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The Darcy's equation consists of the mass conservation equation and the Darcy's law that involves the hydraulic potential (or called pressure) and the fluid velocity, which governs the flow of an incompressible fluid through a porous medium. In this paper, we investigate the Legendre Galerkin spectral collocation least squares method for approximating the problem of Darcy flow in homogeneous medium and non-homogeneous medium, respectively. The proposed scheme can be solved the approximate solutions of the hydraulic potential and the average velocity of the fluid simultaneously. A symmetric positive definite coefficient matrix of the corresponding linear algebra equation is obtained by applying our scheme. Numerical examples are presented to validate the efficiency and accuracy of the proposed scheme. [ABSTRACT FROM AUTHOR]

This paper provides a rudimentary insight into the influence of heat and mass transfer on the magneto-hydrodynamic (MHD) Jeffrey fluid peristaltic flow filling porous space in a symmetric inclined channel using a rotating frame with chemical reaction. In contrast to previous attempts, the flow formulation is based on the impact of a modified Darcy's law porous media on the Jeffrey fluid condition. The derived equations were solved analytically via the standard long wavelength and low Reynolds number assumptions to determine the pressure gradient, temperature, dimensionless velocity, pressure rise, and friction force. Otherwise, the concentration was numerically processed using the ND-Solve built-in command of Mathematica. Such a numerical technique is beneficial in minimizing error and reducing CPU time per evaluation. It chooses an appropriate algorithm for solving the problem. The graph is used to physically interpret the numerical answers for the base flow profiles. For numerous parameters of interest that enter into the issues, graphical findings are developed and tested. The impacts of various involved parameters appearing in the solutions are carefully analyzed. The trapping phenomena are discussed for several parameters. Electromagnetic peristaltic micropumps, for example, are one application of the current study in biomedical engineering. It was claimed that our systematic approach may constitute a basis for accurately examining the impact of heat and mass transfer on the magneto-hydrodynamic (MHD) Jeffrey fluid peristaltic flow filling porous space in a symmetric inclined channel using a rotating frame with chemical reaction, useful for diverse medical applications such as gastric fluid flow through the small intestine. [ABSTRACT FROM AUTHOR]

Fluid flow in isotropic porous media with one porosity scale is a well understood process and a common scenario in numerous simulations published in the literature. However, there exists a class of porous materials that exhibit two porosity scales with strong permeability contrast between the two scales. Examples of such materials are aggregated soils and fractured sedimentary rocks such as shale. In sedimentary rocks, fluid could flow through the micro-fractures at the larger scale as well as through the nanometer-size pores of the rock matrix at the smaller scale. In this paper, we shall refer to the larger and smaller pores of sedimentary rocks as the micro-fractures and nanopores, respectively. Due to preferentially oriented micro-fractures in the rock, fluid could flow predominantly in the direction of the discontinuities, resulting in an anisotropic flow pattern at the larger scale. We idealize such material as a transversely isotropic medium with respect to fluid flow. In addition, the nanopores of sedimentary rocks such as shale are so small that Darcy's law may not hold at this scale. To better understand the impact of non-Darcy flow on the overall flow pattern, we present a hydromechanical model for materials with two porosity scales that accommodates both transverse isotropy at the larger scale and non-Darcy flow at the smaller scale. Even though this study is motivated by shale properties, the discussion revolves around a generic material with two porosity scales whose fluid flow characteristics are similar to those of shale. The overarching goal of this paper is to better understand the impacts of transverse isotropy and non-Darcy flow on the fluid flow pattern in this material. [ABSTRACT FROM AUTHOR]

Non-Newtonian fluids such as asphalt are widely used in engineering field, but their application will also cause environmental pollution. This paper investigates the MHD flow of this kind of non-Newtonian fluid in porous media by using fractional Burgers' model. The effects of first-order chemical reaction, radiation effects and periodic oscillating boundary condition on fluid flow, heat and mass transfer are considered. The governing equations including a multi-term time fractional derivative are obtained by using the modified Darcy's law, fractional Fourier's law and fractional Fick's law. A convergent and stable L-algorithm, is established for governing equations. The influences of model parameters on the velocity, temperature and concentration distributions are analyzed. Numerical simulation results indicate that fractional derivative α and Darcy number Da have significant effect on velocity distribution. The momentum boundary layer becomes thinner remarkably with fractional derivative α. While the influence of Darcy number Da on the velocity performs conversely. [ABSTRACT FROM AUTHOR]

Most numerical approaches for coupling fluid flow with porous media flow rely either on Stokes equations in the fluid part of the domain and Darcy's law in the porous part, or on Brinkman's equation. In both cases, difficulties arise at the boundary between the two parts because the equations used in the porous part are not of Stokes type. In this paper, an alternative to Darcy's law is proposed for modeling flows in porous media. This alternative relies on equations of Stokes type where the permeability tensor is replaced by force and stress derivative tensors. Numerical procedures are presented to compute these tensors from simulations at pore scale. Simulations in domains containing both fluid and porous parts are finally conducted simply assuming continuity of velocity and pressure and hence without imposing any condition at the boundary between the two parts. Results show that the proposed method is accurate and hence a promising alternative to Darcy's law for problems involving both fluid and porous parts. • Alternative to Darcy's law relying on equations of Stokes type for modeling flows in porous media. • Permeability tensor replaced by force and stress derivatives tensors computed from simulations at pore scale. • Flows in domains containing both fluid and porous parts modeled without any boundary conditions between the two parts. [ABSTRACT FROM AUTHOR]

