Application Of Analytical Solutions Research Articles

Time-dependent analytical solutions for tunnels excavated in anisotropic rheological rock masses

The anisotropy combined with the time-dependency of rock mass is commonly observed in tunnel construction. In this study, a comprehensive analytical investigation is performed on analysing time-dependent ground responses induced by tunnel excavation in an anisotropic rheological rock mass.An innovative analytical solution for the anisotropic viscoelastic problems is first proposed, which can be applied to the general cases that the rheological behaviours in anisotropic principal directions are independent and arbitrary. Using the generalized corresponding principle, the analytical model, which can precisely and rapidly address the problem, is then presented for the deformation and stresses around tunnels in transversely isotropic viscoelastic rock mass, considering different viscoelastic characteristics, different anisotropic angles, and circular/elliptical tunnel shapes. The analytical solutions agree very well with the numerical predictions for the consistent models, and can also be reduced to the isotropic viscoelastic case. Furthermore, the application of analytical solutions in practical engineering is validated by the good agreements between analytical solutions and field measurements. A parametric analysis is then performed to investigate the time-dependency of stress, and the effects of anisotropy ratios and anisotropy angles on displacements and stresses.The proposed analytical solution has a valuable contribution to the field of rock rheological mechanics, as it offers a benchmark for anisotropic viscoelastic problems, insights into the complex behaviour of anisotropic rock masses and provides a more accurate prediction of the ground response that may be useful to optimize the design of tunnel excavation in anisotropic rock masses.

Evaluating the performance of the twin tunnel complex in soft soil subjected to horizontal ground shaking

Tunnel construction in soft soil necessitates a thorough evaluation of soil behavior, embedment depth, ground heaves, and tunnel distortions, especially in earthquake-prone areas. This study presents a numerical parametric investigation of an unconventional tunnel complex formed by combining the closely located twin tunnels. The complex is subjected to varying horizontal ground vibrations, and the influence of lining thickness, embedment depth, and interface conditions on seismic-induced thrusts, shear forces, bending moments, tunnel distortions, and ground heaves is assessed. The applicability of analytical solutions from existing literature for singular tunnels is examined through detailed analyses of different embedment ratios. The study reveals that increased tunnel flexural rigidity leads to higher seismic-induced bending moments in the tunnel complex. Comparison of full-slip and no-slip interface conditions shows that the former exhibits reduced overall tunnel distortions. Furthermore, a comparison is made with a conventional-shaped rectangular tunnel complex. The results indicate that the twin tunnel complex behaves more rigidly under a constant embedment ratio and input motion amplitude. It also results in lower ground heaves and suffers lesser induced lining forces during seismic events, making it a superior performer in comparison. Overall, this research provides valuable insights into the behavior of twin tunnel complexes in soft soil under seismic conditions, showcasing their advantages over conventional shaped tunnels in terms of tunnel distortions, ground heaves, and overall structural response.

Application of Artificial Neural Networks for Predicting the Bearing Capacity of Shallow Foundations on Rock Masses

Calculation of the bearing capacity of shallow foundations on rock masses is usually addressed either using empirical equations, analytical solutions, or numerical models. While the empirical laws are limited to the particular conditions and local geology of the data and the application of analytical solutions is complex and limited by its simplified assumptions, numerical models offer a reliable solution for the task but require more computational effort. This research presents an artificial neural network (ANN) solution to predict the bearing capacity due to general shear failure more simply and straightforwardly, obtained from FLAC numerical calculations based on the Hoek and Brown criterion, reproducing more realistic configurations than those offered by empirical or analytical solutions. The inputs included in the proposed ANN are rock type, uniaxial compressive strength, geological strength index, foundation width, dilatancy, bidimensional or axisymmetric problem, the roughness of the foundation-rock contact, and consideration or not of the self-weight of the rock mass. The predictions from the ANN model are in very good agreement with the numerical results, proving that it can be successfully employed to provide a very accurate assessment of the bearing capacity in a simpler and more accessible way than the existing methods.

