Plain strain problem of poroelasticity using eigenvalue approach

A plain strain problem of an isotropic elastic liquid-saturated porous medium in poroelasticity has been studied. The eigenvalue approach using the Laplace and Fourier transforms has been employed and these transforms have been inverted by using a numerical technique. An application of infinite space with concentrated force at the origin has been presented to illustrate the utility of the approach. The displacement and stress components in the physical domain are obtained numerically. The results are shown graphically and can be used for a broad class of problems related to liquid-saturated porous media.     

Surface waves in a cylindrical borehole through partially-saturated porous solid

Propagation of surface waves is discussed in a cylindrical borehole through a liquid-saturated porous solid of infinite extent. The porous medium is assumed to be a continuum consisting of a solid skeletal with connected void space occupied by a mixture of two immiscible inviscid fluids. This model also represents the partial saturation when liquid fills only a part of the pore space and gas bubbles span the remaining void space. In this isotropic medium, potential functions identify the existence of three dilatational waves coupled with a shear wave. For propagation of plane harmonic waves along the axially-symmetric borehole, these potentials decay into the porous medium. Boundary conditions are chosen to disallow the discharge of liquid into the borehole through its impervious porous walls. A dispersion equation is derived for the propagation of surface waves along the curved walls of no-liquid (all gas) borehole. A numerical example is studied to explore the existence of cylindrical waves in a particular model of the porous sandstone. True surface waves do not propagate along the walls of borehole when the supporting medium is partially saturated. Such waves propagate only beyond a certain frequency when the medium is fully-saturated porous or an elastic one. Dispersion in the velocity of pseudo surface waves is analysed through the changes in consolidation, saturation degree, capillary pressure or porosity.     

Wave propagation in thermoelastic saturated porous medium

Biot’s theory for wave propagation in saturated porous solid is modified to study the propagation of thermoelastic waves in poroelastic medium. Propagation of plane harmonic waves is considered in isotropic poroelastic medium. Relations are derived among the wave-induced temperature in the medium and the displacements of fluid and solid particles. Christoffel equations obtained are modified with the thermal as well as thermoelastic coupling parameters. These equations explain the existence and propagation of four waves in the medium. Three of the waves are attenuating longitudinal waves and one is a non-attenuating transverse wave. Thermal properties of the medium have no effect on the transverse wave. The velocities and attenuation of the longitudinal waves are computed for a numerical model of liquid-saturated sandstone. Their variations with thermal as well as poroelastic parameters are exhibited through numerical examples.     

Response of poroelastic halfspace to steady-state harmonic surface tractions

Traditionally, most formulations of dynamic halfspace problems have represented the material as either an elastic or a viscoelastic solid. Herein the counterpart of Lamb's elastodynamic problem is reformulated and solved for a liquid-saturated poroelastic halfspace using Biot's theory of poroelasticity. The responses of the solid and fluid phases are evaluated due to steady-state harmonic concentrated loads applied to each phase at the surface. The solutions are presented over a broad range of permeabilities and are compared to solutions to Lamb's problem for equivalent drained and undrained solids. Methodology is then introduced by which these results are treated as Green functions for the solution of a mixed boundary-value problem. namely, the response of the poroelastic halfspace to steady-state harmonic vertical motion of a rigid. massless plate. It is observed that small differences exist among overall compliance functions for a drained solid, an undrained solid, and a liquid-saturated porous, halfspace. However, use of the poroelastic model permits the distribution between effective skeletal normal stresses and fluid stresses to be determined.     

Flow of non-Newtonian fluids in fractured porous media: Isogeometric vs standard finite element discretisation

A formulation has been derived for the flow of non-Newtonian (power-law) fluids in deformable, fractured porous media. The formulation is enhanced with a subgrid scale model to accurately represent the flow of the power-law fluids inside the cracks. The resulting equations have been discretised using standard (Lagrangian) finite element shape functions and with non-uniform rational B-splines (NURBS), which have been cast into a standard finite element datastructure using Bézier extraction. The effect of the power-law index on the velocity inside the fracture and on the total fluid flow through the porous medium has been analysed for a typical boundary-value problem. It is shown that large differences between non-Newtonian and linearised Newtonian fluids can occur for the fluid velocity inside the fracture. This can significantly influence the total fluid transport through the domain. A mesh sensitivity study has been carried out as well and shows that markedly smaller element sizes are required in order to obtain accurate results for the local flow inside the fracture, compared with the element sizes necessary for obtaining accurate results inside the porous medium away from the fracture. Moreover, a comparison has been made between the results obtained using standard Lagrange polynomials and those obtained using NURBS. It is shown that while both discretisation methods are able to accurately simulate the deformations and pressures in the porous medium, the higher interelement continuity of NURBS is mandatory for obtaining correct values of the fluid velocities inside the fracture, especially near the tips.     

