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.     

Analytical solutions for benchmarking cold regions subsurface water flow and energy transport models: One-dimensional soil thaw with conduction and advection

Numerous cold regions water flow and energy transport models have emerged in recent years. Dissimilarities often exist in their mathematical formulations and/or numerical solution techniques, but few analytical solutions exist for benchmarking flow and energy transport models that include pore water phase change. This paper presents a detailed derivation of the Lunardini solution, an approximate analytical solution for predicting soil thawing subject to conduction, advection, and phase change. Fifteen thawing scenarios are examined by considering differences in porosity, surface temperature, Darcy velocity, and initial temperature. The accuracy of the Lunardini solution is shown to be proportional to the Stefan number. The analytical solution results obtained for soil thawing scenarios with water flow and advection are compared to those obtained from the finite element model SUTRA. Three problems, two involving the Lunardini solution and one involving the classic Neumann solution, are recommended as standard benchmarks for future model development and testing.     

Approximate soil-structure interaction analysis by a perturbation approach: The case of stiff soils

An approximate solution of the classical eigenvalue problem governing the vibrations of a structure on an elastic soil is derived through the application of a perturbation analysis. For stiff soils, the full solution is obtained as the sum of the solution for a rigid-soil and small perturbing terms related to the inverse of the soil shear modulus. The procedure leads to approximate analytical expressions for the system frequencies, modal damping ratios and participation factors for all system modes that generalize those presented by other authors for the fundamental mode. The resulting approximate expressions for the system modal properties are validated by comparison with the corresponding quantities obtained by numerical solution of the eigenvalue problem for a nine-story building. The accuracy of the proposed approach and of the classical normal mode approach is assessed through comparison with the exact frequency response of the test structure.     

Model tests on horizontal pile-to-pile interaction incorporating local non-linearity and resonance effects

Dynamic experimental studies on horizontal interaction factors for laterally loaded model soil–pile systems subjected to low-to-high amplitude pile head loading are reported, focusing on the effect of dynamics and local non-linearity. Results obtained from dynamic experiments with different amplitudes of harmonic lateral loading on instrumented model soil–pile systems conducted on a shaking table indicate that the resonance of soil–pile system shows a profound impact on the horizontal interaction factors. Furthermore, behavior of the soil–pile system becomes highly non-linear with increasing amplitude of loading, as the extent of local non-linearity around the pile increases. Consequently, group effects cannot be predicted by using established approximate equations based on the assumption of linear viscoelastic behavior. To this end, new semi-empirical equations are proposed for obtaining the horizontal interaction factors, as an extension of available approximate equations to incorporate the local non-linear behavior of soil–pile system for both small and large pile spacings.     

The seepage flow through a free boundary aquifer into a river / L'écoulement d'infiltration d'un aquifère a frontière libre vers une rivière

ABSTRACT

This paper presents a model of the groundwater flow into a river from an aquifer beneath the river. The mathematical problem is to solve Laplace's equation with a free boundary and the solution procedure uses a variational inequality which leads to an approximate solution using finite differences. The method can be used to provide for example, inflow conditions in river modelling calculations.     

Domenico solution--is it valid?

The Domenico solution is widely used in several analytical models for simulating ground water contaminant transport scenarios. Unfortunately, many textbook as well as journal article treatments of this approximate solution are full of empirical statements that are developed without mathematical rigor. For this reason, a rigorous analysis of this solution is warranted. In this article, we present a mathematical method to derive the Domenico solution and explore its limits. Our analysis shows that the Domenico solution is a true analytical solution when the value of longitudinal dispersivity is zero. For nonzero longitudinal dispersivity values, the Domenico solution will introduce a finite amount of error. We use an example problem to quantify the nature of this error and suggest some general guidelines for the appropriate use of this solution.     

Hydrologic procedures of storm event watershed models: a comprehensive review and comparison

Currently, many watershed models are available that have various complexities, strengths, and weaknesses. The basic mathematical foundations of these mathematical models are often overlooked due to high demands on convenient applications with graphical user interfaces. Although this and other factors are important while selecting a model, the mathematical foundation should also be taken into account, as performance or efficiency and accuracy of a model depend on its simplicity or complexity. A comprehensive review of 14 storm event watershed models was conducted. Hydrologic procedures (rainfall excess, flow routing, and subsurface flow) of the models are presented and compiled. Among the procedures, flow routing has the most influence on model performances (speed and accuracy). Overland and channel flow routing procedures using different flow‐governing equations, having various approximations and solved by different methods, are compared based on their relative levels of physical bases, complexities, and expected accuracies in simulating the dynamics of water flow. Models using more mathematical terms in the flow‐governing equations are more physically based and expected to be more accurate than models using approximations, however, are more complex due to more intensive but approximate numerical schemes (inefficient). Models using approximate equations with analytical solutions may provide a balance between complexity and accuracy. The review and comparisons are useful to modellers, water resources managers, and researchers in understanding the basic foundations of the models and making informed selections for practical applications or further developments. Other factors such as data intensiveness, user friendliness, and resource requirements are also important considerations. Copyright © 2011 John Wiley & Sons, Ltd.     