Dehomogenization techniques are becoming increasingly popular for enhancing lattice designs of compliant mechanical systems with ultra-large resolutions. Their effectiveness hinges on computing a deformed periodic grid that enable to reconstruct fine-scale designs with modulated and oriented patterns. In this paper, we propose an approach for extending dehomogenization methods to laminar fluid systems. We initiate our methodology by asymptotically deriving Darcy's law on a periodically porous medium deformed by a diffeomorphism. Unlike the mechanical context, we reveal that the homogenized permeability matrix depends not solely on local the orientation but also on the local dilation of the deformed periodic medium. This distinction presents one of the several challenges to be tackled when adapting dehomogenization-based topology optimization techniques to porous media. To accommodate existing methodologies, we formulate a simplified "poor man's" homogenized model, which streamlines various aspects, yet still leans on periodic cell problems to estimate the spatially varying permeability matrix. Specifically, we overlook boundary layer effects, we presume periodic grid deformations, and we neglect local dilation, solely considering the relationship with local cell orientations. Subsequently, we present a numerical approach for designing a system that redistributes an input flow across numerous regularly spaced outlets at an output interface. Leveraging the homogenized model, we deduce optimized geometric arrangements of local channel spacing parameters and orientations. We then use established methods to reconstruct grid deformations and fine-scale designs. The fidelity of these reconstructions is then validated through fine-scale simulations. Our observations indicate that while the proposed designs yield satisfactory performance when subjected to the full-scale model, discernible deviations from the homogenized model persist, appealing to future improvements. [ABSTRACT FROM AUTHOR]

Finite amplitude convection in a inclined porous layer heated from below is studied by using local thermal non-equilibrium (LTNE) as mathematical model which takes into account the heat transferred between the solid phase and the fluid phase. Consequently, in addition to Darcy-Rayleigh number Ra and the inclination angle ϕ , two further non dimensional numbers are introduced: the inter-phase heat transfer parameter H and the porosity modified conductivity ratio γ . In a recent paper (Barletta and Rees, 2015), the linear stability analysis of the basic monocellular flow indicated that the inclination angle promotes the appearance of longitudinal rolls as the preferred mode of convection. The current paper focuses on the nonlinear evolution of longitudinal rolls in a supercritical regime of convection. A weakly nonlinear analysis, using a derived amplitude equation, is adopted to determine the nonlinear effects of the parameters Ra , ϕ , H and γ . The results indicate that in inclined layers (i) the nonlinearity decelerates the mean flow; (ii) the heat transfer, determined by the evaluation of the Nusselt number ( Nu ) at the layer boundary, corresponds to the one obtained for horizontal layers by scaling Ra with cos ϕ , i.e. Nu = Nu ( Ra cos ϕ , H , γ ) ; (iii) in accordance with existing laboratory experiments, the slope of Nu is less than 2, where 2 is the value predicted by the local thermal equilibrium model, and the slope represents the derivative of Nu with respect to the distance of the critical parameter from the threshold value for the onset of instability; (iv) increasing values of both H and γ produce an enhancement of the heat transfer across the layer. Finally, the comparison between the LTNE theoretical predictions and existing experiments conducted with various combinations of solid matrix and fluids suggests a possible alternative way to determine the heat transfer coefficient H . [ABSTRACT FROM AUTHOR]

In this paper, the fractional calculus approach is employed in the constitutive relationship of the fluid model and the Darcy's law. The flow of the fractional Jeffreys' fluid induced by the impulsive motion of a flat plate in a porous half space is studied in form of an initial and boundary value problem with fractional derivatives. Using the Laplace transform method, we obtain an exact solution of the model in term of Fox's H -function. As a byproduct, the solutions of the analogous flow for the generalized second grade fluid, the fractional Maxwell fluid and the Newtonian fluid in the porous half space are also deduced. In addition, the influence of the material parameters and the fractional parameters on the fluid motion is investigated, as well as a comparison among the fractional Jeffreys' fluid, the fractional Maxwell fluid and the Newtonian fluid in porous medium is also analyzed by graphical illustrations. [ABSTRACT FROM AUTHOR]

Pre-Darcy flow is widely found in many naturally porous media, such as low-permeability clay. Nonflowing liquid boundary layers strongly influence fluid flow in low-permeability porous media. There is evidence in the literature that the porosity and permeability of low-permeability porous media change with variations in the effective stress, which is called stress sensitivity. In this paper, a new mathematical model for pre-Darcy flow in low-permeability porous media is developed and validated with experimental data from the literature. The nonflowing boundary layer thickness can be obtained from the proposed model instead of the empirical equation. First, an equation for calculating the pore radius with stress sensitivity is derived. Two parameters ε p and ε r are defined to calculate the nonflowing boundary layer thickness. Second, based on the capillary bundle model, the apparent flow velocity and apparent liquid permeability of the porous media are determined. Finally, the stress sensitivity and boundary-layer effects on the apparent flow velocity and apparent liquid permeability are analyzed. The research results show that the effect of stress sensitivity on apparent flow velocity increases with increasing pore compressibility. When the pore compressibility C p >0 MPa−1, the apparent liquid permeability increases continuously with increasing pressure gradient. When the pore compressibility C p =0 MPa−1, the apparent liquid permeability continues to increase with increasing pressure gradient. The apparent liquid permeability eventually tends to a constant value. The apparent flow velocity decreases with ε p , and ε r decreases at a given pressure gradient. The larger ε p and ε r are, the faster the rate of increase in the apparent liquid permeability with increasing pressure gradient. The proposed model can estimate the apparent flow velocity at different pressure gradients with a high accuracy. This work is important for understanding flow phenomena in low permeability porous media, such as landslides caused by seepage of water in clay. • A low-velocity pre-Darcy flow model in low permeability porous media is developed. • The nonflowing boundary layer thickness was obtained from a new model. • The relationship between the boundary layer and the pressure gradient is established. • The effect of stress sensitivity on velocity and permeability is analyzed. • The effect of nonflowing boundary layer on velocity and permeability is analyzed. [ABSTRACT FROM AUTHOR]

