We propose a mathematical model describing the process of filling the pores of a building material with lime water solution with the goal to improve the consistency of the porous solid. Chemical reactions produce calcium carbonate which glues the solid particles together at some distance from the boundary and strengthens the whole structure. The model consists of a 3D convection–diffusion system with a nonlinear boundary condition for the liquid and for calcium hydroxide, coupled with the mass balance equations for the chemical reaction. The main result consists in proving that the system has a solution for each initial data from a physically relevant class. A 1D numerical test shows a qualitative agreement with experimental observations. © 2021 Elsevier Ltd

First, different porous media theories are presented. Some approaches are based on the classical mixture theory for fluids introduced in the 1960s by Truesdell and Coworkers. One of the first researchers who extended the theory to porous media (thus mixtures containing at least one solid constituent) and also accounting for chemical reactions was Bowen. Another important branch of porous media theory goes back to Biot. In the beginning, he dealt with classical geotechnical problems and set up his model empirically. Mathematicians often use reaction–diffusion equations which are limited in comparison with continuum models by several restrictive assumptions and very often only applicable to special problems. In this paper, the focus lies on approaches based on the mixture theory which incorporate chemical reactions. Different strategies to describe the chemical potential for mixtures are presented, and different opinions about the exploitation of the second law of thermodynamics for mixtures are put forward. Finally, several works of different types including chemical reactions in porous media are summarized. © 2021, The Author(s).

In this study, a 1000 L pilot scale internal loop airlift bioreactor was operated and compared to a mathematical model to determine the best design for optimal supply of oxygen for nitrification and sufficient air for biomass fluidization. The design model is based on parameters such as geometry, carrier density, and airflow of the 1000 L pilot scale bioreactor. The model predicts a range of superficial air velocities (0.009–0.013 m/s) under which the airlift bioreactor was fluidized. Three superficial air velocities (0.009 m/s, 0.011 m/s and 0.013 m/s) were experimentally tested in the pilot plant and the obtained circulation velocities were compared with the predicted design scenarios. The predicted velocity was in agreement with the measured velocity. The aim of the mathematical model and the calculations of different geometry scenarios was to define the optimal geometry design for the physical model. The results show that the ratio of the cross-sectional area between the riser and the downcomer of 1.33 resulted in the lowest superficial liquid velocity of 0.076 m/s in the riser at a relative low superficial air velocity of 0.011 m/s and a carrier density of 1030 kg/m3. This bioreactor design enabled longest retention time of particles in the oxygenated riser. © 2019 Institution of Chemical Engineers

We propose a model for multicomponent flow of immiscible fluids in a deformable porous medium accounting for capillary hysteresis. Oil, water, and air in the soil pores offer a typical example of a real situation occurring in practice. We state the problem within the formalism of continuum mechanics as a slow diffusion process in Lagrange coordinates. The balance laws for volumes, masses, and momentum lead to a degenerate parabolic PDE system. In the special case of a rigid solid matrix material and three fluid components, we prove under further technical assumptions that the system is mathematically well posed in a small neighborhood of an equilibrium. © 2019 John Wiley & Sons, Ltd.

In this article models for porous media containing one, two or three pore fluids, e.g. water, oil and air, are shown. The newly presented model for three pore fluids is an extension of the established ones for one and two pore fluids. The inherent macroscopic material parameters of all models follow from their measurable microscopic counterparts by micro-macro considerations which, in turn, follow from gedanken experiments. The models are intended to describe the propagation of sound waves. Capillary pressures in mixtures of immiscible fluids in the pores are taken into account and relative permeabilities are considered. © 2018 Elsevier Ltd.

