A two-scale model for fluid flow in an unsaturated porous medium with cohesive cracks

A two-scale model is developed for fluid flow in a deforming, unsaturated and progressively fracturing porous medium. At the microscale, the flow in the cohesive crack is modelled using Darcy’s relation for fluid flow in a porous medium, taking into account changes in the permeability due to the progressive damage evolution inside the cohesive zone. From the micromechanics of the flow in the cavity, identities are derived that couple the local momentum and the mass balances to the governing equations for an unsaturated porous medium, which are assumed to hold on the macroscopic scale. The finite element equations are derived for this two-scale approach and integrated over time. By exploiting the partition-of-unity property of the finite element shape functions, the position and direction of the fractures are independent from the underlying discretization. The resulting discrete equations are nonlinear due to the cohesive crack model and the nonlinearity of the coupling terms. A consistent linearization is given for use within a Newton–Raphson iterative procedure. Finally, examples are given to show the versatility and the efficiency of the approach. The calculations indicate that the evolving cohesive cracks can have a significant influence on the fluid flow and vice versa.     

Formulation and sequential numerical algorithms of coupled fluid/heat flow and geomechanics for multiple porosity materials

We extend constitutive relations of coupled flow and geomechanics for the isothermal elastic double porosity model by Berryman [Journal of Engineering Mechanics ASCE 2002; 128(8):840–847] in the previous study to those for the nonisothermal elastic/elastoplastic multiple porosity model, finding coupling coefficients and constraints of the multiple porosity model and determining the upscaled elastic/elastoplastic moduli as well as relations between the local strains of all materials within a gridblock and the global strain of the gridblock. Furthermore, the coupling equations and relations between local and global variables provide well‐posed problems, implying that they honor the dissipative mechanism of coupled flow and geomechanics. For numerical implementation, we modify the fixed‐stress sequential method for the multiple porosity model. From the a priori stability estimate, the sequential method provides numerical stability when an implicit time‐stepping algorithm is used. This sequential scheme can easily be implemented by using a modified porosity function and its porosity correction. In numerical examples, we observe clear differences among the single, double, and multiple porosity systems, and the multiple porosity model can reflect the substantial heterogeneity that exists within a gridblock. We also identify considerably complicated physics in coupled flow and geomechanics of the multiple porosity systems, which cannot accurately be detected in the uncoupled flow simulation. Copyright © 2012 John Wiley & Sons, Ltd.     

A two‐scale approach for fluid flow in fractured porous media

A two‐scale numerical model is developed for fluid flow in fractured, deforming porous media. At the microscale the flow in the cavity of a fracture is modelled as a viscous fluid. From the micromechanics of the flow in the cavity, coupling equations are derived for the momentum and the mass couplings to the equations for a fluid‐saturated porous medium, which are assumed to hold on the macroscopic scale. The finite element equations are derived for this two‐scale approach and integrated over time. By exploiting the partition‐of‐unity property of the finite element shape functions, the position and direction of the fractures is independent from the underlying discretization. The resulting discrete equations are non‐linear due to the non‐linearity of the coupling terms. A consistent linearization is given for use within a Newton–Raphson iterative procedure. Finally, examples are given to show the versatility and the efficiency of the approach, and show that faults in a deforming porous medium can have a significant effect on the local as well as on the overall flow and deformation patterns. Copyright © 2006 John Wiley & Sons, Ltd.     

Homotopy analysis method for fully developed MHD micropolar fluid flow between vertical porous plates

In this paper, the fully developed natural convection of MHD micropolar fluid flow between two vertical porous plates is considered. The coupled system of non‐linear differential equations governing the flow is solved analytically by the homotopy analysis method (HAM). The HAM contains an auxiliary parameter ?, which provides us with a simple way to adjust and control the convergence region and rate of convergence of the series solution. Velocity, microrotation and temperature profiles are presented for several values of the Hartmann number and the micropolar parameter. Copyright © 2008 John Wiley & Sons, Ltd.     