This paper aims to develop a moving-mesh type Finite Element Method for the computation of the transient unconfined seepage flow through the porous medium. The proposed method is based on the time discontinuous Galerkin Space-Time Finite Element Method (ST/FEM). It solves the seepage problem in the saturated region. The primary unknown in ST/FEM is piezometric pressure. Fluid velocities are derived from the pressure using Darcy's law. Further, an iterative algorithm has been proposed in this paper to implement the proposed method. In each iteration step, the computation domain is updated according to the flow velocity on the phreatic boundary. Subsequently, internal nodes are moved using the mesh moving technique to accommodate the newly updated computation domain. The mesh moving technique, which is discussed in this paper, is based on an elasticity problem. ST/FEM is employed to analyze several unconfined seepage flow problems, and results of steady state solutions are compared with those available in the literature to demonstrate the efficacy of the proposed scheme. • Moving mesh type finite element method is developed for transient and unconfined seepage flow analysis. • The proposed scheme requires 2 to 3 iterations in a time step to achieve convergence. • Elasticity equation based automatic mesh moving technique is employed to move the internal nodes. • Initial computation domain significantly affects the deformation characteristics of mesh. • Proposed iterative scheme can be employed with any type of mesh moving technique. [ABSTRACT FROM AUTHOR]

Abstract: This paper studies decoupled numerical methods for a mixed Stokes/Darcy model for coupling fluid and porous media flows. A two-level algorithm is proposed and analyzed in Mu and Xu (2007) . We generalize the two-level algorithm to a multilevel algorithm in this paper and present numerical analysis on the error estimates for the multilevel algorithm. The multilevel algorithm solves the mixed Stokes/Darcy system by applying efficient legacy code for single model solvers to solve two decoupled Stokes and Darcy subproblems on all the subsequently refined meshes, except for a much smaller global problem only on a very coarse initial mesh. Numerical experiments are conducted for both the two-level and multilevel algorithms to illustrate their effectiveness and efficiency, and validate the related theoretical analysis. [Copyright &y& Elsevier]

Abstract In this paper we develop and analyze a unified approximation of the velocity–pressure pair for the Stokes–Darcy coupled problem in a plane domain. It is well known that, stable finite element approximations for the Stokes problem may not be appropriate for Darcy problem and for the coupling of fluid flow (modeled by the Stokes equations) with porous media flow (modeled by the Darcy equation), and therefore, different spaces are commonly used for the discretizations of the Darcy and the Stokes problems. In this work we proposed a modification of the Darcy problem which allows us to apply the classical Mini-element to the whole coupled Stokes–Darcy problem. The proposed method is probably one of the cheapest method for continuous approximation of the coupled system, has optimal accuracy with respect to solution regularity, and has simple and straightforward implementations. Numerical experiments are also presented, which confirm the excellent stability and accuracy of our method. [ABSTRACT FROM AUTHOR]

In this paper a numerical analysis of in situ-biorestoration is presented. Improved stability and error estimates are derived for a P 1 finite-element method applied to coupled system of non linear partial differential equations modeling flow in porous media. These results improve upon previously derived stability and error estimates in two respects: first, a stability of the approximate solution is demonstrated in the case of a non linear diffusive flux λ ( u ) ∇ u with weaker norm assumptions than before and depends only on the initial conditions and time interval, and second, error estimates are optimal as in linear case. Extensions include the finite element approximation of flow field, described by Darcy’s law under the stream function–vorticity formulation because of its multiple advantages. Finally, numerical simulations for a correlation exponential plume are presented. [ABSTRACT FROM AUTHOR]

In this paper, we consider the coupling of fluid flow with porous media flow. A multi-grid finite element method for the coupled Stokes–Darcy problem with the Beavers–Joseph interface condition is proposed and discussed. The optimal error estimates are obtained. Numerical experiment is given to verify the theoretical analysis and indicate the accuracy and efficiency of the multi-grid method. [ABSTRACT FROM AUTHOR]

In vanadium redox flow batteries, an accumulated imbalance of the states-of-charge between the two half-cell electrolytes caused by vanadium ion crossover and gassing side reactions can result in only one half-cell achieving 100% state-of-charge, which is referred to as capacity loss that needs to be corrected online. In order to implement rebalancing control, online states-of-charge have to be monitored. In this paper, the electrolyte viscosity and its use for online state-of-charge monitoring design are investigated. The study firstly measures the viscosities of both V 2+ /V 3+ and VO 2+ /VO 2 + redox couples in sulfuric acid as the negative and positive half-cell electrolytes at different states-of-charge and temperatures, followed by establishing an empirical neural network model that correlates the state-of-charge to viscosity and temperature. To overcome the limitation in online viscosity measurements, Darcy’s law describing the flow of a fluid through a porous medium is further introduced to link the electrolyte viscosity to the pressure drop across a porous medium where the electrolyte solutions flow through. Together with the neural network model, the state-of-charge can be eventually represented as a function of pressure, temperature and flow rate that are readily measurable online, and accordingly an online state-of-charge monitoring design is developed, which could be readily integrated into the online battery control system for automated electrolyte rebalance. Experimental validation is performed on a 15-cell stack system and the results demonstrate the feasibility of the proposed method. [ABSTRACT FROM AUTHOR]

This paper considers a multilevel decoupling method for the coupled Navier–Stokes/Darcy model describing a free flowing fluid over a porous medium. The method utilizes a sequence of meshes on which a low dimensional fully coupled nonlinear problem is solved only on a very coarse initial mesh. On subsequent finer meshes, the approximate solution in each flow region is obtained by solving a linear decoupled problem and performing a correction step. The correction step in each domain is achieved by solving a linear system that differs from the original decoupled system only in the right hand side. We prove optimal error estimates and demonstrate that for a sequence of meshes with spacing h j = h j − 1 2 , the decoupling method is computationally efficient and achieves the same order of approximation as the fully coupled method. [ABSTRACT FROM AUTHOR]

The paper deals with modeling fluid saturated porous media subject to large deformation. An Eulerian incremental formulation is derived using the problem imposed in the spatial configuration in terms of the equilibrium equation and the mass conservation. Perturbation of the hyperelastic porous medium is described by the Biot model which involves poroelastic coefficients and the permeability governing the Darcy flow. Using the material derivative with respect to a convection velocity field we obtain the rate formulation which allows for linearization of the residuum function. For a given time discretization with backward finite difference approximation of the time derivatives, two incremental problems are obtained which constitute the predictor and corrector steps of the implicit time-integration scheme. Conforming mixed finite element approximation in space is used. Validation of the numerical model implemented in the SfePy code is reported for an isotropic medium with a hyperelastic solid phase. The proposed linearization scheme is motivated by the two-scale homogenization which will provide the local material poroelastic coefficients involved in the incremental formulation. [ABSTRACT FROM AUTHOR]

