A multiscale Darcy–Brinkman model for fluid flow in fractured porous media

The aim of this work is to present a reduced mathematical model for describing fluid flow in porous media featuring open channels or fractures. The Darcy’s law is assumed in the porous domain while the Stokes–Brinkman equations are considered in the fractures. We address the case of fractures whose thickness is very small compared to the characteristic diameter of the computational domain, and describe the fracture as if it were an interface between porous regions. We derive the corresponding interface model governing the fluid flow in the fracture and in the porous media, and establish the well-posedness of the coupled problem. Further, we introduce a finite element scheme for the approximation of the coupled problem, and discuss solution strategies. We conclude by showing the numerical results related to several test cases and compare the accuracy of the reduced model compared with the non-reduced one.     

On freezing of a finite humid porous medium with a heat flux condition considering the factor for phase conversion

This work deals with a theoretical mathematical analysis of freezing (desublimation) of moisture in a finite porous medium with heat-flux condition in x=0. The position of phase change front at time t, given by x =s (t), divides the porous body into two regions. In the first region there is no moisture movement, and in the other one the process of the coupled heat and moisture flows is described by the well known Luikov's system, considering that the factor for phase conversion is non zero. Equivalence between this problem and a system of Volterra integral equations is found. The existence of a unique local solution in time for this problem is also obtained. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Thermal effects in orthotropic porous elastic beams

This paper is concerned with the linear theory of anisotropic porous elastic bodies. The extension and bending of orthotropic porous elastic cylinders subjected to a plane temperature field is investigated. The work is motivated by the recent interest in the using of the orthotropic porous elastic solid as model for bones and various engineering materials. First, the thermoelastic deformation of inhomogeneous beams whose constitutive coefficients are independent of the axial coordinate is studied. Then, the extension and bending effects in orthotropic cylinders reinforced by longitudinal rods are investigated. The three-dimensional problem is reduced to the study of two-dimensional problems. The method is used to solve the problem of an orthotropic porous circular cylinder with a special kind of inhomogeneity.      

A two-stage analytical extension for porothermoelastic model under axisymmetric loadings

Porothermoelastic responses of saturated porous media find wide applications in geotechnical engineering. However, the coupled partial differential equations describing the conservation of momentum, mass, and energy in the porous medium posed mathematical difficulties in obtaining analytical solutions. In this paper, we provided a two-stage porothermoelastic model for comprehensive solutions under axisymmetric loadings. At the first stage, the governing equations were decoupled by the Laplace–Hankel transform, which yielded the explicit expressions of the temperature, pore pressure, displacements, and the vertical and shear stresses. At the second stage, the radial and tangential strains were obtained after the numerical inversion of the volumetric strain and displacements. We also found that the volumetric strain played an important role in this model: (1) coupled displacements with the pore pressure and temperature at the first stage; (2) combined the vertical, radial, and tangential strains at the second stage. Results of a finite layer under a disk thermal loading showed that this model could capture the thermal expansion and contraction in terms of displacements, strains and stresses, and such mechanical interactions could give rise to the buildup and dissipation of pore pressures during the thermal conduction.     

On the numerical modelling of initiation of sediment transport using Smoothed Particle Hydrodynamics

Mobilization of solid particles at the interface between a porous and a free flow domain is a relevant subject in many fields of mechanical, civil and environmental engineering. One example is the initiation of sediment transport as it appears in river beds. To approximate this initiation state, various theoretical models exist. Common approaches use two-domain formulations as in [1] or one-domain formulations as in [6]. The named approaches were compared with Direct Numerical Simulations (DNS) using Smoothed Particle Hydrodynamics (SPH) in [3]. The results of these simulations showed that the theoretical models often underestimate the occurring velocities at the interface and therefore critical velocities to initialize the motion of single grains can be lower than predicted by theoretical approaches. In our numerical simulations, we study creeping flow in a free flow domain coupled to flow in a porous media applying various porous structures. To investigate velocities and shear stresses at the interface more intensively we then compare our numerical results to data from experiments that were performed on equivalent microstructures. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A discontinuous method for oil-water flow in heterogeneous porous media

An oil-water flow-transport problem in a heterogeneous porous medium (HPM), given by the Brinkman model coupled with a fractional flow formulation, is solved numerically by a discontinuous finite volume element (DFVE) method coupled with a Runge-Kutta Discontinuous Galerkin (RKDG) scheme. A new (original) numerical example is presented. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Classical solution for a nonlinear hybrid system modeling combustion in a multilayer porous medium