Approximate soil–structure interaction analysis by a perturbation approach: The case of soft soils

An approximate solution of the classical eigenvalue problem governing the vibrations of a relatively stiff structure on a soft elastic soil is derived through the application of a perturbation analysis. The full solution is obtained as the sum of the solution for an unconstrained elastic structure and small perturbing terms related to the ratio of the stiffness of the soil to that of the superstructure. The procedure leads to approximate analytical expressions for the system frequencies, modal damping ratios and participation factors for all system modes that generalize those presented earlier for the case of stiff soils. The resulting approximate expressions for the system modal properties are validated by comparison with the corresponding quantities obtained by numerical solution of the eigenvalue problem for a nine-story building. The accuracy of the proposed approach and of the classical normal mode approach is assessed through comparison with the exact frequency response of the test structure.     

A new saturated/unsaturated model for stormwater infiltration systems

Infiltration systems are widely used as an effective urban stormwater control measure. Most design methods and models roughly approximate the complex physical flow processes in these systems using empirical equations and fixed infiltration rates to calculate emptying times from full. Sophisticated variably saturated flow models are available, but rarely applied owing to their complexity. This paper describes the development and testing of an integrated one‐dimensional model of flow through the porous storage of a typical infiltration system and surrounding soils. The model accounts for the depth in the storage, surrounding soil moisture conditions and the interaction between the storage and surrounding soil. It is a front‐tracking model that innovatively combines a soil‐moisture‐based solution of Richard's equation for unsaturated flow with piston flow through a saturated zone as well as a reservoir equation for flow through a porous storage. This allows the use of a simple non‐iterative numerical solution that can handle ponded infiltration into dry soils. The model is more rigorous than approximate stormwater infiltration system models and could therefore be valuable in everyday practice. A range of test cases commonly used to test soil water flow models for infiltration in unsaturated conditions, drainage from saturation and infiltration under ponded conditions were used to test the model along with an experiment with variable depth in a porous storage over saturated conditions. Results show that the model produces a good fit to the observed data, analytical solutions and Hydrus. Copyright © 2008 John Wiley & Sons, Ltd.     

Time‐domain viscoelastic analysis of earth structures

In this paper two causal models that approximate the nearly frequency‐independent cyclic behaviour of soils are analysed in detail. The study was motivated by the need to conduct time‐domain viscoelastic analysis on soil structures without adopting the ad hoc assumption of Rayleigh damping. First, the causal hysteretic model is introduced in which its imaginary part is frequency independent the same way that is the imaginary part of the popular non‐causal constant hysteretic model. The adoption of an imaginary part that is frequency independent even at the zero‐frequency limit, in conjunction with the condition that the proposed model should be causal, yields a real part that is frequency dependent and singular at zero frequency. The paper shows that the causal hysteretic model, although pathological at the static limit, is the mathematical connection between the non‐causal constant hysteretic model and the physically realizable Biot model. The mathematical structure of the two causal models is examined and it is shown that the causal hysteretic model is precisely the high‐frequency limit of the Biot model. Although both models have a closed‐form time‐domain representation, only the Biot model is suitable for a time‐domain viscoelastic analysis with commercially available computer software. The paper demonstrates that the simplest, causal and physically realizable linear hysteretic model that can approximate the cyclic behaviour of soil is the Biot model. The proposed study elucidates how the dynamic analysis of soil structures can be conducted rigorously in terms of the viscoelastic properties of the soil material and not with the ad hoc Rayleigh damping approach which occasionally has been criticized that tends to overdamp the higher vibration modes. The study concludes that under pulse‐type motions the Rayleigh damping approximation tends to overestimate displacements because of the inappropriate viscous type of dissipation that is imposed. Under longer motions that induce several cycles, the concept of equivalent viscous damping is more appropriate and the Rayleigh damping approximation results to a response that is comparable to the response computed with a rigorous time‐domain viscoelastic finite element analysis. Copyright © 2000 John Wiley & Sons, Ltd.     

Parametric modeling of earthquake response spectra

A mathematical model for the response spectra is determined using statistical analysis. The form of the model is first established using fifty computer simulated accelerograms. The final form is then used on twenty-five accelerograms from fifteen past United States earthquakes. This model smooths out peaks and valleys which are characteristic of the response spectrum of any single earthquake. Thus it serves as a ‘smooth design spectrum’ and can be used to approximate structural response to a future seismic event.     

