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Laplace Domain (laplace + domain)

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Engineering100%

Selected Abstracts

Interpretation of the enhancement of field-scale effective matrix diffusion coefficient in a single fracture using a semi-analytical power series solution

HYDROLOGICAL PROCESSES, Issue 6 2009
Tai-Sheng Liou
Abstract A power series solution for convergent radial transport in a single fracture (PCRTSF) is developed. Transport processes considered in PCRTSF include advection and hydrodynamic dispersion in the fracture, molecular diffusion in the matrix, diffusive mass exchange across the fracture-matrix interface, and mixing effects in the injection and the extraction boreholes. An analytical solution in terms of a power series in Laplace domain is developed first, which is then numerically inverted by de-Hoog et al.'s algorithm. Four dimensionless parameters determine the behaviour of a breakthrough curve (BTC) calculated by PCRTSF, which are, in the order of decreasing sensitivity, the matrix diffusion factor, two mixing factors, and the Peclet number. The first parameter is lumped from matrix porosity, effective matrix diffusion coefficient, fracture aperture, and retardation factors. Its value increases as the matrix diffusion effect becomes significant. A non-zero matrix diffusion factor results in a , 3/2 slope of the tail of a log,log BTC, a common property for tracer diffusion into an infinite matrix. Both mixing factors have equal effects on BTC characteristics. However, the Peclet number has virtually no effect on BTC tail. PCRTSF is applied to re-analyse two published test results that were obtained from convergent radial tracer tests in a discrete, horizontal fracture in Silurian dolomite. PCRTSF is able to fit the field BTCs better than the original channel model does if a large matrix diffusion coefficient is used. Noticeably, the ratio of field-scale to lab-scale matrix diffusion coefficients can be as large as 378. This enhancement of the field-scale matrix diffusion coefficient may be ascribed to the presence of a degraded zone at the fracture-matrix interface because of karstic effects, or to flow channeling as a result of aperture heterogeneity. Copyright © 2009 John Wiley & Sons, Ltd. [source]

Three-dimensional finite elements for the analysis of soil contamination using a multiple-porosity approach

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 7 2006
Abbas El-Zein
Abstract A three-dimensional finite-element model of contaminant migration in fissured clays or contaminated sand which includes multiple sources of non-equilibrium processes is proposed. The conceptual framework can accommodate a regular network of fissures in 1D, 2D or 3D and immobile solutions in the macro-pores of aggregated topsoils, as well as non-equilibrium sorption. A Galerkin weighted-residual statement for the three-dimensional form of the equations in the Laplace domain is formulated. Equations are discretized using linear and quadratic prism elements. The system of algebraic equations is solved in the Laplace domain and solution is inverted to the time domain numerically. The model is validated and its scope is illustrated through the analysis of three problems: a waste repository deeply buried in fissured clay, a storage tank leaking into sand and a sanitary landfill leaching into fissured clay over a sand aquifer. Copyright © 2005 John Wiley & Sons, Ltd. [source]

Modelling poroelastic hollow cylinder experiments with realistic boundary conditions

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 12 2004
S. Jourine
Abstract A general poroelastic solution for axisymmetrical plane strain problems with time dependent boundary conditions is developed in Laplace domain. Time-domain results are obtained using numerical inversion of the Laplace transform. Previously published solutions can be considered as special cases of the proposed solution. In particular, we could reproduce numerical results for solid and hollow poroelastic cylinders with suddenly applied load/pressure (Rice and Cleary, Rev. Geophys. Space Phys. 1976; 14:227; Schmitt, Tait and Spann, Int. J. Rock Mech. Min. Sci. 1993; 30:1057; Cui and Abousleiman, ASCE J. Eng. Mech. 2001; 127:391). The new solution is used to model laboratory tests on thick-walled hollow cylinders of Berea sandstone subjected to intensive pressure drawdown. In the experiments, pressure at the inner boundary of the hollow cylinder is observed to decline exponentially with a decay constant of 3,5 1/s. It is found that solutions with idealized step-function type inner boundary conditions overestimate the induced tensile radial stresses considerably. Although basic poroelastic phenomena can be modelled properly at long time following a stepwise change in pressure, realistic time varying boundary conditions predict actual rock behaviour better at early time. Experimentally observed axial stresses can be matched but appear to require different values for , and , than are measured at long time. The proposed solution can be used to calculate the stress and pore pressure distributions around boreholes under infinite/finite boundary conditions. Prospective applications include investigating the effect of gradually changing pore pressure, modelling open-hole cavity completions, and describing the phenomenon of wellbore collapse (bridging) during oil or gas blowouts. Copyright © 2004 John Wiley & Sons, Ltd. [source]

Poroelastodynamic Boundary Element Method in Time Domain: Numerical Aspects

PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2005
Martin Schanz
Based on Biot's theory the governing equations for a poroelastic continuum are given as a coupled set of partial differential equations (PDEs) for the unknowns solid displacements and pore pressure. Using the Convolution Quadrature Method (CQM) proposed by Lubich a boundary time stepping procedure is established based only on the fundamental solutions in Laplace domain. To improve the numerical behavior of the CQM-based Boundary Element Method (BEM) dimensionless variables are introduced and different choices studied. This will be performed as a numerical study at the example of a poroelastic column. Summarizing the results, the normalization to time and spatial variable as well as on Young's modulus yields the best numerical behavior. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

Material Modelling of Porous Media for Wave Propagation Problems

PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2003
M. Schanz PD Dr.-Ing.
Under the assumption of a linear geometry description and linear constitutive equations, the governing equations are derived for two poroelastic theories, Biot's theory and Theory of Porous Media (TPM), using solid displacements and pore pressure as unknowns. In both theories, this is only possible in the Laplace domain. Comparing the sets of differential equations of Biot's theory and of TPM, they show different constant coefficients but the same structure of coupled differential equations. Identifying these coefficients with the material data and correlating them leads to the known problem with Biot's ,apparent mass density'. Further, in trying to find a correlation between Biot's stress coefficient to parameters used in TPM yet unsolved inconsistencies are found. For studying the numerical effect of these differences, wave propagation results of a one-dimensional poroelastic column are analysed. Differences between both theories are resolved only for compressible constituents. [source]