Evaporation of water from a ceramic layer is a key phenomenon in the drying process for the manufacturing of water-based tape cast ceramics. In this paper we present a coupled free-flow-porous-media model on the Representative Elementary Volume (REV) scale for coupling non-isothermal multi-phase compositional porous-media flow — for the ceramic layer — and single-phase compositional laminar free flow — for the air above it. The preliminary results show the typical expected evaporation behaviour from a porous medium initially saturated with water, and water–vapour transport to the free-flow region in accordance with the available results from the literature. We elaborate on and discuss the characteristic drying-rate curve for a single layer ceramic, and compare it with that of a graded/layered ceramic. We, moreover, show the influence of the mean diameter of particles of the porous medium ( d p ) — which directly affects the intrinsic permeability ( K ) based on the well-known Ergun’s equation — of each single ceramic layer on the drying behaviour of a graded/layered ceramic. [ABSTRACT FROM AUTHOR]

This paper reports a numerical study of natural convection in at all porous enclosure filled with a binary fluid. The Darcy–Dupuis model, which includes effects of the form drag force, is adopted to describe the flow in the porous medium. The two vertical walls of the cavity are subject to constant gradients of temperature while the two horizontal ones are kept adiabatic and impermeable. Concentration gradients are assumed to be induced either by the imposition of constant gradients of solute on the vertical walls of the system ( a = 0 ; double diffusive convection) or by the Soret effect ( a = 1 ) . Governing parameters of the problem under study are the thermal Rayleigh number R T , form drag parameter G , buoyancy ratio φ , Lewis number. Le , normalized porosity ε , and aspect ratio of the cavity A . The case of equal and opposing thermal and solutal buoyancy forces, φ = - 1 , is considered. For this situation, an equilibrium solution corresponding to the rest state is possible and the resulting onset of motion can be either supercritical or subcritical. A semi-analytical solution, valid for an infinite layer ( A ≫ 1 ) assuming parallel flow, is derived. Based on the linear stability theory, the onset of motion from the rest state is predicted for both double diffusive and Soret convection. The onset of Hopf bifurcation, characterizing the transition from a convective steady state to oscillatory state, is also studied. The influence of the governing parameters on the onset of motion and the resulting fluid flow, temperature and concentration fields is discussed in detail. The existence of supercritical, subcritical and oscillatory convective modes is demonstrated. A good agreement is found between the predictions of the parallel flow approximation and the numerical results obtained by solving the full governing equations. The existence of multiple solutions and traveling waves for a given set of the governing parameter is demonstrated and leads to the existence of a bistability phenomenon. Overall, the form drag behaves as a stabilizing effect and is seen to affect considerably the onset of subcritical convection and Hopf bifurcation. [ABSTRACT FROM AUTHOR]

In thermoacoustics, stacks and regenerators are porous media where energy conversion takes place. Modelling full thermoacoustic devices with a CFD approach, in order to capture some nonlinearities, can be extremely expensive from a computational perspective compared to a standard linear approach used in the frequency domain. At the same time, macroscopic models for porous media developed for steady-state flows cannot be directly applied in oscillating flow conditions. Moreover, macroscopic models in the available literature for oscillating flows are inaccurate at high frequencies or require a closure coefficient to be determined numerically (with Direct Numerical Simulations) or experimentally. In this article, a time domain macroscopic model for heat and fluid flow is proposed based on the concepts of complex Darcy and Nusselt numbers in the linear regime. Such coefficients, introduced in the past to describe the oscillatory phenomena, have been used for the first time to build a CFD macroscopic model in terms of their real and imaginary parts. For two different porous media, a parallel plate and a transversal pin array, the developed macroscopic model is verified with the microscopic solution. Furthermore, for a transversal pin array stack, the proposed model is validated against experimental data from the available literature, showing a very good agreement. The findings of this paper can help to strongly reduce the computational costs of oscillatory flow simulations without prior direct numerical simulations of the porous core. [ABSTRACT FROM AUTHOR]

• Hybrid nanofluid flow over a circular elastic surface is studied. • Non-Darcian porous medium of the circular surface is considered. • Numerical results are presented with the help of well-known technique SLM. • The present results are applicable for the solar energy applications. • A comparative study is also performed in the absence of nanoparticles. In recent years, the conversion of solar radiation to thermal energy has received a significant attention as the demand for renewable heat and power increases. Nanofluids can play an important role in improving the performance of solar-thermal systems due to their capabilities for heat transfer enhancement. This paper investigates numerically the flow of a hybrid nanofluid through a porous medium. A water-based hybrid nanofluid is propagating across a circular elastic surface. Water flow is incompressible, irrotational and electrically conducting. The deployment of an external magnetic field ignores the induced magnetic field due to the small magnetic Reynolds number. Cobalt oxide (Co 3 O 4) and Graphene (Go) nanoparticles (NPs) are suspended in the base fluid. For the porous media, the Darcy model is employed, while the viscous dissipation effects are also incorporated in the energy equation. Similarity variables are used to develop the mathematical modeling of momentum and energy equations. The numerical solution of the finalized forms of nonlinear differential equations is accomplished by the use of the Successive Linearization Method (SLM). The simulation results are then validated against the previously published data. [ABSTRACT FROM AUTHOR]