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.     

Ground-Water Mounding Beneath a Large Leaching Bed

An 84 m (276 ft) by 64 m (205 ft) experimental leaching bed was constructed to receive the secondary effluent from the Norwood, Ontario sewage treatment plant. The distribution system consisted of 100 mm diameter perforated pipes laid 2.1 m apart in 0.45 m wide trenches. The main soil layer at the test site was sandy silt. Ground-water levels at the site were measured approximately weekly at 36 piezometers for a period of about 2½ years. An initial hydraulic loading of 122,700 1/day caused surface ponding. A reduced loading of 40,900 1/day was then used for most of the study period. Ground-water level fluctuations beneath and adjacent to the leaching bed were attributed to seasonally varying infiltration and hydraulic loading. The experimental data were simulated using a transient two-dimensional, phreatic integrated depth finite-element model and an analytic solution for ground-water mounding. Results from the study indicated that seasonal infiltration was a controlling factor in determining leaching bed design loads. Both mathematical models were found adequate for design.     

Transient analysis for fluid injection into a dome reservoir

A dome-shaped layer can be selected as a storage site for fluid injection. In this study, we develop a mathematical model for simulating transient head distribution in a heterogeneous and anisotropic dome-shaped layer due to a constant-head injection in a fully penetrating well. In the model, a form of step change is adopted to approximate the upper and lower boundaries of the dome and then the layer is split into two regions. The Laplace-domain solution of the model is developed using the Laplace transform and method of separation of variables. The transient injection rate at wellbore can then be obtained based on Darcy’s law and Bromwich integral method. The predicted head contours from the head solution show significant vertical flow components near the location of step change in the dome reservoir. The results of sensitivity analysis indicate that the hydraulic conductivity is the most sensitive parameter and the specific storage is the least sensitive one to the injection rate after a short period of injection time. In addition, the injection rate for a dome reservoir is also very sensitive to the change of the height for the reservoir near the injection well (first region) at a very early injection time. In contrast, the injection rate is more sensitive to the change of the height of the second region than that of the first region at late time. This analytical solution may be used as a primary tool to assess the capacity of fluid injection to various dome reservoirs.     

Three-dimensional finite-element modelling of magnetotelluric data with a divergence correction

A finite-element method for computing the electric field in a 3-D conductivity model of the Earth for plane wave sources, thus enabling magnetotelluric responses to be calculated, is presented. The method incorporates in the iterative solution of the electric-field system of equations the divergence correction technique introduced for finite-difference solutions by Smith (1996). The correction technique accelerates the development of the discontinuity of the normal component of the approximate electric field across conductivity discontinuities. The convergence rate of the iterative solution is improved significantly, especially for low frequencies. The correction technique involves computing the divergence of the current density for the approximate electric field, computing the static potential whose source is this divergence of the current density, and ‘correcting’ the approximate electric field by subtracting from it the gradient of the potential. This is repeated at regular intervals during the iterative solution of the electric-field system of equations. For the method presented here, the Earth model is discretised using a rectilinear mesh comprising uniform cells. Edge-element basis functions are used to approximate the electric field and nodal basis functions are used to approximate the correction potential. The Galerkin method is used to derive the systems of equations for the approximate electric field and correction potential from the respective differential equations. A bi-conjugate gradient solver was found to be adequate for the system of equations for the correction potential; a generalised minimum residual solver was found to be better for the electric-field system of equations. The method is illustrated using the COMMEMI 3D-1A and 3D-2A models.     

Modified MMF (Morgan–Morgan–Finney) model for evaluating effects of crops and vegetation cover on soil erosion

Modifications are made to the revised Morgan–Morgan–Finney erosion prediction model to enable the effects of vegetation cover to be expressed through measurable plant parameters. Given the potential role of vegetation in controlling water pollution by trapping clay particles in the landscape, changes are also made to the way the model deals with sediment deposition and to allow the model to incorporate particle‐size selectivity in the processes of erosion, transport and deposition. Vegetation effects are described in relation to percentage canopy cover, percentage ground cover, plant height, effective hydrological depth, density of plant stems and stem diameter. Deposition is modelled through a particle fall number, which takes account of particle settling velocity, flow velocity, flow depth and slope length. The detachment, transport and deposition of soil particles are simulated separately for clay, silt and sand. Average linear sensitivity analysis shows that the revised model behaves rationally. For bare soil conditions soil loss predictions are most sensitive to changes in rainfall and soil parameters, but with a vegetation cover plant parameters become more important than soil parameters. Tests with the model using field measurements under a range of slope, soil and crop covers from Bedfordshire and Cambridgeshire, UK, give good predictions of mean annual soil loss. Regression analysis of predicted against observed values yields an intercept value close to zero and a line slope close to 1·0, with a coefficient of efficiency of 0·81 over a range of values from zero to 38·6 t ha^?1. Copyright © 2007 John Wiley & Sons, Ltd.     

