Boundary conditions for porous solids saturated with viscous fluid

Boundary conditions are derived to represent the continuity requirements at the boundaries of a porous solid saturated with viscous fluid. They are derived from the physically grounded principles with a mathematical check on the conservation of energy. The poroelastic solid is a dissipative one for the presence of viscosity in the interstitial fluid. The dissipative stresses due to the viscosity of pore-fluid are well represented in the boundary conditions. The unequal particle motions of two constituents of porous aggre~ gate at a boundary between two solids are explained in terms of the drainage of pore-fluid leading to imperfect bonding. A mathematical model is derived for the partial connec- tion of surface pores at the porous-porous interface. At this interface, the loose-contact slipping and partial pore opening/connection may dissipate a part of strain energy. A numerical example shows that, at the interface between water and oil-saturated sandstone, the modified boundary conditions do affect the energies of the waves refracting into the isotropic porous medium.     

On the static and dynamic stability of thin beam conveying fluid

In this paper, numerical investigation of the statical and dynamical stability of aligned and misaligned viscoelastic cantilevered beam is performed with a terminal nozzle in the presence of gravity in two cases: (1) effect of fluid velocity on the flutter boundary of beam conveying fluid and (2) effect of gravity on the buckling boundary of beam conveying fluid. The beam is assumed to have a large width-to-thickness ratio, so the out-of-plane bending rigidity is far higher than the in-plane bending and torsional rigidities. Gravity vector is considered in the vertical direction. Thus, deflection of the beam because of the gravity effect couples the in-plane bending and torsional equations. The beam is modeled by Euler–Bernoulli beam theory, with the flow-induced inertia, Coriolis and centrifugal forces along the beam considered as a distributed load along the beam. Furthermore, the end nozzle is regarded as a lumped mass and modeled as a follower axial force. The extended Hamilton’s principle and the Galerkin method are utilized to derive the bending–torsional equations of motion. The coupled equations of motion are solved as eigenvalue problems. Also, several cases are examined to study the impact of gravity, beam inclination angle, mass ratio, nozzle aspect ratio, bending-to-bending rigidity ratio and bending-to-torsional rigidity ratio on flutter and buckling margin of the system.

     

Propagation of acceleration waves in incompressible saturated porous solids

Within the framework of the incompressible porous media model, the propagation properties of acceleration waves in liquid-filled porous solids is discussed. The incompressibility of the two constituents in the model forces the amplitudes of the longitudinal waves in the skeleton and in the liquid to satisfy a certain relation. The two propagation speeds are presented by examination for the existence of acceleration waves and only longitudinal and transverse waves are realizable in the incompressible two-phase porous materials.     

A linear dynamic model for a saturated porous medium

A linear isothermal dynamic model for a porous medium saturated by a Newtonian fluid is developed in the paper. In contrast to the mixture theory, the assumption of phase separation is avoided by introducing a single constitutive energy function for the porous medium. An important advantage of the proposed model is it can account for the couplings between the solid skeleton and the pore fluid. The mass and momentum balance equations are obtained according to the generalized mixture theory. Constitutive relations for the stress, the pore pressure are derived from the total free energy accounting for inter-phase interaction. In order to describe the momentum interaction between the fluid and the solid, a frequency independent Biot-type drag force model is introduced. A temporal variable porosity model with relaxation accounting for additional attenuation is introduced for the first time. The details of parameter estimation are discussed in the paper. It is demonstrated that all the material parameters in our model can be estimated from directly measurable phenomenological parameters. In terms of the equations of motion in the frequency domain, the wave velocities and the attenuations for the two P waves and one S wave are calculated. The influences of the porosity relaxation coefficient on the velocities and attenuation coefficients of the three waves of the porous medium are discussed in a numerical example.     