In this paper, a computationally efficient framework capable of modeling well production simulation with general shaped fractures embedded in three dimensional porous matrix is presented. The flow in the matrix is modeled by classical theory of Darcy flow, whereas flow in hydraulic fractures is treated as channel flow. Governing equations of the Darcy flow are formulated in terms of weakly singular, weak-form boundary integral equations, whereas those of channel flow are cast in a weak form using Galerkin method of weighted residuals. We develop a special tip element to capture the dominant O (1 r ) asymptotic field of Darcy flow near the crack tip in porous media. We elaborate a unified transformation technique to overcome the difficulty with singular and nearly singular integrals. The numerical implementation is comprehensively verified through decoupled Darcy flow equation, decoupled channel flow equation and coupled equations, respectively. We show three steady state examples, which are sequential circular cracks case, sequential long cracks case and petal cracks case, to demonstrate the capability of the proposed framework. • Proposes a SGBEM–FEM framework for well production with 3D nonplanar fractures. • The special tip element captures tip asymptotic of Darcy flow around crack tip. • An efficient solution for infinite layer domain as production zone is provided. • Numerical results agree well with analytical solutions and image method solution. [ABSTRACT FROM AUTHOR]

Although the numerical results suggest the optimal convergence order of the two-grid finite element decoupled scheme for mixed Stokes–Darcy model with Beavers–Joseph–Saffman interface condition in literatures, the numerical analysis only gets the optimal error order for porous media flow and a non-optimal error order that is half order lower than the optimal one in fluid flow. The purpose of this paper is to fill in the gap between the numerical results and the theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper discusses the problem of mixed convection in two-sided lid-driven enclosures saturated non-Darcy porous medium. The vertical walls of the cavity were kept thermally insulated. The bottom wall is cooled while the top wall is uniformly heated. The bottom and the top walls are moving in opposite direction. The governing equations were solved using finite volume method with SIMPLE algorithm. A new form for the heat function was derived. The obtained results were presented in contours maps for the streamlines, the isotherms and the heat function. The profiles of the horizontal velocity component and the maximum values of vertical velocity components as well as the mean Nusselt number were presented graphically. It is found that, for the low values of the Richardson number, the forced convection plays a dominant role in the flow region. The increase in inverse Darcy number leads to decrease the mean Nusselt number. [ABSTRACT FROM AUTHOR]

In this paper, we develop a two-scale reduced model for simulating the Darcy flow in two-dimensional porous media with conductive fractures. We apply the approach motivated by the embedded fracture model (EFM) to simulate the flow on the coarse scale, and the effect of fractures on each coarse scale grid cell intersecting with fractures is represented by the discrete fracture model (DFM) on the fine scale. In the DFM used on the fine scale, the matrix-fracture system are resolved on unstructured grid which represents the fractures accurately, while in the EFM used on the coarse scale, the flux interaction between fractures and matrix are dealt with as a source term, and the matrix-fracture system can be resolved on structured grid. The Raviart-Thomas mixed finite element methods are used for the solution of the coupled flows in the matrix and the fractures on both fine and coarse scales. Numerical results are presented to demonstrate the efficiency of the proposed model for simulation of flow in fractured porous media. [ABSTRACT FROM AUTHOR]

DarcyLite is a Matlab toolbox developed for numerical simulations of flow and transport in porous media in two dimensions. This paper focuses on the finite element methods and the corresponding code modules for solving the Darcy equation. Specifically, four major types of finite element solvers are presented: the continuous Galerkin (CG), the discontinuous Galerkin (DG), the weak Galerkin (WG), and the mixed finite element methods (MFEM). We further discuss the main design ideas and implementation strategies in DarcyLite. Numerical examples are included to demonstrate the usage and performance of this toolbox. [ABSTRACT FROM AUTHOR]

Loosening local equilibrium assumptions in two-phase flow in porous media gives rise to new, unknown variables. More specifically, when loosening the local thermal equilibrium assumption, one has to describe the heat transfer between multiple phases, present at the same mathematical point. In this paper, we calibrate a macro-scale mathematical model which is free of local equilibrium assumptions to experimental observations. We emphasize the correct determination and upscaling of necessary input parameters from the experimental data achieved by image analysis. By choosing an appropriate scaling parameter, we are able to reproduce experimental measurements satisfactorily. This is a first step towards quantifying heat transfer in two-phase flow in porous media. Ultimately, our aim is to find the limits of the applicability of local equilibrium assumptions in two-phase flow in porous media. [ABSTRACT FROM AUTHOR]

Conventional Darcy law cannot accurately describe the combined gas transport mechanisms in porous media with free gas only (such as tight gas reservoirs) and porous media with both free gas and adsorbed gas (such as shale gas reservoirs). The gas transport mechanisms in tight porous media including viscous flow, Knudsen diffusion, surface diffusion and molecular diffusion are investigated. Both the equivalent hydraulic radius and transport properties including intrinsic permeability, tortuosity and porosity, are used to build the coupled transport models describing the combined mechanisms of gas transport in ultra-tight porous media with free gas only and porous media with both free gas and adsorbed gas respectively. The effect of the pore-volume occupied by the adsorbed layer and surface diffusion through the adsorbed layer is considered in the transport model of porous media with adsorbed gas. The influences of equivalent hydraulic radius and transport properties on the gas transport mode in ultra-tight porous media are analyzed and the gas transport mode criteria in ultra-tight porous media are obtained. The result shows that viscous flow is the dominant mechanism, Knudsen diffusion can be ignored and Darcy law is applicable in porous media with equivalent hydraulic radius more than 100 nm. Surface diffusion can be ignored in the porous media with equivalent hydraulic radius less than 10 nm, while the effect of volume reduction by adsorption of porous media is less than 100 nm. The critical value of the transport function of intrinsic permeability, tortuosity and porosity also is given in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we consider the mixed Stokes–Darcy problem which describes a fluid flow coupled with a porous media. We present a modified two-grid method for decoupling this mixed model. Stability is proved and optimal error estimates are derived. The numerical results show that the modified two-grid method is effective and has the same accuracy as the coupling scheme when we choose h = H 2 . [ABSTRACT FROM AUTHOR]

