Impermeable Inclusions Research Articles

A peridynamic model for advection–reaction–diffusion problems

In this paper, a bond-based peridynamic (PD) advection–reaction–diffusion model is formulated to simulate processes involving chemical reactions, flows, and diffusion. In the formulation, the reaction rate, represented by a body force-like local term, depends on the local concentrations of the reactants, and the dependence varies according to the reaction type. Both linear and nonlinear dependencies are considered. A system of coupled PD equations for each substance is solved numerically. We verify and validate the PD advection–reaction–diffusion model by comparing results with those from a corresponding classical model and from experiments in both 1D and 2D, for a bimolecular reaction (A+B→AB) in a flowing aqueous environment, similar to purifying polluted water flowing in a channel, via chemical reactions. The model is also used to simulate bimolecular advection–reaction–diffusion in heterogeneous media with impermeable inclusions, and to simulate polymer degradation through hydrolysis and diffusion, similar to how biodegradable polymer implants dissolve.

Study of polymer compositions for formation of impermeable inclusions in rock mass

Laboratory studies of two-component compositions based on polymersilicates and polyurethanes, developed to strengthen destroyed rocks and create impermeable inclusions in the rock mass, were carried out. The rheological properties of liquid components of polymer compositions for the temperature conditions of underground mining of solid minerals, physical and mechanical characteristics of hardened polyurethane and organomineral compositions have been determined.

Peridynamics modelling of coupled water flow and chemical transport in unsaturated porous media

A wide range of phenomena in natural or engineered systems emerge from strongly coupled hydraulic, chemical and mechanical processes involving a mix of clearly discrete and quasi-continuous mechanisms. Non-local formulations of these processes, e.g. based on Peridynamics, offer significant advantages compared to classical local formulations. Existing Peridynamics models for water flow and chemical transport are applicable only to saturated systems and use loosely coupling schemes, such as explicit time stepping approaches. This work advances the non-local approach by developing a bond-based formulation for coupled water flow and chemical transport in partially saturated porous media. An implicit solution is proposed for coupling the PD formulation of chemical transport with water flow formulation. Firstly, the proposed formulation is verified against results from finite element/finite difference transient solutions for 1-D and 2-D coupled problems. The agreement between results demonstrates the accuracy of the proposed methodology. Secondly, a series of case studies are presented to illustrate the model’s capability to capture discontinuities and heterogeneities, including stationary cracks, propagating cracks, and randomly distributed permeable and impermeable inclusions. The results show that the multi-physics Peridynamics-based formulations and computational model (Pyramid) provide clear advantages to classical local formulations for analyses of reactive transport in partially saturated porous media with physically realistic microstructures.

Analysis of inclusion in thermoelectric materials: The thermal stress field and the effect of inclusion on thermoelectric properties

This paper analyses a two-dimensional problem in thermoelectric materials with an inclined elliptic inclusion. We have obtained the closed-form solutions of electric current density and temperature considering the inclusion's electrical and thermal permeability. Based on the derived thermoelectric field, the thermal stresses are given in the explicit form and the effect of inclusion on effective thermoelectric properties is investigated. The electrically impermeable and thermally impermeable inclusions will respectively cause maximum electric current concentration and heat concentration. The thermally impermeable and rigid inclusion will cause maximum stress concentration around the inclusion. Furthermore, we find that the effective electric conductivity (effective thermal conductivity) of the matrix-inclusion system is increased when the inclusion have higher electric conductivity (thermal conductivity) than the matrix. It is possible to enhance the effective figure of merit by inserting inclusions with specific electric conductivity and heat conductivity. It predicts a new way for the design of high-performance thermoelectric devices. The results in this paper can be directly used for reliability consideration in design and optimization of thermoelectric devices in engineering.

Antiplane analysis for an elliptical inclusion in 1D hexagonal piezoelectric quasicrystal composites

An elliptical inclusion embedded in an infinite 1D hexagonal piezoelectric quasicrystal matrix is analysed in the framework of linear piezoelasticity of quasicrystals. Using the technique of conformal mapping, the closed-form solutions of the complex potentials, all the field quantities in the matrix and the inclusion are obtained under far-field antiplane mechanical loads of the phonon and phason fields and an inplane electrical load. Several special cases, such as a homogeneous material, a soft and permeable inclusion, an impermeable inclusion, a line inclusion, a rigid inclusion and a crack are reduced by the present results. Some numerical examples are provided to shows the variations of the stresses of the phonon and phason fields and the electric field with the shape of hole/inclusion, the dielectric permittivity and the distance from the hole/inclusion.