On the limit and applicability of dynamic homogenization

Recent years have seen considerable research success in the field of dynamic homogenization which seeks to define frequency dependent effective properties for heterogeneous composites for the purpose of studying wave propagation. There is an approximation involved in replacing a heterogeneous composite with its homogenized equivalent. In this paper we propose a quantification to this approximation. We study the problem of reflection at the interface of a layered periodic composite and its dynamic homogenized equivalent. It is shown that if the homogenized parameters are to appropriately represent the layered composite in a finite setting and at a given frequency, then reflection at this special interface must be close to zero at that frequency. We show that a comprehensive homogenization scheme proposed in an earlier paper results in negligible reflection in the low frequency regime, thereby suggesting its applicability in a finite composite setting. In this paper we explicitly study a 2-phase composite and a 3-phase composite which exhibits negative effective properties over its second branch. We show that based upon the reflected energy profile of the two cases, there exist good arguments for considering the second branch of a 3-phase composite a true negative branch with negative group velocity. Through arguments of calculated reflected energy we note that infinite-domain homogenization is much more applicable to finite cases of the 3-phase composite than it is to the 2-phase composite. In fact, the applicability of dynamic homogenization extends to most of the first branch (negligible reflection) for the 3-phase composite. This is in contrast with a periodic composite without local resonance where the approximation of homogenization worsens with increasing frequency over the first branch and is demonstrably bad on the second branch. We also study the effect of the interface location on the applicability of homogenization. The results open intriguing questions regarding the effects of replacing a semi-infinite periodic composite with its Bloch-wave (infinite domain) dynamic properties on such phenomenon as negative refraction.     

Non-Darcy buoyancy driven flows in a fluid saturated porous medium: the use of asymptotic computational fluid dynamics (ACFD) approach

Natural convection in a fluid saturated porous medium has been numerically investigated using a generalized non-Darcy approach. The governing equations are solved by using Finite Volume approach. First order upwind scheme is employed for convective formulation and SIMPLE algorithm for pressure velocity coupling. Numerical results are presented to study the influence of parameters such as Rayleigh number (10⁶ ≤Ra ≤10⁸), Darcy number (10⁻⁵ ≤ Da ≤ 10⁻²), porosity (0.4 ≤ ɛ ≤ 0.9) and Prandtl number (0.01 ≤ Pr ≤ 10) on the flow behavior and heat transfer. By combining the method of matched asymptotic expansions with computational fluid dynamics (CFD), so called asymptotic computational fluid dynamics (ACFD) technique has been employed to generate correlation for average Nusselt number. The technique is found to be an attractive option for generating correlation and also in the analysis of natural convection in porous medium over a fairly wide range of parameters with fewer simulations for numerical solutions.     

A numerical investigation into the behavior of ground-supported concrete silos filled with saturated solids

This paper introduces a finite-element solution for simulating the filling process of ground-supported concrete silos filled with saturated granular material. An elasto-plastic axisymmetric finite-element model is used to represent both the granular material and the concrete silo. The interaction between the two materials is modeled using interface elements to allow for relative movement. The filling process is idealized via a multi-stage numerical technique capable of representing both undrained and drained conditions for the granular material. The effects of the relative stiffness between the foundation and wall are examined, as are the boundary conditions at the top of the structure (the roof details).Depending on the drainage properties of the stored material, the effect of the filling process may be time-dependent. The excess pore water pressure resulting from the filling process may cause a substantial increase in the hoop stresses in the wall. The predicted internal forces may be influenced by the foundation rigidity, but not by the boundary condition at the top of the wall.The results of these analyses may be used to design experiments to evaluate existing silos, or to develop filling strategies to minimize loads on existing structures.     

Non-gaussian diffusion modeling in composite porous media by homogenization: Tail effect

In the paper anomalous diffusion appearing in a porous medium composed of two porous components of considerably different diffusion characteristics is examined. The differences in diffusivities are supposed to result either from two medium types being present or from variations in pore size (double porosity media). The long-tail effect is predicted using the homogenization approach based on the application of multiple scale asymptotic developments. It is shown that, if the ratio of effective diffusion coefficients of two porous media is of the order of magnitude smaller or equal O( ²), where is a homogenization parameter, then the macroscopic behaviour of the composite may be affected by the presence of tail-effect. The results of the theoretical analysis were applied to a problem of diffusion in a bilaminate composite. Analytical calculations were performed to show the presence of the long-tail effect in two particular cases.Notations c _i the concentration of chemical species in water within the medium i - D _i the effective diffusion coefficient for the medium i - D _ij ^eff the macroscopic (or effective) diffusion tensor in the composite - ERV the elementary representative volume - h the thickness of the period - l a chracteristic length of the ERV or the periodic cell - L a characteristic macroscopic length - n the volumetric fraction of the material 2 - 1–n the volumetric fraction of the material 1 - N the unit vector normal to - t the time variable - x the macroscopic (or slow) space variable - y the microscopic (or fast) space variable - c _1c ,C _2c ,D _1c ,D _2c the characteristic quantities - T,T _1L ,T _2L ,T _1l ,T _2l the characteristic times - c ₁ ^* ,c ₂ ^* ,D ₁ ^* ,D ₂ ^* ,t ^* the non-dimensional variables - the homogenization parameter - ₁ the domain occupied by the material 1 - ₂ the domain occupied by the material 2 - the interface between the domains 1 and 2 - the total volume of the periodic cell - /x_i the gradient operator - the gradient operator     

