Porous Deformable Media Research Articles

The Simulation of the Nonlinear Consolidation of Porous Media

In this paper, the general formulation of the problem of coupled deformation of a porous deformable medium with a fluid flowing through the pores is formulated, mathematically investigated and numerically implemented within the framework of physical and geometric nonlinearity. We present the formulation of the problem in velocities of solid phase displacements and the rate of pore pressure change in differential and variational forms. A phenomenological approach was used to formulate the mechanical model. The equations of the coupled consolidation model were derived from the general conservation laws of continuum mechanics using spatial averaging over a representative volume element. The consolidation model took into account the change in the porosity and permeability of the medium during deformation. The equations of filtration and porosity change, originally presented in Euler approach, were reformulated in Lagrangian coordinates of the solid phase using the relative fluid velocity according to ALE (Arbitrary Lagrangian - Eulerian) approach. The Gâteaux differentiation technique was used to linearize the variational equilibrium equations. For spatial discretization of the saddle system of equations, the finite element method (FEM) was used: quadratic serendipity elements for approximating the equilibrium equations and Brick type elements for approximating the filtration equation. To solve the system of equilibrium and filtration equations, a generalization of the implicit scheme with internal iterations at each time step by the Uzawa method was used. The results of numerical simulation of elastoplastic deformation of a water-saturated soil under load with fluid outflow are presented. To simulate the constitutive relations of elastoplastic deformation of soil under short-term loads, a generalization of S.S. Grigoryan's model to large deformations is proposed. The calculations were carried out in our own program code. The developed consolidation model can be used to simulate the formation of tracking ruts and unevenness of natural roads, as well as to calculate the uneven settlement of engineering structures.

Two-phase hyperbolic model for porous media saturated with a viscous fluid and its application to wavefields simulation

A new hyperbolic two-phase model of a porous deformable medium saturated with a viscous fluid is presented and some of its features or performances are discussed. The governing equations are derived in the framework of Symmetric Hyperbolic Thermodynamically Compatible (SHTC) systems. The model accounts for such dissipative mechanisms as interfacial friction and viscous dissipation of the saturating fluid. Both dissipative mechanisms are modeled with hyperbolic equations with relaxation type source terms. The linearized version of the presented model is derived under the assumption of instantaneous relaxation of the phase pressures and its application to modeling propagation of small-amplitude waves in a porous medium saturated with a viscous fluid is discussed. We observe that accounting of the fluid viscosity may significantly affect the dispersion of fast and slow compression waves. Additionally, the shear wave attenuates rapidly due to the viscosity of the saturating fluid which makes it difficult to observe this wave in most of the test cases presented. However, some test cases are presented in which shear waves can be observed in the vicinity of interfaces between regions with different porosity.

On the appropriateness of modelling brain parenchyma as a biphasic continuum

Computational methods originally developed for analysis in engineering have been applied to the analysis of biological materials for many years. One particular application of these engineering tools is the brain, allowing researchers to predict the behaviour of brain tissue in various traumatic, surgical and medical scenarios. Typically two different approaches have been used to model deformation of brain tissue: single-phase models which treat the brain as a viscoelastic material, and biphasic models which treat the brain as a porous deformable medium through which liquid can move. In order to model the brain as a biphasic continuum, the hydraulic conductivity of the solid phase is required; there are many theoretical values for this conductivity in the literature, with variations of up to three orders of magnitude.We carried out a series of simple experiments using lamb and sheep brain tissue to establish the rate at which cerebrospinal fluid moves through the brain parenchyma. Mindful of possible variations in hydraulic conductivity with tissue deformation, our intention was to carry out our experiments on brain tissue subjected to minimal deformation. This has enabled us to compare the rate of flow with values predicted by some of the theoretical values of hydraulic conductivity from the literature. Our results indicate that the hydraulic conductivity of the brain parenchyma is consistent with the lowest theoretical published values. These extremely low hydraulic conductivities lead to such low rates of CSF flow through the brain tissue that in effect the material behaves as a single-phase deformable solid.