Numerical modelling of transient contaminant migration problems in infinite porous fractured media using finite/infinite element technique. Part I: Theory

The leakage effect in porous fissured media has been considered in a general sense by introducing a new expression of the leakage term in this paper. The double porosity concept is employed and the related expressions are formulated using the upwind finite element approach. Considering the infinite extension of the problem domain, a mapped transient infinite element has been presented to simulate the far field of the infinite medium. Since the mass transfer function of the present mapped transient infinite element is dependent on both space and time variables, the mechanism of transient contaminant migration problems in infinite porous fractured media can be rigorously simulated because the property matrices of the element are evaluated at any time of interest. By comparing the current numerical results with the analytical ones, the accuracy, correctness and effectiveness of the present method have been established. Three different time discretization schemes were examined and it was found that either the central difference or the backward difference approximation is suitable for the upwind finite element simulation of transient contaminant migration problems.     

Theoretical analyses of nonaqueous phase liquid dissolution‐induced instability in two‐dimensional fluid‐saturated porous media

This paper deals with the theoretical aspects of nonaqueous phase liquid (NAPL)‐dissolution‐induced instability in two‐dimensional fluid‐saturated porous media including solute dispersion effects.After some weaknesses associated with the previous work are analyzed and overcome, a comprehensive dimensionless number, known as the Zhao number, is proposed to represent the main driving force and three controlling mechanisms of an NAPL‐dissolution system that has a finite domain. The linear stability analysis is carried out to derive the critical value of the comprehensive dimensionless number of the NAPL‐dissolution system in a limit case as the ratio of the equilibrium concentration to the density of the NAPL approaches zero. As a result, a theoretical criterion that can be used to assess the instability of planar NAPL‐dissolution fronts in two‐dimensional fluid‐saturated porous media of finite domains has been established. Not only can the present theoretical results be used for the theoretical understanding of the effect of solute dispersion on the instability of an NAPL‐dissolution front in the fluid‐saturated porous medium of either a finite domain or an infinite domain, but also they can be used as benchmark solutions for verifying numerical methods employed to simulate detailed morphological evolution processes of NAPL‐dissolution fronts in two‐dimensional fluid‐saturated porous media. Copyright © 2009 John Wiley & Sons, Ltd.     

Dynamic response of a pile embedded in a porous medium subjected to plane SH waves

In this paper, the frequency domain dynamic response of a pile embedded in a porous medium subjected to SH seismic waves is investigated. The surrounding porous medium of the pile is described by Biot’s poro-elastic theory, while the pile embedded in the porous medium is treated as a beam and described by a beam vibration theory. Using the Hankel transformation method, the fundamental solution for a half-space porous medium subjected to a horizontal circular patch load is established. According to the fictitious pile methodology, the second kind of Fredholm integral equation for the pile is established in terms of the obtained fundamental solution and free wave field. The solution of the integral equation yields the dynamic response of the pile to plane SH waves. Numerical results indicate that the parameters of the porous medium, the pile and incident waves have considerable influences on the dynamic response of the pile and the porous medium.     

Finite element analysis of steady-state natural convection problems in fluid-saturated porous media heated from below

In this paper, a progressive asymptotic approach procedure is presented for solving the steady-state Horton–Rogers–Lapwood problem in a fluid-saturated porous medium. The Horton–Rogers–Lapwood problem possesses a bifurcation and, therefore, makes the direct use of conventional finite element methods difficult. Even if the Rayleigh number is high enough to drive the occurrence of natural convection in a fluid-saturated porous medium, the conventional methods will often produce a trivial non-convective solution. This difficulty can be overcome using the progressive asymptotic approach procedure associated with the finite element method. The method considers a series of modified Horton–Rogers–Lapwood problems in which gravity is assumed to tilt a small angle away from vertical. The main idea behind the progressive asymptotic approach procedure is that through solving a sequence of such modified problems with decreasing tilt, an accurate non-zero velocity solution to the Horton–Rogers–Lapwood problem can be obtained. This solution provides a very good initial prediction for the solution to the original Horton–Rogers–Lapwood problem so that the non-zero velocity solution can be successfully obtained when the tilted angle is set to zero. Comparison of numerical solutions with analytical ones to a benchmark problem of any rectangular geometry has demonstrated the usefulness of the present progressive asymptotic approach procedure. Finally, the procedure has been used to investigate the effect of basin shapes on natural convection of pore-fluid in a porous medium. © 1997 by John Wiley & Sons, Ltd.     