A phase-field modeling approach of hydraulic fracture in saturated porous media

Continuum porous media theories, extended by a diffusive phase-field modeling (PFM) approach, introduce a convenient and efficient tool to the simulation of hydraulic fracture in fluid-saturated heterogeneous materials. In this, hydraulic- or tension-induced fracture occurs in the solid phase. This leads to permanent local changes in the permeability, the volume fractions of the constituents as well as the interstitial-fluid flow. In this work, the mechanical behaviors of the multi-field, multi-phase problem of saturated porous media, such as the pore-fluid flow and the solid-skeleton deformation, are described using the macroscopic Theory of Porous Media (TPM). To account for crack nucleation and propagation in the sense of brittle fracture, the energy-minimization-based PFM procedure is applied, which approximates the sharp edges of the crack by a diffusive transition zone using an auxiliary phase-field variable. Furthermore, the PFM can be implemented in usual continuum finite element packages, allowing for a robust solution of initial-boundary-value problems (IBVP). For the purpose of validation and comparison, simulations of a two-dimensional IBVP of hydraulic fracture are introduced at the end of this research paper.     

Analysis of heat and fluid flow in partially divided fluid saturated porous cavities

Fluid flow and heat transfer phenomena in partially divided cavities filled with porous media have been numerically studied in this research. A non-Darcy generalized formulation is applied to describe the behavior of fluid flow in porous media. A splitting semi-implicit finite element method is adopted to solve the governing equations. The range of Ra involved in this study is between 10⁴ and 10⁶. Three different locations of dividers are investigated to probe the geometrical effect on heat and fluid flow. The results of Da = 10⁻² display a trend similar to the non-porous medium, but those of Da = 10⁻⁴ show dramatic decrease in flow strength, as well as heat transfer rate. A different location of divider may change the local and average Nusselt numbers. Received on 10 August 1999     

On the propagation of waves through porous solids

This note reexamines Biot's model for the propagation of acoustic waves in a material such as cohensionless sand, infused with a fluid, within the context of mixture theory. Instead of the standard entropy equation that is used in mixture theory, an inequality for the viscous dissipation is employed here due to a conceptual difficulty that one encounters in applying the standard equation to a mixture of sand and a fluid. The wave equations are reformulated by taking the velocity field, instead of the displacement, for the fluid as a primary quantity. By recognizing and thereby exploiting the dependence of the stored energy of the sand on the pore fluid pressure and choosing an appropriate form for the rate of dissipation, a set of governing equations are obtained which are equivalent to those derived by Biot [J. Acoust. Soc. Am. 28(1956) 168, 179; J. Appl. Phys. 33(1962) 1482]. A differential equation for the pore fluid pressure is derived and the effects of drag and virtual mass are dealt with in a unified fashion. The procedure allows us to develop generalizations to Biot's equations in a rational manner.     

Linearization of dynamic equations for a saturated porous medium

Equations of the mechanics of a saturated porous medium that account for the deformation tensor of the fluid and the initial state of the medium are derived. Emphasis is on the linearization and justification of the basic nonlinear equations of the theory. The relationship between the pressure (deformation) of the fluid and the thermodynamic and kinematic parameters of the medium is established. The solutions obtained are compared with Biot’s well-known classical results and analyzed     

Throughflow and g-jitter effects on binary fluid saturated porous medium

A non-autonomous complex Ginzburg-Landau equation (CGLE) for the finite amplitude of convection is derived, and a method is presented here to determine the amplitude of this convection with a weakly nonlinear thermal instability for an oscillatory mode under throughflow and gravity modulation. Only infinitesimal disturbances are considered. The disturbances in velocity, temperature, and solutal fields are treated by a perturbation expansion in powers of the amplitude of the applied gravity field. Throughflow can stabilize or destabilize the system for stress free and isothermal boundary conditions. The Nusselt and Sherwood numbers are obtained numerically to present the results of heat and mass transfer. It is found that throughflow and gravity modulation can be used alternately to heat and mass transfer. Further, oscillatory flow, rather than stationary flow, enhances heat and mass transfer.     