A low order dynamical model for runoff predictability

Recent work has identified potential multi-year predictability in soil moisture (Chikamoto et al. in Clim Dyn 45(7–8):2213–2235, 2015). Whether this long-term predictability translates into an extended predictability of runoff still remains an open question. To address this question we develop a physically-based zero-dimensional stochastical dynamical model. The model extends previous work of Dolgonosov and Korchagin (Water Resour 34(6):624–634, 2007) by including a runoff-generating soil moisture threshold. We consider several assumptions on the input rainfall noise. We analyze the applicability of analytical solutions for the stationary probability density functions (pdfs) and for waiting times for runoff under different assumptions. Our results suggest that knowing soil moisture provides important information on the waiting time for runoff. In addition, we fit the simple model to daily NCEP1 reanalysis output on a near-global scale, and analyze fitted model performance. Over many tropical regions, the model reproduces the simulated runoff in NCEP1 reasonably well. More detailed analysis over a single gridpoint illustrates that the model, despite its simplicity, is able to capture some key features of the runoff time series and pdfs of a more complex model. Our model exhibits runoff predictability of up to two months in advance. Our results suggest that there is an optimal predictability “window” in the transition zone between runoff-generating and dry conditions. Our model can serve as a “null hypothesis” model reference against more complex models for runoff predictability.

Mathematical modelling to establish the influence of pesticides on groundwater contamination

Groundwater contamination due to pesticides used during cultivation was modelled mathematically, assuming that both the soil and the aquifer were initially contaminant free and the source of contamination was constant. A plane-type source was considered to show the impact of pesticides in soil domain, and a crack was considered in the upper part of the aquifer to keep a similarity with the physical system. The concentration of contaminants through the soil was used to obtain the concentration of contaminants in the aquifer. Three-dimensional analytical solutions of the dispersion equation for soil contamination and advection-dispersion equation (ADE) for aquifer contamination were obtained using Laplace transform. Application of analytical solution shows that 50–80% of the shallow aquifer, approximately 0.5 km deep and 5–8 km long, is contaminated irrespective of the concentration level. If the aquifer is 2 km deep and 15–20 km long, then the contaminant concentration in the aquifer asymptotically tends to zero after an average distance of 5 km, and 25–33% of the aquifer is contaminated with one or more pesticides irrespective of the concentration level. Result matches exactly the USGS data.

A Comparative Assessment of Analytical Fate and Transport Models of Organic Contaminants in Unsaturated Soils

Analytical models for the simulation of contaminants’ fate and transport in the unsaturated zone are used in many engineering applications concerning groundwater resource management and risk assessment. As a consequence, several scientific studies dealing with the development and application of analytical solutions have been carried out. Six models have been selected and compared based on common characteristics to identify pros and cons as well as to highlight any difference in the final output. The analyzed models have been clustered into three groups according to the assumptions on contaminant source and physico-chemical mechanisms occurring during the transport. Comparative simulations were carried out with five target contaminants (Benzene, Benzo(a)pyrene, Vinyl Chloride, Trichloroethylene and Aldrin) with different decay’s coefficient, three types of soil (sand, loam and clay) and three different thicknesses of the contaminant source. The calculated concentration at a given depth in the soil for the same contamination scenario varied greatly among the models. A significant variability of the concentrations was shown due to the variation of contaminant and soil characteristics. As a general finding, the more advanced is the model, the lower the predicted concentrations; thus, models that are too simplified could lead to outcomes of some orders of magnitude greater than the advanced one.

On the stress state of a pressurised pipe with an initial thickness variation, subjected to non-homogeneous internal corrosion

The paper concerns 2D problem of an elastic thick-walled pipe with an initial thickness variation, subjected to internal pressure and mechanochemical corrosion. The inner perimeter of the pipe cross-section is elliptical, while the outer is circular. The linear Dolinskii corrosion kinetics model is used. In the general case, structural instability of initial boundary value problems with unknown evolving boundaries can cause the divergence of numerical procedures when modelling the processes under study. It is observed that the attempts to circumvent the divergence of numerical procedure can suppress the manifestation of mechanochemical effect and yield inaccurate results. Thus, it is necessary to find a compromise between competing computational processes. Calculations revealed that the variation of the initial pipe wall thickness within the acceptable pipe wall tolerance can noticeably accelerate the growth of stresses and, consequently, reduce the durability of the pipe. The applicability of analytical solutions for a perfect circular pipe with a reduced thickness, equal to the minimum thickness of the imperfect pipe, to the case under study is also discussed.