The variation of the rail support stiffness is an inherent issue of railway tracks. There is still no consensus on the influence of the rail support stiffness variation on the dynamic response of the vehicle-track system. One view indicates that changes of the support stiffness do not have considerable influence on the vehicle dynamic response. The main influence factor is the rail deflection. However, the opposite view presents that the influence of the support stiffness on the system dynamic response is obvious. Reasons that lead to the dispute of previous studies are the neglect of the influence of the excitation frequencies and a lack of understanding of stiffness sensitive zones. In this study a vehicle-track coupling model with equivalent overall support stiffness is employed to investigate the response of the vehicle to changes of the track stiffness and excitation frequencies. Results show that for each of frequencies (1-40 Hz) the dynamic response of the vehicle is only sensitive to a certain range of the support stiffness. A stiffness sensitive zone for each excitation frequency can be observed. In order to further study the influence of the subgrade on the vehicle system dynamic response a vehicle-track-subgrade model is utilised. The subgrade stiffness belonging to the stiffness sensitive zone has a significant influence on vehicle vibrations. For overall support stiffness of the rail higher than 20 kN/mm, the stiffness sensitive zones of low excitation frequencies can be avoided. © 2016 Informa UK Limited, trading as Taylor & Francis Group.

It is well known that the capillary pressure curve of partially saturated soils exhibits a hysteresis. For the same degree of saturation it has different values depending on the initial state of the soil, thus for drying of a wet soil or wetting of a dry soil. The influence of these different values of the capillary pressure on the propagation of sound waves is studied by use of a linear hyperbolic model. Even if the model does not contain a hysteresis operator, the effect of hysteresis in the capillary pressure curve is accounted for. In order to obtain the limits of phase speeds and attenuations for the two processes the correspondent values for main drying and main wetting are inserted into the model separately. This is done for two examples of soils, namely for Del Monte sand and for a silt loam both filled by an air-water mixture. The wave analysis reveals four waves: one transversal wave and three longitudinal waves. The waves which are driven by the immiscible pore fluids are influenced by the hysteresis in the capillary pressure curve while the waves which are mainly driven by the solid are not.

A PDE system consisting of the momentum balance, mass balance, and energy balance equations for displacement, capillary pressure, and temperature as a model for unsaturated fluid flow in a porous viscoelastoplastic solid is shown to admit a solution under appropriate assumptions on the constitutive behavior. The problem involves two hysteresis operators accounting for plastic and capillary hysteresis. © 2016 Society for Industrial and Applied Mathematics.

We propose a model for unsaturated poro-plastic flow derived from the thermodynamic principles. For the isothermal case, the problem consists of a degenerate coupled system of two PDEs with two independent hysteresis operators describing hysteresis phenomena in both the solid and the pore fluids. Under natural hypotheses, we prove the existence of a global strong solution for this system. Copyright © 2015 John Wiley & Sons, Ltd.

Tunnel face stability is important for safe tunneling and the protection of the surrounding environment. Upper bound analysis is a widely applied method to investigate tunnel face stability. In this paper, a tunnel face collapse of Guangzhou metro line 3 is presented. Accordingly, seepage is considered in the upper bound solutions for face stability in layered soils. Steady-state seepage is reached in the first 1200 s of each drilling step. In the crossed layer, the seepage flow is horizontal toward the tunnel face, whereas in the cover layer, the seepage vertically percolates into the crossed layer. By considering the seepage forces on the tunnel face and on the soil particles, the upper bound solution for the support pressure needed for face stability in layered soil with seepage is obtained. Under saturated conditions, the support pressure is influenced by the variation of the depth ratio due to the seepage effect. Moreover, the support pressure depends linearly on the groundwater level. This study provides estimations of the support pressure for face stability in tunnel design. © 2016, Zhejiang University and Springer-Verlag Berlin Heidelberg.

In this review, the phenomenology of hysteresis is discussed, including both empirical and mathematical models, and some examples are presented. The focus lies on soil-moisture hysteresis, where the capillary pressure exhibits different values depending on the initial state of saturation. An historical overview is given of the investigation of this phenomenon, of various empirical models, and also of some mathematical approaches to soil-moisture hysteresis. All these studies are aimed at accurately fitting experimental results - not only the main hysteresis curves but also the inner hysteresis curves that occur upon re-wetting and re-drying. Finally, a comparison is made to another field in which hysteresis appears, the deformation of pseudoelastic bodies such as shape memory alloys. © 2014 Springer-Verlag Wien.

