Solute transport during unsteady, unsaturated soil water flow - the pulse input

Soil Research ◽  
1987 ◽  
Vol 25 (3) ◽  
pp. 223 ◽  
Author(s):  
WJ Bond
Keyword(s):  
Approximate Solution ◽  
Analytical Solution ◽  
Unsaturated Soils ◽  
Approximate Analytical Solution ◽  
Tritiated Water ◽  
Chloride Ion ◽  
Factors Affecting ◽  
Pulse Input ◽  
Structurally Stable ◽  
Good Agreement

A general approximate analytical solution was derived for the movement and dispersion of a solute pulse during unsteady flow of water in unsaturated soils. Two types of pulse were considered: that arising from a non-uniform distribution of solute initially in the soil, and that arising from a change in concentration of solute in the water supply. The movement of pulses of tritiated water and chloride ion was investigated experimentally for the specific case of constant flux horizontal infiltration of water into a strongly aggregated, structurally stable clay soil. Good agreement between the measured distributions of tritium and those predicted using the approximate solution confirmed the validity of the approximate solution. Agreement was much poorer for chloride. The approximate analytical solution was also used to examine factors affecting the shape of solute pulses. It was found that pulses may be asymmetric at short times, but that they rapidly become symmetric as time increases.

scholarly journals An Equal-Strain Analytical Solution for the Radial Consolidation of Unsaturated Soils by Vertical Drains considering Drain Resistance

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Feng Zhou ◽  
Zheng Chen ◽  
Xudong Wang
Keyword(s):  
Analytical Solution ◽  
Unsaturated Soils ◽  
Ground Surface ◽  
Matrix Analysis ◽  
Governing Equations ◽  
Vertical Drains ◽  
Ground Surface Settlement ◽  
Parametric Studies ◽  
Strain Model ◽  
Good Agreement

Developing an analytical solution for the consolidation of unsaturated soils remains a challenging task due to the complexity of coupled governing equations for air and water phases. This paper presents an equal-strain model for the radial consolidation of unsaturated soils by vertical drains, and the effect of drain resistance is also considered. Simplified governing equations are established, and an analytical solution to calculate the excess pore-air and pore-water pressures is derived by using the methods of matrix analysis and eigenfunction expansion. The average degrees of consolidation for air and water phases and the ground surface settlement are also given. The solutions of the equal-strain model are verified by comparing the proposed free-strain model with the equal-strain model, and reasonably good agreement is obtained. Moreover, parametric studies regarding the drain resistance effect are graphically presented.

scholarly journals An Approximate Solution for Flow between Two Disks Rotating about Distinct Axes at Different Speeds

2007 ◽  
Vol 2007 ◽  
pp. 1-16 ◽  
Author(s):  
H. Volkan Ersoy
Keyword(s):  
Reynolds Number ◽  
Approximate Solution ◽  
Angular Velocity ◽  
Analytical Solution ◽  
Viscous Fluid ◽  
Approximate Analytical Solution ◽  
Velocity Difference ◽  
Small Angular Velocity

The flow of a linearly viscous fluid between two disks rotating about two distinct vertical axes is studied. An approximate analytical solution is obtained by taking into account the case of rotation with a small angular velocity difference. It is shown how the velocity components depend on the position, the Reynolds number, the eccentricity, the ratio of angular speeds of the disks, and the parameters satisfying the conditionsu=0andν=0in midplane.

scholarly journals Homotopy perturbation method for a Stefan problem with variable latent heat

2014 ◽  
Vol 18 (2) ◽  
pp. 391-398 ◽  
Author(s):  
R Rajeev
Keyword(s):  
Approximate Solution ◽  
Analytical Solution ◽  
Perturbation Method ◽  
Latent Heat ◽  
Stefan Problem ◽  
Homotopy Perturbation Method ◽  
Homotopy Perturbation ◽  
Variable Latent Heat ◽  
Good Agreement

In this paper, homotopy perturbation method is successfully applied to find an approximate solution of one phase Stefan problem with variable latent heat. The results thus obtained are compared graphically with a published analytical solution and are in good agreement.