Infiltration through Deformable Porous Media

The equations driving the flow of an incompressible fluid through a porous deformable medium are derived in the framework of the mixture theory. This mechanical system is described as a binary saturated mixture of incompressible components. The mathematical problem is characterized by the presence of two moving boundaries, the material boundaries of the solid and the fluid, respectively. The boundary and interface conditions to be supplied to ensure the well-posedness of the initial boundary value problem are inspired by typical processes in the manufacturing of composite materials. They are discussed in their connections with the nature of the partial stress tensors. Then the equations are conveniently recast in a material frame of reference fixed on the solid skeleton. By a proper choice of the characteristic magnitudes of the problem at hand, the equations are rewritten in non-dimensional form and the conditions which enable neglecting the inertial terms are discussed. The second part of the paper is devoted to the study of one-dimensional infiltration by the inertia-neglected model. It is shown that when the flow is driven through an elastic matrix by a constant rate liquid inflow at the border some exact results can be obtained.

Effect of finite deformations of combustible porous media on dynamical processes of internal heat--mass transfer

In the present paper a combustion model for porous deformable medium is developed for the case of finite deformations of solid phase. The effective method of analytico-numerical solving the problem on combustion has been suggested for the case of porous deformable medium with a system of periodically located cracks. With the help of the method, numerical analysis of the effect of the elasticity modulus of solid phase on characteristics of internal heat and mass transfer is conducted.

Effect of finite deformations of a solid propellant on dynamic processes in combustible porous media

A combustion model for porous deformable media that allows for finite deformations of the solid phase is developed. An effective numerical-analytic method of solving the combustion problem for a deformable porous medium with a system of periodically located cracks is proposed. The method is employed for a numerical analysis of the effect of the modulus of elasticity of the solid phase on the characteristics of internal heat and mass transfer.

A Model of Silage Consolidation and Effluent Flow

A mathematical model of silage consolidation and effluent flow is presented. Silage is modelled as a porous deformable medium containing a fluid. Model predictions for silage settlement heights and effluent production are found to be in good agreement with those found in laboratory-scale experiments (6kg of grass silage) and with findings reported in the literature for full-size silos containing several hundred tonnes of silage. The model can provide a quantitative understanding of the processes involved in effluent production so that scientifically based solutions to problems, such as the effective use of absorbents, can be reached. The model has been used to assess the importance of dry-matter content and silo geometry in determining effluent production. The height/diameter ratio of a silo is predicted to have a significant effect on effluent production. Increasing the height/diameter ratio is predicted to prolong the time from initial drainage to the peak flow of effluent.

Mechanics of a porous two-phase strongly viscous medium and some geophysical applications

According to modern geophysical ideas [1], the oceanic crust is formed when the lithospheric plates move apart in the axial regions of the rift zones of the Earth due to the penetration of molten matter from the asthenosphere into cracks and pores of the lithosphere and the subsequent solidification of this matter (the lithosphere and asthenosphere are the upper layers of the Earth). In these regions, which have a width of the order of several tens of kilometers, the liquid magmatic material penetrates the crystalline skeleton of the crust, which means that one must regard this region as a saturated fissured and porous deformable medium that permits phase transitions between the skeleton and the liquid filler. Geophysical data indicate that there is a similar two-phase fissured and porous structure of the medium in the asthenosphere (at least in certain regions) and in regions of high tectonic and volcanic activity. Because geological processes take place slowly, it can be assumed that the skeleton of the two-phase medium is deformed as a strongly viscous incompressible Newtonian liquid with a viscosity much greater than the viscosity of the filler.

Some contributions to the theory of flow of fluids through porous deformable media

The present paper is based onBiot's theory of flow of fluids through a porous deformable medium. The presence of a distribution of sources of fluids in the medium gives rise to an additional term in the generalizedDarcy's law. The present note is concerned with solutions of the displacement equations ofBiot's theory for the case of fluid sources. The regular solution of these equations is given in terms of special functions. The singular solution — connected with the action of fluid sources — is obtained with the aid of the principle of reciprocity of displacements.