The working face of tunnel constructions has to be kept stable during tunneling to prevent large soil deformations or fatal failure. In layered soils with lower cohesion, failures happen more often and more abrupt than in cohesive soils. Therefore, the maintenance of a proper support pressure at the tunnel working face is of high importance. In this paper, an upper bound analysis is introduced to investigate the minimum support pressure for the face stability in layered soils. A three-dimensional kinematically admissible mechanism for the upper bound analysis is improved to model potential failure within different soil layers. An analytical solution for the support pressure assessment is achieved. The influence of the crossing and cover soil on the face stability is analyzed, respectively. This solution provides an analytical estimation of the minimum support pressure for the face stability. It may be used as a reference for projects under similar conditions. © 2013 Springer-Verlag Berlin Heidelberg.

A railway track-subgrade system is modeled using the finite element method (FEM). Two different transition zones between a bridge and an ordinary subgrade, usually used for high speed passenger lines in China, are investigated. Both the calculated vertical displacement and acceleration of the rail and the slab and the calculated vertical displacement and the stress of the subgrade surface of the two transition zones are compared. The dynamic response of the two-part transition section is better than that of the inverted trapezoid transition section, and a 30-m length of both transition sections is recommended. The dynamic response of the track-subgrade system changes abruptly after the first 3. m of the transition section, measured from the bridge abutment. Special attention should be given to this critical zone during construction. © 2012 Elsevier Ltd.

By means of a Biot-like model with isotropic stress-strain relations but anisotropic permeability monochromatic waves in a two-component poroelastic medium are analyzed. The anisotropy is induced by the anisotropy of the tortuosity which is introduced by a second order symmetric tensor. The model describes for a certain choice of orientation of the propagation direction four modes of propagation: a decoupled transversal S1-wave, a pseudo transversal mode S2 and two pseudo longitudinal modes P1 and P2. Phase speeds and attenuations of these waves are shown in dependence on the orientation of the principal directions of the tortuosity. © 2013 American Society of Civil Engineers.

The paper is devoted to the analysis of monochromatic waves in two-component poroelastic materials described by a Biot-likemodelwhose stress-strain relations are isotropic but the permeability is anisotropic. This anisotropy is induced by the anisotropy of the tortuosity which is given by a second order symmetric tensor. This is a new feature of the model while in earlier papers only isotropic permeabilities were considered. We show that this new model describes four modes of propagation. For our special choice of orientation of the direction of propagation these are two pseudo longitudinal modes P1 and P2, one pseudo transversal mode S2 and one transversal mode S1. The latter becomes also pseudo transversal in the general case of anisotropy. We analyze the speeds of propagation and the attenuation of these waves as well as the polarization properties in dependence on the orientation of the principal directions of the tortuosity.We indicate the practical importance of different shear (transversal)modes of propagation in a possible new nondestructive test of geophysical materials. © Springer-Verlag 2011.

The dynamical investigation of two-component poroelastic media is important for practical applications. Analytic solution methods are often not available since they are too complicated for the complex governing sets of equations. For this reason, often some existing numerical methods are used. In this work results obtained with the finite element method are opposed to those obtained by Schanz using the boundary element method. Not only the influence of the number of elements and time steps on the simple example of a poroelastic column but also the impact of different values of the permeability coefficient is investigated.

Macroscopic modeling of soils is based on a number of properties that refer to the mesoscopic morphology. The most fundamental parameters of this art are: 1) coupling parameters between partial stresses of components and deformations of components, 2) porosities, 3) saturation, and 4) permeability and diffusivity, tortuosity. The main aim of this paper is to present in juxtaposition continuous one-, two-, and three-component models of geomaterials appearing in construction of embankment dams. In particular, the above mentioned features, especially saturation with water and seepage problems, modeling of fluidization yielding piping, and generalizations of the Darcy law and changes of porosity, are presented. © 2011 Higher Education Press and Springer-Verlag Berlin Heidelberg.