On the stability of natural convection in a porous vertical slab saturated with an Oldroyd-B fluid

The stability of the conduction regime of natural convection in a porous vertical slab saturated with an Oldroyd-B fluid has been studied. A modified Darcy’s law is utilized to describe the flow in a porous medium. The eigenvalue problem is solved using Chebyshev collocation method and the critical Darcy–Rayleigh number with respect to the wave number is extracted for different values of physical parameters. Despite the basic state being the same for Newtonian and Oldroyd-B fluids, it is observed that the basic flow is unstable for viscoelastic fluids—a result of contrast compared to Newtonian as well as for power-law fluids. It is found that the viscoelasticity parameters exhibit both stabilizing and destabilizing influence on the system. Increase in the value of strain retardation parameter \(\Lambda _2 \) portrays stabilizing influence on the system while increasing stress relaxation parameter \(\Lambda _1\) displays an opposite trend. Also, the effect of increasing ratio of heat capacities is to delay the onset of instability. The results for Maxwell fluid obtained as a particular case from the present study indicate that the system is more unstable compared to Oldroyd-B fluid.     

On solving the problem of reflection of linear waves in a fluid from a porous half-space saturated by this fluid

Solutions of the problem of reflection of a stepwise pressure wave in a linearly compressed fluid from a flat boundary of a porous medium of infinite length saturated by the same fluid are obtained in the acoustic approximation. Based on analytical solutions, a numerical analysis is performed to reveal the specific features of the reflected and incident waves, depending on porosity and permeability of the porous half-space. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 5, pp. 16–26, September–October, 2006.     

Elastic behavior of saturated porous materials under undrained freezing

The elastic behavior of saturated porous materi- als under undrained freezing is investigated by using a poro- mechanical approach. Thermodynamic equilibria are used to describe the crystallization process of the partially frozen solution in bulk state and confined state in pores. By phase transition at freezing, fusion energy, thermal contraction of solid, solution and ice crystals, volume changes of crystallization build up remarkable pore pressure that induces expansion or shrinkage of solid matrix. Owing to the lower chemical potential when pore water mixes with salts, fewer ice forms in pores. Penetration of ice into the porous materials increases the capillary pressure, but limits effect on the pore liquid pressure and the strain of solid matrix. On the contrary, the pore pressure induced by solution density rises as salt concentration increases and causes significant shrinkage of solid matrix.     

Strength homogenization of matrix-inclusion composites using the linear comparison composite approach

A homogenization procedure to estimate the macroscopic strength of nonlinear matrix-inclusion composites with different strength characteristics of the matrix and inclusions, respectively, is presented in this paper. The strength up-scaling is formulated within the framework of the yield design theory and the linear comparison composite (LCC) approach, introduced by Ponte Castaneda (2002) and extended to frictional models by Ortega et al. (2011), which estimates the macroscopic strength of composite materials in terms of an optimally chosen linear thermo–elastic comparison composite with a similar underlying microstructure. In the paper various combinations for the underlying material behavior for the individual phases of the composite are considered: The matrix phase can be a quasi frictional material characterized either by a Drucker–Prager-type (hyperbolic) or an elliptical strength criterion, which predicts a strength limit also in hydrostatic compression, while the inclusion phase either may represent empty pores, pore voids filled with a pore fluid, rigid inclusions, or solid inclusions, whose strength characteristics also maybe described by a Drucker–Prager-type or an elliptical strength criterion. For generating the homogenized strength criterion efficiently in such general cases of matrix-inclusion composites, a novel algorithm is proposed in the paper. The validation of the proposed strength homogenization procedure for selected combinations of strength characteristics of the matrix material and the inclusions is conducted by comparisons with experimental results and alternative existing strength homogenization models.