Stokes–Brinkman–Darcy Solutions of Bimodal Porous Flow Across Periodic Array of Permeable Cylindrical Inclusions: Cell Model, Lubrication Theory and LBM/FEM Numerical Simulations

An analytical study is devised for the problem of bimodal porous flow across a periodic array of permeable cylindrical inclusions. Such a configuration is particularly relevant for porous media systems of dual granulometry, an idealization often taken, e.g. in the modelling of membranes and fibrous applications. The double-porosity system is governed by the Stokes–Brinkman–Darcy equations, the most general description in this class of flow problems characterized by the permeabilities of the surrounding matrix and inclusions, their porosities and the relative volume fraction. We solve this problem with the Kuwabara cell model and lubrication approach, providing analytical solutions for the system effective permeability in closed analytical form. The ensemble of results demonstrates the self-consistency of the bimodal solutions in eight possible limit configurations and supports the validity of the Beavers–Joseph interface stress jump condition for transmission from the open Stokes flow to low-permeable Darcy region. At the same time, these solutions bring further insight on the relative significance of the governing parameters on the effective permeability, with a focus on the role of the effective viscosity (porosity) distribution. Furthermore, although the cell model is restricted to relatively small volume fractions in open flow, its validity extends in less-permeable background flow inside Brinkman/Brinkman description. In turn, the lubrication approximation remains more adequate in the opposite limit of the dense impermeable inclusions. These conclusions are drawn from comparisons with the numerical solutions obtained with the developed lattice Boltzmann model and the standard finite element method. The two methods principally differ in the treatment of the interface conditions: implicit and explicit, respectively. The purpose of this task is therefore twofold. While the numerical schemes help quantifying the validity limits of the theoretical approach, the analytical solutions offer a non-trivial benchmark for numerical schemes in highly heterogeneous soil.

Sequential Homogenization of Reactive Transport in Polydisperse Porous Media

Direct numerical simulations of flow and transport in porous media are computationally prohibitive due to the disparity between the typical scale at which processes are well understood (e.g., the pore-scale) and the scale of interest (the system- or field-scale). Homogenization approaches overcome some of the difficulties of full pore-scale simulations by providing an upscaled representation of fine-scale processes. Real porous systems, e.g., rocks, pose additional challenges since they usually exhibit multimodal distributions in physical and chemical properties. Perforated domains, i.e., domains with impermeable inclusions embedded in a porous matrix, represent one such example. These hierarchical media cannot be approached by a single continuum formulation. Sequential homogenization techniques build a hierarchy of effective equations that sequentially carry the smallest scale information through the intermediate scales up to the macroscale. The advantage of sequential upscaling in handling multimodal distribution in physical and chemical properties lies in its computational efficiency compared to one-step homogenization: the information about smaller-scale heterogeneity is incorporated in the subsequent scales in terms of effective media properties. Yet, existence of one or multiple intermediate scales can significantly decrease the accuracy of multiscale formulations. We show that the accuracy of multiscale methods based on sequential upscaling is strongly influenced by a combination of geometric and dynamical scale separation conditions. In particular, we investigate under which conditions sequential homogenization of reactive solute transport in geometrically and chemically heterogeneous porous domains composed of bidisperse cylinders can accurately describe pore-scale processes. We show that under appropriate conditions, expressed in terms of Peźclet and Damkoźhler numbers and a scales separation parameter, the sequential upscaling method has second-order accuracy. We compare sequential upscaling results with the direct solution of the fully resolved pore-scale problem.

Some mathematical models related to the Zhukovskii problem of the flow around a sheet pile

The seepage under a Zhukovskii sheet pile through a layer of soil underlain by a highly permeable pressurized horizon is considered. The left semi-infinite part of the roof of this horizon is simulated by an impermeable foundation. The flow when the velocity on the edges of the sheet pile is equal to infinity and, on the two water permeable parts of the boundary of the domain of motion, the flow rate takes extremal values, is investigated. The limiting cases, associated with the absence of both a backwater and an impermeable inclusion, are mentioned. The problem of seepage from a foundation pit formed by two Zhukovskii sheet piles is solved within the limits of a flow with a highly permeable pressurized stratum lying below. In the case when there is no infiltration onto the free surface, a solution of the well-known Vedernikov problem is obtained. A contact scheme, arising when there are no such indicated critical points, is considered; it is described outside the scope of the constraints imposed on the unknown conforming mapping parameters ensuring the realization of the basic mathematical model. Solutions are given for two schemes of motion in a semi-inverse formulation. The classical Zhukovskii problem is the limiting case of one of them. The special features of such models are mentioned. The Polubarinova-Kochina method is used to study all the above-mentioned flows. This method enables exact analytical representations of the elements of the motion to be obtained. The results of numerical calculations and an analysis of the effect of all the physical factors on the seepage characteristics are presented.