Dynamical problem of micropolar viscoelasticity

The dynamic problem in micropolar viscoelastic medium has been investigated by employing eigen value approach after applying Laplace and Fourier transformations. An example of infinite space with concentrated force at the origin has been presented to illustrate the application of the approach. The integral transforms have been inverted by using a numerical technique to obtain the displacement components, force stresses, couple stress and microrotation in the physical domain. The results for these quantities are given and illustrated graphically.     

Simulation of saltwater intrusion in a poorly karstified coastal aquifer in Lebanon (Eastern Mediterranean)

To date, there has been no agreement on the best way to simulate saltwater intrusion (SWI) in karst aquifers. An equivalent porous medium (EPM) is usually assumed without justification of its applicability. In this paper, SWI in a poorly karstified aquifer in Lebanon is simulated in various ways and compared to measurements. Time series analysis of rainfall and aquifer response is recommended to decide whether quickflow through conduits can be safely ignored. This aids in justifying the selection of the exemplified EPM model. To examine the improvement of SWI representation when discrete features (DFs) are embedded in the model domain, the results of a coupled discrete-continuum (CDC) approach (a hybrid EPM-DF approach) are compared to the EPM model. The two approaches yielded reasonable patterns of hydraulic head and groundwater salinity, which seem trustworthy enough for management purposes. The CDC model also reproduced some local anomalous chloride patterns, being more adaptable with respect to the measurements. It improved the overall accuracy of salinity predictions at wells and better represented the fresh–brackish water interface. Therefore, the CDC approach can be beneficial in modeling SWI in poorly karstified aquifers, and should be compared with the results of the EPM method to decide whether the differences in the outcome at local scale warrant its (more complicated) application. The simulation utilized the SEAWAT code since it is density dependent and public domain, and it enjoys widespread application. Including DFs necessitated manual handling because the selected code has no built-in option for such features.     

Autorun analysis of sedimentary porous materials

Fundamentals of autorun analysis have been given to describe porous media geometry, including sedimantary rocks. The mathematical abstraction of porous media has been presented on the basis of random fields. Classical parameters of porous media, such as porosity and specific surface, have been expressed in terms of autorun function. Finally, a stochastic model has been proposed for the underlying generating mechanism of the porous medium. This model is capable of producing synthetic porous medium and, on the average, porosity as well as the specific surface. The first autorun coefficient is asymptotically equal to the porosity of the medium concerned. It also has been observed that the porosity together with the autorun function are sufficient to produce the specific surface value of the medium.On leave from the Technical University of Istanbul, Taksim, Turkey.     

The fundamental solution of poroelastic plate saturated by fluid and its applications

In this paper, the numerical model of the transverse vibrations of a thin poroelastic plate saturated by a fluid was proposed. Two coupled dynamic equations of equilibrium related to the plate deflection and the equivalent moment were established for an isotropic porous medium with uniform porosity. The fundamental solutions for a porous plate were derived both in the Laplace transform domain and in the time domain. A meshless method was developed and demonstrated in the Laplace transform domain for solving two coupled dynamic equations. Numerical examples demonstrated the accuracy of the method of the fundamental solutions and comparisons were made with analytical solutions. The proposed meshless method was shown to be simple to implement and gave satisfactory results for a poroelastic plate dynamic analysis. Copyright © 2009 John Wiley & Sons, Ltd.     

Modeling permeability in imperfectly layered porous media. I. Derivation of block-scale permeability tensor for thin grid-blocks

Practical expressions are given for the nine components of the block-scale permeability tensor of a thin block. These expressions are derived from the local-scale continuity equation and Darcy's law in an anisotropic layered porous medium. The flow problem is separated in a bottom-flux problem and a top-flux problem, both of which can be solved in essentially the same way. The bottom-flux problem has been worked out in detail, and has been separated in two parts: a vertical potential difference and a horizontal potential difference part. Each is solved with a different approach specially designed for it. Depth-averaged expressions are obtained first, after which block-scale expressions are obtained by assuming a constant depth-averaged flux. In the zeroth order, this results in the well-known Dupuit approximation in geohydrology, and the vertical equilibrium (VE) approximation in petroleum reservoir engineering. The novelty of the theory presented here stems from the application of a perturbation technique to obtain first-order corrections to these well-known results. The local-scale laws are applied in the coordinate system coinciding with the principal axes of the local-scale permeability tensor. Only in this coordinate system the local-scale permeability tensor has zero off-diagonal components. However, since the porous medium is imperfectly layered, the first-order corrections show that the off-diagonal components of the block-scale permeability tensor are not zero. Furthermore, the block-scale permeability tensor is generally nonsymmetric, which implies that a coordinate system in which the off-diagonal terms disappear does not exist.     