Pseudo Steady State Gas Flow in Tight Reservoir under Dual Mechanism Flow

Gas reservoirs with low permeability (k<0.1 mD) are among the unconventional reservoirs and are commonly termed as Tight Gas Reservoirs. In conventional gas reservoirs that have high permeability, the flow of gas is basically controlled by the reservoir permeability and it is calculated using the Darcy equation. In these reservoirs, gas flow due to gas diffusion is ignored compared to Darcy flow. However, diffusion phenomenon has a significant impact on the gas flow in tight gas reservoirs and the mechanism of gas diffusion can no longer be ignored in comparison to Darcy flow. In this study, a dual mechanism based on Darcy flow as well as diffusion is used for the gas flow modeling in tight gas reservoirs. The diffusivity equation is obtained using this method that it indicates the gas flow in a porous media. The conventional dry gas pseudo pressure function is not able to linearize the diffusivity equation including diffusion effect. Subsequently, a new real gas pseudo pressure function is used and a novel real gas pseudo time function is introduced. These pseudo functions consider changes in gas properties with pressure and linearize the diffusivity equation. The linear diffusivity equation is solved analytically for constant gas flow boundary condition under Pseudo Steady State (PSS) situation. Then, pseudo steady state analytical solution, based on new functions of pseudo pressure and pseudo time, is obtained. The calculation of reservoir parameters such as permeability, effective diffusion coefficient and original gas in place (OGIP) using reservoir data is the first application of analytical solution. Reservoir data is required to analysis the results of application of introduced model in low permeability gas reservoir.

The Application of Analytical Solution for Critical Anchorage Length on the Bolt of Rock Bolt Crane Girder

The anchorage length of the bolt in rock bolt crane girder is obtained by means of rigid limit equilibrium method and the critical the anchorage length of bolt is calculated by analytical solution. Combined with the calculation results, the analysis on the mechanical characteristics is carried on by building the finite element model of rock bolt crane girder with different anchorage length of bolt. The analysis results indicate that the bearing characteristics of rock bolt crane girder are not improved significantly with the increase of anchorage length when the anchorage length of bolt exceeds the engineering critical anchorage length. The existence of critical anchorage length of blot is proved further, and the rationality of analytical solution for critical anchorage length is validated.

Grey Differential Model of Groundwater Pollution

Contaminant transport in underground system usually occurs in varied flow fields and in anisotropic and heterogenous media. Because the applicability of analytical solutions is extremely limited for such conditions, and the distribution and transport of pollutants in groundwater are controlled by physics and chemistry and biology courses, which include advection, mechanical disperse, molecule spreading, adsorption, solve and suck, biodegradation function, numerical techniques are essential for underground pollution modeling. Among the numerical techniques, the gray numerical method has become very popular and is recognized as a powerful numerical tool. The gray numerical method, particularly for pollution modeling, may be preferable to other methods. In this paper, b ased on the grey theory, the velocity and dispersion coefficient and attenuation in groundwater are considered as uncertainty parameters and expressed as gray parameters. A grey differential equation of contaminant diffusion in groundwater is built. And the equation has special structure. The truncation error of finite differential method in solving the model is corrected. According to the model, distribution values of pollutant concentration under sudden pollutant discharged can be obtained directly, which can provide abundant and useful water quality information for the plan and control of water pollution. In this paper, gray numerical model is applied to simulate groundwater pollution. The results indicate that this method provides the references for the forecast of pollutants and the management and evaluation of groundwater resources .

Application of analytical solutions in trapezoidal compound channel flow

AbstractThe accuracy of predictions using an analytical model for a lateral depth varying open channel flow can be affected by the friction factor and the dimensionless eddy viscosity. In this paper, different approaches to determine these two parameters are discussed by comparing analytical results with experimental data obtained from a symmetrical trapezoidal compound open channel. To reflect different hydraulic characteristics along the wetted perimeter, the trapezoidal cross‐section was divided into several sub‐sections, and the friction factor in each sub‐section was calculated with Manning's formula by adopting local hydraulic parameters. The wetted perimeters of main channel and side slope included the interface between panels to reflect the effects of the lateral momentum exchange. This approach is shown to be effective once the roughness coefficients and hydraulic radius in different sub‐regions have been determined. The results show that the friction factor is the major parameter affecting the accuracy of the analytical solution. The effect of dimensionless eddy viscosity on the precision of predictions is much smaller, which means that for practical applications, the dimensionless eddy viscosity can be determined empirically. Inclusion of a term for secondary flows in the model can improve the prediction, especially in the transition from main channel to the floodplain. A reasonable prediction of the lateral distribution of depth mean Reynolds' stress in this region is obtained. Copyright © 2010 John Wiley &amp; Sons, Ltd.