Rocking vibrations of a rigid embedded foundation in a poroelastic soil layer

An approximate analytical method is presented for the dynamic response of a rigid cylindrical foundation embedded in a poroelastic soil layer under the excitation of a time-harmonic rocking moment. The soil underlying the foundation base is represented by a single-layered poroelastic soil based on rigid bedrock while the soil along the side of the foundation is modeled as an independent poroelastic stratum composed of a series of infinitesimally thin layers. The accuracy of the present solution is verified by comparisons with existing solutions obtained from other researchers. Numerical results for the rocking dynamic impedance and dynamic response factor are presented to demonstrate the influence of nondimensional frequency of excitation, poroelastic soil layer thickness, depth ratio of the foundation and internal friction of the poroelastic soil.     

Fluid–foundation interaction in the seismic response of gravity dams

The seismic response of a dam is strongly influenced by its interaction with the water reservoir and the foundation. The hydrodynamic forces in the reservoir are in turn affected by radiation of waves towards infinity, wave absorption at the reservoir bottom, and cross-coupling between the foundation below the dam and the reservoir bottom. The fluid–foundation interaction effect, i.e. the wave absorption along the reservoir bottom, can be accounted for by using either an approximate one-dimensional (1D) wave propagation model or a rigorous analysis of interaction between the flexible soil along the base and the water. The rigorous approach requires enormous computational effort because of (a) cross-coupling between the foundation of the dam and the soil below the reservoir and (b) frequency dependence of the boundary condition along the fluid-foundation interface. The analysis can be simplified by ignoring the cross-coupling and by using the approximate 1D wave propagation model. The effects of each of these two simplifications on the accuracy and computational efficiency of the procedure used for the seismic response analysis of a dam are examined. Analytical results are presented for the complex frequency-response functions as well as the time histories of the response of Pine Flat dam to Taft and E1 Centro ground motions.     

Linear stability analysis for the differentially heated rotating annulus

We use linear stability analysis to approximate the axisymmetric to nonaxisymmetric transition in the differentially heated rotating annulus. We study an accurate mathematical model that uses the Navier–Stokes equations in the Boussinesq approximation. The steady axisymmetric solution satisfies a two-dimensional partial differential boundary value problem. It is not possible to compute the solution analytically, and thus, numerical methods are used. The eigenvalues are also given by a two-dimensional partial differential problem, and are approximated using the matrix eigenvalue problem that results from discretizing the linear part of the appropriate equations.

A comparison is made with experimental results. It is shown that the predictions using linear stability analysis accurately reproduce many of the experimental observations. Of particular interest is that the analysis predicts cusping of the axisymmetric to nonaxisymmetric transition curve at wave number transitions, and the wave number maximum along the lower part of the axisymmetric to nonaxisymmetric transition curve is accurately determined. The correspondence between theoretical and experimental results validates the numerical approximations as well as the application of linear stability analysis.

A linear stability analysis is also performed with the effects of centrifugal buoyancy neglected. Along the lower part of the transition curve, the results are significantly qualitatively and quantitatively different than when the centrifugal effects are considered. In particular, the results indicate that the centrifugal buoyancy is the cause of the observation of a wave number maximum along the transition curve, and is the cause of a change in concavity of the transition curve.     

Tree root systems competing for soil moisture in a 3D soil–plant model

Competition for water among multiple tree rooting systems is investigated using a soil–plant model that accounts for soil moisture dynamics and root water uptake (RWU), whole plant transpiration, and leaf-level photosynthesis. The model is based on a numerical solution to the 3D Richards equation modified to account for a 3D RWU, trunk xylem, and stomatal conductances. The stomatal conductance is determined by combining a conventional biochemical demand formulation for photosynthesis with an optimization hypothesis that selects stomatal aperture so as to maximize carbon gain for a given water loss. Model results compare well with measurements of soil moisture throughout the rooting zone, of total sap flow in the trunk xylem, as well as of leaf water potential collected in a Loblolly pine forest. The model is then used to diagnose plant responses to water stress in the presence of competing rooting systems. Unsurprisingly, the overlap between rooting zones is shown to enhance soil drying. However, the 3D spatial model yielded transpiration-bulk root-zone soil moisture relations that do not deviate appreciably from their proto-typical form commonly assumed in lumped eco-hydrological models. The increased overlap among rooting systems primarily alters the timing at which the point of incipient soil moisture stress is reached by the entire soil–plant system.