因而有利于减少旁侧影响、提高探测的横向分辨率.实测试验表明该方法是浅层探测的一种有效方法.     

A numerical procedure for three-dimensional transient free surface seepage

A three-dimensional procedure based on the finite element method is proposed for transient free surface seepage. It involves solution of the governing equations by using a time integration scheme. The procedure is applied for solution of confined, and transient free surface flow; the latter includes verification with respect to test results from a laboratory model. It is also applied to free surface flow through a dam with a crack.     

Analytical solution for drainage and recession from an unconfined aquifer

One-dimensional transient groundwater flow from a divide to a river in an unconfined aquifer described by the Boussinesq equation was studied. We derived the analytical solution for the water table recession and drainage change process described with a linearized Boussinesq equation with a physically based initial condition. A method for determining the average water table in the solutions was proposed. It is shown that the solution derived in the form of infinite series can be well approximated with the simplified solution which contains only the leading term of the original solution. The solution and their simplification can be easily evaluated and used by others to study the groundwater flow problems, such as drainage and base flow estimation, in an unconfined aquifer.     

An efficient tool for accelerating the numerical solution of the stochastic subsurface flow problem using neural networks

 An efficient numerical solution for the two-dimensional groundwater flow problem using artificial neural networks (ANNs) is presented. Under stationary velocity conditions with unidirectional mean flow, the conductivity realizations and the head gradients, obtained by a traditional finite difference solution to the flow equation, are given as input-output pairs to train a neural network. The ANN is trained successfully and a certain level of recognition of the relationship between input conductivity patterns and output head gradients is achieved. The trained network produced velocity realizations that are physically plausible without solving the flow equation for each of the conductivity realizations. This is achieved in a small fraction of the time necessary for solving the flow equations. The prediction accuracy of the ANN reaches 97.5% for the longitudinal head gradient and 94.7% for the transverse gradient. Head-gradient and velocity statistics in terms of the first two moments are obtained with a very high accuracy. The cross covariances between head gradients and the fluctuating log-conductivity (log-K) and between velocity and log-K obtained with the ANN approach match very closely those obtained by a traditional numerical solution. The same is true for the velocity components auto-covariances. The results are also extended to transport simulations with very good accuracy. Spatial moments (up to the fourth) of mean-concentration plumes obtained using ANNs are in very good agreement with the traditional Monte Carlo simulations. Furthermore, the concentration second moment (concentration variance) is very close between the two approaches. Considering the fact that higher moments of concentration need more computational effort in numerical simulations, the advantage of the presented approach in saving long computational times is evident. Another advantage of the ANNs approach is the ability to generalize a trained network to conductivity distributions different from those used in training. However, the accuracy of the approach in cases with higher conductivity variances is being investigated.     

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 to study substrate-microbial dynamics in soils

We provide an approximate analytical solution for the substrate-microbial dynamics of the organic carbon cycle in natural soils under hydro-climatic variable forcing conditions. The model involves mass balance in two carbon pools: substrate and biomass. The analytical solution is based on a perturbative solution of concentrations, and can properly reproduce the numerical solutions for the full non-linear problem in a system evolving towards a steady state regime governed by the amount of labile carbon supplied to the system. The substrate and the biomass pools exhibit two distinct behaviors depending on whether the amount of carbon supplied is below or above a given threshold. In the latter case, the concentration versus time curves are always monotonic. Contrarily, in the former case the C-pool concentrations present oscillations, allowing the reproduction of non-monotonic small-scale biomass concentration data in a natural soil, observed so far only in short-term experiments in the rhizosphere. Our results illustrate the theoretical dependence of oscillations from soil moisture and temperature and how they may be masked at intermediate scales due to the superposition of solutions with spatially variable parameters.     

The effects of controlled drainage on subsurface outflow from level agricultural fields

Daily outflow frequencies and recession curves were used to identify differences in storage–outflow relationships between two different drainage systems, conventional and controlled drainage. A three‐year (1996–1999) field drainage experiment was carried out on a loamy sand soil in southern Sweden. Plots with an area of 0·2 hectares were drained by conventional subsurface drainage (CD) or by controlled drainage (CWT1 and CWT2). The controlled drainage system allowed the groundwater level in the soil to be varied during the year. It was kept at least 70 cm below the soil surface during the growing season but allowed to rise to a maximum of 20 cm below the soil surface during the rest of the year. Measurements were performed to record precipitation, drain outflow and groundwater levels. Daily values of outflow were divided into 10 categories, based on the size of outflow. Recession curves of hourly measurement of outflow were selected. They behaved like single reservoirs and a linear storage–outflow model was applied. Least squares estimates of the parameters initial outflow, initial storage volume and retention constant were calculated. Controlled drainage had a significant effect on total drain outflow and outflow pattern during the three years of measurement. The total drain outflow was 70% to 90% smaller in CWT than in CD. The analysis revealed that the initial outflows were higher, the retention constant and the temporary storage lower in CWT. The hydrological impacts of the reduction in temporary storage were higher peak flow, shorter lag time and shorter recession time and these effects increased with an increased groundwater level. Copyright © 2003 John Wiley & Sons, Ltd.     

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.     

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。