scholarly journals A General Analytical Solution for Axisymmetric Consolidation of Unsaturated Soil with Impeded Drainage Boundaries

Geofluids ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Ming-hua Huang ◽  
Chang Lv ◽  
Zheng-lin Zhou
Keyword(s):  
Analytical Solution ◽  
Boundary Conditions ◽  
Laplace Transform ◽  
Unsaturated Soil ◽  
Fractured Reservoirs ◽  
Improvement Project ◽  
Soil Stratum ◽  
General Analytical Solution ◽  
Consolidation Behavior ◽  
Excess Pore

The consolidation of soil is one of the most common phenomena in geotechnical engineering. Previous studies for the axisymmetric consolidation of unsaturated soil have usually idealized the boundary conditions as fully drained and absolutely undrained, but the boundaries of unsaturated soil are actually impeded drainage in most practical situations. In this study, we present a general analytical solution for predicting the axisymmetric consolidation behavior of unsaturated soil that incorporates impeded drainage boundary conditions in both the radial and vertical directions simultaneously. The impeded drainage boundary is modeled using the third kind boundary, and it can also realize fully drained and absolutely undrained ones by changing the drainage parameter. A general analytical solution is developed to predict the excess pore-air and pore-water pressures as well as the average degree of consolidation in an unsaturated soil stratum using the common methods of eigenfunction expansion and Laplace transform. The newly developed solution is expressed in the product of the terms of time, depth, and radius, which are derived using Laplace transform, usual Fourier, and Fourier-Bessel series, respectively. The eigenfunctions and eigenvalues are evaluated from the impeded drainage boundaries in both radial and depth dimensions. Then, the correctness of the proposed analytical solution is verified against the existing analytical solution for the case of traditional boundaries and against the finite difference solution for the case of general impeded drainage boundaries, and excellent agreements are obtained. Finally, the axisymmetric consolidation behavior of unsaturated soil involving impeded drainage boundaries is demonstrated and analyzed, and the effects of the drainage parameters are discussed. The results indicate that the larger drainage parameter generally expedites the dissipations of the excess pore pressures and further promotes the soil settling process. As the drainage parameter increases, its influence gradually diminishes and even can be neglected when it is larger than 100. The general analytical solution and findings of this study can help for better understanding the axisymmetric consolidation behavior of the unsaturated soil stratum in the ground improvement project with vertical drains as well as the gas-oil gravity drainage mechanism in the naturally fractured reservoirs.

The Effect of Biot Number on a One-Dimensional General Conduction Solution

2021 ◽  
Author(s):  
Robert L. McMasters ◽  
Filippo de Monte ◽  
James V. Beck
Keyword(s):  
Boundary Condition ◽  
Analytical Solution ◽  
Boundary Conditions ◽  
Biot Number ◽  
Analytical Solutions ◽  
Heat Fluxes ◽  
Graphical Form ◽  
Large Temperature ◽  
Convection Coefficient ◽  
General Analytical Solution

Abstract Analytical solutions for thermal conduction problems are extremely important, particularly for verification of numerical codes. Temperatures and heat fluxes can be calculated very precisely, normally to eight or ten significant figures, even in situations involving large temperature gradients. It can be convenient to have a general analytical solution for a transient conduction problem in rectangular coordinates. The general solution is based on the principle that the three primary types of boundary conditions (prescribed temperature, prescribed heat flux, and convective) can all be handled using convective boundary conditions. A large convection coefficient closely approximates a prescribed temperature boundary condition and a very low convection coefficient closely approximates an insulated boundary condition. Since a dimensionless solution is used in this research, the effect of various values of dimensionless convection coefficients, or Biot number, are explored. An understandable concern with a general analytical solution is the effect of the choice of convection coefficients on the precision of the solution, since the primary motivation for using analytical solutions is the precision offered. An investigation is made in this study to determine the effects of the choices of large and small convection coefficients on the precision of the analytical solutions generated by the general convective formulation. Results are provided, in tablular and graphical form, to illustrate the effects of the choices of convection coefficients on the precision of the general analytical solution.

A General Analytical Solution for the Concentration Profiles of Ion-Exchanged Planar Waveguides

1991 ◽  
Vol 244 ◽  
Author(s):  
Xiaoming Li ◽  
Paul F. Johnson
Keyword(s):  
Ion Exchange ◽  
Analytical Solution ◽  
Boundary Conditions ◽  
Surface Concentration ◽  
Concentration Profiles ◽  
Concentration Boundary ◽  
Glass Waveguides ◽  
General Analytical Solution ◽  
Exchange Processes ◽  
One Step