A fully coupled model for diffusion-induced deformation in polymers

Polymers that mechanically respond to the presence of a diffusing fluid/solvent have found various applications in drug delivery, tissue scaffolding, sensors and actuators. These applications involve understanding of both the diffusion process and the evolution of the deformation of the polymers during the diffusion process. For example, in a polymeric actuator one might be interested in the extent of deformation one can achieve given a solvent environment and the time in which it can be achieved. There are two key aspects in modeling such behavior. First, the displacement gradients involved are usually large, especially in problems such as “self-assembly.” Second, since the diffusion occurs in a deforming polymeric medium, an appropriate diffusion model that includes the effect of the deformed state of the body as well as the interaction between the polymeric medium and the diffusing fluid has to be considered. In effect, this results in the diffusion and equilibrium equation being fully coupled and nonlinear. In this work, we model diffusion-induced deformation in an elastic material including large deformations based on thermodynamics framework. For the chemical potential, we use the Flory–Huggins potential adapted to include the effect of stress in the polymers. Using the model, we simulate folding and bending of a rectangular polymeric strip by simultaneous solution of the diffusion equation as well as the equilibrium equation using the finite element method. Parametric studies are also conducted in order to examine the effect of material parameters on the diffusion and deformation behaviors. Finally, using the coupled diffusion–deformation model we simulate deformations of composite domains comprising of polymeric constituents with different diffusion–deformation behaviors in order to achieve various interesting “self-assembly” shapes.     

A numerical model for immiscible two-phase fluid flow in a porous medium and its time domain solution

The governing equations for the interaction of two immiscible fluids within a deforming porous medium are formulated on the basis of generalized Biot theory. The displacement of the solid skeleton, the pressure and saturation of wetting fluid are taken as primary unknowns of the model. The finite element method is applied to discretize the governing eqations in space. The time domain numerical solution to the coupled problem is achieved by using an unconditionally stable direct integration procedure. Examples are presented to illustrate the performance and capability of the approach.     

A consistent approach for fluid-structure-contact interaction based on a porous flow model for rough surface contact

Simulation approaches for fluid-structure-contact interaction, especially if requested to be consistent even down to the real contact scenarios, belong to the most challenging and still unsolved problems in computational mechanics. The main challenges are 2-fold—one is to have a correct physical model for this scenario, and the other is to have a numerical method that is capable of working and being consistent down to a zero gap. Moreover, when analyzing such challenging setups of fluid-structure interaction, which include contact of submersed solid components, it gets obvious that the influence of surface roughness effects is essential for a physical consistent modeling of such configurations. To capture this system behavior, we present a continuum mechanical model that is able to include the effects of the surface microstructure in a fluid-structure-contact interaction framework. An averaged representation for the mixture of fluid and solid on the rough surfaces, which is of major interest for the macroscopic response of such a system, is introduced therein. The inherent coupling of the macroscopic fluid flow and the flow inside the rough surfaces, the stress exchange of all contacting solid bodies involved, and the interaction between fluid and solid are included in the construction of the model. Although the physical model is not restricted to finite element–based methods, a numerical approach with its core based on the cut finite element method, enabling topological changes of the fluid domain to solve the presented model numerically, is introduced. Such a cut finite element method–based approach is able to deal with the numerical challenges mentioned above. Different test cases give a perspective toward the potential capabilities of the presented physical model and numerical approach.     

An XFEM model for cracked porous media: effects of fluid flow and heat transfer

In this work, a numerical model is developed to investigate the influence of fluid flow and heat transfer on the thermo-mechanical response of a cracked porous media. The fluid flow, governed by the Darcy’s law, is discretized with the nonconforming finite element method. Time splitting is used with the energy conservation equation to solve the fluid and the solid phases separately. A combination of Discontinuous Galerkin (DG) and multi-point flux approximation methods is used to solve the advection-diffusion heat transfer equation in the fluid phase. While the conductive heat transfers equation in the solid phase is solved using the eXtended finite element method (XFEM) to better handle the temperature discontinuities and singularities caused by the cracks. Further, the resulted temperature is used as body force to solve the thermo-mechanical problem using the XFEM. In the post processing stage, the thermal stress intensity factor is computed using the interaction integral technique at each time step and used to validate the obtained results. A good agreement was found when the results were compared with the existing ones in the literature.     