Mathematical Modeling of Filtration Flows from Foundation Pits Fenced with Zhukovskii Sheet Piles

Consideration is given to the filtration of water from foundation pits fenced with Zhukovskii sheet piles through a ground layer underlain by a well-permeable confined aquifer on whose roof there is a water-impermeable portion. A mixed multiparametric boundary-value problem of the theory of analytical functions is formulated, which is solved with the Polubarinova-Kochina method. Based on this model, an algorithm of calculation of the filrtation characteristics is developed for situations where, in water motion from the foundation pits, one has to take into account the joint influence of such important factors, as seepage onto the free surface, upthrust from the water of the underlying highly permeable aquifer, and the presence of the impermeable inclusion on the roof of the latter, on the pattern of the phenomenon. Limiting cases of the model are considered which are related to the absence of one factor characterizing the modeled process (upthrust, an impermeable inclusion, seepage) and to the degeneration of the foundation pits into a submergence band semiinfinite on the left. A solution of the problem is given for the scheme under the assumption of a finite value of the flow velocity at the end of a sheet pile; it is a certain analog of the Zhukovskii classical problem.

On filtration schemes occurring in the problem of flow over a Zhukovskii sheet pile

Consideration is given to fi ltration under a Zhukovskii sheet pile through a ground layer underlain by a highly permeable head horizon the left semiinfi nite part of whose roof is modeled by an impermeable base. A study is made of the case of fl ow where the velocity at the end of the sheet pile is equal to infi nity and the fl ow rate takes on extremum values on both water-permeable portions of the boundary of the region of motion. Limiting cases of fl ow associated with the absence of upthrust and of an impermeable inclusion alike are noted, as is an adjacent scheme due to the absence of the two indicated extremum points. Solutions are given for two schemes of motion in a semiinverse formulation, with the classical Zhukovskii problem being the limiting case of one scheme. Distinctive features of such models are noted. For investigation of these fl ows, use is made of a Polubarinova-Kochina method which allows exact analytical representations for motion elements. Results of numerical calculations and the analysis of the infl uence of all physical factors on fi ltration characteristics are given.

Anchoring of Drifting Spiral and Scroll Waves to Impermeable Inclusions in Excitable Media

Anchoring of spiral and scroll waves in excitable media has attracted considerable interest in the context of cardiac arrhythmias. Here, by bombarding inclusions with drifting spiral and scroll waves, we explore the forces exerted by inclusions onto an approaching spiral and derive the equations of motion governing spiral dynamics in the vicinity of inclusion. We demonstrate that these forces nonmonotonically depend on distance and can lead to complex behavior: (a)anchoring to small but circumnavigating larger inclusions; (b)chirality-dependent anchoring.

On the problem of flow over a Zhukovskii sheet pile in an irrigated ground mass

In the hydrodynamic formulation, the problem is solved of plane steady filtration under a Zhukovskii sheet pile through an irrigated ground mass underlain by a highly permeable pressure horizon, the left semiinfinite part of whose roof is modeled by an impermeable inclusion. Consideration is given to the case of motion where the flow velocity at the end of the sheet pile is equal to infinity, which leads to a multivalence of the relevant region of complex velocity. To study such flow, we formulate and solve, using the Polubarinova-Kochina method, a mixed multiparametric boundary-value problem of the theory of analytical functions. On the basis of this model, an algorithm of calculation of filtration characteristics is developed for the situations where in water filtration, one has to take account of infiltration onto the free surface. Numerical results and the analysis of the effects of all physical parameters on the flow picture are presented. Consideration is given to the limiting cases of motion associated with the absence of both an impermeable inclusion and the hydrostatic upthrust in the well-permeable underlying layer. The solution is compared with results for the case of a finite flow velocity at the end of the sheet pile.

On ground water seepage under hydraulic structures

This present paper continues paper [1] of the authors and is concerned with the following problems regarding seepage flows with unknown boundaries under hydraulic structures, with use being made of the theory of two-dimensional (2D) steady state seepage of incompressible fluid according to Darcy’s law in a uniform isotropic soil. Below we construct a smooth underground contour of a rectangular subsurface dam of constant seepage rate in the case when a pervious abutment is underlain by a blanket consisting of two curvilinear and one (middle) horizontal elements which are also characterized by constant incident velocity. We are concerned with the fluid flow below Joukowski’s cutoff wall through the watered soil mass underlain by a highly permeable aquifer with confined groundwater; the left semi-infinite part of its roof is simulated by impermeable inclusion. These motions are examined using multiparameter problems of the theory of analytic functions. Their solutions are obtained by the semi-inverse method of the velocity hodograph by P.Ya. Polubarinova-Kochina and I.N. Kochina, by Polubarinova-Kochina’s approach dependent on the analytic theory of linear Fuchsian equations, and using the available conformal mappings of domains of special kinds peculiar for underground hydrodynamics. The results of the numerical calculations are given and the hydrodynamic effect of the physical parameters of models on the flow pattern is ascertained. The limiting cases previously examined in [1] are singled out.