Half Sommerfeld Approximation for Finite Journal Bearings

1963 ◽  
Vol 85 (3) ◽  
pp. 435-438 ◽  
Author(s):  
J. V. Fedor
Keyword(s):  
Approximate Solution ◽  
Analytical Solution ◽  
Reynolds Equation ◽  
Approximate Analytical Solution ◽  
Journal Bearings ◽  
Finite Form ◽  
Oil Film

An approximate analytical solution is developed for full journal bearings which includes the effects of bearing finiteness and an incomplete oil film. The approximate solution is obtained by modifying the complete oil film solution to Reynolds equation. The developed equations are in finite form and are simple to evaluate. Calculated values agree well with published computer solutions.

Modified Lanczos Method for Optimal Polynomial Approximation of the Solution of Differential Equations with Polynomial Coefficients

1991 ◽  
Vol 02 (01) ◽  
pp. 243-245
Author(s):  
A.S. BERDNICOV
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Analytical Solution ◽  
Approximate Analytical Solution ◽  
Analytical Form ◽  
Lanczos Method ◽  
Physical Models ◽  
Strict Solution ◽  
Polynomial Coefficients ◽  
Optimal Polynomial

The construction of an approximate analytical solution of a differential equation is an important task for a variety of physical models. A solution in analytical form allows to investigate the properties of a model, and the use of numerical methods allows to construct a parametrized approximate solution when the strict solution is absent.

Numerical Approximations to Solution of Biochemical Reaction Model

2011 ◽  
Vol 9 (1) ◽  
Author(s):  
Ahmet Yildirim ◽  
Ahmet Gökdogan ◽  
Mehmet Merdan
Keyword(s):  
Analytical Solution ◽  
Approximate Analytical Solution ◽  
Transformation Method ◽  
Reaction Model ◽  
Biochemical Reaction ◽  
Graphical Form ◽  
Kutta Method ◽  
Transform Method ◽  
Runge Kutta Method ◽  
Differential Transform

In this paper, approximate analytical solution of biochemical reaction model is used by the multi-step differential transform method (MsDTM) based on classical differential transformation method (DTM). Numerical results are compared to those obtained by the fourth-order Runge-Kutta method to illustrate the preciseness and effectiveness of the proposed method. Results are given explicit and graphical form.

scholarly journals Semi-Analytical Solution to Assess CO2 Leakage in the Subsurface through Abandoned Wells

Energies ◽  
2021 ◽  
Vol 14 (9) ◽  
pp. 2452
Author(s):  
Tian Qiao ◽  
Hussein Hoteit ◽  
Marwan Fahs
Keyword(s):  
Analytical Solution ◽  
Storage Site ◽  
Co2 Leakage ◽  
Transient Pressure ◽  
Flow Paths ◽  
Abandoned Wells ◽  
Diffusivity Equation ◽  
Carbon Dioxide Co2 ◽  
Good Agreement

Geological carbon storage is an effective method capable of reducing carbon dioxide (CO2) emissions at significant scales. Subsurface reservoirs with sealing caprocks can provide long-term containment for the injected fluid. Nevertheless, CO2 leakage is a major concern. The presence of abandoned wells penetrating the reservoir caprock may cause leakage flow-paths for CO2 to the overburden. Assessment of time-varying leaky wells is a need. In this paper, we propose a new semi-analytical approach based on pressure-transient analysis to model the behavior of CO2 leakage and corresponding pressure distribution within the storage site and the overburden. Current methods assume instantaneous leakage of CO2 occurring with injection, which is not realistic. In this work, we employ the superposition in time and space to solve the diffusivity equation in 2D radial flow to approximate the transient pressure in the reservoirs. Fluid and rock compressibilities are taken into consideration, which allow calculating the breakthrough time and the leakage rate of CO2 to the overburden accurately. We use numerical simulations to verify the proposed time-dependent semi-analytical solution. The results show good agreement in both pressure and leakage rates. Sensitivity analysis is then conducted to assess different CO2 leakage scenarios to the overburden. The developed semi-analytical solution provides a new simple and practical approach to assess the potential of CO2 leakage outside the storage site. This approach is an alternative to numerical methods when detailed simulations are not feasible. Furthermore, the proposed solution can also be used to verify numerical codes, which often exhibit numerical artifacts.

New Approximate Analytical Solution of the Diode-Resistance Equation

2021 ◽  
pp. 153665
Author(s):  
José A. Gazquez ◽  
Manuel Fernandez-Ros ◽  
Blas Torrecillas ◽  
José Carmona ◽  
Nuria Novas
Keyword(s):  
Analytical Solution ◽  
Approximate Analytical Solution ◽  
Resistance Equation
Sign in / Sign up

Export Citation Format

Share Document