A 3D finite element approach for the coupled numerical simulation of electrochemical systems and fluid flow

A comprehensive finite element method for three‐dimensional simulations of stationary and transient electrochemical systems including all multi‐ion transport mechanisms (convection, diffusion and migration) is presented. In addition, non‐linear phenomenological electrode kinetics boundary conditions are accounted for. The governing equations form a set of coupled non‐linear partial differential equations subject to an algebraic constraint due to the electroneutrality condition. The advantage of a convective formulation of the ion‐transport equations with respect to a natural application of homogeneous flux boundary conditions is emphasized. For one of the numerical examples, an analytical solution for the coupled problem is provided, and it is demonstrated that the proposed computational approach is robust and provides accurate results. Copyright © 2011 John Wiley & Sons, Ltd.     

A coupled thermo-mechanical bond-based peridynamics for simulating thermal cracking in rocks

A coupled thermo-mechanical bond-based peridynamical (TM-BB-PD) method is developed to simulate thermal cracking processes in rocks. The coupled thermo-mechanical model consists of two parts. In the first part, temperature distribution of the system is modeled based on the heat conduction equation. In the second part, the mechanical deformation caused by temperature change is calculated to investigate thermal fracture problems. The multi-rate explicit time integration scheme is proposed to overcome the multi-scale time problem in coupled thermo-mechanical systems. Two benchmark examples, i.e., steady-state heat conduction and transient heat conduction with deformation problem, are performed to illustrate the correctness and accuracy of the proposed coupled numerical method in dealing with thermo-mechanical problems. Moreover, two kinds of numerical convergence for peridynamics, i.e., m- and \(\delta \)-convergences, are tested. The thermal cracking behaviors in rocks are also investigated using the proposed coupled numerical method. The present numerical results are in good agreement with the previous numerical and experimental data. Effects of PD material point distributions and nonlocal ratios on thermal cracking patterns are also studied. It can be found from the numerical results that thermal crack growth paths do not increases with changes of PD material point spacing when the nonlocal ratio is larger than 4. The present numerical results also indicate that thermal crack growth paths are slightly affected by the arrangements of PD material points. Moreover, influences of thermal expansion coefficients and inhomogeneous properties on thermal cracking patterns are investigated, and the corresponding thermal fracture mechanism is analyzed in simulations. Finally, a LdB granite specimen with a borehole in the heated experiment is taken as an application example to examine applicability and usefulness of the proposed numerical method. Numerical results are in good agreement with the previous experimental and numerical results. Meanwhile, it can be found from the numerical results that the coupled TM-BB-PD has the capacity to capture phenomena of temperature jumps across cracks, which cannot be captured in the previous numerical simulations.     

Spin-up of a source-sink driven flow in a two-layer rotating fluid

Summary In this paper the linearized spin-up process of a two-layer fluid in a rotating annulus is examined. The flow is induced by a source and a sink at the inner boundary of the annulus. The spin-up is controlled by the Ekmansuction velocity as well as the moving interface. On the assumption of vanishing small internal and external Froude numbers, the vorticity in each layer is a function of time only and can be expressed in terms of the hypergeometric functions. The components of the velocity can readily be deduced in terms of the vorticity. Some numerical results are given to illustrate the spin-up process.     

A generalized model for dynamic mixed-mode fracture via state-based peridynamics

A generalized model is developed to investigate dynamic crack propagation in isotropic solids under mixed-mode I/II conditions using state-based peridynamics. The critical stretch and the critical strain energy release rate (ERR) are related within the state-based peridynamic framework to construct a computational model capable of capturing fracture energy of the kinked cracks. A novel formulation is presented to predict crack growth trajectory and pattern by combining the traditional expression of ERR and the peridynamic states of the crack opening and sliding displacements. The proposed model is used to predict dynamic fracture behavior in polymethyl methacrylate (PMMA) and soda-lime glass using various test specimens, including cracked semi-circular bending (SCB), cracked rectangular plate, and single edge-notched tensile (SENT) specimens, and under different dynamic loading conditions. The developed model is examined against the numerical and experimental data available in the literature, and a very good agreement is observed.     