Influence of Hydro-Mechanical Coupling on Tunnel Response in Clays

The design of deep tunnels underneath the water-table in clayey soils should carefully take into account the influence of pore pressures and seepage forces on the stability of the excavation and on the loading conditions of the support systems. This paper focuses on the application of analytical solutions and numerical models to the analysis of stress and deformation around deep tunnels, in the short and long term, considering the idealized situation of axisymmetric conditions. General remarks on the influence of hydro-mechanical coupling, artificial boundary conditions and lining permeability are presented. Theoretical predictions are compared with measurements made in tunnels excavated in the Boom clay formation at Mol (Belgium).

Assessment of the Design of an Experimental Cover with Capillary Barrier Effect Using 4 Years of Field Data

One important step in the design of inclined covers with capillary barrier effect (CCBE) is the determination of the water diversion length (DL). Numerical simulations can predict the DL more precisely than steady-state analytical solutions. Nevertheless, as simplified methods have always been part of engineering design, the application of analytical solutions with conservative boundary conditions, may allow engineers to make reasonable predictions, particularly during the pre-feasibility stage of a project. In this study, a CCBE was designed, constructed and instrumented at the Saint-Tite-des-Caps landfill, Quebec, Canada. This CCBE included a seepage control layer superimposing a sand-gravel capillary barrier. The seepage control layer was made up of deinking by-products (DBP), an industrial by-product that was previously disposed of as waste. The capillary barrier was designed using an adaptation of the Ross analytical solution and the scenario considered was that of steady-state flow during constant seepage flow applied uniformly at the top of the sand-gravel capillary barrier. Although these conditions appear simplistic, they were deemed reasonable because placement of the seepage control layer on the top of the capillary barrier led to very low suctions at the interface, thereby allowing uniform downward seepage rates, limited by the saturated hydraulic conductivity of the DBP. In this paper, a discussion about the behaviour of the cover system based on 4 years of field data from several instruments is presented. The challenge of using DBP, more precisely the settlement of the DBP layer and its impact on ksat, is also assessed. The DL was reassessed considering the new ksat. A discussion on the validity of employing analytical solutions to determine DL is also presented. This paper illustrates how certain variables affect the design of inclined CCBEs that include a highly compressible material as seepage control layer.

Application of Analytic Solution in Relative Motion to Spacecraft Formation Flying in Elliptic Orbit

The current paper presents application of a new analytic solution in general relative motion to spacecraft formation flying in an elliptic orbit. The calculus of variations is used to analytically find optimal trajectories and controls for the given problem. The inverse of the fundamental matrix associated with the dynamic equations is not requiredfor thesolution in the currentstudy. Itis verifiedthat the optimal thrust vector is a function of the fundamental matrix of the given state equations. The cost function and the state vector during the reconfiguration can be analytically obtained as well. The results predict the form of optimal solutions in advance without having to solve the problem. Numerical simulation shows the brevity and the accuracy of the general analytic solutions developed in the current paper.

Steady-state problem of substrate consumption in a biofilm for a square law of microbial death rate

The steady-state problem of substrate consumption by a biofilm is solved over a wide range of biochemical parameters. It is shown that, regardless of the biofilm thickness, it can be completely saturated with the substrate. Ranges of applicability of analytical solutions for various problem parameters are presented.