Due to its propitious material properties sandstone is the most studied porous medium for the investigation of linear wave propagation. However, in practical applications the behavior of other soil types, i.e., especially the three main soil types sand, silt, and clay, are much more important. Therefore, the model for partially saturated soils introduced by Albers (Habilitation Thesis, 2010a) is applied to 11 soil types classified in the German standard DIN 4220 to obtain information on the phase velocities and attenuations of the three longitudinal waves and the shear wave appearing in such media. © 2010 Springer Science+Business Media B.V.

The contributions to the book concern various aspects of extension of classical continuum models. These extensions are related to the appearance of microstructures both natural as well as these created by processes. To the first class belong various thermodynamic models of multicomponent systems such as porous materials, composites, materials with microscopic heterogeneities. To the second class belong primarily microstructures created by phase transformations. Invited authors cover both fields of thermodynamic modeling and mathematical analysis of such continua with microstructure. In particular the following subjects are covered: thermodynamic modeling of saturated and unsaturated porous and granular media, linear and nonlinear waves in such materials, extensions of constitutive laws by internal variables, higher gradients and nonequilibrium fields, stochastic processes in porous and fractal materials, thermodynamic modeling of composite materials, mathematical analysis of multicomponent systems, phase transformations in solids. © 2010 Springer-Verlag Berlin Heidelberg. All rights are reserved.

In this contribution an overview of the continuum mechanical modeling of linear elastic partially saturated porous media and the application of such a model to linear wave propagation is given. First the involved microstructural variables are discussed and the construction of the model is presented. The macroscopic parameters used in the model are obtained by micro-macro-transition procedure from the measurable microscopic quantities. The linear elastic wave propagation analysis is demonstrated exemplarily for sandstone, sand and clayey loam. The properties of the four appearing waves - three compressional and one shear wave - are compared. Phase speeds and attenuations of these waves depend both on the frequency and on the degree of saturation. © 2010 Springer-Verlag Berlin Heidelberg.

Recently, Albers developed a continuum model for the description of wave propagation in partially saturated, threecomponent, porous media. Macroscopic parameters have been obtained by a systematic micro-macro transition procedure. Using this model, acoustic properties of sandstone filled by different pore fillings and of several soil types containing a water-air-mixture have been presented. The soil types are classified in the German standard DIN 4220. Originally, both for rocks and soils the shear modulus had been proposed according to the classical elasticity theory. However, it seems that this approach for granular soils yields, at least for the shear wave, wave speeds which are higher than experimentally observed values. Therefore, in this note, phase speeds and attenuations of the four waves appearing in unsaturated compact granular media are calculated using also another approach for the shear modulus, the Mindlin-Duffy approach. The numerical results of both theoretical approaches are compared to experimentally obtained values. It turns out that the latter approach for granular media is much better while for porous media the first approach is concordant with measurements. © 2010 American Institute of Physics.

In this paper a macroscopic linear model is constructed which is intended to describe wave propagation processes in partially saturated soils under isothermal conditions. A porous medium consisting of a deformable skeleton and compressible pore fluids is investigated. The form of the partial stresses is an extension to the two-component model of Biot. Additionally to the classical coupling between solid and fluid there appear coupling parameters between solid and gas and fluid and gas. Macroscopic material parameters have to be determined from their measurable microscopic counterparts either by a micro-macro-transition or using relations for the three-component body by Santos, Douglas, and Corberó which are analogous to the Gassmann relations for saturated porous media. The transition is illustrated using the examples of oil-water and air-water-mixtures in sandstone. Finally, a glance on the results of the application of the model to wave propagation is given. © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.