可压缩流体饱和孔隙介质中孔隙压力波传播数值分析

首次将高精度时空守恒元/解元方法推广到可压缩流体饱和孔隙介质中孔隙压力波传播的数值计算中。将孔隙度梯度从源（汇）项中分离,直接引入流通量,改进了理论模型。通过对孔隙介质激波问题的数值模拟,验证了方法的精度和有效性。在此基础上,提出了孔隙介质中二维黎曼问题,并揭示了孔隙压力波存在接触间断、激波、膨胀波、压缩波等复杂的结构特征。该成果对二氧化碳地质封存、二氧化碳提高石油采收率、页岩气压裂开采以及地震破裂过程的研究具有重要的理论与应用意义。     

Melt migration and mechanical state in the lower crust of oceanic ridges

Below the melt lens of fast spreading ridges, a low seismic velocity zone has been identified. From the study of ophiolite gabbros, in particular in Oman, this domain has been interpreted as a large magma chamber filled by a melt-poor mush where granular flow controlled by pressure solution-crystallization predominates over plastic flow. Melt migration through the mush is difficult to study in the field because the large magmatic flow taking place in this magma chamber has erased nearly all traces of migration paths. It is, however, still possible to identify sills and former dikes, now largely transposed into the layering. Physical traces for porous flow are rare, but petrological and geochemical evidence suggests that it also contributed to melt migration. Finally, in the lower gabbro horizons large magmatic folds and brecciated zones may bear evidence for magmatic intrusions. The combination of diking, porous flow and large-scale intrusions to carry melt through the magma chamber may be explained by the granular behaviour of the medium. It is suggested that the melt film present between grains and clots of grains reduces the large cohesive forces which characterize a solid, plastic, medium. Melt migration through the mush may thus depend on the size of cohesive clots, evolving through time and space, from porous flow to diking and melt intrusions for increasing larger clots. This process is illustrated by a physical experiment on pressurized air circulation through a granular medium.     

An approximate spring–dashpot artificial boundary for transient wave analysis of fluid-saturated porous media

Practical civil engineering problems are usually formulated in an infinite half-space domain, and a selected finite domain is required to analyze the dynamic responses of a fluid-saturated porous medium by the finite element method (FEM). Devising a method to deal with the boundaries of the finite domain is the key issue for this open system. In this paper, a two-dimensional spring–dashpot artificial boundary (SDAB) for transient analysis in a fluid-saturated porous media is developed. Based on Biot’s dynamic theory of fluid-saturated porous media, the normal and tangential boundary stress formulae are deduced for out-going cylindrical body waves. The boundary stress is proportional to displacement and velocity, thus continuously distributed dashpots and springs can be placed on the artificial boundaries in the normal and tangential directions to simulate the energy absorption of the infinite media outside of the finite domain for the interior distributed source problems. In this paper, the input seismic motion can be realized by applying an equivalent load on the SDAB for the seismic scattering problems of exterior distributed sources. Numerical examples are given and the analyzed results show that the SDAB and the method of wave motion input have good stability and acceptable accuracy.     

水网区区域水位模型及其应用

水网区在宏观尺度上可被视为一种连续介质,可用水在多孔介质中的运动来比拟水网中水的运动.在这种概化条件下,推求出水网区水体运动的表述方法.利用这一方法建立的太湖水网区区域水位模型,与实际情况相比,获得了较好的效果.     

A porosity-gradient replacement approach for computational simulation of chemical-dissolution front propagation in fluid-saturated porous media including pore-fluid compressibility

In dealing with chemical-dissolution-front propagation problems in fluid-saturated porous media, the chemical dissolution front represented by the porosity of the medium may have a very steep slope (i.e., a very large porosity gradient) at the dissolution front, depending on the mineral dissolution ratio that is defined as the equilibrium concentration of the dissolved minerals in the pore-fluid to the solid molar density of the dissolvable minerals in the solid matrix. When the mineral dissolution ratio approaches zero, the theoretical value of the porosity gradient tends to infinity at the chemical dissolution front. Even for a very small value of the mineral dissolution ratio, which is very common in geochemical systems, the porosity gradient can be large enough to cause the solution hard to converge when the conventional finite element method is used to solve a chemical dissolution problem in a fluid-saturated porous medium where the pore-fluid is compressible. To improve the convergent speed of solution, a porosity-gradient replacement approach, in which the term involving porosity-gradient computation is replaced by a new term consisting of pore-fluid density-gradient and pressure-gradient computation, is first proposed and then incorporated into the finite element method in this study. Through comparing the numerical results obtained from the proposed approach with the theoretical solutions for a benchmark problem, it has been demonstrated that not only can the solution divergence be avoid, but also the accurate simulation results can be obtained when the proposed porosity-gradient replacement approach is used to solve chemical-dissolution-front propagation problems in fluid-saturated porous media including pore-fluid compressibility.