Gill’s stability problem is the analysis of the parallel buoyant flow in a vertical porous channel whose parallel walls are kept at different uniform temperatures. Gill’s classical paper [Journal of Fluid Mechanics, 35 (1969) 545–547] provides a rigorous proof that this flow is linearly stable. The aim of our study is to extend Gill’s analysis to the class of non-Newtonian viscous fluids modelled by Ostwald-de Waele power law. The main difference between Newtonian fluids and general power-law fluids is that the basic velocity profile is linear, in the Newtonian case, and nonlinear with an inflexion point at the mid-plane, in the non-Newtonian case. Despite the presence of the inflexion point, this study evidences a stable behaviour of the basic flow versus general normal mode perturbations: longitudinal, oblique and transverse rolls. Stability to longitudinal rolls is proved analytically, while the behaviour of transverse and oblique rolls is investigated numerically. The damping rates of perturbations, evaluated for oblique and transverse rolls, display increasing values as the Darcy–Rayleigh number increases. Numerical data thus suggest that linear stability holds for the whole class of power-law fluids. [ABSTRACT FROM AUTHOR]

• Theoretical models are developed for the dynamic compression of MR fluid-filled porous materials. • Darcy's law and Bingham-plastic model are used for the models. • The two-layer model can predict the results better than the single-layer model. • Viscous effect is dominant for high strain rate cases. • Controllability of the MR fluid-filled materials decreases with the increased of strain-rate. In this paper, the dynamic compression and energy absorption behaviours of porous materials filled with MR fluid are studied theoretically. By means of Darcy's law, Ergun's equation and energy method, the single-layer and two-layer models for the energy absorption and dynamic stress are first derived, in which the compression of the skeletal material, the inertia effect, viscous flowing of the fluid as well as the MR effect are considered. The comparisons between the theoretical results and previous experiments of porous copper specimens show that the two-layer model can predict the dynamic stress before the circumferential failure of specimen very well. Based on the theoretical models, it is found that the energy dissipated by the deformation of the skeletal material and MR effect is nearly insensitive to the strain-rate, while that by the viscous flowing and inertial effect increases linearly and quadratically with the increase of impact velocity, respectively. When the strain rate is low, the energy dissipation due to the MR effect is larger than that by viscous flowing. However, if the strain-rate is higher than a certain value, the viscous energy dissipation will be dominant. Moreover, with the same strain rate, specimens with larger radius/thickness ratio could obtain higher compression stress. To improve the controllability of the MR fluid-filled porous material, higher yield stress induced by magnetic field is preferred, and the strain-rate should be as low as possible. [ABSTRACT FROM AUTHOR]

The permeability of porous media is an important parameter in designing ultra-thin vapor chambers, while there was a little researched on the ultra-thin mesh wick with a free liquid-gas interface. In the paper, the permeability of single layer mesh screen with different mesh numbers and diameters was investigated using experimental and numerical simulation methods. Experimentally, according to the Bernoulli equation with the resistance term, the flow resistance could be balanced by gravity in a sloping placed wick. And when the flow reaches the equilibrium state, the permeability can be obtained through the flow rate and the incline angle based on Darcy's law. In order to further explore the relationship between permeability and mesh structure, scanning electron microscope (SEM) and optical microscope were used to observe the microstructure. Based on the scanning picture, numerical simulation was carried out to study the source of flow resistance. And then the relationship between permeability and structure parameters was obtained. The result shows that there is an approximately linear relationship between fRe (the product of loss coefficient and Reynolds number) and the hydraulic diameter in the position where the longitudinal and horizontal wires intersect. The work can support the calculation of the capillary limit in design of ultra-thin vapor chamber. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we propose a methodology to derive a macro-scale momentum equation that is free from the turbulence model chosen for the pore-scale simulations and that is able to account for large-scale anisotropy. In this method, Navier–Stokes equations are first time-averaged to form a new set of equations involving an effective viscosity. The resulting balance equations are then up-scaled using a volume averaging methodology. This procedure gives a macro-scale generalized Darcy–Forchheimer equation to which is associated a closure problem that can be used to evaluate the apparent permeability tensor including inertia effects. This approach is validated through 2D and 3D calculations. Finally, the method is used to evaluate the tensorial macro-scale properties for a gas flow through structured packings. [Copyright &y& Elsevier]

Abstract: Flow through porous media at low Reynolds numbers has been studied in detail with the Lattice Boltzmann Method (LBM) for applications such as groundwater flow, pollution transport or adsorption processes. In contrast to that, medium to high Reynolds number flow through porous media, which occurs in many areas of industrial engineering, has not yet widely been investigated on a microscopic level by detailed numerical simulations.In this paper, we focus on air flow through a porous medium, because our far goal entails the simulation of acoustic excitations from the turbulent flow leaving the porous medium. We validate the LBM at Reynolds numbers beyond the limit of Darcy''s law, and compare the results of direct numerical simulation with those achieved by applying a Smagorinsky-type large eddy turbulence model. For this, we performed flow simulations through a generic (periodic) porous medium at a variety of resolutions to investigate the effect of LES modelling at lower mesh sizes, where the subgrid scale effects become important. [Copyright &y& Elsevier]

Abstract: This paper presents a comparative study on locally mass-conservative numerical methods for Darcy''s flows. The classical mixed finite element method (MFEM) is compared with the newly developed discontinuous finite volume method (DFVM) with and without weak over-penalization (WOP). These numerical methods are tested on three representative problems in porous media flows. In particular, locality, accuracy of numerical solutions, computational costs, and implementation issues are examined. The study indicates that the discontinuous finite volume methods could be viable alternatives to the classical mixed finite element method for Darcy''s flows. [Copyright &y& Elsevier]

Abstract: This paper develops a theory of mass, momentum, and heat transfer in a tridisperse porous medium. Coupling between three different scales present in this medium is accounted for by introducing momentum and interphase heat transfer coupling coefficients. The developed theory is then applied to solve the classical Rayleigh–Bénard problem, for the onset of convection in a horizontal layer uniformly heated from below, for this new type of a porous medium. The formulation uses the Darcy law, which now results in three different filtration velocities in three porosity scales present in this medium. The linear stability analysis leads to an expression for the critical Rayleigh number as a function of three volume fractions, two permeability ratios, two thermal capacity ratios, two thermal conductivity ratios, two inter-phase heat transfer parameters and two inter-phase momentum transfer parameters. The dependence of the critical Rayleigh number on these parameters is investigated. [Copyright &y& Elsevier]