Combustion processes in porous media have been used by the petroleum engineering industry to extract heavy oil from reservoirs. This study focuses on a one-dimensional nonlinear hybrid system consisting of $n$ reaction–diffusion–convection equations coupled with $n$ ordinary differential equations, which models a combustion front moving through a porous medium with $n$ parallel layers. The state variables are the temperature and fuel concentration in each layer. Coupling occurs in both the reaction function and differential operator coefficients. We prove the existence of a classical solution, first locally and then globally over time, to an initial and boundary value problem for the corresponding system. The proof uses a new approach for combustion problems in porous media. The local solution is obtained by defining an operator in a set of Hölder continuous functions and using Schauder’s fixed-point theorem to find a fixed point as the desired solution. Using Zorn’s lemma, we extend the local solution to a global solution, proving that the first-order spatial derivative of the temperature in each layer is a bounded function.     

On the Stability Analysis of Decoupled Solution Schemes

Many problems in engineering, physics or other disciplines require an integrated treatment of coupled fields. These problems are characterised by a dynamic interaction among two or more physically or computationally distinct components, where the undergoing mathematical model commonly consists of a system of coupled PDE. Considerable progress has been made in the development of appropriate computational schemes to solve such coupled PDE systems. These attempts have resulted in various monolithic and decoupled numerical solution approaches. Despite the unconditional stability offered by implicit monolithic solution strategies, their use is not always recommended. The reason mainly lies in the complexity of the resulting system of equations and the limited flexibility in choosing appropriate time integrators for individual components. This has motivated the elaboration of tailored decoupled solution schemes, which follow the idea of splitting the problem into several sub-problems. But selection of the way of splitting can have a direct influence on the stability of the resulting solution algorithm. This necessitates the stability analysis of such an algorithm. Here, we introduce a general framework for the stability analysis of decoupled solution schemes. The approach is then used to study the stability behaviour of established decoupling strategies applied to typical volume- and surface-coupled problems, namely, coupled problems of thermoelasticity, porous media dynamics and structure-structure interaction. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lubrication analysis of externally pressurized circular porous bearings

The present study introduces a mathematical formulation for externally pressurized circular porous bearings. A porous layer is used to cover one of the bearing surfaces. An empirical boundary condition with a nonzero tangential velocity, which is known as the velocity slip at the interface, is incorporated into the analysis. The effect of pressure on lubricant viscosity is also considered. The mathematical model consists of two coupled partial differential equations; the first governs the pressure distribution in the film and the second governs the pressure distribution in the porous layer. A simultaneous numerical solution of these equations with the boundary conditions is presented. The effects of porous layer permeability parameter, lubricant viscosity parameter, recess radius, and film thickness on pressure distribution and load-carrying capacity are presented and discussed.     

A consistent theory for poroelastic plates

In many fields of engineering thin porous components are used, e.g. as damping elements for noise insulation in cars or walls in buildings. Today these elements are often calculated using a numerical 3-D model. Because of numerical problems which occur using a 3-D model for thin transversly loaded structures a plate theory is advantageous. To take into account the porous structure as well as the damping effect of the porosity of these components a poroelastic plate theory is necessary. Several posibilities exist to establish plate theories. Generally, methods to derive a plate theory require a priory assumptions motivated by engineering intuition (like the classical Kirchhoff normal hypothesis). In this contribution a priori assumptions are not used. Plate theories of different orders are derived from the 3-D poroelastic theory using series expansion. For elastic plates this idea was introduced in [3]. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelling of coupled heat and moisture transfer in porous construction materials

In this paper, a mathematical model is proposed for predicting air temperature and humidity in a two-room region. The model contains a coupled relationship between temperature and humidity within the constructions and can be solved by using the numerical method. However, the two-room region can be reduced to a single region when the region with no ventilation is considered, and then the room temperature and relative humidity can be obtained analytically. The solution obtained in this paper is verified by comparing with the result of the analytical method. It shows that the two results are in agreement. In addition, the proposed model can also be applied to simultaneously obtain the transient temperature and humidity of a two-room region for different porous construction materials.     

Algebraic preconditioning for Biot-Barenblatt poroelastic systems

Poroelastic systems describe fluid flow through porous medium coupled with deformation of the porous matrix. In this paper, the deformation is described by linear elasticity, the fluid flow is modelled as Darcy flow. The main focus is on the Biot-Barenblatt model with double porosity/double permeability flow, which distinguishes flow in two regions considered as continua. The main goal is in proposing block diagonal preconditionings to systems arising from the discretization of the Biot-Barenblatt model by a mixed finite element method in space and implicit Euler method in time and estimating the condition number for such preconditioning. The investigation of preconditioning includes its dependence on material coefficients and parameters of discretization.