Solute transport modeled with Green's functions with application to persistent solute sources

Analytical models can be valuable tools to investigate solute transport in porous media. The application of analytical solutions is limited by the perception that they are too cumbersome to derive while their implementation rests on assumptions that are too restrictive. The Green's function method (GFM) was applied to facilitate analytical solution of the advection–dispersion equation (ADE) for solute transport in uniform porous media with steady one- or two-dimensional flow. The GFM conveniently handles different boundary and initial conditions as well as multi-dimensional problems. Concise expressions are possible for the solute concentration with the GFM. This paper provides a general framework to efficiently formulate analytical solutions for many transport problems. Expressions for the longitudinal and transversal Green's function are presented that can be inserted in the general expression to solve a wide variety of transport problems in infinite, semi-infinite, and finite media. These solutions can be used to elucidate transport phenomena, estimate transport parameters, evaluate numerical solution procedures and simulate the movement and fate of solutes. An illustration of the GFM is provided by the analytical modeling of transport from a planar source of persistent, long-lasting contamination. Such a source may be used to represent dissolution from a pool of a non-aqueous phase liquid (NAPL). Analytical solutions are obtained for a first-, second-, and third-type condition in case of a planar source; the third-type condition is due to downward flow or rate-limited dissolution. Several examples are presented to show the effect of source conditions, the sensitivity of NAPL dissolution to transport parameters included in the Damköhler and Peclet numbers, and upstream dispersion.

DIRECT FORMULATION FINITE ELEMENT METHOD FOR TWO-DIMENSIONAL GROUNDWATER POLLUTION MODELLING WITH A COMPARISON WITH CONVENTIONAL FINITE ELEMENT METHOD

Real world ground water pollution modelling deals with solute transport through anisotropic, heterogeneous media. The applicability of analytical solutions for such a real world system is extremely limited. As an effective tool, numerical models, such as finite difference and finite element methods, are usually employed to model field scenarios. Nevertheless, ground water pollution modelling is a hallenging task and frequently ends up with misleading results. Most of the time insufficient data are blamed for such erratic results. A recent investigation shows that the shortcomings of numerical formulations may be the major cause for many disputes and confusions in numerical analyses. In reality, a point injection of water in a static, homogeneous and isotropic groundwater system shows a radial dissipation of water forming a sphere; and a full-depth line injection shows a radial dissipation forming a cylinder. The finite difference method completely ignores this fundamental flow principles and allows water only to flow along orthogonal directions. To overcome this limitation, the finite element method was developed as a flexible approach in order to connect a node with the neighbouring nodes in various directions where water is assumed to flow in any directions along node connections. In a recent investigation, it has been found that the conventional finite element method does not keep the commitments; and its formulation techniques lead to a global matrix where a solution domain is not connected with all the neighbouring nodes and does not comply with the control-volume mass balance concept. A consistent finite element formulation approach which does not need imaginary mathematical formulation and overcomes the limitations of both the conventional finite difference and finite element methods has been developed. This method allows fluid flow and solute transport in a porous medium in radial directions. The global matrices for flow and transport obtained from this technique are field representative, diagonally dominant and easily convergent. The new method is robust, needs less mathematical computation and has many advantages over the conventional finite difference and finite element methods.

Shortcomings of Existing Finite Element Formulations for Subsurface Water Pollution Modeling and Its Rectification: One-Dimensional Case

Contaminant transport in an underground system usually occurs in varied flow fields and in anisotropic and heterogenous media. Because the applicability of analytical solutions is extremely limited for such conditions, numerical techniques are essential for underground pollution modeling. Among the numerical techniques, the finite element method has become very popular and is recognized as a powerful numerical tool. The finite element method, particularly for pollution modeling, may be preferable to finite difference because it can link a node with the adjacent nodes in various directions. The main disadvantage with the finite element method is that its mathematical formulations are relatively complicated and follow set rules step by step. In a recent investigation, it was found that such prescribed rules are not applicable everywhere; they have many limitations and shortcomings. Shape functions that transform nodal values into element values are frequently used in an integration process to construct elem...

On the Applicability of Analytical Solutions of the Transport Equation for the Migration of Substances in Groundwater-Bearing Horizons

Using experimental data from indicator pump tests, various theoretical migration models for radionuclides in groundwater-bearing horizons were considered. The model parameters necessary for these considerations, especially migration velocity and dispersivity, were obtained by statistical analysis of experimental data. These results show that pulse-like indicator injection best measured values for the model of instantaneous injection in indicator-free flowing groundwater in an infinite strata.