Abstract: This paper developed a mathematical model for the coupled heat and mass transfer in porous media based on the fractal characters of the pore size distribution. According to Darcy’s law and Hagen–Poiseuille’s law for liquid flows, the diffusion coefficient of the liquid water, a function of fractal dimension, is obtained theoretically. The liquid flow affected by the surface tension and the gravity, the water vapor sorption/desorption by fibers, the diffusion of the water vapor and the phase changes are all taken into account in this model. With specification of initial and boundary conditions, distributions of water vapor concentration in void spaces, volume fraction of liquid water, distribution of water molecular content in fibers and temperature changes in porous fibrous media are obtained numerically. Effects of porosity of porous fibrous media on heat and mass transfer are analyzed. The theoretical predictions are compared with experimental data and good agreement is observed between the two, indicating that the fractal model is satisfactory. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this paper is to introduce a new approach to efficiently solve sequences of problems that typically arise when modeling flow in stochastic porous media. The governing equations are based on Darcy’s law with a stochastic permeability field. Starting from a specified covariance relationship, the log permeability is decomposed using a truncated Karhunen–Loève expansion. Multiscale mortar mixed finite element approximations are used in the spatial domain and a nonintrusive sampling method is used in the stochastic dimensions. A multiscale mortar flux basis is computed for a single permeability, called a training permeability, that captures the main characteristics of the porous media, and is used as a preconditioner for each stochastic realization. We prove that the condition number of the preconditioned interface operator is independent of the subdomain mesh size and the mortar mesh size. Computational results confirm that our approach provides an efficient means to quantify the uncertainty for stochastic flow in porous media. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to determine the new exact solution of a magnetohydrodynamic (MHD) and rotating flow of the Maxwell fluid induced by a suddenly moved plate in its own plane. This is accomplished by using the Fourier sine and Laplace transforms. Based on the modified Darcy’s law, the expression for the velocity field is obtained. It is found that similar solutions for Newtonian fluid appear as limiting cases of our solutions. Finally some graphical results of the velocity profiles are presented for different values of the material constants. [Copyright &y& Elsevier]

Abstract: This paper presents an analytic investigation of forced convection in parallel-plate channel partly occupied by a bidisperse porous medium and partly by a fluid clear of solid material, the distribution being asymmetrical. The walls of the channel are subject to an uniform heat flux; the flow is assumed to be hydrodynamically and thermally fully developed. The layer of a bidisperse porous medium is attached to one of the channel walls; it is modeled utilizing a two-velocity two-temperature formulation using Darcy’s law. The Beavers–Joseph boundary condition is employed at the bidisperse porous medium/clear fluid interface. The dependences of the Nusselt number on a conductivity ratio, a velocity ratio, a volume fraction, internal heat exchange parameter, and the position of the porous-fluid interface are investigated. Both cases of symmetric and asymmetric heating are investigated, which is specified by the asymmetry heating parameter introduced here. For the case of asymmetric heating, a singular behavior of the Nusselt number is found and explained. [ABSTRACT FROM AUTHOR]

Abstract: Constructed wetlands are wastewater treatment technologies based in natural systems, and their environmental and hydraulic behaviour is influenced by weather conditions like temperature, solar radiation and precipitation. In this paper, a one-dimensional dynamic model applicable to horizontal flow constructed wetlands is presented. The structure of the hydraulic module considers Darcy''s law for estimating head losses along the porous media and the boundary condition of the outlet structure. For the water budgets, precipitation and evapotranspiration are considered. The model was calibrated and validated with data from a constructed wetland operating in the South of Portugal, and a good agreement between simulated and measured data was obtained. The relevance of considering evapotranspiration in order to obtain good flow estimations was demonstrated, showing a significant influence of that variable on daily flow reductions, especially during summer months. Better simulation results were obtained when considering an evapotranspiration pattern that describes variations during the day, instead of a constant daily evapotranspiration rate. [Copyright &y& Elsevier]

Abstract: This paper introduces and analyzes two mathematical models for coupling the incompressible Navier–Stokes equations with the porous media flow equations. A numerical method that uses continuous finite elements in the incompressible flow region and discontinuous finite elements in the porous medium, is proposed. Existence and uniqueness results under small data condition of the numerical solution are proved. Optimal a priori error estimates are derived. Numerical examples comparing the two models under varying physical parameters are provided. [Copyright &y& Elsevier]

Abstract: The accurate estimation of hydraulic conductivity is important for many geotechnical engineering applications, as the presence of fluids affects all aspects of soil behaviour, including its strength. Darcy’s law is the key experimental (or phenomenological) equation employed to model ground water flow. Yet, this phenomenological equation can be linked to a more fundamental microscale model of flow through the pore spaces of the porous material. This paper provides an experimental verification of the relationships between Darcy’s law (macroscale) and the Navier–Stokes equations (microscale) for actual complex pore geometries of a granular material. The pore geometries are experimentally obtained through state-of-the-art X-ray computer assisted micro-tomography. From the numerical modelling of the microscale flow based on actual pore geometries, it is possible to quantify and visualize the development of pore-scale fluid preferential flow-paths through the porous material, and to assess the importance of pore connectivity in soil transport properties. [Copyright &y& Elsevier]

