Displacement Functions and Linear Transforms Applied to Diffusion Through Porous Elastic Media

McNamee, John; Gibson, R. E.

doi:10.1093/qjmam/13.1.98

Cited by 194 publications

(62 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The models by Domenico and Mifflin [1965] and Lofgren and Klausing [1969] fall into this category. However, as early as 1960, McNamee andGibson [1960] attempted to derive analytical solutions to the Biot consolidation of an isotropic porous medium for various sets of boundary conditions, and even earlier, McCann and Wilts [1951] had developed the first rudimental mathematical model of land subsidence over the oil field of the Long Beach-San Pedro area (CA) by simulating the oil reservoir with a set of hollow cavities (called ''tension centers'') strategically located within the field embedded in a linearly elastic semi-infinite porous medium so as to reproduce the land settlement measured overland. At the internal boundary of each cavity, a pore pressure decline was prescribed equal to the field pressure depletion.…”

Section: Modeling Land Subsidencementioning

confidence: 99%

Geomechanics of subsurface water withdrawal and injection

Gambolati

Teatini

2015

Water Resources Research

133

View full text Add to dashboard Cite

Land subsidence and uplift, ground ruptures, and induced seismicity are the principal geomechanic effects of groundwater withdrawal and injection. The major environmental consequence of groundwater pumping is anthropogenic land subsidence. The first observation concerning land settlement linked to subsurface processes was made in 1926 by the American geologists Pratt and Johnson, who wrote that ''the cause of subsidence is to be found in the extensive extraction of fluid from beneath the affected area.'' Since then, impressive progress has been made in terms of: (a) recognizing the basic hydrologic and geomechanic principles underlying the occurrence; (b) measuring aquifer compaction and ground displacements, both vertical and horizontal; (c) modeling and predicting the past and future event; and (d) mitigating environmental impact through aquifer recharge and/or surface water injection. The first milestone in the theory of pumped aquifer consolidation was reached in 1923 by Terzaghi, who introduced the principle of ''effective intergranular stress.'' In the early 1970s, the emerging computer technology facilitated development of the first mathematical model of the subsidence of Venice, made by Gambolati and Freeze. Since then, the comprehension, measuring, and simulation of the occurrence have improved dramatically. More challenging today are the issues of ground ruptures and induced/ triggered seismicity, which call for a shift from the classical continuum approach to discontinuous mechanics. Although well known for decades, anthropogenic land subsidence is still threatening large urban centers and deltaic areas worldwide, such as Bangkok, Jakarta, and Mexico City, at rates in the order of 10 cm/yr.

show abstract

Section: Modeling Land Subsidencementioning

confidence: 99%

Geomechanics of subsurface water withdrawal and injection

Gambolati

Teatini

2015

Water Resources Research

133

View full text Add to dashboard Cite

show abstract

“…The system (6) is decoupled by the introduction of stress functions (McNamee and Gibson 1960). The stress function S(x, z, t) plays a role in the momentum equations and E(x, z, t) a larger role in the mass equation (Appendix).…”

Section: Decouplingmentioning

confidence: 99%

“…McNamee and Gibson (1960) only consider symmetric loading and therefore only use the symmetric part of Fourier transforms. To be able to consider more general cases, general Fourier and Laplace transformations are used (Booker 1974).…”

Section: Governing Equations In Laplace-fourier Domainmentioning

confidence: 99%

“…Following McNamee and Gibson (1960), the harmonic function S(x, z, t) is introduced. If the chemical potential…”

Section: Stress Function Smentioning

confidence: 99%

“…However, these functions can only be applied when there is compressibility in the model. Booker (1974) calculated the dimensionless stress field along a shear fault for a fully saturated incompressible porous medium by decoupling the equations with stress functions according McNamee and Gibson (1960) (Verruijt 1971;Detournay and Cheng 1993). Rice and Cleary (1976) showed that both methods find the same result in the limiting case of incompressibility.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Singularity solution of Lanir’s osmoelasticity: verification of discontinuity simulations in soft tissues

Huyghe

Kraaijeveld

2010

Biomech Model Mechanobiol

View full text Add to dashboard Cite

The literature characterizes cartilaginous tissues as osmoviscoelastic. Understanding the damage and failure of these tissues is essential for designing treatments. To determine tissue strength and local stresses, experimental studies-both clinical and animal-are generally supported by computational studies. Verification methods for computational studies of ionized porous media including cracks are hardly available. This study provides a method for verification and shows its performance. For this purpose, shear loading of a finite crack is addressed analytically and through a commercial finite element code. Impulsive shear loading by two-edge dislocation of a crack was considered in a 2D plane strain model for an ionized porous medium. To derive the analytical solution, the system of equation is decoupled by stress functions. The shear stress distribution at the plane of the crack is derived using Fourier and Laplace transformations. The analytical solution for the shear stress distribution is compared with computer simulations in ABAQUS version 6.4-5. Decoupling of the equations makes it possible to solve some boundary value problems in porous media taking chemical effects into account. The numerical calculations underestimate the shear stress at the crack-tips. Mesh refinement increases accuracy, but is still low in the neighborhood of the crack-tips.

show abstract

Analytical layer‐element method for non‐axisymmetric consolidation of multilayered soils

Ai¹,

Zeng²

2011

Num Anal Meth Geomechanics

View full text Add to dashboard Cite

SUMMARYA numerically efficient and stable method is developed to analyze Biot's consolidation of multilayered soils subjected to non-axisymmetric loading in arbitrary depth. By the application of a Laplace-Hankel transform and a Fourier expansion, the governing equations are solved analytically. Then, the analytical layer-element (i.e. a symmetric stiffness matrix) describing the relationship between generalized displacements and stresses of a layer is exactly derived in the transformed domain. Considering the continuity conditions between adjacent layers, the global stiffness matrix of multilayered soils is obtained by assembling the inter-related layer-elements. Once the solution in the Laplace-Hankel transformed domain that satisfies the boundary conditions has been obtained, the actual solution can be derived by the inversion of the Laplace-Hankel transform. Finally, numerical examples are presented to verify the theory and to study the influence of the layered soil properties and time history on the consolidation behavior.

show abstract

Displacement Functions and Linear Transforms Applied to Diffusion Through Porous Elastic Media

Cited by 194 publications

References 0 publications

Geomechanics of subsurface water withdrawal and injection

Geomechanics of subsurface water withdrawal and injection

Singularity solution of Lanir’s osmoelasticity: verification of discontinuity simulations in soft tissues

Analytical layer‐element method for non‐axisymmetric consolidation of multilayered soils

Contact Info

Product

Resources

About