ABSTRACTDuring the recent years, a great variety of ion-exchange processes, including one-step or two-step electric field assisted ion-exchange processes, have been developed to fabricate different kinds of passive planar glass waveguides, e.g., surface waveguides, which correspond to surface maximum concentration, or buried waveguides, which correspond to inside maximum concentration [1,2,3]. Theoretical calculation of ionic concentration distribution has been of interest since refractive index is generally a linear function of concentration. A general analytical solution to calculate both surface and buried concentration distributions from different ion-exchange processes, however, has not yet been presented. In addition, traditional ion-exchange has been carried out only with constant surface concentration boundary conditions. Little attention has been paid, either experimentally or theoretically, to ion-exchange processes with variable boundary conditions. In fact, the time-dependent surface concentration is experimentally observed for the ion-exchange of GRIN glass in molten salt bath [4]. Very recently, a novel one-step technique [5,6] involving electric field assisted ion-exchange of Na+ in glass by Ag+ from molten AgNO3 bath with decaying silver concentration has been developed to produce buried Ag+ concentration profiles in glass. As the accurate and reproducible processes are very important for fabricating ion-exchanged glass waveguides, theoretical modeling and analysis on the new process are needed.In this paper, the one-dimensional field-assisted linear diffusion equation has been analytically solved by Laplace transformation to theoretically calculate concentration profiles produced by field enhanced ion-exchange process with exponentially decaying surface concentration boundary conditions. The applications of the solution to a variety of ion-exchange processes with different boundary or processing conditions for optical waveguide fabrication have been discussed. The theoretical results prove that the solution is a general analytical solution which can be used to calculate either surface concentration profiles or buried concentration profiles.

scholarly journals 2D Plane Strain Consolidation Process of Unsaturated Soil with Vertical Impeded Drainage Boundaries

Processes ◽  
2018 ◽  
Vol 7 (1) ◽  
pp. 5 ◽  
Author(s):  
Minghua Huang ◽  
Dun Li
Keyword(s):  
Analytical Solution ◽  
Laplace Transform ◽  
Plane Strain ◽  
Unsaturated Soil ◽  
Analytical Solutions ◽  
Soil Layer ◽  
Permeability Ratio ◽  
Consolidation Process ◽  
The Time Domain ◽  
Excess Pore Pressures

The consolidation process of soil stratum is a common issue in geotechnical engineering. In this paper, the two-dimensional (2D) plane strain consolidation process of unsaturated soil was studied by incorporating vertical impeded drainage boundaries. The eigenfunction expansion and Laplace transform techniques were adopted to transform the partial differential equations for both the air and water phases into two ordinary equations, which can be easily solved. Then, the semi-analytical solutions for the excess pore-pressures and the soil layer settlement were derived in the Laplace domain. The final results in the time domain could be computed by performing the numerical inversion of Laplace transform. Furthermore, two comparisons were presented to verify the accuracy of the proposed semi-analytical solutions. It was found that the semi-analytical solution agreed well with the finite difference solution and the previous analytical solution from the literature. Finally, the 2D plane strain consolidation process of unsaturated soil under different drainage efficiencies of the vertical boundaries was illustrated, and the influences of the air-water permeability ratio, the anisotropic permeability ratio and the spacing-depth ratio were investigated.

scholarly journals One-Dimensional Consolidation of Multi-Layered Unsaturated Soil with Impeded Drainage Boundaries

2020 ◽  
Vol 11 (1) ◽  
pp. 133
Author(s):  
Suhua Zhou ◽  
Jiatao Kang ◽  
Chang Lv ◽  
Minghua Huang
Keyword(s):  
Boundary Condition ◽  
Unsaturated Soil ◽  
Numerical Solutions ◽  
Laplace Domain ◽  
One Dimensional ◽  
Soil Settlement ◽  
The One ◽  
The Time Domain ◽  
Consolidation Behavior ◽  
Excess Pore

In geotechnical engineering, the consolidation of unsaturated soil is a common issue of great interest. Considering the multi-layered property and impeded drainage boundary condition of the soil stratum in real engineering, this study aimed to develop a general semi-analytical solution for assessing the one-dimensional (1D) consolidation behavior of multi-layered unsaturated soil that is subjected to a general impeded drainage boundary condition and a time-dependent loading. To achieve the final solution, the proposed consolidation system is firstly decoupled and solved in the Laplace domain. Then, the semi-analytical solutions for the excess pore-air pressure and excess pore-water pressures as well as the soil settlement are formulated. The Crump method is employed to provide their final results in the time domain. The correctness of the derived solutions was verified against the available analytical and numerical solutions, and excellent agreements were found for the two comparisons. Moreover, two studied examples are presented to illustrate the 1D consolidation behavior of multi-layered unsaturated soil and the influences stemming from the impeded drainage parameters are discussed.

A general semi-analytical solution for predicting axisymmetric consolidation behavior of multi-layered unsaturated soil

2021 ◽  
Vol 14 (19) ◽  
Author(s):  
Minghua Huang ◽  
Chang Lv ◽  
Zhenglin Zhou
Keyword(s):  
Analytical Solution ◽  
Unsaturated Soil ◽  
Consolidation Behavior

scholarly journals General Analytical Solution to Consolidation of Unsaturated Soil with Vertical Drain considering Drain Resistance

2020 ◽  
Vol 794 ◽  
pp. 012039
Author(s):  
Aifang Qin ◽  
Weifang Xu ◽  
Tianyi Li ◽  
Lianghua Jiang
Keyword(s):  
Analytical Solution ◽  
Unsaturated Soil ◽  
Vertical Drain ◽  
General Analytical Solution
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