The flow of a viscoelastic fluid past a porous plate

Summary The flow of a second order viscoelastic fluid past a porous plate is considered. It is characterized by a boundary value problem in which the order of the differential equation exceeds the number of available boundary conditions. The boundary value problem is solved by making a plausible assumption, namely that the variation of the normal derivative of the velocity at the plate withk is sufficiently smooth, wherek is the viscoelastic fluid parameter. Under this assumption it is shown that dual solutions exist for values ofk less than a critical value. Beyond this value, no solution exists.     

A coupled PFEM–Eulerian approach for the solution of porous FSI problems

This paper aims to present a coupled solution strategy for the problem of seepage through a rockfill dam taking into account the free-surface flow within the solid as well as in its vicinity. A combination of a Lagrangian model for the structural behavior and an Eulerian approach for the fluid is used. The particle finite element method is adopted for the evaluation of the structural response, whereas an Eulerian fixed-mesh approach is employed for the fluid. The free surface is tracked by the use of a level set technique. The numerical results are validated with experiments on scale models rockfill dams.     

On boundary conditions for fluid flow in porous media

We continue the study undertaken in [11] of boundary conditions at the surface between a porous medium and a free fluid flow. Two different kinds of phenomena may appear, according to the direction of the averaged gradient of pressure in the porous medium, oblique or normal to the surface. General results about the matching of the different flow regions and boundary conditions are given, as well as examples. The interface layer in the case of a normal gradient of pressure is studied in detail.     

Open-cell porous metals for thermal management applications: fluid flow and heat transfer

Porous metals are produced by a wide range of industrially applied techniques, and the resulting porous structure is a characteristic of the manufacturing method chosen. This review firstly provides an overview of the different techniques to manufacture open-cell porous metals, highlighting the distinction between the resulting porous structures. The effects of the structural parameters on the fluid flow properties and heat transfer in porous metals are also discussed. It was evident from literature that there exist optimum structural parameters which offered maximum thermal exchange performance. Therefore, the last part of this review outlines the current research on porous metals for thermal management applications, which focuses on optimising the design of the porous metal structure to improve the heat transfer performance.

This review was submitted as part of the 2016 Materials Literature Review Prize of the Institute of Materials, Minerals and Mining run by the Editorial Board of MST. Sponsorship of the prize by TWI Ltd is gratefully acknowledged     

Field-to-field coupled fluid structure interaction: A reduced order model study

The standard Fluid-Structure Interaction (fsi ) coupling, that uses as unknowns velocity and pressure for the fluid and displacements for the solid, is compared against two novel types of coupling, the first one a three-field coupling (velocity-pressure-stress/displacement-pressure-stress) introduced by the authors in a recent work, and a two-field coupling (velocity-pressure/displacement-pressure) introduced in this paper, in this way completing our set of Field to Field (f2f ) equations, all stabilized by means of the Variational Multi-Scale (vms ) method using dynamic and orthogonal subscales. The solid two-field fsi coupling formulation is benchmarked statically and dynamically. Proper Orthogonal Decomposition (pod ) is applied to all three fsi formulations to obtain reduced basis and asses their performance in a reduced space. Numerical tests are shown comparing all three formulations. By correctly resolving the Cauchy stress tensor, the three-field fsi coupling proves to provide more accurate results in both Full Order Model (fom ) and Reduced Order Model (rom ) spaces than its counterparts for a similar number of degrees of freedom, making it a reliable formulation. f2f pairing appears to be beneficial, providing more accurate results in all cases shown; mixed pairing with a three-field formulation in the solid appears to produce very precise results as well.     

Incompressible fluid flow and heat transfer through a nonsaturated porous medium

This work studies a nonsaturated flow and the heat transfer associated phenomenon of a newtonian fluid through a rigid porous matrix, using a mixture theory approach in its modelling. The mixture consists of three overlapping continuous constituents: a solid (porous medium), a liquid and an inert gas, included to account for the compressibility of the system as a whole. A set of four nonlinear partial differential equations describe the problem whose hydrodynamical part is approximated by means of a Glimm's scheme combined with an operator splitting technique.