On flow over a zhukovskii sheet pile in an irrigated ground mass

In the hydrodynamic formulation, we solve the problem of plane steady filtration under a Zhukovskii sheet pile through an irrigated ground mass underlain by a highly permeable pressure stratum the left semi-infinite part of whose roof is modeled by an impermeable inclusion. Using the P. Ya. Polubarinova-Kochina method, we give a constructive solution of the problem and accurate analytical representations for characteristic dimensions of the flow pattern. Results of the numerical calculations and an analysis of the effects of all physical parameters of the model on filtration characteristics are presented. Consideration is given to limiting cases of flow associated with the absence of an impermeable inclusion as well as of backwater in the underlying well-permeable layer.

The flow of ground waters around a Zhukovskii sheet pile

The problem of plane steady-state seepage under a Zhukovskii sheet pile through a large irrigated ground mass underlain by highly permeable pressurized layer, the left semi-infinite part of the roof of which is modelled by an impermeable inclusion, is solved in a hydrodynamic formulation. A constructive solution of the problem is given using the Polubarinova-Kochina method and exact analytic representations are obtained for the characteristic dimensions of the flow scheme. The results of numerical calculations and an analysis of the effect of all the physical parameters of the model on the seepage characteristics are presented. Limiting cases of the flow, associated with the absence of both an impermeable inclusion as well as a backwater in the underlying highly permeable layer, are considered.

A BEM for time dependent infiltration from an irrigation channel

In this paper some two dimensional time dependent infiltration problems are considered. The problems involve infiltration from an irrigation channel into a homogenous soil and a soil which contains an impermeable finite inclusion. The problems are reduced to boundary integral equations which may be solved numerically using established procedures. Numerical results are obtained to provide the distribution of the matric flux potential for some particular impermeable inclusions and a particular channel shape. The results indicate how the distance from the channel influences the speed with which the matric flux potential reaches its steady state value. They also illustrate how the presence of an impermeable inclusion can increase the matric flux potential at points below the surface.

Exact solution for elliptical inclusion in magnetoelectroelastic materials

This paper considers the magneto-electro-mechanical coupling between an inclusion and matrix, which are both of magnetoelectroelastic materials. The general cases including the mode I, mode II and mode III are studied. Analytical solutions for an elliptical cylinder inclusion inside an infinite magnetoelectroelastic medium under combined mechanical–electrical–magnetic loadings are formulated via the Stroh formalism. Crack problem is also investigated and the stress, electric and magnetic fields in the vicinity of the crack tip are determined by a complex vector of intensity factors. Various special cases, including an impermeable inclusion, a permeable crack, a rigid and permeable inclusion, a rigid and permeable line inclusion, a permeable cavity, and an impermeable cavity are discussed.

Anti-plane analysis for elliptical inclusion in magnetoelectroelastic materials

This paper considers the multi-field coupling in magneroelectroelastic composite materials consisting of the inclusion and the matrix are magnetoelectroelastic materials. The mechanical, electric and magnetic fields around an elliptical cylinder inclusion are formulated by complex potentials. Inside the inclusion, the strain, electric and magnetic fields are found to be uniform and vary with the shape of the ellipse. When the inclusion is reduced to a crack, along the interface, the strain, electric field strength and magnetic field strength equal the corresponding remote ones, which can be used as the boundary condition. Special cases, such as a rigid and permeable inclusion, a soft and impermeable inclusion, a line inclusion and a crack problem are discussed in detail.

Gravity-driven nonlinear flow in areas with impermeable inclusions

Gravity-driven nonlinear flow in areas with impermeable inclusions

Infiltration from irrigation channels into soil with impermeable inclusions

We consider the effect of impermeable inclusions on the infiltration of water from irrigation channels in a homogeneous soil. An expression for the matric flux potential throughout the soil is obtained in terms of a boundary integral equation. A Green's function derived by Basha [ Water Resources Research , 30:2105--2118, 1994] is suitable for numerical calculations for this class of problems. This Green's function is employed in the boundary integral equation to obtain numerical values for the matric flux potential for a soil with embedded impermeable inclusions of various shapes. The numerical results indicate how impermeable inclusions may be used to effectively direct the flow from irrigation channels to particular regions below the soil surface.