In this paper, we propose a multiscale method for solving the Darcy flow of a single-phase fluid in two-dimensional fractured porous media. We consider a discrete fracture-matrix (DFM) model that treats fractures as one-dimensional objects, and flows in both the matrix and fractures respect the Darcy's law. A multipoint flux mixed finite element (MFMFE) method with the broken Raviart–Thomas (RT 1 2 ) element and the trapezoidal quadrature rule is employed to approximate the matrix velocity and pressure, which results in a block diagonal, symmetric and positive definite mass matrix for the matrix velocity on general quadrilateral grids; the one-dimensional RT 0 mixed finite element method with the one-dimensional trapezoidal quadrature rule is exploited to approximate the fracture velocity and pressure, which leads to a diagonal and positive definite mass matrix for the fracture velocity in each single fracture. All these features of the obtained mass matrices allow for velocity elimination. Multiscale basis functions are constructed for the two-dimensional matrix pressure following the generalized multiscale finite element method (GMsFEM) framework to capture the fine-scale information of heterogeneous fractured porous media and effectively reduce the degrees of freedom for the matrix pressure, while fine-grid basis functions are utilized for the one-dimensional fracture pressure in fractures. Various numerical tests with the oversampling technique for different fracture distributions are performed to show that the proposed multiscale method is effective and able to provide good approximations for the fine-grid solution. • A discrete fracture-matrix model can simulate heterogeneous porous media is studied. • The GMsFEM can capture the fine-scale information of heterogeneous porous media. • The GMsFEM reduces the degrees of freedom for the matrix pressure effectively. • The velocity elimination integrated into the GMsFEM reduces the computational cost. • The proposed method can handle both conductors and barriers. [ABSTRACT FROM AUTHOR]

A dual scale porous medium contains two distinct scales of pores. We will consider the case in which small scale pores are ordered within well-defined sub-regions. Typical examples of such media are textile preforms used in various composite-manufacturing processes. Rigorously, the phenomenon can be modeled by using Darcy''s law to describe flow through porous medium and mass conservation for the flow within the larger pores. The smaller pores can be included within these equations as a sink term. This approach, though straightforward, poses implementation difficulties. In this paper, we suggest an alternative way to model this concept. We use the standard finite element/control volume approach and model the “internal” sink term by appending extra one-dimensional elements to control volumes associated with the control volumes of discretized part geometry. This approach offers two advantages over previously attempted schemes: (1) the problem to be solved remains linear and flow can be calculated explicitly within the time domain and (2) existing simulation packages for RTM filling simulation will be able to incorporate saturation effects to simulate the flow in dual scale media. To illustrate this point we present the implementation of the developed algorithm within the frameworks of an existing simulation package LIMS. The implementation is capable of providing saturation data during filling of arbitrarily shaped part and can capture the influence of saturation on filling pressure or flow-rate. This should prove useful in determining the time needed to completely saturate the fiber tows, which is crucial for part performance and also allow us to explain the variation in injection pressure and difficulties in “assigning” single scale permeability for such porous medium. [Copyright &y& Elsevier]

• Entropy generation analysis is carried out first time for triple diffusive flow. • A new model equation is presented for entropy generation including coupling of species. • A new mass Bejan number is introduced first time for the two salts concentrations. • Entropy generation rates can be decreased up to 4 times for opposing flows in comparison with assisting flows. • A rise of the Darcy number from 0.01 till 0.1 allows to reduce the entropy generation rate for both flows. The combined natural heat and mass transfer, the so-called thermosolutal convective problem, has become an attractive field of research in many diversified areas. In this paper, for the first time, entropy analysis has been carried out for triple diffusive flow. A comprehensive model is developed for the entropy generation due to fluid friction, porous medium, heat, and mass transfer across a finite temperature and concentration difference in the chemical potential of two salts. In this model, a new term showing the entropy generation due to the coupling of concentration gradients of both salts is included. Furthermore, we have introduced a new mass Bejan number for the salts concentrations. The selected salts include sodium chloride and sucrose, which are added in water with different concentrations. The concentrations of both salts are assumed to be higher at the surface. The previously established dimensionless governing equations with Boussinesq approximation and Darcy law were solved numerically. The resulting velocity, temperature, and concentration gradients are employed in the entropy generation model. The impacts of the pertinent parameters and related dimensionless numbers on the dimensionless entropy generation rate, irreversibility ratio and Bejan numbers are examined for both assisting and opposing flows. It has been found that entropy generation rates are higher for assisting flows than opposing flows. At the same time, it is possible to reduce the entropy generation rate with a rise of the Darcy and Lewis numbers while the mass Bejan number increases with Lewis number. [ABSTRACT FROM AUTHOR]

The Brinkman equations can be regarded as a combination of the Stokes and Darcy equations which model transitions between the fast flow in channels (governed by Stokes equations) and the slow flow in porous media (governed by Darcy's law). The numerical challenge for this model is the designing of a numerical scheme which is stable for both the Stokes-dominated (high permeability) and the Darcy-dominated (low permeability) equations. In this paper, we solve the Brinkman model in n dimensions (n = 2 , 3) by using the mixed discontinuous Galerkin (MDG) method, which meets this challenge. This MDG method is based on the pseudostress–velocity formulation and uses a discontinuous piecewise polynomial pair P ̲ k + 1 S - P k (k ≥ 0) , where the stress field is symmetric. The main unknowns are the pseudostress and the velocity, whereas the pressure is easily recovered through a simple postprocessing. A key step in the analysis is to establish the parameter-robust inf–sup stability through specific parameter-dependent norms at both continuous and discrete levels. Therefore, the stability results presented here are uniform with respect to the permeability. Thanks to the parameter-robust stability analysis, we obtain optimal error estimates for the stress in broken H ̲ (div) -norm and velocity in L 2 -norm. Furthermore, the L ̲ 2 error estimate for pseudostress is derived under certain conditions. Finally, numerical experiments are provided to support the theoretical results and to show the robustness, accuracy, and flexibility of the MDG method. [ABSTRACT FROM AUTHOR]

The effect of temperature modulation on natural convection in a horizontal porous annulus is investigated in this paper. The porous medium is confined between the inner and outer cylinders of the annulus which is subjected to time-dependent temperature. It is assumed that the time-dependent modulation is periodic with frequency σ and amplitude λ. The model consists of the heat equation and the equations of motion under the Darcy law. The derived problem with the stream function-temperature formulation is solved numerically using the alternating direction implicit method. It is shown that a stabilizing effect can be gained for appropriate values of the frequency σ and the amplitude λ. [ABSTRACT FROM AUTHOR]